Part V Appendixes

Appendix A Conversion Factors

The following table expresses the definitions of miscellaneous units of measure as exact numerical multiples of coherent SI units, and provides multiplying factors for converting numbers and miscellaneous units to corresponding new numbers and SI units.

The digits of each numerical entry following E represent a power of 10. An asterisk preceding each number expresses an exact definition. For example, the entry "*2.54E-2" expresses the fact that 1 inch=2.54x 10- 2

meter, exactly, by definition. Most of the definitions are extracted from National Bureau of Standards documents. Numbers not preceded by an asterisk are only approximate representations of definitions, or are the results of physical measurements.

This appendix was abstracted from The International Systems of Units-Physical Constants and Conversion Factors. E.A. Mechtly, Second Revision. NASA SP-7012, Washington, D.C. (1973). Permission to use this material was obtained from the Scientific and Technical Informa­tion Office, NASA, Washington, D.C.

Table A.l

To convert from: to: multiply by:

atmosphere newton/meter2 *1.013E5 bar newton/meter2 *l.OOES British thermal unit (mean) joule 1.05587E3 British thermal unit joule 1.054350E3

(thermochemical) British thermal unit (39°F) joule 1.05967E3 British thermal unit ( 60° F) joule 1.05468E3 calorie (International Steam Table) joule 4.1868 calorie (mean) joule 4.19002 calorie (thermochemical) joule *4.184 calorie (15oC) joule 4.18580 calorie (20° C) joule 4.18190 calorie (kilogram, International joule 4.1868E3

Steam Table) continued overleaf

299

300

To convert from:

calorie (kilogram, mean) calorie (kilogram, thermochemical) Celsius (temperature) centimeter of mercury (Oo C) centimeter of water (4° C) electron volt erg Fahrenheit (temperature) Fahrenheit (temperature) fluid ounce (U.S.) foot foot of water (39.2° F) gallon (U.S. dry) gallon (U.S. liquid) horsepower (550 foot lbf/secondJ inch kilocalorie (International Steam

Table) kilocalorie (mean) kilocalorie (thermochemical) kilogram mass kilogram force (kgf) lbf (pound force, avoirdupois) Ibm (pound mass, avoirdupois) liter micron mile (U.S. statute) pascal poise pound force (lbf avoirdupois) pound mass (Ibm avoirdupois) quart (U.S. dry) quart (U.S. liquid) Rankine (temperature) slug ton (long) ton (metric) ton (short, 2000 Jb) Torr (OoC)

Table A.l (Continued)

to:

joule joule kelvin newton/meter2

newton(meter 2

joule joule kelvin Celsius meter 3

meter newton/meter2

meter 3

meter3

watt meter joule

joule joule kilogram newton newton kilogram meter 3

meter meter newton(meter2

newton second(meter2

newton kilogram meter 3

meter 3

kelvin kilogram kilogram kilogram kilogram newton(meter2

Appendix A

multiply by:

4.19002E3 *4.184E3

tK =tc +273.15 1.33322E3 9.80638E1 1.6021917E-19

*l.OOE-7 fK = (5/9)(1F +459.67) tc=(5/9)(tF-32)

*2.957 35295625E5 *3.048E-1 2.98898E3

*4.404 883 770 86E-3 *3.785411784E-3

7.4569987E2 *2.54E-2 4.1868E3

4.19002E3 4.184E3

*1.00 *9.80665 *4.448 221615 260 5 *4.535 923 7E-1 *l.OOE-3 *l.OOE-6 *1.609 344E3 *1.00 *l.OOE-1 *4.4482216152605 *4.535 923 7E-1 *1.101220942 715E-3 9.4635925E-4 (K = (5(9)t R

1.459 39029EI *1.016046908 8E3 *l.OOE3 *9.0718474E2

1.33322E2

Appendix B

Physical Parameters for Prediction of Transport Coefficients

Contents

Table B.l. Intermolecular Force Parameters and Critical Properties, p. 301 Table B.2. Leonard-Jones Patentials as Determined from Viscosity Data,

p. 303 Table B.3. Stockmayer-Potential Parameters, p. 305 Table B.4. Values of the Collision Integral Q v for Viscosity Based on the

Leonard-Jones Potential, p. 306 Table B.S. Values of the Collision Integral !lv Based on the Leonard­

Jones Potential, p. 307 Table B.6. Collision Integrals Q. for Viscosity as Calculated by Use of

the Stockmayer Potential, p. 308 Notation for the Tables, p. 309 References for the Tables, p. 309

Table B.l. Intermolecular Force Parameters and Critical Properties"

Lennard-Jones parametersb Critical constants'

Molecular weight (5 e/K I; p, V, X 103

Substance M (A) (K) (K) (atm) (m 3 kmol- 1)

Light elements H2 2.016 2.915 38.0 33.3 12.80 65.0 He 4.003 2.576 10.2 5.26 2.26 57.8

continued overleaf"

301

302 Appendix B

Table B.l (Continued)

Lennard-Janes parameters• Critical constants'

Molecular ------ ---~---·--

weight (J e/K I; p, V, X 103

Substance M (Al (K) (K) (atm) (m3 kmol- 1)

Noble gases Ne 20.183 2.789 35.7 44.5 26.9 41.7

Ar 39.944 3.418 124. 151. 48.0 75.2

Kr 83.80 3.498 225. 209.4 54.3 92.2

Xe 131.3 4.055 229. 289.8 58.0 118.8

Simple po/yatomic substances Air 28.97d 3.617 97.0 132.d 36.4d 86.6d

Nz 28.02 3.681 91.5 126.2 33.5 90.1

02 32.00 3.433 113. 154.4 49.7 74.4

03 48.00 268. 67. 89.4

co 28.01 3.590 110. 133. 34.5 93.1

col 44.01 3.996 190. 304.2 72.9 94.0

NO 30.01 3.470 119. 180. 64. 57.

N20 44.02 3.879 220. 309.7 71.7 96.3

S02 64.07 4.290 252. 430.7 77.8 122.

Fz 38.00 3.653 112.

Cl 2 70.91 4.115 357. 417. 76.1 124.

Br2 159.83 4.268 520. 584. 102. 144.

12 253.82 4.982 550. 800.

Hydrocarbons CH4 16.04 3.822 137. 190.7 45.8 99.3

CzHz 26.04 4.221 185. 309.5 61.6 113.

CzH• 28.05 4.232 205. 282.4 50.0 124.

CzH6 30.07 4.418 230. 305.4 48.2 148.

C3H6 42.08 365.0 45.5 181.

C3H 8 44.09 5.061 254. 370.0 42.0 200.

n-C.H,o 58.12 425.2 37.5 255.

i-C4 H, 0 58.12 5.341 313. 408.1 36.0 263.

n-C5H,z 72.15 5.769 345. 469.8 33.3 311.

IJ-C6H14 86.17 5.909 413. 507.9 29.9 368.

n-C7H, 6 100.20 540.2 27.0 426.

n-C 8 H, 8 114.22 7.451 320. 569.4 24.6 485.

n-C9 Hzo 128.25 595.0 22.5 543. Cyclohexane 84.16 6.093 324. 553. 40.0 308.

C6 H 6 78.11 5.270 440. 562.6 48.6 260. contmued

Appendix B

Substance

CH4

CH 3Cl CH2Cl 2

CHC13

CC1 4

C 2 N 2

cos CS 2

Molecular weight

M

16.04 50.49 84.94

119.39 153.84

52.04 60.08 76.14

Table B.l (Continued)

Lennard-Jones parameters•

(J B/K 7;, (AJ (K) (K) -----~

Other organic compounds 3.822 137. 190.7 3.375 855. 416.3 4.759 406. 510.0 5.430 327. 536.6 5.881 327. 556.4 4.38 339. 400. 4.13 335. 378. 4.438 488. 552.

"Table used with permisswn from Bird et a1' 11

303

Critical constants'

p, J!; X 103

(atm) (m 3 kmol- 1)

45.8 99.3 65.9 143. 60. 54. 240. 45.0 276. 59. 61. 78. 170.

