Problem 48 - Aircraft Propulsion

Question

In an adiabatic flow of a perfect gas in a constantarea duct with friction, we have the inlet condition as \(M_{1}=\) \(2.0, p_{1}=25 \mathrm{kPa}, T_{1}=250 \mathrm{~K}\). The duct length-to-(hydraulic) diameter ratio is \(L D_{h}=10.262\) and the average friction coefficient at the wall is \(C_{f}=0.005\). Assuming gas properties are \(\gamma=1.4\) and \(R=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), calculate (a) exit Mach number, \(M_{2}\) (b) total temperature of the gas at the exit, \(T_{12}\) in \(\mathrm{K}\) (c) total pressure of the gas at the exit, \(p_{12}\) in \(\mathrm{kPa}\)

Step-by-Step Solution

Answer

The solutions involve the calculation of the exit Mach number as well as the total temperature and pressure of the gas at the exit. Each of these parameters requires knowledge of the initial conditions, the compressibility factor, and the application of the Fanno line equation.

Step 1: Calculate the parameters for inlet condition

First, we need to calculate the compressibility factor \(Z\) at inlet condition, using the given information. In this case, \(Z = \gamma M_{1}^2 = 1.4 \times (2.0)^2 = 5.6\). Next, we can calculate the total temperature at inlet, \(T_{01}\) using the formula \(T_{01} = T_{1}Z = 250K \times 5.6 = 1400K\). Similarly, the total pressure at inlet, \(p_{01}\), can be calculated as \(p_{01} = p_{1}Z = 25kPa \times 5.6 = 140kPa.\)

Step 2: Establish the Fanno line equation

The Fanno line equation for adiabatic flow given by: \(Z - 1 = \frac{4fLD_{h}}{D \pi^2} \left(M^2_{2} - 1\right)\). Here \(f = C_{f} = 0.005\) is the friction coefficient, \(L = 10.262\times D_{h}\) is the length of the duct, and \(D\) represents diameter of the duct, and \(Z\) is the compressibility factor we calculated earlier. We can substitute these values into the equation to solve for \(M_{2}\), the exit Mach number.

Step 3: Calculate the exit Mach number

Upon substituting the values into the Fanno line equation, you will obtain a quadratic equation. Solve for \(M_{2}\), considering only the physically meaningful solution.

Step 4: Determine the exit parameters

To compute \(T_{12}\), and \(p_{12}\), the total temperature and pressure at the exit respectively, we use the formulas: \(T_{12} = T_{01} / Z_2\) and \(p_{12} = p_{01} / Z_2\), where \(Z_2 = \gamma M_{2}^2\).

Step 5: Final checks

Finally, evaluate your solutions for \(M_{2}\), \(T_{12}\), and \(p_{12}\). Ensure they are within logical bounds. This means \(M_{2}\) should be real and positive and \(T_{12}\), \(p_{12}\) values should not be less than absolute zero or the initial pressure respectively.

Key Concepts

Mach Number Calculation

In fluid dynamics, the Mach number is a crucial parameter that indicates the speed of a flow relative to the speed of sound in the medium. A Mach number:

• Less than 1 indicates subsonic flow.
• Exactly 1 is sonic flow.
• Greater than 1 signifies supersonic flow.

Here, to determine the exit Mach number, we must take into account the changes from the inlet to the outlet due to friction within the duct. The Mach number calculation at the inlet is given by the exercise and is utilized in conjunction with the Fanno line equation to determine the exit Mach number. The Mach number not only influences the compressibility factor but also drives changes in entropy and pressure along the flow. By solving the Fanno line equation for the exit condition, we find the Mach number after the effects of duct friction have been accommodated.

Fanno Line Equation

The Fanno line equation describes the relationship between various flow properties in an adiabatic flow within a duct where friction is present. It is expressed mathematically as: \[Z - 1 = \frac{4fLD_{h}}{D \pi^2} (M^2_{2} - 1)\]This equation accounts for:

• The compressibility factor \(Z\).
• The friction coefficient \(f\).
• The duct length \(L\) and diameter \(D\).
• The change in the square of the Mach number \(M^2\).

In this context, the equation is arranged to solve for the exit Mach number \(M_2\), factoring in the compressibility effects and frictional forces. Understanding this equation is key to determining how real gases behave under varying conditions within a confined geometry.

Compressibility Factor

The compressibility factor \(Z\) is a dimensionless number that gives insights into how the physical behavior of a real gas deviates from that of an ideal gas. In the context of adiabatic flow, often assuming perfect gases, it is calculated at the inlet as: \[Z = \gamma M^2\] where \(\gamma\) is the specific heat ratio and \(M\) is the Mach number. The compressibility factor helps in bridging the inlet and outlet states in a duct flow, reflecting the effects of compressibility. For gases with high Mach numbers, \(Z\) becomes significant, reflecting substantial changes in temperature, pressure, and density.

Total Temperature and Pressure

Total temperature and total pressure are pivotal in analyzing duct flow problems. They signify state functions that remain constant along an ideal streamline in adiabatic flows but alter due to friction in real scenarios.

• Total temperature at the inlet is calculated as: \[T_{01} = T_{1} Z\], reflecting how friction impacts energy dissipation.
• Likewise, total pressure is \[p_{01} = p_{1} Z\], showing the impact of velocity and pressure changes.

For the outlet, these are recomputed considering the exit compressibility factor \(Z_2\). Solving these provides insight into the energy conversions and losses along the flow path.

Friction Coefficient in Ducts

The friction coefficient \(C_f\) is an essential parameter in determining how flow behaves in a duct. Defined as a measurement of resistance to flow due to the duct surface's roughness and other factors, it has significant effects on the flow parameters.

• A higher \(C_f\) means more energy is lost to friction, translating into reduced velocity or higher pressure drop.
• In adiabatic flows, the friction coefficient takes center stage in the Fanno line equation, impacting the calculation of parameters like Mach number at exit.

Understanding and calculating the friction coefficient is paramount in designing duct systems for optimal performance, ensuring minimal energy losses while maintaining desired flow conditions.