Now that we've covered the basic theory of pumps and

turbines, we'll do some examples.

Next we have to consider if we have a pump and

we know its characteristics, and we have a particular system.

For example, in this case we have a pump.

We're pumping water from a lower reservoir to an upper reservoir.

And in this system we have friction losses.

We have minor losses for a valve here.

How do we match the system to the characteristics of the pump?

I applied Bernoulli equation between these two free surfaces here, surface one,

the lower reservoir, and surface two, the upper reservoir and

here is our full Bernoulli equation, which looks like this.

But as usual, we can make a lot of simplifications.

The pressure in the lower reservoir is at the surface is atmospheric,

which is zero, so that goes out.

Similarly, the pressure at the upper reservoir is zero, so that goes out.

If the reservoirs are large, the velocities are very small, so V one and

V two go out.

We don't have a turbine in the system, so that goes out.

Re-arranging that equation, I can write it like this.

hP, the head added by the pump

is equal to the elevation difference between the two free surfaces,

z two minus z one plus the summation of all of the head losses in the system.

I can rewrite this equation in this form, hF is f L over D, V squared over two g.

The minor losses are terms which look like this,

loss coefficient times V squared over two g.

Combining those two equations I get hP is z two over z one plus K times Q squared

because each of these terms here are proportional to velocity squared.

However, the velocity squared is proportional to the flow rate, so

this is proportional to Q squared.

So all of those last terms I can replace by this single term here, KQ squared,

where K accounts for the friction loss in the pipe and the minor losses.

If I plot that equation out, I get a curve which looks like this.