THE DESIGN SPACE BOUNDARIES FOR HIGH FLOW CAPACITY CENTRIFUGAL COMPRESSORS. k = impeller inlet shape factor, k r 2

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1 Proceeding of ASE Turbo Expo 0 GT0 June -5, 0, Copenhagen, Denmark GT THE DESIGN SPACE BOUNDARIES FOR HIGH FLOW CAPACITY CENTRIFUGAL COPRESSORS Daniel Ruch Compreor Development (Dept. ZTE) ABB Turbo Sytem Ltd. Bruggertrae 7a CH-540 Baden, Switzerland ichael Caey Intitute of Thermal Turbomachinery (ITS) Univerity of Stuttgart, Germany and PCA Engineer Limited, Lincoln, England ABSTRACT A methodology ha been derived allowing a fat preliminary aement of the deign of centrifugal compreor pecified for high pecific wallowing capacity. The method i baed on one-dimenional (D) deign point value uing claical turbomachinery analyi to determine the inlet geometry for the maximum ma flow function. The key reult are then expreed in a erie of diagram which draw out the nature of the conflicting boundary condition of the deign. In particular it i hown how the inlet caing relative ach number caue the deign flow coefficient to decreae with the total preure ratio and determine the inlet eye diameter. Phyically-baed boundarie of operation are added to the diagram giving guideline for the proper choice of pecification value to the deigner. In addition, link are given to ome well-known impeller efficiency correlation, o that a preliminary etimate of the performance can be made. Comparion are made with a range of compreor data which upport the approach. The derived methodology allow any given pecification to be checked rapidly for feaibility and development rik or can be ued to define a challenging pecification for the deign of a new product. NOENCLATURE A = area of compreor eye (m ) a = peed of ound (m/) c = abolute flow velocity (m/) c = pecific heat at contant preure (J/kg/K) p D = diameter (m) h = pecific enthalpy (J/kg) k = impeller inlet hape factor, k r h r c (-) = abolute ach number at impeller inlet, c c m RT (-) = relative ach number at impeller inlet caing w RT (-) u = tip-peed ach number, u u RT (-) m = ma flow rate (kg/) p = preure (Pa) R = ga contant (J/kg/K) r = radial location (m) = pecific entropy (J/kg/K) T = temperature (K) u = impeller blade peed (m/) V = volume flow rate, V m (m /) V T ref = corrected volume flow rate, V T T (m /) VT ref ref w = relative flow velocity (m/) Greek Symbol = relative flow angle ( ) at caing of impeller inlet, relative to axial direction = ientropic exponent (-) = work input or enthalpy rie coefficient, h t u (-) = ientropic efficiency, total to total, h (-) t h t Copyright 0 by ASE

2 p = polytropic efficiency, (-) = global flow coefficient, m u (-) t t D = ma flow function (equation (7)) = modified ma flow function (equation (7)) = tage total preure ratio (-) = denity (kg/m ) = pecific peed (-) = angular velocity (rad/) Subcript t = total condition = inlet total condition = inlet and inlet condition h = inlet hub m = inlet meridional = inlet caing (hroud) = impeller outlet and outlet condition ref = reference value INTRODUCTION A common trend in centrifugal compreor development i to trive for more compact product and hence for tage with a high pecific wallowing capacity. ore compact tage have a lower weight and inertia, a maller frontal area and a reduced cot. On the other hand, the compactne conflict with the requirement for high preure ratio and efficiency and with the mechanical limit of the impeller material. A uitable compromie ha to be found in order to balance all requirement and to end up with a realizable pecification. In turbocharger and indutrial compreor, a maller ize reult in lower material and machining cot and lower weight of the product. Lower weight and dimenion are alo favorable for the overall cot of the machine and the upporting tructure and alo for the natural frequencie of the machine, provided they do not compromie fuel conumption. The correponding lower rotor inertia i beneficial for fat acceleration and repone of the compreor which i a requirement during engine load increae in turbocharger application. In aircraft engine application, a high wallowing capacity of the centrifugal compreor alo reult in a frontal area reduction. Thi help to decreae the aerodynamic drag on the nacelle and hence reduce fuel conumption. In addition, the lower engine weight directly reduce the fuel conumption a the required aircraft lift decreae. The wallowing capacity alo trongly influence the thermodynamic performance of the compreor. Rodger [] ha preented variou correlation of the efficiency for tage teted in air a a function of pecific peed, defined a follow V () 4 with h t 4 V, u and h D D u t h ot of Rodger publihed correlation are baed on impeller efficiency, but an example of one preented in term of tage efficiency i given in Figure, taken from Rodger []. Thi correlation ugget that an optimum pecific peed exit for a given tage preure ratio and how, by the dahed line, that at higher preure ratio a high inducer ach number i required. The background to thi will be examined in more detail in thi paper. Figure : Dependence of tage adiabatic efficiency on the dimenionle pecific peed ( N ), preure ratio and impeller inlet caing relative ach number ( given by Rodger, [] for high preure ratio tage. t u w ) Figure : Dependence of compreor polytropic efficiency on the flow coefficient and tip-peed ach number ( u u ) derived by Caey and Robinon, []. Copyright 0 by ASE

