Radiation Heat Transfer

Abstract

Thermal radiation is the dominant mode of heat transfer in flames with characteristic lengths exceeding approximately 0.2 m. It is for this reason that quantitative analysis of fire dynamics requires a working knowledge of thermal radiation. This chapter will introduce the fundamentals of thermal radiation and offer several methods for calculating radiant heat transfer in fires. Basic thermal radiation concepts are presented with an emphasis on application to fire phenomena; the reader is referred the literature for specialized topics [1–4].

Nomenclature, Greek Symbols and Subscripts

A

Area (m²)

C

Correction factor for mean beam length

C ₀

Soot concentration parameter

C ₂

Planck’s second constant (1.4388 × 10⁻² m · K)

c

Speed of light in the medium (m/s)

c ₀

Speed of light in a vacuum (2.998 × 10⁸ m/s)

E

Radiative emissive power (W/m²)

F _i−j

Configuration factor from surface i to surface j

f _v

Soot volume fraction

G

Irradiation or radiative heat flux received by surface (W/m2)

H

Height (m)

h

Planck’s constant (6.6256 × 10⁻³⁴ J s)

I

Radiation intensity (W/m²)

$\overrightarrow{\boldsymbol{i}}$, $\overrightarrow{\boldsymbol{j}}$, $\overrightarrow{\boldsymbol{k}}$

Cartesian coordinate direction vectors

J

Radiosity or radiative heat flux leaving surface (W/m²)

k

Boltzmann constant (1.3806 × 10⁻²³ J/K), or infrared optical constant of soot (imaginary component), or thermal conductivity (W/m K)

L

Mean beam length or distance (m)

L ₀

Geometrical mean beam length (m)

n

Index of refraction (c₀/c) or infrared optical constant of soot (real component)

$\overrightarrow{\boldsymbol{n}}$

Unit normal vector

P _a

Partial pressure of absorbing gas (Pa)

P _e

Effective pressure (Pa)

Q

Energy rate (W)

q˙″

Heat flux (W/m²)

$\overrightarrow{R}$

Line of sight vector

r

Radius of cylinder (m)

S

Pathlength (m)

T

Temperature (K)

t

Time (s)

u, v, w

Cartesian components of unit vector $\overrightarrow{n}$

V

Volume (m³)

X

Pressure pathlength, ${\displaystyle {\int}_0^s{P}_ax\left(\xi \right)d\left(\xi \right)}$ (atm-m)

x

Spatial coordinate (m)

α

Absorptivity or thermal diffusivity k/pc_p (m²/s)

β

Angle from normal (radians)

ε

Emissivity

θ

Polar angle (radians)

κ

Extinction coefficient or absorption coefficient (m − l)

λ

Wavelength (m)

μ

Micron (10⁻⁶ m)

μ_λ

Defined parameter, Equation 4.73

ν

Frequency (s^−t)

ξ

Integration dummy variable

ρ

Reflectivity or density (kg/m³)

Ω

Solid angle (steradians)

σ

Stefan-Boltzmann constant (5.6696 × 10^−B W/m²K⁴)

τ

Transmissivity or optical pathlength

ϕ

Azimuthal angle (radians)

χ

Fractional measure

a

Actual

b

Blackbody or base

e

External

f

Flame

g

Gas

i

Initial or i^th surface

j

Summation variable or j^th surface

m

Mean value

0

Original

P

Planck mean

R

Rosseland mean

s

Surface or soot

t

Total

w

Wall

λ

Spectral wavelength

ν

Spectral frequency

∞

Ambient