FLOW OF HEAT
Syllabus:
Classification of heat flow processes :
Conduction: Fourier’s
law, Heat flow through a cylinder
Convection: Natural
convection, forced convection, surface coefficients, overall heat transfer
coefficient. Heat transfer through boiling liquids, and condensing vapors.
Radiation: Black
body, Emissivity, Angle of vision in radiant heat flow, Radiation errors in
thermometry.
Sources of heat energy
Classification of
heat flow process
When
two objects at different temperatures are brought into thermal contact, heat
flows from the object at higher temperature to the object at lower temperature.
The mechanisms by which the heat may flow are three: conduction, convection and
radiation.
Conduction
When
heat flow through a body without any observable motion of matter, the type of
heat flow is called conduction.
Mechanism :
In
metallic solids, thermal conduction results from the unbound electrons (which
is similar to the electrical conductivity).
In
solids that are poor conductor of heat and in liquids the heat is conducted by
the transport of momentum of the individual
molecules along the temperature gradient.
In
gases the conduction occurs by the random motion of the molecules, so that heat
is “diffused” from hotter regions to the colder ones.
Examples:
Heat
flow through the brick wall of a furnace or the metal wall of a tube.
Convection
When
heat flows by the transfer of matter, the type of heat flow is called
convection.
In this case the heat flows by
actual mixing of warmer portions with the cooler portions of the same material.
Convection is restricted to the flow of heat in fluids. Heat flows through
fluid by both conduction and convection and its is difficult to separate the
two methods because of the eddies set up by the change of density with
temperature.
Examples
Transfer
of heat by the eddies of turbulent flow and by the current of warm air from a
room heater flowing across the room.
Natural and forced convection
The
forces used to create convection currents in fluids are two types.
Natural convection:
When
a fluid is heated the warmer part becomes lighter than the cooler part. Due to
this difference in density the cooler (higher density) fluid moves down wards
and the warmer (lighter density) move upwards and thus forming a convection current. Thus heat is
transferred with mass. This method of heat transfer is called natural
convection.
Forced convection:
If
the current (or movement of fluid) is caused not by the density difference but
by some agitator or by some mechanical devices then the type of heat flow
associated with it is called forced convection.
Radiation
When
heat is transferred through the space by electromagnetic waves the type of flow
of heat is called radiation heat flow.
When radiation is passing through
empty space, it is not transformed into heat or any other form of energy nor it
is diverted from its path. If the
radiation falls on a matter the radiation energy will be transmitted (i.e. pass
through the matter), reflected or absorbed. Only the energy that is absorbed is
converted into heat energy.
CONDUCTION
The basic law of heat transfer by
conduction can be written in the form of the rate equation:
The driving force is the temperature gradient.
Fourier’s law
Fourier’s law states that the
rate of heat flow through a uniform material is proportional to the area
perpendicular to the heat flow (A), the temperature drop (dt) and inversely
proportional to the length of the path of flow.
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Consider an area A of a wall of
thickness L. Let the temperature be uniform over the area A on one face of the
wall. Both sides of the wall has a temperature gradient. If a thin thickness dL, parallel to the area
A, be taken at some intermediate point in the wall, with a temperature
difference of dt across such a layer, then Fourier’s law may be represented by
he equation:
eqn
1
Where k = proportionality
constant
If the temperature gradient dt/dL
does not vary with time (this case is observed at steady state of heat flow)
then the rate of heat flow is constant with time and
eqn. 2
Since normally we know only the
temperature at the two faces of the wall hence integrating the Fourier’s
equation: 
eqn.
3
eqn.
3
On integration, if t1
is the higher temperature than t2.
eqn.
4
If A does not vary with L (i.e.
the case of a flat wall) then equation 4 integrates to
or, by rearranging we get 
or, 
eqn. 5
or, eqn. 5
In equation 5 Dt
is the driving force and the resistance is
L / k A..
