Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UF00091083/00001
## Material Information- Title:
- Rheology and rheometry of mud slurry flow in pipes a laboratory investigation
- Series Title:
- UFLCOEL
- Creator:
- Jinchai, Phinai
Jiang, Jianhua Mehta, A. J ( Ashish Jayant ), 1944- University of Florida -- Coastal and Oceanographic Engineering Dept - Place of Publication:
- Gainesville Fla
- Publisher:
- Coastal and Oceanographic Engineering Dept., University of Florida
- Publication Date:
- 1998
- Language:
- English
- Physical Description:
- 1 v. (various foliations) : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Rheology ( lcsh )
Dredging spoil ( lcsh ) Slurry pipelines ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references.
- General Note:
- "February, 1998."
- Statement of Responsibility:
- by Phinai Jinchai, Jianhua Jiang and Ashish J. Mehta.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 42076604 ( OCLC )
## UFDC Membership |

Full Text |

UFL/COEL-98/001
RHEOLOGY AND RHEOMETRY OF MUD SLURRY FLOW IN PIPES: A LABORATORY INVESTIGATION by Phinai Jinchai Jianhua Jiang and Ashish J. Mehta February 1998 UFL/COEL-98/001 RHEOLOGY AND RHEOMETRY OF MUD SLURRY FLOW IN PIPES: A LABORATORY INVESTIGATION By Phinai Jinchai Jianhua Jiang and Ashish J. Mehta Coastal and Oceanographic Engineering Department University of Florida Gainesville, FL 32611 February, 1998 SUMMARY As a response to the need to pump comparatively high density mud slurries though dredged material discharge pipes, this study examined the question of whether relevant mud properties can be tested a priori in order to predict slurry transportation in pipes. To that end, laboratory rheometric experiments were conducted to assess the dependence of slurry flow rate on mud composition and density. Muds consisted of clays and clay mixtures of varying density. The selected clays included a kaolinite, an attapulgite and a bentonite. The choice of these clays was based on the need to vary mud properties widely in terms of their non-Newtonian rheology, as characterized by the threeparameter Sisko power-law model for apparent viscosity variation with shear rate. The overall density range tested was 1,125 kg/m3 to 1,550 kg/m3. Power-law parameters (characterizing ultimate viscosity, consistency and pseudoplasticity) as functions of mud composition and density were obtained by testing the slurries in two types of rheometers. For low shear rates, a coaxial cylinder viscometer (CCV) of the Brookfield type generating annular couette flows was used. For comparatively high shear rates, a specially designed horizontal pipe viscometer (HPV) generating Poiseuille flow was used. The latter apparatus consisted of a 2.54 cm i.d. and 3.1 m long, horizontally positioned, PVC pipe through which mud flow was driven by a piston-diaphragm pump. Rheometric results obtained by combining the CCV and HPV data confirmed the pseudoplastic (shear thinning) behavior of all slurries. For mud of given composition, the ultimate viscosity, consistency and the degree of pseudoplasticity generally increased with density. A slurry cation exchange capacity (CECIu,) is introduced as a cohesion characterizing parameter dependent on the weight fractions of clays and water in the slurry. It is shown that the power-law parameters correlate reasonably well with CECuy, which therefore makes it a convenient measure of the rheology of slurries composed of pure clays and clay mixtures. The steady-state slurry transport equation for the Sisko power-law fluid is solved numerically to yield the velocity distribution in the pipe and the corresponding discharge. To test the model, including the applicability of the derived power-law for the slurries, additional tests were carried out in the HPV at lower pumping pressures than those used to determine the power-law coefficients. It is shown that for most slurries tested the calculated discharge agrees reasonably with that measured. The approach developed in this study lends itself to further exploration as a method designed to test the bottom mud to be dredged for its pumping requirements. Support for this study from the U. S. Army Engineer Waterways Experiment Station, Vicksburg, MS under contract no. DACW39-96-M-2100 is acknowledged. Dr. T. M. Parchure of the Coastal and Hydraulics Laboratory was the project manager. TABLE OF CONTENTS SUM M ARY ............................... LIST OF FIGURES ......................... LIST OF TABLES .......................... 1. PROBLEM STATEMENT ............. 1.1 Need for Investigation ........... 1.2 Objective, Tasks and Scope ....... 1.3 Outline of Chapters ............. 2. SLURRY FLOW IN PIPES ............. 2.1 Rheological Behavior of Materials . 2.2 Equations of Motion in Pipes ...... 2.2.1 General Problem ......... 2.2.2 Poiseuille Flow Problem ... 3. RHEOLOGICAL MODELS ............ 3.1 Viscous M odel ................. 3.1.1 Flow Type ............... 3.1.2 3.1.3 3.1.4 3.1.5 3.2 Linear Apparent Viscosity ................. End and Slip Effects and Corrections .. Slurry Flow Curve ................. Some Empirical Non-Newtonian Models Viscoelastic Models ................ 3.2.1 Boltzmann Superposition Principle ............................ 3-8 3.2.2 Approximate Relationships Among Linear Viscoelastic Functions .... 3-9 4. ANALYTICAL SOLUTIONS FOR SLURRY FLOW 4.1 Poiseuille Flow Problems ................ 4.1.1 Generalized Newtonian Fluid ....... 4.1.2 Bingham Plastic .................. 4.1.3 Power-Law Fluid ................. 4.1.4 Maxwell Model ................... 4.2 Initial Value Problem and Periodic Motion ... 4.2.1 Initial Value Problem .............. 4.2.2 Periodic Motion .................. 5. EXPERIMENTAL SETUP, PROCEDURES AND MATERIALS 5.1 Coaxial Cylinder Viscometer ....................... 5.2 Horizontal Pipe Viscometer ........................ 5.3 Sediments and Slurries ............................ 5.3.1 Sediment and Fluid Properties ............... PAGES . . ii ..... ix ...... 4-1 ...... 4-1 ...... 4-1 ...... 4-1 ...... 4-4 ...... 4-4 ...... 4-4 ...... 4-5 ..... 4-10 5.3.2 Mud Slurries........................................... 5-5 6. EXPERIMENTAL RESULTS......................................... 6-1 6.1 Rheometric Results........................................... 6-1 6.1.1 Power-Law for Mud Flow................................. 6-1 6.1.2 Calculation of Power-Law Coefficients ........................ 6-3 6.2 Power-Law Parameters ........................................ 6-4 6.3 Slurry Discharge with Sisko Model............................... 6-10 6.4 Calculation of Slurry Discharge ................................. 6-11 7. INTERPRETATION OF RESULTS AND CONCLUSIONS ................... 7-1 7.1 Power-Law Correlations with Slurry CEC ........................... 7-1 7.2 Concluding Comments......................................... 7-1 8. REFERENCES.................................................... 8-1 APPENDICES A SLURRY VISCOSITY DATA ....................................... A-1 B DEPENDENCE OF POWER-LAW PARAMETERS ON DENSITY ............B-i LIST OF FIGURES FIGURE PAGE 1.1 Potential for entrainment and spreading of contaminated mud at intake and discharge points during dredging operation .......................................... 1-1 1.2 Schematic drawing of Dry DREdge (after Parchure and Sturdivant, 1997) ......... 1-2 2.1 Poiseuille flow definitions ............................................... 2-4 3.1 Typical pressure-drop/flow rate relationships for slurry flow in pipes ............. 3-2 3.2 Plot of ln-rw versus in]F. Dashed line is tangent to the curve through a data point .... 3-2 3.3 Equivalent "extra length" due to end effects ................................. 3-6 3.4 Length and pressure components in two pipes, where LL and Ls are the fully developed flow sections for longer and shorter pipes, respectively ................ 3-6 3.5 Slurry flow curve for non-Newtonian fluid. Dashed line is tangential extrapolation to obtain the yield stress ................................................. 3-7 3.6 Applied strain in the stress relaxation test .................................. 3-12 3.7 Responses to stress relaxation for Maxwell and Newtonian fluids and an elastic solid. Note that for a Newtonian fluid the instantaneous stress is theoretically infinite. (This is not indicated in the plot, however, for simplicity of depiction of the curves for all three responses.) ...................................................... 3-12 4.1 Velocity profile for a generalized Newtonian fluid with Q=0.003 m 3/s and R=0. 1 m . . . . . . .. . . .. . .. . . .. . .. .. . .. . . .. . . .. . . .. . . 4 -2 4.2 Schematic of velocity profile for a Bingham plastic ........................... 4-2 4.3 Velocity profile for a Bingham fluid, with Q=0.003 m 3/s and R=0. 1 m, for slurry density p=1000, 1099, 1198 and 1314 kg/m3 (from right to left) ................. 4-3 4.4 Velocity profiles for a power-law fluid, with Q=0.003 in 3/s, R=0. Im, for flow index n=0.5, 1.0, 2.0 and 3.0 (from left to right) ................................... 4-5 4.5 Velocity profiles during the generation of Poiseuille flow of a weakly elastic fluid with S =0.04, and S2=0.02, for t, =0.1, 0.18, and 0.34 and -, (from left to right). Broken curves are the corresponding Newtonian profiles ....................... 4-9 4.6 Velocity profiles during the generation of Poiseuille flow of a general viscoelastic fluid with S, =0.04, and S2=0.04, for t1 =0.15, -, 0.85 and 0.5 (from left to right) ... 4-9 4.7 Curves showing the variation of mean square velocity profiles with the parameters (&Xj, &X2)=(4,1), (4,2) and (4,3), (from left to right). Solid line corresponds to a N ew tonian fluid ...................................................... 4-11 4.8 Illustrative curves for V0, V1, V2 as functions of r for rl0 = 10 poises, V10=0.1 poise, 'X1=0.6 s, 1 2 =0.1 s, e=0.25, p=l,000 kg/m3, W=1 s-1, 0=200, R=0.25 cm. Broken line is the velocity profile for Newtonian fluid giving rise to the same rate as VO .... 4-15 4.9 Theoretical (1A) curves for various values of w (s-1), T10=10 poises, p0=1 poises X,1=0.06 s, X2=0.01 sand e=0.25 ........................................ 4-15 4.10 Theoretical (S,D) curves for a Newtonian fluid (broken line) and a viscoelastic fluid with viscosity function for a 1.75% aqueous solution of polyacrylamide and with ).3=2.75 s, X 4=0.3 s, 6o=1 s1, R=0.25 cm and e=0.25 (solid line) ............... 4-16 5.1 Schematic drawing of a coaxial cylinder viscometer ........................... 5-1 5.2 Brookfield viscometer with an attached spindle shearing a clay slurry ............. 5-1 5.3 Schematic drawing of experimental setup for the horizontal pipe viscometer (HPV) ......................................................... 5-2 5.4 Photograph of HPV setup ............................................... 5-3 6.1 Comparison between pseudoplastic (shear-thinning) and Newtonian flow curves. The nature of the shear-thinning curve is such that while at low shear rates its viscosity is higher than the constant value for the Newtonian case (line), with increasing shear rate the pseudoplastic curve becomes asymptotically parallel to the Newtonian line, hence the pseudoplastic viscosity approaches that of the Newtonian case .......... 6-1 6.2 Excess apparent viscosity as a function of shear rate for kaolinite slurry no. 3 ....... 6-7 6.3a. Variation of [t with density of kaolinite slurries .............................. 6-9 6.3b. Variation of c with density of kaolinite slurries ............................... 6-9 6.3c. Variation of n with density of kaolinite slurries ............................. 6-10 6.4 Computed velocity profiles and corresponding discharges for slurry no. 1. Line is numerical solution using Sisko model; dots represent analytic Newtonian solution.. 6-12 6.5 Computed velocity profiles and discharges for slurry no. 24 ................... 6-13 vi 7.1 Variation of g-~ with slurry CEC (CECIu,) for all slurries ..................... 7-2 7.2 Variation of loge with slurry CEC (CECSIU,,,) for all slurries ................... 