Efficiency Increase and Input Power Decrease of Converted Prototype Pump Performance

Transcription

1 International Jornal of Flid Machinery and Systems DOI: Vol. 9, No. 3, Jly-September 016 ISSN (Online): Original Paper Efficiency Increase and Inpt Power Decrease of Converted Prototype Pmp Performance Masao Oshima Retired, Formerly Department of Mechanical Engineering, Kanagawa Institte of Technology Home: 431 Bkko-cho, Hodogaya-k, Yokohama, , Japan, Abstract The performance of a prototype pmp converted from that of its model pmp shows an increase in efficiency broght abot by a decrease in friction loss. As the friction force working on impeller blades cases partial peripheral motion on the otlet flow from the impeller, the increase in the prototype s efficiency cases also a decrease in its inpt power. This paper discsses reslts of analyses on the behavior of the theoretical head or inpt power of a prototype pmp. The eqation of friction-drag coefficient for a flat plate was applied for the analysis of hydralic loss in impeller blade passages. It was revealed that the friction-drag of a flat plate cold be, to a certain degree, sbstitted for the friction drag of impeller blades, i.e. as a means for analyzing the relationship between a prototype pmp s efficiency increase and inpt power decrease. Keywords: Performance conversion, Model pmp, Efficiency increase, Inpt power decrease, Scale effect 1. Introdction The efficiency of a prototype pmp shows a higher vale than that of a model pmp. This is the reslt of a decrease in the flid friction coefficient. The friction force working on impeller blades drags the water in impeller blade passages towards the rotational direction of the impeller, ths increasing the peripheral component of the impeller otlet flow, as well as the impeller inpt power. Compared with the friction coefficient of model impeller blades, that of prototype impeller blades shows a decrease owing to a decrease in relative roghness. Correspondingly, there is a decrease in the ratio of the peripheral component of the impeller otlet flow verss the peripheral velocity at the impeller otlet, and accordingly a decrease in the inpt power coefficient of the prototype impeller [1], which is not well known among engineers handling hydralic machinery. IEC [] specifies that the efficiency increase in the prototype shall be attribted all to the decrease in the power coefficient both for water-trbine and pmp operations. On the other hand, JIS B 837:00 [3] stiplating performance conversion from a model to a prototype pmp specifies that the efficiency increase in the prototype shall be attribted wholly to the increase in the head coefficient increase. Besides, IEC 6097 [4] pblished in 009 specifies that the efficiency increase in the prototype shall be attribted to the head coefficient increase for the pmp operation of reversible water trbine. However, JIS B 837:013 [5] pblished recently specifies that the efficiency increase is attribted half to the head increase and half to the inpt power decrease, based on the reslts reported by the athor [1]. As for the isse on whether an efficiency increase is attribtable to a head or inpt power coefficient, Krokawa [6] states that an increase in an impeller s Reynolds Nmber increases the slip factor of blades and accordingly decreases in trn the theoretical head or inpt power. He proceeds to say that a decrease in inpt power cancels the expected head increase in the impeller de to a decrease in the friction coefficient. He concldes that the head coefficient of a prototype increases only in proportion to the vale of decrease in loss head in the pmp component, not inclding the impeller. Krosawa s view, however, was not reflected in the relevant JIS standard ntil 013. Htton and Fay [7] reported test reslts on a centrifgal pmp, where Reynolds nmber was changed from to , ranging from hydralically smooth to rogh impeller passage srface. The reslts clarify that the shaft power decreases and the delivery head once increases and then decreases when Reynolds nmber increases within the hydralically smooth range. However, when Reynolds nmber changes within the hydralically rogh srface range, both the shaft power and delivery head remain the same. They didn t attempt a detailed analysis on the behavior of the shaft power and delivery head. Recently, Ida [8] stdied changes between the theoretical heads of a model and a prototype on the basis of the relationship Received Agst 6 015; accepted for pblication November : Review condcted by Yon-Jea Kim (Paper nmber O15038K) Corresponding athor: Masao Oshima, mosim@nifty.com 05

