05 Jun

Sizing a heat sink can be daunting tasks for any one who does not have much experience in thermal analysis. There are commercially available software that would allow you to design and analyze a heat sink to meet the thermal requirements of  the device(s) to be cooled. If that type of software is not available to you some quick calculations can be done to get an estimate of the size of the heat sink required.

By making  a few simplifying assumptions you can conduct the heat sink analysis by hand or using a spreadsheet. The output of these calculations will be the dimensions of the heat sink required to maintain the required source temperature.

Figure 1. Plate fin heat sink dimensions

Figure 1 shows a typical plate fin heat sink used to cool common electrical and electronic components such as LEDs used in lighting applications, MOSFET used in digital circuits and microprocessors. There are six dimensions that would need to be determined to design an appropriate heat sink for your needs. In order to reduce the complexity of the calculations the following assumptions will be made:

1. The surface area due to the thickness of the fins t, and thickness of the base b are much much smaller than the total surface area of the heat sink
2. The thermal conductivity of the heat sink is high enough so that the temperature of the surface of heat sink is uniform and approximately equal to the temperature of the heat source
3. The heat source has the same length and width of the heat sink and is centered on the base of the heat sink
4. The source is in perfect contact with the base of the heat sink

The above assumptions will introduce some errors in your calculations. However the purpose of conducting this calculation is to get a rough estimate of the size of the required heat sink. More sophisticated calculation methods, software or testing can then be used to refine the design.

This analysis is for a heat sink whose base is oriented vertically with cooling via natural convection only. The first step of the calculation is to select values for L and H based on your design constraints. The optimum spacing between the fins s that produces the maximum heat transfer due to natural convection is given by equation 1. A detailed explanation of how this equation was derived can be found in [1].

$Latex formula$1
where:
$Latex formula$is the temperature of the heat source
$Latex formula$is the ambient temperature
$Latex formula$is the acceleration due to gravity
$Latex formula$is the expansion coefficient with the temperature units in Kelvin

$Latex formula$
$Latex formula$is the thermal diffusivity of air evaluated at $Latex formula$
$Latex formula$is the kinematic viscosity of air evaluated at $Latex formula$

The convection heat transfer coefficient[1] is given by equation 2,

$Latex formula$2

where $Latex formula$ is the thermal conductivity of air evaluated at $Latex formula$. It is a measure of how well heat is removed from the surface of the heat sink via convection.

The law of conservation of energy dictates that the heat generated by the heat source Q must be equal to the heat dissipated by the heat sink under steady state conditions.  This is represented in equation 3.  The right hand side of equation 3 which accounts for the heat dissipated by the heat sink is derived from Newton’s law of cooling.  The surface area A of the heat sink is given by equation 4. The thickness of the fins, t  and base, b are not included in the surface area calculation since one of the assumptions made is that these dimensions are negligible. The number of fins, N is calculated by combining equations 3 and 4. The width, W of the heat sink is then calculated using equation 5.

$Latex formula$3

$Latex formula$4

$Latex formula$5

This calculation can be easily entered into a spreadsheet allowing you to quickly evaluate the effect of dimensional changes.

This calculation provides some incite into what dimensions should be varied to optimize the size of the heat sink. If you are trying to minimize the volume of the heat sink the length should be made a small as possible. This will maximize the heat transfer from the heat sink as such reducing the surface area required to limit the source temperature below the required value.

References:
[1] A. Bar-Cohen, W. M. Rohsenow  “Thermally Optimum Spacing of Vertical, Natural Convection Cooled, Parallel Plates”, in: Journal of Heat Transfer, Vol 106, p. 116-123, 1984