Space Engineering - Thermal design handbook - Part 9: Radiators

In this Part 9 of the spacecraft thermal control and design data handbooks, view factors of diffuse and specular thermal surfaces are discussed.
For diffuse surfaces, calculations are given for radiation emission and absorption between different configurations of planar, cylindrical, conical, spherical and ellipsoidal surfaces for finite and infinite surfaces.
For specular surfaces the affect of reflectance on calculations for view factors is included in the calculations. View factors for specular and diffuse surfaces are also included.
The Thermal design handbook is published in 16 Parts
TR 17603-31-01 Part 1
Thermal design handbook – Part 1: View factors
TR 17603-31-01 Part 2
Thermal design handbook – Part 2: Holes, Grooves and Cavities
TR 17603-31-01 Part 3
Thermal design handbook – Part 3: Spacecraft Surface Temperature
TR 17603-31-01 Part 4
Thermal design handbook – Part 4: Conductive Heat Transfer
TR 17603-31-01 Part 5
Thermal design handbook – Part 5: Structural Materials: Metallic and Composite
TR 17603-31-01 Part 6
Thermal design handbook – Part 6: Thermal Control Surfaces
TR 17603-31-01 Part 7
Thermal design handbook – Part 7: Insulations
TR 17603-31-01 Part 8
Thermal design handbook – Part 8: Heat Pipes
TR 17603-31-01 Part 9
Thermal design handbook – Part 9: Radiators
TR 17603-31-01 Part 10
Thermal design handbook – Part 10: Phase – Change Capacitors
TR 17603-31-01 Part 11
Thermal design handbook – Part 11: Electrical Heating
TR 17603-31-01 Part 12
Thermal design handbook – Part 12: Louvers
TR 17603-31-01 Part 13
Thermal design handbook – Part 13: Fluid Loops
TR 17603-31-01 Part 14
Thermal design handbook – Part 14: Cryogenic Cooling
TR 17603-31-01 Part 15
Thermal design handbook – Part 15: Existing Satellites
TR 17603-31-01 Part 16
Thermal design handbook – Part 16: Thermal Protection System

Raumfahrttechnik - Handbuch für thermisches Design - Teil 9: Radiatoren

Ingénierie spatiale - Manuel de conception thermique - Partie 9: Radiateurs

Vesoljska tehnika - Priročnik o toplotni zasnovi - 9. del: Radiatorji

General Information

Status
Published
Public Enquiry End Date
19-May-2021
Publication Date
23-Aug-2021
Technical Committee
Current Stage
6060 - National Implementation/Publication (Adopted Project)
Start Date
19-Aug-2021
Due Date
24-Oct-2021
Completion Date
24-Aug-2021

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SLOVENSKI STANDARD
SIST-TP CEN/CLC/TR 17603-31-09:2021
01-oktober-2021
Vesoljska tehnika - Priročnik o toplotni zasnovi - 9. del: Radiatorji
Space Engineering - Thermal design handbook - Part 9: Radiators
Raumfahrttechnik - Handbuch für thermisches Design - Teil 9: Radiatoren
Ingénierie spatiale - Manuel de conception thermique - Partie 9: Radiateurs
Ta slovenski standard je istoveten z: CEN/CLC/TR 17603-31-09:2021
ICS:
49.140 Vesoljski sistemi in operacije Space systems and
operations
SIST-TP CEN/CLC/TR 17603-31-09:2021 en,fr,de

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

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SIST-TP CEN/CLC/TR 17603-31-09:2021
TECHNICAL REPORT
CEN/CLC/TR 17603-31-
RAPPORT TECHNIQUE
TECHNISCHER BERICHT
August 2021
ICS 49.140
English version
Space Engineering - Thermal design handbook - Part 9:
Radiators

Ingénierie spatiale - Manuel de conception thermique - Raumfahrttechnik - Handbuch für thermisches Design -

Partie 9 : Radiateurs Teil 9: Radiatoren

This Technical Report was approved by CEN on 21 June 2021. It has been drawn up by the Technical Committee CEN/CLC/JTC 5.

CEN and CENELEC members are the national standards bodies and national electrotechnical committees of Austria, Belgium,

Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy,

Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Republic of North Macedonia, Romania, Serbia,

Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and United Kingdom.
CEN-CENELEC Management Centre:
Rue de la Science 23, B-1040 Brussels

© 2021 CEN/CENELEC All rights of exploitation in any form and by any means Ref. No. CEN/CLC/TR 17603-31-09:2021 E

reserved worldwide for CEN national Members and for
CENELEC Members.
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Table of contents

European Foreword ................................................................................................... 7

1 Scope ....................................................................................................................... 8

2 References .............................................................................................................. 9

3 Terms, definitions and symbols .......................................................................... 10

3.1 Terms and definitions ............................................................................................. 10

3.2 Symbols .................................................................................................................. 10

4 Diffuse surfaces .................................................................................................... 12

4.1 General ................................................................................................................... 12

4.2 Infinitesimal to finite surfaces .................................................................................. 13

4.2.1 Planar to planar ......................................................................................... 13

4.2.2 Planar to spherical .................................................................................... 19

4.2.3 Cylindrical to spherical .............................................................................. 20

4.2.4 Conical to spherical ................................................................................... 21

4.2.5 Spherical to spherical ................................................................................ 23

4.2.6 Ellipsoidal to spherical ............................................................................... 25

4.2.7 Planar to conical ....................................................................................... 28

4.3 Finite to finite surface .............................................................................................. 31

