FP#6: Microporous hollow fiber membranes for CO 2
mass transfer into liquid water
Pavlo Kostetskyy
1
Table of Contents
Executive Summary ....................................................................................................................................... 1
Introduction and Design Objectives .............................................................................................................. 1
Introduction ............................................................................................................................................... 1
Design Objectives ..................................................................................................................................... 2
Model Schematic ...................................................................................................................................... 2
Results and Discussion .............................................................................................................................. 3
Mesh Convergence.................................................................................................................................... 6
Sensitivity analysis.................................................................................................................................... 6
Conclusions................................................................................................................................................ 7
Appendix A: Preliminary Modeling................................................................................................................ 8
Governing Equations and Boundary Conditions ...................................................................................... 9
Simple Model Validation ........................................................................................................................ 10
Detailed Model Validation of Al-Marzouqi et al. ................................................................................... 11
Mesh Convergence.................................................................................................................................. 13
Appendix B: Mathematical Statement of the Problem ............................................................................... 14
Governing Equations and Boundary Conditions .................................................................................... 14
Appendix C: Solution Strategy ..................................................................................................................... 15
Appendix D: References .............................................................................................................................. 17
2
Executive Summary
The main focus of this study was to model a segment of a single hollow fiber filled with pure CO2 gas 300
µm in diameter surrounded by a liquid water phase. The gas phase was assumed to be stagnant and the
liquid phase following a parabolic velocity profile with an average bulk velocity of 0.00165 m/s. The
model results matched those found in literature [1]. The concentration profiles of a membrane cross
section followed the same trends. A cross section of the membrane at its axial midpoint indicated a
tube-side CO2 concentration of 60% of the initial value. In addition, it was shown that a liquid layer
thickness of 100 µm is sufficient for the given velocity in terms of dissolved gas concentration at the
edge of the liquid layer. Based on these results, the spacing of individual fibers within a bundle can be
determined to be 100 µm or greater. As a result, the concentration gradient between liquid water and
the gas phase would be unaffected.
The model design process consisted of three major phases. First, simple flux calculations were
performed analytically and were validated using CMP. Secondly, a model published by M. H. AlMarzouqi et al. was replicated. Finally, a model based on fiber dimensions and experimental conditions
of an experimental set-up at the South Jersey Technology Park was constructed.
Introduction and Design Objectives
Introduction
Multiphase mass transfer between a gas and a liquid is prevalent in all industries. Conventional gasliquid mass transfer systems most often utilize towers filled with packing in order to maximize surface
area for mass transfer to occur. In a typical packed tower design, gas is fed at the bottom of the tower
and liquid enters at the top. Mass transfer occurs between the two phases on the surface of the packing
within the tower. These units operate within a small range of stream flow rates and become inoperable
when these stream limitations are exceeded. Flooding occurs when gas flow rate exceeds a critical value
and results in liquid backflow or buildup. On the other hand, entrainment occurs when the liquid flow
rate exceeds some critical value and the liquid phase begins accumulating within the tower [8].
Microporous hollow fiber membranes (HFM’s) eliminate the traditional operating limitations as well as
increase the interfacial surface area between the two phases [8]. Hollow fiber membranes provide a
nearly 100% mass transfer efficiency and provide an exceptionally high surface area to volume ratio [4].
Specifically, the use of hollow fibers for CO2 absorption has been considered to be the superior option to
packed towers in many industrial applications such as stack gas treatment.
The use of HFM’s has also been considered for the supply of CO2 into bioreactors for the growth of
algae. These photosynthetic organisms can be used to sequester CO2 produced from combustion of nonrenewable fuel sources. The biomass that is produced has high lipid content. The lipids extracted from
algal biomass can finally be converted into a fuel or used in other industrial applications.
1
Individual hollow fiber membranes can be produced from a variety of materials with physical properties
that best suit the media. Typical fiber diameters range from 200 to 300 µm. In order to be capable of
supplying CO2 on a large enough scale, multi fiber bundles need to be constructed. Individual fibers
within the bundle behave nearly identically and therefore a computational model of a single fiber may
be used to represent all of the fibers within the matrix of fibers.
