problem 3 slow dissolution of the coating on the bottom of a rectangul

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Problem 3. Slow dissolution of the coating on the bottom of a rectangular conduit (duct)
Learning Objective: Use the generalized differential species balance and identify boundary conditions in a system with a
reaction.
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=0 at average velocity Vz.
The conduit has a vertical height of H (y-direction), a
width W (x-direction) and a length L (z-direction).
Note that H<< L and H << W.
Flow
H
CA (2)
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:
дра
VzJz
=
DAB
a² PA
дуг
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
дра
at
=
+vVPA DABV² PA+TA
(not on the exam) Also show that starting with the table from BSL as shown below (noting pa=pA) gives the same result.
§B.11 THE EQUATION OF CONTINUITY FOR SPECIES A
IN TERMS OF WA FOR CONSTANT" PDAB
Cartesian coordinates (x, y, z):
at
awA
მაგ
+ V₂ JZ
JWA + by ay
+ Vx ax
IpDwA/Dt=PDABV² WA + rA]
[BWA +
дал J²WA
+rA
Əz²
= PAB x2
"Briefly note" means briefly stating things such as:
ay²
+
1) If an eliminated term is exactly zero, concisely state why (short phrase is enough).
(B.11-1)
2) 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.