2 between the bondary layer thickness on impeller blades and the theoretical head. A change in the otlet flow angle de to a decrease in the bondary layer thickness on prototype impeller blades is investigated there, the condition being a shock-free inlet flow on a flat blade cascade. It was revealed that even nder a shock-free inlet a slight rotation occrred in the otlet flow from the impeller owing to a drag or friction cased by the blades. Ida states that the change in the direction of relative otlet flow is less than 0.1ºat most, and that the change in performance de to manfactring tolerance is rather larger than that in the theoretical head de to the Reynolds Nmber. His stdy was restricted to the change in otlet flow angle, based on the behavior of the bondary layer only on blades and not on the shrod and disk. The shrod and disk srfaces conjointly work to deliver water to the otlet, and accordingly the friction force on these srfaces also case rotational motion in the flow from the impeller, which was not taken into consideration in his stdy. Pelz and Stonjek [9] proposed a new method for trbomachinery performance prediction, bt this is related solely to hydralic efficiency, not to the change in head and inpt power. Bellary and Samad [10] reported the reslts of nmerical analysis on performance of a centrifgal pmp for for different viscos liqids. The analysis was made on velocity vectors and pressre contors, and not on the relationship between efficiency step-p, head increase and inpt power decrease. The athor [1] investigated previosly the relationship between the efficiency increase from the model to the prototype and the power coefficient decrease on the basis of the relationship between the loss head verss the blade drag force and the theoretical head. This paper analyses how mch efficiency increase from the model to the prototype is broght abot by the head coefficient increase and by the power coefficient decrease, based directly on friction loss analysis of impeller blade passages, where not only blade srfaces bt also shrod and disk srfaces are taken into consideration. Efficiency conversion eqation specified in JIS B 837 [3] [5] is derived based on the friction-drag coefficient eqation on a single flat plate, and the reslts of the conversion provides reasonable reslts within a certain degree of accracy, and therefore, the friction analysis is carried ot in this paper on the basis of the same eqation on a single flat plate.. Change in Blade Drag and Inpt Power Fig. 1 Forces working on impeller blades When a flow arond axial flow impeller blades, as shown in Fig. 1, is considered, the momentm difference of water given by an impeller between inlet and otlet for inlet flow with no pre-rotation eqals to the sm of peripheral components of lift and drag forces working on blades and blade passages, and is shown as follows. ρ v a z b t v = z (W cos β +A sin β ) (1) where the drag W is comprised of the profile drag and friction drag, and the former is cased only by a blade and the latter by all blade passage srfaces. The lift force A on a blade and the drag force W on a blade passage are expressed by sing the lift and drag coefficients, C A, C D and C F, and relevant srface areas, S A and S F, where S A denotes the projected area of a blade and S F the srface area composing a blade passage, see eq. (8). Then, eq. (1) can be written as follows. ρ v a b t v = (ρ/) w {(C D S A + C F S F ) cosβ + C A S A sinβ )} () When expressed in dimensionless eqation, eq. () can be rewritten as follows. v / = (1/) {w /(v a )} {(C D S A +C F S F ) cosβ +C A S A sinβ }/ b t (3) The displacement thickness of the bondary layer on blade passage srfaces for the prototype is thinner than that for the model, and, therefore, strictly speaking, there is no similarity between their main flows throgh blade passages. Nevertheless, their lift and profile drag coefficients, C A and C D, are taken to be the same, the only difference being their friction coefficient. The difference in peripheral force of the prototype and model is cased by the difference in the friction drag coefficient, (C FP C FM ). As sch, the difference in the peripheral velocity component of the prototype and model is expressed as follows. v P / mp