Thermodynamic cycle 3285197 218388567 2008-06-10T12:45:47Z 217.219.170.234 /* Classes */ A '''thermodynamic cycle''' is a series of [[thermodynamic processes]] which returns a system to its initial state. [[Properties]] depend only on the [[thermodynamic state]] and thus do not change over a cycle. Variables such as [[heat]] and [[work (thermodynamics)|work]] are not zero over a cycle, but rather are process dependent. The [[first law of thermodynamics]] dictates that the net heat input is equal to the net work output over any cycle. The repeating nature of the process path allows for continuous operation, making the cycle an important concept in [[thermodynamics]]. Thermodynamic cycles often use [[quasistatic processes]] to model the workings of actual devices [[Image:Stirling Cycle.png|thumb|200px|Example of P-V diagram of a thermodynamic cycle.]] A thermodynamic cycle is a closed loop on a [[Pressure volume diagram|P-V diagram]]. A P-V diagrams ''X'' axis shows volume (''V'') and ''Y'' axis shows pressure (''P''). The area enclosed by the loop is the work (''W'') done by the process: :<math> \text{(1)} \qquad W = \oint P \ dV </math>. This work is equal to the balance of heat (Q) transferred into the system: :<math> \text{(2)} \qquad W = Q = Q_{in} - Q_{out} </math>. Equation (2) makes a cyclic process similar to an [[isothermal process]]: even though the internal energy changes during the course of the cyclic process, when the cyclic process finishes the system's energy is the same as the energy it had when the process began. If the cyclic process moves clockwise around the loop, then it represents a [[heat engine]], and W will be positive. If it moves counterclockwise then it represents a [[heat pump]], and W will be negative. ==Abstract== Two primary classes of thermodynamic cycles are '''power cycles''' and '''heat pump cycles'''. Power cycles are cycles which convert some heat input into a [[mechanical work]] output, while heat pump cycles transfer heat from low to high temperatures using mechanical work input. Cycles composed entirely of quasistatic processes can operate as power or heat pump cycles by controlling the process direction. On a pressure-volume or [[Temperature]]-[[entropy]] diagram, the [[clockwise and counterclockwise]] directions indicate power and heat pump cycles, respectively. ===Thermodynamic power cycles=== [[Image:Carnot_heat_engine_2.svg|200px|thumb|Heat engine diagram.]] {{Main|Heat engine}} Thermodynamic power cycles are the basis for the operation of heat engines, which supply most of the world's [[electric power]] and run almost all [[motor vehicles]]. Power cycles can be divided according to the type of heat engine they seek to model. The most common cycles that model [[internal combustion engines]] are the [[Otto cycle]], which models [[gasoline engine]]s and the [[Diesel cycle]], which models [[diesel engines]]. Cycles that model [[external combustion engines]] include the [[Brayton cycle]], which models [[gas turbines]], and the [[Rankine cycle]], which models [[steam turbines]]. [[Image:Stirling Cycle.png|thumb|200px|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]] done in the heat engine cycle, consisting of 4 thermodynamic processes, is: :<math> \text{(3)} \qquad W = W_{1\to 2} + W_{2\to 3} + W_{3\to 4} + W_{4\to 1} </math> :<math> W_{1\to 2} = \int_{V_1}^{V_2} P \, dV, \, \, \text{positive, gives work} </math> :<math> W_{2\to 3} = \int_{V_2}^{V_3} P \, dV, \, \, \text{zero work if V2 equal V3} </math> :<math> W_{3\to 4} = \int_{V_3}^{V_4} P \, dV, \, \, \text{negative, uses work} </math> :<math> W_{4\to 1} = \int_{V_4}^{V_1} P \, dV, \, \, \text{zero work if V4 equal V1} </math> If no volume change happens in process 4->1 and 2->3, equation (3) simplifies to: :<math> \text{(4)} \qquad W = W_{1\to 2} + W_{3\to 4} </math> ===Thermodynamic heat pump and refrigeration cycle=== {{Main|Heat pump and refrigeration cycle}} Thermodynamic heat pump and refrigeration cycles are the [[Mathematical model|model]]s for [[heat pumps]] and [[refrigerators]]. The difference between the two is that heat pumps are intended to keep a place warm and refrigerators designed to cool it. The most common refrigeration cycle is the [[Vapor compression refrigeration|vapor