CHIEN-BANG WANG A MASTER'S THESIS. requirements for the degree MASTER OF SCIENCE. KANSAS STATE UNIVERSITY Manhattan, Kansas.

Transcription

1 DISPERSION OF NON-NEWTONIAN FLUIDS by CHIEN-BANG WANG B. S., Natinal Taiwan University, 1962 A MASTER'S THESIS submitted in partial fulfillment f the requirements fr the degree MASTER OF SCIENCE Department f Chemical Engineering KANSAS STATE UNIVERSITY Manhattan, Kansas 1965 Apprved by: su)sut#ji *jr -^7^2^- Majr Prfessr

2 LO T* C *- TABLE OF CONTENTS I. INTRODUCTION 1 II. LITERATURE SURVEY k III. CONVECTIVE MODELS 12 Flw thrugh Open Channels Bingham Plastic Mdel Ostwald-de Waele Mdel 17 Flw f the Bingham Plastic with Slip Velcity at the Tube Wall 21 IV. DISPERSION MODEL 40 Flw thrugh a Slit Bingham Plastic Mdel Ellis Mdel 48 Flw thrugh Cylindrical Tubes Bingham Plastic Mdel Ellis Mdel 57 Dispersin f Matter in Turbulent Flw f Nn-Newtnian Fluids Flw thrugh a Slit 76* 2. Flw thrugh Cylindrical Tubes 81 V. EXTENSION OF ARIS'S WORK TO NON-NEWTONIAN FLUID FLOW 87 VI. OUTLINE OF PROPOSED FUTURE WORK 98 ACKNOWLEDGEMENT 101 LITERATURE CITED 102 NOMENCLATURE 105

3 I. INTRODUCTION Cnsiderable interest has been shwn in the study f residence-times in chemical equipment. This is due t the fact that the residence-time distributin gives valuable infrmatin n the nature f the flw thrugh a plant. The ttal quantity f material inside a wrking plant is ften difficult t determine accurately, and a calculatin f the mean residence-time is valuable in giving the material inventry f a plant. Cnsider the case f chemical reactins which, if nt zer rder r autcatalytic, decrease in rate as time prceeds. It is pssible t calculate the average yield f the reactin by means f the fllwing equatin (1) -2_=J^f(8)E(6)dO (1-1) C where C is the mean cncentratin f reagent left in the effluent, C n is the cncentratin f reagent in the feed, f(9) is the kinetic equatin which gives C/C Q as a functin f time in a batch reactin, and E(6) is the residence-time distributin functin. This expressin is nly crrect fr first rder chemical reactins. Hwever, the residence-time study prvides valuable infrmatin fr the explanatin f reactr behavir even in cases where the reactin is nt first rder. Anther field f fluid flw which is increasing in imprtance industrially is the behavir f fluids which d nt bey Newtn's law f viscsity in mtin. Because f this negative definitin f nn-newtnian behavir, n single equatin can describe exactly the shear-stress and shear-rate relatinships f all such materials ver all ranges f shear rates. Numerus empirical equatins have been prpsed t express the steady state relatin between the shear- stress and shear- rate. Amng these the Ostwald-de Waele mdel and the

4 Bingham plastic mdel are the mst generally useful tw-cnstant mdels, and the Ellis mdel is the simplest three-cnstant mdel. Mst f the investigatins f fluid dispersin have been cnfined t Newtnian fluids. But fluid dispersin in the prcessing f nn-newtnian fluids is as imprtant as it is in the prcessing f Newtnian fluids. This thesis will be cncerned mainly with the develpment f mathematical mdels which characterize the dispersin f nn-newtnian fluids in flw systems. A generalized mathematical expressin fr diffusin in a flw system at cnstant temperature and pressure is (2)! -+ V-VG =DV 2 Cf R (1-2) where D is assumed t be independent f C. Suppse that a fluid flw thrugh a cylindrical tube with the symmetrical cncentratin distributin abut the central line f the tube. When there is n chemical reactin in the system, Equatin (2) becmes S-^+iS+S^xWg (1 " 3) It wuld be difficult t find a cmplete slutin f Equatin (3) which gives the value f C fr all values f r, x and t when the distributin C at time t=0 is knwn. Hwever, apprximate slutins can be fund which are valid in certain limiting cnditins. When the effects f bth crss-sectinal and lngitudinal diffusins are negligible, the steady-state velcity prfile becmes the nly factr gverning the apparent fluid dispersin. The flw mdel crrespnding t such a cnditin is usually called the cnvective mdel. When the variatin f axial velcity with crss-sectinal psitin and the crss- sectinal material transprt by mlecular diffusin are assumed t be the dminant dispersin mechanisms, the slutin f Equatin (3) is called

5 the dispersin mdel. Bth mdels are treated in this thesis fr varius nn-newtnian velcity distributin functins, such as Bingham plastic, Ostwald-de Waele and Ellis mdels and varius flw gemetries, such as cylindrical tubes, slits and pen channels. The analysis f turbulent flw f nn-newtnian fluids thrugh smth rund tubes was perfrmed fr the first time by Ddge and Metzner (3). The analysis permitted the predictin f nn-newtnian turbulent velcity prfiles. The dispersin mdel f turbulent nn-newtnian flw is als discussed in this thesis.

6 II. LITERATURE SURVEY Fluid is passed thrugh prcess equipment s that it may be mdified in ne way r anther. T predict the perfrmance f equipment, we must knw the cmplete flw pattern f the fluid within the vessel. Because f the practical difficulties cnnected with btaining and interpreting such infrmatin, an alternate apprach is used, which requires knwledge nly f hw lng different elements f fluid remain in the vessel. Thugh this partial infrmatin is nt sufficient t cmpletely define the nnideal flw within the vessel, it is relatively simple t btain. The field tday can be divided int several areas. One f these is hw residence-time distributin functins can be measured experimentally. Anther area which has had cnsiderable attentin is the theretical derivatin f the relatinships between effective axial dispersin cefficients and dispersin mdels. It is the purpse f this chapter t review the present situatin in this field. The cnvective mdel f dispersin fr Nex-rtnian fluids in laminar flw was prpsed by many investigatrs (4-, 5) t predict residence-time distributin functins fr flw in circular cnduits. The cummulative age distributin at the utlet f the system, ; which crrespnds t the respnse t a step functin input f a tracer withut entrance effect being cnsidered, is expressed as F(e)= i L- fr e^i (20) 2 2 = fr e<i "" 2 (2-1) where Q is the dimensinless time defined by = tv x /L. The exit age distributin, which crrespnds t the respnse t a Dirac delta functin input f

7 a tracer, is 1(9)= 1 fr e^- I? 2 = fr 0^i 2 (2-2) If, hwever, the entrance effect cannt be neglected, the initial distributin f the tracer is unifrm spacewise alng the tube. Fr this case the curaraulative and exit age distributins were fund t be F(9)= 1 - -W JL fr e^i (2-3) = fr 0< I 2 S(9)= -i- fr Q>± ze^ 2 (2-^) = fr 0^- 2 In rder t give a mre precise descriptin f the dispersin characteristics f fluids, axial and radial diffusin and the velcity prfile shuld be cnsidered simultaneusly. In a series f papers (6, 7, 8), Taylr has treated the dispersin f sluble natter in a slvent flwing thrugh a circular tube. Fr a Newtnian fluid in laminar flw, the distributin f cncentratin, C, f the sluble material depends n the balance between cnvectin alng the tube due t variatin in velcity ver the crss sectin and n crss-sectinal mlecular diffusin. The crrespnding partial differential equatin is C(S + S + r 5? )= S? + V» (1 " i M ] (2-5)

