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Laminar flow mass transfer with axial diffusion in a tube with chemical reactions
326 CPL D d E’ H: H2 H I_0 kh L n n,. rrr.0 N ii s t, T v. V z
Shorter
Commulllcahons
heat capacity of hqmd phase, cal kmole-‘” C-l oxygen dnTuslvlty, m* ss’ dmmeter of packmg, m activation energy, kmole-’ kg m2 smz gravltahonal acceleration. m s-* heat of solution, cal kmole-’ heat of reaction, cal kmole-’ enthalpy, cal kmole-’ he&t of packmg, m mth order reaction constant, kmoIe’-” m’“-’ s-l reachon order oxygen molar flux at the mterface, kmole mm2 s-’ = (n In-n-)/In (n’“/n-) super6clal molar velocity, kmole m-* s-’ over-all oxygen absorption rate, kmole ss’ heat losses per umt of packed volume, calm-” s-’ Ideal gas constant, kmole-’ kg m* s-*~K-’ cross-section area of column, m2 temperatur, “C, “K superficml hqmd velocity, m s-’ volumetic flow rate, m3 s-’ stochlometic coefficient = 2
Dlmensronless numbers Re = v.pl(a,vL) We = v.zpl(a,a) Fr = v.‘a#lg Greek symbols p density of the hqutd, kg mm3 g surface tenslon of the hqmd, kgs-’ m= cntlcal surface tenslon of packmg material, kg s-* v, vlscoslty of the hqmd, kg m-’ s-’
Superscripts m mlet condlhons ex outlet comhtions o value obtamed under con&ttons
585 [4] Onda
K , Takeuch
H and Maeda
Y , Chem Bngng Scr
19’7227 449 [S] Ltnek V, Stoy V, Macho5 V And Khvskg Z, Chem Engng SCI 1974 29 1955 {6] Lmek V , Hudec P and Jtiovi V ,Scr Papers Znst Chem
Technol ,m press 171 Onda K,
Takeucin
H and Koyama
Y , Kagaku Kogaku
1%7 31 126 [8] Puramk S S and Vogelpohl A , Chem Engng Scr 1974 29 501 [9] Lmek V and Tvrdik J, Bwtechn Bwengng 1971 13 353 [lo] de Waal K J A and Okeson J C , Chem Engng Scr 1966 21 559 [ll] Lmek V and Mayrhoferova J , Chem Engng Scr 1970 25 787 [12] Wesselmgh J A and van’t Hoog A C , Trans Znstn Chem Bngrs 1970 48 T69 [13] Alper E , Trans Znstn Chem Engrs 1973 51 T159 [14] Lmek V and Mayrhoferova J, Collection Czechoslou Chem Commun 1970 35 688 [15] Czechoslovak Patent No 169 054 [16] Kutateladze S S and Bonshanskn V M, Handbook of Heat Transfer Gosenergo&at Moscow 1959 [17] Rozsnn F D , Selected Values of Chemwal Thermodynamx PropertIes U S Govern Pnt, Office, Washington DC 1952 [18] Blrus P , Carlsson D J , Csulloy G W and Wdes D M , J CON Scg 1974 47 636 El91 Basskm A and Ter-Mmassmn-Sarage L , J Coil See 1973 43 190 [2O] Adamson A W , Physrcal Chemrstry of Surfaces, 2nd Edn InterscIence, Whey. New York 1967 [21] Sahay B N and Sharma M M , Chem Bngng Scr I973 28
[22] gdwans
pH”, T” and c&,,,
REFIZBENCES
[l] Danckwerts P V and Rlzvl S F , Trans Znstn Chem Bngrs 1971 49 124 [2] Andrzeu J and Claude1 B , Chem Engng Scz 1974 29 1263
Lamhar
[S] de Waal K J A and Beek W J , Chem Bngng Scr 1967 22
A. D and Sharma M M , Chem Engng Scr 1967 22 673 [23] Kolev N , Verfahrenstechnrk 1973 7 71 [24] Charpentler J , Prost C , Van SW~J W and Le Goff P , Genre Chrmrque 1968 99 803 [25] Takeuchl H , unpubhshed resuIts
flow maSs transfer with axial dilhsion in a tube with chemical reactiont (Received 6 July 1976)
Mass transfer with chemical reactlon m a fled m lammar pope flow was studled by Cleland and Wdhelm [ 11,Wlssler and Sche-chter 141 and Hsu[3] Cleland and Wdhehn[ l] solved the problem numencally mung fimte dtierence method Wlssler and SchecterM and Hsu131 solved the problem semi-analyhcally usmg elgenfunctton expansion and determmed the coeficlent of the senes expanston by means of the orthogonal propertles of the elgenfunctlons In some chemical processes with low fled veIoclty, axial dtiuslon may become very unportant However, there does not seem to exist a solution for the problem of lammar flow mass transfer m a cucular pipe with axial dtiuslon and chemical reaction The Inherent mculty IS probably due to the nonorthogonal properties of the elgenfunchons and therefore the determmatlon of the coefficients of the elgenfunchon expansion LS
