Effectiveness–mass transfer units (–MTU) model of an exchanger

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Effectiveness–mass transfer units (–MTU) model of an
ideal pressure retarded osmosis membrane mass
exchanger
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Citation
Sharqawy, Mostafa H., Leonardo D. Banchik, and John H.
Lienhard. “Effectiveness–mass Transfer Units (–MTU) Model of
an Ideal Pressure Retarded Osmosis Membrane Mass
Exchanger.” Journal of Membrane Science 445 (October 2013):
211–19.
As Published
http://dx.doi.org/10.1016/j.memsci.2013.06.027
Publisher
Elsevier
Version
Author's final manuscript
Accessed
Thu May 26 21:39:26 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/102433
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Creative Commons Attribution-NonCommercial-NoDerivs
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Effectiveness-Mass Transfer Units (ε-MTU) Model of an Ideal
Pressure Retarded Osmosis Membrane Mass Exchanger
Mostafa H. Sharqawy1, Leonardo D. Banchik2, John H. Lienhard V*, 2
1
Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals,
Dhahran 31261, Saudi Arabia
2
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge,
MA 02139-4307, USA
______________________________________________________________________________
Abstract
Many membrane-based systems, such as reverse osmosis (RO), forward osmosis (FO)
and pressure retarded osmosis (PRO), are being used in desalination, water treatment, and
energy production. These systems work on the basis of mass transfer through a semipermeable membrane which allows for the permeation of water while rejecting salts and
other substances. The membrane-based devices are essentially mass exchangers which
are analogous to heat exchangers. The driving potentials in these mass exchangers are the
concentration and pressure differences, whereas in heat exchangers the driving potential
is the temperature difference. Closed form solutions of the permeation rate through an
ideal PRO mass exchanger are obtained for parallel and counter flow configurations. The
recovery ratio (RR) is obtained as a function of dimensionless parameters such as the
number of mass transfer units (MTU), mass flow rate ratio (MR), and osmotic pressure
ratio (SR). The resulting mathematical expressions form an effectiveness-NTU model for
osmotic mass exchangers. These expressions are analogous to those for heat exchangers
and can be used as an initial design for PRO membrane based mass exchange devices.
Keywords
Membrane mass exchanger, pressure retarded osmosis, effectiveness, mass transfer units
*
Corresponding author
Email address: [email protected] (John H. Lienhard V), Tel. +1 617-253-3790
1
Nomenclature
A
Am
B
C
M
m
P
R
w
Greek symbols
water permeability coefficient
total membrane surface area
molality – moles of solutes per kg of solvent
modified van ’t Hoff coefficient
molar weight
mass flow rate
hydraulic pressure
ideal gas constant
salinity - grams of solutes per kg of solution
kg m-2 s-1 kPa-1
m2
mol kg-1
kPa kg g-1
kg mol-1
kg s-1
kPa
kJ mol-1 K-1
g kg-1



osmotic pressure
density
osmotic coefficient
kPa
kg m-3
per unit area
m-2
Superscripts
’’
Subscripts
d
f
i, in
j
o, out
p
pure
recipe
s
solutes
Abbreviations
CF
ECP
ICP
MR
MTU
NTU
PRO
RO
RR
SR
draw
feed
inlet
jth solute in a solution
outlet
permeate
refers to pure water
corresponds to a reference of seawater constituents
salt
solutes
concentration factor
external concentration polarization
internal concentration polarization
mass flow rate ratio
mass transfer units
number of transfer units
pressure retarded osmosis
reverse osmosis
recovery ratio
osmotic pressure ratio
2
1. Introduction
Pressure retarded osmosis (PRO) systems are mass exchangers which are currently
receiving great attention for their capabilities to produce renewable power from two
streams of different salinities. In a PRO system, the higher salinity solution is called the
draw stream and the lower salinity solution is called the feed stream. For power
production, seawater and river water are typically used as the draw and feed streams
respectively. These streams enter a mass exchanger where a semi-permeable membrane
allows water to pass through, but not salts. The difference in osmotic pressure drives pure
water (permeate) from the feed stream, through the membrane, and into the pressurized
draw stream to dilute it. The pressurized diluted draw stream can be depressurized
through a turbine to produce power. In this paper, we investigate the performance of the
PRO exchanger relative to the amount of permeate and develop a sizing methodology
similar to that commonly used for heat exchangers.
The concept of PRO was first proposed by Loeb [1]. Since then, numerous
mathematical models have been developed and experiments implemented to determine
the work and permeate flux performance of PRO membranes. Mehta and Loeb [2] were
early to recognize the significance of internal concentration polarization, which acts as a
resistance to mass transfer inside the support layer of the membrane. Lee et al. [3] more
rigorously investigated concentration polarization in PRO membranes and developed
equations to determine the maximum power density and flux for zero-dimensional ideal
membranes and membranes with internal concentration polarization (ICP). Two decades
later, McCutcheon and Elimelech [4] developed a model which incorporates internal and
3
external concentration polarization (ECP) for a zero-dimensional FO and PRO
membrane. Most recently, Tan and Ng [5] revisited the FO and PRO transport equations
to include ECP and ICP resistances, which consider the interaction of the solute and the
membrane and consider additional draw solutes. Currently, PRO related research is being
conducted to determine the effect of fouling on flux performance [6-8] and the viability
of novel materials for new membranes and hollow fibers [9-11]. The performance of oneand two-dimensional forward and pressure retarded osmosis exchangers have also been
numerically investigated very recently [12-16].
The operation of an osmotic mass exchanger looks very similar to that of a heat
exchanger when the systems are compared. In the heat exchanger schematic drawing
shown in Fig. 1a, the temperature difference between a hot and cold fluid is the driving
potential for heat transfer. The resistance to heat flow per unit area is the reciprocal of the
overall heat transfer coefficient, U. In the osmotic mass exchanger shown in Fig. 1b, the
osmotic pressure difference between a concentrated and a diluted stream, or draw and
feed stream, is the driving potential for mass transfer. The resistance to mass transfer per
unit membrane area for an ideal membrane is the reciprocal of the water permeability
coefficient, A. For PRO systems, the driving potential, and thus the amount of mass
transfer, is retarded by the hydraulic pressure difference, Pd - Pf which is greater than zero
and less than the maximum osmotic pressure difference in the exchanger. For FO
systems, this hydraulic pressure difference is equal to zero.
4
For heat exchangers, the effectiveness - number of transfer units (ε-NTU) method
developed by Kays and London [17] is a well-known design technique that determines
the required surface area of a heat exchanger for a fixed effectiveness and inlet
conditions. The method uses three dimensionless groups: the effectiveness, which is the
ratio of actual heat transfer to the maximum heat exchange possible; a heat capacity ratio,
which is the lower heat capacity divided by the higher heat capacity; and the number of
transfer units, which is an effective size of the heat exchanger.
This paper proposes an -MTU method for a PRO mass exchanger. The local
transport equation for permeate flow in a zero-dimensional exchanger is combined with
conservation of mass and a linearized equation for osmotic pressure to develop
dimensionless expressions for parallel-flow and counterflow PRO exchangers. The
expressions are closed-form solutions which relate dimensionless performance
parameters of the exchanger to the effective size and input stream properties. The
effectiveness of the PRO exchanger is defined as a novel performance parameter. The
dimensionless groups used in the closed form solutions are discussed, and an analogy to
the heat exchanger groups is made. A numerical model which uses a nonlinear osmotic
pressure function is also implemented in order to assess the errors associated with
linearizing the osmotic pressure function.
5
2. PRO Mass Exchanger Model
Figure 2 is a schematic drawing of a membrane-based mass exchanger device. A feed
solution with low salt concentration (low salinity) flows through a channel where one
side of the channel has a semi-permeable membrane. On the other side of the membrane ,
a draw solution with higher salt concentration flows in the same direction (Fig. 2a,
parallel-flow configuration) or in the opposite direction (Fig. 2b counterflow
configuration). The inlet and outlet conditions of both feed and draw streams are given as
the mass flow rate, osmotic pressure, and pressure as indicated in Fig. 2. The membrane
characteristics are given as the total membrane area (Am) and water permeability
coefficient (A) of the membrane material. The model makes the following assumptions:

Concentration polarization effects are neglected and the salt concentration near
the membrane surface is equal to the bulk concentration of the flow stream.

Pressure drop through the flow channel is negligible on both the feed and the
draw side. Hence the hydraulic pressure difference between the draw side and
feed side (P) is fixed over the length of the exchanger.

The salt rejection is 100% and only pure water flows through the membrane.

Within the operating salinity range of the PRO exchanger, the osmotic pressure
follows van ’t Hoff’s law so that it is linearly proportional to the stream salinity
(see Appendix Section A.1; the constant of proportionality may differ under
different operating conditions).