v M / mm = (1/) (w / v a ) (C FP C FM ) (S F / b t) cosβ (4) where the sbscripts sch as P and M are omitted from those dimensionless terms which are the same for both prototype and model sch as w / v a and S F / b t. The same shall apply hereinafter. Based on the above relation, head coefficient difference Δψ th for the centrifgal prototype and model pmps is obtained in the following. Here it is assmed that: (1) the representative velocity of blade flow w is the vector mean of the relative velocities at the blade inlet and otlet, () the representative impeller diameter D is the arithmetic mean diameter of the inlet and otlet of central blade section dividing the blade passage eqally, on reference to a reslt that the friction loss coefficient for the mixed flow impeller is expressed reasonably by the arithmetic mean diameter of the inlet and otlet [11], (3) the representative blade pitch t, width b and meridian velocity v a are those at the diameter D, and (4) the angle of the representative flow β is the same both for the model and prototype. Based on the following relations, w = v a / sin β w ( va sin β ) va 1 va φ = = (5) v v v v sin β a m a a m a 06

3 Eqation (4) can be rewritten as follows, Δψth = ψthp ψthm 1 = ( CFP C ψ H v g v th m th = = = (6) m g m g m Fm S v F a cos β ) φ bt va sin β l t sin β ( ) v cos β = C C + β a FP F M 1 φ (7) t b va sin where the friction drag working area S F is replaced by S F = l ( b + t sin β ) (8) The velocity casing friction loss at impeller tip periphery is the relative velocity w for closed impellers and the absolte velocity v for open impellers, and the former is ordinarily larger than the latter, especially for high specific speed impellers. Accordingly the second term in eq. (8) is adjsted for the case of open type impellers in this paper, as referred to in Section 4. Eqation (7) gives the difference in the theoretical head, and ths the difference in the inpt power coefficient, between the model and prototype, as the theoretical head denotes inpt power per nit weight flow. Correspondingly, the vale of Δψ th / ψ thm gives the ratio of change in inpt power coefficient verss the coefficient for the model. The skin friction coefficient of a flat plate [1] as shown below is sed in this paper for the coefficient C F in eq. (7) as in the paper by Ida [8] l.5 C F = ( log ) (9) k 3. Hydralic Loss in Impeller In the former section the difference in the impeller work, i.e. the theoretical head, of the model and prototype was examined sing the friction coefficient of a flat plate for estimating the friction loss in the impeller. The effectiveness of the estimation is discssed here. Work done by the drag force on blades is eqal to the energy lost in the impeller [13]. Accordingly, 1 3 ρ g b z t va H L I = z ρ w ( CD b l + CFSF ) (10) The dimensionless head loss in the impeller ψ LI can be expressed as follows. s 3 C FSF m va g H L I 1 ψ L I = Dbl + C w = (11) m bt m va va In eq. (11), blade width b, meridian velocity v a and average relative velocity w are taken at the mean diameter between the inlet and otlet as representative vales of blades, for the prpose of applying the eqation to centrifgal impellers as well. Therefore, the difference in impeller head loss between the prototype and model is written as follows, Δ ψ L I = 1 ( C C ) F P F M 3 S F w bt m 1 v φ v a a 3 l t sin β w 1 va = ( C ) + F P CF M t b m φ va 1 (1) where the profile drag coefficient C D is assmed to be the identical for both the prototype and model. The efficiency conversion eqation specified in JIS B 837 [3] gives reasonable reslts with a certain degree of ncertainty, where the efficiency converted in JIS B 837 [3] is in fact the hydralic efficiency. The difference of impeller head loss between the prototype and model can be obtained also by the hydralic efficiency increase, calclated by the conversion eqation in JIS B 837 [3], based on the following assmptions: (1) Hydralic loss in the impeller is half of the total hydralic loss in the pmp; accordingly, the friction loss decrease in the impeller is also half of the total friction loss decrease in the pmp, where the reslts of tests carried ot on axial flow pmps by Toyokra [14] is referred to. () The ratio of friction loss verss the total hydralic loss and the friction loss decreasing ratio de to Reynolds Nmber are the same for both the impeller and fixed passages. In the previos paper [1] the efficiency conversion formla given in JIS B 837 [3] was referred to, and therefore the same formla is referred to in this paper for the prpose of convenience in the comparison of calclated reslts. The hydralic efficiency of the prototype pmp is given as follows, η hp = η hm [1+δ EM {1-1.08(D P /D M ) 0.18 }] (13) where δ EM = (1-η hm ){1.4Ns } (14) The term (1-η hm ) is replaced by 0.1 in JIS B 837 [3] by asming η hm to be 0.9, bt in this paper the term is given by the exact expression, as there are cases of lower hydralic efficiency. Hydralic head loss coefficient ψ L is given as follows, 07