not strmght forward The objectwe of the present paper 1s to obtam a semi-anal@& solution for the present problem The convective dtiusron equation with axml dtiuslon and first order reversible chemical reaction IS
and the boundary
condltlons
are
C(0, r) = C,
(2) (31, (4)
In solvmg thus set of equations, one fist mtroduces timenslonless vanables consldermg C, = C + C, c
tms work was performed under the ausptces of the U S Energy Research and Development Admlmstratlon, Waslungton, DC , under Contract No E(3&1)_16
(1)
U[l-(~)‘]$=D[~~(r~+$$l-kC+k&
B=
kL‘o k + k, ,~=$,~=--&,Pe=~“,
k Co(-)k+k,
K _ (k + kl)a2 D
the
Shorter Co mmumcations
In general, one can obtam a set of hnear algelmuc equations for A. from eqn (14) as
and then eqns (l)-(4) are transformed to (l-$)$=$(q$+j$-+8
(5)
NO, II)= 1
(6)
~(bO)=~(&I)=O
(7), (8)
where
z”=l’[(1-n2,(K+~)+~o(+~~]qR” (17) dq
Takmg the solution m the followmg form N&q)=
327
_ x A.R=(~)exp [ -(K +j$“Y] n-L
(9)
J.=I’[(l-sl)(.+&)+$$(K+&)‘]nR.2dn
(18)
one can substitute eqn (9) back to eqns (5)-(E) to obtam a set of ordmary tierentml equations for R.(q) d=R. ~+~~+[(l-r17(u+~)8.‘-u dq
+&(K
+$-)‘j3,,4]R_
dR.(O) -= d?
dR.(U dv
=0
(10)
_ o
(ll), (12) (13)
2 A.Rm(q) = 1 n-1
Equations (lOHl2) are a set of elgenvalue problems whuzh can he solved numencally usmg Rungs-Kutta methods m a CDC 7600 computer However the elgenfunctions R,,(q) are not orthogonal so the coefficients A, cannot be determmed by meaus of classlcal methods of orthogonal expansion Instead the coefficients A,, can be determmed by the followmg equation wluch 1s denved from eqn 03)
m*i,
A,, =
(14)
I.‘qR’($)s,drl where &mq)=(1-1')(
In the present study, the mfimte serves IS truncated at m = 18 which 1s folmd to give satisfactory convergence InchuJmg terms more than 18 does not improve the solution &cantly With m = 18, one can obtam a set of 18 smmltaneous lmear equations whtch can be solved for A. by means of Gauss-Se~del method after I,,, J. and r., are evaluated
K+$-)8~-K+&(K+&p
(15)
RESXJL’ISAND DEXXJSSIONS
Elgenvalues and the coefficients A,, for several large values of reactton parameter K and low values of Peclet numbers are reported by Dang et al [2] The present results are very accurate as one can substitute the coefficients A. and the calculated elgenfunctmns back to eqn (13) to check Its val&ty By checkmg eleven pomts of q from 0 to 1 at 0 1 mterval. one finds the deviation of the summation of the senes from 1 ISless than 0 01% m eqn (13) When Pe + 00, the present problem reduces to the problem soIved by Cleland and Wdhelmcl], Wssler and Schecter[4] and Hsu[3] The present results Bve exactly the same results as Hsu[3) or are even more accurate In evaluatmg eqns (17x19) and solvmg eqn (16), one finds that I. 1s always larger than r. _ If one neglects r,,,, m eqn (16), It 1s found that the error m determmmg A. 1s less than 2 -3% However m so domg, the computation for A, IS much sunpler because one can obtam A,, duectly by A, = I,/&, mstead of solvmg a set of sunultaneous equations for A. The average concentration of the reactant at any location of the
Table 1 Comparison of the average concentration of reactant calculated by the present method and that by Cleland and Wdhelm [ 11 for K = 10, Pe im
