Bernoulli Equation

It is easy to see that there is no addition of work or heat between sections (1) and (2) and also between (3) and (4). It is possible to apply Bernoulli equation between (1) and (2) and also between (3) and (4) but not between (2) and (3).

$$p_1~+~{1 \over 2}\rho~V_1^2~ =~p_2~+~{1 \over 2}\rho~V_2^2$$

$$p_3~+~{1 \over 2}\rho~V_3^2~ =~p_4~+~{1 \over 2}\rho~V_4^2$$ (3.104)

Since V₂ = V₃and p₁ = p₄ we have from the above equations

$$(p_3~-~p_2)~=~{1 \over 2}\rho~(V_4^2~-~V_1^2)$$ (3.105)

Eliminating p₃ - p₂ from Eqns.3.105 and 3.103 we have,

$$V_2~=~{{V_1~+~V_4} \over 2 }$$ (3.106)

If the velocities are referred to the freestream air speed, i.e., V₁, we see that the propeller moves at a velocityV₁ . The work done by the propeller on the air stream or the power output is then,

In addition some kinetic energy is added to the air stream, which goes as a waste. The power input therefore is given by

From Eqns. 3.107 and 3.108 the efficiency of the propeller will be

$$\eta_{Fr}~=~{{V_1} \over {V_1~+~{1 \over 2}(V_4~-~V_1)}}$$ (3.109)

The term $\eta_{Fr}$ is called the Froude Efficiency.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney