J. Aerosp. Technol. Manag. vol.4 número2

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LIST OF SYMBOLS

¨P Pressure drop N/m2 V_avg Average velocity m/s a Percentage opening of the valve % h Slurry height in the bowl m

V Volume m3

ȝ 9LVFRVLW\ 3D6

N_Re Reynolds number

INTRODUCTION

Composite propellants (CP) are the most important class of solid rocket propellant used in various missile programmes DQGVSDFHDSSOLFDWLRQV$&3FRQVLVWVRIDQLQRUJDQLFR[LGL]HU such as ammonium perchlorate (65 to 70%), a metallic fuel

such as aluminum powder (15 to 20%), and a polymeric bind-HUVXFKDVK\GUR[\OWHUPLQDWHGSRO\EXWDGLHQHWR along with certain isocyanate based curatives and process aids %R\HUVDQG.ODJHU7KHPDLQREMHFWLYHRISURSHOODQW SURFHVVLQJLVWRSURGXFHÀDZOHVVSURSHOODQWJUDLQWKHUHIRUH special precautions are taken in application of vacuum and FRQWURORIVOXUU\ÀRZUDWHFDVWLQJUDWHGXULQJSURSHOODQW FDVWLQJ6HYHUDOWHFKQLTXHVDUHDGRSWHGZRUOGZLGHIRUWKH FDVWLQJRISURSHOODQWVYL]YDFXXPFDVWLQJSUHVVXUHFDVWLQJ ED\RQHWFDVWLQJHWF'DYHQDV.ULVKQDQDQG*KRNKDOH $PRQJWKHVHDVXLWDEOHFDVWLQJPHWKRGLVFKRVHQEDVHG on parameters like type of propellant, rheological behavior of SURSHOODQWVOXUU\DQGZHEVL]HDQGJUDLQJHRPHWU\RIURFNHW PRWRUV0XWKLDKDQG.ULVKQDPXUWK\

Vacuum/gravity casting is the most widely adopted tech-QLTXHIRUSURSHOODQWFDVWLQJ3URSHOODQWVOXUU\LVLQWURGXFHG LQVLGH D URFNHW PRWRU IURP WKH PL[HU ERZO WKURXJK IHHG SLSHDQGFRQWUROYDOYH7KHGULYLQJIRUFHIRUÀRZRISURSHO-lant slurry is gravitational force and vacuum inside casting FKDPEHU7KHFDVWLQJUDWHRIVOXUU\LVDQLPSRUWDQWSDUDPHWHU LQGHWHUPLQLQJSURGXFWTXDOLW\ZKLFKPDLQO\GHSHQGVXSRQ

Numerical Studies on Flow Behavior of Composite Propellant

Slurry During Vacuum Casting

Bejoy Thiyyarkandy, Mukesh Jain, Ganesh Shankar Dombe, Mehilal Mehilal*, Praveen Prakash Singh, Bikash Bhattacharya

High Energy Materials Research Laboratory - Pune – India

Abstract: Rockets are powered by composite solid propellant, which is a heterogeneous system consisting of solid oxidizer and metallic fuel dispersed in a polymeric fuel binder matrix. The slurry casting technique under vacuum/ gravity condition is well-established for performing a different class of large sized case bonded rocket motors. During propellant casting, the Àow rate of slurry is a very critical parameter as it affects the product quality. The casting rate is governed by suf¿cient degassing and viscosity buildup due to the progress of cure reaction. ,n the present study, casting rate and casting time have been numerically evaluated for ¿xed and varying percentages of valve opening, different viscosity of slurry, and different pressure drop driving force. The velocity pro¿le of propellant slurry inside feeding pipe and valve has also been evaluated. )urthermore, to get a Àawless grain and to predict the slurry casting rate, a microscopic analysis has been carried out to model the Àow behaviour of composite propellant slurry, where the momentum conservation law has been applied to express the mathematical model in an analytical form. The resulting differential and algebraic equations have been solved numerically using MATLAB, computing software. The numerical analysis is useful for designing new casting set-up and for giving the idea of maximum casting rate, which is achievable for given casting set-up and rheological properties of propellant slurry.

