so let us start the second set of ah sessions
solute transfer in weld pool so we are going
to look at the solute segregation profile
and we will do it in two parts the first part
we are going to look at the overall nature
by which we are going to proceed further having
considered already the lever rule and the
scheils equations so here is the situation
that we looked at for the domine so here is
the melt pool the heat source is moving and
we had looking at the longitudinal section
and in the weld pool you have usually a trialing
effect and that means that ahead of the center
of weld pool you have got the melting process
taking place and behind it is a solidification
process taking place and i have drawn a rectangular
domine which is in the direction of maximum
temperature gradient so that it capture the
direction of solidification of the weld pool
and i have enlarged it to show you that in
this particular rectangular shaped small domine
we have almost a directional solidification
that is taking place so therefore it implies
that if we can understand the microstructure
evolution of weld pool within this domine
then we can apply it to the rest of the weldment
because we can imagine the solidified weld
behind has comprising of this tacking of these
kind of rectangular domine ah one after the
other and the domine that we are looking at
is much smaller than the actual weldment and
this must be kept in mind so before we put
the lever rule and the scheils equation into
a prospective let us appreciate one thing
that the ah the length scales are very small
for solute diffusion to take place the diffusivity
as we have written here for the solute element
let us say for example the diffusivity of
copper in the case of aluminum copper belts
for example ah will be very small and when
we take the time available for the process
of the solidification ah then we get what
is called as length scale of diffusion
so this is a length a scaling analysis that
we have shown and here what do we want to
use for the time we can use the time that
is meant for the solidification process which
could be estimated for example the delta t
can be estimated as the width of the pool
divided by the velocity of the torch which
means that that is the amount of time that
is available for the melt pool to solidify
completely ok beyond that the torch has gone
further away and that if that is a time available
and then ah if that you substitute here then
you get the length scales over which the a
diffusivity is going to play a role and the
segregation are going to be (( )) so this
length scale will tell us whether further
features that we are looking at is it sufficient
for the mixing to happen or not in other words
let us say for the feature we are looking
at let us say the side arms spacing or the
primary generate arms spacing etcetera then
if that length scale if we are looking at
is smaller than this number it means that
the mixing is possible in the complete sense
if it is larger that means mixing is not possible
completely so in that we can actually see
what comprises about good mixing or poor mixing
and we can just put that into a map we can
say it is here on the x axis we can say mixing
in solid and on the y axis let us say we put
in mixing in the liquid and what we mean by
this corner is let us say poor mixing good
mixing and here also poor mixing and good
mixing so what we meant by lever rule is we
assumed that the mixing in the solid is perfect
complete and mixing in the liquid is also
complete which means that in this domine it
is here that we are getting the lever rule
ok it is good in both and then we made one
relaxation to this particular set of assumptions
when the derived the scheils equation we said
that the solid has no change in the composition
during the solidification of the weldment
and liquid still has very good mixing so which
means that as far as scheils equation is concern
we are here ok and we would also see that
there is some domine that is not possible
it is very difficult to imagine ah a situation
where in the solid very good mixing is there
but in the liquid it is not there because
that is very unlikely the reason being that
in the liquid the atoms are having transnational
freedom they are jumping around where as in
solid they are sticking around with their
own lattice position so very difficult imagine
a situation so we can say this is unrealistic
and where would the reality be the reality
would be such that the mixing in a solid is
slightly on the poorer side and the mixing
in the liquid is good but not completely available
so such if we can say realities may be here
and what we would like to take up in this
lesson is here where the mixing in the liquid
is not complete but it is also not totally
neglected it is not zero so it is available
and in the solid you just so completely ignore
it
so we are trying to take a some situations
which is quite close to the reality and this
is the place where segregation process are
going to be there as you can see that it is
in the realistic situations that you have
basically some amount of back diffusion in
the solid behind the dendrite and you also
have for example ah some amount of mixing
which is not complete and you can say what
is intermediate it means that diffusivity
is such that it is not fully mixed and it
is determined by the diffusion coefficient
so we can say that we can sake a situation
now which is quite close to the reality and
this is equilibrium situation extremely slow
solidification of welds may be submerged arc
weld for example you may take this kind of
a situation and for all practical proposes
most of the welds like in a palssmark welding
and upwards you may see that it is quite ah
difficult for the liquid to have a enough
time to mix so you may actually take ah scheils
equation regime and that may be quite realistic
and otherwise you would say that this is the
profile that we can use so what do we mean
by the segregation profile is as follows
we are interested in ah as a function of distance
along the domine that we have drawn what would
be the composition of the solute that we are
interested in so if this was the distance
we want to plot the composition of the solute
and that would tell us various things including
what kind of microstructure we will get in
the weldment and also to tell whether there
would be a banding and in which case in those
bands whether there will be possibility of
intermetallic compounds to form excreta so
they large number of things that you derive
from a solute segregation profile that you
can draw and the way we would propose the
solution is as follows in the case of the
derivation for lever rule and scheils equations
we have taken fix length of the box and we
have taken the solute balance but in the case
of the solute segregation profiles we are
already saying that the mixing in the liquid
is not complete it is inadequate so we are
going to assumes that the domine is roughly
about three to four times the length scale
of the diffusion which means that it is the
approximated as a semi infinite domine so
i will just write down the conditions under
which we are going to do the derivation so
the domine we are talking about is like this
this domine size is semi infinite which means
that we are taking a particular length but
we are taking that it is so large compared
to the rate at which the diffusion is going
to take place but it can be considered as
semi infinite and normally if the diffusion
length ah is l then any domine that is of
size three l or more it can be taken as semi
infinite so therefore this is not something
that is abnormal
and we also are going to make some more restrictions
on the way we are going to pose his problem
we want to actually look at the composition
profile only in the liquid so what is sought
composition the liquid as a function of distances
ah where x is the distance in the liquid which
means that we want to have a solution in a
domine which is actually attached to the interface
and is moving which means that we are actually
looking at solute profile in the liquid zone
but as it is solidifying we are only looking
at the remaining liquid so which means that
our distance should be x is in moving coordinate
system 
and at what velocity is this interface moving
or the coordinate system that we have fixed
so at what velocity it is moving at moving
coordinate system which is moving at a velocity
given by velocity of the front and that is
related to velocity of the torch ok through
a trigonometric function which we can apply
for the angle that it is growing with respect
to the vertical and vertical is the direction
of the heat source at an angle the solidification
is happening so in other words we are actually
going to have in a moving coordinate system
the equation return and the solutions are
going to be applied
and what would be the equation that we need
to a solve equation to solve it is nothing
but the solute segregation model that we have
done earlier it is same ah equation as we
have done but that is basically one d generalized
fix law ok which we have already derived or
compared with thermal modeling and done so
that is the equation and it is should be written
in a moving coordinate system also and ah
subject to the boundary conditions the initial
conditions and boundary conditions so that
is the problem statement what we have and
once we have that problem solved then we do
have a grip on how the segregation is going
to happen at a micro scale within a small
domine that is in the melt pool while it is
solidifying so we would draw write the equation
and draw the boundary conditions right away
just give me two seconds to erase this part
so the equations we are going to ah solve
is this normally you would write an equation
like this g a generation term d is a diffusivity
c is the composition of the solute that you
wanted to solve and in one day this is how
the equation is going to be and we want it
to convert the coordinate system to a moving
coordinate system and therefore that means
the time variable is going to be changed to
the distance variable and the distance is
also in the same in x direction which means
that this has to be modified and we already
said that something like this get modified
to if the velocity is along x direction its
going to be modified like that and therefore
we are going to use that here and change the
equation and we would do that 
and if you want to sit on the front and see
which where the material is moving normally
it is moving in the minus x direction so you
would put a minus sign this can be changed
depending upon the way you are going to plot
and if x positive x is going in the forward
direction then the velocity which its the
material is moving is in the backward direction
if it is on the interface and look so therefore
its going to minus v so this is the equation
how we are going to do
and we are doing it in such a small domine
that there is no generation of solute there
is no reaction that is happening so this term
can also be ignored so in other words this
is the equation that you are going to solve
subject to the boundary conditions and for
you to identify what the boundary conditions
are then you must at least kinetically draw
how the profile is going to evolve so that
we can actually identify the boundary conditions
at x is equal to zero and at x is equal to
infinity so for that i would actually ah show
you how we could draw the ah profiles kinematically
and then may the boundary conditions if you
are to look at from a stationary coordinate
system how this profile of composition is
going to be from the solid to liquid as it
is moving then you would notice that if c
naught was the composition initially you will
have k c naught and you would see that the
composition has go to up ok ah and you would
see that 
like this because initial solid to form will
be k c naught average composition is c naught
everywhere and how much ever solute that is
dumped will be in liquid and then liquid composition
is going to be not flat where it is going
to be falling in down approaching c naught
value as you go far away so that the diffusion
of the solute is happening gradually and as
you keep on solidifying so this is going to
be go up and down at the point that the solid
composition reaches c naught then the liquid
composition should reach c naught by k this
we already seen from the phase diagram here
to recollect i will just draw it here 
so we already have this analogy solid solid
plus liquid liquid from the phase diagram
that if the average composition is c naught
then the very first solid that is going to
form is k c naught the last liquid that is
going to solidify is c naught by k and they
are changing along this paths the liquid composition
is going from here to here and the solid composition
is going from here to here ok so that way
we can actually see that the peak is going
to be at c naught by k
now there is an argument to say that if the
study state conditions are going to prevail
ah in the moving coordinate system or in other
words if the profile is going to be stationary
profile in the moving coordinate system then
you will have the liquid composition is always
at c naught by k and solid composition is
always as c naught because these ratio exactly
matches the (( )) coefficient and you may
have a situation where the flux into the liquid
due to the diffusion is matching the segregation
that is happening so under that condition
namely when the solute rejected by the solidification
is balanced by solute taken away due to the
flux diffuse a flux ok so whatever solute
is coming in to the liquid because of segregation
is taken out because of diffusion then you
have a study state condition possible under
that situations what happens is that the composition
of the solid will be c naught the liquid composition
at the beginning will be c naught by k far
away it will be c naught
so how the profile would look after some distance
like that and further more ok so it means
that this amount of solute that is collected
at the interface will be taken along with
the interface as you go along and the profile
is going to look like this so we would then
draw that neatly to identify the boundary
conditions so let us just do that so it means
that 
so as you already we have talked its a semi
infinite domines so we will just keep that
open and ah this is how its going to be the
composition is going to be overall composition
be c naught and this will be distance and
this value is c naught by k and somewhere
in the beginning of the solute ah segregation
profile you have a situation of that nature
and this is called the initial transient and
for a total length that is several tens or
hundreds of the diffusion length scale this
initial transient can be neglected so that
for most of the length you have got a study
state profile
and if this was the profile then what would
be the boundary conditions we need to inspect
if this was the situation than ah what would
be the composition at x is equal to zero and
that will be one boundary condition you can
see that at x is equal to zero the liquid
composition is c naught by k that is one condition
and what would be the boundary condition in
the far away liquid composition at x tends
to infinity is c naught far away it is c naught
so these two are the boundary conditions we
have to apply and this is the equation we
are suppose to solve so the problem is now
completely frozen solve the diffusion equation
in moving coordinate system and one d without
the advection term without the generation
term in the liquid subject to boundary conditions
where at the interface the composition of
the liquid is c naught by k and far away from
the interface it is the same composition as
the bulk of the liquid which is c naught
so we can do that and then we will able to
see how this profile will be having a functional
form so i will do that in a moment so let
us designate this as c dot is equal to d c
double dot ok so dot implies the differentiation
with respect to the distances and i will take
ah this fellow down and this fellow on the
other side ok and we can now see that you
have got when you integrate you have got a
variable whose differentiation is on the top
and therefore went you want to integrate you
get the logarithm of it so integrate with
respect of x 
once and you get minus v x by d plus constant
is equal to logarithm of c dot and you can
then take the logarithm to the other side
and you can see that it is a exponential so
e (( )) to the power of minus v x by d and
plus constant will then become a multiplication
is equal to c dot which is nothing but dau
c by dau x ok and then you can integrate once
more 
and you can see that the integration of e
(( )) to the power of any function will be
the same function and therefore it would look
like a prime e (( )) to the power of minus
v x of by d so i am just absorbing minus d
by v into a and calling it a prime and plus
constant is equal to c
so that is a solution it looks like the composition
of the liquid is going to be b plus a (( )) multiplication
factor a prime into e (( )) to the power of
minus b v plus d now we can substrate that
into the two boundary conditions you can apply
the boundary conditions and you can see that
when you apply the boundary conditions at
x is equal to zero you would see that this
one would then go to unity which means that
a prime plus b is equal to c naught by k and
when you apply the second boundary condition
extends to infinity when extend to infinity
this tends to infinity and it the minus sign
it goes to the denominator and therefore it
goes to zero so it means that b is equal to
c naught and that means that a prime is equal
to c naught by k minus b minus c naught take
c naught into common one minus k by k so you
substitute both of these into that equation
and you will get the solution as follows composition
the liquid is given by a prime that is this
c naught into one minus k by k e (( )) to
the power of minus v x by d plus b which is
c naught
now you can take c naught into the denominator
or on the left hand side so that you can just
look at it is a function as follows so that
it is like a just a profile so you can see
that c liquid by c naught is equal to one
plus one minus k by k e (( )) to the power
of minus v x by d so you can see that ah you
have an exponential function to tell you how
the composition is varying when x is equal
to zero then you have got this unity and you
can see that it will become basically one
by k c naught by k and that is the values
c naught by k here and x is equal to infinity
these goes to zero and that composition will
be just c naught and that becomes c naught
here so we have verified so it is an exponentially
decaying function is something that we have
derived from here ok now what does it implied
to us it implies that the composition profile
in the box towards the weld center is going
to be an exponentially decaying function and
that is going to have some implication on
how the microstructure is going to evolve
ok so we will come to that in a moment and
before we ah windup we can just look at this
function and try to simplify some more aspects
of it ok
so the one thing that we want to simplify
is as follows can we convert this amount of
segregation profile that is there into in
equivalent triangle and inspect what would
be its width ok this is something that i would
give you as a homework problem so the problem
itself i am going to state it like this convert
the exponential decaying function into a rec
a triangle whose width is delta ok everything
else is same ok this value c naught by k this
is c naught this is c naught by k this is
c naught so that the areas they are same and
i am asking what would be the delta value
so we would look at it as a tutorial problem
and see how that comes and that would actually
tell you what would be the distance over which
the segregation is active and that actually
has a very important role in determining how
the microstructure of the weldment will be
whether it will be equiaxed or it will be
columnar and normally in welding we ah would
like to have equiaxed microstructure because
the impact toughness of the weld joint will
be better that way and what parameter governs
that transition this is one such parameter
and we will come to that discussion in a moment
ok so we will then break at this point and
then resume after few minutes to the second
parts
