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 L' AM'C^Kji 4wi? rC RM E54F11 .. ....... . RESEARCH MEMORANDUM * : :: : i" EXPERIMENTAL HEATTRANSFER AND FRICTION COEFFICIENTS FOR AIR FLOWING THROUGH STACKS OF PARALLEL FLAT PLATES By Eldon W. Sams and Walter F. Weiland, Jr. Lewis Flight Propulsion Laboratory Cleveland, Ohio UNIVERSrTY OF FLORIDA :24 DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE UBRARY PRO. BOX 117011 : GAINESVILLE, FL 326117011 USA NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WASHINGTON August 19, 1954 :,i ~: ; i: i..i '' ;;~; Li:;: : . 'I i ' T., ..6i ii: i :. ;' .. :, ." k I I :1 NACA RM E54F11 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS RESEARCH MEMORANDUM EXPERIMENTAL HEATTRANSFER AND FRICTION COEFFICIENTS FOR AIR FLOWING THROUGH STACKS OF PARALLEL FLAT PLATES By Eldon W. Sams and Walter F. Weiland, Jr. SUMMARY An investigation is being conducted at the NACA Lewis laboratory to obtain forcedconvection heattransfer and pressuredrop data for flow of air between electrically heated parallel flat plates stacked to form passages of short lengthtoeffectivediameter ratio. Two such stacks of plates were alined in series in the direction of air flow, with a 1/4inch spacing between plates in each stack. Data were obtained for three gap spacings between stacks of 1/32, 1/8, and 1/4 inch, as well as for various degrees of plate misalinement between stacks, and also with the upstream stack removed from the tunnel. The primary purpose of the investigation is to determine the interference effects of the upstream stack of plates on the heattransfer and friction character istics of the downstream stack. Data were obtained over a range of Reynolds number from 15,000 to 80,000, average surface temperatures from 6610 to 6830 R, heat fluxes up to 8080 Btu per hour per square foot, inlet air temperature of about 5180 R, and inlet pressures up to 45 inches of mercury absolute, and with heat addition to the downstream stack only. The average and local heattransfer coefficients obtained for the downstream stack with the two stacks alined were slightly higher than the values predicted from established data on round tubes for the lengthdiameter ratio used herein. Also, the effects of changes in plate misalinement and gap spacing between stacks were found to be neg ligible. The friction factors for the upstream stack, when fully cor rected for nonfiction losses and lengthdiameter ratio effects, were in good agreement with data for smooth round tubes; for the downstream stack, the friction factors were lower than those for smooth tubes when a uniform velocity profile at the entrance was assumed, and higher than those for smooth tubes when a fully developed velocity profile at the entrance was assumed in the correction of the data for entrance effects. p' NACA RM E54F11 INTRODUCTION The present investigation was undertaken to obtain forcedconvection heattransfer and pressuredrop data for flow of air between electrically heated parallel flat plates forming passages of short lengthtoeffective diameter ratio. More specifically, the program was designed to provide these data for successive stacks of plates wherein the effects of such configuration variables as distance between parallel plates, gap spacing between successive stacks, and degree of misalinement between plates in successive stacks are to be evaluated for a range of Reynolds numbers, plate surface temperatures, and heat fluxes. These data are of interest in the design of heat exchangers of sim ilar geometries (and plain or interrupted surfaces), where advantage can be taken of the increased heat transfer associated with flow in passages of short lengthdiameter ratio. In addition to average heattransfer values, the design of heat exchangers may require a more detailed knowl edge of local surfacetemperature gradients or limiting local tempera tures, in which case recourse must be made to local heattransfer values. Both local and average heattransfer coefficients are reported herein. In the present investigation, heattransfer and pressuredrop data were obtained for two stacks of parallel flat plates with a 1/4inch spacing between plates in each stack, various degrees of plate misaline ment, and various gap spacings (1/32, 1/8, and 1/4 in.) between stacks, as well as with the upstream stack removed. The data reported herein were obtained with heat addition to the downstream stack only. Data were obtained over a range of Reynolds numbers from 15,000 to 80,000, average surface temperatures from 6610 to 6830 R, heat fluxes up to 8080 Btu per hour per square foot, an inlet air temperature of about 5180 R, and inlet pressures up to 45 inches of mercury absolute. The results are presented herein in the form of curves of average values of a NusseltPrandtl number relation (Nu/Pr0"4) against Reynolds number, of local heattransfer coefficients and plate temperatures against distance from leading edge of plate, and of average friction factors against Reynolds number. APPARATUS Air and Electrical Systems A schematic diagram of the test section and related components of the air and electrical systems used in this investigation is shown in figure 1. NACA RM E54F11 Air system. As indicated in figure 1, air at 110 pounds per square inch gage passed through a pressureregulating valve, a filter, and an orifice run consisting of an air straightener and an A.S.M.E. type flatplate orifice where the air flow was measured before entering