Fundamentals of heat transfer lecture notes

1
k
L
)
t
Δ
(
A
Q
kA
L
)
T
Δ
(
kA
L
)
t
Δ
(
Q
kA
L
)
t
t
(
Q
;
kA
L
)
t
t
(
Q
L
)
t
t
(
kA
Q
L
)
T
T
(
kA
Q
)
x
Δ
(
T
Δ
kA
Q
t
t
T
T
T
Δ
L
)
x
Δ
(
T
Δ
kA
)
x
Δ
(
Q
dT
kA
dx
Q
kAdT
Qdx
2
1
1
2
1
2
1
2
1
2
1
2
−
=
−
=
−
=
−
=
−
−
=
−
−
=
−
−
=
−
=
−
=
−
=
=
−
=
−
=
−
=
 
HEAT TRANSFER
By: Engr. Yuri G. Melliza
MODES OF HEAT TRANSFER
1. Conduction: It is the transfer of heat from one part of a body to another part of the same body, or from one body to another in
physical contact with it, without appreciable displacement of the particles of the body.
2. Convection: It is the transfer of heat from one point to another point within a fluid, gas, vapor, or liquid by
the mixing of some portion of the fluid with another.
A. Natural or Free Convection: the movement of the fluid is entirely caused by differences in density resulting from
temperature differences.
B. Forced Convection: the motion of the fluid is accomplished by mechanical means, such as a fan
or a blower.
3. Radiation: It is the transfer of heat from one body to another, not in contact with it, by means of "wave
motion" through space.
CONDUCTION:
From FOURIER'S LAW:
dx
dT
kA
Q −
=
k
Note:
Negative sign is used from Fourier's Equation because temperature decreases in the direction of heat flow.
From Ohms Law;
)
Ω
(
Ohms
in
resistance
electrical
-
R
(V)
Volts
in
Energy
-
V
Amperes
in
current
I
R
V
I
conductor
electric
an
for
flow
heat
Current
−
=

2
1 2
R
Q
W
K
or
W
C
,
resistance
-
R
K
or
C
potential,
temperatue
T
Δ
&
t
Δ
m
area,
Surface
-
Fuid
-
A
K
-
m
W
or
C
-
m
W
e),
conductanc
(surface
t
coefficien
convective
-
h
Watts
flow,
heat
convective
-
Q
where
2
2
2




−
−
−

R
T
Δ
R
t
Δ
Q
hA
1
T
Δ
hA
1
t
Δ
hA
1
)
t
t
(
hA
1
)
t
t
(
Q
Watts
)
t
t
(
hA
Q
Cooling
of
Law
s
Newton'
From
1
2
2
1
2
1
−
=
−
=
−
=
−
=
−
−
=
−
=
−
=
Watts
K
or
Watts
C
kA
L
R
R
T
Δ
-
R
t
Δ
-
Q
flow
heat
conductive
For


=
=
=
where:
(-T) and(-t) - temperature potential in K or C
R - thermal resistance in C/W or K/W
Q - conductive heat flow in Watts
k - thermal conductivity in W/m-C or W/m-K
A - surface area in m2
L - thickness in m
THERMAL CIRCUIT DIAGRAM
CONVECTION:

3
RADIATION:
T1
A
Surface or
(Radiator)
T2
T1 > T2
T1 – Surface temperature, K
T2 – surrounding temperature, K
Q
From Stefan-Boltzmann Law: The radiant heat transfer of a blackbody is directly proportional to the product of the surface area A and
its absolute temperature to the fourth power.
4
4
AT
δ
Q
AT
Q
=

From Fig., the radiant heat flow Q from the surface (or body) to the surrounding (or to other surface or body) is equal to
K
re,
temperatu
surface
other
or
g
surroundin
absolute
-
T
K
re,
temperatu
surface
absolute
-
T
Constant
Boltzmann
-
Stefan
K
-
m
W
,
10
x
5.678
δ
where
Watts
T
T
A
δ
Q
2
1
4
2
8
-
4
2
4
1
→
=





 −
=
Black Body - a hypothetical body that absorbs the entire radiation incident upon it.
Gray Body - are actual bodies or surfaces that absorbs a portion of the black body radiation, because they are not perfect radiators and
absorbers.
EMISSIVITY
Emissivity - is the ratio of the actual body (or surface) radiation at temperature T to the black body (or black surface) radiation at the
same temperature T.
Watts
T
T
A
δ
Q
to
equal
is
)
Radiator
Actual
or
Surface
Actual
(or
Boby
Actual
an
by
radiated
heat
of
rate
the
Therefore
T
@
radiation
Surface
Black
T
@
radiation
Surface
Actual
4
2
4
1 




 −
=
=

4
( )
( )
( ) ( )
( )( )
( )
( ) t
coefficien
radiation
T
T
T
T
δ
h
T
-
T
T
-
T
T
T
T
T
δ
T
-
T
T
T
T
T
δ
T
-
T
T
T
δ
h
T
-
T
h
T
T
δ
T
-
T
A
h
T
T
A
δ
Q
Convection
to
Radiation
Relating
T
T
where
2
1
2
2
2
1
r
2
1
2
1
2
1
2
2
2
1
2
1
2
2
2
1
2
2
2
1
2
1
4
2
4
1
r
2
1
r
4
2
4
1
2
1
r
4
2
4
1
2
1
→
+





 +
=
+





 +

=





 −





 +

=





 −

=
=





 −

=





 −
=

Combined Radiation and Convection heat transfer
Actual surface exposed to the surrounding air involves convection and radiation simultaneously, the total heat transfer Q
Convection
Radiation
combined
2
1
c
h
h
h
)
t
t
(
A
h
Q
+
=
−
=
(Answer: Q = 8400 Watts)
Answer: 8125 W

5
A = 0.5 m2
Q
t1 = 150C
t2 = 25C
 = 0.8
0.50 x 0.25 m
t1 = 300C
h = 250 W/m2
-K
t2 = 40C
QConvection
Answer: 726 W; 547 W
( )
4
2
4
1
4
Emitted
T
T
A
Q
AT
Q
−

=

=
A plate 30 cm long and 10 cm wide with a thickness 0f 12 mm is made of stainless steel (k = 16 W/m-K), the top of the is exposed to
an air stream of temperature 20C. In an experiment, the plate is heated by an electric heater (also 30 cm by 10 cm) positioned on the

