Convection and Air Properties, Altitude Impact

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Introduction

Many specifications include an altitude requirement. This raises two fundamental questions. How should calculations for elevated altitudes be made and how should the designs be verified? The subject has been addressed before but there still seems to be a need for clarifications. This article brings up the background physics and proposes some estimation methods.


Air properties at elevated altitudes

The physics that deals with thermal properties for air is called kinetic gas theory. In essence, it treats the molecules as flying dumb-bells engaged in chaotic and i ntense collisions. The theory is quite complex and this is not the proper place to explain its details. One result is nevertheless very interesting; the only fundamental air property that changes significantly with pressure is density. (Air properties that are combinations of other properties and include the density will naturally also change, for example the kinematic viscosity). This conclusion may be difficult to digest intuitively but it has nevertheless been verified experimentally.

This simple rule wrecks for very high and very low pressures. The latter has been discussed in the article: The Thermal Conductivity of Air at Reduced Pressures and Length Scales. It appears when the mean travelling path for the molecules is of the same order as the distance between the enclosing walls. Extremely low pressures or small pockets of air trapped between two surfaces in contact can provoke this effect. None of these extremes is however interesting in this context.

There are several sources for how pressure varies with altitude. One of them is a daily newspaper! They may differ somewhat because the phenomenon is slightly temperature dependent. The Cina-atmosphere, (Commision International de Navigation Aérienne), will be used here. This organisation does not seem to exist anymore but it has left a collection of easy to use equations, figure 1.


Figure 1 The Cina atmosphere and the density ratio for equal air temperature.

It should be noted that there are two ways to calculate the density. One for see level temperature and one for altitude temperature. For ground based equipment it is mostly the former that is interesting but the latter is important for avionics applications. The diagram shows the density ratio for equal air temperature.


Figure 2
Dimensionless representation of the heat transfer coefficient for flow between two parallel plates.

Convection basics

Figure 2 shows a dimensionless template equation for convection between parallel plates. The constant and the exponents are condition dependent. Each value set can therefore only be used within a given validity range. The density is a part of Reynolds number and does not appear anywhere else. It is therefore only the exponent on the Re-number, m, that is of interest here.


Figure 3
A log-log diagram for the heat transfer coefficient as function of the velocity for a typical PCB case. The slope of the curve reflects the exponent on the Re-number.

If the heat transfer coefficient is drawn as a function of the velocity in a log-log diagram, the slope of the curve reflects the exponent on the Re-number, figure 3. The heat transfer coefficient is in this case the one defined for the inlet temperature difference. The exponent is near 1.0 for low velocities. It then decreases to about 0.5 and for strong turbulent flow it increases to 0.8. For lower air densities the curve is simply pushed to the right. It can be concluded that the impact of altitude must be velocity dependent, or more correctly expressed Re-number dependent.


Figure 4
The air temperature increase varies linearly with the density.

There is also another important effect to consider, air heating, figure 4. It varies linearly with the density. Air heating is therefore more sensitive to altitude changes than convection. Various stacked sub rack arrangements are examples where air heating can be important.


Figure 5
Some examples of how the temperature difference changes with the altitude.

Figure 5 shows some examples of how the temperature difference changes with altitude. They have been calculated for 200x200 mm parallel plates, placed on 20 mm distance. The most sensitive cases are, as expected, the ones that have a strong dependence on the density. For the natural convection case it is because the velocity is low, figure 3, and for the stacked cases it is because of air heating. The temperature difference increase at 3000 m altitude is ~25%. For 25 K this corresponds to a 6 K change, which is significant. This value could be compared with the values proposed in the Adjusting temperatures for high altitude article. They range from 21 - 45 %, (the highest value takes the conservative approach to assume an absolute proportionality with density).


The velocity trick

A very interesting observation is that density and velocity always appear coupled as a product, both for convection and air heating. A consequence of this is that decreasing the air velocity has exactly the same effect as decreasing the density. The impact of altitude can therefore be perfectly simulated at see level by reducing the air velocity with the density ratio.

