## Pump Sizing

There is currently a fair bit of discussion on the aRocket list about amateur electric and turbine powered pumps, so I thought it would be helpful to go over some basic pump sizing work. Then maybe next post I will show a specific pump impeller design example.

First things first, in order to size a pump you need 4 basic numbers: mass flow rate (mdot), pressure rise (deltaP), fluid density (rho), and RPM. If you don’t have these numbers, then go ahead and make them up now. Now, what we want to calculate is pump specific speed (Ns) and pump power.  Pump specific speed is a quasinondimensional number (I’m using English units for Ns and bastard for elsewhere – sorry metric folks!) that should help us identify the pump type and geometry that we should be using and power is useful in selecting our driving motor.

Step 1: calculate volumetric flow rate in GPM

GPM = mdot (kg / s) / rho (kg / m^3)  * 60 * 264

Head rise (ft) = deltaP (psi) /  rho (kg / m^3) * 144  /  (2.205  * 0.305^3)

Step 3: Calculate specific speed

Ns = RPM * 1000 * GPM^0.5  / Head^0.75

And you should get a number between ~200 and 20,000. You then take the number and compare it to the plot below to choose an appropriate pump type. Radial pumps with around an Ns of 1000 and axial pumps with NS of ~10000 are probably the easiest pumps for an amateur to get up and running. In our case, we would rather do the radial because we need the pressure rise.

Now, to calculate pump power, we need efficiency; luckily for us, there is a handy chart for that, also from SP-125. Take your GPM and Ns and you get an efficiency. Pretty easy, right? Well, for an amateur, I would say knock off an extra 10% and, for a damn good pro, you might get an extra 10% as these are just standard values. If you don’t like your efficiency, modify your pressure rise or RPM.

You can see that with any pump for a rocket under 100 GPM, or ~10 klbf, you really want Ns up to 1500-4000 for any sort of efficiency.

Now to calculate power.

Step 1: Fluid Power

Power_Shaft (Watts) = Mdot (kg/s) * 9.8 m/s * Head (ft) / 3.28 / Pump_Efficiency

That is the power that you reed to drive the impeller. If you assume an extra 10% power for bearings and seals, you have your pump roughed out.

## Impeller Sizing

For the sake of this discussion, we will only consider centrifugal pumps – but if you are doing rockets and not using hydrogen, that is a good assumption. The pump on a rocket is broken up into 3 main parts: the inducer, the impeller, and the volute. The inducer is a low pressure rise pump that  limits cavitation in the impeller; it is only necessary for high specific speed pumps, which is most of rocketry. The impeller is the main pumping element and is effectively a paddle wheel that takes slow moving fluid from the core and ejects it at high speed from its sides. The volute takes the high speed fluid and slows it down, converting the high speed to high pressure. All told, they make up a rocket pump, but we will just go over the basic sizing of the impeller and not worry about fun things like velocity triangles today.

Once you have done all of the basic sizing that we talked about last week, you have the basic properties of the pump system but no sizes. If you want to do a detailed pump sizing, I would recommend NASA SP-125 and SP-8109 as it is a bit more complicated than a blog post. But, today, let’s get you the two biggest pieces of information you need: outlet diameter and outlet height. These are useful to determine the mass and manufacturability of the pump, as outlet height is usually the smallest geometry and diameter of the impeller and, thus, the volute determines most of the mass properties for a pump.

The process for exit diameter is iterative and consists of the following equations (everything in ft, GPM, ft/s, and RPM):

Specific_Diameter = Tip_Diameter * Head_Rise^0.25 / Flowrate^0.5

Tip_Diameter = Tip_Velocity / (pi * RPM)

Then use the plot below.

So now iterate on that for a bit, and we have the outlet diameter.

To find the exit height the equation is:

Exit_Height (in) = Flow_Rate(GPM) / (3.12 * pi * exit_diameter(in) * exit_radial_velocity(ft/s) * contraction_factor)

where the contraction factor is usually estimated as 0.9 and the radial velocity can be estimated by:

exit_radial_velocity = flow coefficient * tip_velocity

Flow coefficient is usually between 0.01 and 0.15 and can be found on the plow below.

I hope this has all been useful in determining how to size a impeller; obviously the final design is more complicated with flow angles, but these basics should point you in the correct direction.

## Turbopump Isp

In some previous posts, I have added some Isp’s and, if you have been looking closely, they seem a little lower than expected. This is because I have been looking at gas generator cycles and in these cycles some small percentage of the flow goes thru a gas generator and does not go through the main engine. Due to a bunch of reasons like not wanting to burn the pump up in seconds and needing pressure to drive the turbines, the flow is at a much lower temperature and pressure than the main engine. To account for this we need to:

Estimate the amount of flow in the Turbine – This is a simple energy balance equation that usually comes down to around 3% for a 1000 psi LOX RP-1 engine.

Calculate the Isp from the Engine (easy, lets assume 270 s SL) and the turbopump. Now for temperature ~1000 F and low pressure, ~40 psia, so they have Isp’s of around 80 seconds.

So now you throw them into the equations:

Isptot = (1-%turbine) * IspMainEngine + (%turbine) * IspTurbine.

Or, for our example, = (0.97 * 270 + 0.03 *  80) = 264 s.

As you can easily see, Isp is significantly lower ~2.5% and this is with fairly moderate pressure. As you can image for a certain cycle, there is an optimum pressure; higher isn’t always better as far as gas generator cycles are concerned.  This being said, for LOX RP-1, the tipover point with reasonable efficiencies is about 1500-2000 psi PC, so if you look around you see a lot of engines and studies  for GG at around 1200 psi. Higher than this requires multiple impellers and isn’t worth the extra complexity for the very negligible gains of the next couple hundred psi.