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- http://ufdc.ufl.edu/AA00058822/00001
## Material Information- Title:
- Physicochemical measurements by high precision gas chromatography
- Creator:
- Bowen, Barry Eugene, 1947-
- Publication Date:
- 1973
- Language:
- English
- Physical Description:
- xxiii, 320 leaves. : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Adsorption ( jstor )
Carbon ( jstor ) Ions ( jstor ) Mass transfer ( jstor ) Mathematical moments ( jstor ) Porosity ( jstor ) Propane ( jstor ) Silica gel ( jstor ) Solutes ( jstor ) Velocity ( jstor ) Chemistry thesis Ph. D Dissertations, Academic -- Chemistry -- UF Gas chromatography ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis--University of Florida.
- Bibliography:
- Bibliography: leaves 312-318.
- General Note:
- Typescript.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
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- University of Florida
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- This item is presumed in the public domain according to the terms of the Retrospective Dissertation Scanning (RDS) policy, which may be viewed at http://ufdc.ufl.edu/AA00007596/00001. The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
- Resource Identifier:
- 022925773 ( ALEPH )
14210263 ( OCLC )
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PHYSICOCIEMICAL MEASURITiy-T'S ~BY HIGH PRECISION GAS CiRO ATOGRAPHY By BARRY EUGENE BOWEN A DISSERTATION PRESENTED TO TE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE >LQUIREVLNTS FOR Thl DEOREE OF DOCTOR OF PHILOSOPHY JNIVESMTY OF FLORIDA 1973 The autho: is proud to dedicate this research effort to his wife, Kathy, for her love and devotion during the course of this work, and to his uncle and and aunt, Dr, and Mrs. Ray A. Miller, educators, for financial support during his undergratdua le study and for their en couragement throughout this academic endeavor. ACKNOWLEDC:ZENTS The author wishes to acknowledge his research advisor and supervisory committee chairman, Dr. Stuart P Cram, formerly an assistant professor at the University of Florida, for the initial suggestion of this research problem and for financial support. Other members of the author's supervisory committee, Drs. Roger G. Bates (cochairman), Cerhard M. Schmid, Frank D. Vickers, Arthur W. Westerberg, and James D. Winefordner, are also acknowledged. The efforts of Dr. Cram and Dr. W. Wayne Meinke toward arranging the valuable opportunity of completing this research at the National Bureau of Standards, Gaithersburg, Maryland, are gratefully acknowledged. This experience would not have been possible without the cooperation of the Graduate School of the University of Florida, and the Department of Chemistry, from which much of the chromatograph/computer system was borrowed. Many informative discussions with Dr. John E. Leitner during the development of the computer interface hardware and software are sincerely acknowledged. Discussions with Dr. Robert L. Wade concerning the design of his high pressure sampling valve and with Dr. Stephen N. Chesler concerning statistical moment theory were helpful. iii Thanks go to Mr. Roger V. Krumm for the translation of the papers by Kubin. The author extends sincere appreciation to his wife, Kathy, for her efficiency and cooperation in typing the first draft. The excellent work of Mrs. Edna Larrick on the final draft is acknowledged. iv TABLE OF CONTENTS Page ACKNOWLEDGMENTS . . . . iii LIST OF TABLES . . . . . . . LIST OF FIGURES . . . . . . . LIST OF SYMBOLS . . . . . . . viii X . . xvi ABSTRACT xxii INTRODUCTION . . . . . . . . . Objective of this Research . . . . . . Statistical Moments in Physicochemical Analysis High Precision Gas Chromatographic Measurements 1 1 2 8 THEORETICAL . 13 Development of Theoretical odels for Gas Chromatography Selection of a Gas Chromatographic Model . . . . Mass Transfer in the Interparticle Space . . . . Mass Transfer Through the Thin Film Around the Particle Radial Diffusion Within the Pores . . . . . . Mass Transfer Through the Thin Film at the Pore Wall . Solution of the Mass Balance Equations . . . . Calculation of the Statistical Moments . . . . Chromatographic Significance of the Statistical Moments The Case of Longitudinal, External, and Internal Diffusion and Surface Adsorption . . . . . The Case of Longitudinal, External, and Internal Diffusion and Volumal Adsorption . . . . . The Case of Longitudinal and External Diffusion and Surface Adsorption . . . . . . . . 13 . 16 22 22 23 24 . 24 26 27 . 32 37 42 V TABLE OF CONYTTS (Continued) Assumptions of the Models . . . . . . . . Effect of Pressure Drop on the Statistical Moment Equations . . . I . . I I . EXPERIMN AL . . . . . . . . . . . . Page 48 49 55 Characterization of the Gas Chromatograph/Computer System m . . . . . . . . . . Pneumatic System . . . . . . .. Sampling Valve . . . . . . . . Chromatograph Oven . . . . . . . Chromatograph Detector . . . . . . Electrometer . . . . . . . . Computerized Data Acquisition and Control System Hardware . . . . . . . Computerized Data Acquisition and Control System Software . . . . . . . Characterization of the Gas Chromatographic Columns Preparation of Columns . . . . . . Determination of Particle Diameter . . . . . 55 55 . .60 . 66 68 68 . . 72 . . 80 . . 98 . 100 Determination of Permeability and External Porosity RESULTS AND DISCUSSION . . . . . . . . . . 104 108 Determination of Permeability and External Porosity . Determination of Internal Porosity . . . . . . Determination of Binary olecular Diffusion Coefficients Measurement of Equilibrium, Diffusion, and Rate Constants Silica Gel Columns . . . . . . Measurement of Equilibrium, Diffusion, and Rate Constants OV-101 C3lumns . . . . . . . Measurement of Equilibrium, Diffusion, and Rate Constants Graphitized Carbon Black Columns . . Net Retention Volumes . . . . . . . . . Free Energy Changes . . . . . . . . . 108 118 125 132 166 175 189 194 vi. TADLE OF CO:CEENTS (Continued) Page Effects of Physicochemicnl Parameters on Chromatographic Band Broadening . . . . . . . . . Effects o! Physicochemical Parameters on Chromatographic Dand Asymmetry . . . . . . . . . . Correlation of Expcrinantal with Thworevical Results and Analysis of Eriors . . . . . . . . Summary . . . . . . . . . . . . . APPENDICES . . . . . . . . . . . . . 204 225 241 251 254 A. B. High Precision Sampling in Cas Chromatography . . 255 Effects of Sample Size on Gas Chromatographic Behavior 273 C. Computer Programs . . . . . . . . . 301. 1. Calculation of Ternpcratures Using a Platinum Resistance Thermometer . . . . . . 302 2. Statistical Moment Calculations of Stored Peaks . . . . . . . . . . 303 3. Least Squares Fit to a Polynomial . . . . 306 4. Newton-Raphson Technique for Solvirg the Permeability Equation for External Porosity . . . . . . . . . 308 5. Calculation of Binary Diffusion Coefficients from the hirschfelder Equation . . . . 309 6. Least Squares Iterative Fit to an Equation of the Form E=A+B/v + Cv . . . . . 310 BI BL0OG RAP1{Y BIOGRAPHICAL . . . . . . . . . . . . SKETCH . . . . . . . . . . . 312 319 vii LIST OF TABLES Table Page 1. Identification of Gas Chromatographic Columns . . . 103 2. Permeability and Porosities of Gas Chromatographic Columns . . . . . . . . . . . 116 3. Reproducibility in. Preparation of OV-101 Columns . . 125 4. Comparison of Molecular Diffusion Coefficients in Helium . . . . . . . . . . . . 131 5. Equilibrium, Diffusion, and Rate Constants for Hydrocarbons on Silica Gel . . . . . . . 1.42 6. Absolute and Relative Contributions to Resistance to Mass Transfer for Hydrocarbons on Siiica Gel 7. Equilibrium, Diffusion, and Rate Constants for Propane on OV-101 Impregiiatcd Silica Gl . . 8. Absolute and Relative Contributions to Resistance to Mass Transfer for Propane on OV-101 Impregnated Silica Gel . . . . . . . 9. Equilibrium, Diffusion, and Rate Constants for Hydrocarbons on Graphitized Carbon Black . . 10. Absolute and Relative Contributions to Resistance to Mass Transfer for Hydrocarbons on Graphitized Carbon Black . . . . . . 11. Free Energy of Adsorption for Hydrocarbons on Adsorbents . . . . . . . . . 12. Average Adsorption and Desorption Times for Hydrocarbons on Adsorbents . . . . . . 13. Values of the Rate Coefficients A, B, and C Calculated by Least Squares Using Experimental Data . . . . . . . . 153 S.169 172 180 188 202 203 222 viii LIST ALEXS (Con, iIuued) Table 14. Precision of Statisticul 'Joment Calculations at 2.0% Integration Limits for Eydrocarbons on Silica Gel at 54'C . . . . . . . . . 15. Precision of Statistical Moment Calculatiois at 0.39% Integration Limits for hydrocarbons on Silica Gel at 54'C . . . . . . . . . 16. Flow Rates Used to Measure Sampling Valve Characteristics . . . . . . . . . . 17. Comparison of the Mean Values of the Statistical Moments and Precision for Valve Generated Input Peak Profiles . . . . . . . . . 18. Comparison of the Precision of Several Automated Sampling Valves with Unretained Solutes (Reference) . . . . . . . . . . . 19. Effect of Limits of Integration on the Mean Value and Precision of the Statistical Moment Calculations . . . . . . . . . . . Page 247 248 260 267 270 272 ix LIST OF FIGURES Figure 1. Vodlel particle for silica gel adsorbent . . . . . 2. Model particle for O-101 adsorbent . . . . . . 3. Model particle for graphitized carbon black adsorbent. 4. Schematic diagram of pneumatic, computer control, and data acquisition systems . . . . . . . . 5. High pressure gas sampling valve shown as a cut-away view . . . . . . . . . . . . . 6. Experimental valve injection profile for the high pressure valve showing i 2.0%, 0.5%, and 0.2% integration limits . . . . . . . . . . . . 7. Voltage divider circuit used to measure electrometer time constants . . . . . . . . . . Page 21 39 44 57 62 65 71 8. Device selector logic for computer control of sampling valve . . . . . . . . . . 9. Timing sequence under software control for generating sample valve pulse widths . . . . . . . 10. Driver circuit for automating solenoid actuated sampling valve . . . . . . . . . . 11. Flow diagram for the on-line computer control and calculation program (ADCOM) . . . . . . 12. Flow diagram for ADCOM, continued . . . . . 13. Flow diagram for ADCOM, continued . . . . . 14. Flow diagram for ADCCM, continued . . . . . 15. Flow diagram for ADCOMI, continued . . . . . 16. Flow diagram for ADCOM, continued . . . . . . 74 77 79 83 85 88 90 93 95 x Li ST OF1 FIGURES (Continued) Figure~ 17. 18. Flow diagram for ADCGM, continued . . . . . Particle size distribution for 120/140 mesh silica gel . 19, Pressure transducer calibration plot . . . . 20. Permeability calibration plots for silica gel columns 21. Permeability calibration plots for OV-101 columns . 22. Permeability calibration plots for graphitized carbon black columns . . . . . . . . . . 23. The effect of external porosity on the permeability porosity function . . . . . . . . . 24. Plots of calculated and experimental inert elution times as a function of average column velocity for silica gel columns . . . . . . . . . 25. Molecular diffusion as a function of velocity through Page 97 102 107 113 115 120 124 129 an open tube for methane, ethane, propane, and n-butane in helium . . . . . . . 26. Reduced first moment vs. L/v for methane chromatographed on silica gel at 54'C . . . . 27. Reduced first moment vs. L/v for ethane on silica gel . . . . . . . . . . 28. Reduced first moment vs. L/v for propane on silica gel . . . . . . . . . . 29. Reduced first moment vs. L/v for n-butane on silica gel . . . . . . . . . . 30. Reduced second moment vs. 1/v2 for methane chromatographed on silica gel at 54'C . . . . . 31. Reduced second moment vs. 1/v2 for ethane on silica gel . . . . . . . . . . 32. Reduced second moment vs. 1/v2 for propane on silica gel . . . . . . . . . . 33. Reduced second moment vs. 1/v2 for n-butane on. silica gel . . . . . . . . . . . xi . 135 137 139 141 146 148 150 152 LiST OF FIGURES (Continued) Figure 34. Total mass transfer resistance vs. R2 for methane chromatographed on silica gel at 54'C . . 35. Total mass transfer resistance vs. R for ethane on silica gel . . . . . . . . . 36. Total mass transfer resistance vs. R2 for propane on silica gel . . . . . . . . . 37. Total mass transfer resistance vs. R2 for n-butane on silica gel . . . . . . . . . 38. Reduced first moment vs. L/v for propane chromatographed on OV-101 at 54'C . . . . . . 39. Reduced second moment vs. 1/v2 for propane chromatographed on OV-101 at 54'C . . . 40. Total mass transfer resistance vs. R2 for propane chromatographed on OV-101 at 54'C . . . . Page 157 . . 159 ..61 . . -163 . 168 . 171 . 174 41. Reduced first moment vs. L/v for propane chromatographed on graphitized carbon black at 540C . . . 42. Reduced first moment vs. L/v for n-butane on graphitized carbon black . . . . . . . . 43. Reduced second moment vs. 1/v2 for propane chromatographed on graphitized carbon black at 54'C . . . . . . . . . . . . 44. Reduced second moment vs. 1/v2 for n-butane on graphitized carbon black . . . . . . . . 45. Total mass transfer resistance vs. R2 for propane and n-butane chromatographed on graphitized carbon black 46. Net retention volume as a function of flow rate for methane, ethane, propane, and n-butane chromatographed on silica gel at 541C . . . . 47. Net retention volume as a function of flow rate for propane and n-butane chromatographed on graphitized carbon black at 54'C . . . . . . 48. Plot of log (u1 -ua) vs. log (60 LA0/jFm) for methane, ethane, propane, and n-butane chromatographed on silica gel at 540C . . . . . . . . 177 179 183 185 187 191 196 1,99 xii LIS? OF FIGUITIS (Cont:nuod) Figure 49. Plot of log (u, --u) vs. log (60 LA0/jFm) for propane and n-butane chroLmatographed on graphitized carbon black at 54'C . . . . . . . . 50. Column efficiency in terms of band broadening, V, versus linear velocity, T, for methane chromato.graphed on silica gel at 54'C . . . . . 51. $ vs. v for ethane on silica gel . . . . . 52. 4 vs. v for propane on silica gel . . . . . 53. 4 vs. v for n-butane on silica gel . . . . . 54. Column efficiency in terms of band broadening, ', versus linear velocity, v, for propane chromatographed on OV-101 at 54'C . . . . 55. Column efficiency in ter.s of band broadening, $, versus linear velocity, v, Yor propane chromatographed on graphitized carbon black at 54'C 56. $ vs. v for n-butane on graphitized carbon black 57. Column efficiency in terms of band asymmetry, Z, versus linear velocity, v, for methane chromatographed on silica gel at 540C . . . 201 208 210 212 214 216 . 218 220 227 58. Z vs. v for ethane on silica gel . . . . . . 59. Z vs. v for propane on silica gel . . . . . . 60. Z vs. v for n-butane on silica gel . . . . . . 61. Column efficiency in terms of band asymmetry, Z, versus linear velocity, v, for propane chromatographed on OV-101 at 54'C . . . . . . . . . . 62. Column efficiency in terms of band asymmetry, Z, versus linear velocity, v, for propane chromatographed on graphitized carbon black at 54'C . . . . . 63. Z vs. v for n-butane on graphitized carbon black . . 64. Comparison of experimental data (points) and theoretical data (curves) for propane chromatographed on 100/120 mesh silica gel (bottom), graphitized carbon black (middle), and OV-101 impregnated silica gel (top) 229 231 233 235 237 239 244 xiii LiST 01 1 cGURES (Continued) Figure 65. Nybrid-fluidic valve shown from the front with a cut-away view of the valve assembly. Valve is in the normal position for carrier gas flow onto the column . . . . . . . . . . . Page 258 66. Valve injection profile obtained with a 50 msec valve gate pulse to the hybrid-fluidic valve showing 2.0%, 0.5%, and 0.2% integration limits 67. Valve injection profile obtained with a 100 msec valve gate pulse to the hybrid-fluidic valve showing 2.0%, 0.5%, and 0.2% integration limits . . 68. Experimental valve injection profile for the pneumatically operated Hamilton valve at 100C showing 2.0%, -5%, and -0.2% integration limits . 263 263 265 69. Methane peak profiles obtained on a 714 cm X 0.068 cm i.d. DC-200 open-tube column using hybridfluidic valve sampling times of 10-200 msec . . 277 70. Pentane peak profiles obtained on a 314 cm X 0.172 cm i.d. open stainless steel column using a hybridfluidic valve sampling time of 150 msec and varying sample size by using an exponential dilution flask . . . . . . . . . 277 71. Effect of methane sample size on the statistical moments when retained on a 714 cm X 0.068 cm i.d. column coated with DC-200 at 98C (integration limits = 1.0%) . . . . . . . . . 279 72. Broadening (4) and asymmetry (Z) for methane retained on a 714 cm DC-200 coated open tube . . . . 282 73. Effect of pentane sample size on the statistical moments when retained on a 714 cm X 0.068 cm i.d. column coated with DC-200 at 980C (integration limits = 1.0%.) . . . . . . . . . 285 74. Broadening (4) and asymmetry (Z) for pentane retained on a 714 cm DC-200 coated open tube . . . . 287 75. Effect of the pentane sample size on the statistical moments in a 314 cm X 0.172 cm i.d. stainless steel column at 97' (integration limits = 0.10%) 289 xiv LIST 01 FIGURES (Continved) Figure Page 76. Broadening (") and asymmetry (Z) for pentane in a 314 cm uncoated open tube . . . . . . 291 77. Dispersion in coated and uncoated open tubes . . . 294 xv LIST OF SYMBOLS Symbols, Definitions, and Dimensions a Thermometer calibration constant A Empty column cross-sectional area, cm A Statistical moment eddy diffusion coefficient, cm A Cross-sectional area available to moving carrier e gas, cm2 AD Van Deemter eddy diffusion coefficient, cm B Van Deemter molecular diffusion coefficient, cm 2/sec c A constant defined by equation 17 c Concentration in interparticle space, mole/cc c. Concentration in intraparticle space, mole/cc C Van Deemter coefficient for resistance to mass transfer, sec C Coefficient of mass transfer in mobile phase, sec g C4 Coefficient of mass transfer in stationary phase, sec d Thermometer calibration constant d Average particle diameter, cm or dp/dl Pressure gradient, atm/cm 2 D B Effective (observed) dispersion coefficient, cm /sec ef 2 D Molecular diffusion coefficient, cm /sec g Dk Effective Knudsen diffusion coefficient, cm /sec D Effective coefficient of longitudinal diffusion, cm2/sec xvi D D in column segment i, cm2/sec pi p D Effective coefficient of radial diffusion, cm2 /sec D cDispersion due to secondary flow, cm 2/sec sec 2 D Calculated molecular diffusion coefficient, cm /sec AD D Knudsen diffusion coefficient, cm /sec K E Column efficiency defined by equation 146, cm E .n Value of optimum column efficiency, cm E Effective coefficient of molecular diffusion, A cM2/sec f Pressure correction factor defined by equation 81 f(e ) Porosity function used in permeability equation 101 C . F Carrier gas mass flow rate, cc/min or cc/sec H Average calculated plate height, pressure-corrected, cm H Rate constant of mass transfer, sec1 c H Average plate height from gas phase contributions not pressure-corrected, cm -1 H Rate constant of adsorption, sec n H Defined as OK or OK by equation 150 HETP Height equivalent to one theoretical plate, cm i Transfer through the outer gas film i2 Radial diffusion in the gas i3 Transfer through the pore wall gas film i Transfer through the outer liquid film i5 Radial diffusion in the liquid i Transfer through the pore wall liquid film Carrier gas compressibility factor defined by equation 82 xvii k Columrn permeability, g cm/sec2 atm or cm2 k Mass transfer coecificient, cm/sec K Laplace transform for the effective equilibrium adsorption coefficient K Equilibrium volumal adsorption constant C K Equilibrium surface adsorption constant n L Column length. cm LHETP Local HETP, cm M Amount of adsorbate in a pore volume unit, mole/cc n Mn nth total statistical moment, sec n m Total mass of injected compound, moles/cc m,1A, E Molecular weight of a substance, g/mole n Amount of adsorbate on the surface in a pore volume unit, mole/cc p absolute pressure, atm p. Column inlet pressure, atm p0 Column cutlet pressure, atm P P./p 1 0 P Pressure correction factor defined by equation 93 n q Constant in permeability equation 102 qe External tortuosity factor q. Internal tortuosity factor Q Rate of c. increse, mole/cc sec Q Rate of transfer through a thin stationary film, mole/cc sec r Radial coordinate, cm r Average pore radius, cm or A' xviii R Mean particle radius, cm or 4m R Gas constant, cm 2f/0K mole sec g R Column inside radius, cm 0 Rt Resistance at t0C, ohms RB Resistance at 00C, ohms R CoLumn coil radius, cm Re Reynolds number s Coefficient of Laplace transform t time, sec t Inert solute elution time, sec tads Mean time of adsorption, see tdes Mean time of desorption, see t0 Length of injection time and other system time constants, see T Absolute temperature, 'K u Linear velocity of an inert solute in LHETP equations 78 and 79, cm/sec u Absolute first moment of an inert solute, see n u nth central statistical moment, sec n v Average column cross-sectional linear velocity, j corrected, cm/sec v Average critical v, cm/sec v. Velocity for maximum column efficiency, cm/sec V Sensing volume of FID, cc V. Linear velocity in column segment i, cm/sec V Superficial carrier gas velocity at the column outlet, cm/sec xix VA Inert solute elution volume, cc V9 CVolume of empty colvan, cc VD Extracolumn dead volume, cc w1/2 Width at half height, sec w2 Velocity for minimum band broadening, cm/sec w3 Velocity for minimum band asymmetry, cm/sec W w3 /w2 x Distance along the column from the inlet, cm z Length coordinate, cm Fractional porosity per adsorbent particle y Tortuosity factor in B term 1J Total resistance to mass transfer in liquid, sec A' eInternal and external contributions to yl, sec Total resistance to mass transfer in gas, sec a' i' e Adsorption, internal, and external contributions a 11e to 61, see e Total external porosity fraction Total internal porosity fraction 1 e Total column porosity fraction, e + e. T e Carrier gas viscosity, g/cm sec Multiple path factor in AD term Particle shape factor p Packed column particle density, g/cc a Standard deviation UAB Collision cross section, cm Td Detector time constant, sec xx T EElectrometer timo constant, see T System time eonstani, sec 1 v Sampling value time constant, sec Porosity function in moment equations X (1 + 0(1Kn )) Column efficiency with respect to broadening, cm $f Measured -, not pressure-corrected, cm min Value of optimum 4, em 4, corrected for sampling time, cm 4, corrected for pressure effects, cm Calculated 4 values, cm '2 '2 Z Column efficiency with respect to asymmetry, Cm Z Z Z3 Calculated Z values, cm2 Z' Z, corrected for sampling time, cm2 2 Z Z, corrected for pressure effects, cm A (1 + OK ) 0 Unit of resistance, ohms 0 Function of temperature and potential field in diffusion equation 115 xxi Abstract of Dissertaian Pfrsentnd to iha Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PIIYSICOCIILI CAL MEASURE.MENTS BY HIGH PRECi SION GAS CHROAL1TOGRAPHY By Barry Fugeno Bowen December, 1973 Chairman: Dr. Stuart P. Cram Cochairman: Dr. Roger G. Bates Major Department: Chemistry This research has been directed toward the understanding and the accurate and precise determination of factors affecting gas chromatographic efficiency. Experimental evidence is presented toward the verification of the stochastic theory of gas chromatography. This has been accomplished through the measurement of physicochemical constants, derived from a statistical moment analysis of chromatographic peak shapes. The precision in the resulting permeabilities, porosities, binary molecular diffusion coefficients, equilibrium adsorption constants, radial diffusion coefficients, and adsorption rate constants ranged from A 1-20%. The gas-solid chromatographic model for porous adsorbents is extended to the cases of nonporous adsorbents, liquid-filled porous adsorbents, and to columns with pressure gradients comparable to those encountered in practice. Gaseous hydrocarbons were chromatographed on packed columns of activated silica gel, OV-101 impregnated xxii silica gel, and graphitized carbon black. These columns are characterized with respect to their ')article distribution, permeability, external porosity, and internal porosity. The efficiency of these columns is characterized in terms of peak broadening (Y) and peak asynmnetry (Z) as a function of carrier gas linear velocity and adsorbent particle radius. Experimental and theoretical efficiency curves are compared. A high precision gas chromatograph was assembled in which carrier gas flow rate, sampling valve actuation, column temperature, data acquisition, and statistical moment calculation were controlled by an on-line dedicated laboratory computer. Extracolun dead volumes were minimized and systematic time constants were measured. A comparison of several gas sampling valves is made by a statistical moment analysis of their injection profiles. A precision of 0.05% relative standard deviation for the first moment of the injection profile of one of the valves was achieved. The effect of sample size and length of sampling time on chromatographic behavior is examined for coated and uncoated open tubes. xxiii INTRODUCTION Objective of this Research The objective of this research has been to measure physicochemical Daramete-s in order to develop an experimental verification of the stochastic theory of gas chromatography (GC). A high precision gas chromatographic system was developed to determine equilibrium constants, diffusion constants, and rate constants from a statistical moment analysis of Clution profiles. In addition, extracolumn effects were experimentally investigated, such that their contributions to chromatographic zone broadening and asymmetry were known. In this way, the derived coefficients for the various mass transfer processes were used to completely characterize the on-column dynamics involved in adsorption as well as partition systems. In recent years, GC has become a major technique for chemical analysis. As a laboratory separation techniique, it is recognized for its exceptional resolving power, speed, sensitivity, and ease and versatility of operation. In fact, qualitative precision GC has reached a high degree of reliability such that it actively competes with other identification techniques, for example, spectroscopic alethods [1]. Recent advances in instrumentation and column materials have made it easy to produce experimental data; however, these data have become 1 2 almost ezclusivcly applications oriented, without understanding, and one has to find them interesting rather than informative. It is essential that new methods and techniques be continually developed in GC in order to cope with today' s innumerable separation problems. This can best be accomplished on the basis of understanding of fundamental physical and chemical phenomena, rather than on an empirical basis. The status of current development and applications in GC can perhaps be described as having reached a plateau which is dependent upon a better functional knowledge of the on-column separation process itself in order to quickly choose or predict optimum experimental conditions. Once this knowledge is gained, advantage can be taken of the potentially high precision and accuracy inherent in gas chromatography as an analytical measurement system. Statistical Moments in Physicochemical Analysis Generally, the accurate determination of physicochemical constants from GC data has not been feasible without the use of a statistical moment analysis of elution profiles. This has been necessary because chromatographic mass transport phenomena are random events occurring on a molecular scale. Further, most of the molecules are in a nonequilibrium state most of the time and the molecular distributions obtained are very rarely Gaussian [2]. Peak shapes, which represent distributions of molecules, can be completely defined in terms of their statistical moments since the moments are applicable to all peak profiles as long as the finite integral 3 f~u)dt (1) exists and cam be evaluated in closed form or numerically. The zeroth, first, second, third, an -fourth moiients are measures of thC peak area, mean, variance, asymmetry, and height, respectively [3]. The use of statistical moments to obtain physicochemical constants from experimeotal GC data will be reviewed below. The theoretical development of the dynamics of chromatography will be reviewed in the next chapter of this dissertation. Expressions for the first five statistical moments, in terms of physical constants were first developed independently in 1965 by Kubin [4,5] and Kucera [6]. Chromatographic peak shapes were also treated mathematically by Kaminskii et al. [7], and the first six moments of the peaks were derived on the assumption that diffusion in the mobile phase was the most important column process. The Kubin-Kucera theory for gas-solid chromatography (GSC) was examined initially through the experiments of Grubner and coworkers [8-11]. They investigated the effect of porosity on the retention times of inert gases using columns packed with glass beads, pumice, or coal. The kinetics at high velocities were diffusion controlled for the porous supports, leading to approximate values of the coefficient of radial diffusion [8]. Subsequently, they qualitatively verified their theory for column efficiency with respect to the variance, asymmetry, and excess of the chromatographic curves by studying the adsorption of carbon dioxide on activated charcoal. They did not:, however, report any rate constant values. In all experiments they used 4 relatively short columns with large internal diameters and large particles in order to achieve negligible pressure drop and wall effects [9]. Schneider and Smith [12] described a method for determining adsorption equilibrium constants, rate constants, and intraparticle diffusivities based on analytical expressions from the moment equations of Kubin. They reported experimental values for ethane, propane, and butane on large silica gel particles packed into very short columns. They used the experimentally determined physicochemical constants to predict the shape of breakthrough curves. In most cases, the rate of the overall chromatographic process was controlled by intraparticle diffusion. By using microporous silica, Schneider and Smith [13] determined trends for surface diffusivities when this diffusion was a significant fraction of the total observed intraparticle mass transport. The same moment equations were applied to chemisorption of hydrogen on nickel catalyst [14] and hydrogen on cobalt catalyst [15]. In these experiments, strong chemical as well as physical processes were important. They also reported values for axial diffusion, equilibrium adsorption constants, and adsorption-desorption rates as a function of temperature. The major limitations to the reliability of data were the relatively large samples used, the apparent lack of concern for complete integration of the elution curves, and the simplifications made in the model for the complex catalyst surface. Moreland and Rogers [16] demonstrated the effects of slow mass transfer of small molecules in microporous zeolites in a manner similar to that of Oberholtzer and Rogers [17]. The radial coefficients of 5 internal diffusion were approximately determined from the dependence of the statistical moments of the elution curves on the linear velocity of the carrier gas. They examined the peak mean and variance but not the higher moments. When comparing peak means and maxima, they found a much smaller dependence of retention volujmie on flow rate when using the mean, indicating that the mean is the proper peak parameter to characterize equilibrium conditions. Funk and Rony [18-20] argued that ,Pplication of statistical moment theory to experimental elution curves was difficult due to the infinite time upper limit of the moment integrals. They developed partial statistical moment expressions for relating retention time to the mean, peak shape, and peak area. They did this by defining a single dimensionless group, which is a measure of diffusional mass transfer in the liquid phase relative to the convective gas phase mass transfer in the axial direction. They concluded that only partial normal moments could be experimentally determined since complete moments are limited by system noise and drift. This same conclusion was reached by Chesler and Cram [21] who discussed the effect of locating the limits of integration on the accuracy of measuring and calculating statistical moments from chromatographic data. The effect of noise on the precision of the moment measured was found to correlate with the limits of integration and the number of data points taken per peak. Errors in the calculation of all moments from digital data were dependent on the general shape of the peak; the fastest decrease in errors with lower integration limits and the highest predictability was found for the most Gaussian-like peaks. 6 Grushka eu al. [221 showed the utility of off-line computer calculations for the detailed analysis of peak shape and rapid methods for measuring the moments. eiss [231 extended the stochastic theory of GC to include diffusion effects and multi-site adsorption by developing this simple model in terms of the first and second statistical moments. Grubner [24] has shown how the first four moments can be calculated once the inflection points of an asymmetric elution curve have been determined. A moment analysis was used to sense the presence of two overlapping Gaussian peaks from the skew and excess of the peak profile [25]. An electrical simulator for chromatographic behavior on GSC columns was proposed to simulate linear adsorption kinetics, mass balance in the column, and to calculate the statistical moments [26,27]. Grubner and Underhill [28] developed equations, based on the statistical moments theory for GSC, for determining the bed capacity of porous adsorbents. The principal value of this work is its potential ability to allow the design engineer to recalculate given experimental data to other conditions, thus arriving at the optimum ones. Underhill [29] presented a moment analysis of elution curves from GC columns having a log-normal distribution of adsorbent particles. His equations related number of theoretical plates, skew, and excess to macroporous and microporous column packings for the case where both interparticle and intraparticle diffusions are present. le extended these equations to include corrections for pressure drop effects [29,30] but presented no experimental results. Recently, Chesler and Cram [31] have reported an iterative curvefitting technique for the high accuracy measurement of -total 7 statistical moments from expurimental data. The fitting procedures are designed for experimental peaks with any reasonable degree of asymmetry and real data from a partition column were successfully tested. The ultimate utility of this technique will be realized only when these total moment expressions in terms of eight empirical parameters are finally related to physical phenomena. Concurrently, Dwyer [32] presented a method of Fourier transforms for deconvolution of experimental GC elution curves. He demonstrated from diffusion and adsorption experiments that if the time distribution of one independently occurring column process is known, the other can be resolved into its component time distribution. Subsequently, the statistical moments of the component distributions can be accurately determined by appropriate handling of the digital data. The zeroth moment, the area under a chromatographic peak, has been used for some time for quantitative analysis. This is because, perhaps fortuitously, automatic integrators give a measure of the zeroth moment. However, in practice much information is lost due to the limited variability of starting and stopping techniques provided with hardwired commercial integrators. Retention times are generally still defined simply by the peak maximum, which has no basic physical significance, rather than the more correct value of the mean time [6]. Peak widths are taken as some hybrid of the width at -Alf height or extrapolated tangent baseline width rather than the correct value of the standard deviation derived from the second moment. 8 The major obstacle to the general utilization of statistical moment data has been the nonavailability of a computerized digital data acquisition system and sophisticated software. With such a system at hand, high precision and high accuracy measurements of the chromatographic statistical moments and, thus, fundamental physicochemical and thermodynamic constants, can be made based upon an appropriate model. This is still a rather narrow field in which only a few chromatographers have participated. High Precision Gas Chromatographic Measurements Several authors have recently reported high precision and high accuracy gas chromatographs, and/or systems where the sampling, flow rate, temperature, and data acquisition were computer controlled [33-42]. Oberholtzer [33] first reported a digital programmer capable of performing a series of injections using a pneumatically operated gas sampling valve with a precision of 1.1 msec. Oberholtzer and Rogers [34] designed a chromatograph that determined retention times with a precision of better than 0.02% for times of 26 seconds (slightly retained solutes). This precision allowed them to determine heats of solution of some n-paraffins to at least 10 cal/mole. They also measured the zeroth moment, HETP, and variance with 1 0.05% precision and the skew and excess at precisions of 0. 2% and 0.1%, respectively. These results were achieved by using a sampling value which gave a sample width of 67 msec with 0.22% reproducibility for the elution time of the sample plug. This same 9 system was used later to siudy the effects of rapid repeated injections [35] and ensemble averaging [36]. Burke and Thurman [37] used a dedicated computer for real-time control of GC measurements. It was shown that real-time interaction of a dedicated computer with a GC allowed improved precision of data taken for adsorption studies using porous supports as well as for detector response evaluation. This improvement.stemmed from the ability of the computer to provide an accurate time base for the sampling system. Thurman, Mueller, and Burke [38] elaborated on the above system so that the flow rate and oven temperature could be sampled by the computer except during an actual run. The data acquisition rate was not computer controlled. The purpose of their work was to measure data of sufficient precision to allow meaningful thermodynamic studies of gas-solid adsorption processes. Their system was designed to measure HETP using the peak maximum time and width at half height, then to change to a new flow rate, thus obtaining a van Deemter plot to determine the optimum flow rate. Swingle and Rogers [391 later assembled a high precision gas chromatograph in which the computer controlled sample injection, column temperature, flow rate, and also direct readout of inlet pressure, mass flow rate, and detector response. This system has been used for rapid and precise determination of thermodynamic data, quantitation of peak shapes, and qualitative identification of unknown components. Leitner [40] and Cram and Leitner [41] reported the design of a high precision gas chromatographic computer system. This system is capable of making on-line calculations of the first five statistical 10 moments. These moments an',d their hybrid momtnts were used to make on-line decisions with respect to mass flow and temperature adjustment, sampling time, data acquisition rate, and amplifier gain. On-line optimization has been performed whereby the resolution and skew of adjacent peaks were used as criteria for influencing the resolution and skew of component peaks not yet eluted. This system has been used to study the interaction of flow and temperature programming [40]. It was also used to study the precision of several gas chromatographic sampling valves [42] and to study the effects of sampling time and sample size on column efficiency as determined from the first five statistical moments. Several papers have appeared in recent literature concerning the instrumental limitations for high precision GC data. Sternberg [43] treated the first and second moment contributions from extracolumn sources such as mixing chambers, sampling time, and electrometer time constants. Glenn and Cram [44] developed a digital logic system for the evaluation of instrumental contributions to chromatographic band broadening. They reported experimental second moments contributed by the connecting tubing, mixing volume, sampling valve, electrometer and cable, and detector, Goedert and Guiochon [45] studied the influence of fluctuations of pressure and temperature and time measurement on the precision of the retention time. They established specifications required to achieve precisions of 10% to 0.01% [45,46], concluding that the latter was extremely difficult to achieve in practice, the pressure drop control being the most demanding limitation. For example, for 0.01% relative 11 precision (at the 95% confidence level) on retention times, the outlet pressure, pressure drop, and temperature should be stable to within 0.25%, 0.005%, and 0.001%, respectively, with no more than a 0.005% error in the time measurement itself. Goedert and Guiochon [47] have examined the effect of signal-tonoise ratio on the retention time of the peak maximum. They found that for all signal-to-noise ratios studied that the precision of the peak maximum was better than the precision of the mean time when dealing with nearly symmetrical peaks. They later concluded [48] that it is necessary to have a programmable time constant in electrometers; alternately, the time constant should be small enough such that the smallest peak of the chromatogram is not distorted or high enough to filter out high frequency noise for later broader peaks. Chesler and Cram [49] examined instrumental effects of the analog-to-digital converter (ADC) in the digitization step of data acquisition. They tabulated the errors i.ncurred in the statistical moments due to the ADC and integration limit selection for Gaussian as well as quite skewed curves. In an earlier paper, they showed that large errors in the precise determination of statistical moments could result from choice of integration limits alone [21]. Goedert and Guiochon [50] made an elaborate and comprehensive study of sources of systematic errors in retention time measurement and developed an expression for the overall instrumental correction. They found that the main error in retention times, and thus thermodynamic constants, was due to erroneous determinations of "inert" times. For methane on graphitized carbon black, they were able to detect 1.2 a capacity factor of 0.0031 at 100 C, a value which is normally assumed to be zero. Effect of temperature gradients and fluctuations of temperature on the retention time of an inert compound and on the capacity factor were theorized [51]. The research described in this dissertation has been directed toward the understanding and the accurate and precise determination of factors affecting gas chromatographic efficiency. A physical model of moderate complexity, which closely approximates reality, has been chosen to describe all known on-column processes. Relatively few simplifying assumptions have been made, and experiments have been performed with packed adsorption and partition columns under normal chromatographic conditions. This has been done with respect to column length, support size, solutes, and pressure drop. The above combination of a realistic theory along with realistic experimental situations has not been a common occurrence in recent literature in the field of gas chromatography. THEORETICAL Development of Theoretical Models foi Gas Chromatography The first theory of chromatography was written by Wilson [52] in 1940 and was called "A Theory of Chromatography." This quantitative theory of chromatographic analysis was based on the assumption that equilibrium between solution and adsorbent is instantaneously established and that the effects of diffusion can be neglected. Wilson did, however, describe the processes of diffusion and nonequilibrium as the causes for zone broadening and stated that quantitative agreement between his theory and experiment was unlikely. The plate theory was later developed by Martin and Synge [531 as an extension of distillation theory. They defined the plate height as "the thickness of the layer such that the solution issuing from it is in equilibrium with the mean concentration of solute in the nonmobile phase throughout the layer." The height equivalent to one theoretical plate (HETP) has been used widely to characterize chromatographic zone spreading and chromatographic efficiency. The theory predicts that a sample introduced onto a column begins in a single plate, progresses through a stepped Poisson distribution, and finally to a Gaussian distribution after a large number of sorption-desorption steps. Calculation of the "number of theoretical plates" per column- has 13 14 remained the simplest and most practical method of characterizing column separation efficiency. However, the idea of discrete and discontinuous physical plates on the column has done much to encourage theoreticians to seek better chromatographic models, The theoretical plate model does not in itself account for the basic effects of particle size, molecular structure, sorption phenomena, temperature, pressure, molecular diffusion, or variations in flow patterns. Martin and Synge did, however, deduce the rule that HETP was proportional to flow velocity and the square of the particle diameter (54]. Thomas [55,56] derived from differential mass balance equations expressions for obtaining adsorption and desorption rates from elution curves stated as functions of time. He, too, neglected the longitudinal diffusion term, but he did obtain simplified solutions for low flow rates where close-to-equilibrium conditions were valid. Boyd, Myers, and Adamson [57] described ion exchange kinetics in terms of diffusion through a liquid film and predicted peak shape in terms of independent rate and equilibrium constants. Lapidus and Amundson [58] developed equations for equilibrium as well as nonequilibrium cases. They incorporated the influence of the longitudinal diffusion but assumed that neither the external (migration of material along the particle surface) nor the internal diffusion (migration within the adsorbent pores) comes into consideration. They solved the problem of adsorption on the absorbent surface which follows a linear isotherm and also considered the rate of sorption. Their theory became the foundation for the well-known equation of van Deemter, Zuiderweg, and Klinkenberg [59]. In their paper [59], zone spreading was stated in terms of HELTP which evolved from additive 15 contributions of eddy diffusIo., ,jlecular cdiffusion, and resistance to mass transfer in the liquid phase. This concePt was similar to that proposed by Glueckauf [60] for solute diffusion through ion exchange beads and their surrounding liquid. Glueckauf [61,62] extended the plate height theory by mathematically describing clution and breakthrough curves for linear as well as nonlinear adsorption and exchange isotherms. In 1955 Giddings and Eyring [63] described the chromatographic process as a probability cf molecular events. This paper initialized the statistical concept for GC. They later extended this concept to the multi-site adsorption problem and then to the well-known random walk model of chromatography [64]. which seemed to satisfy a multitude of chromatographic problems. Giddings [65] developed a probability equation to explain the kinetic origin of tailing when sorption isotherms are linear. From this equation adsorption and desorption rate constants could be derived for molecules adsorbed on the 'tailproducing" sites if the Gaussian profile caused by adsorption on "non-tail-producing" sites could first be described. In 1959 Giddings [66] started the development of a generalized nonequilibrium theory for chromatography in order to calculate the effects of any complex adsorption-desorption problem, whether controlled by diffusion or discrete single-step kinetics. He recognized that nonequilibrium and diffusion were a common basis for the three theories of chromatography [67]. He named the material conservation or mass balance approach and the stochastic (random) approach as "rate" theories and considered them different in nature than the theoretical plate model. 