A soldering iron's copper tip is a thin rod of length along the -axis

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Temperature, heat and the macroscopic behaviour of matter

Mathematical Methods for Physics, Ch The classical partial differential equations of physics and their boundary conditions

Problem

A soldering iron's copper tip is a thin rod of length

L = 0.15\,\mathrm{m}

along the

x

-axis, from

x=0

(embedded in the heating element) to

x=L

(the free tip exposed to air). The heating element holds the rod at

x=0

fixed at

T_h = 320\,{}^\circ\mathrm{C}

for all time. At the free end

x=L

, the rod loses heat to the surrounding air, held at a constant

T_{\mathrm{amb}} = 20\,{}^\circ\mathrm{C}

, by convection: Newton's law of cooling states that the heat flowing out through unit area of that end equals

h\,(T(L,t) - T_{\mathrm{amb}})

, with convective heat-transfer coefficient

h = 25\,\mathrm{W\,m^{-2}\,K^{-1}}

. The rod's thermal conductivity is

K = 400\,\mathrm{W\,m^{-1}\,K^{-1}}

(copper), and its temperature obeys Fourier's law

q_x = -K\,\partial T/\partial x

for the heat flux along it. ((a)) Write the boundary condition on

T(x,t)

x=0

, and name its type. ((b)) By equating the conductive flux leaving the rod at

x=L

to the convective loss there, derive the boundary condition on

T

x=L

, and name its type. ((c)) Compute the dimensionless ratio

hL/K

for this rod. Using the two limits

h\to\infty

and

h\to0

of the condition found in part (b), say which of those two limits this rod's tip is closer to, and what that means physically for how effectively the tip's heat is carried away by the air.

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