•values of a and cjK are from J. 0. Hirschfelder. C. F. Curtiss. and R. B. Bird, Molecular Theory of Gases and Liquids, John Wiley and Sons, New York (1954). The above values are computed from viscosity data and are applicable for temperatures above 100 K.

··values of 7;, p, and v; are from K. A. Kobe and R. E. Lynn, Jr., Chern. Rev. 52, 117-236 (1952); and American Petroleum Institute Research Project, Volume 44, (F. D. Rossini, ed.), Carnegie Institute of Technology (1952).

'For air, the molecular weight M and the pseudocritical properties 7;, p" and V,. have been calculated from the average composition of dry air, as given in International Critical Tables, Volume I, p. 393 (1926).

Table B.2. Lennard-Janes Potentials as Determined from Viscosity Dataa

b0 x 103

a(A) Molecule Compound (m 3 kmol- 1) B/k (K)

A Argon 46.08 3.542 93.3 He Helium 20.95 2.551' 10.22 Kr Krypton 61.62 3.655 178.9 Ne Neon 28.30 2.820 32.8 Xe Xenon 83.66 4.047 231.0 Air Air 64.50 3.711 78.6 AsH 3 Arsine 89.88 4.145 259.8 BC13 Boron chloride 170.1 5.127 337.7 BF3 Boron fluoride 93.35 4.198 186.3 B(OCH 3h Methyl borate 210.3 5.503 396.7 Br2 Bromine 100.1 4.296 507.9 CC14 Carbon tetrachloride 265.5 5.947 322.7 CF4 Carbon tetrafluoride 127.9 4.662 134.0 CHC13 Chloroform 197.5 5.389 340.2 CH2 Cl 2 Methylene chloride 148.3 4.898 356.3 CH3 Br Methyl bromide 88.14 4.118 449.2 CH 3Cl Methyl chloride 92.31 4.182 350 CH30H Methanol 60.17 3.626 481.8 CH4 Methane 66.98 3.758 148.6 co Carbon monoxide 63.41 3.690 91.7

continued overleaf

304 Appendix B

Table B.2 (Continued)

b0 x 103

Molecule Compound (m 3 kmol- 1 ) a(A) ejk (K)

cos Carbonyl sulfide 88.91 4.130 336.0 C02 Carbon dioxide 77.25 3.941 195.2 CS 2 Carbon disulfide 113.7 4.483 467

CzHz Acetylene 82.79 4.033 231.8

CzH4 Ethylene 91.06 4.163 224.7

CzH6 Ethane 110.7 4.443 215.7 C2H5Cl Ethyl chloride 148.3 4.898 300 C2H 5 0H Ethanol 117.3 4.530 362.6

CzNz Cyanogen 104.7 4.361 348.6 CH 30CH 3 Methyl ether 100.9 4.307 395.0

CH 2CHCH3 Propylene 129.2 4.678 298.9 CH3CCH Methyl acetylene 136.2 4.761 251.8

C3H6 Cyclopropane 140.2 4.807 248.9

C3Hs Propane 169.2 5.118 237.1 n-C3 H 70H n-Propyl alcohol 118.8 4.549 576.7 CH 3COCH3 Acetone 122.8 4.600 560.2 CH 3COOCH 3 Methyl acetate 151.8 4.936 469.8

n-C4 H 1o n-Butane 130.0 4.687 531.4 iso-C4H 10 !so butane 185.6 5.278 330.1

C 2H 50CzHs Ethyl ether 231.0 5.678 313.8 CH 3COOC2H 5 Ethyl acetate 178.0 5.205 521.3

n-C 5H 1z n-Pentane 244.2 5.784 341.1 C(CH3)4 2,2-Dimethyl propane 340.9 6.464 193.4

C6H6 Benzene 193.2 5.349 412.3

C6H12 Cyclohexane 298.2 6.182 297.1 n-C6H 14 n-Hexane 265.7 5.949 399.3

Cl 2 Chlorine 94.65 4.217 316.0

Fz Fluorine 47.75 3.357 112.6 HBr Hydrogen bromide 47.58 3.353 449 HCN Hydrogen cyanide 60.37 3.630 569.1 HCl Hydrogen chloride 46.98 3.339 344.7 HF Hydrogen fluoride 39.37 3.148 330 HI Hydrogen iodide 94.24 4.211 288.7

Hz Hydrogen 28.51 2.827 59.7 H 20 Water 23.25 2.641 809.1

HzOz Hydrogen peroxide 93.24 4.196 289.3 H 2S Hydrogen sulfide 60.02 3.623 301.1 Hg Mercury 33.03 2.969 750 HgBr2 Mercuric bromide 165.5 5.080 686.2 HgC1 2 Mercuric chloride 118.9 4.550 750 Hgl 2 Mercuric iodide 224.6 5.625 695.6

lz Iodine 173.4 5.160 474.2

NH 3 Ammonia 30.78 2.900 558.3 NO Nitric oxide 53.74 3.492 116.7 NOCl Nitrosyl chloride 87.75 4.112 395.3

continued

Appendix B

Molecule

Nz N 20 02 PH 3

SF6

soz SiF4

SiH4

SnBr4

UF6

Table B.2 (Continued)

b0 x 103

Compound (m 3 kmol-I)

Nitrogen 69.14 Nitrous oxide 70.80 Oxygen 52.60 Phosphine 79.63 Sulfur hexafluoride 170.2 Sulfur dioxide 87.75 Silicon tetrafluoride 146.7 Silicon hydride 85.97 Stannic bromide 329.0 Uranium hexafluoride 268.1

u(A)

3.798 3.828 3.467 3.981 5.128 4.112 4.880 4.084 6.388 5.967

305

o/k (K)

71.4 232.4 106.7 251.5 222.1 335.4 171.9 207.6 563.7 236.8

"R. A. Svehla. NASA Techmcal Report R-132, Lewis Research Center, Cleveland, Ohio (1962); table used with permissiOn from Reid and Sherwood.' 21

'b 0 =~nN0a3 , where N 0 is Avogadro's number. 'The potential a was determined by quantum mechanical formulas.

Table B.J. Stockmayer-Potential Parameters"

Dipole moment <T o/k b0 x 103

Molecule 11 (debyes) (A) (K) (m 3 kmoJ-I) bmax

H20 1.85 2.52 775 20.2 1.0 NH 3 1.47 3.15 358 39.5 0.7 HCl 1.08 3.36 328 47.8 0.34 1-lBr 0.80 3.41 417 50.0 0.14 HI 0.42 4.13 313 88.9 0.029

soz 1.63 4.04 347 83.2 0.42 H 2S 0.92 3.49 343 53.6 0.21 NOCI 1.83 3.53 690 55.5 0.4 CHC1 3 1.013 5.31 355 189 0.07 CH 2Cl2 1.57 4.52 483 117 0.2 CH 3Cl 1.87 3.94 414 77.2 0.5 CH 3 Br 1.80 4.25 382 96.9 0.4 C2H 5Cl 2.03 4.45 423 Ill 0.4 CH 30H 1.70 3.69 417 63.4 0.5 C2H 50H 1.69 4.31 431 101 0.3 n-C3 H 70H 1.69 4.71 495 132 0.2 i-C3 H 70H 1.69 4.64 518 126 0.2 (CH 3 lz0 1.30 4.21 432 94.2 0.19 (CzHslzO 1.15 5.49 362 209 0.08 (CH 3)zCOb 1.20 3.82 428 70.1 1.3

contmued overleaf

306 Appendix B

Table B.J (Continued)

Dipole movement () &/k b0 x 103

Molecule J1 (debyes) (Al (K) (m 3 kmo1- 1 ) <}max

CH 3COOCH 3 1.72 5.04 418 162 0.2 CH 3COOC2H 5 1.78 5.24 499 182 0.16 CH 3NO/ 2.15 4.16 290 90.8 2.3

"L. Monchick and E. A. Mason. J. Chern. Phys. 35. 1676 (1961); table used with permission from Reid and Sherwood.<2> "From G. A. Bottomley and T. H. Spurling. Austral. J. Chern. 16. I (1963). Monchick and Mason show that a=4.50 A; r.fk=549 K.

