WEBVTT
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Welcome to the second week of mechanical operations
course, today we are starting lecture one
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which consists of fine grain size distribution.
If you remember the week one lectures there
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we have discussed particle size distribution
using sieve analysis, here we are covering
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the particle size distribution of very fine
particles.
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So particle size is probably the most important
single physical characteristic of solids,
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it influences the combustion efficiency of
pulverized coal, the setting time of cements,
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the flow characteristics of granular materials,
the compacting and sintering behavior of metallurgical
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powders. These examples illustrate the intimate
involvement of particle size in energy generation,
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industrial processes, resource utilization
and many other phenomena. Now several mathematical
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models and expressions have been developed.
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To obtain the distribution function from experimental
PSD curves. Now basically what we are going
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to do over here is to calculate the size distribution
using mathematical model, therefore when the
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screen analysis or.
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Any other method is not able to distribute,
to give the distribution of particle properly
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there we can utilize the mathematical model
and its functions and these mathematical models
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usually use the experimental PSD curves, so
these functions which are mathematical range
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from well established.
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Normal and log normal distribution to Rosin-
Rammler and Gates- Gaudin- Schuhmann models,
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so here we are having two models Rosin- Rammler
model and Gates- Gaudin- Schuhmann model.
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The RR model that is Rosin- Rammler model
or Rosin- Rammler distribution function has
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long been used to describe the PSD of powders
of various types and sizes.
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This function is particularly suited to represent
powders made by grinding, milling, and crushing
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operations but it has limited applications
due to their greater mathematical complexities.
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On the other hands Gates Gaudin- Schuhmann
distribution is simpler to use so in the present
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lecture.
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We are demonstrating Gates Gaudin- Schuhmann
that we call GGS model for size distribution.
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Now in this slide if you see here I have shown.
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In coarse grain soils by sieve analysis, what
is the meaning of this, that coarse grain
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size distribution we carry out using sieve
analysis. However if you see the data or see
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mesh chart it gives the value up to 40, up
to 40 µm only so sieve analysis does not
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give the size distribution below 40 µm so
when we have to compute the distribution below
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this.
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Then we go for Gates Gaudin- Schuhmann distribution
function which is very much suitable for fine
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grain size particles. So what is the use of
Gates Gaudin- Schuhmann distribution function,
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it calculates or it gives the particle size
distribution when I am handling with very
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fine size particles where sieve analysis is
not suitable, so Gates Gaudin- Schuhmann distribution
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function is widely used function.
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Which is usually applied to evaluate the particle
size distribution data resulted from combination
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processes. It is a two parameter distribution
function which can be expressed by this expression
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where it goes as x=A davg raise to power b,
the parameters over here x is the mass fraction
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or percentage mass davg is average particle
size, A is sized modulus and b is the distribution
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modulus where A and b we obtain from the experimental
data analysis we are having. So while taking
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log of this two we can write it.
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Log x=b x log (davg) + log(A) where b and
A are constants so size modulus is a measure
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of how coarse size distribution is and distribution
modulus is a measure of how broad the size
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distribution is, so A and b are representing
these values which we can obtain from the
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experimental data.
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The Gates Gaudin- Schuhmann plot is the graph
of mass fraction versus average sieve size
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with both the x and y axes being logarithmic
plots. In this type of plot most of the data
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points except for the coarsest sizes measured
should be nearly in a straight line so what
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happens? Here you are aware with this table
where I have shown the Indian standard screen
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mesh number - , + I guess you remember it
– shows under size + shows over size and
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here I am having the average size of particle
davg1 and here I am showing the average size
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of particle which can be obtained by arithmetic
mean.
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Of opening for 480 mesh screen and opening
of 400 mesh screen and x1 is the mass fraction
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which is available on 400 mesh screen / total
mass which we have fed for the screen analysis,
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so you are very well of this data. Now we
are using this data for computation of fine
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grain size distribution, how we will do this,
using Gates Gaudin- Schuhmann plot. Now what
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is that plot, this is the plot.
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Which we call Gates Gaudin- Schuhmann graph,
here the plot is between mass fraction versus
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average size that is davg it is a log- log
plot and this mass fraction as well as this
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average size davg is taking from this table
only. So if you see this figure what it shows
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that when I, I am concentrating on this part
which is basically dealing with the fine particles
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because here davg keeps on decreasing so here
I am having the fine particles so if I am
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considering this particular section and usually
it gives the straight line.
