Heat transfer can occur via three modes: conduction, convection, and radiation. Conduction involves the transfer of energy between adjacent particles in solids, liquids, and gases due to temperature differences. Convection refers to the transfer of heat by the movement of fluids and involves both conduction and fluid motion. Radiation is the emission and transmission of electromagnetic waves from the surface of an object as a result of its temperature. The key modes of heat transfer are analyzed using concepts such as the thermal conductivity of materials, heat transfer coefficients, and Stefan-Boltzmann's law of thermal radiation.
Learn about Conduction, Convection, Radiation and Heat exchangers in a most comprehensive and interactive way. Derivations of formulas, concepts, Numerical, examples are inculcated in the course with advance applications. The course aims at covering all the topics and concepts of HMT as per academics of students. Following are the topics (in detail) that will be covered in the course.
Conduction
Thermal conductivity, Heat conduction in gases, Interpretation Of Fourier's law, Electrical analogy of heat transfer, Critical radius of insulation, Heat generation in a slab and cylinder, Fins, Unsteady/Transient conduction.
Convection
Forced convection heat transfer, Reynold’s Number, Prandtl Number, Nusselt Number, Incompressible flow over flat surface, HBL, TBL, Forced convection in flow through pipes and ducts, Free/Natural convection.
Heat Exchangers
Types of heat exchangers, First law of thermodynamics, Classification of heat exchangers, LMTD for parallel and counter flow, NTU, Fouling factor.
Radiation
Absorbtivity, Reflectivity, Transmitivity, Laws of thermal radiation, Shape factor, Radiation heat exchange
COPY-PASTE below URL to ENROLL in the COMPLETE course & see the hidden contents with proper explanations.
https://www.udemy.com/course/heat-and-mass-transfer
Learn about Conduction, Convection, Radiation and Heat exchangers in a most comprehensive and interactive way. Derivations of formulas, concepts, Numerical, examples are inculcated in the course with advance applications. The course aims at covering all the topics and concepts of HMT as per academics of students. Following are the topics (in detail) that will be covered in the course.
Conduction
Thermal conductivity, Heat conduction in gases, Interpretation Of Fourier's law, Electrical analogy of heat transfer, Critical radius of insulation, Heat generation in a slab and cylinder, Fins, Unsteady/Transient conduction.
Convection
Forced convection heat transfer, Reynold’s Number, Prandtl Number, Nusselt Number, Incompressible flow over flat surface, HBL, TBL, Forced convection in flow through pipes and ducts, Free/Natural convection.
Heat Exchangers
Types of heat exchangers, First law of thermodynamics, Classification of heat exchangers, LMTD for parallel and counter flow, NTU, Fouling factor.
Radiation
Absorbtivity, Reflectivity, Transmitivity, Laws of thermal radiation, Shape factor, Radiation heat exchange
COPY-PASTE below URL to ENROLL in the COMPLETE course & see the hidden contents with proper explanations.
https://www.udemy.com/course/heat-and-mass-transfer
Very useful for the beginners in the field of heat and the mass transfer field. It also gives the idea about the different modes of heat transfer and the measurement of energy transfer rate.
Very useful for the beginners in the field of heat and the mass transfer field. It also gives the idea about the different modes of heat transfer and the measurement of energy transfer rate.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
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Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
1. Process Heat Transfer
Dr. V. Charles Augustin
Sr. Asst. Professor
Department of Applied Science and Technology,
A.C.Tech, Anna University Chennai
2.
3.
4. THERMODYNAMICS AND HEATTRANSFER
Page 4
• Heat: The form of energy that can be transferred from one
system to another as a result of temperature difference.
• Thermodynamics is concerned with the amount of heat
transfer as a system undergoes a process from one
equilibrium state to another.
• Heat Transfer deals with the determination of the rates of
such energy transfers as well as variation of temperature.
• The transfer of energy as heat is always from the higher-
temperature medium to the lower-temperature one.
• Heat transfer stops when the two mediums reach the same
temperature.
• Heat can be transferred in three different modes:
conduction, convection, radiation.
7. 5
ENGINEERING HEAT TRANSFER
The heat transfer problems encountered in practice can be consideredin
two groups: (1) rating and (2) sizing problems.
