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Chapter 4 Transient Heat Conduction 4-55 Transient Heat Conduction in Multidimensional Systems Notes | EduRev

: Chapter 4 Transient Heat Conduction 4-55 Transient Heat Conduction in Multidimensional Systems Notes | EduRev

``` Page 1

Chapter 4 Transient Heat Conduction
4-55

Transient Heat Conduction in Multidimensional Systems

4-69C The product solution enables us to determine the dimensionless temperature of two- or three-
dimensional heat transfer problems as the product of dimensionless temperatures of one-dimensional heat
transfer problems. The dimensionless temperature for a two-dimensional problem is determined by
determining the dimensionless temperatures in both directions, and taking their product.

4-70C The dimensionless temperature for a three-dimensional heat transfer is determined by determining
the dimensionless temperatures of one-dimensional geometries whose intersection is the three dimensional
geometry, and taking their product.

4-71C This short cylinder is physically formed by the intersection of a long cylinder and a plane wall. The
dimensionless temperatures at the center of plane wall and at the center of the cylinder are determined first.
Their product yields the dimensionless temperature at the center of the short cylinder.

4-72C The heat transfer in this short cylinder is one-dimensional since there is no heat transfer in the axial
direction. The temperature will vary in the radial direction only.

Page 2

Chapter 4 Transient Heat Conduction
4-55

Transient Heat Conduction in Multidimensional Systems

4-69C The product solution enables us to determine the dimensionless temperature of two- or three-
dimensional heat transfer problems as the product of dimensionless temperatures of one-dimensional heat
transfer problems. The dimensionless temperature for a two-dimensional problem is determined by
determining the dimensionless temperatures in both directions, and taking their product.

4-70C The dimensionless temperature for a three-dimensional heat transfer is determined by determining
the dimensionless temperatures of one-dimensional geometries whose intersection is the three dimensional
geometry, and taking their product.

4-71C This short cylinder is physically formed by the intersection of a long cylinder and a plane wall. The
dimensionless temperatures at the center of plane wall and at the center of the cylinder are determined first.
Their product yields the dimensionless temperature at the center of the short cylinder.

4-72C The heat transfer in this short cylinder is one-dimensional since there is no heat transfer in the axial
direction. The temperature will vary in the radial direction only.

Chapter 4 Transient Heat Conduction
4-56
4-73 A short cylinder is allowed to cool in atmospheric air. The temperatures at the centers of the cylinder
and the top surface as well as the total heat transfer from the cylinder for 15 min of cooling are to be
determined.
Assumptions 1 Heat conduction in the short cylinder is two-dimensional, and thus the temperature varies in
both the axial x- and the radial r- directions. 2 The thermal properties of the cylinder are constant. 3 The
heat transfer coefficient is constant and uniform over the entire surface. 4 The Fourier number is ? > 0.2 so
that the one-term approximate solutions (or the transient temperature charts) are applicable (this assumption
will be verified).
Properties The thermal properties of brass are given to  be  ? ? 8530 kg/ m
3
, C kJ/kg 389 . 0 ? ? ?
p
C ,
C W/m 110 ? ? ? k , and ? ? ?
?
3 39 10
5
. m / s
2
.
Analysis This short cylinder can physically be formed by the intersection of a long cylinder of radius D/2 =
4 cm and a plane wall of thickness 2L = 15 cm. We measure x from the midplane.
(a) The Biot number is calculated for the plane wall to be
02727 . 0
) C W/m. 110 (
) m 075 . 0 )( C . W/m 40 (
2
?
?
?
? ?
k
hL
Bi
The constants ?
1 1
and A corresponding to this
Biot number are, from Table 4-1,
0050 . 1   and   164 . 0
1 1
? ? A ?
The Fourier number is
?
?
? ?
? ?
? ?
?
t
L
2
5
339 10
0 075
5424 0 2
( .
( .
