Graphical Determination of View Factors

Publisher’s version / Version de l'éditeur:

Technical Translation (National Research Council of Canada), 1961

READ THESE TERMS AND CONDITIONS CAREFULLY BEFORE USING THIS WEBSITE. https://nrc-publications.canada.ca/eng/copyright

Vous avez des questions? Nous pouvons vous aider. Pour communiquer directement avec un auteur, consultez la première page de la revue dans laquelle son article a été publié afin de trouver ses coordonnées. Si vous n’arrivez pas à les repérer, communiquez avec nous à PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca.

Questions? Contact the NRC Publications Archive team at

PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca. If you wish to email the authors directly, please see the first page of the publication for their contact information.

Archives des publications du CNRC

For the publisher’s version, please access the DOI link below./ Pour consulter la version de l’éditeur, utilisez le lien DOI ci-dessous.

https://doi.org/10.4224/20331604

Access and use of this website and the material on it are subject to the Terms and Conditions set forth at

Graphical Determination of View Factors

Johannes, W.

https://publications-cnrc.canada.ca/fra/droits

L’accès à ce site Web et l’utilisation de son contenu sont assujettis aux conditions présentées dans le site

LISEZ CES CONDITIONS ATTENTIVEMENT AVANT D’UTILISER CE SITE WEB.

NRC Publications Record / Notice d'Archives des publications de CNRC:

https://nrc-publications.canada.ca/eng/view/object/?id=d9521eaf-a761-40a2-bb0f-1d43de156ff7 https://publications-cnrc.canada.ca/fra/voir/objet/?id=d9521eaf-a761-40a2-bb0f-1d43de156ff7

PREFACE

One of the difficult aspects of calculating the heat exchanged by radiation between two surfaoes is to

find the appropriate view factor. In this paper and the

accompanying data sheets Herr Johannes has provided a very convenient method of obtaining the view factors which are needed for most radiant heating calculations.

This paper complements the paper by Kollmar which was published in the same issue of Gesundheits-Ingenieur and which is also available as NRC TT-930.

The Division is indebted to Mr. D.A. Sinclair of the TranBlations Section of the National Research Council for preparing this translation.

Ottawa,

January 1961

R.F. Legget, Director

Technical Translation 931

Title: The graphical determination of view factors

(Graphische Errnittlung von Einstrahlzahlen)

Author: W. Johannes

Reference: Gesundheit-Ingenieur, 80 (8): 235-238, 1959

THE GRAPHICAL DETERMINATION OF VIEW FACTORS

Radiation plays an important part in heat transfer since it is always present whenever two solids with different surface tempera-tures are not in direct contact, i.e., are separated by an

inter-vening air space. Indeed even liquids and certain gases at elevated

temperatures radiate heat around them. The physical laws underlying

heat radiation are generally known. With these laws and the view

factor (or angle relation) of the given radiation exchange the theoretical calculation of the heat transfer due to radiation

be-comes a practical possibility. The view ractor, however, introduces

an element of difficulty* because its calculation for other than simple

geometric surfaces and at obtuse angles is troublesome or tiresome. It is not surprising, therefore, that in practice measurements are

often taken. In the heating of living space the situation is

some-what better inasmuch as the heating bodies of panels are generally

of simple geometric form. This is particularly true for panel

heating on ceilings, walls or floors. The boundary surfaces

receiv-ing the radiated heat are also flat and rectangular.

The numerical computation of the view factors (angle relations) of rectangular surfaces that exchange heat ir a prismatic or cubic

space is relatively simple with the aid of the equations(l) ,

al-though it consumes a good deal of time. Where there is great

eccentricity of the reciprocally radiating surfaces the equation

may show & multitude of individual calculation terms. The

calcula-tion then demands concentracalcula-tion as well as time in order to achieve

an accurate result. Recently some data sheets have been

pub-lished(2) from which the view factors can be re&d directly for

simple radiation cases. The further graphical representation of the

view factor equations, beyond point and surface particle radiation, is possible in a simple manner only for parallel and perpendicular

•

The fact that I have an opportunity of gOing into this more fully

rectangular surfaces of equal size. For all other cases the graphi-cal representation of the equations in the form offered is more

difficult, because there are generally more than three variables. Data Sheet 56·

It will be shown here, with reference to one case of radiation, how an equation that is difficult in appearance can nevertheless be represented graphically in a comparatively simple form. This goes back to the recognized order in forming the view factor equations for rectangular surfaces in the parallelepiped obtained through additive and substractive relationshiPs(l). Fig. 1 shows the ir-radiation of a rectangular surface ab by a rectangular surface

(a + w)h standing perpendicular thereto but being of different length and separated therefrom. The equation for this is (reference 1,

equation (30»:

1 [ n a 1£1

qJ142 = 2n

h-n

nlarc tg

-r

+a larc tgT --wiarctg-

_r

n a IV

- nvarctg - - avarc tg - + wv arc

tg-v v v

n _ . n

_ n11_hz+_I"

arc tg - -..-. +nih"+v"arc t g

-Ih"+I" Ih"+v" - - - a . - a _ aIh"+ ["arc tg ｾ］Ｌ +a1h"+v"arc tg ＭｾＭ Yh2_{+ l" Ih"+v"} --- 1£1 .- - 1£1 +1£1I'h2+ ["arc tgｾ］ - 1£11h"+v"arc tgＭ［Ｍｾｾ Ih2+_{I" 1h"}+_v" n2 _{(h"+[2+n 2)(n 2+v") a 2 (a"+h}2_{-+[2)(a"+ＱｊＮｾ} - -4I n(h"+n"+v") (/2-+_nO)-4-1_{n (a}2_-l-[")(a"-+h2+_v") 12 (h"+[2-+nO) (a"+h2+[2)(t"-+1£1") [2

+4 1n (I"+nO) (a 2+Ii)(h"-+12) (h2+12+w 2) v2 (h2 -l- n 2+v2)(a" t h2+v 2) (v2+1£12) v" - 4 / n(n2-+ v2) (a 2+ v2) (h2+v") (h"+v2-l-1(12) h2 (h"+[2+nO) (a 2-+h2-+12) (h2+v2-+1£1")(h"+v2) +4In (h2'+ n2-+v") (a"+h2-+v")(h"-+[2)(h2-+[2-+1£1") w2 (h"+12-+w")(v"+1£12) ] +4 / n(h" -+v"+w")(12+1iJ2) .

The mean view factor ｾ is accordingly a function of the variable

(1 )

-5-edge lengths, i.e.,

(2)

The nomographic representation of the six variables requires an

alignment diagram with six ind1cators. The solution, therefore,

would not be simple and would not always reveal the functional

re-lationship, unlike the ｳＱｾｰｬ･ graphical representation in Cartesian

coordinates. The methOd of determining the view factor ｾ for the

case of radiation where area 1 is parallel to area 2 (in Fig. 1)

has been published in this latter manner as Data Sheet 54(2).

How-ever, owing to the large number of variables this is awkward to obtain and, here as well, no .longer functionally apparent.

In the present case it is possible to represent equation (1) in the summation form:

(3) For this purpose, equation (1) 1s written as follows:

1

ｗｬｾＱ = 2:n;hn X [

a h2+₁2_{- a}2 _a2

- 12

X alarc tgｾ - al/h2+₁2_{arc tg----=o=}+ _{4 In(h}2+₁₂+_a2

)+- 4 - l n (a2+12) I Yh2+₁2 n --- n h2+_{J2 - n}2 _{2 _n}2 - /2 2 2 +nlarctg--n 1/h2+₁2_{arc t g -}+_{- - - - I n ( h}2+₁2+_n)+_-4-ln(l +_n) I ' Yh2+_{J2 4} ｷＬｾ W h2+v2- WI 2 2 w2- v2• ( I + I)

+wvarctg--wrh2+v2arctg-=+ In(hl+v +w-)+-4--t n W v

V Yh2+v2 4

The remain1ng logarithmic terms require a further variable Y1 J

for the determination of which a second network of coordinates is needed.

The view factor ｾＱｾＲ can now be determined from Data Sheet 56. The number of variables has been reduced by one in the data sheet through division of the edee length hy h.

The edge values divided by hare

a',

t',

n! , w' and v'.

The mathematical equation for determining the view factor is

_ 1 fl'l--H = 2--'_:nn (Xl+XI+x8 - X, - Ka - Xa+Yl), where x with a' t' 1

,

x with n ' , t' 2 x₃ with W'

,

_v' x₄ \'t·ith a'

,

v' x with n' v' ｾ

,

xs with w'

,

t'

are obtained from the x-diagram and Y1 is obtained from the y-diagram

with

t',

v'. (Care must be taken with the + and - signs both in

the equation and the diagrams.) The data sheet can be applied to

the cases illustrated in ｆｾＮｧｯＲＮ In u.si.ng diagrams A and B the

x or y value remains as a function value, i.e., a numerical one, even

for cases in which an edge value becomes equal to zero. For example,

f(X₃ } = f(w', v ' ) :: f(O, v'}.

ｅｾ｡ｭｰｬ･Ｚ For Fig. 1 let:

a = 4 a' :: ₂ n == 6 n' :: ₃ t

=

7 _t'

=

3.5 w = 2 w' == 1 v :: _{4 v'}

=

2 h

=

2

-7-Xl = f(a', 1') = 0,80 xa= f(n', 1') = 0,75 xa= f(w',v') = 0,58 x4 = f(a', v') = -0,45 .1'6= f(n', VI= -0,255 x6 = f(w'. 1') =-0,84 YI= f(l', v') = -0,255 and henoe Data Sheet 57* Wi ... ' = 1 = 2n.3 {0,80 + 0.75 +0,58-0,45-0,255-0.84-0,255) = 0,0175 ••• (4)

A seoond data sheet has been worked out for the oase of Fig. 3.

The pertinent equat10n w111 be found in the l1terature (referenoe 1,

equation (28».

The governing equation is derived in the same manner as before and 1s

where x₁ is to be obtained with nt, b', x

2 with a', b

l

x₃ is to be ohtained with Wi, bl and Y1 with b'

from the x- and y-diagrarns of Data Sheet 57, respeotively.

By repeated applioation of this data sheet it is also possible

to oaloulate the two oases in whioh area 1 or 2 or both do not extend.

as far as the oorner edge (equations (50) and (55) of referenoe 1).

Example: Fig. 4 represents a heated oeiling panel for whioh the view

faotor on a window is to be determined.

Let

a

=

1.2 w

=

0.9 t 1.8 v

=

0.3 n

=

2.1 b

=

1.5 h = 3.0.

With Data Sheet 56 we obtain

wl ...a=

= _1_(0,15 + 0,055 +0,12 +0,325-0,17-0,07 -0,175)

2n.·O,7

= 0,053.

See pages 17 and 18"

If the heated area of the ceiling does not extend to the front edge of the room then logically the calculation should be repeated for the unheated part and the result subtracted from the above. For a strip width of less than 005 m the result is not appreciably altered.

According to Fig. 2, Case VIII, the desired view factor can now

be determined from the given equation

(6 ) The derivation of this equation is

ipl,342.4 = q>1.342 +ipl,344= 29J1•3 4 2

(for r-easons of symmetry)

91241,3= 91241+q;243

and hence

because, from symmetry F1

=

F3 and ｾＳｾＲ = ｾＱｾＴＮ

When equation (6) is solved for ｾＱｾＧ we get

91144< = tpl.342.4 - tp141' (6a)

Accordingly it is identical with equation (11) in the literature(3)

for symmetrical, perpendicular and horizontal surfaces according to

the Fig. 3b and 4b there. Hence if an axis of symmetry can be drawn

through the radiating and radiation receiving surfaces, for one radleting section, then the view factor can generally be reduced according to equation (6) or (6a) to two simpler cases.

The view factor still required, ｾＱｾＴＧ is obtainable from Data

Sheet 57. Here the following dimensions Rpply (application of the

case of Fig. 4 to F1go 3):

a

=

1.2, b = 1.8 or b

=

0.3 for the un shaded strips between the

total areas,

-9-For b = 1.8 . 1 9'1-+4' =- 2 - 0_{:t. ,I}- (0,76-0,36-0,48 +0,2) =

°

0273 J , and for b = 0.3 _ 1 9'1-+&" = 2n.0,7 (0,12 - 0,04 - 0,05 +0,02)= 0,0114.

and the total result

'Pl,3-+2,4 = Ij'l-..a+'Pl->&= 0,053 -I-0,0159= 0069_{, .}

Thus about 7% of the total heat from ｴｨｾ heated ceiling panel is

radlated onto the wlndow. Concludipg Remarks

(6 )

It is lntended to present the eases of radiat10n according to equations (25), (26), (51) and (52) of reference 1 also as data

sheets. Equatlon (25) is already dealt with in Data Sheet 54

(Ds. Ztschr. (3/4): 1957), but, as already mentioned, in not so

slmp1e a form of procedure. With these data sheets the view factors

occurring with radiant panel heatlng may be determined in a com-paratively slmple manner wlth adequate practical (sllde-rule) accuracy in every case.

