CompuCom Inc. has developed a new bottle that it claims will store volatile mate

Question

CompuCom Inc. has developed a new bottle that it claims will store volatile materials for a long time without a cover (Figure P4.18). The key is that the cross-sectional area of the device is designed to obey the function A = A_o e^-oz, where z is the distance from the bottom. To test the concept, you must perform the experiment shown in Figure P4.18. Assuming dilute solutions. constant total concentration, constant diffusivity. and steady-state operation, provides answers to the following Derive the differential equation for the mole fraction profile of a. What are the boundary conditions? Solve the differential equation for the mole fraction profile and flux. Can the mole fraction of a exceed y_a0, at any point within the flask? What does this have to say about CompuCom's claims (what is the effect of the bizarre shape)?

Explanation / Answer

As the main secret of the the concept is cross section that means the volatile material haave to cover a large distance before it get the touch of atmosphere.

now we have to assume that diffusivity , total concentration ae constant.

first of all we have to conider two features of volatality and they are:

1) considering equlibrium volatality is the tendency of being in vapour state

2) so we consider vapour pressure here

considering pressure and temperature the tendency increases with increasing pressure and temperature.

again diffusivity means conductivity/volumetric heat capacity

we consider that constant

now if the cross sectional area is a1 then volume is al*l

volumetric heat capacity = a1*l*cp

therefore diffusivity = k/a1*l*cp=P=constant

a) now molar fraction is the ratio of amount of the material in mole/ amount of the total constituents in mole (container)

from the given expression A=A0exp(-aZ)

we can write

ln(A/A0)=-aZ

or a=-ln(A/A0)/Z

lets consider air is the other constituent inside the container at normal temperature

and they are in the ratio 1:n

for the flux total amount= mole fraction of air+mole fraction of material

now at steady state da/dz=0

b) now for boundary condition

a(0)=0 and a(z)=L

c) using steady state da/dz=0

ln(A/A0)d/dz(1/z) =ln(A/A0)*-1/Z^2

using boundary values at z=0 a'= ln(A/A0) and at Z=L we get a'=-ln(A/A0)1/L^2

so the molar concentration is dependent on flux length

d) If the molar fraction exceeds the Yao then total molar concentration will be changed That means the constant diffusity wil be variable and Compucon's claim will no longer be valid as with increasing molar fraction the equilibium of gaseous and condense state will vanish and volatile material will work in a increasing pressure environment.

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