4.3.1 Planar to planar. Two-dimensional configurations ..................................... 31

4.3.2 Planar to planar. Three-dimensional configurations................................... 35

4.3.3 Planar to cylindrical. Two-dimensional configurations ............................... 46

4.3.4 Planar to cylindrical. three-dimensional configurations .............................. 48

4.3.5 Planar to conical ....................................................................................... 54

4.3.6 Spherical to planar .................................................................................... 56

4.3.7 Cylindrical to cylindrical. two-dimensional configurations .......................... 62

4.3.8 Cylindrical to cylindrical. axisymmetrical configurations ............................. 64

4.3.9 Spherical to cylindrical .............................................................................. 69

4.3.10 Conical to conical ...................................................................................... 72

4.3.11 Conical to spherical ................................................................................... 72

4.3.12 Spherical to spherical ................................................................................ 77

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4.4 Additional sources of data ....................................................................................... 80

5 Specular surfaces ............................................................................................... 103

5.1 General ................................................................................................................. 103

5.2 Two planar specular surfaces ............................................................................... 105

5.2.1 Two-dimensional configurations .............................................................. 105

5.2.2 Parallel, directly opposed rectangles of same width and length ............... 109

5.2.3 Rectangles of same width and length with one common edge ................ 115

5.3 Planar specular and planar diffuse surface ........................................................... 118

5.3.1 Two dimensional cavities. Cylinders of quadrangular cross section ........ 118

5.4 Non-planar specular surfaces ............................................................................... 123

5.4.1 Concentric cylinder or concentric spheres ............................................... 123

Bibliography ........................................................................................................... 125

Figures

Figure 4-1: Geometric notation for view factors between diffuse surface. ............................. 13

Figure 4-2: Values of F as a function of x and y. From Hamilton & Morgan (1952) [15]. ..... 15

Figure 4-3: Values of F as a function of x and y. From Hamilton & Morgan (1952) [15] ...... 17

Figure 4-4: F vs. H for different values of dH. Infinitesimal surface to very thin coaxial

annulus with finite radius. Calculated by the compiler......................................... 18

Figure 4-5: Values of F vs. λ for different values of H. The analytical expression (case

I) is only valid in the shadowed region. Calculated by the compiler. ................... 19

Figure 4-6: Values of F as a function of H and λ. Calculated by the compiler. .................... 20

Figure 4-7: Values of F as a function of H and λ, for δ = 10°. Calculated by the

compiler. ............................................................................................................ 21

Figure 4-8: Values of F as a function of H and λ, for δ = 30°. Calculated by the

compiler. ............................................................................................................ 22

Figure 4-9: Values of F as a function of H and λ, for δ = 50°. Calculated by the

compiler. ............................................................................................................ 22

Figure 4-10: Values of F as a function of H and λ, for δ = 80°. Calculated by the

compiler. ............................................................................................................ 23

Figure 4-11: F as a function of H in the case of an infinitesimal sphere viewing a finite

sphere. Calculated by the compiler. ................................................................... 24

Figure 4-12: F as a function of angle λ for different values of the dimensionless

distance H. Calculated by the compiler. ............................................................. 25

Figure 4-13: F as a function of λ and H, for A = 0,5. Calculated by the compiler. ............... 26

Figure 4-14: F as a function of λ and H, for A = 1,5. Calculated by the compiler. ............... 27

Figure 4-15: F as a function of λ and H, for A = 2. Calculated by the compiler. .................. 27

Figure 4-16: Values of F vs. M for different values of L. Configuration 1, β = 10°.

Calculated by the compiler. ................................................................................ 29

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Figure 4-17: Values of F12 vs. M for different values of L. Configuration 1, β = 20°.

Calculated by the compiler. ................................................................................ 29

Figure 4-18: Values of F vs. M for different values of L. Configuration 2, β = 10°.

Calculated by the compiler. ................................................................................ 30

Figure 4-19: Values of F vs. M for different values of L. Configuration 2, β = 20°.

Calculated by the compiler. ................................................................................ 30

Figure 4-20: Values of F as a function of X and Y, for Z = 0. Calculated by the

compiler. ............................................................................................................ 33

Figure 4-21: Values of F as a function of X and Y, for Z = 0,5. Calculated by the

compiler. ............................................................................................................ 33

Figure 4-22: Values of F as a function of X and Y, for Z = 1. Calculated by the

compiler. ............................................................................................................ 34

Figure 4-23: Values of F as a function of X and Y, for Z = 2. Calculated by the

compiler. ............................................................................................................ 34

Figure 4-24: Values of F as a function of X and Y, for Z = 5. Calculated by the

compiler. ............................................................................................................ 35

Figure 4-25: Values of F12 as a function of X and Y. Calculated by the compiler. .................. 36

Figure 4-26: F as a function of L and N for Φ = 30°. Table from Feingold (1966) [11],

figure from Hamilton & Morgan (1952) [15]. ........................................................ 39

Figure 4-27: F as a function of L and N for Φ = 60°. Table from Feingold (1966) [11],

figure from Hamilton & Morgan (1952) [15]. ........................................................ 39

Figure 4-28: F as a function of L and N for Φ = 90°. Table from Feingold (1966) [11],

figure from Hamilton & Morgan (1952) [15]. ........................................................ 40

Figure 4-29: F as a function of L and N for Φ = 120°. Table from Feingold (1966) [11],

figure from Hamilton & Morgan (1952) [15]. ........................................................ 40

Figure 4-30: F as a function of L and N for Φ = 150°. Table from Feingold (1966) [11],

figure from Hamilton & Morgan (1952) [15]. ........................................................ 41

Figure 4-31: Values of F as a function of L for different regular polygons. n is the

number of sides of the polygon. From Feingold (1966) [11]. ............................... 43

Figure 4-32: View factors between different faces of a honeycomb cell as a function of

the cell length, L. From Feingold (1966) [11]. ..................................................... 44

Figure 4-33: Values of F as a function of R and R in the case of two parallel coaxial

12 1 2

discs. Calculated by the compiler. ...................................................................... 46

Figure 4-34: Values of F and F as a function of the parameter K. From Jakob (1957)

12 13

[19]. .................................................................................................................... 48

Figure 4-35: F as a function of T and R. Calculated by the compiler. ................................. 49

Figure 4-36: F as a function of T and R. Calculated by the compiler. ................................. 50

Figure 4-37: F as a function of T and R. Calculated by the compiler. ................................. 50

Figure 4-38: F as a function of Z, for different values of the dimensionless radius R.

