Model Study and Analysis of the Flow Elements of a Recirculation Mixing System

Transcription

1 Brigham Young University BYU ScholarsArchive All Theses and Dissertations Model Study and Analysis of the Flow Elements of a Recirculation Mixing System Albert Warren Berg Brigham Young University - Provo Follow this and additional works at: Part of the Mechanical Engineering Commons BYU ScholarsArchive Citation Berg, Albert Warren, "Model Study and Analysis of the Flow Elements of a Recirculation Mixing System" (1967). All Theses and Dissertations This Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact scholarsarchive@byu.edu, ellen_amatangelo@byu.edu.

2 \ 0 \ QO 'L' mi MODEL STUDY AND ANALYSIS OF THE FLOW ELEMENTS OF A RECIRCULATION MIXING SYSTEM ' A Thesis Presented to the Department of Mechanical Engineering Brigham Young University In Partial Fulfillment of the Requirements for the Degree Master of Science by Albert Warren Berg August, 1967

3 This thesis, by Albert Warren Berg, is accepted in its present form by the Department of Mechanical Engineering of Brigham Young University as satisfying the thesis requirement for the degree of Master of Science. 11 J uly Date

4 To Nelly

5 ACKNOWLEDGMENTS The author wishes to express his sincerest appreciation to Dr. Charles Warner for his personal help and suggestions throughout this entire project, to Farrin West and Don King for their support and encouragement, also to his father Mr. Warren R. Berg who made his schooling possible.

6 TABLE OF CONTENTS Page DEDICATION ACKNOWLEDGMENTS... LIST OF FIGURES... NOMENCLATURE... iii iv vii ix Chapter I. INTRODUCTION... 1 Goals Param eter Evaluation Reynolds Number Effects Analysis H. APPARATUS... 7 Model Instrumentation El. TESTING PROCEDURE IV. DISCUSSION OF RESULTS Blade Shape Velocity Profiles Water Horsepower Blade Spacing Test Results Summary V. FIELD DATA VI. CONCLUSIONS AND RE,COMMENDATIONS Conclusions Recommendations v

7 Page APPENDIX A DIMENSIONAL ANALYSIS APPENDIX B COMPUTER PROGRAMS APPENDIX C TYPICAL DATA SHEETS APPENDIX D INSTRUMENTATION DATA LITERATURE CITED GENERAL BIBLIOGRAPHY ABSTRACT..

8 LIST OF FIGURES Figure Page 1. Reynolds Number Factor for Head in Centrifugal Pumps of Moderate Size 3 2. Schematic of Impeller and Velocity Vector Diagram Significant Dimensions of Water Clarifier Pumping Mechanism Tabulation of a Dimensionless Comparison of Geometric Similarity in the Water Clarifier Pumping Mechanism Model Assembly,, Overhead View Model Assembly, Front View Reaction Well, Bottom View Reaction Well, Front View Tank and Stand Six Blade Impeller Impeller with Adjustable Replacable Blades, Bottom View Impeller with Adjustable Replacable Blades, Side View Position of Velocity Probes in Draft Tube Evaluation of Power Constant Power Equation Flow Rate Equation vii

9 Figure Page 17. Power Based on 4 FPS Tip Speed Flow Rate Based on 4 FPS Tip Speed Impeller Blade Configurations Impeller Blade Configurations Flow Rate vs, Speed Power Number vs. Speed Impeller Exit Velocity Profile, Radial Blades Impeller Exit Velocity Profile, Curved Blades Dimensionless Torque vs. Reynolds Number for a Rotating Flat Disk Flow Rate and Horsepower vs. Number of Blades for 77 RPM or 4 FPS Tip Speed Flow Rate and Horsepower vs. Number of Blades for 100 RPM or 5. 6 FPS Tip Speed Flow Rate and Horsepower vs. Number of Blades for 125 RPM or 6. 5 FPS Tip Speed Q/P vs. Number of Blades, Speed held Constant, Curved Blades Q/P vs. Number of Blades, Speed held Constant, Straight Radial Blades Field Data Tabulation Field Data Comparison viii

10 NOMENCLATURE A Cf Ch CP D d g H h ki N P p ideal p shaft p shear ^ pdyn Ap stat Q Q r area flow coefficient or flow number head coefficient or head number power coefficient or power number a significant linear dimension impeller diameter gravity head vertical impeller blade depth proportionality constant speed power power, assuming no losses shaft work into a control volume shear work in a control volume average change in dynamic pressure average change in static pressure flow rate rate of heat transfer radial distance

11 Re Rg S u V Vr Vrel Vj. 'ID' Z Reynolds Number impeller radius control surface internal energy absolute velocity radial velocity component velocity relative to impeller blade tangential velocity component control volume vertical height c* "is proportional to" ck, P, angle angle 9 mass density ft M specific weight viscosity x

12 MODEL STUDY AND ANALYSIS OF THE FLOW ELEMENTS OF A RECIRCULATION MIXING SYSTEM

13 CHAPTER I INTRODUCTION The term water clarifier as used in this text refers to a versatile water treatment unit which combines flocculation and coagulation, clarification and positive sludge removal in a single tank. Water clarifiers are very compact units vshich are being used in municipal and industrial water treatment for the removal of turbidity, algae, color, iron and silica; lime or lime-soda softening; magnesium precipitation; brine clarification; and waste water clarification with or without chemicals. Water clarifier units have found wide use in the sanitary field, providing high removals in the pre-treatment of sewage and in secondary sewage treatment through addition of coagulants. Goals One criterion used in practice for the design of such units predicted that excessive horsepower might be required to drive the impeller in some of the larger water clarifier units. If In fact, it could be shown that a given impeller could be driven with substani tially less power, the size of the drive could be reduced and the whole unit could be produced more economically. A program of study and testing that would allow more accurate prediction of the 1

14 2 power required was therefore initiated. The initial goal of the study was conformation of the design criteria used for predicting recirculation rate and power. The major goal was to improve the impeller design for the best possible recirculation rate and mixing consistent with a safe fluid shear level and minimum power. The nature of the water clarifier indicated that a model study would provide the necessary information. Parameter Evaluation Through the application of dimensional analysis and the theory of scaled models, a design of a given size and speed may be readily adapted to other speeds and sizes which are geometrically similar. Taking (H, Q, P,>1, f*, D, and N) as the set of important parameters, analysis yielded the following set of dimensionless products (see Appendix A)*: Uf Cu = h NLP H C = P Re = P A PVD *See page ix for nomenclature.