Table B.4. Values of the Collision Integral O..Jor Viscosity Based on the Lennard-Janes Potential"

T*=kT/Eb Qvb kT/£ n,. kT/c Qv

------··~-·---~·------- .. ---- --------

0.30 2. 785 1.65 1.264 4.0 0.9700 0.35 2.628 1.70 1.248 4.1 0.9649 0.40 2.492 1.75 1.234 4.2 0.9600 0.45 2.368 1.80 1.221 4.3 0.9553 0.50 2.257 1.85 1.209 4.4 0.9507 0.55 2.156 1.90 1.197 4.5 0.9464 0.60 2.065 1.95 1.186 4.6 0.9422 0.65 1.982 2.00 1.175 4.7 0.9382 0.70 1.908 2.1 1.156 4.8 0.9343 0.75 1.841 2.2 1.138 4.9 0.9305 0.80 1.780 2.3 1.122 5.0 0.9269 0.85 1.725 2.4 1.107 6.0 0.8963 0.90 1.675 2.5 1.093 7.0 0.8727 0.95 1.629 2.6 1.081 8.0 0.8538 1.00 1.587 2.7 1.069 9.0 0.8379 1.05 1.549 2.8 1.058 10 0.8242 1.10 1.514 2.9 1.048 20 0.7432 1.15 1.482 3.0 1.039 30 0.7005 1.20 1.452 3.1 1.030 40 0.6718 1.25 1.424 3.2 1.022 50 0.6504 1.30 1.399 3.3 1.014 60 0.6335 1.35 1.375 3.4 1.007 70 0.6194 1.40 1.353 3.5 0.9999 80 0.6076 1.45 1.333 3.6 0.9932 90 0.5973

'untmued

Appendix B

1.50 1.55 1.60

Table B.4 (Continued)

1.314 1.296 1.279

kT/e

3.7 3.8 3.9

0.9870 0.9811 0.9755

kT/e

100 200 400

0.5882 0.5320 0.4811

"Table used with permission from Reid and Sherwood.<" •Hirschfelder, Curtiss, and Bird13) use the symbol T* for kT /E and Q<2.2J*

for Q". Bromley and Wilke14l use (kT/e) 112 V/W2(2) forf1(kT/e). More complete tables of these functions are given In the two references cited.

Table B.5. Values ~f the Collision Integral Ov Based on the Lennard-Janes Potential"

kT/eb nvb kT/e nv kT/r. nv

0.30 2.662 1.65 1.153 4.0 0.8836 0.35 2.476 1.70 1.140 4.1 0.8788 0.40 2.318 1.75 1.128 4.2 0.8740 0.45 2.184 1.80 1.116 4.3 0.8694 0.50 2.066 1.85 1.105 4.4 0.8652 0.55 1.966 1.90 1.094 4.5 0.8610 0.60 1.877 1.95 1.084 4.6 0.8568 0.65 1.798 2.00 1.075 4.7 0.8530 0.70 1.729 2.1 1.057 4.8 0.8492 0.75 1.667 2.2 1.041 4.9 0.8456 0.80 1.612 2.3 1.026 5.0 0.8422 0.85 1.562 2.4 1.012 6 0.8124 0.90 1.517 2.5 0.9996 7 0.7896 0.95 1.476 2.6 0.9878 8 0.7712 1.00 1.439 2.7 0.9770 9 0.7556 1.05 1.406 2.8 0.9672 10 0.7424 1.10 1.375 2.9 0.9576 20 0.6640 1.15 1.346 3.0 0.9490 30 0.6232 1.20 1.320 3.1 0.9406 40 0.5960 1.25 1.296 3.2 0.9328 50 0.5756 1.30 1.273 3.3 0.9256 60 0.5596 1.35 1.253 3.4 0.9186 70 0.5464 1.40 1.233 3.5 0.9120 80 0.5352 1.45 1.215 3.6 0.9058 90 0.5256 1.50 1.198 3.7 0.8998 100 0.5130 1.55 1.182 3.8 0.8942 200 0.4644 1.60 1.167 3.9 0.8888 400 0.4170

"From J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, "Molecular Theory of Gases and Liquids, John Wiley and Sons, Inc. New York (1954); table used with permission of Reid and Sherwood.'"

'Hirschfelder used the symbols T* for kT/e and 0°·"* in place of 0 0 .

307

308 Appendix B

Table B.6. Collision Integrals D.vfor Viscosity as Calculated by Use of the Stockmayer Potential"·b

~ 0' 0.25 0.50 0.75 1.0 1.5 2.0 2.5

0.1 4.1005 4.266 4.833 5.742 6.729 8.624 10.34 11.89 0.2 3.2626 3.305 3.516 3.914 4.433 5.570 6.637 7.618 0.3 2.8399 2.836 2.936 3.168 3.511 4.329 5.126 5.874 0.4 2.5310 2.522 2.586 2.749 3.004 3.640 4.282 4.895 0.5 2.2837 2.277 2.329 2.460 2.665 3.187 3.727 4.249 0.6 2.0838 2.081 2.130 2.243 2.417 2.862 3.329 3.786 0.7 1.9220 1.924 1.970 2.072 2.225 2.614 3.028 3.435 0.8 1.7902 1.795 1.840 1.934 2.070 2.417 2.788 3.156 0.9 1.6823 1.689 1.733 1.820 1.944 2.258 2.596 2.933 1.0 1.5929 1.601 1.644 1.725 1.838 2.124 2.435 2.746 1.2 1.4551 1.465 1.504 1.574 1.670 1.913 2.181 2.451 1.4 1.3551 1.365 1.400 1.461 1.544 1.754 1.989 2.228 1.6 1.2800 1.289 1.321 1.374 1.447 1.630 1.838 2.053 1.8 1.2219 1.231 1.259 1.306 1.370 1.532 1.718 1.912 2.0 1.1757 1.184 1.209 1.251 1.307 1.451 1.618 1.795 2.5 1.0933 1.100 1.119 1.150 1.193 1.304 1.435 1.578 3.0 1.0388 1.044 1.059 1.083 1.117 1.204 1.310 1.428 3.5 0.99963 1.004 1.016 1.035 1.062 1.133 1.220 1.319 4.0 0.96988 0.9732 0.9830 0.9991 1.021 1.079 1.153 1.236 5.0 0.92676 0.9291 0.9360 0.9473 0.9628 1.005 1.058 1.121 6.0 0.98616 0.8979 0.9030 0.9114 0.9230 0.9545 0.9955 1.044 7.0 0.87272 0.8741 0.8780 0.8845 0.8935 0.9181 0.9505 0.9893 8.0 0.85379 0.8549 0.8580 0.8632 0.8703 0.8901 0.9164 0.9482 9.0 0.83795 0.8388 0.8414 0.8456 0.8515 0.8678 0.8895 0.9160

10.0 0.82435 0.8251 0.8273 0.8308 0.8356 0.8493 0.8676 0.8901 12.0 0.80184 0.8024 0.8039 0.8065 0.8101 0.8201 0.8337 0.8504 14.0 0.78363 0.7840 0.7852 0.7872 0.7899 0.7976 0.8081 0.8212 16.0 0.76834 0.7687 0.7696 0.7712 0.7733 0.7794 0.7878 0.7983 18,0 0.75518 0.7554 0.7562 0.7575 0.7592 0.7642 0.7711 0.7797 20.0 0.74364 0.7438 0.7445 0.7455 0.7470 0.7512 0.7569 0.7642 25.0 0.71982 0.7200 0.7204 0.7211 0.7221 0.7250 0.7289 0.7339 30.0 0.70097 0.7011 0.7014 0.7019 0.7026 0.7047 0.7076 0.7112 35.0 0.68545 0.6855 0.6858 0.6861 0.6867 0.6883 0.6905 0.6932 40.0 0.67232 0.6724 0.6726 0.6728 0.6733 0.6745 0.6762 0.6784 50.0 0.65009 0.6510 0.6512 0.6513 0.6516 0.6524 0.6534 0.6546 75.0 0.61397 0.6141 0.6143 0.6145 0.6147 0.6148 0.6148 0.6147

100.0 0.58870 0.5889 0.5894 0.5900 0.5903 0.5901 0.5895 0.5885

"L. Monchick and E. A. Mason. J. Chern. Phys. 35, 1676 (1961); table used with permission from Re1d and Sherwood.121

'T*=kT/a.li=(dipole moment) 2/2a0a 3

'The values offl, m this column differ slightly from values in Table B.1 at low values ofT*.

