chapter 13 radiation heat transfer - faculty of...

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DR MAZLAN

CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER

1313131313131313

SME SME 44463 463 HEAT TRANSFERHEAT TRANSFER

LECTURER:

ASSOC. PROF. DR. MAZLAN ABDUL WAHIDhttp://www.fkm.utm.my/~mazlan

Faculty of Mechanical EngineeringUniversiti Teknologi Malaysiawww.fkm.utm.my/~mazlan

FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER INTERNAL FLOW CONVECTIONINTERNAL FLOW CONVECTIONINTERNAL FLOW CONVECTIONINTERNAL FLOW CONVECTIONINTERNAL FLOW CONVECTIONINTERNAL FLOW CONVECTIONINTERNAL FLOW CONVECTIONINTERNAL FLOW CONVECTION

DR MAZLAN

DR MAZLAN


1313131313131313

Chapter 13Radiation Heat Transfer

FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER DR MAZLAN

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1313131313131313ObjectivesWhen you finish studying this chapter, you should be able to:

• Define view factor, and understand its importance in radiation heat transfer calculations,

• Develop view factor relations, and calculate the unknown view factors in an enclosure by using these relations,

• Calculate radiation heat transfer between black surfaces,• Determine radiation heat transfer between diffuse and gray

surfaces in an enclosure using the concept of radiosity,• Obtain relations for net rate of radiation heat transfer between

the surfaces of a two-zone enclosure, including two large parallel plates, two long concentric cylinders, and two concentric spheres,

• Quantify the effect of radiation shields on the reduction of radiation heat transfer between two surfaces, and become aware of the importance of radiation effect in temperature measurements.

DR MAZLAN


1313131313131313The View Factor• Radiation heat transfer between surfaces depends on the

orientationof the surfaces relative to each other as well as their radiation properties and temperatures.

• View factor is defined to account for the effects of orientation on radiation heat transfer between two surfaces.

• View factoris a purely geometric quantity and is independent of the surface properties and temperature.

• Diffuse view factor ─ view factor based on the assumption that the surfaces are diffuse emitters and diffuse reflectors.

• Specular view factor ─ view factor based on the assumption that the surfaces are specular reflectors.

• Here we consider radiation exchange between diffuse surfaces only, and thus the term view factorsimply means diffuse view factor.

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1313131313131313Radiation Exchange between Surfaces

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The View Factor (also known as Configuration or Shape Factor)

� View factor is a purely geometric quantity and is independent of the surface properties and temperature.

� The view factor based on the assumption that the surfaces are diffuse emitters and diffuse reflectors is called the diffuse view factor,and the view factor based on the assumption that the surfaces are diffuse emitters but specular reflectors is called the specular view factor.

� The view factor, Fi,j is a geometrical quantity corresponding to the fraction of the radiation leaving surfacei that is intercepted by surfacej.

DR MAZLAN


1313131313131313• The view factor from a surface i to a surface j is

denoted by Fi→j or just Fij, and is defined as• Fi�j=the fraction of the radiation leaving surface i

that strikes surface j directly.• Consider two differential surfaces dA1 and dA2 on two

arbitrarily oriented surfaces A1 and A2, respectively.• The rate at which radiation leaves

dA1 in the direction of θ1 is:I1cos θ1dA1

• Noting that dω21=dA2cos θ2/r2,

• the portion of this radiation that strikes dA2 is

1 2

2 21 1 1 21 1 1 1 2

coscos cosdA dA

dAQ I dA d I dA

r

θθ ω θ→ = =&

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1313131313131313• The total rate at which radiation leaves dA1 (via emission and reflection) in all directions is the radiosity (J1=πI1) times the surface area:

• Then the differential view factor dFdA1→dA2(the

fraction of radiation leaving dA1 that strikes dA2)

• The view factor from a differential area dA1 to a finite area A2 is:

1 1 1 1 1dAdQ J dA I dAπ= =&

1 2

1 2

1

1 222

cos cosdA dAdA dA

dA

QdF dA

Q r

θ θπ

→→ = =

&

&

1 2

2

1 222

cos cosdA A

A

F dAr

θ θπ→ = ∫

DR MAZLAN


1313131313131313• The total rate at which radiation leaves the entire A1 in all directions is