16 McQuarrie [68] expanOdcd th1e Poisson ranArm walk theory of Giddings and Eyring by using the complex-variable theory of Laplace transforms. lHe described a technique by means of which any case, including mlti-site and delta or Gaussian input functions, could be well approximated by consideration of the first few central statistical moments. He also expanded the asymmetric chromatographic distribution function in terms of a Gram-Charlier series, a method which allows experimental curves to be approximated from the calculated statistical moments. Rosen [69] derived the equation for the time dependence of the concentration for elution curves for packed columns. He considered internal and external diffusion but assumed that molecular diffusion was negligible and that the amount adsorbed at each moment was proportional to the concentration within the porous particle. Finally, Kubin [4,5] and Kucera [6] developed equations for linear nonideal (nonequilibrium) gas-solid chromatography. The statistical moments were derived from mass balance equations for complicated cases of GSC where the processes of external diffusion and adsorption could be considered simultaneously with internal and longitudinal diffusion. Selection of a Gas Chromatographic Model In general, for the system of equations that describe a physical model to be solvable, the model must be relatively simple. Yet it must contain provisions for all of the essential physical and chemical processes that are known to occur. In the case of gas-solid or gasliquid chromatography, the model selected is quite complex due to 17 the several types of diffusion and r:iass transfer -hat may exist simultaneously. The Kubin-Kucera model for porous gas-solid adsorbents F4-6] will be closely followed and explained below. The theory will be developed for the cases of low pressure drop and nonporous adsorbents, and extended to gas-liquid systems. In gas-solid chromatography, adsorption of a gas occurs at the gas-solid interface and is caused by differences in the energetic properties along the interface. The adsorbent bed is composed of small particles whose cross sections are physically and chemically homogeneous. if the particles are porous, each having an internal porosity fraction 3, the column into which they are homogeneously packed will have a total internal porosity fraction, C., = (1-s )E (2) 1 e where e denotes the fraction of free interparticle space (external porosity) in the total column. The total porosity fraction eT will be e = e + e. (3) T e 1 An inert carrier gas enters at the head of the column at a pressure slightly larger than that at the end of the column. Due to the finite pressure drop created, the gas flows through the interparticle spaces toward the open end of the column. Since the pores are quite small (10-100A') the molecules of gas within the pores do not generally move with the carrier gas; rather they diffuse into the pores a certain distance only by a process called Knudsen diffusion whose coefficient is defined by 211 T 4-/ g 2 D = r (4) K 3\ M 18 where r is the average pore radius, R is the gas constant, T the g absolute temperature, and M is the molecular weight of the diffusing substance. Then the molecules diffuse out again into the main stream of the carrier gas. Under usual chromatographic conditions, the carrier gas passes throughout the column with a laminar flow profile with a typical system of streamlines that have zero velocity at the walls of the particles. This thin film of stationary carrier gas coats the entire surface area of the particle, internal as well as external. For highly porous materials, most of the surface area is internal and the external area may often be neglected. When adsorbate molecules are introduced onto the chromatographic bed as a very narrow pulse, the carrier gas transports them through the column. In order for a molecule to adsorb onto the particle surface (internal), it must enter a pore, and undergo four stages of mass transport. The molecule must (a) penetrate the thin film surrounding the particle at a certain rate determined by a concentration gradient. The molecule then (b) diffuses radially within the pore toward the center of the pore and toward the pore wall. It then must (c) penetrate the thin gas film existing on the pore wall and finally (d) be adsorbed onto the particle surface proper at a certain rate. It remains adsorbed for a certain time; then it is desorbed and by the opposite order of processes above, passes from the pore again into the flowing carrier gas. Each of these processes, when viewed as a random operation on the many adsorbate molecules in the sample, contributes to the enlargement of the chromatographic zone. 19 Besides these four contributions to broadening and asymmetry of the originally narrow Lplut profile, at least two other transport phenomena must be considered. In the interparticle space the molecules will be subject to longitudinal diffusion in the direction of the carrier gas flow and also to a diffusion similar to the effects caused by the properties of the packing proper called eddy diffusion. This suggests that the familiar A and B terms of the traditional HETP equation are caused primarily by effects in the interparticle space while the C term is due mainly to intraparticle processes. This model is shown in Figure i where the cross section of a single spherical particle contains a single pore (magnified for illustrative purposes) of diameter 2R. There are, of course, many such pores, which need not be the entire length of the particle, and the particle need not be spherical or smooth. The intraparticle adsorbate concentration is designated by c. and the interparticle concentration by c since they may not be equal at equilibriu-. The internal and external wall gas films are designated by 1 and 2. The processes of mass transfer through the film around the particle, mass transfer by radial diffusion, and mass transfer across the film around the pore wall are depicted by i1, i and i respectively. The adsorbate molecules in the interparticle space diffuse with an effective coefficient of longitudinal diffusion, D p D = D + Av (5) p g where A is a constant containing the effects of eddy diffusion, which may constitute trans-column diffusion, etc. D is the coefficient of molecular diffusion for the adsorbate in the carrier gas. The carrier Figure 1. Model particle for silica gel adsorbent. The external and internal wall gas films are designated by 1 and 2. The particle diameter is 2R, ci is the internal adsorbate concentration, and ce is the external adsorbate concentration. Mass transfer through the gas film around the particle (il), mass transfer by radial diffusion in the pore (i2), and mass transfer across the gas film around the pore wall (3) are shown. 21 2 / _ _ __ 'K Ce 2 R j 'p 22 gas is traveling through the :interparticle :.pvce with an average linear velocity v which is given by the ratio of the outlet flow rate, F to the fraction of crcss-sectional area A available jF m v = --A (6) and is corrected for pressure drop across the column by the gas compressibility factor, j. Mass Transfer in the Interparticle Space This model for GSC is described mathematically by the use of the partial differential equation for the mass balance c ac a 2c e e e -6 Tz where time is denoted by t, the length coordinate by z, and the effective coefficient of longitudinal diffusion, D as in equation 5. Since the interparticle concentration c may differ from that in the pore (c.), there may be an increase or decrease in the concentration in either space. Thus, the rate of concentration increase in the interparticle space due to the flow of adsorbate from the pores is given by Q above. Mass Transfer Through the Thin Film Around the Particle If the rate of transfer Q of the sorbate from the pore into the flowing carrier gas is assumed to be linearly dependent on the difference between the actual and the equilibrium concentrations, then 23 Q = -H(K c *-c. ) (8) C C c e 1/ where 11 is the rate constant and c. is the concentration at the c I/r=R entrance to the pore where r is R. The equilibriuL constant Kc in the equilibrium state is given by C. K =-I m (9) c c E.c e i e where in is the amount of adsorbate in a certain pore volume unit where Kc = 1 if the concentration outside of the particle is the same as that within the pores. Radial Diffusion Within the Pores Further, the model is described by the partial differential equation for the rate of desorption Qn of sorbate molecules from the pore wall 2 ac. 2 c c. D + --- =Q (10) 7 t r( 62 r or Qn r where the concentration of the sorbate in the pore c. is a function of the time t and the coordinate in the direction of the pore radius r. At the particle center, r=0, and at the pore entrance on the particle surface, r= R. WIhen dealing with spherical particles the shape factor V equals three. Concentration c. can increase or decrease due to 1 adsorption or desorption, limited by transfer through the film at the pore wall, and is dependent on the effective rate of radial diffusion, D. r 24 Mass Transfer Through th Thin Film at the Pore Wall If the rate of transfer through the pore wall film is assumed to be linearly dependent on the difference between the actual and equilibrium concentrations as for the film around the particle, then Q = -1 (K c. n) (11) where n is the amount adsorbed on the surface of a certain pore volume unit. The equilibrium constant in the equilibrium state is given by n n K = e= (12) n P.C. m 1 1 and H is the rate constant of adsorption. n Solution of the Mass Balance Eqcations Kucera [6] solved equations 7 through 12, which were later used by Grubner [8-11], with certain boundary conditions by Laplace transformation of the partial differentials to ordinary differentials, For the boundary and initial conditions c (z,t) = c (r,z,t) = n(r,z,t) = 0 Se(z,t) = c e,(z) c.(r,z,t) = c (z) = ,0 n(r,z,t) = n (z) (13) (14) (15) (16) (17) and c (z) = cM a (z) i,0 a the conditions given by equation 13 are for t <0 and z= and those given by equat ions 14-16 are lor t = 0, -- < z <+ and 25 r 0 : r < I. The quantities c (z) c (z) and n (z) describe the e,0 1,0 0 initial distribution of the narrow band of solute introduced onto the colum. The boundary conditions in equation 13 are reasonable, since there are zero moles of solute adsorbed over the whole length of the column before injection (t n assume finite values (equations 14-16). The sample is assumed to enter the column as an infinitesimally narrow plug, ideally described by a constant cNIa times a Dirac function, 8(z) (equation 17). In practice, injection profiles as narrow as a few msec wide have been achieved in recent high precision GC systems [42,50]. The transform f(s) = L(f(t)) = f(t)exp(-st)dt (18) 0 may be applied for c (z,s), c (r,z,s), and n(r,z,s), obtaining the e 1 following time based transformed equations: de d c e e~ sc c (z) + v D Q (19) e e,O dz p d2 c 2~ C 0 V-1 e) ~ sc. c. (z) D r + Q (20) 1 i,O ~ r2 r Cr! n and sn n (z)= -Q (21) 0 n where s is the transform variable for the Laplace coordinate system. IWhen the solution c (z,s) in Laplace coordinates is found for e equations 19-21, it is not easily inverted to normal coordinates in order to obtain even a simple analytical expression for the original 26 distribution profile c (z,t). It is, however, possible to obtain the e statistical moments of the distribution by use of a series of Hermite polynorials for c (t) whose expansion coefficients can be stated in terms of the Laplace transform property dnc (s) tnc (t)dt = (-1) lim e (22) k 0 s-0 ds where n is the order of the desired central moment. The statistical moments can then be arranged in a series which represents the solution of the original set of differential equations. Calculation of the Statistical Moments The total nth absolute moment, u' of the function c (z,t) is n e defined as m U (23) n MO n where m = t nc (z,t)dt for n O (24) n e 0 and the central moments about the mean of a peak, u are defined as u (t-u ) "c (z, t)dt (25) n m 1 e mo 0 for n>1. For chromatographic purposes, the infinite integral of equation 25 may be expressed as (t-uInf M(t) dt u (26) S f(t)dt 27 for n>1 where u is the nth normalized central moment and where f(t) n is the amplitude of the chromatographic signal at time t. Further, with fast data acquisition systems where several hundred points are easily obtained for each peak, the infinite integral of equation 26 may be approximated by the summation / In XE (t.-u')n t iu 1 )Yi U = 1 i (27) n j E Y. 1=1 where Y. is the signal amplitude at time t for j data points. Chesler and Cram [21] have shown that the simple rectangular rule numerical method gives a good approximation of the continuous function (equation 26) if a large number of points are taken with a large percentage of the points in the diffuse edges of the peak. Chromatographic Significance of the Statistical Moments The statistical moments are especially useful, since they describe the position and shape of any distribution, and should therefore be applicable to all chromatographic peaks. Nothing has to be assumed beforehand about the peak shape except that the integral in equation 1 exists and can be evaluated numerically. The Gaussian will be a special limiting case when calculating the moments. The zeroth moment uO, which is the area under a chromatographic peak, can be normalized to one. This coincides with the assumption that all of the input fed into the bed passes through the bed; no fraction of the sample remains permanently or chemically reacted on the bed. 28 The first statistical moment, which is the location of the centroid of the area under the peak, has theoretically been found to be a function mainly of the values of the equilibrium constants K and C K the velocity v, and the porosities e. and e(4,6]. (For simplicn 1 e ity, since there is only the absolute moment u' and the moment around the mean u is zero, U instead of u' will be used to denote the first moment and u will denote the other central moments of the chromatographic peak.) 2D t u =L + (1+O(/K (l+K )+--.(28) 1 v 2 n) 2 v L is the column length, t is the time of duration of the injection Z" 0 (and other system tie constants) and e. (1-e )3 0 = -- (29) C 0 e e There is a very minor dependence of u1 on the effective interparticle longitudinal diffusion coefficient D which is seen only p at low velocities. The time (L/v) (1+0) represents ta or the time required for a nonsorbed inert compound to travel through the column. This time is increased by the amount (L/v)OK (1+K ) for sample comc n ponents which are adsorbed either in the pore space (volumal adsorption) or on the pore wall (surface adsorption). The time of injection, to, can be very important when u1 is small. It is important to point out that the location of the mean does not depend upon any kinetic constants (H nI) and, therefore, not upon the velocity at which equilibrium is established on the column. 29 The second stat i't i moment 2 i roiaied to all of the constants used to describe the particle model and its bed (S. ccR,L), the carrier gas velocity v, the equilibrium consta-ts (K c,K ), and the kincti c constants of adsorption (Ic ) y 2C n u= I + S (1D (1+1)) 2 \ a i/ e n v v 41) +2 2 >0'-K 2 2 4D2 rj (1+K ) f'1+K )-K + 2/ CL D v(v+'2) +-V r C n 2 + ,(30) 1.2 For spherical particles, the shape factor, v, equals three. The second moment is the variance of the probability density curve, and its square root is the standard deviation a- or width U') 0.2 (31) which defines the average deviation of the individual points of the curve from the mean. It can be seen again that usually the D terms divided by a higher power of v will be negligible. The third and fourth moments u3 and u4 get progressively more involved and are extremely sensitive to changes in the leading and tailing edges of chromatographic peaks and to peak symmetry. These moments are as follows. ,12D L 64D p p 3 u 3= ---- + (1+0K c(1.+K)) V V 12D L 481) + -- + OK (1+7tK (1+1 ) + 3 l4 c n v V 30 12 2 I (l+K ) x n D (\ +2) L ((1I ) KI n c n 12D P (GL I ) x~ + c OK 2 r 3 2 (1+K ) n 2 (v (v+2) (v+l) 2 2R (ltK ) n + D) (v-2) r 3 K (1+K ) 20K (1+K ) +) 2' HEI n H cr c K n 2 +1- . (32) 12D L2 U4 + V-- + v 3 16D3L p + 7 V 960D4 4 )(1+OK (1+K ) v 2 /24D L + v 288D2 L + + v 3 960D 2 6 E(1+OKc(1 +K 2 v 9 2 R2(!+K ) n x 4 r 0 (1+K )2 K H cn 2 + 12L + v2 0(1+K ) 2 + Hr c 72D L 144D2 p p 3 4/ V V 2(1+K )2 ~2Kj n11K c D v (v+2) 2 K + T n 48D L +3 + v 196D 2 OKC(1+VK (1+K v 4 X , 2 ID r 2(1+K )3 n 3 2 + (v +2v ) (v+4) 2R2 (1+1K ) n 2 r .(1+K )2 H c K + I n 02(1+K ) 20K (1+K K + + c 11n +112 C n 24L 48D \ + (:---+ 2 ) OKc 31 4 2 6 (5v+12) (K ) R4 ( K ) X3 3 2 2 2 2 D v (v+2)-(\ +10v+24) D (v+4v-) (v+2) r r (5 +12) 0 (1+K ) (6v+12)K X n_ + -_ H c n 2 ~ ,2 4 2 R2 3 (1+K ) 6OK (1+K ) (2 3K )K 2 1 (_ r n n_+ D 2 2 + H H + 2 r v +2v H c n H c n 3 4 2 2 3 (1+K ) 30 K (1+K ) K (2+3K ) K n n n n n n + + 2 -+ 2 + H H H H H H c c n c n n 4 to 120 (33) The third moment will be zero for a symmetrical peak, negative for a fronting peak, and, as is the usual case, positive for tailing peaks. Two hybrid third moments, called specific asymmetry and skew, 31 -3 3 2 are calculated by u3/u and u3/(u2) respectively. These are also useful for characterizing peak asymmetry and have the advantages of being dimensionless and giving numbers which are easily comparable to the values of zero specific asymmetry and zero skew obtained for Gaussian peaks. The fourth moment, which is a measure of the excess or flattening of the peak compared to a Gaussian curve, can be treated as the dimensionless hybrids called specific excess and excess, which are 4 2 calculated by u /u4 and u /u2, respectively. If excess is less than three, the curve is smaller than the normal curve, and if it is greater than three, then the distribution is higher than the normal distribution. 32 All higher odd moments further describe asymmetry, while higher even moments describe characteristics of the peak width. These moments cannot be meaningfully calculated in practice, due to the large errors incurred by raising small signals of large deviation (t-u ) on the tail of chromatographic peaks to a high power (see equation 26). In the next three sections of this chapter, statistical moment expressions for three practical cases of GC will be developed. They are as follows: 1. Gas-solid chromatography (GSC) in which solutes are chromatographed on a porous adsorbent. This is the general case examined by Kubin [5], Kucera [6], and Grubner[l11] where longitudinal diffusion, external diffusion, internal diffusion, and adsorption on the pore wall surfaces may be simultaneously significant. 2. Gas-liquid adsorption chromatography using a porous adsorbent that is coated and impregnated jith a liquid in which the solutes are highly soluble. This case is commonly called gas-liquid partition chromatography(GLC) and the processes of longitudinal diffusion, external diffusion, internal diffusion, and sorption within the pore volume itself must be considered. The equations mentioned above for GSC will be extended to include this particular case of GLC. 3. Gas-solid adsorption chromatography in which solutes are chromatographed on a nonporous adsorbent. In this limiting case of the general example of GSC, there is no internal diffusion, so that only longitudinal diffusion, external diffusion, and external surface adsorption must be considered as simultaneously occurring chromatographic phenomena. The Case of Longitudinal, External, and Internal Diffusion and Surface Adsorption In the general case, as for very porous large surface area silica gel, the processes of longitudinal diffusion, external diffusion, internal diffusion, and surface adsorption may occur simultaneously in the gas chromatographic column. In such a case, there can be few simplifications of the moment equations, and they must be solved exactly. For activated silica gel (Figure 1) it is assumed that only surface, and not volumal, adsorption is occurring, so that K = K and K =1. It will be assumed for the remainder of this disn n cussion that terms with higher powErs of D /v are negligible and that v= 3 (spherical particles). Then simplified expressions may be derived for u1 through u- from equations 28, 30, and 32. These can further be expressed in terms of band broadening by 4 ancd in terms of band asymmetry by Z: u2L $=2 m (34) 2 2 u3L 2 and Z 3 cm (35) u1 The parameter 4 is analogous to the Gaussian-based HETP (height equivalent to one theoretical plate). Temporarily, t0 will be assumed to be unimportant, since it is usually very small compared to retention times of retained adsorbates. It will, however, be included in the analytical moment expressions in the Results and Discussion chapter of this dissertation. Then from equations 28 and 30 L u (1+0(1+K )) (36) v n 2 2 2 2 u 1 L-2 (1+0(1+K )) (37) 2n v 34 2D L P (-, u = ~ (i+f (1+K )) 2 n [2 2 2 2 Rk (1+K ) K(1+ ) 2Ln np8) + 15D (38) r cn whereupon 2 2 u L 2D 2$5R (1+K ) v 2 p n + 2 U 15D (1+(l+K )) 1 r n 2 2 2g(1+K ) v 2PK v + n n (39) H (1+0 (1+K )) H (1+0 (1+K )2 C fl n n Defining D as in equation 5, p 2 2 2D 2OR (l+K ) v *1 = 2A + 2Dg+ 2R21-Kn)2 15D (1+(1+K )) r n 2 2 2 22 (1+K ) v 2Y5K v + n 2+ n (40) H c (1+0 (1+K )) H (1+0 (1+0 (1+K C n n n In the event that for very strong adsorbates K n>>1, say K n>20, then equation 40 simplifies to 2D 2 g 2R v 2v 2v v 15D 0 H -+ (41) r C n This is further simplified to 2D 2 g 2R v 15D= 2A + +D 2 (42) when the rate of mass transfer (H) and the rate of adsorption (II C n are infinitely fast, i.e., H c~ Thus, the C term is determined 35 only by radial diffusion in the pores. In this derivation, the A term is equal to one half of the AD value found for the classical van Deemter equation 43. HETP = A + + Cv D v (43) where A, B, and C are the coefficients for eddy diffusion, molecular diffusion, and resistance to mass transfer, respectively. Using equations 28 and 32, expressions for Z may be found in the following manner: 3 3 L 1 X V (44) 2 3 [2 2 12D 2Ly 12D 10X R (1+K ) u3 5 + V3 13D r S(1+K )2 n H C 610[2R(1+K) 315D r 2 3 2R (1+K ) 0 n 15D1 H r c 2 2R (1+K )K n n +- 15D H r n 2 3 0 (1+K ) + 2 + HC C 20K (1+K ) nn + H n c n K n 21 H n (45) where X will be defined as (1+0(1+K )), and then 12D2 p Z1 ~ 2 + V 4D OR2(1+K ) 2 p n 2 + 5D X 12D 2(1+K )2 p. n 2 + H X C 12D OK p n 2 H X n 4 3 2 40R (1+K ) v n + 2 3 105D X r 2 2 3 2 42 2 (1+K ) v n + 3 5D H X r c 2 2 40R (1+K )K v n n + 3 5D H X r n 3 3 2 2 2 63 (1+K ) v 2 12 K (1+K )v +n + n n + + 23 + IX HH X C C n n . 2 3 11 X n (46) 36 Substitution of the expression for D gives Z1 1 1), 12D 24D A ,+ 1 2 2 v'' 12D 2(14K ) g n + 2 Ii 4 32 40R (1+K )v + 2 3 + 105D X r 2 2 4D iR2 (1+K ) 2 g I 2A + 2 5DrX 2 2 4A R (1+K ) v 2 5D X 12AO (i+K ) v 2 HCX 2 3 2 4 R (1+K ) v2 n 3 + 5D H X r c 12D @K g n , HnX 12AdK ,v n 2 HX 2 2 40R (1+K )K v n n 5D H X r n 3 3 2 65 (1+K ) v + 2n 23 HXl 2 2 1202 K (1+K )v n n 3 1c H nX 2 60K v n 2 3 1nx (47) for the case where diffusion and kinetic processes jointly control the peak shape and K is small. When Kn is large, then X- "K and 2 12D Zy -- + 1- 2 V 24D A --- + v 2 12A * 2 24D R 5D r 2 4AR v + 5DO r 12D 12Av 12D 12Av H H H OK HnOKn c c qnOn1 n n 4 2 4R v + + 22 + 105023)2 r 2 2 4R v 5D I I r c 2 2 4R v r n n 2 6v + 2 H c 2 12v c n n 6v 2 + 222 n n (48) For the case where only radial diffusion is important (H c,n D =D), r r 12D 24D A 4D R2 2 Z' + z-- + 12A + g 5vD v2 v 500 5D 4 2 4R v 2 2 1050 D r (49) 37 The Case of Longitudinal, External, and Internal Diffusion and Volui-al Adsorption If the pores of silica gel are filled with a compound in which the adsorbate is highly soluble, then volunal adsorption will be important rather than surface adsorption. Therefore, K = 0 (and H -) n n and K = K The processes of longitudinal diffusion, external diffuc c sion, and radial internal diffusion may simultaneously be important as shown for a single particle in Figure 2. The thin liquid filmssurrounding the particle and on the pore walls are designated by 3 and 4. The processes of penetration of the thin liquid film around the particle, radial diffusion in the liquidfilled pore, and penetration of the thin liquid film on the pore wall are denoted by i i., and i 6 respectively. The processes for the gas-liquid system are similar to those shown in Figure I for the gassolid system and the concentrations, effective diffusion coefficients, and rate constants in the mass balance equations have analogous meanings. Thus, the moment equations can be derived similarly and the expressions for '2 and Z2 under various conditions are found as follows: L u = (1+OK ) (50) 1 v c 2 2 L 2 = (14-K ) (51) 1 2 c 2D L 2LK r 2 p 2 cg R _u2 =- (1+K ) + + (52) 2 3 c v 15D9 v r c Figure 2. Model particle for OV-101 absorbent. The external and internal wall liquid films are designated by 3 and 4. The particle diameter is 2R and ce and ci designate the external and internal adsorbate concentrations. Mass transfer through the liquid film around particle (i1), mass transfer by radial diffusion in the liquid in the pore (i5) and mass transfer across the liquid film around the pore wall (i6) are shown. 39 . ..4 14 \ L - - -- 1 --15CI I I / I / Ce 4 4- 2R 2 R 40 From these equations u2" 2 2 u1 2 2D 2R K v 15D (1+OK ) 2 r C Q 25 K v (1+0K ) H cC (53) which is the same as 2D 9+ g *2 = 2A + 29R 2K v 15Dr (1+OK ) 2 r C 2#2 C +r 2 (1+95K ) H C C (54) and for the case that Kc is large 2D 2Rv 2v *2 = 2A + + 15D K K2 r c cc which simplifies to 2D 2R2v *2 = 2A + --- + 151r5Kc rO c (55) (56) when the rate of mass transfer through the liquid film is fast (c ). The rate of mass transfer through the stagnant gas film which covers the liquid thin film is relatively very fast in all cases. In the derivation of the various Z2 expressions, 32 3 L 3 u -~ (57) V 2D2L A3 p U3 5 v 12D L9K A p c + 3 v r2 R r H I C 2 2 C] + c 2R v 31 5D 2 r 2 + 15D 111 r c + (58) 41 where f is defined as (1+ ,Kc), and then 2 u 3 z2 -3 u 1 12D 2 v 4D OK I SD A r 2 2 12D 2 K + -- 2 C HA C 2 2 2 42 K R v 2 3. 5D H A r c 4 K R 2 + 23 + 105D A r 3 2 6 K V H A C (59) or with the substitution for D p 12D 24AD zg g 2 2 v v 4D OK F 2 g c + 12A + 2 5D A r 2 2 4AOK R v + A2 2 12D g K, g H A2 C 40 K v C 2,3 1051) r 2 12Ar6 K v C + 2 H A C 2 2 2 42 K R v 3 5D H A r c 3 2 6G K V 2 3 H A C (60) Equation 60 may be simplified if K C>>1, in which case A -OKC, to 2 12D 12AD z g 2 2 v v 2 4D R 4AR 2v +12A + SD OK + 5D rK r c r c 12D 12Av + H + H7K HK HK cc Ccc 4R v + 2 2 2 105D 0 K r C 42 2 4R v 2 5D H OK rc CC 2 6v 2 2 H K C C (61) and to 42 12D~ 12AD 41D R~ 4A v 4 2 Z =- + 12A + +-- +4- (62) 22 v 5D G K 5 D" k 2 2 2 v r c r c 105D 0 K r C when H is fast (H -). c c The Case of Longitudinal and External Diffusion and Surfce Adsoi ti1n When a column is packed with a very low surface area nonpo.Cous adsorbent sucl, as graphitized carbon black, the prir.cipal adsorption will be on the particle surface (K =K ) and the passage through the n n column will be rate controlled at high velocities. Although individual graphitized carbon black crystals are very homogeneous nonporous almost flat polyhedra of about 0.3 micron in diameter, many such polyhedra conglomerate into larger particles. These particles may be sieved and used to pack columns. These particles are quite round (v=3), and they are nonporous (5=0), as evidenced by the very low surface area per gram and by photomicrographs. The model for this particle is shown in Figure 3. In this case, transfer through the thin gas film surrounding the particle i is the same as that in Figure 1, and the process of adsorption i3 is the same as for Figure 1 except that the adsorption is on the external (pore wall) surface. Furthermore, the concept of the thin film associated with the constant H for Figures 1 and 2 is not needed. That is, the rate constant of adsorption Hn has a finite value but it need not n be caused by a thin film. It is assumed that D -- and 8 is nonexistent so that now the r porosity function 0 is simply (1-e )/e It is correct to assume that e e Figure 3. Model particle for graphitized carbon black adsorbent. The external gas film is designated by 1, the diameter by 2R, and the external adsorbate cjncontration by c,. Mass transfers through the .xtcrnal gas film (il) and across the gas film at the particle surface (i3) are shovwn. 44 / I \13 / / / K I I 'I B 2R I- 45 K =1 since we are dealing only with the surface, which is the interC face, for the two previous porous models, where c. = c Since the 1 e original transformation yielded a solution for c (z,s), i.e., the interparticle concentration profile at elution, the resulting moments for the present case should be similar in form to equations 28-33. This assumption should be valid because D and H are always independent of each other. Therefore, either can be independently neglected as presented here without resolving the original mass balance equations. Developing the equations for 3 for this case, we obtain L U (1+0(1+K )) (63) 1 v n 2 2 2 u = (1+0 (1+K 2 (64) v p ~ ~ ~ 22LO 1 2D L 2( +K )2 K2] = -^ (1+@(1+K )) + - (65) 2 3 n v HH v c n which give u2L 3 2 u1 2 2 2 2D 22 (1+K ) v 20K p + n 2 + n 2 (66) H (1+0(1+K )) I1 (1+0(1+K )) C n n and 2D 22 (1+ )2 20K v ~. A __ n n _3 2A + v + n + 2 (67) H (1+0(1+K )) H (1+0 (1+K c n n n 46 for the unsimplified cases with 2D 2D 2v 2 v. 3 = 2A + -- + -- + (68) c n for the special case where K >>1 and n 2D 3 = 2A + + (69) n for the further special case where If -+c and only H is rate limiting. For the expressions describing column efficiency in terms of symmetry, 3 3 L 3 (70) i=3 X v where again X will be defined as (1+(1+K n)) and 2n 12D LX 12D LO (1+K ) K p p n__3 5 3 H H v v c n V V 0 (1+K ) 20K n(1+K ) Kn + 2 + H H (71) H 1 c Hn H2 c n giving 2 u3L Z3 3 u1 2 2 2 12D 12D 2 (1+K ) 12D OK p p n p' n 2 + 2 2 v H IX HnX 3 3 2 2 2 2 60 (1+K ) v 122 K (1+K )v 60K v +n n n n (2 + 2 3 +3 23 (72 H X H I IX H X c c n n 47 and 22 2 2 2 12D 24D A 12D (1+K ) 12P' (14K ) V Z3 2 + v + 12A 2 2n v c X H cX 12D K 12A1K V gn n + + + ..... 3 3 2 2 2 603 (1+K ) v 12f 2K (I+K ) v K n II n + 2 3 + 3 + 2 3 (73) H X H IIX H X for the most complicated case of a weak adsorber (K <10) and where n H and H are simultaneously important. C fl For a very strong adsorber where Kn >>1, 2 12D 12D A 12D g g -E2 g l2Av 3 2 v H +T1 v c c 12D + i + 2Av n 'Tn n 2 2 2 6v 12v 6v + +H3 ~2 (74) H2 H H 1n K n'1 2 2 K2 H c n n H K c n n which is further simplified to 2 12D 12D A Z- g 2 Z = 2 + + 12A 3 2 v 12D 2 g 12Av 6v + +I --++ (75) + FK H1 'A 29 2 2 n n n n H K n n if the rate of mass transfer across the thin film is fast and only H is important. 48 There may also be the case where the resistance to mass transfer terms in *3 and Z3 for a weak adsorber is influenced mainly by the rate of adsorption, H (H -c). Then, from equations 67 and 73 n C 2 2D 20K v 2A + + __ n (76) V H X2 n and 12D 24D A Z= -2 g + + 12A 3 2 v V 12D 9 K 12A4K v 60K v2 g n n n 4- -- -(77) 2 2 2 3 H X HnX I1nX Assumptions of the Models Numerous other assumptions were found necessary to simplify the cases of chromatography mentioned above. The models are regarded as closely approximating reality since all known properties of the bed and transport phenomena are included. Other phenomena, however, are neglected: 1. For the very porous particles, adsorption on the external surface is neglected since the internal surface is typically 103 to 105 times larger than the geometric surface area. For the nonporous particles, all adsorption occurs, of course, on the external surface. 2. It is assumed that the carrier gas is incompressible such that there is a negligible pressure drop across the column, thus no pressure gradient from particle to particle. The experimental data can, however, be corrected for pressure effects as will be discussed in the next section. 3. Forced flow through the pores is neglected since the resistance of such small pores (100A') is so large that probably the carrier only flows in the interparticle space, as stated before. 49 4. It is assumed that all effective diffusion coefficients are independent of the concentration and that all transport phenomena are linearly (first order) dependent on the concentration. That is, Kc,n and H1en are not functions of concentration since very small adsorbate samples are used in which case linear isctherms are observed. 5. All processes occur isothermally so that, for example, the heat of adsorption released by an adsorbing molecule contributes a negligible amount of heat to the overall system. Also, the total kinetic energy of the system and the inherent activity of sorption sites throughout the column remain unchanged. Effect of Pressure Drop on the Statistical Moment Equations Many authors have questioned the role of column pressure drop on the chromatographic peak position and shape and, thus, arises the question of which velocity to use in the rate equations describing chromatographic efficiencies. In any real GC column, the pressure drop across it can affect the interparticle and intraparticle diffusion coefficients, the carrier gas velocity, and the partition coefficient between the moving and stationary phases. The classical form of the local plate height LHETP = (A + B/u + C u) + C u (78) D g 2 as well as the form where the eddy diffusion and mass transfer coefficients are coupled LHIETP -+ B/u + C u (79) (1/A D+ 1/C u D g containsu which is the linear velocity of an inert solute at the point in the column being considered. The AD, B, C and C in the terms are the additive coefficients for eddy diffusion, molecular diffusion, and mass transfer in the gas and liquid phases. Giddings and coworkers 50 [70,71] concluded that the average calculated plate height )i should account for The nature of t'e variation of L4ETP along the column and could be written in the form H =fH + jC 7u (80) where 4 2 9 (P -1) (P -1) 8 3 (P -1) ond 3 (P -1) j (82) (P -1) The hri and jC u are the contributions from the gas and liquid phases. The factor ii is the James-artin tme-average column velocity, applicahl3 to comp-cssible carrier gases and P= p. / the ratio of COlumhn 1 0 inlet pressure to column outlet pressure. The pressure correction factor f was derived from considerations of band spreading due to decompression at the end oi the column where the pressure drop per column segment is smaller but The velocity is larger than at the column head. Underhill 129,30,72] used statistical. moment theory to calculate the pressure drop eflcct in a GC column. He started by considering the column with a pressure drop as consisting of a series of short columns, each ha-ving a nearly constant pressure over its length. Extending McQuarrie's calculations [68] for columns in tandem to calculations for an infinite number of column sections, the pressure corrections for the moments covev:ged to the results of Ciddings, above, for the height of a theoretical plate. 51 The Laplace transform for the response y. to a sample input pulse for one of the column segments i is (AL)V. 4D Ksj(, y.=exp F (1+- P' -1 (83) yi 2D .- 2 Sp,i V. where V 0 (84) [ x (P 21) P L D D P (85) P11 2 x 2p)i i - (p2-1) and 3D 2 2 r sR sR 52 K =e + (K e ) r Fcoth \21 (86) e ne 2D tD 11esR C r r Also, -L is the length of the ith segment, V is the superficial carrier gas velocity at the column outlet, s is the coefficient of the Laplace transform, and K is the transform for the effective equilibrium adsorption coefficient. The quantities D ,P, e ,K ,Dr' and R have been previously defined. In this development, it is assumed that the column is operated isothermally and is packed with uniform spheres of adsorbent, and that the principal mechanisms of mass transfer are interparticle molecular diffusion, interparticle eddy diffusion, and intraparticle diffusion. It can safely be assumed that the adsorption isotherm is linear. 52 Furthermore, the intraparticle diffusion coefficient is pressure independent due to the microporous nature of' the adsorbent (silica gel). integrating the Laplace transform C over the length of the column and with respect to pressure, then using the relation 2 3 s u 2 s U 3 C =u0 -su +L 2 6 (87) where un is the nth statistical moment, the following moment equations can be derived: u = 1 (88) P K L 1 n (89) 0 2 2 2 P K L,2 2P (K -e )R L 2P K D L u n 1 n e 2 n p 2 (V) 15D V 3 o r o V 0 P K L 3 4P 2K (K -e )R2 L2 u = I n- + 1 n n e 3 T 2 o 5D V r o 3 2 4 P P K D L 4P (K -e )R L 1 2 n p 1 n e + 4 + 2 V 105D V 0 r o 2 3 2 P K (K -e )R D L 12P K D L 2 n n e p 3 n p (91) + 3 5 4D V V r o o where L (P 2- (P 21)n/2 dx P 0 (92) nI 53 which can be integrated over the column length to give n+2 P = 1 .(93) nn + 1) (P12_1 2 It can be seen that the u equations 89-91 are only slight variations n of the previously derived equations 28, 30, and 32 for GSC. For n= 1 the correction factor P has the same significance n as the James-Martin j factor, since the effect of pressure on the mean retention time is identical in both cases. The mean retention time effectively increases with increased pressure drop across the column due to the decrease in time-averaged column velocity. For the second and third moments, it is seen that a decrease in D due to p pressure drop is counterbalanced by equal decreases in V such that the D terms increase at a slightly slower rate than expected. The D p r terms are only affected by the decrease in velocity with pressure since D is assumed indeperdent of pressure. r n Very little error is incurred if P is equated to P for P<3 n 1 P = = j (94) n I such that the simple j factor calculated from p./p can be used 1 0 exclusively for pressure corrections. Analogous to equation 80, the corresponding values of H when using the statistical moments are = f$ + jA (95) where is the pressure-corrected value of 4 and 4 is the measured value of *, not corrected for pressure drop effects. All of the other symbols have been defined above. It is clear that, depending on the 54 relative importance of the gas and eddy contributions, the value of 4 could increas- or decrease with pressure drop. This is because f varies from 9/8 for infini e pressure drop to unity when there is very little pressure drop. In contrast, j is always less than one; it approaches one for no pressure drop and zero for infinite. inlet pressures. In the usual case, A will be small such that controls the 'g magnitude of $and, thus, the observed efficiency parameter $ will increase with increased pressure drop. Similarly, at low velocities, for microporous adsorbents where interparticle diffusion controls mass transfer, Z is expected to decrease by a factor between one and 5/4 1r2 with increasing pressure drops as 4 3/2 Z (96) 4/2 (P 5_1)(P -1) At higher velocities, with mass transfer controlled by intraparticle diffusion in microporous adsorbents, and Z are expected to require the following approximate corrections due to pressure drop [30]: 4= j* (97) Z 1=/2 Z (98) These equations predict that the pressure-corrected parameter 4 will -1/2 be smaller by a factor of j and Z will be larger by a factor of j compared to the observed 'r and Z under the influence of a pressure drop. These are, of course, the effects of pressure drop alone, which may not be as significant as the interparticle or intraparticle influences alone. It has also been assumed that the intraparticle diffusion is taking place in spherical particles having a log-normal size distribution. EXPERIMENTAL Characterization of the Gas Chromatograph/Computer System A Ovarian Model 2100 gas chromatograph (CC) equipped with a flame ionization detector (FID) was modified to provide precision temperature and flow regulation under computer control, precision sampling, and automatic data collection. The data system was based on a PDP--S/>L on-line dedicated laboratory computer. In order to obtain reliable information from GC peaks, a critical evaluation of the precisior and accuracy of all the system components was made. Efforts were made to minimize dead volume and mixing volumes so that instrumental effects were small. Thus, systematic influences on the precision and accuracy of the chromatographic data were known and appropriate corrections could be made. Pneumatic System The well-regulated flow system designed for the carrier and sample gases is shown in Figure 4. The balanced pressure regulators (Model PN-4120OG49, Veriflow Corp.) were chosen over standard regulators because they are virtually independent of variations in the inlet pressure at the source: the outlet pressure is changed less than 0.02 psi (0.9 mm Hg) for a 100 psi drop in inlet pressure. 