8 Here, D, the cefficient f mlecular diffusin, is assumed t be independent f C. In general, the transfer f C alng the tube by mlecular diffusin is small cmpared with that prduced by cnvectin. It is thus assumed that 2 2 C is negligible cmpared with -2_Q. a. 1 _?_ The cnditin necessary fr a x 2 ar 2 r Sr. this t be true can be expressed as (8) R 2? 2 D >> _ (2_ 6) 48D The transprt equatin therefre takes the frm subject t the bundary cnditins that the wall f the tube is impermeable and that the cncentratin distributin is symmetrical with respect t the center line f the tube, r JL =0, at r = R and r = (2-8) 5 r It wuld be difficult t find the cmplete slutin f Equatin (7); n the ther hand, certain apprximate slutins d exist which are valid under certain limiting cnditins. One such apprximate slutin cnsiders dispersin by cnvectin alne, which applies t the case when bth the crsssectinal diffusin and lngitudinal diffusin are negligible, and the steady state velcity prfile becmes the nly factr gverning the apparent fluid dispersin. Sme calculated distributins f G alng a tube have been given by Taylr, which can be reduced t the residence-time distributin functins given in Equatins (1) thrugh (4). Anther apprximate slutin suggested by Taylr is that the time necessary fr a radial variatin in C t die dwn wing t radial diffusin is much shrter than the time necessary fr an. appreciable change in C t ccur thrugh lngitudinal cnvectin. This can

9 1 ). be expressed by the cnditin (6) _ >> 2r2 (2-9) V x (3.8) 2 D where V is the mean speed f flw and L is the lngitudinal extent f the x regin in which dc/3x is appreciable. It is cnvenient in the present discussin t define cncentratin and velcity relative t. axes which mve with the mean flw. The velcity relative t these axes is V = 2V (1 - -) - V = V (1-2E- xi x' R 2 x x s R 2 (2-10) Letting ^ = r/r, the equatin f diffusin becmes ~ C SC_R 2 ', ^C, x/ "^2 + 5^"F7t + u " * ; Tx~ n *2nSC (9 T-.N K 1±) where x, = x -V x t. Fr small radial variatins in C, partial equilibrium may be assumed, that is, the rate f change with time is equated t zer. Since the mean velcity acrss planes fr which X-, is cnstant is zer, the transfer f C acrss such planes depends nly n the radial variatin f C. In this calculatin, ^C/^x. is taken t be independent f. The small radial variatin in C can therefre be calculated frm the equatin 2- ±!c + iac = i\ (1 2 2 _c_ ) ^ 2 1*1 V ^x x (2 _ 12) Since ^C^x., is assumed t be independent f, and if C is the mean cncentratin ver a sectin, 3 c/dx, is indistinguishable frm ^C m /&Xn Therefre, Equatin (12) can be slved t give

10 8 The rate at which C is transprted acrss a sectin at x, is -1 Q= 2 2TfR J* V x (l - 2^ 2 )C^d (2-14) Inserting the value f C frm Equatin (13). Equatin (14) is fund t be 2 R 2^2 3fi, Taylr has cncluded that the cmbined effects f lngitudinal cnvectin and radial mlecular diffusin are equivalent t the transfer f the slute acrss planes which mve with the mean speed f the flw and the rate f this transfer is equal t that which diffusivity, K, wuld give t a statinary fluid, as shwn by Tr2 K= (2-16) 48D The analgus prblem f dispersin in turbulent flw can be slved in the same manner (7). Taylr has assumed in his analysis that the universal velcity distributin in a pipe and Reynlds ' analgy that the transfer f matter, heat, and mmentum by turbulence are exactly analgus. The apparent diffusin cefficient K under this case has been fund t be 10.1RV\ r K= 7.14RV If. Here R is the radius f the pipe, V x is the mean flw velcity, f is the Fanning frictin factr and V^ the frictin velcity. In a later paper {9) Aris has presented a new analysis in which the restrictins impsed by Taylr are remved. He btained this by fixing attentin n the mvement f the center f gravity f the distributin f slute and the grwth f its higher mments. It has been shwn that the rate f grwth f the variance is prprtinal t the sum f the mlecular diffusin 2 2 cefficient D, and the Taylr dispersin cefficient KR V /D. The value f K fr the Newtnian system has been given by Taylr t be l/48 fr laminar flw

11 and 10.1 fr turbulent flw (6, 7). Aris (10) has als applied this methd t the case in which the slute can pass int anther fluid phase flwing thrugh a tube. The apparent diffusin cefficient has been shwn t be the sun f the mlecular diffusin cefficient and the Taylr dispersin cefficients in the t;/ phases, and a term due t the finite rate f partitin between them. In anther paper (11), Aris has treated the case f a viscus flw under a pulsating pressure gradient. It has been fund that the Taylr diffusin cefficient cntains terns prprtinal t the square f the amplitude f the pressure pulsatins. But the cefficients f such terms rarely cntribute mre than a fractin f l/l28 t the ttal dispersin cefficient. Taylr's treatment f turbulent flw is valid nly fr high Reynlds numbers, hwever, because the velcity prfile used in the treatment is valid nly when the laminar sublayer and transitin layers are negligibly small. Tichacek and c-wrkers (12) have refined Taylr's methd by including the effect f mlecular diffusin and by intrducing the experimental velcity prfiles rather than a generalized prfile. They have included a first rder apprximatin t 9C/dxas a functin f the prfile. They have als shwn that the effects f axial turbulent diffusin are negligible cmpared t the mixing caused by radial differences in the velcity. Accrding t their analysis, Dj/V d is dependent f the Reynlds number, pipe rughness, and Schmidt number. They have reprted that their theretical data are applicable with less than 25$ errr as shwn by cmparisn with Taylr's experimental data. Giddings and Seager (13) have theretically and experimentally investigated a methd siiailar in peratin t chrmatgraphy techniques fr measuring a wide range f diffusin cefficients. Their experimental wrk deals with

12 10 gaseus diffusin cefficients measured at varius flw velcities, cncentratins, etc. The final equatin is where h is the height equivalent t a theretical plate in a typical gas chrmatgraph and the right-hand side is the sum f the mlecular diffusin cefficient and Taylr's dispersin cefficient. Bailey and Ggarty (1^) have perfrmed experiments in which a dilute slutin f ptassium permanganate is displaced by water. They als develped a numerical methd and btained an accurate slutin f the mixing equatin when lngitudinal diffusin is neglected. They have reprted that gd agreement des results, exist between the numerical slutin and the given experimental that the mixing zne lengths resulting frm their experiment are nearly the same as thse predicted by Taylr fr a narrw range f dimensinless time, 6 = Dt/R. Hwever, experiments at varius flw times with a fixed velcity have shwn that experimental dispersin cefficients increased slwly with time. Burnia and c-wrkers (15) have measured the lngitudinal dispersin f a finite slug f gas at varius velcities by using a gas (1,3-butadiene) that absrbs light in the ultra-vilet regin and passing the dispersed slug thrugh a narrw beam f ultra-vilet light f wave-length 250 m/a. The results indicate that Taylr's apprximatin des nt apply fr values f V x belw 2 cm/sec, and that the criterin O/D» 6.9 is very imprtant while the upper criterin 4L/R» V X R/D is less imprtant because Taylr's apprximatin appears t apply best at the higher velcities. Farrell and Lenard (16) als derived a slutin fr the laminar flw prblem when the axial mlecular diffusin is neglected. Their slutin

13 11 invlved a series f cigen functins f the Laplace transfrmatin f the cncentratin. The mments f the cncentratin distributin can readily be evaluated by making use f a cmputer. Taylr and Aris bth have cncluded that an effective axial dispersin cefficient K can als be used in turbulent flw. This cefficient has been fund t be a functin f the well-knwn Fanning frictin factr. Bischff and Levenspiel (17) have extended Aris' thery t include a linear rate prcess, and have used the result t cnstruct cmprehensive crrelatins f dispersin cefficients. Hawthrn (18) has cnsidered the temperature effect f viscsity n the dispersin cefficient and has fund that it can be altered by a factr f tw in laminar flw, but that there is little effect fr fully develped turbulent flw. The analysis f dispersin accmpanying the flw f a nn-newtnian fluid has recently been cnsidered by Fan and Hwang (19). They have derived cnvective mdels fr Bingham plastic and Ostwald-de Waele fluids. The dispersin mdel fr Ostwald-de Waele fluids flwing thrugh a circular tube has als been derived fr which Taylr's results can be cnsidered as a special case f Fan and Hwang's expressins.