6
2.1 Parallel-flow configuration PRO model
The differential permeate flow rate in a PRO parallel-flow configuration is given by
dmp = A ´ ( Dp - DP) ´ dAm
(1)
where
m p is the permeate mass flow rate through the membrane in kg/s,
A is the modified1 water permeability coefficient of the membrane in kg/m2 s kPa,
 is the osmotic pressure difference between the draw and feed (d – f) in kPa,
P is the hydraulic pressure difference between the draw and feed (Pd – Pf) in kPa, and
Am is the membrane surface area in m2.
Using van ’t Hoff’s law for osmotic pressure
Dp = p d - p f = C ( wd - w f )
(2)
where w is the salt concentration or salinity (mass of solutes per total mass of solution) in
g/kg and C is a modified van ’t Hoff coefficient, it follows that
dmp = A éëC ( wd - w f ) - DPùû dAm
(3)
Under the assumed condition of 100% salt rejection, only pure water permeates though
the membrane; hence, the salinity of the permeate is zero. Applying conservation of
solutes to the feed stream between the inlet and any arbitrary location along the flow
channel yields
ms, f = m f ,in ´ w f ,in = m f ´ w f
(4)
1
It is important to note that the water permeability coefficient (A) is often given in units of m/s-bar, or m/satm, or L/m2-hr-bar [3, 4], which is the permeate water volume flux per unit pressure difference; however,
for the present model, we express this coefficient on a mass basis (equivalent to multiplying it by the
density of pure water and some SI conversion factors).
7
For the same arbitrary control volume, conservation of total mass requires that
m f ,in = m f + mp
(5)
Substitution of Eq. (5) into Eq. (4) yields
wf =
m f ,in ´ w f ,in
m f ,in - m p
(6)
Similarly applying conservation of solutes and total mass on the draw side for a parallel
configuration yields
wd =
md,in ´ wd,in
md,in + m p
(7)
Substituting Eq. (6) and (7) into Eq. (3) yields
é æm ´w
ù
m f ,in ´ w f ,in ö
d,in
d,in
÷ - DPú dAm
dmp = A êC çç
m f ,in - mp ÷ø
êë è md,in + m p
úû
(8)
We now proceed to cast Eq. (8) in a dimensionless form. Four dimensionless parameters
are introduced for this purpose.
Recovery ratio, RR
RR º
mp
m f ,in
(9)
The recovery ratio is a primary performance metric of the PRO mass exchanger as it
represents the amount of pure water recovered from the feed stream. In so far as the inlet
mass flow rate is greater than the maximum amount of permeate that can be recovered,
the recovery ratio should not be confused with the effectiveness which will be described
in Section 3.
8
Mass flow rate ratio, MR
MR º
md,in
m f ,in
(10)
The mass flow rate ratio is the ratio of the mass flow rate of the draw solution to that of
the feed solution at the inlet of the PRO mass exchanger.
Osmotic pressure ratio, SR
For the draw side:
SRd º
p d,in
DP
(11)
For the feed side:
SR f º
p f ,in
DP
(12)
The osmotic pressure ratio is the ratio of the osmotic pressure at the draw or feed inlet to
the hydraulic pressure difference. For PRO exchanger operation, SRd will always be
greater than SRf.
Mass Transfer Units, MTU
MTU º
Am A
m f ,in DP
(13)
The number of mass transfer units (MTU) is a dimensionless parameter for a membrane
mass exchanger similar to the number of transfer units (NTU) used in heat exchanger
design. The total membrane area, Am, is analogous to the total heat exchanger surface
area and A is the overall water permeability coefficient, which is analogous to the overall
heat transfer coefficient in heat exchangers. Therefore, the MTU in the membrane-based
9
mass exchanger will play the same role that NTU plays in the -NTU analysis of heat
exchangers.
Substituting Eqs. (10 – 13) into Eq. (8) yields,
æ MR ´ SRd SR f
ö
dRR = ç
-1÷ dMTU
è MR + RR 1- RR ø
(14)
Equation (14) can be integrated as follows:
RR
ò MR ´ SR
0
1
SR f
-1
MR + RR 1- RR
dRR = MTU
(15)
d
Equation (15) can be simplified into
RR
( MR + RR) (1- RR) dRR = MTU
ò ( RR - a ) ( RR - b )
(16)
0
where
2
ù
1é
a = ê(1+ MR ( SRd -1) + SR f ) - ( -1- MR ( SRd -1) - SR f ) - 4MR ( SRd - SR f -1) ú (17)
û
2ë
2
ù
1é
b = ê(1+ MR ( SRd -1) + SR f ) + ( -1- MR ( SRd -1) - SR f ) - 4MR ( SRd - SR f -1) ú (18)
û
2ë
Therefore the integration of Eq. (16) will be
MTU =
( b -1) ( MR + b ) ln æ b - RR ö - (a -1) ( MR + a ) ln æ a - RR ö - RR
ç
÷
ç
÷
è a ø
è b ø
(a - b )
(a - b )
(19)
Therefore, Eq. (19) can be used in the design of a membrane mass exchanger where the
required mass transfer units (hence the membrane area) is given as an explicit relation.
