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Drag and mass transfer in multiple drop slow motion in a power law fluid
Chemical Engineedng Science, Vol. 41, No. Printed III Great Britain.
DRAG
AND
IO, pp 2569-2573.
1986.
ooo9-250!2/86 Pergamoa
MASS TRANSFER IN MULTIPLE DROP MOTION IN A POWER LAW FLUID
ANDRZEJ
B. JARZEBSKIT
and JANUSZ
53.00 + 0.00 Journals Ltd.
SLOW
J. MALINOWSKI
Polish Academy of Sciences, Institute of Chemical Engineering, 44-100 Gliwice, Poland (Received 7 November 1985; accepted 6 January
1986)
Abstract--Upper and lower bounds on the drag coefficient for a swarm of drops or bubbles moving in a power law fluid are obtained using variational principles and a free surface cell model. The effectsof a wide
range of shearthinning,holdup and viscosityparameterson the drag and mass transferrate are discussed. Recentlypublishedpredictionsof the approximateanalyticalsolution are found to be reasonable if shear thinning
behaviour
is not pronounced.
INTRODUCTION
Motion of, and mass transfer to or from, drops or bubbles moving in non-Newtonian fluids is important in a wide range of chemical and biochemical processes and therefore has attracted considerable attention in the past decade. The papers on this subject published to date are extensively summarized by Bhavaraju et al. (1978), Kawase and Ulbrecht (1981a, b) and Jarzebaski and Malinowski (1986). Most of the studies carried out deal with single drops or bubbles and not their swarms and ensembles, although the latter are more frequently encountered in engineering situations. Gal-Or and Wasio (1968) first extended the free surface cell model of Happel (1958) originally developed for creeping flow of a Newtonian fluid over an assemblage of solid spheres, to study the slow motion of a swarm of drops or bubbles in a Newtonian fluid. More recently, Bhavaraju et al. (1978) adopted the same approach to give the first approximate solution for drag and mass transfer in creeping motion of a swarm of bubbles in a power law fluid. Using the linearization technique of Hirose and Moo-Young (1969), they obtained closedform solutions for both the drag coefficient and external Sherwood number. Shortly afterwards, Kawase and Ulbrecht (1981a), using the same technique, offered a better approximation for the solution of the same problem. However, the results given in Fig. 7 of their paper fall, strangely enough, above the values of Bhavaraju et al. (1978) and hence appear to be suspect. To clarify this fact, both approaches have been recalculated by Jarzebski and Malinowski (1986) and the correct expressions derived. The same paper also provides the approximate expressions for the drag coefficient and Sherwood number for slow motion of a swarm of drops in a power law fluid. To summarize, the open literature provides approximate, closed-form solutions to the problem, obtained using linearization techniques and hence applicable only for weakly pseudoplastic fluids. The purpose of the present paper is to give results valid for a broader +To whom correspondence should be addressed.
range of power law index using the more accurate method of variational principles (Slattery, 1972).
UPPER BOUND CALCULATION
Since the variational principles have been extensively applied to the analysis of the drag coefficient of a slowly moving fluid sphere or a rigid particle in nonNewtonian fluids (Mohan, 1974; Wasserman and Slattery, 1964, Mohan and Venkateshwarlu, 1976; Cho and Hartnett, 1983), only principal expressions will be given here, omitting the more detailed derivations. For steady creeping flow of an incompressible fluid, the velocity variational principle (Slattery, 1972) may be expressed as
P+PI. n)dS
- (P +
(1)
where quantities with an asterisk are evaluated on the basis of a trial function and S - S, is that part of the boundary surface where velocity is not explicitly specified. For a power law fluid, the work function E is given by
For a creeping flow of a power law fluid, eq. (1) takes the form (Mohan, 1974) UF,
< (n+)
[S
ErdV+
s,
E,Z d,].
(3)
“,
The trial stream functions are taken as +r = (C, r2 + CzP)(l I&,*= (Al?
-2’)
+ A*?.*+ A,?,_’ + A/)(1
(4) -22).
