Performance of Stirling Engines * (Arranging Method of Experimental Results and Performance Prediction) Abstract. 1. Introduction

Transcription

1 1 Performace of Stirlig Egies (Arragig Method of Experimetal Results ad Performace Predictio) Shoichi IWAMOTO Koichi HIRATA ad Fujio TODA Key Words: Stirlig Egie, Heat Exchager, Frictio, Egie Desig, Mechaical Loss, Pressure Loss Trasactios of the Japa Society of Mechaical Egieers, No.65, Vol.635, B, p Advaced Research Istitute for Sciece ad Egieerig, Waseda Uiversity, Okubo 3-4-1, Shijuku, Tokyo , Japa Ship Research Istitute, Shikawa , Mitaka, Tokyo , Japa, Saitama Uiversity, Shimo-okubo 255, Urawa, Saitama , Japa Abstract We have developed five kids of high- ad low-temperature differetial Stirlig egies ad their egie performace was ivestigated experimetally. I order to determie the parameters that affect egie performace, experimetal results were discussed ad compared with results calculated usig aalytical methods. We show a arragig method for the experimetal results, ad cosider the performace of geeral Stirlig egies. After usig the arragig method with odimesioal umbers obtaied by a dimesioal aalysis, a predictio method, which is used at the early desig stage, is formulated. Oe of the odimesioal umbers i this predictio method is calculated based o egie specificatios, icludig the properties of the workig gas. The predictio method ca predict egie speed, output power, the effect of workig gas ad operatig coditios. 1. Itroductio Recetly, evirometal pollutio ad eergy utilizatio have become serious problems worldwide. As Stirlig egies are a potetial solutio to the above problems, they have bee developed actively, ad the results of may studies have bee reported (1)-(9). However, to our kowledge, o systematic ivestigatio ad research o the parameters that affect egie performace have bee coducted to date. It is importat that egie performace be predicted accurately from the mai specificatios that are determied from the target performace durig egie desig. Geerally, the calculatio for egie performace is iitiated after the mai specificatios are determied ad the desig for heat exchagers ad mechaical devices is fiished. But, i actual practice, it is ecessary that the egie performace be determied at the same time as egie specificatios, ad also before fiishig the desig for heat exchagers ad mechaical devices. We have developed five kids of high- ad low-temperature differetial Stirlig egies, ad measured their performace. The measured performace is compared with results calculated usig aalytical methods. The parameters that affect egie performace are ivestigated. We suggest a arragig method for the experimetal results, discuss the performace of geeral Stirlig egies ad develop a simple method for predictig egie performace. 2. Previous Performace Predictio Methods As methods for predictig output power, LS, from egie specificatios, two simple methods utilizig the Beale Number (8) ad the West Number (9) are well kow. i) Beale Number, BN; This method for predictig output power was suggested by Beale (8), where BN is defied by Eq. (1). B L P V m S N = (1) SE Here, Pm is mea pressure i the workig space (Pa), VSE is swept volume of a expasio space (m 3 ),

