1. (1.5 points) Helium diffusion through pyrex glass shows an enormously higher diffusion coefficient
(D) than any other gas (25X that of hydrogen). As there is some helium present in natural gas, this
phenomena can be used to separate out helium by flowing the gas through pyrex tubes with an outer
diameter of 2 mm (R2) and an inner diameter of 1 mm (R1). Assume the process is diffusion-limited (i.e.
can ignore momentum transport).
Your job is to specify tube design recommendations.
You first assume that you can unroll the tube, with a concentration of He on one side (Cin = 10° g/cm²)
and a concentration (Caut o) on the other side, which is pumped away to storage tanks. You can use
the term 8 to indicate the thickness of the pyrex.
If the diffusion coefficient for He through pyrex is 2x10 cm2/s what is your flux (J)?
What is your flux if you account for the area of tube array with total length m (i.e. 200
tubes each 10 long)? Note, you should obtain two values, one each if you consider the inner
diameter versus the outer diameter of the tube.
What design criteria (in terms of 6, D, and AC) would you recommend to improve throughput?
2. (1.5 points) Using the same scenario above, derive an expression for the flux times the area if the
tubes were to remain as cylinders, as a function of the radius r. Calculate the flux at R1 and R2.
What new design criteria would you give? Is it better to collect the He on the inside or the outside of the
tubes? Which is easier to clean in an array?
3. (2 points) Liposomes are small spherical lipid bilayer membranes which can be used to enclose high-
molecular weight drugs and deliver them at a controlled rate. However, because the liposome can
interact with the immune system, biomedical engineers often surround the liposome with a protective
"encapsulant" material, like boly-L-lysine.
The liposome "device" can be modeled as a
spherical system with the lipid membrane at radius
(Ri) and the encapsulant between Ri and Re.
Transport through the very thin lipid membrane (J)
can be modeled using a mass transfer coefficient
as:Ja - no (Ci Co). where C is the concentration
inside the membrane and Co is the concentration
between the membrane and the encapsulant.
Diffusion through the encapsulant material has
Necessary data: Ri = 0.01 cm; Re 0.02 cm, D 10-7 cm2/s. You anassume that the drug leaves the
device so slowly, that for all practical purposes, we can assume steady-state behavior.
Start with mass balance in the radial direction, derive the differential equation for steady-state
diffusion of the drug through the encapsulant.
Solve this differential equation, using the boundary conditions:
When Ri, C=Co; When Re C=0
Use the solution above to write expressions for the flux of J, and sphere area A as a function of r.
Note, the product of J, 'Area should be independent of r and depend on Ca.
If the drug delivery is limited by diffusion through the encapsulant, Co can be approximate by C:
What is Jr* 'Area, in terms of C?
If the drug delivery is limited by transport through the membrane, we can assume Co=0. What
is the product of Jr*Area in terms of C? You tanuse ho 1.4 x10 cm/s.
What will the concentration profile look like assuming transport is controlled by the membrane?
What if it is controlled by the encapsulant?
Note that your units for parts above are: Ci (cm³/s) which indicates that these expressions are
for overall rates of delivery. Based on this idea, does the membrane or the encapsulant control
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