FYKOS Year 34 Series 4 Problem 4 — ants · FYKOS 34 #34.4.4

Physics · thermodynamics · optics

Problem

The ants have a peculiar way of keeping the anthill warm – they crawl out, let the sunlight heat them up, and then crawl back in, where the heat is transferred to the anthill. The anthill can be approximated as a cone of height

H=0.8\,\mathrm{m}

with base radius of

R_0=1.5\,\mathrm{m}

. The walls are made of cellulose with heat conductivity

\lambda = 0.039\,\mathrm{W\cdot m^{-1}\cdot K^{-1}}

and are

2\,\mathrm{cm}

thick. Assume that the entire heat exchange between the anthill and its surroundings (which have temperature

T_{\mathrm{o}} = 10\,\mathrm{^\circ\mskip-2mu\mathup{C}}

) is only mediated by the ants and by the conduction of heat through the walls, i.e. neglect the heat exchange with the ground. An ant weighs

m =5\,\mathrm{mg}

and has a specific heat capacity of approximately

4~000\,\mathrm{J\cdot kg^{-1}\cdot K^{-1}}

. How many ants, heated up to

T_{\mathrm{m}} = 37\,\mathrm{^\circ\mskip-2mu\mathup{C}}

, have to enter the anthill every second in order to keep the inner volume of the anthill at constant temperature of

T_{\mathrm{M}} = 20\,\mathrm{^\circ\mskip-2mu\mathup{C}}

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