PHYSICS OF THE ROCKET

Howard S. Seifert, Mark M. Mills
and Martin Summerfield*)

I. PRINCIPLES OF ROCKET PROPULSION

The present article concerns only rocket engines of the “pure” reaction type, which develop thrust without making use of the surrounding atmosphere. Thus such devices as the “turbojet” engine used in rocket-turbine airplanes of the P-80 type are excluded. We shall also not touch upon the aerodynamics of bodies moving at supersonic speed, questions of electronics connected with the remote control of rocket projectiles, the use of rockets for measuring the physical characteristics of the upper layers of the atmosphere, or the special features of rocket artillery. Even after excluding these interesting fields, the science of rocket motion, as will be seen from what follows, comes into contact with many branches of physics and chemistry.

1. Physical Nature of Rocket Motion

A rocket is a certain rigid shell containing a store of matter and energy and equipped with a device that makes it possible to impart kinetic energy to portions of matter and eject them in a definite direction. The matter, initially at rest relative to the shell, is usually discharged in a continuous stream with an “exhaust velocity” \(v\) and a mass flow rate

\[ \dot m = \frac{dm}{dt}, \]

which causes a change in momentum by the amount \(\dot m v\) per unit time. This change in momentum

*) Howard S. Seifert, Mark M. Mills and Martin Summerfield, Amer. Journ. of Physics 15, 1 (1947). Translated by M. L. Antokolsky. The present article is the first of three published by the authors on this subject. The following articles will be printed in subsequent issues of Uspekhi Fizicheskikh Nauk. A historical survey is given in the third article. —Ed.

is imparted to the remaining part of the rocket, having instantaneous mass \(m\), in the form of a reaction force:

\[ F=-\dot{m}v, \tag{1} \]

where \(F\) and \(v\) are directed in opposite directions, since \(\dot{m}\) is negative. Equation (1) is valid if the outflow takes place into a vacuum. Thus the mechanical principle of rocket action is distinguished by exceptional simplicity.

An ideal rocket is one that gives the maximum rate of change of momentum (and, at the same time, the maximum thrust) with the minimum expenditure of mass. The simplest calculation shows that, for a constant thrust, the amount of kinetic energy imparted to the outflowing mass is in inverse proportion to the mass expenditure. Since a technically perfect rocket must expend mass economically, it must impart large amounts of kinetic energy. This leads to the requirement that the expelled substance be heated to a high temperature and, consequently, be in the gaseous state. In what follows it will everywhere be assumed that the outflowing substance behaves as an ideal gas. The purpose of the rocket engine, therefore, is the transformation of the disordered thermal motion of gas molecules into ordered motion, in which all the molecules move in one definite direction. Under such ideal conditions the macroscopic momentum would be maximal, while the temperature and pressure would be equal to zero. Since this requires expansion into a vacuum, a rocket moving in the earth’s atmosphere cannot attain ideal efficiency. The expansion process ends when the pressure in the rocket nozzle becomes equal to the pressure of the surrounding atmosphere. Accordingly, the useful action of the rocket engine has its limit.

2. Criteria for Evaluating Rocket Performance

A special feature of the rocket, as compared with other propulsive devices, is that its thrust does not depend on speed and, for its existence, does not require a surrounding medium, in contrast, for example, to an aeroplane engine, whose tractive force decreases with increasing relative speed and with decreasing atmospheric density. Ordinary engines under normal conditions move their load at constant speed; rocket engines usually impart acceleration to a freely moving body of rapidly decreasing mass. Ordinary engines are designed to develop force over a definite path; rocket engines, to develop it over a certain interval of time, with the aim of imparting a definite final velocity. In view of this, the impulse (or change in momentum) is more

a more essential parameter of a rocket than the energy expended; and the so-called “specific impulse”—the thrust force per unit weight of the outflowing substance:

\[ J_{sp}=\frac{Ft}{m_p g}=\frac{F}{\dot m_{\mathrm{avg.}}g} \tag{2} \]

is a more important characteristic than the power developed. It then turns out that the total expended mass \(m_p\) is the same for a constant impulse, regardless of whether a large force is developed over a short time or a small force over a long time.

The mechanical power developed by a rocket is proportional to its velocity. For example, the German “V-2” rocket, at a maximum speed of \(150\ \mathrm{m/sec}\), develops a power of more than half a million horsepower, whereas immediately after launch its power is relatively small.

The quantity reciprocal to \(J_{sp}\) represents the “specific fuel consumption” \(w_{sp}=\frac{1}{J_{sp}}\). The product \(J_{sp}g\) is the “effective exhaust velocity” \(c\), approximately equal to the velocity \(v\) appearing in equation (1). The difference between the velocity \(c\), defined by the equation

\[ c=gJ_{sp}, \tag{3} \]

and the true exhaust velocity \(v\), is due to harmful effects, such as, for example, the back pressure of the atmosphere. Although \(J_{sp}\) basically characterizes the process of producing thrust, its magnitude is affected by the geometrical shape of the rocket, the combustion pressure, and the external atmospheric pressure. This must be kept in mind when comparing different propellants.

As a rocket parameter, the ratio of impulse to total weight is also used:

\[ \frac{Ft}{(m_p+m_0)g}, \tag{4} \]

in which the unexpended mass \(m_0\) is taken into account. This parameter characterizes the quality of the entire design, consisting of the propellant and the casing. Some propellants that give a relatively high specific impulse may, when compared with others by total weight, lose this advantage. This occurs if the density of such substances is very low.

DYNAMICS OF THE ROCKET ENGINE

3. Basic thermodynamic relations

In this and the following sections, quantitative relations will be derived that determine the velocity of the gases flowing out of the rocket nozzle, as well as the thrust forces that arise in the process. The theory is for the most part derived from the most general initial principles, and

for clarity and consistency of presentation, even questions already well covered in the literature are not omitted.

Up to now we have considered only the relations between thrust force and mechanical energy. Now let us suppose that the propellant possesses a definite heat of combustion, and consider the thermodynamic process of converting this heat into useful mechanical energy.

Let \(dq\) be the incremental heat supplied to a unit mass of gas, \(dE\) the change in the internal energy of the gas, \(dV\) the change in volume caused by this heating, and \(p\) the pressure at which the process takes place. Then \(p\,dV\) is the work performed by the expanding gas. According to the first law of thermodynamics,

\[ dq = dE + p\,dV . \tag{5} \]

The internal energy \(E\), generally speaking, is a function of the temperature \(T\), and also, if van der Waals forces are present, of the specific volume \(V\).

The heat capacity at constant volume \(c_v = \left(\dfrac{\partial q}{\partial T}\right)_v\), according to equation (5), is equal to:

\[ c_v = \left(\frac{\partial E}{\partial T}\right)_{v=\mathrm{const}} . \tag{6} \]

Thus \(c_v\) is equal to the change of internal energy with temperature at constant volume. Transforming equation (5) into another form, we can obtain an expression for the heat capacity at constant pressure. Namely,

\[ dq = d(E + pV) - V\,dp = dH - V\,dp, \tag{7} \]

where \(H = E + pV\) is the enthalpy, or heat content.

From (7) we have:

\[ c_p = \left(\frac{\partial q}{\partial T}\right)_p = \left(\frac{\partial H}{\partial T}\right)_{p=\mathrm{const}} . \tag{8} \]

Consequently, \(c_p\) is equal to the change in enthalpy at constant pressure.

We shall now relate the “mechanical” velocity of a gas to its thermodynamic parameters—pressure, temperature, and density. Mechanical velocity is understood here as the general, or macroscopic, velocity, as distinct from the random thermal motion of the molecules. We shall apply Bernoulli’s equation in a form suitable for a compressible fluid. Consider a jet of gas (Fig. 1) in steady flow, and calculate the acceleration of a gas element enclosed between two nearby cross sections, each of area \(A\). Of course, the fact that the flow is steady does not mean that every element of it moves without acceleration.

Applying Newton’s second law to a gas element of thickness \(ds\), we obtain:

\[ \rho A ds \frac{dv}{dt} = pA - (p + dp)A = - A dp, \tag{9} \]

where \(\rho\) is the density of the gas. In steady flow the velocity of the elements

Fig. 1. Adiabatic expansion of a compressible fluid.

of gas passing through one fixed point of the stream does not change with time; however, it changes from point to point. Therefore

\[ \frac{dv}{dt} = \frac{\partial v}{\partial t} + \frac{\partial v}{\partial s}\frac{ds}{dt} = 0 + v\frac{dv}{ds}. \tag{10} \]

Combining equations (10) and (9), we obtain Bernoulli’s equation

\[ -dp = \rho v dv. \tag{11} \]

The negative sign shows that the velocity increases as the pressure decreases. Since we are considering unit mass, \(\rho = \frac{1}{V}\), where \(V\) is the specific volume. Introducing this substitution into equation (11), which is essentially Newton’s second law, and combining it with the first law of thermodynamics in the form (7), we obtain:

\[ dq = dH + vdv = d\left(H + \frac{1}{2}v^2\right). \tag{12} \]

Whereas \(H\) is the enthalpy of unit mass, the term \(\frac{1}{2}v^2\) represents the kinetic energy of unit mass. Thus thermodynamic quantities are related to the velocity of motion of the gas as a whole by a fundamental and practically important relation. If, further, we assume that the flow shown in Fig. 1 undergoes an adiabatic process, for example expansion in a rocket nozzle, then \(dq = 0\), and integration of (12) gives:

\[ H + \frac{1}{2}v^2 = \mathrm{const}. \tag{13} \]

According to equation (13), in a steady adiabatic process the sum of the enthalpy and kinetic energy of unit mass is constant. In practice this is the case in most instances.

