Thrust Developed by an Aircraft Propeller: 04-BS-7, December 2019, Q6

Question 6: Thrust Developed by an Aircraft Propeller (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

The value is consistent with a large propeller aircraft cruise speed well below the 650 km/hr slipstream exit speed the question also gives.

Given.

Quantity	Value
Propeller diameter, D	2 m
Aircraft (upstream) velocity, V₁	500 km/hr
Slipstream exit velocity, V₂	650 km/hr
Air density (Constants, p.12, 15°C)	1.21 kg/m³

Actuator-disk model: streamtube converges through the propeller as velocity rises from V1 to V2; thrust equals the momentum change of the mass flow through the disk.

Find. The thrust developed by the propeller.

Approach. Model the propeller as an actuator disk: the mass flow rate through the disk uses the AVERAGE of the far-upstream and far-downstream velocities (since the streamtube converges and the velocity at the disk itself is the mean of the two), and thrust is that mass flow rate times the overall velocity change.

Convert velocities and find the disk (mean) velocity. $$V_1=500\times\frac{1000}{3600}=138.9\ \text{m/s}\qquad V_2=650\times\frac{1000}{3600}=180.6\ \text{m/s}$$ $$V_{disk}=\frac{V_1+V_2}{2}=\boxed{159.7\ \text{m/s}}$$
Mass flow rate through the disk. $$A=\frac{\pi}{4}D^2=\frac{\pi}{4}(2)^2=3.1416\ \text{m}^2$$ $$\dot{m}=\rho A V_{disk}=(1.21)(3.1416)(159.7)=\boxed{607.2\ \text{kg/s}}$$
Thrust from the momentum change. $$F_{thrust}=\dot{m}(V_2-V_1)=(607.2)(180.6-138.9)=(607.2)(41.67)$$ $$\boxed{F_{thrust}=25.30\ \text{kN}}$$

Quantity	Value
V₁ / V₂	138.9 / 180.6 m/s
Disk (mean) velocity	159.7 m/s
Mass flow rate	607.2 kg/s
Thrust	25.30 kN

Concept & Theory

Topic. Propeller actuator-disk (momentum) theory.

Key relations. Continuity through a converging streamtube; disk velocity $V_{disk}=(V_1+V_2)/2$; thrust $F=\dot{m}(V_2-V_1)=\rho A_{disk}V_{disk}(V_2-V_1)$.

Why this works. Actuator-disk theory replaces the detailed blade geometry with a uniform pressure jump across an idealized disk; because the flow accelerates smoothly and continuously from far upstream to far downstream, the velocity actually AT the disk (which sets the mass flow rate through it) is the average of the two extremes, not either one alone — using $V_1$ or $V_2$ by itself would misstate the mass flow rate.

Common pitfall. Using the propeller (disk) area with only $V_1$ or only $V_2$ for the mass flow rate instead of the mean disk velocity.

Source. White, Fluid Mechanics, Ch. 11 (Propeller/actuator-disk theory).

Question 6 of 13: Thrust Developed by an Aircraft Propeller

Question 6: Thrust Developed by an Aircraft Propeller (5 marks)