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Welcome to lecture four of module six. In this module we have been discussing on condensation
and boiling heat transfer. First two lectures we have discussed on condensation and heat
transfer, and then last lecture onwards we have discussing on boiling heat transfer,
last lecture we have discussed some basic concept, fundamentals of boiling heat transfer
situations. And now, in this lecture, we will try to concentrate on various correlations
available for boiling heat transfer and also to see, that how the boiling heat, heat transfer
can be calculated by using such correlations in different conditions.
Now, to start with what we are going to discuss is, so in, in this lecture we are going to
discuss mostly, that boiling, boiling
heat transfer correlations.
To start with we are going to discuss first, that how bubble of vapour remains in equilibrium,
mechanical equilibrium in a liquid pool. So, if you see, consider, that there is a liquid
pool and we know, that there is a bubble here and we will assume a spherical bubble and
then, here in this, there will be surface tension forces. So, what happens and this
is the, say diameter of this bubble. So, this is spherical bubble of radius two, sorry,
radius r, surface tension is equal to, say sigma. So, here, this is the surface tension
force. Now, what happens here and this, the liquid at, this is the liquid pool. The, it
is at a pressure of P L, that is the pressure of liquid pool and here P V is the pressure
of the vapour. So, to know the mechanical equilibrium, mechanical
equilibrium we know, that the pressure forces is being balanced, differential pressure forces
is being balanced by surface tension force. So, surface tension force we know, that it
acts in permit length. So, the surface tension force is 2 pi r into sigma, that is the surface
tension force, assuming that is the spherical bubble, that is assumed and that is equal
to P V minus P L. This is the pressure difference and we know, that pressure of the vapour is,
vapour, vapour pressure is, pressure, pressure of the vapour is more and the pressure of
the liquid, and this into pi into r square is the radius of the, this is the area on
which the pressure acts. And therefore, we can say, that P V minus P L.
The differential pressure is balanced by this term, this is 2 sigma by r, so this is a very
standard relationship, that, that if you know pressure in a, in the, of the liquid. And
then, and then we know, that surface tension force, if you know the radius of the bubble,
then we can find out the vapour pressure or if you know the pressure of the vapour, then
we can find out the values of radius of the bubble, that you can find out.
Now, now, next, another point, what we are going to discuss over here is to see, that
how to calculate the, the radius of the bubble by using some theoretical analysis, that is
what we are going to discuss.
Now, some theoretical analysis for calculation of bubble radius, so if we do this we will see, that to start with, we
will think, that we will apply, that Clausius-Clapeyron equation, that you might have studied in thermodynamics,
which is applicable between two phases. So, here we will assume, that vapour phase behaves
like an ideal gas and then what we get is, that according to Clausius-Clapeyron equation,
dlnP, pressure by dT is equal to LV, that is, the latent heat of vaporization, the phase
change equation, and RT square. And then, if you do the integration from P L to P V
at, to pressure conditions dlnP and that is equal to L V, latent heat of vaporization,
by R into dT by T square. And here we will write integration will be T sat and this is
equal to T L. Now, the conditions are here, that T sat is,
we have discussed several times is that boiling point of liquid, boiling point of liquid T
L and P L are temperature and pressure of superheated liquid, pressure of superheated
liquid. As we know, that to have that bubble formation, that liquid has to be, little bit
of superheating is needed, superheated liquid. So, T L will be higher than T sat and then,
another thing is that since the vapour bubble is in thermal equilibrium with the liquid,
so vapour temperature is also, is taken to be T L because the vapour is, vapour in the
bubble is thermal equilibrium with the liquid, surrounding liquid.
And we know, thermal equilibrium means, the temperatures between the two phases has to
be same, vapour in the in bubble is in thermal equilibrium with this surrounding liquid,
otherwise if the surrounding liquid is lower temperature, then there will be condensation, then it will the bubbles may get collapsed.
So, the vapour of the temperature is T L. So, based on that if we just do this, then
we can find out from here, that ln P V by P L by doing the integration we will get L
V by R, latent heat of vaporization by R into 1 by T sat minus 1 by T L.
