A Numerical Method for Solid Propellant Grain Design

In this study, a grain burnback analysis for an Anchor solid propellant grain geometry for a rocket motor was done. The design process involves parametric modeling of the geometry in using PTC Creo software through dynamic variables that define the complex configuration. Grain burnback is achieved by making new surfaces step by step and calculating geometrical properties at each web increment. The surfaces of the grain are determined by PTC Creo 4.0 using grain shapes at every 5 mm thickness of the burned web. This procedure gives the sizes of the remaining grain surfaces, during its burning process in the Solid Rocket Motor. The results obtained using PTC Creo have been checked by the values of burning surfaces calculated using closed form of equations that describe the same geometry, in order to verify the correctness of numerical calculation. The work presented deals with the application of the PTC Creo software tool for design of any arbitrary solid form of a rocket motor propellant grain. This work was concentrated on a numerical description and determination of the mathematical model of the burning process.


Introduction
IFFERENT complex shapes of solid rocket motor propellant grains may be required in accordance with thrust-time profiles depending on the rocket mission.The main parameter affecting the thrust-time profile is the grain geometry [1,2,3].
Different types of thrust-time functions need the use of different shapes of propellant grains.When designing rocket motors, special significance is given to the design of the propellant grain [2][3][4][5][6][7].
According to the motor requirements, designer should choose the appropriate configuration [8].Some recommendations for the choice of the propellant grain shapes can be found in references [2,3,8].
As the propellant burns, the burning surface of the grain moves in a direction normal to the surface.This regression is called burnback.Grain burnback is achieved by making new surfaces at each web increment.It is a pure geometrical analysis of the burning surface distribution depending on the displacement of the flame front through the direction normally to the burning surface.This analysis aims to determine the changes in the grain geometry during the operation of the rocket motor [4].At the end of the burnback analysis, on the basis of the obtained results the distribution of the "burning surface versus burnt layer thickness" can be made.
Analysis of solid propellant rocket motors is progressing in two levels, where, independent of the level, it is needed to assess the following two basic steps.
The steps are, at first, propellant grain configuration selection and defining the geometry which satisfies conditions of internal ballistics.In the second step, structural integrity analysis has to be done.This can be a process that is repeated several times, iteratively.For very complex profiles, especially for the structural analysis, the use of numerical models is required.Nowadays, CAD software programs are used for modeling and drafting the grain geometry [4,9,10,11,12,13].
In this paper PTC Creo 4.0 software was used for this purpose [14].

Design and analysis of propellant grain
The aim of this paper was to determine the change of the burning surface for the complex Anchor grain geometry using the PTC Creo software.The obtained results will be comparing with the results obtained by classical formulas in the closed form.These formulas were not available.Therefore, they derived on this occasion for the Anchor type of grain.
The task was to make burning surface distribution vs. burnt thickness, for the Anchor grain which outer diameter is equal to (C)= 280 mm.The shape of the Anchor was used only as a sample for analyzing the possibilities of applying PTC Creo for this purpose.The Anchor geometry can be categorized as complex geometry [15].
The Anchor grain configuration is defined by seven independent variables.The Anchor grain configuration can be seen in Fig. 1.The parameters which define the Anchor geometry are also given in Fig. 1.

D Figure 1. Anchor grain configuration with geometric parameters
The process of burning surface calculation involves parametric modeling of the geometry in PTC Creo software through dynamic variables that define the complex configuration.
The geometry was modeled parametrically and the parameters which change during the burnback process have been modifing for each burn step using PTC Creo program [16].
Grain burnback was achieved by making new surfaces at each web increment and calculating geometrical properties at each step.The procedure adopted can be applied to any complex geometry in a relatively simple way in the process of grain configuration design.

Results and discussion
The numerical results obtained by the program PTC Creo were compared with the results obtained by using the formulas that describe geometrically each combustion stages.
Table 1 gives the burning surface values in percentages, obtained by the program PTC Creo during the combustion, or after changing the burnt layer thickness or time of combustion.Fig. 2 shows that the combustion of this type propellant grain leads to large changes in the burning surface (24% increase in relation to optimal value) with the burning time.After the Anchor grain shape development of burning surface in the Creo program, a mathematical analysis of the burning surface was done.
In order to better define the mathematical model of the burning surface distribution, the combustion is divided into different phases.
Combustion of the Anchor propellant shape takes place in 3 characteristic phases: Phase I 0 mm < Burnt layer thickness ≤ 17,5 mm At each phase, the values of all individual parts of the perimeter have to be defined.
With equations from 1 to 8, the burning surface in the first phase of combustion is mathematically defined.All the parameters of the given equations are shown in Fig. 3.
In the phase one, the burnt layer thickness (Y) is between the following limits: Perimeter in the phase one (P I ): ( ) Calculation for P I : ( ) sin cos 1 Burn surface area (A P ) in the phase one:

( ) ( )
length of grain, 0,1, 2  With equations from 9 to 15, the combustion surface in the second phase of combustion is mathematically defined.All the parameters of the given equations are shown in Fig. 4.
In the phase two, burnt layer thickness, for the entire range of: ( ) Perimeter in the phase two (P II ): ( ) Exponents of certain sizes (e.g D I , R3 I , R1 I , etc.) represent the values of these sizes at the end of the previous combustion phase.
Calculation for P II ( ) cos sin 3 1 cos sin 1 ( ) ( ) Burn surface area in the phase two: ( ) Phase III 50 mm < Burnt layer thickness ≤ 70 mm With equations from 16 to 20, the combustion surface in the third phase of combustion is mathematically defined.All the parameters of the given equations are shown in Fig. 5.
In the phase three, burnt layer thickness, for the entire range of: Perimeter in the phase three (P III ): ( ) Calculation for P III ( ) ( ) Burn surface area in the phase three: ( ) The resulting mathematical model of combustion was used to obtain graph burning surface change-burnt layer thickness, Fig. 6, which was constructed by changing the burnt layer thickness from 0 to 70 mm (web increment: 1 mm) and obtaining burning surface values.The obtained burning surface values using the Creo software and the classical formulas in the closed form are given in Table 2. Also, in terms of the mean values, the obtained results of burning surface of the Anchor shape based on Creo programe and based on classical formulas in the closed form are consistent, Table 2.
This means that the execution of mathematical formulas of combustion the Anchor shape in closed form was carried out in an appropriate manner.
The dimensions of the Anchor shape propellant are shown in Table 3. Obtained burning surface values for each web increment by using mathematical formulas are compared with PTC Creo solution of Anchor grain shape and can be seen from Fig. 8.The obtained results show the good agreement of the software solution with the obtained mathematical model of combustion (Figure 8), which means that the mathematical description of the combustion process of this propellant type was carried out in an appropriate way, respectively.
Furthermore, the significant correlation between software solution and mathematical model with formulas in closed form, which is 0,99975, confirms that using the PTC Creo software is an appropriate solution for purpose of numerical design of a solid propellant grain with such a complex shape that it is not possible to develop a mathematical model in closed form.

Conclusion
The grain burnback analysis is the one of the most important steps of solid rocket motor design.
In this paper, the increment method is used by PTC Creo software solution to estimate burning surface.The results were compared with the results obtained by classical formulas in the closed form.
An example of the Anchor shaped grain channel has been considered using the PTC Creo software solution, in order to determine the burning surface dependence on burned depth.The obtained results have shown the good agreement of the software solution with the results obtained by classical mathematical model.
The obtained matching results indicate that the selected numerical calculation method can be applied to any complex propellant grain form for which it is not possible to construct the classical formulas for the development of a burning surface in the closed form.
The results obtained show that various grain design requirements can be met using the proposed Creo programe for the grain design process.

Figure 2 .
Figure 2. Burning surface change vs burnt layer thickness graph for Anchor type of grain

Figure 3 .
Figure 3. Appearance of the propellant grain at the start of the first phase of combustion with the dimensions on which the mathematical model of combustion is concerned

8 )n
= the number of burning foreheads, in this case n = 0 Phase II 17,5 mm < Burnt layer thickness ≤ 50 mm

Figure 4 .
Figure 4. Appearance of the propellant grain at the start of the second phase of combustion with the dimensions on which the mathematical model of combustion is concerned

Figure 5 .
Figure 5. Appearance of the propellant grain at the start of the third phase of combustion with the dimensions on which the mathematical model of combustion is concerned

Figure 6 .Figure 7 .
Figure 6.Changes of the burning surface values with the burnt layer thickness for optimized shape of Anchor, based on the mathematical model

Figure 8 .
Figure 8. Changes of the burning surface values vs burnt layer thickness graphs for Anchor type of grain: A -created by using results from the Creo software, Bcreated by using a mathematical model of combustion

Table 1 .
The burning surface values in relation to changing the burnt layer thickness for initial shape of grain I phase [0 ≤ Burnt layer thickness ≤ 0,5(R2 -R1)]

Table 2 .
The burning surface values in relation to changing the burnt layer thickness for Anchor shape propellant

Table 3 .
Dimensions of the Anchor shape propellant