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5 radialivariation of bucket entry and exit angles from root to tip is derived, however, not from the free vortex principle, but from the relation MU=MUR(%) (2) whereMU istheangentialMachnumberatanyradiusr fromtheaxisofrotation,andwhereMmisthetangential MachnumberoftherootradiusrR. ThetangentialMach number is de?ned as the tangential component of the absolute gas velocity VU leaving the nozzle, divided by the local sonic velocity of the gas Vs at the same loca tion. 6 The root pro?le 30 produces an absolute gas velocity 0R1 whilethetippro?le31producesabsolutegasvelocity 0T1. Theaxialandtangentialcomponentsoftheseabso lute gas velocities leaving the nozzle are indicated by the quantities VA and VU respectively, with additional sub— scriptstoindicatetheradius. It will again be observed by those having a knowledge of the usual con?guration of nozzle partitions that the twist or warp of the nozzle partitions of FIG. 5 isreversed from the conventional warp. In other words, the angle subtended between the direction of leaving gas and the tangential line is greater at the nozzle root than at the nozzletip. Thisisaconsequenceofdesigningaccording to the disclosed formula wherein the tangential Mach 3,135,496 It will be observed from FIG. 4 that the relative gas velocityatthebuckettipWTT1indicatesentryangleof15 numbervariesinverselywiththeradius,andWherethe the gas with respect to the entry angle of the gas at the root WRRI, which is just the reverse of the relationship shown in FIG. 3. ReferencetoFIGS.8and9showsplanandelevation temperature is signi?cantly higher toward the tip. An explanation of the design criteria of the invention -as opposed to the constant temperature free vortex design isasfollows. Thefreevortexprincipleachievesradial equilibrium of the fluid by means of certain assumptions, togetherwiththerequirementthattheangularmomentum of the ?uid is constant over the entry plane at any radius. However,amoregeneralexpressionforradialequilibrium canbe.derivedfromEuler’sequationfor?uid?owwhich, metry, and zero external force, reduces to the expression dP pVU2 w—g—r (4) where P isthe staticpressure, pisthegasdensity g is the gravitational constant and the other symbols are as indicated previously. As mentioned previously, the tangential Mach number MU is related to the tangential component of the gas leaving the nozzle VU and the local sonic velocity Vs atthesamepointbytheexpression Vu Recognizing the fact that for a compressible ?uid where vS=\/kgR.T (6). P=RgTp (7)v Equation 4 reduces to the following expression: 11F (1' -P-=kMU2(7) (8) views respectively of a bucket constructed corresponding tothevectordiagramofFIG.4. Therethebucket20is disposed on a base 21 and has a tip pro?le indicated by the closed line 22 and a root pro?le, indicated by the closedline23. Themeandirectionofentryofthegas relativetothebucketattherootisindicatedbyline24 25 byassumingradialequilibrium,steadystate,axialsym and the corresponding direction of entry at the tip by line 25. Similarly,themeandirectionofexitofthegasrela tive to the bucket at the root is indicated by line 26 and atthetipbyline27. Line24correspondstoFIG.4 vector WRR1, line 25 corresponds to vector WTT1, line 26 corresponds to vector WRR2, and line 27 corresponds tovectorWTT2 inFIG. 4. As seen in FIG. 9, the gross cross-sectional area of the bucketiscausedtoincreasewiththeradius. Thegross cross-sectional‘ area is de?ned as the area “seen” by the gasatanyradius. Intheembodimentshown,thegross cross-sectional area is caused to vary as given by the ex pression T2 A=1411(5) (3) 40 were A is the gross cross-sectional area at any radius r, andwhereAR isthegrosscross-sectionalareaattheroot radius rR. Also, for reasons which will be later explained, the net oractualcross-sectionalareaofbucketmaterialiscaused 45 and to vary as a more complex function of the radius by hol lowing out the tip portion of the bucket as indicated by recess 28, thus forming an empty pocket in the tip of the bucket. FIG.12showsagraphofnetcross-sectionalarea variation from root to tip as compared to a conventional bucket. Theareaisnormalizedbyreferringittoaroot areaofunityforbothbuckets. The actual stress on the bucket at various radii depends, in part, upon the radial distribution of bucket material which is measured at any given radius by the net cross-' sectionalarea. ItwillbeobservedfromFIGS.9and12 that the net cross-sectional area increases with the radius until the pocket 28 is reached (due to the gross area varia tion), whereafter the net cross-sectional area decreases withtheradius,theprecisevariationofnetcross-sectional area being shown in FIG. 12. , Referring to FIG. 5 of the drawing, two ‘ofthe nozzle partitions 2 are shown, which provide the absolute gas velocities 0T1 and CR1 which are used to enter the velocitydiagramofFIG.4. Therootpro?lesofnozzle partitions 2 are indicated by contoured line 30, whereas thetippro?lesareindicatedbycontouredline31. As explained previously, the absolute gas velocity 0R1 leav ing the nozzle at the root is calculated to be used as a reference by making assumptions based on previous ex perienceandideal?ow. Thentheremainderofthenozzle exit angles at other radii are calculated according to the formula Now for the special case where Equation 2 applies, i.e., tangential Mach number inversely proportional to radius, the solution for Equation 8 which gives the static pressure P at any radius r is as follows: P kMUR2 'r2 PR_€ 2 l1 (a) l (9) This expression gives the pressure P at any radius r related to the pressure PR at the root radius rR, which is necessary to achieve radial equilibrium of the gas and which is true regardless of the temperature distribu tion of the gas. Hence, Equation 9 is applicable to gas with a slanted temperature pro?le, such as given by the graph of FIG. 2. The conditions ahead of the nozzle can be measured and, since the desired static pressure P at a given radius r at the nozzle exit can be calculated from Equation 9, thetotalvelocityV atthesameradiuscanbecalculated from standard graphs or from the known expression re lating the total absolute velocity to the pressure drop across the nozzle as follows: MU=MU~R(%) (2)75 (10)PDF Image | FLOW TURBINE WITH RADIAL TEMPERATURE GRADIENT
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