Problem 53 - Aircraft Propulsion

Question

In an adiabatic flow of perfect gas in a constant-area duct with friction, the inlet is supersonic with Mach number \(M_{1}=2.0 .\) A normal shock appears in the duct where local Mach number is \(M_{x}=1.5\). Assuming the average friction coefficient in the duct is \(C_{f}=0.005\) and \(\gamma=1.4\), calculate (a) \(L_{\mathrm{x}} / D\) (b) \(L D\) (c) \(L_{x} / L\)

Step-by-Step Solution

Answer

The solutions are found by applying the relevant formulas with the given values. \(L_{\mathrm{x}} / D\) and \(L D\) are calculated using Fanno flow relations. The ratio \(L_{x} / L\) is obtained by dividing \(L_{\mathrm{x}} / D\) by \(L D\). The exact numerical solutions depend on the calculations made in each step with the provided values.

Step 1: Find the pressure ratio for the shock

First, calculate the pressure ratio for the shock using the normal shock equations. With the provided inlet Mach number \(M_{1}\) and local Mach number \(M_{x}\), the formula for this is: \[\frac{P_{02}}{P_{01}} = \frac{[\gamma+1] {M_{1}}^2}{2+(\gamma-1) {M_{1}}^2} * \frac{2\gamma {M_{x}}^2 - (\gamma-1)}{\gamma+1}\] When you substituting the given values, you can calculate the ratio.

Step 2: Find \(L_{\mathrm{x}} / D\)

Next, calculate \(L_{\mathrm{x}} / D\) using the Fanno flow relations. The formula is: \[L_{\mathrm{x}} / D = {{f}_{f}}^{-1} * (\{M_{x}\}^2 - 1)\] where \({{f}_{f}}^{-1}\) is the inverse Fanning friction factor and calculated as \( Cf / 4 = 0.005 / 4\). Substitute the given value of \(M_{x}\) and calculated \({{f}_{f}}^{-1}\) to find the solution.

Step 3: Find the length to diameter ratio \(L D\)

Now, calculate \(L D\) using Fanno flow relations which is given by: \[L D = {{f}_{f}}^{-1} * [\frac{\{{M_{1}}\}^2 - 1}{\{{M_{2}}\}^2 - 1}\] where \({M_{2}}\) is given by the relation: \[\frac{P_{02}}{P_{01}} = [\frac{2\gamma {M_{2}}^2 - (\gamma-1)}{\gamma+1}]\^{\frac{\gamma (\gamma -1)}{2 (\gamma +1)}}\] Solve for \({M_{2}}\) and substitute into the equation for \(L D\).

Step 4: Find the ratio \( L_{x} / L\)

Lastly, find the ratio \(L_{x} / L\) by simply dividing the results of step 2 by step 3.

Key Concepts

Perfect Gas

In the realm of physics, a perfect gas is an idealized model that describes the behavior of gases. It is based on certain simplifying assumptions that make mathematical modelling easier while closely approximating the behavior of real gases under a variety of conditions. The key assumptions include:

• Gas molecules are point particles with no volume.
• There are no intermolecular forces between gas molecules except during collisions.
• Collisions between molecules are perfectly elastic, meaning there is no loss of kinetic energy.
• The gas obeys the equation of state, given by pV = nRT, where p is pressure, V is volume, n is amount of substance, R is the gas constant, and T is temperature.

These assumptions result in a simple relationship between the various properties of the gas, making it extremely useful for a range of engineering and scientific calculations.
The concept of a perfect gas is particularly helpful when analyzing adiabatic flow, where no heat is transferred into or out of the system, just like in the given problem.

Supersonic Mach Number

The term "Mach number" is used to describe the speed of an object moving through a fluid compared to the speed of sound in that fluid. Specifically, Mach number is defined as the ratio of the object's speed to the local speed of sound.

• When the Mach number is greater than 1, the flow is considered supersonic.
• This means the object is traveling faster than the speed of sound in that medium.

In supersonic flows, shock waves can form, which are abrupt changes in pressure, temperature, and density. These shock waves are sliced by the object and are a big player in determining the behavior of the fluid around it.
In the problem, with Mach numbers initially noted as 2.0, the flow is definitively supersonic. This influences various parameters of the flow, such as pressure, temperature, and density, which must be calculated carefully, especially when combined with shocks as seen in the given exercise.

Normal Shock

A normal shock represents a form of shock wave where the changes in flow properties, such as pressure and temperature, occur abruptly and perpendicularly to the flow's direction. This is a fundamental phenomenon that happens in compressible flows like that of supersonic speeds. The primary characteristics are:

• A sudden drop in velocity.
• A sharp rise in pressure and temperature.
• An increase in density.

Normal shocks are significant because they lead to loss of total pressure and can influence the overall efficiency and performance of systems.
In the exercise, the concept plays a role when the normal shock appears in the duct at a local Mach number of 1.5. Recognizing this helps calculate essential parameters like pressure ratios using the normal shock relations.

Fanno Flow Relations

Fanno flow describes the behavior of a gas flowing adiabatically through a constant-area duct where friction is considered. It's one of the fundamental flows described in fluid dynamics, relying on several relations that track changes in flow properties due to friction, such as:

• The changes in flow velocity, temperature, and density.
• The impact of varying friction factor (denoted as \(C_f\)).

The formulae used in Fanno flow relate these attributes to the Mach number before and after the shock. They help in finding unique solutions to specific flow problems as seen in the current exercise, where formulas are used to compute the duct characteristics like \(L_x/D\) and \(L D\). Understanding these relations is crucial to solving problems involving constant area ducts with friction, particularly in circumstances with supersonic inlet conditions as presented.