Jiří Škorpík, skorpik.jiri@email.cz

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.

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).

The internal work of the engine is function the pressure and the working volume:

1:

463 The internal work of the Stirling engine

**p** [Pa] pressure of working gas; **A** [J] internal work of engine; **V**_{TV} [m^{3}] volume of cylinder on hot side; **V**_{SV} [m^{3}] volume of cylinder on cold side; **A**_{T} [J] work of piston on hot side; **A**_{S} [J] work of piston on cold side. Derivation of this equation is shown in the Appendix 463.

Problem 1:

Calculate the internal work of a Stirling engine with parameters which are shown in the Problem 1 in article Stirling engine cycle.

Results:

A_{T} | A_{S} | A | ||

1382.2 | -506.01 | 876.19 |

[J]

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:

2:

The heat balance of the hot side, the cold side and the regenerator of the engine

**Q**_{T} [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; **Q**_{S} [J] heat balance of cold side during one cycle; **Q**_{R} [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 Stirling engine cycle (calculation without losses) or in article Losses in Stirling engines (calculation with losses).

Values of heats *Q _{T}* and

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 *Q _{D}** 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

3:

The input heat and the rejection heat of the Stirling engine during one cycle

**Q**_{D} [J] input heat to working gas from surroundings of engine during one cycle (input heat of engine); **Q**_{OD} [J] rejection heat from working gas to surroundings of engine during one cycle (rejection heat of engine).

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:

4:

An estimate of the internal efficiency of the engine and the heat ΔI

**η**_{t} [1] internal efficiency of engine; **C** [1] Carnot efficiency ratio for Stirling engine* (*C<1*); **τ** [1] temperature ratio on border of regenerator; **η**_{car} [1] Carnot efficiency ratio for temperature ratio. Derivation of these equations is shown in the Appendix 464.

Carnot efficiency ratio usually is in interval

Problem 2:

Estimate (determinate of a probable interval) the internal thermal efficiency and heat *Q*_{T}, *Q*_{S} and *ΔI* of the Stirling engine with parameters indicated in the *Problem 1*.

Results:

η_{t} | Q_{T} | Q_{S} | ΔI | ||

0.4117 | 2128.2 | -1252.0 | 746.03 |

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:

5:

The regenerated heat inside the regenerator of the Stirling engine

**Q**_{R,x} [J] amount of heat transfered inside regenerator of working gas from start of cycle (subscript **0**) to any point of cycle (subscript **x**); **Q**_{Reg} [J] regenerated heat inside regenerator during one cycle; **κ** [1] heat capacity ratio. Derivation of this equation is shown in the Appendix 469. This equations was first published in [3], [4], [5].

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 *I _{T,x}=0*;

The amount of the heat *Q _{Reg}* can be calculated approximately if the amount of the regenerated heat

Problem 3:

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: 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 *Q _{Reg}* and the heat

6:

The ratio between the heat Q_{Reg} and the heat ΔI as function the temperature ratio τ

This curve is true for the cycle from the*Problem 2* and *Problem 3*.

This curve is true for the cycle from the

An equation of a change of specific entropy of the working gas can be derived from First law of thermodynamics for closed system:

7:

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. **c**_{v} [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 [3].

Subscript

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:

8:

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 [6].

Problem 4:

Design of assumed Θ-s diagram of the Stirling engine Stirling engine with parameters which are shown in the Problem 1 in article Stirling engine cycle.

Problem 4: results.

The fact, minimal entropy corresponding with the state*p*_{0}, *V*_{0} is only a random.

The fact, minimal entropy corresponding with the state

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:

Problem 5:

What is amount of the working gas inside of the Stirling engine with parameters which are shown in the Problem 1 in article Stirling engine cycle? The temperature of working gas inside individuals volumes is shown in the Problem 2 in article Stirling engine cycle.

Results:

m | ||

5.9234 |

[g]

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.

ŠKORPÍK, Jiří, 2009, update 2012, Energy balance of Stirling engine cycle, *Transformační technologie*, ISSN 1804-8293.

©Jiří Škorpík, LICENCE