Lecture 20 - Classical Size Effects, Perpendicular Direction - Mechanical Engg. Video Lecture - Mechanical Engineering

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ocw.mit.edu ok ha we're sure you all had
a good weekend and just recap 04 last
lecture is a little bit extra than last
lecture but say what we swapped is a
problem all four heka dashing a long
two-dimensional thin film and they
boards my equation reading for the
deviation function g g is f minus f0
right so g is an f months f0 and we say
G because the boundary in the Y
direction so there are constraints
changing in G in the Y direction so we
have the weight cosine theta way cosine
theta is assured the rushing of the
photon propagation and so we have the on
the left hand side is a function of Y
and then on the right hand side is a
function of X right the f0 because i
have f 0 for fallout is the exponential
i chi omega kg mass one so t is a
function of X because I would consider
heat conduction along the X direction
and if you do electron transport which
we didn't do last time you have to keep
this term the force term and that's when
we did in chapter 6 the thermal actually
transport we treat the both terms but
see the point is if you look at
transport along the exit door
chin and on the right-hand side is a
function of X only so I have a first
order differential equation for the
division functioning in the Y direction
with a right hand side is the X
dependent so because of that the right
hands is essentially constant because
the left hand side is a function of Y so
when I did the integration I find the
heat flux and what I'm given here is
more than I show in the last lecture
because this solution is a folk song
heimer folks song I check that the name
heimer solution okay and a full stone
hammer original soil for the electron
transport so this were in the original
paper electric conductivity ratio of the
sin film divided bulk is say one manís
so it is reduced and this reduction is
this integral mule is cosine theta so
it's a 0 to 1 and the P is a specularity
parameter P specularity parameter so if
it's a p equals 1 it's a specular p
equals 0 is a diffuse so you can see if
it's a specular 1 minus p is 0 so it's
more like a wave guide there's no
reduction if it's a specular interface
but the peak or 0 is diffused so you
will get reduction in the thermal
conductivity or electric conductivity if
the interface scatter electron fallout
diffusely and but say it has been
actually really difficult to say what is
this value of P and so far the best
model people use will still result data
derived by Zeman in 1950s and this P is
related to the search
it's roughness and the wavelength of the
photon or electron so it's an
exponential function and in fact that
you see in the front is 16 pi cubic so
that's a pretty large number typically
in optics when people see a surface is
rough or is a specular for optically
smooth surface is typically roughness is
less than one-tenth of the witness right
so you can see if the roughness less
than one-tenth of the witness this P is
the it's getting close to right this
one's been with very large very small is
close to 1 that's a speculum okay so
this is a focus on de heimer solution
and that's the for diffuse for the
electrical and thermal conductivity a
long of film and we assume that this
this of this film is very long so if you
go to look at the literature there are
lot of papers published you measure and
thermal conductivity of sin films or
nanowires and there are also refine the
model of wires of different cross action
whether it's a circular where there is
rectangle and the people have tried to
solve the board my equations and I
mentioned in the last lecture even now
there's some steel some result on the
wires or puddling and people still
writing what could be division division
from the who fuks and the stone heimer
solution okay and historically this is a
Kazmir catwalk as mere idea that is a
really say he explained without the
solemn boards my equation just the no
internal scattering you have rough
surface and he said okay if you have
rough surface because the scattering
is diffused so you have affected
reduction in the mean free path of the
heat carrier the photons and with that
he was able to explain why thermal
conductivity I low temperature is a
proportional to specific heat that mean
free path is limited by this sample size
and the low temperature see the thermal
conductivity once they receive Lee gamma
so when the mean free passes the sample
size this is proportional to the
specific heat which is tqb so this is
the Casimir say started so that's a last
lecture it's a film conduction along the
along the film just one more comment you
actually could say that recently there
are some paper on the thermal
conductivity of carbon nanotubes and
turns out because some are now to above
the mean free path is even longer the
length of the tube so there is
additional size the fact that either
they two ends of the tube okay and what
we going to do in to this lecture is
located a other direction so he
conduction perpendicular right let's see
transport perpendicular to fail
okay and if I look at ur day now the
problem is and if I have a temperature
difference I have to be a little bit
ambivalent at this stage what's this
temperature really mean so I have a t1
t2 and in the signal to the Russian of a
film and this is my Y right and again I
have the mean free path is a could be
longer than the film thickness and then
what's my effective with thermal
conductivity right and you can ask the
same question what's my electrical
conductivity and also their solution for
radiation photon transport all those are
under the same equation and I write down
the Boltzmann equation on the relaxation
time approximation so here i have d v by
DF dy equals minus f months f0 divided
by tau right because in this case are my
