A conduit $($or duct$)$ with a rectangular cross$-$section

was constructed with a different protective coating on

the bottom than on the sides and the top. While the

coating on the top and sides has been stable, the

bottom coating turned out to be sparingly soluble in

the liquid and thus has been very slowly dissolving

into the solute$-$free liquid that is being fed into the

conduit at $z=O$ at average velocity $V_z.$

The conduit has a vertical height of $H(y-$direction$),$ a

width $W(x-$direction$)$ and a length $L($z$-$direction$).$

Note that $HL$ and $HW.$

This concentration profiles in the conduit can be determined using the generalized species equations of

continuity and assuming constant density and diffusion coefficient to obtain the following PDE:

$v_z\frac{\text{d}\text{e}\text{l}{}_A}{\text{d}\text{e}\text{l}\text{z}}=D_{AB}\frac{de{l}^2{}_A}{del{y}^2}$

a$)$ Briefly note the reasoning for the elimination of terms from the general equation of continuity for species $A$ to

arrive at the simplified equation shown above. You can start with the balance in the form written in WRF as

$\frac{\text{d}\text{e}\text{l}{}_A}{\text{d}\text{e}\text{l}\text{t}}+v**grad{}_A=D_{AB}gra{d}^2{}_A+r_A$

$($not on the exam$)$ Also show that starting with the table from BSL as shown below $($noting ${}_{}={}_{0A})$ gives the same result.

$$B$.11$ THE EQUATION OF CONTINUITY FOR SPECIES $A$

IN TERMS OF ${}_A$ FOR CONSTANT ${?}^a{D}_{AB}$

$D\frac{{}_A}{D}t=D_{AB}gra{d}^2{}_A+r_A$

Cartesian coordinates $(x,y,z)$ :

$(\frac{\text{d}\text{e}\text{l}{}_A}{\text{d}\text{e}\text{l}\text{t}}+v_x\frac{\text{d}\text{e}\text{l}{}_A}{\text{d}\text{e}\text{l}\text{x}}+v_y\frac{\text{d}\text{e}\text{l}{}_A}{\text{d}\text{e}\text{l}\text{y}}+v_z\frac{\text{d}\text{e}\text{l}{}_A}{\text{d}\text{e}\text{l}\text{z}})=D_{AB}[\frac{de{l}^2{}_A}{del{x}^2}+\frac{de{l}^2{}_A}{del{y}^2}+\frac{de{l}^2{}_A}{del{z}^2}]+r_A$

"Briefly note" means briefly stating things such as:

If an eliminated term is exactly zero, concisely state why $($short phrase is enough$).$

If an eliminated term is not exactly zero, state the reasoning that could justify its elimination $($a sentence is enough$).$

b$)$ How many boundary conditions will be required to integrate the simplified equation? Briefly note your reasoning.

c$)$ Write a plausible set of boundary conditions that could be used to integrate this equation $($there may be more

than one plausible option for each BC; you need to give only a minimum consistent set$).$ Briefly note your reasoning.