the test section. The air then passed through the test section which consisted of an approach section, the two stacks of electrically heated flat plates, and a threepass mixing tank having a thermally insulated approach, after which the air discharged to the atmosphere. A window and light source were provided upstream of the test section for visual observation of the flat plates during testing. The temperature of the air entering the test section was measured by an ironconstantan thermocouple upstream of the approach section; the temperature of the air leaving the test section was similarly measured by two thermocouples just downstream of the mixing screens in the mixing tank. Electrical system. Provisions were made for electrically heating both stacks of flat plates, although only the downstream stack was heated for the data reported herein, the electrical system for each stack being separate and independently controlled (see fig. 1). Electric power was supplied to each stack through a variable transformer and a power trans former, the latter being connected by flexible cables to bus bars which were fastened to the two outer plates (top and bottom) of each stack; the plates in each stack were connected in series. The capacity of the electrical equipment for each stack was 12 kilovoltamperes at a maximum of 12 volts across the stack. Test Section Installation. A schematic diagram of the test section is shown in figure 2(a). The two stacks of flat plates were independently mounted in a steel tunnel provided with micrometer screws to permit vertical movement of the stacks for plate alinement and misalinement between stacks. The two stacks could be separated by 1/32, 1/8, or 1/4inch thick transit spacers having a 2 by 3inch opening in the center. Similar spacers of 3/4inch thickness were provided before the upstream stack and after the downstream stack, with a hightemperature rubber gasket recessed into the spacer to provide an air seal between stack and spacer. Transite plates (not shown) running the length of both stacks were provided between the stack and tunnel side walls with sufficient clearance to allow vertical movement of the stacks; these transit plates were grooved vertically to allow plate instrumentation leads to be brought up the side of the stack. A wooden approach section (24 in. long with 2 by 3in. opening) having a rounded entrance was located before the upstream stack. A threepass mixing tank was located after the downstream stack; the mixingtank approach section was thermally NACA RM E54Fll insulated and contacted a rubber gasket in the rear spacer. Two screens were provided in the mixing tank center passage for thorough mixing of air ahead of the exit air thermocouples. The two stacks of plates tested were identical, one such stack being shown in more detail in figure 2(b). The stack consisted of nine flat plates 'stacked vertically with a 0.25inch spacing between plates. This spacing was provided by either one 1/4inch or two 1/8inch spacer strips running the length of the plates along either side. The strips (conductors or insulators) were stacked in such a manner as to provide an electrical series connection between plates. An insulator plate and a steel support plate were provided over the two outside plates in the stack (fig. 2(a)), the entire stack being clamped together by four in sulated bolts through the stack. The bus bars supplying electrical pow er to the plates were silversoldered to the conductor strips of the two outside plates in the stack. Bosses which were provided on the bottom support plate of the stack rested on micrometer screws fastened to the bottom tunnel wall, thereby providing means for vertical movement to ob tain alinement or misalinement of the two stacks. Instrumentation. The instrumentation for each stack of plates is indicated in figure 2(b). The flat plates were 0.018 inch thick, 3 inches wide (exposed to flow), and 3.5 inches long. Ironconstantan thermocouples were imbedded in the plates by cutting grooves 0.014 inch wide and 0.008 inch deep into the plate at right angles to the direction of flow. The ironconstantan wires (0.005in. diameter including insu lation) were then laid in the groove and covered with an insulating ce ment. The thermocouple leads from the edge of the plate were secured to the outer edge of the spacer strips (between plates) and then brought up the sides of the stack. Thermocouples were placed in the various plates of the stack at the locations indicated in figure 2(c). The number of thermocouples was progressively decreased toward the outer plates, giv ing a total of 68 thermocouples in each stack. Voltagedrop leads were also silversoldered to the conductor strips at the edges of each plate (the conductor strips being spot welded to the individual plates) in such a manner as to obtain the voltage drop across each plate. Overall voltagedrop readings were also obtained for the stack. Staticpressure taps were located in each of the three transit spacers as indicated in figure 2(a), the static taps being placed at the center of the bottom and side surfaces of the opening in each case. The various pairs of static Iaps across each stack were read separately on Utube water manometers to obtain the pressure drop across each stack. NACA RM E54Fll A photograph of the complete test section is shown in figure 3. All thermocouple and voltage drop leads were brought out between two rubber gaskets under an access plate on top of