The Determination of Deviation Angles at Exit from the Nozzles of an Inward Flow Radial Turbine

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1 GENTENNIq(r ^,_,. _ r.. /. per opy S1'S01..0 ASME emt#ers ','. 80-GT-147 a THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E 47 St., Ne York, N.Y The Soiety shall not be responsible for statements or opinions advaned in papers or in disussion at meetings of the Soiety or of its Divisions or Setions or printed in its publiations. Disussion is printed only it the n:;per ublished in an ASME Journal or Proeedings Released for general pubirutor vi..n presentation. Full ',^..... redo should be g or.,? I fi T t:rii:alit.^uu.,,nd thi author(.i Copyright 1980 by ASME F. Fairbanks Senior Leturer, The Hatfield Polytehni, Hatfield, Hertfordshire, U.K. The Determination of Deviation Angles at Exit from the Nozzles of an Inard Flo Radial Turbine A method for prediting the deviation angle at exit from the nozzles of an inard flo radial turbine using a streamline urvature method is given. Measurements of flo angles and stagnation and stati pressures aross the nozzle exit span of a small turbine have been made. The auray of the analysis is shon by omparison ith experimental results. Finally, the developed theory has been used to sho the possible effet on the deviation angle at exit from the nozzles of reduing the number of stator blades. NOMENCLATURE A = area m 2 b = blade height m C - veloity m/s C = speifi heat at onstant pressure J/kgK P h = speifi enthalpy J/kg m = distane along streamline m m = mass flo rate kg/s N - number of no^zle blades P = pressure N/m q = distane along quasi-orthogonal m r = radius of urvature of streamline s = speifi entropy J/kgK T = temperature K ' - gas angle measured from radial diretion nozzle setting angle at exit (defined in Fig.1) = angle beteen streamline and x o-ordinate - ratio of prinipal speifi heat apaities = inidene or deviation angle t = nozzle loss oeffiient `F) = polytropi effiieny of diffusion - polytropi effiieny of expansion e - density kg/m 3 1A-N = nozzle setting angle at inlet (defined in Fig.1) Subsripts 0 = stagnation ondition 1 = loation at nozzle inlet 2 = loation at nozzle exit r = radial omponent s = isentropi x = x omponent y - y omponent A symbol surmounted by a bar e.g. P 1 denotes either the mass averaged or spae average value as indiated in the text. INTRODUCTION An aurate method of alulating the effetive flo angle at nozzle exit is neessary in designing an inard flo radial turbine beause of the sensitivity of turbine performane to small hanges in flo angle. Various methods of prediting the deviation have been suggested in the past. Usually these methods have either been empirial (1) or have transformed the nozzles into the equivalent axial asade () and then used axial turbine performane assessmethods (1). A simple approximation to the outlet Contributed by the Gas Turbine Division of The Amerian Soiety of Mehanial Engineers for presentation at the Gas Turbine Conferene & Produts Sho, Ne Orleans, La., Marh 10-13, Manusript reeived at ASME Headquarters January 3, Copies ill be available until Deember 1, flo angle has also been obtained from the osine rule (4). In this rule the outlet flo angle is given by os 1o/s here o is the throat idth and s is the irumferential pith at the trailing edge. This rule suffers from the disadvantage that it gives just a single value of flo angle for all onditions of flo. Experimental evidene has generally been laking although some very detailed results inluding stagnation pressures, Mah numbers and flo angles just upstream and donstream of the nozzles, for a partiular appliation, have been provided (k). In this instane no details of the deviation of the flo at nozzle exit ere inluded. Similarly, there is a sarity of analytial studies of the flo in the nozzle passages. Hoever, the flo in the sroll-nozzle assembly, using a finite element method and assuming inompressible flo, has

