# 11.4.3: The Properties in the Adiabatic Nozzle

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

$dh + U\, dU = 0 \label{gd:iso:eq:energy} \tag{33}$
Differentiation of continuity equation, $$\rho\, A\, U = \dot{m} = constant$$, and dividing by the continuity equation reads
${d\rho \over \rho} + { dA \over A} + {dU \over U} = 0 \label{gd:iso:eq:mass} \tag{34}$

The thermodynamic relationship between the properties can be expressed as
$T\,ds = dh - {dP \over \rho} \label{gd:iso:eq:thermo} \tag{35}$
For isentropic process $$ds quiv 0$$ and combining equations (11.31) with (35) yields
${dP \over \rho} + U\, dU = 0 \label{gd:iso:eq:thermo2} \tag{36}$
Differentiation of the equation state (perfect gas), $$P = \rho R T$$, and dividing the results by the equation of state ($$\rho\, R\, T$$) yields
${dP \over P} = {d\rho \over \rho} + {dT \over T} \label{gd:iso:eq:stateDless} \tag{37}$
Obtaining an expression for $$dU/U$$ from the mass balance equation (34) and using it in equation (36) reads
$\dfrac{dP }{ \rho} - U^{2} \overbrace{\left[ \dfrac{dA }{ A} + \dfrac{d\rho }{ \rho} \right]}^{\dfrac{dU }{ U} } = 0 \label{gd:iso:eq:combine1} \tag{38}$
Rearranging equation (38) so that the density, $$\rho$$, can be replaced by the static pressure, $$dP/\rho$$  yields
$\dfrac{dP }{ \rho} = U^{2}\, \left( {dA \over A} + {d\rho \over \rho}\, {dP \over dP} \right) = U^{2} \, \left( {dA \over A} + \overbrace{d\rho \over dP}^{\dfrac{ 1}{ c^2 }} {dP \over \rho} \right) \label{eq::varibleArea:combine2} \tag{39}$
Recalling that $$dP/d\rho = c^2$$ and substitute the speed of sound into equation (??) to obtain
${dP \over \rho } \left[ 1 - \left(U \over c\right)^2 \right] = U^2 {dA \over A} \label{eq::varibleArea:combine3} \tag{40}$
Or in a dimensionless form
${dP \over \rho } \left( 1 -M^{2} \right) = U^2 {dA \over A} \label{gd:iso:eq:areaChangeVelocity} \tag{41}$
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
$dP = {\rho\, U^{2} \over A} \; {dA \over 1 -M^2} \label{gd:iso:eq:areaChangeMach} \tag{42}$

### Contributors

• Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.