HEAT TRANSFER COURSE MATERIAL (A26C4)
UNIT-I
The science of thermodynamics deals with the amount of heat transfer as a system
undergoes a process from one equilibrium state to another, and makes no reference to how long
the process will take. But in engineering, we are often interested in the rate of heat transfer,
which is the topic of the science of heat transfer. The study of heat transfer has become an
increasingly intense concern in modern technology. In the fields of refrigeration and air
conditioning, computer hardware, electrical engineering civil engineering, geothermal
engineering and in electronic equipment cooling, chemical technology etc… there is a need to
understand and predict how heat energy is carried, distributed and diffused in and by different
materials. This is possible only if we have knowledge of heat transfer. The study of heat transfer
includes the physical process where thermal energy is transferred as a result of a temperature
difference. There are basically two distinct processes for the transport of heat energy viz.,
Conduction and Radiation. If conduction occurs in a medium where there is a relative motion of
particles it is called Convection. Therefore the study of heat transfer requires the knowledge of
the three modes of heat transfer conduction, convection and radiation.
1.1 CONDUCTION
This is the mode of heat transfer, which occurs in a medium with out any relative motion
of particles, though it is impermeable to any other kind of radiation.
This mode is dominant in solids. This
is more effective in metals than in non-
metals. In liquids and gases also we can have
this mode but dominated by the motion of
molecules i.e. mostly by convection.
The rate of heat conduction through a
solid material in a fluid where there is no
relative motion is proportional to temperature
difference across the material. i.e more the
temperature difference we can have more
amount of heat transfer. The rate is also
proportional to the area normal to the
direction of heat flow and inversely
proportional to the thickness of the material.
This dependence was established by Fourier’s law of conduction which states that
Q dT A.T
or Q
A dx x
The rate of heat transfer per unit area is directly proportional to the temperature gradient. The
proportionality is removed by a constant denoted by ‘k’ known as Thermal conductivity.
, dT T
Q = −kA = − kA (Watts)
dx L
The negative sign is introduced since the heat always flows from higher temperature to lower
temperature i.e., opposite to the temperature gradient. The above equation is called Fourier rate
equation.
The value kA/L is known as “conductance” of the plate. And Q /A is known as Heat flux denoted
by ‘q’. From above equation it follows that the rate of heat flow will be the rate along the lines
normal to isothermal surface. If this flow of heat is considered in different directions x,y,z in a
solid we have
dT dT dT
q x = -k x , q y = -k y and q z = -k z
dx dy dz
Where Kx, Kyand Kz represent thermal conductivities in x,y,z directions.
Thermal conductivity is a physical parameter of the substance indicating the substance
capacity to conduct heat.
Q. x
From Fourier equation k =
A. T
When x =1, A= 1, T= 1 unit, k = Q. so we can define the thermal conductivity of a substance
as the quantity of heat it can transfer by conduction per unit area of the isothermal surface per
unit time at a unit temperature gradient. Higher the value of k higher the substance ability to
conduct heat. In general k depends on temperature, pressure and nature of the substance and is
determined experimentally.
1.2 CONVECTION
This is delivered from the latin word ‘convehere’ means bring in or to carry. This is the mode
of heat transfer, which occurs due to the motion of molecules carrying heat from one zone to
another zone. For example if we keep a hot cup of coffee in a stream of air, it gets cooled and we
say heat is convected away by air. The molecules of air while moving over the hot coffee surface
pickup heat and release in a low temperature zone. When we pour the coffee in saucer and
expose it to the same stream of air it gets cooled more quickly because the exposed surface area
of hot fluid increases. Thus convective heat transfer is directly proportional to the exposed
surface area and it is also proportional to the temperature difference between the surface and
stream.
Q A.T
This is known as Newton’s law of
cooling, which states that the rate of heat
transfer by convection is proportional to
exposed surface area and temperature
difference.
If the proportionality is removed we have to
introduce a constant ‘h’ known as convective
heat transfer coefficient.
Q = h. As.(T ) = h. As.(Ts − T ) Watts
Q
h = W / m2 K
AS .T
,1.3 Radiation:
Radiation is an electromagnetic phenomenon in which heat energy is carried by the
waves in quantum. This can happen between bodies without any presence of medium as earth
receiving heat from sun through space. For conduction and convection to transfer heat we need
temperature difference (driving potential which cause heat flow from high temperature to low
temperature) where as in radiation when ever a body is at a temperature greater than 0 K .i.e., -
273 0C it experience molecular motion vibration & collision which result in emitting energy at
various wave lengths. This heat emitted by a body through electromagnetic waves is known as
radiation. Thus all objects will continue to emit radiant heat energy in all directions when their
temperature is above 0K and is governed by Stefan Boltzman law. It states that the amount of
heat energy emitted by a black body is directly proportional to fourth power of its absolute
temperature.
Q b AT 4
or Q b = AT 4 Watts
is called Stefan Boltzman constant and is given as 5.67 10 −8 W/m 2 K 4
Black body is an ideal body, which is a perfect emitter and perfect absorber. Real body
surfaces are not ideal and they emit less energy than a black body. The ratio of energy emitted by
a real body to that of a black body is known as emissivity denoted by .
