## 5 Cooling Calculations and Cooling System Design
### Cooling Profiles  
The heatsink material must show optimum thermal conductivity and heat dissipation (high coefficient of thermal conductivity \(\lambda\)) at reasonable material and processing costs. Aluminum is therefore usually preferred (\(\lambda\) = 247 W/K\(\cdot\)m for pure Al). But copper can also be used to meet specifically high requirements (\(\lambda\) = 398 W/K\(\cdot\)m). The dependence of the heat dissipation on the manufacturing process and the alloy used is striking; in practice, the \(\lambda\)-values of coolers range from 150 W/K\(\cdot\)m (Al-die cast alloy) to 220 W/K\(\cdot\)m (AlMgSi extruded material). The heat dissipation in the material has a significant influence on the thermal efficiency of the heatsink [17].  
Therefore, optimized sizing for root thickness, number of fins, fin height, and fin thickness is critical: The fins of an air cooler are used to dissipate most of the heat to the environment by convection. The root of a cooler is the finless part of the mounting surface for power modules where heat is dissipated [17]. To determine the optimum conditions for forced air-cooled chiller profiles, heat conduction and convection can also be integrated through the fin height pattern and calculated by the following formula:  
\[R_{th(s-a)}=\frac{1}{n\cdot\sqrt{\propto\cdot U\cdot\lambda\cdot A\cdot\left[ \frac{1}{1+e^{-2k}}-\frac{1}{1+e^{2k}}\right]}}\]  
where \(k=h\cdot\sqrt{\frac{s\cdot U}{\lambda\cdot A}}\)  
(\(\alpha\): heat-transfer coefficient, U: fin circumference, \(\lambda\): coefficient of thermal conductivity of the heatsink material, A: cross-section of fins, h: fin height).  
Fans generate the air flow required for air cooling. Different fan types are used depending on the kind of cooler and application (Figure 14):  
Figure 13: Liquid cooling module with IGBTs in place.  
Axial flow fans run with the axis of rotation of the axial rotor parallel to the air flow. The air is moved along the axial rotor, which moves similar to an air screw. The advantages of axial flow fans are their relatively small size relative to the high air flow rate processed. Their disadvantage is the increase in pressure compared to radial flow fans. Radial flow fans (Figure 14) [17], in contrast to axial flow fans, are used where a higher pressure rise is critical for the same amount of air. Air is sucked in parallel or axial to the drive axis of the radial flow fan and blown out in a radial direction, deflected 90\({}^{\circ}\) by the rotation of the radial rotor. In order to minimize pressure losses due to the high outlet velocity of the air exiting the radial flow fan, care must be taken to maintain air channeling, for instance, by using a diffuser. Cross-flow or tangential fans have an inlet and blowing slot along their entire length. Air is sucked from the inlet slot into the interior of the rotor, where it is swirled, deflected, and blown out extremely homogeneously. Cross-flow fans provide a high air flow rate even at low speeds and can therefore be configured to generate relatively low noise. The rotor length and outlet slot are matched to the heatsink width [17]. The energy consumption of the cooling system is associated with two components: the pump and the fan. In conventional cooling systems, the pump and the fan are controlled with a thermostat. In new studies, the thermal loads of all components of the vehicle are calculated and, simulated, and control strategies are developed [14].