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Membrane distillation against a pressure difference

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Membrane distillation is an attractive technology for production of fresh water from seawater. The MemPower® concept, studied in this work, uses available heat (86°C) to produce pressurized water (2.2 bar and 46°C) by membrane distillation, which again can be used to power a turbine for co-production of electricity. We develop a non-equilibrium thermodynamic model to accurately describe the transfer at the liquid-membrane interfaces, as well as through the hydrophobic membrane. The model can explain the observed mass flux, and shows that 85 % of the energy is dissipated at the membrane-permeate interface. It appears that the system's performance will benefit from a lower interface resistance to heat transfer, in particular at the permeate side of the membrane. The nature of the membrane polymer and the pore diameter may play a role in this context.
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Journal of Membrane Science
journal homepage: www.elsevier.com/locate/memsci
Membrane distillation against a pressure dierence
L. Keulen
a,1
, L.V. van der Ham
b
, N.J.M. Kuipers
c
, J.H. Hanemaaijer
c
, T.J.H. Vlugt
a
,
S. Kjelstrup
a,d,
a
Department of Process and Energy, Delft University of Technology, Leeghwaterstraat 39, 2628CB Delft, The Netherlands
b
Department of Process and Instrument Development, Netherlands Organisation for Applied Scientic Research, TNO, 2628CA Delft, The Netherlands
c
Department of Water Treatment, Netherlands Organisation for Applied Scientic Research, TNO, 3704HE Zeist, The Netherlands
d
Department of Chemistry, Norwegian University of Science and Technology, NTNU, NO-7491 Trondheim, Norway
ARTICLE INFO
Keywords:
Membrane distillation
Water desalination
Heat and mass transfer
MemPower
Irreversible thermodynamics
ABSTRACT
Membrane distillation is an attractive technology for production of fresh water from seawater. The
M
emPower
®
concept, studied in this work, uses available heat (86 °C) to produce pressurized water (2.2 bar and 46 °C) by
membrane distillation, which again can be used to power a turbine for co-production of electricity. We develop a
non-equilibrium thermodynamic model to accurately describe the transfer at the liquid-membrane interfaces, as
well as through the hydrophobic membrane. The model can explain the observed mass ux, and shows that 85%
of the energy is dissipated at the membrane-permeate interface. It appears that the system's performance will
benet from a lower interface resistance to heat transfer, in particular at the permeate side of the membrane.
The nature of the membrane polymer and the pore diameter may play a role in this context.
1. Introduction
Fresh drinking water is essential for life on earth. We need water to
survive, not only as drinking water, but also in food production,
washing, industry, etc. According to the United Nations, the increase
in potable water use was more than twice the rate of the population
increase in the last century [1]. By 2025, an estimated 1.8 billion
people will live in areas with water scarcity, and two-thirds of the
world's population will be living in water-stressed regions as a result of
water use, growth and climate change [1]. New solutions are therefore
needed to decrease the scarcity of clean water in the world. Nearly 70%
of the earth is covered by water, but only 2.5% of that water is fresh and
usable for consumption, and only 1% of the fresh water is easily
accessible [2]. The rest is trapped in glaciers or snowelds.
Consequently, fresh water produced from seawater and brackish
water becomes increasingly important. Between 1% and 2% of the fresh
water used as drinking or process water, is extracted from brackish and
saline water [3]. In 2006, the desalination capacity worldwide was
40 million m
3
/day [3]. In 2011 it had increased to almost 70 mil-
lion m
3
/day [3]. Many desalination processes exist, for example multi-
stage ashing, multi-eect distillation, reverse osmosis, electro-dialysis
or membrane distillation. The driving forces for these processes are
either thermal, osmotic or electrical. The challenge in all cases is to
obtain reasonable energy input and equipment costs per amount of
fresh water produced.
Membrane distillation (MD) is attractive in this context, because of
its possibility to use low grade waste heat as energy source in the
production of drinking water. The rst publication on MD dates back to
the sixties of the last century [4]. Water vapor is transported through a
membrane, driven by a temperature dierence. The membrane pores
are lled with water vapor, in contrast to other techniques, where water
is transported in the liquid phase. Presently, MD is nearly commercial.
The technology is competitive with reverse osmosis for low heat costs
and feedstock with high osmotic pressures. The possibility to fully
understand and possibly improve the attractive MD process has
motivated the present study of a new invention, namely the
M
emPower
®
process concept [3,58].Fig. 1 provides a schematic
illustration of the MemPower concept [8], when used for seawater
desalination. It produces fresh water against a hydrostatic pressure
dierence with the help of a thermal driving force. An aqueous
feedstock with hydrostatic pressure P
total,
2
(e.g. seawater) is rst heated
to temperature T
2
, e.g. by utilizing low grade heat. During normal
operation, water is transported against a pressure dierence, P
total,3 -
P
total,
2
, due to the transport of the latent heat of water down the
temperature gradient. The positive temperature dierence,
T
T
2
3
, can
be said to drive the desalination process, producing distilled water on
the permeate side. The gure to the right shows the pressure of the
distillate, P
total,3, maintained by throttling of the euent valve on this
http://dx.doi.org/10.1016/j.memsci.2016.10.054
Received 5 July 2016; Received in revised form 28 October 2016; Accepted 31 October 2016
Corresponding author at: Department of Chemistry, Norwegian University of Science and Technology, NTN, NO-7491 Trondheim, Norway
1
Current aliation: Department of Aerospace Science and Technology, Politecnico di Milano, 20156 Milano, Italy.
E-mail addresses: luuc.keulen@polimi.it (L. Keulen), signe.kjelstrup@ntnu.no (S. Kjelstrup).
Journal of Membrane Science 524 (2017) 151–162
0376-7388/ © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by/4.0/).
Available online 18 November 2016
crossmark
side. The pressure P
total,3 is larger than the hydrostatic pressure on the
feed side P
total,
2
, meaning that water transport takes place against a
pressure dierence. It is indicated in the concept, Fig. 1 on the left, that
the pressurized distillate can be used to drive a turbine to generate
hydroelectric power. The power density is the turbine eciency times
the pressure dierence and the volumetric ow of distillate. The net
eect of this, is that (waste) heat can be used to produce drinking water
as well as hydroelectric power. The process will continue until an upper
pressure, the so-called break-through pressure of the membrane, is
reached. At this pressure, the pores become wetted, causing liquid
water to ow back via the membrane from the distillate to the feed.
Typical temperature and pressure variations under operation are
Fig. 1. Schematic representation of the MemPower concept [8] as applied to seawater desalination. Cold seawater enters the feed side, at (1) to the left in the gure, and ows through a
compartment with non-permeable walls, where it is preheated by a counter-currently owing stream at the permeate side. The preheated seawater is further heated by an external heat
source, which can be waste heat from the industry, solar or geothermal energy. The heated seawater enters the retentate side of the system (2) with pressure P
total,
2
, where the water will
partially evaporate and pass through membrane pores to the permeate side, at (3) in the gure, due to the temperature dierence
T
T
3
2
. The permeate compartment is shown in the
center of the gure to the left, as well as in the enlargement to the right hand side gure. The water vapor condenses to yield distilled water at the permeate side at the hydrostatic
pressure P
total,
. The distilled water is heated by the latent heat freed by condensation and heat conducted via the membrane material.