14 tr 12 III. CONVECTIVE MODELS A general discussin f residence-tine distributins fr sme flw mdels has been made by Fan and Hwang (19). First f all, they have treated the cnvective mdel extensively fr fluids which can be represented by the Ostwald-de Waele and Bingham plastic mdels. In this chapter, the similarity f residence-time distributins between fluid flw thrugh slits (pening between tw parallel plates) and fluid flw thrugh pen channels iri.ll be verified. The cnvective mdel fr the laminar flw f the Bingham plastic fluid which is mre general than that derived by Fan and Hwang (19) will als be derived. Cnstant mlecular diffusin cefficients, steady and is- thermal flw situatins, and cnstant vessel gemetries will be assumed in the present analysis. FLOW THROUGH OPEN CHANNELS 1. Bingham Plastic Mdel The steady- state rhelgical behavir f Bingham plastic fluids can be expressed as (2) =-{w ~ }t fri ($:t)± T* (3-D A = fri("c:r)<r (3-2) 2 where >* and X are parameters which characterize a fluid. Under cnditins fr which a ne-dimensinal rhelgical statement in rectangular crdinates is valid (see Fig. (l)), Equatins (1) and (2) reduce t tyx= T 0->\) J. Xyx >T (3-3)

15 13 CD,_ O c O \_ <D 3 <*- v- C v. 5 JZ *- z) v_ Q) a> c Z) E i c c Q. CO * "* 3 CO u CO Z5 O CO X a i tsl v: > <D Q. JZ O a l- G. Li. O CD c Q. - Ll

16 14 dv -r-2 = fr -c < T A (3-^) dy yx v respectively, and the equatin f mtin reduces t (2) JLr =.pg csf (3-5) Mailing use f the bundary cnditins at the liquid-gas interface X^ - at y = (3-6) The equatin f mtin may be integrated t give "C^-fgy sf. (3-7) Cmbining Equatins (3) and (7), and integrating the resulting equatin subject t the fllwing bundary cnditin V x = at y -S (3-8) ne btains v x = -' cs 4F Li + lr y + c i C3-9) and c = eeii^s^ _ T i s (3 _10) 1 2^ ^0 Hence, the velcity distributin is V x - PES* cs * [l - (g) 2 ] -Zgf (1 - ) fry>y Q (3-11) 'O m 2^ l S J h S ~ where fg cs^f

17 ) 15 Intrducing the dimensinless distance variable, = -, the velcity distributin nay als be written as Z... V n = pcr ^ cs ^ (1-4 r fr y < y (3-13) m ^u u r V = ^2-1 ((l ) -(.4 ) ] 2H Q,y > ^ = V n 1 - I s -li- [ ] d-4> fr y > y (3-14- The mean velcity f the fluid is btained by summing all the velcities ver a crss-sectin and then dividing by the crss-sectinal area, that is, J J V x (y)dydz K-ffi J J dydz 0-15) where W is the width f the channel. Therefre 5l m Jy»l (l-i ) 2J. fk^wy (j-t t { 3 ) (3-16) Dividing Equatin (16) by Equatin (13) yields V x = v m p (3-17) wnere p = i-ii + l* 3. ilia (3 _ 18) v 2

18 16 When the entrance effect is negligible, the cumulative age distributin, which is a respnse t a unit step functin input, is ^ e > = ^r/' f vjy)*&* (3-19) y,_ i r y f, <?-*> 2 = 4- / :y m dy vji-i^iiljdy O-20) "x SV *% ' <l-i > = ^(9) + 5^(6) (3-21) where F (9) = -1- J 7 Vm dy = ^0. U s (0 - p) (3-22) Frm Equatin (1*0 ne has -V^-Mi^ and <L (I) = de * ^ (1 "^L. (3-2*+) 2 Ji- A 2 36 Substituting these equatins int Equatin (21) yields _r 6 #9)«j[ pft-tp) ~ 36 p 3S 3 JTTi ae

19 1? = (i - u fr 9 > r' (3-25) The respnse t a unit step functin input is therefre 34, f(6) = ^.,,, f s (e-s» + (i-4 ) JH 8 3/2 jh. + 1 I/? fr e > p (3-26) = fr 6 < P The 13(8) functin, which is the respnse t a Dirac delta functin input f a tracer, is s( e) = ^.S(9-^ + _ (1 -^- 2j3 ^ ^ 38 fr > = p (3-2?) = fr 8 < p where 5(6 - - ) is a Dirac delta functin defined by S(8- p)= fr e ^ A (3-23) S > fr e = p and /"s<6- h) do-l (3-29) 2. Ostwald-de Waele Mdel The Ostwald-de VJaele Mdel fluid may be characterized by the fllwing

20 18 relatinship between the shear stress tensr and the rate f defrmatin tensr (2) --H/FI7I) }s (3-30) Under cnditins fr which a ne-dimensinal rhelgical statement is valid, Equatin (30) in rectangular crdinates may be written as (see Fig. (1)) I = - m dv. x dy tf-l /dv. dy (3-31) dv, where m and \) are rhelgical parameters. Since - is always negative fr dy the given system, Equatin (31) reduces t dvv i/ (3-32) The mmentum-flux distributin has been fund t be T = fgy cs if (3-7) Substitutin f Equatin (32) int Equatin (7) gives the fllwing differential equatin fr the velcity distributins _ f^ = ( S2L JL) dy ei (3-33) This equatin is integrated subject t the bundary cnditin t give V - 'x at y = S (3-3*0 (3-35) where n = -. The maximum velcity V ccurs at y = 0; that is in n < n+1 m N m ' n+1 (3-36)

21 19 Cmbining Equatins (35) and (36) yields T X W - T (f ) n+1 3 (3-37) The average velcity, V, ver a crss-sectin f the film is btained as fllws: d -vj'll-cr 1 ]^) = ^v (3-33) n+2 m \wi The cumulative age distributin f an Ostwald-de Waele fluid flwing thrugh an pen channel when the entrance effect is negligible is F(8) = 4- f V (y)dy y = i J* [ 1 - ( ) n+1] 'X d(g) (3-39) Frm Equatin (37) ne has z = a - V 1 1 n+1 1 V x (n+2) J

