Calculation of Basic Elements of A Radial-type Centrifugal Pump.

Calculation of Basic Elements of A Radial-type Centrifugal Pump.

A comparative study for blade shape configurations and blade curvature of Impeller and Spiral Shell.

1. tatharas (corresponding author)

   Professor, General Department of Engineering Science , National & Kapodistrian University of Athens , 34400 Psachna Evias, Chalkis Greece

   P.S.Filiousis

   Mechanical Engineer, External Research Fellow

   V.N.Vlachakis

   Mechanical Engineer, University of Thessaly-Greece, M.Sc.Virginia Tech. Research Fellow

   N.W.Vlachakis Professor, Research Fellow

   Note: Except for the first author's name, the names of the other three authors are in alphabetical order

   AbstractA contribution on centrifugal pump characteristics prediction is presented using relations based on performance maps of known pumps developed by some researchers. The introduced modelling equations based on the blade shape configurations, approximate satisfactorily the geometry of industrial centrifugal pumps elements (vane, shell, meridional section etc.). Also a fast calculation of the distribution of the blade angles in the entire blade curvature is developed using smart algorithms eliminating the need of time-consuming tables of classical methodologies.

   KeywordsCentrifugal pump , Volute casing , algorithms for design, flow characteristics, Velocity triangles, blade shape configurations, Blade Profiles, wrap angle, diagram of the impeller.

   1. INTRODUCTION

Pumps are one of the most interesting devices in mechanical engineering and constitute the heart of a large sector of industrial and public networks, piping networks etc. There is a wide range of pump types quite different from each other in terms of operating principle, structure and the applications they cover. A basic category, however, is centrifugal pumps of radial type.

Centrifugal pumps use the centrifugal force created by a rotating disk on which there are vanes of a special shape and which is known as an impeller or impeller.Fluid flow in centrifugal pumps is created by the centrifugal kinetic energy produced by the rotary motion of the impeller. The aspirated liquid reaches the suction opening and is swept into rotation driven by the vanes. The rotary motion of the impeller imparts rotation to the liquid mass which is driven by the vanes imparting centrifugal force to the liquid. The liquid is forced to flow along the vanes and ejected from the impeller. As soon as the liquid escapes from the impeller, it is collected in an internal space of the pump, which has a spiral shape with an ever-increasing cross-section and finally leaves the pump outlet.

Although there is a wide range of centrifugal pumps there are common comparative advantages of all types of centrifugal pumps over other types of pumps. These are their laminar and smooth operation but mainly their operational flexibility in the sense of the possibility of choosing the desired operating range according to the H-Q (Head-Discharge) diagram. At the same time, centrifugal pumps generally require little space compared to other types of pumps for the same pressure gauge or flow rate. Centrifugal pumps, however, have the ability to offer pump types with very high outputs that other types of pumps cannot achieve and for this reason they are usually preferred in high output lines.

The basic parts of centrifugal pumps are the impeller, the volute and the inlet and outlet ducts (Fig.1.1)

Fig. 1.1. Main parts of a radial-type centrifugal pump

For the design of individual parts of centrifugal pumps, one can find in the literature several methods from the simplest to the most complex (e.g. [1],[2],[3]). In this paper we do not intend to review all these methods but to present a relatively simple but at the same time detailed, precise and practical method for the design of the various parts of a centrifugal pump. This method can be used to design any centrifugal pump.

In chapter II, the proposed design method is presented with its simultaneous application to the design of a pump for the purposes of demonstration of the method. The main purpose of this chapter is the detailed and precise design and drawing of the meridional section of the pump, its impeller as well as the configuration of the inlet and outlet sections and the spiral shell.

In chapter III, some empirical relations are proposed for the prediction of basic performance characteristics of a pump. The predictions of these relations are compared with the experimental ones. The comparison concerns the pressure head, the performance and the NPSHR of some pumps from different manufacturers. This comparison is necessary to confirm that the proposed model actually achieves its intended purpose, i.e. for validation purposes.

1. MODELLING PUMP GEOMETRY

   In what follows, the design method of a centrifugal pump will be presented. At the same time, for demonstration purposes, the method will be applied to the design of a pump with given characteristics. The characteristics of the pump to be designed are taken from ([1],[2],[3]) and are shown in Table I.

   TABLE I. characteristics values used for the centrifugal pump to be designed.

   manometric Head H (m)	50
   critical cavitation height Hcr (m)	3
   Flow rate Q(m3/h)	300
   Blade angle at outlet 2 (degrees)	18
   g (m/(s2)	9,81
   Water density, (Kg/m3)	1.000
   Water kinematic viscosity *(m2/s)	0,000001
   Water temperature (0C)	15
   impeller's number of blades z= 2/3	6

   1. Centrifugal Pump Structure Design.

      The procedure for designing the centrifugal pump structure is according to ([5],[7],[9],[15]). By knowing the head pump H and the Thomas' cavitation parameter =1,5, we can calculate the critical cavitation parameter cr:

      The variation of km1 and km2 with nq is shown in Fig.2.1 and Fig.2.2 respectively.

      1,5

      cr= cr= 50 = 0.03

      (2.1)

      0,3

      0,25

      Using Bohls formula, we calculate the rotation specific speed nq:

      3

      0,2

      4 ( )4

      cr= 0.077 nq nq= 0.077

      km1 0,15

      3 cr

      nq=

      3

      (0,03)4

      nq= 29.22 rpm

      0.077

      (2.2)

      0,1

      0,05

      0

      By using the formula of specific speed, the pump rotational speed n is obtained:

      3

      0 10 20 30 40 50 60 70 80 90 100

      q= n

      (Q/ 3600)0,5

      3

      H4

      H4

      n= q

      (Q/ 3600)0,5

      nq

      Fig. 2. 1, Inlet flow velocity coefficient km1

      3

      n= 29,22 50

      4

      n= 1903 rpm

      (300/3600)0,5

      It is known that the parameter is defined by the relationship:

      =

      2 2

      = = 1.01

      (2.3)

      km2

      0,25

      0,2

      0,15

      0,1

      n

      0.2

      q

      29,22

      0.2

      (2.4)

      0,05

      0

      The specific pump work Y is:

      TYhe=refogre thHe i=mp9ell.e8r1out5le0t d=iam4e9ter0D.52 isJ: /kg

      (2.5)

      0 10 20 30 40 50 60 70 80 90 100

      nq

      D2= 0.45

      ( Y/ )0.5

      (n/ 60)

      Fig.2.2. Outlet flow velocity coefficient k

      m2.

      ( 490.5/1.01)0.5

      The meridional flow velocity at the impeller inlet is:

      0.5

      D2= 0.45

      = 0.31 m

      1903/ 60)

      V1m= cm1= km1 (2 Y)

      (2.6)

      V = c

      = 0.12 (2 490.5)0.5= 4.02 m/s

      1m m1

      (2.12)

      The impeller inlet diameter is selected to be:

      The meridional flow velocity at the impeller outlet is.

      D1= 0.48 D2= 0.15 m

      V = c = k

      (2 Y)0.5

      Then the outlet tangential velocity u2 is:

      (2.7)

      2m m2

      V = c

      m2

      = 0.11 (2 490.5)0.5= 3.52 m/ s

      u2= 3.14 D2 ( n/ 60)

      2m m2

      (2.13)

      u2= 3.14 0.31 (1903/ 60)= 31m/ s

      and the inlet tangential velocity u1 is:

      u1= 3.14 D1 ( n/ 60)

      (2.8)

      1. Calculation of blades' widths

   The flow rate factors k1 and k2 that are used to determine the blades' widths, are:

   D1

   1

   Tuhe=spe3ci.a1l 4speed0c.o1e5fficie(n1ts9km01 3an/d6km02 )ar=e: 14.8m / s

   (2.9)

   k1=

   s1

   D1 z

   1

   km1= 0.07+ 0.002 nq

   km1= 0.07+ 0.002 29.22= 0.12

   and

   km2= 0.063+ 0.0017 nq

   km2= 0.063+ 0.0017 29.22= 0.11

   (2.10)

   k1=

   sin ( )

   180

   3.14 0.15 = 1.53

   3.14 0.15 6 0.0099

   (2.11)

   sin(3.14 15 )

   180

   (2.14)

   where: 1=150 and s1=D1/15=0.15/15=0.0099 m, is the blade thickness at the inlet,

   and

   k2=

   D2

   s1

   D2 z

   k2=

   sin ( 2 )

   180

   3.14 0.31

   = 1.1

   3.14 0.31 6 0.0062

   )

   sin(3.14 18

   180

   (2.15)

   where: 2=180 and s2=D2/50=0.31/50=0.0062 m, is the blade thickness at the exit. The blade's width in the impeller inlet is:

   c

   b1=

   m1

   k1 Q

   3.14 D1

   b = 1.53 300 /3600 = 0.044 m

   1 4.02 3.14 0.15

   (2.16)

   where:

   Q= cm1 A1

   and A1 is the section area at the impeller inlet.

   The blade's width in the impeller outlet is:

   c

   b2=

   m2

   k2 Q

   3.14 D2

   2 =

   b = 1.1 300 / 3600 0.024 m

   3.52 3.14 0.31

   (2.17)

   where:

   Q= cm2 A2

   and A2 is the section area at the impeller outlet.

