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JI *— This is to certify that the thesis entitled FLOW DISTRIBUTION OF NON-NEWTONIAN FLUIDS FROM A MANIFOLD SYSTEM presented by Walter F. Salas Valerio has been accepted towards fulfillment of the requirements for M. S . Agricultural degree in Engineering M 75% Major professor Date /0,/362/8? 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution .;-\v PLACE IN RETURN BOX to remove We checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE fif MSU Is An Aflimative Action/Equal Opportunity Inuitutlon chS-o.‘ FLOW DISTRIBUTION OF NON-NEUTONIAN FLUIDS FROM A MANIFOLD SYSTEM BY Walter F. Salas Valerio Submitted to Michigan State University In partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Agricultural Engineering Department of Agricultural Engineering 1988 ABSTRACT FLOW DISTRIBUTION OF NON-NEUTONIAN FEUIDS FROM A.HANIFOLD SYSTEM by Walter Francisco Salas Valerio The flow distribution in a manifold system using non-Newtonian fluids was investigated for different flow rates and different orifice and manifold diameters. Gelatinized corn starch solutions (5, 7.5 and 10%, wet basis) were used as test fluids. Power-law behavior was found for all the solutions. A theoretical model was developed based on the mass balance equation at each orifice and the mechanical energy balance equation between any two orifices in the manifold. Orifice discharge coefficients, determined to be in the range of O - 0.5, were found to be a function of the rheological properties of the fluid (consistency coefficient and flow behavior index) and the orifice diameter. A mathematical expression that correlated the orifice discharge coefficient with the generalized Reynolds number was obtained. Using the theoretical model developed and experimental data, a correction factor for the orifice discharge coefficient was determined to account for flow past the orifice in an actual manifold system. Since the pressure calculated by means of the mechanical energy balance equation was higher than the experimental pressure, it became necessary to include a parameter that accounts for the energy loss due to turbulence at the orifice. Calculated values indicate that the energy loss coefficients due to turbulence increased significantly for decreasing values of the generalized Reynolds number. The theoretical model developed for the manifold distribution was used in conjunction with the corrected orifice discharge coefficient and an energy loss coefficient due to turbulence to simulate fluid flow from a manifold under various conditions. The simulation model for the less viscous fluid (5% starch solution) was inaccurate for several reasons: experimental error in the determination of the flow rate and pressure, the effect of the consistency coefficient in the corrected orifice discharge coefficient, and the system complexity due to the large number of interactive variables present. For highly viscous fluids, the simulation was more accurate; therefore, the model was used to develop general design recommendations to obtain uniform manifold flow. ACKNOWLEDGMENTS The author sincerely appreciates the counsel, encouragement and technical support of his major proffessor Dr. James F. Steffe and wishs to thank to Dr. Robert Ofoli and Dr. L. Segerlind, members of the committee, for their helpful suggestions. Sincerely thanks also to Peruvian Goverment and Universidad Nacional Nacional Agraria -La Molina (Lima-Peru) for its financial support. Thanks to all of those friends and fellows who offered their time and expertise. Finally, thanks for the moral support and encouragement freely given by my wife Elba and my son Julio. iv TABLE OF CONTENTS Page LIST OF TABLES ......................................... viii LIST OF FIGURES ........................................ ix NOMENCLATURE ........................................... xiii 1. INTRODUCTION ........................................ 1 2. LITERATURE REVIEW ................................... 3 2.1 Introduction .................................... 3 2.2 Fluid Models .................................... 3 2.3 The Manifold Problem ............................ 5 2.4 Network System .................................. 8 3. THEORETICAL DEVELOPMENT ............................. 11 3.1 Mechanical Energy Balance ....................... 11 3.2 Energy Losses in the System ..................... 13 3.2.1 Energy Losses Due to Friction in Straight Pipes .................................... 13 3.2.1.1 Friction Factor (f) .............. 14 3.2.1.2 Laminar Transition Criteria ...... 15 3.2.2 Energy Losses Due to Turbulence Induced at the Orifice (bk) ......................... 16 3.3 Kinetic Energy Coefficient ...................... 17 3.4 Application of the Mechanical Energy Balance Equation to the Flow in a Manifold .............. 18 3.5 Manifold and Orifice Equations ............. . ..... 21 3.5.1 Bulk Fluid Velocity in the Pipe ........... 21 3.5.2 Velocity at the Entrance (U) .............. 23 3.5.3 Orifice Flow Rate ......................... 23 3.6 Orifice Discharge coefficient and the Orifice Discharge Coefficient Correction Factor ......... 25 4. MATERIALS AND METHODS ............................... 27 4.1 Experimental Materials .......................... 27 4.2 Determination of the Orifice Discharge Coefficient ..................................... 28 4.2.1 Experimental Orifice System and Data Collection ................................ 28 4.2.2 Calculation of the Orifice Discharge Coefficient ............................... 31 4.3 Manifold Distribution System .............. . ...... 31 4.3.1 Experimental Manifold and Data Collection 31 4.3.2 Calculation of the Energy Loss Coefficient Due to Turbulence at the Orifice ........ 34 4.3.3 Calculation of Orifice Discharge Coefficient Correction Factor ............ 36 4.3.4 Comparison of Simulated and Actual Manifold Performance ............................. 39 5. RESULTS AND DISCUSSION .............................. 42 5.1 Fluid Properties ......................... . ...... 42 5.2 Orifice Discharge Coefficient ................... 42 5.3 Manifold Flow Distribution ..................... 50 5.4 Energy Loss Coefficient Due to Turbulence at the Orifice 5.5 Orifice Discharge Coefficient Correction Factor . 63 5.6 Comparison of Simulated and Actual Manifold Flow Distribution .................................... 68 5.7 Simulation of the Manifold Flow Distribution .... 76 5.8 Strategies for Achieving Uniform Flow ........... 82 . SUMMARY AND CONCLUSIONS .............................. 84 . SUGGESTIONS FOR FUTURE RESEARCH ..................... 86 . REFERENCES .......................................... 87 APPENDICES .......................................... 90 Appendix A .......................................... 91 Appendix B .......................................... 96 Appendix C .......................................... 108 Appendix D .......................................... 