Transcript Heat Transfer in Structures Dr M Gillie

Heat Transfer in Structures
Dr M Gillie
Heat Transfer
Fundamental to Fire Safety Engineering
 Three methods of heat transfer

– Radiation - does not require matter
– Conduction – within matter (normally solids)
– Convection – as a result of mass transfer
CONDUCTION
Some Physics
Heat flow proportional to thermal gradient
 Heat flows from hot to cold
 k thermal conductivity (material property)
 c specific heat capacity give amount of heat
needed to change temperature of mass m by ΔT
as:

E  cmT
Fourier’s Law
Heat flux per unit area
dT
q ' '   k
dx
Heat flow proportional
to temperature gradient
Heat flowing from hot
To cold
Insulated
Constant
Temp
a
q’’
Steady-state conditions
Insulated
Constant
Temp
Steady-state 1-d Heat Flow
dT
q ' '   k
dx
 q ' '0  k T T 1
L
Assume total heat in bar
does not change with
Time – steady state
T2
k
q ' '  (T 1  T 2)
L
Insulated
T1
A
q’’
Steady-state conditions
L
T2
Transient Heat Flow 1-d
T
q x   kAdt
Heat flowing into slice
x
 T  2T 
q x  dx   kAdt
 2 dx  Heat flowing out of slice
 x x

ΔE  ρcAdxT
Change in energy wit hin slice
Insulated
T1
A
T2
Varying heat
flow
x
dx
L
Transient Heat Flow 1-d
 T  2T 
T
ρcAdxT  kAdt
 kAdt
 2 dx 
x
 x x

 2T c T

2
dx
k t
Insulated
T1
A
T2
Varying heat
flow
x
dx
L
CONVECTION
Convection




Heat transfer from solid to fluid as a result of mass
transfer
Can be “forced” or “natural”
First studied by Newton for cooling bodies
Governed by
Fluid at
T2
q ' '  h(T 1  T 2 )
h= convective heat transfer coefficient
Solid at T1
h

Convective heat transfer coefficient depends on
–
–
–
–
–
–

Temperature
Free or forced convection
Turbulence
Geometry
Viscosity
Etc etc
Difficult to determine accurately. “Engineering”
values often used.
Radiation
What is Radiative Heat Transfer?
Electromagnetic radiation emitted on account of a
body’s temperature
 Requires no medium for transfer
 Only a small portion of spectrum transmits heat (0.1100um)

Preliminaries – Absolute Temperature
Absolute temperature needed for radiative
heat transfer problems
 Measured in Kelvin (K)
 0 K at “Absolute 0” - all atomic motion ceases
 A change of 1K equals a change of 1ºC
 0 ºC equals 273.15K

Preliminaries – “Black bodies”
Black bodies are hypothetical but useful for
analysis of radiation
 Absorb all incoming radiation
 No body can emit more radiation at a given
temperature and wavelength
 Are diffuse emitters
 The Sun is very close to being a black body

Stefan-Boltzmann Equation
2c 2 h5
E  ch / KT
for a diffuse black - body
e
1
when integrated wrt  results in
 2 5 k 4  4
T
E  q ' '  
2 3 
 15c h 
or
E  T 4
where  is the Stefan - Boltzmann constant. Grey bodies result in
E  T 4
where  is the emissivity
E  T 4
E  T 4
Stefan-Boltzmann Equation in Action
Question: What is the net incident
radiation arriving at B?
Each “piece” of area emits
uniformly in all directions
according to E=εσT4
A
B
T2
A
T1
Hot surface
Stefan-Boltzmann Equation in Action
Question: What is the net incident
radiation arriving at B?
Answer depends on
-The relative temperatures A and B
radiation is a two way process
B
-The geometry of the system –
configuration factors
T2
d
Some radiation
“escapes” and does
not reach B
A
T1
Configuration Factors

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

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Take account of the geometry of radiating bodies
Allow calculation of net radiation arriving at a surface
Calculation involves much integration – only possible
for simple cases
Details not needed for this course
Two kinds
– Point to surface (eg fire to ceiling)
– Surface to surface (e.g. smoke layer to ceiling)
For compartment fires
Thick layer of hot gas, opaque
Fire compartment
Local fire or
flashed over fire
Hot gases are radiating and so
Ceiling “sees” all of the area of
The room. Therefore configuration
factor~1.
Heat Transfer to Steel Structures
Several cases - insulated, uninsulated etc
 Simple solution methods presented
 More advanced solutions possible but require
LOTS more analysis
 Approach is to make conservative assumptions

Un-insulated Steel

Assume constant temperature in cross-section
– lumped capacitance
Apply energy balance to the problem
 Solve for small time-steps to get approximate
solution
 Involves use of radiative and convective heat
transfer equations

Un-insulated steel
Heat flowing into a unit length of
section in time Δt is equal to the
energy stored in the section
q ' ' Ht *1  cTA *1
Assume steel
at uniform
temperature, Ts
q’’ consists of two parts
Perimeter=H
Area=A
h(Tg  Ts)   (Tg4  Ts4 )
convection
radiation
Substituting and rearranging results

H t
Ts 
hTg  Ts    Tg4  Ts4
A c


Convection and
radiation to steel from
gas at Tg
Section Factors
Give a measure of how rapidly a section heats
 Normally ratio of heated perimeter to area
 Given in some tables
 Various other measures and symbols used

– Area to volume
Insulated Steel-Sections (1)
Insulation has no
thermal capacity (e.g.
intumescent paint)
 Same temp as gas at
outer surface
 Same temp as steel at
inner surface
 Therefore conduction
problem

Perimeter=H
Area=A
Insulated Steel-Sections (1)
Energy balance approach
used again
q ' ' Ht  cAT
q’’ now as a result of conduction only
Perimeter=H
Area=A
k
q ' '  Tg  Ts 
d
Which gives
 k H

Tg  Ts t
T  
 dc A

Insulation thickness d
Insulated Steel-Sections (2)
Insulation has no
thermal capacity (e.g.
cementious spray)
 Assume linear
temperature gradient in
insulation

Perimeter=H
Area=A
Insulated Steel-Sections (2)
Energy balance approach
used again
q ' ' Ht   s cs AT   i ci
q’’ now
Ts
dH
2
Perimeter=H
Area=A
energy in
insulation
k
q ' '  Tg  Ts 
d
Which gives
Insulation thickness d
 s cs
H k

T 
Tg  Ts t
Hd  i ci 
A d s c s 
  s cs 

2A 

Heat Transfer in Concrete

Very large thermal capacity
– Heat slowly
– Lumped mass approach not appropriate
Complicated by water present in concrete
 Usually need computer analysis for nonstandard situation
 Results are published for Standard Fire Tests

Heat
penetration
in concrete
beams
Heat penetration in concrete slabs
(mm)