. .
m / s)(15 min 60 s / min)
m)
2
2

Therefore, the one-term approximate solution (or the transient temperature charts) is applicable. Then the
dimensionless temperature at the center of the plane wall is determined from
869 . 0 ) 0050 . 1 (
) 424 . 5 ( ) 164 . 0 (
1
0
,
2 2
1
? ? ?
?
?
? ?
? ? ? ?
?
?
e e A
T T
T T
i
wall o

We repeat the same calculations for the long cylinder,
01455 . 0
) C W/m. 110 (
) m 04 . 0 )( C . W/m 40 (
2
0
?
?
?
? ?
k
hr
Bi
?
1 1
01704 10038 ? ? . .   and   A
?
?
? ?
? ?
? ?
?
t
r
o
2
5
339 10
0 04
19 069 0 2
( .
( .
. .
m / s)(15 60 s)
m)
2
2

?
? ?
o cyl
i
T T
T T
A e e
,
( . ) ( . )
( . ) . ?
?
?
? ? ?
?
?
? ? 0
1
0 1704 19 069
1
2 2
10038 0577
Then the center temperature of the short cylinder becomes

C 85.1 ? ? ? ? ? ?
?
?
? ? ? ? ? ? ?
?
?
?
?
?
?
?
?
?
?
) , 0 , 0 ( 501 . 0
20 150
20 ) , 0 , 0 (
501 . 0 577 . 0 869 . 0
) , 0 , 0 (
, ,
t T
t T
T T
T t T
cyl o wall o
cylinder
short
i

D0 = 8 cm
L = 15 cm
z
Brass cylinder
Ti = 150 ?C
r
Air
T ? = 20 ?C

Page 3

Chapter 4 Transient Heat Conduction
4-55

Transient Heat Conduction in Multidimensional Systems

4-69C The product solution enables us to determine the dimensionless temperature of two- or three-
dimensional heat transfer problems as the product of dimensionless temperatures of one-dimensional heat
transfer problems. The dimensionless temperature for a two-dimensional problem is determined by
determining the dimensionless temperatures in both directions, and taking their product.

4-70C The dimensionless temperature for a three-dimensional heat transfer is determined by determining
the dimensionless temperatures of one-dimensional geometries whose intersection is the three dimensional
geometry, and taking their product.

4-71C This short cylinder is physically formed by the intersection of a long cylinder and a plane wall. The
dimensionless temperatures at the center of plane wall and at the center of the cylinder are determined first.
Their product yields the dimensionless temperature at the center of the short cylinder.

4-72C The heat transfer in this short cylinder is one-dimensional since there is no heat transfer in the axial
direction. The temperature will vary in the radial direction only.

Chapter 4 Transient Heat Conduction
4-56
4-73 A short cylinder is allowed to cool in atmospheric air. The temperatures at the centers of the cylinder
and the top surface as well as the total heat transfer from the cylinder for 15 min of cooling are to be
determined.
Assumptions 1 Heat conduction in the short cylinder is two-dimensional, and thus the temperature varies in
both the axial x- and the radial r- directions. 2 The thermal properties of the cylinder are constant. 3 The
heat transfer coefficient is constant and uniform over the entire surface. 4 The Fourier number is ? > 0.2 so
that the one-term approximate solutions (or the transient temperature charts) are applicable (this assumption
will be verified).
Properties The thermal properties of brass are given to  be  ? ? 8530 kg/ m
3
, C kJ/kg 389 . 0 ? ? ?
p
C ,
C W/m 110 ? ? ? k , and ? ? ?
?
3 39 10
5
. m / s
2
.
Analysis This short cylinder can physically be formed by the intersection of a long cylinder of radius D/2 =
4 cm and a plane wall of thickness 2L = 15 cm. We measure x from the midplane.
(a) The Biot number is calculated for the plane wall to be
02727 . 0
) C W/m. 110 (
) m 075 . 0 )( C . W/m 40 (
2
?
?
?
? ?
k
hL
Bi
The constants ?
1 1
and A corresponding to this
Biot number are, from Table 4-1,
0050 . 1   and   164 . 0
1 1
? ? A ?
