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II . » .302 ....v 0.23.0.3 $-13}; \I I... .- :3. ..z ...... .. .. ._ ... - . . .. , I- . . I): J; O 44"- 0 . .0. I0 .1 .1 .0 .fi-AKIQDI'HS nLt-Bvl ktia .0 I 90... - cd00000 q. I. I .... . I o . I! 1 \5: It . . I . . .3... ... .... . I... I '51:-.. . , 00 .. a 910' 0 I a 09!». a.” a m I! . n4 5‘ ..O . 1 . 0.10 II . . .. ,... . .. .a0 . .. .3 _ .§.% ‘1‘ I80 -III0:I . I. I0 . D ..00I‘. I... ‘ -0 O L ii‘ . . I v rI...‘0.0.0hI.II Olv‘JJxaIO-I00 Ihi‘tAII. THEth ‘3. 1‘. 3' ~sfd~cufu lt.‘ . O .;.o {ngIPA :L 4 *"¥V’1iwen S-‘r , - .. w . -. 1 Um .21.: .5. x: ,_ w .' n-I warn}, "u, y . _ ”mum-r Inn-Ian ”3‘3 ’ run _ L'BRARY amogas ABSTRACT FLOW OF POLYVINYLPYRROLIDONE SOLUTIONS THROUGH PACKED BEDS By Ramadas U. Acharya The semi-theoretical Ergun equation is well established for flow of Newtonian fluids through packed beds. This definitive study provides a basis for analysis of effects of non-Newtonian fluid flow through packed beds and is based on data for aqueous solutions of Polyvinylpyrrolidone (PVPL'which are viscous non-Newtonian fluids. A number of equations describing flow phenomena in packed beds are derivable for purely viscous non-Newtonian fluids; all of which reduce to the Blake-Kozeny form in the case of Newtonian fluids. Meter's four parameter model was assumed to characterize the rheolo- gical behaviour of aqueous PVP solutions. The Meter's model analog of the Ergun equation was employed to correlate the pressure drop-flow rate data. Both numerical and analytical techniques were employed in this analysis. Packed bed pressure drop versus flow rate data were obtained for aqueous PVP solutions of concentration 0.5, 1.0, 3.0 and 4.0 percent by weight. It was concluded that the modified Ergun equation may be used to describe results for the 0.5 and 1.0 percent by weight PVP solutions. However, wide deviations between the experimental friction factor and those from the Ergun equation were observed for 3.0 and 4.0 percent solutions. It is speculated that this departure from the modified Ergun equation is a result of surface and viscoelastic effects. Modification in the capillary model may be needed to account for such effects. A comparative study between constant flow rate and constant pressure drop experiments is sug- gested in order to further resolve this matter. FLOW OF POLYVINYLPYRROLIDONE SOLUTIONS THROUGH PACKED BEDS By Np; Ramadas UgiAcharya A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Chemical Engineering 1974 ACKNOWLEDGMENTS I wish to express my sincere appreciation to Dr. Martin C. Hawley for his invaluable guidance and constant encouragement to complete this project. I am grateful to the GAF Corporation for their generous donation of Polyvinylpyrrolidone powder. Above all, I remain thankful to friends who helped at all times to complete this project. ii TABLE ACKNOWLEDGMENTS . . . . . . . . LIST OF TABLES . . . . . . LIST OF FIGURES . . . . . . . . NOMENCLATURE . . . . . . . . . . INTRODUCTION . . . . . . . . . . PACKED BED THEORY AND EQUATIONS EXPERIMENTAL PROCEDURE . . . . RESULTS AND DISCUSSION . . CONCLUSIONS . . . . . . . . . . . BIBLIOGRAPHY . . . . . . . . . . OF CONTENTS APPENDIX A - PACKED BED THEORY AND EQUATIONS APPENDIX B - FLOW RATE-PRESSURE DROP DATA FROM EXPERIMENTS, AND COMPUTER PROGRAM . . . . . . iii Page ii iv vi 12 23 24 25 35 Constant Flow Rate Constant Flow Rate Solutions at 21°C Constant Flow Rate Solutions at 21°C Constant Flow Rate Solutions at 21°C Constant Flow Rate Solutions at 21°C Meter Model Parameters for PVP Solutions as Obtained from a Weissenberg Rheogoniometer at 21°C . . . . . . . . . . . Experimental Data Experimental Data Experimental Data Experimental Data Experimental Data LIST OF TABLES iv for Water at 21°C . . for .50 Percent PVP for 1.00 Percent PVP for 3.00 Percent PVP for 4.00 Percent PVP Page 35 36 37 38 39 40 LIST OF FIGURES Figure Page 1. Schematic diagram of the equipment . . . . . . . . . . . . . . 