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**Unformatted text preview: **LECTURE NOTES
ON
HEAT & MASS TRANSFER
BY 1 CHAPTER 1
INTRODUCTORY CONCEPTS AND BASIC LAWS
OF HEAT TRANSFER
1.1. Introduction:- We recall from our knowledge of thermodynamics that heat is a form
of energy transfer that takes place from a region of higher temperature to a region of
lower temperature solely due to the temperature difference between the two regions. With
the knowledge of thermodynamics we can determine the amount of heat transfer for any
system undergoing any process from one equilibrium state to another. Thus the
thermodynamics knowledge will tell us only how much heat must be transferred to
achieve a specified change of state of the system. But in practice we are more interested
in knowing the rate of heat transfer (i.e. heat transfer per unit time) rather than the
amount. This knowledge of rate of heat transfer is necessary for a design engineer to
design all types of heat transfer equipments like boilers, condensers, furnaces, cooling
towers, dryers etc.The subject of heat transfer deals with the determination of the rate of
heat transfer to or from a heat exchange equipment and also the temperature at any
location in the device at any instant of time.
The basic requirement for heat transfer is the presence of a
“temperature difference”. The temperature difference is the driving force for heat
transfer, just as the voltage difference for electric current flow and pressure difference for
fluid flow. One of the parameters ,on which the rate of heat transfer in a certain direction
depends, is the magnitude of the temperature gradient in that direction. The larger the
gradient higher will be the rate of heat transfer.
1.2. Heat Transfer Mechanisms:- There are three mechanisms by which heat transfer
can take place. All the three modes require the existence of temperature difference. The
three mechanisms are: (i) conduction, (ii) convection and (iii) radiation
1.2.1Conduction:- It is the energy transfer that takes place at molecular levels.
Conduction is the transfer of energy from the more energetic molecules of a substance to
the adjacent less energetic molecules as a result of interaction between the molecules. In
the case of liquids and gases conduction is due to collisions and diffusion of the
molecules during their random motion. In solids, it is due to the vibrations of the
molecules in a lattice and motion of free electrons.
Fourier’s Law of Heat Conduction:- The empirical law of conduction based on
experimental results is named after the French Physicist Joseph Fourier. The law states
that the rate of heat flow by conduction in any medium in any direction is proportional to
the area normal to the direction of heat flow and also proportional to the temperature
gradient in that direction. For example the rate of heat transfer in x-direction can be
written according to Fourier’s law as
1.2
2 Qx α − A (dT / dx)
Or …………………….(1.1) Qx = − k A (dT / dx) W………………….. ..(1.2) In equation (1.2), Qx is the rate of heat transfer in positive x-direction through area A of
the medium normal to x-direction, (dT/dx) is the temperature gradient and k is the
constant of proportionality and is a material property called “thermal conductivity”.
Since heat transfer has to take place in the direction of decreasing temperature, (dT/dx)
has to be negative in the direction of heat transfer. Therefore negative sign has to be
introduced in equation (1.2) to make Qx positive in the direction of decreasing
temperature, thereby satisfying the second law of thermodynamics. If equation (1.2) is
divided throughout by A we have
qx is called the heat flux. qx = (Qx / A) = − k (dT / dx) W/m2………..(1.3) Thermal Conductivity:- The constant of proportionality in the equation of Fourier’s law
of conduction is a material property called the thermal conductivity.The units of thermal
conductivity can be obtained from equation (1.2) as follows:
Solving for k from Eq. (1.2) we have k = − qx / (dT/dx)
Therefore units of k = (W/m2 ) (m/ K) = W / (m – K) or W / (m – 0 C). Thermal
conductivity is a measure of a material’s ability to conduct heat. The thermal
conductivities of materials vary over a wide range as shown in Fig. 1.1.
It can be seen from this figure that the thermal conductivities of gases such as
air vary by a factor of 10 4 from those of pure metals such as copper. The kinetic theory
of gases predicts and experiments confirm that the thermal conductivity of gases is
proportional to the square root of the absolute temperature, and inversely proportional to
the square root of the molar mass M. Hence, the thermal conductivity of gases increases
with increase in temperature and decrease with increase in molar mass. It is for these
reasons that the thermal conductivity of helium (M=4) is much higher than those of air
(M=29) and argon (M=40).For wide range of pressures encountered in practice the
thermal conductivity of gases is independent of pressure.
The mechanism of heat conduction in liquids is more complicated due to the
fact that the molecules are more closely spaced, and they exert a stronger inter-molecular
force field. The values of k for liquids usually lie between those for solids and gases.