   B. Calculation of velocity diagrams.

   Now we can calculate the other speeds in order to form the inlet and outlet velocity diagrams. The inlet velocity co at the suction cross-section is:

   o q

   c = 0.02 n0.66

   (2 Y)0.5

   co= 0.02 29.

   c = 5.81m/ s= c 220.66 (2 490.5)0.5

   o 1 (2.18)

   So the inlet relative velocity w1 is:

   2 2 0,5

   w1= (u1+ co)

   2 2 0,5

   w1= (14.8 + 5.81 )

   = 15.92m/ s (2.19)

   The outlet relative velocity w2 can be calculated as follows:

   cm2

   w2u=

   w2u=

   tan( 2 )

   180

   3.52

   tan( 18 3.14 )

   180

   = 12.51m / s

   (2.20)

   and: wm2 = cm2 =3.52 m/s , so:

   2 2 0.5

   w2= (w2u + wm2)

   2 (2.21)

   w = (12.512+ 3.522)0.5= 13m/ s

   We can now calculate the absolute velocities:

   cu2=u2-w2u=31-12.51=18.52 m/s (2.22)

   or

   V1u= V1m cot (a1)= cu2= cm1 cot (a1) (2.23)

   V2u= cu2= u2 cm2cot ( 2) (2.24)

   2 2 0.5

   2 2 0.5

   c2= ( cu2+ cm2)

   = (18.52 + 3.52 )

   = 18.85 (2.25)

   Fig.2.3. Shows the velocity triangles at impeller inlet and Fig.2.4 for the utlet. A schematic reprsentation of the velocity triangles at impeller is shown in Fig.2.5.

   			u1	=14,8m	/sec

   7

   6

   5

   m/sec

   c1=5,8m/sec

   w

   =15,9m/sec

   4

   1

   3

   2

   1

   0

   0 2 4 6 8 10 12 14 16

   m/sec

   Fig.2.3. Inlet Velocity Diagram

   	u2	=31m	/sec

   14

   12

   c

   2

   m/sec

   =13m/sec

   10

   =18,8m/sec

   8

   6

   4

   w

   2

   2

   0

   0 5 10 15 20 25 30 35

   m/sec

   Fig.2.4. Outlet Velocity Diagram

   Fig.2.5. The velocity triangles at the impeller of a centrifugal pump, [9]

   1. Calculation of shaft diameter

      We have first to calculate pump's efficiency. The calculation of the performance efficiencies, is derived according to ([5],[8]) :

      Mechanical efficiency

      nm=

      1 = 1

      = 0.968

      or

      nm=

      1+

      0.287

      n

      0.66

      q

      1 =

      1+

      15.05

      n

      1.8

      q

      1+ 0.287

      29.220.66

      1 = 0.966

      1+ 15.05

      29.221.8

      (2.26)

      (2.27)

      volumetric efficiency

      1 1

      q=

      =

      1+

      0.46

      n

      0.84

      q

      = 0.97 1+ 0.46

      29.220.84

      (2.28)

      while the hydraulic efficiency is:

      h=

      1 0.071

      n

      0.25 =

      q

      3600

      1 0.071

      29.22 0.25 = 0.867

      3600

      (2.29)

      The overall Pump efficiency is: t=qmh=0.8. (2.30)

      The water power N is determined from the relationship:

      N= Y Q

      t

      N= 1000 490.5 (300 /3600 )= 51093Watt

      0.8

      (2.31)

      The methodology for designing the blade geometry and the meridian geometry follows Ref.[6]. According to [6] the torsional moment is estimated to be:

      M = N

      d 2 n

      M = 51093 = 256,47 N m

      d 6.28(1903.3/60)

      The shaft diameter is:

      d = 1000 Md

      w (0.2 )0.33

      w

      d = 1000 256,47

      = 29.4 mm

      (2.32)

      (0.2 50)0.33

      where the volute suction diameter of impeller Ds is:

      Ds=0.42D2=0.13m

      (2.33)

   2. The camber surface of the blade meridional profile of impeller. The meridian section of the pump is calculated for its curved parts (internal and external) from the dimensions of the blades diameters and blades thicknesses for various selected functions shown in Table II.

      TABLE II.

      Walls coordinates and boundary conditions for the camber surface of the blade meridional profile of impeller.

      0,25

      0,2

      0,15

      0,1

      0,05

      A		B		D		C
      xa	ya	xb	yb	xd	yd	xc	yc
      0,000795	0,000795	0	0,05	0,11	0,19885	0,09	0,19885
      wall function 1 (=3)				wall function 2 (=2)		wall algorithms y=aexp(cx)
      c	a			c"	a"
      1988,433	0,052496			3998,65	0,00079
      x	y			x	y
      0,000795	0,052496			0,00079	0,00079
      0,010429	0,052614			0,01307	0,00080
      0,020062	0,053345			0,02536	0,00084
      0,029695	0,055301			0,03764	0,00098
      0,039328	0,059245			0,04993	0,00130
      0,048961	0,066295			0,06221	0,00208
      0,058594	0,078316			0,074	0,00415
      0,068228	0,098717			0,086	0,01
      0,077861	0,134196			0,099	0,038
      0,087	0,19885			0,11	0,19885

      0

      -0,25 -0,2 -0,15 -0,1 -0,05 0 0,05 0,1 0,15 0,2 0,25

      -0,05

      -0,1

      -0,15

      -0,2

      -0,25

      Fig.2.6. The Meridian section of the pump

      For the meridian section of the impeller we use the following wall algorithms:

      y= a exp(c x )

      obtaining the x and y values shown in Table III.

      (2.34)

      TABLE III. IMPELLER'S MERIDIAN SECTION COORDINATES

      b1		b2
      xb1	yb1	xb2	yb2
      0,000795	0,000795	0,087494	0,19885
      0,000795	0,051701	0,111356	0,19885

      The resulting meridian section of the impeller is shown in

      0,25

      0,2

      The resulting meridian section of the pump is shown in Fig.2.6.

      0,15

      y

      0,1

      0,05

      0

      0 0,02 0,04 0,06 0,08 0,1 0,12

      x

      Fig.2.7. Meridional profile of impeller

   3. Design of the blades on the impeller

      The design of the impeller's blades follows the procedure described in [4]. The obtained geometrical results are shown in

      y= a ln(ri )+ c (2.36)

      where:

      Table IV.

      TABLE IV. Blade shape configuration using Sigloch methodology.

      0.8 w

      a=

      r i

      ln ( )

      r 2

      and

      1. ln(ri )

         c= w [ 1 r ]

         ln( i )

         r 2

         (I)

         p o i n t s	rx	rx	bx	wx	tx	x	hx	hx »	h» »	bx »»
         	1	mm	m m	m/ se c
         1	0, 07	0,00 91	44 ,3	15 ,9	0, 07	15 ,7	47, 6	0	0	44 ,3	0
         2	0, 08	0,00 91	42	15 ,4	0, 08	15 ,9	41, 8	0, 41	0, 41	40	23,5
         3	0, 09	0,00 91	39 ,8	14 ,9	0, 09	16 ,2	37, 1	0, 36	0, 77	36 ,6	44,2
         4	0, 1	0,00 91	37 ,6	14 ,5	0, 1	16 ,4	33, 1	0, 32	1, 09	33 ,8	62,7
         5	0, 11	0,00 91	35 ,3	14	0, 11	16 ,7	29, 8	0, 28	1, 38	31 ,5	79,3
         6	0, 12	0,00 91	33 ,1	13 ,5	0, 12	17	27	0, 26	1, 64	29 ,5	94,2
         7	0, 12	0,00 91	30 ,8	13	0, 13	17 ,4	24, 6	0, 23	1, 88	27 ,9	107, 8
         8	0, 13	0,00 91	28 ,6	12 ,6	0, 14	17 ,7	22, 5	0, 21	2, 09	26 ,4	120, 2
         9	0, 14	0,00 91	26 ,3	12 ,1	0, 15	18 ,1	20, 6	0, 19	2, 29	25 ,1	131, 5
         1 0	0, 15	– 0,15	24 ,1	11 ,6	0, 16	18 ,5	18, 9	0, 18	2, 47	24	141, 9

         r ( r )

         r r

         =

         1 1

         r r

         2 ( 2 )

         r 1 r 1

         (2.37)

         where -2 < < 2 ()

         With the first blade showing the calculation arrangement shown in Table V. TABLE V. First blade calculation arrangement

         xa	ya	1	1	r1
         0,0744	0	0°	0°	0,0744
         0,0722	0,0314	Wrap angle	23,514°	0,0787
         0,0601	0,0585	141,99°	44,256°	0,0839
         0,0413	0,0799		62,728°	0,0899
         0,0181	0,0954		79,307°	0,0971
         -0,008	0,1052		94,278°	0,1055
         -0,035	0,1098		107,87°	0,1153
         -0,064	0,1097		120,25°	0,127
         -0,093	0,1053		131,59°	0,1407
         -0,124	0,0968		141,99°	0,1569

         The geometry of the pump we selected to design, also determines the wrap angle w as follows [12] :

         r 2

         • ln( )

      r 1

      D2

      • ln ( )

      D1

      w = ( 2 1)

      ln (cos

      ) = ( 2 1)

      ln ( cos )

      [ 1 ]

      cos 2

      )

      ln ( 0,31

      0,15

      w= (18 15)

      [ ln( cos15) ]

      cos18

      = 1410

      [ 1 ]

      cos2

      (2.35)

      For a fast track calculation the distribution of the blade curvaturecan also be proposed here with the algorithms:

      The parameter can independently determine the blade shape y or the thickness of y. The two methods (I) and (II) are almost identical.

      hese shape forms are preferred because laboratory tests have shown that the blades are safe against impact or friction. The results obtained using these shape forms are shown in Table VI. The corresponding blade shape is shown in Fig.2.8.

      TABLE VI.

      FAST TRACK CALCULATION OF THE BLADE CURVATURE

      Some proposed functions according to ([13],[14]) for the blade shape are:

      r

      xa	ya	1	1	r1
      0,0795	0	0°	0°	0,0795
      0,0818	0,0438	Wrap angle	28,2°	0,0928
      0,071	0,0787	141	47,963°	0,1061
      0,0497	0,1084		65,396°	0,1193
      0,0209	0,1309		80,989°	0,1326
      -0,013	0,1453		95,096°	0,1458
      -0,049	0,1514		107,97°	0,1591
      -0,086	0,1496		119,82°	0,1723
      -0,121	0,1407		130,79°	0,1856
      -0,154	0,1253		141°	0,1988

      ln ( r )

      = 2

      r 2

      ln ( )

      r 1

      = ln(r)+c

      and = J1((r/r1)-1)

      with the boundary conditions: r=r1 for =0

      r=r2 for =w

      (2.38)

      (2.39)

      (2.40)

      0,25

      0,2

      0,15

      0,1

      0,05

      The results are shown in Table VII.

      i Sigloch	i ln	i alnr+c	i besselJ1
      0	0	0	0
      25,79029144	23,74	28,223	21,01241572
      47,44841549	44,304	48,002	41,58808522
      66,05705779	62,444	65,448	61,30034879
      82,32884465	78,67	81,054	79,74247507
      96,75664773	93,348	95,172	96,53701991
      109,6948682	106,75	108,06	111,3444756
      121,4064634	119,07	119,92	123,8710021
      132,0916559	130,49	130,89	133,8750534
      141,9062476	141,11	141,11	141,1727423

      TABLE VII. BLADE SHAPE CONFIGURATION FOR DIFFERENT ALGORITHMS

      0

      -0,05

      -0,1

      -0,15

      -0,2

      -0,25

      -0,25 -0,2 -0,15 -0,1 -0,05 0 0,05 0,1 0,15 0,2 0,25

      Fig.2.8. The blade shape can also be specified with other functions

      Fig.2.9 shows the blade profile and the impeller morphology for 6 blades with blade curvature according to Sigloch methodology.

