1-Steady State Heat Conduction
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Transcript of 1-Steady State Heat Conduction
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Steady state heat
conduction
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Heat Transfer
A background in ODE
Important analogies between heat,
mass, and momentum transfer
Heat transfer is the science that seeksto predict energy transfer that may
take place between material bodies asa result of a temperature difference.
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The science of heat transfer seeks
to predict the rateat which the heatexchange will take place under
certain specified conditions.
Thus, heat transfer rate is the
desired objective
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Difference between Heat Transferand Thermodynamics
Thermodynamics deals with the systems inequilibrium.
Heat transfer predicts how fast change of asystem from one state to another will takeplace.
Heat transfer supplements First and SecondLaws of Thermodynamics with additionalexperimental rules.
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Consider the cooling of a hot steelbar placed in a jar full of water __
Thermodynamics may be used to predictthe final equilibrium temperature of the
steel bar water combination.
Thermodynamics will not tell us how longit takes to reach this equilibriumcondition, or what the temperature ofthe bar will e after a certain length oftime before equilibrium is achieved.
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The three modes of heat transfer are:
Conduction,
convection,
and radiation.
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x
T
A
q
x
TkAq
Conduction heat transfer:When a temperature gradient exists in a
body, there is an energy transfer from the
high-temperature region to the low-temperature region.
The heat transfer rate per unit area isproportional to the normal temperature
gradient,Here, q is the heat transfer rate
and
is temperature gradient in the
direction of heat flow.
k is a positive constant, called
thermal con duct iv i tyof the
material, W/m/OC
x
T
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Three-dimensional analysis:
Heat conduction in and out of a unit volume in all threedirections: Energy balance
Total heat conducted in to the system+ the heatgenerated within the system= the total heatconducted out of the system+ the change ininternal energy of the system
E
qqqqqqq dzzdyydxxgenzyx
x
T
kdydzqx
dydzdx
x
Tk
xx
Tkq dxx
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Cartesian Coordinates:
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y
Tkdxdzqy
dxdzdyy
Tk
yy
Tkq dyy
z
Tkdxdyqz
dxdydz
z
Tk
zz
Tkq dzz
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dxdydzqqgen
T
Cdxdydzd
dE
T
Cqz
Tk
zy
Tk
yx
Tk
x
Substituting the above all in the energy balance equation
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T
kq
zT
yT
xT 1
2
2
2
2
2
2
C
k
For constant thermal conductivity,
Here,
is thermal diffusivity.
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Special Cases:
Steady state one-dimensional heat flow (no heat generation)
02
2
x
T (1.4)
Two-dimensional steady state conduction without heat
sources(1.7)
02
2
2
2
y
T
x
T
Steady state one-dimensional heat flow with heat sources
(1.6)
02
2
k
q
x
T
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Steady state one-dimensional heat flow incylindrical coordinates
01
2
2
dr
dT
rr
T (1.5)
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Thermal conductivity
The Mechanism of Thermal Conduc t iv i ty in aGas
We identify the kinetic energy of a molecule with
its temperature. Thus, in a high-temperatureregion, the molecules have higher velocities than
in some lower temperature region. The
molecules are in continuous random motion,
colliding with one another and exchanging
energy and momentum.
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The molecules have this randommotion whether or not a temperaturegradient exists in a gas. If a moleculemoves from a high-temperature region
to a low-temperature region, ittransports the kinetic energy to thelower temperature part of the system
and gives up this energy thrucollisions with lower-energy molecules.
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In general, thermal conductivity is
strongly temperature-dependent.
The numerical value of thermal
conductivity indicates how fast heat will
flow in a given material.
The faster the molecules move, the
faster they will transport energy.
Therefore, the thermal conductivity of a
gas depends on temperature.
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Thermal conductivity, k, of a gas varies as
Tk
where T is absolute temperature. For most
gases at moderate pressures, k is a function
of temperature alone.
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Thermal energy may be conducted in
so l ids by two modes:Lattice vibrations, and
Transport by free electrons.
In good electric conductors, a large number
of free electrons move about in the lattice of
material. These electrons also carry thermalenergy from a high-temperature region to a
low-temperature region. These electrons are
thus referred to as electron gas.
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Energy may also be transported as
vibrational energy in the latticestructure of a material. But this
component is usually smaller than
electron transport.
Thus, good electrical conductors arealmost always good heat conductors.
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Thermal conduct iv i ty at 0OC
Copper (pure) 385 W/m OC
Diamond 2300
Sawdust 0.059
Glass wool 0.038
Window glass 0.78
Ice 2.22
Hg 8.21Water 0.556
Air 0.024
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Steady state one-dimensional heat flow (no heat generation)
02
2
x
T
Integrating once
dT/dx = C1 >>>>> eqn. 1
Integrating again
T = C1 * x + C2 >>>>>>> eqn. 2
Applying Boundry condtions x=0 , T=T1 and x=L , T=T2
C2 = T1 and C1 = (T2-T1)/L
Q = - KA dT/dx = K A ( T1-T2)/L
or Q = ( T1-T2)/ ( L/ KA)
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1-Dimensional Heat Conduction
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20
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A flat wall is exposed to the environment
temperature of 27C. The wall is covered with twolayers of insulation of 2.5 mm thickness each
whose thermal conductivities are 1.4 and 1.7 W/m-
K respectively. The wall loses heat to the
environment by convection. Compute the value ofthe convection heat transfer coefficient which must
be maintained on the outer surface of the insulation
to ensure that the outer surface temperature does
not exceed 41C. The innermost surface is
maintained at a temp of 70C.
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