Thermodynamic cycle

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In thermodynamics, a cyclic process is a sequence of changes in the state of a working medium (liquid, steam, gas - commonly called fluid ), which occurs periodically, whereby the initial state, characterized by the state variables (see also fundamental equation , thermodynamic potential ), such as u . a. Pressure , temperature and density . They are technical processes, mostly for converting heat into work (e.g. in combustion engines) or for heating and cooling by applying work ( heat pump , refrigerator ).

Two fundamental examples (math)

1. Given a formal expression , e.g. B. with ("absolute temperature") and ("specific liquid volume"). The sequential execution (integration) of such infinitesimal processes defines a thermodynamic process . The "successive execution" happens in a closed way. Nevertheless, one does not speak of a "circular process": ${\displaystyle \delta \omega :=A_{1}\,\mathrm {d} a_{1}+A_{2}\mathrm {d} a_{2}}$ ${\displaystyle a_{1}=T}$ ${\displaystyle a_{2}=v}$

We now ask whether a function exists for - e.g. B. the entropy of the system - so that the above differential expression is the total differential of the given so-called “state function” . Only such processes are called circular processes, more precisely "integrable circular processes". The line integral over any state function f gives yes always zero, calculated on an arbitrary closed path W . However , this does not apply to. As a result, the most important thing is not the closeness of the path, but the integrability of it . ${\displaystyle \delta \omega }$ ${\displaystyle f(T,v)}$ ${\displaystyle f(T,v)}$ ${\displaystyle \oint _{W}\mathrm {d} f}$ ${\displaystyle \oint _{W}\delta \omega }$ ${\displaystyle \delta \omega }$

A cycle is then and only then if always for all closed paths W (the closedness of the path is underlined by the circle symbol next to the integral sign), where and applies. Because of the identity of the mixed 2nd derivatives, this means that must be. ${\displaystyle \oint _{W}\,\delta \omega {\stackrel {!}{\equiv }}0}$ ${\displaystyle \mathrm {A} _{1}={\partial f}/{\partial a_{1}}}$ ${\displaystyle A_{2}={\partial f}/{\partial a_{2}}}$ ${\displaystyle \partial A_{1}/\partial a_{2}=\partial A_{2}/\partial a_{1}}$

So we only need to check whether this so-called “integrability condition” is fulfilled or not: this is usually not the case.

So: circular processes are the exception and not the rule .

For example, there is no need for a cycle for ("heat"), because heat, supplied in different ways, does not produce the same result, even if it is reversibly supplied to the system : ${\displaystyle f=Q}$

${\displaystyle \oint _{W}\delta Q\neq 0}$(see, for example, in the Carnot process )

In contrast, the existence of a cycle is the case with other important variables, e.g. Example, in the entropy S , that is, when a heat energy .DELTA.Q first reversible supplied and discharged, and secondly multiplied by the "integrating factor" 1 / T, the different symbols in the differentials are designed to emphasize here once again that it is once (left Side) is a complete differential, the other time (right side) is an incomplete differential . For the already mentioned "integrating denominator", the "absolute temperature" T , this also means that it is a particularly important quantity (not just a formal number): compared to the usual temperature scales (Celsius, Fahrenheit , Réaumur scale, etc.) it has additional properties that can a. express in the mentioned mathematical relationships. ${\displaystyle \mathrm {d} S={\frac {\delta Q_{\,{\text{reversibel}}}}{T}}\,.}$

2. Instead (see the following example) it can also happen that the closed path breaks down into different sections on which different state functions are considered (e.g. in the next example there are entropy changes in horizontal sections, whereas enthalpy changes occur vertical sections). The result is i. A. the generation of mechanical or electrical work (e.g. steam turbine).

Further description

What is decisive for a circular process (often also called a cycle) is that the return path is different from the path on which the state moves away from the initial state. The most commonly used state diagrams are the pv diagram , the Ts diagram , the hs diagram and the ph diagram (the latter especially for cooling processes). In the first two diagrams mentioned, an area is thereby circled which corresponds to the circular process work in reversible processes . However, this only applies to the ideal comparison processes . The real technical processes are not reversible (see dissipation ) and the area is then enlarged by the dissipated work.

Example: gas turbine process
Comparison process and real process in the hs diagram (h is approximately proportional to the temperature T for gases)	Closed gas turbine process as an example of a cycle

Right and left litigation

There are right-hand processes and left-hand processes, depending on whether the state diagram is run through clockwise or vice versa.

With the clockwise process, part of the heat supplied at a high temperature is converted into work, the other part is removed again at a lower temperature. The difference is the cycle work (see energy balance for cycle processes ). The acquisition of work in the legal process comes about that at low temperature, i. H. at low pressure is compressed (workload) and at high temperature and thus at high pressure the fluid expands with work output. The amount of volume work of expansion is thus greater than that of compression.

In the left-hand process, on the other hand, everything is reversed, so that heat is pumped from a colder reservoir to a warmer one with a lot of work. Particularly large specific cycle work is achieved if the phase change between liquid and gaseous takes place within the process , because then the volume difference is particularly large. This is used in the steam power plant . Since liquid (water) is almost incompressible, there is no compression work and the work involved in pumping the liquid into the boiler at high pressure (boiler feed pump) is relatively low.

Example: steam power plant (legal process)
Cycle process of the Staudinger power plant, block 5 in the Ts diagram (see steam power plant ).

Example: cooling process (left process)
Left process with NH 3 in the hp diagram. The changes in state are: Compression of saturated steam 1-2, heat release up to the condensation point 2-3, heat release through condensation 3-4, throttling 4-5, evaporation 5-1 (see refrigeration machine ).

Open and closed processes

Another differentiation of the cycle processes results from the different heat input. If this takes place internally through the combustion of the fuel introduced, as in the case of an internal combustion engine or aircraft engine, the cycle is open because a gas exchange between exhaust gas and fresh air must take place. There is no fundamental difference from the thermodynamic point of view because the atmosphere can be viewed as a large heat exchanger. The process in the picture example is a closed one with two heat exchangers. Such processes can be used, for example, in a nuclear power plant with gas-cooled reactors (e.g. helium as coolant and working fluid).

With the computational and graphical representation of the processes one has a theoretical aid, both for the formulation of statements, as well as for the technical implementation in the design of thermal engineering machines and systems . For example, the Born-Haber cycle is used in chemistry to calculate the reaction energy (or enthalpy) of a process step or the binding energy of a chemical compound if the energies of the other process steps are known.

The ideal comparison processes are used to assess the efficiency of a circular process . These in turn are compared with the ideal theoretical cycle process, the Carnot process , which has the maximum possible efficiency . It characterizes what is theoretically possible according to the 2nd law of thermodynamics , in practice this efficiency is not (completely) achievable.

Circular processes

• Atkinson cycle
• Born-Haber cycle
• Carnot process
• Clausius-Rankine cycle
• Diesel cycle
• Ericsson cycle
• Joule cycle
• Kalina cycle
• Miller cycle
• Otto cycle
• Seiliger cycle
• Siemens cycle process
• Stirling cycle
• Vuilleumier cycle

literature

• Literature on technical thermodynamics
• Klaus Lüders, Gebhard von Oppen: Mechanics, acoustics, warmth. 12th edition. De Gruyter, Berlin 2008, ISBN 978-3-11-019311-4 ( textbook on experimental physics. Volume 1).

Web links

• Circular processes of thermodynamics
• Uni Duisburg-Essen, basics of technical thermodynamics with exercises (PDF file, 2.50 MB)