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Natural Convection and Chimneys |
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Back ground article....:
A volumetric approach to natural convection
Home page....................:
Frigus Primore home page
Figure 1
The average air density difference is larger for the chimney case
than for the fully filled case.
Introduction
Numerous electronic devices rely on natural convection for their
cooling. It is an attractive technique because it is simple,
reliable and silent. The drawback is that the cooling capacity is
limited. All possible means to enhance natural convection
performance are therefore very interesting for thermal designers.
The issue for this article is the combination of natural
convection and chimneys. The impact of a chimney is by no means
radical but if properly designed it can sometimes contribute with
the boost needed to make a solution attractive.
A former colleague of mine, Hans Westerberg, has developed a
theory for natural convection between isothermal parallel plates
placed in a chimney. It is a simple case and it is therefore
also a good starting point for further studies. Hans was one
of the first to apply CFD to electronics cooling problems. At
the time it was not easy. Each execution could take days and
since there were few codes around he had to develop his own.
He nevertheless had the patience to proceed and his correlation
has been proven to be correct at numerous occasions.
The chimney impact
Figure 1 shows a volume that is fully and partially filled with
parallel plates. The latter will here be referenced as the chimney
case. Weather one case can dissipate more heat than the other is
an interesting question. A correct answer requires extensive
calculations but it is possible to use much simpler means to get
a notion.
The driving force for the convection is the density difference
between the air inside and outside of the volume. The lowest
density is always found at the highest air temperature, which in
both cases is at the outlets from the channels formed by the
plates. For the chimney case however, this happens at a low
section of the volume and the result is a higher average density
difference than for the fully filled case, figure 1. This
advantage can either be used to increase the air velocity or to
increase friction and the plate surface by placing the plates
closer together. Both results in an improved cooling. It is on
the other hand not probable that the chimney case can house as
much plate surface as the fully filled case, which is a
disadvantage.
This type of intuitive reasoning, in which each case has its
pluses and minuses, indicates that there might not be much heat
dissipation difference. This is indeed also the case.
Figure 2
Optimum condition equations for the chimney case.
Figure 2 shows the most important optimum equations for isothermal
parallel plates. The corresponding proportionality parameters are
listed in table 1. The equations are valid for chimneys of modest
height, or more precisely expressed; when the friction forces in
the chimney are small compared with those between the plates. This
is also typically the case for electronics cooling applications.
The equations are almost identical with the ones for pure
parallel plates, see the reference article. The only difference
is that there is a chimney-to-plate height ratio term in some of
them. Another and very interesting phenomenon, is that the total
heat dissipation is independent of chimney-to-plate height
ratio! In more practical terms this means that each volume has
a heat dissipation ceiling that not can be exceeded no matter
how the plate surfaces in that volume are arranged. (There is
possibly an exception from this rule that will be brought up in
a coming article).
|
Air temp [C]
|
Kl
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Ks
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Kv
|
|
-20
|
1.51
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0.0226
|
116
|
|
-10
|
1.50
|
0.0236
|
109
|
|
0
|
1.48
|
0.0247
|
104
|
|
10
|
1.47
|
0.0257
|
98.5
|
|
20
|
1.46
|
0.0268
|
93.9
|
|
30
|
1.44
|
0.0279
|
89.3
|
|
40
|
1.43
|
0.0290
|
85.2
|
|
50
|
1.42
|
0.0301
|
81.5
|
|
60
|
1.41
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0.0311
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78.0
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Parameters for the equations in figure 2.
Equation 4 is a more applied version of equation 3. It has an additional efficiency term that handles all deviations from the ideal case. The temperature difference has also been indexed in order to clarify that it can be used for non-isothermal cases. The volumetric efficiency for actual designs can vary greatly. For heat sinks it is usually in the 70% - 90% range. For natural convection enclosures filled with PCBs its much lower, typically 20% - 30%.
Figure 3
The maximum heat dissipation is independent of the chimney-to-plate height ratio.
Filling a volume with heat transfer surfaces
Figure 3 shows a typical result of a calculation that involves several chimney-to-plate height ratios. The main impact of this parameter is that it tends to decrease the optimum plate spacing. The total heat dissipation is, as expected, the same for all cases.
There are several issues to consider when going from the theoretically perfect world into the real world. A discussion on this level can never be quite scientifically stringent. The problems brought up are however typical for applied thermal design and must therefore be managed somehow.
The first problem is very rudimentary and has to do with the plate spacing. It is always confined to discrete values. It is therefore rarely possible to create a design that operates exactly on the optimum point. The selected spacing should however rather be on the right side of the optimum point than on the left side. The simple reason is that the curves on the right side have a lower slope.