2 2 is egie speed per uit secod (rps). It is kow that BN at the egie speed where maximum output power is realized, is about 0.15 i the case of high-temperature differetial Stirlig egies whose heater wall temperature is about 650 C (8). ii) West Number, WN; This method was suggested by West (9), where WN is defied by Eq. (2) with revisio of BN. LS W N = (2) T E TC PmVSE TE + TC Here, TE is expasio space gas temperature, ad TC is compressio space gas temperature. It was foud that WN is about 0.25 i the case of 5~150 kw class Stirlig egies, ad is about 0.35 i the case of smaller power egies (9). Output power ca be predicted from egie specificatios usig the BN or WN. However, it is ecessary to determie the egie speed at which maximum output power is realized. The, i order to predict egie performace accurately, the relatioship betwee output power, LS, ad egie speed,, ad that betwee the kid of workig gas ad the output power must be determied, with cosideratio of egie type ad operatig coditios. 3. Arragemet for Experimetal Results We have developed high- ad low-temperature differetial Stirlig egies ad measured their performace (10)-(14). Also, egie performace has bee aalyzed ad calculated usig aalytical methods (15)-(17). After compariso ad discussio, the parameters that affect output power sigificatly were clarified. As we have foud that pressure loss of the workig gas ad mechaical loss due to frictio i the mechaical devices affect output power sigificatly, the experimetal results ca be arraged as follows. 3.1 Pressure loss of workig gas The pressure of the workig gas decreases whe the gas flows through heat exchagers cosistig of a heater, a regeerator ad a cooler. The pressure drop is treated as pressure loss. Work doe by the egie depeds maily o the mea pressure of the workig gas, Pm, the egie size, the expasio space gas temperature, TE, ad the compressio space gas temperature, TC. Furthermore, work is affected strogly by pressure loss at heat exchagers. Pressure loss is affected by the egie type ad operatig coditios such as the egie speed, the desity of the workig gas i heat exchagers, ρ, the kietic viscosity, ν, ad the shape of heat exchagers. Thus, pressure loss should be calculated precisely at the fial desig stage, that is, after determiig the detailed sizes of heater ad cooler tubes ad the specificatios of the regeerator matrix. However, it is cosidered that the size of heat exchagers is depedet o egie size, because similar types of heat exchagers have bee adapted i geeral Stirlig egies. Thus, the swept volume of the expasio space, VSE, is used as the typical parameter that expresses as the size of heat exchagers, because VSE is geerally determied at the early desig stage. Also, the gas costat, R, ad the kietic viscosity of the workig gas, ν, are used as the typical parameters that express the properties of pressure loss, based o the fuctio, ρm Pm/(RTE), where ρm is the mea desity i the expasio space, TE is the expasio space gas temperature, Pm is the mea pressure, ad R is the gas costat. 3.2 Mechaical loss by frictio It ca be cosidered that mechaical loss cosists of Coulomb frictio loss ad viscosity frictio loss. The Coulomb frictio coefficiet, µco, ad the viscosity frictio coefficiet, cvi, are parameters that affect output power. The Coulomb frictio coefficiet, µco, is the same for Stirlig egies that cosist of the same type of mechaical elemets. Also, the viscosity frictio coefficiet, cvi, is depedet o the swept volume of the expasio space, VSE, because cvi is affected strogly by the egie size. The, the swept volume of the expasio space, VSE, is determied to be a typical parameter that expresses mechaical loss. 3.3 Derivatio of odimesioal umbers From the above cosideratios, it is assumed that egie performace cosists of the followig fuctio.

3 3 π1 π 2 π 3 π 4 π 5 π 6 π 7 π = Ws Pm VSE R TE ν, (3) where, Ws is output work (J), Pm is mea pressure of workig gas (Pa), VSE is swept volume of expasio space (m 3 ), R is gas costat (J/kgK), TE is expasio space gas temperature (K), ν is kietic viscosity at temperature, TE, ad pressure, Pm, ad is egie speed (rps). The compressio space gas temperature, TC, is assumed to be costat i the case of geeral Stirlig egies, because it was cofirmed that it does ot differ markedly regardless of the egie type. Equatio (3) ca be show as Eq. (4) usig fudametal uits with mass, M, legth, L, time, T, ad temperature, θ. 2 2 π1 1 2 π 2 3 π π 4 π π 6 1 π π = ML T ML T L L T θ θ L T T (4) [ ] [ ] [ ] [ ] [ ] [ ] [ ] 7 The, we focus o leadig odimesioal umbers that symbolize egie speed ad output power. After much discussio ad cosideratio, the followig three odimesioal umbers are suggested. VSE 2 3 (5) ν WS ws (6) P V m SE T RV 2 3 E SE s =, (7) 2 ν where, is defied as odimesioal speed, ad ws is defied as odimesioal output work. I the case of a egie of which cylider bore, D, is equal to pisto stroke, H, Eq. (5) ca be chaged to Eq. (8) with mea pisto speed, u (=2H). = k ud ν, (8) where, k is a fixed umber [=( /4) 2/3 /2]. Namely, the odimesioal speed,, is equivalet to the Reyolds umber, because it is a ratio of iertia force to viscosity force of the workig gas with a typical legth, D. The odimesioal output work, ws, is equal to the Beale Number, BN, defied by Eq. (1). O the other had, the size ad specificatios of seal devices such as a pisto rig ad a rod seal are determied from the mea pressure i the workig space at maximum load, Plim, at the early desig stage. Also, workig pressure ad temperature are set as the operatig coditios of the egie, ad both affect mechaical loss strogly. The, odimesioal pressure, P, ad odimesioal temperature, T, are defied by Eqs. (9) ad (10), respectively. They are the odimesioal umbers that express the operatig coditios. P T P m (9) P T T lim E E T C + T C Here, Pm is mea pressure of workig gas, Plim is limited maximum mea pressure, TE is expasio space gas temperature, ad TC is compressio space gas temperature. 4. Results of Arragemet ad Cosideratios We have developed a 100 W class small egie ad measured its performace (10),(16). The experimetal results are treated usig the odimesioal umbers as defied above. Figure 1 shows the relatioship betwee odimesioal speed,, ad odimesioal output work, ws, at odimesioal temperature, T, of 0.42 (TE=490 C, TC=40 C) usig helium as the workig gas. I the figure, each lie represets the calculated results based o the isothermal aalysis method with cosideratio of pressure loss ad mechaical loss (15) (called the secod-order model). The solid lie represets the result at odimesioal pressure, P =0.73 (Pm=0.8 MPa), the broke lie represets that at P =0.55 (Pm=0.6 MPa), ad the dot-dashed lie represets that at P =0.36 (Pm=0.4 MPa). Symbols represet the experimetal results at P =0.73, 0.64, 0.55, 0.45 or 0.36 (the mea pressure, Pm, is varied from 0.8 MPa to 0.4 MPa with 0.1 MPa steps). From the figure, the odimesioal output work, ws, decreases with icreasig odimesioal speed,. O the other had, it was cofirmed that the output power teds to decrease at higher egie speed, though the (10)