(5)
Introducing eqs (4) and (5) into the standard boundary conditions (Bhavaraju et al., 1978; Kawase and Ulbrecht, 198 1b; Jarzebski and Malinowski, 1986), we
2569
2570
ANDRZEJ
J. MALINOWSKI
and JANUSZ
B.JARZEBSKT
obtain 73 A,+A,+Aj+&=O
(6)
c,+c,=o
(7)
2C,
+ 4C,
= 2A,
+ aA,
- A, + 4A,
A,+A,l”-2+A,1-3+Aq12 (o -
l)(o - 2)A,I”+
=
’ + 6A,
j=;I [6c,X-(~);_1)‘2
-+
+ 6A415
= 0
(8)
=
- [I;(%)"](ax~+A~~x-~~(l
1
(1 -z2)3’2
where
dz = 0
(19)
Following a standard procedure (Mohan and Raghuraman, 1976; Chhabra and Raghuraman, 1984), the following equations are obtained: D=2,
(9)
c’=
(10)
C= -2F’,
E’ = F’,
(((T-I~)((T-~)A~+~A~
+6A,}
-z2)1/2.
-2F
(20)
E=F
(21)
A’ =
-
A” = -AA’Z-B-~, c”=
(11)
-2F”,
1)
(22)
F” = 2A”/3
(23)
3F’/(B
-
E”_F”
C2 = - (A’+
A”)/6X,
(24) C, +C1
= - l/2 (25)
I=
@-1/3
and
X = qJK(U/R)“-‘.
and the drag coefficient is constrained as follows:
Equation (3) may be transformed to
Y DL=
C,Re
-
~ (n + 1)2”-’
24
3 (zz:)cn”
2n+2
(n+
+7
l)C$X
(12)
and the second invariant of the rate of deformation tensor is
+2(1
-z’)[(o-
l)(a-2)A2x3-”
+6A3x4+6A4x-IlZ.
(13)
The system of eqs (6)(12) provides the basis for numerical assessment of the upper bound on the drag coefficient. An algorithm for the numerical solution of eqs (6E(l2) adopted by the authors was similar to that of Mohan (1974) except for a Fibonacci method which was replaced by the golden section method. LOWER
BOUND
_ 2’“- ‘)iZn
lmJx -4dxdz
1 (26)
where ZZ: = ~~?(Fx~+F’x~+F”x-~)~+~(A’x~ + A”x-‘)‘(l
ZZ,* = 6z2[2(2-cr)A2x3-“+6A3x4-4A4x-‘]2
4F _ 1=1X
-2’).
(27)
The lower bound on YD was obtained by maximizing the right-hand side of inequality (26) using the Nelder and Mead procedure (Himmelblau, 1972) and the integral was evaluated using the two-dimensional Simpson’s rule quadrature. A search was made on the three variables F, F’ and B, starting with the converged values for an external fluid of Iower pseudoplasticity. The search was continued until the successively improving values did not alter by more than 1.0 x 10ms. The maximum so obtained gives the lower bound Y,,.
CALCULATIONS
The lower bound on drag is obtained from the stress variational principle (Slattery, 1972) RESULTS
sV
EdZ’>
-j-/fdY+~s~v-([r* - (P +
d)Il.
4 dS
where the complementary work function E, power law fluid is given by cn+ 1,/Z" K-'/n*
(14) for a
(15)
The extra stress profiles are chosen as (Mohan Raghuraman, 1976)
and
(CxD+
(16)
c’xB+
cx-‘)z
ZBe= -[,(~~],,D+E.XA+E”X-l)z
(17)
z& = - [K(~)‘](F~~tF.x~+~~~x-‘)z
(18)
AND
DISCUSSION
Results of the upper and lower bounds on the drag coefficient are given in Tables l-3 for the ratio of viscosities taking the values 0.1, 1 and 10, respectively. The results obtained for X = 1000 agreed well with those given by Mohan and Raghuraman (1976) for the motion of a power law fluid past an assemblage of rigid spheres, and for X = 0.001 the appropriate values of YDU and also Y,, were very close to those obtained using the Hirose and Moo-Young linearization technique (Jarzebski and Malinowski, 1986). However, for larger values of X there was an appreciable discrepancy between the results predicted by variational principles and by the linearization technique, as shown in Fig. 1 for X = 1, especially in the region of small values of the flow index n. In general, variational principles predict a stronger influence of the flow index n on the drag coefficient and this tendency, noted previously for solid spheres by Kawase and Ulbrecht (198la), extends to the whole
