**Analytical Prediction for Grain Burn Time and Burning Area Kinematics in a Solid Rocket Combustion Chamber**

Charles A. Osheku, Oluleke O. Babayomi and Oluwaseyi T. Olawole

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.82822

#### Abstract

This chapter proposes the application of Newtonian particle mechanics for the derivation of predictive equations for burn time, burning and unburnt area propagation for the case of a core propellant grain. The grain is considered to be inhibited in a solid rocket combustion chamber subject to the assumption that the flame propagation speed is constant for the particular solid fuel formulation and formation chemistry in any direction. Here, intricacies surrounding reaction chemistry and kinetic mechanisms are not of interest at the moment. Meanwhile, the physics derives from the assumption of a regressive solid fuel pyrolysis in a cylindrical combustion chamber subject to any theoretical or empirical burn rate characterization law. Essential parametric variables are expressed in terms of the propellant geometrical configuration at any instantaneous time. Profiles from simulation studies revealed the effect of modulating variables on the burning propagation arising from the kinematics and ordinary differential equations models. In the meantime, this mathematical exercise explored the tendency for a tie between essential kernels and matching polynomial approximations. In the limiting cases, closed form expressions are couched in terms of the propellant grain geometrical parameters. Notably, for the fuel burn time, a good agreement is observed for the theoretical and experimental results.

Keywords: solid rocket fuel, tubular rocket propellant, differential equations, burn rate

### 1. Introduction

Since the advent of rocketry, researchers have preoccupied their minds on the development of effective solid fuels for rocket and missile propulsion systems. A compendium of scholarly

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

works in propellant chemistry, aerothermodynamics, flight mechanics, guidance, navigation and control analyses abound in the literature. Solid fuels have been and are still in high demand for space mission and missile development planning. Notwithstanding the progresses in solid fuels physics and the advent of huge numerical studies, analytical conjectures are aptly handy for novel mechanical maneuvering of flight trajectories.

In the meantime, considerable progress was made by Tseng and Yang [1] in investigating the combustion of homogeneous propellants in realistic motor environments. The impact of the dispersion of instability signatures into the burning regions on combustion characteristics of the propellant was investigated. On this note, Roh et al. [2, 3] studied in details the relationship between acoustic oscillations and fast changing propellant burning in laminar flows. While the purpose of the study was to discover the underlying causes of perturbations, the inclusion of chemical characteristics provided a more robust mathematical solution. As a matter of scientific fact, same analysis was extended to incorporate the effect of turbulence [4, 5].

Likewise, a comprehensive numerical analysis was conducted in [6] to study the combustion of a double-base homogeneous propellant in a rocket motor. Emphasis was placed on the motor internal flow development and its influence on propellant combustion. The formulation was based on the Favre-averaged, filtered equations for the conservation laws and took into account finite-rate chemical kinetics and variable thermophysical properties. Nonetheless, results from the study showed that a smoother axial velocity gradient in conjunction with a vertical flow convection have a tendency to prevent or circumvent turbulence regime from deep penetration into the primary flame zone. These turbulence energy spectra have prompted dominant harmonics in a frequency range capable of triggering combustion instabilities. Meanwhile, a methodology for the solution of the internal physics of solid propellant rocket motors was described in [7]. The mathematical problem involved the simulation of a burning surface that dynamically changed the interface between the solid propellant and combustion gas phases.

An additional study in [8] showed how a technique was developed to obtain a burning rate data across a range of pressures from ambient to 345 MPa. It combines the uses of a low loading density combustion bomb with a high loading density closed bomb procedure. Furthermore, a series of nine ammonium perchlorate (AP)-based propellants were used to demonstrate the uses of the technique in comparison to the neat AP burning rate barrier. The effect of plasticizer, oxidizer particle size, catalyst and binder type was investigated. This necessitated an experimental program that was performed at the Space Propulsion Laboratory of the Politecnico di Milano. Notably, within the explored operating conditions and the associated uncertainty bands, a neutral trend for the solid fuel regression rate with increasing pressure was observed. The formulation tested was hydroxyl-terminated polybutadiene in gaseous oxygen at pressures ranging from 4 to 16 bars. A simplified analytical model, which retains the essential physics and accounts for pressure dependency, was developed for hybrid rockets in conjunction with the corresponding numerical simulation reported in [9]. However, the results of its simplified analytical model may not translate directly for use with solid rockets.

Nonetheless, the study reported in [10] was concerned with the prediction of the pressure history during the process of flame-spreading and combustion of solid propellant grains as would occur, for example, in a gun cartridge. Solution of the governing conservation equations for the two-phase media requires the use of empirical relations to account for the physical processes of momentum and energy interaction between the solid grains and hot propellant gas. The results indicated the significance of these interactions for the predictions of pressure and velocity fields. Of note too is the study in [11], where the combustion response of homogeneous and heterogeneous solid propellants to an imposed velocity field was certified to be a viable model for erosive burning mechanism. This leads to an imposed velocity field that has its roots in a multistate analysis of a solid rocket motor combustion processes. In the meantime, for homogeneous solid propellants, it has been shown that for certain realistic choices of the parameters, both positive and negative erosions simultaneously occurred. The underlying mechanism for erosive burning is tied flame stretching. On the hand, for heterogeneous solid propellants, any enhancements of the burn rate are tied to the cross flow velocity, propellant morphology and geometry and chamber pressure.

While information on thermodynamics is readily available in the literature, very clear analytical representation of the burn time of any geometry is rare. For now a gap exists for theoretical closed form results and experiential validation investigations. Theoretical equations that predict analytical burn time, thermal stresses buildup and how they are related ab initio to the solid propellant geometry are rare in literature. It is therefore necessary to have simplified analytical models that reduce computational time and laborious procedures and having reliance on numerically complicated methods such as computational fluid dynamics (CFD) or computational heat transfer (CHT) that would be utilized in the estimation of the burn time.

Traditionally, design and analysis of solid rocket motors have relied on empirical measurements to characterize fuel burn times and other propellant/motor performance quantities. This has been primarily because of the complexity of modeling adequately nontrivial fuel grain geometries and combustion processes. As overall system and vehicle performance models become more advanced and answer greater demands in terms of accuracy and detail, it is increasingly necessary to include more sophisticated models of subsystems such as the rocket motor. On the other hand, improved computational capabilities and better insights gleaned from experimental studies provide the means of achieving these better subsystem models. This chapter therefore covers a topic that is ripe for study and has potential to be of significant use to engineers who need to model burning performance for solid rocket propulsion. It may be of particular interest to those who lack the luxury of pursuing an experimental test campaign for a range of candidate fuel grain designs and parameters.

Several competing approaches exist in recent literature based on different focuses in terms of fundamental physics: analysis of radiation, temperature distribution and a range of coupled fluid flow/combustion approaches of varying complexity from 1D flow models to CFD. The method proposed in this chapter is beneficial in terms of its simplicity and consequently low computational cost, although its significant central assumptions mean that it can be applied only to certain cases (homogeneous propellants, tubular (regressive) grain designs and constant regression rates/steady-state operation). Its focus on only the kinematic viewpoint, without accounting for minutiae of chemical kinetic mechanisms, appears fairly unique among recent studies which have instead delved into the physically dominant processes at work.

Predicting grain burn time and burning area kinematics can be done in three ways: empirically (by experiment), analytically (using approximated mathematical models solved exactly) or numerically (by applying exact mathematical models solved approximately). The method proposed in the chapter falls into two parts: the first (burning time determination) combines analysis for modeling supplemented by empirical test data; the second part (burning/unburnt area determination) only covers an analytical approach without experimental or numerical validation [12].

The chapter is organized as follows: first is the derivation of the burn time equation, followed by an analysis of the effects of multiple points of ignition on the burn time. Analytical models are developed for unburnt and burning area propagation and discussion of results and the conclusion.

## 2. Derivation of burn time equation

In this section, the theory conjured is subject to the under listed assumptions, namely:


The typical tubular propellant and the combustion propagation are illustrated in Figures 1 and 2, while Figure 3 gives an analytical model of the flame particle traversing in the designated axes.

In general, the average value of any time-dependent function F0ð Þt within the time interval tH and tG satisfies any of the equations:

$$F\_{0(avg)} = \frac{1}{(t\_G - t\_H)} \int\_{t\_H}^{t\_G} F\_0(t)dt\tag{1}$$

$$\dot{F}\_{0(avg)} = \frac{1}{(t\_G - t\_H)} \int\_{t\_H}^{t\_G} \left(\frac{dF\_0(t)}{dt}\right) dt\tag{2}$$

Given that, ð Þ W; L; As are the web (thickness), length ð Þ L and the sectorial area ð Þ As of any typical tubular propellant grain, where the following holds:

Analytical Prediction for Grain Burn Time and Burning Area Kinematics in a Solid Rocket Combustion Chamber 53 http://dx.doi.org/10.5772/intechopen.82822

Figure 1. Typical tubular solid fuel.

Figure 2. Illustration of different burning directions at point of ignition.

$$(\mathcal{W}, L, A\_s) \in F\_o(t); \left(\dot{\mathcal{W}}, \dot{L}, \dot{A}\_s\right) \in \dot{F}\_o(t)$$

Consequently, the total time required for the entire burning process specified in Figure 4 must satisfy the following kinematic equation, namely:

$$t\_{b(total)} = \frac{\mathcal{W}}{\dot{\mathcal{W}}} + \frac{L}{\dot{L}} + 2\frac{A\_s}{\dot{A}\_s} \tag{3}$$

From the point of ignition, as illustrated in Figure 2, the following further holds:

$$\mathcal{W} = \mathcal{W}\_o - \dot{r}t\_{b(\text{web})} \colon L = L\_o - \dot{r}t\_{b(\text{axial})} \colon d = \mathbf{d}\_o + 2\dot{r}t\_{b(\text{radial})} \tag{4}$$

$$\left| \left| \mathcal{W} \right| \right| = \left| \left| \mathcal{W}\_o - \dot{r} t\_{b(\text{red})} \right| \right|; \left| \left| L \right| \right| = \left| \left| L\_o - \dot{r} t\_{b(\text{axial})} \right| \right|; \left| \left| d \right| \right| = \left| \left| d\_o + 2 \dot{r} t\_{b(\text{radial})} \right| \right| \tag{5}$$

in conjunction with a constant regression rate ð Þr\_ : Under these circumstances, Eq. (3) now becomes

Figure 3. Regression along burn regions.

