- #1
Master1022
- 611
- 117
- Homework Statement
- Show that the thrust produced by a propeller may be expressed as: [itex] T= \frac{\rho A }{2}(2V \Delta V +(\Delta V)^2 ) [/itex] where V is the forward velocity of the propeller, [itex] \Delta V [/itex] change in the velocity of the slipstream, [itex] A [/itex] the area swept by the propeller, and [itex] \rho [/itex] 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 [itex] R(-->) Force = Change in Momentum Flux [/itex]
[tex] F = \dot{m_{o}} V_{out} - \dot{m_{in}} V_{in} [/tex]
[tex] F = \rho A (V + \Delta V)^2 - \rho A V^2 [/tex]
[tex] F = \rho A (V^2 + 2V\Delta V + (\Delta V)^2 - V^2) [/tex]
[tex] F = \rho A (2V\Delta V + (\Delta V)^2) [/tex]
and thus by Newton's 3rd law, [itex] Thrust = -F [/itex]
thus: [tex] |T| = \rho A (2V\Delta V + (\Delta V)^2) [/tex]
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
So I consider a control volume right around the propeller (in the frame of reference of the propeller) and I apply [itex] R(-->) Force = Change in Momentum Flux [/itex]
[tex] F = \dot{m_{o}} V_{out} - \dot{m_{in}} V_{in} [/tex]
[tex] F = \rho A (V + \Delta V)^2 - \rho A V^2 [/tex]
[tex] F = \rho A (V^2 + 2V\Delta V + (\Delta V)^2 - V^2) [/tex]
[tex] F = \rho A (2V\Delta V + (\Delta V)^2) [/tex]
and thus by Newton's 3rd law, [itex] Thrust = -F [/itex]
thus: [tex] |T| = \rho A (2V\Delta V + (\Delta V)^2) [/tex]
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