Supersonic airflow enters an adiabatic, constant area (inside diameter \(=1 \mathrm{ft}\) ) 30 -ft-long pipe with \(\mathrm{Ma}_{1}=3.0 .\) The pipe friction factor is estimated to be \(0.02 .\) What ratio of pipe exit pressure to pipe inlet stagnation pressure would result in a normal shock wave standing at (a) \(x=5\) ft, or \((\mathbf{b}) x=10 \mathrm{ft},\) where \(x\) is the distance downstream from the pipe entrance? Determine also the duct exit Mach number and sketch the temperature-entropy diagram for each situation.

Short Answer

Expert verified
This problem does not have a numerical short answer as it mainly involves understanding and applying theoretical principles of fluid dynamics and shock waves.End results of the process are the calculated pressure ratios and exit Mach numbers, as well as the temperature-entropy diagrams for both situations.

Step by step solution

01

Derive the Fanning friction factor

Given that the friction factor is 0.02, we need to derive the Fanning friction factor. This is calculated as \(f = F_f / 4 = 0.02 / 4 = 0.005\)
02

Identify given variables

Let's identify the givens: Mach number \(Ma_1 = 3.0\), Diameter = 1 ft, Length (L) = 30 ft, and distance from pipe entrance (x) = 5 ft (Part a) and 10 ft (part b). We also have γ (specific heat ratio) for air which is generally taken as 1.4.
03

Calculate the frictional pressure loss coefficient

For a pipe of constant area, the frictional pressure loss coefficient, \(\phi\), can be calculated by the formula: \[\phi = \frac{2fL}{D} \sqrt{\frac{2}{\gamma}} \sqrt{\frac{(\gamma - 1) Ma_1^2}{1 + (\gamma -1)/2 * Ma_1^2}}\] Substituting the given values, we get the value of \(\phi\).
04

Calculate the pressure ratios

We can use the derived \(\phi\) and given x values (5 ft and 10 ft) to calculate the pressure ratios using the formula: \[P/P_0 = 1 - \phi x / L\] After calculations, assuming the values of \(\phi\) and x be in the acceptable range, we get two pressure ratio values for parts a and b respectively.
05

Determine the duct exit Mach number

Using normal shock relationships and the obtained pressure ratios, we calculate the exit Mach number, \(Ma_2\), for both parts a and b.
06

Sketch the temperature-entropy diagram

A temperature-entropy (T-s) diagram is sketched representing the process. They follow the trajectory of the pressure ratios, and we can observe the points where normal shock occurs at 5ft and 10ft, respectively.

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Most popular questions from this chapter

An ideal gas is to flow isentropically from a large tank where the air is maintained at a temperature and pressure of \(59^{\circ} \mathrm{F}\) and 80 psia to standard atmospheric discharge conditions. Describe in general terms the kind of duct involved and determine the duct exit Mach number and velocity in \(\mathrm{ft} / \mathrm{s}\) if the gas is air.

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