Page 195 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
Where Gr is the Grashof number given as
gβ T L 3 187
Gr = 2 (7.68)
ν
Often, another non-dimensional number called the Rayleigh number is used in the
calculations. This is given as
gβ T L 3
Ra = GrP r = (7.69)
να
On comparing the non-dimensional equations of natural and forced convection, it is
easy to identify the differences. If we substitute 1/P r in place of the Reynolds number
for the forced convection equations, we revert to a natural convection scaling. Obviously,
the extra buoyancy term needs to be added to appropriate component(s) of the momentum
equation for natural convection flows.

7.3.3 Mixed convection

Mixed convection involves features from both forced and natural flow conditions. The
buoyancy effects become comparable to the forced flow effects at small and moderate
Reynolds numbers. Since the flow is partly forced, a reference velocity value is normally
known (Example: velocity at the inlet of a channel). Therefore, non-dimensional scales
of forced convection can be adopted here. However, in mixed convection problems, the
buoyancy term needs to be added to the appropriate component of the momentum equation.
If we replace 1/P r with Re in the non-dimensional natural convection equations of the
previous subsection, we obtain the non-dimensional equations for mixed convection flows.
These equations are the same as for the forced convection flow problem except for the
body force term, which will be added to the momentum equation in the gravity direction.
The body force term is
Gr
T ∗ (7.70)
Re 2
Note that sometimes a non-dimensional parameter referred to as the Richardson number
2
(Gr/Re ) is also used in the literature.

7.4 The Transient Convection–diffusion Problem

An understanding of the fundamentals of the convection–diffusion equations is crucial in
studying fluid-dynamics-assisted heat transfer. The equations governing the combined fluid
flow and heat transfer mainly involve the convection and diffusion components. A typical
scalar convection–diffusion equation may be written as
∂φ ∂φ ∂u i ∂ ∂φ
+ u i + φ − k + Q = 0 (7.71)
∂t ∂x i ∂x i ∂x i ∂x i