Page 189 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
in the x 1 direction and
2
2

∂u 2 ∂u 2 ∂u 2 ∂p ∂ u 2 ∂ u 2 181
ρ + u 1 + u 2 =− + µ + (7.33)
∂t ∂x 1 ∂x 2 ∂x 2 ∂x 2 ∂x 2
1 2
in the x 2 direction. In vector notation, the momentum equations can be written as
∂u

ρ +∇.(u × u) = ∇.[−pI + τ] (7.34)
∂t
or, in indicial form
!
2
∂u i ∂u i ∂p ∂ u i
ρ + u j =− + µ (7.35)
∂t ∂x j ∂x i ∂x 2 i
Note that the above equation is applicable in any dimension.
7.2.3 Energy equation

The energy equation can be derived by following a procedure similar to the momen-
tum equation derivation. However, the difference here is that the temperature, or energy
equation, is a scalar equation. In order to derive this equation, let us consider the control
volume as shown in Figure 7.6. The energy convected into the control volume in the x 1
direction is

ρc p u 1 T x 2 (7.36)
Similarly, the energy convected into the control volume in the x 2 direction is

ρc p u 2 T x 1 (7.37)

∂q 2
q +∆x 2 + ....
2
∂x 2
∂[rc u T]
p 2
rc u T +∆x 2 + ....
p 2
∂x
2
∂q 1
q 1 q +∆x 1 + ....
1
∂x 1
∆x 2
∂[rc u T]
p 1
rc u T ∆x ru T +∆x 1 + ....
p 1
1
1
∂x
1
rc u T
p 2
q 2
Figure 7.6 Infinitesimal control volume in a flow field. Derivation of conservation of
energy