Ind. Eng. Chem. Fundam. 1082, 27, 87-90 emf
= voidage at minimum fluidization
Literature Cited
Davldson, J. F.; Harrlson, D. “Fiuidlzed Partlcles”; Cambridge Universky Press: Cambridge, 1963; p 100. Kobayashl, H.; Arai, F.; chiba, T.; Tanaka, Y. Kagaku K o p k u 1969, 33,274. Kunll, D.; Levensplei, 0. “Fluidlzation Engineering”; Wlley: New York, 1969: p 154. Murray, J. D. J . F/u&jMed,. 1065, 21. 465. Rowe, P. N. “Fluidlzatlon”; Davidson, J. F.; Harrison. D.. Eds.. Academic Press: London, 1971; p 121.
87
Tomb, M.; Adachl, T. J . Cbem. Eng. Jpn. 1973, 6 , 196.
Department of Chemical Engineering Indian Institute of Technology Madras 6oo India
N. H. Hosabettu N. Subramadan*
Received for review August 13, 1980 Revised manuscript received September 9,1981 Accepted September 30, 1981
Effect of Axial Changes in Gas Flow Rate on Values of Mass Transfer and Dispersion Coefficients Computed from Column Measurements The effect of axial changes in the gas flow rate due to absorption is analyzed for systems with constant liqukcside concentration of the solute and with axial dispersion in the gas phase. Whether changes in the flow rate may be neglected in determining the true value of transfer units and/or the gas phase Peclet number from steady-state measurements Is determined by the concentration of the absorbed component In the inlet gas rather than by the absolute amount that has been absorbed.
Phase flow in chemical processing vessels has often been described in terms of a one-dimensional axial dispersion model (or simply a “dispersion model”) in which any contribution of radial dispersion is disregarded. The plausibility of the simplification has been analyzed by examining the effect of mutual interaction between the radial velocity profile and the radial distribution of concentration on the magnitude of neglected dispersion contribution. The extent to which the effect is significant is dependent on the shape of the velocity profile in the vessel: with a flat velocity profile, the axial dispersion model is a satisfactoryrepresentation of the process (Bischoff, 1968). On the other hand, with a sharp velocity profile, occurring for example with laminar or film flow, the Peclet number is a function of the diffusivity of tracer (Converse, 1960). In the latter case the place and the form of the tracer input are of importance. Radial mixing from a point tracer source may create apparent axial mixing, even in the absence of true longitudinal mixing, and may lead to misleadingly low results for the Peclet number derived from experiments (Cairns and Prausnitz, 1960). In packed columns the radial velocity distribution of greatest concern is that in the liquid phase, usually occurring as film flow over the packing. These considerations are substantiated by absorption experiments in which two different tracers were introduced, one directly into the liquid phase using a plane source of tracer (a conductivity method) and the other into the gas phase from which it diffused into the liquid phase (point source of tracer) (Linek et al., 1978). Another source of discrepancies may be the assumption that both phases pass through the vessel at constant rates of flow, i.e., independent of axial position. In most theoretical studies, changes in the gas flow rate arising as a result of absorption of the transferred component have been neglected (Dunn et al., 1977; Mathur and Wellek, 1976; Shioya and Dunn,1978; Stiegel and Shah, 1976; Sater and Levenspiel, 1966). The assumption has also been adopted in several studies analyzing models for absorption processes from more general points of view (Miyauchi and Vermeulen, 1963;Bischoff and Levenspiel, 1962; Sherwood et al., 1975), with the exception of studies by Brittan and Woodburn, who considered axial changes in the flow rate (Brittan, 1967; Brittan and Woodburn, 1966) and in the gas pressure (Woodburn, 1974; Hatton and Woodburn,
