## I

Let d)-0- ifr, tan *j'=i: then d</> = di/f ~ 2 df/( I i r) and

. f f 8 sin if/ cos 0 fi cos •/' sin 0 4 sin 0 sin 0 I , , ,. .

= [ - cos 0 log( 1 4 5 cos .//)]J-<> I Sin 0[«//ft'l„ - sin ^jf , -V^) I

« 2 sin Q/l-«SVj;[(l-t-^/(l-S)*]dl - »n 1» ——^—j j f a

- 2tt sin 0"TfT^IHan '(K 1 - «)*/< I + 5)»)]}I0

In evaluating the definite integral note that f-0 at both limits; lair'(O) 0. 77 etc. It is clear from eqn (4.18), if 5->(), that the cyclic integral must lake Ihe value -277-sin 0 for its third term. Thus

5/ = 2?rsin 0 2tt sin 0/(1-= 2irsin «[1-1/(1 a7)']

Substituting for ptnill gives

When ihe numerator and denominator are both mu Itiplied by |(l -11, this proves to be identical to eqn (4.17).

Compression space The variation in volume of the compression space follows the equation Vc = ¿K Vji[ I + cos(<£ - a)]

and, by a process similar to that above, we can obtain expressions for the pressure and volume in the required form, so that heat transferred in the compression space is niveu bv

~ k(síii 0 cos a cos 0 sin a )/sin t) - k(cos tt - sin a/tan (/), bul tan tf = k sin o/(r l-1< cos-ir) and, therefore, OJO ~ - r.

I he heat transferred in the expansion space is of opposite sign to the heal transferred in the compression space, and is numerically different by the temperature ratio r. By analogy, the work done in the two spaces has the same relationship, Pc = -tPf, and the net power is P - Pr.+Pc~ (l-r)O.

In the case of the machine acting as a prime mover 7"k> Tc, i.e. t< 1. and the thermal efficiency

■q = (heat supplied—heat rejccted)/(heat supplied) = (Q- T0)/0= I-T = (Th-7c)/Tii.

This corresponds to the Carnot efficiency. When the machine acts as a refrigerator, /', > Te, i.e. r>l. and the coefficient of performance = heat lifted/work done

= C?/(Q-Oc)= 1/(1 t> = TE/(7V-T(.). For a heat pump, 7"c> TR, i.e. t> 1. and the coefficient of performance - heat rejected/work done

O AO - Qc) - r/( 1 - r) = TJ(TU - Tc). This corresponds to the inverse thermal efficiency.

Mass distribution in the machine From the characteristic gas equation,

M = pV/RT, where p = Pmeon( I - S2)*/[ 1 5 cos(</> - «)]. (a) Expansion space Vc = \ Vn( I + cos <£).

The instantaneous mass of working fluid in the expansion space is given by

Mc - iVHpnwma( 1 I +eos «(»)/[K7V;(l I 5 eos(,/> - U)|. (4.20)

The rate of change of mass of working fluid in the expansion space is dMJi\<b = Vnpmtan( 1 - ,52)*{fi[sin(</.- íi) sin 0 J-sin <t>}/2RTn

(b) Compression space Vc-)kV,,[I f cos(</> - «)].

The instantaneous mass of working lluul in the compression space is given by

Mc-k VRpmcitn( I - f>2)H 1 + eos(<f> - «)\IRTC{I I- ft cos(4 -0)1 (4.21)

'Hie rate of change of mass of working fluid in the compression space is dMc/d</> = k VEpn,cni)( I - 52)'{5[sin(4> 0) + sin(« - (»1

— sin(«/< - a)}/2R'I I + 8 COS(</> - 0)}2

The instantaneous mass of the working lluid in the dead space is given ## Solar Stirling Engine Basics Explained

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