9 2. Pressure drop-flow rate correlation for flow of 0.5% PVP solutions through packed beds. . . . . . . . . . . . . . . . 13 3. Pressure drOp-flow rate correlation for flow of 1.0% PVP solution through packed beds . . . . . . . . . . . . . . . . 14 4. Pressure drop-flow rate correlation for flow of 3.0% PVP solutions through packed beds. . . . . . . . . . . . . . . . 15 5. Pressure drop—flow rate correlation for flow of 4.0% PVP solution through packed beds . . . . . . . . . . . . . . . . . l6 6. Pressure drop-flow rate correlation for flow Newtonian fluid (water) through packed beds . . . . . . . . . . . . . . . . . l7 7. Comparison of experimental and calculated values of the apparent viscosity for aqueous solutions of PVP . . . . . . . 18 8. Model predicted values of viscosity and wall shear rate for aqugous Solutions Of PVP O O O O O O C O O O O C O O O O O O O 19 NOMENCLATURE Column diameter Particle diameter Experimental friction factor Calculated friction factor Mass flow rate based on superficial velocity Bed length Correction factor for wall effects Effective Reynolds number Flow rate Radius Hydraulic radius Shear stress when n drops down to 1/2(n0+nw) Shear stress Wall shear stress Velocity Average velocity Pressure drop Viscosity Viscosity at zero shear stress Viscosity at infinite shear stress Porosity Density Wall shear rate vi INTRODUCTION A study of non-Newtonian fluid flow through porous media has wide applications in engineering science. Packed bed equations are very useful in individual problems such as industrial filtration of polymer solutions and slurries, movement of aqueous solutions through sand in secondary oil recovery, and design of packed bed reactors and towers. Furthermore, models pr0posed in the literature to describe non-Newtonian fluid behaviour have been tested in most cases only in relatively simple flow geometry. A test of these models for highly complex geometry of packed beds, for example, enhances the reliability and confidence in their application. The most definitive study of flow of Newtonian fluids through porous medium appears to be that of Ergun [1]. Ergun's study provides a basis for similar analysis of non-Newtonian fluid flow. Good discus- sions of this analysis and extension of Ergun equation to viscous non- Newtonian fluids are provided by Bird g£_al.[2], Gaitonde g£_al,[3],‘ Marshall g£_al.[4], Christopher g£_§1.[5], Sadowski g£_gl.[6], and Park 'g£_§l.[7]. Different types of generalized methods to study this problem have been proposed. Aside from dimensional analysis, the main approach to this problem may be categorized as (1) generalized Darcy's law approach, (2) the capillary model combined with a particular rheological equation, or (3) the use of Newtonian equations containing an apparent viscosity evaluated at some appropriate average bed shear rate. So far, 2 the second approach has been most preferred. Bird, Sadowski, Marshall, Gaitonde, and Park are a few among those who used this approach selecting Ellis, Power law, Hershel Bulkley, Spriggs and Meter's rheological equa- tions for a variety of polymer solutions. The capillary model equation for flow in packed bed is a simplifi- cation since the actual flow involves fluids passing through irregular interstices between the particles. Deviations might be expected for non- Newtonian fluid flow through packed beds, due to frequent acceleration and deceleration. A literature survey indicated some evidence of visco- elastic effects and surface adsorption phenomena. Sadowski pointed this out and claimed to have observed viscoelastic and surface adsorption effects for aqueous polymer solutions. Marshall and Metzner also observed