An additional dimensionless parameter which may be useful to a designer is the
concentration factor. The concentration factor is the ratio of the outlet salinity of a stream
10
to the inlet salinity. If a designer is limited to output brine below a certain salinity from
the PRO exchanger, the concentration factor can be useful in determining the maximum
MTU allowable. By considering a pure permeate and applying conservation of solution
and solutes to the draw and feed streams separately, the expressions for the draw and feed
concentration factors can be determined, as given by Eqs. (20) and (21).
For the draw side:
CFd º
wd,out
MR
=
wd,in MR + RR
(20)
For the feed side:
CFf º
w f ,out
1
=
w f ,in 1- RR
(21)
Figures 3 and 4 show the variation of the recovery ratio (RR) and the concentration factor
(CF) with the mass transfer units (MTU), respectively, at different mass flow rate ratios
for the parallel-flow configuration.
2.2 Counterflow configuration PRO model
The transport model Eq. (1) and the van ’t Hoff osmotic pressure model Eq. (2) will again
be used to describe the permeate flow rate in the counterflow configuration shown in Fig.
2b. The differential permeate flow rate is given by Eq. (3) and the conservation of
solution and solute for the feed side between the inlet and any arbitrary location along the
flow channel leads to Eq. (6) of the feed salinity. Applying a similar conservation of
solution and solute on the draw side for the counterflow configuration in the same
arbitrary direction as taken for the feed stream gives a result that differs from that for the
parallel-flow configuration (Eq. 7).
11
wd =
md,out ´ wd,out
md,out - m p
(22)
Substituting Eqs. (6) and (22) into Eq. (3),
é m ´p
ù
m ´p
dmp = A ê d,out d,out - f ,in f ,in - DPú dAm
m f ,in - mp
êë md,out - m p
úû
(23)
Using the same dimensionless parameters as used for the parallel configuration (i.e. RR,
MR, SRf, and MTU), Eq. (23) can be rewritten in a dimensionless form. However, two
additional dimensionless parameters are also required; they are defined as follows.
Outlet mass flow rate ratio, MRo
MRo º
md,out
m f ,in
(24)
Osmotic pressure ratio at draw outlet, SRd,o
SRd,o º
p d,out
DP
(25)
Since we are interested in expressing the recovery ratio (RR) as a function of the inlet
flow conditions, we proceed to develop relations between the outlet dimensionless groups
defined by Eqs. (24 – 25) and the inlet dimensionless groups defined by Eqs. (10 – 12).
The mass flow rate ratio defined in Eq. (10) can be written as a function of the outlet
mass flow rate ratio defined by Eq. (24) and the recovery ratio defined by Eq. (9) as
follows.
MR =
md,in md,out - m p
=
= MRo - RR
m f ,in
m f ,in
(26)
12
Therefore,
MRo = MR + RR
(27)
Similarly, the osmotic pressure ratio of the outlet draw stream can be written as a
function of the osmotic pressure ratio of the inlet draw stream, the mass flow rate ratio,
and the recovery ratio as follows.
SRd,o = SRd
MR
MR + RR
(28)
Using these dimensionless groups, Eq. (23) can be rewritten in a dimensionless form:
æ MR ´ SRd,o SR f
ö
dRR = ç o
-1÷ dMTU
è MRo - RR 1- RR ø
(29)
This result can be integrated
RR
ò MR ´ SR
0
1
SR f
-1
MRo - RR 1- RR
o
dRR = MTU
(30)
d,o
Equation (30) can be simplified into
RR
( MRo - RR) (1- RR) dRR = -MTU
ò ( RR - a¢) ( RR - b ¢)
(31)
0
where
2
ù
1é
a¢ = ê(1+ MRo (1- SRd,o ) + SR f ) - ( -1+ MRo ( SRd,o -1) - SR f ) - 4MRo (1- SRd,o + SR f ) ú
û
2ë
(32)
2
ù
1é
b ¢ = ê(1+ MRo (1- SRd,o ) + SR f ) + ( -1+ MRo ( SRd,o -1) - SR f ) - 4MRo (1- SRd,o + SR f ) ú
û
2ë
(33)
Therefore the integration of Eq. (31) will be
13
MTU =
( b ¢ -1) ( b ¢ - MRo ) ln æ b ¢ - RR ö - (a¢ -1) (a¢ - MRo ) ln æ a¢ - RR ö - RR
ç
÷
ç
÷
è a¢ ø
è b¢ ø
(a¢ - b ¢)
(a¢ - b ¢)
(34)
Therefore, Eq. (34) combined with Eqs. (27) and (28) can be used in the design of a
membrane mass exchanger where the required mass transfer units, effectively the
membrane area, is given as an explicit relation of the form
MTU = f ( RR, MR, SR f , SRd )
(35)
Figures 5 and 6 show the variation of the recovery ratio (RR) and the concentration
factor (CF) with the mass transfer units (MTU) respectively at different mass flow rate
ratios (MR) for the counterflow configuration. The concentration factors for the feed and
draw stream as defined in Eqs. (20) and (21) are applicable to the counterflow
configuration as well.
