1931:
1875:
1939:
271:
1993:
1717:
1963:
527:
513:
1988:
This is special case used for impulse turbine which suggest that entire pressure drop in the turbine is obtained in the stator. The stator performs a nozzle action converting pressure head to velocity head. It is difficult to achieve adiabatic expansion in the impulse stage, i.e. expansion only in
1853:
is the stage loading factor. The diagram shows the optimization of total - to - static efficiency at a given stage loading factor, by a suitable choice of reaction. It is evident from the diagram that for a fixed stage loading factor that there is a relatively small change in total-to-static
523:(Figure 2.) for the flow process within the stage represents the change in fluid velocity as it flows first in the stator or the fixed blades and then through the rotor or the moving blades. Due to the change in velocities there is a corresponding pressure change.
111:
343:
163:
690:
203:
1703:
1975:
Stage having reaction more than half suggest that pressure drop or enthalpy drop in the rotor is more than the pressure drop in the stator for the turbine. The same follows for a pump or compressor. From the relation for degree of
332:
1316:
827:
1214:
1804:
1862:
The degree of reaction contributes to the stage efficiency and thus used as a design parameter. Stages having 50% degree of reaction are used where the pressure drop is equally shared by the stator and the
1042:
518:
Where 1 to 3ss in Figure 1 represents the isentropic process beginning from stator inlet at 1 to rotor outlet at 3. And 2 to 3s is the isentropic process from rotor inlet at 2 to rotor outlet at 3. The
169:
In pumps, degree of reaction deals in static and dynamic head. Degree of reaction is defined as the fraction of energy transfer by change in static head to the total energy transfer in the rotor i.e.
1480:
32:
rise in the rotating blades of a compressor (or drop in turbine blades) to the static pressure rise in the compressor stage (or drop in a turbine stage). Alternatively it is the ratio of static
2056:
Shapiro, A. H., Soderberg, C. R., Stenning, A. H., Taylor, E. S. and
Horlock, J. H. (1957). Notes on Turbomachinery. Department of Mechanical Engineering, Massachusetts Institute of Technology.
1951:
Stage having reaction less than half suggest that pressure drop or enthalpy drop in the rotor is less than the pressure drop in the stator for the turbine. The same follows for a pump or
1851:
265:
1562:
508:{\displaystyle \ R={\frac {\int _{3ss}^{2s}{\textrm {dh}}}{\int _{3ss}^{1}{\textrm {dh}}}}\,\ {\textrm {Or}}\,\ {\frac {\int _{3ss}^{2s}{\textrm {dp}}}{\int _{3ss}^{1}{\textrm {dp}}}}}
83:
1418:
1724:
The Figure 3 alongside shows the variation of total-to-static efficiency at different blade loading coefficient with the degree of reaction. The governing equation is written as
1599:
which again depend on geometrical parameters β3 and β2 i.e. the vane angles of stator outlet and rotor outlet. Using the velocity triangles degree of reaction can be derived as:
1597:
1515:
1094:
135:
540:
175:
1373:
1346:
73:, both impulse and reaction machines, Degree of reaction (R) is defined as the ratio of energy transfer by the change in static head to the total energy transfer in the
1605:
2112:
Kim, T. H., Takao, M., Setoguchi, T., Kaneko, K. and Inoue, M. (2001). Performance comparison of turbines for wave power conversion. Int. J. Therm. Sci.,
129:
heat drop in the moving blades (i.e. the rotor) to the sum of the isentropic heat drops in the fixed blades (i.e. the stator) and the moving blades i.e.
284:
1225:
705:
1102:
1989:
the nozzle, due to irreversibility involved, in actual practice. Figure 8 shows the corresponding enthalpy drop for the reaction = 0 case.
1930:
1730:
2123:
214:
Most turbo machines are efficient to a certain degree and can be approximated to undergo isentropic process in the stage. Hence from
2106:
Dunham, J. and Panton, J. (1973). Experiments on the design of a small axial turbine. Conference
Publication 3, Instn. Mech. Engrs.
842:
2082:
1423:
1874:
2103:
Ainley, D. G. and
Mathieson, G. C. R. (1951). A method of performance estimation for axial flow turbines. ARC R. and M.
