﻿ Energy balance of Stirling engine cycle
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# 35. Energy balance of Stirling engine cycle

Author: Jiří Škorpík, skorpik.jiri@email.cz : updated 2012-04

An energy balance of a Stirling engine cycle can be do through an indicator diagram or trend pressure inside a Stirling engine which is calculated. Results of this energy balance are an input heat, a rejection heat, a regenerated heat, an internal work of the cycle and an internal thermal efficiency.

This article describes analytical calculation through equations, which are derived from the equations of the first law of thermodynamic. Through these derived equations can be calculated energy flows and theirs connection with others quantities.

## Assumptions of solving

The equations of this energy balance are true under certain simplifying assumptions. Calculated results through these equation can vary from reality with increasing of differences between simplified and real processes:

```(1)  Working gas is ideal gas.
(2)  Stirling engine is perfect sealed.
(3)  There is no pressure loss, pressure of working gas is same in whole
working volume.
(4)  Regenerator has perfect thermal isolation.
(5)  Pressure as function of angle of rotation (p(φ)),
working volume on hot side as function of angle of rotation (VT(φ))
and working volume on cold side as function of angle of rotation
(VS(φ)) are known.
(6)  Stirling engine cycle is steady (the same cycle repeats).          ```
1.459 The simplifying assumptions of the solving.
p [Pa] pressure of working gas; VT [m3] volume of hot side of engine; VS [m3] volume of cold side of engine; φ [°] angle of rotation.
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## Internal work of Stirling engine

The internal work of the engine is function the pressure and the working volume: 2.463 The internal work of the Stirling engine.
A [J] internal work of engine; VTV [m3] volume of cylinder on hot side; VSV [m3] volume of cylinder on cold side; AT [J] work of piston on hot side; AS [J] work of piston on cold side. Derivation of this equation is shown in the Appendix 463.
Calculate the internal work of a Stirling engine with parameters which are shown in the Problem 1 [34.].
Problem 1.467
`AT [J] 1382,2113     AS [J] -506,0131     A [J] 876,1982`
Problem 1: summary of entries and results.

## Energy balance

Inside the Stirling engine is transformed heat to work. For this transformation are true the rules of heat cycle, it is means, only a piece of the heat input to the engine is transformed to work and others the heat is necessary rejected from the engine. For input and output of the heat to/from the engine between the working gas and a heat transfer surface must be a temperature gradient. This temperature gradient is vary during one cycle, because working gas temperature is changed also (see chapter 34. Temperature change of working gas inside Stirling engine). Therefore in all internal sections of the engine the heat is inputted and outputted to/from working gas during one cycle. It is evident (from the principle of the Stirling engine) the heat balance during one cycle of the hot side is positive (the heat is inputted to the working gas) and the heat balance during one cycle of the cold side is negative (the heat is outputted from the working gas), the heat balance of the regenerator during one cycle must be neutral: 3.460 The heat balance of the hot side, the cold side and the regenerator of the engine
QT [J] heat balance of hot side during one cycle; ΔI [J] heat and work which needed for change of enthalpy of working gas inside engine during one cycle; QS [J] heat balance of cold side during one cycle; QR [J] heat balance of regenerator during one cycle. Derivation of these equations is shown in the Appendix 460.

The heat ΔI inputs through the cycle on the hot side and it increases of internal energy of the working fluid and other part of the cycle outputts on the cold side from the engine (through the temperature gradient between the hot and cold side). This heat decreasing a dimension capacity-rating of the heat transfer surface and a thermal efficiency of the cycle. On the other hand the heat ΔI develops the temperature gradient between the hot and the cold side even in for case of the regenerator with small capacity or for case it is not installation inside the engine. For cases isothermal processes on the hot and the cold side (e.g. Schmidt theory) must be the change of the enthalpy ΔI equal zero.

The internal work of the engine can be measured indirectly, e.g. by indication of the pressure of the working gas or by measuring work shaft (in this case is necessary to know mechanical losses of the engine). The internal work of the engine can be approximately calculated of the pressure trend by the procedure which is shown in article 35. Stirling engine cycle (calculation without losses) or in article 36. Losses in Stirling engines (calculation with losses).

Values of heats QT and QS can to get from a measuring of heat flow to/from the engine or it can be approximately calculated through an estimate the internal thermal efficiency:

## Internal efficiency of Stirling engine

The internal thermal efficiency of the Stirling engine is ratio between its the internal work A and heat, which inputted to the working gas from a surroundings during one cycle QD* respectively this definition is the same as definition of efficiency of heat cycle, because inside engine is performed complet cycle. For the conditions, which are shown on List 1 can be descripted equality of heat flows: 4.465 The input heat and the rejection heat of the Stirling engine during one cycle.
QD [J] input heat to working gas from surroundings of engine during one cycle (input heat of engine); QOD [J] rejection heat from working gas to surroundings of engine during one cycle (rejection heat of engine).
*Remark
The working gas gets some heat from regenerative surface of the engine (inside the engine) especially from the regenerator, but this heat is saved again to these surface during other part of the cycle, therefore this heat is not component of the input heat.

The internal thermal efficiency can be estimated through similarities of the Stirling engines similar construction: 5.464 An estimate of the internal efficiency of the engine and the heat ΔI.
ηt [-] internal efficiency of engine; C [-] Carnot efficiency ratio for Stirling engine* (C<1); τ [-] temperature ratio on border of regenerator; ηcar [-] Carnot efficiency ratio for temperature ratio. Derivation of these equations is shown in the Appendix 464.
*Remark
Carnot efficiency ratio usually is in interval 0,65..0,75 offers exceptionally for case the best construction 0,8 [1, p. 99]. These values are derived from measurements on several the Stirling engines. An estimate of a value of the ratio C is function of the temperature of the working gas on hot border of the regenerator, the higher can be the expected higher coefficient C [1, p. 45].
Estimate (determinate of a probable interval) the internal thermal efficiency and heat QT, QS and ΔI of the Stirling engine with parameters indicated in the Problem 1.
Problem 2.466
```C  [-] 0,65...0,75   ΔI [J] 746,03...462,42    QS [J] -1252,05...-968,43
ηt [-] 0,41...0,48   QT [J] 2128,24...1844,63                           ```
Problem 2: summary of entries and results.

