Consider the problem of conduction of heat in rods. Assume the rod to be thin so that the

temperature at all points of the section may be taken to be the same. This problem may be

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taken as one of linear flow in which the temperature is specified by the time and the distance,

$x,$ measured along the rod. Assume that each element of the surface of the rod loses heat by

radiation to a surrounding medium. Assume the rod to have the following parameters:

diffusivity $(),$ conductivity $(K),$ density $(),$ constant cross$-$sectional area $(),$ specific heat

$($c$),$ and surface conductance $(H).$ Assume the rod lies along the $x$ axis and consider that the

element of volume is bounded by sections at $x$ and $x+dx.$

i$.$ What is the rate at which heat flows into the element over the face at $x?$

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ii$.$ What is the rate at which heat flows across the face at $x+dx?$

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iii. What is the rate at which the element gains heat by flow across the two faces?

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iv$.$ What is the rate at which heat is lost by radiation at the surface?

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v$.$ Obtain the differential equation that describes the distribution of temperature

across the element.

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vi$.$ Simplify the differential equation at point $v$ for the situation where the rod is

impervious to heat transfer $($Hint: No radiation heat transfer takes place$).$

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vii. Simplify the differential equation at point $v$ for the situation where radiation takes

place into a medium at constant temperature $($Hint: you may take the temperature

of the medium as the zero of your scale$).$

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viii. Simplify the differential equation at point vii by using the substitution $T=$

$u{e}^{-vt},$ where $u$ is temperature which is a function of $x,$ and $v$ is the kinematic

viscosity of the medium.