3 In more recent work, Caey and Robinon [] and Robinon et al. [4] reformulated the Rodger correlation, together with thoe from other ource, in term of the polytropic compreor efficiency a a function of the compreor inlet flow coefficient and the tip-peed ach number a hown in Figure. The background and equation to thi correlation are given in the appendix. The inlet flow coefficient i ometime known a wallowing capacity coefficient or gulp factor but will be referred to here a imply flow coefficient, and the flow coefficient and tip-peed ach number are defined a V m () D u D u u u () RT Thi correlation how that an optimum flow coefficient can be found for a given tip-peed ach number and that the optimum wallowing capacity reduce for a higher tip-peed ach number level. The difference to the Rodger correlation are dicued below. Figure : Dependency of ientropic efficiency on the polytropic efficiency and total preure ratio according to equation (4). The meaure of efficiency predicted by thi correlation i the polytropic efficiency, a defined in equation (4) below, a it i a more uitable meaure of the aero-thermodynamic quality of the deign than the ientropic efficiency, ee Dixon [5]. A a reult of thi, the effect of tip-peed ach number on efficiency appear to be much weaker in Figure than the effect of preure ratio in Figure. Thi thermodynamic effect arie becaue for a contant value of the mall-cale polytropic efficiency the ientropic efficiency fall with the preure ratio a hown in Figure and according to, ee [6]: (4) p The efficiency i correlated a a function of the tip-peed ach number, rather than the preure ratio, a it then become thermodynamically correct for different gae. Different preure ratio occur at the ame ach number level if the ga i changed, a the preure ratio in an ideal ga i given by /( ) ( ) (5) In addition the pecific peed in Rodger correlation, which include flow capacity and head rie in it definition, ee equation (), i replaced by the flow coefficient a a meaure of the non-dimenional wallowing capacity. The jutification for thi i that two tage with the ame non-dimenional flow capacity, but a different non-dimenional head rie, would have different value of the pecific peed, o the pecific peed i clearly not a good meaure of wallowing capacity alone. In fact the head rie i determined mainly by the deign of the impeller outlet (backweep) and the wallowing capacity by the impeller inlet (throat, inlet eye diameter) o the pecific peed confue thee independent feature. In order to avoid thi confuion the pecific peed need to be ued together with a imilar non-dimenional parameter, known a the pecific diameter, and then it ue would be acceptable. Unfortunately mot turbomachinery work doe not pecify both of thee parameter. Other diadvantage of pecific peed are it complexity, in that it include the quare root of the flow capacity, a thi alo mean the value of the pecific peed ha no direct link to the continuity equation. Depite it diadvantage, many publication on centrifugal compreor make ue of the pecific peed rather than the inlet flow coefficient to characterize the non-dimenional wallowing capacity of a centrifugal compreor. It hould be noted however that the ue of a pecific peed i only reaonable for comparing wallowing capacity in thoe cae, like centrifugal compreor, where the non-dimenional head coefficient remain enibly contant over the range of deign being conidered. The efficiency correlation decribed here, and alo other uch a that of Aungier [7], how that there i an optimum non-dimenional wallowing capacity, and that the efficiency decreae toward higher or lower value. In addition, the optimum non-dimenional wallowing capacity i reduced with tip-peed ach number or preure ratio. Thi behavior i the ubject of thi paper. The underlying phyical mechanim for thi increae in loe at high flow capacity and tip-peed ach number are manifold, but are fundamentally related to the need for a higher inlet area with a larger impeller eye diameter for higher non-dimenional flow capacity. Firtly, the ue of a higher u Copyright 0 by ASE

4 impeller inlet eye diameter at high flow, or higher trim, lead to le centrifugal effect along the caing treamline of the impeller and more diffuion i then needed to achieve the ame preure rie with a potential for higher loe. Secondly, the higher trim give an increaingly harp turning of the axial inflow toward the radial direction in the impeller and thi can caue higher hub to hroud loading with an aociated increae in econdary flow loe. Thirdly, limiting the eye diameter to avoid thee problem caue higher axial flow velocitie in the impeller inlet with higher frictional flow loe and kinetic energy leaving loe at high wallowing capacity. Finally, the increaed impeller inlet caing relative ach number aociated with a high eye diameter or a high axial inlet velocity caue poible hock loe and a requirement for a higher diffuion level in the impeller. Hence, triving for more compact centrifugal compreor by increaing the wallowing capacity can reult in inufficient tage performance. The deign of a centrifugal compreor for high flow capacity inevitably require a careful balance between the compactne and efficiency, epecially in high preure ratio tage with high ach number level. OBJECTIVE A method for a rapid aement of thermodynamic performance and compactne requirement i preented in thi paper. A one-dimenional teady (D) analyi i ued and combine tage preure ratio, wallowing capacity, impeller inlet caing relative ach number, tip-peed ach number and the dimenionle impeller eye diameter. It i hown how thee dimenionle number are related and can be diplayed in one diagram. The new method i baed on the ma flow function, ee Dixon [5], which i derived in the next chapter. Baed on the ma flow function, the wallowing capacity and the dimenionle impeller eye diameter are expreed a a function of ga and impeller inlet caing flow propertie, tage efficiency, work input parameter, impeller inlet hape factor and total preure ratio. A method i then decribed to provide a ingle plot for deign parameter aement. A new feature of thi work i that thi plot link the impeller inlet eye dimenion to the overall impeller dimenion and the overall preure ratio. The plot can be directly linked with efficiency correlation both on the bai of pecific peed and flow coefficient. THE ASS FLOW FUNCTION For highly loaded centrifugal compreor with high wallowing