UNCLASSIFIED

Tables of similarity functions defining the flow field behind expanding cylindrical shock waves are presented here for y=5/3 and 7=7/5. A brief discussion includes the differential equations and boundary conditions for these functions together with an analytical solution for them.

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FIR,
The value of an analysis such as this to determine the relative importance of propellant density and specific impulse is limited because I of the interplay of unrelated variables. Quantitative criteriao such as the index n in the expression I can never be applied with generality.
[Analysis can only indicate when density is a major concern and when its Ieffect is minor.

B.
High density is a desirable property of a solid propellant in all I f.applications.

C.
The penalty suffered by a propellant because of low density ranges [ L frcm being completely negligible to very serious. The magnitude of the penalty depends on: 1. the particular mission and design of the stage of the rocket in which the propellant is to be used,

2.
the other properties of the propellant besides density.

D.
A low-density propellant is at its greatest disadvantage in a sealevel, first-stage oooster, regardless of the types of constraints (e.g., constant-weight or constant-volume). A solid propellant with a density as low as 1.0 g/cm 3 is virtually ruled out for this application, at least at present.

E.
The constant-volume-type constraint exacts a greater penalty on a propellant because of low density than does a constant-weight-type --constraint. In terms of equivalent specific impulse, the penalty for a given drop in density will usually be from two to five times as great in the constant-volume case as in the constant-weight case.
On the very tenable premise that new high specific-impulse solid propellants will be used only in upper stages in the foreseeable future, only constant-weight constraint needs to be considered in respect to their use.

G.
The penalty for low-propellant density in an upper stage (with constant-weight constraint) is in direct proportion to the combustion chamber pressure. With present structural materials, and with a pressure of 200 psi, a propellant of density 1.0 g/cm 3 loses about 5 sec of equivalent specific impulse compared to a propellant of density 1.6 g/cm 3 . At 1000 psi chamber pressure, the penalty is five times as great, or 25 sec. It is imperative, therefore, that low-density propellants have good low-pressure burning properties; but if they have, the effect of density can, even now, be slight.

H.
Low combustion temperature and low erosion and corrosion tendencies of the exhaust gases -properties that are often concommitant -are favorable to a low nozzle weight. Good properties in these respects can more than offset the effect of a considerable drop in density.

I.
For space applications, a low absolute burning rate is also desirable. This can serve to mitigate the effect of low density.
J. Future improvements in structural materials will constantly reduce the importance of propellant density for upper-stage use. Eventually, we can expect density to be an insignificant factor in this application.
K. Structure weight reduction, itself, will continually diminish in importance as a research goal. This leaves high specific impulse per se as the outstanding objective of long-term, solid-propellant research.
Nature of the Problem The reason the relative importance of propellant density and specific t limpulse has sometimes been in dispute is that it is extremely sensitive to the context --either hypothetical or real --in which the question $ is examined. To fix the basis of comparison, interest is usually centered on a given rocket stage (i) with a specific mission to perform, e.g., to * -impart a total velocity increment AV to a payload Mui The following question is then asked: "If the size of the stage and the mission requirements are fixed, how many seconds of specific impulse (or +) must be traded for a given change in propellant density (+ or -)?" The answer to this question, it turns out, depends very strongly on what is meant by T size" -whether it is weight or volume -and on what the mission requirements are.

The basic equation is
where g is the acceleration of gravity (constant), I sp, is the specific impulse and R i is thc mass ratio, defined as toLal weight when the i.th stage is ignited (and after separation of the prior stage) divided by the weight when it has burned out (and before separation of the subsequent stage).
The way in which various factors enter Eq. (1) can be seen by writing Ri in an expanded form, as follows: Here, A i is the propellant weight fraction in the ith stage and i is the "payload ratio," the ratio of weight of the payload (  What, therefore, is to be gained by analysis of this problem?
Obviously, only some heavily qualified and rather vague generalities.
But the net conclusion, even of this preliminary examination, is worth attention: The importance of density is bound up with the influence of so many other propellant characteristics, as well as with the mission and system requirements, that it would be very unwise to focus attention on it during the early stages of research on new propellants when these other properties, both favorable and unfavorable, are still unknown, and when the end uses can still only be conjectured on. ?7

