space propulsion lab

7/25/2019 space propulsion lab

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Energetic Problems in Aerospace Propulsion

Notes for Students

Chapter 11 - Appendices and Exercises

Solid Rocket Motors

Adriano AnnovazziAvio - Space Propulsion

22 Corso Garibaldi, I-00034 Colleferro, Rome, Rm, Italy

and

Luigi T. DeLucaSPLab, Department of Aerospace Engineering, Politecnico di Milano

Campus Bovisa, I-20156, Milan, Mi, Italy

Preliminary International Edition


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2 Space Propulsion - DeLuca 2004

Contents

1 EXERCISE No. 1: BOOST - SUSTAINER MISSION 4

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Properties of Combustion Chamber . . . . . . . . . . . . . . . . . . . . 5

1.3 Optimum Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Evaluating Ballistic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Boost or Acceleration Phase . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 Sustainer or Regime Phase . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.7 Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.7.1 Thrust Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.7.2 Propellant Grain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.7.3 Total Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.8 Effects of Initial Temperature . . . . . . . . . . . . . . . . . . . . . . . . 14

1.8.1 Boost or Acceleration Phase . . . . . . . . . . . . . . . . . . . . . . . . 141.8.2 Sustainer or Regime Phase . . . . . . . . . . . . . . . . . . . . . . . 15

2 EXERCISE No. 2: SUSTAINER - BOOST MISSION 17

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Properties of Combustion Chamber . . . . . . . . . . . . . . . . . . . . 192.3 Optimum Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Evaluating Ballistic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Sustainer Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6 Boost or Acceleration Phase . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7.1 Thrust Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7.2 Propellant Grain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7.3 Total Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.8 Effects of Initial Temperature . . . . . . . . . . . . . . . . . . . . . . . . 27

2.8.1 Sustainer or Regime Phase . . . . . . . . . . . . . . . . . . . . . . . . 27

2.8.2 Boost or Acceleration Phase . . . . . . . . . . . . . . . . . . . . . . . . 28

3 EXERCISE No. 3: PERFORATED GRAIN SIZING 30

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Ugello Adattato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.1 Soluzione Grafi

ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.2 Soluzione Analitica (verifica) . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Grano Propellente Solido . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.1 Grano a Combustione Frontale . . . . . . . . . . . . . . . . . . . . . . 35

3.3.2 Grano a Combustione Radiale Progressivo . . . . . . . . . . . . . . . . 36

3.3.3 Grano a Combustione Radiale Neutro . . . . . . . . . . . . . . . . . . 37

3.4 Lunghezza Totale Motore (perdmax= 60cm) . . . . . . . . . . . . . . . . . . 39

3.5 Impulso Totale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6 Accensione . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.7 Erosione (da migliorare) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.8 Velocit di Efflusso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.9 Espansione Isentropica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45


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Chapter 11 - Solid Rocket Engines - AppEx 3

4 CONCLUSIONI 46

5 BIBLIOGRAPHY 47

List of Figures

List of Tables

Chapter 11 - Exercises

SOLID ROCKET MOTORS

This appendix to Chap.11 deals with a variety of specific matters regarding performanceand design of SRMs; matters of general interest regarding thermochemical rockets in generalare discussed in Chap. 09. Principal features of SRMs, with respect to the important classof LREs, concern simplicity and promptness of use as well as economy of realization butto the detriment of modest level and control of performance. Therefore, solid rocket motorsare favorite for all situations where promptness is a premium (e.g., emergency maneuvers),military applications, and in general for civil tasks where maximum performance is not amandatory constraint.

Solid rocket motors are typically distinguished in three main categories: space (for space

access or navigation), ballistic, and tactical motors. This classification reflects sensible differ-ences in terms of size, operating ambient, and design.