Calculated by the compiler. ................................................................................ 52

Figure 4-39: F as a function of R for different values of the sector central angel α.

12 2

Calculated by the compiler. ................................................................................ 57

Figure 4-40: F as a function of Z for different values of R . Calculated by the compiler. ..... 58

12 2
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Figure 4-41: F from a sphere to both sides of a coaxial intersecting disc, vs. H, for

different values of R. Calculated by the compiler. ............................................... 59

Figure 4-42: F12 from a sphere to the upper side of a coaxial intersecting disc, vs. H (-

1 ≤H≤ 1), for different values of R. Calculated by the compiler. .......................... 59

Figure 4-43: Values of F as a function of Z and R. Calculated by the compiler. .................. 60

Figure 4-44: F12 as a function of x in the case of two infinitely long parallel cylinders of

the same diameter. Calculated by the compiler. ................................................. 64

Figure 4-45: Plot of F vs. L for different values of R. From Hamilton & Morgan (1952)

[15] ..................................................................................................................... 66

Figure 4-46: Plot of F , vs. R for different values of L. From Hamilton & Morgan (1952)

[15] ..................................................................................................................... 67

Figure 4-47: F as a function of R for different values of Z. Calculated by the compiler. ...... 70

Figure 4-48: Values of F as a function of H and L for L = 1. Calculated by the
12 2 1

compiler. ............................................................................................................ 71

Figure 4-49: Values of F as a function of S and D, for δ = 15°. From Campbell &

McConnell (1968) [4]. ......................................................................................... 73

Figure 4-50: Values of F12 as a function of S and D, for δ = 30°. From Campbell &

McConnell (1968) [4]. ......................................................................................... 74

Figure 4-51: Values of F as a function of S and D, for δ = 45°. From Campbell &

McConnell (1968) [4]. ......................................................................................... 75

Figure 4-52: Values of F as a function of S and D, for δ = 60°. From Campbell &

McConnell (1968) [4]. ......................................................................................... 76

Figure 4-53: Values of F as a function of S and R. From Jones (1965) [21]. ...................... 79

Figure 4-54: Values of F as a function of S and θ. From Campbell & McConnell

(1968) [4]. ........................................................................................................... 80

Figure 5-1: Values of F as a function of R and H. Calculated by the compiler. ................. 106

s s

Figure 5-2: Values of F /ρ as a function of R and H. Calculated by the compiler. ........... 106

11 2

Figure 5-3: Values of F as a function of R for different values of φ. Calculated by the

compiler. .......................................................................................................... 108

s s

Figure 5-4: Values of F /ρ as a function of R for different values of φ. Calculated by

11 2

the compiler. .................................................................................................... 109

Figure 5-5: Values of F as a function of R and X for Z = 1. Calculated by the

compiler. .......................................................................................................... 110

s s
Figure 5-6: Values of F /ρ as a function of R and X for Z = 1. Calculated by the
11 2

compiler. .......................................................................................................... 111

Figure 5-7: Values of F as a function of R and X for Z = 5. Calculated by the

compiler. .......................................................................................................... 111

s s
Figure 5-8: Values of F /ρ as a function of R and X for Z = 5. Calculated by the
11 2

compiler. .......................................................................................................... 112

Figure 5-9: Values of F as a function of R and X for Z = 10. Calculated by the

compiler. .......................................................................................................... 112

s s

Figure 5-10: Values of F /ρ as a function of R and X for Z = 10. Calculated by the

11 2

compiler. .......................................................................................................... 113

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Figure 5-11: Values of F as a function of R and X for Z = 15. Calculated by the

compiler. .......................................................................................................... 113

s s

Figure 5-12: Values of F /ρ as a function of R and X for Z = 15. Calculated by the

11 2

compiler. .......................................................................................................... 114

Figure 5-13: Values of F as a function of R and X for Z = 20. Calculated by the

compiler. .......................................................................................................... 114

s s

Figure 5-14: Values of F /ρ as a function of R and X for Z = 20. Calculated by the

11 2

compiler. .......................................................................................................... 115

Figure 5-15: Values of F vs. aspect ratio, L, for different values of R. φ = 30°.

Calculated by the compiler. .............................................................................. 116

s s

Figure 5-16: Values of F 11/ρ2 vs. aspect ratio, L, for different values of R . φ = 30°.

Calculated by the compiler. .............................................................................. 117

Figure 5-17: Values of F vs. aspect ratio, L, for different values of R. φ = 45º.

Calculated by the compiler. .............................................................................. 117

s s s

Figure 5-18: Values of F and F /ρ vs. aspect ratio, L, for the limiting values of φ.

12 11 2

Calculated by the compiler. .............................................................................. 118

s s

Figure 5-19: Values of F vs. φ for different values of the specular reflectance, ρ .

Calculated by the compiler. .............................................................................. 120

s s

Figure 5-20: Values of F vs. φ for different values of the specular reflectance, ρ .

Calculated by the compiler. .............................................................................. 121

s s

Figure 5-21: Values of F vs. φ for different values of the specular reflectance, ρ .

Calculated by the compiler. .............................................................................. 121

s s

Figure 5-22: Values of F vs. φ for different values of the specular reflectance, ρ .

Calculated by the compiler. .............................................................................. 122

s s

Figure 5-23: Values of F vs. φ for different values of the specular reflectance, ρ .

Calculated by the compiler. .............................................................................. 122

s s

Figure 5-24: Values of F vs. φ for different values of the specular reflectance, ρ .

