This problem consists of two parts. In Part A there is an homogeneous reaction within the fluid. The

reaction creates species at a constant rate $R_{i,}$ moles $\frac{i}{m^3}s.$ The walls catalytically destroy species $i$

infinitely rapidly.

In Part B there is no homogeneous reaction, but there is a surface reaction at the inside walls s creating

species $i$ at a constant rate of $R_i($surface$)$ moles $\frac{i}{m^2}s.$ Specie i then diffuses into the fluid. For both Parts

A and $B$ assume $($pseudo $)$ binary diffusion because species $i$ is very dilute.

PART A $(40$ POINTS$)$

A reacting incompressible fluid flows between two parallel flat plates. The bottom plate moves with velocity $V.$ The

top plate is stationary. A developing concentration profile of$,$ say species $i$ is formed within the fluid by a constant

volumetric reaction rate $R_i$ moles $\frac{?}{m^{3}s}.$ At $x=0$ the volumetric reaction source is turned on$.$ For $x<mn\; id="EditorElement\_36002">0</mn>$ there is NO

reaction. The walls destroy the species i extremely rapidly. A concentration profile, $C_i(x,y)$ develops, due to the

volumetric reaction source. There is no imposed pressure drop $-$ motion is caused by the bottom sliding plate.. The

concentration at $x=0$ is $C_0.$ Assume there is a steady state, unless you conclude there is not. Then explain. Assume

pseudo$-$binary diffusion, constant total molar concentration, incompressible fluid etc.

Steadv linear velocitv profile

Walls destroy species i infinitely rapidly

I. Prove that the steady velocity profile is given by $v_x=V(1-\frac{y}{H})$

II$.$ Write down the species balance equation with the appropriate boundary conditions.

Scale appropriate variables, non$-$dimensionalize, and derive an expression for the characteristic

concentration difference, and the characteristic length in the $x-$direction. Interpret this length physically.

$(25)$

III. Assume the Peclet $($Mass$)$ Number, $P{e}_m=$ReSc$=\frac{VH}{D}1,$ and use this to simplify the equation above

by eliminating one of the diffusion terms. Provide a physical explanation as to what this means. $(10)$