In part 1 of our 3 part series, we covered why and how natural convection works and how a heat sources geometry can affect natural convection. Today, in part 2, we want to address board orientation and its impact on natural convection cooling and useful equations that can be used for calculating initial and simplistic results based on viable assumptions.
For maximum heat transfer in applications where the desired cooling method is natural convection, boards with power dissipating devices should be stacked in the vertical plane. Packing boards in the vertical direction increases the spatial consolidation of the boards and increases heat transfer per unit area due to optimized natural convection.
For a single plate suspended in the vertical configuration, the resulting velocity profile will be parabolic. There is no interference with any other disturbances; the only controllable variables are adjusting the height, width and power of the board. When multiple boards are stacked to create vertical channels, the velocity profiles interact. Their interaction is a function of the spacing, or channel width, between the boards. Figure 2 is an adapted graph showing the heat dissipation of a set of two parallel isothermal plates as a function of channel width at a temperature difference of 20°C and a constant board area of 0.078m2. A and B are design points where the slope of the line connecting to the origin represents the amount of power per unit of channel width. Point B is shown to have a maximum slope, which represents the maximum possible packing density for a particular configuration. In order to maximize power density for isothermal boards, an engineer should aim to design for point B. Note that past the optimum and toward point A there is no longer any potential gain in power consumption as the channels are widened.
Figure 2. Heat Dissipation from Isothermal Boards as a Function of Board Spacing in Natural Convection .
The relationship discerning board optimization for isothermal boards can be described  as:
S(opt) = optimal board spacing
H = board height
[delta]T = temperature difference (degree C) for a 50 degree C Ambient
For the configuration of two isothermal plates, the average Nusselt number across a plate can be described by the following equation :
R(a) = the Rayleigh number – a non-dimensional number associated with the magnitude of buoyancy flow
H(b) = board height (m)
S(b) = board surface area (m2)
For a channel between two plates with equal heat flux, the hottest — and thus the most critical point of interest — rests at the end of the board nearest the exhaust of the channel. In this particular configuration, the optimal spacing can be defined as :
Ra(s) = the Rayleigh number
S = board spacing (m)
H = board height
Likewise, a calculation for the local heat transfer coefficient at the end of a set of plates with uniform heat flux can be found as :
q”(s) = heat flux generation
T(s) = surface temperature at board edge
T(a) = ambient air temperature
S = board spacing
K(f) = thermal conductivity of the fluid
The above equations are useful for calculating initial and simplistic results based on viable assumptions. A stack of boards can often be modeled as plates where there are less significant heat source points or protrusions from the board, or where the thermal conductivity of the board is high enough to assume isothermal characteristics. For some cases, memory modules aligned in channel-like configurations can be modeled as a set of plates with uniform heat flux – the heat transfer characteristics can be estimated from the above correlations. Similarly, the fins of a heat sink in natural convection can be modeled as isothermal plates which create channels. For most heat sinks, their material conductivity is high enough to negate any significant temperature gradients along the fins.
In part 3 we’ll cover the calculations in generic plate fin heat sinks, and conclude our 3 part series on How to Apply Natural Convection Cooling for the Thermal Management of Electronics.
2. Malhammar, A., Optimum Sized Air Channels for Natural Convection Cooling, Telecommunications Energy Conference, 1987.
3. Incropera, F., Liquid Cooling of Electronic Devices by Single Phase Convection. New York, Wiley. 1999.