To the best of our knowledge, effects of temperature and its gradient on the osmosis phenomena and FO processes have been investigated only phenomenologically without fundamental understanding. The theoretical research is currently in a burgeoning state in explaining the transmembrane temperature gradient effect on the FO performance. In this section, we first briefly review the conventional FO theories [37, 38] based on the solution-diffusion model and van’t Hoff’s law. Then, we revisit statistical mechanics to identify the baseline of the osmosis-diffusion theories, where the isothermal condition was first applied. We then develop a new, general theoretical framework on which FO processes can be better understood under the influence of the system temperature, temperature gradient, and chemical potentials.
The solution-diffusion model is widely used to describe the FO process, which was originally developed by Lonsdale et al. to explain the RO phenomena using isothermal-isobaric ensemble . In the model, the chemical potential of water is represented as a function of temperature, pressure, and solute concentration, i.e. μ w = μ w T P C , and its transmembrane gradient is
Δ μ w = ∫ ∂ μ w ∂ C T , P dC + ∫ ∂ μ w ∂ P T , C dP ,
where the integration is over the membrane region. From the basic thermodynamic relationship,
∫ ∂ μ w ∂ P T , C = V ¯ w
is used where V ¯ w is the molar volume of water. In the isothermal-isobaric equilibrium Δ μ w = 0 , the applied pressure ΔP is balanced with the transmembrane difference of the osmotic pressure, i.e. ΔP = Δπ . This condition gives
0 = ∫ ∂ μ w ∂ C T , P dC + V ¯ w Δπ
and hence we derive Δ μ w = V ¯ w Δp − Δπ . It is assumed that the water transport within the membrane is phenomenologically Fickian, having the transmembrane chemical potential difference of water as a net driving force. The water flux is given as
J w = D w C w RT d μ w dx ≃ D w C w RT Δ μ w δ m ,
J w = A Δp − Δπ ,
where A = D w C w / RT δ m is the solvent permeability that can be obtained experimentally. The solute flux is similarly given as
J s = − D s d C ′ dx ≃ D s Δ C ′ δ m = D s Δ C ′ ΔC ΔC δ m = D s K m δ m ΔC = BΔC ,
where Δ C ′ and Δ C are the concentration differences across the interior and exterior of the membrane, respectively, and K m = Δ C ' / Δ C is the partition coefficient, which is assumed to be constant, and B = D s K / δ m is the solute permeability.
Figure 4(a) shows a schematic representing the PRO and FO modes altogether. Concentrations in the PRO and FO modes are denoted as C and n , respectively. In the PRO mode, C 1 and C 5 are the draw and feed concentrations, and C 2 , C 3 , and C 4 are concentrations at interfaces between the draw solution and the active layer, the active layer and the porous substrate, and the porous substrate and the feed solution, respectively. In the FO mode, n 1 and n 5 are the draw and feed concentrations, respectively, and similarly, n 2 , n 3 , and n 5 have the meanings corresponding to those in the PRO mode. To systematically compare the performances of the PRO and FO modes, we set n 1 = C 1 and n 5 = C 5 , which are the draw ( C d ) and feed ( C f ) concentrations, respectively. Solvent and solute fluxes in the PRO mode are denoted as J w PRO and J s PRO , and those of the FO mode are J w FO and J s FO , respectively. In each mode, solvent and solute fluxes are oriented in opposite directions, influencing each other’s driving forces. The active layer and porous substrate have thicknesses of δ m and δ s , respectively, as located in regions of − δ m < x < 0 and 0 < x < δ s , respectively. Solute molecules migrate with molecular diffusivity D 0 in the porous substrate that is characterized using its thickness δ s , porosity ε , and tortuosity τ .
In the PRO mode, the solvent flux (in magnitude) is
J w = A π 2 − π 3
where π 2 and π 3 are osmotic pressures at concentration C 2 and C 3 , respectively. In a steady state, the water flux J w is constant in both the active and porous regions. The solute flux in the active layer is:
J s = B C 2 − C 3 for − δ m < x < 0
and that in the porous substrate:
J s = − ϵ τ D d C d x − J w C for 0 < x < δ s .
In a steady state, J s of Eqs. (12) and (13) are equal to each other. Flux equations for the FO mode can be easily obtained by replacing subscript 2 by 4 in Eqs. (11) and (12) and replacing C by n in Eqs. (12), (13). Fluxes of the PRO and FO modes are calculated as
J w PRO ≃ 1 K ln B + A π d − J w PRO B + A π f
J w FO ≃ 1 K ln B + A π d B + A π f + J w FO ,
respectively, where π d and π f are the osmotic pressure of the draw and feed concentrations, respectively, and
K = δ s τ D 0 ϵ = S D 0
is interpreted as the characteristic mass transfer resistance, proposed by Lee et al. . Following the convention of standard mass transfer theory, K − 1 can be interpreted as the mass transfer coefficient of FO processes. In Eq. (16), S = δ s τ / ϵ , defined as the structural parameter having units in length, represents the actual path length of molecules passing through the tortuous porous substrate, which is by definition longer than the thickness δ s . For mathematical simplicity, one can write the flux equation for both modes:
J w = 1 K ln B + A π d − φ J w B + A π f + 1 − φ J w
φ = 1 for PRO mode 0 for FO mode
is an integer to toggle between the two modes. Any theoretical development can be initiated from Eq. (17) to consider universally both the FO and PRO modes, and then a proper value of φ can be chosen.
