RPI Project - May11 - Rensselaer Hartford Campus

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Modeling and Optimization of Direct Contact Membrane Desalination
Water Purification Systems Using Computational Fluid Dynamic
Analysis
by
Jeremiah Blair Jones
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING
Major Subject: MECHANICAL ENGINEERING
Approved:
_________________________________________
Norberto Lemcoff, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
May 2015
1
© Copyright 2015
by
Jeremiah Blair Jones
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
NOMENCLATURE ......................................................................................................... vi
LIST OF KEYWORDS .................................................................................................. viii
ACKNOWLEDGMENT .................................................................................................. ix
ABSTRACT ...................................................................................................................... x
1. Introduction.................................................................................................................. 1
1.1
Background ........................................................................................................ 1
1.2
Problem Statement ............................................................................................. 5
2. Theory and Methodology ............................................................................................ 7
2.1
Theory ................................................................................................................ 7
2.1.1
2.2
Theoretical System ................................................................................. 7
Methodology ...................................................................................................... 9
2.2.1
Computational Fluid Dynamic Model ................................................... 9
2.2.2
Model Optimization ............................................................................. 17
3. Results and Discussion .............................................................................................. 19
3.1
Effect of System Flow Input Velocity ............................................................. 19
3.2
Effect of System Flow Inlet Temperatures ...................................................... 22
3.3
Effect of Flow Channel Length ........................................................................ 24
3.4
Effect of Membrane Porosity ........................................................................... 26
4. Conclusion ................................................................................................................. 29
5. References.................................................................................................................. 31
Appendix A. Seawater Fluid Property Equations ............................................................ 33
Appendix B. COMSOL Multiphysics 1D, 2D, 3D Plots ................................................. 34
Appendix C. Constant Water Vapor Diffusion Coefficient Error ................................... 46
iii
LIST OF TABLES
Table 1: Initial Boundary Conditions .............................................................................. 12
Table 2: Initial Material Properties .................................................................................. 13
Table 3: 2-D Model Meshing Boundary Edge Distributions........................................... 16
Table 4: Varying Inlet Velocity Only Results, Constant DF............................................ 19
Table 5: Varying Inlet Velocity Only Results, Varied DF ............................................... 21
Table 6: Varying Inlet Temperature ................................................................................ 23
Table 7: Varying Channel Length ................................................................................... 24
Table 8: Varying Membrane Porosity Only Results........................................................ 26
iv
LIST OF FIGURES
Figure 1: Osmotic and Reverse Osmotic Flow [4] ............................................................ 2
Figure 2: Residential Reverse Osmosis System [5] ........................................................... 4
Figure 3: Industrial Reverse Osmosis System [6] ............................................................. 4
Figure 4: Cylindrical, Spiral Wound Reverse Osmosis Membrane Pass [7] ..................... 7
Figure 5: Simplified Cross-Flow Desalination Membrane Pass [8] ................................. 8
Figure 6: Membrane Mass Flux Resulting from Vapor Pressure Difference .................... 9
Figure 7: Simplified 2-D Model Geometry .................................................................... 12
Figure 8: First Meshing Pattern Using Free Triangular ................................................... 15
Figure 9: Second Mesh using Distribution and Mapping ................................................ 17
Figure 10: Concentrate and Product Concentration as a Function of Inlet Velocity ....... 20
Figure 11: Comparison of Constant DF Results to Varying DF Results .......................... 22
Figure 12: Concentrate and Product Concentrations as a Function of Concentrate Inlet
Temperature ..................................................................................................................... 23
Figure 13: Concentrate and Product Concentrations as a Function of Length ................ 25
Figure 14: Concentrate and Product Concentration as a Function of Porosity................ 27
v
NOMENCLATURE
Symbols
cp
Constant Pressure Heat Capacity [J/(kg*K)]
ci
Concentration of species i (mol/m3)
Di F
Diffusion Coefficient of species i (m2/s)
F
Outside Body Forces (N)
h
Flow Channel height (m)
I
Identity Factor
ji
Molecular mass flux of species I [kg/(m2*s)]
k
Thermal Conductivity [W/(m*K)]
L
Flow channel length (m)
Mi
Molar mass of species i (kg/mol)
Ni
Concentration Flux of species i [mol/(m2*s)]
p
Pressure (Pa)
R
Universal Gas Constant [J/(mol*K)]
T
Temperature (K)
t
Thickness (m)
Q
Heat flux (W)
u
Velocity field (m/s)
vav
Average Inlet Velocity (m/s)
Greek Letters
ρ
Density (kg/m3)
μ
Dynamic viscosity (Pa*s)
θ
Membrane Porosity
ω
Mass fraction
Σ
Summation of
vi
Subscripts
0
Initial
co
Concentrate Stream
p
Porous membrane domain
pr
Product water Stream
w
Water species
vii
LIST OF KEYWORDS
KEYWORD
DEFINITION
CFD
Computational Fluid Dynamics
Cocurrent
Fluid flow condition which all flows travel
in parallel to each other with the same flow
direction.
Convection
Mechanism of heat transfer through liquids
and gases
Countercurrent
Fluid flow condition which all flows travel
in parallel to each other with opposite flow
directions.
Desalination
Process of removing dissolved salts from
water.
Diffusion
Process
by
which
molecules
are
transported as a result of their kinetic
energy of random motion.
FE
Finite Element
Porosity
Percentage of a solid which is open space
Reverse Osmosis
Process which utilizes a semi-permeable
membrane and an applied pressure greater
than the osmotic pressure to remove
dissolved particles from a solution.
Semi-permeable membrane
Membrane which allows the transport of
certain molecules/ions to pass through
while preventing the flow of other
molecules/ions.
viii
ACKNOWLEDGMENT
In the arduous process that was completing this project, there were many influences that
helped ensure success. The most notable of said influences would be my girlfriend
Hannah Frank. Whether it was providing moral support, ensuring that I was adequately
distracted on weekends, or unknowingly giving me something to compete against (e.g.
getting a graduate school GPA of 4.0 in Biomedical Engineering) Hannah was
unwaveringly always there.
