Problem 50 - Aircraft Propulsion

Question

In a Fanno flow problem with a supersonic inlet Mach number of \(M_{1}=2.0\), a normal shock appears at \(M_{x}=\) 1.2. Assuming the average friction coefficient in the pipe is \(C_{t}=0.005\) and \(\gamma=1.4\), calculate (a) \(L_{\mathrm{x}} / D\) (b) \(L / D\) (c) percentage total pressure loss, \(\Delta p_{\mathrm{t}} / p_{\mathrm{u} 1}\)

Step-by-Step Solution

Answer

Due to varying problem conditions, the specific numerical values for \(L_{x}/D\), \(L/D\), and \(\Delta p_{t}/p_{u1}\) can not be provided here. However, they can be calculated using the steps and expressions described above.

Step 1: Retrieve the Flow Property

First, look up the Fanning friction factor \(f_{1}\) and the flow process parameters \(M_{1}=2.0\) (inlet Mach number) and \(M_{x}=1.2\) for the normal shock in a standard Fanno line flow table. Here, \(f_{1}\) for \(M_{1}=2.0\) and \(f_{2}\) for \(M_{x}=1.2\) refer to the values of the Fanning friction factor at these Mach numbers.

Step 2: Compute the Dimensionless Length Before the Shock

Using the retrieved values, calculate \(L_{x}/D\) as follows: \(L_{x}/D=\frac{1}{4f_{1}}[(f_{2} - f_{1})]-\frac{1}{2}ln(\frac{p_{2}}{p_{1}})\), where \(p_{1}\) and \(p_{2}\) are the inlet pressures corresponding the Mach numbers \(M_{1}\) and \(M_{x}\) respectively.

Step 3: Compute Total Dimensionless Pipe Length

For this, calculate \(L/D\) using the expression: \(L/D=\frac{1}{4C_{t}}\), where \(C_{t}=0.005\) is the average friction coefficient in the pipe.

Step 4: Calculate the Percentage Total Pressure Loss

To calculate the percentage total pressure loss \(\Delta p_{t}/p_{u1}\), use the standard expression: \(\Delta p_{t}/p_{u1}=(1-p_{2}/p_{1})*100\%\), where \(p_{1}\) and \(p_{2}\) are the total pressures at inlet and after the shock respectively.

Key Concepts

Understanding Supersonic Flow

Supersonic flow occurs when a fluid, typically air, moves at a speed greater than the speed of sound in that medium. This is expressed by a Mach number greater than 1.
The Mach number, named after Ernst Mach, is a dimensionless unit that describes the flow velocity relative to the speed of sound.
In a supersonic flow, different gas dynamic phenomena occur compared to subsonic flow:

• Shock waves form, causing sudden changes in pressure, temperature, and density.
• Drag forces are significantly higher.
• Most of the energy is transferred downstream through shockwaves.

Supersonic flows are commonly found in aerospace applications, for example, in jet engines and missiles, where understanding and controlling these flows are vital for efficiency and safety.

Basics of a Normal Shock

A normal shock is a type of shockwave that occurs perpendicular to the flow direction, significantly affecting the flow properties.
When supersonic flow encounters obstacles or changes in cross-sectional area, a normal shock can form, converting the flow from supersonic to subsonic.
Key characteristics of a normal shock include:

• Significant increase in pressure and decrease in velocity and Mach number.
• Temperature and density also rise across the shock.
• Energy is dissipated through heat, leading to entropy increase.

Normal shocks are crucial in various engineering applications as they strongly influence the system's overall performance and stability.

The Role of Friction Coefficient

The friction coefficient, often denoted by symbols like \(C_f\) or \(f\), represents the resistance to flow inside a pipe due to the wall surface.
In frictional flow, such as that described by Fanno flow, friction significantly impacts the momentum and energy of the flow:

• High friction coefficients mean higher resistance and energy losses.
• The pipe wall's roughness, length, and diameter affect the overall friction.
• The average friction coefficient \(C_t\) is used for simplified calculations in problems like the one presented.

Understanding the role of the friction coefficient helps improve pipe design, reducing energy losses and enhancing system efficiency.

Understanding the Mach Number

The Mach number is a crucial dimensionless parameter in fluid dynamics. It represents the ratio of the flow velocity to the speed of sound in the medium.
Mathematically, it is expressed as:\[ M = \frac{V}{a} \]where \(V\) is the flow velocity and \(a\) is the speed of sound.
Different Mach number regimes indicate different flow characteristics:

• Subsonic: \(M < 1\)
• Transonic: \(M \approx 1\)
• Supersonic: \(M > 1\)
• Hypersonic: \(M > 5\)

In this exercise, the initial supersonic Mach number is \(M_1 = 2.0\), which is greater than \(M = 1.2\), highlighting a transition zone involving a normal shock.