The Properties in the Adiabatic Nozzle

When there is no external work and heat transfer, the energy equation, reads

$$dh + U dU = 0$$ (4.25)

Differentiation of continuity equation, $\rho A U = \dot{m} = constant$ , and dividing by the continuity equation reads

$${d\rho \over \rho} + { dA \over A} + {dU \over U} = 0$$ (4.26)

The thermodynamic relationship between the properties can be expressed as

$$Tds = dh - {dP \over \rho}$$ (4.27)

For isentropic process $ds \equiv 0$ and combining equations (4.25) with (4.27) yields

$${dP \over \rho} + U dU = 0$$ (4.28)

Differentiation of the equation state (perfect gas), $P = \rho R T$ , and dividing the results by the equation of state ($\rho R T$ ) yields

$${dP \over P} = {d\rho \over \rho} + {dT \over T}$$ (4.29)

Obtaining an expression for dU/U from the mass balance equation (4.26) and using it in equation (4.28) reads

$${dP \over \rho} - U^{2} \overbrace{\left[ {dA \over A} +{d\rho \over \rho} \right]}^{dU \over U} = 0$$ (4.30)

Rearranging equation (4.30) so that the density, dρ, can be replaced by the static pressure, dP/ρ yields

$${dP \over \rho} = U^{2} \left( {dA \over A} +{d\rho \over \rho} {... ...A \over A} + \overbrace{d\rho \over dP}^{1 \over c^{2}} {dP \over \rho} \right)$$ (4.31)

Recalling that $dP/d\rho = c^2$ and substitute the speed of sound into equation (4.31) to obtain

$${dP \over \rho } \left[ 1 - \left(U \over c\right)^2 \right] = U^2 {dA \over A}$$ (4.32)

Or in a dimensionless form

$${dP \over \rho } \left( 1 -M^{2} \right) = U^2 {dA \over A}$$ (4.33)

Equation (4.33) is a differential equation for the pressure as a function of the cross section area. It is convenient to rearrange equation (4.33) to obtain a variables separation form of

$$dP = {\rho U^{2} \over A} \; {dA \over 1 -M^2}$$ (4.34)