Scientific Paper / Artículo Científico  https://doi.org/10.17163/ings.n30.2023.03 pISSN: 1390-650X / eISSN: 1390-860X POLYNOMIAL CROSS-ROOTS APPLICATION FOR THE EXCHANGE OF RADIANT ENERGY BETWEEN TWO TRIANGULAR GEOMETRIES APLICACIÓN DE RAÍCES CRUZADAS POLINOMIALES AL INTERCAMBIO DE ENERGÍA RADIANTE ENTRE DOS GEOMETRÍAS TRIANGULARES
 Yanan Camaraza-Medina1,* 1,*Technical Sciences Faculty, University of Matanzas, Cuba. Corresponding author ✉:: yanan.camaraza@umcc.cu.   Suggested citation: Camaraza-Medina, Y. “Polynomial cross-roots application for the exchange of radiant energy between two triangular geometries,” Ingenius, Revista de Ciencia y Tecnología, N.◦ 30, pp. 29-41, 2023, doi: https://doi.org/10.17163/ings.n30.2023.03.

1. Introduction

It is required to evaluate the thermal radiation between surfaces in thermal engineering. The vision factor establishes what fraction of the radiant energy emitted by one surface is intercepted by another .

The geometrical relationship between two surfaces and its influence on the view factor has been studied for decades, obtaining numerical and analytical solutions for different geometrical configurations [2–5]. For example, Howell extensively compiled view factors with more than 320 different configurations .

The accelerated leap in using computational techniques has generalized the implementation of commercial programs based on the finite element method (FEM) to solve thermal radiation problems [7–10].

Three-dimensional edge problems are reduced to surfaces with common edges and angle θ included. However, shape factor algebra is tedious for these geometries, so numerical solutions such as FEM are preferred [11–13].

In FEM, meshes generally use triangular elements and rarely use rectangles or squares unless the overall geometry is a perfect cube. Determining an analytical solution for the view factor between triangular geometries requires sums of multiple integrals due to changing integration contours. In many cases, the solutions are not elementary functions, requiring the manipulation of inverse trigonometric functions, polylogarithms, and sums of infinite series .

This makes direct integration extremely tedious for unshared or without common edges geometries, so numerical integration is preferred. For this reason, analytical solutions for these types of geometries are lacking .

Using SMR with five intervals, the view factors were plotted for several perpendicular triangular geometries with common edges . However, their graphical interpretation generated mean errors of 12 %, demonstrating that they do not apply to FEM since they cannot be discretized. In the specialized technical literature, only this graphical solution is available to obtain the view factors between triangular geometries [6–13].

The BCR method provides a proper fit during the approximation of complex functions, so it can be used to create the expressions required in the FEM discretization. The BCR method is similar to the FEM because its mathematical conception is based on the formation of nodes, obtaining the polynomial fits from the interconnection of the nodes . Considering the above, it is demonstrated that there is a lack of analytical solutions (exact or approximate) to estimate the view factors between triangular geometries with common edges and angle θ included.

Therefore, this study aims to develop approximate solutions to calculate the view factors between triangular

geometries with common edges and angle θ included, without involving high mathematical complexity and guaranteeing a good fit concerning the SA. Thus, it is possible to establish a new analysis method for use in the FEM.

This research develops the exact analytical solutions for eight basic triangular geometries and their respective BCRs. For comparison, 42 examples with various aspect ratios were calculated for each geometry, using the AS, BCR, and SMR.

The practical nature of the contribution and the reasonable fitting values obtained demonstrate that this proposal is a suitable tool to be applied in thermal engineering and related practices that require thermal radiation calculations between triangular geometries.

2. Materials y methods

2.1.  Definition of the view factor

The view factor F12 depends on the position and geometrical configuration of the emitting surface A1 and

the receiving surface A2, defined as the fraction of the radiation leaving the former and intercepted by the latter, which is expressed as , in equation (1). (1)

Where: O1, O2 angles between the normal vector of the areas dA1 and dA2 and the line connecting the center of the surfaces A1 and A2, respectively. r– distance between the centers of surfaces A1 and A2, (see Figure 1). Figure 1. Basic geometry of the view factor

Equation (1) requires a double integration over the surfaces, which is complex and time-consuming since a large set of immediate integrals must be manipulated and subsequently factorized.

Numerical approximations can simplify the analysis because a suitable fit can be obtained with an appropriate set of intervals. For three-dimensional (3-D) configurations, various solution methods, such as contour integration, are available [19–24].

This work uses contour integration to obtain the view factor of the eight geometries analyzed. To approximate the special functions generated in the integration, the BCR method is used.

2.2.  Mesh creation for surface elements

In modern engineering, triangular elements are widely used to generate meshes. In contrast, rectangular or    square elements are rarely used, except in cases where the overall geometry is a perfect cube. Formulating this type of geometry requires a complex mathematical treatment that includes sums of the quadruple integral equation (1) caused by the variation of the limits in the projection on each coordinate axis. The viewing factor between two rectangular surfaces of the same width, with common edge and angle θ included, is given by equation (2) (see Figure 2). (2)

The following substitutions are used to evaluate equation (2).

