Stationary States under Nonequilibrium Conditions

Let us consider a system of length L in contact with a hot thermal reservoir, at a temperature Th. at one end and a cold thermal reservoirs, at temperature T, at the other (Fig. 17.1). In section 3.5, and in more detail in Chapter 16, we discussed the entropy production due to heat flow but we did not consider entropy balance in detail. We assume here that the conduction of heat is the only irreversible process. For this system, using Table 15.1 for the flows and forces, we see that the entropy production per unit volume is 

If we assume that the temperature gradient is only in the x direction, a per unit length is given by 

NONEQUILIBRIUM STATIONARY STATES AND THEIR STABILITY LINEAR REGIME 

Such a system reaches a state with stationary temperature distribution and a uniform heat flow J. (A stationary temperature T x) implies that the heat flow is uniform otherwise there will be an accumulation or depletion of heat, resulting in a time-dependent temperature.) The evolution of the temperature distribution can be obtained explicitly by using the Fourier laws of heat conduction  

It is easy to see that the stationary state, dT/dt = 0, is one in which T(x) is a linear function of x (Fig. 17.1) and 3q = constant. A stationary state also implies that all other thermodynamic quantities such as the total entropy S of the system are constant  

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This equation describes the pressure difference because of the mass fraction difference when there is no temperature difference. This is called the osmotic pressure. This effect is reversible because AT - 0,, /2 = 0. and at stationary state J = 0. Therefore, Eq. (7.244) yields Jq = 0, and the rate of entropy production is zero. The stationary state under these conditions represents an equilibrium state. Equation (7.263) does not contain heats of transport, which is a characteristic quantity for describing nonequilibrium phenomena. 

Since 8S< 0 under both the equilibrium and nonequilibrium conditions, the stability of a stationary state is accomplished if... 

Under constant external conditions, a nonequilibrium system may reach its stationary state. Specific features of such a state are the time constant values of internal thermodynamic parameters characterizing the system... 

Inequalities (3.2) and (3.3) are generalizations of the principle of the minimal entropy production rate in the course of spontaneous evolution of its system to the stationary state. They are independent of any assump tions on the nature of interrelations of fluxes and forces under the condi tions of the local equilibrium. Expression (3.2), due to its very general nature, is referred to as the Qlansdorf-Prigogine universal criterion of evolution. The criterion implies that in any nonequilibrium system with the fixed boundary conditions, the spontaneous processes lead to a decrease in the rate of changes of the entropy production rate induced by spontaneous variations in thermodynamic forces due to processes inside the system (i.e., due to the changes in internal variables). The equals sign in expres sion (3.2) refers to the stationary state. 

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