Dept. of Chemical & Process Engg
CHP 371: Heat Transfer I
4Moi University
School of Engineering
Department of Chemical & Process Engineering
CHP 371: HEAT TRANSFER I
NOTES
1
Dept. of Chemical & Process Engg
CHP 371: Heat Transfer I
CHP 371 HEAT TRANSFER
CHP 371: Heat Transfer I
Dept. of Chemical & Process Engg
4Moi University
School of Engineering
Department of Chemical & Process
Engineering
CHP 371: HEAT TRANSFER I
NOTES
1
CHP 371: Heat Transfer I
Dept. of Chemical & Process Engg
CHP 371 HEAT TRANSFER
U=
12
Z
L
N(x)2 E(x) A(x)
dx , (3)
and if N, E, and A are constant
U=
12
N2 L E A
.
Alternatively, we may express the strain as a function of the displacements along the bar ux(x), xx =
ux(x)/x, and xx = E ux(x)/x. Again substituting dV = A dx ,
U=
12
Z
L
k3
23
23
3
3 1 R T)(T T)(T x kA
q
=
=
(2.15)
As the system is steadystate and no internal heat generated, the heat flows enter and exit each layer
are equal. Therefore:
3x12 qqqq =
n
03
k3k2k1
03
x
R
TT
RRR
TT
q
=
+
=
(2.17)
(2.16) Then, by combining eq
TT kAq 21 x =
(2.12)
If we define a conductive thermal resistance as:
Ak
L
Rk
=
(2.13) Then equation 2.12 may be written as:
k
21
x
R
TT
q
=
(2.14)
This equation is analogous to the relation for electric current flow:
e
21 R VV
I
=
(2.15)
Based on this an
q
=
(2.14)
This equation is analogous to the relation for electric current flow:
e
21 R VV
I
=
(2.15)
Based on this analogy, the system can be drawn schematically as:
T2 T 1 Rk
Figure 2.4 Conductive thermal resistance.
2.2.2 Composite wall (materials in s
R
T)(T
T)(T
x
kA
q
=
=
(2.14)
Heat transfer rate through material 3
k3
23
23
3
3 1 R T)(T T)(T x kA
q
=
=
(2.15)
As the system is steadystate and no internal heat generated, the heat flows enter and exit each layer
are equal. Therefore:
3x12 qqqq =
n
03
+
=
+
+=
(2.33)
Then
ck RR
1
UA
+ =
(2.34)
2.3 General conduction equation based on Polar Cylindrical Coordinates
q+d
q
qz+dz
qz
dr
dz
rd
qr+dr
qr
Figure 2.9 Volume element in a cylindrical coordinates.
Similar to the case for Cartesian Coordinates, ener
=
+
+
+
+
(2.36)
2.3.1 Heat transfer to/from a circular duct
L
T
T1
T2
R1
R2
r
Figure 2.10 Heat conduction through a cylindrical wall.
Assuming that, the system is steadystate, there is no internal heat generation, temperature varies
only with r. Thus
Figure 2.5 Composite wall with material in series.
Heat transfer rate through material 1
k1
01
01
1
1
1
R
T)(T
T)(T
x
kA
q
=
=
(2.13)
Heat transfer rate through material 2
k2
12
12
2
2
2
R
T)(T
T)(T
x
kA
q
=
=
(2.14)
Heat transfer rate through material 3
T2 T1 T2 T3 Figure 2.11 Heat flow through a cylinder with convection.