Appendix B

Notation for the Tables

T Temperature (K) T* kT/e V Molar volume (m 3 kmol- 1 )

b Dipole moment parameter e/k Potential parameter (K) O" Collision diameter (A)

References for the Tables

309

n Collision integral

Subscripts c critical

D diffusion fl viscosity

I. R. B. Bird, W. E. Stewart, E. N. Lightfoot, Transport Phenomena, John Wiley and Sons, New York (1960).

2. R. C. Reid and T. K. Sherwood, The Properties of Gases and Liquids, 2nd edition, McGraw­Hill Book Co., New York (1966).

3. J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, John Wiley and Sons, New York (1954).

4. L.A. Bromley and C. R. Wilke, Ind. Eng. Chern. 43, 1641 (1951).

Appendix C

Derivation o.f Proposed Four-Flux Radiation Model*

In the following analysis, it was assumed, for the sake of simplicity, that the gas medium is totally transparent, so that the absorption coefficient and scattering coefficient are solvely functions of the number of particles in a unit volume of the gas-particle cloud. The contribution of absorption by gaseous components of the medium, though small compared to the contribu­tion by particles, can be added later as a correction.

In order to formulate a model that represents the actual conditions that exist in a pulverized-coal flame, namely, the presence of "large" particles that scatter radiation, an effort is made to treat scattering as anisotropic. This is achieved by introducing the forward-, backward-, and sidewise-scat­tering components, which represent the fraction of scattered radiation in each of the corresponding directions.

The forward-scattering component is defined as

f+1tj2

f = n P(e) sine de co·s2 e -Tt/2

f1t/2

= 2n 0

P(e) sin e cos2 e de (1)

where the notation is that of Chapter 5. The backward-scattering com­ponent is defined as

b =n f-"12 P(e) sine de cos2 e 1t/2

*Sneh Anjali Varma, Graduate Research Assistant, Department of Mechanical and Industrial Engineering, University of Utah, Salt Lake City, Utah

311

312

= 2n f" P((}) cos2 (} sin (} d(} 1t/2

and the sidewise-scattering component is

s=(l-f-b)/4

Appendix C

(2)

(3)

For the isotropic case, the above fractions degenerate to f= b= s=f>. For detailed discussion, see reference 6 of Chapter 5.

The phase function, P(8), is a function of particle size, index of refraction, and wavelength of radiation. For spherical particles, the angular distribution is symmetrical about the direction of the incident radiation.

To begin with, a six-flux model is based on drawing energy balances for six discrete components of intensity of radiation in six orthogonal directions, and then, by invoking the condition of axial symmetry, reducing the equations to a four-flux model. In order to accommodate the geometry of the combustor of interest, a cylindrical-polar coordinate system is employed.

Consider a small volume element dV located at a point P(R, 0), as shown in Figure 1. The intensity of I(R, 0) is represented by six discrete components in the direction of, and opposite to the direction of, the three major directions, which are axial, radial, and angular.

Now, consider the energy balance for the intensity vector I:, directed in the positive z-direction. Change in intensity during passage through the small volume is given as

U: +di;- /dz)-I;- =di;- /dz (4)

Loss in intensity due to absorption by the matter in the small volume is

(5)

and loss in intensity due to scattering by the matter in the small volume is

(6)

The fraction of scattered radiation in the direction of the forward­directed intensity vector is f K.I: and the rest is scattered in other directions, and thus considered lost. The net lost scattered radiation is then

-(1-.f)K.J: (7)

Similarly, there will be an addition to the forward-directed intensity by the backward-scattering component of intensity vector I;, moving in the opposite direction. This increase is

+bK.I; (8)

The intensities traveling perpendicular to the z-direction will contribute a fraction of their out-scattered radiation to the positive z-direction, given

Appendix C 313

HR • .fi.)

a

Ii

----------b

Figure 1. Volume element for radiative analysis. (a) The angular distribution of intensity. (b) The distribution of intensity in six discrete components.

in terms of s, as

(9)

There will also be a contribution to intensity in the positive z-direction by the radiation emitted by matter in the small volume. Considering this emitted radiation to be uniformly distributed in all directions, it may be represented by six equal, discrete components in the six orthogonal direc­tions. Thus, the contribution of emitted radiation in the positive z-direction is

(10)

Summing up all the above terms, an energy balance for radiative trans­port in the positive z-direction is given as

dl: /dz= -Kai: -(1-f)K.I: +bK.I; +SK.(I: +I; +Iii +Ii) +(Ka/6)Ib(T) (11)

314 Appendix C

or, combining terms:

di; /dz= [Ka+(1-f)K.]I; +bKJ; +SK.(I: +I; +It +Ii)+Kj6[Ib(T)] (12)

Replacing (Ka + K.) by K 1 and K./ K 1 by W0 , the above equation can be rewritten as

(1/Kr)(di: /dz)= -(1- WoJ)I: + Wobi; + WoSUr+ +I;+ It+ Ii) +(1/6)(1- W0 )h(1) (13)

By writing similar energy balance equations in the other directions, the following equations can be obtained for each direction:

-(1/Kr)(di; /dz)= -(1-Wof)I; + W0 bi; + W0 S(I: +I;+ It+ Ii)

+(1/6)(1- W0 )Ib(T) (14)

(1/Kr)[d(I: 0 r)/dr]= r[ -1(1- W0 f)I: + W0 bir- + W0 S(I; +I;+ It+ Ii)

+ (1/6)(1- W0 )I b(T)] (15)

-(1/Kr)[d(I; 0 r)/dr] =r[ -(1- W0 f)I; + W0bi: + W0 S(I; +I;+ It+ Ii)

+(1/6)(1- W0 )Ib(T)] (16)

(1/rKr)(dit /dfJ)= -(1- Wo.f)It + W0 bi0 + W0 S(I; +I;+ I:+ I;)

+ (1/6)(1- W0 )I b(T) (17)

-(1/rKr)(dii /dfJ)= -(1- W0f)Ii + W0bit + W0 S(I; +I;+ I:+ I;)

+ (1/6)(1- W0 )I b(T) (18)

The assumption of axial symmetry results in the following conditions:

(19)

and (20)

By applying these two conditions in the last two equations of the six­flux model, there results

It =Ii =[W0S/(1- W0 f- W0b}](I: +I; +I: +I;) +(1/6)[(1- W0)/(1- W0 f- W0b)]Ib(T) (21)

Substituting these expressions for Ir+ and I; in one of the remaining flux equations gives

(1/K1)(di: /dz)=I: { -(1- W0 f)+ [2W6S2/(1- W0 f- Wob)]} +I; {W0b+ [2W6S2/(1- W0 f- Wob)]} +(I: +I;){W0 S+ [2W6S2/(1- Wof- Wob))}

Appendix C 315

+ Ib[(1/6)(1- W0 )]{1 + [2W0S/(1- W0f- W0 b)]} (22)

(1/K 1)(di: /dz) = C 1I: + C 2I; + C3(1: +I;)+ C4I b(T) (23)

where

C 1 = -(1- W0 f)+ [2W6S2/(1- W0f- W0 b)] (24)

C2 =W0b+[2W6S2/(1-W0 j-W0b)] (25)

C 3 =W0 S+[2W6S2/(1-W0 f-W0b)] (26)

and

C4 = [(1- W0 )/6]{1 + [2W0S/(1- W0f- W0 b ]} (27)

