1. Introduction
A heat spreading angle is one of the concepts to simplify the heat transfer calculation and has been used for thermal design of electronic devices. This angle is applied for the board, e.g. printed circuit board, placed between a heat dissipating element and a relatively large heat sink, and then assumes a heat spreading area inside the board. While high-performance CFD simulators provide detailed numerical information, such a simplified calculation is also effective in a practical situation because the estimation can be made quickly without using high-performance computers. In addition, the rough design can be made easily by changing design parameters such as dimensions, physical properties and operating conditions, etc.
According to Vermeersch and Mey [1] , the early paper about this spreading angle goes back to the publication by Balents et al. in 1969. Balents et al. [2] used the constant value of 45˚ as the heat spreading angle and conducted an approximate heat transfer analysis concerning a heat dissipating element mounted directly on a ceramic substrate which was bonded to a metal baseplate. After a while, the 45˚ heat spreading model of Balents et al. was used by Cook et al. [3] in 1976, and thereafter many attempts have been made to modify the value or extend the theory of the heat spreading angle. Nevertheless, the above-mentioned 45˚ heat spreading angle is still familiar among thermal designers.
The applicability of the 45˚ heat spreading angle was described by several researchers. Masana [4] showed the comparison between the 45˚ heat spreading approach and the Fourier solution for a substrate (thickness: w) on which a square (dimensions: l ´ l) heat dissipating element was mounted. It was described that the common assumption of 45˚ for the spreading angle got close to the Fourier solution for large w/l. Guenin [5] also compared the 45˚ heat spreading analysis to the finite element analysis, and then concluded that the 45˚ heat spreading approximation provided the reasonable accuracy for boards in which the size of the heat source area was greater than the board thickness. Malhammer [6] described briefly that if a 20% error could be accepted, the 45 degree rule was always relevant for small heat sources. Lasance [7] reported that the 45˚ heat spreading angle was acceptable in a single layer board for h/k < 1 and d > 2 mm, where h was the heat transfer coefficient at the cooled section, k and d were the thermal conductivity and the thickness of the board, respectively. Furthermore, concerning the limitation, Lasance [7] and Ha and Graham [8] mentioned that the 45˚ heat spreading approach could not be used for multi-layer boards.
The above descriptions are guides in conducting the heat transfer calculation using the 45˚ heat spreading angle. However, they seem to be incoherent; accordingly, the comprehensive investigation is needed for the rational thermal design using the 45˚ heat spreading angle. In the present study, therefore, the extensive numerical investigation is conducted on the applicability of the 45˚ heat spreading angle. For simplicity, the two-dimensional mathematical model is considered and the calculation is conducted in a cylindrical coordinate system. Based on the numerical results, the validity of the 45˚ heat spreading angle and its relation with design parameters are discussed.
2. Analytical Method
2.1. Mathematical Modeling
Figure 1 shows the analytical system. The heat transfer characteristics inside the board are analyzed in a cylindrical (r − z) coordinate system. This board is essentially desk-shaped, and has the thickness of zb. The center (radius: rh) of the top is heated by a heat source while the bottom (radius: rc) is entirely cooled by a heat sink. The 45˚ heat spreading angle is shown in Figure 2.
The governing equation is given by
, (1)
and the following boundary conditions are applied for the heated section (0 ≤ r ≤ rh, z = 0) and the cooled section (0 ≤ r ≤ rc, z = zb), respectively:
(2)
(3)
where λ is the thermal conductivity of the board, q the heat flux from the heat source, α the heat transfer coefficient at the cooled section and Tf the cooling fluid temperature. The following adiabatic boundary condition is applied at the surfaces except for the heated and the cooled section:
, (4)
where n is the coordinate normal to the boundary surface.
The governing equation is discretized by the control volume method [9] and solved numerically with the boundary conditions. The analytical solution of Equations (1)-(4) having infinite series and Bessel functions was also shown by Lee et al. [10] . However, because the analytical solution is complicated and time-consuming, the numerical procedure is adopted here to obtain the results quickly. The numerical calculation is conducted by changing the parameters as shown in Table 1 at zb = 4.0 mm, q = 1.0 W/cm2 and Tf = 20˚C.
2.2. Dimensionless Parameters
The dimensionless temperature, θ, and the dimensionless coordinates, R, Z, N, are defined as follows:
. (5)
Accordingly, the dimensionless governing equation and the dimensionless boundary conditions for the heated section (0 ≤ R ≤ Rh, Z = 0), the cooled section (0 ≤ R ≤ Rc, Z = 1) and the other adiabatic section are expressed as follows:
, (6)
(7)
(8)
(9)
where Rh, Rc and Bi are the following dimensionless radii and the Biot number:
. (10)
It should be noted that the heat transfer behavior inside the present model is characterized by Rh, Rc, and Bi.
