Essentially a pump adds energy to a system and a turbine takes it away. Therefore typically in the Bernoulli Equation the pump pressure (P_{p}) is added to the left-hand side of the equation and the turbine pressure (P_{t}) is added to the right. So for a system containing a pump and a turbine the Bernoulli equation would look something like this:

(16)

If you know the power (W_{out}) that is delivered by a pump, then it is possible to calculate the pressure it represents (P_{p}).

From Equations (6) and (7):

From Equation (1):

So we can write:

look something like this:

(17)

Now from Equation (4):

so:

Combining this with Equation (17):

From Equation (8):

So we can write:

Thus the energy the pump provides as a pressure (P_{p}) is given by:

(18)

Typically pumps have an efficiency (γ)which is the ratio of the power out (W_{out} ) to the power in (W_{in} ). It represents the losses in the pump due to friction and electrical efficiency. It is usually expressed as a percentage and will always be less than 100%. It should be applied in the equation as a fraction. We can write therefore:

look something like this:

(19)

Combining Equations (18) and (19) gives:

look something like this:

(20)

This can be substituted into the Bernoulli Equation (16) and allows the determination of the pump power requirement or alternatively the flow rate in a system for a given pump power.

Analysis of turbines will not be dealt with in detail here but is very similar to that of pumps (but in reverse).