EXERCISES

EXERCISE No. 1: BOOST - SUSTAINER MISSIONEXERCISE No. 2: SUSTAINER - BOOST MISSIONEXERCISE No. 3: PERFORATED GRAIN SIZING


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1 EXERCISE No. 1: BOOST - SUSTAINER MISSION

Preliminary sizing of solid rocket motor of boost - sustainer type

We wish to get the preliminary sizing of a SRM capable to carry out the two-step propul-sive mission sketched in the following figure:

boost5 s_____| || || || | sustainer 25 s| b_________________________b ______________________________e________ time, sThe assigned input data are: burning times for each flight segment, tb1= 5s and tb2= 25s; thrust under optimum expansion, T1(z0= 2.5104 m) = 104 + 100Ckg; combustion chamber pressure, pc1= 60 + N atm; For sake of simplicity, consider a unique "cigarette" cylindrical grain consisting of

two different unmetallized composite propellants featuring, under the reference conditionsofTref= 300K and pref= 68 atm, the following steady-state ballistic properties:

PROPELLANT 1

densityp1= 1.70g/cm3

thermal sensitivity p1= 0.0021/

Cp, atm 1 10 100

rb(p), cm/s 0.768 2.165 6.102Tf(p), K 2562 2777 3010

PROPELLANT 2

density p2= 1.60g/cm3

thermal sensitivityp2= 0.0051/

Cp, atm 5 50 68

rb(p), cm/s 0.972 2.030 2.240Tf(p), K 2499 2623 2640

1. Please perform a preliminary sizing of the whole motor (nozzle, combustion chamber,and propellant grain) at the optimum expansion altitude ofz0. In particular, pleasededuce the diameter of the "cigarette" cylindrical grain required to achieve the wantedpropulsive mission. Please evaluate how the main ballistic parameters change duringthe two flight segments.

2. How would the whole propulsive mission be affected by a decrease of the initial tem-perature fromT0= Tref toT0= 275K ?

3. Draw and discuss the steady temperature profile in the solid propellant grain.

Consider a monophase gaseous mixture expanding under chemically frozen conditions.When necessary, with due justifications assume typical values for missing properties andmake reasonable assumptions for undefined processes. Assigned data depend on the digitsC and N, that identify the alphabet position of the first letter of respectively the candidate

family name andfirst name.


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1.1 Introduction

Let us assume as typical values

=

M= 25g/gmole or kg/kmole; k = 1.25; g0= 9.807m/s2;


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1.3 Optimum Expansion

z= z0= 25,000 m At this point of the exercise we can only evaluate the nozzle transversalshape !

Forpc/pa = pc/pe = 60/0.02516 = 2384.7, we can determine for chemically frozen expan-sion the ideal values of both cF

(cF)ideal =

r2

k2

k 1 ( 2

k+ 1)k+1k1

s1

pepc

k1k

(1)

=

vuut2

1.252

1.25 1( 2

1.25 + 1)

1.25 + 1

1.25 1

vuuut1

0.02516

60

1.25 11.25 (2)

= 1.8483 (3)

and =Ae/At, being

1/ = At/Ae=

k+ 1

2

1k 1

pepc

1k

vuuuutk+ 1k 1

1

pepc

k 1k

=

1.25 + 1

2

11.25 1

0.02516

60

11.25

vuuuut1.25 + 11.25 11

0.02516

60

1.25 11.25

= 3.1817103

7.1 = 8.4779103

and thus=Ae/At= 1/(8.477910

3) = 117.95

As acheck, plots of optimum expansion - see Fig. 3.7 Sutton VI [1] p. 60 or Fig. 3.6 SuttonVII [2] p.65 - read by interpolation for k = 1.25:

- (cF)ideal= 1.85 very good check;- Ae/At = 120 good check.

