394:
1385:
1008:
Euler’s pump and turbine equations can be used to predict the effect that changing the impeller geometry has on the head. Qualitative estimations can be made from the impeller geometry about the performance of the turbine/pump.
891:
256:
724:
55:(or the “moment of momentum”) which is fundamental to all turbomachines. Accordingly, the change of the angular momentum is equal to the sum of the external moments. The variation of angular momentum
998:
382:
318:
98:
1065:
148:
604:
580:
552:
520:
492:
460:
432:
1087:
118:
741:
31:. These equations govern the power, efficiencies and other factors that contribute to the design of turbomachines. With the help of these equations the
1142: : 185-232. Here, Euler presents his results for maximizing the outputs of windmills and water wheels, among other means of powering pumps.
35:
developed by a pump and the head utilised by a turbine can be easily determined. As the name suggests these equations were formulated by
1370:
1103:
39:
in the eighteenth century. These equations can be derived from the moment of momentum equation when applied for a pump or a turbine.
1235:
163:
628:
1280:
922:
1295:
1201:
1330:
157:
Since no pressure forces are created on cylindrical surfaces in the circumferential direction, it is possible to write:
324:
263:
58:
1340:
1098:
1365:
1310:
1275:
1132:"Maximes pour arranger le plus avantageusement les machines destinées à élever de l'eau par moyen des pompes"
1228:
1325:
1285:
1265:
1255:
522:
are the relative velocities of the fluid with respect to the blade at the inlet and outlet respectively.
1290:
1426:
1270:
1021:
1421:
1300:
1260:
1165:
1131:
1436:
1431:
1389:
1221:
1360:
1134:(Maxims for arranging most advantageously machines intended to raise water by means of pumps),
618:
Based on Eq.(1.13), Euler developed the equation for the pressure head created by an impeller:
126:
48:
1146:
589:
557:
529:
497:
469:
437:
409:
1147:"Théorie plus complette des machines qui sont mises en mouvement par la réaction de l'eau"
8:
1315:
610:
Figures 'a' and 'b' show impellers with backward and forward-curved vanes respectively.
1335:
1305:
1072:
886:{\displaystyle Y_{th}=1/2(u_{2}^{2}-u_{1}^{2}+w_{1}^{2}-w_{2}^{2}+c_{2}^{2}-c_{1}^{2})}
103:
1411:
1197:
400:
52:
1149:(More complete theory of machines that are set in motion by reaction with water),
1193:
28:
1416:
1345:
1244:
913:
36:
32:
462:
are the absolute velocities of the fluid at the inlet and outlet respectively.
1405:
1355:
1350:
1158:
393:
121:
916:
the static component of the head is zero, hence the equation reduces to:
1170:
Mémoires de l'Académie Royale des
Sciences et des Belles Lettres Ă Berlin
1151:
Mémoires de l'Académie Royale des
Sciences et des Belles Lettres Ă Berlin
1136:
Mémoires de l'Académie Royale des
Sciences et des Belles Lettres Ă Berlin
582:
are the velocities of the blade at the inlet and outlet respectively.
399:
The color triangles formed by velocity vectors u,c and w are called
1108:
1013:
151:
1213:
909: : theoretical head pressure; g: gravitational acceleration
24:
251:{\displaystyle \rho Q(c_{2u}r_{2}-c_{1u}r_{1})=M+M_{\tau }\,}
27:
equations are the most fundamental equations in the field of
719:{\displaystyle Y_{th}=H_{t}\cdot g=c_{2u}u_{2}-c_{1u}u_{1}}
20:
1166:"Recherches plus exactes sur l'effect des moulins Ă vent"
993:{\displaystyle H={1 \over {2g}}(V_{1}^{2}-V_{2}^{2}).\,}
1187:
1075:
1024:
925:
744:
631:
592:
560:
532:
500:
472:
440:
412:
327:
266:
166:
129:
106:
61:
1168:(More exact research on the effect of windmills),
42:
1081:
1059:
992:
885:
718:
598:
574:
546:
514:
486:
454:
426:
376:
312:
250:
142:
112:
92:
1403:
377:{\displaystyle c_{1u}=c_{1}\cos \alpha _{1}.\,}
403:and are helpful in explaining how pumps work.
