=
p
density
AP = ( P =
fi
-
pC)/pc
average density of coexisting gas and liquid
Exponents characterizing critical power laws: specific heat coexistence curve = isothermal compressibility = critical isotherm = second derivative of vapor pressure curve. (Primes denote power laws exponents below T,)
= =
a
p y 6
e
SUBSCRIPTS c
=
critical
References
Amirkhanov, Kh. I., Kerimov, A. hI., Teploenergetika9,61 (1s163). r a ~514 ~ (1963a). Baehr, H . D., ~ r e n n s f . - ~ a e r m e - K15, Baehr, H. D., Forsch. Ingenieurw. 29, 143 (1963bj. Baker, G. A., Gaunt, D. S., Phys. Rev. 155, 545 (1967). Barieau, R . E., J . Chem. Phys. 49,2279 (1968). Blank, G., Waerme-Stoffucbertragulzg 2 , 53 (1969). Domb, C., Hunter, D. L., Proc. Phys. Soc. 86, 1147 (196,j). Edwards, C., Lipa, J. A., Buckingham, 11.J , Phys. Rev. Lett . 20, 496 (1968). Fisher, M. E., Rep. Progr. Phys. 30, 615 (1967). Gaunt, D. S.,Proc. Phys. SOC.92, 150 (1967). Goldhammer, D. A., 2. Phys. Chem. 71, 577 (1910). Goodwin, R . B., J . Res. ,?‘at. Bur. Stand. 73A, 585 (1969). Green, AT. S., Vicentini-Missoni, M.,Levelt Sengers, J. 31. H., Phys. Rev. Lrtt. 18, 1113 (1967). Griffiths. R. B.. J . Chem. Phvs. 43. 1958 11965). Griffithsj R . B., Phys. Rev. 158, 176, (1967). Grigor, A. F., Steele, W. A., Phys. Chrm. Liquids 1, 129 (1968). Uhlenbeck, G. E., Hemmer, P. C., J . Math. Phys. 4, 216 Kac, M., (19W.I ,
Levelt Sengers, J. M. H., Chen, W. T., unpublished results, 1968/69. Levelt Sengers, J. Ill. H., Chen, W. T., Vicentini-llissoni, M., Bull. Amer. Phys. SOC.14, 593 (1969). Levelt Sengers, J. 31. H., Straub, J., Vicentini-Missoni, Lf., unpublished data analysis, 1967-1970. Levelt Sengers, J. RI. H., Vicentini-hlissoni, hI., Proceedings of 4th Symposium on Thermophysical Properties, A.S.M.E., u. 79. 1968. RIcCain, W. D., Ziegler, W. T., J . Chem. Eng. Data 12,199 (1967). Metrologia 5 , 35 (1969). Michels, A,, Levelt, J. ht. H., Graaff, W. de, Physica 24, 659 (l9.5RI ,- - -- ,. Moldover, RI. R., Phys. Rev. 182, 343 (1969). hloldover, RI. R., Little, W. A., “Critical Phenomena,” Proceedings of Conference, Washington, D.C., M. S. Green and J. V. Sengers, eds., h-at. Bur. Stand. hlisc. Publ. 273, 79 (1965). Onsager, L. Phys. Rev. 65, 117 (1944). Osborne, N. S., Stimson, H. F., Ginnings, D. C., J . Res. Arat. Bur. Stand. 18, 389 (1937). Pings, C. J., Teague, R . K., Phys. Lett. 26A, 496 (1968). Preston Thomas, H., Kirby, C. G. hI., Metrologia 4, 30 (1968). Reid, C., Sherwood, T. K., “Properties of Gases and Liquids,” Chapt. 4, 11cGraw-Hill, New York, 1966. Rushbrooke, G. S., J . Chem. Phys. 39,842 (1963). Rushbrooke, G. S.,J . Chem. Phys. 43, 3439 (1965). Verbeke, O., Institute of ~lolecularPhysics, Cniversity of Maryland, private communication, 1969. Vicentini-Missoni, M., Joseph, R . I., Green, AI. S., Levelt Sengers, J . h1. H., Phvs. Rev. B1, 2312 (1970). Vicentini-Miss&i, &I., Levelt Sengers, J. 31. H., Green, M. S., J . Res. &Vat.Bur. Stand. 73A. 563 (1969a). Vicentini-hlissoni, M., Levelt ’Sengers, J. hI. H., Green, M. S., Phys. Rev. Lett. 22, 389 (1969b). Voronel, A. V., Chaskin, Yu. R., and Snigirev, V. G., Sou. Phys. J E T P 21,653 (1965). Voronel, A. V., Chaekin, Yu. R., Sou. Phys. J E T P 24,263 (1967). Waals, J. D. van der, 2. Phys. Chrm. 13, 657, 695 (1894). Widom. B.. J . Chem. Phus. 43. 3898 11965’1. Yang, C. N., Yang, C. P:, Phys. Rev. Lett. 13, 303, (1964).