118 vii List of Tables Table Page 1. Properties of gelatinized starch solutions ..... 43 2. Results of the non-linear regression analysis for the orifice discharge coefficient data .......... 49 3. Results of the linear and non-linear regression analysis for the pressure versus mass flow rate viii 10. ll. 12. LIST OF FIGURES Title Page Typical manifold system ........................ 6 Tube with slit attached, functioning as a distributor ................................... 9 Sketch of manifold distribution system with illustration of manifold and orifice flow parameters .................................... 12 Mechanical energy balance between two orifices in the manifold ................................ 19 Mass balance at a orifice in the manifold ..... 22 Definition sketches of the manifold dead end to illustrate Equations (32) and (34) .......... 24 Experimental equipment used to measure the flow rate in the orifice and obtain the orifice discharge coefficient .......................... 29 Experimental equipment used to obtain the manifold flow distribution .................... 33 Procedure to calculate the energy loss coefficient due to turbulence at the orifice .............. 35 Procedure to calculate the orifice discharge coefficient correction factor ................ 37 Procedure to simulate the manifold flow distribution .................................. 4O Orifice discharge coefficient as a function ix 13. 14. 15. 16. 17. 18 19. 20. 21. 22. 23. 24. of the fluid velocity in the orifice for a 5% corn starch solution ....................... Orifice discharge coefficient as a function of the fluid velocity in the orifice for a 7.5% corn starch solution ...................... Orifice discharge coefficient as a function of the fluid velocity in the orifice for a 10% corn starch solution ....................... Orifice discharge coefficient as a function of the generalized Reynolds number ............. Manifold flow distribution for a 5% corn starch solution ...................................... Manifold flow distribution for a 7.5% corn starch solution ...................................... Manifold flow distribution for a 10% corn starch solution ....................................... Pressure as a function of the mass flow rate in the orifice for a 5% starch solution .......... Pressure as a function of the mass flow rate in the orifice for a 7.5% starch solution ........ Pressure as a function of the mass flow rate in the orifice for a 10 % starch solution ........ Energy loss coefficient due to turbulence at the orifice as a function of the generalized Reynolds number ............................... Corrected orifice discharge coefficient as a function of the fluid velocity in the orifice for a S % corn starch solution ................. Corrected orifice discharge coefficient as a 45 46 47 51 52 53 54 56 57 58 62 64 25. 26 27. 28. 29. 30. 31. 32. 33. 34. function of the fluid velocity in the orifice for a 7.5% corn starch solution ................ Corrected orifice discharge coefficient as a function of the fluid velocity in the orifice for a 10% corn starch solution ................. Corrected orifice discharge coefficient as a function of the generalized Reynolds number .... Manifold flow distribution: Simulated versus experimental results for a 5% corn starch solution ...................................... Manifold flow distribution: Simulated versus experimental results for a 5% corn starch solution ...................................... Manifold flow distribution: Simulated versus experimental results for a 7.5% corn starch solution ....................................... Manifold flow distribution: Simulated versus experimental results for a 7.5% corn starch solution ....................................... Manifold flow distribution: Simulated versus experimental results for a 10% corn starch solution ....................................... Manifold flow distribution: Simulated versus experimental results for a 10% corn starch solution ....................................... Simualted manifold flow distribution for different flow rates at the entrance .......... Simualted manifold flow distribution for different orifice diameters in the manifold . xi 65 66 67 69 70 71 72 73 74 77 78 35. 36. Simulated manifold flow distribution for different fluid consistency coefficients ...... Simulated manifold flow distribution for different manifold diameters .................. xii 79 80 W crossectional area of the pipe, m2 crossectional area of the orifice, m2 slit width, m orifice discharge coefficient, dimensionless corrected orifice discharge coefficient, dimensionless pipe diameter, m orifice diameter, m energy losses due to friction, J/kg length of the pipe portion in the manifold, m Fanning friction factor, dimensionless gravitational acceleration, 9.81 m/s2 Hedstrom number, dimensionless energy losses due to friction in the pipe, J/kg energy losses due to fitting, J/kg total number of orifices fluid consistency coefficient, Pa sn average kinetic energy per unit of mass, J/kg energy loss coefficient due to the turbulence at the orifice, dimensionless orifice friction loss coefficient due to contraction and expansion in the orifice, dimensionless. total pipe length, m xiii .0 O .00 n N w Re Re Re number of portions of pipe fluid flow behavior index, dimensionless pressure at any point, Pa pressure at point “a” defined in Figure 6, Pa atmospheric pressure, Pa mass flow rate in the pipe, kg/s mass flow rate at the entrance, kg/s total mass flow rate, kg/s flow rate in z-direction, defined in Figure 2, kg/s mass flow rate at the orifice, kg/s tube radius, m generalized Reynolds number, dimensionless critical generalized Reynolds number, dimensionless generalized Reynolds number for fluid flow in the orifice, dimensionless radius, m bulk or average velocity at any point in the pipe, m/s bulk or average velocity at first portion of pipe, m/s local linear velocity in the x-direction at r, m/s fluid velocity in the orifice, m/s fluid velocity at pipe entrance, m/s direction xiv MM kinetic energy coefficient, dimensionless constant defined by Equation (9), dimensionless constant defined in Equation (17), dimensionless correction factor for the orifice discharge coefficient, dimensionless viscosity coefficient, Pa 5 plug radius, dimensionless critical plug radius, dimensionless fluid density, kg/m3 shear stress, Pa yield stress, Pa shear stress at the wall, Pa constant equal to 3.1415... shear rate, 1/s §ubscripts assumed calculated orifice number 1. IITIODUCTIOI Pumping systems are used in many food processing operations and, in special cases, they pump a fluid into a manifold (perforated pipe with a closed end). Currently, food industries which works with non-Newtonian fluids flowing in a manifold system cannot predict the flow rate in the outlets (orifices) with accuracy, and this causes problems in process and product quality. Little work.has been done in the area of non-Newtonian manifold flow. Previous studies on manifold flow were performed primarily for application to irrigation (drip irrigation) which uses water, a Newtonian fluid. The study of non-Newtonian manifold flow will make an important contribution to the technological advancement of the food industry. The overall focus of this study is to develop a theoretical model that can be usedfor the design of manifold systems for non- Newtonian fluids. To date, analytical expressions to determine flow rate distribution in a manifold system using non-Newtonian fluids are not available. Therefore, the objectives of this study are as follows: 1. Develop a theoretical model to calculate the flow distribution in a horizontal, circular cross-sectional manifold system for non—Newtonian, non-time dependent, non-elastic fluids. 