Another much needed thanks goes to my professor and
academic advisor, Norberto Lemcoff. Without his continual advice, suggestions, and
sometimes much needed constructive criticism on my unorthodox methodology for
utilizing COMSOL Multiphysics, completion of this project would have with much
more difficulty. I would also like to thank my family for their constant support and my
friends and coworkers for the ongoing reminder that life outside of graduate school is a
lot more enjoyable than life in graduate school. Finally I must thank General Dynamics
Electric Boat (GD EB) for providing with the financial motivation to obtain my Masters
of Engineering degree. Without the GD EB academic reimbursement program, returning
to school would likely not have occurred.
ix
ABSTRACT
Water is a very valuable resource both with respect to the continued survival of humans
and to various industrial capabilities. Although water happens to be one of the most
abundant resources, the amount of clean freshwater available for human consumption
and industrial uses is limited. One process which was developed to remedy the limited
supply of freshwater is desalination. This process removes salt and other elements from
seawater thus producing freshwater. In order to keep up with the continually increasing
demand for clean freshwater, the methods and processes available to convert the more
abundant, but less useable, seawater to freshwater must be evaluated and optimized.
This project used a finite element model and computational fluid dynamic analysis to
evaluate a concurrent two-dimensional direct contact membrane desalination system to
determine what system characteristics and operating conditions could be changed to
optimize the system performance. The results of the evaluations performed indicated
that by increasing channel length, increasing porosity, increasing the inlet temperature of
the concentrate stream, or decreasing the inlet velocity condition the salt concentration in
the product water leaving the system could be decreased. The results obtained along
with the literature review conducted in support of the modeling and evaluations also
showed that limitations exist with respect to desalination system optimization. Each
change to the system parameters or characteristics has the potential to affect the
influence of other system characteristics and could ultimately results in a worse system
performance. With this in mind, it is necessary to ensure that the designs and design
considerations used for desalination systems are robust and highly vetted.
x
1. Introduction
1.1 Background
Water is a very important resource and is used for a variety of applications, most notably
sustaining human life. Other applications which utilize water are hydroelectric plants,
nuclear power plants, heating and cooling systems. Even though water is very abundant,
covering approximately 70% of earth’s surface [1], not all of it is able to be readily
utilized. Water’s inability to be directly utilized is a result of the water containing
undesirable ions and compounds. The most common compound found in the water is
Sodium Chloride, or salt. Approximately 97% of the water on earth is considered
seawater, i.e. having a large concentration of salt [2]. The large concentration of salt in
seawater makes it unsafe for human consumption and no longer suitable for applications
which require pure water (e.g. nuclear power plants). In order to combat the continuing
need for purer water, various methods have been developed so that contaminants can be
removed from the water. One of the first uses of such a method dates back to ancient
Egyptian cultures, where a painting on a tomb wall depicts what appears to be siphoning
of liquids using sedimentation [3]. Since then, significant improvements have been
made in the water purification industry. In particular, much advancement has been made
in the capability of removing salt, or salinity, from water. This process is referred to as
desalination, and is often accomplished using one of the following three methods:
distillation, direct contact membrane microfiltration (e.g. reverse osmosis), or
electrodialysis.
Osmosis is a naturally occurring process whenever two solutions with different
concentrations of solute are separated by a semi-permeable membrane1. The difference
in concentration causes the solvent to travel from the solution with the lower solute
concentration, through the semi-permeable membrane, into the solution with higher
concentration of solute.
This process continues until the concentration of the two
solutions has equalized. To counter osmotic flow, some pressure must be applied to the
higher concentration solution in order to prevent pure solvent from going through the
1
A semi-permeable membrane is one which allows the flow of solvent but not the flow of solute.
1
semi-permeable membrane separating the two liquids; this is known as the osmotic
pressure. If the pressure is increased above the osmotic pressure the solvent will pass
from the solution with higher concentration, back through the membrane, and into the
solution with lower concentration. This process is called reverse osmosis and can be
utilized to purify a solution.
Figure 1 depicts side by side the osmosis, osmotic
equilibrium, and the reverse osmosis process. In both the osmosis and reverse osmosis
frames the white arrows indicate the flow of solvent.
The desalination process is
primarily driven by the pressure applied to the system, as indicated by the metallic
plunger in the “Reverse Osmosis” picture in Figure 1, and the difference in solution
concentrations.
Figure 1: Osmotic and Reverse Osmotic Flow [4]
Another method of accomplishing direct contact membrane desalination is with two
solutions of different temperature and concentration flowing in parallel, separated by a
semi-permeable membrane.
Solvent will diffuse from one solution, across the
membrane, into the other solution. The solute is restricted from diffusing across the
membrane. Similarly to reverse osmosis, differences in concentration and pressure are
the driving force of this process.
However, unlike reverse osmosis which uses
mechanical means to apply a backpressure to drive the desalination, this process utilizes
the difference in temperature which results in a vapor pressure gradient which ultimately
drives solvent through the membrane.
To accomplish these purifications, a continuous stream of a solution is passed through a
module containing a semipermeable membrane. As the characteristic pressure of the
2
system is increased, the membrane allows solvent to pass through. Both the solvent and
the now higher concentration solution are extracted from the module containing the
membrane, thereby the process can continuously produce purer water. Most commonly
this is used to remove impurities, such as fluorides or heavy metals, from drinking water
and to reduce the salinity of seawater to produce potable water aboard naval vessels.
This is a very useful application as it allows for naval vessels to utilize the vast amount
of seawater surrounding them to produce water for drinking, showering, dishwashing,
etc.
Direct contact desalination systems can vary significantly in size and complexity.
Systems used in residential applications tend to be fairly small and compact, due to the
size constraints that exist in residential property. Figure 2 shows an example of a
household desalination system which utilizes reverse osmosis. These systems generally
use one membrane accompanied by multiple inline filters (to remove sediments and
chlorine from incoming water which could damage the membrane) and a pressurized
storage tank (to account for fluctuations in demand). Based on the smaller size, these
systems generally are capable of removing up to 1,500 to 1,800 ppm of TDS (Total
Dissolved Solids). Industrial applications are usually larger and more complex. Figure
3 shows an example of a membrane skid of a reverse osmosis system. Generally
industrial applications also have a water pre-treatment skid to remove solutes or
sediment that may be harmful to the membranes (e.g. activated charcoal filter). The pretreatment skid may also utilize heaters/chillers to manipulate the temperature of the
incoming water. One important point that should be mentioned is that the industrial
system utilizes many membranes to accomplish the purification.
This is usually
necessary as industrial applications have stricter product water requirements than those
in household applications, or the incoming water contains more TDS than in household
applications. As a result, the industrial desalination systems are costly to operate and
maintain. Improvements in desalination membrane and/or system performance can help
reduce these costs.