For sake of simplicity, let us neglect nozzle losses (for example, a total loss of2%) andlet us accept as effective values cF = 1.85 and = Ae/At = 118. We can now evaluate thegravimetric specific impulse as

graphical Is(z= z0= 2.5 104 m) = ccFg0

=1507 1.85

9.807 = 284.28s

computed Is(z= z0= 2.5 104 m) = ccFg0

=1507 1.85

9.807 = 284.28s (4)

(a rare agreement, due to the fact that(cF)idealis identical !) and the throat area. based onthe design thrust value, as

cF = T(z= z0= 2.5 104 m)

pcAt

At =

T(z= z0= 2.5

104 m)

pccF =

10000

9.807

60 101325 1.85= 87.196cm2


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Immediately, the nozzle exit area is

Ae= (z= z0 = 2.5 104 m)At= 118 90.09 = 10631cm2

while the corresponding diameters are

dt =

r4 At

=

r4

87.196

= 10.54cm

de =

r4 Ae

=

r4

10631

= 116.34cm


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1.4 Evaluating Ballistic Data

Recall that the Vieille ballistic law can equivalently be written as

rb(p, Tref) = a(Tref) pn

(5)rb(p, Tref) = rb(pref, Tref) (p/pref)n (6)

while the first version is that more commonly used, the second version resorts to the nondi-mensional pressurep/prefinstead of the dimensional value p. Likewise, it may be convenientto write for the flame temperature as well

Tf(p, Tref) = b(Tref) pnTf (7)Tf(p, Tref) = Tf(pref, Tref) (p/pref)nTf (8)

Let us observe that the two constants or multiplicative factors in Eq. 5 and 7 represent thevalues of respectively steady burning rate and steady flame temperature at the unit pressure

(p = 1 atm or 1 bar or 1 MPa...). In general, the two constants can de evaluated at anyselected pressurep of convenience as

a(Tref) = rb(p, Tref)

(p)n

b(Tref) = Tf(p, Tref)

(p)nTf

As a further alternative, comparing the two laws of Eq. 5 and 6 for burning rate or the twolaws of Eq. 7 and 8 for flame temperature, the two constants can also be evaluated at theprecise reference pressurepref as

a(Tref) = rb(p, Tref)(pref)

n

b(Tref) = Tf(p, Tref)

(pref)nTf

- Thus, the steady burning rate of propellant 1 is:notice that a1= 0.768 is already assigned

evaluate n1=ln 6.102 ln 2.1652

ln100 ln10 = 0.45evaluate rb,1(pref, Tref) = 0.768 (68)0.45 = 5.129cm/sverify a1= 6.102/(100)0

.45 = 0.7682(cm/s)/(atmn) assignedverify also a1= rb,ref/(pref)

n1 = 5.129/(68)0.45 = 0.768(cm/s)/(atmn) assignedin this instance let us use the form

rb(p, Tref) = a(Tref) pn =rb,ref (p/68)n rb,1(p, Tref) = 0.7682 p0.45

- Thus, the steady flame temperature of propellant 1 is:notice that b1= 2562K is already assigned

evaluate nTf,1= ln 3010 ln 2777

ln100 ln10 = 0.035evaluate Tf,1(pref, Tref) = 2562 (68)0.035 = 2970Kverify b1= 3010/(100)0

.035 = 2562assignedverify also b1 = Tf,ref/(pref)

nTf,1 = 2970/(68)0.035 = 2562.2assigned

in this instance let us use the form


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Tf(p, Tref) = b(Tref) pnTf =Tf,ref (p/68)nTf Tf,1(p, Tref) = 2970 (p/68)0.035