6
underside of the plate and the temperature of the plate adjacent to the heater is maintained at 100C. A voltmeter and ammeter are
connected to the heater and these read 200 Volts and o.25 Ampere, respectively. Assuming that the plate is perfectly insulated on all
sides except the top surface, what is the convective heat transfer coefficient h if the emittance of the top surface  = 1.
C
-
m
W
7
.
12
h
)
293
75
.
371
(
hA
Q
W
30
20
50
Q
W
20
Q
)
293
75
.
371
)(
03
.
0
)(
10
x
67
.
5
(
1
Q
Q
Q
Q
K
75
.
371
T
16
012
.
0
)
T
373
(
03
.
0
Q
W
50
200(0.25)
Q
K
373
273
100
T
K
293
273
20
T
m
0.03
)
10
.
0
(
30
.
0
A
2
h
h
R
4
4
8
-
R
R
h
1
1
1
air
2

=
−
=
=
−
=
=
−
=
+
=
=
−
=
=
=
=
+
=
=
+
=
=
=
A cold storage room has walls constructed of a 5.08 cm layer of corkboard insulation contained between double wooden walls, each
1.27 cm thick. Find the rate of heat loss in Watts if the wall surface temperature is -12C outside the room and 21C inside the room.
The thermal conductivity of corkboard is 0.04325 W/m-C and the thermal conductivity of the wood walls is 0.10726 W/m-C.
=
+
+
+
=
10726
.
0
0127
.
0
4325
.
0
0508
.
0
10726
.
0
0127
.
0
)
12
21
(
A
Q

7
1
:
Assume =

Example 4
The wall of a house 7 m wide and 6 m high is made from o.3 thick brick with k = 0.6 W/m-K. The surface temperature on the inside
of the wall is 16C and that on the outside is 6C. Find the heat flux through the wall and the total heat loss through it.
A = 7 x 6 m2
k
Q
1
2
L
Example 5
L
20 mm

9
( )
( )
801
7,370,050,
T
293
273
20
T
0
,841.56
45,077,281
-
5.82T
105,820,10
T
0
A
Q
T
hT
T
h
T
0
A
Q
T
hT
hT
T
hT
hT
T
T
A
Q
hT
hT
T
T
A
Q
)
T
T
(
h
T
T
A
Q
)
T
T
(
hA
Qc
T
T
A
Qr
Qc
Qr
Q
4
2
2
1
4
1
4
2
2
1
4
1
4
2
2
1
4
1
2
1
4
2
4
1
2
1
4
2
4
1
2
1
4
2
4
1
2
1
4
2
4
1
=
=
+
=
=
+
=

+

+
−

+
=
−

−
−
+

−
+

−

=
−
+


−

=
−
+
−

=
−
=
−

=
+
=
( )
K
64
.
20
426
T
426
T
formula
quadratic
by
0
0.935
4507760141
-
82T
105820105.
T
s
2
s
s
2
2
s

=
=
=
=
+
Example 8

10
Answer: 1233 KW/m2
Example 9
CONDUCTION THROUGH A COMPOSITE PLANE WALL
Thermal Circuit diagram:

11
2
3
3
2
2
1
1
4
1
3
3
2
2
1
1
4
1
3
3
2
2
1
1
4
1
3
3
3
2
2
2
1
1
1
3
3
2
2
1
1
4
1
3
2
1
4
1
4
1
m
W
k
L
k
L
k
L
)
t
-
(t
A
Q
W
k
L
k
L
k
L
)
t
-
A(t
k
L
k
L
k
L
A
1
)
t
-
(t
Q
W
C
A
k
L
R
W
C
A
k
L
R
W
C
A
k
L
R
W
A
k
L
A
k
L
A
k
L
)
t
-
(t
R
R
R
)
t
-
(t
R
Σ
)
t
-
(t
Q
4
to
1
point
At








+
+
=








+
+
=








+
+
=

=

=

=
+
+
=
+
+
=
=
T)
Δ
(-
t)
Δ
(-
where
k
L
Σ
)
t
Δ
(
A
Q
k
L
Σ
)
t
Δ
(
A
Q
kA
L
Σ
t
Δ
Q
R
Σ
t
Δ
Q
=
−
=
−
=
−
=
−
=
OVERALL COEFFICIENT OF HEAT TRANSFER

12
K
-
m
W
or
C
-
m
W
k
L
Σ
1
U
herefore
t
k
L
Σ
t)
Δ
A(-
Q
From
t)
Δ
UA(-
Q
Transfer
Heat
of
t
Coefficien
Overall
of
basis
On the
2
2







=






=
=
where: U - overall coefficient of heat transfer in W/m2
-C or W/m2
-K
PARALLEL LAYERS OR COMBINED SERIES – PARALLEL LAYERS
HEAT TRANSFER FROM FLUID TO FLUID SEPARATED BY A COMPOSITE PLANE WALL

13
 +
−
=
=

−
=
 +
−
=
=
=
=
=
=
+
+
+
+
−
=
+
+
+
+
−
=
+
+
+
+
−
=
−
=
+
+
+
+
−
=
+
+
+
+
−
=








































k
L
h
1
Δt)
(
A
Q
t
R
ΣR
;
Δt)
(
kA
L
hA
1
Δt)
(
Q
A
k
L
R
;
A
k
L
R
;
A
k
L
R
;
A
h
1
R
;
A
h
1
R
;
h
1
k
L
k
L
k
L
h
1
)
t
(t
A
Q
h
1
k
L
k
L
k
L
h
1
)
t
A(t
Q
;
h
1
k
L
k
L
k
L
h
1
A
1
)
t
(t
Q
)
t
(t
R
R
R
R
R
)
t
(t
Q
;
A
h
1
A
k
L
A
k
L
A
k
L
A
h
1
)
t
(t
Q
R
ΣR
3
3
3
2
2
2
1
1
1
o
o
i
i
o
3
3
2
2
1
1
i
o
i
o
i
o
i
o
i
o
3
2
1
i
o
i
o
i
o
3
3
2
2
1
1
i
o
3
3
2
2
1
1
i
o
3
3
2
2
1
1
i
L1 L2 L3
t1
t2
t3
t4
k1 k2 k3
Q
Fluid
(i)
Fluid
(o)
A
hi
ti
ho
to
Thermal Circuit Diagram:
2
R1
Q
R2 R3
3 4
1
i o
Ri Ro
 +
−
=
 +
−
=
 +
−
=
