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
Multiphysics (CMP).
The main objective of this project is to validate literature and experimental values of mass flux across
the membrane with a computational model of a microporous hollow fiber membrane tube.
Model Schematic
This model was designed to replicate mass transfer an existing experimental set up located in South
Jersey Technology Park. 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 (Figure 1) was set to 12.5 mol/m3.
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
2
r = r1
r=0
r = r2
1.00 E-4 m
0.5 E-4 m
z=L
Axis of symmetry
0.002 m
S1
3
1.0 E-4 m
5
2
1
r = r3
4
10
S2
S3
7
9
8
6
z=0
2.5 E-4 m
Pure CO2 Gas
z
Water
Polypropylene
r
Figure 1: Problem formulation diagram for Experimental Set-Up 1
Results and Discussion
Concentration profiles within the membrane module specified in this section behave as would be
expected in an experimental setting. Figure 1Figure 2 below shows a concentration profile of the
1
Drawing not to scale, numbered boxes represent boundaries in Table B1
3
computational domain. The profiles are consistent with the governing equations and basic mass transfer
theory. Tube-side concentration decreases in the axial direction. The moving liquid water phase is most
saturated at the membrane interface and the degree of saturation increases along the length of the
tube (in downward direction). This is further shown in Figure 3 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.
Convective flux of dissolved CO2 in the liquid phase is indicated by an arrow plot (Figure 2). It can be
seen that the magnitude of the flux vectors increase in the axial direction and decrease in magnitude in
the radial direction. This is consistent with the governing equations where convective flux is a function
of liquid velocity and CO2 concentration.
Figure 2: Concentration Profile of the Membrane system and Arrow plot indicating convective flux in shell side.
4
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 3: Concentration profile in radial cross section at z = 0.5*L 2
Concentration Profile at z = 0
1.2
C/C_init
1
0.8
Tube
0.6
Membrane
0.4
Shell
0.2
0
0.00E+00
1.00E-04
2.00E-04
Arc Length
Figure 4: Concentration profile in radial cross section at z = 0
As shown in Figure 4, the shell-side concentration approaches 0 as the radial distance increases. This
concentration profile may be used for fiber spacing and flow rate optimization when designing a
membrane contactor with multiple membrane fibers. The shell-side concentration profile is largely a
function of superficial velocity and therefore, the minimal liquid velocity may be determined at which
the dissolved gas concentration is close to 0 at the liquid layer edge (r = r3).
2
Initial Concentration = 12.5 mol/m3
5
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. Mesh convergence analysis was performed
on each subdomain. The optimal number of elements for this computational model was found to be
25,400. A free mesh was used in all of the computational subdomains with triangular element geometry.
Graphical representation of mesh convergence analysis can be found in Appendix C: Solution Strategy.
The parameter of interest was the concentration volume integral calculated for each subdomain by
CMP. Due to high mesh density requirements for Subdomain #2 (S2), mesh density of Subdomain #1 (S1)
increased to 2,497 elements. Similarly, S2 mesh density increased to 5,362 elements due to Subdomain
#3 requirements. Subdomain #3 proved to be the most difficult to optimize due to the convective mass
transfer calculations required. Areas of the computational domain closest to boundary #10 failed to
converge at low mesh density. Figure C4 may be misleading in terms of mesh convergence, the volume
integral value stabilizes at a mesh density of 4,473. However, small areas of the domain did not
converge in the area of high instability around boundary #10. As a result, custom mesh refinement was
performed on boundaries #9 and 10 (Figure C1). The maximum element sizes to reach convergence
were found to be 1E-6 for boundary #10 and 2E-6 for boundary #9.