15 Reynolds Number Effects Most texts indicate that Reynolds Number effects for centrifugal pumps, operating with water, are negligible. Further, correction factors can be used to adjust the value of the dimensionless products to account for viscosity. Figure 1 is a plot of correction factors for head number versus Reynolds Number for centrifugal pumps of moderate size, Normal operating Reynolds Numbers for water clarifiers range from 10 to 2 x 10^, with Reynolds Number p; dropping to a minimum of 2 x 10 in a small one foot impeller diameter model. From Figure 1 it can be seen that indeed operation in water is beyond all Reynolds Number effects and therefore Reynolds Number may be deleted from the set of important dimensionless products in the present case [1,28] *. Fig Reynolds Number factor for head in centrifugal pumps of moderate size. *The notation [1,281 is to be read: Page 28 of book number 1 in the list of references.

16 4 Analysis In using the theory of scaled models for the designing of centrifugal pumps, geometric similitude is required as well as geometrically similar velocity diagrams at the entrance to and exit from the impeller [2, 345]. A series of units in which these similarity relationships are observed is said to be homologous. An impeller and velocity diagram is shown schematically in Figure 2. Fig. 2. Schematic of impeller and velocity vectors. The condition of geometric similitude requires that 5 must be the same for all units. The condition of similar streamlines requires thato<(must also be'equal on all units. Thus units in which both c>< and (3 are equal are homologous. It can be seen that Vr = Vt TAN (1) o r = TAN < Vt (2)

17 5 Let h be the impeller depth. For geometrically sim ilar units, h must be proportional to the diameter, hence, H = K D, Since Q = Vr Tt Dh = VrK-]Tf D2 (3) r and Vt = Tf DN (4) (5) Substituting equations 4 and 5 into equation 2 yields Q. NDa = K1rr^TAWo( (6 ) In homologous units c/^is constant and therefore the dimensionless flow rate number becomes = constant (7) NDa Since head is proportional to velocity squared and since flow rate is p proportional to velocity times D, we have H* (8) or Q (9) substitution in equation 7 yields ^ N Da = constant ( 10) thus the second dimensionless ratio is formed H = constant N2D2 Since power is porportional to tf'qh x v Q. v H ( 11) ( 12)

18 6 as the flow rate number and head number are both constant, it is found for homologous units that (^ 3 P 5 = constant (13) which is the third dimensionless product. Only two of these three products are independent. For convenience the flow number and the power number will be used as independent products for determining modeling laws.

19 CHAPTERE APPARATUS Model The selection of a geometrically similar model proved to be difficult because production units are not all exactly geometrically sim ilar. The dimensions of the water clarifier which are significant in the determination of the power and flow rate number are the turbine or impeller dimensions, and the reaction well and baffle dimensions. Since impeller dimensions and baffle dimensions vary directly as the size of the impeller, they are geometrically similar. The reaction well dimensions of several representative units were divided by the diameter of the impeller. These ratios are presented for comparison in Figure 3 and Figure 4. Inspection of Figure 4 shows that the ratios A/d, B/d, and C/d, ratios of the significant dimensions of the reaction well, are relatively constant. In other words, they do not vary much with water clarifier size. Hence, a model similar to one size is reasonably similar to all sizes. A model was built with an impeller (turbine) diameter of one foot which was scaled after size "N". Because size "N" is fairly representative of all other sizes geometrically, it was felt that the experimental results yielded by this model would apply with small error to all other sizes. 7

20 Fig. 3. Significant Dimensions of lfeter Clarifier Ramping Mechanism

21 9 j Unit d (diam) A/d B/d C/d G H I J K L M N P Q R S T U V W X M>del Fig Tabulation of a dimensionless comparison of geometric similarity in the water clarifier pumping mechanism. See Fig. 3 for explanation of dimensionless terms.

22 10 Figure 5 shows an overhead view of the test model. The Baldwin SR-4 torque meter, which was used for torque measurement can be seen in the superstructure under the 3/4 hp dc motor and right i angle gear drive. Also, visible through the front observation window is the window on the bottom of the lank. In Figure 6 the window in the reaction well can be seen. The windows were used extensively for observation and motion picture photography of the flow patterns in the reaction well region. For flow visualization, small colored neutral buoyancy particles were putihtofee water. The arrangement of the reaction well window, baffles, and general construction can be seen in Figures 7 and 8. The tank and stand assembly is shown in Figure 9. The stand was constructed to give adequate ground clearance for observation and photographic work through the bottom observation window. Originally an impeller wife six fixed blades (see Figure 10) was used, however for more versatility a modified impeller with twenty-four adjustable, removable blades was constructed (see Figures 11 and 12). The modified impeller was used for all tests. Instrumentation Instrumentation for measurement of the power and flow rate was designed and constructed. The power was not difficult to measure. I I The signal from the torque meter was first fed through an appro- priate amplifier and then to a strip chart recorder. Thus the torque at any speed could easily be measured. As will be seen, a great

23 11 Fig M odel A ssem bly, O verhead View. Fig M odel A ssem bly, F ro n t View.

24 12 Fig Reaction Well, Bottom View. Fig Reaction Well, Front View.

25 13 Fig Six Blade Impeller. Fig Tank and Stand.

26 14 Fig Impeller with Adjustable Feplacable Blades, Bottom View. Fig. 12. Impeller with Adjustable Replacable Blades, Side View.