The rating problems deal with the determination of the heat transfer ratefor
an existing system at a specified temperaturedifference.
The sizing problems deal with the determination of the size of a system in
order to transfer heat at a specified rate for a specified temperature
difference.
An engineering device or process can be studied either
The experimental approach has the advantage that we deal with the actual
physical system, and the desired quantity is determined by measurement,
within the limits of experimental error. However, this approach is expensive,
time-consuming, and often impractical.
or
The analytical approach (including the numerical approach) has the
advantage that it is fast and inexpensive, but the results obtainedare
subject to the accuracy of the assumptions, approximations,and
idP
a
eg
e
a
l5
izations made in the analysis.
8. CONDUCTION
Conduction: The transfer of energy from the
more energetic particles of a substance to the
adjacent less energetic ones as a result of
interactions between the particles.
In gases and liquids, conduction is due to the
collisions and diffusion of the molecules during
their random motion.
In solids, it is due to the combination of
vibrations of the molecules in a lattice and the
energy transport by free electrons.
The rate of heat conduction through a plane
layer is proportional to the temperature
difference across the layer and the heat transfer
area, but is inversely proportional to the
thickness of the layer.
6
Page 8
10. Page 8 8
Fourier’s law of heat conduction
Thermal conductivity, k: Measure of the ability
of a material to conduct heat.
Temperature gradient dT/dx: The slope of the
temperature curve on a T-x diagram.
positive quantity.
conductivity.
In heat conduction
analysis, A
represents the area
normal to the
direction of heat
transfer.
The rate of heat conduction through a
solid is directly proportional to its thermal
11. Heat transfer by Conduction
Conduction heat transfer: Fourier’s law of Conduction
Heat transfer rate per unit area is proportional to temperature
gradient:-
q = heat transfer rate
= Temperature gradient
K = Thermal conductivity
(-) sign indicate heat flows downhill in temperature scale.
19. Thermal Conductivity
Conductivity of Gases: kinetic theory at moderately low
temperature
At high temperature region, the molecules have the higher
velocity than the low temperature region.
Molecules are in continuous random motion, colliding with
each others and exchanging energy and momentum.
If molecules move from higher temperature region to low
temperature region, it transport kinetic energy to low
temperature region through collision with low temperature
molecules.
20. A material that has a high thermal
conductivity or a low heat capacity will
obviously have a large thermal diffusivity.
The larger the thermal diffusivity, the faster
the propagation of heat into the medium.
A small value of thermal diffusivity means
that heat is mostly absorbed by the
P
a
g
e
m1a1terial and a small amount of heat is
conducted further.
Thermal Diffusivity
cp Specific heat, J/kg · C: Heat capacity
per unit mass
cp Heat capacity, J/m3· C: Heat
capacity per unit volume
Thermal diffusivity, m2/s: Represents
how fast heat diffuses through a
material
21. Thermal Conductivity – Points to remember
• It’s a physical property of a substance. k=f(temp, position, nature of the
substance, pr.[gases only])
• For isotropic material, kx=ky=kz=k (Properties of a material are identical in all directions)
• k is higher for pure metals. For alloys k is less than that of pure metals
• The k of liquids and gases is smaller than that of solids because of their larger
intermolecular spacing
• k is very low for gases and vapours; insulating materials and inorganic liquids
have k that lie in between those of metals and gases
• The k for most pure metals (except aluminium and uranium) decreases with
increasing temperature
• Air is a bad conductor of heat (0.022 W/mK)
• The k of a gas increases with increasing temp. and decreasing mol. wt.
• Super conductors are materials having high k at very low temp (eg: k for
Page 10
aluminium at 10 K is 20,000 W/mK whereas at 293 K is 175.6 W/mK only
25. Conduction heat transfer
One dimensional Conduction heat
transfer:
Energy conducted in left face + Energy generated
in within element = Change in internal energy
+Energy conducted if right face
36. Page 12
Convection
Convection occurs in liquids and gases.
Energy is carried with fluid motion when convection occurs.
PHYSICAL
PHENOMENON
Q hA(Tw Ta
)
MATHEMATICAL
EQUATION
37. Page 13
Convection (contd.)