4. Adiabatic flow

To derive quantitative relations describing adiabatic flow \((dq=0)\), one has to ascribe to the compressible fluid the properties of an ideal gas. By definition, the internal energy \(E\) and the enthalpy \(H\) of an ideal gas are functions of temperature alone. Therefore the first law of thermodynamics, by combining (5) and (6), or, respectively, (7) and (8), is reduced to the form

\[ c_v dT + p\,dV = 0, \tag{14a} \]

\[ c_p dT - V\,dp = 0. \tag{14b} \]

Eliminating \(dT\) from (14a) and (14b) and introducing the ratio of heat capacities

\[ \gamma=\frac{c_p}{c_v}, \]

we obtain:

\[ \gamma \frac{dV}{V}+\frac{dp}{p}=0. \tag{15} \]

Up to this point no restriction has been imposed on \(c_p\) and \(c_v\), which could also have been variable. If we now assume that \(\gamma\) is constant, then equation (15) can be integrated, and we obtain the well-known equation for an adiabatic process

\[ pV^\gamma=\mathrm{const}. \tag{16} \]

The definition of an ideal gas requires not only that \(E\) depend solely on \(T\), but also that the gas obey the equation of state for unit mass:

\[ pV=\frac{R}{M}T=R_sT, \tag{17} \]

where \(R\) is the universal gas constant, \(M\) is the effective molecular weight of the gas*) and \(R_s\) is the specific gas constant.

Taking the difference between (14a) and (14b) and combining with equation (17), we obtain, after rearrangement:

\[ c_p-c_v=R_s. \tag{18} \]

Substitution of \(\gamma=\frac{c_p}{c_v}\) leads to two relations that will be useful later:

\[ c_p=\frac{\gamma}{\gamma-1}R_s, \tag{19a} \]

\[ c_v=\frac{1}{\gamma-1}R_s. \tag{19b} \]

Using equations (16) and (17), one can express any of the quan—

*) The quantity \(M\) must be expressed in units of mass (and not weight).

values: pressure, temperature, and density or specific volume as functions of the other two. Further, the ratio, for example, of the temperatures at two points of an adiabatic cycle can be expressed by a simple power expression in terms of the ratio of the values of any other variable at these points. These relations are especially useful when it is required to compute a parameter at any point of a rocket nozzle as a function of its value in the combustion chamber, where it can more easily be measured. For example,

\[ \frac{T}{T_c}=\left(\frac{p}{p_c}\right)^{\frac{\gamma-1}{\gamma}}, \tag{20} \]

where the subscript \(c\) means that the quantity refers to the combustion chamber. Since in rockets

\[ \frac{\gamma-1}{\gamma}=\frac{1}{5}, \]

equation (20) shows that a comparatively large change in pressure during the adiabatic expansion of the gas in the nozzle corresponds to a small change in temperature. It also shows the importance of the parameter \(\gamma\) in the dynamics of a compressible fluid.

5. Velocities Attained in Adiabatic Expansion

We have derived the necessary relations for obtaining an expression for the velocity of outflow of gases from a rocket nozzle as a function of temperature or pressure. From the defining expression (8) for \(c_p\) and from the constancy of \(c_p\) in an ideal gas, we find by integration:

\[ H=c_pT+H_0, \tag{21} \]

where \(H_0\) is the constant of integration.

Fig. 2. Thermodynamic parameters in the combustion chamber and in an arbitrary cross section of the nozzle.

Let us denote (Fig. 2) the pressure and temperature in the combustion chamber by \(p_c\) and \(T_c\), and in some cross section of the nozzle by \(p\) and \(T\). Since the velocity in the combustion chamber is small, it follows from (13) and (21) that:

\[ \frac{1}{2}v^2-0=c_p(T_c-T). \tag{22} \]

But \(c_p\) can be expressed, according to (19a), in terms of the fundamental constants, whence

\[ v^2=\frac{2\gamma}{\gamma-1}R_sT_c\left(1-\frac{T}{T_c}\right). \tag{23} \]

Using equations (17) and (20), we can replace the temperature by pressure and density, which are easier to determine, and obtain

the final expression for the outflow velocity as a function of the outflow pressure:

\[ v=\sqrt{2\,\frac{\gamma}{\gamma-1}\,\frac{p_c}{\rho_c} \left[1-\left(\frac{p}{p_c}\right)^{\frac{\gamma-1}{\gamma}}\right]} . \tag{24} \]

From equation (24) a number of interesting conclusions can be drawn. The factor containing \(\dfrac{p}{p_c}\) approaches unity when this ratio tends to zero. Thus the outflow velocity increases as the external pressure decreases and reaches a maximum during expansion into a vacuum, when \(p=0\). In this case all the thermal energy of the gas is converted into kinetic energy. For \(v_{\max}\), equation (24) gives:

\[ v_{\max}=\sqrt{2\,\frac{\gamma}{\gamma-1}\,\frac{p_c}{\rho_c}} . \tag{25} \]

In designing a rocket, the ratio of the external pressure to the pressure in the chamber, \(\dfrac{p}{p_c}\), is prescribed. The external pressure varies from one atmosphere to zero depending on the altitude of the rocket’s flight. A typical value of \(\dfrac{p}{p_c}\) for a liquid propellant is \(14.7:300\). For a given value of \(\dfrac{p}{p_c}\), the velocity, according to (24), increases as \(\gamma\) decreases, and therefore it is desirable to use a gas with a small value of \(\gamma\); however, in practice there is little possibility of varying this parameter.

According to (17), \(\dfrac{p_c}{\rho_c}=\dfrac{R}{m}T_c\). Hence it is clear that, in order to attain the greatest outflow velocity \(v\), the temperature \(T_c\) in the combustion chamber must be high, and the molecular weight of the combustion products low. That \(m\) should be low can be shown in another way, with the aid of the principle of equipartition of energy (see \(^{4}\), p. 210). Let us consider two combustion chambers with the same pressure \(p_c\) and temperature \(T_c\), containing ideal gases with molecular weight \(m\) in one chamber and \(100m\) in the other chamber. If the mean kinetic energy of the lighter molecules is \(\dfrac{1}{2}Mv^2\), then, according to the equipartition theorem, for the heavy molecules it must be the same, i.e. \(\dfrac{1}{2}\cdot100M\cdot\left(\dfrac{v}{10}\right)^2\). During expansion of the gas into a vacuum, the average momentum of motion of a light molecule is equal to \(Mv\), and of a heavy one to \(100M\dfrac{v}{10}\), or \(10Mv\). However, if the mass flow rate is the same, then the number of light molecules that have flowed out is 100 times greater than the number of heavy ones, and therefore the total change in the amount of motion, and

therefore the thrust of a [[unclear: word obscured, likely “rocket”]] containing the lighter substance will be ten times greater, despite the sameness of temperature, pressure, and mass flow rate. This circumstance is of great importance in choosing a working substance for which, thus, the highest possible hydrogen content is desirable.

If the two rockets mentioned in the preceding example have equal thrusts, rather than equal mass flow rates, then for the lighter substance the mass flow rate is one-tenth that of the heavier one, but the power imparted to the lighter molecules is ten times greater. This fact is in agreement with the considerations set forth in Section 1.

FLOW FROM A NOZZLE

6. Flow velocity and speed of sound

The factor entering equation (24),

\[ \frac{p_c}{\rho_c}=R_sT_c, \]

is a measure of the thermal energy stored in the gas and has the dimension of energy per unit mass. For an ideal gas it is simply the kinetic energy of molecules moving with a certain mean velocity. Since the velocity of propagation of a wave disturbance in a gas is determined by this same mean velocity of the thermal motion of the molecules, it is natural to expect that the speed of sound waves in a gas is in a simple relation to the velocity acquired by the molecules of this gas during free adiabatic expansion. A good measure of the speed of chaotic molecular motion is the so-called speed of sound \(a\) (see, for example,¹ p. 36 ff.), defined by the equation

\[ a^2=\frac{dp}{d\rho}. \tag{26} \]

For an adiabatic process, using equations (15) and (17), equation (26) is reduced to the form:

\[ a^2=\frac{\gamma p}{\rho}=\gamma R_sT. \tag{27} \]

Thus, if we denote the speed of sound in the combustion chamber by \(a_c\), then equation (25) for the maximum flow velocity is rewritten as

\[ v_{\max}=a_c\sqrt{\frac{2}{\gamma-1}}, \tag{28a} \]

and equation (24) for the flow velocity at any pressure \(p\) becomes

\[ v=a_c\sqrt{\frac{2}{\gamma-1}\left[1-\left(\frac{p}{p_c}\right)^{\frac{\gamma-1}{\gamma}}\right]}. \tag{28b} \]

For many kinds of working substance used in rockets, \(\gamma = 1.25\), so that \(v_{\max} = 2.828\,a_c\). For a typical case of expansion from 30 to \(1\ \mathrm{kg/cm^2}\), the speed of sound at the nozzle exit is \(\frac{1}{2}a_c\), and the ratio of the final outflow velocity at the nozzle exit to the speed of sound at that point, \(\frac{v}{a}\)—the so-called “Mach number” \(M\)—has a value from 5 to 6. Such a fluid flow is called supersonic and can be realized only with a proper nozzle shape.

7. Laval Nozzle

For the calculation of a rocket it is necessary to find the dependence of the thrust force \(\dot{m}v\) on the properties of the combustion products and on the pressure. This has already been done for the outflow velocity in the form of equation (24). To determine the thrust force as a function of pressure and geometrical dimensions, it remains further to find a similar expression for the mass flow rate \(\dot{m}\) and to determine the effect of back pressure on the thrust force.

The geometrical shape of the orifice from which gases under pressure flow out has a strong influence both on the mass flow rate and on the thrust force. It is not obvious a priori what the shape of this “nozzle” should be. Thus, for example, the velocity of an incompressible fluid, in particular water, passed through a Venturi tube first increases and then decreases, reaching a maximum in the narrowest part of the tube. The mass flow rate is proportional to the total pressure drop. On the other hand, a compressible fluid undergoing adiabatic expansion through a similar Venturi tube behaves in the same way only under the condition that the velocity at every point is less than the speed of sound at that same point. As soon as the sonic velocity is reached, which occurs first of all in the narrowest part—the throat of the tube—the behavior of the flow changes completely. Any change of pressure at points of the flow lying downstream from the throat no longer has any influence on the mass flow rate (though it does affect the outflow velocity). This effect is sometimes called “nozzling.” In addition, the gas velocity downstream from the throat increases, becoming supersonic, and reaches a value determined by the pressure at the nozzle exit. The pressure difference necessary to attain sonic velocity is called critical. In all rockets the pressure in the chamber greatly exceeds this critical value, as will be seen from the quantitative relations to whose derivation we now turn.

