Why do diffusive fluxes increase with temperature

At the same time, however, the higher flux also increases both the ICP and ECP, which essentially limit the water flux as a feedback hindrance. This self-hindering effect of the solvent flux is unavoidable in most membrane separation processes. It is similar to the fact that, in RO, applying high pressure initiates increasing permeate flux, which will eventually bring more solutes from the bulk phase to the membrane surface, enhancing the CP.

Therefore, additional gain of the RO permeate flux is not as much as anticipated when the pressure is increased. The change in the temperature influences the CP phenomena in different ways depending on the orientation of the membrane.

This is because the formation of the ICP, which is the most critical factor that limits the driving force, is dependent on the membrane configurations. By reducing the ICP using deionized water as the feed, the water flux was shown to be highly dependent on the temperature, confirming the impact of ICP on the FO process [ 29 ].

Although the increased solute diffusion at higher temperature mitigates the concentrative ICP in the support layer so that the water flux can be increased, such an increased water flux carries more solutes from the feed bulk phase to the vicinity of the support layer surface and enhances the dilutive ECP, thereby reducing the osmotic driving force. Therefore, the two opposing effects on the water transport effectively limit the enhancement of the water flux such that the temperature has a marginal effect on the overall water flux.

If both water and solute diffusivities increase in a similar behavior, the net diffusive transport must be more or less the same. In the FO mode, however, the water flux was shown to be significantly influenced by the temperature.

This was proven mathematically using the method of proof by contradiction [ 32 ]. Such a low water flux effectively suppresses the extent of concentrative ECP in the feed side. Also, the influence of concentrative ECP on the water flux is less important than the dilutive ECP in the draw solution side because the initial solute concentration in the bulk phase is much lower at the feed solution than the draw solution.

Therefore, when the membrane is placed in the FO mode, the water flux is significantly influenced by the temperature since the ICP is the only major factor that determines the driving force.

One assumption McCutcheon and Elimelech had made while analyzing their data were the insignificant solute diffusion across the membrane [ 29 ], which otherwise leads to further ICP. Obviously, commercially available membranes are known to permit diffusion of the solutes, which can impact the formation of the CP effect. Since the solute diffusion is also sensitive to the temperature, the transmembrane solute flux should also lead to a change in the water flux. We discuss the effect of temperature on the solute diffusion and rejection in the following section.

It is of general consensus that the transmembrane solute diffusion increases with temperature. A number of groups have recently investigated experimentally the temperature effect on the transmembrane solute diffusion and the solute rejection [ 26 , 27 , 28 ]. Xie et al. Hydration of charged organic solutes results in an increase in the effective solute size, which directly influences the solute diffusion and rejection rate, as it was well understood that the rejection of the charged organic solutes would be much higher than the neutral organic solutes.

In this regard, neutral solutes were more likely to diffuse across the pores than the charged solutes in both the cellulose triacetate membranes and polyamide membranes. This implies that increasing the temperature leads to higher solute diffusion due to the increased solute diffusivity. Moreover, increasing the temperature leads to faster dissolution of the solutes into the membrane such that even hydrophobic neutral solutes absorb into the membrane at an order of magnitude higher rate at elevated temperatures.

Notably, the ratio between the water flux J w and the solute flux J s was shown to be more or less constant regardless of the system temperature [ 27 ]. However, it is more reasonable to say that the structural properties of FO membranes change with temperature in a way that the ratio between solvent and solute fluxes remain almost constant.

In a solution-diffusion model, permeabilities of solvent and solutes, A and B , respectively, are believed to increase with membrane temperature. This is because although higher T increases both the solute and solvent fluxes, it is only the ratio that influences the concentration of solutes passing through the membrane.

This topic is discussed theoretically in detail in Section 5. Meanwhile, You et al. Membrane scaling occurs when the solute concentration is high enough to initiate precipitation. This is directly related to solute rejection and the CP phenomena, implying that membrane scaling should also be temperature-dependent. Zhao and Zou studied how the temperature influences the membrane scaling over time, which is important in long-term operations [ 24 ].

Due to the fast water flux at elevated temperature, increase in the final concentration of the feed solution i. Concentrative polarization is also enhanced when the water flux is increased, which results in an accelerated membrane scaling. This was confirmed by directly visualizing the fouled membrane by using a scanning electron microscope Figure 3 a — d and also by measuring the decrease in flow rate over time Figure 3 e , showing faster decline of water flux over time at elevated temperatures due to the scaling.

In addition to higher solute concentration near the membrane surface driven by the temperature-enhanced solvent flux, the changes in solubility limits for inorganic species may contribute to the accelerated fouling behavior.

Temperature-dependent membrane fouling and associated water flux decline. One step further, we can also consider a case where the temperature is unevenly distributed across the membrane. In such a case, the temperature gradient may allow independent control of transport on either side of the membrane.

In practice, temperature gradients can occur frequently; temperature of the feed solution can increase due to the heat released from the hydraulic pumping or when the solution is pretreated. Likewise, the temperature of the draw solution may change due to the post-treatment process for recovery and recycling of draw solutes such as thermal and membrane distillation. Since heating only on one side of the solution requires lower energy than heating up the entire system, imposing a temperature gradient across the membrane may offer an energy-efficient control over the osmotic phenomena.

A full theory accounting for the temperature gradient in osmosis may result in highly nonlinear effects on the FO performance. Furthermore, the temperature gradient may provide an additional complexity to the coupled mass and heat transfer phenomena within the membrane.

Although the temperature dependence on the water flux shows an agreeable consensus as shown in Table 1 , the anisotropic temperature effect is shown to differ largely across various studies. When the temperature on either side of the solutions is increased, the water flux becomes higher than that at the base temperature, but lower than when the temperatures of both sides of the solutions are increased.

It is, however, left unclear which side of the solution has more influence on the FO process when heated as this does not have an agreeable consensus. Table 2 provides a summary of the effect of temperature difference on the FO process under various experimental parameters. For simplicity, we define. 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. It is analogous to the property of thermal diffusivity in heat transfer:.

By contrast, diffusion for molecules dissolved in liquids is far slower. As a result, diffusion in liquids is very slow over everyday length scales and is almost always dominated by convection. These units are also clear from a dimensional analysis of Fick's second law also called the Diffusion equation.

Formally, the diffusion coefficient can be understood as parameterizing the area of a spherical surface, defined as the surface of root-mean-square displacement of material diffusing away from an infinitesimal point where a mass is initially concentrated. Since the statistics of diffusion cause this area to grow linearly in time, the diffusion coefficient is a quantity described by area per time.

The diffusion coefficient can be predicted from first principles in some simple cases. By taking the values for the mean free path and average velocity for molecules in an ideal gas from the Maxwell-Boltzmann distribution, it follows that the diffusion coefficient obeys the following relation to temperature and pressure:. For particles or large molecules in a viscous fluid usually a liquid solution , the Stokes-Einstein equation can be applied:.

This equation is derived on the assumption that the particles obey Stokes' law for drag, such that the drag exerted on diffusing molecules, by the solvent molecules, can be computed. Note that solvent viscosity itself strongly depends on temperature, so this equation does not imply a linear relation of solution-phase diffusion coefficient with temperature.

Rather, the diffusion coefficient normally obeys a relation close to an exponential Arrhenius relation:. Here, E diff is an "activation energy of diffusion"; the exponential form of this relation means that diffusion coefficients in the solution phase can grow quickly with temperature. In a porous medium, the effective diffusion coefficient becomes different from the real diffusion coefficient. This is because the available cross section for diffusion is less than for the free fluid and the distance between one point and another in the porous material is less than the distance that a molecule must travel to move between these points since the molecule must navigate between the solid parts of the material.