Scientific Paper / Artículo Científico 



https://doi.org/10.17163/ings.n30.2023.03 


pISSN: 1390650X / eISSN: 1390860X 

POLYNOMIAL CROSSROOTS
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 
Received: 25112023, Received after review:
21042023, Accepted: 08052023, Published: 01072023 
Abstract 
Resumen 
The
view factor between surfaces is essential in heat transfer by radiation.
Currently, there are no analytical solutions available to evaluate the view
factors between triangular geometries with common edges and angle q , due to the high mathematical
complexity associated with their development. For these configurations, the
literature only has Sauer’s graphic solutions, the use of which generates
mean errors of 12 %. In this work an approximate
method is developed that does not give a high mathematical complexity and
that guarantees an adjustment of less than 12 %. For this purpose, 32
different geometric configurations (8 basic and 24 derived) were studied, obtaining the solutions for each of the
evaluated cases. For the validation of the models obtained,
42 different aspect ratios of leaving and reaching surfaces were used, the
view factors being computed in each case by means of the analytical solution
(AS), the numerical solution obtained with Simpson’s multiple rule 1/3 (SMR)
with five intervals and using Bretzhtsov’s crossed
root (BCR), finally comparing the results obtained in each of the eight basic
cases. In all the cases evaluated, the BCR showed the best fits, with
an error of ±6 % in more than 90 % of the samples, while the SMR showed an
average dispersion of ±6 % in 65 % of the data. The practical nature of the
contribution and the reasonable adjustment values obtained establish the
proposal as a suitable tool for use in thermal engineering. 
El factor de visión entre superficies es esencial en la transferencia de calor por radiación. En la actualidad, para evaluar los factores de visión entre geometrías triangulares con bordes comunes y ángulo q no se dispone de soluciones analíticas, debido a la elevada complejidad matemática asociada a su desarrollo. Para estas configuraciones, la literatura solo tiene las soluciones gráficas de Sauer, cuyo uso genera errores medios del 12 %. En este trabajo se desarrolla un método aproximado que no genere una alta complejidad matemática y que garantice un ajuste inferior al 12 %. Para este propósito fueron estudiadas 32 configuraciones geométricas diferentes (8 básicas y 24 derivadas), siendo obtenidas las soluciones para cada uno de los casos evaluados. Para la validación de los modelos obtenidos se usaron 42 dimensiones diferentes de emisor y receptor, siendo computados en cada caso los factores de visión mediante la solución analítica (SA), la solución numérica obtenida con la regla múltiple de Simpson 1/3 (RMS) con cinco intervalos y mediante la raíz cruzada de Bretzhtsov (RCB), comparándose finalmente los resultados obtenidos en cada uno los ocho casos básicos. En todos los casos evaluados, la RCB mostró los mejores ajustes, con un error de ±6 % en más del 90 % de las muestras, mientras que la RMS mostró una dispersión media de ±6 %en el 65 % de los datos. La naturaleza práctica de la contribución y los valores razonables de ajuste obtenidos, establecen a la propuesta como una herramienta adecuada para su uso en la ingeniería térmica.

Keywords: Triangular surfaces, Bretzhtsov
crossroot, View factor 
Palabras clave: superficies triangulares, raíz cruzada de Bretzhtsov, factor de visión 
^{1,*}Technical
Sciences Faculty, University of Matanzas, Cuba. Corresponding author ✉:: yanan.camaraza@umcc.cu. Suggested citation: CamarazaMedina,
Y. “Polynomial crossroots application for the exchange of radiant energy
between two triangular geometries,” Ingenius, Revista de Ciencia y Tecnología, N.◦ 30, pp. 2941, 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 [1]. 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 [6]. 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]. Threedimensional 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 [14]. 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 [15]. Using SMR with five intervals, the view
factors were plotted for several perpendicular
triangular geometries with common edges [16]. 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 [17]. 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 F_{12}
depends on the position and geometrical configuration of the emitting surface
A_{1} and the receiving surface A_{2},
defined as the fraction of the radiation leaving the former and intercepted
by the latter, which is expressed as [18], in equation (1).
Where: O_{1},
O_{2}− angles between the normal vector of the areas dA_{1}
and dA_{2} and the line connecting the center of the surfaces A_{1}
and A_{2}, respectively. r– distance
between the centers of surfaces A_{1} and A_{2},
(see Figure 1).
Figure 1. Basic geometry of the view factor Equation (1)
requires a double integration over the surfaces, which is complex and
timeconsuming 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 threedimensional (3D) 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)). 


