Gas -Turbine Powerplants For Two-Phase Hydropropulsion 



Different from e, \ is not a constant but depends on local conditions. Denoting 

 by ^i = epH/Pgi the volume ratio in the chamber, the previous equations 

 become 



^e / V \' /^h 



. 1 . /3„ - /3, _ - (-j ,--K,/s^(i-/^r<^) . .-^^ -;;-••; (7") 



V= V^ (^1 + /3„-/3, - +-Xi/3,(l-/3r)j ■ (8') 



It will be useful to apply Eq. (8') at the nozzle exit for evaluating the exit ve- 

 locity v^j in two particular cases: 



(i) aligned propulsor (Figs. 1, 3, 4a, 4b, 5): 

 It results Zi = z^, /3hi = /3^„; /S^ = 1. The external pressure is p^, and 

 therefore /S^^ = Pi/p„ = fi^. Therefore, 



1/2 

 V, = V,(/3„ + ^^i^e(l-/^;*)) , : . . , ,,,,,, (10) 



(ii) S-propulsor (Figs. 2a, 2b, 4c, 6): 

 It is /Sf^„ > o and /S^j < o, while the external pressure is now Pa, and there- 

 fore ^i^ = Pi/Pa = ^e': 



-1 1/2 



V. 1 + ^„- -^ +-^Xi/3,(l-/3;-*) . (10') 



1 + /3hi 



and, for the water jet (\. = 0) 



V 





(10") 



It appears from the previous equations that the flow inside the ejector de- 

 pends just on the volume ratio instead of on the actual conditions of the gas 

 injected. Besides, the solution will not depend on the compressibility of the 

 two-phase mixture. On the other hand, it will be useful to introduce the com- 

 pressibility into the flow equations, by the two-phase pseudo-Mach number. 

 In the following an analysis will be outlined which was already developed par- 

 tially in Refs. [7,14,21]. Let us define a mean density of the mixture (this is 

 valid owing to the hypothesis of no slip) as 



PhAh + PgAg Ah A„ 



p ~- X ' ^^ ^ T ^^"" ^^^ - ^g ^ T ^" ' 



where A is the local cross section, and A^ and Ag are the fraction of A oc- 

 cupied respectively by water and gas (A = Ah + Ag). On the other hand, we 

 have \ = Ag/Ajj. Therefore 



1115 