the tunnel. The access plate covered both stacks of heated plates; it also provided seals for the bus bars leaving the tunnel and fittings for insertion of probes into the spacer openings. METHOD OF CALCULATION Evaluation of average plate and stack temperature. As previously indicated in figure 2(c), thermocouples were located down the center line of the plate at various distances from the plate leading edge with a similar row of thermocouples along a line 3/4 inch from the center line toward either side of the plate. The plate temperature gradients normal to the direction of air flow were found to be small compared to gradients in the flow direction; therefore, the temperature gradient for the plate can be well represented by plotting the plate center line tem perature against distance from the leading edge. The gradients for each plate in the stack were plotted in this manner; where center line ther mocouples were not available (see fig. 2(c)), the average of the two side row thermocouples was used. The average temperature for each plate was then obtained by dividing the measured area under the curve of the plate center line temperature against distance from leading edge by the plate length. The average surface temperature for the entire stack was then taken as the arithmetic average of the various average plate tem peratures weighted according to the surface area exposed to flow. (The two outside plates were exposed to flow on one side only.) Average heattransfer coefficients. The average heattransfer co efficient for the stack was computed from the relation (symbols are de fined in appendix A): avt (Q/S)avst WCp(T4 Tl)st/Sst (Ts Tb)av,st (T Tb)av,st The average bulk temperature Tb,av was taken as the arithmetic average of the inletair and exitair total temperatures, T1 and T4, respec tively. The heattransfer surface area S was taken as the total sur face area of the plates exposed to flow. The exposed surface area con tributed by the spacer strips between plates is not included, but would have a negligible effect on h. The values for physical properties of air used herein are presented in figure 4. NACA RM E54Fll Local heattransfer coefficients. Local heattransfer .:effi cients for the center plate (plate 5) in the stack were evaluated in the following manner. The local heattransfer coefficient at any dis tance from the leading edge was taken as (Q/S)z (Q/S)av P/Pav S (Ts Tb) (Ts Tb) ) where all terms in the equation apply to the number 5 plate only. For the number 5 plate, (Q/S)av was obtained by the relation (Q/S)av,P = (Q/S)av,st Ep/Est (3) Tb, was assumed to vary linearly with X (distance from plate leading edge) between TI and T4; Ts was obtained from the plot of local plate temperature against X as explained in the previous section. This curve was also used to evaluate PZ/Pav in equation (2) in the following manner: Inasmuch as the voltage drop across the plate is es sentially constant at any distance from leading edge X, and since heat conduction along the plate can be neglected, the local power generation at any point X is given by P, = Ep2/RM = C/r1 (4) That is, the local power generation at any point (distance X from leading edge) in the plate is inversely proportional to the electrical resistivity r at the point; hence from a curve of electrical resis tivity r against temperature for the plate material and the curve of Ts,Z against X, a curve of l/rz against X was plotted for the plate. By graphically integrating this curve to obtain 1/ray, the ratio of l/r, to 1/ra (which also represents PZ/Pav) was then plotted against distance from leading edge X. The local heattransfer coefficients hz for the number 5 plate could then be computed from equation (2) and plotted against X. Average friction coefficients. The method used for computing the average friction coefficients for the individual stacks is as follows: The friction .:.,ff1icients are based on measured staticpressure drops which were corrected for entrance, exit, vena contract, and momentum losses. The equations used in calculating these losses are derived in asindix B. The values of measured pressure drop, as previously ex plained, were obtained from several pairs of static taps located in the transit spacers on either side of and between the stacks (fig. 2(a)). The pressure drop across the upstream stack was taken as the average of the pressure drops measured by the taps in the two sides of the spacer NACA RM E54F11 opening. Because of slight surface interruptions between stacks, static probes were used in measuring the drop across the downstream stack. The equations derived in appendix B result in the following evaluation of friction factor: The fully corrected average halffriction factor is defined as (f/2) corr corr2 (5) L G De 2gpav where p was evaluated with the total temperature and the static pressure in the center window opening (negligible error for range of conditions investigated); and Pcorr = meanss (en + ex + Apmom + v (6) where: G22 en = (1 F) (7) APex = G12 1 + 1)() 2 G22 P + a (9) A"Pmom gPI (9) pmam= gPl 1 a and: G22 Pvc = K2 (10) 2go1 where Kc is a function of freeflow factor F for the stack, the value of which was obtained from reference 1. The measured average half friction factor, also used herein, is similarly defined as: S means (11) 2/meas L G2 D8 2p De 2gpav NACA RM E54Fll PROCEDURE As previously noted, two stacks of flat plates were used in this investigation. The plates were spaced 1/4 inch apart