Introduction

The purpose of this analysis is to estimate the temperature change of a thermistor attached to an RC car driving through a thermal plume. Students in a new class will be required to design an RC car capable of completing specific challenges within an RC arena, and one such challenge will be to drive over a heat source and detect a change in temperature. The concern is that already purchased 25 x 50 cm heaters that reach 100 degrees Celsius may not be sufficient to produce a large enough temperature difference given an RC car's relative velocity.

Method

The general approach to this problem was to model the temperature distribution of the heater using convection, and subsequently finding the heat transfer of the distribution to the thermistor using forced convection and lumped capacity. Dimensions of the RC car's thermistor height and the thermistor diameter were measured, and thermistor property values were found online. A thermal plume was modeled using an advection diffusion partial differential equation. This plume was solved numerically to create a two dimensional slice of the plume's temperature profile, shown in Figure 1.

From this profile, a one dimensional slice at the thermistor's height was taken and fit with a Gaussian profile to model the temperature versus distance, as shown in Figure 2. This curve fit was integrated into the formula for lumped capacitance and solved numerically for five different car velocities shown in Figure 3.

To calculate the heat transfer coefficient over the thermistor, the Whitaker correlation was implemented using the thermistor property values.

Theory

The steady-state advection-diffusion equation used to model the thermal plume of the heater is

v_z(z,T) \frac{\partial T}{\partial z} = \alpha \left( \frac{\partial^2 T}{\partial r^2} + \frac{1}{r}\frac{\partial T}{\partial r} \right) \tag{1}

Where v_z(z,T) is the upward velocity, \frac{\partial T}{\partial z} is the thermal height gradient, \alpha is the thermal diffusivity, \frac{\partial^2 T}{\partial r^2} is the radial heat curvature which drives the radial diffusion, r is the radius, and \frac{\partial T}{\partial r} is the radial heat gradient. It is important to note that this model assumes that there is no radial velocity due to fluid motion, and the only radial heat transfer comes from diffusion. This is a non-physical assumption, but is a simplification that must be made for the scope of this analysis. Similarly, this model does not account for turbulence.

The approximation used to model the heat velocity based on height is

v_z(z,T) \approx \sqrt{2gz\beta (T-T_{\infty})}\tag{2}

Where g is the gravity, z is the height, \beta is the thermal expansion coefficient, T is the temperature, and T_\infty is the freestream temperature. This equation follows the general form of Torricelli's law but adds temperature as a driving factor for upward velocity.

Results

Because the advection-diffusion equation does not model turbulence, an effective diffusivity \alpha_{eff} was introduced as \alpha_{eff}=\zeta \alpha, where \zeta is a correction factor to account for the effects of turbulence. The assumption of laminarity at the height of z=5 \ cm would be reasonable were it not for the air disruption of the RC car's translation. Results accounting for the higher diffusivity as a result of turbulence are shown below in Figure 1.