Calculated by the compiler. .............................................................................. 123

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European Foreword

This document (CEN/CLC/TR 17603-31-09:2021) has been prepared by Technical Committee

CEN/CLC/JTC 5 “Space”, the secretariat of which is held by DIN.

It is highlighted that this technical report does not contain any requirement but only collection of data

or descriptions and guidelines about how to organize and perform the work in support of EN 16603-

31.

This Technical report (TR 17603-31-09:2021) originates from ECSS-E-HB-31-01 Part 1A .

Attention is drawn to the possibility that some of the elements of this document may be the subject of

patent rights. CEN [and/or CENELEC] shall not be held responsible for identifying any or all such

patent rights.

This document has been prepared under a mandate given to CEN by the European Commission and

the European Free Trade Association.

This document has been developed to cover specifically space systems and has therefore precedence

over any TR covering the same scope but with a wider domain of applicability (e.g.: aerospace).

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Scope

In this Part 9 of the spacecraft thermal control and design data handbooks, view factors of diffuse and

specular thermal surfaces are discussed.

For diffuse surfaces, calculations are given for radiation emission and absorption between different

configurations of planar, cylindrical, conical, spherical and ellipsoidal surfaces for finite and infinite

surfaces.

For specular surfaces the affect of reflectance on calculations for view factors is included in the

calculations. View factors for specular and diffuse surfaces are also included.
The Thermal design handbook is published in 16 Parts
TR 17603-31-01 Thermal design handbook – Part 1: View factors
TR 17603-31-02 Thermal design handbook – Part 2: Holes, Grooves and Cavities
TR 17603-31-03 Thermal design handbook – Part 3: Spacecraft Surface Temperature
TR 17603-31-04 Thermal design handbook – Part 4: Conductive Heat Transfer

TR 17603-31-05 Thermal design handbook – Part 5: Structural Materials: Metallic and

Composite
TR 17603-31-06 Thermal design handbook – Part 6: Thermal Control Surfaces
TR 17603-31-07 Thermal design handbook – Part 7: Insulations
TR 17603-31-08 Thermal design handbook – Part 8: Heat Pipes
TR 17603-31-09 Thermal design handbook – Part 9: Radiators
TR 17603-31-10 Thermal design handbook – Part 10: Phase – Change Capacitors
TR 17603-31-11 Thermal design handbook – Part 11: Electrical Heating
TR 17603-31-12 Thermal design handbook – Part 12: Louvers
TR 17603-31-13 Thermal design handbook – Part 13: Fluid Loops
TR 17603-31-14 Thermal design handbook – Part 14: Cryogenic Cooling
TR 17603-31-15 Thermal design handbook – Part 15: Existing Satellites
TR 17603-31-16 Thermal design handbook – Part 16: Thermal Protection System
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References
EN Reference Reference in text Title
EN 16601-00-01 ECSS-S-ST-00-01 ECSS System - Glossary of terms

TR 17603-31-03 ECSS-E-HB-31-01 Part 3 Thermal design handbook – Part 3: Spacecraft

Surface Temperature

All other references made to publications in this Part are listed, alphabetically, in the Bibliography.

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Terms, definitions and symbols
3.1 Terms and definitions

For the purpose of this Standard, the terms and definitions given in ECSS-S-ST-00-01 apply.

3.2 Symbols
surface area of the i-th surface, [m ]
energy flux leaving surface, i. often called radiosity,
[W.m ]
view factor from diffuse surface, Ai to diffuse surface,
Fij
view factor from the ensemble of diffuse surfaces, Ai1,
F(i1,i2,...in)(j1,j2,...jn)
Ai2,...Ain to the ensemble of diffuse surfaces, Aj1,
Aj2,...Ajn
view factor from specular surface Ai to specular
Fij
surface Aj
energy flux incident on surface i, [W.m ]
term which appears in the expression for the view
Ki2
factor between elements of parallel plates, Ki2 = AiFii'
fraction of the radiative energy leaving Am which
Kmn(i,j,k,p,q,...)
reaches An after i perfectly specular reflections from
surface Ai, j from surface Aj, k from surface Ak,...
distance between two differential elements, [m]
temperature, [K]
angle from normal to surface i, [angular degrees]
hemispherical emittance of a (diffuse-gray) surface
hemispherical diffuse reflectance of a (diffuse-gray)
surface
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specular reflectance of a (gray) surface, it is assumed
to be independent of incident angle
− − −
8 2 4
σ Stefan-Boltzmann constant, S = 5,6697x10 W.m .K

Other symbols, mainly used to define the geometry of the configurations, are introduced when

required.
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Diffuse surfaces
4.1 General

The view factor, F12, between the diffuse surface A1 and A2, is the fraction of the energy leaving the

isothermal surface A1 that arrives at A2.

If the receiver surface is infinitesimal, the view factor is infinitesimal for both infinitesimal and finite

emitting surfaces, and is given by the expression
cos β cos β
1 2
[4-1]
dF = dA
12 2
when both surfaces are infinitesimal, and by
dA cos β cos β
2 1 2
dF = dA
12 1 [4-2]
when A1 is finite.

If the receiver surface is finite, the view factor is finite for both infinitesimal and finite emitting

surfaces, and is given by the expression
cos β cos β
1 2
F = dA
[4-3]
12 2
when A1 is infinitesimal, and by
1 cos β cos β
1 2
F = dA dA
[4-4]
12 2 1
∫ ∫
A πS
1 A A
1 2
when A1 is finite.
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Figure 4-1: Geometric notation for view factors between diffuse surface.

Regardless of which surfaces are considered, their view factors satisfy the following reciprocity

relation:
A1F12 = A2F21

If we consider the diffuse surfaces A1, A2 and A3, the view factor between the surfaces A1 and A2 + A3 is

F1(2,3) = F12 + F13,
when the receiver surface is formed by two surfaces, and
A F + AF
2 21 3 31
F =
[4-5]
(2,3)1
A + A
2 3

when the emitting surface is formed by two surfaces. notice that the notation F1(2,3) and F(2,3)1 will be

used in the following data sheets.