1708:
This relation is again very useful when the rotor blade angle and rotor vane angle are defined for the given geometry.
106:{\displaystyle R={\frac {\text{Isentropic enthalpy change in rotor}}{\text{Isentropic enthalpy change in stage}}}}
1812:
217:
2167:
1520:
1938:
1992:
1378:
270:
1716:
1567:
1485:
1219:
The degree of reaction can also be written in terms of the geometry of the turbomachine as obtained by
1720:
Figure 3. Influence of reaction on total-to-static efficiency with fixed value of stage loading factor
158:{\displaystyle R={\frac {\text{Isentropic heat drop in rotor}}{\text{Isentropic heat drop in stage}}}}
2162:
685:{\displaystyle \,h_{2}-h_{3}={1 \over {2}}(V_{r3}^{2}-V_{r2}^{2})+{1 \over {2}}(U_{2}^{2}-U_{3}^{2})}
198:{\displaystyle R={\frac {\text{Static pressure rise in rotor}}{\text{Total pressure rise in stage}}}}
2157:
1053:
2069:
Gopalakrishnan, G. and
Prithvi Raj, D., A Treatise on Turbomachines, Scitech, Chennai, India, 2012
1962:
2147:
2115:
1351:
1324:
2120:
2047:
Dixon, S. L., Fluid
Mechanics and Thermodynamics of Turbo-machinery, 5th ed. Elsevier,2011.
2109:
Horlock, J. H. (1960). Losses and efficiencies in axial-flow turbines. Int. J. Mech. Sci.,
1698:{\displaystyle R={\frac {1}{2}}+{\frac {V_{f}}{2U}}(\tan {\beta _{3}}-\tan {\alpha _{2}})}
8:
1887:
2142:
2078:
1908:
520:
2097:
Thomson, W.R., Preliminary Design of Gas
Turbines, Emmott and CO. Ltd., London, 1963
1934:
Figure 5. Stage enthalpy diagram for degree of reaction = 1⁄2 in a turbine and pump.
2127:
2094:
Wisclicenus, G.F., Fluid
Mechanics of Turbomachinery, McGraw-Hill, New York, 1947
1920:
1864:
1048:
74:
56:
327:{\displaystyle R={\frac {\Delta H{\text{ (Rotor)}}}{\Delta H{\text{ (Stage)}}}}}
2152:
1952:
1883:
278:
it is easy to see that for isentropic process ∆H ≃ ∆P. Hence it can be implied
122:
63:
17:
2136:
2100:
Traupel, W., Thermische
Turbomachinen, 3rd Edn, Springer Verlag, Berlin, 1978
2091:
Shepherd, D.G., Principles of
Turbomachinery, Ninth Printing, Macmillan, 1969
1311:{\displaystyle R=({\frac {V_{f}}{2U}})(\tan {\beta _{3}}-\tan {\beta _{2}})}
822:{\displaystyle \,h_{01}-h_{03}=h_{02}-h_{03}=(U_{2}\,V_{w2}-U_{1}\,V_{w1})}
118:
48:
126:
1916:
1209:{\displaystyle R={\frac {(V_{r3}^{2}-V_{r2}^{2})}{2U(V_{w3}+V_{w2})}}}
1878:
Figure 4. Velocity triangle for Degree of
Reaction = 1/2 in a turbine
1915:
gets equally distributed in the stage (Figure 5.) . In addition the
526:
1912:
833:
696:
70:
66:
33:
29:
2013:
Peng, William W., Fundamentals of turbomachinery, John Wiley, 2008
1955:
as shown in Figure 6. From the relation for degree of reaction, |
1799:{\displaystyle R=1+{\frac {\Delta W}{2U^{2}}}-{\frac {C_{y2}}{U}}}
1711:
534:
Another useful definition used commonly uses stage velocities as:
1924:
1868:
44:
274:
Figure 1. Enthalpy vs. Entropy diagram for stage flow in turbine
36:
change in the rotor to the static enthalpy change in the stage.
2033:
S.M, Yahya, Turbines, Compressors and Fans, 4th ed. McGraw,2011
1996:
Figure 8. Stage enthalpy for degree of reaction =0 in a turbine
62:
Various definitions exist in terms of enthalpies, pressures or
52:
2116:
http://www.physicsforums.com/archive/index.php/t-243219.html
1980:| α2 < β3 which is also shown in corresponding Figure 7.