Keywords: Microscopic analysis, Composite propellant, Vacuum casting, Casting rate, Casting time.

5HFHLYHG $FFHSWHG

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rheological behavior of slurry, pot life of propellant, and ÀRZSDWKDYDLODEOHIRUFDVWLQJ3URSHOODQWVOXUU\EHKDYHVDVD SVHXGRSODVWLFWKL[RWURSLFÀXLGDQGLWVUKHRORJLFDOEHKDYLRULV PRGHOHGDVWKHSRZHUODZÀXLGPRGHO0DKDQWDet al

Few studies on modeling and simulation of propellant casting DUHUHSRUWHGLQOLWHUDWXUH7KHSUHVVXUHGURSGHWHUPLQDWLRQ during pressure casting was carried out by Dombe et al

7KH'XQVWHDG\ÀRZ¿HOGDQGIUHHVXUIDFHPRYHPHQWGXULQJ VROLGSURSHOODQWFDVWLQJZHUHVLPXODWHGE\5D¿et al

Dormaus et alKDYHGHYHORSHGDFRPSXWHUPRGHOIRU

the distribution prediction of solid particle constituents during FDVWLQJRISURSHOODQWLQURFNHWPRWRU6KLPDGDet al

have also used bidirectional X-ray penetration measurement and digital image analysis for estimation of propellant slurry YHORFLW\¿HOGGXULQJYDFXXPFDVWLQJ

During propellant processing, propellant formulations ZLWKGLIIHUHQWEXUQLQJUDWHVWRPPV#NVFUHVXOW in propellant slurry, with different rheological properties (end RIPL[YLVFRVLW\SVHXGRSODVWLFLW\LQGH[HWF$OVRWKHÀRZ path available for casting (web thickness) of propellant grain YDULHVEDVHGRQWKHJUDLQJHRPHWU\7KHUHIRUHSURFHVVLQJRI wide variety of propellant formulations, with different web WKLFNQHVVHVLVHVVHQWLDOWR¿QGRXWWKHYDULDWLRQLQVOXUU\ÀRZ rate with percentage opening of valve with different rheological FKDUDFWHULVWLFV)XUWKHUPRUHWRSUHGLFWFDVWLQJUDWHDQGGXUD-tion of vacuum casting with % opening of control valve (ball valve), mathematical modeling and simulation of casting set up DUHFDUULHGRXWZKHUHPRPHQWXPEDODQFHHTXDWLRQVDUHDSSOLHG across various components of casting system and related HTXDWLRQVKDYHEHHQVROYHGQXPHULFDOO\XVLQJDFRGHZULWWHQ LQ0$7/$%7KHPDWHULDOXVHGLQVLPXODWLRQIRUSURSHOODQW FRQVLVWRIDPPRQLXPSHUFKORUDWHDOXPLQXPSRZGHU K\GUR[\O WHUPLQDWHG SRO\EXWDGLHQH WROXHQH GLLVRF\DQDWHDQGSURFHVVDLGV

In the following section, we have reported the results of QXPHULFDOVWXGLHVRQYHORFLW\GLVWULEXWLRQPDVVÀRZUDWHFDVWLQJ UDWHSUHVVXUHGURSVOXUU\KHLJKWLQPL[HUERZOGXULQJYDFXXP FDVWLQJDVZHOODVWKHLQÀXHQFHRIYLVFRVLW\RQWKHVHSDUDPHWHUV

MATHEMATICAL MODELING

During the mathematical modeling, momentum balance HTXDWLRQVDUHIRUPXODWHGIRUVHYHUDOFRPSRQHQWVRIFDVWLQJ V\VWHP7KHSUHVVXUHGURSIRUHDFKFRPSRQHQWLVDGGHGWRJHW WKHRYHUDOOSUHVVXUHGURS7KHPRPHQWXPFRQVHUYDWLRQODZ