Unlike gases, the thermal conductivity for most liquids decreases with increase in
temperature except for water. Like gases the thermal conductivity of liquids decreases
with increase in molar mass. 1.3 3 1000 Silver
Copper Solid
metals
100 Sodium Steel Liquid
metals
Oxides 10
k (W/m-K) Mercury NonMetallic
solids 1.0 0.1 Water NonMetallic
liquids
Plastics
Wood Fibres Insul
ating
Mate
rials Oils
Foams He, H
2 NonMeta
llic
gases Evacuated
Insulating
materials CO
2 0.01 Fig. 1.1: Typical range of thermal conductivities of various materials In the case of solids heat conduction is due to two effects: the vibration of lattice
induced by the vibration of molecules positioned at relatively fixed positions , and
energy transported due to the motion of free electrons. The relatively high thermal
conductivities of pure metals are primarily due to the electronic component. The lattice
component of thermal conductivity strongly depends on the way the molecules are
arranged. For example, diamond, which is highly ordered crystalline solid, has the
highest thermal conductivity at room temperature.
Unlike metals, which are good electrical and heat conductors, crystalline solids
such as diamond and semiconductors such as silicon are good heat conductors but poor
electrical conductors. Hence such materials find widespread use in electronic industry.
Despite their high price, diamond heat sinks are used in the cooling of sensitive electronic
components because of their excellent thermal conductivity. Silicon oils and gaskets are
commonly used in the packaging of electronic components because they provide both
good thermal contact and good electrical insulation.
1.4
4 One would expect that metal alloys will have high thermal
conductivities, because pure metals have high thermal conductivities. For example one
would expect that the value of the thermal conductivity k of a metal alloy made of two
metals with thermal conductivities k1 and k2 would lie between k1 and k2.But this is not
the case. In fact k of a metal alloy will be less than that of either metal.
The thermal conductivities of materials vary with temperature. But
for some materials the variation is insignificant even for wide temperature range.At
temperatures near absolute zero, the thermal conductivities of certain solids are extremely
large. For example copper at 20 K will have a thermal conductivity of 20,000 W / (m-K),
which is about 50 times the conductivity at room temperature. The temperature
dependence of thermal conductivity makes the conduction heat transfer analysis more
complex and involved. As a first approximation analysis for solids with variable
conductivity is carried out assuming constant thermal conductivity which is an average
value of the conductivity for the temperature range of interest.
Thermal Diffusivity:- This is a property which is very helpful in analyzing transient heat
conduction problem and is normally denoted by the symbol α . It is defined as follows.
Heat conducted
k
α = -------------------------------------- = -------(m2/s) ……(1.4)
Heat Stored per unit volume
ρCp
It can be seen from the definition of thermal diffusivity that the numerator represents the
ability of the material to conduct heat across its layers and the denominator represents the
ability of the material to store heat per unit volume. Hence we can conclude that larger
the value of the thermal diffusivity, faster will be the propagation of heat into the
medium. A small value of thermal diffusivity indicates that heat is mostly absorbed by
the material and only a small quantity of heat will be conducted across the material.
1.2.2. Convection :- Convection heat transfer is composed of two mechanisms. Apart
from energy transfer due to random molecular motion, energy is also transferred due to
macroscopic motion of the fluid. Such motion in presence of the temperature gradient
contributes to heat transfer. Thus in convection the total heat transfer is due to random
motion of the fluid molecules together with the bulk motion of the fluid, the major
contribution coming from the latter mechanism. Therefore bulk motion of the fluid is a
necessary condition for convection heat transfer to take place in addition to the
temperature gradient in the fluid. Depending on the force responsible for the bulk motion
of the fluid, convective heat transfer is classified into “forced convection” and “natural
or free convection”. In the case of forced convection, the fluid flow is caused by an
external agency like a pump or a blower where as in the case of natural or free convection
the force responsible for the fluid flow (normally referred to as the buoyancy force) is
generated within the fluid itself due to density differences which are caused due to
temperature gradient within the flow field. Regardless of the particular nature of
convection, the rate equation for convective heat transfer is given by
1.5 5 q = h ∆T …………………………………….. (1.5)
where q is the heat flux, ∆T is the temperature difference between the bulk fluid and the
surface which is in contact with the fluid, and ‘h” is called the “convective heat transfer
coefficient” or “surface film coefficient”. Eq.(1.5) is generally referred to as the
Newton’s law of cooling.If Ts is the surface temperature , Tf is the temperature of the
bulk fluid and if Ts > Tf, then Eq. (1.5) in the direction of heat transfer can be written as
q = h [Ts – Tf] ………………………………...(1.6a)
and if Ts < Tf, the equation reduces to
q = h [Tf – Ts] ………………………………...(1.6b)
The heat transfer coefficient h depends on (i) the type of flow (i.e. whether the
flow is laminar or turbulent), (ii) the geometry of the body and flow passage area, (iii)
the thermo-physical properties of the fluid namely the density ρ, viscosity μ, specific heat
at constant pressure Cp and the thermal conductivity of the fluid k and (iv) whether the
mechanism of convection is forced convection or free convection. The heat transfer
coefficient for free convection will be generally lower than that for forced convection as
the fluid velocities in free convection are much lower than those in forced convection.