Figure 4. Illustrations of integral part of multiple ignition points on propellant grain. (a) 2-points, (b) 3-points, and (c) 4-points.

$$t\_{b(total)} = \frac{\mathcal{W}(1+2\eta) + \eta d}{\dot{r}} + 2\frac{A\_s}{\dot{A\_s}}; \quad \eta = \frac{L}{D} \tag{6}$$

From the sector burning area configuration, the following ensues, viz.:

Analytical Prediction for Grain Burn Time and Burning Area Kinematics in a Solid Rocket Combustion Chamber 55 http://dx.doi.org/10.5772/intechopen.82822

$$A\_S = \frac{1}{2} \left( \mathbf{R}^2 - r^2 \right) \mathbf{\dot{f}} = \frac{1}{2} W (W + d) \boldsymbol{\theta} \quad ; \qquad \dot{A}\_S = \frac{1}{2} \left[ \dot{\theta} \mathbf{W} (\mathbf{W} + d) - \theta d \dot{r} \right] \tag{7}$$

where θ is in radians, in conjunction with the following kinematics relation, viz.:

$$
\dot{R} = 0 \\
\therefore \dot{r} = \dot{r}\_0 \\
\forall \text{ } \ddot{r} = 0 \tag{8}
$$

leading to the total segmental burn time for the sectorial propellant grain as

$$\frac{A\_s}{\dot{A\_s}} = \frac{1}{\dot{r}} \cdot \frac{(W+d)\theta}{4\eta \left(1' + \frac{W}{d}\right) \left(\frac{\text{Ad}}{W}\right)} \left( \text{ } \tag{9}$$

From Eq. (9), Eq. (6) becomes

$$t\_{b(total)} = \left(\frac{\mathcal{W}(1+2\eta) + \eta d}{\dot{r}} + \frac{2}{\dot{r}} \left(\frac{(\mathcal{W}+d)\Theta}{4\eta \left(\mathcal{V} + \frac{\mathcal{W}}{d}\right) \ell^{\frac{\Theta d}{\mathcal{W}}}}\right)\right) \quad 0 \le \theta \le 2\pi \tag{10}$$

In the meantime, Eq. (9) in terms of the instantaneous burning time t<sup>ε</sup> takes the form, viz.

$$t\_{\text{b(total)}} = \frac{1}{\dot{r}} \left\{ (1 + 2\eta)\mathcal{W}\_0 + \eta d\_0) - t\_\varepsilon + 2 \left( \frac{(\mathcal{W}\_0 + d\_0)\theta\_0 + (\mathcal{W}\_0 + d\_0)\dot{\theta}t\_\varepsilon + \theta\_0 \dot{r}\_\varepsilon + \dot{\theta} \dot{r}\_\varepsilon^2}{4\eta \left( \mathcal{I} + \frac{\mathcal{W}\_0 - \dot{r}t\_\varepsilon}{4\alpha + 2\dot{r}t\_\varepsilon} \right) - \left( \frac{4\theta\_0 + \left( 2\dot{r}\theta\_0 + \dot{\theta}\dot{d}\_0 \right)t\_\varepsilon + 2\dot{r}\theta\_\varepsilon^2}{W\_0 - \dot{r}t\_\varepsilon} \right)} \right) \right\} \tag{11}$$

arising from the following flame additional kinematics relations, namely, <sup>θ</sup> <sup>¼</sup> <sup>θ</sup><sup>0</sup> <sup>þ</sup> <sup>θ</sup>\_tε, in conjunction with others specified in Eq. (5).

It is significant to examine the limiting case of Eq. (11) as t<sup>ε</sup> ! 0, viz.:

$$\begin{split} \lim\_{t\_{\varepsilon}\to 0} \mathbf{t}\_{b\lfloor \mathrm{total} \rfloor} &= \lim\_{t\_{\varepsilon}\to 0} \frac{1}{\bar{r}} \left\{ \left\{ (1+2\eta)W\_{0} + \eta d\_{0} \right\} - t\_{\varepsilon} + 2 \left( \underbrace{\begin{pmatrix} W\_{0} + d\_{0} \end{pmatrix} \Theta\_{0} + (W\_{0} + d\_{0}) \dot{\Theta}\_{\varepsilon} + \dot{\Theta}\_{0} \dot{r}\_{\varepsilon} + \dot{\theta} \dot{r}\_{\varepsilon}^{2} \\ \mathbf{4}\eta \left( \mathbf{I} + \frac{W\_{0} - \dot{r}\_{\varepsilon}}{d\_{0} + 2\dot{r}\_{\varepsilon}} \right) \left( \mathbf{I} \left( \frac{\not\!\!/ \mathbf{W}\_{0} + (2\dot{r}\_{\varepsilon}\dot{\Theta}\_{0} + \dot{\theta}\dot{r}\_{\varepsilon}) \mathbf{I}}{W\_{0} - \dot{r}\_{\varepsilon}} \right) \right) \right\} \right\} \\ &= \frac{1}{\bar{r}} \left\{ \left[ \left( \not\!\!/ + 2\eta\_{0} \right)W\_{0} + \eta\_{0}d\_{0} \right] \right\} \left\{ \left( \underbrace{\begin{pmatrix} 2(\mathcal{W}\_{0} + d\_{0})\mathcal{O}\_{0} \\ \mathbf{\sqrt{\eta}} \left( \mathbf{I} + \frac{W\_{0}}{d\_{0}} \right) \left( -\frac{\not\!\!/ \mathbf{W}\_{0}}{\ddot{\mathsf{W}}\_{0}} \right) \end{pmatrix} \right) \right\} \end{split} \tag{12}$$

to indicate the closed form burn time prediction in terms of the tubular initial geometrical configuration.

In the meantime, Eqs. (10) and (11) are expressed further as

$$t\_{b(total)} = \frac{d}{\dot{r}} \left\{ \left[ (1 + 2\eta)\chi + \eta \right] + \left( \frac{\left( 2(1 + \chi)\overline{\theta}\_0 \right)}{\eta(1 + \chi) - \frac{\overline{\theta}\_0}{\chi}} \right) \right\} \begin{cases} \dot{r} \\ \dot{\chi} \end{cases}$$

$$\forall \ \chi = \frac{W}{d} = Lt\_{t\_{r=0}} \left( \chi = \frac{W\_0 - \dot{r}t\_\varepsilon}{d\_0 + 2\dot{r}t\_\varepsilon} \right) = Lt\_{t\_{r=0}}(\chi) = Lt\_{t\_{r=0}} \left. \frac{\chi\_0 - \frac{\dot{r}\dot{\varphi}}{d\_0}}{1 + \frac{2\dot{r}t\_\varepsilon}{d\_0 \zeta}} \right| \tag{13}$$

and

$$t\_{b\text{(total)}} = \frac{d\_0}{\dot{r}} \left\{ \left( 1 + 2\eta \right) \chi\_0 + \eta \right\} + \left( \frac{2(1 + \chi\_0)\overline{\rho}\_0}{4\eta (1 + \chi\_0) - \frac{\overline{\rho}\_0}{\chi\_0}} \right) \left\{ \left< \begin{array}{c} \chi\_0 = \frac{W\_0}{d\_0} \\\\ \end{array} \right. \\ \left. \left. \begin{array}{c} 0 \le \overline{\theta}\_0 \le 2\pi \end{array} \right. \\ \left. \end{array} \right. \right. \right. \right\}$$

When a propellant is completely burned out θ<sup>0</sup> ¼ 2π, which corresponds to the case of 1-point ignition, to give the following expression, viz.

$$t\_{b(total)} = \frac{d}{\dot{r}} \left\{ \left( 1 + 2\eta \right) \chi + \eta \right\} + \frac{\pi (1 + \chi)}{\eta (1 + \chi) - \frac{\pi}{2\chi}} \right\} \Bigg(\tag{15}$$

When θ<sup>0</sup> ¼ π, which corresponds to the case of diametric ignition at two opposite sides to give the following expression, viz.

$$t\_{b(total)} = \frac{d\_0}{\dot{r}} \left\{ [(1+2\eta)\chi\_0 + \eta] + \frac{\pi(1+\chi\_0)}{\eta(1+\chi\_0) - \frac{A}{2\chi\_0}} \right\} \Bigg| \tag{16}$$

This is to be further examined in the subsection for multiple ignition points.

#### 2.1. Effect of multiple ignition points (Np)

The effect of multiple ignition points is expected to create multiple sectorial flame propagation kinematics as illustrated in the figures below. Here, hatchings are indicating burning surfaces intersection arising from the sectorial kinematic propagation of the flame in line with the description in Figures 2 and 3.

Here, the matching kinematic equation takes the form

$$\frac{A\_s}{\dot{A}\_s} \left( \mathcal{N}\_p \right) = \frac{1}{\dot{r}} \left( \underbrace{\left( \mathcal{W} + d \right) \theta\_{\binom{N\_r}{0}}}\_{\mathcal{W} \left( 1 + \frac{\mathcal{W}}{d} \right) - \frac{\theta\_{\binom{N\_p}{0}} d}{\mathcal{W}}}\_{\mathcal{W}} \right) \left( \theta\_{\binom{N\_p}{0}} = \frac{2\pi}{\mathcal{N}\_p} \tag{17}$$

It should be noted that propagations in radial and axial directions are expected to be rapid in consonance with the number of ignitions points. The overall effect therefore modulates Eq. (16) as

$$d\_{b\_{\left(N\_{p}\right)}\left(\text{total}\right)} = \frac{d\_{0}}{\dot{r}} \left\{ \left( (1+2\eta)\frac{\chi\_{0}}{N\_{p}} + \frac{\eta}{N\_{p}} \right) \left( + \left( \frac{\frac{\pi}{N\_{p}^{2}}(1+\chi\_{0})}{\frac{\chi}{N\_{p}}(1+\chi\_{0}) - \frac{\pi}{2\chi\_{0}N\_{p}^{2}}} \right) \right\} \right\}$$

Analytical Prediction for Grain Burn Time and Burning Area Kinematics in a Solid Rocket Combustion Chamber 57 http://dx.doi.org/10.5772/intechopen.82822

$$\dot{\lambda} = \frac{d\_0}{\dot{r}} \left\{ \left[ \left( \frac{1 + 2\eta}{N\_p} \right) \chi\_0 + \frac{\eta}{N\_p} \right] + \left( \frac{\pi \left( \frac{1 + \chi\_0}{N\_r} \right)}{\eta \left( \frac{1 + \chi\_0}{N\_r} \right) - \frac{\pi}{2\chi\_0 N\_p^2}} \right) \right\} = \frac{t\_{b\text{(total)}}}{N\_p} \tag{18}$$

#### 3. Unburnt and burning area propagation

The plan and sectional views of the propellant grain geometries are illustrated in the figure below (Figure 5). These views are expected to provide illustrations on how the unburnt propellant grain area is derived.

From the figures above, the tubular grain's surface area is given by

$$A\_{ub} = \frac{\pi}{2} \left( D^2 - d^2 \right) + \pi dL \tag{19}$$

On introducing the aspect ratio, <sup>η</sup> <sup>¼</sup> <sup>L</sup>, where <sup>D</sup> <sup>¼</sup> <sup>d</sup><sup>0</sup> <sup>þ</sup> <sup>2</sup>W0, Eq. (19) becomes <sup>D</sup>

$$A\_{ub} = \pi \left[ 2\mathcal{W}^2 + \eta d^2 + 2(1+\eta)\mathcal{W}d \right] \tag{20}$$

Using parts of Eq. (5), the above equation is further simplified as

$$A\_{ub}(t\_\ell) = A\_0 - \pi \left[ B\_1 \dot{r} t\_\ell + B\_2 \dot{r}^2 t\_\ell^2 + B\_3 \dot{r}^3 t\_\ell^3 \right] \tag{21}$$

where t<sup>ε</sup> is the instantaneous burning time.

Figure 5. Tubular solid fuel geometrical parameters.

$$A\_0 = \pi \left[ 2W\_0^2 + 2\left(1 + \eta\_0\right)W\_0 d\_0 + \eta d\_0^2 \right]; \eta\_0 = \frac{L\_0}{d\_0 + 2W\_0};$$

$$B\_1 = d\_0 \left[ \frac{1}{\left(1 + \frac{2W\_0}{d\_0}\right)} - 4\eta\_0 \right] + 2\left(1 + \eta\_0\right)(d\_0 - W\_0) + \frac{W\_0}{\left(1 + \frac{2W\_0}{d\_0}\right)};$$

$$B\_2 = 2\left[ \left(1 - \eta\_0\right) + \frac{2}{\left(1 + \frac{2W\_0}{d\_0}\right)} \right]; B\_3 = \frac{2}{\left(1 + \frac{2W\_0}{d\_0}\right)}$$

Eq. (21) can be further written as

$$A\_{ub}(t\_\varepsilon) = A\_{ub}(0) - A\_b(t\_\varepsilon) \tag{22}$$

where Aubð0Þ ¼ A0, as illustrated below (Figure 6).

