FP#4: Microporous hollow fiber membranes for CO 2
mass transfer into liquid water
Pavlo Kostetskyy
Table of Contents
Section 1: The Proposal ................................................................................................................................. 3
Design Objectives ..................................................................................................................................... 3
Model Schematic ...................................................................................................................................... 3
Section 2: Preliminary Modeling ................................................................................................................... 4
Governing Equations and Boundary Conditions ...................................................................................... 5
Section 3: Preliminary Results and Discussion .............................................................................................. 6
Simple Model Validation .......................................................................................................................... 6
Detailed Model Validation of Al-Marzouqi et al. ..................................................................................... 7
Mesh Convergence.................................................................................................................................... 9
Section 4: SJTP Experimental Set-Up Problem Formulation .......................................................................... 9
Model Schematic ...................................................................................................................................... 9
Governing Equations and Boundary Conditions .................................................................................... 10
Model Results and Analysis.................................................................................................................... 11
Mesh Convergence.................................................................................................................................. 13
Future Work ............................................................................................................................................ 13
References ............................................................................................................................................... 14
Appendix A: Solution Methods .................................................................................................................... 15
Section 1: The Proposal
Design Objectives
The process of interphase mass transfer of CO2 from pure gas phase into liquid water at standard
temperature and pressure within a single hollow fiber tube membrane will be modeled using COMSOL.
The main objective of this project is to validate literature and experimental values of mass flux across
the membrane with a scale model of a microporous membrane tube bundle in three-dimensional
geometry. The model design process will consist of three major phases.
Model Schematic
The first phase will involve simplified interphase mass transfer. The computational domain will be in 2-D
axial symmetry with dimensions shown in Figure 1. In addition, the boundary conditions can be seen in
Table 1. Initial parameters specified for the model can be found in Table 2. A schematic of the
computational domain can be seen in Figure 1.
Before a computational model was constructed for the geometry proposed for Phase 1, an existing
model published by M. H. Al-Marzouqi et al. was replicated using COMSOL software. The results are
discussed in more detail in Section 2.
The second phase will include a detailed 2-D model that will include mass transfer in multiple liquid
water velocities. The rate of CO2 disappearance due to reactions will be included. The results will be
compared to experimental results and those found in literature resources.
Finally, a 3-D model will be constructed to replicate the full-scale experimental set-up. The results
obtained from the final model will be compared to both experimental results as well as literature values
obtained by peers.
Assumptions:
1.
2.
3.
4.
5.
Fully developed laminar liquid flow with a parabolic liquid velocity profile
Pure carbon dioxide gas phase
Negligible generation term in mass transfer calculations
A constant average velocity around hollow fibers
Negligible initial dissolved CO2 concentration in water
Section 2: Preliminary Modeling
A computational model developed by M. H. Al-Marzouqi et al. was solved using identical geometry,
material properties, initial conditions, governing equations (G.E.’s) and boundary conditions (B. C.’s). A
schematic diagram of the computational domain including dimensions can be found in Figure 1. A
detailed description of the boundary conditions as well as initial values can be found in Table 1. Physical
properties of the materials used in the simulation can be found in Table 2.
r = r1
r=0
r = r2
0.04 E-4 m
0.11 E-4 m
r = r3
0.165 E-4 m
z=L
5
1
10
7
4
3
9
0.002 m
Axis of symmetry
2
6
8
z=0
0.315 E-4 m
Pure CO2 Gas
Water
z
Polypropylene
r
Figure 1: Problem formulation diagram for M. H. Al-Marzouqi et al.1
1
Drawing not to scale, numbered boxes represent boundaries in Table 1
Governing Equations and Boundary Conditions
𝑣𝑧
𝜕𝐶𝐶𝑂2
𝜕 2 𝐶𝐶𝑂2 1 𝜕𝐶𝐶𝑂2 𝜕 2 𝐶𝐶𝑂2
= 𝐷𝐶𝑂2 [
+
+
]
𝜕𝑧
𝜕𝑟 2
𝑟 𝜕𝑟
𝜕𝑧 2
Table 1: Boundary Conditions for M. H. Al-Marzouqi et al.
