Thermodynamic cycle

The net work equals the area inside because it is (a) the Riemann sum of work done on the substance due to expansion, minus (b) the work done to re-compress.

Because the net variation in state properties during a thermodynamic cycle is zero, it forms a closed loop on a PV diagram. A PV diagram's Y axis shows pressure (P) and X axis shows volume (V). The area enclosed by the loop is the work (W) done by the process:

${\displaystyle {\text{(1)}}\qquad W=\oint P\ dV}$

This work is equal to the balance of heat (Q) transferred into the system:

${\displaystyle {\text{(2)}}\qquad W=Q=Q_{in}-Q_{out}}$

Equation (2) is consistent with the First Law; even though the internal energy changes during the course of the cyclic process, when the cyclic process finishes the system's internal energy is the same as the energy it had when the process began.

If the cyclic process moves clockwise around the loop, then W will be positive, and it represents a heat engine. If it moves counterclockwise, then W will be negative, and it represents a heat pump.

The following processes are often used to describe different stages of a thermodynamic cycle:

• Adiabatic : No energy transfer as heat (Q) during that part of the cycle (${\displaystyle \delta Q=0}$). Energy transfer is considered as work done by the system only.
• Isothermal : The process is at a constant temperature during that part of the cycle (T=constant, ${\displaystyle \delta T=0}$). Energy transfer is considered as heat removed from or work done by the system.
• Isobaric : Pressure in that part of the cycle will remain constant. (P=constant, ${\displaystyle \delta P=0}$). Energy transfer is considered as heat removed from or work done by the system.
• Isochoric : The process is constant volume (V=constant, ${\displaystyle \delta V=0}$). Energy transfer is considered as heat removed from the system, as the work done by the system is zero.
• Isentropic : The process is one of constant entropy (S=constant, ${\displaystyle \delta S=0}$). It is adiabatic (no heat nor mass exchange) and reversible.
• Isenthalpic : The process that proceeds without any change in enthalpy or specific enthalpy.
• Polytropic : The process that obeys the relation ${\displaystyle pV^{\,n}=C}$.
• Reversible : The process where the net entropy production is zero; ${\displaystyle dS-{\frac {dQ}{T}}=0}$.

Description of each point in the thermodynamic cycles.

The Otto Cycle is an example of a reversible thermodynamic cycle.

• 1→2: Isentropic / adiabatic expansion: Constant entropy (s), Decrease in pressure (P), Increase in volume (v), Decrease in temperature (T)
• 2→3: Isochoric cooling: Constant volume(v), Decrease in pressure (P), Decrease in entropy (S), Decrease in temperature (T)
• 3→4: Isentropic / adiabatic compression: Constant entropy (s), Increase in pressure (P), Decrease in volume (v), Increase in temperature (T)
• 4→1: Isochoric heating: Constant volume (v), Increase in pressure (P), Increase in entropy (S), Increase in temperature (T)

Heat engine diagram.

Thermodynamic power cycles are the basis for the operation of heat engines, which supply most of the world's electric power and run the vast majority of motor vehicles. Power cycles can be organized into two categories: real cycles and ideal cycles. Cycles encountered in real world devices (real cycles) are difficult to analyze because of the presence of complicating effects (friction), and the absence of sufficient time for the establishment of equilibrium conditions. For the purpose of analysis and design, idealized models (ideal cycles) are created; these ideal models allow engineers to study the effects of major parameters that dominate the cycle without having to spend significant time working out intricate details present in the real cycle model.

Power cycles can also be divided according to the type of heat engine they seek to model. The most common cycles used to model internal combustion engines are the Otto cycle, which models gasoline engines, and the Diesel cycle, which models diesel engines. Cycles that model external combustion engines include the Brayton cycle, which models gas turbines, the Rankine cycle, which models steam turbines, the Stirling cycle, which models hot air engines, and the Ericsson cycle, which also models hot air engines.

The clockwise thermodynamic cycle indicated by the arrows shows that the cycle represents a heat engine. The cycle consists of four states (the point shown by crosses) and four thermodynamic processes (lines).

For example :--the pressure-volume mechanical work output from the ideal Stirling cycle (net work out), consisting of 4 thermodynamic processes, is:

${\displaystyle {\text{(3)}}\qquad W_{\rm {net}}=W_{1\to 2}+W_{2\to 3}+W_{3\to 4}+W_{4\to 1}}$

${\displaystyle W_{1\to 2}=\int _{V_{1}}^{V_{2}}P\,dV,\,\,{\text{negative, work done on system}}}$

${\displaystyle W_{2\to 3}=\int _{V_{2}}^{V_{3}}P\,dV,\,\,{\text{zero work since }}V_{2}=V_{3}}$

${\displaystyle W_{3\to 4}=\int _{V_{3}}^{V_{4}}P\,dV,\,\,{\text{positive, work done by system}}}$

${\displaystyle W_{4\to 1}=\int _{V_{4}}^{V_{1}}P\,dV,\,\,{\text{zero work since }}V_{4}=V_{1}}$

For the ideal Stirling cycle, no volume change happens in process 4-1 and 2-3, thus equation (3) simplifies to:

${\displaystyle {\text{(4)}}\qquad W_{\rm {net}}=W_{1\to 2}+W_{3\to 4}}$

Thermodynamic heat pump cycles are the models for household heat pumps and refrigerators. There is no difference between the two except the purpose of the refrigerator is to cool a very small space while the household heat pump is intended to warm or cool a house. Both work by moving heat from a cold space to a warm space. The most common refrigeration cycle is the vapor compression cycle, which models systems using refrigerants that change phase. The absorption refrigeration cycle is an alternative that absorbs the refrigerant in a liquid solution rather than evaporating it. Gas refrigeration cycles include the reversed Brayton cycle and the Hampson–Linde cycle. Multiple compression and expansion cycles allow gas refrigeration systems to liquify gases.

An illustration of an ideal cycle heat engine (arrows clockwise).

An ideal cycle is simple to analyze and consists of:

1. TOP (A) and BOTTOM (C) of the loop: a pair of parallel isobaric processes
2. RIGHT (B) and LEFT (D) of the loop: a pair of parallel isochoric processes

If the working substance is a perfect gas, ${\displaystyle U}$ is only a function of ${\displaystyle T}$ for a closed system since its internal pressure vanishes. Therefore, the internal energy changes of a perfect gas undergoing various processes connecting initial state ${\displaystyle a}$ to final state ${\displaystyle b}$ are always given by the formula

${\displaystyle \Delta U=\int _{a}^{b}C_{v}dT}$

Assuming that ${\displaystyle C_{v}}$ is constant, ${\displaystyle \Delta U=C_{v}\Delta T}$ for any process undergone by a perfect gas.

Under this set of assumptions, for processes A and C we have ${\displaystyle W=p\Delta v}$ and ${\displaystyle Q=C_{p}\Delta T}$, whereas for processes B and D we have ${\displaystyle W=0}$ and ${\displaystyle Q=\Delta U=C_{v}\Delta T}$.

The total work done per cycle is ${\displaystyle W_{cycle}=p_{A}(v_{2}-v_{1})+p_{C}(v_{4}-v_{3})=p_{A}(v_{2}-v_{1})+p_{C}(v_{1}-v_{2})=(p_{A}-p_{C})(v_{2}-v_{1})}$, which is just the area of the rectangle. If the total heat flow per cycle is required, this is easily obtained. Since ${\displaystyle \Delta U_{cycle}=Q_{cycle}-W_{cycle}=0}$, we have ${\displaystyle Q_{cycle}=W_{cycle}}$.

Thus, the total heat flow per cycle is calculated without knowing the heat capacities and temperature changes for each step (although this information would be needed to assess the thermodynamic efficiency of the cycle).

The Carnot cycle is a cycle composed of the totally reversible processes of isentropic compression and expansion and isothermal heat addition and rejection. The thermal efficiency of a Carnot cycle depends only on the absolute temperatures of the two reservoirs in which heat transfer takes place, and for a power cycle is:

${\displaystyle \eta =1-{\frac {T_{L}}{T_{H}}}}$

where ${\displaystyle {T_{L}}}$ is the lowest cycle temperature and ${\displaystyle {T_{H}}}$ the highest. For Carnot power cycles the coefficient of performance for a heat pump is:

${\displaystyle \ COP=1+{\frac {T_{L}}{T_{H}-T_{L}}}}$

and for a refrigerator the coefficient of performance is:

${\displaystyle \ COP={\frac {T_{L}}{T_{H}-T_{L}}}}$

The second law of thermodynamics limits the efficiency and COP for all cyclic devices to levels at or below the Carnot efficiency. The Stirling cycle and Ericsson cycle are two other reversible cycles that use regeneration to obtain isothermal heat transfer.

A Stirling cycle is like an Otto cycle, except that the adiabats are replaced by isotherms. It is also the same as an Ericsson cycle with the isobaric processes substituted for constant volume processes.

1. TOP and BOTTOM of the loop: a pair of quasi-parallel isothermal processes
2. LEFT and RIGHT sides of the loop: a pair of parallel isochoric processes

Heat flows into the loop through the top isotherm and the left isochore, and some of this heat flows back out through the bottom isotherm and the right isochore, but most of the heat flow is through the pair of isotherms. This makes sense since all the work done by the cycle is done by the pair of isothermal processes, which are described by Q=W. This suggests that all the net heat comes in through the top isotherm. In fact, all of the heat which comes in through the left isochore comes out through the right isochore: since the top isotherm is all at the same warmer temperature ${\displaystyle T_{H}}$ and the bottom isotherm is all at the same cooler temperature ${\displaystyle T_{C}}$, and since change in energy for an isochore is proportional to change in temperature, then all of the heat coming in through the left isochore is cancelled out exactly by the heat going out the right isochore.