      0,2

      0,15

      TABLE VIII.

      BLADE THICKNESS CONFIGURATION FOR

      EXPONENTIAL FORM FOR THE CURVATURE OF THE BLADES.

      ya	1	1	r1
      0	0°	0°	0,074409
      0,033329	Wrap angle	23,51429°	0,083576
      0,064696	141,9887°	44,25575°	0,092743
      0,090556		62,72826°	0,10191
      0,109134		79,30663°	0,111077
      0,119917		94,27784°	0,120244
      0,123208		107,8671°	0,129411
      0,119777		120,2548°	0,138579
      0,110618		131,5881°	0,147746
      0,096785		141,9887°	0,156913

      0,1

      0,05

      0

      -0,05

      -0,1

      -0,15

      -0,2

      -0,2 -0,15 -0,1 -0,05 0 0,05 0,1 0,15 0,2

      Fig.2.9. Blade profile, mpeller morphology for 6 blades with blade curvature according to Sigloch methodology.

      We can obtain an alternative blade thickness by choosing

      an exponential form for the blades' curvature. The results are shown in Table VIII and Fig.2.10.

      0,25

      0,2

      0,15

      0,1

      0,05

      0

      -0,05

      -0,1

      -0,15

      -0,2

      -0,25

      -0,25 -0,2 -0,15 -0,1 -0,05 0 0,05 0,1 0,15 0,2 0,25

      Fig.2.10. Two-dimensional diagram of the impeller [12] , blade thickness morphology by creating another blade of different curvature to the original one -for6 blades- according to the Sigloch or Menny methodology for the same wrap angle.

      With different wrap angle and blade thickness on the outlet side we would have different blade curvature distributions as shown in Tables IX-XI and Figs.2.11-2.12.

      TABLE IX. Blade curvature distribution with different wrap angle and blade thickness on the outlet.

      xa	ya	1	1	r1
      0,0744	0	0°	0°	0,0744
      0,0758	0,0353	Wrap angle	24,9699°	0,0836
      0,0622	0,0688	157,33°	47,9134°	0,0927
      0,0368	0,095		68,8893°	0,1019
      0,004	0,111		87,9635°	0,1111
      -0,031	0,1161		105,206°	0,1202
      -0,066	0,1114		120,689°	0,1294
      -0,097	0,099		134,487°	0,1386
      -0,123	0,0813		146,674°	0,1477
      -0,145	0,0607		157,326°	0,1569

      0,25

      0,2

      0,15

      0,1

      0,05

      0

      -0,05

      -0,1

      -0,15

      -0,2

      -0,25

      -0,3 -0,2 -0,1 0 0,1 0,2 0,3

      Fig.2.11. Morphologies of blade thickness at the exit with the creation of another

      rx	rx	bx	wx	cxm
      0,074409	0	0,04433	15,92	4,02	0	0°
      0,083438	0,009167	0,042088	15,45	3,78	24,96	24,9°
      0,092466	0,009167	0,039846	14,97	3,6	22,94	47,9°
      0,101495	0,009167	0,037605	14,5	3,48	20,97	68,8°
      0,110524	0,009167	0,035363	14,03	3,4	19,07	87,9°
      0,119553	0,009167	0,033121	13,56	3,35	17,24	105,2°
      0,128581	0,009167	0,03088	13,09	3,34	15,48	120,6°
      0,13761	0,009167	0,028638	12,62	3,36	13,79	134,4°
      0,146639	0,009167	0,026396	12,15	3,42	12,19	146,6°
      0,155667	0,009167	0,024155	11,68	3,52	10,65	157,3°

      blade with a different curvature from the original, according to the Menny methodology for another wrap angle TABLE X. MENNY METHOD FOR BLADE SHAPE CONFIGURATION

      The camber line profiles of impellers with different blade

      wrap angle w according to [10], are described in Table XI and Fig.2.12.

      TABLE XI. BLADE SHAPE CONFIGURATION USING DIFFERENT ALGORITHMIC FUNCTIONS AND WITH DIFFERENT WRAP ANGLES IN DETERMINING THEIR

      CURVATURE AS A FUNCTION OF THE BLADE ANGLES B.

      r"	linear		xa	ya
      0,07954	17,29478	0	0,079539939	0
      0,092797	17,64068	27,80325	0,082094294	0,043264
      0,106053	17,98657	50,823124	0,067032551	0,082182
      0,11931	18,33247	70,185765	0,040512377	0,112221
      0,132567	18,67837	86,667782	0,007806968	0,132336
      0,145823	19,02426	100,82799	-0,02726674	0,143251
      0,15908	19,37016	113,08217	-0,06222094	0,146407
      0,172337	19,71605	123,74807	-0,09558327	0,143401
      0,185593	20,06195	133,07401	-0,12658977	0,13572
      0,19885	20,40784	141,25763	-0,15494082	0,12

      0,2

      0,5

      0,15

      0,1

      0,05

      0

      -0,05

      -0,1

      -0,15

      -0,2

      -0,2 -0,15 -0,1 -0,05 0 0,05 0,1 0,15 0,2

      Fig.2.12. Various blades of different algorithmic functions and with different wrap angles in determining their curvature as a function of the blade angles .

   4. Geometry and Coordinates of Spiral Shell and Diffuser For the design of the Spiral Shell and the Diffuser, the procedure described in [6] and [11] is followed. The geometrical results obtained are shown in Tables XII-XIV and Figs. 2.13-2.14.

   0,4

   0,3

   0,2

   0,1

   0

   -0,1

   -0,2

   -0,3

   -0,4

   -0,4 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5

   Fig.2.14. The diffuser geometry

   The Logarithmic-Shaped Spiral Model	The Archimedes" Shaped Spiral Model
   Shell in the form	Shell in the form
   r=aexp(c)	r=ar2 + c
   with the boundary conditions	with the boundary conditions
   r=r2 for =360	r=r2 for =360
   r=2r2 for =720	r=2r2 for =720

   TABLE XII. Geometry and coordinates' models for the Spiral Shell

   0,3

   0,2

   0,1

   0

   -0,1

   TABLE XIII. Spiral shell geometry

   	r	xr	yr		r	xr	yr
   360	0,19	0,19	-0,0006	360	0,19	0,19	-0,00063
   390	0,21	0,18	0,1	390	0,22	0,19	0,107066
   420	0,22	0,11	0,19	420	0,23	0,11	0,200478
   450	0,24	0,0009	0,24	450	0,24	0,0009	0,24856
   480	0,25	0,12	0,21	480	0,26	-0,13	0,230173
   510	0,26	-0,22	0,13	510	0,28	-0,24	0,141951
   540	0,28	-0,28	0,0013	540	0,29	-0,29	0,001425
   570	0,29	-0,26	-0,14	570	0,31	-0,27	-0,15605
   600	0,31	-0,15	-0,27	600	0,33	-0,16	-0,28613
   630	0,33	-0,0018	-0,33	630	0,34	-0,0019	-0,34798
   660	0,35	0,17	-0,30	660	0,36	0,18	-0,31678
   690	0,37	0,323	-0,18	690	0,38	0,32	-0,19258
   720	0,39	0,39	-0,002	720	0,39	0,39	-0,00253

   -0,2

   -0,3

   -0,4

   -0,4 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5

   Fig.2.13. The spiral shell geometry

   TABLE XIV. Geometry and coordinates of the diffuser (Diffuser angle =4)

   0,4

   A		B		C		D
   x	y	x	y	x	y	x
   r2	0	2 r2	0	r2-2r2tan	2 r2	2r2 +2r2 tan
   0,19885	0	0,3977	0	0,1710541	0,3977	0,425495
   	Diffuser walls				Discharge Nozzles
   	AC		BD		CD
   	x	y	x	y	x	y
   	0,19885	0	0,3977	0	0,171054
   	0,171054	0,3977	0,425495	0,3976997	0,425495	0,4

2. MODELLING PUMP PERFORMANCE

   Since in the previous section we obtained to design a pump using our model, it would be nice if we could have an estimate of the usual performance characteristics of this pump, ie efficiency, head and cavitation height in relation to the pump's discharge.

   The best way to have an estimate of these critical parameters of the pump we designed is obviously to construct this pump and test it in the test rig. However, this is a quite costly and time-consuming procedure. An alternative way is to use or develop empirical relationships for the performance of our pump based on the performance measurements of a number of commercial pumps from various manufacturers that have published performance measurements. At the end you should compare our model predictions with these measurements. We do this in subchapter A for head and efficiency and in subchapter B for cavitation height.

   1. Head and efficiency prediction

      Based on the logic we described just before we selected 9 pumps whose design features are shown in Table XV. The published performance maps of these pumps digitized using appropriate software. Using the digitized performance maps we proceeded to develop some useful relationships for the efficiency curve (subsection 1) and head (subsection 2). In subsection 3, the predictions of the new relationships we propose are compared with the measurement data of the 9 pumps.