compression cycle]], which models systems using [[refrigerants]] that change phase. The [[Gas absorption refrigeration|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 [[Linde-Hampson cycle]]. Regeneration in gas refrigeration allows for the [[liquefaction of gases]]. ==Types of thermodynamic cycles== A thermodynamic cycle can (ideally) be made out of 3 or more [[thermodynamic process]]es (typical 4). The processes can be any of these: * [[isothermal process]] (at constant temperature, maintained with heat added or removed from a heat source or sink) * [[isobaric process]] (at constant pressure) * [[isochoric process|isometric / isochoric process]] (at constant volume) * [[adiabatic process]] (no heat is added or removed from the working fluid) ** [[isentropic process]], [[reversible adiabatic process]] (no heat is added or removed from the working fluid - and the [[entropy]] is constant) * [[isenthalpic process]] (the [[enthalpy]] is constant) Some examples are as follows: {| class="wikitable" |+<!-- no title, because it starts a section.--> |- ! Cycle\Process !! Compression !! Heat Addition !! Expansion !! Heat Rejection |- |colspan="5" align="center" | Power cycles normally with [[external combustion]] - or heat pump cycles |- ! [[Ericsson cycle|Ericsson (First, 1833)]]<br>[[Brayton cycle|Brayton]] | adiabatic || isobaric || adiabatic || isobaric |- ! [[Bell Coleman cycle|Bell Coleman]]<br><small>(Reverse Brayton)</small> | adiabatic || isobaric || adiabatic || isobaric |- ! [[Carnot cycle|Carnot]] | isentropic || isothermal || isentropic || isothermal |- ! [[Stoddard cycle|Stoddard]] | adiabatic || isometric || adiabatic || isometric |- ! [[Stirling cycle|Stirling]] | isothermal || isometric || isothermal || isometric |- ! [[Ericsson cycle|Ericsson (Second, 1853)]] | isothermal || isobaric || isothermal || isobaric |- |colspan="5" align="center" | Power cycles normally with [[internal combustion]] |- ! [[Otto cycle|Otto (Petrol)]] | adiabatic ||isometric || adiabatic || isometric |- ! [[Diesel cycle|Diesel]] | adiabatic || isobaric || adiabatic || isometric |- ! [[Brayton cycle|Brayton (Jet)]] | adiabatic || isobaric || adiabatic || isobaric |- ! [[Lenoir cycle|Lenoir (pulse jet)]]<br><small>(Note: 3 of the 4 processes are different)</small> | isobaric || isometric || adiabatic || isobaric |} ===Carnot cycle=== {{main|Carnot 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 temperatures in [[kelvin]]s of the two reservoirs in which heat transfer takes place, and for a power cycle is: :<math>\eta=1-\frac{T_L}{T_H}</math> where <math>{T_L}</math> is the lowest cycle temperature and <math>{T_H}</math> the highest. For Carnot refrigeration cycles the [[coefficient of performance]] for a [[heat pump]] is: :<math>\ COP = 1+\frac{T_L}{T_H - T_L}</math> and for a [[refrigerator]] the coefficient of performance is: :<math>\ COP = \frac{T_L}{T_H - T_L}</math> 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. ===Ideal cycle=== [[Image:Cyclic process.PNG|thumb|250px|An illustration of an ideal cycle heat engine (arrows clockwise).]] An ideal cycle is constructed out of: # TOP and BOTTOM of the loop: a pair of parallel '''isobaric''' processes # LEFT and RIGHT of the loop: a pair of parallel '''isochoric''' processes ===Otto cycle=== {{Mainarticle|Otto cycle}} An Otto cycle is constructed out of: # TOP and BOTTOM of the loop: a pair of quasi-parallel '''adiabatic''' processes # LEFT and RIGHT sides of the loop: a pair of parallel '''isochoric''' processes The adiabatic processes are impermeable to heat: heat flows into the loop through the left pressurizing process and some of it flows back out through the right depressurizing process, and the heat which remains does the work. ===Stirling cycle=== {{Mainarticle|Stirling cycle}} A Stirling cycle is like an Otto cycle, except that the adiabats are replaced by isotherms. # TOP and BOTTOM of the loop: a pair of quasi-parallel '''isothermal''' processes # 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 <math> T_H </math> and the bottom isotherm is all at the same cooler temperature <math> T_C </math>, 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. ==State functions and entropy== If ''Z'' is a state function then the balance of ''Z'' remains unchanged during a cyclic process: :<math> \oint dZ = 0 </math>. If entropy is defined as :<math> S = {Q \over T} </math> so that :<math> \Delta S = {\Delta Q \over T} </math>, then it can be proven that for any cyclic process, :<math> \oint dS = \oint {dQ \over T} = 0. </math> ===Demonstration=== ====Part 1==== Draw a rectangle on a [[P-V diagram]], such that the top and bottom are horizontal isobaric processes and the left and right are vertical isochoric processes. Such a rectangle should be made really small, so that change in temperature can be averaged out, and so that the cycle will enclose an area <math> \Delta \mbox{area} </math>. Let the top left corner be labeled A, then label the rest of the corners clockwise starting from A as ABCD. [[Image:Cyclic process.PNG]] Assume that the system is a monatomic gas. Then :<math> W_{AB} = P_A (V_B - V_A) </math> :<math> Q_{AB} = {5 \over 2} n R (T_B - T_A) </math> :<math> \Delta S_{AB} = {Q_{AB} \over T_{AB, avg}} = 5 n R \left( { T_B - T_A \over T_B + T_A } \right) </math> Process BC: :<math> W_{BC} = 0 </math> :<math> Q_{BC} = {3 \over 2} n R (T_C - T_B) </math> :<math> \Delta S_{BC} = {Q_{BC} \over T_{BC, avg}} = 3 n R \left( { T_C - T_B \over T_C + T_B } \right) </math> Process CD: :<math> W_{CD} = P_C (V_A - V_C) </math> :<math> Q_{CD} = {5 \over 2} n R (T_D - T_C) </math> :<math> \Delta S_{CD} = {Q_{CD} \over T_{CD, avg}} = 5 n R \left( {T_D - T_C \over T_D + T_C } \right) </math> Process DA: :<math> W_{DA} = 0 </math> :<math> Q_{DA} = {3 \over 2} n R (T_A - T_D) </math> :<math> \Delta S_{DA} = {Q_{DA} \over T_{DA, avg}} = 3 n R \left( {T_A - T_D \over T_A + T_D } \right) </math> Process ABCDA (cyclic): :<math> \Delta S_{cyc} = 5 n R \left( {T_B - T_A \over T_B + T_A } \right) + 3 n R \left( {T_C - T_B \over T_C + T_B } \right) + 5 n R \left( {T_D - T_C \over T_D + T_C } \right) + 3 n R \left( { T_A - T_D \over T_A + T_D } \right) </math> :<math> \Delta S_{cyc} = 5 n R \left( {V_C - V_A \over V_C + V_A } \right) + 3 n R \left( {P_C - P_A \over P_C + P_A } \right) + 5 n R \left( {V_A - V_C \over V_A + V_C } \right) + 3 n R \left( { P_A - P_C \over P_A + P_C } \right) </math> :<math> \Delta S_{cyc} = 5 n R \left( {V_C - V_A \over V_C + V_A } \right) + 3 n R \left( {P_C - P_A \over P_C + P_A } \right) - 5 n R \left( {V_C - V_A \over V_C + V_A } \right) - 3 n R \left( { P_C - P_A \over P_C + P_A } \right) </math> :<math> \Delta S_{cyc} = 0 \qquad </math> ====Part 2==== Any loop can be broken up into a rectangular grid of differential areas. The line integral of the entire loop is equal to the sum of the line integrals of each of the constituent differential areas. Let all these integrals be done clockwise. Then any pair of adjacent differential areas will be sharing a process as a common border, but one area will add that process in one direction while the adjacent area adds that process in the reverse direction, so that process is cancelled out. Therefore all processes internal to the loop cancel each other out (see [[Green's theorem]]), and the result of the summation is equal to the line integral of the contour of the loop: :<math> \Delta S ( \mbox{contour} ) = \sum_\mbox{area} \Delta S ( \Delta \mbox{area} ) = \sum_{\mbox{area}} 0 = 0 </math> Q.E.D. ===Conclusion=== The fact that entropy is a state function is what puts [[entropy]] on the map (the P-V diagram). ==References== * Halliday, Resnick & Walker. ''Fundamentals of Physics'', 5th edition. John Wiley & Sons, 1997. Chapter 21, ''Entropy and the Second Law of Thermodynamics''. ==See also== * [[Organic Rankine Cycle]] * [[Entropy]] {{Thermodynamic cycles|state=uncollapsed}} ==External links== {{Commonscat|Thermodynamics cycles}} Educational software links: *[http://www.depcik.com/eduprograms/cycles/index.htm Thermodynamic Cycle Simulation Software] [[Category:Thermodynamic cycles| ]] [[Category:Thermodynamics]] [[ca:Cicle termodinàmic]] [[da:Termodynamisk kredsproces]] [[de:Thermodynamischer Kreisprozess]] [[es:Ciclo termodinámico]] [[fa:چرخه ترمودینامیکی]] [[it:Ciclo termodinamico]] [[nl:Kringproces]] [[nn:Termodynamisk syklus]] [[pl:Obieg termodynamiczny]] [[ru:Термодинамические циклы]] [[simple:Thermodynamic cycle]] [[sk:Tepelný obeh]] [[sl:Krožna sprememba]] [[sv:Termodynamisk cykel]]