1037:{\displaystyle R={\frac {}{(U_{2}\,V_{w2}-U_{1}\,V_{w1})}}}
2121:
https://www.scribd.com/doc/55453233/18/Degree-of-reaction
2063:
1966:
Figure 7. Velocity triangle for reaction more than 50%.
43:) is an important factor in designing the blades of a
1815:
1733:
1608:
1570:
1523:
1488:
1475:{\displaystyle (\tan {\beta _{3}}-\tan {\beta _{2}})}
1426:
1381:
1354:
1327:
1228:
1105:
1056:
845:
708:
543:
530:
Figure 2. Velocity Triangle for fluid flow in turbine
346:
287:
220:
178:
138:
86:
1942:
Figure 6. Stage enthalpy for Reaction less than half
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1798:
1697:
1591:
1556:
1509:
1474:
1412:
1367:
1340:
1310:
1208:
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1036:
836:drop. The degree of reaction is then expressed as
821:
684:
507:
326:
259:
197:
157:
105:
1886:separation from the blade surface avoiding large
1564:The degree of reaction now depends only on ϕ and
2134:
1375:is the vane angle of stator outlet. In practice
1712:Choice of reaction (R) and effect on efficiency
1903:then from the relation of degree of reaction,|
1919:components are also the same at the inlet of
337:The same can be expressed mathematically as:
2077:. New Delhi: PHI Learning Private Limited.
1846:{\displaystyle {\frac {\Delta W}{2U^{2}}}}
260:{\displaystyle Tds=dh-{\frac {dp}{\rho }}}
1970:
1946:
1557:{\displaystyle R=\phi \tan {\beta _{m}}.}
1014:
987:
802:
775:
709:
544:
434:
423:
2072:
1991:
1961:
1937:
1929:
1873:
1854:efficiency for a wide range of designs.
1715:
525:
269:
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2041:
2039:
2029:
2027:
2025:
2023:
2021:
2019:
2135:
1348:is the vane angle of rotor outlet and
28:is defined as the ratio of the static
2064:Further reading and works referred to
1413:{\displaystyle ({\frac {V_{f}}{2U}})}
2036:
2016:
1911:(Figure 4.) is symmetric. The stage
99:Isentropic enthalpy change in stage
96:Isentropic enthalpy change in rotor
13:
1983:
1819:
1749:
310:
297:
14:
2179:
1592:{\displaystyle \tan {\beta _{m}}}
1510:{\displaystyle \tan {\beta _{m}}}
1857:
2075:Fundamentals of Turbomachinery
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2007:
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125:it is defined as the ratio of
1:
2000:
1882:This reduces the tendency of
1089:{\displaystyle U_{2}=U_{1}=U}
188:Static pressure rise in rotor
151:Isentropic heat drop in stage
148:Isentropic heat drop in rotor
69:of the device. In case of
2073:Venkanna, B.K. (July 2011).