      Pump param eters	Ref.[ 19] Pump	Ref.[2 0] Pump	Ref.[2 1] Pump	Ref.[ 22] Pump	Ref.[2 3] Pump	Ref.[2 4] Pump	Ref.[ 25] Pump	Ref.[26 ] Pump	Ref.[27] Pump
      QBEP (m3/h)	40	10,54	200	2,88	4,8	550	50	144	4500
      QBEP (m3/s)	0,011 11	0,002 9	0,0555	0,000 8	0,0013	0,152 7	0,013 88	0,04	1,25
      Qmax	80	21,09	400	5,78	9,6	1100	100	288	9000
      HBEP	30	16	32	4,8	11	50	35	60	60
      Hshut 0ff	35,1	18,7	37,44	5,616	12,87	58,5	40,95	70,2	70,2
      n	2850	2900	1480	1500	2800	1480	2850	1800	600
      nq	23,4	19,62	25,92	13,08	16,92	30,76	23,34	16,69	31,11
      nspec	0,39	0,327	0,432	0,218	46,666	24,66	47,5	30	10
      D	0,17	0,1	0,3146	0,06	0,0707	0,47	0,18	0,2759	1,0933
      D2	168	95	322	120	105	420	174	320	1100
      D1	70	45	148	40	20	200	74	145	450
      z	6	6	6	6	8	6	6	7	6
      b2	12	8	12	15	5,2	34	12	12	100
      2	11	40	25	20	11	26	31	28	37
      1	9,32	33,89	21,18	10	9,32	22,03	37	11,6	21

      TABLE XV. DESIGN PARAMETERS FOR THE SELECTED PUMPS.

      Where:

      QBEP : Design flow rate

      Qmax : Maximum discharge (m3/h) D: Impeller diameter (m)

      nspec : Specific speed ( r/min) n: Rotational speed (r/min)

      D1: Impeller inlet diameter (mm) D2: Impeller outlet diameter (mm)

      b2: Blade's width in the impeller outlet (mm)

      HBEP: Head at BEP (best efficiency point) in meters

      TABLE XVI. SOME CALCULATED PARAMETERS FOR THE SELECTED PUMPS.

      Pump param eters	Ref.[1 9] Pump	Ref.[ 20] Pump	Ref.[21 ] Pump	Ref.[2 2] Pump	Ref.[2 3] Pump	Ref.[2 4] Pump	Ref.[2 5] Pump	Ref.[2 6] Pump	Ref.[27] Pump
      c1m	2,83	1,93	3,05	1,009	1,52	4,11	4,11	3,54	4,53
      c2m	2	1,153	1,995	0,753	1,230	2,602	2,076	2,411	2,763
      	0,986	0,993	0,991	0,983	0,989	0,991	0,994	0,984	0,989
      c2u	14,40	12,94	20,44	7,19	8,88	26,83	22,21	25,15	30,50
      u2	25,05	14,41	24,93	9,42	15,38	32,53	25,95	30,14	34,54
      u1	10,44	6,82	11,46	3,14	2,93	15,49	11,03	13,65	14,13

      Where:

      c2m: inlet flow velocity (m/s), nq<24 r/min c2u: swirl velocity at the outlet (m/s)

      : the slip value

      1. Proposed relations for predicting head curves

         The calculational procedure for predicting the manometric head curves for the 9 selected pumps of Table XV, has as follows: Hshut off = 1,17HBEP (3.1)

         Where the head at BEP (best efficiency point) HBEP is known from the experiment. However HBEP can also be calculated by the following empirical formula:

         H = 0.575 u2/ g

         BEP 2 (3.2)

         By knowing HBEP in the present work we propose the following formulas for predicting the variation of head vs the flow rate:

         0,5

         H= Hshutoff[cos(0,51,57 j)]

         2

         (3.3a)

         H= Hshutoff[1 0.2 j

         ] (3.3b)

         3

         H= Hshutoff[ 1 0.125 j

         ] (3.3c)

         2

         H= Hshutoff[ 2 exp(0.15 j )] (3.3d)

         H= H erfc(0.17 j2)

         shutoff (3.3e)

         0,72

         H= Hshutoff[Besselj0( j)]

         (3.3f)

         shutoff os(0,5 j ) (3.3g)

         Where erfc is the complementary error function given by:

         erfc(z)=1-erf(z) and j is given by:

         j=Q/QBEP (3.4)

         The discharge at BEP (best efficiency point) QBEP is known from the experiment. However QBEP can also be calculated by the following empirical formula:

         3.695 0.49

         QBEP= 3600(0.1 u2 b2 D2 )

         where:

         QBEP in m3/h u2 in m/sec and D2 and b2 in m

         (3.5)

         a) Calculational results for head predictions.

         For predicting the head-discharge variation of the selected pumps of Table XV we use the formulas (3.3a) to (3.3g). The calculational results are shown in Tables XVIII to XXVI and Figs. 3.1 to 3.17.

         TABLE XVII. PRESENT MODEL PREDICTIONS FOR [19] PUMP CHARACTERISTICS

         Ref.[19] Pump			Predictions
         Q (m3/h )	j= Q/QBEP	H (m)	Eq.(3. 3a)	Eq.(3. 3b)	Eq.(3. 3c)	Eq.(3. 3d)	Eq.(3. 3e)	Eq.(3. 3f)	Eq.(3. 3g)
         0	0	35,2	35,1	35,1	35,1	35,1	35,1	35,1	35,1
         10	0,26	35	34,74	34,64	35,03	34,75	34,66	34,69	34,81
         20	0,5	34	33,67	33,25	34,51	33,69	33,33	33,45	33,95
         30	0,75	32	31,85	30,95	33,1	31,84	31,13	31,42	32,54
         40	1	29	29,22	27,72	30,37	29,1	28,09	28,64	30,58
         50	1,25	25	25,67	23,56	25,85	25,29	24,31	25,15	28,13

         TABLE XVIII. PRESENT MODEL PREDICTIONS FOR [20] PUMP CHARACTERISTICS

         Ref.[20] Pump			Predictions
         Q (m3/h)	j= Q/QBEP	H (m)	Eq.(3. 3a)	Eq.(3. 3b)	Eq.(3. 3c)	Eq.(3. 3d)	Eq.(3. 3e)	Eq.(3. 3f)	Eq.(3. 3g)
         0	0	17,5	17,25	17,26	17,26	17,26	17,26	17,26	17,26
         7,2	0,83	15,5	15,35	14,84	15,99	15,34	14,95	15,12	15,77
         14,4	1,67	8,3	8,69	7,58	7,13	8,24	8,63	9,13	11,55

         TABLE XIX. Present model predictions for [21] pump characteristics

         Ref.[21] Pump			Predictions
         Q (m3/h)	j= Q/QBEP	H (m)	Eq.(3. 3a)	Eq.(3. 3b)	Eq.(3. 3c)	Eq.(3. 3d)	Eq.(3. 3e)	Eq.(3. 3f)	Eq.(3. 3g)
         0	0	36,3	37,44	37,44	37,44	37,44	37,44	37,44	37,44
         50	0,25	36	37,07	36,97	37,37	37,09	36,99	37,02	37,15
         100	0,5	35	35,98	35,57	36,86	36,01	35,65	35,77	36,28
         150	0,75	33,5	34,14	33,23	35,47	34,14	33,41	33,71	34,84
         200	1	31	31,48	29,95	32,76	31,38	30,33	30,88	32,86
         250	1,25	26,5	27,91	25,74	28,3	27,55	26,48	27,33	30,36
         275	1,37	24	25,71	23,28	25,27	25,16	24,32	25,3	28,93

         TABLE XX. PRESENT MODEL PREDICTIONS FOR [22] PUMP CHARACTERISTICS

         Ref.[22] Pump			Predictions
         Q (m3/h)	j= Q/QBEP	H (m)	Eq.(3. 3a)	Eq.(3. 3b)	Eq.(3. 3c)	Eq.(3. 3d)	Eq.(3. 3e)	Eq.(3. 3f)	Eq.(3. 3g)
         0	0	6	5,61	5,62	5,62	5,62	5,62	5,62	5,62
         0,72	0,25	5,9	5,56	5,55	5,61	5,56	5,55	5,55	5,57
         1,44	0,5	5,7	5,39	5,34	5,53	5,4	5,35	5,36	5,44
         2,16	0,75	5,4	5,12	4,98	5,32	5,12	5,01	5,06	5,23
         2,88	1	4,9	4,72	4,49	4,91	4,71	4,55	4,63	4,93
         3,6	1,25	4,3	4,18	3,86	4,24	4,13	3,97	4,1	4,55
         4,32	1,5	3,5	3,47	3,09	3,25	3,36	3,31	3,47	4,11

         TABLE XXI. PRESENT MODEL PREDICTIONS FOR [23] PUMP CHARACTERISTICS

         Ref.[23] Pump			Predictions
         Q (m3/h)	j= Q/QBEP	H (m)	Eq.(3. 3a)	Eq.(3. 3b)	Eq.(3. 3c)	Eq.(3. 3d)	Eq.(3. 3e)	Eq.(3. 3f)	Eq.(3. 3g)
         0	0	12,4	12,87	12,87	12,87	12,87	12,87	12,87	12,87
         2	0,42	12,1	12,52	12,42	12,75	12,53	12,44	12,47	12,59
         4	0,83	11,2	11,46	11,08	11,94	11,46	11,16	11,29	11,77
         6	1,25	9,2	9,59	8,85	9,73	9,47	9,1	9,4	10,44
         8	1,66	6	6,55	5,72	5,42	6,22	6,49	6,86	8,65

         TABLE XXII. PRESENT MODEL PREDICTIONS FOR [24] PUMP CHARACTERISTICS

         Ref.[24] Pump			Predictions
         Q (m3/h)	j= Q/QBEP	H (m)	Eq.(3. 3a)	Eq.(3. 3b)	Eq.(3. 3c)	Eq.(3. 3d)	Eq.(3. 3e)	Eq.(3. 3f)	Eq.(3. 3g)
         0	0	61,3	58,5	58,5	58,5	58,5	58,5	58,5	58,5
         200	0,36	60	57,3	56,95	58,15	57,32	57,01	57,11	57,53
         300	0,55	58	55,79	55,01	57,31	55,82	55,15	55,38	56,33
         400	0,73	56	53,64	52,29	55,67	53,65	52,56	52,99	54,66
         500	0,91	53	50,84	48,8	52,98	50,75	49,25	49,96	52,54
         600	1,09	49	47,31	44,53	48,95	47,02	45,28	46,31	49,98
         700	1,28	44	42,97	39,48	43,34	42,34	40,71	42,09	47,01
         800	1,46	38	37,64	33,66	35,87	36,56	35,67	37,32	43,65