Thermal conductivity
The
proportionality in constant k in
equation 
is called the thermal conductivity (also called the
coefficient of thermal conductivity) of the material of which the wall is made.
is called the thermal conductivity (also called the
coefficient of thermal conductivity) of the material of which the wall is made.
If q is expressed
in Btu
q in hr
A in ft2.
t in 0F and
L in ft
the unit of k will be :
The numerical value of the thermal conductivity depends upon
(i) The
material of which the body is made of
The
thermal conductivities of liquids and gases are smaller compared to solids. For
example at 2120F
the thermal conductivity of
silver
is 240 (Btu)(ft) / (hr)(ft2)(0F)
water
is 0.35 (Btu)(ft) / (hr)(ft2)(0F) and
air
is 0.017 (Btu)(ft) / (hr)(ft2)(0F)
(ii) and
upon its temperature.
The
variation of thermal conductivity with temperature is meager (very small) but
it is assumed that the
variation is linear; that is:
k =
a + b t
where
a and b are constants and t is the temperature.
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Compound resistances
in series
Consider a flat wall constructed of a series of layers.
·
L1, L2, L3 are
the thickness of the layers.
·
K1, K2, K3 are
the thermal conductivities of the layers
·
Let the area of the compound wall, at right
angles to the heat flow be A.
Let
t0, t1, t2 and t3 be the
temperatures at the surfaces of the wall and at each junction according to the
figure where t0> t1> t2 > t3
.
Therefore, 
where,
Dt1 = t3
– t0.
Dt1
= t1 – t0.
Dt1
= t2 – t1.
Dt1
= t3 – t2.
Again from Fourier’s law
Since all the heat passing
through the first resistance must pass through the second and in turn, pass
through the third, so q1, q2 and q3 must be
equal and all of them can be represented by q. From equation (1–4):
= q R1 + qR2 + qR3.
= q (R1 + R2 + R3)
\ 
If the equivalent resistance of
the compound wall is R then 
\ R = R1 + R2
+ R3.
HEAT FLOW THROUGH A CYLINDER
Let us consider the hollow
cylinder represented by the figure.
r1 = inside radius
r2 = outside radius
t1 = inside
temperature
t2 = outside
temperature
N = length of the cylinder
km = mean thermal conductivity of
the material of the cylinder
It
is desired to calculate the rate of heat flow through the wall.
Let us consider a very thin
cylinder, concentric with the main cylinder with
a
radius r where r1
< r < r2.
the
thickness of the wall of that cylinder is dr
temperature
difference across dr is dt
Then applying Fourier’s law over
the thin wall will give:
where A = total
area of the thin wall |
\ 
or, 
or, 
or, 
or, 
PRINCIPLES OF HEAT FLOW IN FLUIDS
Heat
transfer from a warmer fluid to a cooler fluid, usually through a solid wall
separating the two fluids, is found in different heat transfer equipment like
heat exchangers, evaporators etc.
The heat transferred may be in
the following forms:
(i) latent heat accompanying a phase change such as condensation
, vaporization
(ii) sensible heat without any phase change.
Mechanisms of heat transfer through a fluid : Both by conduction
and convection.
Heat flow from one fluid to another fluid separated by a solid wall
For
example a liquid is flowing through a pipe and that liquid is heated by a steam
from outside the pipe. In this case heat will be transferred from steam to the
liquid. Both steam and the liquid are having
Reynolds number above 4000 i.e. they are flowing in turbulent motion. In
this case steam will produce a thin film on the outside surface of the pipe and
the liquid will form another film at the inside surface of the pipe wall [because the flow is very slow near the solid
wall hence in the film viscous flow will prevail]. Beyond these films the
steam and the liquid are remaining in turbulent motion i.e. complete mixing is
going on.
