APPENDIX 1

CONVERSION FACTORS AND CONSTANTS

Conversion Factors

Acceleration

(arranged alphabetically)

(L t-2) *

1 m / s e c 2 - 3.2808 ft/sec 2 - 39.3701 in./sec 2 1 ft/sec 2 - 0.3048 m / s e c 2 - 12.0 in./sec 2 go - 9 . 8 0 6 6 5

Area

m / s e c 2 - 32.174 ft/sec 2 ( s t a n d a r d g r a v i t y pull at e a r t h ' s s u r f a c e )

(L2)

1 ft 2 - 144.0 in. 2 - 0 . 0 9 2 9 0 3 m 2 1 m 2 - 1550.0 in. 2 - 10.7639 ft 2 ! in. 2 - 6.4516 x 10 -4 m 2

Density

( M L 3)

Specific gravity is d i m e n s i o n l e s s , b u t h a s the s a m e n u m e r i c a l v a l u e as e x p r e s s e d in g / c m 3 or k g / m 3 1 k g / m 3 - 6.24279 x 10 -2 l b m / f t 3 - 3.61273 x 10 -5 l b m / i n . 3 1 l b m / f t 3 - 16.0184 k g / m 3 1 l b m / i n . 3 - 2.76799 x 104 k g / m 3

density

*The letters in parentheses after each heading indicate the dimensional parameters (L = length, M = mass, t = time, and T = temperature).

727

728

APPENDIX 1

Energy, also Work or Heat (M L 2 t -2) 1.0 1.0 1.0 1.0 1.0

Btu - 1055.056 J (joule) k W - h r - 3.60059 x 106 J ft-lbf - 1.355817 J cal - 4.1868 J kcal - 4 1 8 6 . 8 J

Force (M L t -2) 1.0 lbf = 4.448221 N 1 dyne=10 -SN 1.0 kg (force) [used in Europe] = 9.80665 N 1.0 ton (force) [used in Europe] = 1000 kg (force) 1.0 N = 0.2248089 lbf 1.0 m i l l i n e w t o n (raN) = 10 -3 N Weight is the force on a mass being accelerated by gravity (go applies at the surface of the earth)

Length (L) 1 m = 3.2808 ft - 39.3701 in. 1 ft - 0 . 3 0 4 8 m = 12.0 in. 1 in. -- 2.540 c m = 0.0254 m 1 mile = 1.609344 k m = 1609.344 m = 5280.0 ft 1 nautical mile = 1852.00 m 1 m i l - 0.0000254 m - 1.00 x 10 -3 in. 1 m i c r o n ( ~ m ) = 10 -6 m 1 a s t r o n o m i c a l unit (au) - 1.49600 x 1011 m

Mass (M)

1 slug-

32.174 l b m 1 k g - 2.205 lbm - 1000 g 1 l b m - 16 o u n c e s - 0.4536 kg

Power (M L 2 t -3)

1 Btu/sec -- 0. 2924 W (watt) 1 1 1 1

J/sec = 1.0 W = 0.001 k W cal/sec = 4.186 W h o r s e p o w e r -- 550 ft-lbf/sec = 745.6998 W ft-lbf/sec = 1.35581 W

Pressure (M L -1 t -2) 1 bar-105N/m 2-0.10MPa 1 a t m - 0.101325 M P a 14.696 psia

APPENDIX 1

729

1 m m of m e r c u r y - 13.3322 N / m 2 1 M P a - 10 6 N / m 2 1 psi or lbf/in. 2 - 6 8 9 4 . 7 5 7 N / m 2

Speed (or linear velocity) 1 1 1 1

(L t -1)

ft/sec = 0.3048 m / s e c - 12.00 in./sec m/sec = 3.2808 ft/sec = 39.3701 in./sec k n o t = 0.5144 m/sec m i l e / h r = 0.4770 m/sec

Specific Heat

(L 2 t -2 T -1)

1 g-cal/g-°C- 1 kg-cal/kg-K1.163 x 10 -3 k W - h r / k g - K

Temperature

1 Btu/lbm-°F-

4.186 J / g - ° C -

(T)

1 K-9/5R-1.80R 0°C - 273.15 K 0°F - 459.67 R C - ( 5 / 9 ) ( F - 32)

Time

F - ( 9 / 5 ) C + 32

(t)

1 m e a n solar d a y - 24 hr - 1440 m i n - 86,400 sec 1 c a l e n d a r year = 365 days - 3.1536 x 107 sec

Viscosity

(M L-1 t-1)

1 c e n t i s t o k e - 1.00 x 10 -6 m2/sec 1 centipoise - 1.00 x 10 -3 k g / m sec 1 lbf-sec/ft 2 - 47.88025 k g / m sec

Constants

R

!

