VERIFICATION OF MODEL OF CALCULATION OF INTRA-CHAMBER PARAMETERS IN HYBRID SOLID-PROPELLANT ROCKET ENGINES

On the basis of obtained analytical estimate of characteristics of hybrid solid-propellant rocket engine verification of earlier developed physical and mathematical model of processes in a hybrid solid-propellant rocket engine for quasi-steady-state flow regime was performed. Comparative analysis of calculated and analytical data indicated satisfactory comparability of simulation results. In order to verify mathematical model for calculation of intra-chamber parameters in a hybrid solid-propellant rocket engine [1, 2] adequacy assessment of calculation methods was performed using analytical estimation of engine’s characteristics for quasi-steady-state flow regime, constant flow section area S and perimeter P of cylindrical grain channel with the radius r. Other sufficiently accurate solutions of this problem were not obtained. For abovementioned assumptions the change of oxidizer flow along the grain channel of a hybrid engine can be described by the following equation: Pα , ox f d uS a u dx (1) where ρ,u – oxidizer density and flow rate; a,ν – burning rate law constants; ρf – fuel density; αox=mox⁄mf – ratio of “consumed” oxidizer mass flow mox to the mass of fuel mf in the process of combustion. Solution of equation (1) regarding oxidizer flow at any channel point (section) is as follows 1/ 1 1 ν 0 2 φ , φ α 1 ox f x u u x a r (2) where 0 in ox u m S – mass flow rate of oxidizer injection ( in ox m – total oxidizer consumption). As it is implied from equation (2) oxidizer mass consumer for combustion mox can be determined from the following relation * Corresponding author: bvborisov@tpu.ru DOI: 10.1051/ 0113 (2016) , 2016 72 7201135 HMTTSC MATEC Web of Conferences matecconf/2016 5 © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).

In order to verify mathematical model for calculation of intra-chamber parameters in a hybrid solid-propellant rocket engine [1,2] adequacy assessment of calculation methods was performed using analytical estimation of engine's characteristics for quasi-steady-state flow regime, constant flow section area S and perimeter P of cylindrical grain channel with the radius r.Other sufficiently accurate solutions of this problem were not obtained.For abovementioned assumptions the change of oxidizer flow along the grain channel of a hybrid engine can be described by the following equation: where ϱ,u -oxidizer density and flow rate; a,ν -burning rate law constants; ϱf -fuel density; αox=mox⁄mf -ratio of "consumed" oxidizer mass flow mox to the mass of fuel mf in the process of combustion.Solution of equation ( 1) regarding oxidizer flow at any channel point (section) is as follows As it is implied from equation (2) oxidizer mass consumer for combustion mox can be determined from the following relation , .
where mass fractions of "unused" oxidizer cox and combustion products cf are determined by the following relations Σ Σ , 1 .
The temperature of mixture of combustion products and "unused" oxidizer can be determined from the law of conservation of energy: The first in the part of equation (3) describes energy input due to fuel combustion products with the temperature f T entering the flow.The last member of the right part of this equation represents energy losses due to heating of oxidizer gas from its initial temperature ox T to some average value T T .At a first approximation it can be taken as / 2.
Then from the balance of mass of gas supplied to combustion chamber in ox m and combustion products Σ m taking into account the fact that they are equal to flow rate G through the nozzle block with throat area * S in case of critical flow, pressure p in combustion chamber can be determined RT Comparison of numerical calculation results and analytical estimation data obtained using the abovementioned relationships was performed for various levels of oxidizer supply to combustion chamber of hybrid solid-propellant rocket engine with the following parameters.

Initial data for calculation Engine geometry parameters
Oxidizer parameters Channel radius r = 1 cm; Throat area * S = 0.79 cm 2 ; Fuel grain length L = 50 cm.
Gas temperature ox T = 300 K; Adiabatic index γ ox = 1.4;Gas constant ox R = 296 J/kg/K.Fuel parameters Burning rate law a = 0.07 mm/s, ν = 0.65; Fuel density ϱf = 1000 kg/m3; Gas constant of combustion products Rf = 320 J/kg/K; Adiabatic index of combustion products γf = 1.22;Fuel-oxidizer ratio αox = 2.7; Burning temperature Tf = 3000 K. Fig. 1 illustrates pressure distribution along the relative channel length for in ox m =0.2 kg/s.Relatively smooth pressure profile (power-law type) typical for initial conditions of engine operation is observed.It should be noted that for initial data as well as for other values of oxidizer flow rate the profile is transformed smoothly due to non-uniform fuel burnout and as the front part of fuel grain burns completely a pressure peak is formed and moves together with fuel burnout boundary.Mean difference between values of parameters determining the process (Table 1) does not exceed 3.5% for selected configuration.Thus, an analytical estimate of engine's characteristics for verification of developed physical and mathematical model of processes in a hybrid solid-propellant rocket engine for quasi-steady-state flow regime, constant flow section area and constant perimeter of working surface of cylindrical grain channel is given in this paper.Performed parametric comparative analysis of calculated and analytical data indicated more than satisfactory comparability of results of numerical calculation and analytical estimation -mean difference between values of parameters determining the process does not exceed 3.5% for selected configuration.
the process of combustion is equal to / α combustion products and "unused" oxidizer is equal to Σ the mixture (gas constant R and isobaric heat capacity Cp) are determined additively using mass individual characteristics of oxidizer gas ,

Fig. 2 .
Fig. 2. Gas flow rate distribution along the channel length.

Table 1 .
-Results of comparison of numerical and analytical data.