1512:
1498:
As can be seen, as the Damköhler number increases, the other term must decrease. The ensuing polynomial can be solved and the conversion for the rule of thumb Damköhler numbers found. Alternatively, one can graph the expressions and see where they intersect with the line given by the inverse
136:
189:
1090:
is defined to be the ratio of the reactor volume to volumetric flow rate. It is the time required for a slug of fluid to pass through the reactor. For a decomposition reaction, the rate of reaction is proportional to some power of the concentration of
242:
693:
561:
467:
1614:
1493:
323:
1333:
598:
goes to 0, the residence time is much shorter than the reaction time, so that no chemical reaction has taken place during the brief period when the fluid particles occupy the reaction location. Similarly,
1013:
247:
Since the reaction rate determines the reaction timescale, the exact formula for the Damköhler number varies according to the rate law equation. For a general chemical reaction A → B following the
1211:
1411:
931:
1081:
94:
147:
1607:
761:
823:
594:
goes to infinity, the residence time greatly exceeds the reaction time, such that nearly all chemical reactions have taken place during the period of residency. Otherwise if Da
603:
goes to 0 implies that the energy of the chemical reaction is negligible compared to the energy of the flow. The limit of the Damköhler number going to infinity is called the
472:
It compares the process energy of a thermochemical reaction (such as the energy involved in a nonequilibrium gas process) with a related enthalpy difference (driving force).
200:
141:
In reacting systems that include interphase mass transport, the first Damköhler number can be written as the ratio of the chemical reaction rate to the mass transfer rate
1974:
1600:
853:
379:
1133:
1109:
640:
1969:
652:
1979:
68:) relates particles' characteristic residence time scale in a fluid region to the reaction timescale. The residence time scale can take the form of a
1623:
614:, when Da is less than 0.1 a conversion of less than 10% is achieved, and when Da is greater than 10 a conversion of more than 90% is expected.
481:
1964:
402:
1555:
1419:
261:
1219:
1580:
1087:
643:
81:
1769:
939:
382:
1140:
1341:
131:{\displaystyle \mathrm {Da_{\mathrm {I} }} ={\frac {\text{reaction rate}}{\text{convective mass transport rate}}}}
861:
604:
184:{\displaystyle \mathrm {Da} _{\mathrm {I} }={\frac {\text{reaction rate}}{\text{diffusive mass transfer rate}}}}
1846:
1829:
1021:
1984:
1824:
77:
701:
1724:
1112:
587:
339:
769:
1507:-axis the conversion. The rule-of-thumb Damköhler numbers have been placed as dashed horizontal lines.
237:{\displaystyle \mathrm {Da_{\mathrm {I} }} ={\frac {\text{flow timescale}}{\text{chemical timescale}}}}
1511:
1679:
73:
48:
32:
28:
1749:
44:
1719:
1659:
855:, which is the case for a liquid reactor or a gas phase reaction with no net generation of moles,
1959:
1933:
1115:
may be defined in terms of the limiting reactant, for the simple decomposition that is species
1888:
1853:
1634:
1576:
1551:
336:
252:
85:
36:
1938:
1814:
1809:
1794:
1759:
1669:
1543:
53:
1592:
831:
364:
1903:
1883:
1841:
1836:
1664:
194:
It is also defined as the ratio of the characteristic fluidic and chemical time scales:
1913:
1893:
1878:
1873:
1819:
1804:
1779:
1774:
1764:
1744:
1734:
1699:
1644:
1118:
1094:
625:
357:
1953:
1923:
1918:
1908:
1898:
1863:
1858:
1799:
1739:
1729:
1709:
1704:
1674:
1639:
611:
40:
1928:
1868:
1784:
1754:
1714:
1689:
1649:
688:{\displaystyle {\text{in}}-{\text{out}}+{\text{generation}}={\text{accumulation}}}
1789:
1694:
1654:
51:, who worked in chemical engineering, thermodynamics, and fluid dynamics. The
1547:
69:
1535:
1499:
Damköhler number to see the solution for conversion. In the plot below, the
556:{\displaystyle \mathrm {Da} _{\mathrm {II} }={\frac {kC_{0}^{n-1}}{k_{g}a}}}
248:
462:{\displaystyle \mathrm {Da} _{\mathrm {II} }={\frac {kQ}{c_{p}\Delta T}}}
255:, the Damköhler number for a convective flow system is defined as:
1488:{\displaystyle 0={\frac {(1-X)^{n}}{X}}-{\frac {1}{\rm {Da_{n}}}}}
318:{\displaystyle \mathrm {Da_{\mathrm {I} }} =kC_{0}^{\ n-1}\tau }
1596:
1328:{\displaystyle ((1-X)C_{A0}-C_{A0})=-kC_{A0}^{n}\tau (1-X)^{n}}
64:
In its most commonly used form, the first Damköhler number (Da
47:
rate occurring in a system. It is named after German chemist
1575:(4th ed.). Upper Saddle River, NJ: Pearson Education.