KDVEHHQDSSOLHGRYHUDVKHOORIWKHVOXUU\KDYLQJLQ¿QLWHVLPDO dimension followed by the integration over the whole control YROXPHRIWKHFDVWLQJV\VWHPZKLFKSURYLGHVWKHUHTXLUHG PRPHQWXPHTXDWLRQ)ROORZLQJDVVXPSWLRQVKDYHEHHQPDGH for the mathematical modeling:

VWHDG\VWDWHRQHGLPHQVLRQDOODPLQDUÀRZ FRQWLQXXPÀRZEHKDYLRU

SVHXGRSODVWLFEHKDYLRUREH\LQJSRZHUODZ IXOO\GHYHORSHGÀRZ

QRVOLSERXQGDU\FRQGLWLRQ

FRQVWDQWSURSHUWLHVRIWKHVOXUU\WKURXJKRXWFDVWLQJ FLUFXODUIUHHVXUIDFHRIVOXUU\LQVLGHWKHYDOYH

MOMENTUM BALANCE EQUATION FOR VARIOUS COMPONENTS OF CASTING SYSTEM

7KH PRPHQWXP EDODQFH HTXDWLRQV DUH GHULYHG IRU WKH IROORZLQJIRXUFRPSRQHQWVRIFDVWLQJV\VWHP7KHYDULRXV FRPSRQHQWVRIWKHFDVWLQJV\VWHPDVGHSLFWHGLQ)LJDUH GLVFXVVHGEHORZ

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Slanted and straight pipe-I

)RUÀRZVWKURXJKSLSHZLWKWKHDVVXPSWLRQRIIXOO\GHYHO-RSHG ÀRZ QRVOLS ERXQGDU\ FRQGLWLRQ DQG SVHXGRSODVWLF nature of slurry with power law behavior, slurry velocity, YROXPHWULFÀRZUDWHDQGDYHUDJHYHORFLW\DUHH[SUHVVHGDV (TVWR0DWKHZVDQG)LQN

( / ) ,

V M x R where M

n n KL P _R 1 1 2 / y n n n 1 1 1 1 1 D = - = +

+ c m +

6 @ ₍₁₎

,

Q V dA A x Q

n n

KL P

R

3 1 2

/ y n n sn 2 1 1 &

r r D

= = = + + c m

#

(2) V R Q n n KL P R

3 1 2

/ 1 avg n n n 2 1 r D = = + +

c m (3)

where:

¨P LVRYHUDOOSUHVVXUHGURSLQFOXVLYHRIJUDYLWDWLRQKHDG)RU slanted pipe, cosș FRPSRQHQWRIJUDYLWDWLRQDOKHDGLVFRQVLGHUHG )OXLGVREH\LQJSRZHUODZ5H\QROGVQXPEHULVH[SUHVVHG as (Saleh, 2002):

N

K n n ȡV D 8 4 3 1 Re n n avg n n 1 2 = + -` j Ball valve )RUPRGHOLQJÀRZWKURXJKEDOOYDOYHFRQVLGHUDFLUFXODU VKHOO)LJKDYLQJWKLFNQHVV¨x at a distance of x from the FHQWUHRIVSKHUHEDOOYDOYH0RPHQWXPEDODQFHHTXDWLRQDFURVV WKHWKLFNQHVV¨x LVH[SUHVVHGDV%LUGet al

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If z is the radius of circular shell and r is the distance of shell from the centre of sphere:

x2₊z2₌R2

(6)

7KHQ

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With no-slip boundary condition,Ĳ_xy= ¿nite at x = R, it