Nomenclature
Bmembrane permeability, m
2
K J/K mol s
c
w
concentration of water, mol/m
3
D
w
Fick's diusion coecient for water vapor, m
2
/s
dthickness, m
H
j
molar enthalpy of component j, J/mol
H
j
T
,
molar enthalpy of component jat temperature T, J/mol
HΔ
vap,
j
enthalpy of evaporation of component j, J/mol
Jgeneral symbol for ux
J
j
ux of component j, mol/s m
2
J
q
thermal energy ux, J/s m
2
J
q
measurable heat ux, J/s m
2
k
B
Boltzmann constant, 1.3807·10
23
kg m
2
/s
2
K
Lmean free path, m
m
˙
mass ow, kg/s
M
j
molar mass of component j, kg/mol
nnumber of borders between control volumes, dimension-
less
Nnumber of control volumes, dimensionless
Ptotal pressure, N/m
2
p*
wvapor pressure of water at saturation, N/m
2
q
*
heat of transfer, J/mol
r
m
n
local resistivity coecient, coupling force m to ux n
Runiversal gas constant, 8.3145 J/K mol
R
to
t
resistivity matrix for global description of system
Tabsolute temperature, K
xcoordinate axis for transport, m
X
i
general symbol for driving force no i
Greek symbols
ΔY
ab dierence in property Y:
Y
Y
ba
λthermal conductivity, J/s K m
2
μ
j
chemical potential of component j, J/mol
μ
jT,chemical potential of component jat temperature T,J/
mol
σlocal entropy production, J/K m
3
or J/K m
2
εmembrane porosity, dimensionless
Ωmembrane cross-sectional area, m
2
Sub- and superscripts
0 reference point or ideal gas state
a,b,c,d points on the x-axis
CV control volume
i,j component indices
hhomogeneous face
kcontrol volume index
lliquid
mem membrane
per permeate
qthermal energy or measurable heat
r
et retentate
sinterface
Ttemperature
t
ot
total
va
p
vapor
wwater
L. Keulen et al. Journal of Membrane Science 524 (2017) 151–162
152
schematically illustrated in the right-hand side of Fig. 1. Present state-
of-the-art modeling is not able to compute the pressure variation from
the temperature variation. This is because simple uxes of mass and
heat are used (Fick's and Fourier's law) [9,10]. The heat and mass
uxes are then regarded as independent of one another, a problem
mentioned by some authors [11,12]. The liquid-vapor surface resistiv-
ities on both membrane sides are also neglected in conventional
modeling.
These disadvantages can be dealt with by application of non-
equilibrium thermodynamics theory (NET) [13], a theory which takes
into account the coupling of uxes of heat and mass. The water ux is
then not only driven by a concentration dierence, it can also be driven
by a thermal driving force alone. It is also possible to relax the common
assumption of equilibrium at the phase-boundary water-vapor [14],
using newly published transfer coecients for water evaporation and
condensation [15]. The interfaces will then be included explicitly in the
modeling. A new method of discretization of the relevant equations,
developed by Van der Ham et al. [16,17], provides a practical solution
procedure. The method was successfully used to model waterethanol
distillation columns [18].
The aim of this work is to contribute to a more precise description
of the transport of water vapor and thermal energy through the
membrane in MD systems, using non-equilibrium thermodynamics.
Several new experiments have been done to provide an experimental
basis for the development of the model. These experiments are all done
with pure water, with the same thermal driving force, and against the
same pressure dierence, in order to achieve a measure of the
reproducibility of the processes. We will proceed to show that the
theory can be used to obtain a more-detailed understanding of the
observed mass ux through the membrane, in addition to the heat ux.
The main process irreversibilities can be located with this knowledge.
To know the location and nature of the main process ineciencies, may
give guidelines for system optimization. Such will be pointed out.
2. Coupled transport of mass and heat across a hydrophobic
membrane
The theory of non-equilibrium thermodynamics (NET) provides a
systematic framework for description of transport processes, including
their interdependency or coupling [13,1921]. The local entropy
production is used to dene the sets of conjugate uxes and forces in
this theory. We shall use the procedure of Van der Ham et al. [16,17] to
integrate across the liquid-vapor interfaces of the system. Details of
their derivations can be found in Appendix A. For the interfaces
themselves, we shall use the procedure laid out by Kjelstrup and
Bedeaux [13].
2.1. System lay-out and assumptions
Fig. 2 provides a more detailed representation of the membrane
pore of the system pictured in Fig. 1. The transport in the pore is
considered to be one-dimensional, and we are dealing with operation
under the steady state. The retentate and the permeate, on the left and
right-hand sides, respectively, yield boundary conditions for the
transport of mass and heat across the membrane. The pressure, P
i
,
temperature, T
i
, and salt chemical potential, μ
i
, or concentration C
i
,
are known for the permeate (1) and retentate (2) sides. In the
experiments, to be reported here, pure water is used on both sides,
and C
i
=0. There is a transition from liquid outside the membrane to
vapor in the pore, because the membrane is hydrophobic, see Ref. [22]
for a good illustration of the mechanism. Some air may be trapped in
the pore, without changing the chemical potential of water, or the
description that follows. Heat is owing from the retentate to the
permeate side through the membrane matrix or the vapor-lled pores
of the membrane. The system in Fig. 2 is considered to be isolated from
the surroundings.
2.2. 1D-integrated system description
The aim is to obtain a description using measured properties, i.e.
the water ux, J
w
, and the ux of measurable heat at position n,Jqn,as
variables. Subscript wrefers to water and subscript qindicates that
heat is transported. From the entropy production for the total system
consisting of control volumes 1N, we obtain the following forceux
relationships (see Appendix A):
Δ
TrJ rJ
1=′+
qn w1n qq
tot ,qw
tot
(1)
Δμ T
TrJ rJ()
=′+
wqn w
1n 1
1wq
tot ,ww
tot
(2)
where
r
ij
tot are the total resistivities (see Eqs. (6) to (8). The conjugate
driving forces to the heat and water uxes are
Δ
T
1n 1
and Δμ T
T
()
w
1n 1
1,
respectively. In general, the dierence in the chemical driving force
between points a and b at constant temperature,
T
a
, is equal to
Δμ T
T
μT μT
T
()
=− ()− ()
wT a
a
wT
bawT
aa
a
ab ,,,
(3)
where the symbol
Δ
a
b
refers to the dierence between position a and b.
The chemical potential has a contribution from the vapor pressures at a
and b, and from the dierence in hydrostatic pressure, see e.g. [13].
The total driving force is obtained by adding the driving forces of all n
connected control volumes, cf. Fig. 2. There is a total number of
n
N=+
1
control volume boundaries.
