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SHROFF S. R. ROTARY INSTITUTE OF CHEMICAL TECHNOLOGY (SRICT)
DEPARTMENT OF MECHANICAL ENGINEERING.
Subject: Applied Thermal and Hydraulic Engineering (ATHE)
Prepared by: Mr. Mitesh Gohil
Chapter 2. Heat Transfer
2
Chapter 2. Heat Transfer
2.1 Modes of heat transfer
2.2 Conduction heat transfer
2.3 Convection heat transfer
2.4 Combine conduction and convection heat transfer
2.5 Heat exchangers
2.6 Radiation heat transfer
2.7 Summary
Outline
3
2.1 Modes of heat transfer
The electrical engineering requires the knowledge of heat transfer for designing the
cooling system of motor, generator and transformers etc.
Modes of heat transfer
1) Conduction: In solid the heat energy transferred from one molecular to another
with conduction heat transfer via molecular lattice vibration and free electrons.
2) Convection: In fluid the heat energy is transferred with convection mode and it
is due to the movement of molecular.
3) Radiation: Heat energy due to emission of energy in the form of
electromagnetic waves by all bodies above 0 K temp. It does not required
medium, in fact most efficient in vacuum.
4
2.2 Conduction heat transfer
𝑄 ∝ 𝐴𝑑𝑇
𝑑𝑥
𝑄 = −𝑘 𝐴𝑑𝑇
𝑑𝑥(Watt) (1)
𝑘 = − 𝑄
𝐴
𝑑𝑥
𝑑𝑇 W/m ºC
Fourier’s law of conduction:
“The rate of conduction heat transfer trough a plane layer is proportional to the
temperature difference across the layer and heat transfer area and inversely
proportional to the thickness of the layer.”
Thermal conductivity (k):
Thermal conductivity is rate of heat transfer through a unit thickness of the
material per unit area per unit temperature difference. It depends on the material
and temperature.
5
Heat conduction through plane wall (A Slab)
Fig. 2.1 Temperature
distribution in a plane wall
𝑄𝑐𝑜𝑛𝑑,𝑤𝑎𝑙𝑙 = −𝑘 𝐴𝑑𝑇
𝑑𝑥
(Fourier’s law of
conduction)
𝑥=0
𝑥=𝐿
𝑄𝑐𝑜𝑛𝑑,𝑤𝑎𝑙𝑙 𝑑𝑥 = −
𝑇1
𝑇2
𝑘𝐴 𝑑𝑇
𝑄𝑐𝑜𝑛𝑑,𝑤𝑎𝑙𝑙 = 𝑘 𝐴𝑇1 − 𝑇2
𝐿
𝑄𝑐𝑜𝑛𝑑,𝑤𝑎𝑙𝑙 =𝑇1 − 𝑇2
𝐿𝑘𝐴
(Rearranging above
equation)
𝑄𝑐𝑜𝑛𝑑,𝑤𝑎𝑙𝑙 =𝑇1 − 𝑇2𝑅𝑤𝑎𝑙𝑙
(Thermal Resistance
Concept)
Thermal resistance, 𝑅𝑤𝑎𝑙𝑙 =𝐿
𝑘𝐴
2.2 Conduction heat transfer
6
Fig. 2.2 Analogy between thermal and
electrical resistance concepts
Concept of thermal resistance
2.2 Conduction heat transfer
7
Heat conduction through hollow cylinder
Fig. 2.3 A long cylindrical pipe
𝑄𝑐𝑜𝑛𝑑,𝑐𝑦𝑙 = −𝑘 𝐴𝑑𝑇
𝑑𝑟
(Fourier’s law of
conduction)
𝑟1
𝑟2 𝑄𝑐𝑜𝑛𝑑,𝑐𝑦𝑙
𝐴𝑑𝑟 = −
𝑇1
𝑇2
𝑘𝑑𝑇
𝑄𝑐𝑜𝑛𝑑,𝑐𝑦𝑙 = 2𝜋𝐿𝑘𝑇1 − 𝑇2ln 𝑟2 𝑟1
𝐴 = 2𝜋𝑟𝐿
L= length of cylinder
𝑄𝑐𝑜𝑛𝑑,𝑐𝑦𝑙 =𝑇1 − 𝑇2
ln𝑟2𝑟1
2𝜋𝐿𝑘
(Rearranging above
equation)
𝑄𝑐𝑜𝑛𝑑,𝑐𝑦𝑙 =𝑇1 − 𝑇2
𝑅𝑐𝑦𝑙(Thermal Resistance
Concept)
2.2 Conduction heat transfer
8
Thermal resistance (𝑅𝑐𝑦𝑙)
𝑅𝑐𝑦𝑙 =ln
𝑟2𝑟1
2𝜋𝐿𝑘
2.2 Conduction heat transfer
9
Convection heat transfer
The convection is mode of heat transfer in which heat transfer by two
mechanisms random molecular motion (diffusion) and energy transfer
by bulk or macroscopic motion of the fluid.
