Gupta and Thodos ( 8 ) correlated all the available gas and liquid mass transfer data for a .VR~’ range of 1.0 to 2140. However, the only data used below a I Y Y ( ~ofI 12 were the liquid data of Gaffney and Drew. Their equation is
L Npe NR,’
lvso
= =
liquid flow, lb./sq. ft. empty column cross section-sec.
D,U/D
= D,L/p = p/pD
R
=
+
dilution ratio in active bed, (Xo. benzoic inert spheres)/(So. benzoic spheres) interstitial velocity, ft./sec. packed bed length, ft. void fraction viscosity, lb./ft. sec. density, lb./cu. ft. correction factor to be applied to J defined by Equations 3 and 7
U
=
X
=
E
Some of the data for gases scatter widely, with some deviations of 1OOycfrom this line. .4t a SRe ’ of 1.O, Equation 10 derived primarily from gas data gives Je values 60Yo greater than the liquid correlation (line 1, Figure 6), with the deviations decreasing a t higher values of Bradshaiv and Bennett (7) predict that a t a - V R ~ ’of 1.0, the J value for gases ivith a of 2.57 should be 5795 greater of 2300. Pfeffer also states that than for liquids with a -VB~ because of the higher AYsoand ‘Vpe of liquids, different correlations should be expected for the J values for gases and liquids especially a t low values of .VR,’. Hence, Equations 8 and 9 are recommended for liquids and the correlation of Gupta and Thodos (8) for gases.
p
= =
p
=
N o rnenclature
(11) McCune, L. K., FVilhelm, R. H., Znd. Eng. Chem. 41, 1124 (1949). (12) Pfeffer, Robert, IND. ENG. CHEM.FUNDAMEh-TALS 3, 380 (1964). (13) Stahel, E. P., Geankoplis, C. J., A.I.Ch.E. J . 10, 174 (1964). (14) Steele, L. R., Geankoplis, C. J., Zbid., 5 , 178 (1959). (15) Steinberger, R. L., Treybal, R. E., Ibid., 6 , 227 (1960). (16) FVilliamson, J. E., Bazaire, K. E., Geankoplis, C. J., IND. ENG.CHEM. FUNDAMENTALS 2,126 (1963).
JE = 0.0100
=
b
= Cy:ICs
C1 Cz Cs
= = = = = = =
kL
0.863 (.YRe’)0’58
- 0.483
sq. ft. surface area/cu. ft. volume of bed
Q,
D D, J
+
inlet concentration, lb.~’lb, outlet concentration, lb./lb. saturation solubility, lb./lb. molecular diffusivity, sq. ft./sec. particle diameter, ft. (k,/L)(.Vs,)*;3 mass transfer coefficient, lb. solute//sq. ft.-sec.-iC
=
Literature Cited (1) Bradshaw, R. D., Bennett, C. D., A.Z.CI2.E. J . 7,48 (1961). (2) Bradshaw, R. D., Meyers, J. E., Zbid.,9, 590 (1963). (3) DeAcetis, James, Thodos, George, Znd. Eng. Chem. 5 2 , 1003
(1960). (4) Epstein, Norman, Can. J . Chem. Eng. 36, 210 (1958). (5) Evnochides, Spyros, Thodos! George, -4.Z.Ch.E. J . 7 , 78 (1961). (6) Gaffney, B. J., Drew, T. B., Znd. Eng. Chem. 42, 1120 (1950). ( 7 ) Gottschlich. C. F.. A.I.Ch.E. J . 9. 8 8 (1963). (8) Gupta, A. S., Thodos, George, Zbid., 8: 608’(1962). (9) Zbtd.,9, 751 (1963). (10) Linton, \V. H., Sherwood, T. K., Chem. Eng. Progr. 46, 258 / I ncn\ (1 ’IJV)
.
RECEIVED for review February 15, 1965 ACCEPTED September 7, 1965
MASS TRANSFER FROM A FLAT SOLID
SURFACE T O A FALLING NONNEWTONIAN LIQUID FILM GlANNl A S T A R I T A Zstttuto di Chimica Industriale, UniLersity of .\hples, .Yaples, Italy
The phenomenon of molecular diffusion in the flow field which is established in the neighborhood of a solid surface during steady, laminar flow of a nowNewtonian liquid is discussed, and the solution is given for a velocity distribution of the type u = a x m , This solution is used to interpret experimental results on the rate of dissolution of slabs of benzoic acid into a falling film of aqueous carboxymethylcellulose solution. The diffusivity of benzoic acid in the solution is evaluated from the data as 1.4 X 10-’sq. cm. per second.