2. Determine the validity of the theoretical model by using data from an experimental manifold system. 3. Develop design strategies to achieve uniform flow distribution through a horizontal, circular cross-sectional 2 manifold system for non—Newtonian, non-time dependent, non- elastic fluids. 2. LITERATURE REVIEW 2.1 Introduction In contrast to the concentration of effort on the problem of Newtonian fluid flow (e.g., water in a simple pipe or in network), the problem of non-Newtonian fluids flowing in a manifold system has been almost ignored. Few papers on this subject have been presented and those published deal only with part of the problem. Manifold flow analysis techniques for water systems have been available for many years. The implementation of these techniques by hydraulic engineers have brought improved speed and accuracy to the analysis of irrigation systems (Ramirez-Guzman and Manges, 1971). This chapter will present a short discussion of fluid models and references on manifold systems found in the literature consulted. 2.2 Fluid.Models. Newtonian Model. For a Newtonian fluid, the viscosity (p) is constant. It is convenient to represent the behavior of flowing materials by means of flow curves (shear stress against shear rate), thus the flow curve of a Newtonian fluid is a straight line through the origin, the slope being equal to the viscosity (Whorlow, 1980). The Newtonian fluid model is represented as a - p i (1) Non-Newtonian Model. Non-Newtonian fluids are those for which the flow curve (shear stress versus shear rate) is not linear through the origin at a given temperature and pressure (Bird, et a1. 1987). A great many empirical or semi-empirical equations have been proposed to represent the flow behavior of materials. The choice of an equation for a particular application is to some extent a matter of preference (Whorlow, 1980). Non-Newtonian fluids are commonly divided into three broad groups: 1. Time-independent fluids are those for which the shear rate at a given point is solely dependent upon the instantaneous shear stress at that point. These materials are sometimes referred to as "non-Newtonian viscous fluids" or alternatively as "purely viscous fluids". 2. Time-dependent fluids are those for which the shear rate is a function of both the magnitude and the duration of the shear. 3. Viscoelastic fluids are those which show partial elastic recovery upon the removal of a deforming shear stress. Such materials have properties of both fluid materials and elastic solids (Skelland, 1967). Some of the most common rheological models which have been used in axial laminar flow are the power law, Bingham plastic, and Herschel-Bulkley models. The power-law model, usually attributed to Ostwald but proposed independently by de Waele and others, is used to represent the behavior of many polymer solutions. The equation for the model can be written as 0-K 3)“ (2) Many non-Newtonian fluids are not well approximated by either the Bingham plastic or the power-law model. They are, however, well represented by a combination model known as the Herschel-Bulkley model (H-B) written as (Osorio and Steffe, 1984) a - 00 + K 5“ (3) 2.3. The Manifold problem. A manifold system is a special kind of fluid transport system that is composed of a pump and a manifold as a main pipe (Figure 1). The distribution of flow in a horizontal manifold is determined by the inertia and friction forces (Keller, 1949). The inertia forces correspond to the change in velocity (kinetic energy). The velocity decreases in the direction of the flow as the fluid passes through each outlet (emitter or orifice). The fluid in the manifold decelerates so it increases in pressure as predicted by the mechanical energy balance. On the other hand, there is a pressure drop along the line of the manifold; gaining pressure for down slopes and losing pressure for up slopes. Thus the relative magnitudes at these forces will determine whether the static pressure at the dead end of the manifold increases or decreases. Keller (1949) was one of the first to publish a paper on the rmanifold problem. He took, as an example, a familiar pipe burner for gaseous fuels. Keller stated that there are only two important factors .Emummm paomwcme Hmowexh .H mam QAOMHZ1*“[<1-eoc>2 + zsocu-eoc) ($23) + £3: (Ea—'5 ] (16) If Re is lower than Rec, then the flow is laminar and it can be used Equation (8) is used to calculate the Fanning friction factor in the manifold. Otherwise the relationship for turbulent flow must be used (Garcia and Steffe, 1986). 3.2.2. Energy Losses Due to Turbulence Induced at the Orifice (bk) In each orifice there is a loss of energy due to the turbulence induced in the manifold. The energy loss is (“1 ' “2)2 f 2 ‘ (17) k'k where u1 and u2 are the fluid velocities in the manifold before and after the orifice and kf is a energy loss coefficient due to turbulence. This term was not found in the reviewed literature and is introduced for the first time in this work. It represents a correction factor needed to fit the experimental values with the simulated values by means of a computational program. It also has a physical meaning because when a fluid is flowing in a manifold, then suddenly has two potential flow directions, a turbulence is induced around the orifice which causes a friction loss which must be taken into consideration. The hk could be interpreted as a loss of energy in a fitting like a tee. When the fluid in question is very viscous and has non-Newtonian behavior this value may be significant. The coefficient kf may be 17 function of the fluid properties (flow behavior index and consistency coefficient), orifice diameter and the velocity of the fluid in the orifice. 