Size of marine mammals — Phénomènes de transport 2 : Transfert thermique par conduction

Marine mammals are warm-blooded animals whose temperature remains constant. To maintain this temperature, exothermal reactions happen in their cells and produce a volumetric power

𝑎

. The total power produced by the mammal is

𝑃

To simplify, mammals are described as spheres of radius

𝑅

, immersed in immobile water of thermal conductivity

𝜆

and whose temperature is

𝑇_{\infty}

far away from the animal.

The problem is supposed to be invariant through a rotation about

𝜃

and

𝜑

1/ Establish the thermal diffusion equation in water (in spherical coordinates).

Coup de pouce 1

Faire un bilan d’énergie sur un volume infinitésimal ou une boule creuse.

Corrigé

By applying the first principle of thermodynamics on a spherical shell of radius $𝑟$ and thickness $d 𝑟$ :

d^{2} 𝑈 + d^{2} 𝐸_{𝑐} = 𝛿^{2} 𝑄 + 𝛿^{2} 𝑊

with $d^{2} 𝐸_{𝑐} = 0$ and $𝛿^{2} 𝑊 = 0$ .

\begin{aligned} 𝜇 d 𝑉 (𝑢 (𝑡 + d 𝑡) - 𝑢 (𝑡)) & = 𝑗_{𝑄} (𝑟) 4 𝜋 𝑟^{2} d 𝑡 - 𝑗_{𝑄} (𝑟 + d 𝑟) 4 𝜋 {(𝑟 + d 𝑟)}^{2} d 𝑡 \\ 𝜇 4 𝜋 𝑟^{2} d 𝑟 \frac{𝜕 𝑢}{𝜕 𝑡} d 𝑡 & = - \frac{𝜕 𝑗_{𝑄} 𝑟^{2}}{𝜕 𝑟} 4 𝜋 d 𝑟 d 𝑡 \end{aligned}

We then use Fourier’s law $𝑗_{𝑄} = - 𝜆 \frac{𝜕 𝑇}{𝜕 𝑟}$ and the caloric equation $d 𝑢 = 𝑐 d 𝑇$ to obtain :

\begin{matrix} 𝜇 𝑐 \frac{𝜕 𝑇}{𝜕 𝑡} = 𝜆 \frac{1}{𝑟^{2}} \frac{𝜕 \frac{𝜕 𝑇}{𝜕 𝑟} 𝑟^{2}}{𝜕 𝑟} \\ \frac{𝜕 𝑇}{𝜕 𝑡} - 𝐷_{th} \frac{1}{𝑟^{2}} \frac{𝜕 \frac{𝜕 𝑇}{𝜕 𝑟} 𝑟^{2}}{𝜕 𝑟} = 0 \end{matrix}

with $𝐷_{th} = \frac{𝜆}{𝜇 𝑐}$ the thermal diffusivity of water.

2/ Ascertain the temperature

𝑇 (𝑟)

around the animal in stationary state.

Coup de pouce 1

Résoudre l’équation précédente en régime stationnaire.

Corrigé

In stationary state, the equation becomes :

\begin{aligned} \frac{1}{𝑟^{2}} \frac{𝜕 \frac{𝜕 𝑇}{𝜕 𝑟} 𝑟^{2}}{𝜕 𝑟} & = 0 \\ \frac{𝜕 \frac{𝜕 𝑇}{𝜕 𝑟} 𝑟^{2}}{𝜕 𝑟} & = 0 \end{aligned}

Integrating gives :

\begin{matrix} \frac{𝜕 𝑇}{𝜕 𝑟} 𝑟^{2} = 𝐴 \\ \frac{𝜕 𝑇}{𝜕 𝑟} = \frac{𝐴}{𝑟^{2}} \\ 𝑇 (𝑟) = - \frac{𝐴}{𝑟} + 𝐵 \end{matrix}

with $𝐴$ and $𝐵$ constants to be determined using the boundary conditions :

$𝑇 (𝑟 \to \infty) = 𝑇_{\infty}$ implies $𝐵 = 𝑇_{\infty}$ ,
$𝑇 (𝑟 = 𝑅) = 𝑇_{0}$ implies $𝐴 = (𝑇_{0} - 𝑇_{\infty}) 𝑅$ .

Finally, the temperature profile is :

𝑇 (𝑟) = (𝑇_{0} - 𝑇_{\infty}) \frac{𝑅}{𝑟} + 𝑇_{\infty}

3/ Determine the thermal power lost by the mammal by integrating

{\vec{𝑗}}_{𝑄}

Coup de pouce 1

La puissance perdue par l’animal est le flux thermique sortant de l’animal en

𝑟 = 𝑅

Corrigé

The thermal flux at the surface of the animal is given by Fourier’s law :

𝑗_{𝑄} (𝑅) = - 𝜆 \frac{𝜕 𝑇}{𝜕 𝑟} (𝑟 = 𝑅) = 𝜆 (𝑇_{0} - 𝑇_{\infty}) \frac{𝑅}{𝑅^{2}} = 𝜆 \frac{𝑇_{0} - 𝑇_{\infty}}{𝑅}

The total power lost by the animal is then :

𝑃 = 𝑗_{𝑄} (𝑅) 4 𝜋 𝑅^{2} = 4 𝜋 𝜆 𝑅 (𝑇_{0} - 𝑇_{\infty})

Coup de pouce 1

Déterminer la puissance volumique produite par l’animal et montrer qu’elle est d’autant plus grande que l’animal est petit.

Corrigé

The volumetric power produced by the animal is given by :

𝑎 = \frac{𝑃}{\frac{4}{3} 𝜋 𝑅^{3}} = 3 𝜆 \frac{𝑇_{0} - 𝑇_{\infty}}{𝑅^{2}}

This power increases when the radius of the animal decreases. Small animals thus need to produce more power to maintain their temperature, which is not sustainable for them. This explains why there are no small aquatic mammals.