Summary of One Dimensional Conduction Equations Video Lecture | Heat Transfer - Mechanical Engineering

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okay so what we're doing we're doing
conduction analysis we're looking at
this thing called an alternative method
but recall methods of conduction
analysis there are several
we have the heat diffusion equation
that's the big complex partial
differential equation you need to have
boundary conditions you can solve for
the temperature inside of an object and
sometimes for certain shapes you can do
analysis and solve that analytically
which we will be looking at later in the
course
numerical analysis I've already talked
about what we're going to be doing there
and we will look at numerical analysis
and give you an Excel spreadsheet tool
that enables you to do two-dimensional
numerical analysis but there are other
techniques as well and then finally what
we've been focusing on for the last we
did this last lecture is what we call
this alternate or alternative method and
essentially it's using fourier slaw
under some rather severe assumptions but
turns out these are pretty good
assumptions because for many of the
systems that we study these assumptions
do apply so let me just briefly overview
what that alternate method was and then
what the results were and then we'll
move on looking at how to apply it for
engineering problems so if you recall
the alternate method we started with for
EA's law minus ka DT by DX and if you
know the area as a function of X or it
could be radial location as we saw it
for cylindrical and spherical
coordinates but if you know that then
you can make a substitution into for
EA's law and then what we did is we
rearrange that and we set it up to
enable us to be able to integrate but
typically what we were doing is we were
pulling the area over to the left hand
side in the denominator and then we were
left with minus K DT this is where we
said that we're going to assume that the
thermal
conductivity is not a function of
temperature so that enabled us to pull
it out of the integral which I'll show
you in a moment so we have QX and we
integrate from some boundary condition
and we know what's going on to some
explanation that we're trying to figure
out what the temperature is and you
divide that by a as a function of X you
pull K out of the integral from t1
that's the temperature at the boundary
condition at 1 and then we have the
integral DT now this works it works
subject to a couple of constraints one
of them we had to have steady state the
second constraint that we had was
one-dimensional so that means that all
the heat is flowing only in one
dimension on in two or three dimensions
which we'll look at later on in the
course third one no heat generation
inside of the material that we're
studying and that fourth was that K is
equal to a constant so the thermal
conductivity is not a function of
temperature and what do we get out of
this well you get T of X and you get Q
of X so we were able to get the heat
transfer and the temperature profile in
terms of engineering this is usually the
one that we are most interested in
because we want to figure out what the
heat loss is so what I'm going to do now
I'm going to summarize the three
different systems that we looked at and
we looked at the plane wall well it
wasn't quite a plane wall we looked at
that example problem with a conical
section but this would be what you would
get if you were to look at the plane
wall so I'll draw out a schematic
so that's a schematic now what we were
able to determine was the temperature
distribution we were able to determine
the heat flux and so what we did for all
three cases we plugged Q of X into T of
X in order to give us the temperature
distribution in the solid
okay so that was for a plain wall we
also looked at a cylinder missile that
was the geometry for the cylinder our
eye our outer and what we obtained here
was the temperature distribution
and if you're called for the cylinder
resulted in a natural logarithm we were
able to solve for the heat flux
and just like for the plain wall if you
plug the heat flux into the temperature
distribution we were able to come up
with the temperature distribution
function in the cylinder
okay so that's what we got for a
cylinder pipe flow many many engineering
applications where we would have
cylindrical coordinate systems and we're
trying to determine heat loss but that's
what we got for the cylinder and then
finally spherical there this might be
something like a storage tank you can
have spherical coordinates okay so
that's a scenario for the sphere
temperature distribution and just like
for the other two you plug Q of R into T
of R and you get the temperature
distribution in the wall of this sphere
okay so those are the three different
geometries that we've considered and the
reason why I put them all out is because
we're going to summarize them in the
next segment and what we'll do we're
going to look at the form of them and it
turns out that there is some sort of
commonality to all three of the forms
that we have or for both the heat flux
as well as a temperature distribution
and it's what we'll do we'll come up
with a shortcut method that enables us
to analyze what is going on for heat
transfer either through a plane wall
through a cylinder or a sphere and we'll
be able to do more than just what we're
looking at here we'll be able to do
scenarios where you might have a sphere
with insulation on the outside of this
sphere so you might have some insulating
layer and then there might be convective
heat transfer going on out here so you
have wind blowing and that has an impact
we're going to develop a technique that
will enable us to look at that and that
is referred to as being thermal
resistances so that's where we're going
with all this in case you're wondering
we don't want to have to solve these
equations all the time and write out all
these different terms we'll come up with
shortcut methods and that's why sent
engineers are efficient I said as yours
are lazy but that's no true engineers
are very efficient we like to find
shortcuts and ways to be able to do
things more effectively in official ways
so that's where we're going in the next
segment we're going to work towards
thermal resistances based on the results
that we've got thus far with this
alternative method using 40s law