HEAT TRANSFER MODEL

SUMMARY

Using the 1D heat equation, the developed simulation platform was used to predict the cooking time of different eggs. The model proposes a cooker that keeps water in a pot at boiling point temperature (100 degrees Celsius). The platform is based on a few key simplifications:

  1. Eggs are perfectly spherical with radius R

  2. The ‘material’ of an egg is homogeneous, meaning that the shell, white, and yolk have the same thermal conductivity.

  3. The boiling water is a perfect heat source, it remains at 100 degrees Celsius at the boundary of the egg

Assuming symmetrical heating, the criteria for a cooked egg is defined as keeping the temperature at greater than or equal to 80 degrees across each point of the radius for longer than ten seconds:

The boundary and initial conditions used were as follows:

The 1D heat equation can be written in 3D spherical coordinates to account for the geometry of the eggs:

Assuming symmetrical heat dissipation in every direction, the expression becomes:

Developing the Simulation Platform

Using the above expression and the explicit-time advancement method, a numerical approximation was derived and used as the basis for the model:

MATLAB Implementation and Results

The constructed numerical approximation was implemented into MATLAB, and in conjunction with other parameters, was used to develop a simulation platform. To test the model, the cooking time of three different types of eggs, each varying in radii, was simulated. A table comparing the cooking time gained from the simulation result with the recommended cooking time for different eggs as found online are shown below:

Examples of temperature gradients and plots from different trials are shown in the photos below:

The discrepancy between the simulated cooking times and the recommended cooking times found online can be attributed to a variety of factors:

  • The simulation simplifies the geometry of an egg, in turn making the foundation of the model less precise than it could be. The ovoid shape of an egg is often asymmetrical, thus having notable effects on how heat is transferred through the egg.

  • The eggs in the simulation are assumed to be homogeneous. Not accounting for the difference between the heat diffusivity per egg layer can lead to less accurate results. It was also assumed that each type of egg only differed in terms of radius, and possessed the same heat diffusivity coefficient, which is likely not the case.

  • The value of the heat diffusivity coefficient itself also plays a significant role in the acquired results. Different coefficient values are used across different experiments depending on certain factors and can vary somewhat drastically. In each case, this parameter can have significant effects on the simulated cooking time.

  • Symmetrical heat dissipation was assumed across every direction of the egg, which would most likely not be the case in a real-world scenario.

Finding ways to account for these factors in a feasible manner would ultimately lead to the development of a more accurate model. Additionally, more complex advancement schemes could also be used to improve precision.

Analysis and Future Improvements