After evaluating equation (2), we obtain the following solution f(1), (equation (4)). (3) (4)

 In equations (2), (3) and (4), the angle θ is given in radians. Equation (4) is very complex; for this reason, the last integral was not solved because its solution can be obtained numerically using Simpson’s 1/3 rule (with at least eight intervals). Drawing diagonal lines divides the emitting surface A1 and receiving surface A2 into eight triangular geometries. Applying the shape algebra for the geometry in Figure 3, = =32 combinations of view factors are obtained (see Figure 3). The analyzed geometry is symmetric; therefore, it is possible to define seven basic cases, as shown in Figure 4. Case 1: Right triangle to rectangle, with common side and angle θ between both surfaces. Case 2: Right triangle to right triangle, with common side and angle θ between both surfaces: vertices at a common point. Case 3: Right triangle to right triangle, with common side and angle θ between both surfaces: vertices at opposite ends. Case 4: Isosceles triangle to rectangle, with common side and angle θ between both surfaces. Case 5: Right triangle to right triangle of different size, with angle θ between both surfaces: vertices at a common point. Case 6: Right triangle to right triangle of different size, with angle θ between both surfaces: vertices at opposite ends. Case 7: Perpendicular right triangles with an equal edge and arranged in opposite directions. The view factors for the remaining cases can be obtained using the sum rule. Figure 2. Rectangles of equal width, with common edge and angle θ included Figure 3. Division of rectangular surfaces into triangular elements Figure 4. Basic configurations for triangular geometries

2.3.  Modeling of the view factor. Case 1

In Case 1 (see Figure 5), it is satisfied that the equation (5). (5) Figure 5. Basic Geometry for Case 1

Substituting equation (5) in equation (1), the view factor F12 is given by equation (6). (6)

In equation (6), the change indicated in equation (7) was made to perform the integration. (7)

Equation (6) is first integrated on the emitting surface A1, obtaining a sum of integrals, which is given by equation (8).

After a complex process in which it was necessary to solve nn = 44 = 256 primitive functions, the sum of double integrals of equation (8) was solved, whose solution is given in equation (9). (8) (9)

In equation (9), the term f(1) is obtained by equation (4). Due to the complexity of equation (9), the last integral is not solved, and its solution is obtained numerically using the SMR (twelve intervals are recommended). Equation (9) is transformed as the equation (10). (10)

Equation (10) is then transformed by dividing each dimensional variable by the length of the common edge b. The result is shown below the equation (11). (11)

Applying in equation (9) the change of variables of equation (11), the analytical solution for Case 1 is obtained, which is given by equation (12). (12)

 Equation (12) is a combination of variables (Y ;X). Evaluating this equation may be difficult because it is necessary to solve polylogarithms, sums of infinite series, and inverse trigonometric functions. However, using Bretzhtsov’s cross-root method, it is possible to obtain an approximate result, facilitating the calculation of the view factor. To implement the cross-root method, nodes are constructed using prefixed values (Y ;X), which are joined using diagonal lines forming the families of curves an and bn. In this study, the values Y = (0.1; 0.2; 0.5; 1; 3; 10) and X = (0.1; 0.3; 0.6; 1; 3; 6; 10) are used.

Tables 1 and 2 summarize the combination of variables (Y ;X) for each node and the nodes that integrate each curve an and bn, respectively. Figure 6 plots the families of curves an and bn.

The next step is to compute the vision factor using equation (12) for each of the combinations of variables (Y ;X) in Table 1, plotting them in a F12;X diagram, as shown in Figure 7. The union of the nodes along the x-axis makes it possible to create a third family of curves cn. A particularity is that all the nodes integrating the same curve cn have the same value of the variable Y , as shown in Table 1. Table 3 summarizes the nodes integrating each cn curve.

Table 1. Combinations of variables (Y ;X) for each node Table 2. Nodes integrating each curve an and bn Table 3. Nodes integrating each cn curve  Figure 6. Families of curves an  and bn Figure 7. Scheme for applying cross-roots

Each curve of the families an, bn, cn is approximated individually by the Least Squares Method (LSM), using a third-degree polynomial in the form mX3 +nX2 +oX +p, thus establishing a dependence between the view factor F12 and the X variable. Figure 8 shows the application of the method for curves a5, b5, c4.

Table 4 shows the values of the constants m, n, o, p obtained by applying LMS to all the curves an, bn, cn. The m, n, o, p values are averaged in each curve, thus obtaining the approximate functions An, Bn, Cn.