• considering the radiation that leaves dA1 and strikes dA2, and integrating it over A1,

• Integration of this relation over A2 gives the radiation that strikes the entire A2,

1 1 1 1 1AQ J A I Aπ= =&

1 2 1 2

1 1

1 1 2 212

cos cosA dA dA dA

A A

I dAQ Q dA

r

θ θ→ →= =∫ ∫& &

1 2 1 2

2 2 1

1 1 21 22

cos cosA A A dA

A A A

IQ Q dA dA

r

θ θ→ →= =∫ ∫ ∫& &

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1313131313131313• Dividing this by the total radiation leaving A1 (from Eq. 13–5) gives the fraction of radiation leaving A1 that strikes A2, which is the view factor F1�2,

• The view factor F2�1 is readily determined from Eq. 13–8 by interchanging the subscripts 1 and 2,

• Combining Eqs. 13–8 and 13–9 after multiplying the former by A1 and the latter by A2 gives the reciprocity relation

1 2

1 2

1 2 1

1 21 2 1 22

1

cos cos1A AA A

A A A

QF F dA dA

Q A r

θ θπ

→→ →= = = ∫ ∫

&

&

2 1

2 1

2 2 1

1 22 1 1 22

2

cos cos1A AA A

A A A

QF F dA dA

Q A r

θ θπ

→→ →= = = ∫ ∫

&

&

(13-3)1 1 2 2 2 1A F A F→ →=

DR MAZLAN


1313131313131313

• When j=i:

Fi�i=the fraction of radiation leaving surface i that strikes itself directly.– Fi�i=0: for plane or convex surfaces and

– Fi�i≠0: for concave surfaces

• The value of the view factor ranges between zeroand one.– Fi�j=0 ─ the two surfaces do not have a

direct view of each other,

– Fi�j=1─ surface j completely surrounds surface.

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1313131313131313

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1313131313131313Radiation Exchange between Surfaces

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� View factors for the enclosure formed by two spheres

� The view factor has proven to be very useful in radiation analysis because it allows us to express the fraction of radiationleaving a surface that strikes another surface in terms of the orientation of these two surfaces relative to each other.

� View factors of common geometries are evaluated and the results are given in analytical, graphical, and tabular form (Refer Tables 13.1 & 13.2, Figures 13.4, 13.5 & 13.6)

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1313131313131313

View FactorsTablesfor Selected Geometries (analytical form)

DR MAZLAN


1313131313131313View Factors Figures for Selected Geometries (graphical form)

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1313131313131313View Factor Relations• Radiation analysis on an enclosure consisting

of N surfaces requires the evaluation of N2

view factors.

• Fundamental relations for view factors:– the reciprocity relation,

– the summation rule,

– the superposition rule,

– the symmetry rule.

DR MAZLAN


1313131313131313The Reciprocity Relation• We have shown earlier that the pair of view

factors Fi�j and Fj�i are related to each other by

• This relation is referred to as the reciprocity relation or the reciprocity rule .

• Note that:

(13-3)i i j j j iA F A F→ →=

when

when j i i j i j

j i i j i j

F F A A

F F A A

→ →

→ →

= =

≠ ≠

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1313131313131313The Summation Rule

• The conservation of energy principle requires that the entire radiation leaving any surface i of an enclosure be intercepted by the surfaces of the enclosure.

• Summation rule─ the sum of the view factors from surface i of an enclosure to all surfaces of the enclosure, including to itself, must equal unity.

(13-4)1

1N

i jj

F→=

=∑

DR MAZLAN


1313131313131313• The summation rule can be applied to each surface of an enclosure by varying i from 1 to N.

• The summation rule applied to each of the N surfaces of an enclosure gives N relations for the determination of the view factors.

• The reciprocity rule gives 1/2N(N-1) additional relations.

• The total number of view factors that need to be evaluated directly for an N-surface enclosure becomes

[N2 - N – N (N-1)/2] = N(N-1)/2

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1313131313131313The Superposition Rule• Sometimes the view factor associated with a given

geometry is not available in standard tables and charts.• Superposition rule─ the view factor from a surface i

to a surface j is equal to the sum of the view factors from surface i to the parts of surface j.

