2.1. Mathematical Modelling
An approximate model was developed to characterize multi-component gas separation for SWM modules in conjunction with the mass balance equation. The succession of states approach was selected due to its simple implementation and lower computational time requirement. This method reduces the problem into finite elements by assuming a constant mass transfer at each element [
21]. Each element is independent of one another whereas the outlet condition of the element is computed based on a specific inlet condition that later becomes an inlet condition for the subsequent element. The mass balance and transport equations are computed over every finite element in the matrix to obtain the rate of permeation and composition. The simplicity of this methodology ensures stability and convergence of the algorithm. Furthermore, it enables the non-ideal effects, such as pressure drop and pressure buildup, to be implemented conveniently in conjunction with the mass balance equations.
The solution–diffusion model is used to characterize the transport mechanism of the gas components within the membrane module (see Equation (1)),
where
is the flux of gas component n across the membrane,
is the permeability of gas component
n,
is the thickness of the active layer of the membrane, and
and
respectively are the low-side and high-side pressures of the membrane module.
and
are defined as the local retentate and permeate composition of component n at the boundary of the active membrane, respectively.
The following assumptions were made in the proposed model:
- ○
Local permeation and bulk permeate flow are described by a crossflow pattern [
22,
23];
- ○
There is no permeate mixing in the direction of the bulk permeate flow [
19,
23];
- ○
The pressure variation for flow through the permeate spacer channel is characterized by Hagen–Poiseuille equation [
24];
- ○
The feed-side pressure drop is negligible [
15,
16,
19,
20,
23,
24];
- ○
The pressure drop along the central permeate collector tube is neglected (i.e., pressure is considered as atmospheric) [
25,
26];
- ○
The permeability coefficients are independent of concentration, pressure, and temperature [
15,
16,
19,
20,
23,
24];
- ○
Channel curvature is neglected and the membrane is treated as a flat sheet [
22,
26];
- ○
Operation within the SWM module is isothermal [
16,
18,
19,
20,
27].
To compute the active membrane area () based on a predetermined number of elements, the membrane dimension specifications, as below, are fixed prior to calculation.
Channel spacer,
and membrane,
thickness;
Module, , and collection tube,
diameter;
Number of leaves, ;
Module length, ;
The active membrane area of each element, as computed using Equations (2)–(5), depends upon the number of leaves and dimensions of the membrane itself.
The succession of state methodology divides the SWM module into four types of elements as shown in
Figure 2. The mass computation for each type is different. Type I is referred to as the first element that interacts with the feed, located at the corner of membrane sheet (near the glued side). Type II is located along the glued side’s axial direction, while Type III is situated along the entrance feed on the radial direction. Type IV represents remaining elements in the radial and axial directions in the membrane module. The computation procedures are started at the entrance elements (Type I and Type III, where the initial concentration along the membrane width is identical) to compute the mass transfer across the membrane for the first element before proceeding to the next element. It ends as it covers the entire membrane module.
Figure 3 represents the elemental volume of the cross-flow SWM module elements utilized in the proposed model that is comprised of the feed and permeate sections and membrane layer. The axial direction is represented by the dimension
while the radial direction is represented by the dimension
. The leaf is represented by the dimension
. The components are differentiated in terms of their permeating nature. The fast-permeable component is referred to as
in the retentate stream, while
is the permeable component in the permeate stream.
The characterization of transport across the cross-flow SWM module element using Fick’s law of diffusion, is described in Equation (6):
Coupled with the Newton bisection numerical solution, the guessed permeate compositions are initiated from the low and high sides to ensure quick convergence.
For Type I and Type II cells at various axial positions along the closed-end membrane sheet, the permeate composition of each component,
, is determined using the solution procedure outlined. After determining permeate composition, total flow rate of each component into the permeate stream,
, the flow rate into the permeate stream,
, the retentate stream flow rate,
, and the retentate stream composition,
, are estimated using Equations (7)–(11):
For Type I cells in direct contact with the feed side, the indices and are replaced with the feed condition.
Similarly, for Type III and IV cells, the permeate composition of each component in a gaseous mixture,
, is determined using the algorithm proposed. Later, the total flow rate of each component into the permeate stream,
, the flow rate into the permeate stream,
, the retentate stream flow rate,
, and the retentate stream composition,
, contacting the subsequent cells are calculated using Equations (12)–(16):
For the Type III cells in direct contact with the feed, the indices
and are replaced with the feed condition.
Besides, the permeate spacer channel pressure variation is characterized by the Hagen–Poiseuille equation, as presented in Equation (17):
Where
is the friction coefficient,
is the specific density,
is the linear velocity,
is the length of membrane, and
is the hydraulic diameter. The friction coefficient is a function of Reynolds number as presented in Equation (18):
The classical Reynolds number (
Re) expresses the relationship between turbulence flow and fluid viscosity:
The hydraulic diameter (
) for spacer filled channels on the feed and permeate sides are calculated using Equation (20):
The linear flow velocity (
) is computed as follows:
where
is the linear flow velocity,
is the feed flow rate, and A is area of the feed channel cross section, which is the product of the channel width
, height
, and porosity
of the flow channel.
The viscosity of gas mixture is calculated by adapting Wilke’s method [
28,
29], while the viscosity of the pure components is determined using Lucas’s method [
30,
31].