Convection heat transfer may takes place between solid and fluid or
between two fluids or inside a fluid.
2.3 Convection heat transfer
10
Types:
Natural Convection
In natural (or free) convection the flow of fluid in induced by buoyancy
forces, which arise from density different caused by temperature variation in
fluid.
The layer of fluid which is in immediate vicinity of the plate gets heated by
conduction. This will increase the temperature of that layer and its density
will decrease. So, due to buoyancy force it will try to move upward and mix
with cold fluid.
Force Convection
Heat transfer which takes place in moving fluid is called as forced
convection. In force convection the heat transfer rate is higher as compare to
natural convection.
2.3 Convection heat transfer
11
𝑄𝑐𝑜𝑛𝑣. ∝ 𝐴 𝑇𝑠 − 𝑇𝑓
𝐴 = Surface area
𝑇𝑠 = Temperature of hot surface
𝑇𝑓 = Temperature of cold fluid
Newton’s law of cooling for convection heat transfer
“The rate of heat transfer per unit surface area from surface to fluid is
proportional to temperature difference between surface and fluid.”
𝑄𝑐𝑜𝑛𝑣. = ℎ 𝐴 𝑇𝑠 − 𝑇𝑓
h = heat transfer coefficient
h depends on surface geometry
and fluid properties.
𝑄𝑐𝑜𝑛𝑣. =𝑇𝑠 − 𝑇𝑓1ℎ 𝐴
(Rearranging above equation)
𝑄𝑐𝑜𝑛𝑣. =𝑇𝑠 − 𝑇𝑓
𝑅𝑐𝑜𝑛𝑣.Thermal Resistance Concept
2.3 Convection heat transfer
12
2.3 Convection heat transfer
Thermal resistance (𝑅𝑐𝑜𝑛𝑣.)
𝑅𝑐𝑜𝑛𝑣. =1
ℎ 𝐴
Sr.
No.Heat transfer Case 𝑹𝒕𝒉
1 Conduction through Plane wall𝐿
𝑘𝐴
2Conduction through hollow
cylinder
ln𝑟2𝑟1
2𝜋𝐿𝑘
3 Convection1
ℎ 𝐴
Summary of thermal resistance
13
2.4 Combine conduction and convection heat transfer
Plane composite wall with both side fluids
Fig. 2.4 The temperature drop across a
layer is proportional to its thermal
resistance.
𝑄 =𝑇∞1 − 𝑇∞2
𝑅𝑡𝑜𝑡𝑎𝑙
𝑅𝑡𝑜𝑡𝑎𝑙 = 𝑅𝑐𝑜𝑛𝑣,1 + 𝑅𝑤𝑎𝑙𝑙 + 𝑅𝑐𝑜𝑛𝑣,2
𝑅𝑡𝑜𝑡𝑎𝑙 =1
ℎ1 𝐴+
𝐿
𝑘𝐴+
1
ℎ2 𝐴
14
2.4 Combine conduction and convection heat transfer
Multilayer Plane Walls with both side fluid
Fig. 2.5 The thermal resistance network for
heat transfer through a two-layer plane wall
subjected to convection on both sides.