• Consider the geometry shown in the figure below.• The view factor from surface 1 to the combined

surfaces of 2 and 3 is

• From the chart in Table 13–2:– F1�2 and F1�(2,3)

and then from Eq. 13-13:– F1�3

( ) 1 2 1 31 2,3F F F→ →→ = +

DR MAZLAN


1313131313131313The Symmetry Rule

• Symmetry rule─ two (or more) surfaces that possess symmetry about a third surface will have identical view factors from that surface.

• If the surfaces j and k are symmetric about the surface ithen

• Using the reciprocity rule, it can be shown that

i j i kF F→ →=

j i k iF F→ →=

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1313131313131313View Factors between Infinitely Long Surfaces: The Crossed-Strings Method

• The view factor between two-dimensional surfaces can be determined by the simple crossed-strings method developed by H. C. Hottel in the 1950s.

• Consider the geometry shown in the figure.

• Hottel has shown that the view factor

F1� 2 can be expressed in terms of

the lengths of the stretched strings as

( ) ( )5 6 3 41 2

12

L L L LF

L→

+ − +=

DR MAZLAN


1313131313131313Radiation Heat Transfer: Black Surfaces

• Consider two black surfaces of arbitrary shape maintained at uniform temperatures T1 and T2.

• The net rate of radiation heat transfer from surface 1 to surface 2 can be expressed as

• Applying the reciprocity relation A1F1�2=A2F2�1 yields

• For enclosure consisting of N black surfaces

1 1 1 2 2 2 2 1 (W)b bA E F A E F→ →= −

Radiation leavingthe entire surface 1

that strikes surface 2

Radiation leavingthe entire surface 2

that strikes surface 11 2Q →& -=

( )4 41 2 1 1 2 1 2 (W)Q A F T Tσ→ →= −&

( )4 4

1 1

(W)N N

i i j i i j i jj j

Q Q A F T Tσ→ →= =

= = −∑ ∑& &

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1313131313131313Radiation Heat Transfer: Diffuse, Gray Surfaces

• To make a simple radiation analysis possible, it is common to assume the surfaces of an enclosure are:– opaque (nontransparent), – diffuse (diffuse emitters and diffuse reflectors), – gray (independent of wavelength),– isothermal, and– both the incoming and outgoing radiation are

uniform over each surface.

DR MAZLAN


1313131313131313Radiation exchange in real surfaces: diffuse, gray surfaces

24

Radiosity

Radiosity, J: The total radiation energy leaving a surface per unit time and per unit area (emitted and reflected).

*For a surface i that is grayand opaque(εi = αi and αi + ρi = 1)

For a blackbodyε = 1

Radiation Heat Transfer from a Surface:

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1313131313131313• For a surface i that is grayand

opaque(εi=αi and αi+ρi=1), the

radiosity can be expressed as

where:

• For a surface that can be approximated as a blackbody (εi=1), the radiosity relation reduces to:

(13-21)

( ) 2 1 (W/m )

i i bi i i

i bi i i

J E G

E G

ε ρε ε

= +

= + −

4bi iE Tσ=

4 (blackbody)i bi iJ E Tσ= =

DR MAZLAN


1313131313131313Rate of Energy Loss from a Surface

• The net rate of radiation heat transfer, or called net heat loss, from a surface i of surface area Ai is expressed as

• Solving for Gi from Eq. 13–21 and substituting into Eq. 13–15 yields

( ) (W)i i iA J G= −

Radiation leavingentire surface i

Radiation incidenton entire surface iiQ& -= (13-15)

( ) (W)1- 1-

i i bi i ii i i bi i

i i

J E AQ A J E J

ε εε ε

−= − = −

&

Rate of energy loss from surface i

(13-19)

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1313131313131313

( ) (W)i i iA J G= −

Radiation leavingentire surface i

Radiation incidenton entire surface iiQ& -=

( ) (W)1- 1-

i i bi i ii i i bi i

i i

J E AQ A J E J

ε εε ε

−= − = −

&

J

G

DR MAZLAN


1313131313131313• In an electrical analogy to Ohm’s law, Eq.13-24 can be rearranged as

where surface resistance to

radiation is

• For a blackbody Ri=0 and the net rate of radiation heat transfer in this case is determined directly from Eq. 13–23.

