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So, in the previous lecture we took up the new unit operation drying. In that case, we
said that each solid has a certain characteristics of drying. In other words, if you want to,
you know, design a large scale dryer, we have to start with the the solid which we want
to dry in the the sand or zeolites or aluminum. The more important here to note was that,
for in drying, this characteristics curve, so called the characteristics curve for drying,
not only depends upon the type of the solid, all right, it also depends upon the rate of
the drying. That means, the operating conditions you know, under which you are doing the drying,
that also makes a very significant difference because, we said that the drying is one unit
operations, which is accompanied by both heat transport and mass transport.
So, we have to look at, you know, the different conditions for heating say, are we doing by
microwave heating or have we exposed the entire surface area to heater, some heater is that
the flow through the solids or is there any air flowing just over the surface, all right.
All of these will determine, say equilibrium concentrations some moisture. So, this is
slightly different from the previous say adsorption isotherms, when we said that we fix a temperature,
then the isotherms given the partial pressure in the gas phase or concentration in the liquid
phase, equilibrium concentration is fixed here.
But here, you must have noticed, that there are so many other factors. They also play
a major role. So, essentially what we are saying here that, we have to take a sample
of certain solid and determine its drying characteristics. So, we will do a very simple
experiment; suspend it in air monitor, its weight gain or weight lose. So, drying or
wetting. There also we said that there is a hysteresis. So, the rate of drying is different
from the rate of wetting. More here important here is that, we should ensure that the operating
conditions, Reynolds number or the area exposed for this batch of the solid at lava scale
and you know some plant or the industrial scale is as close as possible.
So, first thing here is establishing the characteristics of drying. We are talking about the batch
drying, to begin with. So, we have suspended solid zealots, accurate carbon fibers, charcoals;
we have given certain Reynolds number. So, the solid dries and we monitor its weight
loss and we get a characteristics curve. This curve is very important. We must understand,
it makes a, you know the basic of your all calculations, as far as a batch drying is
concerned. Before that, if you recall, we also had one more characteristics curve; there
we plotted partial pressure. So, equilibrium vapour pressure of moisture
versus moisture content. And of course, we dimensionalized the equilibrium vapour pressure
by the vapour pressure of pure water.
So, that is one type of curve, very important curve. One, in case of drying and the second
is this drying rate curve, for this batch solids or batch drying. So, let us begin with
this. We are trying to address here the rate of batch drying, all right or we are trying
to establish here drying rate curve. We said that, we have to ensure that our lab scale
or lab conditions are properly scaled or they are similar to the real operating conditions,
under which we want to do this batch drying. So, we are doing here this batch drying. So,
what we will do here, so, we have a solid and we monitor this weight loss.
So, there is certain flow rate, air, temperature, relative humidity, initial weight, certain
diameter, certain type of solid, everything is right fixed here. So, if we monitor its
weight loss, we should expect this type of curve.
So, we have theta, let say time in hour and here, we plot X. How much is the weight loss
per kg of moisture is remaining in this solid? So, kg of moisture, per kg of, say dry solid.
Here also, we said that the two ways of doing this, one is on the weight basis and another
is on the dry solid. So, we have both heat transfer and we have
mass transfer, all right both. So, at certain degree, heat is transported to the solid and
mass is getting out of this solid here, which is moisture here. Now, if you do this experiment,
we will expect that to begin with, if the solid is cold and this temperature of air
is, say very large hot air here, that there will be some initial adjustment time. So,
which means, there will be some rate here, say from A to B where, which would be slightly
ill defined, in the sense, that air could be colder or solid could be colder than this
air or it could be hot, all right. So here, there is some initial adjustment.
Very small amount of moisture will be lost here, so, initial adjustment periods, you
can say. Then typically what happens, it is a very typical curve of most of the solids.
Of course, there are always exceptions there. So then, after certain time, very small amount
of time, when the solid temperature has reached the air temperature steady state, then there
is a weight loss. This weight loss will go linearly like this, for some time, say lets
mark it here C, the change in the moisture content is linear, all right. So, after the
solid is warmed up or cold has reached the temperature, one monitors very typically this
linear change in the weight loss. Then, there is a slow down.
So, let us see, we have till D here D here. So now, we can say that there is a non-linear.
We will see this later that this linear curve will represent constant rate constant rate
of drying and this non-linear or from C onward, we will call it falling rate; will come back
to this later. So, at some rate, this stick reaches non-linearly and then, typically again
things gets slow down. So, we have another decrease, very small decrease
till the moisture content has reached say, certain equilibrium values X C or X star.
So, this is what we call it a characteristics, a typical characteristics, characteristic
with drying curve. Here now, let of course, a different solid will behave differently.
This most common that to begin with, one adsorbs a linear, then non-linear and then again,
it is non-linear for the rate has slow down here.
So now, if you define like in the previous class, rate of drying as N. So, the most common
and practical way of defining this rate of drying would be, so, let say the amount of
the solid, dry solid, so without any moisture, let say S, if its area which is exposed for
the to the drying is a. So, we can define as d X d T. So, in other words kg per second
per meter square. So, that is the rate of drying defined like this. So, you can see
that S is initial weight, which is fixed, A is the area which is exposed.
So, either we have a solid like this, slab like this and you want to dry from here or
you have a solid like this and you are drying, you have some air flow pass this area is of
course, external area; expose area is fixed here. Now, here also we said, that after some
time, when the, say the solid is covered with some thin film will be batches that will be
developed here. So, of course, the wet area will be different.
But, what we are writing here is the initial total area. So, this area is actually fixed,
as per as definition for rate of drying is concerned, it is not the wet area, but, it
is a dry area; it is a total area. So, d X by d T, knowing X here, we can do this del
X over del T. You can take a small small time step, after we have monitored all this rate
and we can calculate this rate here. So, this is the curve two, second type. Remember,
if you had recall, first we plotted relative humidity here, relative saturations right,
B bar over B 0 versus x or versus x here. So, that is the number one in drying curve.
Now, we have the second drying curve, which we have drawn x versus theta and from this,
we calculate n and then again, we plot N versus x.
So, this would be the third curve, which is quite important in our context, will revisit
them quite frequently. So, once we have x versus theta time, now we calculate n here,
which is rate of drying. So now, at what rate moisture, kg of moisture is depleting from
the solid surface R over, let say we have meter square. So, that is the unit of this
rate here. Now, we are starting with a very high amount
of x, let say that to the extent that solid surface is covered with the very thin film
of liquid kg of dry solid. So, that is the unit of this x. So, let us say that we are
starting for very large amount of this moisture. So, schematically you can say that we have
the solid surface and there is a thin film of liquid or moist moist solid. You can also
say completely moist solid to the extent there is a thin film of liquid, all right
So, we can also have a solid which we have suspended and there is a thin film of this
solid here or thin film of liquid, excuse me, all right. So, if you recall the first,
going back to the previous one, we have linear or there is some initial adjustment.
So, when the solid gets heated or warmed up, you can have, may be like this or you can
have like this, starting with a. So, we can call, you know, initial adjustment initial
adjustment and this is purely because of heat transfer; solid either is getting heated up
or it is getting cold up. So that, the rate decreases drastically takes
some adjustment over a very short time and or it increases or the rate increases. Then,
since the x change linearly, d x by d t which we are plotting here, d x by d t, you have
S, S over a. So, since x is decreasing linearly, you will expect expect that d x by d t will
be constant. So A, then we start B. So now, the rate is
constant till we hit the region, which we call earlier as falling rate curve. So, let
us mark this region at C. Why because, X C we will call it as critical moisture content
critical moisture content. So, this represents that kg of moisture, per kg of solid till
or if their moisture content is greater than X C, rate is constant here.
Why rate is constant? You go back to the previous curve or previous discussions we had, we had
thin film of liquids, right. So, just like a constant rate of evaporations, whether you
are drying from a solid surface or you have a pool of liquid, all right, it is all the
same here, critical moisture content. Now, after that, so, we can call it, this
is a constant rate or you can also call as saturated region. So, these are the different
different terminology. All of this, they reflect the same meaning that solid is covered with.
Now, after that we said that X decreases non-linearly. So, the typically what happens here, rate
decreases linearly, all right. So, we are plotting d x by d t. Now, the rate will decrease
linear. So, where you say there is a typical drying
curve and different solids will exhibit different type of behavior, but, this most common rate
is a constant, then rate decreases. So, till here we can say that this is falling rate.