equivalent temperature changes as a
function of Y so T is effectively a
function y and the f0 now through the
temperature dependence exponential H bar
Omega kt minus 1 right so you can say
what the difference from the previous
problem is in the previous problem the
relaxation time is a function of X but
the size if I've happens in the
I direction now my temperature is a
function of Y and the size if Adam
happens in the same direction so my
solution will be more complicated than
the previous one and let's solve this
problem and I will first do wave I again
say any fool on transport the Russian
let's say and on this direction and the
angle to the y direction is a theta so I
will do this way why is way cosine theta
okay and wait times tau I will normalize
it gives the mean free path so mean free
path is a wait times tau and I define a
non-dimensional variable aight i equals
y over mean free path right and in the
previous previous one can say equals D
over mean free path so that so they
normalize the to the thickness but this
is an arbitrary coding the Y and if i do
this i write my divorce my equation now
into meal which is the cosine theta mu
equals cosine theta that's directional
cosine and i have DF dy plus f i move to
the right not why the ida the mean free
path now is wait times tau so say either
is y over mean free paths so a long
dimensional is my equation on the right
hand side then i have f 0 remember f0 is
a function of T and T is a function of Y
i do not know that he so now they are if
i say the function of Y is also
functional either so I have the f is a
function of either f0 is also a function
of either now this equation is a little
harder to solve then
previous one because in the previous one
the right-hand side is a function of X
the left-hand side is a function of Y
now in the current one both left and
right hand sides a function of either
right and to solve this first order
differential equation the trigger you do
is the first use of homogeneous equation
atta DF D either plus F equal to 0 and
here you'll find out this one gives you
f either is a function of a say
exponential mass either over mule right
that's the solution for homogeneous
equation when the right-hand side is 0
ok and now when the right-hand side is
the function of either itself you see
you check that say into an unknown
function you want it's a functional
value so that's the standard way of
solving now rather than C is a constant
say is a function of either you
substitute this back into the original
equation right you'll find out the CD
this term will drop out so what I have
is a mule exponential either mill DC the
at ax equals F 0 which is a function of
either so this is the day when you
substitute this back into the
differential equation this term drops
the ad the other term you can take
derivative of this also drop the
c-terminal so it's it's their right and
now I because this is a funky white hat
is a function at us I can integrate so
what I have
it's a itself d SE di da equals f 0
which is a function of T either right
that's a function of teeth and 10
Vermeil exponential either meal so i
integrate I will get a say is say
integrate another integration constant
right that's depends on where I start so
this is my idol 02 either and one over
mule exponential either prime and the f0
either prime D either prime so this is
my solution for this constant which is
no longer constant is the function of
either and that's my solution and I
combine this say back into the original
my solution for F either but it's also a
direction function of mule is a say 0
exponential mass I'd ha over milk that's
this term now this term is also
multiplied into this term so what I have
because this one is integrated either
prime so I can put in that yen so what I
have is either 0 to either one over mill
exponential must itemize i deprive mule
f0d either prime okay so that's my
general solution for this original
differential equation you say that I
didn't go to find the deviation I just
saw F directly and for the previous
problem
this one I use a deviation function but
you don't have to you can still solve f
directly you get the same result it's
you it's just a math difference okay and
again I need to now use the boundary
condition to solve for C 0 right and for
each meal now for the problem each meal
meal is the cosine theta right so that
for each theta I have a say a different
origin so if we see that between 0 to 90
degree i have y equals 0 this surface if
the diffuse the temperature te one that
goes out is a isotropic so and then for
90 degree to 100 degree it's actually
coming from this surface right and again
say it's diffused the temperature goes
out is te to this is what I call emitted
temperature right so I go to do the
condition while either equals 0 so I
started with the i-20 also at zero and
this one drops out so what I have is the
F for a mule is less than one larger
than 0 that's 0 to 90 degree fi da mu is
f0 te 1 exponential matter I have two
mil plus zero to either one or mill
exponential must either must I deprived
formule f0d either prime okay so
unfortunately the difficult is this is
almost like the simplest heat conduction
problem right becomes a very involved so
this is the solution for 0 2 mu 2 1 I
will put it
a positive that's a day from bottom to
up for the propagating and similarly for
90 degrees to 180 degree I have a
solution right for a 100 90 to 180 so
mule is a lengthy one and 20 that's the
theta less than C 180-degree larger than
90 degree so backward propagating from
top surface 2 down and my solution f
prime item you'll then you'll say f te
to that's the corresponding was an
Einstein distribution for the photons
the coming into the domain with courtesy
temperature te to right so this is a 0
te to an exponential cuz I'm ass either
over meal this is the effective boundary
effect here is at our meal it is a
negative is the exponential decay so
what it says is if my phone is probably
in this direction from the boundary at
this point right because of scatter it
decays and that distance the ratio
dedicada is exposure Cosima is either so
that's the really d minus y so that's
this section and divided by cosine /
Emile's cosine theta so that's this
distance right so it's really because
the mean free path so the how much
distance travel over mean free paths
expelling sure that's they decay because
they scatter out the boundary term scag
ask
other nose to the internal and then the
legs term is a cosi to either and the 10
Vermeil exponential must say here I have
to be careful because this is a classy
to either so any di de prime term I have
to make sure this is a decay so I have
itemized either prime that will be well
let me see whether it's positive will
lick it here so I need term in be 10
this will be naked here with this
negative I should have another negative
sign in the front and the f0 enterprise
so this is how i look at the equation y
i see whether my solution is reasonable
what I have here is I'm looking at a
certain point this point in space I look
at the distribution in this specific
direction and what it comes to one part
is coming from the war the boundary and
that party exponentially scatter out
that so that that's the first term
that's a Decatur write this term is
decaying the other term is along the
path every point is rescare into this
direction right like i say emission
mission into this direction so because
of this i have contribution the
contribution from this point that's f
either prime f 0 either prime that's a
regenerated and then cut the distance is
here the difference so this is a
regenerated a fallen and this is the
distance divided by cosine theta so the
real distance right this is the y
direction coordinate divided by cosine
theta that's a real distance between
this source point and the point I'm
interesting
and again during this trial war there is
a scattering alder so that's why they
exponent I have to make is make sure
this exponent at the end is liquid here
I'd have my side of prime because of my
this is a DA I could say so the other
problem is between iodine can say so
this negative and mu is a negative so I
put a negative sign that should be my
correct solution you can go to actually
check the mathematics right so now i
have the distribution function and my
challenge is remember this one f0 is a
function of either itself I do not know
right it's inside the integral and what
I have is a long local transport it's
not like a for-real law of heat
conduction where Q equals minus K DT DX
right you see this is a local high
school because the heat flux is
proportional to local gradient it
doesn't depend on quantity other place
of course they are all coupled but the
new core flux is the only local
propulsion local gradient what I have
here is the flux at this point depends
on every point along the trajectory
that's why I have the integral path
integral there ok so this is the problem
when you go through nano your transport
becomes a couple and it's long local ok
and now once i have f i can solve for my
flocks same way p 0 equals 1 over way
you sum up or polarization KX KY kz and
we've I H bar Omega F right that's
always the starting point when we try to
find out what's the lead flux here is a
for heat flux again charge flux momentum
flux you do
this is the highway started we sum up
with all possible modes quantum
mechanical classical modes and you get
the day and of course given the quantity
actual mega HTML carry quantum quantum
state kerry and the group velocity so if
you do this and you see we have to
convert this summation into integration
right and my i will actually split this
into two term positive term a negative
term and positive term are the ones that
goes forward negative term is going from
the other boundary going downward right
and then we can look at what the
expression after i convert this into
from integration into summation into
integration so positive term is actually
0 to pi d phi so this is a again I'm
changing this into a spherical
coordinate right and Phi is Adam user
angle and 0 to 90 degrees sine theta D
theta right theta is the angle of the
polar angle under my coordinate system
and then I have integration over
frequency 02 max frequency and H bar
Omega so that's here way cosine theta
that's we buy and see when I converse
amazing in the integration I have
density of states the Omega over 4 pi
because this D Omega is integrate
original for 4 pi isotropic case and
then i have f plus that's the solution i
had their here f+ is a function of both
mule and either and they integrate over
frequency
okay so that's they see the flux here
right and if you plug in your solution
for F either prior plus what we have
here and we can write it into so I give
you the final expression and this one
what I have is a 2 pi I 0 t e1 plus c
times the e3 and this is ray is again
the integral exponential function that
we introduced in the last lecture and 2
pi 0 to either i 0 either Prime and e2 I
Thomas either crime the either Pratt
right so i have to now say introduce
this I 0 I is a transformation so I 0 is
just the integration of H bar Omega and
the way times D over 4 PI and f 0 D
Omega so sometimes say this is called
intensity
but it's just the result coming from the
simplification in rotation and in fact I
say in full count transport this
intensity is widely used but for
electron and photon transport this is
not widely used i remember when i first
wrote the paper and submit a phase
review be and the review come back what
is intensity right and it's a scalar or
vector and i mentioned before the this
relate to f itself is the f itself a
scalar vector it has directionality it
depends on space but it's skyler because
it's a scalar in the face space right
and the directionality came from the
velocity component in the face space so
this is a again it's just a it's a
scalar so this is the positive reflux
and you can integrate the f malice and
together lakeview heat flux and the left
heat flux is the difference of the two
so if i write the left heat flux let me
write it down and you will get the levee
flux q is q plus masculine manners and
what i have here is a 2 pi and I 0 t e1
e 3 either that's the first term and a
plus whether we just wrote it down 02
either i 0 i deprived and the e 2 item s
either prime D either prime so that's
the class term and then I have malice
term malice I 0t to e3 and Chris a much
either malice
either to kiss a and I 0 either prime
and eat ooh I to prime mass either d
either prime so this worthy end result
after you substitute I q+ which we
derived and Q minus which we didn't
write the done ok so this is your left
heat flux this term come from boundary
this term come from boundary these terms
coming from internal points Delano core
part that's a source and how much they
contribute to the lead flux either the
location I'd ha that I'm consider so
this one right this is a queue is a
function of either at different
locations along the Y direction yes I
don't know yet I didn't tell you I know
that's what you want fine right so this
is the complexity of solvent even the
simplest policy problem non-local
problems right so what I'm going to LexA
to this is you ask the question so how
I'm going to find this out right so my
last step is to say to find this out I
have to say the heat flux you know he
flux if there's no internal heat
generation whether there is the internal
heat generation not right remember i'm
doing the steady state problem you will
see that the based on the treatment that
when I do transcend all my announcers
doesn't work ok so well here the RS is a
fine but see when my last step is doing
the
when I say I say there's a case when I'm
dealing with no internal heat generation
right and in fact that we wrote a paper
to compare the case when there is the
internal heat generation was a
difference so I with no internal heat
generation no heat generation and in
this case what I have q is constant
right if Q is constant I can do DQ d
either equals zero like so because I
don't need to find that constant yet
right so with that I take this to kill
di hai equals 0 I will find an equation
that actually has the are none either
they're either 0 SE eyes ero that I want
to find so you go to take the derivative
let me just repeat that they hear what
is the e right en the ends order
integral equation equals say T ms 2
exponential mass t / meal and the d are
now not heed this is a meal and the meal
0 to 1 this is my integral equation
integral function of order n so I have
III and here is the e2 you can
substitute it so if I take a dick here
data equals 0 what I have is the
equation I 0 T 1 and E 2 either and plus
I 20 te to Andy to
the same mass either and the plus zero
to either either 0 either front either
either prime e 1 item must either prime
absolute d either prime equals 0 okay I
didn't go step by step but say when you
take a derivative yes ray becomes a ye
to and AE to become ze one and this to
half integral right this is 0-2 either
either focus a cut is combined right in
20 to kiss a inside there is a zero I
the prime but ye one now has an absolute
sign because the change side when at how
you course I the prime so i put the
absolute to simplify the rotation these
are actually two terms ok so now this is
the equation you need to swap to
actually find given the boundary
condition on one side is tu on on the
other side the te to the mean free path
film thickness that you tell me to say
right and then you need to solve this
equation to find out what is either the
eyes 0 at each point I 0 remember is
related to my temperature right I 0 is
related to F 0 because this is a the
phone are probably things upstairs
frequency Omega which you know from the
solid a dealing with but f0 is a
function of T right he is what I want to
find so you really have to solve this to
find I 0 from I 0u fonty this is the
complication of a heat conduction long
overheating gosh okay and it's an
integral equation and how many you have
works the other way the integral
equation before
one right so what do you need fun is
what's inside and how to swap integral
equations right and i can simplify this
a little bit just to make the equation
lie sir so let me introduce a variable
theta is the function of for either and
that equals lambda measure light so I
have eyes 0 either minus i0 te to and i
have here i 0 t 1 my eyes ero te to so I
introduce this long dimensional
temperature or you can say its
temperature or non dimensional intensity
right and then say it turns out that
there's also learned emission of heat
flux which is they kill the heat flux
over pi I 0 te one must eyes ero t2 so
this is the corresponding on dimension
heat flux and with this seata i can
rewrite the equation you basically you
you substitute this I 0 into here and
using the reverse back and you can
rewrite the integral equation into a
more standard form
this is a what you have is a tool data
either equals e to either plus 0 2
cosine theta either Prime and the e1 8aa
must either Prime and this is d either
prime so this is the integral equation
lie sir phone right aha and this is the
inside the integrand and there is also a
lung function non 0 null function and
this is a called frag home integral
equation of second fat secondhand
integral equation of second pad okay and
when I try to solve this integral
equation i went to numerical recipe and
their people will teach you how to solve
this integral equations there's no
analytical solution and what do you have
here ye one is often called this is the
colonel right that will be multiplied to
the unknown function and in the old days
for this corner people because it when