(13-25) (W)bi ii

i

E JQ

R

−=&

(13-26)1- i

ii i

RA

εε

=

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1313131313131313• Reradiating surface─ an adiabatic surface:– when convection effects is negligible,– under steady-state conditions.

• Reradiating surface must lose as much radiation energy as it gains, thus:

• The temperature of a reradiating surface is independent of its emissivity.

0iQ =&

Eq. 13-25

(13-27)4 2 (W/m )i bi iJ E Tσ= =

DR MAZLAN


1313131313131313Net Radiation Heat Transferbetween Any Two Surfaces

• Consider two diffuse, gray, and opaque surfaces of arbitrary shape maintained at uniform temperatures.

• The net rate of radiation heat transfer from surface i to surface j can be expressed as

(13-28)

Radiation leavingthe entire surface i

that strikes surface j

Radiation leavingthe entire surface j

that strikes surface ii jQ→& -=

(W)i i i j j j j iA J F A J F→ →= −

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1313131313131313

• Applying the reciprocity relation AiFi�j= AjFj�i

yields

• In analogy to Ohm’s law

where space resistance to radiation is

(13-29)( ) (W)i j i i j i jQ A F J J→ →= −&

(13-30) (W)i ji j

i j

J JQ

R→→

−=&

(13-31)1

i ji i j

RA F→

→

=

DR MAZLAN


1313131313131313• In an N-surface enclosure, the conservation of energy

principle requires

• Combining Eqs. 13–25 and 13–32 gives

(13-32)( )1 1 1

= (W)N N N

i ji i j i i j i j

j j j i j

J JQ Q A F J J

R→ →= = = →

−= = −∑ ∑ ∑& &

(13-33)1

(W)N

i jbi i

ji i j

J JE J

R R= →

−− =∑

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1313131313131313Methods of Solving Radiation Problems

• In the radiation analysis of an enclosure, either– the temperature or

– the net rate of heat transfer

must be given for each of the surfaces.

• Two methods commonly used:– surfaces with specified net heat transfer rate (Eq. 13-32)

– surfaces with specified temperature (Eq. 13-33)

(13-34)( )1

N

i i i j i jj

Q A F J J→=

= −∑&

(13-35)( )4

1

1 Ni

i i i j i jji

T J F J Jεσ

ε →=

−= + −∑

DR MAZLAN


1313131313131313• The equations above give N linear algebraic equations for the determination of the N unknown radiosities for an N-surface enclosure.

• Once the radiosities J1, J2, . . . , JN are available, the unknown heat transfer rates can be determined from Eq. 13–34.

• The unknown surface temperatures can be determined from Eq. 13–35.

• Two methods to solve the system of N equations: – direct method

• very suitable for use with today’s popular equation solvers

– network method• not practical for enclosures with more than three or four surfaces

• simple and emphasis on the physics of the problem.

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1313131313131313Radiation Heat Transfer in Two-Surface

Enclosures

• Consider an enclosure consisting of two opaque surfaces at specified temperatures.

• Need to determine the net rate of

radiation heat transfer.

• Known: T1, T2, ε1, ε2, A1, A2, F12.

• Surface resistances:– two surface resistances,

– one space resistance.

• The net rate of radiation transfer is

expressed as 1 212 1 2

1 12 2

b bE EQ Q Q

R R R

−= = = −+ +

& & &

( )( ) ( )

4 41 2

121 1 1 1 12 2 2 21 1 1

T TQ

A A F A

σε ε ε ε

−=

− + + −&

(13-36)

DR MAZLAN


1313131313131313

36

Radiation exchange in real surfaces; diffuse, gray surfaces

Net Radiation Heat Transfer Between Any Two Surfaces

The net rate of radiation heat transfer from surface i to surface j is

*Apply the reciprocity relation

= Space resistance to radiation

where,

�� Eq.(13.16)

*Electrical analogy of space resistance to radiation

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1313131313131313Simplified forms of Eq. 13–36 for some familiar arrangements

DR MAZLAN


1313131313131313Radiation Heat Transferin Three-Surface Enclosures

• Consider an enclosure consisting of three opaque, diffuse, and gray surfaces.