4 ψ L = ψth ( 1 ηh ) (15) H th H ψ L where 1 η h = = H th ψth Hydralic loss in the impeller Δψ L IC is assmed to be half of the loss in the pmp and then, Δψ = ψ ψ = ψ 1 η / ψ 1 η / L I C L I P L I M th P ( ) ( ) Here ψ thp is assmed to be eqal to ψ thm as the difference between them is small compared with (1 η hp ), and then Δψ L IC = ψ thm (η hm η hp )/ (16) Validity of applying the friction loss coefficient of a flat plate to the estimation of hydralic loss in the impeller can be verified by comparing the reslts of eqs. (1) and (16). hp th M hm 4. Verification of the Blade Drag Effect 4.1 Axial Flow Impeller Differences in the theoretical head coefficient, being eqal to the inpt power coefficient, and those of the impeller hydralic loss coefficient between the model and prototype are obtained by referring to test reslts of three axial flow pmps [15][16][17]. The scale ratio of prototype pmps is assmed to be 8, and the roghness of the prototype s blade srface to be twice as rogh as that of the model, as eq. (13) is given on the assmption of the roghness ratio of two. Table 1 shows dimensions, dimensionless performance and chord-spacing ratio of the pmps examined. Table shows the friction-drag coefficient of impeller blades C F M and C F P [eq. (9)], the theoretical head difference between the prototype and model [eq. (7)] and its ratio against the theoretical head of model, calclated based on the flat plate friction drag coefficient. As the impellers are of open type and have no shrod, it is nnecessary to take the friction loss on impeller tip periphery into accont. Accordingly, the area sbjected to friction drag, eq. (8), was calclated with the second term in the parentheses mltiplied by 1/4. Friction loss cased on the casing inner srface arond the impeller tip is generated by absolte velocity, which is mch lower than the relative velocity, and ths the loss there is neglected here. As shown in Table, the decrease in theoretical head, i.e., the decrease in inpt power, between the model and prototype is almost 1% in all cases. Table 3 shows the measred hydralic efficiency η hm, hydralic efficiency increase between the model and prototype, the decrease in impeller head loss calclated by flat plate friction, Δψ LI [eq. (1)], and calclated by efficiency conversion eqation, Δψ LIC [eq. (16)], and the ratio between them. As shown in Table 3, the ratio Δψ LI /Δψ LIC is 1.0~1.1, i.e., almost 1.0, and this spports the idea that the behavior of friction in an axial impeller can be explained by the flat plate friction drag coefficient. Table 1 Dimensions of axial flow pmps stdied No. Ns D o D i D m φ ψ β (l/t) Table Friction-drag coefficients and differences between the theoretical head of the model and prototype, calclated based on flat plate friction drag coefficient No. l (m) k sm (μm) C F M C F P Δψ th Δψ th /ψ thm Table 3 Comparison between the hydralic loss coefficients of the model and prototype, calclated applying the efficiency conversion and flat plate friction decrease No. η hm δ E η hp Δη h Δη h /η hm Δψ LI Δψ LIC Δψ LI /Δψ LIC Centrifgal and Mixed-Flow Impellers The same calclation as done in the previos sb-section was performed on centrifgal and mixed-flow impellers. Here also the scale ratio between the prototype and model is assmed to be 8, while the ratio of the blade roghness srface between them is assmed also to be two, as in the case of the axial flow pmps. Table 4 shows head coefficient ψ, flow coefficient φ, impeller dimensions, nmber of blades z, blade angle at the mean diameter β and inlet velocity v a1. Design constants ψ and φ and inlet-otlet diameter ratio were determined based on the design chart by Stepanoff [18] with blade otlet angle assmed to be 5. Table 5 shows the blade srface roghness of the model k sm, blade width at the mean diameter b, friction-drag coefficients C f M and C f P, head coefficient difference Δψ th and its ratio against head coefficient of the model Δψ th /ψ thm. The blade width at the mean diameter was determined with the tip inlet angle assmed to be

