Page 190 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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As before, a Taylor series expansion may be used to express the energy convected out
of the control volume in both the x 1 and x 2 directions as
∂(u 1 T) CONVECTION HEAT TRANSFER
ρc p u 1 T x 2 + ρc p x 1 x 2 (7.38)
∂x 1
and
∂(u 2 T)
ρc p u 2 T x 1 + ρc p x 2 x 1 (7.39)
∂x 2
Note that the specific heat, c p , and density, ρ, are assumed to be constants in deriving
the above equation. The heat diffusion into and out of the control volume is also derived
using the above approach. The heat diffusing into the domain in the x 1 direction (Fourier’s
law of heat conduction) is
∂T
x 2 (7.40)
x 2 q 1 =−k x 1
∂x 1
and the diffusion entering the control volume in the x 2 direction is
∂T
x 1 q 2 =−k x 2 x 1 (7.41)
∂x 2
Using a Taylor series expansion, the heat diffusing out of the control volume can be
written as
∂T ∂ ∂T
x 2 + (7.42)
−k x 1 −k x 1 x 2 x 1
∂x 1 ∂x 1 ∂x 1
in the x 1 direction and
∂T ∂ ∂T
x 1 + (7.43)
−k x 2 −k x 2 x 1 x 2
∂x 2 ∂x 2 ∂x 2
in the x 2 direction. Finally, the rate of change of energy within the control volume is

∂T
x 1 x 2 ρc p (7.44)
∂t
Now, it is a simple matter of balancing the energy entering and exiting the control
volume. The energy balance can be obtained as

‘heat entering the control volume by convection + heat entering
the control volume by diffusion = heat exiting the control volume
by convection + heat exiting the control volume by diffusion +
rate of change of energy within the control volume’.
Following the above heat balance approach and rearranging, we get

∂T ∂(u 1 T) ∂(u 2 T) 1 ∂ ∂T ∂ ∂T 2
+ + = k x 1 + k x 2 (7.45)
∂t ∂x 1 ∂x 2 ρc p ∂x 1 ∂x 1 ∂x 2 ∂x 2