GRAPHO-ANALYTICAL METHOD FOR CALCULLATING IRREVERSIBILITY PROCESSES WITH FINITE SPEED Prof.Eng. Traian FLOREA PhD 1 Lecturer.mat. Ligia-Adriana SPORIȘ PhD 1 Assist.prof.Eng. Corneliu MOROIANU PhD 1 Eng. Traian Vasile FLOREA Ph.D 2 Prof.Eng. Anastase PRUIU PhD 1 1 Mircea cel Batran Naval Academy of Constanta ROMANIA 2 A.P.M. Agigea of Constanta ROMANIA Abstract: A technique for calculating the efficiency and power of Stirling machines is presented. This technique is based on the First Law of Thermodynamics for processes with finite Speed. A new and novel pv/px diagram is presented that shows the effects of pressure losses due to friction finite speed and throttling processes in the regenerator of the Stirling engine. The method used for the analysis of this irreversible cycle with finite speed involves the direct integration of equations based on the First Law for processes with finite speed to directly obtain the cycle efficiency and power. This technique is termed the Direct Method. The results predicted by this analysis were in good agreement with the actual engine performance data of twelve different Stirling engines over a range of output from economy to maximum power. This provides a solid verification that this analysis can accurately predict actual Stirling engine performance particularly with regard to efficiently and output power. Keywords: power finire speed analysis 1. INTRODUCTION A new technique for calculating the efficiency and power of actual operating Stirling machines is presented. This technique is based on the First Law of Thermodynamics for processes with finite speed [1 14] and a new method for determining the imperfect regeneration coefficient [15]. The analytical results depend upon inclusion of two calibration coefficients (y and z) based on experimental data to accurately predict performance. The thermal efficiency is expressed as a product of the Carnot cycle efficiency and second law efficiency [21 22 31 32] as has been suggested by Bejan [23]: SE CC II irrev T L T L 1 1 TH S TH S 1 CC II irrev T X 1 T L / TH S 3 Pi 1 1 1ln P1 TH S / TL ln II irrevx II irrev Pi with 1 T L / T H S II irrev X 1 (1) 157
11/ 3 The pressure losses and their effect on efficiency and power of the engine depend on the piston speed and hence the speed of the engine. The power output of the engine is: Power mrt ln w/ S (2) SE SE H g 2 The speed for maximum power may be determined since the power output is also a function of the engine speed. Therefore the operating speed of a particular Stirling engine can be selected for either maximum economy or for maximum power. Also knowledge of the nature of these losses can be effectively used in engine design. A major loss in Stirling engines is caused by incomplete regeneration. This is expressed by the coefficient of regenerative losses X. An analysis for determining this loss has been made by Petrescu et al. and Florea [15 33]: 1 1 2M e 21 M mgcv g M m X 1 y X 2 1 y 1 TL / TH S R / c T ln II irrev X (3) v X B 1 B M R C R 1 B M e X 2 (4-5) 1 M har S 1 (6-7) m c w g v g 0424 4P / RT w c T vt 0395 m L g P m m h (8) 0576 2 / 3 1 1 Pr 4 / 1 DR b d where y is one of the adjusting coefficients of the method with the value of 0.72. One objective of this paper is to make a more realistic analysis of the pressure losses through use of a pv/px diagram as will be described below and by Petrescu et al. [14]. Finally the technique for calculating the efficiency and power of Stirling engines is presented and the results predicted by this analysis are compared with performance data taken on twelve actual Stirling engines over a range of operating conditions [24 25 26 27 28]. 2. THE METHOD OF DETERMINING THE PERFORMANCE OF THE STIRLING ENGINE Computation of pressure losses work losses efficiency and power for the processes shown on the new pv/px diagrams [14 15] are made using the first law of thermodynamics for processes with finite speed [1-13]. The first law written to specifically include these conditions is: 0576 158
aw b P f Pf thrott du Q Pm i 1 dv c 2Pmi P (9) mi The irreversible work is: aw b P Pf thrott Wirrev Pmi 1 dv (10) c 2Pmi P mi when applied to processes with finite speed as shown on the pv/px diagram. Computing and summing all pressure losses of the Stirling engine cycle presented above the term from eq. (1) becomes [14 15 33]: II irrev Pi w 3 094 1 ln 5 N w 5 0045w 10 w w S L S L 44 II irrev Pi 1 (11) ln ln The heat input during the expansion process is also irreversible due to finite speed. In order to take account of this influence an calibration coefficient z is introduced: Q z mrt ln (12) 34 H g Finally the real power output of the engine eq. (2) becomes: Power zmrt ln w / 2S (13) SEirrev SE H g where the value of z was evaluated at 08 by comparison with available experimental data for twelve Stirling engines [24-28]. 3. DISCUSSIONS The variation of the coefficient of regenerative losses X with the piston speed for several values of the gas average pressure is shown in Fig. 2. It is an example of the results obtained from the sensitivity study [15] for X. It shows an important increase of the regenerative losses with the gas average pressure. Then the results of computations based on this analysis are compared to performance data taken from a number of operating Stirling engines in Figs. 3-5 and Table I. The high degree of correlation between the analytic and the operational data shown in these figures indicate that the analysis is capable of accurately predicting Stirling engine performance under a wide range of conditions. This capability should be of considerable value in Stirling engine design and in predicting the performance of a particular Stirling engine over a range of operating speed. 2 159
Stirling Engine Actual Power [kw] Calculat ed Power [kw] Actual Efficienc y Calculat ed Efficienc y Table I. Comparison between analytical results and actual engine performance data [24-29] NS-03M regime 1 2.03 2.182 0.359 0.3392 (economy) NS-03M regime 1 (max. 3.81 4.196 0.31 0.3297 power) NS-03T regime 1 3.08 3.145 0.326 0.3189 (economy) NS-03T regime 1 (max. 4.14 4.45 0.303 0.3096 power) NS-30A regime 1 23.2 29.45 0.375 0.357 (economy) NS-30A regime 1 (max. 30.4 33.82 0.33 0.3366 power) NS-30S regime 1 30.9 33.78 0.372 0.366 (economy) NS-30S regime 1 (max. 45.6 45.62 0.352 0.3526 power) STM4-120 25 26.36 0.4 0.4014 V-160 9 8.825 0.3 0.308 4-95 MKII 25 28.4 0.294 0.289 4 275 50 48.61 0.42 0.4119 GPU-3 3.96 4.16 0.127 0.1263 MP1002 CA Free Piston Stirling Engine 200 W 193.9 W 0.156 0.1536 9 9.165 0.33 0.331 RE-1000 0.939 1.005 0.258 0.2285 Fig.1. The pv/px diagram for description of the Stirling cycle processes 160
4. CONCLUSION The objective of this approach was to closely simulate the operation of actual Stirling engines without losing insight to the mechanisms that generate the irreversibilities. Pressure and work losses generated by finite speed of the actual processes were computed as were the power and efficiency of engines. The first law of thermodynamics for processes with finite speed was used to compute the power losses generated by the pressure losses. The analysis presented was applied to specific operating Stirling cycle engines and results were compared to the measured performance of the engines. The strong correlation between the analytical results and actual engine performance data indicates that the Direct Method of using the First Law for Finite Speed is a valid method of analysis for irreversible cycles. Fig. 2. Coefficient of regenerative losses versus the piston speed for several values of the average pressure of the working gas (D C = 60 mm b/d = 1.5 τ = 2 N = 700) Fig. 3. Comparison of the analysis results with actual performance data for the NS-30S Stirling engine 161
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