Sensitivity analysis
Sensitivity analysis was performed on three major parameters involved in the simulation. The diffusivity
of CO2 within the tube, diffusion coefficient of CO2 within the membrane (a function of membrane
porosity) and the diffusivity of CO2 in the shell will be analyzed. A 20% variability of the values for these
parameters was used. The concentration volume integral of CO2 in the shell side was calculated using
CMP. Figure 5 shows the effect of 20% variability of these parameters on the shell-side concentration
integral. Each parameter was varied within the 20% range, while the others remained at the initial (0%)
value. It can be seen that shell-side diffusivity has the largest effect on the total CO2 transferred. The
effect of all parameters varying in the 20% range can be seen in Figure 6. An 11% increase from
“baseline” in the shell-side concentration resulted from a 20% increase in all of the parameters.
6
Parametric Variation
Extreme Variation
6.00E-10
7.00E-10
5.00E-10
6.00E-10
4.00E-10
5.00E-10
3.00E-10
-20%
2.00E-10
0%
1.00E-10
20%
0.00E+00
Tube
Membrane
Shell
Diffusivity Diffusion Diffusivity
Coefficient
4.00E-10
3.00E-10
2.00E-10
1.00E-10
0.00E+00
Min
Starting
Max
Figure 6: Sensitivity analysis of extreme variation of all
parameters between -20% and 20%
Figure 5: Sensitivity analysis on shell-side concentration integral
with diffusion parameters varied by 20%
Conclusions
Based on the results obtained from the model, it can be concluded that steady state operation of a
single hollow fiber membrane module can be predicted using computational methods. The
concentration profiles calculated by the software indicate a significant rate of mass transfer that match
similar studies in the literature [1].
During analysis of the model results it was determined that a 20% variation in diffusivity parameters had
a fairly significant effect on the steady state shell-side gas concentration with a maximum increase of
11% predicted by CMP. In addition, the model may be used in membrane contactor design and
optimization. The spacing of individual fibers may be determined by specifying a velocity or the optimal
liquid velocity may be found by specifying the spacing the thickness of the outer liquid layer.
A computational model is a useful tool for experimental validation as well as prediction of results in
experiments that have not been completed. The speed of the computational software allow for fast
estimates of a physical process that can save both time and resources in the physical world.
7
Appendix A: 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 A1. A
detailed description of the boundary conditions as well as initial values can be found in Table A1.
Physical properties of the materials used in the simulation can be found in Table A2.
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 A1: Problem formulation diagram for M. H. Al-Marzouqi et al.3
3
Drawing not to scale, numbered boxes represent boundaries in Table 1
8
Governing Equations and Boundary Conditions
𝑣𝑧
𝜕𝐶𝐶𝑂2
𝜕 2 𝐶𝐶𝑂2 1 𝜕𝐶𝐶𝑂2 𝜕 2 𝐶𝐶𝑂2
= 𝐷𝐶𝑂2 [
+
+
]
𝜕𝑧
𝜕𝑟 2
𝑟 𝜕𝑟
𝜕𝑧 2
Table A1: 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 4
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 A2: 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
4
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]
9
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 A2
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 AlMarzouqi, meaning that Equation 1 is solved for a continuously changing concentration gradient.
Figure A2: Mean flux based on concentration gradient and diffusion coefficient
𝑁𝐶𝑂2 = 𝐷𝐶𝑂2 ,𝑚𝑒𝑚
𝑁𝐶𝑂2 = (3.71 ∗ 10
−6
𝜕𝐶
𝜕𝑥
𝑚𝑜𝑙
(5.312 3 − 0)
𝑚2
𝑚𝑜𝑙
𝑚
= 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 A3. It is important to
note that the concentration profile is linear, which is consistent with Equation 1.
10
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 A3: 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 A1. 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 A4).
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
0.0002
Arc Length (m)
Figure A4: Parabolic fluid velocity profile
11
Figure A5 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 A5 agree with results published by Al-Marzouqi et al. shown in Figure A6. The
concentration profiles at the axial midpoint are identical.