The Fourier number is
?
?
? ?
? ?
? ?
?
t
L
2
5
339 10
0 075
5424 0 2
( .
( .
. .
m / s)(15 min 60 s / min)
m)
2
2

Therefore, the one-term approximate solution (or the transient temperature charts) is applicable. Then the
dimensionless temperature at the center of the plane wall is determined from
869 . 0 ) 0050 . 1 (
) 424 . 5 ( ) 164 . 0 (
1
0
,
2 2
1
? ? ?
?
?
? ?
? ? ? ?
?
?
e e A
T T
T T
i
wall o

We repeat the same calculations for the long cylinder,
01455 . 0
) C W/m. 110 (
) m 04 . 0 )( C . W/m 40 (
2
0
?
?
?
? ?
k
hr
Bi
?
1 1
01704 10038 ? ? . .   and   A
?
?
? ?
? ?
? ?
?
t
r
o
2
5
339 10
0 04
19 069 0 2
( .
( .
. .
m / s)(15 60 s)
m)
2
2

?
? ?
o cyl
i
T T
T T
A e e
,
( . ) ( . )
( . ) . ?
?
?
? ? ?
?
?
? ? 0
1
0 1704 19 069
1
2 2
10038 0577
Then the center temperature of the short cylinder becomes

C 85.1 ? ? ? ? ? ?
?
?
? ? ? ? ? ? ?
?
?
?
?
?
?
?
?
?
?
) , 0 , 0 ( 501 . 0
20 150
20 ) , 0 , 0 (
501 . 0 577 . 0 869 . 0
) , 0 , 0 (
, ,
t T
t T
T T
T t T
cyl o wall o
cylinder
short
i

D0 = 8 cm
L = 15 cm
z
Brass cylinder
Ti = 150 ?C
r
Air
T ? = 20 ?C

Chapter 4 Transient Heat Conduction
4-57
(b) The center of the top surface of the cylinder is still at the center of the long cylinder (r = 0), but at the
outer surface of the plane wall (x = L). Therefore, we first need to determine the dimensionless temperature
at the surface of the wall.
857 . 0 ) 164 . 0 cos( ) 0050 . 1 ( ) / cos(
) , (
) , (
) 424 . 5 ( ) 164 . 0 (
1 1
2 2
1
? ? ?
?
?
?
? ?
?
?
e L L e A
T T
T t x T
t L
i
wall
? ?
? ?

Then the center temperature of the top surface of the cylinder becomes

C 84.2 ? ? ? ? ? ?
?
?
? ? ? ? ?
?
?
?
?
?
?
?
?
?
?
) , 0 , ( 494 . 0
20 150
20 ) , 0 , (
494 . 0 577 . 0 857 . 0 ) , (
) , 0 , (
,
t L T
t L T
t L
T T
T t L T
cyl o wall
cylinder
short
i
? ?

(c) We first need to determine the maximum heat can be transferred from the cylinder

? ?
kJ 325 C ) 20 150 )( C kJ/kg. 389 . 0 )( kg 43 . 6 ( ) (
kg 43 . 6 m) 15 . 0 ( m) 04 . 0 ( ) kg/m 8530 (
max
2. 3 2
? ? ? ? ? ? ?
? ? ? ?? ? ? ?
?
T T mC Q
L r V m
i p
o

Then we determine the dimensionless heat transfer ratios for both geometries as
135 . 0
164 . 0
) 164 . 0 sin(
) 869 . 0 ( 1
) sin(
1
1
1
,
max
? ? ?
?
?
? ? ?
?
?
?
?
?
?
?
?
wall o
wall
Q
Q

427 . 0
1704 . 0
0846 . 0
) 577 . 0 ( 2 1
) (
2 1
1
1 1
,
max
? ? ? ? ?
?
?
?
?
?
?
?
?
?
?
?