3
Figure 2: Residential Reverse Osmosis System [5]
Figure 3: Industrial Reverse Osmosis System [6]
4
1.2 Problem Statement
Although this is a very useful process, there are various factors which can affect the
performance of a direct contact membrane desalination system. An increase in the
pressure applied to the membranes or solution flow will result in an improved
performance (product solution concentration will decrease, and rate at which product
solution is created will increase). Depending on the solution being purified, fouling of
the membranes can occur, which will negatively affect performance. In residential
applications, where water is not as abundant as in naval vessel applications (surrounded
by seawater), an issue can arise since significant amounts of reject water are required to
produce useable amounts of product water. This can drain available resources and/or be
very costly (e.g. running a well dry, significant electrical cost to have the desalination
system constantly running). In order to prevent adverse effects, improvements can be
made to the desalination system design.
Although naval vessels have an abundant source of water, the high salt concentration of
the ocean requires the water be processed through many membranes to produce
acceptable product water. The introduction of more membranes results in an increase in
both manufacturing and maintenance costs. If, however, each of the membranes/passes
were designed to be more efficient there would no longer be a need for as many
membranes/passes. Possible improvements in membrane/pass design are increasing the
pressure of the system, choosing a better membrane material such as a composite, or
changing the incoming or product water flow characteristics.
Another problem that occurs with the use of desalination systems is the development of
waste. The more efficient the process is, the more concentrated the solution being
rejected by the membrane becomes. This can create problems with the disposal of said
concentrated output. In the example of a residential reverse osmosis system being used
to remove total dissolved solids from drinking water, if the water rejected by the
membrane becomes too concentrated with TDS it may not be allowed, per government
regulations, to be disposed of in sewer/septic lines. This project plans to focus on the
performance of the direct contact desalination membranes. Also, a follow-up study
5
could analyze the waste created by more efficient systems, and how this waste can be
handled/diluted in accordance with regulations.
6
2. Theory and Methodology
This project will analyze the performance of a single membrane direct contact
desalination system. Most commercially available direct contact desalination systems
utilize a single membrane pass. Although larger systems use multiple membrane passes
in series, investigating a single membrane could easily be extrapolated to determine the
larger system performance results. Therefore the results of modeling a single pass can
be utilized in the design and operation of a wide variety of direct contact membrane
desalination systems.
2.1 Theory
2.1.1
Theoretical System
One conventional desalination system is a reverse osmosis system utilizing spiral wound
membrane configurations to filter salt out of water. Figure 4 depicts a partially extruded
membrane pass. The salt water flows in through the feed channel into the membrane
assembly. Backpressure applied to the system causes pure water to pass through the
semi-permeable membrane into the permeate carrier (a spiral wound channel which is
wrapped in between layers of the membrane). The permeate carrier allows the product
water to travel from the outside into the center where a “punctured” product water tube
collects and flows the product water out. Simultaneously, the salt that is rejected by the
membrane causes an increase in concentration in the feed stream which is ultimately
pushed through the feed channels and leaves the system as concentrate.
Figure 4: Cylindrical, Spiral Wound Reverse Osmosis Membrane Pass [7]
7
Although the spiral wound membrane configuration is commonly used for a wide range
of applications, to accommodate the time constraints of this project a simpler model
ultimately had to be chosen. To accomplish this, a direct contact membrane desalination
model was chosen (Figure 5). In this model, seawater (feed) is introduced on one side of
the system and flows across the semi-permeable membrane. Concurrently, a second
stream is introduced to flow in parallel to the feed stream across the other side of the
membrane (product/permeate). The driving mechanism for mass transfer in this system
is the difference in vapor pressure between the feed and permeate streams. Since the
vapor pressure is a function of the stream temperature, the ultimate driving force of the
mass transfer becomes the temperature difference between the feed and permeate stream.
With a sufficient temperature difference, water is transferred across the membrane while
salt is rejected. This process is depicted in Figure 6. As may be evident by comparing
the system products, this simplified model has strong correlation to the spiral wound
reverse osmosis filtration system.
Figure 5: Simplified Cross-Flow Desalination Membrane Pass [8]
8
Mass Flux
Hot/Concentrate
Cold/Product
Stream
Stream
Figure 6: Membrane Mass Flux Resulting from Vapor Pressure Difference
2.2 Methodology
2.2.1
Computational Fluid Dynamic Model
Computational Fluid Dynamics (CFD) is a method of analysis which utilizes finite
element models to solve problems relating to the flow of fluids. CFD can be used to
verify analytical results in basic situations or tackle more complicated problems that
cannot be solved analytically. The CFD program used to accomplish this project was
COMSOL Multiphysics.
In order to utilize CFD, first the physics and FE (Finite
Element) models need to be created. Initially a two-dimensional model is created.
2.2.1.1 CFD Physics Models
In order to develop an accurate representation of the desalination process, three existing
physics models contained in COMSOL had to be used simultaneously.
The three
physics models were Laminar Single-Phase Fluid Flow, Heat Transfer in Fluids and
Transport of Diluted Species. The Laminar Single-Phase Fluid Flow physics module is
used to model the solution flows through the system.
Since the mass transport
mechanism is a result of the difference in water vapor pressures, fluid flow is assumed to
9
only occur in the two solution channels (i.e. no fluid flow through the membrane). As a
result, the Laminar Flow physics module only applies to the solution channel domains.
For the flow of the solutions, the module utilizes the continuity and Navier Stokes
equations for incompressible flow, shown in Equations 1 and 2.
𝜌∇ ∗ 𝒖 = 0
(1)
𝜌 ∗ (𝒖 ∗ ∇)𝒖 = ∇ ∗ [−𝑝 ∗ 𝑰 + 𝜇 ∗ (∇𝒖 + (∇𝐮)𝑇 )] + 𝐹
(2)
The Transport of Diluted Species physics module is used to model the mass transfer of
diluted species through the membrane.
