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🔬 The Physics of Perfect Eggs
Welcome to the wonderful realm of physics-informed egg cooking, just like 2nd year thermal physics questions! This app uses the exact solution to the thermal diffusion equation for a spherically symmetric egg, following the derivation by Dr Charles D. H. Wnilliams.
The Thermal Diffusion Equation
The temperature distribution in the egg follows the spherical heat equation:
\[\frac{\partial T}{\partial t} = \frac{\kappa}{C\rho} \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial T}{\partial r}\right)\]
Where:
- T = temperature (K)
- t = time (s)
- κ = thermal conductivity (W/(m·K))
- C = specific heat capacity (J/(kg·K))
- ρ = density (kg/m³)
- r = radial distance from centre (m)
The Solution Method
Using the substitution \(T(r,t) = T_1 + B(r,t)/r\), we transform this into a 1D diffusion equation:
\[\frac{\partial B}{\partial t} = D \frac{\partial^2 B}{\partial r^2}\]
where \(D = \kappa/(C\rho)\) is the thermal diffusivity. The boundary conditions are:
- \(B(0,t) = 0\) (finite temperature at centre)
- \(B(R,t) = 0\) (surface at water temperature)
- \(B(r,0) = r(T_0 - T_1)\) (initial condition)
Fourier Mode Expansion
We expand in sine waves: \(B = \sin(kr) \exp(-Dk^2t)\). The boundary condition \(B(R,t)=0\) requires \(kR = n\pi\), giving:
\[B(r,t) = \frac{2R}{\pi}(T_1-T_0) \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sin\left(\frac{n\pi r}{R}\right) \exp\left(-D\left(\frac{n\pi}{R}\right)^2 t\right)\]
Temperature at the Centre
Taking the limit \(r \to 0\), using \(\lim_{r\to 0} \sin(n\pi r/R)/r = n\pi/R\):
\[T(0,t) = T_1 + 2(T_1-T_0)\sum_{n=1}^{\infty} (-1)^n \exp\left(-\frac{n^2\pi^2 t}{\tau_0}\right)\]
where the thermal time constant is \(\tau_0 = C\rho R^2/\kappa = R^2/D\).
The "Williams Formula" (first Fourier mode)
For \(t > 0.1\tau_0\), the first Fourier mode dominates, and solving for when the yolk-white boundary (at \(r=0.69R\)) reaches temperature \(T_{\rm yolk}\) gives:
\[t_{\rm cooked} = \kappa M^{2/3} \ln\left[0.76 \frac{T_{\rm egg}-T_{\rm water}}{T_{\rm yolk}-T_{\rm water}}\right]\]
where \(\kappa = C\rho^{1/3}/[\pi^2 \kappa (4\pi/3)^{2/3}]\). This is the formula used by this app!
Key Physical Insights
🔹 Cooking time scales as \(M^{2/3}\) (radius squared), not volume!
🔹 The egg acts as a low-pass thermal filter - high-frequency modes decay rapidly
🔹 Temperature approaches equilibrium exponentially with time constant \(\tau_0\)
🔹 Smaller eggs equilibrate faster (shorter \(\tau_0\))
Altitude Effects
At higher altitudes, atmospheric pressure decreases, lowering the boiling point of water approximately as:
\[T_{\rm boil} \approx 100°\text{C} - \frac{h}{300°\text{C/m}}\]
This means eggs cook slower at altitude since the water temperature is lower!
Egg Thermal Properties
Values from Polley et al. (1980) via Williams:
- Yolk: c = 2.7 J/(g·K), κ = 3.4×10⁻³ W/(cm·K), ρ = 1.032 g/cm³ → κ_Williams = 31 s·g⁻²/³
- White: c = 3.7 J/(g·K), κ = 5.4×10⁻³ W/(cm·K), ρ = 1.038 g/cm³ → κ_Williams = 27 s·g⁻²/³
- Average: c ≈ 3.2 J/(g·K), κ ≈ 4.4×10⁻³ W/(cm·K), ρ ≈ 1.035 g/cm³ → κ_Williams ≈ 29 s·g⁻²/³
This app uses κ = 29 s·g⁻²/³ (average), which targets the yolk-white boundary temperature.
References
This implementation follows the derivation in:
C.D.H. Williams, "The Science of Boiling an Egg", University of Exeter (1998-2006)
Made with ❤️ and ∂/∂t by a physics enthusiast!