Pump Power related queries

I know that pump power can be calculated using this formula

$$P_p = \frac{\rho \ Q \ \Delta H \ g_0}{\mu_p}$$

Source: Rocket Propulsion Elements by Sutton 8th Ed. Equation 10-16, Chapter 10. Page 386. (screenshot)

How can we plot the performance curves if we know all these parameters given in the formula?

Also, how would performance curved changes if the number of RPMs increases? is there any other formula through which we can calculate pump power that relates power with RPMs?

The data given is just flow rate, density of propellant, pump efficiency, head and RPM.

How can I plot pump performance curves and then how would changing the RPM from 5000 to 10000 affect the curves?

• What kind of pump, centrifugal or axial flow? – Organic Marble Jun 1 '20 at 20:04
• Centrifugal Pump @Organic Marble – Adeel Ahsan Jun 1 '20 at 20:06
• You need to give us the source of this formula so that we can read the context and understand what each letter stands for. – user3528438 Jun 1 '20 at 20:42
• I redited the question kindly check – Adeel Ahsan Jun 1 '20 at 20:45
• @user3528438 OP has added the source. – uhoh Jun 2 '20 at 0:27

First of all, let's specify the meaning of all the terms along with respective units (S.I units) - $$P_{p} = \frac{\rho Q\Delta H g_{0}}{\eta_{p}}$$ $$P_{p}$$ = Pump Power ($$W$$)
$$\rho$$ = Propellant density ($$Kg/m^{3}$$)
$$Q$$ = Volumetric flow rate ($$m^{3}/s$$)
$$\Delta H$$ = Differential Head ($$m$$)
$$g_{0}$$ = Acceleration due to gravity ($$9.8\thinspace m/s^{2}$$)
$$\eta_{p}$$ = Efficiency of the pump

Now, let's tackle your questions one by one.

1. How can we plot the performance curves if we know all these parameters given in the formula?

Let's take a comprehensible document for reference - Understanding Pump Curves

Pump performance is usually plotted as Differential Head Vs. Flow at a specific RPM for specific inlet and outlet diameters. I assume the later would be given when you choose a specific pump. You would plot the performance curve after deciding which RPM you want it for. It looks something like this (presented in imperial units) -

Read it like this (for red lines) - For an impeller size of 7.9inch (variable) and the pumping rate of 140 gallons per minute ($$Q$$; given), you need 40ft of differential head ($$\Delta H$$). The area enclosed by faint lines which are intersecting the bold lines (impeller size) represents the efficiency of the operating pump (also given).

Now, corresponding to this Head Vs. Flow performance curve, there is a Power consumption Vs. Flow curve which looks like this (you can substitute imperial units with metric) -

Read it like this (for red lines) - For an impeller of size 7.9'' and the flow rate of 140 gallons per minute, we need 2 HP (1 Horse Power $$\approx$$ 746 Watts).

For plotting the performance curve, you would have to keep one of those specified physical quantities as a variable like take an impeller of specified size, fix the power, and vary the head to determine the flow rate. As you can see this is highly tedious, hence the performance curve is given by the manufacturer when you buy a pump.

1. Also, how would performance curve changes if RPM increases? is there any other formula through which we can calculate pump power that relates it with RPM?

I had to revise my Turbomachinery knowledge for this one. Refer to this short book - Basic Concepts in Turbomachinery page 127, pump power is given by - $$P = \rho \thinspace \left[\frac{\omega tan\beta_{2}}{2\pi b_{2}(1 - t_{2})}Q^{2} + \omega^{2}r_{2}^{2}Q\right]$$

Let's quickly define what each term means here (other than those defined previously)-
$$b_{2}$$ = Guide vane height ($$m$$)
$$t_{2}$$ = Vane blockage
$$\beta_{2}$$ = Blade angle
$$\omega$$ = Rotational speed ($$RPM = \frac{\omega}{2\pi} * 60$$)

Subscript 2 means the values are at the impeller exit.

So for calculating the power, first you need to have an impeller size fixed with vane height, vane blockage, and blade angle (forward curved/backward curved/radial). Then after determining the power at a specific RPM and keeping it constant, you vary the head and determine the flow rate and subsequently plot the performance curve (as done in part 1).

Evidently, the relation is not that straight forward and the performance curve is a convoluted function of RPM but if you are ready to take the pain, this is the way to go. Hope this helps!

• Thanks. yeah, it helped @OrangeDurito – Adeel Ahsan Jun 3 '20 at 14:43