few data are available ( 7 , 2, 6 ) on mass transfer phenomena in non-Newtonian liquids, and only one paper (6) reports a n investigation of mass transfer in the nonzero shear rate region of a flowing non-Newtonian liquid. The mass transfer experiment discussed in this paper consisted of the dissolution of the inner wall of a tube into a liquid flowing through it in laminar flow; the order of magnitude of the shear rates in the concentration boundary layer was 100 sec. Kramers and Kreyger (70) have made experiments on the dissolution of a flat solid surface into a falling Neivtonian ERY
14
I&EC FUNDAMENTALS
liquid film. The shear rates encountered in the flow of a viscous liquid along an inclined flat surface are generally lower than those encountered in the flowthrough capillary tubes; this suggests that an extension of the technique of Kramers and Kreyger to non-Xewtonian liquids may make it possible to investigate the low-shear-rate region. Such an extension requires the understanding of the fluid mechanics involved. The fluid mechanics of the laminar steady flow of a non-Kewtonian liquid along an inclined plane surface have been discussed extensively by Astarita, Marrucci, and Palumbo ( 3 ) . I n the present paper, the phenomenon of
molecular diffusion in a flow field such as described in (3) is discussed. The fallirlg film technique has been applied to the system benzoic acid-aqueous carboxymethylcellulose (CMC) solution, and the diffusivity of benzoic acid in the solution has been evaluated from the data. The application shculd be regarded as a n example of a more widely applicable analysis of a diffusion problem. Theory Consider a non-Newtonian liquid which is flowing under the influence of gravity along a flat solid surface in steady laminar flow. Let be the direction of flow and x the direction normal to the flat surface. Downstream of a position) = 0, where the velocity profile is fully developed, the solid surface consists, for a length Y , of a soluble material. The differential equation governing the diffusion phenomenon is, when the diffusional transport in the )-direction is neglected as compared to the convective transport
b2c u b= U(X) X2
dC -
dY
where U(X) is the velocity at distance x from the solid surface. The boundary conditions to be satisfied are: x = 0, =I
0,
y + a,
c = c,’
(2)
c=c,
(3)
c is bounded
(4)
Lt‘hen a constant value of u is assumed (plug flow), the solution of the problem is well known (9). The solution is also known for the case of a linear velocity profile (7, 70), which corresponds to the case of a Newtonian liquid when the concentration boundary layer is much thinner than the velocity boundary layer. I t has been shown (J) that in the case of non-Newtonian liquids, minor wall-slip effects cannot be excluded from consideration. Thus, even if a very thin concentration boundary layer is assumed, there is some doubt of the validity of a linear velocity profile assumption. In reality, there is probably no true wall slip effect, buit a layer of low-viscosity solvent may form in the immediate. neighborhood of the solid surface during the flow of a non-Xewtonian solution of a high polymer. The thickness, A, of this layer is so small that its effect is equivalent to a wall slip as far a s the macroscopic hydrodynamics are concerned ( 3 ) . If the hypothesis is made that X is also much smaller than the concentration boundary layer’s thickness, A, (this is of course a more severe hypothesi:; because A, is in turn assumed to be much smaller than the liquid film’s thickness, 6 ) , the problem of diffusion could also be treated as if a wall slip were present. Beek and Bakker’s solution (4) of the analogous free interface problem could be used, but their model appears to be unrealistic for the case considered here because a true interface nonzero velocity is assumed. Fortunately, when thr. problem is formulated more realistically, it also turns out to be amenable to analytic solution. In fact, it is unbelievable that a solvent layer of well defined thickness exists, so that, at L = A, the rheological properties of the fluid change abruptly from those of the Newtonian solvent to those of the non-Newtonian solution; furthermore, the A, 6 seems to be rather restrictive. double assumption X It is physically more realistic to assume that, in the neighborhood of the solid surface, there is a gradual decrease in polymer concentration whose over-all effect is equivalent to that of a solvent layer.
If such is the case, the velocity distribution can be approximated by a n equation of the form
(5)
~ ( x )= amxm
with m < 1. When there is no wall slip, Equation 5 holds true with m = 1; in the other extreme case-when the velocity of slip is so high that the velocity profile may be taken to be flat-Equation 5 holds with m = 0, where a. is the (constant) slip velocity. Substitution of Equation 5 into 1 gives:
Equation 6, subject to Boundary Conditions 2, 3, and 4, can be integrated ( 5 ) . The solution of this problem is believed to have a more general interest than the problem discussed in this paper. In fact, Equation 6 may be considered to govern diffusion in a liquid moving with a n arbitrary velocity profile because Equation 5 can in practice approximate any realistic velocity profile within the concentration boundary layer. Definition of the Laplace-transformed concentration, g, g = s
J0
exp(--sy) (c
- co) dy
(7)
reduces Equation 6 with Boundary Condition 3 to
subject to Boundary Conditions 2 and 4. Equation 8 is a form of the Bessel equation; the integral which satisfies Conditions 2 and 4 is:
(9) where t =
2 q(ams/D)1/2x1/2q
(10)
and
q = l/(m
+ 2)
(11)
In order to calculate the average mass transfer rate, only the concentration gradient at x = 0 needs to be known. The derivative (dg/dx),,o is obtained from Equation 9 as
and, by straightforward inverse-transformation, (bc/bx),,o
=
- (to’ - co)
qzq-l
__ (a,/Dy)Q
r(4)
(13)
The average mass transfer coefficient is, by definition,
Substitution of Equation 13 into 14 gives:
