303
Ind. Eng. Chem. Fundam. 1988, 25,303-305
C , = solute composition of flowing solvent, mol/cm3
DL = liquid diffusivity, cmz/s
E = dimensionless composition (defined by eq 4) E = fraction of solute unextracted (eq 7) g = acceleration due to gravity, cm/sz H z ( t ) = height from base of feed reservoir, cm H , ( t ) = height from base of exit reservoir, cm Hzo = initial height of feed reservoir, cm L = length of flow coils or length of capillary, cm R = inner radius of coiled tubing, cm s = dimensionless Laplace variable t , = exposure time for extraction, s t = time, s z = capillary tube coordinate, cm Creek Letters = liquid viscosity, g/(cm.s) [ = dimensionless capillary coordinate ( z / L )
j~
= liquid density, g/cm3 0 = dimensionless time Registry No. rn-Xylene, 108-38-3; o-xylene, 95-47-6.
p
Literature Cited Anderson, J. S.; Saddington, K. J. Chem. SOC. 1949, 381. Caldweli, C. S.; Babb, A. C. J. Fhys. Chem. 1956, 6 0 , 14. Goodner, M. A. M.S. Thesis, Louisiana State University, Baton Rouge, LA, 1984. Hammond, 0. R.; Stokes, R. H. Trans. Faraday SOC. 1955, 51, 1955. Mickley, M. S.; Sherwood, T. K.; Reid, R. C. "Applied Mathematics in Chemical Engineering"; McGraw-Hill: New York, 1957; p 308. Milis, R.; Woolf, L. A. "The Diaphragm Cell"; Diffusion Research Unit: A.N.U., Canberra, Australia, 1988. Piret, E. L.; Ebel, R. A.; Kiang, C. T.; Armstrong, W. D. Chem. Eflg. Bog. 1951, 47, 405. Scheibel, E. G. Ifld. Eng. Chem. 1954, 4 6 , 2007.
Received for review September 26, 1984 Accepted May 17, 1985
COMMUNICATIONS Reaction-Diffusion in Suspended Particles with Limited Supply of Reactant The concentration decay of a species from the bulk phase of a suspension is examined. This species has limited presence in the fluid and disappears by means of diffusion and first-order reaction in the suspended spherical particles. Equations are presented which relate the decay of the reactant in the bulk of the fluid to the diffusivity of the reactant in the particle, its reaction rate constant, the mass-transfer coefficient, and the ratio of the volume of the particle to the volume of the fluid that surrounds it.
There are many situations of practical significance where an ionic or molecular species reacts with particles in a liquid suspension and where the concentration of this species in the bulk fluid changes with time due to its limited mass. One example is the time dependence of the stability of a suspension. If a species reacts with the suspended particles, then the surface chemistry and charge characteristics of those particles will be altered with time. Moreover, if the concentration of that species decreases in the bulk fluid of the suspension, then its solution chemistry will be altered as well. The macroscopic result of such changes is a time dependence of the suspension's stability and settling properties. Other examples include the uptake of drugs and nutrients by cells in suspension, the diffusion-reaction in suspended catalysts, etc. In the absence of chemical reaction and surface resistance to mass transfer, the model that describes the rate of disappearance of a species from a fluid phase by means of diffusion in a solid (Crank, 1956) has provided the basis for the experimental determination of diffusivities in porous solids (Satterfield and Margetts, 1971; Park and Kim, 1984). In the present study, the species, which diffuses in the solid, reacts with the solid and may experience resistance to mass transfer in the fluid layer around the suspended particles. Therefore, measurements of fluid concentration vs. time may be used in conjunction with the equations in order to determine, in addition to the diffusivity, the values of the reaction rate constant and the mass-transfer coefficient. Also, note that these equations can be adapted to the cooling of a sphere which is immersed in a well-stirred or poorly stirred fluid, when there
is heat generation inside the sphere with a linear dependence on temperature. The approach is similar to the one followed by Paterson (1947) for the heating or cooling of a solid sphere which is immersed in a well-stirred fluid. That method is reviewed by Carslaw and Jaeger (1959) and Bird et al. (1960). The general problem dealt with in this communication has also been studied by Sotirchos and Villadsen (1981), Do and Bailey (1981), and Do (1984). However, each of these studies addressed different aspects of diffusion-reaction with limited reactant and also presented different mathematical treatments toward either the formulation or the solution of the model.
Theory In a suspension, there is an amount of fluid which corresponds to each individual particle, and therefore the following analysis may be confined to a single particle. The suspended particle is assumed to have spherical shape of radius R and be immersed in a fluid whose volume is V (Figure 1). For first-order reaction kinetics the equation that describes the concentration profile inside the particle is
where E = r / R , the dimensionless radial position; T = t D / R 2 , the dimensionless time; and k = k l R 2 / D ,the dimensionless first-order reaction rate constant and also the square of the Thiele modulus. The concentration C is
0196-4313/86/1025-0303$01.50/0 0 1986 American Chemical Soclety
304
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986
7Figure 2. Decay of the bulk concentration, as affected by the particle to fluid volume ratio. Figure 1. Representation of a spherical particle suspended in a fluid of volume V.
represented by C([,T),and the diffusivity, a, is assumed constant. Initially, the concentration is zero, and suddenly a certain amount of reactant is introduced in the surrounding fluid. In addition to (l), another differential equation is needed to describe the disappearance and thus decreasing concentration of reactant in the fluid. Such an equation is derived by balancing the rate of transfer of the reactant from the fluid to the particle
20
40
50
7Figure 3. Decay of the bulk concentration, as affected by the dimensionless reaction rate constant.
where 6 is the ratio of the particle’s volume to the fluid’s volume. The initial conditions for (1) and ( 2 ) and the boundary conditions for (1)are C(E,7=0) = 0
(3)
co
(4)
Cf(7=0) =
C(E=0,7) = finite
(5)
where m = D/k,R (the inverse of the Nusselt number Nu) and k , is the mass-transfer coefficient. By taking the Laplace transform ( L : 7 s) of (1) and (2)’ we obtain
pc” + 2@’-
-
(s
+ k)pC = 0
(7)
and
Solving (7) and (8) with the use of the boundary conditions gives
Cf/CO = l / s - (36[(s+ k ) 1 / 2cosh (s + k ) 1 / 2- sinh (s + k ) 1 ’ 2 ] ] / { ~ { (+ m3~6 ) ( ~+ k)’/2 cash (S + k ) l l z - [ ( m1)s + 361 sinh (s + k ) 1 / 2 ) (9) ) If one uses the method of residues (Churchill, 1944), after a series of tedious manipulations (9) may be inverted (L-I: s T ) and the concentration in the bulk is found
-
m
Cf/Co = 66
+
[(P,2e-(”*+klT)/(Pn2{m(Pn2 k ) - 36) X il=l
(rn(P,*
+ k ) - 36 + 2(m - 111- (2m/3,2+ pn* + k ) ( ( m1)(Pn2 + k ) - 36111 (10)
The eigenvalues, P,, are the roots of the transcendental equation tan P, =
Pn(m(Bn2 +
k ) - 361 ( m - l)(Pn2+ k ) - 36
(11)
Results and Discussion When there is no surface resistance to mass transfer, k , = ~0 and thus m = 0. In this case eq 10 and 11 are simplified to
with P, being the roots of
This result may also be obtained by altering the surface boundary condition as expressed by (6) to one that equates the fluid concentration to the surface concentration, i.e.
C([=1,7) = Cf (14) It is instructive to observe the effects of changes in key parameters such as 6, k , D,and m. For a particle radius R = cm, diffusivity 33 = cm2/s, k , = 1 s-l, and infinite mass-transfer coefficient, the effect of increasing the fluid’s volume on the bulk concentration decay is studied by means of Figure 2. When the volume of the fluid equals the particle’s volume (6 = 11, the half-life of the reactant in the bulk is 0.1 s (7 = 1). If the volume of the fluid is 3 times the volume of the particle (6 = 0.331, the corresponding half-life is 1.6 s ( 7 = 16). The next comparison is shown in Figure 3. The particle radius and the diffusivity remain the same as before, and the reaction rate constant is allowed to increase from 1to 5 s-l. The half-life of the reactant in the bulk is then reduced from 1.6 s ( 7 = 16) to 0.35 s ( T = 3.5). Note that the value of k may be changed from 0.1 to 0.5 by changing from to 0.2 X cm2/s, keeping k , constant. As a result, the half-life of the reactant in the bulk will only increase from 1.6 s (T = 16) to 1.75 s (T = 3.5). Next, a comparison was made to examine the effect of the mass-transfer coefficient on the fluid concentration. For this purpose a selection was made of Nu = 2 (m = 0.5), which corresponds to a mass-transfer coefficient k , = 2 X lo-’ cm/s. From Figure 4,it is noted that such a masstransfer coefficient has little effect on the concentration decay of the reactant. If h , is further reduced to a value
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986 305
calculations were carried by Hsien-Ming Liu, and the typing of the manuscript was done by Renee Ladmirault.