k
L
h
1
Δt)
A(
Q
k
L
h
1
A
1
Δt)
(
; Q
kA
L
hA
1
Δt)
(
Q

14





 +
=
−
=






+
−
=
k
L
h
1
Σ
1
U
)
t
Δ
(
UA
Q
k
L
h
1
Σ
)
t
Δ
(
A
Q
OVERALL COEFFICIENT OF HEAT TRANSFER COMPOSITE PLANE WALL
SAMPLE PROBLEMS
Problem No. 1
A composite plane wall is made up of an external thickness of brickwork 10 cm thick, followed by a layer of fiber glass 8 cm thick
and an insulating board 2.5 cm thick. The thermal conductivity for the three are as follows;
Brickwork - 1.5 W/m-C
Fiber Glass -0.04 W/m-C
Insulating Board - 0.06 W/m-C
The fluid film coefficient of the inside wall is 3.1 Wm2
-C while that of the outside wall is 2.5 Wm2
-C. Determine the overall
coefficient of heat transfer and the heat loss through such a wall 3.5 m high and 15 m long. Take the internal ambient temperature as
15C and the external temperature as 28C.
L1 L2 L3
k1 k2 k3
1
2
3
4
o
ho
to
i
hi
ti
Q

15
Problem No. 2
A furnace wall is constructed with 25 cm of firebrick, k = 1.36 W/m-K, 8 cm of insulating brick, k = 0.26 W/m-K, and 15 cm of
building brick, k = 0.69 W/m-K. The inside surface temperature is 600C and the outside air temperature is 30 C. The convective
coefficient for the outside air is 3.6 W/m2
-K. Determine
a. The total heat transferred W/m2
b. The interface temperature
c. The overall coefficient of heat transfer in W/m2
-K
Problem No. 3
A composite wall is made up of an external thickness of brickwork 11 cm thick, inside which is a layer of fibre glass 7.5 cm thick.
The fibre glass is faced internally by an insulating board 2.5 cm thick. The coefficient of thermal conductivity for the three are as
follows;

16
The surface transfer coefficient of the inside wall is 3.1 Wm2
-C while that of the outside wall is 2.5 Wm2
-C. Determine the overall
coefficient of heat transfer and the heat loss through such a wall 6 m high and 10 m long. Take the internal ambient temperature as
10C and the external temperature as 27C.
HEAT TRANSFER
QUIZ NO. 1
NAME ________________________________________ RATING ______________________
1. A wall 3.00 m by 2.44 m is made up of a thickness of 10.0 cm of brick (k = 0.649 W/m-C), 10.0 cm of glass wool (k =
0.0414 W/m-C), 1.25 cm of plaster (k = 0.469 W/m-C), and 0.640 cm of oak wood paneling (k = 0.147 W/m-C). If the
inside temperature of the wall is ti =18 C and the outside temperature t0 =−7C, Determine
a. The total Resistance of the wall in C/Watt
b. The overall coefficient of heat transfer in W/m2
-C
c. The total conductive heat flow in Watts
d. The interface temperature in C
2. A composite plane wall consisting of two layers of materials, 38 mm steel and 51 mm aluminum, separates a hot gas at ti =
93 C; hi = 11.4 W/m2
-C, from a cold gas at to = 27C; ho = 28.4 W/m2
-C. If the hot fluid is on the aluminum side, Find
a. the transmittance U in W/m2
-C
b. the total resistance R in C -m2
/W
c. the interface temperature at the junction of two metals in C
d. the heat through 9.3 m2
of the surface under steady-state conditions.

17
L1 L2
k1 k2
1
2
3
Q




















−
=
−
=
+
−
=
1
1
1
2
1
1
2
1
2
2
1
1
3
1
k
L
A
Q
t
t
k
L
t
t
A
Q
k
L
k
L
)
t
t
(
A
Q
2
x
1
3
1
x
2
2
2
1
1
1
R
R
R
t
t
A
Q
W
C
95
.
0
R
A
k
L
R
A
k
L
R
+
+
−
=

=
=
=
QUIZ NO. 2
I
An insulated steam pipe run through a dark warehouse room. The pipe outside diameter is 60 mm and its surface temperature and
emissivity are 165C and 0.95, respectively. The warehouse and air is kept at 5C. If the coefficient of heat transfer by natural
convection from the outside surface to the air is 11.4 W/m2
-C and the pipe surface may be treated as a gray body, what is the rate of
heat loss from the surface per meter of pipe length.
II
A flat furnace wall is constructed of a 11 cm layer of sil-o-cel brick with a thermal conductivity of 0.14 W/m-C, backed by a 23 cm
layer of common brick of conductivity 1.4 W/m-C. The temperature of the inner face of the wall is 760 C, and that of the outer face
is 77C.
a. What is the heat loss through the wall
b. What is the temperature of the interface between the refractory brick and common brick
c. Supposing that the contact between the two brick layers is poor and that a contact resistance of 0.95 C/W
is present, what would be the heat loss.

18
III
A furnace wall consists of 20 mm of refractory fireclay brick, 100 mm of sil-ocel brick, and 6 mm of steel plate. The fire side of the
refractory is at 1150C and the outside of the steel is at 30C. An accurate heat balance over the furnace shows the heat loss from the
wall to be 300 W/m2
. It is known that there may be thin layers of air between the layers of brick and steel. To how many mm of sil-o-
cel are these air layers equivalent.
k = 1.52 W/m-C (fireclay)
k = 0.138 W/m-C (sil-ocel)
k = 4.5 W/m-C (steel)
IV
A carpenter builds an outer house wall with a layer of wood (k = 0.080 W/m-K) 2 cm thick on the outside and a layer of Styrofoam (k
= 0.01 w/m-K) insulation 3.5 cm thick as the inside wall surface. What is the temperature at the plane where the wood meets the
Styrofoam? Interior temperature is 19C; exterior temperature is -10C.

