# Introduction he problem of mathematical simulation of fluid flow in the fuel consumption control system is reviewed. In the course of the rocket travel the fuel from the oxidizer tank and fuel tank enters to the rocket combustion chamber. Synchronous fuel entry provides efficient operation. In real conditions this requirement is violated due to various reasons [1], resulting in inefficient fuel consumption. Residual fuel should have a minimum volume. Accomplishment of this objective depends on accurate measurement of fuel level in the tank. The problem is non-stationary, and is described by parabolic equation of motion. Solution of unsteady-state equation of motion for one-dimensional problem was found by a number of researchers. Solutions reviewed in [2,3] may be considered as classical. Paper [2] investigates laminar flow development from the rest state, work [3] reviews the pulsating flow. In [4] calculation results are compared with experimental records. Operational calculus methods are used to resolve parabolic equations in [2][3][4]. Research paper [5] presents oscillatory flow mathematical model. The solution is obtained using numerical method, obtained results are compared with experimental data. The authors [6] review non-Newtonian fluid throbbing stream in cylindrical channel Author: Department of Mechanics, Samara State Technical University, Russian Federation. e-mail: nikolay_klyuev@mail.ru with immediate valve closing. Method of Runge-Kutta was used to resolve the motion equation. Paper [7] contains the results of incompressible liquid flow in micro-tube at pressure jump research. The problem solution was obtained analytically, using Laplace transformation, and numerically, using Boltzmann method. Stationary flows and pulsating streams in slightly bent tube for a wide range of Reynolds numbers are reviewed in [8]. Numeric methods were used to resolve the problem. Work [9] presents pulsed incompressible flow through the pipeline. The flow is generated by periodical pressure gradient. The results show good compliance between analytical and numerical solutions. The study [10] represents method of characteristics for fluctuating streams simulation in the pipeline. It provides convergence estimate and method accuracy. Article [11] contains analysis of dynamical interference between the pipe and non-stationary flow on the basis of experiments and numerical models. Method of characteristics for determination of one-dimensional model of fluctuating fluid stream in the pipeline is used in [12]. Paper [13] provides experimental study of characteristics of non-stationary oscillatory flow in cylindrical channel. Obtained results comparison with known experimental results confirms good compliance. Work [14] reviews incompressible liquid non-steady laminar flow in expanding (convergent) channel with porous walls. Analytical solutions are compared with numerical solutions. In [15] the authors study nonstationary fluctuation problems related to non-viscous and low viscosity fluid in extensive network. # II. Physical Statement of The Problem Liquid level metering system is provided in the tank to control propellant consumption. For this purpose, vertical cylindrical channel, with fuel surface level indicators, is installed in the tank. Due to tank design features, the vertical channel may not match the tank centre line. Besides, short-period oscillations may occur at liquid free surface. In order that liquid level in the vertical channel reflects the liquid level in the tank, the metering system is supplemented by two horizontal channels located at the tank bottom. Horizontal channels outlets are located at one tank diameter. Horizontal channels overall length may exceed the tank diameter (Fig. 1). In case of fuel level reduction in the tank, the fuel level in the vertical channel is also reduced. When the propellant level in the channel reaches the indicator, the indicator is activated. The signal comes to the fuel consumption control system. As a result, fuel consumption may by changing. Thus, fuel level in the tank is determined on the basis of the fuel level in the channel. The channel and the propellant tank are communicating vessels. The problem is that in case of fuel consumption free surface levels in the channel and in the tank do not match. The error in the fuel level measurement results to inefficient fuel consumption. As a result, rocket motor is operated not with the optimum performance, and "excessive" fuel volume is left in the tanks. At the initial moment the tank and the channel are filled with the fuel with level 0 H . Free upper end of the cylindrical channel is above the fuel level in the tank, therefore the fuel overflow from the tank to the channel at this point is excluded. Fuel is free communicating between the tank and the channel. Constant pressure 0 p is maintained above free fuel surface in the tank and in the channel. From the time point 0 t > fuel is taken from the tank, so that the liquid level in it is reduced in linear fashion t V H t H 0 0 ) ( ? = , where ? 0 V fuel level depression rate in the tank. Therefore, liquid level in the channel is changing. # III. # Mathematical Model Of Liquid Flow The equation of viscous incompressible liquid non-steady motion in horizontal cylindrical channel is used as the flow model )( ) ? ? ? ? ? ? + ? ? ? = ? ? dr du r dr d r x p t u ? ? 1 ,(1) where ( ) t ,R u t t V H t dt du 4 ) ( 07 , 0 8 , 9 2 0 0 ? ? ? ? ? ? + ? = ? , 0 , 0 , 0 H u t = = = ? , (2) where ? -friction coefficient, ? -horizontal channel ? = t dt t u R R H t 0 2 1 2 0 ) ( 2 ) ( ? ,(3) and the problem will be determined by system of equations ( 2) and ( 3). As a result, we obtain Cauchy problem. For numerical solution of set problem, formulate system of 2 1 2 1 ) ( 2 ) ( R R t u t u = , where 1 R -vertical channel radius. Then equations ( 2), (3) as standard form. For that purpose take derivative with time from equation ( 3) 2 1 2 2 R u R dt d ? = ? . (4) Now the problem will be determined by the system (2) and (4). # IV. Numerical Solution And Results Problem solution is obtained numerically for , 02 , 0 , 039 , 0 1 m R m R = = , 2 , 8 0 m H = 2 10 5 ? ? = ? , m s m V 2 , / 039 , 0 0 = = ? . Using Mathcad application software package, solution results are given at diagrams (Fig. 2-Fig. 5). Fig. 2 illustrates liquid levels in the tank and in the vertical channel, Fig. 3 illustrate under damping oscillations of liquid average velocity in vertical channels. # V. Discussion Can be seen (Fig. 3), that the average velocity of the fluid in the vertical channel has synchronous damped oscillations. Fluctuations in a vertical channel are attenuated through 100 seconds. We can see (Fig. 4), that the magnitude of the error is a periodic function, in which the amplitude of oscillations # Conclusion Executed study proves that it is impossible to completely exclude liquid oscillations. Measurement error reduction may be expected in case of changing fuel consumption measurement system design features (introduction of holes on the vertical channel or dampers installation in the horizontal channels). To ensure zero error the indicators should be located at the points, corresponding to functions intersection nodes ) (t H and ) (t ? . references références referencias 1![Figure 1: Second diagram of a fuel tank. 1-tank, 2-fuel, 3-measurement vertical channel, 4-horizontal channels, 0 p -gas pressure, ? -liquid level in the channel, H -liquid level in the tank, x -coordinate axis](image-2.png "TFigure 1 :") 2![in the horizontal channel, p -pressure, ? -density, t -time, ?kinematic viscosity.Find approximate solution of equation(1). Enter average channel section longitudinal flow velocity -Multiply left and right sections of equation (1) by r and integrate each additive component from 0 to R , where R -horizontal channel radius.Taking into account, that in the process motion the acceleration of rocket increases in linear fashion ( obtain equation (the oblique brackets are omitted in the following).](image-3.png "2 .") ![](image-4.png "(") ![](image-5.png "?") ![Using volumetric flow rate conservation law, we write down:](image-6.png "D") Year 2017 D © 2017 Global Journals Inc. (US) Mathematical Model of Fluid Flow in Rocket Fuel System © 2017 Global Journals Inc. (US) © 2017 Global Journals Inc. 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