- Thus, the steady burning rate of propellant 2 is:

notice that rb,2(pref, Tref) is already assigned

evaluate n2=ln 2.240 ln 0.97166

ln68 ln 5 = 0.32evaluate 0.972 =a2(p= 5atm)0.32 a2= 0.972/(50.32) = 0.581(cm/s)/(atmn)verify a2= rb,ref/(pref)

n2 = 2.240/(68)0.32 = 0.5806 (cm/s)/(atmn) assignedin this instance let us use the form

rb(p, T ref) = a(Tref) pn =rb,ref (p/68)n rb,2(p, T ref) = 0.581 p0.32

- Thus, the steady flame temperature of propellant 2 is:

notice that Tf,2(pref, Tref) = 2640K is already assignedevaluate nTf,2=

ln 2640 ln 2499ln68 ln 5 = 0.021

evaluate b2 = 2499/(5)0.021 = 2416K/(atmn) assigned

verify b2= Tf,ref/(pref)nTf,2 = 2640/(68)0.021 = 2416.1K/(atmn) assigned

in this instance let us use the form

Tf(p, T ref) = b(Tref) pnTf =Tf,ref (p/68)0.035 Tf,2(p, T ref) = 2640 (p/68)0.021

1.5 Boost or Acceleration Phase

For the assigned boost duration tb1= 5s, the cigarette cylindrical grain of propellant can be

designed as follows.The propellant flow rate (weight in kgfor mass in kgm) is

Is(z = z0) =T(z= z0)

.w

graphical .w= TIs

= 10000

284.28= 35.177kg/s

computed .w= TIs

= 10000

284.28= 35.177kg/s (9)

while the ideal total propellant amount (weight or mass) neglecting unburnt residuals, tran-

sients, and various inefficiencies

(wp)ideal,1= .w tb,1= 35.1775 = 175.89kg (10)

For sake of simplicity, let us neglect an excess of, say4%and thus keep for the amount (weightor mass) of propellant required by the boost phase

wp,1= (wp)ideal,1= 175.89kg (11)

wherefrom the volume of the boost propellant grain (wp is the weight in kgfor mass in kgmof propellant)

Vp,1= wp,1/p = 175.89/1.70 = 103.46dm3

= 103460cm3


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Mass conservation, between the combustion surfaceAb(produced mass) and nozzle throatAt (discharged mass), requires for the motor operation a combustion surface of

Abprb = CDAtpc=Atpc

c

= .w

Ab = 1

cAtpcprb

= 87.196104 (60101325)

1507.01.7010+3 4.849102 = 0.42673 m2 = 4267.3cm2

Just as acheck, let us verify the produced flow rate.

w= prbAb= 1.7010+3 4.849102 0.42673 = 35.177kg/s

or, asanothercheck, let us verify the combustion surface from the produced flow rate.

w = prbAb (12)

graphical Ab=.

w

prb=

35.17710+3

1.704.849 = 4267.3cm2 (13)

computed Ab=.w

prb=

35.17710+3

1.704.849 = 4267.3cm2 (14)

The most important check regards the combustion pressure in the boost phase

pc =

AbAt

pa1c

11 n1 =

4267.3

87.196 1.7010+3 0.7682 10

2

(101325)0.45 1507.0

110.45

=

= 6.0793106 Pa= 6.0793MPa = 6.0793/0.101325 atm = 59.998atm OKwherefrom the characteristic velocityccan be verified as

.

w = prbAb= CDAtpc=

Atpcc

c = Atpc

.w

=87.196 104 60 101325

35.177 = 1507.0m/s

Since a cigarette grain is required, the requested boost surface combustion is easily pro-vided by an ideally neutral cylindrical grain having diameter

db=

r4

Ab

=

r4

4267.3

3.1415= 73.712cm

while the klemmung ratioKbt is

Kbt=Ab

At=

4267.3

87.196

= 48.939 (15)

For the boost phase the length or web thickness of the (ideally) neutral grain is

b1= rb1tb1= 4.8495 = 24.245cm (16)

wherefrom the boost grain volume can be verified as

Vp= Ab b= 4267.3 24.245 = 103.46dm3 = 103.46103 cm3 (17)Assuming a total thickness of1.144cm for the liner (between propellant grain and com-

bustion chamber case) + possible ablative protection + case wall, the external diameter dcof the combustion chamber (max cross-section clutter) is

dmax= dc= db+ 2tliner= 73.712 + 21.144 = 76.0cm (18)