191:Total pressure rise in stage
7:
1977:
1956:
1904:
209:
10:
2184:
1368:{\displaystyle \beta _{2}}
1341:{\displaystyle \beta _{3}}
1420:is substituted as ϕ and
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1971:Reaction more than 50%
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1947:Reaction less than 50%
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2168:Hydraulic engineering
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2126:5 March 2016 at the
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39:Degree of reaction (
1888:stagnation pressure
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1907:| α2 = β3 and the
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22:degree of reaction
2084:978-81-203-3775-6
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2158:Gas compressors
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2128:Wayback Machine
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57:turbo-machinery
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2148:Steam turbines
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971:
968:
963:
958:
954:
950:
945:
940:
936:
932:
926:
922:
917:
914:
909:
904:
901:
897:
893:
888:
883:
880:
876:
872:
866:
862:
857:
851:
848:
830:
829:
818:
813:
810:
806:
799:
795:
791:
786:
783:
779:
772:
768:
764:
761:
756:
752:
748:
743:
739:
735:
730:
726:
722:
717:
713:
693:
692:
681:
676:
671:
667:
663:
658:
653:
649:
645:
639:
635:
630:
627:
622:
617:
614:
610:
606:
601:
596:
593:
589:
585:
579:
575:
570:
565:
561:
557:
552:
548:
516:
515:
492:
487:
484:
481:
477:
462:
459:
454:
451:
448:
444:
410:
405:
402:
399:
395:
380:
377:
372:
369:
366:
362:
355:
352:
335:
334:
315:
312:
302:
299:
293:
290:
254:
250:
247:
241:
238:
235:
232:
229:
226:
223:
211:
208:
207:
206:
184:
181:
167:
166:
144:
141:
115:
114:
92:
89:
18:turbomachinery
9:
6:
4:
3:
2:
2180:
2169:
2166:
2164:
2161:
2159:
2156:
2154:
2151:
2149:
2146:
2144:
2141:
2140:
2138:
2129:
2125:
2122:
2119:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2086:
2080:
2076:
2071:
2068:
2067:
2053:
2044:
2042:
2040:
2030:
2028:
2026:
2024:
2022:
2020:
2010:
2006:
1994:
1990:
1981:
1964:
1960:
1954:
1940:
1932:
1928:
1926:
1922:
1918:
1914:
1910:
1891:
1889:
1885:
1876:
1872:
1870:
1866:
1855:
1835:
1831:
1827:
1822:
1791:
1786:
1783:
1779:
1773:
1765:
1761:
1757:
1752:
1743:
1740:
1737:
1734:
1727:
1726:
1725:
1718:
1709:
1686:
1682:
1677:
1674:
1671:
1665:
1661:
1656:
1653:
1644:
1641:
1635:
1631:
1625:
1620:
1617:
1612:
1609:
1602:
1601:
1600:
1583:
1579:
1574:
1571:
1551:
1545:
1541:
1536:
1533:
1530:
1527:
1524:
1501:
1497:
1492:
1489:
1463:
1459:
1454:
1451:
1448:
1442:
1438:
1433:
1430:
1401:
1398:
1392:
1388:
1360:
1356:
1333:
1329:
1299:
1295:
1290:
1287:
1284:
1278:
1274:
1269:
1266:
1254:
1251:
1245:
1241:
1232:
1229:
1222:
1221:
1220:
1195:
1192:
1188:
1184:
1179:
1176:
1172:
1165:
1162:
1152:
1147:
1144:
1140:
1136:
1131:
1126:
1123:
1119:
1109:
1106:
1099:
1098:
1097:
1083:
1080:
1075:
1071:
1067:
1062:
1058:
1050:
1023:
1020:
1016:
1009:
1005:
1001:
996:
993:
989:
982:
978:
961:
956:
952:
948:
943:
938:
934:
924:
920:
915:
907:
902:
899:
895:
891:
886:
881:
878:
874:
864:
860:
849:
846:
839:
838:
837:
835:
832:is the total
811:
808:
804:
797:
793:
789:
784:
781:
777:
770:
766:
759:
754:
750:
746:
741:
737:
733:
728:
724:
720:
715:
711:
702:
701:
700:
698:
674:
669:
665:
661:
656:
651:
647:
637:
633:
628:
620:
615:
612:
608:
604:
599:
594:
591:
587:
577:
573:
568:
563:
559:
555:
550:
546:
537:
536:
535:
528:
524:
522:
490:
485:
482:
479:
475:
460:
457:
452:
449:
446:
442:
408:
403:
400:
397:
393:
378:
375:
370:
367:
364:
360:
353:
350:
340:
339:
338:
318: (Stage)
313:
305: (Rotor)
300:
291:
288:
281:
280:
279:
272:
268:
252:
248:
245:
239:
236:
233:
230:
227:
224:
221:
182:
179:
172:
171:
170:
142:
139:
132:
131:
130:
128:
124:
120:
90:
87:
80:
79:
78:
76:
72:
68:
65:
60:
58:
54:
50:
46:
42:
37:
35:
31:
27:
23:
19:
2074:
2052:
2009:
1987:
1974:
1950:
1892:
1881:
1861:
1858:50% reaction
1808:
1723:
1707:
1320:
1218:
1046:
831:
694:
533:
517:
336:
277:
213:
168:
116:
61:
40:
38:
25:
21:
15:
119:gas turbine
49:compressors
2137:Categories
2001:References
1976:reaction,|
1953:compressor
127:isentropic
123:compressor
55:and other
1820:Δ
1774:−
1750:Δ
1683:α
1678:
1672:−
1662:β
1657:
1580:β
1575:
1542:β
1537:
1531:ϕ
1498:β
1493:
1460:β
1455:
1449:−
1439:β
1434:
1357:β
1330:β
1296:β
1291:
1285:−
1275:β
1270:
1137:−
1002:−
949:−
892:−
790:−
747:−
721:−
662:−
605:−
556:−
476:∫
443:∫
394:∫
361:∫
311:Δ
298:Δ
253:ρ
240:−
2143:Turbines
2124:Archived
1925:diffuser
1913:enthalpy
1890:losses.