Heat
is transferred through this stagnant films by conduction and when it reaches the bulk of the fluid heat is mixed
by forced convection. Since the bulk
of the fluids are in great motion hence the heat transfer within the bulk is
very rapid. Since the thermal
conductivities of the fluids are low hence, although the films are very
thin, the resistance offered by it to
the flow of heat is very large.
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Temperature gradients in forced convection
Let
us consider that heat is flowing from a hot fluid through a metal wall to a
cold fluid.
·
The dotted lines F1F1 and
F2F2 on each side of the solid wall is representing the boundaries of the films in viscous flow;
all parts of the fluids to the right of F1F1 and to the
left of F2F2 are in turbulent flow.
·
The temperature gradient from the bulk of the
hot fluid to the metal wall is represented by the curved line tatbtc.
The temperature ta is the maximum temperature
in the hot fluid (i.e. in the bulk of the fluid).
The temperature tb is the temperature at the
boundary between the viscous and turbulent flow.
The temperature tc is the temperature at the
actual interface between the fluid and the solid wall.
Similarly for fluid 2 the curved line is tdtetf.
·
When a thermometer (or any heat sensing
instrument) is inserted into the bulk of the fluid it will show a temperature t1
and t2 respectively for the two fluids. This t1 is
neither ta nor tb but this will be an average temperature
and ta < t1 < tb . t1 is shown as a straight line MM.
The same remarks will apply for fluid 2 also whose
average temperature t2 is represented by the line NN.
·
The temperature gradient tdtc
is caused by the flow of heat in pure conduction through the solid wall and
this (tc – td) smaller than (t1 – t2).
Surface coefficients
Since
the thickness of the film is not known, the simple equation of conduction
cannot be applied in this case. The difficulty is circumvented by the use of surface coefficient.
The surface coefficient on the hot side is defined by the relation
where, h1 = surface coefficient of the fluid on the hot
side [Btu/(ft2 0F s)]
q =
amount of heat flowing from hot to cold fluid. [Btu / s]
A1
= area of the metal wall on the hot side in a plane at right angle to the heat
flow.[ft2]
(t1
– tC) = temperature gradient. [0F]
If we compare 
equation with 
then it is evident that
equation with then it is evident that
h1 is analogous to (k / L) and (1/h1A1
) is the resistance term same as that of (L / kA) and h1 contains the effect of
both the viscous film and of the thermal resistance of the turbulent core that causes the temperature difference
(t1 – tC).
In the same way h2 may be defined as 
So the resistance imparted by the hot side = 1 / h1A1.
Resistance imparted by the metallic wall = L / kAm.
Resistance imparted by the cold side = 1 / h2A2.
Overall heat transfer
coefficient
If this resistances are substituted in the equation for
compound resistance in series 
then 
If the numerator and denominator are multiplied with A1
then
So the overall heat
transfer coefficient U1 is defined by the equation
Therefore q = U1A1Dt states that the rate of heat transfer is the product of three factors: overall heat
transfer coefficient (U1), temperature drop (Dt), and area of heating surface(A1).
For a tubular pipe A1 = pD1 l where
D1 and l are the
inner diameter and length of the pipe respectively. Similarly A2 = pD2
l and Am = pDm
l .
So another form of overall heat transfer coefficient :
Analogous equation can be written for Um and U2.
Fluids in natural
convection
If
a fluid is in contact with a heated surface. The fluid immediately adjacent to
the tube will tend to rise because of its decreased density and to be replace
by colder fluid. This fluid circulation caused by density differences due to
the temperature differences in the fluid is termed natural convection.
The velocity of circulation of
the fluid is dependent
on
the density differences
on
the geometry of the system, i.e. to the size, shape, arrangement of the heating
surface
and
the shape of the heating vessel in which the fluid is enclosed.
