Page 191 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
Differentiating the convection terms by parts and substituting Equation 7.11 (continuity)
into Equation 7.45, we obtain the simplified energy equation in two dimensions as
∂T ∂T ∂T 1 ∂ ∂T ∂ ∂T 183
+ u 1 + u 2 = k x 1 + k x 2 (7.46)
∂t ∂x 1 ∂x 2 ρc p ∂x 1 ∂x 1 ∂x 2 ∂x 2
, the energy
If the thermal conductivity is assumed to be constant and k = k x 1 = k x 2
equation is reduced to
!
2
2
∂T ∂T ∂T ∂ T ∂ T
+ u 1 + u 2 = α + (7.47)
∂t ∂x 1 ∂x 2 ∂x 2 ∂x 2
1 2
where α = k/ρc p is called the thermal diffusivity. The energy equation in vector form is
∂T
2
+ u.∇T = α∇ T (7.48)
∂t
and in indicial form
2
∂T ∂T ∂ T
+ u i = α (7.49)
∂t ∂x i ∂x 2
i
The above equation is applicable in any space dimension.

7.3 Non-dimensional Form of the Governing Equations

In the previous section, we discussed the derivation of the Navier–Stokes equations for
an incompressible fluid. In many heat transfer applications, it is often easy to generate
data by non-dimensionalizing the equations using appropriate non-dimensional scales. To
demonstrate the non-dimensional form of the governing equations, let us consider the
following two-dimensional incompressible flow equations in dimensional form:
Continuity equation

∂u 1 ∂u 2
+ = 0 (7.50)
∂x 1 ∂x 2
x 1 momentum equation

2 2 !
∂u 1 ∂u 1 ∂u 1 1 ∂p ∂ u 1 ∂ u 1
+ u 1 + u 2 =− + ν + (7.51)
∂t ∂x 1 ∂x 2 ρ ∂x 1 ∂x 2 ∂x 2
1 2
x 2 momentum equation

2 2 !
∂u 2 ∂u 2 ∂u 2 1 ∂p ∂ u 2 ∂ u 2
+ u 1 + u 2 =− + ν + (7.52)
∂t ∂x 1 ∂x 2 ρ ∂x 2 ∂x 2 ∂x 2
1 2