984:(diffusive transport will be comparable to reaction in this zone). In this thin reaction sheet, both fuel and oxygen are consumed and nothing leaks to the other side of the sheet. Due to the instantaneous consumption of fuel and oxidizer, the normal gradients of scalars exhibit discontinuities at the reaction sheet.
709:
345:
36:. One important conclusion of infinitely fast chemistry is the non-co-existence of fuel and oxidizer simultaneously except in a thin reaction sheet. The inner structure of the reaction sheet is described by
184:
824:
565:
458:
982:
523:
413:
782:
593:
878:
i.e., fuel and oxidizer cannot coexist, since far away from the reaction sheet, only one of the reactant is available (non premixed). On the fuel side of the reaction sheet,
736:
873:
942:
909:
585:
485:
375:
242:
104:
73:
48:
In a typical non-premixed combustion (fuel and oxidizer are separated initially), mixing of fuel and oxidizer takes place based on the mechanical time scale
254:
1003:
1108:
1073:
116:
993:
791:
532:
785:
567:
guarantees that chemical reaction dominates over the other terms. A typical conservation equation for the scalar
418:
998:
37:
33:
75:
dictated by the convection/diffusion (the relative importance between convection and diffusion depends on the
1139:
947:
490:
380:
741:
210:
194:
704:{\displaystyle {\mathcal {L}}(\psi )=\mathrm {Da} _{b}Y_{F}Y_{O}e^{-{\frac {E}{RT}}+{\frac {E}{RT_{b}}}}}
245:
29:
944:. Fuel and oxygen can coexist (with very small concentrations) only in a thin reaction sheet, where
717:
1134:
832:
460:. On the other hand, the shortest chemical time is found at the flame (with burnt gas temperature
1104:
1069:
202:
914:
881:
1096:
1031:
570:
463:
353:
220:
82:
51:
1089:
Liñán, A. (2001). "Diffusion-controlled combustion". In Aref, H.; Phillips, J. W. (eds.).
526:
76:
1129:
1123:
107:
350:
Due to the large activation energy, the Damköhler number at unburnt gas temperature
340:{\displaystyle \mathrm {Da} ={\frac {t_{m}}{t_{c}}}=t_{m}Be^{-{\frac {E}{RT}}}.}
32:), named after S.P. Burke and T.E.W. Schumann, due to their pioneering work on
17:
1100:
28:, is the limit of infinitely fast chemistry (or in other words, infinite
1035:
79:) terms. Similarly, chemical reaction takes certain amount of time
106:
to consume reactants. For one-step irreversible chemistry with
1022:
Burke, S. P.; Schumann, T. E. W. (1928). "Diffusion flames".
723:
599:
587:(species concentration or energy) takes the following form,
179:{\displaystyle t_{c}=\left(Be^{\frac {E}{RT}}\right)^{-1}}
1090:
950:
917:
884:
835:
794:
788:
of fuel and oxidizer, respectively. Taking the limit
744:
720:
596:
573:
535:
493:
466:
421:
383:
356:
257:
223:
119:
85:
54:
819:{\displaystyle \mathrm {Da} _{b}\rightarrow \infty }
560:{\displaystyle \mathrm {Da} _{b}\rightarrow \infty }
244:appropriate for particular flow configuration. The
976:
936:
903:
867:
818:
776:
730:
703:
579:
559:
517:
479:
452:
407:
369:
339:
236:
178:
98:
67:
217:is the temperature. Similarly, one can define
8:
453:{\displaystyle {\frac {E}{RT_{u}}}\sim 100}
1095:. Dordrecht: Springer. pp. 487–502.
951:
949:
922:
916:
889:
883:
850:
840:
834:
804:
796:
793:
768:
749:
743:
738:is the convective-diffusive operator and
722:
721:
719:
690:
677:
659:
655:
645:
635:
625:
617:
598:
597:
595:
572:
545:
537:
534:
503:
495:
492:
471:
465:
435:
422:
420:
393:
385:
382:
361:
355:
317:
313:
300:
285:
275:
269:
258:
256:
228:
222:
167:
146:
124:
118:
90:
84:
59:
53:
1014:
1024:Industrial & Engineering Chemistry
977:{\displaystyle \mathrm {Da} \sim O(1)}
518:{\displaystyle \mathrm {Da} _{b}\gg 1}
408:{\displaystyle \mathrm {Da} _{u}\ll 1}
1068:. New York: Oxford University Press.
7:
826:in the above equation, we find that
777:{\displaystyle Y_{F}\ \&\ Y_{O}}
1064:Linan, A.; Williams, F. A. (1993).