22 20 and ± (Z) 1 (n+2) 8' 1 - n+1 (nt2)e -ft n+i (3-^1) Substituting Equatins (40 ) and (41) int Equatin (39) shws that n+2j n+l fl 3 I (n+2) 8 J n+2 (3-42) Therefre "" f(9) = 4p n+2 n+1 n _ 3 n+2 n+1 L,n+2v J d8 fr Q > SJi u - n+2 Wi ; (3-43) fr e < n+2 The residence-time distributin, which is the respnse t a Dirac delta functin input, is btained by taking the derivative f F(0) with respect t 0, that is, E(0) = 1 n+2 e 3f 1 - /n+2xnj (ntl )0 n n+1 fr 9 > - n+1 n+2 (3-44) fr < 2±1 n+2 VJhen the entrance effect is cnsidered, the tracer is unifrmly distributed in the entrance sectin. The cumulative age distributin at the " utlet f the system, which crrespnds t the respnse t a step functin input f a tracer, is Z(0)= J I J dy

23 i 21 Therefre.0 F(9) - 1 J %, 1 db, fr. e >?il ".+1 (n+2)? " n+2 r, rrr ] = fr 9<~ n+2 The respnse t a Dirac delta functin input is then s(e) = JL. ~ n+2 n ~ nt2 I fr a > ^nti ;e = fr fl ^ nt2 Cmparing the distributin functins derived in this chapter fr fluids flwing thrugh an pen channel with thse derived by Fan and Hwang (19) fr fluids flwing thrugh a slit, it is seen that the frms f the velcity and cncentratin distributins are exactly identical. The differences arise frm the fact that extensive prperties, such as crss-sectinal area, flw rate, and ttal tracer injected in this analysis are reduced t ne half f thse in Fan and Hwang's analysis. Thus, it is nt surprising t find that bth flw gemetries result in the same residence-time distributin functins. FLOW OF THE BINGHAM PLASTIC WITH SLIP VELOCITY AT THE T:J3E WALL The cnvective mdel fr Bingham plastics will be treated by using a generalized velcity prfile fr which the slippage at the tube wall is cnsidered. The mmentum flux distributin fr the flw f fluid thrugh a circular tube has been given as (2) (see Fig. (2)) T ra -( -?0 ~ pl 2L >r n

24 22 O p 2 ' c en c 3 <4- CO

25 23 in which P = p - pgz and where p is the pressure acting n the given system. The ne-dimensinal rhelgical statement f the Bingham plastic in cylindrical crdinates is fund t be (2) dv T = t - u ; fr "C > T (3- L i-6) c rx r dr ** - w ' Cmbining Equatins (^5) and (^6) and applying the bundary cnditin V.= Vp at r R, x it the velcity distributin is d-i) where "~ % and V is the maximum velcity f the fluid. The average velcity f the fluid is calculated by summing all the velcities ver a crss-sectin and then dividing by the crss-sectinal area as 2 R J / V x (r)rdrd9 J J rdrdb (i -l r

26 24 = y«+ (1 - \) v R (3-50) where l = 3? = i (3 4 2^ + ^j (>5l) The minimum residence time f the fluid will be defined as min v m ~ vm ( 2(l-(l- )v] (3-52) where V is a slip velcity factr and is defined by V = vr/v x * ^ ' explici t expressin f in terms f V can be btained frm Equatin (48) as R X = L + (1 :- $ ) / 5l1-!s (3-53) V m R Substituting Equatin (52), V = v r/^x and V x /V x = int ^uat 10*1 (53) yields -* ^ I (3-54) r t, 2 1- l-^yl-cj/e I = L + (1-1 ) R JN 2(1 -V) and j &(f)- - j^ (3-55) 46 Jl-/J[l- (1- )4<A/2S In respnse t a unit step functin input, the cumulative age

27 25 distributin is?(0)= -4- I V x (r)rdrd0 (3-56) (3-57) - i^w + z 2 (0) (3-58) where h. = ir 1.-, *V V n rdr!^(i-a. )yj. Us {e-- l - (l. )V] l (3-59) and z 2 ce> fn^-vl 1 -?^!] rdr x u-u' ^/^^-'"^^KJ^lU (3.60) Substitutin f Equatins (5*0 and (55) int Equatin (60) reveals that Z 2(8)=J G 2[l- d> (1- )V) i..+ (i -<*)V- A 2 29 i-i i a-$ ) «a/t-v 0-1 d6 JT^v

28 3* 26./iv I /T^7.rrz^z I VI~^2 "-'",. /T7«> ' a 3/2 ^ii^2 3 * e ^[i-u- %)i% A /2[i.(i. )yje-d} 3^ 1/2 fr a _ ^ e < i (3-61) r 2[i-(i- f)v] The cmplete expressin f F(9) is then F(9)«!kiuj. * (i-$> < f (l4 n )A 2 1-J( 2V,,,71^7 //i^v<* Wi-Ve 2 1 rrr~ /2 3eC,1/2 = ^ &<& $ * * **& (3-62) = 1 fr > i where A=l- (i- )V (3-63) The residence-time distributin, which is the respnse t a Dirac delta functin input, E(e) is 2 V^<,* l.^^i^ f ^0 ^ _ "*<) s<e- >+ (' )

29 27 fr ^9 ^ i 2A ^ (3-6*0 = elsewhere When the entrance effect cannt be neglected, the F- functin can be calculated by -i r 2TT r r F(0)= -i- I rdrd0 (3-65) Thus ^ S 2A ijliv l <* 29A/I3 7 Jl~7 J f0p f^ " J - fr e < ^ (3-66) = 1 fr e > ~ y and the crrespnding S- curve is ^ 2A / _2 I / r n~z J /it/ (3-67) = elsewhere When y= 0, the slip velcity at the wall f the tube is zer, Equatins (62), (6^0, (66) and (67) respectively becme a/^1q J 29-( ^J 2 *Q ^29-01 I /2 3^2 e 1/2 J

30 28 fr > - ~ 2 = fr 6^ (3-68) 2, 5 (6> = %S(6- f ) + i^.(-k= + l. ) fr e > (3-69) = fr < 2 m, %! Us(9. ) + (1. J), fks. hk + Lk ] fr > (3-70) = fr e < 2 L s) = % $. %) + id^d^ + i-i) * ^ fr 9 > ^ - 2 (3-71) * fr < - 2 When V=0 and jj Q 0, the abve distributins reduce t Equatins (2-1), (2-2), (2-3), and (2-4) respectively, -which have been derived fr the Newtnian flw (4,5). A family f numerically cmputed F and E curves with flw behavir index Q and slip velcity factr V as parameters are shwn in Figures (3) thrugh (12). Since mlecular diffusin has been neglected, certain pints f discntinuity may ccur in the distributin curves due t the irregular

31 29 & J _ O O O II XXr^ II II II VVN Y^"s? VV^ il CV a u-l,*0 O CD d CO d a J 5? LL (9)d

32 lo 30 II ns* GO ' CM. " d d u it n i: 11 O O O V^P» VM^» V^*- VM""s )C^\ CM m m ' 00) 3

33 31 O CO '3 c CD Si \-> ca CD O U D "c c «.^_ «.%_ j CJ E h-.03 c «*- O c u< s> c O r CO «*- r n r - b.= CO > k-\ O 5 2 Ifi

34 32 Q CM CD. CO A3 r < CD i_ CD - O* t " C\J CO. CD O i; ii i! ii ii O Wn \^ \^j^ v^tn \i^>» E sz.e b O CD «.; CD c1 CD c c U. Q cd CO ml -c c 00 O' CD st ro (M CO) 3

35 33 * ji C" vz O "3 0) g - *} c CO c c LI c * CO & «. _ «*>- c CD 'E CO > D HI u CJ O (0) d

36 ^ y± sfr O M c V -CO. CD. - CO r y ~J <- >. ^v CI w CO -E cm. c c CG O CD CD O c O c CO CD r V- c CD Q c c 5 s CO c CD 3 O J CM 3 O iii CD c [7 r CM OJ lo. O) 3 '