So, we have constant rate and now we have a falling rate and here we say, it is a saturated
region. Now, we are saying that this region, so, let us mark here, till it falls linearly
we will call it, so, C. So, we have x c and d. Let us call give some number here, let
us say x d here. So, this is a region which is a part of this falling rate, except, now
we are calling it unsaturated region unsaturated region.
So, this is the region. Now, you are start seeing the dry patches. So, some surface has
been dried up and some surface still there is a thin film of liquid. Now, you recall
also, that it is possible for, in fact, it is like this. That, if you measure this kg
per meter square of wet area per second, then whether you are in this region or in this
region, the range would be the same, which is same as the rate of typical evaporations.
Since, we are refining to be consistent area, which is the total area; there is a decrease
in this. So, as far as the mechanism is concerned, whether the saturated region or unsaturated
region, most fundamental, you know, the two mechanisms are the same. What we are seeing
here, the decrease is the artifact of this area, which you have chosen, total area. Mind
you, the actual area, since because of their patches, actual wet areas are smaller, getting
smaller. So, if you measure per wet area, the two will
be the same. So, we have saturated region, unsaturated region, constant rate, falling
rate, and go back to this now, because there was earlier when we measure x versus time.
There was one more region where the drying rate has gone very very slow. So, you will
expect some other rate that will fall like this. When now, it was reached x star or x
E, so, this we define; now the solid has reached equilibrium concentrations.
So, this is also, equilibrium concentrations, should go back and recall from our first figure.
When we drew p bar over p 0 versus T vapour pressure of pure water and the partial pressure
or equilibrium vapour pressure in this, there if you recall, we had drawn like this, typical
curve like this, till it reach at some point X star equilibrium concentrations moisture
contents, which is in equilibrium with the drying conditions or quality of air, which
we said let say p some air over p 0 pure water. So, this X star corresponds to this X star.
It is the same line. Now, this solid cannot be dried below this. It has reached a concentrations,
which corresponds to the partial pressure partial pressure which moisture exerts at
the surface. That equals your, what about the quality of air relative humidity R H or
relative saturation you have in the air. So, now, you cannot go below this.
If you want to go below this X star, you will have to lower, you will have to drier use
at drier what will. Now, come back to this this drying curve here. Let us go back, constant
rate, falling rate, so, right here, till here it is both are falling rate. So, C to X star,
if you call it E for equilibrium. So that, we can mark this curve B C D E. So,
B C is a constant rate, C E is a falling rate and C E has two components. C D, it is a linear
decrease. So, rate decreases linear and this we call it unsaturated region. Then, now we
have D E; that is more important. Now, here also, we should discuss that what is the mechanism?
Here, the mechanism is the same. Whatever mechanism we had here, is the same mechanism,
except now, since there are wet batches, so, the area is different. Since, we are plotting
with the ‘a’, total moisture content divide by their total area, will be decreasing, will
be smaller here. Now, here also it is a falling rate. So, second
component, except this region, where now internal movement of moisture internal movement of
moisture or you can say that, now this is a region where now pore diffusion capillary
effects. See, all these are they reflect the same meaning
here, capillary effects. All of them, they start taking place here. That is why we have
different characteristics from here to here and from here to here.
So, what we are trying to say here that, if you have the solid surface, bring in contact
with some quality of air, dry air as long as filled with, covered with some thin film
of liquid. It is just like a common evaporations. Either you are evaporating from a solid surface
or from a lake or from some river, pool of water, the two will be the same. Same rate
of evaporation you will expect here and you will expect here.
This water will not see the solid here, all right. So, we have plenty of pool of liquid
here. So, the two rates are same as long as the temperature is same and the drying air
quality of the drying air is same. However, when now you have patches, dry patches
appears and then, there are thin films here. Here also this rate of evaporation is same
as this. The two are the one and the same, that all three, in fact, are the same.
But, why we see a decrease here, falling rate because we are expressing our rate based on
the total area. Had we expressed based on the wet area, then of course, this would have
happened, since would have been the same here. So, since a total area and this rate decreases
because, we have evaporations; moisture evaporates only from here, all right.
Then comes third region, where still it is a falling rate. However, now the capillary
forces become a part. So now, we are saying that these solids surfaces are deep solid
surfaces, there are capillaries and the moisture has to be supplied. So, this moisture has
to diffuse. Moisture has to diffuse by pore diffusion, by Knudsen diffusion. All this
we have talked, when we took up the top previous unit operations, adsorption, desorption same
mechanism holds good. The moisture has to be, has to come to the
top of the surface, then it will it will be evaporate, all right. Now, we are talking
of the hidden moistures. These are the surface moisture, these are the moisture which comes
from inside, from within inside. So, we have seen the third type of falling region, where
the rate was, you know non-linear here, the rate was linear, and here the rate was constant.