Since the mass flow rate is constant in any transverse section of the nozzle, we have the continuity condition

\[ \dot{m} = f\rho v = \mathrm{const.}, \tag{29} \]

where \(f\) is the area of any cross section, and \(\rho\) and \(v\) are the density and velocity in this section. These latter quantities can be expressed by means of the ratio \(\dfrac{p}{p_c}\) and one can obtain a relation between the pressure \(p\) in some section and the area of the section \(f\) for a given mass flow rate \(\dot m\). The value of \(\rho\) can be expressed through the pressure with the aid of an equation similar to equation (20), namely:

\[ \rho=\rho_c\left(\frac{p}{p_c}\right)^{\frac{1}{\gamma}}. \tag{30} \]

Substituting this value of \(\rho\) and the value of \(v\) from equation (24) into (29) and solving for \(f\), we find:

\[ f=\frac{\dot m}{\rho_c}\left(\frac{p}{p_c}\right)^{-\frac{1}{\gamma}} \left\{ \frac{2\gamma}{\gamma-1}\frac{p_c}{\rho_c} \left[ 1-\left(\frac{p}{p_c}\right)^{\frac{\gamma-1}{\gamma}} \right] \right\}^{-\frac{1}{2}}. \tag{31} \]

If the value of \(f\) is calculated for a series of decreasing values of \(\dfrac{p}{p_c}\), it turns out that \(f\) passes through a minimum value². This shows that, in order for \(p\) to decrease (and, consequently, \(v\) to increase) continuously, the nozzle must be given a definite shape, shown in Fig. 3. Nozzles of this form were called Laval nozzles after the Swedish engineer Carl de Laval, who first used them for obtaining supersonic gas velocity.

Fig. 3. Laval nozzle for supersonic flow. Area of the throat \(f_t\) and area of the exit opening \(f_e\).

The fact that a supersonic nozzle must have a shape which first narrows and then widens can be shown in another way. The continuity equation (29), written in differential form, is:

\[ \frac{df}{f}+\frac{d\rho}{\rho}+\frac{dv}{v}=0. \tag{32} \]

The differential form of Bernoulli’s law [equation (11)] can be rewritten, with the aid of equation (26), in the form:

\[ \frac{dp}{\rho}=\frac{dp}{d\rho}\frac{d\rho}{\rho}=a^2\frac{d\rho}{\rho}=-v\,dv. \tag{33} \]

Substituting this expression for \(\dfrac{d\rho}{\rho}\) into equation (32) and recalling the definition of the Mach number \(M=\dfrac{v}{a}\), we find:

\[ \frac{df}{f}=-\frac{dv}{v}\left(1-M^2\right). \tag{34} \]

Equation (34) shows that, in order for the velocity to increase continuously along the nozzle, i.e., for \(dv\) to remain positive at all times, it is necessary that:

if the velocity \(v\) is below the speed of sound, \(M<1\),

\[ \frac{df}{f}<0, \]

i.e., \(f\) decreases;

if the velocity \(v\) is sonic, \(M=1\),

\[ \frac{df}{f}=0, \]

i.e., \(f\) is minimal;

if the velocity \(v\) is supersonic, \(M>1\),

\[ \frac{df}{f}>0, \]

i.e., it increases.

The place where the cross-sectional area is minimal is called the throat. To find the value of the pressure in the throat \(p_t\), one must differentiate equations (31) with respect to \(\dfrac{p}{p_c}\) and set the derivative equal to zero. In this way we find:

\[ \frac{p_t}{p_c}=\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma}{\gamma-1}} . \tag{35} \]

Combining (35) and (20), we find for the temperature in the throat

\[ \frac{T_t}{T_c}=\frac{2}{\gamma+1}. \tag{36} \]

The pressure ratio determined by equation (35) is called the critical ratio. If the ratio is below this initial value, then the nozzle must have a converging shape; otherwise it must be converging and then diverging again. Substitution of the value of the critical ratio (35) into equation (24) gives, with the aid of (20) and (27), the expression for the velocity in the throat \(v_t\):

\[ v_t=\sqrt{\frac{2\gamma}{\gamma+1}\frac{p_c}{\rho_c}} =\sqrt{\frac{2\gamma R_sT_c}{\gamma+1}} =\sqrt{\gamma R_sT_t}=a_t . \tag{37} \]

This is precisely the speed of sound under the conditions obtaining in the throat.

The cross-sectional area of the throat \(f_t\), required for a given value of the flow rate \(\dot m\) and for a given pressure ratio, can be calculated by substituting into equation (31) the value \(\dfrac{p_t}{p_c}\) from equation (35), on the assumption that the pressure ratio exceeds the critical value. Thus the values of the pressures outside the throat have no effect on the flow rate. This, in appearance anomalous, phenomenon does not follow from Bernoulli’s equation, but is in contradiction with it³. It is a consequence of finite-

...of the speed of sound and the fact that the pressure drop in the liquid after passing the throat cannot be transmitted in the reverse direction to the flow of liquid if the latter is moving at supersonic velocity. This explanation was first given by Osborne Reynolds[^4].

It is interesting to note that, although equations (31) and (34) show the necessity of a minimum in the cross section of the nozzle if the gas velocity is required to increase continuously, and although equation (31) uniquely determines the pressure \(p\) and the area \(f\), it gives no other indications regarding the geometrical shape of the nozzle. Indeed, this shape is not uniquely determined, and its choice is governed by other, less fundamental considerations, such as, for example, weight and heat-transfer conditions. Two typical nozzle contours are shown in Fig. 4. Of these, (a) is intended for liquid propellant and low pressure in the combustion chamber, and (b) for solid propellant and high pressure in the chamber.

Fig. 4. Two typical nozzle outlines. The upper one is designed for a thrust of 135 kg with expansion of the gas from an initial pressure of \(18 \text{ kg}/\text{cm}^2\) (liquid working substance); the lower one is for the same thrust at an initial pressure of \(120 \text{ kg}/\text{cm}^2\) (solid working substance).

A brief qualitative description of the behavior of a Laval nozzle at different pressure ratios is of interest. Maintaining the pressure in the chamber constant, we shall gradually reduce the pressure at the nozzle exit, beginning with a value only slightly less than the pressure in the chamber. At first the velocity is everywhere below sonic, and the nozzle operates like a Venturi tube. The velocity increases at the entrance to the nozzle and then, beyond the throat, falls again. As the external pressure is reduced, the velocity in the throat increases and, together with it, the mass flow rate \(\dot m\) increases. After the speed of sound has been reached in the narrowest section, a further decrease in the external pressure does not increase \(\dot m\), because the velocity in the throat section remains constant and equal to the speed of sound. However, the velocity at the nozzle exit continues to increase, the outflow being accompanied by the formation of shock waves of complex form[^5]. The increase continues until the pressure near the very exit of the nozzle reaches the value determined by the area ratio of the nozzle*). A further fall

*) This refers to the pressure attained in continuous adiabatic expansion for a given area of the nozzle exit, i.e., to the pressure calculated from formula (31) if the area of the exit opening is substituted for \(f\). The area ratio \(\varepsilon\) is considered in Section 9.

“atmospheric pressure” does not increase the velocity of efflux from the nozzle opening, since the additional expansion of the gas no longer takes place in the nozzle. This circumstance may result in an increase of the thrust force \(F\), caused by the influence of pressure, as will be shown in Section 9. Figure 5 shows how the parameters \(p\), \(\rho\), \(v\), and \(T\), referred to their values in the narrowest section, vary along the nozzle.

8. Mass flow rate in the nozzle

The magnitude of the mass flow rate \(\dot m\) can be expressed in terms of the parameters characterizing the conditions in the chamber and the area of the throat section, by combining the continuity equation and the relations derived earlier. According to equation (29),

\[ \dot m=f_t \rho_t v_t, \tag{38} \]

Fig. 5. Graph of supersonic velocity and thermodynamic parameters in the nozzle, referred to the values of these quantities in the throat, as a function of relative area. The direction of gas flow is from left to right.

where the subscript \(t\) indicates that the quantities refer to the throat. The speed of sound in the throat \(v_t\) is obtained from equation (37), and the density in the throat \(\rho_t\) can be expressed in terms of \(p_t\) and \(T_t\) with the aid of equation (17). But \(p_t\) and \(T_t\), by means of equations (35) and (36), can be expressed through the parameters characterizing the conditions in the chamber. Making all the necessary substitutions in (38), we obtain:

\[ \dot m=\Gamma' \frac{f_t p_c}{\sqrt{\gamma R_s T_c}} =\frac{\Gamma' f_t p_c}{a_c}, \tag{39} \]

where \(\Gamma'(\gamma)\) is a constant determined by the equation

\[ \Gamma'=\gamma\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}}. \tag{40} \]

From equation (39) it is seen that \(\dot m\) does not depend on the characteristics of the flow after it has passed through the throat. The speed of sound \(a_c\) cannot be measured directly; however, as will be seen from Section 10, the quantity \(a_c\) is eliminated from the formulas used for the practical calculation of \(\dot m\). The mass flow rate \(\dot m\) is used in calculating the thrust of a rocket.

DETERMINATION OF THE REACTION FORCE

9. Calculation of the total thrust force of a rocket

For a rocket moving in a vacuum, the thrust force is calculated simply by the formula \(F=\dot m v\), with a correction due to the fact that the emerging jet is in reality not parallel, but slightly divergent. But since ordinarily the rocket is in the atmosphere and is subjected to external pressure, a more detailed analysis of the processes taking place is necessary.

