# 11.4.3: The Properties in the Adiabatic Nozzle

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

$dh + U\, dU = 0 \label{gd:iso:eq:energy}$

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

${d\rho \over \rho} + { dA \over A} + {dU \over U} = 0 \label{gd:iso:eq:mass}$

The thermodynamic relationship between the properties can be expressed as

$T\,ds = dh - {dP \over \rho} \label{gd:iso:eq:thermo}$

For isentropic process $$ds quiv 0$$ and combining equations (11.31) with (35) yields

${dP \over \rho} + U\, dU = 0 \label{gd:iso:eq:thermo2}$ Differentiation of the equation state (perfect gas), $$P = \rho R T$$, and dividing the results by the equation of state ($$\rho\, R\, T$$) yields

${dP \over P} = {d\rho \over \rho} + {dT \over T} \label{gd:iso:eq:stateDless}$ Obtaining an expression for $$dU/U$$ from the mass balance equation (34) and using it in equation (36) reads

$\dfrac{dP }{ \rho} - U^{2} \overbrace{\left[ \dfrac{dA }{ A} + \dfrac{d\rho }{ \rho} \right]}^{\dfrac{dU }{ U} } = 0 \label{gd:iso:eq:combine1}$ Rearranging equation (38) so that the density, $$\rho$$, can be replaced by the static pressure, $$dP/\rho$$ yields

$\dfrac{dP }{ \rho} = U^{2}\, \left( {dA \over A} + {d\rho \over \rho}\, {dP \over dP} \right) = U^{2} \, \left( {dA \over A} + \overbrace{d\rho \over dP}^{\dfrac{ 1}{ c^2 }} {dP \over \rho} \right) \label{eq::varibleArea:combine2}$ Recalling that $$dP/d\rho = c^2$$ and substitute the speed of sound into equation (??) to obtain

${dP \over \rho } \left[ 1 - \left(U \over c\right)^2 \right] = U^2 {dA \over A} \label{eq::varibleArea:combine3}$ Or in a dimensionless form

${dP \over \rho } \left( 1 -M^{2} \right) = U^2 {dA \over A} \label{gd:iso:eq:areaChangeVelocity}$ Equation (41) is a differential equation for the pressure as a function of the cross section area. It is convenient to rearrange equation (41) to obtain a variables separation form of

$dP = {\rho\, U^{2} \over A} \; {dA \over 1 -M^2} \label{gd:iso:eq:areaChangeMach}$