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Apart from the coarsest section the rest of
the section or where the fine particle lie
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it gives the straight line section. Now what
is my purpose over here? To plot this, my
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purpose is to compute the fine grain size
distribution, now how I will obtain this?
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If you consider this – 8 mesh screen what
is the meaning of this, that the material
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which is passed through 8 mesh screen and
it is retained on the pan so what is the distribution
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of particles which are available on pan that
we can obtain using this curve.
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That is the purpose of this Gates Gaudin-
Schuhmann plot, so this is the Gates Gaudin-
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Schuhmann plot which I have just shown.
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Now my area of work in this graph is this
section only where I am getting fine particles.
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And which is slightly straight as far as its
nature is concerned, so to understand this
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properly I will extra pull at this in another
graph where I have re plotted this particular
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section as you can see over here so this is
again the same plot but only a section I have
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shown over here. Now what I have to do, I
have to extra pull at the straight line by
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passing these points till it will reach to
the end okay.
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So here you see this red line I have shown
which is basically passing through the data
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points falling at below section or below screens
or bottom screens, so for this line we have
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to compute the slope of it. For this line
I have to compute the slope to know the value
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of b. If you remember b in Gates-Gaudin-Schumann
distribution functions can be obtained by
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slope of a straight line, if I plot the graph
of log law plot. So finding the b from this
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graph is easier, once I know the value of
b, d avg m and xm what is d avg m ? If you
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see the value at the end what is the value
of d avg as well as what is the value of xm.
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What is the last value of a mass fraction
as well as d avg available to me that I can
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see from the table or I can see from this
chart also. So once I have the value of b,
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d avg m and xm I can calculate ‘a’ using
this expression.
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This is the Gates-Gaudin-Schumann distribution
function, here I know the value of x, I know
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the value of d avg, I know the value of b
while seeing the slope of a straight line
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which is shown in red color and ‘a’ I
can found from this. So once I know the value
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of a and b I can have the expression of x
as a function of d avg. Further if I want
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to calculate distribution in this particular
region what I have to find is the d avg below
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to this.
I know the lower most d avg from the particle
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size distribution, what is the next lower
value of d avg, how I can find it by using
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the factor by which the two consecutive screens
are defined. For example if you remember lecture
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4 of week 1 there we have discussed that between
two successive screen there is a gap of, there
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is a difference of 21/4 in the opening, in
the similar line the same statement we will
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use over here that d avg just below.
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2 d avg m should be calculated as d avg m/1.41,
now why this 1.41 comes? Because instead of
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taking value 21/4 we have taken over here
20.5 so that value may vary.
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But the successive difference between opening
of a screen can be defined by these vectors
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only, here I have taken that vector as 1.41
so once I am having the lower value of davg.
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I can put that value of d avg over here, I
know a and b value so I can find next value
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of xm. So using this values of d avg x can
be computed so further we will find the next
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lower value of davg by using the same expression
and then I can find next value of x.
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In a similar line we keep on calculating the
x values till all the values of x would be
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equal to xp.
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What is xp? Is the mass of total feed available
in the pan. So if x= xp it means I am having
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the distribution of material.
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Which is available in the pan. If the size
distribution of particles from a crushing
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or grinding operation does not approximate
a straight line it suggests that there may
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have been a problem with a data collection
and there is something unusual happening in
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the combination process, so again you have
to repeat the process till we are getting
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the straight line because usually the data
should be like that.
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When I am going to lower particle size it
will follow the straight line. So to illustrate
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the computation, to illustrate how the GGS
calculates the particle size distribution
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I have taken this example. Here if you see
this in this table I am having the mesh number
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from 4 to 200 and below 200 I am having the
pan, here we have the sieve opening and mass
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retain on each screen is shown over here.
So you can see in pan I am having 60 gram.
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So what is the purpose of this problem is
to find the size distribution of material
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present in the pan for following experimental
PSD data using GGS model. So let us start
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the computation of this, here this is the
same table which just I have shown.