capacity it i neceary to limit the inlet relative ach number at the caing inlet. Stage deigned for other application may require other deign trategie. In tranonic inducer, the entropy generation due to the hock ha to be controlled for high efficiency tage. The lo directly related to the entropy production acro the hock i mall but increae teeply with the uptream ach number roughly a [8]: (6) ore importantly, the preure rie at the hock can caue boundary layer eparation and be a caue of further loe. According to Bölc, [9] eparation due to hock interaction in an axial compreor cacade occur if the hock uptream ach number exceed approximately.5. The eparated boundary layer itelf caue additional loe due to the related econdary flow and mixing loe. Careful deign of the impeller blade profile at inlet i required for higher relative uperonic inlet ach number, ee Lohmberg et al. [0]. It i well known that an optimum inlet blade metal angle at the impeller inlet caing exit which minimize the relative flow ach number at the impeller inlet caing for a given ma flow, rotational peed and inlet total condition. A derivation of the governing equation can be found in Dixon, [5] and a ueful dicuion can be found in Lohmberg et al. [0]. Additional relevant and imilar derivation can be found in Whitfield and Baine [] and in Stanitz []. The et of equation for zero inlet wirl i derived here in a lightly different form uing a clearer definition of the relevant nondimenional parameter. We define the ma flow function of the tage a the ma flow relative to that which can pa through an area of D with a ga velocity equal to the inlet total peed of ound with the denity at inlet total condition a m m u u (7) a D u D a Axi of rotation Figure 4: Velocity triangle at impeller inlet caing. Thi formulation of the ma flow function differ to that ued by Dixon. Thi i conitent with the conventional definition of reduced ma flow and replacing the denity from the ideal ga relation and peed of ound by the uual ideal ga equation lead to the uual definition of the non-dimenional ma flow function a m RT /( D p t ). In thi form, however, the equation immediately highlight the phyic that a high ma w c m u 4 Copyright 0 by ASE

5 flow function i related to a combination of both a high flow coefficient and a high tip-peed ach number. Note alo that the correlation of Caey and Robinon [] given in Figure and the appendix ue the product of flow coefficient and tippeed ach number for the effect of ach number on efficiency. If we ubtitute the ma flow at the impeller inlet from the continuity equation a m Ac m the ma flow function can be expreed a follow: Ac m (8) ad Following Dixon, the inlet area of the compreor eye i expreed uing an impeller inlet hape factor k which repreent the hub blockage area: A ( r r h ) k( / 4) D with (9) k r h r and D / D u / u For uniform axial inflow, the velocity triangle yield c m co and u in (0) where c m denote the inlet axial abolute velocity, w the relative flow velocity and the angle between the relative and abolute flow at the impeller inlet caing, ee Figure 4. The axial velocity i aumed to be uniform over the inlet area. Auming zero incidence angle, which i realitic at high tranonic ach number, the angle correpond to the blade metal angle at the impeller inlet caing. With the above equation, the ma flow function can now be expreed a follow: k in co 4 a tu () a a k in co 4 a a u The definition of the relative and abolute inlet ach number c c m a co a co with () w a are ued to rewrite the ma flow function a: a k in co () 4 a u In the next tep, the tatic flow quantitie at the impeller inlet are replaced by the total one uing the following expreion /( ) c / (4) a a t c The abolute inlet ach number i replaced by the relative ach number a follow: c w co (5) to relate the relative ach number at the impeller inlet caing and the relative flow angle to the ma flow function. The final reult i: in co k (6) 4 u /( ) co And thi can be reformulated to yield the modified ma flow function a: 4 u in co /( ) (7) k co which i the ame expreion a derived by Dixon [5] for hi modified form of ma flow function, and imilar to that given by Whitfield and Baine []. The modified ma flow function w, i plotted over the relative inlet flow angle for a erie of impeller inlet relative ach number and two different value of the ientropic exponent in Figure 5. Thee repreent air (.4) and a high molecular weight refrigerant ga (.). The important feature of thi curve i that for a given inlet ma flow function there i a certain inlet angle leading to the minimum relative inlet ach number. Thi can be explained phyically by the fact that a maller inlet flow angle i accompanied by a low inlet caing diameter and a higher inlet ach number due to the mall inlet flow area. A large inlet angle i accompanied by a large caing radiu and in thi cae the inlet ach number rie due to the increaed inducer caing blade peed. Between thee extreme there i an optimum flow angle that lead to a maximum inlet relative ach number. Figure 5: odified ma flow function for a centrifugal compreor with zero inlet wirl and for two different value of the ientropic exponent baed on equation (7). 5 Copyright 0 by ASE

