Thermodynamic cycle

A particularly interesting type of process in Thermodynamics is what is called a cyclic process or simply cycle. What makes this type of process interesting is that it is the underlying principle for both heat engines and heat pumps.

A cyclic process is a process in which a thermodynamic system (its working fluid in the case of a heat engine and a heat pump) starts from an initial state A then undergoes a series of processes before returning to this initial state A, as shown in the figure below. A cycle can be operated clockwise (left of the figure) or counterclockwise (right). In these pages we will consider that the working fluid is a gas.

The work done by the gas in a cycle is the area enclosed by it in the PV diagram (to which the sign must be added if necessary). When a cycle is operated clockwise, the net work done by the gas is positive, since the work AB is positive (the gas expands) and the work BA is negative (the gas compresses) but smaller in absolute value. Therefore the sum of the two is positive.

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The working fluid of a heat engine passes through a clockwise-operated cycle, because a heat engine is a device that provides a net work output (therefore, positive). On the other hand, the working fluid of a refrigerator (also called heat pump), goes through a counterclockwise-operated cycle, since it is necessary to supply energy in the form of work (therefore negative) in order for the heat pump to extract heat from the cold reservoir and discharge it to the hot bath.

Mathematically, the net work is the sum of the work done by the gas in each process of the cycle:

Remember that the work done by a gas is given by:

Therefore, in order to calculate the work done by the working fluid in each process of the cycle it is necessary to know the pressure p as function of V.

The variation of internal energy in a cycle is always zero. Regardless of the shape of the cycle, whether it is reversible or not, the type of substance, the internal energy of the thermodynamic system remains constant because this quantity is a state function and therefore its value depends only on the initial and final states of the system. And in a cycle the initial and final states coincide.

The total heat exchanged during a cycle equals the net work done by the system:

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The efficiency of a cycle is given by:

Where W is the net work done by the gas on its surroundings and Q1 is the heat absorbed by it. That is, W is the sum of the work done by the gas in each process of the cycle (on the left in the upper figure WABA) and Q1 is the sum of the heats absorbed (only the positive ones).

The change in entropy for a cycle is always zero, regardless of whether it is reversible or not, because entropy is a state function and therefore, its change depends only on the initial and final state of the system. And in a cycle the initial and final states coincide.

We will apply these results to the study of heat engine.

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