3.3 Kinetic Energy Coefficient (a). Solving design problems of non-Newtonian fluids flowing in circular tubes requires a knowledge of the energy requirements related to the changes in kinetic energy. An expression for kinetic energy is generally presented as a separate term in the mechanical energy balance equation. The average kinetic energy per unit mass (KB) of any fluid stream moving in a round pipe is (Skelland, 1967) KE - g I r u'3dr (18) R u 7 The KE in laminar flow can be expressed in terms of a kinetic energy correction factor as KE - DIG (19) where a is the kinetic energy correction factor. For non-Newtonian fluids (Herschel- Bulkley model) in laminar flow, Osorio and Steffe (1984) found a as 18 2 a - [(2 (1+3n+2n +2n2go+2ngo +2n2£°2)3 (2+3n) (3+5n) 3+4n)]/ [((l+2n)2(l+3n)2) (18+n(105-66£o) + n2(243+30650+35502) + 3 2 a 2 n (279+52250+3soso ) + n (159+39os0+477eo ) +(n5 (36+10350+216e§ )] (20) 3.4 Application of the Mechanical Energy Balance Equation to Flow in a.Manifold. Recalling Equation (5) and considering the assumptions made in Figure 3 (no difference in height between points 1 and 2, and equal bulk velocities between two orifices), Equation (6) may be written as - h p p -;2 -;l - h (21) f k In this equation, hf and hk are functions of u1 and u2, where u1 and u2 are the bulk average velocities of the fluid in the manifold. Equation (21) permits calculation of the pressure at orifice number 2 in the manifold, so it is possible to determine the mass flow rate in each orifice by this relationship. Equation (21) may be generalized as (Figure 4) Pi h h p1+1 ' ”I p ' f1+1' k1+1] (22) where: l9 .tHomHCmE onu CH moowmfiuo ozu cooaumn mocmamn mwuoco Hmowcmcomz .q wwm H+H 7L MMMZDZ monHmO 20 2 f e u hf _ igl 1+1 (23) 1+1 and 2 (u -u ) 1+1 1+1 A special case is when the mechanical energy balance is applied between the entrance and the first orifice (number 1) in this case the value of‘i equals zero, which represents the pressure and the velocity at the manifold entrance. Another special case is when the mechanical energy balance is applied in the last orifice where the fluid velocity is zero at the dead end of the system. Substituting Equations (23) and (24) into Equation (22) gives 2 iii; _ _p.i_ _ 2 f1+1e u1+1 - k p p D (Sui+l’“1+2)'2 f 2 (25) i+1 Equation (25) gives the pressure at any orifice in the manifold as a function of fluid velocity in the manifold preceding the orifice (u), manifold diameter (D), Fanning friction factor, energy loss coefficient due to the turbulence induced by the orifice and the static pressure at the preceding orifice. 21 3.5 Manifold and Orifice Equations. 3.5.1 Fluid Bqu'Velocity in the Manifold. The velocity is maximum at the manifold entrance and zero at the dead end. Velocity in the manifold diminishes as the flow passes each orifice. If a uniform orifice distribution exists, then the velocity distribution will be a straight line; however, the form of this relationship for a non-Newtonian fluid is unknown. In this work the velocity distribution will be predicted by applying mass and mechanical energy balances in each orifice. The evaluation will be done using Equation (5) and the following relationship (Figure 5): Q1+1" Q1 ' ‘11 (26) where: p vld2 ql - 4 (27) or 519 “1+1" “1 ' A V1 (28) If the value of i is equal to I (total number of orifices), then Q1+1 is zero and the value of QI should be equal to qI this case occurs in the last orifice. 22 ORIFICE NUMBER ——\3 C? r Q Q 1 1+1 _.__., --——-’ u u 1 1+1 -ai le-d “'llll‘ q l i V i Fig 5. Mass Balance at an orifice in the manifold. I<-——==-——aI 23 3.5.2 velocity at the Entrance (U). The average velocity at the entrance of the manifold is given by U - A (29) 3.5.3 Orifice Flow Rate. Consider a manifold as shown in the Figure 6-A under the conditions existing (single orifice pipe with dead end). Applying the mechanical energy balance between the point "a" and the point "b" gives the'following relationship: p u2 pb vi k vfi _8 + J + Z 8 - — + — + Zbg + O (30) p (18 a p ab 2 Rearranging terms and assuming that the distance 2a is equal to Zb, gives 2 2 (p /p + ua/a ) 1v - -—JL——————-——9— (31) b (kc/2 g + l/ab) or qb- p Ao C -;— + (32) a where: k0 1 ]-1/2 C - 2 g + -;; (33) 24 U ‘-—- DEAD END (Al. fl ‘— DEAD END - ,mfiv I 1 (B) Fig 6. Definition sketches of the manifold dead end to illustrate Equations (32) and (34). 25 Generalizing this relationship for any orifice in the manifold, we have the following expression (Figure 6-B) {.1 Pi qi- p A0 C1 p + (34) 9 H 2 ]1/2 This expression is for the discharge of a fluid through an orifice, where p1 is the static pressure at orifice i. Notice that ui represents the fluid velocity in manifold section before the orifice. The coefficient of discharge represent the losses resulting from the friction in the orifice. The C values for water vary from 0.96 to 0.98 (Eskinazi, 1962). The discharge coefficient may be function of the fluid properties, velocity, orifice diameter, so on, then it is necessary to experimentally determine exact values. 3.6 Orifice Discharge Coefficient and the Orifice Discharge Coefficient Correction Factor. To obtain the orifice discharge coefficient presented in Equation (34), it is necessary to collect experimental pressure data at different fluid velocities in the orifice. For this purpose, it is necessary to set up a manifold system and have a well defined fluid. The manifold and the orifice cross-sectional areas, length of the manifold, and fluid rheological properties are known. Using the pressure data obtained by experimentation the orifice discharge coefficient for a closed end system is calculated by means of Equation (34). Since this experiment is not the same as the actual process in the manifold (except for the last orifice) it is necessary to introduce 26 a correction factor (5) for the orifice discharge coefficient which accounts for energy losses due to the fluid flowing past a discharging orifice. This correction factor will modify the calculated flow rate using the orifice discharge coefficient obtained from Equation (34) 21 :1 qi - p AOCi p + “1 e (35) or P1 EL qi - p AOCi [ p + a1] (36) 4. MATERIALS AND.METHODS 4.1 Experimental Materials. A modified waxy maize food starch (National 150: National Starch and Chemical Co.,Bridgewater, New Jersey) containing erythorbic acid was used in the experiments. The starch is a white powder containing 11% moisture (wet basis). Tap water (pH - 7.5) was used to prepare aqueous solutions of 5, 7.5 and 10% (wet basis) starch. A Haake RV-12 concentric cylinder viscometer was used to measure the rheological properties of the starch solution. The inner cylinder, the bob (MVI), was rotating, while the outer cylinder, the cup, was stationary. The height of the bob was 0.020 m and the cup radius was 0.021 m. The torque was measured and transformed into a proportional electrical signal by the measuring drive unit (M150). A Haake PG-12 was connected to the measuring drive unit to manually control the bob speed. Data were acquired using an HP-3497A data acquisition system, which was connected to a HP-85 computer via a 82937a HP-IB interface. All samples were obtained directly from the orifice or manifold system. Once the product was in the cup, temperature control was established with a temperature vessel (Haake FC-3) built around the cup. Tests were performed at 22 i 1 00 over a speed range of 10-150 1 rpm, resulting in a shear rate range of approximately 10 - 250 5. depending on the product