/ cos

C 2 P R g R

2 2 3

1 3

2r 3

t i D = - + (9) )RUREWDLQLQJUYDOXHDFLUFXODUVKHOORIÀXLGLQWKHEDOOYDOYHLV FRQVLGHUHG)LJLQZKLFKYROXPHLQWKH¿UVWRFWDQWLVJLYHQDV (10)

Volume of slurry in the valve is V = × vol,ISHUFHQW-age opening of the valve is Į, this volume will be Į / of

(4)

the total volume:

[ 3 2 ]

( / ) ,

r R r R R

r R r R where r R

3 100 3

4

3 2 25 0 0 < <

3 2 3 3

3 2 3

r a

r

a

#

- + =

- + - =

(11)

Straight pipe II

7KHVWUDLJKWSLSH,,LVDSDUWLDOO\ILOOHGSLSH)LJ For modeling flow through this pipe, consider a shell of slurry at a distance x from the centre of pipe having thick-QHVV¨x,Ir is the distance of shell from the centre of SLSHPRPHQWXPEDODQFHHTXDWLRQFDQEHH[SUHVVHG%LUG

et alDV

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:LWKQRVOLSERXQGDU\FRQGLWLRQLHĲxy= ¿nite at x = R:

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)RU REWDLQLQJ WKH UYDOXH D FLUFXODU VKHOO RI ÀXLG LQ VWUDLJKWSLSH,,LVFRQVLGHUHG)LJ,ISHUFHQWDJHRIWKH SLSH ÀRZ DUHD ZLOO EH RFFXSLHG E\ VOXUU\ DQ DUHD RI WKH shaded part will be:

Modeling of slurry height in bowl

Slurry height in the bowl is modeled using continuity HTXDWLRQDQGDVVXPLQJFRQVWDQWSURSHUWLHVRIWKHVOXUU\IRU the casting setup, by considering one surface as a free one of VOXUU\LQWKHERZODQGDQRWKHURQHDVÀRZVXUIDFHRIVOXUU\LQ straight pipe after the ball valve:

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(TXDWLRQSURYLGHVDYDULDWLRQRIWKHSURSHOODQWVOXUU\ KHLJKWLQDERZODWDQ\WLPH

RESULTS AND DISCUSSION

$ FRPSXWDWLRQDO VRIWZDUH FDOOHG 0$7/$% KDV EHHQ used to carry out numerical simulations of differential and DOJHEUDLF(TVWRJRYHUQLQJWKHSURSHOODQWÀRZGXULQJ YDFXXP FDVWLQJ 0DWKHZV DQG )LQN 1HZWRQ 5DSKVRQPHWKRGZDVXVHGWRVROYHWKHHTXDWLRQVIRUZKLFK initial values were obtained from the bisection method &KDSUD DQG &DQDOH )RU WKH QXPHULFDO DQDO\VLV SURSHOODQWSURSHUWLHVXVHGZHUHVSHFL¿FJUDYLW\RISURSHO-ODQWVOXUU\RIJFFLQLWLDOKHLJKWRISURSHOODQWVOXUU\LQ WKHERZORIPXQLWVKHDUYLVFRVLW\RISURSHOODQWVOXUU\ RI3DVSVHXGRSODVWLFLW\LQGH[QRIDQGYDFXXP OHYHOLQVLGHWKHPRWRURIWRUU&DVWLQJWLPHZDVHYDOXDWHG EDVHG RQ WLPH UHTXLUHG IRU NJ FDVWLQJ RI SURSHOODQW VOXUU\LQWRWKHURFNHWPRWRU

R

x Z

¨x r o

)LJXUH6HFWLRQRIVWUDLJKWSLSHDIWHUEDOOYDOYH

R x

¨[

r o

ș

y

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7KHYDULRXVÀRZSDUDPHWHUVVXFKDVFDVWLQJUDWHDYHUDJH DQGPD[LPXPYHORFLW\YHORFLW\SUR¿OHVLQVLGHYDULRXVFRPSR -nents, were obtained numerically for a given % opening of ball YDOYHDQGSUHVVXUHGURS7KHVHSDUDPHWHUVZHUHREWDLQHGIURP VHYHUDORSHQLQJRIYDOYHYDU\LQJIURPWR

