SHMIDT THOY FO STILING NGINS KOIHI HIATA Musashin-jjutaku 6-10, Gakuen -6-1, Musashimurayama, Tky 08, Japan Phne & Fax: +81-45-67-0086 e-mail: khirata@gem.bekkame.ne.jp url: http://www.bekkame.ne.jp/~khirata 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
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)
{ } 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)
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 + + 1- (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
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
1 v sindx a tan t + sdx + 1 (48) 4t S t + t + + v + + 1 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 1 1 1 1 1 (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 1 1 1 1 1 (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 1 1 1 1 1 (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
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). 0 + 7 t 450 400 + 7 68 10 v 1000. 68 10 10 68 10 10 68 10 10 68 10 ah effiient is alulated with equatin (14) - (16) and (19). 1 sin 90 1 a tan 6577. 45 + s90 4 S + + 45 45 45 +. 767 450 π 45 + 45 s + 1 1097. 1097. 96. 767 ngine pressure is alulated with equatin (0). When rank angle - x0deg: 101. 10 96 P 101988. 10 ( Pa) 101988. ( kpa) 96s( 0 65. 77) Similarly, when x10deg: P 109. 89( kpa) When x0deg: P 118. 011( kpa) Next eah mmental vlume is alulated with equatin () ~ (4). When rank angle, x0deg: 68 10 ( s 0 ) + 00 10 ( m ) 00( m ) 7
Pressure P kpa 160 140 10 100 80 60 40 5 1.0 1.5.0 lume m Fig. 4 P- Diagram 68 10 { s( 0 90 )} + 0514 10 ( m ) 514( m ) + + 0514. 914( m ) When x10deg: 0864. ( 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). 101. 10 68 10 14. 96 sin 65. 77 W 760. 10 ( J) 96 101. 10 68 10 14. 96 45 sin 65. 77 W 169. 10 ( J) 1 96 Wi 760. 10 169. 10. 068 10 ( J) The indiated pwer f this engines is alulated with equatin (59). 5. 45 10 000 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