When an enclosure of N surfaces A1,A2,....,An is considered, their view factors satisfy the relation

F = 1
[4-6]
∑ ij
j=1

for any surface Ai. This relationship results from the fact that the overall heat transfer in the enclosure

should be zero.
4.2 Infinitesimal to finite surfaces
4.2.1 Planar to planar
4.2.1.1 Two-dimensional configurations

A plane point source dA1 and any surface A2 generated by an infinitely long line moving parallel to

itself and to the plane dA1.
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Formula:
F = (cosθ − cosω) [4-7]
References: Hamilton & Morgan (1952) [15], Moon (1961) [26], Kreith (1962) [22]
Comments:
Notice that F12 is independent of the shape of A2 for given values of θ and ω.

View factors for several configurations may be obtained as a particular case of this one. An example is

shown in the next page.

A plane point source dA1 and any infinite plane A2 with the planes of dA1 and A2 intersecting at an

angle θ.
Formula:
F = (1+ cosθ ) [4-8]
References: Hamilton & Morgan (1952) [15], Moon (1961) [26], Kreith (1962) [22]
4.2.1.2 Point source to rectangle

A plane source dA1 and a plane rectangle A2 parallel to the plane of dA1 (see sketch). The normal to dA1

passes through one corner of A2.
x = a/c
y = b/c
Formula:
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...

SLOVENSKI STANDARD
kSIST-TP FprCEN/CLC/TR 17603-31-09:2021
01-maj-2021
Vesoljska tehnika - Priročnik za toplotno zasnovo - 9. del: Radiatorji
Space Engineering - Thermal design handbook - Part 9: Radiators
Raumfahrttechnik - Handbuch für thermisches Design - Teil 9: Radiatoren
Ingénierie spatiale - Manuel de conception thermique - Partie 9: Radiateurs
Ta slovenski standard je istoveten z: FprCEN/CLC/TR 17603-31-09
ICS:
49.140 Vesoljski sistemi in operacije Space systems and
operations
kSIST-TP FprCEN/CLC/TR 17603-31- en,fr,de
09:2021

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

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kSIST-TP FprCEN/CLC/TR 17603-31-09:2021
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kSIST-TP FprCEN/CLC/TR 17603-31-09:2021
TECHNICAL REPORT
FINAL DRAFT
FprCEN/CLC/TR 17603-
RAPPORT TECHNIQUE
31-09
TECHNISCHER BERICHT
February 2021
ICS 49.140
English version
Space Engineering - Thermal design handbook - Part 9:
Radiators

Ingénierie spatiale - Manuel de conception thermique - Raumfahrttechnik - Handbuch für thermisches Design -

Partie 9: Radiateurs Teil 9: Radiatoren

This draft Technical Report is submitted to CEN members for Vote. It has been drawn up by the Technical Committee

CEN/CLC/JTC 5.

CEN and CENELEC members are the national standards bodies and national electrotechnical committees of Austria, Belgium,

Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy,

Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Republic of North Macedonia, Romania, Serbia,

Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and United Kingdom.

Recipients of this draft are invited to submit, with their comments, notification of any relevant patent rights of which they are

aware and to provide supporting documentation.

Warning : This document is not a Technical Report. It is distributed for review and comments. It is subject to change without

notice and shall not be referred to as a Technical Report.
CEN-CENELEC Management Centre:
Rue de la Science 23, B-1040 Brussels

© 2021 CEN/CENELEC All rights of exploitation in any form and by any means Ref. No. FprCEN/CLC/TR 17603-31-09:2021 E

reserved worldwide for CEN national Members and for
CENELEC Members.
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Table of contents

European Foreword ................................................................................................... 7

1 Scope ....................................................................................................................... 8

2 References .............................................................................................................. 9

3 Terms, definitions and symbols .......................................................................... 10

3.1 Terms and definitions ............................................................................................. 10

3.2 Symbols .................................................................................................................. 10

4 Diffuse surfaces .................................................................................................... 12

4.1 General ................................................................................................................... 12

4.2 Infinitesimal to finite surfaces .................................................................................. 13

4.2.1 Planar to planar ......................................................................................... 13

4.2.2 Planar to spherical .................................................................................... 19

4.2.3 Cylindrical to spherical .............................................................................. 20

4.2.4 Conical to spherical ................................................................................... 21

4.2.5 Spherical to spherical ................................................................................ 23

4.2.6 Ellipsoidal to spherical ............................................................................... 25

4.2.7 Planar to conical ....................................................................................... 28

4.3 Finite to finite surface .............................................................................................. 31

4.3.1 Planar to planar. Two-dimensional configurations ..................................... 31

4.3.2 Planar to planar. Three-dimensional configurations................................... 35

4.3.3 Planar to cylindrical. Two-dimensional configurations ............................... 46

4.3.4 Planar to cylindrical. three-dimensional configurations .............................. 48

4.3.5 Planar to conical ....................................................................................... 54

4.3.6 Spherical to planar .................................................................................... 56

4.3.7 Cylindrical to cylindrical. two-dimensional configurations .......................... 62

4.3.8 Cylindrical to cylindrical. axisymmetrical configurations ............................. 64

4.3.9 Spherical to cylindrical .............................................................................. 69

4.3.10 Conical to conical ...................................................................................... 72

4.3.11 Conical to spherical ................................................................................... 72

4.3.12 Spherical to spherical ................................................................................ 77

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4.4 Additional sources of data ....................................................................................... 80

5 Specular surfaces ............................................................................................... 103

5.1 General ................................................................................................................. 103