(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 A_{1} and receiving surface A_{2} 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).
Figure 5. Basic Geometry
for Case 1 
Substituting
equation (5) in equation (1), the view factor F12 is given
by equation (6).
In equation (6), the change indicated in
equation (7) was made to perform the integration.
Equation (6) is first integrated on the
emitting surface A_{1}, obtaining a sum of integrals, which is given
by equation (8). After a complex process in which it was
necessary to solve n^{n} = 4^{4} = 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).

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).
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 crossroot method,
it is possible to obtain an approximate result,
facilitating the calculation of the view factor. 
To implement the crossroot 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 a_{n} and b_{n},
respectively. Figure 6 plots the families of curves an and
b_{n}. 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 F_{12};X
diagram, as shown in Figure 7. The union of the nodes along the xaxis
makes it possible to create a third family of curves c_{n}.
A particularity is that all the nodes integrating the same curve c_{n}_{ }have the same value of
the variable Y , as shown in Table 1. Table 3
summarizes the nodes integrating each c_{n}
curve. Table 1. Combinations of variables (Y
;X) for each node
Table 2. Nodes
integrating each curve a_{n} and b_{n}
Table 3. Nodes
integrating each c_{n} curve

Figure 6. Families of
curves a_{n } and b_{n} _{} Figure 7. Scheme for
applying crossroots Each curve of the families a_{n},
b_{n}, c_{n}
is approximated individually by the Least Squares Method (LSM), using a
thirddegree polynomial in the form mX^{3} +nX^{2} +oX +p, thus establishing a dependence between
the view factor F_{12} and the X variable. Figure 8 shows the application
of the method for curves a_{5}, b_{5}, c_{4}. Table 4 shows the values of the constants m,
n, o, p obtained by applying LMS to all the curves a_{n}, b_{n}, cn. The m,
n, o, p values are averaged in each curve, thus
obtaining the approximate functions A_{n}, B_{n},
C_{n}. For each curve, the apparent angle of
transmissibility (see Figure 8) is given by the equation (13)
Therefore, Bretzhtsov’s
crossroot is given by the equation (14).

Table 4 shows the constants m, n, o, p for
the polynomials A_{n}, B_{n}, C_{n}.
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 A_{n}, B_{n},
C_{n}.

Figura 8. Approximation by
Least Squares: (a) curve a_{5}, (b) curve b_{5}, (c) curve c_{4} _{ } Substituting equations (15) through (17) into
equation (14), we obtain that Bretzhtsov’s
crossroot 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) [25].
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 F_{12} 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; 
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 0.5n^{n−1 }= 0.5 · 4^{4−1
}= 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 
4. Conclusions This
study developed an approximate method to determine the view factor for 32
combinations of triangular geometries with common edges and angle θ
included, located in a 3D space. To validate the proposed models, 42 examples
with various aspect ratios were evaluated for each
geometry of the eight basic cases, comparing the results obtained by the AS
with those of the SMR with five intervals and those computed by the proposed
method with BCR. In all the cases evaluated, the RCB showed
the best fits with an error of ±6% in more than 90% of the samples, and the
SMR showed an average dispersion of ±6% in 65% of the data, confirming the
validity of the hypothesis on its use. For the remaining 24 geometric
configurations studied, the basic relations for calculating the view factor
from the expressions obtained for the eight basic cases were
presented. 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 radiation heat transfer calculation tasks. Due to the lack of similar precedents in the
literature, the proposed method highlights this research’s scientific and
practical value. The solutions provided could be
incorporated into the available catalogs for calculating view factors. Acknowledgments The
author gratefully acknowledges the assistance and recommendations from
Professor Dr. John R. Howell, Department of Mechanical Engineering,
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