1978) along the column length. Nevertheless, even these studies fail to bring out the effect of the changes, since only the relevant differential equations rather than particular results were presented. In a number of applications of practical interest, the effect of longitudinal mixing in the liquid phase is suppressed. Such a situation arises when the concentration of the transferred component in the liquid phase is nearly invariant in the axial direction (or when its variation has no effect on the driving force profile between phases). An example is absorption accompanied by chemical reaction in the liquid phase with the reacting liquid component in large excess. Another example may be the situation where the liquid flow rate is so high as to prevent any appreciable saturation of the liquid phase. As a starting point of our contribution we have taken the simple situation where the effect of axial mixing in the liquid phase may be disregarded. This approach enables us to assess separately the errors incurred by neglecting the axial change in the gas flow rate. The derivation of the steady-state balance equations is based on the following assumptions: (i) the gas flow rate along the column may be represented by the dispersed piston-flow model; (ii) the volumetric mass transfer coefficient is constant along the column and absorption occurs under isothermal and isochoric conditions; (iii) the concentration of the transferred component in the liquid phase is constant along the column and equal to ciao; (iv) the liquid concentration at equilibrium with the gas is described by the approximation c + d Z ) = mcga(z) + n. The differential balance for the concentration of the component a in the gas phase is of the form
and the overall gas phase balance is
With the use of dimensionless variables and parameters, eq 1 and 2 may be rewritten as
o i ~ ~ ~ ~ i ~ 1 ~ ~ / ~ o ~0~ 1982 - o American o ~ ~ $ Chemical o 1 . ~ Society ~ 1 o
(3)
88
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982
gdz+ y N x = O
(4)
where N is the true number of overall liquid-phasetransfer units, x is a dimensionless expression for liquid-phase bulk concentration, w is proportional to the local gas flow rate, and y is the volume fraction of a in the gas. The boundary conditions are of the same type used by Danckwerts and may be written in dimensionless form as z = 0 w = 1; l/Pgdx/dz - x + 1 = 0; z = 1: dx/dz = 0 (5)
1 3.1
ilnr
Y
-
0
If no change in the gas phase volume rate of flow is considered (Le., if y = 0), the set of eq 3 to 5 simplifies to a form, the analytical solution of which has been presented by Miayuchi and Vermeulen (1963) x 2 = exp(-N,)
= (4b exp(Pg/2)/[(1 + bI2 exp(Pgb/2) (1- bI2 ex~(-P,b/2)11 (6)
where b = (1 + 4N/Pg)lI2. In his original work, Brittan (1967) presented a differential equation describing a more involved situation where the concentration of the absorbed component in the liquid phase also varies along the column. Introducing the above assumption (iii), this equation may be converted into the form taken by eq 3 and 4. The set of differential eq 3 to 5 represents a nonlinear boundary value problem for an ordinary differential equation. Since it cannot be solved analytically,numerical integration was performed. After some manipulation, the set (3) to (5) takes a form more convenient for numerical solution dx/& = Pg(l- y - w + X Y W ) / Y (7) dw/& = - ~ N x
(0 < y
< 1)
(8)
The boundary conditions are w(0) = 1; w(1) = (1 - y ) / ( l - yx(1))
(9)
The set (7) to (9) was solved for assumed values of x(1) = x2 and of the parameters y and PB,searching by iterative procedure for the value of N satisfying the set of equations. Owing to the character of the eigenvalues of the differential equation, it was expedient to integrate the equation in the direction from z = 1 to z = 0 (Coste et al., 1960). The results of the calculations for different values of the Peclet number and for y = 0.21 are s h p n in Figure 1in the form of the ratio N / N p as a function of N p , the number of transfer units computed from compositions at the top and bottom of the column and neglecting dispersion and changes in gas flow rate. Also included in the figure are lines corresponding to asymptotic values of the Peclet number, Pr Assuming piston flow of the gas through the vessel and Pg m , we can derive the relation N _
-
NP
Here and in the following, N p = -In x2. For intense mixing of the gas phase, Pg= 0 and the analogous dependence is of the form
It can be demonstrated, however, that the limiting value
I
2
I
I
I
,
1
1
5
1
1
I
1.
2.
I
1
1
,
5.
1
1
1
IO.
NP
Figure 1. Comparison of plots of NIN, vs. N p calculated with (y = 0.21) and without Cy = 0) considering changes in the gas flow rate due to absorption.