2 SIN( No OF NOZZLES=14 been analysed (6). The present paper ombines experimental evidene ith a theoretial predition of the nozzle performane using a streamline urvature method. Streamline urvature analysis using quasi-orthogonals (7,8) has already been used to design inard flo radial turbine rotors ith satisfatory results (Z). EXPERIMENTAL, INVESTIGATION Fig. 1 Nozzle dimensions and pressure tappings The prinipal omponent of the test rig as part of a turboharger. The ompressor as removed leaving only the turbine and the turbine rotor as replaed by a vaneless dummy rotor hih as lamped so that it as unable to rotate. The inlet sroll split the flo into to equal halves, eah half supplying an equal number of nozzles. The nozzle stator blades ere riveted to the nozzle bak plate hih as bolted to the turbine asing. The prinipal dimensions of the nozzles are shon in Fig.1. A valve upstream of the turbine asing ontrolled the pressure to the turbine stator nozzles. The rig as supplied ith filtered, dry air from the laboratory ompressed air main. A flo straightener as fitted to take the sirl out of the air as it left the turbine asing. Instrumentation One of the nozzle passages as seleted as the test nozzle for reording the pressures. Stati pressure tappings in the nozzle bak plate ere provided to reord the pressures in this nozzle passage. These tappings, fifteen in number, ere arranged as shon in Fig.1 to give the pressure distribution through the nozzle passage and also aross the inlet and exit setions. A alibrated 3-hole ya probe hih ould be rotated to fae diretly into the airstream as the air entered the nozzles as fitted to the turbine asing. All the pressure tappings ere onneted to merury filled manometers. The dummy rotor as fitted ith eight pitot pressure probes equally spaed around its periphery as shon in Fig.2. The tip of eah probe oinided ith the nozzle exit radius and the angles of the probes, measured from the radial diretion at the nozzle exit radius, varied from to The probes ere emented into radial holes drilled in the dummy rotor and onneted ith axially drilled holes along the rotor Fig.2 Traversing pitot probe angles hub. These axial holes ere then onneted to merury filled manometers. The probe hih oinided approximately ith the flo diretion as traversed aross the test nozzle passage exit setion, noting the pressure indiated by the manometer at eah measuring station. Probes either side of the first probe ere then traversed aross the exit setion and the manometer readings ere again obtained at the same measuring stations. The pressures obtained at eah measuring station ere then plotted against the probe angle. The gas angle and stagnation pressure ould thus be obtained anyhere along the span. Air mass flo rate as measured ith a standard orifie meter using pipe tappings hih ere onneted to ater filled manometers. The meter as sited donstream of the turbine asing. Inidene Angles Inidene angles at nozzle inlet have been obtained by redution of the experimental data using equations (1) and (2) and by diret measurement using the alibrated 3-hole ya probe

3 CK 1 = os -1 (m ) ( 1) /0 1 A 1 C Y-N ( 2 ) N 0 L The onditions at nozzle inlet, at pressure tapping number 1, have been assumed to be the average and have been used to derive«the alulated values obtained in this ay ere onfirmed by the ya probe. The inidene angles ranged from 17.2 at the loest mass flo rate of 0.38 kg/s to at the highest mass flo rate of kg/s through the turbine. Deviation Angles Deviation angles have been obtained by diret observation and also by redution of the experimental data, using spae averaged quantities and equations (3) and (4). OC = os 1 ( _ m ) ( 3) /' 2A2C2 S2 =«N - «2 (4) The o alulated values using equation (3) ranged from 64.2 at the loest mass flo rate to 63.1 at the highest flo rate. The orresponding inlet stag 2 nation pressures ere N, /m and N/m Thus, the alulated flo angles are greater than the nozzle setting angle at Other investigators (4) have reported the angle of flo as being less than the blade setting angle. This ontradits the present findings. A typial distribution of the diretly observed flo angles aross the exit span is shon in Fig.3. 6 ANGULAR DISTANCE ALONG NOZZLE EXIT SPAN, DEGREES Fig.3 Experimentally observed flo angles at nozzle exit It an be seen that the exit flo angle is not uniform aross the span. The physial interpretation is shon in Fig.4. At positive inidene angles a ake is formed on the nozzle passage loer surfae as shon in Fig.4. The flo at the nozzle passage upper surfae follos Fig.4 Probable flo pattern in nozzle passage more losely the nozzle profile and the diretion of flo beteen the ake and the nozzle passage upper surfae at nozzle exit is ditated by mass flo rate onsiderations. The mass averaged value of diretly observed flo angle for the example shon in Fig.3 as 59.1 and ourred at the highest mass flo rate. This angle is onsiderably less than the value of 63.1 using equation (3). There are several possibilities to explain the disrepany beteen the to results: (1) the mass flo rate through individual nozzle passages as not obtained experimentally beause of the pratial diffiulties involved. Gradients in the flo around the irumferene hoever ould be expeted; (2) the effetive area of flo at nozzle exit as less than the atual area beause of boundary layer groth in the nozzle passage; (3) the presene of the ake on the nozzle passage loer surfae hih has been disussed earlier. The first possibility is less ritial from a design point of vie than numbers (2) and (3) sine an inrease in flo angle in one nozzle ill be ompensated for by a derease in flo angle in another nozzle. On the other hand, all nozzles ill be affeted by boundary layer groth and ake effets and some alloane ould be made for this at the design stage. The presene of the boundary layer hoever ompliates the analysis still further as not only does the effetive area hange, but also the flo beomes three dimensional. The flo angle at inlet o< 1 varied from 56 at the loest mass flo rate to 54.7 at the highest mass flo rate. The orresponding flo angles at exit varied from 58.4 to It is interesting to ompare these results ith those hih ould be obtained in a vaneless spae. For isentropi inompressible flo the flo angle ould remain onstant and pratial tests have shon (9) that for inompressible flo the flo angle dereases through a vaneless spae ith parallel end alls. Hene it ould seem that the nozzled stator performs better than the vaneless stator and that the performane improves as the mass flo rate through the turbine is inreased. Nozzle Loss Coeffiients Using mass averaged values at nozzle exit, values of nozzle loss oeffiient, defined in equation (5), ere found. T2 - T 2s () T O - T 2 _ 2r here T 2s = TO (F2/) The values of nozzle loss oeffiient obtained experimentally ranged from at the loest mass 3

4 flo rate to at the highest mass flo rate. ANALYSIS Derivation of the Veloity Gradient Equation Although the veloity gradient equation has been derived elsehere (1) the analysis is presented so that it applies diretly to the stator nozzles and also in order to inorporate the polytropi effiieny terms. The equations governing to dimensional invisid steady flo are: (a) Momentum. Euler's momentum equations in Cartesian o-ordinates are as follos: Introduing the slope,6 and the radius of urvature r of the streamlines, the veloity gradient equation along the quasi-orthogonal an be put in the form: a dc = 1.dh + (os.dc - C.sin,Q)dx qc d,i dm r dq + r C. os + sin6. dq^ r / dm dq (15) Poiytropi Effiieny A onstant polytropi effieny of expansion may be defined along a streamline here: -1.aP=C.aC + C.aC x y x = /max ax a y e (6) and similarly a polytropi effiieny of diffusion - 1.aP = C x.a + C.^ y may be defined along a streamline here: ay a x a y = dh Along an arbitrary urve of length q: S dh dp =ap.dx +a P. (7) dq a x dq a y dq Combining equations (6) and (7): -1. dp =(C x.)c x + Cy.aCxldx,dq ax ay/dq +(C a +Cya ) x ax a yl dq here dh is the speifi isentropi enthalpy hange and dh is the atual speifi enthalpy hange orresponding to the small pressure hange dp. If the nozzle is adiabati: dh = -C.dC (8) along the atual expansion line and dh = -C.dC s s (b) Energy. along the isentropi expansion line. 2 + h = h (9) Hene 2 dc = dc s loe (16) Differentiating along the urve of length q: C.dC + dh = dh 0 (10) dq dq dq () Entropy. Along the urve of length q: T.ds = dh - 1.dP dq dq yo dq Sine along the quasi-orthogonal the relationship is assumed to be isentropi: (11) dh = 1.dP (12) dq,.o dq Eliminating h from equations (10) and (12): -1.dP = C.dC - dh,o dq dq dq (13) and eliminating P beteen equations (8) and (13) gives the veloity gradient equation along the quasiorthogonal: 4 dc = 1. dh + Cx.a Cx +C.^ Cx dx _ o ( dq C dq 'C a x C a y dq x.a + 2Z a C ax C dy/dq l (14) Similarly if the proess is one of diffusion: dc = ').dc s Sine the nozzle is assumed to be adiabati and there is no ork transfer throughout the nozzle passage: dh = 0 0 The polytropi effiieny is introdued into equation (15) along the streamlines. Hene equation (15) an be ritten for an expansion proess: dc =/os.dc - C.sin/3 dx dq l^ dm r /dq e + C.osA + 1.s=nQ.dC) C r 71 edm dq (1'7) and for a diffusion /proess: (18) dc = (^ os dc - C sin% dx dq \ odm r / dq + (C osa+'j sin,q.ddc)d q here C no approximates to the atual veloity of flo instead of the isentropi variety.