Vmole e SO

M e c h a n i c a l e q u i v a l e n t of heat - 4.186 joule/cal - 777.9 ft-lbf/Btu = 1055 j o u l e / B t u U n i v e r s a l gas c o n s t a n t - 8314.3 J / k g - m o l e - K 1545 f t - l b f / l b m - m o l e - R M o l e c u l a r v o l u m e o f an ideal gas - 22.41 liter/kg-mole at s t a n d a r d conditions E l e c t r o n charge - 1.6021176 x 10 -19 c o u l o m b P e r m i t t i v i t y of v a c u u m - 8.854187 x 10 -12 f a r a d / m G r a v i t a t i o n a l c o n s t a n t - 6 . 6 7 3 x 10 -11 m3/kg-sec Boltzmann's constant 1.38065003 x 10 -23 J / ° K E l e c t r o n mass 9.109381 x 10 -31 kg Avogadro's number 6.022142 x 1026/kg-mol S t e f a n - B o l t z m a n c o n s t a n t 5.6696 x 10 -8 W / m Z - K -4

APPENDIX 2 I

I

II

PROPERTIES OF THE EARTH'S STANDARD ATMOSPHERE

Sea level pressure is 0.101325 M P a (or 14.696 psia or 1.000 atm). Altitude (m)

T e m p e r a t u r e (K)

0 (sea level) 1,000 3,000 5,000 10,000 25,000 50,000 75,000 100,000 130,000 160,000 200,000 300,000 400,000 600,000 1,000,000

288.150 281.651 268.650 255.650 223.252 221.552 270.650 206.650 195.08 469.27 696.29 845.56 976.01 995.83 999.85 1000.00

Pressure R a t i o

Density ( k g / m 3)

1.0000 8.8700 6.6919 5.3313 2.6151 2.5158 7.8735 2.0408 3.1593 1.2341 2.9997 8.3628 8.6557 1.4328 8.1056 7.4155

1.2250 1.1117 9.0912 x 10 -1 7.6312 x 10 -1 4.1351 x 10 -1 4.0084 x 10 .2 1.0269 x 10 .3 3.4861 x 10 .5 5.604 x 10 .7 8.152 x 10 .9 1.233 x 10 .9 2.541 x 10 -~° 1.916 x 10 -11 2.803 x 10 -12 2.137 x 10 -13 3.561 x 10 -15

x x x x x x x × x × x x

10 -1 10 -1 10 -1 10 -I 10 .2 10 .4 10 .5 10 .7 10 .8 10 .9 10 -10 10 -11

× 10 -11

x 10 -13 x 10 -14

Source. U.S. Standard Atmosphere, National Oceanic and Atmospheric Administration, National Aeronautics and Space Administration, and U.S. Air Force, Washington, DC, 1976 (NOAA-S/T1562).

730

APPENDIX 3

SUMMARY OF KEY EQUATIONS FOR IDEAL CHEMICAL ROCKETS

Parameter Average exhaust velocity, Vz (m/sec or ft/sec) (assume that v~ = 0)

2-16

v2 -- c - (Pe - p 3 ) A 2 / r h

When Pz = P3, Vz = c V/[2k/(k -

'0 2 =

= , ~

Effective exhaust velocity, c (m/sec or ft/sec) Thrust, F (N or lbf)

Equation Numbers

Equations

1)]RTI[1 - ( p z / p l ) (k-l)/k]

3-16 3-15 3-32 2-16 2-17 3-31 2-14

- h2)

c = c~'-" = F / r h = Isgo

c = /)2 nt- (P2 -- p 3 ) A 2 / r h F -- cth -- C m p / t p F = C F p 1A t

F - - rh'o 2 q- (P2 - p 3 ) A 2 F = rhlsgo = Is~i'