         TABLE XXIII. PRESENT MODEL PREDICTIONS FOR [25] PUMP CHARACTERISTICS

         Ref.[25] Pump			Predictions
         Q (m3/h)	j= Q/QBEP	H (m)	Eq.(3. 3a)	Eq.(3. 3b)	Eq.(3. 3c)	Eq.(3. 3d)	Eq.(3. 3e)	Eq.(3. 3f)	Eq.(3. 3g)
         0	0	40	40,95	40,95	40,95	40,95	40,95	40,95	40,95
         10	0,2	39,5	40,7	40,62	40,91	40,7	40,64	40,66	40,75
         20	0,4	39,2	39,94	39,64	40,62	39,96	39,69	39,78	40,13
         30	0,6	38,5	38,66	38	39,84	38,68	38,13	38,32	39,12
         40	0,8	37,5	36,84	35,71	38,33	36,82	35,94	36,31	37,72
         50	1	35,5	34,44	32,76	35,83	34,32	33,17	33,77	35,94
         60	1,2	32	31,41	29,16	32,1	31,08	29,86	30,73	33,8
         70	1,4	28	27,61	24,9	26,9	26,95	26,11	27,21	31,32

         TABLE XXIV. PRESENT MODEL PREDICTIONS FOR [26] PUMP CHARACTERISTICS

         Ref.[26] Pump			Predictions
         Q (m3/h)	j= Q/QBEP	H (m)	Eq.(3. 3a)	Eq.(3. 3b)	Eq.(3. 3c)	Eq.(3. 3d)	Eq.(3. 3e)	Eq.(3. 3f)	Eq.(3. 3g)
         0	0	65	62,01	62,01	62,01	62,01	62,01	62,01	62,01
         72	0,5	63	59,61	58,91	61,04	59,64	59,04	59,24	60,08
         144	1	54	52,15	49,61	54,26	51,97	50,23	51,14	54,42
         216	1,5	39	38,39	34,11	35,85	37,12	36,5	38,29	45,37
         252	1,75	27	27,44	24,03	20,47	25,85	28,62	30,25	39,75

         TABLE XXV. PRESENT MODEL PREDICTIONS FOR [27] PUMP CHARACTERISTICS

         Ref.[27] Pump			Predictions
         Q (m3/h)	j= Q/QBEP	H (m)	Eq.(3. 3a)	Eq.(3. 3b)	Eq.(3. 3c)	Eq.(3. 3d)	Eq.(3. 3e)	Eq.(3. 3f)	Eq.(3. 3g)
         0	0	68	70,2	70,2	70,2	70,2	70,2	70,2	70,2
         2000	0,46	67	67,88	67,21	69,34	67,92	67,33	67,52	68,34
         3000	0,69	66	64,95	63,47	67,29	64,97	63,76	64,22	66,04
         4000	0,92	63	60,75	58,24	63,3	60,63	58,81	59,68	62,86
         5000	1,15	57	55,15	51,51	56,73	54,69	52,58	53,96	58,84
         6000	1,38	50	47,89	43,29	46,92	46,82	45,28	47,15	54,04
         7000	1,62	40	38,34	33,57	33,23	36,58	37,24	39,3	48,53

         Head Prediction for Ref.[19] Pump

         40

         35

         30

         Head (m)

         25

         20

         Measured Eq.(3.3a)

         15 Eq.(3.3b) Eq.(3.3c)

         Eq.(3.3d) Eq.(3.3e)

         10 Eq.(3.3f) Eq.(3.3g)

         5

         0

         0 5 10 15 20 25 30 35 40 45 50

         /h)

         ict [19] pump head

         Q (m^3

         Head Prediction for Ref.[20] Pump

         20

         18

         16

         14

         Head (m)

         12

         10 Measured Eq.(3.3a)

         8 Eq.(3.3b) Eq.(3.3c)

         6 Eq.(3.3d) Eq.(3.3e)

         Eq.(3.3f) Eq.(3.3g)

         4

         2

         0

         0 2 4 6 8 10 12 14 16

         Q (m^3/h)

         Fig.3.2. Using present model to predict [20] pump head

         Fig.3.1. Using present model to pred

         Head Prediction for Ref.[21] Pump

         40

         35

         30

         Head (m)

         25

         20

         Measured Eq.(3.3a)

         15 Eq.(3.3b) Eq.(3.3c)

         Eq.(3.3d) Eq.(3.3e)

         10 Eq.(3.3f) Eq.(3.3g)

         5

         Head Prediction for Ref.[22] Pump

         7

         6

         5

         Head (m)

         4

         Measured Eq.(3.3a)

         3 Eq.(3.3b) Eq.(3.3c)

         Eq.(3.3d) Eq.(3.3e)

         2 Eq.(3.3f) Eq.(3.3g)

         1

         0

         0 0,5 1 1,5 2 2,5 3 3,5 4 4,5

         Q (m^3/h)

         0

         0 50 100 150 200 250 300

         Fig.3.4. Using present model to predict [22] pump head

         Q (m^3/h)

         Fig.3.3. Using present model to predict [21] pump head

         Head Prediction for Ref.[23] Pump

         14

         12

         10

         Head (m)

         8

         6 Measured Eq.(3.3a)

         Eq.(3.3b) Eq.(3.3c)

         4 Eq.(3.3d) Eq.(3.3e)

         2 Eq.(3.3f) Eq.(3.3g)

         0

         0 1 2 3 4 5 6 7 8

         Q (m^3/h)

         Fig.3.5. Using present model to predict [23] pump head

         Head Prediction for Ref.[24] Pump

         70

         60

         50

         Head Prediction for Ref.[25] Pump

         45

         40

         35

         30

         Head (m)

         25 Measured Eq.(3.3a)

         20 Eq.(3.3b) Eq.(3.3c)

         15 Eq.(3.3d) Eq.(3.3e)

         10 Eq.(3.3f) Eq.(3.3g)

         5

         0

         0 10 20 30 40 50 60 70

         Q (m^3/h)

         Fig.3.7. Using present model to predict [25] pump head

         Head Prediction for Ref.[26] Pump

         70

         60

         50

         Head (m)

         40

         Head (m)

         40

         30 Measured Eq.(3.3a)

         Eq.(3.3b) Eq.(3.3c)

         20 Eq.(3.3d) Eq.(3.3e)

         10 Eq.(3.3f) Eq.(3.3g)

         0

         0 100 200 300 400 500 600 700 800

         Q (m^3/h)

         Fig.3.6. Using present model to predict [24] pump head.

         30 Measured Eq.(3.3a)

         20 Eq.(3.3b) Eq.(3.3c)

         Eq.(3.3d) Eq.(3.3e)

         10 Eq.(3.3f) Eq.(3.3g)

         0

         0 50 100 150 200 250 300

         Q (m^3/h)

         Fig.3.8. Using present model to predict [26] pump head

         80 Head Prediction for Ref.[27] Pump

         70

         60

         Head (m)

         50

         40 Measured Eq.(3.3a)

         30 Eq.(3.3b) Eq.(3.3c)

         20 Eq.(3.3d) Eq.(3.3e)

         10 Eq.(3.3f) Eq.(3.3g)

         0

         0 1000 2000 3000 4000 5000 6000 7000

         Q (m^3/h)

         Fig.3.9. Using present model to predict [27] pump head

      2. Proposed relations for predicting efficiency curves

         The calculational procedure for predicting the pump efficiency curves for the 9 selected pumps of Table XV, has as follows:

         The mechanical efficiency is:

         m=

         1

         1.1+ 1

         nq (3.6)

         The volumetric efficiency for nq < 23,5 r/min is:

         1

         q=

         1+

         2

         n

         0.66

         q

         while the volumetric efficiency for nq >23,5 r/min is:

         (3.7)

         The hydraulic efficiency is:

         1

         q=

         1

         1+

         0.2

         n

         0.66

         q

         (3.8)

         h=

         1.1+

         0.9

         n

         0.66

         q

         (3.9)

         As a result the overall efficiency is:

         max=m qh (3.10)

         Summary of calculations is shown in Table XVII.

         TABLE XXVI. EFFICIENCY CALCULATIONS

         Ref.[21] Pump			Predictions
         Q	j	H	Eq. 3.11a	Eq. 3.11b	Eq. 3.11c	Eq. 3.11d	Eq. 3.11e	Eq. 3.11f	Eq. 3.11g
         0	0	0	0	0	0	0	0	0	0
         50	0,25	0,35	0,32	0,34	0,37	0,37	0,38	0,39	0,37
         100	0,5	0,6	0,59	0,63	0,63	0,65	0,63	0,63	0,63
         150	0,75	0,75	0,77	0,8	0,78	0,78	0,79	0,79	0,78
         200	1	0,8	0,84	0,85	0,84	0,8	0,84	0,84	0,84
         250	1,25	0,76	0,77	0,77	0,78	0,73	0,79	0,79	0,78
         275	1,38	0,72	0,7	0,69	0,72	0,68	0,72	0,72	0,72

         TABLE XXVIII. Present model predictions for [21] pump characteristics

         Pump param eters	Ref.[1 9] Pump	Ref.[ 20] Pump	Ref.[21 ] Pump	Ref.[2 2] Pump	Ref.[2 3] Pump	Ref.[2 4] Pump	Ref.[2 5] Pump	Ref.[2 6] Pump	Ref.[27] Pump
         q	0,8	0,78	0,977	0,731	0,731	0,979	0,975	0,762	0,979
         h	0,96	0,943	0,975	0,932	0,873	0,938	0,901	0,920	0,974
         m	0,88	0,868	0,878	0,850	0,862	0,882	0,875	0,862	0,883
         max	0,67	0,639	0,837	0,579	0,551	0,811	0,769	0,604	0,843

         In the present work we propose the following formulas for predicting the variation of pump efficiency vs the flow rate:

         = max sin(1,57 j ) (3.11a) = 3.18 max [ 0.49 j + 1.02 erf (0.89 j )] (3.11b) max (2 j ) (3.11c)

         = [erf (1. j)] [1.14 erf ( 0.12 j3)]

         max (3.11d)

