# Phenomenological Thermodynamics of Irreversible Processes

^{1}

^{2}

^{*}

_{2}O refrigeration; Tsallis thermodynamics in 2D turbulence; Variational nonequilibrium thermodynamics

## 1. How Does Early Thermodynamics Evolve into a Rigorous Theory of Irreversibility

**q**/T. As a rule, the external entropy supply is tacitly ignored, which corresponds to a conceptional inconsistency since the physical balance laws were thought to be valid as an open system with non-vanishing body forces, ρ

**f**and energy supply term, ρr. Therefore, by mere analogy with Duhem’s relation, Truesdell in 1957 [10] postulated that the entropy supply ρs was connected to the heat supply ρr by s = r/T. Therefore, the nascent a priori estimate for the physical balance laws with the SL was:

## 2. The Clausius-Duhem Inequality as the Expression of the Second Law

**f**and the energy supply r may at any material point and any time take values such that they outbalance all other terms in the remainder of these equations. Moreover, the energy equation may be used to eliminate the source term in the Clausius-Duhem inequality because of Equation (1). This leads to the energy and entropy equations.

**1**, in which ${\mathbf{\sigma}}^{\mathit{E}}$ is the extra-stress and p the pressure, it is straightforward to show that Equation (4) takes the form below.

- The free energy is neither a function of $\mathit{D}\text{}\mathrm{nor}\text{}\mathrm{of}\text{}\mathit{g}:\text{}\partial \psi /\partial \mathit{D}=\mathbf{0},\text{}\partial \psi /\partial \mathit{g}=\mathbf{0}.$
- The free energy is a potential for $-\eta \text{}\mathrm{and}\text{}p/{\rho}^{2}\text{}$, i.e., $-\eta =\partial \psi /\partial T\mathrm{and}$ $p/{\rho}^{2\text{}}$=$\text{}\partial \psi /\partial \rho .$ This implies$$\text{}\left(-\eta ,\text{}\frac{p}{{\rho}^{2}}\right)={\nabla}_{T,\rho}\psi \left(T,\text{}\rho \right)$$
- These results lead quite naturally to the Gibbs relation.$$\mathrm{d}\eta =\text{}\frac{1}{T}\left\{\mathrm{d}\epsilon +p\mathrm{d}\left(\frac{1}{\rho}\right)\right\}.$$

## 3. The Müller-Liu Procedure of the Exploitation of the Second Law

#### Entropy Priciple (EP) (by Müller in 1971 [19])

- 1.
- In every material body, there exists an additive quantity, the specific entropy η, which obeys a balance equation.$$\rho \frac{d\eta}{dt}=-div\mathit{\varphi}+\rho s+\rho \gamma ,$$
- 2.
- The specific entropy η and the entropy flux$\mathit{\varphi}$are material quantities for which, according to the rule of equipresence, the same material laws hold as for the remaining constitutive quantities.
- 3.
- The entropy production must be a non-negative quantity.$$\text{}\gamma \ge 0$$
- 4.
- The supply terms, which appear in the balance equations, cannot influence the material behavior.
- 5.
- There exist special material surfaces, the so-called ideal walls between two continual walls, across which the empirical temperature and the tangential velocity are continuous.

- -
- Entropy is an additive quantity for static processes. In modern studies, however, entropy is often postulated to be non-additive. In these cases, the above EP must be changed.
- -
- Thermodynamic processes for a certain material class are in the context of the applicability of the equipresence rule for all solutions of the field equations.
- -
- External supply terms cannot influence the material behavior. This means that influences of the SL to the constitutive relations can be deduced from the supply free physical balance laws and paired with the supply free entropy balance. This means that the EP applies for closed systems thermodynamics. The same inferences can be drawn for open systems thermodynamics, according to item 4 of the EP since this statement means that the entropy supply must, at most, depend on the remaining supply terms of the physical balance laws.
- -
- In item 5 of the EP, the concept of an ideal wall is introduced. This concept is needed because a quantity that may replace the absolute temperature is not incorporated in this SL (as it is in the Clausius-Duhem inequality). Using temperature as a variable requires an empirical measure θ (in instruments a pressure, volume expansion or electrical conductivity etc., monotonically changing with our sensation of hotness/coldness). Similar to a Duhem relation, $\mathit{\varphi}=\mathit{q}/T$, can be proven by the EP. Then this divisor T should be a universal functional, e.g., $T=T(\theta ,\dot{\theta},\dots )$ that is materially independent and reduces to the absolute temperature under thermostatic processes. Item 5 in the EP achieves this result.