Radial Concentraion Profile
0.50
C/C_init
0.45
0.40
0.35
Tube
0.30
Membrane
Shell
0.25
0.20
0.E+00
1.E-05
2.E-05
3.E-05
Arc Length (m)
Figure A5: Concentration profile in radial cross section at z = 0.5*L 5
Figure A6: Radial Concentration Profile at z = 0.5*L (Taken from Al-Marzouqi et al.)
5
Initial Concentration = 5.312 mol/m3
12
Mesh Convergence
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 A7.
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 A7: Mesh Convergence analysis in aqeous region
Table A3: 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
13
Appendix B: Mathematical Statement of the Problem
Governing Equations and Boundary Conditions
𝑣𝑧
𝜕𝐶𝐶𝑂2
𝜕 2 𝐶𝐶𝑂2 1 𝜕𝐶𝐶𝑂2 𝜕 2 𝐶𝐶𝑂2
= 𝐷𝐶𝑂2 [
+
+
]
𝜕𝑧
𝜕𝑟 2
𝑟 𝜕𝑟
𝜕𝑧 2
Table B1: Boundary Conditions for SJTP Model
Boundary
Value
Boundary Condition
𝜕𝐶𝐶𝑂2
1
Symmetry
2
Flux
3
Concentration
4
Continuity
5
Flux
6
Flux
7
Continuity 6
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 B2: 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
6
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]
Concentration in shell was calculated using Henry’s Law Conversion 𝐶𝐶𝑂2 ,𝑚𝑒𝑚𝑏 =
𝐶𝐶𝑂2 ,𝑠ℎ𝑒𝑙𝑙
𝐻
for H = 0.8314 [1]
14
Appendix C: Solution Strategy
To solve the algebraic equations constructed by CMP, a direct UMFPACK method was used.
Figure C1: Mesh Used for Solution
Subdomain #1 Mesh Convergence
Table C1: S1 Mesh Convergence
Volume Integral
(mol)
4.92E-10
4.95E-10
4.97E-10
4.96E-10
4.96E-10
4.96E-10
4.98E-10
Volume Integral (mol)
Number of
Elements
242
245
307
353
619
1253
4.97E-10
4.96E-10
4.95E-10
4.94E-10
4.93E-10
4.92E-10
4.91E-10
0
500
1000
1500
Number of Elements
Figure C2: Mesh Convergence analysis
15
Table C2: S2 Mesh Convergence
Number of
Elements
Volume Integral
(mol)
210
608
1084
2408
5152
10406
5.68E-10
5.76E-10
5.80E-10
5.88E-10
5.91E-10
5.92E-10
Volume Integral (mol)
Subdomain #2 Mesh Convergence
5.95E-10
5.90E-10
5.85E-10
5.80E-10
5.75E-10
5.70E-10
5.65E-10
0
2000
4000
6000
8000
10000 12000
Number of Elements
Figure C3: Mesh Convergence Analysis
Table C3: S3 Mesh Convergence
Number of
Elements
Volume Integral
(mol)
2294
3920
4473
5348
8078
11326
4.98E-10
4.98E-10
4.97E-10
4.97E-10
4.97E-10
4.97E-10
Volume integral (mol)
Subdomain #3 Mesh Convergence
4.98E-10
4.98E-10
4.98E-10
4.98E-10
4.97E-10
4.97E-10
4.97E-10
4.97E-10
4.97E-10
4.97E-10
0
2000
4000
6000
8000 10000 12000
Number of Elements
Figure C4: Mesh Convergence Analysis
16
Appendix D: 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
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Final Presentation Outline
1. Introduction
a. Conventional Mass Transfer Equipment
b. Bioreactors
c. Hollow Fibber Membranes for Mass Transfer
2. Problem Formulation
a. Geometry
b. Assumptions
c. GE’s
d. BC’s
e. Properties
3. Problem Solution
a. Solver Type
b. Mesh
c. Validation
i. Analytical
ii. Numerical
4. Major Findings
a. Concentration Profiles
b. Velocity Profiles
c. Flux values
5. Conclusions
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