5.2 Two planar specular surfaces ............................................................................... 105

5.2.1 Two-dimensional configurations .............................................................. 105

5.2.2 Parallel, directly opposed rectangles of same width and length ............... 109

5.2.3 Rectangles of same width and length with one common edge ................ 115

5.3 Planar specular and planar diffuse surface ........................................................... 118

5.3.1 Two dimensional cavities. Cylinders of quadrangular cross section ........ 118

5.4 Non-planar specular surfaces ............................................................................... 123

5.4.1 Concentric cylinder or concentric spheres ............................................... 123

Bibliography ........................................................................................................... 125

Figures

Figure 4-1: Geometric notation for view factors between diffuse surface. ............................. 13

Figure 4-2: Values of F as a function of x and y. From Hamilton & Morgan (1952) [15]. ..... 15

Figure 4-3: Values of F as a function of x and y. From Hamilton & Morgan (1952) [15] ...... 17

Figure 4-4: F vs. H for different values of dH. Infinitesimal surface to very thin coaxial

annulus with finite radius. Calculated by the compiler......................................... 18

Figure 4-5: Values of F vs.  for different values of H. The analytical expression (case

I) is only valid in the shadowed region. Calculated by the compiler. ................... 19

Figure 4-6: Values of F as a function of H and . Calculated by the compiler. .................... 20

Figure 4-7: Values of F12 as a function of H and , for  = 10°. Calculated by the

compiler. ............................................................................................................ 21

Figure 4-8: Values of F as a function of H and , for  = 30°. Calculated by the

compiler. ............................................................................................................ 22

Figure 4-9: Values of F as a function of H and , for  = 50°. Calculated by the

compiler. ............................................................................................................ 22

Figure 4-10: Values of F as a function of H and , for  = 80°. Calculated by the

compiler. ............................................................................................................ 23

Figure 4-11: F as a function of H in the case of an infinitesimal sphere viewing a finite

sphere. Calculated by the compiler. ................................................................... 24

Figure 4-12: F as a function of angle  for different values of the dimensionless

distance H. Calculated by the compiler. ............................................................. 25

Figure 4-13: F as a function of  and H, for A = 0,5. Calculated by the compiler. ............... 26

Figure 4-14: F as a function of  and H, for A = 1,5. Calculated by the compiler. ............... 27

Figure 4-15: F as a function of  and H, for A = 2. Calculated by the compiler. .................. 27

Figure 4-16: Values of F vs. M for different values of L. Configuration 1,  = 10°.

Calculated by the compiler. ................................................................................ 29

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Figure 4-17: Values of F vs. M for different values of L. Configuration 1,  = 20°.

Calculated by the compiler. ................................................................................ 29

Figure 4-18: Values of F vs. M for different values of L. Configuration 2,  = 10°.

Calculated by the compiler. ................................................................................ 30

Figure 4-19: Values of F vs. M for different values of L. Configuration 2,  = 20°.

Calculated by the compiler. ................................................................................ 30

Figure 4-20: Values of F as a function of X and Y, for Z = 0. Calculated by the

compiler. ............................................................................................................ 33

Figure 4-21: Values of F as a function of X and Y, for Z = 0,5. Calculated by the

compiler. ............................................................................................................ 33

Figure 4-22: Values of F as a function of X and Y, for Z = 1. Calculated by the

compiler. ............................................................................................................ 34

Figure 4-23: Values of F as a function of X and Y, for Z = 2. Calculated by the

compiler. ............................................................................................................ 34

Figure 4-24: Values of F as a function of X and Y, for Z = 5. Calculated by the

compiler. ............................................................................................................ 35

Figure 4-25: Values of F as a function of X and Y. Calculated by the compiler. .................. 36

Figure 4-26: F as a function of L and N for  = 30°. Table from Feingold (1966) [11],

figure from Hamilton & Morgan (1952) [15]. ........................................................ 39

Figure 4-27: F as a function of L and N for  = 60°. Table from Feingold (1966) [11],

figure from Hamilton & Morgan (1952) [15]. ........................................................ 39

Figure 4-28: F as a function of L and N for  = 90°. Table from Feingold (1966) [11],

figure from Hamilton & Morgan (1952) [15]. ........................................................ 40

Figure 4-29: F12 as a function of L and N for  = 120°. Table from Feingold (1966) [11],

figure from Hamilton & Morgan (1952) [15]. ........................................................ 40

Figure 4-30: F as a function of L and N for  = 150°. Table from Feingold (1966) [11],

figure from Hamilton & Morgan (1952) [15]. ........................................................ 41

Figure 4-31: Values of F as a function of L for different regular polygons. n is the

number of sides of the polygon. From Feingold (1966) [11]. ............................... 43

Figure 4-32: View factors between different faces of a honeycomb cell as a function of

the cell length, L. From Feingold (1966) [11]. ..................................................... 44

Figure 4-33: Values of F as a function of R and R in the case of two parallel coaxial

12 1 2

discs. Calculated by the compiler. ...................................................................... 46

Figure 4-34: Values of F and F as a function of the parameter K. From Jakob (1957)

12 13

[19]. .................................................................................................................... 48

Figure 4-35: F as a function of T and R. Calculated by the compiler. ................................. 49

Figure 4-36: F as a function of T and R. Calculated by the compiler. ................................. 50

Figure 4-37: F as a function of T and R. Calculated by the compiler. ................................. 50

Figure 4-38: F as a function of Z, for different values of the dimensionless radius R.

Calculated by the compiler. ................................................................................ 52

Figure 4-39: F as a function of R for different values of the sector central angel .