𝑄 =𝑇∞1 − 𝑇∞2
𝑅𝑡𝑜𝑡𝑎𝑙
𝑅𝑡𝑜𝑡𝑎𝑙 = 𝑅𝑐𝑜𝑛𝑣,1 + 𝑅𝑤𝑎𝑙𝑙,1 + 𝑅𝑤𝑎𝑙𝑙,2 + 𝑅𝑐𝑜𝑛𝑣,2
𝑅𝑡𝑜𝑡𝑎𝑙 =1
ℎ1 𝐴+
𝐿1𝑘1 𝐴
+𝐿2𝑘2 𝐴
+1
ℎ2 𝐴
15
2.4 Combine conduction and convection heat transfer
Hollow composite cylinder with both side fluids
Fig. 2.6 The thermal resistance
network for heat transfer through a
three layered composite cylinder to
convection on both sides.
𝑄 =𝑇∞1 − 𝑇∞2
𝑅𝑡𝑜𝑡𝑎𝑙
𝑅𝑡𝑜𝑡𝑎𝑙 = 𝑅𝑐𝑜𝑛𝑣,1 + 𝑅𝑐𝑦𝑙,1 + 𝑅𝑐𝑦𝑙,2 + 𝑅𝑐𝑦𝑙,3 + 𝑅𝑐𝑜𝑛𝑣,2
𝑅𝑡𝑜𝑡𝑎𝑙 =1
ℎ1 𝐴1+ln
𝑟2𝑟1
2𝜋𝐿𝑘1+ln
𝑟3𝑟2
2𝜋𝐿𝑘2+ln
𝑟4𝑟3
2𝜋𝐿𝑘3+
1
ℎ2 𝐴4
Where,𝐴1 = 2𝜋𝑟1𝐿 and 𝐴4 = 2𝜋𝑟4𝐿
16
2.4 Combine conduction and convection heat transfer
Critical Radius of Insulation
Fig. 2.7 An insulated cylindrical pipe
and the thermal resistance network
associated with it.
𝑄 =𝑇1 − 𝑇∞
𝑅𝑖𝑛𝑠 + 𝑅𝑐𝑜𝑛𝑣=
𝑇1 − 𝑇∞
ln𝑟2𝑟1
2𝜋𝐿𝑘+
1ℎ (2𝜋 𝑟2𝐿)
𝑄 =𝑇1 − 𝑇∞𝑅𝑡𝑜𝑡𝑎𝑙
Fig. 2.8 The variation of heat transfer
rate with the outer radius of the
insulation 𝑟2
17
2.4 Combine conduction and convection heat transfer
Above equation shows that if we decrease the outer radius 𝑟2, it decrease the
conductive resistance of wall(𝑅𝑖𝑛𝑠), but increase convective resistance (𝑅𝑐𝑜𝑛𝑣).
The heat transfer from the pipe may increase or decrease, depending on which
effect dominates.
The variation of 𝑄 with the outer radius of the insulation 𝑟2 is plotted in Fig.
The value of 𝑟2 at which 𝑄 reaches a maximum is determined from the
requirement that 𝑑 𝑄 𝑑𝑟2 = 0 (zero slope).
Performing the differentiation and solving for 𝑟2 yields the critical radius of
insulation for a cylindrical body to be
𝑟𝑐𝑟,𝑐𝑦𝑙 =𝑘
ℎ
18
2.4 Combine conduction and convection heat transfer
Fins and their application
We are interested to improve the heat dissipation from the surface of
motor, generator etc.
From Newton’s law of 𝑄𝑐𝑜𝑛𝑣. = ℎ 𝐴 𝑇𝑠 − 𝑇𝑓
So, it is possible to increase the heat dissipation (heat removal) by
increase in surface area or heat transfer co-efficient.