Effect of pressure drop on mass Àow rate

7KHUHVXOWVRIQXPHULFDOVLPXODWLRQLQWHUPVRIRYHUDOO SUHVVXUH GURS DYHUDJH DQG PD[LPXP YHORFLW\ LQ SLSHV DQGFDVWLQJWLPHDUHSUHVHQWHGLQ7DEOH,WLVFOHDUWKDWWKH RYHUDOOSUHVVXUHGURSYDULHVOLQHDUO\ZLWKWKHPDVVÀRZUDWH IRUDFRQVWDQWYDOYHRSHQLQJ7KHYDULDWLRQRISUHVVXUHGURS ZLWKPDVVÀRZUDWHLVJLYHQLQ)LJZKLFKLQGLFDWHVOLQHDU YDULDWLRQRISUHVVXUHGURSZLWKPDVVÀRZUDWH,WLVIRXQGWKDW LQFUHDVHLQPDVVÀRZUDWHIURPWRNJKULQFUHDVHVSUHV -VXUHGURSIURPWR3D 7DEOH9DULDWLRQRISUHVVXUHGURSZLWKPDVVÀRZUDWH Mass ÀRZUDWH (kg/min) Pressure drop (Pa) Average velocity (m/s) 0D[YHORFLW\ (m/s) 7LPH (minutes)

10 95

11 50121

12

13 57155 73

60615

15

53

20

25

Velocity distribution inside various components of casting system 7KHYHORFLW\GLVWULEXWLRQLQVLGHVWUDLJKWSLSHV,DQG,,EDOO YDOYHDUHJLYHQLQ)LJVWR7KHYHORFLW\GLVWULEXWLRQLQVWUDLJKW SLSH,IRUSURSHOODQWVOXUU\LQGLFDWHVWKDWWKHFXUYHLVQRWH[DFWO\ SDUDEROLFZKLFKLVDWWULEXWHGWRWKHSVHXGRSODVWLFLW\QDWXUHQ RIWKHSURSHOODQWVOXUU\7KHYHORFLW\GLVWULEXWLRQLQVLGHEDOOYDOYH RISURSHOODQWVOXUU\LQGLFDWHVWKDWWKHPD[LPXPYHORFLW\RIWKH propellant slurry is at the free surface of slurry inside the valve, and decreases towards the valve surface, whereas velocity of the VOXUU\LV]HURDWWKHVXUIDFHRIWKHYDOYH7KHYHORFLW\GLVWULEXWLRQ of propellant slurry in the pipe after ball valve also indicates that WKHPD[LPXPYHORFLW\LVDWWKHIUHHVXUIDFHRIVOXUU\LQVLGHWKH SLSH,WLVIRXQGWKDWLQFUHDVHLQPDVVÀRZUDWHIURPWRNJKU LQFUHDVHVDYHUDJHYHORFLW\IURPî-3WRî-3PV5H\Q -old’s number for the propellant slurry during casting is calculated, which is of the order of 10-3LQGLFDWLQJÀRZWREHFUHHSLQJÀRZ

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

Distance from centre of pipe (m)

Velocity of slurry (m/s)

)LJXUH9HORFLW\GLVWULEXWLRQLQVLGHSLSH,

0.02 0.03 0.04 0.05 0.06 0.07 0.08

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Distance from centre of Ball Valve (m)

Ve lo c it y o f sl u rr y ( m /s ) )LJXUH9HORFLW\GLVWULEXWLRQLQVLGHEDOOYDOYH

10 15 20 25

4 5 6 7 8 9 10

Flow rate (Kg/min)

Pressure Drop (Pa)

× 104

(6)

