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By placing a sample between a heated and a cooled rod, a thermal conductivity of the sample can be evaluated easily with the assumption of a one-dimensional heat flow. However, a three-dimensional constriction/spreading heat flow may occur inside the rods when the sample is a composite having different thermal conductivities. In order to investigate the thermal resistance due to the constriction/spreading heat flow, the three-dimensional numerical analyses were conducted on the heat transfer characteristics of the rods. In the present analyses, a polymer-based composite board having thermal vias was sandwiched between the rods. From the numerical results, it was confirmed that the constriction/spreading resistance of the rods was strongly affected by the thermal conductivity of the rods as well as the number and size of the thermal vias. A simple equation was also proposed to evaluate the constriction/spreading resistance of the rods. Fairly good agreements were obtained between the numerical results and the calculated ones by the simple equation. Moreover, the discussion was also made on an effective thermal conductivity of the composite board evaluated with the heated and the cooled rod.

A simple way to evaluate a thermal conductivity of a sample is a steady-state method using two reference rods of known thermal conductivity [

In this method, a one-dimensional heat flow from the heated to the cooled section of the rod is assumed. Therefore, an attention is required when this method is applied for the measurement of an effective thermal conductivity of a composite sample. This is because a difference in thermal conductivity between components of the composite sample may cause a constriction/spreading heat flow inside the rods. The effect of the constriction/spreading heat flow on the measurement results would be conspicuous when the difference in thermal conductivity between the components of the composite sample is very large. Concerning the constriction/spreading heat flow and thermal resistance, a detailed review article was presented by Yovanovich and Marotta [

The thermal via is one of the options to enhance heat transfer through a board of low thermal conductivity. As mentioned in the authors’ previous paper [

This paper describes numerical analyses on the heat transfer characteristics of the two rods having the composite board in between. The present analyses dealt with a polymer-based composite board having the thermal vias. As mentioned earlier, sufficient information has not been published on the constriction/ spreading heat flow inside the rods caused by such a composite board. Like the authors’ previous study [

_{r}. The composite board was a model of a polymer board having thermal vias. As shown in _{h}, and a low, λ_{l}, thermal conductivity. A uniform heat flux, q_{h}, was applied on a top surface of the upper rod while a bottom surface of the lower rod was maintained at a uniform temperature, T_{c}. Outer surfaces of the model were thermally insulated except the heated and the cooled section. The temperature distributions of the model were obtained by solving a heat conduction equation given by

∇ ⋅ ( λ j ∇ T ) = 0 ( j = h , l , r ) (1)

with boundary conditions of

− λ r ( ∂ T / ∂ z ) = q h for the heated section (2)

T = T c for the cooled section (3)

∂ T / ∂ n = 0 for the adiabatic section (4)

where n was a coordinate normal to a boundary surface. The governing equation was discretised by a control volume method.

As shown in _{h} = 5.0 W/cm^{2} and T_{c} = 20˚C.

From the numerical results of the temperature distribution of the model, the thermal resistance of the board, R_{b}, was evaluated by

R b = R t − 2 R r (5)

where R_{t} was the total thermal resistance of the model and R_{r} the thermal resistance of each rod. R_{t} and R_{r} were expressed respectively as

R t = T h − T c q h W 2 (6)

R r = L λ r W 2 (7)

where T_{h} was the temperature at the top surface of the upper rod. L and W were the length (=45 mm) and the width (=32 mm) of the rod, respectively.

Type | Number | Size |
---|---|---|

1 | 4 | 8 mm × 8 mm |

2 | 16 | 4 mm × 4 mm |

3 | 64 | 2 mm × 2 mm |

4 | 256 | 1 mm × 1 mm |

_{h} = 400 W/(m∙K), λ_{l} = 0.40 W/(m∙K) and λ_{r} = 113 W/(m∙K), which correspond to the thermal condutivities of a copper, a polymer and a brass, respectively. The thickness of the composite board, δ, was 2.0 mm. The temperature distributions at the cross section of y = x for Type 1 (N = 4) to Type 4 (N = 256) are compared in this figure. Since the top surface of the upper rod was heated while the bottom surface cooled, the heat flowed downward from the heated to the cooled section. One- dimensional heat flows were found inside the upper and the lower rods. However, because of a large difference in thermal conductivity between the thermal via, λ_{h}, and the polymer, λ_{l}, in the board, the heat flow was constricted in the upper rod and spread in the lower rod near the board. It was also found that the constriction/spreading of the heat flow became smaller with N, resulting in a smaller temperature difference between the heated and the cooled section of the rod.

board, R_{b}, and the number of the thermal vias, N, at λ_{h} = 400 W/(m∙K), λ_{l} = 0.40 W/(m∙K), λ_{r} = 113 W/(m∙K) and δ = 2.0 mm. It is noted that R_{b} includes the constriction/spreading resistance of the rods. Moreover, when a one-dimensional heat flow inside the composite board was assumed and an equivalent thermal circuit for parallel two thermal resistances was considered, the thermal resistance of the board, R_{b}_{,1D}, was calculated by

R b ,1D = ( λ h A h δ + λ l A l δ ) − 1 (8)

where A_{h} and A_{l} were the cross-sectional areas of the thermal vias and the polymer in the board, respectively. The thermal resistance obtained by Equation (8) is also shown in _{b}_{,1D} does not consider the constriction/ spreading resistance of the rods, R_{b}_{,1D} was smaller than R_{b} and not affected by N. Moreover, it was also found that the difference between R_{b} and R_{b}_{,1D} became smaller with the increase in N. The constriction/spreading resistance of the rods, R_{cs}, was evaluated by

