NUMERICAL SIMULATION OF THE GRAIN BURNBACK

IN SOLID PROPELLANT ROCKET MOTOR

Cengizhan YILDIRIM2

Senior Research Engineer

Turkish Scientific and Technical Research Council

Defense Industries Research and Development Institute

Ankara, TURKEY

and

M. Halûk AKSEL1

Professor of Mechanical Engineering

Middle East Technical University

Department of Mechanical Engineering

06531 Ankara, TURKEY

2 Engineer, Internal Ballistic Group at Defense Industries Research and Development Institute.

1 Professor, Mechanical Engineering Department.

41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit10 - 13 July 2005, Tucson, Arizona

AIAA 2005-4160

Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

ABSTRACT

Determination of the grain geometry is one of the critical steps during the design

of a solid propellant rocket motor. Because, performance prediction of the solid

propellant rocket motor is achieved if burning surface area of the propellant grain is

known as it recedes. In this work, a technique which is called “Level-Set” is adapted to

solid propellant regression. Therefore, the simulation of the change of the burning surface

of the propellant for related burnback steps is performed. Initial surface of the solid

propellant is modeled by using commercial solid modeling software. After that,

tetrahedral elements are generated on solid model to construct initial geometry of the

grain by using newly developed algorithms. The most important advantage of “Level-Set

Method” is that a general computer code can be prepared in order to perform a burnback

analysis for different grain configurations and it handles discontinuity of the burning

surface of the solid propellant grain. It imposes no limitations either two-dimensional or

three dimensional grain configurations. In this study some of the conventional two or

three dimensional grain burnbacks are analyzed and pressure time histories are predicted

by making an assumption of constant pressure along the motor. These graphs are

validated by comparing the static test data obtained by Ballistic Test Motor.

Key words: Solid Propellant, Grain, Grain Burnback Analysis, Internal Ballistic

NOMENCLATURE

φ : Level-Set functionF : Regression speednr

: Unit normal vectorA : Speed of soundU : Velocity in x directionV : Volume of the gap∆t : Time incrementa0 : Burning rate constantr& : Burning ratePc : Chamber pressureρp : Density of the propellantSb : Burning surface area of the propellantc& : Characteristic velocityAth : Nozzle throat areaN : Burning rate constant∆x, ∆y, ∆z : Grid sizes

1. Introduction:

First part of the design process of the solid propellant rocket motor is the

determination of the flight-mission requirements. After that, thrust-time curve is prepared

according to requirements. At this point, internal ballistic calculations are performed to

estimate chamber pressure of the rocket motor. Composition of the propellant, grain

geometry and nozzle geometry is so adjusted that desired chamber pressure of the motor

is achieved [1, 2]. However, these parameters are depending on chamber pressure itself.

Therefore, an iterative procedure is applied.

As explained above, grain geometry should be known to simulate pressure-time

history of the rocket motor, because, burning area of the grain depends on grain geometry

of the propellant. Grain geometry changes as solid propellant burns. This change causes a

change in burning area of the solid propellant. For this reason, burning area of the solid

grain should be known at each burning steps to predict pressure-time history of the rocket

motor. The analysis for determination of the burning area is called grain burnback

analysis. In this analysis, propellant burning is modeled by considering each burnback

steps to find the burning surface area.

In literature, both analytical and numerical burnback codes can be found. As a

well-known analytic burnback code SPP [3], it computes burning area of the propellant

by creating a model consists of simple three dimensional shapes such as prisms, cone,

pyramid, etc. Subtracting, adding and cutting procedures are applied to these shapes to

construct the model. In one of the burnback code, ELEA [4], initial grain geometry is

constructed by a solid modeling CAD program. Some numerical techniques are then used

for modeling of regression of the propellant.

In Defense Industries Research and Development Institute, a numerical burnback

code is developed by using Level-Set[5, 6] technique.

2. Technical Approach:

A grain burnback code, written in Fortran 90 computer language under Developer

Studio editor is developed. Thanks to Fortran 90, dynamic array is used so the size of the

arrays can be limited and can be fixed at any programming line of the code. Some array

processing algorithms are developed to prevent unnecessary computation on arrays.

