INCREASING OF ACCURACY FOR ENGINEERING CALCULATION OF HEAT TRANSFER PROBLEMS IN TWO LAYER MEDIA

Abstract. In this paper we study the simple algorithms for modelling the heat transfer problem in two layer media. The initial model which is based on a partial differential equation is reduced to ordinary differential equations (ODEs). The increase of accuracy is shown if instead of first order ODE initial value problem ([4, 5]) the second order differential equations is taken. Such a procedure allows us to obtain a simple engineering algorithm for solving heat transfer equations in two layered domain of Cartesian, cylindrical (with axial symmetry) and spherical coordinates (with radial symmetry). In a stationary case the exact finite difference scheme is obtained.


The Mathematical Model
We shall consider the partial differential equations [4]: in multilayered domain Ω with N layers .K ] are corresponding constants of specific thermal capacity, density and coefficients of heat conductivity in every layer, t[s] is the time, q k = q k (x, t) is the function of thermal sources, x[m] is the space coordinate, p = p(x) is given function depending on the system of coordinates: p = 1 in the Cartesian coordinates, p = x in cylindrical coordinates with an axial symmetry, p = x 2 in spherical coordinates with a radial symmetry.
Adding continuity conditions on surfaces x = x k , k = 1, N − 1 boundary conditions on the surfaces x = x 0 = 0, In the case of homogeneous media we consider the following partial differential equation where the constants of heat transfer parameters c, ρ, λ are normalizing magnitudes and cρ/λ is used as appropriate factor to the time t and function q.
In every layer the heat equation (1.1) can be presented in the following form where F k = γ kuk + q k ,u k = ∂u k ∂t .

The Exact 3-Points Finite-Difference Scheme
We use the method of finite volumes [3] for approximation of the differential problem. We consider N + 1 grid points in the x− direction Then the exact finite-difference scheme for a given function F k is defined in the form [2] x k+1 x dψ p(ψ) dx, If p(0) = 0 (cylindrical and spherical coordinates), then a 1 = α 0 = 0 and the difference equation (2.1) is defined as 4) and the equation (2.2) for k = 1 is given as where Summarizing expressions (2.2), (2.3) we obtain equation This equation has a positive root u N > 0, if σT 4 * + α L T L > Q. Next we consider only two layers, then N = 2, Then the unknown values are u 1 , u 2 (p(0) = 1, α 0 = ∞) or u 0 , u 1 , u 2 (p(0) = 0). In this case (see fig. 1)

The Cartesian Coordinates
Increasing of Accuracy for Engineering Calculation of Heat Transfer Problems 177 and the finite-difference scheme (α 0 = ∞) is given in the form (3.1) In the stationary case, if Therefore the value u 2 can be obtained from the equation This equation have a positive root for b > 0, e.g., q 1 ≤ 0, q 2 ≤ 0. The exact solution of the stationary problem (1.1)-(1.3) satisfies expressions (3.2).
In the non-stationary case (u k = 0), initial-value problems for ODE are used and we compute integrals R ± i approximately by the following quadrature formulas where The solutions of these systems are given as Using the difference equations (3.1) and the right hand side integrals approximating by (3.3), (3.4) with neglected error terms r m , m = 1, 2, 3, the approximate numerical solution u 1 (t), u 2 (t) at every time step t > 0 can be found by solving the following stiff system of two nonlinear ODEs of the second order Here one should take into account that from (1.1), (1.3) it follow that The initial conditions for ODEs (3.6), (3.7) are the following . . , 6 in the expressions (3.11) we obtain two systems of 7 linear algebraic equations for determination of A .
(the method of lines with a high order approximation).
The results of calculations are presented in Table 1. Here L = 2, u 1 * is the analytical solution, u 1pp is a O(h 8 ) order approximation, u 1p is O(h 6 ) order approximation, u 1 is a usual approximation, u 1t is obtained by using the method of lines, u 1tt is computed by the method of lines of high approximation.  In the cylindrical coordinates (see fig.2) we get the following coefficients:

The Cylindrical Coordinates
x k+1 x k x ln x k+1 xu k+1 (x, t) dx, x k+1 x k x ln x k+1 x q k+1 dx .
The finite difference scheme (N = 2, where In the stationary case, if q k = const,u k = 0, then and the values u 0 , u 1 , u 2 can be obtained from the system of equations In the non-stationary case integrals J 1 , J 2 , R + 1 , R − 2 can be approximated by the following quadrature formulas .

The unknown coefficients
can be determinated by using W (x) =x i , i = 0, 1, 2, 3, and solving the system of linear algebraic equations with parameter β :
The following stiff system of three ODEs of the second order is obtained for finding u 0 (t), u 1 (t), u 2 (t): The initial conditions are For the equation (1.6) (h 1 = h 2 = h = L/2, α L = ∞) the finite difference scheme (4.1) is defined as If V (x) =x i , i = 0, 5, then we get the following results The following system of two ODEs of the second order is obtained for finding u 0 , u 1 : 1 (q 0 +ü 0 ) + B The results of calculations are presented in Table 2, where L = 2, u 0 * , u 1 * are the exact values, u 0p , u 1p approximate values.  In the non-stationary case the results are presented in tables 3 (with radiation) and 4 (without radiation). We can see the effect of radiation. Table 3. The values u0 = u1, u2,u0,u2 in time t with radiation.  Table 4. The values u0 = u1, u2,u0,u2 in time t without radiation. In the spherical coordinates (see Fig. 3) we get the following coefficients x k+1 x k+1 x k The finite-difference scheme (N = 2) is given by (4.1), where In the stationary case (q k = const) we get: From (4.1) it follows that

Conclusions
1. The proposed method allows us to reduce 1D heat transfer problem in Cartesian, cylindrical and spherical coordinates to the system of the ordinary differential equations of the second order.
2. The described methods make it possible to find the distribution of temperature in the case of different layers with the heat source in between the layers and on layers borders.
3. In different coordinates it is possible to enlarge the accuracy of the given algorithm, when second order derivatives are used instead of first order derivatives.
4. Such formulations have a big practical importance as compared to Cartesian coordinates, e.g. for analysis of heat transfer in cylindrical wire-metal (coper) conductor with insulation.