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11.4.3.2: Relationship Between the Mach Number and Cross Section Area

The equations used in the solution are energy (47), second law (??), state (37), mass (34). Note, equation (41) isn't the solution but demonstration of certain properties of the pressure profile. The relationship between temperature and the cross section area can be obtained by utilizing the relationship between the pressure and temperature (45) and the relationship of pressure with cross section area (41). First stage equation (47) is combined with equation (45) and becomes

\[
    {(k - 1) \over k}\, { dP \over P } = - { (k -1) \, M \,dM
                        \over 1 + \dfrac{ k - 1 }{ 2 } M^{2} }
    \label{gd:iso:eq:press-M}   \tag{48}
\]
Combining equation (48) with equation (??) yields
\[
     {1 \over k}  {\dfrac {\rho \,U^{2}} { A} \, \dfrac{dA }{ 1 - M^2}  \over P }
         = - \dfrac{  M \, dM }{ 1 + \dfrac{ k - 1 }{ 2 } M^{2} }
    \label{gd:iso:eq:M-A-0}  \tag{49}
\]
The following identify, \( \rho \, U ^{2} = k\,M\, P\) can be proved as  
\[
    k\, M^2\, P =
    k \overbrace{ U^{2} \over c^2}^{M^2} \overbrace{\rho R T}^{P} =
        k  { U ^{2} \over  k\,R\,T} \overbrace{\rho\, R \, T}^{P}
    = \rho U ^{2}
    \label{gd:iso:eq:rU-kMP}  \tag{50}
\]
Using the identity in equation (50) changes equation (49) into
\[
    \dfrac{dA }{ A} = { M^2 -1 \over  
            M  \left( 1 + \dfrac{k-1 }{ 2} M^2  
            \right)} dM
    \label{gd:iso:eq:M-A}  \tag{51}
\]

Fig. 11.8 The relationship between the cross section and the Mach number on the subsonic branch.

Equation (51) is very important because it relates the geometry (area) with the relative velocity (Mach number). In equation (51), the factors \(M \,\left( 1 + \dfrac{k-1 }{ 2} M^2 \right)\) and \(A\) are positive regardless of the values of \(M\) or \(A\). Therefore, the only factor that affects relationship between the cross area and the Mach number is \(M^2 -1\). For \(M <1\) the Mach number is varied opposite to the cross section area. In the case of \(M > 1\) the Mach number increases with the cross section area and vice versa. The special case is when \(M=1\) which requires that \(dA=0\). This condition imposes that internal flow has to pass a converting-diverging device to obtain supersonic velocity. This minimum area is referred to as "throat.'' Again, the opposite conclusion that when \(dA=0\) implies that \(M=1\) is not correct because possibility of \(dM=0\). In subsonic flow branch, from the mathematical point of view: on one hand, a decrease of the cross section increases the velocity and the Mach number, on the other hand, an increase of the cross section decreases the velocity and Mach number (see Figure (??)).

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.