Grain burnback analysis is performed in three steps. The construction of a gap or

perforation of the grain geometry is the first step. Then, as a second step, this geometry is

moved towards to its surface normal. Finally, a new surface or volume is constructed. In

these steps, Auto-CAD solid modeling software, a commercial mesh generator and Tec-

Plot animations are used as pre and post processors.

2.1 Propellant Regression by Using Level-Set Numerical Method:

As the propellant burns its shape changes. Therefore, the distance values, which

are explained in Section 2.3, should be updated to simulate formation of the new surface

after the propellant burning. This is achieved by Level-Set numerical method.

Let a function φ which is a distance value or Level-Set value of zero at time t=0

as given in (1). The time rate of change of this function is given in (2) by using chain

rule.

0)),(( =ttxφ (1)

0)(').),(( =∇+∂∂

txttxt

φφ

(2)

Introducing a regression speed, F, towards to surface normal and n, a unit normal

vector, one can obtained (3).

x’(t) . n = F (3)

From the unit normal vector definition,

φφ

∇∇=n

(4)

If (3) and (4) are substituted into (2), then Level-Set equation is obtained given in(5).

0=∇+∂∂ φφ

Ft (5)

To understand the physical meaning of (5), linear wave equation is considered.

0x

ua

t

u =∂∂+

∂∂

(6)

It is noticed that (6) and (5) are the similar form except absolute value sign. One

can simulate a boundary moving with a speed of “a” towards a certain direction by using

linear wave equation. However, a boundary moving with a speed of “F” towards its

normal is simulated by using the Level-Set equation. By using the property of absolute

value given in (7) and some adjustments (8), (9), and (10) are obtained. The distance

values for each cartesian grid points are updated according to (8), (9) and (10).

( ) ( ) ( )( )2z2y2x222

DDDzyx

++=

∂∂+

∂∂+

∂∂=∇

φφφφ (7)

( ) ( )[ ]−++ ∇+∇∆−= kjikjin

kjin

kji FFt ,,,,,,1

,, ,0min,0maxφφ(8)

( ) ( )( ) ( )( ) ( )

2

1

2

,,

2

,,

2

,,

2

,,

2

,,

2

,,

0,min0,max

0,min0,max

0,min0,max

+

++

++

=∇+−

+−

+−

+

zkji

zkji

ykji

ykji

xkji

xkji

DD

DD

DD

(9)

( ) ( )( ) ( )( ) ( )

2

1

2

,,

2

,,

2

,,

2

,,

2

,,

2

,,

0,min0,max

0,min0,max

0,min0,max

+

++

++

=∇−+

−+

−+

−

zkji

zkji

ykji

ykji

xkji

xkji

DD

DD

DD

(10)

As far as the notation in (7), (8), (9) and (10) is concerned, forward differences in

x direction is given in (11), backward differences towards y direction is given in (12).

Other notations can be understood from these examples.

( ) ( )x

D k,j,ik,j,1ixk,j,i ∆

φφ −= ++ (11)

( ) ( )y

D k,1j,ik,j,iyk,j,i ∆

φφ −− −= (12)

nk,j,iφ represents the original Level-Set whereas 1n

k,j,i+φ is the updated Level-Set.

One can consider the numerical scheme of (6), given in (13).

( ) ( )( )ni

xni

xni

1ni uDa,0minuDa,0maxtuu +−+ +−= ∆ (13)

When the speed, a, is positive, then information travels from left to the right. To catch

this information, backward difference is needed. This requirement is satisfied by using

(13), because, for positive speed, only backward difference term is left.

� As a result, in (8), (9) and (10), the maximum, minimum and zero terms

determine the direction of the information taken.

2.2 Construction of a model:The computer code needs an input file containing information about the locations

of the meshes and their connectivities. This work is performed first by constructing a

three dimensional solid model and then generating meshes on this model.

As a model, rather than solid propellant grain, its gap or perforation is considered.

A three dimensional solid model is drawn by using AutoCAD-AMD software. The cross-

section of the propellant grain and its model is shown in Figure 1 and 2.

