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.