6 A curve connecting the maxima of the ma flow function for different impeller inlet caing relative ach number i included. Thi curve can be interected with a given deign ma flow function value to yield the optimum deign location with repect to inducer hock related loe a it repreent the lowet impeller inlet relative ach number poible for thi deign value. An analytical expreion for the optimum inlet angle ha previouly been given by Whitfield and Baine [], but from equation (7) a more imple expreion for the optimum relative flow angle can be derived a: co, opt (8) A can be een, at low abolute inlet ach number, c w co, the term on the bottom line of equation (7) become unity o that the maximum ma flow function then occur at the peak value of in co which i tan or The optimum angle increae with increaing ach number and i about 60 for a typical relative inlet ach number of unity at the impeller inlet caing. For a given relative ach number the dependence of the ma flow function on the choice of angle i weak o that other angle within ±5 from the optimum do not ubtantially change the ma flow function. The throat area elected i conidered to be an apect of the detailed deign where other tradeoff may be appropriate, but thee are only poible when the correct inlet ach number and inlet angle have already been elected to enure low relative inlet ach number. The deigner may chooe lightly lower value than the optimum to increae the throat if he i concerned with choke, or higher value if he i intereted in decreaing incidence for higher urge margin. It ha to be tated that in the derivation of equation (6) effect due to metal blockage, boundary layer blockage, incidence angle and non-uniform pan-wie ditribution of axial velocity have been neglected. The effect of flow nonuniformity and blockage can be included by a mall modification of the hape factor k. FLOW COEFFICIENT AND DIAETER RATIO To derive an expreion for the wallowing capacity, equation (6) i further proceed. Firtly we ue equation (7) and (7) to derive an expreion for the flow coefficient, a follow: k (9) 4 u Thi i plotted in Figure 6 which how the variation of tip-peed ach number with flow coefficient and impeller inlet caing relative ach number for the optimum deign at the peak ma flow function in Figure 5. For a deign at a given impeller inlet relative ach number, the flow coefficient decreae a the tip-peed ach number i increaed. Deign for high flow coefficient and high tip-peed ach number are penalized by a higher inlet relative ach number. The olution to thi problem i alo hown in the diagram in that deign for higher preure ratio automatically require a lower flow coefficient (or a lower trim) to avoid high impeller inlet relative ach number. Figure 6: Dependence of u on k and different value of according to equation (9). In practical application it i ueful to expre the tippeed ach number in term of the preure ratio, leading from equation (5) to: ( ) (0) u ( ) o that finally the flow coefficient can be expreed a k ( ) t () 4 The actual impeller inlet relative ach number required in equation (6) and (7) for the modified ma flow function i however a function of the impeller inlet caing diameter, o that we alo need to derive an expreion for thi in term of the other parameter. From the velocity triangle at the impeller inlet, hown in Figure 4, an expreion for the dimenionle impeller inlet diameter can be derived by expreing the inducer tip-peed a a function of inlet total condition, relative ach number and relative flow angle. The relation c co( ) a RT / T T t c, c c m a c, a u a in( ) () 6 Copyright 0 by ASE

7 are combined to expre the inducer caing velocity a a function of relative ach number, flow angle and total inlet temperature: u co ( in( ) RTt ) () Uing the definition of the angular velocity u u (4) D D and combining thi with equation (5) rewritten a c p T t u (5) and () finally yield the expreion for the non-dimenional impeller inlet caing diameter: D in( ) (6) D co ( ) w Equation () i viualized in Figure 7 for a given parameter et of the total-to-total tage efficiency, the work input parameter and the inlet hape factor and for a total preure ratio of 4.5 in air. The overall hape of the curve i the ame a for the modified ma flow function hown in Figure 5. Thi can be confirmed by examining equation (9) a the inlet flow coefficient cale linearly with the ma flow function for a fixed preure ratio. A minimum impeller inlet relative ach number can be found for any pecified inlet flow coefficient. The correponding point in the figure are the optimum impeller with a maximum ma flow function which lie on the maximum of each curve indicated with the circle ymbol. Figure 7: Flow coefficient according to equation (). The diameter ratio value are given for an example deign at an inlet flow coefficient of in air. The diameter ratio according to equation (6) are alo given in Figure 7 for a typical value of flow coefficient of With thi value, the minimum impeller inlet caing relative ach number i about. for a tage with deign total-to-total preure ratio of 4.5, a flow coefficient of 0.09 and the given parameter et according to the preented D theory. The correponding diameter ratio i roughly 0.65 and the relative flow angle at the hroud i about 6.6. Deigner of radial compreor impeller will recognize thee value a being typical for the mot common tyle of deign. Before continuing it i worthwhile clarifying the utility of thee equation and ummarizing the reult o far. Firtly the analyi ha been derived in a new form which give clearer indication of the relevance of the term and the relationhip of the ma flow function to the wallowing capacity and tippeed ach number. For a given impeller inlet caing relative ach number, which may be conidered to repreent the difficulty of the impeller inlet deign or the technology level of the deign, we can elect an appropriate blade inlet angle (near to 60 ) which minimize the impeller inlet relative ach number and calculate the modified ma flow function. The new analyi then how that from equation () the inlet flow coefficient of the deign i related to the required preure ratio and the cale factor k. A higher preure ratio or a higher tip-peed ach number require a lower flow coefficient. So thi equation provide u with a way of examining how the non-dimenional wallowing capacity of the tage varie with ach number, trim hape factor, and preure ratio. Equation () and (6) how that both the wallowing capacity a well a the impeller diameter ratio are independent of the inlet total condition and can be expreed a function of the inlet hape factor k, the ientropic exponent (ga property), the tage ientropic efficiency, the work input factor, the tage preure ratio and of the impeller inlet relative ach number and flow angle, which can be expreed functionally with the new equation a: F, k,,,, w,, D D F,,, w, (7) ALTERNATIVE FLOW CAPACITY EXPRESSIONS For turbocharger centrifugal compreor map, the ma flow rate can be corrected for total inlet temperature uing a corrected volume flow rate, V T ref. The ma flow i then expreed a: T VT ref t (8) Tref m The converion of the different flow rate definition are a follow: 7 Copyright 0 by ASE