Drag and mass transfer
2571
Table 1. Upper and lower bounds on the drag coefficient for X = 0.1 Q = 0.2
Q = 0.01
?I
YDIJ 0.919
0.5
0.6 0.7 0.8 0.9 1.0
0.909 0.903 0.900 o.a99 0.899
yDL
0.796
0.839 0.869 0.888 0.896 0.899
Q = 0.4
Q = 0.6
‘DU
yDL
yDU
yDL
‘DU
yDL
0.96
0.92
1.22 1.46 1.74 2.08 2.50 3.00
1.18 1.43 1.72 2.07 2.50 3.00
1.88 2.33 2.88 3.58 4.46 5.59
1.83 2.29 2.86 3.57 4.46 5.59
1.08 1.23 1.39 1.58 1.79
1.06 1.21 1.38 1.57 1.79
Table 2. Upper and lower bounds on the drag coefficientfor X = 1 Q = 0.2
Q = 0.01 n
0.5 0.6 0.7 0.8 0.9 1.0
Q = 0.4
Q = 0.6
YDU
yDL
yDU
yDL
yDU
‘DL
yDU
‘DL
1.15 1.14 1.14 1.14 1.14 1.14
1.05 1.09 1.12 1.13 1.14 1.14
1.53 1.74 1.98 2.26 2.57 2.91
1.50 1.72 1.97 2.25 2.56 2.91
2.63 3.17 3.79 4.49 5.31 6.25
2.60 3.15 3.77 4.48 5.30 6.25
5.99
5.95
7.50 9.20 11.13 13.34 15.90
7.48 9.13 11.11 13.33 15.90
Table 3. Upper and lower bounds on the drag coefficientfor X = 10 Q =
n 0.5 0.6 0.7 0.8 0.9 1.0
20 1.0
-----_____ __
Q = 0.2
yDL
yDU
yDL
yDtJ
1.39
1.29
2.08 2.49 2.97 3.53 4.18 4.94
2.05 2.46 2.95 3.52 4.18 4.94
3.99 5.27 6.91 8.99 11.58 14.75
and
1.34 1.37 1.39 1.41 1.41
Malinowski
o.z------_-__---o,()t-----0.8 n
0.7 I-l
0.6
‘DL
3.94 5.20 6.73 8.98 11.57 14.75
Q = 0.6 ‘DU
9.86 14.49 20.94 29.62 40.90 54.97
‘DL
9.66 14.41 19.68 29.55 40.56 54.97
range of finite values of the ratio of viscosities. A theoretical prediction of the Sherwood number for a swarm of drops moving in a power law fluid can be obtained using the well-known thin boundary layer solution of Baird and Hamielec (1962):
(1985)
I
0.9
Q = 0.4
‘DU
1.39 1.40 1.40 1.41 1.41
Jarzebski
0.01
0.5
Fig. 1. Effect of pseudoplastic anomaly on the drag coefficient.
Substituting appropriate values of Al, . , A4 into eq. (5), calculating us/U and then putting it into eq. (28X theoretical results for the mass-transfer rate can be achieved, provided the controlling resistance is in the continuous phase. Figure 2 shows the effect of pseudoplasticity and the ratio X on the mass-transfer rate and Fig. 3 gives the influence of holdup of a dispersed phase for various values of the flow index. It may be concluded that the greater the values of Q and X, the more marked is the effect of the pseudoplactic anomaly, and this is intuitively in agreement with the physical picture of the process. This is shown in Fig. 3.
2572
ANDRZEJ
6. JARZEBSKI and JANLJSZ J. MALINOWSKI
D,
E, E’, E”
E
EC F, F', F”
FLi I II
II, K
1 n
n I-l
n
Fig. 2. Effect of pseudoplastic anomaly on the mass-transfer rate to a swarm of drops.
P Pe r R Re
S
Xl
ST Sh u V
V x X y DL? yD” 2
Greek
0.5
0.6
0.7 n
0.8
L-1
0.9
1.0
Fig. 3. Effect of holdup and pseudoplastic behaviour on the mass-transfer rate.
letters
tl P u T 4
T NOTATION A,,
AZ,
A39
-44 A’,
A”, B, C,
c’, c” CI> c2 G
i
=q.