A “sat” extension file is created from the model and then this file is opened in ANSYS

software. Tetrahedral elements are generated on this model by using ANSYS software as

shown in Figure 3. After element generation procedure is completed, information about

these elements is obtained. In detail, this information includes x, y and z coordinates of

the grid points, their connectivities and their boundary locations. From this information,

an input file is generated.

2.3 The determination of distances:An equally spaced overlapping cartesian grid is generated around the solid

propellant grain by the code. This grid is shown in Figure 4.

For each point on cartesian grid, a procedure is applied such that the minimum distance

between the point under consideration and model surface are computed by the code.

Since x, y and z coordinates of the unstructured grid points is given as an input, the code

computes the distances between a point on structured grid and all points on unstructured

grid. The minimum distance is taken as the distance between the point under

consideration and model surface. The procedure explained above is shown in Figure 5.

In this figure, point P is a cartesian grid point. The distance between point P and all nodes

on tetrahedral elements are computed as shown in Figure 5. In the figure, the distances

noted as 1, 2, 3, 4 and 5 are computed and the minimum one is taken as the distance value

of point P. If cartesian grid point is located inside of the model, the distance value is

multiplied by -1, otherwise it is taken as it is. Therefore, cartesian points located inside of

the model have negative distance values, whereas, cartesian points located outside of the

model have positive distance values. Therefore, cartesian grid points on the surface of the

model have a distance value of zero.

In order to determine weather cartesian point is located inside of the model or not, vector

cross-product feature is used and a 3-D algorithm is developed. For simplicity, only 2-D

algorithm of this feature is explained in Figure 6. Three dimensional algorithm is nothing

but the expansion of 2-D algorithm explained. In Figure 6, point P represents one of the

cartesian node. A triangle composed of 1, 2 and 3 represents one of the triangle element

generated in solid model in 2-D space. In 3-D space, the triangle element becomes

tetrahedral element. For a cartesian node, P, checking procedure given in Figure 6 is

performed for every triangle element. If all of the inequalities are satisfied, it is said that

point P is located outside of the model. Otherwise, point P is located inside of the model

therefore, the distance value is multiplied by -1.

2.4 Determination of Burning Area:

The most important output of the grain burnback codes is the burning surface area

for each burnback steps. In this code, burning surface area can be obtained by using

distance value on each cartesian node. The calculation procedure is as follows:

• One can checked whether the distance value of a cartesian grid point is smaller

than zero. (n

k,j,iφ < 0)

• The number of cartesian grid points those level-set values smaller than zero iscomputed.( ( )∑ <k,j,i k,j,i 0φ )

• The volume of the gap or perforation of the grain can be computed by multiplyingvolume of a cube element by the number of cartesian grid points those level-setvalues is smaller than zero. (V= ( )∑ <k,j,i k,j,i 0φ ) (∆x ∆y ∆z))

• Volume computed at corresponding burnback step is subtracted from previously

computed volume and this value is divided into burnback distance as formulated

in (14) to find the burning surface area of the propellant. In (14), Sb stands for

burning surface area, F is boundary regression speed and ∆t is the unit time.

)t∆(F

VVS

n1n

b−

≅+

(14)

Distance values which are equal to zero simulate burning surface of the

corresponding burnback step of the propellant grain. By using the TecPlot software,

burnback of the circular perforated grain is drawn as shown in Figure 7. The left view

represents original grain geometry and the right view shows its geometry after 3

burnback steps.

3. Results and Discussion:

Some of the conventional 2-D grain shapes as well as 3-D shapes are analyzed by

using code developed.

For 2-D burnbacks, both port area and burning surface area values are compared

with the exact values. The exact values are computed by using Auto-CAD drawing

software for each burnback steps. In 3-D burnbacks, pressure-time or thrust time histories

are calculated by using previously developed flow solver codes. These histories are

compared with the static test data of the real motors.

3.1 Two Dimensional Burnback Results

As a first attempt for verification purpose, a simple shape, circular slotted grain, is

analyzed. Because, exact port and burning surface area values are easy to calculate

analytically. Burnback of this grain shape is given in Figure 8. The values in x and y

coordinates represent grid number. Port area and burning surface area values obtained by

code are compared with the exact values. These are tabulated in Table 1.

