SCHMIDT THEORY FOR STIRLING ENGINES

Transcription

1 SHMIDT THOY FO STILING NGINS KOIHI HIATA Musashin-jjutaku 6-10, Gakuen -6-1, Musashimurayama, Tky 08, Japan Phne & Fax: url: 1. INTODUTION The Shmidt thery is ne f the isthermal alulatin methds fr Stirling engines. It is the mst simple methd and very useful during Stirling engine develpment. This thery is based n the isthermal expansin and mpressin f an ideal gas.. ASSUMPTION OF SHMIDT THOY The perfrmane f the engine an be alulated a P- diagram. The vlume in the engine is easily alulated by using the internal gemetry. When the vlume, mass f the wrking gas and the temperature is deided, the pressure is alulated using an ideal gas methd as shwn in equatin (1). P mt (1) The engine pressure an be alulated under fllwing assumptins: (a) There is n pressure lss in the heat-exhangers and there are n internal pressure differenes. (b) The expansin press and the mpressin press hanges isthermal. () nditins f the wrking gas is hanged as an ideal gas. (d) There is a perfet regeneratin. (e) The expansin dead spae maintains the expansin gas temperature - T, the Table 1 Symbls mpressin dead spae maintains the Name Symbl Unit mpressin gas temperature - T during the ngine pressure P Pa yle. (f) (g) The regeneratr gas temperature is an average f the expansin gas temperature - T and the mpressin gas temperature - T. The expansin spae - and the mpressin spae - hanges arding a sine urves. Table 1 shws symbls used the Shmidt Thery.. ALPHA-TYP STILING NGIN Figure 1 shws the alulatin mdel f Alphatype Stirling engine. The vlumes f the expansin- and mpressin ylinder at a given rank angle are determined at first. The mmental vlumes is desribed with a rank angle - x. This rank angle is defined as x0 when the expansin pistn is lated the mst tp psitin (tp dead pint). The mmental expansin vlume - is desribed in equatin () with a swept vlume f 1 Swept vlume f expansin pistn r displaer pistn m Swept vlume f mpressin pistn r pwer pistn S m Dead vlume f expansin spae m egeneratr vlume m Dead vlume f mpressin spae m xpansin spae mmental vlume m mpressin spae mmental vlume m Ttal mmental vlume m Ttal mass f wrking gas m kg Gas nstant J/kgK xpansin spae gas temperature T H K mpressin spae gas temprature T K egeneratr spae gas temperature T K Phase angle deg Temperatuer rati Swept vlume rati Dead vlume rati ngine speed n Hz Indiated expansin energy W J Indiated mpressin energy W J Indiated energy W i J Indiated expansin pwer L W Indiated mpressin pwer L W Indiated pwer L i W Indiated effiieny

2 xpansin spae (, T, P) xpansin pistn H egeneratr spae (, T, P) mpressin spae (, T, P) mpressin pistn H: Heater : egeneratr : ler the expansin pistn -, an expansin dead vlume - under the nditin f assumptin (g). ( 1- s x) + () The mmental mpressin vlume - is fund in equatin () with a swept vlume f the mpressin pistn - S, a mpressin dead vlume - and a phase angle - dx. S { 1- s( x - dx) } + () The ttal mmental vlume is alulated in equatin (4). + + (4) y the assumptins (a), (b) and (), the ttal mass in the engine - m is alulated using the engine pressure - P, eah temperature - T, eah vlume - and the gas nstant -. m P P P + + (5) T T T The temperature rati - t, a swept vlume rati - v and ther dead vlume ratis are fund using the fllwing equatins. T t (6) T v S The regeneratr temperature - T is alulated in equatin (11), by using the assumptin (f). T + T T (11) When equatin (5) is hanged using equatin (6)-(10), the ttal gas mass - m is desribed in the next equatin. P m T t ( t + t + ) (1) Fig. 1 Alpha-type Stirling ngine quatin (1) is hanged in equatin (1), using equatin () and (). (7) (8) (9) (10)

