Cooking Time Of Chickens

Relevant wiki: Dimensional Analysis

Conceptual solution

Suppose the bigger chicken has a radius that is $\alpha$ times bigger than the smaller chicken.

Since the volume is $\alpha^3$ times as big, we have to heat up $\alpha^3$ times more mass, requiring $\alpha^3$ times more heat, and therefore $\alpha^3$ as much time if the heat flow rate were the same.

However, the heat flow rate will be greater: because the area of the chicken is $\alpha^2$ times bigger, heat will flow $\alpha^2$ times faster, if the heat flow density were the same.

But because all distances are $\alpha$ times bigger, the temperature gradient will be less by a factor $\alpha$ , and since the heat flow density is proportional to the temperature gradient, heat will flow $\alpha$ times slower.

Together, the time needed to heat up the chicken is multiplied by $\alpha^3\cdot \alpha^{-2}\cdot \alpha = \alpha^2;$ in this problem clearly $\alpha = \sqrt[3]2$ , so that the answer is $60\cdot (\sqrt[3]2)^2 \approx \boxed{95}$ minutes.

Dimension analysis solution

The constants dimensions in this problem are

• the density $[\rho] = \text{kg}/\text{m}^3$ ;

• the specific heat capacity $[c] = \text{J}/\text{kg}\cdot\text{K}$ ;

• the conductivity $[\kappa] = (\text{J}/\text{m}^2\cdot\text{s})/(\text{K}/\text{m}) = \text{J}/\text{K}\cdot\text{m}\cdot\text{s}$ .

Multiplying and dividing to cancel all units except kg and s, we find that $\frac{[\rho]\,[c]^3}{[\kappa]^3} = \frac{\text{s}^3}{\text{kg}^2}$ is a constant relating time and mass. This shows that for cooking the chicken, $t^3 \propto m^2$ so that if the mass doubles, the cooking time is multiplied by $(2^2)^{1/3}$ .

Heat equation solution

This problem is described by the heat equation, $\frac{\partial T}{\partial t} =k\nabla^2 T,$ with $k = \kappa / \rho c$ . Multiplying space coordinates by $\alpha$ would cause the second space derivative to be divided by $\alpha^2$ ; to compensate, the time coordinates must be multiplied by $\alpha^2$ . In this problem, of course, $\alpha = \sqrt[3]{2}$ .