E E
= real = real4
Eblack T
Qreal = AT 4 Watts
If a black body at temperature T1 having area A1 is completely surrounded by an environment
(black body) at temperature T2 then the net Heat Exchange between them is given by
( )
Qb1−2 = A1 T14 − T24 watts.
Since the value of is very low, radiation heat is dominant only when the bodies are at
high temperature. More detailed discussion on radiation is presented in chapter on Radiation.
1.4 COMBINED MODES OF HEAT TRANSFER
So far we have discussed the three fundamental modes of heat transfer treating them
separately. However in practice must of the devices experience more than one mode. From a hot
surface convention & radiation are both common. If we consider a car radiator heat transfer from
hot water to outer surface of tubes is by conduction & from the tubes surface to cold stream of
air is by convection. Not only that the hot tube emitter radiation heat into surrounding. Same
thing in the case of an IC engine cylinder. How ever the entry of heat energy from a hot surface
into fluid through the stagnant film at its surface is by conduction only. So there is need to
analyze these three modes combinedly rather than dealing them separately. The following
example illustrates the combined analysis.
, 1.5 General Conduction equation in – Cartesian System.
Let us consider a volume element as
shown in figure 3 having dimensions x, y
and z. We assume that the element has an
internal heat generation of qG W/m3 due to
chemical reaction or electrical resistance
heating or nuclear reaction etc…
We further consider the conduction
heat transfer in 3directions x, y, z as Qx, Qy
and Qzentering the element and Q x + dx , Q y + dy
and Qz + dz as leaving. From the Fourier’s law
of conduction
T
Q x = - k x . y.z - (2.1)
x
T
Q y = - k y . x.z
y
T
Q z = - k z . y.x
z
Further the heat leaving in x direction at x + x is mathematically expressed as
Q x +Δ x = Q x + Q x .x
x
Similarly in y and z directions
Q y + y = Q y + Q y .y
y
Q z + z = Q z + Q z .z
z
The amount of total heat generated in the elemental volume = q G vol
= qG x y z
If we assume the density of the element ρ then the mass of the element m = x.y .z
Now applying law of Conservation of energy
Heat energy entering + Heat generated = Heat energy leaving
+Accumulation rate of energy in the element
The energy accumulated is responsible for increasing the internal heat capacity of the
T
element given by m.C. Where ‘C’ is the specific heat of the material in J/kg k and ‘t’
t
represents time.
UNIT-I
The science of thermodynamics deals with the amount of heat transfer as a system
undergoes a process from one equilibrium state to another, and makes no reference to how long
the process will take. But in engineering, we are often interested in the rate of heat transfer,
which is the topic of the science of heat transfer. The study of heat transfer has become an
increasingly intense concern in modern technology. In the fields of refrigeration and air
conditioning, computer hardware, electrical engineering civil engineering, geothermal
engineering and in electronic equipment cooling, chemical technology etc… there is a need to
understand and predict how heat energy is carried, distributed and diffused in and by different
materials. This is possible only if we have knowledge of heat transfer. The study of heat transfer
includes the physical process where thermal energy is transferred as a result of a temperature
difference. There are basically two distinct processes for the transport of heat energy viz.,
Conduction and Radiation. If conduction occurs in a medium where there is a relative motion of
particles it is called Convection. Therefore the study of heat transfer requires the knowledge of
the three modes of heat transfer conduction, convection and radiation.
1.1 CONDUCTION
This is the mode of heat transfer, which occurs in a medium with out any relative motion
of particles, though it is impermeable to any other kind of radiation.
This mode is dominant in solids. This
is more effective in metals than in non-
metals. In liquids and gases also we can have
this mode but dominated by the motion of
molecules i.e. mostly by convection.
The rate of heat conduction through a
solid material in a fluid where there is no
relative motion is proportional to temperature
difference across the material. i.e more the
temperature difference we can have more
amount of heat transfer. The rate is also
proportional to the area normal to the
direction of heat flow and inversely
proportional to the thickness of the material.
This dependence was established by Fourier’s law of conduction which states that
Q dT A.T
or Q
A dx x
The rate of heat transfer per unit area is directly proportional to the temperature gradient. The
proportionality is removed by a constant denoted by ‘k’ known as Thermal conductivity.
, dT T
Q = −kA = − kA (Watts)
dx L
The negative sign is introduced since the heat always flows from higher temperature to lower
temperature i.e., opposite to the temperature gradient. The above equation is called Fourier rate
equation.
The value kA/L is known as “conductance” of the plate. And Q /A is known as Heat flux denoted
by ‘q’. From above equation it follows that the rate of heat flow will be the rate along the lines
normal to isothermal surface. If this flow of heat is considered in different directions x,y,z in a
solid we have
dT dT dT
q x = -k x , q y = -k y and q z = -k z
dx dy dz
Where Kx, Kyand Kz represent thermal conductivities in x,y,z directions.
