Equation for Angular Momentum

Many of the flow devices and machinery involve rotating components. Examples are Centrifugal pumps, Turbines and Compressors. The analysis of such systems is facilitated by the Reynolds Transport theorem written for angular momentum. we have from Eqn.3.41,

$$T~ =~{{dH} \over {dt}}$$

$$~~H~=~\int_{system} (r\times V)\delta m = \int_{system} (r \times V) \rho d \forall$$

It becomes necessary now to calculate the angular momentum about some point, say O. Then we have,

$$N~=~H_O ~~\texttt{and}~~\eta~=~{{dH_O} \over {dm}}~=~r \times V$$ (3.54)

Substitution into the equation for Reynolds theorem (Eqn.3.27) gives,

$$\left. {{dH_O} \over {dt}} \right)_s~=~~{\partial \over \partial ... ...\forall~~+~~ \int_{CS}~(r \times V) \rho \overrightarrow{V}.d\overrightarrow{A}$$ (3.55)

Further, the LHS of the above equation is the sum of all the moments about the point $O$, ie., $\sum (r \times F)$. Accordingly we have,

$$\sum (r \times F)~=~~{\partial \over \partial t} \int_{CV} (r \t... ...\forall~~+~~ \int_{CS}~(r \times V) \rho \overrightarrow{V}.d\overrightarrow{A}$$ (3.56)

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