Now, if we do now little bit of modifications or manipulations, so ln P V by P L, we will
write as ln 1 plus P V minus P L by P L and that can be approximated as, P V minus P L,
sorry, P V minus P L by P L and that is becoming equal to, according to previous equation,
L V by R into T 1 minus T sat by T L into T sat. Now, as we know, that T L is nearly
equal to T sat, so we can write, that T L into T sat is nearly equal to T sat square.
And then, we can get, that T L minus T sat and that is equal to twice sigma, is equal
to T L minus T sat, will be equal to P V minus P L by P L and then into R by L V, I am sorry.
So, this will be equal to, if you just put over here, T L minus T sat is equal to P V
minus P L by P L into R by L V into T sat square. And we know, that P V minus P L is
equal to twice sigma by R. So, we will get twice sigma by R into P L. Already we had
into R by L V into T sat square, so, so this is approximated as twice sigma and this is
equal to, we write again twice sigma into R into T sat square by R into P L into L V. So, this is an equation.
From this equation, if we know the definite values of say, sigma T sat and T L and all
of the things, we can easily calculate the value of bubble radius R. So, this relationship
can be used to calculate the radius of the bubbles.
Now, Fritz has given an expression, empirical relation, has proposed an empirical relation
for bubble dia, which is denoted as d b here, at detachment from the heating surface. We
have to understand, that the bubble is started forming already, we have discussed in the
last lecture and then, it gradually grows and then finally, it detaches from the surface.
So, the, with time the diameter of the bubble changes, size of the bubble changes. So, during
the time of detachment, at the time of detachment what happens to the diameter of the bubble,
that is being calculated by the relationship given by Fritz and this relationship is, that
d b equals to c d into beta root over twice sigma g into rho L minus rho v, where c d
equals to some factor and that is 0.0148 for water and beta is equal to angle of contact.
Now, Zuber also proposed a relationship for bubble diameter and that is f into d b equals
to 1.18 tau c by tau c plus tau d into g sigma rho l minus rho v by rho l whole to the power
1 by 4 and in this case, tau c is time for which the bubble is attached to the surface.
So, it is basically depending upon the time it will find out the bubble diameter. Then,
tau d is the time that elapses before, time that elapses before the next bubble is formed.
And then, and f is called the frequency of bubble formation. So, this is a relation,
this is, these are the two relationships, which are used. These are empirical relations
to calculate the bubble diameter at different situations.
Now, we will this see, that there are some correlations for calculating the pool boiling
heat transfer, so, which is very important for us correlations for calculating pool boiling
heat transfer. Now, in this case, the most important one is, that we should consider
is, that nucleate boiling situation first, because we have already discussed, we have
already discussed in the last lecture, that nucleates boiling is the phenomena, that we
should try to work on, we should prefer for any kind of boiling situations because that,
that time it is giving, giving us the ***, very good value of a transfer coefficient
at the time. At the same time, it, it, it occurs and this having or reaching to boiling
crisis phenomena or burnout phenomena. So, for nucleate boiling there is a very well-known
relationship, that has been given by Rohsenow relation. According to Rohsenow relation,
that q dot s that is the heat flux equals to mu l into l v into g into rho l minus rho
v by sigma whole to the power half and c Pl into T e, excess temperature, c s f is the
constant L v into l whole to the power n, so this whole to the power 3. So, this is
a Rohsenow relation and here the different parameters of given like this, that q dot
s is called heat flux in nucleate boiling. Then, we have mu l liquid viscosity, l indicates
the liquid and it is in its Pascal second. Then, L V, latent heat of vaporization or
enthalpy vaporization unit is joule per kg. Then, we have g is gravity acceleration due
to gravity, acceleration due to gravity, it is meter per second square. Then, c Pl is
specific heat liquid, specific heat, it is given as joule per kg per degree centigrade.
Then, rho liquid density rho l, rather liquid density and it is definitely, kg per meter
cube.
