Fortran IV programs to calculate radiant energy interchange factors

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Fortran IV programs to calculate radiant energy interchange factors

Mitalas, G. P.; Stephenson, D. G.

.zqoO-;)

loMrlJl\.

ff\o&RA(YI

NO

J.5

ｾ

Ｍ ｰ［ｱｾ＠

ｦＮｾ＠

y :-

Jv\

-L '( - \

q.\o

\c, .

I.

National Research Conseil

national

Council Canada

de

recherches Canada

DIVISION OF BUILDING RESEARCH

FORTRAN IV PROGRAMS TO CALCULATE RADIANT

ENERGY INTERCHANGE FACTORS

BY

G.P. MITALAS AND D.G. STEPHENSON

DBR COMPUTER PROGRAM NO. 25

OTTAWA JULY 1966

This publication series has been initiated by the Division of BuildiJ?-g Research as a convenience in the listing and exchange of computer programs which are developed

in the course of i ts ｷｯｲｫ ｾ＠ Programs

sub-mitted by the ｾ ｩｶｩｳｩｯｮ＠ to user groups and

available elsewhere will be lis ted in the series as well as those of less general in-terest which will be available only fro .m

tl:J,e Division. A list of all . programs in

the series will be made available on

re-quest. ｃ ｾ ｰｩ･ｳ＠ of the program tapes or

car ds are· also available for some of the propams in. this series.

>

t(tr

,J

.

, .

...

..!? t)

?

I • I

NATIONAL RESEARCH COUNCIL CANADA

DIVISION OF BUILDING RESEARCH

FORTRAN IV PROGRAMS TO CALCULATE RADIANT ENERGY INTERCHANGE FACTORS

by

G.P. Mitalas and D.G. Stephenson

Computer Program No. 25 of the

Division of Building Research

Ottawa July 1966

FORTRAN IV PROGRAMS TO CALCULATE RADIANT ENERGY INTERCHANGE FACTORS

by

G. P. Mitalas and D. G. Stephenson

The calculation of energy interchange by radiation between the surfaces that enclose a room is an important step in the design of both air-conditioning and lighting systems for buildings. These calculations require the evaluation of special factors that depend on the shape and the radiation properties of the room surfaces. The short-wavelength radiation from lights and the sun, and the long-wavelength radiation that is emitted by the surfaces have to be considered separately because the radi-ation properties of the surfaces can be very different for the different wavelengths. The programs presented in this report allow both the short-wave and long-wave factors to be evaluated by a digital computer. There can be up to fifty surfaces forming the enclosure, and it is assumed that over each of these surfaces the surface reflectivity, emissivity, temperature and illumination are uniform.

DEFINITION OF TERMS

In this report the suffix 11_{-ivity" is used to denote the}

dimensionless ratio of two radiant fluxes; the suffix 11

-ance" denotes a flux density.

FORM FACTOR, f

In the case of radiation transfer between two finite

surfaces identified by the subscripts, i and j, the form factor, f. . I

1' J

is the ratio of the flux incident on surface j from surface i, to

the total flux leaving surface i. (If one of the surfaces has area

dA, the factor is called a configuration factor I c. . • )

1, J

SHORT-WAVELENGTH FACTORS

Excitance I G

The excitance is the short-wave radiant-energy

flux-density incident on a surface from

a

primary source su ch as a lamp

2

-Luminance, L

The luminance is the short-wave radiant-energy flux

density leaving a surface*

L.

=

p. {G.

+

EN f₁• J. LJ. }

1 1 1 j=

1

t

••• (1)

where

is the reflectivity of surface i for short-wave

pi

radiation

N is the number of surfaces forming the enclosure, The set of equations for the luminance of all the surfaces can be combined into the following matrix equation.

(L)

= [H) •

1 •

(G)

_{•·• • (Z)}

where

(H]

= [

1/

p) ..

(£]

••• ( 3)

The

[t/

p]

matrix has zero elements everywhere except on the diagonal,

and the diagonal element that is common to row i and column i is 1/p ..

The element in row i and column j of the form factor matrix,

[£],

is 1

simply f ..•

l,J

Illuminance, I

. . The illuminance is the total radiant-energy flux-density

1nc1dent at a surface

I.

1

=

G

_i

and in matrix form:

[I]

=

where

+

I:

N f. . •

j=

1

1, J

L.

_J

[l]

is a unit matrix of order N

••• (4)

••• (5)

•

The illumi · t that

it i _{s expressed m po} . nance ls the same as the illumination excep _{. .}

wer umts mstead of light units.

*

In this repo t 't ·

r 1 ls assumed that this radiation is perfectly diffuse.