But h is constant and fix for air or according to fluid.
So, only way to increase the heat dissipation is to increase the surface area
by making extended surface or fins.
Cross sectional area of the extended surface which is called as profile of
fin may find in different form depending upon application.
19
2.4 Combine conduction and convection heat transfer
Table: 2.1 Fins type and its application
Sr.
No.Name Shape Application
1Straight
rectangular fins
IC Engine, Compressor,
Electric Motor2
Straight triangular
fins
3Straight parabolic
fins
4Circular fins of
rectangular profile
Heat Exchanger, Heat
Pipe, Refrigeration
Evaporator and Condenser
20
2.4 Combine conduction and convection heat transfer
5Pin fins of rectangular
profile
6Pin fins of triangular
profile
7Pin fins of parabolic
profile
8Pin fins of parabolic
profile (blunt tip)
Plate type heat
exchanger, fin
type heat
exchanger
21
2.5 Heat exchangers
Definition
A Heat Exchanger may be defined as a Mechanical device which transfer the
heat from hot fluid to cold fluid, with maximum rate, with minimum
investment and with running cost.
Energy Balance: Enthalpy lost by hot fluid = enthalpy gained by cold fluid
(Assume negligible heat losses to surroundings)
Example:-
Condenser, Evaporator,
Boilers, super heater and reheaters,
Regenerative Heat Exchanger
Radiators of automobiles,
Oil coolers of heat engine,
Evaporator and condenser of
refrigeration system ,
Water and air coolers or heaters
Application:
Power production, process,
chemical, food and manufacturing
industries,
Electronics,
Waste heat recovery,
Refrigeration and air conditioning,
Space applications.
22
2.5 Heat exchangers
Classification of Heat Exchangers
Recuperators / Regenerators
Recuperative: In these type of heat exchanger, the cold and hot fluid
simultaneously through the device and the heat is transferred through the wall
separating therm. (most commonly used)
Regenerative: In these type of heat exchanger, the hot and cold fluids
alternatively on the same surface. Thus same surface is subjected to periodic
heating and cooling.
Fig. 2.9 Recuperative and Regenerative
23
2.5 Heat exchangers
According to Heat transfer mechanism:
single phase and two phase
Heaters or Cooler (sensible heat changes)
Condensers
In condensers, The hot fluid condense
(Gas to Liquid) at constant temperature
whilst temperature of cold fluid
gradually increases from inlet to outlet.
The hot fluid reject latent heat which is
absorbed by the cold fluid.
In case of condenser of thermal power
plant, the hot fluid is steam and the cold
fluid is cooling water which is gaining
the heat and rejecting the heat in cooling
towers.
Fig. 2.10 Temperature distribution
for condenser
24
2.5 Heat exchangers
Evaporators
In evaporator cold fluid evaporates (Liquid
to Gas) at constant temperature whilst
temperature of hot fluid gradually
decreases from inlet to outlet.
In case of thermal power plants, the flue
gases (hot fluids) reject the heat and the
same is gain by the cold fluid (i.e. water) in
boilers.
In evaporator of refrigeration system, the
cold fluid (refrigerant) change its phase at
constant temperature by absorbing the heat
from hot fluid (air or water).
Fig. 2.10 Temperature distribution
for boiler (or evaporator)
25
2.5 Heat exchangers
According to Transfer process:
Direct contact (mixing of fluid): In this type of heat exchanger, the two fluids
at different temperatures are come and mixed with direct contact. Example:
cooling tower
Indirect contact (No mixing) : In this type of heat exchanger, the two fluids at
different temperatures are exchange heat without direct contact. Example:
shell and tube heat exchanger.
26
2.5 Heat exchangers
According to Flow arrangement:
Parallel flow/unidirectional flow:
The hot and cold fluids enter at
same end of the heat exchanger flow
through in same direction and leave
at other end. Since, low heat transfer
compare to counter flow generally
not used.
