A Volumetric Approach to Natural Convection


Calculator.....: Natural convection between plates 3
Calculator.....: Natural convection heat sink
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Introduction

Natural convection was widely used a couple of decades ago. Nowadays it is different. The heat densities are often so high that fan cooling must be used. There are nevertheless still several applications for which natural convection is the best choice, mainly because it is simple and safe. The subject has been treated before in this magazine. There are also several other web publications about it, so there is no lack of information. Here are some references:

Estimating Natural Convection Heat Transfer for Arrays of Vertical Parallel Flat Plates
Heat Sinks in Natural Convection: Optimum Fin Spacing
Natural convection modelling of heat sinks using web-based tools

The purpose of this article is not to repeat the content of those excellent publications. It is to deal with a concept that probably is new to most readers: the volumetric efficiency for natural convection. The background needed will also be rapidly scanned.

Conventional heat transfer theory is based on defining a heat transfer coefficient that operates on surfaces. This approach is always correct and it has also been considerably refined over the years. There are however cases when it does not really provide a good overview. Natural convection is one example. It can then be an advantage to use a more volume oriented approach.



Figure 1
Typical velocity profiles.

Some basics

Figure 1 shows some typical velocity profiles for natural convection. There are clearly two distinct cases. One in which the velocity profile does not interact with any nearby surface and one in which this is indeed the case. All heat transfer correlations for natural convection between parallel plates are based on this observation and they can all simply be said to map a weighted average between these two extremes.


Figure 2
Typical heat transfer coefficient.

A typical case is shown in Figure 2. The heat transfer coefficient increases sharply with the spacing when it is small but it then levels out and approaches the one for single plates. The slope of a straight line drawn from any point on that curve to the origin is a measure of the heat transfer coefficient per spacing unit. The optimum spacing is found where this line has its maximum slope. If a given volume is filled with heated plates this is also the spacing for which the dissipated heat has its maximum value.


Figure 3
Some basic equations put in an applied format.

Air temp [C]
Kl
Kq
Kt
Ks
Kv
-20
1.51
1.39
1.73
0.0226
116
-10
1.50
1.38
1.69
0.0236
109
0
1.48
1.37
1.65
0.0247
104
10
1.47
1.36
1.61
0.0257
98.5
20
1.46
1.35
1.58
0.0268
93.9
30
1.44
1.34
1.54
0.0279
89.3
40
1.43
1.33
1.51
0.0290
85.2
50
1.42
1.32
1.48
0.0301
81.5
60
1.41
1.32
1.45
0.0311
78.0
Table 1
Parameters for the equations in figure 3.

Some basic equations for the isothermal plates are given in figure 3 and their corresponding proportionality parameters are listed in table 1. The parallel plate case is based on the correlation given by Bar-Cohen. It should be noted that these equations are valid for convection only and that radiation can have an impact which is well as large, particularly for single surfaces.


Figure 4
The volumetric approach.

The volumetric approach

There is nothing new about the equations in figure 3. They have been used with good results for many years. It is however possible to go one step further and look at natural convection in a given volume. Equation 6 is the result of combining the expressions for optimum spacing, optimum heat transfer coefficient and Newton's cooling equation. The value of its proportionality parameter is listed in table 1. This equation is based on the highly theoretical assumptions that the plates are absolutely isothermal and negligibly thick. The result is nevertheless interesting. First of all, it sets a landmark for what is possible to do with natural convection; (there is possibly an exception from this that might be dealt with in a coming article). Secondly, it reveals the relative importance of the cross section, the height and the temperature difference.

An interesting observation is that the temperature difference, dT, is raised to the power of 1.5 and not 1.25, which normally is given for natural convection. The reason is that when dT increases it does not only increase the air velocity but it also decreases the optimum spacing and therefore increases the optimum plate count. A consequence of this is that dT must be regarded as the most important parameter of all.

Equation 7 is a more applied version of equation 6. It has the same general form but there is an additional term, the volumetric efficiency. This efficiency is actually a definition and it handles all deviations from the ideal case. The temperature difference term has also changed somewhat and is now indexed "ref", which stands for reference. The purpose of this is to clarify that the equation also can be used for non-isothermal plates. For heat sinks it is natural to base the temperature difference on the maximum temperature. For PCBs it is convenient to base it on the maximum plate temperature and not the maximum component case temperature. The many possibilities to define the temperature difference is evidently a dilemma. All applications of equation 7 should therefore ideally be followed by a declaration of the temperature difference definition used.

The volumetric efficiency concept can be applied to all kinds of different devises. The values for heat sinks are usually in the 70% - 90% range. The values for telecom racks are much lower, typically 20% - 30% and small boxes of the consumer equipment type can have even lower efficiencies. In spite of this large variation it is however always possible to use the concept to compare devices that have different measures and operating temperatures.


Some examples

Example 1
How much heat can a heat sink with the measures 200x80x200 mm (width, depth, height) dissipate by convection if the room temperature is 50 C and the temperature difference is 20 C?

Solution
The cross section, Ac, is 0.016 m2. Assume 75% volumetric efficiency and apply equation 7. The answer is 39 W.

Note
A check with in the referenced calculator results in 42 W because the assumed volumetric efficiency was slightly too low. It could also be noted that radiation in this case could boost the result to 56 W, (+33%). ____________________________________________________________

Example 2
A closed box must be cooled by natural convection. The vertical side that is available for a heat sink is 400x200 mm (width x height). The total heat dissipation is 40 W. The temperature difference must not exceed 15 C and the maximum outside air temperature is 40 C. Estimate the fin height need for a natural convection heat sink.

Solution
Assume 70% volumetric efficiency. Use equation 7 to estimate the cross section, which results in Ac=0.026 m2. The width is 0.4 m, which yields the approximate fin height 0.064 m or 64 mm.

Note
A real world problem of this type would also need to include a convection calculation for all other unit surfaces and a radiation calculation.

____________________________________________________________

Example 3
The current design rules for a particular natural convection application states a maximum PCB plate temperature of 75 C. The maximum room temperature is 50 C. How much could the heat dissipation be increased if the maximum plate temperature was increased to 85 C?

Solution
The temperature difference is increased with a factor 1.4. If the PCBs are looked at as single plates the impact of the temperature difference comes raised to the power 1.25, which would result in 52% increase. If however there was a possibility to combine the increased temperature with a smaller PCB pitch, one could possibly look at the problem from a volumetric point of view in which case that power is 1.5. The potential gain would then be 65%.