Now, of course, before this we talked of initial adjustment, where the solid gets warmed up
or comes to equilibrium with this air. So, one we call in, second is a constant rate,
third is a falling rate except linear and the fourth is non-linear falling rate. Till
at the end we have reached X star equilibrium concentrations, you cannot dry your solid
below this moisture content will be at the most X star. The more important here, this
X star is one quantity which depends upon temperature of course, depends upon the type
of solid that is one thing, all right. It also depends upon the air quality. So,
look at this equilibrium. This is, you cannot say that truly this is your thermodynamic
equilibrium which we have seen in case of adsorption desorption. Here, the drying has
to do a lot depending upon the heating arrangement, rate of heating, all of this they decide the
heat of rate of heating, the rate of drying, all right.
So, the more important here is of course, the third curve, which we obtain, which is
a characteristics of the solid. One more important here is that, we must be very careful that
what we do in the batch, in the lab, must also replicate or as close as possible the
operating condition on the drying condition on a large, for a on large scale, all right.
So, what we do now, we will like to re emphasize here, that there are three characteristics
curves. Traditionally, when you say that characteristics curve, one represents this rate versus x n
versus that is very important. But, we must also understand that, we have also have one
more curve when you talked of bound moisture and unbound moisture, free moistures, all
right. Here also you have X minus X star which is free moisture.
So, X minus X star or X C is your free moisture, all right. So, going back to the previous
discussions, now we have three curves. One partial pressure over vapour pressure of pure
water versus X, that was the one kg moisture per kg of solid. There we talked about bound
moisture, unbound, free moistures, remember the curve we have. Second one is X versus
time, that is a true experimental data for that type of solid and that type of operating
conditions. Cross flow, through flow, surprisingly, you
know this X star remains that quantity which has not been quantified exactly, you know,
for the type of the solids. For the reasons here, that things are so complicated, four
diffusions Knudsen diffusion or the heating effects, heat of radiations or heating by
microwave heating or heating by conductions to have the flow through, cross through or
you just have a flow over the surface equilibrium concentration is a strong function of the
operating conditions unlike any other previous, you know, thermodynamic equilibrium we had
we talked of absorptions, given partial pressure, how much is a solubility, Henry’s law. So,
if you fix a solute ammonia and water, temperature, equilibrium curve is fixed.
Then, we talk about the relative volatility distillations. We have y versus x curve for
benzene water, all right. If you fix the temperature, then we have y versus x is fixed, d x y is
fixed. There is no operating conditions here. Then, we talked of extractions, solubility’s
immiscible fluids. We fix the temperature, we fix a system and we have this triangular
equilibrium diagram A B C. They are distributed by some phase diagram. Then, we talked about
adsorptions. We fix a temperature, then moisture content, moisture loading or any solute loading,
given the partial pressure in the gas phase or given the concentration in the liquid phase.
We have isotherm, Langmuir, freundlich, whatever we have. Here, X star because of this X star,
which is strongly depended upon, you know the operating condition. One has to ensure
that you have established this equilibrium curve, which is X, which is rate versus X
before you design a real system. And, for that you have to choose that the two conditions
are batch and the lab conditions or the pilot plant or the industrial scale, they are as
close as possible. So now, we, what we do, we will setup some
equation. We like to study, how long will it take given this N versus X curve for certain
solid, certain system. How long will it take to dry this solid from one moisture content
to another moisture content. So, we will setup the governing equations, then we will like
to take an example and we will put some numerical numbers there and we will compute some quantities,
So, let us take this, the second topic is time of drying. So, again we have to start
from here. We have N versus x, if you leave aside, you know, this is start initial adjustment.
Then, you have p, then, you have C decrease linearly then, non-linearly. So, C D E, you
can mark this x C critical moisture. Corresponding to this, we have N c, critical rate of drying,
but, notice that this is a flat. So, critical rate of drying is constant between B and C,
and then it decreases. So, this is the falling region. Let say, we reach till x t, then we
have reach till X star or x C. So, will follow this nomenclature, B C D E.