-
-
of the ratio N / N p for x 2 1 (i.e., for N p 0) is independent of the intensity of longitudinal mixing since
For piston flow of gas and total absorption the ratio N / N p takes the value N lim - = 1 - y x2-0 Np Pg-m
-
In performing experiments under conditions where only a small amount of gas is absorbed ( N , 0), it is usually assumed that the influence of axial mixing and of changes in the flow rate due to absorption on the determination of the true number of transfer units (or the true value of KLa)may be ignored. As can be seen from Figure 1 and eq 12, the first assumption is fulfilled. A change in the flow rate, on the other hand, gives rise to a constant error in determination of N even under these conditions. The magnitude of the error depends on the concentration of the absorbed component, y , in the inlet gas. This may be illustrated by the process in which oxygen is absorbed from air into a sulfite solution in the presence of a cobalt catalyst. For an oxygen concentration fall from 21% to 19% during passage through the column, it may be estimated from Figure 1that the true KLu value is 25% higher than that estimated on the basis of the piston-flow or dispersion models. Neglecting changes in the gas flow rate may lead to considerable distortion in KLu values when evaluating results of an experiment performed on columns of different lengths. The distortion is evident from Figure 2, a plot of the ratio of the KLu value obtained by solving the set (3) to (5) for y = 0.21 to KLu calculated from Miyauchi's relation 6 against the outlet concentration x2. For short packing lengths, i.e., at higher values of x2, the simpler relation 6 yields underestimates of KLa,while with greater lengths of packing, i.e., for low x2, the reverse may be true. The magnitude of the error may be illustrated by the following example: let us consider absorption of carbon dioxide (21% mixture with air) into a basic solution in two columns with the same packing but of different lengths. Let the depth of the packing in the second column be four
Ind. Eng. Chem. Fundam., Vol. 21,
No. 1, 1982 89
----------------------- - -- -- -- ---N = 0.13
2
.0 0
Figure 3. Axial gas-side concentration profiles calculated with (solid line, y = 0.21) and without (dashed line, y = 0) considering changes in the gas flow rate.
.5 x2
Figure 2. The ratio of KLa values calculated with and without considering changes in the gas flow rate as a function of the outlet gas concentration, x2, for y = 0.21.
times that in the first. The other conditions are the same for both experiments. Now, if the longitudinal mixing in the fmt column can be characterized by the Peclet number PSl= 10, then the Peclet number for the second column wll be PB2= 40. Also, it holds that N2 = 4Nl. Suppose that 35.2% of the supplied carbon dioxide is absorbed in the short column, corresponding to Npl= 0.357. Measurement on the column of 4-fold length then yields Np2 = 1.68, indicating that 86% of the supplied COz has been absorbed. From Figures 1and 2 it may be found that the ratio of (KLu), values evaluated from both experiments with allowance for the change in the flow rate is equal to unity [ ( K ~ u z/(KLul)y ) = 11whereas neglecting the change U ~ This means leads to a vafue of 1.19 [ K L U ~ / K=~1.191. that the KLu value attained with the higher packing is 19% greater than that found with the shorter packing. Another case where neglecting changes in the gas flow rate due to absorption may lead to considerable distortion of the measured values of parameters is the determination of the Peclet number, P,, by the steady-state method. In this method such values of the model parameters (P, and N ) are sought for which the experimental data ( x vs. z ) match the gas-side concentration profile calculated from the relevant model. The measurement of P, has often been carried out on a system involving absorption into water of carbon dioxide from a 20% mixture with nitrogen (Brittan and Woodburn, 1966; Mathur and Wellek, 1976). The possible differences can be seen from Figure 3, which shows the concentration profiles calculated from the dispersion model both with (solid lines) and without (dashed lines) allowance for changes in the gas flow rate during passage through the column. The profiles drawn in the figure were calculated for the values y = 0.21 and P, = 15, approximately the same as those obtained in the above mentioned studies of Brittan and Mathur. In view of the marked parametric sensitivity of the resultant values of P, and N (or KLu)to the shape of the profile, we can expect conflicting results in studies comparing results obtained with and without considering changes in the gas flow rate. A good illustration of this discrepancy may be found in Figure 7 of the paper of Mathur and Wellek (1976). Their figure provides a comparison between two plots of the Peclet number, Pg, for the gas phase against the gas phase
'PSIY 3
1 L
I
,'l
.E'
'
1.