8 V T Tref V T (9) ref D u D u Tref D u Tref u With the equation (), (5) and (9) an expreion for the pecific flow rate can be derived a: V / T ref RTref ( ) k D 4 (0) Thi definition of pecific flow rate ha the dimenion of a velocity and i often ued in matching calculation of the turbocharger compreor with the turbine. Another common alternative formulation of the wallowing capacity would be in term of the pecific peed in which cae equation () and equation () would need to be combined. PRESSURE RATIO VERSUS SWALLOWING CAPACITY In thi ection, chart howing the tage total preure ratio againt the tage wallowing capacity are elaborated. To come up with a figure to repreent the total-to-total preure ratio veru flow coefficient, it i aumed that the optimum point for centrifugal tage deign lie on the maxima-curve in the F, k,,,, w, diagram a thi minimize the impeller inlet relative ach number and it aociated loe. In the firt intance, the value of the hape factor k, and the efficiency and the work coefficient ( and ) are kept contant. V T Uing thee aumption, the optimum point plotted in Figure 7 uing circular ymbol can be determined for a range of preure ratio and plotted againt the flow coefficient a hown in Figure 8 to yield the io-line of impeller inlet relative ach number (colored line). A for each point and are known, the diameter ratio can be computed according to equation (6) and alo plotted a olid io-line. The line of contant tip-peed ach number are evaluated according to equation (5) and indicated with dahed line. The exemplary deign point ued in Figure 7 i highlighted in Figure 8 with the red dot and labeled a DP. The plot can be tranlated to the dimenionle pecific peed formulation preented by Rodger, [] uing equation (), and a hown in Figure 9, a follow: 4 t () Thi diagram ha the diadvantage over Figure 8 that the band of curve hift horizontally when the work coefficient i varied. Thi identifie the clear advantage of flow coefficient over pecific peed a a meaure of flow capacity. Figure 9: Stage total preure ratio v. wallowing capacity expreed a pecific peed, auming contant efficiency. Figure 8: Stage total preure ratio v. wallowing capacity expreed a inlet flow coefficient, auming contant efficiency. Intead of auming contant efficiency over the entire, pace, the efficiency correlation hown in Figure can be ued. In thi cae a parameter et of, k, and initial value for (, t ) have to be pecified. (, u ) i then computed with equation (5). Knowing the tip-peed ach number, a new etimate for the ientropic total-to-total efficiency (, ) can be derived from Figure and equation t (4). The iteration i performed until convergence in (, ) t 8 Copyright 0 by ASE

9 i reached. Afterward, (, ) i computed from equation t (9) which in turn i ued to derive (, ) and t ( t, ) from equation (7) by interecting the optimum line, hown in Figure 5, at the given value of ( t, ). The diameter ratio i then computed uing equation (6). An example of uch a reult i given in Figure 0. Note that the line of contant tip-peed ach number are now no longer line of contant preure ratio, but become lower at high and low flow coefficient, ee equation (5). impeller ize and other effect uing correlation in the open literature. Intead of uing the inlet flow coefficient, the invetigation can alo be performed in term of pecific peed by comparing Figure 9 and the efficiency correlation preented in Figure. DISCUSSION A a reult of the aumption made during the derivation of the preure ratio veru wallowing capacity chart, Figure 0, each point in the chart repreent a compreor geometry pecially deigned for thi point. The chart hould not be confued with a performance map for a given impeller geometry a the off-deign effect of flow on efficiency and work are not included, which i not a realitic aumption for a whole compreor map. Limitation of the method The underlying aumption, for the derivation of the wallowing capacity plot, i that the optimum deign point i characterized a the point of minimum inlet caing relative ach number. Clearly, the method i limited to deign that are carried out with thi objective, but a thi i uually done in high flow capacity compreor thi i not a eriou limitation. In addition, the following parameter need to be pecified: impeller hape factor, k work input coefficient, ientropic tage efficiency, In Figure 8 and Figure 9 the efficiency i pecified a a contant value for the whole map, which i not entirely realitic at very low or high flow coefficient a vicou or ach number effect will contribute to an efficiency deficit. Figure 0 take the efficiency to be that of the correlation hown in Figure and the efficiency then fall to high and low flow coefficient, a expected. It i worthwhile to note that no information about total inlet flow condition need to be pecified. Neverthele Figure 8 and Figure 9 remain ueful, a the expected achievable ientropic efficiency can be ued to repreent the deign point, o the diagram i correct in the neighborhood of thi point. For the example given in Figure 8 and Figure 9 with 4. 