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Calculus of regenerative losses coefficient in Stirling engines.


This paper reveals a new technique used for calculus of power and efficiency of actual Stirling engines. This technique relies on first law of termodynamics for processes with finite speed (Atrey et al., 1993). It is also used in conjunction with a new PV/Px diagram and a new calculus method for regeneration coefficient (Tanaka et al., 1990).

The first objective of this paper is to develop a method to determine the X coefficient of imperfect regeneration and to use it for calculus of efficiency and power output of the Stirling engine.

Finally, the power and efficiency determined by this analysis (which involves the computation of X coefficient) are compared with performance data from twelve actual Stirling engines working in a large range of operating conditions.


The analysis requests the integration of differential equations. This integration is based on either a lump analysis, which leads to pessimistic results, [X.sub.1], or on a linear distribution of temperature in the regenerator matrix and gas--fig. 1--which leads to optimistic results, [X.sub.2].

The expressions of coefficient are:




where [m.sub.g] is the mass of gas passing through regenerator, [m.sub.R] the mass of regenerator screens, [A.sub.R] the area of the wires in regenerator, v the viscosity of working gas and h is the convective heat transfer coefficient in the regenerator (based on correlation given in).


It was determined the sensitivity of [X.sub.1] and [X.sub.2] to changes in operating variables such as the piston speed. The computed values of [X.sub.1] and [X.sub.2] were compared with values of X determined from experimental data available in the literature (Chen & Yan, 1989), (Incropera et al., 2001), (Walker et al., 1994). The results based on theory were found to predict the values from experimental data by using the following equation:

X = y[X.sub.1] + {1-y)[X.sub.2], (4)

where y is an adjusting parameter with the value of 0.72.

The loss caused by incomplete regeneration, as determined using the eq. (4), is the final loss to be considered in the analysis. The second law efficiency due to irreversibilities from incomplete regeneration is:

[[eta].sub.II,irrev,X] =

[1 + (0.72[X.sub.1] + 0.28[X.sub.2](1 - [square root of [T.sub.0]/[T.sub.HS])/R/[c.sub.v)(T)lne].sup.-1]. (5)

Fig. 2 presents the convective heat transfer coefficient dependence of the piston speed; [D.sub.R] = 50, b/d = 1.5, [tau] = 2.

Fig. 3 reveals the variation of the coefficient of regenerative losses with the piston speed for several values of analysis parameters (d, S, porosity).



In fig. 3 d is the wire diameter, S is the piston stroke, [D.sub.c] = 60mm, [D.sub.R] = 60mm, [P.sub.m] = 50bar, d = 0.05mm, N = 700 and [tau] = 2.


The results of computation efficiency and power output based on this analysis are compared to performance data taken from twelve operating Stirling engines in fig. 4 and in tab. 1.



Fig. 4 and table 1 reveal a high degree of correlation between this analysis and the operational data. This indicates that this analysis can be used to accurately calculate X coefficient and of other losses. Therefore, this analysis can be used to accurately predicting Stirling engine performance under a wide range of conditions. This capability is a valuable tool in Stirling engine design and in performance prediction of a particular Stirling engine over a range of operating speed.

The Direct Method of using the first law for processes with finite speed is an analysis valid method for irreversible cycles based on correlation between analcal and experimental results.

We intend to develop further research onto implementing the determined ecuations into a software application.


Atrey, M. D.; Bapat, S. L. & Narayankedkar, K. G. (1993). Optimization of Design Parameters of Stirling Cycle Machine, Cryogenics, Vol. 33, No.1, February 1993, 18-24, 0011-2275

Chen, L. & Yan, Z. (1989). The Effect of Heat Transfer Law on Performance of a Two-Heat Source Endoreversible Cycle, Journal of Chemical Physics, Vol. 90, 120-126, 0021-9606

Incropera, F.; De Witt, D. David, P. (2001). Introduction to Heat Transfer, John Wiley & Sons, 9780471386490, Australia

Tanaka, M.; Yamashita, I. & Chisaka, F. (1990). Flow and Heat Transfer Characteristics of the Stirling Engine Regenerator in an Oscillating Flow, The Japan Society of Mechanical Engineers International Journal, Vol. 33, Series B, No.3, August, 1993, 380-386, 1340-8054

Walker, G.; Reader, G.; Fauvel, O.R. & Bingham, E.R. (1994). The Stirling Alternative, Gordon and Breach Science Publishers, 978-2-88124-600-5, Amsterdam
Tab. 1. Analytical results and actual engines performances

 Stirling engine Actual Calculated
 power [kW] power [kW]

NS-03M, max. power 3.81 4.196
NS-03T, economy 3.08 3.145
NS-03T, max. power 4.14 4.45
NS-30A, economy 23.2 29.45
NS-30A, max. power 30.4 33.82
NS-30S, economy 30.9 33.78
NS-30S, max. power 45.6 45.62
STM4-120 25 26.36
V-160 9 8.825
4-95 MKII 25 28.4
4-275 50 48.61
GPU-3 3.96 4.16
MP1002 CA 200W 193.9W
Free Piston Stirling Engine 9 9.165
RE 1000 0.939 1.005

 Stirling engine Actual Calculated
 efficiency efficiency

NS-03M, max. power 0.31 0.3297
NS-03T, economy 0.326 0.3189
NS-03T, max. power 0.303 0.3096
NS-30A, economy 0.375 0.357
NS-30A, max. power 0.33 0.3366
NS-30S, economy 0.372 0.366
NS-30S, max. power 0.352 0.3526
STM4-120 0.4 0.4014
V-160 0.3 0.308
4-95 MKII 0.294 0.289
4-275 0.42 0.4119
GPU-3 0.127 0.1263
MP1002 CA 0.156 0.1536
Free Piston Stirling Engine 0.33 0.331
RE 1000 0.258 0.2285
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Author:Florea, Traian; Dragalina, Alexandru; Florea, Traian Vasile; Bejan, Mihai; Pruiu, Anastase
Publication:Annals of DAAAM & Proceedings
Article Type:Report
Geographic Code:4EUAU
Date:Jan 1, 2009
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