387:
504:
702:
1240:
1284:
as time-dependent and then applying the derivative product rule. A correct description of such an object requires the application of Newton's second law to the entire, constant-mass system consisting of both the object and its ejected mass.
341:
411:
722:- ignoring the area spanned by the holes in the filter/membrane. The spaces would be cross-sectional areas. For liquids passing through a pipe, the area is the cross-section of the pipe, at the section considered. The
607:
1123:
374:
crossing the boundary for some time duration, not the initial amount of mass at the boundary minus the final amount at the boundary, since the change in mass flowing through the area would be zero for
1148:
1031:
801:
872:
1336:
249:
908:
761:
120:
934:
539:
202:
65:
710:
required to calculate the mass flow rate is real or imaginary, flat or curved, either as a cross-sectional area or a surface, e.g. for substances passing through a
1356:
1050:
1033:
These results are equivalent to the equation containing the dot product. Sometimes these equations are used to define the mass flow rate.
390:
Illustration of volume flow rate. Mass flow rate can be calculated by multiplying the volume flow rate by the mass density of the fluid,
966:
600:
The above equation is only true for a flat, plane area. In general, including cases where the area is curved, the equation becomes a
499:{\displaystyle {\dot {m}}=\rho \cdot {\dot {V}}=\rho \cdot \mathbf {v} \cdot \mathbf {A} =\mathbf {j} _{\text{m}}\cdot \mathbf {A} ,}
370:
quantity, the mass flow rate (the time derivative of mass) is also a scalar quantity. The change in mass is the amount that flows
697:{\displaystyle {\dot {m}}=\iint _{A}\rho \mathbf {v} \cdot d\mathbf {A} =\iint _{A}\mathbf {j} _{\text{m}}\cdot d\mathbf {A} .}
1521:
1442:
1408:
1291:
1036:
Considering flow through porous media, a special quantity, superficial mass flow rate, can be introduced. It is related with
766:
1610:
1496:
824:
1235:{\displaystyle \rho _{1}\mathbf {v} _{1}\cdot \mathbf {A} _{1}=\rho _{2}\mathbf {v} _{2}\cdot \mathbf {A} _{2}.}
1655:
1482:
Lindeburg M. R. Chemical
Engineering Reference Manual for the PE Exam. – Professional Publications (CA), 2013.
910:
and the velocity of mass elements. The amount passing through the cross-section is reduced by the factor
884:
737:
88:
940:
increases less mass passes through. All mass which passes in tangential directions to the area, that is
81:
1532:
derive a general expression for Newton's second law for variable mass systems by treating the mass in
1126:
576:
913:
1645:
1640:
1413:
515:
178:
41:
1491:
Essential
Principles of Physics, P. M. Whelan, M. J. Hodgeson, 2nd Edition, 1978, John Murray,
1249:
1248:, such as a rocket ejecting spent fuel. Often, descriptions of such objects erroneously invoke
336:{\displaystyle {\dot {m}}=\lim _{\Delta t\to 0}{\frac {\Delta m}{\Delta t}}={\frac {dm}{dt}},}
1455:
394:. The volume flow rate is calculated by multiplying the flow velocity of the mass elements,
1418:
1388:
1245:
1037:
546:
359:
20:
8:
1378:
1138:
815:
715:
243:
148:
1469:
1437:
Fluid
Mechanics, M. Potter, D. C. Wiggart, Schaum's Outlines, McGraw Hill (USA), 2008,
1341:
1616:
1606:
1517:
1492:
1438:
726:
is a combination of the magnitude of the area through which the mass passes through,
719:
172:
1244:
In elementary classical mechanics, mass flow rate is encountered when dealing with
601:
367:
1393:
363:
1383:
215:
1129:
or mass transfer coefficient calculation for fixed and fluidized bed systems.