37 35 c CD *^? "t O c c i_ - II II II II It CM " u_ O II _ (6)d CD. ^. *. CJ O Ll

38 36 "]. CM Si " t. &>. <f> c fluid ed nde! nidr - <-. c "<5 CD ^ " OJ. CD I- CO Q lw entr cu " *^_ L. -'J-> i- <_ 3 Q O c. c L) c Q a CO CD O 3 1 O CO cj c CM O" wl J O si- ro CM (0)

39 ' _ 37 - II v^ II w? [I v*? II II v^ w$ <* C " CO c I! C OL) a -a CD f.o c (9)" Q u CO 2 CO V- c *C/> c: v. O v. c CD.i 5 "^ c O.Q > O -> ^3 O "r_ - 11 ii «ii» l*r? \v? iv? **»*"> CO O" Pi > J* ^ i d CM. O JO

40 38 CD O" m CO II 10^ " >^? ii VX> II Wn It VVN ^ '5 d i! a e S-L "^ c CD t O HŌ >; c (0)3 CO CO 0) c # CO {^ 0) E Q c LLI1 C O D -> CO <D 5 *- C LU.b cd LO CO / is Si.-? -C II I! 11 _i. O". 1 0J, O., CD, C. ^r., CM O a O " (0)d Ll

41 39 Thelgical behavir f a Bingham plastic fluid and the slip velcity at the tube wall, l&en, becmes 0, the flw system reduces t the Newtnian case and the resulting cncentratin distributin are analgus with thse prpsed by ther investigatrs (4,5). Hwever, the E-curve and F-curve are defrmed in actual bservatins since the effect f mlecular diffusin des exist in actual systems.

42 40 IV. DISPERSION MODEL A generalized mathematical expressin fr diffusin in the flw system at cnstant temperature and pressure has been given (2) where D, the mlecular diffusin cefficient, is assumed t be cnstant. Suppse that there is n chemical reactin ccurring in a fluid which flws thrugh a circular tube and that the cncentratin distributin is symmetrical abut the central line f the tube. The diffusin equatin becmes' (see Fig. (2)) S 2 C,,. C C,& 2 C, ldc, it - D^2 + r S7 + ^2> - V x^) a3e (4-1) Taylr (8) has shwn that under the cnditin E >^ D (4-2) the transfer f C alng the tube by mlecular diffusin is small cmpared C with that prduced by cnvectin and thus the ^ term may be neglected. & He has als shwn that, when the time necessary fr a radial variatin in C t die dwn wing t radial diffusin is much shrter than the time necessary fr an appreciable change in C t ccur thrugh lngitudinal cnvectin, an apprximate slutin can be btained after certain simplificatins. The mdel crrespnding t such a slutin f the partial differential equatin f diffusin is called the dispersin mdel. Taylr's assumptin can als be expressed as (6) V m (3.8) 2 D This is applicable t all types f flw mdels as lng as fluids flw thrugh circular tubes and the ~ term can be taken as zer in the derivatin (6). dx If rectangular crdinates are used (see Fig. (13) )» a crrespnding simplified expressin f the partial differential equatin fr diffusin is

43 UL O CO *3 <-: O 3 O ^. 00 CO CD ^, O C Q. O u_ O CL g>

44 42 In rder t find the cnditins under which Taylr's limiting cnditin maybe valid, it is necessary t calculate hw rapidly cncentratin becmes unifrm due t mlecular diffusin. The gverning differential equatin is du 2 *y 2 subject t the bundary cnditins = at y = and ± % (4-6) sy It shuld be nted that the velcity distributin f the fluid ver the crss sectin des exist. Intrducing the diraensinless variables tv j ne btains Equatin (5) in diraensinless frm as 3C _ Dt a 2 C (4-7) The technique f separatin f variables is used t slve this equatin, r in ther wrds, C = F(9)G(4). Substitutin f this relatin int Equatin (7) gives Fde s 2G de 2 This equatin has the fllwing slutins: F(6) = Ax e" a G (4-9) i 2? I 2 2~ G(0 = A 2 cs (J^i) + A3 sin (J ^4-4) (^-10)

45 , 43 -I» ^2 an<^ ^3 are cnstants ^ Dc determined by the bundary cnditins. JU must be zer because the cncentratin distributin is.etrical arund the center line f the tube. Equatin (6) requires that Therefre i 2 sin / JC = a/ Dt /^X = r.tt. n = 0, 1,... n (4-11) n Dt The rt f this equatin crrespnding t the lwest value f d, is <C = n 2 4 (4-12) 1 r and the slutin f Equatin (7) takes the frm C = A e~v cs ( JSLL. ) (4-13) w lit The fact that the cncentratin at 6 = and = is 1 ensures that A = 1. Frm Equatin (13 ) the time necessary t decay the central cncentratin dwn t e~ f its initial value is Jr Dlf =1 u, 101 D (4-14) i implies that L Vm >> D (4-15) in rder that Taylr's secnd limiting cnditin may be applicable. Equatin (15) means that the time necessary fr appreciable effects t appear, wing t cnvective transprt, is lng cmpared with crss-sectinal variatins f cncentratin thrugh the actin f the mlecular diffusin. The systems studied in this chapter are thse in which the dispersin

46 44 in steady flw is due t the cmbined actin f cnvectin parallel t the plates and mlecular diffusin in the directin perpendicular t the plates. Restrictins suggested by Taylr are basically fllwed in the present analysis, FLOW THROUGH A SLIT 1. Bingham plastic mdel The diffusin equatin becmes I!» D i-c v(y) -^ (4-4) Vfen a ne-dimensinal rhelgical statement is valid, the velcity distributin f the Bingham plastic fluid in rectangular crdinates is where ( 1 > ^ (-y1 = \[ 1 -^ ^ ]. (1 - i ) fr Irl S y (3-14) 2 V = PZ* cs ^ (1 - I )' ^0 (3-13) and v x= 3 f\ (3-1?) Cmbining Equatins (4) and (3-14) yields g-^-^j^ji, 2., ttz\ ; 2 \ fr yl, 7l ^ = D a!s. v (4-18) it = u 73" v»di fot iri < yfl Intrducing the dimensihless variables

47 k 5 tv ft x _ t > I V t I b T. I 5. %.- X I L ne btains Equatin (18) in dimensinless frm as ac = Dt a C J3_ []_ ^ ' S2 zf" 2 M 2 (t-4nri 0' (l-^) 2j - frj I _ Dt a> c 3 a c 2 ' a^ 2 "' 2?' M fr ^ < (4-19) with the bundary cnditins H = at = and 1 (4-20) since the plates are impermeable. Fr cnvenience the derivative fllwing the mean speed f flw is intrduced. Relative t these axes, the velcity distributin is V, = V v - xl x V x -l$-«1 ( ^-*> 2 " 2p a- 2 ] V. = %-^\ *» $ >l * 4< 4 (4-21) r V xl= V x (* + 2) (l-4 )' ((l^) 3-3(^^) 2 J fr J >l a &-i >- * n <i (4-22) '0 "' >0' Let 7i= 7-9 Equatin (19) becmes