3 { } m P T S s( x a ) (1) Nw; 1 v sindx a tan t + sdx 4t S t + t + t (14) + v + (15) t + tv sdx + v The engine pressure - P is defined as a next equatin using equatin (1). mt P S s( θ a) { } The mean pressure - P mean an be alulated as fllws: 1 mt Pmean Pdx (18) π S is defined in the next equatin. (19) S As a result, the engine pressure - P, based the mean engine pressure - P mean is alulated in equatin (0). Pmean S Pmean P (0) S s( x a) s( x a) On the ther hand, in the ase f equatin (17), when s(x-a)-1, the engine pressure - P bemes the minimum pressure - P min, the next equatin is intrdued. mt Pmin (1) (S + ) Therefre, the engine pressure - P, based the minimum pressure - P min is desribed in equatin (). Pmin (S + ) Pmin ( ) P () S s( x a) s( x a) Similarly, when s(x-a)1, the engine pressure - P bemes the maximum pressure - P max. The fllwing equatin is intrdued. Pmax (S ) Pmax ( ) P () S s( x a) s( x a) The P- diagram f Alpha-type Stirling engine an be made with abve equatins. 4. TA-TYP STILING NGIN Similarly, the equatins fr eta-type Stirling engine are delared. Figure shws a alulatin mdel f a eta-type Stirling engine. The expansin mmental vlume - and the mpressin mmental vlume - are desribed in the fllwing equatins, with a swept vlume f a displaer pistn -, a swept vlume f a pwer pistn - S and a phase angle -dx between the displaer pistn and pwer pistn. ( s x) + (4) S ( 1 s x ) + { 1 s( x dx )} + (5) In the ase f the eta-type Stirling engine, the displaer pistn and the pwer pistn are (16) (17)

4 lated in the same ylinder. When bth pistns verlaps there strke, an effetive wrking spae is reated. The verlap vlume - in equatin (5) an be alulated in the next equatin. + S + S S - - s dx (6) 4 Then the ttal mmental vlume - is fund in equatin (7). + + (7) The engine pressure - P based the mean pressure - P mean, the minimum pressure - P min and the maximum pressure - P max are desribed in the fllwing equatins like the Alpha-type Stirling engine. Pmean 1- Pmin ( ) Pmax ( 1- ) P 1- s( x - a) 1- s( x - a) 1- s( x - a) Several ratis and effiients are defined as fllws. T t T S v -1 v sin dx a tan t + sdx -1 4t S t + t + t (8) (9) (0) (1) () () (4) (5) + v (6) t + ( t - 1) v s dx + v - t + 1 (7) Displaer pistn egeneratr spae (, T, P) H xpansin spae (, T, P) mpressin spae (, T, P) Pwer pistn H: Heater : egeneratr : ler Fig. eta-type Stirling ngine 4

5 S The P- diagram f eta-type Stirling engine an be made with abve equatins. (8) 5. GAMMA-TYP STILING NGIN Figure shws a alulatin mdel f a Gamma-type Stirling engine. Similar alulatin equatins are made as the Alpha- and eta-type engine. The expansin mmental vlume - and the mpressin mmental vlume - are desribed in the fllwing equatins with a swept vlume f a displaer pistn -, a swept vlume f a pwer pistn - S and a phase angle - dx between the displaer pistn and the pwer pistn.. ( 1- s x) + (9) S ( 1 - sx ) + { 1 - s( x - dx )} + (40) The ttal mmental vlume - is desribed a next equatin. + + (41) The engine pressure - P based the mean pressure - P mean, the minimum pressure - P min and the maximum pressure - P max are fund in the fllwing equatins. Pmean 1- Pmin ( ) Pmax ( 1- ) P 1- s( x - a) 1- s( x - a) 1- s( x - a) Nw, T t T S v a tan -1 v sin dx t + sdx -1 (4a) (4) (44) (45) (46) (47) egeneratr spae (, T, P) H xpansin spae (, T, P) Displaer pistn mpressin spae (, T, P) Pwer pistn H: Heater : egeneratr : ler Fig. Gamma-type Stirling ngine 5

