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
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) |
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expresses the heat balance. Now |
(3.3a) |
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units: Watts |
while |
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(3.3b) |
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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,
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so that |
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From Fourier's law,
, we have |
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Therefore |
(3.4) |
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Substituting (3.3a) and (3.4) into (3.2), gives |
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Dividing by
, and then letting
, we get |
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and Equation (3.1). |
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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).
Copyright
1998-2001
Rensselaer Polytechnic Institute. All Rights Reserved.