313:{\displaystyle c_{2u}=c_{2}\cos \alpha _{2}\,}
93:{\displaystyle \rho \cdot Q\cdot r\cdot c_{u}}
1229:
1236:
1222:
1104:List of topics named after Leonhard Euler
989:
571:
543:
511:
483:
451:
423:
373:
309:
247:
16:Turbomachinery Equation in Fluid Dynamics
613:
100:at inlet and outlet, an external torque
51:of mechanics is the conservation of the
1404:
905: : theoretical specific supply; H
1217:
387:
1089:is constant across the rotor blade.
13:
1296:Euler's continued fraction formula
1243:
1157: : 227-295. An analysis of
14:
1448:
1321:Euler's pump and turbine equation
1384:
1383:
1341:Euler equations (fluid dynamics)
1331:Euler's sum of powers conjecture
1188:Johann Friedrich GĂĽlich (2010).
1099:Euler equations (fluid dynamics)
1012:This equation can be written as
392:
43:Conservation of angular momentum
1281:Euler–Poisson–Darboux equation
1181:
1121:
1060:{\displaystyle I=h_{0}-uc_{u}}
983:
947:
880:
772:
225:
173:
1:
1114:
1311:Euler's four-square identity
120:and friction moments due to
7:
1366:Euler–Bernoulli beam theory
1092:
10:
1453:
1379:
1276:Euler–Mascheroni constant
1251:
143:{\displaystyle M_{\tau }}
1326:Euler's rotation theorem
1192:(2nd ed.). Berlin:
1003:
1286:Euler–Rodrigues formula
1266:Euler–Maclaurin formula
1256:Euler–Lagrange equation
599:{\displaystyle \omega }
575:{\displaystyle u_{2}\,}
547:{\displaystyle u_{1}\,}
515:{\displaystyle w_{2}\,}
487:{\displaystyle w_{1}\,}
455:{\displaystyle c_{2}\,}
427:{\displaystyle c_{1}\,}
1361:Euler number (physics)
1291:Euler–Tricomi equation
1083:
1061:
994:
887:
720:
600:
576:
548:
516:
488:
456:
428:
378:
314:
252:
144:
114:
94:
1301:Euler's critical load
1271:Euler–Maruyama method
1084:
1062:
995:
888:
721:
614:Euler's pump equation
601:
577:
549:
517:
489:
457:
429:
379:
315:
253:
145:
115:
95:
1261:Euler–Lotka equation
1073:
1022:
923:
742:
629:
606:is angular velocity.
590:
558:
530:
498:
470:
438:
410:
325:
264:
164:
127:
104:
59:
982:
964:
879:
861:
843:
825:
807:
789:
49:Newton's second law
1079:
1057:
990:
968:
950:
912:For the case of a
883:
865:
847:
829:
811:
793:
775:
716:
596:
572:
544:
512:
484:
452:
424:
401:velocity triangles
388:Velocity triangles
374:
310:
248:
140:
110:
90:
1399:
1398:
1203:978-3-642-12823-3
1190:Centrifugal Pumps
1176: : 166-234.
1082:{\displaystyle I}
945:
113:{\displaystyle M}
47:A consequence of
1444:
1427:Ventilation fans
1387:
1386:
1316:Euler's identity
1238:
1231:
1224:
1215:
1214:
1208:
1207:
1185:
1179:
1125:
1088:
1086:
1085:
1080:
1066:
1064:
1063:
1058:
1056:
1055:
1040:
1039:
999:
997:
996:
991:
981:
976:
963:
958:
946:
944:
933:
892:
890:
889:
884:
878:
873:
860:
855:
842:
837:
824:
819:
806:
801:
788:
783:
768:
757:
756:
725:
723:
722:
717:
715:
714:
705:
704:
689:
688:
679:
678:
657:
656:
644:
643:
605:
603:
602:
597:
581:
579:
578:
573:
570:
569:
553:
551:
550:
545:
542:
541:
521:
519:
518:
513:
510:
509:
493:
491:
490:
485:
482:
481:
461:
459:
458:
453:
450:
449:
433:
431:
430:
425:
422:
421:
396:
383:
381:
380:
375:
369:
368:
353:
352:
340:
339:
319:
317:
316:
311:
308:
307:
292:
291:
279:
278:
257:
255:
254:
249:
246:
245:
224:
223:
214:
213:
198:
197:
188:
187:
149:
147:
146:
141:
139:
138:
119:
117:
116:
111:
99:
97:
96:
91:
89:
88:
53:angular momentum
1452:
1451:
1447:
1446:
1445:
1443:
1442:
1441:
1422:Gas compressors
1402:
1401:
1400:
1395:
1375:
1336:Euler's theorem
1306:Euler's formula
1247:
1242:
1212:
1211:
1204:
1194:Springer-Verlag
1186:
1182:
1126:
1122:
1117:
1095:
1074:
1071:
1070:
1051:
1047:
1035:
1031:
1023:
1020:
1019:
1006:
977:
972:
959:
954:
937:
932:
924:
921:
920:
908:
904:
874:
869:
856:
851:
838:
833:
820:
815:
802:
797:
784:
779:
764:
749:
745:
743:
740:
739:
710:
706:
697:
693:
684:
680:
671:
667:
652:
648:
636:
632:
630:
627:
626:
616:
591:
588:
587:
565:
561:
559:
556:
555:
537:
533:
531:
528:
527:
505:
501:
499:
496:
495:
477:
473:
471:
468:
467:
445:
441:
439:
436:
435:
417:
413:
411:
408:
407:
390:
364:
360:
348:
344:
332:
328:
326:
323:
322:
303:
299:
287:
283:
271:
267:
265:
262:
261:
241:
237:
219:
215:
206:
202:
193:
189:
180:
176:
165:
162:
161:
154:or a diffuser.