48 ; 46 E - l Jl _ i 2L V 7 0' "^ '0' J 3fl, ^ - s f n-m 3-3^- n 2 1 * c fr > > «St ^fc 1 (1 > )3 _^C_ fr < < (^-23) Since the mving axes which mve with mean fluid velcity are intrduced, the transfer f tracer cncentratin acrss the plane at which ^ is cnstant depends nly n the crss-sectinal variatin f tracer cncentratin. A partial equilibrium may be assumed fr a small crss-sectinal variatin in C, If _il2 is cnsidered t be independent f ', the small crss-sectinal variatin in C can be calculated frm * 2 H 2 (^+2)(l-0»7l (i +2)(l-^) 2 (I-^O) 3 -^- fr 4 < i (4-24) " **l The cncentratin distributin is fund as 2(4 Q + 2)(l-^ ) 2 ^ ^1 fr, >.2 =r c n + 1 ~K (- -) (i - J 4 2 fr > <<> 2(4 + 2)(l-t ) 2 Dt ^i * ' * * S (4-25) The rate t which C is transprted acrss a sectin at */-, is Q = 2S:// CV ({) d (4-26) Therefre

49 17 ^=I (c + M(i-i )V]«H )3d * + / c + M 2 [ f CH>>Y" N((i-i ) «-«0^J 3-3(4-^> j d < C*-2?) v.-here,2 Dt ^Y 1 2(^+ 2)(l-* )' Ha ^^ ( + 2)(i-4 ) 2 and JL = - -L (i + 22 z + a > 2 } (H)2 <^x_ J^ 2SW *0' v +2 DL >?, -L (4-23) Since the crss-sectinal variatins in C are assumed t be small cmpared with thse in the lngitudinal directin, ^CwiA7i ^s near-*-y ^ e same as ^Cm/^l if Cjjj is the dimensinless mean cncentratin ver a crss sectin. Equatin (23) may be written as Q = -2$wJL (1+ 33 /. 21 > 2 n ^""V 2 $>\ * C m (^ 29) Cmparing this with Fick's lav: f diffusin, it can be seen that C_ is dispersed relative t a plane which mves with velcity V with a diffusin cefficient s = -I- (i 22 / + ZLf ) (ilk) 2 Li (MO) Q +2 D which is called the effective r apparent dispersin cefficient. It shuld be remembered that tw cnditins, Equatins (2) and (15). are

50 48 necessary fr the given apprximate slutin f the equatin fr diffusin t be valid t interpret the lngitudinal dispersin f a slute in a stream. Frm the first cnditin and Equatin (30), ne knx-rs that 105 ( 16 * VV 1 ~ > D (4 " 31) in rder that the lngitudinal mlecular diffusin may be negligible cmpared with the dispersive effect represented by S. The secnd cnditin ensures that the time necessary fr a crss-sectinal variatin in C t die dwn wing t crss-sectinal diffusin is much shrter than the time necessary fr any appreciable change in C t ccur thrugh lngitudinal cnvectin. Frm this cnditin and the fact that V = -r-2-. V Y (4-32) m + 2 x ne btains JL >> ( + 2) JL (4-33) v x Cmbining Equatins (31) and (33) gives the cnditins under which the given slutin is valid. 2. Ellis Mdel It has been shwn that the mmentum flux is related t the velcity gradient accrding t (2) X = ( yx "?L )y (4-3*0 l and the fluid which has the shear-stress and shear-rate relatin -fs.-^ + <f_ -r I dy 11"1 ) X (4-35) J J 1 1 yx ' yx

51 49 is called the Ellis mdel. y> ^> and n are measurable parameters. The Ellis mdel is the simplest and mst generally useful three-cnstant mdel. It describes prperly the lwer limiting viscsity >J. This mdel seems t have sufficient flexibility t fit the data fr varius types f fluids. Substituting Equatin (3^) int Equatin (35). ne btains -im^it^r 1)^ dv. =y C= *)y + ^ (^) n - y* (4-36) This equatin can be integrated subject t the bundary cnditin V x *, at y = 1 S The resulting velcity distributin is \ ' ^ Pf) 4 (l - (f) 2 j + *, (=f >* -^ ll - (f) n+1 ](^-37) The maximum velcity is the velcity at y = 0, that is, V = if (= ) i! + tf (rap)n _*^ B I 2 1 L ntl = \l + V (4-38) Expressing V x and V x in terms f V^ and V n» ne btains V = % V - + ii V (4-39 and Cmbining Equatins (40 ) and (4) yields the gverning partial differential equatin, that is

52 = " * 2 h 2 v»*? m2 s J (a r (4-41) with the bundary cnditins ac =, at = and 1 (4-42) If ne defines cncentratin and velcity relative t axes which mve with the mean velcity f fluid flw, the velcity distributin is V xl= V x- V x -V -r>+ V *2 ( nk-* > (4-43) and the transprt equatin becmes 2 2 n+1 -, The distributin f C under the cnditins, <S>C/ >8 = and ^c/^'/-i = cnstant, is fund t be in the frm c c + J- M ft * '1 v^3 s v^ + v *2 ti& ".^ 3). ]](«5) where C is the cncentratin f the tube at 2, = 0. The rate at which C is transprted acrss a sectin is = &!$J CV^d* (4-46) Inserting values f C and V -, int Equatin (46), Q is fund t be

53 51 Q= ^ ("^H^ ^ + 27<n+ 4>(2n+5> ^ l6(n+l? (n+?) 405(n+2)(n+4)(n+6) Vnl V a2 1 (4-47) J Since BC/^ yj 1 has been assumed t be independent f, C may be replaced by C The rate f transfer f matter in a slit due t a diffusivity 2 is 2^3 f^a (4JW) L ^1 Cmparing Equatins (47) and (48), it can be seen that 2 2 = 11.8 v 2 _ j. ' v^ + stc^w v^ + w(l >B& ^ v] (*-*?) 945 It shuld be recalled that S > > D in rder that the axial mlecular diffusin be negligible and that the fllwing cnditin must be satisfied s that Taylr's secnd assumptin may be true: L s. >.ii -i. where When if, is zer the Ellis mdel reduces t the Newtnian case and the

54 52 effective dispersin cefficient becmes because Y becmes zer. Fr the case 4> = 0, V, becmes zer, that is, m2 J ml fr an Ostwald-de Waele mdel, the effective dispersin cefficient is E= 3(n- 4)(2n+5) "D^ <^2) FLOW THROUGH CYLINDRICAL TUBES If the transfer f a tracer alng the tube by the mlecular diffusin is small cmpared with that prduced by cnvectin, the transprt equatin becmes (see Fig. (2)) The bundary cnditins are = at r = and R (^-53) because the cncentratin distributin is symmetrical arund the tube and the wall f the tube is impermeable. Intrducing the fllwing dimensinless variables t x u r x Equatins (1) and (53) becme 5C Dt / a C, 1 *e r 2 w s ±Cs _ \ ^C (i -5^) n vv ^7 'X

55 53 ana JL2 =, at NOandl &-55) The cnditin under which Taylr's apprach may be valid can be expressed as (6) Jl >> _2_ (4-56) V m (3.8) 2 D Velcity distributins which can be represented by the Bingham plastic and Ellis mdel will be cnsidered in this sectin. 1. Bingham Plastic Mdel The velcity distributin f Bingham plastic fluid has been derived as (see Equatin (3-^8)) as V fr fc <: L m * ->0 (^-57) and ; r ; e are nw cnsidering the cnvectin acrss a plane which mves with the mean speed f flw. Writing fi-t-fl (^-59) the velcity distributin relative t these axes is V, = v - V xl x x

