FLOW TURBINE WITH RADIAL TEMPERATURE GRADIENT

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FLOW TURBINE WITH RADIAL TEMPERATURE GRADIENT ( flow-turbine-with-radial-temperature-gradient )

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7 where k is the adiabatic exponent, and where P1 and p1 are the gas pressure and density ahead of the nozzle at the same radius r. Now thatwe know theabsolutevelocityatthechosen radius r, the static temperature T at that radius can be calculated, since the total temperature or stagnation tem perature T, ahead of the nozzle is known from the temperature pro?le of FIG. 2. The static temperature T at the nozzle outlet at the chosen radius 1'may be cal culatedbytheexpression T=T,—- (11) whereJistheheatenergy-mechanicalenergyequivalency factor, and 0D is the constant pressure speci?c heat. Bytheinitial?xingoftangentialMachnumberin versely proportional to radius, we recall from Equation 2 that 8 the spacing between adjacent buckets, or pitch, is pro portional to the radius and hence is greatest at the tip. Hence the “solidity,” which is the ratio of chord to pitch and which is used in the art as an aerodynamic de~ sign parameter related to the bucket entrance and exit angles, is considerably smaller at the tip than at the rootinpriorartdesign. In contrast, in the present invention the bucket angles de?ned by the aforementioned method of velocity dia 10 gram calculation are such as to require greater turning of the ?uid at the tip than at the root, mainly as a con sequence of the reversed entrance angle warp, as illus tratedinFIG.8. Inthiscase,theaerodynamicrequire ments for minimum bucket losses are better suited by Now since <2) (5) 25 Equation 2 may be expressed as _ Vs Tn VH4“ 73X?) And since Equation 12 may be further expressed as It is found that causing the gross area to vary in this manner results in a bucket surface having radial elements whichsubstantiallyconvergeattherotoraxis. Forexam ple, if the bucket cross-section were a circle, the bucket MU=MUR(%) 15 constant “solidity” from root to tip, and hence a chord proportionaltoradius. Inaddition,thebucketmaximum thickness is most suitably made to increase proportional to the chord and hence the radius. This causes the gross cross—sectional area of the bucket to be approximately proportional to the radius squared. The gross area A ispreferablycausedtovarybythepreviouslystated formula where T and TR are the static temperatures of the gas at the nozzle exit at any radius r, and at the root radius rR respectively. 3,135,496 T2 A=AR (3) (12) The increase of gross area with increased distance from (13) rotoraxishasanotherimportanteffectwhichisutilized indesigningthebuckettobeusedwithradialtemperature gradients. When employed in conjunction with the hol 30 would have a frustoconical shape, with the apex of the coneattheaxisofrotation. Ofcourse,theactualbucket shape is much more complicated, but the shape is such that radial line elements in the surface pass substantially throughtheaxisofrotation. 35 lowed-outpocketinthetipofthebucket,thenetcross 40 sectional area AN can be caused to vary in a complex manner (FIG. 12) in order to make the actual radial stressdistributioninthebucketconformcloselytothe variationofallowablestresswithtemperature. Referring to FIG. 10 of the drawing, the curve 35 indicates a typical which accounts for the difference in static temperature 45 variation of allowable stress on a given bucket material We should examine the signi?cance of Equation 13 by comparingitwiththefreevortexEquation1. Itwillbe observed that the form is similar to the free vortex equation, with the addition of a compensating factor, atdifferentradii. Thus,Equation13canbeusedto calculate the absolute tangential velocity component VU at each chosen radius r, since the static temperature T at that radius has been obtained from Equation 11. Now since the total velocity V and the tangential component VU of the total velocity is known, the vector diagram can be completed for the bucket entry condi tions, either graphically or by solving for the axial com ponentVA ofthevelocitybytheexpression Thus it will be seen that knowing the conditions at the root, the bucket entry angles can be calculated at eachradiusrbytheforegoingprocedure. The‘bucket exit angles for the corresponding entry angles may then be calculated, as will be understood by those skilled in the art, by the use of the energy and flow continuity equations. Even though the blade angles have been chosen, there stilremains considerablelatitudeindesigningtheblades, and the cross-sectional shape and dimensions of the blades at each radius must be designed for minimum frictionlossandminimum?owseparation. Inthecase of the prior art, free vortex design, as illustrated in FIG. 6, the amount of turning of the fluid is smaller at the tip than at the root, and to minimize the blade losses, the bucket chord is generally made smaller at thetipthanattheroot. Inaddition,itshouldbeob served that due to the radial extension of the buckets, atdifferenttemperatures. Thecurveindicatesthatfor a particular material, the allowable local stress on the material, which may be expressed as allowable p.s.i, for a constant given rupture life of the material, decreases rather rapidly with increase in temperature, at ?rst, and then less rapidly as the temperature becomes higher. Theoretically, as the temperature reached the melting point of the material, there would be, of course, zero allowable tensile stress, since the metal would be a liquid. 55 Curve 35, as shown in FIG. 10, has been reduced by appropriate safety factors in order to compare it with actual bucket stress, and itmay therefore be assumed that the actual stress on the bucket may be as high as the allow able stress curve 35, but no higher, without exceeding the preselectedsafetyfactor. Under conventional uniform temperature assumptions, the design might be predicated on a preselected root tem— perature, wherein the stress might vary from a maximum allowablestresssuchasC attheroottoazerostresssuch asD atthetip. ItwillbereadilyobservedfromFIG.10 that the bucket material has capability for withstanding higher temperatures, moving from root to tip, without exceeding the allowable stress, than are actually being employed with the uniform temperature design. Suppose now that a radial temperature gradient were provided by the arrangement of FIG. 1 so that the ab scissaofFIG. 10alsorepresentsradiusfromroottotip. Suppose also that the bucket area decreased from root to tip in a manner similar to the conventional free vortex bucketofFIGS.6and7. Theactualstresscalculatedin

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