~ _I,. I
Kadanoff, L. P., Physics 2, 263 (1966). Laar, J. J. van, Comm. Kon. Akad. Amsterdam 14 (II), 1091 (1911/12). Lee, T. D., Yang, C. N., Phys. Rev. 87, 410 (19j2j. Levelt Sengers, J. 11.H., unpublished calibration, 1963.
RECEIVED for review March 4, 1970 ACCEPTEDJuly 6, 1970 American Institute of Chemical Engineers hIeeting, Washington, D.C., November 1969.
Mass Transfer from Spherical Gas Bubbles and Liquid Droplets Moving through Power-Law Fluids in the Laminar Flow Regime Robert M. Wellek and Cheng-Chun Huang Department of Chemical Engineering, Cniversity of Missouri-Rolla, Rolla, 310.66401
Calculated mass transfer coefficients for the continuous phase are presented for internally circulating spheres of a Newtonian fluid traveling through a power-law-type continuous phase in the so-called creeping-flow region. The Nakano and Tien stream functions allow the Sherwood number to be computed as a function of the Peclet number, power-law index, and a viscosity ratio parameter. The Hirose and Moo-Young relation is shown to be a limiting case of this numerical solution. Mass transfer rates increase as the fluids become more pseudoplastic and/or the continuous phase consistency index increases, all other factors held constant.
BECAUSE
OF its importance in many chemical engineering processes, mass transfer from bubbles and droplets has been investigated by numerous workers. I n almost all previous theoretical and experimental studies, the fluids under consideration have been Newtonian in character. I n one recent study, a theoretical relation was developed which described
480
Ind. Eng. Chem. Pundom., Vol. 9, No. 3, 1970
the continuous phase mass transfer mechanism for a fully circulating gas bubble moving through a non-Kewtonian liquid continuous phase described by the power-law rheological model (Hirose and 1100-Young, 1969). This paper presents a solution for the continuous phase mass transfer coefficient for a wide range of dispersed and
continuous phase properties. The primary investigation is based upon the use of the Xakano and Tien (1968) continuous phase stream functions which were developed for a powerlam type, non-Newtonian fluid in the continuous phase and a Newtonian fluid in the dispersed phase. The solution is applicable for both a gas and liquid h'ewtonian dispersed phase and for a range of non-Kewtonian continuous phase characteristics. The results are compared with the continuous phase mass transfer coefficients obtained using the more restrictive nonNewtonian stream function of Hirose and Moo-Young (1969) for fully circulating gas bubbles rising in a power-law fluid, and also with the approximate non-Sewtonian stream functions by Tomita (1959) [or the case of no internal circulation within the dispersed phase and a power-law type, continuous phase fluid. I n all cases the models are compared with existing model3 based upon a Newtonian continuous phase. Mass Transfer Model
The mathematical model for mass transfer from spheres in the creeping flow regime is described by the following dimensionless partial differentiid equation and boundary conditions, using spherical coordinates with origin a t the center of the sphere :
The following assumptions have been made: The flow is axially symmetric and isothermal, and steady state has been reached. The dispersed phase iri a Newtonian fluid, while the continuous phase obeys a power-law model-Le.,
r3' = -K
These coefficients were so chosen that for the Selvtonian case (n = 1) the Xakano and 'rien stream functiuns would become identical with the Hadamard (1911) stream functions for Newtonian continuous and dispersed phase fluids. Nakano and Tien (1968) were concerned with the prediction of drag coefficients for a fluid sphere. Drag coefficients determined by Nakano and Tieii agreed very well with the experimental results of Fararoui and Kintiier (1961) for liquid droplets of a nitrobenzene and tetrachloroethane mixture falling through an aqueous solution of carboxymethylcellulose. The velocity components based on the Hadaniard A ream functions were also used in this work as a check upon the mass transfer rates determined using the Nakano and Tien stream function when the continuous phase was Newtonian. They are as follows:
-
where p d and p c are the viscosity of the dispereed phase and the continuous phase, respectively. B y use of Equations 3a and 3b, the boundary value problem of Equations 1 and 2 can be solved numerically for the solute concentration profiles in the continuous phase around the bubble or droplet. The Sherwood number is then calculated as follows:
for dispersed phase
= -pAij
(: A p A k )
(4)
In this case (n = l ) , the viscosity ratio parameter of Nakano and Tien, X, becomes
with boundary conditions
7,'
for different values of flow behavior index n and a viscosity ratio parameter, X,defined iii Equation 4.
