Natural Convection Cooled PCBs and Equivalent Velocity

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Introduction

Natural convection is not that easy to deal with. It is particularly the interdependence between velocity and temperature that makes it complex. Good natural convection correlations are therefore only available for a few elementary cases while forced convection theory cover thousands of cases.

It is therefore almost impossible to find correlations that work well for natural convection cooled PCBs in enclosures. One solution to this problem is to use CFD. It is a good way but it is not fast. Many questions of the type "what happens if" may therefore be left aside and never answered. A more analytical and faster approach could therefore be an attractive alternative. The method presented in this article is based on the equivalent air velocity concept. It is not a universal remedy for every problem in the world but it can facilitate the solution of quite a few of them.


Figure 1
Two examples of natural cooling arrangements in telecom racks.

Figure 1 shows the main airflow paths in a couple of natural convection cooled telecom racks. In actual applications there are in addition many side flows and numerous flow obstacles. Both arrangements are therefore far away from any of the elementary cases covered by natural convection theory. The calculation problem would have been much simpler if the air velocity had been known.

A very direct approach could therefore be to measure the air velocity in an enclosure filled with resistor PCBs and then correlate the result with the heat dissipation. One problem with this line of attack is that velocity measurements inside enclosures are complicated. Another problem is that the velocity retrieved eventually must be interpreted as a heat transfer coefficient. From what follows, it will be clear that this sometimes can be difficult. Another method is therefore proposed here.


Figure 2
The maximum velocity for forced convection is always found in the middle of the flow channel. This is not always the case for natural convection.

The equivalent velocity

The equivalent air velocity is defined as the virtual velocity that results in the same heat transfer coefficient as forced convection. It is a simple definition but there are a couple of issues that need to be clarified. The first issue is the velocity definition. It is here defined as the average velocity in the gap between two PCBs, if there had been no components. The second issue is the difference between the physical velocity and the equivalent velocity. Figure 2 intends to show that the velocity profile for forced convection always has its maximum value in the middle of the flow channel. This is also true for natural convection in narrow channels but definitely not for wide ones.


Figure 3
The equivalent and the physical air velocity start to diverge at optimum plate spacing.

The physical and the equivalent air velocity can therefore be expected to deviate considerably for wide channels. This is also the case. Figure 3 shows a comparison for an ideal parallel plate case. The two velocities follow each other quite well up to about optimum spacing but they then diverge significantly.

The flow conditions in actual enclosures are more complex. There may be stacked sub racks or chimney arrangements. Stacked sub racks can be expected to create velocity profiles that resemble those for long and narrow flow channels. The discrepancy between the physical and the equivalent velocity should therefore be small. Chimneys will always result in a draft that create near forced convection conditions. The equivalent velocity concept can therefore be expected to function well also for those cases.

It must in addition be pointed out that the method proposed is quite tolerant to differences between the equivalent and the physical velocity. The reason is that the extraction process is reversed when the concept is used. The resulting error is therefore much lower than the error in the process itself. Experience has shown that differences as high as 50% very well can be tolerated.

This is also the reason why the direct velocity measurement method not is recommended. That method would essentially mean that a measured physical velocity at calculation time would be used as an equivalent velocity. As a consequence there would be no error reduction.


Figure 4
Measurement PCB and a typical result for stacked sub racks.

Evaluation method

The first step in the evaluation method is to measure the maximum PCB temperature as a function of the heat load. The enclosure is for this purpose filled with heat generating PCBs and one or several measurement PCBs are inserted at critical places. The heat dissipation is then increased in steps and the maximum temperature on the measurement PCBs is registered. Figure 4 shows a typical result for stacked sub racks.


Figure 5
Evaluation equations.

The second step is to extract the equivalent velocity. The maximum PCB-to-room temperature difference has two parts. One is the temperature increase of the air in the sub racks below the top sub rack. The other is convection in the top sub rack. Figure 5 shows a set of equations that describe this physics.

There are a couple of difficulties to overcome when solving this equation system. The first is that the equivalent velocity is implicit. It must therefore be extracted by iteration. The second is that the heat transfer coefficient must be taken from a correlation that models the non-fully developed laminar flow region well. A reasonably good correlation can be found at the site: Heat transfer coefficients for parallel plates.


Figure 6
An example of how the equivalent velocity varies with the heat dissipation.

The third step is to correlate the equivalent velocity with the heat dissipation. This is just a matter of curve fitting. Figure 6 shows an example. The velocities for the cases displayed are strikingly similar. This effect is typical for long and narrow natural convection chimneys. The reason is that the additional driving force created when the chimney height is increased, is exactly consumed by an increase of the pressure losses. It should however be noted that a fixed maximum PCB-to-room temperature target results in a higher heat dissipation level for the 4-sub rack case than for the 8-sub rack case. An optimally charged 4-sub rack case will therefore always sense a higher equivalent velocity than the corresponding 8-sub rack case.

An alternative to extract the equivalent velocity is to extract the heat transfer coefficient. The evaluation process is basically the same, except that the final result is a heat transfer coefficient correlation. There are some advantages with this line of approach. One is that the correlation can be given a formulation, with regards to the PCB height and the surface flux, which resembles that for parallel plates. The result is a more general expression that can be expected to function also for conditions that do not map the measurement conditions exactly.

Software application

A straightforward employment of the equivalent velocity would be to use it as an input value to a thermal PCB code. It is a good way but it requires a new input for each heat dissipation level used. Since applied calculations often must cover a range of alternatives this method is not that smooth. A more flexible solution is to store the correlations in a database. This solution requires an internal line compiler or possibly a script interpreter but that is not a big obstacle in the computer age.

To use a database with heat transfer coefficients rather than equivalent velocities may seem to be a roundabout. The actual difference is however small. All that is needed to surface the equivalent velocity from a heat transfer coefficient is a simple iteration. Such operations are done in a snap on modern computers.

@TBYB	// title
Mag,1,Buf,0,Mag,1,Buf,0,Mag,1,Buf,0,Mag,1	// image script
F1=1.31+0.66*Nracks^0.076	// h script
F2=1	// --- "-----
Alfa0=1.06*F1*(Psurf/Bheight)^0.20	// --- "-----
MinSubRackCount=4	// low validity limit
MaxSubRackCount=8	// high validity limit
BoardSize=K2	// PCB size
Valid for 4 to 8 sub racks in cabinet	// comment

Figure 7
An example of an item in a database.

Figure 7 shows an example of an item in a database for heat transfer coefficients. Three lines describe the heat transfer correlation. All of them are sometimes needed it the expression is complex. There are in addition lines for a simple image script and various validity limits.

Discussion

The method described has been used for many years with surprisingly little problems. The measurements do to take some time to execute but once the correlations are in place they are both simple and smooth to use.

There is however one problem that sometimes can be a bother. The measurement method includes radiation. The radiation exchange between the measurement PCB and the side PCBs is zero because they have the same temperature. There is however a net contribution from the surfaces around the PCB edges. It is negligible for densely packed PCBs but for large gaps it can become significant. The remedy is to make sure that this radiation path not is accounted for when calculating.