27 15 deal of effort was put into finding a reliable flow measurement system. It was thought that by using a pitot tube, a velocity traverse across the exit of the impeller could be made. Since the area of the impeller was known, the flow rate could be found. However, it was found that there was a significant variation in dynamic pressure as each blade of the impeller passed the pitot tube. This, coupled with the fact that the flow had only a small radial component of velocity at the edge of the impeller, made accurate flow measurements very difficult. To overcome the problem of the pulsing pressure, a very sensitive pressure transducer was installed which would record the dynamic variation of pressure at the pitot tube. The pressure transducer was connected to the strip chart recorder through an appropriate amplifier. After eliminating some minor electrical problems the recording instrument worked very well. However, it was found that good results could not be obtained at high speeds of operation because the range of the pressure transducer was limited. Delivery time of a new pressure transducer was excessive so another means of measuring flow rate was sought. The obvious place for flow measurement is in the draft tube; however, the installation of pitot-static tubes created some problem. Therefore it was decided to first attempt to measure the velocity profile of the horizontal flow across the entrance to the draft tube. After building and installing appropriate instrumentation it was found that the dynamic pressure was so low that accurate measurements could not be made even with a sensitive micro-manometer. There-

28 16 fore, the water clarifier model was disassembled and pitot and static pressure tubes were installed in the draft tube (see ELgure 13). These pressure tubes were installed such that their position (both radial and angular) could be controlled from outside the water clarifier, thus enabling the alignment of the pressure tubes with the flow. In this system of flow measurement the results were repeatable. Further, it proved to be easy to align the pitot tube and the static tube with the flow. By recording both the dynamic pressure and the angle at which the flow was traveling in the draft tube (see data sheets, Appendix C) the flow rate could be determined. Before making each test, the torque meter was carefully calibrated by placing a known torque on the impeller and noting the read-out on the torque meter indicator. Also, the micro-manometer which was used for the dynamic pressure measurement was carefully adjusted before every test. The pressure measurements were highly reliable, as the micromanometer was capable of measuring a pressure to the nearest 1/1000 of an inch of water. It was felt that less time variation of pressure in the flow at the impeller outlet would be obtained by using an impeller with more blades. Also, since the optimum number of blades that should be used was not known, it was logical that tests should be run to determine the optimum number of blades. In order to make these tests as simple as possible, a modified impeller was designed and built. On the modified impeller, as many as twenty-four blades could be installed. Each

29 17 Fig. 13. Position of Velocity Probes in Draft Tube.

30 18 blade was mounted on one screw so that it could easily be adjusted or exchanged. The water clarifier model, fully instrumented with the new impeller, was a very helpful tool for finding flow and performance characteristics.

31 CHAPTER HI TESTING PROCEDURE In the first tests that were run, the impeller was fitted with twenty-four equally spaced straight radial blades. It was felt that with twenty-four straight radial blades the maximum flow for the straight radial blade configuration would be obtained. It was also felt that the pressure fluctuation would be minimized. The goal in this first set of tests was to find a value for the constants in the power and flow rate numbers. To do this, torque, speed, and draft tube velocity measurements were taken. In this and all succeeding tests, the pitot and static tubes were in the central region of the draft tube (see Figure 13). The draft tube on the model was eight inches in diameter. In order to get an accurate flow rate measurement, velocity measurements were taken at 1/2 inch radial increments from the center of the draft tube. In other words, the magnitude and direction of the velocity was measured at radii of 1/2 inch, 1 inch, 1 1/2 inches, 2 inches, 2 1/2 inches, 3 inches, 3 1/2 inches, and as close to 4 inches as possible (near the wall of the draft tube). Thus, with the velocity and direction of the fluid known at these points in the draft tube, the total flow rates could accurately be found by summing the flow through each 1/2 inch thick annular ring in the draft tube. In 19

32 20 other words, the flow was found in the ring from radius 3 1/2 to 4 inches, and 2 1/2 to 3 inches, etc. Then all these individual flow rates were added, giving the total flow rate for a particular impeller speed. In taking an individual velocity measurement, the pitot tube was first adjusted to the desired radial depth and then rotated until a maximum pressure reading was obtained. This gave the angle where the pitot tube was in line with the direction of the flow at any given point. The static tube was then aligned to the same position and. then the position of both tubes was re-checked. In this way the direction of the flow was found. In calculating the flow, only the vertical component of velocity in the draft tube was used, of course. There was a little concern that the flow rate measured might actually be somewhat in error. In order to check this, flow rate measurements made using the above method were checked against those taken from a velocity traverse across the exit of the impeller. This comparison was made using curved blades. In the test, it was found that the flow rates agreed within 1%. Though 1% accuracy might have not been typical it indicated a high degree of reliability in the method of flow rate measurement that was used. In order to determine the values of the constants in the power and flow rate equations, many many tests were run. The water clarifier was run with twenty-four straight radial blades at speeds of 2. 6, 4. 0, 5. 2, 6. 5, and 7. 8 feet per second tip speed. By evaluating the power constant and the flow rate constant at each of these speeds an

33 21 accurate average value could be found. At each speed, several (sometimes as many as six or eight runs) were made, in order that sufficient data would be obtained for accurate calculation of the average power and flow rate constants.

34 CHAPTER IV DISCUSSION OF RESULTS Great care was taken to assure the accuracy of all calculations. The flow rate constant was evaluated for each run and speed. The value of the torque required, along with the speed and the characteristic dimension, were used to evaluate the power equation, and the value for the power constant was found. When the values of the power and flow rate coefficients were all plotted versus speed, it was found that they were very nearly constant (within 1 or 2%) and that the results were repeatable (see Figure 14). It was found that when P was expressed in horsepower, N in revolutions per minute, and D in feet of impeller diameter, and Q in gallons per minute, that the power reduced to: P =.465 x 10"7 x NSD5 and the flow rate reduced to: Q = ND3 These equations are graphically represented in Figures 15 and 16, and should hold for all water clarifiers, which are geometrically sim ilar to this model, at all normal operating speeds. It is felt that these numbers are probably conservative. When the size of a homologous unit is increased the efficiency generally increases. In other words, in an actual model the flow rate would be slightly higher and the power will be 22

35 >: ' ' * *. i ' ; f JV V< 1 < UN O *)* CU.0050 co CO SPEED (RPM) F ig. 1U. -E valuation o f Power Constant

36 24 Fig Power Equation.

