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11.4.3: The Properties in the Adiabatic Nozzle

  • Page ID
    799
  • 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.