Equation (5) is the rate expression at steady state for a first order reaction in terms of bulk concentration when significant mass transfer limitation is present.

Now ,

Or.

Case 1: When diffusion controls the process that is reaction is very fast then . Hence from (5) it can be written as

Or ---

(6)

Case 2 : When surface reaction controls the overall process that is diffusion is very fast then

Hence from (5) it can be written as

(7)

Mass transfer coefficient in packed bed for solid-fluid system

Average transport coefficient between bulk stream and particle surface can be determined from following correlation in terms of dimension less groups characterizing the flow conditions.

Where a_m is the effective mass transfer area and a_t is the total external area of the particles. For Reynolds number greater than 10, the following relationship can be used.

Where,

G = superficial mass velocity based upon cross sectional area of empty reactor = uρ

d_p = diameter of spherical catalyst particles

µ = viscosity of fluid

ρ = density of fluid

D = molecular diffusivity of component being transferred

ε_B = void fraction of the bed

Other correlations are also available in the literature.