in each stack with either a 1/32, 1/8, or 1/4inch gap between stacks. The two stacks of plates were first alined in the direction of air flow, and data were ob tained at various Reynolds numbers. The average stack temperature level reported herein was that which resulted from limiting the highest local temperatures (trailing edge of outside plates) to a safe operating value for the ironconstantan thermocouples used. Data were similarly ob tained for two degrees of plate misalinement, which were obtained by mov ing the upstream stack vertically. For the case of slight misalinement, the upstream stack was moved a distance equal to one plate thickness (top of plate in upstream stack level with bottom of plate in downstream stack); while for complete misalinement, the upstream stack was moved by onehalf the distance between plates. RESULTS AND DISCUSSION Plate Temperature Gradients Typical plate temperature gradients obtained for the downstream stack are presented in figure 5, where local surface temperature Tsz is plotted against distance from plate leading edge X. The plates are numbered from top to bottom of the stack, but are listed in an order of symmetry starting from the center plate and moving toward the outer top and bottom plates. The curves for the three cen termost plates are essentially coincident, while the next two plates on either side of the three centermost plates show slightly higher tempera tures at the trailing edges. The curves for the two outside plates fall considerably above the other curves, particularly at the trailing edge; this would be expected inasmuch as the top and bottom plates are cooled on one side only (see fig. 2(a)). The dashed lines, for the four outer most plates, represent an approximation of the temperature gradient, since thermocouples were located only at the leading and trailing edges of these plates. The average plate and stack temperatures were obtained from these curves as indicated in the section METHOD OF CALCULATION. HeatTransfer Data Average heattransfer coefficients for stack. The average heat transfer coefficients obtained for the downstream stack are [resented in figure 6 where the NusseltPrandtl number relation Nub/PrbO.4 is plotted against Reynolds number DeG/4b. Included for comparison is NACA RM E54F11 the McAdams line (solid) which was found to best represent the data of various investigators (ref. 1) for fully developed turbulent flow in smooth tubes; the equation is: Nub DeG 0.8 N"b =0.023 (. (12) Prb0.4 b (b The predicted line (dashed) in figure 6 includes a correction to the conventional McAdams equation (eq. (12)) to account for the increase in average heattransfer coefficient to be expected with passages of short L/De (7.6). This correction was made in reference 2 by inclusion of a power function of L/De in equation (12), the equation being: Nub 0.034 ( 0.8 L0.1 (13) =0.034 (13) Pr 0.4 \ ) De which, for L/De = 60, becomes equation (12) and, for L/De = 7.6, becomes Nub /DeG\0"8 b 0.028 (e\) (14) Prb0.4 4b I represented by the dashed line in figure 6. The magnitude of this in crease in the heattransfer coefficient is further verified in reference 3 (investigation of average and local heattransfer coefficients as functions of Re and X/De for smooth entrance flow through tubes and between parallel flat plates) for the case of uniform heat flux with uniform initial temperature and velocity distributions. The average heattransfer coefficients for the downstream stack (fig. 6) are shown for the case of stacks (plates) alined in the di rection of air flow with various gap spacings (1/32, 1/8, and 1/4 in.) between stacks, as well as with the upstream stack removed from the tun nel. The data for all configurations are in reasonably good agreement (about 15 percent high) with the predicted line. The effect of gap spacing and removal of the upstream stack is seen to be negligible. The effect of plate misalinement is shown in figure 7. With the two stacks completely misalined as described in PROCEDURE, the top in sulator plate of the upstream stack (see fig. 2(a)) partly blocks off the top passage of the downstream stack. Hence, in order to compare the data for various degrees of misalinement, the average heattransfer coefficients for the three center plates of the downstream stack were NACA RM E54F11 used in each case. The results are shown in figure 7, with the same co ordinates as in figure 6. The data for the alined condition fall slightly higher than in figure 6 inasmuch as the electrical heat input was used in evaluating Q for the three center plates. These data are compared (fig. 7) with those for the completely misalined and slightly misalined conditions for the various gap spacings investigated. The effect of plate misalinement also appears to be negligible. Visual observation of the plates during testing indicated that the center seven plates of the heated stack remained flat and perfectly alined with the first stack; the two outside plates, which were cooled on one side only, showed slight bending. Local heattransfer coefficients for center plate. The local heattransfer coefficients obtained for the center plate of the down stream stack are presented in figures 8 to 11. In figure 8, hi is plotted against X for the case of plates alined with various gap spacings between stacks, as well as with the upstream stack removed; this comparison is shown for two values of Reynolds number. As seen in figure 8, gap spacing and removal of the