I
I
I
Figure 4. P,/(P,)and N / N , vs. Ny for y = 0.2. The values of Pg and N were calcdated from eq 14 by curve-fitting axial profiles N y and y = calculated by integrating the set of eq 3 to 5 for (P,),, 0.2.
Reynolds number, one based on the data of Brittan and Woodburn (1966) evaluated with allowance for changes in the gas flow rate, and the other on the data of Mathur and Wellek (1976), who ignored the changes. The P, values found by Mathur and Wellek are higher than those obtained by Brittan and Woodburn under very similar experimental conditions, in some cases by as much as a fador of 4. In order to examine whether the difference between the results of the two research teams was due, at least in part, to neglecting changes in the gas flow rate, we calculated values of the parameters P, and N in the relation x = [A2 exphz exp(hlz) - X1 expXl exp(Xzz)]/[Xz(l X,/P,) exph - h(1- X2/P,) expX11 (14) by curve-fitting axial profiles of the concentration x vs. z calculated from the set of eq 3 to 5 for chosen values of the parameters (PJY,N y , and y = 0.2. Changes in the volume flow rate were considered. Relation 14 was derived by Miyauchi and Vermeulen (1963) for axial concentration profile without considering changes in the flow rate. The
Ind. Eng. Chem. Fundam. l982, 21, 90-93
90
values are depicted in Figure 4 in the form of plots of Pg/(P) y vs. Ny and N / N y vs. N,,. The figure also includes lines for the constant outlet concentrations, x 2 = 0.8,0.5, and 0.1. The use of Miyauchi's relation 14 leads to systematic overestimation of the Peclet number along with underestimation of the number of transfer units. The difference between Pg and (Pa is largest when the outlet gas concentration, x 2 , lies within the interval (0.5, O.l), which is just the region involved in the experiments of both research teams. Therefore, we believe that the difference between the results of Brittan and Woodburn and those of Mathur and Wellek may, in part, be accounted for in this way. Nomenclature
a = interfacial area related to volume of packing, m-l cga = molar concentration of component a in the gas phase c,,, = equilibrium molar concentration: ch = mcga+ n , kmol
m-3 E = coefficient of axial dispersion, m2 s-l KBL = overall mass transfer coefficient, m s-1 L = depth of packing, m m, n = constants of the equilibrium relation: cla = mega + n N = KLamL/Q1,true number of transfer units Np= -In x2, apparent number of transfer units based on piston flow of gas phase through column p = barometric pressure, Pa p w = water vapor tension, Pa Pg= QL/E,, Peclet number Q = gas phase interstitial velocity, m R = universal gas constant w = Q/Q1, dimensionless gas flow rate x = (mc, + n - cln)/(mcgal), dimensionless concentration of component a in the gas phase y = volume fraction of component a in the inlet gas z = Z/L, dimensionless axial coordinate 2 = axial coordinate
A1
= P,/2
+ [(Pg/2)2+ N P 1112 + NP#2
X2 = Pg/2- [(Pg/2),
Subscripts a = absorbed component g = gas phase 1 = liquid phase p = calculated on the assumption of piston flow of the gas
phase through column
y = calculated with considering changes in the gas flow rate 1 = inlet 2 = outlet L i t e r a t u r e Cited Blschoff, B. K. A I C M J . 1988, 14, 820. Blschoff, 8. K.; Levenspiel, 0. Chem. Eng. Scl. 1982, 17, 245. Blschoff. B. K.; Levenspiel, 0. Chem. Eng. Scl. 1982. 17. 257. B r b n , M. I. Chem. Eng. Sci. 1967, 2 2 , 1019. Brittan, M. 1.; Woodburn, E. T. A I C M J. 1968, 12, 541. Cairns, E. J.; Prausnltz, J. M. Chem. Eng. Sci. 1960, 12, 20. Converse, A. 0. AIChE J. 1980, 6 , 345. Coste. J.; Rudd, D.; Amundson, N. R. Can. J . Chem. Eng. 1980, 39. 149. Dum, W. E.; Vermeulen. T.; Wllke. Ch. R.; Word, T. T. Ind. Eng. Chem. Fundam. 