B. The Constraints
The effect of density is usually derived under either a constant-5 weight or constant-volume constraint.
In the constant-weight case, only the term A in Eq. (3) is sensibly affected by a change in density; in the constant-volume case, both A and M will be subject to change. m Mathematically, there is no difficulty in deriving the results.
But to do so, further assumptions -or constraints -have to be imposed. Due to the important way in which propellant characteristics besides density affect A (as outlined in the foregoing section), these ad hoc assumptions may be just as important in their own right as the main constraint of fixed volume or fixed weight.
One implicit constraint is chamberapressure.
Since the only way in which propellant density enters the propulsion equation is through its effect on the proportional distribution of weight as between propellant and structural hardware, some assumption must be made about the chamber pressure because pressure is the major criterion for the strength of the motor casing required. For want of a more general or more plausible assumption, pressure is usually considered to remain unchanged when the propellant density is altered.
Another implicit assumption is that the total weight of all the inert parts other than the motor case remains constant.
Since the weight of these other parts often totals to two or three times that of the motor case (in upper stages), and since these components are strongly affected by the propellant characteristics (other than density), 8 fixed inerts weight can almost never be maintained in practice when one propellant is substituted for another. The constant-inerts-weight i lconstraint is therefore highly artificial; but --as in the case of chamber pressure --there seems to be no better alternative, since I the propellant characteristics that influence these weight factors are not functionally related to density in any general way.
For the sake of brevity, and to be consistent with conventional practice, we shall further perpetrate the use of the terms "weightlimited" and "volume-limited" to designate the two major constraint groups. But it must be borne in mind that these are merely labels.
The risk taken in using these terms is that they might be construed as if to cover important practical situations in rocket design.
Unfortunately, situations as simple as this seldom arise in the world of reality.

C. The Density-Exponent Expression
For purposes of a performance criterion or a tradeoff relation, specific impulse I and density p are often combined in an expression sp The meaning here is that if two propellants of different I and p sp are considered, their relative performance can be gauged by the value n of Isp n . Or, from another standpoint, if the value of this quantity is equal for two propellants, they will give equal performance under conditions for which the expression is valid. Even within the narrow range of such restrictions, the expression 7 1pn cannot be used to compare propellants of widely differing density.

SP
This is because n is derived on the basis of differential changes in I and p as follows: sp n" V where AV is the velocity increase of the rocket due to the burning of the propellant in the particular stage in question. (Equation (4) is derived from the significance ascribed to I p n as a figure of merit.

SP
It may be regarded as a definition of n consistent with such an ascription.) In general, the expression for n as defined in Eq. (4) involves parameters that themselves depend on p; hence only an instantaneous value of n at a particular density can be obtained.
Equation (4), it may be incidentally noted, demonstrates the "tradeoff" significance of n. Here n appears as the ratio of the change in velocity for a given percentage increment in density to the change in velocity for an equal percentage increment in impulse.
In this sense, n may be regarded as the ratio of a density "index" to an impulse index. However appealing it is to have a single number representing the relative importance of impulse and density, the * practical use of the concept is severely limited. In fact, generalizations on the basis of a particular value ior n are always misleading and can result in seriously erroneous conclusions.
The important information to be gained from the Isp n concept is not a precise knowledge of the significance of propellant density, but rather a general idea of when density plays a prominent role and when it plays a minor one.  factor. When 0 is zero, n is also zero. The factor 0 depends inversely on the strength-to-density ratio of the case material; hence as lighter and stronger materials are found, the effect of density will be reduced.
The most important effect on the value of 0 (and hence on n), however, comes from the chamber pressure Pc, since 0 depends on Pc directly. Aside from its dependence on chamber pressure, the exponent n is also seen to depend on the mass ratio R of the rocket stage in question.
This again illustrates why it is impossible to assign to n any fixed numerical value. When n is as high as 0.25, the effect of density becomes fairly sizeable.
In Table I, the specific impulse tradeoff for a density change of 0.1 g/cm 3 is 4 sec when n = 0.25. Thus, for large changes in density, the effect could be quite serious.
The value n = 0.25, however, is extreme for this case. If an average" value can be considered to have meaning, one would be justified to take, say, n = 0.10 to 0.15. This would mean a tradeoff of about 2 sec in specific impulse for a density change of 0.1 9/cm 3 . 9@ jI IV. DIFFERENTIAL EXPONENT n FOR A VOLUME-LIMITED SYSTEM * It is to be expected that density will be more important in a volume-limited than in a weight-limited system. Here again, however, For a volume-limited, first-stage booster, the expression for n is found to involve only the mass ratio R as follows: For an upper stage (which we shall designate the i t h stage), n is less th the value given by Eq. (7)  to be equal to 3.0, and the payload ratio was assumed to be the same in all the lower stages and to have a value 0.25.