7.3 Variation of n with slurry CEC (CEC,,IU,) for all slurries ......................7-3 A-1 A-2 A-3 A-4 A-5 A-6 A-7 A-8 A-9 A-i10 A-11 A-12 A- 13 A- 14 A-i15 A-16 A- 17 A- 18 A- 19 A-20 A-21 A-22 A-23 A-24 A-25 A-26 A-27 A-28 A-29 A-30 A-3 1 A-32 A-33 A-34 A-35 A-36 A-37 A-3 8 A-39 Viscosity data for slurry no. 1 Viscosity data for slurry no. 2 Viscosity data for slurry no. 3 Viscosity data for slurry no. 4 Viscosity data for slurry no. 5 Viscosity data for slurry no. 6 Viscosity data for slurry no. 8 Viscosity data for slurry no. 9 Viscosity data for slurry no. 10 Viscosity data for slurry no. 11 Viscosity data for slurry no. 12 Viscosity data for slurry no. 13 Viscosity data for slurry no. 14 Viscosity data for slurry no. 15 Viscosity data for slurry no. 16 Viscosity data for slurry no. 17 Viscosity data for slurry no. 18 Viscosity data for slurry no. 19 Viscosity data for slurry no. 20 Viscosity data for slurry no. 21 Viscosity data for slurry no. 22 Viscosity data for slurry no. 23 Viscosity data for slurry no. 24 Viscosity data for slurry no. 25 Viscosity data for slurry no. 26 Viscosity data for slurry no. 27 Viscosity data for slurry no. 28 Viscosity data for slurry no. 29 Viscosity data for slurry no. 30 Viscosity data for slurry no. 31 Viscosity data for slurry no. 32 Viscosity data for slurry no. 33 Viscosity data for slurry no. 34 Viscosity data for slurry no. 35 Viscosity data for slurry no. 36 Viscosity data for slurry no. 37 Viscosity data for slurry no. 38 Viscosity data for slurry no. 39 Viscosity data for slurry no. 40 . . . . . . . . . . .. A -2 .. . . . . . . . . . .. A -2 .. . . . . . . . . ... A -3 .. . . . . . . . . . ... A -3 .. . . . . . . . . . .. A -3 .. . . . . . . . . . ... A -3 .. . . . . . . . . ... A -4 .. . . . . . . . . . ... A -4 .. . . . . . . . . . ... A -4 .. . . . . . . . . . ... A -4 . . . . . . . . . . .. A -5 .. . . . . . . . . .. A -6 .. . . . . . . . . . .. A -6 .. . . . . . . . . . .. A -6 .. . . . . . . . . ... A -7 .... ... ... .... ... ... .... ... ... ... A -7 .. . . . . . . . . . ... A -7 .. . . . . . . . . . .. A -8 .. . . . . . . . . ... A -8 .. . . . . . . . . . ... A -9 ... ... ... ... ... ... ... ... ... ... .. A -9 .. . . . . . . . . . A -10 .. . . . . . . . . . A -10 .... ... ... ... ... .... ... ... ... ... A -10 ... ... ... ... ... .. ... ... ... ... .. A -10 .... ... ... ... ... .... ... ... ... ... A -11 A-40 Viscosity data for slurry no. 41 ......................................... A-11 A-41 Viscosity data for slurry no. 42 ......................................... A-11 A-42 Viscosity data for slurry no. 43 ......................................... A-11 B-1 Variations of power-law coefficients with density for kaolinite slurries. Top: P_; m iddle: c; bottom : n .................................................... B -1 B-2 Variations of power-law coefficients with density for 75% kaolinite + 25% attapulgite slurries. Top: p; middle: c; bottom: n ............................ B-2 B-3 Variations of power-law coefficients with density for 50% kaolinite + 50% attapulgite slurries. Top: [t_; middle: c; bottom: n ............................ B-3 B-4 Variations of power-law coefficients with density for 25% kaolinite +75% attapulgite slurries. Top: p ; middle: c; bottom: n ..................................... B-4 B-5 Variations of power-law coefficients with density for attapulgite slurries. Top: V_; m iddle: c; bottom : n .................................................... B -5 B-6 Variations of power-law coefficients with density for 90% kaolinite + 10% bentonite slurries. Top: Vo; middle: c; bottom : n ..................................... B-6 B-7 Variations of power-law coefficients with density for 65% kaolinite + 25% attapulgite + 10% bentonite slurries. Top: [t; middle: c; bottom: n .............. B-7 B-8 Variations of power-law coefficients with density for 40% kaolinite + 50% attapulgite + 10% bentonite slurries. Top: V_; middle: c; bottom: n .............. B-8 B-9 Variations of power-law coefficients with density for 15% kaolinite + 75% attapulgite + 10% bentonite slurries. Top: [t; middle: c; bottom: n .............. B-9 B-10 Variations of power-law coefficients with density for 90% attapulgite + 10% kaolinite slurries. Top: p_,; middle: c; bottom: n ............................. B-10 LIST OF TABLES TABLE PAGE 5.1 Chemical composition of kaolinite ........................................ 5-3 5.2 Chemical composition of bentonite ........................................ 5-4 5.3 Chemical composition of attapulgite (palygorskite) ........................... 5-4 5.4 Chemical composition of water ........................................... 5-4 5.5 Size distribution of kaolinite ............................................. 5-5 5.6 Size distribution of bentonite ............................................. 5-6 5.7 Size distribution of attapulgite ............................................ 5-6 5.8 Properties of mud slurries tested .......................................... 5-7 6.1 Pressure drop, discharge, shear rate and wall stress data from HPV tests ........... 6-5 6.2 Sisko model coefficients and HPV flow Reynolds number ...................... 6-7 6.3 Low pressure HPV test parameters for selected slurries ....................... 6-11 1. PROBLEM STATEMENT 1. 1 Need for Investigation As coastal navigation channels are deepened by dredging due to requirements for greater draft vessels calling at ports, the need to maintain these channels against sedimentation has also risen. Disposal of dredged material has become problematic in many urbanized estuaries where the number of sites at which the material can be discharged has been restricted for ecological reasons (Marine Board, 1985). Added to this is the problem of contaminated bottom sediments, which can pose a threat to habitats both at the intake point and at the discharge point, as a result of the potential for the dispersal of sediment and associated pore fluid into the ambient waters. Referring to Fig. 1. 1, it is a common practice to cut and loosen bottom mud at the intake point mechanically or by highpressure water jets in preparation for the suction of the diluted slurry and its transport to the out-take point. In this process the water content of mud may increase from, say, 20% for a compacted bed to as much as 400% in a diluted slurry (Parchure and Sturdivant, 1997). It is therefore evident that this procedure can exacerbate contaminant dispersal at both ends, especially where strong currents or waves are present. The question therefore arises whether it is feasible to transport the slurry at in situ bottom density, as this would be a "cleaner" operation and therefore would be highly desirable for areas where sediment at toxic "hot spots" needs to be removed and transported as safely as possible. Although this seems like an obvious solution, the reason for diluting the mud at the intake point in the first place is that it is often difficult to transport undiluted, relatively dense bottom mud without a very high-pressure pump. Discharge Intake A Entrainment and Spreading J# Mud Fig. 1. 1 Potential for entrainment and spreading of contaminated mud at intake and discharge points during dredging operation. To address the above issue, at the U. S. Army Engineer Waterways Experiment Station in Vicksburg, Mississippi, a collaborative research program was undertaken by WES and DRE Technologies of Brentwood, Tennessee, to design, construct and operate a dredge which can transport bottom mud at in situ density. Concurrently, a laboratory experimental program was carried out at WES as part of the same investigation to explore the effect of mud composition and density on its transportability (Parchure and Sturdivant, 1997). The Dry DREdge (Fig. 1.2) had a specially designed sealed clashell mounted on a rigid, extendable boom. The open clamshell could be hydraulically driven into the bed sediment at a low speed, thus minimizing sediment disturbance, resuspension and spreading. The clamshell could be closed hydraulically and sealed, trapping a plug of sediment at its in situ water content. The sediment could then be deposited in the hopper of a positive displacement pump which could deliver it through a pipeline to the disposal site. Based on the results of prototype tests conducted in a shallow lake with a muddy bottom it was found that this dredge was useful for the following operating conditions: 1) shallow water depth (less than about 4.5 m), 2) small quantity of dredging, low output rate (less than about 30 m3/hr), 3) sediment with a high proportion of fines and low amount of sand, 4) short pumping distance, and 5) no significant wave or current action (Parchure and Sturdivant, 1997). IHIS DRAWNIG ANID IHE WFORMATIONH IT CONTAINS, INCLIIG THE PRINCIPAL OF DESIGN, IS PATENT PENDING THE PROPERTY OF DRE ENRONMENTAL SERMCES, RENOTWOOD TENNESSEE, AND IS SUSMITIED I WIH THE ENOPRESS AGREEMENT 1HAT IT WLL HOT SE REPRODUCED. COPIED, OR OHERN SE V.M. 2100 DISPOSED OF, DIRECTLY OR WDIRECLY. AND WI.l NOT BE USED AS A BASIS FOR MNUFAClURE oa 11 3-92 a I o 2 1 ENVIRONMENTAL OF EQUIPMENT OR APPARATUS W1HOUT WRIMEN PERMISSION OF DRE ENMRONHMENTAL SERMCES NS ,., FIRST OBTAINED AND SPECIFIC AS TO EACH CASE. w, NONE FIGURE 1 I REHRIOO, IENNESEE Fig. 1.2 Schematic drawing of Dry DREdge (after Parchure and Sturdivant, 1997). The main objective of the laboratory tests at WES was to ascertain the validity of the assumption that a high-pressure positive displacement pump is capable of transporting dredged sediments through a discharge pipe at near in situ water content. A 12.7 cm diameter, 274 mn long steel pipe was laid out for this purpose, and various mixtures in water of two clays: a kaolinite and a bentonite, and 0.4 mm diameter sand were transported with the help of a piston-type concrete pump. One noteworthy conclusion based on twelve tests carried out was that it was possible to transport a dense slurry with a specific gravity as high as 1.75. Also, in general, it was found easier to transport clays alone, than clay mixed with sand. In general, because bottom mud composition and density can vary widely, the pumping requirements can also vary accordingly. It is therefore a matter of considerable engineering interest to have the know-how to determine a priori what the pumping requirements will be in a given situation, in order to design the dredging operation and its execution. Since the transport of a slurry of given composition and density depends on the rheological behavior of the slurry (Heywood, 199 1; Wasp et al., 1977), by determining this behavior of mud to be dredged in a rheometer should make it feasible to ascertain: 1) if the bottom mud can or cannot be transported without dilution, and 2) what the rate of discharge of transportable will be. This laboratory study was therefore concerned with a two-step procedure, namely: 1) relating mud discharge in a pipe to mud rheology, and 2) relating mud rheology characterizing coefficients to mud composition and density. Accordingly, the study objective, tasks and scope were accordingly as follows. 1.2 Objective, Tasks and Scope The objective of this study was to correlate mud discharge in a pipe with mud composition and density through mud rheology, as a basis for developing a predictive tool for assessing pumping requirements for transportation of relatively dense dredged bottom mud. The associated tasks were as follows: 1. To select muds of widely varying composition. 2. For mud of given composition and density, to determine its rheological behavior. 3. To pump each mud of given composition and density through a horizontal pipe, and measure the discharge and pressure loss. 4. For mud of given composition, to determine the relationship between discharge, density and rheology characterizing parameters. 5. To explore the feasibility of using the above relationship as a predictive tool for assessing the transportability of a given bottom mud. The experimental scope of the work was defined by the choice of muds selected and the facility used for experiments. Since this work was of an exploratory nature, it was decided to select pure clays and their mixtures in water at different densities as mud slurries. The rheological behavior of each slurry was determined by combining the data from two types of rheometers: a coaxial cylinder viscometer (CCV) and a horizontal pipe viscometer (HPV). A Brookfield (model LVT) viscometer was used as CCV for testing the slurries at low shear rates. For relatively high shear rates, an HPV was designed specially. This particular type of viscometer was chosen because it also served as the arrangement used to measure slurry discharge at different pump pressures. The bench-top apparatus consisted of a 2.54 cm i.d. and 3.1 long PVC pipe through which mud was made to flow due to pressure applied by a diaphragm pump. The objectives of the study were met entirely by data obtained using these two types of viscometers, as described in the subsequent chapters. 1.3 Outline of Chapters In Chapter 2, the slurry transport problem is formulated in terms of Poiseuille flow in horizontal pipes. Chapter 3 covers rheological models for slurry flow, and in Chapter 4 several analytic solutions of the Poiseuille flow problem are explored, as well as the initial value problem and periodic motion. The experimental setup, materials used and test procedures are described in Chapter 5. Results of rheometry along with rheological simulations are presented in Chapter 6. These are further interpreted with reference to the overall objective of the study in Chapter 7. Finally, relevant references are given in Chapter 8. 