19
k1 k2
L1 L2
t3
t1
A
Q
V
An industrial freezer is designed to operate with an internal air temperature of -20C when the external air temperature is 25C and
the internal and external heat transfer coefficient are 12 W/m2
-K and 8 W/m2
-K, respectively. The walls of the freezer are composite
construction, comprising of an inner layer of plastic (k = 1 W/m-K, and thickness 3 mm), and an outer layer of stainless stell (k = 16
W/m-K, and thickness of 1 mm). Sandwiched between these two layers is a layer of insulation material with k = 0.07 W/m-K. Find
the width of the insulation that is required to reduce the convective heat loss to 15 W/m2
.
CONDUCTION THROUGH CYLINDRICAL COORDINATES

20
)
T
T
(
kL
π
2
r
r
Qln
)
T
T
(
kL
π
2
r
r
Qln
;
dT
kL
π
2
-
r
dr
Q
kLdT
π
2
-
r
dr
Q
;
,
dr
dT
)
rL
π
k(2
-
Q
rL
π
2
A
,
dx
dT
kA
-
Q
is;
Q
sfer
heat tran
the
direction,
radial
a
in
flows
heat
since
,
dx
dT
kA
-
Q
:
Equation
s
Fourier'
From
2
1
1
2
1
2
1
2
2
1
2
1
−
=
−
−
=
=
=
=
=
=
=


2
1
t
t 
R
t
Δ
kL
π
2
ri
ro
ln
t
Δ
Q
−
=
−
=
o – refers to outside
i – refers to inside
L
r1
r2
1 2
k
Q
L - length of the cylinder perpendicular to the paper
where: Q in Watts
k in W/m-C or W/m-K
L in meters
resistance
-
R
where
kL
π
2
r
r
ln
R
R
)
t
(t
Q
;
kL
π
2
r
r
ln
)
t
(t
Q
t
-
t
T
-
T
where
kL
π
2
r
r
ln
)
T
(T
Q
;
r
r
ln
)
T
T
(
kL
π
2
Q
1
2
2
1
1
2
2
1
2
1
2
1
1
2
2
1
1
2
2
1
=
−
=
−
=
=
−
=
−
=

21
r in m
-t in C or K
R in W/C or W/K
CONDUCTION THROUGH A COMPOSITE CURVED WALL
Let
HEAT TRANSFER FROM FLUID TO FLUID SEPARATED BY A COMPOSITE CURVED WALL
( )
( )
L
k
π
2
r
r
ln
t
t
Q
3
to
2
at
L
k
π
2
r
r
ln
t
t
Q
2
to
1
at
2
2
3
3
2
1
1
2
2
1
−
=
−
=
( )
( )
L
k
π
2
r
r
ln
L
k
π
2
r
r
ln
L
k
π
2
r
r
ln
)
t
(t
Q
kL
π
2
r
r
ln
t
Δ
-
Q
L
k
π
2
r
r
ln
L
k
π
2
r
r
ln
L
k
π
2
r
r
ln
)
t
(t
Q
:
4
to
1
at
L
k
π
2
r
r
ln
t
t
Q
:
4
to
3
at
3
3
4
2
2
3
1
1
2
4
1
i
o
3
3
4
2
2
3
1
1
2
4
1
3
3
4
4
3
+
+
−
=













=
+
+
−
=
−
=
 











=










+
+
−
=










+
+
−
=










+
+






−
=
k
r
r
ln
)
t
Δ
(-
π
2
L
Q
k
r
r
ln
k
r
r
ln
k
r
r
ln
)
t
(t
π
2
L
Q
k
r
r
ln
k
r
r
ln
k
r
r
ln
)
t
L(t
π
2
Q
k
r
r
ln
k
r
r
ln
k
r
r
ln
L
π
2
1
)
t
(t
Q
i
o
3
3
4
2
2
3
1
1
2
0
i
3
3
4
2
2
3
1
1
2
0
i
3
3
4
2
2
3
1
1
2
0
i

22
L
k
2
r
r
ln
)
t
-
(t
Q
2
2
3
3
2

=
i
i
1
i
A
h
1
)
t
t
(
Q
−
=
L
k
2
r
r
ln
)
t
-
(t
Q
3
3
4
4
3

=
at i to 1: at 1 to 2: at 2 to 3:
Q = Aihi(ti - t1)
at 3 to 4: at 4 to 0:
at i to 0:
0
0
3
3
4
2
2
3
1
1
2
i
i
0
i
h
A
1
L
k
2
r
r
ln
L
k
2
r
r
ln
L
k
2
r
r
ln
h
A
1
)
t
t
(
Q
+

+

+

+
−
=
0
4
3
3
4
2
2
3
1
1
2
i
1
0
i
Lh
r
2
1
L
k
2
r
r
ln
L
k
2
r
r
ln
L
k
2
r
r
ln
Lh
r
2
1
)
t
t
(
Q

+

+

+

+

−
=










+
+
+
+

−
=
0
4
3
3
4
2
2
3
1
1
2
i
1
0
i
h
r
1
k
r
r
ln
k
r
r
ln
k
r
r
ln
h
r
1
L
2
1
)
t
t
(
Q












+
+
+
+
−

=
0
4
3
3
4
2
2
3
1
1
2
i
1
0
i
h
r
1
k
r
r
ln
k
r
r
ln
k
r
r
ln
h
r
1
)
t
t
(
L
2
Q
L
k
2
r
r
ln
)
t
-
(t
Q
1
1
2
2
1

=
L
r
2
A
h
A
1
)
t
t
(
Q
)
t
t
(
h
A
Q
4
0
0
0
0
4
0
4
0
0

=
−
=
−
=
L
r
2
A 1
i

=

23












+
+
+
+
−

=
0
4
3
3
4
2
2
3
1
1
2
i
1
0
i
h
r
1
k
r
r
ln
k
r
r
ln
k
r
r
ln
h
r
1
)
t
t
(
2
L
Q
General Equation:
kL
2
r
r
ln
Ah
1
)
t
(
Q
i
o


+


−
= ;
R
)
t
(
Q


−
= ; where
kL
2
r
r
ln
Ah
1
R i
o


+

=

o - refers to outside
i - refers to inside













+



−
=
k
r
r
ln
rh
1
L
2
1
)
t
(
Q
i
o













+


−

=
k
r
r
ln
rh
1
)
t
(
L
2
Q
i
o













+


−

=
k
r
r
ln
rh
1
)
t
(
2
L
Q
i
o
Cylindrical coordinates sample Problems
1. Water through a cast steel pipe (k = 50 W/m-K) with an outer diameter of 104 mm and 2 mm wall thickness.
a. Calculate the heat loss by convection and conduction per meter length of insulated pipe (OD = 104 mm ; ID = 100
mm when the water temperature is 15C, the outside air temperature is -10C, the water side heat transfer coefficient
is 30,000 W/m2
-K and the outside heat transfer coefficient is 20 W/m2
-K.
b. Calculate the corresponding heat loss when the pipe is lagged with insulation having an outer diameter 0f 300 mm,
and thermal conductivity of k = 0.05 W/m-K.
r1
r2
i 1 2
Q
hi
ti
ho
to
t1 t2