1650:
1634:
1403:
1142:
941:
566:
1620:
711:
168:
144:
1288:
Mass flow rate can be used to calculate the energy flow rate of a fluid:
952:
the area, so the mass passing through the area is zero. This occurs when
807:
731:
723:
718:, the real surface is the (generally curved) surface area of the filter,
579:
375:
219:
164:
132:
386:
1398:
1362:
592:
227:
1366:
152:
556:
128:
71:
156:
1118:{\displaystyle {\dot {m}}_{s}=v_{s}\cdot \rho ={\dot {m}}/A}
239:. In this article, the (more intuitive) definition is used.
707:
140:
1589:
having parts among which there is an interchange of mass.
814:
the cross-section is the amount normal to the area, i.e.
160:
1456:"Mass Flow Rate Fluids Flow Equation - Engineers Edge"
1344:
1294:
1151:
1053:
969:
916:
887:
827:
769:
740:
610:
518:
414:
252:
181:
91:
44:
1350:
1330:
1234:
1117:
1025:
928:
902:
866:
795:
755:
696:
533:
498:
335:
196:
114:
59:
1026:{\displaystyle {\dot {m}}=\rho vA\cos(\pi /2)=0.}
894:
796:{\displaystyle \mathbf {A} =A\mathbf {\hat {n}} }
787:
747:
16:Mass of a substance which passes per unit of time
1632:
269:
1511:
867:{\displaystyle {\dot {m}}=\rho vA\cos \theta ,}
1603:Thermodynamics : an engineering approach
1601:Çengel, Yunus A.; Boles, Michael A. (2002).
1600:
1507:
1505:
210:, pronounced "m-dot"), although sometimes
406:Mass flow rate can also be calculated by
385:
381:
1502:
1331:{\displaystyle {\dot {E}}={\dot {m}}e,}
1633:
398:, by the cross-sectional vector area,
105:
102:
98:
94:
1605:(4th ed.). Boston: McGraw-Hill.
1409:Standard cubic centimetres per minute
1358:is the unit mass energy of a system.
881:is the angle between the unit normal
810:is as follows. The only mass flowing
225:Sometimes, mass flow rate is termed
1516:. Vol. 1. Wiley. p. 199.
1047:, with the following relationship:
903:{\displaystyle \mathbf {\hat {n}} }
818:to the unit normal. This amount is
756:{\displaystyle \mathbf {\hat {n}} }
237:Schaum's Outline of Fluid Mechanics
115:{\displaystyle {\mathsf {MT^{-1}}}}
13:
298:
290:
273:
14:
1667:
1581:to analyze variable mass systems
1361:Energy flow rate has SI units of
242:Mass flow rate is defined by the
1528:It is important to note that we
1219:
1204:
1179:
1164:
891:
784:
771:
744:
687:
670:
651:
640:
489:
475:
466:
458:
347:through a surface per unit time
143:of a substance which passes per
1594:
1587:entire system of constant mass
1485:
1476:
1462:
1448:
1431:
1137:In the elementary form of the
1014:
1000:
279:
1:
1424:
1125:The quantity can be used in
929:{\displaystyle \cos \theta }
7:
1372:
10:
1674:
1512:Halliday; Resnick (1977).
