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:

Fourier’s Law Heat flux per unit area Heat flow proportional to temperature gradient Heat flowing from hot To cold Insulated a Constant Temp Constant Temp q’’ Steady-state conditions Insulated

Steady-state 1-d Heat Flow Assume total heat in bar does not change with Time – steady state Insulated A T1 q’’ Steady-state conditions T2 L

Transient Heat Flow 1-d Insulated A T1 T2 Varying heat flow x dx L

Transient Heat Flow 1-d Insulated A T1 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 Solid at T1 h= convective heat transfer coefficient

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 (0.1-100um)

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

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 B T2 A 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 -The geometry of the system – configuration factors B T2 d Some radiation “escapes” and does not reach B A T1

Configuration Factors 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 Hot gases are radiating and so Ceiling “sees” all of the area of The room. Therefore configuration factor~1. Local fire or flashed over fire

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 q’’ consists of two parts Heat flowing into a unit length of section in time Δt is equal to the energy stored in the section Assume steel at uniform temperature, Ts Perimeter=H Area=A q’’ consists of two parts convection radiation Convection and radiation to steel from gas at Tg Substituting and rearranging results

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’’ now as a result of conduction only Perimeter=H Area=A Which gives 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 Perimeter=H Area=A q’’ now energy in insulation Insulation thickness d Which gives

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 non-standard situation Results are published for Standard Fire Tests

Heat penetration in concrete beams

Heat penetration in concrete slabs (mm)