# 11.4.3: The Properties in the Adiabatic Nozzle

- Page ID
- 799

\[
dh + U\, dU = 0

\label{gd:iso:eq:energy}

\]

\[
{d\rho \over \rho} + { dA \over A} + {dU \over U} = 0

\label{gd:iso:eq:mass}

\]

\[
T\,ds = dh - {dP \over \rho}

\label{gd:iso:eq:thermo}

\]

\[
{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}

\]

## Contributors and Attributions

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.