Question

A rocket operates at an absolute chamber pressure of 20 bar to produce thrust \(F_1\). The hot exhaust is optimally expanded to 1 bar (absolute pressure) using a convergent-divergent nozzle with exit to throat area ratio \(\left(\frac{A_e}{A_t}\right)\) of 3.5 and thrust coefficient, \(C_{F,1} = 1.42\). The same rocket when operated at an absolute chamber pressure of 50 bar produces thrust \(F_2\) and the thrust coefficient is \(C_{F,2}\). Which of the following statement(s) is/are correct?

• A. \(\frac{F_2}{F_1} = 2.5\)
• B. \(\frac{F_2}{F_1}>2.5\)
• C. \(\frac{C_{F,2}}{C_{F,1}} = 1\)
• D. \(\frac{C_{F,2}}{C_{F,1}} > 1\)

Hint:

Thrust and thrust coefficient are influenced by chamber pressure and nozzle expansion. For a fixed nozzle expansion ratio, thrust increases with the square root of chamber pressure, while the thrust coefficient remains constant.

Answer 1

The Correct Option is B, D

Step 1: Relationship between chamber pressure and thrust.
The thrust \(F\) produced by a rocket is related to the chamber pressure \(P_c\) and exhaust conditions. The thrust coefficient \(C_F\) is defined as: \[ C_F = \frac{F}{P_c A_t} \] where \(A_t\) is the throat area of the nozzle. When the chamber pressure increases, the exhaust velocity increases, leading to higher thrust production.
Step 2: Comparison of thrusts \(F_1\) and \(F_2\).
For the first case, with a chamber pressure of 20 bar, the rocket produces thrust \(F_1\). For the second case, with a chamber pressure of 50 bar, the rocket will produce a thrust \(F_2\). The thrust is proportional to the square root of the chamber pressure: \[ \frac{F_2}{F_1} = \sqrt{\frac{P_{c2}}{P_{c1}}} = \sqrt{\frac{50}{20}} = \sqrt{2.5}>2.5. \] Thus, statement (B) is correct.
Step 3: Comparison of thrust coefficients \(C_{F,2}\) and \(C_{F,1}\).
The thrust coefficient \(C_F\) is primarily determined by the nozzle expansion and is influenced by the chamber pressure. Since the expansion ratios are the same for both cases, and assuming optimal expansion, the thrust coefficient remains constant across both cases: \[ C_{F,2} = C_{F,1}. \] Thus, \(\frac{C_{F,2}}{C_{F,1}} = 1\), making statement (C) correct.