Evaluate your solution using python As a homework assignment (problem set #3, first problem), you generated a full solution for transient heat transfer in a sphere. The dimensionless solution that you (hopefully) found is given by (r, t) = _n=1 [2(1)ⁿ¹/n] * (sin (nr)/ r) * exp (n22t) where (r, t) = (T(x, t) T1) / (T0 T1), r = r/R, t = t/R2, R is the sphere radius, and is its thermal diffusivity. Evaluate your solution using python and answer the following questions:

1) Generate a python code to calculate and plot the value of (r, t) as a function of time near the center (r = 0.1), midway (r = .5), and near the edge (r = 0.9) of the sphere. Make the number of modes (maximum value of n) a parameter that can be easily changed within your code. Output your graph at intervals of t no larger than 0.01 and plot your solutions over a time from t = 0 to 0.6. Show your plot and answer the following questions: How many terms did you need to use to generate an accurate solution? Give the number and explain the definition of accurate that you utilized.

2) Modify your python code to calculate the time required at each radial position for the temperature () to drop 5% from its initial value of 1. Plot the time (tp for time of penetration) versus position for at least 20 positions between r = 0.1 and 1.0.

3) Your book argues that a one-term approximation is sufficient for evaluating the temperature as a function of position and time for a Fourier number (Fo) greater than 0.2. Recalling that the dimensionless time is the Fourier number (Fo= t), modify a code to compare and plot a comparison of the full solution and one-term approximation. Show the comparison for the case where the comparison is most likely to fail (r = 0.1, 0.5, or 0.9). State which position is approximated the worst and then comment on the argument that a Fourier number (Fo) greater than 0.2 is sufficient.