LectureNotes03-2012 (Student Version)

8/2/2019 LectureNotes03-2012 (Student Version)

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HEAT

TRANSFER

Course Materials are based on

Heat and Mass Transfer: Fundamentals & Applications

Fourth Edition in SI Units

Yunus A. Cengel, Afshin J. GhajarMcGraw-Hill, 2011

Lecture Notes 3BDA 3063/30603


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MAIN CONTENTS OF LECTURE 3

CONDUCTION I (continued)

I. Uniform Heat Generation

II. 1-D Steady Conduction

CONDUCTION II

I. Thermal Resistance ConceptII. Composite Bodies

CONDUCTION III

I. Heat Transfer from Curved SurfaceII. Critical Thickness of Insulation


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Heat

Generation

Examples:

electrical energy being converted to heat at a rate ofI2R

fuel elements of nuclear reactors

exothermic chemical reactions

Heat generation is a volumetric phenomenon.

The rate of heat generation unit: W/m3 or Btu/hft3.

The rate of heat generation in a medium may vary with time as well as

position within the medium.


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ONE-DIMENSIONAL HEAT CONDUCTIONEQUATION

Consider heat conduction through a large plane wall such as the wall of ahouse, the glass of a single pane window, the metal plate at the bottom ofa pressing iron, a cast-iron steam pipe, a cylindrical nuclear fuel element,an electrical resistance wire, the wall of a spherical container, or aspherical metal ball that is being quenched or tempered.

Heat conduction in these and many other geometries can beapproximated as being one-dimensionalsince heat conduction throughthese geometries is dominant in one direction and negligible in otherdirections.

Next we develop the one-dimensional heat conduction equation inrectangular, cylindrical, and spherical coordinates.


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(2-6)

Heat ConductionEquation in a LargePlane Wall


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HeatConductionEquation in a

Long Cylinder


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Heat Conduction Equationin a Sphere


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Combined One-Dimensional Heat ConductionEquation

An examination of the one-dimensional transient heat conductionequations for the plane wall, cylinder, and sphere reveals that allthree equations can be expressed in a compact form as

n =0 for a plane wall

n =1 for a cylinder

n =2 for a sphere

In the case of a plane wall, it is customary to replace the variabler by x.

This equation can be simplified for steady-state or no heatgeneration cases as described before.


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STEADY HEAT CONDUCTION IN PLANE WALLS

for steadyoperation

In steady operation, the rate of heat transferthrough the wall is constant.

Fouriers law of

heat conduction

Heat transfer through the wall of a house can bemodeled as steadyand one-dimensional.

The temperature of the wall in this case dependson one direction only (say the x-direction) andcan be expressed as T(x).


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Under steady conditions, thetemperature distribution in a planewall is a straight line: dT/dx =const.

The rate of heat conduction througha plane wall is proportional to theaverage thermal conductivity, thewall area, and the temperaturedifference, but is inverselyproportional to the wall thickness.

Once the rate of heat conduction isavailable, the temperature T(x) atany location xcan be determined byreplacing T2 by T, and L by x.


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Analogy between thermal and electricalresistance concepts.

rate of heat transfer electric currentthermal resistanceelectrical resistancetemperature difference voltage difference

Thermal Resistance Concept

Conduction resistance of thewall:Thermal resistanceof thewall against heat conduction.

Thermal resistance of a mediumdepends on the geometryand the

thermal propertiesof the medium.

which is theelectric resistance.


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Schematic for convection resistance at a surface.

Newtons law of cooling

Convection resistance of thesurface:Thermal resistanceof thesurface against heat convection.

When the convection heat transfer coefficient is very large (h ),the convection resistance becomes zeroand TsT.

That is, the surface offers no resistance to convection, and thus itdoes not slow down the heat transfer process.

This situation is approached in practice at surfaces where boiling

and condensation occur.


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Radiation resistance of the

surface:Thermal resistanceof asurface against radiation.

Radiation heat transfer coefficient

where hcombined is thecombined heat transfercoefficient.

Schematic for convection and

radiation resistances at a surface.


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Thermal Resistance Network

The thermal resistance network for heat transfer through a plane wall subjected toconvection on both sides, and the electrical analogy.


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Uis the overall heattransfer coefficient

Once Qis evaluated, thesurface temperature T1 canbe determined from

Temperature drop

The temperature drop across a layer isproportional to its thermal resistance.


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The thermal resistancenetwork for heat transferthrough a two-layer planewall subjected toconvection on both sides.

Multilayer Plane Walls (Composite Bodies)


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THERMAL CONTACT RESISTANCE

Temperature distribution and heat flow lines along two solid plates

pressed against each other for the case of perfect and imperfect contact.


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A typical experimentalsetup for the

determination of thermalcontact resistance.

When two such surfaces arepressed against each other, thepeaks form good materialcontact but the valleys form

voids filled with air.

These numerous air gapsofvarying sizes act as insulationbecause of the low thermalconductivity of air.

Thus, an interface offers someresistance to heat transfer, andthis resistance per unit interfacearea is called the thermalcontact resistance, Rc.


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hc is called the thermal

contact conductance

The value of thermalcontact resistancedepends on:

surface roughness material properties temperatureand

pressureat theinterface

type of fluidtrapped

at the interface

Thermal contact resistance is significant and can even dominate theheat transfer for good heat conductors such as metals, but can be

disregarded for poor heat conductors such as insulations.


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Effect of metallic coatings on

thermal contact conductance.

The thermal contact resistance canbe minimized by applying

a thermal greasesuch as silicon oil

a better conducting gassuch ashelium or hydrogen

a soft metallic foilsuch as tin, silver,

copper, nickel, or aluminum


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The thermal contact conductanceis highest(and thus the contactresistance is lowest) for soft metalswith smooth surfacesat high pressure.


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GENERALIZED THERMAL RESISTANCE NETWORKS

Thermalresistance

network for two

parallel layers.


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Thermal resistance network forcombined series-parallel

arrangement.

Two assumptions in solving complexmultidimensional heat transferproblems by treating them as one-dimensional using the thermalresistance network are

(1) any plane wall normal to the x-axis is

isothermal(i.e., to assume thetemperature to vary in the x-directiononly)

(2) any plane parallel to the x-axis isadiabatic(i.e., to assume heat transferto occur in the x-direction only)

Do they give the same result?


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HEAT CONDUCTION IN CYLINDERS AND SPHERES

Heat is lost from a hot-water pipe tothe air outside in the radial direction,and thus heat transfer from a longpipe is one-dimensional.

Heat transfer through the pipecan be modeled as steady

and one-dimensional.

The temperature of the pipedepends on one direction only(the radial r-direction) and canbe expressed as T =T(r).

The temperature isindependent of the azimuthalangle or the axial distance.

This situation is approximated

in practice in long cylindricalpipes and sphericalcontainers.


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A long cylindrical pipe (or sphericalshell) with specified inner and outersurface temperatures T

1and T

2.

is the conduction resistanceof the cylinder layer.


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is the conduction resistanceof the spherical layer.

A spherical shellwith specifiedinner and outer

surfacetemperatures T1and T2.


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The thermal resistancenetwork for a cylindrical (orspherical) shell subjectedto convection from both theinner and the outer sides.

for a cylindricallayer, and

for a sphericallayer

where


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Multilayered Cylinders and Spheres

The thermal resistancenetwork for heat transferthrough a three-layeredcomposite cylindersubjected to convectionon both sides.


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Once heat transfer rate Qhas beencalculated, the interface temperatureT2 can be determined from any of the

following two relations:


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CRITICAL RADIUS OF INSULATION

Adding more insulation to a wall orto the attic always decreases heat

transfer since the heat transfer areais constant, and adding insulationalways increases the thermalresistance of the wall withoutincreasing the convectionresistance.

In a cylindrical pipe or a sphericalshell, the additional insulationincreases the conductionresistance of the insulation layerbut decreases the convectionresistance of the surface because

of the increase in the outer surfacearea for convection.

The heat transfer from the pipemay increase or decrease,depending on which effect

dominates.

An insulated cylindrical pipe exposed toconvection from the outer surface and

the thermal resistance networkassociated with it.


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The critical radius of insulationfor a cylindrical body:

The critical radius of insulationfor a spherical shell:

The variation of heat transferrate with the outer radius of theinsulation r2 when r1 < rcr.

We can insulate hot-water orsteam pipes freely withoutworrying about the possibility ofincreasing the heat transfer by

insulating the pipes

The largest value of the criticalradius we are likely toencounter is

LectureNotes03-2012 (Student Version)

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