25:
1561:
279:
1388:
821:
703:
For air with a pressure of 1 bar, the
Prandtl numbers in the temperature range between −100 °C and +500 °C can be calculated using the formula given below. The temperature is to be used in the unit degree Celsius. The deviations are a maximum of 0.1% from the literature values.
1239:
1344:
846:
The
Prandtl numbers for water (1 bar) can be determined in the temperature range between 0 °C and 90 °C using the formula given below. The temperature is to be used in the unit degree Celsius. The deviations are a maximum of 1% from the literature values.
1075:
155:
919:
1556:{\displaystyle \mathrm {Nu} _{x}={\frac {0.3387\mathrm {Re} _{x}^{\frac {1}{2}}\mathrm {Pr} ^{\frac {1}{3}}}{\left(1+\left({\frac {0.0468}{\mathrm {Pr} }}\right)^{\frac {2}{3}}\right)^{\frac {1}{4}}}},\quad \mathrm {Re} \mathrm {Pr} >100.}
548:
are subscripted with a scale variable, the
Prandtl number contains no such length scale and is dependent only on the fluid and the fluid state. The Prandtl number is often found in property tables alongside other properties such as
709:
589:
is approximately constant. Therefore, it can be used to determine the thermal conductivity of gases at high temperatures, where it is difficult to measure experimentally due to the formation of convection currents.
1764:
1140:
982:
is small, it means that the heat diffuses quickly compared to the velocity (momentum). This means that for liquid metals the thermal boundary layer is much thicker than the velocity boundary layer.
1245:
991:
1757:
423:
944:, so thermal diffusivity is dominant. However, engine oil with its high viscosity and low heat conductivity, has a higher momentum diffusivity as compared to thermal diffusivity.
274:{\displaystyle \mathrm {Pr} ={\frac {\nu }{\alpha }}={\frac {\mbox{momentum diffusivity}}{\mbox{thermal diffusivity}}}={\frac {\mu /\rho }{k/(c_{p}\rho )}}={\frac {c_{p}\mu }{k}}}
343:
1372:
852:
1105:
2114:
1750:
841:
2119:
1125:
369:
530:
504:
445:
305:
471:
964:
816:{\displaystyle \mathrm {Pr} _{\text{air}}={\frac {10^{9}}{1.1\cdot \vartheta ^{3}-1200\cdot \vartheta ^{2}+322000\cdot \vartheta +1.393\cdot 10^{9}}}}
1773:
1623:
1733:
985:
In laminar boundary layers, the ratio of the thermal to momentum boundary layer thickness over a flat plate is well approximated by
108:
1234:{\displaystyle \mathrm {Nu} _{x}=0.339\mathrm {Re} _{x}^{\frac {1}{2}}\mathrm {Pr} ^{\frac {1}{3}},\quad \mathrm {Pr} \to \infty ,}
42:
1676:
1339:{\displaystyle \mathrm {Nu} _{x}=0.565\mathrm {Re} _{x}^{\frac {1}{2}}\mathrm {Pr} ^{\frac {1}{2}},\quad \mathrm {Pr} \to 0,}
89:
46:
1919:
61:
936:, the momentum diffusivity dominates the behavior. For example, the listed value for liquid mercury indicates that the
68:
1070:{\displaystyle {\frac {\delta _{t}}{\delta }}=\mathrm {Pr} ^{-{\frac {1}{3}}},\quad 0.6\leq \mathrm {Pr} \leq 50,}
35:
75:
1996:
1979:
1572:
968:
2124:
1974:
1577:
640:
378:
57:
1874:
1643:
974:
In heat transfer problems, the
Prandtl number controls the relative thickness of the momentum and thermal