4 idicated power icreases at higher egie speed i this experimet (15). The decrease i ws with icreasig is caused by the icrease i pressure loss i the heat exchagers ad mechaical loss caused by viscosity frictio at higher, because pressure loss ad mechaical loss are affected by egie speed as described above. Also, the odimesioal output work, ws, decreases with decreasig odimesioal pressure, P. Because the specificatios of seal devices such as a pisto rig are determied based o P 1 at the desig stage, the rate of Coulomb frictio loss of the seal devices icreases with decreasig odimesioal pressure, P. Figure 2 shows the relatioship betwee odimesioal speed,, ad odimesioal output work, ws, at odimesioal pressure, P, of 0.73 (Pm=0.8 MPa) usig helium as the workig gas. I the figure, the solid lie represets the calculated result at T =0.42 (TE=490 C, TC=40 C), the broke lie represets that at T =0.38 (TE=430 C, TC=40 C), ad the dot-dashed lie represets that at T =0.35 (TE=370 C, TC=40 C). Symbols represet the experimetal results i the case of T at 0.42, 0.40, 0.38, 0.37 or 0.35 (the expasio space gas temperature, TE is varied from 490 to 370 C with 30 C steps). From the figure, the odimesioal output work, ws, decreases with icreasig odimesioal speed,, ad decreasig odimesioal temperature, T, similar to the case i Fig. 1. This is due to the fact that the mechaical loss is ot strogly affected by the expasio space gas temperature, TE, ad the odimesioal temperature, T ; furthermore the rate of mechaical loss per uit output power icreases with icreasig odimesioal speed,. Figure 3 shows the relatioship betwee odimesioal speed,, ad odimesioal output work, ws, at odimesioal pressure, P, of 0.73 (Pm=0.8 MPa) usig itroge as the workig gas. I the figure, the solid lie represets the calculated result at T =0.40 (TE=460 C, TC=40 C), the broke lie represets that at T =0.37 (TE=400 C, TC=40 C), ad the dot-dashed lie represets that at T =0.32 (TE=340 C, TC=40 C). Symbols represet the experimetal results whe T is set at 0.40, 0.38, 0.37, 0.35 or 0.32 (the expasio space gas temperature, TE is varied from 460 to 340 C with 30 C steps). From the figure, the odimesioal output work, ws, decreases with icreasig odimesioal speed,, ad decreasig odimesioal temperature, T, similar to the case i Fig. 2, though the effect becomes small at higher values of odimesioal egie speed,. This may be due to the strog effects of pressure loss i the heat exchagers. The calculated results agree with experimetal oes very well. Also, the rage of odimesioal egie speed,, is much higher, about 2000~6000, tha the rage of of about 100~800 whe helium is used as the workig gas, as show i Figs. 1 ad 2. This is because the kietic viscosity, ν, strogly affects odimesioal speed,, ad because is equivalet to the Reyolds umber, as described above. Figures 1, 2 ad 3 show two cases of the relatioship betwee odimesioal speed,, ad odimesioal output work, ws, as defied by Eqs. (5) ad (6), for the 100 W class Stirlig egie. 4

