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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.