Ind. Eng. Chem. Res. 2003, 42, 2033
2033
CORRESPONDENCE Comments on “An Analytical Solution for the Analysis of Zero-Length-Column Experiments with Heat Effects” Stefano Brandani† Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, U.K.
Sir: Silva et al.1 recently presented an analytical solution to the nonisothermal model for the zero-lengthcolumn (ZLC) experiment. The authors also suggest that this solution should be used to interpret experimental results and obtain simultaneously information on the mass- and heat-transfer characteristics of porous materials. The analytical solution does have the advantage of showing in an elegant way that the dimensionless group that governs whether the system is isothermal is
R)
( )
∆H 2 q0RF RT0 K0haVs
(1)
which coincides with the result obtained by Brandani et al.2 from repeated numerical simulations. However, when applying their solution, Silva et al.1 obtain a criterion for isothermal behavior of the ZLC system which is an order of magnitude more stringent than the value previously established by Brandani et al.2 The purpose of this paper is to point out that the model of Silva et al.1 is based on the same assumptions as the model of Brandani et al.,2 with the exception that in order to obtain a closed analytical solution the equilibrium relationship is linearized.1 The observed difference between the predictions from the two models comes from this linearization. If the system is sufficiently dilute, we can assume that the isotherm is linear:
q ) K(T) c
(2)
Linearization of eq 2 yields
q ≈ K0c + (T - T0)
|
∂K ∂T
T0c0
(3)
For a linear approximation to hold, it must be valid over the full range of sorbate concentration encountered in the desorption response. In this case, if we use for c0 the initial concentration in the carrier gas, we obtain a different result from what we would have when using the final value of the gas concentration, i.e., c0 ) 0, where the last term in eq 3 cancels out. This observation leads to the conclusion that the linearization of the system of equations is not applicable over the required range.2 Furthermore, when the concentration is sufficiently low, the nonisothermal system will inevitably revert to the isothermal ZLC solution, shifted in time. This means that the slope of the long-time asymptotic exponential decay will be the same as that of an isothermal system.2 Therefore, the real solution to the nonisothermal ZLC †
Holder Royal Society Wolfson Research Merit Award.
Figure 1. Comparison between the full numerical solution2 and the analytical approximation1 for a system under kinetic control (L ) 10). R ) 0.1, 1, and 10. The full solution calculated using R ) 0.1 and 1 is the same as the isothermal result (R ) 0).
model is always characterized by a double curvature2 on a semilog plot of the dimensionless gas concentration versus time. This key feature is completely lost in Silva’s analytical approximation,1 and it is therefore not surprising that this leads incorrectly to a more stringent criterion of isothermal behavior. Figure 1 shows the comparison between the numerical solution of the full model2 and the analytical approximation1 for a system under kinetic control (L ) 10), confirming all of the above intuitive observations. The important conclusion is that, as a result of the linearization, the criterion suggested to assess the validity of the isothermal model is overly restrictive and the analytical approximation should not be used to interpret experimental ZLC data. Notation
a ) surface-to-volume ratio, m-1 c ) gas-phase concentration, mol/m3 F ) volumetric flow rate, m3/s h ) heat-transfer coefficient, W/m2‚K ∆H ) heat of adsorption, J/mol K ) Henry constant L ) dimensionless ZLC parameter: L ) Fr2/3KVsD q ) adsorbed phase concentration, mol/m3 r ) radius of a solid particle, m R ) ideal gas constant, J/K‚mol T ) temperature, K Vs ) solid volume, m3
Literature Cited (1) Silva, J. A. C.; Da Silva, F. A.; Rodrigues, A. E. An Analytical Solution for the Analysis of Zero-Length-Column Experiments with Heat Effects. Ind. Eng. Chem. Res. 2001, 40, 3697-3702. (2) Brandani, S.; Cavalcante, C.; Guimaraes, A.; Ruthven, D. Heat Effects in ZLC Experiments. Adsorption 1998, 4, 275-285.
IE0210053
10.1021/ie0210053 CCC: $25.00 © 2003 American Chemical Society Published on Web 03/29/2003