The heat transfer coefficients for some typical applications are given in table 1.2.
Table 1.2: Typical values of the convective heat transfer coefficient h
-----------------------------------------------------------------------------------------------------------Type of flow
h ,W / (m2 – K)
Free convection
Gases
2 – 25
Liquids
50 – 1000
Forced Convection
Gases
25 – 250
Liquids
50 – 20,000
Convection with change of phase
Boiling or condensation
2500 – 100,000
1.2.3. Thermal Radiation:- Thermal radiation is the energy emitted by matter (solid,
liquid or gas) by virtue of its temperature. This energy is transported by electromagnetic
waves (or alternatively, photons).While the transfer of energy by conduction and
convection requires the presence of a material medium, radiation does not require.Infact
radiation transfer occurs most effectively in vacuum.
Consider radiation transfer process for the surface shown in Fig.1.2a.Radiation that 1.6 6 G ρG E q s q surr Surface of emissivity ε, absorptivity α, and
Surface of emissivity ε, area temperature T
s A, and temperature T
s
Surroundings (black) at T
surr (a) (b) Fig.1.2: Radiation exchange: (a) at a surface and (b) between a surface
and large surroundings
is emitted by the surface originates from the thermal energy of matter bounded by the
surface, and the rate at which this energy is released per unit area is called as the surface
emissive power E.An ideal surface is one which emits maximum emissive power and is
called an ideal radiator or a black body.Stefan-Boltzman’s law of radiation states that the
emissive power of a black body is proportional to the fourth power of the absolute
temperature of the body. Therefore if Eb is the emissive power of a black body at
temperature T 0K, then
Eb α T 4
Or Eb = σ T 4 ………………………………….(1.7) σ is the Stefan-Boltzman constant (σ = 5.67 x 10 − 8 W / (m2 – K4) ). For a non black
surface the emissive power is given by
E = ε σ T 4…………………………………(1.8)
where ε is called the emissivity of the surface (0 ≤ ε ≤ 1).The emissivity provides a
measure of how efficiently a surface emits radiation relative to a black body. The
emissivity strongly depends on the surface material and finish.
Radiation may also incident on a surface from its surroundings. The rate at which
the radiation is incident on a surface per unit area of the surface is calle the “irradiation”
of the surface and is denoted by G. The fraction of this energy absorbed by the surface is
called “absorptivity” of the surface and is denoted by the symbol α. The fraction of the
1.7 7 incident energy is reflected and is called the “reflectivity” of the surface denoted by ρ
and the remaining fraction of the incident energy is transmitted through the surface and
is called the “transmissivity” of the surface denoted by τ. It follows from the definitions
of α, ρ, and τ that
α+ ρ + τ = 1 …………………………………….(1.9)
Therefore the energy absorbed by a surface due to any radiation falling on it is given by
Gabs = αG …………………………………(1.10)
The absorptivity α of a body is generally different from its emissivity. However in
many practical applications, to simplify the analysis α is assumed to be equal to its
emissivity ε.
Radiation Exchange:- When two bodies at different temperatures “see” each other, heat
is exchanged between them by radiation. If the intervening medium is filled with a
substance like air which is transparent to radiation, the radiation emitted from one body
travels through the intervening medium without any attenuation and reaches the other
body, and vice versa. Then the hot body experiences a net heat loss, and the cold body a
net heat gain due to radiation heat exchange between the two. The analysis of radiation
heat exchange among surfaces is quite complex which will be discussed in chapter 10.
Here we shall consider two simple examples to illustrate the method of calculating the
radiation heat exchange between surfaces.
As the first example’ let us consider a small opaque plate (for an opaque
surface τ = 0) of area A, emissivity ε and maintained at a uniform temperature Ts. Let
this plate is exposed to a large surroundings of area Asu (Asu >> A) whish is at a uniform
temperature Tsur as shown in Fig. 1.2b.The space between them contains air which is
transparent to thermal radiation.
