## P

3 lk + kcosqj k = swept volume ratio Vcj V'P

£ = ratio of the dimensionless engine output with compound working fluid to the output with a gaseous working fluid r - temperature ratio TC/TE <l> = crank angle («/) u> = angular speed

Note: The suffix a or v refers to the air or vapour component. An upper case suffix refers to a constant or maximum value. A lower case suffix refers to an instantaneous value.

Principal design parameters

The principal independent design parameters of a regenerative Stirling engine, according to the Schmidt analysis, arc:

1. the temperature ratio t = Tc/Tn, ratio of temperatures in the compression and expansion spaces

2. the swept volume ratio k= Vc/V,.. ratio of swept volumes in the compression and expansion spaces

4. some characteristic pressure of the working fluid of which the maximum pressure was the preferred criterion

5. the dead space ratio X= VD/W: the porous volume in the regenerator and associated heat exchangers expressed in multiples of the swept volume in the expansion space

### 6. engine speed.

In the case of the compound working fluid there was an additional independent design parameter, the mass ratio /J = mjma and freedom of choice of the selection of fluids having different thermodynamic characteristics.

Design charts lor Stirling engines acting as prime movers and refrigerating machines were given in Chapter 5, prepared from an optimization study based on the Schmidt analysis. For a particular case with T,. - 1000 K (1800 aRl and Tc = 300 K (540 °R) so that r - 0.3 and with the dead space ratio X specified as 1.0 the optimum values of phase angle and swept volume ratio were found to he 0.54 ,-r radians (97°) and 0.74 respectively. These values were taken as the standard design configuration for most of the cases reported here.

Effect of compound working fluid, work diagrams

In Fig. S.7 a comparison is drawn between the work diagrams obtained for three Schmidt cycles having mass ratios of {S=Q, I. and 2, and the standard design configuration of r = 0.3. X= 1.0, a = 0.54 and k -=(J.74. Three work diagrams for each case were obtained. The diagrams show the simultaneous variation of pressure and volume in the expansion and compression space and in the total working space.

The area of these diagrams represents the expansion and compression work of the expansion and compression spaces. 'Hie area of the diagram

yjy-x vdvx v,vi

FlG. 8.7. Work dmgrunis (or Schmidt-cycle system with compound working fluid (after

Walker und Aghi t974).

yjy-x vdvx v,vi

FlG. 8.7. Work dmgrunis (or Schmidt-cycle system with compound working fluid (after

Walker und Aghi t974).

for the lotal working space represents the indicated power output of the engine and is in fact exactly equal to the difference in the areas of the diagrams for the expansion and compression spaces.

For the purpose of comparison the diagrams are rationalized so that pressures and volumes are expressed as fractions of the maximum value, thereby permitting comparison of machines similar to a first approximation in size, weight, and cost. 1 he area of the diagram for the total working space was thought to be the principal criterion: the bigger the area, the better the engine. It was clear from Fig. 8.7, that the cycles with a compound working fluid, j3 = 1.0 and 2.0, had work areas for the total working space much greater than that for the single fluid system 0 = 0. For the particular cases considered, values of the work ratio £ (or ratio of the diagram areas for (i - 1.0 and 2.0 compared with the diagram area for 0 = 0) were 1.6 and 1.76, respectively.

### Effect of mass ratio

Further study of the effect oi the mass ratio /3 resulted in the generation of the data shown in Fig. 8.8. 'l"his figure shows the work ratio £ as a r-0-5

Fig. 8.8. Degree of improvement in the specific output (work ratio) of Schmidt-cycle svstems with a comnound workinc fluid (after Walker and Auhi l<)74i function of Ihe mass ratio ft. A family of curves was presented for a series of temperature ratios ranging from 7 = 0.1 (7,= 3000 K, (5400 °R), Tc= 300 K (540 °R)) to r = 0.5 (Tn = 600K (I080°R). TC = 300K (540 nR)). It was interesting to note from this figure that at very high temperatures (r = 0.1) the addition of a phase change component above a mass ratio of 2 was actually detrimental and resulted in a work ratio less than 1 In every other case an improvement in the work ratio was obtained by the addition of a phase change component, with marked benefits at the lower temperatures. According to big. S.N an improvement in specific output of over three times was gained with a temperature ratio 7 - 0.5 (J'E - 600 K (1080 R), Tc = 300 K (540 °R)). This may have particular significance for future developments of low temperature 'bottoming cycle* power systems utilizing engine-exhaust energy, or fiat-plate solar collectors, as the thermal source.

### System variables

The effect of the other system design variables 7. k. a. and X were investigated in systematic fashion by the independent variation of one parameter, while maintaining the others constant at the arbitrarily selected standard design configuration r = 0.3. a -0.54 77 (97°), /< = 0.7-4, and X = 1.0 for a range of ft values 0, 1.0, and 2.0. The results of the study are presented in Fig. 8.9. The engine output was represented in dimensionless terms as (P!p„tax VT).

Fig. 8.9(a) shows the effect of variation in the phase angle a on the power output for three dillerent mass ratios. Increase in ft causes substantial increase in the power output and a slight change in the angle ai which the maximum power occurs.

Fig. 8.9(b) shows the effect of variation in the swept volume ralio k on the power output for three different mass ratios. Increase in (3 causes substantial increase in the power output and a marked reduction in the swept volume ratio at which the maximum power occurs.

Fig. 8.9(c) shows the elTect of variation in temperature ratio r on engine output. A decrease in t corresponds to an increase in the expansion space temperature T, with the compression space temperature held constant. With a gaseous working fluid the engine output increases progressively with increase in expansion space temperature. This is not true for the compound working fluid. For a mass taiio of ft - 1.0 there is a progressive increase in the output power towards an apparent maximum. For a mass ratio of ft 2.0 the engine output increases slightly to a maximum value at t = 0.2 and then actually decreases with further

Tcmpcratuer inlo.» Tempcuiurc ntto, .V

Fi«i. 8.9. Composite diagram illustrating the eilccts on the power output of Schmidt-cycle systems with n compound working fluid as a consequence of changes ir. the temperature ratio, dead volume ratio, phase angle and swept volume ratio (aftci Walker and Agbi 1974).

Tcmpcratuer inlo.» Tempcuiurc ntto, .V

Fi«i. 8.9. Composite diagram illustrating the eilccts on the power output of Schmidt-cycle systems with n compound working fluid as a consequence of changes ir. the temperature ratio, dead volume ratio, phase angle and swept volume ratio (aftci Walker and Agbi 1974).

characteristic for a power system which may find application in some control and monitoring functions.

Fig 8.9(d) shows the effect of variation in the dead space ratio X on the power output of the engine. It has long been recognized that for gaseous working fluids the dead space is an important design criterion and should be reduced to the minimum in order to maximize power output. It is therefore of substantial interest that the power output with a compound working fluid is much less sensitive to variation in the dead space. Indeed, as shown in big. 8.9(d), the power output actually increases with increase in dead volume ratio up to X= 1.0 and thereafter declines along a shallow curve. This virtual independence of power output to the dead space is important for it will permit the use of heat exchange components of enhanced design with large internal surface area.

### Distribution and mass flow rate

It was of interest to study the mass distribution of working fluid in a Stirling engine with a compound working fluid. The mass distribution characteristics obtained for the standard design configuration with 0 = 0

Fig. 8.10. Mass distribution in the compression ami expansion spaces of Schmidt-cycle systems with a gaseous and compound working fluid (after Walker and Agbi 19741.

compression space. Fluid in the dead space is not included- The fluid masses are expressed in dimensionless terms by the arbitrary definition of a reference mass m*.

It was of interest to note from Fig. 8.10 that with a compound working fluid there was a concentration of mass in the compression space and a reduced mass in the expansion space, compared with a gaseous working fluid. The concentration of mass in the compression space was explained by the assumption that the vapour component at low temperatures was assumed to have zero vapour pressure and volume so that the gas component pressure in the compression space was greater than in the expansion space. This eliect was additional to the normal densilication of the working fluid due to the temperature difference. Concentration of mass in the compression space was thought to explain the reduction observed in Fig. 8.9(b) in the optimum value of the swept volume ratio k with increasing (1.

Cyclic mass velocity characteristics lor ihe How of working fluid to and from the expansion and compression spaces were derived for the standard design configuration. These curves are presented in Tig. 8.11(a) for the case (} - 0 and Fig. 8.11(b) for the case 0 = 2.0.

In both Figs. 8.1 1(a) and (b) the curve G-C represents the rate of fluid flow out of the expansion space, the curve C-K-E represents fluid flow rates into the expansion space, and the curve E-Vl represents fluid flow out of the expansion space. The curve l l-B represents fluid How into the compression space, the curve B-J-D represents flow out of the compression space, and the curve D-L-N represents flow into the compression space.

Plotted m this inverse fashion the curves are of interest when overlaid as in Figs. 8.11(a) and (b). for Ihe shaded areas then revealed the rate and duration of flow through the regenerator. Thus, the shaded areas A-G-B plus F.-M-F represents the net flow of fluid through the regenerator in the direction from the expansion space towards the compression space. Similarly, the shaded area C-K-D represents net flow through the regenerator in the direction from the compression space towards the expansion space. In the part of cycle. B-C, fluid was flowing into the regenerator from both expansion and compression spaces and for the cyclic fraction D-F., fluid was flowing out of the regenerator into both the compression and expansion spaces.

Fig. 8.11. Mass velocity characteristics for the compression and expansion spaces of a Schmidt-cycle system with i gaseous and compound working fluid (after Walker and Agbi

Fig. 8.11. Mass velocity characteristics for the compression and expansion spaces of a Schmidt-cycle system with i gaseous and compound working fluid (after Walker and Agbi

Comparison of Figs. 8.10(a) and (b) shows that the rates of mass flow for the compression space were substantially increased with a compound working fluid and the rates of flow for the expansion space marginally decreased, a result that could be deduced from Fig. 8.10. Increase in the fluid mass velocities could be expected to increase the fluid friction effects thereby increasing the difference in pressure between the expansion and compression spaces causing a deleterious effect on the engine power output.

### Optimum design

Subsequently Walker and Agbi <1973) found that the optimum combination of swept volume ratio k and phase angle o for given values of temperature ratio r and dead volume A' varied with the mass ratio {3. With a compound working fluid the optimum ratio of swept volumes in the compression and expansion space was appreciably less than the optimum value with a gaseous working fluid. This arose out of the difference in mass distribution in the engine such as that discussed above with reference to Fig. 8.10.

The effect of the modification of optimum swept volume ratio was beneficial. When the change in optimum swept volume ratio was made and the proportions were adjusted such that the same maximum working space volumes were compared, the result was as shown in Fig. 8.12. This figure shows work diagrams for the expansion, compression, and expansion spaces for three cases:

(a) the optimum design configuration for a gaseous working fluid, 7 = 0.3, X= 1. « = 0.54tt radians (97°) and k = 0.74. In this case 0 = 0.

(b) same parameter configuration but with a compound working fluid, (1 = 2.

(c) optimum design configuration for a compound working fluid with (1 = 2. t = 0.3 and x= 1. In this case a =0.53ir radians (95°) and k =0.35.

The net work-diagram area for case (b) was 1.76 times that for case (a), and was given previously in Fig. S.7. The net work-diagram area for case (c) (same size machine as case (b) but with the proportions adjusted for maximum power) was 2.0 times that for case (a).

Walker and Agbi (1973) presented optimum design charts similar to those for the gaseous working fluid given in Chapter 5. For the compound working fluid separate charts were developed for different values of the mass ratio

Fluid combinations

The thporv available w:k en híahlv íItj» ih« nniu M.

weights Mc and Mv. Studies were made of fluid combinations having a range of values of the ratio S - Ai,/Mv. Light gases (helium and hydrogen. Mb-4 and 2) were combined with heavy liquids (l-'reons Mv> 100): heavy gases (carbon dioxide M„ = 44) were combined with light liquids (water Mv = 18) and intermediate combinations were investigated.

For the particular configuration studied, r = 0.3. X 1.0, k - 0.74, a = 0.547T (97°), (i ^ 2 the best results in terms of the maximum work ratio, were gained with a molecular weight ratio N corresponding to that for air and water i.e. i\' = 29/18= 1.6. Insufficient work was done for this result to be generally applicable to all machine configurations. Furthermore, the idealizations of the theory are such that any conclusions drawn must remain tentative.

The elements of the theory thought to be particularly suspect are the assumptions of isothermal processes, constant mixing of the fluids for uniform mass-ratio {i and neglect of aerodynamic-flow loss. We have seen

## Solar Stirling Engine Basics Explained

The solar Stirling engine is progressively becoming a viable alternative to solar panels for its higher efficiency. Stirling engines might be the best way to harvest the power provided by the sun. This is an easy-to-understand explanation of how Stirling engines work, the different types, and why they are more efficient than steam engines.

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