Since water is being transferred from the
“concentrate” stream to the “product” stream, water was chosen as the “diluted species”,
and therefore all of the model concentrations refer to water. The mass transfer of the
diluted species utilizes Equations 3 and 4:
∇ ∗ (−𝐷𝑖 ∇𝑐𝑖 ) + 𝒖 ∗ ∇𝑐𝑖 = 𝑅𝑖
(3)
𝑁𝑖 = (−𝐷𝑖 ∇𝑐𝑖 ) + 𝒖𝑐𝑖
(4)
These equations model the flow of water in the fluid phase through the porous
membrane as well as the dilution of water in the concentrate and product streams. As
mentioned before, the primary transfer mechanism for water in direct contact membrane
distillation is in the vapor phase. Therefore, a flux discontinuity had to be created at
each side of the membrane domain to account for the transfer of water vapor from the
concentrated stream to the product stream. The flux discontinuity is modeled using
Equations 5, 6, and 7:
𝑁𝑖 =
𝐹
(𝐷𝑣𝑎𝑝𝑜𝑟,𝑖
∗𝜃2 )
𝑅∗𝑇𝑎𝑣𝑒
∗ (𝑃𝑣𝑎𝑝𝑜𝑟,2 − 𝑃𝑣𝑎𝑝𝑜𝑟,1 )
(5)
Where,
𝑃𝑣𝑎𝑝𝑜𝑟,𝑖 = 133.3 ∗
𝑒
5132
20.386−
𝑇
[9]
(6)
𝐹
𝐷𝑣𝑎𝑝𝑜𝑟,𝑖
= −2.775𝑒 −6 + 4.479𝑒 −8 ∗ 𝑇 + 1.656𝑒 −10 ∗ 𝑇 2 [10]
(7)
ℎ𝑚
10
Equation 5 is developed integrating Equation 8, which represents the flux through a
porous solid according to Fick’s Law, over the length of the membrane, and using
Equation 9 for the effective diffusivity of a porous membrane [11].
𝑑𝑐
𝐷
𝑑𝑝
𝑁𝑖 = −𝐷𝑒 ∗ 𝑑𝑥 = − 𝑅∗𝑇𝑒 ∗ 𝑑𝑥
(8)
𝐷𝑒 = 𝐷𝐴,𝐵 ∗ 𝜃 2
(9)
𝑎𝑣𝑒
It is important to note the dependency of the mass transport on only the temperatures of
the domains. Therefore, changes in inlet concentrations will not affect the amount of
mass flux that occurs at the membrane boundaries.
In order to account for the temperature effects on the mass transfer, the Heat Transfer in
Fluids physics module was used to model the thermal behavior of the three domains
throughout the process. The heat transfer module utilizes the conservation of energy for
conductive and convective heat transfer, as shown in Equation 10.
𝜌 ∗ 𝑐𝑝 ∗ 𝒖 ∗ ∇𝑇 = ∇ ∗ (𝑘𝑒𝑞 ∗ ∇𝑇) + 𝑄
(10)
After choosing the physics models in COMSOL, the geometry and initial conditions
were defined. For this model, two 0.2 meter by 0.001 meter flow channels were stacked
on top of each other, separated by a 0.0001 meter by 0.2 meter membrane, as shown in
Figure 7.
11
10
9
7
8
5
6
4
1
3
2
Figure 7: Simplified 2-D Model Geometry
These dimensions were chosen for the initial model to satisfy the condition that the
length of the flow channels be significantly larger than the thickness of the membrane
layer (in this model, the length of the flow channels is 2000 times larger than the
membrane thickness). Once the geometry was determined and coupled appropriately,
the boundary conditions and material properties were defined, as shown in Table 1 and
Table 2.
Table 1: Initial Boundary Conditions
Boundary Label
Parameter Name
Parameter Value/Condition
Normal Inflow Velocity = .1 m/s,
1
Concentrated Inlet
Ti = 313 K,
C0,w=55,000 mol/m3
2
Wall
No-Slip
3
Concentrate Outlet
4
Wall
No-Slip, Flux Discontinuity
5
Wall
No-Slip
6
Wall
No-Slip
Pressure = 0 Pa,
Outlet heat flux
12
Normal Inflow Velocity = .1 m/s,
7
Diluted Inlet
Ti = 293 K,
C0,w = 55,000 mol/m3
8
Wall
9
Diluted Outlet
10
Wall
No-Slip, Flux Discontinuity
Pressure = 0 Pa,
Outlet heat flux
No-Slip
Table 2: Initial Material Properties
Material Property
Value
Porous Medium Porosity
0.83 [12]
Membrane Thickness, t
0.0001 m
Molar Mass, Water
0.018 kg/mol
Molar Mass, Salt
0.058 kg/mol
Diffusion Coefficient, Concentrate, Water
2.68 10-5 m2/s
Diffusion Coefficient, Product, Water
2.484 10-5 m2/s
Diffusion Coefficient, Membrane, Water
10-12 m2/s
Average Diffusion Coefficient, Water
Vapor
2.6017 10-5 m2/s
Average Temperature for Concentration
Flux
303.5 K
The inlet concentration value for both the concentrated and product streams was
assumed to be that of the incoming seawater, which has a salinity of approximately 3.5%
and contains numerous ions other than sodium and chloride [13]. Therefore, an inlet
water species concentration of 55,000 mol/m3 was chosen. The inlet flow condition and
concentrate outlet pressure condition were chosen to allow the flow to pass through the
system without restriction, ultimately isolating the driving mechanism of the desalination
process to the temperature gradient. The membrane porosity for the base case was
chosen to coincide with previous research done for direct contact membrane distillation
13
processes [12]. This porosity value will be varied in this project to determine its effect
on the system performance and efficiency.
When calculating the concentration flux, a constant water vapor diffusion coefficient and
average temperature were assumed.
The water vapor diffusion coefficient was
determined by calculating the average of the diffusion coefficient at the inlet
temperatures of the hot and cold streams. The assumed average temperature is simply
the average of the two inlet temperatures. These assumptions were made to simplify the
equations that were simultaneously solved by COMSOL Multiphysics, therefore
decreasing the run time of the program. The constant average temperature assumption
was validated calculating the average temperature for the membrane domain after each
run.
The constant water vapor diffusion coefficient assumption was validated by
carrying out one set of runs without a constant coefficient and comparing the results.
The values for the density, dynamic viscosity, thermal conductivity, and specific heat
capacity all depend on both temperature and salt concentration. The equations used to
determine these values are shown in Appendix A and derived from Sharqawy et al. [14].
Since the Appendix A equations are dependent on salt concentration, the model output
water concentration has to be converted to be used as an input. To do so, Equations 11,
12 and 13 below were used.
