Effective Diffusivities in Biological Gels

The analysis of intraparticle mass-transfer resistance requires the knowledge of the effective diffusivity Ds of a substrate in an immobilized matrix, such as agarose, agar, or gelatin. Gels are porous 

Various techniques are available for determining the effective diffusivity of solute in gel (Itamunoala, 1988). One of the most reliable techniques is the thin-disk method which uses a diffusion cell with two compartments divided by a thin gel. Each compartment contains a well-stirred solution with different solute concentrations. Effective diffusivity can be calculated from the mass flux verses time measurement (Hannoun and Stephanopoulos, 1986). A few typical values of effective diffusivities are listed in Table 3.2. 

When the rate of diffusion is very slow relative to the rate of reaction, all substrate will be consumed in the thin layer near the exterior surface of the spherical particle. Derive the equation for the effectiveness of an immobilized enzyme for this diffusion limited case by employing the same assumptions as for the distributed model. The rate of substrate consumption can be expressed by the Michaelis-Menten equation. 

After replacing rs with the Michaelis-Menten equation in Eq. (3.42), it can be simplified to 

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