2.5 Heat exchangers
Counter flow/ Counter current heat
exchanger: The hot and cold fluid enter
at the opposite ends of heat exchanger
flow through in opposite direction and
leave at opposite ends.
Cross flow heat exchanger: The two
fluids flow at right angel to each other.
27
2.5 Heat exchangers
According to Geometry of construction;
Concentric tube type heat exchangers: It
is also called tubular or double pipe heat
exchanger.
Shell and tube type heat exchangers:
These are most commonly used for
heating, cooling, condensation or
evaporation where large heat transfer
required.
Fig. 2.13 Double pipe heat exchanger
Fig. 2.14 Shell-and-Tube Heat Exchanger28
2.5 Heat exchangers
Finned type: When an enhanced heat transfer rate is required, the
extended surfaces are used on one side of the heat exchanger.
Example: Liquid-Gas (Fins are always added on gas side)
Fig. 2.15 (a) Tube-Finned & (b) Plate -Finned plate heat exchanger29
2.5 Heat exchangers
Compact heat exchangers: These are special class of heat exchangers in
which the heat transfer surface area per unit volume is very large (>
700𝑚2/𝑚3)
Fig. 2.16 Plate and Frame Heat Exchanger 30
2.5 Heat exchangers
Overall heat transfer co-efficient
A heat exchanger typically involves two flowing fluids separated by a solid
wall. Heat is first transferred from the hot fluid to the wall by convection,
through the wall by conduction, and from the wall to the cold fluid again by
convection. Any radiation effects are usually included in the convection heat
transfer coefficients.
The thermal resistance network associated with this heat transfer process
involves two convection and one conduction resistances, as shown in Fig.
Here the subscripts i and o represent the inner and outer surfaces of the inner
tube.
For a double-pipe heat exchanger,
𝑄 =𝑇𝑖 − 𝑇𝑜𝑅𝑡𝑜𝑡𝑎𝑙
31
2.5 Heat exchangers
Fig. 2.17 Thermal resistance network
associated with heat transfer in a double-
pipe heat exchanger
Total thermal resistance,
𝑅𝑡𝑜𝑡𝑎𝑙 = 𝑅𝑐𝑜𝑛𝑣,𝑖 + 𝑅𝑤𝑎𝑙𝑙 + 𝑅𝑐𝑜𝑛𝑣,𝑜
𝑅𝑡𝑜𝑡𝑎𝑙 =1
ℎ𝑖 𝐴𝑖+ln
𝑟𝑜𝑟𝑖
2𝜋𝐿𝑘+
1
ℎ𝑜 𝐴𝑜
In the analysis of heat exchangers, it is
convenient to combine all the thermal
resistances in the path of heat flow from the
hot fluid to the cold one into a single
resistance R, and to express the rate of heat
transfer between the two fluids as,
𝑄 =∆𝑇
𝑅𝑡𝑜𝑡𝑎𝑙= 𝑈𝐴𝑠∆𝑇 = 𝑈𝑖𝐴𝑖∆𝑇 = 𝑈𝑜𝐴𝑜∆𝑇
32
2.5 Heat exchangers
Where 𝐴𝑠 is the surface area and U is the overall heat transfer coefficient,
whose unit is W/𝑚2·K.
From above equation
1
𝑈 𝐴𝑠=
1
𝑈𝑖 𝐴𝑖=
1
𝑈𝑖 𝐴𝑖= 𝑅𝑡𝑜𝑡𝑎𝑙 =
1
ℎ𝑖 𝐴𝑖+ln
𝑟𝑜𝑟𝑖
2𝜋𝐿𝑘+
1
ℎ𝑜 𝐴𝑜
33
2.5 Heat exchangers
The Log Mean Temperature Difference Method (LMTD)
The temperature difference between the hot and cold fluids varies along the
heat exchanger.