In the theory, there are several key assumptions during derivations of Eqs. (14) and (15). These assumptions are summarized in the following for the PRO mode for simplicity, but conceptually are identical to those in the FO mode.
1. Mass transfer phenomena are described using the solution-diffusion model in which the solvent and solute transport are proportional to the transmembrane differences in the osmotic pressures and solute concentrations, respectively . If one sees these combined phenomena as diffusion, the solvent transport can be treated as semibarometric diffusion. In other words, under the influence of pressure, the solute transport can be treated as Fickian diffusion, driven by the concentration gradient. In a universal view, the net driving forces of the solvent and solutes are their chemical potential differences.
2. In the flux equations, π d and π f are, respectively, overestimated and underestimated because their ture values are those at the draw-membrane and feed-membrane interfaces, i.e. π 2 and π 4 , which are difficult to obtain. This approximation does not cause obvious errors if the flow velocities of the draw and feed solutions are fast enough to suppress formation of any significant external concentration polarizations. A necessary condition, which is less discussed in theories, is the high diffusivity or low molecular weight of solutes.
3. The osmotic pressure is presumed to be linear with the solute concentration C . In the PRO mode, one can indicate
π 2 − π 3 = π 2 − π 3 π 2 − π 4 π 2 − π 4 = 1 − C 3 / C 2 1 − C 4 / C 2 π 2 − π 4
using π 2 − π k = π 2 1 − C k / C 2 for k = 3 , 4 . Eq. (19) can be erroneous if the draw concentration is extraordinarily high or pair-wise interactions between solutes are very strong so that the weak solution approach fails. A study on nonlinearity of π with respect to C can be found elsewhere [37, 38].
4. Rigorously saying, mass transport phenomena are assumed to be in a steady state and equilibrium thermodynamics are used to explain the filtration phenomena. Although the FO phenomenon occurs in an open system, transient behavior is barely described in the literature.
5. In the porous substrate, the bulk porosity is assumed to be uniform, which implies isotropic pore spaces. Moreover, the interfacial porosity between the active and porous layers is assumed to be equal to the bulk porosity. An in-depth discussion on the interfacial porosity can be found elsewhere . In the same vein, the tortuosity is a characteristic geometric constant of the substrate, which is hard to measure independently. More importantly, tortuosity is included in the definition of the structural parameter S , which is used to fit the experimental data to the flux equations.
6. The solute diffusivity D 0 is assumed to be constant, that is, independent of the solute concentration such that the concentration profile is further implied to be linear within the porous substrate.
7. Finally, temperatures of the draw and the feed streams are assumed equal although hydraulic and thermal conditions of these two streams can be independently controlled. As a consequence, heat transfer across the membrane is barely discussed in the literature.
In practice, solvent and solute permeability A and B are measured experimentally in the RO mode using feed solution of zero and finite concentrations, respectively. The applied pressure is selected as a normal pressure to operate the RO, and the solute concentrations are usually in the range of that of a typical brackish water. Variations in A and B with C d and C f are presumed to be negligible, similar to those of RO cases. In Eq. (17), J w is directly related to the interfacial concentration, i.e. C 3 and n 3 in the PRO and FO modes, respectively, and therefore it can be predicted only if K is known. Mathematically, one FO flux equation has two unknowns, which are J w and K . In most cases, the permeate flux J w is measured experimentally and then used to back-calculate K . This experiment-based prediction often results in an imbalance of mass transfer [41, 42]. A recent study assumes that the interfacial porosity between the active and porous layers is different from the bulk porosity of the porous substrate, which successfully resolves the origin of the imbalance between theoretical and measured K values .
This chapter aims to explain how the temperature across the FO membrane, which consists of the active and porous layers, may affect the performance of the mass transfer at the level of statistical physics. The transmembrane temperature gradient prevents from using the abovementioned assumptions and approximations, which are widely used in the FO analysis. First, the SD model is purely based on isothermal-isobaric equilibrium in a closed system. Second, the external concentration polarizations in the draw and feed sides cannot be neglected at the same level because the temperature gradient causes a viscosity difference across the membrane. Third, the weighting factor connecting π 2 − π 3 and π 2 − π 4 cannot be represented only by concentrations but instead should include temperatures at the interfaces. Fourth, even if one can achieve a perfect solute rejection, i.e. B = 0 , steady heat transfer across the membrane should be included since porous membrane is not a perfect thermal insulator. Fifth, the temperature gradient may change the (effective) properties of the active and porous layers such as A , B , ϵ , and τ in principle and the molecular diffusivity D 0 → D T . Sixth, Fick’s law should include additional thermal diffusion or temperature effects for determining the collective diffusion. Seventh, of great necessity is a novel, quantitative equation to calculate the osmotic pressure under the gradients of concentration as well as temperature, which generalizes van’t Hoff’s equation (1).