(7)
n-1 2 ~
Aj for continuous phase
Natural convection, molecular diffusion in the angular direction, and resistance to mass transfer inside the sphere are negligible. The solute is dilute. The solute diffusivity and density of both phases are constant. T o complete the description of the forced-convective mass transfer model, velocity components V e and V , have to be specified. The following continuous phase velocity components were derived in this work from the Nakano and Tien (1968) stream functions:
+ B2/y4)(2cos% - sin%) Ve = (1 - Aluy0-2 + A2/y3) sine + (B1/y3
(.Bl/ya
(3a)
- 2 B2/y4) case sine (3b)
Coefficients A I , AP,B1,and BP)and exponent
u were
Continuous phase mass transfer coefficients were also calculated based upon velocity components derived from the works of Hirose and Moo-Young (1969) and Toniita (1959). These velocity components are given below (Equations 14 and 16), with a discussion of their conditions of applicability. Numerical Solution
Because of its strong stability, the Crank-Yicolson implicit numerical method (1947) was used to solve the mass transfer models described above. T o initiate the solution, boundary condition Equation 2c was first applied t o obtain the solution for 8 = 0. I n this case, Equation 1 becomes 2 d2C V dC _ = - + - -2 d C dy Pe (dyP y dy)
'
or written in the finite difference form using central difference approximations :
tabulated I d . Eng. Chem. Fundam., Vol. 9, No. 3, 1970
481
where the second subscript', 1, represents the solution for 8 0, and 2 +2 v -2 at,l = PeAy 2 Pe y
b(J =
=
lV"V
L-
1
- The
-
Present Work
-__... Bowman, Ward, Johnson, 0
A
4 -Pe Ay
Trass (1961) Friedlander (1957 for Pa 1100, 1961 for Per 100) Yugo(1956) Levich (1952)
Sh
Applying boundary condition Equations 2a and 2b, Equation 9 can be reduced to a system of linear, homogeneous, simultaneous equations having a triangular coefficient matrix. Thus the solution is readily obtained for e = 0 by use of any well established numerical technique-e.g., the Gauss elimination method or the Gauss-Jordan method with pivotal condensation. For e > 0, Equation 2 is written in the following finite difference form using central difference approximations :
10
2 10.~
IO'*
io"
IO
10
io2
10'
lo4
Pe Figure 1 . Sherwood number as a function of Peclet number for selected Newtonian systems
where
1. 2. 3
Equations 11 and 12 represent an implicit numerical scheme obtained by use of the Crank-Sicolson method. T o solve the equations for any step in the angular direction] the solution for the previous step is needed. Therefore] with the known solution of Equation 8 or 9 for 8 = 0, the solution for any 8 > 0 can be obtained by solving Equations 11 and 12 repeatedly with boundary condition Equations 2a and 2b. Results and Discussion
The boundary value problem of Equations 1 and 2 for the forced-convective mass transfer problem was solved numerically for various values of Peclet numbers ranging between 10-3 and lo5. The results are presented in terms of Sherwood numbers as a function of the Peclet number, the continuous phase flow behavior index, n, and the viscosity ratio parameter, X. The upper Pe limit, 105, was the largest value of Pe possible for ordinarily encountered systems which still satisfy the creeping flow criterion Re < 1. This wide range of Peclet numbers creates a stability problem in the numerical solution of the partial differential equation, if a n attempt is made to use one specific value of the radial increment, Ay, for all values of the Peclet number. Therefore, it was found essential to use different values of Ay for different values of the Peclet number; it was found by trial that a smaller radial increment has to be used for a larger Peclet number in order to obtain the most stable concentration profiles. The Stokes stream function was used as a standard for this preliminary search for optimum values of Ay a t each Peclet number level. The values of Ay used in this work was presented in Table 111 (deposited with ASIS). 482
Ind. Eng. Chem. Fundam., Vol.
