Fluid Mechanics for industrial (Pump).pptx

Fluid Mechanics Pump

1. Mechanics of Fluids

2. CHAPTER 12 TURBOMACHINERY

3. Chapter Objectives The objectives of this chapter are to: ▲ Develop fundamental pump theory using the moment-of-momentum principle ▲ Describe radial-flow, mixed-flow, and axial-flow pumps, including presentation of prototype data and deviation from ideal conditions ▲ Introduce dimensionless pump coefficients and show how they are used in conjunction with similarity rules to develop data for turbomachinery design and analysis ▲ Demonstrate how pumps are incorporated into piping design and analysis ▲ Present introductory material on turbines: basic theory, types of turbines, and their selection and operation in conjunction with piping systems

4. 12.1 Introduction The bulk of this chapter is devoted to pumps, in particular, those that convey liquids such as water or gasoline. The centrifugal, or radial-flow pump is presented in detail, and to a lesser extent, mixed-flow and axial-flow pumps are studied. Dimensionalanalysis, an important tool for the selection and design of turbomachines, is presented next, followed by a disc - ussion of the proper selection and implementation of pumps in piping systems. Finally, turbines are introduced by considering their fundamental characteristics.

5. 12.2 Turbopumps A turbopump consists of two principal parts: an impeller, which imparts a rotary motion to the liquid, and the pump housing, or casing, which directs the liquid into the impeller region and transports it away under a higher pressure. Fig. 12.1 shows a typical single-suction radial-flow pump.

6. In a radial-flow pump, the impeller vanes are commonly curved backward and the impeller is relatively narrow. As the impeller becomes wider, the vanes have a double curvature, becoming twisted at the suction end. Such pumps convey liquids with a lower-pressure rise than radial-flow pumps and are called mixed-flow pumps. At the opposite extreme from the radial-flow pump is the axial-flow pump; it is characterized by the flow entering and leaving the impeller region axially, parallel to the shaft axis. Typically, an axial-flow pump delivers liquid with a relatively low pressure rise. For axial-flow pumps and some mixed-flow pumps, the impellers are open; that is, there is no shroud surrounding them.Various types of impellers are shown in Fig. 12.2.

7. 12.2.1 Radial-Flow Pumps Fig.12.3a defines a control volume that encompasses the impeller region. Flow enters through the inlet control surface and exits through the outlet surface. Note that a series of vanes exists within the control volume,and that they are rotating about the axis with an angular speed ω .

8. A portion of the control volume is shown at an instant in time in Fig.12.3b. The idealized velocity vectors are diagrammed at the inlet, location 1, and the outlet, location 2. In the velocity diagrams, V is the absolute fluid velocity, is the tangential component of V, and Vn is the radial, or normal, component of V. The peripheral or circumferential speed of the blade is u = ω r, where r is the radius of the control surface. The angle between V and u is α . The fluid velocity measured relative to the vane is v . The relative velocity is assumed to be always tangent to the vane; that is, perfect guidance of the fluid throughout the control volume takes place. The angle between v and u is designated as β ; since perfect guidance along the vane is assumed, β designates the blade angle as well.  

9. The moment-of-momentum relation, Eq. 4.6.3, can be written for steady flow in the form Applied to the control volume of Fig. 12.3, this becomes in which T is the torque acting on the fluid in the control volume, and the right - hand side represents the flux of angular momentum through the control volume. The power delivered to the fluid is the product of ω and T: (12.2.1) (12.2.2) (12.2.3)

10. From the velocity vector diagrams in Fig. 12.3b,Vt1 = V1cos a1 Vt2 = V2 cos a2so that Eq. 12.2.3 can be written as For the idealized situation in which there are no losses, the delivered power must be equal to gQHt, in which Ht is the theoretical pressure head rise across the pump (see Eq. 4.5.26). Then there results Euler’s turbomachine relation, Insight on the nature of flow through an impeller region can be obtained using Eq. 12.2.5. From the law of cosines we can write

11. These can be substituted into Eq. 12.2.5 to provide the relation The first term on the right-hand side represents the gain in kinetic energy as the fluid passes through the impeller; the second term accounts for the increase in pressure across the impeller. This can be seen by applying the energy equation across the impeller and solving for Ht: Eliminating Ht between Eqs. 12.2.6 and 12.2.7 yields the expression

12. This relation has historically been called the Bernoulli equation in rotating coor - dinates. Since z2 - z1 is usually much smaller than (p2 - p1)/γ，it can be eliminated, and thus the pressure difference is Returning to Eq. 12.2.5, we see that a “best design” for a pump would be one in which the angular momentum entering the impeller is zero, so that maximum pressure rise can take place. Then in Fig. 12.3b, α1= 90, Vn1 = V1, and Eq.12.2.5 becomes From the triangle geometry of Fig. 12.3, V2 cos α 2 = u2 - Vn2 cot β 2, so that Eq. 12.2.10 takes the form

13. Applying the continuity principle at the outlet region to the control volume provides the relation in which b2 is the width of the impeller at location 2. Introducing Eq. 12.2.12 into Eq. 12.2.11, and recalling that u2 = ω r2, one has the relation For a pump running at constant speed, Eq. 12.2.13 takes the form

14. in which α 0 and α 1 are constants. Equation 12.2.13 is the theoretical head curve and is seen to be a straight line with a slope of - α 1, as shown in Fig. 12.4a. The effect of the blade angle β 2 is shown in Fig. 12.4b. A forward curving blade ( β 2＞ 90°) can be unstable and cause pump surge, where the pump oscillates in an attempt to establish an operating point. Backward-curving vanes ( β 2 ＜90°) aregenerally preferred.

15. Head-Discharge Relations: Performance Curves. For real fluid flow, the theoretical head curve cannot be achieved in practice, and it is necessary to resort to experimentation to determine the actual head-discharge curve. The energy equation written across a pump from the suction side (location 1, Fig. 12.3) to the discharge side (location 2) is in which HP is the actual head across the pump, and hL represents the losses through the pump. The actual power delivered to the fluid, designated as ˙Wf, is

16. while the power delivered to the impeller is ˙WP, often termed the brake power, and is given by If there were no losses, ˙Wf would be equal to ˙WP. Since in actuality ˙Wf ˙WP, the pump efficiency1 hP is defined as

17. Example 12.1： A radial-flow pump has the following dimensions: β1=44 ° r1=21mm b1=11mm β2=30 ° r2=66mm b2=5mm For a rotational speed of 2500 rev/min, assuming ideal conditions (frictionless flow,negligible vane thickness, perfect guidance), with a1 = 90° (no prerotation), determine(a) the discharge, theoretical head, required power, and pressure rise across the impeller, and (b) the theoretical head-discharge curve. Use water as the fluid.

18. Solution： (a) Construct the velocity diagram at location 1, as shown in Fig. E12.1a. The rotational speed is converted to the appropriate units as The impeller speed at r1 is then

19. From the velocity diagram we see that and since The discharge is computed to be The normal component of velocity at location 2 is and the impeller speed at the outlet is The velocity diagram at location 2 is now sketched as shown in Fig. E12.1b. From the velocity diagram we see that

20. Therefore, The theoretical head is computed with Eq. 12.2.10: Hence the theoretical required power is

21. The pressure rise is determined from the energy equation as follows: (b) The theoretical head-discharge curve is Eq. 12.2.13. For the present example we have The curve is shown in Fig. E12.1c.

22. 12.2.2 Axial- and Mixed-Flow Pumps The velocity diagram for an axial-flow pump is shown in Fig. 12.7. In the axial-flow pump, there is no radial flow and the liquid particles leave the impeller at the same radius at which they enter, so that u1 = u2 = u. Furthermore, assuming a uniform flow, continuity considerations give Vn1 = Vn2 = Vn. Equation 12.2.5, which is valid for an axial-flow pump as well as a radial-flow pump, can be combined with the identities V2 cos α 2 = u - Vn cot β 2 and V1 cos α 1 = Vn cot α 1 to produce This form of the turbomachine relation is useful when the ideal absolute velocityentrance angle α 1 is established by a fixed vane, or stator. If there is no prerotation , α 1 = 90° and the theoretical head relation, Eq. 12.2.19, becomes

23. This relation is identical to Eq.12.2.11, which relates the theoretical head to the impeller outlet parameters for a radial flow pump. Note, however, that Eq.12.2.11 is valid at the periphery of the impeller for the radial-flow pump, and that all fluid particles reach the maximum head at that location. By contrast,Eq. 12.2.20 pertains only at a specified radius for the axial-flow pump, since fluid particles enter and leave the control volume at their respective radii. In this case the head varies from a minimum at the axis to a maximum at the periphery, and the total pump head is an integrated average. Axial-flow impellers are designed to maintain a constant axial velocity throughout the impeller region; this requires that the vane angles increase gradually from the periphery to the axis, as well as from the inlet to the outlet region. Fig. 12.8 on the next page shows a representative axial-flow pump, and its performance curves.

24. Example 12. 2 ： An axial-flow pump is designed with a fixed guide vane, or stator blade, located upstream of the impeller. The stator imparts an angle α 1 = 75° to the fluid as it enters the impeller region. The impeller has a rotational speed of 500 rpm with a blade exit angle of β 2 = 70°. The controlvolume has an outer diameter of Do = 300 mm and an inner diameter of Di = 150 mm. Determine the theoretical head rise and power required if 150 L/s of liquid (S = 0.85) is to be pumped.

25. Solution： First, the normal velocity component Vn is The peripheral speed u of the impeller is based on an average radius: The theoretical head Ht is computed with Eq. 12.2.19 to be Finally, for the assumed ideal conditions, the required power is

26. 12.2.3 Cavitation in Turbomachines Cavitation refers to conditions at certain locations within the turbomachine where the local pressure drops to the vapor pressure of the liquid, and as a result, vaporfilled cavities are formed. As the cavities are transported through the turbomachine into regions of greater pressure, they will collapse rapidly, generatingextremely high localized pressures. Those bubbles that collapse close to solid boundaries can weaken the solid surface, and after repeated collapsing, pittingerosion, and fatigue of the surface can occur. Signs of cavitation in turbopumps include noise, vibration, and lowering of the head-discharge and efficiency curves.Regions most susceptible to damage in a turbomachine are those slightly beyondthe low-pressure zones on the back side of impellers (see Fig. 12.10). In general,sudden changes in direction, sudden increases in area, and lack of streamlining arethe culprits causing cavitation damage in turbopumps .

27. The proper design of turbopumps will have minimized the possibility for cavitation to occur. Under adverse operating conditions, however, the pressures may decrease and cavitation may occur. Two parameters are used to designate the potential for cavitation: the cavitation number and the net positive suction head.Their interrelationship and use are now discussed.

28. Consider a pump operating in the manner shown in Fig. 12.11. Location 1 is on the liquid surface on the suction side, and location 2 is the point of minimum pressure within the pump. Writing the energy equation from location 1 to location 2 and using an absolute pressure datum results in in which hL is the loss between location 1 and location 2, Δ z = z2 - z1, and thekinetic energy at location 1 is assumed negligible. The minimum allowable pressure at location 2 is the vapor pressure pv. Substituting this into the expression,one can say that the left-hand side of Eq. 12.2.21 represents the maximum kineticenergy head possible at location 2 when cavitation is imminent. Thus the net positive suction head (NPSH) is defined as

29. The NPSH is also used for turbines; however, the sign of the hL term in Eq. 12.2.22 changes, and location 1 refers to the liquid surface on the discharge side of the machine. The design requirement for a pump is thus established as follows: The performance data supplied by turbomachinery manufacturers usually includes NPSH curves; these are developed by testing a given family in a laboratory environment. Figures 12.6, 12.8, and 12.9 show NPSH curves. The NPSH curve enables one to specify the required maximum value of Δ z to be used for a given turbomachine; note that it is necessary to estimate hL to obtain this. The right-hand side of Eq. 12.2.22 can be divided by HP, the total head across the pump to yield

30. Example 12. 3 ： Determine the elevation at which the 240-mm-diameter pump of Fig. 12.6 can be situated above the water surface of the suction reservoir without experiencing cavitation. Water at 15°C is being pumped at 250 m3/h. Neglect losses in the system. Use patm = 101 kPa.

31. Solution： From Fig. 12.6, at a discharge of 250 m3/h, the NPSH for the 240-mm-diameter impeller is approximately 7.4 m. For a water temperature of 15°C, pv = 1666 Pa absolute, andγ = 9800 N/m3 . Equation 12.2.22 with hL = 0 is employed to compute △z to be Thus the pump can be placed approximately 2.7 m above the suction reservoir water surface .

32. 12.3 DIMENSIONAL ANALYSIS AND SIMILITUDE FOR TURBOMACHINERY The development and utilization of turbomachinery in engineering practice has benefited greatly from the application of dimensional analysis ， probably to an extent more than any other area of applied fluid mechanics. 12.3.1 Dimensionless Coefficients The following parameters can be considered significant ones for a turbomachine: power, rotational speed, outer diameter of the impeller, discharge, pressure change across the impeller, density of the fluid, and viscosity of the fluid. (12.3.1)

33. Using v, r, and D as repeating variables, a set of convenient dimensionless groupings can be derived. They are given as follows: Because it is customary to equate p to , where represents either the pressure head rise in the case of a pump or the pressure head drop for a turbine.   (12.3.2) (12.3.3) (12.3.4) (12.3.5) (12.3.6)

34. TABLE 12.1 Turbomachinery Parameters Parameter Symbol Dimensions Power W Rotational speed Outer diameter of impeller Discharge Pressure change Fluid density Fluid viscosity TABLE 12.1 Turbomachinery Parameters Parameter Symbol Dimensions Power W Rotational speed Outer diameter of impeller Discharge Pressure change Fluid density Fluid viscosity

35. Another dimensionless parameter for a pump can be found by grouping , , and to form the ratio.   Hence the efficiency is also a similitude parameter. For a turbine the efficiency grouping is given by in which W is the output power . (12.3.7) (12.3.8)

36. Fig. 12.12 Dimensionless radial-flow pump performance curves for the pump presented in Fig. 12.6; .  

37. Fig.12 . 13 Dimensionless axial-flow pump performance curve for the pump presented in Fig.12.8；  

38. Fig.12.14 Dimensionless mixed-flow pump performance curve for the pump presented in Fig.12.9；  

39. 12.3.2 Similarity Rules Following the principles outlined in Chapter 6 for similitude, similarity relationships between any two pumps from the same geometric family can be developed. They are: (12.3.9) (12.3.10) (12.3.11)

40. These equations, called the turbomachinery similarity rules, are used to design or select a turbomachine from a family of geometrically similar units.Another use of them is to examine the effects of changing speed, fluid, or size on a given unit. They could also be used to design a pump to deliver flow on the moon or on a space station. In light of Eq.12.3.7, one should also expect that , but as stated previously, larger pumps are more efficient than smaller ones of the same geometric family.An acceptable empirical correlation (Stepanoff, 1957) relating efficiencies to size is   This relation can also be used for turbines by replacing by   （ 12.3.12 ）

41. Example 12.4 Determine the speed, size, and required power for the axial-flow pump of Fig. 12.13 to deliver of water at a head of .  

42. Solutio n The pump data are obtained from Fig. 12.13. Reading from the figure we find that at the design, or maximum, efficiency of   Equations 12.3.4 and 12.3.6 are employed to determine the two unknowns and . Rearrange Eq.12.3.6 to solve for :   Equation 12.3.4 is rearranged to include as a known quantity to provide :   Substituting into the relation , one finds that  

43. The density for water is r .Then the pump power is determined by using Eq. 12.3.2:   Thus the speed, size, and required power are approximately , , and , respectively.  

44. Determine the performance curve for the mixed-flow pump whose characteristic curves are shown in Fig.12.14 . The required discharge is with the pump operating at a speed of . At design efficiency, what are the power and the NPSH requirements ? Water is being pumped .   Example 12.5

45. Solutio n From Fig. 12.14 the design efficiency is and the corresponding design values for the coefficients are   The rotational speed in radians per second is The pump diameter is computed with Eq. 12.3.4: With Eqs. 12.3.2 and 12.3.6, the required power and are computed:  

46. Hence, the required power is and the required is . The performance curve is constructed using Eqs. 12.3.10 and 12.3.11:  

47. 12.3.3 Specific Speed It is possible to correlate a turbomachine of a given family to a dimensionless number that characterizes its operation at optimum conditions.Such a number is termed the specific speed , and it is determined in the following manner. The specific speed of a pump is a dimensionless parameter associated with its operation at maximum efficiency, with known , , and . It is obtained by eliminating between Eqs.12.3.4 and 12.3.6 and expressing the rotational speed to the first power :   (12.3.13)

48. (12.3.14) In Eq.12.3.13, is usually based on motor requirements,and the values of and are those at maximum efficiency. A preliminary pump selection can be based on the specific speed. Fig. 12.15 shows how the maximum efficiency and the discharge vary with for radial-flow pumps. The specific speed can be correlated with the type of impeller shown in Fig.12.2 ; it is given as follows:  

49. For a turbine, the specific speed is a dimensionless parameter associated with a given family of turbines operating at maximum efficiency, with known ， ， and . The diameter is eliminated in Eqs.12.3.2 and 12.3.6，and is expressed to the first power to obtain   A comparison of for different turbines will be made in Section 12.5.   (12.3.15)

50. Fig.12.15 Maximum efficiency as a function of specific speed and discharge for radial-flow pumps. (PUMP HAND-BOOK, 2ND EDITION (DURO) by Karassik. Copyright 1986 by McGraw-Hill Comp - anies, Inc. -Books. Reproduced with permission of McGraw-Hill Companies, Inc. -Books in the format Textbook via Copyright Clearance Center.)

51. Another measure of cavitation is . Analogous to the formulation of specific speed of a pump，Eq. 12.3.13，suction specific speed for either a pump or turbine is given by   (12.3.15) Here, two geometrically similar units will have the same when operating at the same flow coefficient . Conversely，equal values of indicate similar cavitation characteristics when the units are operating differently. Design values of are determined by experiment；when cavitation is not present，Eq. 12.3.16 is no longer valid.  

52. Select a pump to deliver 500 gal/min of water with a pressure rise of . Assume a rotational speed not to exceed .   Example 12. 6

53. Solutio n To estimate the specific speed we need the following: Use Eq. 12.3.13 to find the specific speed to be

54. From Eq.12.3.14,this would indicate a radial-flow pump.The pump of Fig.12.12 could be used, even though the specific speed (0.61) would result in a lower efficiency.With (Fig. 12.12), the speed is estimated with Eq.12.3.13 as   Hence the required speed is ，which does not exceed . The required diameter is determined with use of Fig. 12.12, where at maximum efficiency， .This is substituted into Eq.12.3.4 to determine the diameter:  

55. 12.4 USE OF TURBOPUMPS IN PIPING SYSTEMS The appropriate selection of one or more pumps to meet the flow demands of a piping system requires, in addition to a fundamental understanding of turbo-pumps, a hydraulic analysis of the pumps integrated into the piping system. A single pipe and pump arrangement is analyzed in Section 7.6.7. In Example 7.18 a system demand curve is obtained,and a trial solution to find the discharge and required pump head is shown. The technique is extended to simple piping systems in Section 11.3, wherein it is shown that one can employ either a pump performance curve or assume a constant power input to the pump. Section 11.4 deals with piping networks.In those systems,the pump performance curve can be represented by a polynomial relation (Eq. 11.4.14) and incorporated into the lin-earized energy equations, or alternatively, a constant pump power requirement can be employed (Eq. 11.4.15).

56. 12.4.1 Matching Pumps to System Demand Consider a single pipeline that contains a pump to deliver fluid between two reservoirs.The system demand curve is defined as (12.4.1) where atmospheric pressure is assumed at each reservoir，and the upstream and downstream reservoirs are at elevations and respectively . Note that need not be greater than and that the friction factor can vary as the discharge (i.e.,Reynolds number) varies. The term represents the minor losses in the pipe，see Section 7.6.4.Equation 12.4.1 is represented as curve (a) in Fig.12.16.The first term on the right-hand side is the static head and the second term is the head loss due to pipe friction and minor losses. The steepness of the demand curve is  

57. Fig. 12.16 Pump characteristic curve and system demand curve.

58. dependent on the sum of the loss coefficients in the system; as the loss coefficients increase,signified by curve (b) in Fig.12.16,the pumping head required for a given discharge is increased.Piping systems may experience short-term changes in the demand curve such as throttling of valves,and over the long term，aging of pipes may cause the loss coefficients to increase permanently. In either case the system demand curve could change from (a) to (b) as shown. For a given design discharge , Eq.12.3.14 allows the selection of a pump based on the specific speed. Once the type of pump is determined, an appropriate size is selected from a manufacturers characteristic curve; a representative curve is shown in Fig.12.16.The intersection of the characteristic curve with the desired system demand curve will provide the and .It is desirable to have the intersection occur at or close to the point of maximum efficiency of the pump, designated as the .  

59. 12.4.2 Pumps in Parallel and in Series In some instances, pumping installations may have a wide range of head or dis- charge requirements, so that a single pump may not meet the required range of demands. In these situations, pumps may be staged either in series or in parallel to provide operation in a more efficient manner. In this discussion, it is assumed that the pumps are placed at a single location with short lines connecting the sep- arate units. Where a large variation in flow demand is required, two or more pumps are placed in a parallel configuration (Fig. 12.17). Pumps are turned on individually to meet the required flow demand; in this way operation at higher efficiency can be attained. It is not necessary to have identical pumps, but individual pumps, when running in parallel, should not be operating in undesirable zones. For par- allel pumping the combined characteristic curve is generated by recognizing that the head across each pump is identical,and the total discharge through the pump- ing system is , the sum of the individual discharges through each pump. Note  

60. Fig. 12.17 Characteristic curves for pumps operating in parallel.

61. the existence of three operating points in Fig.12.17,in which pump A or pump B is used separately, or in which pumps A and B are combined. Other design operating points could be obtained by throttling the flow or by changing the pump speeds.The overall efficiency of pumps in parallel is (12.4.2) in which is the sum of the individual power required by each pump. For high head demands, pumps placed in series will produce a head rise greater than those of the individual pumps (Fig. 12.18). Since the discharge through each pump is identical, the characteristic curve is found by summing the head across each pump.Note that it is not necessary that the two pumps be iden- tical. In Fig. 12.18 the system demand curve is such that pump A operating alone cannot deliver any liquid because its shutoff head is lower than the static system head.There are two operating points,either with pump B alone or with pumps A and B combined.The overall efficiency is  

62. (12.4.3) in which is the sum of the individual heads across each pump.   Fig. 12.18 Characteristic curves for pumps operating in series.

63. Example 12.7 Water is pumped between two reservoirs in a pipeline with the following characteristics: The radial-flow pump characteristic curve is approximated by the formula   where is in meters and is in m ³ /s. Determine the discharge and pump head for the following situations； (a) , one pump placed in operation； (b) , with two identical pumps operating in parallel; and (c) the pump layout, discharge, and head for  

64. Fig. E12.7

65. Solution (a) The system demand curve (Eq. 12.4.1) is developed first: To find the operating point, equate the pump characteristic curve to the system demand curve, Reduce and solve for :  

66. Using the system demand curve, is computed as   (b) For two pumps in parallel, the characteristic curve is Equate this to the system demand curve and solve for :   The design head is calculated to be

67. (c) Since is greater than the single pump shutoff head , it is necessary to operate with two pumps in series.The combined pump curve is   The system demand curve is changed since . It becomes   or and

68. Fig. 12.19 Multistage centrifugal pump. (Courtesy of Sulzer Pumps Ltd.)

69. 12.4.3 Multistage Pumps Instead of placing several pumps in series, multistage pumps are available (Fig. 12.19). Basically, the impellers are all housed in a single casing and the outlet from one impeller stage ejects into the eye of the next. Such pumps can provide extremely high heads.For the pump shown in Fig.12.19,the pressure head range is to and the discharge can vary from down to . U p to eight stages can be selected and the maximum speed is about .  

70. 12.5 Turbines There are two types of turbines.. The reaction turbine utilizes both flow energy and kinetic energy of the liquid; energy conversion takes place in anenclosed space at pressures above atmospheric conditions.The impulse turbine requires that the flow energy in the liquid beconverted into kinetic energy by means of a nozzle before the liquid impacts onthe runner; the energy is in the form of a high-velocity jet at or near atmospher - ic pressure .Turbines can be classified according to turbine specific speed, as shown in Fig. 12.20.

71. 12.5.1 Reaction Turbines In the Francis turbine, the incoming flow through the guide vanes is radial, with a significant tangential velocity component at the entrance to the runner vanes (Fig. 12.22).

72. The theoretical torque delivered to the runner is developed by applying Eq. 12.2.1 to the control volume shown in Fig. 12.22; the same assumptions leading to the pump torque relation, Eq. 12.2.2, apply. The resulting relation is Multiplying the torque by the angular speed v gives the power delivered to the shaft: The fluid power input to the turbine is given by in which HT is the actual head drop across the turbine.Thus the overall efficiency is given by

73. The action of the guide vanes can be described by considering the velocity vector diagrams in Fig. 12.22b. Assume perfect guidance of the fluid along the guide vane; then the tangential velocity at the entrance to the runner is From the velocity vector diagram the tangential velocity is also given by The radial velocity component can be expressed in terms of the discharge Q and the width of the runner b1: Equations 12.5.5 to 12.5.7 are combined to eliminate Vt1 and Vn1. There results

74. T he important quantities are the variations of discharge, speed, and efficiency. The interrelationships among the three parameters are shown in the isoefficiency curve of Fig. 12.23.

75. A representative dimensionless performance curve of a Francis turbine is shown in Fig. 12.24.The speed and head are kept constant, and the guide vanes are automatically adjusted as the discharge varies in order to attain peak efficiency.

76. Example 12. 8 A reaction turbine, whose runner radii are r1 = 300 mm and r2 = 150 mm, operatesunder the following conditions: Q = 0.057 m3/s, w= 25 rad/s, α 1 = 30°, V1 = 6 m/s, α 2 = 80°, and V2 = 3 m/s. Assuming ideal conditions, find the torque applied to the runner, the head on the turbine, and the fluid power. Use p= 1000 kg/m3.

77. Solution： The applied torque is computed using Eq. 12.5.1: Under ideal conditions, the power delivered to the shaft is the same as the fluid power input to the turbine (i.e., ηT= 1). Thus The head on the turbine is found with the use of Eq. 12.5.3:

78. Eqs. 12.2.21 to 12.2.23 the sign of the loss term becomes positive. The Thoma cavitation number, defined by Eq. 12.2.24 for a pump, is given in the following form for a turbine: Typically, location 2 is defined at the outlet of the runner, and location 1 refers to the liquid surface at the draft tube outlet (Fig. 12.26a).Fig. 12.26b shows a representative plot of the cavitationnumber versus turbine efficiency, which is obtained experimentally by the turbine manufacturer.

79. 12.5.2 Impulse Turbines The Pelton wheel (Fig. 12.27e) is an impulse turbine that consists of three basic components: one or more stationary inlet nozzles, a runner, and a casing.

80. The moment of momentum equation, illustrated in the preceding sections, can be applied to the control volume shown in Fig. 12.28. Neglecting friction, the torque delivered to the wheel by the liquid jet is in which Q is the discharge from all jets, and u _x0002_ rv, r is the wheel radius as shown in Fig. 12.27b. The power delivered by the fluid to the turbine runner is Usually, β 2 varies between 160 and 168°. Differentiation of Eq. 12.5.11 with respect to u and setting it equal to zero shows that maximum power occurs when u = V1/2.The jet velocity can be given in terms of the available head HT:

81. The velocity coefficient C v accounts for the nozzle losses; typically，0.92 ≤ Cy ≤ 0.98.The efficiency is Substituting Eqs. 12.5.11 and 12.5.12 into this relation and rearranging produces in which the speed factor Φ is defined as

82. Fig. 12.29 shows Eq. 12.5.14 plotted for C v= 0.94 and β2= 168°.

83. Example 12. 9 ： A Pelton turbine rotates at an angular speed of 400 rpm, developing 67.5 kW under a head of 60 m of water. The inlet pipe diameter at the base of the single nozzle is 200 mm. The operating conditions are C v= 0.97, Φ = 0.46, and ηT = 0.83. Determine (a) the volumetric flow rate, (b) the diameter of the jet, (c) the wheel diameter, and (d) the pressure in the inlet pipe at the nozzle base.

84. Solution： (a) The discharge is computed from Eq. 12.5.13 to be (b) From Eq. 12.5.12, the velocity of the jet is The area of the jet is the discharge divided by V1, or Hence the jet diameter D1 is

85. (c) Use Eq. 12.5.15 to compute the wheel diameter D to be (d) The area of the inlet pipe is The piezometric head just upstream of the nozzle is equal to HT, so that the pressure at that location is

86. 12.5.3 Selection and Operation of Turbines A preliminary selection of the appropriate type of turbine for a given installation is based on the specific speed. Fig. 12.20 shows how the turbine runner varies with T. Impulse turbines normally operate most economically at heads above 300 m,but small units can be used for heads as low as 60 m. Heads up to 300 m are possible for Francis units, and propeller turbines are normally used for heads lower than 30 m. Fig. 12.32 illustrates the ranges of application for a variety of hydraulic turbines.

87. Example 12. 10 ： A discharge of 2100 m3/s and a head of 113 m are available for a proposed pumped - storage hydroelectric scheme. Reversible Francis pump/turbines are to be installed; in the turbine mode of operation, = 2.19, the rotational speed is 240 rpm, and the efficiency is 80%. Determine the power produced by each unit and the number of units required.

88. Solution： The power produced by each unit is found using the definition of specific speed,Eq. 12.3.15. Solving for the power, we have From Eq. 12.5.13, the discharge in each unit is The required number of units is equal to the available discharge divided by the discharge in each unit, or 2100/351 = 5.98. Hence, six units are required.

89. 12.6 Summary This chapter focuses on pumps or turbines that supply or extract energy by means of rotating impellers or vanes. First we emphasized the radial-flow, or centrifugal,pump and made use of the moment-of-momentum principle to derive an idealized head-discharge relation. We then contrasted that with the actual head-discharge performance curve, one that is obtained experimentally and is required for use in pump analysis and design. Axial-flow and mixed-flow pumps were introduced, and the differences between these and the radial-flow pump were discussed. The significance of pump efficiency and net positive suction head (relating to cavitation) diagrams were shown for the proper design and selection of pumps.

90. Dimensional analysis and similitude were applied to both pumps and turbines; the three significant dimensionless numbers that were developed are the power coefficient, flow coefficient, and head coefficient. These coefficients were combined to yield pump or turbine efficiency, or to produce the specific speed for a pump or turbine. It was shown that a family of pumps or turbines could be represented in dimensionless form by a single curve. The grouping of pumps—either singly, in parallel, or in series—to meet appropriate head and discharge requirements for piping was illustrated. Finally, a brief introduction to turbines was provided, by relating fundamental theory along with the application of reaction, axial-flow, and impulse turbines to various hydraulic design situations.