## 4. The Use of Nonlinear Entropic Functions

## 5. The 12 Contributions Published in This Special Issue

- (i)
- The introduction of additional space state variables consists of position and time plus additional variables accounting for the internal structure of the complex material, e.g., porosity in porous media. This work modeled solid mechanical structures, ice sheets of the Earth, and more. Internal variables may also be incorporated in a material theory with no other explicit definition than equations that are written down for them. For instance, the climate relevant gases of the air trapped in the air bubbles of glaciers and ice sheets serve as parameters for the ice fluidity and ice age in climate relevant studies [44].
- (ii)
- The second approach is the so-called mesoscopic method whose domain of the field quantities $\left(\mathit{x},\text{}t\right)$ is enlarged by mesoscopic quantities, which gives rise to the mesoscopic space ${\mathbb{R}}_{3}^{3}\times {\mathbb{R}}_{t}\times M$ in which M consists of variables defining the complex local behavior of the material. This procedure often entails partial or full embedding of statistical components in the theory.

^{−3}.

- (i)
- Filling an insulated tank with a pressurized gas,
- (ii)
- Flowing fluid carrying kinetic energy into a piston device,
- (iii)
- Diffusion between two components,
- (iv)
- Diffusion through a series array of membranes,
- (v)
- Diffusion through a parallel array of membranes,
- (vi)
- Heat conduction and diffusion between two compartments,
- (vii)
- Heat conduction and diffusion through a composite membrane.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Roman Symbols | |

ARMA | Archive for Rational Mechanics and Analysis |

D = $\frac{1}{2}(\mathrm{grad}\text{}v+\text{}{\left(\mathrm{grad}\text{}v\right)}^{\mathrm{T}})$ | Stretching tensor |

EEI | Extended Entropy Inequality |

EP | Entropy production |

f | Body force per unit mass |

$\mathrm{grad}\text{}T$ | Temperature gradient |

p | Pressure |

q | Heat flux vector |

r | Heat supply rate per unit mass |

Re | Reynolds number |

SL | Second Law |

s | Entropy supply per unit mass |

${s}_{\mathrm{i}}$ | Supply rate density of the i-th physical balance law |

[${s}_{\mathrm{i}}=f$ or ${s}_{\mathrm{i}}=r$ for momentum and internal energy] | |

T | Absolute temperature |

t | Time variable |

${w}_{\mathrm{st}}$ | Stress power |

Greek Symbols | |

γ | Entropy production per unit mass |

ε | Internal energy density per unit mass |

η | Entropy density per unit mass |

θ | Empirical temperature |

${\mathsf{\Lambda}}_{\mathrm{i}}$ | Lagrange par. of the i-th phys. Balance law |

${\mathsf{\lambda}}_{\mathrm{i}}$ | Prop. factor of the i-th supply rate density ${\mathrm{s}}_{\mathrm{i}}$ |

ρ | Mass density |

$\mathsf{\Sigma}$ | Summation sign |

$\mathsf{\sigma},{\text{}\mathsf{\sigma}}^{\mathrm{E}}$ | Cauchy stress tensor, Extra stress tensor |

$\mathit{\varphi}$ | Entropy flux vector |

ψ | Helmholtz free energy |

Miscellanea | |

$\frac{\mathrm{d}}{\mathrm{dt}}(\xb7)=\text{}{(\xb7)}^{\xb7}$ | $\frac{\mathrm{d}}{\mathrm{dt}}(\xb7)=\text{}{(\xb7)}^{\xb7}$ |

$\frac{\partial}{\partial \mathrm{x}}(\xb7)$ | $\frac{\partial}{\partial \mathrm{x}}(\xb7)$ |

x, y, z | x, y, z |

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Wang, Y.; Hutter, K.
Phenomenological Thermodynamics of Irreversible Processes. *Entropy* **2018**, *20*, 479.
https://doi.org/10.3390/e20060479

**AMA Style**

Wang Y, Hutter K.
Phenomenological Thermodynamics of Irreversible Processes. *Entropy*. 2018; 20(6):479.
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**Chicago/Turabian Style**

Wang, Yongqi, and Kolumban Hutter.
2018. "Phenomenological Thermodynamics of Irreversible Processes" *Entropy* 20, no. 6: 479.
https://doi.org/10.3390/e20060479