To calculate the thrust generated by the engine, we will use the following formula for thrust in an isentropic flow through a nozzle:
\[ F = \dot{m} \cdot V_e + (P_e - P_0) \cdot A_e \]
Where:
- \( F \) is the thrust,
- \( \dot{m} \) is the mass flow rate,
- \( V_e \) is the exit velocity of exhaust gases,
- \( P_e \) is the exit pressure,
- \( P_0 \) is the ambient pressure,
- \( A_e \) is the exit area.
Step 1: Calculate the mass flow rate (\( \dot{m} \))
The mass flow rate can be calculated using the equation:
\[ \dot{m} = P_0 \cdot A_t \cdot \frac{y}{R \cdot T} \]
Where:
- \( A_t \) is the throat area,
- \( y \) is the specific heat ratio,
- \( R \) is the specific gas constant,
- \( T \) is the temperature in the combustion chamber.
Assuming that the throat area is \( A_t = 0.05 , m^2 \) (since it was not provided, we will use this as an approximation).
First, we need to calculate the exit velocity \( V_e \) using the isentropic relation:
Step 2: Calculate the exit velocity (\( V_e \))
Using the isentropic flow equations, we can find the exit velocity. The exit velocity can be expressed as:
\[ V_e = \sqrt{2 \cdot R \cdot T \cdot \left(1 - \left(\frac{P_e}{P_c}\right)^{\frac{y-1}{y}}\right)} \]
Given values:
- \( P_c = 9.7 , \text{MPa} = 9700000 , \text{Pa} \)
- \( T = 3685 , \text{K} \)
- \( P_e = P_0 = 1 , \text{atm} = 101325 , \text{Pa} \)
Calculate the ratio:
\[ \frac{P_e}{P_c} = \frac{101325}{9700000} \approx 0.01045 \]
Now, substitute these into the equation for exit velocity:
\[ V_e = \sqrt{2 \cdot 378 \cdot 3685 \cdot \left(1 - \left(0.01045\right)^{\frac{1.14-1}{1.14}}\right)} \]
First, calculate \( \left(0.01045\right)^{\frac{0.14}{1.14}} \):
\[ = 0.01045^{0.1228} \approx 0.4185 \]
Now substitute this value in:
\[ V_e = \sqrt{2 \cdot 378 \cdot 3685 \cdot (1 - 0.4185)} \approx \sqrt{2 \cdot 378 \cdot 3685 \cdot 0.5815} \]
Calculate the inner term:
\[ = 2 \cdot 378 \cdot 3685 \cdot 0.5815 \approx 1.5904 \times 10^6 \]
Finally, calculate \( V_e \):
\[ V_e = \sqrt{1590400} \approx 1261.12 , \text{m/s} \]
Step 3: Calculate the mass flow rate (\( \dot{m} \))
Using the mass flow equation:
\[ \dot{m} = \frac{P_c \cdot A_t}{R \cdot T} = \frac{9700000 \cdot 0.05}{378 \cdot 3685} \]
Calculating this gives:
\[ \dot{m} \approx \frac{485000}{1383790} \approx 0.350 , \text{kg/s} \]
Step 4: Calculate the thrust (\( F \))
Using the thrust equation:
\[ F = \dot{m} \cdot V_e + (P_e - P_0) \cdot A_e \]
Substituting the values calculated into this equation:
\[ F = 0.350 \cdot 1261.12 + (101325 - 101325) \cdot A_e \]
The second term becomes zero, leading to:
\[ F = 0.350 \cdot 1261.12 = 442.39 , \text{N} = 0.442 , \text{kN} \]
Final Result
Thus, the thrust generated by one engine is approximately 0.442 kN.