## Regenerated heat inside regenerator

The regenerated heat inside the regenerator can be calculated from a function which describes an amount of a heat transfered inside the regenerator of the working gas from a start of the cycle to any point of the cycle. The regenerated heat is equal of a difference between the maximum and the minimum of this function: 6.469 The regenerated heat inside the regenerator of the Stirling engine.
QR,x [J] amount of heat transfered inside regenerator of working gas from start of cycle (subscript 0) to any point of cycle (subscript x); QReg [J] regenerated heat inside regenerator during one cycle; κ [-] heat capacity ratio. Derivation of this equation is shown in the Appendix 469. This equations was first published in , , .

The enthalpy of the working gas inside the hot or the cold side in case isothermal processes is not changed respectively in Equation 5 is true IT,x=0; IT,x=0.

The amount of the heat QReg can be calculated approximately if the amount of the regenerated heat QReg is much more than the amount of the heat ΔI:

Calculate approximately the amount of a regenerated heat inside the regenerator of a Stirling engine with parameters which are shown in the Problem 2. Calculate a ratio QReg/ΔI (assume the highest value of ΔI from Problem 2).
Problem 3.470
```QReg    [J] 6421,6506                           QReg/ΔI [-] 8,6077

φ [°] V [cm3]  QR,x [J]                 φ [°] V [cm3]  QR,x [J]
--------------------------              --------------------------
0       377,1489 0                      190     508,8472 2477,7114
10      365,4078 359,5118               200     519,8974 2005,3991
20      355,8134 818,4803               210     528,337  1577,1375
30      348,5937 1377,206               220     533,859  1193,4843
40      343,9179 2026,5137              232,5   536,3626 773,8312
52,5    341,8098 2931,351               240     535,4585 551,9241
60      342,5702 3495,834               250     531,4778 288,1827
70      345,9296 4228,6122              259,647 524,7618 66,2469
80      351,8947 4883,2127              270     514,6757 -138,2063
87,0244 357,5704 5264,0468              280     502,4646 -303,3848
100     371,0266 5733,5592              285     495,5851 -373,7416
105     377,1489 5822,4453              300     472,5932 -532,2328
120     398,1411 5777,4868              310     455,9721 -588,8656
130     413,8632 5516,86                320     438,9569 -599,2053
140     430,4707 5117,7967              340     405,8632 -447,5454
150     447,4797 4626,2816              350     390,7495 -265,9646
160     464,3663 4085,5746              360     377,1489 0
180     495,5851 2989,632                                         ```
Problem 3: summary of entries and results. Problem 3: summary of entries and results.
φ [°] angle of rotation.

During the solving of previous problem was neglected an influence of the change of enthalpy of the working gas on the hot and cold side respectively the heat ΔI. For actually case the influence of the change enthalpy is increased with decrease of the temperature ration τ respectively the ration between the heat QReg and the heat QD significantly decreasing and this procedure of the calculate used in Problem 3 increases its inaccuracy. This fact must be taken into account when designing the size of the regenerator: 7.197 The ratio between the heat QReg and the heat ΔI as function the temperature ratio τ.This curve is true for the cycle from the Problem 2 and Problem 3.

## Entropy of working gas

An equation of a change of specific entropy of the working gas can be derived from First law of thermodynamics for closed system: 8.474 The change of specific entropy of the working gas in relation of the start of the cycle.
Subscript 0 denotes the start of cycle and subscript x denotes any point of cycle. cv [J·kg-1·K-1] specific heat at constant volume; r [J·kg-1·K-1] gas constant of working gas; s [J·kg-1·K-1] specific entropy. Derivation of this equation is shown in the Appendix 474. This equation was first published in .

Through last Equation can be constructed a T-s diagram of the cycle. From T-s diagram of cycle can be identified losses of the cycle. In praxis can be measured perfectly only pressure of the working gas as function of angle of rotation. Exact amount of the working gas inside of the engine and its exact mean temperature of the working gas is impossible to measure. Therefore is constructed Θ-s diagram*, where Θ is ratio between the mean temperature of working gas inside of the engine and maximum temperature of the working gas inside of the engine. The Θ-s diagram can be constructed only from the measured pressure.

Θ-s diagram, which is designed from a measurement, can show losses and weaknesses of the engine: 9.471 Θ-s diagrams of the Stirling engine cycles.
a ideal Θ-s diagram; b Θ-s diagram of an engine with small capacity of regenerator; c Θ-s diagram of an engine with leakage of piston rings. This equation was first published in .
Design of assumed Θ-s diagram of the Stirling engine Stirling engine with parameters which are shown in the Problem 1 [34.].
Problem 4.472
```cv [J·kg-1·K-1] 3116,168   r  [J·kg-1·K-1] 2077,22