         0.8

         TABLE XXIX. PRESENT MODEL PREDICTIONS FOR [22] PUMP CHARACTERI

         = max[erf (6.28 j)][sin (1.57 j)]

         max

         = [sin(1,57 j)]0.8

         2

         Ref.[22] Pump			Predictions
         Q	j	H	Eq. 3.11a	Eq. 3.11b	Eq. 3.11c	Eq. 3.11d	Eq. 3.11e	Eq. 3.11f	Eq. 3.11g
         0	0	0	0	0	0	0	0	0	0
         0	0,25	0,25	0,22	0,24	0,25	0,26	0,26	0,27	0,25
         0,72	0,5	0,43	0,41	0,43	0,43	0,45	0,44	0,44	0,43
         1,44	0,75	0,53	0,54	0,55	0,54	0,54	0,54	0,54	0,54
         2,16	1	0,56	0,58	0,59	0,58	0,55	0,58	0,58	0,58
         2,88	1,25	0,52	0,54	0,53	0,54	0,51	0,54	0,54	0,54
         3,6	1,5	0,4	0,41	0,41	0,43	0,42	0,44	0,44	0,43

         (3.11e)

         (3.11f)

         = max[1 (1 j) ] (3.11g)

         Where erf is the error function and the factor j is given by:

         1. Calculational results for efficiency predictions.

            For predicting the efficiency-discharge variation of the selected pumps of Table XV we use the formulas (3.11a) to (3.11g). The calculational results are shown in Tables XVIII to XXVI and Figs. 3.1 to 3.17. The symbols in the first three columns in Tables XVIII to XXVI are:

            Q: Flow rate (m3/h) j=Q/QBEP

            H: Head (m)

            Ref.[19] Pump			Predictions
            Q	j	H	Eq. 3.11a	Eq. 3.11b	Eq. 3.11c	Eq. 3.11d	Eq. 3.11e	Eq. 3.11f	Eq. 3.11g
            0	0	0	0	0	0	0	0	0	0
            10	0,26	0,29	0,26	0,28	0,3	0,3	0,31	0,32	0,3
            20	0,51	0,5	0,48	0,51	0,51	0,53	0,52	0,52	0,51
            30	0,77	0,63	0,63	0,65	0,63	0,63	0,63	0,63	0,63
            40	1,03	0,65	0,67	0,67	0,67	0,64	0,67	0,67	0,67
            50	1,28	0,6	0,61	0,6	0,62	0,58	0,62	0,62	0,62

            TABLE XXVII. PRESENT MODEL PREDICTIONS FOR [19] PUMP CHARACTERISTICS

            TABLE XXX. PRESENT MODEL PREDICTIONS FOR [23] PUMP CHARACTERISTICS

            Ref.[23] Pump			Predictions
            Q	j	H	Eq. 3.11a	Eq. 3.11b	Eq. 3.11c	Eq. 3.11d	Eq. 3.11e	Eq. 3.11f	Eq. 3.11g
            0	0	0	0	0	0	0	0	0	0
            2	0,42	0,34	0,34	0,36	0,36	0,38	0,37	0,37	0,36
            4	0,83	0,53	0,53	0,55	0,54	0,53	0,54	0,54	0,54
            6	1,25	0,51	0,51	0,51	0,52	0,48	0,52	0,52	0,52
            8	1,67	0,28	0,28	0,29	0,31	0,34	0,32	0,32	0,31

            TABLE XXXI. PRESENT MODEL PREDICTIONS FOR [24] PUMP CHARACTERISTICS

            Ref.[24] Pump			Predictions
            Q	j	H	Eq. 3.11a	Eq. 3.11b	Eq. 3.11c	Eq. 3.11d	Eq. 3.11e	Eq. 3.11f	Eq. 3.11g
            0	0			0	0	0	0	0	0
            200	0,36			0,47	0,48	0,5	0,5	0,5	0,48
            300	0,55	0,68	0,61	0,65	0,64	0,66	0,65	0,65	0,64
            400	0,73	0,77	0,74	0,77	0,75	0,75	0,75	0,75	0,75
            500	0,91	0,81	0,8	0,82	0,81	0,78	0,81	0,81	0,81
            600	1,09	0,82	0,8	0,81	0,8	0,76	0,8	0,8	0,8
            700	1,28	0,77	0,74	0,73	0,75	0,7	0,75	0,75	0,75
            800	1,46	0,67	0,61	0,61	0,64	0,62	0,65	0,65	0,64

            TABLE XXXII. PRESENT MODEL PREDICTIONS FOR [25] PUMP CHARACTERISTICS

            Ref.[25] Pump			Predictions
            Q	j	H	Eq. 3.11a	Eq. 3.11b	Eq. 3.11c	Eq. 3.11d	Eq. 3.11e	Eq. 3.11f	Eq. 3.11g
            0	0	0	0	0	0	0	<>0	0	0
            10	0,2	0,3	0,24	0,26	0,28	0,28	0,28	0,3	0,29
            20	0,4	0,5	0,45	0,48	0,49	0,51	0,5	0,5	0,52
            30	0,6	0,64	0,62	0,65	0,65	0,66	0,65	0,65	0,68
            40	0,8	0,73	0,73	0,75	0,74	0,73	0,74	0,74	0,78
            50	1	0,77	0,77	0,78	0,77	0,73	0,77	0,77	0,81
            60	1,2	0,75	0,73	0,73	0,74	0,69	0,74	0,74	0,78
            70	1,4	0,67	0,62	0,62	0,65	0,61	0,65	0,65	0,68

            TABLE XXXIII. PRESENT MODEL PREDICTIONS FOR [26] PUMP CHARACTERISTICS

            Ref.[26] Pump			Predictions
            Q	j	H	Eq. 3.11a	Eq. 3.11b	Eq. 3.11c	Eq. 3.11d	Eq. 3.11e	Eq. 3.11f	Eq. 3.11g
            0	0	0	0	0	0	0	0	0	0
            72	0,5	0,46	0,44	0,46	0,46	0,48	0,47	0,47	0,46
            144	1	0,6	0,62	0,62	0,62	0,59	0,62	0,62	0,62
            216	1,5	0,45	0,44	0,44	0,46	0,45	0,47	0,47	0,46
            252	1,75	0,3	0,24	0,26	0,27	0,34	0,29	0,29	0,27

            TABLE XXXIV. PRESENT MODEL PREDICTIONS FOR [27] PUMP CHARACTERISTICS

            Ref.[27] Pump			Predictions
            Q	j	H	Eq. 3.11a	Eq. 3.11b	Eq. 3.11c	Eq. 3.11d	Eq. 3.11e	Eq. 3.11f	Eq. 3.11g
            0	0	0	0	0	0	0	0	0	0
            2000	0,46	0,6	0,56	0,59	0,6	0,62	0,61	0,61	0,6
            3000	0,69	0,78	0,75	0,78	0,76	0,77	0,76	0,76	0,76
            4000	0,92	0,87	0,84	0,85	0,84	0,81	0,84	0,84	0,84
            5000	1,15	0,86	0,82	0,82	0,82	0,77	0,82	0,82	0,82
            6000	1,38	0,76	0,69	0,69	0,72	0,68	0,72	0,72	0,72
            7000	1,62	0,59	0,48	0,5	0,52	0,55	0,54	0,54	0,52

            0,8

            0,7

            0,6

            Efficiency

            0,5

            0,4

            Efficiency Prediction for Ref.[19] Pump

            Efficiency Prediction for Ref.[22] Pump

            0,7

            0,6

            0,5

            Efficiency

            0,4

            0,3

            0,2

            0,1

            0

            Measured Eq.(3.11a)

            Eq.(3.11b) Eq.(3.11c)

            Eq.(3.11d) Eq.(3.11e)

            Eq.(3.11f) Eq.(3.11g)

            0 5 10 15 20 25 30 35 40 45 50

            Q (m^3/h)

            0,3

            0,2

            0,1

            0

            Measured Eq.(3.11a)

            Eq.(3.11b) Eq.(3.11c)

            Eq.(3.11d) Eq.(3.11e)

            Eq.(3.11f) Eq.(3.11g) 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5

            Q (m^3/h)

            Fig.3.10. Using present model to predict [19] pump efficiency.

            Fig.3.12. Using present model to predict [22] pump efficiency.

            Efficiency Prediction for Ref.[21] Pump

            0,9

            0,8

            0,7

            0,6

            Efficiency

            0,5

            0,6

            0,5

            Efficiency

            0,4

            0,3

            Efficiency Prediction for Ref.[23] Pump

            0,4

            0,3

            0,2

            0,1

            0

            Measured Eq.(3.11a)

            Eq.(3.11b) Eq.(3.11c)

            Eq.(3.11d) Eq.(3.11e)

            Eq.(3.11f) Eq.(3.11g)

            0 50 100 150 200 250 300

            Q (m^3/h)

            0,2

            0,1

            0

            Measured Eq.(3.11a)

            Eq.(3.11b) Eq.(3.11c)

            Eq.(3.11d) Eq.(3.11e)

            Eq.(3.11f) Eq.(3.11g)

            0 1 2 3 4 5 6 7 8

            Q (m^3/h)

            Fig.3.11. Using present model to predict [21] pump efficiency.

            Fig.3.13. Using present model to predict [23] pump efficiency.

            0,9

            0,8

            0,7

            0,6

            Efficiency

            0,5

            0,4

            0,3

            Efficiency Prediction for Ref.[24] Pump

            Measured Eq.(3.11a)

            Eq.(3.11b) Eq.(3.11c)

            Efficiency Prediction for Ref.[26] Pump

            0,7

            0,6

            0,5

            Efficiency

            0,4

            0,3

            0,2

            0,1

            0

            Eq.(3.11d) Eq.(3.11e)

            Eq.(3.11f) Eq.(3.11g)

            0 100 200 300 400 500 600 700 800

            Q (m^3/h)

            Fig.3.14. Using present model to predict [24] pump efficiency.

            0,2

            0,1

            0

            Measured Eq.(3.11a)

            Eq.(3.11b) Eq.(3.11c)

            Eq.(3.11d) Eq.(3.11e)

            Eq.(3.11f) Eq.(3.11g)

            0 50 100 150 200 250 300

            Q (m^3/h)

            Efficiency Prediction for Ref.[25] Pump

            0,9

            0,8

            0,7

            0,6

            Efficiency

            0,5

            0,4

            Fig.3.16. Using present model to predict [26] pump efficiency.

