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We revisit the problem of optimal power extraction in four-step cycles (two adiabatic and two heat-transfer branches) when the finite-rate heat transfer obeys a linear law and the heat reservoirs have finite heat capacities. The heat-transfer branch follows a polytropic process in which the heat capacity of the working fluid stays constant. For the case of ideal gas as working fluid and a given switching time, it is shown that maximum work is obtained at Curzon-Ahlborn efficiency. Our expressions clearly show the dependence on the relative magnitudes of heat capacities of the fluid and the reservoirs. Many previous formulae, including infinite reservoirs, infinite-time cycles, and Carnot-like and non-Carnot-like cycles, are recovered as special cases of our model.

Curzon-Ahlborn efficiency,

In this paper, we revisit the problem of optimal performance with “linear” irreversibilities of finite time and finite heat reservoirs in classical models of engines. This question was addressed in [

The present analysis of the problem is based on the following features: (a) the working fluid is a classical ideal gas; (b) the total cycle time as well as switching time is given. We consider a generic four-step cycle with two adiabatic and two heat-transfer branches, which follow a polytropic process with a constant heat capacity

Our key results may be listed as follows. For ideal gas as the working fluid and the generic four-step cycle (including Carnot and non-Carnot heat cycles), the power of the engine is maximum at CA efficiency, even for finite reservoirs. Many special cases like infinite time and infinite reservoirs [

First of all, we calculate how the temperatures of the two bodies change, when kept in contact with one another for some time. The heat transfer between the bodies is Newtonian, and during heat transfer, their heat capacities remain constant.

Let the two bodies denoted as

At time

We have two finite heat reservoirs at initial temperatures

Working fluid is brought in contact with the hot reservoir from time

The fluid is allowed to expand adiabatically. This step is assumed to be instantaneous and takes negligible time. As a result, the temperature of the working fluids jumps from

The fluid is brought in contact with the cold reservoir from time

Working fluid is allowed to contract adiabatically. This step is also assumed to take negligible time. As a result, the temperature of the working fluids jumps from

Since

Heat cycle of the working fluid.

We will now use the above temperatures to evaluate the work performed by the engine. Due to the cyclic process, the first law of thermodynamics implies that the net heat exchanged by the working fluid during the cycle equals the work performed. Thus

Here, it should be noticed that no heat leakage or entropy production is considered in this model.

In this section, we will first optimize the work per cycle over the initial temperature

Under the infinite reservoirs condition

Using (

We can further maximize the work with respect to the switching time:

When we take the nonadiabatic paths to be isothermal, that is,

In this section, we carry out the optimisation of work under the constraints of both finite reservoirs and finite cycle time. Solving similarly as in Section

Optimal work as function of

We may further maximize the work with respect to the switching time:

Optimal switching time versus the heat capacity of the hot reservoir. Here,

Further, if in (

In this paper, the maximum power point characteristics of a generalized four-step heat engine are studied. The heat cycle has two adiabatic steps and in the remaining two steps the working fluid follows a polytropic process with a constant heat capacity. It is observed that some common heat cycles such as Otto cycle, Joule-Brayton cycle, and Carnot cycle can be incorporated as special cases of this model. Curzon and Ahlborn in their seminal paper derived the expression for efficiency at maximum power for a Carnot-like engine with infinite reservoirs. Later, Gordon proved that even with finite reservoirs, Carnot-like engines show this efficiency at maximum power. Here we have analysed this result for the generalised heat cycle and shown that CA efficiency is obtained by optimizing the initial temperature of the working fluid and is independent of the switching time. However, we can further maximize this work over the switching time. Finally, the effect of finiteness of cycle time alone, the finiteness of the reservoirs alone, and the finiteness of both the cycle time and the reservoirs together on the maximum work per cycle can be respectively attributed to the following factors:

The factor in (

Since

Using similar procedure as to time irreversibility, we can show that

This work was initiated during the summer program at IISER Mohali. A. Khanna expresses his gratitude towards IISER Mohali, for hospitality and financial support. R. S. Johal acknowledges financial support from the Department of Science and Technology, India, under the Research Project no. SR/S2/CMP-0047/2010(G).