5-42 A plane wall with surface temperature of 350°C is attached with straight rectangular fins (k 235 W/m K). The fins are exposed to an ambient air condition of 25°C, and the convection heat transfercoefficient is 154 W/m2-K. Each fin has a length of 50 mm, a base 5 mm thick, and a width of 100 mm. For a single fin, using a uniform nodal spacing of 10 mm, determine (a) the finite difference equations,(b) the nodal temperatures by solving the finite difference equations, and (c) the heat transfer rate and compare the result with the analytical solution.FIGURE P5-42Air, 25°Ch= 154 W/m2KT,-350°C=5 mmk= 235 W/m Kw= 100 mmL=50 mm

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Asked Nov 14, 2019
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5-42 A plane wall with surface temperature of 350°C is attached with straight rectangular fins (k 235 W/m K). The fins are exposed to an ambient air condition of 25°C, and the convection heat transfer
coefficient is 154 W/m2-K. Each fin has a length of 50 mm, a base 5 mm thick, and a width of 100 mm. For a single fin, using a uniform nodal spacing of 10 mm, determine (a) the finite difference equations,
(b) the nodal temperatures by solving the finite difference equations, and (c) the heat transfer rate and compare the result with the analytical solution.
FIGURE P5-42
Air, 25°C
h= 154 W/m2K
T,-350°C
=5 mm
k= 235 W/m K
w= 100 mm
L=50 mm
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5-42 A plane wall with surface temperature of 350°C is attached with straight rectangular fins (k 235 W/m K). The fins are exposed to an ambient air condition of 25°C, and the convection heat transfer coefficient is 154 W/m2-K. Each fin has a length of 50 mm, a base 5 mm thick, and a width of 100 mm. For a single fin, using a uniform nodal spacing of 10 mm, determine (a) the finite difference equations, (b) the nodal temperatures by solving the finite difference equations, and (c) the heat transfer rate and compare the result with the analytical solution. FIGURE P5-42 Air, 25°C h= 154 W/m2K T,-350°C =5 mm k= 235 W/m K w= 100 mm L=50 mm

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Expert Answer

Step 1

Calculate the number of nodes:

М
1
Ax
50
1 51
10
М -6
Draw the node diagram as shown below:
Т.
5
0
2
4
1
3
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М 1 Ax 50 1 51 10 М -6 Draw the node diagram as shown below: Т. 5 0 2 4 1 3

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Step 2

Write the general finite difference relation for each node:

T
Т.
kA
Т.
m+1
kA
T
+ КрАr)(T, — Т,
Ar
m-1
= (0
CO
Дr
Пр(Дк)?
Т,) %3D0
(Т,
- Т,) + (Т. —т,) +
— Т.) + (T.
m+1
(Т.
kA
т-1
Пp(Дr)?
-Т,
Пp(Дr)?
Т,
-T 0
— Т, + Т.
m-1
m+1
kA
kA
hp(Ar)
-То
Ip(Aк)?
-T 0
Т, — 27, + Т,
m-1
m+1
kKA
kA
hp(Ax)
hp(Ar)
2+
|T+T
(I)
m-1
m+1
kA
kA
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T Т. kA Т. m+1 kA T + КрАr)(T, — Т, Ar m-1 = (0 CO Дr Пр(Дк)? Т,) %3D0 (Т, - Т,) + (Т. —т,) + — Т.) + (T. m+1 (Т. kA т-1 Пp(Дr)? -Т, Пp(Дr)? Т, -T 0 — Т, + Т. m-1 m+1 kA kA hp(Ar) -То Ip(Aк)? -T 0 Т, — 27, + Т, m-1 m+1 kKA kA hp(Ax) hp(Ar) 2+ |T+T (I) m-1 m+1 kA kA

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Step 3

Calculate the area of fi...

A wt
A 0.005 x 0.1
=0.0005 m2
Calculate the perimeter ofthe fin for convection:
p 2(w
= 2 (0.005+0.1)
+t
=0.21m
hp(Ar)?
Calculate the value of
ww.
kA
hp(Ar)2
154 x0.21 x (0.01)
kA
235x 0.0005
=0.0275
The temperature at the node 0 is 350°C
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A wt A 0.005 x 0.1 =0.0005 m2 Calculate the perimeter ofthe fin for convection: p 2(w = 2 (0.005+0.1) +t =0.21m hp(Ar)? Calculate the value of ww. kA hp(Ar)2 154 x0.21 x (0.01) kA 235x 0.0005 =0.0275 The temperature at the node 0 is 350°C

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