The total thrust force \(F\), for which the direction opposite to the gas velocity is taken as positive, is the (vector) sum of all pressure forces acting on the inner and outer surfaces of the solid shell:

\[ F=\int p\,dS=\int_{S_i} p_i\,dS_i+\int_{S_0} p_0\,dS_0, \tag{41} \]

where \(p\) is the magnitude of the pressure on the shell, \(dS\) denotes an elementary area, and the indices \(i\) and \(0\) denote, respectively, the inner and outer surfaces. By symmetry it is evident that the resultant vector is directed along the axis of the rocket.

Fig. 6. Pressure forces acting on the rocket shell and producing thrust:
a) external forces produced by atmospheric pressure,
b) internal forces produced by the expanding gases.

The integral over the outer surface \(S_0\) can be evaluated as follows. The resultant force produced by uniform atmospheric pressure and acting on a completely closed shell at rest is equal to zero. If we divide this force into 1) the force acting on the plane of the nozzle opening and 2) the remaining force of the external pressure acting on the rocket, we obtain (Fig. 6a):

\[ p_0 f_e+\int_{S_0} p_0\,dS_0=0. \tag{42} \]

The action of the open aperture of area \(f_e\) consists in producing an unbalanced force, directed opposite to the thrust force, having magnitude \(-p_0 f_e\) and, consequently, equal to the integral in expression (42).

To estimate the term \(\int_{S_i} p_i\,dS_i\) in expression (41), it is necessary to consider the mass of gas contained in the body of the rocket. The theorem of conservation

of momentum requires that the integral of all forces acting on the surface surrounding this mass be equal to the decrease of momentum through this surface, or \(\dot m v_{ex}\), where \(v_{ex}\) is the mean velocity along the axis of symmetry for the entire area of the nozzle opening. The pressure acting on the gas is equal to \(p_i\) (the reaction of the engine walls to the pressure \(p_i\) exerted on them) and the mean pressure opposing the outflow of the gas from the opening of area \(f_e\) is equal to \(p_e\) (we recall that the pressure \(p_e\) at the nozzle exit is not necessarily equal to \(p_0\)). Equating the total force acting on the gas to the rate of change of momentum, we obtain:

\[ -\int_{S_i} p_i\, dS_i + p_e f_e = -\dot m v_{ex}. \tag{43} \]

The negative sign on the right is introduced because it is customary to consider the absolute value of the quantity \(\dot m\), which in reality is a substantially negative quantity, representing a decrease of the total mass.

The mean velocity \(v_{ex}\) in the axial direction is less than the true outflow velocity \(v_e\), which has a component perpendicular to the axis. To take into account the correction for the divergence of the flow, a factor \(\lambda\) is introduced, so that we write \(v_{ex}=\lambda v_e\). The value of \(\lambda\) depends on the nozzle divergence angle and can be calculated. For example, if half the nozzle divergence angle is \(15^\circ\), then \(\lambda=0.985\).

The thrust equation (41) can now be transformed with the aid of (42) and (43); we then obtain:

\[ F=\lambda \dot m v_e + (p_e-p_0)f_e. \tag{44} \]

The two terms on the right-hand side are sometimes denoted as the thrust due to velocity and due to pressure. If \(p_e=p_0\), then the entire thrust depends on velocity, and it can be shown, determining \(v_e\) by means of equation (24) and differentiating \(F\) with respect to \(p_e\), that under this condition \(F\) reaches its maximum. A nozzle for which \(p_e=p_0\) is said to give “perfect” expansion; the corresponding area ratio \(\varepsilon=\dfrac{f_e}{f_t}\) can be calculated. It is equal to:

\[ \varepsilon=\frac{f_e}{f_t} = \frac{\Gamma}{ \gamma\left(\dfrac{p_0}{p_c}\right)^{1/\gamma} \sqrt{ \dfrac{2}{\gamma-1} \left[ 1-\left(\dfrac{p_0}{p_c}\right)^{(\gamma-1)/\gamma} \right] } }. \tag{45} \]

This equation is obtained from equation (31) for the area \(f\) by substituting the value of \(\dot m\) from equation (39) and simplifying under

by means of equation (17), written in the form \(\dfrac{p}{\rho}=R_sT\). Equation (45) is useful in practical calculations.

If \(p_e<p_0\), then the gases undergo “overexpansion,” and the thrust due to pressure is negative; this, however, is partly compensated by an increase in thrust due to velocity. If \(p_e>p_0\), then the thrust due to pressure is positive, i.e., directed in the same direction as the velocity thrust, but it does not fully compensate for the losses caused by the decrease in velocity \(v_e\) owing to incomplete expansion. Since the changes in thrust due to pressure and due to velocity partially compensate each other, the value of the total thrust \(F\) is rather little sensitive to changes in the area ratio \(\varepsilon\). Thus, for example, a nozzle having complete expansion at sea level gives, at an altitude of \(12{,}000\ \text{m}\), a thrust approximately 6% less than a nozzle calculated for operation at this altitude.

10. Design parameters for construction

a) Effective exhaust velocity

Experimental determination of \(v_e\) and \(p_e\) in equation (44) is difficult. Moreover, in reality the expansion is not strictly adiabatic, frictionless, and “perfect,” as a result of which the effective exhaust velocity \(c\) is introduced, defined by the equation

\[ c \equiv \lambda v_e + (p_e-p_0)\frac{f_e}{\dot{m}}, \tag{46} \]

so that equation (44) is rewritten in the form

\[ F=\dot{m}c. \tag{47} \]

The quantity \(c\) has already been discussed from another point of view in Section 2. This parameter is used in all practical calculations, although it may differ noticeably from the true velocity. In practice it is determined by measuring the thrust \(F\) and the mass flow rate \(\dot{m}\) by means of equation (47). The thrust \(F\) appearing in (47) includes the effects of pressure, friction, and flow divergence, and differs from the force produced by reaction alone, which is determined by equation (1).

b) Thrust coefficient

It has been found convenient to define the rocket thrust in the form

\[ F=C_{FX}\,p_c f_t. \tag{48} \]

Since all quantities entering into (48) are easily measured, the experimental value of the thrust coefficient \(C_{FX}\) can be found, with the aid of which one can select the throat cross-sectional area for

required thrust force and pressure in the chamber. The theoretical expression for the thrust coefficient, which can be compared with the experimental one, is obtained as follows.

Expressing the thrust force [equation (44)] as a function of pressure by means of (24) and (39), we obtain, under the assumption of parallel flow \((\lambda=1)\),

\[ F=\Gamma' \left\{\frac{2}{\gamma-1}\left[1-\left(\frac{p_e}{p_c}\right)^{\frac{\gamma-1}{\gamma}}\right]\right\}^{\frac12} p_c f_t+(p_e-p_0)f_e . \tag{49} \]

Dividing this equation by \(p_c f_t\), we arrive at an expression for the theoretical value of the thrust coefficient without taking account of the correction for flow divergence,

\[ C_F=\frac{F}{p_c f_t} =\Gamma' \sqrt{\frac{2}{\gamma-1}\left[1-\left(\frac{p_e}{p_c}\right)^{\frac{\gamma-1}{\gamma}}\right]} +\frac{p_e-p_0}{p_c}\frac{f_e}{f_t}. \tag{50} \]

This coefficient has its maximum value for “perfect” expansion, when \(p_e=p_0\).

In Fig. 7 a graph is given of the maximum value of \(C_F\) and of the corresponding optimal area ratio \(\varepsilon\) for the typical

Fig. 7. Graph of the theoretical maximum nozzle thrust coefficient \(C_F\) and the area ratio \(\varepsilon\) as a function of the pressure ratio for an ideal gas at \(\gamma=1.25\). \(C_F\) approaches the value 2.1 asymptotically as \(\varepsilon\) increases.

case \(\gamma=1.25\). These are the values obtained when \(p_e=p_0\). For some fields of application, as, for example, for long-range rocket projectiles, the pressure ratio \(\frac{p_c}{p_0}\) changes considerably. For a given nozzle it is impossible to obtain the correct area ratio at different pressure ratios, since the desig-

to design a “rubber” nozzle with variable $\varepsilon$ proved impossible without introducing undesirable complications. As a compromise, the value of $\varepsilon$ is often chosen to be optimal at the altitude reached after half the supply of propellant has been expended.

In actual nozzles, in the presence of friction and flow divergence, the measured value $C_{FX}$ in equation (48) is lower than $C_F$ as determined from (50). In the case of proper expansion and absence of friction we should have $C_{FX}=\lambda C_F$ [see equation (46)]; however, experimentally measured values of $C_{FX}$ are lower than those calculated in this way by 2–4%, depending on friction and other losses.

c) Characteristic velocity. In addition to the thrust coefficient $C_{FX}$, it is useful to introduce an empirical parameter that makes it possible in practice to calculate the propellant consumption, since equation (48) gives no indications with regard to $\dot m$. We rewrite equation (39), which determines $\dot m$, introducing a new parameter $c^*$, called the characteristic velocity, as follows:

\[ \dot m=\frac{\Gamma'}{a_c}\,p_c f_t=\frac{p_c f_t}{c^*}, \tag{51} \]

where

\[ c^*\equiv \frac{p_c f_t}{\dot m}=\frac{a_c}{\Gamma'}=\frac{1}{\Gamma'}\sqrt{\frac{\gamma R T_c}{M}}. \tag{52} \]

The introduction of the quantity $c^*$ is convenient because the effective exhaust velocity $c$ is expressed through it and through the thrust coefficient $C_{FX}$, namely:

\[ c=\frac{F}{\dot m}= \frac{F}{p_c f_t}\, \frac{p_c f_t}{\dot m} =C_{FX}\cdot c^*. \tag{53} \]

Thus, if for a given propellant, at a given chamber pressure, the quantity $c^*$ has been measured experimentally, then both the effective velocity $c$ and the mass flow rate $\dot m$ can be calculated. As is seen from (52), $c^*$ is determined only by the properties of the propellant and by the throat diameter. The characteristic velocity, therefore, does not depend on the conditions at the nozzle exit and characterizes the efficiency of the process of gas generation, or combustion. Usually the quantity $c^*$ is used as a characteristic of the propellant, despite the fact that its magnitude is also affected by the design of the combustion chamber. Using physicochemical methods for calculating $T_c$, $\gamma$, and $M$ entering equation (52), one can find the theoretical value of $c^*$; this value is approximately 10% higher than the experimental values calculated by measuring $p_c f_t/\dot m$.

d) Other operating parameters of a rocket. Besides the three parameters considered, \(c\), \(C_F\), and \(c^*\), there are two more, introduced in Section 2, which we can now express by means of the newly introduced quantities. The consumption of propellant by a rocket engine is determined by the specific consumption, in kilograms per second per 1 kg of thrust:

\[ w_{sp}=\frac{\dot m g}{F}=\frac{g}{c}=\frac{g}{c^* C_F}. \tag{54} \]

The reciprocal of the specific consumption is called the specific impulse, or efficiency index, and is expressed in kilograms of thrust obtained per 1 kg per second of propellant consumption:

\[ J_{sp}=\frac{F}{\dot m g}=\frac{c}{g}=\frac{c^* C_F}{g}. \tag{55} \]

Table I gives typical values and ranges of variation of all these parameters. This table provides an idea of the operation of existing types of rockets.