It is convenient to have a mean temperature difference ∆ 𝑇𝑚 for use in the
relation
𝑄 = 𝑈 𝐴𝑠∆𝑇𝑚
LMTD for Counter flow
Consider the parallel-flow double-pipe heat exchanger.
Fig. 2.18 LMTD for counter flow 34
2.5 Heat exchangers
𝑑𝑄 = −𝑚ℎ𝑐𝑝ℎ𝑑𝑇ℎ = −𝑚𝑐𝑐𝑝𝑐𝑑𝑇𝑐 = 𝑈 𝑑𝐴 𝜃U=overall heat transfer co-efficient
𝜃 = 𝑇ℎ − 𝑇𝑐
𝑑𝑄 = −𝐶ℎ𝑑𝑇ℎ = −𝐶𝑐𝑑𝑇𝑐 = 𝑈 𝑑𝐴 𝜃(∵ 𝑚ℎ𝑐𝑝ℎ = 𝐶ℎ ;
𝑚𝑐𝑐𝑝𝑐 = 𝐶𝑐)…(1)
𝑄 = 𝐶ℎ(𝑇ℎ1 − 𝑇ℎ2) = 𝐶𝑐(𝑇𝑐1 − 𝑇𝑐2) …(2)
Energy balance (counter flow) on element shown
Energy balance from inlet to outlet
35
Fig. 2.18 LMTD for counter flow
2.5 Heat exchangers
From equation (1) 𝑑𝑇ℎ − 𝑑𝑇𝑐
𝑑𝑇ℎ − 𝑑𝑇𝑐 = 𝑑𝑄1
𝐶𝑐−
1
𝐶ℎ
𝑑𝜃 = 𝑈 𝑑𝐴 𝜃1
𝐶𝑐−
1
𝐶ℎFrom eq. (1)
1
2𝑑𝜃
𝜃=
1
2
𝑈𝑑𝐴1
𝐶𝑐−
1
𝐶ℎ
Integrate from
end 1 to 2
ln𝜃2𝜃1
= 𝑈𝐴1
𝐶𝑐−
1
𝐶ℎ
ln𝜃2𝜃1
= 𝑈𝐴(𝑇𝑐1 − 𝑇𝑐2)
𝑄−(𝑇ℎ1 − 𝑇ℎ2)
𝑄From eq. (2)
Fig. 2.18 LMTD for counter flow
36
2.5 Heat exchangers
ln𝜃2𝜃1
= 𝑈𝐴(𝑇ℎ2 − 𝑇𝑐2)
𝑄−(𝑇ℎ1 − 𝑇𝑐1)
𝑄(Rearranging terms)
ln∆𝑇2∆𝑇1
= −𝑈𝐴∆𝑇2𝑄
−∆𝑇1𝑄
∆𝑇1= (𝑇ℎ1 − 𝑇𝑐1)
∆𝑇2=(𝑇ℎ2 − 𝑇𝑐2)
𝑄 = 𝑈𝐴∆𝑇2 − ∆𝑇1
ln∆𝑇2∆𝑇1
𝑄 = 𝑈𝐴 𝑇𝑙𝑚
Fig. 2.18 LMTD for counter flow
37
2.5 Heat exchangers
LMTD for Parallel flow
Consider the parallel-flow double-pipe heat exchanger.