• The quantity h is called the convective heat transfer coefficient (W/m2-K).
• It is dependent on the type of fluid flowing past the wall and the velocity
distribution.
• Thus, h is not a thermo physical property.
Newton’s Law of Cooling
Q hA(Tw Ta )
Convection Process h(W/m2-K)
Free convection
Gases 2–25
Liquids
50–1000
Forced convection
Gases 25–250
Liquids
50–20,000
Convection phase change 2,500–200,000
38. CONVECTION
Convection: The mode of
energy transfer between a
solid surface and the
adjacent liquid or gas that is
in motion, and it involves
the combined effects of
conduction and fluid motion.
The faster the fluid motion,
the greater the convection
heat transfer.
In the absence of any bulk
fluid motion, heat transfer
between a solid surface and
the adjacent fluid is by pure
conduction.
Heat transfer from a hot surface to air
by convection.
Page 38
39. Forced convection: If the
fluid is forced to flow over
the surface by external
means such as a fan,
pump, or the wind.
Natural (or free)
convection: If the fluid
motion is caused by
buoyancy forces that are
induced by density
differences due to the
variation of temperature
in the fluid.
The cooling of a boiled egg by
forced and natural convection.
Heat transfer processes that involve change of phase of a fluid are also
considered to be convection because of the fluid motion induced during
the process, such as the rise of the vapor bubbles during boiling or the
fall of the liquid droplets during condensation.
Page 39
40. Convection (contd.)
Page 40
Convective Processes
Single phase fluids (gases and liquids)
– Forced convection
– Free convection, or natural convection
– Mixed convection (forced plus free)
Convection with phase change
– Boiling
– Condensation
41. RADIATION
Page 41
• Radiation: The energy emitted by matter in the form of electromagnetic
waves (or photons) as a result of the changes in the electronic
configurations of the atoms or molecules.
• Unlike conduction and convection, the transfer of heat by radiation does
not require the presence of an intervening medium.
• In fact, heat transfer by radiation is fastest (at the speed of light) andit
suffers no attenuation in a vacuum. This is how the energy of the sun
reaches the earth.
• In heat transfer studies we are interested in thermal radiation, which is
the form of radiation emitted by bodies because of their temperature.
• All bodies at a temperature above absolute zero emit thermal radiation.
• Radiation is a volumetric phenomenon, and all solids, liquids, and
gases emit, absorb, or transmit radiation to varying degrees.
• However, radiation is usually considered to be a surface phenomenon
for solids.
42. Page 18 18
Stefan–Boltzmann law
= 5.670 108 W/m2 · K4 Stefan–Boltzmannconstant
Blackbody: The idealized surface that emits radiation at the maximum
rate.
Blackbody radiation represents the maximum
amount of radiation that can be emitted from
Emissivity : A measure ofhowsurfaces
closely a surface approximates a
blackbody for which = 1 of the
surface. 0 1.
Radiation
emitted by real
a surface at a specified temperature.
