Problem 8 - Aircraft Propulsion

Question

A rocket has a mass ratio of \(\mathrm{MR}=0.10\) and a mean specific impulse of \(365 \mathrm{~s}\). The flight trajectory is described by a constant dynamic pressure of \(q_{0}=50 \mathrm{kPa}\). The mean drag coefficient is approximated to be \(0.25\), the vehicle initial mass is \(m_{0}=100,000 \mathrm{~kg}\), and the vehicle (maximum) frontal cross- sectional area \(A_{\mathrm{f}}\) is \(5 \mathrm{~m}^{2}\). For a burn time of \(100 \mathrm{~s}\), calculate the rocket terminal speed while neglecting gravitational effect.

Step-by-Step Solution

Answer

The terminal speed of the rocket is 23,916 m/s.

Step 1: Calculate final mass

Using the mass ratio (MR), calculate the final mass of the rocket. By definition, MR = m_final / m_initial. Hence, the final mass (m_final) can be obtained by multiplying the mass ratio by the initial mass. Substitute MR = 0.1 and m_initial = 100,000 kg into the formula, the calculation for m_final becomes \(m_{final} = MR \times m_{initial} = 0.10 \times 100,000 = 10,000 kg\)

Step 2: Determine the change in velocity

Apply the rocket equation to calculate the change in velocity. The rocket equation is \(\Delta v = I_{sp} \times g \times \ln(\frac{m_{initial}}{m_{final}})\). Here, the specific impulse Isp = 365 s (which is a measure of the efficiency of the rocket), g = 9.81 m/s^2 (acceleration due to gravity), the initial mass m_initial = 100,000 kg, and the final mass m_final = 10,000 kg. Substitute these values into the equation, the change in velocity \(\Delta v = 365 \times 9.81 \times \ln(\frac{100,000}{10,000}) = 84,265 m/s\)

Step 3: Determine drag force

The drag force is determined using the dynamic pressure and drag coefficient. The drag equation is \(F_{drag} = q \times A_f \times Cd\), where dynamic pressure q = 50 kPa = 50000 Pa, the frontal cross-sectional area A_f = 5 m^2, and the drag coefficient Cd = 0.25. Substituting these values, the drag force becomes \(F_{drag} = 50000 \times 5 \times 0.25 = 62500 N\)

Step 4: Determine terminal speed

The terminal speed is found when the force from the rocket engine equals the drag force. We have the thrust force by multiplying the specific impulse by the value of g and the mass flow rate (mass ratio multiplied by the initial mass divided by burn time). Thus, \(F_{thrust} = I_{sp} \times g \times \frac{MR \times m_{0}}{t_{burn}} = 365 \times 9.81 \times \frac{0.10 \times 100,000}{100} = 357,735 N\). The terminal speed is determined by setting thrust force equal to drag force, then solving for the change in velocity (burn speed). We get: \(F_{thrust} = F_{drag} \Rightarrow 357,735 = 62500 + 0.5 \times 1.2 \times A_f \times v^2 \times Cd\). Solving the equation gives the terminal speed, v = 23916 m/s

Key Concepts

Mass Ratio

The mass ratio ( \( MR \) ) of a rocket is a crucial parameter in rocket propulsion. It determines how much of the rocket's initial mass is composed of propellant versus the rest of the rocket. Specifically, it's calculated using the formula:

• \( MR = \frac{m_{\text{final}}}{m_{\text{initial}}} \)

where \( m_{\text{final}} \) is the final mass of the rocket and \( m_{\text{initial}} \) is the initial mass. A mass ratio of 0.10, as given in the exercise, indicates that only 10% of the rocket's initial mass remains after the propellant is spent. This means 90% of the initial mass was used as fuel. This concept is vital for understanding how efficient a rocket engine is during flight. The lower the final mass concerning the initial mass, the more work the rocket can perform in altering its velocity.

Specific Impulse

Specific impulse ( \( I_{sp} \) ) measures the efficiency of a rocket engine. It's expressed in seconds and gives an idea of how much thrust is produced per unit of propellant. Essentially, it tells us how effectively a rocket uses its fuel. Specific impulse is calculated using the equation:

• \( I_{sp} = \frac{F_{thrust}}{\dot{m} \cdot g} \)

where \( F_{thrust} \) is the thrust force, \( \dot{m} \) is the mass flow rate (mass of fuel used per second), and \( g \) is acceleration due to gravity (approximately 9.81 m/s² on Earth). A specific impulse of 365 seconds, which is mentioned in the exercise, indicates a highly efficient engine, as the thrust remains for 365 seconds using a given amount of propellant. This measurement is key to determining the effectiveness of the rocket's fuel consumption.

Dynamic Pressure

Dynamic pressure ( \( q \) ) is an essential element in aerodynamics and refers to the pressure caused by the motion of the fluid (in this case, air) around the rocket. It is calculated using the equation:

• \( q = \frac{1}{2} \cdot \rho \cdot v^2 \)

where \( \rho \) represents the air density and \( v \) is the velocity of the rocket relative to the air. This pressure affects the forces experienced by the rocket, notably influencing the drag force. A constant dynamic pressure of 50 kPa in the exercise signifies a steady force acting on the rocket's surface as it travels through the atmosphere, impacting its speed and trajectory. Understanding dynamic pressure is crucial in designing rockets to withstand the stresses of launch and passage through the atmosphere.

Drag Coefficient

The drag coefficient ( \( C_d \) ) quantifies how streamlined an object is as it moves through a fluid, such as air. It influences the drag force acting on the rocket, which opposes motion. The drag force is calculated using:

• \( F_{drag} = q \cdot A_f \cdot C_d \)

where \( q \) is dynamic pressure, \( A_f \) is the frontal cross-sectional area, and \( C_d \) is the drag coefficient. A drag coefficient of 0.25, as mentioned in the exercise, suggests that the rocket is relatively aerodynamic. The smaller this value, the less resistance the rocket faces as it moves. Understanding the drag coefficient is crucial to minimizing energy losses and maximizing the efficiency of the rocket during ascent.