293:
and composition of the sphere), and the second for convection at the surface of the sphere. If the thermal resistance of the fluid/sphere interface exceeds that thermal resistance offered by the interior of the metal sphere, the Biot number will be less than one. For systems where it is much less than one, the interior of the sphere may be presumed to be a uniform temperature, although this temperature may be changing with time as heat passes into the sphere from the surface. The equation to describe this change in (relatively uniform) temperature inside the object, is a simple exponential one described by
584:
The study of heat transfer in micro-encapsulated phase-change slurries is an application where the Biot number is useful. For the dispersed phase of the micro-encapsulated phase-change slurry, the micro-encapsulated phase-change material itself, the Biot number is calculated to be below 0.1 and so it
309:
The value of the Biot number can indicate the applicability (or inapplicability) of certain methods of solving transient heat transfer problems. For example, a Biot number smaller than about 0.1 implies that heat conduction inside the body offers much lower thermal resistance than the heat convection
292:
The physical significance of Biot number can be understood by imagining the heat flow from a small hot metal sphere suddenly immersed in a pool, to the surrounding fluid. The heat flow experiences two resistances: the first for conduction within the solid metal (which is influenced by both the size
300:
In contrast, the metal sphere may be large, so that the characteristic length is large and the Biot number is greater than one. Now, thermal gradients within the sphere become important, even though the sphere material is a good conductor. Equivalently, if the sphere is made of a poorly conducting
318:
may be used to evaluate a body's transient temperature variation. The opposite is also true: a Biot number greater than about 0.1 indicates that thermal resistance within the body is not negligible, and more complex methods are need in analyzing heat transfer to or from the body (such bodies are
62:
In general, problems involving small Biot numbers (much smaller than 1) are analytically simple, as a result of nearly uniform temperature fields inside the body. Biot numbers of order one or greater indicate more difficult problems with nonuniform temperature fields inside the body.
363:
of the body is directly proportional to the temperature of the body, and the difference between the body temperature and the fluid temperature is linearly proportional to rate of heat transfer into or out of the body. Combining these relationships with the
301:(thermally insulating) material, such as wood or styrofoam, the interior resistance to heat flow will exceed that of convection at the fluid/sphere boundary, even for a much smaller sphere. In this case, again, the Biot number will be greater than one.
916:
Delgado, Mónica; Lázaro, Ana; Mazo, Javier; Zalba, Belén (January 2012). "Review on phase change material emulsions and microencapsulated phase change material slurries: Materials, heat transfer studies and applications".
331:
must be solved to determine the time-varying and spatially-nonuniform temperature field within the body. Analytic methods for handling these problems, which may exist for simple geometric shapes and uniform material
457:
340:. Examples of verified analytic solutions along with precise numerical values are available. Often such problems are too difficult to be done except numerically, with the use of a computer model of heat transfer.
681:
520:
359:
The simplest type of lumped capacity solution, for a step change in fluid temperature, shows that a body's temperature decays exponentially in time ("Newtonian" cooling or heating) because the
277:
128:
978:
237:
The characteristic length in most relevant problems becomes the heat characteristic length, i.e. the ratio between the body volume and the heated (or cooled) surface of the body:
623:
348:
As noted, a Biot number smaller than about 0.1 shows that the conduction resistance inside a body is much smaller than heat convection at the surface, so that temperature
716:
356:
of transient heat transfer can be used. (A Biot number less than 0.1 generally indicates less than 3% error will be present when using the lumped-capacitance model.)
575:
544:
778:
211:
184:
156:
746:
971:
1323:
964:
1328:
374:
846:
Cole, Kevin D.; Beck, James V.; Woodbury, Keith A.; de Monte, Filippo (2014). "Intrinsic verification and a heat conduction database".
240:
987:
289:, is used to denote that the surface to be considered is only the portion of the total surface through which the heat passes.
631:
43:
for conduction inside a body to the resistance for convection at the surface of the body. This ratio indicates whether the
465:
368:
leads to a simple first-order linear differential equation. The corresponding lumped capacity solution can be written
314:
are negligible inside of the body (such bodies are sometimes labeled "thermally thin"). In this situation, the simple
92:
889:
Ostorgorsky, Aleks G. (January 2009). "Simple
Explicit Equations for Transient Heat Conduction in Finite Solids".
1133:
365:
596:
294:
719:
187:
1333:
1210:
1193:
523:
353:
315:
1188:
1088:
578:
75:
28:
692:
214:
593:
An analogous version of the Biot number (usually called the "mass transfer Biot number", or
1043:
956:
926:
855:
553:
529:
333:
226:
159:
1048:
8:
761:
194:
167:
139:
1113:
930:
859:
1297:
1083:
1023:
867:
731:
40:
36:
1252:
1217:
998:
942:
871:
1302:
1173:
1158:
1123:
1033:
934:
898:
863:
749:
1178:
1267:
1247:
1205:
1200:
1028:
800:
585:
can be assumed that thermal gradients within the dispersed phase are negligible.