12 2

Calculated by the compiler. ................................................................................ 57

Figure 4-40: F as a function of Z for different values of R . Calculated by the compiler. ..... 58

12 2
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Figure 4-41: F from a sphere to both sides of a coaxial intersecting disc, vs. H, for

different values of R. Calculated by the compiler. ............................................... 59

Figure 4-42: F12 from a sphere to the upper side of a coaxial intersecting disc, vs. H (-

1 H 1), for different values of R. Calculated by the compiler. .......................... 59

Figure 4-43: Values of F as a function of Z and R. Calculated by the compiler. .................. 60

Figure 4-44: F12 as a function of x in the case of two infinitely long parallel cylinders of

the same diameter. Calculated by the compiler. ................................................. 64

Figure 4-45: Plot of F vs. L for different values of R. From Hamilton & Morgan (1952)

[15] ..................................................................................................................... 66

Figure 4-46: Plot of F , vs. R for different values of L. From Hamilton & Morgan (1952)

[15] ..................................................................................................................... 67

Figure 4-47: F as a function of R for different values of Z. Calculated by the compiler. ...... 70

Figure 4-48: Values of F as a function of H and L for L = 1. Calculated by the
12 2 1

compiler. ............................................................................................................ 71

Figure 4-49: Values of F as a function of S and D, for  = 15°. From Campbell &

McConnell (1968) [4]. ......................................................................................... 73

Figure 4-50: Values of F as a function of S and D, for  = 30°. From Campbell &

McConnell (1968) [4]. ......................................................................................... 74

Figure 4-51: Values of F as a function of S and D, for  = 45°. From Campbell &

McConnell (1968) [4]. ......................................................................................... 75

Figure 4-52: Values of F as a function of S and D, for  = 60°. From Campbell &

McConnell (1968) [4]. ......................................................................................... 76

Figure 4-53: Values of F as a function of S and R. From Jones (1965) [21]. ...................... 79

Figure 4-54: Values of F as a function of S and . From Campbell & McConnell

(1968) [4]. ........................................................................................................... 80

Figure 5-1: Values of F as a function of R and H. Calculated by the compiler. ................. 106

s s

Figure 5-2: Values of F / as a function of R and H. Calculated by the compiler. ........... 106

11 2

Figure 5-3: Values of F as a function of R for different values of . Calculated by the

compiler. .......................................................................................................... 108

s s

Figure 5-4: Values of F / as a function of R for different values of . Calculated by

11 2

the compiler. .................................................................................................... 109

Figure 5-5: Values of F as a function of R and X for Z = 1. Calculated by the

compiler. .......................................................................................................... 110

s s
Figure 5-6: Values of F / as a function of R and X for Z = 1. Calculated by the
11 2

compiler. .......................................................................................................... 111

Figure 5-7: Values of F as a function of R and X for Z = 5. Calculated by the

compiler. .......................................................................................................... 111

s s
Figure 5-8: Values of F / as a function of R and X for Z = 5. Calculated by the
11 2

compiler. .......................................................................................................... 112

Figure 5-9: Values of F as a function of R and X for Z = 10. Calculated by the

compiler. .......................................................................................................... 112

s s

Figure 5-10: Values of F / as a function of R and X for Z = 10. Calculated by the

11 2

compiler. .......................................................................................................... 113

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Figure 5-11: Values of F as a function of R and X for Z = 15. Calculated by the

compiler. .......................................................................................................... 113

s s

Figure 5-12: Values of F / as a function of R and X for Z = 15. Calculated by the

11 2

compiler. .......................................................................................................... 114

Figure 5-13: Values of F as a function of R and X for Z = 20. Calculated by the

compiler. .......................................................................................................... 114

s s

Figure 5-14: Values of F / as a function of R and X for Z = 20. Calculated by the

11 2

compiler. .......................................................................................................... 115

Figure 5-15: Values of F vs. aspect ratio, L, for different values of R.  = 30°.

Calculated by the compiler. .............................................................................. 116

s s

Figure 5-16: Values of F / vs. aspect ratio, L, for different values of R.  = 30°.

11 2

Calculated by the compiler. .............................................................................. 117

Figure 5-17: Values of F vs. aspect ratio, L, for different values of R.  = 45º.

Calculated by the compiler. .............................................................................. 117

s s s

Figure 5-18: Values of F and F / vs. aspect ratio, L, for the limiting values of .

12 11 2

Calculated by the compiler. .............................................................................. 118

s s

Figure 5-19: Values of F vs. for different values of the specular reflectance,  .

Calculated by the compiler. .............................................................................. 120

s s

Figure 5-20: Values of F vs. for different values of the specular reflectance,  .

Calculated by the compiler. .............................................................................. 121

s s

Figure 5-21: Values of F vs. for different values of the specular reflectance,  .

Calculated by the compiler. .............................................................................. 121

s s

Figure 5-22: Values of F vs. for different values of the specular reflectance,  .

Calculated by the compiler. .............................................................................. 122

s s

Figure 5-23: Values of F vs. for different values of the specular reflectance,  .

Calculated by the compiler. .............................................................................. 122

s s

Figure 5-24: Values of F 34 vs. for different values of the specular reflectance,  .

Calculated by the compiler. .............................................................................. 123

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European Foreword

This document (FprCEN/CLC/TR 17603-31-09:2021) has been prepared by Technical Committee

CEN/CLC/JTC 5 “Space”, the secretariat of which is held by DIN.
This document is currently submitted to the Vote on TR.

It is highlighted that this technical report does not contain any requirement but only collection of data

or descriptions and guidelines about how to organize and perform the work in support of EN 16603-

31.

This Technical report (FprCEN/CLC/TR 17603-31-09:2021) originates from ECSS-E-HB-31-01 Part 1A .

Attention is drawn to the possibility that some of the elements of this document may be the subject of

patent rights. CEN [and/or CENELEC] shall not be held responsible for identifying any or all such

patent rights.

This document has been prepared under a mandate given to CEN by the European Commission and

the European Free Trade Association.

This document has been developed to cover specifically space systems and has therefore precedence

over any TR covering the same scope but with a wider domain of applicability (e.g.: aerospace).

