PARAMETRIC IDENTIFICATION OF THE HEAT CONDITION OF RADIO ELECTRONIC EQUIPMENTIN AIRPLANE COMPARTMENT

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Introduction. To design the aircraft avionics with a given reliability is necessary to determine the required where C (x) = ìCcompo , x Î compo ; thermal characteristics of the insulated body of avionics and thermal control system. To do this, the study of the cv í î Cair , x Î air , thermal state of avionics in the aircraft compartment, us- ing a mathematical model of their thermal state is conducted. The model should take into account the unsteady l cv ( x) = ìl compo , í î l air x Î compo ; , x Î air , heat transfer of the insulated housing, the transfer of heat It means, that the coefficients Ccv , lcv depend on energy from one part of the equipment to another by the air flow, the convective heat transfer of the equipment which layer the heat transfer is considered. In equations (1)-(4) the following notations are used: elements. Tcv (t, x) - the temperature of the honeycomb panel; The mathematical model will be a system of partial Tcv,t - the first derivative Tcv of t ; Tcv, x - the first derivadifferential equations and ordinary differential equations, the number of which can reach tens or hundreds for real tive Tcv of x ; Tcv,x,x - the second derivative Tcv of x ; equipment. Therefore, it is necessary to develop effective methods for solving direct and inverse problems in the study of heat transfer of elements of onboard equipment Ccv (x) - volumetric heat capacity of the case honeycomb panel, determined by the heat capacity of the composite С and air capacity С ; l (l) - the thermal conducand estimation of the error of parametric identification. compo air cv A physical model of the thermal state of onboard equipment in the compartment of the aircraft. The tivity of the honeycomb panel, determined by the thermal conductivity of the composite lcompo and air thermal conthermal condition of the onboard equipment in the aircraft compartment is formed by external and internal factors. ductivity lair ; αcv,out - heat transfer coefficient of the External factors include heat exchange between the outer outer surface of the equipment housing; αcv,in - heat surface of the avionics and the air environment, radiation transfer coefficient of the inner surface of the equipment heat exchange of the outer surface of avionics and other housing; Fcv - the area of the equipment body for extersurfaces in the compartment. Inside the onboard equip- ment, thermal energy is released by the elements of the onboard equipment and is withdrawn or supplied by the heat supply system [1]. Mathematical model of the thermal state of on- board equipment in the aircraft compartment. The body of the onboard equipment in the aircraft compart- ment is a structure that includes a heat-protective honey- comb panel made of carbon fiber composite filled with air. The process of heat transfer in honeycomb panels is described by the boundary value problem for the heat equation with discontinuous coefficients. To solve this boundary value problem, the Monte-Carlo method based on stochastic differential equations is used in combination nal and internal heat exchange; Qcv,out - heat energy of external sources; Tair ,out - the air temperature in the com- partment; t - time; Tair ,in - air temperature in onboard equipment or its part; l - the thickness of the honeycomb panel. The process of heat transfer of the elements of the on- board equipment is presented in the form of an ordinary differential equation describing the convective-radiation heat transfer with the surrounding structures: Tm,t = αair,m (t) Fair,m / Cm (Tair (t) -Tm ) + + å g / C T 4 (t) / T 4 - c ε F / C T 4 + Q / C , (5) with the method of wandering in moving spheres [2]. In j ,m m j ms m 0 m m m m m m general, the heat transfer process in the carbon fiber panel is described by the equations [1; 3]: where Tm - the temperature of m-element of onboard equipment; Tm,t - the first derivative Tm of t ; αair ,m - Ccv (x) Tcv,t = (lcv ( x) Tcv , x )x , 0 < x < l, 0 < t £ tk ; (1) heat transfer coefficient of m-element of onboard equipment; Fair ,m - the area of m-element of onboard equiplcv (x) Fcv Tcv ,x = ment in convective heat exchange; Cm - heat capacity of = α (t) F (T (t, x) - T (t)) + Q , x = 0; (2) m-element of onboard equipment; g j ,m - radiation heat cv,out cv cv air ,out cv,out transfer coefficient of the system “j-element - m-element of onboard equipment”; εm - emission black ratio of mlcv (x) Fcv Tcv,x = element; Qm - heat dissipation or heat absorption energy = αcv,in (t) Fcv (Tair ,in (t) - Tcv (t, x)) + Qcv,in , x = l; (3) of m-element by onboard equipment from the air condi- tioning system and converted from electrical energy. Tcv (0, x) = T0 (x), 0 < x < l, (4) The equation of air heat exchange in the unpressurized blown onboard equipment is presented in the form of an ordinary differential equation describing the convective heat transfer of the inner surface of the housing of the on- board equipment, the elements of the on-board equipment and the enthalpy transfer from one part of the on-board equipment to another: Tair,k,t = αcv,in (t) Fcv / Cair,k [Tcv (t, x) -Tair,k + + åαair, j Fair, j / Cair,k (Tj -Tair,k ) + j (figure) the housing of the onboard equipment is consid- ered as a homogeneous medium with averaged coefficients of volumetric heat capacity and thermal con- ductivity, heat transfer through the housing of the onboard equipment is described by