and temperature. Twenty data points were taken in this range for each test. A computer program on the HP-85 calculated shear stress and shear rate values for each test; the Krieger method 27 28 (Krieger, 1968) was used to calculate the shear rate. A power-law model was then fitted to obtain the rheogram which gives the consistency coefficient, flow behavior index, correlation index of the regression analysis, and the data standard deviation. The total solid contents (used to verify starch concentration) were determined with a drying oven at 103 °C for 24 hours. Fluid density was measured using a graduate cylinder and an analytical balance. In this study, three different fluids with different rheological characteristics were examined. These fluids were prepared by first weighing the correct amount of water into the mixing tank, staring a mixer, and slowly adding starch until the required amount was added. The mixture was heated (68°C) until starch gelatinization was obtained, and the mixture had the appropriate thickness. After this period, the mixer was shut off and the solution was allowed to cool down to room temperature overnight. 4.2 Determination of the Orifice Discharge Coefficient. 4.2.1 Experimental Orifice System.and.Data Collection. Flow tests were carried out in an experimental manifold system (Figure 7). The experimental manifold system included a Waukesha Model 10 rotary drive with variable speed drive. The displacement of this pump was 0.0133 gal/rev with a pressure range of 0 - 200 psia. Two tanks made out of stainless steel were used. The bigger tank (diameter - 0.8 m and height - 0.7 m) contained the product was equipped with a mixer. The small tank (diameter - 0.7 m and height - 0.7 m) was used to hold water for cleaning purposes. A bypass was constructed, using an 29 .OCoHonmooo owumcomwu mowmwuo use samuno can oonHuo amaze; / \ F monHmo \ I I \ 1 A mmHm o>m oSu Cw oumu 30am one ousmmoa ou pom: ucoaeasvo Hmucoefiuomxm .n mam A with 4v a ¥Z4<> 30 air-to-close valve, just after the pump to allow for a lower flow rate. When air was applied, the valve was physically more closed, allowing less fluid through the bypass. Polyvinyl chloride (PVC) and stainless steel pipes were used to build the system. Threaded PVC pipes, schedule #40 (ASTM D17 85), with an inside diameter 0.0157, 0.0409 and 0.0525 m were used as a main pipe when taking single orifice measurements. The PVC pipe was 0.5 m long and the orifice was at the end of this pipe. Also, a manometer was installed opposite the orifice, on the wall, to measure the pressure at the orifice. Three orifice diameters (0.00318, 0.00476 and 0.07838 m) and different flow rates were used in the experiments. Tests were performed at room temperature (22 i 1 °C). According to the literature consulted, the orifice diameter, fluid velocity in the orifice and the fluid consistency coefficient are very important variables in the determination of the orifice discharge coefficient. To study the effect of orifice diameter, and the effect of flow rate on the orifice discharge coefficient, the following steps were performed: 1. A power-law fluid was selected and pumped through the experimental system. 2. An orifice diameter was selected and the mass flow rate was varied using the variable speed rotary pump. 3. The mass or volumetric flow rate in the orifice was measured by collecting and weighing samples after a fixed period of time. The pressure drop was collected by reading the manometer. The readings were in meters of mercury. Samples were taken at this point to measure the fluid rheological properties and fluid density. Step three was repeated for different orifice flow rates. 31 4.2.2 Calculation of the Orifice Discharge Coefficient. The orifice discharge coefficient was calculated from Equation (34), using the data collected: mass flow rate, pressure, density, diameter of the orifice and kinetic energy coefficient. The values of the orifice discharge coefficients were plotted versus mass flow rate. A mathematical expression that fit the data was obtained. This results in the orifice discharge coefficient being a function of the mass flow rate in the orifice when the orifice diameter and the rheological properties of the fluid are kept constant. In this way, three mathematical functions for each orifice diameter and three mathematical functions for each fluid for a total of nine mathematical functions were obtained. 4.3 Mbnifold Distribution System. 4.3.1 Experimental Manifold and Data Collection. Studying the theoretical model, it can be observed that the diameter of the manifold, orifice diameter, flow rate at the entrance and fluid properties are the most important variables in the flow distribution from a manifold system. The same laboratory pump system (Figure 7) described previously was used to collect manifold data, but a longer PVC pipe with 10 orifices was used as the manifold. A schematic view of the total system is given in Figure 8. The main pipe was threaded and had fittings enabling the changing of manifold pipe diameters. In this part of the experiment, the same three fluids used in experimentation with the single orifice (Section 4.2) were tested. Since the objective was 32 to determine the response of the flow rate at each orifice in manifold, three orifice diameters (0.00318, 0.00476 and 0.00794 m) and two mass flow rates at the entrance of the manifold were tested. Tests were performed at 25 i 1 °C. The PVC manifold pipe was 1 m long, and the space between each orifice was 0.1 m. All orifices were aligned on the manifold, and the wall thickness of the manifold pipe at each orifice was 0.0017 m; hence the effect of the pipe thickness was constant through the experiments. One manometer was installed at the end of the manifold beside the last orifice, and a second manometer was installed at the entrance of the manifold to measure the pressure at the first orifice. To study the effect of the rheological characteristics of the fluid, orifice diameter and flow rate at the entrance of the manifold on the distribution of flow in the manifold, the following steps were performed: 1. Power-law fluid with known rheological characteristics was selected. 2. A constant orifice diameter for all orifices in the manifold was selected. 3. Different flow rates, at the entrance, were obtained using the bypass valve and variable speed rotary pump. 4. The mass or volumetric flow rates were measured by collecting and weighing samples of fluid at each orifice after a fixed period of time. The flow rate at the entrance was kept constant during the collection of the data enabling measurement of the flow rate at each orifice under the same conditions. 33 .coausnwuumwu 30am uHowficmE on» Camuno ou new: uCoEeHSVo HmOCwEHuooxm .w mam | mZDm MZm ... m N. ,Mmhmzoz Es: 9: .6 c0305: 0 no 320580 motocofln motto N. 9.... aka md EN in Na: o.« b —I p — IP b b — b b b — p woe: c2883”. b_ee_e> 2:: o.“ i— m.