Table I

Summary of operating parameters of a rocket

Parameter	Symbol	Definition	Units	Typical value	Range of variation
Specific propellant consumption	\(w_{sp}\)	\(\dfrac{\dot w}{F}\)	\(\sec^{-1}\)	0.0051	0.0036—0.0100
Effective exhaust velocity	\(c\)	\(\dfrac{Fg}{\dot w}\)	\(m\cdot \sec^{-1}\)	1890	1000—2700
Specific impulse	\(J_{sp}\)	\(\dfrac{Ft}{W}=\dfrac{c}{g}\)	\(\sec\)	196	110—380
Thrust coefficient	\(C_F\)	\(\dfrac{F}{p_c f_t}\)	—	1.36	1.1—1.8
Characteristic velocity	\(c^*\)	\(\dfrac{p_c f_t g}{\dot w}\)	\(m\cdot \sec^{-1}\)	1389	900—1500

\[ \begin{aligned} \dot w&\text{ — fuel consumption }(kg/\sec),\\ F&\text{ — thrust force }(kg),\\ g&\text{ — }9.81\ (m/\sec^2),\\ W&\text{ — total weight of propellant }(kg),\\ t&\text{ — total duration of existence of the thrust force }(\sec),\\ p_c&\text{ — pressure in the chamber }(kg/cm^2),\\ f_t&\text{ — throat area }(cm^2). \end{aligned} \]

II. ROCKETS WITH SOLID PROPELLANT

CHARACTERISTICS OF ROCKETS WITH SOLID PROPELLANT

11. Application of rockets with solid propellant

For persons who during the war had no direct dealings with rockets, this word is associated with the idea of what in technical language are called artillery rockets—in other words, projectiles with a rocket principle of motion. The majority of artillery rockets are set in motion by rocket engines using solid propellant. This term means that before ignition the propellant has the properties of a solid body. The widely known “Bazooka” is an artillery rocket of this type. The German “V-2” is also an artillery rocket, but with a liquid propellant.

The military application of rockets is not new. However, as a result of the successful development of artillery guns in the second half of the nineteenth century, the latter acquired a considerable advantage over rockets both in range and in accuracy of fire. The question arises: what caused the considerable efforts toward the development of the artillery rocket in the Second World War? The answer is that the increased requirements for mobility and firepower, and the appearance of aircraft, once again made rocket artillery an important branch of weapons.

Another area that has recently attracted serious attention is the use of rockets on aircraft, either as an auxiliary or as the sole means of propulsion. Rockets with solid propellant are of little use for the latter purpose, since they provide significant thrust only for a short time. However, special types of such rockets have been developed to facilitate the takeoff of heavy and high-speed aircraft, which require ever larger takeoff areas. Auxiliary takeoff rockets (designated JATO—jet assisted take-off) are of particular importance in operations from aircraft carriers, where the length of takeoff areas is necessarily limited.

Rocket artillery is an incomparably more mobile weapon than ordinary artillery. The devices for launching a rocket are basically reduced to an aiming appliance; they have no device for directing powder gases, like a gun barrel, nor for absorbing a large recoil. Accordingly, the launching device has approximately the same weight as the projectiles of one salvo. This constitutes a sharp contrast, for example, with a 75-millimeter gun, which, with a total weight of 1170 kg, sends a projectile weighing 6.3 kg. Despite the fact that rocket artillery still yields

to guns in range and accuracy; thanks to its greater mobility, it can be moved to considerably more forward positions, which to a large extent compensates for the indicated shortcomings.

At the present time infantry is for the first time able to carry with it a “Bazooka” weapon capable of stopping the advance of a tank at a reasonable distance. A medium-sized airplane is capable of delivering a barrage of fire from 5-inch rockets equivalent to the broadside salvo of a destroyer, and, after launching the rockets, of having the same flight qualities as an airplane of the same type not equipped with rocket artillery at all. Meanwhile, the problem of absorbing the recoil of a 5-inch gun and of protecting the structural elements of an airplane is very difficult, and the reduction in the flight qualities of an airplane carrying such armament would be very considerable. From this it is clear what an important role rocket artillery plays as a supplementary weapon to ordinary artillery in modern maneuver warfare.

The principal property of a rocket with a solid propellant, in comparison with a liquid-fuel rocket, is its great simplicity in manufacture and handling. This advantage of simplicity leads to the fact that, if the required flight duration is less than approximately 30 seconds, then a solid-propellant rocket can, at a lower weight, give the same thrust as a liquid-fuel one. It is precisely the simplicity of manufacture and handling that gives definite advantages to this type of rocket as an auxiliary engine for takeoff and in many other cases where a device of purely auxiliary purpose, convenient to handle, is required.

The advantage of simplicity is somewhat weakened by the sensitivity of the solid-propellant rocket to climatic conditions, especially to the temperature of the surrounding air—an circumstance that usually does not exist for liquid-fuel rockets. Another disadvantage is the impossibility of regulating the rocket’s operation. For artillery rockets with a flight time not exceeding 1 second, regulation of the thrust during flight is of no importance. For JATO, whose operating time reaches up to 30 seconds, it would be very desirable to have means of regulation, up to and including the possibility of switching off and then switching on again the action of the rocket. However, such devices are not used, although a device for switching off an ignited rocket can be made sufficiently simply to prove practical. Regulation of the thrust during the operation of a solid-propellant rocket is practically impossible.

As we see, the most important and valuable characteristic of a rocket with a solid propellant is simplicity. Under conditions where control of the rocket’s flight is required, the liquid-fuel type is preferable. In this case any “improvements” of the solid-

substances leading to mechanical complications are usually unsuitable. As we shall see, the main direction of further development is the search for better propellants and materials for the structure and (for artillery rockets) the improvement of firing accuracy.

12. Principle of Operation

Rockets with a solid propellant consist of a solid charge placed in a combustion chamber and of a nozzle serving for the outflow of the combustion products. The reaction arising during this outflow is equal to the rocket’s thrust. The propellant, as in artillery guns, does not detonate but burns at a definite rate on its surface exposed to the action of the hot gases and flame in the combustion chamber. The distance \(r\) by which the surface of the propellant recedes in the direction normal to it is called the burning rate and is expressed in \(\text{cm}/\text{sec}\). This quantity depends on the pressure in the chamber and increases with it; in most modern rockets, at a pressure of about \(120\ \text{kg}/\text{cm}^2\), it amounts to from \(2.5\) to \(5\ \text{cm}/\text{sec}\).

Since the thrust, according to equation (1), is equal to the product of the exhaust velocity \(v\) and the mass flow rate, a large burning surface is needed to obtain a large thrust, giving a large outflow flux. In the same way, the duration of the existence of the thrust is determined by the burning rate. Since each combustion chamber can contain a definite quantity of substance, the thrust can be made very considerable but short-lived by increasing the burning surface, and conversely. The methods of arranging the charge in the combustion chamber are very varied. Here we shall confine ourselves to considering only two limiting types: rockets with a limited and an unlimited burning area, and we shall describe in relative detail the simplest type—a rocket with limited burning. Below, in Section 20, we shall qualitatively consider certain special questions connected with the operation of a rocket with an unlimited burning area. A schematic representation of these two limiting types is given in Fig. 8.

In a rocket with a limited burning area the charge is given the form of a circular cylinder. The cylindrical side surfaces and one of the end surfaces are protected from ignition by a corresponding coating or casing, and burning takes place only from one end. This type of rocket is often called a “cigarette-type” rocket or one with “end burning.” The duration of thrust in these rockets is proportional to the length of the charge and also depends on the pressure in the chamber and on the substance used. The thrust is proportional to the area of the circular burning surface and also depends on the pressure in the chamber, the nature of the substance, and the perfection of the rocket design.

In a rocket with an unrestricted burning area, the charge is often given the form of a hollow circular cylinder (a tubular charge). This charge is fastened at the end facing the nozzle or the head of the rocket, and has several centering supports along its length, but has no coating on the remaining surface. (The annular ends are sometimes protected from ignition by means of applied gaskets.)

Fig. 8. Two types of rockets with solid propellant

After ignition of the charge, combustion spreads over the entire surface, with the exception of a very small area at the points of attachment. The thrust is proportional to the total burning area and depends on the pressure in the chamber, the type of propellant, and the shape of the rocket and charge. The duration is proportional to the thickness of the cylindrical walls and depends on the pressure in the chamber, the type of propellant, the internal geometry of the combustion chamber, and the geometry of the charge.

For a rocket of any type, the pressure in the chamber depends on the ratio of the burning area of the charge to the cross-sectional area of the nozzle at its narrowest part (the throat). This ratio is sometimes called the “area ratio” \(K\) of the rocket. For rockets with an unrestricted burning area, the pressure varies considerably from one end of the charge to the other and depends to a noticeable degree on the geometry of the charge and the internal geometry of the chamber.