1272:by treating both the mass
534:{\displaystyle {\dot {V}}}
197:{\displaystyle {\dot {m}}}
60:{\displaystyle {\dot {m}}}
18:
80:
70:
33:
28:
1246:objects of variable mass
1132:
1127:particle Reynolds number
175:. The common symbol is
19:Not to be confused with
1414:Thermal mass flow meter
343:i.e., the flow of mass
1352:
1332:
1236:
1119:
1027:
930:
904:
868:
797:
757:
698:
535:
500:
403:
337:
198:
116:
61:
1656:Mechanical quantities
1585:if we apply it to an
1353:
1333:
1237:
1120:
1028:
931:
905:
869:
798:
758:
699:
569:of the mass elements,
536:
501:
389:
382:Alternative equations
338:
199:
117:
62:
1419:Volumetric flow rate
1389:Mass flow controller
1342:
1292:
1149:
1051:
1038:superficial velocity
967:
944:to the unit normal,
914:
885:
825:
767:
738:
734:normal to the area,
608:
516:
412:
250:
179:
89:
42:
21:Volumetric flow rate
1379:Continuity equation
1250:Newton's second law
1139:continuity equation
806:The reason for the
354:The overdot on the
1348:
1328:
1232:
1115:
1023:
926:
900:
864:
793:
763:. The relation is
753:
694:
531:
496:
404:
366:. Since mass is a
333:
286:
235:, see for example
194:
173:US customary units
112:
57:
1523:978-0-471-03710-1
1443:978-0-07-148781-8
1351:{\displaystyle e}
1319:
1304:
1278:and the velocity
1104:
1064:
979:
897:
837:
790:
750:
677:
620:
528:
482:
445:
424:
360:Newton's notation
328:
305:
268:
262:
191:
125:
124:
54:
1663:
1625:
1624:
1598:
1592:
1591:
1509:
1500:
1489:
1483:
1480:
1474:
1473:
1470:"Mass Flow Rate"
1466:
1460:
1459:
1452:
1446:
1435:
1357:
1355:
1354:
1349:
1337:
1335:
1334:
1329:
1321:
1320:
1312:
1306:
1305:
1297:
1283:
1277:
1271:
1241:
1239:
1238:
1233:
1228:
1227:
1222:
1213:
1212:
1207:
1201:
1200:
1188:
1187:
1182:
1173:
1172:
1167:
1161:
1160:
1124:
1122:
1121:
1116:
1111:
1106:
1105:
1097:
1085:
1084:
1072:
1071:
1066:
1065:
1057:
1032:
1030:
1029:
1024:
1010:
981:
980:
972:
962:
935:
933:
932:
927:
909:
907:
906:
901:
899:
898:
890:
873:
871:
870:
865:
839:
838:
830:
802:
800:
799:
794:
792:
791:
783:
774:
762:
760:
759:
754:
752:
751:
743:
703:
701:
700:
695:
690:
679:
678:
675:
673:
667:
666:
654:
643:
635:
634:
622:
621:
613:
602:surface integral
547:volume flow rate
540:
538:
537:
532:
530:
529:
521:
505:
503:
502:
497:
492:
484:
483:
480:
478:
469:
461:
447:
446:
438:
426:
425:
417:
357:
350:
346:
342:
340:
339:
334:
329:
327:
319:
311:
306:
304:
296:
288:
285:
264:
263:
255:
203:
201:
200:
195:
193:
192:
184:
121:
119:
118:
113:
111:
110:
109:
108:
66:
64:
63:
58:
56:
55:
47:
26:
25:
1673:
1672:
1666:
1665:
1664:
1662:
1661:
1660:
1631:
1630:
1629:
1628:
1613:
1599:
1595:
1524:
1510:
1503:
1490:
1486:
1481:
1477:
1468:
1467:
1463:
1454:
1453:
1449:
1436:
1432:
1427:
1394:Mass flow meter
1375:
1343:
1340:
1339:
1311:
1310:
1296:
1295:
1293:
1290:
1289:
1279:
1273:
1252:
1223:
1218:
1217:
1208:
1203:
1202:
1196:
1192:
1183:
1178:
1177:
1168:
1163:
1162:
1156:
1152:
1150:
1147:
1146:
1135:
1107:
1096:
1095:
1080:
1076:
1067:
1056:
1055:
1054:
1052:
1049:
1048:
1045:
1006:
971:
970:
968:
965:
964:
953:
915:
912:
911:
889:
888:
886:
883:
882:
829:
828:
826:
823:
822:
782:
781:
770:
768:
765:
764:
742:
741:
739:
736:
735:
720:macroscopically
686:
674:
669:
668:
662:
658:
650:
639:
630:
626:
612:
611:
609:
606:
605:
598:
590:
577:cross-sectional
520:
519:
517:
514:
513:
488:
479:
474:
473:
465:
457:
437:
436:
416:
415:
413:
410:
409:
384:
364:time derivative
355:
348:
344:
320:
312:
310:
297:
289:
287:
272:
254:
253:
251:
248:
247:
183:
182:
180:
177:
176:
101:
97:
93:
92:
90:
87:
86:
46:
45:
43:
40:
39:
36:
24:
17:
12:
11:
5:
1671:
1670:
1659:
1658:
1653:
1648:
1646:Temporal rates
1643:
1641:Fluid dynamics
1627:
1626:
1611:
1593:
1522:
1501:
1484:
1475:
1461:
1447:
1429:
1428:
1426:
1423:
1422:
1421:
1416:
1411:
1406:
1401:
1396:
1391:
1386:
1384:Fluid dynamics
1381:
1374:
1371:
1365:per second or
1347:
1327:
1324:
1318:
1315:
1309:
1303:
1300:
1231:
1226:
1221:
1216:
1211:
1206:
1199:
1195:
1191:
1186:
1181:
1176:
1171:
1166:
1159:
1155:
1134:
1131:
1114:
1110:
1103:
1100:
1094:
1091:
1088:
1083:
1079:
1075:
1070:
1063:
1060:
1043:
1022:
1019:
1016:
1013:
1009:
1005:
1002:
999:
996:
993:
990:
987:
984:
978:
975:
948:actually pass
925:
922:
919:
896:
893:
875:
874:
863:
860:
857:
854:
851:
848:
845:
842:
836:
833:
789:
786:
780:
777:
773:
749:
746:
693:
689:
685:
682:
672:
665:
661:
657:
653:
649:
646:
642:
638:
633:
629:
625:
619:
616:
597:
596:
588:
583:
570:
560:
550:
527:
524:
510:
495:
491:
487:
477:
472:
468:
464:
460:
456:
453:
450:
444:
441:
435:
432:
429:
423:
420:
383:
380:
332:
326:
323:
318:
315:
309:
303:
300:
295:
292:
284:
281:
278:
275:
271:
267:
261:
258:
190:
187:
171:per second in
167:per second or
137:mass flow rate
123:
122:
107:
104:
100:
96:
84:
78:
77:
74:
68:
67:
53:
50:
37:
35:Common symbols
34:
31:
30:
29:Mass Flow rate
15:
9:
6:
4:
3:
2:
1669:
1668:
1657:
1654:
1652:
1649:
1647:
1644:
1642:
1639:
1638:
1636:
1622:
1618:
1614:
1612:0-07-238332-1
1608:
1604:
1597:
1590:
1588:
1584:
1580:
1576:
1573:
1569:
1565:
1561:
1557:
1554:
1550:
1546:
1542:
1539:
1535:
1531:
1525:
1519:
1515:
1508:
1506:
1498:
1497:0-7195-3382-1
1494:
1488:
1479:
1471:
1465:
1457:
1451:
1444:
1440:
1434:
1430:
1420:
1417:
1415:
1412:
1410:
1407:
1405:
1404:Orifice plate
1402:
1400:
1397:
1395:
1392:
1390:
1387:
1385:
1382:
1380:
1377:
1376:
1370:
1368:
1364:
1359:
1345:
1325:
1322:
1316:
1313:
1307:
1301:
1298:
1286:
1282:
1276:
1270:
1266:
1263:
1259:
1255:
1251:
1247:
1242:
1229:
1224:
1214:
1209:
1197:
1193:
1189:
1184:
1174:
1169:
1157:
1153:
1144:
1143:hydrodynamics
1141:for mass, in
1140:
1130:
1128:
1112:
1108:
1101:
1098:
1092:
1089:
1086:
1081:
1077:
1073:
1068:
1061:
1058:
1046:
1039:
1034:
1020:
1017:
1011:
1007:
1003:
997:
994:
991:
988:
985:
982:
976:
973:
960:
956:
951:
947:
943:
942:perpendicular
939:
923:
920:
917:
880:
861:
858:
855:
852:
849:
846:
843:
840:
834:
831:
821:
820:
819:
817:
813:
809:
804:
778:
775:
733:
729:
725:
721:
717:
713:
709:
704:
691:
683:
680:
663:
659:
655:
647:
644:
636:
631:
627:
623:
617:
614:
603:
594:
587:
584:
581:
578:
574:
571:
568:
567:flow velocity
564:
561:
559:of the fluid,
558:
554:
551:
548:
544:
525:
522:
512:
511:
509:
506:
493:
485:
470:
462:
454:
451:
448:
442:
439:
433:
430:
427:
421:
418:
407:
401:
397:
393:
388:
379:
377:
373:
369:
365:
361:
352:
330:
324:
321:
316:
313:
307:
301:
293:
282:
276:
265:
259:
256:
245:
240:
238:
234:
230:
229:
223:
221:
217:
213:
209:
208:
188:
185:
174:
170:
166:
162:
158:
154:
150:
146:
142:
138:
134:
130:
85:
83:
79:
75:
73:
69:
51:
48:
38:
32:
27:
22:
1602:
1596:
1586:
1582:
1578:
1574:
1571:
1567:
1563:
1559:
1555:
1552:
1548:
1544:
1540:
1537:
1533:
1529:
1527:
1513:
1487:
1478:
1464:
1450:
1433:
1360:
1287:
1280:
1274:
1268:
1264:
1261:
1257:
1253:
1243:
1136:
1041:
1035:
958:
954:
949:
945:
937:
878:
876:
811:
805:
727:
705:
599:
585:
572:
562:
552:
542:
507:
408:
405:
399:
395:
391:
371:
353:
241:
236:
233:mass current
232:
226:
224:
211:
206:
205:
145:unit of time
136:
126:
72:SI unit
808:dot product
732:unit vector
724:vector area
580:vector area
376:steady flow
222:) is used.
163:units, and
133:engineering
1635:Categories
1425:References
218:lowercase
1399:Mass flux
1363:kilojoule
1317:˙
1302:˙
1215:⋅
1194:ρ
1175:⋅
1154:ρ
1102:˙
1090:ρ
1087:⋅
1062:˙
1004:π
998:
986:ρ
977:˙
924:θ
921:
895:^
859:θ
856:
844:ρ
835:˙
788:^
748:^
681:⋅
660:∬
645:⋅
637:ρ
628:∬
618:˙
593:mass flux
582:/surface,
526:˙
486:⋅
463:⋅
455:⋅
452:ρ
443:˙
434:⋅
431:ρ
422:˙
299:Δ
291:Δ
280:→
274:Δ
260:˙
228:mass flux
189:˙
103:−
82:Dimension
52:˙
1621:45791449
1560:variable
1373:See also
1367:kilowatt
816:parallel
730:, and a
716:membrane
153:kilogram
1558:) as a
1514:Physics
950:through
946:doesn't
812:through
557:density
555:= mass
147:. Its
139:is the
129:physics
1619:
1609:
1562:. We
1530:cannot
1520:
1495:
1441:
1338:where
877:where
712:filter
508:where
368:scalar
362:for a
157:second
1133:Usage
936:, as
714:or a
372:after
244:limit
216:Greek
169:pound
1651:Mass
1617:OCLC
1607:ISBN
1583:only
1566:use
1518:ISBN
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165:slug
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149:unit
141:mass
131:and
76:kg/s
1564:can
995:cos
918:cos
853:cos
541:or
358:is
270:lim
231:or
159:in
151:is
127:In
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959:π
955:θ
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892:n
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862:,
850:A
847:v
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785:n
779:A
776:=
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745:n
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684:d
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671:j
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325:t
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