9,No. 3, 1970
Carbon dioxide bubble in water Ethyl acetate droplet in water Any noncirculating system
As expected for larger Peclet numbers, the concentration boundary layer is thinner. The outer limits of the boundary layer used in the present work are 0.5440 dinieiisionless radii for Pe = l o 4 (and 0.1176 dimensionless radii for Pe = 106) from the sphere surface. For such high Peclet numbers, these are believed to be sufficiently far away from the sphere surface where the concentration is unity. For the angular direction, an increment of 3 degrees was used. A reduction of the angular increment by a factor of 2 rewlted in a change of only about 0.5% in the calculated value of Sh when Pe was equal to lo4. By choosing n equal to unity in the Nakano and Tien streani function (or directly using the Stokes and Hadaniard stream functions) the Sherwood number relations obtained numerically by the present model are firqt compared with previous studies of circulating droplets and bubbles moving through Sewtonian fluids. This preliminary study is followed by the presentation of results for a non-Sewtonian continuous phase. Newtonian Continuous Phase. For the noncirculating spheres, the above numerical technique when combined with the Stokes velocity components ~
gave Sherwood numbers almost identical with those obtained by Friedlander (1957, 1961) using boundary layer theory and the Stokes (1880) stream functions. Bowman et al. (1961) obtained nearly identical values for the Sherwood number using the same technique as Friedlander's but with a fourterm polynomial used for the concentration profile in contrast to the two-term profile used by Friedlander. In comparison with Friedlander's early work (1957) Friedlander's later work (1961) and that of Levich (1952) agree more closely with the results of the present work for the thin boundary layer ~
Table I. Analytical Solutions for a Noncirculating Sphere, Newtonian Continuous Phase Equation No.
Region o f Validity
Equation for Sherwood No.
17a 17h
2 $ -9P e + - P e92 + 16 64
, . .
18a 0 89 Pella 18b 0.9'78 Pella 19 0,991 0 997 Pe1j3 20a 20b 0 64 B.L. = boundary layer. Table II. Equation No.
21 22
23a
24
25
Reference
Pe
lo3 > 102 > lo2
Thin B.L. Thin B.L. Thin B.L. Thin B.L. Thin B.L.
Friedlander 1957 Bowman et al. (1961) Friedlander (1961) Levich (1952) Levich (1962)
>> 1
Analytical Solutions for a Circulating Sphere, Newtonian Continuous Phase, Thin Concentration Boundary layer Region of Validity, Re
Eiquation for Sherwood No.
