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:

	At = Wt/Pt SQRT(R Tt/(gamma)g_c)	(7)

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 oxygenhydrocarbon fuel is about 24, so that
R is about 65 ft-lb/lb(deg)R.
	Gamma, (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.
	g_c 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

(gamma) = 1.2

g_c = 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

	Tt = Tc (1/(1+((gamma)-1)/2))	(8)

For (gamma) = 1.2

	Tt = (.909)(Tc)			(9)

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

	T (degR) = T (degF) + 460	(10)

	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

	Pt = Pc(1+((gamma)-1)/2)^((gamma)/((gamma)-1))		(11)

For (gamma) = 1.2

	Pt = (.564)(Pc)		(12)

	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

	Me^2 = 2/((gamma)-1)((Pc/Patm)^(((gamma)-1)/gamma) - 1)		(13)

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 

Ae = At/Me * ((1 + ((gamma)-1)/2 Me^2)/(((gamma)+1)/2))^(((gamma)+1)/2((gamma)-1))	(14)

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, (gamma) = 1.2, Patm = 14.7 psi

<table>
</table>

	Ae = At (Ae/At)		(15)

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

	Te = Tc (Te/Tc)		(16)

The nozzle throat diameter is given by

	Dt = SQRT(4At/(pi))	(17)

and the exit diameter is given by

	De = SQRT(4Ae/(pi))	(18)

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.