1834:
1829:
914:{\displaystyle \mathrm {Pr} _{\text{water}}={\frac {50000}{\vartheta ^{2}+155\cdot \vartheta +3700}}}
554:
474:
314:
134:
2109:
1899:
1352:
372:
146:
1869:
1809:
2083:
1379:
1083:
955:
dissipate through the fluid at about the same rate. Heat diffuses very quickly in liquid metals (
826:
2038:
2003:
1784:
1729:
1682:
1672:
1619:
605:
448:
1110:
354:
2088:
1964:
1959:
1944:
1909:
1819:
691:
82:
1742:
515:
482:
2053:
2033:
1991:
1986:
1814:
1375:
975:
937:
650:
541:
430:
290:
2063:
2043:
2028:
2023:
1954:
1929:
1924:
1914:
1894:
1884:
1849:
1794:
1131:
565:
561:
545:
456:
138:
2103:
2073:
2068:
2058:
2048:
2013:
2008:
1949:
1889:
1879:
1859:
1854:
1824:
1789:
507:
2078:
2018:
1934:
1904:
1864:
1839:
1799:
569:
1378:. These two asymptotic solutions can be blended together using the concept of the
1939:
1844:
1804:
24:
941:
671:
1686:
967:
is much thicker for liquid metals and much thinner for oils relative to the
614:
550:
308:
142:
948:
657:
618:
533:
932:, means the thermal diffusivity dominates. Whereas with large values,
947:
The
Prandtl numbers of gases are about 1, which indicates that both
699:
Formula for the calculation of the
Prandtl number of air and water
687:
646:
952:
1746:
631:
627:
585:
For most gases over a wide range of temperature and pressure,
18:
346:
1700:
1698:
1696:
1705:
Lienhard IV, John Henry; Lienhard V, John Henry (2017).
560:
The mass transfer analog of the
Prandtl number is the
190:
185:
1391:
1355:
1248:
1143:
1113:
1086:
994:
855:
829:
712:
518:
485:
459:
433:
381:
357:
317:
293:
158:
1130:For incompressible flow over a flat plate, the two
49:. Unsourced material may be challenged and removed.
1555:
1366:
1338:
1233:
1119:
1099:
1069:
913:
835:
815:
524:
498:
465:
439:
417:
363:
337:
299:
273:
1709:(4th ed.). Cambridge, MA: Phlogiston Press.
1758:
1637:
1635:
8:
1107:is the thermal boundary layer thickness and
564:and the ratio of the Prandtl number and the
1765:
1751:
1743:
1614:Coulson, J. M.; Richardson, J. F. (1999).
1127:is the momentum boundary layer thickness.
660:(At 0 °C and 20 °C respectively)
1669:Heat transfer : a practical approach
1609:
1607:
1605:
1603:
1601:
1599:
1597:
1595:
1593:
1539:
1531:
1514:
1498:
1484:
1479:
1450:
1442:
1430:
1425:
1417:
1410:
1401:
1393:
1390:
1356:
1354:
1319:
1304:
1296:
1284:
1279:
1271:
1258:
1250:
1247:
1214:
1199:
1191:
1179:
1174:
1166:
1153:
1145:
1142:
1134:correlations are asymptotically correct:
1112:
1091:
1085:
1050:
1028:
1024:
1016:
1001:
995:
993:
884:
874:
865:
857:
854:
828:
804:
773:
754:
737:
731:
722:
714:
711:
517:
490:
484:
458:
432:
406:
391:
380:
356:
327:
316:
292:
256:
249:
231:
219:
206:
200:
183:
170:
159:
157:
109:Learn how and when to remove this message
16:Ratio of kinematic to thermal diffusivity
2115:Dimensionless numbers of fluid mechanics
1774:Dimensionless numbers in fluid mechanics
1728:(3rd. ed.). New York: McGraw-Hill.
2120:Dimensionless numbers of thermodynamics
1589:
418:{\displaystyle \alpha =k/(\rho c_{p})}
1671:(2nd ed.). Boston: McGraw-Hill.