Boundary
Value
Boundary Condition
𝜕𝐶𝐶𝑂2
1
Symmetry
2
Convective Flux
3
Concentration
4
Continuity
5
Flux
6
Flux
7
Continuity 2
8
Convective Flux
9
Flux
𝜕𝐶𝑂2
𝜕𝑟
10
Concentration
𝐶𝐶𝑂2 = 0 at z = L
𝜕𝑟
= 0 (symmetry) at r = 0
-
𝐶𝐶𝑂2 = 𝐶𝐶𝑂2 ,𝑖𝑛𝑖𝑡 = 𝐶0 at z =0
𝜕𝐶𝐶𝑂2
𝜕𝑟
𝜕𝐶𝐶𝑂2
𝜕𝑧
𝜕𝐶𝐶𝑂2
𝜕𝑧
𝜕𝐶𝐶𝑂2
𝜕𝑟
|
𝑔𝑎𝑠
=
𝜕𝐶𝐶𝑂2
𝜕𝑟
|
𝑚𝑒𝑚
at r = r1
= 0 at z = L
= 0 at z = 0
|
𝑚𝑒𝑚
=
𝜕𝐶𝐶𝑂2
𝜕𝑟
|
𝑠ℎ𝑒𝑙𝑙
at r = r2
= 0 at r = r3
Table 2: Physical Properties of Fluids
Propertiy
Diffusivity of CO2 inside lumen
Diffusivity of CO2 in membrane
Diffusivity of CO2 in water
Density of water
Viscosity of water
2
Value
Units
Reference
1.86E-05
3.71E-06
1.92E-09
9.98E+02
9.31E-04
m2/s
m2/s
m2/s
kg/m3
Pa*s
[1]
[1]
[1]
[5]
[5]
Concentration in shell was calculated using Henry’s Law Conversion 𝐶𝐶𝑂2 ,𝑚𝑒𝑚𝑏 =
𝐶𝐶𝑂2 ,𝑠ℎ𝑒𝑙𝑙
𝐻
for H = 0.8314 [1]
Section 3: Preliminary Results and Discussion
Simple Model Validation
Before specifying the model by Al-Marzouqi et al., a much simpler process was modeled using Comsol
Multiphysics (CMP) as well as analytically. The process of diffusion through the membrane was modeled
assuming Fick’s law of diffusion. Knowing the thickness of the diffusion medium, concentration gradient
and the mean diffusion coefficient, the total flux through the material can be calculated. Figure 2 shows
the mean flux value in a 2D geometry represented in CMP. The analytical solution, shown in equation 1,
matches the numerical solution. A constant concentration boundary condition was assumed for both
models. The gas phase changes in concentration in the z-direction in the full model by Al-Marzouqi,
meaning that Equation 1 is solved for a continuously changing concentration gradient.
Figure 2: Mean flux based on concentration gradient and diffusion coefficient
𝑁𝐶𝑂2 = 𝐷𝐶𝑂2 ,𝑚𝑒𝑚
𝑁𝐶𝑂2
𝜕𝐶
𝜕𝑥
𝑚𝑜𝑙
2
(5.312 3 − 0)
𝑚
𝑚𝑜𝑙
𝑚
= (3.71 ∗ 10−6
= 4.927 2
)∗
−4
𝑠
0.04 ∗ 10 𝑚
𝑚 𝑠
(1)
(2)
The concentration profile of carbon dioxide in the membrane can be seen in Figure 3. It is important to
note that the concentration profile is linear, which is consistent with Equation 1.
Membrane Concentration Profile
Concentration (mol/m3)
6
5
4
3
2
1
0
0.00E+00
1.00E-06
2.00E-06
3.00E-06
4.00E-06
5.00E-06
Arc Length (m)
Figure 3: Diffusion through the membrane
Detailed Model Validation of Al-Marzouqi et al.
After the simple model was constructed and validated using a numerical solution, a steady state model
published by Al-Marzouqi was replicated using CPM. The geometry of the numerical model was
identical to that shown in Figure 1. The gas phase was assumed to follow a parabolic velocity profile with
an average bulk velocity of 1.075*10-2 m/s and an initial inlet concentration of 5.312 mol/m3. The fluid
flow outside of the membrane was assumed to follow a parabolic velocity profile with an average bulk
velocity of 3.407*10-3 m/s (shown in Figure 4).
Axial Velocity (m/s)
Velocity Profile
0.004
0.0035
0.003
0.0025
0.002
0.0015
0.001
0.0005
0
0
0.00005
0.0001
0.00015
Arc Length (m)
Figure 4: Parabolic fluid velocity profile
0.0002
Figure 5 shows the cross-section plot of the concentration at z = 0.5*L. It can be seen that the
concentration remains nearly constant in the regions of the stagnant gas phase and the membrane.
However, the concentration drops as the radial position changes in the water phase. The results
graphed in Figure 5 agree with results published by Al-Marzouqi et al. shown in Figure 6. The
concentration profiles at the axial midpoint have matching trends and are of similar magnitude. The
small discrepancy in the values could be attributed to calculated average bulk velocities based on
flowrate values provided in the publication.
Concentration Profile at z = 0.5*L
0.45
C/C_initial
0.4
0.35
0.3
Gas
Membrane
0.25
Water
0.2
0.15
0.00E+00
1.00E-05
2.00E-05
3.00E-05
Arc Length (m)
Figure 5: Concentration profile in radial cross section at z = 0.5*L 3
Figure 6: Radial Concentration Profile at z = 0.5*L (Taken from Al-Marzouqi et al.)
3
Initial Concentration = 5.312 mol/m3
Mesh Convergence
In order to maximize the accuracy of the model and minimize the computing time and power
requirements a mesh convergence analysis was performed. A free mesh was used in all of the
computational subdomains. The maximum element size was varied in order to obtain an optimal mesh
density. The concentration volume integral was calculated for several mesh densities using CMP in the
aqueous phase. Optimal mesh density was achieved at 26,938 elements with a maximum element size
of 2.5E-6, indicated by the volume integral not changing significantly with increasing mesh size. The
graphical representation of these results can be seen in Figure 7.
Concentration Integral in Aqeous Phase
(mol)
Mesh Convergence
8.8950E-12
8.8850E-12
8.8750E-12
8.8650E-12
8.8550E-12
8.8450E-12
8.8350E-12
0
5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000
Number of Elements
Figure 7: Mesh Convergence analysis in aqeous region
Section 4: SJTP Experimental Set-Up Problem Formulation
Model Schematic
The computational domain will be in 2-D axial symmetry, a schematic diagram of the computational
domain including dimensions can be found in Figure 8. The numbered boxes in Figure 8 represent
specific boundaries in the computational domain, a detailed description of the boundary conditions as
well can be found in Table 3. Physical properties of the materials used in the simulation can be found in
Table 4. A parabolic fluid velocity profile was assumed for the aqueous phase outside of the membrane
fibers with an average bulk velocity of 0.00165 m/s. The initial CO2 concentration at boundary #3 (Fig. 8)
was set to 12.5 mol/m3.
r = r1
r=0
r = r2
1.00 E-4 m
0.5 E-4 m
z=L
1.0 E-4 m
5
2
Axis of symmetry
0.002 m
r = r3
1
10
3
9
7
4
8
6
z=0
2.5 E-4 m
Pure CO2 Gas
z
Water
Polypropylene
r
Figure 8: Problem formulation diagram for Experimental Set-Up 4
Governing Equations and Boundary Conditions
𝑣𝑧
4
𝜕𝐶𝐶𝑂2
𝜕 2 𝐶𝐶𝑂2 1 𝜕𝐶𝐶𝑂2 𝜕 2 𝐶𝐶𝑂2
= 𝐷𝐶𝑂2 [
+
+
]
𝜕𝑧
𝜕𝑟 2
𝑟 𝜕𝑟
𝜕𝑧 2
Drawing not to scale, numbered boxes represent boundaries in Table 1
Table 3: Boundary Conditions for M. H. Al-Marzouqi et al.
Boundary
Value
Boundary Condition
𝜕𝐶𝐶𝑂2
1
Symmetry
2
Flux
3
Concentration
4
Continuity
5
Flux
6
Flux
7
Continuity 5
8
Convective Flux
9
Flux
𝜕𝐶𝑂2
𝜕𝑟
10
Concentration
𝐶𝐶𝑂2 = 0 at z = L
𝜕𝑟
𝜕𝐶𝐶𝑂2
𝜕𝑧
= 0 (symmetry) at r = 0
= 0 at z = L
𝐶𝐶𝑂2 = 𝐶𝐶𝑂2 ,𝑖𝑛𝑖𝑡 = 𝐶0 at z =0
𝜕𝐶𝐶𝑂2
𝜕𝑟
𝜕𝐶𝐶𝑂2
𝜕𝑧
𝜕𝐶𝐶𝑂2
𝜕𝑧
𝜕𝐶𝐶𝑂2
𝜕𝑟
|
𝑔𝑎𝑠
=
𝜕𝐶𝐶𝑂2
𝜕𝑟
|
𝑚𝑒𝑚
at r = r1
= 0 at z = L
= 0 at z = 0
|
𝑚𝑒𝑚
=
𝜕𝐶𝐶𝑂2
𝜕𝑟
|
𝑠ℎ𝑒𝑙𝑙
at r = r2
= 0 at r = r3
Table 4: Physical Properties of Fluids
Propertiy
Diffusivity of CO2 inside lumen
Diffusivity of CO2 in membrane
Diffusivity of CO2 in water
Density of water
Viscosity of water
Value
Units
Reference
1.39E-05
2.32E-06
1.92E-09
9.98E+02
9.31E-04
m2/s
m2/s
m2/s
kg/m3
Pa*s
[6]
[7]
[5]
[5]
[5]
Model Results and Analysis
Figure 9 below shows a concentration profile of the computational domain. The profiles are consistent
with the governing equations and basic mass transfer theory. Tube-side concentration decreases in the
axial direction and the moving liquid water phase is most saturated at the membrane interface. This is
further shown in Figure 10 which is a radial cross section of the axial model at z = 0.5*L. It is clear that
the dissolved CO2 concentration in the aqueous phase approaches 0 as the radial distance increases.
