Heat Transfer in StructuresHeat Transfer in Structures

Dr M GillieDr M Gillie

Heat TransferHeat Transfer

Fundamental to Fire Safety EngineeringFundamental to Fire Safety Engineering Three methods of heat transferThree methods of heat transfer

– Radiation - does not require matterRadiation - does not require matter– Conduction – within matter (normally solids)Conduction – within matter (normally solids)– Convection – as a result of mass transferConvection – as a result of mass transfer

CONDUCTIONCONDUCTION

Some PhysicsSome Physics

Heat flow proportional to thermal gradientHeat flow proportional to thermal gradient Heat flows from hot to coldHeat flows from hot to cold kk thermal conductivity (material property) thermal conductivity (material property) cc specific heat capacity give amount of heat specific heat capacity give amount of heat

needed to change temperature of mass needed to change temperature of mass mm by by ΔΔTT as: as:

TcmE

Fourier’s LawFourier’s Law

dx

dTkq ''Heat flux per unit area

Heat flowing from hotTo cold

Heat flow proportionalto temperature gradient

a q’’ConstantTemp

ConstantTemp

Insulated

Insulated

Steady-state conditions

A q’’T1 T2

Insulated

Steady-state conditions

Steady-state 1-d Heat FlowSteady-state 1-d Heat Flow

)(''

''

''

21

2

10

TTL

kq

Tkq

dx

dTkq

T

T

L

L

Assume total heat in bardoes not change with Time – steady state

Transient Heat Flow 1-dTransient Heat Flow 1-d

AT1T2

Insulated

L

Varying heatflow

x dx

slicehin energy witin Change

slice ofout flowingHeat

slice into flowingHeat

2

2

TρcAdxΔE

dxx

T

x

TkAdtq

x

TkAdtq

dxx

x

Transient Heat Flow 1-dTransient Heat Flow 1-d

AT1T2

Insulated

L

Varying heatflow

x dx

t

T

k

c

dx

T

dxx

T

x

TkAdt

x

TkAdtTρcAdx

2

2

2

2

CONVECTIONCONVECTION

ConvectionConvection

Heat transfer from solid to fluid as a result of Heat transfer from solid to fluid as a result of mass transfermass transfer

Can be “forced” or “natural”Can be “forced” or “natural” First studied by Newton for cooling bodiesFirst studied by Newton for cooling bodies Governed byGoverned by

)21('' TThq

Solid at T1h= convective heat transfer coefficient

Fluid atT2

hh

Convective heat transfer coefficient depends Convective heat transfer coefficient depends onon– TemperatureTemperature– Free or forced convectionFree or forced convection– TurbulenceTurbulence– GeometryGeometry– ViscosityViscosity– Etc etcEtc etc

Difficult to determine accurately. Difficult to determine accurately. “Engineering” values often used.“Engineering” values often used.

RadiationRadiation

What is Radiative Heat Transfer?What is Radiative Heat Transfer? Electromagnetic radiation emitted on account Electromagnetic radiation emitted on account

of a body’s temperatureof a body’s temperature Requires no medium for transferRequires no medium for transfer Only a small portion of spectrum transmits Only a small portion of spectrum transmits

heat (0.1-100heat (0.1-100uum)m)

Preliminaries – Absolute Preliminaries – Absolute TemperatureTemperature

Absolute temperature needed for Absolute temperature needed for radiative heat transfer problemsradiative heat transfer problems

Measured in Kelvin (K)Measured in Kelvin (K) 0 K at “Absolute 0” - all atomic motion 0 K at “Absolute 0” - all atomic motion

ceasesceases A change of 1K equals a change of 1A change of 1K equals a change of 1ººCC 0 0 ººC equals 273.15KC equals 273.15K

Preliminaries – “Black bodies”Preliminaries – “Black bodies”

Black bodies are hypothetical but useful Black bodies are hypothetical but useful for analysis of radiationfor analysis of radiation

Absorb all incoming radiationAbsorb all incoming radiation No body can emit more radiation at a No body can emit more radiation at a

given temperature and wavelengthgiven temperature and wavelength Are diffuse emittersAre diffuse emitters The Sun is very close to being a black The Sun is very close to being a black

bodybody

Stefan-Boltzmann EquationStefan-Boltzmann Equation

emissivity theis where

inresult bodiesGrey constant.Boltzmann -Stefan theis where

or

15

2''

in results wrt integratedwhen

body-black diffuse afor 1

2

4

4

432

45

/

52

TE

TE

Thc

kqE

e

hcE

KTch

4TE

4TE

Stefan-Boltzmann Equation in Stefan-Boltzmann Equation in ActionAction

Hot surface

Each “piece” of area emitsuniformly in all directions according to E=εσT4

Question: What is the net incident radiation arriving at B?

AA

B

T1

T2

Stefan-Boltzmann Equation in Stefan-Boltzmann Equation in ActionAction

Question: What is the net incident radiation arriving at B?

A

B

T1

T2

Answer depends on

-The relative temperatures A and Bradiation is a two way process

-The geometry of the system – configuration factors

d

Some radiation“escapes” and doesnot reach B

Configuration FactorsConfiguration Factors

Take account of the geometry of radiating Take account of the geometry of radiating bodiesbodies

Allow calculation of net radiation arriving at a Allow calculation of net radiation arriving at a surfacesurface

Calculation involves much integration – only Calculation involves much integration – only possible for simple casespossible for simple cases

Details not needed for this courseDetails not needed for this course Two kindsTwo kinds

– Point to surface (eg fire to ceiling)Point to surface (eg fire to ceiling)– Surface to surface (e.g. smoke layer to ceiling)Surface to surface (e.g. smoke layer to ceiling)

For compartment firesFor compartment fires

Fire compartment

Local fire or flashed over fire

Thick layer of hot gas, opaque

Hot gases are radiating and soCeiling “sees” all of the area of The room. Therefore configurationfactor~1.

Heat Transfer to Steel Heat Transfer to Steel StructuresStructures

Several cases - insulated, uninsulated Several cases - insulated, uninsulated etcetc

Simple solution methods presentedSimple solution methods presented More advanced solutions possible but More advanced solutions possible but

require LOTS more analysisrequire LOTS more analysis Approach is to make conservative Approach is to make conservative

assumptionsassumptions

Un-insulated SteelUn-insulated Steel

Assume constant temperature in cross-Assume constant temperature in cross-sectionsection– lumped capacitancelumped capacitance

Apply energy balance to the problemApply energy balance to the problem Solve for small time-steps to get Solve for small time-steps to get

approximate solutionapproximate solution Involves use of radiative Involves use of radiative andand convective convective

heat transfer equationsheat transfer equations

Un-insulated steelUn-insulated steel Heat flowing into a unit length of Heat flowing into a unit length of

section in time section in time ΔΔt it is equal to the s equal to the energy stored in the sectionenergy stored in the section

)()( 44sgsg TTTTh

Assume steelat uniformtemperature, Ts

Convection andradiation to steel fromgas at Tg

Perimeter=HArea=A

1*1*'' TActHq

Substituting and rearranging results

44sgsgs TTTTh

c

t

A

HT

q’’ consists of two parts

convection radiation

Section FactorsSection Factors

Give a measure of how rapidly a section Give a measure of how rapidly a section heatsheats

Normally ratio of heated perimeter to Normally ratio of heated perimeter to areaarea

Given in some tablesGiven in some tables Various other measures and symbols Various other measures and symbols

usedused– Area to volumeArea to volume

Insulated Steel-Sections (1)Insulated Steel-Sections (1)

Insulation has no Insulation has no thermal capacity thermal capacity (e.g. intumescent (e.g. intumescent paint)paint)

Same temp as gas at Same temp as gas at outer surfaceouter surface

Same temp as steel Same temp as steel at inner surfaceat inner surface

Therefore conduction Therefore conduction problemproblem

Perimeter=HArea=A

Insulated Steel-Sections (1)Insulated Steel-Sections (1)

Energy balance Energy balance approach used againapproach used again

TcAtHq ''

q’’ now as a result of conduction only

sg TTd

kq ''

Which gives

tTTA

H

cd

kT sg

Perimeter=HArea=A

Insulation thickness d

Insulated Steel-Sections (2)Insulated Steel-Sections (2)

Insulation has no Insulation has no thermal capacity thermal capacity (e.g. cementious (e.g. cementious spray)spray)

Assume linear Assume linear temperature gradient temperature gradient in insulationin insulation

Perimeter=HArea=A

Insulated Steel-Sections (2)Insulated Steel-Sections (2)

Energy balance Energy balance approach used againapproach used again

dHT

cTActHq siiss 2

''

q’’ now

sg TTd

kq ''

Which gives

tTT

AcHd

c

c

cd

k

A

HT sg

iiss

ss

ss

2

Perimeter=HArea=A

Insulation thickness d

energy ininsulation

Heat Transfer in ConcreteHeat Transfer in Concrete

Very large thermal capacityVery large thermal capacity– Heat slowlyHeat slowly– Lumped mass approach not appropriateLumped mass approach not appropriate

Complicated by water present in concreteComplicated by water present in concrete Usually need computer analysis for non-Usually need computer analysis for non-

standard situationstandard situation Results are published for Standard Fire Results are published for Standard Fire

TestsTests

Heat Heat penetration penetration in concrete in concrete

beamsbeams

Heat penetration in concrete Heat penetration in concrete slabsslabs

(mm)