Radiation Heat Transfer

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

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Radiosity• The radiosity of a surface is the rate at which

radiation energy leaves a surface per unit area.

sincos,, )(2

0

2/

0

,

ddIJ re

Spectral Radiosity:

dsincos,, 0

2

0

2/

0

,

ddIJ re

Total Radiosity

Radiative Heat Transfer Consider the heat transfer between two black surfaces, as shown in Figure. What is the rate of heat transfer into Surface B? To find this, we will first look at the emission from A to B. Surface A emits radiation as described in

4, AAAemittedA TAq

This radiation is emitted in all directions, and only a fraction of it will actually strike Surface B. This fraction is called the shape factor, F.

The amount of radiation striking Surface B is therefore:

4, AAABAincidentB TAFq

All the incident radiation will contribute to heating of Surface B :

4, AABAabsorbedB TAFq

Above equation is the amount of radiation gained by Surface B from Surface A. To find the net heat transfer rate at B, we must now subtract the amount of radiation emitted by B:

4, BBemittedB TAq

The net radiative heat transfer (gain) rate at Surface B is

emittedBabsorbedBB qqq ,,

44BBAABAB TATAFq

Similarly, the net radiative heat transfer (loss) rate at Surface A is

44AABBABA TATAFq

What is the relation between qA and qB ?

Shape Factors • Shape factor, F, is a geometrical

factor which is determined by the shapes and relative locations of two surfaces.

• Figure illustrates this for a simple case of cylindrical source and planar surface.

• Both the cylinder and the plate are infinite in length.

• In this case, it is easy to see that the shape factor is reduced as the distance between the source and plane increases.

• The shape factor for this simple geometry is simply the cone angle (θ) divided by 2π

Geometrical Concepts in Radiation Heat Transfer

Human Shape Factors

Wherever artificial climates are created for human occupation, the aim of the design is that individuals experience thermal comfort in the environment.

Among other factors thermal comfort depends on mean radiant temperature.

Flame to Furnace Wall Shape Factors

Radiative Heat Exchange between Two Differential Area Elements

• The elements dAi and dAj are isothermal at temperatures Ti and Tj respectively.

• The normals of these elements are at angles i and j respectively to their common normal.

• The total energy per unit time leaving dAi and incident upon dAj is:

iiiibji dAdIQd cos,2

2

cos

r

dAd jj

i

r

jdA

idA

i

j

in

jn

di is the solid angle subtended by dAj when viewed from dAi.

r

jdA

idA

i

j

in

jn

The monochromatic energy per unit timeleaving dAi and incident on dAj is

ddAdIQd iiiibji cos,,3

•The total energy per unit time leaving dAi and incident upon dAj is:

iiiibji dAdIQd cos,2

The monochromatic energy per unit time leaving dAi and incident on dAj is:

ddAdIQd iiiibji cos,,3

0

,

0

,3

,2 cos ddAdIQdQd iiiibjiji

The monochromatic energy per unit time leaving A real body element dAi and incident on dAj is:

ddAdIQd iiiibiji cos,,3

0

,

0

,32 cos ddAdIQdQd iiiibijiji

2

cos

r

dAd jj

i

2

,,

2 coscos

r

dAdAIQd ijjiib

jib

2

,,

2 coscos

r

dAdAIQd jiijjb

ijb

r

jdA

idA

i

j

in

jn

The fraction of energy leaving a black surface element dAi that arrive at black body dAj is defined as the Geometric configuration Factor dFij.

ib

jibji dAe

QddF

,2

4TeI b

b

For a diffusive surface

ii

ijjiib

ji dATr

dAdAI

dF4

2, coscos

ii

ijjii

ji dATr

dAdAT

dF4

2

4 coscos

2

coscos

r

dAdF jji

ji

2

coscos

r

dAdF jji

ji

Configuration Factor for rate of heat Exchange from dAi to dAj

Configuration Factor for Energy Exchange from dAj to dAi

2

coscos

r

dAdF iji

ij

Reciprocity of Differential-elemental Configuration Factors

2

coscos

r

dAdAdFdA jji

ijii

Consider the products of :

2

coscos

r

dAdAdFdA iji

jijj

2

coscos

r

dAdAdFdAdFdA jiji

jiiijj

jibjibjib QdQdQd ,2

,2

,2

2

,,,

2 coscos

r

dAdAIIQd jijijbib

jib

The net energy per unit time transferred from black element dAi

to dAj along emissive path r is then the difference of i to j and j to i.

Net Rate of Heat Exchange between Two differential Black Elements

Ib of a black element =

4TeI b

b

2

44

,2 coscos

r

dAdATTQd jijiji

jib

Finally the net rate of heat transfer from dAi to dAj is:

jijjiijijijib dAdFTTdAdFTTQd 4444,

2

Configuration Factor between a Differential Element and a Finite Area

dAi, Ti

i

Aj, Tj

dAi

j

j

2

coscos

r

dAdF jji

dAdA ji

Integrating over Aj to obtain:

j

ji

A

jjiAdA r

dAF

2

coscos

j

ji

A

jji

j

iAdA r

dA

A

dAF

2

coscos

Configuration Factor for Two Finite Areas

Ai, Ti

i

Aj, Tj

dAi

j

i

A A

ijji

AA A

r

dAdA

F i j

ji

2

coscos

i

A A

ijji

AA A

r

dAdA

F i j

ji

2

coscos

j

A A

ijji

AA A

r

dAdA

F i j

ij

2

coscos

ijji AAjAAi FAFA

Radiation Exchange between Two Finite Areas

jiiiji FATQ 4

ijjjij FATQ 4

The net rate of radiative heat exchange between Ai and Aj

ijjjjiiiijjiji FATFATQQQ 44

ijjjjiiiji FATFATQ 44

Using reciprocity theorem:

44jijiiji TTFAQ

44jiijjji TTFAQ

Configuration Factor Relation for An Enclosure

2 2

,

0 0 0

( , , ) cos sineJ I d d d

T1,A1

T2,A1

Ti,Ai

TN,AN

.