As realized from x and y axes of Figure 8, 100x100 cartesian grids are used. Rather than

modeling whole grain, only circular perforation is modeled and analyzed. As far as the

results in Table 1 are considered, error in computing port area or burning surface area is

less than 2 percent. Therefore, one can say that percent error is in acceptable limits and

code is able to handle circular slotted grain burnbacks. After this simple grain shape,

four-arm star shape is analyzed. Burnback of this grain shape is given in Figure 9. Port

area and burning surface area values obtained by code are compared with the exact

values. The exact values are computed by using Auto-CAD drawing software. These

comparisons and percent error for each burnback step are tabulated in Table 2.

Similar to previous case, deviation from exact values is less than 1 percent in port area

computations. As far as burning surface area is concerned, deviation is less than 2

percent. Therefore, one can say that error is in acceptable limit and code performs 4-arm

star shaped grain burnback with an acceptable accuracy.

Similar burnback analyses are performed for a multi-perforated circular, a dendrite and

an anchor type of the grains as shown in Figures 10, 11 and 12, respectively.

3.2 Three Dimensional Burnback Results:

Three dimensional burnback is first applied to ballistic research motor, generally

used to determine burning rate of the propellant. Perforation of the propellant of the

motor is shown in Figure 2. Dimensions of the motor are tabulated in Table 3. Motor

burning time is approximately 0.8-1 second. Since length to diameter ratio is

approximately 2.25, one can easily assume that the flow inside of the motor chamber is

steady and non-erosive. Therefore, the chamber pressure of the rocket motor is constant

along the length of the chamber. In that case it is possible to find pressure-time history of

the rocket motor by using well known Equation (15) [1, 2] given below.

)n1/(1

n5th

bp0c 10A

CSρaP

−

=

&(15)

Note that the term of 105 comes from the conversion of units, Pascal to bar. Since

pressure is known, it is possible to calculate burning rate and elapsed time by using

Equations (16) and (17), respectively.

( )n5c0 10/Par =& (16)

r

x∆t∆

&= (17)

Therefore, it is possible to find pressure of the rocket motor by using constant

pressure assumption. From the burning area results obtained by the code, one can find

pressure for corresponding time. In Figure 12, pressure time graph obtained by simulation

is compared with the one obtained by static motor test. In Figure 7, burnback steps of this

grain is shown. Since burning area at time t=0 exists and not equal to zero, simulation

starts at a certain pressure value. This causes a deviation between simulation and static

test initially. The areas under the curves are calculated. One can say that areas are

approximately equal to each other and trend of the static test curve is simulated.

Not only circular slotted grain, but also finocyl type of grain burnback analysis is

performed by the code. The verification is performed by using Ballistic Research Motor

as well. Motor grain is modified from circular slotted shape to finocyl shape as shown in

Figure 13.

As its name implies, finocyl is composed of a circular slotted grain and star

shaped grain. It is considered as 3-Dimensional grain due to 3-D formation of fins as it

burns back. Both the original grain geometry and the new geometry 10 burnback steps

later are given in Figure 14 for this grain.

By using Equations (15, 16, 17), the prediction of pressure-time history of the Ballistic

Research Motor are calculated and compared with a static test data shown in Figure 15.

In Equation (15), burning area values are computed by using 3-D grain burnback code.

Grain burnback analysis with the variable burning rate is also performed by code.

Especially, variable burning rate burnback is difficult to model for analytical burnback

code. With the help of Level-Set method, this analysis is performed by just giving

burning speed, Fi,j,k,, for each cartesian node. When two different propellant are used,

analysis considering variable burning rate should be made. In Figure 16, having

considered variable burning rate, burnbacks for two different cylindrical grains are

performed. The original propellants and their burnbacks are shown in the Figure 16.

4. Conclusions:

In this study, both 2-D and 3-D grain burnback codes, developed in Turkish

Defense Industries Research and Development Institute, use the first order Level-Set

Method and they are found to be a strong tool for determining solid propellant regression.

Therefore, one can concluded that after necessary modifications, Level-Set method can

be used to simulate solid propellant burnback.