0
constants in the trial extra stress profile, eqs (16) and (19) constants in the trial stream function, =q. (4) drag coefficient
dynamic viscosity, mm ’ kg s-l density, kg mm3 constant in the trial stream function, eq. (5) extra stress tensor, m-’ kg s-’ body force potential per unit mass of fluid, m2 s-’ holdup dimensionless stream function
Subscripts
constants in the trial stream function (5)
constants in the trial extra stress profile, eq. (17) work function defined by eq. (2), m-l kg s-~ complementary work function defined by eq. (15), m-l kg sm3 constants in the trial extra stress profile, eq. (18) drag force, m kg s-’ unit tensor dimensionless second invariant of the rate of deformation tensor defined by eq. (13) dimensionless second invariant of the extra stress profile defined by eq. (27) consistency index in the power law model, m-l kg s”-’ dimensionless radius of a hypothetical sphere flow index in the power law model normal vector in surface integrals pressure, m-l kg se2 Peclet number dimensionless radial distance radius of fluid sphere, m Reynolds number, 2R”U2-“/K bounding surface of the flow domain, m2 the part of the bounding surface on which the velocity is explicitly stated, m2 the part of the bounding surface on which the stress is explicitly stated, m2 Sherwood number terminal velocity of fluid sphere, m s-l velocity vector, m s- 1 volume domain of the flow, m3 reciprocal r viscosity ratio parameter, vi/K U/R”-1 lower and upper bounds on drag coefficient cos 8
internal fluid (dispersed phase) external fluid (continuous phase) REFERENCES
Baird, M. H. I. and Hamielec, A. E., 1962, Forced convection transfer around spheres at intermediate Reynolds number. Can. J. &em. Engng 40. 119-124.
Drag and mass transfer Bhavaraju, S. M., Mashelkar, R. A. and Blanch, H. W., 1978, Bubble motion and mass transfer in non-Newtonian fluids: Part II. Swarm of bubbles in a power law fluid. A.I.Ch.E. J. 24, 107&1076. Chhabra, R. P. and Raghuraman, J., 1984, Slow nonNewtonian flow past an assemblage of rigid spheres. Chem. Engng Commun. 27, 23-46. Cho, Y. I. and Hartnett, J. P., 1983, Drag coefficients of a slowly moving sphere in non-Newtonian fluids. J. NonNewtonian Fluid Mech. 12, 243247. Gal-Or, B. and Waslo, S., 1968, Hydrodynamics of an ensemble of drops or bubbles in the presence or absence of surfactants. Chem. Engng Sci. 23, 1431-1446. Happel, J., 1958, Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles. A.1.Ch.E. J. 4, 197-201. Himmelblau, D., 1972, Applied Nonlinear Programming. McGraw-Hill, New York. Hirose, T. and Moo-Young, M., 1969, Bubble drag and mass transfer in non-Newtonian fluids: creeping flow with power-law fluids. Can. J. them. Engng 47, 265267. Jarzebski, A. B. and Malinowski, J. J., 1986, Drag and mass
2573
transfer in slow non-Newtonian flows over an ensemble of Engng Newtonian spherical drops or bubbles. Chem. Commun. in press. Kawase, Y. and Ulbrecht, J. J., 198 la, Drag and mass transfer in non-Newtonian flows through multiparticle systems at low Reynolds numbers. Chem. Engng Sci. 36, 1193-1202. Kawase, Y. and Ulbrecht, J. J., 1981b, Newtonian fluid sphere with rigid or mobile interface in a shear-thinning liquid: drag and mass transfer. Chem. Engng Commun. 8.213231. Mohan, V., 1974, Creeping flow of a power-law fluid over a Newtonian fluid sphere. A.I.Ch.E. J. 20, 180-182. Mohan, V. and Raghuraman, J., 1976, A theoretical study of pressure drop for non-Newtonian creeping flow past an assemblage of spheres. A.1.Ch.E. J. 22, 259-264. Mohan, V. and Venkateshwarlu, D., 1976, Creeping flow of a power law fluid past a fluid sphere. Int. J. Multiphase Flow 2, 563569. Slattery, J. C., 1972, Momentum, Energy and Mass Transfer in Continua. McGraw-Hill, New York. Wasserman, M. L. and Slattery, J. C., 1964, Upper and lower bounds on the drag coefficient of a sphere in a power law fluid. A.I.Ch.E. J. IO, 383-388.