The complete four-flux model can now be written as

(1/K1)(di: /dz)= C 1I: + C2I; + C3(1: + Ir-)+ C4Ib(T) (28)

-(1/K1)(di; /dz)= C 1I; + C 2I: + C3(1: + Ir-)+ C4 Ib(T) (29)

(1/K1)[d(Ir+ · r)/dr] = r[ C 1I: + C2I; + C 3(J,: +I;)+ C4 Ib(T)] (30)

and

Appendix D

Derivation of Eulerian Finite-Difference Equations*

In this appendix, the general Eulerian finite-difference equation is derived in two-dimensional, steady, cylindrical coordinates. The flow field of interest is subdivided into computational cells by some grid pattern. Figure 1 displays a typical internal main cell where the 4>-equation [Eq. (2), Chapter 14] can be integrated over the volume obtained by rotating the area represented by the dashed lines about the symmetry axis to give r: rn s: {(a;ax)(pur4>)+(8/8rXpvr4>)-(a;ax)[rr(84>/8x)]

-(8/8r)[rr(84>/8r)]-rS<1>} dx dr d~ 3 =0 (1)

where the third coordinate, ~ 3 , has the integration limits of 0 to 1 radians for convenience instead of 0 to 2n radians, because of assumed axial sym­metry.

Considering the first convection term in Eq. (1)

fXe f'n e (8/8xXpur4>) dx dr d~3 Xw r8 J 0

and noting that all properties are uniform in the third direction, performing two formal integrations gives

frn [purc/> J: dr rs

where e and w represent the expression to be evaluated at the east and west faces, respectively. As with any finite-difference development, the derivation

*John J. Wormeck, Senior Engineer, Department of Mechanical and Industrial Engineering, University of Utah, Salt Lake City, Utah

317

318

NW •

we

• sw

N •

nw n ne r-------------~

I I I I I

I I I

:e

L ----- - ---- - - - _J sw s se

• 5

Appendix D

NE •

• SE

Figure 1. Illustration of the grid symbols for a computational cell.

is somewhat arbitrary; the following method yields the most accurate results. From the mean-value theorem:

f+t.r f(r) dr~J(f)(ilr) (2)

where r<r<r+ilr, and as r---+0 convergence is assumed. Applying this theorem to the first convection term yields

(puro/).,(rn -r.)-(puro/)w(rn -r.)

where each subscript represents evaluation at that particular face. Grouping the geometric terms gives

(puo/ ).,A., - (puo/ )wAw

with the following definition.s

A.,= r.(rn -r.)

Aw=rw(rn-rs)

(3)

which are the areas of the east and west faces of the cell, respectively, as

Appendix D 319

shown by considering

f rn Jl A,= rdrdC 3 =(r;-r;)/2

'n 0

or

A,= (rn -r.)(rn + r.)/2 = rp(rn -r.)

Furthermore, since re = r w• these areas are equal and only one symbol will be used:

(4)

The numerical procedure TEACH< 1l employs a staggered grid system,<2 l

where the velocities are stored midway between the grid lines; that is, at the exact locations which are required. The first convection term [Eq. (3)] becomes

Convection coefficients are defined as

CE = PeUEA.:w

Cw = PwUpA,w

which gives the mass flux through the face corresponding with the subscript. Both p and 4> are defined at the main grid nodes and some sort of inter­

polation is needed to determine their values at the faces midway between node points. The practice with TEACH is to linearly interpolate dependent variables and use simple averaging for fluid properties; thus

Pe = ( PP + PE)/2,

and

P<¢<E: ¢=(1-fE)4Jp+fE¢E, fE=(X-Xp)j(XE -Xp)

W<¢<P: 4>=0-fw)¢w+fw¢P, fw=(x -xw)/(xp-Xw) P<¢<N: 4> = (1-fN)4Jp + fNcf>N, !N= (r -rp)j(rN -rp) S <4> <P: ¢=(1-fs)4>s +fs¢p, fs =(r-rs)/(rp-rs)

Using these relationships, the convection coefficients are

CE = (pp + PE)UEAew/2

and the first convection term becomes, upon substitution:

(5)

(6)

(7)

(8)

320

Similarly, the second convection term in Eq. (1) is

fxefrn f1 (o/or)(pvr¢) dx dr d~ 3 Xw rs J 0

Again, two integrations can be performed formally to give

r: [pvr¢ ]~ dx

and from the mean value theorem

(pvc/J)nrn(Xe -xw) -(pvc/J),r,(Xe -Xw)

where the geometric terms are

An= rn(Xe- Xw) A,=r,(Xe-Xw)

Appendix D

(9)

and in this case are different, and hence the convection coefficients are defined as

CN = PnVnAn = (PN + pp)VNAn/2 Cs = p,v,A, = (Ps + pp)VpA,/2

(10)

Therefore the final form of a second convection term of the ¢-equation becomes

CNfN¢N-Cs(1-fs)¢s + [CN(l-fN)-Csfs]cPP

Considering the diffusion terms in Eq. (1) separately

(11)

r: rn f { -(8/ox)[rr(o¢/ox)] -(o/or)[rr(o¢jor)]} dx dr d~3 (12)

and integrating twice gives

rn - [rr(o¢jox)]~ dr-r: [rr(o¢/or)]~ dx

Using the same technique as presented for the convection terms, these last integrals can be evaluated as

-re(o¢/ox)ere(rn -r,)+r w(o¢jox)wrw(rn -r,)

- rn(oc/Jjor)nrn(Xe- Xw) + r,(oc/Jjor),r8(Xe- Xw)

As expected, the same geometric quantities appear as the convection terms; substituting Eqs. (4) and (9) yields

- re(o¢/ox)eAew + r w(o¢/ox)wAew- rn(o¢/or)nAn + r.(o¢/or).A.

The derivatives at the four faces must be expressed in terms of variables at

Appendix D 321

main node points. Employing central differences (which are second-order accurate)<3> gives

- r e[(cj>E -cj>p)j(DXpE)]Aew + r w[(cj>p -c/>w)/(DXpw)]Aew

- rn[(cf>N -cj>p)/(D.YNp)]An + r.[(cj>p -c/>s)/(byps)]A.

where the bx and by stand for coordinate distance between the node points indicated by their corresponding subscripts.

Diffusion coefficients can be defined as

DE~ re(Aew)/(DXpE)=(r p+ r E)Aew/(2 DXpE)

Dw= r w(Aew)/(DXpw)=(rp+ r w)Aew/(2 DXpw)

DN = rn(An)/(D}'Np) = (r p + r N)An/(2 D}'Np)

Ds = r.(A.)/(Dyp5 ) =(r p+ r s)A./(2 Dyps)

Thus the diffusion terms can be expressed as

-DE(cJ>E -cj>p) + Dw(cJ>p -c/Jw) -~(cf>N -cj>p)+ Ds(cJ>p-cJ>s) (13)

where the similarity with the control volume formulation is noted. The exchange coefficients and geometric quantities are contained in the diffusion coefficients D, while the difference in 4> which drives the diffusion is explicitly shown with the correct sign, such that 4> enters the cell when the ¢-difference is negative.

Finally, considering the source terms in Eq. (1):

fxe frn il rS<I> dx dr d~ 3

xw r5 0

One of the major techniques responsible for success of the TEACH formulation is to express this source term as linear in the dependent variable. Thus

fxe frn il rS<I> dx dr d~ 3 =St+SicJ>P

Xw rs 0 (14)

which defines two source term coefficients, st and Si. If the source term happens to be nonlinear in terms of the dependent variable, cj>p, the technique calls for just cj>p to be factored from the expression (if possible) and to appear with the st coefficient in Eq. (14); i.e., cj>p appears implicitly, while the remaining factored expression involving cj>p will be considered as known (explicit-based on old values) and lumped together in the st coefficient of Eq. (14). Furthermore, as shown in the stability analysis,<2 > st must be nega­tive to guarantee stable convergence.

