4.8 Tank Design
Tank design is an application of the principles of mechanics of materials. Tanks are a kind of pressure vessel, and are designed with two major criteria in mind: minimize weight and maximize allowable pressure. A third criteria, ever present in design, is to minimize cost. As is typical in engineering design, these criteria conflict, resulting in a compromise optimum design.
If the ratio of wall thickness to radius of the tank is small (less than 1/10) the pressure vessel is thin-walled and the following formulas for tank design will apply. In this case, the radial stress is small relative to the tangential stress, and the tangential stress is fairly uniform across the wall thickness. Both maximum radial and tangential stress occur at the inside of the wall. Tanks and pipes are usually thin-walled. Gun barrels are not and the assumptions of thin-wall analysis do not apply.
To begin the stress analysis of pressure vessels, we will start with the typical cylindrical tank shape. The fluid pressure inside the tank is distributed along the wall, so that for a cylinder, as shown below, the shear stresses in the walls equal the fluid force.
Along the longitudinal (long) axis of the tank a plane at the diameter has an area of 2× r× L, where r is the radius (and 2×r the diameter), and L is the tank length. Fluid pressure, p, over this area results in a tangential force in the tank wall of
P = p× 2× r× L
Unless the tank is rupturing and the cylinder walls are accelerating, the net force on a motionless wall must be zero. Fluid force, P, must be equal to opposing forces within the opposing tank walls, Q. The area of the wall of thickness, t, and length L is L×t, and the tangential wall stress is s. The total reactive force to P = 2×Q. By equating and solving, the wall thickness can be calculated from:
Applying the same kind of analysis to a spherical tank, the area inside the tank at the diameter is p ×r2 and the fluid force is
P = p× p × r2
The total reaction force in the wall of the tank – the perimeter of the sphere – is
Q = 2× p × r× s × t
Again, equating these static forces and solving for stress,
The normal stress in the wall of a sphere is half that of a cylinder at the same pressure. In other words, if walls are made of materials with the same stress ratings, then the spherical tank can be used to twice the pressure of the cylindrical tank, or at the same pressure, the tank wall can be half the thickness, thereby reducing tank weight. Consequently, the pressure-source tank is sometimes spherical.
Propulsion system design
The previously derived design equations can now be organized into a propulsion-system design procedure. Rocket mission requirements determine the burn time, which is also the propellant flow time, tb. To produce the required thrust, the engine design equations (see chapter 3) can be solved for fuel and oxidizer flow rates, pressures, and mixture ratio (O/F). Of course, the fuel and oxidizer substances are also selected, and their densities known.
To illustrate the design procedure by means of a concrete example, the Great Lakes Rocket Society Huron rocket propulsion system will be used as an example. At one time during development, the burn time was determined to be 15 s. The engine is a Rocketdyne RL-101 Atlas-missile vernier engine, which burns kerosene (or Jet A fuel, to be precise) and liquid oxygen (LOX). The engine is designed for a mixture ratio of
(though r = 3.24 is stoichiometric – it burns fuel-rich), and the inlet pressure is 3.52 MPa (510 psi). The propellant mass flow rate that produces 4.45 kN (1000 lbf) of thrust is
This value is calculated from the basic equation (section 3.20) relating engine performance variables to flow rate:
Fuel density is
rf = 800 kg/m3 = 50 lbf/ft3
(or a specific gravity of 0.8) and LOX density at a vapor
pressure of one atmosphere (at a temperature of –173 °C
= –280 °F) is
ro = 1100 kg/m3 = 68.6 lbf/ft3
(Conversion factor: kg/m3 = 16.03 lbf/ft3) Using these numbers, the equation for r above, and the mass-flow balance equation,
the fuel and oxidizer flow rates are calculated (see section 3.11) as:
The total mass of propellants is calculated next, from flow rates and the burn time:
The volumetric flow rates are calculated from the propellant densities:
The total propellant volumes can now be calculated:
Propellant masses can now be calculated from the volumes, given their densities:
mo = 18.0 kg, mf = 9.96 kg
In specifying tank volumes, some margin or ullage is required, especially for cryogenic propellants. If an ullage fraction of u is used (such as 10 %, or u = 0.1) the required tank volumes are then:
Vtank = (1 + u)× Vx
where x is f or o. For the Huron,
Vo = 18.0×10–3 m3, Vf = 13.7×10–3 m3
We are now ready to size the tanks. The rocket outside diameter of 0.305 m (1 foot) must exceed tank diameter. Aluminum tubing of 8 inches (0.203 m) in outside diameter are commercially available, with a 0.125 in (3.175 mm) wall thickness, from which tanks can be fabricated. The ratio of inside tank diameter to wall thickness is
(Do – 2× t)/t = 31 > 20
and thin-wall analysis applies.
Ordinarily, we would choose a tank material, find its maximum stress value, and calculate the minimum wall thickness. But in practice, available materials limit the choice. In this case (which is not unusual), we can obtain aluminum tubing with the given dimensions. We therefore need to check that the thickness is adequate for our application, and to also assess how much safety margin we have.
From the previous maximum-stress values of given materials, we find that for 6061-T6 aluminum, the yield stress is 276 MPa. Calculating wall stress,
The safety factor is about 1.27, which results in a narrow safety margin but minimal tank weight.
The inside tank volume is:
V = p × (r – t)2×L
Substituting and solving for L with an inside radius, r – t = 98.33 mm for both tanks,
Then Lo = 0.593 m (23.3 in) and Lf = 0.451 m (17.8 in).
Tank mass can be approximated by calculating the tank wall volume and multiplying it by the material density. For a cylindrical tank, neglecting end mass,
where r is the tank outside radius. The propellant tank masses are thus
mcylo = 3.27 kg, mcylf = 2.49 kg
The total weight of tanks and propellants is 34.72 kg = 76.38 lb.
The remaining tank is the pressure source for the propellant tanks. Because this tank must sustain a higher pressure than the propellant tanks, it might best be a spherical tank, with interior volume of
and tank mass of
The volume of the pressure feed tank can be derived assuming no blowdown occurs. Then if the pressure-source fluid is polytropic,
where Ps and Vs are the feed-tank pressure and volume, PR is the regulator pressure applied to the propellant tanks, and V2 is the combined volume of the propellant tanks. Then
Ps is often chosen based on the pressure ratings of available tanks. If a tank is designed, then tank volume is constrained by the inside diameter of the airframe.
The design approach taken here is first-order in that it omits refinements such as pressure loss in the pipes, changes of propellant density with temperature, and other complicating factors. Unless highly accurate modeling is required (due to costly experimental testing), the above procedure should be adequate in accuracy, subject to testing.