And this way there are some more, T e, excess temperature and then we know, that T is equal
to T w minus T sat and this is in degree centigrade or degree Kelvin or Kelvin and then, we have
T w, we know wall temperature, Kelvin and T sat, saturation liquid, saturation temperature
of liquid, of liquid, that is, it is in Kelvin and say, P r l, liquid Prandtl number, sorry,
then, and it is unitless and q dot s is Watt per meter square. So, unit of q dot s is in
watt per meter square and then, what happens if the values of C s f, C s f and n are constants,
C s f and n are constants. It depends on the geometry, I am sorry, material of construction
and the liquid material are, say free surface depends on the fluid, rather liquid and surface
liquid and surface combination for boiling operation.
Now, typically, if we say, that say, water-stainless steel and C s f and n, this is 0.01 and this
is 1.0, then water, then brass, it is 0.006, 1.0, like that. Then, one interesting is water-copper,
0.013 and this 1.0. Similarly, benzene and chromium, it is 0.010 and it is 1.7. So, these
are the typical ranges of the values, that is what I just wanted to highlight, that when
we would be solving these problems or then, we have to, we may have to make use of the,
of these values of these constants before applying the equations.
Now, in addition to this correlation by manipulate boiling given by Rohsenow, that another useful
correlation for that, which is nothing, but h b is boiling heat flux, which given as 0.00341
into P c to the power 2.3 T e to the power 2.33 Prandtl to the power 0.566 and its unit
is Watt per meter square per degree centigrade. This is the boiling heat transfer, it is heat
transfer coefficient, boiling heat transfer coefficient and then P c is critical pressure
in bar, P r is reduced pressure and T e is equal to excess temperature T w minus T sat,
it is integration centigrade or Kelvin. So, now, in addition to this, whatever we
have discussed, that is another important thing, that we should know, that is called
critical heat flux, heat flux. So, there are certain relationships we have discussed in
the last lecture, that there is a critical heat flux. So, if you move more than critical
heat flux, then we may, or that, that is the critical point where we see, that, which is
the more or less metastable point, which is, after which, that film boiling becomes important.
So, we have to be operating before this critical heat flux condition that is the first term,
the top point of the maximum of the first term in case of boiling curve, whatever we
have seen. So, that critical heat flux we should find out and our operations should
be always below the critical heat flux conditions. So, to find out the critical heat flux, which
is also called the maximum heat flux calculation. We have some correlations; first one is given
by Lienhard et al and in 1973, here the q dot max. So, critical heat flux, which is
nothing, but the maximum heat flux. Also, we are telling, because if we go beyond that,
then it can, because if you do not have the control, then may it may as switchover to
the burnout conditions. So, q dot max is equal to 0.149 into L V into rho V and then we have
sigma, then g into rho l minus rho v by rho v square whole to the power 1 by 4 and it
is in Watt per meter square. And the terms of the significant and units are all the,
terms are exactly similar to, it is the same terminologies whatever use in case of Rohsenow
equations. So, all the terminologies and their units are as discussed in case of Rohsenow
equation, Rohsenow equation.
Now, there is another relationship, that is to give the peak boiling flux, so this critical
heat flux also is called maximum heat flux, also it is called peak boiling heat flux because
that boiling happening at the peak condition. So, if I just draw again here, that we had
the boiling probably like this, so this is the region, this is the region and here this
is called the critical heat flux condition. So, q dot max is the critical heat flux condition
and this is the excess temperature for that. Now, this is the boiling curve. Now, in this
case, again another relation, that is given by sun and Lienhard relation, according to
this relation, that q dot max is equal to q dot max dashed, then multiplied by 0.89
plus 2.27 exponential minus 3.44 root over r dashed and then, here this is for r dashed
greater than 0.15 and r dashed is dimensionless, radius is equal to r into g into rho l minus
rho v by sigma whole to the power half and q dot max dashed is equal to peak heat flux
on an infinite horizontal plate. And q dot max dashed is given by 0.131 L V and root
over of rho v into sigma g into rho l minus rho v whole to the power 1 by 4 and this is
Watt per meter square. So, this is happening above that different correlation for critical
heat flux.