37 N (RPM) DO CJl F ig Flow Rate Equation.

38 26 slightly lower than these equations predict. Comparing the power equation with values found in a particular water clarifier standardization data tabulation used by one company, it was found that the power equation predicted that about 20% or 30% less power and about 20% less flow rate predicted on the water clarifier standardization sheet was required. These correlations indicate that the power values on the standardization sheet are conservative. The correlations are shown graphically in Figures 17 and 18. Blade Shape There are two major categories of inefficiency in centrifugal pumps in addition to friction. The first is not an actual loss but a failure of a finite number of blades to fully impart the desired motion to the fluid. Without perfect guidance which theoretically requires an infinite number of blades, the fluid is discharged as if the blades had an angle some small amount different than they actually do. For the same power input, the flow is reduced. The second loss which must be considered is that of turbulence. It is the loss due to an improper relative velocity angle at the blade inlet which causes undesirable viscous effects and separation: "The pump can be designed for one discharge (at a given speed) at which the relative velocity is tangent to the blade inlet. This is the point of best efficiency, and shock or turbulence losses are negligible. For other discharges, the loss varies about as the square of the dis- crepency in the approach angle." [2,99b]

39 SHAFT HP DO -a Fig. 17. Power Based on 4 FPS Tip Speed. DIAMETER (ft)

40 Q (GPM) eo CO Fig. 18. Flow Rate Based on 4 FPS Tip Speed DIAMETER (ft)

41 29 Because as indicated the major inefficiencies are at the inlet and outlet of the impeller it was decided to investigate the effect of the blade design in the water clarifier model. Performance of four different blade designs was investigated. (Figures 19 and 20). After running many tests, the flow rates and powers were determined. The first blade employed was the straight radial blade presently being used in the field. The power and flow rate correlations associated with this blade have been discussed above. It was found that by turning the straight blades to an angle of 45 backwards, the power required was cut by a factor of three while the flow rate dropped only slightly. Further, it was suggested that the drop in flow rate was due to the decrease in radial depth of the blade. To test this theory, a third blade was built and installed. This blade had the same radial depth as the straight radial blades and was bent about 10 in the center. With the third blade, the flow rate actually increased above that of the radial blades, while the power required was only about one half as much. Enough data was then available to design and build a blade of near optimum curvature. The blade design was of circular section which approximates the ideal blade. The curvature was calculated such that the flow theoretically entered and left the impeller tangent to the blades. With this curved blade, the power dropped to about 25% of that of the radial blade. Also, the curved blade had nearly the same flow rate as the radial blade at the same speed. The performance of the blades tested is shown in Figures 21 and 22. Thus

42 30 Fig Im p e lle r Blade Configurations.

43 31 Fig Im p e lle r Blade C onfigurations.

44 Q (GFM) co CO SPEED (RPM) Fig,, 21. Flow Rate vs Speed

45 ( P / f N ^ 5) co CO SPEED (RFM) Figo 2 2. Power Number vs. Speed

46 34 the curved blade had by far the highest flow rate per unit power of all blades tested. It must be pointed out that insufficient data exists to insure that because the power requirement was decreased by 75% in the model, the power requirement could be cut 75% in the field. However, a substantial reduction in power should be obtained in the field as well. Examination of the data indicated that the flow rate for the curved blades could be increased by optimizing the blade spacing. Velocity Profiles Another very interesting and perhaps the most important result of the study was the influence of the blade shape on the velocity profile in the draft tube and especially at the impeller exit. It was found that in the case of curved blades the velocity profile was near that of plug flow. In other words, the velocity at the exit of the blade was very nearly uniform along the height of the blade. (See Figures 23 and 24.) However, with the straight radial blades, only the top half of the impeller was really effective; the velocity there being more than twice that in the curved blade case at the same flow rate. It was found that by using curved blades, the flow rate in the draft tube was also closer to plub flow than in the straight blade case. The fact that plug flow can be obtained through the impeller is very important. The fluid velocities are now much lower for the same flow rate. Also, there is far less fluid shear across the impeller exit

47 35 Fig. 23. Impeller Exit Velocity Profile, Radial Blades. -*>- * > >- *- >- > > Fig. 24. Impeller Exit Velocity Profile, Curved Blades.

48 36 which is the critical region for floe break-up, * Thus the impeller can be run at higher speeds and for a given flow rate without increasing the rate of floe destruction and the torque that must be transmitted for a given power input is much smaller. It is therefore possible to reduce the size of the gear drive required. Further^ the process can be improved by taking advantage of the more even velocity distribution. It was also discovered that the curved blades operated even more efficiently when straightening vanes were installed in the draft tube, encouraging the flow to have the proper entry conditions to the blades. The efficiency was thus improved. Water Horsepower The water horsepower is that power actually imparted to the fluid by the impeller to cause the flow. Thus? the ratio water horsepower to shaft horsepower is actually a measure of the overall efficiency of the system. It was determined that since the average velocity exiting the impeller in the straight radial blade case was much higher than that in the curved blade case at the same flow rate, that part of the energy being supplied was consumed in the establishment of harmful velocity gradients. In order to determine the portion of the energy that was being used to accelerate the fluid to a higher velocity,, the first law of thermodynamics was employed [3,39]. *The flocculation process consists of a chemical reaction in which a precipitate is formed. The precipitate is gently recirculated and mixed with new flow thus using the fine precipitate particles as nucleation sites for building large floe particles which may be destroyed in regions of high fluid shear.

49 37 SHEAR which can be reduced to For an approximate solution average pressures and velocities (see Figure 13) may be used, and the equation becomes f or, using Bernoulli's equation The outlet area of the impeller is three halves of the inlet area. If uniform plug flow could be maintained throughout the impeller area, the outlet velocity would be lower than the inlet velocity and thus the outlet dynamic pressure would be lower than the inlet dynamic pressure. From the first law of thermodynamics it can be seen that the required water horsepower, P(ideal), would be minimized for a given static head. Even though the mass flow rate remains constant, an increase in the outlet velocity would increase the power required. Thus with the first law of thermodynamics, it can be determined from static and dynamic pressure measurements that part of the power input actually used for increasing the re-circulation rate, and that part wasted in establishing harmful velocity gradients. With the curved blades the component of the power absorbed in establishing velocity gradients

50 38 was very small. With the straight blades (like those that are now being used in the field)the velocity profile was not uniform and there was a very high velocity, small area,, jet coming out of the top of the impeller. Thus the portion of the power going to kinetic energy was much higher than in the curved blade case. This high kinetic energy could cause floc break-up. In the straight blade case,, a very large proportion of the actual horsepower was put into accelerating the water into a small area high speed jet which made small contribution to the re-circulation rate. This accounted for a good part of the reduction in power between the straight radial blades and the curved blades. With the straight blades a large portion of the shaft power is going into accelerating the fluid into the high speed narrow jet. This is an obvious waste of power. Thus, it can be seen that perhaps the biggest benefit of using the curved blades is in the smooth uniform velocity profiles produced. The fact that the efficiency increases is not only due to the improved blade shape but mainly to the improved velocity distribution. The velocity distribution therefore not only decreases the amount of shear in the water going through the impeller and allows faster flow rates, but also very much increases the water horsepower and re-circulation efficiency. In other words, with the curved blades, much more flow, mixing, and re-circulation per unit power is obtained without exceeding the safe velocity limit for maintaining flocculation.