It can be seen by comparing Figs. 3 and 5 that, for each contour of MR, the recovery
ratio is higher for the counterflow case than that of parallel-flow case. This is an expected
result which is found in PRO literature [11, 12, 15] and is analogous to similar results for
heat exchangers.
3. PRO Mass Exchanger Effectiveness ( – MTU Model)
The effectiveness of the PRO system can be defined as the ratio of the permeate flow rate
to the maximum permeate flow rate, which occurs when MTU is increased to infinity.
The effectiveness can also be defined as the recovery ratio divided by the maximum
recovery ratio. Note that the maximum permeate flow rate is not the inlet feed flow rate.
The maximum accumulated permeated water occurs when the net driving pressure ( P) to draw water from the feed stream has decreased to zero at one end of the mass
14
exchanger. The following is a derivation of the maximum permeation flow rate and,
hence, the effectiveness of the PRO exchanger.
3.1 Parallel-flow PRO Effectiveness
Using Eq. (1), the maximum permeate in the case of parallel flow configuration will
occur when the hydraulic pressure difference is equal to the osmotic pressure difference
at the outlet.
Dp out = p d,out - p f ,out = DP
(36)
Using the van ’t Hoff model
C ( wd,out - w f ,out ) = DP
(37)
Applying conservation of solutes and solution on the draw side, one can find that
wd,out =
MR
wd,in
MR + RR
(38)
Similarly on the feed side, one can find that
w f ,out =
1
w f ,in
1- RR
(39)
Substituting Eq. (38) and (39) into Eq. (37) and using the dimensionless groups defined
earlier
SR f
MR× SRd
=1
MR + RR 1- RR
(40)
Solving Eq. (40) to find the maximum recovery ratio, one can find that there are two
solutions
RRmax,1 = a
(41a)
15
RRmax,2 = b
(41b)
where  and  are given by Eqs. (17) and (18), respectively. We notice from Eqs. (17)
and (18) that  is always less than 1 and  is always greater than 1. The recovery ratio
must be less than one. Therefore, the maximum recovery ratio is equal to . Now the
effectiveness is defined as
e=
RR
RRmax
(42)
Hence,
RR = e RRmax = e a
(43)
By substituting Eq. (43) into Eq. (19), an expression for MTU as a function of the
effectiveness can be obtained:
MTU =
( b -1) ( MR + b ) ln æ b - a e ö - (a -1) ( MR + a ) ln 1- e - a e
( )
ç
÷
è b ø
(a - b )
(a - b )
(44)
Figure 7 shows the effectiveness changing with the mass transfer units for varying mass
flow rate ratios.
3.2 Counterflow PRO Effectiveness
Using Eq. (1), the maximum permeate in the case of counterflow configuration will occur
when the hydraulic pressure difference is equal to the osmotic pressure difference at the
right side or the left side of the exchanger schematic shown in Fig. 2b. Therefore, there
are two conditions at which the driving potential for permeate flow will become zero.
16
p d,out - p f ,in = DP
(45)
Using the van ’t Hoff model and applying conservation of solution and solute this
condition will lead to
RRmax,1 =
MR SRd
- MR
1+ SR f
(46)
The other condition will lead to
p d,in - p f ,out = DP
(47)
And
RRmax,2 =1-
SR f
SRd -1
(48)
Since there are two solutions for the maximum recovery ratio, we should take the
minimum value, hence:
RRmax = min RRmax,1, RRmax,2
(49)
The effectiveness defined by Eq. (42) and the recovery ratio can be written in terms of
the effectiveness and maximum recovery ratio as given by the first equality of Eq. (43).