J
Q
Q
cyl o
cyl

The heat transfer ratio for the short cylinder is
504 . 0 ) 135 . 0 1 )( 427 . 0 ( 135 . 0 1
max max max max
? ? ? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
wall
plane
cylinder
long
wall
plane
cylinder
short Q
Q
Q
Q
Q
Q
Q
Q

Then the total heat transfer from the short cylinder during the first 15 minutes of cooling becomes
kJ 164 ? ? ? kJ) 325 )( 504 . 0 ( 503 . 0
max
Q Q

Page 4

Chapter 4 Transient Heat Conduction
4-55

Transient Heat Conduction in Multidimensional Systems

4-69C The product solution enables us to determine the dimensionless temperature of two- or three-
dimensional heat transfer problems as the product of dimensionless temperatures of one-dimensional heat
transfer problems. The dimensionless temperature for a two-dimensional problem is determined by
determining the dimensionless temperatures in both directions, and taking their product.

4-70C The dimensionless temperature for a three-dimensional heat transfer is determined by determining
the dimensionless temperatures of one-dimensional geometries whose intersection is the three dimensional
geometry, and taking their product.

4-71C This short cylinder is physically formed by the intersection of a long cylinder and a plane wall. The
dimensionless temperatures at the center of plane wall and at the center of the cylinder are determined first.
Their product yields the dimensionless temperature at the center of the short cylinder.

4-72C The heat transfer in this short cylinder is one-dimensional since there is no heat transfer in the axial
direction. The temperature will vary in the radial direction only.

Chapter 4 Transient Heat Conduction
4-56
4-73 A short cylinder is allowed to cool in atmospheric air. The temperatures at the centers of the cylinder
and the top surface as well as the total heat transfer from the cylinder for 15 min of cooling are to be
determined.
Assumptions 1 Heat conduction in the short cylinder is two-dimensional, and thus the temperature varies in
both the axial x- and the radial r- directions. 2 The thermal properties of the cylinder are constant. 3 The
heat transfer coefficient is constant and uniform over the entire surface. 4 The Fourier number is ? > 0.2 so
that the one-term approximate solutions (or the transient temperature charts) are applicable (this assumption
will be verified).
Properties The thermal properties of brass are given to  be  ? ? 8530 kg/ m
3
, C kJ/kg 389 . 0 ? ? ?
p
C ,
C W/m 110 ? ? ? k , and ? ? ?
?
3 39 10
5
. m / s
2
.
Analysis This short cylinder can physically be formed by the intersection of a long cylinder of radius D/2 =
4 cm and a plane wall of thickness 2L = 15 cm. We measure x from the midplane.
(a) The Biot number is calculated for the plane wall to be
02727 . 0
) C W/m. 110 (
) m 075 . 0 )( C . W/m 40 (
2
?
?
?
? ?
k
hL
Bi
The constants ?
1 1
and A corresponding to this
Biot number are, from Table 4-1,
0050 . 1   and   164 . 0
1 1
? ? A ?
The Fourier number is
?
?
? ?
? ?
? ?
?
t
L
2
5
339 10
0 075
5424 0 2
( .
( .
. .
m / s)(15 min 60 s / min)
m)
2
2

Therefore, the one-term approximate solution (or the transient temperature charts) is applicable. Then the
dimensionless temperature at the center of the plane wall is determined from
869 . 0 ) 0050 . 1 (
) 424 . 5 ( ) 164 . 0 (
1
0
,
2 2
1
? ? ?
?
?
? ?
? ? ? ?
?
?
e e A
T T
T T
i
wall o

We repeat the same calculations for the long cylinder,
01455 . 0
) C W/m. 110 (
) m 04 . 0 )( C . W/m 40 (
2
0
?
?
?
? ?
k
hr
Bi
?
1 1
01704 10038 ? ? . .   and   A
?
?
? ?
? ?
? ?
?
t
r
o
2
5
339 10
0 04
19 069 0 2
( .
( .
. .
m / s)(15 60 s)
m)
2
2

?
? ?
o cyl
i
T T
T T
A e e
,
( . ) ( . )
( . ) . ?
?
?
? ? ?
?
?
? ? 0
1
0 1704 19 069
1
2 2
10038 0577
Then the center temperature of the short cylinder becomes

C 85.1 ? ? ? ? ? ?
?
?
? ? ? ? ? ? ?
?
?
?
?
?
?
?
?
?
?
) , 0 , 0 ( 501 . 0
20 150
20 ) , 0 , 0 (
501 . 0 577 . 0 869 . 0
) , 0 , 0 (
, ,
t T
t T
T T
T t T
cyl o wall o
cylinder
short
i

D0 = 8 cm
L = 15 cm
z
Brass cylinder
Ti = 150 ?C
r
Air
T ? = 20 ?C

Chapter 4 Transient Heat Conduction
4-57
(b) The center of the top surface of the cylinder is still at the center of the long cylinder (r = 0), but at the
outer surface of the plane wall (x = L). Therefore, we first need to determine the dimensionless temperature
at the surface of the wall.
857 . 0 ) 164 . 0 cos( ) 0050 . 1 ( ) / cos(
) , (
) , (
) 424 . 5 ( ) 164 . 0 (
1 1
2 2
1
? ? ?
?
?
?
? ?
?
?
e L L e A
T T
T t x T
t L
i
wall
? ?
? ?

Then the center temperature of the top surface of the cylinder becomes

C 84.2 ? ? ? ? ? ?
?
?
? ? ? ? ?
?
?
?
?
?
?
?
?
?
?
) , 0 , ( 494 . 0
20 150
20 ) , 0 , (
494 . 0 577 . 0 857 . 0 ) , (
) , 0 , (
,
t L T
t L T
t L
T T
T t L T
cyl o wall
cylinder
short
i
? ?

(c) We first need to determine the maximum heat can be transferred from the cylinder

? ?
kJ 325 C ) 20 150 )( C kJ/kg. 389 . 0 )( kg 43 . 6 ( ) (
kg 43 . 6 m) 15 . 0 ( m) 04 . 0 ( ) kg/m 8530 (
max
2. 3 2
? ? ? ? ? ? ?
? ? ? ?? ? ? ?
?
T T mC Q
L r V m
i p
o

Then we determine the dimensionless heat transfer ratios for both geometries as
135 . 0
164 . 0
) 164 . 0 sin(
) 869 . 0 ( 1
) sin(
1
1
1
,
max
? ? ?
?
?
? ? ?
?
?
?
?
?
?
?
?
wall o
wall
Q
Q

427 . 0
1704 . 0
0846 . 0
) 577 . 0 ( 2 1
) (
2 1
1
1 1
,
max
? ? ? ? ?
?
?
?
?
?
?
?
?
?
?
?
J
Q
Q
cyl o
cyl

The heat transfer ratio for the short cylinder is
504 . 0 ) 135 . 0 1 )( 427 . 0 ( 135 . 0 1
max max max max
? ? ? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
wall
plane
cylinder
long
wall
plane
cylinder
short Q
Q
Q
Q
Q
Q
Q
Q

Then the total heat transfer from the short cylinder during the first 15 minutes of cooling becomes
kJ 164 ? ? ? kJ) 325 )( 504 . 0 ( 503 . 0
max
Q Q

Chapter 4 Transient Heat Conduction
4-58
4-74
"!PROBLEM 4-74"

"GIVEN"
D=0.08 "[m]"
r_o=D/2
height=0.15 "[m]"
L=height/2
T_i=150 "[C]"
T_infinity=20 "[C]"
h=40 "[W/m^2-C]"
"time=15 [min], parameter to be varied"

"PROPERTIES"
k=110 "[W/m-C]"
rho=8530 "[kg/m^3]"
C_p=0.389 "[kJ/kg-C]"
alpha=3.39E-5 "[m^2/s]"

"ANALYSIS"
"(a)"
"This short cylinder can physically be formed by the intersection of a long cylinder of radius r_o
and a plane wall of thickness 2L"
"For plane wall"
Bi_w=(h*L)/k
"From Table 4-1 corresponding to this Bi number, we read"
lambda_1_w=0.2282 "w stands for wall"
A_1_w=1.0060
tau_w=(alpha*time*Convert(min, s))/L^2
theta_o_w=A_1_w*exp(-lambda_1_w^2*tau_w) "theta_o_w=(T_o_w-T_infinity)/(T_i-T_infinity)"
"For long cylinder"
Bi_c=(h*r_o)/k "c stands for cylinder"
"From Table 4-1 corresponding to this Bi number, we read"
lambda_1_c=0.1704
A_1_c=1.0038
tau_c=(alpha*time*Convert(min, s))/r_o^2