2. SLURRY FLOW IN PIPES 2.1 Rheological Behavior of Materials In this chapter we refer to the works by Huilgol (1975) and Darby (1976). This chapter briefly describes the theoretical basis for the rheologial behavior of materials, such as mud, in pipes. For high density flow in pipes, the classic Newtonian rheological model cannot describe flow behavior appropriately, because here the elastic as well as viscous characteristics of the mixture of water and soil are also very important. The general relationship between an applied stress and the resulting deformation of a material is a unique function of the material. This relation may be represented symbolically by the general constitutive equation P(Tij,tij ...... )=a(yij, ij .....) (2.1) where t,, P"t ,... y, ij ..... are the shear stress, strain and their first and higher order derivatives with respect to time. Generally, a real material has non-linear characteristics, and it is difficult to accurately construct its theological model. In practice, we therefore make use of some simple empirical models for non-Newtonian behavior, e.g., the Bingham plastic model, power-law model, generalized Newtonian fluid model, etc., and also linear rheological models, e.g., Maxwell model, to express the theological properties approximately. The constitutive Eq. (2.1) implies that a unique property of viscoelastic fluids, such as mud slurries, in contrast with purely viscous fluids, is the time-dependence of material response. If a material undergoes a large-amplitude or continuous deformation, the coordinate positions of a given material point (with reference to a fixed origin) will not be "fixed". As a result, any measure of deformation based on infinitesimal displacements of fixed coordinate positions will be meaningless, since this would not always refer to the same material element. Thus, in order to describe the deformation of a viscoelastic material under these circumstances, it is necessary to follow a given material element or point with time as it moves from place to place in the system. One way in which we might follow the deformation of a material element as it moves through a system is by means of a reference frame which is defined by a set of base vectors that move and deform with the material. The position vectors in this frame are referred to as convective coordinates. These coordinates follow the path line of material particle, defined by S i =Xi -f vi[ j(t-t ),t-t/]dt/ (2.2) 0 where xi is the absolute coordinate; V is the absolute velocity; j is the position of a particle at time t-s, with 0__s_< oo and s is the time history of a particle. Under convective coordinates, the strain tensor yij becomes yij(xkts) =gi(xk)-c.(xkts)g(xk) g(Xk n k) (2.3) Note that y, is called the covariant convected strain tensor, where cu is referred to as the Cauchy deformation tensor and g4, is called the metric tensor. c. andg are defined as 8x m an c (xkt, S)= _- a g(Xk) (2.4) axi ax m ax m ax n g j- 8(2.5) g ax. ax. nm where xi and xi are the absolute Cartesian and curve coordinates, respectively. In order to determine the flow behavior of a given material in a pipe, the rheological equations of the material, which relate the force at all points in the material to the local deformation, must be solved simultaneously with the equations of conservation of slurry mass and momentum. We will derive the general equations of motion in next section. 2.2 Equations of Motion in Pipes 2.2.1 General Problem The general equations of motion in a pipe in cylindrical coordinates (z, r, 0) are as follows: Incompressible continuity equation: SarVr + aVe =0 (2.6) r ar r A azz Momentum equations: r-component: (VrV aVr Ve aVr V62 av, ap + 1 arr 1 atre ee rz p V + V g+-- +-- -+ (2.7) at r r r a r az) ar r r r r A r az 0-component: avo 8Ve Va Ve VrVo ave 1 p 1 ar2 Tro 1 aOO aoz V + --+V -- .. +pgo+- +-at r'ar r O r Z z r a0 r2 r0 z-component: aV +r avz Ve a K 8t r Br r 86 Z z p 1 arrz 1 aez - +pgz + + a + z 8z r Br r 8O 8z where rzz, Trr, OO are the normal stresses which contain the elastic effect; ,rz'r, TrO Oz are shear stresses; Vz, V Veare the velocity components in z, r and 0 directions, gz' gr' geare the corresponding gravity acceleration components; t is the time; p is the pressure and p is the density of mixture of water and sediment. 2.2.2 Poiseuille Flow Problem In this section we will describe the steady, isothermal, axial and laminar flow of an imcompressible fluid in a pipe (Fig. 2.1), known as Poiseuille flow. It is assumed that the flow is symmetric and that the axial (z) velocity component is the only non-zero component. These conditions can only be satisfied if Vr=V6=O' Vz f(0) (2.10) Under the above conditions, the continuity Eq. (2.6) reduces to z -0 (2.11) so that Vz ftz), i.e., Vz = Vz(r) only. The rate of strain tensor therefore becomes 0 dV/dr j = dV/dr 0 0 0 0 0 0 0 1 0 = 1 0 0 0 0 0 (2.8) (2.9) (2.12) so that the shear stress tensor has, at most, the following non-zero components ZZ t. = t rz 0 tI 0 rz rr 0 0 roe The three component momentum equations are simplified as z-component: r-component: Op 10 az r arr -p Pr 1 8(r0 ) tee Br r ar r 1 op P=0 r 00 0-component: Po PL L I --* V/(r) - 30 Z Fig. 2.1 Poiseuille flow definitions. (2.13) (2.14) (2.15) (2.16) Considering the fact that the velocity varies only in the r direction, all internal stresses which depend on flow deformation must also be functions only of r. Using this condition, differentiating Eq. (2.15) with respect to z leads to a ( fi ~ ( ~(2.17) ak erP-) = r 0 d ,(z) In other words, the pressure gradient (Op/dz) is independent of r. So we can calculate the pressure gradient along the pipe as plOz= (PL -po)/L. Considering Eq. (2.14), the left hand side is independent of r, whereas the right side is a function only of r. This can be true only if both are equal to a constant (say, 0): -- -pgz = 1 a (rr) (2.18) aZ r ar Integrating Eq. (2.18) leads to ro C1 z=-+- (2.19) 2 r The integration constant, C1, must be zero, since otherwise an infinite stress would be predicted at the center (r=0). Considering the stress exerted by the fluid on the pipe wall 7tR 2( RD -U (2.20) 27rR 2 then Eq. (2.19) becomes rI r 'r rZ- =2 ----U w (2.21) rz2 RW which is the final form of motion equation for Poiseuille flow. It is valid for either laminar (Newtonian or non-Newtonian) or turbulent flow. 3. RHEOLOGICAL MODELS 3.1 Viscous Model 3.1.1 Flow Type In this chapter we refer to the works by Alfrey and Doty (1945), Williams and Ferry (1953), Ferry (1970), Darby (1976) and Mehta (1996). In terms of flow properties for homogeneous, nonsettling slurries considered here under steady state in pipes, the flow type, the apparent viscosity and the flow curve (j, versus r) must be obtained. The experiment data required include the pressure drop zdp over the fully developed flow length L and the volumetric flow rate Q (or mean velocity Vm). From the plot of ln(.Ap/L) versus lnQ or lnVm shown in Fig. 3.1, where n is the power-law index, the flow type can be determined. It can be laminar or turbulent, Newtonian or NonNewtonian, etc. 3.1.2 Apparent Viscosity For a non-Newtonian fluid, the ratio of shear stress and shear rate is not constant, i.e., doubling the shear stress will not result in twice the shear rate, or vice versa. Thus the viscosity is not independent of the shear rate. We therefore define a function called the apparent viscosity as: ' = (3.1) Now, from Eq. (2.21) it is evident that measurement of the pressure gradient (D provides a direct means of determination of the shear stress at any point in the pipe. The calculation of the apparent viscosity of the fluid also requires the determination of the shear rate at some point in the pipe. An expression for the shear rate can be derived by considering the following relation for the volumetric flow rate, Q: R Q =f2rrrV(r)dr (3.2) 0 Integrating Eq. (3.2) by parts, with the condition that V =0, at r=R, leads to R R( V Q=-7Efr2dV,=-7frj -J dr(3) dr (3.3) 0 oo0 InQ or InVm Fig. 3.1 Typical pressure-drop/flow rate relationships for slurry flow in pipes. InF Fig. 3.2 Plot of lnL versus InF. Dashed line is tangent to the curve through a data point. Equation (3.3) can now be used to change the variables from r to r (for a given wall shear stress tC and R) to give Q=tR 3 2 (3.4) W 0 By taking the derivative of Eq. (3.4) with respect to Tw, we obtain d( Tw 3) =4-cwZ 2w (3.5) d'cw where --4Q/htR3 and j w is the shear rate at the wall. Solving Eq. (3.5) for 'w leads to - d w T (3.6) If we let n /=dln('Cw)/dlnP, Eq. (3.6) can also be written as w = -n /3n/+l r1- (3.7) Based on Eq. (3.7), the apparent viscosity takes the form TWT,4n / 3n(/+ I )(3.8) Yw r> 3n'+l) From the laboratory data (measurements of discharge and pressure drop), lntr can be plotted against lnJ, as shown in Fig 3.2. For that purpose, the wall shear stress, tw, is calculated from Eq. (2.20). Values of the coefficient, n are obtained manually from tangents drawn to the curve, as shown by the example in Fig.3.2. If necessary, end and slip corrections can be applied to correct for the measured values of the pressure drop, dp. These corrections are described next. 3.1.3 End and Slip Effects and Corrections A. End Effects: A major error which may arise in pipe flow measurements is due to end effects. Near the entrance and exit regions of the pipe, the velocity profile is not constant along the pipe but is in a state of transition between the flow configurations outside and inside the pipe, and the pressure gradient is not constant over these regions. For viscoelastic fluids, there is an additional end pressure drop due to elastic or normal stress effects. Therefore, if the measurement of pressure drop dp is not carried out within the fully developed flow section, a correction for d~p becomes necessary. Entrance effects can be corrected for experimentally in various ways. One approach is to determine an equivalent "extra length" (Le) of the pipe that would have to be added to the actual length if the total measured dp were that for an entirely fully developed flow region. This can be done as follows. Consider the total pressure gradient due to friction in fully developed flow in the pipe (of length L), plus an extra pressure drop due to entrance effects which would be equivalent to friction in fully developed flow over an additional length Le: (D= 'P pg = -r (3.9) L+Le R If the pipe is horizontal (g=O) and noting that -r is a unique function ofFr as shown in Eq. (3.6), Eq. (3.9) can be rearranged to give: 1&p = -2 r +I) =f(F)r (3.10) Hence, if several pipes of different L'R ratios are used, and dp is plotted against LII? for the same value of FP in each pipe, the plot should be linear if the flow becomes fully developed within each pipe, and the intercept at dp=O determines Le (Fig. 3.3). The intercept on the zdp axis at L/R=O is the pressure drop (APe) due to the combined end effects. Since a different value of Le would be obtained for each value of IF, L/IR can be empirically correlated with FP. An alternate procedure involves the use of two pipes of the same diameter, operating at the same flow rate (Q or F). Using subscript S for the shorter pipe and L for the longer, the various lengths and pressure components are defined in Fig. 3.4. Care must be taken in choosing the pipe lengths so that errors in pressure measurement are not unduly compounded by taking differences of large numbers. Assume the pressure gradient in the fully developed flow sections of the longer and shorter pipes are identical, i.e., L =I0ScI. Then following relationships are satisfied apa= aPe+(La-Le) APe +LL' ( Aps= Ape +(LS-Le)0= Ape+Ls' (3.12) Subtracting Eq. (3.12) from Eq. (3.11), the true pressure gradient in the fully developed flow section reads D = -APS (3.13) LL-LS B. Slip Effect An error in the measurement of Q can arise from an apparent slip between the fluid and the solid wall. This effect is actually due to the general inhomogeneity of the fluid near the wall. However, the extent of the region affected is often very small, so that the effect may be accounted for by assuming an effective slip velocity (us) superimposed upon the fluid in the pipe, and modifying Eq. (3.4), i.e., Q=tusR 2 + 7tR 3 f2dT (3.14) w 0 or +- [I_2dv (3.15) 4 t R 174 ) W W 0 where P3 =us-w is a slip coefficient. This coefficient can be evaluated as follows: 1. Using various pipes of the same length but different radii, plot P/4tw versus Iw for each pipe. If 3=0, these curves should coincide. If not, the curves will be distinct, in which case proceed as follows: 2. At constant Tw, plot P/4Tw versus 1/R from the above curves. This plot should be linear with a slope =P3; (3.11) 3. Repeat step 2 for various values of tw, and then plot P3 versus r. The appropriate value of F to use in evaluating ,w is then a "corrected" value corresponding to no slip: rno-slip s (slip)measured R(3.16) F=const. I, Le IR UR Fig. 3.3 Equivalent "extra length" due to end effects. SShort pipe, radius R LPs --- Q L, Ls' I p Long pipe, radius R --0 Q Le L Fig. 3.4 Length and pressure components in two pipes, where LL' andLs' are the fully developed flow sections for longer and shorter pipes, respectively. 3.1.4 Slurry Flow Curve After completing the above calculations, the slurry flow curves characterizing the rheological behavior of the slurry can be drawn, i.e., plots of c, and rI versus ,w (Fig. 3.5). The yield stress of the m ud, 'ry is obtained by extrapolating the curve of tw versus ,w. N ext w e w ill attem pt to determine the empirical relationships between rw and Tj versus i'w. V 77 rw r, versus y YW Fig. 3.5 Slurry flow curve for non-Newtonian fluid. Dashed line is tangential extrapolation to obtain the yield stress. 