5 5 Oe is the case of chagig the mea pressure, Pm, while maitaiig the same gas temperature, TE ad TC. The other is the case of chagig the gas temperature, TE, while maitaiig the same mea pressure, Pm. Furthermore, we cofirmed that similar figures were obtaied i the case of other operatig coditios ad differet types of egies icludig low-temperature differetial Stirlig egies. Thus, it is cosidered that the experimetal results ca be arraged with ad ws. Also, it is cofirmed that the results of the aalytical method agree well with the experimetal results, ad that the method has high accuracy. 5. Performace of Prototype Egies ad Predictio Method From the above cosideratios, the odimesioal output work, ws, ca be arraged well accordig to the odimesioal speed,, with odim esioal pressure, P, ad temperature T, as parameters. I order to determie the relatioship betwee egie specificatios ad output power, aother odimesioal output work, WS, ad odimesioal output power, LS, are defied by Eqs. (11) ad (12), respectively. W L P V W S m SEP S s (11) S T = W (12) O the other had, assumig that the egie is operated at the limited maximum gas temperature, Tlim, Eq. (13) is derived by revisig Eq. (7), where the expasio space gas temperature, TE, is chaged to Tlim, ad the kietic viscosity, νlim is used at temperature, Tlim, ad pressure, Plim. 2 3 TlimRV S = SE (13) 2 ν lim Here, S is defied as a odimesioal egie specificatio, because it is a odimesioal umber which is calculated from the egie specificatios at the early desig stage. Table 1 shows the egie specificatios ad the operatig coditios of five kids of prototype egies. They have differet egie types, operatig temperatures, pressures ad output power levels. I the table, Egie A is the 100 W class gamma type (see Figs. 1~3), Egie B is a alpha type of a similar power ad temperature level as Egie A, ad Egie C is a 2 kw class beta type. They are the high-temperature differetial egies. Egie D is a 1 kw class alpha type, ad Egie E is a 300 W class gamma type, both of which are low-temperature differetial egies. Figure 4 shows the relatioship betwee odimesioal speed, ad odimesioal output power, LS based o the experimetal results of the prototype egies listed i Table 1. Each odimesioal egie specificatio, S, is show i the figure. I the case of Egie A, two types of operatig coditios, i which either helium or itroge is used as the workig gas, are idicated. From the figure, each egie exhibits a maximum value of the odimesioal output power, LS,max at a give ; i.e., the maximum value of LS,max is the optimal coditio. The odimesioal maximum output power, LS,max, is obtaied whe the egie reaches the maximum output power, LS,max, i each experimet. LS,max icreases ad appears i the rage of higher odimesioal speed,, with

6 6 icreasig odimesioal egie specificatio, S. As a result, the odimesioal speed, opt at which the odimesioal maximum output power, LS,max, is obtaied accordig to the odimesioal egie specificatio, S. Table 2 shows the results of odimesioal values, LS,max, opt ad S, based o the measured maximum output power, LS,max, ad egie speed, opt. Namely, the values of LS,max are the peak values (the optimal coditio) for each egie show i Fig. 4. Figure 5 shows the relatioship betwee opt ad LS,max give i Table 2. I the figure, circles represet the experimetal results, ad the solid lie represets the calculated result usig the method of least squares. From the figure, it is cosidered that the relatioship betwee maximum output power ad egie speed ca be arraged usig the odimesioal maximum output power, LS,max ad the odimesioal speed, opt. As the relatioship betwee opt ad LS,max is a liear oe, it is expressed by Eqs. (14) ad (15). L S, max al opt = (14) The coefficiet, al, is determied to be the value give below from Fig. 5. a = 0.24 (15) L Figure 6 shows the relatioship betwee odimesioal egie specificatio, S, ad odimesioal speed, opt. I the figure, circles represet the experimetal results, ad the solid lie represets the calculated result usig the method of least squares. From the figure, the relatioship betwee S ad opt is expressed by Eqs. (16) ad (17) i the same maer as i Fig.5. opt m = a S (16) The coefficiets, a ad m, are determied to be the followig values from Fig. 6. a m = = (17) Next, the relatioship betwee odimesioal egie specificatio, S, ad odimesioal maximum output power, LS,max, is derived as show i Eq. (18) from Eqs. (14)~(17). L S max, = S 0.60 (18) Also from Eq. (14), Eq. (12) ad the defiitios of the odimesioal umbers, we ca see that the coefficiet al=0.24 is equivalet to the West Number, WN, at P =1. I order to cofirm the propriety of Eqs. (14)~(18) based o the experimetal results of the five kids of egies, other experimetal results that have already bee published are arraged by the same method, ad they are compared with Figs. 5 ad 6. Table 3 lists the egie specificatios, the experimetal results ad the values of the odimesioal umbers of previously reported egies. I the table, there are several egies that do ot have the measured value of the expasio space gas temperature, TE. I the case of high-performace egies, the heater wall temperature is estimated