𝜌𝑤 = 838.466 + 1.40051 ∗ 𝑇 − 3.01121𝑒 −3 ∗ 𝑇 2 + 3.71821𝑒 −7 ∗ 𝑇 3
𝜌
𝑐𝑠𝑤 = (𝑀𝑤 − 𝑐)
(11)
(12)
𝑤
𝑀
𝑠𝑎𝑙𝑡
𝑆𝑠𝑤 = 𝑐𝑠𝑤 ∗ (1000∗𝜌
)
(13)
𝑠𝑤
Equation 11 calculates the density of pure water as a function of temperature. Equation
12 converts this calculated pure water density to a concentration by dividing by the
molar mass of water, and then subtracting the concentration being solved for as part of
the evaluation. c is the dependent variable of the Transport of Diluted Species physics
module. Since the density of the seawater solution is being calculated simultaneously
14
with Equation 12, which is also dependent on the concentration, a density value of 1.025
kg/L was chosen to reduce computational complexity and model run time.
2.2.1.2 CFD Model Meshing
Before the aforementioned calculations could be performed, a mesh had to be created for
each of the three domains (two flow channels and one membrane layer).
Since
COMSOL Multiphysics allows for a variety of meshing patterns, two meshes were
created and then compared to determine the more accurate FE model. The first mesh
utilizes the free triangular meshing function with a sizing calibrated for general physics
of normal sizing. This meshing pattern results in a total of 24,249 meshing elements.
Figure 8 is a zoomed in picture of the first mesh.
Figure 8: First Meshing Pattern Using Free Triangular
The second mesh utilizes the distribution function of COMSOL Multiphysics. This
function allows the user to input how many rows of meshing elements each boundary
should be broken into.
The user can then use the mapping feature of COMSOL
15
Multiphysics to extrude the mesh across the remainder of the domain. Table 3 shows the
distribution used for each boundary edge of the model. The boundary edge numbers in
the table correspond to Figure 7. Since the flow channels are of matching geometries,
they can both utilize the same mesh. However, since the membrane layer is significantly
thinner than the flow channels, a finer mesh had to be used.
Table 3: 2-D Model Meshing Boundary Edge Distributions
Boundary Edge
Distribution Value
1
10
2
200
3
10
4
200
5
10
6
10
7
10
8
200
9
10
10
200
Figure 9 is a zoomed in picture of the 2-D model and meshing described above. These
meshing patterns result in a meshing element total of 6,000, all of which are
quadrilateral elements.
16
Figure 9: Second Mesh using Distribution and Mapping
Although the second mesh with quadrilateral elements utilizes a mesh which is finer in
the vertical direction for the membrane domain, the first mesh with triangular elements
utilizes a mesh finer in the horizontal direction for all three domains near the membrane
boundaries. Since the model uses flux discontinuities at the top and bottom membrane
boundaries, the number of meshing elements within the membrane domain is not as
significant as the number of meshing elements used for the flow channels near the
membrane boundaries. Therefore, the meshing with triangular elements was chosen.
2.2.2
Model Optimization
Once a baseline model was established, the model parameters were varied to optimize
the system performance. The system performance was based on the outlet concentration
for both the concentrate and product channels. The parameters chosen for this project to
be varied in the optimization process were the overall channel length, channel average
velocity inlet condition, membrane porosity and inlet temperature conditions.
To
accomplish the optimization, all of the aforementioned parameters were held constant
except for one. This one parameter was varied and the resulting system performance
17
was tabulated. This process was then repeated for each of the optimization parameters.
Since the species being transferred from the hot concentrate stream to the colder product
stream was water, it is necessary to verify that the product stream concentration is
maintained below that of pure water at the specified inlet temperature (i.e. 293 K results
in a pure water concentration of 55,536.41 mol/m3). Any solutions with outlet product
concentrations higher than this value were either discarded or included and prefaced
appropriately for discussion purposes.
Another model limitation was the inlet temperature of the hot stream. If the temperature
is too high, the membrane performance and integrity can begin to diminish. In order to
prevent this, most systems maintain a very tight control of both the upper and lower
temperature limits. In this project, a maximum temperature of 54.85 °C and a minimum
temperature of 19.85 °C were assumed. These assumptions are consistent with previous
research of different desalination membrane systems [15].
18
3. Results and Discussion
The results from the optimization runs described above were compiled in tables and
plots to better show trends and general system behavior as a result of each optimization
trial. These tables and plots are included and discussed below. It should be pointed out
that all the concentration values indicated below are of water. Therefore an increase in
resultant product stream concentration refers to a decrease in the salt concentration.
Appendix B contains the 1D, 2D, and simulated 3D plots created by COMSOL
Multiphysics for the case with a channel length of 0.1 m, channel height of 0.001 m, a
membrane thickness of 0.0001 m, an initial velocity of 0.25 m/s, and inlet temperatures
of 293 K to 328 K. These plots depict the concentration, temperature, pressure and
velocity profiles throughout the model. These plots are intended to be visual aids to the
data included and discussed herein.
3.1 Effect of System Flow Input Velocity
Table 4 shows the results when the inlet velocity condition is varied while maintaining
constant the channel length, inlet temperatures, and membrane porosity for a channel
height of 0.001 m. Figure 10 is a graphical representation of the data contained in Table
4.
Table 4: Varying Inlet Velocity Only Results, Constant DF
Inlet Velocity Effect (L = 0.2, h = 0.001, hp = 0.0001)
vavg =
vavg =
vavg =
vavg =
0.05
0.1
0.15
0.2
305.31
307.45
308.54
309.19
Outlet Temp (K)
Concentrate Outlet Conc
54,579
54,788
54,858
54,893
(mol/m3)
300.67
298.45
297.33
296.66
Outlet Temp (K)
Product
Outlet Conc
55,421
55,212
55,142
55,107
(mol/m3)
19
vavg =
0.25
309.63
54,914
296.23
55,086
Case 1 - Varying Inlet Velocity
55500
Outlet Concentration (mol/m3)
55400
55300
55200
55100
55000
Concentrate
54900
Product
54800
54700
54600
54500
0
0.05
0.1
0.15
0.2
0.25
0.3
Velocity (m/s)
Figure 10: Concentrate and Product Concentration as a Function of Inlet Velocity
As is evident by analyzing the curves shown on Figure 10 and comparing sequential
columns in Table 4, as the velocity is increased the system performance is decreased. As
the velocity increases, the product stream outlet concentration decreases and that of the
concentrate increases. Similarly, as the velocity increases, the outlet temperature for the
product stream decreases and the outlet temperature for the concentrate stream
decreases. This behavior is to be expected. As the velocity increases, the residence time
decreases.
Therefore, further increases in velocity will drive the system outlet
characteristics to approach the inlet conditions, as the relative amount of fluid capable of
being desalinated goes to zero.
Figure 10 shows that, as velocity increases, both
concentrate and product stream concentrations approach 55,000 mol/m3, the inlet
concentration value.
It is important to note that although decreasing velocity in this case resulted in improved
system performance, there is a limit to how much the velocity can be lowered. This
limit is dependent on other system characteristics such as channel length and inlet initial
20
temperatures. What should be avoided are the concentrate and product stream outlet
temperatures reaching equilibrium. When this temperature equilibrium is reached, the
system no longer has the ability to force more pure water from the concentrate stream
(i.e. the system loses the vapor pressure differential driving force). This can occur with
low velocities in long channels and systems with an initial small temperature difference
between concentrate and product streams.
As discussed above, the data shown in Table 4 and Figure 10 was calculated assuming
the diffusion coefficient of water vapor was constant throughout the process.
To
confirm this assumption the same evaluation was performed using Equation 7 to
calculate the diffusion coefficient based on temperature with each calculation. The data
collected during this evaluation is summarized in Table 5. The concentrations from
Table 5 as well as the constant diffusion coefficient concentrations from Table 4 are
plotted concurrently in Figure 11.
Table 5: Varying Inlet Velocity Only Results, Varied DF
Inlet Velocity Effect (L = 0.2, h = 0.001, hp = 0.0001)
vavg =
vavg =
vavg =
vavg =
0.05
0.1
0.15
0.2
305.31
307.45
308.54
309.19
Outlet Temp (K)
Concentrate Outlet Conc
54,578
54,788
54,858
54,893
(mol/m3)
300.67
298.45
297.33
296.66
Outlet Temp (K)
Product
Outlet Conc
55,419
55,210
55,141
55,106
(mol/m3)
21
vavg =
0.25
309.63
54,914
296.23
55,085
Case 1 - Varying Inlet Velocity
55,500
Outlet Concentration (mol/m3)
55,400
Concentrate Varied Df
55,300
55,200
55,100
Product - Varied
Df
55,000
54,900
Concentrate Const. Df
54,800
54,700
Product - Const.
Df
54,600
54,500
0
0.05
0.1
0.15
0.2
0.25
0.3
Velocity (m/s)
Figure 11: Comparison of Constant DF Results to Varying DF Results
The comparison of results shown in Figure 11 confirms the validity of assuming a
constant water vapor diffusion coefficient calculated based on the two inlet conditions.
The difference between results at each data point is very small. The largest error
between Table 4 and Table 5 values is 0.00902%, which corresponds to the outlet
concentration for the product stream at a velocity of 0.05 m/s. Appendix C contains the
tabulated errors for each run shown in Table 4 and Table 5.
3.2 Effect of System Flow Inlet Temperatures
Table 6 shows the results when the inlet temperature conditions are varied while
maintaining constant the channel length, membrane porosity, and inlet velocity. Table 6
shows the results for a channel length of 0.2 m, membrane porosity of 0.45, and inlet
velocity of 0.25 m/s. Figure 12 is a graphical representation of the data contained in
Table 6.
22
Table 6: Varying Inlet Temperature
Inlet Temperature Effect (L = 0.2, h = 0.001, hp = 0.0001, vavg = 0.25)
Tco=313,
Tco=318,
Tco=323,
Tco=328,
Tpr=293
Tpr=293
Tpr=293
Tpr=275
Outlet Temp
309.63
313.74
317.84
321.96
(K)
Concentrate
Outlet Conc
54,914
54,876
54,828
54,770
(mol/m3)
Outlet Temp
296.23
297.03
297.84
298.65
(K)
Product
Outlet Conc
55,086
55,124
55,172
55,229
(mol/m3)
Case 2 - Varying Inlet Hot Temperature
Outlet Concentration (mol/m3)
55,300
55,200
55,100
55,000
Concentrate
Product
54,900
54,800
54,700
310
315
320
325
330
Inlet Hot Temperature (K)
Figure 12: Concentrate and Product Concentrations as a Function of Concentrate
Inlet Temperature
As is evident by comparing all four columns in Table 6, an increase in the inlet
temperature of the concentrate stream, while maintaining the product water stream inlet
temperature constant, results in improved system performance.
For each run, the
increase in inlet temperature results in a decrease in concentrate stream outlet
concentration as well as an increase in product water outlet concentration (lower
23
salinity).
By increasing the concentrate inlet temperature, the pressure differential
within the membrane is increased, therefore the driving force for the water vapor
through the membrane is increased causing more of the water vapor to flow through.
3.3 Effect of Flow Channel Length
Table 7 shows the results when the channel length is varied while maintaining constant
the membrane porosity, inlet temperature condition and inlet velocity. Table 6 shows
the results for membrane porosity of 0.45, inlet velocity of 0.25 m/s, and inlet
temperature difference of 293 K to 328 K. Figure 13 is a graphical representation of the
data contained in Table 7.
Table 7: Varying Channel Length
Channel Length Effect (h = 0.001, hp = 0.0001, vavg = 0.25)
L=
L = 0.1
L = 0.15 L = 0.2
0.225
323.84
323.53
321.96 321.73
Outlet Temp (K)
Concentrate Outlet Conc
54,883
54,822
54,770 54,742
(mol/m3)
296.94
297.98
298.65 299.10
Outlet Temp (K)
Product
Outlet Conc
55,117
55,173
55,229 55,258
(mol/m3)
24
L = 0.25
321.2992
54,714
299.47
55,286
Case 3 - Varying Channel Length
Outlet Concentration (mol/m3)
55,400
55,300
55,200
55,100
55,000
Concentrate
54,900
Product
54,800
54,700
54,600
0
0.05
0.1
0.15
0.2
0.25
0.3
Channel Length (m)
Figure 13: Concentrate and Product Concentrations as a Function of Length
The results in Table 7 show that as the length of the flow channel increases, the product
water outlet concentration increases and the concentrate water concentration decreases.
Intuitively these results make sense. By increasing the length of the channel, the length
of the membrane and therefore the area of filtration increase. An increase in filtration
area allows for more salt to be rejected, decreasing the concentrate water outlet
concentration, and more product water to be produced, increasing the product water
outlet concentration.
Similar to the limitations when reducing the system inlet velocity, limitations exist in the
effectiveness when increasing the channel/membrane length. As the channel length is
increased further, the increases in performance will start to diminish until further
increases result in no improvements at all. The physical meaning of this stabilization of
product water outlet concentration is that for the given parameters the output of the
membrane has been maximized. This maximization occurs when the concentrate stream
and product stream temperatures reach equilibrium. When the temperature difference no
longer exists, the system lacks a driving force for the water vapor to be transported
25
through the membrane. Therefore, if the membrane length was increased further the
product water concentration will remain constant.