The most important problem for grain burnback codes are the initial construction

of the boundary or surface. A new algorithm is established for the determination of the

location of the cartesian points on the solid model of the propellant. Therefore, initial

construction of a boundary or surface of the solid propellant is performed with ease. To

simplify flow solver, constant pressure along the rocket motor assumption is made and it

is concluded that this assumption works for the motors which have a small length to

diameter ratio.

Even the definition of the burning surface is discontinuous, grain burnback model

works. This situation is observed in burnbacks of the multi-perforated grain. As the

burning surface progress, grain is divided into many parts, however, burnback code does

not crash in that case.

As the most important advantage of the burnback code developed, variable speed

burnback is achieved just by giving corresponding speed values to the nodes.

5. References:

[1] Barrere. M, Joumatte. A, Vandenkerckhove. J, “Rocket Propulsion”, The AdvisoryGroup for Aeronautical Research and Development of NATO, 1960.

[2] “Design Methods in Solid Rocket Motor”, AGARD-LS-150, 1988

[3] “French J. C., Dunn S. S, “New Capabilities in Solid Rocket Motor Grain DesignModeling (SPP 02)”, Software and Engineering Associates, 2002

[4] Saintout E., Ribereau D, Perrin P, “ELEA: A Tool for 3-D Surface RegressionAnalysis In Propellant Grains”, AIAA-89-2782, 1992

[5] Sethian J. A., “Level Set Method”, Cambridge Monographs on Applied andComputational Mathematics, Cambridge, 1996.

[6] Sethian J. A., “Level Set Methods and Fast Marching Methods”, CambridgeUniversity Press, 1999.

Figure 1. Circular perforated case bounded grain.

Figure 2. The Auto-CAD model of the circular perforation.

Figure 3. Tetrahedral elements generated on the model.

Figure 4. Equally spaced grid.

Figure 5. The procedure for determination of distances.

Figure 6. Procedure for the determination of the location of the cartesian point.

Figure 7. Original perforation and perforation after 3 burnback steps.

Figure 8. Burnbacks of a circular slotted grain.

X

Y

50 100

10

20

30

40

50

60

70

80

90

100

Frame 001 09 May 2003 | | | | | | | | | |Frame 001 09 May 2003 | | | | | | | | | |

Figure 9. Burnbacks of 4-armed star grain.

X

Y

100 200

50

100

150

200

250

Frame 001 12 May 2003 | | | | | | |Frame 001 12 May 2003 | | | | | | |

Figure 10. Burnbacks of a multi-perforated circular grain.

X

Y

50 100

10

20

30

40

50

60

70

80

90

100

Frame 001 14 May 2003 | | | | | | | | | |Frame 001 14 May 2003 | | | | | | | | | |

Figure 11. Burnbacks of a dendrite-shaped

X

Y

50 100 150 200

25

50

75

100

125

150

175

200

Frame 001 20 May 2003 | | | | | |Frame 001 20 May 2003 | | | | | |

Figure 11. Burnbacks of an anchor-shaped grain.

X

Y

50 100 150 200

25

50

75

100

125

150

175

200

Frame 001 15 May 2003 | | | | | | | | | |Frame 001 15 May 2003 | | | | | | | | | |

Figure 12. Comparison of pressure-time history of the Ballistic Research Test Motorwith simulation.

0102030405060708090

100

0 0.2 0.4 0.6 0.8 1 1.2

Time (s)

Pre

ssu

re(b

ar)

Simulation

Statik Test

Figure 13. Ballistic Research Test Motor with finocyl grain.

Figure 14. Burnback of a finocyle grain.

0

10

2030

40

50

60

7080

90

100

0 0.5 1 1.5 2 2.5 3

Time (s)

Pre

sure

(bar

)

Simulation

Static Test

Figure 15. Comparison of pressure-time histories of the Ballistic Research Test Motorwith finocyl grain.

Figure 16. Variable speed burnbacks.

Table 1. Results and comparison of burnbacks of a circular slotted grain.

Table 2. Results and comparison of burnbacks of 4-arms star grain.

Table 3. Dimensions of the ballistic research motor.

Motor Length (mm) Motor Diameter (mm) Perforation diameter (mm)

150 66.7 50