Integral relations for Non Inertial frame of reference for rotating bucket
I am trying to apply Integral relation of Angular Momentum for a non inertial frame of Reference to a rotating bucket, basically a Pelton Wheel made up of just one bucket. All of the books I've consulted just think of the bucket as moving at constant velocity, which is pretty accurate if the radius of the Pelton is long enough compared to that of the bucket. I want to get the solution for non inertial frame of reference and compare the solutions when the radius of the Pelton is comparable to the diameter of the bucket and when it is not. 782
Here's my try.
$$\sum (\vec{r} \times \vec{F})O - \int{VC} (\vec{r} \times \vec{a}{rel})dm= \int{SC} (\vec{r} \times \vec{V}) \rho (\vec{V}r \cdot \vec{n})dA $$ $$ \vec{a}{rel} = \frac{d^2\vec{R}}{dt^2}+\frac{d\vec{\Omega}}{dt} \times \vec{r} + 2 \vec{\Omega} \times \vec{V} + \vec{\Omega} \times (\vec{\Omega} \times \vec{r}) $$
Here is a sketch, the inertial and non-inertial frame of reference lay on the same point, the point of rotation, but the non-inertial is rotating around z-axis, the point O. The non-inertial frame is (x',y',z') and the inertial one is (x,y,z)
So $ \vec {R} = 0 $, angular velocity is constant, that is, $ \frac{d\vec{\Omega}}{dt} =0 $, and $\vec{r} \times \vec{\omega} \times (\vec{\omega} \times \vec{r} = 0)$, so the only term left will be Coriolis Force. Thereforce $$ 2 \vec{\Omega} \times \vec{V} = 2\omega (V_{jet}-\omega R)$$ $$ \int_{VC} (\vec{r} \times \vec{a}{rel})dm = \int{VC} [2\omega(V_{jet}-\omega R)r]\rho dV $$ $$ dV = A \bar{r} d\theta $$ $$ r^2 = R^2 +\bar{r}^2 - 2R\bar{r}cos\theta $$
Therefore
$$ \int_0^{\pi} [2\omega(V_{jet}-\omega R) \sqrt{R^2+\bar{r}^2-2R\bar{r}cos\theta}] \rho A \bar{r} d\theta $$
Regarding the other term
$$ \int_{SC} (\vec{r} \times \vec{V}) \rho (\vec{V} \cdot \vec{n})dA =\rho A (R-\bar{r})(V_{jet}-\omega R)^2 + \rho A (R+\bar{r})(V_{jet} - \omega R)^2 = 2\rho A R (V_{jet} - \omega R)^2 $$
So, I solved the first integral with Mathematica, and I plotted in Matlab, in comparison with the solution for constant velocity Control Volume.
It is reasonable that when the radius of the bucket is comparable to that of the Pelton there is a notorious difference (second image), and when radius of Pelton >> radius of bucket, both solutions are pretty similar (first image).
Basically, I wanted to see what other people think of this result. I haven't found a book that adresses this problem, or at least a similar one, except for "Mechanics of Fluids" of Irving Shames.
Thanks. 804