You took, you can have a dotted line, which remains ill defined, for most of the cases.
Initial adjustment, we ignore this time. Here, rate is defined N, as minus S s, the amount
of solid dry basis, total area which has been expose for drying over d x over d theta. Theta
is the time in hour. So, what is the total time? We just integrate theta 0 to theta,
let say d theta equals S s, over ‘a’, since it is a minus sign, here we can say
x 2 over x 1 d x over N. So, we want to dry this solid from x 1 to
x 2. Typically, you can have a start from x 1 here; you can go all the way till x e.
But here, this x 1 and x 2, you need two quantities in between, may be, they are from the falling
linear here, may be they are here, may be x 1 is here and x 2 is all the way till down
here. So, it is a very general expression for calculating
this total time of drying, which is integral of this quantity. So, first case is that immediately
once you know N, you can find it by area under the curve, from n you calculate this curve,
1 over N, plot x and whatever you have, this we calculate the area under the curve and
then calculate this quantity and theta. But, realizing that the area here, this rate
remains constant and the rate decreases linearly, we can have a analytical expressions as well.
We can avoid this numerical, you know, integrations, whatever we have, we can have here.
So, the first thing is that, constant rate period. So, we are talking of B C right. We
neglected the initial adjustment. So, we have this B C. B C, the rate is constant at N c.
So, when you integrate, this very simple integrations, theta will be S s over A, N is a constant.
So, N C comes here and you have x 1 minus x 2.
So, this x 2, now, will correspond to actually whatever concentration you have between B
and C. So, this x 2 is greater than x 1 greater than x c, but, this is smaller than x 1, all
right. So, you understand. So, x 1 and x 2 is any quantities between A and c, x 1 of
course, the starting. So, this is the very simple expression for
constant rate period. There is no need for numerical integrations. You read N c from
the graph and x 1 to all the way till x c, you can do this calculations. We will come
back to this. We will take the example number two comes falling rate period.
So, falling rate period, now we are talking of C to E, all right. All the way till its
falling of course, they are two regions. ‘a’ where the curve is linear starting from critical
moisture content, its c to this d, which is x d.
So, here the rate is N equal to m x plus B, all right. You can assume N equal to m x plus
B, then you can integrate linearly, you put it back there. Very simple integrations, theta
equal to S s over A. We have x 2, we have x 1, d x over m x plus B, which if you integrate,
you will get S s. You have m slope of this curve here, ‘a’, l N m x 1 plus b over
m x 2 plus B, all right. You must realize that this m is a slope of
the curve. So, any point between 2 N 1 minus N 2, x 1 minus x 2, mind you this x 1 x 2,
we now we are talking of N here. So, now we are trying to integrate between
this. This is the slope of the curve N 1 is nothing, but, m x 1 plus b n 2 is nothing,
but, m x 2 plus B, all right. You can just put it there, if you like to substitute this
m here, to obtain this expression for S s over a x 1 minus x 2, N 1 minus N 2, l N 1
over N 2. So, now we are talking of x 1 and x 2, which
is greater than x d, but, it is less than x c. So, we are in this ratio. So, x 1 and
x 2 let us say, this is x 1, this is x 2, this is the rate N 2, this is the rate N 1.
So, we got this expressions for linear falling curve. This can also be rearranged to, if
you notice that you have l N term here from our some previous context here. This x s over
A x 1 minus x 2 and we can call it N m. What is the n m? You should recall, this is
nothing, but, log-arithmetic average, all right. So, now, we have these expressions
for this and now the third curve, which is also a falling rate. So, we will call it the
region B A, this D C. Now we are talking of this, now D and E till we have reached equilibrium
C. So, from here to here of course, we do not
have any analytical expressions. All it means, we will have to integrate numerically. So,
one we like to plot one over N and then this x, all right. Whatever curve we have, take
that curve, and find the area under the curve to obtain this theta here.
So, the same quantity, which we have here, has to be solved analytically, all right.
Very often, what happens that one makes assumptions that, after this constant rate, which starts
decreasing at critical moisture content, one makes this assumption that this curve is all
the way till x c. Now, again this depends upon type of solid,
where we can say that, well this region is quite smaller than the total region, total
theta, or it is possible that there is not much of change here, whenever you have this
inflection at d. In that case, one can make a linear in assumptions,
right. From here to here to, say that the rate is an approximations, since m slope of
the curve x minus x star. Since, we know this n c, which will be satisfying the equation
of this line, right. From, you know or this, whatever you have here, we can say that this
is nothing, but, slope is nothing, but, N c x minus x star minus x c minus x star.
So, this another approximations one can do for this N c here. So, if you have N, N c
is known, x c is known, x star is known, equilibrium concentrations or x c, same here. Here theta
can also be integrated to show that S s, x c minus x star over N c critical rate corresponding
to critical moistures into l N x 1 minus x star over x 2 minus x star.
So, it is an approximation. One can make or you can make to the numerical integration
as well, all right. So, we have non analytical expressions, as well as these numerical expressions
to calculate total time of drying here. So, starting with this N versus x N versus
critical, this moisture content, we identify all the points, constant rate decreasing,
then further decreasing till it is x c, one can start with this general example, very
basic definition for rate for drying and see which range is constant. We can do analytically
analytically numerically or you can make these approximations, you can do the numerical integration
So, with that, now we take an example for this time of drying for a patch system. So,
we take this example here, very simple example, very basic. Now, this is about batch batch
drying. Problem reads like this, something like this; that we want to decrease the moisture
content from 25 to 6 percent, all right. So, we have to decrease the moisture content
here. The more important here, you reads that what condition is this; we want to decrease
under conditions identical to those for which the given drying curve applies.
So, what we have been trying to say earlier, it is very important that we understand this.
Here, first one has to start with the drying curve, all right. So, this is again very,
it is different to what we had, you know, different under the unit operations like adsorptions.
There the isotherms is fixed for the system, not for the operating conditions. Here, the
problem says that I have I have the solids, it has the moisture content 25 percent, all
right. I want to decrease from 25 percent to 6 percent; however, the conditions are
the same under which, I have a drying curve available.
So, this drying curve are generated for a priory. It was generated before I want to
do design a real system and we are hoping that this drying curve will represent the
two conditions which are identical. Of course, it is not practical, you know, doing the experiment
in the lab laboratory or in the pallet plant or in the industrial scale.
But, that is the way it is drying is there, you know, it is a more of an engineering drying,
that some approximations here. So, may be, we have kept the two Reynolds numbers same.
Here, we have the same heating conditions, heating by radiations, same level of radiations
convection, same Reynolds number, same heating by conductions. So, that means, we have the
thickness of the solid same as what we are going to face there in the real world, all
right. So, with that, so, we are starting with that
drying curve. The drying curve is looks like this. So, we have x, we have N, the unit of
N is kg meter square per second. This n is reported as N into 10 to the power 3. Let
us put some number 0.3. We have 0.2, all right. Let us put a 0.1 here and we have 0 here.
So, you have this B C starting with B. We are ignoring A. The rate is a constant till
it hits C. At this level of 0.3, it decreases linearly, it decreases all the way till D.
Let say, this D is 0.15. So, this is a falling linear rate from C to d. Then, it starts decreasing
non-linearly, till it goes to all the way till concentration. Here, which is x star
and let us put this number at 0.05. From the graph, let us put this number as 0.1. Let
us put this number as 0.2 and the first number, here we have this and the graph is 0.3 or
maybe, let us put it as 0.35. So, this is the 0.35 to 0.2. This is a constant.
This is your constant rate here, then it is a decreasing rate, linearly decreasing day
rate, non-linearly, all these numbers are known to us.
Now, we say that moisture content is 25 to 6 percent moisture. Solid weight is total
solid weight is 160 kg to surface area or the area for drying available is 1 meter square
per 40 kg of dry weight. So, from this, you should be able to make a guess or make a compute
that, what we require is S s, over A total weight that was the expressions for N.
So, the number outside the integral over S s, over A it is nothing but 40, all right.
So, kg of total weight, dry weight, no moisture there, per area, so, that is equal to 40.
So, this is given to us; total weight is 160 kg, moisture content is 25.
So, the initial moisture content on dry basis, all this is all dry basis right. This is nothing
but, 0.25 over 1 minus 0.25. This, we have done dry basis, weight basis earlier in previous
lectures. It is always advantageous most of the time to work on this dry basis.
So, 0.333 per kg of moisture, per kg of dry solid. What is x 2? 6 percent moisture, so,
0.06 over 1 minus 0.06, which is 0.064 same unit as before. So, where are we right now,
0.33? So, we are in this range. Very close to this that would be our starting x 1 and
wherever we have to go till 0.064, this is 0.1. So, we are going till here.
