Heat transport solutions in rectangular shields using harmonic polynomials

The search for the temperature field in a two-dimensional problem is common in building physics and heat exchange in general. Both numerical and analytical methods can be used to obtain a solution. Here a method of initial functions, the basics of which were given by W.Z. Vlasov i A.Y. Lur’e were adopted. Originally MIF was used for analysis of the loads of a flat elastic medium. Since then it was used for solving concrete beams, plates and composite materials problems. Polynomial half-reverse solutions are used in the theory of a continuous medium. Here solutions were obtained by direct method. As a result, polynomial forms of the considered temperature field were obtained. The Cartesian coordinate system and rectangular shape of the plate were assumed. The governing are the Fourier equation in steady state . Boundary conditions in the form of temperature (τ(x) ,t(y)) or/and flux (p(x), q(y)) can be provided. The solution T(x, y) were assumed in the form of an infinite power series developed in relation to the variable y with coefficients Cn depending on x. The assumed solution were substituted into Fourier equation and after expanding into Taylor series the boundary condition for y = 0 and y=h were taken into account. Form this condition a coefficients Cn can be calculated and therefore a closed solution for temperature field in plate. 1 Formulation of two dimensional temperature problem The search for the temperature field in a two-dimensional problem is common in building physics and heat exchange in general. Both numerical and analytical methods can be used to obtain a solution. Here a Method of Initial Functions, the basics of which were given by [1,2] were adopted. The approach used in MIF allowed to derive the form of harmonic polynomials, which form the basis of solutions in this paper. These polynomials are 4 * infinity which is the unique value of this article. An important value of this article is the use of these polynomials to determine the temperature in a rectangular area. Originally MIF was used for analysis of the loads of a flat elastic medium [3]. Since then it was used for solving concrete beams, plates and composite materials problems [4,5]. A solution in the form of a power series with coefficients depending on x was assumed. Then these coefficients were found by solving the differential equation. Harmonic polynomials were obtained that satisfy the Laplace equation in the area. The coefficients of the linear combination of these functions were determined by approximating the boundary conditions. The values of the approximation function for the edge of considered area are here initial functions. As a result, polynomial forms of the considered temperature function were obtained. Primary concepts are: boundary components, heat transfer flux , governing equation and characteristic operators of solutions. 1.1 Governing equation The heat transfer equation derived from the energy balance in the infinitesimal volume (1.1) Where T (x, y) is temperature, ρ, density, cpspecific heat capacity, λ coefficient of thermal conductivity. In steady state heat exchange, no internal heat sources and isotropic body, equation (1.1) become: (1.2) In this consideration, expressions were found for the temperature that satisfies the Laplace equation in the area. This solution has the form of a sum of polynomials. These polynomials exist in products with constant coefficients. The * Corresponding author: mariusz.owczarek@wat.edu.pl p xx yy v T T T c q t x x y y                             2 2 2 2 2 2 ( , ) 0 T x y x y                , 0 (2019) https://doi.org/10.1051/matecconf /201928 MATEC Web of Conferences 282 CESBP 2019 202064 2064 © 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/). obtained solution was divided into four independent states according to symmetry features of the temperature function. These are the following : -symmetry-symmetry SS (x and y even), -symmetry-antisymmetry SA (x even, y odd), -antisymmetry-symmetry AS (x odd, y even), -antisymmetryantisymmetry AA (x and y odd). 2 Boundary conditions The coordinate system and geometrical parameters of the medium were assumed as in Figure 1. Fig. 1. Coordinate system adopted in the analysis. The variable t is the set temperature at the edge of the shield (approximated), and variable T is the interior temperature (approximating). The following boundary conditions apply: 1. ( ), 2. ( ). 3. ( ). 4. ( ). y h Ut x y h LWt x x b Rt y x b LFt y       (2.1) Preceding the variable t in capital letters U, LW, R, LF, the temperature of the upper, lower, right and left edges respectively was assigned. The task was divided into four independent groups corresponding to four independent thermal states of the symmetry of the shield. The boundary conditions for the entire shield were split over four states specified in the first quadrant of the coordinate system. The following indices have been assigned to shorten the write to these states:. The temperature in the SS state is preceded by the letter S, in the SA state with the letter B and the AS with the letter C, and in the state AA with the letter A. The quantities in these states are determined by the initial state with the following formulas: SS state 1 1.2. , [ ( ) ( ) ( ) ( )] 4 1 3.4. , [ ( ) ( ) ( ) ( )]. 4 S S y h T SUt SLWt Ut x Ut x LWt x LWt x x b T SRt SLFt Rt y Rt y LFt y LFt y                       (2.2) SA state 1 1.2. , [[ ( ) ( )] [ ( ) ( )]] 4 1 3.4. , [ ( ) ( ) [ ( ) ( )]]. 4 B B y h T BUt BLWt Ut x Ut x LWt x LWt x x b T BRt BLFt Rt y Rt y LFt y LFt y                        (2.3) AS state 1 1.2. , [[ ( ) ( )] [ ( ) ( )]] 4 1 3.4. , [[ ( ) ( )] [ ( ) ( )]]. 4 C C y h T CUt CLWt Ut x Ut x LWt x LWt x x b T CRt CLFt Rt y Rt y LFt y LFt y                        (2.4) AA state 1 1.2. , [[ ( ) ( )] [ ( ) ( )]] 4 1 3.4. , [[ ( ) ( )] [ ( ) ( )]]. 4 A A y h T AUt ALWt Ut x Ut x LWt x LWt x x b T ARt ALFt Rt y Rt y LFt y LFt y                         (2.5) For example, boundary conditions for the whole disc on the lower edge will be equal to the sum ) ( ) ( ) ( ) ( x ADt x CDt x BDt x SDt    Example 1. The boundary conditions can be decomposited and described by functions: , 0 (2019) https://doi.org/10.1051/matecconf /201928 MATEC Web of Conferences 282 CESBP 2019 202064 2064