55 Figure 4. Pneumatic, computer control, and data acquisition systems. Gas lines are denoted by (t), electrical lines by (-- ) are pressure dauges, are flow dciverters, are threeway switching valves, and are fine metering VaLVeS. RE GUL ATO f-GULATOR TRAPR DEVCE VALV SAFING VALVE 11 VENT PDP-8/L SEE R REITRj COMPUTER ETCO (8K) sF-EE IR E CT ADCR-SVERMjP GAIN AL MAGEIC 71E4 G, TC"PE, TTYP 01 58 The regulators were followed by 500 cc traps of 5A and 13A Molecular Sieves. These were followed by standard single stage pressure regulators (Model 41.300451, Veriflow Corp.) which offer precise control when the inlet pressure is held constant: the outlet pressure varies less than 0.5 psi (20 mm Hg) for a 100 psi change in the inlet pressure. Thus, they were used as a second stage of regulation. In the sample line an extra fine constant upstream flow controller (Model PN-42300080, Veriflow Corp.), which had a repeatability of flow setting of better than 2%, was used. The helium carrier gas (Grade A, Gardener Cryogenics), which had a stated purity of 99.995% and was oil and moisture free, was regulated by a computer controlled mass flow controller (Model FCS-100, Tylan Corp.). The flow controller operates by providing a mass flow rate from 0 to 100 standard cubic centimeters per minute (SCCM), which will subsequently be referred to as cubic centimeters per minute (cc/miin). This flow rate is proportional to a 0 to 5V command voltage set through the computer interface. The controller compares a reference voltage to the output of a mass flow transducer and adjusts a value until the two voltages are equal. A pressure drop (50 psi) was maintained across the flow controller at all times to insure rapid response and correct flow rate. The controller was periodically calibrated by displacing one to five liter quantities of water from a volumetric flask while the controller was set at various flow rates. For most of this work the 4-25 cc/min flow range was used. It was found that precision on the mean flow rate was 0.08% relative standard deviation for the slowest rates and t 0.06% at the higher 59 rates. The repeatability ol a given setting over several months' time was i 0.087 for all parts of the flow range used, which was considered excellent. The voltage output of the controller was continuously monitored using a digital multimeter (Model 160, Keithley Instr.) and was always stable to at least + 0.85 mv, which corresponds to 0.017 cc/min. This is a stability of 0.42% at the lowest flow rates and 0.05% at the highest rates used. The response time of the mass flow controller was observed to be approximately three seconds for a flow increase of 5 cc/min for rates greater than 8 cc/min. This implies that the 0.017 cc/min variations in the stability could be corrected in about 12 msec. The response was slightly slower at 4 cc/min and extremely slow, around 20 seconds, at 2 cc/min. This device was fortunately the most reliable part of the flow system and indeed control of the flow rate was found to be a suitable substitute for independent regulation of inlet and outlet pressure [45,46]. At no time during an experiment was it necessary to interrupt the carrier gas flow at the injection or detector end in order to take a flow rate reading. It was found that maintaining the controller in Styrofoam at room temperature was sufficient to achieve the above stability. The Tylan mass flow controller was factory calibrated to deliver constant mass flow rates of helium. For example, the number of molecules of helium in a 10 cc volume at standard Iemperature and pressure is allowed through the controller when 0.5 v is applied. This mass flow is independent of pressure, since an electronic feedback circuit 60 maintains the balance between actual and standard density and mass flow of helium. PactFact QstdFstd- (99) However, the 10 cc volume is compressible such that the corresponding volumetric flow rate is slower at the high pressure end of the column and faster at the outlet, which is normally at atmospheric pressure. Therefore, the set "mass flow rate" must be corrected for compressibility due to pressure drop. since most chromatographic calculations are based on the volume of carrier gas passed, rather than on its mass. Pressure gauges, flow transducers, or flow diverters (Model 51-000146-00, Varian Aerograph) were inserted into either the carrier or sample lines. Compressed gases or liquids could be sampled directly or diluted through an exponential dilution flask (EDF) before sampling. A description of the EDF and a study of the effects of sample size on chromatographic peak broadening and asymmetry has been included as Appendix B. Sampling Valve A cut-away cross-sectional view of the sampling valve used for most of this work is shown in Figure 5 in the sampling position. This valve was designed to maintain a pressure differential of 2,000 psi, to have minimum dead volume, to be amenable to complete automation, for high precision and minimum injection time, and to avoid interruption of the carrier gas flow while in the sampling position so that steady state pneumatic equilibrium is maintained on the column. Figure 5. High pressure sampling valve shown from the front with a cut-away view of the valve assembly. Valve is in the sample injection position. Sample inlet ports are perpendicular to the plane of the drawing. SOLENOID -- IIOV AC He IN ~~ L SHAFT SEALING NUT -- COLUMN SUPPORT TUBE TENSION SPRING SAMPLE CHAMBER COLUMN TEFLON SEAL INJECTION PORT 1- I!Ov AC 63 A detailed description of the valve has been reported elsewhere [73]. However, it should be mentioned that the sample volume, defined by an annular groove 0.015 in. X 0.005 in. on the flash-chromeplated (0.0002 in., National Bureau of Standards) plunger rod (0.2562 in., National Bureau of Standards) was 1.0 .1 for all of this work. Capillary columns or 1/8 in. packed columns could be placed within 0.094 in. of the plunger shaft to reduce the dead volume. This chamber represents a mixing volume of only 22 pl. When 1/8 in. columns were used, the column support tube in Figure 5 was not necessary. A preliminary study involved the comparison of the performance of this valve with two others by an analysis of their statistical moments for respective column input profiles [42]. A complete description of this study has been compiled as Appendix A. An inspection of Table 17 in Appendix A shows that the valve referred to in the previous paragraph (high pressure) has a very acceptable injection repeatability (area, 0.16,; first moment, 0.33%) and gives a very sharp -3 2 sample profile (variance, 3.095 X 10 sec ; skew, 0.9762) using 2% integration limits [21]. The variance corresponds to a sigma value of 55.64 msec, T, and the skew compares well with symmetrical peaks (skew for a Gaussian= 0.0). Figure 6 illustrates a typical injection profile obtained with this valve. Although this valve did not give the best precision, it was selected for this work because there were virtually no problems in column alignment, column changing, or valve overheating as experienced with the others. At all times the valve body (304 stainless steel) was maintained at 70'C with a 200 watt cartridge heater (Model HS3725, Hotwatt). Figure 6. Experimental valve injection profile for the high pressure valve with integration limits at 2.0%, 0.5%, and L 0. 2% of the peak height. Time base of display is 0-2000 msec. 65 f(t) t 66 The travel distance between the sample chamber and the injection por.. was 0.726 in. This is sufficiently long to provide a large sealing surface with the Teflon seal and yet short enough to retain the pulling power of the solenoids for high speed operation. Fifty-pound solenoids (No. 447-1, Dorneyer Ind.) were used to activate the valve under computer control. The narrow injection profile can be explained in terms of the short effective injection time. Only during the last 2.1% of travel of the annulus will the sample volume come into alignment with the column; thus, the effective sampling time is on the order of 0.3 msec, since the total distance takes about 14 msec. Chromatograph Oven The large air mass oven of the Varian 2100 GC was used for column temperature regulation. Both long term and short term temper-ature stabilities of the oven were measured using a 36 cm platinum resistance thermometer (Serial #1691079, Leeds and Northrup Co.) which had an absolute resistance of 25.5475 ohms, as calibrated at the National Bureau of Standards. Four-terminal measurements were made by wiring the four leads of the thermometer directly to a precalibrated digital voltmeter (Model 8400A, John Fluke Co.) which had milliohm resolution on the 100 ohm scale. In this way milliohm changes in the DVM output were seen as approximately 0.01'C changes in oven temperature. The actual oven temperature T was calculated by the relationship + 2 T -R 0 100) 1(1+d) 41 t 0 T 10 (100) 214 10 67 where a and d are calibration constants specific to the thermometer. R and R are the resistances of the platinum resistor at T' and 00C, respectively. Program 1 in Appendix C shows that the negative root is applicable. The oven, injection port, and detector block temperatures were equilibrated for 10 hours at set points of 50'C, 50'C, and 1201C, respectively. These settings were maintained throughout this work. Over a 28-hour period, 16 temperature measurements 30 seconds apart were made at each of 64 locations in the upper one third of the oven. The 64- locations formed 2 imaginary vertical planes, one in the oven front and one in the back, and 8 horizontal planes. Since the coiled analytical columns used in this research were in the upper third of the ovcn, the temperature stability was only determined for this region. Evaluation of this temperature map indicated that the stability at a single point for 8 riinutes was always at least 0.020C at the 95% confidence interval. There was a gradient of about 0.10C from the top of the oven, next to the detector oven, to the horizontal plane one third of the way down. Stability within each of the 2 vertical planes and all 8 horizontal planes was about 0.03'C except the planes one centimeter from the oven top or back where it was about 0.08 C. Sixteen locations in the oven were selected as being nearest and encompassing the region where the analytical columns were later placed. The average temperature within the 95% confidence interval was 54.005-0.030 C. The precision on this mean temperature is slightly less than the precision at any one location. In fact, the average temperature for all locations, including the gradient but not the warm 68 points nearest the top and back, was 54.091 t0.027' C. These 0.03'C stabilities very nearly approximate one of the specifications needed to achieve at least a 0.1% precision in retention time measurements [45]. Chromatograph Detector The Varian Model 2100 FID contributes a time constant, Td = V/'F due to the effective sensing volume V and the flow rate, F M, of carrier gas [43]. As a worst case example, if the sensing volume of the detector is taken as 1% of its volume (28.8 pl), then d = 19 msec at F = 12. Glenn [74] attributed the finite detector response time to the speed of production and transport of ionic species and to the detector itself acting as a resistor-capacitor network. Although some authors [50] have dismissed this time constant as negligible, it has been added to the system time constant T in accordance with others [43.44]. The hydrogen and oxygen (technical grade, Air Products) flows were regulated by two-stage regulators (Model No. 8-350 and Model No. 8-540, Matheson Co.), passed through 500 cc Molecular Sieve traps, and finely controlled by micrometering valves (Model PN-43000285, Veriflow Corp.). Optimum flow rates for the least flame noise and maximum response were found to be 35 cc/min for hydrogen and 150 cc/min for oxygen for helium flows less than 30 cc/min and could be measured through flow diverters. Electrometer A Barber-Colman Model 5044 electrometer was used to amplify -14 -5 FID currents in the range 1 x10 to 1 X10 amp to a 0 to 10 volt signal. While offering high sensitivity for small signals, this 69 electrometer also had a very low noise level on its most sensitive scale. With the 4 it. grapinU-Jined coaxial cable (Chester Cable Co.) from the FID attached and the flame off, the electrometer noise was less than 0. 5 x 10 14 amp with no detectable drift over a twohour Deriod. The electrometer time constant,7 e, was measured for two ranges (X O.1 range = 10-12 amp full-scale; X 1. O range = 10-11 amp full-scale) using the simple voltage divider circuit in Figure 7. The resistor network was placed in a shielded can made of copper. In order to measure T, SWJ1 was closed, several seconds elapsed until a steady state current was established through RI, SWi was disconnected, and the decay of the electrometer output was monitored with the data acquisition system described below. The constant ,e was defined as the time required to decay to 37% of the steady state level. Values for Rl were 51.11M2 and 4.5 M. in order to simulate FID currents of 0.6 X10 12 and 0.67 X10~11 amp (60% and 67% of full scale) for the XO.l and Xi.0 ranges, respectively. The decaying electrometer output was sampled every 20 msec for the most sensitive range and an average Te of 290 msec was obtained. The X1.0 range, which was used for sharper, taller peaks, was sampled every 2 msec giving an average Te of 9 msec. These times include the time constants for the FID connecting cable. On the x0.1 range, where most of this research was done, the electrometer was by far the largest contributor to the overall system time constant. Figure 7. Voltage divider circuit used to measure electrometer time constant. s(1) SW 3V __ 1.0 M Q RI TO ELEC TR0MVETER r j ImLm GND 72 Computerized Data Acquisition and Control System-Hlardw.are Figure 4 also illustrates the computer interface used in these studies. The system also included a computer controlled mass flow controller described above which is not shown. The details of the hardware are presented elsewhere [40] but it is important to point out here that the system does include a 10-bit successive approximation ADC (Model A811, Digital Equipment Corp., DEC) with sample and hold amplifier (Model A400, DEC), a programmable gain amplifier, and a programmable clock. The computer is a PDP-8//L (DEC) with 8K of core. A four-tape magnetic tape cassette system (Model 4096, Tri-Data Corp.), a high speed paper tape reader (Mark V, Data Scan Corp.), and ASR-33 teletype (Teletype Corp.), and a dual channel oscilloscope (Type 547-.1A2, Tektronix, Inc.) were the principal peripherals used. The PDP-8/L computer controls external devices by selecting its 6-bit device code and then sending control pulses. For example, the device selector for control of the sampling valve is shown in Figure 8 as the device code and the input/output control pulses (IOP) were brought out from the buffered memory buffer (BM) register of the computer. The device code is used to enable NAND gates 3, 4, and 5 upon receipt of a 12-bit 614X instruction, where X is an IOP 2, 4, or 6 pulse. The valve gate to switch the valve to the sampling position is generated by the IOP 2 pulse which clocks flip flop no. 1 (FF1) to a logical "1" at the Q output. Then a 6146 command enables NAND gate 5 and resets FF1 and FF2. The next instrnetion generates an IOP 4 which puts a logical "O" on the clocked input of FF2 and thereby Figure 8. Device selector logic for computer control of the sampling valve. Component 1 2,3,4 5 FF1,FF2 Description NAND Gate, 9007, NAND Gate, 9002, NAND Gate, 7400, Flip flop, 7410, Fairchild Fairchild Fairchild Fairchild 0 DEVICE CODE 0FROM CPU I 0I loP 2 0Fo VALVE GATE I IOP 4 0 o VALVE GATE 2 BMB09 5 BM B 10 -:1 75 fires the second solenoid to return the valve to its original position. The timing sequence for actuation of the valve is shown in Figure 9. The duration of the valve gate pulses is variable in the software. The time fo- the valve gate 1 pulse for the sampling valve described earlier determines the amount of time during which the sample volume is flushed with the carrier gas in the sample injection position. The valve gate 2 pulse width corresponds to the time needed to return the valve to its normal position (about 12 msec). The solenoids were fired and switched by the driver circuit in Figure 10. The positive going input pulse from the device selector is driven through the emitter coupled transistors in a Darlington configuration in order to increase the current gain at the output and to reduce the loading on the device selector by the high impedance input. The RF interference filter (Filtron 9605) serves to keep the electrical noise, generated by switching the inductive load, out of the digital logic system. The diode around the solenoid acts as a low impedance path to allow the magnetic field to collapse when it is cut off without damping the current in the coils through the 2N5877. The 18 ohm resistor is a damping resistor. The use of the programmable gain amplifier allowed gains of 1, 2, 4, and 8, giving the 10-bit ADC an effective 13-bit resolution. This was extremely useful for determination of the very diffuse edges of chromatographic peaks. The sample and hold amplifier required 12 psec to track to 0.025% accuracy, and the ADC required a conversion time of 10 psec. These times were provided by software delays which added a negligible contribution to the overall system time constant. Figure 9. Timing sequence under software control for generating a pulse width for sample injection (valve gate 1) and for returning the sampling valve to the normal position (valve gate pulse 2). RESET FF SET CLOCK IOP 2 VALVE GA IOP 4 FE I I I I I I I I Ii I, I I I I II I I _____________LI I ____________________I, I F III I. I I I, I I I I ~ I I I I II I It I ___IF-f VALVE GATE 2 |

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PAGE 1 PHYSICOCHE1\1 ICAL ~ !EASUR mlEN TS BY HIGH PRECISIOK GAS CHR miATOGRAPHY By BARRY EUGENE BOWEN A DIS SERTAT IO N PRESENTED TO THE GRAD UATE COUNCIL OF T H E UNIVEHSITY OF FLORIDA IN PAHTIAL FULFILL MHlT OF THi: lLEQUR E:1 1 1:NT S FOR TH: E J EGll E E OF ooeroR OF PHILOSOPHY UNI .' EI'.SI'l'Y OE' PLORIDA 1973 PAGE 2 The autho1 j s proud to dedicate this r esea rch effort to hi s w if e, Kathy for h e r love an d d evot ion durin g the course of t h:l.s work and to hi s u ncle an d and aunt Dr. E nd Mrs Ra y A. M ill er e du cato rs, for financi a l support du r i ng hi s und erira du2~e study and for th e ir encou-ragemeat throughout this nc aclE:mi.c e nd eav or. PAGE 3 Th e author wishes to acknowle d ge his r esearc h adv i sor and su pervisory committee c hairman Dr Stuart P C r am formerly a n a ss i stant profes s or at the Univers ity of Florida, for the initi al sug gestion o f th i s r esea r c h prob l em an d for financial s~pport. Oth e r m em b e rs of the a uthor' s superviscry corn;nitte "' Drs. Roge r G. Bates ( cochairman), Gerh ar d M S chmi d, Fra nk D. Vi ckers, Arthur W Westerberg and JameR D. W inefordn er are a l s o acknowledged. The e fforts of Dr Cram and Dr W Wayne Me irike towar d arr anging the valuable opportunity of com p l et in ~ this r esea rch at the :N"ai.:ional Bureau of Stanc! ar ds, Gaither sb urg, Mar yland, a re gr a te fully acknowledged This experience w ould not h a ve be e n possibl e w it h out the cooperation of the Graduate School of the Univer sity of Flo rid a and the Dep a rtm e nt of Chemistry, from which much of th e chromatograph / co mp ut er system was borro w ed. M any infor mat ive di sc u ss ions with Dr. John E. Leitner during the dev e lopm en t of the comp t !ter interfac e hardware a nd software are since re ly acknowl e dged. Discuss io n s with Dr Robert L. Wade conc er n ing the de sig n of his high press ure sa m plin g valve and w ith Dr. Step hen N. Chesler conc erning st at istical moment th e ory we re he l pful iii PAGE 4 Thanks go to Mr. noger V. Krumm fo1 1.:he translation of the papers by Kubin. The author extends si n cere app r eciation to his wife Kathy, f or her efficiency and cooperation in typin g the first drnft The exce llent work of hlrs. Edna Larrick on the final dr aft is acknowl e dged. iv PAGE 5 TABLE OF CO NTENTS ACK~O\VL1::DG]\1E?IT S LI ST OF TABLES LI ST OF FI GlTRES LI S T OF SYMBOLS ABS TR<'i CT I:NTROD UCT ION Obje ctive of thi s R esear ch Statisti. ca l },bme n ts in Physicochemica l Analysis High Precision Gas Chromatogr apJ1i .<.:: Me::,suTements 'l' IIEORETI CAL Develop ment of Theo reti cal l\Iode l s for G as Chromatography Selec tion of a G as Chromatograp h ic Mode l Mass Transfer in the Interparticle Space Mass Transfer T hro ugh the Thin Fi lm Around the Particle Radial Diffusion Within the Por e s Mass Transfer Through the Thin Film at th e Po~e WaJl Solution of the Mass Balance Equations Calcul atio n of the Sta tistical M oments Chromatographic Significance of th e Stati .sti cal Mome nt s The Case o:f Longitudin91, Exter n a l, and In terna l Dif f u ai on and Surfa ce Ad s orp ti on The Case of Lon gitudina l, Externa l and Intern a l Diffu si on a nd Vo l umal Ad sorptio n The C ase of Lon g itudinal and Exte rnal Di ff u si on and Surfac~ Ad s orption .... V Page Hi viii X xvi xxii l l 2 8 13 13 16 22 22 23 24 24 26 27 37 42 PAGE 6 TABL E O T-' CO:'iTE:Yn, (Co:1tinued) Assu mptions of the M odels Effect of Pressure D rop on the Statistical t.!o m ent E quatio ns EXPERi. MT~ NT f,.L Ciar:icterization of the G as C hromut o g rc i ph / Computer System Pn euma tic System Sa mp ling Valve Chromatogr a ph Oven Chr omatograph Detectcr Electromet e r Computerized D 2. ta Acquisition and Control System H ard.war e Co mputer i zed Data A cquisiti o n and Control System So ftw are Charact e:ri.z ation of the Gas Chro ma"'.:: o g raphic C ol u mns Preparation of Columns D etermin ation of Pa rticle Diameter Det erm ination of Permeability and Externa l Porosity RESULTS A ..l'IT) DISCUSSION Determination of Permeability and External Porosity Det erminatio n of Internal Porosity. Determination of Binary Mo lecul ar Diff:1sion Coefficients Measure me nt of Equilibri um, Diffusion, and Rate Constants Silica G e l Columns Me as ure ment of Equilibrium, Diffusion, a n d Rate Con sta nts OV-101 C.:ilumns Measurem e nt of Equilibri um, Diffusion, and Rate C onstants Graphitized Carbon Black Columns Net Ret e ntion Volumes Fre e En e rgy Cha nges vi Page 48 49 55 55 55 60 66 68 68 72 80 98 98 100 104 108 108 118 125 132 166 175 189 194 PAGE 7 TABLE OF CO it fENTS ( ( \rn ti m1 e d) Effects of Ph ys icochemicc1l 1-'arametcrs on Chro matog raphic J3and B:coad e ning. Eff ec ts o .f Phys icoch emi cal Parameters cm Chrcmato g raphi c Ear:d Asy1u m ctry Co :r -rcl ati0n of I; :--:p c ri r11enta l w: th Thcore-i;ic a l R e s ul ts 8nd A naly sis of Errors Summary APP EN71! o:s A. High Precision Sa:mpl i ng in C-as Chromatogra p l1y B Effects of Samp l e Size on Ga s Chromatographic Behavio r c. Computer Prog;raJils 1. Calculation of Temperat ures Using a Platinum Resi .sta nce Tl ::. e1mometer 2. Statisti cal Momen t Calcu l ations of Stored Peaks 3. L east Squares Fit to a Polynomial 4. Newton-Raphson Tec1rnique Jor Solving tbe Perm ea bility Equa tion fo r External Porosity 5. Calculation of Bina.ry Diffu sion Coefficients from the H i r sc hf e ld er Equation 6. L east Sq ua res Iterative Fit to an Equation of the Form E = A+ B / v + Cv BIBLIOG RAPHY BIOGRAPHICAL SKETCH Vii Page 2 0 4 225 24 1 251 254 255 273 30 1. 302 303 306 308 309 310 312 319 PAGE 8 . LIST OF TABLES Table 1. Identific at ion of G as Chromatographic Colus ... 2. Perm eabi lity and Poro sit i e s of Gas Chromato g raphic 3. 4. 5. Col umns . . . ...... Reproducibility in Preparation of OV-101 Colum:.'1s Comparison of Molecular Diffusion Coefficients in Helium . . . . . . Equilibrium, Diffusion, and Rate Constants for Hydrocarbons on Silica Gel ........ 6. Absolute and Relative Contribution s to Re s istance to Mass Transfer for Hydrocarbons on Silica Gel 7. Equilibrium, Diffusion, and Rate Constants for Propane on OV-101 Im pregt 1at c.:d Silica G e l . 8. Absolute an d R e l ative Contributions to Resistan ce to ~Tass Tr a nsfer for Propane on OV-101 Impr egnated S ilic a G e l 9 Equilibrium, Diffusion, and Rate Constants for Hydroc arbo ns on Graphitized Carbon B l ack 10. Absolute and Relat i v e Contri buti ons to Resistance to Mass Transf e r for Hydroc a rbons on 11. 12. Gr aphitized Carbon Black ..... Free Energy of Adsorption for Hydrocarbons on Adsorbents .... Average Adsorption and Desorption Times for Hydrocarbons on Adsorbents. 13. Values of the Rate Coefficients A, B, and C Calculated by Least Squares Using Page 103 116 125 131 142 153 169 172 180 188 202 203 Experimental Data . . . . . . . 222 viii PAGE 9 LI S 'f OF TA GLE S ( C oi 1U nued ) Table 14 P reci sio n of S t a t j st:i c a J ',fo me nt C al culat i on s a t 2 .0 % Int eg r ation Li mits f or Hyd roca r b o n s on Silic a Ge l a t 5 4 c ......... 15. P r e cisio n o f S ta tis t i cal Mome nt C a lculatio ns a t 0.39 % Int eg r at i on Li m j ts f o 1 H ydroc ar b o ns on 16. Sili ca Ge l at 54 C Flo w R ates U s e d to 1\ Ie a s ur e Sampli ng Valve Charact er istics ..... 17. Co m parison of tl,e l\I e an Va lues of the Statistical M om e nts an d Pr e ci s io n fer Valve G en e ra te d Input P eak Profiles ........ 1 8 Comp c1 ri s on o f the Pr ec i s ion of S e veral Automated Sam p lj ng Va lves w it h U n r e t ain ed So lutes ( Referenc e ) ......... 19. Eff e ct of Limits of Int eg ration on t h e Mean V a lue and Pr e ci s1 on of th e S tatistic a l I\I oment Calcul at ions .. ix Page 247 248 260 267 270 272 PAGE 10 LIS T OF FIGURES Figu re 1. Mc1de l particl e for s ilica ge l adsorbent 2. fllodel particle for O V -101 a dsorb ent 3. Mo d e l p article for graphiti z ed c a rbon black adsor bent 4. Sch emati c d iag ram of pn e umatic, com p ut er contro l, and data acquis i tion system s .. 5. High pr ess u re gas sampling va l ve sho w n as a cut a w ay view 6. Experimental v a lve injection profil e for the high pressure valve sh ow in g 2 0 % 0.5 % and 0.2 % in tegr ation limits 7. V olt age divid e r c ir cuit u s ed to measure e l ectro m ete r time con stants 8. D e vice selector lo g ic for comput er control of samplin g va l ve .. 9. Timin g seq u e nce und er s oftware con t rol for generat ing sample v a lve pulse w idths ..... 10. Driver circuit for a u tomati ng so l e n oi d actuated sampli ng valve . ... 11. Flow di agram fo r the on-line co m puter control an d calcula tion pro g r am ( ADC O M ) .... 12. Flow dia gram for AD C O M, continu ed 13. F l ow d i agram fo r ADCOM continu ed 14. F l ow di agram for ADCCM, continu e d 15. Flow di agram for ADCO M c onti nu ed 16. Flow diagram for ADCO lll continu e d X Page 21 39 57 62 65 71 74 77 79 83 85 88 90 93 95 PAGE 11 LIST C;F FIGURES (Co nti nu ed ) Figur e 17. Flow di ag ram f or ADCOI.1, cont i nued .... 18. Pa rticle size distributi o n for 120 / 140 me s h silicn gel 19. Pressur e tr ans duc er c a l ibratio n plot 20. P e rme a bility calibration plots for silica gel columns 21. P e rme a bility calibratio!l plots for OV-101 columns .. 22. Permeabi lity c a libration plots for graph itized carbon black col urnns . . . . . . . . 23. The effect of external porosity on the permeability porosity function 24. Plots of calculat ed a nd experimental inert eluti on times as a function of average colunm ve locity for silica gel columns . . . . . . . . 25. Mo lecul ar diffusion as a function of velocity thro ugh an open tube for m et hane, eth ane propane, and n-but ane in helium ........... 26. Reduced fir s t moment vs L / v for methane chroma tographed on silica gel at 54C 27. Reduced first moment vs. L / v for ethane on silica gel . . . . 28. Reduced first moment vs. L / v for propane on silica gel . . . . 29. Reduc ed first moment vs. L/v for n-butane on silica gel . . . . 30. 2 Reduced second moment vs. 1 / v for methane chromato. graphed on silica gel at 54 C . . ... 31. 2 Reduced second moment vs. 1 / v for ethane on silica gel . . ...... 32. 2 Reduced second moment vs. 1 / v for propane on silica gel .... 33 . Reduced second moment vs 1 / v 2 for n-butane on silica gel . . ..... xi . Page 97 102 107 111 113 115 120 124 129 135 137 139 141 146 148 150 152 PAGE 12 L IST O F F IG URES (Continued) Figure 34. 2 Tol a l mass transf e r resistance vs. R for methane chr omatographed on sil ica gel a t 54 C 2 35. Total mass transfer resistance vs R for e than e 36 3 7 38 39. 4 0. 41. 42. 43. 44. 45. on silica gel Tota l mas s transfer resistance vs. 2 for R propa ne on silic a gel . T o ta l mass transfer resistance vs. 2 R for n-but ane on silica ge l Reduced first mome nt vs. L / v for propane chromatograph e d on OV-101 at 54C ......... 2 R educe d secon d moment vs. 1 /v for propan e chromat ographe d on OV -101 at 54 C 2 Total ma ss transfer r e s ista nce v s R for prop ane chromatogr aphed on O V -101 at 54 C Reduced first moment vs L / v for propane chromato graphed on gr aphi t i ze d carbon black at 54 C Reduced first moment vs L / v for n-but an e on graphitizcd c arb on black ..... 2 Reduced second moment vs. 1 / v for propan e chromato g rap hed on g r aphi ti zed carbon black at 54 C ........ 2 Reduced second moment vs 1 / v for n-butane on graphitized carbon black ... 2 Total m a ss tran s fer r e sistance vs. R for propane and n-bu tane chromatogra phe d on graphitized c a rbon black 46 Net r etention volume as a function of flow rate for methane, ethane propane, and n-butane chromato grap hed on silic a gel at 54 C 47. Ne t retention volume a s a function of flow rate for prop ane and n-butane chromato gra ph e d on graphitized carb o n black at 54 C .. 48 Plot of lo g (u 1 ua) vs. lo g (60 LA e/ j F m) for methane, ethane, propane, and n-butane chro mat ograph e d on silica ge l a t 54 c ..... xii Page 157 159 161 163 lGS 171 174 177 179 183 185 1 87 191 196 199 PAGE 13 LI. S '~ OF FIGURES (C o11t: ~ m1 e d ) Fi g ure 4 9 Plot of lo g (u 1 uci) vs. lo g (60 LAe / jFm) for propane and n-bu tane c} 1rornat o g r :1phed or: g raphj_ tiz ed carbon black at 5 C. . . 50. Colu 1ru1 efficiency in terms of band broad e nin g t versu s line ar v e loci~y v for metha n e chronrato graphed on silica g el a t 54 C 51. 52. 54. 55 56 v vs. V for e than e on silic a gel 'V vs V for prop a n e on silica gel . vs. V for n-butan e on silic a gel . Co l umn efficiency ir1 terms of band broad en in g 1 versu s linear velocity, v, for pr o pane chrom a to gra ph ed on OV-101 at 54 ~ c Column efficiency in tern-,s of band broadening, ~ ver sus l ine ar velocity, v, f or propan e ch r omat graphed on graphitized carbon bl a ck at 54 C t vs v for n-butane on graphiti ze d carbon bl a ck 57 Column efficiency in terms of b a nd asymmetry, Z, versus line a r v elo city, v, for methane chromatographed on silica gel at 54C 5 8. Z vs v for ethane on silica gel 59 Z vs. v for prop a ne on silica gel 60 Z vs. v for n butane on silica ge l . 61. Col unm ef ficiency in terms of band asymmetry, Z, versus linear velocity, v, for propane chrom atog raphed on Page 201 208 210 212 214 216 218 220 227 229 231 233 o v 101 at 54c . . . 235 62. Column efficiency in te r ms of band a symm et ry, Z, versus linear v e locity, v, for propan e chro mat o g raphed o n graphitized carbon black at 54 C 237 63 Z vs v for n butan e on graphitized carbon bla c k 23 9 64 Comp a ri son of e :x -p er im e nta l d ata (poi nts ) and theoretical data (cur ve s) for propane c hromatog r aphed on 100 / 1 2 0 mesh silica gel (bottom), graph iti zed carbon black (middl e ), and OV -101 impr eg nated silica gel (top) 244 xiii PAGE 14 LI ST O F FI GURES ( Continued) Fi gu re 65. Hybrid-fluidic valve show n fr om the front with a cut-aw a y view of the valve assembly Va lve is j n the normal position fo r carrier gas flow onto the column . . . . . 66. Valve injection profile obtained w ith a 50 msec valve gate pulse to the hybrid-fluidic valve s ho w ing 2.0 % 0 5 % and 0.2 % inte g ration limits 67. Valve injection profile obtained w ith a 100 msec va lv e gate pulse to th e hybrid-fluidic valve showing 2.0 % 0.5 % and 0.2 % integration li m its 68. Experimental valve injection profile for the pneum at ically operated Hamilton valve at 100 C sho w ing 2 .0 % -5 % a nd -0.2 % int eg ration limi ts 69. Methane p eak profil es obtained on a 714 cm X 0.068 cm i. d. DC-200 op 2 n tube column usin g hybri fluidic valv e sampling tim es of 10-200 msec 70. Pentane peak profiles obtained on a 314 cm x 0.172 cm i. d. open stainless steel colu mn usin g a hybrid flu idic valve sampling time of 150 mscc and vary in g sample si ze by using an exponential dilution flask ............... 71. Effect of methane sample size on the statistica l moments when r etained on a 714 cm x 0. 068 cm i. d. column coated with DC -200 at 98 C (inte g ration limits= 1.0 % ) ................ 72. Broadening (11.r ) and asymmet r y (Z) for methane retain ed on a 714 cm DC-200 coated open tube ..... 73. Effect of pentane samp le size on the sta~isti c a l momen ts when re ta ined on a 711 cm x 0. 06 8 cm i. d. col u n;n cor..ted wit h DC -2 00 at 98 C (integration 1 imi ts = 1. 0 % ) . . . . . . . . 74. Broadening ( ) and asymmetry (Z) for pentane retained on a 714 cm DC-200 coated open tube ...... 75. Effect of the pentane sample size on the statistical m0ments in a 314 c ~ 1 X 0 .1 72 cm i. d. stainless steel column at 97 (int egration limi ts= 0.10 % ) xiv Page 258 263 263 265 277 277 279 282 285 287 289 PAGE 15 Fi g ure 76. 77. LIS T O F' F IGUl-:ES (Conth : n cc! ) Broadenj ng ( ~ ) and a sym f'letry (Z) for pentanc in a 314 cm tmcoated oen tub e Disp e r sion in co ated a nd unco ated open tubes xv Page 291 291 PAGE 16 a A A A e C C e c. ]. C C g Ct d d p dp / dl LI ST O F SY1\ 1B OLS Symbol s D e fin it i o n s and Di mens ion s Thermom e t e r c a libration constant Empty colum n cross-se ct ional ar e a, 2 cm Statistical mom e nt eddy diffusion co e fficient, cm Cross-sectional area available to m o ving carrier gas, cm 2 Van Deemter e ddy diffusion coeffici e nt, cm 2 Van Deemter mol e cular diffusion coefficient, cm / sec A constant d e fined by e quation 17 Concentration i n int e rp a rticl e spac e, mole / cc Concentration in intraparticle spac e mole / cc Van Deemter co e fficient for resistance to mass transfer, sec Coefficient of mass transfer in mobile phase, sec Coefficient of mass transfer in stationary ph a se, sec Thermometer calibration constant Average particle diam e ter, cm or m Pressure gradient, atm / cm 2 Effective (observed) dispersion coefficient, cm / sec Molecular diffusion coefficient, en? / sec 2 Effective Knudsen diffusion coefficient, cm / sec Effective co e fficient of longitudinal diffusion, cm 2 / sec xvi PAGE 17 D P, 1 D r D sec E E min E A f f ( ) e F m H H C H g H n H 0 HETP ).3 j 2 D in column segment i, cm / s e c p 2 Effective co e ffi cien t of radial diffusion, cm / sec 2 Dispersion due to seconda ry flow cm / sec 2 C a lculated molecu l ar di ff usion coefficient, cm / sec ') Knudsen ciif fust on coefficient, err ~ / sec Colu m n efficiency define d by equation 1 4 6, cm Value of opt im'-lm column eff ici enc y, cm Effective coefficient of molecular dif f usion, cm 2 / sec Pr essu re corr ecti on factor d e fi ne d by equation 81 Porosity function used in permeability equation 101 Carrier gas mass flow rate, cc / mi n or cc / sec Average calculated plate height, pressure-corrected, cm -1 Rate constant of mass transfer, sec Average plat e h e ight from gas phase contributions not pressur ecorrected, cm Rate constant of adsorption, -1 se c Defined as K or K by equation 150 C n Height equivalent to one theoretical p late, cm Transfer th ro ugh the outer gas film Radial diffu si on in the gas Transfer through the pore wall gas film Transfer through the outer liquid film Radial diffu sio n in the liquid Transfer through the pore wall liquid film Carrier gas compres si bility factor defined by equation 82 xvii PAGE 18 k K C K n L LE ET P m m n m a M,MA M ,-, B n p p p n q r r Colll iri:1 perin c:> b ili ty, g cm / sc c 2 atm or cm 2 Mass tr ansfer co e f f i cient, cm / sec L a place tr a n sform for the effective equilibrium adsorption co effi cient Equ ili brium volu ma l adsorp tion const a nt Equilibrium s u riace ad sorp t ion constant C o lumn len g th c m Local H E TP, cm Amount of adso rb ate in a p ore volume unj_t, mole / cc nth tot a l statistic a l moment, n se c Total mass of inj e cted co mpo und, moles / cc Mo lecular weight of a substance, g / mole Amount of adsorba te on th e surface in a pore volwne un it mole / cc absolute pr es sure, atm Colunm inlet pr essur e, atm Colunm outlet pr essur e, atm P. / p 1 0 Pressure correction factor defin e d by equat ion 93 Constant in perme a bility equa t ion 102 External tortuosity factor Int e rn a l tortuo s ity fac tor Rate of c incre se mole / cc sec 1 Rate of transfer through a thin stationary film, mole / cc sec Radial coordina te cm Av er age pore radiu s cm or A 0 xviii PAGE 19 R R g R 0 Re s t t a t ads t des T u u a u n V V C V min V V. 1 V 0 Mean p art icl e r n dius cm or ;.m1 t 2f 0 2 Gas constan, cm / K mo l e s e c Colu mn insid e r adius, cm R esista nce at t C, ohms Resistance at 0 C, ohms Colu mn coil radiu s, cm Reynolds number Coefficient of Laplace transform tim e sec Inert solute elution time, sec Mean time of ad so rption, sec Mean time of d esor ption, sec Length of inj e ct io n time and o ther system time const a nts, sec Absolute temperature, K Linear v e loci ty of an inert so lute in LHETP equations 78 a nd 79, cm / sec Absolute first moment of an inert solu te sec nth central statistical mom ent seen Average colu.7111 cross-sectional linear velocity, j corrected, cm /se c Aver ag e critical v, cm / sec Velocity for maximum column efficiency, cm / sec Sensin g volume of FID, cc Linear velocity in column segment i, cm /s ec Superficial carrier gas velocity at the column outlet, cm /se c xix PAGE 20 V A VC V D wl / 2 w2 w3 w X z y Y1 yi,Ye 01 6 5 a' 6 i' e e 1 eT A \) Pp CJ CJAB Td In e rt s olu te elu t io n volume cc Volum e of emp ty cnI 1 .rn m cc Extracolu cm dend v o l u me cc Wid t h at h a lf hei gh t sec Velo c ity f or minimum b a nd bro a d e nin g cm / s ec Velocity for mjni mum b a nd as ym me try, cm / se c Dist a nce a long th e c o lumn from the inl e t, cm Length c o ordinat e cm Fractional poro s i t y pe r adsorbent particle Tortuosity facto r in B term Total r es i s tance to ma ss tran s fer in liquid, s ec Inte r nal and e x t e rnal contribu t ions to y 1 sec Total re s istanc e to mass tran s fer in gas, s ec Adsorption, internal, and ex t ernal contribu t ions to 6 1 s ec Total ex t e rnal porosity fraction Total internal porosity fraction Total column porosity fraction, e + e. e 1 Carrier g a s viscosity, g / cm s ec Multiple path fac t or in~ term Particle shape factor Pack e d column particle density, g / cc Stand a rd d e viation Collision cross section, cm Det e ctor time c o n s tant, sec xx PAGE 21 T e T s T V X ;Ir 'g z z /\ 0, mi.n Electro mete r time c ons t a nt, sec System time Cl : nstan L sec Samplin g va lue time constant sec Porosity function L1 moment e( 1 uacions (1 + (1+K )) n Column ef ficiency with respect to broadenin g cm Measured f not pressure-corrected, cm Value of optimum ijr cm t corr e cted for sampling time, cm ijr, corr e cted for pr ess ure effects, cm Calculat e d ijr values, cm 2 Column efficiency w ith respect to asymmetry cm Calcula te d Z values, cm 2 2 Z, corrected for samp ling time, cm Z, corrected for pressure effects, (1 + c/JK ) C Unit of resistance, ohms 2 cm Function of temperature and pot ent i a l field in diffusion equation 115 xxi PAGE 22 Ab stract of D::. s::; ert ; : ,t ion rrcse:1t i:,d to the Gr aduate Counci l of the University of Florida in Partial Fulfillment of the R e quir eraents fo r the Degree o f Doctor of Ph:i.losophy PIIY SICOCHEAHC.c\L I,lli ASURE !v lE?\'TS i3Y HIGH PRECJSIO?-, GA S C.f-IRO?ll: .! \.TOGH. A PHY By B a r r y Eugene Bowen Dece m b e r, 19 73 Chair man : Dr. Stuart P. Cr am C o c hain n a n: Dr Roge r G Bates Major Dep a rtment : Cher.1ist r y This res ea rch has b een direct e d toward the und ersta nd ing and th e accurat e and precise d etermina tion of factors a ffec tin g gas chrom a tograpi.-:i.c efficiency. Experimental evidence i fi present e d toward the verification of the stoch as tic theory of gas chro matag raphy. Th is has b ee n accompli shed through the me as ur emen t of phy sicoch enical constants, derived from a statistic a l mome nt an a l ys i s of chrom atograph ic peak shapes. The preci sio n in th e resultin g penie abiliti es, porosities bina 1 y mc,lecular diffusion coefficients, equili brium ~dsorpt i on con stan ts, radial diffusion coefficients, and adsorption r ate co nst an ts r ange d fro m 1-20 % Th e gas-so lid chromato g raphic mod e l for porous a dsorb en t s is ext ernl.ed to the c ases of no nporo us ad sorbents l:i .q uid-filled p oro us ~dsorbents, and to columns w ith pr essure g r a dients co mp arabl e to t hos e en coun tere d i n practice. Gaseou s hydrocarb o ns were chr omat ograp hed on packed columns of act i 1 ted silica gel, OV-101 im pregnate d )(Xii PAGE 23 silic a ge l, a nd g r aph iti ze d c arbon black. These colu mns a re charac terized with r es pect to th ei r pa rticle di s t r i b ution, pe r mea bility, extern al porosity, and in te rn a l porosity The eff ici e ncy of th es e colu mns is characterized in terms of peak broad e ni ng ( 1~ ) and pe ak asyrr unetry (Z) as a f unction of carrier gas 1 inear velocity and ad sorbent p a rticl e r a dius. Experimental and theoretical effici e ncy curv es are co mpa r e d A hig h precision gas chromatograph was ass emb l ed in which ca r rj_e r gas flow r ate sampling valve actuation, column temperature, d a ta acquisition, a nd statistical moment calculation were controlled by an on-line dedic a ted laboratory computer Extracolumn dead volumes were minimized and systematic time constants we r e measured A comp a rison c,f se veral gas samplin g val ves is made by a stati st ic al moment analysis of the i r injection p rof iles. A precisi o n of 0 05 % relative standard deviation for the first moment of the injection profile of one of tl1e va l ves was achieved. The effect of sample size and l ength of sampling time on chro m ato g raphic behavior is examined for coated and uncoated open tubes xxiii PAGE 24 INTROD U CTION Ob jective of this Research Th e objective of th is research ha s been to measure physico c :i-wmica l para :nete1 s in order to d eve lop an experimenta l ver ific ation of the stochastic U eory of gas chron mt ography (GC). A hi gh pr e cision gas chrom atog r aphic system w&s develope d to det ermine e qu il ibrium con stant s, diffu .sion cons ~a n ts 2 nd r ate constants from a statistical moment ana lysis of elution profil es. In addition, extracolurnn effects were e xperim enta l ly inves ti gate d, such that the ir con tributi o ns to ch r omatogra phic zone b roaden ing and a symmet ry were known In this way, the deriv ed coefficients for the va riou s mass transfer pr o c esses we re u se d to complet e ly characterize th e on-colu1m1. dynamics involved in ad sorp tion as well as p ar ti tio n systems. In 1ecent years, GC ha s b ec o me a major te chnique for ch e mical analysis. As a l a bora tor y separation technique, it is r e co g nized for it s exceptio na l res olvin g power, speed sen s itivity a nd ease and versa tility o f operstion In fact, qu a litative precision GC h a s reached a high d egree of reliability s u ch that it act iv ely co mpetes with other identifi catio n te c hniqu e s, for exampl~, s p ect ro s copic ,netho ds [l] Recent adva n c es in iu str umenta t:i.o n and colu m n materia ls have made it easy to pr od uc e experimental data; ho weve r, t h es e data have b ec om e 1 PAGE 25 2 almost exc J. us:;.vcly applic at i ons orient e ci, wi thotd ; und e rstanding, and one h a s to find t h em in te :cesti ng rat h er than in Jormat i ve It is essentia l tJiat n ew met hod s a nd te c hniques be continu a lly dev elope d in GC in order to cope w ith today's innumerable separat ion probl em s This can bes t be accom plished on the b as is of understm1d ing of fundamental p h y s ic a l an d chemi cal phenomena, rather than on an empirical ba si s. The sta t u s of current develop me nt and applications in GC can p erhaps be de scr ibed as having reached a plateau which i s depend ent upon a b e tter func t ional kno w ledge of the on-column separa tion process itself in ord er to quickly choose or pred i ct optinrum experimental co n ditions. Once this knowledge is g ained, advantage can be taker, of the potentially high precision and accuracy inherent in gas chromato graphy as an analyti6al m e asur e ment system Statistica l Mome nts in Ph ys icoch emi cal An alysis Generally, tr.e accurate determination of physicochemical cons ta nts from GC data has not been feasible without the us e of a st atistical mom e n t analysis of elution profiles. Th is has been neces sary because chromatographic mass transport pheno me na are r andom events occurring on a molecular scale Further, most of the molecules are in a nonequilibriu m s tate n~ st of the tim e and the molec ular distributions obtain ed are very r arely Gaussian [2]. Peak shapes, whic h rep res ent distributions of mol ec ules, can be comple tely defin e d in terms of their statistical moments since the moments are applicable to all peak profile s as long as the finite integral PAGE 26 3 uo J f(t)clt (1) existr, anc! c &n be evaluateci in closed form or nu mer ically. The zeroth, fir st second, third, an d fourth mo ~e nts are mo as ures of the peak are :c:. mcrrn, variance, asymm et ry, 2 nd height, re spe ctively [ 3]. The use of statj _stica l m,>m ents to obtain physicoch em i cal constants from e1>.1)cr im e Dtal GC data wil l be r ev i ewe d below. The th e ore t ical de ve lopment of th' = dynami .cs of cliro rnatograph y wi 11 be revie we d in th e next chapt er of this diss ertat ion. Expr ess ions for the first five statistic a l moments, in terms o f physical constants were first dev elo p ed ind ependen tly in 1965 by Kubin [4,5] and Ku ce ra [6]. Ch romatograp h i c peak shapes we re also treated m athem atically by Kaminskii e t a l. [7], and the first six moments of the peaks were der ived on the assumption that diffusion in the me.bile pha s e was the most important column process. The Kubin-Kucera theory for gas -solid chromatography (GSC) was exa mi ned initially through the eA-periments of Grubn er ar.d co w orkers [8 11]. They investigated the effect of porosity on the ret e ntion ti :11c s of in ert gases usin g columns packed with g lass bead s pumice, or coal. The kinetics at high velocities were diifu si on con t rolled for the porous supports, leading to approximate values of the co eff icient of radial diffusion [8]. Subsequently, they qualitatively v e rified their theory for column efficiency with respect to the v ar ian ce asym metry, and exces s of th e chromatographic curv es by studying the adsorp tion of carb on dioxide on activated charcoal. They did not:, however, report any r ate constant values. In all experiments th ey used PAGE 27 rel. atj vely short columns w ith J.arze inte: : r.21 PAGE 28 int er11a l diffusion we r e approximate ly d ete rmined from the dep e nd e nce of the stati s tical moments of the e lution curves on the lin ear ve lo c i ty o:[ the carri er gas They examined the peak mean ancl varian ce but no t the high e r mom e nts ifaen comparin g pe a k means and maxima they found a much sma ller depend e nce of retention volu me on flow r ate w hen us ing the mean, indicating that th e mean is the prop e r peak parameter to ch a racterize e quilibrium condit i ons Funk and Rony [1820] 2 r gued th at app licati on of s tati stic al mo ment t h eory to e:;,.