7 geerally istead of the expasio gas temperature, TE. Thus, the value of TE is estimated by a simple calculatio of the heat trasfer based o the specificatios of the heat exchagers, the heater wall temperature ad other measured values. Figure 7 shows the relatioship betwee odimesioal speed obtaied from the maximum output power, opt, ad odimesioal maximum output power, LS,max. Figure 8 shows the relatioship betwee odimesioal egie specificatio, S, ad odimesioal speed, opt. I these figures, white circles represet the experimetal results of the previously reported egies listed i Table 3, ad black circles represet those of our prototype egies listed i Table 2. The broke lies were obtaied usig Eqs. (14) ad (16), respectively. From the figures, most of the experimetal results of the previously reported egies lie o the opt - LS,max ad S -opt lies, i the same maer as that of our prototype egies listed i Table 2. As a result, it is cofirmed that the performace of geeral Stirlig egies ca be estimated usig Eqs. (14)~(18), because the experimetal results ca be arraged by the odimesioal umbers, opt, LS,max ad S i egies of differet types, sizes, workig gas ad operatig coditios. 7

8 8 6. Flowchart of Predictio Method As the relatioship betwee egie specificatios ad output power is clarified as described above, egie performace ca be predicted at the early desig stage. Figure 9 shows a flowchart of the performace predictio method for the egie desig. First, the odimesioal egie specificatio, S, ca be calculated usig Eq. (13) based o the desig coditio ad egie specificatios. After the odimesioal speed, opt, ad the odimesioal maximum output power, LS,max, are derived from Eqs. (14)~(18), the maximum output power, LS,max, ad the egie speed at a particular time, opt, ca be derived. Namely, the predictio method suggested i this paper ca simply predict the maximum output power ad egie speed at a particular time based o egie specificatios, though it was impossible to do so usig previous predictio methods. Also, this predictio method ca estimate the effects of the kid of workig gas ad the operatig coditios by the derivatio of the odimesioal egie specificatio, S ; thus, it is possible to predict the egie performace i detail. 7. Coclusio I this paper, we measured the performace of high- ad low-temperature differetial Stirlig egies, ad derived the odimesioal umbers i order to estimate egie performace. Fially, the performace predictio method based o egie specificatios was suggested. This study is summarized as follows. (1) The relatioship betwee output work ad egie speed ca be arraged usig the odimesioal speed,, ad the odimesioal output work, ws, with odimesioal pressure, P, ad temperature, T, as parameters (see Figs. 1~3). Also, because the calculated results agree well with the experimetal results, the aalytical method is prove to have sufficietly high accuracy. (2) From the relatioship betwee odimesioal speed,, ad odimesioal output power, Ls, the optimal value of the odimesioal speed, opt, is derived based o the odimesioal egie specificatio, S (see Fig. 4). Thus, there is a optimal egie speed, opt, at which maximum output power, LS,max, is obtaied. (3) The maximum output power, LS,max, ad the optimal egie speed, opt, are derived from Eqs. (5)~(18) whe the mai egie specificatios are determied. Egie performace ca be predicted by the flowchart show i Fig. 9. Additioally, as egie performace is affected by pressure loss i the heat exchagers ad mechaical loss caused by frictio i the mechaical devices, the effects of these should be clarified. However, egies with differet types of mechaical devices ad differet heat exchage situatios such as high- ad low-temperature heat source, were ivestigated i this study. I particular, the heater of a high-temperature differetial egie exchages heat betwee hot gas ad workig gas, i.e., from oe gas to aother gas, ad similar types of heat exchagers are used i such cases. I cotrast, the heat exchagers ad the mechaical devices of a low-temperature differetial egie are differet from those of the high-temperature differetial egie. The heater of the low-temperature differetial egie described i this paper exchages heat betwee hot water ad workig gas, i.e., from liquid to gas. I spite of the fact that various egies were estimated, egie performace agrees well, as show i the diagrams (see Figs. 4~8), based o the odimesioal umbers defied i this paper. The relatioship betwee egie specificatio ad output power is expressed by Eqs. (14)~(18). Namely, it is cosidered that the type of heat exchager ad the mechaical loss deped o the swept volume of the expasio space. As a result, it was foud that the predictio method suggested i this study has sufficietly high accuracy for use at the desig stage of the geeral Stirlig egies.

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