theta_o_c=A_1_c*exp(-lambda_1_c^2*tau_c) "theta_o_c=(T_o_c-T_infinity)/(T_i-T_infinity)"
(T_o_o-T_infinity)/(T_i-T_infinity)=theta_o_w*theta_o_c "center temperature of short cylinder"
"(b)"
theta_L_w=A_1_w*exp(-lambda_1_w^2*tau_w)*Cos(lambda_1_w*L/L) "theta_L_w=(T_L_w-
T_infinity)/(T_i-T_infinity)"
(T_L_o-T_infinity)/(T_i-T_infinity)=theta_L_w*theta_o_c "center temperature of the top surface"
"(c)"
V=pi*r_o^2*(2*L)
m=rho*V
Q_max=m*C_p*(T_i-T_infinity)
Q_w=1-theta_o_w*Sin(lambda_1_w)/lambda_1_w "Q_w=(Q/Q_max)_w"
Q_c=1-2*theta_o_c*J_1/lambda_1_c "Q_c=(Q/Q_max)_c"
J_1=0.0846 "From Table 4-2, at lambda_1_c"
Q/Q_max=Q_w+Q_c*(1-Q_w) "total heat transfer"
Page 5

Chapter 4 Transient Heat Conduction
4-55

Transient Heat Conduction in Multidimensional Systems

4-69C The product solution enables us to determine the dimensionless temperature of two- or three-
dimensional heat transfer problems as the product of dimensionless temperatures of one-dimensional heat
transfer problems. The dimensionless temperature for a two-dimensional problem is determined by
determining the dimensionless temperatures in both directions, and taking their product.

4-70C The dimensionless temperature for a three-dimensional heat transfer is determined by determining
the dimensionless temperatures of one-dimensional geometries whose intersection is the three dimensional
geometry, and taking their product.

4-71C This short cylinder is physically formed by the intersection of a long cylinder and a plane wall. The
dimensionless temperatures at the center of plane wall and at the center of the cylinder are determined first.
Their product yields the dimensionless temperature at the center of the short cylinder.

4-72C The heat transfer in this short cylinder is one-dimensional since there is no heat transfer in the axial
direction. The temperature will vary in the radial direction only.

Chapter 4 Transient Heat Conduction
4-56
4-73 A short cylinder is allowed to cool in atmospheric air. The temperatures at the centers of the cylinder
and the top surface as well as the total heat transfer from the cylinder for 15 min of cooling are to be
determined.
Assumptions 1 Heat conduction in the short cylinder is two-dimensional, and thus the temperature varies in
both the axial x- and the radial r- directions. 2 The thermal properties of the cylinder are constant. 3 The
heat transfer coefficient is constant and uniform over the entire surface. 4 The Fourier number is ? > 0.2 so
that the one-term approximate solutions (or the transient temperature charts) are applicable (this assumption
will be verified).
Properties The thermal properties of brass are given to  be  ? ? 8530 kg/ m
3
, C kJ/kg 389 . 0 ? ? ?
p
C ,
C W/m 110 ? ? ? k , and ? ? ?
?
3 39 10
5
. m / s
2
.
Analysis This short cylinder can physically be formed by the intersection of a long cylinder of radius D/2 =
4 cm and a plane wall of thickness 2L = 15 cm. We measure x from the midplane.
