q x
y
∂y
q x
dx
∂q x
+ ∂x
q
z
x
y
E st
E g
q y
∂q z
z
q +
dz
∂q y
z
∂z
q +
dy
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7
Heat Conduction Equations
FIGURE 1.6
The volume element for deriving the heat conduction equation.
conservation in a 1-D system (x-direction only),
∂q x
q x − q x +
dx + E g = E st
(1.9)
∂x
The conduction heat rates can be evaluated from Fourier’s Law,
∂T
q x = −kA x
(1.10)
∂x

With control surface area A x = dy dz, Equation 1.9 can be written as

∂
∂T
−
−k dy dz
dx + E g = E st
(1.11)
∂x
∂x
The thermal energy generation can be represented by
E g = q ˙ dx dy dz
(1.12)
where q ˙ is the energy generation per unit volume dx dy dz.
The energy storage can be expressed as
∂(ρ dx dy dz · C p · T)
E st =
(1.13)
∂t
Substituting Equations 1.12 and 1.13 into Equation 1.11, we have
∂
∂T
∂(ρ dx dy dz · C p · T)
−
−k
dx dy dz + q ˙ dx dy dz =
(1.14)
∂x
∂x
∂t