Chapter 12, Problem 148P

To determine

To plot: the mass flow parameter.

Answer to Problem 148P

Explanation of Solution

Given:

Mach number $k=1.2,1.4and\text{ 1}\text{.6 range of 0}\le \text{Ma}\le \text{1}\text{.}$

Calculation:

Equation is given by

  $\begin{array}{l}\frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}\frac{\text{PAMa}\sqrt{\frac{\text{k}}{\text{RT}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}}{\text{A}{\text{P}}_{\text{0}}}\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=Ma}\sqrt{\text{k}}\left(\frac{\text{P}}{{\text{P}}_{\text{0}}}\right)\left(\sqrt{\frac{{\text{T}}_{\text{0}}}{\text{T}}}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=Ma}\sqrt{\text{k}}\left({\left(\text{1+}\frac{\text{k-1}}{\text{2}}\text{M}{\text{a}}^{\text{2}}\right)}^{\text{-k/k-1}}\right){\left(\text{1+}\frac{\text{k-1}}{\text{2}}\text{M}{\text{a}}^{\text{2}}\right)}^{\text{1/2}}\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=Ma}\sqrt{\text{k}}\left({\left(\text{1+}\frac{\text{k-1}}{\text{2}}\text{M}{\text{a}}^{\text{2}}\right)}^{\text{-}\left(\text{k+1}\right)\text{/2}\left(\text{k-1}\right)}\right)\end{array}$

  $\begin{array}{l}k=1.4\\ Ma=0.2\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=0}\text{.2}\sqrt{1.4}\left({\left(\text{1+}\frac{\text{1}\text{.4-1}}{\text{2}}\text{0}\text{.}{\text{2}}^{\text{2}}\right)}^{\text{-}\left(\text{1}\text{.4+1}\right)\text{/2}\left(\text{1}\text{.4-1}\right)}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}0.231\end{array}$

  $\begin{array}{l}k=1.4\\ Ma=0.4\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=0}\text{.4}\sqrt{1.4}\left({\left(\text{1+}\frac{\text{1}\text{.4-1}}{\text{2}}\text{0}\text{.}{\text{4}}^{\text{2}}\right)}^{\text{-}\left(\text{1}\text{.4+1}\right)\text{/2}\left(\text{1}\text{.4-1}\right)}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}0.4306\end{array}$

  $\begin{array}{l}k=1.4\\ Ma=0.6\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=0}\text{.6}\sqrt{1.4}\left({\left(\text{1+}\frac{\text{1}\text{.4-1}}{\text{2}}\text{0}\text{.}{\text{6}}^{\text{2}}\right)}^{\text{-}\left(\text{1}\text{.4+1}\right)\text{/2}\left(\text{1}\text{.4-1}\right)}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}0.57\end{array}$

  $\begin{array}{l}k=1.4\\ Ma=0.8\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=0}\text{.8}\sqrt{1.4}\left({\left(\text{1+}\frac{\text{1}\text{.4-1}}{\text{2}}\text{0}\text{.}{\text{8}}^{\text{2}}\right)}^{\text{-}\left(\text{1}\text{.4+1}\right)\text{/2}\left(\text{1}\text{.4-1}\right)}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}0.65\end{array}$

  $\begin{array}{l}k=1.4\\ Ma=1.0\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=1}\text{.0}\sqrt{1.4}\left({\left(\text{1+}\frac{\text{1}\text{.4-1}}{\text{2}}\text{1}\text{.}{\text{0}}^{\text{2}}\right)}^{\text{-}\left(\text{1}\text{.4+1}\right)\text{/2}\left(\text{1}\text{.4-1}\right)}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}0.68\end{array}$

  $\begin{array}{l}k=1.6\\ Ma=0.2\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=0}\text{.2}\sqrt{1.6}\left({\left(\text{1+}\frac{\text{1}\text{.6-1}}{\text{2}}\text{0}\text{.}{\text{2}}^{\text{2}}\right)}^{\text{-}\left(\text{1}\text{.6+1}\right)\text{/2}\left(\text{1}\text{.6-1}\right)}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}0.246\end{array}$