586:
The value of Da provides a quick estimate of the degree of
1008:{\displaystyle (C_{A}-C_{A0})=r_{A}{\frac {V}{v_{0}}}}
1422:
1344:
1222:
1143:
1121:
1097:
1024:
942:
864:
834:
772:
704:
655:
628:
484:
405:
367:
264:
203:
150:
97:
61:) is related to the Damköhler number by Da = 1/Ka.
1487:
1405:
1327:
1205:
1127:
1103:
1075:
1007:
925:
847:
817:
755:
687:
634:
555:
461:
392:On the other hand, the second Damköhler number (Da
373:
317:
236:
183:
130:
16:Dimensionless numbers used in chemical engineering
618:Derivation for decomposition of a single species
1206:{\displaystyle (C_{A}-C_{A0})=-kC_{A}^{n}\tau }
1503:-axis is the inverse Damköhler number and the
622:From the general mole balance on some species
1608:
1529:
1527:
1406:{\displaystyle X=kC_{A0}^{n-1}\tau (1-X)^{n}}
646:steady state and perfect mixing are assumed,
8:
1615:
1601:
1593:
926:{\displaystyle (C_{A}-C_{A0})v_{0}=r_{A}V}
1573:Elements of Chemical Reaction Engineering
1476:
1468:
1463:
1448:
1429:
1421:
1397:
1366:
1358:
1343:
1319:
1294:
1286:
1261:
1245:
1221:
1194:
1189:
1164:
1151:
1142:
1120:
1096:
1076:{\displaystyle (C_{A}-C_{A0})=r_{A}\tau }
1064:
1045:
1032:
1023:
997:
988:
982:
963:
950:
941:
914:
901:
885:
872:
863:
839:
833:
828:Assuming a constant volumetric flow rate
806:
790:
777:
771:
738:
725:
709:
703:
680:
672:
664:
656:
654:
627:
541:
523:
518:
508:
495:
494:
486:
483:
444:
429:
416:
415:
407:
404:
366:
297:
292:
274:
273:
265:
263:
224:
213:
212:
204:
202:
171:
161:
160:
152:
149:
118:
107:
106:
98:
96:
1975:Dimensionless numbers of fluid mechanics
1624:Dimensionless numbers in fluid mechanics
576:is the global mass transport coefficient
1523:
1111:. In addition, for a single reaction a
756:{\displaystyle F_{A0}-F_{A}+r_{A}V=0}
7:
76:through the reactor for continuous (
818:{\displaystyle F_{A}-F_{A0}=r_{A}V}
1970:Dimensionless numbers of chemistry
1477:
1473:
1469:
499:
496:
490:
487:
450:
420:
417:
411:
408:
275:
270:
266:
214:
209:
205:
162:
156:
153:
108:
103:
99:
14:
1980:Eponymous numbers in mathematics
1536:"Mechanics of Flow Similarities"
1510:
1445:
1432:
1394:
1381:
1316:
1303:
1270:
1238:
1226:
1223:
1173:
1144:
1054:
1025:
972:
943:
894:
865:
124:convective mass transport rate
1:
1965:Chemical reaction engineering
475:In terms of reaction rates:
396:) is defined in general as:
177:diffusive mass transfer rate
590:that can be achieved. If Da
2001:
1630:
1548:10.1007/978-3-030-42930-0
1534:Weiland, Claus (2020).