Above is the closed form expression for the instantaneous burning propellant area. Using Eq. (12), the kernel in the unburnt area AubðtεÞ are rearranged to effect erosive regressive burning process, where

$$A\_{ub}(t\_\ell) = A\_0 \left[ 1 - \mathsf{C}\_1 \dot{r} t\_\ell - \mathsf{C}\_2 \dot{r}^2 t\_\ell^2 - \mathsf{C}\_3 \dot{r}^3 t\_\ell^3 \right] \tag{23}$$

� � � � where <sup>C</sup><sup>1</sup> <sup>¼</sup> <sup>B</sup><sup>1</sup> ; C<sup>2</sup> <sup>¼</sup> <sup>B</sup><sup>2</sup> ; C<sup>3</sup> <sup>¼</sup> <sup>B</sup><sup>3</sup> ; <sup>Λ</sup><sup>0</sup> <sup>¼</sup> <sup>2</sup>W<sup>2</sup> <sup>0</sup> <sup>þ</sup> 2 1 <sup>þ</sup> <sup>η</sup><sup>0</sup> <sup>W</sup>0d<sup>0</sup> <sup>þ</sup> <sup>η</sup>d<sup>2</sup> Λ<sup>0</sup> Λ<sup>0</sup> Λ<sup>0</sup> 0 <sup>2</sup> <sup>0</sup> <sup>13</sup> <sup>2</sup>ðW<sup>0</sup> <sup>þ</sup> <sup>d</sup>0Þθ<sup>0</sup> <sup>Ψ</sup> <sup>¼</sup> <sup>4</sup>½ð<sup>1</sup> <sup>þ</sup> <sup>2</sup>ηÞW<sup>0</sup> <sup>þ</sup> <sup>η</sup>d0� þ @ � � A5; <sup>∀</sup> <sup>0</sup> <sup>≤</sup> <sup>θ</sup><sup>0</sup> <sup>≤</sup> <sup>2</sup>π: <sup>4</sup><sup>η</sup> <sup>1</sup> <sup>þ</sup> <sup>W</sup><sup>0</sup> � <sup>θ</sup>0d<sup>0</sup> d<sup>0</sup> W<sup>0</sup>

By introducing <sup>Χ</sup><sup>0</sup> <sup>¼</sup> <sup>W</sup> d0 0 , Eq. (23) results to

$$A\_{ub}(t\_\varepsilon) = \overline{A}\_0 \left[ 1 - \overline{\mathsf{C}}\_1 \tau - \overline{\mathsf{C}}\_2 \tau^2 - \overline{\mathsf{C}}\_3 \tau^3 \right] \tag{24}$$

Figure 6. (a) Illustration of a point ignition at the commencement of burning propagation. (b) Cross section of burning propagation. (c) Complete burning process.

Analytical Prediction for Grain Burn Time and Burning Area Kinematics in a Solid Rocket Combustion Chamber 59 http://dx.doi.org/10.5772/intechopen.82822

$$\text{where } \overline{\mathbf{A}\_{0}} = \pi d\_{0}^{2} [2\chi\_{0}^{2} + 2(1 + \eta\_{0})\chi\_{0} + \eta\_{0}]; \overline{\mathbf{C}\_{1}} = \frac{\overline{\mathbf{a}\_{0}}\overline{\overline{\mathbf{V}}}}{\Lambda\_{0}}; \overline{\mathbf{C}\_{2}} = \frac{\overline{\mathbf{a}\_{0}}\overline{\overline{\mathbf{V}}}}{\Lambda\_{0}}; \overline{\mathbf{C}\_{3}} = \frac{\overline{\overline{\mathbf{a}}\_{0}\overline{\overline{\mathbf{V}}}}}{\Lambda\_{0}}$$

$$\text{where } \Lambda\_{0} = 2\mathcal{W}\_{0}^{2} + 2\left(1 + \eta\_{0}\right)\mathcal{W}\_{0}d\_{0} + \eta d\_{0}^{2}; \overline{\overline{\mathbf{V}}} = \left[(1 + 2\eta)\chi\_{0} + \eta\right] + \left(\frac{\overline{n(1 + \chi\_{0})}}{\eta(1 + \chi\_{0}) - \frac{\chi\_{0}}{\overline{\chi\_{0}}}}\right)$$

$$\text{Now at } t\_{\varepsilon} = t\_{b} \quad ; \quad A\_{ab}(t\_{b}) = A\_{0}(1 - \mathsf{C}\_{1} - \mathsf{C}\_{2} - \mathsf{C}\_{3})\tag{25}$$

From Eqs. (24) and (25), the non-dimensionalized unburnt propellant grain area is evaluated as

$$\frac{A\_{ub}(t\_\ell)}{A\_{ub}(t\_b)} = A\_{ub}(\tau) = \frac{\left[1 - \overline{\mathsf{C}}\_1 \tau - \overline{\mathsf{C}}\_2 \tau^2 - \overline{\mathsf{C}}\_3 \tau^3\right]}{\left[1 - \overline{\mathsf{C}}\_1 - \overline{\mathsf{C}}\_2 - \overline{\mathsf{C}}\_3\right]}\tag{26}$$

Next, we return to the instantaneous burning area propagation via the following expression, namely:

$$A\_b(t\_\varepsilon) = \overline{A}\_0 \left[ \overline{\mathbf{C}}\_1 \tau + \overline{\mathbf{C}}\_2 \tau^2 + \overline{\mathbf{C}}\_3 \tau^3 \right] \quad \forall \quad \tau = \frac{t\_\varepsilon}{t\_b} \tag{27}$$

$$\text{Now at } t\_\varepsilon = t\_b.\\ A\_b(t\_b) = \overline{A}\_0 \left[ \overline{\mathsf{C}}\_1 + \overline{\mathsf{C}}\_2 + \overline{\mathsf{C}}\_3 \right] \tag{28}$$

From Eqs. (25) and (28), the non-dimensionalized form of the instantaneous burning area becomes

$$\frac{A\_b(t\_\varepsilon)}{A\_b(t\_b)} = \overline{A}\_b(\tau) = \left(\frac{\overline{\mathbb{C}\_1}}{\overline{\mathbb{C}\_1} + \overline{\mathbb{C}\_2} + \overline{\mathbb{C}\_3}}\right)\tau + \left(\frac{\overline{\mathbb{C}\_2}}{\overline{\mathbb{C}\_1} + \overline{\mathbb{C}\_2} + \overline{\mathbb{C}\_3}}\right)\tau^2 + \left(\frac{\overline{\mathbb{C}\_3}}{\overline{\mathbb{C}\_1} + \overline{\mathbb{C}\_2} + \overline{\mathbb{C}\_3}}\right)\tau^3 \tag{29}$$

#### 3.1. Effect of multiple ignition points

The effect of multiple ignition points (Np) is expected to fractionalize the unburnt area timedependent equations as follows, viz.:

$$A\_{\rm ub}(\tau) = \begin{pmatrix} \frac{1}{N\_p} - \overline{\overline{\mathsf{C}}\_1}\tau - \overline{\overline{\mathsf{C}}\_2}\tau^2 - \overline{\overline{\mathsf{C}}\_3}\tau^3\\ \left(\frac{1}{N\_p} - \overline{\overline{\mathsf{C}}\_1} - \overline{\overline{\mathsf{C}}\_2} - \overline{\overline{\mathsf{C}}\_3}\right) \end{pmatrix}; \forall \, \overline{\mathsf{C}}\_1 = \overline{\mathsf{C}}\_1(\mathcal{N}\_p); \overline{\overline{\mathsf{C}}\_2} = \overline{\mathsf{C}}\_2(\mathcal{N}\_p); \overline{\overline{\mathsf{C}}\_3} = \overline{\mathsf{C}}\_3(\mathcal{N}\_p) \end{pmatrix} \tag{30}$$

$$\overline{\mathsf{C}}\_1(\mathcal{N}\_p) = B\_1 \overline{\overline{\mathcal{W}}}(\mathcal{N}\_p); \overline{\mathcal{C}}\_2(\mathcal{N}\_p) = B\_2 \overline{\overline{\mathcal{W}}}^2(\mathcal{N}\_p); \overline{\mathcal{C}}\_3(\mathcal{N}\_p) = B\_3 \overline{\overline{\mathcal{W}}}^3(\mathcal{N}\_p)$$

$$\overline{\overline{\mathcal{W}}}(\mathcal{N}\_p) = \left[ \left[ \frac{(1 + 2\eta\_0)\mathcal{X}\_0}{N\_p} + \frac{\eta\_0}{N\_p} \right] + \left( \frac{\frac{\pi(1 + \eta\_0)}{N\_p^2}}{\eta\_0 \left(\frac{1 + \chi\_0}{N\_p}\right) - \frac{\pi}{2\chi\_0 \lambda\_p^2}} \right) \right];$$

while for the burning area, the modification is

$$A\_b(\tau) = \left(\frac{\overline{\overline{\mathbf{C}}}\_1 \tau + \overline{\overline{\mathbf{C}}}\_2 \tau^2 + \overline{\overline{\mathbf{C}}}\_3 \tau^3}{\left(\overline{\overline{\mathbf{C}}}\_1 + \overline{\overline{\mathbf{C}}}\_2 + \overline{\overline{\mathbf{C}}}\_3\right)}\right) = \overline{\overline{B}}\_1 \tau + \overline{\overline{B}}\_2 \tau^2 + \overline{\overline{B}}\_3 \tau^3 \tag{31}$$

where 
$$
\overline{\overline{B}}\_1 = \left[ \frac{1}{1 + 2\chi\_0} - 4\eta\_0 + 2\left(1 + \eta\_0\right)(1 - \chi\_0) + \frac{\chi\_0}{1 + 2\chi\_0} \right] \overline{\overline{\overline{\mu}}}\_2^\*
$$

$$
\overline{\overline{B}}\_2 = 2\left[ (1 - \eta\_0) + \frac{2}{1 + 2\chi\_0} \right] \overline{\overline{\overline{\mu}}}\_2^2; \quad \overline{\overline{B}}\_3 = \frac{2}{1 + 2\chi\_0} \overline{\overline{\overline{\mu}}}^3
$$

For <sup>χ</sup><sup>0</sup> <sup>¼</sup> <sup>W</sup> d0 <sup>0</sup> ≪ 1, Eqs. (26) and (31) can be further written as

$$A\_{ub}(\tau) = \begin{pmatrix} 1 - \overline{\overline{\overline{\phantom{v}}}} \tau - \overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\tau}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$$
}

Note that these closed form propagation profiles are kinematically derived. From the kernels, they exhibit third-order polynomial equations. To enable us study these profiles further, the following ordinary differential equation (ODE) modeling follows in the subsection.

#### 4. Ordinary differential equation (ODE) modeling

As previously done, the modeling is taking off from the unburnt propagation problem. Given that AubðtεÞ and Aubðt<sup>ε</sup> þ ΔtεÞ are unburnt propellant grains at time ðtε) and ðt<sup>ε</sup> þ ΔtεÞ, respectively, it is apparent that Aubðt<sup>ε</sup> þ ΔtεÞ < AubðtεÞ; ∀ 0 < t<sup>ε</sup> < ðt<sup>ε</sup> þ ΔtεÞ:

Consequently,

$$\lim\_{\Delta t\_{\varepsilon} \to 0} \left( \frac{A\_{\text{ub}}(t\_{\varepsilon} + \Delta t\_{\varepsilon}) - A\_{\text{ub}}(t\_{\varepsilon})}{\Delta t\_{\varepsilon}} \right) = \frac{-dA\_{\text{ub}}(t)}{dt\_{\varepsilon}} \tag{34}$$

leading to a simple linear ODE of the form

$$\frac{dA\_{ub}(t\_\varepsilon)}{dt\_\varepsilon} = -\lambda\_b A\_{ub}(t\_\varepsilon) \tag{35}$$

where λ<sup>b</sup> ¼ propagation constant=s.

