PROBLEM $4$

Given: A polysaccharide Pn is convectively transported with chyme through the small intestine where it is hydrolyzed to produce glucose according to the reaction:

$P_n+(n-1)H2OnG.$

Glucose is produced from the polysaccharide at a molar rate per unit volume $\frac{R_G^{}}{V}$ and is absorbed across through the mucosal wall of the intestine at a molar rate per unit surface $\frac{M_G^{}}{S}.$ The polysaccharide is not absorbed in the intestine. Assume that the intestinal lumen is a rigid cylinder with a constant volume$-$to$-$length ratio A and a constant surfaceto$-$volume ratio of $=\frac{S}{}V.$ Also, assume that chyme flows through the lumen at a constant flow rate $Q.$ The concentrations of polysaccharides $C_P$ and glucose $C_G$ in chyme will be functions of time $t$ and position $z$ along the intestine.

Find: $($a$)$ Sketch a control volume consisting of a tube section of length $z$ showing convective inputs and outputs of $P$ and $G,$ absorption of $G$ at the tube wall and internal chemical reaction.

$($b$)$ Using this control volume, derive the dynamic molar balances for glucose and polysaccharide to describe the concentrations $C_P(z,t)$ and $C_G(z,t).$ Start with $)=(G,P.$

$($c$)$ By taking the limit of an infinitesimal control volume, derive the differential concentration equations for polysaccharide and glucose. Under what conditions are the same final equations obtained directly from Eq$.2\mathrm{.}4-25$ in the textbook?