similar deviations and argued that the viscoelastic effect is the single cause of deviations. On the contrary, Christopher, Middleman, and Gaitonde, in similar studies of polymeric fluids, observed no such deviations, hence reported absence of viscoelastic effects. ChristOpher correlated Sadowski's data successfully with capillary model equation combined with Power law and surmised that Sadowski's conclusion is an artifact of his modification of Blake-Kozeny equation. More recently Park g£_al. reported occurrence of viscoelastic effects for dilute polymeric solutions. In particular for PVP solu- tions departure from Ergun equation occurred at effective Reynolds number greater than one. It was noted that the deviations were larger for most dilute solution in contrast to the other polymeric solutions. The deviations decreased with an increase in concentration. Thus it was decided to investigate PVP solutions further and extend the data to higher Reynolds numbers. 3 The following assumptions were made in developing the capillary model equations for packed beds in this analysis: (1) The fluid is incompressible. (2) The porous medium is isotropic and of regular geometry. (3) Inertial terms from the equation of motion are deleted. (4) The fluid is homogeneous. PACKED BED THEORY AND EQUATIONS Analyses of non-Newtonian fluid flow through packed bed generally fall into three categories: (1) generalized scaleup, (2) the capillary model combined with a particular rheological equation, and (3) the use of Newtonian equation containing an apparent viscosity evaluated at some average bed shear rate. Approach (1) is an extension of Darcy's law and does not require a rheological model. Darcy's law simply states that for a given bed and Newtonian fluid, flow rate is proportional to the pres- sure drop. As discussed by Park g£_gl.[7], the inherent complexities involved in the solution are quite cumbersome. The use of Newtonian equations containing an apparent viscosity evaluated at some appropriate average bed shear rate is not generally an acceptable solution, although it may turn out to be reliable for a particular solution. The second method as the choice made by the earlier workers, is both reasonable and supported by extensive experimental measurements. It is chosen as the basis for analyzing measurements of this study. In the second method, the packed bed is regarded as a bundle of capillary tubes of complex cross sections and shape. The theory is then developed by applying the results of hydrodynamic analysis of straight tube to the collection of crooked capillaries. The tortuous shape of the capillaries is accommodated by making an appropriate correction to the length by a factor 25/12, as in the Newtonian case. Of course, it becomes necessary to make a suitable choice of rheological equation. 5 Apparently the choice is between a constitutive equation which describes viscoelastic or normal stress effects and equations describing purely viscous flow behaviour. An appropriate choice would be that one which fits the experimental viscometric data for all ranges of shear rate; at least in the range of average bed shear rate. It has been shown that Meter's[8] model given below may be used to predict the non-Newtonian properties of the PVP solutions. no - nan n = 7100+ T (H (1) 1* H T 'm where n = viscosity, no,nm = viscosities at shear rates approaching zero and infinity, T m shear rate when viscosity drops down to 1/2, and o = constant exponent. Park[9] investigated this model for PVP solutions and determined the parameters n”, no, Tm and a. Table 6 contains the parameters for the relevant solutions. Equation (1) can be rewritten as: 0-1 a- -l T]: all :0 Izl m The equations derived in Appendix A based on the hydrodynamic Tlo “.fi—= 1+ analysis of the capillary model of the packed bed for Meter's model may be summarized as follows. The average velocity Vz through the avail- able cross section for flow of the capillary is [In] o-l T 1 +- -—- 2Rh w Tm (V2) = 3 O Trz T] T a-l dTrz (3) TIOTw 1 + -21 -£2' “0 {an 6 where Tw = 2RhAP/2L, shear stress at the wall of the capillary, AP = pressure drop, and L = length of the bed. The effective viscosity “eff is: * 1 * 1 * 1 1 = .1. 