In general, s<~> will be functions of all the dependent variables and other fluid properties as well as various types of derivatives involving these quantities. When integrating this source term, the value prevailing at the cell center will be used for all quantities and any derivatives will be evaluated

322 Appendix D

by central differencing. Therefore, the source term is considered constant, giving

(15)

where .1 V is the volume of the cell. Upon substitution of these newly defined coefficients [Eqs. (8),(11 ),

(13), and (15)] into Eq. (1) the general ¢-equation becomes

[CE/E -DE]cf>E -[Cw(1-fw)+ Dw]cf>w+ [CN/N -~]cf>N-[Cs(1-fs)+ Ds]cf>s

+ [ CE(l-/E)+ DE- Cwfw+ Dw+ CN(1 -/N) + ~-Csfs + Ds]cf>P

=Su+Spcf>p

Adding and subtracting CE-Cw+CN-Cs from the expression in brackets preceding cf>p and rearranging to obtain common expressions gives

[(CE-Cw+CN-Cs)+Dw+(1-fw)Cw+DE -fECE

+Ds +(1-fs)Cs +~-fNCN]cf>p

= [Dw+(1-fw)Cw]cf>w+ [DE -fECE]cf>E + [Ds +(1-fs)Cs]cf>s

+ [DN-fNCN]cf>N+Su+ Spcpp

It is convenient to define new total coefficients to replace these common expressions:

AE=DE-/ECE

Aw=Dw+(1-fw)Cw

AN=~-fNCN

As =Ds -(1-fs)Cs

(16)

In terms of these total coefficients, the finite-difference form of the ¢-equation becomes

[CE -Cw+ CN-Cs +AE + Aw+ AN+ As]cf>p=AEcf>E + Awcf>w+ AN<f>N +Asc/>s+Su+Spcpp (17)

Numerical stability considerations lead to two further modifications of the above equation, which finally reduces to

[ AE + Aw+ AN+ As]cf>p= AEcf>E + Awcf>w+ AN<f>N + Asc/>s + Su+ Spcpp (18)

The above analysis is for illustrative purposes only; the interested reader is referred to a lengthy but exhaustive treatment< 2> of the procedure including the formulation of both velocity and pressure equations, boundary condi­tions, solution of finite-difference equations, underrelaxation, grid definition, and extensions to three-dimensional time-dependent, general orthogonal coordinates systems.

Appendix D 323

References l. A.D. Gosman and W.M. Pun, Lecture notes for course entitled "Calculation of Recirculating

Flows," Imperial College of Science and Technology, London, Report No. HTS/74/2 (1974). 2. J.J. Wormeck, Computer Modeling of Turbulent Combustion in a Longwell Jet-Stirred

Reactor, Ph.D. Dissertation, Washington State University, Pullman, Washington (1976). 3. P.J. Roache, Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, New

Mexico (1972).

Index

Absorption coefficient definition, 87 for gas, 100 for gas-particle cloud, 100 for monodisperse particles, 95 overall, 101 for p·_,lydisperse particles, 101 relation to absorption index, 92

Absorption efficiency definition, 93, 99 vs. size parameter of char, 94

Absorption index, 92 Acceleration modulus, 114 Activation temperature, 66 Advection of turbulent kinetic energy,

61 Alkali metals

calcium CaC03 , 196, 198 CaS04, 196, 198 gypsum, 196 reactions of, 197-198 sulfur retention by, 198, 204

sodium NaCl, evaporation and condensation

of, 201 NaOH, 201 Na2 S04, formation and condensation

of, 201 reactions of, 201, 204 sulfur retention by, 198, 204

Ammonia in coal combustion, 186 from devolatilization, 190 gas-phase kinetics, role in fuel-NO,

192-194 gasification, 208 in industrial coking, 188

325

Ammonia (cont.) from pyrolysis of model nitrogen

compounds, 192 in stirred-reactor combustion, 194-195

Anisotropic scattering, 89, 97, 99 Anisotropy, 60,62 Ash

composition, 129 definition, 133 effect on overall absorption coefficient,

101 effect on overall scattering coefficient,

101 effect on radiative transport, 100 equations, 228-229 model assumptions, 220 phase function, 100 shape, 92 trace elements, 130 as a tracer, 13 9

Attenuation: see also Extinction, Interception

defmition, 87, 92, 93 efficiency, 93 efficiency vs. size parameter of char, 94

Benzonitrile, 192 Blowing parameter

heat transfer, 225, 226, 282 mass transfer, 226, 282

Bouger-Lambert's law, 87 Boundary conditions, 102, 241, 268 Boussinesq assumption, 59 Buoyancy, 60

Calcium: see Alkali metals Calcium sulfate: see Alkali metals

326

Carbon monoxide, 183 Carbon-to-hydrogen ratio, 179 Char

appearance, 140 composition,128,137, 220 definition, 131 density, 228 effect on overall absorption coefficient,

101 effect on overall scattering coefficient,

101 enthalpy, 229 formation, 221, 223 index of refraction, 92 oxidation, 217-221, 223 phase function, 100 product enthalpy, 230 radiation coefficient vs. size parameter,

94 SEM photographs, 141 shape, 92 size distribution, 131 size parameter, 100

Char-carbon dioxide reaction activation energy, 157-159 Arrhenius factors, 157-158 controlling resistances, 160 mechanism, 156 "poisoning" due to carbon monoxide,

156 rate expressions

exponential, 158 Langmuir, 156

reaction order, 156 recommendation, 160

Char-hydrogen reaction, 164 Char-nitrogen: see Nitrogen-in-coal Char-oxygen reaction

activation energy, 155 Arrhenius factors, 155 controlling resistances, 152 internal surface area effects, 154 mechanism, 153 rate constants

chemical, 153-155 diffusional, 153

reaction order, 153, 155 recommendation,154

Char-steam reaction activation energy, 161-163 Arrhenius factors, 162-163 controlling resistances, 16 3

Char-steam reaction (cont.) mechanism, 161 "poisoning," 160 rate expressions

exponential, 161 Langmuir, 160

reaction order, 161 recommendation, 163 water-gas shift reaction, 163

Chemical equilibrium, 3-13 Chemical potential, ideal gas, 4 Chlorine

condensation of, 201 evaporation of, 201 HQ, 201 reaction of, 201

aosure problem, 59 Qosure schemes

mean Reynolds stress, 62 mean turbulent energy, 61-62 mean velocity field, 59-61

CN, kinetic role in fuel-NO, 192-194 Coal

ASTMrank anthracite, 124,126, 129 bituminous, 124, 126, 129 lignite, 123-124, 126, 129 meta-anthracite, 124 subbituminous, 124, 126, 129

characteristics, 123 classification, 123-125 composition, 123 formation, 123 heating value, 124 properties

density, 228 enthalpy, 229

reaction kinetics, 221-224 model assumptions, 220 modeling, 217-232 observations regarding, 218-219

SEM photograph, 141 size distribution, 131 specific gravity, 126 specific heat, 126 structure, 131-132 swelling index, 127 thermal conductivity, 126

Index

Coal-·dust flames: see Propagating flames Coal-mine explosions, 235-236, 248 Coal particle: see Particle

Index

Coalescence/dispersion mixing model, 7 5 Collision functions, 2 84 Combustion

of methane-air, 172 of volatiles, 169, 179

Condensed phases, 9 Continuity equation, differential

discrete phase, 22-23 gas-phase

in gas-particle mixture, 21-22 mixture of species, 20 species, 23

overall, 23 Continuity equation, macroscopic

gas-phase, in gas-particle mixture, 40-41 overall, 41 particulate phase, 41

Contact index, 68, 73, 75, 250 Convergence criteria, 12, 241, 254, 291 COS,197,202,203,205 Cross sections

for angular scattering, 93, 99 for total scattering, 93, 99

cs2 ,202,205 Cyanogen, 190

Density bulk density

gas, 21 particles, 22

gas mixture, 19 gas-particle mixture, 23 mass density of species i, 19 molar density of species i, 19

Devolatilization, 217, 218, 222, 239, 242, 244,246,247, 252;see also Pyrolysis

Diffuse particles, 100 Diffuse surface, 87, 102; see also Lambert

surface Diffusion

as a rate-limiting process, 245-24 7 transport of enthalpy, 30, 3 7 transport of kinetic energy, 30