For Eq. (35) to be well posed, the following conditions are specified, viz.:

$$\begin{aligned} \text{(i)} \quad t\_{\ell} &= 0; \; A\_{\text{ub}}(0) = A\_0 \\\\ \text{(ii)} \; t\_{\ell} &= t\_{\text{b}}; \; A\_{\text{ub}}(t\_{\emptyset}) = -A\_0(\mathbb{C}\_1 + \mathbb{C}\_2 + \mathbb{C}\_3) \forall \; A\_{\text{ub}}(t\_{\emptyset}) + A\_{\text{b}}(t\_{\emptyset}) = 0 \end{aligned} \tag{36}$$

Analytical Prediction for Grain Burn Time and Burning Area Kinematics in a Solid Rocket Combustion Chamber 61 http://dx.doi.org/10.5772/intechopen.82822

In the meantime, the solution to Eq. (35) is given by

$$\dots A\_{ub}(t\_\varepsilon) = A\_{ub}(0)e^{-\lambda\_b t\_\varepsilon} \tag{37}$$

On imposing the conditions in Eq. (36), the following closed forms ensue, namely:

$$\dots A\_{ub}(t\_\varepsilon) = A\_{ub}(\mathbf{0})e^{-\Lambda\_b t\_\varepsilon} \tag{38}$$

$$A\_{ub}(t\_\varepsilon) = A\_0 e^{-\lambda\_b t\_\varepsilon} = A\_0(\mathbb{C}\_1 + \mathbb{C}\_2) \tag{39}$$

Meanwhile, Eq. (38) satisfies the following form:

$$A\_{ub}(t\_\varepsilon) = A\_0 e^{-\beta} \left(\frac{t\_\varepsilon}{t\_b}\right) = A\_0 e^{-\beta \tau}; \forall \, \beta = \ln\left(\mathcal{C}\_1 + \mathcal{C}\_2 + \mathcal{C}\_3\right); \tau = \frac{t\_\varepsilon}{t\_b} \tag{40}$$

to give a simple relation of the form

$$\overline{A}\_{ub}(\tau) = e^{\beta(1-\tau)} \,\forall\,\beta = \ln\left(\overline{\mathcal{C}}\_1 + \overline{\mathcal{C}}\_2 + \overline{\mathcal{C}}\_3\right) \tag{41}$$

The foregoing represents the generalized unburnt propellant area propagation as a function of the dimensionless timeð Þτ : To enable us generate semi-infinite polynomial models, the following series approximation suffices, namely:

$$\begin{split} \varepsilon^{\theta(1-\tau)} &\simeq \sum\_{n=0}^{\infty} \frac{\beta^n (1-\tau)^n}{n!} \\ &\simeq \left\{ 1 + \beta (1-\tau) + \frac{\beta^2 (1-\tau)^2}{2!} + \frac{\beta^3 (1-\tau)^3}{3!} + \frac{\beta^4 (1-\tau)^4}{4!} + \dots + \frac{\beta^n (1-\tau)^n}{n!} + R\_N(\beta, \tau) \right\} \end{split} \tag{42}$$

From Eqs. (43) and (44), linear to infinite order profiles can be further deduced. A few illustrations follow, namely:

a. Linear unburnt area profile

$$A\_{ub}^{(1)}(\tau) = \left( (1+\beta) - \beta \tau \right) \tag{43}$$

b. Secondary degree unburnt area propagation profile (quadratic)

$$A\_{ub}^{(2)}(\tau) = \left( \left( 1 + \beta + \frac{\beta^2}{2} \right) - \left( \beta + \beta^2 \right) \tau + \frac{\beta^2}{2} \tau^2 \right) \tag{44}$$

c. Third-degree unburnt area propagation profile (cubic)

$$A\_{ub}^{(3)}(\tau) = \left( \left( 1 + \beta + \frac{\beta^2}{2} + \frac{\beta^3}{6} \right) - \left( \beta + \beta^2 + \frac{\beta^3}{2} \right) \tau + \left( \frac{\beta^2 + \beta^3}{2} \right) \tau^2 - \frac{\beta^3}{6} \tau^3 \right) \tag{45}$$

d. Fourth-degree unburnt area propagation profile

$$A\_{ub}^{(4)}(\tau) = \left( \left( \left\{ +\beta + \frac{\beta^2}{2} + \frac{\beta^3}{6} + \frac{\beta^4}{24} \right\} \left( -\left( \left\{ +\beta^2 + \frac{\beta^3}{2} - \frac{2\beta^4}{3} \right\} \left( +\left( \frac{\beta^2 + \beta^3}{2} + \frac{\beta^4}{4} \right) \right\}^2 \right)^{-1} \right) \right) \right)$$

$$- \left( \frac{\beta^3}{6} + \frac{\beta^4}{6} \right) \left( \frac{3}{4} + \left( \frac{\beta^4}{24} \right) \tau^4 \right) \left( \right)$$

e. Nth degree unburnt area propagation profile

$$A\_{ub}^{(\text{Nth})}(\tau) = \left\{ \left( \begin{matrix} + \beta(1-\tau) + \frac{\beta^2(1-\tau)^2}{2!} + \frac{\beta^3(1-\tau)^3}{3!} + \frac{\beta^4(1-\tau)^4}{4!} \\\\ & \dots + \frac{\beta^N(1-\tau)^N}{N!} \end{matrix} \right) \right\} \tag{47}$$

#### 4.1. Burning propellant area modeling

The burning area AbðtεÞ can be deduced from the following equation, namely:

$$A\_b(t\_\varepsilon) = A\_{ub}(0) - A\_{ub}(t\_\varepsilon) \tag{48}$$

resulting to an expression of the form

$$A\_b(t\_\varepsilon) = A\_0 \left(\mathbb{Y} - \varepsilon^{-\lambda\_b t\_\varepsilon}\right) \tag{49}$$

subject to the following conditions, viz.:

$$\begin{aligned} \text{(i)} \quad t\_{\varepsilon} &= 0; A\_b(t\_{\varepsilon}) = 0\\ \text{(ii)} \quad t\_{\varepsilon} &= t\_{b\prime}; A\_b(t\_b) = \overline{A}\_0 (\overline{Q}\_1 + \overline{C}\_2 + \overline{C}\_3) \left( \\ &\tag{50} \end{aligned}$$

Note that Eq. (47) from the second part of Eq. (48) becomes

$$A\_{\mathbb{B}}(t\_{\mathbb{B}}) = A\_0 \quad 1 - e^{-\beta \left(\frac{t\_{\mathbb{B}}}{t\_{\mathbb{B}}}\right)} \Big(\Big) \Big(\Big) \tag{51}$$

such that at

$$t\_{\varepsilon} = t\_{b\prime} \colon A\_b(t\_b) = A\_0 \left( \bigvee -\varepsilon^{-\beta} \right) \tag{52}$$

from which the following equation ensues:

$$\frac{A\_b(t\_\varepsilon)}{A\_b(t\_b)} = \overline{A}\_b(\tau) = \left(\frac{\mathbb{Y} - e^{-\beta \tau}}{\mathbb{I} - e^{-\beta}}\right) \tag{53}$$

upon using the following series approximations, viz.:

Analytical Prediction for Grain Burn Time and Burning Area Kinematics in a Solid Rocket Combustion Chamber 63 http://dx.doi.org/10.5772/intechopen.82822

$$e^{-\beta \tau} \approx \sum\_{n=0}^{\infty} \frac{\left(-\beta \tau\right)^{n}}{n!} \; ; \; e^{-\beta \tau} \approx \sum\_{n=0}^{\infty} \frac{\left(-\beta\right)^{n}}{n!} \tag{54}$$

The following hold, namely:

$$\left(1 - e^{-\beta \tau}\right) \approx \ 1 - \sum\_{n=0}^{\infty} \left(\frac{-\beta \tau}{n!}\right) \left(\left(1 - e^{-\beta}\right) \not\equiv \ 1 - \sum\_{n=0}^{\infty} \left(\frac{-\beta \beta}{n!}\right)\right) \tag{55}$$

The use of the above equations transforms Abð Þτ as

$$\overline{A}\_b(\tau) = \frac{\sum\_{N=0}^{\infty} \frac{(-1)^{N+1} \beta^N t^N}{N!}}{\sum\_{N=1}^{\infty} \frac{(-1)^{N+1} \beta^N}{N!}} \tag{56}$$

The following approximated profiles can be generated, viz.:

$$A\_b^{(1)}(\\\tau) = \frac{\rho \cdot \tau}{\sum\_{N=1}^{\infty} \frac{(-1)^{N+1} \beta^N}{N!}} \qquad \qquad \qquad A\_b^{(2)}(\\\tau) = \frac{\tau \cdot \tau \cdot \frac{N!}{\sum\_{N=1}^{\infty} \frac{(-1)^{N+1}}{N!}}}{\sum\_{N=1}^{\infty} \frac{(-1)^{N+1}}{N!}}$$

$$\begin{array}{ll} \text{(a) First-degree profile (linear)} & \text{(b) Second-degree profile (quadratic)}\\ \overline{A}\_{b}^{(1)}(\tau) = \frac{\beta \tau}{\sum\_{N=1}^{\infty} \frac{(-1)^{N+1} \beta^{N}}{N!}} & \overline{A}\_{b}^{(2)}(\tau) = \frac{\sum\_{N=1}^{2} \frac{(-1)^{N+1} \beta^{N} \tau^{N}}{N!}}{\sum\_{N=1}^{\infty} \frac{(-1)^{N+1} \beta^{N}}{N!}} \end{array}$$

� � ð Þc Third-degree profile ðcubicÞ ð Þ d Fourth-degree profile biquadratic Nþ1 β<sup>N</sup>τ<sup>N</sup> <sup>N</sup>þ<sup>1</sup> <sup>β</sup><sup>N</sup><sup>τ</sup> <sup>P</sup> N <sup>3</sup> ð�1<sup>Þ</sup> <sup>P</sup><sup>4</sup> ð�1<sup>Þ</sup> ð Þ<sup>3</sup> <sup>N</sup>¼<sup>1</sup> ð Þ<sup>4</sup> <sup>N</sup>¼<sup>1</sup> (57) <sup>N</sup>! <sup>N</sup>! <sup>A</sup> ð Þ¼ <sup>τ</sup> <sup>A</sup> ð Þ¼ <sup>τ</sup> <sup>b</sup> <sup>N</sup>þ<sup>1</sup> β<sup>N</sup> <sup>b</sup> <sup>N</sup>þ<sup>1</sup> <sup>β</sup> <sup>P</sup> <sup>N</sup> <sup>∞</sup> ð�1<sup>Þ</sup> <sup>P</sup><sup>∞</sup> ð�1<sup>Þ</sup> <sup>N</sup>¼<sup>1</sup> <sup>N</sup>¼<sup>1</sup> N! N!

$$\begin{array}{c} \text{(e) Mth order degree profile} \\ \hline \mathbf{A\_b^{(M)}} (\mathbf{r}) = \frac{\sum\_{N=1}^{M} \frac{(-1)^{N+1} \beta^N \mathbf{r}^N}{\mathbf{N!}!}}{\sum\_{N=1}^{\infty} \frac{(-1)^{N+1} \beta^N}{\mathbf{N!}!}} \end{array}$$

#### 5. Effect of multiple ignition points

The effects of multiple ignition points are accounted for in the following equations, viz. (unburnt area propagation):

$$
\overline{A}\_{ub}^{\{N\_p\}}(\tau) = e^{\beta \binom{N\_p}{} (1-\tau)} ; \overline{A}\_b^{\{N\_p\}}(\tau) = \frac{1 - e^{-\beta \binom{N\_p}{} \xi}}{1 - e^{-\beta \binom{N\_p}{}}} \bigg) \tag{58}
$$

� � � � � � � � � � �� where <sup>β</sup> Np <sup>¼</sup> ln C1 Np <sup>þ</sup> C2 Np <sup>þ</sup> C3 Np For the various associated polynomials, the matching kernel α Np applies.