360:
71:
1277:
1257:
1242:
1237:
1183:
1168:
1143:
1138:
1128:
1108:
1098:
1063:
1008:
938:
795:
222:
1317:
1287:
1282:
1272:
1262:
1227:
1222:
1163:
1103:
1093:
1073:
1068:
1038:
1003:
946:
875:
337:
328:
67:
32:
1292:
1232:
1148:
1118:
1078:
1053:
1013:
547:
1153:
1058:
44:
790:
452:{\displaystyle {\frac {T-T_{\infty }}{T_{0}-T_{\infty }}}=e^{-t/\tau }}
902:
56:
16:
Ratio of the thermal resistances of a body's interior to its surface
349:
311:
52:
35:
calculations, named for the eighteenth-century French physicist
230:
47:
inside a body varies significantly in space when the body is
986:
285:
48:
322:
845:
820:
915:
764:
734:
695:
676:{\displaystyle \mathrm {Bi} _{m}={\frac {k_{c}}{D}}L}
634:
599:
556:
532:
468:
377:
352:
are negligible inside of the body. In this case, the
243:
197:
170:
142:
95:
327:
When the Biot number is greater than 0.1 or so, the
515:{\displaystyle \tau ={\frac {\rho c_{p}V}{hA_{Q}}}}
772:
740:
710:
675:
617:
569:
538:
514:
451:
271:
205:
178:
150:
122:
221:(The Biot number should not be confused with the
39:(1774–1862). The Biot number is the ratio of the
1315:
272:{\displaystyle L={\frac {V}{A_{\mathrm {Q} }}}}
972:
343:
123:{\displaystyle \mathrm {Bi} ={\frac {h}{k}}L}
625:) is also used in mass diffusion processes:
888:
839:
979:
965:
848:International Journal of Thermal Sciences
588:
1324:Dimensionless numbers of fluid mechanics
988:Dimensionless numbers in fluid mechanics
919:Renewable and Sustainable Energy Reviews
1329:Dimensionless numbers of thermodynamics
66:The Biot number appears in a number of
1316:
827:. University of Nebraska. January 2013
336:, are described in the article on the
323:Heat conduction for finite Biot number
319:sometimes called "thermally thick").
960:
310:at the surface, so that temperature
825:Exact Analytical Conduction Toolbox
13:
868:10.1016/j.ijthermalsci.2013.11.002
640:
637:
605:
602:
417:
392:
261:
100:
97:
14:
1345:
618:{\displaystyle \mathrm {Bi} _{m}}
304:
233:rather than that of the body.)
86:The Biot number is defined as:
909:
882:
813:
70:problems, including transient
1:
806:
780: : characteristic length
726:of the heat transfer problem)
81:
78:heat transfer calculations.
7:
784:
366:First law of thermodynamics
217:of the geometry considered.
10:
1350:
939:10.1016/j.rser.2011.07.152
344:Heat conduction for Bi ≪ 1
994:
756:of heat transfer problem)
720:mass transfer coefficient
188:heat transfer coefficient
891:Journal of Heat Transfer
354:lumped-capacitance model
316:lumped-capacitance model
711:{\displaystyle {k_{c}}}
295:Newton's law of cooling
774:
742:
712:
677:
619:
589:Mass transfer analogue
579:specific heat capacity
571:
540:
516:
453:
273:
207:
180:
152:
124:
29:dimensionless quantity
775:
743:
713:
678:
620:
572:
570:{\displaystyle c_{p}}
541:
539:{\displaystyle \rho }
524:thermal time constant
517:
454:
274:
215:characteristic length
208:
181:
153:
125:
762:
732:
693:
632:
597:
554:
530:
466:
375:
334:thermal conductivity
279:Here, the subscript
241:
227:thermal conductivity
225:, which employs the
195:
168:
160:thermal conductivity
140:
93:
931:2012RSERv..16..253D
860:2014IJTS...78...36C
773:{\displaystyle {L}}
718: : convective
206:{\displaystyle {L}}
179:{\displaystyle {h}}
151:{\displaystyle {k}}
1134:Keulegan–Carpenter
770:
752:(analogous to the
738:
722:(analogous to the
708:
673:
615:
567:
536:
512:
449:
269:
203:
176:
148:
120:
41:thermal resistance
37:Jean-Baptiste Biot
1311:
1310:
903:10.1115/1.2977540
741:{\displaystyle D}
668:
510:
423:
267:
115:
1341:
981:
974:
967:
958:
957:
951:
950:
913:
907:
906:
886:
880:
879:
843:
837:
836:
834:
832:
817:
779:
777:
776:
771:
769:
750:mass diffusivity
747:
745:
744:
739:
717:
715:
714:
709:
707:
706:
705:
682:
680:
679:
674:
669:
664:
663:
654:
649:
648:
643:
624:
622:
621:
616:
614:
613:
608:
576:
574:
573:
568:
566:
565:
545:
543:
542:
537:
521:
519:
518:
513:
511:
509:
508:
507:
494:
490:
489:
476:
458:
456:
455:
450:
448:
447:
443:
424:
422:
421:
420:
408:
407:
397:
396:
395:
379:
278:
276:
275:
270:
268:
266:
265:
264:
251:
212:
210:
209:
204:
202:
186:is a convective
185:
183:
182:
177:
175:
157:
155:
154:
149:
147:
129:
127:
126:
121:
116:
108:
103:
59:at its surface.