— 0.. ad _ L _ t b . _ t _ -U'---DU----E'U'I----"l--'-'-"" wd cm on mopd c0329... £83m Eco Rm II II X: U--"-'U' lli'l'lll"|“ E #05006 .0”; 3an a E unvood .cm; 3&5 o E $250.0 .0”; «£ch 0 iuegonpoo ablouosgo eouuo 46 .5328 £0.66 See .NmN e 28 8:20 e5 5 seem; 2:: 2: to c0305: 0 no 330580 motocomfi 85.5 2 9m Am\EV b_oo_m>..n_:_n_ en as ea 3.. as. 2.. 3 3 so 2 so we e.o _L_.r.__._r_ L_._t_._.o.o e a: 1.6 N} I .c . . a two . o \I .. I s woe: coBmoemo .. \. Ind e \ x 4. \\\ \ o \s \ led O \sx \ a 11s\\. \ I .......................................................... or-.. \ \ . lllllllllllll Polllllllauollllsllllllll loo . I a 1.6.0 the mud n c lwd em on. n. u x - . E 3.26.0 ea 855 a I . cozeom cocoa...” Eoo Rm h E 2.86 65 game o mo E Snead duo osto o u 9.— 1uegogneoo efuoqosgo 83113.10 47 .n QN 9N «N Nd 0d mm? 0.. v; 60:28 5.6% 500 N OF 0 toe. motto of E b_oo_m> 2:: m5 .6 cozoce o no EmBEooo mmuocomfi motto 3 0E a}; b_eo_e> es; N.« 0.9 0.0 0.0 v.0 . — p h n — s b p P . «.0 P P — L — b .— P by B . p m b p — h moc: commmoumoa ‘U U‘ ‘ ‘- U. a. ‘0‘0...‘ . mod n... c an on we u x .8328 £834. See N. 9 “WWW...” Mm ”Hum u E 8306 68 8:25 o 111310144903 96.10140ng 90111.10 , 48 discharge coefficient was not found in the published literature. Perry et a1. (1963) mention that, for different fluids with discharge coefficients in the range of 0.6 to 0.95, C values increase as the orifice diameter increases and decreases as the fluid density decreases. This is consistent with the lower C values being found for the smallest orifice diameter and for more viscous fluids (see Figures 12, 13 and 14). No references for the orifice discharge coefficients for non-Newtonian fluids were found. Each set of orifice discharge coefficient data was analyzed as a function of the fluid velocity in the orifice. Considering the distribution of the data in Figures 12, 13 and 14, it can be deduced that the data follows an exponential mathematical model. This mathematical model does not have any particular physical interpretation and is presented only as a compact representation of the experimental data. The orifice discharge coefficient versus the fluid velocity in the orifice were pooled and the following equation was found to fit the data: C - 13(1) (1 - eXP( -B(2) V)) (39) The coefficients 8(1) and 8(2) are parameters to be determined. Table 2 shows the parameter estimates found by means of non-linear regression analysis. A comparison of the experimental data and the regression curves shown in Figures 12, 13 and 14 indicates that the experimental orifice discharge coefficients fit well within the regression line in the range of fluid velocities studied. Another way to analyze the data is to relate the calculated C values with Reo (Equation 37), Tables A1 to A3 of Appendix A present the calculated values of Reo. If the orifice discharge coefficient is 49 mmmm.~ Hoom.a mmme.o Noon.o m-mnmmn.o mm.o mHo.N moms.o ammoo.o ammo.a mmow.o mmmm.o come.o m-mwae.o mm.o Hmw.o meme.o mucoo.o OH eme.H emom.o owmm.o onm~.o m-mammm.o mm.o mqm.o mmms.o meoo.o Hemm.oH anae.w eqmm.o qmmq.o a-mmmqa.o om.o omm.o mHNm.o emnoo.o wmom.m omom.o mwom.o Heoq.o H-m~mme.o Hm.o mmo.¢ mome.o ouqoo.o m.m momn.m Nqu.o mmom.o mamm.o m-memnm.o qm.o mem.a omoq.o mamoo.o mmom.m mmwm.m mumm.o ammo.o m-mwomn.o em.o eem.m mamo.o «mnoo.o oeom.c mmem.H sumo.o maem.o m-moomq.o mm.o mom.m smmm.o mueoo.o m mmow.~ mNoN.H mmmm.o NHo<.o a-mwwHH.o mm.o mno.m wmam.o mamoo.o Home: Hm3oA .umemp noSOA 8: SE 8.23% a}. e w mumumemumm ammo: Edm up Amvm Aavm Houseman Scandaom new mOMEMA mucmnHMCOU mumEHfibna< HmSpwmmm mnoumemumm . mowwwuo Loumum 2> Genes... - 0.: Aavm I 0 "Honor .OCoHonmeoo mwumnomwc oonHuo mo some new mwm%Hmcm :onmouMou umocHH-co: 0:» mo muHSmom N oHan 50 plotted as a function of Reo (Figure 15), one finds that most of the Reo values are in the laminar region and C values increase with Reo until C is almost a constant. Results suggest an exponential relationship could exit between the orifice discharge coefficient and Reo that is similar to the relationship between C and the fluid velocity in the orifice. This mathematical expression was determined as C - 0.494 (1 - exp(-0.011 Reo)) + 0.086 (40) Equation (40) had an r2 value of 0.77 and represents the value of the coefficient that provides the best fit to the experimental data. Equation (40) is very important for two reasons. First, the fact that the discharge orifice coefficient is a function of the generalized Reynolds number in the range studied means that it is possible to find the orifice discharge coefficient for a non-Newtonian fluid (power-law model) with any fluid property values (consistency coefficient and flow behavior index) and any orifice diameter. Second, and just as important, it enables one to transfer this model to analysis of manifold system subjected to the same or similar conditions. 5.3 Manifold Fluid Flow Distribution. Using the manifold system (Figure 8), the data included in Tables Bl to B3 of Appendix B were obtained. These tables report the fluid flow rate at each orifice from the manifold for different orifice and pipe diameters for different fluids. Figures 16, 17 and 18 are manifold flow distribution example for 5% starch solution (low viscous fluid), 7.5% starch solution 51 .umeEsc mEoEAmm nmNzoumcoo 9: to cozoczc 0 mo Lcm_o_tooo motosomfi motto m. 9.1 59:32 mEocxom noNzotmcoo oo...~ comm 000m com. com. 00v. 009. 000. com com 8». com o h . _ _ _ _ _ . _ h _ F L e _ _ h _ b _ P h _ p 0.0 = I ”— .LI ..0 .. Ind or: coimouooa I 1.. Rd l ad I m6 0.. iuagoweog 3610143530 30an A3380. BotcoE 832.com so ooEco cc 3385. 88 so: 82.: 8.865 .an Son 2003 c0323 nouoem Eoo Nm 0 to. cozs£bm6 30: 20:52 m. 0E as £83 seizes WA or no md no so no no no no to 0.0 e .. x e I I O o o r .. .. 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M .. $00 a D I a E 0500.0 1 .20 02:5 . . ) 8005.0 1 .20 use .. W05 0 W 802.3820 20:52 - V 1000.0 00.0 n e M 0m 00 me 1 x N sea .6: e. - .cozaom couoem N o. . 80m 32... o lvwod 55 (medium viscous fluid) and 10% starch solution (high viscous fluid) and Figures 19, 20 and 21 are plots of pressure at the first orifice versus the mass flow rate for the same fluid at different orifice diameters. From the data and figures the following can be observed: 1. The fluid flow rate in each orifice decreases along the manifold. This can be observed at any fluid flow rate distribution reported in Tables B1 to B3 of the Appendix B. Even though most of the data follows this tendency, there are some data points which do not. One explanation for this is that fluids with high viscosity or high concentrations sometimes formed starch clumps which acted as plugs causing a reduction of the flow in the orifice, especially in the smaller orifices. 2. The discharge at any orifice in the manifold is controlled by the pressure at that orifice, and the orifice and pipe diameter. The pressure profiles are similar to the pressure profile along the manifold reported by Dow (1950) and we and Glitin (1974). The fluid flow rate in the orifice is function of the square root of the pressure (Equation 38) and the orifice discharge coefficient which depends on on fluid properties (consistency coefficient and flow behavior index) and flow rate in the orifice. 