## 6. Discussion of results

Having shown the details of mathematical analyses, which leads to the derivation of closed form equations for both burning and unburnt propellant grain areas, subject to treating a flame particle arising from one to multiple ignition points theoretically using Newtonian mechanics, we shall now shift focus to discussing parametric modulations of these closed form equations. The interest here is to match theoretical simulated burn time results to those of conducted static burning tests of the propellant as demonstrated in Figures 7 and 8. Firstly, a cache of experimental set-ups for measuring the burn time are illustrated in Figures 7 and 8.

The static test rig holder as shown in Figures 7 and 8 has an in-built sensor system which captures the burning propagation signature in the form of a digitized time signal that is fed into a transducer for a real-time display. A redundant system that has a stop-watch is also utilized for comparative purposes. After a number of static test experiments as demonstrated in Figures 7 and 8, the experimental results and theoretical comparisons are contained in Tables 2 and 3. The parameters of the solid propellant and combustion chamber are highlighted in Tables 1 and 4.

Meanwhile, the associated generalized chemical combustion equation for two classical composite formulations as illustrated below, namely,

Dextrose-based composite combustion equation:

$$\begin{aligned} &46.09 \text{KNO}\_{\text{\textdegree}(\text{s})} + 1.438 \text{C}\_6 \text{H}\_{12} \text{O}\_{6(\text{s})} + 4.11 \text{M}\_{\text{\textdegree}(\text{s})} + 1.07 \text{C}\_{(\text{s})} + 0.094 \text{Fe}\_2\text{O}\_{\text{\textdegree}(\text{s})} = 5.17 \text{CO}\_{(\text{\textdegree})} \\ &+ 4.94 \text{KOH}\_{(\text{\textdegree})} + 4.74 \text{H}\_2\text{O}\_{(\text{\textdegree})} + 4.11 \text{CO}\_{2(\text{g})} + 4.11 \text{MgO}\_{(\text{s})} + 3.12 \text{N}\_{2(\text{g})} + 1.42 \text{H}\_{2(\text{g})} + 1.15 \text{K}\_{(\text{\textdegree})} \end{aligned}$$

Figure 7. Static test rigs with single motor.

Analytical Prediction for Grain Burn Time and Burning Area Kinematics in a Solid Rocket Combustion Chamber 65 http://dx.doi.org/10.5772/intechopen.82822

Figure 8. Static test rigs with tri-cluster rocket motors.


Table 1. Experimental parameters.

Sorbitol-based composite combustion equation:

$$\begin{aligned} &5.53\text{KNO}\_{\text{3(s)}} + 1.42\text{C}\_6\text{H}\_{14}\text{O}\_{6(s)} + 4.11\text{M}\_{\text{g(s)}} + 0.378\text{C}\_{(s)} + 0.377\text{Fe}\_2\text{O}\_{3(s)} = 5.59\text{CO}\_{(\text{g})} + 5.44\text{H}\_2\text{O}\_{(\text{g})} \\ &+ 4.48\text{KOH}\_{(\text{g})} + 4.11\text{MgO}\_{(\text{s})} + 3.32\text{CO}\_{2(\text{g})} + 3.12\text{N}\_{2(\text{g})} + 2.28\text{H}\_{2(\text{g})} + 1.05\text{K}\_{(\text{g})} \end{aligned}$$

Figure 9 depicts the behavioral pattern of the burn time as a function of the burn rate in conjunction with the modulating role of number of ignition points. It is noted from the closed form expression Eq. (16) that an inverse or semi-hyperbolic relationship holds for each of the curves asymptotically. From design consideration, ab initio prediction can be conjured for appropriate ballistic suitability (Tables 2–4). Secondly, reduction in burn time is noted with higher ignition points for any burn rate, by having a hold on other variables as couched in Eq. (16). Very significantly, the role of the ignition points is essential for controlling the amount of transient buildup of the combustion chamber pressure in such a manner that is helpful to fasten the occurrences of explosion if hollow cylindrical explosives are desired for military purposes. This transient pressure can be built up very rapidly and reach high levels for a very

Figure 9. Plot of burn time against regression rate.


tb, total burn time calculated; ta, experimental burn time result; ts, burnout time; θ, total sectorial angle covered by flame.

Table 2. Experimental results.

short burn time. It is very important to state here that such pressure value preferences must take into cognizance of the ultimate tensile strength of the combustion chamber material to forestall thermal rupturing of the walls.

In the meantime, Figure 10 indicates the characteristic profiles of the burn time as a function of web thickness to core diameter ratio. As seen clearly, two zones are exhibited with a jump tendency in each case. Notably too, the effect of multiple ignition points is copiously observed to be very central here. As an option, a preferred burn time to govern the fuel ballistic characteristics can be selected to match desired ignition points by holding other parametric values of the grain geometry. Meanwhile, in the first zone, the burn time is noted to be fairly constant before transiting through an impulsive spark to a local peak. Beyond these points, Analytical Prediction for Grain Burn Time and Burning Area Kinematics in a Solid Rocket Combustion Chamber 67 http://dx.doi.org/10.5772/intechopen.82822


Table 3. Experimental results (burn time).


Table 4. Table of simulation parameters.

Figure 10. Plot of normalized burn time against χ:

Figure 11. Plot of burn time against pressure index.

Figure 12. Plot of burn time against pressure.

slight droppings are noted till sharp turning points are initiated to prompt monotonic increasing linear profiles.

In the meantime, we illustrate in Figure 11 the characteristics of the burn time versus the propellant characterization index. Expectedly, all the curves originate from a common point irrespective of the value or quantum of the combustion chamber pressure. It is a direct consequence of Saint Robert Veille's law adopted for this study. In general, inverse relationships for any kernels must hold as can be inferred from the nature of the closed form equation tying the burn time with other parametric values deducible from empirical relations as published in literature.

Figure 12 illustrates the plot of burn time against combustion chamber pressure for any index (n). Here, we note two zones in each case where increasing pressure has decreasing effects on the burn time consistently up to a common crossover point before exhibiting reverse ordering to prompt fairly constant horizontal curves for any index value. Parts of these characteristics are previously noted in Figure 11. We expect these profiles to be generic irrespective of the propellant formulation and chemistry for this class of solid geometry. In grain design exercise, a number of choices are handy starting from index selection to consideration of stress tolerance of the chamber wall and aerothermodynamic properties arising from a fuel compositional chemistry and reaction kinetics.

Having examined the characteristic profiles of the burn time as modulated by specified parameters in the previous figures, we next shift focus in observing the commutative effects it is having on the burning and unburning propellant grain areas. Firstly, the behavioral pattern of

Figure 13. Plot of burnt area against normalized time.

Figure 14. Plot of normalized unburnt area against normalized time.

Figure 15. Plot of burnt area against χ:

Analytical Prediction for Grain Burn Time and Burning Area Kinematics in a Solid Rocket Combustion Chamber 71 http://dx.doi.org/10.5772/intechopen.82822

Figure 16. Plot of normalized burnt area against normalized time.

Figure 17. Comparison of a third-order polynomial approximation with kinematic prediction of burn area against time.

the burning area against the normalized burn time is illustrated in Figure 13. As can be seen, ordering is in consonance with increasing number of ignition points. Note clearly too that profiles are curvilinear for all the ignition points. We now proceed further to the case of the ordinary differential equation model in the illustration. Next, as seen, ordering is in reverse consonance with increasing number of ignition points. Note clearly that the pattern of profiles is curvilinear for all the ignition points. These distinct appearances are apparently visible from the onset, while overlapping tendencies are exhibited from the midpoint of the normalized time. This is possibly sequel to the segmental placement of the ignition points, to reduce the unburnt areas proportionally as indicated in the vertical axis of the plotted figure.

At this point, we shift focus to studying the ordinary differential equation model for predicting the behavioral pattern for a temporal dependent closed form unburnt grain depreciation conjectural result. The mathematical functional relation is a geometrically decaying exponential kernel that is tied to the parametric variables linking a number of factors ranging from the aspect ratio to the web thickness and the number of points at which the propellant bate is ignited. Figure 14 depicts the behavioral pattern of the unburnt area as a function of normalized time in conjunction with the modulating role of ignition points. As can be seen, all the curves exhibit a decaying exponential characteristic. This is expected, as Eq. (41) is an exponential function. Next, as seen, ordering occurs in the order of increasing ignition points.

Figure 15 is a plot of Eq. (58) to demonstrate the modulating roles of web thickness to core diameter ratio (χÞ on the burning area propagation for one to multiple (six) ignition points in

Figure 18. Plot of polynomial approximations of normalized unburnt area.

Analytical Prediction for Grain Burn Time and Burning Area Kinematics in a Solid Rocket Combustion Chamber 73 http://dx.doi.org/10.5772/intechopen.82822

Figure 19. Plot of polynomial approximations of normalized burnt area: Eq. (54).

the range of χ hyperbolic profiles as noted for Np = 1 and Np = 2. A turning point is initiated at the maximum point and thereafter decreases asymptotically.

The profile of non-dimensionalized burnt area with respect to normalized time is depicted in Figure 16. Firstly, for the different numbers of ignition points simulated, the normalized burnt area has a characteristic increasing curvilinear signature. Secondly, the curves are ordered with increasing ignition points. There is also an overlap of the curves at the beginning and end points.

The kinematic prediction of burn area against time is illustrated in Figure 17. As can be seen, the profiles are curvilinear for all the points of ignition. With increasing number of ignition points, the curvilinear signature tends toward a linear profile. Next, as seen, ordering is in reverse consonance with increasing number of ignition points. Note clearly that the profiles are curvilinear for all the ignition points. These distinct appearances are apparently visible from the onset, while overlapping tendencies are exhibited from the midpoint of the normalized time. The third-order burnt area propagation profile also exhibits a curvilinear profile that is similar to the kinematic prediction.

The polynomial approximations of unburnt area propagation of the propellant are depicted in Figure 18. For all orders of the polynomial approximations, profiles are fairly linear with monotonic decreases. The exact solution on the other hand is a decreasing curvilinear profile with a steeper slope than the approximations. All the polynomial approximations have approximately equal values of unburnt area at onset of the time period and decrease to the same value at the end of the burning. The value at the end of the period can be regarded as the unburnt propellant residue.

The profiles of first- to fourth-degree approximate burnt area propagation and the exact polynomial solution are shown in Figure 19. The exact solution manifests an initial increasing curvilinear feature for the first half of the burning period. During the second half of burning, the burnt area is constant until the end of the period. The second- and fourthdegree approximations have similar profiles. They rise to a maximum and fall curvilinearly to zero. The first-degree polynomial increases linearly throughout the period, while the third-degree approximation rises curvilinearly to a value above the exact solution.

## 7. Conclusion

This chapter proposes the derivation of equations to predict burn time, burning and unburnt area propagation of a tubular propellant grain. A regressive solid fuel pyrolysis in a cylindrical combustion chamber is assumed to hold. The behavioral patterns of simulated results reveal the modulating impact of variables on the burning propagation due to the kinematic and mathematical models. Closed form expressions are couched in terms of the propellant grain geometrical constraints. In addition, for the burn time, a close conformity between theoretical models and experimental results is shown.

