A summary of governing equations and conditions considering both ECP and ICP .
Forward osmosis (FO) as an osmotically driven membrane process is severely affected by the concentration polarization phenomenon on both sides of the membrane as well as inside the support layer. Though the effect of internal concentration polarization (ICP) in the porous support on the draw solution side is far more pronounced than that of the external concentration polarization (ECP), still the importance of ECP cannot be neglected. The ECP becomes particularly important when the feed flow rate is enhanced to increase the permeation flux by increasing the agitation and turbulence on the membrane surface. To capture the effect of ECP a suitable value of mass transfer coefficient must be determined. In this chapter, an FO mass transport model that accounts for the presence of both ICP and ECP phenomena is first presented on the basis of solution-diffusion model coupled with diffusion-convection. Then, three methods for the estimation of mass transfer coefficient based on empirical Sherwood (Sh) number correlations, pressure-driven reverse osmosis (RO), and osmosis-driven pressure retarded osmosis (PRO) are proposed. Finally, a methodology for the prediction of water flux through FO membranes using the theoretical model and calculated/measured parameters (hydraulic permeability, salt resistivity of the support layer, and mass transfer coefficient) is presented.
- forward osmosis
- concentration polarization
- mass transfer coefficient
- reverse osmosis
- pressure retarded osmosis
With the increasing application of membrane-based separation processes in desalination and wastewater treatment, vast efforts have been devoted to making them more energy efficient. In the hunt of more economical and efficient method, forward osmosis (FO) has been developed as an alternative to the conventional pressure-driven separation processes like reverse osmosis (RO) and nanofiltration (NF) [1, 2, 3]. FO is an osmotically driven membrane separation process, where water molecules are transferred from a dilute feed solution to a more concentrated draw solution through a semi-permeable membrane which selectively rejects a broad range of dissolved contaminants in the wastewater . The driving force for water transport is the chemical potential difference between the draw and feed solutions, thus eliminating the use of hydraulic pressure and consequently enhances energy efficiency [4, 5, 6].
Besides being energy efficient, FO process is less prone to fouling as compared to pressure-driven NF and RO processes. However, FO suffers from an enhanced concentration polarization effect inside the support layer known as internal concentration polarization (ICP), where the solvent (commonly water) permeates through the support and dilutes the draw concentration at the inner side of the active layer. The ICP reduces the real driving force for mass transfer, thereby reducing the performance of the FO process, significantly [7, 8]. In addition to ICP, FO suffers from an external concentration polarization (ECP). In fact, in a typical pressure driven process, ECP occurs on one side of the membrane (feed side), whereas in the FO, this phenomenon happens on both sides (feed and draw). The polarization that occurs on the feed side is concentrative and is different in nature from the dilutive polarization on the draw side due to incoming permeate flux. The first polarization is called concentrative ECP and the second one that takes places in the draw side is termed as dilutive ECP. The ICP is not affected by the hydrodynamics of the flow and is more severe than the ECP which makes the theoretical study of transport phenomena in an FO process very challenging.
Early attempts to model the mass transfer through an FO membrane was conducted by Lee et al. . They considered the ICP inside the porous support and developed a model to predict the performance of a pressure retarded osmosis (PRO) process. In the PRO process, which is used for energy generation from an osmotic pressure difference, the membranes are oriented in the exact opposite configuration of FO with the active layer facing the draw solution. Later, Loeb et al.  followed the same approach and developed a model for the FO process. McCutcheon et al.  coupled the boundary layer film theory to capture the effect of ECP on the active layer as well as the ICP in the porous support for both FO and PRO processes. Suh and Lee  fine-tuned this model by considering the dilutive ECP phenomenon on the draw side which was neglected by previous researchers. They suggested that the effect of diluted draw solution on the ECP must be taken into account, particularly for low cross-flow velocities and high water flux. Even though the above models provide a comprehensive framework of relationships for the ICP and ECP on both sides of FO membranes and can predict the flux satisfactorily at a particular flow rate, they are not sufficiently sensitive to a change in the feed flow rate.