5 the deign point ientropic efficiency i computed a about 8%. Thi value i conitent with one choen to plot the wallowing capacity diagram. The efficiency value given in Figure are value for compreor with an impeller diameter of 450mm and low clearance. ore detailed information can be found in the appendix. Depending on the invetigation, the value have to be corrected for Reynold number effect due to different Figure 0: Stage total preure ratio v. wallowing capacity expreed a inlet flow coefficient, but uing the efficiency correlation according to Figure. A further limitation of the method i that it i onedimenional and it aume a contant meridional velocity acro the pan. In fact the curvature of the impeller caue an acceleration of the flow at the caing of the inlet which i tronger for hort tage with a higher curvature. To take thi into account the parameter k in equation (9) need to be adjuted by an additional acceleration factor for the effect of the curvature, which need to vary for tage of different axial length. Phyical limit in compreor deign Before dicuing the phyical limit it i ueful to examine the value of the flow coefficient and the modified ma flow function for a cae with a few imple approximation. The modified ma flow function, equation (7), ha a peak cloe to 60 (ee Figure 5), o thi angular value i ued in the approximation. In addition, the correction for the change from total to tatic condition in the term on the bottom line of equation (6) i cloe to unity for low inlet ach number c (ee equation (5)). Equation (7) and (6) then implify to 4 u in co k () 9 Copyright 0 by ASE

10 D w in( ) () D Combining equation () and () and ubtituting by equation () give k D k D co( ) (4) 4 D in( ) 4 D in( ) Approximating 60 and applying equation (7) lead to k D (5) 4 D k D u 4 D (6) Two ditinct phyical wallowing capacity limit can be identified from thi and are hown in Figure. For low deign total preure ratio, the wallowing capacity i limited by the impeller diameter ratio and not by the impeller inlet caing relative ach number, indicated with the bluih zone in Figure ( D D ). Turbocharger tage and indutrial proce compreor tage with preure ratio below (or a tip-peed ach number below.) tend to be limited in flow capacity by the impeller diameter ratio. The impeller inlet caing relative ach number i low and the increae in loe with high flow coefficient effectively determine the deign flow capacity limit. Thi mechanim can alo be identified in Figure following a curve of contant u toward high inlet flow coefficient. A the diameter ratio approache, the geometry would tranit toward a mixed flow deign or an axial compreor. A typical upper limit for low preure ratio proce centrifugal compreor i D D and with a large hub diameter equation (5) lead to a maximum flow coefficient of around 0., which i the typical maximum flow coefficient found in uch machine. Higher value might be acceptable if a tradeoff between wallowing capacity and tage efficiency can be found for a given application. For high deign total preure ratio, the wallowing capacity i limited by the impeller inlet relative ach number, indicated with the reddih region in Figure (. ). In thi region it i the ma flow function that i limited by the value of the impeller inlet relative ach number. The phyical background for the maximum impeller inlet relative ach number i the hock boundary layer interaction, a dicued in the introduction. High value of the impeller inlet relative ach number are inconitent with highly efficient centrifugal compreor, a repreent an averaged value jut uptream of the inducer reulting from D correlation. In reality, the flow will further accelerate in the inducer at the uction ide of the blade due to blade metal blockage, treamline curvature and uncovered paage turning reulting in even higher pre-hock ach number. An impeller inlet caing relative ach number maller than.5 i uggeted for good performance. If we et thi limit it effectively limit the ma flow function and Figure then how that an increae in preure ratio (or tip-peed ach number) can only be achieved by decreaing the impeller diameter ratio. If the pecification require a higher impeller inlet relative ach number then the diagram ugget the degree of difficulty of the deign, and the deigner may then make ue of pecial tranonic flow deign feature to combat thi, uch a extremely thin blade with reduced uction urface curvature, and leading edge weepback, a decribed by Rodger []. Figure : Phyical limit of the centrifugal compreor deign pace. The hading i explained in the text and indicate the region where deign become more difficult. A third phyical border which can be identified from Figure i the achievable total preure ratio due to material limitation. A hown in equation (5) the total preure ratio i linked to the tip-peed ach number, u and hence to mechanical tree within the impeller. Thi region i chematically indicated with the greenih area in Figure and depend on the available impeller material and geometry. The geometry combined with the required rotational peed define the tre level in the compreor which in turn have to be within a afety margin compared to the maximum allowable tre defined by the impeller material choice. A further phyical border i given by blade vibration iue: Deigning for no interference up to a given harmonic require ufficient tiffne of the blade which can be demanding for high D D and k value. Increaing the natural frequency of the blade by increaing blade thickne 0 Copyright 0 by ASE

