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The egg boiling problem is a seemingly simple problem posted from the blog of UChicago's physicist Dam Thanh Son.

egg

A chicken's egg has a length of 5 cm and takes 6 minutes to boil. An ostrich's egg has a length of 15 cm. How much time does it take to boil an ostrich's egg? (Photo for illustrative purpose only)

Since the blog doesn't offer further clarification, I made some assumptions and arrived at the following interpretation:

Assume two spheres made of the same materials and have the same initial temperature $T_0$. At time $t_0$ the surrounding temperature is $T > T_0$. If sphere 1 has diameter $D_1 = 5\, \text{cm}$ and takes $t_1 = 5\, \text{min}$ for the center of the sphere (thanks knzhou) to reach temperature $T_0 < T_2 < T$. What is the time for the center of sphere 2 which has diameter $D_2 = 15\, \text{cm}$ to reach temperature $T_2$?

According to the blog, the answer is $(\frac{15}{5})^{2} \times 6 = 54$ minutes. How did they arrive at this?

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  • $\begingroup$ It neglects shell thickness, which anyone who has cooked them knows is substantial. Also, the yokes are very large, more than two thirds of the volume. I would have thought that like cooking anything, it is about time for heat to get to the middle. That would probably be proportionate to the cube of the radius..? $\endgroup$ – CriglCragl Jun 28 '18 at 20:54
  • $\begingroup$ Your formulation is not quite right; the center will never reach the same temperature as the surroundings. It's like how exponential decay never quite gets to zero. $\endgroup$ – knzhou Jun 28 '18 at 21:33
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    $\begingroup$ Still, if you put some fixed standard like, "the egg is done when the center is 80% heated up to the surrounding temperature" then the problem works out, and the solution is as given. $\endgroup$ – knzhou Jun 28 '18 at 21:34
  • $\begingroup$ @knzhou I changed the formulation a bit. According to the blog, eggs are considered boiled if the center reaches temperature ~80Celcius. The results should still hold, correct? $\endgroup$ – Minh Tran Jun 28 '18 at 22:19
  • $\begingroup$ Actually it takes 90 minutes to hard boil an ostrich egg. $\endgroup$ – MaxW Jun 29 '18 at 13:38
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The temperature distribution within each sphere as a function of time and radial position is governed by the transient heat conduction equation (in spherical coordinates): $$\frac{\partial T}{\partial t}=\alpha\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial T}{\partial r}\right)$$where $\alpha$ is the thermal diffusivity. Boundary and initial conditions are:

$T=T_0$ at t = 0, all r

$T=T_1$ all t, at r=R

$r^2\frac{\partial T}{\partial r}=0$, all t, r = 0 These equations can be reduced to dimensionless form by means of the following substitutions: $$\theta=\frac{T-T_0}{T_1-T_0}$$ $$\rho=\frac{r}{R}$$ $$\tau=\frac{\alpha t}{R^2}$$ With these substitutions, the equations become:$$\frac{\partial \theta}{\partial \tau}=\frac{1}{\rho^2}\frac{\partial}{\partial \rho}\left(\rho^2\frac{\partial \theta}{\partial \rho}\right)$$ $\theta = 0$ at $\tau=0$, all $\rho$

$\theta = 1$ all $\tau$, at $\rho=1$

$\rho^2\frac{\partial \theta}{\partial \rho}$, all $\tau$, $\rho = 0$

Note that there are no adjustable parameters in the dimensionless differential equation and boundary conditions. So, to achieve a specific value of the dimensionless temperature $\theta$ at the center of both spheres requires a specific value of dimensionless time $\tau=\tau^*$. In terms of the actual time, this would be $$t=\frac{R^2}{\alpha}\tau^*$$Assuming that the thermal diffusivities of the two egg materials are the same, this means that the time is proportional to the square of the egg radius R.

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    $\begingroup$ We must assume that the egg contents is semi-solid, so that diffusion rather than convection determines the heat profile. Diffusion constant is length squared per second, and dimensional analysis fills in the rest. $\endgroup$ – Whit3rd Jun 28 '18 at 23:59
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It looks like the assumption is being made that the time for the egg to cook is proportional to the square of the diameter.

$$T=pd^2$$

Where $p$ is the proportionality constant. Therefore

$$T_{chicken}=pd_{chicken}^2$$

$$T_{ostrich}=pd_{ostrich}^2$$

Or solving both for $p$

$$p=\frac{T_{chicken}}{d_{chicken}^2}=\frac{T_{ostrich}}{d_{ostrich}^2}$$

Therefore

$$T_{ostrich}=(\frac{d_{ostrich}^2}{d_{chicken}^2})*T_{chicken}=54\space min$$

So the question then is why are we assuming this proportionality? The easiest thing I can think of is that they are assuming this time is proportional to the surface area of the egg. So then $p=a*\pi$ where $a$ is some other factor dealing with the heat transfer from the water to the egg. This makes sense. More surface area allows for more heat transfer.

I would think that the volume should also come into play too though. It would take longer time to heat up something with more volume. If anyone can give guidance to this point I can adjust my answer accordingly.


Thanks to @BowlOfRed:

The time to cook is proportional to $\frac{energy\space needed\space to\space boil}{rate\space of\space heat\space exchange}$ (energy/(energy/time))->time).

The energy needed is going to proportional to the volume of the object (more stuff means more energy needed). This brings in a $d^3$ dependency.

The rate of heat exchange is proportional to both the area ($d^2$) of the surface as well as the temperature difference per unit length ($1/d$) (faster energy transfer if we have more energy and larger temperature differences over shorter lengths).

Therefore our time $T$ for heating is proportional to $\frac{d^3}{d^2/d}=d^2$

Then we can go through the above work to get to the final answer.

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  • $\begingroup$ I would assume this is a heat flow problem. If the time for heat to reach the center of a chicken egg is X, what is the time for the center of an ostrich egg to reach the same temperature? I'm bad at thermodynamics so I can't answer, but I think that's the idea. (Often the answer for questions like these is "ask someone who knows." Engineers should rely on proven technique first, their own testing second and fall back to pure calculation only as last resort.) $\endgroup$ – markspace Jun 28 '18 at 20:03
  • $\begingroup$ @BowlOfRed Thanks! I just added this to the answer $\endgroup$ – Aaron Stevens Jun 28 '18 at 21:15
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We want to show the time for cooking is proportional to $R^2$, where $R$ is the radius; then the ostrich egg will take $9$ times as long to cook.

You can get this intuitively by considering how the heat flow scales with the object size, but it also can be proven by dimensional analysis. The only things that can matter are $$\text{radius} \ R \sim [\text{m}], \quad \text{heat capacity } C \sim [\text{J/K}], \quad \text{thermal conductivity } \kappa \sim \left[ \frac{\text{J}}{\text{K} \cdot \text{m} \cdot \text{s}} \right].$$ I'm not including temperatures here, because the initial temperature, the temperature of the boiling water, and the "goal" temperature are the same for both eggs. Since the heat equation is linear that means any dependence on these temperatures will drop out when we take a time ratio.

The only way to get a time period $T$ is to have $$T \sim \frac{C}{R \kappa}$$ but since $C$ is proportional to the volume, $T$ is proportional to $R^2$ as desired.

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protected by Qmechanic Jun 29 '18 at 3:17

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