### DESIGN EQUATIONS

The following section will detail simplified equations for the design of small liquid-fuel rocket motors. The nomenclature for the motor design is shown in Figure 6.

#### Nozzle

The nozzle throat cross-sectional area may be computed if the total propellant flow rate is known and the propellants and operating conditions have been chosen. Assuming perfect gas law theory:

where R = gas constant, given by R = R_bar/M. R_bar is the universal gas constant equal to 1545.32 ft-lb/lb(deg)R, and M is the molecular weight of the gas. The molecular weight of the hot gaseous products of combustion of gaseous oxygen/hydrocarbon fuel is about 24, so that R is about 65 ft-lb/lb(deg)R.

Gamma, , is the ratio of gas specific heats and is a thermodynamic variable which the reader is encouraged to read about elsewhere (see Bibliography). Gamma is about 1.2 for the products of combustion of gaseous oxygen/hydrocarbon fuel.

gc is a constant relating to the earth's gravitation and is equal to 32.2 ft/sec/sec.

For further calculations the reader may consider the following as constants whenever gaseous oxygen/hydrocarbon propellants are used:
R = 65 ft-lb/lb(deg)R
= 1.2
gc = 32.2 ft/sec^2

Tt is the temperature of the gases at the nozzle throat. The gas temperature at the nozzle throat is less than in the combustion chamber due to loss of thermal energy in accelerating the gas to the local speed of sound (Mach number = 1) at the throat. Therefore

For = 1.2

Tc is the combustion chamber flame temperature in degrees Rankine (degR), given by

Pt is the gas pressure at the nozzle throat. The pressure at the nozzle throat is less than in the combustion chamber due to acceleration of the gas to the local speed of sound (Mach number = 1) at the throat. Therefore

For = 1.2

The hot gases must now be expanded in the diverging section of the nozzle to obtain maximum thrust. The pressure of these gases will decrease as energy is used to accelerate the gas and we must now find that area of the nozzle where the gas pressure is equal to atmospheric pressure. This area will then be the nozzle exit area.

Mach number is the ratio of the gas velocity to the local speed of sound. The Mach number at the nozzle exit is given by a perfect gas expansion expression

Pc is the pressure in the combustion chamber and Patm is atmospheric pressure, or 14.7 psi.

The nozzle exit area corresponding to the exit Mach number resulting from the choice of chamber pressure is given by

Since gamma is fixed at 1.2 for gaseous oxygen/hydrocarbon propellant products, we can calculate the parameters for future design use; the results are tabulated in Table III.

#### TABLE III

Nozzle Parameters for Various chamber pressures, = 1.2, Patm = 14.7 psi
Pc Me Ae/At Te/Tc
100 1.95 1.79 0.725
200 2.74 2.74 0.65
300 2.55 3.65 0.606
400 2.73 4.6 0.574
500 2.83 5.28 0.55

The temperature ratio between the chamber gases and those at the nozzle exit is given by

The nozzle throat diameter is given by

and the exit diameter is given by

A good value for the nozzle convergence half-angle (beta) (see Fig. 3) is 60 deg. The nozzle divergence half-angle, (alpha), should be no greater than 15 deg to prevent nozzle internal flow losses.

#### Combustion Chamber

A parameter describing the chamber volume required for complete combustion is the characteristic chamber length, L*, which is given by

where Vc is the chamber volume (including the converging section of the nozzle), in cubic inches, and At is the nozzle throat area (in2). For gaseous oxygen/hydrocarbon fuels, an L* of 50 to 100 inches is appropriate. L* is really a substitute for determining the chamber residence time of the reacting propellants.

To reduce losses due to flow velocity of gases within the chamber, the combustion chamber cross sectional area should be at least three times the nozzle throat area. This ratio is known as "contraction ratio".

The combustion chamber cross-sectional area is given by

The chamber volume is given by

For small combustion chambers the convergent volume is about 1/10 the volume of the cylinrical portion of the chamber, so that

The chamber diameter for small combustion chambers (thrust level less than 75 lbs) should be three to five times the nozzle throat diameter so the injector will have usable face area.