1349:
1348:
1344:
1343:
1342:
1340:
1339:
1338:
1334:Heat conduction
1314:
1313:
1312:
1307:
990:
985:
955:
954:
914:
910:
887:
883:
844:
840:
830:
828:
819:
818:
814:
809:
801:Heat conduction
787:
765:
763:
760:
759:
733:
730:
729:
701:
697:
696:
694:
691:
690:
659:
655:
653:
644:
636:
635:
633:
630:
629:
609:
601:
600:
598:
595:
594:
591:
561:
557:
555:
552:
551:
531:
528:
527:
503:
499:
495:
485:
481:
477:
475:
467:
464:
463:
439:
432:
428:
416:
412:
403:
399:
398:
391:
387:
380:
378:
376:
373:
372:
361:internal energy
346:
325:
307:
260:
259:
255:
250:
242:
239:
238:
236:
198:
196:
193:
192:
171:
169:
166:
165:
143:
141:
138:
137:
107:
96:
94:
91:
90:
84:
72:heat conduction
55:over time by a
17:
12:
11:
5:
1347:
1337:
1336:
1331:
1326:
1309:
1308:
1306:
1305:
1300:
1295:
1290:
1285:
1280:
1275:
1270:
1265:
1260:
1255:
1250:
1245:
1240:
1235:
1230:
1225:
1220:
1215:
1214:
1213:
1203:
1198:
1197:
1196:
1191:
1181:
1176:
1171:
1166:
1161:
1156:
1151:
1146:
1141:
1136:
1131:
1126:
1121:
1116:
1111:
1106:
1101:
1096:
1091:
1086:
1081:
1076:
1071:
1066:
1061:
1056:
1051:
1046:
1041:
1036:
1031:
1026:
1021:
1016:
1011:
1006:
1001:
995:
992:
991:
984:
983:
976:
969:
961:
953:
952:
925:(1): 253–273.
908:
881:
838:
811:
810:
808:
805:
804:
803:
798:
796:Fourier number
793:
786:
783:
782:
781:
768:
757:
737:
727:
704:
700:
684:
683:
672:
667:
662:
658:
652:
647:
642:
639:
612:
607:
604:
590:
587:
564:
560:
535:
506:
502:
498:
493:
488:
484:
480:
474:
471:
460:
459:
446:
442:
438:
435:
431:
427:
419:
415:
411:
406:
402:
394:
390:
386:
383:
345:
342:
324:
321:
306:
303:
263:
258:
254:
249:
246:
223:Nusselt number
219:
218:
201:
190:
174:
163:
146:
131:
130:
119:
114:
111:
106:
102:
99:
83:
80:
15:
9:
6:
4:
3:
2:
1346:
1335:
1332:
1330:
1327:
1325:
1322:
1321:
1319:
1304:
1301:
1299:
1296:
1294:
1291:
1289:
1286:
1284:
1281:
1279:
1276:
1274:
1271:
1269:
1266:
1264:
1261:
1259:
1256:
1254:
1251:
1249:
1246:
1244:
1241:
1239:
1236:
1234:
1231:
1229:
1226:
1224:
1221:
1219:
1216:
1212:
1209:
1208:
1207:
1204:
1202:
1199:
1195:
1192:
1190:
1187:
1186:
1185:
1182:
1180:
1177:
1175:
1172:
1170:
1167:
1165:
1162:
1160:
1157:
1155:
1152:
1150:
1147:
1145:
1142:
1140:
1137:
1135:
1132:
1130:
1127:
1125:
1122:
1120:
1117:
1115:
1112:
1110:
1107:
1105:
1102:
1100:
1097:
1095:
1092:
1090:
1087:
1085:
1082:
1080:
1077:
1075:
1072:
1070:
1067:
1065:
1062:
1060:
1057:
1055:
1052:
1050:
1047:
1045:
1044:Chandrasekhar
1042:
1040:
1037:
1035:
1032:
1030:
1027:
1025:
1022:
1020:
1017:
1015:
1012:
1010:
1007:
1005:
1002:
1000:
997:
996:
993:
989:
982:
977:
975:
970:
968:
963:
962:
959:
948:
944:
940:
936:
932:
928:
924:
920:
912:
904:
900:
897:(1): 011303.
