The 45-degree rule

Home page........: Frigus Primore home page
Calculator..........: Thermal resistance for a bottle neck

Introduction

Conduction problems are common in thermal design. An solution approach that always works is numerical analyses. An alternative approach is to subdivide the problem into elements that each can be treated analytically. This method is not very accurate but it is usually simpler. It is therefore often preferred for overview purposes.


Figure 1
The 45-degree rule.

This article is about the 45-degree rule. It is an approach to the problem of calculating the thermal resistance between a small heat source and a large surface separated by some material, figure 1. There are sometimes claims that the angle should be chosen differently. Could it be that 45 degrees is recommended simply because it is easy to work with? The purpose of this article is to clarify this issue.


Figure 2
Temperature profiles for the two cases uniform temperature and uniform heat flux.

Theory

The general problem is quite complex. Some simplifications are therefore made here. The large surface is always considered to be isothermal. The small surface is either considered to be isothermal or to have a uniform heat flux. As shown in figure 2 it is the isothermal assumption that results in the lowest temperature difference. The discrepancy is in the 10%-20% range.

In actual applications it is often not evident which of these two cases that should be used. In general one is therefore forced to accept this difference as an uncertainly associated with the method itself.


Figure 3
A general solution to the bottleneck problem.

There are several ways to approach the conduction problem analytically. They are all quite complex. The one used here, figure 3, is probably the simplest! It is a solution for the temperature distribution in a rectangular rod that has a rectangular heat source in one end and a uniform temperature in the other. This problem is often refereed to as the bottleneck problem. The solution is however general and it is therefore also valid for rods with small heights, or in other words plates.

The advantage with the solution is that it is accurate. The disadvantage is that it requires some lines of program code. It can therefore never replace simple rules. Apart from that, there is an additional difficulty. The solution is based on a double Fourier series. Both series involved serve to model the heat source as a rectangular square wave function. The term count must therefore be pushed to the degree that the wavelength of the last term included is smaller than the heat source side. A 1x1 mm source on a 10x10 mm plate will require at least 3*10/1=30 terms in each series, all in all 900 terms. That is not a problem but if the heat source is decreased to 0.1x0.1 mm the total term count increases to 90 000, which definitely is a lot for high level program languages of the spreadsheet type.


Figure 4
The thermal resistance solution.

Thermal resistance

The average temperature on the heat source can be found by integration. The equation for the associated thermal resistance is shown in figure 4. It has two terms. One is the thermal resistance caused by a uniform heat flow over the entire cross section of the rod and the other the contribution caused by the bottleneck.

The size and location of the heat source can be freely chosen in this equation. For the purpose of this article it is however convenient to only deal with the symmetrical case. The set up is shown in figure 5. Two heat source types are considered and they will be referred to as the quadratic and the line heat source.


Figure 5
The comparison case.

Examining the equation reveals that the thermal conductivity only is a scaling parameter and therefore does not have any impact on relative comparisons. For comparisons there are actually only two important parameters, the ratios d/H and t/d.

There are a couple of cases of interest. The first is a single source on a large plate. It can be simulated by letting t>>d. The second is when several sources are so closely spaced that their spreading angles start to intercept, which can be simulated by letting t=d2.


Figure 6
Comparison with the 45-degree rule for the uniform heat flux case, t>>d.

Figure 6 shows a comparison with the 45-degree rule for the single source case. The d/H ratio range from 0.1 to 10, which should cover most applications. The maximum error for the quadratic heat source is about 12%. The line heat source has its maximum error at the low end of the scale where it reaches levels that are difficult to accept.

The calculated spreading angle oscillates around the 45-degree line. For small sources it is a bit higher and for large sources it is significantly lower. The error for the latter case is however not that important because the extra surface created by the spreading angle is small compared with surface of the heat source itself.


Figure 7
Comparison with the 45-degree rule for the uniform temperature case, d << t.

The uniform temperature case

It is difficult to find a corresponding analytical solution for the uniform temperature case. Finite element analyses was therefore used here. Figure 7 shows the comparison. The results are about 15% - 20% higher than for the uniform heat source case.


Figure 8
Impact of side sources.

Side sources

Figure 8 shows what happens when two heat sources are approached. The data was simulated for t/d2 ratios ranging from 0.5 to 10. The side source effect is apparently negligible unless the spreading angles overlap.

Conclusions

A major difficulty with the 45-degree rule is that there are two cases, uniform heat flux and uniform temperature. The difference is in the 10%-20% range.

If the accepted error is on the 10% level, the 45-degree rule can be used for cases where d/H>1, figure 6 & 7.

Side sources have no impact unless their spreading angles overlap.