The change in water flux with a change in the flow rate is captured by the mass transfer coefficient (
Although extensive research has been carried out on the derivation of empirical and semi-empirical Sh number correlations for pressure driven membrane processes (at various operating conditions and spacer geometries) [14, 16, 17], no such efforts have been made to better understand the boundary layer phenomena in an FO process. It is worth mentioning that, based on the film theory the severity of the ECP depends upon the value of mass transfer coefficient. Since the concentration profile in the boundary layer is exponential in nature, a small error in the value of mass transfer coefficient may magnify error to a large extent. Hence, to develop a robust model for the FO process, there is crucial need to find an appropriate correlation of mass transfer coefficient for each specific membrane process with certain hydrodynamic properties of channel and membrane characteristics.
In this chapter an attempt has been made to provide (i) the theoretical background of internal and external concentration polarization phenomena, and (ii) different methods that can be used for the estimation of mass transfer coefficient in the FO process. Since the support layer of a thin film composite FO membrane is made from a porous material (e.g., polysulfone, PSf), having a similar structure and porosity as that of a UF membrane, the literature Sh number correlations might be valid on this side of the membrane. But for the selective layer of the membrane, which is smooth and non-porous, these relationships are not necessarily usable. Hence, more practical methods to get an estimate of the value of mass transfer coefficient on the active side of the membrane in an FO process by (i) RO and (ii) PRO experiments are proposed. These mass transfer coefficients can then be used in the theoretical model to predict the water flux with a change in the feed flow velocity.
2.1. Water flux in FO
Water flux in a pressure-driven membrane separation process is directly proportional to the applied pressure (
2.2. Concentration polarization
In an FO operation, the actual flux is far less than the theoretical flux obtained from Eq. (2) which shows a decline in driving force. On the feed side, where the solvent permeates through the membrane, the solutes are retained by the membrane increasing their concentration on the membrane surface that is referred to as concentrative ECP. The permeate entering the draw side dilutes the draw solution at the membrane surface that is known as diluted ECP. Figure 1(a) and (b) depict concentrative and dilutive ECP, as well as ICP, occurring in FO and PRO processes. Both these phenomena contribute to a decrease in the net osmotic driving force across the membrane and hence lowering the flux.
The ECP can be mitigated by inducing turbulence which enhances the mixing and consequently levels the concentration difference between the bulk and adjacent solution to the membrane surface. However, the concentration polarization in FO is not just limited to ECP. The structure of FO membranes is typically asymmetric, i.e. a thin active layer which governs the molecular transport rate is coated on a porous support that provides mechanical strength. In the FO mode (when active layer and support are facing the feed and draw solutions, respectively), a more severe concentration polarization takes places inside the porous support layer of the membrane, known as ICP. The enhanced dilution of the draw solution inside the porous support contributes to a massive decline in the osmotic pressure difference, thereby decreasing the flux more severely.
2.2.1. Internal concentration polarization
At steady state, the salt leakage (
Applying these boundary conditions a relation for the concentration of solution inside the porous support near the active layer (
Considering a perfect membrane with 100% salt rejection, the value of
The salt concentration ratio is approximately proportional to the osmotic pressure ratio of the solution, which gives:
2.2.2. External concentration polarization
The concentrative ECP occurring on the feed side of the membrane can be captured by using the same differential equation and applying appropriate boundary conditions between the membrane surface and bulk solution on the feed side [1, 7]:
Again for a high solute rejecting membrane
The effect of ICP in the support layer and ECP on the feed side are accounted in Eq. (12). By considering the effect of dilutive ECP on the draw side, a concentration boundary layer forms on the support layer of the membrane and
The following equation is derived for the concentration of solution on the support layer (
Finally, the modified flux equation by incorporating the ICP in the support layer and ECP on both sides of the membrane is acquired as follows:
A summary of the main mass transfer equations, boundary conditions, and concentration relations is presented in Table 1.
