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HEAT TRANSFER
PROBLEMS
Equipo docente:
Alfonso Calera Belmonte
Antonio J. Barbero
Departamento de Física Aplicada
UCLM
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PROBLEM 1.
The differential equation giving the profile of temperature through a hollow cylinder
infinitely long at steady state is
d  dT 
Where r is the radious
r
0
dr  dr 
Find out the heat flux per unit of length in a cylindrical pipe with the boundary
conditions T=T1 at r = r1, T=T2 at r=r2 (the border effects are neglected, and it is
assumed that conductivity k is constant).
d  dT 
r
0
dr  dr 
From boundary
conditions
T1 
r
dT
 C1
dr
dT 
T1  C1 ln r1  C2

dT  C1
T1  T2  C1ln r1  ln r2 
T2  C1 ln r2  C2
T1  T2
ln r1  C2
ln r1 / r2 
C1
dr
r
C2  T1 
T1  T2
ln r1
ln r1 / r2 
T  T1 
T

dr
 C2
r
C1 
T  C1 ln r  C2
T1  T2
ln r1 / r2 
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T1  T2
T T
ln r  T1  1 2 ln r1 P
ln r1 / r2 
ln r1 / r2 
h
T1  T2
ln r / r1 
ln r1 / r2 
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PROBLEM 1 (continued)
Heat flux
Note that
From Fourier’s equation
dT C1

dr
r
q  kAr
dT
dr
1 T1  T2
q  k 2rL
r ln r1 / r2 
Here Ar means the curved surface
of a cylindrical area, given by
q
T T
 2k 1 2
L
ln r1 / r2 
Ar  2rL
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PROBLEM 2.
A metallic cylindrical pipe (thermal conductivity km) having an inner radious r1 and
an outer radious r2 is covered by a d-cm thickness insulation of thermal conductivity
ki. The pipe carries a fluid at T1 temperature, and the external temperature is T’.
Calculate the losses of heat per meter of isolated pipe.
From the solution of the prior problem: we use the concept of thermal resistence
The heat flux is given by a set of terms having each the form
 2kij
Ti  T j
ln ri / rj 
Thermal resistences are given in this case by terms of the form
ln ri / r j 
kij
q
2 T3  T1 

ln r1 / r2  k12   ln r2 / r3  k23 
L
d
E
T’ = T3 n
v
T
i
r2
T 3
r
T1 2
o
n
r1
m
r3
e
km
n
ki
t
a
r3  r2  d
l
km  k12
ki  k23
Numerical application: km = 40 W/mºC, ki = 0.75 W/mºC, r1 = 5 cm, r2 = 7 cm,
d = 3 cm, T1 = 80 ºC, T’ = 25 ºC. Find out the intermediate temperature T2.
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PROBLEM 3.
A metallic cylindrical pipe (thermal conductivity km= 40 W/mºC) having an inner
radious r1 = 5 cm and an outer radious r2 = 7 cm carries a fluid at T1 = 80 ºC. The
inner and outer heat transfer coefficients are 2500 W m-2 ºC and 1600 W m-2 ºC.
The external temperature is 25 ºC. Find out the losses of heat per meter of pipe.
PROBLEM 4.
Consider three panes of glass, each of thickness 5 mm.
The panes trap two 2.5 cm layers of air in a large glass door. How much power leaks
through a 2.0 m2 glass door if the temperature outside is -40 ºC and the temperature
inside is 20 ºC? Data: kglass= 0.84 W/mºC, kair= 0.0234 W/m ºC (discuss if we can
consider air in the same way we consider glass).
PROBLEM 5.
If the temperature of the Sun fell 5%, and the radius shrank 10%, what would be the
percentage change of the Sun’s power output?
PROBLEM 6.
We know that the sun radiates 3.74·1026 W. We also know that the distance from Sun to
Earth is 1.5·1011 m and the radius of Earth is 6.36·106 m.
•What is the intensity (power/m2) of sunlight when it reaches Earth?
•How much power is absorbed by Earth in sunlight? (assume that none of the sunlight is
reflected)
•What average temperature would allow Earth to radiate an amount of power equal to the
amount of sun power absorbed?
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PROBLEM 5.
If the temperature of the Sun fell 5%, and the radius shrank 10%, what would be the
percentage change of the Sun’s power output?
Pr  AT
4
P0  A0T04
Pr AT 4
4R 2T 4
2
4



0
.
90

0
.
95
 0.66
P0 A0T04 4R02T04
The power output would decrease 34%
PROBLEM 6.
We know that the sun radiates 3.74·1026 W. We also know that the distance from Sun to
Earth is 1.5·1011 m and the radius of Earth is 6.36·106 m.
•What is the intensity (power/m2) of sunlight when it reaches Earth?
•How much power is absorbed by Earth in sunlight? (assume that none of the
sunlight is reflected)
•What average temperature would allow Earth to radiate an amount of power equal
to the amount of sun power absorbed?
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PROBLEM 6.
We know that the sun radiates 3.74·1026 W. We also know that the distance from Sun to
Earth is 1.5·1011 m and the radius of Earth is 6.36·106 m.
•What is the intensity (power/m2) of sunlight when it reaches Earth?
•How much power is absorbed by Earth in sunlight? (assume that none of the
sunlight is reflected)
•What average temperature would allow Earth to radiate an amount of power equal
to the amount of sun power absorbed?
From the radiated power Pr  AT 4 we get the power of sunlight at the Earth’s orbit
I received 
Pr
2
4d Earth
 Sun

3.74 1026

4 1.5 1011

2
 1320 Wm-2
From the Sun, the Earth appears as a tiny disk whose radious is 6.36·106 m
2
The power intercepted by Earth is Preceived  I received  REarth
 1.67 1017 W
If Earth would behave as a perfect blackbody PEarth  Preceived  ATEarth
4
 5.67051 x
10-8
W/m2K4
4
TEarth

Preceived
A
TEarth = 276 ºK
 1
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