Irom
FiegUrC it -I .ccn that,. .... te rge of R that is crcmn for most practical rockets (i.e., R = 2 to 5), the differential density exponent n for the booster lies between 0.7 and 0.5. For a second-stage motor, the figure is equal to about one-half the value for a booster stage, i.e., n ranges from 0.35 to 0.25. For a third-stage motor, it is still lower, lying between 0.26 and 0.19. For stages above the third, the n-values differ little from those for the third stage. (See the curve for the fourth stage in Figure 1.) For volume-limited systems, therefore, density has a pronounced influence on performance in a booster (first) stage; but it is not so important in the second and higher stages. However, as reference to Table I will illustrate, the effect of propellant density in the volume-limited case is never by any means negligible.
An examination of the n-curves for constant-weight and those for constant-volume in Figure 1 may create the impression --since the two sets of curves overlap --that the difference between the two constraint conditions becomes small for the upper stages. Actually, this is probably never true because the value of 0 is usually smaller, the higher the stage. This is because chamber pressure is generally lower in the higher stages, and 0 bears a direct relation to pressure. In most actual cases, it is probable that n would be anywhere from two to five times larger for the constant-volume case than for the constantweight case.
In view of the range in values of n seen in Figure 1, it is difficult to see a reason for choosing n = 1 to represent any actual situation in rocketry. And yet often when the density factor is considered in making comparisons of propellants, it is introduced through the quantity known as "density-impulse" or "impulse-density" I p, i.e., the exponent n is assumed equal to one. From the foregoing analysis it is clear that n = 1 is an upper limit; this is the case for constant-volume constraint when R = 1, i.e., when the propellant represents a negligible fraction of the total weight.
The quantity Is p is proportional to the total impulse available from a fixed volume of propellant. When the propellant weight-fraction approaches zero, the velocity increment AV is just proportional to the total impulse, and, therefore, to Is p; but when the propellant comprises an appreciable fraction of the total weight, AV is proportional to I spPn where n is less than one. The reason is as follows: When the density of the propellant changes, the weight of the (constant-volume) rocket also changes by an amount that depends on the proportion of propellant in the total weight (this proportion being reflected in the magnitude of the mass ratio R). Therefore, when density changes either up or down the propellant has either more or less mass (respectively) to accelerate. As a result, the effec; on AV of a propellant density change is actually always less than it is for the limiting case when the weight of the propellant is negligible compared to the total. This is indeed the rationalization for the descending trend of the curves representing constant-volume constraint in Figure 1.
The "density-impulse" expression Is p, therefore, considerably exaggerates the importance of density in all real situations. It is hard to justify any value of n as being 'representative or average ; but, certainly, the value should never be greater than about 0.7. In 18 fact, insofar as a quantity of the type I pn is used to compare SP advanced high specific-impulse propellants, very much lower values of n., lying between, say, 0.1 and 0.2 would have more validity.
The reasons for this will be made clear in the following section.

V. EFFCT Or GROSS DENSITY CHANGES OF THE SOLID PROPELLANT ON PERFORMANCE OF A 1WEIGHT-LIMITED ROCKET ENGINE
The foregoing sections dealing with the exponent n in Isp show that a solid propellant overy density, say g/cm 3 , will probably never be of interest for booster stages, either weight-limited or volumej limited. This is not so great a handicap as it might seem. The major rocket systems of the future are likely to be multi-stage, each stage making approximately the same contribution to the final payload velocity.
Since advanced high-performance propellants are likely to be expensive or in short supply -probably both -for some time yet to come, economy will dictate they be used only where the greatest payoff per pound is to be obtained; and this, of course, is in upper stages.* Therefore, at least * As a specific example, consider a three-stage rocket of which the third stage is the ultimate useful payload; that is to say, there are two rocket motors comprising stages No. 1 and No. 2, and No. 3 stage is inert. For simplicity, assume the mass ratio R and the propellant loading fraction A are the same for stages No. 1 and No. 2 and equal respectively to 2.72 and 0.90. Also, assume that the same solid propellant is used in both these stages. If a new propellant becomes available that has a 10 per cent higher specific impulse, and other properties essentially the same as the old propellant, it could be added to the first stage, second stage, or.both. We wish to calculate for each case the increased weight of payload that could be delivered with the same final velocity, under the restriction that the total weight of the rocket be unchanged.
Analysis shows that the increase in payload under these conditions is the same when either the first-stage motor or the second-stage motor is filled with the new propellant. The efficiency, pounds of added payload per pound of propellant, is therefore in inverse proportion to the weight of propellant in the motor. In this example, the first stage contains 3.33 times as much propellant as the second, and the efficiency is therefore less than one-third as much when the propellant is put in the first stage as in the second. The actual figures are in the first applications, the new high-I 5 s propellants will be used only in upper stages regardless of density. And since n is smaller for upper stages, slight density differences between various advanced propellants will be of little consequence.
The remaining question is to determine the effect of gross differences. For this purpose, a new approach to the problem is necessary because the Isp n concept applies, in principle, only to infinitesimal changes.
Where upper stages are concerned, weight is of much greater significance than volume. For example, so far as the performance of lower stages is affected, the weight of the penultimate stage is indistinguishable from the weight of the ultimate useful-payload stage.
The weight at this point is therefore a prime consideration. The volume, on the other hand, is usually small for upper stages anyway,

I
In the case of a similar four-stage rocket, the efficiency drops by another factor of 1/3.33 when the new propellant is put in the first stage, that is to 0.011 lb added payload per pound of the new propellant. The efficiency in the first stage here, therefore, is only about 9 per cent of what it is when the propellant is put in the third, or penultimate stage of the same rocket.
(These calculations do not, of course, take account of the effects of gravity or drag. The comparisons would be essentially the same, however, if these refinements were added.)

