International Journal of Aerospace Engineering

Volume 2019, Article ID 2974537, 9 pages

https://doi.org/10.1155/2019/2974537

## An Approach to Analysing Erosive Characteristics of Two-Channel Combustion Chambers

Science and Technology on Combustion, Internal Flow and Thermo-Structure Laboratory, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

Correspondence should be addressed to Weihua Hui; moc.361@oacnauygnohz

Received 28 November 2018; Revised 12 February 2019; Accepted 3 March 2019; Published 17 April 2019

Academic Editor: Angelo Cervone

Copyright © 2019 Yanjie Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To acquire the erosive characteristics of two-channel combustion chambers, a quasi-one-dimensional internal ballistics model is proposed. Combining the model with two different erosive burning models, the modified L-R expression, and the Mukunda-Paul expression, the flow parameters and the internal ballistics performance of a test solid rocket motor are computed. The results show good agreement with experimental data. According to the results, the more severe the erosion is, the earlier, longer, and gentler the tail-off stage becomes. During the tail-off stage, the dramatic drop of pressure leads to a low normal burning rate and makes it easier for erosion burning to occur. For this reason, notable erosive burning might appear during tail-off if the case-to-throat area ratio is extremely low. The results also show that flows in the inner channel and the outer channel are similar but not identical. This leads to different erosive burning behaviours in the two channels. Also, the two erosive burning rate models involved in this paper are compared. It seems that the M-P expression provides better results than the modified L-R expression does, since it reveals the threshold phenomena. Besides, the M-P expression has great advantages for its universality for most propellants and different SRM geometries.

#### 1. Introduction

In a solid rocket motor (SRM) with a high load fraction, the high flow velocity often induces an augmentation of the burning rate, which is called “erosive burning.” Erosive burning has been the subject of many studies, yet a satisfactory interpretation has not been proposed because of the complexity of its nature. Nevertheless, some empirical models have been derived by studies involving experimentation and theoretical analysis.

Among these theories, the expression that gives a linear relationship between erosive burning rate and velocity might be the simplest one. Vandenkerckhove [1] proposed that the erosive burning rate is proportional to the difference between the crossover velocity and the “threshold velocity.” Based on the same erosive expression, Rout et al. [2] carried out one-dimensional (1D) numerical computation for rectangular and cylinder grains and discussed the influences of different factors on erosive burning. Many studies [3, 4] were analysed by Landsbaum [5], and the erosive burning rate was processed as a function of mass velocity (i.e., or ) beyond a threshold value.

Lenoir and Robillard [6] discussed the sources of heat to the burning surface and classified them into two categories, which are dependent on pressure and velocity, respectively. Correspondingly, the total burning rate of the propellant was written as the expression , where is the normal burning rate and is the erosive burning rate. Afterward, an implicit expression for the erosive burning rate was proposed, in which the distance to the head end of the grain is used as the characteristic length. Though the L-R expression is a rough theory which does not properly interpret the mechanism of propellant burning, it is widely accepted and used due to its good prediction.

Due to the L-R expression’s overestimation of erosive burning in large SRMs, some researchers have come up with a revised version of the L-R expression, where the hydraulic diameter is chosen to be the characteristic length instead of the length to the head of the grain [7–10]. Yet, to the authors’ knowledge, a rigorous derivation for the modification has not been given.

To validate the models referred to for a specific SRM, there are always constants to be confirmed. This makes it difficult to obtain a reliable internal ballistics prediction of an SRM before firing tests. Mukunda and Paul [11] proposed a dimensionless model in 1997. The ratio of total burning rate to normal burning rate was expressed as a function of the nondimensional flux. After vast comparisons with experimental data, parameters in the model were determined. This makes the model valid for most practical propellants. The model provides a reliable way to predict the internal ballistics performance for SRMs, with good accuracy even without a firing test.

In this paper, a two-channel combustion chamber with a noninhibitor grain is studied. A quasi-1D model is proposed, and, by combining this model with two different burning rate models (modified L-R expression and Mukunda-Paul expression), the authors calculate and discuss the internal ballistics performance of the combustion chamber and draw some interesting conclusions.

#### 2. Mathematical Models

##### 2.1. Quasi-1D Model for One-Channel Combustion Chambers

First, a steady, one-dimensional situation is considered for a combustion chamber with a single channel. As shown in Figure 1, a 1D control volume of the chamber has a length of , and the main flow goes from left to right. The transpiration flow has a velocity denoted as and a mass flow rate expressed as where is the density of the propellant, is the total burning rate of the grain, and is the area of the burning surface of this control volume. At the left section of the control volume, the mass flow rate, port area, pressure, velocity, and temperature of the flow are denoted as , , , , and , respectively. Due to the port area difference and mass flow from transpiration, these variables at the right section change into , , , , and .