Membrane Air Separation Theory
Microporous membranes perform essentially like sieves and allow small molecules to
move through the pores while large molecules are blocked (Towler and Sinnot, 562). Membrane
separation works on the basis of relative permeation rates of the components being separated.
Each component transporting through the membrane has a permeation rate that is unique and is a
function of the ability to dissolve and diffuse. The two main relationships for this separation are
Fick’s Law (1) which relates diffusion and Henry’s Law (3) which relates solubility. The gas
being separated in this experiment is dry air. This fluid is assumed to be an ideal binary perfect
mixture of oxygen and nitrogen. The permeate in this experiment is oxygen whereas nitrogen is
the retentate. The first relation to show the diffusive flux through the membrane is expressed by
Fick’s Law in equation (1) below.
𝐽𝑖 =
𝐷𝑖
𝐿
∗ (𝐶𝑖𝑛1 − 𝐶𝑖𝑛2 )
(1)
Where:
Ji = Flux of component i in mol/m2s
Di = Diffusivity of component i in m2/s
L = Thickness of membrane in meters
Cin1 = Concentration of component i inside membrane wall on feed side in mol/m3
Cin2 = Concentration of component i outside the membrane wall on permeate side in mol/m3
The total flux of component i, Ji, can then be calculated.
𝐽𝑖 =
𝑄𝑖𝑝 𝜌
𝑛𝐴
(2)
Where:
Qip = Volumetric flow rate of species i in the permeate in m3/s
ρ = Density of permeate in mol/m3
A = Area of membrane, (in this case 2.7m2 per module)
n = Number of modules used, (in this case 2)
Using equation (1) and (2) the only unknown turns out to be the diffusivity of component i, Di,
which can be solved for when the two equations are equated through the calculated flux of
component i, Ji .
From Henry’s Law.
𝐶𝑖𝑚 = 𝑆𝑖 𝑝𝑖
(3)
Where:
Cim = Concentration of component i inside the membrane wall in mol/m3
Si = Solubility constant for component i in the membrane in mol/m3Pa
pi = Partial pressure of component i in the gas phase in Pa ( Pascal)
Permeation through the membrane is a function of diffusivity and solubility therefore:
𝑃𝑖 = 𝐷𝑖 𝑆𝑖
(4)
Where:
Pi = Component i membrane permeability in mol/m3sPa
Di = Diffusivity of component i in m2/s
Si = Solubility constant for component i in the membrane in mol/m3Pa
Figure. XXX. General values on the permeability of species through different membranes
(Wankat, 545)
The graph above shows permeability’s for oxygen and nitrogen through four types of membranes
calculated at standard pressure and temperature. Using equation (3) and (4) along with the
collected data allows the calculation of the solubility constant for component i in the membrane,
Si.
Next the separation efficiency can be determined based off of different permeation rates of the
gas components.
𝑃
𝛼𝑖𝑗 = 𝑃 𝑖
(5)
𝑗
Where:
αij = Separation factor
Pi = Component i membrane permeability in mol/m3sPa
An experimental separation factor is frequently used to quantify the separation of a binary
system using mole fractions.
′
𝛼𝑖𝑗
=(
𝑥𝑖𝑝
𝑥𝑗𝑝
𝑥𝑖𝑓
) (6)
𝑥𝑗𝑓
′′
𝛼𝑖𝑗
=(
𝑥𝑖𝑝
𝑥𝑗𝑝
𝑥𝑖𝑓
) (7)
𝑥𝑗𝑓
Where:
α’ij = Separation factor based on the non-permeate composition
α’’ij = Separation factor based on feed composition
Next the recovery of oxygen and nitrogen can be determined.
𝑄𝑝 𝐶𝑂2𝑝
𝑂2 𝑅𝑒𝑐𝑜𝑣𝑒𝑟𝑦 = 𝑄
𝑓 𝐶𝑂2𝑓
𝑄 𝐶
𝑁2 𝑅𝑒𝑐𝑜𝑣𝑒𝑟𝑦 = 𝑄 𝑟 𝐶𝑁2𝑟
𝑓 𝑁2𝑓
Where:
Qp = Volumetric flow rate of permeate in m3/s
Qf = Volumetric flow rate of feed in m3/s
CO2f = Molar concentration of oxygen in feed in mol/m3
CO2p = Molar concentration of oxygen in permeate in mol/m3
(8)
(9)
CN2f = Molar concentration of nitrogen in feed in mol/m3
CN2r = Molar concentration of oxygen in permeate in mol/m3
Finally the stage cut can be determined which is the fractional amount of the total feed entering
the membrane that passes through as the permeate.
𝑆𝑇𝐴𝐺𝐸𝐶𝑈𝑇 = 𝑄
𝑄𝑝
𝑝 +𝑄𝑓
Where:
Qp = Volumetric flow rate of permeate in m3/s
Qf = Volumetric flow rate of feed in m3/s
(10)