# On the Validity of a Linearity Axiom in Diffusion and Heat Transfer

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

**J**) are linear functions of thermodynamic forces (X) (linearity axiom: $J=L.X$). The quasi-equilibrium axiom states that equilibrium thermodynamics relations are still valid in a differential (local) form. Equilibrium thermodynamic laws apply to those systems which are not in equilibrium, as long as the gradients are not too significant, according to quasi-equilibrium. The correlation between entropy production, fluxes, and forces is described by the axioms of quasi-linear thermodynamics of irreversible processes.

## 2. Theoretical Section and Results

- Close to equilibrium: ${X}_{q}=-\frac{\nabla \mathrm{T}}{{\mathrm{T}}^{2}}$
- Extended thermodynamics region: ${X}_{q}=-\frac{\nabla \mathrm{T}}{{\mathrm{T}}^{2}}-\mathrm{A}\frac{\partial {J}_{q}}{\partial t}$

_{0}. It is reasonable to take as reference temperature the calibration temperature T

_{0}. From the theoretical point of view, one can simply establish the necessity of reference quantity by using the Taylor expansion of a continuous flux around a reference quantity located at system II.

_{0}:

- Close to equilibrium: ${X}_{\mathrm{i}}=-\frac{{\left(\nabla {\mu}_{\mathrm{i}}\right)}_{\mathrm{T},\mathrm{P}}}{\mathrm{T}}$
- Extended thermodynamics region: ${X}_{\mathrm{i}}=-\frac{{\left(\nabla {\mu}_{\mathrm{i}}\right)}_{\mathrm{T},\mathrm{P}}}{\mathrm{T}}-{\mathrm{b}}_{\mathrm{i}}\frac{\partial {J}_{i}^{*}}{\partial \mathrm{t}}$

_{i}is the chemical potential of the i-th component,

**v***stands for the velocity of the center of mass, b

_{i}is a thermodynamic parameter.

**J**or

_{2}**J**from the above entropy invariance equation, the following equations have been derived:

_{1}_{i}parameter depends neither on ${J}_{i}^{\ne}$ nor on ${X}_{i}$.

_{12}

^{2}= L

_{21}

^{2}= L

_{11}× L

_{22}), which is valid for binary solutions [7,8,9]. The Fick’s law or the Maxwell–Stefan equation for diffusion could be directly derived from Εquation (4) by further using the linearity axiom and the quasi-equilibrium axiom.

_{0}as reported by L. Onsager [33] has been chosen. The entropy production for this ternary isothermal diffusing system is written as: $\sigma ={\displaystyle \sum _{\mathrm{i}=1}^{3}{J}_{\mathrm{i}}^{\ne}{X}_{\mathrm{i}}}$.

_{ij}are written as:

_{2}0 placed on a vehicle moving with a constant and uniform velocity. The conductivity coefficients L

_{ij}by using the transformed thermodynamic forces (please see Equations (9) and (10)) were reported by D.G. Miller [28] in his landmark work for the validity of the Onsager reciprocal relations as: L

_{11}= −0.200, L

_{12}= 0.0605, L

_{21}= 0.0586, L

_{22}= −0.283.

^{−8}/RT and have the units molescm

^{−1}s

^{−1}. R is the universal gas rate constant. Strongly related to the entropy production invariance principle is the uniqueness axiom of phenomenological resistance, or conductivity coefficients with respect to frames of reference in the Euclidean space [35,36,37,38]. In other words, the conductivity coefficients of the diffusing system placed on a vehicle moving with a constant and uniform velocity are identical to the conductivity coefficients measured at the ground.

_{1}and m

_{2}parameters, one could directly estimate the pro-exponential parameters (see Equation (17)) by using standard methods of non-linear regression analysis. In particular, by assuming a constant uniform velocity ν

_{0}equal to 3 m/s and by setting the m

_{1}and m

_{2}parameters equal to 0.3, the conductivity coefficient data for the system LiCl/KCl/H

_{2}O [28] was exactly reproduced for the following adjustable parameters values:

^{−8}/RT.

## 3. Conclusions

- The linearity postulate is a reasonable approximation for both diffusion and heat transfer in the Euclidean space.
- The linearity postulate for heat transfer and diffusion is compatible with the entropy production invariance in the Euclidean space.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A,b | thermodynamic parameter |

C_{i} | molar concentration of the i-th substance |

k | integration constant |

J | flux |

L | phenomenological coefficients relating fluxes with thermodynamic driving forces |

R | universal gas constant |

T | absolute temperature |

T_{0} | reference absolute temperature |

t | time |

v* | velocity of the center of mass |

v_{0} | constant and uniform velocity |

v_{i} | velocity of the i-th substance |

v^{≠} | arbitrary reference velocity |

${\overline{V}}_{i}$ | partial molar volume of the i-th substance |

w_{i} | weighting factors whose sum is equal to unity |

X | thermodynamic driving force |

X_{1T} | transformed thermodynamic driving force |

Greek Letters | |

λ | parameter of the system |

${\mathsf{\mu}}_{i}$ | chemical potential of the i-th substance |

σ | entropy production rate per unit volume |

Ψ | dissipation function |

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**MDPI and ACS Style**

Arya, R.K.; Thapliyal, D.; Verros, G.D.; Singh, N.; Singh, D.; Kumar, R.; Srivastava, R.K.; Tiwari, A.K.
On the Validity of a Linearity Axiom in Diffusion and Heat Transfer. *Coatings* **2022**, *12*, 1582.
https://doi.org/10.3390/coatings12101582

**AMA Style**

Arya RK, Thapliyal D, Verros GD, Singh N, Singh D, Kumar R, Srivastava RK, Tiwari AK.
On the Validity of a Linearity Axiom in Diffusion and Heat Transfer. *Coatings*. 2022; 12(10):1582.
https://doi.org/10.3390/coatings12101582

**Chicago/Turabian Style**

Arya, Raj Kumar, Devyani Thapliyal, George D. Verros, Neetu Singh, Dhananjay Singh, Rahul Kumar, Rajesh Kumar Srivastava, and Anurag Kumar Tiwari.
2022. "On the Validity of a Linearity Axiom in Diffusion and Heat Transfer" *Coatings* 12, no. 10: 1582.
https://doi.org/10.3390/coatings12101582