(a) The Biot number is calculated for the plane wall to be
02727 . 0
) C W/m. 110 (
) m 075 . 0 )( C . W/m 40 (
2
?
?
?
? ?
k
hL
Bi
The constants ?
1 1
and A corresponding to this
Biot number are, from Table 4-1,
0050 . 1   and   164 . 0
1 1
? ? A ?
The Fourier number is
?
?
? ?
? ?
? ?
?
t
L
2
5
339 10
0 075
5424 0 2
( .
( .
. .
m / s)(15 min 60 s / min)
m)
2
2

Therefore, the one-term approximate solution (or the transient temperature charts) is applicable. Then the
dimensionless temperature at the center of the plane wall is determined from
869 . 0 ) 0050 . 1 (
) 424 . 5 ( ) 164 . 0 (
1
0
,
2 2
1
? ? ?
?
?
? ?
? ? ? ?
?
?
e e A
T T
T T
i
wall o

We repeat the same calculations for the long cylinder,
01455 . 0
) C W/m. 110 (
) m 04 . 0 )( C . W/m 40 (
2
0
?
?
?
? ?
k
hr
Bi
?
1 1
01704 10038 ? ? . .   and   A
?
?
? ?
? ?
? ?
?
t
r
o
2
5
339 10
0 04
19 069 0 2
( .
( .
. .
m / s)(15 60 s)
m)
2
2

?
? ?
o cyl
i
T T
T T
A e e
,
( . ) ( . )
( . ) . ?
?
?
? ? ?
?
?
? ? 0
1
0 1704 19 069
1
2 2
10038 0577
Then the center temperature of the short cylinder becomes

C 85.1 ? ? ? ? ? ?
?
?
? ? ? ? ? ? ?
?
?
?
?
?
?
?
?
?
?
) , 0 , 0 ( 501 . 0
20 150
20 ) , 0 , 0 (
501 . 0 577 . 0 869 . 0
) , 0 , 0 (
, ,
t T
t T
T T
T t T
cyl o wall o
cylinder
short
i

D0 = 8 cm
L = 15 cm
z
Brass cylinder
Ti = 150 ?C
r
Air
T ? = 20 ?C

Chapter 4 Transient Heat Conduction
4-57
(b) The center of the top surface of the cylinder is still at the center of the long cylinder (r = 0), but at the
outer surface of the plane wall (x = L). Therefore, we first need to determine the dimensionless temperature
at the surface of the wall.
857 . 0 ) 164 . 0 cos( ) 0050 . 1 ( ) / cos(
) , (
) , (
) 424 . 5 ( ) 164 . 0 (
1 1
2 2
1
? ? ?
?
?
?
? ?
?
?
e L L e A
T T
T t x T
t L
i
wall
? ?
? ?

Then the center temperature of the top surface of the cylinder becomes

C 84.2 ? ? ? ? ? ?
?
?
? ? ? ? ?
?
?
?
?
?
?
?
?
?
?
) , 0 , ( 494 . 0
20 150
20 ) , 0 , (
494 . 0 577 . 0 857 . 0 ) , (
) , 0 , (
,
t L T
t L T
t L
T T
T t L T
cyl o wall
cylinder
short
i
? ?

(c) We first need to determine the maximum heat can be transferred from the cylinder

? ?
kJ 325 C ) 20 150 )( C kJ/kg. 389 . 0 )( kg 43 . 6 ( ) (
kg 43 . 6 m) 15 . 0 ( m) 04 . 0 ( ) kg/m 8530 (
max
2. 3 2
? ? ? ? ? ? ?
? ? ? ?? ? ? ?
?
T T mC Q
L r V m
i p
o

Then we determine the dimensionless heat transfer ratios for both geometries as
135 . 0
164 . 0
) 164 . 0 sin(
) 869 . 0 ( 1
) sin(
1
1
1
,
max
? ? ?
?
?
? ? ?
?
?
?
?
?
?
?
?
wall o
wall
Q
Q

427 . 0
1704 . 0
0846 . 0
) 577 . 0 ( 2 1
) (
2 1
1
1 1
,
max
? ? ? ? ?
?
?
?
?
?
?
?
?
?
?
?
J
Q
Q
cyl o
cyl

The heat transfer ratio for the short cylinder is
504 . 0 ) 135 . 0 1 )( 427 . 0 ( 135 . 0 1
max max max max
? ? ? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
wall
plane
cylinder
long
wall
plane
cylinder
short Q
Q
Q
Q
Q
Q
Q
Q

Then the total heat transfer from the short cylinder during the first 15 minutes of cooling becomes
kJ 164 ? ? ? kJ) 325 )( 504 . 0 ( 503 . 0
max
Q Q

Chapter 4 Transient Heat Conduction
4-58
4-74
"!PROBLEM 4-74"

"GIVEN"
D=0.08 "[m]"
r_o=D/2
height=0.15 "[m]"
L=height/2
T_i=150 "[C]"
T_infinity=20 "[C]"
h=40 "[W/m^2-C]"
"time=15 [min], parameter to be varied"

"PROPERTIES"
k=110 "[W/m-C]"
rho=8530 "[kg/m^3]"
C_p=0.389 "[kJ/kg-C]"
alpha=3.39E-5 "[m^2/s]"

"ANALYSIS"
"(a)"
"This short cylinder can physically be formed by the intersection of a long cylinder of radius r_o
and a plane wall of thickness 2L"
"For plane wall"
Bi_w=(h*L)/k
"From Table 4-1 corresponding to this Bi number, we read"
lambda_1_w=0.2282 "w stands for wall"
A_1_w=1.0060
tau_w=(alpha*time*Convert(min, s))/L^2
theta_o_w=A_1_w*exp(-lambda_1_w^2*tau_w) "theta_o_w=(T_o_w-T_infinity)/(T_i-T_infinity)"
"For long cylinder"
Bi_c=(h*r_o)/k "c stands for cylinder"
"From Table 4-1 corresponding to this Bi number, we read"
lambda_1_c=0.1704
A_1_c=1.0038
tau_c=(alpha*time*Convert(min, s))/r_o^2
theta_o_c=A_1_c*exp(-lambda_1_c^2*tau_c) "theta_o_c=(T_o_c-T_infinity)/(T_i-T_infinity)"
(T_o_o-T_infinity)/(T_i-T_infinity)=theta_o_w*theta_o_c "center temperature of short cylinder"
"(b)"
theta_L_w=A_1_w*exp(-lambda_1_w^2*tau_w)*Cos(lambda_1_w*L/L) "theta_L_w=(T_L_w-
T_infinity)/(T_i-T_infinity)"
(T_L_o-T_infinity)/(T_i-T_infinity)=theta_L_w*theta_o_c "center temperature of the top surface"
"(c)"
V=pi*r_o^2*(2*L)
m=rho*V
Q_max=m*C_p*(T_i-T_infinity)
Q_w=1-theta_o_w*Sin(lambda_1_w)/lambda_1_w "Q_w=(Q/Q_max)_w"
Q_c=1-2*theta_o_c*J_1/lambda_1_c "Q_c=(Q/Q_max)_c"
J_1=0.0846 "From Table 4-2, at lambda_1_c"
Q/Q_max=Q_w+Q_c*(1-Q_w) "total heat transfer"
Chapter 4 Transient Heat Conduction
4-59

time [min] T [C] T [C] Q [kJ]
5 119.3 116.8 80.58
10 95.18 93.23 140.1
15 76.89 75.42 185.1
20 63.05 61.94 219.2
25 52.58 51.74 245
30 44.66 44.02 264.5
35 38.66 38.18 279.3
40 34.12 33.75 290.5
45 30.69 30.41 298.9
50 28.09 27.88 305.3
55 26.12 25.96 310.2
60 24.63 24.51 313.8

0 10 20 30 40 50 60
20
40
60
80
100
120
50
100
150
200
250
300
350
time  [min]
T
o,o
[C]
Q  [kJ]
temperature
heat

```
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