A small rocket motor, fueled with hydrogen and oxygen, is tested on a thrust stand at a simulated altitude of 10 km. The motor is operated at chamber stagnation conditions of 1500 K and 8.0 MPa (gage). The combustion product is water vapor, which may be treated as an ideal gas. Expansion occurs through a converging-diverging nozzle with design Mach number of 3.5 and exit area of 700 mm². Find the pressure at the nozzle exit plane. Calculate the mass flow rate of exhaust gas. Determine the force exerted by the rocket motor on the thrust stand.
To find the pressure at the nozzle exit plane, we can use the isentropic flow relations for an ideal gas. We can use the stagnation conditions of the motor's chamber, which are given as 1500 K and 8.0 MPa (gage).
Step 1: Convert the pressure to absolute pressure:
P1 = 8.0 MPa + atmospheric pressure
= 8.0 MPa + 101.325 kPa (assuming standard atmospheric pressure)
= 9089.325 kPa
Step 2: Calculate the sonic velocity at the chamber stagnation conditions:
Using the equation for the speed of sound in an ideal gas:
a1 = sqrt(gamma * R * T1)
= sqrt(1.4 * 287 * 1500)
≈ 405.96 m/s
Step 3: Calculate the area ratio (A/A*) at the nozzle exit:
M2 = 3.5 (given)
A2 = 700 mm² = 700 * 10^(-6) m²
A* = A2 / M2
= (700 * 10^(-6)) / 3.5
= 200 * 10^(-6) m²
Step 4: Calculate the pressure at the nozzle exit plane:
P2 = P1 * ((1 + ((gamma - 1) / 2) * M1^2)^(gamma / (gamma - 1)) / ((1 + ((gamma - 1) / 2) * M2^2)^(gamma / (gamma - 1))))
= 9089.325 * ((1 + ((1.4 - 1) / 2) * (3.5)^2)^(1.4 / (1.4 - 1)) / ((1 + ((1.4 - 1) / 2) * (1)^2)^(1.4 / (1.4 - 1))))
≈ 3776.42 Pa
The pressure at the nozzle exit plane is approximately 3776.42 Pa.
To calculate the mass flow rate of the exhaust gas, we can use the ideal gas law and the known conditions.
Step 1: Calculate the density of the exhaust gas:
Using the ideal gas law: PV = mRT
where P is the pressure, V is the volume, m is the mass, R is the specific gas constant, and T is the temperature.
ρ = P / (RT)
= 3776.42 / (287 * 1500)
≈ 0.011 kg/m³
Step 2: Calculate the mass flow rate:
The mass flow rate is given by the equation m_dot = ρ * A * V
where ρ is the density, A is the nozzle exit area, and V is the velocity.
Using the continuity equation: A1 * V1 = A2 * V2
V2 = (A1 / A2) * V1
= (700 * 10^(-6) / (200 * 10^(-6))) * 405.96
≈ 1423.66 m/s
m_dot = ρ * A2 * V2
= 0.011 * (200 * 10^(-6)) * 1423.66
≈ 0.00309 kg/s
The mass flow rate of the exhaust gas is approximately 0.00309 kg/s.
To determine the force exerted by the rocket motor on the thrust stand, we can use Newton's third law of motion, which states that for every action, there is an equal and opposite reaction.
The force exerted by the rocket motor is equal to the rate of change of momentum of the exhaust gas, which can be calculated using the equation F = m_dot * V2.
F = 0.00309 * 1423.66
≈ 4.399 N
The force exerted by the rocket motor on the thrust stand is approximately 4.399 N.
To solve this problem, we need to use the principles of thermodynamics and gas dynamics. Here's how you can find the pressure at the nozzle exit plane, calculate the mass flow rate of exhaust gas, and determine the force exerted by the rocket motor on the thrust stand:
Step 1: Find the exit Mach number (Mₑ):
The exit Mach number can be calculated using the design Mach number (Mₔd) and the ratio of specific heats (γ).
Mₑ = Mₔd / √(1 + (γ-1)/2 * Mₔd^2)
Step 2: Find the exit temperature (Tₑ):
The exit temperature can be calculated using the chamber stagnation temperature (Tₛ) and the exit Mach number (Mₑ) by the following equation:
Tₑ = Tₛ / (1 + (γ-1)/2 * Mₑ^2)
Step 3: Find the exit pressure (Pₑ):
To find the pressure at the nozzle exit plane, we will use the isentropic flow relations:
Pₑ = Pₛ / (1 + (γ-1)/2 * Mₑ^2)^(γ/(γ-1))
Step 4: Calculate the mass flow rate (ṁ):
The mass flow rate can be calculated using the ideal gas law:
ṁ = Aₑ * ρₑ * Vₑ
where Aₑ is the exit area, ρₑ is the density of the exhaust gas, and Vₑ is the exit velocity. Assuming the exhaust gas can be treated as an ideal gas, ρₑ can be calculated using the ideal gas law:
ρₑ = Pₑ / (R * Tₑ)
where R is the specific gas constant for water vapor.
Step 5: Find the exit velocity (Vₑ):
The exit velocity can be calculated using the ideal gas law and the specific heat capacity at constant pressure (Cp):
Vₑ = √(γ * R * Tₑ)
Step 6: Calculate the force exerted by the rocket motor (F):
The force can be calculated using Newton's second law of motion:
F = ṁ * Vₑ + (Pₑ - P₀) * Aₑ
where P₀ is the surrounding atmospheric pressure.
Now, let's plug in the given values:
- Chamber stagnation conditions (Tₛ): 1500 K
- Chamber stagnation pressure (Pₛ): 8.0 MPa
- Design Mach number (Mₔd): 3.5
- Exit area (Aₑ): 700 mm²
Using these values and the equations mentioned above, you can find the pressure at the nozzle exit plane, the mass flow rate of the exhaust gas, and the force exerted by the rocket motor on the thrust stand.
I have helped you with two of these nozzle flow problems already and would like to see you make an effort. I suggest you use isentropic flow tables and a specific heat ratio of 1.3 for water vapor, but you should confirm that. There is a formula for the thrust that involves the mass flow rate the exit velocity, as well as P(exit) and
P(ambient). You can get P(ambient) from the simulated altitude.