Fig. 2.19 LMTD for parallel flow
𝑑𝑄 = −𝑚ℎ𝑐𝑝ℎ𝑑𝑇ℎ = 𝑚𝑐𝑐𝑝𝑐𝑑𝑇𝑐 = 𝑈 𝑑𝐴 𝜃U=overall heat transfer co-efficient
𝜃 = 𝑇ℎ − 𝑇𝑐
𝑑𝑄 = −𝐶ℎ𝑑𝑇ℎ = 𝐶𝑐𝑑𝑇𝑐 = 𝑈 𝑑𝐴 𝜃(∵ 𝑚ℎ𝑐𝑝ℎ = 𝐶ℎ ;
𝑚𝑐𝑐𝑝𝑐 = 𝐶𝑐)…(1)
𝑄 = 𝐶ℎ(𝑇ℎ1 − 𝑇ℎ2) = 𝐶𝑐(𝑇𝑐2 − 𝑇𝑐1) …(2)
Energy balance (parallel flow) on element shown
Energy balance from inlet to outlet38
2.5 Heat exchangers
𝑑𝑇ℎ − 𝑑𝑇𝑐 = −𝑑𝑄1
𝐶𝑐+
1
𝐶ℎ
𝑑𝜃 = −𝑈 𝑑𝐴 𝜃1
𝐶𝑐+
1
𝐶ℎFrom eq. (1)
1
2𝑑𝜃
𝜃= −
1
2
𝑈𝑑𝐴1
𝐶𝑐+
1
𝐶ℎ
Integrate from end
1 to 2
ln𝜃2𝜃1
= −𝑈𝐴1
𝐶𝑐+
1
𝐶ℎ
ln𝜃2𝜃1
= −𝑈𝐴(𝑇𝑐2 − 𝑇𝑐1)
𝑄+(𝑇ℎ1 − 𝑇ℎ2)
𝑄From eq. (2)
From equation (1) 𝑑𝑇ℎ − 𝑑𝑇𝑐
Fig. 2.19 LMTD for parallel flow
39
2.5 Heat exchangers
Fig. 2.19 LMTD for parallel flowln𝜃2𝜃1
= 𝑈𝐴(𝑇𝑐1 − 𝑇𝑐2)
𝑄−(𝑇ℎ1 − 𝑇ℎ2)
𝑄
ln𝜃2𝜃1
= 𝑈𝐴(𝑇ℎ2 − 𝑇𝑐2)
𝑄−(𝑇ℎ1 − 𝑇𝑐1)
𝑄(Rearranging terms)
ln∆𝑇2∆𝑇1
= 𝑈𝐴∆𝑇2𝑄
−∆𝑇1𝑄
∆𝑇1= (𝑇ℎ1 − 𝑇𝑐1)
∆𝑇2=(𝑇ℎ2 − 𝑇𝑐2)
𝑄 = 𝑈𝐴∆𝑇2 − ∆𝑇1
ln∆𝑇2∆𝑇1
𝑄 = 𝑈𝐴 𝑇𝑙𝑚𝑡𝑑 40
2.5 Heat exchangers
Temperature distribution and ∆𝑇1&∆𝑇2 for condenser and Evaporator
(a) Condenser (b) Evaporator or Boiler
41
42
2.6 Radiation heat transfer
Radiation heat transfer: “The transfer of heat across a system boundary by
means of electro-magnetic waves which is caused solely by a temperature
difference.”
Does not required medium and occurs most effectively in vacuum and occurs
above 0 K temperature.
Radiation heat transfer occurs by electro-magnetic waves or quanta of energy
called photons.
The photons are propagated through space as ray (trance with light speed)
which unchanged frequency.
Spectrum of electro-magnetic radiation contain γ-ray, X-ray, UV, Infra-red,
thermal radiation (Include Visible), and micro waves etc.
Out of all above radiation only thermal radiation (λ=10−1𝜇𝑚 𝑡𝑜 102𝜇𝑚) is of
our interest.
Example: Solar, electric bulb, furnaces, nuclear explosions etc.
43
2.6 Radiation heat transfer
𝐸𝑏 = 𝜎𝐴𝑇4Unit of 𝐸𝑏 𝑖𝑠 Watt &
𝜎 = 5.67 × 10−8𝑊/𝑚2𝐾4
= Stefan Boltzmann constant
𝐸 =
0
𝜆
𝐸𝜆 𝑑𝜆 W/𝑚2
2) Monochromatic (Spectral) emissive power
The amount of radiation energy emitted from a surface at a given
temperature varies with variation in wavelength (λ).