equations (1)-(4). However, the averaged thermophysical properties of inhomogene- ous medium can vary with a change in the direction of heat flow [6]. For this reason, we also consider the deter- mination of the thermal state of a honeycomb panel as a + cp Jair ,k Fk / Cair ,k (Tair ,k -1 - Tair ,k ); x = l, (6) solution to a three-dimensional boundary value problem for the thermal conductivity equation with a discontinuwhere Tair ,k -1 , Tair ,k - air flow temperatures respectively in ous thermal diffusivity. Due to the peculiarities of the ( k -1) and k parts of onboard equipment; Jair ,k - the method used, it is assumed that the thermal diffusivity coefficients of the composite and the air are constant. mass rate of the air flow in k part of onboard equipment; Fk - the total area of the air channels in k part of onboard equipment; cp - specific heat capacity of air; Cair ,k - heat capacity of air in k part of onboard equip- ment. Summation in equation (6) is carried out according to the j-element included in the k-part of the on-board There is a description of the boundary value problem below. The area in which the boundary value problem under consideration is defined is a rectangular parallele- piped G = (-l1, l1 ) × (-l2 , l2 ) × (0, l3 ). Where G is the union of two disjoint subsets: G = G1 È G2 , where G1 - is a subset, corresponding to the frame and plates limiting the panel, G2 is the union of subsets, corresponding to the cells with equipment. Heat capacity of air tion: Cair ,k is calculated by the equaair. The considered heat transfer process takes place on the time interval [0,T] and is described by the following boundary value problem for the heat equation: C = с ρ (W F Dt + V ), (7) ¶Tcv = 3 ¶ æ ¶Tcv ö air ,k p air ,k air ,ent air ,ent air ,k ¶ t å ¶ x ç a (x) ¶ x ÷ , (8) where ρair ,k - air density in k-part of onboard equipment; i=1 i è i ø Wair ,ent - air velocity at the inlet to the on-board equipwhere ìa , x Î G ment; F - the area of air channels at the inlet a (x) = í compo 1 ; air ,ent to the first part of the on-board equipment; Dt - time î acair , x Î G2 discretization interval in solving a system of differential equations; Vair ,k - air volume in k-part of the on-board equipment. x1 =- l1 = 0, x1 =l1 = 0; (9) Heat transfer coefficients of surfaces αcv,out , αcv,in , = 0, = 0; (10) αair ,m in equations (2)-(6) will be calculated using the methods described in [3; 4]. x2 =- l2 x2 =l2 The coefficients of radiation heat transfer in equation -l = α (t) (T - T (t)) ; (11) (5) are determined by the Monte-Carlo method [5]. Application of the Monte Carlo method to solve the cv x3 =l cv,out cv air ,out direct problem of the thermal state of l = α (t) (T - T (t)). (12) the honeycomb structure of the housing of onboard equipment. In the case where the honeycomb panel cv x3 =l cv,in cv air ,in Honeycomb housing design of onboard equipment Сотовая конструкция корпуса бортового оборудования In (8)-(12) equations the following designations are At each step of the algorithm, the step parameter is used: acompo , aair - thermal diffusivity coefficients of automatically adjusted ρk . If F(θk+1) > F(θk ) , so the composite and air, respectively; αcv,out , αcv,in - heat ρk +1 = ρk / γ , where γ > 1 - is a fixed parameter. If transfer coefficients of the panel surface and the air envi- ronment outside and inside the onboard equipment, F(θk+1 ) < F(θk ) , so the following sequence of actions is respectively; Tair ,out , Tair ,in - air temperature at the outer performed: i = ρ γ , ρ k ,i k θk +1,i = θk - ρ k ,i Hk Ñk F , and the side of the panel and the inner, respectively. In [7] the existence of generalized solutions of bound- ary value problems with discontinuous coefficients is calculation F(θk+1,i ), i = 0,1,K. These actions are performed until the value of the function F decreases and the conditions are met: proved. Moreover, these solutions can be approximated by solutions of boundary value problems, in which the ρmin £ ρk ,i £ ρmax (ρmin , ρmax - minimum, maximum step coefficients are the approximations of the initial disconlength, respectively) and i < imax (imax - the specified tinuous coefficients. For example, it is possible to obtain maximum number of iterations to increase the step). The an approximate solution of the original problem by solving the problem with smoothed coefficients based values θk +1, ρ k +1 are assumed to be equal to the values on integral averaging [8]. In this paper we propose to de- θk +1,i , ρ k ,i that are equal to the minimum of the obtained termine the approximate solution of the problem as a problem with smoothed coefficients by the Monte-Carlo method based on the probability of representation of the solution in the form of a mathematical expectation of the functional of the diffusion process. Initially, in work [9] the estimates of the mathematical expectation of this func- tional were determined on the basis of the numerical solu- tion of stochastic differential equations by the Euler method. The disadvantage of this method is its great com- plexity. A significant acceleration of the calculation was obtained using the combined method proposed in [2], in which the calculation of the trajectories of the diffusion process in air - filled cells (G2) was carried out by the method of wandering through moving spheres, and along the frame, bounding the plates (G1) and in their close area - by the Euler method. Note that the use of the com- bined method is possible only in the case of constant thermal properties