Substituting Eq. (43) into Eq. (34), an expression for MTU as a function of the
effectiveness is obtained:
MTU =
( b ¢ -1) ( b¢ - MRo ) ln æ b ¢ - e RRmax ö - (a¢ -1) (a¢ - MRo ) ln æ a¢ - e RRmax ö - e RR
ç
÷
ç
÷
max
è
ø
b¢
a¢
è
ø
(a¢ - b ¢)
(a¢ - b ¢)
(50)
Figure 8 shows the effectiveness changing with the mass transfer units for varying mass
flow rate ratios.
17
It should be noted that by varying the modified van ’t Hoff coefficient of each stream, the
closed-form solutions given in the above sections allow for the two streams entering the
PRO exchanger to have different compositions, such as seawater and ammonia-carbon
dioxide, sodium chloride and pure water, or flowback water and ammonia-carbon
dioxide.
4. Numerical PRO mass exchanger model
The closed form solutions derived in the previous sections required the assumption
that osmotic pressure is a linear function of salinity. While this assumption is valid for
relatively dilute solutions, the variation becomes increasingly nonlinear as the mixture
salinity increases. In this section, a numerical model of a one-dimensional PRO mass
exchanger is developed using a nonlinear function for the osmotic pressure of seawater,
so as to determine the accuracy of the approximate analytical results given earlier. The
model applies a discretized version of the transport equation, Eq. (1), and conservation of
solution and solutes to N membrane elements in series. The number of elements was
increased to 50 at which point the results were grid independent. The total amount of
permeate is calculated by numerically integrating the permeate mass flow rate produced
by all elements. The equations comprised by the numerical model were solved using
Engineering Equation Solver [18], a simultaneous equation solver which iterates to find a
solution to sets of coupled nonlinear algebraic equations. The details of the nonlinear
osmotic pressure function used in this numerical model is given in Banchik et al. [19]
whereas the modified van ’t Hoff coefficient are given in the Appendix.
18
The numerical model is used to estimate the error in the analytical solutions that
results from using a linearized osmotic pressure function (the van ’t Hoff equation). All
other assumptions made for the analytical model are also made for the numerical model.
An additional assumption is that the PRO membranes can withstand arbitrary net driving
pressures. Two representative uses of a PRO system are for power production at a river
delta [20] and for recovering the chemical energy which exists between the rejected brine
of a desalination system and the available seawater [21-23]. Therefore, two numerical
cases are considered: (1) a power production case with seawater and river water,
wd,in = 35 g/kg and w f ,in =1.5g/kg; (2) and an energy recovery case with brine and
seawater, wd,in = 70 g/kg and w f ,in = 35g/kg. The inputs for the numerical model are
given in Table 1. The water permeability coefficient used is representative of a typical
spiral wound forward osmosis membrane [24].
The percent error of the analytical model is given by Eq. (51):
error = 1-
analytical
´100%
numerical
(51)
The inlet salinities given in Table 1 are used to calculate the inlet osmotic pressures
using the nonlinear osmotic pressure function and the modified van ’t Hoff coefficient.
For both cases, the hydraulic pressure difference is set to equal one-half of the maximum
osmotic pressure difference. This assumption requires that SRd - SR f = 2 . To improve the
accuracy of the linear case, we use different values of the modified van ’t Hoff coefficient
to cover ranges of interest. Table 2 lists modified van ’t Hoff coefficients for three ranges
with the coefficient of determination given in the third column. The first and second
19
value of C from Table 2 will be used in the linear model to compare the two numerical
cases.
We first consider the seawater-river water stream combination with a counterflow
exchanger configuration. Figure 9 shows the recovery ratio versus mass transfer units for
contours of the mass flow rate ratio, MR, and for fixed inlet draw and feed salinities. The
inlet salinities are used to calculate the nonlinear and linear osmotic pressure for the draw
and feed stream and the hydraulic pressure difference is determined by assuming that
SRd - SR f = 2 as described in the previous section. It can be seen in Fig. 9 that the errors
associated with linearization are highest for low mass flow rate ratios and for high values
of MTU. For MR = 0.5 and for MTU > 2, the largest error associated with linearization of
the osmotic pressure function is 4.62%. Figures 10 and 11 show the concentration factor
and effectiveness plotted versus MTU for the same contours of mass flow rate ratios and
inlet salinity. The maximum error for the draw and feed stream concentration factors is
2.12% and 20.3%, respectively. The relatively high error of 20.3% results from the
contour of MR = 1 and corresponds to an outlet salinity difference between the numerical
model and the analytical solution of about 1.9 g/kg. The errors are higher for the feed
stream concentration factor because the feed stream salinities are lower, which lends to a
smaller denominator in the error calculation even though the overall difference in outlet
feed salinities may be low. The maximum error for the effectiveness is 4.41%.
