Transport equations force-flux relations

An examination of the derived flux equations [18] leads to the following conclusions (i) Onsager reciprocity relation (ORR) is obeyed, (ii) All coefficients are scalar, (iii) Higher powers of single force do not occur, (iv) Space derivatives of forces occur in the transport equations, (v) In none of the X, Xj terms, the tensorial order of X and Xj term is the same, (vi) All the X,X terms have the same tensorial order as the fluxes, (vii) Non-linearity arises on account of gradient of barycentric velocity. 

We shall try to cover all the membrane processes within one model at the end of fliis chapter, in order to relate the various membrane processes with each other in terms of driving forces, fluxes and basic separation principles. To do so, the starting point must be a simple model, such as a generalised Pick equation [41 or a generalised Stefan-MaxweU equation [42]. In order to describe transport through a porous membrane or through a nonporous membrane, two contributions must be taken into account, the diffusional flow (v) and the convective flow (u). The flux of component i through a membrane can be described as the product of velocity and concentration, i.e. 

Non-equilibrium thermodynamics is a systematic theory of transport, which satisfies the second law, because the transport equations are always derived from the entropy production. When forces and fluxes are constructed according to eq 14.2, the theory gives relations which are linear. The theory can also handle non-linear processes, discussed in 14.3.2. When the prescription to find the entropy production is followed, the advantages listed in a to e above follow. 

The three formulas (2.22)-(2.24) are examples of linear transport equations they relate the response of a system (the flux) to a small perturbing force (the gradient). The transport coefficients Z), rj, and x are the parameters of proportionality, to be determined experimentally. A familiar transport equation is Ohm s law. Here voltage is the force, current the response, and conductivity (the reciprocal of resistance) the transport coefficient. In general, equations of transport are not as simple as these. In a two-component system with a temperature gradient, Fourier s law states that there is only heat flow. However, if the masses of the components... 

The description of coupled flow and transport phenomena is usually based on non-equilibrium thermodynamics [2], Application of this theory leads to a set of linear equations, relating all thermodynamical fluxes Jj to all thermodynamical forces Xj in a system ... 

In other words, under realistic conditions (/ 0), entropy is produced, with the positive entropy production being given by fluxes and forces related to the process j. Equation (5) assumes that all these processes are in series, which is mostly correct. The most obvious contributions are transport resistances, due to the finite conductivities of Li+ and e in electrolyte and electrodes. For usual geometries, these resistances are constant to a good approximation, while for resistances stemming from impeded charge transfer and phase boundaries, the dependence on current can be severe. 

However, if convective transport of heat and species mass in porous catalyst pellets have to be taken into account simulating catal3dic reactor processes, either the Maxwell-Stefan mass flux equations (2.394) or dusty gas model for the mass fluxes (2.427) have to be used with a variable pressure driving force expressed in terms of mass fractions (2.426). The reason for this demand is that any viscous flow in the catalyst pores is driven by a pressure gradient induced by the potential non-uniform spatial species composition and temperature evolution created by the chemical reactions. The pressure gradient in porous media is usually related to the consistent viscous gas velocity through a correlation inspired by the Darcy s law [21] (see e.g., [5] [49] [89], p 197) ... 

Onsager s thermodynamic equation of motion rests on the linear flux-force relations of transport theory [38]. Thus, to start, we beiefly review these phenomenological equations. 

Thus, the flux-force equations linearly related n thermodynamics fluxes Jy. Ji,.. Ao n conjugate external thermodynamic forces Fi. Fi-- - with the proportionality constants Lij being the n transport coefficients for the process. Note that Eq. (A.9) is an example of Eq. (A. 10) for n = 1 with Ji=J,Lii — a, and Fi—E. Thermoelectric conduction and thermal diffusion are examples of processes that obey Eq. (A. 10) for n > 1. 

Onsager relations - An important set of equations in the thermodynamics of irreversible processes. They express the symmetry between the transport coefficients describing reciprocal processes in systems with a linear dependence of flux on driving forces. 

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