Our findings include:


The above find application in the use of variable number of ignition points for controlling the amount of transient buildup of the combustion chamber pressure. This helps to fasten the occurrences of explosion if hollow cylindrical explosives are desired for military purposes. Also, preferred burn time to govern the fuel ballistic characteristics can be selected to match desired ignition points by holding other parametric values of the grain geometry constant. In grain design exercise, different parameters can be altered, namely, pressure index selection to consideration of stress tolerance of the chamber wall and aerothermodynamic properties arising from a fuel compositional chemistry and reaction kinetics.

## Nomenclature


## Author details

Charles A. Osheku\*†, Oluleke O. Babayomi and Oluwaseyi T. Olawole

\*Address all correspondence to: charlesosheku2002@yahoo.com

Centre for Space Transport and Propulsion, National Space Research and Development Agency, Federal Ministry of Science and Technology, Lagos, Nigeria

† On Leave of Absence from the Department of Systems Engineering, Faculty of Engineering, University of Lagos.

## References


## **Ballistic Testing of Armor Panels Based on Aramid**

Catalin Pirvu and Lorena Deleanu

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.78315

#### Abstract

Industry and market of ballistic protection materials and systems are characterized by a dynamic and competing succession of inventions for projectiles and protective systems. The requirements for the ballistic panels are many and complex, varying depending on the threat type, the required mobility in the tactical theater, and protection level. The safety degree, the price, and the dynamics of research in the field are also taken into account. This chapter underlines the necessity of testing ballistic protection panels made of LFT SB1 plus (multidirectional fiber fabrics, supplied by Teijin) against a certain threat in order to assess their resistance to this specific threat and the investigation of failure mechanisms in order to improve their behavior at ballistic impact. The models for ballistic impact are useful when they are particularly formulated for resembling the actual system projectile, target, and can be validated through laboratory experiments. Tests made on panels made of LFT SB1plus, according to NIJ Standard-0101.06-2008 gave good results for the panels made of 12 layers of this fabric, and the backface signature (BFS) was measured. The BFS upper tolerance limit of 24,441 mm recommends this system for protection level IIA, according to the abovementioned standard.

Keywords: ballistic impact, aramid fabrics, damage investigation

### 1. Introduction

Industry and market of ballistic protection materials and systems are characterized by a dynamic and competing succession of inventions for projectiles and protective systems. The requirements for the ballistic panels are many and complex, varying depending on the threat type, the required mobility in a tactical theater, and protection level. The safety degree, the price, and the dynamics of research in the field are also taken into account. The research on hit targets by a projectile characterized by more than 120 m/s is of high interest nowadays in several important domains, like army, navy, space systems, and nuclear one. The intensive

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competition on polymeric, woven or not, fabrics and the growth of their market at a global level are factors that bust research toward efficient innovations, including the assembling technologies here.

The initial design of a protective system is supported by simulations using sophisticated codes that take into account the material characteristics under a high strain rate and particular processes characterizing the impact (friction, heating, material modifications concerning phase and structure, stratifications, and/or the fibers arrangements, etc.). Simulations by the help of finite element method (FEM) or smoothed-particle hydrodynamics method (SPH) [1] of the impact have become an initial stage in designing new systems, but the experimental validation of models is asked by the particular use of the protective system. The tests on ballistic systems or panels are strictly necessary for evaluating the impact level and for identifying factors influencing penetration and failure mechanisms that could help improve the already existing systems.

Starting from these considerations, the main goal of the study "Ballistic testing of armor panels based on aramid fibers" is the concern of balancing innovations of destructive systems (projectiles, fragments, bullets, etc.) with those that are designated to protect personnel and equipment, using materials like fabrics, woven or multidirectional, stratified and complex composites. At a low speed, even glass fiber fabrics could have satisfactory results [1], but for protecting personnel and equipment at a higher impact velocity, aramid and ultrahigh density polyethylene fibers are more efficient (Figure 1) [1, 2].

Even if simulation and modeling offer results closer to the actual processes by using new principles and performance codes and computers in solving impact issues, the experimental work is the final and main stage for the approval of new and improved protective systems. In terminal ballistics, experimental complex techniques, specific equipment, and a testing methodology are required to determine the performance of both the projectile and the protection system and, ultimately, by analyzing the results, to characterize and improve the

Figure 1. An increase in ballistic performance as a function of the fiber type used in manufacturing body armor [1].

protection system. If, in a classic war, the splinters and fragments are the most dangerous, in other conflicts, bullets remain a major threat to lives and physical integrity of fighters and civilians, representing the main cause of human loss, including during peacetime.

Bullets differ in caliber, initial velocity, mass, jacket, core and shape, and so on. A bullet must have a considerable kinetic energy when reaching the target to penetrate it. In terms of terminal ballistics, mechanical work involves many aspects of bullet-target interaction. Part of the energy is transformed into heat, a part is lost by friction, and the other part by rotating the bullet, by elastic and plastic deformation of both bullet and target. It is a practical impossibility to produce individual ballistic protection equipment that provides total security for the whole arsenal, given the drastic constraint imposed by the limitation of the total mass of protective equipment that a combatant or a user can wear.

In order to assess the impact resistance of protective panels, there are reference standards that offer test methods and procedure, as found in [3]. The test results give the possibility of including ballistic panels in a level of protection. For individual armor front panels, the standards require the absence of perforation for a determined number of repetitions under the same firing conditions, plus a condition related to the depth of the trace generated in a support material (ballistic clays) after impact [4].

The purpose of this chapter includes the process of manufacturing specific flexible ballistic panels made of quatro-axial fiber fabrics in layered composites, tested at fire with 9-mm bullet and 400 m/s (II and IIA protection levels, according to [4]), followed by an investigation of processes and stages of deterioration using scanning electron microscopy (SEM) and macrophotography of the failed zones of the panels after the bullet recovery. Also, a statistical analysis of the depth in the support material [backface signature (BFS)] is presented.

## 2. Manufacturing and testing the flexible panels

Personnel armor for ballistic protection includes both body systems and helmets. The threats for which this armor is designed are small-caliber projectiles, including bullets and fragments. The level of ballistic protection is taken as the total kinetic energy of a single round that the armor can stop [5].

For polymeric, carbon, ceramic, and glass fibers, researchers reported that the tensile strength increases with decreasing their diameter. For polymeric fibers (Table 1), diameters are in the range of 10–15 μm, greater than those for carbon fibers (4–10 μm), but smaller than fibers obtained by chemical vapor deposition, such as boron fibers (100–150 μm). The probability of defects decreases with decreasing the fiber diameter.

Fabrics from fibers and yarns by weaving allow for designing panels that can face both ballistic and blast events. Most ballistic fabrics have two-dimensional plain weave yarns in two orthogonal directions, although some work is being done on three-dimensional weaves and on nonwoven and knitted fabrics [6, 7]. Polymers, glass, and ceramic fibers have high stiffness and high strength-to-weight ratios. From less a decade, the unidirectional and multidirectional


Table 1. Typical properties of fibers [2].

fabrics made of polymeric and glass fibers tend to replace the woven ones for certain applications, including panels for the ballistic protection of personnel and equipment.

Figure 2 and Table 2 present characteristics of threats, as classified by NIJ Standard-0101.06.2008, Ballistic Resistance of Body Armor, and it is obvious that the kinetic energy is the key parameter that will destroy a body armor, for each level of protection.

Many parameters influence the response of fabrics to ballistic impact. These include material properties of and yarn, fabric architecture, boundary conditions, inter-yarn friction, friction between the projectile and the yarn, projectile geometry and velocity, impact direction, and environmental conditions.

The designers of body armor take into account the specific threat that has to face the vest and the helmet. Depending on these threats, the protective body system could be made of polymeric, metallic, and/or ceramic materials, and engineers have to select them. Vests made of

Figure 2. Characteristics of threats as given by NIJ Standard-0101.06. 2008.


Table 2. Levels of ballistic resistance [5].

only polymeric materials are intended for protecting human body against fragments and lower velocity projectiles. Glass fibers and polyamide fibers were firstly used, but polyaramid fibers, introduced by DuPont [8] and later by Teijin [9], make the armor lighter and more reliable for a greater ballistic limit V50. In the 1980s, fibers made of high-molecular-weight polyethylene (UHDWPE) and polybenzobisoxazole (PBO) have also been used.

Body armors have to fulfill two types of requirements:


There is no universal method to design an armor system, but the report "Opportunities in Protection Materials Science and Technology for Future Army Applications" [10] gives a flow chart of activities for homologating a new or a redesigned armor, pointing out the place and importance of simulation based on the actual material properties when bearing a high strain rate as in ballistic impact (Figure 3). Certainly, the main stage in evaluating the armor is the shoot test.

New materials are developed, but these are infrequently selected for protective systems because their behavior in actual events and configurations is not directly related to the laboratory tests. Moreover, most non-armor application materials are chosen according to their bulk quasi-static properties, even though such properties do not always predict their ballistic performance. If the constitutive relations for properties characterizing new materials needed for running simulations are not known, then the engineer has to use information from the most similar existing material [11], making the result uncertain. This is another reason why armor designers do not consider using new-entry materials that have not yet been sufficiently characterized under particular dynamic conditions. The simulation is often done as a guide to identify trends due to design modifications than as a source of practical results. The modeling of a protective system or a panel is difficult to do at different levels [12, 13]. For instance, the panel made of polymeric fibers of an armor could be considered as a stratified material at macro-level (Figure 4), but each layer is a fabric, woven or unidirectional, that contains yarns, their cross-dimensions being thousand times smaller than their length. This could be a

Figure 3. Flow chart of new or redesigned armor [10].

mezzo-level in modeling [14]. And there is the micro-level: each yarn contains hundreds of fibers, their diameter also being smaller by several orders of magnitude. The simulation could not cover all these levels at the same time, in one model, and the designers have to rely on their experience and ingenuity to generate a model that could help restraining the feasible solutions.

The panels were made of layers of Twaron LFT SB1plus, as supplied by Teijin Aramid [9], a new entry on the market in 2012. The four sublayers of LFT SB1plus, laminated together with a very thin sheet of resin, are visible as shown in Figure 5: the angle positioning of the

Figure 4. A simplified isothermal macro-model of a flexible panel (v = 400 m/s, friction contact between bodies: Friction coefficient between layers 0,4, friction coefficient between layers and bullet: 0.3 [14], bilinear hardening constitutive models for materials) (see Table 3). (a) 8 layers (t = 10˜<sup>4</sup> s), 7 broken layers. (b) 12 layers (t = 10˜<sup>4</sup> s), 4 broken layers.

2 unidirectional yarns of sublayers being 0, 90, 45, and ˜45° and the specific area density 430 g/ m . The layers in a panel were secured by sewing on two edges to maintain the integrity and order of the layers. The sewing line had a length of approx. 200 mm, made at 25 mm from the panel edge, with a step of 2–2.5 mm.

The manufacturing of the panels consists of cutting squared layers of <sup>500</sup> ˛ <sup>500</sup> mm, having an <sup>2</sup> area of 0.25 <sup>m</sup> , from fabrics with the width of <sup>1200</sup> mm. The area value positioned these panels between NIJ-C-4 (0.23 m2 ) and NIJ-C-5 (0.3 m2 ) for large and very large surfaces. After cutting the squared layers, these were arranged in three types of panels, each one containing a different number of layers: 4, 8, and 12, respectively (Table 4). The number of layers was selected after a simulation [14] at macro-level (Figure 4), with layer properties similar to aramid fiber, as given in the study [15].

Taking into account the standard Ballistic Resistance of Body Armor, NIJ Standard-0101.06, U.S., 2008, the test plan for flexible ballistic protection panels based on aramid fibers included the fire with 9-mm bullet for level II and level IIA (Figure 6).

Figure 5. Cross section on a LFT SB1plus fabric (4 sublayers (0, 90, 45, ˜45), each one with unidirectional yarns).


Table 3. Material properties for the impact model bullet—Stratified pack [11].


Table 4. Calculated mass of a flexible panel.

