Homework 2
1. When the permeate pressure is negligible in comparison with the retentate pressure in
gas separation, the separation factor can be expressed as the permeability ratio as
follows:
Pi
ij* =
Pj
This is the so-called ideal separation factor.
1) For cases in which the permeate pressure (P’’) cannot be ignored, prove that the
true separation factor ij is related to ij* in the following equation:
xi ( ij − 1) + 1 − r ij
ij = ij* [ ]
xi ( ij − 1) + 1 − r
Where r = p’’ /p is the ratio of the permeate pressure to the retentate pressure.
Note: ij is equal to ij* for r = 0.
2) For polyimide membrane process to recover hydrogen from a refinery waste
stream from a hydrotreating operation, i.e., to separate hydrogen from methane, a
methane-rich product contains 60 mol % methane and 40 mol% hydrogen. The
ideal separation factor of hydrogen vs. methane ij* is 125. If p’’ / p = 0.5, find
the value of the true separation factor ij and the composition of the hydrogen-
rich product. Explain physically why ij is so much less than ij* and propose a
way to increase ij .
3) Show that the value of ij is required to provide a given recovery of component j
in the retentate (a given composition x j ), with fixed compositions of the binary
feed and the permeate, is given by the following expression:
− Rj
ij =
1− Rj
, xj
Where R j = is moles of component j in the retentate divided by the moles
x jF
component j in the feed and
x j xiF
=
xi x jF
xiF is the mole fraction of component i in the feed.
Moles of Component i permeated p ' xiF − p ' xi (x iF − xi )
[Hint: = = ]
Moles of Component j permeated p ' x jF − p ' x j x jF − x j
1. When the permeate pressure is negligible in comparison with the retentate pressure in
gas separation, the separation factor can be expressed as the permeability ratio as
follows:
Pi
ij* =
Pj
This is the so-called ideal separation factor.
1) For cases in which the permeate pressure (P’’) cannot be ignored, prove that the
true separation factor ij is related to ij* in the following equation:
xi ( ij − 1) + 1 − r ij
ij = ij* [ ]
xi ( ij − 1) + 1 − r
Where r = p’’ /p is the ratio of the permeate pressure to the retentate pressure.
Note: ij is equal to ij* for r = 0.
2) For polyimide membrane process to recover hydrogen from a refinery waste
stream from a hydrotreating operation, i.e., to separate hydrogen from methane, a
methane-rich product contains 60 mol % methane and 40 mol% hydrogen. The
ideal separation factor of hydrogen vs. methane ij* is 125. If p’’ / p = 0.5, find
the value of the true separation factor ij and the composition of the hydrogen-
rich product. Explain physically why ij is so much less than ij* and propose a
way to increase ij .
3) Show that the value of ij is required to provide a given recovery of component j
in the retentate (a given composition x j ), with fixed compositions of the binary
feed and the permeate, is given by the following expression:
− Rj
ij =
1− Rj
, xj
Where R j = is moles of component j in the retentate divided by the moles
x jF
component j in the feed and
x j xiF
=
xi x jF
xiF is the mole fraction of component i in the feed.
Moles of Component i permeated p ' xiF − p ' xi (x iF − xi )
[Hint: = = ]
Moles of Component j permeated p ' x jF − p ' x j x jF − x j