40
20
60
7-
Figure 4. Decay of the bulk concentration, as affected by the inverse of the Nusselt number. of 0.5 X cm/s, the results are practically the same. It may thus be concluded that the ratio of particle volume to fluid volume and the reaction rate constant are the most important parameters for the rate of uptake of a reactant from the bulk by particles in suspension. This conclusion is independent of the choice of parameters in the above discussion, and it may also add a new dimension in interpreting experiments on very rapid uptake of substances by suspended particles. For example, in explaining the varying rate of oxygen uptake by red blood cells, Merchuk et al. (1983) focused their attention on the effect of the mass-transfer coefficient,which, according to results of this investigation, is the least sensitive parameter. Finally, if interest lies in the time depeqdence of the concentration profile inside the particle, C is found as
c/c, =
(l/[)[sinh [(s + k ) 1 / 2 ] / [ ( m+s 36)(s + k)1/2 X cosh (s k)1/2 - ((m- 1)s + 361 sinh (s + k)1/2] (15)
+
and then inverted _C C,
+
((m- l)(Pn2 k) - 36}(2mpn2+ 02 + k where 0,are the eigenvalues of (11).
+ 36)]
t
(16)
Acknowledgment Partial support by the National Science Foundation (Grant PE-8307488) is gratefully acknowledged. The
Nomenclature C = concentration profile inside a particle, mol/cm3 C = C in the Laplace domain Co = initial concentration in the bulk, mol/cm3 Cf = time-dependent concentration in the bulk, mol/cm3 Cf = Cf in the Laplace domain Zl = diffusivity, cm2/s 12, = first-order reaction rate constant, s-l 12 = dimensionless first-order reaction rate constant (= kiR2/a) It, = mass-transfer coefficient, cm/s I = Laplace operator I-'= inverse Laplace operator m = inverse Nusselt number Nu = Nusselt number (= 12,Rl.B) r = radial coordinate, cm R = particle radius, cm s = Laplace variable t = time, s V = volume of the fluid, cm3 & = eigenvalue 6 = ratio of the particle volume to the volume of the fluid E = dimensionless radial coordinate (= r / R ) T = dimensionless time (= t Z l / R 2 )
Literature Cited Bird, R. B.; Stewart, W. E.: Llghtfoot, E. N. "Transport Phenomena"; Wiley: New York, 1960. Carslaw, H. S.; Jaeger, J. C. "Conduction of Heat in Solids", 2nd ed.; Oxford Unlverslty Press: London, 1959. Churchlll, R. V. "Modern Operational Mathematics In Engineering"; McGrawHill: New York, 1944. Crank, J. "The Mathematics of Diffusion"; Oxford University Press: London, 1956. Do, D. D. Biotechnol. Bioeng. 1984, 26, 1032. Do, D. D.; Bailey, J. E. Chem. Eng. Commun. 1981, 12,221. Merchuk, J. C.; Tzur, 2.; Lightfoot, E. N. Chem. Eng. Sci. 1983, 38, 1315. Park, S. H.; Kim, Y. G. Chem. Eng. Sci. 1084, 39, 533. Paterson, S. Proc. Phys. SOC.,London 1947, 59, 50. Satterfield, C. N.; Margetts, W. G. AIChE J. 1971, 17,295. Sotirchos, S. V.; Villadsen, J. Chem. Eng. Commun. 1081, 13, 145.
Department of Chemical Kyriakos D. Papadopoulos* Engineering Raymond V. Bailey Tulane University New Orleans, Louisiana 70118 Received for review October 15, 1984 Revised manuscript received October 21, 1985 Accepted December 26, 1985 To whom correspondence should be addressed.