Fig. 9. Typical curves of thrust variation with time for a rocket with a restricted burning area (top) and with an unrestricted area (bottom). Note the “initial peak” in the latter

Typical curves of the dependence of thrust on time for both types of rockets are shown in Fig. 9. The small thickness and large burning area ...

...the burning of rockets with an unrestricted area contributes to the development of a short-duration but large thrust; for the same reasons, the opposite is true for rockets with a restricted area.

13. Special Characteristics

For solid propellants there exists a whole series of characteristics, knowledge of which is necessary for evaluating their behavior. These characteristics chiefly determine the applicability of one or another of the compositions currently in use. Such characteristics are: 1) sensitivity to temperature, 2) temperature limits, 3) ignition limit, 4) limiting pressure, 5) decomposition during storage.

Temperature sensitivity. When rockets of one type are stored at different temperatures, it subsequently turns out that those which were stored at higher temperatures develop a higher pressure in the chamber and greater thrusts than those stored at a low temperature. The duration for the high-temperature group is shorter, so that the total impulse for both groups proves to be almost the same. From this it may be concluded that temperature has a large effect on the rate of progress of the process in time, but very little on the total liberated energy and on the impulse. The temperature sensitivity \(\alpha\) is quantitatively determined by the equation

\[ \alpha=\frac{1}{p_c}\left(\frac{dp_c}{dT_p}\right)_k, \tag{56} \]

where \(p_c\) is the pressure in the chamber, and \(T_p\) is the temperature of the charge before ignition. For ballistite in a rocket with a restricted burning area, the temperature sensitivity is \(0.104\,(\mathrm{deg.\ C})^{-1}\); with an unrestricted burning area it is much higher—\(0.238\,(\mathrm{deg.\ C})^{-1}\). The compositions developed in our laboratory with asphalt and potassium perchlorate*) and certain special compositions developed by the National Defense Research Committee (NDRC) during the war have a temperature sensitivity of only \(0.043\,(\mathrm{deg.\ C})^{-1}\). The temperature sensitivity of ballistite is so great that, when it is used in rockets with an unrestricted burning area, the dangerous high pressures that develop at temperatures above \(49^\circ\mathrm{C}\), and the weak combustion at temperatures below \(-18^\circ\mathrm{C}\), limit the applicability of such rockets by the indicated temperature limits. The temperature sensitivity of the GALCIT and NDRC compositions is so small that no such difficulties arise. In the case of an unrestricted burning area

*) This composition is usually designated GALCIT; the name arose from the abbreviation Guggenheim Aeronautical Laboratory California Institute of Technology.

The great temperature sensitivity of rockets has a harmful effect on the accuracy of rocket-artillery fire.

Temperature limits. A special restriction of the temperature interval in which a rocket may be used sometimes arises in connection with changes in the mechanical properties of the charge. Certain grades of GALCIT soften at high temperature, and the charge changes its shape, with the formation of an abnormally large burning surface. As a result, when the rocket is ignited, a high pressure develops which may lead to the bursting of the rocket.

At low temperatures certain grades of GALCIT become so brittle that they crack upon ignition. The large burning area thereby exposed may entail an instantaneous explosion. NDRC compositions have no temperature limits associated with changes in the mechanical properties of the charge.

Ignition limit. By studying rockets having different nozzle-throat diameters, but otherwise identical, one can construct a graph of the dependence of the pressure in the chamber on the throat diameter (Fig. 10). As was to be expected, it turns out that the pressure falls as the diameter increases. However, when the throat diameter exceeds a certain definite limit, the pressure in the chamber falls to values much lower than those which could have been predicted on the basis of extrapolation along the curve corresponding to higher pressures. The pressure corresponding to this critical diameter is called the ignition limit. In rockets with the same and, moreover, large throat diameter, the pressure in the chamber varies from rocket to rocket in a random fashion. Finally, with a very wide nozzle, combustion is no longer continuous; instead, a series of irregular flashes occurs, accompanied by a characteristic crackling noise.

Fig. 10. Curve showing the value of the ignition limit

Thus, a propellant of a given composition cannot be used at an arbitrary pressure, and in designing rockets it is necessary to take care that the chamber pressure exceed this lower limit, in order that reproducible results be obtained for all rockets of the same series. If a small rocket weight is required, the existence of this limit constitutes a serious inconvenience, since the thickness of the walls of the combustion chamber is directly proportional to the volume of the latter and to the calculated pressure in the chamber.

The ignition limit for ballistite is about \(30\ \mathrm{kg/cm^2}\), for GALCIT about \(60\ \mathrm{kg/cm^2}\). NDRC compositions have a limit not exceeding \(6\ \mathrm{kg/cm^2}\).

The ignition limit for ordinary nitrocellulose powders used in artillery guns exceeds \(300\ \mathrm{kg/cm^2}\), as a result of which these powders are unsuitable for rockets. The ballistite used for the latter consists of approximately equal parts nitrocellulose and nitroglycerin and is a simple modification of the artillery powder of the same name.

Limiting pressure. Some compositions are safe in use only below a certain critical pressure, beyond which they give a violent explosion, and the result of the rocket’s action cannot be predicted. Cracking compositions with a granular structure are especially subject to this effect. Most commonly used compositions have this limit so high—above \(300\ \mathrm{kg/cm^2}\)—that the possibility of exceeding it need not be considered.

Decomposition during storage. Double-base powder compositions*) (ballistite and substances similar to it) slowly decompose during prolonged storage. This decomposition is autocatalytic. To suppress this process, an addition of diphenylamine is introduced, which neutralizes the catalytic action of the first portions of the decomposition products. It is not recommended to store ballistite for long periods under conditions of high temperature.

The group of compositions developed by NDRC, containing ammonium picrate and sodium nitrate, may soften and lose mechanical strength under the influence of absorption by the sodium nitrate of moisture from the atmosphere. Such compositions must be transported in waterproof packaging and protected from dampness during loading. GALCIT compositions, apparently, can be stored indefinitely without any signs of chemical decomposition.

THEORY OF THE OPERATION OF A ROCKET WITH A SOLID SUBSTANCE

14. Formula for the Burning Rate

An ideal rocket should have a columnar curve of the dependence of thrust on time. After ignition, the pressure in the chamber rapidly but uniformly rises to a value that remains constant throughout the entire burning time of the charge, after which it falls when the high-pressure gas remaining from combustion flows out of the chamber. For one and the same rocket, the thrust and the pressure in the cham—

*) By the term “double-base compositions” are meant substances whose principal components are nitrocellulose and nitroglycerin, as opposed to simple compositions, whose principal constituent is nitrocellulose.

are almost proportional over wide limits, so that one may speak of a “pillar-shaped” curve as identical for pressure and for thrust.

An irregular and improper course of the pressure curve in time is undesirable not only because of the difficulties of using a rocket with thrust that is not constant in time, but also because the strength, and with it the weight of the chamber walls, are determined by the need to withstand the maximum pressure. The velocity of gas outflow in a rocket with a solid propellant depends comparatively little on the pressure in the chamber, whereas the weight of the rocket is directly proportional to the pressure which it must withstand. The total weight of most modern rockets can be considerably reduced when propellants with a lower ignition limit appear.

We shall now set forth, in a very condensed form, the theory of the burning process in a rocket. We shall confine ourselves exclusively to the consideration of a rocket with a limited burning area, since it is simpler in principle; later (Section 20) the special problems connected with the use of rockets with an unlimited area will be described qualitatively. In the exposition we shall follow the work of Kármán and Malina[^6].

As we have already indicated, in practice very little can be done in the way of regulating a rocket with a solid propellant after it has been ignited. The parameters governing the operation of the rocket must be chosen during its very design. The formula given below for the burning rate describes, in a purely empirical way, the factors affecting the rate of burning of successive layers of the charge parallel to the burning surface. Attempts have been made to obtain theoretical formulas for the burning rate proceeding from the heat of combustion, heat capacity, and other fundamental characteristics of the substance. The combustion of a solid propellant composition is a very complex process, including a reaction in the solid phase, a reaction in the liquid phase, if such exists, and in the gaseous phase. In existing solid compositions the reaction even in each separate phase is very complex and proceeds under high pressure; moreover, the reactions in all simultaneously existing phases are not mutually independent. Years of empirical observations were required in order to create a suitable theoretical model for the kinetics of the simplest combustion process of a mixture of hydrogen and oxygen in the gaseous phase under low pressure. One may therefore expect that theories of the combustion of solid powder compositions will be of a wholly preliminary character and will contain many approximations. Although these theories often give a valuable conception of the combustion process, they are not sufficiently quantitative to serve as a satisfactory basis in investigations of the internal ballistics of a rocket.

Special experimental investigations have shown that the burning rate of a solid charge is a function of the pressure in the chamber \(p_c\), the temperature \(T_p\) of the charge before ignition, and the velocity

\(v\) the gas-flow velocity parallel to the burning surface (this factor is not always present in rockets with a restricted burning area), and the time \(t\) elapsed since ignition. Thus

\[ r = r(p_c, T_p, v, t). \tag{57} \]

For rockets with restricted burning, the dependence on \(v\) and \(t\) is small, and we shall use the expression

\[ r = a p_c^n, \tag{58} \]

where \(a\) and \(n\) are experimental constants depending on the type of working substance. The parameter \(a\) is assumed to depend on temperature, in contrast to \(n\), which does not depend on temperature. The value of \(n\) for commonly used compositions ranges from 0.4 to 0.8, while \(a\) has such an order of magnitude that \(r\) is \(2.5\ \text{cm/sec}\) at a pressure of \(120\ \text{kg}/\text{cm}^2\).

15. Stability of the Shape of the Burning Surface

Let us consider the diagram of a rocket with a restricted burning area shown in Fig. 11. Suppose that the burning surface, having area \(f_c\), is not flat, as shown in the drawing, but, owing to imperfections in manufacture of the charge or to nonuniform burning, has become concave or irregular in shape. The question is whether this surface, in the course of burning, will become leveled out or, on the contrary, will become more and more curved, leading to an increase in the burning area and, ultimately, to rupture of the rocket? The importance of this question follows from the fact that, as we shall see below, an increase of the burning area by 10% entails an increase of pressure by 70%.