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1.6 Sustainer or Regime Phase

Sizing the sustainer propellant grain requires first determining the new operating conditions(sustainer phase or phase 2) in the combustion chamber, including the newpc ! If we neglect

the (minor) effect ofpc on the ratio (Tc/=

M), we find again

c 1CD

=pcAt

.m

=Isg0

cF=

c

cF=

q(


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calculate the new non optimum gasdynamic expansion or force again optimum expansion bya variable geometry limited to the nozzle exit area Ae. Let us take this second approach,leading to higher performance. We need to calculatethe new values of both (cF)ideal

(cF)ideal =

r2

k2

k 1 ( 2

k+ 1)k+1k1

s1

pepc

k1k

=

r2

1.252

1.25 1( 2

1.25 + 1)1.25+11.251

s1

0.02516

16.638

1.2511.25

= 1.775

and =Ae/At

1/ = At/Ae=k+ 1

2

1

k

1 pepc

1

k vuuuutk+ 1k 1

1pepc

k 1k

=

1.25 + 1

2

11.251

0.02516

16.638

11.25

vuut1.25 + 11.25 1

"1

0.02516

16.638

1.2511.25

#

= 8.8776103

6.544 = 2.271102

and thus=Ae/At= 1/(2.27110

2) = 44.033

To be consistent and for sake of simplicity, let us again neglect nozzle losses (for example,

a total inefficiency of 2%) and let us accept as effective values for the thrust coefficientcF = 1.775and for the expansion geometric ratio = Ae/At = 44.0. We can now evaluateduring the sustainer phase the new (ideal) values of:

- thrust

T2(z = z0= 2.5 104 m) = cFpcAt= 1.775 16.638 1.01325 105 87.196 104 = 26092.0N = 26092.0/9.807 = 2660.5kg

- nozzle exit area

Ae= (z= z0= 2.5 104 m)At= 44.033 87.196 = 3839.5cm2 (22)

- gravimetric specific impulse (for c assumed invariant)

Is(z= z0 = 2.5 104 m) = ccFg0

=1507.0 1.775

9.807 = 272.76s (23)

- and relevant diameters (for the taken approach, only the exit divergent is affected)

db = dmax= db+ 2tliner = 76.0cm (24)

dt =

r4 At

=

r4

87.196

= 10.537cm (25)

de = r4 Ae

= r4 3839.5

= 69.92cm (26)


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1.7 Motor

1.7.1 Thrust Nozzle

For the taken approach, the total nozzle length (convergent + divergent) changes dependingon the flight segment. In the boost phase, assuming for the combustion chamber dc = dmaxand for the gasdynamic thrust nozzle a45 convergent and a 15 divergent, we find

Lconv = (dc dt)/2

tan 45=

(76 10.54)/2tan45

= 32.73cm (27)

Ldiv = (de dt)/2

tan 15=

(116.34 10.54)/2tan15

= 197.43cm (28)

Ltot = Lconv+Ldiv= 32.73 + 197.43 = 230.16cm (29)

In the sustainer phase, under identical assumptions for the relevant geometrical parameters,

wefi

nd

Lconv = (dc dt)/2

tan45=

(76 10.54)/2tan45

= 32.73cm (30)

Ldiv = (de dt)/2

tan 15=

(69.92 10.54)/2tan15

= 110.80cm (31)

Ltot = Lconv+ Ldiv = 32.73 + 110.80 = 143.53cm (32)

Thus, in the sustainer phase the gasdynamic thrust nozzle results less cluttering in termsof both total length (but the convergent has not changed) and exit diameter, thanks to thereduced pressure in the combustion chamber.

