## Transcribed Text

1. (2 points) Consider the diffusion of species a from
source located at the top of alarge tank of length (i.e.
at 0, xa=xxo). The lower surface of the tank made
of catalytic material such that a is converted to at the
surface at rate proportional to the concentration of
:-
right above the surface. In other words, at the
flux N,is:
/moles
D.
where Tas is the rate of surface reaction of species and
k,is the reaction rate constant (units of m/s). You may
Aram-A,
assume one mole of reacts to form one mole o
then diffuses back through the fluid. sowe have equimolar counterdiffusior throughout the
system. We also assume that the total concentration (c) does not change.
Since this particular reaction occurs at the boundary, we can consider mass balance around a
small control volume, similar to what we did in Unit II
Part A: Write the differential equation for the mole fraction of species a (i.e xa)
Part B: What is the general solution?
Part C: What are the boundary conditions?
Part D: Using the boundary conditions in Part C. solve for the constants in the general expression
you wrote for part B. What is the mole fraction profile of species. a? Draw rough graphot X3
versus :
Part E: What is the molar flux of species a? Draw rough graph of Na versus z.
Part Discuss how your result in part Ecompares with the now familiar driving force -resistance
concept. How does the chemical reaction effect either the driving force or resistance to transport?
What is the net effect on mass transfer?
Part G: When is the system mass transfer controlled? When is the system reaction rate
controlled?
2. (1.25 points) Consider heat transfer through a fluid with evaporation
from the surface (see picture for clarification). In the next problem,
you will be asked to determine the temperature profile within the fluid,
the temperature of the upper surface. and the rate at which the free
surface recedes. The overal outcome of the next three problems isto
,--
relate heat and mass transfer together
Part A: Write the differential equation for the temperature
Part B: What is the general solution?
Part C: Note that this is convection problem
Also. the rate of heat removal by evaporation depends on the
temperature of the free surface and the rate of mass transfer there. At z=L, we can relate
the
flux
with the mass flow rate of evaporation m, where each kilogram of evaporating fluid removes he
joules heat:
dT
-k Area
dz
mhrg
Evaporation (negative generation) at the surface of the liquid into the gas dependent on the
fact
that to change from liquid to gas requires heat input from the denser material while the
opposite phase change liberates heat. Hence. we couple heat and mass transfer since we need to
know the total heat flux determine how much material may evaporate from the surface. Thus.
in order to specify the boundary condition at the top, we need torecall from class that for phase
change:
dz
hrgMw
where his is the heat of vaporization and g' is the area independent heat flux (i.e. -kdT/dz)
Also, we have convection coefficient at the surface (k) and driving force for mass transfer
(defined as the difference between the partial pressure of water the free stream (Pi,ao) and the
partial pressure of water in equilibrium with the fluid, prat Finally, the total pressure P. Show
that the mass flow rate, mis:
m =
Hints:
You could solve for Tiat the interface. then rewrite the expression to yield T. for species
x, (in solution).
Likewise, you could use your knowledge of phase diagrams to convert the mole fraction
of species a in liquid (xa) into the mole fraction in the gas phase (ya) via Henry's law (y,
The partition coefficient my changes as function of pressure, temperature, and
concentration. So. we can also define y, in terms of overall pressure and the vapor
pressure y.*P).
3. (0.5 points) From your work in problem #2,
Part Use this expression for m above to determine the expression for the temperature profile
(T). Determine the temperature at the interface (i.e atz=L). Sketch rough plot of temperature
versus distance z.
Part B: Determine the heat flux (in terms of area). What the driving force? What the
resistance? Sketch rough flux profile versus z
Part C: Determine the rate at which the interface recedes. (Hint: this is given by the evaporation
rate divided by the density and area of the interface).
4.
(1.25 points) From the problems above we assume that we are evaporating intodry air and heat
transfer takes place only by conduction (how this different if there was moisture in the air?)
Assume the following for the system:
L=0.:
A,= 0.001 m²
TO =60PC
k= 1x102/m/s
k 10 W/mK
Pi,co .0
In
general, we can consider all physical parameters for the water in the tank as function of
temperature. Specifically the heat of vaporization and vapor pressure from 300 to 325 K are
given by:
Since we have problem that has two parameters that depend upon the temperature of the
interface, it can be solved in one of two ways. The first isto use an iteration approach The
second to reduce the problem to a single non-linear equation whose root the temperature atz
= L. Combine all of your data to write partial differential equation. (Hint what can you
assume P?)
Part A: Determine the final surface temperature.
Part B: Determine the mass flow rate of evaporation fluid (m).
Part C: Determine the velocity of the receding evaporating surface.
Part D: How is the mass transport in problem coupled to your answer?

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