I
K5
r
1
e,(9) resent
Method
Cleland
and
Q
1.0000
1
0 20
0
0.6863
0.40
0.4809
0.4803
0.60
0
0
0.80
0.2411
0.2407
1 00
0
1715
0.1709
1 20
0
1222
0.1222
1 40
0.0872
0.0071
l-60
0
O-0621
1.80
0.0444
0.0443
2.00
0.0317
0.0316
6866
3397
0622
3392
Wxlhelm
328
Shorter Comnuuucations
reactor can be detined as
c
=I.‘U[l-(:~zl~~d~ ’
Substitute
x/a
c
0
05
IO
15
20
I
’
’
’
’ ----z,
F
(20)
U[l-(;)*]rdr
---
25
3.0
’
’
Ic
-
z
35 ’
40 ’
45 ’
50 ’
55 ’
‘I
1
,o
Kc’100
1
t
eqn (9) mto eqn (20) and perform some mampuiafion, IO 08 06
”
xexp[-(x+~)8.‘5]~‘rlR,drl
04
(21)
Comparison of the present results wdh fimte ddference method of Cl@nd and Wtielm[l] 1s lpven m Table 1 It is seen that agreement 16 good &ure 1 shows the &menslonless concentratin at various locations of the reactor for K = 1. 10, 100, and Pe = 1,5, 10 One can see that mcrease the reaction parameter K wdl decrease the average concentration of the reactor more drastically at the same locahon of the reactor The effect of axud dtiuslon Hrlll intend to retard the parabohc concentration dntibutron due to the flow field and hence the average dnnenslonless concentmtion remams to be higher value at smaller Peclet number of any constant value of x/a Thus problem IS also solved analykally m terms of confluent hypergeometnc functions which give exactly the same results
Acknowledgement-The computational
assistance
authors wish to acknowledge by Mr T Y Chang
Department of Applred Science Brookhauen Natumal Laboratory Upton, NY 11973, USA
02 005
‘0
01
015
020025
Q30035040045050055
060
x/a Fg
1 DImensionless
concentration
radial coordmate
Rr elgenfunctions fi maximum velocity
vs axml distance of reactor
of tube of fhud m tube
X axial coordmate of reactor tube
Greek symbols B. ergenvalues r “)n integral defined by eqn (19) k,Co
some
e
=-k+k,
VI-DUONG DANG* MEYER STEINBERG
k,C, eb
=ktk,
NOTATION I)
radius of tube coefficient of senes expansion in eqn (9) concentration of reactant average concentration of reactant inlet concentration of reactant molecular dfiuslvlty function defined by eqn (15) mtegral defined by eqn (17) mtegraI defined by eqn (18) forward reaction rate constant backward reactlon rate constant
D
[1] Cleland F A and WdszA I Ch E J 1956 2 489 [2] Dang V D , Stemberg M , Lazareth 0 W and Powell I R , Paper No 122d, 69th AIChE Meetmg, Chicago, u1 Nov 1976 and BNL report 21495, May, 1976 [33 Hsu C J, AIChEJ 1%5 11 938 [41 Wlssler E H and Schechter R S , Appl &t Res 1%1 A10 198
Pe
A flow method
for measuring
r/a
E *IaPe (k + k,)a* K
diffusion
coefficients in power law fluids
(Received 22 October 1975, accepted 4 August 1976) INTRODUCTION
Mass transfer m non-Newtonmn flluds IS of consrderable prachcal Interest m view of their preponderance m mdustrml practice and m blologcal systems Yet the measurement of tiuslon coefficients m non-Newtoman thuds has received attention only m recent years and the avadable data are meagre A possible reason for meagre data IS that the avadable methods are comparatively tious, speck and some of them reqmre rather sophisticated eqmpment The conventional dmphragm cell cannot be used for
the lugh vlscoslty non-Newtoman flurds [ 1.21 Nlshpma and Oster [3] employed an mterferometic mlcromethod whch IS rapid but reqwres specmhzed eqmpment and mvolved data processmg Further, the method cannot be used for studymg the mfluence, d any, of fimte shear rate on the tiuslon wet&lent Methods have therefore been devised for the measurement of dtiuslon coefficient under flow cond&ons The method used by Lmton and Sherwood[4] for Newtoman flluds was extended by Clough et al [ 11 to non-Newtoman flu& The dissolution rates m power law
