heat transfer (good too)

8/3/2019 Heat Transfer (Good Too)

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Weeks 5-6

Chapters 17, 18-1CONDUCTION

ME 354 Kumpaty


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STEADY HEAT CONDUCTION IN PLANE WALLS

Heat transfer through a wallis one-dimensional whenthe temperature of the wall

varies in one direction only.

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 resistance electrical 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 thethermal propertiesof the medium.

Electrical 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 thesurface against radiation.

Schematic forconvection and radiationresistances at a surface.

Radiation heat transfer coefficient

Combined heat transfercoefficient


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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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U 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.

MultilayerPlaneWalls


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THERMAL CONTACT RESISTANCE

Temperature distribution and heat flow lines along two solid platespressed against each other for the case of perfect and imperfect contact.


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hc 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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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 can

be 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 outer

surface temperatures T1 and T2.

Conduction resistanceof the cylinder layer


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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

for a sphericallayer


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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 INSULATIONAdding more insulation to a wall orto the attic always decreases heattransfer since the heat transfer areais constant, and adding insulationalways increases the thermalresistance of the wall withoutincreasing the convectionresistance.

In a 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 effectdominates.

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


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HEAT TRANSFER FROM FINNED SURFACES

When Tsand T are fixed, two waysto increase the rate of heat transfer are To increase the convection heat transfer coefficient h. This may require the

installation of a pump or fan, or replacing the existing one with a larger one, butthis approach may or may not be practical. Besides, it may not be adequate.

To increase the surface area Asby attaching to the surface extended surfacescalled finsmade of highly conductive materials such as aluminum.

Newtons law of cooling: The rate of heat transfer froma surface to the surrounding medium

The thin plate fins of a car radiator greatlyincrease the rate of heat transfer to the air.

Some innovative fin designs.


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Fin Equation

Volume element of a fin at location xhaving a length of x, cross-sectional

area of Ac, and perimeter of p.

Differentialequation

Temperature

excess


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The general solution of thedifferential equation

Boundary condition at fin base

Boundary conditions at the finbase and the fin tip.

1 Infinitely Long Fin(Tfin tip = T)

Boundary condition at fin tip

The variation of temperature along the fin

The steady rate of heat transferfrom the entire fin


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A long circular fin of uniformcross section and the variation

of temperature along it.

Under steady conditions, heattransfer from the exposed surfacesof the fin is equal to heat conductionto the fin at the base.

The rate of heat transfer from the fin could alsobe determined by considering heat transfer froma differential volume element of the fin andintegrating it over the entire surface of the fin:


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2 Negligible Heat Loss from the Fin Tip(Adiabatic fin tip, Qfin tip = 0)

Boundary condition at fin tip

The variation of temperature along the fin

Heat transfer from the entire fin

Fins are not likely to be so long that their temperature approaches the

surrounding temperature at the tip. A more realistic assumption is forheat transfer from the fin tip to be negligible since the surface area ofthe fin tip is usually a negligible fraction of the total fin area.


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3 Convection (or CombinedConvection and Radiation)from Fin Tip

A practical way of accounting for theheat loss from the fin tip is to replacethe fin length L in the relation for theinsulated tipcase by a correctedlength defined as

Corrected fin length Lc is defined suchthat heat transfer from a fin of length Lcwith insulated tip is equal to heat transferfrom the actual fin of length L with

convection at the fin tip.

tthe thickness of the rectangular finsDthe diameter of the cylindrical fins


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Fin Efficiency

Fins enhance heattransfer from

a surface byenhancing surfacearea.

Ideal and actualtemperature

distributionalong a fin.


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Zero thermal resistanceor infinitethermal conductivity(Tfin = Tb)


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Efficiency of straight fins of rectangular, triangular, and parabolic profiles.


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Efficiency of annular fins of constant thickness t.


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Fins with triangular and parabolic profiles contain less materialand are more efficient than the ones with rectangular profiles.

The fin efficiency decreases with increasing fin length. Why?

How to choose fin length? Increasing the length of the finbeyond a certain value cannot be justified unless the addedbenefits outweigh the added cost.

Fin lengths that cause the fin efficiency to drop below 60 percentusually cannot be justified economically.

The efficiency of most fins used in practice is above 90 percent.


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FinEffectiveness

Theeffectiveness

of a fin

The thermal conductivitykof the finshould be as high as possible. Usealuminum, copper, iron.

The ratio of the perimeterto the cross-sectional areaof the fin p/Acshould beas high as possible. Use slender pin fins.

Low convection heat transfer coefficienth. Place fins on gas (air) side.


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Various surface areas associatedwith a rectangular surface with

three fins.

Overall effectiveness for a finned surface

The overall fin effectiveness dependson the fin density (number of fins per

unit length) as well as theeffectiveness of the individual fins.

The overall effectiveness is a bettermeasure of the performance of afinned surface than the effectiveness

of the individual fins.

The total rate of heat transfer from afinned surface

P L th f Fi


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Proper Length of a Fin

Because of the gradual temperature dropalong the fin, the region near the fin tip makeslittle or no contribution to heat transfer.

mL =5 an infinitely long finmL = 1 offer a good compromisebetween heat transfer

performance and the fin size.


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Heat sinks: Speciallydesigned finned surfaceswhich are commonly used inthe cooling of electronicequipment, and involve one-of-a-kind complexgeometries.

The heat transferperformance of heat sinks isusually expressed in terms of

their thermal resistances R. A small value of thermal

resistance indicates a smalltemperature drop across theheat sink, and thus a high finefficiency.


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TRANSIENT: LUMPED SYSTEM ANALYSIS

Interior temperature of somebodies remains essentially

uniform at all times during aheat transfer process.

The temperature of suchbodies can be taken to be afunction of time only, T(t).

Heat transfer analysis thatutilizes this idealization isknown as lumped systemanalysis.

A small copper ballcan be modeled as a

lumped system, but

a roast beef cannot.


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Integrating withT = Ti at t =0T = T(t) at t = t

The geometry andparameters involved in thelumped system analysis.

timeconstant


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The temperature of a lumped systemapproaches the environmenttemperature as time gets larger.

This equation enables us todetermine the temperatureT(t) of a body at time t, oralternatively, the time trequired for the temperatureto reach a specified value T(t).

The temperature of a bodyapproaches the ambienttemperature T

exponentially.

The temperature of the bodychanges rapidly at thebeginning, but rather slowlylater on. A large value of bindicates that the bodyapproaches the environmenttemperature in a short time


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Heat transfer to or from abody reaches its

maximum value when thebody reaches the

environment temperature.

The rateof convection heattransfer between the bodyand its environment at time t

The total amountof heat transferbetween the body and the surroundingmedium over the time interval t =0 to t

The maximumheat transfer betweenthe body and its surroundings

Criteria for Lumped System Analysis


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Criteria for Lumped System Analysis

Lumped system analysisis applicableif

When Bi 0.1, the temperatureswithin the body relative to thesurroundings (i.e., TT

) remain

within 5 percent of each other.

Characteristiclength

Biot number


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Small bodies with highthermal conductivitiesand low convection

coefficients are mostlikely to satisfy thecriterion for lumpedsystem analysis.

When the convection coefficient hishigh and kis low, large temperature

differences occur between the innerand outer regions of a large solid.

heat transfer (good too)

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