For a pressure-fed upper stage vehicle, operating in a vacuum, what is the optimal tank pressure for lowest weight? In this article we shall compute a first-order approximation of the answer to this question. It will be informative for designers of upper stages in selecting operating pressures for their upper stage designs.
Pressure fed upper stages are a good solution for small launch vehicles. They are the simplest kind of rocket stages to build and their performance can be high enough to meet most mission objectives. Schematically, they are relatively simple as shown in the diagram below. The major components are the rocket engine, two propellant tanks, and a pressurization tank.
With several simplifying assumptions, we can quickly estimate the weights of these components as a function of propellant tank pressure and thereby originate a first-order approximation of system weight.
A NOTIONAL UPPER STAGE
For this study, a notional upper stage is selected which has the following parameters
PRESSURANT TANK WEIGHT
The pressurant tank supplies the pressurizing gas for expelling the propellants from their tanks. We shall presume that the pressurant tank holds 3000 PSI, has a fixed performance factor and varies in its contained volume to hold 2 times the volume of the propellant tanks with a 10% ullage (empty space).
Pressurant Volume = 2 * 1.1 * ( Oxidizer Volume + Fuel Volume )
Pressurant Volume = 2 * 1.1 * ( 1.027 + 0.990 ) cubic feet
Pressurant Volume = 4.437 cubic feet
The Performance Factor of a pressure vessel is computed as:
PF = P * V / W
PF = performance factor
P = Pressure
V = Volume of compressed gas
W = weight of the pressurant vessel (without the pressurant gas)
So a pressure vessel with a PF of 1.6 Million which can hold 1 cubic foot (1728 cubic inches) of gas at 3000 PSI will weigh approximately:
w = 3000 PSI * 1728 cuin / 1600000
w = 3.24 lbs
For the sake of this discussion, we shall presume that pressurant tanks with a performance factor of 250,000 are used. This is a conservative, realistic value to have, although much better performance is possible. The amount of pressurant required increases with pressure for the same volume (think PV=nRT) and so does the weight of the pressurant itself and the weight of the vessel required to hold it.
The following graph shows the relationship between pressurant system mass versus the propellant tank pressure. It shows an obvious trend that increasing pressure requires increasing mass.
ROCKET ENGINE WEIGHT
We shall take a very simple model for calculating the weight of the rocket engine. The simplifying assumption about the design of the rocket engines considered is that it has the same thrust and Isp as the chamber pressure varies. In order for each to have the same Isp, we must vary the nozzle expansion ratio. Lower pressure engines will require larger nozzles and combustion chambers to get comparable thrust and Isp to higher pressure designs.
The rocket engine is presumed to be composed of 4 components made of a non-specific steel with a density of 0.285 lbs/cuin and an annealed yield strength of 81000 PSI.
The engine sub-components are:
1. A flat plate which is the injector
2. A cylindrical portion which is the combustion chamber
3. A conical frustrum converging part of the nozzle with a half-angle of 45 degrees
4. A conical frustrum diverging part of the Nozzle with a half-angle of 18 degrees
For the purposes of this calculation, we shall presume that all materials are the same thickness where the thickness is determined by the hoop stress of the combustion chamber which provides a 1.5 safety factor over the combustion chamber pressure for steel in its annealed state.
The combustion chamber diameter shall be 3 times the diameter of the throat. The length of the combustion chamber shall be such as to provide 60 times volume the area of the throat (an L* of 60 inches).
The combustion chamber pressure is 25% below the tank pressure to be a reasonable approximation of actual pressure drops and proper operating pressure for stability. By selecting a chamber pressure, we can then calculate the motor dimensions and nozzle expansion that will give a certain thrust and Isp. From these dimensions, we can estimate the engine weight using the formulas above.
We can derive a formula that relates the two independent factors: the chamber pressure and the nozzle expansion ratio to the dependent factor: the weight of the engine.
The equation of the area of a conical frustrum representing the nozzle converging and diverging sections is:
a = pi * (r1+r2)*sqrt((r1-r2)^2 + h^2)
The equation of the area of a cylindrical tube representing the combustion chamber is:
a = pi * d * l
The equation of the area of a circular plate is:
a = pi * d^2/4
The following table shows the engine combustion parameters used for this study. This table presumes Lox and Propane propellants with a mixture ratio of 2.2 required to obtain an Isp of 325 seconds for a thrust of 333 lbf. I used RPA [http://www.propulsion-analysis.com/] to make the following table.
The following graph shows the relationship between Propellant Tank pressure and Combustion Chamber weight. As can be seen, there is an inverse relationship: a higher chamber pressure results in a smaller and lighter combustion chamber.
PROPELLANT TANK WEIGHT
The propellant tanks are simple pressurized spherical tanks made from the same steel as the combustion chamber. The same maximum stress limit was used.
The following graph shows the weight of the tanks as a function of their pressure. It shows the obvious trend that the tank weight increases with pressure.
The following graph shows the combination of weights from the various components and their sum. As can be seen, propellant tank weights and pressurant system weights dominate the overall weight. Although the engine weight decreases with propellant pressure, it's such a small amount that it is overshadowed by the effects of propellant tanks and pressurant system weights.
The resulting summary is that the pressure that will result in the lightest overall stage is the lowest pressure feasible.