Lecture 37 Species transport equation
In the last lecture, we basically derived
a relationship for mass flux for particular species with respect to stationary coordinate
system, right for a methane, right. Ah this is mass flux is equal to Y CH 4 into
mass flux minus rho D CH 4 minus H 2, and ah D Y CH 4 dx, right. Now I couldn’t really
cover in the last lecture, something which need to be discussed now, right. Because this
is very important to appreciate the mass transfer process, right. and ah similarly of course,
I can write down for that hydrogen is equal to hydrogen m dot minus rho D hydrogen CH
4 D Y hydrogen dx. Let say this is I am considering now as equation
one in this todays lecture, right. We let look at what is happening to this bulk flow,
right. Let us have a for that let us you know have a better feel of the bulk flow and the
diffusional fluxes, right. If you look at these are the basically diffusional fluxes
this thing is known as fluxes for methane, right. This for methane and similarly for
hydrogen in equation 2, right. Let us consider total mixture mass flux, right mixture mass
flux and this mixture is binary in nature, right. What is that? That is basically m dot
triple ah double dash is equal to CH 4 plus m dot hydrogen. What I will do I will basically
use this equation 1 and 2 right. So, what is that? Y CH 4 minus rho D CH 4 H 2 D Y CH
4 D x plus Y hydrogen minus rho D H 2 CH 4 Y hydrogen by dx , right.
So, if I will add this together, what it would be? What it would be I can write down this
as Y CH 4 plus Y H 2, yes or no? Can I not write down this portion ? And that Y CH 4
plus Y hydrogen, what is that? One, right? That means, you can note that Y CH 4 plus
Y hydrogen is equal to 1, right. That we know for a binary mixture of methane and methane
and hydrogen. And then I can write down this as m is equal to this portion will be becoming
what m dot is equal minus rho D CH 4 hydrogen dy CH 4 dx minus rho DH 2 CH 4 . So, this
will cancel it out, right. Yes or no? That means, this term total diffusional term is
mass diffusional term is 0, yes or no? Right that means, I can write down this as, right.
Row D CH 4 hydrogen dy CH 4 D x minus rho D CH 4 is equal to 0 , right. And this is
for the binary mixtures you can also derive that basically for what for the I s species,
right. And for what is this thing this is basically? Mass flux due to diffusion.
So, for nth species, right summation of m dot i, and this is I am saying diffusion ok
. i is equal to 1 to n is equal to 0; that means, this is basically is what you call?
The total mass flux is equal to 0, for what? diffusion.
For the diffusion; that means, total diffusional mass flux will be 0 in a mixture, right. Therefore,
the bulk will be coming, you know like, are you getting the very interesting thing you
should appreciate; that means, somebody will be moving fast somebody will be moving slow
altogether diffusion will be. 0.
Not affecting the bulk flow that is very important point I am making. And this is due to only
molecular diffusion ok, or the due to the you can say, but there will be some thermal
diffusion, and this diffusion whatever I am talking about equation 4 is basically for
what? Concentration due to concentration gradient
there will be molecular diffusion. There will be also some molecular diffusion due to temperature
that is thermal or the soret effect. There will be also the duffer effect, there will
be also pressure diffusion all those thing will be that we are not considering, we will
be not considering in also later on because it is quite complex in nature.
But, however, when you do actual modeling particularly using the computational tools
you will have to consider at least soret effect in the combustion whenever you are taking
the hydrogen oxygen system . You cannot afford to say no ok . Otherwise things would not
be good prediction would not be good. So, with armed with these expression particularly
keep in mind that ah equation 3, and also it ah will be using and for ith species will
be using, and ah we will be now using this for ah deriving the species conservation equation,
right. Let us do that. So now, what we will be doing basically looking
at the species transport equation we will be deriving this, right.
Let us ah consider that, I mean as usual we will take a infinitesimally control volume
in a 2-dimensional situations. This is 0, 0, and this is delta x 0, and you know delta
x delta Y this is 0 delta Y. And ah this is in x direction this is in Y direction, right.
We are taking infinite decimally control volume, this is our control volume, it is having face
A B and this is face C and this is D, right. This is our control volume basically.
Now, what we will have to consider we will have to consider for ith species, right. We
are considering ith species; that means, the mass flux mass which will be entering mass
flux will be what? For due to the ith species is this much ok, and what will be coming out
of here? And this is in x direction, right. We will be also considering x direction plus
x into into delta x ok ah noise], right. This will be delta x. Now in the similar fashion
in the Y direction, right. In the Y direction, this will be this will be is ok i ith means
ith species ok, y mass flux . So, these are the things we are doing, but
we will have to go back to the main things. What is that? This is the basically species
conservation. And species conservation if you look at we know the rate of ith species
mass accumulation is equal to rate of ith species entering into the control volume and
rate of ith species leaving the control volume, right. plus what it would be there? There
will be mass production or congestion of the species, isn’t it? There will be something
which we didn’t considered in the last lecture. So now, then only we can you know look at
this conservation. So, let us consider, let us carry out mass
balance for i th species. Ith species means it can be methane it can be hydrogen it can
be anything you know, like, if it is a binary mixture it will be methane and hydrogen. So,
that what is that that is basically rate of ith species mass accumulation is equal to
rate of I th species entering into species mass, right. Into CV minus rate of ith species
mass out of CV plus production and destruction, ok, I am just production and ok, should I
write destruction also of ith species mass in control volume . Keep in mind that this
equation is what we call 4 or 5 just check. 5
Ah 5 ah. 5 makes sense.