-perimental. e lution curves was difficult due to the inf inite tim e u pp e T l imit of the mom ent i n te grals They dev e lop ed par tia l statistical mome n t ex pre s sions for rela tj_ n g retention time to the mean, peak shape, and p ea k a r e::i Th e y di d this by definin g a single di mens i onless group, wh i c:-1 is a measure of diffusional m as s 5 transf e r in th e liquid ph as e r e lativ e to the conv e ctive gas phase mass trans fer in the axial direction. They concluded that only partial no rma l moments could be e x perimentally d e termined since compl e te mome n ts ar e l imited by system noi se an d drift. This same conclusion was reach e d by Che s l e r and Cram [21.] who discu ssed the e ffect of loc at ing the limit s of integration on the accuracy of meas uring and cal culating statist i cal moments from chro mato graph ic da ta The eff ect of no ise on the precision of the mo men t m easured w as found to corr e late with th e limit s of integration a nd the numb er of data poin t s t ake n per peak. Er rors in the calculation of all mo me n ts from di gita l dat a we r e depend ent on the gene ral shape of the peak; the fast est decrease in errors w ith lo we r integration l imits and th e h ig hest pr e dicta bj li ty was found for the most Gaussian-like peaks. PAGE 29 6 Gru shka et al. [ 22 ] sh owed th e utili .t y of offl ine compute r calculations for the d etailed analysis of peak shape nnd rapid methods for measuring the moments. Weiss [2 3 ] exten ded the stochastic Theory of GC to includ e diffusion effects and multi -s ite adsorption by developi ng this si mp l e model in t e rms of the first and second stat i stica l mo ments Grubner [ 24 ] has shown how the first four mom e nts can be calcula ted once the inf l ec tion points of an as ymm etric e lu tion curve have b een d ete r mi n e d. A moment ana lysis was used to sense the presence of two ov er l apping Gaussian p eaks from the skew and excess of the peak profil e [25]. An electrical sim ulator for chro matographic behavior on GSC colu mn s was prop osed to s i m ulate lin ea r adsorption k in et ics, ma ss bal a nc e in the column, and to c a lcula te the statistical moments [26,27]. Grubner an d Underhill [28] d eveloped equations, based on the s t atis t ical moments theory for GSC, for determinin g the be d capacity of p orou s adsorb en ts. The principal valu e of this work is its potential ability to allow tte design engin ee r to r e calculate given experimental data to other condjtions, thus arriving at the optimum ones. Und er hill [29] pr ese nted a moment analysis of eJ..ution curv es from GC columns having a log normal distribution of ad s orbent particl es His equations r elated number of theoretical plates, skew, and exc ess to macr opor ous and m:i cropo ro u s column p a ck ings for the case where both interparticle and intraparticle diffu s i on s are present. He extended these equations to includ e corrections for pressure drop eff e cts [29,30] but presented no experimental result s Recently, C he s l er and Cram [31] have repo rte d an it erat ive curv ef itting techni qu e for the high acc uracy me as ur e m en t of total PAGE 30 statis t ical moments from exp e rimen tal data The fitting proc e dures are d esigne d for ex--pe rimental peaks with any reason ab le degree of asym met ry and real data fro m a partition column were successfully tested. Th e ultimate utili ty of this technique wi ll be r ealized only when th e s e total moment exp re ss ions in terms of eight e m pirical para meters are finally related to physical phenomena. Concurrently, D w yer [32] presen te d a method of Fourier trans forms for deconvolution of experimental GC elution curves. H e demon strated from diffusion and adsorption experiments that if the time distr ibut ion of on e independ e ntly occurring column process is known, the other can be resolved into its component time distribution. Sub sequent ly, the statistical moments of the component distributions can be accurately d e termined by appropriate handling of the digital data. 7 The zeroth moment, the area under a chromato gra phic p eak has been used for some time for quantitative analysis. This is because, perhaps fortui to u s ly, automatic integrators gi~e a measure of the zeroth moment. llowever, in practice much information is lost due to the limited variability of starting and stopping techniques provided with hardwired commercial integrators. Retention times are generally still d e fined simply by the peak maximum, which has no basic physical sig nif icance, rather than the more correct value of the mean time [6]. Peak widths are taken as some hybrid of the width a t a lf hei g ht or extrapolated tan ge nt baseline width rather than the correct v a lue of the standard deviation derived from the second moment. PAGE 31 The major ob sta cle to t he general utilization of stati st ical mo me nt da ta has been the nonavailability o f a co m puterized digital data acquisition system &nd sophisticated software With such a system at hand, high preci sion and high accuracy meas ur ements of the 8 chr omatographic statistical m01aents and, thus, fundamental ph ys j_co chemical and thermody namic consta nts can be made based upon an appro priate m o d e l. This is still a rather narrow field in which only a few chroma tographe rs have participated. High Pr ec ision Gas Chromatographic Mea sur ements Several authors have rec en tly report e d high precision and high rtccur:::.c y gas chroma togra phs, and / or systems where the sampling, flo w rate, t empera ture, and data acquisition were computer controlled [334 2]. Oberholt zer [33] first report e d a digital pro gramme r capable of per formin g a series of injections using a pneu mat ically operated gas sampling valve with a precision of 1.1 msec. Oberholtzer and Rogers [3 4 ] designed a chromatograph that determined retention times with a precision of better than 0.02 % for times of 26 seconds (slightly r et ained solutes). This precision allowed them to d e termine heats of solution of some n-paraffins to at least 10 cal /mo le. They also measured the zeroth moment, IIETP, and var iance with 0.05 % precision and the skew and excess at precisions of 0.2 % and 0 1 % respectively. These results were achieved by using a sampling val ue which gave a sample width of 67 msec with 0.22 % reproducibility for the el u tion time of the sample plu g This sam e PAGE 32 9 sy stem w a s u se d lat e r to s t udy the e ffec t s o f r a pid r e p eat ed injections (35] and en s emble av e r a giL g [36]. Burk e and Thu rm an ( 3 7 ] used a dedic a ted comput e r for real time control of GC measur eme nts It was shown that real-tim e interaction of a dedi c a te d co m put er w ith a GC allo w ed improved precision of data taken for adsorp ti on studi e s using porous supports as well as for detector response e v a luation. This im p 1ove m ent stem me d from the abi l ity :)f the computer to provide an accura t e t:i.me base for the s a mplh1g sysi:rnn. Thurman, M u e ller, and Burke [38] elahorated on the above sy s tem s o that the flow rate and oven temperature could be sampled by the computer except during an actual run. The data acquisition rate was not computer controlled. The purpose of their w d rk was to measure data of sufficient pr e cision to allow m eaningfu l thermodynamic s tudies of eas-solid a d s orption process e s. Thei r system was d esi g ned to measure HETP usin g the peak maximum time and width at half height, then to chan ge to a new flow rate, thus obtaining a van Deemter p lot to determine the optimum flow rate. Swingle and Rogers [39] later assembled a hi gh precision gas chro matogr a ph in which the computer controll ed sample injection, colmm tem pe r a ture, flow rate, and also dir ect readout of inl et pressure, mass flo,v r a te, and detector r sponse. This system has been u sed for rapid and precise determination of thermodynamic data, quantitation of peak shap e s, and qualitative identification of unknown components. L eitner [ 40 ] and Cram and Leitner (41] reported the design of a h igh precision gas chromatographic computer system. This system is c apable of making on-line calculations of the first five statistical PAGE 33 10 mom e n ts. T hese moments and the ir hybrid rnomP.nts w ere u sed to make on-1 ine d e cisi ons w ith rt :! s ect to mass flow and temp erature adjustment, samp li ng t:l.rne, d ata acqui s iti on rate, and amplifi er g ain. On-line optimi zatio n has b ee n perfor me d whe reby t he resolution and skew of adjacent peaks were u sed as criter ia for influ encing the re so lution and skew of co mpo nent peaks not yet elut e d. This sy stem has b een used to s t udy t he in te raction of flow and temperatur e prograrrm1ing [ 40]. It was also used to study the precision of sever a l gas chrom a tographic samplin g valves [42] and to study the effects of sam pling time and sample si z e on column efficiency as determin e d from the first five stati st ical moments Se vera l papers h ave appeared in recent lit er ature concernin g the instrument al limitation s for high precision GC data Sternberg [4 3] treated the first and secon d moment contribu tions from extracolumn sourc e s such as mixing ch amber s, sampling time, an d electrometer time cons tant s. Gl e nn and Cram [ 44 ] develop ed a digit al logic system for the evaluation of instrum enta l contributions to chromatographic band broadening. They reported experimental second moments contributed by the connecting tubing, mixing volume, sampling valve, electrometer and cable, and detector, Go ede rt and Guiochon [45] studied the influence of fluctuations of pressure and temperature and time measurement on the precision of the retention time. They established specifications requir e d to achieve precisions of 10 % to 0.01 % [45,46], concludin g that th e latter w as extr eme ly di ffic ult to achieve in practice, the p :ce ssure drop control bein g the most d e mandin g limitation. For example, for 0.01 % relative PAGE 34 11 precisio n (a t the 95 % con fidence level) on retention ti mes the outlet pressure p r ess ur e drop, a nd t empe r at ure should be stabl e to within 0.25 ~ 0.00 5 % and 0.001 % respectively, with no mor e th an a 0.005 % error in t he t i me meas ur eme n t its e lf. Goede r t and Guiochon [ 4 7] have examined th e eff e ct of signal-to noise ratio on the r e t ent ion tim e of the p ea k maximum. They found that for all s igna l-t o -n oise ratio s stud ied that th e pr e cision of the pe ak maximum wa s be tter than the pr e cision of th e mean time w hen dealin g w ith nearly symmetrical p ea ks. They later conclud e d [48] that it is neces sary to have a progr amma ble time constant in el e ctrometers; alternat e ly, the tim e cons ta nt sho~ld be small enough su ch that the smal lest peak 0f the chro matog ram i s not distorted or hi g h en ou g h to filter out hi g h frequ ~ncy noise for l a t er bro a der peaks. Chesler and Cram [49] examined in strumental effects of the analog-to-di g ital convert e r (ADC) in the digiti z ation step of data acquisition. They tabulated th e errors j ncurred in the statistical moment s du e to the ADC and int egration limit selection for Gaussian as well as quite skewed curves. In an earlier paper, they showed that large errors in the precise d e termination of statistical moments could result from choice of integr a tion limits alone [21]. Goedert and Guiochon [50] made an e laborat e a1 : d comprehensive study of sources of systematic errors in retention time measurement and dev e loped an expression for the ove1all instrumental correction. They found that the main error in retention times, and t hus thermody namic co ns tants, was due to erroneous det enn ination s of "inert" times. For methane o n g r ap hitized ca rbo n black, th e y were able to det e ct PAGE 35 1.2 a capacity fa cto r of 0.0 03 ~ at 100 C, a value ~tic~ is normally essu med to be zero E ffect of temperat u re g radi e nts a nd fluctuatio ns of t em p erat ur e on t he retention time of an inert compound a nd on the capacity :f acto r were t h eo riz e d [51.]. The r es earch de.scribed i .n t h is dissertation has be en directed toward th e u n de rstan d ing and th e accurate and precise d eternti nation of f ac tors a f fecting ga s ch roma tographic efficiency A phy s ical model of moderate c omple xity, which closely approximates r ea lity ; has be e n cho sen to describe a ll kno w n on-colu m n processes. Relatively few si m plifyin g assumptions h a ve been made, and experiments have been perform e d with pack ed adsorption and p a rtition colu mns und er normal chromatographic conditions. This has b ee n don e with respect to column l ength, support size, solu tes and pr ess m e drop. The above co m bination of a re al istic th e ory along w ith r ealistic experimental situations has not been a common occurr e nce in r e c e nt literature in the field of gas chromatography. PAGE 36 THEORETICAL Dev e lop m e nt of Theoretica l M od e ls fo r Gas Chroma t o g raphy The f i r st theory of chro mat ography w as written by Wilson [52] in 1940 a nd was called "A Theory of Chromatography." T his quantitative theory of chr omat ographic analysis was based on the assumption that equilibrium b etwee n solution and adsorbent is instantan eo usly establi s h e d and th at the effects of diffusion can be negle cted. Wilson did, how eve r, describe the proc ess es of diffusion and nonequilibrium as the causes for zone broadenin g and stated that quant itat ive agreement betwe en his theory and e)..l)eriment was unlikely. The plate theory was later develop e d by Martin and Synge [53] as an extens i on of distillation theory. They d e fined the plate height as "the thickne ss of the layer such that the solution issuing from it is in equilibrium with the mean concentration of solute in the nonmobile phase throughout the layer." The height equivalent to one theo retical plat e ( HET P) has been used widely to ch a racterize chromatographic zon e spreadin g and chroma tog raphic efficiency. The theory predicts that a sample introd uce d onto a column be g ins in a single plate, progresses through a stepped Poisson distribution, and finally to a Gaussian distribution after a lar ge number of sorption-d es orption steps. Calculation of th e '' number of theor et ical plates" per column has 13 PAGE 37 14 rem a ined the sim plest and most practical method uf characterizing column s epcirat ion efficiency H owev er, the ide a of discrete and dis continuou s phy s ical plat es on the column has d one much to encourage theoreticians to seek bet ter chro matog r a phic models. The theoret ic a l plat e mod e l does not in itself accoun t for the basic effects of particle si z e, molecular structure sorption phenomen a temperature, pr ess ure, molecular diffusion, or variations in f low patt erns Martin and Synge did, however, d educe the rule that HE TP was proportional to flow velocity and the square of the particle di ame ter [5 4 ] Thomas [5 5, 56] d erive d fro m dif f erential mass balance equations expressions for obtaining adsorption and desorption rates from elution curves stated as functions of time. He, too, ne g lected the longitudin al diffusj on term, but he did obtain simplified solutions for low flow rates where close-to-equilibrium conditions were valid Boyd Myers, and Adamson [57] describ ed ion exchange kinetics in terms of diffusion through a liquid film and predicted peak shape in terms of independent r ate and equilibrium constants. Lapidus and Amundson [58] developed equations for equilibrium as we ll as nonequilibrium cases. They incorpor ate d the influence of the longitudin a l diffusion but assumed that neither the external (migration of material along the particle surface) nor the internal diffusion (migration within the adsorbent pores) comes into consider ation. They solved the problem of adsorption on the absorbent surface which follows a linear isotherm and also consid e red the rate of sorp tion. Their theory became the fo~ndation for the well-known equation of van Dee mt er, Zuiderw eg and Klinkenberg [ 59] In their paper [59], zone spread ing was stated in terms of HETP which evolved f rom additive PAGE 38 contribution s of eddy dif fa s:\ cm, rr.o lecul a r cli.f f ,i sio n, and resistance to mass transfer in the liq uid phase Thi s con cept was similar to that proposed by Glueckauf [GO] for s olu te diffusjon through ion exchan ge beads a nd their surrounding liquid. Glueckauf [ 6 1,62] 15 exte11cled the plate hei g ht th eo ry by mathematically d e scribing e lution and breakthrough cur, e s for lin ea r as we ll as nonlin e ar adsorption and e x cbangc i sotherms. In 1955 G~cldings and Ey ring [63] described the chromatogr2phic proc ess as a probability of mulec ular events This p ape r initialized the statistical concept for GC. They lat er extend e d this conc e pt to the multi-site adsorption problem and then to the we ll-known random walk model of chromatography f64], which seemed to sati sfy a multitude of chromatographic problems. Giddings [65] develop e d a probability equation to explain the ki,1 et ic origin o f tailing when sorpt:J.on isotherms are lin ear From this equation adsorption and desorpti on rate constants could be derived for molecules ad s orbed on the "ta il producing" sites if the Gaus s ian profile caused by adsorption on "non-tail-producin g" sites could first be described. In 1959 Giddings [66] started the development of a generalized nonequilibrium theory for chromatography in ord er to calculate the effects of any complex adsorption-desorption problem, whether con trolled by diffusion or di screte single-step kinetics. He recognized that nonequilibrium and diffusion were a common basis for the three theori es of chromatography [67]. He named the material conservation or ma s s balance approach and the stochastic (random) approach as "rate" theories and considered th em different in nature than the theoretical plat e mode l. PAGE 39 i\icQuarr i e L68 ] e::q ;a~,t~c d t he Pois s o n n1. nfl0m walk theory of Giddin gs and Ey rin g by u si n g the compl ex -variable ch e ory of Laplac e trar. sforms He d escribed a te chnique by m e ans of which any ca se includin g multi-si te and delta or G a us s i a n input functions, could be we ll approxj_ mate cl by cor .s i derat io n of the first fe\\' c entra l stati s tical m ome nt s He a l so expr-mded the asynu ne tr ic chro mat ogra p hic dis16 t1ibuti o n function in t erm s cf a Gram-Ch a rli er se1ies, a 111ethod wh ich allo ws experimental curves to be appro ximated from the calculat e d stati stica l moments Rosen [69] derived the equation for the time d e p e nd e nce of the concentr a t ion for e luti on curv es for p acked columns. He consid ere d in ternal and ex t e: cn a l diffu sion but assu me d that molecular diffu s ion was n eg lig ib le a nd t hat the amount ad sorbe d at each moment was pro portional to t h e c oncentration w jthin the porous pa:i .'ticle. Fi n a lly, Kubin [4,5] and Kucera [6] d e veloped e qu at ions for li near nonid ea l (noneq11ilibrium) gas -solid chromat o graphy The stati s tical moments were derived from ma ss balance equations for complicated cases of GSC where th e proc e sses of external diffusion and adsorption could be con sidered simultaneously with internal and longitudinal diffusion. Sel ec tion of a Gas Chro matograp hic Mo del In general, for the system of equations that describ e a physical mod el to be solvable, the mod e l must be relatively simp l e Yet it must con tain provisions for all of the essenti a l physical and chemical proces ses that a re known to occ ur. In the case of gas-solid or gas liqu i d chrom a tc grn phy the mode l selec te d is quit e complex du e to PAGE 40 17 the several t yp es of diffu s i o n a n d m ass tr ansfe r t h ... t may exist sj mul tan e ousl y Th e Kubinucera model for porous gas-sol id adsorbents [4-6] w ill be closely follow e d and explain e d b e low. Th e theory will be dev e l o p e d for the ca s es of low pressure drop and nonporous ad s orbents, a nd extended to gas-liquid systems. In gas-solid chromatography, adsorption of a gas occurs at the gas-solid int e rface and is caused by differ e nc e s in the energetic properties along the interface. The adsorbent bed is composed of small particles whose cro s s sections are physically and chemically homogeneous. If the particles are porous, e ach having an internal porosity fraction S, the column into which they are homogeneously packed will have a total internal poro s ity fraction, e. = c1-e )S 1 e e. l where denot e s the fraction of free interparticle space (external e (2) porosity) in the total column. The total porosity fraction eT will be An inert carrier gas enters at the head of the column at a pressure slightly larger than that at the end of the column. Due to the finite pressure drop created, the gas flows through the interparticle sp a ces toward the open end of the column. Since the pores are quite small (10-100A 0 ), the molecules of gas within the pores do not generally move with the carrier gas; rather they diffuse into the pores a certain distance only by a process called Knudsen diffusion whose coefficient is defined by D = i r (2RgT)f K 3 \ nl\I (4) PAGE 41 where r is the average pore radius, R is the gas c onstan t, T th e g absolute temperature, and M i s the molecular weight of the diffusing sub stan ce. Then the molecules diffuse out a ga in into the main stream of 18 the car ri er gas Under u sua l chro matograp hic condi t ion s th e carri er gas passes throughout the column with a laminar fl ow profile with a ty p ical system of strea m l ine s that have zero velocity at the w alls of the particl es This thin film of stationary c a rrier gas coats the entire surface a rea of the particle, internal as we ll as external. For highl y porous materials, most of the surface area is in te rnal and the external area may oft e n be neglected. When ad s orbate molecules are introduced onto the chromatographic bed as a very narro w pulse, th e carrier gas tran s ports them throu g h the colu m n. In ord e r for a molecule to adsorb onto the particle surface (internal), it must enter a pore, and undergo four stages of mass transport. The molecule must (a) penetrate the thin film surrounding the particle at a certain rate determined by a concentration gr adient. The molecule then (b) diffuses radially within the pore toward the center of the pore and toward the pore w all. It then must (c) penetrate the thin gas film existing on the pore wall and finally (d) b e adsorbed onto the particle surface proper at a certain rate. It remains adsorbed for a certain time; then it is desorbed and by the opposite order of processes above, passes from the pore again into the flo w ing carri e r gas. Each of these processes, when viewed as a random operation on the many adsorbate molecules in the sample, contributes to the enlargement of the chro m_at ographic zone. PAGE 42 19 Besid es these four c o r!.tributions to broad e ning and a s y m metry of th e ori g inally n a rro w inp u t pr ofile, at le a s t t wo other transport pheno me na mu s t b e consid e r e d. In th e interp a rticl e space the molecule s will be s ubject to lo ng itudinal di f fusion in the dir e ction of the c a rri e r ga s flo w and a l. s o to a dif f u s ion si m ilar to the eff e cts caused by the properti e s of the packin g proper call e d eddy djffusion. This s ugg es ts that the f a mili a r A and B terms of the traditi o nal HETP equation are caused primarily by eff e cts in the interparticle space while the C t e rm is due mainly to intraparticle processes. This model is shown in Figur e 1 where the cross section of a single spheric a l particle contains a single pore (magnified for illustrative purposes) of diameter 2R. There are of course, many such pores, which n ee d not be the entire length of the particle, and the particle need not be sph e rical or smooth The intraparU cle adsorbate conc entration is designated by c. and t h e interparticle concentration l. by c since they may not be equal at equilibrium The internal and e external wall gas films are designated by 1 and 2. The processes of mas s transfer through the film around the particle, mass tr ansfer by radial diffusion, and mass transfer across the film around the pore wall are depicted by i 1 i 2 and i 3 respectively. The adsorbate molecules in the interparticle space diffuse with an effective coefficient of longitudin a l diffusion, D p D = D + Av p g (5) where A is a constant containing the effects of eddy diffusion, which may constitute trans -colu nm diffusion, etc D is the coefficient of g molecular diffu s ion for the adsorbate in the carrier gas. The carrier PAGE 43 Figure 1. Model particle for silica ge l adsorbent. The exte rnal and int e r na l wall ga s fil ms are designated by 1 and 2. The par ticle diameter is 2R, ci is the internal adsorb at e concentr ation and ce is the exte rnal adsorbate con centrati on Ma ss transf e r t hrou g h the gas film aro und th e parti c le (i 1 ), mass transfer by r ad ial diffu s ion in the pore (i2), and mass transfer across the gas f ilm around the pore wa ll (i 3 ) are shown PAGE 44 21 2R PAGE 45 22 ga s i s tr ave li n g throu g h tht i ;:1i: erp a rticl e s pa c e w ith an average l i near v e locity v which is g iv e n by the ratio of the outlet flow rate, F to th e fra c tion of c r os s -s e ctional area eeA available JU (6) and i s corr e ct e d for pr e ssure drop across the column by the gas compressibility factor, j. Mass Tran s f e r in the Interparticle Space This model for GSC is described mathematically by the use of the partial diffe1~ential equation for the mass balance oc e Tt" + oc e V--;-oz = Q C where time is denoted by t, the length coordinate by z, and the (7) effective co e fficient of longitudinal diffusion, D as in equation 5. p Since the interparticle concentration c may differ from that in the e pore (c.), there may be an increase or decrease in the concentration l in either space. Thus, the rate of concentration increase in the interparticle space due to the flow of adsorbate from the pores is given by Q above. C Mass Transfer Through the Thin Film Around the Particle If the rate of transfer Q of the sorbate from the pore into C the flowing carrier gas is assumed to be linearly dependent on the difference betw e en the actual and the equilibrium concentrations, then PAGE 46 = H (K c C C e (8) where H j_s th e rat e con stan t ar,d c. is the concentration at the c 1 / r=R entranc e to the por e where r is R. The c quilib :::iun: c onstan t K in th e C equilibriu m state is given by c 1 m K = = C C e c ( 9 ) e 1 e where mis th e amount of adsorbate in a c e rt ai n pore volume unit w h ere K = 1 if th e concentration outside of th e particl e is the same as that C within t he por es Radial Diffusion Within the Pores Further, the mod e l is described by the partial diff e r e ntial equation for th e rate of desorption Q of sorbate mol e cules from the n pore wall oc. 1. di; 2 ( 0 C. l OC.) 1 v1 Dr _o_r_2_ + -;dr = (10) where the concentration of the sorbate in the pore c. is a function of 1 the time t and the coordinate in the direction of the pore radiu s r. At the particl e c enter r = 0, and at the pore entrance on the particle sur:f H c e, r = R. Whe n d e aling with spherical particl es the shape factor v equal s three. Concentration c. can incr eas e or decrease due to 1 adsorption or desorption, limited by tran sfe r throu g h the film at the pore wall, and is d epe ndent on the effective rate of radial dif f usion, D. r PAGE 47 24 Mass Tr ansfer Throu g h th e Thin ri lm at the Pore Wa J 1 If the rate of tr a nsf er through the pore wall film is assumed to be linearly d e pend e nt on the diff e r e nce b etween the a c t u a l and equilibr i u m conc e ntration s as for the film arou n d the particle, then = -H (K c. n) n n J (11) wh e re n is the amount ad s orbed o n the surface of a certain pore volume unit The equilibrium constant in the equilibrium st a t e is given by K n n --= c. 1. 1. n m and H is th e rate constant of adsorption n Solution of the Mass Balance Equatio n s (12) Kucera [6] solved eqnation s 7 through 12, which we re later u se d by Grubner [8-11], with certain boundary conditions by Laplace trans formation of the partial differ e ntials to ordinary differentials. For the boundary and initial conditions C (z,t) = c (r,z,t) = n(r,z,t) = 0 e 1. (13) C (z, t) = C 0 (z) e e, (14) c.(r,z,t) = C Q (z) ]. l. (15) n(r,z,t) = n 0 (z) (16) and c. 0 (z) = cM O(z) ]. a (17) th e condi tions given by equation 13 are for t PAGE 48 25 O s r < R. The quantiti es c O (z ), c. 0 ( z ), and n 1 ) (z) describe t he e, 1, initial dis trib ution of the narrow band of sol u te introduced onto th e column. Th e boundary conditions in equation 13 are r easonab le, since th ere are ze ro mo les of so lu te ad s orbed over the wh ole l ength of the colu mn befor e injection (t < 0) Only pure carrier gas is flowing through the column. I mme diately up on inj ection ho wever, c c., and e 1 n as sume finit e values (equ a tions 14-16). The sample is assumed to enter the c o lu mn as an in f initesimally narro w plug, ideally describ ed by a const ant c M times a Dir a c function, 6 ( z ) (equation 17). In D. practice, inj ec tion profiles as narro w as a few msec wide ha.ve been achieved in recent high precision GC systems [42,50]. The transform f(s) = L(f(t)) = J f(t)exp(-st)dt 0 may be applied for C (z,s), C (r,z,s), and n(r,z,s), obtaining the e 1 following time based tr a nsformed equations: and SC. 1 c. (z) 1,0 sn no(z) = D p ,le oc D (~ + v-1 Tr)= r or2 r r -Q n (18) (19) (20) (21) wheres is the transform variable for the Laplace coordinate system. When the solution c (z,s) in Laplace coordinat es is found for e equations 19-21, it is not easily inverted to normal coordinates in order to ob t ain even a simple analytical expression for the original PAGE 49 26 distribution p ro f ile c (z,t) It i s however, possible to obtain the e statis t i cal moments of the distribu tion by u s e o f a series of Hermit e polyn om i a ls for c (t) whose expa n s i o n c oeff icient s can be stated in e te rms of the L ap lace transform property co (-1) lim s-o n d C ( s ) e k ds (22) wher e n is th e ord e r of th e d e sir e d ce ntr a l moment T he sta tist ical momen ts can then be arranged in a seri es wh ich r e pre sent s the solution o f th e ori g inal set of dif fe rential equations. Calcula tion of the Stat i st ic a l Mome n ts I The tot a l nth a bsolut e moment, u of the function c ( z ,t) is n e defin e d as m I n u n mo (23) 00 wh ere m = J tnc ( z ,t)dt for n ~ O n 0 e (24) and th e c e ntral mome nts about the mean of a peak, un' are defined as for n > l. u n 00 1 n = J (t-u 1 ) c ( z t)dt mo O e For chro mat ographic purposes, the infinite integral o f equ at ion 25 ma y b e expres se d as +oo u = n f (t u 1 ) 11 f(t)dt _ro +oo f f(t)dt -00 (25) ( 26) PAGE 50 27 for n > 1 where u i s the nth norm a lized centra l m om ent and where f (t) n is the amp li t ud e of the chromatograp hic signa l at time t. Further, w ith fa s t data acquis iti o n sy s tems whe r e several hundred po i nt s are easil y obtained for each peak, the infinite int e g ral of equation 26 may be approxim a ted by the su m matio n u = n j / 11 L (t.-u 1 ) Y i = l 1 1 j Y. 1 (27) where Y is t he signal amplitude at ti .me t for j d ata points. Chesler i and Cram [21] have sho w n that t h e simpl e rectan g ul ar rul e nu mer ical me t hod gives a goo d approximation of the continuous function ( eq uation 26) if a la rge numb er of points are taken with a lar ge percentage of th e points in the diffu se edge s of t he p e ak. Chrom ato g raphic Si g n i ficance of the Statistical ~ bments The statistical moments ar e especially u sef ul, since they describe the po s ition and shape of any distribution, and should therefor e be applicabl e to all chromatogr ap hic peaks. N othing ha s to be assu med befor ehan d about the p eak shape e x cept that the int egra l in equation 1 exists and can be evaluated numerically. The Gaus sia n w ill be a special li mi ting case whe n calculatin g the momen ts. The zeroth moment u 0 w hich i s the a rea under a chro ma togr a phic peak, can be normalized to one. Thi s coincides with the assumption that all of the input fed into the b e d passes through the be d ; no fr action of the sam ple remains perm anently or chemically reacted on t he b e d. PAGE 51 28 The first statist.i.c al :n ome nt, which is the location o f the c en tro id of the area un d er the peak has theoret i ca l ly been found t o be a function main l y of the valu es of the eq uilib r iu m constants K and C K th e ve locity v, and the porosit ies E and e [ 4 ,6]. ( F or s irnplicn 1. e I ity, since there is only the abso lute mom e nt u 1 and the mom e nt ar ou nd the me a~ u 1 i s ze ro, I u 1 i.nstead of u 1 will be used to denote the f ir st mom ent 8.Dd u w ill denot e the other c e ntr al moments of t he chromato n grap hi c peak ) 2D : ) <1H/ > Kc (l+Kn)) + V (28) L is the column len gth t 0 is the time of durr, tio ~ of the injection (and other system ti me constants) and = ]. e (1-e ) 8 e = ---e (29) There is a very minor dependence of u 1 on the ef fectiv e int e rp a rticl e lo ng itudinal diffusion co effi ci e nt D which is see n only p at low ve locities. The tim e (L / v)(1+ ) r epres en ts t or the time a reqt ir e d for a nons orbe d in ert compo und to travel through the co lumn. Thj s tim e is incr eased by the amo unt (L /v ) K (l +K ) for sample come n ponent s w hich are adsor b e d either in th e pore space ( vo lu ma l ad sor ption) or on the pore wall (surface adsorption ). The ti m e of i n j ect ion, t 0 c an be very important wJ1en u 1 is sm a l l It is i mpo rtant to pc int out that the location of th e mea n does not depend upon any kin eti c con stants (H ,H ) an d, therefo re not up o n the velocity at which eq ui c n libriu m is estab li she d on the column PAGE 52 29 The s e co nd s t ati sti er : D'.O !n e nt u n i s 1e l 2tec1 -:: o a l 1 of the L., con stants us e d to d es crib e the particle model and its bed ( S,e ,R,L), e the ca r rier gas velocity v, the eq u i libr ium con st d nt s (K ,K), and the C n kin e tic c onstants of adsorption (H ,H) by C n 1 2D L = I _E..__ \ 3 V \ / 2; + + V 2 + )( 1 +K (l + I< ) ) 2 4 / ~ n V 4D P } -, -;:.,j\, .:, C V ? 2 [ R~(l+ K ) n l) \) ( \!+ 2) + r 2 K ] 11 + HJ 11 For spherical p a rticles, the shape factor, v e qual s three (30) The Scond mom ent is the v a riance of the prob ab ility density curve, and its square root is the sta ndard de v i at ion cr or wid t h 2 u,., o ,, (31) which defi.n e s the average deviation of the individual points of the curve :from t he mean. It can be seen again that u s ually the D terms p divid e d by a hi .g her pow e r of v will be negligibl e The third and fourth moments u 3 and u 4 get pro g ressively mor e involved and are extremely sensitive to cha nge s in the leading and tailing e dges of chromatographic peaks and to peak symmetry. These moment s are as follo w s: 3 64D) 3 6 p (1+ K (l+K ) ) C n V 48D + 4 P) K (1+ K (l +K ) ) C C n V PAGE 53 R(l +K ) 1 + K ) ~ K n n n l ? 2 ,, l X -2+ ~-+ iT D ( v +2v ) c n r ; ( GL J. 2 D p) R4 2 (l + K 0 ) 2R (l + Kn) [ 3 2 + \-v + 2 Kc 2 2 -. -+ D \ I ( \!! 2 ) v D v (v+2) ( v+4 ) r r +K ) 2 X (H n K (l+K ) + Hn) + n n H 2 3 K (l + K ) K J n n n + H H + 2 c n H C 3 16D L 4 C p ~+ V 960D 4 -8 -P 1 l(1+K (l + K )) C n V 288D 2 L 3 n 1 2 4 D L 2 + \ + p --5-+ 960D 2 -6 -p ) (1 +K (l+K ) ) c n V V V PAGE 54 l R 6 X 3 D r 2 (l + K ) n : 3 2 2 (v + 4 v ) ( v+ 2) 2 ( (5 v+ 12) (l +K n) (6 v+ 1 2)K ___ H ___ n) X \ H + C n t2(1:Kn )4 (l +K 2 R2 1 6K ) (2+3K n n n + D 2 + H H -+ 2 r \) + 2\) H C n H C n 3 4 2 2 (l+K ) 3 K (l +K ) Y. (2+ 3K ) Kn] n n n n n + 3 + + + 2 H n 2 3 H H H H C C n C n n 31 )K n ) ( 33) Th e third moment will be ZP. ro for a symmetrical peak, ne g at ive for a fronting peak, and, as is the usual c ase positiv e for taiU.ng peaks. Two hybrid third moments, called specific asy mmetr y and skew 3 3 are calcul ate d by u 3 ; u 1 and u 3 / (u 2 ) respectively. T h e se are also useful for characteri z ing peak asymmetry and have the a dvantages of being dime n sionless and giving nu m bers which are easily comparable to the valu es of zero s pecific asymmetry and zer o skew obtained for Gaussian peaks The fourth m ome nt, w hich is a mea s ur e of the excess or flattening of the peak compa red to a G a us sia n curve, can b e treated as the dimensionl ess hybrid s called specific excess and exce s s, which are 4 2 ca l culated by u 4 / u 1 a nd u 4 / u 2 respectively. If exc ess is less than three, the curve is s ma l ler than the normal curve, and if it is g r ea ter than thr ee, then t he distribu t ion is hi g h e r than th e normal distribution PAGE 55 32 All higher odd mo mcn t s further desc r ibe asynl'll.etry, while higher even moments describe charact e ristics of th e peak width These moments c annot be m e aningfu ll y c a lcul at e d in prac tice, due to the lar ge erro rs incurr ed by raising sma ll signals of larg e d ev i ation ( t -u 1 ) on the tail o f chromatographic peaks to a hi g h power (see equ a tio n 26). In the n ext three secti ons of this chapter, statistical moment expr essions for three practical cases of GC w ill be d e v elo p ed T h ey are as follo ws: 1. Gas-solid chromato g r aphy ( GSC ) in ~11ich solutes are chromato graphed on a porous ad s o rbent. This is the genera l case examined by Kubin [5], K u ce r a [6], an d Grubner[11] w here longitudinal diffusion, external diffus i on i nt e rnal diffu sion, and adsorption on the pore wall surfac es ma y be si m ul taneously significant. 2. Ga s -liquid ads orption chroma to g r aphy using a porous a dsorbent th at is coated and impr eg nated v it h a liquid in wh ich the solu tes ar e highly soluble. Thi s c ase i.s com m o nly call e d gas-liquid par t ition chromatog : .cap hy (GLC), and the processes of lon g itud ina l diffusion, external diffu s ion, internal dif fusion, and sorption with in the pore volume it se lf must be consid e r ed The equations mention e d above for GSC w ill be extended to include ~1is particular case of GLC 3. Gas-solid ad sorption chro1:rntography in wh ich solutes are chromato graphe d on a nonporous adsorbent. In th is limiting case of th e general example of GSC, ther e is no internal diffusion, so that only longitudi nal diffusion, external dif fusion, and external surface adsorption must b e considered as simultaneou sly occurrin g chromatographic phenomena. The Case of Lon gi tudinal, External, and Int e rnal Diffusi on and Surface Ad s orption In the general case, as for very porous lar ge surface area silica gel, the processes of longitudinal diffusion, external diffusion, internal diffusion, and s urface ad s orption may occur simultan eously in the gas chromatogra p hic column. In such a case, there can PAGE 56 33 be f ew si m plific a tions of the mom e n t equation s a nd th ey must be solved ex act ly. For activated si l ica g el ( F i gure 1), it is assumed that only s urface, and not vo lu ma l, adsor pt i o n i s o c curri ng so that K :=-: K and K = 1. It will b e assumed for the r em ai nder of this dis n n c cussion that terms with hi g her powe r s of D / v a re negli g ible an d that p v = 3 ( spherical particles). The n simplified ex pr e ssions m ay be deriv ed for u 1 th rou g h u 3 from equa t ions 28, 30, and 32. These can furthe:::b e e:-.. --pressed in terms of band broad e ning by t ai'icl in terrr, s of band asym met ry by Z : t u 2 L -2 ul T2 and z U3.u = -3ul cm 2 cm The param ete r t is a n a lo gous to the Gaussian-ba se d H E TP (height equivalent to one theoretical plate). (34) (35) T emporarily, t 0 wi ll be assumed to be unimportant, since it is usually very small compared to retention times of retained adsorbates. It wi ll however, be included in the analytical moment expres sions in the Resu lts and Discussion chapter of this dissertation Then from equations 28 and 30 L __ (1+ (l+K ) ) V n (36) (37) PAGE 57 whereupon Definin g D p 2D L p ? u =~ 2 ..., (1+~(}. JK )) ~ 11 ~ 1 = as in ~ 1 = V 2 0 (l + I{ ) ::-1 2 n, [H2 (i +K_ ) 2 L 'P il + 1 5D + H 2D ___E, + V r C 2 2 2 (/J R (l + K ) v n i 15D (l 1 +K )) r n 2 2(1 +K ) V n 2 2 K V n + --------=+ H (1+(1 +K )) 2 2 H (1 + (l+K ) ) C n n n equation 5 2D 2 2 (l+K )2v 2A + g n -+ ))2 + V 15D (1+ (l+K r n 2 2 2 (l +K ) V n + 2 H (1+ (1 + K )) C n 2 2 K V n H (l+'tJ (1 + (l +K ) ) 2 n n In the ev e nt that for very strong adsorbates K >> 1 say K > 20, n n then equation 40 simplifies to 2D 2R2v __[ + 2v 2v ~1 = 2 A + v i5D + H + !-I r c n This is further simplified to 2D 2R 2 v = 2A + __[ + 1 V 15D r 34 (3 8 ) (39) (40) ( 41) (42) when the rate of mass tran s fer (H) and the rate of adsorp tion (H) C n are infin ite ly fast, i e H _, oo c,n Thus, the C term is determin ed PAGE 58 only by radial diffusion in tie pores. In tb i s d e rivation, the A term is equal to one half of the A value found for the clas s ical D van Deemter eq ua t ion 43. B HETP = A + + Cv D V 35 (43) where A, B, and Care the coeffici en ts :for eddy diffusion, molec ular di ffusion, and resi st ance to mass transfer, respectively. Using equati<:,ns 2 8 and 32, expressions for z 1 may be found in the follo w ing manner: u,., = .:, 2 3 r 2 2 12D LX 12D r ~x R (l+K) P + __ P.uy., l n 5 3 v v lvDr + 2 3 2R (l+K) n __ c_1_:K-c_n_) 2 + :: l 2 2R (l+K )K n n __ 1_5_D_H __ + 15D H r n r C 2 (l+K ) 3 2 K (l+K ) K l n n n n +---2--+ HH +2 H c n H C n where x will be defined as (1+(1+K )), and then n + + 4D 2 (l + K ) 2 P n 12D 0 2 (1+K ) 2 12D K p n p n _____ 2 ___ + 5D X r ---=---2--+ 2 H X H X 4 3 2 4R (l+K ) v n ------+ 105D 2 x 3 r C n 2 2 3 2 4 R (l +K ) v n ------3--+ 5D H X r C 12 2 K (l+K ) v 2 n n -----3---+ H H X C n 4R 2 (1+K )K v 2 n n 5D H x 3 r n 2 6 K V n (44) (45) (46) PAGE 59 Substitution of the e xpre ssion f o r D gives p 12D 2 2 ? 2 2 2 4D A 2 4D 0 R (l +K ) 4A R (l +K ) v g O" g n 11 zl = ~+ __ E.,_ + 1 2A + + V 2 2 V 5D X 5D X r r 12 2 (1 + K ) 2 2 2 1 2D K 12 AK v 1 2A (l +K ) v g n n g n 11 + n x2 + 2 + 2 'T 2 H X H X H V C C n n 4 ,:, ? 2 2 )\-2 2 2 4R (l +K ) ""v" 4 R (l + K 4R (l+K ) K v n n n n + 2 3 + 3 + 3 105D X 5D H X 5D H X r r c r n 3 (1 +K ) 3 v 2 12 2 K (l + K )v 2 6 K V 2 36 n n n n (47) + 2 3 + 3 + 2 3 HcX HcHnX H X n for the c a se w h e r e diffusi on and kinetic processes jointly control the peak zl shape and K i s sma ll. When K is 11 ::: n 12D 2 2 4D A 12A 2 24D R2 _[+ g + + g + 5D 2 V V r 12D 12 A v 12D 12Av g g +~+ + H K + H K C C n n n n 4R 4 v 2 4R 2 v 2 4R 2 v 2 + 105 2 1/ + 5D H + 2 r C 5D H K r r n n 6v 2 + -+ 2 H C 12v 2 H H K C 11 n + 6v 2 2 2 2 K K n n l arge, then X ->ciJK and n 1 4.AR 2 V 5 D (/J r For the case where only radial diffusion is important (H -+ co D = D ) r r z1 = 12D g --T 2 V 24D A 12 A 2 + __ g _+ V c,n 4D R 2 2 4 2 g 4AR v 4R v + 5D + 5D dJ 105 2 D 2 r r r (48) (49) PAGE 60 The Ca se o f Lon g itu d i na l, Ext ernal and In ~e rn a l Dif f u s i o n a nd Voluma l Adsor p tion 37 If the pores of s ilica gel are filled w ith a c o mpound in w hich the adsorbate is hi g hly so lu bl e then volurnal ad s orption wil l be importan t rath e r than surface ad s orption Ther e fore, K = 0 (and H ..... 00 ) n n and K = K C C The proc e s se s of lon g itudinal diffusion, externa l diffusion, and rudial intern a l diffusion may s imultaneously b e important as shown for a s in g le part icl e in F igur e 2. T he thin liquid filn1ssurrounding the particle and on the pore walls are d es i gnated by 3 an d 4. The pr ocesses of penetrat ion of the thin l iq ui d film around the particle, radial diffusion in the liquidf illed pore, and penetration of the thin liq uid film on the pore wal l are denoted by i 4 i 5 and i 6 respectively. The processes f or the gas-liquid system 2re simjlar to those shown in Figur e 1 for the gas solid system and the concentrations, effectiv e diffusion coefficients, an d rate constants in the mass balance e quations have analogo u s meanin gs Thus, the mome nt eq u ations can be derived sim il ar ly a nd the eh-pressions for t 2 and z 2 under various cond it i ons are found as fo llo ws : L ul = (1 + K) V C (50) 2 2 (l +K ) 2 L ul = 2 C (51) V 2D L (1+K ) 2 2LKc [ R 2 :J p u2 = -3+ V 15D + C V r (52) PAGE 61 Figure 2. Model particle for OV-101 adsorbe nt. The external and intern a l wall liquid films are designated by 3 and 4. The particle di amete r is 2R and ce and ci designate the external and in ternal adsorbat e co ncentrations. Mass tra nsfe r throu gh the li quid film a rou nd partic le (i 4), mass t1 ansfer by r n dial diffusion in the 1 iquid in the pore (i 5 ), and mass transfer across the liquid film around the pore wall (i 6 ) are shown. PAGE 62 39 -----~ / ---"""' / y:3 / 0\ I 4 \\ \ I \ I \ I CA / -----/ '-..... -----C 2R PAGE 63 F rom ~1ese eq u at i o n s 1jr u 2 L = 2 2 ul 2D 2 J? K v 2 2 K v p C C = -+ + V 15D (1 + K )2 r C w hich is the sa m e as 2D g 2R 2 K v C (l + lil K 1j1 2 A + 2 ---1v -----+ 15D (1+ K / r C and for t h e c as e th a t K is lar ge C 2D 2 2 R v 2v 1lr 2 = 2A + + 1 5 D K + ~ V r C C C which si mp lifi e s to 2 2D 2 R v ,I, = 2A + _[ + o/2 V 15 D K r C 2 ) H C C 2 (1 + K ) H C C 4 0 (53) ( 54) ( 55 ) (56) when the r a t e of mas s t r ansfer through the liquid film is fast (H ...., 00 ) C The rate of mass tran s fer throu g h the sta g n a nt g as film w hich covers t he liquid thin fil m is relativ e ly ve r y f as t in all ca ses. In the derivat i on o f the v a rio u s z 2 e x pres s ions, ( 5 7) 1 2D L K /\ [ 2 p C R + 3 15D + V r : J (58 ) PAGE 64 where f, is defined as (1+ K ) an i then C r) 12n= p -y + 2 4D QJK R p C ---.-2+ V + or with the 5D ii 4K R 2 v 2 C r ___ 2_3_ + 105D /\ r 2 K r/v 2 C ----3-+ 5D II /\ r C s ubs t itution for D p 3 2 6 KV C 2 3 H /\ C 12D 2 24AD 4D K R2 2 __ g + ___ g + g C 12A + + 4AK 2 R V C z2 2 V 2 2 V 5D /1. 5D /1. r r 2 2 12D K 12A K v g C C + H /1.2 + 2 H / 1 C C 4 2 ? 2 2 3 2 4K R v 4'" K R V 6 KV C C C + 105D 2 t? + t? + 2 3 5D H H A r r C C Equation GO may be simplified if K >> 1, in w hich case /1. ... K to C C + and to ? 12AD 2 ___ g + 12A + V 4D R~ _.;:::.g __ + 4 4 2 R V 2 2 2 + 105D rj) K r C 5D K 5D (/)K r c r c 6v 2 ----2 + -2-2 5D H K HK 4 2 2 R V r C C C C 41 (59) (60) (61) PAGE 65 42 l? 2 ? 2 ~D 12AD :JD R ~ 4A R 1 4 2 ? 4H g g g V z2 -2, + + 1 21-\ ~ + 5D dJK + 5D .,K + 2 2 2 (62 ) V V y C r C 105D K r C when H is fas t (H .. 00 ). C C The Case of Lan g i t udj_ na l an d Externa l Diffus i .o n and S ur face Adsortion When a column i s pack ed v t t h a very l ow s urfac e area nonpo co us ad s orb e nt s uc h as graphiti z ed car b on black, tl;e p 1 n cip a l adsorption will be on the p a rticl e :::; nrface ( K = K ) and the passa g e through tll. e n n column w ill be rate controll e d a t high vel o cities Although individual graphi tlzed carb on bl a ck cry s tal s are ve r y hoir ,ogeneo us nonporous almost fl a t polyhed ra of about 0 .. 3 micron in di2 rne ter, many such poly hedra con g lo merate into larger p a rticle s These particl e s may be sieved an d used to p a ck colun ms T he s e p a r t icl es are quite round (v = 3), and they are nonporous ( 13 = 0), as evidenced by th e very lo w surface ar e a per g :ram and by phot om icrograph s The model for this particle is shown in Figure 3. In this case, transfer throu gh the thin g a s film surrounding the particle i 1 is the sam e as that in Figure 1, and the proc e ss of adsorption i 3 is the same as for Figure 1 except that the ad s orption is on the external (pore wall) surface Furthermore, the conc ep t of the thin film associated with the con s tant H for Figures 1 and 2 is not n ee ded. That is, n the rate constant of adsorption H has a finite valu e but it need not n be caused by a thin film. It is assum ed that D 00 and S is nonexistent s o that now the r porosity function i s simply (1) / e e It is correct to assum e that PAGE 66 Figure 3. Mode l particle for g r aphitized c ::i rbon blac k adsorbent. The external gas film is designated by 1 the diameter by 2R, and th e external adsorb ate c 0n c ontration by Ce Mass tran sfe rs through the e xt e rn a l g as fil m (i 1 ) and across tr.e gas film at the particle surface (i 3 ) are sho w n. PAGE 67 44 4-------.