Unit Operation $($III$)$

Midterm Exam

April $30,2022$

Consider the drug treatment system shown in the figure. A hemispherical cluster of unhealthy cells is surrounded by a larger hemisphere of stagnant dead tissue $($species $B),$ which is in turn surrounded by a flowing fluid. The bulk, well$-$mixed fluid contains a drug compound $($species $A)$ of constant but dilute bulk concentration $c_{A0},$ Drug $A$ is also soluble in the unhealthy tissue but does not preferentially partition into it relative to the fluid. The drug $($species $A)$ enters the dead tissue and targets the unhealthy cells. At the unhealthy cell boundary $)=(R_1,$ the consumption of drugA is so fast that the flux of $A$ to the unhealthy cells is diffusion limited$.$ All metabolites of drug A produced by the unhealthy cells stay within the unhealthy cells. However, drug A can also degrade to inert metabolite $D$ by a first$-$order reaction dependent on $C_A-$ i$.$e$.,AkD-$ that occurs only within the stagnant dead tissue.

$($a$)$ State at least four reasonable assumptions and conditions that appropriately describe the system of mass transfer.

$($b$)$ Develop the differential form of Fick's flux equation for drug A within the multicomponent system without the "dilute system" assumptions. Then, simplify this equation for a dilute solution. State all additional assumptions as necessary.

$(4\%)$

$($c$)$ Appropriately simplify the general differential equation for mass transfer for drug $A.$ Specify the final differential equation in two ways: in terms of $N_A,$ and in terms of concentration $c_A.(6\%)$

$($d$)$ Specify the bounda