DRAG
AND
IO, pp 2569-2573.
1986.
ooo9-250!2/86 Pergamoa
MASS TRANSFER IN MULTIPLE DROP MOTION IN A POWER LAW FLUID
ANDRZEJ
B. JARZEBSKIT
and JANUSZ
53.00 + 0.00 Journals Ltd.
SLOW
J. MALINOWSKI
Polish Academy of Sciences, Institute of Chemical Engineering, 44-100 Gliwice, Poland (Received 7 November 1985; accepted 6 January
1986)
Abstract--Upper and lower bounds on the drag coefficient for a swarm of drops or bubbles moving in a power law fluid are obtained using variational principles and a free surface cell model. The effectsof a wide
range of shearthinning,holdup and viscosityparameterson the drag and mass transferrate are discussed. Recentlypublishedpredictionsof the approximateanalyticalsolution are found to be reasonable if shear thinning
behaviour
is not pronounced.
INTRODUCTION
Motion of, and mass transfer to or from, drops or bubbles moving in non-Newtonian fluids is important in a wide range of chemical and biochemical processes and therefore has attracted considerable attention in the past decade. The papers on this subject published to date are extensively summarized by Bhavaraju et al. (1978), Kawase and Ulbrecht (1981a, b) and Jarzebaski and Malinowski (1986). Most of the studies carried out deal with single drops or bubbles and not their swarms and ensembles, although the latter are more frequently encountered in engineering situations. Gal-Or and Wasio (1968) first extended the free surface cell model of Happel (1958) originally developed for creeping flow of a Newtonian fluid over an assemblage of solid spheres, to study the slow motion of a swarm of drops or bubbles in a Newtonian fluid. More recently, Bhavaraju et al. (1978) adopted the same approach to give the first approximate solution for drag and mass transfer in creeping motion of a swarm of bubbles in a power law fluid. Using the linearization technique of Hirose and Moo-Young (1969), they obtained closedform solutions for both the drag coefficient and external Sherwood number. Shortly afterwards, Kawase and Ulbrecht (1981a), using the same technique, offered a better approximation for the solution of the same problem. However, the results given in Fig. 7 of their paper fall, strangely enough, above the values of Bhavaraju et al. (1978) and hence appear to be suspect. To clarify this fact, both approaches have been recalculated by Jarzebski and Malinowski (1986) and the correct expressions derived. The same paper also provides the approximate expressions for the drag coefficient and Sherwood number for slow motion of a swarm of drops in a power law fluid. To summarize, the open literature provides approximate, closed-form solutions to the problem, obtained using linearization techniques and hence applicable only for weakly pseudoplastic fluids. The purpose of the present paper is to give results valid for a broader +To whom correspondence should be addressed.
range of power law index using the more accurate method of variational principles (Slattery, 1972).
UPPER BOUND CALCULATION
Since the variational principles have been extensively applied to the analysis of the drag coefficient of a slowly moving fluid sphere or a rigid particle in nonNewtonian fluids (Mohan, 1974; Wasserman and Slattery, 1964, Mohan and Venkateshwarlu, 1976; Cho and Hartnett, 1983), only principal expressions will be given here, omitting the more detailed derivations. For steady creeping flow of an incompressible fluid, the velocity variational principle (Slattery, 1972) may be expressed as
P+PI. n)dS
- (P +
(1)
where quantities with an asterisk are evaluated on the basis of a trial function and S - S, is that part of the boundary surface where velocity is not explicitly specified. For a power law fluid, the work function E is given by
For a creeping flow of a power law fluid, eq. (1) takes the form (Mohan, 1974) UF,
< (n+)
[S
ErdV+
s,
E,Z d,].
(3)
“,
The trial stream functions are taken as +r = (C, r2 + CzP)(l I&,*= (Al?
-2’)
+ A*?.*+ A,?,_’ + A/)(1
(4) -22).
