A room across has a radiator against one wall and a single-glazed window in the…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Convection, rotation and geophysical flows

Fluid Dynamics, Ch Vorticity: its sources, its field and the motion of vortices

Problem

A room

L=6\,\mathrm{m}

across has a radiator against one wall and a single-glazed window in the opposite wall, with the ceiling

H=3\,\mathrm{m}

above the floor. Take a material circuit of perimeter

2(L+H)

that runs up the radiator wall, across the ceiling, down the window wall and back across the floor. At every height the air next to the radiator is

\Delta T = 8\,\mathrm{K}

warmer than the air next to the window, both otherwise near

T=293\,\mathrm{K}

. The pressure is

p_1=101325\,\mathrm{Pa}

at floor level and

p_2=101290\,\mathrm{Pa}

at ceiling level. Treat the air as an ideal gas with specific gas constant

R=287\,\mathrm{J\,kg^{-1}\,K^{-1}}

and use Bjerknes's circulation theorem in the same form used for the sea breeze,

\mathrm{d}\Gamma/\mathrm{d}t = R\ln(p_1/p_2)\,\Delta T

. ((a)) Find the rate at which circulation is generated around the circuit. ((b)) Express this as an acceleration averaged around the perimeter. ((c)) If nothing opposed the motion, what speed would be reached after two minutes?

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