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PREFACE
This manual/web page contains the solutions to many (but not all) of the problems that are given at the end
of each chapter, in particular for problems on topics that are commonly covered in a first (or, at least, second)
graduate course on radiative heat transfer. Thus, solutions to problems of Chapters 1 through 6, 9 through
11, 13, 14 and 18 are almost complete; for other chapters (7, 15, 16, 19) only around half of solutions are
given, for problems on the more basic aspects covered in that chapter. Quite a few solutions, together with
Fortran90 codes, are also given for Chapter 20, not so much intended as homework solutions, but perhaps
useful for semester projects and/or to aid the researcher to write his/her first Monte Carlo code. At this date
no problem solutions are provided for the remaining chapters. However, it is the intent to periodically update
this web-based solution manual, at which time additional solutions may be posted.
Most solutions have been checked, and some have been rechecked. Still, only a major miracle could have
prevented me from making an occasional error or overlooking an occasional typo. I would appreciate it very
much if any such errors in the manual (or, for that matter, in the text) would be reported to me. In addition,
for the benefit of other instructors who will receive future editions of this manual, any reasonably written up
solutions to presently unsolved problems would also be appreciated.
This manual is posted as a pdf-file with hyperreferences enabled, including a Table of Contents and a
Bookmark column listing all chapters and problems. This enables the reader to immediately jump back and
forth within the document to locally referenced items (chapters, problems, figures, equations). References to
equations, figures, etc., in the book itself can, of course, not be reached in this way. Users of Adobe Acrobatrcan
easily assemble individual problem set solutions from this manual. To make this also possible for instructors
with access to only the shareware postscript/pdf viewer Ghostview c
, each individual problem is started at the
top of a new page.
State College, PA
December 2002
Michael F. Modest
Department of Mechanical Engineering
The Pennsylvania State University
University Park, PA 16802
e-mail: mfm6@psu.edu
i
CONTENTS
Chapter 1 2
Problem 1.1 .................................................... 2
Problem 1.2 .................................................... 3
Problem 1.3 .................................................... 4
Problem 1.4 .................................................... 5
Problem 1.5 .................................................... 6
Problem 1.6 .................................................... 7
Problem 1.7 .................................................... 8
Problem 1.8 .................................................... 9
Problem 1.9 .................................................... 10
Problem 1.10 ................................................... 11
Problem 1.11 ................................................... 12
Problem 1.12 ................................................... 13
Problem 1.13 ................................................... 15
Problem 1.14 ................................................... 16
Problem 1.15 ................................................... 17
Chapter 2 19
Problem 2.1 .................................................... 19
Problem 2.2 .................................................... 20
Problem 2.3 .................................................... 22
Problem 2.4 .................................................... 23
Problem 2.5 .................................................... 24
Problem 2.6 .................................................... 25
Chapter 3 27
Problem 3.1 .................................................... 27
Problem 3.2 .................................................... 28
Problem 3.3 .................................................... 29
Problem 3.4 .................................................... 30
Problem 3.5 .................................................... 31
Problem 3.6 .................................................... 33
Problem 3.7 .................................................... 34
Problem 3.8 .................................................... 35
Problem 3.9 .................................................... 36
Problem 3.10 ................................................... 37
Problem 3.11 ................................................... 38
Problem 3.12 ................................................... 39
Problem 3.13 ................................................... 40
Problem 3.14 ................................................... 41
Problem 3.15 ................................................... 42
Problem 3.16 ................................................... 43
Problem 3.17 ................................................... 44
Problem 3.18 ................................................... 45
Problem 3.19 ................................................... 46
ii
CONTENTS iii
Problem 3.20 ................................................... 47
Problem 3.21 ................................................... 48
Problem 3.22 ................................................... 49
Problem 3.23 ................................................... 50
Problem 3.24 ................................................... 52
Problem 3.25 ................................................... 53
Problem 3.26 ................................................... 54
Problem 3.27 ................................................... 55
Problem 3.28 ................................................... 56
Problem 3.29 ................................................... 57
Problem 3.30 ................................................... 58
Problem 3.31 ................................................... 59
Problem 3.32 ................................................... 60
Problem 3.33 ................................................... 61
Problem 3.34 ................................................... 62
Problem 3.35 ................................................... 63
Problem 3.36 ................................................... 65
Chapter 4 66
Problem 4.1 .................................................... 66
Problem 4.2 .................................................... 68
Problem 4.3 .................................................... 69
Problem 4.4 .................................................... 70
Problem 4.5 .................................................... 72
Problem 4.6 .................................................... 73
