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AbstractThis page contains a short rehash of the Stirling cycle for heat engines. Some math and knowledge of thermodynamics ( conservation of energy , internal energy , specific heat capacities at constant pressure and constant volume ) is required for understanding. Major finding, beyond what can practically be found in any text book, is that the heat transfer load on the regenerator is influenced by the gas being used through the value of the gas' ratio of specific heats. The heat load on the regenerator increases as we go from monatomic to diatomic gases by a factor of 1.67 and by a factor of more than 2 as the gas molecules contain 3 or more atoms.
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IntroductionThe ideal Stirling Cycle consists of 4 individual processes with end points as marked in the p-V diagram ( p=pressure, V=volume). Its analysis is a standard item in all text books of engineering thermodynamics dealing with gas powered heat engines. So why revisited ? Apart from summarizing the underlying thermodynamic laws and the resulting equations we look at the heat transfer needed for the two isochoric processes of the cycle and the amount of heat needed to facilitate them. Surprisingly, the amount of heat needed for these processes depends strongly on the ratio of the specific heats of the working fluid favoring monatomic ideal gases to be used as working fluid. The term "fluid" is used here in the same way physicists use it : it can mean either a gas or a liquid.
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Description of Cycle
The Regenerator and Efficency of Stirling Cycle
Basis of the analysis of a cycle is the 1. law of thermodynamics of all its sub processes. In this case we have a working fluid of constant mass which for a process with starting point i and end point j is :
Qij - Wij = Uj - Ui
Qij = heat transferred into the working fluid when fluid changes from state i to state j
Wij = ∫ p dV = work gained during the process i→j
Ui = internal energy of working fluid at state i
In particular, for the process 2→3 the volume does not change and therefore W23 is zero and :
Q23 = U3 - U2
and similarily for the process 4→1 :
Q41 = U1 - U4
If we choose a working fluid for which the internal energy depends only on temperature ( liquids and ideal gases for example ) the internal energies U3 and U4 are equal ( same temperature TH ). Along the same lines : U2=U1. Therefore :
Q41 = - Q23
Or in other words, the heat we have to remove from the gas during the process 4→1 can be used to heat the gas during the process 2→3, which is exactly the function of the regenerator.
Once a regenerator is employed, the interaction of a Stirling engine with its surrounding is restricted to receiving heat at temperature TH and rejecting heat at temperature TC. With that its thermodynamic efficiency ( = ratio of work gained to heat provided ) must be equal to that of the Carnot cycle :
η = 1 - TC/TH
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Influence of working fluid on work and heat transfer rates
We assume that our working fluid is an ideal gas with constant specific heat capacities.Hence for the process 3→4 the internal energy of the gas remains constant and :
Q34=W34= ∫ p dV = ∫ n Ru TH /V dV = n Ru TH ln ( V4/V3 )
In the same fashion for the process 1→2 :
Q12=W12= ∫ p dV = ∫ n Ru TC /V dV = n Ru TC ln ( V2/V1 )
And the network per cycle :
Wnet = W34 + W12 = n Ru ( TH - TC ) ln ( V4/V3 ) ;
The heat supplied at temperature TH ( remember we employ a regenerator ) is equal to Q34 which then gives us a thermodynamic efficiency ( ratio of benefit to costs ) of
η = 1 - TC/TH
This is standard thermodynamic stuff and nothing new.
We wish to take a look though at the influence of changing the working fluid, for example He vs. H2 versus air. Of course we have to keep the temperatures TC and TH the same. Furthermore, the volume of our engine, V0, remains constant and we load the engine to the same initial pressure, p0, at the same initial temperature, T0. In this case, the number of moles of gas , n , loaded into the engine remains constant :
n = (p0 V0)/(Ru T0)
With that, the numerical values for the above works, heats, and the efficiency are not affected by a change of gas.
The story is different though for the regenerator. The heat transferred from the regenerator material into the gas during the process 2→3 and removal from the gas during 4→1 is equal to the change in internal energy of the gas which for constant specific heat capacities is :
Q23 = m cv ( TH - TC ) = 1/(κ-1) n Ru ( TH - TC )
Q23 is influenced by the ratio of specific heats which changes as we go from monatomic, to diatomic and then to gases with even more atoms per molecule.
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