We see now from Eq. (2) how the water ux depends on the heat
ux, which again depends on the thermal driving force. At zero water
ux, there is a balance of forces, obtained by dividing the two
equations. This will dene the upper pressure one can possibly reach
on the permeate side when J
w
=0. While the resistivities on the
diagonal are related to Fick's and Fourier's laws (see below), the
coupling coecients are characteristic for non-equilibrium thermo-
dynamics. In the bulk (homogeneous) phases, they are sometimes
small, and can be neglected. This will be done also here. At the
interfaces, however, they are large, and must be taken along [15]. The
overall coupling coecients will therefore have nonzero contributions
from the interfaces.
According to Van der Ham et al. [16,17] (see Appendix A for more
details), it is possible to choose as variables the measurable heat ux,
J
q
, referred to position c, and the chemical potential dierence,
Δ
μ
w
,
evaluated at the temperature at position d, for a control volume
between positions a and b. The equations become
Δ
TrJ r H J
1=′+(+ )
qc wT wab qq
ab ,qw
ab qq
ab ac , (4)
Δμ T
TrJ r H J ΔH ΔT
()
=′+(+ )+ 1
wd
d
qc wT w wT
ab
wq
ab ,ww
ab wq
ab ac , ab , db (5)
Fig. 2. Schematic representation of volume elements across a membrane pore of
diameter dpor
e
. The whole rectangular box in the gure is a close-up of the pore between
two black rectangles in the right-hand side of Fig. 1. The control volume to the left
represents the boundary of the pore to the permeate, and the control volume to the right
represents the boundary of the pore to the retentate side. The membrane is seen as a
homogeneous phase of
N
2
control volumes. Boundaries are numbered by
in=1,
.
Pressure, P
i
, temperature, T
i
, and salt chemical potential, μ
i
, or concentration C
i
, are
known for the permeate (1) and retentate (2) sides.
L. Keulen et al. Journal of Membrane Science 524 (2017) 151–162
153
The coecient
r
ij
ab is the resistivity that is coupling driving force i
with ux j. The resistivity
r
qq can be related to the Fourier thermal
conductivity, while the resistivity
r
w
w
can be related to the diusion
coecient for water, see Eqs. (11) and (12) below. The resistivities
r
r=
wq q
w
describe the coupling between uxes of water and heat.
The symmetry of the problem means that the same solution must be
found when boundary conditions are switched. The number of control
volumes, N, in Eqs. (2) and (4) is thus an even number. We show in
Appendix A how the force ux equations are derived with four
control volumes (N=4). The general expressions for the total resistiv-
ities that can be derived, are:
r
r=
k
N
k
qq
tot
=1
qq
(6)
r
rrrΔH== +
k
N
kk wT
qw
tot wq
tot
=1
/2
qw qq k,n ,
(7)
∑∑rrΔHr r rΔH rΔ
H
++ =+(+2·)
kN
N
kk wT
k
N
kkwT k
wT
=( /2)+1
qw qq k+1,n , ww
tot
=1
/2
ww qq k,n , wq k,n
,(8)
rrΔH rΔH
+
+( +2· )
kN
N
kk wT kwT
=( /2)+1
ww qq k+1,n , wq k+1,n ,
In the expansion across the system, the enthalpy dierence of water
between control volume number n and k, taken at temperature Tis
introduced,
Δ
HwTn,k , .
For the bulk part of the membrane, the contributions to Eq. (6)
come from Fourier's law. The contributions to Eq. (8) come from Fick's
law. In the NET formulation, these laws take the form
T
xTr J
=− ′
q
2qq (9)
T
μ
xrJ
1
=−
wT
w
,ww (10)
The membrane resistivity coecients could then be obtained from
r
λTd=1
w
qq
mem
2mem (11)
r
TD d=1
w
ww
mem me
m
(12)
The membrane resistance to heat transfer is given by the thermal
conductivity of water vapor in the membrane pore, λ
w
and the
membrane thickness,
d
me
m
. The resistance to mass transfer is given
by the diusion coecient for vapor in the pore, D
w
. We have assumed
that the vapor is ideal. In the membrane pores, the coupling coecients
are neglected,
r
r==0
qw
mem wq
mem . For the two surfaces, the full set of
interface transfer coecients provided by Wilhelmsen et al. [15]
contributed to the overall resistivities. In setting up these equations,
we have assumed that mass transport takes place by vapor transport
only. In reality there may be air trapped inside the membrane.
Stationary air will not alter the driving forces much (the vapor pressure
or temperature), but it may alter the transport coecients. This has
been neglected.
3. Experimental
The proof of the MemPower principle was reported earlier [8]. The
experimental set-up was a simplied version of Fig. 1, see Fig. 3. For
the purpose of comparing theoretical and experimental results under
the simplest possible conditions, only pure water was used on both
sides of the membrane unit. Several experiments were performed.
Among them, seven repeats of the same conditions were selected as a
basis for the model development. The inlet and outlet temperatures of
the water on the left and right hand sides were measured, along with
the retentate and permeate pressures and mass ows.
The average temperature of the retentate (right hand) side was on
the average always higher than the average temperature on the
permeate (left hand) side. As the average temperature on each side,
we took the average of the measured inlet and outlet temperatures. The
pressure dierence was on the average 1.2 ± 0.1 bar. Due to the
temperature dierence, a water ux was set up across the membrane,
in spite of a pressure dierence in the opposite direction, again proving
the system's concept. The amount of water passing the membrane was
computed from the systematic dierence between the permeate and
retentate streams (see below), given that the streams at the inlets were
the same.
The hydrophobic membrane used in all experiments was reported
earlier [8]. Such a membrane has been realized in the laboratory [22].
In this case, the membrane consisted of a selective layer and a support
layer, that makes it able to withstand a pressure dierence. The
selective layer had pore diameters near 0.2 µm and a thickness of
5 µm. Due to its hydrophobicity it was permeable to water vapor only.
The support layer had pore sizes larger than 10 µm and a thickness of
80 µm. It was lled with liquid water. The large pores in the support
layer mean that it plays an insignicant role for the transport of heat
and mass and it was therefore neglected in the simulations. The
properties of the selective layer are listed in Table 1.Here, εis the
porosity, the void volume fraction available for vapor transport in the
membrane. In view of other uncertainties the membrane is assumed to
have cylindrical perpendicular pores with a tortuosity, τ, set to unity.
More complex membrane structures can have a tortuosity higher than
unity [23].
Fig. 3. Schematic representation of the experimental cell. Water is owing from the
retentate to the permeate sides through membrane pores against a xed pressure
dierence of 1 bar. Temperatures are measured at the inlet and outlet on both sides. The
water ux across the membrane was delivered from the retentate to the permeate, and
was computed from the dierence in the mass ows at steady state. A constant pressure
dierence was maintained across the membrane.
Table 1
Membrane properties.
Property Symbol Value Unit
Membrane cross-sectional area Ω6.00·10
3
m
2
Membrane thickness
d
mem
5.00·10
6
m
Pore diameter dpor
e
2.00·10
7
m
Porosity ε0.8
Thermal conductivity
λ
me
m
0.19
W
m·K
L. Keulen et al. Journal of Membrane Science 524 (2017) 151–162
154
3.1. Data reduction
The results from all seven experiments are presented in Table 2.
