# Sizing a Heat Exchanger for Cold Plate Applications

## Calculating the Thermal Resistance of a Heat Exchanger

In many liquid cooling loops, the heat that is picked up by a cold plate is rejected to ambient air via a heat exchanger. Figure 1 shows a typical liquid cooling loop, consisting of a cold plate (CP), pump, and heat exchanger (HX) connected by hoses or tubing. Since the components are part of a system, it is important to select them together to ensure proper component sizing for your application. Manufacturers typically provide performance data for cold plates and heat exchangers individually, with cold plate performance in thermal resistance and heat exchanger performance in thermal capacity. So how do you select the optimal heat exchanger and cold plate for the complete system? It is easier than you might think, since the equations needed to determine the right cold plate and heat exchanger combination reduce to a very simple format: To arrive at this equation, the first step is to calculate the cold plate thermal resistance, θCP, which is defined as the difference between the maximum required surface temperature, TS, MAX, and fluid exit temperature, TH, divided by a heat load, Q, evenly distributed over the entire cold plate surface:  Similarly, heat exchanger thermal capacity, CHX, which is defined as the heat load, Q, divided by the temperature differential between two incoming fluids, TH -TAIR, is described by the following equation: Thermal capacity is also equal to the inverse of thermal resistance: Assuming no heat gains from the pump or heat losses through connecting hoses or tubing between the cold plate and heat exchanger (since these are usually minor), equations (2), (3) and (4) can be combined into one simple equation: Hot process fluid temperature TH has dropped out of the formula because liquid temperature has been removed from the equation, we do not have to calculate flow rates and heat capacities of the liquid. We are just left with the desired surface temperature of the cold plate, as well as the temperature of ambient air cooling the heat exchanger, and the performance is fully characterized by the thermal resistances of the cold plate and heat exchanger. Therefore, we no longer have to analyze the individual components of the system. Instead we determine the thermal resistance of the entire system. Note that the effect of flow is not excluded from the results because it is already incorporated within thermal resistance values.

A customer wants to use a CP12 a 12" (30.48 cm) cold plate (plate side), to remove 1200 W of heat from a 12"x5" (30.48 cm x 12.70 cm) electronic device. The coolant is 1 gpm (3.785 LPM) of water and room temperature is 20°C. The customer wants the smallest heat exchanger that will remove 1200 W of heat generated by this device, while maintaining a maximum surface temperature of 80°C.

Step 1: First we determine system thermal resistance, θSYSTEM:

Step 2: Any combination of cold plates and heat exchangers that provide a thermal resistance less than or equal to the total system requirement will work. In other words:

Step 3: Table 1 shows the resistance and flow rates of the CP12 cold plate and two different heat exchanger/fan combinations:  #### Table 1:

 Flow Rate (gpm) θCP (CP12) (°C/W) θHX (6110 w/Kona fan) (°C/W) θHX (6210 w/Marin Fan) (°C/W) 0.5 0.013 0.049 0.019 1.0 0.009 0.046 0.017 1.5 0.007 0.044 0.016 1.5 0.007 0.044 0.016 2.0 0.006 0.042 0.016

Table 1 shows that the CP12/6110 combination satisfies the 0.050 °C/W condition at 2 gpm (0.006 +0.042 = 0.048

By looking at the system as a whole, we start to see trade offs between the components, including how flow rate can impact heat exchanger selection. At low flow rates, cold plate thermal resistance increases. This requires a larger heat exchanger with more thermal capacity, and therefore lower thermal resistance. At higher flow rates, it is possible to use a smaller heat exchanger.

Liquid-to-air heat exchangers and cold plates are often combined in a fluid circuit, so it is important to understand how to select the components simultaneously to optimize your system's performance. With accurate specifications and a simplified equation, selecting the components in your liquid cooling loop can be relatively straightforward. In addition, by selecting components from the same thermal vendor, you use components that are tested in a similar manner and are more likely to work well as a system.

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