For the simple case of a fluid
outside a single horizontal cylinder with a large extent of fluid surrounding
the cylinder, the heat-transfer coefficient for natural convection has been
correlated by an equation containing three dimensionless groups the Nusselt
number (Nu), the Prandtl number (Pr) and the Grashof number (Gr):
Nu = y (Gr,
Pr)
where 
, 
, 
, ,
where h = average heat transfer coefficient, based on the entire pipe
surface
kf
= thermal conductivity of fluid
cp
= specific heat of fluid at constant pressure
rf
= density of fluid
b =
coefficient of thermal expansion of fluid
g
= acceleration of gravity
DTo
= average difference in temperature between outside of pipe and fluid distant
from wall
mf
= viscosity of fluid
When hot bodies lose heat to
their surroundings they do so both by radiation and convection. In the lower
temperature range convection is more important, in higher temperature range
radiation is more important.
Heat transfer through boiling liquids
Consider
a horizontal tube or a group of horizontal tube immersed in a pool of pure
liquid with steam or other source of heat inside the tubes. DT is
the difference in temperature between the tube wall temperature and the temperature
of the liquid (under the pressure of the vapour space above the liquid).
·
When the DT is very small the rate
of heat transfer is nearly similar to that of a non-boiling liquid.
·
As the DT is increased, the
coefficient of heat transfer is increased rapidly because the stirring effect
of the increasing number of bubbles released produces currents in the liquid
that accelerates the heat transfer. This increasing coefficient multiplied with
the increasing DT,
results in an even more rapid increase in the total heat transferred per unit
area.
·
However, if the temperature of the surface is
continually increased, a point is found where the heat-transfer coefficient
reaches a maximum.
·
At higher DT beyond the maximum value
the heat transfer rate is sharply lowered. Actually the bubbles of vapour
formed on the heating surface are discharged rapidly to rise through the
liquid. This type of boiling is called nucleate
boiling. At critical DT, these bubble coalesce into a continuous film of
vapour that insulates the tube, and this reduces the heat transfer rate with
increasing DT.
With polished horizontal tube in
reasonably pure water, this critical point is reached at relatively modest
value of DT,
possibly 45 to 500F. With rougher commercial steel tubes the critical
DT
is much higher.
In
case of nucleate boiling how easily the bubbles will leave the surface depends
on the following factors:
·
roughness
of the tube and the type of roughness: For instance a rough surface with
small sharp projections makes it possible to detach bubbles from points more
easily than from a smooth surface.
·
the
tendency of the liquid to wet the tube: A liquid that wets the tube
strongly tends to pinch off the bubbles of gas and liberate them more quickly
than a liquid that does not wet the surface easily.
·
the
difference in density between the bubble and the liquid
·
the
physical arrangement of the surface: For instance, a vertical tube with
bubbles rising inside it, will always show a much higher critical DT
than a horizontal tube, with bubbles formed on the outside.
Heat flow through condensing vapours
When
a saturated vapour, such as steam, transmits its heat to a metal surface and is condensed, the condensation may take place in either of two entirely
distinct forms. One is film type condensation
and the other is drop-wise condensation.
Film-type condensation:
In
this case the condensed liquid wets the surface on which it is condensing and
forms a continuous film of condensate. If the condensate is occurring on the
outside of the tube then this film of condensate drops off the underside of the
tube. If the tube is vertical then the condensate runs down the whole length of
the tube.
For the case of a horizontal tube in true-film type
condensation on which there is condensing a saturated vapour, free from any
non-condensed gas and moving at low velocities, Nusselt has derived the
following equation:
where l
= latent heat of vaporisation of vapor, Btu/lb
r =
density of condensate, lb/ft3.
k
= thermal conductivity of condensed vapor 
g
= acceleration due to gravity (ft/hr2)
m =
viscosity of condensate film, (ft-lb-hr) units
D
= outside pipe diameter, ft
DT =temperature difference between vapor and
metal, 0F.
For the case of a vertical tube
with all other conditions same as above Nusselt has given the equation:
where L =
the length of the tube.