the computer will become powerful before
that for this corner people try to get
an analytical solution by approximately
the corners and but nowadays you can
solve this integral equation relatively
easily and the way to solve the integral
equation is going back to think if you
want to do an integral of a function
what method do you use right the
simplest method of solving doing a
numerical integral
is trapezoidal method right so whatever
my function in this seat is this is my
either this is the unknown function i
want the sign right let's say this is my
c de ida and i divide this domain into n
point oh say here i have a say you can
do the trapezoidal is equally divided
right and for each of this say trapezoid
you take a trapezoid you have your area
is the waist x height so i can do my
approximate this integral by
discretization 0 plus a seat ha i
deprive any one item s either prime T
either prime equals and trapezoid
typically have two ends special because
it's only half sale right so I have a
seat at zero and e1 8aa and this is a
win see the prime equals 0 so see Adam
90 that's the idea itself is positive so
I don't need a the absolute sign the
other either over 2 so the other idea is
the separation of the point right and I
have a similarly I the other endpoint
see taka say so this is a woman either
prime of course this endpoint I have
this half see only so I have an e 1 and
i the prime equals kosei so i have the
same as either and then say theta either
the sea see the point
and then the rest is Delta Ida and one
to n minus one point in the center of a
theta I de aya say I sit I at ie1 I
Thomas I to I so I'm discretizing either
prime right so either is the point I
want to find right see that either the
either si si iodized any point of this
equation is satisfied okay so this is my
district wise term here and I want to if
I look at this how many points I have I
want to find i have c 20 sit either
let's see the I 1 to n minus 1 point so
i have n plus 1 point right I want to
find a seat ha I either i equals 0 1 to
n point so i have a total n plus 1
points ok now you look at your equation
you substitute this discretized term
into an equation and this should be
valid for any either point and then you
know you need any equation because you
have you have a you need n plus 1
equation because you have n plus 1
unknowns right so how you get that an
plus 1 equation you let your idea equals
0 Delta either to the data anterior to
the total end point so you said that
you're either equals either I that's the
district realization point and then
you'll get a n plus 1 numerical equation
and with that you can
word the matrix to find out whether you
see the eyes and this matrix is a
different thing then you solve for your
heat conduction equation or live in
stokes equation the in that case your
matrix is mostly zero except the
diagonal and the two lines to say
adjacent to the argument direction this
matrix is a typical of full matrix
because the every term is a nonzero okay
so you need to invert a matrix to find
the temperature distribution or land in
this case is lambda mission or intensity
distribution okay but there is one more
trick you put in there one more problem
that's this corner most of time when you
solve integral equation this corner has
a problem it's a singular and this is
the case if you think when you do you're
here when I taw equals either i right
what is ye 10 when I course I'd I write
a 10 if you look at my definition for
the integral function here right n
equals 1 so mule and at 0 T equals 0 so
it's a 1 over bill team you
so what I have is 0 to 1 and 10 ml de
miel y our milk give you is a log meal
what's the value at dogg xero blows up
so i have a singular corner and i have
to deal with that problem when I saw
will integral equations now yeah you
don't need to do the others there is a
trick to do it okay so the trick to do
this one is actually 0 to kiss a state
ha Ida prime e 1 either match either
prime T either prime right that's what
we want to do numerically and what I
what I'm going to do is actually i'm
going to do because this is Sita i'm
going to subtract this point out ok so i
don't do that integration i do say da
prime minus theta either you one either
match either prime D either prime so
this i subtract that see the ida and of
course that means i will need to add
that back so Sid iodized nada is not an
integration variable so Lucita either 0
cos i ii 1 i'll have much either prime D
either prime ok so this is just an
equivalency right because this term i
subtract i add and because i'm doing
this and see that i like and proud this
is the integral the corner itself can be
analytical done and if you do this
integration you find that it doesn't
blow up ok so that's the trick and this
one if you do it this term in fact you
will say this is
meal two months exponential must either
over meal and the mass exponential
cosine must either / meal so this
integral doesn't blow up so I solve this
singularity problem and that means your
discretization you it's a little bit
more complex you have to write this
carefully to deal with this point but
overall now you have by incorporating
those tricks you can solve this integral
equation and you have to do it yourself
to really there's nothing you all know
matlab matlab is good at the inverting
matrix when I do it I have to write a
fortune code that's what I know and but
there are also libraries to call
forwarding matrix so so this is the
trapezoidal method and if you use
trapezoidal method you need a lot of
point this will try you for integration
trapezoidal is the most straightforward
but it's mostly Union fishing right so
as i said that i am not good at either
really new marks the coding so but the
one that i can reduce the number of
integration i know is if I use this
quadrature method so if I want to use
the quadrature method F X DX and
integration and there is one popular
