Problem 4 - Aircraft Propulsion

Question

A rocket engine has a chamber pressure of \(p_{\mathrm{c}}=1000\) psia and the throat area is \(A_{\mathrm{th}}=1.5 \mathrm{ft}^{2}\). Assuming that the nozzle is perfectly expanded with the gas ratio of specific heats \(\gamma=1.2\) and the ambient pressure of \(p_{0}=14.7 \mathrm{psia}\), calculate (a) optimum thrust coefficient \(C_{\mathrm{F}, \mathrm{opt}}\) (b) thrust \(F\) in lbf (c) nozzle exit Mach number \(M_{2}\) (d) nozzle area expansion ratio \(A_{2} / A_{\mathrm{th}}\)

Step-by-Step Solution

Answer

The short answers are: (a) Optimum thrust coefficient \(C_{F, opt}\) will be calculated as per Step 1 (b) Thrust \(F\) will be calculated as per Step 2 (c) Nozzle exit Mach number \(M_{2}\) will be calculated as per Step 3 (d) Nozzle area expansion ratio \(A_{2} / A_{th}\) will be calculated as per Step 4 Specific numerical values depend on the calculations done in each step.

Step 1: Calculate the Optimum Thrust Coefficient

The optimum thrust coefficient \(C_{F,opt}\) is calculated using the formula: \[ C_{F,opt} = \sqrt{2 \gamma^{2} / (\gamma - 1) *(1 - (p_{0}/p_{c})^{(\gamma - 1)/\gamma})}\]Substitute the values \(\gamma = 1.2\), \(p_{0} = 14.7 \, \mathrm{psia}\), and \(p_{c} = 1000 \, \mathrm{psia}\) to calculate \(C_{F,opt}\).

Step 2: Calculate the Thrust

The thrust \(F\) is calculated using the formula: \[ F = C_{F,opt} * p_{c} * A_{th}\]Substitute the values \(C_{F,opt}\) (calculated in previous step), \(p_{c} = 1000 \, \mathrm{psia}\), and \(A_{th} = 1.5 \, \mathrm{ft}^{2}\) to calculate \(F\). Convert psia to lbf/ft² when calculating.

Step 3: Calculate the Nozzle Exit Mach Number

The nozzle exit Mach number \(M_{2}\) is calculated using the formula: \[ M_{2} = \sqrt{(2/(\gamma - 1)) * ((p_{c}/p_{0})^{(\gamma - 1)/\gamma} - 1)}\]Substitute the values \(\gamma = 1.2\), \(p_{c} = 1000 \, \mathrm{psia}\), and \(p_{0} = 14.7 \, \mathrm{psia}\) to calculate \(M_{2}\).

Step 4: Calculate the Nozzle Area Expansion Ratio

The nozzle area expansion ratio \(A_{2} / A_{th}\) is calculated using the formula: \[ A_{2} / A_{th} = (1 / \gamma) * (2 / (\gamma + 1))^{(\gamma + 1) / (2(\gamma - 1))} * M_{2}\]Substitute the values \(\gamma = 1.2\) and \(M_{2}\) (calculated in previous step) to calculate \(A_{2}/A_{th}\).

Key Concepts

Thrust Calculation

Thrust in rocketry refers to the force produced by the rocket engine to move the rocket forward. It's essentially the push that propels the vehicle. Calculating thrust involves using the formula:

• \[ F = C_{F,opt} \times p_{c} \times A_{th} \]

Here, - \( C_{F,opt} \) is the optimum thrust coefficient,- \( p_{c} \) represents the chamber pressure, - and \( A_{th} \) is the throat area.Convert the chamber pressure from \( \, \) psia to \( \, \) lbf/ft² for accurate results.This formula helps in understanding how different parameters affect the thrust produced by the rocket engine. By modifying the chamber pressure or the throat area, engineers can design different engines to meet specific mission requirements.

Nozzle Expansion Ratio

The nozzle expansion ratio is the ratio between the exit area of the nozzle and the throat area. The formula is:

• \[ A_2 / A_{th} = (1 / \gamma) \times (2 / (\gamma + 1))^{(\gamma + 1) / (2(\gamma - 1))} \times M_2 \]

This ratio impacts how efficiently the gas expands in the nozzle and transforms thermal energy into kinetic energy.

• A larger expansion ratio allows for higher conversion efficiency at the cost of increased nozzle size and weight.
• A smaller ratio could lead to inefficient expansion.

Choosing the optimum expansion ratio is crucial for maximizing the efficiency and effectiveness of the rocket engine during its flight mission.

Mach Number

The Mach number is a dimensionless unit that describes the speed of an object compared to the speed of sound. In rocketry, it's used to analyze flow speeds in the nozzle, especially at the exit:

• \[ M_2 = \sqrt{\left(\frac{2}{\gamma - 1}\right) \times \left(\left(\frac{p_{c}}{p_{0}}\right)^{(\gamma - 1)/\gamma} - 1\right)} \]

Here, \( \gamma \) is the ratio of the specific heats, \( p_{c} \) is the chamber pressure, and \( p_{0} \) is the ambient pressure.Understanding the Mach number in the rocket's nozzle helps in designing nozzles to ensure that the flow is optimized for maximum exit velocity and, hence, maximum thrust.

Specific Heats

Specific heats are critical for understanding thermodynamics in gases used in rocket propulsion. The ratio of specific heats, \( \gamma \), is defined as:

• \( \gamma = c_p / c_v \)

Where \( c_p \) is specific heat at constant pressure, and \( c_v \) is specific heat at constant volume.This ratio affects:

• The speed of sound in the gas.
• Performance and efficiency of rocket nozzles.

A higher \( \gamma \) often indicates that the gas can expand more effectively, contributing to producing thrust more efficiently. It's a key parameter in determining how the rocket engine will perform under various conditions.

Thrust Coefficient

The thrust coefficient, \( C_{F,opt} \), is an essential parameter that relates the actual thrust produced to the theoretical maximum thrust:

• \[ C_{F,opt} = \sqrt{\frac{2 \gamma^2}{\gamma - 1} \times \left(1 - \left(\frac{p_{0}}{p_{c}}\right)^{(\gamma - 1)/\gamma}\right)}\]

This coefficient informs engineers how well the rocket engine uses the available energy to produce thrust. A higher thrust coefficient suggests that the engine is performing efficiently and that the nozzle design is optimal for converting pressure into motion. Adjustments in \( \gamma \), \( p_{c} \), and \( p_{0} \) can all impact this coefficient, affecting the engine's overall thrust capability.