+ 1.33)
( R + 1.58 O.!% Pel/* ( ) 0.58 Pel12 ( R + 1 1 0 fi7 pel12 (-) R + l 0 . 8 9 pel13
R
L,
1 < Pe
+
113
0 . 6 1 pel12
(2) R + 1
2: R + 1
< 2.4(3R
lo3 < Pe
+
2.4(3R + 4 ) 2 ( R+ 1)
Pe
> 2.4(3R
Pe
> 2.8(12R + 19)2(R + 1)
Ward (1961)
Pe
>> 1
Levich (1962)
1/2
23b
Physical Description
case, Pe > 102. [Yuge (1956) obtained Sherwood numbers for the case of Pe = 0.3, 1.0, 3.0, and 10.0 n-hich are nearly identical with the result's in this paper, except' a t Pe = 0.30.1 A comparison of the Sherwood number-Peclet number relations discussed above is indicated as case 3 on Figure 1, and formulas representing these relations are summarized in Table I. For internally circulating spheres, the present numerical scheme when used with the Hadamard stream functions for Xewtonian fluids gave results in very good agreement with the Sherwood numbers c,slculated by Bowman et al. (1961) using a four-term concentration profile, except when Pe is greater than about 5 X Comparisons between these two approaches are presented in Figure 1 for a circulating gas bubble of carbon dio.xide rising in water (case 1) and a circulating liquid droplet of ethyl acetate rising in water (case 2 ) . As indicated in Figure 1, the ratio of the dispersed to the continuous phase viscosity has a definite effect on the intensity of internal circulation and, therefore, the mass transfer rat'es. T h e Sherwood number relation is inversely proportional to the value of t'he viscosity ratio. -1s expected, the value of the Sherwood number approaches that of noncirculating spheres when the viscosity ratio approaches infinity or {vhen the Peclet number becomes very small. T h e former case implies that the spheres become essentially rigid; the latter case, Sh equal to 2, implies a very thick concentra-
+ 4)2(R + 1) > 1
Griffith (1958) Griffith (1960)
tion boundary layer or diffusion into an essentially stagnant medium. In either case, the effect of internal circulation is negligible. Analytical solutions for mass transfer from circulating spheres have been obtained by a number of previous workers, assuming a very thin concentration boundary layer, which is usually encountered in the high Peclet number region. Formulas for the Sherwood number are listed in Table 11, and a comparison of the present model with the relat'ions of previous workers is illustrated in Figure 2 for this Peclet number region. Each of the analytical expressions was derived through several stages of approximation-for example, Griffith (1960) and Ward (1961), using different' approximate forms for the concentration distribution, arrived a t different relations for predicting the Sherwood number. Also, because of different assumptions made in one stage of the approximation procedure, their relations have different regions of validity. The Peclet number criteria for the range of applicability of these relations are considered first. T o simplify the calculations, the case of a zero viscosity ratio (R = 0) is considered as a n illustration. For this case, Griffith's (1958, 1960) development suggests l < Pe < 38.4 and Pe > 38.4 for Equation 21 and 23, respectively, whereas the ranges for the development of Ward (1961) are l < P e < 1010.8 and P e > 1010.8 for Equations 22 and 24, respectively. The close similarity of their Sherwood number relations and yet the Ind. Eng. Chem. Fundom., Vol.
9, NO. 3, 1970 483
70 .
(bl (c) (d) (e)
60
-.
Ward (1961) this work Levich (1962) Griffith 11958)
50
40
Sh 30
20
IO a.
-
' 1 - _
-.
-1 -.-
(a$
0 10'3
10-2
10-1
io
IO
to2
io3
Figure 2. Comparison of current model with previous studies for Newtonian continuous phase Effect of viscosity ratio in high Peclet number region
wide difference in their region of validity combine to suggest the uncertain, approximat,e nature of t,he analytical solutions. The inconsistency of the results of the previous workers can be seen from another point of view. For the case of Pe = 104, Figure 2 indicates that, Griffith (1958, 1960) predicts a switchover of the power to which the Peclet' uuniber is raised (one half to one third) a t R E 6.55, whereas Ward (1961) predicts that this occurs at R G% 1.55. This accounts, in part, for t,he wide discrepancies between their results in t,he region
1 100, as it reduces essentially to Equation 19 when R approaches infinity, which Friedlander (1961) claimed to be more realistic than Equation 18a. For the same reason, Griffith's Equation 21 cannot be m. used, as it reduces identically to Equation 189 when R For the region R < 1, the present model a t high Peclet numbers predicts lower transfer rates than Griffith (1960) and Ward (1961). I n an article coaut,hored ivith Ward, 1 3 0 ~ man et al. (1961) suggested the use of a numerical niet'hod for --f
484
Ind. Eng. Chem. Fundam., Vol. 9, No.