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Which is the problem table and here I have
to calculate the average size and mass fraction
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to draw GGS plot. So how I can compute davg
? Because if you see the value of mesh number
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4 over here correspond to this no mass is
retained on this, it means all mass is passed
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through 4 mesh screen and here if I consider
6 mesh screen it has 30 gram.
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So how I can write this -4 + 6 and davg I
can calculate by doing the arithmetic mean
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of these two value, that is 4760 + 3353/2.
So it gives me the value 4056.5 µm. Mass
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fraction how can obtain? By joining all this,
by adding all these value and then 30 would
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be divided by that added value it gives the
mass fraction of these particular fraction.
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In a similar line I can calculate the value,
so here you see in the pan.
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I have written -200 instead of pan because
the material which is passed through 200 mesh
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screen is collected on the pan. So -200 corresponding
to this I am having the value 0.1, so I have
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to find the distribution of this particular
section using GGS plot, so next step is to
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draw the GGS plot, so here we have the GGS
plot.
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How I have obtained this, using this davg
as well as mass fraction. So this mass fraction
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will give the negative value on logarithmic
axis that is why I have considered its percentage.
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So here you see this is davg in µm and here
I am having percentage of mass and both these
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axis are logarithmic axis as you can identify
while seeing the grids of the diagram so this
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is the GGS plot. Now to find the value of
A and V what we have to do, we have to extrapolate
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the data which is available at the end of
this graph.
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Where the fine size is available, N is also
here but here I am having the cosset size
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but here I have to find fine grain size distribution,
so I am concentrating on this particular end.
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So here I have taken the straight line which
is not very much suited to this, so if I consider
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a straight line like this which is shown with
red color the slope of this straight line
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you can find very well that is 5 if I am having
1 over here, then 2,3,4,5 5-2/200-100, so
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it gives me the value 0.03.
Therefore the value of b is 0.03. For computation
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of A I will use davg.
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And x of last screen which is available at
this point, so it is you can take this value
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from the table only, it is having davg as
89.5µm and x as 1.7 percent. So here x is
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basically mass fraction as well as percent
mass if you remember the definition of it.
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So here I can write the x and davg value as
well as b value and find the value of A which
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comes out as this much, so here to compute
the size distribution of mass present in pan.
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One has to find lower values of davg then
89.5 µm, to find this the interval between
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two subsequent values of davg should be known.
Now what I have done over here if you see
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the value using these value I have plotted
the, I have considered the end section of
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the graph. Now what happens if I consider
these two value, the ratio of this two is
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coming 1.42458, the ratio of these two is
1.419, the ratio of these two 1.4005, so if
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I consider the interval between these the
average interval I am finding 1.4149 so this
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value I will use to find the value of davg
below to the 89.5 µm value.
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And in this particular slide if you see here
to compute the size distribution of mass present
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in pan different values of davg and x are
required. These are found using values of
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average interval A and b, so if you see.
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The average interval it is coming as 1.41495
A value and b value we have just seen how
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we have computed this. So the davg below 89.5
µm is 89.5/1.414914 that is this value and
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it is coming as 63.2547. Once I am having
this value I can use this value along with
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value of A and b in this expression to calculate
the value of x, so x is coming like this.
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So next value of davg I can find like 63.2547/1.414914
and further I will use that davg value along
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with A and b value in this expression to calculate
x. So x I am finding as 1.664994, so here
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if you see here I am having different davg
values and here corresponding x values are
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given.
So if you add all these x it is coming as
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9.8367 which is close to 10% value which we
are having at the pan. So it is close to 10%,
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now why this difference is there, I have to
achieve 10 and I have achieved only 9.8367
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because of this slope and slope value.
As well as value of A, so further difference
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between total percent mass as well as that
is in pan can be reduced by extending graph
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of x and davg with other slope. So once I
am changing the slope I can find new value
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of b and A.
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And similarly I am finding more suitable values
of A and b which gives the total mass equal
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to 10%.
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So the summary of this lecture is fine grain
size distribution was discussed using Gates-Gaudin-Schumann
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distribution function. The log-log plot of
mass fraction versus average particle size
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for GGS was discussed. Worked example was
considered to illustrate the computation of
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GGS. So these are the three summaries, three
main points of the present lecture.
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These are the references and that is all for
now. Thank you.