Self-Assessment Quiz: Nozzle Theory and Thermodynamics

Twenty questions on the de Laval nozzle, compressible flow, choked flow, exhaust velocity, and expansion ratio. Answer each before opening the key. Aim for 16 or more. Use $g_0 = 9.81\ \text{m/s}^2$ and $R_u = 8.314\ \text{J/(mol·K)}$.

Question 1

The diverging section of a de Laval nozzle exists in order to:

A) slow the exhaust down before it leaves B) accelerate the already-supersonic flow to higher Mach number C) increase the chamber pressure D) cool the gas before combustion

Question 2

The area–velocity relation is $\dfrac{dA}{A} = (M^2 - 1)\dfrac{dV}{V}$. For subsonic flow ($M < 1$), to speed the gas up you must:

A) widen the duct B) narrow the duct C) keep the area constant D) add heat

Question 3

Mach 1 in a nozzle can occur only:

A) at the exit B) in the combustion chamber C) at the throat (minimum area) D) anywhere in the diverging section

Question 4

"Choked flow" means:

A) the nozzle is blocked by debris B) the flow has reached $M = 1$ at the throat and mass flow no longer depends on back pressure C) the exhaust has stopped D) the chamber pressure has dropped to ambient

Question 5

Once a nozzle is choked, lowering the ambient (back) pressure further will:

A) increase the mass flow rate B) decrease the mass flow rate C) not change the mass flow rate D) reverse the flow

Question 6

The ideal exhaust velocity scales with chamber temperature and molar mass as:

A) $v_e \propto T_c / \mathcal{M}$ B) $v_e \propto \sqrt{T_c / \mathcal{M}}$ C) $v_e \propto \sqrt{\mathcal{M} / T_c}$ D) $v_e \propto \mathcal{M} \, T_c$

Question 7

LOX/LH2 gives a higher specific impulse than LOX/RP-1 mainly because hydrogen exhaust has:

A) a higher flame temperature B) a lower exhaust molar mass C) a higher chamber pressure D) a larger expansion ratio

Question 8

The expansion ratio of a nozzle is defined as:

A) $A_t / A_e$ B) $A_e / A_t$ C) $p_c / p_e$ D) $T_c / T_e$

Question 9

A nozzle whose exit pressure is below the ambient pressure ($p_e < p_a$) is:

A) under-expanded B) perfectly expanded C) over-expanded D) choked

Question 10

An upper-stage engine uses a much larger expansion ratio than a first-stage engine because:

A) it needs more thrust at liftoff B) in vacuum it can expand to very low $p_e$ without flow separation, gaining $v_e$ C) hydrogen requires a bigger throat D) the chamber is colder

Question 11

The Mach number is defined as:

A) the flow speed divided by the local speed of sound B) the flow speed divided by $g_0$ C) the exit pressure divided by chamber pressure D) the area ratio

Question 12

Isentropic flow assumes the flow is:

A) adiabatic and reversible (constant entropy) B) constant pressure C) constant velocity D) incompressible

Question 13

Chamber conditions ($T_c$, $p_c$) are treated as the stagnation conditions because:

A) the gas is at rest in the chamber B) the chamber is the coldest point C) combustion has not finished D) the pressure is lowest there

Question 14

For a supersonic flow ($M > 1$), a converging duct will:

A) accelerate it B) decelerate it C) leave its speed unchanged D) choke it

Question 15 (True/False, justify)

"A longer nozzle (larger expansion ratio) always produces more thrust, at every altitude." True or false? Justify in one sentence.

Question 16 (True/False, justify)

"Because kerosene burns hotter than hydrogen, a kerosene engine has a higher ideal exhaust velocity." True or false? Explain briefly.

Question 17 (True/False, justify)

"The nozzle accelerates the exhaust mainly by friction between the gas and the nozzle wall." True or false? Say why.

Question 18 (Short answer)

In one or two sentences, explain why a purely converging nozzle (like a garden-hose nozzle) can never produce supersonic exhaust, no matter how high the chamber pressure.

Question 19 (Short answer)

A nozzle expands from $p_c = 100\ \text{bar}$ to $p_e = 0.1\ \text{bar}$ with $\gamma = 1.20$, $T_c = 3600\ \text{K}$, $\mathcal{M} = 0.013\ \text{kg/mol}$. Write the exhaust-velocity formula and estimate $v_e$ (the chapter computed it).

Question 20 (Short answer)

In your own words, describe what "altitude compensation" is and why no single fixed nozzle is optimal for an entire launch from sea level to vacuum.

Answer Key

Q Ans Note
1 B Supersonic flow accelerates only in a diverging duct — the whole point of the bell.
2 B $M<1$: $(M^2-1)<0$, so shrinking area speeds the flow up (garden hose).
3 C $dA=0$ at $M=1$; the accelerating-nozzle extremum is the minimum-area throat.
4 B Sonic throat; mass flow set by $p_c, A_t, T_c$ alone.
5 C Back-pressure information can't travel upstream past a sonic throat.
6 B From $v_e = \sqrt{\frac{2\gamma}{\gamma-1}\frac{R_u T_c}{\mathcal{M}}[\dots]}$.
7 B Light molecules move faster at a given temperature; $v_e \propto 1/\sqrt{\mathcal{M}}$.
8 B Exit area over throat area.
9 C $p_e
10 B Vacuum allows low $p_e$ with no separation, capturing more $v_e$.
11 A $M = V/a$, with $a=\sqrt{\gamma R T}$ local.
12 A Adiabatic + reversible = constant entropy.
13 A Near-zero chamber velocity means static ≈ stagnation.
14 B $M>1$: $(M^2-1)>0$, shrinking area decelerates (supersonic diffuser).
15 False Only true up to separation; an over-expanded long nozzle at sea level loses thrust and can separate destructively.
16 False $v_e \propto \sqrt{T_c/\mathcal{M}}$; hydrogen's far lower $\mathcal{M}$ beats kerosene's slightly higher $T_c$.
17 False Acceleration comes from the pressure gradient (the gas expanding); friction only slows it. The wall just sets the area.
18 — A converging duct can raise the flow only to $M=1$ at its narrowest point; going supersonic requires a diverging section afterward. Extra chamber pressure raises mass flow, not exit Mach past 1.
19 — $v_e = \sqrt{\frac{2(1.2)}{0.2}\cdot\frac{8.314(3600)}{0.013}\,[1-(0.001)^{0.1667}]} = \sqrt{12 \cdot 639.5 \cdot 3600 \cdot 0.684} \approx 4346\ \text{m/s}$.
20 — Ambient pressure falls from ~1 bar to 0 during ascent, but a fixed nozzle expands to one $p_e$; it is perfectly expanded at only one altitude, over-expanded below and under-expanded above. Altitude compensation is the (unsolved-in-practice) goal of a nozzle that adjusts to ambient.

Topics to review by question

Questions Topic Section
1, 2, 3, 14, 18 Area–velocity relation & nozzle shape §19.1–19.2
11, 12, 13 Mach number & isentropic flow §19.2–19.3
4, 5 Choked flow §19.3
6, 7, 16, 19 Exhaust velocity & molar-mass effect §19.4, §19.6
8, 9, 10, 15, 20 Expansion ratio & altitude compensation §19.5
17 How a nozzle really accelerates gas §19.2