11 increae blockage and hence reduce both efficiency and wallowing capacity by trend. A final phyical border i the drop off in efficiency to low flow coefficient, and thi i identified clearly in Figure 0. Thi i not the ubject of thi paper and i not conidered further here. The preented value for the different phyical limit are not harp boundarie but jut rough indication. During the deign proce a tradeoff ha to be found between wallowing capacity and efficiency for a given application. The range of achievable total preure ratio can be extended toward higher value by changing the impeller material. Toward high preure ratio tage, not only the impeller hock related loe but alo the diffuer loe due to high diffuer inlet ach number increae which in turn reduce the overall achievable ientropic tage efficiency. The dicued limit can alo be identified in Figure 0 where the correlation from Figure i ued intead of the aumption of contant efficiency. Peak efficiency value are obtained roughly along the contant diameter ratio line of D D 0.6 for the example with a work input coefficient of 0.75, an ientropic exponent of.4 and a hape factor of 0.9. For low preure ratio, the io-efficiency zone are more aligned to the D D io-line wherea for high preure ratio, they tend to align with the io-relative-ach curve in the region of high efficiency. The different limit given in the figure preented above are identified by thi analyi in a clear and logical way. It i intereting to note that a deign chart of a imilar nature can alo be found in the book by Eckert and Schnell [4]. DEONSTRATION OF THE ETHODOLOGY FOR COPRESSOR DESIGN The equation and methodology given here have been incorporated into a preliminary deign and aement ytem for radial compreor. A D deign flow diagram of thi i preented in Figure. It tart with two pecification block, an inner pecification part A and an outer pecification block, called part B. In the inner pecification block, part A, only dimenionle propertie are pecified wherea the pecification part B repreent the total inflow condition of the compreor. The deign preure ratio, the target efficiency, the fluid property and the material choice have to be taken in the pecification part A. The work input coefficient ha to be pecified due to matching retriction or material trength conideration. In addition, the impeller hape factor i ued a an input, which can be choen from experience on other available deign. Baed on the pecification part A, the wallowing capacity diagram can be computed. For the deign total-to-total preure ratio, a uitable trade of between the deign flow coefficient, the impeller inlet caing relative ach number and diameter ratio ha to be found baed on experience. Efficiency plauibility check Specification part B: volume flow rate, V total inlet temperature, T Specification part A: deign preure ratio, target efficiency, Fluid: ientropic exponent, aterial choice: required work input factor, Empirical data: Impeller hape factor, k Figure : D deign flow chart., F k,,,, w, D D F,,,, w, Tradeoff for given : flow coefficient, relative ach number,, relative flow angle, diameter ratio, D D r h from known k : Cae : known r and D D : D from known D Cae : known D form known D D : from known k : r h D and Check flow rate and material tre blade tip peed, u volume flow rate, V r Copyright 0 by ASE

12 The relative flow angle at the inducer caing i a reult of the choen deign point. Uing equation (5) the tip-peed ach number can be computed which i ued together with the flow coefficient to check the target efficiency for plauibility baed on efficiency correlation uch a given in Figure. In cae of a mimatch, the target efficiency or the tradeoff parameter have to be changed until convergence i reached in term of efficiency value. Alternatively, Figure 0 can be directly plotted for a known et of, k, which include the inner efficiency loop. In the next tep, geometry information i required: In cae, the inducer hub radiu i pecified from which the inducer caing radiu can be computed a the impeller hape factor i known. The impeller outer diameter follow from the known diameter ratio. In cae, the impeller outlet diameter i known and the inducer caing radiu i computed from the known diameter ratio. The inducer hub radiu follow form the known hape factor. From the pecified total inlet temperature (pecification part B), the blade tip velocity can be computed from the definition of ientropic total-to-total tage efficiency and work input coefficient according to equation (5). Knowing the inlet flow coefficient thi information i ued to both check the material tre level and the pecified volume flow rate according equation (9). Validation It i difficult to provide exact validation for the deign methodology given here, a it would require detailed commercially enitive information of tage type, pecification and deign performance. Figure how an attempt to do thi for a range of commercially inenitive tage deign available to the author, covering variou proce compreor, ga turbine, automotive turbocharger and large turbocharger application, including data from variou publihed ource. The point in Figure how the deign point preure ratio and the flow coefficient of ome ucceful deign from a range of ource, and the background of thi figure i baed on Figure 0. Firtly it can be een that the pread of the data point overlap with the peak efficiency region of Figure 0, demontrating that mot deign fall within the range of the approach uggeted here. Not all of thee tage in the figure have been deigned for maximum poible flow capacity a ome are clearly deign for maximum efficiency. In addition many of the tage have a lower work coefficient than 0.75, the bai of Figure 0. Neverthele the tage on the right hand of thi diagram demontrate a good agreement with the deign limit for high flow capacity uggeted in thi paper. In addition there i evidence from thi that the low preure ratio tage are limited by the ratio of the inlet eye to impeller diameter and the high preure ratio tage by the inlet caing relative ach number, a dicued above. Figure : Stage total preure ratio v. wallowing capacity for a range of compreor tage in comparion to Figure 0. Specification aement The method preented can be ued for the deign of new tage, but can be alo ued to quickly analyze exiting tage or ae new pecification data. The dicued procedure allow the deign point to be judged and the aociated development rik to be aeed by deriving Figure 8 for the pecification which can be compared againt the efficiency correlation. The incluion of ound phyically-baed limit not only help to judge the pecification but alo help to explain the rik to take holder and iterate the pecification to come up with realitic and feaible