3

-Short-Wave Absorptance, B

The short-wave absorptance is the short-wave radiant-energy flux density absorbed by a surface,

• • • ( 6)

Short-Wave Absorptivity Matrix, S

The absorptivity matrix,

[s] ,

relates the short-wave

absorptance,

[.BJ,

to the excitance matrix, [G] , viz:

From equations (2}, (5}, (6) and (7) it follows that

and

[s]

[r]

=

[lj

+

=

[CsJ

+

LONG-WAVELENGTH FACTORS Emittance, W • • • ( 7)

...

(8) (Sa)

The emittance of a su.rface is the radiant-energy flux density emitted by a surface:

where

w

_i

=

E:. a T . 4

1 1

E:. is the emissivity of surface i

1

• • • (9)

a

T4 is the emissive power of a black body at absolute

temperature, T.

Radiance, R

The radiance is the long-wave radiant-energy flux density leaving a surface':c (1- e .) N R . R =

w

+

r.

f. 1 i 1 _j=l 1, J _J • • • ( 1 0} In matrix form,

LRJ

= [Mj-1

[yv]

• • • ( 11)

::: It is assumed that the surface emits and reflects long-wavelength radiation diffusely.

4

-where

ll] - [£]

+

[e] •

• • • ( 12)

[.M]

=

. has zero elements everywhere except on the diagonal;

The matnx, E: • ••

h d. 1 element in row i and column I Is

e. •

t e 1agona I

Long-Wave Absorptance, C

The long-wave absorptance is the long-wave radiant-energy flux density absorbed by a surface

[c]

₌

LRJ

• • • ( 13)

Long-Wave Absorptivity Matrix, [F]

The absorptivity matrix, [F), relates the long-wave absorptance,

[C],

to the emittance and emissivity matrices, viz:

••• ( 14)

From equations (11), (13) and (14) it follows that

[f]

_[MJ-1

_{• ( e]}

••• (15)

The elements, Fi,j, in this matrix are exactly the same as script, Fi,j• introduced by Hottle for gray enclosures (I). The foregoing formulae show that the luminance and absorptance and illuminance

ｶ｡ｬｾ･ｳ＠ for all the room surfaces can be evaluated very easily for any

･ｸ｣ｬｴｾｮ｣･＠

values if the [Hj-l and [S] matrices are known. These matnces are character1' t' _{s 1cs o a particular room but are Indep-endent} f · · of the radiation densit · th _{Y ln e room. Similarly, the radiance and long·} . .

wave absorptance values b f ·

r·

rl-1

[ _{FJ matnces are known.} ,1 . can e ound very easily when the ｾｊ＠ and

EXPRESSIONS FOR FO

RM AND CONFIGURATION FACTORS

given by The form factor between two finite areas' A₁ and A₂ lS .

and the configuration factor

cos 'i'l cos 2

nr

...

Ref. ( 2)

( 16)

between an infinitesimal area, dA , and

0

5

-finite area, A

1, is given by

where

cos 'i'o cos 'i'l

COl = JAl 2 dAl

nr Refo (2)

• • • ( 17)

r

=

distance between dA

1 and dA2 or dA0 and dA1 'i' = angle that the line joining dA

1 and dA2 or dA0

makes with the normal to dA.

The area integral in expressions {16) and (17) can be replaced by a line integral around the perimeter of A; i.e.,

and where 1

!

1 cOl = 2n

J

1 2 r + In r dz 1 dz2} ••• (18) Refo ( 3) { [- rn 0 ( z 1 - z 0)

+

n 0 ( y 1 - y 0)

J

dx 1

+ [ +

.to (

zl - zo) - no (xi - xo>] dy 1

+[-.tO (yl -yO)+ rnO (xl -

ｸｏｾ､ｺｬｽ＠

• • • ( 19)

r = _{distance between point {x 1, y 1 , z 1 ) on the}

perimeter of A1 and (x2, Y2• z2) on the perimeter of A 2 , or distance between a point (x

1, y1, z1 ) and dA0

.t

0, rn0, n0 = direction cosines of the normal to dA0• When A is enclosed by straight lines, the expres sian under the integral can be simplified by expressing the equations of the boundary lines in parametric form; i.e., for the line from the point, Lk,to point, Lk+l'

xl = yl

=

zl = where

6

-+ a,ke dx₁ = a.kde xk yk + eke dy1 = ｾ､･＠ • • • (20) zk + yke dz1 = ykde

= coordinates of the point, Lk, from which the distance, e, is measured.

