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Describe isentropic pressure?
Isentropic pressure refers to the pressure that results from a thermodynamic process that occurs at constant entropy. In other words, the entropy of a system does not change during an isentropic process, and therefore the pressure changes in a specific manner based on the initial and final states of the system. Isentropic processes are typically adiabatic, meaning that no heat is added or removed from the system during the process. These processes can occur in many different forms, such as in a compressor, a nozzle, or a thermodynamic cycle. In each of these cases, the isentropic pressure change can be calculated using the first and second laws of thermodynamics, along with the specific conditions of the process.
a. The inlet to the compressor can be analyzed using the isentropic relationship between pressure and temperature. Given the ambient conditions of P = 11.0 kPa and T = 216.7 K, the stagnation pressure and temperature can be found using the equation:
P0 = P * (1 + (γ - 1) / 2 * M^2)^(γ / (γ - 1))
where γ = 1.4 is the ratio of specific heats for air.
Solving for P0, we find that the stagnation pressure is 22.0 kPa. To find the stagnation temperature, we use the equation:
T0 = T * (1 + (γ - 1) / 2 * M^2)
Solving for T0, we find that the stagnation temperature is 533.3 K.
Next, the pressure ratio of the compressor is given as 10 and the isentropic efficiency is 90%, so the actual pressure rise can be calculated as:
P2/P1 = η * P2/P1,isentropic
where η is the isentropic efficiency. Solving for P2, we find that the pressure at the exit of the compressor is 86.1 kPa.
The temperature rise in the compressor can be calculated using the equation:
T2 = T1 * (P2 / P1)^((γ - 1) / γ)
Solving for T2, we find that the temperature at the exit of the compressor is 793.3 K.
b. In the combustor, the velocities are low, so the stagnation and static pressures are equal, and the stagnation pressure falls by 5%. The fall in stagnation pressure can be expressed as:
P0,comb = P0 * (1 - ΔP / P0)
where ΔP is the fall in stagnation pressure. Solving for P0,comb, we find that the stagnation pressure in the combustor is 20.38 kPa.
Next, the stagnation temperature at turbine entry is given as 1400 K, and the turbine efficiency is 90%. The temperature at turbine exit can be found using the equation:
T3 = T2 * ηturbine
Solving for T3, we find that the temperature at turbine exit is 996.7 K.
The pressure at turbine exit can be found using the equation:
P3 = P2 * (T3 / T2)^(γ / (γ - 1))
Solving for P3, we find that the pressure at turbine exit is 0.212 MPa.
c. The velocity of the jet can be found by finding the velocity of an isentropic expansion from the stagnation conditions at the exit of the propulsive nozzle to the ambient conditions. The velocity can be found using the equation:
V = (2 * (γ / (γ - 1)) * R * T0 * (1 - (P / P0)^((γ - 1) / γ)))^0.5
where R is the specific gas constant for air. Solving for V, we find that the velocity of the jet is 1069 m/s.
The gross thrust per unit mass flow can be found using the equation:
Fa = V * (1 - (P / P0))
Solving for Fa, we find that the gross thrust per unit mass flow is 1069 N kg
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Solving for Fa, we have find that the gross thrust per unit mass flow is 1069 N·kg.
Describe isentropic pressure?
Isentropic pressure refers to the pressure that results from a thermodynamic process that occurs at constant entropy. In other words, the entropy of a system does not change during an isentropic process, and therefore the pressure changes in a specific manner based on the initial and final states of the system. Isentropic processes are typically adiabatic, meaning that no heat is added or removed from the system during the process. These processes can occur in many different forms, such as in a compressor, a nozzle, or a thermodynamic cycle. In each of these cases, the isentropic pressure change can be calculated using the first and second laws of thermodynamics, along with the specific conditions of the process.
a. The inlet to the compressor can be analyzed using the isentropic relationship between pressure and temperature. Given the ambient conditions of P = 11.0 kPa and T = 216.7 K, the stagnation pressure and temperature can be found using the equation:
P0 = P * (1 + (γ - 1) / 2 * M^2)^(γ / (γ - 1))
where γ = 1.4 is the ratio of specific heats for air.
Solving for P0, we find that the stagnation pressure is 22.0 kPa. To find the stagnation temperature, we use the equation:
T0 = T * (1 + (γ - 1) / 2 * M^2)
Solving for T0, we find that the stagnation temperature is 533.3 K.
Next, the pressure ratio of the compressor is given as 10 and the isentropic efficiency is 90%, so the actual pressure rise can be calculated as:
P2/P1 = η * P2/P1,isentropic
where η is the isentropic efficiency. Solving for P2, we find that the pressure at the exit of the compressor is 86.1 kPa.
The temperature rise in the compressor can be calculated using the equation:
T2 = T1 * (P2 / P1)^((γ - 1) / γ)
Solving for T2, we find that the temperature at the exit of the compressor is 793.3 K.
b. In the combustor, the velocities are low, so the stagnation and static pressures are equal, and the stagnation pressure falls by 5%. The fall in stagnation pressure can be expressed as:
P0,comb = P0 * (1 - ΔP / P0)
where ΔP is the fall in stagnation pressure. Solving for P0,comb, we find that the stagnation pressure in the combustor is 20.38 kPa.
Next, the stagnation temperature at turbine entry is given as 1400 K, and the turbine efficiency is 90%. The temperature at turbine exit can be found using the equation:
T3 = T2 * ηturbine
Solving for T3, we find that the temperature at turbine exit is 996.7 K.
The pressure at turbine exit can be found using the equation:
P3 = P2 * (T3 / T2)^(γ / (γ - 1))
Solving for P3, we find that the pressure at turbine exit is 0.212 MPa.
c. The velocity of the jet can be found by finding the velocity of an isentropic expansion from the stagnation conditions at the exit of the propulsive nozzle to the ambient conditions. The velocity can be found using the equation:
V = (2 * (γ / (γ - 1)) * R * T0 * (1 - (P / P0)^((γ - 1) / γ)))^0.5
where R is the specific gas constant for air. Solving for V, we find that the velocity of the jet is 1069 m/s.
The gross thrust per unit mass flow can be found using the equation:
Fa = V * (1 - (P / P0))
Solving for Fa, we find that the gross thrust per unit mass flow is 1069 N kg
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