3, 1970
circulating spheres-Le., for spheres in the low visc0sit.y ratio region. This numerical method predicts lower mass transfer rates than Ward's (1961) analyt'ical solutions and compares more favorably with the present model. Six points from the Bowman et al. (1961) results taken from Figure 1 are replotted in Figure 2. For this region ( R < 1) Griffith's earlier work (1958) results in the closest' agreement with the present model for Pe = lo4, but it is Griffith's later work (1960) that predicts almost t'he same transfer rate as the present model for Pe 5 lo3. On the other hand, Levich (1962) used a different approach from Friedlander (1957) and arrived at an analytical solution which predicts lower mass transfer rates than the present model for all values of Peclet numbers. However, in his derivation Levich used an approximate form of the stream funct'ion and also neglected the second term on the right-hand , side of Equation 1. I n Figure 2 for Pe = lo4, the results of the present model are between those of Griffith (1960) and Levich (1962). Levich's criterion for a thin concentration boundary layer is Pe > 1.0, whereas Friedlander's (1961) is Pe > 100. Therefore, the exact region of validity for the t'hin boundary layer assumption is still open to further investigation. Because of the various aspects of uncertainty in the analgtical solutions in Table 11,it is suggested that the present model, free from any limitations (except t'he numerical, finitedifference approximation) be used for predicting mass transfer from circulating spheres. For approximate purposes, Ward's (1961) relat'ionship can be used for R > 100 and Pe > 1000, and Griffit'h's (1960) results can be used for R < 1.0 and Pe < 1000. For R < 1.0 and Peclet numbers higher than about' 5 X both Ward's (1961) and Griffith's (1960) relations tend to predict high mass transfer rates which are t'oo large when compared with the present model; thus, these relations should not be used in this region-Le., R < 1.0 and Pe 2 5 x The above discussion and comparison of continuous phase mass transfer models for Newtonian fluids in both phases indicate that the numerical technique used in t,his \vork leads to a Sherwood number-Peclet number relation which agrees well with the various solutions of previous workers, particularly for Peclet numbers less than about lo3.Thus, this invest,igation was extended with confidence to situations involving a non-Newtonian continuous phase. Non-Newtonian Continuous Phase. The results obtained using the velocity component,s derived from the Kakano and Tien (1968) stream functions are discussed first, followed by comparisons and studies of Sherwood number relations based upon more restrictive stream functions (Hirose and MOOYoung, 1969; Tornita, 1959). As described earlier, t'he Nakano and Tien (1968) stream functions are applicable to either liquid droplets or gas bubbles (assumed to be Yewtonian fluids) traveling under laminar flow conditions through a non-Newtonian continuous phase described by a power-law constitutive equation. The continuous phase Sherwood number (kC2a/D)was calculated as a function of the Peclet number, Pe, flow behavior index, n, and a viscosity ratio parameter, X (see Table 111, XSIS). The Sherwood number relation obtained by considering t'he flow behavior index, n, equal to unity was discussed above for specific examples (COz bubbles in water; ethyl acetate droplets in water; and rigid spheres in any Seivtonian fluid). Mass transfer results using the Nakano and Tien stream functions are presented as a funct'ion of Pe in Figure 3 for values of n equal to 1.0,0.8,and 0.6, respectively. Threevalues of the viscosity ratio parameter [X = ( p d / K ) ( ~ / v ~ ) ~ - ' ]
2 00
too 60 40
Sh 20
n=l.o
IO 6
4
2
10-3
10-2
10-1
10
1.0
io2
io4
io3
I os
Pe
100
60
1
10'~
IO-^
IO-'
ID
IO
io2
lo3
io4
ios
Pe Figure 3. Sherwood number as a function of Peclet number and viscosity ratio parameter, based on Nakano and Tien stream functions Upper. Center. Lower.