1+_Z_EE_)Q- _T_‘2(Es °” _L+_2_(Iv_z)“‘} neff “0 0+1 TIn T10 TIn 0+3 0+1 Tm * a . .- T] 2 1? 2 f* 0 l +, .2 .Ja .+ 4‘ .4! + _ + - - - 1] 0 T 0+1 30+l '1‘ m m * where Tw = 12 DPAP / 25 6M(1-e)L, is a measure of wall shear stress in the bed. The effective viscosity so defined is useful only when 2 * 2 2 * 2 (Um/110) (Tw/Tm) < l . When (nab/110) (Tw/Tm) Z l, superficial velocity V0 based on the column diameter, expressed as a product of average velo- city and porosity e, = c = ‘ 1“ V0 "(V2 *3 I Trz T] T 0-1 dTrz (5) TI T 0 co rz 0 w 1 + — T “0 m 63D:AP n = (6) eff 150 M?(1-.)2v0 and 150(l-e)MT]eff f = (7) D G p 0 This result is good when the void fraction is less than 0.5 and is valid only in the region given by DPGO/(l-e)Mneff <'10, where Co = pVb. For highly turbulent flow in packed beds the friction factor is a func- tion of roughness only, and remains fairly constant and is, f = 1.75 (a) 7 Hence the modified Ergun equation as in Appendix A is: "D - 3. e.__AP .2 :2_ ... 15° +1.75 (9) , 2 I, l-s D G CO M 4p 0 (l'e)me£f If we define f = 2.93. 32. iii. expt 2 I. l-e G M O . and DPGO f calc = 150 / 1 + 1.75 (1-6)Mneff it can be seen that in the low flow regions the logarithmic plot of vs will be a straight line with a slope of -l. fexpt ' (NRe)eff Equation (5) is numerically solved to obtain effective viscosity. Appendix A contains the details and the techniques involved in the analysis. Hence in principle one can modify the capillary tube approach to account for departure from Newtonian flow. In this study the Meter's model equation for flow through a tube packed with spherical particles was applied. EXPERIMENTAL PROCEDURE The schematic diagram (Figure 1) depicts the experimental set up. The equipment consisted of two glass columns of inner diameter 1 inch and 1/2 inch and spherical glass beads of average diameter 0.1621, 0.0597 and 0.0432 centimeters. Segments of glass columns are assembled together with aluminum.flanges. Two stainless steel screens at either end con- tained the glass beads in the column. Extensive care is needed while packing to avoid any possible air entrapment. On either end, at least 6 inches of extra packing above the test section were provided to avoid end effects and foreign particles influencing the flow pattern in the test section. Two pre-calibrated rotameters for low and high flow rates respectively were used. "U" tube mercury manometers served to measure the pressure drop. The packed glass column and the tank containing the solution were immersed in constant temperature water jacket and bath respectively. Extreme care was taken to maintain the temperature at 21 : 0.5°C in the baths. Water was recirculated for the entire period in the bath and the jacket. The average molecular weight of PVP (k-90) used was 360000. Aqueous solutions of PVP of 0.5, 1.0, 3.0, 4.0 percent by weight were prepared in distilled water and filtered to avoid gel. The solution was forced to flow through the bed by a constant nitrogen pressure in the tank. Prior to recording any data the solution was allowed to flow for some time until the refractive index of the xcsa nowouuaz .ucofiawsvo we? we Emummwo oHumEosom 4 .mwm ...ouofimuom 0H. aaaaoo voxuum xawa oonOum cowuoflom uoahfiom r} «VAH coau5H0n HoahHom emu NZ. nouoaoamz nusonoz .IIIIIVAIIIV o>~m> woman 10 solution matched with that of the sample. Constant flow rate and pres- sure drop were the indicators of steady state flow. Pressure drop and flow