24
2. Saturated steam at 500 K flows in a 20 cm ID; 21 cm OD pipe. The pipe is covered with 8 cm of insulation with a thermal
conductivity of 0.10 w/m-C. The pipes conductivity is 52 W/m-C. the ambient temperature is 300 K. The unit convective
coefficient are hi = 18 000 W/m2
-C and ho = 12 W/m2
-C. Determine the heat loss from 4 m of pipe. Calculate the overall
coefficient of heat transfer base on the outside area.
r1
r2
r3
ti
hi
1 2 3
k1
k2
t0
ho
Q

25
L
Q
k1
k2
r1
r2
r3
1 2 3
C
61
.
213
ti 
=
C
22
t0 
=
ms ms
hs1 hs2
Q
QUIZ NO. 3
1. Steam initially saturated at 2.05 MPa, passes through a 10.10 cm (ID =10 cm; OD =12 cm) standard steel pipe (k = 50 W/m
– K) for a total distance of 152 m. The steam line is insulated with a 5.08 cm thickness of 85% magnesia (k = 0.069 W/m-
K). For an ambient temperature of 22C, what is the quality of the steam which arises at its destination if the mass flow rate
is 0.125 kg steam/sec.
hsteam = 550 W/m2
– K
hstill air = 9.36 W/m2
– K
tsat at 2.05 MPa = 213.61C
r1 0.0500 k1 50
r2 0.06 k2 0.069
r3 0.1108 hi 550
r2/r1 1.2 ho 9.36
r3/r2 1.85 m 0.125
ln(r2/r1) 0.182 hs1 2796.3
ln(r3/r2) 0.613 L 152
1/hir1 0.036 2PiL 955.044
1/hor3 0.964 ti 213.61
ln(r2/r1)/k1 0.0036 to 22
ln(r3/r2)/k2 8.8896 (ti -t0) 191.61
Q 18495.94 Watts
hs2 2648.3 Enthalpy
hf 913.75
hg 2796.3
hfg 1882.55
x 92.14 Quality

26
r1
r2
r3
r4
k1
k2
k3
ti
hi
1 2 3 4
to, ho
Q
( )
%
14
.
92
x
)
h
h
(
x
h
hs
r
h
1
k
r
r
ln
k
r
r
ln
r
h
1
to
-
ti
L
2
Q
Watts
1000
)
h
-
(h
m
Q
2
s
f
g
2
s
f
2
3
o
2
2
3
1
1
2
1
i
s2
s1
s
=
−
+
=
+
+
+

=
=
2. A steel pipe 100 mm bore and 7 mm wall thickness, carrying steam at 260C is insulated with 40 mm of a moulded high-
temperature diatomaceous earth covering, this covering in turn insulated with 60 mm of asbestos felt. The atmospheric
temperature is 15C. The heat transfer coefficients for the inside and outside surfaces are 550 and 15 W/m2
-K, respectively
and the thermal conductivities of steel, diatomaceous earth and asbestos felt are 50, 0.09, and 0.07 W/m-K respectively.
Calculate:
a. the rate of heat loss by the steam per unit length of pipe
b. the temperature of the outside surface
c. the overall coefficient of heat transfer based on outside surface
( )
( )
( ) ( ) ( )( )
C
-
m
W
481
.
0
U
t
-
t
L
r
2
U
t
-
t
A
U
r
h
1
k
r
r
ln
k
r
r
ln
k
r
r
ln
r
h
1
t
-
t
L
2
Q
k
r
r
ln
k
r
r
ln
k
r
r
ln
r
h
1
t
-
t
2
L
Q
r
h
1
k
r
r
ln
k
r
r
ln
k
r
r
ln
r
h
1
t
-
t
2
L
Q
2
o
o
i
4
o
o
i
o
o
4
o
3
3
4
2
2
3
1
1
2
1
i
o
i
3
3
4
2
2
3
1
1
2
1
i
4
i
4
o
3
3
4
2
2
3
1
1
2
1
i
o
i

=

=
=
+
+
+
+

=
+
+
+

=
+
+
+
+

=

27
r1
r2
r3
k1
k2
ti
hi
1 2 3
to, ho
Q
r1 0.050 ti 260 ln(r2/r1) 0.1310
r2 0.057 t0 15 ln(r3/r2) 0.5317
r3 0.097 hi 550 ln(r4/r3) 0.4815
r4 0.157 ho 15 ln(r2/r1)/k1 0.0026
k1 50 (ti - to) 245 ln(r3/r2)/k2 5.9073
k2 0.09 2*PI() 6.28 ln(r4/r3)k3 6.8791
k3 0.07 Ai 0.314 1/hir1 0.0364
L 1 Ao 0.986 1/hor4 0.4246
Q 116.18 SUM 13.2500
Uo 0.481 SUM2 12.8254
t4 22.85
QUIZ NO. 4
1. Steam at 280C flows in a stainless steel pipe (k = 15 W/m-K) whose inner and outer diameters are 5 cm and 5.5 cm,
respectively. The pipe is covered with 3 cm thick glass wool insulation (k = 0.038 W/m-K). Heat is lost to the surroundings
at 5C by natural convection and radiation, with a combined natural convection and radiation heat transfer coefficient of 22
W/m2
-K. Taking the heat transfer coefficient inside the pipe is 80 W/m2
-K. determine the rate of heat loss from the steam per
unit length of pipe. Also determine the temperature drops across the pipe shell and the insulation.
2. Hot water (hi = 6895 W/m2 -C) flows in a 2.5 cm ID; 2.66 cm OD smooth copper pipe (k = 400 W/m -C). The pipe is
horizontal in still air ( ho = 3.56 W/m2
-C )and covered with a 1-cm layer of polystyrene foam insulation (k =0.038 W/m-
C) . For a 65°C water temperature and 20°C air temperature, Calculate
a. the heat loss per unit length of pipe

28
b. the interface temperature in C
c. Overall coefficient of heat transfer U

32
OVERALL COEFFICIENT OF HEAT TRANSFER

33
Q = UA(-t)
A. For a composite Plane Wall









+
=






+
=

−
=






+
−
=
k
L
h
1
1
U
kA
L
hA
1
k
L
h
1
Δt)
A(
Q
Rt
Rt
t
B. For a Composite Curved Wall
( )
kL
2
r
r
ln
hA
1
UA
kL
2
r
r
ln
hA
1
kL
2
r
r
ln
hA
1
Δt
-
Q
i
o
i
o
i
o

+
=

+
=

−
=

+
=



t
R
Rt
t
UA = UiAi = UoAo
where:
Ui - overall coefficient of heat transfer based on
inside surface, W/m2
-C or W/m2
-K
Uo - overall coefficient of heat transfer based on
outside surface, W/m2
-C or W/m2
-K
HEAT EXCHANGER OR HEAT TRANSFER EQUIPMENT
TYPES OF HEAT EXCHANGERS
1. DIRECT CONTACT TYPE: The same fluid at two different states is mixed.
2. SHELL AND TUBE TYPE: One fluid flows inside the tubes and the other one on the outside.

