Diffusion Through a Solid
Approximating the permeability of a gas through the
wall of a balloon
Balloon
Gas B
CBi
PBi
Rubber
CBo
PB=PBo
SB = Solubility of Gas B in Rubber cm3 B at 0oC & 1 atm/cm3 Rubber/Atm PHe
PB = Partial Pressure of gas B (atm)
CB = Concentration of Gas B in the Rubber (gm Moles B/cm3 of Rubber)
CB 
S B PB
Reasonable to Assume Solubility is constant
22,414
r2
r1
1
What is(are)the models ?
Initially apply steady state model.
N Ber
What are the assumptions?
UUD but since dilute Conc of B in
Rubber. Therefore, use ECD
approximation
dC Be
dr
4r
N He r  RadialMass TransferRa te( gmMoles / Sec)
2
  DHeinRubber
CB = SBPB; SBe=constant
PB=PBo outside the balloon & PBi is
constant pressure inside balloon.
Mass Transfer Rate in Radial
Direction is Constant
Diffusivity Not Dependent on PB
Now integrate the above equation across the balloon wall between the inside radius
(r1) and the outside radius (r2).
N Br
4 r
2

DB  Rubber
dP
S He Be
22, 414
dr
22414cm3  1.gmmole( STP)
cm 2
DB  Rubber ( Diffusion.B.in.Rubber..
)
sec
cm3 ( He)
S B (Solubility.B.in.Rubber 3
)
cm ( solid )atm
dPB
atm
 Pressure Gradient Across Balloon Wall(
)
dr
cm
S B DB  Rubber ( PBii  PBo )
N Br r2 dr


4 r1 r 2
22, 414
4 r2 r1 S B DB  Rubber ( PBii  PBo )
(1) N B 
*
r2  r1
22, 414
PM  S B DB  Rubber  Permeability
cm3 ( STP)
sec.cm 2 ( Solid . Area )atm / cm( Solid .thickness )
PM = Permeability SI Units
M2/Sec M3 Gas @STP
M3 Solid Atm
Permeability is Temperature Dependent
(0)PM = Po exp (-Qp/(RT))
2
where Po & Qp/R are constants & T is the absolute temperature of gas inside the
balloon.
Now Consider the unsteady state performance of the a gas (e.g.
helium) diffusing through the wall of a spherical balloon which will decrease
its radius with time. Also since the balloon wall thickness is very thin it is
reasonable to use the following approximations. During this period we will
assume that only r2 is varying.
r2 r1  r22
M Bal  Balloon .Mass (Cons tan t )
M Bal  4r22 (r2  r1 )  Bal ;  Bal  Balloon .material.density
(2)( r2  r1 ) 
M Bal
4r22  Bal
M Bal  Balloon ..Mass..Portion.that..is..contracting..with..time
 Bal  Density ..of ..balloon
B d
4 / 3r  
dt
4r22
PM ( PBi  PBo )
Mw B
22,414
 M Bal 


2
 4r2  Bal 

(4r22 ) 2 PM ( PBi  PBo )  Bal
d
 B 4r22 r2  
Mw B
dt
22,414 M Bal
(3) 

r2
r2 o
3
2
dr2 (4 ) PM ( PBi  PBo )  Bal M wB

t
B
22,414 M Bal
r22
1
1
(4 ) PM ( PBi  PBo )  Bal M wB


t
r2 r20
22,414 M Bal  B
By integrating this equation you should be able to linearize the resulting
equation so that you will generate an algebraic form (1/r2 vs t) that enable you to
determine the value of the permeability PM from the slope of the appropriate
straight line. Note that at t = 0 the initial value of r2 will be r2o
and at time t the radius of the balloon will be r2 .
Develop an experiment to determine the value of PM for two different
balloon materials. Your balloon & its contained gas (e.g.,O2,N2, Argon, CO2, He)
should be in a fairly constant temperature room. Please refer to eqn . (0) for the
effect of temperature on permeability. Also note that your balloon may not be a
perfect sphere . You might consider how you can include the fact that the balloon
is an ellipsoid in both your experimental procedure & data analysis.
Compare the permeability you measured with published values.
3
Report Format
1) Title Page + Table of Contents (10%)
2) Research and summarize in your report the permeability of materials to
various gases & solubility of gas in solids. Summarize this research with proper
referencing.(10%)
3) Permeability measurement: Research & summarize in your report
methods for measuring the permeability of materials to gases and the solubility of
gases in solids. Use proper referencing.(15%)
4) Experimental Plan: Your report should include an appropriate drawing
and description of the procedure that you used to determine the permeability of
balloon material to the gases you are measuring. Your experimental plan must
include studying the following: different balloon materials & different gases.
What about the effect of temperature? What was the ambient pressure and the
pressure within the balloon during this experiment? Is it important to know these
pressures & the temperature? Explain how you assured that temperature was held
constant or you accounted for the variation in temperature during an experiment
in your data analysis. (15%)
5) Tabulate your experimental data (an appropriate linear measurement &
temperature & ambient pressure vs time) in a spreadsheet & apply appropriate
regression analysis . Based on this statistical analysis what is the +/-%
uncertainty in your reported permeability for each balloon material and each gas.
(35%)
6) Recommendation: Describe how you could improve your experimental
procedure. How should you account for the effect of gases outside the balloon
diffusing into the balloon while the gases in the balloon are diffusing out. (15%)
4
Title Page
California State Polytechnic University, Pomona
Chemical and Materials Engineering
CHE 313- MASS TRANSPORT SPRING QUARTER, 2012
EXPERIMENT #2
Measurement of Permeability through balloon wall
INSTRUCTOR:
T. K. Nguyen
SECTION 01
GROUP #1
MEMBERS: Lau, Lee
McCaffrey, Charles
DUE DATE: May 10 2013
Title Page
(10) ______
Research binary gaseous diffusivity
(10) ______
Diffusivity measurement
(15) ______
Experimental Plan
(15) ______
Tabulated Data & Analysis
(35) ______
Recommendation
(15) ______
TOTAL
(100) ______
5