Thermal conductivity is a physical parameter of the substance indicating the substance
capacity to conduct heat.
Q. x
From Fourier equation k =
A. T
When x =1, A= 1, T= 1 unit, k = Q. so we can define the thermal conductivity of a substance
as the quantity of heat it can transfer by conduction per unit area of the isothermal surface per
unit time at a unit temperature gradient. Higher the value of k higher the substance ability to
conduct heat. In general k depends on temperature, pressure and nature of the substance and is
determined experimentally.
1.2 CONVECTION
This is delivered from the latin word ‘convehere’ means bring in or to carry. This is the mode
of heat transfer, which occurs due to the motion of molecules carrying heat from one zone to
another zone. For example if we keep a hot cup of coffee in a stream of air, it gets cooled and we
say heat is convected away by air. The molecules of air while moving over the hot coffee surface
pickup heat and release in a low temperature zone. When we pour the coffee in saucer and
expose it to the same stream of air it gets cooled more quickly because the exposed surface area
of hot fluid increases. Thus convective heat transfer is directly proportional to the exposed
surface area and it is also proportional to the temperature difference between the surface and
stream.
Q A.T
This is known as Newton’s law of
cooling, which states that the rate of heat
transfer by convection is proportional to
exposed surface area and temperature
difference.
If the proportionality is removed we have to
introduce a constant ‘h’ known as convective
heat transfer coefficient.
Q = h. As.(T ) = h. As.(Ts − T ) Watts
Q
h = W / m2 K
AS .T
,1.3 Radiation:
Radiation is an electromagnetic phenomenon in which heat energy is carried by the
waves in quantum. This can happen between bodies without any presence of medium as earth
receiving heat from sun through space. For conduction and convection to transfer heat we need
temperature difference (driving potential which cause heat flow from high temperature to low
temperature) where as in radiation when ever a body is at a temperature greater than 0 K .i.e., -
273 0C it experience molecular motion vibration & collision which result in emitting energy at
various wave lengths. This heat emitted by a body through electromagnetic waves is known as
radiation. Thus all objects will continue to emit radiant heat energy in all directions when their
temperature is above 0K and is governed by Stefan Boltzman law. It states that the amount of
heat energy emitted by a black body is directly proportional to fourth power of its absolute
temperature.
Q b AT 4
or Q b = AT 4 Watts
is called Stefan Boltzman constant and is given as 5.67 10 −8 W/m 2 K 4
Black body is an ideal body, which is a perfect emitter and perfect absorber. Real body
surfaces are not ideal and they emit less energy than a black body. The ratio of energy emitted by
a real body to that of a black body is known as emissivity denoted by .
E E
= real = real4
Eblack T
Qreal = AT 4 Watts
If a black body at temperature T1 having area A1 is completely surrounded by an environment
(black body) at temperature T2 then the net Heat Exchange between them is given by
( )
Qb1−2 = A1 T14 − T24 watts.
Since the value of is very low, radiation heat is dominant only when the bodies are at
high temperature. More detailed discussion on radiation is presented in chapter on Radiation.
1.4 COMBINED MODES OF HEAT TRANSFER
So far we have discussed the three fundamental modes of heat transfer treating them
separately. However in practice must of the devices experience more than one mode. From a hot
surface convention & radiation are both common. If we consider a car radiator heat transfer from
hot water to outer surface of tubes is by conduction & from the tubes surface to cold stream of
air is by convection. Not only that the hot tube emitter radiation heat into surrounding. Same
thing in the case of an IC engine cylinder. How ever the entry of heat energy from a hot surface
into fluid through the stagnant film at its surface is by conduction only. So there is need to
analyze these three modes combinedly rather than dealing them separately. The following
example illustrates the combined analysis.
, 1.5 General Conduction equation in – Cartesian System.
Let us consider a volume element as
shown in figure 3 having dimensions x, y
and z. We assume that the element has an
internal heat generation of qG W/m3 due to
chemical reaction or electrical resistance
heating or nuclear reaction etc…
We further consider the conduction
heat transfer in 3directions x, y, z as Qx, Qy
and Qzentering the element and Q x + dx , Q y + dy
and Qz + dz as leaving. From the Fourier’s law
of conduction
T
Q x = - k x . y.z - (2.1)
x
T
Q y = - k y . x.z
y
T
Q z = - k z . y.x
z
Further the heat leaving in x direction at x + x is mathematically expressed as
Q x +Δ x = Q x + Q x .x
x
Similarly in y and z directions
Q y + y = Q y + Q y .y
y
Q z + z = Q z + Q z .z
z
The amount of total heat generated in the elemental volume = q G vol
= qG x y z
If we assume the density of the element ρ then the mass of the element m = x.y .z
Now applying law of Conservation of energy
Heat energy entering + Heat generated = Heat energy leaving
+Accumulation rate of energy in the element
The energy accumulated is responsible for increasing the internal heat capacity of the
T
element given by m.C. Where ‘C’ is the specific heat of the material in J/kg k and ‘t’
t
represents time.