3.4 Effect of Membrane Porosity
Table 8 shows the results when the membrane porosity is varied while maintaining
constant the channel length, inlet temperature and inlet velocity. Table 8 shows the
results for a channel length of 0.2 m, inlet velocity of 0.25 m/s and inlet temperatures of
293 K and 328 K. Figure 14 is a graphical representation of the data contained in Table
8.
Table 8: Varying Membrane Porosity Only Results
Membrane Porosity Effect (L = 0.2, h = 0.001, hp = 0.0001, vavg = 0.25)
θ=
θ=
θ=
θ=
θ=
0.25
0.35
0.45
0.55
0.65
Outlet Temp
321.96 321.96 321.96 321.97 321.97
(K)
Concentrate
Outlet Conc
54,929 54,861 54,770 54,657 54,521
(mol/m3)
Outlet Temp
298.64 298.64 298.65 298.66 298.67
(K)
Product
Outlet Conc
55,071 55,139 55,229 55,343 55,479
(mol/m3)
26
θ=
0.83
321.98
54,219
298.69
55,780
Outlet Concentration (mol/m3)
Case 4 - Varying Membrane Porosity
56,000
55,800
55,600
55,400
55,200
55,000
54,800
54,600
54,400
54,200
54,000
Concentrate
Product
0
0.2
0.4
0.6
0.8
1
Porosity
Figure 14: Concentrate and Product Concentration as a Function of Porosity
As stated in the initial conditions above, the first porosity that was modeled was θ =
0.83. However, the resulting product outlet concentration was 55,780 mol/m3, which is
greater than the pure water concentration limit of 55,536 mol/m3, and therefore not a
valid data point. For this reason, the data point associated with θ = 0.83 on Figure 14
and the line leading from the next valid run is shown as dotted. The remaining four runs
for this case resulted in valid product outlet concentrations. These four runs indicate that
as the membrane porosity is decreased, the system performance also decreases. The
porosity of the membrane is the percent of the membrane which is open space.
Therefore by decreasing the porosity of the membrane, the pores and open space within
the membrane layer decrease which decreases the ability of the water vapor to pass
through the membrane. As the porosity approaches zero, the amount of water vapor
capable of passing across the membrane layer also goes to zero. Although the opposite
is true, that increasing porosity will allow more water vapor to pass across the membrane
layer, increasing porosity too much could also allow salt and seawater to flow across the
membrane more readily. It is important to note though that the ability of salt and
seawater to flow across the membrane is not only a function of porosity.
Other
membrane characteristics also have an effect on the overall mass transfer, e.g. the
membrane pore size and membrane permeability. Based on the limited scope of this
27
project, the increase in capability for salt and seawater to flow across the membrane as
porosity is increased was not modeled.
28
4. Conclusion
The purpose of this project was to model a direct contact membrane desalination system
and determine how said model could be optimized. This was accomplished by varying
initial parameters or system characteristics such as channel length, inlet velocity, inlet
initial temperatures, and membrane porosity. As discussed above, an improvement in
the system performance can be realized with specific changes to each of the
aforementioned parameters. The following changes resulted in increased performance
(i.e. increase in the product water outlet concentration):
-
Increasing channel length
-
Increasing membrane porosity
-
Increasing the inlet concentrate temperature while maintaining the inlet product
temperature constant
-
Decreasing inlet concentrate and product velocities
Although initially it may seem possible to just increase or decrease one or more of the
above parameters until the desired results are obtained, this is not always the case. Due
to system performance limitations, which were not all capable of being captured by this
project, some parameters are only capable of being optimized a certain amount before
either no further increases in performance are noted or negative performance effects are
realized. Also, it is important to note that changes to any of the four parameters noted
above could have an impact, either positive or negative, on one of the other four. For
example, significantly increasing the channel length could result in a necessary increase
in inlet velocities to prevent reaching the equilibrium scenario briefly discussed in Case
1. This interdependency of all of the system parameters makes the system design a very
involved process.
As such, it is very important that all parameters, even those not
discussed and evaluated herein, to be modeled and considered. It is also important to
consider all of the system limitations during the design phase.
Based on the timeframe in which this project had to be completed in and the resources
available during the completion, the scope of this project had to be limited. Therefore,
there are still beneficial evaluations which can be performed to improve the design and
29
performance of desalination systems. The following serves as a summary of such follow
up evaluations which can be performed:
-
Evaluating cases where the concentrate and product inlet velocities were not
equal to each other.
-
Evaluating counter-current flow cases.
-
Determining appropriate modeling conditions that would allow an increase of
salt and seawater to permeate the membrane with an increase of membrane
porosity.
-
Modeling a three-dimensional desalination system and compare to the twodimensional results obtained herein.
The three-dimensional model could
evaluate a rectangular channel and compare that to the results of a cylindrical
model.
30
5. References
[1] V. I. Grover, Water: Global Common and Global Problems, Enfield, NH: Science
Publishers, 2006.
[2] I. Shiklomanov, "World Fresh Water Resources," in Water in Crisis: A Guide to the
World's Fresh Water Resources, P. H. Gleick, Ed., New York, Oxford University
Press, 1993.
[3] M. N. Baker, The Quest for Pure Water, New York: The American Water Works,
Inc., 1948.
[4] ESP Water Products, "Water Filtration and Purification Products; Reverse Osmosis
Systems," 2009. [Online]. Available: http://espwaterproducts.com/reverse-osmosissystem.htm. [Accessed 18 February 2015].
[5] Water King, "Water King's Genesis Reverse Osmosis System," 2012. [Online].
Available:
http://www.waterkingwater.com/el_paso_reverse_osmosis.htm.
[Accessed 16 April 2015].
[6] Degrémont Technologies Ltd., "REVERSE OSMOSIS SKIDS," Suez Enviroment,
2015.
[Online].
Available:
http://www.degremont-
technologies.com/dgtech.php?article458. [Accessed 16 April 2015].
[7] The Purchase Advantage, "Tory RO Membranes; TM-Series Element," 2015.
[Online]. [Accessed 18 February 2015].
[8] General Electric, Cross Flow Filtration Method Handbook, Piscataway: General
Electric Company, 2014.
[9] M. Adaramola, Solar Energy: Applications, Economics, and Public Perception,
Boca Raton: CRC Press, 2014.
[10] R. E. Bolz and G. L. Tuve, Handbook of Tables for Applied Engineering Science,
New York: CRC Press, 1976.