            Efficiency Prediction for Ref.[27] Pump

            1

            0,9

            0,8

            0,7

            0,3

            0,2

            0,1

            0

            Measured Eq.(3.11a)

            Eq.(3.11b) Eq.(3.11c)

            Eq.(3.11d) Eq.(3.11e)

            Eq.(3.11f) Eq.(3.11g)

            0 10 20 30 40 50 60 70

            Q (m^3/h)

            Efficiency

            0,6

            0,5

            0,4 Measured Eq.(3.11a)

            0,3 Eq.(3.11b) Eq.(3.11c)

            0,2 Eq.(3.11d) Eq.(3.11e)

            0,1 Eq.(3.11f) Eq.(3.11g)

            0

            Fig.3.15. Using present model to predict [25] pump efficiency.

            0 1000 2000 3000 4000 5000 6000 7000

            Q (m^3/h)

            Fig.3.17. Using present model to predict [27] pump efficiency.

   2. NPSHR Predictions

   NPSH (Net Positive Suction Head) is a measure of the pressure experienced by a fluid on the suction side of a centrifugal pump. It is used to avoid running a pump under conditions which favour cavitation. NPSHR (NPSH Required) and NPSHA (NPSH Available) are two key NPSH values:

   NPSHR is a pump property quoted by pump manufacturers as the suction pressure at which cavitation has already reduced pump performance by 3%.

   NPSHA is a system property calculated from the suction-side system configuration. It is essentially the suction-side pressure less the vapour pressure of the pumped fluid at that point.

   To avoid cavitation, it is necessary to ensure that NPSHA exceeds NPSHR by a sufficient safety margin, for example: NPSHA>NPSHR + 0.5m.

   This margin depends on the type of pump and application and may be quoted as a ratio or a head difference.

   So in order to avoid cavitation problems in a pumping assembly, it is important to calculate the values of NPSHA and NPSHR. In what follows, we present methodologies for estimating both NPSHA and NPSHR as accurately as possible.

   NPSHA calculation [1]

   NPSHA can easily be calculated by the following relation:

   NPSHA = hp hvpa + hst hfs ha (3.12) Where:

   hp: absolute pressure head in meters on the surface of the liquid supply level hvpa: Vapor pressure of the liquid converted to meters

   hst: Static height in meters, difference between liquid level and established pump datum.

   hfs: All suction line losses converted from pressure to meters including piping entrance losses and friction losses through the pipe, valves, etc.

   ha: Acceleration head

   = NPSH/H, Thomas cavitation number

   NPSHR calculation

   Using the pump's affinity law, the cavitation problem is transformed into the following empirical formula used in the present model for estimating the variation of NPSHR,x in relation to the pump's flow rate:

   NPSHR,x=NPSHR,ref (Qx/Qref)0.1 (3.13)

   where the reference values are known from the experiments in the test rig.

   Using Eq. (3.13) we can calculate the variation of NPSHR with respect to Q for the pump we designed in II. The results are shown in Table XXVII and Figure 3.18.

   Q/QBEP	NPSHR,r
   0	0
   0,6	1,67
   0,8	2,3
   1	2,9
   1,2	3,7

   TABLE XXXV. MMMM

   4

   3,5

   3

   NPSHR(m)

   2,5

   2

   1,5

   1

   0,5

   0

   0 0,2 0,4 0,6 0,8 1 1,2 1,4

   Q/QBEP

   Fig. 3.18. Present model test rig centrifugal pump NPSHR variation with discharge flow.

   In order to check the validity of Eq.(3.13) we selected 5 commercial pumps ([29]-[33]) for which the manufacturers provided values for the NPSHR for each pump. We then proceeded to calculate the variation of NPSHR for each pump separately. The results of the calculations are shown in Tables XXVIII to XXXII and Figures 3.19 to 3.23.

   Comparison of cavitation prediction performances for some centrifugal pumps

   Q/QBE	NPSHR.ref	Experiment	Prediction
   0	0
   0,6	2,1	2	2,09
   0,8	2,8	2,65	2,79
   1	3,5	3,3	3,49
   1,2	4,2	4	4,2

   TABLE XXXVI. PREDICTING NPSHR FOR REF.[29] EXPERIMENT

   NPSHR(m)

   TABLE XXXVII. PREDICTING NPSHR FOR REF.[30] EXPERIMENT

   Q/QBE	Experiment	Prediction
   0,67	1,9	2,12
   1	2,65	2,86
   1,3	3,44	3,59
   1,49	4,1	4,29

   TABLE XXXVIII. PREDICTING NPSHR FOR REF.[31] EXPERIMENT

   4,5

   4

   3,5

   3

   2,5

   2

   1,5

   1

   0,5

   0

   NPSHRprediction for Ref.[29]pump

   Measured

   Predicted

   Q/QBE	Experiment	Prediction
   0,6	2	2,09
   0,8	2,7	2,79
   1	3,4	3,49
   1,2	4,1	4,19

   0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3

   Q/QBEP

   Fig. 3.19. Predicting NPSHR for [29] pump.

   TABLE XXXIX. PREDICTING NPSHR FOR REF.[32] EXPERIMENT

   Q/QBE	Experiment	Prediction
   0,8	2	2,16
   0,92	2,63	2,83
   1	3,3	3,49
   1,05	3,94	4,14

   TABLE XL. PREDICTING NPSHR FOR REF.[33] EXPERIMENT

   Q/QBE	Experiment	Prediction
   0,8	2,4	2,16
   0,9	3,05	2,83
   1	3,7	3,49
   1,1	4,4	4,16

   5

   4,5

   NPSHR(m)

   4

   3,5

   3

   2,5

   2

   1,5

   1

   0,5

   0

   NPSHRprediction for Ref.[30]pump

   Measured Predicted

   4,5

   4

   NPSHR(m)

   3,5

   3

   2,5

   2

   1,5

   1

   NPSHRprediction for Ref.[32]pump

   Measured Predicted

   0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5 1,6

   Q/QBEP

   Fig. 3.20. Predicting NPSHR for [30] pump.

   NPSHRprediction for Ref.[31]pump

   0,5

   0

   0,75 0,8 0,85 0,9 0,95 1 1,05 1,1

   Q/QBEP

   Fig. 3.22. Predicting NPSHR for [32] pump.

   			Measured Predicted

   4,5

   4

   NPSHR(m)

   3,5

   3

   2,5

   2

   1,5

   1

   0,5

   0

   0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3

   Q/QBEP

   5

   4,5

   4

   NPSHR(m)

   3,5

   3

   2,5

   2

   1,5

   1

   0,5

   0

   NPSHRprediction for Ref.[33]pump

   Predicted

   Measured

   Fig. 3.21. Predicting NPSHR for [31] pump.

   0,75 0,8 0,85 0,9 0,95 1 1,05 1,1 1,15

   Q/QBEP

   Fig. 3.23. Predicting NPSHR for [33] pump.

3. ESTIMATING PRESENT PUMP PERFORMANCE

   As we said in the previous chapter, the best way to have accurate knowledge of the performance of the pump we designed would be to build it and then measure its performance in the test room. But this is quite difficult, costly and time-consuming. lternatively we can make an estimate of the performances by making use of the empirical relationships we proposed in the previous chapter. This i what we attempt below in A and B.

   1. Estimating present pump head Eq.(2.8) gives: u2=31 m/s.

      By knowing that g=9.81 m/s2, Eq. (3.2) gives:

      BEP 2 2

      H = 0.575 u2/ g= 0.575 312 /9.81= 56.32m

      Therefore, Eq.(3.1) finds:

      Hshut off = 1,17HBEP=1,1756.32=65.9 m

      Eq.(2.6) gives: D2=0.31 m. Eq.(2.17) gives: b2=0.024 m. Therefore, Eq.(3.5) finds:

      3.695 0.49

      QBEP= 3600(0.1 u2 b2 D2 )

      QBEP

      2

      = 3600(0.1 3.1431 0.024 0.313.695)0.49

      3

      QBEP= 211.86m /h

      We can now calculate the pump head via equations (3.3a) until (3.3g). The results of the calculations are shown in the Table XLI where j is calculated according to Eq.(3.4):

      j=Q/QBEP

      In the Table XLI we have divided the interval 0-300 m3/h into intervals of 50 m3/h.

      TABLE XLI. PRESENT PUMP HEAD PREDICTION

      j	Q	Eq.(3. 3a)	Eq.(3. 3b)	Eq.(3. 3c)	Eq.(3. 3d)	Eq.(3. 3e)	Eq.(3. 3f)	Eq.(3.3 g)
      0	0	65,9	65,9	65,9	65,9	65,9	65,9	65,9
      0,24	50	65,34	65,17	65,8	65,35	65,2	65,24	65,45
      0,47	100	63,63	62,97	65,04	63,66	63,09	63,28	64,08
      0,71	150	60,74	59,3	62,98	60,76	59,58	60,04	61,82
      0,94	200	56,61	54,16	58,97	56,48	54,72	55,58	58,7
      1,18	250	51,08	47,55	52,37	50,6	48,62	49,98	54,76
      1,42	300	43,88	39,48	42,52	42,78	41,5	43,31	50,06

      A graphical representation of the results of Table XLI is shown in Figure 4.1.

      Head prediction for present pump

      70

      0,9

      0,8

      Efficiency prediction for present pump

      60

      50

      Head (m)

      40

      Eq.(3.3a) Eq.(3.3b)

      30 Eq.(3.3c) Eq.(3.3d)

      Eq.(3.3e) Eq.(3.3f)

      20 Eq.(3.3g)

      10

      0

      0 50 100 150 200 250 300

      Q (m^3/h)

      0,7

      0,6

      Efficiency

      0,5

      0,4

      0,3

      0,2

      0,1

      0

      Eq.(3.11a) Eq.(3.11b)

      Eq.(3.11c) Eq.(3.11d)

      Eq.(3.11e) Eq.(3.11f) Eq.(3.11g)

      0 50 100 150 200 250 300

      Q (m^3/h)

      Fig. 4.1. Predicting head for present pump.

   2. Estimating present pump efficiency

   Fig. 4.2. Predicting efficiency for present pump.