Problem 4.7 .................................................... 74
Problem 4.8 .................................................... 75
Problem 4.9 .................................................... 76
Problem 4.10 ................................................... 77
Problem 4.11 ................................................... 78
Problem 4.12 ................................................... 79
Problem 4.13 ................................................... 80
Problem 4.14 ................................................... 81
Problem 4.15 ................................................... 82
Problem 4.16 ................................................... 83
Problem 4.17 ................................................... 84
Problem 4.18 ................................................... 85
Problem 4.19 ................................................... 87
Problem 4.20 ................................................... 88
Problem 4.21 ................................................... 89
Problem 4.22 ................................................... 90
Problem 4.23 ................................................... 92
Problem 4.24 ................................................... 93
Problem 4.25 ................................................... 94
Problem 4.26 ................................................... 96
Chapter 5 100
Problem 5.1 .................................................... 100
Problem 5.2 .................................................... 101
Problem 5.3 .................................................... 103
Problem 5.4 .................................................... 105
Problem 5.5 .................................................... 106
Problem 5.6 .................................................... 107
Problem 5.7 .................................................... 108
Problem 5.9 .................................................... 110
Problem 5.10 ................................................... 112
Problem 5.11 ................................................... 113
Problem 5.12 ................................................... 114
Problem 5.13 ................................................... 115
Problem 5.14 ................................................... 118
Problem 5.15 ................................................... 120
Problem 5.16 ................................................... 122
Problem 5.17 ................................................... 124
Problem 5.18 ................................................... 125
Problem 5.19 ................................................... 126
Problem 5.20 ................................................... 127
Problem 5.21 ................................................... 129
Problem 5.22 ................................................... 130
Problem 5.23 ................................................... 131
Problem 5.24 ................................................... 132
Problem 5.25 ................................................... 134
Problem 5.26 ................................................... 135
Problem 5.27 ................................................... 136
Problem 5.28 ................................................... 137
Problem 5.29 ................................................... 139
Problem 5.30 ................................................... 141
Problem 5.31 ................................................... 143
Problem 5.32 ................................................... 146
Problem 5.33 ................................................... 154
Problem 5.34 ................................................... 155
Problem 5.35 ................................................... 159
Problem 5.36 ................................................... 160
Problem 5.37 ................................................... 161
Problem 5.38 ................................................... 164
Problem 5.39 ................................................... 165
Problem 5.40 ................................................... 166
Chapter 6 168
Problem 6.1 .................................................... 168
Problem 6.2 .................................................... 169
Problem 6.3 .................................................... 170
Problem 6.4 .................................................... 172
Problem 6.5 .................................................... 174
Problem 6.6 .................................................... 176
Problem 6.7 .................................................... 178
Problem 6.8 .................................................... 179
Problem 6.9 .................................................... 180
Problem 6.10 ................................................... 182
Problem 6.11 ................................................... 183
Problem 6.12 ................................................... 184
Problem 6.13 ................................................... 185
Problem 6.14 ................................................... 187
Problem 6.15 ................................................... 188
Problem 6.16 ................................................... 189
Problem 6.17 ................................................... 191
Problem 6.18 ................................................... 192
Problem 6.19 ................................................... 193
Problem 6.21 ................................................... 195
Problem 6.22 ................................................... 198
CONTENTS v
Problem 6.24 ................................................... 200
Problem 6.26 ................................................... 202
Problem 6.27 ................................................... 203
Problem 6.29 ................................................... 205
Problem 6.30 ................................................... 208
Problem 6.31 ................................................... 210
Problem 6.34 ................................................... 212
Problem 6.35 ................................................... 213
Chapter 7 214
Problem 7.1 .................................................... 214
Problem 7.2 .................................................... 216
Problem 7.5 .................................................... 218
Problem 7.6 .................................................... 219
Problem 7.8 .................................................... 221
Problem 7.10 ................................................... 224
Problem 7.11 ................................................... 225
Problem 7.13 ................................................... 227
Problem 7.14 ................................................... 232
Problem 7.15 ................................................... 234
Problem 7.16 ................................................... 237
Chapter 8 241
Problem 8.1 .................................................... 241
Problem 8.2 .................................................... 242
Problem 8.3 .................................................... 243
Problem 8.4 .................................................... 245
Problem 8.5 .................................................... 246
Problem 8.6 .................................................... 247
Problem 8.7 .................................................... 248
Problem 8.8 .................................................... 249
Problem 8.9 .................................................... 251
Problem 8.10 ................................................... 254
Problem 8.14 ................................................... 255
Problem 8.15 ................................................... 260
Chapter 9 265
Problem 9.1 .................................................... 265
Chapter 10 267
Problem 10.1 ................................................... 267
Problem 10.2 ................................................... 269
Problem 10.3 ................................................... 270
Problem 10.4 ................................................... 272
Problem 10.5 ................................................... 273
Problem 10.6 ................................................... 275
Problem 10.7 ................................................... 276
Problem 10.8 ................................................... 278
Problem 10.9 ................................................... 279
Problem 10.10 ................................................... 280
Problem 10.11 ................................................... 281
Problem 10.12 ................................................... 283
Problem 10.13 ................................................... 285
Problem 10.14 ................................................... 287
Problem 10.15 ................................................... 288
vi CONTENTS
Chapter 11 289
Problem 11.1 ................................................... 289
Problem 11.2 ................................................... 290
Problem 11.3 ................................................... 291
Problem 11.4 ................................................... 292
Problem 11.5 ................................................... 293
Problem 11.6 ................................................... 294
Problem 11.7 ................................................... 295
Problem 11.8 ................................................... 296
Problem 11.9 ................................................... 297
Problem 11.10 ................................................... 298
Problem 11.11 ................................................... 299
Problem 11.12 ................................................... 300
Problem 11.13 ................................................... 301
Problem 11.14 ................................................... 302
Problem 11.15 ................................................... 303
Problem 11.16 ................................................... 304
Problem 11.17 ................................................... 305
Problem 11.18 ................................................... 306
Problem 11.19 ................................................... 307
Problem 11.20 ................................................... 309
Problem 11.21 ................................................... 310
Problem 11.22 ................................................... 311
Problem 11.23 ................................................... 313
Problem 11.24 ................................................... 314
Problem 11.25 ................................................... 316
Problem 11.26 ................................................... 317
Problem 11.27 ................................................... 318
Problem 11.28 ................................................... 319
Problem 11.29 ................................................... 320
Problem 11.30 ................................................... 321
Problem 11.31 ................................................... 323
Problem 11.32 ................................................... 324
Problem 11.33 ................................................... 325
Problem 11.34 ................................................... 327
Problem 11.35 ................................................... 328
Problem 11.37 ................................................... 329
Problem 11.38 ................................................... 330
Problem 11.39 ................................................... 331
Problem 11.40 ................................................... 332
Problem 11.41 ................................................... 334
Problem 11.42 ................................................... 337
Chapter 12 339
Problem 12.1 ................................................... 339
Problem 12.2 ................................................... 340
Problem 12.3 ................................................... 341
Problem 12.4 ................................................... 342
Problem 12.5 ................................................... 343
Problem 12.6 ................................................... 344
Problem 12.7 ................................................... 345
Problem 12.8 ................................................... 346
Problem 12.9 ................................................... 347
Problem 12.10 ................................................... 348
Problem 12.11 ................................................... 349
CONTENTS vii
Problem 12.13 ................................................... 350
Problem 12.14 ................................................... 351
Problem 12.15 ................................................... 352
Problem 12.16 ................................................... 353
Problem 12.17 ................................................... 354
Problem 12.18 ................................................... 355
Problem 12.19 ................................................... 356
Problem 12.20 ................................................... 358
Problem 12.21 ................................................... 359
Chapter 14 360
Problem 14.1 ................................................... 360
Problem 14.2 ................................................... 361
Problem 14.3 ................................................... 362
Problem 14.4 ................................................... 364
Problem 14.5 ................................................... 366
Problem 14.6 ................................................... 367
Problem 14.7 ................................................... 368
Problem 14.8 ................................................... 369
Problem 14.9 ................................................... 370
Problem 14.10 ................................................... 371
Problem 14.11 ................................................... 372
Problem 14.12 ................................................... 373
Problem 14.13 ................................................... 374
Problem 14.14 ................................................... 376
Chapter 15 377
Problem 15.1 ................................................... 377
Problem 15.2 ................................................... 378
Problem 15.3 ................................................... 379
Problem 15.4 ................................................... 380
Problem 15.5 ................................................... 382
Problem 15.6 ................................................... 383
Problem 15.7 ................................................... 384
Problem 15.8 ................................................... 386
Problem 15.9 ................................................... 387
Problem 15.10 ................................................... 388
Problem 15.11 ................................................... 390
Problem 15.12 ................................................... 392
Problem 15.13 ................................................... 394
Problem 15.14 ................................................... 396
Problem 15.15 ................................................... 397
Problem 15.16 ................................................... 399
Problem 15.18 ................................................... 401
Problem 15.19 ................................................... 403
Chapter 16 405
Problem 16.1 ................................................... 405
Problem 16.2 ................................................... 406
Problem 16.3 ................................................... 407
Problem 16.4 ................................................... 408
Problem 16.5 ................................................... 410
Problem 16.6 ................................................... 412
Problem 16.7 ................................................... 413
Problem 16.9 ................................................... 417
Problem 16.10 ................................................... 419
Problem 16.11 ................................................... 420
Problem 16.12 ................................................... 421
Problem 16.13 ................................................... 423
Problem 16.14 ................................................... 424
Problem 16.15 ................................................... 426
Problem 16.18 ................................................... 428
Chapter 17 429
Problem 17.1 ................................................... 429
Problem 17.2 ................................................... 430
Problem 17.3 ................................................... 431
Problem 17.4 ................................................... 432
Problem 17.5 ................................................... 434
Problem 17.6 ................................................... 436
Problem 17.7 ................................................... 438
Problem 17.8 ................................................... 439
Problem 17.9 ................................................... 441
Problem 17.10 ................................................... 443
Chapter 19 445
Problem 19.1 ................................................... 445
Problem 19.2 ................................................... 447
Problem 19.3 ................................................... 448
Problem 19.4 ................................................... 449
Problem 19.5 ................................................... 450
Problem 19.6 ................................................... 451
Problem 19.7 ................................................... 452
Chapter 20 454
Problem 20.1 ................................................... 454
Problem 20.2 ................................................... 455
Problem 20.3 ................................................... 456
Problem 20.4 ................................................... 457
Problem 20.5 ................................................... 459
Problem 20.7 ................................................... 464
Problem 20.8 ................................................... 466
Problem 20.9 ................................................... 468
Problem 20.10 ................................................... 469
Problem 20.11 ................................................... 470
Problem 20.15 ................................................... 472
Problem 20.20 ................................................... 474
Chapter 21 477
Problem 21.1 ................................................... 477
Problem 21.8 ................................................... 479
CHAPTER 1
1.1 Solar energy impinging on the outer layer of earth’s atmosphere (usually called “solar constant”) has been
measured as 1367 W/m2. What is the solar constant on Mars? (Distance earth to sun =1.496 ×1011 m, Mars
to sun =2.28 ×1011 m).
Solution
The total energy emitted from the sun Q=4πR2
SσT4
sun goes equally into all directions, so that
2
1.2 Assuming earth to be a blackbody, what would be its average temperature if there was no internal heating
from the core of earth?
1.3 Assuming earth to be a black sphere with a surface temperature of 300 K, what must earth’s internal heat
generation be in order to maintain that temperature (neglect radiation from the stars, but not the sun) (radius
of the earth RE=6.37 ×106m).
1.4 To estimate the diameter of the sun, one may use solar radiation data. The solar energy impinging onto the
earth’s atmosphere (called the “solar constant”) has been measured as 1367 W/m2. Assuming that the sun
may be approximated to have a black surface with an effective temperature of 5777 K, estimate the diameter
of the sun (distance sun to earth SES ≃1.496 ×1011 m).
=1.496 ×1011 m 1367 W/m2
RS=6.96 ×108m
6RADIATIVE HEAT TRANSFER
1.5 Solar energy impinging on the outer layer of earth’s atmosphere (usually called “solar constant”) has been
measured as 1367 W/m2. Assuming the sun may be approximated as having a surface that behaves like a
blackbody, estimate its effective surface temperature. (Distance sun to earth SES ≃1.496 ×1011 m, radius of
sun RS≃6.96 ×108m).
Tsun =5777 K
