PROBLEM 4.56
KNOWN:
Rectangular air ducts having surfaces at 80
°
C in a concrete slab with an insulated bottom
nd upper surface maintained at 30
°
C.
a
FIND:
Heat rate from each duct per unit length of duct,
q .
′
SCHEMATIC:
ASSUMPTIONS:
(1) Steady-state conditions, (2) Two-dimensional conduction, (3) No internal
olumetric generation, (4) Constant properties.
v
P
ROPERTIES:
Concrete (given):
k = 1.4 W/m
⋅
K.
ANALYSIS:
Taking advantage of symmetry, the
nodal network, using the suggested grid spacing
Δ
x = 2
Δ
y = 37.50 mm
Δ
y = 0.125L = 18.75 mm
where L = 150 mm, is shown in the sketch.
To
evaluate the heat rate, we need the temperatures T
1
,
T
2
, T
3
, T
4
, and T
5
.
All the nodes may be treated as
interior nodes ( considering symmetry for those nodes on
insulated boundaries), Eq. 4.29.
Use matrix notation, Eq.
.48, [A][T] = [C], and perform the inversion.
4
The heat rate per unit length from the prescribed section of

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