Journal of Applied Mathematics

Volume 2015 (2015), Article ID 627351, 11 pages

http://dx.doi.org/10.1155/2015/627351

## Numerical Study on Similarity of Plume’s Infrared Radiation from Reduced Scaling Solid Rocket

South China University of Technology, Wushan Road, No. 381, Guangzhou 510641, China

Received 7 January 2015; Revised 22 March 2015; Accepted 22 March 2015

Academic Editor: Yogesh Jaluria

Copyright © 2015 Xiaoying Zhang and Rui Li. 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

Similarity of plume radiation between reduced scaling solid rocket models and full scale ones in ground conditions has been taken for investigation. Flow and radiation of plume from solid rockets with scaling ratio from 0.1 to 1 have been computed. The radiative transfer equation (RTE) is solved by the finite volume method (FVM) in infrared band 2~6 *μ*m. The spectral characteristics of plume gases have been calculated with the weighted-sum-of-gray-gas (WSGG) model, and those of the Al_{2}O_{3} particles have been solved by the Mie scattering model. Our research shows that, with the decreasing scaling ratio of the rocket engine, the radiation intensity of the plume decreases with 1.5~2.5 power of the scaling ratio. The infrared radiation of the plume gases shows a strong spectral dependency, while that of the Al_{2}O_{3} particles shows grey property. Spectral radiation intensity of the high temperature core of the solid rocket plume increases greatly in the peak absorption spectrum of plume gases. Al_{2}O_{3} particle is the major radiation composition in the rocket plume, whose scattering coefficient is much larger than its absorption coefficient. There is good similarity between spectral variations of plumes from different scaling solid rockets. The directional plume radiation rises with the increasing azimuth angle.

#### 1. Introduction

As for base heating and thermopneumatic property of the solid rocket, plume radiation has been the focus of the investigation in thermal protection design of solid rockets in past decades [1]. Much related experimental and theoretical study has been carried out to study radiation characteristics of the solid rocket plume. As a limitation of ground test, most experiment study of rocket plume radiation used reduced scaling model, and also some of the theoretical studies used scaling model. Considering the application of research results with reduced scaling models in radiation knowledge and thermal protecting design of full scale rockets, similarity between plume radiation of reduced scaling model and full scale rocket must be obtained. However, to the authors’ best knowledge, there has not been experimental study about that problem published till now. Only several numerical analysis studies can be found out.

Rozanov and Lyapustin had studied a new integral form of similarity conditions to the error analysis of truncation techniques for the forward peak scattering and deduced the integral similarity method, that is, Delta-M [2]. Jun and Wenbing had studied the self-similarity of transmittance depth and radiation energy in a cool medium, with changing time, boundary temperature, and medium density [3]. Research of Duracz and Mccormick was focused on the ratio of two similarity parameters, radiation intensity and irradiance, and the influence of the single-scattering albedo and asymmetric coefficients within weakly absorbing clouds [4]. Mitrescu and Stephens had studied a new approach for determining the scaling parameter of the radiative transfer equation and proposed that the key assumption regarding the angular dependence of the radiative field is essential [5]. Bril et al. had studied the similarities between temperature field and concentration field in the near nozzle area; this study also proposed a method for determining a nondimensional radiance intensity with the outlet parameter, radiation wavelength, and temperature gradient of absorptivity [6].

Calculation of solid rocket plume radiation is highly complicated as the RTE is a high ordered nonlinear equation with differential term and integral terms, considering the strong spectral properties of plume gases and radiative absorption and scattering of groups of Al_{2}O_{3} particles with different temperatures and diameters. The plume gases consist of several polarity gases: H_{2}O, CO_{2}, CO, HCL, and OH, whose infrared radiative spectrum is made up with ten thousands of lines [7]. The common detailed calculating method for gas radiation uses a band model [8, 9] or the weighted-sum-of-gray-gas (WSGG) model [10]. The band model has a higher precision for nonhomogeneous gases but is somewhat time lasting for large volume gases and is not very suitable for solid rocket plume gases. The WSGG model always shows a compromise of precision and calculation time, which has been widely used to calculate radiation of clouds or combustion gases. As for spectral property of Al_{2}O_{3} particles, effects of particle temperature, concentrations, and diameters are needed to be considered [11, 12], and the Mie scattering theory for calculating particle radiation can also be used for Al_{2}O_{3} particles [13]. Many numerical solution methods have been developed to calculate radiation in nonhomogenous absorptive/emissive/scattering medium by solving RTE numerically, including Monte Carlo (MC) methods [14], the streaming method (SM) [15], discrete-ordinate method (DOM) [16], and finite volume methods (FVM) [17]. To improve the solution accuracy with less execution time, the FVM has often been used to calculate the spectral radiation of the high temperature two-phase plume with highly expanding volume.

To study the similarity of infrared radiation from plumes with different scaling ratio, the Trident D5 solid rocket and a series of reduced scaling models with similar geometry have been taken for investigation. CFD code has been used to compute the axial symmetric flowing field inside the rocket engine and in plume of the full scaled rocket and reduced scaling model, with same combustion chamber pressure and plume’s flowing Mach number. The spectral absorption coefficient of plume gases: H_{2}O, CO_{2}, CO, HCL, and OH, will be computed with WSGG model. Eight groups of Al_{2}O_{3} particles with different diameters between 2 and 16 *μ*m have been taken into consideration. The Mie scattering model has been used to compute the spectral absorption, scattering coefficient, and phase function of Al_{2}O_{3} particles. To calculate the spectral radiance of the plume, a finite volume method has been used to solve the RTE. Integrating spectral radiance on exterior surface of plume volume in one direction will give the plume radiation intensity. Ratio of radiation intensity between reduced scaling model and full scaled one has been computed to study the similarity rules of plume radiation.

#### 2. Calculation of Plume Flowing Field

The Trident D5 solid rocket used a composite propellant NEPE; there is 10% weight of Al in the composite propellant. Plume’s flowing data of the full scaled rocket and reduced scaling models are calculated with the same input data for the rocket nozzle, pressure is 9 Mpa, and temperature is 3750 K. Input concentrations of major species are = 0.192, = 0.299, = 0.027, = 0.146, and = 0.0008, on the inlet of nozzle. The full scaled rocket has a nozzle with 1.55 m length, 0.35 m diameter, and 9.7 ratio of area expansion. The reduced scaling model has the same geometry but with a minimized diameter and length. To obtain the flowing data of plume, the CEA program has been adopted to compute the chemical-equilibrium compositions of the propellants and the inlet parameters.

Then, CFD code is used to compute the united flowing in rocket nozzle and plume. The time-marching method and the advection upstream splitting method (AUSM) have been used for numerical discretization of the 2D axis symmetric N-S equations. The - turbulence model is adopted to simulate blending of the plume and atmosphere. A finite rate chemical reaction model with 12 components and 17 reactions has been taken for consideration, the detailed reaction equations and related coefficients can be seen in [18]. For the Al_{2}O_{3} particles, the Lagrangian particle trajectory model has been used to simulate exchange of energy and momentum between particles and plume gases. The distribution of the particle diameter is determined by Braithwaite’s proposed functions [19].

The plume’s flowing data of the full scaled Trident D5 rocket and the 9 reduced scaling models have been calculated in our study. Only two groups of result are given in Figure 1. One is for the full scaled rocket and the other is for the 0.5 scaling model. Considering the axial symmetric characteristics of the plume’s flowing field, only flowing field map on the upper - plane has been plotted. Figure 1 shows pressure and temperature of plume gases and volume fractions of the 3 major components: H_{2}O, CO_{2}, and CO. Volume fractions of HCL and OH molecule have not been shown here as their fractions are very small, less than 0.05. Temperature and number concentration of the 8 groups of Al_{2}O_{3} particles have been calculated in our study, only results of 3 groups with diameters being 6 *μ*m, 8 *μ*m, and 10 *μ*m are given here.