Derivation of the differential equation for the steady state temperature distribution in the rod

Let us assume that the surface r = R is perfectly insulated. That makes
 
 
We also assume that heat flows only in the axial (z) direction and that the thermal conductivity k is constant.

It will be shown that the temperature distribution is governed by the equation
(3.1)
 
where Qgen denotes the rate at which heat is generated per unit volume (W/m3) within the rod.

To begin the derivation choose an arbitrary value of z, and slice the rod with planes perpendicular to the axis at positions z and , where is assumed small. Now consider the solid circular cylinder shown, of radius R and thickness


Figure 1

We shall now derive equation (3.1) by formulating the "heat balance", which relates Qgen to the rate at which heat is conducted along the rod.

Let Hgen denote the rate at which heat is generated within the finite cylinder of length previously shown, and let Hb denote the net rate of heat flow outward through the surface of the cylinder. Then under steady state conditions
(3.2)
 
expresses the heat balance. Now
(3.3a)
units: Watts
while    
(3.3b)
units: Watts
where q(z) is the heat flux (in W/m2) at z in the positive - direction. Remember that no heat flows through the lateral surface r = R because it is insulated.

By Taylor's theorem,
 
 
 
 
so that
 
From Fourier's law, , we have
 
 
Therefore
(3.4)
 
Substituting (3.3a) and (3.4) into (3.2), gives
 
 
Dividing by , and then letting , we get
 
 
and Equation (3.1).  

Analysis Link I presents an alternative derivation to equation (3.1) using vector calculus. It gives the derivation of the partial differential equation for heat conduction with sources and sinks located within the region where heat conduction is under study. (Analysis Link I - equation 20). That equation is then simplified to obtain equation (3.1).


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