Figures 12-14 display the same dimensionless groups as Figs. 9-11 except that the
inlet salinities are now representative of brine and seawater as defined by the energy
20
recovery case 2. The maximum error between models for the recovery ratio versus MTU
of Fig. 12 is 5.37% and corresponds to the lowest mass flow rate ratio of MR = 0.5 at
MTU = 5. For the case 2 concentration factor and effectiveness versus MTU shown in
Figs. 13 and 14, the maximum error is less than 2% and 5%, respectively. Table 3
summarizes the maximum error incurred for each dimensionless variable and for each
case and configuration.
5.
Conclusions
The major conclusions of this paper are as follows:

Closed form analytical solutions for a one-dimensional pressure retarded osmosis
mass exchanger using an ideal membrane were developed. The equations express
the recovery ratio of the membrane as a function of the configuration of the
membrane and four dimensionless groups: two osmotic pressure ratios, a mass
flow rate ratio, and the number of mass transfer units.

A robust analogy exists between heat exchangers and osmotic mass exchangers in
which the effectiveness can be expressed by four dimensionless groups. The new
-MTU model developed for the osmotic mass exchanger can be used as a design
tool for PRO systems using a linearized osmotic pressure function and ideal
membrane characteristics. Combinations of stream compositions can be analyzed
by selecting the appropriate modified van ’t Hoff coefficient to be used in the
dimensionless osmotic pressure ratio.

In order to develop closed-form analytical solutions of the PRO exchanger
performance, osmotic pressure must be approximated as a linear function of
21
salinity. Modest errors are incurred in doing so. Two cases were modeled to
determine the errors associated with linearization: a power production case with
inlet seawater and river water, and a desalination energy recovery case with inlet
brine and seawater. For both cases the hydraulic pressure was set to equal half of
the maximum osmotic pressure difference. Also for each case, the modified van ’t
Hoff coefficient is varied to more closely approximate the nonlinear osmotic
pressure for the range of salinity. For the power production case, the maximum
error is less than 5% for the recovery ratio and effectiveness of a counterflow
PRO exchanger. For the energy recovery case, the maximum error is less than
5.5%.
Acknowledgments
The authors would like to thank King Fahd University of Petroleum and Minerals in
Dhahran, Saudi Arabia, for funding the research reported in this paper through the Center
for Clean Water and Clean Energy at MIT and KFUPM.
22
Appendix
A.1 The Modified van ’t Hoff Coefficient
A nonlinear seawater osmotic pressure function is developed using the relation with
the osmotic coefficient of seawater. The osmotic coefficient of seawater is given by a
piecewise function which uses Brønsted’s equation for 0 ≤ w < 10 g/kg and an empirical
correlation given by Sharqawy et al. [25] for 10 ≤ w < 120 g/kg. For details regarding the
development of the nonlinear seawater osmotic coefficient function, please refer to the
Appendix of Banchik et al. [19].
The nonlinear osmotic pressure of seawater as a function of salinity in g/kg is shown
in Fig. (A.1). The modified van ’t Hoff coefficient (C) can be determined as the slope of
the nonlinear osmotic pressure between specific ranges of interest. The osmotic pressure
can be approximated as the product of the modified van ’t Hoff coefficient and the
salinity of the solution using Eq. (A.1).
p » Cw
(A.1)
Using a best-fit least squares method between the range of 0 and 35 g/kg, the modified
van ’t Hoff coefficient is determined to be 73.07 kPa-kg/g at a temperature of 25oC. This
linear model represented by Eq. (A.1) can be used for a salinity range of 0 to 35 g/kg,
which is the typical range for power production at a river delta with a PRO exchanger.
For salinities between 35 and 70 g/kg, the modified van ’t Hoff coefficient is determined
to be 76.76 kPa-kg/g which is the typical range for power production using seawater and
disposed brine from a seawater desalination plant. For salinities between 70 and 105
g/kg, the modified van ’t Hoff coefficient is determined to be 82.65 kPa-kg/g.
23
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25
Fig. 1a Schematic drawing of a counterflow heat exchanger. A hot and cold stream
enter the exchanger where a temperature difference drives heat transfer.
26
Fig. 1b Schematic drawing of a counterflow osmotic mass exchanger. A concentrated
and dilute stream, draw and feed, enter the exchanger where a salinity difference
drives mass transfer.
