CHE151 homework solutions 9.pdf - CHE151 Introduction to...

CHE151 Introduction to Heat Transfer
Fall Semester, 2019
Homework Assignment #9
Due Thursday November 14, 2019 at the start of class
In Introduction to Heat Transfer 6 th and 7 th Edition
NOTE: Problem numbers have changed between editions; the problem number
are given for the 6 th and 7 th editions. I am also providing the first few words to
help find the correct problem if you have the 8 th edition. Also note that this in
Introduction to Heat Transfer not Fundamentals of Heat Transfer
Problem 7.66 begins with “An aluminum transmission line with diameter of 20
mm…”
Problem 7.84 begins with “Tissue engineering involves the development…”
Problem 8.4 begins with “An engine oil coller consists of a bundle…”
Problem 8.12 begins with “Water enters a tube at 27 ° C with a flow rate…”
Problem 8.30 begins with “Air at p =1 atm enters a thin-walled ( D =5-mm
diameter) …
Problem 8.49 begins with “Colling water flows through the 25.4-mm…”

PROBLEM 7.66
KNOWN: Aluminum transmission line with a diameter of 20 mm having an electrical resistance of
R
′
= 2.636 × 10
-4
ohm/m carrying a current of 700 A subjected to severe cross winds. To reduce
potential fire hazard when adjacent lines make contact and spark, insulation is to be applied.
FIND: (a) The bare conductor temperature when the air temperature is 20 ° C and the line is subjected
to cross flow with a velocity of 10 m/s; (b) The conductor temperature for the same conditions, but
with an insulation covering of 2 mm thickness and thermal conductivity of 0.15 W/m ⋅ K; and (c) Plot
the conductor temperatures of the bare and insulated conductors for wind velocities in the range of 2 to
20 m/s. Comment on the features of the curves and the effect that wind velocity has on the conductor
operating temperatures.
SCHEMATIC:
ASSUMPTIONS : (1) Steady-state conditions, (2) Uniform surface temperatures, (3) Negligible solar
irradiation and radiation exchange, and (4) Constant properties.
PROPERTIES: Table A-4 , Air (T
f
= (T
s
+ T
∞
)/2, 1 atm): evaluated using the IHT Properties library
with a Correlation function; see Comment 2.
ANALYSIS: (a) For the bare conductor the energy balance per unit length is
in out gen
E E E 0
′ ′ ′
− + =
& & &
cv c
0 q q A 0
′
− + =
&
(1)
where the cross-sectional area of the conductor is A
c
= π D
2
/4 and the generation rate is
( ) ( )
(
2 2
2 4
e c
q I R / A 700A 2.636 10 / m 0.020m / 4 π
−
′
= = × × Ω
&
(2)
5 3
q 4.111 10 W / m = ×
&
The convection rate equation can be expressed as
( )
( )
cv c,bare t t D
q T T / R R 1/ h D π
∞
′ ′ ′ = − = ×
(3,4)
and the convection coefficient is estimated using the Churchill-Bernstein correlation, Eq. 7.46, with
Re
D
= VD/ ν ,
( )
L
4/5
1/ 2 1/3 5/8
D D D
1/ 4
2/3
0.62 Re Pr
h D Re
Nu 0.3 1
k 282,000
1 0.4/ Pr
⎡ ⎤
⎛ ⎞
⎢ ⎥
= = + +
⎜ ⎟
⎢ ⎥
⎝ ⎠
⎡ ⎤
⎣ ⎦
+
⎢ ⎥
⎣ ⎦
(4)
(b) For the conductor with insulation thickness t = 2 mm, the energy balance per unit length is
in out gen
E E E 0
′ ′ ′
− + =
& & &
( )
2
c,ins t e c
0 T T / R I R / A 0
∞
′ ′
− − + =
(5)
Continued …

PROBLEM 7.66 (Cont.)
where
t
R
′
is the sum of the insulation conduction and convection process thermal resistances,
( ) ( ) ( )
t D 2t
R n D 2t / D / 2 k 1/ h D 2t
+
′ ⎡ ⎤ ⎡ ⎤ = + + +
⎣ ⎦ ⎣ ⎦
l π π