3. At constant fluid properties, pipe diameter and flow rate at the entrance, a manifold with a small orifice diameter required more pressure than the manifold with a bigger orifices diameter (see Figure 19). Also the flow distribution was different. This occurs because the energy loss in the small orifice diameter are larger and consequently larger pressure drop is needed to keep a 56 00.0 00028 cououm N m o co.— mozto 9: E 88 so: 89.: 0.10 00 c0083 0 me @5390 0. 0E Am\ov_v Boa 30E mmoE 000.0 000.0 0v0.0 0N0.0 000.0 _ _ _ _ _ _ _ _ . 0 . \ \ T \ \ 0 0” \\\\ XXX \Hl000 1‘ \ \ \ \\\\ 0m \ 1 \ \\\ ”I000— \ l \ \\ .I \ \ \ \\\ O H \ _\ \ \\ W000. 1 . \ 1| \ 0 1 1 110000 \ II as O .l .wc: commmmtmmm .. rlooH \ n Q H.002 H T l00mm 1 0.0 H C 1l.0 on c m OQ 00.0 H v. 0 o o H #05000 .2 mutt loan... U . 0 0000 .3 I 1 m L... « LU U. _.m mhvood .20 00:10 D 1 0.0000 .05 oucto O H 0 O (‘1 in (0d) eJnsseJd S7 :83.» NmN 0 L8 motto 9: E 38 .co::_om 26: mmoE 9: 00 c0305: 0 mo mSmmmi 0m mi ONTO Am\oxv 30m 30E mmoE mc: co_mmm.mmm Ed I c cm on n; .. x 85.3855 25 380.0 .05 3.53 4 0200.0 .20 023.5 0 0300.0 .20 oucto o (0d) aJnSSBJd ¢0+mn 58 .8328 Box 32... mmofi 583m N 0_ 0 L8 80:0 9: E 38 so: mmoE 9.: 00 c0383 0 mo mSmmmi _m 9... ON 3 09.0 omod 80.0 9.0 o 80.0 08.0 H _ . H L _ . _ _ 2 . a M, o \ n. 1 l t I 9%. mmc: commmmhmmm C c x 85.3885 vs: 00.0 m ca 04» wmnood .20 3:10 mnvood .20 motto D 0300.0 .20 30:0 d 0 II «041w, I ¢0+mm (Dd) GJnssaJd 59 higher flow rate. 4. At constant pipe and orifice diameter but different fluid properties, the more viscous fluid requires more pressure at the manifold entrance than the less viscous fluid to produce the same flow rate (see Figures 19, 20 and 21). This occurs because the more viscous fluid produces more friction loss in the pipe and consequently it needs more pressure to produce the same manifold flow distribution. 5. At constant fluid properties and orifice diameter, the pressure needed to pump the same flow rate in the manifold is higher in the smaller pipe diameter than in the bigger one. This can be observed from the data when a flow rate of 0.1 kg/s of 10% corn starch solution was pump in manifold diameters 0.0158, 0.0409 and 0.0525 m. The pressures necessary were approximately 40000, 15000 and 11000 Pa, respectively. This is because in a small pipe diameter, the velocity was higher and the energy loss due to the friction in the straight pipe (Equation 7) is increased due to the fluid velocity that is proportional to the square of the pressure. 5.4 Energy Loss Coefficient Due to Turbulence at the Orifice. Using the procedure described in Section 4.3.2 and the relationship between the mass flow rate and the pressure at the orifice shown in Figures 19, 20 and 21; the energy loss coefficient due to turbulence at the orifice (kf) was calculated. Table 3 shows the mathematical model parameters used to calculated the pressure at the orifice as a function of the mass flow rate. Tables Cl and C2 of the 60 Table 3 Results of the non-linear regression analysis for data of pressure versus flow rate in the orifice. Model: p - 8(1) exp( 8(2) q) Starch Orifice Parameters solution Diameter 3(1) 8(2) r2 % m Pa ms/kg 0.00318 407.4 160.3 0.89 5.0 0.00476 287.9 70.3 0.98 0.00318 1037.2 86.1 0.98 7.5 0.00476 652.0 52.5 0.99 0.00794 113.3 32.5 0.99 0.00318 1746.1 52.2 0.90 10.0 0.00476 592.8 18.4 0.91 0.00794 189.8 19.4 0.98 61 Appendix C present the calculated kf. A comparison of the kf calculated for different starch solutions indicates that kf is consistently present in the less viscous fluid while in the higher viscous fluid did not follow a defined pattern. It is important to note, for 5% starch solution (low viscous fluid), the pressure calculated by means of the mechanical energy balance is larger than the calculated pressure using the mathematical model shown in Table 3. This means that the energy loss due to the turbulence at the orifice is important and it is necessary to consider in the simulation model to get the actual pressure at each orifice. For more viscous fluids, the data indicates that the friction loss due to turbulence was insignificant; because without using this factor it was possible to obtain good results for the pressure at the last orifice compared with the experimental data. To include this coefficient into the model it is necessary to find a mathematical relationship between kf and the generalized Reynolds number for the fluid in the straight pipe (Figure 22). Considering the above, the energy loss coefficient kf versus the generalized Reynolds number data were pooled and the following equation (determined by non—linear regression) was found to fit the data: f - 281.2 Re’°°97+ 148.4 (42) k Equation (42) had an r2 value of 0.77 and represent the value of the coefficient that provides the best fit to the data. The equation has no particular theoretical significance and is present only as a representation of the data; however, it should be mentioned that a similar form of the equation was suggested by Steffe et al. (1984) and similar results were obtained for a tee, valve and 62 .8926 mEocxom Dwfifiamcmm or: 00 c0525.. 6 mo oocto 9: mo 88393 8 mac Eo_o_:moo $0. x 6:0 mm 9m tmnEsz mEocxmm 820850 00 om 0.9 0_m 0_N 01 . O mo:_o> 0223200 C 0 CC 00 COCJOO 018! cc: c2mm$0mm II 0 T‘IIIIIIIIIIITII'ITII A ( ’1) wepmeog SSO'] KbJaug 63 elbow. 5.5 Orifice Discharge Coefficient Correction Factor. It was explained in Section 3.6 that the orifice discharge coefficient does not represent the actual coefficient present in the manifold orifice because there is a "pass by flow" (this is not present in the experiment to find the orifice discharge coefficient). This coefficient underestimates the discharge in the orifices. Using the manifold system, as explained in Section 4.3.1, the data included in Tables Cl to C3 of Appendix C were generated. The tables report the flow rate at each orifice from the manifold for different orifice and pipe diameters, and different fluid properties. Pressure drop at each orifice was calculated by means of mathematical function found in Section 5.4 (see Table 3) that related the pressure with the mass flow rate in the orifice. Results indicate that the correction factor generally is larger than 1, and follows the same pattern as the orifice discharge coefficient versus fluid velocity in the orifice. This leads one to believe that the orifice discharge coefficient times its correction factor is another coefficient that can be called ”the corrected orifice discharge coefficient, C' " defined in Equation (36). Figures 23, 24 and 25 show the relationship between the corrected orifice discharge coefficient and fluid velocity in the orifice for different orifice diameters and fluid properties. The calculated corrected orifice discharge coefficient (C') may be considered a function of the generalized Reynolds number (Rec). Tables Cl to C3 of Appendix C present the calculated values of Rec. If the C' values are plotted as a function of the Rec (Figure 26), it can 64 003200 @806 Eco x0 0 t8 000:0 05 E >:0o_0> 23: 00 c0305: 0 00 #:2000000 00.65020 motto 0309200 mm or. Am}; 062$ 23: _ _ _ . L h p _ _ b p b _ . o 0 on 1 I . O o r 00 O o d N. u 8 0 “w m I 11 o m I n0 w. 0 B I a I v.0 MU. o a a I H... o .