For each curve, the apparent angle of transmissibility (see Figure 8) is given by the equation (13) (13)

Therefore, Bretzhtsov’s cross-root is given by the equation (14). (14)

Table 4 shows the constants m, n, o, p for the polynomials An, Bn, Cn. For the approximations, the X variables were used, keeping the Y variables constant; therefore, to apply the cross roots, the Y variables were alternated by X, obtaining the following equations (15) to (17) for the polynomials An, Bn, Cn. (15) (16) (17)   Figura 8. Approximation by Least Squares: (a) curve a5, (b) curve b5, (c) curve c4

Substituting equations (15) through (17) into equation (14), we obtain that Bretzhtsov’s cross-root for Case 1 is given by equation (18). (18)

 Table 4. Constants m, n, o, p obtained by applying LMS  Substituting equation (18) in equation (10), the view factor for Case 1 is obtained, which is given by equation (19). (19)

3.      Results and discussion

For practical engineering use, equation (19) is much simpler than the analytical solution (SA) of equation (12). The percentage deviation (error) is computed with respect to the analytical solution and is obtained by the following relation in equation (20) . (20)

Where: D% is the percentage of deviation. SA is the view factor obtained by the analytical solution. V al is the view factor obtained by approximate methods.

To calculate the D% values, the view factors are computed for the 42 combinations of variables (Y ;X) in Table 1, using the AS, the SMR with five intervals, and the view factors obtained with the BCR.

Figure 9 plots the D% values obtained with equation (18) for the view factors calculated by SMR and BCR, adjusted in error bands of ±3% and ±6%. Figure 9. D% obtained with equation (18) for Case 1

For Case 1, Figure 9 shows that BCRs have a better fit with respect to SA, with a mean error of ±3% for 100% of the (Y ;X) points analyzed. On the contrary, the view factors obtained with SMR have a lower fit with respect to the AS, with mean errors of ±3% and ±6% for 54,8% and 85,7% of the (Y ;X) points evaluated, respectively.

3.1.  Modeling and validation of Cases 2 to 7

For Cases 2 to 7 (see Figure 4), mathematically, the

view factor F12 is given by equations (21) to (26). (21) (22) (23) (24) (25) (26)

 The analytical solutions of equations (21) to (26) are long and complex because they require the handling of Spence functions, Gamma function, sums of polylogarithms, modified Bessel functions of first species and zero, one, and two orders; for this reason, they are not presented in this study. For the solution of equations (21) to (26), the same procedure for Case 1 is used, obtaining the following approximations to calculate the view factor for Cases 2 to 7. (27) (28) (29) (30) (31) (32)

 Figure 10 plots in ±3% and ±6% error band the D% obtained with equation (18) for the view factors calculated with SMR and BCR for Cases 2 to 7. For Case 2, Figure 10 shows that BCRs have the best fit with respect to AS, with a mean error of ±3% in 97.6% of the (Y ;X) points analyzed. On the contrary, the view factors obtained with SMR have a lower fit with respect to AS, with mean errors of ±3% and ±6% in 28,5% and 64.3% of the (Y ;X) points evaluated, respectively. For Case 3, Figure 10 shows that BCRs have a better fit with respect to AS, with mean errors of ±3% and ±6% in 92.9% and 100% of the (Y ;X) points analyzed. The view factors obtained with SMR have a lower fit with respect to AS, computing mean errors of ±3% and ±6% in 38.1% and 69.0% of the (Y ;X) points evaluated, respectively.  Figure 10. D% values obtained with equation (18) for the cases analyzed. (a) Case 2; (b) Case 3; (c) Case 4; (d) Case 5; (e) Case 6; (f) Case 7

 For Case 4, Figure 10 shows that BCRs have a better fit with respect to AS with mean errors of ±3% and ±6% in 90.5% and 100% of the (Y ;X) points analyzed. In contrast, the view factors obtained with SMR have a lower fit with respect to AS, with mean errors of ±3% and ±6% in 21.4% and 61.9% of the (Y ;X) points evaluated, respectively. For Case 5, Figure 10 shows that BCRs have a better fit with respect to AS with mean errors of ±3% and ±6% in 95.2% and 100% of the (Y ;X) points analyzed. The view factors obtained with SMR have a lower fit with respect to AS, computing mean errors of ±3% and ±6% in 26.2% and 71.4% of the (Y ;X) points evaluated, respectively. For Case 6, Figure 10 shows that BCRs have a better fit with respect to AS with mean errors of ±3% in 100% of the (Y ;X) points analyzed. On the contrary, the view factors obtained with SMR have a lower fit with respect to AS, with mean errors of ±3% and ±6% in 31.0% and 81.0% of the (Y ;X) points evaluated, respectively. For Case 7, Figure 10 shows that BCRs have a better fit with respect to AS with mean errors of ±3% in 100% of the (Y ;X) points analyzed. The view factors obtained with SMR have a lower fit with respect to AS, computing mean errors of ±3% and ±6% in 23.8%and 73.8% of the (Y ;X) points evaluated, respectively.   3.2.  Other geometric configurations   In Figure 3, the emitting and receiving surfaces are divided into four triangular surfaces, resulting in 0.5nn−1 = 0.5 · 44−1 = 32 possible combinations (see Figure 11). Using the view factors f(1) to f(8), it is possible to obtain the view factors for the remaining configurations by applying the rule of sums and the algebra of form factors. Table 5 shows the relationships for computing the view factor for the configurations in Figure 11.

 Table 5. View factor settings for triangular surfaces  Figure 11. View factor settings for triangular surfaces