136 Chapter 2 The Heat Equation
Figure 1 Rodofheat-conductingmaterial.Figure 2 Slice cut from rod.The rate of heat storage in the slice of material is proportional to the rate of
change of temperature. Thus, ifρis the density andcis the heat capacity per
unit mass([c]=H/mT), we may approximate the rate of heat storage in the
slice by
ρcA x∂u
∂t(x,t),whereu(x,t)is the temperature.
There are other ways in which heat may enter (or leave) the section of rod
we are looking at. One possibility is that heat is transferred by radiation or
convection from (or to) a surrounding medium. Another is that heat is con-
verted from another form of energy — for instance, by resistance to an elec-
trical current or by chemical or nuclear reaction. All of these possibilities we
lump together in a “generation rate.” If the rate of generation per unit volume
isg,[g]=H/tL^3 , then the rate at which heat is generated in the slice isA xg.
(Note thatgmay depend onx,t,andevenu.)
We have now quantified the law of conservation of energy for the slice of
rod in the form
Aq(x,t)+A xg=Aq(x+ x,t)+A xρc∂u
∂t. (1)After some algebraic manipulation, we have
q(x,t)−q(x+ x,t)
x +g=ρc∂u
∂t.The ratio
q(x+ x,t)−q(x,t)
x