|Assumption||Mass transfer equation||Boundary condition||Concentration relations|
|ICP in the support layer|
|ECP on the draw side|
|ECP on the feed side|
A similar analogy can be applied when the process is operated in the PRO mode. In this mode, the feed and draw solutions face the support and active layers, respectively. Hence, the ECP occurs on the draw side and is dilutive in nature, i.e. the draw solution becomes diluted near the membrane surface by the incoming permeate that leads to a decrease in osmotic driving force. The dilutive ECP phenomenon provides the following relation for the ratio of draw solution concentration on the membrane surface (
On the feed side of the membrane, the ICP occurs that increases the concentration of salt inside the porous support and makes it concentrative in nature, thus decreasing the driving force. The modulus for concentrative ICP is given by the following relation:
3. Standard experiments to use the data analysis model
Draw solutions with various concentrations of a particular salt in deionized water are first prepared. Then, the properties of both feed and draw solutions like viscosity, density, diffusion coefficient, and osmotic pressure are measured or taken from literature . The Osmotic pressure of unknown feed and draw solutions can be found experimentally by using automatic osmometers. This instrument estimates the osmotic pressure by measuring the depression in the freezing point of the solution. The osmotic pressure of at least three solutions is measured, and a linear relationship is obtained between the osmotic pressure and the concentration.
The FO experiments are conducted by a cross-flow filtration setup. The schematic diagram of a typical setup is shown in Figure 3. The membrane cell has channels on both sides of the membrane for the flow of feed and draw solutions. The length, width, and depth of the channels should be measured precisely for the calculation of mass transfer coefficient. The effective filtration area of the membrane is measured to calculate the water flux. Feed and permeate spacers are typically used on both draw and feed channels in the cell to provide mechanical support to the membrane. The feed and draw solution flow rates and the temperature of the experiment are maintained at certain values and are used for the calculation of mass transfer coefficients and osmotic pressure, respectively. The water flux through the membrane can be calculated by recording the change in the weight of the draw solution with time using a weighing scale. During the experiment, the conductivity and temperature of both feed and draw solutions must be monitored.
The same setup but opposite configuration is utilized for the PRO experiments. For both FO and PRO experiments, membranes are needed to be kept in the DI water for about 24 hours before experiments. After mounting the membranes in the module, the flow rates of the feed and the draw solutions are adjusted to desired values. The system is allowed to stabilize, and then the change in weight of the draw solution is recorded over time. Due to the change of draw solution concentration, a certain amount of concentrated draw solution needs to be gradually added to the solution. The conductivity of the draw solution is monitored online, and the addition of concentrated solution stopped when the conductivity of the draw solution reaches to the desired concentration of the solution. A similar procedure needs to be followed to increase the concentration the feed solution.
The pure water permeability of the membrane should be acquired using an RO setup with DI water as feed solution. The water flux is measured at different transmembrane pressures, and the slope is obtained as pure water permeability (
4. Estimation of mass transfer coefficient
The value of mass transfer coefficient depends on the hydrodynamics of the flow, applied driving force, water flux through the membrane, characteristics of the membrane (roughness and porosity) and the type of
4.1. Empirical equations based on Sh number
Mass transfer coefficient is a parameter which describes the ratio between the actual mass (molar) flows of species into or out of a flowing fluid and the driving force that creates that flux. Mass transfer coefficient depends on module configuration, solute diffusion coefficient, viscosity, density, and velocity of feed solution . It is related to the Sh number which shows the ratio of the convective mass transfer to diffusive rate and can be defined as follows :
In these equations,
|Flow geometry||Laminar regime (Re < 2000)-(a)||Turbulent regime (Re > 2000)-(b)||Equation|
|Rectangular channels w/o spacers||(22)|
|Rectangular channels w/spacers||(23)|
|Radial cross flow system||(25)*|
Eqs. (22)(a) and (b) are widely used to calculate the mass transfer coefficient in both feed and draw side of FO membrane. However, the implementation of these empirical equations in forward osmosis process has brought some controversial debates. These equations were derived based on ultrafiltration (UF) process which suffers more severely from concentration polarization phenomenon as compared to FO process. Hence, they are not necessarily valid for the evaluation of dilutive and concentrative ECP in FO . Moreover, UF membranes differ from FO ones structurally as the former is porous while the latter is mainly dense composite membranes. Besides, the Sh number is correlated to the frictional factor which might be different for FO and UF processes [14, 19].