--2R PAGE 68 45 K = 1 since we are d ea lin g only with the surface, which is the int er e face, for the two previous po; ~o us models, wh e re c = c 1 e Since the ori ginal transformation yield e d a solution for~ (z,s), i.e., the e interparticle conc entrat ion profile at e lu tion, the resu l ting moments for tl1e present case should be similar in form to equations 28-33. This assu mpt ion should be valid because D and H are always inder c,n pendent of each oth er Therefore, either c a n ba independently neglected as presented here without resolving the original mass balance equations. Developing the equations for ~ 3 for this case, we obtain T, = :::. (1 + (l+K ) ) V Il 2D L u 2 -+ (1+(1+Kn)) 2 + V which give and 2D = __p_ + V 2 2 2(/J (l+K ) v n ------+ H (1+ (l+K ) ) 2 C Il 2 H (l+(1+K )) 11 n 2D 2 (1+K ) 2 2K 2 v W3 = 2A + __[ + _____ n_, __ + n v H (1+(1+K )) 2 H (1+(1+K )) 2 c n n n (63) (64) (65) (66) (67) PAGE 69 for the unsi mp lified ca ses w ith ~ 3 2A + 2D er __ o + V 2v 3v H + I-I C n for the sp ecia l case where K >> 1 and n ~3 = 2D 2A + -g + V 2v H n for the furth er spec i a l c ase whe r e H -+ 00 and onl y H is rate C r ; limitin :;. F'or the expressions descri bing column effici en cy in terms 46 (68) (69) of symmetry, 3 L 3 3 u 1 = 3 X (70) V where again X w ill be d efine d as (1 (1 +K )) and n 12DPL X 12DPL X l(1 +K n) 2 HKnnl u3 = 5 + 3 H + V V C giving 12D 2 = __ P_ + 2 V + 2K (l+K ) K J _n n + __:: H H H2 C n n 2 2 12D (l+K) P n ----2---+ 12D 0 K p n 2 H X H X C n 2 2 12 K (l +K )v n n -----3---+ H H X C n 2 6 K V n 2 3 H X n (71) (72) PAGE 70 47 and ') ') 2 2 2 12D"' 24D A ') 1 2D ~ "' (l+ K ) 12 A' ~ ( l + K ) v __ g_ + g 1 2A"' G !l n z3 = + + 2 + 2 2 V V H ex H e x + 3 3 2 12 2 K (l+K ) v 2 2 6 (l +K ) V GK v n 11 n n + 2 3 + 3 + 2 3 H X H H X H X C C n n (73) for the most cornpl icated case o f a weak adsorb e r (K < 10) an d where n H and H are simult aneous ly important. C n For a very strong adsorber where K >> 1 n which z3 is = 12D A ---"-g_ + 12A 2 + V 12D 12 g Av + H !~ + H K n n n n 12v 2 H H K c n n fur the r simplified 12D 2 12D A 12A 2 g __ g_+ --2+ V V 12D to 12D g -+ H C o12 Av + __ "--+ -+ H K H K 6v 2 2 2 2 H K n n n n n n 12Av H C if the rat e of mass transfer acro s s the thin film is fast and only H is import an t. n (74) (75) PAGE 71 48 Ther e may also b e the case wh ere th e re sistance to rnnss transfer terms in ~ 3 a nd z,, for f; wea k adsc.rber t s influenced m ainly .., by the r ate of adso r ption, H (H 00 ). Then from eq u ations 67 and 73 n C and 2D 'V = 2A + __[ + 3 V 2 2 K V n ? 12D~ 24D A __ g...._ + 12A 2 2 3 = 2 g + V V 12D K g n + 2 H X n 12 A/JK v n + --z+ H X n 2 6K V n 2 3 H X n Assu mpt ion s of the :vro d e ls (76) (77) N um e rous other assumptions were found nec ess ary to si mp l if y the cas e s of chromatography me ntion e d above. The models are r egar ded as closely approximating r ea lity since all known proper t ies of the bed and transport phen omena are included. Other ph en omena, ho w ever, are neglected: 1. For the very porous p a rticl es adsorption on the external surfac e js ne g lec ted s i n ce t h e internal s u rfa ce is typically 10 3 to 10 5 times la rge r than the geometric surface a rea. For the nonporous parti cl es, all adsor ption occurs, of course, on the external s urf ace 2. It i s assumed that the carri er gas is incompressibl e s uch that th ere is a negli gible pressure drop ac ro ss the colu mn thus no pres s ur e gradient from particle to particle. The exper im ent al data c a n, ho wever, b e corr ected for pressure effects as will be d isc u ss ed in the n ex t s ec tion. 3. Forc ed flow throu g h the pores is neglect e d since the resis tan ce of s uch small por es (100A 0 ) is so lar ge that probably th e carr ier only flows in th e interp a~ticle space, as stat ed b e fore. PAGE 72 4. It i s assumed that a ll affective dif fusio n coefficients are indep endent of the conc PAGE 73 50 [70 71] ccncl udecl that th e a v e~ age c:1lc: : u lr: ted pldte height H shtluld ,:ccount for the r.. atu re of t: :e variat ion of LE ETP along the co l umu and c o uld be w ri tte n in the f or~ H = fH + jc .. e,ll g (80) where 4 2 f 9 (P l) (P -l) 8 3 l ) 2 (P (81) ond. ? 3 (P~--1) j = -3-2 (P 1 ) (82) The 1i f '.3 nd :iC nu are th e contributions f ro m the gas and J.iql:id [::hases p.; .(., The fa ct or ju is the Ja r;ies-;1-ia rtin t ~ rne average co lu mn ve loc i ty, app li c abL, t o cornp :i : essib l e carr:ier f; Bse s and P = pi /p 0 the ratio o f -.:o 1u n.n i nlet pr e ssu. :r e to coJ.u:nn outlet pr e ssure The p ress u re cor rection fac tor f was derived f r om c onsiderations of band spreading due to clecompressio:1 at the en d of the c ol umn w here th e pressure drop per colu mn segment is sma l l er but he ve locity is l a rger t !:an at the colum r. head. Und erhil l [29,3 0 ,72] used s tati s ti ca l m ome nt theory to c a lculate th e pr ess ur e drop ef :l'e ct in a GC column He started b y consid er in g the col unrn w i ~h a pre ss ur e drop as con sis ti ng of a series of s hor t column s eac:n ha .r i.n~ R ne ar ly con s tant pre s.s ur e over i ts le ngt h. Extending Mc~urn-ri e s c~ ; lc u latjon s [ 68] for col uim1s in ta nd em to calculation s for an i n f i.ni te ,:nmber of colunm scci.:i.ons the pressure correctio::1s for the mor.,e nt s c o we ::. ;c-;e d to the re s ul ts o f Giddings above for the h e i. gh t of a th eoreti cal plate PAGE 74 51 The Lapl a ce transform for the re sponse y. to a sample input l puls e for one of the colu m.n segment s i i s y i = exp 4D .Ks .;\,. -l ') ] p,1.) 2 ? v~ l. where and V i = D = p,1 V 0 D p X 2 ].1. (p -1) 2 L K = e: + (K -e: ) r sR 2 2 ( 0 ) coth 3D L 2 .1. e n e sR r 2 2. ( SR \ 2 nJ r Also, 6 L is the length of the ith segment, V is the superficial 0 carrier gas velocity at the column outlet, sis the coefficient of the Laplace tr a nsform, and K is the transform for the effective equilibrium adsorption co e fficient. The quantities D ,P,e ,K ,D, p e n r and R have b ee n previously defined. (83) (84) (85) (86) In thjs develop ment it is assum e d that the column is operated isothermally and is packed with uniform spheres of adsorb ent, and that the princip a l mechanisms of mass transfer are interparticle molecular diffusion, int erpa rticl e eddy diffusion, and intraparticle diffusion. It can safely be assumed that the adsorption is othe rm is line ar PAGE 75 Furth ermo re th e in tr2 p a r t icl e diffusion coeff i cient i s pressure ind epend e n t d u e to t he rnicr o porous nature o f th e adso rbent ( si lica g e l). Int eg ratin g the Lapl a ce transform Cover the l eng th of the colu mn a nd w i th r espe ct to pre s sure then using th e relation where u is the nth statistic a l m oment the follo w in g moment equa n tions c an b e derived: uo = 1 p lKnL ul = --V 0 (1;nL ) 2 2P 1 (K ) R 2 L 2P 1<2D L 2 n e 2 n P u2 -+ 15D V + 3 0 r o V 0 /1KnL '{ 2 2 2 4P K (K )TI L 1 n n e u3 = \ V I + 5D v 2 0 r o 3 2 4Pl (Kn 4 P 1 P 2 KnD/, -e: )R L e + 4 + 105D 2 V V 0 r o P 2 K (K )R 2 D L 3 2 12P 3 K D L n n e p + n P + 3 vs 4D V r o 0 where L (P2 (P2-l)n / 2 J X L dx p 0 = n L 52 (87) (88) (89) (90) (91) (92) PAGE 76 53 whi ch can b 8 int eg rat e d ov e r the c olumn l e n gth to give p = n (93) I t c an be seen that the u eq uation s 89-91 are only sli g ht variations n of the previously de r iv e d equa t ions 28, 30, and 32 for GSC. For n = 1 the corr e c tion fsctor P has the same s i g nificance n as the Ja mes-M a rtin j fac tor, si n ce the effect of pressure on t he mean r etent io n time is identical in both cases. The mean retention time effectively increa ses with increased pressure drop across the column due to the d e cre ase in time-averaged col umn velocity. For the second an d third moments, it is seen that a decrease in D due to p pre ss ure dr op is coun terba lanc e d by equal decreases in V such that the 0 D te::-ms incr eas e at a slightly slo we r rate than expected. The D P r terms a re only a f fected by the d e crease in velocity with pressure since D is ass umed ind epen d e nt of pre ssure. r n Very little error i s incurred if Pn is equated to P 1 for P<3 P = Pn = n 1 j -n such that the si mple j factor calculated from pi / p 0 can be used exclusively for pressur e corrections. (94) Analogous to equatio n 80, the c6rre sp onding valu es of H when using the statistical moments are = ft + jA g (95) where is the pressure-corrected value of ~ and+ is the measured g value of t not corr e ct e d for pressure drop effects. All of the oth e r symbols have been d ef in e d above. It is clear that, d ep e nding on th e PAGE 77 54 rel ative i mpo rt ance of th e g as and eddy con t ributi o ns, the value of i c ould incr e as e or decre as e with pressure drop. T his is because f varie s f r om 9 / 8 for infinit e press ure drop to uni ty wh en there is very littl e pressure drop In contrast, j is al ways l ess than on e ; it appro aches one for no pr ess u re drop and ze ro for inf inite inlet pr e ssures. In th e u s u a l case, A wi ll be small such tha t t controls the g magn itu d e of t and, th us, the ob se rved efficiency parameter t w ill g incre ase with incre ase d pre ss ure drop. Simil a rly, at lo w velocities, for mi croporo us adsorbents where int e r pa1 ticle diffus i on controls mass transf er Z is expected to dec rease by a factor betw ee n one an d 5 / 4 ./2 with increasing pressure drops as 5(P 4 -1 )3 / 2 z = ------z 4 ,.,_0,( P 5 -1) ( P 2 -1) l / 2 (96) At hi g her velocities, with mass transfer controlled by intraparticle diffu sion in microporous adsorbents t and Z are e:>,.l)ected to require the follo w ing approximate corrections due to pressure drop [30]: (97) .-1 / 2 z = J z (98) These equations predict that the pressure-corr .:.e t e d parameter tw ill b 11 b f t f d Z 11 b 1 b a factor of J.-l / 2 e sma er ya ac or o Jan w 1 e arg,er y comp ared to the observed ~ and Z und er the inf ~ u e nce of a pressure drop. T hes e are, of cours e, the effects of pr PAGE 78 EXPERIMENTAL Char acterizat io n of the Ga s Chrom at o g r aph / Comp uter System A Varian Mo d e l 2100 gas chro mat o g raph (CC) eq uipped with a flam e ionization detector (FID) was modified to provide precision tempera tur e and flow regulation under computer control, precision samplin g and a utomatic dnta collection. The data system was based on a PDP --S / L on-lin e d e dicated ll'lboratory computer. In order to obtain reli able infor mation from GC p ea ks, a critical evaluation of the precisio n ari d accuracy of all the system compo nents was made. Efforts were rr:ade to m inimi ze dead volu m e and mi x i_ng volum es so that in stru mental e ffects were small. Thus syst ema tic influ e nc e s on the precision and accur a cy of the chromatographic data were known and appropri ate correction s could be made Pne . um at ic System The well-re g ulated flow system designed for the carrier and sample gases is shov.-"TI in Fi g ure 4. The balanced pressure re g ulators (Mod el P N -41200G 19, Veriflow Corp.) wer e chosen ov e r standard re g ulator s b e csuse they are virtually ind epen de R t of variations in the inlet pressure at the sour ce : the outl e t pressure is changed l ess than 0.0 2 p si (0.9 mm Hg) for 2. 100 psi drop in inlet pressu re. 55 PAGE 79 Figure 4. Pneumatic, compl.lter control, an d data a cqui s ition systems G as lines are denote d by ( :=) electricz. l l ines by ( -), are pr essure d rn ges @ are flow cU.verters, /7) are three way switching valves, and J are fine m e tering v~ c>s PAGE 80 B U F f E ~ R IS TR P DP-8/L CR Y STAL C1 l LA l OR (SK ) BUf FER R E GIS TEil Tt..P [ S COPE O E V I CE SE L E C T O R ML PX FP. E O UC NC Y O IVI O E R FF 't._ _J.~; x TTY P EXP O I L FLA SK VA L VE CR IVER VA RIABLE GA I N AM PL V A L VE DR I H R VENT SY H COf! D ER PAGE 81 The regulators were followed by 500 cc traps of 5A and 13A Molecular Sieves These were followed by standard sing le stage pressure r eg l ators (Model 4 1300451, Verif l ow Corp.) which off er precise control when the inlet pressure is h e ld constan t : the outlet pressure varies le ss than 0.5 psi (20 mm Hg ) for a 100 psi chan ge in the inlet pressure. Thus they were used as a second stage of re g ulation. In the s am pl e line an extra fine con sta nt upstre am flow con tro l le r ( M odel PN-4 2300 0 80 Verif low Corp.), w hich had a repeatability of flo w set ting of better t han 2 % was used. 58 The heliu m c ar ri er gas (Grade A, Gardener Cryogenics), which had a stated purity of 99.995 % and was oil and moisture free, was reg ulated by a computer controlled mass flow controller ( M odel FCS-100, Tylan C or p.). The f lo w controller op e r ates by pro vi ding a mass flow rat e from Oto 100 standard cubic centimeters per minute (SCC M ), which will subsequently be referred to as cubic centimeters per minute (cc / min). This flow rate i s proportional to a Oto 5V command voltage set thr ough the comp ute r in terfa ce. The controller compares a reference vo ltage -:.:o the output of a mass flo w transducer and adjusts a valu e until the two voltages a re equal. A pressure drop (50 psi) was main t ained across th e flo w controller at all times to insure rapid respon se and correct flow rate. The controller was periodically calibrated by displacing one to five liter qu a ntities of water from a volumetric flask wh ile the controller was set at various flow rates. For most of this work the 4-25 cc/min flow range was used. It was found that precision on the mean flo w ra te was 0.08 % relative stand a rd d ev ia tion for the slowest rat es and 0.06 % at the hi ghe r PAGE 82 r ates The repeatability o :f a gi v en setting over seve ral 11!onths' ti me was 0.0 8 % for a ll parts of the flow ran ge use d, wh ich w as co nsjdercd excellerit Th e voltage output of the co ntr o ll e r was continu o u s ly mon i tore d u s in g a digital multimeter (J\I ode l 160, Keith l ey 59 In s tr.) and w 2 s always stable to at le ast 0. 85 m v, which corr espo nds to 0 0 17 cc / mi n. This i s a stabi lit y of 0.4 2 % at the lo we s t flow rates a nd 0.05 % at the h ighest rates used. The re s ponse time of the mass flow controller w as observ e d to be approximat e ly three seconds for a flow incr ease of 5 cc / min for rat es greater than 8 cc / min. This implies that the 0.017 cc / min var iat ions in th e stabi lity could be corrected in about 12 msec. The r esponse was slightly slo wer a t 4 cc/min and extreme ly slow, around 20 seconds at 2 cc / min. This d ev ice was fortunately th e most reli a ble part of the flow system and i ndee d control of the flo w rate was found to be a suitable substitute for independ ent regulation of inlet an d outlet pressure [45,46]. At no time durin g an experim e nt was it: necessary to inter rupt the c arrier gas flow at the injection or detecto r end in order to take a flo w r a te readin g It was found th at maintaining the controller in Styrofoam at room temperature was sw:fficient to achieve the above stability. The Tylan mass flow controller was factory calibr ated to deliver constant mass flow rates of helium. For example, the numb er of mole cules of heliu m in a 10 cc volume at standard te mperature and pressure is al lo we d throu g h the controller when 0.5 vi applied. This mass flow is ind epen dent of pr es sure, since a n electi.ronic feedback circuit PAGE 83 maintain s the balance bet w een actual and standard density and mass flow of helium. PactFact -PstdFstd" 60 (99) However, the 10 cc volume is compressible such that the corresponding volumetri c flow rate is slow e r at the high pressur e end o f the column and f aster at the outlet, w hich is normally at atmospheric pressure. Therefor e the set "m a ss flow rate" must be corrected for compre ss ibil ity due to pressure drop, since most chromatogr a phic calculation s are based on the volume of carrier gas passed, rather than on its mass. Pressure gauges, flow transducers, or flo w diverters (Model 51-000146-00, Varian Aerograph) were inserted into either the carrier or sample lines. Compressed gases or liquids could be sampled directly or diluted through an exponential dilution fl as k (EDF) b e fore samplin g A d es cription of the EDF and a study of the effects of sample size on chro mat ographic peak broadening and asymmetry has been included a s Appendix B. Samplin g Valve A cut-away cross-sectional view of the sampling valve used for most of this work is shown in Figure 5 in the sampling position. This valve was designed to maintain a pressure differential of 2,000 psi, to have minimum dead volume, to be amenable to complete automation, for high pr ec ision and minimum injection time, and to avoid interruption of the carrier gas flow while in the sampling position so that steady state pn E:: umatic equilibrium is maintained on the column. PAGE 84 Figure 5. High pressure sampling valve shown from the front with a cut-away view of the valve assembly. Valve is in the sample injection position. Sample inlet ports are perpendicular to the plane of the drawing. PAGE 85 SOLENOID II O v AC ___ ___. SHAFT TENSION TEFLON SEAL INJECTION PORT r He IN i J COLUMN SUPPORT TUBE ; l !C Y AC COLUMN PAGE 86 63 A detaj_l e d description of the valve has been reported elsewhere [7 3 ]. However, it should be mentio ned that the sam ple volume, defined by an annular groov e 0.015 in. x 0.005 in. on the flash-chromeplated (0. 0002 in ., N ational Bureau of Stan dards), plun g er rod (O. 2562 in., N ational Bu reau of Stan d ar ds) was 1.0 l for all of t his work. Capilla ry colu m n s or 1 / 8 in. pack e d colunms could be placed within 0.094 i n of the plu n ge r shaft to reduce the d ea d volum e This cha m ber represe nts a mixing vo lume of only 22 When 1 / 8 in. columns were used, th e column support tube in Figure 5 was not necessary. A pr e limin ary study involv e d the comparison of the performance of this valve with tw o others by an analysis of their statistical moments for resp e cti ve column input profil es [ 42] A complete d es cription of th is study has been compiled as Appendix A. An inspection of Table 17 in Appendix A sho w s that the valve referred to in the previous para g raph (high pressure) has a very acceptable injection repeatability ( a rea, 0.16 % ; first moment, 0.33 % ) and gives a very sharp sample p rof ile (variance, 3.095 x 103 2 sec ; sk ew 0.9762) using 2 % int eg ration limits [21]. The variance corr es ponds to a sigma value of 55.6 4 msec. T and the skew compares we ll with symmetrical peaks V (skew for a Gaussian= 0. 0). Figure 6 illust rates a typical injection profile obtained with this valve. Althou g h t his valve did not give the best precision, it was selected for this w ork because there were virtually no problems in column alignment, column chan g ing, or valve overheating as experienced with the others. At all times the valve body (30 4 stainl ess steel) was maintained at 70 C with a 200 watt cartrid ge he ate r ( l\Io del HS3725, IIotwatt). PAGE 87 Figure 6. Experirr,ental valve injection profile for the high pressure valve with int eg ra t ion limits at 2.0 % 0.5 % and 0 .2 % of the p ea k hei g ht T ime base of display is 0-2000 mse c PAGE 88 6 5 f(t) t PAGE 89 66 The travel dj_star,ce b etwee n the sa mple ch ambe r and the inj e ction por t was 0.726 in. This is suff ici ently lon g to provide a l arge sealin g surface with the Teflon seal and yet short enou g h to retain the pull ing power of the solenoids for high speed operation. Fifty-pound solenoids (No 147 -1, Dorm e yer Ind.) were used to activate the valv e und e r computer control. The narrow injection profjle can be explained in terms of the short effective injection time. Only duri ng the l ast 2.1 % of travel of the an nulu s wi ll the sample volume come into alignm ent with the column; thus, the effective sampling time is on the ord er of 0.3 ms e c, sinc e the total distance takes about 14 msec Chro m atograph Ov en TI1e lar ge air mass oven of the Varian 2100 GC was used for colu mn t emperat ur e re g ul ation Both lo ng term and sho rt term temperature stabilitie s c,f the oven were measured usin g a 36 cm platinum resi stan ce thermometer (Seri.al # 1691079, Leeds a nd No rthrup Co.) which had an a bsolut e r esistance of 25 5475 ohms, as calibrated at the National Bureau of Standard s Four-terminal measurements were made by wiring the four leads of the thermometer directly to a precalibrated digital voltm eter ( Mode l 8400A, John Fluke Co.) w hich had milliohm res o lution on the 100 ohm scale. In this way milliohm chan ges in the DVM output were see n as approximately 0.01C chan ges in oven temperature. The actual ov e n temperature T was calculated by the r e lationship T = 2d 10 4 4 d 10 4 T -R ] t 0 aR 0 (100) PAGE 90 67 where a a nd d are calibratio n constants specific to th e thermometer. R an d R ar e the resistances of the pl a tinum resi st or at T 0 and 0 C, t 0 resp e ctively. Program 1 in Appendix C sho w s that the negative root is a ppl ica ble. The o ven injection port, and det e ctor block temperatures were equili brate d for 10 h o urs at set points of 50C, 5 0 C, and 120 res pectively These settings were maintained throughout this wo rk. Over a 28-hour period, 16 temper a ture mea s urements 30 seconds apart wer e made at ea ch of 64 loc Rtions in the upper one third of the oven The 64. locati ons forme d 2 im ag inary vertical planes, one in the oven fron t a nd one in the back, and 8 hori zonta l planes. Since the coiled analytic a l coJ.u :nns u se d in this res e arch we re in the upper third of the o ven the t em p era tur e s tability was only deter mi ned for this region Evaluatio n of this temp eratur e map indicat e d that the stab ili ty at a sin g l e point for 8 minutes w as a lways at le as t 0.02 C at the 95 % con fidence inter va l. There was a gradient of about 0.1C from the top of t he oven next to the det e ctor oven, to the horizontal plane one third of the war PAGE 91 68 points nea rest the top and back was 54.091 0.027 C. Th ese 0.0 3C stabiliti es v e ry nearly approximate one of the spec i f ica tions needed to ach i eve a t l east a 0.1 % precision in retention time measurements [45]. Ch romatograph D e tector The Varian Mode l 2100 FID contributes a t ime constant, Td = V /F m' du e to the effec ti ve sensing v o lu me V and the flow rate, F of c a rri er gas [43]. As a worst ca se example, if th e sensing v o m' u me of the d etect or is taken as 1 % o f its volume (2 8 8 ), then T d = 19 msec at Fm= 12. Glenn [7 4 ] attributed the finite d ete ctor re sponse time to the spe e d of production and tr anspo rt of ionic species and to th e detector itself acting as a resistor-c apaci tor network Although some aut h ors [5 0 ] hav~ di sm issed this time constant as ne g ligib le, it h as been added to the system time con s tant T in accordance s with oth ers [ 43,44 ]. The hydrogen and oxy gen (technical g r a de, Air Products) flows we re r egu lated by two-stage regulators ( Mo del No. 8-350 and Mo d e l No 8-540, Mathes on Co.), passed thr o ugh 500 cc Mo l ec ular Sieve tr aps an d fin e ly controll ed by microm eter ing valves ( Mo del P N -430002 85 Ve riflo w Corp.). Optimum flo w rates for the least flam e noise and maximum re sponse we re f ound to be 35 cc / min for hydrogen and 150 cc / min for oxygen for helium flows less than 30 cc / min and could be measured t hro u gh flo w diverters Electromet e r A Barber-Colm a n Mo del 50 44 e lectrom eter was used to amplify 14 -5 FID curr ents in the range 1 X 10 to 1 x 10 am p to a O to 10 volt signal. Whi l e offering high s e nsitivity for sma ll si g nal s, this PAGE 92 electro meter also had a very lo w noise level on its mo s t sensitive scale. With the 4 ft graphj_t e-li n ed coa xial :::able (Chester Cable Co.) from the FID at tached and the flame off, the electrometer noise -14 was l ess than 0.5 x lO amp w ith no detect ab le drift over a twohour period 69 The electrometer time constant,T was measured for two ran ge s e 12 -11 (xO.lran g e=-=10 amp full-s ca le; x l.Oran ge= 10 amp full-scale) using the simple voltage divid e r circuit in Figure 7. 1~e resistor network was placed in a shielded can made of copper. In order to measure T SWl was closed sever a l second s e lapsed until a steady e state current was estab li shed through JU, SWl was disconnected, and the decay of the e l ectrometer output was monitor e d with the data acquisition system described below. The constant T was defin e d as the time e required to decay to 37 % of the steady state level. Values for Rl were 51.1 MO and 4. 5 MO in order to simulate -12 -11 FID currents of 0.6 x lO and 0.67 x lO amp (60 % and 67 % of full scale) for the x 0.1 and x 1. 0 ranges, respectively. The decaying electrom e ter output was sampled every 20 msec for the most sensitive range and an average T of 290 msec was obtained. The x 1. 0 ran ge, e which was used for sharper taller peaks, was sampled every 2 msec giving an aver age T of 9 msec These times include the time constants e for the FID connecting cable. On the X 0.1. range, where m ost of this research was done, the electrometer was by far the largest contributor to the overall system time constant. PAGE 93 Figure 7. Voltage di vide:r circuit used to mea st!re electrometer time constant PAGE 94 l.O M.Q S W I RI TO ELECTRO ~ /lETER 3V I Oil r--------0 G N D PAGE 95 Comput e ri ze d Data Acquis i t ion and C ont rol System H ard w a r e 72 Figure 4 a l so illu s trates the computer interface used in these stuc:ies. The system also included a computer controlled mass flow controll er described above which is not shown. The details of the hard ware are pr esente d else w h ere [40] but it is import an t to point out here that the system does include a 10-bit succes s ive approximation ADC ( Mode l A811, Dj_gital Equipment Corp., DEC) with sample and hold amplifier ( Mo d e l A400, DEC), a programmable gain amplifier, and a progr amma ble clock. The computer is a PDP-8 / L (DEC) with SK of core. A four-tape magn eti c tape cassette system ( &b del 4096, Tri-Data Corp.), a high speed paper tape re ade r ( Ma rk V, Data Scan Corp.), and ASR-33 teletype (Teletype Corp.), and a dual chann e l oscilloscope (Type 547-A2, Tektronix, Inc.) were the principal peripherals used. The PDP-8 / L computer controls external de vices by selecting its 6-bit device code and then sending control puls e s. For example, the device selector for control of the sampling val ve is shown in Figure 8 as the device code and the input / output control pulses (!OP) were brou g ht out f rom tl,e buffered memory buffer (Bl VlB ) regi.::;ter of the computer. The device code is used to enable NA. 'ID gates 3, 4, and 5 upon receipt of a 12-bj_t 61 4X instruction, where Xis an !OP 2, 4, or 6 pulse. The valve gate to switch the valve tm the sampling position is generated by the IOP 2 pulse whi ch clocks f l ip flop no. 1 (FFl) to a logic8l "1" at the Q output. Then a 6146 con..'Tland enables NAND gate 5 and resets FFl and FF2. The next instr mtio n generates an IOP '1 which put s a logical "o" on the clocked li nput of FF2 and thereby PAGE 96 Figure 8. Device selector logic for computer control of the sa m pling valve. Compon e nt Description 1 N A N D G a te, 9007, Fairchild 2,3,4 NA:'.'l D G ate 9002, F a irc h ild 5 N A N D G a t e 7 4 00, :F a ir ch il d F F1,FF2 Fl i p flo p, 7 11 0, Fa i rchi ld PAGE 97 DEVICE CODE FROM CPU ~7 IOP 2 IOP 4 BM B 09 BMB lO F Fl R K FF 2 R Q Sl. VALVE GATE I i I I I 0 1.rL. VALVE GATE 2 PAGE 98 75 fires th e se cond so le no id to r e turn the valv e to its original po s i tio n. The timin g s eq u e nc e f or ac t uati o n o f the va lv e i s shown i n Fi g ure 9. The dura tion of th e: valve gate pulses is varinble in the software. '!11e ti m2 fo : the valve g ate 1 pulse for the sampl ing valve describ e d earlier det e rmines the amo unt of time durin g wh i ch th e sample vo :.i..ume is flush e d wit h th e c ar ri er gas in the sample injectio n po si tion. The valve gate 2 pulse width c o rr esp onds to the time needed to return the valve to its normal po si tion (about 12 msec). The solenoids were fired and switched by the driver circuit in Figure 10. The positive going input pulse from the device selector is driven thr o ugh the emitter coupled transisto rs in a Darlington configu ration in order to increase th e current gain at the output and to reduce th e lo a d i ng on the d e vic e selector by the hi g h i m pedance input. The RF int e rference filter (Filtron 9605) serves to keep the electrical noise, gen e r a ted by switching th e inductive load, out of the digital logic system. The diode around the solenoid acts as a low impedance path to allow the ma gnet ic field to coll apse w h e n it is cut off w j thout damping the current in the coils through th e 2N5877. The 18 ohm resis tor is a d am ping resistor. The use of the programm a ble gain amplifier allowed gains of 1, 2, 4, and 8, givin g the 10-bit ADC an effec-tive 13-bit resoluti o n. This was ext remely useful for determination o f the very diffuse ed ge s of chromato g raphic peaks. The s a mple and hohli a mp lifi er required 12 sec to track to 0.025 % accuracy, and the .&DC required a conver sion time of 10 sec. These times were provi d!e d by so ftwa re del a ys which add e d a n eg li g ible contribution to the ave r a ll sys t e m time constant. PAGE 99 Figure 9. Timing sequence under software control for g e ~ 1 e rating a pulse width for sample injection (valve gate 1) a nd for returning the sampling valve to the normal position (valve gate pulse 2). PAGE 100 RESET FF I I I I I I l l I SET CLOCK I I I I I I I IOP 2 : I i I I I I I I I I 'f I I : j :, I VA L VE GATE I I I I I I : I IO P 4 I I I I I t1: l I :,., VALVE GATE 2 PAGE 101 Figure 10, Driver circuit for automating the solenoid actuated sampling valve. PAGE 102 + 5 voe -'r50VDC 9 I D40CI ,< ~ l __ !~ ~ ~ k' I ~-0:~ ~v 1 r--~ - PAGE 103 80 The 1 M Hz crystal oscHlator used as the time base for the pro g ram m able clock was co mpar e d to Nati o nal Bure a u of Standards stan dard fr e qu e ncie s using a digi ta l counter (Model 8 030B, Dana Laboratories) in the A / B ratio mod e With A being the NBS 100 KHz and Bas the progr a mmed oscillator 5 cps, the average A / B ratio for ten 10 second runs w as 20,000.33 0. 02. The crystal clock is apparently slo w at this 5 cps rate. At the 1 cps rate, in straight count mode, the accuracy was 1 part in 10 5 with a precision for 10 runs of 2 parts 1' 5 in u The o s cillator probably performs better than these specifica tions indicate, its apparent accuracy and precision being limited by the counter accuracy. Data were generally taken faster than 1 cps and rates as fast a s 500 Hz Wl!re used, in which case s these time measurement errors w ere surely n eg ligible. Clock rates from 1 M Hz to 0.1 Hz in decade and h a lf decade steps were available through the pro g r a mmable clock. Computeri z ed D a t a Ac q ui s ition and C o ntrol System S o f t w are This research utilized computer programs for both on-line control and c om putation and for off-line calculations. The on-line r o utines were mo difications of the computer program called ADCOM used previously r.40], written in DEC's PAL III assembly languag e Basically, the foreground program contains routines to service the ADC when the PDP-8 / L interrupt facility is enabled, acquires data, alternates the signal throu g h the programmable gain amplifier, and senses the peaks. Most of real time is spent in the background pro g ram which has routines for moment c a lculations, control of e :h rperiment a l conditions, data PAGE 104 81 stor age on magnetic tape, and data displ ay The off-line programmin g u tilized DEC s conver s ational. mode p rob:ram FOCAL. The method of statist ical mome n ts c a lcul ntion was a ccord ing to th e Slum1w.t i on equatio n 27. Limits of inte g ration for indivi dua l pe akE wer e d efined as a p e r centage of the maxim u m peak height an algorithm j ustified by Chesler a n d Cr am [ 2 1]. The limits of ir.tegration were from t he base l ine on th e p eak front to 0.39 % or 0.19 % on the p eak tail for th e on-lin e c a lcul a tion ( p hysicoc hem ical constants studi es ). For the off-line calculations ma d e for peaks sto r e d on ma g netic tape (b inary dif f usion, valve input profil e and sample size studi es ) the l imits varied fro m 2.0 % to 0.1 % of the pe a k maximum. Because these two programs we re of such g r eat imp ortance throughout this work, a description of them in some detail is included h e re. Figures 11-17 represent t he flow of the on-line ex per imenta l pro g r am At th e st a rt (Fi g ur e 11) the experimenter has the option of selectin g mag n e tic tape storag e clock rate, nu m ber o f expected peaks, flo w r ate a nd inhibition of clock r a te ch anges Upon restarting the comput e r (Figure 12) a 1 KHz clock rate is assumed for the sampling valve timi ng, th en the pre se lect erl d a ta acquisition r at e is r es umed Th e v a l ve gate 1 and 2 tim es for the sampling valve, as well as T d and T men t ioned before we re s u mmed to give T for the total syst em time e s con stant Gen eral ly T varied dependin g u po n the prevailing flow rate s and electrometer range but it was given a value of 394 msec and 113 ms e c when using the :i< O.l aud x l.O ran ges, respectiv e ly. PAGE 105 Figure 11. Flow di ag ram for the on-line co m puter control and calculation program(ADCO :V I). PAGE 106 GET f H. G TAPE RECO P. O # A N D P R EP A RE TO RECO R D FIX S UBROUT I NE S TO R ECG R O GET H EW VALUES YES GET C HAN GES TO fiE I HHIB ITE D & F I X S U S RO ll Wi E S YES HO START GET MA G ~ ~ P E// RECO R D t ;__,; SET TO S K IP RECG R D lt G SUBROUTI N ES YES SET FL ON TO ltllTI A L V/\ LUE STOP Al!O i'l fdT U N TIL ST R T IS D EPRE SS ED _____ __, 83 PAGE 107 Figure 1 2 Flo w di a g r am for ADCO!\'l (cor,tinued) PAGE 108 YES ST.! ~ T JHE ( (X Ft ~l~ H t ;f A L '-. R U k ClH R B U FFE R S SH HP. !A 3 l S ~ r.D P Q::HER S ~ RES ~ VALVE SET Cloer TO NITIAL RA TE nE? AR[ TC START SC R AiCH PAD m : o R Y m INTER R U PT S Y /IT CH DI SPLAY P[H 8UIHR 85 PAGE 109 86 At this poin t (Figur e 12, point B) provisions for a 21-point baselin e s c ratch pad memnry (S PM ) w ere made and the program interrupt switch w as en a bl e d. At the start, the interrupt had b e en off so that the p e al~ buf fe r wa s si m ply displ a yed. Thus the peak could be dis pl a yed a s it was pa s s in g th rou g h the FID. The ADC service routine (Figur e 13) w:::. s e nter e d w h e n the clock set its fl a g, the signal was sampl e d and ci ig i t i zed (in 22 ~i.s ec), the time counter incremented, and the gain for the next data point adjusted according to the magnitude of the current point. Figure 14 illustrat e s the operations of the moving SPM buffer. This routine continuously updates the baseline prior to an eluting peak so that when a p e ak is sensed by 5 consecutive positive first derivative points, 16 data points of baseline plus the rising edge of the peak are saved as part of the peak Upon storing these 21 points, the initial parameters are stored, the 16 baseline points averaged, the interrupt enabled, and the peak buffer dis~layed. After the next interrupt, the peak end routine (Figure 15) was entered in order to determine the peak maximum, which was defined lny 5 consecutive negative first derivative points Since signal noise was always less than the least significant bit of the effective 13-bit filJC, even on the most sensitive electrom e t e r range this simple first derivative technique proved quite adequa t e. All experimental chronutograms mentioned in this dissertation involved only single or very well resolved peaks, so overlapping peak routines were unnecessary Next (Figure 15, left) a predetermined percentage of the peak maximum amplitude is calculated and declared fue cutoff value (limit PAGE 110 Figure 13 F lo w dia gram for A.TJCOJ\l ( continu e d). PAGE 111 NO DEC RE A SE GA IN m m I N TE R R UP T QUESHO AOO TI M E IM C R [ M E HT TO TOTAL HO C m YES I NCP. EAS[ HI~ 88 PAGE 112 Figure 1 4 Flo w diagram for ADCO M (Continued). PAGE 113 ST OP. P OIN T I~ S P MrnOR Y SET UP J UM P TO PEAK SEARCH SU BR OUTI / H ~o t BR~~Cit A CCCROl~G T O SlG k[O PO l~ TEP. RESTO R E ll ~~-. ACCU l!.U LAT OR. MEH Cil Y f1LOS ENABLE mmRUPT RH U~ N TO BACKG ROUH O PROGRA M NO lES STO R E PO I X T AT HD or S P MEM OH Y A H O DROP 1ST P OIN T CAl CULA!E I ST DERIVATIVE or CU RR ENT POI NT STO R E TI M E. CLOCK P.m, FL OW HT UO SI SELIHE SET UP J U) \ P TO PEH ENO SEARCH SUB ROUT IH E ENABLE INTERRUPT DISPLAY PEAK 90 PAGE 114 91 of int egration) and this v a lue was compared to each successive data point. If th e numb e r of d ata points in the peak ex ceeds 150 (one third of t he maximum allo wed points per total peak) before th e max irimm was r E:ached t:ien the d at a acquisition rate could b e low ere d to the next lowr :r p r ogramm able rate This was done to collect th e whole peak sinc e the time on t he ta iling side of the maximum was generally 2 to 4 ti me s t h a t fo r the le a ding side. When the ri g ht-h and i nteg1 ation limit for the peak is re a ched (Figur e 15, cent e r), the elapsed time is recorded, and the initial parame te rs re se t. If another peak is e xpe cted (Figure 15, right), the baseline monitorin g routine is again entered. I f the last pe a k has eluted, th e interrupt switch is cleared and the moment calculations begun (Figur e 16). These parameters are calculated using DEC's three word floating point package routines which have a 7-digit accuracy. After the zeroth moment, absolute first moment, standard deviation, third moment, and specific asymmetry are calculated, the peak is given a code number, saved on magn et ic tape and then di sp layed (Figure 17). The initial conditions and peak parameters are printed when th e zero bit switch on the computer console is set equal to one. The off-line FOCAL program for calculatin g the statistical moments is listed as Program 2 in Appendix C. It begins by id e ntifying the peak that is to be read from magnetic tape and placed into the common storag e ar ea called FNEW (1, ). These two steps are con trolled by th e user defined functions FCO!VI(U,V,W). The peak is dis played (FNEW(4, )) until switch re g ister zero equals one. The baseline is average d from th e first 16 points of the peak and this PAGE 115 Figure 15 Flow diagram for ADCOilI (contin ued) PAGE 116 REST OKE ll :i K ACCU I-IU lAT CR IIIEM OR Y FIEL DS RETU R N TO UC KGROU~O PRO GRAM NO NO 0 Li rO l. l[ S P f.\ i PAGE 117 Figure 16. Flow di ag ram for A DC O M (continued). PAGE 118 CHCULHE 1 1 y 'r.Yt/ -r. Y l CALCUL A TE 7 ,_ = A/'r y I >'3 == 8/L Y I i ---, CALCULATE SP A SS = t ;/1-113 , STO R E M O f.1EH TS A N D SP A SS 0 95 PAGE 119 Figure 17. F lo w diagram for .I \DC 0/1! (continued). PAGE 120 NO ClHR PAK BUffER DISPLAY CHRG MATOGRH l BUFFER OISPLA Y PEAK BUFFER 97 PAGE 121 a ver age subtracted from th e entire peak u sing FKEW(2 , ) and FNEW(l, ). The in tegration limi t s of the p ea k a re th e n fow1d from the ordin ate limits s p e cifi ed i .n line 1.10. The exper im e ntal times for the p eak start peak maximum, and peak end are printed and the width at half height is calculat e d (statem en ts 2). Th e n the first five statistical mom e nts are calculated a lon g w ith the standard 98 devi ation, skew, and e xcess and printed. If desired the proi;ram will autom a tically r ead the next peak into FNEW and calculate its moments. An example of the output follows Program 2. Char acterization of the Gas Chrom a to g raphic Columns Prepar a tion of Colum ns Three sets o f ana lytic a l GC columns w ere prepar e d from sing le lengths of 1 / 8 in. o.d. shiny interior precision bore stainless stee l tubing (Superior Tub e Co.). All tubing was thoroughly cleaned with succe ssive solutions of water, methylene chloride, methanol, acetone, methanol, and then air dried for one hour. The tube l engt hs were measured to the nearest millimeter and the volume per foot determined by fillin g a 4 ft. length of tubing, cut from the same piece from which the analytic n l columns were made, with mercury. In this way the tube radii were calculated to within 0.3 % of the manufacturer's quot atio n. The three GC experiments involv ed three different support material s : silica gel (60 / 200 mesh, Mathe s on, Coleman, and Bell), silica gel filled and coat ed with OV-101 (Varian Aerograph), and graphitiz 2 d carbon black (Sterlin gJ\,IT, Cab ot Corp.). Eight accurately measured (121.9-122.1 cm) colu m ns (0.206 cm i.d.) were p a cked with PAGE 122 diff e rent f r8ct ion s of s ilic a g e l m e chanic a lly sieved for more than 2 hour s usin g A SThl Sta nd a rd Si e v es The eight separated s ieve fractions r anged f rom 50 / 6 0 t o 170 / 200 mesh. A 60 / 200 mesh mixture of the silica was quot e d b y the manuf a cturer to have a surface area of 2 750-800 m /g bu t 60 / 100 a nd 100 / 200 mesh mixtur e s had ar e as of 99 2 2 550-625 mg and 900-97~ m / g, re s pectively The average pore diameter wa s quot e d t o be 22A 0 The avera ge packed d e nsi t y for all columns w as found to b e 0.721 g / cc. Special care was taken to pack the columns uniformly and in the s a me way, i.e., successive repetitions of filling, bouncing, a n d vibratin g The ends were covered with 300 mesh copper screens to hold in the p a cking. Each column was then coiled (14 cm coils) and condition e d at 185C for 18 hours with about 5 cc / min of dry helium passing through the column. After conditionin g the ends were capped in order to keep the silica gel activated. Six 61.0 cm columns (0.163 cm i.d.) were filled with OV-101 impregnated silica gel. OV-101 is a nonpolar dimethylsilicone fluid with density 0.975 g / cc. There were columns of three different mesh ranges, one range (100 / 120) being done in quadruplicate as a check on reproducibility of preparation technique. From the known external porosity internal porosity ., the tube volume Ve, and the packe 1 ing density p it was calculat e d that a mixture of approximately 30 % p (g OV-101 / g silica) would fill the silica pores. In this way the 2 entire surface area (750-800 m / g by N 2 adsorption) would be coated. The silica was impr e gnated by adding an accurately weighed amount of OV-101 dissolved in chloroform to a known weight of silica gel, evap orating th e chloroform, and heatin g for one hour at 110C. Initial PAGE 123 100 50 J;: :ni d 35 ~'., p ackings we r e still sticky after dryin g Using 25 % 0'. :L yi clded a d y pack:i.ng tha t was easily siev e d and packed. The a v e ra ge packed d ens ity was abo u t 0.92 g / cc. The tube ends we re blocked wi tl 2 very s ;;rn ll piece o:f silanize d glass wool and the co lunms bent int c '"' 1 4 cm L'shape They we re then condition ed a t 300 C, which is be:i.o w t h e 32: 5 -375 C stabil i ty limit for OV-101, for two hours w ith a 10 cc / min str ea m of h eli um. Three 61. 0 cm graphiti z ed carbon black (GCB, s urf ac e area 2 9.6 m ;g ) c ol11mns (0.163 cm i.d.) were prepared using three average pa : rt:'Lc} e diam e tei s d et cr : nin cd by sieving. The average packed d ensit y was Cl. 82 g / cc. There was no appare nt problem o f partic l e c r ushin g wl1ile :-,ie vi ng o r pack i ng. These columns we re conditio ne d at 250 C for on e hour before us e Det erm i nat ion of Partic l e D iame ter The avera g e particle d iameter was generally taken as the mesh r ange m ea n exc e pt for the 5 smallest bare silica fractions. Their dia mete rs were d ete rmined by a particle counter ( Mode l Z, Coulter Co.) h:t erface d to a 1024 chann e l signa l averager (Model 1074, Fabri-Tek In str .). A pro gramma bl e cal c ulator ( M odel HP9100B, Hewlett Packard Co.) plotted tl1e particle distributions and the mean diameter was calculated. It was found that even after prolon g ed mechanical sieving, a broad distribution of particles rema i n ed but th at the mean particle diam e ter wa s f ai rly close to that deter m ined from the s creen sizes. For example Figure 18 show s 1~e norm a li ze d dist r ibution for the 120 / 1 1 0 f raction. The mean diameter comput e d from the inflection PAGE 124 Figure 18. Particle size distribution for 120 / 140 mesh silica gel. PAGE 125 Cl) w _J u ._ 0:: <( a.. LL 0 ._ z w (.) 0:: w a.. 0 w N _J <( (C 0 z 100 50 25 50 75 PARTICLE DIA M ETER, microns 100 I-' 0 t -:> PAGE 126 103 p o j nt of th e in t egra l lin e is 11~ .3 microns co mpared to 115 microns calcula te d as the average o f the sieve s i ze s. TI1 e inte grn l also shows that about 90 % of the particl e s a re within 10 % c f th e mea n va lu e Th "' dia me t e r s of the O V -s il ica and GCB particl E: s could not be de termined by part icle cou n tin g due to t e chnical difficulties or pos si bility of pa rticle cru s hin g Th e m e a n dia mete rs u sed for all of the columns pr epa red are U sted in Table 1. TABLE 1 I DENT I Fl CATIO :-T OF GAS CHROfl l A T OGRA.PHI C COLUMNS Adsorbent Mean and Particl e Diameter Mesh R a nge Column d p' m Silica Gel 50 / 60 A 273 60 / 70 B 230 70 /8 0 C 19 4 80 / 100 D 161.1 100 / 120 E 127.5 120 / 140 F 112. 