Characteristic velocity, ¢* (m/sec or ft/sec)

c* = c~ C F = p ! A t / r h

3-32

c* =

3-32

/

~T1

k~/[2/(k + 1

Thrust coefficient, CF (dimensionless)

3-31, 3-32

CF -- c / c * = F / ( p I A , )

F-sSF-T-f q_P2 -- P3 A2

Total impulse Specific impulse, I s (sec)

3-32, 3-33

c* -- I s g o / C g -- F / ( m C F )

cE=

'

)]~k+l)/Ck'l)

Pl At I t - f F dt = Ft = Isw I s -- c / g o = c* C F / g o I s = F / r h g o -- F / ~ v Is -- v z / g o -k- (P2 - P 3 ) A z / ( r h g o ) Is = I t / ( m p g o ) = I t / w

1-

3-30 2-1, 2-2, 2-5 2-5 2-16 2-4, 2-5

731

732

APPENDIX 3

Parameter Propellant mass fraction, (dimensionless)

Equations mo

--- m p / m

-

mf

2-8, 2-9

0 -mo

4-4 2-7

~'= I - M R Mass ratio of vehicle or stage, MR (dimensionless)

IVIR =

mr--2 =

mo

IH 0

-

mr

DI 0

= m l , / ( m t- + mp) mo = m f + mp

Vehicle velocity increase in gravity-free vacuum, Av (m/see or ft/sec) (assume that Vo = O) Propellant mass flow rate, rh (kg/sec or lb/sec)

Equation Numbers

Au = - - c l n ~ = c

Into° mf

2-10 4-6 4-5, 4-6

= c in m o / ( m o - r a p ) = c ln(mp + m f ) / m f rh = A v / V

= Alvl/V1

- - A t v t / Vt --" A 2 v 2 /

V2

rh = F / c = p l A , / c *

3-24 2-17, 3-31

+ 1)](k+l)/(k-l) rh = p, A , k _

2/(

3-24 v/kRTl

Mach number, M (dimensionless) Nozzle area rate,

tn = m p / t p M =v/a

3-11

= v / k,/FUT At throat, v = a and M = 1.O

3-19

¢ = A2/A I

3-14

Isentropic flow relationships for stagnation and free-stream conditions Satellite velocity, us, in circular orbit (m/see or ft/see) Escape velocity, Ve (m/see or ft/sec) Liquid propellant engine mixture ratio r and propellant flow rh

T o / T = (po/p) (k-l)/k = ( V / V o ) (k-l) = (p,./pr) (k-1)/k = (V,./V,.) k-l

3-7

v~ = R o v / g o / ( R o + h)

4--26

v e = R o v / 2 g o / ( R o + h)

4-25

r = rho/rh/. rn = rho + m r my = rh/(r + 1)

6-1 6-2 6-4 6-3 7-2

T,./T,.

m o = rrh/(r + 1t

Average density Pay for (or average specific gravity) Characteristic chamber length L* Solid propellant mass flow rate rn Solid propellant burning rate r Ratio of burning area A b to throat area A f Temperature sensitivity of burning rate at constant pressure Temperature sensitivity of pressure at constant K

PoPr(r + J ) tOav

--

r&, + Po

L*= V,/A,

8-9

rh = Ahrpo

11--1

r = ap~'

11-3

K = Ab/A ,

11-14

I(~T ) ~7P--" r

yrK - - ~pl

11-4 P

-g-f K

11-5

APPENDIX 4

DERIVATION OF HYBRID FUEL REGRESSION RATE EQUATION IN CHAPTER 15 Terry A. Boardman

Listed below is an approach for analyzing hybrid fuel regression, based on a simplified model of heat transfer in a turbulent boundary layer. This approach, first developed by Marxman and Gilbert (see Ref. 15-9), assumes that the combustion port boundary layer is divided into two regions separated by a thin flame zone. Above the flame zone the flow is oxidizer rich, while below the flame zone the flow is fuel rich (see Fig. 15-7). An expression is developed to relate fuel regression rate to heat transfer from the flame to the fuel surface. For the definition of the symbols in this appendix, please see the list of symbols in Chapter 15. Figure A4-1 illustrates a simplified picture of the energy balance at the fuel grain surface. Neglecting radiation and in-depth conduction in the fuel mass, the steady-sate surface energy balance becomes