.

.

.

...

.

.

.

Ji

JN J2

J1

Radiosity of a black surface i

For each surface, i

11

N

jijF

The summation rule !

T1,A1

T2,A1

Ti,Ai

TN,AN

.

.

.

.

...

.

.

.

Ji

JN J2

J1

•The summation rule follows from the conservation requirement that al radiation leaving the surface I must be intercepted by the enclosures surfaces.

•The term Fii appearing in this summation represents the fraction of the radiation that leaves surface i and is directly intercept by i.

•If the surface is concave, it sees itself and Fii is non zero.

•If the surface is convex or plane, Fii = 0.

• To calculate radiation exchange in an enclosure of N surfaces, a total of N2 view factors is needed.

Real Opaque Surfaces

Kichoff’s Law: substances that are poor emitters are also poor absorbers for any given wavelength

At thermal equilibrium• Emissivity of surface ( = Absorptivity(

• Transmissivity of solid surfaces = 0

• Emissivity is the only significant parameter

• Emissivities vary from 0.1 (polished surfaces) to 0.95 (blackboard)

Complication

• In practice, we cannot just consider the emissivity or absorptivity of surfaces in isolation

• Radiation bounces backwards and forwards between surfaces

• Use concept of “radiosity” (J) = emissive power for real surface, allowing for emissivity, reflected radiation, etc

Radiosity of Real Opaque Surface

• Consider an opaque surface.

• If the incident energy flux is G, a part of it is absorbed and the rest of it is reflected.

• The surface also emits an energy flux of E.

GEJ b Rate of Energy leaving a surface: J A

Rate of Energy incident on this surface: GA

Net rate of energy leaving the surface: A(J-G)

Rate of heat transfer from a surface by radiation: Q = A(J-G)

)( GGEAq b

Enclosure of Real Surfaces

T1,A1

T2,A1

Ti,Ai

TN,AN

.

.

.

.

...

.

.

.

Ji

JN J2

J1

Gi

EiiGi

For Every ith surfaceThe net rate of heat transfer by radiation:

iiiiiiiii GJAGGEAq )(

)( , iiibii GEJ

For any real surface: 1 iii

For an opaque surface: iiii 11

If the entire enclosure is at Thermal Equilibrium, From Kirchoff’s law:

iiiii 11

Substituting all above:

i

biiiiiiibii

EJGGEJ

1)1( ,

i

biiiiii

EJJAq

1

i

i

i

ibiii

A

JEAq

1

Surface Resistance of A Real Surface

Real Surface Resistance

ii

i A1

Ebi JiBlack body Actual Surface

Ebi –Ji : Driving Potential

ii

i A1

:surface radiative resistance

Gi

EiiGi

Qi

qi

Ji

Radiation Exchange between Real Surfaces

• To solve net rate of Radiation from a surface, the radiosity Ji must be known.

• It is necessary to consider radiation exchange between the surfaces of encclosure.

• The irradiation of surface i can be evaluated from the radiosities of all the other surfaces in the enclosure.

• From the definition of view factor : The total rate at which radiation reaches surface i from all surfaces including i, is:

N

jjjjiii JAFGA

1

From reciprocity relation

N

jjiijii JAFGA

1

iiii GJAq

N

jjiji JFG

1

N

jjijiii JFJAq

1

N

jjij

N

jiijii JFJFAq

11

N

jij

N

jjiijiii QJJFAAq

11

N

jij

N

jjiijiii QJJFAAQ

11

This result equates the net rate of radiation transfer from surface i, Qi to the sum of components Qij related to radiative exchange with the other surfaces.

Each component may be represented by a network element for which (Ji-Jj) is driving potential and (AiFij)-1 is a space or geometrical resistance.

i

i

i

ibii

N

jjiijiii

A

JEAJJFAAQ

11

Geometrical (View Factor) Resistance

Relevance?

• “Heat-transfer coefficients”: – view factors (can surfaces see each other?

Radiation is “line of sight” )– Emissivities (can surface radiate easily? Shiny

surfaces cannot)

Basic Concepts of Network Analysis

Analogies with electrical circuit analysis

• Blackbody emissive power = voltage

• Resistance (Real +Geometric) = resistance

• Heat-transfer rate = current

Resistance Network for ith surface interaction in an Enclosure

T1,A1

T2,A1

Ti,Ai

TN,AN

.

.

.

.

...

.

.

.

Ji

JN J2

J1

Gi

EiiGi

Qi

JiEbi

i

i

1

J1

Qi1

1

1

ii FA

J2

Qi2

2

1

ii FA

J3

Qi3

3

1

ii FA

JN-1

QiN-1

1

1

Nii FA

JN

QiN

Nii FA

1