A Solution of Fourier's Equation for a Sphere and Comparison with the Experimental Cooling Data

  We can compare out cooling data with the way a sphere of water of comparable size will cool according to Fourier's Equation. To simplify our calculations we need to convert our data to the cgs system of units so we can use the thermodynamic constants for water.

From the fit of the cooling data we can determine some of the constants that we need to solve Fourier's Eqn.

Since we are doing a rough we can arbitrarily set hR=2 which makes the Λ in Fourier's boundary condition equal to -1. The diameter of the sphere is taken to be 10 cm so R=5 cm. From the values of ε which satisfy the boundary condition we get the values of μ for the terms of the series and can expand the function ΔT(r,0)=ΔT₀, the initial temperature difference from T_a, to obtain the coefficients of the terms. Because the ε_k were not regularly spaced the coefficients were obtains by evaluating the table of functions sin(μr)/μr for each term and using a least squares method to minimize the deviation of the series from f(r,0)=1.

We can calculate the solutions for T(r,Δt) with steps of Δt=20 min to get the following plot. In the equation we have to use Δt given in seconds because we're working in cgs units. The solution at Δt=0 sec is a little wiggly because of the Gibbs phenomenon which prevents getting the series for ΔT(r,0)=ΔT₀ to match exactly at r=5 cm.

The best match of our cooling data with the Fourier solution is with the thermometer readings taken at a position that is about equal to 4.2 cm from the center of the sphere and the assumption of an ambient temperature, T_a, equal to about 23 °C.

In our cooling experiment the measuring cup holding the water would have offered some resistance to the heat trying to escape. The measuring cup was microwave safe so its composition was probably similar to that of Pyrex. In the sphere of water some of the water would be needed to model the resistance to the flow of heat by the measuring cup.

Edit (Mar 27): Corrected the misstatement T(r,0)=T₀ with ΔT(r,0)=ΔT₀ which was used in the calculation of the set of cooling curves.