Case of the metal bar: revised version

The case of the metal bar is revised from this post. and this one. The main error made in these previous posts is that the the heat flux q was chosen proportional to ∂_xT instead of ∂_x(1/T) according to the phenomenological law. Thus, the profile corresponding to the minimum entropy principle did not correspond to the profile of a steady state (∂_tT=0, q is constant uniform along x). By using the correct phenomenological law, the profile of minimum entropy production is also the profile of a steady state; but it is not the traditional linear profile in T, it is a profile such as (1/T) is linear. The linear profile we learnt is only an approximation of the real steady state profile.

First, we need the temperature equation:

where ρ is the density and c_v is the specific heat at constant volume. The phenomenological law gives:

In that case, the steady state ∂_tT=0, q uniform in x gives a linear profile in 1/T.

The equation corresponding to the internal entropy production ∂_tS of the system is:

We want to minimize the internal entropy production inside the volume. Using the calculus of variation and the phenomenological law Eq.(2), we need to resolve:

Writing F(1/T,∂_x(1/T)) = q², we obtain:

The last term is zero because 1/T is fixed at the boundary conditions. We thus obtain the Euler-Lagrange equation:

The term on the lhs gives:

after using Eq.(2). Thus

But

using Eq.(2), so that the solution corresponds to q constant in x that is it is uniform all along the bar. We have seen that such state corresponds also to the steady state. Thus the steady state corresponds to a state of minimum entropy production. Calculation of the time rate of change of the entropy production &part_ttS would show that &part_ttS ≤ 0 so that the entropy production is always decreasing and indeed at steady state, the entropy production has reached a minimum.

The case presented above works also for a motionless one-component fluid.

See de Groot and Mazur, Non-equilibrium thermodynamics, p.47-48.

Temperature profile at which the entropy production is minimum

The temperature profile at which the entropy production is minimum has been computed here for the case of a metal bar heated at both ends by two difference heat sources. It is not the linear profile corresponding to the steady state but it is a profile exponentially increasing from the cold to the warm source.

The entropy production for any temperature profile T(x) is:


where K is the heat diffusion coefficient and is considered constant.
Varational calculus shows that the entropy production is an extremum for T(x) satisfying:

the solution of which is simply


Such profile fits the profile found for the family of broken linear profiles (see Entropy production in a metal bar for steady and non-steady states). Fig. 1 shows the present profile (dashed blue), the profile found for the family of broken linear profiles found previously (dotted red), and the linear steady-state profile (dashed black):

Figure 1

Entropy production in a metal bar for steady and non-steady states

The entropy production of a metal bar heated on each side by a different heat source is computed for a family of temperature profiles. Except for the linear profile which corresponds to the steady state solution, all the other profiles do not correspond to a steady state. The calculation was performed to see if the steady-state profile corresponds or not to the maximum entropy production among all possible (steady and non-steady) states.

The result is that the steady-state profile does not correspond to such state. It is rather close, but not exactly, to a state where entropy production is the minimum among all possible states. This result is discussed in contrast to Prigogine's "minimum entropy production principle".

The family of profiles consists of a linear profile between x=0 and x=a and another one with a different slope between x=a and x=L, where L is the length of the metal bar (Figure 1). The bar is heated with temperature T₁ on one end and T₂ on the other. Among all these profiles, the one corresponding to a steady state is linear throughout the bar length.

Figure 1

The temperature profile T(x) for this family is thus:



The time rate of change of T(x) is governed by the following equation:


where K is the heat diffusion coefficient and is considered constant here. The equation governing the production of entropy is:


where the last expression is positive in accord with the second law of thermodynamics. The entropy production is plotted in Fig. 2 for different values of a and T_a.

Figure 2

The entropy production can be as large as you want. For instance, the entropy production goes to infinity for a going to x=0 or L with a fixed T_a. On the other hand, there is a band of minimum entropy production. Derivating Eq. (8) with respect to T_a with fixed a, and equating it with zero gives the value of T_a for which the entropy production is minimum for a given a:


Such profile is plotted in red in Fig. 2 and it has to be compared with the linear profile plotted in dash black corresponding to the steady state solution.

Thus, in this simple example, the steady solution does not correspond to a state of maximum entropy production and is rather closer to a state of minimum entropy production. Ozawa et al (2003) states that linear systems, such as the present one, should be governed by the principle of minimum entropy production suggested by Prigogine (1947). Why we did not find exactly that? Well, there are different definitions of "minimum": the one we computed is the minimum among a family of possible states, while Prigogine's one is the minimum over the evolution of the system. Basically, Prigogine's principle simply states that the time evolution of the entropy for the forced-dissipative system we study is:

Figure 3

Because the system is constantly forced and dissipative, entropy is produced all the time even at steady state (defined here as d²S/dt²=0). At steady state the rate of production is constant. Prigogine's principle says that dS/dt is always decreasing and at steady state, it has reached a minimum.

Ozawa et al. (2003) argue that for a nonlinear system however, dS/dt reaches a maximum at steady state.