rate were measured simultaneously under steady state conditions. The bed porosity was calculated from the packed bed data for dis- tilled water by employing Ergun equation and known particle diameter before each run of solution. Only data for which (N less than Re)eff 10 have been used for such bed calibrations. Experimental data were obtained for water in the same range as that of the solutions. RESULTS AND DISCUSSION Results of the experiments are summarized in Figures 2 through 6 as a set of plots depicting the f and fca for each expt vs. (N 1c Re)eff aqueous solution. Effective viscosity vs. wall shear rate plots (Figure 7 and Figure 8) are presented to make further comparison and to gauge the usefulness of Ergun equation to predict viscosity of non-Newtonian fluids. The Newtonian data presented in Figure 6 serves to calibrate the bed and technique. Figures 2 and 3 show the general agreement of 0.5 and 1.0 percent solutions and water with Ergun equation. This agreement is an indication of accuracy and consistency of sets of data. This was felt especially important in view of Park's[9] contrasting conclusions. Having observed greater deviations for 0.5 and 1.0 percent solutions, he concluded that the viscoelastic effects are significant above (NRe)eff equal to 1. The check between the Newtonian fluid and PVP solutions confirm no such viscoelastic effects. These results are consistent with those reported by Gaitonde et a1. and Christopher et a1. even though the flow regions are of considerably higher Reynolds number. The data for 3.0 and 4.0 percent solutions when plotted (Figure 4 and Figure 5) fall well above the theoretical curve. In fact, the data fall on a line somewhat parallel to the theoretical curve. The ratio of the experimental values of friction factor and those given by the Ergun 12 10 ...: O expt f Friction Factor, H 0 1C) N 13 Figure 2. 7 f = £2. 32 .5. , expt 2 I. 1-e . G0 fcalc = WEE—— + 1.75 Re eff - (N ) = EPGO Re eff (1-e)Mneff p Theoretical line 0 0 Experimental points [3 Newtonian fluid , L. I r \ '- \ \\ E \ \ 1o1 102 4 I n 1 L 1 L 1 I I I I L A l A L L Effective Reynolds Number (NRe)eff Pressure drop-flow rate correlation for flow of 0.5% PVP solutions through packed beds. 10 H O N Friction Factor, fexpt .... O 10 11" T f T 10 Figure 3. 14 D _ . f 92.1.2.2. expt 2 L l-e Go 150 Re)eff ,=E.€g Re eff (l-e)Mneff fcalc = (N + 1.75 (N Theoretical line 0 Experimental points [5 Newtonian fluid 1o0 191 L inimjl‘ i L ILALII Effective Reynolds Number, (NRe)eff Pressure drop-flow rate correlation for flow of 1.0% PVP solution through packed beds. 10 T—l TT‘ 1 gi_ 15 10 - u . Q _ x 0 P ‘H 11‘ o . u u (U h. n o o H U .. o H In In 102 - D "‘ g é—Efl _B - expt G 2 L - _ 0 ’ f = _ 150 K calc (NRe e - (NRe)eff = - Theoretical line 0 Experimental points -2 -l 0 1 10 10 I L L j 4 1 11101 j A l : L L l 118 Effective Reynolds Number, (NRe eff Figure 4. Pressure drop-flow rate correlation for flow of 3.01 PVP solutions through packed beds. 10 H O U expt Friction Factor, f 10 10 16 _L L Effective Reynolds Number, (NRe) p- t ' o p L. 1- - . D __ _ f = 92!. .2. .2. . expt 2 I. 1-e - Co E fc IC = (N1? + 1.75 r a Re eff (N ) = EPGO ' Re eff (l-e)MT]eff Theoretical line 0 Experimental points -1 -2 10 L l A k4 _A_1 Lao Figure 5. Pressure drop-flow rate correlation for flow of 4.0% PVP solution through packed beds. 10 H O N Friction Factors, fexpt H O H 10 17 T f .1132 32.1. ' expt 2 I. 1-s . G 0 . f 1 =-(-N—13)°—-+1.75 ca c Re eff (N) = DpGO Re eff (l-s)M1‘|eff . ____ Theoretical line 0 Experimental points -2 0 10 L j 111.1}q 1 A #AJalL&o ._ L - Effective Reynolds number, (NRe)eff Figure 6. Pressure drop-flow rate correlation for flow of Newtonian fluid (water) through packed beds. l8 Viscosity n. qr/cm sec H H Ho Ho 0 o 1H I~ m . I 3003 unennnnno: . O ..3 v5. . 0 r3 3:. D v.2. 