In order to evaluate the impact resistance of the protective panels, there are reference standards proposing testing methods, the results being ranked as a protection level, characterized especially by the projectile mass and velocity. For panels used as body armor, these standards require the absence of total perforation and a supplementary condition of limiting the back deformation as human body could not face a high deformation without critical injuries, even fatal, even if the bullet does not penetrate this one. As testing directly on the human body is not recommended neither ethical, panels are required to have a maximum value in a support material that could be similar (in a closer manner) to the human body response, as, for instance, the ballistic clay.

The impact velocity (just before hitting the target) was measured with the help of a system including a chronograph Oehler model 43, stable for the temperature range of 5–40˜C and having an accuracy of 0.3%. Other measurement devices used for these tests were a ballistic barrel for bullets of 9-mm FMJ (fulfilling the requirements of NIJ 0101.04/2000) and a fire arm table with a blow-back compensation [4], a hygrometer with an accuracy of 1%, a barometer with an accuracy of 1 mm Hg, a thermometer with an accuracy of 1˜C, a box for the ballistic clay, an oven allowing for tempering at 20 ° 5˜C, a climate enclosure allowing for maintaining a temperature of 20 ° 5˜C, and an oven for ammunition conditioning. A partial view of the test facilities is presented in Figure 7.

The shooting was done in the Scientific Research Center for CBRN Defense and Ecology laboratory, by specialized personnel in order to fulfill the requirements of reference [4], including the arrangement shown in Figure 8, the distance between fires and the distance from

Figure 6. Tested panel made of 12 layers of LFT SB1plus.

Figure 7. Fire laboratory at the scientific research center for CBRN defense and ecology, Bucharest, Romania.

the panel edges and the regulations of protection, specific to this type of laboratory, being observed [16].

The framed box has the dimensions 610 ˜ 610 ˜ 140 mm (°2 mm). The back of the box is detachable, and it was made of wood (19.1 mm). The frame is made of steel and helps the ballistic clay to be leveled. As recommended by [4, 5], the clay grade was Roma Plastilina no. 1, which has a durability of about 1 year. This clay must have no voids, a smooth-free surface, and it has be easy to level with a ruler, the free surface being determined by the metal frame.

The panel behavior was evaluated by the number of failed (broken) layers and by the values of backface signature (BFS). Figure 9 presents the method of measuring the depth of the impact deformation within the support material.

The determination of ballistic resistance of protection materials and equipment at the action of infantry bullets is carried out according to NIJ Standard-0101.06 [4] (Figure 6 presents one of the tested panels). The samples were tested with a ballistic pipe (with a measured velocity of

Figure 8. Shooting arrangement [4].

Figure 9. Measuring the BFS according to ballistic resistance of body armor, NIJ Standard-0101.06, U.S., 2008 [4].

430 ˜ 10 m/s), with a projectile of 9-mm full metal jacket (FMJ) bullet. The deformation remained in the ballistic clay (backface signature or BFS) that was measured according to [4], with a depth caliper, with an accuracy of ˜0.1 mm. After each measurement, the calipers were cleaned to avoid any traces of clay on the measuring area. The evaluation of the total penetration of a package is in many cases simple, when a hole with a diameter at least equal to the size of the bullet is found and the entire bullet passes through it. When testing individual ballistic protection equipment, the trauma to the human body is evaluated by the depth of the print that is formed in the clay on which the sample is fixed.

Environmental parameters inside the fire enclosure were as follows: temperature: 20°C (˜5°C), relative humidity: 50–70%, atmospheric pressure: 760 mm Hg (˜15 mm Hg).

After the fire, the projectile or their fragments were removed from the clay. The clay was added anytime needed, after measuring the backface signature (BFS).

The fire procedure has the following steps:


## 3. Evaluation of ballistic resistance for the flexible panels made of aramid fibers

At molecular level, variables in polymers include chemical makeup, the length and degree of branching of molecular chains, the degree of alignment and entanglement, and the extent of cross-linking. The types and strengths of bonds in chains and among chains influence the polymeric fiber strength and strain and influence the failure mechanisms.

The factors affecting the ballistic performance may be grouped into three categories:


When fabrics are impacted by a projectile, the target size, its clamping conditions are important. A longer yarn can absorb more deformation energy than a shorter one before failure. Thus, a larger target area will lead to a higher energy dissipation. However, this is not true when the velocity of the projectile is very high as compared to the velocity of the shock wave in the fibers since only a small zone of the target can dissipate the kinetic energy of the projectile. The boundary conditions of the target also play an important role. Shockey et al. [17] observed that a two-edge gripped fabric absorbs more energy than a four-edge gripped fabric, and fabrics with free boundaries absorb the least energy. Chitrangad et al. [18] observed that when pretension is applied on aramid fabrics, their ballistic performance is improved. Zeng et al. [19] observed that for four-edge gripped fabrics, energy absorbed is improved if the yarns are oriented at 45˜ relative to the edge.

The number of fabric plies or sublayers also affects the ballistic performance (note that typically there may be 20–50 plies). Shockey et al. [17] observed an increased specific energy absorbed for multi-ply targets due to friction forces between layers. The influence of interplay materials and the distance on ballistic performance have also been investigated [7]. The influence of a projectile geometry also becomes less important with an increased number of plies [20].

Frictional effects between a projectile and a fabric are observed at a low-velocity impact, but they diminish at a higher velocity [17]. Friction does help maintain the integrity of fabrics in the impact region by allowing more yarns to be involved in the impact and it increases energy absorption by increasing yarn strain and kinetic energy. Dischler [21] applied a thin polymeric film on Kevlar (20-ply), which increased the coefficient of friction from 0.19 to 0.27 and reported a 19% improvement in ballistic performance in stopping a flechette. Carrillo et al. [22] investigated the ballistic behavior of a multilayer Kevlar aramid fabric/polypropylene (atactic PP films of 0.032 mm and a density of 910 kg/m3 ) composite laminate and simply plain-layered aramid panel (plain-woven Hexcel aramid 720 fabric, Kevlar129 fiber, 1420 denier), under a sphere impact (with a diameter of 6.7 mm and a mass m of 1.11 g), at a velocity of 274.5 m/s and found that the improved performance of composite laminate is due to the fact that the thermoplastic matrix enables energy-absorbing mechanisms, such as fabric/matrix debonding and delamination.

There is a tendency to combine high-resistance fabrics with lower cost ones, but the results are still indecisive. Yahaya et al. [28] presented ballistic properties of non-woven kenaf fibers/ Kevlar epoxy-hybrid laminates with thicknesses ranging from 3.1 to 10.8 mm, when impacting with a 9-mm full metal jacket bullet at speeds varying from 172 to 339 m/s, at normal incidence, but hybrid composites recorded a lower ballistic limit (V50) and energy absorption than the Kevlar/epoxy composite.

The processes evidenced by macrophotography and SEM images help for understanding the failure mechanisms specific for the designed panels with layers of LFT SB1 plus a quatro-axial fabric.

Taking into account reference works [23], several stages for this type of panels were identified:

Stage I is dominated by deformation, yarn breakage, and energy dissipation mechanisms; the moment transfer between the projectile and the fabric leads to an increase in the kinetic energy of the fabric, which initially leads to the production of the pyramidal deformation, less evident on flexible panel with unidirectional fibers (see Figure 10 with photographs 1F-1 and 2F-1). Simultaneously, the yarns begin to stretch as the longitudinal wave propagates along the thread, leading to an increase in internal energy and/or wire deformation (statistical process). The sheet of resin, even very thin, keeps the yarns in positions, being more difficult to be laterally impelled.

Stage II is characterized by friction produced by pulling the yarns; one or more yarns can be pulled out of the fabric and a large amount of energy dissipates through this sliding friction; the rate of deceleration is lower at this step than in Stage I, but excessive yarn drawing promotes the fabric opening mechanism, the bullet pushing several yarns laterally. The fabric pattern or the way of arranging and maintaining the unidirectional yarn compaction (by sewing or a rare weaving with other types of fibers that maintain the surface density of yarns) influences the opening mechanism.

Figure 10. Posttest images of fabric damage from a panel made of a layer of LFT SB1plus, showing yarn and fibers' breakage characteristics. (a) 1F-1. (b) 2F-1. (c) Fiber damage on the front of 3-rd layers (3F-1). (d) Fiber damage on the back of the 12-th layer of a panel (12B-1).

Stage III corresponds to the postimpact region for impact without penetration, and the projectile can be arrested in the fabric. The bullet is strongly flattened, remaining with the typical aspect of mushroom (see Figure 15). Depending on material, projectile, and impact parameters, these steps may differ in duration and appearance.

## 4. Failure mechanisms of panels, yarns, and fibers by SEM and macrophotography investigations

The study of ballistic impact of fabrics includes residual velocity, stroke response, energy absorption, and tensile properties of yarns and failure mechanisms [17].

Mechanisms of energy dissipation are breaking the fibers/yarns, fibers' deformation (stretching, twisting) (see Figures 11a and 16c), fiber fibrillations (Figures 12 and 16c), bullet deformation and cracking, local heating, acoustic energy, air entrainment, cross-sectional deformation of yarn (Figure 16b) and friction among yarn fibers, yarns, and also friction between these ones and the bullet.

Types of damage in filaments, yarns, and fabrics may be noticed at both micro- and macrolevels. The micro-level involves breaking bonds that are involved in the structure of the

Figure 11. Fiber break in the third layer of the panel (front view, code 3F-1); less fibers with fibrillation.

Figure 12. Fibrillation of fibers (front view, code 2F-1).

filaments, while at the macro-level, the destruction may be characterized by mechanisms such as yarn pulling or bowing.

Higher magnifications show that the fibrils in broken fibers are also stretched (see Figure 10b). Fiber material very likely has plastic strain but, more obviously, localized plastic ones, failure also occurring due to nucleation of voids, cracks, and shear bands. Failure initiators are thought to originate in material defects such as tiny voids, foreign particles, and chain entanglements, resulting from chemical non-homogeneities or manufacturing procedures.

Fiber failure modes other than tensile failure are also observed. The influence of the structure of polymeric fibers at nano (molecular)-level on failure behavior is not well understood, especially at high strain rates and high pressures [3].

When a projectile hits a fabric or a panel made of layers of fabrics, it is caught by the yarn network (woven or not). Kinetic energy is transferred to the fabric as the stress wave spreads outward from the point of impact. The energy is partially dissipated by fiber deformation and breakage and by friction caused by inter-fiber slippage. A projectile with a sufficiently high mass and velocity may penetrate the fabric and cause it to fail.

Figure 10 indicates that tensile fracture first occurred at defects such as voids and kinks and was assisted by the residual stresses that are induced during processing. Similar mechanisms were reported by Allen et al. [24].

For example, a projectile impact on fabric compresses the fabric against the backing layers and causes transverse loads on the yarns and fibers that can result in deformation and failure (breakage). When compressed fibers are examined by SEM, they and the fibrils show flattening, kinking, and buckling (see Figures 11 and 16c).

When a projectile hits the individual fiber or a yarn [14, 16], longitudinal and transverse waves propagate from the impact point. Most of the kinetic energy transfers from the projectile to the principal yarns (those that come directly into contact with the projectile). The orthogonal yarns, which intersect the principal yarns, absorb less energy. The transient deformation within the fabric was simulated by Grujicic et al. [12]. The transverse deflection continuously increases until it reaches the breaking strain of the fibers and causes failure. Failure mechanisms characterizing the fabrics under ballistic impact include


In accordance with the kinetic theory of rupture, breaking the bond occurs when it is excited beyond its activating energy. When activation energy or stress for a particular type of destruction is reached, the failure mechanism is triggered.

