Quandt 



Integrating the particle equation at constant a, p^, c^, and D yields 



3 Cq ^g (l-a)2 



In (V^/V.) = ^ 



= pL/X 



or 



V 



e 



V. 



where 



L/;v - - — ^ ^^"^^' . 

 4 D Pf ct2 



Combining these equations to eliminate nozzle length allows calculation of 

 momentum mean velocity increase for given r, a, and v . Results of this calcu- 

 lation are shown in Fig. 5. Figure 6 illustrates the variation of pressure and 

 velocity with distance along the nozzle for a pressure ratio of 1.4 with a = 0.85. 

 Figure 7 shows the axial variation of nozzle area ratio with length for v = 1.4; 

 a = 0.85; and r = 5, 10, and 20. Here it can be seen that the proper nozzle 

 shpae may be converging, converging-diverging, or diverging, depending upon 

 the water -to -air mixture ratio. 



Nozzle Thrust 



Using the law of conservation of axial momentum it is possible to compute 

 an ideal propulsion system thrust by taking a control volume around the entire 

 ship. In that case, neglecting free -stream diffusion for the moment, 



W_ Wp 



= — {V„ - Vj +-i {Wo -V) . 



The major losses in this system are those associated with the water scoop, duct, 

 and nozzles. Internal flow losses such as friction drag, area, or form losses 

 and discharge losses will be lumped into one loss coefficient based on craft 

 speed. Therefore, the liquid static pressure after injection into the air stream 

 is given by 



p? Pf ^^ 



° 2g^ ^ 2g^ 



When the nozzle pressure ratio is specified this equation is used to compute an 

 effective injection velocity for use in the thrust analysis: 



It is of interest to note here that the scoop system being analyzed will always 

 operate with a free-stream compression because of internal losses. 



1066 