134:
130:
128:
125:
124:
105:
102:
101:
84:
80:
60:
57:
56:
45:
17:
12:
11:
5:
1450:
1440:
1439:
1437:Leonhard Euler
1434:
1432:Fluid dynamics
1429:
1424:
1419:
1414:
1397:
1396:
1394:
1393:
1380:
1377:
1376:
1374:
1373:
1368:
1363:
1358:
1353:
1348:
1346:Euler function
1343:
1338:
1333:
1328:
1323:
1318:
1313:
1308:
1303:
1298:
1293:
1288:
1283:
1278:
1273:
1268:
1263:
1258:
1252:
1249:
1248:
1245:Leonhard Euler
1241:
1240:
1233:
1226:
1218:
1210:
1209:
1202:
1180:
1178:
1177:
1162:
1159:Segner's wheel
1143:
1119:
1118:
1116:
1113:
1112:
1111:
1106:
1101:
1094:
1091:
1078:
1054:
1050:
1046:
1043:
1038:
1034:
1030:
1027:
1005:
1002:
1001:
1000:
988:
985:
980:
975:
971:
967:
962:
957:
953:
949:
943:
940:
936:
931:
928:
914:Pelton turbine
906:
902:
899:
898:
897:
896:
895:
894:
882:
877:
872:
868:
864:
859:
854:
850:
846:
841:
836:
832:
828:
823:
818:
814:
810:
805:
800:
796:
792:
787:
782:
778:
774:
771:
767:
763:
760:
755:
752:
748:
732:
731:
730:
729:
728:
727:
713:
709:
703:
700:
696:
692:
687:
683:
677:
674:
670:
666:
663:
660:
655:
651:
647:
642:
639:
635:
615:
612:
608:
607:
595:
584:
583:
568:
564:
540:
536:
524:
523:
508:
504:
480:
476:
464:
463:
448:
444:
420:
416:
389:
386:
385:
384:
372:
367:
363:
359:
356:
351:
347:
343:
338:
335:
331:
320:
306:
302:
298:
295:
290:
286:
282:
277:
274:
270:
259:
244:
240:
236:
233:
230:
227:
222:
218:
212:
209:
205:
201:
196:
192:
186:
183:
179:
175:
172:
169:
137:
133:
122:shear stresses
109:
87:
83:
79:
76:
73:
70:
67:
64:
44:
41:
37:Leonhard Euler
29:turbomachinery
15:
9:
6:
4:
3:
2:
1449:
1438:
1435:
1433:
1430:
1428:
1425:
1423:
1420:
1418:
1415:
1413:
1410:
1409:
1407:
1392:
1391:
1382:
1381:
1378:
1372:
1369:
1367:
1364:
1362:
1359:
1357:
1356:Euler numbers
1354:
1352:
1349:
1347:
1344:
1342:
1339:
1337:
1334:
1332:
1329:
1327:
1324:
1322:
1319:
1317:
1314:
1312:
1309:
1307:
1304:
1302:
1299:
1297:
1294:
1292:
1289:
1287:
1284:
1282:
1279:
1277:
1274:
1272:
1269:
1267:
1264:
1262:
1259:
1257:
1254:
1253:
1250:
1246:
1239:
1234:
1232:
1227:
1225:
1220:
1219:
1216:
1205:
1199:
1195:
1191:
1184:
1175:
1171:
1167:
1164:Euler (1756)
1163:
1160:
1156:
1152:
1148:
1145:Euler (1754)
1144:
1141:
1137:
1133:
1130:Euler (1752)
1129:
1128:
1124:
1120:
1110:
1107:
1105:
1102:
1100:
1097:
1096:
1090:
1076:
1067:
1052:
1048:
1044:
1041:
1036:
1032:
1028:
1025:
1017:
1015:
1010:
986:
978:
973:
969:
965:
960:
955:
951:
941:
938:
934:
929:
926:
919:
918:
917:
915:
910:
875:
870:
866:
862:
857:
852:
848:
844:
839:
834:
830:
826:
821:
816:
812:
808:
803:
798:
794:
790:
785:
780:
776:
769:
765:
761:
758:
753:
750:
746:
738:
737:
736:
735:
734:
733:
711:
707:
701:
698:
694:
690:
685:
681:
675:
672:
668:
664:
661:
658:
653:
649:
645:
640:
637:
633:
625:
624:
623:
622:
621:
620:
619:
611:
593:
586:
585:
566:
562:
538:
534:
526:
525:
506:
502:
478:
474:
466:
465:
446:
442:
418:
414:
406:
405:
404:
402:
397:
395:
370:
365:
361:
357:
354:
349:
345:
341:
336:
333:
329:
321:
304:
300:
296:
293:
288:
284:
280:
275:
272:
268:
260:
242:
238:
234:
231:
228:
220:
216:
210:
207:
203:
199:
194:
190:
184:
181:
177:
170:
167:
160:
159:
158:
155:
153:
135:
131:
123:
107:
85:
81:
77:
74:
71:
68:
65:
62:
54:
50:
40:
38:
34:
30:
26:
22:
1388:
1351:Euler method
1320:
1189:
1183:
1173:
1169:
1154:
1150:
1139:
1135:
1123:
1068:
1018:
1016:invariance:
1011:
1007:
911:
900:
617:
609:
398:
391:
156:
46:
18:
1406:Categories
1115:References
150:act on an
19:The Euler
1371:Namesakes
1042:−
966:−
863:−
827:−
791:−
691:−
659:⋅
594:ω
362:α
358:
301:α
297:
243:τ
200:−
168:ρ
136:τ
78:⋅
72:⋅
66:⋅
63:ρ
1412:Turbines
1390:Category
1109:Rothalpy
1093:See also
1014:rothalpy
152:impeller
25:turbine
1200:
1069:where
258:(1.13)
1417:Pumps
1127:See:
1004:Usage
1198:ISBN
554:and
494:and
434:and
33:head
23:and
21:pump
893:(2)
726:(1)
355:cos
294:cos
1408::
1196:.
1174:12
1172:,
1155:10
1153:,
1138:,
903:th
1237:e
1230:t
1223:v
1206:.
1161:.
1140:8
1077:I
1053:u
1049:c
1045:u
1037:0
1033:h
1029:=
1026:I
987:.
984:)
979:2
974:2
970:V
961:2
956:1
952:V
948:(
942:g
939:2
935:1
930:=
927:H
907:t
901:Y
881:)
876:2
871:1
867:c
858:2
853:2
849:c
845:+
840:2
835:2
831:w
822:2
817:1
813:w
809:+
804:2
799:1
795:u
786:2
781:2
777:u
773:(
770:2
766:/
762:1
759:=
754:h
751:t
747:Y
712:1
708:u
702:u
699:1
695:c
686:2
682:u
676:u
673:2
669:c
665:=
662:g
654:t
650:H
646:=
641:h
638:t
634:Y
567:2
563:u
539:1
535:u
507:2
503:w
479:1
475:w
447:2
443:c
419:1
415:c
371:.
366:1
350:1
346:c
342:=
337:u
334:1
330:c
305:2
289:2
285:c
281:=
276:u
273:2
269:c
239:M
235:+
232:M
229:=
226:)
221:1
217:r
211:u
208:1
204:c
195:2
191:r
185:u
182:2
178:c
174:(
171:Q
132:M
108:M
86:u
82:c
75:r
69:Q
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