27
Fig. 2a Schematic diagram of a PRO mass exchanger in parallel-flow configuration
28
Fig. 2b Schematic diagram of a PRO mass exchanger in counterflow configuration
29
Fig. 3 Recovery ratio vs. mass transfer units at different mass flow rate ratios for a
parallel-flow configuration
30
Fig. 4 Concentration factor of the feed (dashed curves) and draw (solid curves)
stream vs. mass transfer units at different mass flow rate ratios for a parallel-flow
configuration
31
Fig. 5 Recovery ratio vs. mass transfer units at different mass flow rate ratios for a
counterflow configuration
32
Fig. 6 Concentration factor of the feed (dashed curves) and draw (solid curves)
stream vs. mass transfer units at different mass flow rate ratios for a counterflow
configuration
33
Fig. 7 Effectiveness vs. mass transfer units at different mass flow rate ratios for a
parallel-flow configuration
34
Fig. 8 Effectiveness vs. mass transfer units at different mass ratios for a counterflow
configuration
35
Fig. 9 Recovery ratio vs. mass transfer units for varying mass flow rate ratios,
counterflow configuration, and fixed inlet salinities representative of seawater and
river water. Lines are from analytical Eq. (34), and the points are the results of a
numerical model using a nonlinear osmotic pressure function.
36
Fig. 10 Concentration factor vs. mass transfer units for varying mass flow rate
ratios, counterflow configuration, and fixed inlet salinities representative of
seawater and river water. Solid and dashed lines are from analytical Eqs. (20) and
(21), respectively, and the points are the results of a numerical model using a
nonlinear osmotic pressure function.
37
Fig. 11 Effectiveness vs. mass transfer units for varying mass flow rate ratios,
counterflow configuration, and fixed inlet salinities representative of seawater and
river water. Lines are from analytical Eq. (50), and the points are the results of a
numerical model using a nonlinear osmotic pressure function.
38
Fig. 12 Recovery ratio vs. mass transfer units for varying mass flow rate ratios,
counterflow configuration, and fixed inlet salinities representative of brine and
seawater. Lines are from analytical Eq. (34), and the points are the results of a
numerical model using a nonlinear osmotic pressure function.
39
Fig. 13 Concentration factor vs. mass transfer units for varying mass flow rate
ratios, counterflow configuration, and fixed inlet salinities representative of brine
and seawater. Solid and dashed lines are from analytical Eqs. (20) and (21),
respectively, and the points are the results of a numerical model using a nonlinear
osmotic pressure function.
40
Fig. 14 Effectiveness vs. mass transfer units for varying mass flow rate ratios,
counterflow configuration, and fixed inlet salinities representative of brine and
seawater. Lines are from analytical Eq. (50), and the points are the results of a
numerical model using a nonlinear osmotic pressure function.
41
Fig. A.1 Seawater osmotic pressure versus salinity for T = 25 °C shown as a solid
curve. The points indicate the linear osmotic pressure approximation using varying
modified van ’t Hoff coefficients.
42
Table 1 Data input for numerical model
Input
Value/Range
Ambient temperature, To
25 °C
2
Modified water permeability coefficient, A
3.07´10-6 kg/m -s-kPa
Feed mass flow rate, m f ,in
1 kg/s
Inlet draw salinity, wd ,in
70 g/kg and 35 g/kg
Inlet feed salinity, w f ,in
Mass flow rate ratio, MR
35 g/kg and 1.5 g/kg
0.5, 0.75, 1, 2, 3
Hydraulic pressure, P
(p d,in - p f ,in ) / 2
Trans-membrane pressure difference, P
Membrane area, Am
1.24 – 1.47 MPa
0 – 9.45´105 m2
43
Table 2 Modified van ’t Hoff coefficients over three ranges for determining osmotic
pressure as a function of salinity at T=25°C
C [kPa-kg/g]
73.07
76.76
82.65
Range [g/kg]
0 - 35
35 - 70
70 - 105
R2
0.9997
0.9926
0.9691
44
Table 3 Maximum error resulting from linearized osmotic pressure
Case 1: Seawater + River Water
RR
CFd
CFf
ε
Parallel-flow
Counterflow
Case 2: Brine + Seawater
Parallel-flow
Counterflow
3.69%
4.62%
RR
2.34%
5.37%
1.61%
2.12%
CFd
0.43%
1.26%
3.41%
20.03%
CFf
0.45%
1.68%
3.36%
4.41%
ε
1.71%
4.95%
45
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