Truesdell's formulation of second law for homogeneous systems in Rational Thermodynamics

This is an extract of Lecture 1 (page-67) of Rational Thermodynamics by Truesdell.

The irreversibility of natural processes is represented by the existence of an a priori least upper bound $$\mathfrak{B}$$ for the heating $$\mathfrak{Q}$$. The term "irreversibility" is justified because $$\mathfrak{Q}$$, the rate of increase of energy not accompanied by mechanical working, is bounded above but not necessarily below. There is no limit to the magnitude of negative values of $$\mathfrak{Q}$$ which represent conversion of energy into heating without performance of work, but there certainly is to positive ones. Work and energy may always be converted into heat, but each body is limited in the rate at which it can convert heat into energy without doing work. That is the content of the second axiom: $$\mathfrak{Q} \leq \mathfrak{B}$$.

The first axiom was (the first law) : $$\dot{\mathfrak{E}} = \mathfrak{W} + \mathfrak{Q}$$, where $$\mathfrak{E}$$ is the energy, $$\mathfrak{W}$$ is the rate of working.

My question is, how does saying there is an upper bound for positive values of $$\mathfrak{Q}$$ correspond in any way to the second law ? Loosely, the second law is supposed to imply that you cannot completely convert $$\mathfrak{Q} >0$$ into working $$\mathfrak{W}$$.

Furthermore, the statement

Work and energy may always be converted into heat, but each body is limited in the rate at which it can convert heat into energy without doing work

seems incorrect. I do not see why there is a limit on converting heat into energy without doing work. Surely, one can simply keep supplying heat and raising the energy without doing work ?

This second axiom is critical to the rest of the chapter, so I am very confused.

The important idea here is not that there is some upper bound but that there is a constitutive upper bound. This idea is similar to Clausius's inequality but in a different guise.

You surely learned in your elementary thermodynamics course, the kind that Truesdell detests, that the quantity of heat, say, $$\delta Q$$ that is absorbed between two infinitesimally close equilibrium states satisfies the inequality $$\delta Q \le TdS$$ where $$dS$$ is the infinitesimal entropy difference between the two states. Notice the different notations: $$\delta$$ v. $$d$$ where $$\delta$$ just means "small" while $$d$$ refers to an exact differential of a state function. Truesdell (Coleman, Noll, Gurtin, etc.,) generalizes and turns it around this idea with two changes

1. Replaces both exact and inexact infinitesimals with time rates, that is in the inequality he replaces $$dS$$ with the time derivative of the entropy that is assumed to exists not just in equilibrium but also during an arbitrary irreversible process always and everywhere, say $$\frac{dS}{dt}$$. He also introduces the heating rate $$\mathfrak Q$$, not the quantity of heat $$\delta Q$$ as being basic; $$\int \delta Q = \int \mathfrak Q dt$$

2. Assumes that there is a basic constitutive quantity $$\mathfrak B$$ characteristic of the thermodynamic system instead of the product $$T\frac{dS}{dt}$$ such that the inequality $$\mathfrak Q \le \mathfrak B$$ holds for all processes. The upper bound $$\mathfrak B$$ itself being related to a rate (entropy rate) limits how fast can heat be absorbed by the body and it is as constitutive as any other equilibrium "state" variable.

To see better its meaning think of the conventional Gibbs formulation of the 1st and 2nd laws. Between two neighboring equilibrium states we have the equation $$dU=TdS+\sum_k Y_kdX_k$$ when the internal energy change is described by its internal equilibrium sate variables and the $$dU=\delta Q + \delta W$$ where $$\delta Q$$ is the absorbed heat from the external environment and $$\delta W$$ is the work done on the system by its external environment. Now you know that per Clausius $$\delta Q \le TdS$$, then, per force, $$\delta W \ge \sum_k Y_kdX_k$$. Now take a deep breath and replace the infinitesimals with time rates by dividing both sides with $$dt$$: $$\dot Q =\frac{\delta Q}{dt} \le T\frac{dS}{dt} \tag{1}$$ and $$\dot W =\frac{\delta W}{dt} \ge \sum_k Y_k\frac{dX_k}{dt}\tag{2}$$ So far we have not done anything new or radical as long as all the ends of the $$dt$$ time steps do represent equilibrium states.

Truesdell, and actually much earlier Bridgman, Eckart, Bronsted, and others, just stated that something like this should hold no matter what the speed of process is, that is it should hold for any irreversible process at all rates. This is a radical departure form the way Clausius saw this because even the meaning of an expression such as $$\frac{dX_k}{dt}$$ can be questioned. More importantly, and first Bridgman was to state it explicitly, the difference $$T\dot S - \dot Q = \dot D \ge 0$$ representing the dissipation is a constitutive quantity characteristic of the thermodynamic body that is subjected to an arbitrary process and $$\dot D = T\dot \sigma$$ where $$\dot \sigma$$ is the rate entropy production a result of dissipation. For example, $$T\dot \sigma = I^2R$$ in a resistor $$R$$ at temperature $$T$$ when current $$I$$ goes through it and it is obviously never negative. This much entropy production is in the very nature of the resistor and all processes by that very nature will automatically satisfy that its entropy production be nonnegative. This is a constraint on the constitutive law and not on the processes.

The way a thermodynamic body and its process are viewed by Clausius and Truesdell are radically different. Clausius sees that the 2nd law limits the kind of processes a body may participate in and the process must satisfy the 2nd law. In contrast, Truesdell sees the processes are arbitrary but the body's own constitutive relationships are limited by the 2nd law. This is why Clausius deals with reversible processes while for Truesdell the process can be anything.

Regarding your last question, set $$\mathfrak W=0$$, then $$\dot {\mathfrak E} = \mathfrak Q \le \mathfrak B$$ showing that the rate of internal energy increase in workless heat exchange is also limited by the same bound.