·
The film coefficient between condensing vapors
and metal walls increases with increasing temperature of the vapor, because of
decreased viscosity of the film condensate.
·
Coefficient (h) decreases with increasing
temperature drop, because increasing temperature drops cause faster
condensation and hence thicker liquid films.
·
The presence of non-condensable gas accumulate
near the heating surface and add to their resistance to that of the liquid
film.
Drop-wise condensation
In
this case the condensed liquid does not wet the surface, but collects in drops
that may range from microscopic size up to drops easily seen with naked eye.
This drops grow for a while and then fall of the surface, leaving an apparently
bare area in which new drops form again.
These
two types of condensation give widely different film coefficients of heat
transfer. Coefficients in the case of drop-wise condensation is much more
greater than those obtained in the film type condensation provided all other
characteristics of the surface is the same.
RADIATION
Any solid
body at any temperature above absolute zero radiates energy. This radiation is
an electromagnetic phenomenon and takes place without the necessity of any
medium.
The
approximate range of wavelengths for infra-red radiation (or heat rays) is 0.8
to 400 mm.
In
industries, most of the cases, the thermal radiation corresponds to wavelengths
from 0.8 to 25 mm.
The Black Body
Not all
substances radiate heat at same rate at a given temperature. So a theoretical
hot body is defined, which is called ‘black body’.
Definition: A ‘black body is defined as that body
which radiates maximum possible amount of energy
at a given temperature.
Example: It has been shown that the inside of
an enclosed space, at a constant temperature throughout,
viewed through an opening so small that the amount of energy escaping through the opening is
negligible, corresponds to a black body.
Practical e.g. In practice, a convenient black body is made
from a tube of carbon plugged at both ends
with a small observation hole in the center of one end.
Industrial e.g. The inside of furnace at completely uniform
temperatures, viewed through a small opening,
is a black body. The interior of the furnace and all the objects within the furnace can also
be considered black bodies.
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Rates of radiation
If the
radiation energy emitted by a hot body is plotted against the wave lengths
emitted then graphs of the nature shown in the fig will be obtained.
The total
amount of radiation emitted by a black body would be given by integrating the
curves of the figure (Effect of temperature on amount and distribution of
black-body radiation) The result is the Stefan-Boltzmann law
q = b A T4.
where
q = energy radiated per hour (BTU/hr)
A = area of radiating surface (sqft)
T = absolute temperature of the radiating surface 0R
(Rankine) = t 0F + 460
For black bodies the value of b = 0.174 x 10 –8
Btu / (hr sqft 0F4)
No actual body radiates quite as much as the black body. The
radiation by any actual body can be expressed as
q = e b A
T4.
where e is the emissivity of the body.
·
The emissivity is a fraction less than 1 and is
the ratio of the energy emitted by the body in question to that emitted by a
black body at the same temperature.
·
When a radiant energy falls on a cooler body
either the energy is reflected or transmitted or absorbed. The fraction of the radiation energy falling on a body
that is absorbed is represented by a, the absorptivity,
which is always less than 1. If the reflected energy can be neglected then the
energy absorbed by any body is equal to the radiation falling on it.
·
It can be shown that the absorptivity of a given
substance at a given temperature and its emissivity at the same temperature
must be equal.
That is, e
= a
Since the e of a black body is 1 hence the a of the black body will
also be equal to 1. Therefore the black body absorbs all the radiation falling
on it – an important property of black body.
·
The value of a for a given surface at a
given temperature varies some what with the wavelength of the radiation
involved. To avoid complication due to this the concept of gray body has been introduced. The absorptivity of a gray body at
a given temperature is constant for
all wavelengths of radiation.
