Quandt 



Approximate Solution to Nozzle Equations 



Examination of the energy equations revealed that over the range of air- 

 water mixtures of interest it is permissible to approximate the expansion as 

 isothermal at the liquid temperature. Additionally it will be assumed that noz- 

 zle friction is associated with the gas phase and will depend upon the total mo- 

 momentum flux. With these assumptions the total momentum equation becomes 



w 



-^dV= -AdP - T^ttD^ dx . 



where a momentum average velocity is defined as 



■ ~ 1 r 



V = v„ + 



V = V. 



1 + r s 1 + r 



1 + FCT 



1 + r 



where the slip ratio is defined as 



Using the mass continuity equation allows definition of a liquid-gas area ratio as 



A = A(l + Ap/\) = A^(l+ a) , 



where 



or, more simply, 



. /. *^ ^^ ^^ 



^ ^ W Pf V^ 



JLli 



from which it may be seen that a is very small for a near one and r less than 

 ten. 



Now, the gas area may be related to pressure through the ideal gas equation 



W„ W„ RT 



' \ P, \ P"^ 



Substituting a, r, and a into the total momentum equation yields 



1064 