InÀuence of viscosity on pressure drop and casting time

,QÀXHQFHRIYLVFRVLW\RQSUHVVXUHGURSIRU¿[HGPDVV ÀRZUDWHLVJLYHQLQ)LJ,WZDVIRXQGWKDWDVYLVFRVLW\ LQFUHDVHVWKHSUHVVXUHGURSOLQHDUO\LQFUHDVHV7KHRYHUDOO SUHVVXUHGURSYDULHVIURPWR3DIRUYLVFRV

-LW\YDOXHVLQWKHUDQJHRIWR3DV)XUWKHULWZDV

found that as viscosity increases, casting time increases for

FRQVWDQWYDOYHRIRSHQLQJ

InÀuence of percentage opening of valve on pressure drop

,QÀXHQFHRISHUFHQWDJHRSHQLQJRIYDOYHRQSUHVVXUHGURSLV JLYHQLQ)LJ,WZDVIRXQGWKDWSUHVVXUHGURSLQFUHDVHVZLWK DGHFUHDVHLQSHUFHQWDJHRSHQLQJRIYDOYH7KHSUHVVXUHGURS REVHUYHGIRURSHQLQJRIYDOYHLVî_{3D7KHSUHVVXUH} GURSIRUDQGDERYHRSHQLQJRIYDOYHLVî_3D

CONCLUSIONS

7KHÀRZEHKDYLRURI&3VOXUU\GXULQJYDFXXPFDVWLQJ

has been numerically modeled in terms of differential and

DOJHEUDLFHTXDWLRQVZKLFKDUHIRUPXODWHGXVLQJQXPHUL

-FDOO\PLFURVFRSLFDQDO\VLV7KHJRYHUQLQJHTXDWLRQVDUH VLPXODWHG WR VWXG\ WKH EHKDYLRU RI VOXUU\ ÀRZ UDWH RQ

overall pressure drop and the time taken by the slurry to

UHDFK LQVLGH PRWRU IURP WKH ERZO IRU ¿[HG DQG YDU\LQJ SHUFHQWDJHVRIYDOYHRSHQLQJ7KLVQXPHULFDODQDO\VLVRI

casting system is useful in designing new casting setup and

JLYHVWKHLGHDRIPD[LPXPFDVWLQJUDWHDFKLHYDEOHIRUD FHUWDLQFDVWLQJVHWXSDQGUKHRORJLFDOSURSHUWLHVRIVOXUU\ 7KLVUHODWLRQLVYHU\XVHIXOIRUGHFLGLQJRSHQLQJRIYDOYH

for attaining given casting rate for propellant slurry, with

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REFERENCES

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-RKQ:LOH\DQG6RQV

%R\HUV&.ODJHU.³3URSHOODQW0DQXIDFWXULQJ+D]DUG DQG7HVWLQJ´$PHULFDQ&KHPLFDO6RFLHW\:DVKLQJWRQ'&

&KDSUD&6&DQDOH35³1XPHULFDO0HWKRGVIRU (QJLQHHUV´th HGLWLRQ0F*UDZ+LOO1HZ<RUN

'DYHQDV$³6ROLG5RFNHW3URSXOVLRQ7HFKQRORJ\´ 1HZ<RUN3HUJDPRQ3UHVV

90000

80000

70000

60000

50000

40000

30000

20000

500 700 900

Viscosity, Pa.s

Pressure drop, Pa

1100 1300 1500

)LJXUH9DULDWLRQRISUHVVXUHGURSZLWKYLVFRVLW\

15 20 25 30 35

4.5 5 5.5 6 6.5 7 7.5 8 8.5 ×104

% Valve Opening

Pressure Drop (Pa)

)LJXUH9DULDWLRQRISUHVVXUHGURSZLWKYDOYHRSHQLQJ

0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Distance from centre of pipe (m)

Ve

lo

c

it

y

o

f

sl

u

rr

y

(m

/s

)

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