Problem

As an egg cooks, the proteins first denature and then coagulate. When the temperature exceeds a critical point, reactions begin and proceed faster as the temperature increases. In the egg white, the proteins start to coagulate for temperatures above $63^\circ\rm{C}$, while in the yolk the proteins start to coagulate for temperatures above $70^\circ\rm{C}$. For a soft boiled egg, the white needs to have been heated long enough to coagulate at a temperature above $63^\circ\rm{C}$, but the yolk should not be heated above $70^\circ\rm{C}$. For a hard-boiled egg, the center of the yolk should be allowed to reach $70^\circ\rm{C}$.

The following formula expresses the time $t$ it takes (in seconds) for the center of the yolk to reach the temperature Ty (in Celsius degrees):

\[t = \frac { M^{2/3} ~ c ~ \rho^{1/3} } { K\pi^2 ~ (4\pi/3)^{2/3} } ~\ln \bigg[ 0.76 \frac{T_0 - T_w}{T_y-T_w} \bigg]\]

where $M$ is the mass of egg, $\rho$ is the density, $c$ is the specific heat capacity, and $K$ is thermal conductivity. Relevant values are $M=47~[g]$ for a small egg and $M=67~[g]$ for a large egg, $\rho=1.038 ~[g~cm^{-3}]$, $c = 3.7 ~[J~g^{-1}~K^{-1}]$, and $K = 5.4\times10^{-3} ~[Wcm^{-1}K^{-1}]$. Furthermore, $T_w$ is the temperature (in C degrees) of the boiling water, and $T_0$ is the original temperature (in C degrees) of the egg before being put in the water. Implement the formula in a Python/MATLAB program, set $T_w = 100^\circ C$ and $T_y = 70^\circ C$, and compute $t$ for a large egg taken from the fridge ($T_0 = 4^\circ C$) and from room temperature ($T_0 = 20^\circ C$).

Python

Hint: Note that for this problem you will need to import log, pi from math module in Python, although these are intrinsic in MATLAB.

Solution

MATLAB

A MATLAB version of the code can be downloaded from here.

Python

A Python version of the code can be downloaded from here.

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