Diffusion approximation method, 84, 98 Diffusivity, gas

basis, 46 binary, 51 equations, 51 mixtures, 51, 283 multicomponent, 51-52

Dissipation function, 273 of turbulent kinetic energy, 61, 273

Distribution of particles, 245

327

Drag coefficient, particle, 110, 219;see also Particle drag

Eddy viscosity, 59-62 Efficiency factor

for absorption, 93, 94, 99 for extinction, 93, 94, 99 for scattering, 93, 94, 99

Emission coefficient, 86 Emissivity, 100, 102 Energy density, 86 Energy equation, differential

discrete-phase, 33-38 gas-particle mixture, 31-33 gas-phase

mixture of gaseous species, 28-31 thermal, 31

overall, 38 Energy equation, macroscopic

gas phase, in gas-particle mixture, 43-44 overall, 45 particulate phase, 44-45

Enthalpy ideal gas, 4, 10 zero of, 11

Equations of motion, 58 Equilibrium

chemical, 3-13 multicomponent, 3-13

Equilibrium temperature, 101 Equilibrium theory of turbulence, 61 Equivalence ratio, 242 Eulerian formulation, 15-18 Eulerian reference frame, 266 Explosions: see Coal-mine explosions Extinction, 87; see also Attenuation Extinction coefficient, 94

"f-g" equations, 71 FeS, 197 Film theory, 225 Finite-difference equations, 268 Fixed carbon, 134 Flame thickness, 236,241, 243, 251 Flame velocity: see Propagation velocity Fluorine, 183

328

Flux method basis, 98 four-flux, 98, 99 for furnaces, 84 six-flux, 99 two-flux, 98

Fluctuations, 59-62,67, 75 Fly ash

formation mechanisms cenospheres, 199-200 dense flyash, 198-199 submicron flyash, 200-201

gasification, 208 size distribution, 195

Force aerodynamic, 25, 27, 32, 34,44 body,24,26,27,44 surface, 24

Four-flux radiation model, 285 Fuel-nitrogen: see Nitrogen-in-coal Fuel-NO: see Nitric oxide

"Garbage" estimates, 13 Gas absorption coefficient, 101 Gas emissivity, 101 Gas-phase

density, 226, 227 diffusivity, 226, 227 flames: see Propagating flames heat capacity, 226, 227 mixture properties, 226 oxidation: see Hydrocarbon oxidation thermal conductivity, 226, 227

Gasification entrained-flow, 207 K-T process, 208 pollutantformation,207

Gauss-Seidel iteration, 269 Gibbs function

ideal gas, 4 minimization of, 3-9

Global reactions, 178-179 Gray body, 87 Gfay boundary, 102 Gray medium, 96, 100

Heat transfer conductive, in gas, 30, 32,44 gas and particle, 225 moisture vaporization control, 224

Heat transfer (cont.) radiative, 32, 44,45 on separated surface, 35, 36

Heating rates rapid, 169-171 slow, 169-171

Heavy metals condensation of, 195, 199, 200 list of, 195, 200 toxicity, 183 vaporization of, 199, 200

Index

Heterogeneous reactions of carbon (char) activation energy

apparent, 150 true, 150

with carbon dioxide: see Char-carbon dioxide reaction

controlling resistances bulk phase mass transfer, 150 chemical reaction, 150 pore diffusion, 150

diffusional effects, minimization of, 150 with hydrogen: see Char-hydrogen

reaction impurities, catalytic effect of, 152 model assumptions, 149 with oxygen: see Char-oxygen reaction reaction order, 150 relative rates, 150 with steam: see Char-steam reaction temperature regimes, 150 variables influencing rates, 151

HOCN, 193 Hottel's zone method, 84, 97, 98 HS, 204,205 HS02, 204 H2S04, 205 Hydrocarbon, 239, 245 Hydrocarbon oxidation, 217, 219-221,

224 Hydrogen cyanide

in coal combustion, 186 from devolatilization, 190-191 gas-phase kinetics, role in fuel-NO,

192-194 gasification, 208 from pyrolysis of model nitrogen

compounds, 192 role in prompt NO, 187 in stirred-reactor combustion, 194-195

Hydrogen sulfide decomposition of, 204-205

Index

Hydrogen sulfide (cont.) from devolatilization, 203, 204 gasification, 209 oxidation of, 204-205 from pyrite, 197 reaction with tars, 204

Ignition, 219 Input parameters, 242, 243 In-scattering, 97, 99 Integral scale of turbulence, 62 Intensity

average, 86 definition, 85 homogeneous, 85 integrated, 86 isotropic, 85, 89

Interception, 87; see also Attenuation Intermittency, fuel-air, 7 3 Iron reactions

C02 formation, 200 from pyrite: see Pyrite

Isotropic scattering, 97 Isotropic turbulence, 249 Isotropy, 60,62

Kirchhoff's law, 87 Knudsen number, 111

Lagrange multipliers method of undetermined, 5 nondirnensional, 6

Lagrangian formulation, 15-18 Lagrangian reference frame, 266 Lambert surface, 102; see also Diffuse

surface Laminar flames: see Propagating flames Laminar flow, 57 Lennard-Jones potential, 284

Macromixing, 69 Macroscale of turbulence, 57, 249 Macroscopic equations of change, 253 Mass flux of gaseous species, 19-20,40 Mass transfer

diffusion, 245-247 gas particles, 226

Mean beam length, 101

Mean free path, 46, 60 Mean speed, 4 7 Methane-air flames: see Propagating

flames Methyl cyanide, 192 Micromixing, 69 Microscale of turbulence, 57, 249 Mie coefficient, 93 Mie scattering, 89 Mie theory, 89, 99, 100 Mineral matter

composition of CaC03 , 196 CaS04, 196 clay, 196 elemental, 196 ferric and ferrous sulfates, 196 gypsum, 196 kaolinite, 196 pyrite, 195-196 quartz, 196 total inorganic sulfur, 202

concentration of, 195 size of, 195

Mixing length, 60 Models

multidimensional, 263 one-dimensional, 235 pulverized-coal, 217 pyrolysis, 140 quasi-global, 17 8-179 reaction, 172 theoretical, 16 9

Modified scattering efficiency, 100 Modules, program, 264-266 Moisture

density, 228 enthalpy, 229-230 gas-particle, 226 vaporization, 221, 224

Momentum equation, differential discrete-phase, 26-28 gas-phase

in gas-particle mixture, 25-26 mixture of gaseous species, 23-25

overall, 28

329

Momentum equation, macroscopic gas-phase, in gas-particle mixture, 41-42 overall, 43 particulate phase, 42-43

Monte Carlo method, 84, 97 Multiple scattering, 89

330

Newton-Raphson correction equations, 6-9,76

Newton-Raphson functionals, 6 NHand NH2

in coal, 188 gas-phase kinetics, role in fuel-NO,

192-194 Nitric oxide

exhaust levels, 186 formation from atmospheric N2

flame radicals, effect of, 187 general behavior, 186 from nitrous oxide mechanism,

186-187 prompt NO, 187-188 sulfur, effect of, 206 thermal NO, 187 turbulent fluctuations, effect of, 187 from Zeldovich mechanism, 186-187

formation from coal-nitrogen (fuel-NO) gas-phase kinetics, 192-194 in stirred-reactor combustion,

194-195 Nitrogen

bimolecular from coal-nitrogen, 190, 194

gasification, 207 Nitrogen dioxide, 187-188 Nitrogen-in-coal

concentration of, 188 oxidation of

air mixing, effect of, 191 of char-nitrogen, 190-191 of volatile nitrogen, 190-191

structure of, 188 volatilization of

in industrial coking, 188 by rapid heating, 189-190 by slow heating, 189 by very rapid heating, 191

Nitrogen tars, 190 Nitrous oxide

in atmosphere, 186 from atmospheric N2 , 187 exhaust levels of, 186 from fuel-nitrogen, 194