5 , Table 4 Dimensions of centrifgal and mixed flow pmps stdied No. Ns ψ φ D 1o (mm) D 1o /D D 1i /D 1o D (mm) z β (l/t) v a Table 5 Friction-drag coefficients and differences between the theoretical head of the model and prototype calclated based on the flat plate friction drag coefficient No. Ns k sm (μm) b (mm) C F M C FP Δψ th Δψ th /ψ thm Table 6 Comparison between the hydralic loss coefficients of the model and prototype calclated applying efficiency conversion and flat plate friction decrease No. Ns η hm δ E η hp Δη h Δη h /η hm Δψ LI Δψ LIC Δψ LI /Δψ LIC Table 6 shows the assmed model hydralic efficiency η hm,, friction loss ratio δ E, converted prototype hydralic efficiency η hp, hydralic efficiency difference Δη h, its ratio Δη h /η hm, hydralic loss difference between model and prototype impellers Δψ LI, hydralic loss difference calclated by converted efficiency Δψ LIC and its ratio Δψ LI /Δψ LIC. Friction-drag working area sed for calclating differences in head coefficients and impeller head losses was obtained by mltiplying 1/ or 1/3 to the second term in eq. (8) for the cases of pmps of the specific speed of 800 and 110, with consideration given on the difference between the diameters of the tip and hb. As seen in Table 5, the decreasing ratio of theoretical head Δψ th /ψ thm was 1.1% for low specific speed impellers and 0.6% for high specific speed impellers. The ratio Δψ LI /Δψ LIC in Table 6 is 0.31~0.397 in contrast to approximately 1.0 for axial impellers in Table 3, which will be discssed in the following section. 5. Discssions 5.1 Difference in Hydralic Impeller Loss of the Model and Prototype Examination is made to verify whether the flat plate friction loss eqation, eq. (9), is able to be dly sbstitted for the friction loss eqation for estimating impeller blade passage losses. This is done by comparing the reslts calclated based on the flat plate friction drag, eq. (1), with those calclated based on hydralic efficiency conversion, eq. (16). As seen in Table 6 for centrifgal and mixed flow impellers, the vales of Δψ LI /Δψ LIC are below 0.4 for all cases, althogh the vales are between 1.0 and 1.1 for the cases of axial flow impellers (see Table 3). This indicates that the friction loss calclated by the flat plate friction eqation is small, almost 1/3 of the actal friction loss. Shirakra [19] carried ot a test for the prpose of clarifying the effect of the impeller blade srface roghness on the hydralic efficiency of the pmp. He states that the difference in impeller head cased by different blade srface roghnesses is in average abot 3.3 times as large as the hydralic loss difference calclated by applying flat plate trblent bondary layer eqation on for srfaces composing impeller passages. The hydralic loss cased by crvatre and divergence in the blade passages is assmed in his calclation to be independent of the srface roghness. Frthermore, Ida [11] reports in his paper on the characteristics of mixed flow pmps that the friction loss coefficient of impeller passages is abot.8 times as large as the friction loss coefficient of a straight pipe. This is based on the analysis on hydralic loss at shock-free flow, where the loss comprised of only friction loss (Normally the hydralic loss incldes losses in bends and divergence, bt in his stdy the losses are classified into shock and friction losses; therefore, in a shock-free flow the loss is all friction loss.). Ths, the vales of Δψ LI /Δψ LIC presented in Table 6 show abot the same reslts as is given by Shirakra [19] and Ida [11], and can be claimed to be reasonable. 09

6 In the case of axial flow impellers, Table 3, however, the vales of Δψ LI /Δψ LIC are 1.0~1.1. In axial impellers, the blades compose straight cascades, the normal spacing between blades t sinβ is nearly eqal to blade width b which is the same from inlet to otlet, and accordingly the cross sections of flow passages remain almost nchanged. In contrast, blades compose circlar cascades in centrifgal and mixed flow impellers; therefore, the peripheral spacing between blades expands remarkably from inlet to otlet, althogh the expansion ratio is abot the same as that of the axial flow impellers. That is, the passage crosssections change from a nearly sqare section at the inlet to a slender rectanglar section at the otlet. This is considered to be the main reason for the difference in the vales of Δψ LI /Δψ LIC for centrifgal and mixed flow impellers verss that for axial impellers. As a conclsion, the flat plate friction eqation can be applied for estimating friction loss in axial flow impellers which have no serios problems. As for its application for centrifgal and mixed flow impellers, de consideration mst be given to that the eqation does not offer a complete soltion. Table 7 Converted Performance of Prototype pmp for a case of D P /D M = 8 and β =0º, reported previosly by the Athor [1] Ns η M η hm ψ th /ψ thm η hp ψ P /ψ M η h /η hm Difference in the Theoretical Head Coefficient Difference in the ratio of theoretical head coefficient of the model and prototype, calclated by eq. (7), verss the theoretical head coefficient of the model, Δψ th /ψ thm, gives the change in inpt power coefficient of the prototype. As far as axial impellers are concerned (Table ), the vales of Δψ th /ψ thm are ~ 0.01, which correspond to 45 ~ 48% of hydralic efficiency increase Δη h /η hm ( = 0.0~0.05), i.e. the decrease in the inpt power coefficient. The reslts of hydralic efficiency conversion reported in the previos paper[1] are shown in Table 7. For the case of Ns = 1600, the vale of Δψ th /ψ thm is and abot a half of hydralic efficiency increase Δη h /η hm ( = 0.017) in negative vale. In contrast, for the cases of centrifgal and mixed flow impellers (Ns = 00~110), Table 5, the vales of Δψ th /ψ thm are ~ , which correspond to abot 38 ~ 1% of the vales of hydralic efficiency increase Δη h /η hm ( = 0.081~0.05), i.e. the decrease in the inpt power coefficient. However, the reslts of the previos paper [1] give the vales of Δψ th /ψ thm of ~ 0.009, corresponding to abot 80 ~ 50% of hydralic efficiency increase Δη h /η hm ( = 0.0 ~ 0.018) in negative vale. As a conclsion, the flat plate friction eqation can be applied for estimating a decrease in the inpt power coefficient of prototype pmp for axial flow impellers withot any serios problem. The eqation can also be sed for centrifgal and mixed flow impellers; however, it is noted that the amont of decrease is abot a half of that calclated by the efficiency conversion formla. 5.3 Working Area of Friction-Drag In the analysis of the measred hydralic loss, the impeller passage is sbstitted by a channel comprising 4 flat srfaces [19] or a pipe with an eqivalent hydralic mean depth [11]. When the impeller work is discssed based on the forces working on blades, not only the friction forces on both pper and lower blade srfaces bt also the forces on both disk and shrod srfaces shold be taken into accont (see eq. ()). This is becase the power is transmitted to water throgh the impeller passage comprising 4 srfaces. The impeller work cannot be carried ot only by the blade srfaces. Withot disk and shrod srfaces power cannot be transmitted to the water discharged from the impeller. When the slip factor of blades is examined, with consideration given on change in friction-drag related with scale effect, it is not reasonable to stdy the effect of friction-drag referring to the friction forces only on blade srfaces. The effect of disk and shrod srface friction in the stdy of scale effect becomes larger with decreasing vale of Ns, as the blade height becomes lower compared with impeller otlet diameter. 6. Conclsions The difference in theoretical head coefficient (eqal to inpt power coefficient) of a model and a prototype with a scale ratio of 8 and srface roghness ratio of was obtained, where the flat plate friction coefficient eqation was applied for estimating the friction-drag on impeller blades. In addition, the difference in the hydralic loss coefficient of the model and prototype impellers was obtained based on the efficiency step-p calclated by the conversion formla. The validity of sbstitting a flat plate friction-drag eqation for estimating the friction loss in an impeller was investigated based on a comparison of these differences. The reslts are smmarized as follows. (1) The hydralic efficiency of the prototype pmps shows higher vales than that of the models by ~ depending on Ns of 00 ~ () The theoretical head coefficient, eqal to inpt power coefficient, of the prototype shows a decrease of abot a half of hydralic efficiency increase between the model and prototype for axial flow impellers, bt a decrease of abot 40 ~ 3% of the hydralic efficiency increase for centrifgal and mixed flow impellers. 10

7 (3) The difference in impeller hydralic loss between the model and prototype, calclated sing the flat plate friction-drag eqation, agrees with that derived by the hydralic efficiency conversion for axial impellers, bt is abot 1/3 for centrifgal and mixed flow impellers. (4) The difference shown above among axial flow, centrifgal and mixed flow impellers is considered to be cased by the difference between linear and circlar cascades. (5) The flat plate friction-drag eqation can be dly applied for discssion of the relation between hydralic efficiency increase and inpt power coefficient decrease of the prototype pmp, provided that the hydralic friction loss calclated gives nderestimated vales for a circlar cascade. (6) When examining impeller work with hydralic friction considered, not only the blade srfaces bt also the disk and shrod srfaces shold be taken into accont, as these 4 srfaces work conjointly to deliver water throgh the impeller. A b C D g H k s l Ns Q S t v w W z β δ E η ρ φ ψ Nomenclatre Lift force working on a blade Passage width Coefficient Impeller diameter Acceleration de to gravity Head Srface roghness Blade length Specific speed Flow rate Area Peripheral spacing of blades Peripheral velocity Absolte flow velocity Relative flow velocity Drag force working on a blade passage Nmber of blades Blade angle or angle of relative flow Friction loss ratio (total friction loss head / pmp total head) Efficiency Flid density Flow coefficient = v a / m Head coefficient = gh / m A a C D F h I i L M m o P th v 1 Sbscripts Lift force Meridian component Calclated by efficiency conversion eqation Profile drag Friction drag hydralic Impeller Hb or disk Hydralic loss Model Sqare mean of vales at disk and shrod Tip or shrod Prototype Theoretical Peripheral component Volmetric Impeller inlet Impeller otlet Representative vale (vale at arithmetic mean diameter of inlet and otlet of central blade section dividing blade passage eqally) References [1] Oshima, M., 010, A Stdy on Performance Conversion from Model to Prototype, IJFMS, Vol. 3, No. 3, Jly-Sept., pp. 1-6 [] IEC 60193:1999, Hydralic trbines, storage pmps and pmp-trbines Model acceptance tests, nd ed. [3] JIS B 837: 00, Testing method for performance of pmp, sing model pmp [4] IEC 6097:009, Hydralic machines, radial and axial Performance conversion method from model to prototype, 1st ed. [5] JIS B 837: 013, Testing method for performance of pmp, sing model pmp [6] Krokawa, T., 1959, Efficiency conversion of centrifgal pmp from model to prototype based on test reslts, Jornal. JSME, Vol. 6, No. 485, pp (in Japanese) [7] Htton, S. P. and Fay, A., 1974, Scaling p head, flow and power crves for water trbines and pmps, Proc. IAHR Symp. on Hydralic Machinery and Cavitation, Vienna, pp. X4-1~X4-19 and [8] Ida, T., 00, Scale effect on actal Eler s head in hydralic trbomachinery, Trbomachinery, Vol. 30. No.6, pp (in Japanese) [9] Pelz P. F. and Stonjek, S. S., 013, A Second Order Exact Scaling Method for Trbomachinery Performance Prediction, IJFMS, Vol. 6, No. 4, pp [10] Bellary, S. A. and Samad, A., 015, Nmerical Analysis of Centrifgal Impeller for Different Viscos Liqids, IJFMS, Vol. 8, No. 1, pp [11] Ida, T., 1964, On the Characteristics of Mixed-Flow Propeller Pmps, Blletin JSME, Vol. 7, No.8, pp [1] Schlichting, H., 1979, Bondary Layer Theory, 7th ed., McGraw Hill, New York, p.654 [13] Toyokra, T., 1960, Stdies on Characteristics of Axial Flow Pmps (Part 5, Experimental Investigation on the Impeller Performance), Blletin JSME, Vol. 4, No. 14, pp [14] Toyokra, T., 1960, Stdies on Characteristics of Axial Flow Pmps (Part 6, Experimental Stdies on Hydralic Losses), Blletin JSME, Vol. 4, No.14, pp [15] Oshima, M., 197, Blade Characteristics of Axial flow Impellers, Proc. nd Intern. JSME Symp. Flid Mach. and Flidics, 11

8 Tokyo, pp [16] Oshima, M., 1976, Streamlines throgh axial flow impeller and axial thrst, Trbomachinery, Vol. 4, No., pp (in Japanese) [17] Horie, C. and Kawagchi, K., 1959, Cavitation Tests on Axial Flow Pmp, Blletin JSME, Vol., No. 5, pp [18] Stepanoff, A. J., 1957, Centrifgal and Axial Flow Pmps, John Wiley and Sons, New York, nd ed. p.79 [19] Shirakra, M., 1959, Effect of srface roghness of impeller passages on volte pmp performance, Jornal. JSME, Vol. 6, No.485, pp (in Japanese) 1