• Known: T1, T2, T3, ε1, ε2, ε3, A1, A2, A3, F12.

• Since the temperatures are known Eb1, Eb2, and Eb3 are considered known.

• Need to determine

the radiosities

J1, J2, and J3.

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1313131313131313• The three equations for the determination of these three unknowns are obtained from the requirement that the algebraic sum of the currents (net radiation heat transfer) at each node must equal zero.

• Once the radiosities J1, J2, and J3 are available, the net rate of radiation heat transfers at each surface can be determined from Eq. 13–32.

1 1 3 12 1

1 12 13

2 2 3 21 2

12 2 23

1 3 2 3 3 3

13 23 3

0

0

0

b

b

b

E J J JJ J

R R R

E J J JJ J

R R R

J J J J E J

R R R

− −−+ + =

− −− + + =

− − −+ + =

(13-41)

DR MAZLAN


1313131313131313Radiation Shields and the Radiation Effects

• Radiation heat transfer between two surfaces can be reduced greatly by inserting a thin, high-reflectivity (low-emissivity) sheet of material (radiation shields) between the two surfaces.

• Radiation heat transfer between two large parallel plates of emissivities ε1 and ε2

maintained at uniform temperatures T1 and T2 is given by Eq. 13–38: ( )4 4

1 2

12,no shield

1 2

1 11

A T TQ

σ

ε ε

−=

+ −&

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1313131313131313• Consider a radiation shield placed between these two plates.

Radiation network:

• The rate of radiation heat transfer is

1 212,one shield

3,1 3,21 2

1 1 1 13 3 3,1 3 3,2 3 32 2 2

1 11 11 1b bE E

Q

A A F A A A F A

ε εε εε ε ε ε

−= − −− −+ + + + +

&(13-42)

DR MAZLAN


1313131313131313• Noting that F13=F23=1 and A1=A2=A3=A for infinite parallel plates, Eq. 13–42 simplifies to

• The radiation heat transfer through large parallel plates separated by N radiation shields

(13-43)( )4 4

1 2

12,one shield

1 2 3,1 3,2

No shield resistance shield resistance

1 1 1 11 1

A T TQ

σ

ε ε ε ε

−=

+ − + + −

&

1442443 1442443

(13-44)( )4 41 2

12,N shield

1 2 3,1 3,2 ,1 ,2

No shield resistance shield 1 resistance shield N resistance

1 1 1 1 1 11 1 1

N N

A T TQ

σ

ε ε ε ε ε ε

−=

+ − + + − + + + −

&

L

1442443 1442443 1442443

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1313131313131313Radiation Exchange with Emitting and Absorbing Gases

• Nonparticipating medium─ medium that is completely transparent to thermal radiation (no emission, absorption, or scattering).

• Examples of nonparticipating medium:– air at ordinary temperatures and pressures,– gases that consist of monatomic molecules (e.g., Ar and He).– gases that consist of symmetric diatomic molecules (e.g., N2

and O2).

• Examples of participating medium– asymmetric molecules such as H2O, CO2, CO, SO2, and

hydrocarbons HmCn• by absorption at moderate temperatures, and • by absorption and emission at high temperatures.

DR MAZLAN


1313131313131313The presence of a participating medium complicates the radiation analysis considerably:

• A participating medium is a volumetric phenomena, • Gases emit and absorb radiation at a number of narrow

wavelength bands, and the gray assumption may notalways be appropriate,

• The emission and absorption characteristics of the constituents of a gas mixture also depends on the temperature, pressure, and composition of the gas mixture,

• Scattering─ the change of direction of radiation due to reflection, refraction, and diffraction.

• We limit our consideration to gases that emit and absorb radiation.

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1313131313131313Radiation Properties of a Participating Medium

• Consider a participating medium of thickness L.• A spectral radiation beam of intensity Iλ,0 is incident

on the medium, which is attenuated as it propagates due to absorption.

• The decrease in the intensity of radiation is proportional to the – intensity Iλ,– thickness dx.