NOx: see Nitric oxide, Nitrogen dioxide Nusselt number, 116, 282;see also

Particle-gas heat transfer

OCN, 192

1-DICOG model assumptions, 253 auxiliary equations, 253-256 comparisons, with experiment, 257 differential equations, 254 predictions, 254-258 solution technique, 254

One-dimensional models, 235-262 Optical properties of components, 99 Overall absorption coefficient, 101 Oxidation

of acetylene, 176-177 of carbon monoxide, 178 of ethane, 176, 178

Index

of ethylene, 176, 178 high-molecular-weight hydrocarbons,

177 hydrocarbons, 169, 176 of hydrogen, 170 of methane, 170-177 of propane, 177

Particle absorption coefficients, 94 ash, 92, 97, 99,100, 103;seealso Ash char, 92, 97, 99, 100, 103; see also Char diameter, 217, 218, 220 efficiency factor, 93 gray, 96,100 large, 97, 98 mass, 228 nonspherical, 92 phase function, 100 products enthalpy, 229 properties

density, 228 enthalpy, 229 heat capacity, 228

refractive index, 92 scattering coefficient, 94 shape, 92 size, 95, 96, 99 size parameter, 91 soot, 91, 95 swelling, 218, 220

Particle diffusion, 107-110 Particle drag

Basset force, 114 buoyancy, 113 steady-state aerodynamic drag

Mach number effects, 110

Index

Particle drag (cont.)

mass transfer effects, 111 particle asphericity effects, 112 rarefaction effects, 111 Reynolds number effects, 111

virtual mass, 113 Particle-gas heat transfer

convective Mach number effect, 116 mass transfer effect, 116 Reynolds number effect, 116 rotation effects, 117

radiative heat transfer, 117 Particle-gas mass transfer, 11 7 Particle lag, 239, 253 Particle lift forces

Magnus effect, 114 Saffman lift, 115

Particle scattering, 89-94 Particle sphericity, 113 Perfectly stirred reactor, 70, 289 Phase function

for char and ash particles, 100 definition, 95 in radiative transport equation, 95

Pollutants combustion, 184-208 gasification

inorganics, 208 nitrogen, 207 particulate, 208 sulfur, 209 trace elements, 208

mechanisms, 170 Polydispersed: see Distribution of

particles Polynuclear aromatics, 184 Prandtl hypothesis, 60 Prandtl-Kolmogorov relation, 61, 272 Prandtl number, 116, 289 Prandtl-Schmidt numbers, 62 Predictor-corrector, 254 Premixed flames: see Propagating flames Pressure

partial, 29 stress, 24

Probability density function, 70, 275, 290 Production of turbulent kinetic energy, 61 Propagating flames

laminar, 235-250 mechanism, 248-251

Propagating flames (cont.) model

assumptions, 237 comparisons, with experiment,

242-245 equations, 238-240 solution, 240-241

turbulent, 248-251 Propagation mechanism, 248-251 Propagation velocity

331

laminar, 236, 241, 244-245 turbulent, 250

Proximate analysis, 126 -128 Pyridine

in coal, 188 combustion of, 194-195 from devolatilization, 192 pyrolysis of, 192

Pyrite: see also Mineral matter oxidation of, 197 reaction with CaC03 , 197-198 reductive decomposition of, 197, 203 thermal decomposition of, 197

Pyrolysis: see also Devolatilization data, 135-140 experiments, 134 fast, 133 kinetics, 143 methods, 134 models

kinetic, 142 series, 143 single-step, 14 2 two-step, 142-143

observations, summary of, 144 products, 139-140 rate of, 135 two-step, 135

Pyrolysis products: see Pyrolyzate composition

Pyrolytic graphite, 185 Pyrolyzate composition, 169-171, 176 Pyrolyzates: see Volatiles Pyrrole

in coal, 188 from devolatilization, 192 pyrolysis of, 192

Quasi-laminar flow, 63 Quinoline, 192

332

Radiation basis equation, 84, 95 definition, 85, 95 effects of, 245 energy density, 86 flux,86 heat transfer, 93-106 intensity, 85 modeling,239,242,244,252

Radiation coefficients vs. size parameter for char, 94

Rate global reaction, 178-179 kinetic data, 170, 172-175, 177-178 limiting step, 178

Rate-resolution reactor combustor, 256-257, 267 gasifier, 256, 257-258, 267

Rayleigh scattering, 90 Reaction mechanisms: see also Oxidation

global, 169, 177 overall, 180

Reaction rate gaseous species, 20 particles, 22, 40

Reflectivity, 1 02 Refractive index, 92, 99, 100 Reynolds decomposition, 58 Reynolds equations, 59 Reynolds number

particle, 279 relative, 111, 114, 116, 117 turbulent, 62, 249

Reynolds stress, 59 Reynolds transport theorem, 15-18

sand s2, 205 "S" coordinates: see Streamline Scattering

coefficient, 87, 88, 100 definition, 89, 90 efficiency, 103

Schmidt number, 108, 282 Segregation, scale of, 6 9 Shear stresses, 57, 59-61 Sherwood number, 11 7; see also Particle-

gas mass transfer SiO, 200 Six-flux radiation model, 286 so, 204,205

Sodium: see Alkali metals Soot

formation in flames, 184-185 oxidation of, 185 properties of, 184

Index

SOx: see Sulfur dioxide, Sulfur trioxide Specific heat capacity

ideal gas, 10, 227 mixtures, 227

Stockmayer potential, 285 Streamline,238-240 Stress

"apparent," due to diffusion, 24, 25, 27, 28

on particle due to local strain rate in fluid, 26

on species, i, 29 Sulfur dioxide

exhaust level, 204 formation

gas-phase kinetics, 204-206 oxidation of pyrite: see Pyrite reaction with alkalis, 198, 204 reaction with NOx, 206

Sulfur gasification, 209 Sulfur, gas-phase reactions of, 204-206 Sulfur-in-coal

inorganic CaS04 , 196 ferric and ferrous sulfates, 196 gypsum, 196 pyrite, 195-196 total concentration, 202

organic disulfides, 202 mercaptans, 202 sulfides, 202 thiophene structures, 202 total concentration, 202

volatilization of in industrial coking, 202-203 inorganic decomposition, 202-203 organic, 202-203 pyrite decomposition: see Pyrite residual char-sulfur, 203

Sulfur trioxide behavior, 205 concentration, 206 gas-phase kinetics, 206

Suppressant, 248 Surface tension, 34, 35,44 Swelling: see Particle, swelling

Index

Thermal conductivity, gas basis, 46 equations, 49-50 mixtures, 50, 227 single species, 50, 227

Thermal radiation, 85 wavelength, 99

Trace elements ash, 129-130 combustion, 196 gasification, 208

Trajectories, particle, 279 Transmissivity, 102 Transport coefficients, 45 Transport properties, 283 Tridiagonal matrix algorithm, 269, 287 Turbulence coefficients, 62-63 Turbulent flames, 248-251 Turbulent intensity, 249 Turbulent kinetic energy, 61, 272 Turbulent macroscale, 57 Turbulent microscale, 57 Turbulent Reynolds number, 62 Two-phase flow, 278

Ultimate analysis, 123, 127-128 Unburned hydrocarbons

as pollutants, 183 reaction with NO, 194

Underrelaxation parameters, 12

Vaporization: see Moisture, vaporization

Velocity diffusion, 19, 24 mass-averaged

gas, 19 particles, 22, 27

mixture, 23 mole-averaged, 19

Vinyl cyanide, 192 Viscosity

basis, 46 eddy,59-62 equations, 47-48, 227 mixtures, 48-49, 227 molecular, 57 turbulent effective, 59-62

Volatile nitrogen, 190-191 Volatiles

combustion, 169, 179 composition,137, 169-171, 176 defmition, 133 product enthalpy, 230

Work rate due to aerodynamic forces, 34 due to body forces, 29 due to particle dilatation, 34 due to surface forces, 29 flow, 34,38

333

Zone method, 84, 97, 98; see also Hottel's zone method