896:
892:
885:
877:
873:
869:
865:
861:
857:
853:
849:
842:
826:
822:
816:
812:
802:
799:
797:
794:
792:
789:
788:
766:
758:
755:
751:
735:
728:
725:
721:
702:
698:
689:
688:
687:
670:
665:
660:
656:
650:
645:
628:
627:
626:
610:
586:
582:
580:
562:
558:
549:
533:
526:of the body,
525:
504:
500:
496:
491:
486:
482:
478:
472:
469:
444:
440:
436:
433:
429:
425:
413:
409:
404:
400:
388:
384:
381:
371:
370:
369:
367:
362:
357:
355:
351:
341:
339:
338:heat equation
335:
330:
329:heat equation
320:
317:
313:
302:
298:
296:
290:
288:
287:
282:
256:
252:
247:
244:
234:
232:
228:
224:
216:
199:
191:
189:
172:
164:
161:
144:
136:
135:
134:
117:
112:
109:
104:
89:
88:
87:
79:
77:
73:
69:
68:heat transfer
64:
60:
58:
54:
50:
46:
42:
38:
34:
33:heat transfer
30:
26:
22:
1018:
922:
918:
911:
894:
890:
884:
851:
847:
841:
829:. Retrieved
824:
815:
753:
723:
685:
592:
583:
550:(kg/m), and
548:mass density
461:
358:
347:
326:
308:
305:Applications
299:
291:
284:
280:
235:
220:
162:of the body
132:
85:
65:
61:
24:
20:
18:
1298:Weissenberg
45:temperature
21:Biot number
1318:Categories
1218:Richardson
999:Archimedes
831:24 January
807:References
791:Convection
581:(J/kg-K).
82:Definition
1303:Womersley
1194:turbulent
1174:Ohnesorge
1159:Marangoni
1124:Iribarren
1049:Damköhler
1034:Capillary
947:1364-0321
876:1290-0729
854:: 36–47.
534:ρ
479:ρ
470:τ
462:in which
445:τ
434:−
418:∞
410:−
393:∞
385:−
350:gradients
312:gradients
57:heat flux
1278:Suratman
1268:Strouhal
1248:Sherwood
1211:magnetic
1206:Reynolds
1201:Rayleigh
1189:magnetic
1029:Brinkman
785:See also
748: :
158:is the
31:used in
1258:Stanton
1253:Shields
1243:Scruton
1238:Schmidt
1184:Prandtl
1169:Nusselt
1144:Laplace
1139:Knudsen
1129:Kapitza
1114:Görtler
1109:Grashof
1099:Galilei
1064:Deborah
1009:Bagnold
927:Bibcode
856:Bibcode
821:"EXACT"
686:where:
546:is the
522:is the
229:of the
133:where:
27:) is a
1288:Ursell
1283:Taylor
1273:Stuart
1263:Stokes
1228:Rossby
1223:Roshko
1179:PĂ©clet
1164:Morton
1104:Graetz
1094:Froude
1084:Eötvös
1074:Eckert
1069:Dukhin
1039:Cauchy
1004:Atwood
945:
874:
283:, for
53:cooled
49:heated
1293:Weber
1233:Rouse
1149:Lewis
1119:Hagen
1089:Euler
1079:Ekman
1054:Darcy
1014:Bejan
231:fluid
213:is a
1154:Mach
1059:Dean
1024:Bond
1019:Biot
943:ISSN
872:ISSN
833:2015
286:heat
74:and
19:The
935:doi
899:doi
895:131
864:doi
577:is
76:fin
51:or
1320::
941:.
933:.
923:16
921:.
893:.
870:.
862:.
852:78
850:.
823:.
297:.
25:Bi
980:e
973:t
966:v
949:.
937::
929::
905:.
901::
878:.
866::
858::
835:.
767:L
754:k
736:D
724:h
703:c
699:k
671:L
666:D
661:c
657:k
651:=
646:m
641:i
638:B
611:m
606:i
603:B
563:p
559:c
505:Q
501:A
497:h
492:V
487:p
483:c
473:=
441:/
437:t
430:e
426:=
414:T
405:0
401:T
389:T
382:T
281:Q
262:Q
257:A
253:V
248:=
245:L
200:L
173:h
145:k
118:L
113:k
110:h
105:=
101:i
98:B
23:(
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