56 54 ($. i) - s ki>!] <* d v u X fr > ) (4-60) fr ^ < 1 and Equatin (54) becmes 2 a_c _ Dt / ^ C 1 ^Cn d^2 i ** l<* (4-D 3^ fr U> Dt R 2 2 a c 1 C< C TT' (2. d ^c fr * < (4-61) The small radial variatin in C can be calculated frm Equatin (6l) by letting ac/<3>0 and cnsidering ac/^>/-i ^ ^ e independent f, that is, Dt / C 1 SC ) r 2 *i 2 + s n = Dt (1 _±_ (t±c\] r2 L'g» v 3 *fc'j I (H ) : ^(1-V 2-1 jl2 ^1 fr i*l = (2-i) -i-2 lr t<t (4-62) Equatin (62) has the bundary cnditin ac = at I = and 1 (4-63) The slutin f Equatin (62) can be btained as 2 r S0, f*0, = 0+^(^2.)/ Id J (i-djdj Dt ^7i J Q I > J Q & *

57 55 R- /^C Sid* r^-i)-2<iiklf d 7i 4 $< s l t) > (i- 2 s 5 ) J Dt fr ^ ^ = c n+ (±2-)( l idj (i-l)u I fr $ < (4-64) Inserting c* = = (3+2J + Q ) int Equatin (6k) yields C = C + ac 2 Dta(l-I )' L 4 3 ^0 12 ^7-l s ^ 9 *0 ^ 8 $ + 12t + ir ln-il *>* * > t = c +?, _ C ^c ri _ 2 1 i^ _ 4 1,2 i_r ir DtcJ(l4 2 ) ^7 ^U 3 5 2^ J^, fr < (^-65) where C Q is the value f C at \ = 0. The rate f transfer f C acrss the sectin at ^ ^s Q = 2ITR i C \ll d ^C CV xl <J)*d! (4-66) '0 ' Substituting Equatins (60) and (65) int Equatin (66), the rate f transfer f C acrss the sectin at ^ is fund t be Q = L3( 3 t2^) 2 (1^ ) '2., 44 l6.2,4 + U $ ^0

58 56 The rate f transfer f matter acrss a crss sectin with the diffusivity E is q *. EO (± B) (4.68) L ^3/ It shuld be nted that <^c /5 1 / is indistinguishable frm «5c/^7-j as lng as ^C/^7 ]_ is independent f \. Thus it will nt be unreasnable t define the effective dispersin cefficient E f Bingham plastic fluids flwing thrugh a cylindrical tube as _2-2 E _ R v x f 3 t + I6fr c 5 + fc V 2v 6 " * l " 15 " 5 S 2D(3+2 +^) 2 (l4) U8 " Fr the case jn = 0, Equatin (69) becmes E = -^SF (^ 70) This agrees with the result fr the Newtnian flx* derived by Taylr (6). Since the final expressin fr S appears t be peculiar, it may be desirable t evaluate the value f E at 5 = 1. As $ «- 1, In ^ n may be expressed as m ^ = - (i4 ) - I (i-l) 2 - \ a-^) 3 - I (i-!) V*r7i) Cmbining Equatins (71) and (69). it can be seen that the effective dispersin

59 57 cefficient S becmes zer as «- 1. This is reasnable since we have neglected lngitudinal diffusin in the analysis. The cnditins under which the given apprximate slutin can be used t interpret lngitudinal dispersin f a slute are, as stated previusly* (4-72) where S is given in Equatin (69) and ( &L R 2 V n 6v x (3.8) 2 D 2. Ellis Mdel The mmentum flux distributin fr flw f a fluid thrugh a circular tube is P - P Tw = ( v -) rx 2L r (4-7*0 where (P^-Pr ) ^ s "^e result f a pressure gradient and/r gravitatinal acceleratin. Fr the case when a ne-dimensinal rhelgical statement is valid, the relatinship between the shear-stress and shear-rate written in cylindrical crdinates is (see Fig. (2)) *V V,,.,,n-l x _ dr = <* + *ik*l >t«<*-«> where j* ; ^, and n are parameters t be determined by experiment, Substitutin f Equatin (74) int Equatin (75) gives dv v..,. A v.,_ap *n - ^ = J (=* ), + y (-"^r) (4-76) ar 2i» j 1 2i_.

60 58 The bundary cnditin that V = at r-r can be used t evaluate the integratin cnstant. Then the velcity distributin becmes The maximum velcity ccurs at r = 0; thus it has the value -AP<4> _2 AP n^r n+1 v = ( 2) - + (= ) li (4-78) m *> 2L ' 2 T v 21/ n+1 v r ; = \l+ V m2 ^-79) Cmbining Equatins (77) and (79) yields v-*-ji-<i> 2 J + %2 [ i - n+1 ]- (*- The average velcity V v is calculated by summing up all the velcity ver Jib a crss sectin and then dividing by the crss-sectinal area, -which yields V = 1 V, + S& V (4-81) x 2 fill nt3 m2 The mathematical expressin f the diffusin equatin fr an incmpressible fluid flwing in a circular tube with the assumptin that the axial diffusin may be neglected is 4f=M^ + ^)-V x (,-83) It is cnvenient in this discussin t define cncentratin and velcity relative t axes mving with the mean flw. Relative t these axes, the velcity V -, is

61 59 V - V - V - V 1-2 f ( ) + ] V n+;l I 2. l ( ) (4-83) xl x x mll«c R J n2^ n+-3 R J Substituting Equatin (83) int the gverning partial differential equatin and writing it in dimensinless variables, ne btains C _ Dt, a C.li.Cv 1 (, r A «. 2v, n+lv N 1 _ C C.., 2. (4-84) The distributin f C in the case when & C/«3 & = and - ~- is independent d "l f \ is fund as c - c 0+ -fl ( f-)( Vnl < f- k *> + ^( f - i*>)l (^85) by applying the bundary cnditins that C/^ = at ^ =1. The rate f transfer at which C is transprted acrss a sectin at 7n is 9 r 1 (4-86) Replacing C and V.by Equatins (85) and (83), it is fund that q=2.tr 2 K 2 f^c, f\ v,i s 2 1 iv\2,n 2 1 t n+ 3 I = - 2TTR 2 R^, ^C (n+1)' C4*-> Dt Vx *Ti (n+3) 3 (n+5) V. ra2 (Yl+l)(ri4ll) "J /a N

62 60 Cmparing this with ml Q = -ttr 2 1 L d7 1 (4-88) the effective dispersin cefficient E is D V^ + 2 (nt3) 3 (nt5) 12(n+3)(n+5)(n +7) ^ m2 J ^ ^ It shuld be nted that the cnditins E => > D 0-90) and L >;> R 2 \l+ V ia2 ( 3.8) 2 D ^91) must be satisfied in rder fr the analysis t be valid. When ^ n = 0, V _ becmes zer and the Ellis mdel reduces t the J l m2 Newtnian case and the effective dispersin cefficient becmes 2-2 R V E - tst e*-92 > If y = 0, that is, fr the Ostwald-de Waele mdel, V- is zer and the effective dispersin cefficient becmes R V E " 2(n.3)(n^5) IT (^93) which is identical t that derived by Fan and Hwang (19) fr the Ostwald-de Waele fluid. The limiting cnditin fr the Ostwald-de Waele mdel, as derived by Fan and Hwang, are 2(n+3)(n<-5) 2-2 R V7 D 2 > > D (4*94) and

63 = 61 V x (ntl)(3-8)td The equatin f cntinuity fr C in the given prcess is ^Q, _ ay -2 (4-96) in which A is the crss-sectinal area f the system. Substituting fr Q frm Equatin (38) the dispersin relative t axes mving with speed V x is gverning by the equatin a C. a 2 CL Since the cncentratin distributin has been reduced t ne dimensinal statement, ne can replace C by C in Equatin (97)» that is, This has t be slved subject t the bundary cnditins B.C.I. C=0 fr 8=0 and Y[ *j* (4-99) B.C. 2. C = 6(8) fr 6=0 and *J = (4-100) B.C. 3. f C dl = 1 (4-101) The slutin subject t the given bundary cnditins is (20, 21) *? C = 1 exp c Ti(H~)e ^ VYL 'X J J VVL :: A 2 Since ^ = f - ft has been defined in this analysis, ne may replace 1-, by