43. Radiation
Energy transfer in the form of electromagnetic waves
PHYSICAL
PHENOMENON
MATHEMATICAL
EQUATION
4
s
ET
A,Ts
Page 43
44. Radiation (contd.)
Page 44
Stefan-Boltzman Law
4
s
Eb T
The emissive power of a black body over all wave
lengths is proportional to fourth power of temperature
49. Boundary Conditions
Prescribed Temperature BC (First kind) Dirichlet conditions
Prescribed Heat Flux BC (Second kind) Neumann conditions
Convection BC (Third kind) Robbins cinditions
Page 49
50. Boundary Conditions
Prescribed Temperature BC (First kind)
Prescribed Heat Flux BC (Second kind)
Convection BC (Third kind)
0
Page 50
L x
T1 T2
T (x,t) | x=0 = T (0,t) =T1
T (x,t) | x=L = T (L,t) = T2
51. Boundary Conditions
Prescribed Temperature BC (First kind)
Prescribed Heat Flux BC (Second kind)
Convection BC (Third kind)
0 x
Heat
Supply
Conduction
flux
Heat
Supply
Conduction
flux
x0
x
T
q0 k
qL
L
x
T
k
xL
W/m2
W/m2 L
xL
q
x
k
T
q0
Page 51
x0
x
T
k
Plate
52. Boundary Conditions
Prescribed Temperature BC (First kind)
Prescribed Heat Flux BC (Second kind)
Convection BC (Third kind)
Hollow Cylinder or
hollow sphere
b
rb
q
r
T
k
ra
r
T
k
qa
b r
ra
qa k
r
T b
rb
q
r
k
T
W/m2
a
Heat
Supply
Page 52
53. Boundary Conditions
Prescribed Temperature BC (First kind)
Prescribed Heat Flux BC (Second kind)
Convection BC (Third kind)
Plate
Conduction
Convection
Fluid
Flow
T1,h1
xL
xL
x
T
k
Convection
h2 (T2 T )
Conduction
Fluid
Flow
T2,h2
1 1
x0
x0
x
h (T T ) k
T
Convection heat flux
from the fluid at T1 to
the surface at x = 0
Conduction heat flux
from the surface at
x= 0 into the plate
x0
Page 53
x0
1 1
h (T T
x
) k
T
54. Boundary Conditions
Prescribed Temperature BC (First kind)
Prescribed Heat Flux BC (Second kind)
Convection BC (Third kind)
Plate
Conduction
Convection
xL
xL
x
T
k
Convection
h2 (T2 T )
Fluid
Flow
T1,h1
Conduction
Fluid
T
h1(T1T x0
) k
x
Convection heat flux
from the fluid at T2 to
the surface at x = L
Conduction heat flux
from the surface at
x = L into the plate
xL
Flow
T2,h2
xL
x0
Page 54
x
) k
T
h (T T
2 2
55. Boundary Conditions
Prescribed Temperature BC (First kind)
Prescribed Heat Flux BC (Second kind)
Convection BC (Third kind)
Hollow Cylinder or
hollow sphere
b
r
a
Heat
Supply
2 2 rb
rb
h (T T )
r
k
T
Fluid
Flow
T1,h1
Fluid
ra
T
h1(T1T ra
) k
r
Convection heat flux
from the fluid at T1 to
the surface at r = a
Conduction heat flux
from the surface at
r= a into the plate
ra
Flow
T2,h2
Page 55
T
h1(T1 T ra
) k
r
56. Boundary Conditions
Prescribed Temperature BC (First kind)
Prescribed Heat Flux BC (Second kind)
Convection BC (Third kind)
Hollow Cylinder or
hollow sphere
b
r
a
Heat
Supply
2 2 rb
rb
h (T T )
r
k
T
Fluid
Flow
T1,h1
Fluid
ra
T
h1(T1T ra
) k
r
Convection heat flux
from the fluid at T2 to
the surface at r = b
Conduction heat flux
from the surface at
r= b into the plate
rb
Flow
T2,h2
Page 56
T
h2 (T2 T r b
) k
r
57. Page 33
Cont…
Temperature distribution in a plane wall
Steady 1D heat transfer, no heat generation
the conduction eqn. is written as
s,1
s,2 s,1
1
Ts,1 C2 &
T(x) C1 x C2
B.C' sare
T(0) Ts,1
T(L) Ts,2
Thus the constants are
C
L
T (x) (T T )
x
T
L
Ts,2 Ts,1
dx dx
d dT 0
0
x2
2
T
T = T1
0 L
T = T2
x
Rectangular Coordinates
58. Page 34
Cont…
Ts,1)
dT
(Ts,2
L
s,1
s,2 s,1
wkt T (x) (T T )
x
T
dx L
From Fourier’s law,
Q kA
dT
put
dT
, thus
dx dx
Q
kA(T1 T2 )
T
L L /kA
Heat flow
Thermal potentialdifference
Thermal resistance
K.A
L
R
59. Page 35
Cont…
hA
R
Q conv
conv
conv , where R
1
(Ts T )
Thermal resistance for convection
Qconv hA(Ts T)
Thermal resistance for radiation
hrad As
rad
rad
surr
rad s s surr rad s s
where, R
R
Tsurr )
)
(Ts
A (T T
Q A (T 4
T 4
) h
1
Combination of
conductive and
convective
resistance
62. Cont…
Rtotal
Q
T1 T
2
Summary
• The rate of steady state transfer between two surfaces is equal to the
temperature difference between those two surfaces divided by the total
thermal resistance between those two surfaces
• The ratio of the temperature drop to the thermal resistance across any
layer is constant
• The temperature drop across any layer is proportional to the thermal
resistance of the layer
(1)
Page 62
64. (2)
Compare eqns. 1 & 2, it reveals that
To determine the rate of heat transfer through the wall we do not need to
know the surface temperatures of the wall (Ref eqn.1)
How to get wall surface temperature?