51 39 Blade Spacing The flow induced by the impeller can be considered to arise from two sources. The first is the flow caused simply by the presence of the rotating plate in the fluid. This has been thoroughly discussed by Schlichting [4,84-89, The relation between moment coefficient Cm = T / (w3r3/2) and Reynolds Number Re = R^w/n is sketched in Figure 25. The Reynolds Number employed in the tests to be discussed shortly are indicated on the figure. Schlichting gives empirical formulas for the flow rate for laminar and turbulent flow. For a disk of radius R and turning at angular velocity w in a fluid with kinematic viscosity, they are for a disk wetted on one side: QLAM = 886 R^wdte)"1/ 2, 0 = Re - 3xl05 Qtu rb = R3w(Re)~1/5, 105 Re 107 The revelance of this data to the present study is that it provides the zero-blade limiting case for the present impeller design. * The addition of blades can be expected to have two types of effects. Firstly, the obvious effect of better motion transfer from impeller to fluid can be expected, together with the attendant increases in power and flow rate. A second, more subtle effect, is the tendency toward transition to turbulence introduced by the presence of the blade. * Although the model impeller is wetted on both sides the actual measured flows are aided only by the action of the lower side of the disk. The flow above the disk, although it absorbs power, makes no contribution to the measurable flow rates.

52 DIMENSIONLESS TORQUE Fig, 2 5. Dimensionless Torque v g e Reynolds Number for a Rotating Flat Disk.

53 41 We shall return to this phenomenon in the discussion of the experimental results which follows. Test Results The results of model tests of impellers with varying numbers of straight and curved blades are reported for three speeds in Figures 26, 27 and 28. The effect of additional blades can be seen to increase the flow and power transferred from the impeller to the fluid in the case of both straight and curved blades. It will also be seen in almost every case that while the curved blades impart slightly less flow rate to the fluid than the straight blades, the power absorbed is also significantly less and the kinetic energy better distributed for the curved blades, as discussed above. This is reflected in Figures 29 and 30, where the ratio of flow rate to power is given as a function of speed and number of blades. Here the real merit of curved blades is clearly shown. One of the difficulties encountered in this study is also shown in these figures. Due to the low power transfer rates for the lower numbers of blades, the torquemeter, used in the tests was operating in the lower 10% of its range. Further, the relative magnitude of the fluctuations to the average increases for the fewer-blade measurements. This made precise readings quite elusive, increasing the uncertainty of the power data in this region. To a similar, but less severe degree, the flow measurement uncertainty also is larger for few blades. Figures 29 and 30 therefore show uncertainty bands

54 42 Hp x 10' NUMBER OF BLADES Fig Flow Rate and Horsepower vs. Number of Blades for 77 RPM or 4 FPS Tip Speed.

55 43 Hp x 10 Fig Flow Rate and Horsepower vs. Number of Blades for 100 RPM or 5. 6 FPS Tip Speed.

56 44 FLOW RATE (GAL/fem ) i x da NUMBER OF BLADES Fig Flow Rate and Horsepower vs. Number of Blades for 125 RPM or 6. 5 FPS Tip Speed

57 45 50K 1 40K j ) 77RPM 30K Q/p (Gal/Hp) i i I ) iu U RrM 10K Li f ) 125RPM j 0 i NUMBER OF BLADES Fig. 29. Q/P vs, Number of Blades, Speed held Constant, Curved Blades.

58 46 50K 40K 30K Q/P (Gal/Hp) 20K 10K i A A ) 77RPM > 100 RPM ) 125 RPM NUMBER OF BLADES Fig. 30, Q/P vs. Number of Blades, Speed held Constant, Straight Radial Blades.

59 47 into which the majority of experimental data points are expected to fall, together with the data taken. As a means of comparison, the theoretical blade-free Q/P ratio has been included at the intercept for each speed. To determine this point, the lower, laminar ratio was calculated for each speed from the Schlichting data discussed above. The trends shown in Figures 29 and 30 are especially interesting. In Figure 29 the curved blade case, the Q/P ratio or efficiency increases as the number of blades increases. The slope of the curves indicates that the fluid guidance through the impeller and hence, efficiency, is approaching optimum for the 24 blade case. An increase in the number of blades over 24 would probably show a decrease in the Q/P ratio as turbulence and separation losses through the blades would become significant. There is some uncertainty as to the shape of the curve between the zero and four blade case. Although what happens to the flow in this region is not fully understood, some interesting possibilities present themselves. It will be noted in Figure 25 that all of the speeds employed in the tests correspond to Reynolds Numbers in the transition range. It is quite possible, that the addition of one blade to a rotating flat disk would have two effects. A significant departure from the power and flow rate behavior of the free disk would be expected. A more significant effect is the encouragement to transition provided by the presence of the blade.

60 48 Unfortunately, the limited range of the present instrumentation precluded a complete study of this effect. Summary It is felt that the trends shown are quite realistic and that the data presented are reliable. Not only have curved blades shown a general increase in efficiency over the straight radial case, but the trend of increasing efficiency with number of blades is completely in line with centrifugal pump theory. Figures 26 through 28 demonstrate a significant increase in flow rate from 4 to 12 blades at all operating speeds. Of course, the optimization of blade spacing on an actual operating unit requires considerably more attention to geometric detail than could be given here. These tests, rather than providing specific design information, should serve as a guide to meaningful, specific tests more directly related to actual equipment.

