# Aerodynamics Questions and Answers – Momentum Theory

This set of Aerodynamics Multiple Choice Questions & Answers (MCQs) focuses on “Momentum Theory”.

1. Who developed the momentum theory?
a) Daniel Bernoulli
b) Ludwig Prandtl
c) Osborne Reynolds
d) W.J.M. Rankine

Explanation: The momentum theory was developed by W.J.M. Ranking initially for marine propellers in the year 1865 along with R.E. Froude in the same year. This theory was later extended in 1920 by A.Betz who incorporated the slipstream rotation.

2. For momentum theory analysis, what is rotor modelled as?
a) Thin actuator disk
b) Thick actuator disk
d) Two dimensional airfoil

Explanation: For conduction momentum theory analysis of the rotor, it is modeled as an actuator disk. It is considered to be thin circular disk of negligible thickness which can support pressure difference and accelerate air. This works as an approximate.

3. The actual induced power loss result is less than the momentum theory result.
a) True
b) False

Explanation: While formulizing momentum theory, the rotor blade is modeled as an actuator disk which is only an approximation. The flow field in an actuator is assumed to be steady whereas in real life it isn’t. Hence, the actual induced power loss is higher than the calculated value from the momentum theory due to the presence of unsteady vortices.

4. What is the formula for induced velocity for a helicopter while it hovers?
a) vh=$$\frac {T}{2\rho A}$$
b) vh=$$\sqrt {\frac {T}{2\rho A}}$$
c) vh=$$\frac {T^2}{\rho A}$$
d) vh=2ρAT

Explanation: Momentum equation helps in giving us a relation between the thrust of the rotor and the induced velocity. It is given by:
T=2ρAvh2
Which is rearranged to obtain the induced velocity:
vh=$$\sqrt {\frac {T}{2\rho A}}$$
Where, T is the total thrust
A is area of actuator disk
vh is induced velocity at rotor disk.

5. In order to increase the hovering efficiency, which parameter must be taken care of?
a) High power by thrust ratio
b) High thrust by area ratio
c) Low induced drag
d) High rotor tip speed

Explanation: According to the momentum theory, the induced power by thrust ratio is given by:
$$\frac {P}{T}=\sqrt {\frac {T}{2\rho A}}$$
Thus, for low inflow velocity and low induced power loss, air must be accelerated through the rotor by virtue of pressure difference. For improving hovering efficiency, the value of P/T must be small thus improving fuel efficiency and decreasing engine’s weight.

a) True
b) False

Explanation: Based on the relation between induced power per unit thrust and the disk loading, $$\frac {P}{T}=\sqrt {\frac {T}{2\rho A}}$$ we notice that in order to have a small value of P/T which is a requirement for high hovering efficiency, we need to have a small value of T/A which is known as disk loading.

7. What is the ideal range of disk loading for high hover performance?
a) 10-100 $$\frac {N}{m^2}$$
b) 100-500 $$\frac {N}{m^2}$$
c) 1000-4000 $$\frac {N}{m^2}$$
d) 5000-10,000 $$\frac {N}{m^2}$$

Explanation: Based on the momentum theorem for the hovering helicopter, the disk loading must be of the minimum value to have a low value of power to thrust. Disk loading value generally ranges between 100-500 $$\frac {N}{m^2}$$ for the best performance.

8. Which of these is not an assumption made while formulating momentum theory for hover and clib of a helicopter?
a) Blade is modeled as an actuator disk
c) Smooth slipstream
d) Slipstream rotation velocity is negligible

Explanation: There are various assumptions made while formulating the momentum theory for the helicopter in hover and climb condition. These are-the blades are modeled to be a thin circular actuator disk, there is uniform loading over the disk, the slipstream is well defined, the rotational velocity in the slipstream is neglected and there is uniform induced velocity.

9. The momentum conservation equation for a helicopter in climb state is independent of which term?
a) Climb velocity
b) Induced velocity
c) Wake velocity
d) Mass flux

Explanation: The momentum equation for the helicopter in flight condition is given by:
T=$$\dot {m}$$(V+w)-$$\dot {m}$$V
T=$$\dot {m}$$w
Where, V is the climb velocity
w is the induced velocity in the wake region
T is the thrust
$$\dot {m}$$ is mass flux
On observing the equation, we see that the momentum equation is independent of the climb velocity.

10. What is the relation between the induced velocity in far wake region (w) and that at the rotor disk (v) for hovering condition?
a) w=v
b) w=v2
c) w=2v
d) w=v/2

Explanation: The momentum equation for the rotor is given by:
T=$$\dot {m}$$w
Which can be rearranged as:
$$\frac {T}{\dot {m}}$$=w (equation 1)
And the energy equation is given by:
Tv=$$\frac {1}{2}\dot {m}$$
Which is rearranges as:
$$\frac {T}{\dot {m}}=\frac {1}{2}$$vw2 (equation 2)
On equating equation 1 and 2 we get w=2v
Thus, the induced velocity in the wake region is twice that of the velocity at the rotor disk.

11. How is induced velocity in far wake region and actuator disk related in climb condition?
a) w=√v
b) w=v2
c) w=2v
d) w=3v

Explanation: The momentum equation for the rotor in climb condition is given by:
T=$$\dot {m}$$(V+w)-$$\dot {m}$$V=$$\dot {m}$$w
And the energy equation for climb condition rotor is given by:
T(V+v)=$$\frac {1}{2}\dot {m}$$(V+w)2–$$\frac {1}{2}\dot {m}$$(V)2=$$\frac {1}{2}\dot {m}$$w(w+2V)
On rearranging and eliminating T/$$\dot {m}$$ term, we get w=2v which is similar to the condition for hovering helicopter. Thus, the induced velocity in the wake region is twice that of the velocity at the rotor disk.

Sanfoundry Global Education & Learning Series – Aerodynamics.

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