Nonequilibrium thermodynamics. II. Application to inhomogeneous systems

We provide an extension of a recent approach to study nonequilibrium thermodynamics [Gujrati, Phys. Rev. E 81, 051130 (2010), to be denoted by I in this work] to inhomogeneous systems by considering the latter to be composed of quasi-independent subsystems. The system $\Sigma$ along with the (macroscopically extremely large) medium $\tilde{\Sigma }$ form an isolated system ${\Sigma }_0$. The fields (temperature, pressure, etc.) of $\Sigma$ and $\tilde{\Sigma }$ differ unless at equilibrium. We show that the additivity of entropy requires quasi-independence of the subsystems, which results from the interaction energies between different subsystems being negligible so the energy also becomes additive. The thermodynamic potentials such as the Gibbs free energy that continuously decrease during approach to equilibrium are determined by the fields of the medium and exist no matter how far the subsystems are out of equilibrium, so their fields may not even exist. This and the requirement of quasi-independence make our approach differ from the conventional approach used by de Groot and others, as discussed in the text. We find it useful to introduce the time-dependent Gibbs statistical entropy for ${\Sigma }_0$, from which we derive the Gibbs entropy of $\Sigma$; in equilibrium this entropy reduces to the equilibrium thermodynamic entropy. As the energy depends on the frame of reference, the thermodynamic potentials and the Gibbs fundamental relation, but not the entropy, depend on the frame of reference. The possibility of relative motion between subsystems described by their net linear and angular momenta gives rise to viscous dissipation. The concept of internal equilibrium introduced in I is developed further here and its important consequences are discussed for inhomogeneous systems. The concept of internal variables (various examples are given in the text) as variables that cannot be controlled by the observer for nonequilibrium evolution is also discussed. They are important because the concept of internal equilibrium in the presence of internal variables no longer holds if internal variables are not used. The Gibbs fundamental relation, thermodynamic potentials, and irreversible entropy generation are expressed in terms of observables and internal variables. We use these relations to eventually formulate the nonequilibrium thermodynamics of inhomogeneous systems. We also briefly discuss the case when bodies form an isolated system without any medium to obtain their irreversible contributions and show that this case does not differ from when bodies are in an extremely large medium.

Figure 1

Schematic representation of a macroscopically large system $\Sigma$ and the medium $\tilde{\Sigma }$ surrounding it to form an isolated system ${\Sigma }_0$. The system is a very small part of ${\Sigma }_0$. The medium is described by its fixed fields $T_0$, $P_0,$ etc., while the system, if in internal equilibrium (see text), is characterized by $T\left(t\right),P\left(t\right),$ and so on.

Figure 2

We show schematically the two subsystems ${\sigma }_1$ and ${\sigma }_2$ [$T_2\left(t\right)>T_1\left(t\right)$] forming the system $\Sigma$ in an extensively large medium in (a) and by themselves forming an isolated system ${\overline{\Sigma }}_0$ without an extensively large medium in (b). The heat output $dQ\left(t\right)$ in (a) by ${\sigma }_2$ is the sum of $d{Q}^{\prime }\left(t\right)$ and $d{Q}_2\left(t\right)$, while the heat intake by ${\sigma }_1$ is the sum of $d{Q}^{\prime }\left(t\right)$ and $d{Q}_1\left(t\right)$. We take $T_0$ to be the equilibrium temperature for the isolated system in (b). As we are dealing with isolated systems in both cases, the heat input and output must be equal. Therefore, we must have $d{Q}_1\left(t\right)\equiv d{Q}_2\left(t\right)$. The equality of the heat input and output is also true in (b). As the heat transfers between objects do not occur isothermally, there is irreversible entropy generation due to each heat transfer. We will study this issue later in Sec. 10c.

Figure 3

We show in (a) a simplified situation in which two bodies 1 and 2 in thermal contact are alligned along the $x$ axis. Initially, the two bodies are in internal equilibrium at temperatures $T_1$ and $T_2>T_1$. Their contact interface is taken to be a plane orthogonal to the $x$ axis as shown in the figure, but the discussion is valid for any shape of the interface. In reality, however, any interface is an interface region of some very small width $\Delta x$ over which the temperature continuously changes between $T_1\left(t\right)$ and $T_2\left(t\right)$ so there is a very narrow region of width $dx\ll \Delta x$ around a point E on the $x$ axis, where the temperature is exactly $T_0$, the equilibrium temperature of the two bodies, as shown in (b) for the simple case shown in (a).

Figure 4

A schematic representation of the isolated system ${\Sigma }_0$ consisting of the two bodies 1 and 2, not in thermal contact, surrounded by an extensively large medium $\tilde{\Sigma }$, which is so large that the presence of 1 and 2 does not alter its internal temperature, which, therefore, remains constant. We choose $\tilde{\Sigma }$, as explained in the text, to be at the equilibrium temperature $T_0$ of ${\overline{\Sigma }}_0$. In the case $\tilde{\Sigma }$ is at a different temperature $T_0^{\prime }$, then the two $dQ\left(t\right)$ should be replaced by $d{Q}_{1\text{,net}}$ and $d{Q}_{1\text{,net}}\ne$ $d{Q}_{1\text{,net}}$; see text.

Figure 5

Schematic form of the entropy $S\left(t\right)$ (solid curve) and $S_{\text{eq}}\left(t\right)$ (dotted upper curve; see text) in a cooling experiment. Both curves start from $S^{\prime }$ at $t=0$ and end at $S^0$ as $t\to \infty$. The entropy difference $\Delta S\left(t\right)=S\left(t\right)-S^{\prime }$ is shown by the leftmost downward-pointing arrow and the cumulative reversible entropy change ${\Delta }_{e}S\left(t\right)$ by the rightmost-pointing arrow. The cumulative irreversible entropy generation ${\Delta }_{\mathrm{i}}S\left(t\right)>0$ is shown by the small upward-pointing arrow.