3 140 / 170 G 95.5 170 / 200 H 86.6 ov~ 101 60 / 70 I 230 100 / 120 Jl-J4 137 14 0 / 170 K 97 Graphitized Carb0n Black 60 / 170 L 230 100 / 120 M 137 140 / 170 N 97 PAGE 127 D ete n :i.i n ation of Perme ::i bili t y a n d Exte r na l Po1os i ty Colur r ..n p e r me ab i lj t y w a s d et e rm in e d by a computer controlled e~J e ri me nt w h e r e th e inl et p r essure, indicated by a pr ess ure trans duc e r ( M od e l UC3, Sta t ham, Hon e ywell Instr .) was monitored as a 104 function of the c a rrier ga s f lo w rat e through the column. Th e program was a simplific a tion of the data acquisition sof tw ar e d e scrib e d earlier. The transducer w as coupled to a Oto 100 psi pre s sure diaphragm ( l\Io d e l UGP4, Honey w ell Instr.) and the signal was am plified to Oto 10 V by a modified brid g e a m plified (Mod e l SCllOO, P.oneywell Instr.). This signal was further amplified if necessary and then fed to the ADC, w hich w a s pro g r rn mr.ed to sample the signal every 2 to 10 seconds. M1en the transduc er output was stable to within 1 least significant bit of resolu t jon for 20 data points, the computer commanded an in c r eas e or decr e ase in th e mass flo w t o the next lever, gen e rally a 2 cc / min step. After 10 to 15 different levels the transducer ou t put levels and flow rates were printed out. The experimental flo w rates were in the range 4 -25 cc / min, which represents inlet pressures up to 2.5 atm. In order to establish the transducer lev e l in terms of mm Hg, several cali~ration runs were made by placing an open ended U-tube (each arm= 105 cm) as well as the transducer head at the head of the column. A 120 cm column with 140 / 170 particles served as a typical resistance. The average and precision at the 95 % confid e nce level of eight calibration runs established th a t the conversion factor was 828.9.3 digital uni ts/atm. PAGE 128 On e of the trials is plott ed in Figure 19 wh ere F was m increa sed fro 1, 1 0-16 cc / min i n 1 c c / min incr ements and w h e re 105 tip= p p p wa s ntmosph er ic pressure This corr esponds to changes !. o' o o f 18-113 8 0-103 40 c m, and 1.001-2.356 atm for the values of the di gita l level, the differ ence in the l eve ls of the mercury arms of t he U-tub~, and the absolute inlet pressure, r e sp e c t iv el y. I t c an be seen that the t ra nsd uc e r response is quite lin ea r and in this pa rticular case the in terce pt is 5.9 v:h ich corr espo nds very n ea rly to ze ro mass flow. In all cases, the mass flow rate wa s chan ge d in incr ea sing step s only, sinc e only a v e ry sma ll hyster esi s effect was observed when d e cr eas in g steps w e re m a de. The outlet pr essu re wa s atmospheric which was re a d from a di g i ta l barometer (B e ll a nd Howell In s tr.) aft e r each flo w rate chan ge PAGE 129 Figure 19. Pr ess ur e tr ansducer ca li bration plot usin g a Oto 100 psi pr essur e diaphra gm PAGE 130 107 1200 r-----------------------...... l000 (/) 800 Iz ::> .J <{ (!) 600 -D 400 200 0.2 0.4 0.6 0.8 1.0 I. 2 1.4 6 p, atm PAGE 131 RESUL TS A:'ID DISC USS ION Deter minat ion of P ermea bility and External Po ro s ity According to Darcy's Law [75], the gener a l relationship describing flow of gases through porous media is V = -k f ( e ) dp 1f e dl (101) where vis the average carrie r g as velocity, k is the column permeability,~ is the carrier gas viscosity, f(e) is an external porosity e function, and dp / dl designa tes a pressure gradient. If it is assumed that k is indep e ndent of f low rate, as is normally done for n ea rly id ea l gases, then the equation can be integrat e d over the l eng th of the column to give [761 2 2 qAk(p. -p ) F = m l 0 11 L p I 0 (102) where F is the carrier gas mass flow rate, q = 2. 95 x 10 7 A is the m empty colunm cross-sectional area, pi is the colunm inlet pressure, and p 0 is the column outlet pressure. Perme abi lity can also be calculated from the Konzeny-Carmen equation [77] k d2 e3 P e 180 2 (1-e ) e (103) 108 PAGE 132 109 where p e r meability is relat ed t o t:1e a v e ::: age particJ .c diameter and an extern a l porosity function. The fraction 1/180 is d e riv ed by assuming spheric a lly shaped p a rticl es and capill ar ic pore spaces, a model upon which most chrom atog raphic theories rest. It can be seen that pJ.ots of F versus m 2 2 (p p ) l 0 should be linear, have a slope proportional to k, and have a zero interc ept This was in fact the case for all columr.s investi gated and these plots are sho w n in Figures 20-22. The flow rate Fm was re g ulated ~ya Ty l an mass flo w controller which received control signals from a PDP-8 / L -4 computer. Usin g a value of 'Tl= 1.86 xlO g / cm sec (78] for helium at 0C (which is the reference temperature for the helium mass flow controller), k was c a lculated and these values are tabul ate d (Table 2). Tbey agree well w ith expected permeabilities for columns packed with 0.01 to 0.03 cm particles. All slopes and intercepts were calculated by the method of least squar es by using a FOCAL program which i s reproduced as Program 3 in Appendix C. The intercepts for Figures 20-22 are in the range -0.01 to 0.09 cc / min. By equating equations 102 and 103 and using the known k and d values, the value of the interparticle porosity E can be calP e cu:! ated from equation 103. This is possible because f(E) in equation e 101 ref er s only to void space available to the moving gas, not the stagnant gas within the particle pores; 8 was found numerically using the Newton Raphson iteration technique for solving for roots of equ ations. This is listed as computer Program 4 in Appendix C. PAGE 133 Figure 20. Permeability cali bration plots for silica gel columns. The columns and average particle diameter s in microns are a s follows: A 273 E 127 5 B 220 F :..12. 3 C 194 G 95.5 D 1 6 1.1 H 84 .6 PAGE 134 2s oo I~ 2QQQ~ (.) (.) E LL 0 D C 2 0~___J__----=..L__ I 3.0 4 0 _(_ __ I ( 2 2 ~o P-p ) at 2 1 0 m 6.0 _J 7.0 PAGE 135 Figure 21. Permeability calibration plo t s for O V -101 impregnated silica gel columns. The columns and average par t icle diameters in microns are as follo w s: I 2: 3 0 Jl D7 J2 137 K 9 7 PAGE 136 35.0.------------------------------. 30.0 J ~ I 25.0 I 20.0 C .E .. 0 0 15.0 E lL 10.0 5.0 O ....._ __._ _________ _.._ ___. __ __.___ __.__ _._ __ 0 1.00 2.00 3 00 4.00 (pi 2 -po 2), atm2 PAGE 137 Figure 22. Permeability calibration plots for graphit i zed c arbon black columns. ': 'he colum n s and ::i. v e r a ge p a r t icl8 diameters in mic r ons nr~ as follows: L M N 230 137 97 PAGE 138 35.0------------------------------. 0 0 30.0 25.0 E 15.0 LL 10.0 5.0 7 N .--. _,-----0 ~----'------'-----L------'------.1...---1...---'-----'-----' 0 1.00 2.00 3.00 4.00 ( Pi 2 -po 2)' atm2 PAGE 139 Adsorbent and Colu mn 2 Sil i ca Gel A B C D E F G H O V-101 I Jl K Graphitized L M N TABLE 2 PERMEABILITY AND P OROSITIES OF GAS CHROMATOGMPHIC COLUMNS Permeability k 10 7 2 X cm 3 859 3 206 2 577 1 735 1.166 0 910 0.680 0.597 3 093 1.185 0 666 Carbon Black 2.884 0.768 o. 574 External P o r os ity e 0 343 0. 356 0 364 0 363 0.358 0.367 0 372 0.3 92 0.353 0 36 0 0 370 0. 347 0 323 o. 357 Inte rna l Poro sity e l 0.200 0.199 0 196 0 189 0 198 0 197 0.195 Total Poros it y E:T 0 556 0.557 0.563 0.5 8 1 0.5 5 1 0. 557 0.565 0. 347 0 32 3 0 357 a The columns denoted by letters are in order of decrea s ing particle dia mete r according to Table 1. PAGE 140 117 Table 2 sh ow s that r e:.is o! 1.a b l e valu es of p o rosj ty were ob t a :l n e d, and it g enerally d e cr e as e d inv e rsely with particle si ze indicatin g that lar ge r particl es eff e cU vely pack more den s ely This means thnt there arc relat i v e ly few er lar g e interp a rticle sp a ce s for the lar Ge r particl e s u se d in t h i s wor k co m pared to the many small s paces in a smaller pa rticl e be d. Thu s the total is larger for the small e r e particl e colu rrms T h e se r e sults support those of some [12,79]. They are, however, con t rary to others who found that e: 1 emained con e stant [80,81] or de c r ea sed [8] with particle size for particles 4-10 times larger than those r e ported here. Th e re is a possibility th a t for smaller particl e s the internal pores of the very porous bare silica mor e eff e cttv c ly connect with the interstitial void spaces to give an apparent increase in external void volume However, similar values and the same incre a sing trend in e w ere observ e d for the e OV-101 c o lumn s where the internal void spaces of the silica were presumed to be fill e d with liquid. Porosity trends apparent ly dep e nd on particle size and shape Further, for the bar e s ilic a columns ( A -H) there is a 6.5-fold decrease ink with only a corresponding 11.4-fold decrease d2 1n p In order for equation 103 t o hold, there must be a 1.7-fold increase in E. Th e porosity function plotted in F i gure 23 sho w s such e a n incr ease for the calculated E range found here. e If the decrease in k had been due solely to the decrease ind th e n the k decrease p would hav e been only 5.7-fold, i.e., only one half of that observed. Also from the c alibr a tion plots 20 22, one can calculate the inl et pr e ssure P. for any giv e n flow rate F and outlet pressure p 0 1 m PAGE 141 118 and thu s obtain th e p r e s s u re ~rnd ient correction factor, j, used in the calcul ation of the avera g e column lin ear velocity v. This is done in th e follo w in g w a y: th e slopes S L of the plots are SL= (F F ) m,2 m,l (10 4 ) w here the s ubscrip ts 1 and 2 refer to conditions for any two points on the s lope If 2 refers t o an actual c a libration point used in 2 2 dr aw in g th e slope, t hen Fm, 2 and (pi -p 0 ) 2 are known exact ly. If F is a differ en t mass flow set in some lat e r experiment with the m,1 2 2 same column, then :F'm,l is also known exactly. Only (pi p 0 ) 1 is unknown 2 now, exc e pt for p wh ich can b e gotten from the barometric reading 0 of p. Thus, equation 10 4 c a n be solved for (p ), by 0 l 2 2 SL (pi po) 2 -SL 2 -1. SL(p ) ] 2 0 1 Subsequ e ntly, the calculation of j was very crucial in acco untin g for pre ss ure effects on the statistica l moment values. Determination of Intern a l Porosity (105) The total porosity T of a packed column can be defined as the sum of the tot a l external and total internal porosities of the column = e; + T e i (106) and can also be related to the carrier gas volume VA needed to elute an in e r t: compou n d from a colu mn by [76] PAGE 142 Figure 23. The effect of external porosity e: of the e 3 2 permeability porosity function / (1-e:) e e PAGE 143 2 {I-Ee) 0.30 0.20 0.10 0.30 0.35 0. 4 0 0. 4 5 PAGE 144 1 21 (107) wh e 1e VD is tlle extra colu w n dead volum e a !1d Ve is the volu me of t he empty column A m e th od mentioned in r ef erence 82 was u sed to determ i ne the inert time with an FID w~ ich i s traditionally a fr ustra ting problem The oxygen flo w rate to the F ID was ma intained at 1 50 c c / min while the F ID hydr oge n flo w was lo we r ed to about 2 cc / min and the he lium mass flo w c o ntroller was conv e rt e d to a hydro gen mass flo w con troller. Helium at abo u t 4 5 psi was samp led t h rou g h the sampling va lv e U p on reachin g t h e det e ctor, the h e l i u m pro du ced a d ecrease in current, a -12 very sm a ll (1 x 10 amp ) n ega tiv e peak. The mini mum in the p ea k was take n as the inert ti me t and s ub st itu ted into the eq u at ion a t = a to c a lcul a te VA 1. 017 j ---F 1. 446 m Then, from equations 106 and 107, e; and were T i (10 8 ) found a nd are li sted in Ta ble 2. The hydro ge n compressibility f a ctor j was calculat ed from the measured inlet a nd outlet pre ss ures. Since the flow contro ller h a d b ee n factory calibr ate d for contr o ll ing H e mass flow, the r atio 1.017 / 1. 44~ based on the relative he a t capac itie s, was used to conv e rt to H 2 mass flow. The internal porosity is also defined [ 28] b y ]. e = (1-e )B (10 9 ) l. e where Si s the average fraction of void volume per indiv i dual particle. For silic a ge l, this va lue was quoted to b e 0.29 by the ma nufactur er PAGE 145 1.22 Usir.g this value, solving fer ei fo r each column, calcula tin g eT from equation 1.06, and calculating VA for hydrogen at various flow rates from eq u ation 107, the expecte d t values for hydrogen were found. a Thes e are plotted i n the lo w er portion of Figure 24 for 6 silica gel colu mns versus the ave r age v e locity, d e fi ned by equation 6. Figure 24 shows that po ir..ts for all colu mns r8gardl ess of their eT value s, fall on che same line with a slope of 230 and an intercept of 0 0 sec The experimental t values as a function of a velocity for three of the same columns are plotted in the upper partion of Figure 2 1. This 1 ine has a slope of 225 and an intercept of 0.8 sec Although the precision of replicate t values was only about a 0 4 % the calcul a ted and experimental curves are almost superimpos able. Values of 13 calculat ed by equation 109 for each column sho we d S to be constant, as anticip ate d [28] at 0.31. This was the value used throughout this work. Values of e. for the OV 101 columns were similar to the bare 1. silica columns since the He samples were not retained by the OV 101, and S was almost the same. The GCB particles, being nonporous (S = 0 ) should have equal total and interpartic l e porosities as shown in Table 2 The reproducibility in terms of percent r e lative standard deviation for four OV1 01 columns with d = 0.0137 cm is shown in p Table 3 Columns J l and J2 and then J3 and J4 were prepared and anal yzed on diff e r e nt days about two weeks apart Although these preci sions may be typical for the other ana l ytical columns, they are most l ikely not as good, due to the additional complic at ions of liquid loading. PAGE 146 Figure 24. Plots of calculated (lower) and experim e ntal (upper) inert elution times as a function of the average column velocity. The sili c a gel columns are id e ntified as follows with reference to Table 1: AO B 6 ED PAGE 147 100~----------------------~ (.) 75 _j 50 I.JJ 2 0::: w 25 o._ X w 75 (/) .._o 0 50 PAGE 148 125 TABLE 3 REPRODUCIBILITY IS PHEP ARAT IO:N" OF O V -101 COLUM~S Packed P e r mea b ility Exte rn a l D e n s ity C olumn % OV-101 Pp g / cc k X 107 cm JJ. J2 J3 J4 M ean 25 4 25.4 Rel S td Dev. 0.90 5 1.186 0 918 1 1 -1 0 0.918 1. 137 0.917 1.121 o. 915 1. 1 46 o.35 % .22 % Deter min a tion of Binary M olecular Diffu s i on Co eff icients 2 Porosity e 0.360 0 356 0.356 0.355 0. 357 0 31 % The m o l e cular diffu s ion coefficients D for the low molecular g weight hydroc arbon s in helium used in this study we re measured from o p e n tube chro matographi c data [83]. One microliter of hydrocarbon gas (>99 5 % pur e, Ma th eson Gas Products) w as inj ec ted into a 4 51 6 cm (8 52 1 cc) stainless st ee l t ube usin g the gas sampling valve described e a rlier The resulting e luti on profile was sampled w ith the computer i z e d data acq ui s ition suc h t~at there we re 250-350 points per p eak The s tati sti c al moments of t he peak we r e c a lcul ated on lin e as w ell as off -li n e u s in g 0.19 % integration limits PAGE 149 The sample hydrocarbon pulses diffu se with an ef f e ctive disper s i on coefficient Deff 2 2 R V = D + g 48D g where R is the tube inside r ad ius and vis the average cross o 126 (110) sectional linear velocity of the carrier gas defined by equation 6. 'rhis dispersion can be related to the second central moment u 2 of the peak by 2D L u = __ g_ + 2 V 2 R vL 0 24D g and since the broad e ning efficiency tis related to u 2 in length units by then D can be measured by g where the positive root will be a pplicable Qt velocities below a certain critical velocity v [84] C V = C 1 4(3) 2 D g R 0 (111) (112) (113) (114) For this work v was greater than 35 cm / sec for the smallest D value C g and all experimental velocities were between 1 and 10 cm / sec. PAGE 150 127 It was found tirnt for all hydr ocarb ons the re was an increase in the calcula te d D values wit h incre as ing velocity. This phenomenon g has been observed by others [85-87] and was explained as a consequence o f flow eddies at the tube inlet w here the flo w :from the sampling valv e capill ary into the d if fusion tube expands and produces turbulence at the he a d of the tube [87]. In the present w ork, the valve exit had about the sa me diam e ter as the colu mn inlet s o that this eddy effect probably was not present. I nstea d, the rise in D w ith v was very g linear, and the data fitted by l east squares analysis (Pro gram 3, App endix C) were e xtrapol at ed to zero velocity to obtain the appropriate D value. This meth od is presumed to be more a ccurate than arbi g trarily selecting a lo w velocity as previous workers have done. The data are shown in Figure 25; it can be seen that the plots are quite par a llel which may validate the extrapolation procedure. Even thou gh no theoretical reasoning has been report ed for the a ppar ent increase of D with v, in this case a rather predictable systematic effe c t is g occurrin g The contribution of radial diffusion ( eq uation 111) to u 2 was only about 0.1 % at all flow rates. This effect is not lar ge enough to account for th e l a rge deviations in D with velocity. The possibility g of a dsorptior. on the tube wa ll was dismi ss ed for two re ason s. First, adsorption, a kinetic phenomenon, would be more appa rent at higher vel oci ties an d, thus, the peaks would beco m e more asymmetric. However, 3 the specific asymmetry (u 3 / u 1 ) of the peaks was a lmost con s tant at 0.01 for all flow ra tes and for all compounds. A symmet ry= O. O for Gaussi an pe aks PAGE 151 Figure 25. Molecular diffusion as a function of velocity through an open tube. From top to bottom, the lines refer to methane, ethane, propane, and n-butane in helium. PAGE 152 I 0 G) ff) (\J E 0 1.000---------------------------------, 0.800 0.600 0.400 0 --...-0,_,,, ------~ --_-e ---. --_ ... --2.0 4.0 6.0 8 0 10.0 VELOCITY, cm sec1 PAGE 153 130 Secondl y ad s orpti or1 w ould be more r,ppa r ent f o r higher molecula r weigh t c ompo unds Th e me[:n time f or n-bu tan e was sli gh tly lon ger tha n for t he others. Ho we v e r, t h e me a n r etent i on time u 1 fo r all four com pour:ds at 1.9 6 c m/se c was 229 29 0.43 sec ( 0.18 % ) and 50.931.055 sec ( 0 .11 % ) at 8. 87 cm /s ec Since all c omp ounds are unretained, thl i y are seen to h ave equa l ret en tion tim e s wi thin e::,,_rperimental error. Sample volu me dead vo lum e, detecto r volume and coilin g contribution s to ~ were calculated by Gid din g s' equations in referenc e 84 to be less than 0.01 % each. There was a negligible pressure drop across the empty tub e. Table 4 li st s the extranolated D values at 5 C and one g atmospher e a lon g with the precision t ake :::i as the average of the precisions of the indivi d u a l points from Figure 25. Each point is the average of 5-9 measur-em u 1ts The precision ran ge s from 0. 6 % for methane to 1.2 % for but a ne. This co mpares favorably with others who found it difficult to achi e ve better than 1-2 % precision u si n g the op e n tube meth od due to problems in measuring the second moment [84,87]. Table 4 al so lists the D values for 54 C and one atmosphere g calculat e d from Hirschfelder's equation 115 (Program 5, Appendix C) Dg =DAB= 0.001 8583 (115) In this e quation, the molecular diffusion coefficients of compounds A and Bare a function of temper at ure T, molecular weight MA and MB, pressure p, colli s ion cross section crAJ3' and a dimen sio nless function PAGE 154 TA B LE 4 CO M PARISON OF MOLECULAR DIF F USION COEFFICIENTS IN HELI UMa Molecule This w ork b Methane o. 726 o. 004 Et han e 0. 5 4 1 0 006 Prop a ne 0 447 o. 005 n-Butane o. 408 0. 005 2 D, cm / se c (Ref erence ) g Hirsc hfe ld e r Equation 115 Sawyer 0.791 0.6 92 [76] 0.602 0,491 0.472 0.382 [89] a Referenced D valu es have been c a lculated for 54 C. g Gid dings 0.804 [90] bExperimental values at 54 C, extrapolated to zero linear velocity. Grushlrn 0. 812 [ 87 ] PAGE 155 132 0 of the temp e r a ture nnd intermo l e cul ar otential f i el d fo r one D,A B mo l ec ule of A a n d one of B Va l ues of crAB an d D D,AB we re calculat ed from c ompi l ed d ata in refer e nc e 88 The equat ion 115 D v a lu es a ll appear hi ghe r than the g experimental on es wh ich should b e expected s inc e the accuracy of the Hi rschfelder equat ion u se d here is only abo u t 6 % [88]. S aw y e r [76, 89 ] found D == 0.865 0.009 cm 2 / sec a nd 0. 477 .010 crr? / sec fo r g methan e and n--but ane r espective ly, in helium at 100 C a nd one atmo spher e 2 Gidd ings [90] r eported 1.00t c m / s e c a nd Grushka [87] reported 1. 014 c m 2 / se c. Since these we re o p en tub e expe ri ments and D is pro g 1.7 ] portional t o T [89,91 the se literatur e values c a n b e extrapolated to 54C for com paris on w ith t hi s w ork. Th e se values are listed in Table 4. Ma rr ero [91] has stated that the best static measurements of D (t wo -bulb and clos ed t ub e expe riments) a re "r e li a ble," compar e d to g other metho ds, to w ithin 2 % a nd a re comparable to 2 % for dyn ami c (GC) methods, This work has shown that D valu es can be found quickly g with 1 % precision. Ho w ever, it is be s t to experim e n ta l]y det e r mi ne D valu es w hen required, since precision for D usin g a given op e n g g tube GC system is a cceptabl e but the accuracy is questionable. Measur e ment of E quilibrium, Diff u sion, and Rate Con stants Silica Gel Col u mns The fir s t abs olute mome nt u 1 of a chro m at ograph ic elution profile is determined by the equilibrium a dsorption con s tant K. n Since u 1 can be me as ured very a ccurat e ly from di g i ta l data, Kn can be fou nd eas ily by re ar ranging the first moment equa tions tor the silica gel c-olumns. PAGE 156 133 Equations 28, 29, a n d 3G g iv e ( u 1 u a) 0 5 t 0 (116) where u is the absolute first moment of an inert solute (K = 0) a n calculated fro m L u = (1+) a V and vis the pre s sure-cor r ected lin e ar velocity calculated using equation 6 or 89. (117) A plot of the left-hand side (LHS) of equation 116, called the reduced first moment, vs. L / v should yield a straight line passing through the origin with a slope equal to K n Data from colu11U1s filled with silica gel of different particle diameters 2R should lie on the same line since u 1 is not a function of R. It is, however, a function of the internal porosity (l-), external porosity colu11U1 tempere e ature, and column pressure, as well as the d e gree of adsorbent surface activation, the average pore size, and the amount of active surface area available for adsorption. Figur e s 26-29 show that methane, ethane, propane, and n-butane adsorbed on four particle si z es of activated silica gel follow equation 116 in genera l. Each point on each plot is the average of 3-8 experimental runs. Upon close inspection the intercepts are not always zero, nor are the s lopes for different d equal. Table 5 shows p that there is a definite decrease in K for all solutes as R is n decreased, especia lly for the longest retained solutes Reasons for this will be discussed la t er in terms of net retention volume. All data pres e nte d in the R e sults and Discussion chapter ref e r to the e:;-,.l)erimental t e r.1 p er a t ur c-~ 5 4 06 :i.: 0. 30 C PAGE 157 Figt! r e 26 Reduced f i rst m o ment (u 1 u 0 5t ) / as a function a o o f L / v for methane, chro matog r aphed on silica gel at 54C The symbols refer tc colu mns filled with particles of me an radius accordin g to T ab l e 1: 0 Column A :?article radius = 13 6 p m A Column B Parti cle radius = 115 pm a Col t m m E Partj cle rndi .1s 6 4 um V Col urnn F Particle radius 56 m 0 Col umn G Partjcle radi us = 49 m 0 Col m1111 H Particle r adius = 43 rn PAGE 158 250-----------------------g 2oorV) I 1-~ z w 150 1(J) n:: G::: 100 0 w u ) 0 w n:: 50 O..__ __ _..._ ___ _._ ___ ..._ __ __,_ ___ __._ ___ ...._ __ __.._ ___ __, 0 10 20 LENGTH/VELOCITY, sec 30 40 PAGE 159 Figure 27. Reduced first moment vs. L / v for ethane on silica gel. Symbols are the same as for Figure 26. PAGE 160 900.---------------------~ 800 7 00 60 0 I-" z Lu 2 50 0 (/) 0::: LL 40 0 0 w u :J 0 ~ 300 200 / 0 OL-.. __ ......__ __ __,_ __ ___. ___ _._1 ___ 1__ --1-.. __ ___,J 0 10 20 30 LENGTH/VELOCITY, sec PAGE 161 Figure 28. Reduced first moment vs. L / v for propane on silica ge l. Symbols are the same as for Figure 26. PAGE 162 3500~-----------------, 3000(.) Q) 2500 (/) f--'." z w 2: 0 2000 2: f(f) 0::: LL 1500 0 w u :::) 0 w 1000 0::: D 500 OL._ __ .,L_ __ _L._ __ -1.... __ _i_ __ ---1.. __ ___. __ _, 0 10 20 30 LENGTH/VELOCITY, sec PAGE 163 Figure 29. Reduced first moment vs L / v for n-buta n e on s ilica ge l. Symbols are the sa r;ic as for Figure 26. PAGE 164 sooo~----------------------7000 6000 0 u Cl) V, f-" 5000 z w 0 2 I4000 (J) cc LL 0 w u 3000 ::) 0 w cc 2000 1000 0 L__ ___J __ _J_ __ _J. __ __._ __ ___,_ __ ,_._ _____ :"'. 0 5 10 15 20 LENGTH/VE LO CITY, sec PAGE 165 TABLE 5 EQUILIBRIUM, DIFFUSIO N, AND RATE CONSTANTS FOR HYDROCARBONS O N SILICA GEL 2 Xl0 3 3 -6 So lute D H x 10 D H DK x lO D x lO an d p C r n s 2 -1 2 -1 2 2 Column (R,cm) K c m / sec sec cm / se c sec q cm / s e c qi cm / s e c n e Me than e A (0.0136) 5.30 0. 493 39.0 1. 17 50 9 1. 4 7 4 .12 B (0.0115) 4 83 0.60 4 54.9 1.17 48.1 1.20 4 .11 E (0 .0064) 4 64 0.656 1 64 1.12 4 7. 9 1.11 4 .31 F (0.005 6 ) 5.19 0.600 223 1. 20 46 .1 1. 21 4 02 G (0.00 4 8) 4 8 3 o. 597 308 1.11 44 9 1. 22 4 34 H (O 00 4 3) 5 04 0.538 4 12 1. 02 4 1. 4 1. 35 ~ .71 Average 4 .90 1.13 4.6.G 4.82 4 27 0.1 Eth a ne A 26.0 3.37 245 1. 04 B 2 4 .0 0.503 41. 2 2. 73 198 1. 28 1. 23 E 23.7 0.391 12 2 2.59 190 1. 64 1. 3 6 F 25.1 2,82 204 1.25 G 24 .0 0.395 230 2.44 183 1. 62 1. 44 H 23 .8 0.377 30 7 2 29 165 1. 70 1. 54 Average 24 4 2.71 214 3.52 1. 32 1 6 .5 PAGE 166 TABLE 5 (continued ) Solute 2 3 -3 -6 D H x l 0 D x l0 H DK xlO D x l0 and p C r n s 2 1 2 -1 2 2 Column K cm /se c sec cm / sec sec qe cm / secq_ cm /se c n 1 Propane: B 93.4 0 506 33 8 2 26 706 1. 08 1. 29 E 85 8 0.458 101 1.91 591 1.19 1 52 G 87. 8 0.457 190 1. 35 583 1. 20 1. 57 II 83.3 o. 576 253 1. 57 48 3 0 95 1. 86 Average 87. 6 1. 90 591 2.91 1. 56 2.59 n-Butane B 393 0.834 30 9 2.65 1900 0.49 0.95 E 359 0.684 92 3 2.18 1570 0.60 1.16 G 356 0 479 173 2.01 1 45 0 0 85 1. 26 H 339 o. 574 231 1. 68 1 2 10 0.70 1. 51 Average 362 2.13 1530 2.53 1. 32 0. 9 6 PAGE 167 144 quationd for th e s e cond centn:i l mornents of the e lu t ion curves can be arranged to gi v e ef fe ctive longitudinal diffus ion a nd rate con sta nt s For ::ei lica gel equations 30 and 38 m a y be used in the form 2 2 u2 o.o s33t 0 D X 61 p 2L / v = + ~(1.18) V where 61 = 6 + 6 + 6 a e L (119) K2 6 n = a H (120) n 6 (X-1) (l +K n) = e H (121) C (X-l)(l+K )R2 6 n i = 15D (122) r and X = (1+(1+K ) ) n (123) Since u 2 t 0 L, v, and Kn are kno w n, the LHS of equation 118, c a lled the second reduced moment, may be plotted against 1 / v 2 for each R to obtain values of D from the slopes and p values of 6 1 from the intercepts. These plots are shown as Figures 30-33 for methane, ethane, propane, and n-butane, and the respective D and 6 constants are tabulated in Tables 5 and 6. p 1 The quantity 6 1 can be considered as the sum of the resistances to mass transfer which contribute to the width of the chromatographic zone: adsorptiorl resistance 6 intraparticle diffusion resistance 6. a i and interparticle r es istance 6 While the D term is compared to the e P PAGE 168 Figure 30. Reduced second moment (u 2 -0.0833t~)/(2L /v ) as a function of l /v 2 fer methane chrornatographed on silica gel at 54C The symbols refer to columns of different mean particle radius and are d e fined i n the legend for Figure 26 PAGE 169 ' I.000L--------u 0.750 i_:z w 2 0 --=:-L 0 ? 6 Gj (f) 0 LlJ u :=J 0 0.250 0 o~~~~~=---------1-100 200 300 400 500 (VELOCITYf2 x 10-4, sec2 cm-2 PAGE 170 Figure 31. 2 Reduced second mom ent vs. 1 / v for ethane on silica g e l. The symbols are de fine d in the l egen d of Figure 26. PAGE 171 700.-------------------~~----~ 6 00 // i-:5 00 ..,.-0 z w 2 0 ... 2 4.00 0 z 0 u 3 00 w ( /) 0 w u 2.00 :J 0 w 0::: !.00 0 ..... __ ....._ __ .___ __._ __ _.._ __ __.__ __ ....._ __,_ __ _._ __ ~---0 100 200 300 400 500 (VELOCITYf 2 x 10-~ se c 2 cm2 PAGE 172 Figure 32. Reduced second moment vs. 1/v 2 for :ropane on silica gel. The symbols were defined on the legend of Figure 26. PAGE 173 ' 0 Cl) V) f--'" 60 .0 z l.tJ 0 50-0 0 z 0 40 0 0 w 0 =:) 0 w a: 20.0 0 L__ _L __ .J_ ___.L __ _j__ __! __ _: 1 .__ _J_ __ .,__ __,_ __ 0 100 200 300 400 500 (VELOCITYr 2 x 1 04 sec 2 cm2 :-< ( )l 0 PAGE 174 Figure 33. 2 Reduced second moment vs. 1 / v for n-buta~e on silica gel. The symbols were def i ned in the le gend for Figure 26. PAGE 175 600 u (l) ff) 1-~ 500 z Lu 2 0 2 0 z 400 0 u w (/) 0 w u 300 ::::> 0 w 0::: 200 PAGE 176 TABLE 6 ABSOLUTE A N D RELATIVE CO:fffRI BUT IONS TO RESI STAl'lCE TO MASS TRANSFER FOR HYDROCAR BONS ON SILICA GEL Solute 61 6 6 x lo3 6. and a e l Column sec sec sec sec % 6 % 6 % 6 a (;, l Methane A 0 488 0 59 0.206 57.4 0. 4 4' ., ., 4 B 0. 425 o. 6:3 0.145 65.8 0.1 3 4 .1 E 0. 33:3 0 13 0 053 84. .0 0.1 15 .9 F 0 317 0.09 0.037 8 8. :3 0.1 H.6 G 0.303 0 06 0.0 26 91. 5 0.1 8 4 H 0.299 0.04 0.019 93.G 0.1 6.3 Average 0.280 Ethane A 3.27 13 l 1.60 50. 4 0. 4 49.2 B 2.79 7.5 1.13 59.0 0.3 40 .7 E 2 06 2.6 0 41 79. 7 0.2 20.1 F 2 94 2 0 0.28 8 4 .9 0 2 14 .9 G 1. 85 1. 3 0 20 8 8 .8 0.1 11. 1 H 1. 79 1.0 0.14 91. 9 0.1 8.0 Average 1.65 PAGE 177 TABLE 6 (continued) S ol ute 6~ 0 6 103 6 and J. a ex 1 Column sec sec sec sec %o % 6 % 6. a e 1 Propane B 26.6 125 19.4 26 1 0 2 73 7 E 1 4. 0 38 7 0 49 .6 0.3 50.1 G 10 5 19 3.5 66 1 0 2 33 7 H 9 4 11 2 4 73 5 0. 2 26 .3 Average 6 9 4 n-B u tane B 3 38 2910 289 13 5 0.9 8 5 G E 152 860 105 30.0 0.7 69 .3 G 9 8 7 400 52 7 4 6 2 0.5 5 3 3 H 83 3 240 37 4 5 4 8 0.4 t14 8 Average 45 6 PAGE 178 familiar AD and B t e rms of t he r a te equ at ion 7 8 6 1 is analogous to the C te~m for H ET P. 155 For low R e ynolds numbers, w hich were j_n the rnn g e 0. 02 to 0. 5 for this work, it has been sho w n that the mass tran s fer coefficient kf can be defined by k = D /R = H R f g C (124) that is, it depends on the radius of the particles [12] and the molecular diffusion coefficient for the solute in the carrier gas If He in eq u atio n 121 is replaced by kfR / R~, then both 6. and 6 1 e are ?. a function of R Then from equation 119, if the above values of 6 1 2 are plotted against R a straight lin e of s lope (6 +6.) and in tercept e 1 6 will result (s e e Figures 34-37) a Since the thi n gas film penetration rate constant kf is known from equation 12 4, H is known and 6 c an be calculated from equat ion C E: 121. Subsequently, usin g the slope value (6 .+6 ) 6. will be known by 1 e 1 differe nce and D can be calculated from equation 122. Also, knowledge r of the intercept 6 will a llo w calculation of the adsorption rate con a stant H from equation 120. Resulting values of H, D, H 6 6 n c r n a e and O. with appropriate dimensions are shmm in Tables 5 and 6. 1 If it is assumed t hat the interparticle voids of the silica are a series of parallel cylindrical capill aries of the same average diameter and runnin g in the direction of the carrier gas flo w [77], then a tortuosity factor qe can be found by [12] (125) PAGE 179 Figure 34. Total mass transfer resi stan c e 61 as a function of mean particle r 2 diu s squared for meth ane chrornato g raphed on silica ge l a t 5 4c The symbols were identified in the l ege nd of Fig ure 26. PAGE 180 0,60r---------------------~ 0 8 1 ,sec <> 0.20 0.10 o __ ,___....___..___....___....___....___~-~-~~ 0 0.4 o e 1. 2 1. 6 2 0 (PARTICLE RA D I US )2 x 10 4 cm2 PAGE 181 . 35 1 f A 2 Figure Poto ~ 1 vs. R for eth a ne on silic a gel. Symbols are the sa m e as defin e d fo~ Figure 26. PAGE 182 3.50...---------------------, 3 00 2.50 0 2 00 81, s ec 1.50 1.00 0.50 O~_....__....__....__....___....___......__.....__.....__.....___. 0 0.4 0 8 1.2 1.6 2.0 (PARTICLE RA0IUS)2 x ,o4 cm 2 PAGE 183 Figure 36. 2 Plot of o 1 vs. R for propane on silica gel. Symbols are the same as defined in Figure 26. PAGE 184 25.0 20.0 15,0 !O.O 5.0 O~_....__....__....__....__..___..__ _.___....___.....___.. 0 0.4 0.8 1.2 1.6 2.0 (PARTICLE RADIUS)2 x 104 cm 2 PAGE 185 2 Figure 37. Plot of u1 vs. R for n-butane on silica gel Symbols ar e the same as defined for Fig ure 26. PAGE 186 350.-------------------~ 300 250 200 .. 150 100 50 0 ~-----~-_.__ _.__ _.___....__....___..,____, 0 0;4 0 8 1.2 1.6 2.0 (PARTICLE RADIUS) 2 x t04 cm 2 PAGE 187 where E A is the ef f ective mo l ec ular diffusion co ef ficient. This equa tion assumes that lon g itu din s l diff usion is only due to molecular diffusi o n in the interparticle sp a ce. If this i s true, then DA= De = (D + Av)e p e g e (126) wh e re A= 0 and q c a n be obtained from equation 125. The quanti t y e q which is the reciprocal of the tortuosity f a ct o r in the van De em ter e HETP equation, expresses the average l e n g th of the path follo wed by the movin g gas relativ e to the column length a nd must be at least unity. The effective q valu e s from Table 5, however, are very close to e unity and sometim e s less than one. It is expected to r e main con s tant for all compounds and incr ease with sma l ler particles since there are more and smaller interparticl e spaces for smaller particles, which causes a larger number of individual path len g ths [12]. This was seen to be true in some cases. However, the low q values found could be e a result of callin g the eddy diffusion co e fficient A negligible; if, indeed, A is significant, EA would decrease and qe w ould increase. It appears that A is a function of the solute us e d a nd perhaps the particle size since D is in most cases either higher or lower than D. p g Further, an internal tortuosity factor q can be defined by [13] 1 D = r (127) where S = 0. 31 for the silica used here, and DK is defined by equation 4. The average pore radius, for a mixture of particles from columns A-H, was quoted by the manufacturer to be 11 xlo8 cm. Calculated valu es of DK and qi are listed in Table 5. The internal PAGE 188 165 tort.uo.sities are greater than one, but were expected to approach 2-3 a nd r ema i n constant [13,92]. If the possibility of surface diffusion is considered, seve r a l anomalies can b e explained Surface diffusion is a beneficial m a ss transfer process whi ch occur s when molecules are partially d esorbed such that they slide along the a dsorb e nt surface rath e r than advance thr o u g h the column only via the carri e r g as. If this is t he case, its effect will be more noticeable for n-butane than for ethane, for ex&mp le, as shown below. The co e fficient of surface diffusion in the pores, D can be s defined by [13] DS-Dk+KD r n s where the effective Knud sen diffusion coeffici en t D 1 = D S/q .. ( K i Values for D calculated from th e aver&ge tabulated values of D, s r D and K (with q. = 2. 5) are shewn in Table 5. The contribution K n 1. K D I D S of surface diffu s ion to the observed r a dial diffusion D n s r r values is 0 % 48 % 39 % a nd 52 % for methane, e thane propane, and (128) n-butane, respectively. When these D contri lbutions are subtracted s from D the actual D values r r -3 -3 1.13 X 10 (met ih;a ne), 1. 41 x 10 (ethane), -3 -3 2 1. 38 X 10 (propane) and 1. 01 X 10 cm / sec (( n-butane) are obtained. Except for meth a ne, these intraparticle diffusion coefficients are in the e xpe cted order: they decrease wi "Uh molecular si z e. The above D values and their effective contributi on s to observ e d D s r values are of approximately the same magnitu as those extrapolated from Figure 8 of reference 13 for 54C. Sinm D is naturally a func s tion of surface concentr at ion, the surface c me rage must be very low PAGE 189 166 to achiev e acc ur ate r e sults. For this wor k the s urf a c e coverage was 6 about 0.6 x 10 % of a mono l aye r, w hich is e xtreme ly low. The cover-4 age for r efe rence 1 2 was abou t 10 % and D values obtained ther e wer e s 5 2 around 10 c m sec which w as lo we r th an the D v a lues derived from s static methods where covera g e was from 1-10 % Meas ur eme nt of Equilibrium, Diffusion, and R a te C onstants O V -101 Columns For the OV-101 colu m ns, adso rption is an intr a p art icle volumal phenom e non r at her th a n a n in t rap a rticle surface proc ess. The equilibrium adsorption con s tant K can be determin ed by rearran g ing equations C 28 and 50 into the form = (K -1) L (129) C V Values f or K w ere obt a ined from the slop es in Figure 38 for propane C chromato g r a p he d on silica ge l co at ed and i mp r eg n a ted w ith OV-101 and are listed in Table 7. The s e K value s for pro pane reflect an equi c librium condition where intraparticle concentration c is about 30 1 times gre ate r than th e interparticle concentr a tion c (equation 9); e K values for propane on bare silica were around 90. n The second mom e nt equations 30 and 52 c a n be rearranged in order to d e termine D D and H as a function of particle radius R: p r C 2 u 2 o. 0833t 0 2L / v = Y1 + D t-.2 p -2v (130) PAGE 190 Figure 38 Reduced first moment (u 1 ua 0 5t 0 ) / as a function of L / v for prop a ne chrom at o g raphed on OV-101 im p r eg nat e d silica gel a t 5 C. Th e sy m bols re fe r to c o lu nms filled with particles of mean radius a ccording to T a bl e 1: Column I 0 Column Jl Colu1m1 K Part i cle Particle Particl8 r a dius r a diu s r a diu s 115 m 68 = m 48 = m PAGE 191 300 /' 0 u 250 Q) CJ) f-"' z w 200 2 0 2 I150 (f) er LL o 100 U J u :) 0 w er 50 O...__ ___. _____ _... __ __.__ __ __,_ __ __._ __ __._ __ _,__ __ __.__ ___. 0 I 2 3 4 5 6 7 8 9 LENGTH/VELOCITY, sec PAGE 192 whe:i:-e V = Y e + Y i 1 (A-1) Ye H C ( A -1) R 2 Yi = 15D r and fl. = (1 +K ) C The D va lues obtained fro m the slopes of the LHS of e quation 1 30 p ? 169 (131) (132) (133) (134) vs. 1 /v~ Figure 39, are li sted in Table 7. The int er c ep ts give the total of the contributions to ma ss tr a n sfer resis tance y 1 (Tabl e 8). TABLE 7 EQUILIB RIUl\I DIF FUSION, AND RATE CONSTANTS FOR PROPANE ON OV-101 IMPREGNATED SI LICA GEL D H D 106 p C rx (R,cm) 2 -1 2 Column K cm / sec sec cm / sec C I (0. 0115) 28.1 9.77 0.51 4 0.67 Jl (0.00 68 ) 36.2 9.45 0.633 0.84 K (O. 00 48 ) 32. 4 4.12 0.511 0.72 Average 32.2 PAGE 193 Figure 39. Reduce? second mom e nt (u 2 -0.0833t~) / (2L / v) as a function of l / v 2 for prop ane chro mat o g r aphed on OV-101 at 54C The symbols are the same as tho se in the l ege nd of Figure 38. PAGE 194 300.-------------------------0 250 PAGE 195 TABLE 8 AB S OL UTE A.\1) RELATIVE CO NT RI BU T IO ~S TO R ES STA.i~CE T O MASS T RANSFER FOR PROP r~~E O N OV-101 I ~IP REGNATED SILICA G E L Y1 Ye Yi Column sec sec sec %y e I 232 214 7.6 Jl 7 9,9 62 2 22 1 K 67 3 49 .7 26 2 Average 17.7 The external resistance y is found from the e o/ay i 92 4 77.9 73.8 intercept of 172 2 a plot of y 1 vs R (Figure 40) Substitution of y into equ a tion 132 e yields the rate of mass tran sf er through the l iquid film on the pore walls, H which is much slower than gas phase p e netration The c' slope in Figure 40 is equal to (A. 1) / 15D from which the value of r internal diffusion in OV-101 can be ca l culated, since interval resis tance yi was found by substituting y 1 and ye i nto equat i on 131. Com~arison of Tables 5 and 7 s hows that radial diffusio n in th e O V-101 impregnat e d silica ge l pores is at least 1 000 times slower than f o r the helium-filled pores. Such a large differ en ce cou l d be anti ipated from the ratios o f 10 5 obtained for the molecu l ar diffusion of solutes in gases D c ompar e d t o molecular diffusion in liquids [93,94 ] g PAGE 196 Figure 40, Total mass tr an sfer resi sta nce Y l as a function of mean particl e r a dius squ a r e d for propane on OV-101 at 54 C Th e symbols are the s am e as those in the l egend of Fi g ure 38 PAGE 197 17 4 300~-----------------------~ 250 200 Yi, sec 150 100 50 O.__ __ .._ __ _.__ __ -'-__ _.__ __ __.__ __ _.__ __ _.._ __ __, 0 0.4 0.8 1.2 1.6 (PARTICLE RADIUS) 2 x 10-4, crn 2 PAGE 198 Ho weve r, D for propane is fairly con s t ant in OV-101 for r diff e rent R, wh ile it d ecreased w ith R for gas fi lled pores. This is due t o the over w h e lmi ng influ e nce of the liquid rather than the 175 fact th a t average pore size decre as es w ith R, in w hich c a se the propane mo lecul e wo uld diffu se mo re slowly in the sma ll er pores. Meas ure ment of Eq uili bri um Diffusion and Rate Con stants Graph i tizecl Carbon Bl ack Col u mns External surface adsorption on graphitized carbon black (GCB) can b e charact e ri zed by the equilibrium adsorption constant K n obtained from the mean retention time u 1 In con t rast to the experiments with porous silic a gel, propane and n-butane on nonporous GCB, gave th e r es ults shown in :Figures 4 1 and 42 deriv ed fro m equa t ion s 28 and 63 in the form Values for K were n (u 1 u 8 ) O. 5t 0 _ K L (135) n V where= (1e ) / E: e e Table 9 sho w s a slight increase in K w ith n decre asin g d a nd that the equilibrium concentration of the adsorb e d p solute is only 2-6 times larger than the int erpartic le concentr a tion at equilibrium. Again, the le ast squares lines do not pass through the ori g in as predicted by equation 135. Transformation of equations 30 a nd 65 into equation 136 u 2 0. 0833t~ 2L / v D 2 PX -2v (136) PAGE 199 Figure 41. Reduced first moment (u1 ua O. 5to) / as a function of L/v for propane chromatographed on graphitized carbon black a t 54C. The symbols represent columns filled with particles with mean radius according to Table 1: A Column L D Column M (t Column N Partic le radius= 115 m Particle radius= Particle radius= 68 ,m 48 m PAGE 200 20.--------------------------~ u 15 Cl) en z w 0 2 f10 CJ) 0::: LL 0 w u :::> 0 w 0::: 5 O...._ ______ __.. __ _.._ __ __,_ __ __._ __ ___._ __ __._ __ __._ __ ___.. 0 2 3 4 5 6 7 8 9 LENGTH/VELOCITY, sec PAGE 201 Figure 42. Reduced first moment vs. L/v for r.-bu t.anc o f graphitized carbon black The symbo ls a re the sa me as those identified for Figure 4 1. PAGE 202 70---------------------(.) Q) (/) 60 r-=50 z w 2: 40 1(j) 0::: LL 30 0 w 0 5 20 w 0:: 10 o':::---~--~---:1::,----__.-,---_ __.__ __ --1__ _;, ___ __.___----' ~ 0 2 3 4 5 6 7 8 9 LENGTH/VELOCITY, sec PAGE 203 TABLE 9 EQUILIBRIUM, DIFFUSION, AND RATE CONSTA.i'ITS FOR HYDROCARBONS ON GRAPH ITIZ ED CARBON BLA C K Solute 2 D H X lO H and p C n 2 1 -1 Column (R, cm) K cm /sec sec sec q n e Propane L (0.0115) 1. 67 0 349 33.7 27 1 1. 64 M (0 .00 68) 1. 89 0 359 95 3 20.2 1. 58 N (0.0048) 1.91 0. 3 14 190 33 8 1. 74 Avera ge 1. 82 27 .0 nButane L 6.32 0 170 30 .9 289 1. 42 M 6.54 0. 309 87 .0 345 2.23 N 6.60 0.207 1 73 301 2 .1 8 Average 6. 48 3 12 PAGE 204 181 where 5i 6 + 0 (137) a e K2 6 n = a }I (138) n 1) 2 6 l X= e H (139) C where X is defined as in equ a tion 123, allows the calculation of D p fro m the s lop e of the I.JI S of equation 136 vs. 1 / v 2 (Figures 43 and 44) In thi s case, the intercept 6 1 is the sum of geometrical surface adsorption and exte rn a l gas film penetration only, s ince 6 =0. l. 2 Sub sti h,ting He= kfR / R into equation 139, 6 is a fu nct ion e of R 2 as for t h e silica g e l studies Values fo r kf c a n be calculat e d from equation 12 4 so that a plot of the o 1 values v s R 2 (Figure 45) ') will yield a str a ight lin e of s lope (X-1)'"' / kfR and interc ep t li a. EA and qe were calculated from equation s 125 and 12 6 The results for propan e and n-butane adsorbed on GCB a re give n in Ta bl e s 9 and 10. The D values are smaller than the D values, indicatin g a n p g appar e nt negativ e e ddy diffu si on coeffici e nt A according to equation 5. The 6 results indicate th at p ropane and n-butan e zon e bro ade nin g due a to adsorption i s from 50-1000 times le ss i m port an t on GCB than for silic a This r es ulted in very sharp e x p e rimental e lution profil e s. As for silic a the mass tran sfe r throu g h the gas film on GCB is fast ( H large) and therefore 6 is small. The adsorp tio n rate H on the c e n GCB su r f a ce i s constant for a ll d for eac h solut e but is about 35 p time s s lo ~ er tha n for the a d so rption r ates of pr opane and n-butane on silica. This will be explained below in term s o f s urface th e rmodynamics PAGE 205 Figure 43 2 Red u ced s e c on d mom e nt (u 2 0.0 8 33 t 0 ) / (2L / v) a s a function of 1 / v 2 for prop a n e o n gr a phiti z ed carbon bl a c k a t 5 C. Th e symbols are defin e d in the l egend on Fi g ure 41. PAGE 206 0 (!.) (/) z w ::i: 0.40 0.30 0 z 0 0 w (J) 0 w 0 0 20 w er: 183 20 40 60 (VELOCITY )2 x 104 sec 2 cm2 PAGE 207 Figur e 4 4 Red u c.ed seco 1 ,.d mo r :i e nt vs 1 / v 2 f or n-b u t ane gr a ph it i zed ca r bon b l ack The symb ols are defin ed i n t he l e ge nd o f F i ~ u r e 41 on PAGE 208 185 ~---------------------, 0.80() Q) VJ i-=z w 0 0.601 0 z 0 0 w Cf) 0 I.JJ u :::> OAO 0 w 0::: 0.20 .__ __ ....._ __ _,___ __ __,__ __ __,_ ___ .._ __ ......_ __ ___, 0 20 40 60 PAGE 209 Figure 45. Total mass transfer r es i s ta11ce 6 1 as a function of mean particle r adi u s squ a r ed i or prop ane a nd n-but ane chroma tographe d on graph iti zed carbon bl ack The s ymbo ls are the same as identifi ed in the le gen d of Figure 41 PAGE 210 187 OAO 0.30 o.20L_-o ___ __,,_-----------~ 0.1 0.._ __ __ .....__ __ _.__ __ ...._ _______ 0 OA 0.8 1.2 (PARTICLE RADIUS) 2 x 104 cm 2 PAGE 211 TABLE 10 ABSOLUTE AND RELATIVE CONTRIBUTIONS TO RESISTANCE FOR HYDROCAR B O N S ON GHAPHITIZED C AR3 0N Solute 8 6 6 x io3 and a e e Column sec sec sec Propane L 0.195 1.1 M 0.194 0.5 N 0.194 0.2 Average 0.193 n-Butane L 0.271 10.9 M 0.264 4.6 N 0.262 2.0 Average 0.260 TO WiASS BLA CK %6 a 99 4 99.7 99.9 96.0 98 .3 99.2 TRANSFER % 8 e 0.6 0.3 0.1 4.0 1. 7 0.8 ...... 00 00 PAGE 212 189 Values for q were in the expected 1-2 r a nge a nd we re larg e r e than for the s am e solut e s en s ilica. The larger tortuosity factors for n-but ane on the smaller particles (Table 9) can be explained by the lar ge si ze of the molecule The factor is smaller for n-butane on the lar ge GCB particl es since the interp a rticle spaces are larger and the flow path is les s obstructed than it is for the small interparticle spaces of the smaller d fractions. p Net Retention Volum e s Figure 46 shows that the net retention volume VN defined by (140) where the inert volume (141) incre ases with pressure-corrected flow rate jF for all solutes on m all silica gel fractions. Since the intern al surface area of the larg e r particl es is abou t one half that of the larger particles (se e Experimental), yet the fractional porosity per particle was found to be 0.31 for all d the mean pore opening diam eter is greater than p 22A 0 for the l arge r particles but smaller than 22A 0 for the smaller particles For example, there are many small pores in the column H particles, whi l e there are r e lativ e ly fewer and larger pores in the column B particles. This is in line with the results of Kiselev et a l. [95], who found that silica gels with surface areas of 715, 650, 520, 2 and 185 m /g had mean pore di ameter s of 32, 46, 70 and 82 A 0 respectively. Therefore, all of the solutes have access to the internal PAGE 213 Figure 46, Net retention volume as a function of flow rate, for methane (C1), ethane (C2), propane (C3), and n-butane (C4) chrom at ogr ap h e d on silica ge l at 54C TI 1 e symbols r efe r t o the mean p a rticl e radius according to Table 1: 0 Colurnn A Particle radius = 136 m A Col mm1 B Particle radius = 1.15 m Cl Column E Particle radius = 61 m V Column F Particle r adi us = 56 m Column G Particle r adius = 49 :.m 1 Column H Particle radius = 43 n PAGE 214 . 34~0 ~ ~ ----=~~~~==. ___,.