upstream stack have no appreciable effect on the local coefficients, as was also shown for the average coefficients in figure 6. As explained in METHOD OF CALCULATION, the local heattransfer co efficients were calculated with the assumption of no heat conduction along the plate. The effect of heat conduction can be taken into ac count by use of the following equation (all terms are for the center plate): (+ bkm 2T (15) S )Z = (g dX2 where the first term (right side, eq. (15)) is defined in equation (2), and the second term accounts for heat conduction along the plate. The corrected values of h, can then be computed; the uncorrected values of hz obtained at the highest Reynolds number with the upstream stack removed from the tunnel are compared with the corresponding corrected values of h1 in figure 9. The effect of conduction along the plate is seen to be small; hence, the effect of conduction is neglected in subsequent discussion. A comparison of the local heattransfer coefficients obailed with the upstream stack removed, and the predicted values of reference 3 is shown in figure 10 as a variation of local Nusselt number Nuj with NACA RM E54F1l X/De. The values obtained from reference 3 (dashed lines) are for the case of a gas (Pr = 0.73) flowing between parallel flat plates with the conditions of uniform heat flux, constant properties, and uniform tem perature and velocity distribution at the entrance. The predicted curves are shown for essentially the same values of inlet Reynolds num ber Rei and heatflux parameter as obtained in the present investiga tion. At the lowest Reynolds number, the experimental values (solid lines) are in good agreement with those predicted, although slightly higher near the leading edge. With increase in Reynolds number, however, the experimental values are somewhat higher throughout as also indicated by the average coefficients (downstream stack only) in figure 6. The effect of plate misalinement on the variation of local heat transfer coefficient along the plate is shown in figure 11 where hZ/hav is plotted against X. The data are shown for the cases of plates alined, slightly misalined, and completely misalined at a Reynolds number of about 40,000 and a 1/8inch gap between stacks. Figure 11 indicates that plate misalinement results in slightly higher values of hj/ha at the leading edge and slightly lower values at the trailing edge. The average heattransfer coefficient for the center plate was essentially the same for all degrees of misalinement, as was also shown in figure 7 for the three center plates. Friction Data Average friction factors for upstream stack. The average half friction factors obtained for the upstream stack are presented in fig ure 12 wherein halffriction factor f/2 is plotted against Reynolds number DeG/ib. Included, for comparison, is the KarmanNikuradse line (solid), representing the relation between friction factor and Reynolds number for turbulent flow in smooth pipes, which is S1 = 2 log (Re V f/) 0.8 (16) In comparing the data presented herein with those for fully devel oped turbulent flow (reference line), the effect of L/De must be taken into account; this effect consists of (1) a momentum loss in the stack associated with transition from a flat velocity profile at the entrance to some degree of fully developed turbulent velocity profile at the exit, and (2) an increase in average friction factor associated with short L/De wherein the entrance effect (region of high local shear stress and friction) may exist over a considerable portion of the pas sage length. These effects are accounted for in the analysis of ref erence 3 which predicts friction factors for flow between parallel flat NACA RM E54F11 plates with uniform velocity distribution at the entrance as a function of X/De and Reynolds number. Reference 3 indicates that the ratio of average friction factor for X/De = 7.6 to the friction factor for fully developed turbulent flow is about 1.54 for the range of Reynolds numbers investigated herein. The predicted line (dashed, fig. 12) represents the reference line (solid) corrected by this ratio. Two different values of friction factor are presented in figure 12 for the present data: (1) friction factor based on measured static pressure drop, and (2) friction factor based on measured pressure drop fully corrected for entrance, exit, momentum, and vena contract losses as explained in the section METHOD OF CALCULATION. The measured values of f/2 fall slightly above the predicted line. When the various losses are taken into account, the data fall within 8 percent of the predicted line. Average friction factors for downstream stack. The average half friction factors for the downstream stack are presented in figure 13 in the same manner as that used for the upstream stack in figure 12. The friction data presented are for a 1/8inch gap between stacks. The friction factors based on measured pressure drops fall near the pre dicted line; but when the various losses and the same L/De correc tion used for the upstream stack is taken into account, the data for the downstream stack fall considerably below the predicted line. The L/De correction, as previously indicated, is for the case of uniform velocity distribution at the stack entrance. This condition is satisfied for the upstream stack (fig. 12), but it is questionable as to whether the 1/8inch gap between stacks is adequate to completely flatten out (be fore it enters the downstream stack) what