834:enthalpy
697:enthalpy
210:Relation
71:turbines
67:geometry
34:enthalpy
30:pressure
1898:⁄
1869:turbine
1096:, then
695:is the
45:turbine
2081:
1893:If R=
1867:for a
1809:where
1517:giving
1321:where
436:
425:
348:
117:For a
2153:Pumps
1921:rotor
1917:whirl
1865:rotor
77:i.e.
75:rotor
53:pumps
2079:ISBN
1923:and
1047:For
64:flow
1871:.
1675:tan
1654:tan
1572:tan
1534:tan
1490:tan
1482:as
1452:tan
1431:tan
1288:tan
1267:tan
267:,
121:or
24:or
16:In
2139::
2038:^
2018:^
1927:.
755:03
742:02
729:03
716:01
498:dp
468:dp
430:Or
416:dh
386:dh
59:.
51:,
47:,
20:,
2087:.
1978:C
1957:C
1905:C
1900:2
1896:1
1836:2
1832:U
1828:2
1823:W
1792:U
1787:2
1784:y
1780:C
1766:2
1762:U
1758:2
1753:W
1744:+
1741:1
1738:=
1735:R
1693:)
1687:2
1666:3
1651:(
1645:U
1642:2
1636:f
1632:V
1626:+
1621:2
1618:1
1613:=
1610:R
1584:m
1552:.
1546:m
1528:=
1525:R
1502:m
1470:)
1464:2
1443:3
1428:(
1408:)
1402:U
1399:2
1393:f
1389:V
1383:(
1361:2
1334:3
1306:)
1300:2
1279:3
1264:(
1261:)
1255:U
1252:2
1246:f
1242:V
1236:(
1233:=
1230:R
1201:)
1196:2
1193:w
1189:V
1185:+
1180:3
1177:w
1173:V
1169:(
1166:U
1163:2
1158:)
1153:2
1148:2
1145:r
1141:V
1132:2
1127:3
1124:r
1120:V
1116:(
1110:=
1107:R
1084:U
1081:=
1076:1
1072:U
1068:=
1063:2
1059:U
1029:)
1024:1
1021:w
1017:V
1010:1
1006:U
997:2
994:w
990:V
983:2
979:U
975:(
970:]
967:)
962:2
957:3
953:U
944:2
939:2
935:U
931:(
925:2
921:1
916:+
913:)
908:2
903:2
900:r
896:V
887:2
882:3
879:r
875:V
871:(
865:2
861:1
856:[
850:=
847:R
817:)
812:1
809:w
805:V
798:1
794:U
785:2
782:w
778:V
771:2
767:U
763:(
760:=
751:h
738:h
734:=
725:h
712:h
680:)
675:2
670:3
666:U
657:2
652:2
648:U
644:(
638:2
634:1
629:+
626:)
621:2
616:2
613:r
609:V
600:2
595:3
592:r
588:V
584:(
578:2
574:1
569:=
564:3
560:h
551:2
547:h
491:1
486:s
483:s
480:3
461:s
458:2
453:s
450:s
447:3
409:1
404:s
401:s
398:3
379:s
376:2
371:s
368:s
365:3
354:=
351:R
314:H
301:H
292:=
289:R
249:p
246:d
237:h
234:d
231:=
228:s
225:d
222:T
205:.
183:=
180:R
165:.
143:=
140:R
113:.
91:=
88:R
41:R
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