R c s = R b − R b ,1D (9)

The relations between the constriction/spreading resistance, R_{cs}, and the number of the thermal vias, N, are shown in _{r}, and the thermal vias, λ_{h}, were changed respectively at λ_{l} = 0.40 W/(m∙K) and δ = 2.0 mm. From _{cs} and N was strongly affected by λ_{r} confirming that the thermal conductivity of the rod is a factor dominating the constriction/ spreading resistance of the rods. From _{cs} and N was hardly affected by λ_{h} in the present calculation range from λ_{h} = 100 W/(m∙K) to λ_{h} = 400 W/(m∙K).

As shown in

to evaluate the constriction/spreading resistance of the rods, R ′ c s :

R ′ c s = a − b N π a b λ r (10)

where a and b were the side lengths of the square control volume and the thermal via, respectively (see

good agreements were obtained between R_{cs} and R ′ c s confirming the validity of Equation (10) for the evaluation of the constriction/spreading resistance of the rods.

The relation between R_{cs} and N is shown in _{h} = 400 W/(m∙K), λ_{l} = 0.40 W/(m∙K) and λ_{r} = 113 W/(m∙K). Moreover, the ratio of R_{cs} to R_{b} was calculated and the relation between R_{cs}/R_{b} and N is shown in _{cs} was hardly affected by δ while R_{cs}/R_{b} decreased with the increase in δ implying that

the effect of R_{cs} on R_{b} became smaller with the increase in δ. This is because of the increase in R_{b} with δ. The effective thermal conductivity of the composite board, λ_{eff}, was calculated by

λ e f f = δ R b W 2 (11)

Moreover, when a one-dimensional heat flow inside the composite board was assumed, the effective thermal conductivity of the board, λ_{eff}_{,1D}, was calculated by

λ e f f , 1 D = δ R b ,1D W 2 (12)

It is noted that R_{b} in Equation (11) was obtained from the numerical results while R_{b}_{,1D} in Equation (12) was calculated by Equation (8).

_{eff} and N when δ was changed at λ_{h} = 400 W/(m∙K), λ_{l} = 0.40 W/(m∙K) and λ_{r} = 113 W/(m∙K). The effective thermal conductivity obtained by Equation (12) is also shown in this figure. In case of the thinner board of δ = 2.0 mm, a relatively large difference was found between λ_{eff} and λ_{eff}_{,1D} due to the constriction/spreading heat flow inside the rods. However, because the effect of R_{cs} on R_{b} was reduced, it was also found that the difference between λ_{eff} and λ_{eff}_{,1D} became smaller with the increase in δ. In the present calculation range, the difference between λ_{eff} and λ_{eff}_{,1D} was minimum (λ_{eff} = 92.4 W/(m∙K), λ_{eff}_{,1D} = 100 W/(m∙K)) when N = 256 and δ = 20 mm.

Numerical analyses were conducted on the heat transfer characteristics of the heated and the cooled rod having the composite board in between. The present analyses dealt with the polymer-based composite board having the thermal vias.

From the numerical results, the constriction/spreading heat flow inside the rods caused by the composite board was clarified. It was found that the constriction/spreading resistance of the rods was strongly affected by the thermal conductivity of the rods as well as the number and size of the thermal vias. A simple equation was also proposed to evaluate the constriction/spreading resistance of the rods. The validity of the simple equation was confirmed by the comparison with the numerical results. The attention is required when the effective thermal conductivity of the composite board is evaluated with the heated and the cooled rod. This is because of the constriction/spreading heat flow inside the rods. It was found that the effect of the constriction/spreading resistance on the effective thermal conductivity became smaller with the increase in the thickness of the

composite board as well as the decrease in the size of the thermal vias.

Koito, Y. and Tomimura, T. (2017) Numerical Study on Heat Transfer Characteristics of Heated/Cooled Rods Having a Composite Board in between: Effect of Thermal Vias. Journal of Electronics Cooling and Thermal Control, 7, 91-102. https://doi.org/10.4236/jectc.2017.74008

A constriction heat flow in a semi-sphere from the radius r = r_{1} to r = r_{2} (r_{1 }> r_{2}) was considered. The heat flow rate, Q_{1}, was expressed as

Q 1 = 2 π r 1 r 2 λ r Δ T r 1 − r 2 (A1)

where ΔT was the temperature difference between the points at r = r_{1} and r = r_{2}. When the number of the constriction heat flow was N, the total heat flow rate, Q_{N}, was obtained by

Q N = N Q 1 (A2)

From Equations (A1) and (A2), the constriction resistance, R ′ c , was obtained as

R ′ c = Δ T Q N = Δ T N Q 1 = r 1 − r 2 2 N π r 1 r 2 λ r (A3)

The spreading resistance, R ′ s , from r = r_{2} to r = r_{1} was also expressed by Equation (A3). Because the cross sections of the control volume and the thermal via were square in shape (see _{1} and r_{2} were evaluated respectively by

π r 1 2 = a 2 , π r 2 2 = b 2 (A4)

Therefore, the constriction/spreading resistance, R ′ c s , was obtained as

R ′ c s = R ′ c + R ′ s = a − b N π a b λ r (A5)