The average quantities relevant for the development of the model are
extracted to Table 3. The mass ow observed through the membrane
gives the membrane water ux
J
me
m
from
Jm
Ω
=˙/
mem mem (13)
where
m
˙
me
m
is the mass ow across the membrane and Ωis the cross-
sectional area of the MemPower system. The water ux can also be
referred to the area accessible to vapor transport, via the membrane
porosity, ε. The relation between this water ux, J
w
, and
J
me
m
, for
cylindrical pores is
JJ
ε
=/
wmem
(14)
4. Numerical data input and solution procedures
All simulations refer to Fig. 2 and use the membrane properties
presented in Table 1 and below. The boundary conditions of the
retentate and permeate sides, as given in Table 3 could then be used to
compute the mass and heat uxes through the membrane.
4.1. Numerical data input
Thermodynamic properties of water were calculated using the IF97
model in FluidProp [24]. The inputs to this model are the temperature
and pressure of the control volume in question. Surface resistivities for
water were taken from Wilhelmsen et al. [15], assuming that the
surface is at.
For the membrane thermal conductivity, we used the vapor value,
λ
= 2.30·10 W/(mK
)
vap −2 , and the polymer matrix value from Table 1.
The partial derivative
μc∂/
wT
w
,
was taken to be unity (ideal vapor). The
value of D
w
was modeled with the Knudsen diusion model, following
[22,9,10], as the pore diameter was always smaller than the mean free
path (calculated from the kinetic theory-formula to 2.6·10
7
m).
D
dRT
πM
=3
8=4.27·10 m
s
w
w
pore −5
2(15)
The result did not vary much with temperature around this average
value.
4.2. Solution procedure. NET
The equations that constitute the NET model were presented in
Section 2. An overview of the solution procedure for the one-dimen-
sional problem is given in Fig. 4. The procedure which follows [16],
starts by dening the retentate and permeate boundary temperature
and pressure, using Table 3, and computing the corresponding overall
driving forces Xj
to
t
Eqs. (1) and (2). In the next step (2), the number of
membrane control volumes is chosen, N. There are now
n
N=+
1
control volume boundaries. The initial temperature and pressure
proles, and the molar enthalpies at the control volume boundaries
can then be obtained. From this information, the resistivities
r
ij
C
V
of all
control volumes for the membrane and for the surface are computed
(step 3). With the molar enthalpies,
H
j
T
,
, at all control volume
boundaries, and the initial temperature and pressure proles, the total
resistivities
r
ij
tot are next computed (step 4) using
r
ij
C
V
.
From the overall driving forces, Xj
to
t
, and
r
ij
tot, one can next compute
the water ux J
j
and the measurable heat ux at boundary n,Jqn,(step
5). The measurable heat ux prole, Jq
i
,, follows (step 6) using the
energy balances JJ ΔHJ′=′ +
qi qn w w,,i,n
.
Table 2
Results from seven measurements with the experimental setup reported in Section 3,
showing the temperatures at inlets and outlets of the retentate and permeate ows, the
pressures on the permeate side, and the mass ows on the two sides. The pressure on the
retentate side, P
re
t
, is always 1.00·10
5
N/m
2
. The averages (see bottom row) are used in
the analysis of the model, see Section 3.
T
inret,
T
outret,
T
inper,
T
outper, P
per
m
˙
re
t
m
˙per J
w
°C °C °C °C 105N
m2
10
−3kg
s
10
−3kg
s
10−2 kg
m2s
93.09 76.83 38.36 53.96 2.0 3.42 3.49 1.52
94.10 77.27 38.14 54.30 2.2 3.31 3.38 1.50
94.05 77.12 38.01 53.97 2.2 3.28 3.35 1.48
94.05 76.87 37.91 53.85 2.3 3.25 3.33 1.46
94.02 76.79 37.90 53.70 2.3 3.17 3.24 1.41
94.07 77.31 38.43 54.42 2.3 3.36 3.44 1.47
94.09 77.29 37.75 53.70 2.3 3.39 3.46 1.41
94.1 77.0 38.1 54.0 2.2 3.31 3.38 1.47
± 0.1 ± 0.2 ± 0.2 ± 0.3 ± 0.1 ± 0.09 ± 0.07 ± 0.05
Table 3
Average measured temperatures, pressures and water flux, computed from Table 2.
Uncertainties are within
0.1 °C
, 0.1 bar and 2 10
3
kg/(m
2
s).
T
ret,avg
T
per,avg
P
re
t
P
per J
w
°C °C 105N
m2105N
m2
10
−2 kg
m2s
Average 85.5 46.0 1.0 2.2 1.5
Fig. 4. Flow sheet illustrating how the NET model is solved. For more details, see text.
L. Keulen et al. Journal of Membrane Science 524 (2017) 151–162
155
With this information the local driving forces of the control
volumes, XΔj
C
V
, are available (step 7) using
r
ij
C
V
,J
j
and
J
q
. A new
temperature prole T(x) and a new pressure prole p(x) is next
available.
Steps 37 are repeated until the uxes have converged, and the
total entropy production can be computed using Xj
CV,
J
q
and J
j
, as well
as the entropy balance (step 8). The number of membrane control
volumes can then be raised (step 9). Steps 39 are repeated until the
entropy production is constant and the uxes have converged. The
solution will then obey consistency checks as described in Appendix B.
The resistivities of the control volumes (step 3 in the gure) were
calculated for the local temperature and pressure. Membrane resistiv-
ities were multiplied with the thickness of the control volume in the x-
direction,
d
d
N
=/
CV mem . The surface resistivities already include the
surface thickness. The total resistivities,
r
ij
tot, were calculated using
equations in Section 2.2. The uxes J
j
and Jqn,were obtained by
dividing the overall driving force vector Xj
to
t
by the total resistivity
matrix composed of
r
ij
tot.
The side with the highest temperature (T
n
, see Fig. 2) was used as
the reference temperature for enthalpy calculations. This was done in
order to make sure that the enthalpies inside the membrane always
apply to water in the vapor phase. The enthalpies used in the
calculation of the heat ux through the membrane were rst taken at
constant temperature. The energy balance over the membrane is then
not obeyed. The measurable heat ux at the retentate side, Jq,
1
, was in
the end recalculated for the control volume boundaries to conform with
the energy balance;
JJJΔH′= +
qi q w w,,1 i,1
(16)
The total energy ux through the membrane pore:
JJ JH=′ +
qqiw
i
,(17)
is constant in the calculations.
When heat conduction in the membrane is considered, the measur-
able heat ux through the membrane cross-sectional area is calculated
as:
J ελ
dT
d
=′+(1)
qi qi,mem, , mem
mem (18)
where
λ
me
m
is taken from Table 1. Otherwise,
λ
=0
mem
.
Two iteration loops were included in the calculations, for which the
following convergence criterion was used:
Ji Ji(|1 − ( + 1)/ ( ) | < 0.0001
)
.
Five iteration steps were normally needed to obtain the nal pressure,
temperature prole and entropy production, with 10 membrane
control volumes, and two surface control volumes. The solution
procedure was tested for inversion of the boundary conditions. An
inversion did not change the outcome of the calculations, as required.
In order to ensure that the model was thermodynamically consistent,
several additional validation tests were performed. They are explained
in detail in Appendix B, and reported as Results.
