Not surprisingly, the aquaporin proteins that facilitate water movement play a large role in osmosis, most prominently in red blood cells and the membranes of kidney tubules. Osmosis is a special case of diffusion. Water, like other substances, moves from an area of high concentration to one of low concentration. An obvious question is what makes water move at all?
Imagine a beaker with a semipermeable membrane separating the two sides or halves. On both sides of the membrane the water level is the same, but there are different concentrations of a dissolved substance, or solute, that cannot cross the membrane otherwise the concentrations on each side would be balanced by the solute crossing the membrane.
If the volume of the solution on both sides of the membrane is the same but the concentrations of solute are different, then there are different amounts of water, the solvent, on either side of the membrane. If there is more solute in one area, then there is less water; if there is less solute in one area, then there must be more water.
To illustrate this, imagine two full glasses of water. One has a single teaspoon of sugar in it, whereas the second one contains one-quarter cup of sugar.
Overstreet , R. Jacobson : Mechanisms of ion absorption bv roots. Annual Rev. Plant Physiol. Scarth , G. Lloyd : An elementary course in general physiology. Levitt There are no affiliations available. Personalised recommendations. Cite chapter How to cite? ENW EndNote. Buy options. Phuntsho et al. In general, regardless of either the feed or draw, raising the temperature on either side leads to increase in both the water flux and the solute flux. However, the degree to which the water flux and solute flux are increased varies across the literature [ 10 , 26 , 27 , 28 , 31 , 35 , 36 ] see Table 2.
Their membrane was oriented in the PRO mode where the active layer was facing the draw solution. They argued that increasing the draw temperature led to reduced solution viscosity and increased draw solute diffusivity.
This change resulted in the reduction of dilutive ICP on the draw side, thereby increasing the water flux. Again, such a behavior is attributed to the fact that the dilutive ICP plays a more significant role than the concentrative ECP in determining the water flux [ 26 ].
Such a preferential water flux increase due to the increased draw temperature was also observed by Xie et al. You et al. They also showed that this is indeed true for concentrating anthocyanin, which is a large sugar molecule. As mentioned in the preceding section, Xie et al.
In this sense, transmembrane temperature differences barely influenced the solute rejection rate for the charged solutes, whereas the neutral solutes were significantly influenced by the temperature difference. The reason being is that raising the draw temperature leads to increased water flux, which contributes to the increased solute rejection.
At the same time, keeping the feed temperature low reduces the deposition of the solutes on to the membrane, thus preventing the neutral feed solutes from dissolving into the membrane and diffusing across the membrane [ 27 ]. 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. 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. In the model, the chemical potential of water is represented as a function of temperature, pressure, and solute concentration, i. From the basic thermodynamic relationship,.
This condition gives. 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. The solute flux is similarly given as. 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. A schematic representation of a concentration polarization across a skinned membrane during FO process in the PRO and FO modes, represented using the solid and dashed lines, respectively and b arbitrary temperature profile increasing from the active layer to the porous substrate.
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:. In a steady state, J s of Eqs. Flux equations for the FO mode can be easily obtained by replacing subscript 2 by 4 in Eqs. In Eq. For mathematical simplicity, one can write the flux equation for both modes:. Any theoretical development can be initiated from Eq. In the theory, there are several key assumptions during derivations of Eqs. These assumptions are summarized in the following for the PRO mode for simplicity, but conceptually are identical to those in the FO mode.
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 [ 39 ].
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. 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. The osmotic pressure is presumed to be linear with the solute concentration C. In the PRO mode, one can indicate. 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.
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 [ 40 ]. 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. 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.
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. 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 [ 40 ].
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.
Fourth, even if one can achieve a perfect solute rejection, i. Figure 4 b shows an arbitrary temperature profile across the FO membrane, increasing from the active layer side to the porous layer side.
In bulk phases of the active and porous sides, temperatures are maintained at T 1 and T 4 , respectively. Stream temperature on the active side increases to T 2 , and within the membrane, temperature elevates from T 2 to T 3. Since the active layer is often made thin, a linear variation of temperature can be readily assumed. From the active-porous interface to the porous layer surface to the solution, the temperature increases from T 3 to T 4.
A similar external temperature polarization occurs in the PL-side bulk phase, generating the temperature change from T 4 to T 5. The overall temperature profile is conceptually akin to the concentration profile in the FO mode. Having the same bulk temperatures, i. T 1 and T 5 , the flow direction can noticeably change values from T 2 to T 4.
For logical consistency, a steady state is assumed while investigating the heat transfer across the FO membrane in this chapter. Thus, heat fluxes of the four regions are. Note that Eq. In the FO process with the transmembrane thermal gradient, Eqs. For example, for the temperature profile shown in Figure 4 , the FO concentration profile has the same trend to that of the temperature, and therefore signs in Eqs.
In this case, Eq. This heat balance analysis is very similar to that of membrane distillation [ 43 , 44 ], but the FO process does not have any solvent phase transition so that the latent heat is not considered. In statistical mechanics, Gibbs energy is the master function of the isothermal-isobaric ensemble. Consider a box in which two regions are separated by a semipermeable membrane. In equilibrium, the maximum entropy condition requires that the chemical potential divided by the temperature should be constant, i.
From Eq. Here we assume that the membrane properties do not change significantly with solute concentration C and local temperature T. This is due to the water chemical potential being higher in the lower C region.
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