61 CHAPTERV FIELD DATA As a means of evaluation of model results, a comparison was attempted with power data reported from field installations of several water clarifier units. The field data referred to is summarized in Figure 31. Several obstacles stand in the way of a meaningful comparison. First, examination of the geometries suggested in Figure 31 reveals that there is not a good case for dynamic flow similarity with the model in any instance. The complex arrangements of baffles, blades, and dimensions renders such a comparison dubious, to say the least. Secondly, the field data itself is not too meaningful. It is in places self-contradictory. It represents only total electrical input, leaving open to question the motor and drive efficiencies and their variation with speed. Perhaps even more important, no power factor measurements were reported. * Taken together, this means that the field data itself, except for the maxima indicated, gives little information which may be used for comparison. A third factor in such a comparison is the size effect, or * Power factor (^) = EI/P can vary for common induction motors from near 1. 0 at design load to as low as 0. 2 for low-load conditions. 49

62 Fig Field Data Tabulation.

63 51 the increase in efficiency that attends a scaling-up of a hydraulic machine, due primarily to an improved configuration for viscous effects. Although this is surely not a large effect, it could be expected to give some improvement of performance. Bearing these limitations in mind, consider the comparisons illustrated in Figures 32. As may be observed, the electrical (P = V"3EI)* "power" curve from the actual units, because of the efficiency and power factor considerations, is consistently higher than the theoretical behavior (P = kn^). However, the degree of correspondence is very striking in the light of these considerations. If suitable allowances are made for the efficiencies (gear and mechanical) and power factor, the actual and predicted powers are in very close correspondence. It can thus be seen that even with some geometric dissimilarity, a reasonable power prediction is afforded from this simple model. This means that the power requirements for any water clarifier may be predicted from the study of a model satisfying dynamic sim ilarity requirements. The economic impact of this simple design tool on the design of very large hydraulic units is obvious. *For a 3-phase motor, P = t/3ei cos 0. For these calculations, cos 0 was taken as unity.

64 52 2 Hp 1 0 Fig Field Data Comparison.

65 CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS Conclusions The design, construction, and testing of a model water clarifier pump have been discussed. In light of the information presented several conclusions may be drawn. 1. The power and flow rate of the model obeys the expected simple functional dependence upon speed. Comparison with field tests data, even in the absence of complete dynamic similarity, indicate that the model properly predicts the performance of units almost thirty times its size. This successful prediction is very impressive. 2. By using curved rather than straight radial blades, significant improvements in power and flow rate characteristics were obtained for all speeds tested. In some instances, power dropped to one-fourth of its straight radial blade value. Mixing of the fluid in the reaction well is an important part of the water clarifier process. The reduction of power achieved through the use of curved blades does not come at the expense of the mixing process. In arriving at this conclusion the mechanism of losses must be considered. It is true that by using curved blades, the tangential velocity at the blade exit is reduced and thus some mixing is eliminated. However, only a small part of the power economy comes from the elimination 53

66 54 of the tangential velocity. The main saving comes from improving the blade entry conditions. In other words, there is gross separation as the fluid enters the straight radial blades because of difference in direction of flow and blade. Thus turbulence and separation losses are very high. The power saving from this source much more than makes up for the loss in mixing by having essentially radial flow. Because the radial component of velocity is essentially the same for the straight and curved blade cases, the re-circulation rate is the same and increasing the power for the curved blade case to the level of the straight blade case means more mixing and a much higher recirculation rate. 3. Measured velocity profiles at the impeller exit and in the draft tube indicate better floe conveyance properties for the curved blades, due to more nearly uniform velocity distributions. 4. Data relative to blade spacing indicates there is a practical limit to the number of blades to be used. Tests on large scale models are needed to determine the optimum blade spacing for a given size water clarifier. Recommendations Based on the results of these tests, several recommendations are appropriate. 1. Future model testing, as the testing of large scale models, could yield significant advance data as proposed clarifier designs. 2. The use of curved or shaped blades is recommended

67 55 because not only is the process improved,, but the cost of the process and the hardware (gear drives) can be reduced. 3. At sometime in the process of development of a successful water clarifier, a program for evaluation of optimum blade spacing should be undertaken, not only on dynamically similar models, but also on full-scale units where possible.

68 56 APPENDIX A DIMENSIONAL ANALYSIS

69 DIMENSIONAL ANALYSIS A general procedure for dimensional analysis using matrix notation* [5,7] will be used to find a complete set of dimensionless products. The number (N) of variables for this problem is 7. The variables are H (head), Q (flow rate), P (power), A (viscosity), f (density), D (diameter), and N (speed). These variables are first arranged in a dimensional matrix with one column for each parameter and one row for each dimension such as mass, length, time, etc H Q P M. P D N M L T It can be seen that the rank of the matrix is 3, thus the number of *This matrix notation method of organizing dimensionless numbers produces two results: (1) It always gives a complete set of dimensionless products (of course depending on the parameters used); (2) It places one of a predetermined group of the original parameters in each of the products obtained, as long as it is possible [5,7]. 57

70 58 dimensional products will be N-R = 7-3 = 4. The rows of the rearranged matrix are next expressed as three homogenous simultaneous equations. k 3 + K4 + k 5-0 2K4 + 3Kg + 2Kg - K4-3K5 + K0 = 0 -K4 - K2-3Kg - K4 - = 0 solving for Kg, Kg, K^ k 5 = - k 3 - k4 Kg = -2K1-3Kg - 5K3-2K4 K7 = -2K 4 - Kg - 3Kg - K4 A solution matrix is prepared with (N - R = 4) rows corresponding to the first four parameters and (R = 3) columns in which the elements are coefficients in the solutions obtained in the previous step f D N 1 H Q P A

71 59 The four dimensionless products are written as the four row parameters times the three column parameters raised to the powers given by the elements in the rows of the solutions matrix ^ 1 = 1$ d 2 ^ 3 = fn3d5 ^ M, A % CD'/N * f DV It can be seen that IT4 is simply the inverse of Reynolds Number.