4.3. Solution procedure. Conventional model
The conventional model of membrane distillation described the
transport of water by Fick's law after introduction of vapor pressures
via the ideal gas law RTdc dP=
*
ww[9,10].
JεB
p
d
==
Δ*
ww
mem
mem (19)
where the driving force is the gradient in the saturation pressure of
water vapor. The water is aected by the temperature gradient only
indirectly in this description, via the temperature dependence of
p*
w
along the x-axis [25,12]. The ux refers to the accessible area, the
cross-sectional area of the pores. The membrane permeability
B
=D
RT
w,
where D
w
is the diusion coecient as before.
The heat ux through the membrane has contributions from
transport of latent heat and from conduction. Conduction takes place
across the membrane matrix as well as the pores. The combined
contributions lead to [26,27,11,3]
JJH λ
T
d
=Δ −
Δ
qw
per mem vap,
mem
(20)
Where
J
q
pe
r
is the total measurable heat ux, λthe thermal
conductivity of the membrane; pores and polymer network combined,
TΔis the temperature dierence across the membrane, and HΔvap w,the
enthalpy of evaporation of water at the entrance to the pores. The
eective thermal conductivity is given by
λ
ελ ε λ=+(1)
vap mem (21)
A one-dimensional system was constructed to compute results.
Only one control volume was used for the membrane pore, N=1 in
Fig. 2. There are no surface control volumes in this model. The Eqs.
(19) and (20) were solved in the following stepwise manner. The
boundary conditions were dened for the retentate and permeate,
similar to step 1 in the NET model. The overall driving forces were
calculated, using the temperatures and vapor pressures at the mem-
brane boundaries. With knowledge of thermophysical properties, D
w
,
λ
and HΔ
vap,
w
, the mass ux J
w
becomes available. The heat ux,
J
q
pe
r
,is
nally calculated from Eq. (20).
5. Results and discussion
The experimental results are reported in Table 2, cf. Section 3 for
data analysis. The temperatures and pressures from this table were
used as boundary conditions in the simulation (see Table 3).
Simulation results are reported and discussed in the sections that
follow this one. A rst aim is to compute the water ux across the
membrane and compare to the experimental water ux across the
membrane (last column of Table 2).
The average water ux, J
w
, was computed from Eq. (14) and the
observed systematic dierence between the permeate and the retentate
mass ows (resulting in
m
˙
me
m
), and related to the dierence in the
average of the inlet and outlet temperatures in Table 2. The results are
given in Table 3. The temperatures and pressures in this table are used
as boundary conditions for the calculations reported in the next
section. The water ux of Table 3 will be compared to the simulated
one (see next section).
5.1. NET simulation results
Table 4 (top row) presents results of the NET model simulations for
the given boundary conditions given from the experiments. The model
was described in Section 4.2.
The simulated mass ux, 1.1·10 kg/(m s
)
−3 2 , is smaller than the
experimental mass ux, 1.5·10 kg/(m s
)
−2 2 by one order of magnitude.
The experimental value has a large uncertainty, however. It appears as
adierence between two large numbers, but obtains reliability because
the permeate side value is systematically larger than the retentate value
in all seven experiments. A lower limit for J
w
can be obtained from the
uncertainty of the experimental ux;
2
·10 kg/(m s
)
−3 2 . This is close to the
experimental value. Lee et al. [22] report an experimental value for the
Table 4
Simulation results for the 1D-NET model. The membrane conductivity is finite (top row)
or zero (middle row). The surface resistivities also set to a negligible value (bottom row).
J
w
J
q,ret
J
q,per
J
q,mem
kg
m2s
J
m2s
J
m2s
J
m2s
Finite
λ
me
m
1.1·10
3
3.1·10
5
3.1·10
5
3.0·10
5
λ
=0
mem
1.1·10
3
8
.0·10
3
8
.2·10
3
0
No surface eect 1.4
4.1·10
5
4.4·10
5
0
L. Keulen et al. Journal of Membrane Science 524 (2017) 151–162
156
mass ux in osmotic experiments with a hydrophobic membrane,
against a small pressure dierence, of the same order of magnitude as
we obtain. This gives support to the lower estimates.
The simulated value is rather sensitive to the surface resistivities
(see below). We have used surface transfer coecients for pure water.
With pore diameters in the nanometer range, the presence of the
hydrophobic membrane material may enhance the water transport
through the phase boundary. It is known that a concave shape of the
liquid-vapor interface will reduce the resistivities [15]. This will make a
better t to the experimental value. Given the uncertainties, however,
we prefer to not attempt to tailor the model to experimental results, but
continue with observing trends. The meaning is to obtain knowledge
that can lead to further research.
A sensitivity analysis was therefore carried out for the resistivities,
to see their impact. The results are shown in Fig. 5. The resistivities
were varied one by one, and their inuence on the water ux and on the
measurable heat ux was computed. The x-axis shows the new
resistivity divided by the reference; which is the resistivity according
to Wilhelmsen et al.,
r
r/
ij ij,re
f
. The y-axis show the impact on the two
uxes. The thermal resistivity of the surface,
r
s
qq, has the most
important inuence on the mass ux and heat ux. Also
r
s
q
w
is
signicant. The mass ux can increase or decrease by an order of
magnitude, should the resistivity decrease or increase. This is impor-
tant, and should be tested experimentally. In this context, it is
interesting that the mass ux observed by Lee et al. [22] for small
osmotic pressure dierences across hydrophobic pores, is order of
magnitude the same as our experimental values. This gives support to
our eort to take surface coecients into account. Surface resistivities
are therefore central for accurate information on the mass ux. The
coecients are less important for the heat ux, however. A change in
the heat ux is within 20% for similar variations in the coecients. The
results show that the membrane resistivities have a negligible impact
on both uxes. This could mean that our assumption of a negligible
coupling coecient in the membrane is good.
The heat ux on the permeate and retentate sides in Table 4 dier
from the heat ux in the membrane, due to the latent heat carried by
the mass ux, but this is a small contribution. The heat ux is largely
aected by the thermal conductivity of the membrane polymer. When
the membrane conductivity is equal to zero (
λ
=0
mem
, center row
results), the heat ux drops more than one order of magnitude. For the
heat ux, we have no measurements to compare to. If also this is
measured, there would be a second useful handle for model develop-
ment. A membrane that is less conductive to heat is an advantage for
the mass ux, see Fig. 5a.
5.2. Properties of the NET model
The model constructed with NET was subject to several validation
tests, see Appendix B. All were conrmed, as described below. This
does not only give credibility to the model. It provides also a solid base
for further development.
Total and local entropy production. In order to test for agreement
with the second law of thermodynamics, we computed the total entropy
production of the system in two ways, from the entropy balance over
the total system, σ
balanc
e
tot
, and from the global uxes and forces,
σ
JX
tot
.
Agreement was obtained within less than 0.1%,
σσ= = 3.53 J/m K
s
JXbalance
tot tot 2.