(5)
Introducing eqs (4) and (5) into the standard boundary conditions (Bhavaraju et al., 1978; Kawase and Ulbrecht, 198 1b; Jarzebski and Malinowski, 1986), we
2569
2570
ANDRZEJ
J. MALINOWSKI
and JANUSZ
B.JARZEBSKT
obtain 73 A,+A,+Aj+&=O
(6)
c,+c,=o
(7)
2C,
+ 4C,
= 2A,
+ aA,
- A, + 4A,
A,+A,l”-2+A,1-3+Aq12 (o -
l)(o - 2)A,I”+
=
’ + 6A,
j=;I [6c,X-(~);_1)‘2
-+
+ 6A415
= 0
(8)
=
- [I;(%)"](ax~+A~~x-~~(l
1
(1 -z2)3’2
where
dz = 0
(19)
Following a standard procedure (Mohan and Raghuraman, 1976; Chhabra and Raghuraman, 1984), the following equations are obtained: D=2,
(9)
c’=
(10)
C= -2F’,
E’ = F’,
(((T-I~)((T-~)A~+~A~
+6A,}
-z2)1/2.
-2F
(20)
E=F
(21)
A’ =
-
A” = -AA’Z-B-~, c”=
(11)
-2F”,
1)
(22)
F” = 2A”/3
(23)
3F’/(B
-
E”_F”
C2 = - (A’+
A”)/6X,
(24) C, +C1
= - l/2 (25)
I=
@-1/3
and
X = qJK(U/R)“-‘.
and the drag coefficient is constrained as follows:
Equation (3) may be transformed to
Y DL=
C,Re
-
~ (n + 1)2”-’
24
3 (zz:)cn”
2n+2
(n+
+7
l)C$X
(12)
and the second invariant of the rate of deformation tensor is
+2(1
-z’)[(o-
l)(a-2)A2x3-”
+6A3x4+6A4x-IlZ.
(13)
The system of eqs (6)(12) provides the basis for numerical assessment of the upper bound on the drag coefficient. An algorithm for the numerical solution of eqs (6E(l2) adopted by the authors was similar to that of Mohan (1974) except for a Fibonacci method which was replaced by the golden section method. LOWER
BOUND
_ 2’“- ‘)iZn
lmJx -4dxdz
1 (26)
where ZZ: = ~~?(Fx~+F’x~+F”x-~)~+~(A’x~ + A”x-‘)‘(l
ZZ,* = 6z2[2(2-cr)A2x3-“+6A3x4-4A4x-‘]2
4F _ 1=1X
-2’).
(27)
The lower bound on YD was obtained by maximizing the right-hand side of inequality (26) using the Nelder and Mead procedure (Himmelblau, 1972) and the integral was evaluated using the two-dimensional Simpson’s rule quadrature. A search was made on the three variables F, F’ and B, starting with the converged values for an external fluid of Iower pseudoplasticity. The search was continued until the successively improving values did not alter by more than 1.0 x 10ms. The maximum so obtained gives the lower bound Y,,.
CALCULATIONS
The lower bound on drag is obtained from the stress variational principle (Slattery, 1972) RESULTS
sV
EdZ’>
-j-/fdY+~s~v-([r* - (P +
d)Il.
4 dS
where the complementary work function E, power law fluid is given by cn+ 1,/Z" K-'/n*
(14) for a
(15)
The extra stress profiles are chosen as (Mohan Raghuraman, 1976)
and
(CxD+
(16)
c’xB+
cx-‘)z
ZBe= -[,(~~],,D+E.XA+E”X-l)z
(17)
z& = - [K(~)‘](F~~tF.x~+~~~x-‘)z
(18)
AND
DISCUSSION
Results of the upper and lower bounds on the drag coefficient are given in Tables l-3 for the ratio of viscosities taking the values 0.1, 1 and 10, respectively. The results obtained for X = 1000 agreed well with those given by Mohan and Raghuraman (1976) for the motion of a power law fluid past an assemblage of rigid spheres, and for X = 0.001 the appropriate values of YDU and also Y,, were very close to those obtained using the Hirose and Moo-Young linearization technique (Jarzebski and Malinowski, 1986). However, for larger values of X there was an appreciable discrepancy between the results predicted by variational principles and by the linearization technique, as shown in Fig. 1 for X = 1, especially in the region of small values of the flow index n. In general, variational principles predict a stronger influence of the flow index n on the drag coefficient and this tendency, noted previously for solid spheres by Kawase and Ulbrecht (198la), extends to the whole