3. Results and Discussion
3.1. Temperature Distribution
The representative numerical results are shown in Figures 3 and 4. The effect of λ is shown at α = 100 W/(m2×K) and α = 5000 W/(m2∙K). Figure 3 is the temperature contour inside the board, and Figure 4 the temperature distribution at the cooled section (the top surface of the board). It should be noted that in Figure 4, the scales of the vertical axis for α = 100 W/(m2×K) and α = 5000 W/(m2×K) are different. In addition, if the conductive thermal resistance inside the board is neglected, the temperature of the board becomes uniform and then the uniform temperature, 
, is obtained by the following equation:
. (11)
The value of 
is also shown in Figure 4. When α = 100 W/(m2×K) and λ = 0.4 W/(m×K), it is observed that the contour lines inside the board are crowded near the heated section. Accordingly, the temperature at the cooled section changes conspicuously near the heated section and then approaches the cooling temperature at Tf = 20˚C. In this case, because the conductive thermal resistance inside the board is relatively large, the heat flow is not spread out but concentrated near the heated section. When α = 100 W/(m2×K) and λ = 400 W/(m×K), on the other hand, it is observed that except for the region near the heated section, the contour lines are almost perpendicular to the top/bottom surfaces and then the temperature at the cooled section is very close to the uniform temperature at
. In this case, the heat is spread out over the board. It is also confirmed that the temperature distribution becomes flatter and approaches 
with the increase in λ.
When α = 5000 W/(m2×K), it is observed that the temperature inside the board is entirely lower than that in the case of α = 100 W/(m2×K). Furthermore, the temperature at the cooled section is very close to the uniform temperature at 
even in the case of λ = 0.4 W/(m×K). This is attributed to the decrease in the convective thermal resistance at the cooled section, which enhances the heat transfer in z direction.
The temperature distribution at the cooled section is moreover shown in Figure 5 in the dimensionless form. The numerical results are shown by changing all dimensionless parameters of Rh, Rc and Bi. It should be noted that the scales of the vertical axis for Rh = 1, Rh = 2, Rh = 4 are different. In addition, the value of Bi is also indicated in Figure 3 for reference. Based on Bi, the heat transfer behavior inside the board is characterized as follows. When Bi is very small, namely Bi = 0.001, because the conductive thermal resistance inside the board is very small compared to the convective thermal resistance at the cooled section, the temperature is almost uniform irrespective of Rh and Rc. When Bi = 0.1, the temperature distribution is conspicuous due to the increase in the conductive thermal resistance inside the board. Furthermore, when Bi = 10, because of the decrease in the convective thermal resistance at the cooled section, the temperature is entirely lower than that in the case of Bi = 0.1.
Because the heated and the cooled area are prescribed by Rh and Rc respectively, it is expected that the temperature entirely increases with the increase in Rh or the decrease in Rc. However, due to the concentration of heat flow near the heated section, it is found that the temperature distribution is hardly affected by Rc, namely, the board width when Bi is large (Bi = 10).
3.2. 45˚ Heat Spreading Characteristics
In order to evaluate the 45˚ heat spreading characteristics, the index, η, is defined here by the ratio of the heat transfer rate inside the 45˚ heat spreading area to the total heat transfer rate through the board. η is calculated by,
Figure 3. Temperature distribution inside board (rh = 8 mm, rc = 40 mm).
Figure 4. Temperature distribution at cooled section (rh = 8 mm, rc = 40 mm).
Figure 5. Dimensionless temperature distribution at cooled section.
. (12)
Figure 6 shows the relation between η and λ changing α and rc as parameters. As reference, the values of η = 0.80 and η = 0.90 are indicated by lines in this figure. As expected, it is observed that η increases with the decrease in λ because of the concentration of heat flow near the heated section; η increases with α because of the enhancement of heat transfer in z direction. It is also confirmed that the high value of η such as η = 0.80 and η = 0.90 is obtained for small λ and large α, which corresponds to relatively large Bi.
The numerical calculation is moreover conducted by using the following isothermal boundary conditions:
. (13)
This boundary condition is applied at the cooled section and then the numerical result of η is also shown in Figure 6. It is found that irrespective of the numerical conditions, the high value of η (η » 0.91) is obtained under Equation (13), implying that the 45˚ heat spreading angle is applicable when the isothermal boundary condition is used at the cooled section.