582:is the interfacial area
350:= initial concentration
1571:Fogler, Scott (2006).
1489:
1407:
1329:
1207:
1129:
1105:
1077:
1009:
927:
849:
819:
757:
689:
636:
557:
463:
375:
340:reaction rate constant
319:
238:
185:
132:
1490:
1408:
1330:
1208:
1130:
1106:
1078:
1010:
928:
850:
848:{\displaystyle v_{0}}
820:
758:
690:
637:
558:
464:
376:
374:{\displaystyle \tau }
320:
239:
186:
133:
29:dimensionless numbers
1420:
1342:
1220:
1141:
1119:
1095:
1022:
940:
862:
832:
770:
702:
653:
626:
605:Burke–Schumann limit
482:
403:
365:
262:
201:
148:
95:
88:chemical processes:
74:volumetric flow rate
72:time scale, such as
33:chemical engineering
1377:
1299:
1199:
534:
311:
45:transport phenomena
1770:Keulegan–Carpenter
1485:
1403:
1354:
1325:
1282:
1203:
1185:
1125:
1101:
1073:
1005:
923:
845:
815:
753:
685:
632:
553:
514:
459:
371:
315:
288:
234:
230:chemical timescale
181:
128:
1947:
1946:
1557:978-3-030-42929-4
1483:
1458:
1128:{\displaystyle A}
1104:{\displaystyle A}
1003:
683:
675:
667:
659:
635:{\displaystyle A}
551:
457:
300:
251:kinetics of n-th
232:
231:
228:
179:
178:
175:
126:
125:
122:
49:Gerhard Damköhler
37:chemical reaction
21:Damköhler numbers
1992:
1617:
1610:
1603:
1594:
1587:
1586:
1568:
1562:
1561:
1531:
1514:
1494:
1492:
1491:
1486:
1484:
1482:
1481:
1480:
1464:
1459:
1454:
1453:
1452:
1430:
1412:
1410:
1409:
1404:
1402:
1401:
1376:
1365:
1334:
1332:
1331:
1326:
1324:
1323:
1298:
1293:
1269:
1268:
1253:
1252:
1212:
1210:
1209:
1204:
1198:
1193:
1172:
1171:
1156:
1155:
1134:
1132:
1131:
1126:
1110:
1108:
1107:
1102:
1082:
1080:
1079:
1074:
1069:
1068:
1053:
1052:
1037:
1036:
1014:
1012:
1011:
1006:
1004:
1002:
1001:
989:
987:
986:
971:
970:
955:
954:
932:
930:
929:
924:
919:
918:
906:
905:
893:
892:
877:
876:
854:
852:
851:
846:
844:
843:
824:
822:
821:
816:
811:
810:
798:
797:
782:
781:
762:
760:
759:
754:
743:
742:
730:
729:
717:
716:
694:
692:
691:
686:
684:
681:
676:
673:
668:
665:
660:
657:
641:
639:
638:
633:
562:
560:
559:
554:
552:
550:
546:
545:
535:
533:
522:
509:
504:
503:
502:
493:
468:
466:
465:
460:
458:
456:
449:
448:
438:
430:
425:
424:
423:
414:
380:
378:
377:
372:
324:
322:
321:
316:
310:
298:
296:
281:
280:
279:
278:
243:
241:
240:
235:
233:
229:
226:
225:
220:
219:
218:
217:
190:
188:
187:
182:
180:
176:
173:
172:
167:
166:
165:
159:
137:
135:
134:
129:
127:
123:
120:
119:
114:
113:
112:
111:
54:Karlovitz number
2000:
1999:
1995:
1994:
1993:
1991:
1990:
1989:
1950:
1949:
1948:
1943:
1626:
1621:
1591:
1590:
1583:
1570:
1569:
1565:
1558:
1533:
1532:
1525:
1520:
1472:
1444:
1431:
1418:
1417:
1393:
1340:
1339:
1315:
1257:
1241:
1218:
1217:
1160:
1147:
1139:
1138:
1117:
1116:
1093:
1092:
1060:
1041:
1028:
1020:
1019:
993:
978:
959:
946:
938:
937:
910:
897:
881:
868:
860:
859:
835:
830:
829:
802:
786:
773:
768:
767:
734:
721:
705:
700:
699:
651:
650:
624:
623:
620:
602:
597:
593:
574:
537:
536:
510:
485:
480:
479:
440:
439:
431:
406:
401:
400:
395:
363:
362:
349:
269:
260:
259:
208:
199:
198:
151:
146:
145:
102:
93:
92:
67:
17:
12:
11:
5:
1998:
1996:
1988:
1987:
1985:Fluid dynamics
1982:
1977:
1972:
1967:
1962:
1952:
1951:
1945:
1944:
1942:
1941:
1936:
1931:
1926:
1921:
1916:
1911:
1906:
1901:
1896:
1891:
1886:
1881:
1876:
1871:
1866:
1861:
1856:
1851:
1850:
1849:
1839:
1834:
1833:
1832:
1827:
1817:
1812:
1807:
1802:
1797:
1792:
1787:
1782:
1777:
1772:
1767:
1762:
1757:
1752:
1747:
1742:
1737:
1732:
1727:
1722:
1717:
1712:
1707:
1702:
1697:
1692:
1687:
1682:
1677:
1672:
1667:
1662:
1657:
1652:
1647:
1642:
1637:
1631:
1628:
1627:
1622:
1620:
1619:
1612:
1605:
1597:
1589:
1588:
1581:
1563:
1556:
1522:
1521:
1519:
1516:
1496:
1495:
1479:
1475:
1471:
1467:
1462:
1457:
1451:
1447:
1443:
1440:
1437:
1434:
1428:
1425:
1414:
1413:
1400:
1396:
1392:
1389:
1386:
1383:
1380:
1375:
1372:
1369:
1364:
1361:
1357:
1353:
1350:
1347:
1336:
1335:
1322:
1318:
1314:
1311:
1308:
1305:
1302:
1297:
1292:
1289:
1285:
1281:
1278:
1275:
1272:
1267:
1264:
1260:
1256:
1251:
1248:
1244:
1240:
1237:
1234:
1231:
1228:
1225:
1214:
1213:
1202:
1197:
1192:
1188:
1184:
1181:
1178:
1175:
1170:
1167:
1163:
1159:
1154:
1150:
1146:
1124:
1100:
1084:
1083:
1072:
1067:
1063:
1059:
1056:
1051:
1048:
1044:
1040:
1035:
1031:
1027:
1016:
1015:
1000:
996:
992:
985:
981:
977:
974:
969:
966:
962:
958:
953:
949:
945:
934:
933:
922:
917:
913:
909:
904:
900:
896:
891:
888:
884:
880:
875:
871:
867:
842:
838:
826:
825:
814:
809:
805:
801:
796:
793:
789:
785:
780:
776:
764:
763:
752:
749:
746:
741:
737:
733:
728:
724:
720:
715:
712:
708:
696:
695:
679:
671:
663:
642:, where for a
631:
619:
616:
600:
595:
591:
584:
583:
577:
572:
564:
563:
549:
544:
540:
532:
529:
526:
521:
517:
513:
507:
501:
498:
492:
489:
470:
469:
455:
452:
447:
443:
437:
434:
428:
422:
419:
413:
410:
393:
390:
389:
383:residence time
370:
360:
358:reaction order
351:
347:
342:
326:
325:
314:
309:
306:
303:
295:
291:
287:
284:
277:
272:
268:
245:
244:
227:flow timescale
223:
216:
211:
207:
192:
191:
170:
164:
158:
155:
139:
138:
117:
110:
105:
101:
65:
35:to relate the
15:
13:
10:
9:
6:
4:
3:
2:
1997:
1986:
1983:
1981:
1978:
1976:
1973:
1971:
1968:
1966:
1963:
1961:
1958:
1957:
1955:
1940:
1937:
1935:
1932:
1930:
1927:
1925:
1922:
1920:
1917:
1915:
1912:
1910:
1907:
1905:
1902:
1900:
1897:
1895:
1892:
1890:
1887:
1885:
1882:
1880:
1877:
1875:
1872:
1870:
1867:
1865:
1862:
1860:
1857:
1855:
1852:
1848:
1845:
1844:
1843:
1840:
1838:
1835:
1831:
1828:
1826:
1823:
1822:
1821:
1818:
1816:
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
1758:
1756:
1753:
1751:
1748:
1746:
1743:
1741:
1738:
1736:
1733:
1731:
1728:
1726:
1723:
1721:
1718:
1716:
1713:
1711:
1708:
1706:
1703:
1701:
1698:
1696:
1693:
1691:
1688:
1686:
1683:
1681:
1680:Chandrasekhar
1678:
1676:
1673:
1671:
1668:
1666:
1663:
1661:
1658:
1656:
1653:
1651:
1648:
1646:
1643:
1641:
1638:
1636:
1633:
1632:
1629:
1625:
1618:
1613:
1611:
1606:
1604:
1599:
1598:
1595:
1584:
1582:0-13-047394-4
1578:
1574:
1567:
1564:
1559:
1553:
1549:
1545:
1541:
1537:
1530:
1528:
1524:
1517:
1515:
1513:
1508:
1506:
1502:
1465:
1460:
1455:
1449:
1441:
1438:
1435:
1426:
1423:
1416:
1415:
1398:
1390:
1387:
1384:
1378:
1373:
1370:
1367:
1362:
1359:
1355:
1351:
1348:
1345:
1338:
1337:
1320:
1312:
1309:
1306:
1300:
1295:
1290:
1287:
1283:
1279:
1276:
1273:
1265:
1262:
1258:
1254:
1249:
1246:
1242:
1235:
1232:
1229:
1216:
1215:
1200:
1195:
1190:
1186:
1182:
1179:
1176:
1168:
1165:
1161:
1157:
1152:
1148:
1137:
1136:
1135:
1122:
1114:
1098:
1089:
1070:
1065:
1061:
1057:
1049:
1046:
1042:
1038:
1033:
1029:
1018:
1017:
998:
994:
990:
983:
979:
975:
967:
964:
960:
956:
951:
947:
936:
935:
920:
915:
911:
907:
902:
898:
889:
886:
882:
878:
873:
869:
858:
857:
856:
840:
836:
812:
807:
803:
799:
794:
791:
787:
783:
778:
774:
766:
765:
750:
747:
744:
739:
735:
731:
726:
722:
718:
713:
710:
706:
698:
697:
677:
669:
661:
649:
648:
647:
645:
629:
617:
615:
613:
612:rule of thumb
608:
606:
589:
581:
578:
575:
569:
568:
567:
547:
542:
538:
530:
527:
524:
519:
515:
511:
505:
478:
477:
476:
473:
453:
445:
441:
435:
432:
426:
399:
398:
397:
388:
384:
368:
361:
359:
355:
352:
346:
343:
341:
338:
334:
331:
330:
329:
312:
307:
304:
301:
293:
289:
285:
282:
258:
257:
256:
254:
250:
221:
197:
196:
195:
174:reaction rate
168:
144:
143:
142:
121:reaction rate
115:
91:
90:
89:
87:
83:
79:
75:
71:
62:
60:
56:
55:
50:
46:
42:
41:reaction rate
38:
34:
30:
26:
22:
1684:
1572:
1566:
1540:SpringerLink
1539:
1509:
1504:
1500:
1497:
1085:
827:
682:accumulation
621:
609:
585:
579:
570:
565:
474:
471:
391:
386:
353:
344:
332:
327:
246:
193:
140:
82:stirred tank
63:
58:
52:
24:
20:
18:
1934:Weissenberg
39:timescale (
1954:Categories
1854:Richardson
1635:Archimedes
1518:References
1113:conversion