This possibility can reduce aeronautic chamber testing considerably. The only concern is natural convection. It always plays a part. The impact is often negligible but for some cases there are reasons to be cautious. A PC may for example appear to be a fan-cooled device. Some designs might however have internal pockets of almost still air in which natural convection could be important.

Most thermal PCB programs have an option for adjusting the air properties. To change them may therefore seem to be a straightforward method to compensate for non-see level altitudes. There is however a problem with heat sinks. If they are specified as thermal resistances they do not sense the air properties. A trick out of this dilemma is to specify these thermal resistances for a velocity reduced with the density ratio.


Figure 6
Fan and system curves for a case involving a heat sink.

Fans and flow rate

The fan laws predict that the pressure gain increases linearly with density but that the flow rate remains constant. For the pressure losses in the system it is more complicated. There are several phenomena to consider. The first is that there is a basic linearity with density. All losses caused by re-linking or sudden cross section changes are therefore proportional to the density. When these types of losses are dominating the net result of an altitude increase is an unchanged flow rate. This is most often the case but there is a complication, friction.

The friction factor increases when the density decreases, particularly for laminar flow. This tendency counteracts the basic linearity and may in extreme cases cancel it. Heat sinks and air filters are examples of devices where friction losses are dominant. If those losses make up the major part of the total loss there is reason to be cautious. The net result could be a significant lower flow rate at higher altitudes. The article Cooling of electronics at high altitudes made easy covers this issue more in depth. A somewhat different and very interesting approach can be found in the article Quick and easy fan/sink characterization.


Some examples

Example
An equipment is designed for 1.5 m/s forced convection cooling. What air velocity should be used to simulate the impact of 3000 m altitude.

Solution
Assume that the air velocity remains unchanged. Calculate the density ratio for equal temperature at 3000 m.
(density ratio)=((288+6.5e-3*3000)/288)^5.255=0.692.
The velocity should be reduced to
(reduced velocity)=1.5*0.692=1.04 m/s.

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Example
A maximum PCB temperature of 82 C has been measured on a PCB at 50 C in the room. The PCB has a cooling efficiency of 70%, low profile components, no heat sinks and it is the only heated object in the air flow path. Estimate the average temperature rise that can be expected at 3000 m altitude.

Solution
Assume that the air velocity remains unchanged. Since there are no heat sinks and the components have low profiles it can be assumed that the major part of the cooling efficiency is caused by non-uniform temperature on the PCB. The maximum PCB temperature difference is:
(max dT)=82-50 = 32 K
The average temperature difference for convection is:
(average convection dT)=32*0.7=22.4 K
The density ratio, see above, is:
(density ratio)=0.692.
Assume that the exponent on the Re-number is 0.7, figure 3. The heat transfer coefficient is reduced by:
(heat transfer coefficient ratio)=0.692^0.7=0.773
The corresponding temperature difference ratio is:
(dT ratio)=1/0.773= 1.29
The average temperature increase is:
(average dT increase)=(1.29-1)*22.4=6.5 K

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Example
The most critical component in the example above has the footprint 10x10 mm and dissipates 1.2 W. It has Rjb=15 K/W and Rjc=12 K/W. The PCB measures 200x200x1.6 mm, is placed on 20 mm spacing, cooled by 1.5 m/s air velocity and has a thermal conductivity of 15 W/mK. The PCB temperature below the component is 82 C when the room temperature is 50 C. Estimate the temperature increase at 3000 m altitude.

Solution
Assume that the air velocity remains unchanged. Open the heat transfer coefficient calculator and determine the coefficient for the inlet temperature difference definition. It reads 14 W/m2K. This is an average value and it is approximately valid for a position in the centre of the PCB. Assume it can be used.
Open the thermal territory calculator. Use the heat transfer coefficient above and a top factor of 2. The thermal efficiency for the thermal territory is 85%.
(therm terr maximum dT)=82-50 =32 K
(therm terr average dT) =32*0.85=27.2 K
Display the component image. The case temperature is 96.5 C and 11% of the heat is dissipated from the top.
(component top dT)=96.5-50=46.5 K
(average convection surface dT)=46.5*0.11+27.2*(1-0.11)=29.3 K
Accept the altitude multiplier above, 1.29.
(dT increase)=(1.29-1)*29.3=8.5 K