Drag and mass transfer
2571
Table 1. Upper and lower bounds on the drag coefficient for X = 0.1 Q = 0.2
Q = 0.01
?I
YDIJ 0.919
0.5
0.6 0.7 0.8 0.9 1.0
0.909 0.903 0.900 o.a99 0.899
yDL
0.796
0.839 0.869 0.888 0.896 0.899
Q = 0.4
Q = 0.6
‘DU
yDL
yDU
yDL
‘DU
yDL
0.96
0.92
1.22 1.46 1.74 2.08 2.50 3.00
1.18 1.43 1.72 2.07 2.50 3.00
1.88 2.33 2.88 3.58 4.46 5.59
1.83 2.29 2.86 3.57 4.46 5.59
1.08 1.23 1.39 1.58 1.79
1.06 1.21 1.38 1.57 1.79
Table 2. Upper and lower bounds on the drag coefficientfor X = 1 Q = 0.2
Q = 0.01 n
0.5 0.6 0.7 0.8 0.9 1.0
Q = 0.4
Q = 0.6
YDU
yDL
yDU
yDL
yDU
‘DL
yDU
‘DL
1.15 1.14 1.14 1.14 1.14 1.14
1.05 1.09 1.12 1.13 1.14 1.14
1.53 1.74 1.98 2.26 2.57 2.91
1.50 1.72 1.97 2.25 2.56 2.91
2.63 3.17 3.79 4.49 5.31 6.25
2.60 3.15 3.77 4.48 5.30 6.25
5.99
5.95
7.50 9.20 11.13 13.34 15.90
7.48 9.13 11.11 13.33 15.90
Table 3. Upper and lower bounds on the drag coefficientfor X = 10 Q =
n 0.5 0.6 0.7 0.8 0.9 1.0
20 1.0
-----_____ __
Q = 0.2
yDL
yDU
yDL
yDtJ
1.39
1.29
2.08 2.49 2.97 3.53 4.18 4.94
2.05 2.46 2.95 3.52 4.18 4.94
3.99 5.27 6.91 8.99 11.58 14.75
and
1.34 1.37 1.39 1.41 1.41
Malinowski
o.z------_-__---o,()t-----0.8 n
0.7 I-l
0.6
‘DL
3.94 5.20 6.73 8.98 11.57 14.75
Q = 0.6 ‘DU
9.86 14.49 20.94 29.62 40.90 54.97
‘DL
9.66 14.41 19.68 29.55 40.56 54.97
range of finite values of the ratio of viscosities. A theoretical prediction of the Sherwood number for a swarm of drops moving in a power law fluid can be obtained using the well-known thin boundary layer solution of Baird and Hamielec (1962):
(1985)
I
0.9
Q = 0.4
‘DU
1.39 1.40 1.40 1.41 1.41
Jarzebski
0.01
0.5
Fig. 1. Effect of pseudoplastic anomaly on the drag coefficient.
Substituting appropriate values of Al, . , A4 into eq. (5), calculating us/U and then putting it into eq. (28X theoretical results for the mass-transfer rate can be achieved, provided the controlling resistance is in the continuous phase. Figure 2 shows the effect of pseudoplasticity and the ratio X on the mass-transfer rate and Fig. 3 gives the influence of holdup of a dispersed phase for various values of the flow index. It may be concluded that the greater the values of Q and X, the more marked is the effect of the pseudoplactic anomaly, and this is intuitively in agreement with the physical picture of the process. This is shown in Fig. 3.
2572
ANDRZEJ
6. JARZEBSKI and JANLJSZ J. MALINOWSKI
D,
E, E’, E”
E
EC F, F', F”
FLi I II
II, K
1 n
n I-l
n
Fig. 2. Effect of pseudoplastic anomaly on the mass-transfer rate to a swarm of drops.
P Pe r R Re
S
Xl
ST Sh u V
V x X y DL? yD” 2
Greek
0.5
0.6
0.7 n
0.8
L-1
0.9
1.0
Fig. 3. Effect of holdup and pseudoplastic behaviour on the mass-transfer rate.
letters
tl P u T 4
T NOTATION A,,
AZ,
A39
-44 A’,
A”, B, C,
c’, c” CI> c2 G
i
=q.