curvature not always good i equals n say
equals one to n so this quadrature does
not divide the integration domain
uniformly you actually choose fan on the
number of rule in this case i have i
equals 1 to n so any number you take and
it will find the optimal route of this
curvature and
giveaway so this is the ex I that I ex I
this solution depends on what quadrature
you choose and how many order your any
is okay and then it gave us a weight to
that point and so the popular way is the
council agenda quadrature so when I
wrote on my code was written use the
gods council agenda coalition and
sometimes it works well other times it
may not be that efficient and but there
are other quadrature you go to look at
the numerical integration books okay and
so with all this the math power let's
look at the result that you are supposed
to get look at the dissolution so what
this solution looked like so I have the
problem at dealing with in the current
assay the original configuration I draw
this is my Y direction right and you
know I use along the mission of
coordinate x equals y / mean free paths
and so if I plot this solution this for
this long dimension or intensity città
either ok and I'm going to actually not
use the idea here is a double confusion
what I'm going to do is y over D so that
I can plot all the lines between 0 to 1
ok d is the thickness of the plate so I
have a 0 to 1 okay and this long
dimension or so the first of the
one the maxima is one a minimum is zero
so one solution this is a Fourier
solution when the mean free path is much
smaller than the thick sickness so what
I have here is this one is a Class A
equals d / min free pass right so can
say is the non-dimensional signity or
mean free path when this one goes to
infinity that's a sickness a much larger
than the mean free path and this profile
is linear and that's what you would
expect from fuller law of heat
conduction this is a diffusive the local
diffusion a price of a solution now in
the other case when this cos a goes to 0
what it means there's no scattering
inside right you will be ballistic so
you'll have a solution flat is going to
0 point five ok so this is the utter
limit when I say goes to zero you have
ballistic transport so why is that and
although I see here I'm solving using a
full foulards example but see this is a
valid for electron and the form article
for full long and fulfilled huh so let
me take it a example I talked before
this is the case when I have radiation
transfer you turn to parallel play in
vacuum right there's no scattering
inside four-fold hung because could say
is this a goes to zero that means the D
is a very small compared to mean free
path there's no internal scan when
there's no internal scattering what you
have here
recall we said here is a sigma t 1 4th
power here is Sigma T 2 4th power right
that's what it will gather if you do I
zero integration right recall with
either the eyes 0 if you put a day for
black body radiation integration from 0
to infinity you get Sigma Chi 4th power
that's the Stefan board manual ok so in
this case I do not have any scattering
in between I do not have equilibrium
between very long in pune bro but I'm
forcing my boards my solution to get a
temperature right this temperature is
the equivalent temperature when I have
in this imaginary absorber that absorb
this on this side absorbs Sigma chief to
4th power on the other side see absorb t
sigma t 1 4th power so equivalently the
absorber temperature will be the average
of this to 4th power / to right and it's
everywhere it said so that's why you see
in this limit my solution is a flat ok
and now in between I have a jump a
discontinuity at both sides and this
will be almost linear not exactly this
is where it's in between when you go
between zero and the infinite that
depends on how large is the cosine so
what's interesting out of this is at the
boundary there's a temperature
discontinuity temperature jump ok so
this is an original ray this is SE te
one is here that's normalize the one but
say what I see here is inside the
material when I use the Fourier law to
find a
coolant temperature when they are
scattering you want at the war the
temperature local temperature is
different from te 1 so this temperature
discontinued the temperature jump is a
very tricky point I and when we discuss
chapter 5 I mentioned this and here it
reappear because my solution now is more
general I can solve for Annie's in the
chapter 5 there's no internal schedule I
see there's a jump that's point five
from 12.5 right so what I'm dealing here
is a first that depends on what problem
you you're solving if it's a radiation
problem this is physical if the
conduction problem what I'm talking is
on physical it's a wrong definition okay
so why I see that now let's go back to
think about the radiation I mentioned
this before here again right so i have
this imaginary detector I can put there
on one side of the flow of one type of
fall on a photon as a tee fool and the
other side t1 so once I go to the war is
still absorbed in this side and the
other side and so far I have assume the
war is a black what does black means is
on this side the photon comes in d just
go no reflection right that's a black
black is no reflection everything is
coming is absorb so it's also
transmission that means the transmission
is one right when transmission is one so
of course this is solid and sorry that I
don't have a way to get into it I put a
detector but I imagine you got this
magic at detector I have and if moves
into solid but it doesn't interact with
the folin doesn't interact with solid
they only measure the light photon
right what would be the temperature in
measure ah so incoming is so here is
here on this side and on this side both
side right so you will be continues in
fact that there will be no jump if you
can do that at Magic detector and of