Fig. 11. Diagram of a rocket with a restricted burning area, showing the notation adopted in the discussion of the stability problem.

Fig. 12. Diagrams illustrating the stability of the burning surface.

If we assume that the burning surface moves parallel to itself, with the same velocity at every point,

as is shown by the diagrams in Fig. 12, the irregularities tend to level out. The assumption that the velocity of displacement of the surface is the same at all points appears plausible on grounds of symmetry. For rockets with a limited burning area the velocity of the gas flow along the burning surface is close to zero and the pressure is almost uniform, so that, on the basis of (58), one may expect the same burning velocity at all points. Rockets with unlimited burning are considered in Section 20. The general conclusion is that the burning surface is stable, in other words, tends to remain plane during burning.

16. The basic differential equation

We shall now show that in a rocket with a solid propellant there exists an equilibrium pressure, in the sense that if the burning area of the charge remains constant, then the pressure in the chamber also remains constant throughout the entire period of burning. Assuming that the burning area remains unchanged, we may write the following equation of conservation of mass:

\[ \left\{ \begin{array}{c} \text{Mass burning}\\ \text{per unit}\\ \text{time} \end{array} \right\} = \left\{ \begin{array}{c} \text{increase of the mass}\\ \text{of gas in the}\\ \text{combustion chamber}\\ \text{per unit time} \end{array} \right\} + \left\{ \begin{array}{c} \text{mass flowing out}\\ \text{through the nozzle in}\\ \text{the same time} \end{array} \right\}, \]

or, with the aid of equation (39):

\[ r f_c \rho_p=\frac{d}{dt}\left(\rho_c V_c\right)+\Gamma \sqrt{k_s T_c}\, f_t \rho_c . \tag{59} \]

Here

\[ \Gamma=\sqrt{\gamma}\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}}=\frac{\Gamma'}{\sqrt{\gamma}}; \tag{60} \]

\(f_c\) is the burning area, assumed constant; \(\rho_c\) is the density of the gas in the combustion chamber; \(\rho_p\) is the density of the solid propellant; \(V_c\) is the volume of gas in the combustion chamber, increasing as the charge burns; \(R_s\) is the technical gas constant, equal to the universal gas constant divided by the mean molecular weight of the gas that sets the rocket in motion; \(\gamma\) is the ratio of specific heats at constant pressure and constant volume for the gaseous products of combustion; \(T_c\) is the temperature of the gas in the combustion chamber, often called the “flame temperature”; and \(t\) is the time elapsed since the beginning of burning. In deriving equations (59) and (60) it was assumed that the laws relating to an ideal gas remain valid for the products of combustion and that \(\rho_c\) exceeds the critical pressure for the nozzle discharging the gas.

If the burning in the rocket now takes place at constant pressure, then the temperature \(T_c\) retains the constant value given by the equation

\[ c_p T_c=H_p, \tag{61} \]

where \(c_p\) is the mean heat capacity for the gas at constant pressure and \(H_p\) is the heat of combustion of the working substance at constant pressure.

We neglect the influence of \(T_c\), the initial temperature of the charge. For ballistite \(T_c\) is about \(2760^\circ\mathrm{C}\), and for other compositions it varies from \(1600\) to \(2200^\circ\mathrm{C}\). A sufficiently complicated calculation shows that, \(10\) milliseconds after ignition, \(T_c\) differs by no more than \(5\%\) from the value given by equation (61), and does not deviate from it by more than a few percent under large pressure fluctuations. In view of this, we may assume \(T_c\) to be constant in time. Under this assumption we have:

\[ \frac{d}{dt}\left(R_s T_c \rho_c V_c\right) =R_s T_c \frac{d}{dt}\left(\rho_c V_c\right). \tag{62} \]

Multiplying equation (59) by \(R_sT_c\), taking into account the equation of state

\[ p_c=R_s\rho_cT_c, \tag{63} \]

we define the quantity \(p_p\), constant for the given working substance:

\[ p_p=\rho_p R_sT_c, \tag{64} \]

and finally obtain the equation

\[ r f_c p_p=\frac{d}{dt}\left(p_cV_c\right)+\Gamma\sqrt{R_sT_c}\,f_t p_c, \tag{65} \]

which coincides with equation (59), with the difference that the densities have been replaced by pressures.

The constant \(p_p\), defined by equation (64), has the dimensions of pressure, and since the density of the solid charge is much greater than the density of the gas in the combustion chamber, the quantity \(p_p\) has a numerical value from \(6000\) to \(12000\ \mathrm{kg}/\mathrm{cm}^2\). High values of \(p_p\) indicate a large specific impulse of the given working substance and, to a certain degree, \(p_p\) characterizes the quality of this substance.

For the geometry shown in Fig. 11, and assuming \(f_c\) to be constant, we obtain:

\[ \frac{d}{dt}\left(p_cV_c\right) =V_c\frac{dp_c}{dt}+p_c\frac{dV_c}{dt} =V_c\frac{dp_c}{dt}+p_c r f_c. \tag{66} \]

We can now write equation (65) in the form

\[ \frac{dp_c}{dt} =\frac{1}{V_c}\left[ r\left(p_p-p_c\right) -\Gamma\sqrt{R_sT_c}\,\frac{f_t}{f_c}\,p_c \right]. \tag{67} \]

In addition, \(V_c\) is subject to the equation

\[ V_c=V_c^0+f_c\int_0^t r\,dt, \tag{68} \]

where \(V_c^0\) is the “free volume” of the combustion chamber, remaining unoccupied

upon placing a charge in it. If \(r\) is known as a function of \(p'_c\), then equations (67) and (68) can be solved jointly by numerical methods, which gives the law of variation of the pressure in the combustion chamber with time.

17. Stability of the Pressure in the Chamber

We note that if the bracket in the right-hand side of equation (67) is equal to zero, then the pressure in the chamber does not change with time. Let us call the corresponding value of the pressure in the chamber \(p_s\). Since \(1/V_c\) is always positive, the algebraic sign of \(\dfrac{dp_c}{dt}\) is the same as that of the bracket in the right-hand side.

Let us denote the two terms entering into this bracket as follows:

\[ m_i(p_c)=r(p_p-p_c)=(p_p-p_c)\alpha f_c^n, \tag{69} \]

\[ m_0(p_c)=\Gamma \sqrt{R_s T_c}\,\frac{f_t}{f_c}\,p_c. \tag{70} \]

In most cases the mechanical strength of rockets does not permit pressures in the chamber exceeding \(600\ \mathrm{kg/cm^2}\); recalling the order of magnitude of \(p_p\), we see that (69) can be rewritten in the form

\[ m_i(p_c)=p_p\alpha p_c^n. \tag{71} \]

Equation (67) shows us that if \(m_i\) exceeds \(m_0\), then \(p_c\) increases; in the opposite case \(p_c\) decreases. In Fig. 13, \(m_i\) and \(m_0\)

Fig. 13. Stability curve of the pressure in the chamber for \(n=0.76\).

Fig. 14. Stability curve of the pressure in the chamber for \(n=2\).

are plotted as functions of \(p_c\) for a working substance having \(n=0.76\), which is an ordinary value. In Fig. 14, by way of example, these same curves are plotted for \(n=2\).

In Fig. 13 we see that if the pressure in the chamber exceeds \(p_s\) by a small amount, then \(m_0\) exceeds \(m_i\), and \(\dfrac{dp_c}{dt}\) becomes negative, and the pressure returns to the value \(p_s\). Similarly,

Thus, if \(p_c\) falls below \(p_s\), then \(m_1\) becomes greater than \(m_0\), and the pressure again rises to \(p_s\). In this way the pressure in the chamber is stable with respect to small deviations.

On the other hand, from Fig. 14 we see that if \(p_c\) exceeds \(p_s\), then \(m_1\) exceeds \(m_0\), \(\dfrac{dp_c}{dt}\) is positive, and the pressure continues to increase. Conversely, if \(p_c\) falls below \(p_s\), then the pressure tends to fall to zero. Thus, for \(n=2\), \(p_s\) represents a point of unstable equilibrium.

These different cases occur for all values of \(\dfrac{f_c}{f_t}\) in equation (70), since varying this ratio gives a family of straight lines passing through the origin. An elementary calculation shows that for \(n>1\) the curve \(m_i\) is convex in the direction of the \(p_c\) axis, while for \(n<1\) it is concave in the direction of the same axis. In view of this, our conclusion is general in character, and we may conclude that the pressure in the chamber is stable if the exponent \(n\) in the burning-rate formula is less than 1.0, and unstable if \(n\) is greater than 1.0. If the exact value \(p_p-p_c\), which we replaced by \(p_p\), is taken into account in the theory, it turns out that the limiting case \(n=1.0\) is also stable. In practice, however, rockets with a propellant for which the exponent \(n\) exceeds 0.85 prove to be so sensitive to small deviations in the method of manufacture that they become unreliable and dangerous in handling.

18. Equilibrium Pressure in the Chamber

If we assume that \(n\) is less than 0.85, then, as we know, there exists a stable chamber pressure \(p_s\) (which hereafter we shall denote by \(p_c\), in order to conform with the standard notation\(^*\)). This stable pressure is found if, in equation (67), we put \(\dfrac{dp_c}{dt}=0\), which gives, for the stable pressure, the equation

\[ \frac{f_c}{f_t} = \frac{\Gamma \sqrt{R_s T_c}}{r(p_p-p_c)}\,p_c = \frac{\Gamma \sqrt{R_s T_c}}{a(p_p-p_c)}\,p_c^{\,1-n}. \tag{72} \]

A typical curve describing the variation of the pressure in the chamber as a function of the area ratio \(\dfrac{f_c}{f_t}\), sometimes denoted by \(K\), is shown—

\(^*\) We have said nothing about the rapidity with which the chamber pressure recovers after the deviation that has occurred. A simple approximate solution of equation (67), under the assumption that \(V_c\) is constant, gives

\[ \Delta p_c(t) = \Delta p_c \exp\left[ - \frac{\left(\dfrac{p_p}{p_c}\right)Kt}{\dfrac{V_c}{f_c}} \right], \]

where \(\Delta p_c\) is the initial deviation, and \(K\) is a constant of the order of \(0.5\ \text{cm/sec}\). The recovery time is of the order of \(0.2\) seconds.

is shown in Fig. 15 together with the burning-rate curve. As could have been foreseen, higher pressures are obtained when the area ratio is increased, i.e., when the nozzle opening is small.