of the substances that make up the honeycomb panel. A detailed description of the combined method is given in work [2]. Algorithm of parametric identification of mathe- matical model of thermal condition of onboard equipment. To determine the vector of the coefficients θ of the model of the thermal state of the honeycomb panel, the minimum of the function F(θ) of the weighted sum of squares of residuals [10] using an iterative minimiza- tion algorithm with the derived functions F(θ) should be defined. For this purpose, it is suggested to use a variant of the stochastic quasigradient algorithm with variable metric [8], in which approximations to the minimum point are constructed according to the rule: values F (θ) . Parametric identification of a mathematical model of the thermal state of the other elements of onboard equip- ment proposed to carry out the composition method of the steepest descent method, Newton method and quasi- Newton method of Broyden - Fletcher - Goldfarb - Shanno [11]. When solving a rigid system of ordinary differential equations, it is proposed to use the implicit Rosenbrock method of the second order [12]. Estimation of the parameters of the mathematical model of the avionics compartment of the aircraft. Verification of the proposed theoretical method was per- formed for onboard equipment in the aircraft compart- ment, which is a block of onboard equipment in the body with a honeycomb design. The unit is blown with air from the thermal control system. The air cools or heats the elements of the onboard equipment located in the com- partment. The elements of the block are separated by air layers. At the same time, the thickness of the honeycomb structure, temperature and air consumption of the system for ensuring the thermal regime of the onboard equipment unit were optimized. The main criterion for optimizing the thickness of the honeycomb structure, temperature and air flow of the thermal control system is the air temperature in the equipment unit within the limits of 283.15-293.15 K. The air temperature in the compartment is adjustable from 253.15 K to 328.15 K [13; 14]. At the same time, the values of the airflow in the thermal control system where Hk θk +1 = θk - ρ Hk Ñk F, k = 0, 1,K , - k a random square matrix of size l ´ l; (13) Ñk F - must be within the range of 1.5-2.0 kg/s. The coefficient of thermal conductivity of the honeycomb structure of the block body is l = 8·10-2 W/(m·K). k gradient of the objective function at a point θk ; ρ - step parameter. Matrix sequence Hk is calculated by the scheme: cv The thickness of the honeycomb structure block body took 2·10-2-5·10-2 m. The vector of coefficients of the model lcv of the H0 = I , Hk +1 + (I - μ Ñk +1F × (Dk +1θ)T ) Hk , Подпись: k Q= [ l T G ]T (15) Dk +1θ = θk +1 - θk . (14) cv stm stm Parameter μ is chosen from the equality includes the thickness of the honeycomb structure lcv in k k μ = μ /( Ñk +1F × Dk +1θ , where μ - is such a constant, that 0 < μ < 1. m, the necessary characteristics of the system to ensure the thermal control (the values of air temperature Tstm in K and air flow consumption Gstm in kg/s). ˆ r Estimates of the coefficients of the model Q thick- 7. Ladyzhenskaya O. A., Solonnikov V. A., Ural't- seva N. N. Linear and quasilinear equations of parabolic ness of the honeycomb structure, for temperature and airflow consumption, respectively, are equal: ˆ r Q = [0.003 287.4 1.9 ]T . type. Moskow, Nauka Publ., 1967, 736 p. 8. Sobolev S. L. Applications of functional analysis in mathematical physics. Moscow, Nauka Publ., 1988, 336 p. ˆ Joint confidence intervals DQ* of uncertainty of coef- 9. Gusev S. A. Application of SDEs to Estimating Soficient estimates (15) r Q with confidence probability lutions to Heat Conduction Equations with Discontinuous Coefficients. Numerical Analysis and Applications. 2015, β = 0.99 are, respectively, equal to DQ* = [0.0004 5.0 0.08]T . Joint confidence intervals DQ* of each of the sought coefficients are obtained by the method given in [15]. Conclusion. A theoretical method for determining the parameters of the housing of avionics with honeycomb structures and the system of providing the thermal control of avionics on the basis of the developed mathematical model of the thermal state of avionics of the aircraft is proposed. The determination of the thermal state of the honey- comb structure is carried out by a combined method, in which the calculation of the trajectories of the diffusion process in the cells filled with air is based on the method of wandering through the moving spheres, and along the frame, limiting plates and in their close area - by the Euler method. A stochastic quasigradient algorithm with a variable metric is used for parametric identification of the mathe- matical model of the thermal state of the honeycomb structure of the housing of the onboard equipment. The proposed method makes it possible to optimize the thermal parameters of the housing of the onboard equipment and the system of ensuring the thermal control in the design of onboard equipment.

About the authors

S. A. Gusev

Institute of Computational Mathematics and Mathematical Geophysics SB RAS; Novosibirsk State Technical University

6, Academika Lavrent'eva Av., Novosibirsk, 630090, Russian Federation

V. N. Nikolaev

Siberian Aeronautical Research Institute Named After S. A. Chaplygin

Email: sag@osmf.sscc.ru
21, Polzunova St., Novosibirsk, 630051, Russian Federation

References

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