72 60 APPENDIX B COMPUTER PROGRAMS

73 COMPUTER PROGRAMS Included in this appendix are two computer programs. The first program was used for data reduction. A computer analysis was suggested as the flow rate in over fifteen hundred cases had to be calculated. A simple program was written. The program and sample output are included in this appendix. The second program included was written to put the results, for the straight radial blade case, in > tabular form. The results were extended to include water clarifiers whose size ranged to an impeller diameter of 35 feet. Also included is a sample of the output from this program. 61

74 DIMENSION Q (50,50),P (50,50),A (50,50),R (50,50) WRITE (6,1 ) FORMAT (1 HI,26HWATER CLARIFIER FLOWRATE, / / / / ) DO 200 1=2,51 r ( i, i )=o DO 100 N=2,9 READ (5,1 0 ) R (I,N ),A (I,N ),P (I,N ) X =(A (I,N )*( /180.0)), x. Q(I,N)=22.87*(R(I^T)"W*2-R(I,N-1 )**2)-»(P(l»*)*C0S(X) )*» 5 WRITE (6,2 ) I,N,Q (I,N ) ' A FORMAT (1,13H FLOWRATE Q (,I2,1 H,,I2,2 H )=,1 0 X,1 F t0.3,/) FORMAT (F 10.1,F 10 1,F 10.3) CONTINUE Q (l,t)aao DO 50 N=»9 Q (l,n)=q (l,n-1 )4Q(I,N) CONTINUE WRITE (6,3) I,Q (I,N ), ^ 1Mn o /o FORMAT (1H0,19H TOTAL FLOWRATE Q(,I2,8H,TOTAL)-,1F10,3,/0 CONTINUE STOP END

75 63 FLOW RATE Q(2,2)= FLOW RATE Q(2,3)= FLOW RATE Q( 294)= FLOW RATE Q(2,5)= FLOW RATE Q(2,6)= FLOW RATE Q(2,?)= FLOW RATE Q( 2 s8)= FLOW RATE Q(2,9)= TOTAL FLOW RATE Q(2»T0TAL)= FLOW RATE Q ( 3 S2)= FLOW RATE Q(3,3)= FLOW RATE Q(3»4)= FLOW RATE Q(3*5) FLOW RATE Q(356)= FLOW RATE Q(3,7)= FLOW RATE Q(3s8)= FLOW RATE Q(3,9)» TOTAL FLOW RATE Q( 3sTOTAL>

76 64 C TABULATION OF RESULTS D=1.0 B=1.0 BB=1.0 SM=100«0 1=1 1 S=BB 2 L=1 WRITE (6,3) D,I 3 FORMAT (1 H t,///,1 5 X,1 8 H IM P E L L E R DIAMETER, F 5» 1,1 X,2 H F IT. 26X,5H P A G E,I3, / /, 5HSPEED,9X,9 H T IP SPEED, 11X,5HP0WER, 21tX,9H FL0W R A T E,/,1 5 X?5H (R F M ),11X,5H (F P S ),14X,4 H (H P ), 314X,5K(GFM), / ) 1= IF (LoGToJJO) GO TO 6 HP=( (S**3 0)*(D**5.0)*(4*65))/((10 o0)**(8.0)) Q=( (2.915) MS*(D**3»0)) TS=4.1415*D*S/60 0 WRITE (6,5) S,TS,HP Q 5 FORMAT (fh,f19.2,f1663,f19.4,f18 2) L=Lf1 S=S+BB GO TO 4 6 IF(S ol E 0SM) GO TO 2 D=D4-.5 IF (D«LEo4.5) GO TO 1 SM=50o0 B B = 05 IF (DoLE.5.0) GO TO 1 IF (D.LE.9.5) GO TO 7 SM=25.0 BB=*25 IF (D«LE«10»0) GO TO 1 IF (D»LEo14»5) GO TO 7 SM=15.0 BB=.15 IF (D.LE T5.0) GO TO 1 IF (D.LEo19.5) GO TO 7 SM=10 0 BB=»1 IF (D.LE.20.0) GO TO 1 IF (D.LEo24.5) GO TO 7 SM=5.0 IF (DeLE*25.0) GO TO 1 IF (DaLEo29.5) GO TO 7 IF (DoLEo30o0) GO TO 1 IF (D.LE.35.0) GO TO 7 GO TO 8 7 IF (S.LT.SM) GO TO 2 GO TO 1 8 CONTINUE END

77 IMPELLER DIAMETER 16.5 FT PAGE 64 SPEED TIP SPEED POWER FLOW RAT (RPM ) (FPS ) (HP ) {GPM ) e CO C CO G CO

78 66 IMPELLER DIAMETER 17.0 FT PAGE 65 SPEED TIP SPEED POWER FLOW RATE (RPM ) (FPS ) (HP ) (GPM ) o.eo G CO eo C

79 67 APPENDIX C TYPICAL DATA SHEETS

80 TYPICAL DATA SHEETS Included in this appendix are a few typical data sheets which show a representative cross section of the data for seventy-seven revolutions per minute or 4 FPS tip speed. These data sheets are included to show how the data was taken and to help the reader reach a general understanding of the type of data which was available for analyzing the problem and formulating conclusions. 68

81 69 BLADES: SHAFT SPEED: TIP SPEED: TORQUE: 24 - Straight Radial 77 (RPM) 4 (FPS) 17.5 (IN-LBS) SYMBOLS R = radius from c e n te r of d raft tube (in.) 9 = angle of velocity from vertical (degrees) Pd= dynamic pressure (in. of water) Q = flow in annular ring from (R) to (R-l/2) (GPM) (from computer solution) DATA R 9 Pd Q' POWER: FLOW total: (HP) (GPM)

82 70 BLADES: SHAFT SPEED: TIP SPEED: TORQUE: 12 - Straight Radial 77 (RPM) 4 (FPS) 20 (IN-LBS) SYMBOLS R = radius from center of draft tube (in.) 9 = angle of velocity from vertical (degrees) P^ = dynamic pressure (in. of water) Q* = flow in annular ring from (R) to (R-l/2) (GPM) (from computer solution) DATA R 9 P d Q ' POWER: FLOW total: (HP) (GPM)

83 71 BLADES: SHAFT SPEED: TIP SPEED: TORQUE: 6 - Straight Radial 77 (RPM) 4 (FPS) 16 (IN-LBS) SYMBOLS R = radius from center of draft tube (in.) 0 = angle of velocity from vertical (degrees) P^ = dynamic pressure (in. of water) Q1= flow in annular ring from (R) to (R-l/2) (GPM) (from computer solution) DATA R 0 Pd Q' POWER FLOW total: (HP) (GPM)

84 72 BLADES: SHAFT SPEED: TIP SPEED: TORQUE: 4 - Straight Radial 77 (RPM) 4 (FPS) (IN-LBS) SYMBOLS R = radius from center of draft tube (in.) 9 = angle of velocity from vertical (degrees) P^= dynamic pressure (in. of water) Q' = flow in annular ring from (R) to (R-l/2) (GPM) DATA R 9 Pd Q' o o LO t POWER: FLOW total:. 015 (HP) (GPM)