3:. w -- o.wa v ererer b F » LP>»»-» » >f>bu>~ >>.>Lh 3-... 3L use 3 S S mason ”one «. noon arcane Q nosvunpoo: 0n mxvonnaonnop nan newnawunoa cause. an n30 >ppanoan 25 26 Park [9] studied extensively the rheological behavior of PVP solutions and determined the parameters conforming to Meter's model. The parame- ters are listed in Table l. 2. Packed bed equations: It will be shown that the packed bed equations for non-Newtonian fluids reduces to the Newtonian form of Blake - Kozney and Ergun equa- tions. The following assumptions are made in the derivation of the relevent equations. 1. The fluid is incompressible. 2. The porous medium is isotropic and of regular geometry. 3. Inertial terms from the equation of motion are deleted. 4. Fluid is homogeneous. 5. Temperature is constant. Consider the flow of non-Newtonian fluid through a circular tube of radius R. I ‘Y 12 O V; .-.I'. 'I'.’I'I'll’l..l Making a momentum balance over the shell of a thickness dr and length L ; the following differential equation is obtained. 27 where Trz = shear stress, r = radius of the shell, and AP = pressure drop. Integrating, T = ég'r +'§ r rz 2L (A-4) The constant C must be zero if the momentum flux is not to be infinite at r=0. Hence _ a - Trz - 2L r (A 5) At the wall, RAP ' (T ) = T = --' (A-6) rz r=R w 2L Trz = Tw(r/R) (A-7) It is assumed that (l) the fluid is in steady state laminar flow, (2) the fluid is time independent, and (3) there is no slip between the fluid and the tube wall. The volumetric flow rate through the cylindri- cal shell of thickness dr and length L is dQ = Vz 2W r dr - (A-8) where Vz = velocity at radius r. R Q = n J) Vz 2r dr 0 org 2 = n J VZ d(r ) (A-9) 0 Integrating by parts, 28 Q =~w J r2 dvz (A-lO) 2 2 R T 2 rz _ R 2 T rz T w w Making these substitutions into equation (A-lO), j?w R2T:z de Q = -" ( )d'r 0 T 2 dr rz W (A-ll) T w dV <6 = s = -—%r Ms)“- nR T d w where (V?) = average velocity. From Meter‘s equation, rz 0-1 dvz Trz 1-+ Tm '- dr = ‘nOJU = Trz n T 0-1 (A-12) n + °° 1+—°'3- 33 00 T 0-1 “0 Tm 14—5—1) T m Hence T 2 0 1 T 1+ r R ['w 3 T111 1 _ 3 J Trz “a (T 0-1 dTrz (A- 3) T 0 rz w 1 + T— 71,—) 0 m_ If no.é> nm, the denominator can be expressed in terms of powers of (nu/n0)(Trz/Tm)' Equation (A-13) can be integrated analytically. 29 Analytical Technique: R Tw 3 Trz)0-1 Tim Trz 0-1 J 4V2) = "'3' I Trz 1 + T— - :fi— "1:— d Trz (A-14) 0 m 0 n1 J: Integrating term by term, T 4 0- -1 nm -1 T 0-1 (V) . ..13__w_ 1+__ _)_ ... 0.9.4.1..(1 z 3 4 0+3 T 0+3 0+1 T T “0 m m w (A-15) 11m 2(TW 2"“2 2 4 (Tw)0-l + 11—0 T" 311+E‘fi 7r" +'+'+' m m Imagine that the packed bed is a bundle of tubes of very complicated cross section with hydraulic radius Rh. The average velocity in the available cross section for flow in a single tube is: 2RhTw 0-1 11000-1 4 2 Tw -1 = «no 1+El'5()() 33+;I(F) ‘m 20_2 (A-16) 2 + (fig) (a) 0+1 + 30+1\T 4:.) + ' + ' The hydraulic radius may be expressed in terms of the void fraction "e" and the wetted surface "a" per unit volume of the bed in the following way 0 = cross section available for flow Rh wetted perimeter = volume available for flow Rh total wetted surface volume of voids) ( volume of bed _ e wetted surface) - ( volume of bed ml 30 where "a" is related to specific surface aV (the total particle surface/the volume of the particle) by a = av(l—e) The quantity av defined in terms of the mean particle diameter DP is a = 6/D V P (A-17) L Rh 6(l-e) Mehta and Hawley [8] modified hydraulic radius for Newtonian fluid flow through packed bed as = wetted surface of spheres +~wetted surface of wall Rh volume of bed ..‘il’z. [l + 6(l-c)] 6(l-e) 6D 4D = .—__;E__ .____IL__ 6(l-e)M where M 6Dc(1-e) + 1 For packed beds the superficial velocity V is given by 0 VO = (V2) 6 substituting for R, Tw in (A-16), 31 v = e 233912 AP 1+ iCRh—j—PTd 12(2RhAPY-l o 6(1-e)M| 21.4110 0+3 211m no 2mm _&_.