Shorter
Commulllcahons
heat capacity of hqmd phase, cal kmole-‘” C-l oxygen dnTuslvlty, m* ss’ dmmeter of packmg, m activation energy, kmole-’ kg m2 smz gravltahonal acceleration. m s-* heat of solution, cal kmole-’ heat of reaction, cal kmole-’ enthalpy, cal kmole-’ he&t of packmg, m mth order reaction constant, kmoIe’-” m’“-’ s-l reachon order oxygen molar flux at the mterface, kmole mm2 s-’ = (n In-n-)/In (n’“/n-) super6clal molar velocity, kmole m-* s-’ over-all oxygen absorption rate, kmole ss’ heat losses per umt of packed volume, calm-” s-’ Ideal gas constant, kmole-’ kg m* s-*~K-’ cross-section area of column, m2 temperatur, “C, “K superficml hqmd velocity, m s-’ volumetic flow rate, m3 s-’ stochlometic coefficient = 2
Dlmensronless numbers Re = v.pl(a,vL) We = v.zpl(a,a) Fr = v.‘a#lg Greek symbols p density of the hqutd, kg mm3 g surface tenslon of the hqmd, kgs-’ m= cntlcal surface tenslon of packmg material, kg s-* v, vlscoslty of the hqmd, kg m-’ s-’
Superscripts m mlet condlhons ex outlet comhtions o value obtamed under con&ttons
585 [4] Onda
K , Takeuch
H and Maeda
Y , Chem Bngng Scr
19’7227 449 [S] Ltnek V, Stoy V, Macho5 V And Khvskg Z, Chem Engng SCI 1974 29 1955 {6] Lmek V , Hudec P and Jtiovi V ,Scr Papers Znst Chem
Technol ,m press 171 Onda K,
Takeucin
H and Koyama
Y , Kagaku Kogaku
1%7 31 126 [8] Puramk S S and Vogelpohl A , Chem Engng Scr 1974 29 501 [9] Lmek V and Tvrdik J, Bwtechn Bwengng 1971 13 353 [lo] de Waal K J A and Okeson J C , Chem Engng Scr 1966 21 559 [ll] Lmek V and Mayrhoferova J , Chem Engng Scr 1970 25 787 [12] Wesselmgh J A and van’t Hoog A C , Trans Znstn Chem Bngrs 1970 48 T69 [13] Alper E , Trans Znstn Chem Engrs 1973 51 T159 [14] Lmek V and Mayrhoferova J, Collection Czechoslou Chem Commun 1970 35 688 [15] Czechoslovak Patent No 169 054 [16] Kutateladze S S and Bonshanskn V M, Handbook of Heat Transfer Gosenergo&at Moscow 1959 [17] Rozsnn F D , Selected Values of Chemwal Thermodynamx PropertIes U S Govern Pnt, Office, Washington DC 1952 [18] Blrus P , Carlsson D J , Csulloy G W and Wdes D M , J CON Scg 1974 47 636 El91 Basskm A and Ter-Mmassmn-Sarage L , J Coil See 1973 43 190 [2O] Adamson A W , Physrcal Chemrstry of Surfaces, 2nd Edn InterscIence, Whey. New York 1967 [21] Sahay B N and Sharma M M , Chem Bngng Scr I973 28
[22] gdwans
pH”, T” and c&,,,
REFIZBENCES
[l] Danckwerts P V and Rlzvl S F , Trans Znstn Chem Bngrs 1971 49 124 [2] Andrzeu J and Claude1 B , Chem Engng Scz 1974 29 1263
Lamhar
[S] de Waal K J A and Beek W J , Chem Bngng Scr 1967 22
A. D and Sharma M M , Chem Engng Scr 1967 22 673 [23] Kolev N , Verfahrenstechnrk 1973 7 71 [24] Charpentler J , Prost C , Van SW~J W and Le Goff P , Genre Chrmrque 1968 99 803 [25] Takeuchl H , unpubhshed resuIts
flow maSs transfer with axial dilhsion in a tube with chemical reactiont (Received 6 July 1976)
Mass transfer with chemical reactlon m a fled m lammar pope flow was studled by Cleland and Wdhelm [ 11,Wlssler and Sche-chter 141 and Hsu[3] Cleland and Wdhehn[ l] solved the problem numencally mung fimte dtierence method Wlssler and SchecterM and Hsu131 solved the problem semi-analyhcally usmg elgenfunctton expansion and determmed the coeficlent of the senes expanston by means of the orthogonal propertles of the elgenfunctlons In some chemical processes with low fled veIoclty, axial dtiuslon may become very unportant However, there does not seem to exist a solution for the problem of lammar flow mass transfer m a cucular pipe with axial dtiuslon and chemical reaction The Inherent mculty IS probably due to the nonorthogonal properties of the elgenfunchons and therefore the determmatlon of the coefficients of the elgenfunchon expansion LS
