Advanced thermal solutions – innovations in thermal management

R s stands for the spreading resistance that is non-zero when the heat sink base is larger than the component. The next few sections show the full analytical solution for calculating spreading resistance, followed by an approximate simplified solution and the amount of error from the full solution and finally the use of these solutions to model and optimize a heat sink.

Lee et al. [2] derived an analytical solution for the spreading resistance. Figure 2 shows a cross-section of a circular heat source with radius a on the base with radius b and thickness t. The heat, q, originates from the source, spreads out over the base and dissipates into the fluid on the other side with heat transfer coefficient, h. For heat transfer through finned heat sinks, the effective heat transfer coefficient is related to thermal resistance of the fins, R f as shown in Equation 3.


For square heat source and plates, the values of a and b can be approximated by finding an effective radius as shown in equations 4 and 5.

The derivation of the analytical solution starts with the Laplace equation for conduction heat transfer and applying the boundary conditions. Equation 6 shows the final analytical solution for spreading resistance. The values for the eigenvalue can be computed by using the Bessel function of the first kind at the outer edge of the plate, r=b as shown in Equation 7.

Simons [3] compared the full solution (Equations 6 and 7) with the approximations shown in (Equations 8 and 9). The problem contained a 10 mm square heat source on a 2.5 mm thick plate with a conductivity of 25 W/mK, 20 mm width and varying length, L as shown in Figure 3. Figure 4 shows that the percentage error increases with length but stays relatively low. Less than 10% error is expected for lengths up to 50 mm; five times the length of the heater. This is acceptable for most engineering problems since analytical solutions are first-cut approximations that should later be verified through empirical testing and/or CFD simulations. However, the full analytical solution should be used if the heater-to-heat sink base area difference gets much larger or if a more accurate solution is desired.

In the Qpedia Issue 99 article, “Minimizing Thermocouple Errors in Electronics Thermal Characterization,” we examined how the use of thermocouples can alter the measurements that are made, whether it is because the thermocouple itself acts as a fin that dissipates heat, or any of a number of other reasons [1]. We saw that even use of a very fine 36-gauge thermocouple wire can introduce measurement error of more than 5%.

The influence of the sensor on the measurement can be even more challenging as the item under test becomes smaller and more sensitive to ambient influences. As an example, a thin film made of carbon nanotubes presents an extreme challenge, as the film is only 100 nm thick, or about 1/1000 of the diameter of an average human hair. At this scale, the material has practically no thermal mass, so the temperature of the film can easily be affected by most temperature transducers. In order to create a sensor small enough to measure temperatures on the thin film, Shrestha et al. chose to use a glass micropipette as a base [2]. A sensor was then fabricated by creating a junction of dissimilar metals, using the same thermoelectric principles as a thermocouple.

Thermocouple sensors had been made using micropipettes previously, but past examples used materials that were much more expensive, and the construction was very complex. The platinum and gold materials were chosen because of their use in a biological environment, but as a thermocouple junction, they had a low voltage output in response to temperature differences [3,4]. The sensors that Shrestha et al. created were meant to address these issues.

Glass micropipettes can be made with tips so small that they are commonly used to inject substances into individual living cells. The tip diameters that Shrestha et al. created ranged from 2 to 30 µm. The micropipettes were made using a micropipette puller which heated a 1.5mm glass tube and then pulled the tube apart, creating a finely tapered section (see Figure 1a). The micropipettes were then filled with a tin based soldering alloy (Figure 1b), and a micropipette beveler was used to sharpen and achieve the finished tip shape (Figure 1c).

The next step in the construction was to coat the outside of the micropipette with a nickel film using a sputtering process (Figure 1d). On the end of the tip where the tin was exposed, the deposition of nickel created the working thermocouple junction. The deposition conditions were varied to create different film thicknesses, which we will examine later. Finally, lead wires made of tin and nickel were attached to the respective materials in the sensor (Figure 1e).

The sensors were calibrated in a water bath at temperatures from 21 to 40°C, while the cold junction was maintained at a constant 24.5°C. The voltages generated by the sensors at each temperature were recorded using a Nano voltmeter, as the measurement scale was in the range of 100µV. The calibration curves for some of the sensors are shown in Figures 2 and 3 below, where a linear response is clearly demonstrated. It can also be seen that the thicker layer of nickel significantly increased the sensitivity of the sensor, more than doubling its voltage output for a given temperature.

Clearly these are not quite like the standard thermocouples that we use for standard thermal measurements, and the setup and fabrication of the sensors is more involved. The payoff is sensors that have an extremely fast response time to temperature changes, only a few microseconds in air [3], as well as very little thermal influence on the samples they are testing, which is critical at the small scales that Shrestha et al. examined. In addition, Shrestha et al. determined that the measurement accuracy of their sensors was 0.01°C, which compares very favorably to a standard J-type thermocouple. Even with the “special limits of error” grade wire, J-type accuracy is only about 1ºC. [5]

This approach may be excessive for most electronics thermal management purposes, but it does illustrate that even something as familiar as a thermocouple can be approached in many different ways. The sizes of these sensors allow them to make temperature measurements with pinpoint precision, and to identify very small heat sources and heat flow paths. One such application could be localizing hot spots on a CPU die, for example. Such small sensors could generate a temperature map by traversing the surface of a die.

Another important factor is the amount of liquid in the heat pipe which is commonly called the fill ratio or inventory. If there is too much liquid, evaporation will not happen or delayed, and if there is not enough liquid, the dry out condition will happen. The rule of thumb is the volume of the liquid should be higher than the volume of the pore volume of the wick.

Figure 3 shows that as the pressure decreases from 10 Torr to 1 Torr the Q max increases. The graph also shows that as the inventory (fill ratio) increases from 0.7 ml to 1.1 ml, Q max peaks at 0.8 ml. This corresponds to a fill ratio of 26.4%, which is the ratio of the liquid volume to total volume of the heat pipe when it is empty. This graph shows the importance of fill ratio. If the fill ratio is not optimized as is shown for example for 1 Torr, Q max drops from 8 W to 4 W, a 50% drop that can be catastrophic for the application. Mozumder et al. [3], in their experiment, measured the thermal resistance of a heat pipe for different fill ratios and power.