Localized fracture of the yarn occurs when all fibers of the yarn break almost in the same location, usually at the sharpest point of a penetrator. This type of failure is accompanied by a popping sound and a sudden decrease in the measured load. The two causes of yarn breakage are the traction of yarns along their length and the shear in their thickness. The fiber in this yarn will break when the induced strain exceeds the strain at breakage that depends also on the strain rate but the strain at breakage generally decreases as the strain rate increases.

The breakage of yarns could occur at different points along their length and not necessarily at the point of impact. Also, if the penetrator is not too sharp, it compresses the superficial material between its front and the bulk material, outside its contact to the target, and the yarns could be pulled up and could break in traction.

In multilayered (stratified) systems, friction between layers is important in reducing damage [11]. All projectiles penetrating through a fabric, with semispherical, ogival, or conical shape, cause a splitting of the yarns [25]. Martinez et al. [26] have stipulated that pulling or rubbing is involved during the fabric-woven manufacturing and that its severity depends on the contact pressure between the layers.

The yarn pulling occurs when none of the yarn fibers break, but the yarn is pulled out of the fabric mesh. This type of failure can happen to lose or unfixed yarns (on the edge). The force required to pull the yarn from the net depends on the frictional force on the contact area between the yarn in question and the other perpendicular yarns with which it intersects (for woven fabrics) or on the friction among yarns and sublayers when the fabric is made of unidirectional yarns. As the yarn is pulled out, the number of yarns intersecting constantly decreases, resulting in a gradual decrease in the measured impact load.

Splitting of fibers along their length or fibrillation is a type of destruction favored by the abrasive action on the fiber length (but also uneven traction along the fiber and their local defects play an important role in fibrillation).

The bowing is dominant in the back layers of a multilayer panel, where the projectile attempts to penetrate through a tip-edge approach, after that, it is considerably slowed down by the back layers that did not fail. The passing process of the projectile through the layers usually produces a hole less than the diameter of the projectile in the first layers, a smaller number of yarns being broken as compared to the number of threads intersecting or contacting the projectile [27]. Typical aspects of the aramid fiber failure are given in Figures 12 and 13: micro-fibrillation, peeling, and shear, but also fiber twisting and thinning zones along the fiber. An obvious less strain rate could be noticed in Figure 13 as compared to that in Figure 12.

A study at the macro- and micro-level was done for pointing out the failure processes characterizing each layer. Figure 14 shows the front and back views of the panel made of 12 layers LFT SB1plus. On this type of package, a partial penetration was obtained, that is, the destruction of the first four layers of the panels. The photographs show the entire pack after testing with three shots. It is obvious from Figure 14 that the design of the unidirectional fabrics helps yarns to develop a better resistance against pulling out.

Figure 13. A fiber broken on the back of the last layer (the 12th layer).

The layers may be grouped as follows (Figure 14):


the last layer with pull-out yarns and disorder yarns, especially on the back of the last layer (layer 12).

An investigation of the arrested bullet offers details on how the yarns are broken. Figure 15 shows that the projectile attack makes the yarn to break laterally from the direct impact, mainly from tensile solicitation. On the top of the projectile, the fragment of the yarn remains. One may notice an orientation similar but not exactly as the orientation of yarns in the four sublayers (0, 90, 45, ˜45), meaning the bullet is forced to change the initial position due to yarn resistance and break unevenly. The jacket of the bullet is split like a flower petal and migrates toward the boundary of the lead core. When the core hits the target, it is strongly compressed and laterally expanded, some of the yarn fragments being embedded into the lead alloy.

The holes in layers 1 and 2 are similar, resulting that the process of yarn destruction is also similar, which argues that the perforation of the first two layers is made approximately with the same parameters (the velocity of the bullet through the first two layers is not significantly reduced and the shape of the bullet is not modified too much because it does not face yet the resistance of the other layers and it only cuts the yarns, as it is presented in a FE model in [11, 14]. It is worth mentioning that the tests were carried out under conditions of a small variations in the initial bullet velocity (410–430 m/s). The impact angle is normal on the target surface, with deviations of less than 5% at the mouth of the pipe.

Figure 14. Macrophotographic study of a panel made of 12 layers of LFT SB1plus.

Starting from layers 3 and 4, the widening of holes and the pulling-out process of the yarns are noticed. Layer 4 is the last layer in the LFT SB1plus panels through which bullets have passed or stopped (arrested).

Layer 5 shows more uniformly circular shapes of crushing/compression, imparting a tendency to uniformize the response of the material.

Figure 15. The bullet extracted from the sample panel presented in Figure 14. (a) Front view. (b) back view of the same bullet.

Figure 16. SEM images of the bullet with fragments of yarns on its front. (a) SEM image of the flattened bullet, as extracted between the fourth and the fifth layer. (b) Magnification of embedded fibers, very probably from the first yarn touching the bullet. (c) A—Fibrillation, B—Break by tensile loading with twisting of the fiber end, C—Necking of the fiber without break.

Figure 16a shows the fragmentation of fibers and the embedding of the yarn fragments remained under the bullet, with details shown in Figure 16b. Figure 16c points out the types of failures on the fibers remained on the projectile.

## 5. A statistical analysis of backface signature

The values of BFS for panels made of layers LFT SB1plus are given in Table 5 and they were measured according to Ballistic Resistance of Body Armor, NIJ Standard-0101.06, 2008 [4].

NIJ Standard-0101.06 [4] asks for having fires complying with requirements concerning the shot-to-edge distance and shot-to-shot distance (minimum of 51 mm). For armor types subjected to a single threat and for the lighter weight threat round when two threats are


Table 5. BFS for panels made of layers of LFT SB1plus.

specified, the minimum shot-to-edge distance shall not be greater than 51 mm. For the heavier threat round when two threats are specified, the minimum shot-to-edge distance shall not be greater than 76 mm.

Each test panel must withstand the appropriate number of fair hits and may not experience any perforations. Any complete perforation by a fair hit constitutes a failure. Each new size of a body armor model shall either have no BFS depth measurements that exceed 44 mm (Figure 17), or for each threat round, an estimated probability of a single BFS depth measurement exceeding 44 mm of less than 20% with a confidence of 95%.

The armor model shall be deemed to meet these requirements if no BFS depth measurement due to a fair hit exceeds 50 mm, and either


In this case, the upper tolerance limit, YU, and the sample standard deviation, s, of all recorded BFS measurements for body armor sample panels of a particular model, size, condition, and test threat shall be calculated, and

$$Y\_{ll} = \overline{Y} + K\_1 \mathbf{s} \tag{1}$$

where Y is the average of all BFS measurements for armor samples of that particular model, size, condition, and test threat; s is the sample standard deviation of the same set of BFS measurements; and K1 is a factor that must be determined such that the interval covers the appropriate proportion, with a confidence of γ.

Figure 17. BFS results, based on the number of shoots/panel, for panels made of 12 layers of LFT B1plus.

The average Y is simply calculated as

$$\dot{Y} = \frac{1}{N} \sum\_{i=1}^{N} Y\_i \tag{2}$$

where N is the number of BSF measurements and Yi is the BFS value for the i-th shot. The standard deviation of the sample population, s, is calculated with the relationship:

$$s = \sqrt{\frac{1}{N-1}} \sum\_{i=1}^{N} \left( Y\_i - \dot{Y} \right)^2 \tag{3}$$

The approximate factor, k1, for a one-sided tolerance interval can be calculated as

$$K\_1 = \frac{z\_{1-p} + \sqrt{z\_{1-p}^2 - ab}}{a} \tag{4}$$

where z<sup>1</sup>�<sup>p</sup> is the critical value of the normal distribution which is overpassed with a probability 1 � p. The factors a and b are defined as

$$a = 1 - \frac{z\_{1-\gamma}^2}{2(N-1)}\tag{5}$$

$$b = z\_{1-p}^2 - \frac{z\_{1-\nu}^2}{N} \tag{6}$$

where z<sup>1</sup>�<sup>p</sup> is the critical normal distribution which is overpassed by a probability 1 � γ.

In order to analyze the BFS measurements in accordance with [4], the probability for no BFS measurement to be higher than 44 mm has to be at least 80%, thus p = 0.80 and the confidence coefficient is 95%

$$
\gamma = 0.95\tag{7}
$$

The critical values for the normal distribution for this case study are

$$z\_{1-\gamma} = z\_{0.05} = 1.645$$

$$z\_{1-p} = z\_{0.20} = 0.842$$

Using these data, the factors a and b may be calculated for an imposed number of BFS measurements, N. For N = 15, the factors a and b are

$$a = 1 - \frac{1.645^2}{2(15 - 1)} = 0.903, \qquad b = 0.842^2 - \frac{1.645^2}{15} = 0.528.1$$

And the factor k1 is

$$k\_1 = \frac{0.842 + \sqrt{0.842^2 - 0.903 \cdot 0.528}}{0.903} \approx 1.466$$

The allowable excessive BFS probability, 20%, may appear to be high; however, this value is intended to account for both the variation in the armor's performance, which should be small, and the variation in the BFS measurement due to the backing material and the backing material preparation. While careful treatment and preparation of the backing material by the test laboratory can minimize the variation due to the backing material, there will always be some inherent variation introduced into the test results by the backing material. The required probability is chosen to reduce that possibility that an acceptable armor design will fail the PBFS test due to reasonable variation in the backing clay.

The average value of BFS Y˜ was calculated for panels made of 12 layers of LFT SB1plus, with N = 15 (the values of BFS are given in Table 5)

$$\acute{Y} = \frac{1}{N} \sum\_{i=l}^{N} Y\_i = 19.4 \text{ mm}$$

where N is the number of measured BFSs and Yi is the value of measurement i for BFS.

The standard deviation of the sample population, s, for panels made of 12 layers LFT SB1 plus is equal to

$$s = \sqrt{\frac{1}{N-1}} \sum\_{i=1}^{N} \left( Y\_i - \dot{Y} \right)^2 = 3.4391$$

The upper tolerance limit, Yu, for panels made of 12 layers LFT SB1plus is

Yu <sup>¼</sup> <sup>Y</sup>˜ <sup>þ</sup> <sup>k</sup>1<sup>s</sup> <sup>¼</sup> <sup>19</sup>:<sup>4</sup> <sup>þ</sup> <sup>1</sup>:466∙3:<sup>439</sup> <sup>¼</sup> <sup>24</sup>:<sup>441</sup> mm

## 6. Conclusions

This chapter underlines the necessity of testing ballistic protection packs made of LFT SB1 plus against a certain threat in order to assess their resistance to this specific threat and the investigation of failure mechanisms in order to improve their behavior at ballistic impact.

Ballistic testing of the LFT SB1 plus panels can provide reliable information about the new material.

Tests made on packs made of LFT SB1 plus according to NIJ Standard-0101.06-2008 gave good results for the packs made of 12 layers of this fabric and the BFS was measured.

The uppertolerance limit of 24,441 mm obtained for backface signature recommends this panel of 12 layers of LFT SB1 plus for protection level of IIA, according to the abovementioned standard.

## Author details

Catalin Pirvu<sup>1</sup> \* and Lorena Deleanu<sup>2</sup>

\*Address all correspondence to: pirvu.catalin@incas.ro

1 National Institute for Aerospace Research "Elie Carafoli"—INCAS, Bucharest, Romania

2 Department of Mechanical Engineering, "Dunarea de Jos" University of Galati, Galati, Romania,

## References


## *Edited by Charles Osheku*

Te edited volume *Ballistics* is a collection of reviewed and relevant research chapters, ofering a comprehensive overview of recent developments in the feld of engineered mechanics. Te book comprises single chapters authored by various researchers and edited by an expert from the respective research area. Each chapter is complete in itself but united under a common research study topic. Tis publication aims to provide a thorough overview of the latest research eforts by international authors on engineered mechanics and opens new possible research paths for further novel developments.

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