Page 219 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
The convection time step t c is identical to that of Equation 7.120. The diffusion time
steps contain two parts. One due to the kinematic viscosity and another to the thermal
diffusivity of the fluid. The diffusion time step may be expressed as 211
2 2
h h
t d = min , (7.165)
2ν 2α
where ν is the kinematic viscosity and α is the thermal diffusivity. The local element size
may be calculated using the same procedure as that discussed in Section 7.5. However, a
more advanced method of the calculation of element size, for example, an element size in
the streamline direction, is possible and readers are referred to the appropriate publication
(Tezduyar et al. 2000).

7.6.3 Boundary and initial conditions

The two main boundary conditions prevalent in heat convection problems are the prescribed
temperature, pressure and velocity (Dirichlet conditions) and flux boundary conditions
(Neumann conditions). Other possibilities may be derived from these conditions.

Prescribed values If a value of the velocity components, temperature or pressure is given
at a boundary node, the value will be ‘forced’ at these nodes. The implementation is easy
and straightforward.
Flux conditions In a heat transfer calculation, it is possible to have prescribed heat flux
conditions, which are normally given as
∂T
−k = q (7.166)
∂n
where n is the normal direction to the surface on which the prescribed flux boundary
is imposed. The heat flux condition is imposed by rearranging {f 4 } (Equation 7.157) as
follows:
1
 

{f 4 }= q 1 (7.167)
 
2
0
Often, symmetric (or zero flux) boundary conditions are employed in convection heat
transfer calculations. In such cases, the forcing vector terms disappear. Other relevant
boundary conditions will be discussed along with appropriate examples later in this chapter.
In many industrial heat transfer applications, convection heat transfer boundary condi-
tions are common. If a boundary, as shown in Figure 7.14, is convecting to the atmosphere,
then the boundary condition on this wall can be expressed as
∂T
−k = h c (T − T a ) (7.168)
∂n
where the wall temperature T is unknown. The implementation is carried out by replacing
q (Equation 7.167) by the right-hand side of the above equation. However, T must be
treated as an unknown and should be evaluated at each time step.