Now, we will be discussing on some correlation of stable film boiling. In case of stable
film boiling, it is also called as, also called as film pool boiling, stable pool boiling,
Bromley has given a relation. Bromley relation can be used, it says that h b, the boiling
heat transfer coefficient equals to 0.62. Now, K v is the thermal conductivity of the
vapour, this into rho v density into rho l minus rho v into g.
Now, here is the modified latent heat or enthalpy of vaporization, 0.4 C p v into T e is the
excess temperature divided by d is the diameter of the tube and this is happening for horizontal
tube into mu v into T e whole to the power 1 by 4 and Watt per meter square per degree
centigrade. The heat transfer coefficient and boiling heat transfer coefficient were,
suffix v implies vapour phase and d is diameter outside of tube and the equation is applicable
for film boiling outside horizontal tubes and the equation can be used for, but the,
these, these equation also can be used for vertical plates after replacing d by L, the
plate height and the constant 0.62 by 0.70. So, this is, this can be done and now sometimes
this is the film boiling situation. Many times we have already seen that during
film, film boiling, that because the temperature is very high. There is a possibility, that
radiation conclusion is also quite strong.
So, if radiation contribution is strong, is significant, then we can use, that h is equal
to the equivalent heat of the coefficient and that is equal to h b, boiling heat transfer
coefficient into h b by h whole to the power one-third plus h r. So, h is that equivalent,
so h is the resultant heat transfer coefficient. You can understand, that is an implicit coefficient,
so it is some method is required for getting a solution to this and h r is called radiation
heat transfer coefficient, h b is boiling heat transfer coefficient. This is from Bromley
relation. And h r is obtained as say, sigma is b epsilon
T w to the power 4 minus T sat to the power 4 by T w minus T sat. Here, sigma s b is called
Stefan-Boltzmann constant, fine, and epsilon is called emissivity of the surface, fine,
and properties are to be evaluated at T f, fine, and except L V at saturation temperature,
this is for Bromley equation, all the parameters has to be calculated.
So, this is about the correlations regarding, that convective heat transfer, boiling heat
transfer coefficient, where we have nucleate boiling and, and film boiling. Now, there
is another point that is called forced convection boiling. In the forced convection boiling,
that we can see, that it happens when a liquid, when a liquid is forced through a channel
or a tube or over a surface and maintain at a, the surface is maintained at a temperature
higher than the fluid liquid basically. Then, there will be boiling phenomena, that will
happen and that vapour and the liquid vapour bubble and the liquid. That means, there will
be a two phase flow, that will happen through the device or may be over the device and that
kind of situation is called the forced convection boiling, and forced convection boiling is
very important from industrial point of view, from applications point of view.
A typical, a forced figure is being shown here regarding the forced convection boiling.
If you see here, that there are several regimes: one, two, three, four, five, six, there are
several regimes, six regime regimes are there in this particular case, which says, that
in the very first and the very first is the subcooled liquid in, and the subcooled boiling.
So, in this regime what is happening, that there will, is that, that vapour is, bubble
is formed in that region, but this bubble cannot go to this region, it is getting collapsed.
And then, in this region, that, we that more amount of bubbles are getting formed and the
bubbles, there is a mixture of the bubble in the liquid, so this is called bubbly flow.
So, this flow is happening in the same direction of the flow and it is called a bubbly, bubbly
flow. And then, there will be the, the, with further increase in temperature.
What is happening, that bubbles, some of the bubbles will, some of the bubbles will collapse
and the bigger bubbles will form and then it is, it is kind of relation, this is called
a slug flow situation, where that it is mostly, that big vapours are coming up in the form
of bubbles and then after that you can see, that there is annular region. In this annular
region, there will be the little, there will be the most of the liquid will be flowing
and in this region, this, this is the vapour, which is flowing and then, after that what
is happening in this, there will be fine particles of the, fine particles of the droplets will
be there in the, and then after that here it is called mist flow. That means, all the
liquids, whatever is present, it will be like small particles into this and then finally,
in this region there will be no liquid, only the vapour and they will come out.