References

[IJ Kollmar, A.: Die Strahlu ngsverhaltnisse im bcheizten Wohnraum. Miinchen 1950.

[2J GI-Arbeitsblatter zur Berechnung von Einstrahlzahlen I brs VII. Nr. 26,27,28, 29, 40, 48 und 54. Verlag R. Olden-bourg, Miinchen.

[3J Kollmar, A.: Zur Theorie und Praxis der Einstrahlzahlen. Ges.lng. 75 (1954) S. 309.

[4JSquassi, F.: Die Einstrahlzahlen in Wohnraurne n. Ges.lng. 78 (1957) S.69.

[5J Kollmar, A.: Erganzende Betrachtungen zu F. Squassi : Einstrahlzahlen in Wohnrau men. Gcs. Ing. 78 (1957) S. 73.

[6J Krischer, 0.: Die wissenschaftlichen Grundlagen der Trock-nungstechnik. BerlinjGottingenjHeidelberg 1956.

[7J Raber, E.F., II.Hutchinson,F. IV.: Panel heating and

cool-ing analysis. New York 1957.

These references deal with view factors as they occur with surface

heating in a room. In addition, a general bibliography on radiant

heating is ｣ｯｾｴ｡ｩｮ･､ in papers quoted in references (1) and (4).

Fig. 1

-11-Fig. 2

Cases for applicat10n of Data Sheet 56 and equation (4) as given& The equations cited in brackets relate to reference 1

III (a)

I (Equat1on 24)

(a) According to Data Sheet 40

(b) According to Data Sheet 56

wi th w and v = 0

(Equat1on 27.app11ed twice)

According to Data Sheet 40

w1th ｡ｰｾｬｩ｣｡ｴＱｯｮ four

t1mes(3)

W1-+1=

(1

+ ;:) (W1.S -+ 2•• - W1.S-->.)

- ; : (W1-+2•• - (Ps-+.)

(b) Accord1ng to Data Sheet 56 with application twice

(w

=

0)

II (Equation 27)

(a) According to Data Sheet 40

applied twice

(b) According to Data Sheet 56

with w

=

0 i j / 1 / .rJi) ｾ ｾ IV (Equation 29)

According to Data Sheet 56 with

Fig. 2 (continued)

V (Equation 44)

According to Data Sheets 56 and

57 with v

=

0

VII (Equation 30)

According to Data Sheet 56

VI (Equation 46)

According to Data Sheet 56 with

v

=

0

1f1 ... 2= -;: (0/1.3 ... 2-0/...) +9'1.3 ...

@J

!'---r-VIII (Equation 45)

According to Data Sheets 56 and

57 with the equation according

-13-Fig. 2 (continued)

IX (Equation 49)

According to ｄ｡ｴｾ Sheet 56 with

the equation according to Case VI

X (Equation 49a)

According to Data Sheets 56 and

57 with the equation according

to Case V

XI

From Data Sheet 40:

.'P1.S ->- 2.4.5

According to Data Sheets 56 and

57 with Case VIII and the equa-tion according to Case V:

I/'1.S ->- 5= 'PI,S ->-2.4.5 - /PI.S ->- 2,4

= q71,3->- 2,4.5 - qJ1->-2 - qJ1->-4 F1+Fs · -1/'5->-1.3= -p--1/'1'3->-5 5 XII From Data like Ca.se similarly Sheet 40: XI for area F 1 for area, FｾＧＶ

Fig.. ::3

Irradiation of Area 2 by Area 1

n I I I J-) - 7 - = I / / tCr!- - -ＭｊＮｩ］ｨｾ / / / Fig. 4 bfv=l a

-15-Data Sheet of Gesundh.-Ing. Director:

E. Sprenger

Data Sheets for Calculation of View Factors VIII

Data Sheet

56

Determination of the view factor ｾ of a ｲ･｣ｴ｡ｮｌｾｬ｡ｲ area F1 ; nh on

a r-ec t angu'Lar- area F₂ := ab 't!hich is per-pend i cu Lar- to F1 hut is

shorter on both edges (If F₂ radiates on F1 then the IBw of

recipro-cal effect ｾＱｆＱ ｾ ｾＲｆＲ is to be applied)

0.0

_____ area sides -tI and Vi

HS't3,S3,025 20 15 12 1.0 0.8 as as il't 0,3 0,2 ill

7.0 O,S 0 O.S 7.0 I,S 2.0 2,S 3,0 3,S '1,0 'l,S 5,0 5,S 6,0 +x • ｾ .. -x function values x

ij

I

I / /

1/

!

y

j /' ; /

V' V'

/I'

- -

I

F;--- / / / / /

---L

'II I

I /

I

.

