# What does the author mean? Turbomachinery - Reaction

I am reading a book of Fundamental Energy Systems. The author describes the rate of change in head for a turbomachine as:

$$\frac{1}{2}[(V_1^2-V_2^2)+(U_1^2-U_2^2)+(V_{R2}^2-V_{R1}^2)] = H =U_1V_{u1} - U_2V_{u2}$$ and the static effect as: $$SE =(U_1^2-U_2^2)+(V_{R2}^2-V_{R1}^2)$$

However he later says:

His wording states $$R = SE/H$$. But he actually wrote it as we see above: $$R = \frac{SE}{2H} = \frac{(1/2)SE}{(1/2)2H} = \frac{(1/2)SE}{H}$$

I tried deriving his result from $$R = SE/H$$ but it was not possible, I just want to make sure my conclusion is correct (R = (0.5*SE)/H ; instead of R = SE/H ) as the author does not seem to mind his own wording.

Thanks for illustrating me.

• The highlighted part says it is a ratio... and if you divide the other halves of the expressions it comes out. – Solar Mike Jan 15 at 19:03
• Yes, the ratio would be SE/H. Which is the ratio of the total SE, to the total head H. But that is not SE/2H = (1/2)SE/H which is what the author wrote. – RSM Jan 15 at 19:23
• No, multiply first eqn on both sides by 2 gives 2H and the U and Vr terms then cancel with the lhs terms for SE leaving the velocity difference which is head. – Solar Mike Jan 15 at 19:27
• So you are saying R = SE/2H ? SE/2H = [ (U1^2- U2^2) + (Vr1^2 - Vr1^2) ] /[ (U1^2- U2^2) + (Vr1^2 - Vr1^2) ] [1 + (V1^2-V2^2)/((U1^2- U2^2) + (Vr1^2 - Vr1^2)) ] = 1/(1 + (V1^2-V2^2)/((U1^2- U2^2) + (Vr1^2 - Vr1^2)) ). That is not what he got, I am not sure I am following you. His words say SE/H. With SE/H and the definitions provided please show how to reach that form. I appreciate the help. – RSM Jan 15 at 19:40
• I think this is just a typo, and the definition of SE should be $\frac 1 2[(U^2_1−U^2_2)+(V^2_{R2}−V^2_{R1})]$. Compare with the previous formula. The total head should be the SE + some extra terms. – alephzero Jan 15 at 20:21