Page 19 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

INTRODUCTION
A Taylor series expansion results in
∂Q x 11
Q x+dx = Q x + x
∂x
∂Q y
Q y+dy = Q y + y
∂y
∂Q z
Q z+dz = Q z + z (1.32)
∂z
Note that the second- and higher-order terms are neglected in the above equation. The
heat generated in the control volume is G x y z and the rate of change in energy storage
is given as
∂T
ρ x y zc p (1.33)
∂t
Now, with reference to Figure 1.4, we can write the energy balance as

inlet energy + energy generated = energy stored + exit energy
that is,
∂T
G x y z + Q x + Q y + Q z = ρ x y z + Q x+dx + Q y+dy + Q z+dz (1.34)
∂t
Substituting Equation 1.32 into the above equation and rearranging results in

∂Q x ∂Q y ∂Q z ∂T
− x − y − z + G x y z = ρc p x y z (1.35)
∂x ∂y ∂z ∂t
The total heat transfer Q in each direction can be expressed as

∂T
Q x = y zq =−k x y z
x
∂x
∂T
Q y = x zq =−k y x z
y
∂y
∂T
Q z = x yq =−k z x y (1.36)
z
∂z
Substituting Equation 1.36 into Equation 1.35 and dividing by the volume, x y z,
we get
∂ ∂T ∂ ∂T ∂ ∂T ∂T
k x + k y + k z + G = ρc p (1.37)
∂x ∂x ∂y ∂y ∂z ∂y ∂t
Equation 1.37 is the transient heat conduction equation for a stationary system expressed
in Cartesian coordinates. The thermal conductivity, k, in the above equation is a vector. In
its most general form, the thermal conductivity can be expressed as a tensor, that is,

 
k xx k xy k xz
k =  k yx k yy k yz  (1.38)
k zx k zy k zz