85 73 BLADES: SHAFT SPEED: TIP SPEED: TORQUE: 24 - Straight, Turned (RPM) 4 (FPS) 6 (IN-LBS) SYMBOLS R = radius from center of draft tube (in.) 9 = angle of velocity from vertical (degrees) Pd= dynamic pressure (in. of water) Q' = flow in annular ring from (R) to (R-l/2) (GPM) (from computer solution) DATA R 9...-Pd Q.! POWER: FLOW total: (HP) (GPM)

86 74 BLADES: SHAFT SPEED: TIP SPEED: TORQUE: 24 - Bent 77 (RPM) 4. 0 (FPS) 11 (IN-LBS) SYMBOLS R = radius from c en ter of d raft tube (in.) 9 - angle of velocity from v e rtic a l (degrees) Pd= dynamic pressure (in. of water) Q' =.flow in annular ring from (R) to (R-l/2) (GPM) (from computer solution) DATA R 9 p<a Q' POWER FLOW total:.0161 (HP) (GPM)

87 75 BLADES: SHAFT SPEED: TIP SPEED: TORQUE: 24 - Curved 77 (RPM) 4 (FPS) 5 (IN-LBS) SYMBOLS R = radius from center of draft tube (in.) 9 = angle of velocity from vertical (degrees) P(j= dynamic pressure (in. of water) Q' = flow in annular ring from (R) to (R-l/2) (GPM) (from computer solution) DATA R 9 p<a Q' , POWER: FLOW total: (HP) (GPM)

88 76 BLADES: SHAFT SPEED: TIP SPEED: TORQUE: 12 - Curved 77 (RPM) 4 (FPS) 5.33 (IN-LBS) SYMBOLS R = radius from center of draft tube (in.) 9 = angle of velocity from vertical (degrees) P^= dynamic pressure (in. of water) Q* = flow in annular ring from (R) to (R-l/2) (GPM) (from computer solution) DATA R 9 Pd Q' POWER FLOW total:.0061 (HP) (GPM)

89 77 BLADES: SHAFT SPEED: TIP SPEED: TORQUE: 6 - Curved 77 (RPM) 4 (FPS) 5 (IN-LBS) SYMBOLS R = radius from c e n te r of d raft tube (in.) 9 = angle of velocity from v e rtic a l (degrees) P^= dynamic pressure (in. of water) Q' = flow in annular ring from (R) to (R-l/2) (GPM) (from computer solution) DATA R 9 Pd Q' CO o POWER: FLOW total:.0061 (HP) (GPM)

90 BLADES: SHAFT SPEED: TIP SPEED: TORQUE: 4 - Curved 77 (RPM) 4 (FPS) 4. 8 (IN-LBS) SYMBOLS R = radius from center of draft tube (in.) 9 = angle, of velocity from vertical (degrees) P^= dynamic pressure (in. of water) Q' = flow in annular ring from (R) to (R-l/2) (GPM) (from computer solution) DATA R 0 Pd Q! POWER: FLOW total: (HP) (GPM)

91 79 APPENDIX D INSTRUMENTATION DATA

92 80 Torque Meter: Baldwin SR-4 Torque pickup Model D1196 Range in #. Strip Chart Recorder: Sanborn, 4 channel, Model B with Model 1300 DC Coupling Preamplifier Oscilliscope: Tektronix Type 561A, equipped with: Type 3B3 plug in time base Type 3C66 plug in carrier amplifier Pressure Transducer: Northam Model DP-7 Range 0-15 psi.

93 LITERATURE CITED

94 82 1. Csanady, G. T. Theory of Turbomachines. New York: McGraw Hill Book Co., Streeter, Victor L. Fluid Mechanics. New York: McGraw Hill Book Co., Shapiro, Ascher H. The Dynamics and Thermodynamics of Compressible Fluid Flow. New York: The Ronald Press Co., Schlichting, Herman. Boundary Layer Theory. New York: McGraw Hill Book C o., Simonsen, John M. Unpublished Text on Fluid Dynamics. Utah: 1967.

95 83 GENERAL BIBLIOGRAPHY

96 84 1. Albertson, Maurice L., Barton, James R, and Simons, Daryl B. Fluid Mechanics for Engineers. New Jersey: Prentice-Hall, Inc., Binder, R C. Fluid Mechanics. 3rd ed. New York: Prentice- Hall, Inc., Brenkert, Karl Jr. Elementary Theoretical Fluid Mechanics. New York: John Wiley and Sons, Inc., Daugherty, R L., and Ingersoll, A. E. Fluid Mechanics. 5th ed. New York: McGraw Hill Book Co., Lazarkiewicz, Stephen, and Troskolanski, Adam T. Impeller Pumps. New York: Pergamon P ress, Jones, Jacob O. Introduction to Hydraulics and Fluid Mechanics. New York: Harper and Brothers, Kline, Stephen J. Similitude and Approximation Theory. New York: McGraw Hill Book Co., Olson, Reuben M. Essentials of Engineering Fluid Mechanics. Pennsylvania: International Textbook Co., Pao, Richard H. F. Fluid Mechanics. New York: John Wiley and Sons, Inc., Shepherd, Dennis G. Elements ol Fluid Mechanics. New York: Harcourt, Brace and World, Inc., Shepherd, Dennis G. Principles of Turbomachinery. New York: The Macmillan Co., Stephanoff, A. J. Centrifugal and Axial Flow Pumps. New York: John Wiley and Sons, Inc., 1956.

97 MODEL STUDY AND ANALYSIS OF THE FLOW ELEMENTS OF A RECIRCULATION MIXING SYSTEM An Abstract of a Thesis Presented to the Department of Mechanical Engineering Brigham Young University In Partial Fulfillment of the Requirements for the Degree Master of Science by Albert Warren Berg August, 1967

98 ABSTRACT The purpose of this thesis was to confirm the water clarifier design criteria being used in the field, and to improve the impeller design so that more efficient water treatment units could be built at a lower cost. The principles of dimensional analysis were employed and the resulting dimensionless groups were analyzed for significance. From the resulting significant groups it was seen that a model study would yield the desired information. Thus a small scale model and appropriate instrumentation vere designed and built. The model study yielded results which were verified by field data and showed that the design criteria being used are slightly conservative; also it was shown that by using backward curved blades in the impeller the operational characteristics of the water clarifier were greatly improved. APPRCWED: