Hindawi Publishing Corporation International Journal of Rotating Machinery Volume 2013, Article ID 473512, 19 pages http://dx.doi.org/10.1155/2013/473512 Research Article A One-Dimensional Flow Analysis for the Prediction of Centrifugal Pump Performance Characteristics Mohammed Ahmed El-Naggar Department of Mechanical Power Engineering, Mansoura University, Mansoura 35516, Egypt Correspondence should be addressed to Mohammed Ahmed El-Naggar; naggarm@mans.edu.eg Received 25 June 2013; Accepted 29 August 2013 Academic Editor: Shigenao Maruyama Copyright © 2013 Mohammed Ahmed El-Naggar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A one-dimensional flow procedure for analytical study of centrifugal pump performance is done applying the principle theories of turbomachines. Euler equation and energy equation are manipulated to find pump performance parameters at different discharge coefficients. Fluid slippage loss at impeller exit and volute loss are estimated. The fluid slippage is modeled by the slip factor approach using Wiesner empirical expression. The volute loss model counts friction loss associated with the volute throw flow velocity, diffusion friction loss due to circulation associated with volute flow, loss due to vanishing of radial flow at volute outlet, and loss inside pump volute throat. Models for impeller hydraulic friction power loss, disk friction power loss, internal flow leakage power loss, and inlet shock circulation power loss are considered by suitable models. Pump internal volumetric flow leakage and volumetric efficiency are related to pump geometry and flow properties. The procedure adopted in this paper is capable of obtaining performance characteristic curves of centrifugal pump in a dimensionless form. Pump head coefficient, manometric efficiency, power coefficient, and required NPSH are characterized. The predicted coefficients and obtained performance curves are consistent with experimental characteristics of centrifugal pump. 1. Introduction Centrifugal pumps are used in various applications and are integral to many industries. Yet, in spite of their prevalence and relatively simple configurations compared to other turbomachines, designing an efficient and durable pump remains a challenge. The design of centrifugal pumps is still determined empirically because it relies on the use of a number of experimental and statistical rules. However, during the last few years, the design and performance analysis of turbomachinery have experienced great progress due to the joint evolution of computer power and the accuracy of numerical methods. The one-dimensional performance analysis has proved to be an effective and important approach on pump design [1]. Analytical calculations of pump characteristics depend on geometrical dimensions of pump and losses models in different parts of pump. A series of formulae for calculating losses exist [2–5], but they lack accuracy when applied to centrifugal pumps. In this work, suggested models for calculating several losses in pump are introduced to examine its validity in evaluating pump performance. This paper is an effort towards theoretically obtaining accurate centrifugal pump performance characteristics. Pump characteristics and parameters are presented in dimensionless forms. It presents a onedimensional flow analysis procedure towards obtaining optimum centrifugal pump design parameters. 2. Theoretical Analysis The pump flow coefficient π and pump speed coefficient π are defined as π= ππ2 √2ππ» , π= π’2 , √2ππ» (1) where π» is the pump manometric head, ππ2 is the flow radial velocity at impeller outlet, and π’2 is the tangential velocity at outlet of impeller. 2 International Journal of Rotating Machinery Q in Vs D2 D1 Q + QL π½b2 π½ b2 t u2 W1 L V1 d1 π½ b1 Deye yc πD2 /Z t/ s Veye V2 W2 π½σ³°b2 d2 QL D1 u1 b1 D2 N Q b2 Figure 1: Pump impeller notations. 2.1. Pump Leakage. Due to the difference between the outlet pressure and the inlet pressure of the impeller, a portion of impeller outlet flow rate, ππΏ , returns to the impeller inlet from the existing clearances between the impeller and the casing, Figure 1. This internal discharge leakage, ππΏ , causes some losses as the flow rate through the impeller (π + ππΏ ) is greater than the pump useful outlet discharge π. The volumetric efficiency πvol of pump is defined as the ratio of pump outlet discharge to the impeller discharge: πvol = π π = ππ π + ππΏ or ππΏ 1 − 1, = π πvol ΔWu2 W W2 πΌ2 V2 2 eq V2eq Vr2 π½b2 u2 Vu2act (2) ΔVu2 Vu2 in which ππ = π + ππΏ = ππ2 ⋅ ππ·2 π2 π2 = ππ1 ⋅ ππ·1 π1 π1 , (3) where π1 is the blade width at inlet, π·1 is the impeller diameter at inlet, π1 is the blade thickness coefficient at impeller inlet, π1 = 1 − (π/π)((π‘/ sin π½π1 )/π·1 ), π2 is the blade width at outlet, π·2 is the impeller diameter at outlet, π2 is the blade thickness coefficient at impeller outlet, π2 = 1−(π/π)((π‘/ sin π½π2 )/π·2 ), π‘ is the blade thickness, π is the number of blades, π½π2 is the blade angle at impeller outlet, and π½π1 is the blade angle at impeller inlet. The value of π2 is about 0.95 [6]. The relationship between π1 and π2 with constant blade thickness is given as π1 = 1 − (1 − π2 ) (π·1 /π·2 ) (sin π½π1 / sin π½π2 ) . (4) The impeller inlet flow velocity coefficient πΆπ1 = π1 /√2ππ» is calculated from (3) dividing both sides by √2ππ» and assuming that the flow enters the impeller without swirl (π1 = ππ1 ): πΆπ1 π = . (π·1 /π·2 ) (π1 /π2 ) (π1 /π2 ) The pump discharge coefficient is πΆπ = and substituting for π from (2) and (3); π π πΆπ = πvol π2 π 2 ⋅ . π·2 π 2 (5) π/((π/60)π·23 ), (6) Figure 2: Velocity diagram at impeller outlet. 2.2. Outlet Velocity Diagram. A fluid slippage occurs at impeller exit due to the relative rotation of fluid in a direction opposite to that of impeller. A slip factor π defined as π = ππ’2 act /ππ’2 could be estimated later in Section 2.3. From the velocity diagram at the impeller outlet, Figure 2, the tangential component of the outlet flow absolute velocity is given as ππ’2 = ππ2 cot πΌ2 = π’2 + ππ2 cot π½π2 . (7) The ratio of outlet swirl velocity to outlet tangential velocity is ππ ππ’2 π (8) = 1 + 2 cot π½π2 = 1 + cot π½π2 , π’2 π’2 π and thus, ππ’2 act π’2 = π ⋅ ππ’2 π’2 = π (1 + π cot π½ π2 ) . π (9) The outlet tangential velocity (from (7)) and the pump speed coefficient π = π’2 /√2ππ» are given as π’2 = ππ2 (cot πΌ2 − cot π½π2 ) , (10) π = π (cot πΌ2 − cot π½π2 ) . (11) International Journal of Rotating Machinery 3 Since the relative eddy angular velocity = 2π’2 /π·2 , then π½ /sin b2 /Z-t πD 2 −π π½σ³°b2 Δππ’2 = π2 π½b2 d2 ππ’2 sin π½π2 , π π= −π d1 = π2 π sin π½π2 . π (16) π=1− From the outlet velocity diagram (Figure 2) and (7), the outlet velocity and the outlet velocity coefficient defined as πΆπ2 = π2 /√2ππ» are 2 π22 = ππ22 + ππ’22 = ππ22 + (π’2 + ππ2 cot π½π2 ) , 2 ππ’2 act =1− ππ’2 Δππ’2 ππ’2 =1− Δππ’2 /π’2 ππ’2 /π’2 . (17) Substituting for Δππ’2 /π’2 from (16) and for ππ’2 /π’2 from (8), the formulae proposed by Stodola, cited in [7] for the calculation of slip factor, π, are obtained which are π Figure 3: Flow model for Stodola slip factor. 2 π’2 The slip factor is given by Wav 2 πΆπ 2 Δππ’2 or π2 (π/π) sin π½π2 =1− 1 + (ππ2 /π’2 ) cot π½π2 At π = 0 (π = π0 ) (12) 1−π= = π + (π + πcot π½π2 ) . 1 + (π/π) cot π½π2 . (18) π sin π½π2 . π (19) 1 − π0 , π¦ (20) π cot π½π2 . π (21) 1 − π0 = π2 Therefore, π2 (π/π) sin π½π2 where 2.3. Slip Factor 2.3.1. The Relative Eddy (Eddy Circulation). A simple explanation for the slip effect in an impeller is obtained from the idea of a relative eddy. Suppose that an irrotational and frictionless fluid flow is possible which passes through an impeller. If the absolute flow enters the impeller without spin, then at outlet the spin of the absolute flow must still be zero. The impeller itself has an angular velocity π so that, relative to the impeller, the fluid has an angular velocity of −π; this is termed the relative eddy. At outlet of impeller, the relative flow, π2 , can be regarded as a through flow on which a relative eddy is superimposed. The net effect of these two motions is that the average relative flow emerging from the impeller passages is at an angle to the vanes and in a direction opposite to the blade motion. One of the earliest and simplest expressions for the slip factor was obtained by Stodola, cited in [7]. Referring to Figure 3, the slip velocity due to relative eddy, Δππ’2 = Δππ’2 = ππ’2 −ππ’2 act , is considered to be the product of the relative eddy and the radius (π2 /2) of a circle which can be inscribed within the channel. Thus, Δππ’2 = π π2 . 2 (13) An approximate expression for π2 can be written if the number of blades π is not small: ππ·2 ππ·2 (14) sin π½πσΈ 2 − π‘ = π2 sin π½π2 , π2 = π π π2 π = π2 sin π½π2 . π·2 π (15) π¦=1+ Wiesner [8] introduced an empirical expression which extremely well fits the experimental results of slip factor for wide range of practical blade angles and number of blades. It is used in this work and given as 1 − π0 = √sin π½π2 π0.7 for π·1 ≤ πlimit , π·2 (22) where πlimit is the limiting diameter ratio, πlimit = π−(8.16 sin π½π2 /π) . (23) It is assumed that the water entering the pump impeller is purely in the radial direction. Relative to the impeller, the fluid has an angular velocity of −π, relative eddy, and thus the relative flow at blade inlet acquires an additional component ΔσΈ ππ’1 opposite to rotational direction, as seen in Figure 4, which is ΔσΈ ππ’1 = π where π1 = π1 π1 , 2 ππ·1 sin π½π1 , π (24) (25) π1 π π·1 = π1 sin π½π1 . π·2 π π·2 (26) π ΔσΈ ππ’1 = π1 π’1 sin π½π1 . π (27) Hence, 4 International Journal of Rotating Machinery Δσ³° Wu1 V1∗ = ∗ Vr1 ∗ V1∗ = Vr1 ∗ Vr1 ∗ W1b = sin π½b1 W1∗ V1 = Vr1 π½b1 Δσ³° Wu1 π sin π½b1 )u1 Z Δ Wu1 Δσ³°σ³° Wu1 W1b = W1 π½∗f1 (1 − π1 ∗ W1b W1∗ σ³° Vr1 sin π½b1 π½b1 π½f1 u1 (1 − π1 u1 π sin π½b1 )u1 Z Figure 4: Velocity diagram at impeller inlet without shock. Figure 5: Velocity diagram at blade inlet with shock eddy. When the pump that operates at a discharge π differs from that at designed condition π∗ , the relative flow velocity at blade inlet tends to acquire an additional component in counter of the rotational direction, ΔσΈ σΈ ππ’1 . So, the flow enters the blade passage tangent to the blade surface, and a shock eddy or a shock circulation exists prior to the blade leading edge inside pump eye. As could be noticed from Figure 5, the relative velocity additional rotational speed at blade inlet equals point. With a uniform rate of increase of the volute area, the volute angle is ΔσΈ σΈ ππ’1 = π’1 (1 − π1 π sin π½π1 ) − ππ1 cot (180 − π½π1 ) . (28) π 2.4. Euler Equation of Turbomachines. In the case that the fluid entering the pump impeller is purely in the radial direction without swirl, the pump Euler head is given as [6, 7] π»∞ = π’2 ππ’2 π tan πΌπ = π’2 ⋅ ππ’2 act π = π π΄ πth ππ·2 π2 (π·3 /π·2 ) (π3 /π2 ) . π4 = πvol ⋅ ππ2 ππ·2 π2 π2 ππ·3 π3 ⋅ tan πΌπ πΆπ4 = πvol π2 π·2 π2 ⋅π , tan πΌπ π·3 π3 π2 π2 πvol = ⋅ π. (π·3 /π·2 ) (π3 /π2 ) tan πΌπ (29) = π ⋅ π»∞ . (30) 2.5. Pump Volute. The flow that discharges from the impeller requires careful handling in order to preserve the gains in energy imparted to the fluid. This requires the conversion of velocity head to pressure head by means of a diffuser, and this inevitably implies hydraulic losses. The application of mass conservation to a volute element, [9], reveals that the discharge flow from impeller is matched to the flow in the volute if ππ΄π/ππ = ππ3 /ππ’3 π3 π3 . This requires a circumferentially uniform rate of increase of the volute area π΄ π over the entire development of the spiral (π). Consequently, for a given impeller, there exist a specific volute angle πΌπ and a specific volute throat inlet area π΄ πth for the volute geometry. The volute angle πΌπ, Figure 6, is chosen to match the angle of flow entering the volute πΌ3eq at a certain pump operating (34) π·th = π tan πΌπ + tan πΌth . π·3 (35) The throat outlet flow velocity π5 = π/π΄ th , and using (2) and (3), then π5 = 4π2 πvol (31) (33) With the assumption that the throat height πΏ th = π·3 , thus And so, the slippage head loss is βπslp = π»∞ − π»0 = (1 − π) π»∞ . (32) The throat is assumed to have an expanding angle πΌth , and hence the throat diameter equals π·th = ππ·3 tan πΌπ + πΏ th tan πΌth . . π’2 ⋅ πππ’2 ππ·3 π3 = The volute outlet flow velocity π4 = π/π΄ πth and the volute outlet flow velocity coefficient πΆπ4 = π4 /√2ππ» are Due to the fluid slippage at impeller exit, the actual head given to fluid by the impeller, π»0 , is calculated from [6, 7]: π»0 = π΄ πth π·22 π2 ⋅ ππ2 . 2 π· π·th 2 (36) The throat outlet flow velocity coefficient πΆπ5 (= π5 /√2ππ») equals πΆπ5 = 4π2 πvol (π2 /π·2 ) 2 2 (π·th /π·3 ) (π·3 /π·2 ) ⋅ π. (37) It is assumed that the throat diameter π·th equals the eye diameter π·eye ; that is, π·eye /π·th = 1, which equals the suction pipe diameter π·π . Thus, the throat outlet flow velocity, π5 , equals the flow velocity at suction pipe, ππ . Therefore, π5 = ππ = πvol ⋅ πeye . (38) And the eye velocity coefficient is πΆπeye = πeye √2ππ» = πΆπ5 πvol = 4π2 (π2 /π·2 ) 2 2 (π·th /π·3 ) (π·3 /π·2 ) ⋅ π. (39) International Journal of Rotating Machinery 5 Case πΌ3eq > πΌV : V3d V3eq V3d + V3p ≡ V 3p Dth V5 D2 πth V3 V4 yth = πD3 tan πΌV D1 D3 Impeller N πΌV V 3p V3eq V3u π xc dπ V3u 2∗πD3 sin πΌV /Z b3 Volute πΌV xc Volute V 3p V3 πD3 Vr2 V3dσ³° V3 Vr3 = π2 πvol Vu2act V4 = V3pu V3r V2 πΌ3eq < πΌV πΌ3eq πΌV πΌV V2eq ΔVu 2 ΔVu3 V3d b2 D2 ·V b3 D3 r2 Vu2 V4 Dth Throat πΌth Case πΌ3eq < πΌV : V3eq ≡ V3d + V3p πD3tan πΌV L th πΌV V5 y th Throat L th πD2 Figure 6: Pump and volute notations. The eye diameter relative to the impeller inlet diameter is therefore π·eye π·1 = π·eye π·th π·3 π·2 π·th π·3 π·2 π·1 = (π·3 /π·2 ) (π tan πΌπ + tan πΌth ) . (π·1 /π·2 ) (40) The ratio (π·eye /π·1 ) must not exceed 1, and thus an upper limit is imposed on the volute angle: tan πΌπ < (π·1 /π·2 ) tan πΌth − . π π (π·3 /π·2 ) (41) The throat has a cone angle πth , where tan ( (π·th − π3 ) /2 πth . )= 2 πΏ th (42) volute velocity π3 is decomposed into a velocity parallel to the direction of volute, π3π , and another one in the tangential σΈ (Figure 6). The velocity component direction of impeller, π3π σΈ π3π is the motive of a second circulatory motion given to the volute flow in direction of impeller motion. Thus, the net σΈ − Δππ’3 in circulation velocity of flow in volute is π3π = π3π direction of impeller motion. The component of the velocity π3π in the tangential direction denoted as π3ππ’ equals the volute outlet velocity π4 . The volute loss model counts friction loss associated with the volute throw flow velocity, diffusion friction loss due to circulation associated with volute flow, loss due to vanishing of radial flow at volute outlet, and loss inside pump volute throat. Consequently, the volute head loss and the volute head loss relative to the pump head are written, respectively, as βππ = πΆππ Substituting from (35), tan ( (π /π ) (π /π· ) πth 1 ) = [π tan πΌπ + tan πΌth − 3 2 2 2 ] . 2 2 (π·3 /π·2 ) (43) 2.6. Volute Loss Model. Prediction models that account for the main features of the swirling flow in volutes, reviewed in [4], do not account for the circulatory flow initiated in volute at off-design discharge operation of pump. In this work, a model is proposed to account the volute head loss at offdesign pump operation. The flow enters the volute with a through-velocity π3 at an angle πΌ3 (which may differ from that of the volute angle πΌπ) on which an eddy of a tangential velocity Δππ’3 = Δππ’2 is superimposed opposite to impeller motion. This inlet 2 π3π 2π + πΆππ 2 π3π π2 + 3π + βπth , 2π 2π βππ 2 2 2 2 + πΆππ ⋅ πΆπ + πΆπ + πΆπth ⋅ πΆπ , = πΆππ ⋅ πΆπ 3π 3π 3π 4 π» where πΆππ is the volute friction loss coefficient: πΆππ = ππ πΏπ πΏ 1 = ππ ( π ) ( ), π·βπ π·2 π·βπ /π·2 (44) (45) (46) where ππ is the volute friction coefficient, π·βπ is the volute hydraulic diameter, and πΏ π is the average volute length: πΏπ = 1 ππ·3 , 2 cos πΌπ πΏπ 1 π π· = ( 3). π·2 2 cos πΌπ π·2 (47) (48) 6 International Journal of Rotating Machinery The average volute hydraulic diameter relative to impeller diameter is 2.7. Pump Eye Head Loss βπeye . The head loss in pump eye βπeye , [2], and the eye head loss relative to the pump head are 1 1 1 = . + π·βπ /π·2 2 (π3 /π2 ) (π2 /π·2 ) 8 (π/π) (π·3 /π·2 ) sin πΌπ (49) The volute friction coefficient ππ which corresponds to a pipe flow is function of the volute Reynolds number Reπ and the roughness π , [10]: ππ = 0.3086 , 1.11 2 {log [(6.9/Reπ) + ((π /π·βπ ) /3.7) ]} (50) (π /π·2 ) π . = π·βπ (π·β /π·2 ) π (51) Reπ = ] =2 π ( π·βπ π·2 ) ⋅ Re2 , (52) where Re2 = π’2 ⋅ π·2 /2 . ] π4 , cos πΌπ or πΆπ3π = πΆπ4 cos πΌπ . (54a) In the second term of (44) and (45), πΆππ is the volute diffusion loss coefficient which could be assumed to have the value πΆππ = 0.8, and π3π is the volute circulatory velocity component: π3π = ππ’2 act − π4 πΆπ3π (54b) π2 πvol π3π = ⋅ π. (π·3 /π·2 ) (π3 /π2 ) √2ππ» (54c) In the fourth term of (44) and (45), the volute throat head loss βπth could be calculated as π5 − πeye πΎ . (58) With the assumption that the throat diameter π·th equals the eye diameter π·eye , the flow velocity is the same at pump suction pipe and pump delivery pipe (ππ = π5 ) and neglecting the difference in elevation head across the pump, then π» = π»0 − βππ − βπeye (59) π» = π»∞ − βπslp − βππ − βπeye . (60) or Define a parameter π₯ as π₯= or π» 1 = , π»0 [1 + (βππ /π») + (βπeye /π»)] βππ βπeye (61) 1 =1+ + . π₯ π» π» 1 2 2 2 2 + πΆππ ⋅ πΆπ + πΆπ + πΆπth ⋅ πΆπ = 1 + πΆππ ⋅ πΆπ 3π 3π 3π 4 π₯ (62) The sum of volute and eye head losses, from (59), is written as βππ + βπeye = (1 − π₯) π»0 = (1 − π₯) ππ»∞ . (63) Using (9), (29), and (59), the pump manometric head π» is π2 βπth = πΆπth 4 , 2π (55) where πΆπth is the volute throat friction loss coefficient assumed to be, [10], where πth is the throat cone angle. 2 , = πΆeye ⋅ πΆπ eye 2 + πΆeye ⋅ πΆπ . eye The third term of (45) is πΆπth = 0.5 + 2.6 ∗ sin ( 2ππ» Using (45) as well as (57) and (61) then π3π = = π (π + π cot π½π2 ) − πΆπ4 . √2ππ» πΆπ3π = = πΆeye (57) where πeye is the flow velocity in pump eye and πΆeye is the eye loss coefficient defined by (39). (53) In the first term of (44) and (45), π3π is the volute throw flow velocity (Figure 6) and given as π3π = π» , 2 πeye π»= The volute Reynolds number is calculated as πΆπ3π βπeye 2π 2.8. Pump Manometric Head π». The pump manometric head is the difference in static pressure heads between pump outlet and pump eye: where π3π ⋅ π·βπ βπeye = πΆeye 2 πeye πth ), 2 (56) π» = π₯ππ»∞ = π₯π π’2 ⋅ ππ’2 π , π’2 π π» = π₯π 2 (1 + cot π½π2 ) . π π (64) Coefficients. There are two additional groups of coefficients, namely, the pump head coefficients and head loss coefficients. International Journal of Rotating Machinery 7 The pump head coefficients are the pump manometric head coefficient, πΆπ», Euler head coefficient, πΆπ»∞ , and the head coefficient at impeller outlet, πΆπ»0 . They are defined and given, respectively, as 2.9. Pump Shaft Power πsh , and Pump Shaft Head π»sh . The total shaft power required to drive the impeller is σΈ + ππσΈ π + ππσΈ cir + πππ· , πsh = πsh 0 (70) in πΆπ» = π π» = π₯π (1 + cot π½π2 ) , 2 π π’2 /π (65a) π π»∞ = 1 + cot π½π2 , 2 π π’2 /π (65b) π π»0 = π (1 + cot π½π2 ) . 2 π π’2 /π (65c) πΆπ»∞ = πΆπ»0 = According to (21) and (65b), π¦ = πΆπ»∞ . The head loss coefficients include the slippage head loss coefficient, πΆβπ , the volute-eye head loss coefficient, πΆβπ , π βπslp πΆβπ = π’22 /π slp π+eye πΆβπ eye πΆβπ = π = (1 − π) πΆπ»∞ , βππ + βπeye = π’22 /π βπeye = βππ impeller, ππσΈ cir = πΎ(π + ππΏ )βπcir is the power needed for given in in circulation to flow at impeller inlet, and πππ· is the power lost in friction on outside surface of impeller disks. The total shaft power and pump shaft head can be simplified, respectively, to πsh = πsh0 + πππ + ππcir + ππvol + πππ· , (71) in π+eye slp the eye head loss coefficient, πΆβπ eye , and the volute head loss coefficient, πΆβπ . They are given, respectively, as πΆβπ σΈ where πsh = πΎ(π + ππΏ )π»0 is the impeller power given to 0 σΈ water, πππ = πΎ(π + ππΏ )βππ is the power lost in friction inside π’22 /π π’22 /π = (1 − π₯) ππΆπ»∞ , 2 = πΆeye ⋅ πΆπ ⋅ ππ¦π 1 , 2π2 + πΆπth ⋅ 1 ⋅ 2. 2π πsh0 = πΎππ»0 , πππ = πΎπβππ , ππcir = πΎπβπcir , in in ππvol = πΎπβπvol = πΎππΏ (π»0 + βππ + βπcir ) , (65d) in (72) πππ· = πΎπβππ· , (65e) (65f) where βππ is the impeller skin friction head loss, βπcir is in the inlet shock circulation head loss, βπvol is the volumetric (leakage) head loss, and βππ· is the disk friction head loss. The shaft head π»sh = πsh /(πΎπ) and the shaft head coefficient πΆπ»sh = π»sh /(π’22 /π) are given, respectively, as (65g) π»sh = π»0 + βππ + βπcir + βπvol + βππ· , 2 2 2 = (πΆππ ⋅ πΆπ + πΆππ ⋅ πΆπ + πΆπ 3π 3π 3π 2 πΆπ ) 4 in which in πΆπ»sh = πΆπ»0 + πΆβπ + πΆβπ cir + πΆβπ vol + πΆβπ π· , π (73) in The relationship between the pump speed coefficient π and the pump flow coefficient π is derived as follows. Using (11), (64) becomes where πΆβπ = βππ /(π’22 /π) is the impeller skin friction head loss π’2 cot πΌ2 . π» = π₯π 2 π cotπΌ2 − cot π½π2 head loss coefficient, πΆβπ vol = βπvol /(π’22 /π) is the volumetric head loss coefficient, and πΆβπ π· = βππ· /(π’22 /π) is the disk friction head loss coefficient. in (66) Dividing both sides by π», noting that π’22 /(2ππ») = π2 , and using (11), (64) becomes a quadric equation for cotπΌ2 : cot2 πΌ2 − cot π½π2 ⋅ cot πΌ2 − 1 = 0, 2π₯ππ2 π coefficient, πΆβπ cir = βπcir /(π’22 /π) is the inlet shock circulation (67) 2.10. Pump Efficiency (Manometric Efficiency) π. The pump manometric efficiency is the ratio of gained water power (ππ€ ) to the pump shaft power (πsh ) supplied to pump impeller. According to its definition, it takes the following forms: which has a solution (since cot πΌ2 should be greater than cot π½π2 , whence only the +ve sign is considered) 1 1 2 cot πΌ2 = cot π½π2 + √cot2 π½π2 + . 2 2 π₯ππ2 π= ππ€ ππ€ = , πsh πsh0 + πππ + ππcir + ππvol + πππ· in (68) Multiplying (68) by π and then subtracting π cot π½π2 from both sides and using (11) yield the following relation for π: 1 2 1 . π = − πcot π½π2 + √ π2 cot2 π½π2 + 2 2 π₯π in (69) π= π»∞ − βπslp − βππ − βπeye π» = , π»sh π»0 + βππ + βπcir + βπvol + βππ· in π= πΆπ»∞ − πΆβπ slp − πΆβπ π+eye πΆπ» = . πΆπ»π β πΆπ»0 + πΆβπ + πΆβπ cir + πΆβπ vol + πΆβπ π· π in (74) 8 International Journal of Rotating Machinery 2.11. Pump Shaft Power Coefficient πΆππ β . The pump shaft power coefficient and the pump water power coefficient are given, respectively, as πsh πΆπsh = π(π/60)3 π·25 πΆππ€ = ππ€ 3 π(π/60) π·25 2 The impeller friction coefficient ππ is function of the average impeller Reynolds number Re and the roughness π [10]: ππ = = π πΆππΆπ»sh , (75) = π2 πΆππΆπ». (76) πΏ π πav2 , π·hyd 2π (77) where πΆππ is the dissipation coefficient, πΏ π is the blade length, π·hyd is the hydraulic diameter, and πav is the average relative velocity. Therefore, the impeller skin friction head loss coefficient could be calculated from πΆβπ π βππ (πΏ π /π·2 ) 1 πav 2 = 2 = 4πΆππ ). ( π’2 /π (π·hyd /π·2 ) 2 π’2 2 (π1 π1 + π2 π2 ) 4 ∗ Area , = Perimeter π1 + π1 + π2 + π2 π /π ππ /π = πav = π . π΄ av (π1 π1 + π2 π2 ) /2 π·2 = (79) Re = πav ⋅ π·hyd π2 (π/π) (π2 /π·2 ) π πav = ⋅ . π’2 1/2((π1 /π·2 ) (π1 /π2 ) (π2 /π·2 )+(π2 /π·2 ) (π2 /π·2 )) π (81) The blade length is πΏ π = ((1/2)(π·2 − π·1 ))/ sin π½bm , and thus (82a) 2 (82b) . The impeller dissipation coefficient is given as, GuΜlich [11] 4πΆππ = (ππ + 0.006) (1.1 + 4 π2 ). π·2 (84) π·hyd πav )( ) ⋅ Re2 , π’2 π·2 (85) (86) 2.13. Disk Friction Head Loss βππ· . The disk friction power loss πππ· is the power loss in the fluid between external surfaces of the impeller disks and internal walls of the pump casing, Figure 7. The πππ· can be estimated as π2 πππ· = 2 ∗ ∫ π ⋅ π ⋅ π ⋅ 2ππ ππ 0 π= ππ· 1 2 ππ· 1 ⋅ ππ’ = ⋅ π (ππ)2 , 4 2 4 2 (87) (83) (88) where π is the shear stress in circumferential direction and ππ· is the disk skin friction coefficient, and it is assumed constant along the disk surface. Thus, π 3 π2 ππ ∫ ππ·π4 ππ. 2 0 (89) Therefore, the final expression after the integration for the disk friction power loss and, consequently, the impeller disk friction head loss are πππ· = π5 π π ππ· ππ3 2 = ππ·ππ’23 π·22 , 2 5 40 3 2 π ππ· π’2 π·2 βππ· = = . πππ 40 π π πππ· (90) The impeller disk friction head loss coefficient πΆβπ is given π· as π· (π½π1 + π½π2 ) , where Re2 is defined by (53). πΆβπ ≡ where π½bm = = 2( ] πππ· = 2 ((π1 /π·2 ) (π1 /π2 ) (π2 /π·2 ) + (π2 /π·2 ) (π2 /π·2 )) , (π1 /π·2 ) + (π1 /π2 ) (π2 /π·2 ) + (π2 /π·2 ) + (π2 /π·2 ) (80) πΏπ 1 π· 1 = (1 − 1 ) , π·2 2 π·2 sin π½bm 2 ]} with Substituting for π1 , (26), π2 , (15), and ππ , (3), yield: π·hyd 1.11 (π /π·2 ) π . = π·hyd (π·hyd /π·2 ) (78) The hydraulic diameter and the average relative velocity are given, respectively, as, GuΜlich [11] π·hyd = {log [(6.9/ Re) + ((π /π·hyd ) /3.7) where 2.12. Impeller Skin Friction Power Head Loss βππ . The impeller hydraulic friction head loss βππ is estimated by the theory of flow through pipes and is given by [11]: βππ = 4πΆππ 0.3086 βππ· π’22 /π = πΎπ· ⋅ 1 π , πvol π (91) where πΎπ· is the impeller disk loss coefficient, πΎπ· = ππ· 1 . 40 π2 (π2 /π·2 ) (92a) It is derived by substituting for π from (2) and (3) and using the definition of π which yields π = πvol π2 ⋅ π√2ππ» ⋅ ππ·2 π2 . International Journal of Rotating Machinery 9 Correlations for ππ· were obtained by Kruyt, cited in [12]. Four different regimes were identified, Figure 8: Regime I (laminar flow, boundary layers have merged), Regime II (laminar flow with two separate boundary layers), Regime III (turbulent flow, boundary layers have merged), and Regime IV (turbulent flow with two separate boundary layers). These regimes are characterized by the Reynolds number, Re2 = π’2 ⋅ (π·2 /2)/], and a nondimensional gap parameter, πΊ = π¦0 /(π·2 /2), where π¦0 is the axial gap between impeller disk and casing (Figure 7). The equations of curves 1 to 5 in , πΊ = 188Re−9/10 , πΊ = Figure 8 are πΊ = 1.62Re−5/11 2 2 −3/16 5 , Re = 1.58 ∗ 10 , and πΊ = 0.402Re , 0.57 ∗ 10−6 Re15/16 2 2 respectively. The disk friction coefficient for each regime is given as Regime I : ππ· = 10πΊ−1 Re−1 2 18.5 1/10 −1/2 Regime II : ππ· = πΊ Re2 π 0.4 −1/6 −1/4 Regime III : ππ· = πΊ Re2 π 0.51 1/10 −1/5 Regime IV : ππ· = πΊ Re2 . π βπvol πΆβπ vol = βπvol π’22 /π (92b) Deye r2 = D2 /2 Figure 7: Disk and wearing ring clearances. IV 4 0.075 5 1 3 0.025 III I 2 10000 100000 1000000 Figure 8: Disk friction flow regimes [12]. (93) The pressure distribution along the radial direction of impeller shroud is parabolic [13], and thus the pressure head difference between impeller outlet and before clearance is given by (94) where ππ is the pressure before clearance and ππ is the pressure after clearance at suction side of the impeller (Figure 7). The pressure head difference between volute and pump suction side equals π3 − ππ π5 − ππ π3 − π5 = + πΎ πΎ πΎ π42 π52 − ). 2π 2π y0 Re2 where ππ is the clearance area of wearing ring (= ππ·eye π¦π ), π¦π is the clearance of wearing ring, Figure 7, πΆdL is the leakage discharge coefficient (≈0.6), and Δπ»π is the pressure head drop across the clearance: π − ππ π3 − ππ π3 − ππ = − , Δπ»π = π (95) πΎ πΎ πΎ = π»0 − βπeye − ( w 0.000 1000 The leakage flow rate ππΏ can be estimated using orifice formula [13]: π2 π2 = π» + [βππ − ( 4 − 5 )] 2π 2π ps G 0.050 1 − 1) ⋅ (πΆπ»0 + πΆβπ π + πΆβπ cir ) . in πvol ππΏ = πΆdL ⋅ ππ √2πΔπ»π , yc II ππvol =( pc 0.100 2.14. Volumetric Head Loss βπVol . The volumetric (leakage) head loss and the volumetric head loss coefficient are given as follows after using (2): π = = πΏ (π»0 + βππ + βπcir ) in πΎπ π 1 − 1) ⋅ (π»0 + βππ + βπcir ) =( in πvol p3 2 π·eye π3 − ππ π’22 = (1 − 2 ) . πΎ 8π π·2 (97) Thus, the leakage flow rate and the leakage flow rate coefficient, πΆππΏ , are ππΏ = πΆdL ⋅ ππ·eye π¦π ⋅ π’2 √2 ∗ (πΆπ»0 − 2 2 2 πΆπ − πΆπ + πΆeye ⋅ πΆπ 4 5 eye 2π2 1/2 2 π·eye 1 − (1 − 2 )) 8 π·2 (96) πΆππΏ = , π¦π √2π2 πΆdL ππΏ = (π/60) π·23 π·eye (π· /π· )2 2 eye 10 International Journal of Rotating Machinery ∗ (πΆπ»0 − − 2 2 2 πΆπ − πΆπ + πΆeye ⋅ πΆπ 4 5 eye Therefore, the pump required net positive suction head is 2π2 1/2 2 π·eye 1 (1 − 2 )) . 8 π·2 Using (2), ππΏ /π = πΆππΏ /πΆπ, and (8) for πΆπ, the pump volumetric efficiency can be deduced after some mathematical manipulations: πΆNPSHπ = 2 2 2 πΆπ − πΆπ + πΆeye ⋅ πΆπ 4 5 eye 2 π·eye π· 2 1 − [1 − ( ) ( 1 ) ]) 8 π·1 π·2 πΆNPSHπ (99) 2π2 1/2 . in ΔσΈ σΈ ππ’1 ⋅ π’1 π . (100) πΆNPSHπ = 0.02 + ππ = π√π [π (rpm) , π» (m) , π (lit/s)] . π»3/4 (108) Substitute for π 60 π√2ππ», and π2 = π·2 ( 2 ) yield : ππ·2 π·2 3/4 ππ = 60 ∗ √1000 ∗ (2π) √π ∗ √π2 √( π2 ) ∗ √πvol ⋅ π√π. π·2 (109) Therefore, ππ = 9977√π2 √ ( π2 ) ⋅ √πvol ⋅ π√π. π·2 (110) (102) where π1 , π1 are the absolute pressure and absolute velocity of fluid at impeller inlet and πV is the absolute vapour pressure of fluid at the corresponding fluid temperature. In the vicinity of the leading edge of the impeller blades, the fluid has to accelerate in order to follow the rotating movement of the blades. This acceleration leads to a drop of the static pressure, which results in a local minimum pressure at blade inlet: π2 π2 πmin π1 = − ( 1π − 1 ) . πΎ πΎ 2π 2π (π/π) 1 . (107) 2sin2 π½π1 (π·1 /π·2 )2 (π1 /π2 )2 (π1 /π2 )2 2.17. Pump Specific Speed ππ . The pump specific speed is defined with suitable units as π= 2.16. Required Net Positive Suction Head ππππ»R . The pump net positive suction head NPSH is defined as the difference between the fluid inlet stagnation pressure head and vapour pressure head [13]: π π1 π12 + )− V, πΎ 2π πΎ (106) π = 1000πvol π2 ⋅ ππ·2 π2 ⋅ π√2ππ» (lit/s) , π· 2 cot (180 − π½π1 ) π π ⋅ . sin π½π1 ) ( 1 ) − π π·2 (π1 /π2 ) (π1 /π2 ) π (101) NPSH = ( 2 (πmin − πV ) /πΎ 1 ππ1 + = . sin2 π½π1 2π’22 π’22 /π The term ((πmin − πV )/πΎ)/(π’22 /π) is a design parameter for the pump and could be assumed to be 0.02. From (3) ππ1 = ππ2 (π·2 /π·1 )(π2 /π1 )(π2 /π1 ), and therefore the NPSHπ coefficient is Thus, the inlet shock circulation head loss coefficient is βπcir πΆβπ = 2 in cirin π’2 /π = (1 − π1 (105) 2 2.15. Inlet Shock Circulation Head Loss βπcir . At impeller inlet, in a shock circulation exists inside pump eye when the flow discharge differs from that of designed one. As discussed before in Section 2.3, the tangential velocity responsible for this shock eddy is ΔσΈ σΈ ππ’1 , which could be estimated from (28). Therefore, the inlet shock circulation head loss equals βπcir = NPSHπ . π’22 /π Referring to Figure 5, π1π = (ππ1 / sin π½π1 ), and thus π·eye 2 π·1 2 √2πΆdL π π¦π )( )( ) π·eye π·1 π·2 (π2 /π·2 ) π2 π ∗ (πΆπ»0 − (104) The pump NPSHπ coefficient is defined as (98) πvol = 1 − ( 2 (πmin − πV ) π1π + . πΎ 2π NPSHπ = (103) 3. Calculation Procedure and Results An iterative procedure using Microsoft Office EXCEL program is performed to calculate the pump design parameters described by the preceding derived equations. The input parameters are the blades number, π, impeller inlet diameter to outlet diameter ratio, (π·1 /π·2 ), volute diameter to impeller outlet diameter ratio, (π·3 /π·2 ), ratio of blade width at impeller outlet to impeller outlet diameter, (π2 /π·2 ), blade width at impeller inlet to blade width at impeller outlet ratio, (π1 /π2 ), volute width to blade width at impeller outlet ratio, (π3 /π2 ), blade angle at impeller outlet, π½π2 , blade angle at International Journal of Rotating Machinery 11 Table 1: Pump input-parameters. Impeller Blades Volute Input parameter π·1 π·2 π2 π·2 π1 π2 π2 π¦0 πΊ= (π·2 /2) π’ ⋅ (π·2 /2) Re2 = 2 ] (π·2 = 0.209 m) (π = 1450 rpm) (] = 10−6 m2 /s) water π π·2 (π = 0.0001 m) π π½π2 π½π1 π·3 π·2 π3 π2 πΌπ πΌth πΆππ π·eye Eye π·th π¦π π·eye πΆdL πΆeye 0.8 π Value 0.5 0.121 1 0.95 0.7 π 0.6 CH CH 0.5 0.4 0.05 0.3 1.16 ∗ 106 0.2 0.1 0.000478 4 164β 164β 1.05 1.05 7β 4β 0.8 1 0.005 0.6 1.0 impeller inlet, π½π1 , volute angle, πΌπ, throat angle, πΌth , ratio of wearing ring clearance/impeller eye diameter, (π¦π /π·eye ), leakage discharge coefficient, πΆdL , volute diffusion loss coefficient, πΆππ , eye loss coefficient, πΆeye , blade thickness coefficient at impeller outlet, π2 , with constant-thickness blades, ratio of axial gap between impeller disk and casing to impeller outlet radius, π¦0 /(π·2 /2), Reynolds number at impeller outlet, Re2 , and relative roughness, (π /π·2 ) approximated values are used by putting π·2 ≈ 0.209 m, π ≈ 1450 rpm, π ≈ 0.0001 m, ] ≈ 10−6 m2 /π : Re2 = 1.16 ∗ 106 , and π /π·2 = 0.000478. For a certain flow coefficient, π, all pump performance parameters and coefficients are calculated after iterations. Case Study. For reason of comparison between results of present analytical study and experimental pump performance obtained by Baun and Flack [14], calculations of centrifugal pump performance are performed with the following input parameters given in Table 1. Other pump constant-parameters are calculated according to the present procedure as shown in Table 2. The sequence of calculations by using the procedure derived equations of pump variable parameters is listed in Table 3. 