-320 0 270 0 8 4 0 c,~-.-_,_ --~0 2 2.0 G ~ ~--------~-2, .0 C _ 2 -2 00 ------1 90 55 -I jFm,ccmm l I l PAGE 215 192 surface a r en of tile l arge r particles while only sma ller solu t es have easy a cc es s to the int ern~l surfa ce area of th e sma ller particl e s. The calcul ate d siL e s o f the so lu tes, as calculated by the me tho d of Sn yd er [ 96 J Hl'e 18, 27, 38, and 4 9 A 0 2 for methane, ethane, prop ane an d n-butane, res9ect ivel y A pore op ening of r2.dius equal to 11A 0 has an a r ea of 3 8 0 A 0 2 The relatively lar ge r solute molecules have a lo w prob ab ility of e nt eri ng the sma ll er p o r e s. \\11en molecules do enter pores, they radially diffuse to ward the pore wall and may be adsorb ed lf the eq u i librium conc e ntr a tion gr eat ly favors the amount ad s orbed on the pore wall compared to the amo unt in the intraparticle space (lar g e K ), then r ete ntion time of the solute will increase with n K according to eq uation 116. n It is apparent fro m F i g ure 46 th a t adsorption on the pore walls increases with molecular weight of the solute, since l arger vollunes of c ar rier gas are required to elute the sample. Adsorption is larger for propane and n-butane when using large silica particles. Since ethane and methane h a ve access to small as we ll as large pores, their K valu e s (ther e fore VN) ar e determined solely by available n internal surf a c e area and molecular weight. Since the smaller particles have g r ea ter internal surface area, adsorption is greater when using small partic l e colu mn s and small molecul es Thus, a reversal of the order of the curves wh e n progressing from n-butane to methane is observed. Kiselev [97] u se d similar arguments to e>..-plain increases in retention volu me with decr eas ing pore diameter for normal paraffins. It was expected that VN w ould be ind e pend e nt of jFm such that curves of zero slope would be obtained for each particle size [98]. PAGE 216 However, a small but st ea dy i ncr e a s e in V N occurred (Figure 46). Since adsorpti o n a nd in t e rna l d if f u s ion a re kinet ic proc e s s es, they 193 will be m or e notic e abl e as F is incr e a se d s1:ch t hat the elution prom files will b e come more a s y m metric. Thi s tends to increase u 1 which is proportional to VN This do es not, ho w ever, e xplain w hy there is a slightly faster rate of incr ease at lower flow rates compared to higher flow rates. These arguments can b e e x tended to explain the relative importance of the three broadening resistances oe' o i, and e a by normalizing each 6 by o 1 (See T a ble 6 for % 6 % 8 and %8 .. ) a e 1 For silica, the resistance caused by transfer through the pore wall gas film is very insignificant. This is becaus e H is very large and C a small 6 value results from equation 121. Re s istance from intra e particle diffusion 6. is always more important for larger particles 1 and for larger molecules. This is because the small pores in smaller particles are not accessible to large molecul e s. From equation 120, adsorption resistance 6 will depend on the relative rates of increase a 2 or decrease of the equilibrium adsorption cons t ant K and the adsorp n tion rate constant H. Table 6 shows that 6 is inversely proportional n a to particle size. Oberholtzer and Rogers [17] observed sli g ht decreases in retention volumes with increased flow rates fo1r the same solutes on 4A and SA Mo l ecular Sieves. They attributed t E! is effect entirely to a slow intraparticle diffusion process rather t han to accessibility of surface are or equilibrium concentrations. T h:ii s explanation may not be entirely valid because u 1 (and VN) is not ~ function of Dr as shown PAGE 217 by equation 28 How e ver, u 3 is a funct ion of D, so that increased I' peak tailing due to slow inte r n a l diffu s io n could effectively increase u 1 and therefore V. N 194 Figure 47 jllu s trates a slow increase in net retention volume with flo w rate for propane and n-butane on graphitized carbon black. In contra s t to the porous silica adsorbent, the GCB particles are nonporous such that only the external surface is important for adsorption. Column N (R = 0. 00 48 cm) has considerably more GCB surface area than column L (R = 0. 0115 cm). Solute mo l ecules wi ll have a higher e quilibrium adsorption constant on the smaller particles and will be retained lon ger than on the larg er particle columns. Since retention i s controlled by a n adsorption (kinetic) process, faster carrier gas flow rates will increase asymmetry (tailing) of the elut ion profiles. In this event, the mean retention times u 1 will be shifted to longer times. These effects are shown in Figure 47. Free Energy Changes The differences in the nature of the adsorbate-adsorbent inter actions for hydrocarbon solutes on silica gel and graphitized carbon black can further be emphasized by examining the changes in free energy of adsorption It is we ll kno w n that the free energy change in going from the adsorbed state to the gaseous state, AG, is appro x imately related to the logarithm 0 the mean retention time by [99] /1,G 2.30 3R T g (142) PAGE 218 Figure 47. Net retention volume as a function of flow rate for propane (C3) and n-butane (C 4 ) chromatogrnphed on graphitized carbon black at 54c. The mean particle radiu s i s symbo li zed according to TablG 1: .ie.. Column L C Column M e Column N Particle radius Particle radius= = 115 m 68 m Particle radius = 48 m PAGE 219 70.---------------------------------------, 6 0 5 0 .._ _____________________________________ __. 2.2 1.8 1.4 1.0 L_ ___ ----1 ____ -1... ____ _i_ ____ ..J_ ____ ..J..._ ____ .L..._ ____ ,.._ ___ _, 0 4.0 8.0 12 0 16 0 i F mln1 m cc PAGE 220 197 AJ.l of the symbols have b een previously defined except for the crosssection al area a v ai lable to ~he mo ving gas A A. e e If equation 1 42 is put in t o the form 1-l G 60Ll, e 2. 303R T + F g J m then a plot of lo g (u 1 u ) vs log 60LA / jF should be linear. a e m (143) The ch ange in fr ee e nergy of adsorption can be derived from the intercept 6G / 2.303R T, which is 6 G / l.497 and 54C. A series of plots for g various homologs w ill give a ch a racteristic additive contribution to 6G for a certain adsorbent. Figures 48 and 49 sho w the results obtained for the hydrocarbons chromato g raphed on silic a gel and GCB at 54 C. The least squares lines for all points for a single solute are shown. The u 1 data are the same data us ed previou s ly to deriv e values for K and V as a function of n N particle size. On a logarithmic scal e the least squares lines are par a llel and equally spaced as expected. The values of 6G calculated from the intercepts are listed in Table 11 along with the 6(6G) values for adjacent homologs. Table 11 gives -0.843 and -0.681 kcal/mole as the aver age methyl e ne contributions, 6(6G), to the fre e energy changes on silica and GCB, respectiv e ly, at 54C. This indicates that there is a stronger adsorbate-adsorbent interaction when silica is the adsorbent. This means that there is a faster desorption on GCB, resulting in narrow and symmetrical peaks. PAGE 221 Figure 48. Plot of log ('lq ua) vs. leg (60LAe / jFm) for methane (C 1 ), ethane (C 2 ) propane (C 3 ), and n-butane (C 4 ) chromato gr aph e d on silica gel a t 54 C. The m e an parti c l e rad i u s r e f er r e d to by the symb o ls are d e fin ed in th e l ege nd for Figur e 46. PAGE 222 t.') 9 0 A A~ ~o 0-~~~_l_____________ 0.50 1.00 LOG ( 60 L Ae j F ) sec m 1.50 2.00 PAGE 223 Figure 49 Plot of log (u 1 ua) vs. lo g (60LA e/jFm ) for propane (C 3 ) and n-butane (C 4) chr omatographcd on g r a phitized carb on b l a c k at 54C TI1e m~a n particle r adi i r efer r ed to by the symbols are defined i n the l egend f or Figur e 4 7. PAGE 224 2.20--------------------------1.90 (.) PAGE 225 TABL E 11 FREE E: N""EH.GY OF ADSORPTI O N FOR HYDROCARBONS O N ADSORBEtrrsa Adsorbent /:. G 6 ( 6 G) and Hydrocarbon kcal / mo le kcal / mole Silica Gel Methane -0. 982 -0. 836 Ethane -1. 818 -0.808 Propane -2.626 -0.8 84 n-But an e -3.510 Graphiti zed Carbon Black Propane -1. 132 -0.681 n-Butane -1. 8 13 The times for so lute adsorption t d and desorption td as es c a n be ca lc ulated using the relationships and t ads K = n 1 H n t des t ads From the average va lu es of H and K in T a bles 5 and 9 for each n n 202 (1 44 ) (1 45 ) solute, tads and tdes have been calculated and tabu l ated (Tabl e 12). Using silica gel, td / t d is 8 7 for prop ane a nd 358 for n-but a ne es a s or a factor of 4.2. These ratios a re 1.8 an PAGE 226 TABLE 1 2 AV E RAGE ADSORPT IO N A ND D E S O RPT IO N TI M ES FOR HYD ROCARB O NS O N ADS O RBE:r-r:rs Adsorb e nt t ads t and d es Hydr o c a rbon sec sec Silica G e l Methane 0.022 0.105 Ethane 0.051 x 101 0.125 Prop ane 0.011 x 101 0.150 n-Butane o 0 67 x 102 0.2 40 Gr aphitized Carb o n Black Propane 0.055 0.101 nButane 0 .0 21 0.1 35 Since the differenc e in the td / t d ratio is controll e d es z. s 203 by the successive additions of methylene gro ups, then the correl a tion oft / t a nd 6G w ill h ave some predictive potential for different d es ads solut es in a homologous ser i es at a given temperature It can be seen that the ratio of the 6 ( 6G ) v a lue for silica over that for GCB is 1.23. An almost id entica l va lu e of 1.19 is obtained for ratio of the t /t values for silica over that for GC& des ads Table 12 shows that the absolute desorption times are quite similar. How ever, the apparently la rge retentions on silic a a re govern e d by the very fas t adsorption rates F or example, at the in stant of equilibri u m (u 1 of the elution curve) in an n-butan e sample population, the molec ul es a d so rb at a rate of 1530 times per se c ond (H ) for sili ca but only 48 .1 times p er second for GCB (Tabl es 5 and 9). n PAGE 227 The rate of desorption, however, indicated by the magnitude of the equilibrium adsorption cons tant K by equation 145, is 4.2 and 7.4 11 204 times per second for butan e on silica and carbon black, respectively The above r es ults can be explained on the b asis of surface charact er istic s Since the GCB s urface is highly energetic due to the hi gh density of carbon atoms on the n onpo lar surface [100,101], it ad sorbs and desorbs molecules quj_ckly. The silica gel, however, is covered w ith strongly adsorbing (reactive) and other weakly adsorbin g (bound) hydroxyl g roups as a result of conditioning below 200 C [102]. Appare nt ly, the activation energy of adsorption is lo w while that for desorption is relativ e ly much hi g her than for GCB. While the adsorbate-adsorbent interactions are a sequence of nonspecific (mainly disper s ion) interactions for graphitized c a rbon black, specific interaction s (hydrogen bonding or induction) occur for solutes chro mate graphed on activated sil ica gel [103,104]. There is some evidence that the silica surface causes an induced dipole mom e nt in methane and ethane and other hydrocarbon gases that adsorb [105]. The surface has a loc alized positive charge since the silicon atoms act as electron sinks and the hydroxyls are partially protonated. This charge induction of the solutes would, indeed, cause t d to decrease and, likewj se, a s cause td to decrease. es Effec~of Physicochemical Parameters on Chromat o g raphic Band Broadening Keulemans [106] has pointed out that classification of chromatographic theories re su lts from the assumption that one of several sets of conditions exists The usual classification for gas PAGE 228 205 adsorp t i on chr omatog r ap hy is "line a r non:\.de a l" chr oma to g raphy. A plot of conc e n tration of sol ute jn the mobile p hase versus the co ncentra tion in the stationary phase (di st ribu tio n i sotherm) wi ll b e "linear." T his is the case in this work, since extremely small (1 l) samples were used. The conditions arc classed as "nonide al" (nonequilibrium) becau se the ini t ially narro w sample pulse bro adens and becomes asymmetr i c as it passes throu gh the chrom atographic column. Band br oadening occurs because of irregular flow patt erns in the mobi le phas e mol ec ular diffusion in the mobile and stationary phases, inconsistencies in the ratio of mobile to stationary phase throu g hout the c ol umn, and due to a finite r ate of mass transfer between mobile and stab onary pha s es. The "pla te" theory views the chromatographic process as a series of discrete events, analogous to distillation or extraction a series of one-plate events. The "rate" or "nonequilibrium" theories of GC take a more re a listic approach in that they consid e r mass transfer kinetics and diffusion phenomena as continuous processes. However, the fund ament al id ea of the H E TP has been retained as a conv en ient way to indicate column efficiency. In statistical moment theory, w is analogous to HETP. It is generally accepted th a t the rate ( e fficiency) equation for GC has the form B E = A + + Cv (146) V 2 where Eis either HETP or w = u 2 L / u 1 The bes t effi ciency is achieved when the coefficients A ( e ddy diffu s ion), B (.nolecular diffu sio n), PAGE 229 206 and C (r es istanc e to mass tr ansfer ) are s m al l s u c h that Eis sma ll. The exact natur e of the se co ef ficients, especi a ll y A and C, have been disputed and st udi ed by m any chro matographe r s and the ma ny contributions to Care the m a jor concern in this dissertat ion A plot of E vs. the l inea r velocity through th e column, v, will give a parabolic curve with E = A + 2(BC) min (147) and V = (~) min (148) Plots of t vs. v as a function of me a n part icle radius, R, for lo w molecular weight hydrocarbons chromato grap hed on silica gel, OV-101, and graphitized carbon black are shown as Figures 50-56 iv was calculated directly from the exp er im e ntal u 1 u 2 and L values, while v was corrected for pressure drop and poro s ity according to equations 6 and 82. The experimental data were fitted by the method of iterative least squares to equation 1 46 to obtain the curv es shown. This method has been discu s sed by several authors [31,107,10 8 ] and the FOCAL pro gr a m used is reproduced as computer Program 6 in Appendix C. The pro gram asks for the initial t vs. v data set and iteratively solves a 3 by 3 matrix until ne w values of iv are found which give an acceptable SM value. S M is the sum of the squares of the d e viations of the calculated points from the initial experimental points. Sinc e the least squares technique was not self-starting, the "v ariab les" A, B, and C were initially given th e values 0.1, 0.1, and PAGE 230 Figure 50. Column efficiency in terms of band bro adening 1v versus linear velocity, v, for methane chromatographed on silica gel at 54 C. The symbols refer t o columns wi th a given average particle size Ras follows: A Column B R = 115 m CJ Column E R = 64 m 0 Column G R = 49 m Column H R 43 m PAGE 231 0.900--------------------------, 0 800 0.700 0.600 1/J,Cm 0.500 0.400 0.300 0.200 0.1 00 .___ __ .._ __ .._ __ ..__ __ ..__ __ ..__ __ .,____ __ ---__ ...:...I __ __, 0 5 0 10.0 15.0 20 0 VELOCITY, cm sec-I PAGE 232 Figure 51. Column efficiency* vs. velocity for ethane on silica gel. Symbols are defined in the legend of Figure 50. PAGE 233 0.500~-----------------------1 0 450t 0.400 0.350 ._ \/f,Cm 0.300 0.250 0 200 0.150 0.100.._ __ .._ __ .._ __ .,__ __ .,___ __ _.__ __ ..,__ __ ..,__ __ _.___ ___, 0 5.0 10.0 15.0 20.0 VELOCITY, cm sec-I PAGE 234 Figure 52. Column efficiency vs. velocity for propane on silica ge l. Symbols a re defined in th e l ege nd of F i g ur e 50. PAGE 235 0.400,------------------------0.350 0.300 '/1,Cm 0.250 0.200 0.1500. I 00 .__ __ ..._ __ .._ __ __ _.__ __ _.__ __ _._ __ ....._ __ ...,__ __... 0 5.0 10.0 15.0 20 0 VEI_0CITY, cm sec-I PAGE 236 Figure 53. Column efficiency t vs. v e locity for n-butane on silica gel. Symbols are defined in the le ge nd for Figure 50. PAGE 237 0.450.--------------------------------i 0.400 0.350 0.300 'f',Cm 0.250 0 200 0.150 0 .100 ..._ ____._ __ ..__ ____._ __ ..__ ___._ __ ..__ ___._ __ ..__ __._ __ ...._ ___. 0 5.0 10.0 15.0 20 0 25.0 VELOCITY, cm sec-I PAGE 238 Figure 54. Column ef fic i ency in terms of b a nd broadenin g ~ versu s line&r velocity, v, for propane chromato grep hed on OV-101 at 54C The symbols refer to column s w ith particle s of mean radius n as follows: Column I 1J Colun m Jl 0 Colunm K R = 115 m R = 68 m R = 48 m PAGE 239 216 30.0 25.0 20.0 'If.cm 7.0--5.0 3.0--~------------0 5.0 10.0 15.0 20.0 25.0 30.0 VELOCITY, cm sec-I PAGE 240 Figure 55, Column effici.ency in terms of band broadening, t, versus linear velocity, v, for propane chromatographed on graphitized c a rbon black at 5 C. The symbols refer to columns with parti cles of mean radius Ras follows: A Column L tJ Column M @ Column N R 115 m R = 68 m R = 48 m PAGE 241 0.550-----------------------, 0.450 ',cm 0.350 / fr,.. D /0 -......... D --/ 0 ........_ ____ ___ 0.250.._ __._ __ __._ __ '--_ __,__ __ __,__ __ ....__ _,i._ ___ 0 IQ0 2Q0 3Q0 4Q0 VELOCITY, cm sec-I PAGE 242 Figure 56. Column efficiency t vs. v for n-butane on graphitized carbon black. 'I11e symbols are the same as thos e defined in the legend of Figu r e 55. PAGE 243 0.150-------------------~ 0.130 ',cm 0.110 0.090 0.070 ._ ____._ __ _.,__ __ ....i,____-1. __ ____.___ __ _,__ ___. __ ___ 0 10.0 20.0 30.0 40.0 VELOC IT Y, cm sec-I PAGE 244 221 0.1, r espective ly. A l thou g h some of t h e data was scattered, the fi ts were good with an av erage of only 0.001-0.00 5 cm deviation for indivictual t valu e s A l so calculated v a lu e s of ,I, a nd v corresponded nn n min well with thos e ob s e rv e d eJ\.-pe rim en t a lly. The resulting A, B, C, SM, t and v values for propane c hromatographed on si lica gel min min (Figur e 52) and graphiti ze d carbon black (Figure 55) are li ste d in T a ble 13. The calculation of A, B, and C for the O V -101 d a ta (Fi g ure 54) was not s ucc ess ful because there were no exp e ri me ntal points on the left sid e of the expected p ara bol a. Eddy diffusion (A) i s usually found by extrapolating the hi g h velocity t values to v = 0, and it has had repor ted values r a n g in g from negativ e numbers by Bohemen a nd Purnell [10 9 ] to 0.1 cm by Grubner [11]. The hi gh vel oc ity portion of the silica ge l and GCB curv es (Figur es 5053, 55, and 56) generally extrapolate to abou t 0.1 cm regardle s s of the p ar t i cle radius used. This supports the cont enti on of Grubn e r [11] that A is ind ee d a si gn ificant constant in the t equation for gas-solid chromato g raphy, but that it is not a function of Rand is not influ enced by the roughness or shape of the adsorbe nt This notion is contrary to th e classic al van Deemter equation for eddy diffusion, = 2M p (149) where A is a multiple p a th term dependent on p h ysical charact e ristics of the particle and i ts bed and r a nges from 0. 5 t.o 8, and d = 2R [110]. p However, calculated v a lu e s of A were often very small or negative as for prop a ne on silica gel (Table 13). PAGE 245 TABLE 1 3 VALUE S O F THE Rl\.TE COEFFICIENTS A, B, AND C CALCULATED BY LEAST SQUARES USING EXPERIMENTAL DATAa Adsorbent A B C SM 'f'm in and 2 x105 Partic l e Radius, cm cm cm / sec sec cm Silica Gel 0.0115 0. 03 1 0 907 0 014 1 0 256 0.0064 0.042 1 .073 0 014 5 0.203 0 00 1 19 -0.046 1 007 0 010 6 0.151 0 00 4 3 -0.058 0 923 O. 068 X 10 2 6 0.108 Graphitized Car b on Black -1 0.0115 0 141 2 112 0 088 >, 10 18 0.27 7 0 0068 0 157 2 332 0.061 X 10 l 6 o. 276 0 0048b 0 4 7 4 0 510 0 076 X 10 -1 11 V ;nin cm /s ec 8.05 8 75 10.2 36.8 15.5 1 9 6 a Experimenta l data f or propane (Figures 53 and 56) were used as input to Pro g ram 6 in App e ndix C. b T he fit to these data was not success f ul becau s e no exp erimental d ata w ere a v aj_ labl e for the right hand s ide o f this parabola (Fi g ure 56, botto m curv e) PAGE 246 The mo l e cular diffusion coeffi cien t (B) has be e n eq uat e d to 2 y D i n the c l ass ical van D eent er eq uation [ 110], wh ere y is a tor g tuo si ty factor It r e flect s t he labyrinth str uc ture of the in te rparticl e ads orb e nt spa ce and the unifor m ity of the packin g a nd ha s 223 be en found to r a n g e f rom abo u t 0 7 to 1. 0. Table 13 sho w s that th e le ast squares B val u e s for propane on silic a a re abo ut 2D (D = 0. 44 7 g g cm /s ec for propane in He) ; the D values for prop a ne fro m Table 5 are p a l so c onsistent w ith B / 2 from Tab l e 13. For prop a ne chfomatograph cd on gr a phiti ze d c a rbon bl ack ho w eve r, B appro x imately equa ls 4D g The C coeffic ients c alculated by le ast squa res a re small and are in the normal ran g e found by o t h e r chr omatographe r s Substitution of th e Table 13 co effi ci e nts into equation 1 46 for v shows th at the min Band C t e r ms have abo ut the same magnitud e Ho we ver, at velocities less than v the B t e rm ( m ol ec ular diffu sio n) makes the major contri mJ n bution to ,jr At v > v ,jr results primarily from resi sta nce to mass min transfer (C term) The position and magnitude of ,jr is a function of the solute, min adsorb ent and particle size. For all ,jr vs. v plots (Fi g ures 50-56), smaller p ar ticl es gave more efficient chrom atograph ic behavior. This agrees wit h the stoch a stic theory of Giddin gs w hich states that b a nd bro adenin g decreases a s the len gt h of th e random wa lk step decr eases and the num be r o f steps increases [63,64]. This w ould be true for mo lec ul es passing through the sm a ll interparticle chann e ls of small particles. Like w ise, band bro a d e ning decre ases as the number of combinations w ith the surface (ad sorptio ns) incr eases since the numb e r of individu a l steps b efor e elution is incre ased PAGE 247 224 In this li ght T a bl e 1 2 showed that the a v era ge ti me of adso rption d ecreased with mo l e cular we i ght o f th e so lute For ex2 mple, at equilibrium on silica ge l an n butane mo lecul e i s a dsorbed 8 times more often t han an et hane mo lecu l e. F i gure s 50-53 s how a lo we ring of the vs v cur v e s for all d w hen progress in g from m etha ne to n-but a ne. p This indic ates a r e l at iv e decrease in band bro a d e ning for lon ge r ret ained (hi gher mo lecul a r weigh t) solu te s. Similarly, at e qu il ibriu m on graphitized c a rbo n black, Table 12 sho ws tha t n-butan e a dsorb s 2.5 times per second, w hi le pr op a n e a dsorbs only once. This is illu strated by Figures 55 and 56 w hich indic a te rel at i ve ly n arro w er (more effi ci e nt) pe aks for n-butane t han for propane. For prop ane adsorb ed by O V -101, the chro ma to g r ap hic elution profil es had mean retention times com parab le to metha ne on silica ge l. Ho wever, th e second moments o f t he solute bands were on the ord e r of thos e for pr opane on silica ge l. This resulted in ex ceptionally large valu e s ( F i gure 54). Extrapol at ion of th e high velocity side s of the curves for OV-101 to v = 0 gives an intercept (2A) of about 4 cm for all p a rticle sizes. This is a factor of about 40 greater th ~ n for propane chro m tograph e d on b a r e silica or on GCB. The coeffi u: i e nt A is cle a rly a function of the difficul ty of passage of the s cl ut e through the interparticl e space as a ssumed before. In this case, a liquid partly fills the int e rparticle space r at her than a gas. Agai.n, A app e ars not to be a function of R or adsorbent rou gh ness, since ::the same silica was used in both the silic a gel and OV-101 exp er iments. PAGE 248 225 The short ret e ntion times of propa n e on th e OV-101 columns can be e xplained by th e l a rg e so lulJil.i ty o f propanEJ in OV -101 and the rel atively fast adsorpt i o n-d esor ption process compared to propane on b a re silic a ge l. T h e exaggerat e d exte nt of b an d br oade nin g can be explained b y s lo w int rapar-t icl e dif fusi on ( sma ll D) w hich accounts for most of r th e r es i s tanc e to mass tra ns fer, as seen in Tab l e 8. For sma ll R how e ver Tabl e 8 shows that diffusion through the external f ilm o f OV 101 ( % 6 ) was si g nificant. This is prob a bly b e c a u se e pockets of OV-101 are mor e lik e ly to form in the interparticle spac es of small-particle b eds than for those made fro m lar ge particles. P e ne tr at ion of the external OV -101 fil m is a much f aster mass transfer process tha n d iffu sion in th e OV 101 that is in the pores. This is apparent from the si g nific ant ly l owe r curv es for t he t wo colunms wit h small R co mpared to the cu rve for the large particles (Fj g ure 5 4 ). Effec ts of Physicochemica l Param ete rs on Chrom atographic Band Asymmetry Equation 32 s hows that the third momen t, w hich ch a racteri zes peak synun e try, is dep e nd e nt on the same physicochemic a l parameters (D ,H ,H ,K ,K) as the second mom e nt, which ch arac terizes peak broad r c n c n 2 3 ening. Th e statistical moment qu a ntity Z = u 3 L / u 1 can be used as a measure of c hroma tographic efficiency with r espe ct to a sy mme try In Figures 57-63, Z has b ee n plotted as a function of pressure corrected velocity and adsorbent particle radius. All of the Z vs. v data follow e d parabolic paths with minima occurring at approximately the sam e v e locity as for the corr espo nding t curves. F o r this rea so n, the expe r imenta l data were fitted to equation 1 46 by th e least squares PAGE 249 Figure 57. Column effj_ciency in terms of b an d asymmet ry Z, versus lin ear velocity, v, for methane c hroma to grap hed on silic a ge l at 5 C. Th e sy mbo ls, which refer to g iv en pa:rt icl e radi.i, a r e define d in the l egend of Figure 5 0. PAGE 250 s.oo~--------------------------, 7 00 6.0 0 5 00 Z,cm 2 4. 00 3 0 0 2.00 1.00 Q..__ __ ......_ __ ___._ ___ .._ __ ......_ _________ .._ __ ......_ __ 0 5.0 10 0 15 0 20.0 VELOCITY cm sec-I PAGE 251 Figure 58. Column efficiency Z vs v for ethane on silica gel. Symbols are defined in the legend of Figure 50 PAGE 252 5.00 400 Z,cm 2 3.00 2.00 1.00 O.___--'__ ___.__ __ _,_ __._ __ _._ __ _.__ __ .___ _..__ __ 0 5.0 10.0 15.0 20.0 VELOCITY, cm sec-I PAGE 253 Fig u re 59. Column efficiency Z vs. v for propane on silica gel. Symbols are defined in the l ege nd of :Figure 50. PAGE 254 3.50r-----------------3.00 2 50 200Z,cm2 1.50 1.00 0.50 0 ._ ____._ ___.~ __._ ___. __ _.,__---1. __ ....J-_--1-_,_,...:._ _J 0 5.0 10.0 15.0 20.0 25 0 VELOCITY, cm sec-I PAGE 255 Figure 60. Column efficiency Z vs. v for n-butane on silica g e l Symbols are defined in the leg e nd for Figur e 50. PAGE 256 3 50~---------------------------, 3 00 2.50 Z,cm 2 2 00 1.50 1.50 0.50 a 0"---1---L---__JL-_ __J __ __,1 __ __,1 __ ---1, __ ---1, __ ---1. __ __._ __, 0 5 0 10 0 15 0 20 0 25.0 VELOCITY, cm secI PAGE 257 Figure 61. Colu m n e: ffici en cy in ter m s of ban d asymmetry Z versus line ar velo city, v, for prop ane chromato graphed on O V -101 a t 54 C The symbo l s are defin ed in F igu re 54 PAGE 258 235 20 0 0 1500 l000 Z, cm 2 500 500 250 o...._ ____ _.__ ____ ~-----...L....------'------...L....-----' 0 10.0 20.0 30.0 VELOCITY, cm sec I PAGE 259 Figure 62. Column efficiency in terms of band asymmetry, Z, versus linear velocity, v, for propane chromato graph e d on graphiti z ed carbon black a t 54 C. Th e symbols ar e d ef in e d i n the l e ge nd for Figure 55. PAGE 260 1.40--------------------------, 1.20 1.00 2 Z,cm 0.80 0.60 0!40 0.20 O._ __ .._ __ .._ __ .._ __ ....._ __ _.___----'-....__ __ _.__ __, 0 10.0 20.0 30.0 40.0 VELOCITY, cm sec-I PAGE 261 Figure 63. Column efficiency Z vs. v for n-butane on graphitized carbon bl a ck. Symbols are defin e d as in the le gend for Figure 55. PAGE 262 I .40---------------------------, 1.20 1.00 Z,cm 2 0.80 0.60 0.40 0.20 _-o--: t::,. I 0-,: --O.__ ______ __.._ ____________ _._ _____ ___. 0 10.0 20.0 VELOCITY, cm sec-I 30.0 40.0 PAGE 263 210 technique ( P ro gra m 6, App e n dix C) to obt a in the c a lcul a ted curv e s sho \ m in Fi g ur es 57 -63. The coinci de nce of v for corr e spondin g+ and Z curves was m in antic i p ate d fro m s t a t em e nt s by K ubin [4]. He postulated thnt the sel e ction of t he opti m al v a lue of v ( w here ~ occurs) from the min vie w point of th e w idth of th e elution curve w ould a lso result in approxi m ately the v giving minimum peak asymmetry if K or K is> 0.9. n C Kubin [4] d e riv e d the equation (150) for gas-solid chrom & t o gr a phy by forming the deriv a tives du 2 / dV O = 0 and clu 3 / dV = 0 and then solving for v w ith re s pect to u 2 and u 3 o min V is the superficial outlet velocity. 0 In equation 150, Wis the ratio of the minimum velocity for u 3 w 3 over that for u 2 w 2 and H is either K or K 0 n C For the results reported in Figures 50-63, H was in the range 2-180 such that more than 8 % according to 0 equation 150. Due to the difficulty in precisely determining u 3 from chromatographic data at low integration limits, scattered results were sometimes obtained for Z. The Z vs. v curves do, however, show sev e r a l trends in common with the corresponding sets of ~ curves. For instance, greater asymmetry occurs for all solutes chromatographed on large particles than for the same solutes chromatographed on smaller particles. Asymmetry is inversely proportional to the mean retention time. For ex a mple, ethan e chromatographed on silica gel (Figure 58) PAGE 264 241 result s in Z v s v plots w l1i ch are lo we r than for methane (Figure 57) but hi ghe r than fo r propane a nd n-butane (Figures 5 9 a nd 60). The b as ic diff e r ence between th e t and Z curves is the beh avior at values of v > v .. min t increa s es by a f acto r of less than 2 for l arge Rat v>v wh : ll e very little incr ease occurs for smaller nun p a rticl es In contr as t, Z increases by a f a ctor of 2-5 for all R for v>v .. These results indic a te the relatively g reater sensitivity of min peak symmetry compared to that of peak broadenin g when both are influenced by the sa me m as s tr a nsf e r processes. At hi g h flow rates, the major contribution to both broadening and asymmetry result e d from slow intrap art icle diffusion for the silica gel and OV-101 experiments (see o and y. in Tables 6 and 8, respectively). For solutes chromatographed 1 1 on graphi ti zed c ar bon black, t and Z increased wi th v > v due to the min slow d e sorption step (see Table 10, % 6 ). a The first moment, indicating the tempor a l position of the peak mean, is in turn even les s sensitive to delays caused by the s a me slow diffusion or kinetic proce ss es at high linear velocities. This was sho w n by only slight increases in the net retention volume when plotted against flow rate for individual R values (Figures 46 and 47). Correlation of Experimental with Theoretical Results and Analysis of E rr o r s In the preceding chapters and sectiorn;, theoretic a l equations were derived and experimentally determined p1sicochemical constants were presented. Some of the constants( ,.,.D ,L,R,t 0 ) were found e 1 g from independ e nt e xperim e nts. Other physico"d l emical constants (K ,K n C D ,H ,H and ~ G) were found from an analysi of the first and second r c n PAGE 265 242 s t at i s t ica l moment s a s a fu n ction of ave r n ge colu mn l i n ea r velocity and a ds or b e nt p arti c le ra d ius Th e e f f ic iency ex p r e ssio n s t a nd Z we re cal c ul a t e d from the exp er imen t a l fi r .st sec ond, and th ird m o m e n ts. T he validity of the st a ti s tica l m o m e nt e xp r ess i on s a nd ~ and Z e xpr es si o n s derived in the Th e or eti c a l ch apt e r, ma y b e te s te d by s ub st i t u t in g t h e e xperim e nt a lly d e ter mined qu a nti t i es in t o th e t h e oretic a l equati o n s In this way, experi me nt a l and c a l c ul a ted e f f iciency curve s for hy d roc a rbons chrom a to g r a ph e d o n v a rious a d s orbents should coincid e Sever a l a d s orbat e a d s orbent combinations w er e arbitrarily chosen in ord e r to compare ex p e rimental a nd theor e tic a l data. These are sho w n in Figur e 6 4 wher e the experim e ntal d ata (points) ar e the same a s those plott e d previously for prop a ne u s in g 100 / 120 mesh adsorb e nt par t icl e,; a nd equati o ns 34 and 35. The th e oretical valu e s (curves) wer e c a l cu l a ted from the theoretical equations using the previou s ly tabulat e d physicoch e mical constants. In Figur e 6 4 (bottom) the experim e ntal broadening points ( i 1 ) for prop a ne chro m a t ographed on silica gel particles of radius 0.0064 cm corresp o nd well w i t h the curve calculated from e q u a tion 39. Equation 4 0 was a lso satisfactory by using Dg for p ropan e and valu e s of A= 0. 01 cm. An analysis of the individ ua l terns in equation 39 showed the eventual drn n i nan ce of the third and fifth t er m s a t higher flow rates. In contr as t, since H is large, the f a:r rth t e rm is small at C all flow rates. For prop a ne chroma t o g raphed on gr a phitiz e d carbon black particle s of R=0.0068 (Figur e 64, middle), the~ e ri m ental asymmetry PAGE 266 Figure 64. Compar is on of experimental data (points) and theor etical d ata (curves) for propa ne chro matographed on various adsorbents: Bottom: Efficiency with respect to bro adening, 1 for prop ane on silica ge l w ith particles of radius 0.0064 cm, Colunm E. Middle: Efficiency w i th re spect to asymmetry, z 3 for prop an e on graphitized carbon blac k particl es of R=0.0068 cm, Column M. Top: Efficiency with respect to broadening, t 2 for prop a ne on OV-101 impregnated silica gel particles of R = 0. 0068 cm, Column Jl. PAGE 267 8.0 0 "1 2 ,cm 6.00 "--0 4 00 0.300 ~, cm 0 200 0 10.0 20 0 VELOCITY, cm sec-I ' 0 7 2 44 0.600 2 3 cm 2 0.400 0.200 30 0 PAGE 268 ( Z_) a g re e s we ll with th e curv e calculated from equation 73 u sing ;j D = 0. 4 47 cm / sec and A= 0 0 8 cm Asymm e try is increased at higher g lin e a r velociti es mainly due to the lar ge v a lu es of the 3rd, 7th, and 10th t e rms of equat i on 73. The finite rate of de sorptio n (H) n 245 c au s es a delay in the elution of th e chromatographic b and and thus, more asymm e tric peaks a r e ob se rved at h ig her flo w rates. In th is regard, Gidd ing s [ 65] h as stated tha t a sorption s ite whi ch holds molec ul es for a t ime equal to that n ecessa ri for one quar te r of the z one to pass the site will cause tailing. This is consist ent with the observati ons for GCB (Figure 64). For e xam ple, peaks r ep resented by the point at v = 22. 9 cm / se c eluted in 15. 88 sec. Since th e colu m n was 61. 0 cm lon g the peak me a n was travelin g at 3. 83 c m / sec. Ho weve r, the aver a ge time 0 desorption for a molecule of propane adsorb e d on an a dsorption s ite is 0.101 sec (Table 12) It is clear that peak asym.'netry will increase wit h carrier gas velocity. The corr esp ond ence of the experimental points and the w 2 curve calculated fro m equation 53 for prop a ne adsorbed by OV-101 (Figure 64, top) is not satisfactory. This is probably due to the uncertainty in the value of D derived from the thr e e-point slope of Figure 40. r Radi a l diffusion (equation 53, 2nd term) is the dominant cause of bro ade nin g at v > v while transfer through the OV-101 film on the min silica is also a significant cause of broadenin g (II in equation 53, C 3rd term). This is consistent with % y e and % y i from Table 8. A similar curve was obtained using D = 0. 447 cm / sec and 2A = 3. 5 cm in g equation 54. PAGE 269 246 The d eg r ee of coincid e nc e between exp e rim e ntal and calculated d ata ca n b e es timated fr o m the r e liability of the statistical moments data. Sinc 8 the r elationsh ips of the t and Z curves to the physicochemical c onstants are quite complex, a nd Zin terms of the moments thems e lv es w ill be examined below. Table 14 illu str a tes the precision of the statistical moments for hydrocarbons chrom atog r aph ed on silica gel usin g 2.0 % integration limits. The avera ge and ercent relative standard devi a tion for 3-6 runs on column Eat F = 8 cc / min are sho wn The relative molar m response for norm a l FID operation is 1.00, 2.04, 3.12 and 4.08 for methane, ethane, propane, and n-butane, respectively [111]. Table 14 shows that the experimental relative proportions w ere 1.00, 1.79, 4.05, and 6.35 for the four hyd r ocarbons such that approximately 1 xlo10 mole (1 l, 54 C, 15 psi) samples for each were used. It is apparent from Table 14 that precision decreases with increasing order of the statistical moment and that a precision in the range 0.01 to 0.07 for u 1 can be expected with the GC system used in this research using 2.0 % int eg ration limits. It is important to point out that the errors in t and Z are almost totally reflected by the errors in u 2 and u 3 respectively, such that the small error of the first moment measurement is not significant in the propagation of errors [112]. Table 15 shows the same d a ta as Table 14 calculated at the 0,39 % integration limits. All experimental data in Figures 26-63 were plotted using the 0.39 % limits. Table 15 shows that there is an increas e in the absolute value of all the moments, si nce more of the PAGE 270 Moment or TABLE 14 PRECISION OF STATISTICAL MOMENT CALCUL l\ TIONS AT 2. 0 % INTEGRATION LI MITS FOR HYDROCARBONS ON SILICA GEL AT 54 Ca Hydrocarbon f Hybridb C d e Methane Ethane Propane n Butane uo 15065 2 08 % 26591 0 51 % 61016 0.05 % 95640 0 26 % ul, sec 65.93 0.07 % 222 6 0 01 % 727.5 0.03 % 2 957.3 0.02 % 2 u2 sec 12 55 2.62 % 70.2 1 0.33 % 721. 5 0.7 4 % 1 3960 1 0 6% 3 6 u3, se c 36 7 7 8.74 % 421.4 2.15 % 13271 0.71 % 1 207 x lO 1.16 % 4 6 6 u4, sec 640 9 8 12 % 1 8395 0 70 % 1. 835 x lO 2.75 % 6. 290 x lO 3 12 % skew 0.822 4 96 % 0.701 1. 85 % 0.685 1 77 % 0.7 3 1 0 77 % excess 4 042 3 08 % 3.69 7 0. 96 % 3.5 44 1.32 % 3 334 2.52 % 'V cm 0.352 2 55 % 0 173 0.33 % 0.166 0 62 % 0.192 0 90 % z: 2 1,913 8.65 % 0.557 2.15 % 0.513 0 68 % 0.695 1.18 % em aM oments ca l cul ate d by the off-line FOCAL Program 2 in App endix C. The valu e s r ep orted are the mean and percent relative standard devi ation for 3 6 r e p l icate samples on column E at F m= 8 cc / min b The hybrid mom e nts 1V and Z were calculated using L = 121. 9 cm. cData acquisition rate (DAR) was 10 H z The a ppro x imate number of d a ta points per p ea k can be calcu l ated from 6. 2 /4"" DAR. dDAR = 5 Hz. eDAR = 1 Hz fDAR = 0. 5 Hz. PAGE 271 M oment or TABLE 15 PRECISION OF STATISTICAL MOMENI' CALCULATIONS AT 0. 39 % INTEG R A T ION LI M ITS FOR HYDROCARBONS ON SILICA GET, AT 54 C a C H ydrocarbon b Hybrid Methane Ethane Propane n B ut a n e uo ul, sec u2, s e c u3, sec u4, sec skew excess t, cm 2 Z, c m 15172 66 .05 ..I.. ...L. 2 14.39 3 61. 49 4 1754 1.118 8.646 0. 402 3 176 ..I.. ...L. 2.09 % 27197 1.55 % 0.07 % 223.8 0.04 % 3.22 % 9 7.15 0.86 % 9.72 % 970.0 2.14 % 4.65 % 38450 3 .50 % 5.22 % 1.01 3 o. 87 % 3.61 % 4 .0 82 5.00 % 3.18 % 0.236 0.88 % 9.65 % 1. 288 2.20 % a Moments c a lcula ted by the on-line pro g r a m ADCOM. relative standard deviation for the same d a ta set as 61500 0.71 % 9 68 30 0. 57 % 731. 6 0 08 % 2 98 7 5 0 05 % 926 2 1 60 % 1 6 0 3 0 3 7:. % 305 4 0 2 73 % 2. 5 6 0 x1 o 6 2. 4 5 % 3. 86o x 10 6 2.52 % l.52 3 x lO 9 8 1 4% 1. 083 0 77 % 1. 2 71 6.21 % 4 .51 7 5.40 % 5.918 4. 8 0 % 0.211. 1. 70 % 0 219 L 3 80 % ...L. 1.160 2.93 % 1. 42 9 2. 33 % The values reported are the me a n and p e rc e nt in Table 14 bThe v a lues of the hybrid moments t and Z are those points in Figures 50-53 and Fi g ures 57-60 at v = 8 38 cm / sec cThe data acquisit ion rates (DAR) were the same as those for Table 14. PAGE 272 249 peak is includ e d in the calculation. The precision of the statistical moments has not decreased sign ific ant ly, and in some cases, precision has improved ov e r the 2 % limit s The pr ecisions of t and Z range bet ween 1-3 % for the lon ger retain ed solutes and 3-9 % for methane. It will be assumed that the er rors in the statistical moments are du e mostly to instrumental or d eterm inate errors, rather than random (indeterminate) errors, since the values in Tables 14 and 15 are based on only 3-6 replicate chromatographic peaks. Also, Guiochon [45,50] has sho w n that it is possible to measure retention times with -4 a relative precision of about 10 ( 0.005 % ) at the 95 % confidence level, provided suitable experimental conditions are realized. This relative precision improves with the number of replicate runs according to equation 151. Relative precision= at (151) where a is the standard deviation of the average retention time, tR, and tis the student's t value for n replicate measurements [45]. The average retention time tR in equation 151 can be replaced by u 1 such that, from Table 15, the relative precisions on u 1 for -3 -3 -3 methane, ethane, propane, and n-butane are 1.6 x lO 1.2xlO 2.6 x lO -3 and 1.4x10 respectively, at the 95 % confidence interval using 0.39 % int eg ration limits. The precisions could be improved 4-fold simply by making at least 60 replicate measurements according to equation 151. However, the resulting precisions with the GC system described above compare well with that reported by Guiochon [50] who me as ured peak maximum times, rather than peak mean times, so that PAGE 273 250 integration limit s w ere no t con s idered. The precisions for the OV-101 and graphiti z ed carbon bl a c k e x periments w e re similar to those for methane on silica ge l (Tables 14 and 15). This was either because of the relatively short retention times or because of the exaggerated broadenin g and ske w in g of the chromatographic peaks. It is known that for addition or subtraction, the absolute determinate errors are transmitted directly into the results, while for multi pli cation and division, the relative determinate errors are transmitted directly into the result [112,113]. In view of these facts in conjunction with Table 15 and the close fits of the experimental and theoretical data in Figure 64, it can be estimated that the precision on any one physicochemical constant is not greater than 20 % The precision on D, e S, and L were previously shown to be g e 1% or better. The values of Rare mean values derived from a rather large distribution of particles, either from particle counting exper iment s or by sieving. The values of K and K are probably reliable n C to within 1 % since they were derived from the first statistical moments and knowledge of the porosities of the columns (calculation of u ). The rates of external diffusion, H and surface adsorption, a C H, have larger errors, since they are very large numbers calculated n 2 from small interc e pts of 6 1 or y 1 vs. R plots, the 6 1 or y 1 having been previously obtained from reduced second moment plots. Table 15 shows that the error propagated by the second moment alone is 1-3%. The values of D are probably most in question, since they r were derived from the slopes of lines with only 3-6 points, which PAGE 274 were scatt e red in most c ases (e.g., Figures 36 and 40). The error in D coul d ea s ily be as m u ch as 20 % r 251 The accuracy of the abo v e physicoch em ic a l constants can only be estim ated from the sparse literature valu es reported from static and chro matog r a phic meas urements. Previ ous discu ssio ns have shown that so m e physicochemical con st ants determined in this r es earch (D, e R K) are very close to those reported by other chromatog e' n graphers for similar c onditions Several physicochemic a l constants (K H H, D) are of the same order of ma g nitude as those observed c c n r by others or are being r e ported for the first time. Summary This research has shown the applic a bility of statistical moment analysis in determining physicochemic a l constants from gas c hromato grap hic data. The stochastic theory of GC, using a realistic model, has been semi-quantitatively verified by examining adsorbateadsorbent systems which were previously examined with GC equipment which was not highly precise, and the theory was extended to new adsorbate-adsorbent systems Some system a tic contributions to band broadening and asymmetry were kno w n and the st a tistical moments could be corrected. However the errors in the s ta tistical moments in Table 15 were assumed to be due largely to other systematic errors of unknown origin. As a result, the precision of the measured physicochemic a l constants was limited by the propagation of errors These errors are magnified due to the complex relationship between the moments describing peak sh a pe and the physicoc hem ical ph en omena that control the peak shape. PAGE 275 252 Precisi o n on the st a tist i ca l mom ents was fo und t o ran ge from 0.0 4 % t o 3 % rel a tive s tan d a rd deviation in m ost ca se s, wi t h the wors t c as e abo ut 9.7 % for the first fiv e stati sti c a l mo m ents calculat e d at th e 0 .3 9 % int eg r a tion limits. G e nerally, a 0.3 % to 10 % reliability in t h e resultin g ph ys icoch em ical con sta nt s w a s observed, with a 20 % reliability in the worst case. Extensions of this re s earch should includ e s tudie s of the physicoc h emic a l constants D, K K H, H and D as functions of p c n c n r temper a ture a nd pressure. With this kno w led g e, it s hould be feasible to pr e dict, from the appropri a te theoretical equations, the efficiency cur v es ( ~r Z vs. v) for any compound w ithin a homol o gous series given one complete s e t of conditi o ns for one compound in the series. It is expected that more reliable values of D could be r obtain e d from chromatographic methods ~1ich do not involve propa ga tion of lar g e errors. The ideal adsorbate-adsorbent system for this w ould re q u ir e molecul e s of precisely known proportions and part:i cles of known radiu s and pore s i z e, which are not distributed, s uch that internal diffusion is the only resistance to mass transfer. In this case, D r for solutes in various gases and liquids could be obtained directly from the slope of the high v e locity side of the t vs. v parabola. The model applied in this research did, indeed, assume some kno w led g e of t h ese variables in order to render the statistical moment equations solvable. It would be more fruitful to develop chromategraphic column s w hose char a cteristics are exactly kno w n in all respects in ord e r to obtain more accurate and precise physicochemical data, since i t i s cl ea r that exp e rim e ntal re s ults are in good agr ee m e nt with th e stoch as tic theory of g as chroma t o gra phy. PAGE 276 253 Finally, the full p ote ntial of computeri ze d deconvolut:i.on techniques fer gas chromat ogr aphic curves has not been explored. Future research sho ~ ld disclose the relationship of empirical curve fitting para mete rs [31] to the physicochemical processes which inf luence the p e ak shape. Likewise, Fourier analysis [32] should further be developed such that individual on-column phenomena can be characterized and predicted. PAGE 277 APPENDICES PAGE 278 APPENDIX A HIGH PR EC I SIOK SAMP LI N G I N GAS CHRO W \TOGRAPHY The pr ecis ion of several chromatographic sam plin g valves is sho w n to appro a ch 0.05 % for unret a in e d solutes. Hybricl-fluidic, high pressur e, and com me rc ia l valves have been ch a r a cterized by mea surin g the pr e cision of their column input profil es and statistical mom e nts. Introduction High precision sampling in gas and liquid chromato g r ap hy is prerequisite to the development of ne w chromato grap hic t e chniques, for the measurement of fund ame nt a l paramet e rs in chromatographic systems, and for quantitative an a ly ses High precision gas chromato g raphy, high resolution separations, automated and computeri ze d chromato g r a phic sys tems, qualit a tive identification from retention data, studies of column properties, 2nd extra-column contributions to band bro a dening all require a reproducible inj e ction peak profile. Further, the input pro file should b e of known functional form in order to obtain the maximum resolution and. column efficiency, and to measure the transfer functions of the column. However, the exact nature of the transfer function of the column itself has not been previously determin e d. The prerequisites for such a study are (a) a high preci sio n sampling system with a 255 PAGE 279 256 well-defi n ed inp;t function w hich can b e treated ma thematically, (b) a high pre ci s ion, lo w noise dig ital d ata acquisition system, and (c) a n accurate method of d et e r m inin g the instru menta l contributions to band broadenin g a nd a definition of their func tio n a l form so that r aw data may be corrected for these effects. In the work presented here, the i n put pro files of three gas chromato g r aphi c samplin g valves ar e co mpare d. Peak profiles for all of the valv e s were measured on a chrom atog raphic system w here the instrumental effects on the peak shape we re minimized and were accurately l ~nm v n, and with a co m puter based d a ta acquisition and control system for hi g h precision and speed, described in the te x t of this dissertation. Experimental A cut-aw ay, cross-sectional view of the "hybrid-fluidic valve" is sho w n in Fi g ure 65 in the normal position. The push-pull s olenoids are Gu a rdian typ e 14 with an 11 ohm coil. The carrier gas and inlet tubes were 0.040 in. i.d. stainless steel tubing mounted in a block attached to th e a rm a tures of the solenoids. The centerline distance between the tubes is 0.188 in., which is also the total travel distance of the solenoid armatures. The column is mount e d in a brass bellows beneath the inlet tubes so that it is normally aligned with the He carrier gas inlet. In this manner, there are no sliding seals in the valve and the dead volume is negligible. However, the coaxial alignment of the column w ith the carrier and sample inlets is critical in order to achieve good reproducibility a nd to prevent sample bleed into the PAGE 280 Figure 65. Hybrid-fluidic valve shewn from t h e front w ith a cut-away view of the valve ass e m'!:lly. Valve is in normal position for carrier gas flow cnto the column. PAGE 281 -<--HEIN r SA M PLE IN ' I ~----rf1 ~ ~ "-' _) ~------1-l-C~ I [ '-""--: ... u ll : --=:: ;;..,__,-L 1--.