So, this is what is happening in case of the different regions of flow situation, in case
of boiling heat transfer through a channel. And this is a typical case we have seen, that
there are six zones, one to subcooled bubbly flow, slag flow to annular flow, mist flow,
single phase vapour flow. These are the six, six zones, that can be observed and of which
we can very easily understand, that particularly, 2nd, 3rd and maybe, little bit of 4th, these
three regions gives us a very good heat transfer phenomena, whereas 5th and 6th, 5th and 6th,
they are not at all good for any heat transfer phenomena because is to mostly covered by
the vapour. So, this is what is called forced convection boiling, that, that takes place.
And we can see, that when there is a forced convection boiling, there are two kind of
things is happening, it is a, one is flow of the fluid and this flow can be a two phase
flow kind of scene, situation, and there is a boiling also, that the bubbles come into
picture.
So, in that case there is a recommendation, that the heat, Chen suggested, so in case
of forced convection boiling Chen suggested, that the total h T, total heat transfer coefficient
can be written as h b, bubbling heat transfer coefficient plus h c, heat transfer coefficient
are the two convection due to flow. So, now, this h t is that, so that means the total,
this is boiling and this is convective, that means, due to flow, so these are all heat
transfer coefficients. And then, a relationship is also being given, that is, that for convective
heat transfer coefficient it is like, because of the flow of the fluid through a tube kind
of situation, in that case it will be something like Dittus-Boelter equation kind of things
is used, it is like 0.023 with little modifications to that R e l to the power 0.8 and then Prandtl
L to the power 0.4 into K l by d whole to the power into f, this is in Watt per meter
square per Kelvin. Here, we can say that R e l is that liquid
only, Reynolds number, it is defined as G mass velocity into 1 minus w into d by mu
l, here G is fluid mass velocity, fluid mass velocity, which may have, so it is basically
kg per meter square per second, which may have the vapour component into it. So, w is
called the mass fraction of vapour in fluid and the subscript l stands for liquid. We know that P r is the Prandtl number, K
l is the formal conductivity of the liquid and then d is tube inner diameter. Now, the
question is K l by F and F, F is a factor, we will discuss these things, F is a factor.
And then, h b, the relationship for h b is given as 1.218, it is a very big relation,
this into K l thermal conductive to the power 0.79, then C Pl to the power 0.45 into rho
l to the power 0.49 by sigma to the power 0.5, is the 45, sorry, 45, it is 0.79 sigma
to the power 0.5 mu l to the power 0.29 L V to the power 0.24 and rho v to the power
0.24, so v stands for vapour, and then, T e, the excess temperature to the power 0.24
and delta p saturation, this to the power 0.75 into s.
Now, here, the delta p saturation is given as T into L V by T sat into L, sorry, T sat
into V l v, where T e, we know it is T w, wall temperature minus saturation temperature
V L v is V L v v minus V L, this in meter cube per kg. That means, this is specific
volumes, specific volumes of vapour and liquid. And delta P sat is, is, it is in Newton per
meter square, it is the difference of vapour pressure between the superheated liquid at
T w and the saturated liquid is this, at T sat and sigma is surface tension and it is
Newton per meter.
And then, we know, that S is given as, it is factor 1 plus 0.12 R e bar to the power,
sorry, 1.14 whole to the power minus 1.0, for R e bar less than 32.5 and this is equal
to 1 plus 0.42 R e bar to the power 0.78 whole to the power, this is 8 minus 1.0, for 32.5
less than R e bar less than 70. And this is equal to 0.1, for R e bar is greater than
70. So, here R e bar is again defined as 10 to the power minus 4 R e l liquid Reynolds
number into F to the power 0, sorry, 1.25 F to the power 1.25 and F is equal to 1.0,
for 1 by xi less than 0.10 and F is equal to, this is also a factor, 2.35 into 1 by
xi plus 0.231 to the power 0.736, for 1 by xi greater than 0.10. And 1 by xi is equal
to w by 1 minus w. I have discussed already, it is 0.9 into rho l by rho v whole to the
power 0.5 and mu v by mu l whole to the power 0.1 per, F are the factors, S is called the
boiling.
