Hello, today, we're going to talk
about the Boltzmann equation and the Chapman-Enskog solutions to
the Boltzmann equation really to develop a more complete theory. We have to have an accurate
picture of the time and spatial variation of velocity distribution
function, and a method for doing this involves solving an equation that
describes the evolution of the function, and this equation is called
Boltzmann's equation. Because obviously,
it was developed by Boltzmann. So more formally,, Boltzmann's
equation describes the evolution of the Velocity distribution
function in phase space. Phase space is a multi-dimensional space, which includes physical space
velocity space and time. So generally, f is a function of position,
velocity and time. If we know f, then all the transport
properties can be calculated. Boltzmann's equation is of this
form the terms on the left describe the evolution of the distribution
function in time, physical space and velocity space and
the term on the right accounts for collisions that change
the velocity of a particle. So the left-hand side, the first two terms should be familiar
from the Navier-Stokes equations. They represent the rate of
change with time and convection. The third term represents convection
in velocity space due to a applied body forces, F. In other words, due to F=ma if
there's an applied body force, then this will cause
molecules to accelerate. And therefore,
change their position in velocity space. The collision integral, one may show that the collisional term
can be written in the following fashion. HCB stands for Hirschfelder, Curtiss and
Bird, also known as the green Bible and you can find that reference in
the reference list provided. In this integral, prime indicates after
a collision no prime before a collision. So we are talking about the collision
between a k molecule and a j molecule, and g is called is
the relative collision velocity. B is called the impact factor and
it's just shown in the little sketch. That is, it's the offset between lines
of centers and molecules that collide. Epsilon is the energy and
see the velocity. So Boltzmann's equation is
an integral differential equation. Meaning that it has both differential and
integral terms. It's not a trivial equation to solve. It was first solved
independently by Chapman and Enskog using asymptotic methods. And in the small departure
from equilibrium limit, the equilibrium limit the solution results
in the Navier-Stokes equations and provides expressions for
the transport properties. So these are the transport terms
that arise out of the Chapman and Enskog solution for viscosity,,
thermal conductivity and diffusivity, a mass diffusivity. You'll know that lambda, the thermal conductivity is
written in terms of viscosity. This is only precisely true for
monatomic gases and also the diffusion coefficient
is only precisely true when one species is in a much
smaller concentration than the background gas and
we'll dress that in a minute. Now the terms and those equations
should all be familiar,, except for capital omega. Capital omega is the so-called
collision integral and the first equation shows
the form of that integral, the second two equations
components of the integral and these can actually be
calculated numerically. I've done it in Matlab, for example. However, you have to know
the intermolecular potential and that's typically
the Lennard- Jones potential. The other thing that's normal is to
normalize the collision integrals and typically you normalize energy
into a temperature characteristic temperature called T star,
the potential energy well-depth in terms of the well-depth epsilon
of the potential function. And then a distance, r star in
terms of the collision distance and then one can tabulate the integral
values and this is an example. This is a table from my textbook that
came from Hirschfelder, Curtiss and Bird, and it's also appears in birds do it in
life foot an excellent book on transport. This is just a little plot
of to of the collision of the rules omega 1 and omega 2 2. So it's the non-dimensional version
of the collision cross-section versus non-dimensional temperature. Lennard-Jones parameters,
one can find these in a number of sources. This is one example,
although the Lennard-Jones parameters are determined experimentally
from from experiments. And so
depending on the experimental environment, they could be they could be somewhat
different and I was there determined for measuring the viscosity and so on. Now mixtures,
mixtures add complexity to the theory. And as a result in calculating viscosity,
thermal conductivity or thermal conductivity various
empirical mixture rules are used. For example, this shows the Wilke
formula for viscosity and one has to choose
an appropriate mixing rule for the situation at hand and
the literature is full of them now. Diffusion is a little more complex. It turns out that when Fick's
law of diffusion isn't valid and we call Fix's law is the version of the
diffusion coefficient that applies when one component is in a much, much
smaller concentration than the other or the rest of the mixture
then multi-component. It turns out that when Fick's law
of diffusion isn't valid than multicomponent diffusion depends
on the concentrations of all of the species in the mixture commonly
called Maxwell-Stefan diffusion, calculating diffusion full coefficients or
flexes require simultaneous solution of the multi component
Navier-Stokes equations. This slide shows
the expression that relates the diffusion coefficients
to molar mass fractions and to the mole fraction gradients, and temperature gradients in
a multicomponent mixture. And as you can see, it's everything
depends on everything else. That's the basic problem here. So in summary for
pure substances, viscosity and thermal conductivity can be measured or
calculated once and for all. For mixtures, one can use,
either empirical mixture rules for viscosity and thermal conductivity or there is more complex theory which is
beyond beyond the scope of our course. For mass diffusion and mixtures, simple
fixed law can be used for minor species. Otherwise, for multicomponent or
some approximation must be used and depending on the software
system you're using. Often, they will have embedded
various approximations very various levels of theory for
multi-component diffusion. So that's it for this video. Thanks for listening. Have a great day.