sx-s0                  sx-s0                  sx-s0
[J·kg-1·K-1] Θ [-]     [J·kg-1·K-1] Θ [-]     [J·kg-1·K-1] Θ [-]
0            0,6853    1269,3741    0,9934    712,0368     0,6852
29,309       0,7066    1296,7358    0,9767    634,0198     0,674
88,6894      0,733     1300,7945    0,9526    546,6878     0,6639
177,8893     0,7647    1285,939     0,9239    459,8105     0,6561
294,1768     0,801     1256,5622    0,8929    415,9258     0,6529
469,1112     0,8507    1168,7142    0,8313    285,7881     0,6463
583,793      0,8813    1115,3026    0,8029    203,8984     0,6447
739,4541     0,9204    1057,5191    0,7769    130,0204     0,6458
888,758      0,9546    995,9703     0,7536    23,1234      0,6575
984,6345     0,974     930,7996     0,7329    -0,9958      0,6691
1132,4227    0,9965    844,0397     0,7105    0            0,6853
1177,3692    1         789,0023     0,6989                       ```
Problem 4: summary of entries and results. Problem 4: summary of entries and results.The fact, minimal entropy corresponding with the state p0, V0 is only a random.

## Amount of working gas inside engine

The amount of the working gas inside of the Stirling engine can be computed by the equation of state [2, p. 67 (cz)] for each working volumes. The amount of the working gas is computed for a known state of the working gas. If real mean temperature of the working gas inside individual volumes is not known, then can be use of computed of temperature change of the working gas inside Stirling engine:

What is amount of the working gas inside of the Stirling engine with parameters which are shown in the Problem 1 [34.]. The temperature of working gas inside individuals volumes is shown in the Problem 2 [34.].
Problem 5.885
 `m [kg] 5,92337E-3` Problem 5: summary of entries and results.

## References

1. MARTINI, William. Stirling engine design manual, 2004. Přetisk vydání z roku 1983. Honolulu: University press of the Pacific, ISBN: 1-4102-1604-7.
2. KALČÍK, Josef, SÝKORA, Karel. Technická termomechanika, 1973. 1. vydání, Praha: Academia.
3. ŠKORPÍK, Jiří. Příspěvek k návrhu Stirlingova motoru, VUT v Brně, Edice PhD Thesis, 2008, ISBN 978-80-214-3763-0.
4. ŠKORPÍK, Jiří. An energy balance of the Stirling engine cycle, článek vyšel ve sborníku vědeckých prací Taвpiйcькoгo дepжaвнoгo aгpoтexнoлoгiчнoгo yнiвepcитeтy, 2008, YДК 621.311:631, UDC 621.412:621.5.01.
5. ŠKORPÍK, Jiří. The Amount of Regenerated Heat Inside the Regenerator of a Stirling Engine, Acta Polytechnica, 2009, roč. 2008, č. 6, s. 10-14. ISSN 1210 – 2709.
6. ŠKORPÍK, Jiří. Stirling engine cycle-supplement, The 15th International Stirling Engine Conference, in Dubrovnik–Croatia, 2012, ISBN: 978-88-8326-019-3.

## Citation this page

This document is English version of the original in Czech language: ŠKORPÍK, Jiří. Energetická bilance oběhu Stirlingova motoru, Transformační technologie, 2009-07, [last updated 2012-04]. Brno: Jiří Škorpík, [on-line] pokračující zdroj, ISSN 1804-8293. Dostupné z http://www.transformacni-technologie.cz/35.html. English version: Energy balance of Stirling engine cycle. Web: http://www.transformacni-technologie.cz/en_35.html.

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