Monochromatic emissive power is defined as the amount of radiant energy
emitted by a surface at given temperature per unit area, per unit time.
Surface Emission Properties:
1) Total emissive power (E)
It is the total amount of radiation emitted by a black body per unit area and time.
44
2.6 Radiation heat transfer
𝐸 = 𝜀𝜎𝐴𝑇4Unit of 𝐸 𝑖𝑠 Watt &
ε = emissivity of the material
𝜀 =𝐸
𝐸𝑏
3) Emission form a real surface emissivity
4) Emissivity (ε) is defined as the ability of the surface of a body to radiate
heat.
It is the ratio of emissive power of any body to black body of equal
temperature.
It may varies with temperature and wavelength.
5) Intensity of radiation or Irradiation
It is defined as the total incident radiation on a surface from all the
directions per unit area per unit time expressed in W/𝑚2.
6) Radiosity:
It is defined as the total radiant energy leaving from the surface from all the
directions per unit area per unit time expressed in W/𝑚2.
45
2.6 Radiation heat transfer
Concepts of different bodies
𝐺 = 𝐺𝑟 + 𝐺𝑎 + 𝐺𝑡
1 =𝐺𝑟𝐺+𝐺𝑎𝐺+𝐺𝑡𝐺
1 = 𝜌 + 𝛼 + 𝜏
(1) Black Body: Perfect Absorbing body with unit absorptivity (𝛼 = 1) is called as black body.
(2) White Body: Perfect reflective body with unit reflectivity (𝜌 = 1)is called as White body.
(3) Opaque Body: Perfect transparent body with zero transmissivity(𝜏 = 0) is called as Opaque
body.
(4) Grey Body
If the radiative properties, 𝛼, 𝜌, 𝜏 of a body are assumed to be uniform over the entire
wave length spectrum, then such a body is called Grey body.
One whose absorptivity does not vary with temperature and wavelength of the incident
radiation (𝛼 = (𝛼)𝜆= 𝑐𝑜𝑛𝑠𝑡.)
46
2.6 Radiation heat transfer
Laws of radiation
𝐸𝑏 = 𝜎𝐴𝑇4Unit of 𝐸𝑏 𝑖𝑠 Watt &
𝜎 = 5.67 × 10−8𝑊/𝑚2𝐾4
= Stefan Boltzmann constant
Stefan Boltzmann
The Stefan Boltzmann states that the emissive power of black body is directly
proportional to the fourth power of its absolute temperature.
47
2.6 Radiation heat transfer
Kirchhoff’s law
This law states that at any temperature the ratio of total emissive power (E) to
the total absorptivity (𝛼) is constant for all substance which are in thermal
equilibrium with environment.𝐸
𝛼= 𝑐𝑜𝑛𝑠𝑡.
𝐸
𝛼= 𝐸𝑏
Now we know, 𝐸 = 𝜀𝜎𝐴𝑇4 & 𝐸𝑏 = 𝜎𝐴𝑇4
𝜀𝜎𝐴𝑇4
𝛼= 𝜎𝐴𝑇4
𝜀 = 𝛼
Emissivity (ε) of a body is equal to Absorptivity (𝛼) of the body in thermal
equilibrium with environment.
48
2.7 Summary
1) Conduction:(Fourier Law)
2) Convection: (Newton’s law of cooling)
3) Radiation:
𝑄 = −𝑘 𝐴𝑑𝑇
𝑑𝑥
(Watt)
k = thermal conductivity
k depends on material
𝑄𝑐𝑜𝑛𝑣. = ℎ 𝐴 𝑇𝑠 − 𝑇𝑓
h = heat transfer coefficient
h depends on surface geometry
and fluid properties.
𝐸 = 𝜀𝜎𝐴𝑇4
Unit of 𝐸 𝑖𝑠 Watt &
ε = emissivity of the material
𝜎 = 5.67 × 10−8𝑊/𝑚2𝐾4
= Stefan Boltzmann constant
Modes of heat transfer
49