Optimal Cooling Fin Array Design of a Transient RC Car Speed Controller

Objective

For transient heat cycles and given spatial constraints, design a passive cooling fin array for an RC car speed controller by optimizing surface temperature and array weight.

Outcome

Using the lumped capacity approximation model, a 4th-order Runge-Kutta scheme, and Monte Carlo methodology, an optimal aluminum rectangular-straight fin array was selected. The maximum temperature of the fin array is 46.5 degrees Celsius and the weight is 0.30 kilograms.

Constraints & Parameters

The proposed RC vehicle will create peak power for a short burst, followed by a rest time. The vehicle's speed controller will generate heat due to the internal resistance, and it is important that the controller be kept below a maximum operating temperature; preliminary tests indicate that an unmodified speed controller will overheat. The maximum surface temperature is given to be 50 degrees Celsius, and the freestream temperature is assumed to be 25 degrees Celsius.

For this model, the full-power burst will last 30 seconds, and subsequently rest for 30 more seconds to complete one cycle. There will be 10 cycles required for the model to remain under the maximum surface temperature, but reaching a heat limit underneath the maximum is highly desired. It is also highly desirable to keep the controller as cool as possible, while simultaneously keeping the weight as low as possible for the quickest RC car acceleration.

There is a maximum fin clearance above the speed controller of 45 mm, and the fins have a maximum base coverage of 30%. The fins also have a minimum thickness of 1.2 mm for structural integrity. The convection coefficient will depend on the fin base area coverage, given by this assumed model:

h = h0 - C(% coverage)^2, where h0 = 50 W/m^2 K and C = 500 W/m^2 K

Solution Model

Lumped capacity is an effective approximation for this design problem because fins are thin and chosen materials are likely to have very high thermal conductivity. Calculating many configurations of the Biot number often returned values of less than 0.1, signifying that lumped capacity is valid; this was reaffirmed with the final design once chosen.

Included in the lumped capacity model is the heat source (from the speed controller), radiation, and convection from the fins and base area, as shown below.

It should be noted that conduction is not included in this model; the core assumption of lumped capacitance is that conduction through the object is negligible. However, radiation is included and its heat equation contains temperature to a fourth-order power. Consequently, solving this ordinary differential equation analytically is currently impossible and a numerical technique must be utilized and fourth order Runge-Kutta was the selected technique due to its proven accuracy.

Optimization Method

To obtain quick results before analyzing the physics, the Monte Carlo method was utilized. The Monte Carlo method leverages statistics by simply setting bounds within all desired parameters, and runs a specified number of iterations to assign each parameter a random variable within those bounds. Since the objective is to minimize the fin temperature, the minimum solution is selected. The plot below shows 10,000 iterations of the Monte Carlo method:

Note that this Monte Carlo plot only considered aluminum as the fin material, although other Monte Carlo plots were generated with many different material types. The minimum solution shown quickly reaches a steady-state oscillation, and could handle an infinite number of the given cycles without increasing the maximum temperature. Shown below are the quantities for the optimized parameters and the dimensions of the fin for context:

Among the lowest values of Monte Carlo outputs, the base coverage remained tightly grouped from about 17 to 20%, while the fin thickness and depth seemed to vary greatly. The fin length remained constant, around its maximum possible length. Of course, the Monte Carlo method here optimized for temperature and not weight, so it was important to scrutinize the output fin dimensions and also the fin material.

Conclusion

Out of the materials tested, copper and aluminum were the only two realistic materials able to keep the maximum temperature under 50 degrees Celsius. It also became clear that triangular fins were not as effective at removing heat as rectangular straight fins.

It can be seen from the plot above that copper is clearly superior to aluminum, and rectangular configurations are more temperature-efficient. However, copper is over three times as dense as aluminum. For weight optimization, it would be most ideal to select the aluminum triangular fins, but the maximum temperature of 50 degrees Celsius is exceeded. Therefore, the aluminum rectangular fins were selected to attain the temperature goal and to reduce the fin array weight as much as possible. Ultimately, the array weighed 0.3 kilograms, or 0.66 pounds.

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