0
constants in the trial extra stress profile, eqs (16) and (19) constants in the trial stream function, =q. (4) drag coefficient
dynamic viscosity, mm ’ kg s-l density, kg mm3 constant in the trial stream function, eq. (5) extra stress tensor, m-’ kg s-’ body force potential per unit mass of fluid, m2 s-’ holdup dimensionless stream function
Subscripts
constants in the trial stream function (5)
constants in the trial extra stress profile, eq. (17) work function defined by eq. (2), m-l kg s-~ complementary work function defined by eq. (15), m-l kg sm3 constants in the trial extra stress profile, eq. (18) drag force, m kg s-’ unit tensor dimensionless second invariant of the rate of deformation tensor defined by eq. (13) dimensionless second invariant of the extra stress profile defined by eq. (27) consistency index in the power law model, m-l kg s”-’ dimensionless radius of a hypothetical sphere flow index in the power law model normal vector in surface integrals pressure, m-l kg se2 Peclet number dimensionless radial distance radius of fluid sphere, m Reynolds number, 2R”U2-“/K bounding surface of the flow domain, m2 the part of the bounding surface on which the velocity is explicitly stated, m2 the part of the bounding surface on which the stress is explicitly stated, m2 Sherwood number terminal velocity of fluid sphere, m s-l velocity vector, m s- 1 volume domain of the flow, m3 reciprocal r viscosity ratio parameter, vi/K U/R”-1 lower and upper bounds on drag coefficient cos 8
internal fluid (dispersed phase) external fluid (continuous phase) REFERENCES
Baird, M. H. I. and Hamielec, A. E., 1962, Forced convection transfer around spheres at intermediate Reynolds number. Can. J. &em. Engng 40. 119-124.
Drag and mass transfer Bhavaraju, S. M., Mashelkar, R. A. and Blanch, H. W., 1978, Bubble motion and mass transfer in non-Newtonian fluids: Part II. Swarm of bubbles in a power law fluid. A.I.Ch.E. J. 24, 107&1076. Chhabra, R. P. and Raghuraman, J., 1984, Slow nonNewtonian flow past an assemblage of rigid spheres. Chem. Engng Commun. 27, 23-46. Cho, Y. I. and Hartnett, J. P., 1983, Drag coefficients of a slowly moving sphere in non-Newtonian fluids. J. NonNewtonian Fluid Mech. 12, 243247. Gal-Or, B. and Waslo, S., 1968, Hydrodynamics of an ensemble of drops or bubbles in the presence or absence of surfactants. Chem. Engng Sci. 23, 1431-1446. Happel, J., 1958, Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles. A.1.Ch.E. J. 4, 197-201. Himmelblau, D., 1972, Applied Nonlinear Programming. McGraw-Hill, New York. Hirose, T. and Moo-Young, M., 1969, Bubble drag and mass transfer in non-Newtonian fluids: creeping flow with power-law fluids. Can. J. them. Engng 47, 265267. Jarzebski, A. B. and Malinowski, J. J., 1986, Drag and mass
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transfer in slow non-Newtonian flows over an ensemble of Engng Newtonian spherical drops or bubbles. Chem. Commun. in press. Kawase, Y. and Ulbrecht, J. J., 198 la, Drag and mass transfer in non-Newtonian flows through multiparticle systems at low Reynolds numbers. Chem. Engng Sci. 36, 1193-1202. Kawase, Y. and Ulbrecht, J. J., 1981b, Newtonian fluid sphere with rigid or mobile interface in a shear-thinning liquid: drag and mass transfer. Chem. Engng Commun. 8.213231. Mohan, V., 1974, Creeping flow of a power-law fluid over a Newtonian fluid sphere. A.I.Ch.E. J. 20, 180-182. Mohan, V. and Raghuraman, J., 1976, A theoretical study of pressure drop for non-Newtonian creeping flow past an assemblage of spheres. A.1.Ch.E. J. 22, 259-264. Mohan, V. and Venkateshwarlu, D., 1976, Creeping flow of a power law fluid past a fluid sphere. Int. J. Multiphase Flow 2, 563569. Slattery, J. C., 1972, Momentum, Energy and Mass Transfer in Continua. McGraw-Hill, New York. Wasserman, M. L. and Slattery, J. C., 1964, Upper and lower bounds on the drag coefficient of a sphere in a power law fluid. A.I.Ch.E. J. IO, 383-388.