The following correlating equation of state has proven very useful for describin

Question

The following correlating equation of state has proven very useful for describing a flashing homogeneous two - phase mixture undergoing an isentropic depressurization. v/v_0 - 1 = a{p_0/p - 1} Using the given correlating equation of state for a flashing homogeneous two-phase mixture, derive an equation for the ideal mass flux through a frictionless nozzle in terms of the correlating parameter (a), the stagnation pressure (P_0) and the stagnation specific volume (V_0). Using the given correlating equation of state for a flashing homogeneous two-phase mixture, derive an equation for the critical pressure ratio (P_c/P_0) and the ideal critical mass Using the given correlating equation of state for a flashing homogeneous two-phase mixture, derive an equation for the critical pressure ratio (Pc/P0) and the ideal critical mass flux (G_c) through a frictionless nozzle.

Explanation / Answer

solution:

1)when pure liquid is undergone pressure chnage below its vaccum pressure point then liquid get converted to its vapor and this process occure in nozzle, as liquid start to enhance their velocity due to reduced area and as result their pressure fall below vapor pressure and vapor form and then this two phase mixture shows compressible characteristics and this kind of flow is know as homogeneous two phase flow.

2)by applying steady flow energy equation to inlet as staganation 0 and outlet ,we get that

c=(2(h-ho))^.5

ho=enthalpy at inlet

h=enthalpy at exit

where for isentropic Pv^k=constant, for liquid gas law is not obey,hence value k is different from 1.4,process enthalpy change is given as

h-ho=(k/k-1)[PoVo-PV]

so putting value from our relation we get that

P/Po=[(v/vo-1)(1/a)+1]

velocity is given as

c=[2(k/k-1)PoVo[1-(P/Po)(v/vo)]]^.5

c=[2(k/k-1)PoVo[1-([(v/vo-1)(1/a)+1])(v/vo)]]^.5

where mass flux in nozzle section is given as

m/A=m'=c/v=(1/v)[2(k/k-1)PoVo[1-([(v/vo-1)(1/a)+1])(v/vo)]]^.5

5)critical pressure is pressure in nozzle at which velocity of fluid is equal to sonic velocity

P/Po=Pc/Po=(2/k+1)^(k/k-1)

and mass flux is as follows

m/A=m'=c/v=(1/v)[2(k/k-1)PoVo[1-(2/k+1)^(k/k-1)(v/vo)]]^.5

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