So, we want to dry this till 0.064, which is our x 2. So, from x 1 to x 2; that means,
going to cover all three range, constant falling rate, linear falling rate and this non-linear
falling rate, till this locations here, which is 0.064.
So, it is a very straight forward problem here. x c is one quantity we should note immediately
because, we are going to use this. So, this x c is 0.2 and corresponding to this x c is,
we have N c, which is 0.310 to the power minus 3. So, the graph was reported like this.
So, it is 10 for minus 3 kg per meter square per second. So, these are the two quantities
of importance. Important parameters of a drying curve is a critical moisture content that
reflects the range beyond which or below which, you will have the falling rate, beyond which,
you have constant rate, pool of liquid, etcetera. We have talked those and critical, this constant
rate is 0.3 into 10 to the power minus 3 which remains constant between B and C.
So, with these two, all we have do, we can go back and substitute in our equations which
we have developed. You can start from the first principle. You should not take much
time to start, to obtain the final expressions, which is x 1 minus x 2 over N c. So, S s over
A is 40 x 1 minus x 2. So, we are starting with 0.333, constant rate
goes till 0.2. So, this is the first constant rate we are trying to calculate theta between
B and C, all right. So, 0.33 minus 0.22 divide by N c, we have rate 0.3 10 to the power minus
3, we have 16000 seconds. So, it takes 16000 seconds for the moisture
to dry from 0.33 initial concentrations, initial content to theta C or x C, where now, you
start to see decrease in the falling rate. in the rate. So now, your heat falling rate
period. Already we have said, N c is 0.3 10 to the power minus 3 corresponding to which,
x c is 0.2 units you have for both and now, you want to, you have to dry till 0.065. So,
you have to go till N d, all right. So, now we are going starting from C. We go
till d, till you have the linear falling rate period. So, this N d if you read again from
the graph, this is 0.15. Go back to the previous, here this N d is 0.15 10 to the power minus
3, we have this x d; we can read 0.1 units. So, theta F 1, 1 to denote the first region,
falling rate period is same as S s a x d x c.
So, we are decreasing from x c to x d. First, we decrease from x 1 to x c, now, we are decreasing
moisture content from x c to x d. We have d x over m x plus b, all of those we integrated;
you can go back and integrate here. We can just write down the final expressions. When
you substitute the slope of the curve, we have x c minus x d N c minus N d l N N c over
N d. All these quantities are known to us. All
you have to do is to substitute S s by A is 40, x c minus x d 0.2 minus 0.1, N c minus
N d 0.15 10 to the power minus 3, l N 0.3 divide by 0.15. Put all of these numbers to
obtain approximately, say 18480 seconds, all right. So, now we are done with this C to
d, where the rate is falling linearly, it is a linear. Here, the rate was constant.
Now, we come to the third rate. Let us plot N versus x for last stage of drying. So, we
have this rate curve, drying curve like this, N versus x for the last stage of the drying,
when the rate decreases non-linearly. So, already we did the calculation till d,
when the moisture content x d was 0.1. So, this is the non-linear part of the drying
curve, the last stage. Corresponding to this d, we have the rate described as or calculated
as 0.15. We have been asked to calculate, how long will it take moisture content decreases
from 0.1 to x 2, which is 0.066 equilibrium concentration; that means, the rate becomes
0, x c is 0.05. So, question asked is, x d to x 2, how long
will it take; what is the time. Here, we have the general expressions for drying theta equal
to S s, over A amount of solids, over the area of drying x d integrated to x 2, we have
d x over N. So, this is the general expressions for drying
here. We are discussing here this non non-linear part of this curve right, from here to here.
This extended till x c becomes 0.05, at which the rate is 0.
So, to integrate this, you can see, since this is a non-linear part here, we require
numerical integrations. So, some numerical integration technique, you take as many as
point possible here. So, let say we we have x moisture content 0.1 0.08 decreases to 0.64.
So, we have to go till it is below x d, which was given as 0.066.
We are starting from 0.1. We have that, which is your x d equal to this, for this the rate
N 10 to the power minus 3 is given as 0.15. These are the data 0.07 0.04. You are supposed
to take as many as data point in between, for a very smooth curve here, all right.
