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in the last class we defined heat transfer
we defined the objectives of the course and
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what are the different modes of heat transfer
let us start with conduction from this lecture
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so what is conduction
whats conduction conduction refers to transfer
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of energy from the more energetic to the
less energetic particles of a substance due
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to interactions between the particles moving
randomly
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so the more energetic particles would actually
have the tendency because of this interaction
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between them to transfer the energy to a a
less energetic particle so this process is
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what is referred to as conduction mode of
heat transfer
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so suppose a particle is at higher temperature
suppose a particle is at a higher temperature
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then they naturally have higher energy higher
molecular energy
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higher molecular energy and then due to collision
they may experience collision and due to collision
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they actually transfer energy to a
energy to those particles whose temperature
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is actually smaller then the temperature of
itself so this process is what is called as
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conduction so now in the presence of a temperature
gradient so clearly in the presence of temperature
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gradient
temperature gradient is now present so in
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the presence of temperature gradient what
happens is that the energy transfer energy
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transfer must occur
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must occur in the direction of decreasing
temperature
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so direction of energy transfer is very important
while quantifying the heat transfer process
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so its important to recognize that when there
is a temperature gradient the energy transfer
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must actually happen in the direction of the
decreasing temperature so in various quantification
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process we will actually take into account
the direction in which the heat transfer occurs
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and that actually is crucial in quantifying
the heat transport process and the ah various
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methods that we will develop in this course
so this process of energy transfer that occurs
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in the direction of decreasing temperature
is what is called as thermal diffusion
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so the conduction or the thermal diffusion
in liquids or solids or solids it typically
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occurs through molecular interactions
however in a non conductor in a non conducting
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material the typical transport occurs because
of the lattice waves its primarily because
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of lattice waves the primary mechanism that
governs the thermal diffusion in a non conductor
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is the lattice waves and in the case of conductors
it is essentially governed by the translational
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motion of electrons its primarily governed
by the translational motion of the electrons
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that is actually present
so so different types of materials the mechanism
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of thermal diffusion is actually different
and accordingly the properties of these materials
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would actually be different so the next aspect
we are going to look at is how to [quantife/quantify]
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quantify the thermal diffusion process so
we are going to look at quantitation of thermal
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diffusion process now in order to quantify
one need to look at the rate equations so
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what we are going to see is we are going to
develop how to write the rate equations for
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these processes in particular we are referring
to how to estimate the heat transfer rate
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in a given system due to conduction
so suppose i want to estimate a heat transfer
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rate for a particular system then we need
to have a certain governing loss that connects
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the rate or that connects the rates with the
corresponding driving force and other properties
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so such a rate law is what is called as the
fourier s rate law or fourier s law and the
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fourier s law essentially relates the flux
of heat transport
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it relates the flux of heat transport and
the temperature gradient
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because the heat transport is occurring in
the direction of the decreasing temperature
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the fouriers laws states that the flux of
heat transport is proportional to the negative
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temperature gradient so suppose we take a
slab let us assume that the slab is actually
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having a ah is a is in the is in a cuboid
form let us assume that its a cuboid slab
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and let us assume that the cross sectional
area of the cuboid is a and let us also assume
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that the temperature of one end of the slab
is t one and the temperature of the other
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end of the slab is t two and let us assume
that t one is actually greater than t two
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so what will be the temperature profile inside
the slab there will be a a certain profile
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inside the slab and the heat is actually transferred
from the end whose temperature is t one to
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the end whose temperature is actually t two
which is lower than t one so this is the direction
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in which the heat is actually transported
or the flux of energy that is being transported
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from one end to the other end is in this direction
so suppose if i draw coordinates and lets
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say that this direction is x direction then
the flux that is actually flux of heat that
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is actually transported can actually be written
as flux in the x direction or generally referred
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to as q x double prime and so henceforth in
this course the ah the quantity double prime
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superscript for q usually you will refer to
as flux and so without a superscript would
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be rate and q with a with a double prime superscript
would actually be referred to flux so the
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fouriers law says that the flux of heat transport
in the x direction for this case is actually
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given by minus k which is the conductivity
of the slab into the temperature gradient
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in that direction
so thats what is actually relation between
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the flux of heat transport and the corresponding
temperature gradient which is the driving
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force suppose a is the area of the heat transport
remember that the heat transport is actually
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a is actually from one end to the other end
which is actually across the cross sections
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of the slab and so the rate of heat that is
actually transported from one end to the other
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end via the cross section of the slab is essentially
given by the flux multiplied by the cross
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sectional area so therefore the heat transport
rate q is essentially given by a times in
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the x direction is given by a times q x double
prime which is essentially given by minus
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k times the cross sectional area into the
temperature gradient in that direction
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so from here we can clearly see that we can
see that the heat transport rate q is essentially
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proportional to the area of heat transport
it is also proportional to the driving force
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it is also proportional to the driving force
and of course the rate depends on depends
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on k which is the thermal conductivity of
the material which is the thermal conductivity
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of the material now note that thermal conductivity
is actually an intrinsic property so the moment
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a material is specified then the thermal conductivity
of that material has actually fixed thermal
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conductivity of that material is very specific
to a given property and its an intrinsic property
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of the system that is actually being considered
so let us look at what are the units of these
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different quantities that we have looked at
q double prime which is the flux is essentially
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the units of this is watt per meter square
and the units of heat transport rate is one
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which is nothing but joules per second
so now we know that heat transfer rate is
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given by minus k a times d t by d x so if
i now set the units here so the units of q
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is watts units of a is area is meter square
and the units of d t by d x is kelvin per
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meter ok so therefore from here we can easily
see that the units of thermal conductivity
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as units of watt per meter kelvin that s the
units for the thermal conductivity of any
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material suppose suppose if the system is
non isotropic so there is no reason why the
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temperature gradient has to be same in all
directions the temperature gradient can in
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principle be different and different directions
in a given material so which also automatically
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implies that the flux of heat transport in
different directions can in principle be different
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so suppose if it is a non isotropic system
suppose if it is a non isotropic then the
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flux of heat transport is essentially given
by minus k into some gravity which is theta
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temperature gradient and thats equal to minus
k into the unit factor in the ith direction
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d t by d x plus j d t by d y gradient in the
y direction and gradient in the z direction
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which automatically implies that this is also
equal to i q x double prime which is the flux
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in the x direction plus j q y double prime
plus k q z double prime
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so this is the flux in x direction this is
the flux in y direction and this is the flux
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in z direction ok so from here it is very
clear that the temperature temperature in
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the slab or in the material that is being
considered is clearly a function of all three
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positions so its a function of both x y and
z with x y and z being the position inside
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the solid let us look at what is thermal conductivity
so thermal conductivity
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thermal conductivity is a property which essentially
captures the ability of the system to actually
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conduct heat so the thermal conductivity of
solids is typically greater than the thermal
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conductivity of liquids which is typically
greater than the thermal conductivity of gases
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so to give you a sense of numbers the thermal
conductivity of gas systems typically is in
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the order of magnitude of point zero one to
point one watt per meter kelvin and the thermal
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conductivity of liquid is typically in the
range of point two to five watt per meter
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kelvin and the thermal conductivity of gas
systems solid systems is typically ten watt
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per meter kelvin
so one may ask a question as to why there
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is difference in the thermal conductivities
across different types of the materials why
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solid thermal conductivity of solids is high
why k solid is high what could be the reason
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so typically the thermal conductivity in solids
has two components and note that in solids
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the the part molecules are actually packed
very close to each other and so the propensity
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for these molecules to collide with each other
is very high and therefore the ability to
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actually conduct heat is at much higher in
solid so which explains why the thermal conductivity
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of solids is much higher than that of liquids
and gases now the thermal conductivity in
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a solid typically has two components k e and
k l so this is essentially the component that
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is because of lattice waves and this is essentially
because of the electron movement component
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thats because of the free electron movement
now if you take metals if you take metals
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then the thermal conductivity component due
to free electrons is actually much higher
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that of the lattice so typically for such
systems k is approximately equal to the component
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that comes from the ah free electrons movement
and k e which is typically a function of one
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by rho e which is the electrical resistivity
which is the electrical resistivity of the
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metal if we take non metals for example ceramics
then typically k e is much smaller than k
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l and in fact this is the reason why the insulation
systems typically have low conductivities
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k l actually depends on strongly on the lattice
arrangement
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so one can play with the lattice arrangement
of the non conducting material in order to
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change the conductivity and thereby thereby
consider those materials as actually an insulating
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material
so for example the conductivity of quartz
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which actually is a more regular lattice arrangement
is actually much higher than the conductivity
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of lets say amorphous graphite so thats because
the arrangement if there is regular arrangement
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then there is a a strong potential for conducting
heat better than if there is lattice arrangement
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is regular in the case of fluids the particles
are much more random particles are more random
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and therefore the conductivity of the liquids
is actually smaller than conductivity of the
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solids or else the propensity or possibility
of the molecules to collide with each other
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actually goes down because they are more randomly
packed and therefore the conductivity of the
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ah fluid ah conductivity of liquids is actually
smaller than that of conductivity of the solids
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so let us consider gas systems so in a gas
system the conductivity of gas is once again
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much smaller than the liquids thats because
the particles are even more randomly packed
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and typically depends on depends on temperature
pressure and the chemical species itself
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chemical species itself
so the conductivity of the gas is typically
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proportional to the number of particles per
unit volume that is present the mean molecular
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speed c bar and the mean free path lambda
and so the conductivity of the gas is typically
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a function of these three and ah different
gases will have a different mean free path
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and the different molecular speed so as a
result different gases will have different
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conductivity so we said that the conduction
occurs through the the process of energy transfer
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in the decreasing temperature direction is
typically what is called as the thermal diffusion
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process so the thermal diffusion is actually
characterized by a quantity called thermal
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diffusivity
thermal diffusivity and thermal diffuse the
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symbol that is typically used is alpha which
is given by k by rho c p where k is the conduct
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thermal conductivity of the material rho is
the density of the material and c p is the
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specific heat capacity of the material so
what will be the units of thermal conductivity
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thermal conductivity units are alpha a the
units of ah thermal conductivity is watt per
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meter kelvin units of density is kilogram
per meter cube and the units of specific heat
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capacity is joule per kilogram kelvin which
is equal to joules per second meter kelvin
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divided by kilogram per meter cube joules
per kilogram kelvin
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so if you cancel out the light terms so the
units of of thermal diffusivity is meter square
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per second so what is thermal diffusivity
thermal diffusivity essentially characterizes
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or captures the ability of captures the alpha
essentially captures the ability of the material
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to conduct thermal energy relative to its
ability to store heat so with these definitions
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what we will start from the next lecture is
looking at how to write heat balance in order
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to estimate or find the temperature distribution
in the system that is being considered