=

direction cosines of the line, Ek, from Lk

to Lk+

1, which forms part of perimeter of area 1;

similarly, x , y , z _{q q q} = coordinates of the point _,

L , from which the distance, u, is measured.

q

=

direction cosines at the line, U , from L to

q q

Lq+l' which forms part of perimeter of area 2.

Substituting in expressions (18) and (19) the equations for

ｾｨ･＠ boundary lines in parametric form, and noting that the line

lntegral around the entire perimeter is just the sum of the integrals for the straight-line segrnents gives

1 K Q fl2 = _2rrA I: I:

₀

_(k,q)f

f

1 k=l q=l ln r de du Ek U and q • • • { 21) 1 K COl = I: Dk

f

de 2rr _k=1 2 Ek r where ••• ( 22) K and Q = number of st . h 1'

. ra1g t- me segments forming the

penmeter of A and A

cjJ

(k, q) = 1 _{2 , respectively.}

ｾ＠

aq

+

ｾ＠

8 _q

+

yk y

_q

Dk =

Ｈｾ＠

xO), tO,

ｾ＠

(yk - yo), mo, [\

(zk - zo), n

0' yk

The

Ｎ･ｸｾｲ･ｳｳｩｯｮｳ＠

(21) and (22)

the lndlcated integrations. can be further simplified by carrying out

Thus, £12 = where and 7 -l K Q

r.

r.

2nA 1 k=l q=l

(/J

(k, q)

J

U [(T cos ;>.. ln T q + J cos 8 ln J

+

V w - E) du] k

T, J, ;>.., 8, V and w are functions of u;

1 col = 2rr K L: k=l ,q

The new symbols are defined in Figure 1.

• • • ( 2 3)

••• ( 24)

The sign of each term in the summation of expression (24) depend·s on the direction in which the line integral is evaluated. As configuration factor is always a positive number, the integral can be evaluated by following the boundary in either direction and taking the

absolute value of the sum. It should be noted also, that expression (24)

does not hold when the dA 0 is on any line segment or .extension of a line segment that forms part of the perimeter of A 1, because then the Sin 8 is zero. Ambiguity may arise when the surface is defined by a contour because the same contour defines two different surfaces. For example, the contour of the ceiling defines the boundaries of the ceiling as well

as the boundaries of the rest of the room enclosure. Therefore, it

should be noted that the value given by expression (24) is the configuration factor from an infinitesimal area, dAo, and area, A1, where A 1 does

not include dA_0•

The integrals in expressions {23) and (24) mU£t be evaluated only over the parts of A 1 that are directly visible from A 2 or dA0 •

These equations are valid even when A1 or A 2 are not plane surfaces.

The integral in expression (23) can be approximated by dividing the line U into equal segments and using the average of the

values at the

ｭｩ､Ｍｾｯｩｮｴｳ＠

of each segment. The error due to finite

difference integration can be estimated by noting the convergence of

the form factor value as the number of divisions of the U line segment

q

is increased.

U,

q

When the point, Lk, is on the line, U g' or the extension of

the integral,

J

U J cos 8 ln J du, can be evaluated exactly because

8

-cos 8 = f2f (k, q) and is independent of u, so

r

J cos 8 ln J du J

u

-

})

q where dl d2 = = - dl 2 (ln2dl

_{- 4) }}

1\

_¢

_{(k, q)}

distance between points, Lk and L q

distance between points, Lk and L q+ ₁

• • • ( 25)

Similarly, the integral,

Ju

T cos ;\. ln T du, can be evaluated exactly

q

when the point, L , is on the line, U , or the extension of the line,

k+l q

u .

q

The area of a plane surface bounded by K straight-line

segments and defined by K coordinate points is given by

A = 1/2 V(A )2 + {A )2 + (A )2 X y z ••• {26) where K-2 A = 2: _{{ (yk+ 1}

-

_{y 1} (zk+2} zl) X k=l - {zk+ 1

-

zl) {yk+2 - y l) } • • • ( 27} K-2 A _{= 2: {(zk+ 1}

-

_{zl) (xk+2 xl)} y k=l - {xk+l - xl) {zk+2 - X ) _l

}

• • • ( 28) K-2 A = 2: _{z {(xk+ 1- X 1) ( y k+ 2- y 1)} k=l - {yk+l - y 1) (xk+2

-

XI)

}

' • • • { 2 9)

9

-GENERAL DESCRIPTION OF THE PROGRAMS

These Fortran IV programs are for an IBM - 360 computer with a line printer.

A. Program to calculate the configuration factors between an

infinitesimal surface and surfaces enclosed by straight lines

The coding sheets and a sample of input-output of this program are given on pages A-1 to A-3 of Appendix A.