The local entropy production, σ
T
, was also calculated. The σ
T
(not
shown) was positive in all control volumes, as it should be. Almost all
entropy production took place at the membrane boundaries, the
permeate side more than the retentate side, see Table 9 for more
details. This reects again the importance of the surface resistivities.
Most of the dissipation of energy takes place here (86%). This nding is
supported by the ndings of Lee et al. [22], that the surface played a
central role for the rate of water transfer.
Overall and local driving forces. The overall thermal driving force
is shown in Table 5. The overall thermal driving force and the sum of
the local thermal driving forces are equal. The overall chemical driving
force, evaluated at T
1
, is shown in Table 6. Again, the overall driving
force is equal to the sum of the local chemical driving forces; meaning
that the model is thermodynamically correct.
Resistivity coecients. It was next veried that the local, as well as
the total resistivity matrices obey Onsager symmetry [13]. Coecients
in the total resistivity matrix are shown in Table 7. The main
coecients are positive as they should be. The coupling coecients
are in this case negative, meaning that heat is transported in the same
direction as mass. Because the coupling coecient in the membrane is
set to zero; the contribution from the surfaces dominate completely the
value of the coupling coecients.
5.3. Transport properties. Numerical results
Table 8 provides detailed insight into the coecients that con-
Fig. 5. The dependence of the uxes on the overall Onsager resistivities.
Table 5
The overall thermal driving force (in 1/K), calculated from the overall boundary
conditions, from the sum of local driving forces and from the force-flux relations.
Δ
T
1n 1
Δ
k
nk
T
=1
−1 ab 1
r
JrJ′+
qn w
qq
tot ,qw
tot
3.45·10
4
3.45·10
4
3.45·10
4
Table 6
The chemical driving force (in J/(K mol)), calculated from the overall boundary
conditions, from the sum of local driving forces and from the force-flux relations.
Δμ
wT T
T
1n ,(1)
1
k
nΔkμwT T
T
=1
−1 ab ,(1)
1
r
JrJ′+
qn w
wq
tot ,ww
tot
−6.3·10−3 −6.3·10−3 −6.3·10−3
L. Keulen et al. Journal of Membrane Science 524 (2017) 151–162
157
tribute to the overall coecients of Table 7. Results for four control
volumes are shown. The surface resistivities at the permeate side are
somewhat higher than on the retentate side surface, explaining the
higher entropy production on this side. This is due to the lower
temperature on the permeate side of the system. The resistivities of
the membrane near the two surfaces, increase slightly, going from the
retentate side to the permeate side. The surface resistance is about two
orders of magnitude larger than the membrane resistance. The negative
coupling coecient of the surface means that uphill transport of water
can take place.
5.3.1. Contributions to the entropy production
The contributions to the global entropy production are shown in %
in Table 9. The thermal contribution dominates completely, and that
the dominating part is at the permeate side. The contribution from the
chemical driving force is small, but not zero. The surface facing the
permeate side is clearly a target for material optimization and further
studies. By lowering the surface resistivity to heat transport, the
dominating thermal term can be much reduced. An increase in the
pressure dierence across the membrane did not markedly aect the
results.
5.4. Results from the conventional model
The results from the conventional model, using input from Table 3
are presented in Table 10. The uxes were calculated as described in
Section 4.3. A scenario with an insulating membrane,
λ
=0
mem
, is also
presented in this Table 10. We see that the measured mass ux is now
over-predicted by 3 orders of magnitude. This can be somewhat
mended by including a tortuosity factor, which is usual higher than
unity due to a more complex membrane structure; by reducing the
vapor diusion coecient (due to presence of air in the voids), or by
including larger membrane resistances. This may reduce the mass ux,
maybe as much as an order of magnitude. It is dicult to see how the
overall resistivities could be much larger, without including a surface
resistance. This suspicion was conrmed when we replaced the present
surface resistivities with values that were similar to bulk values. By
neglecting the special properties of the surface, we recovered in our
model the large water ux of the conventional model, see bottom row of
Table 4.
Also the support layer may play a role in reducing the temperature
dierence across the membrane, thereby reducing the driving force.
Such a reduction will increase the gap between the NET model and the
conventional model, however.
The two ways to describe the system, produces water uxes which
dier by orders of magnitude. Taking the experimental result as a
reference, the conventional model largely over-predicts this ux, while
it is somewhat under-predicted by NET.
The conventional model oversimplies the description, and is not
able to describe the entropy production. The NET model has been
checked for consistency and gives further insight into where energy is
dissipated (at the membrane surfaces, in particular on the permeate
side), and can provide a basis for further work through that. Eorts
should be made to improve conditions in particular for heat transport
on this side. From the sensitivity of the results to the choice of
resistivities, demonstrated above, a meaningful step forward seems to
be the determination of membrane surface resistivities. How does the
membrane surface limit the transport, and how precisely does the
hydrophobic nature of the membrane aects the resistivities? The
precise location and value of the driving forces at this location is clearly
important for the eciency of the unit.
6. Conclusion
We have used non-equilibrium thermodynamics to describe mem-
brane distillation of water against a pressure dierence, as in the
MemPower concept, assuming that the transport takes place in 1
dimension. The theory can be used to predict a water ux directly
driven by a thermal driving force, unlike in most other models, where
the temperature-dependent vapor pressure drives the ux. Using
recently published transport coecients for pure water at a at surface,
and driving forces of the experiment, we compute the water ux, and
argue that it is reasonable compared to the experimental result. By
taking into account the hydrophobic nature of the membrane, better
agreement seems within reach. The conventional model, on the other
hand, over-predicts the mass ux largely. Better knowledge of surface
Table 7
Overall resistivities for mass and heat transport in the hydrophobic membrane of
MemPower.
r
qq
to
t
r
qw
to
t
r
wq
to
t
r
w
w
tot
m2s
J·K
m2s
mol·K
m2s
mol·K
J·m2s
mol·K
4.95·10
8
−2.08·10
−3
−2.08·10
−3 2.75·10
2
Table 8
Average resistivities of the retentate surface CV, the CV next to this surface (1), the CV
next to surface (2) and permeate surface CV.
Control volume (CV)
r
qq
r
qw
r
ww
m2s
J·K
m2s
mol·K
J·m2s
mol·K
Retentate surface CV 5.50·10
9
−2. 15·10
4
27.44
CV next to retentate surface 1.74·10
10
0 3.31·10
5
CV next to permeate surface 1.76·10
10
0 3.33·10
5
Permeate surface CV 4.23·10
8
−1. 80·10−3 2.44·10
2
Table 9
Contributions in percent to the overall entropy production from the surfaces and the
membrane.
σ
Ti
′(%)
qT
1
J(%
)
w
ΔμwT
T
()
σ(%
)
Ti
tot
σ
Ti
s,
1
12 0 12
σ
Ti
h
20 2
σ
Ti
s,
2
85 1 86
σTi
tot 99 1 100
Table 10
Results for the conventional 1D model. The membrane thermal conductivity is finite (top
row) or zero (bottom row).
J
w
J
q,ret
J
q,per
J
q,mem
kg
m2s
J
m2s
J
m2s
J
m2s
2.12
5
.40·10
6
5
.40·10
6
3.00·10
5
2.12
5
.10·10
6
5
.10·10
6
0
Fig. A.6. A model of the symmetrical system consisting of 4 control volumes. Boundary
conditions are given in terms of pressure and temperature,
p
i
,
T
i
, at position i=1 and 5.
