## Its Limit Expressions and Approximation

Mean <k> Over the Full Length of the Channel

In some cases it is necessary to find the average adsorption constant over the full sample length where laminar convection occurs. When the process is fully transport-controlled, the integration of the local Lévêque equation over the channel length L leads to <kLev> = 0.808(D2y/L)1/3. When the contribution of the interfacial reaction is taken into account, the mean kinetic constant <k> over the channel length L is given by (Valette et al. 1999):

where:

Hence:

ka i 3A 3A

Close to the conditions of transport-controlled process, we have:

while close to the conditions of control by the interfacial reaction

with the total resistance being the sum of the two resistances (one due to the transport and the other one due to the interfacial reaction).

As the numerical coefficients in both linear approximations are close to 1, the simple approximation

is rather good for the whole range of ka. The same route as that for the local adsorption constant led to the following approximation of <k>/ka as a function of U =<k>/< kLev>:

where A = 0.203127 and B = -0.272759, to satisfy Eqs. 10d-e. The greatest relative variation between Eq. 12 and the complete calculation is 0.03% around U = 0.8. It can be used in the form <k>= kaF(U) for a two-parameter fit (D and ka) to the experimental value of <k> as a function of U' =<k>(L/y)1/3/0.808 = UD2/3.

ka (U' - D2/3)(AU' - D2/3) {k} = D2/3 BU' + D2/3 (13)

Mean [fr] Over a Restricted Length

In experiments, the adsorption kinetics is always integrated over some length Ax of the channel, between x - Ax/2 and x + Ax/2. We shall estimate the influence of Ax on the numerical coefficients a and b in Eqs. 8 or 9.

Referring to Eq. 8, the variable u = k (x)/kLev (x) becomes u* = [k]/[kLev], where the star superscript and the brackets mean that the average is taken between x1 = x - Ax/2 and x2 = x + Ax/2. [k] is the actual measured average kinetic constant, which was assumed to be k(x) in Eq. 8, and [kLev] = Ax-1(x2<kLev>2 - x1<kLev>1). <kLev>j is the mean value of the Leveque constant between x = 0 and x = xj.

We adopt the same procedure looking for two limit expressions for a transport-limited process and interfacial reaction control (see Appendix). For small values of £ = Ax/x we obtain:

a « a0 (1 + 0.078 £2) ; b « b0 (1 + 0.044 £2) (14)

where a0 and b0 are the values of the numerical coefficients determined in Eq. 8, which corresponds to Ax ^ 0.