/1///////// / /

l

-I /

_-7

1

[7 /

Ir///

V/ /

/ V

ｾ Ｍ

III /

//1/ )!/l0//VV

---

_f-/

_{ＷＯＷＷｾＯＯＯｙＺｖ}

_.

I / /

III/

ＭｖｲＯ［ｾｖ

/o-b--..'v.<o-/YI,1-j/

1

II

II;

ｾｾ

V

w I 1

j

III

!/d

ｾ ｾ

/ＯｾＭｉｾ ' / / / Q+W:<fl

ＱＱｾＧ］

! t _A b-v-L x-diagre.rn. _I I 11 /

(1

r1,5 aI cD ｾ 2,0 2,S 0,5 1,0

-

_aI 3,5 CIl GJ '"C 3,0 ..-f CIl

-

s::; ____ area s i.de Vi

5,0 H 'to 35 30 25 20 15 1.21.0 0.8 as a't aeao

0 ｾ , , . r r

I

II

/

1/ 1/

_II

_{1/ /}

/

1/

1/1

1/

1 - - - - 1---

1//

1//

1//

_{I /}

_/

_/

_II

1/ /

I II

B

/

III

_VI

V

II

1/ 1//

1/

1/

I/!/ y-diagram / /

I

/

1/

'I)

:1

/

1/1/

II

II

1/

1//

V

l -

f---VI

/

7

I I

--I---___ __ _ _ _ ,---. L.. ...

T

;(

II

II

.:

11

I

'/

/

---ｾ

);

_ｾ

r/

V

ｾ

L::Vj

v/

'/

/;

/

Ｏｾｾ

r-

/

ｾ

/'

ｾＯ

ｾ

ｾ

ｾ - 1/ ＭＭＯ［ｾｾｾ

_ｾ

/

-:

vr

_{ｾ Ｐｾ}

_ｾ ,--- /

ｾ

ｾ

ｾｾｖｖＯ

. / ./'" ./'

-:

ｾ

ｾ

ｾ

ｾ

ＳＧｬＺＯｾＯ

_Ｏｾ

_ｾ

_/

ｾ

/

?r./

i 1,0 o,s 2,0

L,s

5, - 't,0

"

ｾ 3.5 ..-f CIl 3.0 aI ｾ 2,S aI 0,9 0.8 a7 as as a't 0.3 0.2 0,1 0 0,1 0,2 0.3 a't as as 0,7 0,8 0,9 1,0 +y I } iii-y function value y Explanations ｯｶ･ｲｬ･｡ｦｾ

Data Sheet of Gesundhe-Ing. Director:

E. Sprenger

Data Sheets for Calculation of View Factors VIII

Data Sheet

56

_

I[

n a w n a w

'1',-••= 2n hn nl arctgT -/-a l arctg_{T --}url arctg_{T -} n7) arctgV - av arctg

v

+av arctg

v--11Vh2+I' arctg Vht--/- I' -i- n V h'+-v' arctg VhO;'--;. - a Vh' -/-i'arctg VhOa/-i; -/-aｾＧｨＧＫｩ［Ｇ arctg Vh'a-/-i,'

-/-,;--- w ,/---- w n' (h' -i- I'+n', (n' -i- v')

ＭＯＭｷｲｨＧＫＯＧ｡ｲ｣ｴｧＭＭＭ］ＭｷｲｨＧＫｶＧ｡ｲ｣ｴｧｾＭＭ -Ln .

-Vh' -i- I' Vh'+v' 4 (h' +n'+v')(1'+n')

a' (a'+h' -/- {')(a'+v') I' (h' -/- I'+_n')(a'+h'+1')(/'+w') I' v' (h' -/-_{n' -/-}v')(a'+h'+v')(v'+w')v' ---4I n(a' +I')(a' -/- h'+v') -/-4 1n(1'+_n')(a'+I')(h'

+

I')(h'+I'+w') - 41n(n'/- v')(a_'+v')(h'+ v')(II'+v'+ w') +

h' (h'+I'+n')(a'+h'+I') (h'-1- v'+w') (h'+v') w' (h' +I'+w')(v'+w') ] +4 1n (hI+ ,,'+ｶＧＩＨｾｩＭＫｨＧＭＫ v')(h'+l')(h'+I' +w')+4 1n (h'+v'

+

w')(12+w')

Equation 30, p.19 from

Kollmar, A.: Die StrahlungsverhAltnisse im beheizten Wohnraum

(Radiation conditions in the heated room). R. Oldenbourg, Mu.nich, 1950.

Determination equation:

1

ql,....= 2n n'(x,+x.+x. -x. - x. - x. +y,) x-values from diagram A y-values from diagram B

X.'W'Iｴｾ n ', VI Xflwrth.w', r

The '-values follow from the edge dimensions of the drawing overleaf

each to be divided by h (n' =

E

etc.).

Numerical example (see reference below)

Reference:

Johannes, W.: Graphische Ermittlung von Einstrahlzahlen (Graphical

determination of vieN factors). Ges.-Ing. 1959. p.236

Detailer:

w.

Johannes

Data Sheet of

Oesundh--InES.

Director:

E. Sprenger

-17-D9ta Sheets for Calculation of View Factors IX

Data Sheet

57

Determination of the view factor ｾ of a rectangular area F1 ｾ wh on

an adjacent perpendicular rectangular area F2 = ab (If F2 radiates towards F then the reciprocal law _ｾ F

=

_ｾ F is to be applied)

1 1 1 2 2 5.0

1/

[f) Q) ::s rl ro7,2

----r---

ＭＭｾ : 7,0

--+___

--+-__ I _--+--='---:==--1 o I ｾｯＮｂ o

§M

r.-.

t

M 0.2ｉＭＭＭＭＭｾＭＫＭＭＭ 5,0 Explanation overleaf!

Data Sheet of Gesundh.-Ing. Dlrector:

E. Sprenger

Data Sheets for calculatlon of Vlew Factors IX

Data Sheet

57

_ 1 [ a+w . . a+w . a a w

'1'1-.,= 2:1<hw (a+w)barct g - -_b- +(a+w)Itarc tg ----:;;- -ab arctgt;--aharc tgJl - wbarc tg

iJ-- wharctg'j;' -·(a+w)Vb'+h'arcｴｧＺ［ｾ +w +a Vb'+h'arcｴｧ｟ｾＭ -\- w Vb'+h'arc tg_ w _

r b'-I-h' Vb'+h' Vb'+h'

_ (a+w.l in [(a-l-w>'+b'+h'](a+w)' +ｾ In Ｈ｡ＧＭ｜Ｍ｢ｾﾱｨＧＩ｡Ｇ I !!..In (a+w)'-\-b'+h'](a'+b')(b'+h')(b'+w')

4 [(a-I-w)'+b'][(a+w)'+hi] 4 (a'+b')(a'+ h')T 4 (a'-I-b'+h')(b'+h'+w·) [(a+w)'+b']b' +

":' In [(a+w)'+b'+h'J(a' +h')(b'+h')(h'-I-w·) w· (b'+h'+w·) w' ]

+ 4 (a' +b'+II') (b'+II'+w.) [(a+w)'+h'J h' +Tin (b'+w') (h'+w·)

Equation ＲＸｾ p.18 from

Kollmar, A.: Dle ｓｴｲ｡ｨｬｵｮｧｳｶ･ｲｨｾｬｴｮｬｳｳ･ 1m behelzten Wohnraum

(Radlatlon condltions in the heated room).

R. Oldenbourg, Munich, 1950

Determination equat10n

_ 1

Ｇｐｾ ....,= 2:1<W' (Xl - X, - X,+Yl)

x-values from diagram A y-values from diagram B

The '-values follow from the edge d1mensions of the drawing overleaf

each d1vided by h (nl

; ｾ etc.).

Numerical example (see reference below)

Reference:

Johannes, W.: Graph1sche Erm1ttlung von Elnstrahlzahlen (Graphical

determlnation of vlew factors). Ges.-Ing. 1959. p.238

Detaller: