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APPENDIX A. VIEW FACTORS FOR

SELECTED CONFIGURATIONS

In this appendix, the view factors for three configurations are given:

1.

radiation from a sphere,

2. radiation from a vertical cylinder,

3. radiation from a vertical plane surface.

For other configurations, refer to Love (1968) and Buschman and Pittmann

(1961). A view factor depends on the shapes of the emitter and receiver.

Consider the receiver to be a small, plane surface at ground level with a given

orientation with respect to the emitter. The angle between the normal to the

surface and the connection between the surface and the center of the emitter (Θ)

must be known.

A-1. VIEW FACTOR OF A SPHER ICAL EMITTER E.G., FIREBALL)

If the distance from the receiver to the center of the sphere is L, and Θ is the

angle between the connection of the surface to the center of the sphere and the

tangent to the sphere, then, for Θ < π/2 - Φ, the view factor F is given by:

r

2

F = — c o s ( 0 )

L

(Eq .A. l )

where:

L = distance between receiving surface and sphere's center (m)

r = radius of sphere (m)

Θ = orientation angle (rad)

In this case, the sphere is in full sight.

397

Guidelines for Vapor Cloud Explosion Pressure Vessel Burst

BLEVE and Flash Fire Hazards Second Edition

by Center for Chemical Process Safety

Copyright © 2010 American Institute of Chemical Engineers Inc.



398

GUIDELINES FOR VCE, PVB, BLEVE AND FF HAZARDS

When, on extension, the receiving surface intersects with the sphere (Θ > id

2 - < Φ), the receiver cannot "see" the total emitter. The view factor F is then

given as:

+

- \

cos Θ

cos"

1

[-

L)

- \)

m

cot θ ]

--Ul*-1)

, /2

(1-Z?cos

2

0)

1/2

(Eq. A.2)

where:

r = fireball radius (r = D/2) (m)

D = fireball diam eter (m)

L = distance to center of sphere (m)

Θ = angle between the normal to surface and connection of

point to center of sphere (rad)

2Φ = view angle (rad)

L

r

= reduced length L /r (-)

^ 1 l - i

F = sin

2 2

{L]-\r

The view factor for incom plete visibility is given in Figure A-2 .



APPENDIX A

399

fireball

receptor

fireball

I

£

receptor

<

K/2

φ

B

Figure A-1. View factor of a fireball.

(A) Receiver "sees" the sphere completely. (B) Receiver "sees" the sphere partially.

F ^

0 . 50

\ \

2 0

\ \ Θ degrees

~~

1.00

1.50 2.00

L/r

2 .50

Figure A-2. View factors of

a

sphere as function of the dimensionless distance

(distance/radius) (incomplete view).



400 GUIDELINES FOR

VCE,

PVB, BLEVE AND FF HAZARDS

A-2. VIEW FACTOR OF A VERTICAL CYLINDER

A pool fire's flame can be represented (under no wind conditions) by a

vertically placed cylinder with a height h and a ground surface radius r. The

view factor of a plane surface at ground level whose normal lies in one vertical

plane with the axis of the cylinder is given by the following equations:

π

h

r

= h /r

X

r

= X / r

A

=

X

t

+\f

+

h

2

B =

X

r

-l)

2

+ h

2

horizontal surface (Θ

π/2):

tarT

<

1/2

X

2

-\

+

h

2

_,

r

,

r

tan -

JAB

-l)A)

+

\)Bj

(Eq

.A.3)

(Eq. A.4)

(Eq. A.5)

(Eq. A.6)

1/2

(Eq. A.7)

cylindrical flame

s/1

Ä

Figure

A-3.

View factor of cylindrical flame.



APPENDIX

401

And

for a

vertical surface

(Θ = 0):

F„

=

1

7CX

r

A.

7lX

r

t n

{

tan

Ί

ί

1/2

h

r

{A-2X\ _,

TtX^AB

X,-\

1/2

v ^

r

+ly

*,-iM

,1/2

(Eq.

A.8)

The maximum view factor

is

given

by

•fmax

- (-̂ h + ^ v )

(Eq.

A.9)

For

the

view factor

of a

tilted cylinder, refer

to Raj

(1977).

The

view factors

generated

by

Eqs.

(A.7)

and (A.8)

are

given

in

Table A-1 and Figure

A-4.



402 GUIDELINES FOR VCE , PVB, BLEVE AND FF HAZARDS

Table

A-1.

View Factors of a Vertical Cylindrical Emitter h

r

= 2L(/df·; X

r

= 2X/df

K

X

r

0.1 0.2 0.5 1.0 2.0 3.0 5.0 6.0 10.0 20.0

1.

Horizontal target

(1000 χ F

h

)

1 1

1 2

1 3

1 4

1 5

2 0

3 0

4 0

5 0

10 0

20 0

132

44

20

11

6

1

242

120

65

38

24

5

332

243

178

130

97

27

5

1

354

291

242

203

170

73

19

7

3

360

307

268

238

212

126

50

22

11

1

362

310

272

246

222

145

71

38

21

3

362

312

177

250

228

158

91

57

37

7

1

362

312

278

251

229

160

95

62

43

9

1

363

313

278

252

231

164

103

73

54

17

3

363

314

279

153

232

166

107

78

61

26

8

1 1

1 2

1 3

1 4

1 5

2 0

3 0

4 0

5 0

10 0

20 0

330

196

130

94

71

28

9

5

3

415

308

227

173

135

56

19

10

6

1

449

397

344

296

253

126

47

24

15

3

453

413

376

342

312

194

86

47

29

6

1

454

416

383

354

329

236

132

80

53

13

3

454

416

384

356

332

245

150

100

69

19

4

454

416

384

356

333

248

161

115

86

29

7

454

416

384

357

333

249

163

119

91

32

9

454

416

384

357

333

249

165

123

97

42

14

455

417

385

357

333

250

167

125

100

48

21

Continued next page)



APPEND IX A 403

Table A-1. View Factors of

a

Vertical Cylindrical Emitter,

continued)

K

Xr

0.1

0.2 0.5

1.0 2.0

3.0

5.0

6.0

10.0 20.0

3. Maximum view factor (1000 χ F

max

)

1 1

1 2

1 3

1 4

1 5

2 0

3 0

4 0

5 0

10 0

20 0

356

201

132

94

72

28

9

5

3

481

331

236

177

138

56

19

10

6

1

559

466

287

323

271

129

48

24

15

3

575

505

448

398

355

208

88

47

29

6

1

580

517

468

427

392

267

141

83

54

13

3

581

519

472

433

400

285

166

106

73

19

4

581

520

474

436

404

294

185

129

94

30

7

581

521

474

436

404

296

189

134

100

34

9

581

521

475

437

405

299

195

143

111

45

14

581

521

475

437

406

300

197

147

117

55

22

E 0.1

0.01

\Vwv

\ \

\ W X N

h/r o 2 0)5 i 2

iv

3 5 10 20

10

x/r

Figure A-4. Maximum view factors of

a

cylindrical flame as function of

dimensionless distance to flame axes.



404

GUIDELINES FOR VCE PVB BLEVE AND FF HAZARDS

A-3.

VIEW FACTOR OF A VERTICAL PLANE SURFACE

In the case of a vertical plane surface, it is assumed that the emitter and

receiver are parallel to each other. The view factor is calculated from the sum of

view factors from surface I and surface II (see Figure A-5). Surfaces I and II are

defined as those to the left and the right of a plane through the center of the

receiver and perpendicular to the intersections of the receiver with the ground.

receiver

Figure

A-5.

View factor of

vertical plane surface.

For each of

the

two surfaces:

h

r

h/b

X

r

X/b

A = l/ h

r

2

+

X

r

2

)

0

·

5

5 = /z

r

/ ( l + X

r

2

)

0 5

( E q . A . 1 0 )

( E q . A . l l )

( E q . A . 1 2 )

( E q . A . 1 3 )



APPEND IX A 405

For a horizontal target on ground level (Θ = π/2), the view factor is given by:

2 π

f 1 Λ

tan"

K

X

r J

-Α Χ

τ

\ΖΆ~\Α)

( E q . A . l 4 )

and, for a vertical surface (Θ = 0):

2 π

( E q . A . l 5 )

The maximum view factor is given by:

•fmax - (Fh + F

v

) ' (Eq . A . 16)

It must be noted that, unless b\ = b

u

, Fmax is not the maximum view factor

for any distance C from the emitter.

The view factors F

h

, F

v

, and

F

m ax

can easily be found by summing the view

factors calculated from the surfaces I and II. Values of the view factor

F

m ax

as

function of X

r

are given in Table A-2 and Figure A-6.



406 GUIDELINES FOR VCE, PVB, BLEVE AND FF HAZARDS

Table A-2. View Factors of a Vertical Plane Surface Emitter h

r

= h/b; X

r

= X/b

(See Figure A-5)

K

X

r

0.1 0.2 0.3 0.5 1.0 1.5 2.0 3.0 5.0

1. Horizontal tar get

(1000 χ F

h

)

0 1

0 2

0 3

0 5

1 0

1 5

2 0

3 0

5 0

146

53

25

9

2

1

276

146

83

34

8

3

1

341

221

144

68

17

6

3

1

400

310

236

137

42

17

8

3

1

443

389

337

249

111

53

28

10

2

456

413

371

296

161

88

51

20

5

461

423

386

318

190

114

71

31

9

465

430

397

336

219

146

100

50

16

467

435

403

346

238

170

126

75

31

0 1

0 2

0 3

0 5

1 0

1 5

2 0

3 0

5 0

353

223

156

94

41

22

14

7

2

447

352

274

178

80

44

17

13

5

474

414

349

245

117

65

41

20

7

489

461

421

335

180

105

66

32

12

496

484

466

416

277

179

120

62

24

497

488

474

435

318

222

157

86

35

497

489

All

442

335

245

180

105

45

497

490

478

445

347

264

204

129

61

498

490

479

447

352

274

218

148

80

Continued on next page)



APPEND IX A 407

Table A-2. View Factors of a Vertical Plane Surface Emitter, continued)

K

X

r

0.1 X

r

0.1 X

r

0.1 X

r

0.1 X

r

0.1

3. Maximum view factor (1000 χ F

m ax

)

0 1

0 2

0 3

0 5

1 0

1 5

2 0

3 0

5 0

382

229

158

94

41

22

14

7

2

525

381

286

181

80

44

27

13

5

584

469

377

255

188

66

41

20

7

632

555

483

362

185

106

67

33

12

665

621

575

484

299

187

123

62

24

674

639

602

526

356

239

165

88

36

678

647

613

544

385

270

194

109

46

681

652

622

558

410

302

227

138

63

682

655

626

565

425

322

252

165

86

n)

E 0.1

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