The heat transfer between the fluids can be obtained similar to the case for composites wall (series),
thus:
c2k2k1c1
12 r RR RR TT
q
+ =
(2.49)
and the thermal resistances are:
RLh2
1
23c2
32
2 233c22
3
hR 1
k R)R/ln(
hR1/kR1/T)(T2
0
dR L)q/d(
+
=
(2.56)
solving for R3cri gives:
c2
2
3cri
h
k
R
=
(2.57)
2.3.4 Cylinder with internal heat generation
Tw
T
R
dz
d
)kTTA(hd cs =
T)(T
dz
dA
k
h
dZ
dT
A
dz
dT
dz dA cs 2 2 =+
)0TT(
dz
dA
A
1
k
h
dz
dT
dz
dA
A
1
(3.8)
(3.7) Then divide by kdz :
dZ dT cs 2 2 =+
If we define
dz
d
(3.9)
(3.10) = TT Then
d
kA
=
=
(3.29)
[
]z0
w
z)(m)coshmLsinh(m
).mzsinh(L)(mmcosh
L)(mcosh kA
=
=
(3.30)
as sinh 0 = 0 and cosh 0 = 1 then L) (mmtanhkAq zw =
(3.31)
3.3 Fin efficiency and effectiveness The fin efficiency is defined as:
temperatureatwallfinisifentiretransferr
Matrix Analysis of Frame Structures
8.24 ENGINEERING MECHANICS FOR STRUCTURES
posing corresponding components at each of the two nodes, produces the stiffness matrix for the
whole structure. We obtain:
X1 Y1 M1 X2 Y2 M2
AE a 12EI
b
3 + 0
6EI
b
2 AE a 
=
kL2
R)R/ln(
R
2
32
k2
=
Based on the inner surface area, the heat transfer rate may be obtained as:
T)(TRL(2qU 1211r =
surface area is:
c23
1
2
321
1
211
c1
1
Rh
R
k
R)R/ln(R
k
R)R/ln(R
h
1
1
(2.50) The overall heat transfer coefficient based on the i
.List down the various types of AnalogtoDigital converters and explain how each functions.
Q2.Explain with aid of a diagram how a ballistic Galvanometer functions and particularly how
damping is achieved.
Q3.Explain with aid of a well labeled diagram, h
4
d
dv
2
x
2
2
d
dv
+=0
where 2 P EI =v=0;
xd dv
=0
vx () c
1 =c 2 xc 3
sinxc
4
cosx+
Problems  Stesses/Deections, Beams in
8.38 ENGINEERING MECHANICS FOR STRUCTURES
Stainless steel (316)
14.4 Glass
0.81 Concrete
0.128 Fiberglass wool
0.0262 Water
0.540 Wood
0.17

0.040 Air
Convection
Convection is the mode of heat transfer between a solid surface and the adjacent fluid that
is in motion, and it involves the combined
Maximum Shear Stress = max = Greatest of
( 1  3 ) / 2
( 1  2 ) / 2 : ( 2  3 ) / 2 : ( 1  3 ) / 2 =
The factor of safety selected would be
FoS = Sy / ( 2 . max ) = Sy / ( 1  3 )
The theory is conservative especially if the yield strength is more then
equation 2.41 can be used to calculate temperature at r when R1 < r < R2. The heat transfer through
the pipe wall can be obtained by applying Fouriers law of heat conduction, thus:
r
T
L)2rk(q r
=
(2.44)
by differentiating equation 2.43 and combining wit
thus for a prescribed geometry, the dimensionless velocity has the following dependencies () * * * ,
,Re Lu f x y =
*
*
*00
s
yy
uVuyLy
=
=
and the shear stress that the surface can be expressed as
Similarity Considerations (cont.)
The localfriction co
The difference between the rates of radiation emitted by the surface and the
radiation absorbed is the net radiation heat transfer. If the rate of radiation absorption is
greater than the rate of radiation emission, the surface is said to be gaining energ
Example 4.1 A heattreating furnace has outside dimensions of 15 mm 150 mm200 mm. The walls
are 6 mm thick and made of fireclay brick. For an inside wall temperature of 550C and an outside
wall temperature of 30C, determine the heat lost through the walls
ENGINEERING MECHANICS FOR STRUCTURES 8.17
or we can set
sin [PL2/(EI)]1/2= 0
which has roots PL2/(EI) = 2, 42, 92,.
The critical eigenvalue will be the lowest one, the one which gives the lowest value for the end load
P. We see that this depends upon the
and take a totally unmotivated step, multiplying both side of this equation by a function of x which
can be anything whatso ever, we integrate over the length of the beam:
This arbitrary function bears an asterisk to distinguish it from the actual bendin
Signal processing
minimmn change in level to be detected is 0.1 m. The empty vessel has a weight of 50 kg. The acid
has a density of 1050 kg/ml
Because of the corrosive nature of the acid there could be problems in using a sensor which is
inserted in the
Convection Heat Transfer
Forced convection is achieved by subjecting the fluid to a pressure gradient (e.g., by a fan or pump),
thereby forcing motion to occur according to the laws of fluid mechanics.
Convective heat transfer rate is calculated from Neut