course the salt itself does a full tank
will have absorption and remission in
the solid so what are happening is that
this is whether you measure the photon
temperature you would measure but there
you don't have that detector what do you
have with the thermocouple right and
when you put a thermocouple into this
solid wooden measure is a solid
temperature that's a full-on lattice
vibration okay so this is my t1 what I
say is this is the full on temperature
and you've proven Anthony when they emit
I say this is a half of this space that
you meet as a black black body so that's
where my physical temperature jump
coming from in the case of photon now
when I do full-on right and if the
surface again in the black eyes what I'm
saying that's the immediate at the
temperature t1 equilibrium temperature
so transmissivity is one there's no so
if there's no reflection this is a
material one mature to mature three
right install it that I typically can do
multi-layer structure and in that case
there should really be no temperature
jump okay it's a similar you can
actually measure you can actually
measure the temperature suppose you have
lights way to measure the lattice
temperature so there should be no jump
and this jump is only because I'm
assuming the fall on coming out is at
this temperature what happens is that
this is the interface i discussed
for right if you look at what comes to
this side is a te to and what ey is
going out like this but the te2 it goes
out like this because the lower
temperature right so if you define
temperature consistently you should not
have the jump this is a highly long
equilibrium situation we there's no good
temperature definition okay and however
the other solution this is again you can
go back to read arrived everything that
is when your surface is not a black when
your surface has a fan of the
transmission reflection right and this
is where you actually have a physicker
temperature discontinuity a tattoo
interface and this is a this wall if you
do boards my equation you can
consistently including everything okay
and in chapter 5 I discuss this it's
really how you have to be careful how
you define temperature when you go to
this ballistic or quasi bodies regime
otherwise you may be comparing apple an
orange ok any questions
it takes a lot of kind people still
continue making those mistakes okay so
if you go to read papers and then many
of them don't distinguish the start of
difference okay so that's the case so if
we think about the way we talked is a
transport along the film transport the
perpendicular to a feel ok I'm going to
next to tell you a story of a mixture
the transport the boats along and
perpendicular to the film what would be
the case like that huh this is a fluid
flow think about that you push your
fluids through a pipe air molecules
right a small pipe and this is the post
air flow so let's look at in the case
simplest case verify puts a flow ok so
high if so this is the pipe flow that
you are familiar with but in the limit
opposite flow i say two parallel plate
between two power-play that I'm flew in
fluids through and if this if one plate
is moving the other plate is stationary
that's quad flow if with the pressure
driven both plates are stationary that's
pecet flew right you learn that in fluid
mechanics so I'm going to deal with the
case where I have stationary plates I
have a fluid flow inside and in this
case we wrote down the boards my
equation DF DX right this is my X
direction and my YY
BF d y equals to F months FD over tau
that's my relaxation time and there is
another f-zero because I have a long 0
fluid velocity average velocity so you
displaced Maxwell distribution we said
before right so this displaced Maxwell
distribution is n m 2 pi k bt 3 over 2x
pollen sure m way that's the random
velocity my co which is a function in in
the case of laminar flow is the tool yo
is a function only you is changing in
the x direction but it's a function of Y
right that's what a pipe flow in normal
is the parable parabola and if the why
is my coordinate you is a flow energy
flow direction so U is a function of Y
butter is in the X direction right so
flow is in the X direction my constraint
is again in the Y direction that's a fun
at thickness right and it's a pressure
driven so where does the pressure came
in it doesn't come into my boards by
equation pressure in fact is a
statistical average just like computers
mechanical rights strategic average on
the momentum but in this case here it
can relate to n is a function of X and
then you can use ideal gas AE as a PV
equals NRT right so this is a you relay
this molecular density to the pressure I
have to say that I'm not satisfied with
this approximation so ideally I will not
even do that I was
say okay my pressure is just a momentum
average I tried to do that I was not
successful okay so this is the classical
you'll put in the boards my equation now
it's a boost in the Z X direction I have
you and then you divide the right and
the NEX I have the source term in the
x-direction pressure and the in the
right direction I have the boundary and
if you plug in into the equation you
will find out that you need to do the
same set as we did for the perpendicular
transport is the because you that's a
velocity distribution is we do not know
average velocity distribution just like
when you try to find temperature
distribution you is the function of Y
and so it's you is the inside the
integral it's an integral equation over
you when you do everything derive just
and follow the same way okay so what is
the interesting is out of this problem
is a loose and minimum so this was
observed experimentally in 1905 1906 the
spirit by loosing doing rarefied gas
flow and what they observed is if you
plot your flocks your say mass flux
right which is normalized say to our
ideal gas constant zero and the D Square
t is the thickness of the plate and the