Now , what is this ah rate of ah species of mass accumulation ? Rate of ith species mass
accumulation in CV will be rho Y i into delta x into delta Y. Into of course, one therefore,
volume is taken care, right. And rate of ith species mass entering into CV along what?
X direction , what it would be? It will be x into delta Y, is this along with this space
A. And rate of ith species mass out of control volume along x direction will be ix m dot
i x into delta x into delta Y . So, if you look at net rate of ith species
mass, you know, along x direction what it would be? It will be nothing but you I have
done earlier same thing. So, it will be minus dou by dou x m dot x delta x into delta Y,
yes or no? Is it ok? And similarly, net flux net rate of ith species mass along Y direction,
what it would be? It will be minus. Y m dot flux i Y delta x into delta Y , right.
mass of production of ith species is basically in CV, what it would be? Into i delta x delta
y, right. into one basically, right. Now in equation 5, all the terms we have derived.
If we will just include that in the equation 5 , right. If I say this is may be I can say
ah this is your 6 this is 7 8 and 9. So, using equation 6 7 8 9 in equation 5, right.
I can write down that as equation 6, 7, 8, 9 in equation 5, we can have rho Y i dt plus
i i x D x plus i Y D Y is equal to this is equation 10. Am I right?
So, if you look at what is this term? This is known as source term , right. And how will
get this thing? You know, we have already derived that from the chemical kinetics, right.
We can find out what it would be using arrhenius laws and other things we have derived, you
remember that ? So, now this is the non-linear in nature, this will be non-linear in nature,
because you it will be dependent on the temperature. Dependent on the other factors concentration
because you do not know that. Now coming back to these, right we have already derived a
relationship for mass flux of I th species along the x direction, right. , right. That
is mass flux of ith species along x direction, what it would be ? Y i m dot i x minus rho
Dij , right. This we have already derived .
And similarly , mass flux of ith species, similarly, the mass
flux of ith species along Y direction would be , and keep in mind that, these theme if I consider,
what is this one? Any idea? This will be nothing but your m x, which is nothing but your rho
v x, yes or no? Similarly, if I sum sum means this is i is equal to n, all the species,
it will be binary 2 if it is you know whatever the number 10 plus species you are considering
ten. So, this also will be equal to m dot i Y is equal to rho v Y , yes or no? Right?
So, what I will write down, I can write down here itself as Y i rho v x minus ij D x. Similarly,
I can write down rho v Y minus rho D ij . Let us say this is 11, and this is 12 by using
equation 11 and 12 , in equation 10 , we can have plus Y di is a Y by dy . Plus, this is
some might these 3 dash means, per unit volume ok that you should keep in mind.
Now this is your expression for ith species . Are you getting? And what is saying? This
is basically the unsteady term, this term is unsteady , and this is mass flux of ith
species along x direction, this portion, right. Right this is mass flux of ith species along
x direction . Of course, the similar thing is here, this portion, right. And this is
due to diffusion, right. This mass diffusion term along x direction, right. Gradient diffusion
gradient basically . And this is a source term, and this is the
mass diffusion along the . Y direction.
Y direction, this is basically mass diffusion in Y direction .
So, for ah axis symmetric, you know, the species conservation equation would be , geometry
will be using basically ah what we call r z theta coordinate system, but; however, we
will be considering rz, ok, the species equation becomes i by dt, plus 1 by r dou by dou r,
r rho v r Y I, right. plus are is equal to D z rho D
i j Y i by D z by r rho r D ij Y i by D r plus m dot, thus i . Keep in mind that this
is a thing, and generally this ah what you call axial diffusion; that means, z is your
axial, right. Z is your axial direction that is very, very small, right. Is neglected as
compared to what? Radial diffusion, this approximation will be making in whenever we are dealing
with what you call z diffusion flame because z diffusion flame will be using this axisymmetric
equations. So, therefore, it is very important to look at it and similarly you can write
down for spherical coordinate system also, yes.
triple dash triple dash, you know. Tripled dash yes is, right. This will be ok
thank you very much. We will stop over here. And in the next ah class we will be discussing
about energy equation. And also, a little bit about term terminology , about turbulent
flow, and then that will be the end of your thing ok.
Thank you very much .