## Heat Engines and Finite-Time Thermodynamics

If you completed the standard curriculum of physics, you should be familiar with the concept of a heat engine:

Moreover I bet you know that the maximum efficiency is achieved by a Carnot heat engine, for which the efficiency is:

$\displaystyle{\eta = 1 - \frac{T_c}{T_h}. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (1)}$

You should also be aware that Carnot heat engine implies the processes involved to be reversible, which by turn means that the engine proceeds infinitely slowly. But if you are like me, chances are you haven’t made the final step to conclude that such engine is absolutely useless! Indeed, if it operates infinitely slowly then it has zero power. Since heat engines are made to do some actual work, zero power is not what we are looking for.

Thus we need a simple model which would take into account the finite time required by an engine to perform a cycle. Fortunately there is such a model by Curzon and Ahlborn. For an introduction to a finite-time thermodynamics I refer you to a book “Understanding non-equilibrium thermodynamics: foundations, applications, frontiers” by Georgy Lebon, David Jou and José Casas-Vázquez. The rest of the post follows it.

In essence Curzon and Ahlborn heat engine operates in a Carnot cycle, with two isotherms and two adiabats. However, the heat flux between various systems (cooler, working substance, heater) is governed by Newton’s law:

$\displaystyle{\frac{\text d Q_{12}}{\text d t} = \alpha_{12} (T_2 - T_1).}$

It is also assumed that these heat transfers are the only source of irreversibility in a system and their combined duration almost equals to the length of the whole cycle. Leaving aside calculations, we get for the power:

$\displaystyle{P(T_c, T_c, T_{h \to w}, T_{w \to c}) = \frac{T_{h \to w} - T_{w \to c}}{\dfrac{T_{h \to w}}{\alpha_{h,w}(T_h - T_{h \to w})} + \dfrac{T_{w \to c}}{\alpha_{w,c}(T_{w \to c} - T_c)}}, }$

the notation used is explained in this figure:

To find the maximum possible power given fixed $T_c$ and $T_h$ one have to solve the following system:

$\displaystyle{\frac{\!\!\! \partial P}{\partial T_{c \to w}} = 0, \;\;\;\; \frac{\!\!\! \partial P}{\partial T_{w \to h}} = 0.}$

If you are patient enough, you’ll ultimately get the efficiency, corresponding to $P_\text{max}$:

$\displaystyle{\eta_{\text{max power}} = 1 - \sqrt{\frac{T_c}{T_h}}. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (2)}$

Surprisingly, the result does not depend on $\alpha_{c,w}$ or $\alpha_{w,h}$. As an illustration of differences between Carnot’s result (1) and the result of Curzon and Ahlborn (2) I’ll quote “Understanding non-equilibrium thermodynamics”:

As an illustration, consider a power station working, for instance, between heat reservoirs at 565 and 25 °C and having an efficiency of 36%. We want to evaluate this power station from the thermodynamic point of view. It follows from (1) that the Carnot’s maximum efficiency is 64.1%; our first opinion on the quality of the power station would be rather negative if Carnot’s efficiency is our standard for evaluation. However, according to (2), its efficiency at maximum power is 40%. Thus, if the objective of the power station is to work at maximum power output, it is seen that the efficiency is not bad compared to the 40% efficiency corresponding to this situation. We can therefore conclude that, to have realistic standards for evaluation of an actual heat engine, one needs to go beyond Carnot’s efficiency and to incorporate finite-time considerations.

If you got a taste for finite-time thermodynamics, I suggest you to read the book. Keep in mind, that power maximization might be not the only goal, for example one probably would like to minimize pollution or running cost of the plant. These topics are also discussed in that book.

I’ll conclude with a thought (or a question) of my own. It is obvious, that power, unlike efficiency, is an additive quantity. Hence one could combine several (let’s say n) low power, but efficient heat engines to produce a single engine. Roughly, the efficiency of such engine would remain high, but it’s power would increase n times. So I wonder is that the reason why to produce a powerful car engine one adds cylinders instead of increasing a power production of a single cylinder?