We are interested in the water ux, J
w
and the varying measurable heat ux across the
system.
L. Keulen et al. Journal of Membrane Science 524 (2017) 151–162
158
transfer coecient for pores in hydrophobic membranes may help
improve the MemPower concept. Modeling of more realistic operating
conditions is an interesting next step.
Acknowledgment
This work was sponsored by NWO Exacte Wetenschappen (Physical
Sciences) for the use of supercomputer facilities, with nancial support
from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek
(Netherlands Organisation for Scientic Research, NWO). TJHV
acknowledges NWO-CW (Chemical Sciences) for a VICI grant. SK is
grateful to TU Delft for an adjunct professorship until 2015. This
technology is developed by Nederlands Organisation for Applied
Scientic Research (TNO). There is no conict of interests between
the parties.
Appendix A. Local and global forces and conjugate uxes
Consider for the sake of illustration that the system consists of four control volumes, a volume including the surface (I), two volumes in the
homogeneous membrane (II and III) and a volume containing the other surface (IV). The symmetrical arrangement is shown in Fig. A.6.
Following Kjelstrup and Bedeaux [13] the forceux relations for each control volume are:
Δ
TrJ rJ
1=′+
qa
w
ab qq
ab ,qw
ab
(A.1)
Δμ T
TrJ r J()
=′+
wb
b
qa
w
ab
wq
ab ,ww
ab
(A.2)
where a and b are the boundaries of the control volume. The measurable heat ux
J
q
refers to point a, while the chemical potential dierence refer
to point b, see Kjelstrup and Bedeaux [13] for more details. For the four control volumes in Fig. A.6, we obtain:
Control volume I
Δ
TrJ r J
1=′+
IqIw12 qq ,1 qw (A.3)
Δμ T
TrJ r J()
=′+
wIqIw
12 2
2wq ,1 ww (A.4)
Control volume II
Δ
TrJ r J
1=′+
II qII w23 qq ,2 qw (A.5)
Δμ T
TrJ r J()
=′+
wII qII w
23 3
3wq ,2 ww (A.6)
Control volume III
Δ
TrJ rJ
1=′+
III qIII w34 qq ,4 qw (A.7)
Δμ T
TrJ rJ()
=′+
wIII qIII w
34 3
3wq ,4 ww (A.8)
Control volume IV
Δ
TrJ rJ
1=′+
IV qIV w45 qq ,5 qw (A.9)
Δμ T
TrJ rJ()
=′+
wIV qIV w
45 4
4wq ,5 ww (A.10)
A.1. From the local to the overall description
Boundary conditions are only known at position 1 and 5 (=c). In order to make use of these, we introduce as variables, the measurable heat ux
at position 5 and the chemical potential dierence at temperature T
1
. The heat ux at any position is related to that of position 5 via the energy
balance:
JJΔHJ′=′+
qc qa wT w,,ca,
(A.11)
Δμ T
T
Δμ T
TΔH Δ T
()
=− ()
1
wd
d
wb
b
wT
ab ab
ab , bd (A.12)
The chemical potential dierences in the last line were related by the van't Hoequation, see [17] for more details. Applying these equations, we
rewrite all measurable heat uxes in Eqs. (A.3) to (A.10) to contain
J
q,
5
and all chemical potential dierences to refer to T
1
. This yields:
Control volume I
Δ
TrJ r rΔH J
1=′+(+ )
IqIIwT w12 qq ,5 qw qq 15 , (A.13)
L. Keulen et al. Journal of Membrane Science 524 (2017) 151–162
159
Δμ T
TrJ r H J ΔH Δ
T
()
=′+(+ )+ 1
wIqIIwT w wT
12 1
1wq ,5 ww wq 15 , 12 , 12 (A.14)
Control volume II
Δ
TrJ r rΔH J
1=′+(+ )
II qII II wT w23 qq ,5 qw qq 25 , (A.15)
Δμ T
TrJ r H J ΔH Δ
T
()
=′+(+ )+ 1
wII qII II wT w wT
23 1
1wq ,5 ww wq 25 , 23 , 13 (A.16)
Control volume III
Δ
TrJ r H J
1=′+(+ )
III qIII III wT
w
34 qq ,5 qw qq 45 , (A.17)
Δμ T
TrJ r H J ΔH Δ
T
()
=′+(+ )+ 1
wIII qIII III wT w wT
34 1
1wq ,5 ww wq 45 , 34 , 13 (A.18)
Control volume IV
Δ
TrJ r H J
1=′+(+ )
IV qIV IV wT w45 qq ,5 qw qq 55 , (A.19)
Δμ T
TrJ r H J ΔH Δ
T
()
=′+(+ )+ 1
wIV qIV IV wT w wT
45 1
1wq ,5 tot wq 55 , 45 , 14 (A.20)
We see that the dierences in the inverse temperatures on the right-hand sides of Eqs. (A.14), (A.16), (A.18) and (A.20) are given by (sums of)
Eqs. (A.13), (A.15) and (A.17). We execute these summations and substitutions, and obtain:
Δ
TrrJ
1=( + ) ′
III
q13 qq qq ,
5
(A.21)
rrΔH rrΔHJΔ
TrrrJ
+
(+ + + ) 1=( + + ) ′
IIwT II II wT w I II III q
qw qq 15 , qw qq 25 , 14 qq qq qq ,
5
(A.22)
rrΔH rrΔH rrΔHJ
Δμ T
TrrΔHJ
+
(+ + + + + ) ()
=( + ) ′
IIwT II II wT III III wT w wII wT q
qw qq 15 , qw qq 25 , qw qq 45 ,
12 1
1wq qq 12 , ,
5
(A.23)
r r ΔH r r ΔH ΔH J Δμ T
TrrrΔHJ
+
(+ +(+ ) )()
=( +( + ) ) ′
II wT IIwT wT w wII I II wT q
tot wq 15 , qw qq 15 , 12 ,
23 1
1wq qq qq 23 , ,
5
(A.24)
r r ΔH r rΔH r rΔH ΔH J ΔμT
TrrrΔHJ
+
(+ +(+ + + ) ) ()
=( +( + ) ) ′
II II wT IIwT II II wT wT w w
wq
III I II wT q
tot wq 25 , qw qq 15 , qw qq 25 , 23 ,
34 1
1qq qq 34 , ,
5
(A.25)
rrΔH rrΔH rrΔHΔHJ
Δμ T
Tr rrrΔHJ
+
(+ +(+ + + ) ) ()
=( +( + + ) ) ′
III III wT IIwT II II wT wT w wIV I II III wT q
tot wq 45 , qw qq 15 , qw qq 25 , 34 ,
45 1
1wq qq qq qq 45 , ,
5
(A.26)
rrΔH rrΔH rrΔH rrΔHΔHJ
+
(+ +(+ + + + + ) )
IV IV wT II wT II II wT III III wT wT w
tot wq 55 , qw qq 15 , qw qq 25 , qw qq 45 , 45 ,
We are now in a position to sum forces across the system, keeping in mind that
r
r=
qw wq and
Δ
H=
0
wT55 ,
. The expressions for the total
resistivities
r
ij
tot are:
r
rrr r=++ +
I II III IV
qq
tot qq qq qq qq (A.27)
r
rrΔH rrΔH=+ ++
IIwT II II wT
qw
tot qw qq 15 , qw qq 25 , (A.28)
rrΔH rrΔHrrrΔH r rrΔH
+
+++ =+++(+)
III III wT IV IV wT II wT II I II wT
qw qq 45 , qw qq 55 , wq
tot wq qq 12 , wq qq qq 23 , (A.29)
r r r ΔH r r r r Δ H r r r r ΔH r r ΔH ΔH
+
+( + ) + +( + + ) = = + +( + )
III I II wT IV I II III wT II wT II wT w
T
wq qq qq 34 , wq qq qq qq 45 , qw
tot ww
tot tot wq 15 , qw qq 15 , 12 , (A.30)
r rΔH rrΔH rrΔHΔH r rΔH rrΔH rrΔHΔH r
rΔH rrΔH rrΔH rrΔHΔH rrrr
rΔHrΔHrΔHrΔH rΔH rΔH
H rΔH
+ + +( + + + ) + + +( + + + ) +
+ +(+ ++ ++ ) =+++
+2·( + + + )+ ( ) + ( )
+( )+( )
II II wT IIwT II II wT wT III III wT II wT II II wT wT IV
IV wT IIwT II II wT III III wT wT I II III IV
IwT II wT III wT IV wT IwT II wT