not strmght forward The objectwe of the present paper 1s to obtam a semi-anal@& solution for the present problem The convective dtiusron equation with axml dtiuslon and first order reversible chemical reaction IS
and the boundary
condltlons
are
C(0, r) = C,
(2) (31, (4)
In solvmg thus set of equations, one fist mtroduces timenslonless vanables consldermg C, = C + C, c
tms work was performed under the ausptces of the U S Energy Research and Development Admlmstratlon, Waslungton, DC , under Contract No E(3&1)_16
(1)
U[l-(~)‘]$=D[~~(r~+$$l-kC+k&
B=
kL‘o k + k, ,~=$,~=--&,Pe=~“,
k Co(-)k+k,
K _ (k + kl)a2 D
the
Shorter Co mmumcations
In general, one can obtam a set of hnear algelmuc equations for A. from eqn (14) as
and then eqns (l)-(4) are transformed to (l-$)$=$(q$+j$-+8
(5)
NO, II)= 1
(6)
~(bO)=~(&I)=O
(7), (8)
where
z”=l’[(1-n2,(K+~)+~o(+~~]qR” (17) dq
Takmg the solution m the followmg form N&q)=
327
_ x A.R=(~)exp [ -(K +j$“Y] n-L
(9)
J.=I’[(l-sl)(.+&)+$$(K+&)‘]nR.2dn
(18)
one can substitute eqn (9) back to eqns (5)-(E) to obtam a set of ordmary tierentml equations for R.(q) d=R. ~+~~+[(l-r17(u+~)8.‘-u dq
+&(K
+$-)‘j3,,4]R_
dR.(O) -= d?
dR.(U dv
=0
(10)
_ o
(ll), (12) (13)
2 A.Rm(q) = 1 n-1
Equations (lOHl2) are a set of elgenvalue problems whuzh can he solved numencally usmg Rungs-Kutta methods m a CDC 7600 computer However the elgenfunctions R,,(q) are not orthogonal so the coefficients A, cannot be determmed by meaus of classlcal methods of orthogonal expansion Instead the coefficients A,, can be determmed by the followmg equation wluch 1s denved from eqn 03)
m*i,
A,, =
(14)
I.‘qR’($)s,drl where &mq)=(1-1')(
In the present study, the mfimte serves IS truncated at m = 18 which 1s folmd to give satisfactory convergence InchuJmg terms more than 18 does not improve the solution &cantly With m = 18, one can obtam a set of 18 smmltaneous lmear equations whtch can be solved for A. by means of Gauss-Se~del method after I,,, J. and r., are evaluated
K+$-)8~-K+&(K+&p
(15)
RESXJL’ISAND DEXXJSSIONS
Elgenvalues and the coefficients A,, for several large values of reactton parameter K and low values of Peclet numbers are reported by Dang et al [2] The present results are very accurate as one can substitute the coefficients A. and the calculated elgenfunctmns back to eqn (13) to check Its val&ty By checkmg eleven pomts of q from 0 to 1 at 0 1 mterval. one finds the deviation of the summation of the senes from 1 ISless than 0 01% m eqn (13) When Pe + 00, the present problem reduces to the problem soIved by Cleland and Wdhelmcl], Wssler and Schecter[4] and Hsu[3] The present results Bve exactly the same results as Hsu[3) or are even more accurate In evaluatmg eqns (17x19) and solvmg eqn (16), one finds that I. 1s always larger than r. _ If one neglects r,,,, m eqn (16), It 1s found that the error m determmmg A. 1s less than 2 -3% However m so domg, the computation for A, IS much sunpler because one can obtam A,, duectly by A, = I,/&, mstead of solvmg a set of sunultaneous equations for A. The average concentration of the reactant at any location of the
Table 1 Comparison of the average concentration of reactant calculated by the present method and that by Cleland and Wdhelm [ 11 for K = 10, Pe im
I
K5
r
1
e,(9) resent
Method
Cleland
and
Q
1.0000
1
0 20
0
0.6863
0.40
0.4809
0.4803