1.7.2 Propellant Grain

The total length of the (ideally) neutral cylindrical grain, summing up the two burning phases,is

Lp = Lboost+ Lsustainer = 24.245 + 35.70 = 59.945' 60 cm (33)

1.7.3 Total Impulse

The total impulse for each of the burning phases is by definition

Itot,1(z = z0) = T1(z= z0)tb1= 100005 = 50.0103 kg s (34)

Itot,2(z = z0) = T2(z= z0)tb2= 2660.525 = 66.513103 kg s (35)

and thus, summing up the two burning phases, we find

Itot(z= z0) = T1(z= z0) tb1 + T2(z= z0) tb2= 50.0 103 + 66.513 103 = 116.51 103 kg s

As acheck, an alternative way (Sutton VI p. 24) to evaluate the total impulse Itotresortsto the gravimetric specific impulseIs

Itot,1(z = z0) = Is,1(z= z0)(wp,1)ideal = 284.28 175.89 = 50.002103 kg s (36)

Itot,2(z = z0) = Is,2(z= z0)(wp,2)ideal = 272.76 243.75 = 66.485103

kg s (37)


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1.8 Effects of Initial Temperature

1.8.1 Boost or Acceleration Phase

We need to evaluate the new combustion chamber pressure at the assigned initial temperatureofT0= 275 K- via mass balance

pc =

AbAt

p1a1c

11n1

= (38)

=

4267.3

87.196 1.7010+3 0.7682 10

2

(101325)0.45exp[0.002 (275 300)] 1507.0

110.45

(39)

= 5.551106 Pa= 5.551MPa = 5.551/0.101325 atm = 54.784atm (40)

- or via the thermal sensitivity parameter k

k,1 = p,11 n1

= 0.0021 0.45= 3.636410

3 (41)

pc(To) = pc(Tref)exp[k,1 (To Tref)] = (42)pc(275) = 60 exp[3.636410

3 (275 300)] = 54.786atm (43)Knowing the new combustion chamber pressure, for the boost phase at low initial temperaturewe can immediately evaluate the new

- steady burning rate

rb1(pc1 = 54.784atm) = a1(Tref) exp[p,1 (To Tref)] pn1 (44)= 0.7682 exp[0.002 (275 300)] 54.7840.45 = 4.4275 cm/s (45)

- burning time

tb1(pc1= 54.784atm) =b1/rb1 = 24.245/4.4275 = 5.476s (46)

- steady flame temperature (accounting only for the dominating pressure effect)

Tc1= Tf1(pc1= 54.784) = 2970 (54.784/68)0.035 = 2947.6K (47)- characteristic velocity

c 1CD

=pcAt

.m

=Isg0

cF=

c

cF=

q(


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- gravimetric specific impulse

Is,1(z= z0 = 2.5 104 m) = ccFg0

= 1504.6 1.8438

9.807 = 282.88s

- thrust

T1(z= z0= 2.5104 m) = cFpcAt = 1.8438 54.7841.01325 105 87.196104 = 89244N

1.8.2 Sustainer or Regime Phase

Again, we need to evaluate the new combustion chamber pressure at the assigned initialtemperature ofT0= 275 K

- via mass balance

pc = AbAt

p,2a2c

1

1n2

=

=

4267.3

87.196 1.6010+3 (0.581 10

2

1013250.32) exp[0.005 (275 300)] 1507.0

110.32

= 1.4028106 Pa= 1.4028MPa = 1.4028/0.101325 atm = 13.845atm

- or via the thermal sensitivity parameter k

k,2 = p,21 n2 =

0.005

1 0.32= 7.3529103 (48)

pc(To) = pc(Tref) exp[k,2 (To Tref)] = (49)pc(275) = 16.638 exp[7.352910

3

(275

300)] = 13.844atm (50)

Knowing the new combustion chamber pressure, for the sustainer phase at low initial tem-perature we can immediately evaluate the new

- steady burning rate

rb2(pc2 = 16.638atm) = a2(Tref) exp[p,2 (To Tref)] pn2 (51)= 0.581 exp[0.005 (275 300)] 13.8440.32 = 1.189cm/s (52)

