Heat Equation Derivation Video Lecture | Heat Transfer - Mechanical Engineering

hey everybody so in this webcast I'll be
talking about the derivation of this
equation which is known as the heat
equation and my goal is to make it maybe
a little bit more approachable and less
intimidating to some of you and that's
because this equation is simply based on
an energy balance what we see is that
the the rate of accumulation is equal to
the flow of energy entering the system
minus the flow of energy leaving the
system plus the net rate of energy
generation so the box to the left is our
energy box and at any moment in time it
contains some amount of energy and the
analogy that I'm making is compared to
conservation of mass for example you
might have you have water flowing into
the system and water leaving the system
and if the water is flowing into the
system more quickly than the water
leaving it what we'd observe is the
courses that the box would fill up with
mass over time just conservation of mass
now conservation of energy is a little
bit more nebulous but the fact is that
it's just the same we've got energy
entering the system and energy leaving
so energy can get into and out of the
system by the flow of heat so if we look
at this box and think about all the ways
that energy could flow into and out of
the system it helps to define a
coordinate axis where we've got the X
direction in the horizontal direction
the Y in the vertical direction and the
Z Direction into and out of the screen
so if you look at our control volume we
can define a few points so for example
we've got position x right here and a
position X plus a really small amount X
plus DX on the right-hand side of the
cube and we can look at heat flowing
into the system from left to right and
the heat flowing out of the system on
the right hand side of the cube and we
can give these variable names
they're both q x2 the flow of heat in
the X direction and this one on the left
will evaluate it at position X and on
the right this is also Q X and in this
case its evaluated at x plus DX and of
course we can have energy entering and
leaving the system not only from left to
right but from top to
and into and out of the screen so both
in the Y and in the Z direction so I
draw if we look at energy leaving your
entering the bottom of it and enter
energy leaving the top of it so we can
find similar names for the flow of
energy in from the bottom of the cube in
the flow of energy leaving it so in this
case we've got a location Y and a
location a little bit above Y which is y
plus dy so we can evaluate the flow of
heat into the system at position Y and
the flow of heat leaving the system at
position y plus dy and of course we can
do the exact same thing for the flow of
energy into the screen and out of the
screen by defining a position Z and a
position a little bit further into the
screen at Z plus DZ so of course I've
got the flow of Heath the flow of heat
coming in is Q Z evaluated at position Z
and Q Z evaluated at position Z plus DZ
so I've got green arrows pointing into
the cube and a green arrow leaving the
cube into the screen so of course all
that makes it pretty cluttered so I've
gotten rid of energy entering and
leaving in the Z and the y directions we
can really just look at the flow of
energy entering and leaving in the X
direction so what we've got is a flow of
energy entering the cube in the X
Direction is simply Q of X evaluated at
and the flow of energy leaving the X
Direction is Q X evaluated at X plus DX
so the question that remains now is how
do we relate Q of X evaluated position X
to that evaluated at position X plus DX
so in other words how do we make the
relationship between these two and we
can do that by using a Taylor series
expansion and all we're really doing is
just looking at the first order term so
what we'll get is the heat flux in the X
Direction u evaluated at x plus DX so
you evaluate it out here is simply the
flow of heat into the left hand side so
Q of X Q X evaluated at X
Plus this rate at which the heat changes
is a function of X multiplied by DX and
really all that's saying is if we know Q
X evaluated at position X and we know
the slope of a line
DQ X DX and we know this distance which
is DX we can now figure out the heat
flux at X plus DX so simply we're just
looking at the first term in a Taylor
series expansion which just suggests a
linear relationship between these two so
at this point we're able to start
thinking about the net flow of energy
into the system which is a flow of
energy entering the system minus the
flow of energy leaving the system and
that net flow of energy is simply Q X
evaluated at X minus Q X evaluated at X
plus DX so we take this variable and
plug it into here what we're left with
is this expression minus the expression
in parentheses and at this point it's
not hard to see that the differences
here at Q X will cancel out and the
negative sign will just be propagated
into here so we're left with a
relatively simple expression that
describes the net flow of energy
entering the system in the X direction