  $\begin{array}{l}k=1.6\\ Ma=0.4\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=0}\text{.4}\sqrt{1.6}\left({\left(\text{1+}\frac{\text{1}\text{.6-1}}{\text{2}}\text{0}\text{.}{\text{4}}^{\text{2}}\right)}^{\text{-}\left(\text{1}\text{.6+1}\right)\text{/2}\left(\text{1}\text{.6-1}\right)}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}0.457\end{array}$

  $\begin{array}{l}k=1.6\\ Ma=0.6\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=0}\text{.6}\sqrt{1.6}\left({\left(\text{1+}\frac{\text{1}\text{.6-1}}{\text{2}}\text{0}\text{.}{\text{6}}^{\text{2}}\right)}^{\text{-}\left(\text{1}\text{.6+1}\right)\text{/2}\left(\text{1}\text{.6-1}\right)}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}0.607\end{array}$

  $\begin{array}{l}k=1.6\\ Ma=0.8\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=0}\text{.8}\sqrt{1.6}\left({\left(\text{1+}\frac{\text{1}\text{.6-1}}{\text{2}}\text{0}\text{.}{\text{8}}^{\text{2}}\right)}^{\text{-}\left(\text{1}\text{.6+1}\right)\text{/2}\left(\text{1}\text{.6-1}\right)}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}0.69\end{array}$

  $\begin{array}{l}k=1.6\\ Ma=1.0\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=1}\text{.0}\sqrt{1.6}\left({\left(\text{1+}\frac{\text{1}\text{.6-1}}{\text{2}}\text{1}\text{.}{\text{0}}^{\text{2}}\right)}^{\text{-}\left(\text{1}\text{.6+1}\right)\text{/2}\left(\text{1}\text{.6-1}\right)}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}0.71\end{array}$

  $\begin{array}{l}k=1.2\\ Ma=0.2\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=0}\text{.2}\sqrt{1.2}\left({\left(\text{1+}\frac{\text{1}\text{.2-1}}{\text{2}}\text{0}\text{.}{\text{2}}^{\text{2}}\right)}^{\text{-}\left(\text{1}\text{.2+1}\right)\text{/2}\left(\text{1}\text{.2-1}\right)}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}0.214\end{array}$

  $\begin{array}{l}k=1.2\\ Ma=0.4\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=0}\text{.4}\sqrt{1.2}\left({\left(\text{1+}\frac{\text{1}\text{.2-1}}{\text{2}}\text{0}\text{.}{\text{4}}^{\text{2}}\right)}^{\text{-}\left(\text{1}\text{.2+1}\right)\text{/2}\left(\text{1}\text{.2-1}\right)}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}0.401\end{array}$

  $\begin{array}{l}k=1.2\\ Ma=0.6\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=0}\text{.6}\sqrt{1.2}\left({\left(\text{1+}\frac{\text{1}\text{.2-1}}{\text{2}}\text{0}\text{.}{\text{6}}^{\text{2}}\right)}^{\text{-}\left(\text{1}\text{.2+1}\right)\text{/2}\left(\text{1}\text{.2-1}\right)}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}0.541\end{array}$

  $\begin{array}{l}k=1.2\\ Ma=0.8\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=0}\text{.8}\sqrt{1.2}\left({\left(\text{1+}\frac{\text{1}\text{.2-1}}{\text{2}}\text{0}\text{.}{\text{8}}^{\text{2}}\right)}^{\text{-}\left(\text{1}\text{.2+1}\right)\text{/2}\left(\text{1}\text{.2-1}\right)}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}0.62\end{array}$

  $\begin{array}{l}k=1.2\\ Ma=1.0\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=1}\text{.0}\sqrt{1.2}\left({\left(\text{1+}\frac{\text{1}\text{.2-1}}{\text{2}}\text{1}\text{.}{\text{0}}^{\text{2}}\right)}^{\text{-}\left(\text{1}\text{.2+1}\right)\text{/2}\left(\text{1}\text{.2-1}\right)}\right)\\ \frac{\stackrel{\text{.}}{\text{m}}\sqrt{\text{R}{\text{T}}_{\text{0}}}}{\text{A}{\text{P}}_{\text{0}}}\text{=}0.64\end{array}$

The graph for the above values is given below:

Conclusion:

Therefore, for the values Ma and k are plotted but only one value of $\text{k and Ma}$.