So, S is called boiling suppression factor and F is a factor, that modifies, that modifies
R e l of liquid to Reynolds number for the mixture.
Now, as we have seen, that forced convection, there are some more relations being given
Mc Adams et al suggested, that q dot heat flux is equal to 2.253 delta T to the power
3.96 Watt per meter square, where 0.2 less than P less than 0.7 mega Pascal. So, it is
a pressure dependency, so this, for this is a relationship for boiling of water at low
pressure; for boiling of water at low pressure. And then, for higher pressure, q dot is equal
to 283.2 P to the power 4 by 3 and then delta T to the power 3 Watt per meter square and
here, 0.7 less than P, less than P 14 mega Pascal. And delta is the temperature difference,
delta T is T w minus T sat. Then, another relation, see for vertical tubes,
for forced convection boiling on vertical tube, h is equal to 2.54 delta T whole cube
into e to the power P by 1.551 Watt per meter square per degree centigrade and 5 less than
P less than 170 atmosphere. So, this is the relationship.
Now, in addition to this, whatever we have discussed, so these are the different relationships
for the boiling heat transfer. In addition to this, I am going to discuss one important
aspect that is called heat pipe.
So, heat pipe is a device, it is, it is a device that makes use of change of phase heat
transfer; so, it makes use of change of phase heat transfer. So, what happens in case of
heat pipe is that if you draw, it is like this, we have a pipe over here and then it
is like this and then, fine and it is like this. Then, what is happening is that here
the heat addition at one end, heat addition at one end takes place, so that there is pipe,
say this the pipe and the pipe is being covered, so we have a circular pipe, so a layer of
wicking material covering the inside surface of pipe. So, this is the, we can say, that
this is the wicking material here. This is the insulation, the heat addition.
So, what is happening from the, there will be a, a, vapour, therefore evaporation. The
vapours will move like this and then, this there will, the vapour will be condensing.
So, here you can say condensation, condensation and here heat rejection; that means it is
heat taken out, heat rejection. So, one end is the heat addition, therefore there will
be an evaporation; in another end the heat will be rejected, therefore the condensation.
And that the vapour, which is coming in this direction is the vapour flow, it will come
to this end and the heat rejection, that will be condensing and then it will be the, condense,
it will come into this direction, condensate, this is a condensate flow. So, condensate
will come to this direction, this is a typical of a heat pipe. So, one end we are heat, it
is getting evaporated and another end it is getting condensed and it is coming down.
So, this kind of, so there will be an heat exchange, so this is used, very commonly used
for extraction of energy from hot waste gases, like flue gas and then it is used, the same
energy is used to preheat some other fluid. So, this is being done using the heat pump
and many times it can be tilted, tilted. So, one is called favorable tilt. In this case,
elevation of condenser and condensation end is higher than the evaporation end, then what will happen?
The liquid will try to come down due to gravity, so gravity flow, flow of condensate through
the wicky materials and adverse tilt the opposite arrangement; the opposite arrangement is called
the adverse tilt. And another thing is, that the typical wicky materials used are mesh screen, S.S. fibers, sintered
fibers and sintered fibers, then etched micro groups and channels.
So, this heat pipe is of very important to us and it is being, nowadays the optimum heat
design and the configuration of the heat pipe is a major, major research issue and people
working in this particular area. So, heat pipe is also related to phase change phenomena,
various type of fluids can be used depending upon the temperature and other conditions.
It can be ammonia, it can be liquid ammonia, methanol, water and so on, different liquids
can be used for the situation. This is all for today, we will be having some
exercise for, in future we will be adding some exercise associated with it, particular
section in the particular module to have to calculate some problems in this particular
areas. In next lecture we will start with a new module.
Thank you very much.