0.60
0
0
0.80
0.2411
0.2407
1 00
0
1715
0.1709
1 20
0
1222
0.1222
1 40
0.0872
0.0071
l-60
0
O-0621
1.80
0.0444
0.0443
2.00
0.0317
0.0316
6866
3397
0622
3392
Wxlhelm
328
Shorter Comnuuucations
reactor can be detined as
c
=I.‘U[l-(:~zl~~d~ ’
Substitute
x/a
c
0
05
IO
15
20
I
’
’
’
’ ----z,
F
(20)
U[l-(;)*]rdr
---
25
3.0
’
’
Ic
-
z
35 ’
40 ’
45 ’
50 ’
55 ’
‘I
1
,o
Kc’100
1
t
eqn (9) mto eqn (20) and perform some mampuiafion, IO 08 06
”
xexp[-(x+~)8.‘5]~‘rlR,drl
04
(21)
Comparison of the present results wdh fimte ddference method of Cl@nd and Wtielm[l] 1s lpven m Table 1 It is seen that agreement 16 good &ure 1 shows the &menslonless concentratin at various locations of the reactor for K = 1. 10, 100, and Pe = 1,5, 10 One can see that mcrease the reaction parameter K wdl decrease the average concentration of the reactor more drastically at the same locahon of the reactor The effect of axud dtiuslon Hrlll intend to retard the parabohc concentration dntibutron due to the flow field and hence the average dnnenslonless concentmtion remams to be higher value at smaller Peclet number of any constant value of x/a Thus problem IS also solved analykally m terms of confluent hypergeometnc functions which give exactly the same results
Acknowledgement-The computational
assistance
authors wish to acknowledge by Mr T Y Chang
Department of Applred Science Brookhauen Natumal Laboratory Upton, NY 11973, USA
02 005
‘0
01
015
020025
Q30035040045050055
060
x/a Fg
1 DImensionless
concentration
radial coordmate
Rr elgenfunctions fi maximum velocity
vs axml distance of reactor
of tube of fhud m tube
X axial coordmate of reactor tube
Greek symbols B. ergenvalues r “)n integral defined by eqn (19) k,Co
some
e
=-k+k,
VI-DUONG DANG* MEYER STEINBERG
k,C, eb
=ktk,
NOTATION I)
radius of tube coefficient of senes expansion in eqn (9) concentration of reactant average concentration of reactant inlet concentration of reactant molecular dfiuslvlty function defined by eqn (15) mtegral defined by eqn (17) mtegraI defined by eqn (18) forward reaction rate constant backward reactlon rate constant
D
[1] Cleland F A and WdszA I Ch E J 1956 2 489 [2] Dang V D , Stemberg M , Lazareth 0 W and Powell I R , Paper No 122d, 69th AIChE Meetmg, Chicago, u1 Nov 1976 and BNL report 21495, May, 1976 [33 Hsu C J, AIChEJ 1%5 11 938 [41 Wlssler E H and Schechter R S , Appl &t Res 1%1 A10 198
Pe
A flow method
for measuring
r/a
E *IaPe (k + k,)a* K
diffusion
coefficients in power law fluids
(Received 22 October 1975, accepted 4 August 1976) INTRODUCTION
Mass transfer m non-Newtonmn flluds IS of consrderable prachcal Interest m view of their preponderance m mdustrml practice and m blologcal systems Yet the measurement of tiuslon coefficients m non-Newtoman thuds has received attention only m recent years and the avadable data are meagre A possible reason for meagre data IS that the avadable methods are comparatively tious, speck and some of them reqmre rather sophisticated eqmpment The conventional dmphragm cell cannot be used for
the lugh vlscoslty non-Newtoman flurds [ 1.21 Nlshpma and Oster [3] employed an mterferometic mlcromethod whch IS rapid but reqwres specmhzed eqmpment and mvolved data processmg Further, the method cannot be used for studymg the mfluence, d any, of fimte shear rate on the tiuslon wet&lent Methods have therefore been devised for the measurement of dtiuslon coefficient under flow cond&ons The method used by Lmton and Sherwood[4] for Newtoman flluds was extended by Clough et al [ 11 to non-Newtoman flu& The dissolution rates m power law