A second problem is that the orientation of the heat transfer surfaces sometimes not even is near a parallel plate arrangement. A reasonable assumption here is that the general tendencies in figure 3 are valid for all kinds of vertical surfaces, regardless of their orientation. A volume would in this line if thinking rather be characterised by an optimum total heat transfer surface than by an optimum plate spacing. An issue that can be approached in this way is heat sinks mounted on natural convection cooled PCBs. It is apparent that it is problematic to introduce a heat sink between two PCBs that already are on an optimum distance. The result will in most cases be an increased temperature! Natural convection is however very sensitive to dT and if that heat sink could be given a temperature considerably above that of the PCB, there could possibly possible be some gains. This does not mean that heat sinks not can be used for natural convection cooled PCBs. It simply means that the PCBs in that case must be placed on a larger than optimum distance. This distance is quite difficult to calculate but simple considerations based on the parallel plate case can at least give an indication in which range it can be found.
A third problem is that plates always have a thickness. For flat plates with a thickness of a millimetre or two, it is seems reasonable to interpret the spacing as a wall-to-wall distance rather than as a centre-to-centre distance. For PCBs it is more complicated. The components must in that case be simulated as an extra surface layer with a thickness that by no means is simple to calculate. Typical PCB pitches are however above 15 mm and if the height of the components only are a couple of millimetres, it is apparent that an exact thickness compensation not is crucial. It can in this context also be added that the boundary layer thickness for natural convection typically is in the 4 - 10 mm range. Most components will be embedded in that layer. Their physical surface enhancements will therefore only be slightly sensed on the heat transfer level.
A fourth problem has to do with natural convection cooled enclosures. They can be designed in a many different ways, with and without chimneys. From an overview perspective and just considering the total heat dissipation, it can be concluded that none of these designs has any advantage over the other. The difference is that chimney arrangements usually result in a higher allowed heat density on each single PCB.
Figure 4
Laminar flow in the chimney creates a re-circulation region at the air outlet, which completely eliminates the benefits of the chimney.
Making a chimney work
There are also some difficulties associated with the chimney impact. One of them is the flow characteristic in the chimney. If it is laminar it will also be non-uniform and tend to create a region of re-circulation at the outlet, figure 4. The phenomenon is closely related to the cohanda effect, (adherence of a jet to a wall). This impact totally neutralises the benefits of a chimney. A coarse estimate, based on 0.3 m/s velocity, indicates that the critical size limit is somewhere around a hydraulic diameter of 140 mm.
A similar phenomenon also appears for very large cross sections. An example of this is convectors in cold stores. If there is a chimney it is in this case rather a skirt but the physics is the same. That skirt will nevertheless only function well provided that it is subdivided into a number of smaller sections.
There is also a problem associated with heat sinks. They can be looked at as parallel plates if the fins are high enough. The difficulty is that the fins must be made sufficiently thick to keep the fin temperature on a reasonable level. Placing a heat sink in a chimney always results in a reduction of the total heat transfer surface, which increases the heat density and thus require thicker fins. Very high chimney-to-plate height ratios are therefore excluded simply because the fins would become too thick and choke the airflow. It is difficult to give any general guide lines for this effect but chimney-to-plate height ratios as high as 4 have been successfully tested.
An example
Example
A side of a box must be cooled by natural convection, The available volume is 200x200x80 mm, (width, height, depth). The allowed temperature difference is 20 C and the maximum room temperature is 40 C.
A)
Estimate how much heat that can be dissipated by natural convection from this volume?
B)
Estimate how much heat that can be dissipated if the heat sink is embedded in an 800 mm high chimney?
Solution
A)
The cross section, Ac, is 0.016 m2. Assume 75% volumetric efficiency and apply equation 4. The answer is 41 W.
B)
Assume an air velocity of 0.3 m/s. Calculate the hydraulic diameter of the chimney as (4)x(cross section)/(circumference)= 4x0.08x0.2/(2x(0.08+0.2))=0.11 m. Get the kinematic viscosity from a table, 16.9E-6 m2/s. Calculate Reynolds number as (velocity)*(hydraulic diameter)/(kinematic viscosity) =0.3x0.11/16.9E-6=1952. Compare this number with the laminar limit, which is 2300. It is clearly on the laminar side. The velocity assumption was on the other hand coarse and the slightly higher assumption of 0.35 m/s would result in a design that is on the limit to the possible.
If the flow in the chimney is turbulent and based on the same data as above except the height, which now is 800 mm, the answer is 82 W.
Note
This design has actually been tried. An elaborate calculation of the velocity in the chimney resulted in 0.36 m/s. At testing the chimney flow was laminar during the warming up period but then shifted to turbulent. The volumetric efficiency was found to be 85%.