!a
limitations will restrict the size of the booster but will not usually appreciably affect upper stages. It will, therefore, be necessary to treat only the weight-limited case when we are concerned primarily with the use of new advanced propellants in upper-stage motors.
Consider the problem that a "reference" solid-propellant engine is to be replaced by one containing a propellant of substantially different density. To compare the propellant performance, we shall Limpose the restriction that the total weight of the rocket be held constant. We shall then calculate the change in specific impulse A sp (tradeoff) necessary to compensate for the change in density, i.e., to keep the velocity change of the rocket AV. due to the burning of 1 the propellant in this engine the same.
The basic formula of rocket engine performance (in the absence of gravity and drag) is where we now use I instead of I to stand for specific impulse. In sp Eq. (10) AV,, as before, is the increase in velocity due to burning of propellant in the 1th stage rocket motor, g is the acceleration of gravity, and R is defined by  (21) and (22) are given in Table II Table II is based on an S-value of 1.1 x 10 -4 (g/cm/(lb/in2) and the (b) set of 1.65 x l0-4 . These numbers were arrived at in the following way: The safety factor F was taken as 1.2 (20% safety factor), and the geometrical factor G as 3, which, being the number for a sphere, is the lowest value this factor can have. The value S = 1.1 x 10 4 corresponds to a combination of these values of F and G with a strength-to-weight ratio a = 0.6 x l06.  value of S represents a more conservative choice, which could allow for a larger safety factor and/or a less favorable vessel shape. Although this second figure might seem unduly conservative, it is thought to be realistic in consideration of the fact that the propellant chamber liner has not been considered as part of the density-sensitive structure weight.

M+ M+ M
In practice, one should probably include an allowance for the liner in calculating 0. Therefore, the two numbers chosen as a basis for S probably represent, for these purposes the extremes of the range *of this factor that will apply in the next two or three-year period.
When they are examined for a particular value of density, the figures in Table II  The total spread in Al in this case is thus seen to be from 2 to 35 see, that is, from a truly negligible figure to one that is so great as to put the propellant entirely out of serious consideration.
n Pressure is by far the most important factor affecting AI. The effectiveness of a propellant with density 1.0 g/cm 3 would be seriously eroded if the pressure were much above 300 psi. Therefore, a propellant with this low density must be able to burn satisfactorily at pressures below 300 psi if it is to maintain essentially unimpaired the advantage it may have in specific impulse. In practice, this is probably not a severe requirement.

29
VI. SYNOPSIS 1 jor goals in solid rocket propulsion development. t 'W high-specific impulse propellants and high propellant loading I fractions C Progress in the second direction (b) will come through developmet of materials with higher strength/weight ratio, by the I discovery \ improved nozzle designs and throat materials, and -possibly --y development of cool-burning propellants with non-erosive, non-corrosive product gases and good low-pressure burning properties.
It is apparent that as A approaches more closely the limiting value of 1.0, that is, as the relative weight of structure compared to that of propellant approaches the vanishing point -the effect of propellant density tends to disappear entirely. (This statement applies in the case of constant-weight constraint, which is the one of most concern where upper stages are concerned.) Actual calculations show (cf Table II) that even with presently attainable tankage-structure factors and combustion-chamber pressure levels the penalties for low propellant density are not large. Therefore, since the future trend will be to reduce the density effect still farther, there will come a time --if it is not already here --when concern for propellant density will be very sliCht indeed.
In fact, a point of steeply diminishing returns will also be reached in future efforts to increase the loading factor A; and this will mean more concentrated attention on the single goal of high specific 30 " i impulse to the exclusion of most other concerns. As new materials with better strength/weight are found, and as other improvements are 3 made to reduce the structure factor (1 -A), further improvements will obviously become increasingly difficult to make. Besides, a Ii given percentage reduction in the structure fraction (1 -A) has a 1 smaller and smaller payoff in terms of equivalent specific impulse as A approaches closer and closer to 1.O.* Eventually, therefore, I efforts in this direction will not be worth the candle.
We are therefore led to conclude that high specific impulse