Constitutive Relations of Thermal and Mass Diffusion

Antonio Bertei; Andrea Lamorgese; Roberto Mauri

doi:10.1515/jnet-2019-0055

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Constitutive Relations of Thermal and Mass Diffusion

Antonio Bertei , Andrea Lamorgese und Roberto Mauri

Veröffentlicht/Copyright: 9. November 2019

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Aus der Zeitschrift Journal of Non-Equilibrium Thermodynamics Band 45 Heft 1

Abstract

Non-equilibrium thermodynamics provides a general framework for the description of mass and thermal diffusion, thereby including also cross-thermal and material diffusion effects, which are generally modeled through the Onsager coupling terms within the constitutive equations relating heat and mass flux to the gradients of temperature and chemical potential. These so-called Soret and Dufour coefficients are not uniquely defined, though, as they can be derived by adopting one of the several constitutive relations satisfying the principles of non-equilibrium thermodynamics. Therefore, mass diffusion induced by a temperature gradient and heat conduction induced by a composition gradient can be implicitly, and unexpectedly, predicted even in the absence of coupling terms. This study presents a critical analysis of different formulations of the constitutive relations, with special focus on regular binary mixtures. It is shown that, among the different formulations presented, the one which adopts the chemical potential gradient at constant temperature as the driving force for mass diffusion allows for the implicit thermo-diffusion effect to be strictly absent while the resulting Dufour effect is negligibly small. Such a formulation must be preferred to the other ones since cross-coupling effects are predicted only if explicitly introduced via Onsager coupling coefficients.

Keywords: constitutive relations; chemical potential; regular mixtures; thermo-diffusion

Appendix A Generalization to non-local forces

In eqs. (3), (4), (6), (7), (11), and (12), ψ12 represents the difference between the potential of any conservative force field acting on the two species (see eq. (2)). A particularly important example is the Korteweg force, which can be derived assuming that the generalized free energy g′ depends also on the gradients of the molar fraction, ϕ, that is, g′ϕ,∇ϕ,T=gϕ,T+gnl∇ϕ,T. Here, g is the thermodynamic free energy (eq. (14a)), while gnl is its so-called non-local component. Expressing gnl in terms of a power series of ∇ϕ and assuming that the mixture is locally isotropic, we find at leading order gnl=12RTa2∇ϕ2, where a is a characteristic length [15]. It must be clear that this non-local term is relevant only in regions where ∇ϕ is very large, that is, when the mixture is separated in two coexisting phases, so that there is an interphase region, where the molar fraction ϕ changes rapidly. Macroscopically, the two coexisting phases are separated by a sharp interfacial surface which, at equilibrium, is characterized by a surface tension, expressing the energy stored per unit interfacial area [15], [26]. Therefore, imposing that at equilibrium the line integral of gnl across the interfacial region equals the surface tension σ, we see that the characteristic length a is proportional to the surface tension, i. e., a≈σ/ρRT. Now, we define a generalized chemical potential difference μ12′,

(A.1)μ12′=δg′δϕ=∂g∂ϕ−∇·∂gnl∂∇ϕ=μ12+μ12nl;μ12nl=−RTa2∇2ϕ,

where μ12 is the thermodynamic chemical potential difference in eq. (14b), while μ12nl is its non-local component. In the same way, we obtain the so-called Korteweg force F′, i. e., a generalized body force (per unit mole) that can be obtained formally as

F′=δg′δr=μ12′∇ϕ=∇p′−ϕ∇μ12nl,

with p′=g+ϕμ12nl. Here the first term is the gradient of a scalar which, as in eq. (2), can be reabsorbed into the pressure term and therefore in our case it has no effect, while the second term can be interpreted as the conservative force of eqs. (2) and (3), where the potential difference ψ12 is equal to μ12nl. Therefore,

(A.2)F′=−ϕ∇ψ12;ψ12=μ12nl=−RTa2∇2ϕ.

For regular mixtures, assuming that, as for van der Waals fluids, μ12nl does not depend on T, we find that a=aˆTc/T, where aˆ is a constant characteristic length, so that ψ12=−RTcaˆ2∇2ϕ [15].

These non-local effects contribute an additional non-local term in the material flux and in the heat source term of eq. (1). In fact, adopting either CR1 in eq. (17) or CR2 in eq. (25) we find the same result, namely,

(A.3a)Jϕnl=−ρDϕ1RT∇ψ12=ρD0ϕ1−ϕa2∇∇2ϕ,

(A.3b)q=−Jϕ·∇ψ12.

Comprehensive discussions about these assumptions can be found in Lamorgese et al. [15], [26].

Adopting CR2 (eq. (25)), we saw that, in the absence of coupling terms, the thermo-diffusion effect is strictly absent and the Dufour effect is negligibly small. Accordingly, adding the coupling terms to CR2 as in eq. (12), eq. (25) becomes

(A.4a)Jϕ=−ρDϕ,T1RT∇μ12T+∇ψ12−Lϕq″∇TT2;

(A.4b)Jq″=−Lqϕ″1T∇μ12T+∇ψ12−k(ϕ,T)∇T,

where

Lϕϕ″=ρDϕ,TR;Lϕq″=Lqϕ″;Lqq″=kϕ,TT2.

Substituting eqs. (14b) and (A.3a) into eq. (A.4) and defining the thermo-diffusion coefficient DT′,

(A.5)Lϕq″=Lqϕ″=DT′Tρϕ1−ϕ,

we obtain

(A.6a)Jϕ=−ρD01−2Ψϕ1−ϕ∇ϕ−ρϕ1−ϕDT′∇lnT+ρD0ϕ1−ϕa2∇∇2ϕ,

(A.6b)Jq=−ρRTDT′1−2Ψϕ1−ϕ∇ϕ−k∇T+ρDT′ϕ1−ϕa2RT∇∇2ϕ.

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Received: 2019-07-17

Revised: 2019-10-17

Accepted: 2019-10-23

Published Online: 2019-11-09

Published in Print: 2020-01-28

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Artikel in diesem Heft

https://doi.org/10.1515/jnet-2019-0055

Schlagwörter für diesen Artikel

constitutive relations; chemical potential; regular mixtures; thermo-diffusion