Qc - pfi.h,,

(A4--1)

where Qc is the energy transferred to the fuel surface by convection, and pf, i', and hv are respectively the solid fuel density, surface regression rate, and overall fuel heat of vaporization or decomposition. At the fuel surface the heat transferred by convection equals that transferred by conduction, so that

OT

(A4-2) y=0

where h is the convective heat transfer film coefficient, AT is the temperature difference between the flame zone and the fuel surface, Xg is the gas phase conductivity, and OT/Oyly=o is the local boundary layer temperature gradient evaluated at the fuel surface. The central problem in determining the hybrid fuel regression rate is thereby reduced to determining the basic aerothermal 733

734

APPENDIX 4

Combustion port oxidizer flow

> Boundary layer edge

Te pe, U

,

~

---'---xgOT

(2r T F, eg

.....

[

.... /~//////////////Fla

'/,~//////////'[////"

/,

i//./////////,~~//f/////./Z (pV)S •

1"

I I

FT-T-

{

Fh7

m e z o ne

]

s.,s

I.

.aT

I

Fuel grain

L

t x f - ay -

I

General steady-state energy balance:

Energy input fuel surface = Energy out of fuel surface econvection + Oradiationin - Oconductionout + Ophasechange + eradiation out 0 T[ y=o + aegaT 4 = tcf -0~T + Pfihv hAT or Xg ~---~

+ esoT 4

Neglecting radiation and solid phase heat conduction leg aT I

ly=o

= pfrh v

FIGURE A4--1. Energy Balance at Fuel Grain Surface.

properties of the boundary layer. Approximate solutions to the flat plate boundary layer problem are well established (Ref. A4-1) and show that the heat transfer coefficient at the wall (in this case, the fuel surface) is related to the skin friction coefficient via the following relationship (called Reynolds' analogy)

c,, - yC f pr_2/3

(A4-3)

where CUis the skin friction coefficient with blowing (defined in this case as the evolution of vaporized fuel from the fuel surface and proportional to pv evaluated at the fuel surface), Ch is the Stanton number, and Pr is the Prandtl number (Stanton, Prandtl, and Reynolds number definitions are summarized in Table A4-1). Furthermore, the Stanton number can be written in terms of the heat flux to the fuel surface as

Qs

C h -- AhloeU e

(A4-4)

APPENDIX 4

735

TABLE A4-1. Dimensionless Numbers Used in Hybrid Boundary Layer Analysis Parameter

Definition

Comment

Stanton number, Ch

Dimensionless heat transfer coefficient Ratio of momentum transport via molecular diffusion to energy transport by diffusion Ratio of gas inertial forces to viscous forces (x is distance from leading edge of fuel grain)

,De He Cp

Prandtl number, Pr

CP#egO

Xg PeUe x

Reynolds number, Rex

go#e

where Ah is the enthalpy difference between the flame zone and the fuel surface, and Pe, Ue are the density and velocity of oxidizer at the edge of the boundary layer. Combining Equations A4-1, A4-3, and A4-4, the regression rate of the fuel surface can be written as

i" - Cf Ah PeUepr_2/3 2 hv pf

(A4-5)

From boundary layer theory, one can show that the skin friction coefficient without blowing (Cfo) is related to the local Reynolds number by the relation

Cf°-0.0296Rex°2

(5 x 105 < Re~ < 1 x 107) m

.