1977, 16, 116. Hatton, T. A.; Woodburn, E. T. AIChE J . 1978, 2 4 , 187. Llnek, V.; Bend, P.; Slnkule, J.; Dhskq. 2. Ind. Eng. Chem. Fundam. w- - -a . 17. 29s. M a t h ~ V. , K.; Wellek, R. M. Can. J . Chem. Eng. 1978, 54, 90. Miyauchl, T.; Vermeulen, T. I d . Eng. Chem. Fundam. 1983, 2 , 113. V. Sater. E.;Levenspiel. 0. I d . Eng. Chem. Fundam. 1968, 5 , 86. Sherwood, T. K.; Plgford, R. L.; Wllke, C. R. "Mass Transfer", McGraw-Hill: New York, 1975; pp 609-614. Shioya,S.; Dunn, I. J. Chem. Eng. Sci. 1976. 33, 1529. Stlegel. 0. J.; Shah, Y. T. Ind. Eng. Chem. Process Des. Dev. 1977. 16, 37. Woodburn, E. T. A I C M J. 1974, 2 0 , 1003.
-. .
Department of Chemical Engineering Institute of Chemical Technology 166 28 Prague 6, Czechoslovakia
V&clavLinek* Ji5i Sinkule Petr PetfiEek ZdenGk KSvskg
Received for review November 20, 1980 Accepted August 26, 1981
Integral Equation Solution for the Effectlveness Factor of Partially Wetted Catalysts Evaluation of the catalyst effectiveness factor of a partially wetted catalyst slab, for a first-order liquid reactant limited reaction, is formulated as a mixed (spllt) boundary value problem. A dual series representation of the problem is presented and an equivalent integral equation is obtained. The effort necessary to transform the kernel to a form that can be accurately computed is described. The integral equation is solved using quadratures, and the results are compared with those obtained by application of the method of weighted residuals to the original dual serles. The advantages of the integral equation solution are outlined.
Evaluation of effectiveness factors for partly wetted catalyst pellets has recently received considerable attention (Mills and DudukoviE, 1979, 1980; Ramachandran and Smith, 1979; Goto et al., 1981; Levec et al., 1980; Tan and Smith, 1980; Martinez et al., 1980). These results are of importance in assessing catalyst utilization in trickle-bed reactors. The original problem for nonvolatile reactants was formulated as a mixed (split) boundary value problem (DudukoviE and Mills, 1978). Such problems arise frequently in fracture mechanics and other engineering disciplines (Sneddon, 1966). In chemical engineering, the problems of this type which appeared in the literature dealt with diffusion (Meyer et al., 1976) and diffusion with reaction (Mills and DudukoviE, 1979). Since no general approach of finding the solution which is assured of success is available at present, it is of interest to present possible methods of solution for a particular case. In a recent publication in this journal (Mills and DudukoviE, 1979) we have shown that the mixed (split) 0196-4313/82I1021-0Q90$01.25/0
boundary value problem for the catalyst effectiveness of partly wetted pellets of various geometries can be reduced to a dual series problem which can be effectively solved by the method of weighted residuals. This has been extended to the application of the method of weighted residuals to general dual series (Mills and DudukoviE, 1980) and general triple series (Mills and DudukoviE, 1980b). An early development of the least-squares method for dual series was given by Kelman and Koper (1973). Some results in the literature for the effectiveness factor problem (Ramachandran and Smith, 1979) indicate that finite difference methods may lead to values of insufficient accuracy. It has been shown (Whiteman, 1967) that finite difference methods give the least accurate results in the region of the singularity, which occurs as a result of the mixed (split) boundary conditions,when compared to dual series methods. For this reason, the solution of mixed (split) boundary value problems using standard finite difference methods is not recommended. At the same time 0 1982 American Chemical Society