963:) relative to momentum. Consequently
7:
928:Small values of the Prandtl number,
47:adding citations to reliable sources
601:0.003 for molten potassium at 975 K
137:, named after the German physicist
1543:
1540:
1535:
1532:
1488:
1485:
1446:
1443:
1421:
1418:
1397:
1394:
1360:
1357:
1323:
1320:
1300:
1297:
1275:
1272:
1254:
1251:
1225:
1218:
1215:
1195:
1192:
1170:
1167:
1149:
1146:
1054:
1051:
1020:
1017:
861:
858:
718:
715:
163:
160:
14:
610:0.065 for molten lithium at 975 K
149:. The Prandtl number is given as:
940:is more significant compared to
613:around 0.16–0.7 for mixtures of
23:
1530:
1318:
1213:
1043:
843:is the temperature in Celsius.
338:{\displaystyle \nu =\mu /\rho }
34:needs additional citations for
1327:
1222:
412:
396:
307: : momentum diffusivity (
240:
224:
1:
1616:Chemical Engineering Volume 1
1367:{\displaystyle \mathrm {Re} }
1100:{\displaystyle \delta _{t}}
959:) and very slowly in oils (
670:between 100 and 40,000 for
2141:
1642:tec-science (2020-05-10).
1618:(6th ed.). Elsevier.
836:{\displaystyle \vartheta }
451:, (SI units: Pa s = N s/m)
141:, defined as the ratio of
1780:
1667:Çengel, Yunus A. (2003).
1707:A Heat Transfer Textbook
1573:Turbulent Prandtl number
679:10,000 for polymer melts
636:1.38 for gaseous ammonia
1578:Magnetic Prandtl number
1120:{\displaystyle \delta }
969:velocity boundary layer
924:Physical interpretation
364:{\displaystyle \alpha }
1557:
1368:
1340:
1235:
1121:
1101:
1071:
965:thermal boundary layer
915:
837:
817:
540:Note that whereas the
526:
510:, (SI units: J/(kg·K))
500:
467:
441:
419:
365:
339:
301:
275:
1724:White, F. M. (2006).
1558:
1369:
1341:
1236:
1122:
1102:
1072:
916:
838:
818:
527:
525:{\displaystyle \rho }
501:
499:{\displaystyle c_{p}}
477:, (SI units: W/(m·K))
468:
442:
420:
366:
340:
302:
276:
1389:
1353:
1246:
1141:
1111:
1084:
992:
853:
827:
710:
639:between 4 and 5 for
617:or noble gases with
555:thermal conductivity
516:
483:
475:thermal conductivity
457:
440:{\displaystyle \mu }
431:
379:
355:
315:
300:{\displaystyle \nu }
291:
187:momentum diffusivity
156:
143:momentum diffusivity
135:dimensionless number
43:improve this article
1440:
1294:
1189:
593:Typical values for
576:Experimental values
536:, (SI units: kg/m).
373:thermal diffusivity
309:kinematic viscosity
192:thermal diffusivity
147:thermal diffusivity
1920:Keulegan–Carpenter
1726:Viscous Fluid Flow
1553:
1416:
1380:Norm (mathematics)
1364:
1336:
1270:
1231:
1165:
1117:
1097:
1067:
911:
833:
813:
522:
496:
463:
437:
415:
361:
335:
297:
271:
194:
189:
2097:
2096:
1625:978-0-7506-4444-0
1525:
1522:
1506:
1492:
1458:
1438:
1312:
1292:
1207:
1187:
1036:
1010:
909:
868:
811:
725:
676:1000 for glycerol
656:13.4 and 7.2 for
604:around 0.015 for
466:{\displaystyle k}
449:dynamic viscosity
425:, (SI units: m/s)
269:
244:
195:
193:
188:
178:
119:
118:
111:
93:
2132:
1767:
1760:
1753:
1744:
1739:
1711:
1710:
1702:
1691:
1690:
1664:
1658:
1657:
1655:
1654:
1644:"Prandtl number"
1639:
1630:
1629:
1611:
1562:
1560:
1559:
1554:
1546:
1538:
1526:
1524:
1523:
1515:
1513:
1509:
1508:
1507:
1499:
1497:
1493:
1491:
1480:
1461:
1460:
1459:
1451:
1449:
1439:
1431:
1429:
1424:
1411:
1406:
1405:
1400:
1373:
1371:
1370:
1365:
1363:
1345:
1343:
1342:
1337:
1326:
1314:
1313:
1305:
1303:
1293:
1285:
1283:
1278:
1263:
1262:
1257:
1240:
1238:
1237:
1232:
1221:
1209:
1208:
1200:
1198:
1188:
1180:
1178:
1173:
1158:
1157:
1152:
1126:
1124:
1123:
1118:
1106:
1104:
1103:
1098:
1096:
1095:
1076:
1074:
1073:
1068:
1057:
1039:
1038:
1037:
1029:
1023:
1011:
1006:
1005:
996:
981:
962:
958:
935:
931:
920:
918:
917:
912:
910:
908:
889:
888:
875:
870:
869:
866:
864:
842:
840:
839:
834:
822:
820:
819:
814:
812:
810:
809:
808:
778:
777:
759:
758:
742:
741:
732:
727:
726:
723:
721:
685:
645:around 7.56 for
641:R-12 refrigerant
626:around 0.71 for
596:
588:
531:
529:
528:
523:
505:
503:
502:
497:
495:
494:
472:
470:
469:
464:
446:
444:
443:
438:
424:
422:
421:
416:
411:
410:
395:
370:
368:
367:
362:
344:
342:
341:
336:
331:
306:
304:
303:
298:
280:
278:
277:
272:
270:
265:
261:
260:
250:
245:
243:
236:
235:
223:
214:
210:
201:
196:
191:
186:
184:
179:
171:
166:
114:
107:
103:
100:
94:
92:
58:"Prandtl number"
51:
27:
19:
2140:
2139:
2135:
2134:
2133:
2131:
2130:
2129:
2100:
2099:
2098:
2093:
1776:
1771:
1736:
1723:
1720:
1718:Further reading
1715:
1714:
1704:
1703:
1694:
1679:
1666:
1665:
1661:
1652:
1650:
1641:
1640:
1633:
1626:
1613:
1612:
1591:
1586:
1569:
1475:
1474:
1467:
1463:
1462:
1441:
1412:
1392:
1387:
1386:
1376:Reynolds number
1351:
1350:
1295:
1249:
1244:
1243:
1190:
1144:
1139:
1138:
1109:
1108:
1087:
1082:
1081:
1015:
997:
990:
989:
979:
976:boundary layers
960:
956:
938:heat conduction
933:
929:
926:
880:
879:
856:
851:
850:
825:
824:
800:
769:
750:
743:
733:
713:
708:
707:
701:
683:
630:and many other
623:0.63 for oxygen
594:
586:
583:
578:
542:Reynolds number
514:
513:
486:
481:
480:
455:
454:
429:
428:
402:
377:
376:
353:
352:
313:
312:
289:
288:
282:
252:
251:
227:
215:
202:
154:
153:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
2138:
2136:
2128:
2127:
2125:Fluid dynamics
2122:
2117:
2112:
2102:
2101:
2095:
2094:
2092:
2091:
2086:
2081:
2076:
2071:
2066:
2061:
2056:
2051:
2046:
2041:
2036:
2031:
2026:
2021:
2016:
2011:
2006:
2001:
2000:
1999:
1989:
1984:
1983:
1982:
1977:
1967:
1962:
1957:
1952:
1947:
1942:
1937:
1932:
1927:
1922:
1917:
1912:
1907:
1902:
1897:
1892:
1887:
1882:
1877:
1872:
1867:
1862:
1857:
1852:
1847:
1842:
1837:
1832:
1827:
1822:
1817:
1812:
1807:
1802:
1797:
1792:
1787:
1781:
1778:
1777:
1772:
1770:
1769:
1762:
1755:
1747:
1741:
1740:
1734:
1719:
1716:
1713:
1712:
1692:
1677:
1659:
1631:
1624:
1588:
1587:
1585:
1582:
1581:
1580:
1575:
1568:
1565:
1564:
1563:
1552:
1549:
1545:
1542:
1537:
1534:
1529:
1521:
1518:
1512:
1505:
1502:
1496:
1490:
1487:
1483:
1478:
1473:
1470:
1466:
1457:
1454:
1448:
1445:
1437:
1434:
1428:
1423:
1420:
1415:
1409:
1404:
1399:
1396:
1362:
1359:
1347:
1346:
1335:
1332:
1329:
1325:
1322:
1317:
1311:
1308:
1302:
1299:
1291:
1288:
1282:
1277:
1274:
1269:
1266:
1261:
1256:
1253:
1241:
1230:
1227:
1224:
1220:
1217:
1212:
1206:
1203:
1197:
1194:
1186:
1183:
1177:
1172:
1169:
1164:
1161:
1156:
1151:
1148:
1132:Nusselt number
1116:
1094:
1090:
1078:
1077:
1066:
1063:
1060:
1056:
1053:
1049:
1046:
1042:
1035:
1032:
1027:
1022:
1019:
1014:
1009:
1004:
1000:
925:
922:
907:
904:
901:
898:
895:
892:
887:
883:
878:
873:
863:
860:
832:
807:
803:
799:
796:
793:
790:
787:
784:
781:
776:
772:
768:
765:
762:
757:
753:
749:
746:
740:
736:
730:
720:
717:
700:
697:
696:
695:
680:
677:
674:
668:
661:
654:
643:
637:
634:
624:
621:
611:
608:
602:
582:
581:Typical values
579:
577:
574:
566:Schmidt number
562:Schmidt number
546:Grashof number
538:
537:
521:
511:
493:
489:
478:
462:
452:
436:
426:
414:
409:
405:
401:
398:
394:
390:
387:
384:
360:
350:
334:
330:
326:
323:
320:
296:
268:
264:
259:
255:
248:
242:
239:
234:
230:
226:
222:
218:
213:
209:
205:
199:
182:
177:
174:
169:
165:
162:
151:
139:Ludwig Prandtl
123:Prandtl number
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
2137:
2126:
2123:
2121:
2118:
2116:
2113:
2111:
2108:
2107:
2105:
2090:
2087:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2062:
2060:
2057:
2055:
2052:
2050:
2047:
2045:
2042:
2040:
2037:
2035:
2032:
2030:
2027:
2025:
2022:
2020:
2017:
2015:
2012:
2010:
2007:
2005:
2002:
1998:
1995:
1994:
1993:
1990:
1988:
1985:
1981:
1978:
1976:
1973:
1972:
1971:
1968:
1966:
1963:
1961:
1958:
1956:
1953:
1951:
1948:
1946:
1943:
1941:
1938:
1936:
1933:
1931:
1928:
1926:
1923:
1921:
1918:
1916:
1913:
1911:
1908:
1906:
1903:
1901:
1898:
1896:
1893:
1891:
1888:
1886:
1883:
1881:
1878:
1876:
1873:
1871:
1868:
1866:
1863:
1861:
1858:
1856:
1853:
1851:
1848:
1846:
1843:
1841:
1838:
1836:
1833:
1831:
1830:Chandrasekhar
1828:
1826:
1823:
1821:
1818:
1816:
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1782:
1779:
1775:
1768:
1763:
1761:
1756:
1754:
1749:
1748:
1745:
1737:
1735:0-07-240231-8
1731:
1727:
1722:
1721:
1717:
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1663:
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1649:
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1550:
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1494:
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1471:
1468:
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1280:
1267:
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1242:
1228:
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1201:
1184:
1181:
1175:
1162:
1159:
1154:
1137:
1136:
1135:
1133:
1128:
1114:
1092:
1088:
1064:
1061:
1058:
1047:
1044:
1040:
1033:
1030:
1025:
1012:
1007:
1002:
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988:
987:
986:
983:
977:
972:
970:
966:
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945:
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923:
921:
905:
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755:
751:
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744:
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734:
728:
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675:
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669:
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629:
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580:
575:
573:
571:
567:
563:
558:
556:
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543:
535:
519:
512:
509:
508:specific heat
491:
487:
479:
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460:
453:
450:
434:
427:
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403:
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392:
388:
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374:
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332:
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294:
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285:
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266:
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257:
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220:
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211:
207:
203:
197:
180:
175:
172:
167:
150:
148:
144:
140:
136:
132:
131:Prandtl group
128:
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
1969:
1725:
1706:
1668:
1662:
1651:. Retrieved
1647:
1615:
1348:
1129:
1079:
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973:
946:
927:
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702:
664:
592:
584:
570:Lewis number
559:
539:
283:
152:
130:
126:
122:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
2084:Weissenberg
1648:tec-science
615:noble gases
349:units: m/s)
99:August 2014
2110:Convection
2104:Categories
2004:Richardson
1785:Archimedes
1678:0072458933
1653:2020-06-25
1584:References
942:convection
672:engine oil
69:newspapers
2089:Womersley
1980:turbulent
1960:Ohnesorge
1945:Marangoni
1910:Iribarren
1835:Damköhler
1820:Capillary
1328:→
1226:∞
1223:→
1115:δ
1089:δ
1059:≤
1048:≤
1026:−
1008:δ
999:δ
900:ϑ
897:⋅
882:ϑ
831:ϑ
798:⋅
789:ϑ
786:⋅
771:ϑ
767:⋅
761:−
752:ϑ
748:⋅
551:viscosity
520:ρ
435:μ
400:ρ
383:α
359:α
333:ρ
325:μ
319:ν
295:ν
263:μ
238:ρ
212:ρ
204:μ
176:α
173:ν
2064:Suratman
2054:Strouhal
2034:Sherwood
1997:magnetic
1992:Reynolds
1987:Rayleigh
1975:magnetic
1815:Brinkman
1687:50192222
1567:See also
949:momentum
823:, where
682:around 1
667:-butanol
658:seawater
619:hydrogen
532: :
506: :
473: :
447: :
371: :
2044:Stanton
2039:Shields
2029:Scruton
2024:Schmidt
1970:Prandtl
1955:Nusselt
1930:Laplace
1925:Knudsen
1915:Kapitza
1900:Görtler
1895:Grashof
1885:Galilei
1850:Deborah
1795:Bagnold
1374:is the
978:. When
686:10 for
663:50 for
649:(At 18
606:mercury
568:is the
534:density
284:where:
83:scholar
2074:Ursell
2069:Taylor
2059:Stuart
2049:Stokes
2014:Rossby
2009:Roshko
1965:Péclet
1950:Morton
1890:Graetz
1880:Froude
1870:Eötvös
1860:Eckert
1855:Dukhin
1825:Cauchy
1790:Atwood
1732:
1685:
1675:
1622:
1482:0.0468
1414:0.3387
1349:where
1080:where
961:Pr ≫ 1
957:Pr ≪ 1
934:Pr ≫ 1
930:Pr ≪ 1
783:322000
692:mantle
85:
78:
71:
64:
56:
2079:Weber
2019:Rouse
1935:Lewis
1905:Hagen
1875:Euler
1865:Ekman
1840:Darcy
1800:Bejan
1268:0.565
1163:0.339
877:50000
867:water
795:1.393
688:Earth
647:water
632:gases
597:are:
133:is a
129:) or
90:JSTOR
76:books
1940:Mach
1845:Dean
1810:Bond
1805:Biot
1730:ISBN
1683:OCLC
1673:ISBN
1620:ISBN
1551:100.
1548:>
953:heat
951:and
906:3700
764:1200
553:and
544:and
121:The
62:news
1045:0.6
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745:1.1
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1003:t
903:+
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886:2
872:=
862:r
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806:9
792:+
780:+
775:2
756:3
739:9
729:=
719:r
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694:.
684:×
665:n
653:)
492:p
488:c
461:k
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408:p
404:c
397:(
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322:=
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225:(
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217:k
208:/
198:=
181:=
168:=
164:r
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125:(
112:)
106:(
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97:(
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39:.
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