+__Z__2RhAP 0-1 +.fl2 2 2RhAP 20-2 (A-18) 0+3 0+4. 2LT n 2LT m 0 m -2-+ (ZRhAP)+-+-+-- 0+1 30+1 2LTm A second assumption implicitly made in the foregoing development is that the path of the fluid going through in the bed is of length L; that is the same as the length of the bed. Actually, of course, the liquid tra- verses a very tortuous path, the length of which may be half again as long as the length L. For Newtonian fluids experimental measurements indicated the length be changed to‘25/12/L. It is quite logical to assume the same value for non-Newtonian fluids. Insertion of this value into Equation (A-18) gives 3 2 Ti: 6 D AP T:)0-1)0-1-l V0 = 2 1 + +31" +3+ +1117:- 150 M2 (1- 6:) L110 0’ a c” (A-19) * +'-"_l-_002-:—EZa-22+4 :I‘E-l +-- +110 0+1 30+1T * where Tw = 12 Dp e AP/25 6M(1-e)L. Equation (A-l9) is Meter's model analog of Blake-Kozeny equation for Newtonian fluids; i.e., €3D2AP P v = (A—ZO) O 150 MZL(1-e)2'neff where 32 T* 1 T* 1 ( * 1 1' = .1; 1.+.JE_(JEYI -‘nm _E)a _&_.+ 2 E!) - 'ne f f T10 0+3 Tm 110( Tm 0+3 0+1 Tm (A-2l) n 2 T? 20-2 T? 0-1 1. .:fl .1: .JL_., 4 .11 + ..... “o T 0+1 30+1 T m m However Equation (A-19, (A-21) are applicable when n 2 T* 2 .JE .11 (n0) (Tm) S 1 Numerical technique discussed later in this section solves this problem and has no bounds as above. For highly turbulent flow, friction factor is only a function of roughness. For the flow of fluid through a bed of spheres, the pressure drop AP is given by AP = F/A (A-22) where F is the force exerted on the solid surface and A is the cross sectional area. The friction factor f, a dimensionless quantity, is also called a drag coefficient. It is approximately a constant at higher Reynolds numbers. Consider the fluid flowing through a cylindrical tube as before. The fluid will exert a force F on the solid surface which is equal to: F = A' K f where A' is the surface area of the column or the wetted surface, K is the kinetic energy per unit volume, and f is the friction factor; therefore, 1 .2 K — 2 . (v2) . 33 and A' 2nR L '11 II (21111 L)( % p 022) f (11-23) substituting for R in terms of hydraulic radius, ZRhAP 214-:- o (v; 2) Experimental data for Newtonian fluids indicated that: 6 f = 3.5 Hence 1.75 p(1-€)V3 9.2 (A-24) L 3 e D P Combining Equation (A-20 and Equation (A-24), 150 n sz (14;)2 1.75 p(1-e)V2M AP eff O 0 T = 2 3 + 3 “'25) D e e D P P which is Ergun equation for non-Newtonian fluids. Rewriting in terms of GO’ mass flow rate, and in the dimensionless groups: AP D 63 Mgl-ez —-1 i ——- = 150 +1.75 (A-26) MG 2 I. 1- D Go 0 _JL__ “eff and effective Reynolds number is: DPGO (N ) = _ . (A-27) Re eff (1 e)Mneff 34 Numerical Technique: The Ergun equation so derived is only applicable when (“a/“0)2 (T:/ T1192 5 1 A numerical integration of Equation (A-13) does not impose any such bounds on Ergun equation. With proper substitution for R, Tw * — 1 rz 0-1 q 26Dr Tw 3 +- Tm Vz = 6M(l-e) T*3 J, Trz n T '0-1 w w rz MT) 0 u1 . From Equation (A-20) and (A-28) €3D2AP TI = P - eff 150 M2(l-e)2[ (v2) 8] D c _ p 0 (NRe)eff ‘ M(1-e)'fleff in Equation (A-13) gives dT z (A-28) (A-29) (A-30) With these two definitions Ergun equation is versatile and is applicable at all ranges. The method involves application of Simpson's three point integra- tion formula. The salient feature of the computer program is the Sub- routine Sizsimp which divides the intervals applies Simpson's rule over each interval. . -10 n number of times and This is repeated until APPENDIX B FLOW RATE-PRESSURE DROP DATA FROM EXPERIMENTS AND COMPUTER PROGRAM TARLE - 1 o CONSTANT FLOW DATE EXPERIMENTAL DATA FOR WATER AF 21°C AND FLUID PROPERTIES HE) G ’4 row the-10‘ INONO mhqoo O O o O O "Hm-4 3 .1621 u uuunuun a _1<»C” 35 calc expt NRe,eff AP,Psia r 0 cmzsec cc min QO. 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