PROB

$26\mathrm{.}22$ Living cells homogeneously distributed $($immobilized$)$ with an agarose gel require glucose to survive. An important aspect of the biochemical system design is the effective diffusion coefficient of glucose $($species A$)$ into the cell$-$immobilized gel. Consider the experiment shown below where a slab of the cell$-$immobilized gel of $1\mathrm{.}0cm$ thickness is placed within a

well$-$mixed aqueous solution of glucose maintained at a concentration of $50$mmo$\frac{l}{L}.$ The glucose consumption within the cell$-$immobilized gel proceeds by a zero$-$order process given by

$R_A=-m=-0\mathrm{.}05$mmo$\frac{l}{L}$min

The solubilities of the glucose in both water and the gel are the same; that is$,$ the concentration of glucose on the water side of the water$-$gel interface is equal to the concentration of glucose on the gel side of the water$-$gel interface. A syringe mounted at the center of the gel carefully excises a tiny sample of the gel for glucose analysis.

Develop a model in final integrated form to predict the concentration profile of glucose within the gel. Be careful with specification of boundary conditions, so that your model is truly predictive and is based only on process input parameters, not measured parameters.