Impulse Density

How do we measure the performance of a rocket propellant? A propellant has a number of different characteristics and several of them have direct impact on a rocket's design. Let’s review the idea of Impulse Density and its influence on propellant performance. I have heard about it as a way of measuring the performance of propellants and I want to explore its impact..

DEFINITION
Impulse Density relates the amount of impulse (force over time) one gets from a propellant per unit of volume. Since all propellants vary with their density from one another, it is often difficult to compare propellants' effectiveness. Impulse density allows us to make comparisons of propellants without considering their relative densities. One formula for impulse density is:

Impulse Density = Ve * d

 Where:
  Ve = exhaust velocity of propellant
  d  = density of the propellant

Remembering the relationship between Ve and Isp:

Ve = Isp * g

We see that there is a relationship between the impulse density and the specific impulse. Whereas specific impulse allows one to judge a propellant for its impulse per unit mass of propellant, Impulse Density allows one to compare propellants for their impulse per unit volume.

So another way to write the Impulse Density is:

Impulse Density = Isp * d

 where:
  Isp = propellant specific impulse
  d   = density of the propellant

The result of this form of the equation will be different from the other representation merely by a constant, g.

Understanding impulse density in a comparative manner allows us to compare different propellants for their relevant benefits in a different way from specific impulse. Essentially, there is a relationship between the impulse density of a propellant combination and the size of the vehicle to produce a certain impulse. A higher impulse density means that a smaller volume is required to contain that impulse.

COMPARATIVE IMPULSE DENSITY
The following table illustrates the impulse densities of various propellants ordered from highest impulse density towards the lowest.

As one can see, a solid propellant like AP-HTPB-Al has one of the highest impulse densities (although in reality it will be less than listed because the core will reduce the average propellant density). Second in the table is nitric acid with furfuryl alcohol. Hydrogen peroxide with kerosene comes next. A number of denser liquid propellant combinations like N2O4-Hydrazine, Nitric Acid-Kerosene, AK27-T185 (Russian Scud missile propellant) and Lox-Kerosene follow. Liquid Oxygen and liquid hydrogen has the lowest impulse density of the propellants examined.

VISUALIZING THE RESULTS
The best way to understand the results of impulse density on rocket size is to see what different rockets with different propellants will be sized to for the same delta V performance. I will first look at upper stages and the result of impulse density on their weight and size and then look at booster stages for their comparative weights and sizes. The examination that I will do in no way considers the complications and complexities associated with making each of the propellants meet the defined requirements. It’s merely to provide a minimum number of variables so that comparisons can be made

UPPER STAGES WITH DIFFERENT PROPELLANTS
First, let's examine what impact these propellant densities have on the size of an upper stage vehicle. For this exercise, let's presume that we want to have a delta V of 15000 feet per second (fps) for each of several different propellants. We'll calculate the amount of propellant required and the volume of the tanks for them. I'll show the resulting values in a table with a graphical representation of what each of the vehicles looks like.

Looking at the table of characteristics and the resultant vehicle design images, one can see that the smallest and lightest vehicle is the solid rocket motor, which also has the impulse density; the largest vehicle is Lox-Hydrogen vehicle which has the lowest impulse density. The obvious trend is that with decreasing impulse density, the upper stage vehicles get dimensionally larger (generally). Another important characteristic is that the denser propellants also result in heavier stages.

The following graph lists each of the propellants and charts their resultant Isp, weight and volume.

The obvious general trend is that decreasing the impulse density from left to right generally results in larger tank volume. In fact, the tank volume of the liquid oxygen-liquid hydrogen vehicle is almost double that of the solid propellant vehicle. Additionally, it can be seen that the weight of the stage generally decreases for the same performance as the impulse density gets smaller.

BOOSTER STAGES WITH DIFFERENT PROPELLANTS
Now let's look at the impact of density impulse on the size of booster stages. For this examination, I will take the lightest upper stage, the Lox-LH2 stage and place it into a 250 mile circular orbit. To do this, the second (upper) stage will produce the full 24161 feet per second (fps) necessary to put itself into orbit and the first (booster) stage will only be responsible for lofting its payload to altitude. I've selected this scenario because it allows me to ensure that each vehicle is subjected to nearly identical flight trajectory characteristics. This means that the only variable which will affect the vehicle will be the effects due to density impulse and its size effect on vehicles.

To do this comparison, I created a single design which is parametric with propellant volume (which is proportional to delta V). This means that the vehicle is larger or smaller depending on the amount of propellant but all other parameters scale equally (i.e. diameter to length ratio, the size of the fins relative to the body size, etc). I then calculated the size of these vehicles and simulated their flight to get to a 250 mile +/- 1 mile altitude. I adjusted the delta V to get the vehicle that met the altitude criteria. The thrust of the booster stage was the same scaled value for each: 1.333 g’s at takeoff, with compensation of the thrust for the Isp change with altitude. Now, for the solid motor, this is not the most realistic thrust profile but end burners might be able to produce something similar. Solid motors often have trouble with longer burn times and usually accelerate faster, but I needed to control the thrust to have the aerodynamic effects similar across all vehicles.

There are some obvious differences with these vehicles and their size related to their impulse density (although it could be argued that the differences are mostly superficial). The most obvious difference is design NO7 which is the Lox-LH2 propellant. Because of Lox-LH2’s performance, it requires less propellant mass to produce the same amount of delta V (in a vacuum) but its geometric size is larger. But in the atmosphere, the lower mass causes increased aerodynamic losses so the vehicle must compensate by adding more propellant. This relationship eventually balances out with a vehicle that is still physically lighter but which is substantially larger.

One other way to look at the performance of the various propellants is to compare them by their various stage characteristics. The following table shows the length, diameter, weight and delta V for each booster propellant approach.

We seem to be able to make some generalizations from this data. Generally, the lower the impulse density, the lighter the vehicle, the more delta V is required and higher gravity and aerodynamic losses are incurred (but not necessarily drastically so) for the same overall stage performance. Again, Lox-LH2 (NO7) is an outlier in the trends.

SUMMARY
So the implications of Impulse Density generally means that a lower impulse density is a larger, lighter vehicle and vice versa. For a booster stage, higher impulse density can result in a heavier, but smaller vehicle (generally).