So, take as many as points here. Since, we have 1 by N d x, we require 1 by N rate here,
all right. Which if you do it, you will get 6.67, 1 over 0.5 will get 14.3 and then, we
have this 25 and some intermediate data points. So, all you have suppose to do is to plot
1 over N. We have x here, all right and your trend is like this. You have x d, you have
x 2 essentially; this integration d x by N between x d to x 2 is nothing, but, the area
under this curve. So, calculate this area under the curve. Take
this as an exercise. Put the values of S a S s to calculate theta. The time taken to
reduce the moisture content from 0.1 to 0.066 is approximately 24000 second.
So, we have three, all three times, total time for drying. First, we had number for
constant falling, constant rate which was 16000 seconds. So, this is the rate at, this
is the time over which the rate which remains constant moisture dries at a constant rate.
Then we have falling rate, but, we have the linear falling rate which was 18480 seconds.
The third, which we just now we calculated for non-linear rate. So, this is linear, this
is non-linear rate from 24 around 24000 seconds, all right.
So, total time is around 58480 seconds to dry the solid under batch conditions. So,
it is a very simple example and here, we have made use of certain analytical expressions
or we have made use of certain numerical calculations to calculate the last quantity.
We also said, if you recall that one very good approximation, which can also work. If
you assume that from B to C, which is a constant rate, when you hit this critical moisture
content then the rate decreases linearly. So, we ignore this D point here or we are
at least, we are approximating that the rate which decrease non-linearly is very close
to this, very approximately, we can say that it is a linear throughout. So, we hits till
x c so; that means, till x 2 x c to x 2, we can do, we can assume it is a linear. Right
here, throughout it is a linear. So, this is approximations. We also discussed;
we can use the expression directly for theta. All you have to do is, now you have the same
m x plus b N, N c satisfies this here. We have x c satisfying here. We know this x c
at which the rate is 0, that is equilibrium rate is 0 here.
So, we know this quantity, we know this quantity, we know this quantity here, x c, one can also
make use of this to find the slope. What is the slope here? We can go back calculate this
n, put in the integration equations, we showed that this number assuming that entire rate
of period is linear, falling rate is linear, you have S s x C x star N c, which is also
known to us l N x c minus X star over x 2 minus x star.
So, this is another expressions, approximate approximations assuming that entire falling
rate is linear. So that, we avoid this numerical integrations, of course, depends upon type
of the curve here. May be, in this case, one can do, one has to plot and convince himself
that the error is not much here. So, S s by a, is known to us. This is 40, x c is 0.2,
x star 0.05, 05 equilibrium moisture contents, x 2 you have to dry till 0.065.
S s by a, the whole quantity was 40. Put all these numbers to obtain, that you are getting
this very very close to 18480 plus 24000, the two numbers which is 42480 seconds, all
right. So, in today's lecture, we have taken this
example and before that, we had developed analytical expressions to calculate different
time of drying for constant rate, falling rate, linear, non-linearly for a batch system.
Before that, we said that there are three very important drying characteristics curves.
One is partial pressure versus x, then you have x versus time and from that you calculate
rate versus x. All three of them have important meanings, in terms of, you know, understanding
the mechanism of drying. What is the bound moisture, what is the unbound moisture, what
is the falling rate, when does it happen, why it is a linear or de fact of the area,
why it is linear mechanism of evaporation is the same in two cases. Then, there is a
non-linear curve, where we said that the drying is controlled by capillary forces or capillaries
effects. There is a Knudsen diffusivity. Then we talked of this equilibrium concentration,
equilibrium content which depends upon, not only the type of the solids, also the operating
condition. So based on, we marked all those regions.
All three curves, it should be familiar that play a major role in designing a real system,
real drier. So, that is about the batch drying. We had the example. Next time, we, when we
meet we talked up, now flow through. So, we have the solid batch of solids through
which, now the drying it, hot air passes through. Here we have the solid and over which, the
air falls air flows upon. So, it is a more like a semi batch kind of thing.
The second one is also semi batch. So, all it is a stationary, but, now air will pass
through this. Before we taking that, we will also talk, talked about heat transport. How
we can, we are heating the drier; remember, we said that drying, we have to give due considerations
to heat transport. Solid has to be heated by conduction, by convection, by radiations.
So, we will make use of how understanding, very basic understanding of heat transfer
coefficients and convective heat transfer coefficients, radiation etcetera, to address
those issues here.