Dufour effect

Remember that this is a heat flux resulting from a concentration gradient. The corresponding law can be written as follows  

---

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... 

Conversely, when mass transfer is occurring as a result of a constant concentration gradient, a temperature gradient may be generated this is known as the Dufour effect. 

Since variations In the pressure Induced by fluid dynamic effects are negligible for MOCVD reactor flows, the Inlet pressure, Pq, Is used. In formulating the energy balance, the contributions from pressure changes, viscous dissipation and Dufour effects may neglected for MOCVD conditions (14.15) so the equation becomes ... 

By Dalton s law, Equation (2.9), the mixture pressure, p, is Y i= Pi- The ternl Y I PiVjhj is sometimes considered to be a heat flow rate due to the transport of enthalpy by the species. (This is not the same as q" arising from VT which is called the Dufour effect and is generally negligible in combustion.) With the exception of the enthalpy diffusion term, all the sums can be represented in mixture properties since ph = Ya i Pihi However, it is convenient to express the enthalpies in terms of the heat of formation and specific heat terms, and then to separate these two parts. 

The first term on the right-hand side of eq. (5) describes the heat conduction and the second term accounts for the Dufour effect. The convective contribution of the energy flux is given by... 

Dufour effect, the total energy flux E is given by... 

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. 

These equations are called the Navier-Stokes equations, and when supplemented by the state equation for fluid pressure and species transport equations, they form the basis for any computational model describing the flows in fires. For simplicity, several approximations are inherent (see Equation 20.3) (no Soret/Dufour effects, no viscous dissipation, Fickian diffusion, equal diffusion coefficients of all species, unit Lewis number). 

Fluid Thermal osmosis, Jlq = CjkjVT, where Att is the thermoosmotic permeability (m2/(K s)) Dufour effect Electric osmosis Advection, adv CjKVh... 

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. 

The phenomenological coefficients are important in defining the coupled phenomena. For example, the coupled processes of heat and mass transport give rise to the Soret effect (which is the mass diffusion due to heat transfer), and the Dufour effect (which is the heat transport due to mass diffusion). We can identify the cross coefficients of the coupling between the mass diffusion (vectorial process) and chemical reaction (scalar process) in an anisotropic membrane wall. Therefore, the linear nonequilibrium thermodynamics theory provides a unifying approach to examining various processes usually studied under separate disciplines. 

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. 

The heat flow due to the Dufour effect arises only from a concentration gradient, and is expressed by... 

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). 

Example 7.3 Total energy flow and phenomenological equations For mixtures, the energy flow contains the conductive flow qc, and the contributions resulting from the interdiffusion q, of various substances and the Dufour effect q we therefore express the total energy flow relative to the mass-average velocity... 

When we express the energy flow e with respect to fixed stationary coordinates by disregarding the Dufour effect, the viscous effect, and kinetic energy, we have... 

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... 

DS e effective diffusion coefficient for the substrate S Dy)c coupling coefficient related to the Dufour effect DTe coupling coefficient related to the Soret effect e energy... 

The equation for conservation of energy will be written under the assumptions that the Mach number is small and that the work done by body forces is negligible, both of which are accurate approximations for the deflagration problems considered. In addition, we shall neglect the Dufour effect (and later the Soret effect), so that by use of equation (1-6) in equation (3-72), in which h = Yj= i we obtain... 

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. 

Note that the standard temperature (77°F or T0 = 298.15 K) is used in this definition. cp i is the specific heat and Ah p is the enthalpy of formation at the standard state, both for species i. The heat flux, q, includes contributions from conduction, radiation, differential diffusion among component species, and concentration gradient-driven Dufour effect. For combustion applications, the most important contributions come from conduction and radiation. As discussed in Section 4.3, conduction heat flux follows Fourier s law (Equation 4.27) and radiation heat flux is related to the local intensity as... 

The second term on the right-hand side of Eq. 11.2.5 gives the contribution of the mass fluxes to the heat flux q this contribution is commonly referred to as the Dufour effect. 

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). 

---