0.0 0.00 0.02 0.04 0.06 π/π 0.08 0.10 0.12 Experimental head coeff. [14] Present procedure head coeff. Efficiency Experimental [14] Present procedure Figure 9: Comparison between present procedure results and experimental results by Baun and Flack [14]. The manometric head coefficient and pump efficiency of both present procedure and experimental measurements by Baun and Flack [14] are plotted in Figure 9 versus the ratio (π/π) (defined as flow coefficient in [14]). It shows a good similarity between the present procedure results and the experimental ones. Only in range of low ratios of π/π, the calculated efficiency is relatively high. This is attributed to the uncounted mechanical power loss in the prediction of pump shaft power and hence in efficiency. The pump performance characteristics are presented by the calculated pump head coefficient, efficiency, power coefficients, and required NPSH as shown in Figure 10. From the figure, the pump best efficiency point (BEP) occurs at discharge coefficient πΆπ ≈ 0.0675 giving an efficiency π ≈ 75.3%, a manometric head coefficient πΆπ» ≈ 0.468, a shaft power coefficient πΆπsh ≈ 0.414, a water power coefficient πΆππ€ ≈ 0.312, and a pump required net positive suction head coefficient πΆNPSHπ ≈ 0.129. The maximum shaft power coefficient πΆπsh ≈ 0.428 occurs at πΆπ ≈ 0.085, and the maximum water power coefficient πΆππ€ ≈ 0.3165 occurs at πΆπ ≈ 0.075. The variations of pump flow coefficient π and pump speed coefficient π with the discharge coefficient are shown in Figures 11 and 12, respectively. Figure 11 shows also the variation of the ratio π/π with the discharge coefficient πΆπ. The linear relationship between πΆπ and π/π is evident as given by (6). The dotted lines in the figures correspond to the value of πΆπ that gives the pump best efficiency. At pump best efficiency point (πΆπ ≈ 0.0675), π ≈ 0.063 (Figure 11) and π ≈ 1.034 (Figure 12). Also, it is indicated from Figure 12 that 12 International Journal of Rotating Machinery Table 2: Pump constant-parameters. Constant parameter equation 1 − π0 = √sin π½π2 π0.7 π1 = 1 − Impeller π·hyd π·2 Blades Volute Eye for π·1 ≤ πlimit = π−8.16 sin π½π2 /π π·2 (1 − π2 ) (π·1 /π·2 ) (sin π½π1 / sin π½π2 ) π2 π = π2 sin π½π2 π·2 π π1 π π·1 = π1 sin π½π1 π·2 π π·2 2 (((π1 /π·2 ) (π1 /π2 ) (π2 /π·2 )) + ((π2 /π·2 ) (π2 /π·2 ))) = (π1 /π·2 ) + ((π1 /π2 ) (π2 /π·2 )) + (π2 /π·2 ) + (π2 /π·2 ) π /π·2 π = π·hyd π·hyd /π·2 ππ· 1 πΎπ· = 40 π2 (π2 /π·2 ) ππ· = ππ· (Re2 , πΊ) πΏπ π· 1 1 = (1 − 1 ) π·2 2 π·2 sin π½bm 1 π½bm = (π½π1 + π½π2 ) 2 1 1 1 = + π·βπ /π·2 2(π3 /π2 )(π2 /π·2 ) 8(π/π)(π·3 /π·2 ) sin πΌπ πΏπ 1 π π· = ( 3) π·2 2 cos πΌπ π·2 (π /π·2 ) π = π·βπ (π·βπ /π·2 ) (π /π ) (π /π· ) π 1 tan ( th ) = [π tan πΌπ + tan πΌth − 3 2 2 2 ] 2 2 (π·3 /π·2 ) πth πΆπth = 0.5 + 2.6 ∗ sin ( ) 2 π·eye (π·3 /π·2 ) = (π tan πΌπ + tan πΌth ) π·1 (π·1 /π·2 ) the speed coefficient takes a minimum value of π ≈ 0.932 at πΆπ ≈ 0.024, which corresponds to the maximum πΆπ» in Figure 10. Figure 13 shows the theoretical dimensionless headdischarge curve (Euler head) which is a straight line, and πΆπ»∞ decreases with the increase of πΆπ for the proposed outlet blade angle π½π2 = 164β > 90β . The actual impeller outlet dimensionless head-discharge curve πΆπ»0 is obtained taking into consideration the slip or eddy circulation of flow inside impeller which is nearly constant. Actually, the effect of slip is not a loss but a discrepancy not accounted by the basic assumptions. The second loss shown is the volute-and-eye loss, which is minimum at πΆπ ≈ 0.051. Reduced or increased πΆπ, from the value 0.051, increases the volute-and-eye loss. Figure 14 gives the variation of different head loss coefficients with πΆπ in addition to the πΆπ» and πΆπ»0 curves. Generally, these head loss coefficients decrease with the increase of πΆπ. At very low discharge coefficients, both the disk head loss coefficient and volumetric head loss coefficient become very big. The inlet circulation head loss is big at low Equation no. (22) (4) (15) (26) (80) (85) (92a) (92b) (82a) (82b) (49) (48) (51) (43) (56) (40) πΆπ and decreases until it reaches zero at πΆπ ≈ 0.06, and then, its direction is inverse (becomes negative) which means that the inlet circulation adds power to impeller and does not bleed power from impeller. The variation of the flow velocity coefficients and the pump volumetric efficiency πvol with the pump discharge coefficient is shown in Figure 15. The coefficients πΆπ1 , πΆπΔππ’ , 2 and πΆπ4 have the trends of increasing with the increase of πΆπ, whereas the two coefficients πΆπ3π and πΆπ3πσΈ have the trends of decreasing with the increase of πΆπ. These two later coefficients have maximum values at πΆπ = 0. At πΆπ ≈ 0.085, the coefficient πΆπ3π becomes zero, which indicates that at this point of operation the flow inside volute is without circulation and π4 = ππ’2 act . At increased πΆπ, the πΆπ3π becomes negative which means that the flow circulation inside the volute changes its direction of rotation opposite to impeller motion whereas π4 > ππ’2 act (Figure 6). The pump specific speed is presented in Figure 16, which shows that ππ ≈ 870 at pump best efficiency point. International Journal of Rotating Machinery 13 Table 3: Pump variable-parameters. Order Variable parameter equation π≡ ππ2 √2ππ» step = 0.001 π = π0 , π₯ = 1, πvol = 1 Start of iteration 1 1 2 π = − π cot π½π2 + √ π2 cot2 π½π2 + 2 2 π₯π Initial 1 π¦ = πΆπ»∞ ≡ 2 π π»∞ = 1 + cotπ½π2 π’22 /π π π=1− 3 4 πΆβπ 5 πΆπ»0 ≡ slp πΆπ4 ≡ 7 βπslp π’22 /π (21), (65b) (1 − π0 ) π¦ (20) = (1 − π) πΆπ»∞ (65d) (65c) 4π2 πvol (π2 /π·2 ) π5 ⋅π = √2ππ» (π·π‘β /π·3 )2 (π·3 /π·2 )2 (37) π2 πvol π4 ⋅π = √2ππ» (π·3 /π·2 ) (π3 /π2 ) tan πΌπ (33) πΆπeye ≡ 8 ≡ πeye √2ππ» = 4π2 (π2 /π·2 ) 2 2 (π·π‘β /π·3 ) (π·3 /π·2 ) πΆπ3π = 9 ⋅π πΆπ4 cos πΌπ πΆπ3π = π (π + πcotπ½π2 ) − πΆπ4 10 πΆπ3π ≡ 11 πvol = 1 − ( (69) π π»0 = π (1 + cotπ½π2 ) π’22 /π π πΆπ5 ≡ 6 12 = 0.002 to 0.12 Eq. no. π2 πvol π3π = ⋅π (π· /π· √2ππ» 3 2 ) (π3 /π2 ) 2 2 πΆπ2 − πΆπ2 5 + πΆeye ⋅ πΆπeye π·eye 2 π·1 2 1 √2πΆdL π π¦π π·1 2 π·eye 1 )( )( ) − ) ( ) ] ∗ √ πΆπ»0 − 4 [1 − ( π·eye π·1 π·2 (π2 /π·2 ) π2 π 2π2 8 π·2 π·1 (39) (54a) (54b) (54c) (99) 14 International Journal of Rotating Machinery Table 3: Continued. Order 13 14 15 16 Variable parameter equation Reπ = ππ = π3π ⋅π·βπ ] π» 22 23 24 25 π·2 ) ⋅ Re2 (52) 1.11 ]} 2 π+eye (57) βπeye βπ 1 =1+ π + π₯ π» π» (61) End of iteration βππ + βπeye ≡ = (1 − π₯) ππΆπ»∞ π’22 /π (65e) π π» = π₯π (1 + cot π½π2 ) π’22 /π π (65a) π π π = πvol π2 π2 ( 2 ) ⋅ ( ) 3 π· π (π/60) π·2 2 (6) πΆπ» ≡ πΆπ ≡ (45) = πΆeye ⋅ πΆπ2 eye π» πΆβπ π2 (π/π) (π2 /π·2 ) ⋅ (π/π) πav = π’2 1/2 (((π1 /π·2 ) (π1 /π2 ) (π2 /π·2 )) + ((π2 /π·2 ) (π2 /π·2 ))) Re = ππ = (50) (46) = πΆππ πΆπ2 3π + πΆππ πΆπ2 3π + πΆπ2 3π + πΆππ‘β πΆπ2 4 18 21 π·βπ πΏπ πΏ 1 = ππ ( π ) ( ) π·βπ π·2 π·βπ /π·2 βπeye 20 π )( {log [(6.9/Reπ ) + ((π /π·βπ ) /3.7) 17 19 πΆπ3π 0.3086 πΆππ = ππ βππ = 2( Eq. no. πav ⋅ π·hyd ] = 2( π·hyd πav )( ) ⋅ Re2 π’2 π·2 (86) 0.3086 {log [(6.9/ Re) + ((π /π·hyd ) /3.7) 4πΆππ = (ππ + 0.006) (1.1 + 4 1.11 π2 ) π·2 (81) 2 (84) ]} (83) International Journal of Rotating Machinery 15 Table 3: Continued. Order Variable parameter equation πΆβππ ≡ 26 = 4πΆππ π’22 /π πΆβππ· ≡ 27 28 βππ πΆβπcir in ≡ βπcir in π’22 /π 29 πΆβπvol ≡ 30 πΆπ»sh ≡ = (1 − π1 βπvol π’22 /π βππ· π’22 /π Eq. no. (πΏ π /π·2 ) 1 πav 2 ) ( (π·hyd /π·2 ) 2 π’2 (78) 1 π πvol π (91) = πΎπ· ⋅ π· 2 cot (180 − π½π1 ) π π ⋅ sin π½π1 ) ( 1 ) − π π·2 (π1 /π2 ) (π1 /π2 ) π (101) 1 − 1) ⋅ (πΆπ»0 + πΆβππ + πΆβπcir ) in πvol (93) =( π»sh = πΆπ»0 + πΆβππ + πΆβπcir + πΆβπvol + πΆβππ· in π’22 /π π≡ 31 ππ€ πΆ = π» πsh πΆπ»sh (73) (74) 32 πΆπsh ≡ πsh = π2 πΆππΆπ»sh π(π/60)3 π·25 (75) 33 πΆππ€ ≡ ππ€ = π2 πΆππΆπ» π(π/60)3 π·25 (76) 34 πΆNPSHπ = 0.02 + 2 35 (π/π) 1 2 sin2 π½π1 (π·1 /π·2 )2 (π1 /π2 )2 (π1 /π2 )2 (107) π2 ) ⋅ √πvol ⋅ π√π π·2 (110) ππ = 9977√π2 √( Figure 17 shows the procedure results of centrifugal pump performance when the pump handles fluids with different kinematic viscosities. The head and efficiency for the pump when handling oils are lower than those when handling water. But the required power when handling oil is higher than that when handling water. The pump head decreases slightly due to the increase in volute friction loss as fluid viscosity increases, while the increase in pump power is high due to the increase in both hydraulic friction inside impeller and disk friction power losses. Hence, the drop in pump efficiency is very high as the fluid viscosity increases. This result is in accordance with experimental results by Shojaee Fard and Boyaghchi [15]. 4. Conclusions A one-dimensional flow procedure for analytical study of centrifugal pump performance is accomplished applying the principle theories of turbomachines. The procedure is capable of providing the performance characteristic of centrifugal pump in a dimensionless information form. The predicted coefficients and performance curves obtained have been found to be in a reasonable agreement with experimental measurements. The present procedure is also capable of predicting the effects of handling viscous fluids on the centrifugal pump performance. The input form for this procedure of pump flow analysis makes it an effective tool analysis and can be used in the pump conceptual design. Notations π΄: ππ : π΄ π: π΄ th : π΄ πth : Impeller net area, m2 Clearance area of wearing ring, m2 Volute area, m2 2 Throat outlet area, ππ·th /4, m2 Volute throat inlet area, m2 16 International Journal of Rotating Machinery 0.8 1.6 BEP 1.5 π 0.7 1.4 π 0.6 π CH CH 1.3 1.2 0.5 1.1 0.4 1.0 CPsh 0.3 0.9 CPπ€ CPπ€ BEP CPsh 0.8 0.2 CNPSH π 0.1 0.7 0.00 π PSH CN 0.0 0.00 0.02 0.04 0.02 0.04 0.06 0.08 0.10 0.12 CQ 0.06 0.08 0.10 Figure 12: Speed coefficient variation with pump discharge coefficient. 0.12 CQ Figure 10: Characteristics of centrifugal pump obtained by present study. 1.4 1.3 1.2 1.1 Head and head loss coefficients 0.20 0.18 π/π 0.16 0.14 0.12 0.10 π/π π 0.08 0.06 0.02 0.00 0.00 0.02 0.04 0.06 Slip loss 0.7 0.6 CH∞ CH0 Volu te an d CH 0.5 0.4 0.2 0.08 0.10 0.1 0.12 0.0 0.00 CQ Figure 11: Variation of flow coefficient and ratio of flow coefficient to speed coefficient with discharge coefficient. π: πΆππ : πΆdL : πΆππ : πΆeye : πΆπth : πΆππ : πΆπ»: πΆπ»0 : 0.8 head 0.3 BEP 0.04 Eule r 0.9 Impeller width, m Impeller dissipation coefficient Leakage discharge coefficient Volute diffusion loss coefficient Eye loss coefficient Volute throat friction loss coefficient Volute friction loss coefficient Manometric head coefficient, π»/(π’22 /π) Head coefficient at impeller outlet, π»0 /(π’22 /π) πΆπ»∞ : Euler head coefficient, π»∞ /(π’22 /π) πΆπ»sh : Shaft head coefficient, π»sh /(π’22 /π) BEP π 1.0 0.02 0.04 0.06 eye l oss M an om etr ic he ad 0.08 0.10 0.12 CQ Figure 13: Variation of head coefficients and head loss coefficients with discharge coefficient. πΆβπ : Inlet shock circulation head loss coefficient, βπcir /(π’22 /π) in πΆβπ π : Impeller skin friction head loss coefficient πΆβπ : Eye head loss coefficient, βπeye /(π’22 /π) cirin eye πΆβπ : π· πΆβπ slp : πΆβπ : π πΆβπ : π+eye πΆβπ : vol Disk friction head loss coefficient, βππ· /(π’22 /π) Slip head loss coefficient, βπslp /(π’22 /π) Volute head loss coefficient, βππ /(π’22 /π) Volute-eye head loss coefficient, (βππ + βπeye )/(π’22 /π) Volumetric head loss coefficient, βπvol /(π’22 /π) International Journal of Rotating Machinery 17 1.4 2000 1.3 1800 1.2 0.9 0.8 1400 1200 C H 1000 sh CH0 Impeller hydraulic friction loss 0.7 CH 0.6 1600 800 600 400 0.5 200 0.4 0.3 BEP 0.2 0.1 0.0 0.00 0.02 0.04 0.06 0 0.00 M an om etr ic he ad 0.08 0.10 BEP Head and head loss coefficients 1.0 ns ad he loss ss o . aft cir c l Sh tr i let In me s lu os Vo isk l D 1.1 0.02 0.04 0.06 0.08 0.10 0.12 CQ Figure 16: Pump specific speed values at different discharge coefficient. 0.12 CQ 0.8 Figure 14: Head loss coefficients affecting pump power at different discharge coefficients. π 0.7 π 1.0 CH πvol 0.9 0.5 CV2 πvol 0.8 0.4 C V CV CPsh 3π σ³° 0.7 CV2 0.6 0.3 act CH 0.2 C 0.5 V 3π 0.1 0.4 0.3 4 V 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 CQ Figure 15: Values of volumetric efficiency and different pump velocity coefficients. Coefficient of pump NPSHπ , NPSHπ /(π’22 /π) Shaft power coefficient, πsh /π(π/60)3 π·25 Water power coefficient, ππ€ /π(π/60)3 π·25 Pump discharge coefficient, π/(π/60)π·23 Leakage flow rate coefficient Velocity coefficient, π/√2ππ» Impeller outer diameter, m 0.02 0.04 0.06 0.08 0.10 CQ 0.12 Water = 1 ∗ 10−6 m2 /s Oil = 100 ∗ 10−6 m2 /s = 200 ∗ 10−6 m2 /s Oil BEP CV1 0.1 0.0 0.00 C CVΔVu2 0.2 πΆNPSHπ : πΆπsh : πΆππ€ : πΆπ: πΆππΏ : πΆπ : π·2 : CPsh 0.6 Figure 17: Effect of pumped fluid viscosity on the performance of centrifugal pump. π·eye : π·βπ : π·th : π: ππ·: π: πΊ: βπcir : in βππ· : Pump eye diameter, m Average volute hydraulic diameter, m Throat diameter, m Hydraulic friction coefficient Disk friction coefficient Acceleration of gravity, m/s2 Impeller disk gap parameter, π¦0 /(π·2 /2) Inlet shock circulation head loss, m Disk friction head loss, m 18 International Journal of Rotating Machinery βππ : βπslp : βππ : βπvol : π»: π»0 : π»sh : π»∞ : πΎπ·: πΏ π: πΏ th : πΏ π: π: NPSH: NPSHπ : ππ : π: ππ : ππ : πsh : ππ€ : πsh0 : ππcir : in πππ : πππ· : ππvol : π: ππ : ππΏ : π: π‘: π’: π: π3π : π3π : π: π₯: π¦: π¦0 : π¦π : π: Impeller skin friction head loss, m Slippage head loss, m Volute head loss, m Volumetric head loss, m Manometric head of pump, m Water head at impeller outlet, m Pump shaft head, m Euler pump head, m Impeller disk loss coefficient Blade length, m Throat length, m Volute length, m Pump rotational speed, rpm Pump net positive suction head, m Pump required net positive suction head, m Pump specific speed Pressure, N/m2 Pressure before wearing ring clearance, N/m2 Pressure at pump suction side, N/m2 Pump shaft power, W Pumped water power, W Impeller Euler power, W Inlet shock circulation power loss, W Impeller friction power loss, W Disk friction power loss, W Volumetric power loss, W Pump discharge, m3 /s Impeller discharge, m3 /s Pump internal discharge leakage, m3 /s Radius, m Blade thickness, m Tangential flow velocity, m/s Absolute flow velocity, m/s Throughflow component of volute velocity, m/s Circulation velocity of flow in volute, m/s Relative velocity of flow, m/s Ratio, π»/π»0 Group parameter, (π¦ = πΆπ»∞ ) Axial gap between impeller disk and casing, m Clearance of wearing ring, m Number of blades. Greek Letters πΌ: πΌπ : π½π : πth : π: π: π: πvol : ]: π: π: Theoretical absolute velocity angle, degrees Volute angle, degrees Blade angle, degrees Volute throat angle, degrees Blade thickness coefficient Pump speed coefficient, π’2 /√2ππ» Manometric efficiency of pump Volumetric efficiency Fluid kinematic viscosity, m2 /s Density of fluid, kg/m3 Slip factor, ππ’2act /ππ’2 π: Pump flow coefficient, ππ2 /√2ππ» π : Roughness height, m. Subscripts 1: 2: 3: 4: 5: av: act: eye: π: hl: π: π: π: π : sh: th: π’: vol: Impeller inlet Impeller outlet Volute inlet Volute outlet (throat inlet) Pump outlet (throat outlet) Average Actual Pump eye Clearance Head loss Impeller Loss Radial direction Suction Shaft Volute throat Tangential direction Volumetric. References [1] M. Asuaje, F. Bakir, S. Kouidri, F. Kenyery, and R. Rey, “Numerical modelization of the flow in centrifugal pump: volute influence in velocity and pressure fields,” International Journal of Rotating Machinery, vol. 2005, no. 3, pp. 244–255, 2005. [2] H. W. Oh and M. K. Chung, “Optimum values of design variables versus specific speed for centrifugal pumps,” Proceedings of the Institution of Mechanical Engineers A, vol. 213, no. 3, pp. 219–226, 1999. [3] H. W. Oh and M. K. Chung, “Design and performance analysis of centrifugal pump,” World Academy of Science, Engineering and Technology, vol. 36, pp. 422–429, 2008. [4] R. A. van den Braembussche, “Flow and loss mechanisms in volutes of centrifugal pumps,” in Design and Analysis of High Speed Pumps, pp. 121–12-26, Educational Notes RTO-EN-AVT143, Paper 12, Neuilly-sur-Seine, France, 2006. [5] J. Tuzson, Centrifugal Pump Design, John Wiley & Sons, New York, NY, USA, 2000. [6] E. Logan Jr. and R. Roy, Handbook of Turbomachinery, Marcel Dekker, New York, NY, USA, 2003. [7] S. L. Dixon, Fluid Mechanics and Thermodynamics of Turbomachinery, Elsevier Butterworth-Heinemann, New York, NY, USA, 5th edition, 2005. [8] F. J. Wiesner, “A review of slip factors for centrifugal impellers,” Journal for Engineering for Power, vol. 89, no. 4, pp. 558–572, 1967. [9] C. E. Brennen, Hydrodynamics of Pump, Oxford University Press, Oxford, UK, 1994. [10] B. K. Hodge, Analysis and Design of Energy Systems, PrenticeHall, New York, NY, USA, 2nd edition, 1990. [11] J. F. GuΜlich, Centrifugal Pumps, Springer, Berlin, Germany, 2010. [12] N. P. Kruyt, Lecture Notes: Fluid Mechanics of Turbomachines II, Turbomachinery Laboratory, University of Twente, Amsterdam, The Netherlands, 2009. International Journal of Rotating Machinery [13] K. M. Srinivasan, Rotodynamic Pumps (Centrifugal and Axial), New Age International, New Delhi, India, 2008. [14] D. O. Baun and R. D. Flack, “Effects of volute design and number of impeller blades on lateral impeller forces and hydraulic performance,” International Journal of Rotating Machinery, vol. 9, no. 2, pp. 145–152, 2003. [15] M. H. Shojaee Fard and F. A. Boyaghchi, “Studies on the influence of various blade outlet angles in a centrifugal pump when handling viscous fluids,” American Journal of Applied Sciences, vol. 4, no. 9, pp. 718–724, 2007. 19 International Journal of Rotating Machinery Engineering Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Distributed Sensor Networks Journal of Sensors Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Control Science and Engineering Advances in Civil Engineering Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Submit your manuscripts at http://www.hindawi.com Journal of Journal of Electrical and Computer Engineering Robotics Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 VLSI Design Advances in OptoElectronics International Journal of Navigation and Observation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Chemical Engineering Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Active and Passive Electronic Components Antennas and Propagation Hindawi Publishing Corporation http://www.hindawi.com Aerospace Engineering Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 International Journal of International Journal of International Journal of Modelling & Simulation in Engineering Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Shock and Vibration Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Advances in Acoustics and Vibration Hindawi Publishing Corporation http://www.hindawi.com Volume 2014