is essentially a fully devel oped velocity distribution at the exit of the upstream stack (indicated by ref. 3). The results of figure 13 overcorrectionn of the data) indi cate that complete flattening out does not occur. Average friction factors for both stacks combined. It might be of interest to compare the results in figure 13 with those for the case where no flattening out of the flow at the entrance to the downstream stack is assumed to occur, that is, where the two stacks are considered as a continuous passage. The data for the latter case are presented in figure 14, where the fully corrected friction factors are based on meas ured pressure drops across both stacks corrected for entrance, exit, mo mentum, and vena contract losses, and with the reference line corrected for an L/De = 15.2 (dashed line). Friction factors based on measured pressure drops are again included. The fully corrected halffriction factors now fall slightly above the predicted line (data undercorrected). Hence, a comparison of figures 13 and 14 indicates that the velocity distribution at the entrance of the downstream stack is neither uniform nor substantially fully developed as at the exit of the upstream stack. NACA RM E54Fll SUMMARY OF RESULTS The results of tests to obtain heattransfer and pressuredrop data for flow of air between electrically heated parallel flat plates, with various gap spacings and degrees of misalinement between stacks, can be summarized as follows: 1. Average heattransfer coefficients obtained for the downstream stack fell considerably above the conventional McAdams line representing average coefficients for fully developed turbulent flow in smooth tubes of lengthdiameter ratio L/De approximately equal to 60. When the McAdams line was corrected to account for L/De effects (L/De = 7.6), the data gave reasonably good agreement with the predicted line (about 15 percent high). The effects of gap spacing and plate misalinement between stacks were found to be negligible. 2. Local heattransfer coefficients obtained for the center plate of the downstream stack (with upstream stack removed) were compared with those predicted by analytical methods for flow of a gas (Pr = 0.73) be tween parallel flat plates for the case of uniform heat flux and initial temperature distribution, and uniform velocity distribution at the en trance. The experimental and predicted values were in fairly good agreement at the lowest Reynolds number, the experimental values becom ing somewhat higher with increase in Reynolds number. 3. Average halffriction factors for the upstream stack, when fully corrected for entrance, exit, momentum, and vena contract losses were in good agreement with a predicted line; which was based on the Karmin Nikuradse line representing average friction factors for fully devel oped turbulent flow in smooth pipes with suitable correction for L/De effects. The data fell within +8 percent of the reference line. 4. Average halffriction factors for the downstream stack fell be low the predicted line when a uniform velocity profile at entrance was assumed and above the predicted line when a fully developed velocity profile at the entrance was assumed. Lewis Flight Propulsion Laboratory National Advisory Committee for Aeronautics Cleveland, Ohio, June 14, 1954 1ACA RM E54F11 APPENDIX A SYMBOLS The following symbols are used in this report: A freeflow area, sq ft b plate thickness, ft C constant c specific heat of air at constant pressure, Btu/(lb)(OF) D effective diameter of stack passage, 4A/Z, ft E potential difference, volts F freeflow factor (free flow area/total frontal area) (f/2) average halffriction factor based on measured static meas pressure drop (f/2) average halffriction factor based on measured static corr pressure drop corrected for entrance, exit, momentum, and venacontracta losses G mass velocity (mass flow per unit crosssectional freeflow area) W/A, lb/(hr)(sq ft) g acceleration due to gravity, 4.17x108 ft/hr2 h heattransfer coefficient, Btu/(hr)(sq ft)(OF) Kc vena contract pressureloss coefficient k thermal conductivity of air, Btu/(hr)(sq ft)(F/ft) ~ thermal conductivity of plate material, Btu/(hr)(sq ft)(0F/ft) L length of stack passage, ft P power .erLe ration, watts p static pressure, lb/sq ft abs NACA RM E54Fll 15 Pmeas measured staticpressure drop across stack, lb/sq ft 4corr measured staticpressure drop across stack corrected for entrance, exit, momentum, and venacontracta losses APen entrance pressure drop, Ib/sq ft Apex exit pressure drop, lb/sq ft Pmomn momentum pressure drop, Ib/sq ft APVc vena contract pressure drop, lb/sq ft Q rate of heat transfer to air, Btu/hr Q' rate of heat transfer to air (corrected for heat conduction along plate), Btu/hr R gas constant for air, ftlb/(lb)(oF) R' electrical resistance, ohms r electrical resistivity, ohmcm S heattransfer surface area, sq ft T air total temperature, OR Tb air bulk temperature, OR Ts surface temperature, OR t static temperature, OR At statictemperature difference between entrance and exit of stack, OR W air flow, lb/hr X distance from plate leading edge, in. Z wetted perimeter of stack passage, ft Pmeas/P1 3 At/t1 16 NACA RM E54F11 . absolute viscosity of air, lb/(hr)(ft) p density of air, lb/cu ft Dimensionless parameters: Nu Nusselt number, hDe/k Pr Prandtl number, cp_/k Re Reynolds number, DeG/p Subscripts: av average value b physical properties evaluated at bulk temperature i physical properties evaluated at inlet temperature 1 local value P plate st stack The following diagrammatic sketch of the stack defines the subscripts pertaining to various stations: 12 34 I Air flow _' I Plates NACA RM E54F1l APPENDIX B DERIVATION OF PRESSURE LOSS EQUATIONS USED HEREIN A schematic diagram of the test section (stack) showing the various stations is given below: ,rF Pmeas Air flow !Ii ~i'l ! Entrance pressure loss APen" A pressure loss occurs in the flow stream in the entrance region between stations 1 and 2. Assuming no friction and fluid incompressibility (pl = p2), the general energy equa tion between these stations can be written: PlV12 Pl + 2g PlV22 = 2 2g Also, Al V2 A2 V hence, 2g A2 1 Apen = PI P2 = Rewriting equation (B2) in terms of the mass velocity G flow factor F (defined as A2/Al) results in: lpen 2 (1 F2) ,1 and the free (B3) (Bl) (B2) NACA RM E54Fll Exit pressure drop Apex. The momentum change between stations 3 and 4, where an expansion occurs, can be stated as follows: W V3 + 3A4 W V4 + 4A4 (B4) g g Assuming incompressibility, equation (B4) becomes: 3 3 A A_ pex = P3 P = g(5) As before, A2/A1 = F and A3/A4 = F; therefore, 2 Apex= (F2 F) (B6) APex gp3 In order to write equation (B6) in terms of entrance conditions at sta tion 1, the following was assumed: P3 P A7) 3= Rt3 R(t + At) (B7) The Ap used in equation (B7) was approximated by using APmeas to avoid a trialanderror solution. The error incurred by this assumption is negligible. From equation (B7), equation (B6) can now be written: G2 Ap = G 1 + (1 1/F) (B8) ex gp 1 a where a = Ap/pl (B9) and P = At/t (B10) NACA RM E54F11 Venacontracta pressure loss Apvc. The venacontracta pressure loss was taken as a factor Kc times the dynamic pressure in the test section as given by 2 G2 pv = Kc (Bll) e 2gp1 where Ke is a function of the freeflow factor of the test section; values of K were obtained from reference 1. Momentum pressure loss APmom. Using the general momentum equa tion and assuming pl = p2 result in the following equation G2 /1 1\ pmo = ( (B12) ;nom g \Pg P3 Rewriting equation (B12) with the aid of equations (B7), (B9), and (B1O) gives G22 3 + Pmom g gpl 1 REFERENCES 1. McAdams, William H.: Heat Transmission. Second ed., McGrawHill Book Co., Inc., 1942. 2. Humble, Leroy V., Lowdermilk, Warren H., and Desmon, Leland G.: Measurements of Average HeatTransfer and Friction Coefficients for Subsonic Flow of Air in Smooth Tubes at High Surface and Fluid Temperatures. NACA Rep. 1020, 1951. (Supersedes NACA RM's E7L31, E8L03, E50E23, and E50H25.) 3. Deissler, Robert G.: Analysis of Turbulent Heat Transfer and Flow in the Entrance Pegicns of Smooth Passages. NACA TN 3016, 1953. 04~.04 0i4040 m m3 a' a a> C)I  EC O a r .I t" Q. Q  A * 04 0u4.4. *> 1i( 040404 040i40 >4m>4> .0.0.0 400404 (04040 444444 040404 0.0.0. 0.0.0. 0C40404 020202 .0.a.0 000n 444DG 444444 ai Uai Sdi O? 000 ouoo 040404 040404 0)0)0) >4>4>4 040404 040404 0.... 0)0)2 000 040404 0 U 0 c c c 40404 0.0.0 04... .0.0.0 000r a', o M rc r. E.0.0. 040404 020X02 .0.0o. 040404 E E E .0. 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Da ijiai 3>y>^T ci I0)0 ' 0) 000'i .0 000 000 000 000 >00 0 0 000 000 0  5. 04 <. i 1 04<)  0< '0' 0o 00* "" )cr C Q, y r ". 9 m if r na r ~ vo rfi s . c rs~ rt o ~ F< *0 r .0) 44024, .K 0 '> .. 000 "'. 40 1,,.. 44j .~ ,p44 0C~ Q \ V 3 C ~ UM 0404 i 04 i 44> < . u) u Di .ni LR. U .0 024 4>4 u .) <4,.4 n L o i 0 > O 040 4. 4i .000 4> 0 000 0 4t)040 044j im I, Oiu B3VC I~iDi iffltDU r~ (iSUtO > 0.0.&* 24o 044 4 o w o mo ',.ao o o4 <"0. '04004 '020> 40 'o 044.04l i .<000 4>0 4 0 4 4>40 i o00 00~0 40440 0 041o04.. 044 000 2 < 0~m 0i4 04, m,.  4...a 4 '444 u 02. 04) 04. '004. .2 iO l 0 .2 00 000 0204044 1 C liO 0 000 000 044440 02.4.rrl '4U ~ &~O0 0' i7~ 02020 )ic Ir :" 4 4 447 4447047i(D~ 0404.I~ r7 04: 4. NACA RM E54011 NACA RM E54F11 o 1 0 ) ,4 I P4 m 0 4 o 44 ] r o 0 4 $ 0 p 0 r4 0 21 Z0 g  41) $4 20 o 0) / a * P, 0 $4 0 P, v 1 Cl 20 0 0 cl ,0 I  0) 21 I\ 1 4'4'\  22 NACA EM E54F11 o r a 0 FO 'q a f  4 9 0 00 0 0 E4.) m 0, x 0 030 0 vi a 43 4 Id, m ( 1 0 0O mo o ;4 (o 4) ct 0 4' 4 4 0 00 a4  04 C 0 S 4 l 4  4 4 I 03 ( O 0, t t I4 4 :% j 4 0 Pg NACA RM E54F11 o 0) 0) p p4 0 C, (0 o E (d CD 0~C. H) 1 CoT  t & kMf 24 NACA lM E54FII g0 r d'l IC~ r r ,4, 15 4, p4 ... 4. ~4, rH y0 I ,0 p4 '4 r1 ~4JiL V '1* .ri. " '1 .. NACA RM E54F11 .026 .022 .014 .010  .070 .0601 .050  .040 _______ 400 500 600 700 800 900 1000 Temperature, 'R .060      ^   . 0 5 0           .040 * Figure 4. Physical properties of air over temperature range 4000 to 1000' R. NACA RM E54F11  _ __ /  Plate Tsav'       ^ / OR 0 5 581 __ 0 4 649 0 6 650 V 3 660 ___ 7 654 / 2 662 S 8 656 ___1 748 17 9 728 SThermocouples located _only at leading and trailing edges of plate 1// ___ 2 o  /K  1 2 3 Distance from plate leading edge, X, in. Figure 5. Typical temperature gradient curves for plates in downstream stack. Average surface temperature for stack, Ts,av, 6650 R; air bulk temperature, Tb, 5280 R; heattransfer rate to air, Q, 5890 Btu per hour; Reynolds number, Re, 25,800; stacks alined. Tb,av (stack), OR 524532 526532 526534 524532 Condition Stacks alined; 1/8in. gap Stacks alined; 1/32in. gap Stacks alined; 1/4in. gap Second stack only McAdams reference line  Predicted line (for L/D = 7.6) 40 DeG/o 400x103 Figure 6. Average heattransfer coefficients for downstream stack as affected by gap between stacks. NACA RM E54F11 s,av (stack), OR 665672 664669 668675 670681 ,pI NACA RM E54F11 s,av (stack), OR 0 665672 0, 667673 ,0 666683 o 664669 0% 661669 )3 662674 0 668675 0(. 668674 ,, 665668 Tb, av (stack), OR 524532 526532 525531 526532 524531 524530 526534 526532 526532 Predicted line (for 8 10 40 DeG/io Condition Alined Slightly misalined Completely misalined Alined Slightly misalined Completely misalined Alined Slightly misalined Completely misalined L/De = 7.6) 100 Figure 7. Average heattransfer coefficients for three center plates of downstream stack as affected by plate misalinement and gap between stacks. Gap spacing, in. 1/8 1/8 1/8 1/32 1/32 1/32 1/4 1/4 1/4 600 400 200 z  100 80 60 406 6 /001  7 EE __ _  200x106 s,av (plate), oR Re (stack) 41,400 41,400 41,500 41,600 25,800 26,000 26,100 26,100 Condition Alined; 1/8in. gap Alined; 1/32in. gap Alined; 1/4in. gap Downstream stack only Alined; 1/8in. gap Alined; 1/32in. gap Alined; 1/4in. gap Downstream stack only 1 2 3 4 Distance from plate leading edge, X, in. Figure 8. Local heattransfer coefficients for center plate of downstream stack as affected by gap between stacks. NACA BM E54F11 Ts,av (stack), OR 665 667 669 670 665 664 668 673 648 651 649 649 651 649 654 656 O 0 A 0 D 0 A 200 160 120 NACA RM E54F11 O Uncorrected O Corrected for 180 conduction  140 O4O 0 3 Distance from leading edge, X, in. U 4 Figure 9. Local heattransfer coefficients for center plate of downstream stack, with upstream stack removed from tunnel, as affected by heat conduction along plate. Average surface temper ature, Tsa for stack, 6700 R; for plate 5, uav> 6490 R; average bulk temperature for stack, 5240 R; Reynolds number, Re, 41,600. 0C Distance from leading edge, X, in. Figure 9. Local heattransfer coefficients for center plate of downstream stack, with upstream stack removed from tunnel, as affected by heat conduction along plate. Average surface temper ature, T Sav ; for stack, 6700 R; for plate 5, 6490 R; average bulk temperature for stack, 5240 R; Reynolds number, Re, 41,600. NACA RM E54F11 500 400 X/De Figure 10. Local values of plate of downstream stack, as compared with predicted Nusselt number for center with upstream stack removed, curves of reference 3. Re s,av s,av b,av (stack) (stack), (plate), (stack), R OR OR 0 41,600 670 649 524 0 26,100 673 656 527 A 16,400 681 673 532 __Rei 040,000 25,000 3001 200 NACA RM E54F11 Re Tsav (stack) (stack) OR 0 41,400 O 41,500 A 41,700 665 669 666 Ts,av (plate) OR 648 653 640 b, av (stack) oR Condition 524 Alined 526 Slightly misalined 524 Completely misalined Distance from leading edge, X, in. Figure 11. Ratio of local to average heattransfer coefficients for center plate of downstream stack for various degrees of plate mis alinement and 1/8inch gap between stacks. 35.0 p 0 S2.6 18 0 0 0 o 2.2 sp o 4 a! 0 , aS NACA RM E54F11 KarminNikuradse line Predicted line (for L/De = 7.6) [] Based on measured pressure drop .01 O Based on measured pressure drop corrected for entrance, exit, Z momentum and venacontracta 0 losses 006 002  rn ^ __ ____     ____ __ _ 2 4 6 10 20 40 60 100 200 DeG/ub Figure 12. Average halffriction factors for upstream stack, based on measured values of staticpressure drop, and measured pressure drops corrected for entrance, exit, momentum, and venacontracta losses.  KarmgnNikuradse line  Predicted line (for L/De = 7.6) 0 Based on measured pressure drop .01 0 Based on measured pressure drop corrected for entrance, exit, momentum and venacontracta S losses .006  .004  .002 nmni   2 3 4( DeG/Hb 200 400X103 Figure 13. Average halffriction factors for downstream stack, based on measured values of static pressure drop, and measured pressure drops corrected for entrance, exit, momentum, and venacontracta losses. 4 6 10 20 DeGG/1 40 60 100 200 Figure 14. Average halffriction factor for upstream and downstream stacks com bined, based on measured values of staticpressure drop, and measured pressure drops corrected for entrance, exit, momentum, and venacontracta losses. NACALagley 6955 75 .UuF rfAI Ka'rmanNikuradse line  Predicted line (for L/De = 15.2) Q Based on measured pressure drop O Based on measured pressure drop corrected for entrance, exit, momentum and venacontracta losses  I I; IB I I I I I I 1 I I I 2 UL1  OO . 0   "%1,^    . l 400x103 T)+ r I`\ 1 1111I 400xlO3 i 5 9 r 0 0 c. e M 1. E L) CO f W< zaf, <. zi H ! o a.0 C i Q~~j w a: = 'd C,; CU Ci 1. c? J 0 Eo U PH4 (l^"lZ '^M dal ni i.a S'S a~a a& z 'U N >, a 3 ~z o o Ud. 0 U > > 0 0 c eo go co F. v W C ) co0" :QSS^ S r" M C 0C S OFrZ Z o o 2 0 13 w : C1. w U0 <~ '016 1..0~44~C ~. > z Z > r o~~~ 4.W~L ,o~~a ZZ~J 0,vrj2C 0 w YaS0 a0g, M U3& zz~ut o z W &0;9 z 0. I 0 Z o o 0 a o . OUy o 0 0. liFZa.. I^^J2~3 s~s50*o 0 4:WIS~as,, ,< "hg B u f a ~crf ct C m A 23 M t > M >,' s. 0 c S "0 0 tm 5 u UPa .0a to a.. m .~~ C CMr Cd (0 o s . p Z; 0 > U .a .0 4. 0.04 1 0 S c CE O r( C  0 ba W U 4 > E cL m E6(U3 0 0< (a. 60 . e 2"o. (4 4 CJ '~ C o .0 >fl. 0 S Snz oa, J.Saa g S? " "3 uS oO S S '1' CS ^.cO (c o (E~ (4~~U .. ^ r 8<, S i S, iL. l~E~liLP, 111 i5S1li 4m 13 r to (D sc S. h  *00 g 0. ' E Ct 0 0a V o jc4 4 44 0 0 c 9z a ii 5 hjjfjj o ~ .. 0 E w C; >4 0, *U 0 ci WU AW M < & S * >1Btc, 2.2 0 m .0 z aO m tv 06'2 t3!2 "a w EP 0 Z S:ul S _a o to  a  Sc5Vgv i p 0U 4. s o* "F a 0 4 0 1 0 1A S ZS Z B W ^ uS. ul 0 W^ v w 0 ==g .s~ s 0 4.~iS.  ~.0  lp"' !b ~ .~0. 4 . ^ipei; is!li sil, l sl l i Z~rz~cnco UNIVERSITY OF FLORIDA 3 1262 08106 565 7 UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE UBRARY P.O. BOX 117011 fAINESVII.LE, FL 326117011 USA :l "' r, 