(A4-6)

Experiments (Ref. A4-2) conducted to determine the effect of blowing on skin friction coefficients have shown that Cf is related to Cf0 by the following

C f = 1.27/3_0.77

(5 < / 3 < 100)

cj0

(A4--7)

where the blowing coefficient/3 is defined as fl--

(PV)s

PeUeCf/2

(A4--8)

In a turbulent boundary layer, the Prandtl number is very nearly equal to 1. It can be shown that for Pr = 1, /7, as defined in Eq. A4-8, is also equal to Ah/hv (see Appendix 5). Noting that peUe is the definition of oxidizer mass velocity (G), Eq. A4-5 can be written in the final form as

? -- 0.036

G08 (~) °2fl0.23 PS

(A4-9)

736

APPENDIX 4

The coefficient 0.036 applies when the quantities are expressed in the English Engineering system of units as given in the list of symbols at the end of Chapter 15. In some hybrid motors, radiation may be a significant contributor to the total fuel surface heat flux. Such motors include those with metal additives to the fuel grain (such as aluminum) or motors in which soot may be present in significant concentrations in the combustion chamber. In these instances, Eq. A4-1 must be modified to account for heat flux from a radiating particle cloud. The radiative contribution affects surface blowing, and hence the convective heat flux as well, so that one cannot simply add the radiative term to Eq. A4-1. Instead, one can show (Ref. A4-3) that the total heat flux to the fuel surface (and hence the fuel regression rate) is expressed by

Q, - pzi'h~ - Qce-Qr'd/Q" -F Qrad

(A4-10)

which reduces to Eq. A4-1 if Q r a d - 0. The radiation heat flux has been hypothesized to have the following form Qrad - o-orT4(1 - e -Ac-)

(A4-11)

where the term 1 - e -ACz is Sg, the emissivity of particle-laden gas. Here, a is the Stefan-Boltzmann constant, ot is the fuel surface absorptivity, A is the particle cloud attenuation coefficient, C is the particle cloud concentration (number density), and z is the radiation path length. By assuming that the particle cloud concentration is proportional to chamber pressure and the optical path length is proportional to port diameter, experimenters (see Ref. 15-14) have approximated the functional dependencies of Eq. A4-11 for correlating metallized fuel grain regression rates with expressions of the following form

i.

-

-

?{Go, L, (1

-

e-P/Pref), (1

-

e-n/Z)'ef)}

(A4-12)

REFERENCES A4-1. H. Schlichting, "Boundary Layer Theory," Pergamon Press, Oxford, 1955. A4-2. L. Lees, "Convective Heat Transfer with Mass Addition and Chemical Reactions," Combustion and Propulsion, Third A GARD Colloquium, New York, Pergamon Press, 1958, p. 451. A4-3. G. A. Marxman, E. E. Woldridge, and R. J. Muzzy, "Fundamentals of Hybrid Boundary Layer Combustion," AIAA Paper 63-505, December 1963.

APPENDIX 5

ALTERNATIVE INTERPRETATIONS OF BOUNDARY LAYER BLOWING COEFFICIENT IN CHAPTER 15 Terry A. Boardman

The blowing coefficient/3 is an important parameter affecting boundary layer heat transfer. It is interesting to note that, although it is defined as the nondimensional fuel mass flow rate per unit area normal to the fuel surface, it is also a thermochemical parameter equivalent to the nondimensional enthalpy difference between the fuel surface and the flame zone. In terms of the fuel mass flux,/3 is defined as

fl--

(PV)s

Pe"eCyl2

(A5-1)

For the definition of the letter symbols please refer to the list of symbols of Chapter 15. Noting that C U / 2 - Ch Pr -2/3, Eq. A5-1 can be rewritten as fl -

(pV)s

pr-2/3

PeUe

Ch

(A5-2)

Recalling that the heat flux at the fuel surface is

Q~ - h(Tf - L )

(A5-3)

and that the definition of Stanton number is

Ch ~

(A5-4) pel,teCp

Eq. A5-4 can be rewritten as 737

738

APPENDIX5

Q, Ch = AhPeUe

(A5-5)

From energy balance considerations, heat flux to the fuel surface in steady state is equivalent to

Q,- pf~h~

(A5--6)

so that Eq. A5-2 becomes

A__hhpr_2/3

fl _ (pv)_.____A ~

(A5-7)

pfr hv Since (Pv)s = pfi, at the fuel surface, the fuel regression rate, Eq. A5-7, becomes Ah n~

fl - - -7-- p r - 2 / 3

As has been previously stated, the Prandtl number in a turbulent boundary layer is very nearly equal to 1 so that the final form for the blowing coefficient is Ah hv Thus, the blowing coefficient is shown to describe the nondimensional enthalpy difference between the fuel surface and flame zone, as well as the nondimensional fuel surface regression rate.