5
Concentration in shell was calculated using Henry’s Law Conversion 𝐶𝐶𝑂2 ,𝑚𝑒𝑚𝑏 =
𝐶𝐶𝑂2 ,𝑠ℎ𝑒𝑙𝑙
𝐻
for H = 0.8314 [1]
Figure 9: Concentration Profile of the Membrane system
Concentration Profile at z = 0.5*L
0.7
0.6
C/C_init
0.5
0.4
Tube
0.3
Membrane
0.2
Shell
0.1
0
0.00E+00
1.00E-04
2.00E-04
Arc Length (m)
Figure 10: Concentration profile in radial cross section at z = 0.5*L 6
6
Initial Concentration = 12.5 mol/m3
Mesh Convergence
As well as in the validation model in Section 3, a free mesh was used in all of the computational
subdomains. The maximum element size was varied in order to obtain an optimal mesh density. The
concentration volume integral was calculated for several mesh densities using CMP in the aqueous
phase. Optimal mesh density was achieved at 12,820 elements with a maximum element size of 1E-5,
indicated by the aqueous-phase volume integral not changing significantly with increasing mesh size.
The graphical representation of these results can be seen in Figure 11.
Volume Integral (mol)
Mesh Convergence
4.95E-10
4.90E-10
4.85E-10
4.80E-10
4.75E-10
4.70E-10
4.65E-10
4.60E-10
4.55E-10
4.50E-10
4.45E-10
4.40E-10
0
5000
10000
15000
20000
25000
Number of Elements
Figure 11: Mesh convergence analysis of aqueous phase
Future Work
Based on the current model and the validation model, the next steps will include:
1. A thorough mesh convergence around boundary #8
2. Thorough accuracy analysis, looking for other possible sources of error
3. Perform sensitivity analysis on Diffusivity of CO2 in water, Diffusion coefficient of CO2 in the
membrane (porosity dependent) and shell-side water velocity
4. Attempt to increase the size of the computational domain to reach to scale dimensions
References
1. M. H. Al-Marzouqi, M. H. El-Naas, S. A. M. Marzouk, M. A. Al-Zarooni, N. Abdulatif, R. Faiz.
“Modeling of CO2 absorption in membrane contactors”. Separation and Purification Technology
59 (2008) 286–293
2. E.L. Cussler, "Diffusion. Mass Transfer in Fluid Systems", 2nd edition, Cambridge University
Press, 1997. pp 111-115
3. E.L. Cussler, Q. Zhang. "Microporous Hollow Fibers for gas absorption” Department of Chemical
Engineering, University of Minnesota, 1984 p. 327
4. B.S. Ferraira, H. L. Fernandes, A. Reis and M. Mateus, “Microporous Hollow Fiber Membranes
for Carbon Dioxide Absorption: Mass Transfer Model Fitting and the Supplying of Carbon
Dioxide to Microalgal Cultures” October 1996. pp 66-68
5. A. Datta and V. Rakesh. “An Introduction to Modeling of Transport Processes” Cambridge Texts
in Biomedical Engineering. 2010. pp. 439-450
6. Extraction of Carbon Dioxide from the atmorspeher through engineered chemical sinkage. M. K.
Dubey, H. Ziock, G. Rueff, S. Elliott and W. S. Smith Fuel Chemistry Division Preprints 2002, 47(1),
82
7. Celgard X40-200 Microporous Hollow Fiber Membrane Spec Sheet. Membrana – Charlotte A
Division of Celgard, LLC. 13800 South Lakes Drive Charlotte, North Carolina 28273
Appendix A: Solution Methods
Table B1: Mesh Convergence Analysis for Al-Marzouqi et al.
Concentration Integral (mol)
Maximum
Element Size
Number of
Elements
Subdomain #1
Subdomain #2
Subdomain #3
1.00E-05
7.00E-06
5.00E-06
3.00E-06
2.50E-06
2.30E-06
2.20E-06
2.15E-06
3,781
5,400
7,291
18,712
26,938
29,598
34,566
37,250
1.7951E-12
1.7950E-12
1.7965E-12
1.7985E-12
1.8001E-12
1.8003E-12
1.7995E-12
1.7996E-12
1.5427E-12
1.5427E-12
1.5440E-12
1.5457E-12
1.5470E-12
1.5472E-12
1.5465E-12
1.5466E-12
8.8474E-12
8.8470E-12
8.8563E-12
8.8684E-12
8.8779E-12
8.8790E-12
8.8744E-12
8.8748E-12