This document is currently submitted to the CEN CONSULTATION.
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Scope

In this Part 9 of the spacecraft thermal control and design data handbooks, view factors of diffuse and

specular thermal surfaces are discussed.

For diffuse surfaces, calculations are given for radiation emission and absorption between different

configurations of planar, cylindrical, conical, spherical and ellipsoidal surfaces for finite and infinite

surfaces.

For specular surfaces the affect of reflectance on calculations for view factors is included in the

calculations. View factors for specular and diffuse surfaces are also included.
The Thermal design handbook is published in 16 Parts
TR 17603-31-01 Thermal design handbook – Part 1: View factors
TR 17603-31-02 Thermal design handbook – Part 2: Holes, Grooves and Cavities
TR 17603-31-03 Thermal design handbook – Part 3: Spacecraft Surface Temperature
TR 17603-31-04 Thermal design handbook – Part 4: Conductive Heat Transfer

TR 17603-31-05 Thermal design handbook – Part 5: Structural Materials: Metallic and

Composite
TR 17603-31-06 Thermal design handbook – Part 6: Thermal Control Surfaces
TR 17603-31-07 Thermal design handbook – Part 7: Insulations
TR 17603-31-08 Thermal design handbook – Part 8: Heat Pipes
TR 17603-31-09 Thermal design handbook – Part 9: Radiators
TR 17603-31-10 Thermal design handbook – Part 10: Phase – Change Capacitors
TR 17603-31-11 Thermal design handbook – Part 11: Electrical Heating
TR 17603-31-12 Thermal design handbook – Part 12: Louvers
TR 17603-31-13 Thermal design handbook – Part 13: Fluid Loops
TR 17603-31-14 Thermal design handbook – Part 14: Cryogenic Cooling
TR 17603-31-15 Thermal design handbook – Part 15: Existing Satellites
TR 17603-31-16 Thermal design handbook – Part 16: Thermal Protection System
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References
EN Reference Reference in text Title
EN 16601-00-01 ECSS-S-ST-00-01 ECSS System - Glossary of terms

TR 17603-31-03 ECSS-E-HB-31-01 Part 3 Thermal design handbook – Part 3: Spacecraft

Surface Temperature

All other references made to publications in this Part are listed, alphabetically, in the Bibliography.

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Terms, definitions and symbols
3.1 Terms and definitions

For the purpose of this Standard, the terms and definitions given in ECSS-S-ST-00-01 apply.

3.2 Symbols
surface area of the i-th surface, [m ]
energy flux leaving surface, i. often called radiosity,
[W.m ]
view factor from diffuse surface, Ai to diffuse surface,
Fij
view factor from the ensemble of diffuse surfaces, Ai1,
F(i1,i2,...in)(j1,j2,...jn)
Ai2,...Ain to the ensemble of diffuse surfaces, Aj1,
Aj2,...Ajn
view factor from specular surface Ai to specular
Fij
surface Aj
energy flux incident on surface i, [W.m ]
term which appears in the expression for the view
Ki2
factor between elements of parallel plates, Ki2 = AiFii'
fraction of the radiative energy leaving Am which
Kmn(i,j,k,p,q,...)
reaches An after i perfectly specular reflections from
surface Ai, j from surface Aj, k from surface Ak,...
distance between two differential elements, [m]
temperature, [K]
angle from normal to surface i, [angular degrees]
hemispherical emittance of a (diffuse-gray) surface
hemispherical diffuse reflectance of a (diffuse-gray)
surface
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specular reflectance of a (gray) surface, it is assumed
to be independent of incident angle
8 2 4
 Stefan-Boltzmann constant, S = 5,6697x10 W.m .K

Other symbols, mainly used to define the geometry of the configurations, are introduced when

required.
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Diffuse surfaces
4.1 General

The view factor, F12, between the diffuse surface A1 and A2, is the fraction of the energy leaving the

isothermal surface A1 that arrives at A2.

If the receiver surface is infinitesimal, the view factor is infinitesimal for both infinitesimal and finite

emitting surfaces, and is given by the expression
cos  cos 
1 2
dF  dA [4-1]
12 2
when both surfaces are infinitesimal, and by
dA cos  cos 
2 1 2
dF  dA
[4-2]
12 1
 2
A S
when A1 is finite.

If the receiver surface is finite, the view factor is finite for both infinitesimal and finite emitting

surfaces, and is given by the expression
cos  cos 
1 2
F  dA
[4-3]
12 2
 2
when A1 is infinitesimal, and by
1 cos  cos 
1 2
F  dA dA
[4-4]
12 2 1
  2
A S
A A
1 2
when A1 is finite.
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Figure 4-1: Geometric notation for view factors between diffuse surface.

Regardless of which surfaces are considered, their view factors satisfy the following reciprocity

relation:
A1F12 = A2F21

If we consider the diffuse surfaces A1, A2 and A3, the view factor between the surfaces A1 and A2 + A3 is

F1(2,3) = F12 + F13,
when the receiver surface is formed by two surfaces, and
A F  A F
2 21 3 31
F 
[4-5]
 2 , 31
A  A
2 3

when the emitting surface is formed by two surfaces. notice that the notation F1(2,3) and F(2,3)1 will be

used in the following data sheets.

When an enclosure of N surfaces A1,A2,....,An is considered, their view factors satisfy the relation

F  1 [4-6]
j1

for any surface Ai. This relationship results from the fact that the overall heat transfer in the enclosure

should be zero.
4.2 Infinitesimal to finite surfaces
4.2.1 Planar to planar
4.2.1.1 Two-dimensional configurations

A plane point source dA1 and any surface A2 generated by an infinitely long line moving parallel to

itself and to the plane dA1.
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Formula:
F  cos   cos   [4-7]
References: Hamilton & Morgan (1952) [15], Moon (1961) [26], Kreith (1962) [22]
Comments:
Notice that F12 is independent of the shape of A2 for given
...

Questions, Comments and Discussion

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