955:
952:
813:
800:
797:
758:
621:
618:
554:
541:
538:
499:
496:
389:
386:
262:
259:
14:
1066:Fundamental Aspects of Combustion
110:, this chemical time is given by
1092:Mechanics for a New Millennium
1004:Liñán's diffusion flame theory
971:
965:
810:
731:{\displaystyle {\mathcal {L}}}
610:
604:
551:
1:
994:Activation energy asymptotics
868:{\displaystyle Y_{F}Y_{O}=0,}
26:large Damköhler number limit
1156:
911:and on the oxidizer side,
1101:10.1007/0-306-46956-1_31
1049:Williams, F. A. (2018).
937:{\displaystyle Y_{F}=0}
904:{\displaystyle Y_{O}=0}
978:
938:
905:
869:
820:
778:
732:
705:
581:
561:
519:
481:
454:
409:
371:
341:
238:
211:universal gas constant
195:pre-exponential factor
180:
100:
69:
979:
939:
906:
870:
821:
779:
733:
706:
582:
580:{\displaystyle \psi }
562:
520:
482:
480:{\displaystyle T_{b}}
455:
410:
372:
370:{\displaystyle T_{u}}
342:
239:
237:{\displaystyle t_{m}}
181:
101:
99:{\displaystyle t_{c}}
70:
68:{\displaystyle t_{m}}
948:
915:
882:
833:
792:
742:
718:
594:
571:
533:
491:
464:
419:
381:
354:
255:
221:
117:
83:
52:
34:Burke–Schumann flame
22:Burke–Schumann limit
1036:10.1021/ie50226a005
974:
934:
901:
865:
816:
774:
728:
701:
577:
557:
515:
477:
450:
405:
367:
337:
234:
176:
96:
65:
1110:978-0-7923-7156-4
1051:Combustion Theory
763:
757:
697:
672:
442:
330:
291:
203:activation energy
159:
44:Limit description
1147:
1115:
1114:
1086:
1080:
1079:
1061:
1055:
1054:
1046:
1040:
1039:
1030:(10): 998–1004.
1019:
999:Liñán's equation
983:
981:
980:
975:
958:
943:
941:
940:
935:
927:
926:
910:
908:
907:
902:
894:
893:
874:
872:
871:
866:
855:
854:
845:
844:
825:
823:
822:
817:
809:
808:
803:
783:
781:
780:
775:
773:
772:
761:
755:
754:
753:
737:
735:
734:
729:
727:
726:
710:
708:
707:
702:
700:
699:
698:
696:
695:
694:
678:
673:
671:
660:
650:
649:
640:
639:
630:
629:
624:
603:
602:
586:
584:
583:
578:
566:
564:
563:
558:
550:
549:
544:
525:. Regardless of
524:
522:
521:
516:
508:
507:
502:
486:
484:
483:
478:
476:
475:
459:
457:
456:
451:
443:
441:
440:
439:
423:
414:
412:
411:
406:
398:
397:
392:
376:
374:
373:
368:
366:
365:
346:
344:
343:
338:
333:
332:
331:
329:
318:
305:
304:
292:
290:
289:
280:
279:
270:
265:
246:Damköhler number
243:
241:
240:
235:
233:
232:
216:
208:
200:
192:
185:
183:
182:
177:
175:
174:
166:
162:
161:
160:
158:
147:
129:
128:
105:
103:
102:
97:
95:
94:
74:
72:
71:
66:
64:
63:
38:Liñán's equation
30:Damköhler number
1155:
1154:
1150:
1149:
1148:
1146:
1145:
1144:
1120:
1119:
1118:
1111:
1088:
1087:
1083:
1076:
1063:
1062:
1058:
1048:
1047:
1043:
1021:
1020:
1016:
1012:
990:
946:
945:
918:
913:
912:
885:
880:
879:
846:
836:
831:
830:
795:
790:
789:
764:
745:
740:
739:
716:
715:
686:
682:
664:
651:
641:
631:
616:
592:
591:
569:
568:
536:
531:
530:
527:Reynolds number
494:
489:
488:
467:
462:
461:
431:
427:
417:
416:
384:
379:
378:
357:
352:
351:
322:
309:
296:
281:
271:
253:
252:
224:
219:
218:
214:
206:
198:
190:
151:
142:
138:
134:
133:
120:
115:
114:
86:
81:
80:
77:Reynolds number
