# Propellant Density Calculation for LOX/Kero

I am trying to make sense of the equation for propellant density calculation, rho=(rw-1)/(rw/bo+1/bf).

Sutton, 7th edition, gives this equation (7-2) as:

## $$\ \ \ \rho_{av} = \frac {\rho_o\rho_f(1+r)} {\rho_fr + \rho_o}$$

Where

• $$\rho_{av}$$ = average propellant density
• $$\rho_o$$ = oxidizer density
• $$\rho_f$$ = fuel density
• $$r$$ = mixture ratio

What would the bulk density of the LOX be? I have so far assumed 1,141 kg/L. However, this needs to be at a pressure higher than atmospheric pressure, correct? Or at a very low temp.

In this case, then what is the density of the fuel? Meaning, how do we select the P and T at which the density of the kero should be determined?

And would T and P other than standard conditions affect the O/F ratio, rw?

• What is the source of the equation, and what do the terms in it mean? Sep 27 at 11:36
• The equation comes from standard books like Sutton's Rocket Propulsion Elements, or papers from the Wright-Patterson Air Force (i.e. Tim Edwards). bo and bf are the bulk densities of the oxidizer and fuel, respectively.
– Mrr
Sep 27 at 13:30
• Is there a reason you are reluctant to give a specific source of the equation (link, page reference, etc)? Sep 27 at 13:56
• I just had to search some of the literature I have. It takes some time :)
– Mrr
Sep 27 at 14:42
• I did think you were asking about equation 7-2 from Sutton but you give it in a different form. Thanks for the info. Sep 27 at 14:44

## 3 Answers

I'm a bit confused about what you are actually asking about.

The things you mention, oxidizer and fuel density, are inputs into the equation. So you should use the densities for whatever system you doing the calculation for. This equation will not help you pick those.

If you are asking for densities for a particular vehicle, please clarify that.

The mixture ratio is generally determined by the engines used on the vehicle. So that, too, is an input. For vehicles with autogenous pressurization, the loaded mixture ratio may be different from the engine inlet mixture ratio. See Why do the contents of the Space Shuttle External Tank not match the mixture ratio of the engines? for an example.

Again, if you are asking for mixture ratio for a particular vehicle, please clarify.

• Ok. In Table 5-5 (Sutton)- examples for liquid propelant combinations. "The specific gravity at the boiling point was used for those oxidizers or fuels that boil below 20°C at 1 atm pressure". Values given are 2,24 for r and 1,01 for the avg specific gravity, which I take as being the propellant density, as the density for a kero type liquid is typically around 0,8-0,9 kg/L. So I just try to use these values and the 1,141 kg/L for LOX to replicate these results. It does not work. So I am trying to understand the values that have to be input which I guess will depend on the P, T conditions.
– Mrr
Sep 28 at 9:00
• @Mrr It really helps out people trying to answer your questions if you supply all the needed information about what you are asking. What exactly are "these results" that you are trying to replicate? In other words, exactly what numbers are you putting in exactly what equation and exactly what result are you getting and exactly what are you comparing it to? Also note specific gravity and density are not the same thing. Sep 28 at 13:02
• Organic Marble, as silly as it may sound, I have recalculated the avg spec gravity with Sutton form of the equation and density values at standard conditions for the fuel and got the right result. So it seems I've been beating around the bush due to a calculation error. Hey, it happens sometimes! Nevertheless, thank you and the others for the support. You guys kept it very interesting and I sure have learnt from it.
– Mrr
Sep 28 at 14:38
• @Mrr glad you got it figured out! Sep 28 at 15:16

I also do not see what is the question that is being asked. These parameters are just inputs to a basic chemistry relationship to compute the bulk density of a mixture of two liquids of differing densities at a defined mixture ratio by mass. Density of a liquid does vary with temperature and pressure that the fluid sees but not greatly. Unless you are looking for a precise value of bulk density that factor can be ignored and standard reference values used. Not sure how this equaton got into Sutton as in a rocket the fuel and oxidizer are stored separately; not mixed together. Lost my Sutton over the years of many travels.

• "Not sure how this equaton got into Sutton as in a rocket the fuel and oxidizer are stored separately; not mixed together." I am not an expert but this equation appears also in the Appendix 3 - "Summary of Key Equations for Ideal Chemical Rockets"
– Mrr
Sep 28 at 9:03

To address the part about temperature and pressure, these are engineering considerations, based on what can practically be achieved. In general, liquids have rather small variations in density, so standard reference values are usually good enough for a wide range of rocket configurations.

What would the bulk density of the LOX be? I have so far assumed 1,141 kg/L. However, this needs to be at a pressure higher than atmospheric pressure, correct? Or at a very low temp.

Yes. High pressure is impractical, since pressure vessels are heavy. That would cause the dry weight of the rocket to be much higher, decreasing performance. So the pressure is close to atmospheric for practical reasons. Instead low temperatures are used, typically a little below the boiling point in order to be able to have the rocket stand by for a little while before the propellant boils off.

In this case, then what is the density of the fuel? Meaning, how do we select the P and T at which the density of the kero should be determined?

The same pressure argument exist here. To compress the liquid for a very slightly better density, the container must be much stronger and heavier. A small boost in kerosene density can be achieved by chilling it down slightly, but this is limited by the freezing point of kerosene. You can't pump solids, another engineering concern.

And would T and P other than standard conditions affect the O/F ratio, rw?

The intake condition of the propellants could affect their rate of reaction somewhat, causing the engine to have a different optimal O/F ratio.

• Nice and easy to read, thanks! And I agree. However, it still does not resolve my doubts regarding the worked examples in Sutton (see my reply to Organic Marble). Searching for actual values for LOX/Kero propellant density values did not return many useful links to work with. So, in general, my focus lies in understanding how to properly interpret this (at first sight simple) equation. Are the conditions in the tanks giving us the values? Or the condition after entry on the engine? Are injectors affecting the outcome of this equation?...
– Mrr
Sep 28 at 9:25