6 1 v sindx a tan t + sdx + 1 (48) 4t S t + t + + v t + 1 (49) t + ( t 1) v s dx + v t + 1 (50) (51) S The P- diagram f Gamma-type Stirling engine an be made with abve equatins. 6. INDIATD NGY, POW AND FFIINY The indiated energy (area f the P- diagram) in the expansin and mpressin spae an be alulated as an analytial slutins with use f the abve effiients. The indiated energy in the expansin spae (indiated expansin energy) - W (J), based n the mean pressure - P mean, the minimum pressure - P min and the maximum pressure - P max are desribed in the fllwing equatins. Pmeanπ sina Pminπ sina P a W Pd + + max π sin (5) The indiated energy in the mpressin spae (indiated mpressin energy) - W (J) are desribed in the next equatins. Pmeanπt sin a Pminπt sin a P t a W Pd + + max π sin (5) The indiated energy per ne yle f this engine - W i (J) is demanded in the next equatins. Wi We + W Pmeanπ( t) sin a Pminπ( t)sin a + + Pmaxπ( t) sina (54) elatins between P mean, P min and P max are determined in the fllwing equatins. Pmin (55) P P P mean max mean The indiated expansin pwer - L (W), the indiated mpressin pwer - L (W) and the indiated pwer f this engine - L i (W) are defined in the fllwing equatins, using the engine speed per ne send, n(rps, Hz). L Wn (57) L W n (58) L i Wi n (59) The indiated expansin energy - W fund equatin (5) means an input heat frm a heat sure t the engine. The indiated mpressin energy - W alulated by equatin (5) means a rejet heat frm the engine t ling water r air. Then the thermal effiieny f the engine - e is alulated in the next equatin. Wi e W t (60) (56) 6

7 This effiieny equals that f a rnt yle whih is the mst highest effiieny in every thermal engine. 7. AMPL OF ALULATION I: Make a P- diagram and alulate the indiated pwer f an Alpha-type Stirling engine under fllwing nditins. Swept vlume f an expansin pistn: 68 m, swept vlume f a mpressin pistn: 68 m, dead vlume f the expansin spae: m, dead vlume f the mpressin spae: m, regeneratr vlume: m, phase angle: 90deg, mean pressure: 101. kpa, expansin gas temperature: 400deg, mpressin gas temperature: 0deg, engine speed: 000 rpm. A temperature rati - t, a swept vlume rati - v and ther dead vlume rati are alulated with the equatin (6) - (10) t v ah effiient is alulated with equatin (14) - (16) and (19). 1 sin 90 1 a tan s90 4 S π s ngine pressure is alulated with equatin (0). When rank angle - x0deg: P ( Pa) ( kpa) 96s( ) Similarly, when x10deg: P ( kpa) When x0deg: P ( kpa) Next eah mmental vlume is alulated with equatin () ~ (4). When rank angle, x0deg: ( s 0 ) ( m ) 00( m ) 7

8 Pressure P kpa lume m Fig. 4 P- Diagram { s( 0 90 )} ( m ) 514( m ) ( m ) When x10deg: ( m ) When x0deg: 086. ( m ) epeat abve alulatin t ne mplete yle and plt the vlumes - and pressures - P n a graph paper. An example f the P- diagram is shwn in Fig. 4. The indiated energy is alulated with equatin (5), (5) and (54) sin W ( J) sin W ( J) 1 96 Wi ( J) The indiated pwer f this engines is alulated with equatin (59) L i 689( W) 60 The indiated pwer f this engine is 689 W. AKNOWLDGMNT I wish t thank J. H. de aat fr many helpful suggestins during the translatin f this sript. FFNS 1) G. Walker., Stirling ngines, (1980),17, Oxfrd Univ. Press. 8