Dufour effect, thermal diffusion

Chapters 5 and 6 deal with systems where interaction between temperature gradient, concentration gradient and potential gradients without any barrier are involved. In these chapters, theoretical and experimental studies relating to thermal diffusion, Dufour effect, Soret effect, thermal diffusion potential, thermo-cells, precipitation and dissolution potential have been described. Physical implications of the experimental results have also been described. 

The coefficients, L., are characteristic of the phenomenon of thermal diffusion, i.e. the flow of matter caused by a temperature gradient. In liquids, this is called the Soret effect [12]. A reciprocal effect associated with the coefficient L. is called the Dufour effect [12] and describes heat flow caused by concentration gradients. The... 

Equation (56) states that the effect of a thermal gradient on the material transport bears a reciprocal relationship to the effect of a composition gradient upon the thermal transport. Examples of Land L are the coefficient of thermal diffusion (S19) and the coefficient of the Dufour effect (D6). The Onsager reciprocity relationships (Dl, 01, 02) are based upon certain linear approximations that have a firm physical foundation only when close to equilibrium. For this reason it is possible that under circumstances in which unusually high potential gradients are encountered the coupling between mutually related effects may be somewhat more complicated than that indicated by Eq. (56). Hirschfelder (BIO, HI) discussed many aspects of these cross linkings of transport phenomena. 

Another well-known example is the coupling between mass flow and heat flow. As a result, an induced effect known as thermal diffusion (Soret effect) may occur because of the temperature gradient. This indicates that a mass flow of component A may occur without the concentration gradient of component A. Dufour effect is an induced heat flow caused by the concentration gradient. These effects represent examples of couplings between two vectorial flows. The cross-phenomenological coefficients relate the Dufour and Soret effects. In order to describe the coupling effects, the thermal diffusion ratio is introduced besides the transport coefficients of thermal conductivity and dififusivity. 

Simultaneous heat and mass transfer plays an important role in various physical, chemical, and biological processes hence, a vast amount of published research is available in the literature. Heat and mass transfer occurs in absorption, distillation extraction, drying, melting and crystallization, evaporation, and condensation. Mass flow due to the temperature gradient is known as the thermal diffusion or Soret effect. Heat flow due to the isothermal chemical potential gradient is known as the diffusion thermoeffect or the Dufour effect. The Dufour effect is characterized by the heat of transport, which represents the heat flow due to the diffusion of component / under isothermal conditions. Soret effect and Dufour effect represent the coupled phenomena between the vectorial flows of heat and mass. Since many chemical reactions within a biological cell produce or consume heat, local temperature gradients may contribute in the transport of materials across biomembranes. 

These equations obey the Onsager reciprocal relations, which state that the phenomenological coefficient matrix is symmetric. The coefficients Lqq and Lu arc associated with the thermal conductivity k and the mutual diffusivity >, respectively. In contrast, the cross coefficients Llq and Lql define the coupling phenomena, namely the thermal diffusion (Soret effect) and the heat flow due to the diffusion of substance / (Dufour effect). 

Here, Ds and Dd are the coefficients representing the Soret and Dufour effects, respectively, Du is the self-diffusion coefficient, and Dik is the diffusion coefficient between components / and k. Equations (7.149) and (7.150) may be nonlinear because of, for example, reference frame differences, an anisotropic medium for heat and mass transfer, and temperature- and concentration-dependent thermal conductivity and diffusion coefficients. 

We may define two new effective diffusion coefficients, DJe and Z)D e, which are related to the thermal diffusion and the Dufour effect, respectively... 

Here, DSqX and Z)SqY are the cross coefficients representing the temperature gradient-induced mass flows (thermal diffusion) of X and Y, respectively, and Z)DYq and Z)DXq are the cross coefficients representing the Dufour effects. Under steady-state conditions, the temperature is related to concentration by Eq. (9.18), we have... 

Onsager s reciprocal relations of irreversible thermodynamics [27-30] imply that if temperature gradients give rise to diffusion velocities (thermal diffusion), then concentration gradients must produce a heat flux. This reciprocal cross-transport process, known as the Dufour effect, provides another additive contribution to q. It is conventional to express the concentration gradients in terms of differences in diffusion velocities by using the diffusion equation, after which it is found that the Dufour heat flux is [5]. 

In most cases the Dufour effect is so small that it apparently often is negligible even when thermal diffusion is not negligible. Although it is omitted in the applications, this term is retained in the general equations for completeness. 

The inverse of the Dufour effect is the production of mass fluxes due to temperature gradients this is referred to as thermal diffusion or the Soret effect. To account for this effect, we need to augment the generalized Maxwell-Stefan diffusion equations in the following manner ... 

Develop the film model for simultaneous mass and energy transfer including Soret and Dufour effects. Use the Toor-Stewart-Prober linearized theory in developing the model. An example of a process where thermal diffusion effects cannot be ignored is chemical vapor deposition. Use the model to perform some sample calculations for a system of practical interest. You will have to search the literature to find practical systems. To get an idea of the numerical values of the transport coefficients consult the book by Rosner (1986). 

The diffusion-thermal effect or the Dufour energy flux eff describes the tendency of a temperature gradient under the influence of mass diffusion of chemical species. Onsager s reciprocal relations for the thermod3mamics of irreversible processes imply that if temperature gives rise to diffusion velocities (the thermal-diffusion effect or Soret effect), concentration gradients must produce a heat flux. This reciprocal effect, known as the Dufour effect, provides an additional contribution to the heat flux [89]. 

The latter serves as a reminder that the kinetic theory predicts the cross effects like the transport of mass resulting from a temperature gradient (thermal diffusion). It can also be shown that the theory predicts transport of energy resulting from a concentration gradient (the diffusion-thermo effects). These second-order effects are often referred to as the Soret - and Dufour effects. Unfortunately, no shortcuts are available as these terms do not appear when applying simple kinetic theory, only the more rigorous solution methods resolve these properties. 

This expression accounts for the transfer of heat by conduction, in which k is the effective thermal conductivity of the solid, and transport of energy due to the mass diffusion. We have neglected any viscous heating effects, and the flux of energy due to concentration gradients (Dufour energyO and radiation. So the energy balance becomes... 

By definition of the mass-average velocity v of the mixture, all diffusional mass fluxes with respect to v must sum to zero. Hence, Ja = —jn for binary mixtures. The final expression for the molecular flux of thermal energy in binary mixtures, neglecting the diffusion-thermo (i.e., Dufour) effect, is... 

Thermophoresis is defined as the migration of a colloidal particle or large molecule in a solution in response to a macroscopic temperature gradient. The inverse effect, i.e., the formation of a temperature gradient as the result of the mixing of different molecular species, is referred to as the Dufour effect. The Soret coefficient is defined as the ratio of the thermal diffusion coefficient and the normal diffusion coefficient it is a measure for the degree of separation of the species. These concepts are the same as for a molecular mixture. 

In the earlier chapters, transport phenomena involving a barrier have been discussed from the angle of (i) basic understanding of the physico-chemical phenomena and (ii) test of the linear thermodynamics of irreversible processes. Similar phenomena in continuous systems such as thermal diffusion (Soret effect)/Dufour effect are of equal... 

Systems involving thermal diffusion and Dufour effect are continuous systems without a barrier. For investigating continuous systems, the local variation of properties has to be considered. We shall first consider a general case where mass flux, heat flux and chemical reactions are occurring [4, 5]. 

We now consider the phenomena of thermal diffusion, Soret effect and Dufour effect in a specific system. We consider an isotropic system consisting of two components 1 and 2. The concentration and temperature are non-uniform in the system. The pressure is supposed to be different at all points in the system so that mechanical equilibrium is rapidly established. There are no viscous forces and we shall neglect the viscous phenomena. Furthermore we assume that no chemical reactions are occurring. Since the system is non-uniform, local quantities have to be considered. 

Later on, improved technique was employed by Rastogi and Yadava [12, 13] for investigation of Dufour effect in several liquid mixtures. Diffusion coefficient was also measured. D" estimated from the data has been compared with those for thermal diffusion coefficients estimated from the known data on Soret coefficient using measured values of diffusion coefficients. Although AT is found to be >= 0.3°C, the values of D" and D are on reasonable agreement. 

For simpler phenomena such as thermo-osmosis, electro-kinetic phenomena, thermal diffusion and Dufour effect, the linear thermodynamics of irreversible processes is valid in a wide range as indicated by the experimental results discussed in Chapters 3-5. It may be noted that Onsager relations for thermal diffusion can be proved by ETT [2]. 

An even more potent concept comes from the Onsager reciprocal relations, which states that there is a coupling between conjugate force-flux pairs. For example, mass transfer of species by diffusion in an aqueous solution causes a change in concentration, which is accompanied by heat consumption or release due to the heat of dilution. This sets up a thermal gradient, which causes heat flow. The resulting link between heat flux and isothermal diffusion is the Dufour effect. Its conjugate is the Soret effect, which is the diffu-sional mass flux linked to heat flow. The Soret effect has been coupled with... 

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