When thermally stacking several heatsinks, in particular in combination with larger power electronics assemblies, the reduction in coolant flow rate resulting from the increased pressure drop and pre-heating of the coolant for the "backward" units has to be considered in the calculations.

**Figure 1.** *a) Individual cooling; b) Thermal stacking with forced air cooling*

The following two methods are suitable for calculating pre-heating:

a) Determining a thermal impedance Z _{th(a-a’)} between measuring points at the heatsink 1, 2 and 3

b) Calculating coolant pre-heating; the coolant outlet temperature from the first module is the inlet temperature at the second one, etc.

__Determining an additional thermal impedance__

Pre-heating is determined by total power dissipation P_{tot(n)} of the heat source, the stationary thermal resistance R_{th(a-a*)}, or the transient thermal impedance Z_{th(a-a*)} between two adjacent heatsinks (Figure 1b). To this end, the temperature differences between the heatsink temperatures must be determined at a given power dissipation. As time passes, the second and any other heatsink will get warmer than the unit directly in front. This temperature difference divided by the component power dissipation results in Z_{th(a-a*)}. For this part of the transient thermal impedances, 1 R/τ element is sufficient in most cases. The following known correlation applies to component 1 in the direction of coolant flow:

To cater for component 2, an additional element for the temperature difference between heatsink 1 ("a") and 2 ("a*") is introduced. This pre-heating depends on the losses of component 1, which is why all of the losses must be weighted. If the losses of all heat sources are identical, this may be omitted:

For component 3 and any additional one, this applies by analogy:

__Calculating pre-heating for air cooling__

The basic idea behind this method is to retain the well proven basic equations of temperature calculation and redetermine the coolant inlet temperature for the "n-th" element only: This can easily be done using the heat retention capability of the coolant, if even through-heating is assumed. Both the specific weight and the heat retention capability are temperature dependent, which is why there is a temperature coefficient for pre-heating.

**Figure 2.** *Thermally stacked, air-cooled layout, split into sectors with different air pre-heating*

The general formula is:

**c _{p}:** Specific heat capacity of air [kJ/K/kg]

**ρ:**Density of air [kg/m³]

**TC**Temperature coefficient of the specific heat capacity

_{c}:**T**Coolant temperature for the second heat source

_{a}*:**P**Power dissipation of heat source 1

_{tot1}:Adapted to an average atmospheric pressure of 1 bar, a basic temperature of 0°C and the conversion to a specification for the volumetric flow rate in [m³/h], the following results:

**P _{tot1} [W]:** Power dissipation of heat source 1

**V**Volumetric flow rate through the heatsink

_{air}[m³/h]:**K**Correction factor for uneven heat profile at the outlet of heatsink 1

_{ing}:**Figure 3.** *Uneven temperature profile of the exiting air over the fan cross section if heat sources are arranged centrally*

**Figure 4.** *Temperature difference between incoming and outgoing air as a function of power dissipation P _{V1} (W) and the volumetric flow rate, T_{a }= 40°C*

__Calculating pre-heating for water cooling__

In principle, the same basic equation applies to thermal stacking for liquid cooling as for air cooling. However, it must be borne in mind that the dynamic viscosity changes as a function of the temperature. The heat retention characteristic for a 50:50 water/glycol mixture and a conversion to the volumetric flow rate in [l/min] results in:

For pure water, the specific heat capacity and the temperature coefficient will change to:

**Figure 5.** *Temperature difference between water/glycol mixture flowing in/out as a function of the power dissipation and the volumetric flow rate, T _{a} = 25°C, 50% glycol*

**For more information, please read:**

Heat Transfer in Power Semiconductor Devices

Cooling Methods for Power Semiconductor Devices

Thermal Modeling of Power Module Cooling Systems

Heat Dissipation and Thermal Resistance in Power Modules

Thermal Equivalent Circuit Diagrams for Junction Temperature Calculations