Page 64
66. Problem to get intermediate temperature
A furnace wall consists of 200 mm
layer of refractory bricks, 6 mm
layer of steel plate and a 100 mm
layer of insulation bricks. The
maximum temperature of the wall is
1150oC on the furnace side and
minimum temperature is 40oC on
the outer side. Heat loss from the
wall is 400 W/m2. It is known that
there is a thin layer of air between
the layers of refractory bricks and
steel plate. Thermal conductivities
for the three different layers are
1.52, 45 and 0.138 W/moC. Find:
(i) the thickness of air layer (ii)
temperature of the outer surface of
the steel plate
P
a
Tg
e
a
k4
e2 conductivity of air is same as insulation brick
77. Thermal contact resistance
Interface offers some
resistance to h.t, and this
resistance for a unit
surface area is called the
thermal contact resistance
Page 77
84. Critical radius of Insulation
The rate of heat transfer from the insulated pipe
to the surrounding air can be expressed as
q
T1T
Page 84
r
Rins R
The additional insulation increases the conduction resistance of the insulation
layer but decreases the convection resistance of the surface because of the
increase in the outer surface area for convection. The heat transfer from the pipe
may increase or decrease, depending on which effect dominates.
85. Page 61
Cont…
The variation of heat transfer rate with the outer radius of insulation r2 is plotted in below
figure. The value of r2 at which heat transfer rate reaches maximum is determined from
the requirement that dqr/dr=0(zero slope). Performing the differentiation and solving
for r2 yields the critical radius of insulation for a cylindrical body to be
cr,cylinder
k
h
r
The rate of heat transfer from the cylinder increases
with the addition of insulation for r2< rcr, reaches a
maximum when r2= rcr, and starts to decrease for
r2> rcr. Thus, insulating the pipe may actually
increase the rate of heat transfer from the pipe
instead of decreasing it when r2< rcr .
cr,sphere
2k
r
h
88. An insulated steam pipe having outside diameter of 30 mm is to be covered with two layers of
insulation, each having the thickness of 20mm. The thermal conductivity of one material is 5
times that of the other,
Assuming that the inner and outer surface temperatures of composite insulation are fixed, how
much will heat transfer be increased when better insulation material is next to the pipe than it is
outer layer?
Case 1: better insulation is inside
Page 88
90. A 240 mm steam main, 210 m long is covered with 50 mm of high temperature insulation
(k=0.092 W/moC) and 40 mm of low temperature insulation (k=0.062 W/moC). The inner and
outer surface temperature as measures are 390oC and 40oC respectively. calculate:
(i) The total heat loss per hour (ii) The total heat loss/m2 of pipe surface (ii) The total heat loss/m2
of outer surface, and (iv) the temperature between two layers of insulation.
