Because there will be no forced convection, the velocity at the boundary sur-
face s in eq 69 and 70 will be zero, thus these two terms drop out. In the global
formulation, the equations representing velocity boundary nodes will be set to zero
and no other velocity boundary condition will be allowed.
The buoyancy β is defined by
1 ρref - ρ
β=
(72)
ρ T - Tref
where Tref is a reference temperature at which buoyancy has no effect. Gebhart
et al. (1988) suggested using the minimum boundary surface temperature for the
reference temperature, and that suggestion was followed.
The N j , Q and the ∫ Ni kn ∇T e ds terms of eq 71 represent heat generated
s
within an element and the thermal boundary conditions. For this application it is
assumed that there is no heat generated within an element, thus this term is elimi-
nated. Expanding the remaining term to account for specified heat flux and con-
vective boundaries yields
∫s Ni kn ∇T e ds = ∫s hN jT∞ ds - ∫s hN j NiTi ds - ∫s φN jds
(73)
where h and T∞ are the convective heat transfer coefficient and associated tem-
perature and φ is the heat flux for the boundaries s.
Summarizing the integrals required for eq 68, 69, 70, 71, and 73, the following
list is obtained:
Ni
p
(74)
Nj ,
x
Ni
p
(75)
Nj ,
y
Ni
p
(76)
, Nj
x
Ni
p
(77)
, Nj
y
Ni
um , N j ,
(78)
i
x
Ni
v im , N j ,
(79)
y
Nj
Ni
(80)
,
y
y
Nj
Ni
(81)
,
x
x
Nj
Ni
(82)
,
x
y
15
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