3.1.5 Some Empirical Non-Newtonian Models The most successful attempts at describing the steady rate of shear-stress behavior of nonNewtonian fluids have been largely empirical. The following represents some of the more common empirical models which have been used to represent the various classes of experimentally observed non-Newtonian behavior. A. Bingham Plastic: Given and j both positive, this model is "= -T +p for TH (3.17) j=0 for T v=p" (3.19) This is also a two-parameter model, with n as the flow index, p as the consistency. The apparent viscosity for this model is rI = n-1. C. Ellis Model In this model, the apparent viscosity is obtained as S7 1 (3.20) 1 1 (/1/2a1 The three parameters in Eq. (3.20) are p, t1/2 and a. Here 1/2 is the value of u at which rj =0.5p, and a is an empirical constant. 3.2 Linear Viscoelastic Models 3.2.1 Boltzmann Superposition Principle For any linear viscoelastic system, the constitutive equation has the general form d d2 dl=" d +q2d 2 d y (3.21) PoPI*2 ""P 90 1 9 "+.^M Y (3.21) o+pdt2 dt2 +pNdt N dt -dt2 dt 3-8 where the parameters po, P' ..... PN and q0, ql ...... qM are material properties and is the jth dtJ derivative. According to the Boltzmann superposition principle for a linear system, the total effect will be a linear combination of the separate contributions: T(t)=E VTi=E 41(t-ti)Vy(ti) (3.22) i=0 i=0 where j is a ralaxation function. When the system is continuous, the sum can be replaced by an integral: t T(t)=- fP(t-t')?(t')dt' f i(s)9(t-s)ds (3.23) - 0 Therefore, in the convective coordidate system the stress tensor Ti takes the form TU(xkt) (t-t )-' mn(zt')dt' (3.24) f Xkt axi ax. Under a steady state, 9 =const., so that we obtain the apparent viscosity from Eq. (3.23) as TI =ft(s)ds (3.25) Yo which implies that the motion of any general linear viscoelastic fluid falls into the category of "viscometric flow" at steady state. 3.2.2 Approximate Relationships Among Linear Viscoelastic Functions A. Dynamic Experiments: In dynamic experiments to determine the specific type of viscoelastic behavior of a material it is found that, if the viscoelastic behavior is linear, the strain also alternates sinusoidally but is out of phase with the stress. This can be shown from the constitutive equation (3.21) as follows: Let (3.26) where yo is the maximum strain amplitude and ) is the angular frequency of oscillation. Then =Y0cos(Gt) (3.27) Substituting Eq. (3.27) into Eq. (3.23) we have t(t) =f J(s)oYocos[w(t-s)]ds 0 (3.28) =Yo[W f*(s)sin(s)ds]sin(wt) +yo[wf *(s)cos(ws)ds]cos(wt) 0 0 It is evident that the term with sin(ot) is in phase with y0 and the term with cos(wt) is 900 out of phase. In other words, r is periodic in w but out of phase with y to a degree depending on the relative magnitudes of these two terms. The quantities in brackets are functions of frequency but not of elapsed time, so that Eq. (3.28) can be conveniently written as t =Y0[G'sin(wt) +G"cos(ot)] (3.29) thereby defining two frequency-dependent functions the shear storage modulus G' and the shear loss modulus G". It is instructive to write Eq. (3.29) in an alternative form displaying the amplitude to of the stress and the phase angle 6 between stress and strain. From trigonometric relations, t =t0sin(ot +8) =t0cosisin6)t+t0osinicost (3.30) Comparison of Eqs. (3.27) and (3.28) shows that G'=('orYo)cos6 G"=(t0/Y0)sin6 (3.31) Thus it is evident that each periodic, or dynamic, measurement at a given frequency provides simultaneously two independent quantities, G' and G". 3-10 y =Yosin(wt) B. Relaxation Function: To better illustrate the relaxation function, let us consider a Maxwell fluid response to a stress relaxation test in simple shear, in which a shear strain of magnitude y0 is suddenly applied at time zero, and is then held constant (Fig. 3.6). The Maxwell model has the form T+Xt=vtj, (3.32) where =pl/G is the relaxation time and V is the Newtonian viscosity. In the stress relaxation test, the stress response will be (Fig. 3.7) -r(t) =Gyoe -/.(3.33) where G is the constant shear modulus. It is seen that the initial stress response, (t0- Gy0 as t-0 +) is purely elastic. It then decays exponentially with time, reaching 37% [=(1/e)x 100] of its initial value Gy0 at the time t=-X. Thus, X is a characteristic time constant of the material, representative of the time scale for stress relaxation, and j(t) =Ge 11 is called the relaxation function. Note also in Fig. 3.7 the corresponding responses of a Newtonian fluid and an elastic solid. For a generalized linear viscoelastic model defined in Eq. (3.21), the appropriate relaxation function corresponding to an infinite series of relaxation times is *i(t)= Gi e d' (3.34) If the spectrum of relaxation times in continuous rather than discrete, Eq. (3.34) can be written as it)= f F()e -'I;'d). (3.35) 0 Let us set F(X)=H(X)/X and s=t. Then, the relaxation function for continuous linear viscoelastic system takes the form lV(s) =fH ) e -s/ldx = (fX)e (-s/ )dlnX (3.36) 0 0 3-11 Fig. 3.6 Ap yOG Y(t)=Yo t=O t strain in the stress relaxation test. - Elastic solid Newtonian fluid , Maxwell fluid t=O Xt Fig. 3.7 Responses to stress relaxation for Maxwell and Newtonian fluids and an elastic solid. Note that for a Newtonian fluid the instantaneous stress is theoretically infinite. (This is not indicated in the plot, however, for simplicity of depiction of the curves for all three responses.) where H(X) is a distribution function of relaxation times X. Theoretically, H can be obtained from G'or G". In practice, the functional forms of G' or G" are so complicated that no attempt is made to represent them by analytical expressions. For this reason, a variety of approximation methods have been developed for performing such calculations. Eq. (3.36) shows that H is multiplied by the kernel function e (-S/I), which varies from 0 at 3-12 X=0 to 1 when X-oo. If the latter were approximated by a step function varying from 0 to 1 at X=s, we would have (3.37) (s) fH(X)dlnk Ins and the integral would not be grossly different Eq. (3.36). Moreover, by differentiating Eq. (3.37) with respect to the limit in Ins, we obtain di(s) =H(X) dlns (3.38) Hence the relaxation spectrum at X=s is obtainable in the first approximation as the negative slope of the relaxation modulus. This is called Alfrey's rule. Williams and Ferry (1953) provided two formulas to calculate the relaxation spectrum from the storage modulus, depending on whether m, the negative slope of H against r on a doublelogarithmic plot, is greater or less than 1. If m<1, H(.) =AG'dlogG'/dlog I/o=, (3.39) A- sin(mir/2) ms/2 (3.40) If, on the other hand, 1 H(A) =AG'(2-dlogG'/dloge) lI/=x. (3.41) A- sin(mit/2) u(1 -m/2) 3-13 where where (3.42) The calculation is carried out in two stages. First A is set equal to unity, and a preliminary calculation is made with each point at a given value of w yielding a value of H at X= 1 /w. From the tentative graph of H versus 6), the value of m is measured at each point, and the appropriate correction factor A is applied. Then, through the calculated values of H and X, one can construct an empirical formula for H. 3-14 4. ANALYTICAL SOLUTIONS FOR SLURRY FLOW 4.1 Poiseuille Flow Problems 4.1.1 Generalized Newtonian Fluid In this chapter we refer to the works by Oldroyd (1950, 1958), Jones and Waters (1967), Waters and King (1970, 1971) and Huilgol (1975). As stated earlier, any linear viscoelastic system which falls in the category of viscometric flows is known as a generalized Newtonian fluid. The apparent viscosity of such a fluid is ri. For convenience, we will use V to represent the fluid viscosity, so that the rheological model is t=pj' By using the boundary condition V,=O at r=R (4.1) The solution for the velocity profile is v= wR 1 _( r )21 (4.2) The corresponding volumetric flow rate is Q = w(4.3) 4p and the ratio between V, and mean velocity V is V =2[1( r)2] (4.4) At steady state V,, is constant, and the velocity profile only depends on R. A sample of velocity profile in the pipe is shown in Fig. 4.1. 4.1.2 Bingham Plastic For a material that conforms to the Bingham plastic model, the rheological formulas are given in Eqs. (3.17) and (3.18). If ITrjI< try, the material will behave like a rigid solid. Therefore, from the pipe centerline to the point at which I -cr, I = -, the material moves as a" solid plug", as shown in Fig. 4.2. Solving Eqs. (2.21), (3.17) and (3.18) leads to: 0.5 rIR 0 0 0.5 1 1.5 2 V/V. Fig. 4.1 Velocity profile for a generalized Newtonian fluid with Q=0.003 m3/s and R=O. 1 m. ro f Fig. 4.2 Schematic of velocity profile for a Bingham plastic. for ror R, where ro=(r/Tw)R: and for r-ro: The corresponding volu VR (R 2 1 R 2 1R T plug 2p w metric flow rate is: 14 Q=Qfluid +Qplug 4 Ty +3 3 Ty 4 4V 3 w 3w Examining of above solution indicates that the velocity profile depends on iy, p, R and Q. A sample of velocity profile is shown in Fig. 4.3. 1 0.5 r/ 0 --------------------------0.5 0.5 1 1.5 2 V/V., Fig. 4.3 Velocity profile for a Bingham fluid, with Q=0.003m3/s, R=0.lm, for slurry density p=1000, 1099, 1198 and 1314 kg/m3 (from right to left). (4.5) (4.6) (4.7) 4.1.3 Power-Law Fluid The rheological equation of state for a power-law fluid takes the form (3.19). Combining Eqs. (2.21) and (3.19) leads to the following solutions: Velocity profile: ,j n+n) V- =n R 1- r n (4.8) z n+1 R Volumetric flow rate: = nTR 3 (4.9) 3n+1p Ratio between Vz and mean velocity Vm: Vz =( 3n+l)1(r)-]n (4.10) n+1 Therefore, when Vm is constant, the velocity profile only depends on n and R. Examples of velocity profiles in the pipe is shown in Fig. 4.4. 4.1.4 Maxwell Model From Eq. (3.32), the rheological formula for a Maxwell fluid is of the form + 1d-t dV v + Pt dz (4.11) rG dt dr For steady flow, dTddt=O, so that in this case the Maxwell fluid is a viscometric flow, as noted earlier. The solution is identical to that given in Section 4.1.1. 4.2 Initial Value Problem and Periodic Motion 0 0.5 1 1.5 2 2.5 V/I. Fig. 4.4 Velocity profiles for a power-law fluid, with Q=0.003m3/s, R=0. im, for flow index n=0.5, 1.0, 2.0 and 3.0 (from left to right). Thus far, we have mentioned some analytical solutions of viscometric flows at steady state. However, in so doing we lack knowledge of the dynamic behavior of viscoelastic fluids in the unsteady state, e. g., the initial value problem and periodic motion which commonly exist in real situations. Therefore, in this section we will examine the initial value problem and periodic motion in pipes for a linear viscoelastic material. To solve initial value problem, one is required to consider the integral model for finite linear viscoelasticity, i.e., Eq. (2.24), in which the only unknown quantity is the relaxation function. As an illustration, we will us consider the so-called "start up" problem. Here the fluid, which is initially at rest, is set in motion by a suddenly applied pressure gradient. The transitional velocity field as well as the final velocity field are of interest in the analysis of this flow. Next, the motion of a viscoelastic fluid under a periodic pressure gradient will also be examined. Here we will concentrate on the results obtained for the theory of finitely linear viscoelasticity for flow due to a pulsating pressure gradient superposed on a mean pressure gradient. 4.2.1 Initial Value Problem Consider the axial flow of a finitely linear viscoelastic fluid in the cylindrical coordinates. The known path lines are given by Vr=Vo=O, Vz=Vz(r,t) (4.12) From Eq. (2.12) it is seen that, for Poiseuille flow, the only non-zero strain tensors g are frz=izr=i'=dVz/dr. Hence from Eq. (3.24) the stress tensor reduces to t O2 Oa1 Oa1 2 Vz/ r'i= l~tt-/)Ox. Ox. O x. Ox. J Or Tj=1(J(t-t )+- dt (4.13) Therefore, the only non-zero stress components are trz' tzr and the normal stress trr where the first two components, using the initial problem condition, i.e., =0, for t<0, take the form Trz =Tzr ,fqj(t-t)a dt' = fi(s)Vz(rt-s)ds (4.14) - o 0 Then, the z-component of the equation of motion, (2.9), combined with the Poiseuille flow properties, Eq. (4.12) and (4.14), reduces to t p =( +v2f (s)Vz(r't-s)ds (4.15) 0 where v _=2 + 1 and #=const. is the pressure gradient as in Eq (3.18). The boundary conditions are Or2 r Or av Vz(R,t) -=0, r (0,t) =0 (4.16) Or Let W denote the Laplace transform of Vz W(r,o) =f Vz(r,t)e "tdt (4.17) 0 Then, by using Eq. (3.36) and (4.17), Eq. (4.15) becomes where 82W +1 8W-q 2w= q 2D -+---ew q2 a2r r ar pA)2 2 2__ 2 q f H(1) A S1 +W1 0 which is subject to the conditions W(R,G)) =0, A solution of Eqs. (4.18) and (4.20) is WD pW2 Let the distribution function H(A) be assumed to be H(;1) =o X2 ) +o io)- _l X X1 (4.22) where 8 is the Dirac delta function. It is assumed that X1 >2>0, where )1 is the relaxation time and X2 is called the retardation time. Using Eq. (4.19) and (4.22) we have q 2G( +)X) To(1 +012) (4.23) (4.18) (4.19) 8W -(0,) = 0 Br (4.20) SJo(iqr) Jo(iqR) (4.21) If we set r 1ot rI- t1 , R pR2 R2 Vm -- 0 R8o2 Boa S a =1,2, a pR 2 p R2 11o The complex inverse integral of Eq. (4.21) is given by 1c.,"i Vz(r,'tl)= i f Wl(r,o)exp(otl)do, C-i1 8V, Jo(iqr_ ) a (ro)= jo(iq) Through singularity analysis of W,(rl,o)exp(ot1), and using Eq. (4.24), the solution becomes (4.26) Vz(rt)=(1 r2)8E Jo(rZ) exp 2Vn z J,(Z )Zn3 where (4.27) Gn(tl)=cosh Pntl + 1+Z2(S2-2S) sinh Pntl 2S1 P, 2S, and Zn=iqI, an =1+S2Z2, Pn= (1 +S2Zn2)-4Sn21]/2, s=0, sn -n 2S For a Newtonian fluid, S, =S2 =0 and thus Vz rzt J(~o(rjZ,) =(1-rl2)-8E exp(-ZiZt) n Z4 Jl (Zn)Zn 3 (4.28) (4.29) Note that J0, J1 are the Bessel functions of the first kind and order 0 and 1, respectively. It can be shown that a weakly elastic fluid behaves like a Newtonian fluid as the velocity increases from zero to its final steady state value gradually (Fig. 4.5). However for a general viscoelastic fluid, its velocity at the center oscillates about the Newtonian value before reaching it (Fig. 4.6). 2 o(1+SIo) q1(1+S2o) (4.24) (4.25) 2Sl) G n(ti) 2S, rIR -0.2 -0.4 -0.6/ -0.8 -it 0 0.2 0.4 0.6 0.8 1 VI2V Fig. 4.5 Velocity profiles during the generation of Poiseuille flow of a weakly elastic fluid with S,=0.04, and S2=0.02, for tj =0.1, 0.18, 0.34 and (from left to right). Broken curves are the corresponding Newtonian profiles. I. 0 0.5 1 Vj2Vm 1.5 2 Fig. 4.6 Velocity profiles during the generation of Poiseuille flow of a general viscoelastic fluid with S,=0.4 and S2=0.04, for t1=0.15, o, 0.85 and 0.5 (from left to right). 4.2.2 Periodic Motion To examine flow behavior under a periodic pressure gradient, let us assume that the pressure and velocity fields are given by -p =Re [zAe iG'q, (4.30) Now, the equation of motion for the axial flow of an incompressible fluid [Eq. (4.15)] becomes d2dfldf +k2f=_ Ak2, dr2 r dr ipo k P{ H-1 k2=-ipe f H i) A f 1 +iG)el 0 The solution of Eq. (4.31), satisfying the conditions f(R)=0 and f(0) ip[ Jo(kR) whereJ0 is the Bessel function of the first kind of order 0. For small frequencies w, that is, small k, one can approxiamate Eq. (4.32) and obtain Vz(r,t) Vz(r,t) A (R 2 _r 2)cos(Wt+6) (4.33) where rj *(0) is the zero shear rate complex viscosity, written as 1 *(ao)= |1 exp(-i6). Thus, for a slowly varying pressure gradient the flow is equivalent to that of a Newtonian fluid, in which the velocity field is parabolic at each instant, but leading the applied pressure gradient by a phase angle 8. When I kr is very large (except near the centerline of the pipe, where r=O), Jones and Walters (1967) obtained an asymptotic expansion for Jo(kr) in powers of (kr)-I (4.34) f(r)- A (R) 13 i(R-r) (R-r)(9R+7r) e-ik(R-r) f(r)~ 1- 18kRr 128k2R2r2 4-10 (4.31) Vz(r,t)=Re [f(r)e ist] Near the centerline, for large values of kr, Eq. (4.32) yields A Vz(r,t) = -sinwt pwo (4.35) This is a noteworthy result, indicating that at large frequencies the velocity at the centerline of the pipe lags the applied pressure gradient by 7t/2 in phase. The mean square velocity Vm2 is given by Vm2= 1 [(ReVz)2 +(ImV)2] A 2 1/2 iS S(16B-7S) e-iKS 2W2f2 B-S 8B(B-S)K 128B 8B(BS)K 28B 2 (B -S)2K 2 (4.36) where K=(6)/v0)1/2, vo=rl0/p, B= (/v0)R and S=(/vo)"l/2(R-r). For a fluid with the distribution function H(X) given by Eq. (4.22), the plot of 2p2 2Vm2/A 2 versus S (Fig. 4.7) shows the following: 0.8 3 Al A Fig. 4.7 Curves showing the variation of mean square velocity profiles with the parameters (01, )X2)=(4,1), (4,2) and (4,3), (from left to right). Solid line corresponds to a Newtonian fluid. 4-11 S=(w/Jo)li2(R-r) (1) the mean square velocity has a higher value than for the corresponding Newtonian value; (2) this higher value occurs closer to the wall than in the Newtonian fluid; (3) the thickness of the boundary layer in which the rapid velocity fluctuations occur is reduced from that for the Newtonian fluid. For flow under a small periodic pressure gradient superposed on a constant mean value, the above problem becomes considerably more complicated. If the pressure gradient dp/dz is given by --(1 +ee i0t) (4.37) dz The equation of motion, (4.15), becomes p z (1+eei)+a---(rTrz) (4.38) t ( r Br In view of the form of the pressure gradient producing the flow, one can write the axial velocity and the shear rate as Vz(r,t) = Vo(r) +eVl(r)eist, +2[V2(r) + V)re Wi't+ V(22)(e i20,t] (4.39) (1) cat (2) i2(4 Vz(r,t)=,o(r)+eV1(r)e it+e2[V2(r)+V2 (r)e +V2 (r)e t](4.40) where terms of order 83 and higher order have been ignored. Note that Vo and o'0 correspond to the case of flow under a constant pressure gradient D. The shear stress rz is given by t"rz=1(io)'o +f1(o,s)y(s)ds+ ffoP2o,,ss2)XYy(S1)y(s2)dszds2 (4.41) 0 00 where yo=dVo/dr and we let 4-12 s y(s) fV(rt-o)do -yos (4.42) 0 as the oscillatory shear part, correct to O(e2). Here, 9p, Y2 are called memory functions. Note that these kernels obey the consistency relations fy(o,s)sds= ((M())|=Yo D2( 0,S1,S2)S1S2dSldS2-1 = ( t (4.43) 0 d 00 2 dj2 It is impossible to determine the kernel functions p, p2. However, qualitative predictions may made by making plausible assumptions concerning the unknown functions. Barnes et al. (1971) obtained following approximation relation (1 +02X32) I=/ o (4.44) (1 +W2 42) where I=100 --1 (4.45) Q and Qs are the mean flow rates with and without the pressure fluctuation, respectively, and R Q =f(Vo+e2V2)27trdr (4.46) 0 Eq. (4.44) implies that a non-zero w will give rise to a constant magnification of the (I0o,) curve by a factor depending on the frequency of the fluctuation and the values of X3, X4' Barnes et al. (1971) also employed the Oldroyd (1958) model through a numerical method to arrive at the following conclusions (Figs. 4.8 and 4.9): (1) the mean flow rate Q is not affected if the viscosity is constant; 4-13 R (2) let I denote the percentage fluctuation due to the ,2 in Eq. (4.46) over fVo27trdr. Then for low values of 0, I is positive and for high value 0, I is negative; 0 (3) there is also a strong indication of a "resonance" effect where a large percentage increase in mean flow rate can be expected for a given w. This could be useful in increasing the mean flow rate Q for a given mean pressure gradient D. In order to determine any practical significance of above conclusion (3), Barnes et al. (1971) examined the energy required to produce a given mean flow rate. The mean energy E required to produce the flow in the case of a fluctuating pressure gradient is given by EP =27If { V0+2( I +V2J ]dr (4.47) 0 The mean energy Es required to produce the same flow rate in the case of a constant pressure gradient is given by R Ep=27c Dsfr(Vo +e2V2)dr (4.48) 0 where Is is the constant pressure gradient required to produce a flow rate Q. Fig. 4.10 shows the theoretical curves of S against (D, where S is the percentage increase in the energy =100/ i 1 (.9 It can be observed that for low flow rates, S is lower in the case of the viscoelastic fluids, but the opposite is the case at higher flow rates. Of considerable interest is the observation that under some conditions, S can take fairly large negative values, which implies that a significant saving of energy by the use of a fluctuating pressure gradient is possible. This may be due to the retardation of the development of a fully developed, steady flow, boundary layer as a result of periodic flow reversal and associated interruption of the developmental process due to momentum transfer. The behavior at high flow rates is also noteworthy, indicating that energy can be wasted in pulsatile flow in this range. 4-14 Fig. 4.8 Illustrative curves for Vo, V1, V2 as functions of r for rjo =10 poises, P =0.1 poise, X1=0.6 s, X2=0.1 s, e=0.25, p=l,000 kg/m3, c=l S-, D=200, R=0.25 cm. Broken line is the velocity profile for Newtonian fluid giving rise to same rate as Vo. 40 30- Inelastic 1 20CO=10 10- 6=20 01 10 20 0 (Pa) Fig. 4.9 Theoretical (I, ID) curves for various values of w3 s-, i10=10 poises, [b=1 poises X,=0.06 S, X2=0.01 s and c=0.25. 4-15 400 D (pa) Fig. 4.10 Theoretical (S, D ) curves for a Newtonian fluid (broken line) and a viscoelastic fluid with viscosity function for a 1.75% aqueous solution of polyacrylamide and with X3=2.75 s, X4=0.3 s, w=1 s-', R=0.25 cm and e=0.25 (solid line). 4-16 5. EXPERIMENTAL SETUP, PROCEDURES AND MATERIALS 5.1 Coaxial Cylinder Viscometer The coaxial cylinder viscometer (CCV) (Fig. 5. 1) used was of the Brookfield (model LVT) type (Fig. 5.2). The general procedure for using the CCV involves rotating a metallic bob (a rightcircular cylinder) or a spindle at a selected rate in a beaker containing the mud of known density. In the present case the spindle could rotate at fixed speeds, giving a shear rate range of 0.063 to 20.4 Hz. The torque generated in mud by the rotation of the spindle was recorded from a readout meter. The shear stress, which is proportional to the torque, was calculated directly from the torque using a formula supplied by the maker (Brookfield Dial Viscometer, 198 1). Outer Torque Inner Cylinder Cylinder (Bob) Fig. 5.1 Schematic drawing of a coaxial cylinder viscometer. Fig. 5.2 Brookfield viscometer with an attached spindle shearing a clay slurry. The Brookfield viscometer is actually equipped with a series of spindles. The spindle required to shear a particular slurry depends on the density and viscosity of the slurry. Use of these spindles along with the charts provided results in values of the apparent viscosity, pt, which is inherently corrected for end-effects in the viscometer (Brookfield Dial Viscometer, 198 1). 5.2 Horizontal Pipe Viscometer The horizontal pipe viscometer (HPV) constructed at the Coastal Engineering Laboratory of the University of Florida is shown schematically in Fig. 5.3, and a photographic view is given in Fig. 5.4. The 3.1 mn long, 2.54 cm i.d. PVC pipe was clamped on to the a work-bench, with one end attached to a piston-diaphragm pump and the other end open, with a bucket receptacle to collect the slurry. The ARO model 666 1A3-344-C non-metallic double-diaphragm pump was operated at a nominal pressure of 40 psi (276 kPa). Pressurized air required at the pump inlet was supplied by a compressed air line observed in Fig. 5.4. Over the central 2.46 mn length of the pipe the pressure drop was measured by two flush-diaphragm gage pressure sensors. The positions A and B of these sensors were selected to be 32 cm from the pipe ends in order to minimize end-effects. Note that end-effects tend to become significant when the ratio of pipe length to diameter exceeds about 100. In the present case this ratio was 3.1/0.0254 = 122, i.e., well over 100. The pressure readings at A and B were recorded by a PC using Global Lab software. Mud slurry could be fed through a hopper above the pump and connected to it. Any slip effects that might have occurred during slurry flow were likely to have been small compared to the high velocities that were generated during the tests, and therefore were not explored further. Hopper Pump Pipe Fig. 5.3 Schematic drawing of experimental setup for the horizontal pipe viscometer (HPV). 5.3 Sediments and Slurries 5.3.1 Sediment and Fluid Properties Three types of commercially available clays: a kaolinite, a bentonite, and an attapulgite, which together covered a wide range of cohesive soil properties, were selected. Kaolinite (pulverized kaolin), a light beige-colored powder, was obtained from the EPK Division of Feldspar Corporation in Edgar, Florida. Its Cation Exchange Capacity (CEC), as given by the supplier, was 5.2-6.5 milliequivalents per 100 grams. Its granular density was 2,630 kg/m 3. Bentonite was obtained from the American Colloid Company in Arlington Heights, Illinois. It was a sodium montmorillonite (commercial name Volclay) of a light gray color. Its CEC was 105 milliequivalents per 100 grams, and its granular density was 2,760 kg/m 3. Attapulgite, of greenish-white color, was obtained from Floridin Company in Quincy, Florida. Also called palygorskite, its CEC was 28 milliequivalents per 100 grams, and its granular density was 2,300 kg/m 3. Tables 5.1 through 5.3 respectively give the chemical compositions of the three clays (provided by the suppliers). Fig. 5.4 Photograph of HPV setup. Table 5.1: Chemical composition of kaolinite Chemical % Chemical % SiO2 46.5 MgO 0.16 Al203 37.62 Na20 0.02 Fe203 0.51 K20 0.40 TiO2 0.36 SO3 0.21 P205 0.19 V205 < 0.001 CaO 0.25 Table 5.2: Chemical composition of bentonite Chemical % Chemical % SiO2 63.02 A1203 21.08 Fe203 3.25 FeO 0.35 MgO 2.67 Na20 & K20 2.57 CaO 0.65 H20 5.64 Trace Elements 0.72 Table 5.3: Chemical composition of attapulgite (palygorskite) Chemical % Chemical % SiO2 55.2 AI203 9.67 Na20 0.10 K20 0.10 Fe203 2.32 FeO 0.19 MgO 8.92 CaO 1.65 HO 10.03 NHO 9.48 Table 5.4 gives the results of chemical analysis of the tap water used to prepare mud, whose pH value was 8 and conductivity 0.284 millimhos. This analysis was conducted at the Material Science Department of the University of Florida (Feng, 1992). The procedure was as follows: firstly, an element survey of both the tap water and double-distilled water was performed, which determined the ions in tap water. Secondly, standard solutions of these ions contained in the tap water were made, and the tap water was analyzed against the standard solutions to determine the concentrations of the ions by an emission spectrometer (Plasma II). Table 5.4: Chemical composition of water Chemical Concentration (ppm) Si 11.4 Al 1.2 Fe 0.2 Ca 24.4 Mg 16.2 Na 9.6 Total Salts 278 The particle size distributions of kaolinite, bentonite and attapulgite are given in Tables 5.5, 5.6 and 5.7, respectively. The procedure for determination was: firstly, a particular suspension was prepared at about 0.5% by weight concentration, and run for at least 15 minutes in a sonic dismembrater (Fisher, model 300) to breakdown any agglomerates. Secondly, the suspension was analyzed in a particle size distribution analyzer (Horiba, model CAPA 700), and allowed to gradually settle down to the bottom. Particle concentration and fall velocities were determined with an X-ray, which could be converted to Stokes equivalent diameters. The median particle sizes of kaolinite, bentonite and attapulgite were 1.10 pm, 1.01 pn and 0.86 pm, respectively. Table 5.5: Size distribution of kaolinite Frequency distribution (%) 0.0 0.0 2.9 4.0 2.6 4.1 4.0 6.0 5.7 6.2 5.5 6.2 5.8 5.0 10.4 11.2 13.6 6.8 Cumulative frequency distribution (%) 0.0 0.0 2.9 6.9 9.5 13.6 17.6 23.6 29.3 35.5 41.0 47.2 53.0 58.0 68.4 79.6 93.2 100.0 5.3.2 Mud Slurries Mud slurries of different densities were prepared by thoroughly mixing the selected dry clays and clay mixtures with tap water at the ambient temperature, allowing these mixtures to stand for a minimum of 24 hours before testing them in the CCV and the HPV. Mud composition, density and water content are given in Table 5.8. Also given, in the last column, is the CEC of the slurry, calculated as follows: CECslurry fkaoliniteCECkaolinite CECattapulgite +fbentoniteCECbentonite (5.1) where f represents the weight fraction of the subscripted sediment, and subscripted CEC are the Diameter (vtm) 5.00< 5.00-3.20 3.20-3.00 3.00-2.80 2.80-2.60 2.60-2.40 2.40-2.20 2.20-2.00 2.00-1.80 1.80-1.60 1.60-1.40 1.40-1.20 1.20-1.00 1.00-0.80 0.80-0.60 0.60-0.40 0.40-0.20 0.20-0.00 corresponding cation exchange capacities. Note that given fwater as the weight fraction of water in the slurry, we have: fkaoiite attapulgite be.t.. water 1. The CEC values (in milliequivalents per 100 g) were selected to be: 6 (nominal) for kaolinite, 28 for attapulgite and 105 for bentonite. The weight fractions depend on the composition of each slurry given in Table 5.8. Table 5.6: Size distribution of bentonite Frequency distribution Cumulative fr (%) 5.9 1.9 2.3 2.5 3.0 3.0 4.9 5.3 8.1 4.5 9.3 9.1 equency distribution (%) 5.9 7.8 10.1 12.6 15.6 18.6 23.5 28.8 36.9 41.4 50.7 59.8 71.2 82.4 93.3 100.0 Table 5.7: Size distribution of attapulgite Diameter Frequency distribution Cumulative frequency distribution (pm) (%) (%) 2.00< 11.8 11.8 2.00-1.80 4.1 15.9 1.80-1.60 4.9 20.8 1.60-1.40 5.3 26.1 1.40-1.20 5.6 31.7 1.20-1.00 5.8 37.5 1.00-0.80 17.4 54.9 0.80-0.60 25.5 80.4 0.60-0.40 12.3 92.7 0.40-0.20 6.1 98.8 0290-000 12 100.0 Diameter (pm) 3.00< 3.00-2.80 2.80-2.60 2.60-2.40 2.40-2.20 2.20-2.00 2.00-1.80 1.80-1.60 1.60-1.40 1.40-1.20 1.20-1.00 1.00-0.80 0.80-0.60 0.60-0.40 0.40-0.20 0.20-0.00 .... . . Slurry No. Table 5.8: Properties of mud slurries tested Sediment Density Water Content (kg/m') 100%K 1,250 210 100%K 1,300 167 100%K 1,350 139 100%K 1,400 117 100%K 1,450 100 100%K 1,500 86 100%K 1,550 75 75%K+25%A 1,243 210 75%K+25%A 1,291 169 75%K+25%A 1,339 139 75%K+25%A 1,387 117 50%K+50%A 1,236 210 50%K+50%A 1,283 169 50%K+50%A 1,306 153 50%K+50%A 1,329 139 25%K+75%A 1,175 289 25%K+75%A 1,200 253 25%K+75%A 1,225 215 25%K+75%A 1,250 189 100%A 1,125 409 100%A 1,150 333 100%A 1,175 280 100%A 1,200 239 90%K+10%B 1,200 273 90%K+10%B 1,250 211 90%K+10%B 1,300 169 90%K+10%B 1,350 140 CEC, IU (meq/100 g) 1.94 2.25 2.51 2.84 3.00 3.23 3.44 3.71 4.28 4.80 5.29 5.48 6.33 6.72 7.10 5.79 6.38 7.14 7.77 5.50 6.46 7.38 8.26 4.26 5.12 5.90 6.63 28 65%K+25%A+10%B 1,225 231 6.47 29 65%K+25%A+10%B 1,250 204 7.04 30 65%K+25%A+10%B 1,275 182 7.59 31 65%K+25%A+10%B 1,300 163 8.12 32 40%K+50%A+10%B 1,175 299 6.74 33 40%K+50%A+10%B 1,200 257 7.54 34 40%K+50%A+10%B 1,225 224 8.31 35 40%K+50%A+10%B 1,250 197 9.05 36 15%K+75%A+10%B 1,125 423 6.20 37 15%K+75%A+10%B 1,150 345 7.28 38 15%K+75%A+10%B 1,175 290 8.31 39 15%K+75%A+10%B 1,200 248 9.30 40 90%A+10%B 1,125 415 6.93 41 90%A+10%B 1,150 339 8.13 42 90%A+10%B 1,175 284 9.29 43 90%A+10%B 1,200 244 10.39 From Table 5.8 we note that the density range covered was from a low 1,125 kg/m3 to a high 1,550 kg/m3. The water content varied from a high 423% to a low 75%. Finally, the CECSIU values ranged from 1.94 meq/100g for a kaolinite slurry (no. 1) to 10.39 meq/100g for a slurry (no. 43) composed of attapulgite and bentonite. 6. EXPERIMENTAL RESULTS 6.1 Rheometric Results 6.1.1 Power-Law for Mud Flow Previous work on flocculated bottom muds in the coastal environment has established their the pseudoplastic (shear thinning) flow behavior (e.g., Parker and Kirby, 1982). Subsequent work by, among others, Feng (1992) has revealed that the well-known Sisko (1958) power-law provides a reasonable fit to the measured decrease in apparent viscosity, p, with increasing shear rate, ',, a behavior that is consistent with the pseudoplastic flow curve (Fig. 6.1). Note that for convenience we will use the symbol pi for the apparent viscosity in place of ri used for example in Chapter 3. Pseudoplastic shearthinning flow curve Newtonian flow curve Fig. 6.1 Comparison between pseudoplastic (shear-thinning) and Newtonian flow curves. The nature of the shear-thinning curve is such that while at low shear rates the viscosity is higher than the constant value for the Newtonian case (straight line), with increasing shear rate the pseudoplastic curve becomes asymptotically parallel to the Newtonian line, hence the pseudoplastic viscosity approaches that of the Newtonian case. With reference to the Sisko model, we begin by noting that general power-law equations that predict the shape of the curves representing the variation of viscosity with shear rate typically need at least four parameters. One such relation is the Cross (1965) equation given by (Cj~y(6.1) where j'o and p_ refer to the asymptotic values of the viscosity at very low and very high shear rates, respectively, c, is a constant parameter having dimensions of time, p is a dimensionless constant, and pi is the apparent viscosity. It is generally found that V<< [to, hence the above equation can be simplified as PO = Cj~y(6.2) which can be rewritten as 11 11 + [t (6.3) or V =V.+ Cyn (6.4) Equation (6.4) is the Sisko model, where p_ is the constant ultimate viscosity at the limit of high (theoretically infinite) shear rate, c is a measure of the consistency of the material, and n is a parameter which indicates whether the material is shear-thinning or shear-thickening, that is, when n > 1 the material exhibits shear-thickening, and n < 1 denotes a shear-thinning behavior. When n = 1 the behavior is Newtonian, with a with a constant viscosity equal to p_~ + c. Note also that when p_=O, Eq. (6.4) becomes consistent with the power-law given by Eq. (3.19). Note that we are conveniently using the symbol c denoting consistency in place of P used in Eq. (3.19). It is important to recognize that the coefficients of Eq. (6.4) must be derived from measurements conducted under a laminar flow. The laminar limit for Newtonian slurries is given by the well-known Reynolds number criterion: Re = pVD < 2100 (6.5) It For a power-law [Eq. (3.19)] slurry, the limit is given by (Ryan and Johnson, 1959): Re' pVD <404 (n +2) n+3+1 ) 8 1 n-( 3n+1) n3 ~ 66 D4n) In a strict sense, condition (6.6) is applicable to the Sisko power-law fluid only when pi= 0. It can perhaps be used to represent the flow of a Sisko fluid in general in an approximate way in cases where V_ is small, e.g., for many of the slurries in the present investigation, for which the value of [twas found to be close to that of water. 6.1.2 Calculation of Power-Law Coefficients To solve for the three Sisko parameters, [t_, c and n, the least squares method can be used for fitting the curves obtained from Eq. (6.4) to the experimental data on the apparent viscosity, V, as a function of the shear rate, j, obtained from the measured relationship between stress (-t) versus j, such as shown qualitatively in Fig. 6.1. For this least squares method it is required that the viscosity difference between the model [Eq. (6.4)] and data, D, be minimized, that is, N D=E (i-[t)2 = minimum (6.7) i=1 or N D=E (pi-pt.-cjn-1)2 minimum (6.8) i=1 where Ai is the mud viscosity obtained from the experiment, and N is the number of data points. Setting -=0; -- =0; -0 (6.9) Oa- an ac from Eq. (6.8) we obtain the following by differentiation: N E (pi_- [t-cjn-l) =0 (6.10) i=1 N E {.fn-l (pi c'n1)} =0 (6.11) i=1 and N S{cn-1log'(1-i-[a-cjn-')} =0 (6.12) i-l In this way, V_, c and n can be determined by solving Eqs. (6.10), (6.11) and (6.12). A requirement for the determination of these coefficients is that each slurry be tested over a comparatively wide range of the shear rate j, so that the low-shear rate non-Newtonian and high shear rate Newtonian behaviors are identified. 6.2 Power-Law Parameters All 43 slurries noted in Table 5.8 were tested in the CCV and the HPV; the CCV for data at low shear rates, and the HPV for high shear rates. Slurry no. 7 did not flow through the HPV due to insufficient pump pressure. For this slurry therefore the Sisko power-law coefficients were not calculated. The overall rage of shear rates covered in the CCV was 0.063 Hz to 20.4 Hz, whereas the in the HPV they were considerably higher, in the range of 150.7 to 1094.5 Hz. Note that in the CCV the shear rate is an independent parameter which is inputted, whereas in the HPV it depends on the pipe diameter, length, pressure drop and slurry rheology. As discussed further in Section 6.3, at the high shear rates in the HPV the behavior of the slurry was close to Newtonian, hence the shear rate, j, and the corresponding shear stress, -rw, both at the pipe wall, could be calculated from the following Newtonian flow equations: 8V =- (6.13) D DAp (6.14) w 4L here V is the mean flow velocity in the pipe, D = 2R is the pipe diameter, R is the pipe radius and L is the distance over which the pressure drop, Ap, occurs. Then Vtjj/. For each slurry the measured pressure drop, Ap, the measured discharge, Q, and the calculated wall stress, -rw, are given in Table 6.1. Note that while in the CCV each sample was tested only once after the correct spindle was selected, in the HPV each sample was tested three times. The reported pressure reading and the discharge (obtained by timing the rate of flow of the slurry out of the pipe, weighing the mass accumulated in the bucket placed to receive the slurry, converting this weight to volume knowing the density and dividing the volume by the measured time) are means of the three measurements. Table 6. 1: Pressure drop, discharge, shear rate and wall stress data from HPV tests Slurry no. Pressure drop, zip Discharge, Q Shear rate, j'Wall stress, 'r,, (Pa) (me/s) (Hz) (Pa) 1 76,065.9 0.00150 931.8 196.3 2 54,452.4 0.00140 869.7 140.6 3 150,503.0 0.00136 844.9 388.5 4 192,067.5 0.00130 807.6 495.8 5 205,348.3 0.00124 770.3 530.1 6 144,997.7 0.00086 534.3 374.3 7 N. D.a N. D. N. D. N. D. 8 119,050.5 0.00144 898.7 307.3 9b177,736.9 0.00129 803.4 458.8 10 312,957.7 0.00076 472.1 807.8 11 54,381.6 0.00024 150.7 140.4 12 93,544.0 0.00149 927.7 241.5 13 85,376.3 0.00123 766.2 220.4 14 78,436.3 0.00104 647.1 202.5 15 73,784.9 0.00064 399.7 190.5 16 95,384.3 0.00158 980.9 226.2 17 137,272.7 0.00141 874.8 354.3 18 168,589.8 0.00121 754.2 435.2 19 179,701.1 0.001 17 728.2 463.9 20 75,907.4 0.00156 907.0 195.9 21 106,010.3 0.00141 873.2 273.6 22 128,008.6 0.00130 809.2 330.4 23 167,889.2 0.00123 764.1 433.4 25 73,888.0 0.00153 953.5 190.7 26 95,130.6 0.00141 872.9 245.6 27 126,692.1 0.00130 810.1 327.0 28 95,659.2 0.00166 1030.1 246.9 29 113,064.7 0.00157 974.4 291.9 30 115,010.4 0.00145 898.7 296.9 31 249,492.7 0.00134 829.6 644.0 32 58,314.3 0.00161 1002.2 150.5 33 33,372.3 0.00150 931.8 86.1 34 55,968.5 0.00146 907.9 144.5 35 96,128.4 0.00130 809.2 248.1 36 71,254.7 0.00157 976.1 183.9 37 57,862.2 0.00144 895.5 149.4 38 54,352.8 0.00153 952.2 140.3 39 121,193.6 0.00131 811.6 312.8 40 82,942.3 0.00156 968.8 214.1 41 45,127.2 0.00147 911.3 116.5 42 134,873.8 0.00141 878.0 348.1 43 160,813.2 0.00084 520.6 415.1 'No data obtained as slurry did not flow at this high density. 'For the particular Brookfield spindle which had to be used for this slurry, the torque reading was found to be below the minimal value at which the measurements were considered to be reliable by the maker of the viscometer. An example of the Sisko relationship [Eq. (6.4)] based on the combined CCV and HPV data is shown in Fig. 5.5, in which the eight points within the lower shear rate range were obtained by the CCV, and the single value at the higher shear rate from the HPV. Best-fit coefficients p_,, c and n for all the slurries obtained in the same way are listed in Table 6.2, which also gives the characteristic HPV flow Reynolds number, Ref, calculated according to Eq. (6.5). Thus, in other words, the Newtonian Reynolds number is given as the characteristic value in lieu of Re' from Eq. (6.6) because the flow regime in the HPV was practically Newtonian (see Section 6.3). All plots of excess apparent viscosity, [,. -p., as a function of shear rate are given in Appendix A. As seen in Table 6.2 from the range of Reynolds numbers experienced, all tests were carried under non-turbulent conditions, as required for the rheological analysis. The power-law coefficients 0.00176 1094.5 209.2 24 81,047.4 show considerable variability with slurry composition and density. Note that the lowest value of pwas chosen to be 0.001 Pa, the viscosity of water. In other words, in the least squares analysis V_, was not allowed to have values lower than the viscosity of water. Kaolinite, density = 1350 kg/m3 - 10 3 10' 010 EU a ,610 -2 'U 101 10 0 a. 0-1 ., 10.2 1 0-2 100 102 1 Shear rate (Hz) Fig. 6.2 Excess apparent viscosity as a function of shear rate for kaolinite slurry no. 3. Table 6.2: Sisko model coefficients and HPV flow Reynolds number Slurry No. P c n Ref (Pa.s) 1 0.1904 1.99 0.308 223 2 0.0024 3.79 0.520 282 3 0.2813 9.75 0.403 100 4 0.0011 6.81 0.610 74 5 0.4633 8.19 0.448 66 6 0.5837 14.46 0.228 46 7 N.C.1 N.C. N.C. N.C. 8 0.0015 2.43 0.581 132 9 0.4983 8.02 0.296 73 10 1.6880 9.98 -0.003 15 p_ = 0.2813 Pa.s c = 9.75 n = 0.403 41.31 4.31 13.21 21.67 86.17 3.14 9.61 22.07 34.01 1.27 3.36 13.55 21.38 2.97 9.43 15.86 47.92 9.46 18.29 23.12 66.70 8.05 12.71 11.37 40.29 2.31 6.55 9.23 23.96 4.20 -0.083 0.230 -0.022 -0.116 -0.047 0.223 0.212 -0.124 -0.205 0.559 0.068 0.148 -0.236 -0.152 0.199 0.289 0.149 0.318 0.239 0.216 0.188 0.274 0.297 0.280 0.174 0.538 0.398 0.251 0.160 0.318 9 130 45 45 31 185 105 65 58 229 129 94 65 277 240 163 109 213 164 140 56 317 488 282 133 236 249 307 102 199 0.7513 0.2380 0.2723 0.2965 0.3140 0.2357 0.3585 0.5632 0.6242 0.0019 0.3070 0.3628 0.5604 0.1899 0.1608 0.1530 0.2432 0.1544 0.2012 0.2179 0.4926 0.0970 0.0013 0.0719 0.1473 0.0012 0.0566 0.0912 0.2990 0.1787 0.0950 0.3570 0.5263 15.24 0.100 17.95 104.00 0.095 0.048 a Not calculated, because slurry did not flow. In Fig. 6.3, p_, c and n are plotted as functions of kaolinite slurry density (for slurry nos. 1 through 6; slurry no. 7 with a density of 1,550kg/m 3 did not flow in the HPV). Analogous data for all ten mud types tested are given in Appendix B. Observe in Fig. 6.3 that P_ shows an overall increasing trend with increasing density. The curve seems to suggest that this increase may not be monotonic, although it should be recognized that the number of data points is too few to arrive at a firm conclusion in this respect. In any event, it is logical to expect p_ to increase with density. The consistency, c, is seen to increase linearly with density. This trend can also be expected as it implies that for a given shear rate (and holding n invariant), slurry viscosity increases with density. Finally, n seems to increase at first with density, go through a maximum, and then decrease. Since n<1 throughout, over the entire density range the slurry behavior is seen to be pseudoplastic. Reviewing the data for the other nine muds in Appendix B we note that the trends are qualitatively akin to those for kaolinite, especially with regard to consistent pseudoplastic behavior. The non-monotonic effects of density on the coefficients in many cases may reflect the complex physical and physicochemical interactions between the particles and the pore fluid at different shear rates. Kaolinite 0.8 0.6 - 0.4 0.2 0 1 P0o 1!of1 1400 1500 161AM3 Kaolinite ig.6.8 yaratio of ithdenisity of ktl~t-lnis 1200 1300 1400 1500 1600 Density (kg/m^3) Fig. 6.3b Variation of c with density of kaolinite slurries. Fig. 6.3c Variation of n with density of kaolinite slurries. 6.3 Slurry Discharge with Sisko Model Given horizontal velocity of the slurry v(r), we have the shear rate =av/ar, and the steadystate momentum equation for pipe flow is r p y = --_T (6.15) Rw where Iw is the wall shear stress. For expressing the apparent viscosity, g, in terms of the shear rate, the Sisko model is given by Eq. (6.4). Next, we let S= 1 -- (6.16) R Then, combining Eqs. (6.4), (6.15) and (6.16) we obtain - + (nl -1) = 0 (6.17) which must satisfy the no-slip boundary condition at the wall, i.e., v(l=0) = 0. Equation (6.17) can next be written in the finite difference form as 6-10 B s on Eq.-1 i+ 1 was _Ied Based on Eq. (6.18) the following iterative relation was used: m+1 A i+1 i+ ili+Tli+i m n-1 1+C V~i+l-Vi where vi+l, (i = 1,2 ...... N) is the layer velocity, N is the total number of layers into which the distance from the wall to the centerline is divided, and m is the iteration index. Equation (6.19) was solved with the initial condition v, -0 at the wall. The chosen criterion for convergence was IVi +1ViMlI < 10-6 Finally, the discharge, Q, is obtained from the summation Q i=N Vi V+l) 27i=1 ( i+2 rd (6.20) (6.21) 6.4 Calculation of Slurry Discharge In order to test the applicability of the Eq. (6.21) for numerical determination of the discharge, Q, several slurries were pumped again through the HPV. The data are given in Table 6.3. These tests were deliberately carried out at pressures lower than those used to obtain the data points for determining the power-law relationships. (Compare the pressure drops in Table 6.3 with the corresponding ones in Table 6.1). Table 6.3: Low pressure HPV test parameters for selected slurries Slurry no. Pressure drop, Zip Measured Shear rate, j' Wall shear stress, T. Computed (Pa) discharge, Q (Hz) (Pa) discharge, Q (m3/s) (m3/s) 1 68,465.5 0.00135 836.6 176.7 0.00133 12 82,678.5 0.00136 844.9 213.4 0.00128 16 67,697.0 0.00128 794.5 174.8 0.00108 6-11 (6.18) (6.19) 28 67,464.2 0.00133 825.4 174.2 0.00093 32 73,927.5 0.00129 800.9 189.2 0.00200 40 74,920.0 0.00137 850.0 193.4 0.00136 Using the power-law coefficients for these slurries form Table 6.2, Eqs. (6.19) and (6.2 1) were solved along with the convergence criterion of Eq. (6.20). The number of layers, N, into which the pipe radius was divided was 20. As an example, the calculated result for slurry no. 1 is shown in Fig. 6.4, which plots the computed velocity profile and gives the corresponding discharge. Observe that this discharge agrees well with that measured in Table 6.2. Also plotted is the velocity profile assuming the slurry to be Newtonian [Eq. (4. 1)], and the corresponding discharge is calculated from Eq. (4.2). It is seen that the Newtonian assumption is reasonable at the high shear rate (836.6 Hz) at which the data were obtained, and justifies the Newtonian assumption for calculating the viscosity from HPV for determining the power-law coefficients in Table 6.2. In fact, for each slurry the data point for calculating the power-law coefficients was obtained at a higher shear rate (931.8 Hz) than in Table 6.3, thus making it even more acceptable to assume Newtonian flow. Another illustrative plot (for slurry no. 24) is given in Fig. 6.5. Measured and computed discharges for all slurries tested are given in Table 6.3. The degree of agreement varies, and can be shown to be sensitive to the power-law approximation of the rheological data, i.e., to the extent to which the power-law fits the measured data from the viscometers. QNewtonian =00 3 m 0.5 Qsko = 0. 00 133 m31s *- 0 --- - - 0.5 0 1 2 4 5S Veloity nil6 Fig. 6. Compued veloity profile ad corrsnigdshre Fig.sen 6.4aompted veloiy proiond.orsodngdshre 6-12 24 71,668.8 0.00133 823.2 185.0 0.00155 7 0.5 Qsisko = 0.00155 m31S Quewtonian = 0.00156 m3/s -0.5 -1 0 2 4 6 8 Velocity (m/s) Fig. 6.5 Computed velocity profiles and discharges for slurry no. 24. 6-13 7. INTERPRETATION OF RESULTS AND CONCLUSIONS 7.1 Power-Law Correlations with Slurry CEC The slurry CEC (CECIu J), as defined by Eq. (5.1), potentially lends itself as a measure of slurry cohesion, hence its rheology, at least to the extent to which cohesion and rheology are likely to be physico-chemically related. In Figs. 7.1, 7.2 and 7.3, data from Table 6.2 have been used to plot the power-law coefficients p_, (logarithm of) c and n against CECSIury given in Table 5.8. (Slurry nos. 7 and 9 have been omitted for reasons cited in Table 6.2.) Observe that while there is considerable data scatter, not all of which is likely to be "random", correlations indeed seem to exist in the mean (lines). We observe in Fig. 7.1 that in the mean p_ increases with CECIu J, which can be expected since greater cohesion would imply greater inter-particle interaction, hence viscosity. Similarly, in Fig. 7.2 the consistency, c, is seen to increase with CEClur, which is consistent with the trend in Fig. 7.1, given that consistency can be expected to vary directly with viscosity. Finally, in Fig. 7.3, n is seen to decrease with CECIU. Note that since n = 1 would mean a Newtonian fluid, and for a shear thinning material n < 1, the observed trend of variation of n with CECIu implies increasingly nonNewtonian, shear-thinning behavior of the slurries with increasing cohesion. The mean trend lines in Figs. 7.1, 7.2 and 7.3 respectively correspond to the following relations: V_ = 0.0125CECsIf + 0.181 (7.1) logc = 0.125CECSIun + 0.27 (7.2) n = -0.047CECslury + 0.50 (7.3) 7.2 Concluding Comments The experiments and analyses presented in the previous chapters essentially highlight a method which may be explored further in future for assessing pumping requirements. A draw-back of course is that without knowing the rheology of a given mud its transportation characteristics cannot be determined. Secondly, the use of CEC as a measure of rheological behvior of slurry cannot be extended to sediments that are not clayey. Given these two limitations, it will be necessary to: 1) examine a wide range of natural muds for their rheological behavior, and 2) develop correlations between rheology characterizing parameters and readily determinable parameters perhaps including, but not limited to, CEC. Finally it should be added that slurries of densities higher than those tested must be studied, using pumps which can supply higher pressures, in order to fulfill the need to quantify the understanding of transportability of high density muds at in situ densities. In any event, the following procedure, developed as part of this study, can serve as a guide for future efforts in this regard. 1. For the site to be dredged determine the required pipe discharge. 2. Collect the bottom mud sample to be discharged. 3. With mud rheology known, back-calculate the pressure drop required to achieve the discharge for a pipe of known dimensions. This can be done by matching the required discharge with that calculated from Eqs. (6.19) and (6.21), for a given (calibrated) value of Ap. 4. This value of Ap should be considered to be the minimum pressure drop required for the pump to be selected. 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Slurry CEC Fig. 7.1 Variation of [t with slurry CEC (CEC,,,,,J) for all slurries. 4 4 4 4 4 * 4 4 * 4 4 4 p 4 4 4 4 4 4 * p 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 Slurry CEC Fig. 7.2 Variation of logc with slurry CEC (CECslury) for all slurries. 0.8 0.6 0.4 0.2 0 -0.2 -0.4 Slurry CEC Fig. 7.3 Variation of n with slurry CEC (CECsIuy) for all slurries. 4 4 4 * 02 4. 6 *"* 8 10 1 4 8. REFERENCES Alfrey, T., and Doty, P., 1945. The method of specifying the properties of viscoelastic materials, Journal of Applied Physics, 16, 700-713. Barnes, T. N. G., Townsend, P., and Walters, K., 1971. On pulsatile flow of non-Newtonian liquids, Rheologica Acta, 10, 517-527. Brookfield Dial Viscometer, 1981. Operating Manual, Brookfield Engineering Laboratories, Stoughton, MA. Cross, M. M., 1965. Rheology of non-Netonian fluids: a new flow equation for pseudoplastic systems, Journal of Colloidal Science, 20, 417-437. Darby, R., 1976. Viscoelastic Fluids, Marcel Dekker, New York. Feng, J., 1992. Laboratory experiments on cohesive soil bed fluidization by water waves, M. S. Thesis, University of Florida, Gainesville, FL. Ferry, J. D., 1970. Viscoelastic Properties of Polymers, Wiley, New York. Heywood, N. I., 1991. Rheological characterisation of non-settling slurries, In: Slurry Handling Design of Solid-Liquid Systems, N. P. Brown and N. I. Heywood (eds.), Elsevier, Amsterdam, 53-87. Huilgol, R. R., 1975. Continuum Mechanics of Viscoelastic Liquids, Wiley, New York. Jones, J. R., and Walters, T. S., 1967. Flow of elastico-viscous liquids in channels under the influence of a periodic pressure gradient, Rheologica Acta, 6, Part 1, 240-245; Part 2, 330-338. Marine Board, 1985. Dredging Coastal Ports: An Assessment of the Issues, National Research Council, Washington, DC. Mehta, A. J., 1996. Interaction between fluid mud and water waves, In: Environmental Hydraulics, V. P. Singh and W. H. Hager W H (eds.), Kluwer, Dordrecht, The Netherlands, 153-187. Oldroyd, J. G., 1950. On the formulation of rheological equations of state, Proceedings of the Royal Society (London), A200, 523-541. Oldroyd, J. G., 1958. Non-Newtonian effects in steady motion of some idealized elasto-viscous liquids, Proceedings of the Royal Society (London), A245, 278-297. Parchure, T. M., and Sturdivant, C. N., 1997. Development of a portable innovative contaminated sediment dredge. Final Report CPAR-CHL-97-2, Construction Productivity Research Program, U. S. Army Engineer Waterways Experiment Station, Vicksburg, MS. Parker, W. R., and Kirby, W. R., 1982. Time dependent properties of cohesive sedment relevant to sedimentation management European experience, In: Estuarine Comparisons, V. S. Kennedy (ed.), Academic Press, New York, 573-589. Ryan, N. W., and Johnson, M. M., 1959. Transition from laminar to turbulent flow in pipes, American Institute of Chemical Engineers Journal, 5, 433-435. Sisko, A. W., 1958. The flow of lubricating greases, Industrial Engineering Chemistry, 50, 17891792. Wasp, E. J., Kenny, J. P., and Gandhi, R. L., 1977. Solid-Liquid Flow Slurry pipeline Transportation. Trans Tech Publications, San Francisco. Waters, N. D., and King, M. J., 1970. Unsteady flow of a elastic-viscous liquid, Rheologica Acta, 9, 345-355. Waters, N. D., and King, M. J., 197 1. The unsteady flow of an elastico-viscous liquid in a straight pipe of circular cross section, Journal of Physics D : Applied Physics., 4, 204-211. Williams, M. L., and Ferry, J. D., 1953. Second approximation calculations of mechanical and electrical relaxation and retardation distributions, Journal of Polymer Science, 11, 169-175. APPENDIX A: SLURRY VISCOSITY DATA Kaolinite, density = 1250 kg/m^3 0 2 4 10 10 10 Shear rate (Hz) Fig. A-1 Viscosity data for slurry no. 1. Kaolinite, density = 1350 kg/m^3 Mu (inf) 0.2813 Pa.s c = 9.75 n = 0.403 -2 0 2 102 10 102 Shear rate (Hz) Fig. A-3 Viscosity data for slurry no. 3. 2 10 .10 0 0. 0 00 ._O 0 4/) X10 w 102 210 10 Kaolinite, density = 1300 kg/m^3 Mu (inf)= 0.0024 Pa.s c = 3.79 n = 0.520 0 2 10 10 1 Shear rate (Hz) Fig. A-2 Viscosity data for slurry no. 2. 2 10 10 10 0 > 0 U 10-L -2 10 Kaolinite, density = 1400 kg/m^3 SMu (inf)= 0.0011 Pa.s c=6.81 n = 0.610 0 2 10 10 Shear rate (Hz) Fig. A-4 Viscosity data for slurry no. 4. 2 10 -.10 10 (U x. 10 0 0 w 0" 0 ,,, 102= Mu (inf)= 0.1904 Pa.s c = 1.99 n = 0.308 103 -210 10 3 10 2 ,10 0 0 o p101 0 0 0 "3 0 010 x 10-2 10 ' 101 10. m 0 0 10 (4 X10 LU 2 Kaolinite, density = 1450 kg/m^3 102 -2 0 2 10 10 10 Shear rate (Hz) Fig. A-5 Viscosity data for slurry no. 5. Kaolinite, density = 1500 kg/m^3 3 10 2 -,10 o 0 U '> 0 10 10 0 U 1 -21 -2 0 2 102 10 10 1 Shear rate (Hz) Fig. A-6 Viscosity data for slurry no. 6. A-2 Mu (inf) = 0.4633 Pa.s c=8.19 = 0.448 X $ Mu (inf)= 0.5837 Pa.s c = 14.46 n = 0.228 I i I I i I 2 75% K + 25% A, density = 1243 kg/m^3 10 0 10 10 0) 02 o 10. Iw -2 10 21 -2 0 2 10 10 10 1 Shear rate (Hz) Fig. A-7 Viscosity data for slurry no. 8. 75% K + 25% A, density = 1291 kg/mA3 10 r 2 102 a) a 10 0 0 000 x ) 100 102 -2 10 0 2 10 10 1 Shear rate (Hz) Fig. A-8 Viscosity data for slurry no. 9. 4 10 102 a 10) O 0 ( 100 a) 0 -2 x10 w10' 75% K + 25% A, density = 1339 kg/m^3 -2 0 2 10 10 10 10 Shear rate (Hz) Fig. A-9 Viscosity data for slurry no. 10. 102 0 0 0 10 .w a) 0 -2 x10 wJ 75% K + 25% A, density = 1387 kg/m^3 10-41 I 104 -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-10 Viscosity data for slurry no. 11. Mu (inf) = 0.0015 Pa.s c = 2.43 n = 0.581 x * Mu (inf)= 0.4983 Pa.s c = 8.02 n = 0.296 Mu (inf)= 1.688 Pa.s c = 9.98 n = -0.003 Mu (inf)= 0.7513 Pa.s c = 41.31 n = -0.083 2 50% K + 50% A, density = 1236 kg/mA3 10 1 . 50% K + 50% A, density = 1283 kg/mA3 2 0.10 a. 0 0 10 0 -2 X10 -2024 10 10 102 10 Shear rate (Hz) Fig. A-11 Viscosity data for slurry no. 12. 50% K + 50% A, density = 1306 kg/mA3 -2 0 24 10 10 10 10 Shear rate (Hz) Fig. A-12 Viscosity data for slurry no. 13. 4 10 02 a. 10 0 0 0100 0 0) x U 1 0 4 1.1.4 -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-13 Viscosity data for slurry no. 14. 50% K + 50% A, density = 1329 kg/mA3 -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-14 Viscosity data for slurry no. 15. A-4 ,10 iL (U Z' 0 0 0 0 0210 0 U x UJw " Mu (inf) = 0.2723 Pa.s c = 13.21 n = -0.022 t0 4- 4 10 ('.102 a 00 00 0 2 w o 0-2 Mu (inf) = 0.2965 Pa.s c = 21.67 n = -0.116 I I 2 25% K + 75% A, density = 1175 kg/m^3 10 2 0 2 4 10 10 10 104 Shear rate (Hz) Fig. A-15 Viscosity data for slurry no. 16. 10 102 100 00 0 -2 0 X10 wL 10-4' 25% K + 75% A, density = 1225 kg/m^3 2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-17 Viscosity data for slurry no. 18. 25% K + 75% A, density = 1200 kg/m^3 10 2 Mu (inf)= 0.3585 Pa.s 10 c = 9.61 .n = 0.212 .101 10 0 0X *100 W 10-1 -2 102 -2 0 2 4 102 10 10 10 Shear rate (Hz) Fig. A-16 Viscosity data for slurry no. 17. 4 10 2 . O 00 ulO 01 0 -2 X10 10 - 25% K + 75% A, density = 1250 kg/m^3 -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-18 Viscosity data for slurry no. 19. A-5 Mu (inf) = 0.5632 Pa.s c = 22.07 n = -0.124 Mu (inf) = 0.6242 Pa.s c = 34.01 n = -0.205 I I i I i I 2 Attapulgite, density = 1125 kg/m^3 10 101 0. 0 0 0 a) X10 .U 102 - i 0 2 I -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-19 Viscosity data for slurry no. 20. 4 10 2 S102 0 0 o10 0 0 ' 0 w 102 10I - Attapulgite, density = 1175 kg/m^3 -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-21 Viscosity data for slurry no. 22. Attapulgite, density = 1150 kg/m^3 2 0 010 x10 U) o -2 ' to 10 -2 10 0 2 10 10 Shear rate (Hz) Fig. A-20 Viscosity data for slurry no. 21. A Attapulgite, density = 1200 kg/m^3 -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-22 Viscosity data for slurry no. 23. Mu (inf)= 0.0019 Pa.s c = 1.27 n = 0.559 Mu (inf)= 0.307 Pa.s c = 3.36 n = 0.068 Mu (inf)= 0.3628 Pa.s c = 13.55 n = 0.148 i I i I 90% K + 10% B, density = 1200 kg/m^3 10 -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-23 Viscosity data for slurry no. 24. 90% K + 10% B, density = 1300 kg/m^3 10 2 Mu (inf) = 0.153 Pa.s '10 c = 15.86 .n = 0.289 =10 >oo S10 S10 S-20 10 0 10,2 -2 0 2 4 102 10 10 10 Shear rate (Hz) Fig. A-25 Viscosity data for slurry no. 26. S 90% K + 10% B, density = 1250 kg/m^3 10 2 Mu (inf) = 0.1608 PR ,10 c = 9.43 2n = 0.199 -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-24 Viscosity data for slurry no. 25. 90% K + 10% B, density = 1350 kg/m^3 -' S c = 47.92 ~02 n = 0.149 -10 > 0 uJ 10 x -2 *2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-26 Viscosity data for slurry no. 27. Mu (inf)= 0.1899 Pa.s c = 2.97 n = -0.152 102 0. 0 10 01 0 -2 x 10 wI Mu (inf) = 0.2432 Pa.s I 1U I 3 65% K + 25% A + 10% B, density = 1225 kg/m^3 10 2 Mu (inf)= 0.1544 Pa.s ~10 c= 9.46 10 O n =0.318 0 m 10-1 -2 10 A 2 0 20 10 10 10 10 Shear rate (Hz) Fig. A-27 Viscosity data for slurry no. 28. 1 65% K + 25% A + 10% B, density = 1275 kg/m^3 10 2 Mu (inf)= 0.2179 Pa.s 10= c = 23.12 n = 0.216 o10 X X 0 "> 0 x n10 ,,, 10-1 10 2 -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-29 Viscosity data for slurry no. 30. 3 65% K + 25% A + 10% B, density = 1250 kg/m^3 10 2 Mu (inf)= 0.2012 Pa.s .10 c= 18.29 10 0 10 10 -2 x LU10 10,2 10 10 10 10 Shear rate (Hz) Fig. A-28 Viscosity data for slurry no. 29. 4 65% K + 25% A + 10% B, density = 1300 kg/m^3 10 Mu (inf)= 0.4926 Pa.s c = 66.7 12 n = 0.188 -10 0 0 me $10. , 2 0 110 x -2 0 2 LU 0-2. 10 10 10 10 Shear rate (Hz) Fig. A-30 Viscosity data for slurry no. 31. A-8 340% K + 50% A + 10% B, density = 1175 kg/m^3 10 2 Mu (inf) = 0.097 Pa.s -10 1 c = 8.05 n = 0.274 >10, 0 3 0 (10 m 10-1 10-2 10,2 -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-31 Viscosity data for slurry no. 32. 40% K + 50% A + 10% B, density = 1225 kg/m^3 10 2 x Mu (inf) = 0.0719 Pa.s ,10 x c = 11.37 n = 0.280 101 0X CO 0 8X CIb " 10 x 10 10 -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-33 Viscosity data for slurry no. 34. S40% K + 50% A + 10% B, density = 1200 kg/mA3 10 2 Mu (inf) = 0.0013 Pa.s 10 c = 12.71 n = 0.297 10 S0 10 x w -1 10 -2 10'' -2 0 24 10 10 10 10 Shear rate (Hz) Fig. A-32 Viscosity data for slurry no. 33. 40% K + 50% A + 10% B, density = 1250 kg/m^3 101 1 10 102 2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-34 Viscosity data for slurry no. 35. A-9 2 10 10 a. 0 -1 o 10 U) O X10 15% K + 75% A + 10% B, density = 1125 kg/m^3 Mu (inf) = 0.0012 Pa.s S c = 2.31 X n = 0.538 02 , 1021 -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-35 Viscosity data for slurry no. 36. 15% K + 75% A + 10% B, density = 1175 kg/m^3 10 2 x Mu (inf) 0.0912 Pa.s 10 c = 9.23 a.n = 0.251 0 X -2 -2 0 2 010 10 10 10 10 Shear rate (Hz) Fig. A-37 Viscosity data for slurry no. 38. 2 15% K + 75% A + 10% B, density = 1150 kg/m^3 10 Mu (inf)= 0.0566 Pa.s S c = 6.55 a. n = 0.398 0 @2 x 10 uO -2 X10 w 10,2 -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-36 Viscosity data for slurry no. 37. S15% K + 75% A + 10% B, density = 1200 kg/m^3 10 X 2 Mu (inf) = 0.299 Pa.s ,10 c = 23.96 . n = 0.160 0 X, 0 010 .' 101 -2 10 x w10' 10,2 -2 0 2 4 102 10 10 10 Shear rate (Hz) Fig. A-38 Viscosity data for slurry no. 39. A-10 2 90% A + 10% B, density = 1125 kg/m^3 10 Mu (inf) 0.1787 Pa.s 110 c = 4.120 X n = 0.318 00 0 -1 10 10., w 1 -2 10 -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-39 Viscosity data for slurry no. 40. 90% A + 10% B, density = 1175 kg/m^3 -2 102 0 0 010 2 10 (310 L} 0. 0l 90% A + 10% B, density = 1150 kg/m^3 10 Mu (inf) 0.095 Pa.s -.2 c = 15.24 0110 0n = 0.100 100 0 -2 10 -4 0 2 10 1 -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-40 Viscosity data for slurry no. 41. 90% A + 10% B, density = 1200 kg/m^3 4A 4U, 0 -102 0 o 0 "3 :10 0 x uJ -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-41 Viscosity data for slurry no. 42. -21 10 .. -2 0 2 4 10 10 10 10 Shear rate (Hz) Fig. A-42 Viscosity data for slurry no. 43. A-11 Mu (inf) 0.357 Pa.s c = 17.95 n = 0.095 Mu (inf) = 0.5263 Pa.s c= 104 n = 0.048 I 0 IU APPENDIX B: DEPENDENCE OF POWER-LAW PARAMETERS ON DENSITY Kaolinite 20 15 o 10 5 0 1200 1300 1400 1500 1600 Density (kg/m^3) Kaolinite 0.8 0.6 C 0.4 0.2 0 1200 1300 1400 1500 1600 Density (kg/m^3) Fig. B-1 Variations of power-law coefficients with density for kaolinite slurries. Top: P; middle: c; bottom: n. Kaolinite 0.8 0.6 0.4 S0.2 0 * 1200 1300 1400 1500 1600 Density (kg/m^3) 75%K +25%A 0.8 0.6 0.4 0.2 0 1 -- -400 Density (kg/m^3) Fig. B-2 Variations of power-law coefficients with density for 75% kaolinite + 25% attapulgite slurries. Top: p_; middle: c; bottom: n. B-2 75%K + 25%A 2 1.5 1 0.5 0 , -0.5 -e 3400 Density (kg/m^3) 75%K + 25%A 50 40 30 20 10 0, 1200 1250 1300 1350 1400 Density (kg/m^3) 50%K + 50%A 0.3 0.2 0.1 0 -0.11 150 -0.2 Density (kg/m^3) Fig. B-3 Variations of power-law coefficients with density for 50% kaolinite + 50% attapulgite slurries. Top: gm; middle: c; bottom: n. 50%K + 50%A 0.4 S0.3 S0.2 2 0.1 0, 1200 1250 1300 1350 Density (kg/m^3) 50%K + 50%A 100 501 0 12O0 1250 1300 150 -50 Density (kg/m^3) 25%K + 75%A 40 30 o 20 10 0 1160 1180 1200 1220 1240 1260 Density (kg/m^3) 25%K + 75%A 0.4 0.2 C 0' -0.1 60 1180 1200 122 10 -0.4 Density (kg/m^3) Fig. B-4 Variations of power-law coefficients with density for 25% kaolinite + 75% attapulgite slurries. Top: p; middle: c; bottom: n. 25%K + 75%A 0.8 0.6 *- 0.4 0.2 0 1160 1180 1200 1220 1240 1260 Density (kg/m^3) Attapulgite 0.6 0.4 0.2 -0.21 4 6 0 1 20 -0.4 Density (kg/m^3) Fig. B-5 Variations of power-law coefficients with density for attapulgite slurries. Top: p_; middle: c; bottom: n. Attapulgite S0.6 0.4 ~0.2 0 1120 1140 1160 1180 1200 1220 Density (kg/m^3) Attapulgite 25 20 15 10 5 0 1120 1140 1160 1180 1200 1220 Density (kg/m^3) Fig. B-6 Variations of power-law coefficients with density for 90% kaolinite + 10% bentonite slurries. Top: g_; middle: c; bottom: n. B-6 90%K + 10%B 60 40 20 0 1150 1200 1250 1300 1350 1400 Density (kg/m^3) Fig. B-7 Variations of power-law coefficients with density for 65% kaolinite + 25% attapulgite + 10% bentonite slurries. Top: p; middle: c; bottom: n. 65%K+25%A+10%B 8060 o 40 20 0 1220 1240 1260 1280 1300 1320 Density (kg/m^3) 40%K+50%A+10%B 0.4 0.3 C 0.2 -. 0.1 0 1160 1180 1200 1220 1240 1260 Density (kg/m^3) Fig. B-8 Variations of power-law coefficients with density for 40% kaolinite + 50% attapulgite + 10% bentonite slurries. Top: i; middle: c; bottom: n. B-8 40%K+50%A+10 O%B 0.2 0.15 0.1 0.05 0 1160 1180 1200 1220 1240 1260 Density (kg/m^3) 40%K+50%A+10%B 50 40 o 30 20 10 0 1160 1180 1200 1220 1240 1260 Density (kg/m^3) Fig. B-9 Variations of power-law coefficients with density for 15% kaolinite + 75% attapulgite + 10% bentonite slurries. Top: p; middle: c; bottom: n. B-9 15%K+75%A+10%B 0.4 0.3 0.2 0.1 . 1120 1140 1160 1180 1200 1220 Density (kg/m^3) 15%K+75%A+10%B 30 o 20 10 0 1120 1140 1160 1180 1200 1220 Density (kg/m^3) 90%A + 10%B 0.4 0.3 c 0.2 0.1 0 1120 1140 1160 1180 1200 1220 Density (kg/m^3) Fig. B-10 Variations of power-law coefficients with density for 90% attapulgite + 10% kaolinite slurries. Top: [t; middle: c; bottom: n. B-10 |