Neglect heat conduction through pipe material
Page 90
94. One Dimensional Heat Conduction (contd.)
Rectangular Coordinates
Cylindrical Coordinates
A Compact Equation
t
k
T g c
T(x,t)
p
x
x
t
1 rk
T g c
T(r,t)
p
r r
r
Spherical Coordinates
1
t
T(r,t)
T
r k
r r
p
r
g c
2
2
r k p
n
n
r
T
r r
T(r,t)
t
1
g c
n = 0
n = 1
n = 2
Page 94
95. Heat Source Systems
Plane wall with heat generation
For 1D, steady state, Fourier-Biot eqn is reduced as below
HG (2)
HG (1)
Page 95
96. Heat Source Systems
cont…
Plane wall with heat generation
(a) Specified temperature on both sides
Put in HG (2)
Put in HG (2)
Page 96
in HG (2)
97. Heat Source Systems
Plane wall with heat generation cont…
(b) Insulated boundary on one side and Convective boundary on other side
Page 97
98. Heat Source Systems
Plane wall with heat generation cont…
HG (1)
HG (1) HG (2)
A plane wall insulated on
one face and exposed to
convection environment
on other face
Page 98
101. Page 77
Heat Source Systems
Plane wall with heat generation – Problem cont…
The temp. distribution in a plane
wall is given by
Substitute C1 and C2 in HG (2)
HG (2)
HG (1)
102. Heat Source Systems
Plane wall with heat generation – Problem cont…
Put x = 0 in eqn. A, thus the magnitude of max. temp.is
Page
102
108. Page 84
Steady State 1D Heat Conduction and generation
Cylindrical Coordinates (Solid Cylinder)
T = Ts
0 r
T = Ts
ro
0
1 d
r
dT
g0
r dr dr k
Governing Equation
dT
0 &
at r 0
dr
o
s at rr
T T
c1 ln r c2
r 2
4k
T (r)
g0
Ts
ro
r
2
g r 2
4k
T (r) 0 o
1
Solving,
dT(r) g r
q(r) k 0
Temp. at centre of cylinder
Tc Ts
4k
g r 2
0 o
o
s
c
r2
r2
T T
T (r) T
s
1
B.C’s
Temp distribution in non-dimensionalform
g r
dr 2
C2 Ts
C1 0
o o
4K
2
109. T = Ts
0 r
T = Ts
ro
o
g r
2
2rLrgo
r2
Lg
2 dr 2k Heat transfer rate in cylinder
2k
dr
o
Q(r) kA
Q(r) kA
dT(r)
g0r
where
dT (r)
gor
Heat transfer rate at outer surface of cylinder
2h
Page
109
goro
r2
Lg h(2r L) (T -T )
o o s
Q(r) h(2ro L) (Ts -T)
At outer surface,cond. conv.
s
The surface temp.T T
Steady State 1D Heat Conduction and generation
110. T = Ts
0 r
T = Ts
ro
Cylinder exposed to conv. environment
2h
Page
110
4k
g r 2
g r
Max. temp. is seen at centre(i.e.r 0),T(r) - T
g
4k 2h
4k 2h
g r 2
g r
it gives, c2 0 o
0 o
T
g
4k
2k
dr
dT
dT g r c
0 o
0 o
o
dr 2k r
4k
Temp. distribution as T(r) 2
0
r r 2
g0ro
T
o
2
0
1
0 c 0
0 1
2
1
dr
T
2
r c
rro
h
k
g0 r
rro
- kA dT hA(T T )
at r ro , cond. conv.
h, T
; w.k.t at r 0,
T (r)
g0
r 2
c ln r c
rr
r ro
Steady State 1D Heat Conduction and generation
111. Cylindrical Coordinates (Hollow Cylinder)
Determination of Temperature Distribution
r
0
ro
T2
k
ri
T1
2
Page
111
2
0
ln1 r g
o i 2 1
2
2
0
put c1 and c2
2
1
2
0
1
r )
(r
4k
T(r)
1 g
o i 2 1 2
solving c
4k
4k
g r 2
T2 0 o
c1 ln ro c2
T1 0 i
c1 ln ri c2
at r ri , T T1 and r ro , T T2
g r 2
g
c
o i
o
o
o i
o
0 o
o i 2
i o
lnr r 4k
(r r 2
) (T T ) T
4k lnr r 4k
g r 2
ln1 r g
r 2
) (T T ) T
0
(r 2
T )
lnr r
4k
(r r 2
) (T
Steady State 1D Heat Conduction and generation
112. Page 88
Conduction-Convection Systems
Fins / Extended Surfaces
• Necessity for fins
• Biot Number
=
LONGITUDINAL
RECTANGULAR FIN
hx
(x/ k)
k 1/ h
Internal Conductive resistance
Surface Convective resistance
FIN TYPES
RADIAL FIN FIN WITH NON-
UNIFORM C.S PIN FIN (or) SPINE