III wT IV wT
ww wq 25 , qw qq 15 , qw qq 25 , 23 , ww wq 45 , qw qq 15 , qw qq 25 , 34 , ww
wq 55 , qw qq 15 , qw qq 25 , qw qq 45 , 45 , ww ww ww ww
wq 15 , wq 25 , wq 45 , wq 55 , qq 15 , 2qq 25 ,
2
qq 45 , 2qq 55 , 2
These equations enable us to construct the global resistivities to transport of mass and heat, from knowledge of local properties. We observe that the
Onsager relations apply also to the global description.
Appendix B. Thermodynamic consistency
Several computations can be done to check the model for thermodynamic consistency. It follows a detailed description of the conditions reported
in the paper.
L. Keulen et al. Journal of Membrane Science 524 (2017) 151–162
160
1. The entropy balance and the entropy production The entropy production, as calculated from the entropy balance, Eq. (B.1) must be the same as
calculated from the overall uxes and forces, Eq. (B.2).
σJ
T
J
TST STJ=+( ( )− ( ))
balance
qn
out
q
in
wn w w
tot ,,1
,1 ,11 (B.1)
σΔ
TJΔμ T
TJ=1′− ()
JX qn ww
tot 1n ,
1n 1
1
(B.2)
The equations express the entropy production accumulated until volume no n. For the whole system, this amounts to.
σσ σσ=+ +
TT
s
k
n
Tk
h
T
s
tot
=2
−2
,
III
1111
(B.3)
σσ σσ=+ +
TT
s
k
n
Tk
h
T
s
tot
=1
−2
,
III
(B.4)
2. The local entropy production. The local entropy production is everywhere non-negative, meaning that:
σJX=≥0
k
n
ii
=1
(B.5)
The uxes and driving forces refer here to the control volume in question.
3. The overall driving forces. The thermal and chemical driving forces across a control volume must add to the overall driving forces over the
membrane. The thermal driving forces are calculated from Eqs. (B.6) to (B.8). Points a and b indicate the left and right side, respectively, of a CV.
Δ
TT T
1=11
n
1n
1
(B.6)
Δ
TrJ rJ
1=′+
qn w1n qq
tot ,qw
tot
(B.7)
Δ
TΔT
1=1
k
n
k
1n
=1
−1
ab (B.8)
The total chemical driving force over the membrane is calculated from Eq. (5). It must be equal to the sum of the local chemical driving forces of all
CV's calculated from the proper resistivities and uxes. The chemical potential dierence is calculated at constant temperature T
1
.
Δμ T
T
μT μT
T
()
=− ()− ()
wwn w
1n 1
1
,1,1 1
1(B.9)
Δμ T
TrJ rJ()
=′+
w
wq qn
w
1n 1
1
tot ,tot
tot
(B.10)
Δμ T
T
Δμ T
T
()
=− ()
w
k
nk
w
1n 1
1=1
−1
ab 1
1
(B.11)
4. The resistivity coecients. The resistivity matrix on any level of description must obey the Onsager relations and have a positive denite
determinant [13].
r
rrr rr=−
0
qw wq qq ww qw wq
(B.12)
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Membrane reactors are increasingly replacing conventional separation, process and conversion technologies across a wide range of applications. Exploiting advanced membrane materials, they offer enhanced efficiency, are very adaptable and have great economic potential. There has therefore been increasing interest in membrane reactors from both the scientific and industrial communities, stimulating research and development. The two volumes of the Handbook of membrane reactors draw on this research to provide an authoritative review of this important field. Volume 1 explores fundamental materials science, design and optimisation, beginning with a review of polymeric, dense metallic and composite membranes for membrane reactors in part one. Polymeric and nanocomposite membranes for membrane reactors, inorganic membrane reactors for hydrogen production, palladium-based composite membranes and alternatives to palladium-based membranes for hydrogen separation in membrane reactors are all discussed. Part two goes on to investigate zeolite, ceramic and carbon membranes and catalysts for membrane reactors in more depth. Finally, part three explores membrane reactor modelling, simulation and optimisation, including the use of mathematical modelling, computational fluid dynamics, artificial neural networks and non-equilibrium thermodynamics to analyse varied aspects of membrane reactor design and production enhancement. With its distinguished editor and international team of expert contributors, the two volumes of the Handbook of membrane reactors provide an authoritative guide for membrane reactor researchers and materials scientists, chemical and biochemical manufacturers, industrial separations and process engineers, and academics in this field. Considers polymeric, dense metallic and composite membranes for membrane reactors. Discusses cereamic and carbon for membrane reactors in detail. Reactor modelling, simulation and optimisation is also discussed.
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General Theory: The Entropy Production for a Homogeneous Phase The Excess Entropy Production for the Surface Flux Equations and Onsager Relations Transport of Heat and Mass Transport of Mass and Charge Applications: Evaporation and Condensation A Non-Isothermal Concentration Cell Adiabatic Electrode Reactions The Formation Cell Modeling the Polymer Electrolyte Fuel Cell The Impedance of an Electrode Surface The Non-Equilibrium Two-Phase van der Waals Model and other chapters.
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The framework of non-equilibrium thermodynamics (NET) is used to derive heat and mass transport equations for pervaporation of a binary mixture in a membrane. In this study, the assumption of equilibrium of the sorbed phase in the membrane and the adjacent phases at the feed and permeate sides of the membrane is abandoned, defining the interface properties using local equilibrium. The transport equations have been used to model the pervaporation of a water-ethanol mixture, which is typically encountered in the dehydration of organics. The water and ethan