34
mh, hh
mc, hc
m, h
Transfer
Heat
Total
)
h
h
(
m
)
h
(h
m
Q
)
h
h
(
m
)
h
(h
m
h
m
h
m
h
m
h
m
h
)
m
m
(
h
m
h
m
h
m
h
m
h
m
)
negligible
are
PE
Δ
and
KE
Δ
(
Balance
Energy
By
m
m
m
Balance
Mass
By
c
c
h
h
c
c
h
h
c
h
c
c
h
h
c
h
c
c
h
h
c
c
h
h
c
h
→
−
=
−
=
−
=
−
+
=
+
+
=
+
=
+
=
+
Example:
Shell and Tube Type Heat Exchanger

35
By energy balance:
Heat Rejected by hot fluid = Heat Absorbed by cold fluid
Fluid
Cold
)
t
t
(
C
m
Q
Fluid
Hot
)
t
t
(
C
m
Q
Q
Q
1
c
2
c
pc
c
c
2
h
1
h
ph
h
h
c
h
→
−
=
→
−
=
=

36
2
AMTD
ln
LMTD
1
2
2
1
2

+

=



−

=
1
difference
e
Temperatur
Mean
c
Arithmeti
on
Based
)
AMTD
(
UA
Q
difference
e
Temperatur
Mean
Log
on
Based
)
LMTD
(
UA
Q
Q
Q
Q c
h
→
=
→
=
=
=
Where:
A – total heat transfer area, m2
LMTD - Log Mean Temperature Difference, C or K
AMTD - Arithmetic Mean Temperature Difference, C or K
U – Overall Coefficient of Heat Transfer,
K
m
KW
or
C
m
KW
2
2
−

−
where:  - terminal temperature difference

38
If the design of the heat exchanger is more complex, the LMTD is modified by a correction factor F.
factor
correction
-
F
:
Where
)
LMTD
(
UAF
Q =

39
passes.
of
number
the
by
tubes
of
number
the
multiply
and
passes
of
number
by
length
the
divide
passes,
tube
Multiple
For
tables
or
charts
from
determine
be
can
F
factor
correction
-
F
meters
sheet,
tube
of
thickness
-
t
2t
L
L
tube
of
length
actual
-
L
meters
tube,
of
length
efdfective
-
L
tubes
of
number
total
-
n
area)
surface
inside
on
based
U
(For
diameter
tube
inside
-
d
dLn
π
A
area)
surface
outside
on
based
U
for
(
tube
of
diameter
outside
-
D
m
DLn
π
A
sec
m
tubes
of
Number
velocity x
x
Area
V
t
t
2
3
flow
+
=
=
=
=

40
Example No. 1
Exhaust gases flowing through a tubular heat exchanger at the rate of 0.3 kg/sec are cooled from 400 to 120C by water initially at
10C. The specific heat capacities of exhaust gases and water may be taken as 1.13 and 4.19 KJ/kg-K respectively, and the overall
heat transfer coefficient from gases to water is 140 W/m2
-K. Calculate the surface area required when the cooling water flow is 0.4
kg/sec;
a. for parallel flow (4.01 m2
)
b. for counter flow (3.37 m2
)
C
3
.
53
66.7
-
120
t
-
t
θ
C
390
10
-
400
t
-
t
θ
Flow
Parallel
For
C
7
.
66
t
)
10
t
)(
19
.
4
(
4
.
0
)
120
400
)(
13
.
1
(
3
.
0
Qc
Qh
C
10
t
C
120
t
;
C
400
t
C
-
KJ/kg
1.13
C
;
kg/sec
0.3
m
C
-
KJ/kg
4.19
C
;
kg/sec
4
.
0
m
Given
c2
h2
1
c1
h1
2
2
c
2
c
c1
h2
h1
ph
h
pc
c

=
=
=

=
=
=

=
−
=
−
=

=

=

=
=
=
=
=
)
LMTD
(
U
Q
A
)
LMTD
(
UA
Q
C
43
.
201
110
3
.
333
ln
110
3
.
333
LMTD
C
110
10
120
t
t
θ
C
3
.
333
7
.
66
400
t
-
1
t
θ
Flow
Counter
For
C
2
.
169
3
.
53
390
ln
3
.
53
390
θ
θ
ln
θ
-
θ
LMTD
1
c
2
h
1
c2
h
2
1
2
1
2
=
=

=
−
=

=
−
=
−
=

=
−
=
=

=
−
=
=
2
2
Q 0.3(1.13)(400 120) 94.92 KW
Q 94920 Watts
94,920
A 4.01 m Parallel Flow
140(169.2)
94,920
A 3.4 m Counter Flow
140(201.43)
= − =
=
= = →
= = →

41
SAMPLE PROBLEMS
1. A composite wall is made up of an external thickness of brickwork 11 cm thick, inside which
is a layer of fiber glass 7.5 cm thick. The fiber glass is faced internally by an insulating
board 2.5 cm thick. The coefficient of thermal conductivity for the three are as follows;
The surface transfer coefficient of the inside wall is 3.1 Wm2
-C while that of the outside
wall is 2.5 Wm2
-C. Determine the overall coefficient of heat transfer and the heat loss
through such a wall 6 m high and 10 m long. Take the internal ambient temperature as 10C
and the external temperature as 27C.
2. A furnace is constructed with 20 cm of firebrick, k = 1.36 W/m-K, 10 cm of insulating
brick, k = 0.26 W/m-K, and 20 cm of building brick, k = 0.69 W/m-K. The inside surface
temperature is 650C and the outside air temperature is 32 C. The heat loss from the
furnace wall is 0.56 W/m2
. Determine
a. the unit convective coefficient for the air W/m2
-K (3.545)
b. the temperature at 25 cm from the outside surface in C. (460 C)
3. A furnace wall consists of 20 mm of refractory fireclay brick, 100 mm of sil-ocel brick, and
6 mm of steel plate. The fire side of the refractory is at 1150C and the outside of the
steel is at 30C. An accurate heat balance over the furnace shows the heat loss from the
wall to be 300 W/m2
. It is known that there may be thin layers of air between the layers of
brick and steel. to how many mm of sil-o-cel are these air layers equivalent. (400 mm)
k = 1.52 W/m-C (fireclay)
k = 0.138 W/m-C (sil-ocel)
k = 4.5 W/m-C (steel)
4. A composite plane wall consisting of two layers of materials, 38 mm steel and 51 mm
aluminum, separates a hot gas at ti = 93 C; hi = 11.4 W/m2
-C, from a cold gas at to = 27C;
ho = 28.4 W/m2
-C. If the hot fluid is on the aluminum side, Find
a. the transmittance U in W/m2
-C (8)
b. the total resistance R in C -m2
/W (0.124)
c. the interface temperature at the junction of two metals in C (45)
d. the heat through 9.3 m2
of the surface under steady-state conditions. (4937)
5. A heat exchanger is to be designed to the following specifications:
Hot gas temperature = 1145C
Cold gas temperature = 45C
Unit surface conductance on the hot side = 230 W/m2
-K
Unit surface conductance on the cold side = 290 W/m2
-K
Thermal conductivity of the metal wall = 115 W/m-K
Find the maximum thickness of metal wall between the hot gas and the cold gas, so that the
maximum temperature of the wall does not exceed 545C. (20.115 mm)
6. A composite furnace wall is made up of a 300 mm lining of magnesite refractory brick, a
130 mm thickness of 85% magnesia and a steel 2.54 mm thick. Flue gas temperature is
1205 C and the boiler is at 27 C. Gas side film coefficient is 85 W/m2
-C and the air side
is 23 W/m2
-C.
Determine:
a. the thermal current Q/A in W/m2
b. the interface temperatures in C
c. effect on thermal current and the inside refractory wall temperature if the
magnesia insulation were doubled.
k for magnesite = 296 W/m-C
k for 85% magnesia = 0.0692 W/m-C
k for steel = 43.3 W/m-C
7. Determine the thermal conductivity of a wood that is used in a 1.5 m2
test panel, 25 mm
thick, if during a 4 hours test period there are conducted 190 KJ through the panel with a
temperature differential of 6C between the surfaces. Express answer in W/m-C. (0.0244)
8. The walls of a cold storage are composed of an insulating material (k = 0.065 W/m-C)

42
10.16 cm thick held between two layers of concrete ( k = 1.04 W/m-C) each 10.16 cm thick.
The film coefficients are 22.7 W/m2
-C on the outside and 11.4 W/m2
-C on the inside. Cold
storage temperature is -7C and the ambient temperature is 32C. Determine the heat
transmitted in KW through an area of 56 m2
. (1.2)
9. A 12 in thick furnace wall with a dimensions of 5 m x 2 m has temperature difference of
60C. The wall has a thermal conductivity of 0.75 BTU/hr-ft-F. Calculate the heat
transmitted across the wall. (2554 W)
10. A 15 cm thick wall has a thermal conductivity of 5 W/m-K. If the inside and outside surface
temperature of the wall are 200C and 30C, respectively. Determine the heat transmitted.
( 5.67 W/m-K)
11. Two walls of cold storage plant are composed on an insulating material (k = 0.07 W/m-K),
100 mm thick at the outer layer and material (k = 0.97 W/m-K), 15 cm thick at the inner
layer. If the surface temperature of the cold side is 30C and hot side is 250C, find the
heat transmitted in W/m2
. (138)
12. An insulated steam pipe runs through a dark warehouse room.The pipe outside diameter is
60 mm and its surface temperature and emissivity are 165 C and 0.95, respectively. The
warehouse and air is kept at 5C. If the coefficient of heat transfer by natural convection
from the outside surface to the air is 11.4 w/m2
-C and the pipe surface maybe treated as
a gray body, what is the rate of heat loss from the surface per meter of pipe length.
( 657 W/m)
13. Saturated steam at 500 K flows in a 20 cm ID; 21 cm OD pipe. The pipe is covered with 8
cm of insulation with a thermal conductivity of 0.10 w/m-C. The pipes conductivity is 52
W/m-C. the ambient temperature is 300 K. The unit convective coefficient are hi = 18 000
W/m2
-C and ho = 12 W/m2
-C. Determine the heat loss from 4 m of pipe. Calculate the
overall coefficient of heat transfer base on the outside area. (822 W; 0.9 W/m2
-C)
14. A tube 60 mm OD is lagged with a 50 mm layer of asbestos for which the conductivity is
0.21 W/m-C, followed with a 40 mm layer of cork with a conductivity of 0.05 W/m-C. If
the temperature of the outer surface of the pipe is 150C and the temperature of the
outer surface of the cork is 30C, Calculate the heat loss in Watts per meter length of
pipe.(59)
15. An economizer receives hot gas (Cp = 1.13o6 KJ/kg-K) and water in the ratio 1.5 kg gas/kg
water. The gas enters at 455C and leaves at 180C; the water enters at 50C. Find the
exit temperature of the water and the LMTD:
a. for parallel flow ( 125.425C)
b. for counter flow (200.81C)
Assume no energy losses external to the system.
16. A flat furnace wall is constructed of a 11 cm layer of sil-o-cel brick with a thermal
conductivity of 0.14 W/m-C, backed by a 23 cm layer of common brick of conductivity 1.4
W/m-C. The temperature of the inner face of the wall is 760 C, and that of the outer
face is 77C.
a. What is the heat loss through the wall
b. What is the temperature of the interface between the refractory brick and
common brick
c. Supposing that the contact between the two brick layers is poor and that a
contact resistance of 0.95 C/W is present, what would be the heat loss.
17. Steam initially saturated at 2.05 MPa, passes through a 10.10 cm standard steel pipe for a
total distance of 152 m. the steam line is insulated with a 5.08 cm thickness of 85%
magnesia. For an ambient temperature of 22C, what is the quality of the steam which
arises at its destination if the mass flow rate is 0.125 kg steam/sec. (x = 93%)
At 2.05 MPa: hf = 914.52 KJ/kg; hfg = 1885.5 KJ/kg; hg = 2800 KJ/kg
18. The hot combustion gases of a furnace are separated from the ambient air and its
surrounding which are at 25C, by a brick wall 15 cm thick.The brick has a thermal
conductivity of 1.2 W/m-C and a surface emissivity of 0.8. Under steady-state
conditions and outer surface temperature of 100C is measured. Free convection heat
transfer to the air adjoining this surface is characterized by a convection coefficient of