1088:space-time
1086:where the
674:generation
588:conversion
387:space-time
70:convection
1960:Catalysis
1939:Womersley
1830:turbulent
1810:Ohnesorge
1795:Marangoni
1760:Iribarren
1685:Damköhler
1670:Capillary
1461:−
1439:−
1388:−
1379:τ
1371:−
1310:−
1301:τ
1277:−
1255:−
1233:−
1201:τ
1180:−
1158:−
1071:τ
1039:−
957:−
879:−
784:−
719:−
662:−
528:−
451:Δ
369:τ
313:τ
305:−
249:Power law
86:semibatch
78:plug flow
43:) to the
1914:Suratman
1904:Strouhal
1884:Sherwood
1847:magnetic
1842:Reynolds
1837:Rayleigh
1825:magnetic
1665:Brinkman
337:kinetics
31:used in
1894:Stanton
1889:Shields
1879:Scruton
1874:Schmidt
1820:Prandtl
1805:Nusselt
1780:Laplace
1775:Knudsen
1765:Kapitza
1750:Görtler
1745:Grashof
1735:Galilei
1700:Deborah
1645:Bagnold
381:= mean
328:where:
1924:Ursell
1919:Taylor
1909:Stuart
1899:Stokes
1864:Rossby
1859:Roshko
1815:Péclet
1800:Morton
1740:Graetz
1730:Froude
1720:Eötvös
1710:Eckert
1705:Dukhin
1675:Cauchy
1640:Atwood
1579:
1554:
566:where
299:
27:) are
1929:Weber
1869:Rouse
1785:Lewis
1755:Hagen
1725:Euler
1715:Ekman
1690:Darcy
1650:Bejan
610:As a
253:order
84:) or
1790:Mach
1695:Dean
1660:Bond
1655:Biot
1577:ISBN
1552:ISBN
644:CSTR
19:The
1544:doi
666:out
607:.
385:or
80:or
1956::
1550:.
1542:.
1538:.
1526:^
658:in
601:II
599:Da
394:II
356:=
335:=
59:Ka
25:Da
1616:e
1609:t
1602:v
1585:.
1560:.
1546::
1505:x
1501:y
1478:n
1474:a
1470:D
1466:1
1456:X
1450:n
1446:)
1442:X
1436:1
1433:(
1427:=
1424:0
1399:n
1395:)
1391:X
1385:1
1382:(
1374:1
1368:n
1363:0
1360:A
1356:C
1352:k
1349:=
1346:X
1321:n
1317:)
1313:X
1307:1
1304:(
1296:n
1291:0
1288:A
1284:C
1280:k
1274:=
1271:)
1266:0
1263:A
1259:C
1250:0
1247:A
1243:C
1239:)
1236:X
1230:1
1227:(
1224:(
1196:n
1191:A
1187:C
1183:k
1177:=
1174:)
1169:0
1166:A
1162:C
1153:A
1149:C
1145:(
1123:A
1099:A
1066:A
1062:r
1058:=
1055:)
1050:0
1047:A
1043:C
1034:A
1030:C
1026:(
999:0
995:v
991:V
984:A
980:r
976:=
973:)
968:0
965:A
961:C
952:A
948:C
944:(
921:V
916:A
912:r
908:=
903:0
899:v
895:)
890:0
887:A
883:C
874:A
870:C
866:(
841:0
837:v
813:V
808:A
804:r
800:=
795:0
792:A
788:F
779:A
775:F
751:0
748:=
745:V
740:A
736:r
732:+
727:A
723:F
714:0
711:A
707:F
678:=
670:+
630:A
596:I
592:I
580:a
573:g
571:k
548:a
543:g
539:k
531:1
525:n
520:0
516:C
512:k
506:=
500:I
497:I
491:a
488:D
454:T
446:p
442:c
436:Q
433:k
427:=
421:I
418:I
412:a
409:D
354:n
348:0
345:C
333:k
308:1
302:n
294:0
290:C
286:k
283:=
276:I
271:a
267:D
222:=
215:I
210:a
206:D
169:=
163:I
157:a
154:D
116:=
109:I
104:a
100:D
66:I
57:(
23:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