55:
50:
49:
46:
12:
11:
5:
1153:
1151:
1143:
1142:
1140:Fluid dynamics
1137:
1132:
1122:
1121:
1117:
1116:
1109:
1081:
1074:
1056:
1041:
1013:
1011:
1008:
1007:
1006:
1001:
996:
989:
986:
973:
970:
967:
964:
961:
957:
954:
933:
930:
925:
921:
900:
897:
892:
888:
876:
875:
864:
861:
858:
853:
849:
843:
839:
815:
812:
807:
802:
799:
786:mass fractions
771:
767:
760:
752:
748:
725:
712:
711:
693:
689:
685:
681:
676:
670:
667:
663:
658:
654:
648:
644:
638:
634:
628:
623:
620:
615:
612:
609:
606:
601:
576:
556:
553:
548:
543:
540:
514:
511:
506:
501:
498:
487:), leading to
474:
470:
449:
446:
438:
434:
430:
426:
404:
401:
396:
391:
388:
364:
360:
348:
347:
336:
328:
325:
321:
316:
312:
308:
303:
299:
295:
288:
284:
278:
274:
268:
264:
261:
231:
227:
187:
186:
173:
170:
165:
157:
154:
150:
145:
141:
137:
132:
127:
123:
108:Arrhenius rate
93:
89:
62:
58:
45:
42:
13:
10:
9:
6:
4:
3:
2:
1152:
1141:
1138:
1136:
1133:
1131:
1128:
1127:
1125:
1112:
1106:
1102:
1098:
1094:
1093:
1085:
1082:
1077:
1075:0-19-507626-5
1071:
1067:
1060:
1057:
1052:
1045:
1042:
1037:
1033:
1029:
1025:
1018:
1015:
1009:
1005:
1002:
1000:
997:
995:
992:
991:
987:
985:
968:
962:
959:
931:
928:
923:
919:
898:
895:
890:
886:
862:
859:
856:
851:
847:
841:
837:
829:
828:
827:
805:
787:
769:
765:
750:
746:
691:
687:
683:
679:
674:
668:
665:
661:
656:
652:
646:
642:
636:
632:
626:
613:
607:
590:
589:
588:
574:
546:
528:
512:
509:
504:
472:
468:
447:
444:
436:
432:
428:
424:
402:
399:
394:
362:
358:
334:
326:
323:
319:
314:
310:
306:
301:
297:
293:
286:
282:
276:
272:
266:
251:
250:
249:
247:
229:
225:
212:
204:
196:
171:
168:
163:
155:
152:
148:
143:
139:
135:
130:
125:
121:
113:
112:
111:
109:
91:
87:
78:
60:
56:
43:
41:
39:
35:
31:
27:
23:
19:
1091:
1084:
1065:
1059:
1053:. CRC Press.
1050:
1044:
1027:
1023:
1017:
877:
713:
529:, the limit
349:
188:
47:
25:
21:
15:
1135:Combustion
1124:Categories
1010:References
415:, because
18:combustion
960:∼
814:∞
811:→
759:&
657:−
608:ψ
575:ψ
555:∞
552:→
510:≫
445:∼
400:≪
315:−
169:−
988:See also
784:are the
248:is then
209:is the
201:is the
193:is the
1107:
1072:
762:
756:
714:where
189:where
24:, or
1130:Fire
1105:ISBN
1070:ISBN
213:and
1097:doi
1032:doi
448:100
377:is
16:In
1126::
1103:.
1028:20
1026:.
205:,
197:,
40:.
20:,
1113:.
1099::
1078:.
1038:.
1034::
972:)
969:1
966:(
963:O
956:a
953:D
932:0
929:=
924:F
920:Y
899:0
896:=
891:O
887:Y
863:,
860:0
857:=
852:O
848:Y
842:F
838:Y
806:b
801:a
798:D
770:O
766:Y
751:F
747:Y
724:L
692:b
688:T
684:R
680:E
675:+
669:T
666:R
662:E
653:e
647:O
643:Y
637:F
633:Y
627:b
622:a
619:D
614:=
611:)
605:(
600:L
547:b
542:a
539:D
513:1
505:b
500:a
497:D
473:b
469:T
437:u
433:T
429:R
425:E
403:1
395:u
390:a
387:D
363:u
359:T
335:.
327:T
324:R
320:E
311:e
307:B
302:m
298:t
294:=
287:c
283:t
277:m
273:t
267:=
263:a
260:D
230:m
226:t
215:T
207:R
199:E
191:B
172:1
164:)
156:T
153:R
149:E
144:e
140:B
136:(
131:=
126:c
122:t
92:c
88:t
61:m
57:t
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.