Formulation of two dimensional temperature problem
The search for the temperature field in a two-dimensional problem is common in building physics and heat exchange in general. Both numerical and analytical methods can be used to obtain a solution. Here a Method of Initial Functions, the basics of which were given by [1,2] were adopted. The approach used in MIF allowed to derive the form of harmonic polynomials, which form the basis of solutions in this paper. These polynomials are 4 * infinity which is the unique value of this article. An important value of this article is the use of these polynomials to determine the temperature in a rectangular area.
Originally MIF was used for analysis of the loads of a flat elastic medium [3]. Since then it was used for solving concrete beams, plates and composite materials problems [4,5]. A solution in the form of a power series with coefficients depending on x was assumed. Then these coefficients were found by solving the differential equation. Harmonic polynomials were obtained that satisfy the Laplace equation in the area. The coefficients of the linear combination of these functions were determined by approximating the boundary conditions. The values of the approximation function for the edge of considered area are here initial functions. As a result, polynomial forms of the considered temperature function were obtained. Primary concepts are: boundary components, heat transfer flux , governing equation and characteristic operators of solutions.

Governing equation
The heat transfer equation derived from the energy balance in the infinitesimal volume (1.1) Where T (x, y) is temperature, ρ, -density, c p -specific heat capacity, λ -coefficient of thermal conductivity. In steady state heat exchange, no internal heat sources and isotropic body, equation (1.1) become: (1.2) In this consideration, expressions were found for the temperature that satisfies the Laplace equation in the area. This solution has the form of a sum of polynomials. These polynomials exist in products with constant coefficients. The * Corresponding author: mariusz.owczarek@wat.edu.pl

Boundary conditions
The coordinate system and geometrical parameters of the medium were assumed as in Figure 1. The variable t is the set temperature at the edge of the shield (approximated), and variable T is the interior temperature (approximating). The following boundary conditions apply: Preceding the variable t in capital letters U, LW, R, LF, the temperature of the upper, lower, right and left edges respectively was assigned. The task was divided into four independent groups corresponding to four independent thermal states of the symmetry of the shield. The boundary conditions for the entire shield were split over four states specified in the first quadrant of the coordinate system. The following indices have been assigned to shorten the write to these states:. The temperature in the SS state is preceded by the letter S, in the SA state with the letter B and the AS with the letter C, and in the state AA with the letter A. The quantities in these states are determined by the initial state with the following formulas:

Fig. 2. Example of boundary conditions of temperature in the shield
The task is split into four states: The boundary conditions in the SS and AS states are shown in the Fig. 3 Example 2. The boundary conditions can be decomposited and described by functions: The picture of boundary conditions is shown in (2.14)

Solution of the area problem
The solution functions T(x, y) were assumed in the form of an infinite power series developed in relation to the variable y with coefficients C n depending on x.
and with respect to y were calculated. To sum the series (3.3) from n = 0, the substitution n = n + 2 was performed where n is a new variable with the same designation and (3.3) was rewritten in the form (3.5) C n (x) are treated as n-dependent terms of progression and as such can be derived from the differential equation. Putting the sum into a common sign and introducing the operator D: The equation (3.6) will be met if: Multiplying both sides by (n!/n!) a differential equation on C n (x) is obtained After applying the shift operator Equation (12) can be solved as equation with constant coefficients, the general solution is (3.11) Variables A and B were determined by substituting n = 0, 1 for equation (3.11). In this way expressions were obtained on C 0 and C 1 , although their value is unknown, one can express A and B: Determining A and B from equations (14), it was obtained: (3.13) Substituting (3.13) to (3.11) and introducing divalent functions j(n), j(n + 1) with values (0,1) and trivalent J(n) and J(n + 1) with values (-1, 0, + 1) an expression for C n term is as follows 17) The Taylor series expansion of sin() and cos() function is: XS before the function T (x, y) means that the function T (x, y) is arbitrary with respect to x and even with respect to y, likewise the designation XA means that the function T(x, y) is odd in relation to the variable y. Taking into account in (3.20)(3.21) the relationship j(i) + j(i + 1) = 1 and compounds (3.18) were obtained: Expressions (3.22)(3.23) can be easily separated into states symmetry-symmetry (SS), symmetry-antisymmetry (SA), antisymmetry-symmetry (AS) and antisymmetry-antisymmetry (AA).
The temperature in individual states is expressed in the following formulas: The polynomials specified in expressions (3.24-3.27) are harmonics, i.e. they satisfy the Laplace equations. By accepting a sufficient number of polynomials and their corresponding constant factors, boundary conditions can be met precisely enough.
In this work, the method of approximating the temperature at the edge of the shield was used to determine the constant coefficients.  The following are given in individual states of temperature symmetry: five polynomials corresponding to the following "n" rank: SS -8, SA-9, AS-9, AA-10.

SS
As can be seen, these polynomials can be easily written, because their coefficients are equivalents from the Pascal triangle.

Solution to the examples formulated in point 2.
We solve the task of meeting boundary conditions assuming fixed temperatures at the edges x=b, T=t z , y=h, T= Fig  3. This is symmetric part of Example 1, thus SS state.
We adopt harmonic functions Assuming an approximating function in the form: This function on the right edge takes the form: We will set parameters a 1 , a 2 from the approximation condition; meeting the minimum deviation on the right and upper edges. The minimum deviation condition has the form: By calculating derivatives relative to a 1 and a 2 , we obtained: By equating derivatives to zero, 2 equations were obtained to determine a 1 , a 2 constants : The temperature field in this case is the constant temperature equal to that shown in Figure 3.  We accept for the solution of this example the solving function is: 14) The boundary conditions are described by the equations: Where F, t are fixed numbers The square of deviation on the right edge is equal to: The deviation square on the upper edge is: As a criterion of approximation, we accept the minimum sum of deviations: Hence the deviation derivatives relative to a 1 and a 2 are equal to zero: 19)   1   3  2  2  3  2  2  1  2  1  2  0  0   3  2  3  2  1  2  1  2  0  0  1   3  3  5  3  3  3  2  1  2  1  2  1 , 0 The equations determining a 1 and a 2 have the form: or in an alternative form: 3  3  3  5  2  2  4  2  2  1  2   5  7  4  3  2  2  3  6  5  2  4  3  3  4  2  5  1 2 We write set of equations in the form: 11  1  12  2  1   21  1  22  2  2   3  3  3  5  2  2  4  2  2  11  12  1   5  7  4  3  2  2  3  6  5  2  4  3  3  4  2  5  21  22   4  2  2  3  3  2   3  3  ,  3  3  3  5  6  6  9  2 3 , 5 7 5 5