heat transfer in structures dr m gillie. heat transfer fundamental to fire safety engineering...
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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)