Chapter 12, Problem 119P

To determine

The plot for mass flow parameter and Mach number.

Answer to Problem 119P

The plot for mass flow parameter and Mach number is

Explanation of Solution

Given information:

The Mach number varies from $0$ to $1$.

Relation for mass flow parameter and Mach number.

   $\frac{\dot{m}\sqrt{R{T}_0}}{A{P}_0}=\text{Ma}\sqrt{k}\left[{\left(1+\frac{k-1}{2}{\text{Ma}}^2\right)}^{-\left(k+1\right)/2\left(k-1\right)}\right]$     ...... (I)

Here, mass flow rate is $\dot{m}$, gas constant is $R$, specific heat ratio is $k$, Mach number is $\text{Ma}$.

Calculation:

Substitute $1.4$ for $k$ in Equation (I).

   $\begin{array}{c}\frac{\dot{m}\sqrt{R{T}_0}}{A{P}_0}=\text{Ma}\sqrt{1.4}\left[{\left(1+\frac{1.4-1}{2}{\text{Ma}}^2\right)}^{-\left(1.4+1\right)/2\left(1.4-1\right)}\right]\\ =1.18321\text{Ma}\left[{\left(1+0.2{\text{Ma}}^2\right)}^{-3}\right]\end{array}$

Substitute $1.4$ for $k$ and $0$ for $\text{Ma}$ in Equation (I)

   $\begin{array}{c}\frac{\dot{m}\sqrt{R{T}_0}}{A{P}_0}=\text{0}\sqrt{1.4}\left[{\left(1+\frac{1.4-1}{2}{\text{0}}^2\right)}^{-\left(1.4+1\right)/2\left(1.4-1\right)}\right]\\ =0\left[{\left(1+\frac{1.4-1}{2}{\text{0}}^2\right)}^{-\left(1.4+1\right)/2\left(1.4-1\right)}\right]\\ =0\end{array}$

Table-(1) represents the value of flow rate parameter for different Mach numbers at specific heat ratio of $1.4$

     Table-(1)

$\text{Ma}$	$\frac{\dot{m}\sqrt{R{T}_0}}{A{P}_0}$
$0$	$0$
$0.2$	$0.23105$
$0.4$	$0.4306$
$0.6$	$0.57$
$0.8$	$0.65$
$1$	$0.68$

Substitute $1.6$ for $k$ in Equation (I).

   $\begin{array}{c}\frac{\dot{m}\sqrt{R{T}_0}}{A{P}_0}=\text{Ma}\sqrt{1.6}\left[{\left(1+\frac{1.6-1}{2}{\text{Ma}}^2\right)}^{-\left(1.6+1\right)/2\left(1.6-1\right)}\right]\\ =1.2649\text{Ma}\left[{\left(1+0.3{\text{Ma}}^2\right)}^{-2.16667}\right]\end{array}$

Substitute $1.6$ for $k$ and $0$ for $\text{Ma}$ in Equation (I)

   $\begin{array}{c}\frac{\dot{m}\sqrt{R{T}_0}}{A{P}_0}=\text{0}\sqrt{1.6}\left[{\left(1+\frac{1.6-1}{2}{\text{0}}^2\right)}^{-\left(1.6+1\right)/2\left(1.6-1\right)}\right]\\ =0\left[{\left(1+\frac{1.6-1}{2}{\text{0}}^2\right)}^{-\left(1.6+1\right)/2\left(1.6-1\right)}\right]\\ =0\end{array}$

Table-(2) represents the value of flow rate parameter for different Mach numbers at specific heat ratio of $1.6$

     Table-(2)

$\text{Ma}$	$\frac{\dot{m}\sqrt{R{T}_0}}{A{P}_0}$
$0$	$0$
$0.2$	$0.246527$
$0.4$	$0.4570$
$0.6$	$0.607$
$0.8$	$0.69$
$1$	$0.71$

Substitute $1.2$ for $k$ in Equation (I).

   $\begin{array}{c}\frac{\dot{m}\sqrt{R{T}_0}}{A{P}_0}=\text{Ma}\sqrt{1.2}\left[{\left(1+\frac{1.2-1}{2}{\text{Ma}}^2\right)}^{-\left(1.2+1\right)/2\left(1.2-1\right)}\right]\\ =1.09544\text{Ma}\left[{\left(1+0.1{\text{Ma}}^2\right)}^{-5.5}\right]\end{array}$

Substitute $1.2$ for $k$ and $0$ for $\text{Ma}$ in Equation (I)

   $\begin{array}{c}\frac{\dot{m}\sqrt{R{T}_0}}{A{P}_0}=\text{0}\sqrt{1.2}\left[{\left(1+\frac{1.2-1}{2}{\text{0}}^2\right)}^{-\left(1.2+1\right)/2\left(1.2-1\right)}\right]\\ =0\left[{\left(1+\frac{1.2-1}{2}{\text{0}}^2\right)}^{-\left(1.2+1\right)/2\left(1.2-1\right)}\right]\\ =0\end{array}$

Table-(3) represents the value of flow rate parameter for different Mach number at specific heat ratio of $1.4$

     Table-(3)

$\text{Ma}$	$\frac{\dot{m}\sqrt{R{T}_0}}{A{P}_0}$
$0$	$0$
$0.2$	$0.214$
$0.4$	$0.401$
$0.6$	$0.541$
$0.8$	$0.62$
$1$	$0.64$

  Figure-(1) represents the plot of flow rate parameter with Mach number for different value of specific heat ratio.

        Figure-(1)