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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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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1-steady state heat conduction

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