• Beer’s law

• kλ is the spectral absorption coefficient (units m-1).( ) ( )dI x k I x dxλ λ λ= − (13-47)

DR MAZLAN


1313131313131313• Separating the variables and integrating from

x=0 to x=L gives (assuming kλ =constant)

• The spectral transmissivity

• Radiation passing through a nonscattering (and thus nonreflecting) medium is either absorbed or transmitted � αλ+τλ=1

• spectral absorptivity of a medium of thickness L

,

,0

L k LIe

Iλλ

λ

−= (13-48)

,

,0

L k LIe

Iλλ

λλ

τ −= = (13-49)

1 1 k Le λλ λα τ −= − = − (13-50)

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1313131313131313• From Kirchoff’s law, the spectral emissivity of the medium is

• The spectral absorption coefficient of a medium (and thus ελ, αλ, τλ), in general, vary with:– wavelength, – temperature, – pressure, and – composition.

• Optically thick medium ─ a medium with a large value of kλ L.

• For optically thick mediumελ≈αλ≈1.

1 k Le λλ λε α −= = − (13-51)

DR MAZLAN


1313131313131313Emissivity and Absorptivityof Gases and Gas Mixtures

• The band nature of absorption of most gases are strongly nongray.

• The nongray nature of properties should be considered in radiation calculations for high accuracy.

• Satisfactory results can be obtained by assuming the gas to be gray, and using an effective total absorptivity and emissivity determined by some averaging process.

• An approach which assumes the gas to be gray, was developed by Hottel (1954) and is presented below.

Spectral absorptivity of CO2 at 830 K and 10 atm for a path length of 38.8 cm.

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1313131313131313

• The emissivity and absorptivity of a gas component in a mixture depends primarily on its density.

• Emissivity at a total pressure Pother than P=1 atm is determined by multiplying the emissivity value at 1 atm by a pressure correction factor Cw obtained from Figure 13–37a for water vapor

Emissivity of H2O in a mixture of nonparticipating gases at a total pressure of 1 atm for a mean beam length of L.

Fig. 13-37 Correction factors for the emissivities of H2O at pressures other than 1 atm

, 1 atmw w wCε ε= (13-52)

DR MAZLAN


1313131313131313• When CO2 and H2O gases exist together in a mixture with nonparticipating gases

• ∆ε is the emissivity correction factor.

Emissivity correction ∆ε∆ε∆ε∆ε for use in εεεεg=εεεεw+εεεεc-∆ε∆ε∆ε∆ε when both CO2 and H2O vapor are present in a gas mixture.

, 1 atm w, 1 atm g c w

c c wC C

ε ε ε εε ε ε

= + − ∆

= + − ∆(13-53)

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1313131313131313• Hottel and his coworkers considered the emission of radiation from a hemispherical gas body to a small surface element located at the center of the base of the hemisphere.

• It is certainly desirable to extend the reported emissivity data to gas bodies of other geometries.

• Mean beam length L ─ represents the radius of an equivalent hemisphere.

• The mean beam lengths

for various gas

geometries are listed in

Table 13–4.

DR MAZLAN


1313131313131313• The absorptivity of a gas that contains CO2 and H2O gases for radiation emitted by a source at temperature Ts can be determined from

• The absorptivities of CO2 and H2O can be determined from the emissivity charts (Figs. 12–36 and 12–37) as

• CO2:

• H2O:

g c wα α α αα ε

= + − ∆

∆ = ∆(13-54)

( ) ( )0.65, /c c g s c s c s gC T T T P LT Tα ε= × × (13-55)

( ) ( )0.45, /w w g s w s w s gC T T T P LT Tα ε= × × (13-56)

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1313131313131313• The rate of radiation energy emitted by a gas to a bounding surface of area As

• The net rate of radiation heat transfer between the gas and a black surface surrounding it becomes

• For surfaces that are nearly black with an emissivity εs>0.7 (Hottel)

4,g e g s gQ A Tε σ=& (13-57)

( )4 4 4 4

energy emitted energy emittedby the gas to by the surfacethe surface to the gas

net g s g g s s s g g g sQ A T A T A T Tε σ α σ σ ε α= − = −&

14243 14243(13-58)

( )4 4, ,

1 1

2 2s s

net gray net black s g g g sQ Q A T Tε ε σ ε α+ += = −& & (13-59)

chapter 13 radiation heat transfer - faculty of...

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