It is worth mentioning that Eq. (22)a is only valid where the length of the entry region is equal to 0.029
Laminar boundary layer
Turbulent boundary layer
Hence, mean mass transfer coefficient,
4.2. Evaluating mass transfer coefficient by RO experiment
The film theory is generally applied to capture the effect of the ECP on a membrane surface. Using this theory, the concentration profile near the membrane surface is obtained as a function of permeation flux and mass transfer coefficient:
4.3. Evaluating mass transfer coefficient in the PRO mode
Using DI water as the feed solution in the PRO mode, the water flux through the membrane can be calculated by a reduced form of Eq. (18) as follows:
The mass transfer coefficient can be calculated by rearranging this equation.
5. Flux prediction
The current models developed are mainly focused on finding an accurate value of solute resistivity (
Investigating the current models developed for FO, it was also observed that they are insensitive to a change in the feed flow rate, while our earlier investigations demonstrated that the flux changes moderately with the flow rate . In the previous sections, it was shown that the mass transfer coefficients could be obtained by three methods. Hence, it is suggested that the researchers critically compare the results obtained from the three sets of mass transfer coefficient and utilize the one that increases the sensitivity of the flux results to the feed flow rate.
To start with the modeling of the FO, the hydraulic permeability (
|Draw concentration (M)||Osmotic pressure (draw side) (bar)||Feed concentration (M)||Osmotic pressure (feed side) (bar)||Flux (LMH)|
Obtaining hydraulic permeability (
It is well known that increasing the feed flow rate increases the water flux through the membrane by enhancing the mixing near the membrane surface, thereby reducing the effect of ECP (concentrative ECP in the case of FO). The change in the flow rate is reflected through the change in the mass transfer coefficient. Hence it is recommended to test the sensitivity of the FO developed model to the variation of feed flow rate. As a case study, the experimental results and the model predictions obtained using two mass transfer coefficients, one from Eq. (22) and the other one from RO tests, are presented in Table 4. As can be observed, mass transfer coefficients yield results that match well with experimental data. However, using the values of
|Draw concentration||Feed concentration||Experimental flux (LMH)||Theoretical flux (LMH) (
||Theoretical flux (LMH)* (
|Feed flow 1 LPM||Feed Flow 3 LPM||Feed flow 1 LPM||Feed flow 3 LPM||Feed flow 1 LPM||Feed flow 3 LPM|
|0.25 M||0.05 M||3.9||4.5||3.8||4.2||4.1||4.1|
|0.5 M||0.05 M||5.6||7.7||5.8||6.4||6.1||6.1|
|1 M||0.05 M||8.7||10.1||8.2||9.0||8.5||8.5|
|1.5 M||0.05 M||9.9||11.6||9.6||10.6||9.9||9.9|
In this chapter, the governing equations of transport through an FO membrane were presented based on the mass balance in the concentration boundary layers on both sides of the membrane (ECP) and inside the support layer (ICP). Although ICP is reported in the literature to play a significant role in the reduction of the effective osmotic driving force, the impact of ECP is usually underestimated. The ECP primarily depends upon the value of mass transfer coefficient (
The authors gratefully acknowledge the financial support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) and Canada’s Oil Sands Innovation Alliance (COSIA).
|A||pure water permeability (Lm−2 h−1 bar−1)|
|B||solute permeability (Lm−2 h−1)|
|c||concentration of solute (mol L−1)|
|dh||hydraulic diameter (m)|
|D||diffusion coefficient (m2 s−1)|
|n||van’t Hoff factor|
|J||flux (Lm−2 h−1)|
|k||mass transfer coefficient (ms−1)|
|K||solute resistivity (m)|
|L||length of channel (m)|
|p||hydraulic pressure (bar)|
|Rg||universal gas constant (Jmol−1 K−1)|
|ReL||Reynolds number at the end of membrane channel|
|Ret||transition Reynolds number|
|T||absolute temperature (K)|
|δ||thickness of ECP boundary layer (m)|
|ε||porosity of membrane support|
|μ||dynamic viscosity (Pa.s)|
|ν||kinematic viscosity (m2 s−1)|
|ρ||density of water (kg m−3)|
|τ||tortuosity of membrane support|
|π||osmotic pressure (bar)|
|i||interface between support layer and active layer of membrane|
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