1.
Assumptions 1 The fins
are sufficiently long so that the temperature of the fin at the tip is nearly T∞ . 2
Heat transfer
from the fin tips is negligible.
Analysis Taking
the temperature of the fin at the base to be Tb and
using the heat transfer relation for a long
fin, fin
efficiency for long fins can be expressed as
η fin=
if the entire
fin were at base temperature
=
=
=
This relation
can be simplified for a circular fin of diameter D
and rectangular
fin of thickness t and width w to be
η fin,circular
=
=
=
η fin,rectangular
=
=
=
2.
A hot plate is to be cooled by attaching
aluminum pin fins on one side. The rate of heat transfer from
the 1 m by 1 m
section of the plate and the effectiveness of the fins are to be determined.
Assumptions 1 Steady
operating conditions exist. 2 The temperature along the fins varies in
one direction
only (normal to
the plate). 3 Heat transfer from the fin tips is negligible. 4 The
heat transfer coefficient is
constant and
uniform over the entire fin surface. 5 The thermal properties of the
fins are constant. 6 The
heat transfer
coefficient accounts for the effect of radiation from the fins.
Properties The
thermal conductivity of the aluminum plate and fins is given to be k =
237 W/m⋅°C.
Analysis Noting
that the cross-sectional areas of the fins are constant, the efficiency of the
circular fins can
be determined to
be
a =
=
=
=
= 15.37
η fin=
=
= 0.935
The number of
fins, finned and unfinned surface areas, and heat transfer rates from those
areas are
n =
= 27.777
Afin = 27.777
= 6.68 m2
Aunfinned = 1- 27.777 -
= 0.86 m2
Qfinned = η fin. Qfinmax = 15300
Qunfinned = hAunfinned(Tb-To) = 2107
Then the total
heat transfer from the finned plate becomes
Qtotal = Qfinned + Qunfinned = 15,300 + 2107 = 1.74×104 W = 17.4 kW
The rate of heat
transfer if there were no fin attached to the plate would be
Anofin = 1 (1) =
1
Qno fin =
hAnofin (Tb-To) = 2450 W
Then the fin
effectiveness becomes
ɛfin =
=
= 7.10
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