so now that we've got a handle on the
net flow of energy in and out in the X
direction we can begin to look at the
rate of energy generation and how we
might describe that mathematically and
it turns out this is relatively
straightforward so what we've got is the
rate at which energy is being generated
is equal to a variable we'll call it Q
dot and this Q dot is the amount of
energy per unit volume so we need to
actually multiply by the volume the
volume of our box and the volume of our
box of course is simply it's a cube so
we've got DX multiplied by dy multiplied
by DZ so we've got the rate of energy
generation is equal to Q dot times DX dy
DZ and again I should emphasize are not
really generating energy we're just
looking at the creation of thermal
energy from some different source and
some examples of this include a chemical
electrical and nuclear sources a
chemical you can imagine
the reaction either being endothermic or
exothermic either cooling off or heating
up the local region which it's occurring
so all we're left with now we know that
the net flow of energy in and the rate
of energy generation all we are left
with now is to figure out the rate of
the accumulation of energy and the rate
of energy accumulation is simply the
rate at which internal energy increases
or decreases with time so the rate at
which internal energy changes over time
and I use a symbol you here for the
internal energy and U is the internal
energy per unit mass so you have to
multiply by the mass in the cube so if
you remember from your course in thermo
the internal energy per unit mass is
equal to the heat capacity multiplied by
the temperature difference
from some reference state so we've got
this reference state is completely
arbitrary I mean it just depends on on
how you're evaluating U or how you're
defining it so the derivative D u DT is
simply the heat capacity times the rate
at which temperature changes and we've
also got the mass of material is equal
to the material density multiplied by
the volume of that cube so the mass is
equal to the density times DX dy DC so
if we multiply the two together take D u
DT multiplied by its mass take this with
the product of that and we're left with
this expression on the right-hand side
so before we can put it all together we
need to think about a relationship for
the thermal conductivity so the next
thing we need to figure out is some way
to relate the heat flux to the
temperature and this flow of energy I
may have call it a heat flux but it's
actually the flow of energy and might
have units of joules per second or watts
or horsepower if you're so inclined but
this is a total flow of energy entering
the left and leaving the right side of
the cube so if you remember this
expression we've got the flow of energy
into the left side of the cube is equal
to negative K the thermal conductivity
multiplied by the area through which
it's it's
traveling so you've got in this case
we've got dy the vertical axis and DZ is
the axis going into the screen so we've
got K times dy times DZ and it's also
multiplied by the local temperature
gradient in the X direction and this
negative sign of course says that the
heat will flow downhill it will flow
towards a region of reduced temperature
so if the temperature is getting smaller
as you move from left to right that
means DT DX is negative and your Q of X
would then be a positive value
indicating that heat is flowing from
left to right or flowing in the positive
x direction so now we've got what we've
got is a nice way to relate the thermal
the flow of thermal energy to the
thermal conductivity the area through
which it's traveling and the temperature
gradient now we want to be able to
evaluate this expression by plugging a Q
of X into this differential so if we
plug our expression for Q into this
differential we've come up with this
expression and this expression can be
simplified so at this point you might be
able to see where I'm heading with this
is we've now got a nice definition for
this first term and we'll find along the
way that all these differentials DX dy
and DZ are going to cancel out and of
course we can develop this expression
not only in the X direction but of
course also in the Y and the Z direction
so we've got now we've got the net flow
of energy into the system in all three
directions and I think I wrote I wrote
down the wrong thing up here I did it
correctly earlier but the rate of energy
accumulation is simply Rho times the
heat capacity the rate at which
temperature is changing multiplied by
the volume of our differential element
so now what we're left with is that the
rate of energy accumulation this term on
the right hand side is equal to the sum
the net rate of energy entering and
leaving in the x y&z directions plus the
rate of energy generation and it's
pretty obvious now the relationship
between these expressions in the
original equations so what we'll see
when you sum it all up the differential
DX dy and DZ z-- all cancel out and
we're left with the final expression if
in all heat equation