n = 1.0 n = 0.8 n = 0.6
Ind. Eng. Chem. Fundom., Vol. 9, No. 3, 1970
485
"fully" circulating gas bubbles or liquid droplets. The effect of the viscosity ratio parameter asymptotically diminishes as X increases beyond a value of about 10; this reflects the decrease of circulation velocity within the dispersed phase, partly as a result of the relative increase in dispersed phase viscosity which in a practical sense would most likely be encountered with certain liquid droplet systems. Hirose and Moo-Young (1969), in a recent study of drag and mass transfer characteristics of a spherical gas bubble moving in creeping flow through a power-law non-Newtonian fluid, derived the following continuous phase velocity components:
n=
vu= {[I -
0.6 0.8 0.9
--
-
-0.9
1 ('>"-}]1)A(
1
5 Y n z 0 . 6 - 1.0
6 Y
}I)!(
+6
cos0
(14a)
sin0
(14b)
Y
In the derivation of the above relations, it was assumed that the gas phase may be regarded as inviscid relative to the external liquid phase. At one point in their derivation, it was also assumed that the 8eviation from Newtonian flow is small-Le., 1 % - 1' 5 x (lO)a]. But, this modified expression (or any other simple modification) will not accurately estimate the effect on the Sherwood number of either the power law index, n (when n is not equal to l), or the Peclet number when the Peclet number is less than about 103. I t is concluded that
the Hirose and Moo-Young relation (Equation i5) may be used only when the Peclet number is less than about 5 X (10) and when X is less than about 0.01. Almost a11 gas bubble systems will satisfy the criterion for X; however, not all liquid droplet systems will satisfy this criterion. I n an attempt to study further and possibly explain this discrepancy of the Hirose-Moo-Young model, the authors directly used the velocity components developed by Hirose and Moo-Young (1969)-i.e., Equations 14a and 14b-with the differential mass balance and boundary conditions used in this investigation (Equations 1 and 2) t o solve for the Sherwood number numerically a t various values of the Peclet number and power-law index, n (Figure 5 ) . I n contrast to Hirose and Moo-Young's analytic solution, which was based upon some approximations in the integration procedure, the Sherwood number relation obtained numerically using their velocity components agrees very closely with the numerical results obtained in this work using the Nakano and Tien (1968) velocity components (for X 5 0.01) a t all levels of the Peclet number. It is tentatively concluded that the Hirose and Moo-Young velocity components are correct, but there probably is an inaccurate approximation in their integration procedure which becomes important when Pe > 5 x (10)3. Tomita (1959) studied the drag force on a very slowly moving sphere (with no internal circulation) moving through a power-law type fluid by assuming an expression for the continuous phase stream function which satisfied certain boundary conditions. The corresponding velocity components based upon these stream functions are given as follows:
1.1
I
1.0
0.9
Sh +/Sh s
Shs:Sherwood number obtained using Stokes stream functions
0.7
Y
1
0.6I 0.6
0.7
0.8
1.0
0.9
1.2
1.1
1.3
1.4
n Figure 6. Effect of non-Newtonian behavior for noncirculating spheres using Tomita stream functions 80
70
60
The approximate nature of the stream function is demonstrated by the fact that when n is equal to unity, the above velocity components do not reduce identically to those derived from the Stokes stream function (let R + m in Equation 5) which were based on Newtonian fluid behavior. Despite this difficulty, Sherwood numbers were calculated numerically using Equations 1, 2, and 16. The results are presented in Figure 6 as the ratio of the Sherwood numbers obtained using the present model based on the Tomita stream functions to those Sherwood numbers based on the Stokes stream function plotted as a function of the power-law index, n, with the Peclet number as a parameter. At n equal to unity, all curves should coincide a t the point [1.0, 1.01; however, as seen in Figure 6, the curves do not coincide a t this point, indicating the error in the Tomita analysis. I n fact, the discrepancy increases as Pe increases. I n the pseudoplastic region the Sherwood number based on Tomita's stream functions decreases as n decreases, contrary to the results obtained in this work for all values of X or by the development of Hirose and Moo-Young (1969). Conclusions and Summary
The Nakano and Tien (1968) stream functions which were developed for spheres of a Newtonian fluid moving in powerlaw type, non-Newtonian fluid were used to solve for the continuous phase rates of mass transfer in the creeping-flow region. The Sherwood number relation was developed for a wide range of the Peclet number, power-law index, and viscosity ratio parameter. The solution applies to both liquid droplets and gas bubbles. I n the pseudoplastic region, the
---_-Present
study using Nakano and Ten$ stream function ( X = O . O I l
50
-
Sh
Present study using Hirose and Moo-Young's stream functions
40
- .-
Hirose and Moo-Young's results (19691
30
20
IO
_--__-_ - . -.-. ----_._.
