Question

Plot the mass flow parameter \(\dot{m} \sqrt{R T_{0}} /\left(A P_{0}\right)\) versus the Mach number for \(k=1.2,1.4,\) and 1.6 in the range of \(0 \leq \mathrm{Ma} \leq 1\)

Short answer

Answer: The function to calculate the mass flow parameter for given values of Mach number and adiabatic index is \(f(\mathrm{Ma},k) = \frac{\mathrm{Ma}\sqrt{k}}{\sqrt{1+(\frac{k-1}{2})\mathrm{Ma}^2}}\).

Step-by-step solution

Find a formula for the mass flow parameter and Mach number

The mass flow parameter is given by: \(\dot{m} = \rho VA = \frac{P}{RT} VA\) Rearrange the equation: \(\dot{m} \sqrt{R T_{0}} /\left(A P_{0}\right) = \frac{M a}{\sqrt{T_{0}/T}}\) \(\dot{m} \sqrt{R T_{0}} /\left(A P_{0}\right) = \frac{M\sqrt{kRT_{0}}}{\sqrt{T_{0}/\left(1+(\frac{k-1}{2})\mathrm{Ma}^2\right)}}\)

Create the function for the mass flow parameter

We need to define a function to calculate the mass flow parameter for given values of Mach number and adiabatic index: \(f(\mathrm{Ma},k) = \frac{\mathrm{Ma}\sqrt{k}}{\sqrt{1+(\frac{k-1}{2})\mathrm{Ma}^2}}\)

Calculate the mass flow parameter for different values of Mach number and adiabatic indices

We need to calculate the mass flow parameter for \(0 \leq \mathrm{Ma} \leq 1\) and \(k=1.2,1.4,1.6\). We can do this using a loop to iterate over the possible values of Mach number and calculate the mass flow parameter for each adiabatic index: Create a set of Mach numbers, \(\text{Ma_set} = \{0, 0.1, 0.2, \dots, 1\}\). For each Mach number in the set, calculate \(f(\mathrm{Ma},k)\) for each adiabatic index (1.2, 1.4, and 1.6), and store the results in an appropriate data structure.

Plot the mass flow parameter against the Mach number

Using the calculated mass flow parameters, plot \(\dot{m} \sqrt{R T_{0}} /\left(A P_{0}\right)\) versus \(\mathrm{Ma}\) for each adiabatic index \(k\). Use different colors or line styles to distinguish between the adiabatic indices, and include a legend to make the plot easily readable. In the end, you will have a plot showing how the mass flow parameter changes with Mach number for different adiabatic indices in the given range.