-------~---i. -------.....-1 50v r 0 V _J =======l--L __ 7 50v ._ --- Ov ~ --BELLO W S -VE N T PAGE 282 259 ca rri er gas line The i n l et t u b es an d t he co lu mn a r e sepa r a t e d b y about one ha l f t h e i n side di a meter of the t u b es o r 0.0 2 in. so th at the d y n ami c l ine s o f s a mp l e gas fl ow int e r sect the colu m n axi s onl y w hen th e:: va l ve is sw i t ch e d for sa m p lin g T hi s mini m i zes t he e nd e ff e c on the l ami n a r f l o w th r oug h t h e tu bes an d a ll ows f l ow of g a s into the c o lu mn onl y from th e i nlet c a pill ar y w ith w hich t h e column i s ali g n e d. Th u s t he sepa r ati on dis tanc e and t he a li g n men t a re crit ical, and t he c a rri er gas flo w ra te m ust b e 10 to 2 0 ti mes the colu m n carri e r g as flo w rat e The s a mpl e gas and c a rri e r g as a re allo we d to escap e throu g h a ve nt v a lve (l\I od e l S S -2 M A, N upro) w hich i s mount e d in the base o f the f le xi bl e bell ow s as s em bly. This v a lv e a llo w s control of the r a te o f v entin g of s amp le a nd carrier ga ses with c o lumn o f different p n e um a ti c r es i s t a nce. In order to ch a n g e the sam ple si z e, the valve gate puls e w id th t h e s a mple g as flo w r a te, or th e d e lay tim e in s a mplin g fr om the e xp onenti a l dilu t ion fl a sh c a n be vari e d. Th e "high pres s ure" valve e v a lu a t e d in this stud y w as described in the Exp e rimental se ction in the text of thi s dissertation. The third v a lv e s tudied wa s a pneumatic a lly op e r ate d valve design e d for gas and liquid chrom a tography and is commercially avail able from H am ilton Co m pany. This valve is also designed around an annulus c u t on the slidin g shaft A 1 s a mple volum e w as used for the work r epo rt e d here. The valve was a ctuat e d by a pn e umatic pi s ton in the valve which was driven by air pressur e controll e d by t w o three way electric switching valves ( Mo d e l 250El-3-10-21-36-61, Humphrey Products). The iwitching pressure was nominally 120 psi of N 2 PAGE 283 260 G as Ch romato g r a ph / Co mp ut er S y stem A Varian i\Iode l 2 1 00 gas chromatograph with a lin ea r t em perature programmer and dual flame ionization detector was used in this w ork. The samp lin g v a l ves we re mounted above the injec tio n ports and nominally operate d at room temper a ture. The other t e mperatures were as follo ws : injection port, 80C; column oven, 96 C; and detector, 85 c c. The volumetric f lo w rates used in the system are given in Table 16. The pn e umatic s ystem and data acquisition and control system were d e s cribed earlier a nd are shown in Figure 4. The flow controller in the helium line was a constant upstre a m flo w controller (Model PN-42300080, Veriflow Corp.) rather than the Tylan mass flo w controll er d esc r i bed earlier. TABLE 16 FLOW RATES US E D TO MEASURE SA M PLI N G VALVE CHARACTERISTICS Valve Hybrid-fluidic 100 msec Hybrid-fluidic 50 msec High Pressure Hamilton 100 ca Hamilton 50c He cc / min 770 770 114 113 113 CH 4 cc / min 76.8 76.8 182 116 H2 cc / min 54.0 54.0 41. 6 98.3 95.6 Air cc / min 263 263 468 568 589 a Temperature r e f e rs to column oven te mpe rature. Valve was heated by conduc t ion to a temperature l ess than that o f the oven. PAGE 284 261 Result s and Di s c u ss ion F igur es 6 6, 6 7, 6, and 68 are th e va lve samp l e injection profiles for a 50 msec valve gate pulse width on the hy brid-flui dic valve, the hybrid-fluidic va lv e with a 100 msec gat in g pulse, the high pr ess u re valve, and the Hamilton valve operated at l0 0C, resp e ctiv e ly. All of the data were taken w ith a n ADC con versio n rate of 500 Hz and ar e di splayed on the same time scale. The amp litude of the peaks a nd their lo ca tion s hav e be e n normali ze d for display purposes. The marker indicator dots d e no te the limits of in tegrat i o n and are locat e d a t the point whe re the signal is 2.0 % 0.5 % and 0.2 % of the maxi mu m peak amp lit u d e T h e descriptio n and error analysis o f this tech n i qu e has b een d es crib e d e lsewhere [ 2 1]. Figures 66 and 67 are virtually id ent ical in p eak shap e so that valve gate pulse widths do w n to 50 msec c an be expected to introduce uniform input profiles to the column. The sample shown in Figure 67 is appro x imately twice as larg e as the one in Figure 66 but was divided by two for display purposes and thus appe ars to be the same size as Figure 66. The narrow injection profile in Figure 6 was previously explained in terms of the short effective injection time of the high pressure valve. Figure 68 sho ws a peak shape similar to those in Figure s 66 and 67, except that it has a much longer tail due to a small amount of leakage and/or the presence of diffusion ch am bers in the valve. Thus the value of the higher order moments is consid e rably larg e r for this valve. The methane sample size was adjusted so that the range in sample sizes was less than a factor of 2.5 for t he thr ee diff ere nt PAGE 285 Figure 6 6. Valve injection profile obtained w i th a 50 msec valve g ate pulse to the hybrid fluidic valve. M arker dots denot e inte g ra ti on limits at 2.0 % 0 5 % and 0.2 % of the peak he i gh t T i me ba s e of di sp l ay is 0-2000 rnsec Figure 67. Valve inj ect ion profi le obtain ed w ith a 100 m sec va lve gate puls e to the hybrid fluidic v a lve. M arker dots deno te int e 6 ~a tion li mits at 2 .0 % 0.5 % and 0.2 % o f the peak height Time base of display is 0-2000 msec PAGE 286 263 f(t) t f(t) t PAGE 287 Figure 68. Experimental va lve injection profile for the pneumatically op e rated Hamilton valve at 100 Limits of inte g ration are at 2 0 % -0. 5 % and -0.2 % of the pe ak hei g ht Last data poi nt on the tai l of the p eak is 1 25 % of the peak amp litude. Time base of display is 0-2000 msec. PAGE 288 265 f(t) t PAGE 289 266 valves in ord e r to a void in troducing additional v a riables. The ratio of the p eak a r ea s for th e h y brid-fluidic valve was 2.03 8 when the val ve puls e width ratio was 2.00 so that it is n ear ly lin ea r. The deviation from 2.00 was probably due to some r ec oil and ringin g of the inl e t t ub e s which w ill introduce incr easi n g ly lar ge r errors as the valve ga te pul se width is d e cr ease d. The difference in sample si z es of th e Hamilton valve at t w o diff e rent te m p e r at ur es is only and can be explained as a change in the leaka ge rate through the '> 0/ ._, ,o Teflon seals and the trav e l time of the sample shaft as it is affected by frictional forces. The high pr es sure valve minimizes these prob lems because it has a constant tension spring and thus can be aperated over much wider temperature ranges. Table 17 summ a ri zes the ch a racteristics of each of the peak profiles in t e rms of their statistical moments and their precision for at least 10 d e terminations. A relative precision of 0.1-0.2 % in the peak area is particul ar ly encouraging when one considers the errors associated with locating the peak start and finish (limits of integration) and lack of regulation on the sample gas supply. The method of peak truncation previously described by Chesler and Cram [21] was used to minimize the first of these errors, and because the relative errors at the 2 % integration limits can be easily determined. Thermostating the pressure and flow regulators would also increase the precision of the pneumatic control system. As the order of the moment (n) incr eas es, we would expect to realize an increase in the errors bec a use of the uncertainty in the time of each data point raised to the nth power. However, the data PAGE 290 Parameter Mea s ured uo b ul, sec 2 u2, sec wl/2 max 3 u3, sec 4 u4, sec skew excess TABLE 17 COMPARIS O N OF THE MEAN VALUES OF THE STATISTICAL MOMENTS AND PRECISION FOR VALVE GENERATED I:NI UT PEAK PROFILESa sec Hybrid-fluidicc Valve, 100 msec 2146 0 15 % 4 0.2466 0 12 '1~ 4 x io:... 4 l.33Sx102 o.07% 1x105 0 ,2482 0.12 % :: o. 4x103 1.101x103 0.15% 2 x 106 6.273xlo 4 o 16 % 1x106 0 7114 0.12 % 1x103 3 502 0 08 % 3x103 Hybrid-fluidicc Valve, 50 msec 10 5::l 0. 36 % 5 0.2276 0.18 % 5 x io4 1.2o x 102 o.22 % 3 x 10 5 0 2313 0.21 % 3 0. 6 x lO 1. 01sx103 0.79% 1 x 105 5.212 x 104 0.79 % 5xio 6 0.7706 0.45 % 4xio3 3.595 0.35 % 1 x io2 High Pressure Valved 903 7 0 16 % 2 0.1177 0.33 % lG x J.o4 3.9o x io3 o.10 % 2 x 10 5 0.1123 0.31 % -3 0 5 x lO 1 6S1x104 1. 4 4% 3 x 106 4. 4 13 x io5 1 4 6 % 9 x 107 0 9762 0.86 % 1 x io2 4.605 0 69 % 4 x 102 PAGE 291 Parameter Measured ul b ul, sec 2 u2, sec wl/2 max, sec 3 u3, sec 4 u4, sec skew excess TABLE 17 (Continued) Hamilton Valve 30Ce 1367 0.80% 13 0.1805 0.75 % 16 x 104 2.481 x 102 2.96% 8x104 0.1316 0.48 % o.1 x 103 8.649x103 4.81~ 4 4xlO 4.989 x 103 6.15 % 4x10 4 2.210 0. 42% 1 x1 02 8.080 0.3 9 % 4x10 2 Hamilton Valve 100 ce 1.325 0 86% 32 0.1987 0.83% 22 x 104 2.1oo x 102 1.95 % 7 x10 4 0.1480 0.95 % 2x103 9.586 x io3 3.61% 5x10 4 5.535 x 103 4 .62 % 3x10 4 2.142 0.14 % 4x103 7.843 0 30 % 3 x 102 aMoments are calculated for unretained methane in a 46 5 cm X 0.10 cm i .d. tube for 10 determinations. Errors are given as percent relative stan d ar d deviation followed by the 90 % confidence interval about the mean Limits of integration are 2.0 % of peak amplitude. bCalculated as a central moment from peak start. c ::c: 320 data points per peak d ::c: 175 data points per peak. e ::c: 450 data points per peak. PAGE 292 269 taken for th e hybrid-fluidic valve with a puls e wi dth of 100 msec indic a t e th at the preci sion is constant t o w i thi n 0.1 % for all of the p eak par a m e t er s a nd as mu ch as 28 tim es b e tter th a n some of the other valves. Th e n ear ly Ga u ss i a n ch ara c te r of this v a lve i s a l s o signif ic ant and uniqu e ( s ke w= 0.71 vs. 0.0 for a Gau ss i a n and e x cess= 3.50 vs. 3.00 for a Gaussian). It should be p ointed out th a t valve gate pulse wi dth s as short as 9 msec h ave b ee n us e d w i th the hybrid-fluidic valve. Ho weve r, the best prec i si on for this value wa s obtained at 100 msec and the increas e d pulse width is not d e trimental for most chrom atog raphic experiments. Thus the hybrid-fluidic v a lve is the v a lve of choice for the best precision and for gene ratin g ne a r G a ussian input profiles. The pr e cision of t he high pressure valve for the zer oth and first m o ments is comp a r ab le to that of the hybrid-fluidic valve. The f a ct that the former giv e s n a rro we r pulse profiles and that columns can be chang ed much more readily currently mak es it the valve of choice for implem en t a tion in the labor at ory. Althou gh the excess for this valve profile is markedly increased, the ske w is not. The larger error in th e first moment is due to variation in the sealing pressure, caused by frictional heating when the valve is fired. T his time rep resents a lon ge r trav e l time than the first valve a nd do e s require sliding se a ls. The Hamilton valve was originally d es i g ned for samp ling in liquid chromatography at pressures to 5,000 p sig Any va lve of this nature can b e expected to vary in performanc e wit h temp e rature chan ge s. The relativ e ly good pr e cision m akes it well suited for liquid PAGE 293 270 chro ma to g r a phy. Th e most s e 1 io us disad va n t a g e is th e tailing of the input profil e which arises from leaks in the Teflon seals (because they c a nnot b e too tight w i t h a pneumatic drive), diffusion chambers, and th e perturbation in the b a nd shape cau s ed by t he carrier gas pressure sur g e follo w ing shut of f during the s a mpling time. Bec a use pr e cision s am pling data are very s carce in the c h romategraphic literature, only the relative precision in the elution time can be comp a red directly. By defining elution time as the time at which data acquisition is started t o the center of gravity of the peak and the percent r e lative standard deviation as a me as ure of precision, several valves can be compared as shown in Table 18. TABLE 18 COMPARISON OF THE PRECISIO N OF SEVERAL AUTO M ATED SAMPLING VALVES WITH U NRET AINED SOLUT E S (REFERE N CE) Valve Hybrid-fluidic [42] High Pressure [42] Hamil ton [ 42] Seiscor [34] Carle [34] Kieselbach [ 34] Precision in Elution Time 0.05-0.08 % o. 64% 1. 36 % 0.22 % 1. 08-1. 17 % 1. 00-2. 00 % PAGE 294 271 The depe1"dence of the mo1r.ents and t h eir precision on the location of the pe ::t k li mi ts is s l 10,m in Table 19 for the hybridfluidic val v e with a 100 n, sec valve gate pulse width. Peak sensing by thresholding and the first derivative method are subject to intolerable errors w h e r m a king mom e nt calcul2ti0ns and in the presence of noise. Thus the method of limits in terms of a fraction of the peak amplitude is used here and has been previou s ly justified [21]. It is of par t icular interest to note that the precision of the peak area and !:lean time are about constant for the limits tested. This is most encouraging because these are the moments of general interest. As expected, the precisj_on decreases by as much as 50-fold as the order of the moment increases and as the limits are extended to lower signal amplitudes. The peak area naturally increases as the limits are extended to include more of the peak are~. At 0.2 % limits, the peak area is 2 % larger than at limits of 2.0 % This error is nontrivial and not readily apparent from the valve profile in Figure 67. The high order momen t s also increase in a regular manner by as much as 1900 % (for the fourth moment) with the extended limits while the reproduc ibility decreases by a factor of about 50 for the same reasons. These data clearly show the importance of specifying the limits of integra tion, since they affect both the absolute value of the statistical moments of chromatographic peaks and their precision. PAGE 295 TAI3LE 1 9 EFFE CT OF LI M I TS OF INTEGMTIO N O N TI-IE MEAN VALUE AND PRECISIO N OF THE STATISTICAL M O ME NT CALCULATIO N Sa I n t e g r a ti o n Lim i t s Mom ent 2 .0 % 0.5 % 0.2 % uo 2 1 45 0 1 5 % 2171 0.16 % 2 1 89 0.16 % ul, sec 0.51 32 0 .08 % 0. 51 8 9 0. 0 8 % 0 5 2 7 3 0.12 % 2 u 2 s e c 2 -2 -2 1. 3 8 8 x 10 0 0 7 % 1. 6 5 6 x lO 0 .30 % 2 538 x lO 1. 95 % 3 u3, s e c 1 lO lxlO -3 0 .15 % 2. 687 x 10 -3 1. 22 % 1 1. 99 x lo -3 5 52 % 4 u4, s ec 3 3 3 0. 6 2 73 X 10 0 1 6 % 1. 672 X 1 0 1. 5 2 % 12. 63 x l O 7. 44 % skew 0.71 44 0 1 2 % 1. 216 0.7 9 % 2.996 2.76 % exces s 3 502 0 08 % 6 097 0 95 % 19. 4 8 3.86 % a D a t a were taken with the hybrid fl u idi c samplin g valve a t a va l ve g a te pulse width of 1 00 msec All cal c ulations were mad e on the s a me da ta set and under the s ame condi t ions as for Tab l e 17 PAGE 296 APP E NDIX B EFFECT S O F SA M PLE SIZ E ON GAS CH R O MATOGRAPHI C BEHAVIOR In thr ee op e n tub e experiments, evidence was found for loss of chro ms.tog r a phic e ffici e ncy c a used by band bro ade nin g and asymmetry as sa mp ling time or sample size was increased. Introduction High precision sa m plin g in gas chromatogr a phy was discuss e d in Appendi x A, where the input profiles of t w o valves of original d es ign and on e commercial va lve w ere compared. One of the d esig ns, the hybridfluidic, g a ve the most Gaussi a n-like profiles and w as the best example of an on-column injection system reported to date. Since it was sho w n that this samplin g valve gave a superior precision (bet w een 0.07 and 0.16 % RSD) for all of the first five statistical moments, it was selected for further study of sampling effects on chromatographic performance. Experimental The pneumatic system, gas chromato g raph, computerized data acquisition and control system, and the hybrid-fluidic valve were describ e d in det a il in Appendix A and the experimental section of this dissertation. 273 PAGE 297 274 Two ope n tubular ec-lu :c ;_, ts w e re u se d :i..-1 t } ds work The fir st was a 71 4 cm x 0.06 8 c m i .d tube wal l coated with DC 200 The se c ond was an un coat e d 3 14 cm X 0 :t.72 cm i. d. tube Bot h had br i gh t interior f inishes and we re m3de fro ~ p r ec i s i o n b are s tainless s te e l (Superior Tube Co.). Si:::ice the inj ectio n encl of the c ol umn was a n in teg ral part of the valve d es ign (Figure 65) a 11d th e detector e nd of the colu mn w as b utted to th elo w vo lu me F ID bnse t he tota l d ea d vo lu me at both ends (3. 2 J ~) was ca lcula ted to be negligible co m pared to the column volu m e s ; th e meas ur ed va lt ,e was s ma ller than e :Kpe rir.1ental er ror. Met h ane was inje cte d o nt o the coated colu mn u si n g s amp ling times of 10, 25, 50, 100, 150, and 200 msec, thus incre as in g the sampl e size. The resultant p e2 k profiles are supe-ri mposed in Figure 69. The inc rease in ret e ntion t im e with incr eas in g sample size is readily ap par ent, and in all cases, nearly Gaussian curv es are obta i n ed. The dif fer en c e in ret e ntion times over this 20-fold concentration range is 513 msec in total retention time of about 1. 76 sec. This incre:::: se can be a cc o unt e d for in p a rt by the incr ea se in b an d width on the column bec a use th e time base is referenced to the valve a ctuation software routine rath er than th e c e nter of g ravity of the peak profile. The rel at ively lar ge chang e in the ret e ntion time a lso indic a te s that the isotherm is concave up wa rd rather t han a La ngm uir isoth erm Since the pe ak symm et ry is unchan ged a nd th e variance in the sample size (u 2 ) is only a factor of 5, it is assumed that the column w as not overloaded Figure 69 illustrates the i mportanc e of usin g central moments for pe a k shap e ch a racteri za tion and th e value cf normalized mome nts in order to account for sample size ef f e cts. Since the minimum valve PAGE 298 275 sampU ng ti n e i s 9 m [; ec it is not surprisin g th a t the 10 mse c sample gave unpredic ta bl e r es ul t s. T hi s is l argely a me cha nic al artifa ct w h ich can be attr i bu te d to the a c t i o n and desi g n of the valve. Secondly, sampli n g si z e was vari e d as p e ntane was samp l ed fr o m a 650 cc exponenti al dilution f l ask (ED F ), using a constant sampling g a te w idth o f 1 00 mse c, on to the s a me coated colu mn In the third c a s e, pentane was aga in sampled from a n EDF, u si n g a con stant sample gate width o f 150 msec, into the uncoated tube. Seve ra l of the peak p rofile s are s uperi mposed in F igure 70 Re s ults and Discussion It is i m portan t that the e f fe ct o f sam p le s i ze on all of the mom ent s be known qua1ni tative ly. Discu ss i ons of sample size effects in GC a r e plentiful [ e g ., 114-116] but 3. co mp l ete a nal ysis of such effects using the first five s t atist i cal momen ts of the r es ultin g peaks has not been r epo rted. Figure 71 shows this effect for C1-I 4 injected w ith the hybrid fl uidic valv e The zeroth moment is seen to be line a r as the valve gate pulse widt h (and therefore the s amp le size) is incre ased. This is the expect e d result and certainly is characteristic of a good samplin g valve. The nonzero intercept on the abscis s a i s j n good ag reem e nt wit h the minimum value g ate pulse w idth. The change in the first moment has previously been discussed, and u 1 is seen to be constant for sampling tim es greater than 75 m sec The pe ak va riance and wid t h at half-height cur ves tend to parallel each other and in c r ease in a nonJ ine ar manner with samp le size. The peak PAGE 299 Fi g ur e 69 I\lethan e peak pro.files o b ta i ned on 714 cm x 0.0 68 cm i d. DC 200 column after inj e ction w i th t he hybrid fJuid i c valve V nl ve gate pulse w idths= 10-20 0 msec ADC cl ock rr, t e w n s 1 KHz and t:l"le time int e rval b e-cween d a ta points dis p l ., ye d on each pe a k iE 12 mse c Ol'igin repr esents 1.1 3 7 ,,ec a:fter star t of da t a acq uisition. Figure 70. Pentane peak profiles obtained on a 311 cm X 0.172 cm i. d. open stainless s te el cclumn aftel' injection w:i.th the hybrid-fluid valve. V alve gate pulse width = 15 0 msc c. ADC cl0ck ra te w;-is 250 Hz a nd the t tme interv a l bet wee n dat e po i ~ t ~ displayed o n eaci1 peak is 20 msec. All p eaks ar e normalized to an e1ution time of 24 .50 sec at the c ri g in in ord e r to compar e peak shap e s and first mo me nts T he first, filth, ninth, thirte e nth, and sixteenth samples taken from the EDF are sh own. PAGE 300 277 f(t) t f(t) t PAGE 301 Figure 71. Effec t of th e met han e sample size on the stati st ical m oments w h en retained on a 7 1 4 c m X 0. O G8 cm i. d. colu mn co ated w ith DC -20 0 at 98C (in tegration limits = 1. 0 % ). Ordinate is scaled to a fu ll scale value of 12 50 0 fo r tile peak heig ht (ph), 2500 for uo, 2500 msec fo r u 1 0 .01 25 sec 2 for u 9 0 .12 5 sec for the s t andard d eviation (sc), 0.250 sec~for the wi dth at half h e i g ht ( w 1 ; 2) 250 x io 6 sec 3 fo l' u 3, 25 0 x 10-G sec 4 f or u4, 0.625 for the skew (sk), and 5.0 for the exces s (ex). PAGE 302 279 10 -----9 8 7 1 u1 (1 1 Uo G ph o sd ...... "'/ / .._ / ex 0 5 // U2 4 J. /y 0 I / /2 ~ 3 ;r U4 0 -o ,,,., / I I I I 2 I 0 I ,' / U3 I -/_ ' sk I / er / I / 0 .,,,,,. ., I . --~ -1(..--0 I 0 50 100 150 200 VALV E GATE, M SEC PAGE 303 280 as yr::rr.etr y appears to incr ease only slightly. It would, however, incr e ase m arkedl y as th e sample siz e app r o a c h es the point of over10 <1. din g the col ttmn T h e fourth moment increases more rapidly with incre asing s amp le size becau s e of the incr ea sed band width and the iner eas e in the nu mbe r of molecules competing for sorption sites Thu s the sorptiondes orption rate effects are enhanced with larg er sar. rple sizes and the result wi ll be a rapid incre a se in u 4 The skew anc! excess are seen to remaii1 relatively constant because they are normali z ed to the first and s e cond moments, respE:cb vely. Any changes in these two p aramete rs, as 3how n in Figure 71 reflects the mann e r in which the data were r ed uc ed s uch as the choice of limits, and does no t repr esent any r eal trend. The fact that t h e peak mean (u 1 ) reaches a constant value, while the peak variance (u 2 ) a nd asymmetry (u 3 ) continue to increase, results in decr ease d column efficiE::ncy. This is shown in Figure 72, where valve gate time has been translated into concentration units and the first, second, and third moments are reduced to measures of ? bro adeni ng and a3ymmetry efficiency, and Z. is u 2 L / u~, analogous 2 3 to HETP, while Z is u 3 L / u 1 where Lis the colurrm length. t uses the ra w data from Figure 3 and* is the same curve recalculated with the sampling time contributions to u 1 and u 2 subtracted [43]. has leveled to a constant value, which is the expected result for HETP at a constant flow rate. However, when sampling time contrj_butions to u 1 and u 3 are subtracted from the raw data [4], giving the Z curve, effi ciency with resp e ct to asymmetry has in fact worsened. PAGE 304 Figure 72. Broadening(~) and asymmetry (Z), for methane retained on a 714 cm DC-200 coated open tube. PAGE 305 2.0 1.5 'I', cm 1.0 0 5 .1 ~-:1.0 2 0 cone moles 3 0 -6 10 /z 4 0 / z I 3.0 I 'ft Z, cm 2 2.0 1.0 ,\ 4.0 PAGE 306 Figur es 7 3 a r : d 7 4 ::, ho w plots of t h e: ':lorw : m ts and column effici c 1 :c y w he n pen t an e i s inj e cte d onto the s a me co a ted column. Tl 1e m e an tim e for all 12 sam ples taken w as 7 4 .7 0.03 sec, at the 1.0 :{ int eg ra t i o n l im j_ i : s Alth o u g h the r ete nt i on tim e re m ained constant, t her e were subtle, yet si g nific a nt v ariations in the second and t hird mo ment s In this case, absolut e values of and Z are an order of m ag ni t ude better tha~ for CH 4 and there was a negligibl~ correction in both curv e s due to the 100 msec sampling time, since it is small 283 corr p a r e d to the ret ~ ntion time. It should be noticed that both t and Z ar e at limitin g values until the sample has decayed to about 0.1 micro mo le (a b out 2 l), where efficiency markedly i m proves a s smaller samples are taken from the ED F Jn contrast to the coated tube runs, pentane injected into the lmcoated tube gave the results shown in Figures 75 and 76. In this case, the mean retention time for all 16 samples was 25.02 0.02 sec at the 0.1 % integration limits. It should be pointed out that the cha ng e in sample size from a minimum delay or dilution time to 20 min utes is a factor of 28 in Figure 75 and only 19 in Figure 73. This is because different carrier gas flow rates through the dilution flask were used and thus the concentration coordinates in terms of dilution time are different in the two figures. In both columns where the EDF was used, the peak area is seen to decay logarithmically within the experimental limits of the EDF. PAGE 307 Figure 73. Effect of the pentane sample size on the statistical mo ments w hen r etained on a 714 cm X 0 .0 68 cm i.d colu mn coated with DC 200 at 98 C (j_ntegration 11.mits = 1. 0 % ). ADC clock rate was 100 Hz. Or dina te is sc a l ed to a full scale value of 5000 for the pe ak 9 height (ph), 9800 for u 0 1 25 sec for u 1 1. 250 secfo r u2, 1.250 sec for the standard deviation (sd), 3 5. 0 sec for the widt h at hal:: he i ght (w1 ; 2) 1. 25 se c for u3, 5 0 0 sec 4 fo r u4, 1.25 fo r the skew (sk), a nd t.00 for the excess (ex). PAGE 308 1r1 ..-------.., I I 9 8 7 6 4 3 2 0 .. .. .. .. ... .... .. .. .. .. sd 'U :4 0 200 400 600 800 1000 1200 DILUTIO N TI ME SEC 285 PAGE 309 Figure 74. Broadening (t) and asymmetry (Z) for pentane retained on a 714 cm DC-200 co at ed open tube. PAGE 310 I I 0 15 ~030 I e 'It 't',cm /~ 0.25 .. ,., "m2 LI t" 0.10 ----=z 0. 20 0 15 0.05 0 10 0 05 I I 5.0 10.0 15 0 20.0 I 10 -8 cone., mo es x PAGE 311 Figur e 75. Effect of the p e n tane sample size on the statistical mom ents i n a 314 cm X 0.172 c m i.d. open stainless ste el co lumn at 97 C after injection w ith tJ-w Lybr id fluidi c valve Va lv e gate puse w idth= 15 0 msec (in tegration li mits= 0.10 % ) Ordinate is scaled to a full scal e value of 4 00 0 fo r the peak heig~t, 2000 for uo, 40 sec for u1, 0.0 2 sec 2 for u 2 0.20 fo r t he standard deviation (S D! J 0. 4 00 sec for the at h alf h eight ( w 1 ; 2) 20 X 10 sec for u3, 20 X sec 4 for u4, 0.20 for the skew (sk), and 4 for the excess (ex). sec wi dth 104 PAGE 312 2 8 9 1 0 .----m ~ _ ______ _____ _ 9 6 5 4 2 sk 0 0 20 0 400 600 8 00 1000 1200 DILUTIO N TI ME s SEC PAGE 313 Figure 76. Broadening(~) and asymmetry (Z) for pentane in a 314 cm uncoated open tube. PAGE 314 z 0.020 0.0020 0.015 00015 't', cm Z, cm 2 0 010 ____:_-:-_--.;..r:..-=--~=-=---=---=---:.--=---_----====:===::==.,,,_:___--:_-_--A-~ t 0.0010 0.005 0 0005 10.0 20.0 30 0 40 0 50 0 I I 0 -8 cone mo es PAGE 315 292 Figure 76 shows this to be an extremely effi cient column, generatin g 31.-52,000 theoretical plates, bu t it is most likely of littl e pract i cal use in s e pa ratj ons. It is, however, useful in examining samp l e size e ff ec ts an d flow phenomena. There is a small I constant corr e ction to the + curve due to the 150 msec sampling time being a signific a nt part of the 25 sec residence time of the pentane in th e tube. But Z is not signific;:;ntly affected in this respect. Broadenin g aga in beco mes constant at higher sample siz es but skewing cont i nues to increase markedly; even though i ts absolute value is small, Z is incre as ing at a faster rate than for pentane in the coated tube. It should be noted that only upon statistical moment analysis of the individual data p G int::0 is an increase in asymmetry observed, since it is not readily :ipparent from Figure 70. The different phenomena observed for the coated and uncoated tubes can be explained with the aid of Figure 77. Effective disp ersi on, D in open tubes is generally a summation of ordinary axial diffueff sion, D i.e., g longitudinal diffusion of the solute and the carrier, alcng with Taylor diffusion, which accounts for dispersion of the sample pulse caused by the parabolic velocity profile as modified by diffusion in the lateral, or radial, direction [117]. Listed in Figure 77 are the three previously mentioned experiments, along with the binary diffusion coefficients calculated from Hirschfelder's equation for l00C and 1.0 atm the tube radius, the average linear velocity of the sample through the tube, and the contribution of the second term to the observed dispersion equation. PAGE 316 Figure 77. Dispersion in co ate d and uncoated open tubes. Arro ws pointing to the r i ght, awa y from the paraboli c flo w ~rofil e a nd t ~w ard the c e nt e r of the tube r e present d i recti0n of carrier gas flow, viscous flow disp e r s j o n, and radial dis persion, r e sp e ctivel y D o ubl e headed arrows in the coat ed tub e ind ic a te di i fu s ion in and out of the liquid coati ng PAGE 317 294 Disp ersion in Open Tubes D = Dg + eff 2 -I cm sec Sam p le Column V c, coated 0. 98 0 0 034 500 6.1 C5 coated 0.4SO 0.03 4 10 0.005 C5 uncoated 0.490 0.086 13 0.053 Coated Uncoated PAGE 318 \ 295 In the first exper iment, the second t e r m of the dispersion equation in Fi g ure 77 is ov e r w h e lm i ngl y lar g e c.iue to th e fast li near velocity Th e rat e of radi a l diffusion is much less than the gas flo w so d is pe r s i o n due to vi~cous flow is ~ccentuated. Viscous flow has t he a ction of di s persing molecule s in the most concentrated p a rt of the ve locity profile, to w ard the e d g e s of the tub e and radial dif fusion w orks in the opposite direc t ion. The fHct that the tub e is coated m a ke s no difference sinc e the 1-2 sec retention time does not allow time enough for the peak to broaden by moelcular diffusion in the He or by diffusion in the DC 200. Thus, p was almost constant with sampling time aft e r the raw data was corrected for sampling time (Figure 72). Ho w ever, Z got cont i nually worse with larger samples du e to skewing caused by the velocity profile. The lar g er the sample, the more concentrated is the front of the veloc ity profile, and the greater the tendency for molecules to disperse toward the walls where the concentration (pressure) is less. In the second experiment, the s e cond term in the observed diffusion is negligible. This is because the linear velocity is sig nificantly lower and here the rate of radial diffusion is about the same as the speed of the carrier gas and its viscous dispersion effect, so that these two processes cancel each other. Thus, any broadening or skewing that occurs will be due to molecular diffusion and diffusion into the liquid pha s e only. It was seen that wand Z increased up to 2 l sample sizes but for samples greater than 2 l, w and Z were constant (Figure 71). This was because the column capacity was exceeded, i.e., no additional PAGE 319 296 pentan e molecules could bE. accommodated by the liquid phase, a nd since radi a l and viscous dispersion ~ere equal, n o fu rther degradation of efficiency occurred. In the third e:>..-perime n t w i th pent a ne in the uncoat ed tube, the increase in tube r adius squared by a factor of 7 as well as an incr ease in ve locity sq u are d by a factor of 1.7 m ~ kes the second term in the equation for observed diffusion significant, since it makes up about 10 % of the observed d ispers ion. In this case, r a dial diffusion is apparently less than the zone flow rate and the action of viscou s flow dispersion cre ate s an increasingly mo re pronou nce d skewing as sample size increases (Figure 76). There is no liquid phase to retain part of the sample. Bro a dening again increased but only very slowly for sample greater than 2 l, being due mostly to binary diffusion of pentane in He and in part due to the lag caused by viscous flow dispersion of some molecules to the tube sides. Other causes for dispersion and skewing of the solute band were considered. For example, turbulence is not usually encountered in open tube s for Reynolds numbers < 1800 [118). It was calculated that the linear velocity would have to be 499 cm / sec to achieve Re= 30 for the experiments with methane a nd pentane in the 0.068 cm i.d. open tube. For pentane in the 0.172 cm i.d. open tube, a linear velocity of 198 cm / sec would be necessary to achieve Re= 30. Values of v from Figure 77 indicate that the experimental conditions were well out of the turbulent region. PAGE 320 297 Since both th e 0.068 cm an ct 0.17 2 cm t ub es we r e coiled (1 4 coil s of r adi u s 7.5 c m and 5 coils of radius 8 5 c m, respectiv e ly) the magn itud e of effect s due to un e qual pat h l engt hs and pressure g radients a l ong the in s ide and o ut side stream pa ths w ere in spected Gid dings [8 4 ] has show n that contribu t ions o f sample 4 2 b roa d ening d u e to c oil in g in op e n tubes is proportional to (vR ) / (D R 1 ), 0 g whe re vis the aver age cro ss secti on a l lin ea r velocity, R is the 0 tube r adi us, D is the binary 1nolecular diffusion co effic ient of the g solute a nd carri e r gas, and R 1 is the coil radius. Foi the studies with methane and p e n ta ne in the 71 4 cm open tube, the coiling contri butions to ~ we r e 1.5 x lo6 cm and 5 x l08 cm, re s pectively. Coiling -6 incre ase d by 2. 2 X 10 cm for the 314 cm tube with p e ntane. Indeed, these are negligible contributions, and these obs ervations are in line with others r s4,119] w ho have o bse rved enh a nced efficiency for columns with relatively large coil r a dii and rel a tively small internal r adii Here we see that even for extremely fast 1 inear velocities, "race-track" coil ing effects on the primary flow are insi g nific a nt. It should be mentioned that the pressure drop through the 714 cm tube was only a few mm Hg for methane sampling and negligible for both studies with pentane so that solute br oadening d~e to decompression of the carrier gas was not important. Several authors [119-121] have r eported the influence of second ary flow when using liquids or gases in packed columns or open tubes. Secondary flow is a flow perpendicular to the direction of lon gitudina l flo~ through a tube caused by centrifugal force in a coil. It h as the action of dispersing molecule s outwardly in a lateral PAGE 321 298 direc t ion, a way from the outer bend w all, toward the inner bend wall. However, the flow patten cr eat ed consis t8 ) f 2 symmet rical secticns, both originating at the outer t~ 1he bend and flowing toward th e inner bend, such that s w irling eff e cts an actu a d e creas e in solute b a nd bro adening This is bec a,:se the so l ute mol e cul es cro ss the variable column velocity streamlines w ith equal frequency, but at dif fe r ent times. From experimental o bserva tions ~1211, it c an be deduced that secondary flow would increase skewing of the diffu se edges of the solute band, but in the op p o si te direction from that seen in usu a l chromato g raphic peaks. That is, molecule s in the center secondary flow streams are about t w ice as far ahead of the band mean as the molecules in the outer secondary flow stre am s are behind the band mean. Thus, o ne would expect ske w values < 0. 0 (ske w for G a ussian= 0. 0) instead of > 0. 0 as normally encountered in chromatography. It is clear that secondary flow tends to counteract diffusional broadening in the gas or liquid phase as well as skewing caused by viscous flow dispersion. There may be some evidence of secondary flow effects in this work, since the solute bands measured here had skew values which were very small (about 0.2). In fact, they were smaller than the skew (about 1-3) for the valve input profile (hybrid-fluidic, 100 m se c) which was established using a very short U-shaped tube (see Appendix A) and calculated at the same integration limits. This indicates that after retention times which are long relative to the injection period, the original valve input pulse can beco m e relatively narrower and more symmetric a l as column proce sse s occur, and it is probably not important PAGE 322 299 to deterrnir, e moments of th e injecti o n profile b ey ond the second. However, va lu e s for e x c ess we r e calculated to be between 2.5 and 2.9 in o. 1 1 cases, w hich indica tes th a t the pe aks we re flatter than for a norm a l distrib11tion of solute molecules ( excess= 3.0 for Gaussian di str ibution; va lve input val ues for excess (Table 17) were al ways > 4. 0). This is contr ary to Koutsky' s observations with 1 iquids at Reyno lds numb e rs > 300, where pea ks became narrower and higher, with increased m ea n times, for increased flow rates, thus exhibiting the beneficial effe cts of seconda :i. ay flow. Giddings [85] observed no sign ificant secondary effec ts for velocities less than three times a certain critical velocity v defin e d by equation 114, C for a coil radius of about 30 cm. In the thre e CA'J)eriments in this sturly, the critical velocities were 300, 99.8, and 39.5 cm / sec, while our e xpe rimental velocities were 500, 10, and 13 cm /se c (Figure 77), which are all well below 3v C According to Tijssen [120], the magnitude of the diffusion . 3 /4 1 / 4 coefficient due to secondary flow D is proportional to R 0 / R 1 sec In our cases with pentane in the co a ted and uncoated tubes, this ratio is 0.0014 and 0.0021, respectively (R 1 / R 0 was 220 and 99 for the 2 coils). Figures 74 and 76 show that t and Z are 10-20 and 100-200 times larg er respectively, for pentane in the coated tube compar ed to pentane in the uncoated tube. The factor of 1.5 for the above contribution to D can hardly account for such large differences in t and sec Z (the velocities are nearly equal). Apparently, secondary flow effects may be present to some extent but molecular diffusion a nd radial and viscous diffusion from the primary flow patterns, as explained by PAGE 323 300 Figure 7'1, account for most of t he pre sent ob s er vat i ons It is ass umed, th e n, that all trend s in peak sha pe as in dicated by the statistical mom e n ts were due to cha nges in sam ple s ize only. It appear s t hat the u 1 or mean retention ti me, the u 2 u se d in calcula ting bro ade nin g and the u 3 used in calculatin g skew in g a re increas ing ly se nsitive to di spe rsion due to open tube geometry a nd to sam p le si z e. While the me a n c a n remain constant w ithin 1-2 parts per thousand a s sample size increa se s as for the pentane experiments, broad ening 'Nill worsen until a limiting sample size is reached as in all three experiments. Skewing will w orsen continuously, as for per.tane in the uncoated tube, or until a limiting sample si ze is reached, as for p e ntane in the coated capillary. Information such as this is important for qu a ntitative analysis, especially when retention times are short and p eaks are narrow, as for fast capill a ry runs. In the deter mina tion of equilibrium, mass transfer adsorption-desorption, diffusion, and ther mo dynamic constants, which depend on accurate mea sure ment s of an inert time (1-200 sec), it is equ a lly important that instrum ent al contributions to peak shape a nd position be measured. PAGE 324 APPEl'.1JI X C CO MP UT E R PHOGRA J \'IS The follo wing off-line computel' prog r ams were w ritten in DEC s conversation a l mode ca ll e d FOCAL Reference to specific pro grams and their function s h as been marie in the text of this disser tation 30 1 PAGE 325 Program 1 Calculation of Temperatures Using a Platinum Resistance Thermometer C8K FOCAL g l969 06 06 5 06 06.t0 06 06 06-41 06 C C C s s s s T 12/1 2 /72 C~LC OF DE G CENT USING L&N PT RESISTANCE THE~ M S#l691079 AND OHMS VALUES READ FROM FLUKE 8400A DVM A=S;s DE=1SS;s K0=85;A RT X=l+DE/lOOJS RA=C~T-RO)/CA*~O) SQ=FSQTCXt2-4*DE*~A/18000) T1=CX+5Q)/C2 8 ~ /10000>;s f2=CX-S J )/C2*DE/10000l %6 Tl,T2JT ,!;GT 6-t 302 PAGE 326 Program 2 Statistical M om e nt CalculatiorE of S t ored Peaks C-B K F OCAL @ 1969 01 E 01 .t 0 4 01 l 5 s 01 1 6 s 01 1 7 T 01 1 8 s 01 w20 s 01 s 01 F 01 s Olol-42 G 01 ~4 '-! s 01oLl6 s ! P K" t. J; A C H i {" V ; A TAP td S P = U 1 ; A 0 :i. D L I M C%)"0L ... PT=F AB SCFCO M CU,V,W>> P=P + l; T % 3!" P K I! "? (PT-1000>1;5 PT=PT-1000 Z = F N E vJ < 4 i 6 0 0 ) B L = O;F I=to,2s;s Z=FN EW (2,l,W)JS BL=BL+Z B L= B L/16JT % 3 3 0 BL" B L,! !=10,PTJS X= FN E W C2,l, W )-dLJS Z=F N EW PAGE 327 03 S Ul=(Ul/YS);f 7.9 9 "1Jl"ut," Oq"dl+PS+(L1-26)*IN,!; S X=IN 03.34 s u2=o;s uJ=o;s U4=o 03-40 F I=Ll,Lr;s T=X-01;5 Z=FNEWC2,l,~)*Tt2JS U2=U2+ZJD 03 4 : 2 s 03 1-'4 s 03 9 Lj 6 T 03.4g s 3.9 U2=J : ~/YS;S SD=F'SQTCU2>JT "U2"U2," SD"SD,! U3=U3/fSJS AS=u3/CJ113)JS SK=J3/F'SQTCJ2f3) "UJ" U3," SP l\S y, 1'' AS," S.K" .SK, LJ4=J4/YSJS E X= J4/(U2t2)JT ".J4"i.J4," EX"EX,!;G 4 0 03 S Z=Z*TJS U3=03+z;s Z=Z*TiS U4=U4+ZJS X=X+IN 04 S ?T=FABSCFCOMC1,0,0))JG 1 304 PAGE 328 Example of Output from Program 2 PK~2 CH ~ :39 TAPE:1 OHD L li'1 C I. ) : 2 PK # = 2 BL= 124 FLO W = 12 TEMP= 54 FREQ= 0 SEC PK MAX= 907 # PTS = 271 START= 188 MAX= 204 END= 243 CUTOFF%= 2 WHH= 15 UO= 16094 700 U1= 19 U2 = 76 447 5000 U3= 709 U4= 272 8 5 O K = 208 SD= 8-43000 SP ASYM= 0~08971860 EX= 4 SK= 1 305 PAGE 329 Progr am 3 Lea se Sq u ares Fit to a Po l ynomial C-SK FOC4L @ 1969 01 i,' "' 01 .07 T 01 F 0 t 1 0 T 01 1 l F' 01 1 5 F 01.20 Q 02 l 7 s 02 F 02.22 F !!;A ?NA?,?N B ?,?Lt?;T !! J 1=1,L1; T !;A X (Jl),Y(Jl) !" DA.TA POINTS:", !!," NO Y", Jl=t,Lt;T % 2 ,Jl," ",%,XCJt)," L=N A ,N 8 ; DO 2 N2=2*L-1 Jl=t,N2;s SX(Jl>=O Jl = t,L; S YX PAGE 330 307 1 5 l G 15 15 :r K=KK, L; S B C N ) =B C N>-A CN+K L) *B CK)/ACN+ N* L) 15 R PAGE 331 Program 4 Newton-Raph so n Techn ique for So lvi ng the Perme a bility Equ ati on for External Porosity C-8K FOC A L @ 1969 01 $10 E 308 01 A "INITI/'.\L PSILON ",E," Pl::rhwlEA ci lLITY",B," AVE P A:'?.T SIZE",DP, ! 01 T "IT ERATIO,\J EPSILON",!! 02e05 S N=l 02 S K=CDP )t2/(180 *B ) 02 S ER= Cl-E>t2 03 03 03 0 03 50 03 s s s T s 'f=< ,K*Et3)/E i D-l DY~ PAGE 332 Program 5 Calcula t ion of Binary Dif fus io n C oeffi cients frcm the Hirschfeld e r Eq uation 309 C-81 { FOCAL @ 1 969 01 ~20 E 01 30 A 01 diO A. 01 5 0 A 01 T 01 070 i\ 01 80 s S IGl" S liA "SIG 2 s2;s S=(Sl+S2)/2 E 1 / K .C:: 1; A E2 / i\" E : 2; S EK=?SJT < *E :2) "D EG cri ;s T2 =273 2 + T US Kf=T2/EK;T %-4o04 "Sl2" S ," E: 1 2 / K' t~K-" KT / "KT, ''O t1.2:GC\ Oi'1H\ 1' 1 <\l"l'11 ;A M2 ,l2 ;A AT('i "P, A8=*CFSQT CT 2t3* CC1/ M l>+Cl/ M 2))))/CP*St2*0 M) 01 T %11 AB, "S Q C i'1 /SEC",!;GT 1 PAGE 333 Program 6 Least Squar es I te rati v e Fit t c an E1uation of the form. E = A + B / v + Cv C-8K F'OC A L @ 1969 01'1 6 C 01 .07 C 01 8 G 01 .io s 01 1 5 F 0 l 5 s 0 1 0 l'~ 01 -35 s 01 F 01 .45 F' 01 a 5G s 01 55 I 01 F 01 !' 01 T 01 F 5/ 1 3/7 3 ITE ~A TIVE LINEA rl L E AST S QUAR ES FIT TO EQ N OF GEN E RAL F OR M Y= A +B/X +C*X 2 03 L=3JS N=L-lJS I=-1 K=O,N;S R.(K) = K+l iv J=tE-6 J=O,NJF K=O NJD 4 R (P) =O K=O,LJS Y =Y(P+L*K+t)/M J=O,N;D 5 I=I+l < I N) 1 2 5, 1 60,, 1 2 5 J=O,NJF K=O,NJD 7 K=O,NJT ll,"tC("K+l,"' >", 1. 8,XCK) !!JT ?S M ?,!!JS S M =O V=l.,3JS C C V )=C( V)+X |