
174 Heat Transfer: Thermal Management of Electronics
obtained by taking the limit of Equation 8.10 as both Δx and Δt approach zero.
Note that
lim lim
∆
∆
∆
∆
∆
∆
x
xx xx
t
QQ
x
Q
x
T
t
T
→
+
→
−
=
∂
∂
=
∂
00
and
∂∂t
. (8.11)
Therefore, if both Δx and Δt approach zero, Equation 8.10 becomes
(8.12)
Finally, substituting Fourier’s law of heat conduction for
gives
−
∂
∂
−
∂
∂
+=
∂
∂
1
Ax
kA
T
x
gC
T
t
xx p
ρ .
(8.13)
Note that A
x
= A and is constant in this one-dimensional analysis. Also, since heat
conduction is in one direction only, the index x is removed from thermal conductivity
and k will be used to represent thermal conductivity in the direction ...