·
When a small black body of area A and
temperature T2 is completely surrounded by a hotter black body of
temperature T1, the net amount of heat transferred from the hotter
body to the colder body is, therefore the algebraic sum of the radiation from
the two bodies, so that Stefan’s law may be written for this case as
q = b A (T14– T24)
ANGLE OF VISION
Let us
consider a hot plate of indefinitely large extent is radiating heat. An element
on this hot plate having an area of A is radiating heat at the rate of q Btu/hr in every direction. In order to
intercept all this radiation, a cooler body would need to subtend an angle of
approximately 180 spherical degree (or 2p steradians), i.e. an
hemisphere with the hot element as the centre.
So angle of vision is the solid angle subtended by a finite surface at
the radiating element.
The solid
angle subtended by a hemisphere is 2p steradians. This is the maximum angle of vision that
can be subtended at any area element by a plane surface in sight of the
element. If the angle of vision is less than 2p steradians, only a fraction
of the radiation from the area element will be intercepted by the receiving
area and the remainder will pass on to be absorbed by other surfaces in sight
of the remaining solid angle.
Figure shows several typical radiating surfaces:
(Fig. a) Two large
parallel planes, one is a hot surface and the other is a cold surface. An area
element on either plane is subtended by a solid angle of 2p
steradians by the other. The radiation from either plane is intercepted by the
other completely.
(Fig. b) A point on the hot body sees only the cold surface, and the angle of
vision (of the element on hot body) is again 2p steradians for the cold
surface. Elements of cold surface can see the hot body and at the same time
other portions of the cold body hence the angle of vision (of the element on
cold body) is smaller for the hot body.
(Fig. c) A point
on the hot surface can see the cold body as well as portions of the hot body,
hence the angle of vision (of the element of hot body) is smaller than 2p
steradians for the cold surface.
(Fig. d) The cold
surface subtends a small angle at the hot surface, and the bulk of the
radiation from the hot surface passes on to some undetermined objects situated
at the background.
Square of the
distance effect
The heat flow can be measured in two terms: (1) Total heat flow (Btu/hr) and
(2)
Heat flow per unit area (Btu/hr/sqft)
The energy from a small surface that is intercepted by a
large surface depends only upon the angle of vision. In the above figure the cold surface may
remain in any of the positions (I, II, III). The total amount of heat (Btu/hr)
received by cold surface I will be equal to the cold surfaces II and III, since
the angle of vision is same (2p steradians) in all the cases. But the amount of heat
received per unit area of cold surface (Btu/hr/sqft) will differ and will
decrease as the cold surface goes far away from the hot element.
Practical example
 |
When a kettle (cold surface) is placed over a hot plate (hot
surface) the angle of vision (of an element on the hot plate) will decrease
with the distance of the two surfaces.
i.e f > q So the amount of
heat received per unit area of the kettle will be greater in the first case
than that of the 2nd case and hence the temperature of the kettle in first case
will be much more than the 2nd one.
Projecting
fins or ribs that tend to increase the cold surface do not absorb an increased
amount of heat by radiation unless they increase the angle of vision of the
cold surface.
Radiation error in
thermometry
Ordinary
gases, free from smoke or visible flames, are practically transparent to
radiation. When such gases are flowing through any pipe (or conduit) the
temperature of the wall of the pipe is usually much cooler than the average gas
stream.
Now
if any type of temperature measuring instrument is inserted into the gas stream
, as the temperature of the instrument approaches the temperature of the gas it
(temp. of the instrument) becomes higher than the temperature of the wall of
the pipe. The instrument, therefore, immediately starts radiating heat.
The
instrument is heated by convection, and loses heat by radiation. At higher
temperature, rate of heat loss by radiation is much more greater than that of
convection. Hence, the instrument always shows a temperature less than the
actual average gas temperature.
Remedy Various
means are available to reduce this loss by radiation though it never can be
eliminated completely.
·
Radiation to the wall can be reduced by using
polished metal for the wall of the pipe.
·
The measuring instrument itself, the bulb of the
thermometer, the junction of the thermocouple, or other devices should be made
of bright metal to make the device deviate as far as possible from a black
body.