[11] N. Wakao and J. M. Smith, "Diffusion in Catalyst Pellets," Chemical Engineering
Science, vol. 17, pp. 825-834, 1962.
[12] H. J. Hwang, K. He, S. Gray, J. Zhang and I. S. Moon, "Direct Contact Membrane
31
Distillation (DCMD): Experimental Study on the Commercial PTFE Membrane and
Modeling," Journal of Membrane Science, vol. 371, no. 1-2, pp. 90-98, 2011.
[13] P. Castro and M. Huber, Marine Biology, McGraw-Hill Companies, 2012.
[14] M. H. Sharqawy, J. H. Lienhard V and S. M. Zubair, "Thermophysical Properties of
Seawater: A Review of Existing Correlations and Data," Desalination and Water
Treatment, no. 16, pp. 354-380, 2010.
[15] A. Alkhdhiri, N. Darwish and N. Hilal, "Membrane Distillation: A Comprehensive
Review," Desalination, vol. 287, pp. 2-18, 2012.
[23] J. H. Nam and M. Kaviany, "Effective Diffusivity and Water-Saturation
Distribution in Single- and Two-Layer PEMFC Diffusion Medium," International
Journal of Heat and Mass Transfer, no. 46, pp. 4595-4611, 2003.
[24] M. Flury and T. F. Gimmi, "Solute Diffusion," in Methods of Soil Analysis, Part 4,
Physical Methods, J. H. Dane and C. Topp, Eds., Madison, Soil Science Society of
America, 2002, pp. 1323-1351.
[25] R. E. Zeebe, "On the Molecuar Diffusion Coefficients of Dissolved, CO2, HCO3-,
and CO2- and Their Dependence on Isotopic Mass," Geochimica et Cosmochimica
Acta, no. 75, pp. 2483-2498, 2011.
[26] F. Franks, Water: A Matrix of Life, Cambridge: The Royal Society of Chemisty,
2000.
[27] H. Yasuda, C. E. Lamaze and L. D. Ikenberry, "Permeability of Solutes through
Hydrated Polymer Membranes," Die Makromolekulare Chemie, vol. 118, no. 2858,
pp. 19-35, 1968.
32
Appendix A. Seawater Fluid Property Equations
Seawater Density [kg/m3]:
𝜌𝑠𝑤 = (𝑎1 + 𝑎2 ∗ (𝑡 − 273.15) + 𝑎3 ∗ (𝑡 − 273.15)2 + 𝑎4 ∗ (𝑡 − 273.15)3 + 𝑎5
∗ (𝑡 − 273.15)4 )
+ (𝑏1 ∗ 𝑆 + 𝑏2 ∗ 𝑆 ∗ (𝑡 − 273.15) + 𝑏3 ∗ 𝑆 ∗ (𝑡 − 273.15)2 + 𝑏4 ∗ 𝑆
∗ (𝑡 − 273.15)3 + 𝑏5 ∗ 𝑆 2 ∗ (𝑡 − 273.15)2 )
Where,
𝑎1 = 9.999𝑒 2 , 𝑎2 = 2.034𝑒 −2 , 𝑎3 = −6.162𝑒 −3 , 𝑎4 = 2.261𝑒 −5 ,
𝑎5 = −4.657𝑒 −8 , 𝑏1 = 8.020𝑒 2 , 𝑏2 = −2.001, 𝑏3 = 1.677𝑒 −2 , 𝑏4 = −3.060𝑒 −5 ,
𝑏5 = −1.613𝑒 −5
Seawater Dynamic Viscosity [kg/(m*s)]:
𝜇𝑠𝑤 = 𝜇𝑤 ∗ (1 + 𝐴 ∗ 𝑆 + 𝐵 ∗ 𝑆 2 )
Where,
𝐴 = 1.541 + 1.998𝑒 −2 ∗ (𝑡 − 273.15) − 9.52𝑒 −5 ∗ (𝑡 − 273.15)2
𝐵 = 7.974 − 7.561𝑒 −2 ∗ (𝑡 − 273.15) + 4.724𝑒 −5 ∗ (𝑡 − 273.15)2
𝜇𝑤 = 4.2844𝑒 −5 + (0.157 ∗ ((𝑡 − 273.15) + 64.993)2 − 91.296)−1
Seawater Specific Heat [kJ/(kg*K)]:
𝑐𝑠𝑤 = 𝐴 + 𝐵 ∗ 𝑡 + 𝐶 ∗ 𝑡 2 + 𝐷 ∗ 𝑡 3
Where,
𝐴 = 5.328 − 9.76𝑒 −2 ∗ 𝑆 + 4.04𝑒 −4 ∗ 𝑆 2
𝐵 = −6.913𝑒 −3 + 7.351𝑒 −4 ∗ 𝑆 − 3.15𝑒 −6 ∗ 𝑆 2
𝐶 = 9.6𝑒 −6 − 1.927𝑒 −6 ∗ 𝑆 + 8.23𝑒 −9 ∗ 𝑆 2
𝐷 = 2.5𝑒 −9 + 1.666𝑒 −9 ∗ 𝑆 − 7.125𝑒 −12 ∗ 𝑆 2
Seawater Thermal Conductivity [mW/(m*K)]:
log10 (𝑘𝑠𝑤 ) = log10 (240 + 0.0002 ∗ 𝑆) +
0.333
343.5 + 0.037 ∗ 𝑆
𝑡
0.434 ∗ (2.3 −
) ∗ (1 −
)
𝑡
647 + 0.03 ∗ 𝑆
33
Appendix B. COMSOL Multiphysics 1D, 2D, 3D Plots
34
35
36
37
38
39
40
41
42
43
44
45
Appendix C. Constant Water Vapor Diffusion Coefficient Error
Error Between Constant DF and Varied DF
v = 0.05
v = 0.1 v = 0.15 v = 0.2 v = 0.25
2.247E- 1.283E- 6.409E- 3.203E- 3.203EOutlet Temp (K)
05
05
06
06
06
Concentrate
Outlet Conc
-7.329E- -3.651E- -3.646E- -1.822E- -1.821E(mol/m3)
03
03
03
03
03
-2.717E- -1.020E- 3.335E- -6.808E- -3.405EOutlet Temp (K)
05
05
04
06
06
Product
Outlet Conc
-9.023E- -7.245E- -5.441E- -3.629E- -3.631E(mol/m3)
03
03
03
03
03
*All values are percent error.
46
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