Membrane Gas Separation

16.2.2
Explicit Sweep Distribution Simulations 
To explicitly calculate the shell and lumen fl ow distributions, the shell - side and lumen -
side spaces are treated as bi - continuous porous media. Volume averaging the conservation 

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The Effect of Sweep Uniformity on Gas Dehydration Module Performance
339
equations for the lumen and shell spaces yields conservation equations in terms of volu-
metric average values of the fi eld variables (velocity, pressure, and concentration) where 
the averaging volume is small compared to the macroscopic dimensions of the module 
but larger than fi bre dimensions. 
For low Reynolds number fl ows, volume average of the conservation of mass equations 
yields Darcy ’ s law as the relationship between superfi cial velocity and pressure:
u
K
p
= − ∇
η
(16.15)
where K denotes the hydraulic permeability of the porous medium and u is the volume 
average superfi cial velocity. Note that for anisotropic porous media K is a tensor. However, 
previous work [23] shows that for suffi ciently high module aspect ratios (i.e. module 
length to diameter ratios) the effect of anisotropy on module performance is negligible 
so one may assume the porous media are isotropic. 
The steady - state volume average conservation of mass equation for component i is 
given by Equation (16.16) 
∇⋅
( )
= ±
ρ
i
i
u
J
(16.16)
where  
ρ
  is the molar fl uid density and J the permeation fl ux defi ned in Equation (16.4) . 
For a lumen - fed module, the positive sign is used for the shell fl ow and the negative for 
the lumen fl ow. Note that mass transfer due to molecular diffusion and Taylor dispersion 
is neglected relative to convection, as suggested in the literature [24] . 
Summing Equation (16.16) over all of the components yields the continuity equation 
for the porous media.
∇⋅
( )
= ±
=
∑
ρ
u
J
i
i
n
1
(16.17)
The shell and lumen velocity fi elds are obtained by substituting Darcy ’ s law, Equation 
(16.15) for the velocity and applying appropriate boundary conditions. An appropriate 
equation of state also is required to calculate density from pressure. The ideal gas law is 
used for the relatively low pressure dehydration process considered here. 
Shell and lumen boundary conditions for external sweep ports are illustrated in Figure 
16.2 for an axis symmetric module cross - section. Symmetry is applied along the module 
centreline while the radial velocity and radial mass fl uxes are set to zero along the external 
case. The pressures along the inlet and outlet are specifi ed for the lumen and shell regions. 
The lumen and shell fl ow rates are controlled by the magnitude of the pressure drop 
between inlet and outlet. Note that specifi cation of uniform pressures along the inlet and 
outlet boundaries assumes uniform gas distribution in the header that connects these 
regions to the external plumbing that delivers/removes the gas fl ow.
Equations (16.14) – (16.16) are solved using COMSOL Multiphysics ® [32] . This simu-
lation environment solves the governing conservation equations using the fi nite element 
method [33] . It also readily accommodates introduction of the appropriate form for the 
permeation fl ux and its dependence on gas partial pressure.

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Membrane Gas Separation