By changing all dimensionless parameters of Rh, Rc and Bi, η is rearranged as shown in Figure 7 in the dimensionless form. It should be noted that η = 1 is obtained irrespective of Bi when Rh = 4 and Rc = 5, because
Figure 6. 45˚ heat spreading characteristics (rh = 8 mm).
the cooled section is completely included within the 45˚ heat spreading area. This trivial case is not shown in this figure. The values of η = 0.80 and η = 0.90 are also indicated by lines in this figure. As shown, η is well characterized by Rh, Rc and Bi. It is observed that irrespective of Rh and Rc, η increases with Bi and then relatively large increase in η is obtained between Bi = 0.01 and Bi = 10. Furthermore, it is also found that the value of η is not affected by Rc when Bi is large. This is due to the concentration of heat flow near the heated section.
From Figure 7, the values of Bi when η = 0.80 and η = 0.90 are obtained, and then these values denoted by Bi|η=0.80 and Bi|η=0.90 respectively are shown in Figure 8. Since η is not affected by Rc for large Bi as mentioned above, Bi|η=0.80 and Bi|η=0.90 depend on only Rh. Moreover, the relations between Bi|η=0.80, Bi|η=0.90 and Rh are expressed as follows:
(14)
(15)
Equations (14) and (15) are also shown in Figure 8. It is confirmed that the 45˚ heat spreading angle is appli-
Figure 7. 45˚ heat spreading characteristics in dimensionless form.
Figure 8. Biot number index to use 45˚ heat spreading angle.
cable when Bi is relatively large as expected, and then the specific numerical values of Bi are obtained by Equations (14) and (15) when 20% and 10% errors are accepted, respectively.
3.3. Thermal Resistance
When the 45˚ heat spreading angle is used, the thermal resistance of the board, R45, is simply obtained as
. (16)
The derivation of Equation (16) is shown in Appendix. From the numerical results, on the other hand, the thermal resistance of the board, Rb, is also obtained by subtracting the convective thermal resistance at the cooled section from the total thermal resistance. Rb is calculated by,
, (17)
where Th,ave is the average temperature at the heated section.
By using the ratio of R45 to Rb, the comparison between these thermal resistances is shown in Figure 9 changing all dimensionless parameters of Rh, Rc and Bi. Irrespective of the numerical conditions, it is found that the value of R45/Rb is less than 1, implying that R45 is smaller than Rb. This is owing to the fact that the effect of the conductive thermal resistance inside the board in r direction is neglected in Equation (16). It is also observed that the change in R45/Rb with Bi and Rc is similar to that in η; while in the range of R45/Rb > 0.8, the effect of Rh on R45/Rb is smaller than the case of η. Therefore, Bi = 10 is the index to use Equation (16) when 20% error is accepted.
Figure 9. Thermal resistance of board: comparison between 45˚ heat spreading approach and numerical results.
4. Conclusions
The numerical analysis is conducted on the applicability of the 45˚ heat spreading angle inside the board. In the present calculation range, the main findings can be summarized as follows:
1) The heat transfer behavior inside the board is well characterized by the three dimensionless parameters: the two dimensionless widths of the heated and the cooled section, and the Biot number.
2) However, when the Biot number is large, the heat transfer behavior is hardly affected by the dimensionless width of the cooled section due to the concentration of heat flow near the heated section.
3) The validity of the 45˚ heat spreading angle is confirmed when the isothermal boundary condition is used at the cooled section.
4) The 45˚ heat spreading angle is applicable when the Biot number is large. The equations are proposed to calculate the Biot number index to use the 45˚ heat spreading angle.
5) The thermal resistance of the board obtained by the 45˚ heat spreading angle can be used with 20% error when the Biot number is 10.
Appendix
The thermal resistance, R, is defined by
, (18)
where Q(z) is the heat transfer rate inside the board at the position of z. When the 45˚ heat spreading angle is used, Q(z) is expressed as
(19)
Substituting Equation (19) into Equation (18) and then integrating from z = 0 to z = zb, R45 is obtained as shown in Equation (16).
Nomenclature
Bi : Biot number (-);
N : dimensionless coordinate normal to surface (-);
n : coordinate normal to surface (m);
Q : heat transfer rate (W);
q : heat flux (W/cm2, W/m2);
R : thermal resistance (K/W); dimensionless radial coordinate (-); dimensionless radius (-);
r : radial coordinate (mm, m); radius (mm, m);
T : temperature (˚C);
Tf : cooling temperature (˚C);
: uniform temperature (˚C);
Z : dimensionless axial coordinate (-);
z : axial coordinate (mm, m); thickness (mm, m).
Greek Symbols
a : heat transfer coefficient (W/(m2×K));
h : index defined by Equation (12) (-);
l : thermal conductivity (W/(m×K));
q : dimensionless temperature (-).
Subscripts
45 : 45˚ heat spreading angle;
ave: average;
b : board;
c : cooled section;
h : heated section.