- burning time

tb2(pc2= 16.638atm) = b2/rb2= 39.38/1.189 = 33.120s (53)

- steady flame temperature (accounting only for the dominating pressure effect)

Tc1= Tf1(pc1 = 13.844) = 2970 (13.844/68)0.035 = 2809K (54)

- characteristic velocity

c 1CD

=pcAt

.m

=Isg0

cF=

c

cF=

q(


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- thrust coefficient

(cF)ideal = r2 k2

k 1(

2

k+ 1

)k+1k1s1

pe

pc

k1k

(55)

=

r2

1.252

1.25 1( 2

1.25 + 1)1.25+11.251

s1

0.02516

13.844

1.2511.25

(56)

= 1.762 (57)

- gravimetric specific impulse

Is,2(z= z0 = 2.5 104 m) = ccFg0

=1468.8 1.762

9.807 = 263.90s (58)

- thrust

T2(z= z0= 2.5104 m) = cFpcAt= 1.762 (13.8441.01325105)87.196104 = 21552N


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2 EXERCISE No. 2: SUSTAINER - BOOST MISSION

Preliminary sizing of solid rocket motor of sustainer - boost type

We wish to get the preliminary sizing of a SRM capable to carry out the two-step propul-sive mission sketched in the following figure:

boost 5 s| _____| | || | || | || sustainer 25 s | || _________________________c |b ______________________________b________ time, s

The assigned input data are: burning times for each flight segment, tb1= 25s e tb2= 5s; thrust under optimum expansion, T1(z0= 2.5104 m) = 5, 000 + 100Ckg; combustion chamber pressure, pc1= 30 + N atm; For sake of simplicity, consider a unique "cigarette" cylindrical grain consisting of

two different unmetallized composite propellants featuring, under the reference conditionsofTref= 300K and pref= 68 atm, the following steady-state ballistic properties:

PROPELLANT 1

densityp1 = 1.60g/cm3

thermal sensitivity p1 = 0.0041/C

p, atm 10 50 68

rb(p), cm/s 1.213 2.030 2.240

Tf(p), K 2536 2623 2640

PROPELLANT 2

density p2= 1.70g/cm3

thermal sensitivityp2= 0.0001/C

p, atm 1 30 68

rb(p), cm/s 0.831 3.026 4.130

Tf(p), K 2562 2886 2970

1. Please perform a preliminary sizing of the whole motor (gasdynamic thrust nozzle,combustion chamber, and propellant grain), at the optimum expansion altitude ofz0,forcing the nozzle to always work under optimum expansion by using a divergent withconstant aperture d = 15

but variable length. In particular, please deduce the di-ameter of the "cigarette" cylindrical grain required to achieve the wanted propulsivemission.

2. Please evaluate how the main ballistic parameters, in particular the nozzle exit areaAt, change during the two flight segments.

3. How would the whole propulsive mission be affected by an increase of the initial tem-perature from T0= Tref toT0= 350K ?

Consider a monophase gaseous mixture expanding under chemically frozen conditions.When necessary, with due justifications assume typical values for missing properties andmake reasonable assumptions for undefined processes. Assigned data depend on the digitsC and N, that identify the alphabet position of the first letter of respectively the candidate

family name andfirst name.


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2.1 Introduction

Let us assume as typical values:

=

M= 25g/gmole or kg/kmole; k = 1.25; g0= 9.807m/s2;


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2.2 Properties of Combustion Chamber

At the nominal combustion chamber pressure ofpc1= 30atm, we can evaluate at once: Tc1= Tf1(pc1= 30) = 2970

(30/68)0.035 = 2886 K;

rb1(pc1= 30atm) = 0.581 300.32 = 1.7253 cm/sorrb1(pc1= 30 101325 Pa) = 0.581

1

(101325)0.32 (30 101325)0.32 = 1.7253cm/s;

ac1(pc1 = 30) =

sk

space propulsion lab

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