The radiation energy emitted by the plate is given by
Qem = A ε σ Ts4
The large surroundings can be approximated as a black body in relation to the small plate.
Then the radiation flux emitted by the surroundings is σ Tsur4 which is also the radiaton
flux incident on the plate. Therefore the radiation energy absorbed by the plate due to
emission from the surroundings is given by
Qab = A α σ Tsur4.
The net radiation loss from the plate to the surroundings is therefore given by
Qrad = A ε σ Ts4 − A α σ Tsur4.
1.8
Assuming α = ε for the plate the above expression for Qnet reduces to 8 Qrad = A ε σ [Ts4 – Tsur4 ] ……………….(1.11)
The above expression can be used to calculate the net radiation heat exchange between a
small area and a large surroundings.
As the second example, consider two finite surfaces A1 and A2 as shown in Fig. 1.3. Surroundings A2, ε2, T2
A1, ε1, T1 Fig.1.3: Radiation exchange between surfaces A1 and A2 The surfaces are maintained at absolute temperatures T1 and T2 respectively, and have
emissivities ε1 and ε2. Part of the radiation leaving A1 reaches A2, while the remaining
energy is lost to the surroundings. Similar considerations apply for the radiation leaving
A2.If it is assumed that the radiation from the surroundings is negligible when compared
to the radiation from the surfaces A1 and A2 then we can write the expression for the
radiation emitted by A1 and reaching A2 as
Q1→2 = F1− 2 A1ε1σ T14……………………………(1.12)
where F1 – 2 is defined as the fraction of radiation energy emitted by A1 and reaching A2.
Similarly the radiation energy emitted by A2 and reaching A1 is given by
Q2→1 = F2− 1 A2 ε2 σ T24 …………………………..(1.13)
where F2 – 1 is the fraction of radiation energy leaving A2 and reaching A1. Hence the net
radiation energy transfer from A1 to A2 is given by
Q1 – 2 = Q1→2 − Q2→1
1.9 9 = [F1− 2 A1ε1σ T14] − [F2− 1 A2 ε2 σ T24]
F1-2 is called the view factor (or geometric shape factor or configuration factor) of A2
with respect to A1 and F2 - 1 is the view factor of A1 with respect to A2.It will be shown in
chapter 10 that the view factor is purely a geometric property which depends on the
relative orientations of A1 and A2 satisfying the reciprocity relation, A1 F1 – 2 = A2 F2 – 1.
Therefore Q1 – 2 = A1F1 – 2 σ [ε1 T14 − ε2 T24]………………….(1.13) Radiation Heat Transfer Coefficient:- Under certain restrictive conditions it is possible
to simplify the radiation heat transfer calculations by defining a radiation heat transfer
coefficient hr analogous to convective heat transfer coefficient as
Qr = hrA ΔT
For the example of radiation exchange between a surface and the surroundings
[Eq. (1. 11)] using the concept of radiation heat transfer coefficient we can write
Qr = hrA[Ts – Tsur] = A ε σ [Ts4 – Tsur4 ]
Or ε σ [Ts4 – Tsur4 ]
ε σ [Ts2 + Tsur2 ][Ts + Tsur][Ts – Tsur]
hr = --------------------- = ----------------------------------------------[Ts – Tsur]
[Ts – Tsur] Or hr = ε σ [Ts2 + Tsur2 ][Ts + Tsur] ………………………(1.14) 1.3.First Law of Thermodynamics (Law of conservation of energy) as applied to
Heat Transfer Problems :- The first law of thermodynamics is an essential tool for
solving many heat transfer problems. Hence it is necessary to know the general
formulation of the first law of thermodynamics.
First law equation for a control volume:- A control volume is a region in space bounded
by a control surface through which energy and matter may pass.There are two options of
formulating the first law for a control volume. One option is formulating the law on a
rate basis. That is, at any instant, there must be a balance between all energy rates.
Alternatively, the first law must also be satisfied over any time interval Δt. For such an
interval, there must be a balance between the amounts of all energy changes.
First Law on rate basis :- The rate at which thermal and mechanical energy enters a control volume, plus the rate at which thermal energy is generated within the control
volume, minus the rate at which thermal and mechanical energy leaves the control
volume must be equal to the rate of increase of stored energy within the control volume.
Consider a control volume shown in Fig. 1.4 which shows that thermal and
1.10 10 . mechanical energy are entering the control volume at a rate denoted by Ein, thermal and .
. Eg . Est . Eout Ein Fig. 1.4: Conservation of energy for a control volume on rate basis . mechanical en...

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- Spring '14
- Heat Transfer, Joseph Fourier