real measure the flux and DP DX so this
is a long dimensional and woody what it
has is and this is drawn as a functional
loosen number which is a universe loosen
number is a mean free path / d
that Co say I mentioned before is d /
mean free pass right in different feel
they're different name for example
immerse of this is called that you
objects the call the up your thickness
acoustic we also call acoustic signals
but the this universe mean free path
where D is a nuisance number that's why
I has the longest history so if you say
loosen number 44 long loose and number
four electrons that's all fine you just
need to follow the same mean free path
over the thickness / the carriage penis
there people actually call the run run
universe resolution number that's wrong
ok so what they find experimentally is
that it goes this way and this is the
one this is about one this what happens
around the one and when you go to verse
oh here very small loosen number that's
a continual very large loosen number
that's a rarefied gas flow and the
in-between there is a minimum and this
minimum causing Newson minimum this was
the experimental Fonda okay so I said
the line around 1900 and that there was
no solution people try to do approximate
solution boys my equation and it was
really unsure 1960 online in 50 and
there's a great mathematician and I
solved the problem it's a search
signally he saw the world this problem
boards my equation this way I just
written down like why the Russian you
find that a velocity and they found that
indeed this solution has a loosening and
the reason for this I try to come up
with a very in director picture right
and the reason is this you have a
pressure driven flux going in the X
direction and we see in heat conduction
is always at the roughly so
decrease the thermal connectivity right
and by say here actually you got a
regime where in the very dilute regime
your pressure is the pressing this
fluids much faster so the boundary
effect is actually the scattering at the
boundary is less important and that's
why you have the recovery of this C flux
the fluid mass flux dear ok and then
corresponding to this solution you go to
look at your velocity distribution in
the pipe or in two parallel plate you'll
actually find this is where the leg
actually come from right we know that
the typically you will do fluid problem
you see velocity is a non slip boundary
condition but the board's my equation
will tell you actually this is zero the
velocity has a discontinuity at the war
so this is the velocity there is
actually sleep happen between the fluid
at the wall ok so this is a valid they
see a sleep velocity sleep and in the
next lecture we'll actually discuss
people I tried the worse let me see if
you do verify flu or you do flu
mechanics and micro fluidics that you
will add a artificial to velocity sleep
and in fact you add the first order and
second order velocity sleep to try to
accommodate and see it turns out that
the first order velocity sleep typically
can go to this regime cannot go to
loosen equals one or even higher second
order veloci sleep you will or predict
the velocities they loosen say this this
range so you in those days there are
people still try to write the boundary
condition and I'll tell you a story how
I derive a new boundary
next lecture but that's the this is the
loosen flow and the right fight
flu lucent minimum and let me tell you a
story I mentioned many times that addy
stripe right the disk drive industry and
the disk drive is a lubrication problem
so what it has is this is a disk
lubrication probably is tubular reloads
equation well I'm the reloads equation
how it is it's basically this rotating
da gizmag live in here and this magnetic
heather has a inclination right so it's
like an airplane you have angle so that
gives you lift and the lift is very
important because otherwise are your
slide your head that will crash the disk
right and this disk a rotating about
10,000 RPM very fast it's about 10 years
ago I have a friendly said that can you
look at this problem we have one
nanometer separation that we cannot
predict but that one nanometer they
already long for we have now used that
they continued the actually use the
boards my equation solution for the flux
so the used search engine on a solution
because this problem is say when so
you're stationary this is so you can
roughly say this is your move-in play
it's a combination of a tool it's a
pressure because when you have an angle
if you don't have anger that's a quad
flow we have an angle you have a
pressure difference build up so is the
pressure we're driving pase flows it's a
superposition of wet and and post a flu
so they have now used them they have
been using the search signaling solution
and there was a famous paper for this
described problem that decomposing but
they say there is a 1 nanometer problem
I said what can happen one nanometer by
you need turns out the data because at
the time I mentioned they were about 20
nanometer fly height right and now this
high
I high-dimensional also to you it's
about 1.5 nanometer okay that's why it's
a it's a cinch important problem and we
will ever find out what's the reason you
can see it could be the molecular it
could be absorbed the molecule it could
be the molecular fiber size of molecules
right and but it's a very relevant
technology problem and noticed like see
this still concern because you still
have a disk and that you have a slider
there and finally there is another
problem this is a you can think about
this if I have tea hot aunty code and
have a small opening so this opening is
smaller than molecular mean free path
the question is what's the flocks
direction which what are the mass flux
going from cold to hot or hot cocoa
maybe we'll discuss in the next lecture
you can think about this I'll ask you at
the beginning huh but let's hello I want
the left flux