   Now using MAX=0.8 [from Eq.(2.30)] and Equations 3.11 we can construct a Table similar to Table XLI for predicting pump efficiency. The results are shown numerically in Table XLII and graphically in Fig.4.2.j

   TABLE XLII. Present Pump Efficiency Prediction

   j	Q	EQ.(3.1 1A)	EQ.(3.1 1B)	EQ.(3.1 1C)	EQ.(3.1 1D)	EQ.(3. 11E)	EQ.(3. 11F)	EQ.(3.1 1G)
   0	0	0	0	0	0	0	0	0
   0,24	50	0,29	0,31	0,33	0,34	0,34	0,35	0,33
   0,47	100	0,54	0,57	0,58	0,59	0,58	0,58	0,58
   0,71	150	0,72	0,74	0,73	0,74	0,73	0,73	0,73
   0,94	200	0,8	0,81	0,8	0,77	0,8	0,8	0,8
   1,18	250	0,77	0,77	0,77	0,72	0,77	0,77	0,77
   1,42	300	0,64	0,64	0,66	0,63	0,67	0,67	0,66

4. DISCUSSION OF RESULTS

Observing the comparison results in Chapter III, we can make the following findings: The present model predicts the efficiency and manometric head curves quite well showing a slight underestimation for large dischargerates.

The prediction of NPSHR is equally satisfactory, showing however a systematic trend of slight overestimation in the majority of cases.

CONCLUSIONS

In the present study, a radial centrifugal pump was numerically investigated. Design models of impeller, volute, and combined impeller-diffuser geometry were developed. The impeller and diffuser geometry was developed in order to analyze the effects of the design parameters, including the blade shape, the blade height, the outlet blade angle, the blade width, the blade number, the impeller outer diameter and the overall efficiency. A good agreement was observed, when comparing the model numerical results with the experimental data obtained by a pump test rig.

Moreover, a group of empirical equations was developed in order to estimate the performance curves (H-Q, Q, NPSHR-Q) of a centrifugal pump, using quite simple geometric data. The development of these equations prove that the prediction of centrifugal pumps performance curves could be accomplished without using complex mathematical models.

Furthermore, all-basic parameters that affect the pump behavior are taken into account resulting on very good approach to some original pump performance curves. The developed empirical formulas satisfactorily provide the ability to approximate the potential performance head, efficiency and cavitation curves of some commercial centrifugal pumps.

Nomenclature

English symbols
A	section area	m2
a1	absolute flow angle	degrees
b	Blade section width	m
D	Impeller diameter	m
d	Shaft diameter	mm
g	gravitational acceleration	m/s2
H	head	m
Q	discharge	m3/h or m3/s
Ms	shaft torque	Nm
N	power	Watts
n	rotational speed	rpm
nq	specific speed
r	impeller radius	m
u	peripheral (or tangential) velocity	m/s
Vm = cm	meridional flow velocity	m/s
Vu=cu	peipheral components of the absolute velocity	m/s
w	angular velocity of rotation	rad/s
z	Number of blades
Greek symbols
	Blade angle	degrees
	efficiency
	temperature	Celcius degrees
	Kinematic viscosity	m2/s
	flow density	kg/m3
	Wrap angle	degrees
Subscripts
1	impeller inlet
2	impeller outlet
Abbreviations
BEP	best efficiency point
NPSH	Net Positive Suction Head	m

MEMORANDA

In memory of our professors and teachers in special education and science:

Prof. D. Papailiou, Department of mechanical engineering, University of Patras, Greece.

Prof. H.Marcinowski, Department of mechanical engineering, University of Karlsruhe,Germany.

[1] K.Menny, Stroemungsmaschinen, Teubner Verlag 1985.

REFERENCES

[2] M.A., El-Naggar, A One Dimensional Flow Analysis for the Prediction of Centrifugal Pump Performance Characteristics, International Journal of Rotating Machinery, Vol. 2013.

[3] J.F. Gülich, Centrifugal Pumps, 2nd edition, Springer: Berlin,2010. [4] .Sigloch, Stroemungsmaschinen ,Hanser Verlag 1984.

[5] W. Bohl, Strömungsmachinen 2, 5th edition, Vogel:Würzburg,1995.

[6] M. Teodor, Impeller Design Using CAD Techniques and Conformal Mapping Method Polytechnic University of Timisoara, Romania, 2012.

[7] V. A.Grapsas, J. S. Anagnostopoulos and D. E. Papantonis, PARAMETRIC STUDY AND DESIGN OPTIMIZATION OF RADIAL FLOW PUMP IMPELLER, Laboratory of Hydraulic Turbomachines School of Mechanical Engineering / Fluids Section National Technical University of Athens, Greece.

[8] J.Kurokawa, Simple formulae for volumetric efficiency and mechanical efficiency of hydraulic machinery, Conference on fluid machinery Osaka 1990. https://ynu.repo.nii.ac.jp/?action=repository_action_common_download&item_id=3858&item_no=1&attribute_id=22&file_no=1

[9] H. Li, H. Lin, W. Huang, J. Li, M. Zeng, J. Ma and X. Hu, A New Prediction Method for the Complete Characteristic Curves of Centrifugal Pumps, Energies 2021, 14.

[10] X. Zhou, Y. Zhang, Z. Ji, and H. Hou, The Optimal Hydraulic Design of Centrifugal Impeller Using Genetic Algorithm with BVF, Hindawi Publishing Corporation ,International Journal of Rotating Machinery Volume 2014.

[11] N. Athanassiadis, Potential flow through spiral Casings, Dissertation 1961, ETH Zuerich.

[12] H. Zhang, L. Tang, and Y. Zhao, Influence of Blade Profiles on Plastic Centrifugal Pump Performance, Hindawi ,Advances in Materials Science and Engineering ,Volume 2020,

[13] Hydraulic design method for low specic speed overload free centrifugal pump impeller, google patent ,2014- 11-26 CN 201410712052 .

[14] J.C. Statharas, E. S. Valamontes, P. Filiousis and N. W. Vlachakis, Strategies for Predicting Centrifugal Pump Performance Characteristics by Validating Blade Shape Configurations. Introducing the Trojan Horse Method, International Journal of Engineering Research & Technology (IJERT), Vol. 8 Issue 06,

June-2019

[15] J. F. Glich, Centrifugal Pumps, Springer Berlin Heidelberg New York 2007

[16] V.N. Vlachakis, Turbulent characteristics in stirring vessels: a numerical investigation, Virginia Tech 2006.

[17] N. Vlachakis, Vergleich zweier Geschwindigkeitansätze für di radiale Spaltstroemung in Bezug auf das Drehmoment der Rotierenden Scheibe, Bericht Universitaet Karlsruhe 1974.

[18] .Siegle, Die Radialspaltstroemung fuer Fluessigkeiten mit grossen Prandtl-Zahlen und Temperaturahhaengiger Viskositaet, Dissertation Universitaet Karlsruhe 1974.

[19] L. Zhou, W. Shi, and S. Wu, Performance Optimization in a Centrifugal Pump Impeller by Orthogonal Experiment and Numerical Simulation, Hindawi Publishing Corporation Advances in Mechanical Engineering Volume 2013.

[20] K. C. Thin, M. M. Khaing, and K. M. Aye, Design and Performance Analysis of Centrifugal Pump, World Academy of Science, Engineering and Technology 46 2008.

[21] T. Lei, Z. Baoshan, C. Shuliang, B. Hao, and W. Yuming, Influence of Blade Wrap Angle on Centrifugal Pump Performance by Numerical and Experimental Study. CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 27, No. 1, 2014.

[22] A. K. Omar, A. Khaldi and A. Ladouani, Prediction of centrifugal pump performance using energy loss analysis, Australian Journal of Mechanical Engineering 2017.

[23] W. Li, X. Jiang, Q. Pang, L. Zhou and W. Wang, Numerical simulation and performance analysis of a four-stage centrifugal pump, Advances in Mechanical Engineering, 2016, Vol. 8(10) 18.

[24] X. Han, Y. Kang, D. Li and W. Zhao, Impeller Optimized Design of the Centrifugal Pump: A Numerical and Experimental Investigation, Energies 2018. [25] H. Wang, B. Long, C. Wang, C. Han and L. Li, Effects of the Impeller Blade with a Slot Structure on the Centrifugal Pump Performance, Energies 2020. [26] M. Djerroud, G. D. Ngoma, and W. Ghie, Numerical Identification of Key Design Parameters Enhancing the Centrifugal Pump Performance: Impeller,

Impeller-Volute and Impeller-Diffuser. International Scholarly Research Network ISRN Mechanical Engineering Volume 2011.

[27] J.-H. Kim, H.-C. Lee, J.-H. Kim, S. Kim, J.-Y. Yoon and Y.-S. Choi, Design techniques to improve the performance of a centrifugal pump using CFD, Journal of Mechanical Science and Technology 29 (1) (2015).

[28] A. ALMASLAMANI, M. CONSTANTIN1, N. BARAN and M. M. STOICAN, EXPERIMENTAL STUDY OF THE PERFORMANCE AND PREDICTION OF CAVITATION PHENOMENON FOR A VARIABLE SPEED CENTRIFUGAL PUMP, Journal of Science and Arts Year 20, No. 1(50), pp. 225-238, 2020.

[29] Md Rakibuzzaman, S.-H. Suh, H.-H. Kim and Y.-H. Jang, Relationship between Cavitation Incipient and NPSH Characteristic for Inverter Drive Centrifugal Pumps, The KSFM Journal of Fluid Machinery: Vol. 18, No. 6, December, 2015.

[30] B. Schiavello and F.C.Visser, Pump cavitation, Pump symposium 2009.

[31] J. Pei, M. K. Osman, W. Wang, D. Appiah, T. Yin and Q. Deng, A Practical Method for Speeding up the Cavitation Prediction in an Industrial Double- Suction Centrifugal Pump, Energies 2019, 12.

[32] R. Kliem, Experimentelle und numerische Untersuchung zum Einfluss von Einbaubedingungen auf das Betriebs- und Kavitationsverhalten von Kreiselpumpen, Dissertation 2006, Technische Universität Darmstadt.

[33] M. Kaya and E. Ayder, Prediction of Cavitation Performance of Radial Flow Pumps, Journal of Applied Fluid Mechanics, Vol. 10, No. 5, pp. 1397-1408, 2017.