In older propellant compositions the ignition limit is situated so that the point characterizing the rocket falls on a very steep part of the area-ratio curve. The approximate solution of equation (72), expressing \(p_c\) in terms of \(\dfrac{f_c}{f_t}\), shows that \(p_c\) varies as \(\left(\dfrac{f_c}{f_t}\right)^{\frac{1}{1-n}}\), and for \(n=0.8\) a very steep increase according to a fifth-power law is obtained. This illustrates the delicate balancing between the amount of gas developed in the rocket as a result of combustion and the outflow of gas through the nozzle. As is easy to understand, what is involved here is a dynamic equilibrium depending on the successive course of combustion. Small changes in the properties of the substance may exert a large influence on the burning rate and at the same time have no noticeable effect on the combustion temperature, specific gravity, and other properties used for product control in the manufacture of ordinary powders. Consequently, special test methods are required for product control in the manufacture of rocket compositions.

Figure labels:
Top graph: ordinate—“Pressure in the chamber \(p_c\)”; abscissa—“Area ratio \(f_c/f_t\)”; curve label—\(p_c \sim A\left(\dfrac{f_c}{f_t}\right)^{\frac{1}{1-n}}\).
Bottom graph: ordinate—“Burning rate \(r\)”; abscissa—“Pressure in the chamber \(p_c\)”; curve label—\(r = a p_c^n;\ 0<n<1\).

Fig. 15. Curves of equilibrium pressure as a function of the area ratio and of burning rate as a function of pressure.

Assuming the ratio \(\dfrac{f_c}{f_t}\) to be constant and neglecting \(p_c\) in comparison with \(p_p\), we differentiate equation (72) with respect to the charge temperature \(T_p\) (recalling that \(a\) is assumed to depend on \(T_p\)). We then obtain:

\[ \frac{1}{p_c}\left(\frac{dp_c}{dT_p}\right)_k = \frac{1}{1-n}\,\frac{1}{a}\,\frac{da}{dT_p}. \tag{73} \]

Here the factor \(\dfrac{1}{1-n}\) appears again. Indeed, one might regard the quantity \(\dfrac{1}{a}\dfrac{da}{dT_p}\) as a measure of the temperature sensitivity of the propellant, but in a loaded rocket, for \(n\), for example, equal to 0.8, this effect is increased fivefold. Some success-

known chemical modifications of solid compositions, currently known, reduce the value of \(n\) from 0.76 to 0.45.

Thus, a solid composition that operates satisfactorily in rockets must have an exponent in the burning-rate formula lying between 0 and 0.85, and a stable burning surface.

19. Designing Rockets with a Limited Burning Area

Let us give a brief description of the process of designing a rocket, which may provide a certain intuitive idea of a rocket with a solid propellant. The designer specifies the required thrust and the duration \(t_b\), the so-called “burning duration.” He then selects a suitable propellant composition, for which he has at his disposal an experimental burning-rate curve, as well as an experimental curve of the dependence of pressure on the area ratio, like that shown in Fig. 15*). Knowing the ignition limit for the propellant, the designer chooses a pressure slightly exceeding this limit. It is desirable that the pressure be as low as possible, since this leads to a reduction in weight and increased safety. From the experimental data the specific impulse \(J_{sp}(p_p)\) of the propellant is obtained for the selected pressure.

Fig. 16. A typical rocket with a solid propellant (JATO), having a thrust of 90 kg and a duration of 8 sec, intended for accelerating aircraft during takeoff. The nozzle, igniting device, and safety disk are visible.

The weight of the required propellant is then equal to

\[ W_p=\frac{Ft_b}{J_{sp}}. \tag{74} \]

From the experimental burning-rate curve (Fig. 15) the designer obtains the value \(r(p_p)\), whence he finds the charge length \(l_p\):

\[ l_p=rt_b. \tag{75} \]

Knowing the density of the propellant composition \(\rho_p\), he calculates the end area \(f_c\) and the diameter \(d_c\) of the charge from the equations

\[ l_p f_c \rho_p = W_p \tag{76} \]

and

\[ \frac{\pi d_c^2}{4}=f_c. \tag{77} \]

* Although the simplified theory set forth correctly predicts the general character of the curve for the area ratio, it is entirely based on the hypothesis of an ideal gas, in particular for the outflow of gas from the nozzle. In view of this, it remains necessary to use the experimental curve.

From the design pressure \(p_c\), the designer knows the area ratio \(K\) required to create such a pressure (Fig. 15). Then

\[ f_t = \frac{f_c}{K} \tag{78} \]

and

\[ \frac{\pi d_t^{\,2}}{4}=f_t . \tag{79} \]

After this, the problem reduces to designing the nozzle shape, as discussed in Sections 8 and 9, and then to designing the metal parts. Special problems arise in connection with the high temperature of the combustion products, but we cannot deal with them here. A typical auxiliary take-off rocket is shown in Fig. 16.

20. Special Questions Relating to Rockets with an Unlimited Burning Area

In rockets with an unlimited burning area, combustion occurs over the entire surface of the charge; therefore the stream of issuing gases must flow along the burning surface. Obviously, the pressure in the chamber then varies along the length of the charge and has its highest value at the front end and its lowest at the rear end of the charge, adjacent to the nozzle. The average velocity of the gas stream, parallel to the charge, is zero at the front end and increases in the direction of the nozzle. This is shown in the diagram of Fig. 17. In addition, as the charge burns away, the average pressure in the chamber falls, since more space is freed for the outflow of gases past the charge. Thus we are dealing with a non-stationary problem.

In a well-designed rocket, the burning rate is almost the same at all points of the charge, since although the higher pressure increases the burning rate at the front end of the charge, the increased velocity of the gas flow at the rear end also accelerates burning, almost compensating for the effect of pressure. Obviously, correct

Fig. 17. Distribution of velocity and pressure in a rocket with an unlimited burning area shortly after take-off and after half of the charge has burned.

designing such a rocket is a difficult task. The necessity of proper design is connected with the fact that nonuniform burning of the charge leads to its cracking and to the possibility of its breaking up into separate unburned fragments, and to incomplete development by the rocket of its design impulse. Even good rockets thus lose up to 5% of their charge.

Long-range artillery rockets must have a small diameter in order to reduce drag. But when the charge is placed in a narrow long tube, the outflow of the developing gases is impeded because of the narrowness of the external and axial channels—the effect of “choking” of the charge. If it comes to the point that the gas flow somewhere on the surface of the charge reaches sonic velocity, then the rocket will almost certainly function poorly, if it does not burst. Thus the requirements of external and internal ballistics prove to be diametrically opposed.

Finally, since the weight of the casing is proportional to the volume of the chamber and to the design pressure in it, the fullest possible filling of the combustion chamber with the charge is desirable. The choking effect that appears when too large an amount of charge is placed limits the volumetric utilization coefficient (the ratio of the volume of the working substance to the total volume) to a value somewhat below 75%.

The effect of temperature sensitivity is greatly increased by the geometry of a rocket with unrestricted burning area, and although a well-thought-out geometrical form of the charge can reduce this effect, nevertheless a rocket with a restricted burning area generally has less temperature sensitivity than a rocket with unrestricted area, for the same working substance.

In the final analysis, the principal problems in designing rockets with solid working substances of all types are, above all, a deeper understanding of the mechanism of combustion, which would make it possible to control the burning rate; then, improvement of the geometrical forms of the rocket and reduction of temperature sensitivity.

21. Chemistry of solid working compositions

Solid working compositions may be roughly divided into two classes: 1) homogeneous, 2) composite. The most important homogeneous compositions are ballistites, consisting of approximately equal amounts of nitrocellulose and nitroglycerin. The composite compositions include two types of compositions developed by NDRC. One of them is a ballistite with a considerable addition of inorganic salts to reduce temperature sensitivity; the other, existing in many modifications, is a powdery mixture of ammonium picrate and sodium nitrate, fused under high pressure with the addition of 10% artificial resins.

as a binder. The original composition of GALCIT consisted of 75% powdered potassium perchlorate (oxidizer), mixed with 25% asphalt (fuel). These materials are mixed and poured into the combustion chamber while hot, and are then cooled to a solid state resembling asphalt road pavement. General characteristics of solid rocket compositions are given in Table II.

Table II

General characteristics of solid rocket compositions

Composition	Flame temperature (°C)	Exhaust velocity (m/sec)	Specific impulse (sec)	Burning rate at a pressure of up to 90 kg/cm² in cm/sec	Temperature sensitivity	Density kg/l
Ballistite	2800	2100	200	1.75	high	1.61
NDRC*)	1700—2200	1650	180	0.5—2.5	low	1.85
GALCIT*)	1700—2200	1650	180	3.5	"	1.85

REFERENCES CITED

1. Millikan, Roller and Watson, Mechanics, molecular physics, heat and sound (Ginn, 1937), p. 264.
2. Roberts, Heat and thermodynamics (Blackie, 1910, ed. 3), p. 293.
3. O’Brien and Hickox, Applied fluid mechanics (Mc Grow-Hill, 1937), p. 43.
4. O. Reynolds, Phil. Mag., (5), 21, 185 (1886).
5. Durand, Aerodynamical theory, Springer, Berlin, 1934—1936, vol. III, pp. 213—222.
6. Th. v. Karman and F. Malina, Unpublished report, 1940.

*) Large quantities of smoke emitted present an inconvenience in some cases.