#### Chamber Wall Thickness

The combustion chamber must be able to withstand the internal pressure of the hot combustion gases. The combustion chamber must also be physically attached to the cooling jacket and, therefore, the chamber wall thickness must be sufficient for welding or brazing purposes. Since the chamber will be a cylindrical shell, the working stress in the wall is given by

where P is the pressure in the combustion chamber (neglecting the effect of coolant pressure on the outside of the shell), D is the mean diameter of the cylinder, and tw is the thickness of the cylinder wall. A typical material for small water-cooled combustion chambers is copper, for which the allowable working stress is about 8000 psi. The thickness of the combustion chamber wall is therefore given by

This is the minimum thickness; actually the thickness should be somewhat greater to allow for welding, buckling, and stress concentration. The thickness of the chamber wall and nozzle are usually equal.

Equation (22) can also he used to calculate the wall thickness of the water cooling jacket. Here again, the value of tw will be the minimum thickness since welding factors and design considertions (such as O-rings, grooves, etc.) will usually require walls thicker than those indicated by the stress equation. A new allowable stress value must be used in Equation (22), dependent on the jacket material chosen.

#### Engine Cooling

The amateur should not consider building uncooled rocket engines since they can operate for only a short time and their design requires a thorough knowledge of heat and mass transfer engineering. Cooled rocket motors have provision for cooling some or all metal parts coming into contact with the hot combustion gases. The injector is usually self-cooled by the incoming flow of propellants. The combustion chamber and nozzle definitely require cooling.

A cooling jacket permits the circulation of a coolant, which, in the case of flight engines is usually one of the propellants. However, for static tests and for amateur operation, water is the only coolant recommended. The cooling jacket consists of an inner and outer wall. The combustion chamber forms the inner wall and another concentric but larger cylinder provides the outer wall. The space between the walls serves as the coolant passage. The nozzle throat region usually has the highest heat transfer intensity and is, therefore, the most difficult to cool.

The energy release per unit chamber volume of a rocket engine is very large, and can be 250 times that of a good steam boiler or five times that of a gas turbine combustion chamber. The heat transfer rate of a rocket engine is usually 20 to 200 times that of a good boiler. It is apparent, therefore, that the cooling of a rocket engine is a difficult and exacting task. The complete heat transfer design of a rocket engine is extremely complex and is usually beyond the capabilities of most amateur builders. Some important empirical design guidelines are available, however, and are listed below:

1. Use water as the coolant.
2. Use copper for the combustion chamber and nozzle walls.
3. Water flow velocity in the cooling jacket should be 20-50 ft/sec.
4. Water flow rate should be high enough so that boiling does not occur.
5. Extend the water cooling jacket beyond the face of the injector.
6. A steady flow of cooling water is essential.

#### Heat Transfer

The largest part of the heat transferred from the hot chamber gases to the chamber walls is by convection. The amount of heat transferred by conduction is small and the amount transferred by radiation is usually less than 25%, of the total. The chamber walls have to be kept at a temperature such that the wall material strength is adequate to prevent failure. Material failure is usually caused by either raising the wall temperature on the gas side so as to weaken, melt, or damage the wall material or by raising the wall temperature on the liquid coolant side so was to vaporize the liquid next to the wall. The consequent failure is caused because of the sharp temperature rise in the wall caused by exessive heat transfer to the boiling coolant.

In water-cooled chambers the transferred heat is absorbed by the water. The water must have in adequate heat capacity to prevent boiling of the water at any point in the cooling jacket. The total heat tranferred from the chamber to the cooling water is given by

where
Q = total heat transferred, Btu/sec
q = average heat transfer rate of chamber, Btu/in^2-sec
A = heat transfer area, in2
w_w = coolant flow rate, Ib/sec
c_p = specific heat of coolant, Btu/lb(deg) F
T = temperature of coolant leaving jacket, deg F
Ti = temperature of coolant entering jacket, deg F
the use of this equation will be illustrated in the section Example Design calculation.

