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# Pump and Turbine Cavitation

The cavitation within pumps and turbines
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### Key Facts

Gyroscopic Couple: The rate of change of angular momentum ($\inline&space;\tau$) = $\inline&space;I\omega\Omega$ (In the limit).
• $\inline&space;I$ = Moment of Inertia.
• $\inline&space;\omega$ = Angular velocity
• $\inline&space;\Omega$ = Angular velocity of precession.

## Pump Cavitation

##### MISSING IMAGE!

Apply Bernoulli's equation at the suction flange and at the water supply surface.

$0&space;=&space;\frac{P}{\rho}&space;+&space;\frac{v^2}{2g}&space;+&space;z&space;+&space;h_l$

Therefore the total head at the suction flange with the centre line as the datum is given by:
$\frac{P}{\rho}&space;+&space;\frac{v^2}{2g}&space;=&space;-&space;(z&space;+&space;h_l)\;\;[\text{m&space;of&space;water}]$
$=h_{Atmos}&space;-&space;(z&space;+&space;h_l)\;\;[\text{m&space;of&space;water&space;absolute}]$

The Net Positive Suction Head (NPSH) is defined as the total head at the suction flange minus the vapour pressure of water at the prevailing temperature, $\inline&space;h_v$. It is also known as the dynamic depression head. Therefore NPSH is the net head available at the suction flange to supply the increased velocity head and the losses at entry to the impeller.

$NPSH&space;=&space;\left(\frac{P}{\rho&space;g}&space;+&space;\frac{v^2}{2g}&space;\right)&space;-&space;h_v$

$=&space;h_{Atmos}&space;-&space;(z&space;+&space;h_l&space;+&space;h_v)\;\;[\text{m&space;of&space;water}]$

### Thoma's Cavitation Factor

$\sigma&space;&space;=&space;\frac{NPSH}{\text{Pump&space;Head&space;H}}$

When the NPSH falls to the point where Cavitation occurs then:
$\sigma_{crit}&space;&space;=&space;\frac{NPSH_{crit}}{H}$

For suction conditions to be the same for model and prototype the value of $\inline&space;\sigma$ must be equal.

## Turbine Cavitation

##### MISSING IMAGE!

Let $\inline&space;v_1$ be the water velocity at entry to the draft tube. This equals the velocity at outlet from the runner (assuming that there is no whirl)

Applying Bernoulli at the entry to the draft tube and the tail race surface.

$\frac{P_1}{\rho&space;g}&space;+&space;\frac{v_1^2}{2g}&space;+&space;z&space;=&space;0&space;+&space;h_l+\frac{v_d^2}{2g}$

$\frac{P_1}{\rho&space;g}&space;=&space;-z&space;-&space;\left(\displaystyle\frac{v_1^2}{2g}&space;-&space;h_l&space;-&space;\frac{v_d^2}{2g}&space;\right)\;\;[\text{m}]$
or
$=&space;h_{Atmos}&space;-&space;z&space;-&space;\left(\displaystyle\frac{v_1^2}{2g}&space;-&space;h_l&space;-&space;\frac{v_d^2}{2g}&space;\right)\;\;[\text{m&space;absolute}]$

The Velocity head is converted into Pressure head in the draft tube. i.e. Pressure recovery in the draft tube is
$\left(\frac{v_1^2}{2g}&space;-&space;h_l-\frac{v_d^2}{2g}&space;\right)$

The draft tube efficiency or recovery factor
$\eta_d=&space;\left(\frac{v_1^2}{2g}&space;-&space;h_l-\frac{v_d^2}{2g}&space;\right)\times&space;\frac{1}{\displaystyle\frac{v_1^2}{2g}}$

Therefore
$\frac{P_1}{\rho&space;g}&space;=&space;h_{atmos}&space;-&space;z&space;-&space;\eta&space;_d\times&space;\frac{v_1^2}{2g}\;\;[\text{m&space;absolute}]$

NPSH or the Dynamic depression head at the entry to the draft tube minus the vapour pressure of water at the prevailing temperature i.e. The amount by which pressure at the point of lowest pressure P may be below $\inline&space;\frac{P_1}{\rho&space;g}$ and still avoid cavitation.

$NPSH&space;=&space;\frac{P_1}{\rho&space;g}&space;-&space;h_v&space;=&space;h_{atmos}&space;-&space;\left(z&space;+&space;\eta&space;_d\times&space;\frac{V_1^2}{2g}&space;+&space;h_v&space;\right)$

The Thoma Cavitation Factor, $\inline&space;\sigma$ = NPSM / Turbine Head $\inline&space;H$,

$\inline&space;\sigma$ will be the same for similar machines running under dynamically similar conditions.