The program assumes that the finite surfaces are defined by a set of co-ordinate points (x, y, z) joined by straight-line segments and that the infinitesimal surface is defined by the centre point

(x , y , z ) and two additional points that lie in the plane of the

. 0. . 0 . 0 l f

1nfln1tes1ma sur ace.

The program can handle any number of surfaces provided only that the total number of defining points is not more than 200. The

areas of the surfaces are calculated if it is indicated that the surfaces

are planes. If more than one surface is used in calculation, each

surface must be defined by the same number of straight-line segments and the line segment cannot be of zero length.

Input:

(1)

Card 1 ( 2) Card 2 ( 3) Card 3 (4) Card 4 (5) Cards

Co-ordinate points defining the infinitesimal area

Format: Floating point, 8 Col. The centre point,

Col. 1{8)24. The other two points, Col. 25(8)72

Number of surfaces

Format: Fixed point, Col. 1, 2 and 3

- Number of points defining each surface

Format: Fixed point, Col. 1, 2 and 3

Code Number

=

1 or 0

CN = 1, plane surfaces

CN = 0, curved surfaces

Format: Fixed point, Col. 1

Co-ordinate points defining the regions enclosed. by straight-line segments. Each point must be d_eflned

by x y z co-ordinates. The points must be 1n

con-' ' . t

( 6) The last card Output: ( l) { 2) Format: ｾ＠ 10

-Floating point, nine numbers per card, Col. 1{8)72

Terminator record (Exit control number)

>

1000

Format: Floating point, 8 Col.

Configuration factor from infinitesimal area to the surface, Area of the surface,

B. Program to calculate form factors between surfaces enclosed by

straight lines

The coding sheets and a sample of input-output are given on

pages A-4 to A-7 of Appendix A.

The regions can be defined by three or four points (i.e., surfaces enclosed by three- or four-line segments). The program assumes that the surfaces are plane areas and uses the equality,

A. f . .

=

A.£ ..•

1 lJ J Jl

The surfaces must form a complete enclosure.

The maximum number of surfaces that can be handled by this

program is 50.

The program adjusts the calculated form factors so the sum

ｾｦ＠

the form factors from any one of the surfaces to the rest of the enclosUI

18

equal to one, This adjustment is required because the form factors calculated, using the finite difference integration, are not exact. Input:

Card 1

( 1)

- Number of surf _{aces orm1ng the enclosure} f .

Format: Fixed point number, Col. 1, 2 and 3

- Number of p · t d . .

F . .om 5 ehnmg the surfaces { 3 or 4)

ormat. Flxed point number C

1 ol. 3

( 2) _{Card 2}

(3) _Cards

ｾ＠ Co-ordinate point d f' .

st · h . 5 e mmg the surfaces enclosed by

ralg t-hne se

b gments • Each point must be defined

y x, y, z ｣ｯｾｯｲ､ｩｮ＠ t T

consecut· a es. he points must be in

lve order d'

perimet 1 _{accor mg to the position on the}

er. 3 I 4 I o o o (4) The second last card (5) The last card Output: ( l) (2) ( 3) >:C ( 4) :;c Format: 11

-Floating point, nine numbers per card, Col. l (8)72

The number of divisions on line segment, U,

Format: Floating point, Col. 1 to 8

Terminator record (Exit control number)

>

50.

Format: Floating point, Col. 1 to 8,

Surface areas

Matrix of calculated form factors printed in row form The number of areas {printed and punched)

Matrix of adjusted form factors printed and punched in row form.

Format: Floating point, eight numbers per card,

Col. 2(9)73

C. Program to calculate [H] -l,

[_s] ,

[M] -1 and

[F]

The coding sheets and a sample of input-outl(ut are given on pages A-8 to A-13 of Appendix A.

The program reads a matrix of form factors for an enclosure, combines these with emissivity and reflectivity factors for the surfaces and calculates the four characteristic matrices. It assumes that the matrix of form factors is for a complete enclosure.

The maximum matrix dimensions that can be handled by the program is 50 x 50.

Input:

( 1 )':' Card 1

(2}':' Cards

2 tom

The number of surfaces

Format: Fixed point, Col. 1, 2 and 3

The matrix of form factors

Format: Floating point, eight numbers per card,

Col. 2(9}7 3. Matrix must be in row form,

':' The inputs (1) and (2) are of the same format and sequence as the

outputs (3) and (4) of the program to calculate form fa ctors between regions enclosed by straight lines (program B).

(3) Card m + 1 (4) Cards m + 2 to n (5) Card n

+

1 ( 6) Cards n

+

2 to p {7) Last Card Output: ( 1) ( 2) ( 3) (4) ( 5) ( 6) (7) 12 -w Code Number

=

1 or 0

Format: Fixed point, Col. 1 . .

The Code Number = 0 md1cate s that there

is no emissivity data. The reading of the

emissivity values, ｾｦ･ｲ･ｦｯｲ･Ｌ＠ and the

calculations of

[MJ

and

[F

J

are deleted if

Code Number is zero.

_ Emissivities of the enclosure surfaces. The. emissivity

values must be in sequence {i.e.,

e

_{1 ,}

e

₂ • o.) and one

value per card.

Format: Floating point, Col. 1 to 5.

- Code Number

=

1 or 0

Format: Fixed point, Col. 1

The Code Number = 0 indicates that there

is no reflectivity data. Therefore, the reading of the reflectivity values and the

calculations of [HJ1 and

[s]

are deleted if

Code Number is zero.

- Reflectivities of the enclosure surfaces. The reflectivity

values must be in sequence (i.e., p , p , p , ••• ) and

one per card, If any value is zero i1

ｳｨｾｵｬ､＠

3be

repre-sented as lo- 10

Format: Floating point, Col. 1 to 5

- Terminator Record (Exit Control number)

>

50.

Format: Floating point, Col. 1 to 8.

The number of areas

The ｶｾｬｵ･＠ of determinant of ｛ｾ＠ and the test number

ｾｳｳｯ｣ｬｾｴ･､＠ with the inver sian of matrix [M] • (If the

mverslOn of

[M]

is completed, the test number is zero.)

The matrix _•

[M]-

1 _{, in row form}

The matrix [Fl _{' .J ' in row form}

The value of determinant of [H]

associated with . . and the

the lnverslon of _{rna r1x,} t .

The matrix

[I{)

-1 .

' ' ln row form test number

[H]

The matrix

[s

1 . ' :..J ' ln row form 13 -REFERENCES

(1} McAdams, W. H. Heat Transmission. McGraw-Hill Book

Company, 1954.

(2) Kreith, F. Principles of Heat Transfer. International

Textbook Co., Scranton, 1959.

(3) Sparrow, E. M. A New and Simpler Formulation for

Radiative Angle Factors. ASME Journal of Heat Transfer, May 1963, p.81.

ACKNOWLEDGEMENT

The authors gratefully acknowledge the preparation of the

coding sheets and the checking of the programs by Mr. J. G, Arseneault.

The many helpful suggestions made by Dr. Gosta Brown of the Royal

Line segment forming

part of the perimeter

of the region A,

Lk=(Xk,Yk,Zk)

ek /

ｾｾ＠

L ktr

FIGURE 1

•a---... ...__

APPENDIX A

Coding Sheets and Input-Output Samples for Programs A, B and

c

c.

I NP= l ! 0U T= 3

ｇｉｾｏｏｏｃ Ｗ＠

;IJR- i 1 E ( i

ocr-;-i-r·)·---

--- --·

·-··--· --- -- ·- -· ---·

··-11 FORMAT I 3 1H PROGRAM BR 7 G. ARSEN EAULT M20l

R Ｑｾ＠ tAD I I NP, 3 1 I I PI I, J l • J ;;-r -;··"':)·y ;T;:·1-; "3T·

-1 1-J ;{"'A r < 9 f' B • 1 1

· Tn Vi r-;-rr= 10 o1J.CT9·v 2; 'i o-.r; 9··o 3 --·· ·· · · .. - --- .. -- --- - - - ---

·-9..; 2 v L

= (

p ( 3 • 3 ) - p ( 2 • 3 J )

* (

p ( 1 • 2 ) - p ( 2 I 2 ) ) - ( p ( 3' 2 ) -p ( 2 '2 ) )

* (

p ( 1 • 3 ) -p ( 2 • 3 J J - -- ---v;,;- = 1 PT3-;-n-=PTZ;T))*TPTrt- 3T=Pl .2;

:n:r-.:·(

P ｮ Ｍ ［ＳＩＭｐｔｔｴｊｬｾｦｔｐｔｩｴ Ｍ ｬｬ＠ :.1JT29Tl l - · ----· v N = ( p ( 3 ' 2) -P ( 2 '2) ) lf ( p ( l ' 1 J -P ( 2 I 1) ) - ( p (

3'

1) ..!p ( 2 I l ) )

* (

p ( 1. 2) -P ( z '2) ) X-VL • VL+VM * VM+VN*VN X=SQRT IXJ - --.,-,V-:-L-= V L I X ｶｲｾ］ｖｩ Ａｉｘ＠ VN = VNIX ·- -- - - -- -- - - ----·-

-,'-1 Ｑ｜｜ｩ ﾷｾ＠

= ().

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