*.-.-.Pe ol : =
-.-.-
- -._
-._,_,
d e = 1.0
0 5
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
n Figure 5. Effect of non-Newtonian behavior on mass transfer from circulating gas bubbles Comparison of various models
Sherwood number decreases as the power-law index increases; but this effectsignificantly decreases a t low Peclet numbers. The Hirose and Moo-Young analytic mass transfer solution for fully circulating bubbles and the numerical results of this investigation (for X < 0.01) are in close hgreement except However, for a t very high Peclet numbers [Pe > 5 X viscosity ratio parameters greater than about 0.05, the results Ind. Eng. Chem. Fundam., Vol. 9,
No. 3, 1970 487
of this investigation rather than the Hirose and Moo-Young relation must be used to consider the relative viscous nature of the two phases. The Tomita stream functions for noncirculating spheres cannot be used accurately to predict continuous phase mass transfer coefficients.
P U
7;
A;
AY
A0
= = = = = =
density exponent in Equations 3a and 3b stress tensor rate of deformation tensor radial increment in finite difference equations angular increment in finite difference equations
SUBSCRIPTS spherical radius of dispersed phase coefficients in Equation 9, defined in Equations 10a to 1Oc coefficients in Equations 3a and 3b concentration of solute transferred from sphere into continuous phase, dimensionless concentration of solute in finite difference Equations 9 and 11 diffusivity of solute in continuous phase coefficients in Equation 11, defined in Equations 12a to 12d continuous phase mass transfer coefficient consistency index of power law fluid flow behavior index Peclet number, 2aV,/D, dimensionless p d / p c , dimensionless Reynolds number, 2a V,p/p, dimensionless Sherwood number, k,V,/D, dimensionless coefficients in Equation 12d, defined in Equation 12e radial velocity, dimensionless tangential velocity, dimensionless relative velocity between sphere and continuous uhase fluid radial displacement from center of sphere
GREEKLETTERS 0
=
P
=
488
angular displacement from front stagnation point viscosity
Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
C
= continuous phase
d
=
dispersed phase
literature Cited
Baird, M. H. I., Hamielec, A. E., Can. J . Chem. Eng. 40, 119 (1962). Bowman, C. W., Ward, D. M., Johnson, A. I., Trass, O., Can. J . Chem. Eng. 29,9 (1961). Crank, J., Nicolson, P., Proc. Cambridge Phil. SOC.43, 50 (1947). Fararoui, A., Kiptner, It. C., Trans. SOC.Rheol. 5 , 369 (1961). Friedlander, S.K., A.1.Ch.E J. 3,43 (1957). Friedlander, S. K., A.I.Ch.E.J. 7, 347 (1961). Griffith, R. M., Chem. Eng. Sci. 12, 198 (1960). Griffith, R. M., Ph.D. dissertation, University of Wisconsin, 1958. Hadamard, J., Compt. Rend. 152, 1734 (1911). Hirose, T., Moo-Young, M., Can. J . Chem. Eng. 47,265 (1969). Levich, F. G., “Physicochemical Hydrodynamics,” PrenticeHall, Englewood Cliffs, N. J., 1962. Levich, V. G., “Physicochemical Hydrodynamics,” Moscow, 1952. Nakano, Y., Tien, C., A.1.Ch.E J . 14, 145 (,1,968). Stokes, G. G., “Math and Phys. Papers, Vol. 1, Cambridge University Press, 1880. Tomita, Y., Bull. SOC.Mech. Engrs. 2,469 (1959). Ward, D. M., Ph.D. dissertation, University of Toronto, 1961. Yuge, T., Rep. Inst. High Speed Mech. Tohoku Univ. 6, 143 (1956). RECEIVED for review October 17, 1969 ACCEPTED June 22, 1970 For supplementary material, order NAPS Document 01069 from
ASIS National Auxiliary Publications Service, c/o CCM Information Sciences, Inc., 909 Third Ave., New York, N. Y. 10022, remitting $2.00 for microfiche or $5.00 for photocopies.