#### Materials

The combustion chamber and nozzle walls have to withstand relatively high temperature, high gas velocity, chemical erosion, and high stress. The wall material must be capable of high heat transfer rates (which means good thermal conductivity) yet, at the same time, have adequate strength to withstand the chamber combustion pressure. Material requirements are critical only in those parts which come into direct contact with propellant gases. Other motor components can be made of conventional material.

Once the wall material of an operating rocket engine begins to fail, final burn-through and engine destruction are extremely rapid. Even a small pinhole in the chamber wall will almost immediately (within one second) open into a large hole because the hot chamber gases (4000-6000 deg F) will oxidize or melt the adjacent metal, which is then blown away exposing new metal to the hot gases.

Exotic metals and difficult fabrication techniques are used in today's space and missile rocket engines, providlng a lightweight structure absolutely required for efficient launch and flight vehiclcs. These advanced metals and fabrication techniques are far outside the reach of the serious amateur builder. However, the use of more commonplace (and much less expensive) metals and fabrication techniques is quite possible, except that a flightweight engine will not result. Since almost all amateur rocket firing should be conducted on a static test stand, this is not a severe restriction to the amateur builder. Experience with a wide variety of rocket engine designs leads to the following rcconmendations for amateur rocket engines:

1. The combustion chamber and nozzle should be machined in one piece, from copper.
2. Those injector parts in contact with the hot chamber gases should also be machined from copper.
3. The cooling jacket and those injector parts not in contact with the hot propellant gases, should be fabricated from brass or stainless steel.
4. Expert machine and welding work is essential to produce a safe and useable rocket engine. Shoddy or careless workmanship, or poor welds, can easily cause engine failure.

#### Injectors

The function of the injector is to introduce the propellants into the combustion chamber in such a a way that efficent combustion can occur. There are two types of injectors which the amateur buider can consider for small engine design. One of these is the impinging stream injector win which the oxidizer and fuel are injected through a number of sepaate holes so that the resulting strams intersect with each other. The fuel stream will impinge with the oxidizer stream and both will break up into small droplets. When gaseous oxygen is used as the oxidizer, and a liquid hydrocarbon is used as fuel, the impingement of the liquid stream with the high velocity gas stream results in diffusion and vaporisation, causing good mixing and efficient combustion. A disadvantage of this type of injector is that extremely small holes are required for small engine flow rates and the hydraulic characterisitcs and equations normally used to predict injector parameters do not give good results for small orifices. The small holes are also difficult to drill, especially in soft copper.

However, to provide a complete picture of the equations used in rocket engine design, we present below the equation for the flow of liquid through a simple orifice (a round drilled hole, for example)

where
w = propellalnt flow rate, lb/sec
A = area of orifice, ft2
(deltaP) = pressure drop across orifice, lb/ft^2
(rho) = density of propellant, lb/ft^3
g = gravitational constant, 32.2 ft/sec2
Cd = orifice discharge coefficient
The discharge coefficient for a well-shaped simple orifice will usually have a value between 0.5 and 0.7.

The injection velocity, or vclocity of the liquid stream issuing from the orifice, is given by

Injection pressure drops of 70 to 150 psi, or injection vclocities of 50 to 100 ft/sec are usually used in small liquid-fuel rocket engines. The injection pressure drop must be high enough to eliminate combustion instability inside the combustion chamber but must not be so high that the tankage and pressurization system used to supply fuel to the engine are penalized.

A second type of injector is the spray nozzle in which conical, solid cone, hollow cone, or other type of spray sheet can be obtained. When a liquid hydrocarbon fuel is forced through a spray nozzle (similar to those used in home oil burners) the resulting fuel droplets are easily mixed with gaseous oxygen and the resulting mixture readily vaporized and burned. Spray nozzles are especially attractive for the amateur builder since several companies manufacture them commercially for oil burners and other applications. The amateur need only determine the size and spray characteristics required for his engine design and the correct spray nozzle can then be purchascd at low cost. Figure 7 illustrates the two types of injectors.

The use of commercial spray nozzles for amateur built rocket engines is highly recommended.