64 . 62 *[ - 0, which gives C = -p exp f- -Cffi).. ] (4^103) 1 V X L J V X L An experiment designed by setting»7 * 1 yields the utlet tracer cncentratin distributin functin t be C= E(9) exp (l-9> 2 (4-104) 1 V- J V- where E/VJL is a functin which varies with dimensinless parameter ft 2 2 (r R V^D) and the flw behavir index describing different flw mdels. vz/d Integrating the given distributin functin frm t 9, we can btain the cncentratin distributin crrespnding t a step functin input, that is, C«F(B) 9? : f 1 _ exp ( - Sk&L 1 d0 (4-105) J zk(j-)6) l/z l 4(^)6 J 1 V_T. J V T. A family f E(0) and F(0) curves, which have been numerically cmputed and shwn in Figures (14) and (15)» are pltted with E/V^L as parameters. The crrelatins amng E/V L, the dimensinless parameter $ V x /D (r RyD) and different flw behavir indexes are shwn in Figures (16) and (17). When E/VL is very small, the value f E(0) are essentially zer except fr the values f 9 clse t 1.0. Fr this small E/V X L, the residence time

65 ) 63 u/jy JJ 0.4 i Di.TSGnsinlecs TirrsG ; 9 Fig. 14. Gsneralizsd E- curves with G3 the parameter. ^ e/v l

66 (0)d 64

67 65 -. cf>. V- CO c Jfm c 13 «c I s -. v^ CD CO O t y c O \_ -: O > v>- r* lo < CD C JD <- * ^. «.;_ b». _ d «*- c CO ro a O* CO Q. c Q. ;_; a < OJ td. 0> 1_ ZL ffl >» CD C5.C a vx

68 66 1 O CO c c O CO O si-. r O C\J X CO - a r- c v_ *t CJ a i_ *> <-3- jc -^-.Q c ^ QJ *" ^> O C :- Vi- «.J- V- <D CD O c: E t ~^ \- s» D. CO " \_ O CO l_ Q. ID G) ~> z*- >» O jc 4- h- CM O. O. 8- t *

69 67 distributin functins can apprximately be reduced t 3(9) = Jirfcrr)] 8XP (4-i6) which is a nrnal distributin functin with mean (4-107) and variance cs - 2C-&-) VXL (4-108) The apprximate cumulative age distributin can therefre be expressed by (6) F(8)= - 1 erf [ (l-0)(e/v x L)*] fr 6 < 1 (4-109) = l + lerf[ (9-l)(E/V x L) l J fr 6 > 1 where erf z = 2TT *L z,2 e" Z dz If # ne differentiates Equatin (104) with respect t 8 and sets the derivative equal t zer, the slutin shws that the maximum value f E(8) ccurs at time Q, that is, m' at 8=8 m (4-110) r 8 + 2(= E -)9 -i=

70 68 Therefre, m t E v2 ( -) ( -) (4-111) L V * /B V- r ( (JL) 2 + il 1/2 + ( JL) This equatin shws that the maximum pint is clse t 1 when E/V.JL is far smaller than ne. The value f the residence time distributin at the maximum pint is btained by substituting Equatin (111) int Equatin (104). The stimulus-respnse analysis dne by Taylr prvides a cnvenient mathematical technique fr finding the age distributin f a tracer passing thrugh clsed vessels. The usual methd f finding the effective dispersin cefficient is t inject a tracer int the system. The tracer cncentratin is then measured dwnstream, and the dispersin cefficient may be fund frm an analysis f the cncentratin data. The first and secnd mments are usually needed t characterize a tracer distributin curve. The first mment abut the rigin, which lcates the center f gravity f the tracer curve with respect t the rigin, is usually called the mean f a distributin curve and the secnd mment abut this mean which measures the spread f the curve, is called the variance f a distributin. One can find the functinal relatinship between the mean and variance f the tracer distributin curve and the effective dispersin cefficient. The frmulas used t 2 evaluate the mean, )^, and variance, (^, f a distributin are p = f E(6) d0 (4-113) j and

71 69 2 = f (e-^) 2 e(8) de r ex 2 = f e 2 s(b) de-fpe bcg) del 2 (^-^) ^0 Using a perfect delta- functin input, the tracer distributin evaluated at y) = i has been fund t be 1 "_ (1-6) 1 -r (4-10*0 - exp V. 7r t J V T. Frm this equatin the mean and variance are fund t be (22, 23), U = i + 2(- 2 -) V X L (4-115) ^2 =2(= E_ )+8( JL) 2 (^-116 > Slving fr E/V L gives JL = -L [ (8*? + l) 2 - l] (4-117) If the velcity distributin f the fluid passing thrugh the system is knwn, the dispersin cefficient can be btained by injecting an impulse f tracer int the system and finding the variance frm the experimental data. When S/V^ is far smaller than 1, Equatins (115) and (116) apprximate 1**1

72 70 (A. = 2(= -) V X L which are the sane as Equatins (107) and (108). Cnsider a system f steady state flw in a straight pipe frm the pint f view f their dimensins. It has been fund (26) that the diraensinless grup E/V d is a functin f the Reynlds number, the Schmidt number and a relative rughness number. The rughness factr has been shwn t be imprtant nly in turbulent flw. The functinal relatinship fr laminar flw in a pipe may therefre be expressed as V*4 ( dv P y, )t( f>j)) (4-118) The generalized expressin f the effective dispersin cefficient is E= -^ (4-119) r Vxd * >* ^ D This relatin is applicable nly when the system satisfies Taylr's limiting cnditins, that is, 2 i- >> v m fr fluid flw in a cylindrical tube and L Z y" > > % m fr fluid flw thrugh a slit. A family f curves with E/V d as a functin

73 71 f Reynlds number and Schmidt number. 8 parameters is shwn in Figure (18). Equatin (119) can als be writ as Frm this a family f curves f slpe 1 and intercept K/4- is resulted as pltted in Figures (19) thrugh (21). DISPERSION OF MATTER IN TURBULENT FLOW OF NON- NEWTONIAN FLUIDS. In a later paper Taylr (7) has shwn that the analgus prblem f dispersin in turbulent flw can be slved in the same way as that used fr laminar flw. In the case f Newtnian flw, the effective dispersin cefficient E is fund t be 10.1 RV* r S = 7.14 RVjJT (26) in which f is the Fanning frictin factr and V* the frictin velcity. The effective dispersin cefficient is almst entirely due t the variatin f the time-average velcity with radius and the effect f turbulent axial diffusin is almst negligible in this cnnectin. Taylr's calculatin assumed the validity f Reynlds ' analgy, and the universal velcity distributin. The turbulent flw regin is f the greatest practical imprtance in the fact that mst fluid flw in industrial systems is turbulent. It is the purpse f this sectin t extend the analysis t nn-newtnian turbulent flw systems. The theretical analysis fr turbulent flw f nn-newtnian fluids thrugh smth rund tubes was perfrmed fr the first time by Ddge and Metzner (3). By analysis and experiments, they shwed that a purely viscus nn-newtnian fluid prvides a small amunt f drag reductin. Varius bservatins and cnclusins regarding the different trends in frictinal drag were made by many investigatrs. Hwever, the turbulent velcity prfile suggested by