43
20 W/m2
-C. what is the brick inner surface temperature inC. (352.5C)
19. A counter flow heat exchanger is designed to heat fuel oil from 28 C to 90 C, while the
heating fluid enters at 138C and leaves at 105C. The fuel has a specific gravity of
21API, a specific heat of 2.1 KJ/kg-K and enters the heat at the rate of 3 000 L/hr.
Determine the required heating surface area in m2
if the overall coefficient of heat
transfer is 465.2 W/m2
-K. ((3.5 m2
)
20. Brine enters a circulating brine cooler at the rate of 5.7 m3
/hr at -10C and leaves at
-16C.Specific heat of the brine is 1.072 KJ/kg-C and the specific gravity is 1.10. The
refrigerant evaporates at -25 C. What is the required heat transfer area if U = 454
W/m2
-C. (2.1 m2
)
21. A steel pipe 100 mm bore and 7 mm wall thickness, carrying steam at 260C is insulated
with 40 mm of a moulded high-temperature diatomaceous earth covering, this covering in
turn insulated with 60 mm of asbestos felt. The atmospheric temperature is 15C. The
heat transfer coefficients for the inside and outside surfaces are 550 and 15 W/m2
-K,
respectively and the thermal conductivities of steel, diatomaceous earth and asbestos felt
are 50, 0.09, and 0.07 W/m-K respectively. Calculate:
a. the rate of heat loss by the steam per unit length of pipe (116 W)
b. the temperature of the outside surface (22.8C)
22. A furnace wall consists of 125 mm wide refractory brick and a25 mm wide insulating
firebrick separated by an air gap. The outside wall is covered with a 12 mm thickness of
plaster. The inner surface of the wall is at 1100C and the room temperature is 25C. The
heat transfer coefficient from the outside wall surface to the air in the room is
17 W/m2
-K and the resistance to heat flow of the air gap is 0.16 K/W. The thermal
conductivities of refractory brick, insulating brick, and plaster are 1.6, 0.3, and
0.14 W/m-K, respectively. Calculate:
a. the rate of heat loss per unit area of wall surface (1344 W)
b. the temperature of each interface throughout the wall
(995, 780, 220, 104 C)
c. the temperature at the outside surface of the wall ( 104.1C)
23. Exhaust gases flowing through a tubular heat exchanger at the rate of 0.3 kg/sec are
cooled from 400 to 120C by water initially at 10C. The specific heat capacities of
exhaust gases and water may be taken as 1.13 and 4.19 KJ/kg-K respectively, and the
overall heat transfer coefficient from gases to water is 140 W/m2
-K. Calculate the
surface area required when the cooling water flow is 0.4 kg/sec;
a. for parallel flow (4.01 m2
)
b. for counter flow (3.37 m2
)
24. A properly designed steam heated tubular pre heater is heating 5.7 kg/sec of air from 21C
to 77C when using steam at 0.14 MPa. It is proposed to double the rate of air flow
through the heater and yet heat the air from 21C to 77C; this is to be accomplished by
increasing the steam pressure. Calculate the new steam pressure required to meet the
change condition expressed in KPa. (661.5 KPa)
25. A 404.34 m2
heating surface counter-flow economizer is used in conjunction with a 72,000
kg/hr boiler. the inlet and outlet water temperature are 100C and 155C. The inlet and
outlet gas temperature are 340C and 192C. Find the overall coefficient in W/m2
-C.
(85.6 w/M2
-C)
26. A boiler tube for steam at 8.2 MPa (ts = 296.79C; h = 2574.82 KJ/kg) is 9 cm OD and 7 cm
ID. assume an internal film coefficient of 11,350 W/m2
-C and a thermal flow of 157,640
W/m2
based on the outside area, calculate
a) Outside tube temperature
b) Allowable boiler scale thickness if the metal is not to exceed 482C. Assume k for
steel = 43.26 W/m-C anf for scale k = 0.52 W/m-C.
27. An 8" steel pipeline (OD = 22 cm; ID = 19 cm) carries steam at 232C. An 85% magnesia
(k = 0.07 W/m-C) pipe covering is to be applied of such a thickness so as to limit the
surface temperature to 50C with room temperature of 16C. Assume inside and outside
coefficients of 1700 W/m2
-C and 0.011 W/m2
-C and k for steel = 41 W/m-C, calculate
the magnesia thickness in cm,
28.Calculate the energy transfer rate across 6 in wall of firebrick with a temperature difference across the wall

44
of 50C. The thermal conductivity of the firebrick is 0.65 BTU/hr-ft-F at the temperature interest.
C
. 

m-
W
73
1
F
-
ft
-
hr
BTU
=
k = 0.65(1.73) = 1.1245 W/m-C
L = 0.6" = 0.01524 m
2
W/m
3
3689.
L
)
t
Δ
(
k
A
Q
=
−
=
29. A carpenter builds an outer house wall with a layer of wood (k = 0.080 W/m-K) 2 cm thick on the outside
and a layer of styrofoam (k = 0.01 w/m-K) insulation 3.5 cm thick as the inside wall surface. What is the
temperature at the plane where the wood meets the Styrofoam? Interior temperature is 19C; exterior
temperature is -10C.
L1 L2
1
2
3
Q k1 k2
1
1
2
1
2
2
1
1
3
1
k
L
)
t
t
(
k
L
k
L
)
t
t
(
A
Q −
=
+
−
=
t2 = - 8C
Exams
1. An 8" steel pipeline (OD = 22 cm; ID = 19 cm) carries steam at 232C. An 85% magnesia (k = 0.07 W/m-C) pipe
covering is to be applied of such a thickness so as to limit the surface temperature to 50C with room temperature of
16C. Assume inside and outside coefficients of 1700 W/m2
-C and 0.011 W/m2
-C and k for steel = 41 W/m-C,
calculate the magnesia thickness in cm,
2. The emissivity of tungsten is 0.35. A tungsten sphere with a radius of 1.5 cm is suspended within a large enclosure
whose walls are at 290K. What power input is required to maintain the sphere at a temperature of 3000K if heat
conduction along the supports is neglected? (Area of sphere = 4r2
)
3. A copper cylinder is initially at 20C. At what temperature will be its volume be 0.150%larger than it is at 20C.
Coefficient of linear expansion  of copper is 1.7 x 10-5
.
t1 = 19C
t3 = -10C
k1 = 0.01
k2 = 0.08
L1 = 0.035 m
L2 = 0.02 m

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