value. Figure 0 can alo be ued for pecification aement. Intead of uing abolute value it might be more ueful to compare different deign baed on efficiency difference. Hence, baed on a meaured deign, the trend on efficiency of a new deign can be etimated and judged. CONCLUSIONS A method ha been preented allowing a fat preliminary aement of the deign of centrifugal compreor with high wallowing capacity. A diagram of total-to-total tage preure ratio veru inlet flow coefficient i derived and thi include phyical boundarie that provide ueful guideline for the deigner. The diagram i expreed in non-dimenional term and both the inlet flow coefficient and the impeller diameter ratio are independent of total inlet condition. Phyical border which limit the maximum pecific flow rate of centrifugal compreor are given in the diagram. At low total-to-total deign preure ratio, the impeller inlet eye to outlet diameter ratio limit the wallowing capacity wherea at high preure ratio, the impeller entry relative ach number at the blade caing i critical. The maximum Copyright 0 by ASE

13 achievable preure ratio i alo limited by the impeller geometry and material. A guideline including a flow chart i given how to ue the derived equation for preliminary D centrifugal compreor deign and aement. It how that the pecification can be plit in to dimenionle and non-dimenionle number and how they interact. Experimental data from a range of compreor deign upport the ue of thi guideline for deign. An additional and important concluion of the work i that it how clearly that the ue of pecific peed a a meaure of wallowing capacity cannot be recommended. ACKNOWLEDGENTS The firt author thank ABB Turbocharging for permiion to publih thi paper. The many valuable dicuion with Gerd undinger of ABB Turbo Sytem Ltd, Baden, Switzerland, are gratefully acknowledged. Ueful comment on an early draft of thi paper from Colin Rodger, Peter Came, Chri Robinon and the reviewer were very helpful. REFERENCES [] Rodger, C., (99), The Efficiencie of Single-Stage Centrifugal Compreor for Aircraft Application, ASE 9-GT-77, Orlando, USA. [] Rodger, C., (99), Centrifugal compreor deign: tate of the art performance, Cranfield Univerity hort coure on centrifugal compreor, arch, Cranfield Univerity. [] Caey.V., Robinon C.J. (006), A guide to turbocharger compreor characteritic, in Dieelmotorentechnik, 0th Sympoium, 0- arch, 006, Otfildern, Ed.. Bargende,, TAE Elingen, ISBN [4] Robinon, C. J., Caey,. V., Wood, I. (0) An integrated approach to the aero-mechanical optimiation of turbo compreor, publihed in 0 Current Trend in Deign and Computation of Turbomachinery, conference organied by KD Nové Energo & TechSoft Engineering in Prague, Czech Republic, ay 0. [5] Dixon, S. L., (997), Thermodynamic of Turbomachinery, rd Edition, Butterworth-Heinemann, Oxford, England. [6] Hill, P. and Peteron, C., (99), echanic and Thermodynamic of Propulion, nd Edition, Addion-Weley Publihing Company, Reading, aachuett, USA. [7] Aungier, R. H., (000), Centrifugal compreor a trategy for aerodynamic deign and analyi, ASE Pre, New York, USA. [8] Denton, J. D. (99), Lo mechanim in Turbomachine, Tran. ASE Journal of Turbomachinery October 99, Volume: 5, pp [9] Bölc, A., (986), Tranoniche Turbomachinen, Braun, Karlruhe, Germany. [0] Lohmberg, A., Caey,., Ammann S., (00), Tranonic Radial Compreor Inlet Deign, Proc. Intn. ech. Engr., Vol. 7, J. Power and Energy, 00. [] Whitfield, A. and Baine, N.C., (990), Deign of radial turbomachine, Longman Scientific and Technical, UK. [] Stanitz, J. D., (95), Deign conideration for mixed flow compreor with high flow rate per unit frontal area, NACA R E5A5. [] Rodger, C. (00) High pecific peed, high inducer tip ach number centrifugal compreor, Paper GT , ASE TurboExpo 00. [4] Eckert, B. and Schnell, E. (96), Axial- und Radialkompreoren, Springer, Berlin. [5] Caey,.V., and arty, F., (985), "Centrifugal compreor - performance at deign and off-deign condition", Proceeding of the Intitute of Refrigeration, Vol. 8, , page APPENDIX The correlation of in Figure are dicued in Caey and Robinon [] and Robinon et al [4] and are baed on experimental data publihed by Rodger [] and Caey and arty [5]. The equation have not been previouly publihed and are given below. The performance level i conidered to be typical of that which a good deigner with good deign tool can be expected to achieve, and may even be a bit conervative up to % point. The value predicted are generally in line with other correlation of experimental data uch a thoe of Aungier [7]. Level of performance higher than thee can be expected if a thorough development program with CFD analyi and an experimental teting program i undertaken to evaluate different deign choice. The correlation are for typical large turbocharger tage with tate of the art back-wept open impeller (circa 40 backweep), impeller diameter of D = 450mm with a mall tip clearance of ay 0.5 mm in operation and roughne typical of milled component, and with a well-deigned and wellmatched vaned diffuer. Reduction in efficiency for the following effect are needed: Increaed tip clearance Shroud friction in hrouded tage Vanele diffuer Smaller ize Lower Reynold number Increaed relative roughne Le back-weep Fewer blade The individual correction require pecific correlation for each effect and lead to much lower efficiencie of 75 to 80% for typical turbocharger automotive tage. The equation for the variation in efficiency at a tip-peed ach number of 0.8 and below are: Copyright 0 by ASE

14 0.08 : 0.08 : max 0.86, p p max max max k k 0.08, max t k max t max k 7, k 5000, 4 k 0 (7) At ach number above a tip-peed ach number of 0.8 an efficiency deficit i impoed a: 0.8: 0.0 u u k 4 0.8: 0.05, p k P k P p k, P ( u 0.8) (8) Note that the trength of the deficit i baed on the product of tip-peed ach number and flow coefficient conitent with the finding of thi paper and equation (5). 4 Copyright 0 by ASE