Fluid Mechanics: Momentum Equation Propeller Question

[Master1022]

Homework Statement

Show that the thrust produced by a propeller may be expressed as: $T= \frac{\rho A }{2}(2V \Delta V +(\Delta V)^2 )$ where V is the forward velocity of the propeller, $\Delta V$ change in the velocity of the slipstream, $A$ the area swept by the propeller, and $\rho$ is the density of the air which may be considered to remain constant.

Relevant Equations

Force = Change in Momentum Flux

For this question, this is my working. The main issue is I cannot figure out where the factor of \frac{1}{2} comes from.

So I consider a control volume right around the propeller (in the frame of reference of the propeller) and I apply $R(-->) Force = Change in Momentum Flux$
$$F = \dot{m_{o}} V_{out} - \dot{m_{in}} V_{in}$$
$$F = \rho A (V + \Delta V)^2 - \rho A V^2$$
$$F = \rho A (V^2 + 2V\Delta V + (\Delta V)^2 - V^2)$$
$$F = \rho A (2V\Delta V + (\Delta V)^2)$$

and thus by Newton's 3rd law, $Thrust = -F$
thus: $$|T| = \rho A (2V\Delta V + (\Delta V)^2)$$

However, this is not the same as the answer, which seems to use Bernoulli's principle. However, I cannot see what is wrong with this method. I would appreciate any help.

Thanks in advance

 

[haruspex]

How can the mass flow rate out exceed the mass flow rate in?

 

[Master1022]

How can the mass flow rate out exceed the mass flow rate in?

Thank you for your response. Yes, that is something that confused me as I thought this was supposed to be a steady flow process. Hmm, perhaps that is where this method falls apart.

Is there a way to attempt this problem using the momentum equation and not just Bernoulli's equation?

 

[haruspex]

Thank you for your response. Yes, that is something that confused me as I thought this was supposed to be a steady flow process. Hmm, perhaps that is where this method falls apart.

Is there a way to attempt this problem using the momentum equation and not just Bernoulli's equation?

You can try the method you had, but make the mass flow rate constant. What does that give you?

 

[Master1022]

You can try the method you had, but make the mass flow rate constant. What does that give you?

Thanks for responding. The next part of the question is to "Hence, show that the mass flow rate is:" $\dot{m} = \frac{\rho A}{2}(V + \frac{\Delta V}{2})$. If I take the first part to be correct, I can get here just fine. However, I don't know how to arrive here otherwise. Looking at this expression, it seems to be the average of the mass flow rates in and out. Would I be able to get to the first expression by making this assumption about the mass flow rate and then following through? I thought about potentially changing the frame of reference of the propeller to get here, but that doesn't work.

I am wondering if this is an actual method in these momentum equation questions- i.e. taking the average of the mass flow rates?

 

[haruspex]

it seems to be the average of the mass flow rates in and out.

There can only be one mass flow rate, in and out. The problem is how to write it in terms of the other variables. What options do you have?