Page 193 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

185
CONVECTION HEAT TRANSFER
Where Re is the Reynolds number defined as
Re = u a L (7.59)
ν
and Pr is the Prandtl number given as
ν
Pr = (7.60)
α
Once again, note that the density, kinematic viscosity and thermal conductivity are
assumed to be constant in deriving the above non-dimensional equations. Appropriate
changes will be necessary if an appreciable variation in these quantities occurs in a flow
field. Another non-dimensional number, which is often employed in forced convection
heat transfer calculations is the Peclet number and is given as Pe = ReP r = u a L/α.For
buoyancy-driven natural convection problems, a different type of non-dimensional scale
is necessary if there are no reference velocity values available. The following subsection
gives the natural convection scales:

7.3.2 Natural convection (Buoyancy-driven convection)

Natural convection is generated by the density difference induced by the temperature differ-
ences within a fluid system. Because of the small density variations present in these types
of flows, a general incompressible flow approximation is adopted. In most buoyancy-driven
convection problems, flow is generated by either a temperature variation or a concentration
variation in the fluid system, which leads to local density differences. Therefore, in such
flows, a body force term needs to be added to the momentum equations to include the effect
of local density differences. For temperature-driven flows, the Boussinesq approximation
is often employed, that is,
g(ρ − ρ a ) = gβ(T − T a ) (7.61)

2
where g is the acceleration due to gravity (9.81 m/s )and β is the coefficient of thermal
expansion. The above body force term is added to the momentum equation in the gravity
direction. In a normal situation (refer to Figure 7.7), the body force is added to the x 2
momentum (if the gravity direction is negative x 2 ), that is,
2 2 !
∂u 2 ∂u 2 ∂u 2 1 ∂p ∂ u 2 ∂ u 2
+ u 1 + u 2 =− + ν + + gβ(T − T ∞ ) (7.62)
∂t ∂x 1 ∂x 2 ρ ∂x 2 ∂x 2 ∂x 2
1 2
In practice, the following non-dimensional scales are adopted for natural convection in
the absence of a reference velocity value:
x 1 x 2 tα
∗ ∗ ∗
x = ; x = ; t = ;
2
1
L L L 2
u 1 L u 2 L pL 2
∗ ∗ ∗
u = ; u = ; p = ;
1 2 2
α α ρα
T − T a
∗
T = (7.63)
T w − T a