Hello. In this video, we will discuss the transport of momentum, energy, and mass. So starting out
here with momentum, the average momentum carried
per particle would be m, the mass of the particle times the velocity as a function
of position, say, y. It's transport results
in a shear stress. So the flux of momentum is equal to the negative of the shear stress which is going to be minus
Alpha over two, m times n times c bar
times l times du dy. mn is density and Alpha over 2 mn or
density times c bar, we define as the viscosity. So the viscosity is one-half Alpha and we put
a subscript on Alpha, subscript Mu to remind
us that the Alpha is for the different transport
properties may not be the same. So Mu is one-half Alpha sub
mu times rho times c bar l. One can show from
more advanced theory that Alpha Mu is
about of order unity. If we use the ideal gas law, we can explore the
temperature dependency of the viscosity. From this simple theory, the viscosity should
be proportional to t to the one-half, and it should be
inversely proportional to the diameter of the molecule. So not surprisingly, the
bigger the molecule, the less momentum transport. A better approximation for viscosity is a formula given by Sutherland and it's shown there at the bottom of the slide, S is a fitting constant, and you can find values
for the fitting constant. But needless to say, there are many empirical formulas for the viscosity of
various substances. Just a little bit about units, the viscosity of Mu is
mass over length time, which in SI units is
kilograms per meter second. A common unit for viscosity is the poise which is the CGS
version of the SI version, it's grams per centimeter second. The center poise is
10 to minus 2 poise and you will see viscosity is expressed in terms of poise, or center poise, or
middle poise, etc. The kinematic viscosity
which is defined as Mu over Rho has dimensions of
length squared over time. So in the SI system of units, it would be meters
squared over seconds, in the CGS system, it will be centimeters
squared per second etc. This slide shows a
table taken from Engineering toolbox
for the viscosity of various gaseous substances and their dependency upon
the temperature. It's an interesting
exercise to plot these and see how close the
temperature dependency is to one-half, t
to the one-half. So that's momentum. Now energy, what we are transporting
as internal energy and that can be written
for a monatomic gas is mc_v T. So the heat flux
becomes minus one-half alpha, mc bar l times m times c_v dT dy. If we gather up everything
in front of the dT dy, that becomes the thermal
conductivity which here, we have called Lambda. Lambda is then equal
to one-half Alpha times Rho times c bar
times l times c_v. Or it can be written as Alpha for thermal conductivity
divided by Alpha for viscosity times the
viscosity times c_v. So it is interesting
that for monatomic gas, the thermal conductivity and the viscosity are
so closely related. So recall various numbers,
the Prandtl number, C_p over Lambda, and the Gamma which is the
ratio of specific heat. So the Prandtl number can
be written as Alpha Mu over Alpha Lambda times Gamma
for a monatomic gas. For a monatomic gas, the ratio of Alpha Mu over Alpha Lambda is
equal to 5 halves. So if Alpha Mu is one, then Alpha Lambda
would be 5 halves. There is a correction called
the Eucken correction for non-monatomic gases
and basically, what one does is write
the internal energy as the translational contribution plus the internal contribution. Likewise, for the specific heat. Then, assume that the
thermal conductivity can be written as the sum of a translational and
internal portion and then you get an expression at the bottom which is
Lambda is equal to Mu times 5 halves times the translational contribution to the specific heat plus the internal contribution
to the specific heat. There are many
empirical expressions for thermal conductivity. If you search the literature
for some specific substance, you will undoubtedly find some. But also, again, we'll be looking at more complete theory, at least, for gases. So here's a table, this is for my CRC Handbook of Chemistry and Physics but the table
is generated by Marcia Huber and Allan
Harvey who work for NIST. The table is actually much
more extensive than that. But if I put it on a slide here with me, we
wouldn't read any of them. But this gets for a
number of substances of varying molecular weight, the thermal conductivity
at various temperatures. Finally, how about mass? For mass, the flux
is a given species, and for multicomponent mixture, it turns out that mass
transport is quite complicated. But if we don't have
at least two species, then mass flux
doesn't really mean anything because there's
only one species and there'll be no net flux
of that species other than through the
motion of the gas. But we'll look at a simple binary mixture
whether two species, species one and two, so the sum of the
number densities of the two species has to equal
the total number density. Assume that's constant.
If that's the case, then the gradient one has to be the negative of the
gradient and the other and so the flux of one has to be the negative of the
flux of the other. So if we take phi
as the fraction, the mole fraction of species one, so that would be n the number
density one divided by the total number density and that is transported
by nc by over four. Then, we get the first
expression which is when we group everything turns out to
be minus one-half Alpha. So then, for mass transport, c bar l for species
one, dn and 1 dy. Now, if you'll get an expression, a similar expression for two, that is also shown. Then, the diffusion
coefficient is defined by the relation J equals
minus D dn dy and that's just a gathering
of one-half Alpha c bar l. So because of the mass balance where the
fluxes have to be equal to each other of one and two
but of opposite sign, then it turns out that the diffusion coefficients of one into two has to
equal two into one. In fact, that is not the case
because that implies that the square root of m pi d
squared is the same for the two gases which may not be true unless they're
nearly identical. Our result which is called
Fick's law of diffusion is only valid in the limit
of n_1 much less than n_2. In general, the equations of multi-component diffusion must
be used and we obtain them from the Chapman-Enskog's
solution to Boltzmann's equation which we'll get to a couple of videos. So here are just some
diffusion coefficient numbers for common gases and this is an assumption that the gases in small
concentration in air. So it's the diffusion
coefficient of these gases in air and the numbers are given at the table and for a
range of temperatures and these are from
engineeringtoolbox.com. Just a reminder that there are some important non-dimensional
numbers which are ratios of transport properties. So the Prandtl
number is the ratio of momentum to thermal diffusion. For air, it's about 0.7. The Schmidt numbers are the ratio of momentum to mass diffusion. In air, the Schmidt
number is about 0.55. The Lewis number is the
ratio of mass diffusion to thermal diffusion and
then there, it's about 1.3. Alpha, in this case, is not the Alpha that we
were discussing earlier, it's the thermal
diffusivity which is defined as the thermal
conductivity divided by the mass density times
the specific heat times pressure specific heat. So that's it for this video. Thanks for listening.
Have a great day.