5 I, - Finally equation (19) must be integrated along the quasi-orthogonal are in order to satisfy ontinuity: a N = b r o p. C. do (19) J r Equations of this type an be solved using a digital 8 omputer (Z). Analytial Proedure Equations (18) and (19) are solved along a series of onentri irular ars ith their entres on the turbine axis. For simpliity the ars are spaed so that the radial distanes separating them are equal. Streamlines are positioned approximately, initially, by dividing eah irular are into equal segments. The number of segments hosen depends on the number of streamlines required in the solution. The streamlines, together ith the onentri irular ars, form the grid system a 0 r 0.4 (INLET) _ P/ Pot (EXIT) fpm Input Data The streamline urvature analysis has been tested using some of the experimental data obtained in the pratial tests. Basi design data for evaluation of the flo pattern at exit from the nozzle passage ould inlude mass lo rate of fluid, inlet stagnation =:ress.zre and temperature, as ell as inidene angle of flo at inlet, the prinipal dimensions of the nozzles and the thermodyna-i properties of the fluid. This data, most of hih as obtained from the experimental tests, has been used as input data for the omputer program and the omputed flo pattern has then been ompared ith the pattern obtained experimentally. The Kutta-- Joukoski ondition has been assumed at exit and the deviation angle at exit has been found by trial and error until the K-J ondition has been satisfied. Some of the input variables used in the ase study ere as follos: m = kg/s N =14 P01 = N/m 2Cp J/kgK T 01 = K 6 1 = 15.9 '1) e =9. = 0.95 DISCUSSION Ui RESULTS S= 1.4 Nozzle Inlet Pressures The omputed and observed stati to inlet stagnation pressure ratio along the nozzle inlet span is shon in Fig.5. The effet of several simplifying assumptions hih ere made in the analysis an be seen. At inlet the omputed and experimental urves ross. The omputed pressures, from the nozzle passage loer surfae to approximately tapping number 9, are less than the observed pressures. This is probably due to negleting the radius on the leading tip of the nozzle blade. Beyond pressure tapping number 9, the omputed pressures are higher than the experimentally observed pressures. Nozzle Outlet Pressures The omputed and experimentally observed pressure ratios are also shon in Fig.5. It an be seen that the to urves have the same general form TEST RESULT I ---- COMPUTED RESULT M d 0-2r 6i=15.9' POSITION OF PRESSURE TAPPING No (SEE FIG.1) No 12 No No9 No1No7 No10 No14 No 11 No15 I.1 II II ANGULAR DISTANCE FROM BLADE CENTRE LINE, DEGREES. Fig.5 Pressure distribution along nozzle inlet and exit spans i.e. the pressure is loest at approximately mid span. Probably the greatest ontributory fators to inauray are the neglet of the boundary layer groth on the nozzle blade surfaes and the to nozzle passage end faes and the absene in the alulation, of a ake effet on the nozzle blade. The effet of taking these fators into aount ould be to derease the effetive flo area at outlet and hene inrease the veloity, thereby dereasing the stati pressure. This diffiulty ould be obviated by assuming a boundary layer groth shape and a breakaay shape on the sution surfae. This latter effet ould beome less pronouned as the turbine operates more nearly to the design onditions. Flo Pattern Through the Nozzle Passage The omputed streamline pattern is shon in Fig.6. The program alulates the Mah number at eah grid intersetion point and from this information ontours of onstant Mah number have been obtained. A ompliated flo pattern emerges. The flo an be divided into three regions. At inlet the pattern is very onfused. The flo then settles don before starting to beome more onfused one again in the exit region. Thus it is emphasised that the flo in the nozzle passages an only be effetively analysed on the basis of at least to dimensional flo. The only part of the nozzle passage here there is uniformity of flo is in the middle portion here the 5

6 STREAMLINES 32 N = 14 b2 5 0 Fig.6 Computed flo pattern through the nozzle passage fluid reeives maximum guidane. The omputed angle of flo at mid span of the nozzle exit is 58 0 giving a deviation angle of 5 0. The mass averaged value obtained experimentally as A non-dimensional veloity potential and a non-dimensional veloity distribution through nozzle passages, for both ambered and symmetri nozzle blades, have been provided using finite element analysis assuming inompressible flo (6). Diret omparison of these results ith the results given in the present paper is diffiult sine the nozzle designs are obviously different and also the flo onditions upstream of the nozzles are not knon. Hoever, sine the analytial approah is different from the method adopted in the present paper it ould be interesting to ompare results ith similar designs and flo onditions. Optimum Number of Stator Blades The results from the to dimensional streamline urvature analysis hen the number of blades has been redued from 14 to 12 are shon in Fig.7. The purpose of the nozzle stator blades is to guide the flo so that it enters the rotor at the orret angle Too many blades ill introdue exessive blokage and therefore the ideal solution is to provide the minimum number of blades hih ill, at the same time, provide adequate guidane. It an be seen that the region of settled flo in the middle region has been redued. Hoever possibly the most signifiant result of reduing the number of blades is an inrease in the angle of deviation at exit from the nozzles from 5 to 70 CONCLUSIONS 1. A method of prediting the flo pattern and the deviation of the flo at exit from the stator nozzles of an inard flo radial turbine is presented. A streamline urvature method is used and the solution involves the use of a digital omputer. 2. The omputed stati pressures at nozzle exit are higher than those obtained from the experimental tests. This is due to assuming that the effetive flo area at exit is the same as the nozzle exit area. Alloanes for boundary layer groth and the presene of the ake on the sution surfae of the blade ould improve the auray of the analysis. N=12 62 =7O Fig.7 Effet of number of blades on omputed flo pattern through the nozzle passage 6

7 3. The alulation of the devia!;ion of the flo from the nozzle setting angle gives good agreement ith the experimentally observed value. 4. The analysis an be used to predit the effets on the flo pattern and the deviation of the flo at nozzle exit of varying the number of nozzle blades. It is therefore possible to optimise the number of nozzles. 5. As there is little published information on the design of nozzles for the inard flo radial turbine the investigation makes a ontribution toards orreting this omission. ACKNOWLEDGMENTS The ork as arried out at The Hatfield Polytehni and the author ould like to aknoledge the interest shon in the projet by Mr. R V Watts, Diretor of Studios, and Dr. F S Bhinder, Diretor of Researh, of the Mehanial and Aeronautial Engineering Department, The Hatfield Polytehni. REFERENCES 1. Benson, R.S.,"An Analysis of the Losses in a Radial Gas Turbine," Proeedings of the Institute of Mehanial Engineers, Vol.180, Wisleanus, G.F., Fluid Mhanis of Tu;'bomahinery, MGra-Hill, Ne York, Ainley, D.G.,and Mathieson, G.C.R.,"A Method of Performane Estimation for Axial Flo Turbines," A.R.C.R. and M. 2974, Hiatt, G F, and Johnston, I H, "Experiments Conerning the Aerodynami Performane in Inard Flo Radial Turbines," Proeedings of the Institute of Mehanial Engineers, Vol.174, Khalil, I.M.,Tabakoff, W.,and Earned, A., "Losses in Radial Inflo Turbines," Journal of Fluids Engineering, Trans. ASME, Vol.98, Sept.1976, pp Homed, A.,Baskharone, E.,and Tabakoff, W., "A Flo Study in Radial Inflo Turbine Sroll-Nozzle Assembly," Journal of Fluids Engineering, Trans. ASME, Vol.100, Marh 1978, pp Katsanis, T.,"Use of Arbitrary Quasi- Orthogonals for Calulating Flo Distribution in the Meridional Plane of a Turbomahine," NASA TN D2546, Wilkinson, D.H.,"Calulation of Blade to Bla4e Flo in a Turbomahine by Streamline Curvature," Teh. Report R and M No.3704, 1972, Ministry of Defene Aeronautial Researh Counil, U.K. 9. Benson, R.S.,and Jakson, D.C.,"Flo Studies in Radial Inflo Turbines Interspae Beteen Nozzles and Rotors,"Heat and Fluid Flo in Steam and Gas Turbine Plant. (Conferene Publiation 3), (Institution of Mehanial Engineers, London), 1973, pp