- . 3 nWBB. m C e g l . mu. 0 o a 00 8 CB 8 C q o I W m I No a o . 6 c l w o 8 u 3 0.0 u c 1 as m c... f 09.0 n x - M... 002200 50.66 Km E 3800 .30 35.0 o r . W. E 0200.0 .20 80:0 .- I P — 1. 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E 03000 .20 8:00 a I _ _ E 03000 .20 30:0 0 r N; iuagomaog) abmqosgg aoIIIJO papauog 66 000200 000000 500 N 2 0 :00 000:0 00.: E >0_00_0> 050 00 000002 0 00 000600000 09000000 000:0 00000050 mm 90 A0\Ev b_00_0> 03E 00.0 n c cm on. 3. ll X 000200 £0.60» N op E .2000 .20 8000 + E 0:80 .20 80:0 .- E Snood .20 03:00 o «.0 0.. 0.0 _ _ _ p _ . p r _ . . . 3T 0.0 l—é N.— 1uepmeog 361003340 03!;00 papauog 67 00:50 020530 0030:0000 05 00 0000000 00 000600000 09000000 000:0 00000050 mm 9... 00:52 00.0500 00~=E0c00 can 000 000 _ _ _ 00m om— _ on: 50000600 |.. 0.0 v 91-... I 9.0 .I 0.0 o... wagomeog 0&1qule 93!;310 papeuog 68 be observed that C' values increase with Reo until C' is almost a constant. Results suggest an exponential relationship between the corrected orifice discharge coefficient and the generalized Reynolds number. Similar results were obtained between the orifice discharge coefficient (C) and the generalized Reynolds number (Rec). The mathematical model that best fit the data points was determined as 0.093 C' - 0.905 Reo - 0.7742 (43) with an r2 value of 0.77. The low correlation index is due to variation of the C' values which may depend on the fluid rheological properties. Equation (43) is important because this is going to be used in the simulation of the flow distribution in the manifold for different conditions. 5.6 Comparison of Simulated and.Actua1 thifold Distribution. Using the theoretical model, the energy loss coefficient due to the turbulence at the orifice (Section 5.4) and the corrected orifice discharge coefficient developed in Section 5.5, the manifold flow distribution at any fluid flow rate at the entrance may be predicted. To test the theoretical model, two sets of experimental flow rates for each type of fluid were plotted with the simulated flow rates. Figures 27, 28, 29, 30, 31 and 32 show the experimental and the simulated manifold flow distribution predicted by the theoretical model for 5% (low viscosity), 7.55 (medium viscosity) and 10% corn starch solutions (high viscosity). The theoretical model underestimated the flow rate in the first orifices and then overestimated the flow rate in the last ones 69 :002om 00:30 Eoo .Nm 0 :00 0:000: .300E000x0 mamt0> 00:03:05 “00030505 30: 0.00002 mm 9... EV £003 20:82 . . . . 0.0 0.0 «.0 . 0.0 0.0 0.0 0.0 0%. M0 000 90 0 E _ 0 _ . _ 000.0 a u 4 r . a o w... v00 0 a . V0000 1 Q m 1 ”IN—0.0 c 0 .. o 0 mum—0.0 . W080 . $00.0 W080 E 0500.0 n .20 85.0 m. . E 00.0.0 :20 20052 n ~00 0 .mozm_._£oo:._u 0.00:0: Wonod W030 0.0 n c ”r . . 3.0 0 m 0 mu 1 c a 00. 0 0. 32220 < n .co32ow 0.9.0.5 No .300 .3005:an 4 W0¢0.0 (s/bx) 0103 M0” ssow 70 002200 50:30 500 ”mm o to» 0:000: .300E:00x0 . 2.29, 02230 “0203520 30: 28202 00 2... 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W030 .cozaom c820 N 0. 300 62308.0 0 w (s/bx) 9103 MOL-J ssow 75 for the 5% solution (Figures 27 and 28). This behavior is repeated in each case. However, the simulated flow rates followed the trend of the experimental data. It is important to note that experimental flow rate and pressure at the first orifice of the manifold were used to initiate the simulation procedure and, for that reason the predicted and the experimental data are always the same for the first orifice. The model also predicted flow rates that were lower and higher than the experimental values. A good agreement between the experimental predicted values for the 10% solution (Figures 31 and 32). A good agreement, however, between the experimental and predicted values was obtained. The error between the estimated and the experimental flow rate in the orifices ranges from O to 15 %. The flow rate distribution was overestimated in both cases for the 10% solution and agreement was poor between the simulated and the experimental values. As showed in Figures 27 to 32 the theoretical model does not always accurately predict the experimental data. The inaccuracy of the model may be related to many factors: a. The effect of the experimental errors in the determination of the flow rate in each orifice and the determination of the pressure at the first and last orifices in the manifold. b. The effect of the consistency coefficient appears to be important in the determination of the corrected orifice discharge coefficient and this means that there is not a single mathematical expression to obtain this coefficient. c. The energy loss coefficient due to turbulence does not have a well defined pattern in the highly viscous fluids, and this does not always allow it to be incorporated into the simulation model. 76 d. One of the major factors which causes the model inaccuracy is the system complexity due to the large number of interacting variables presented. 5.7 Simulation of the Hanifold Plow Distribution. As shown in Section 5.6 the theoretical model, using the coefficients found in Section 5.4 and 5.5, fit the experimental data in same cases; hence it is instructional to simulate the manifold distribution for different conditions to observe the behavior of the manifold when different parameters are varied: the flow rate at the entrance, manifold diameter, orifice diameter, fluid consistency coefficient and so on. To simulate the manifold flow distribution using the theoretical model and the coefficients found above, a manifold system with a manifold length of l m and with 10 orifices was selected. The simulation was done for different flow rates at the entrance (0.12, 0.14, 0.16, 0.18 and 0.20 kg/s), manifold diameters (0.020, 0.025, 0.030 and 0.035 m), orifice diameters (0.00318, 0.004, 0.00476, 0.005 and 0.00525 m) and consistency coefficients (0.3, 0.7, 1.0, 1.4 and 3.0 Pa 8“). Figures 33, 34, 35 and 36 show the simulation values of the flow rate in each particular orifice. When the flow rate at the entrance is 0.2 kg/s, the difference between the flow rate in the first orifice and the last orifice is larger than for the case when the flow rate at the entrance is 0.12 kg/s (Figure 33). This means that when everything is constant and the flow rate at the entrance decreases the flow rate value for each orifice tend to be closer. This behavior is due to the energy loss due to the friction along the manifold which decreases when the flow rate 77 0000000 05 00 030.. 20: Emtmtfi 3. 803500 :2. 2225 0323.0 00 0: “EV 505.. 0.8.5: 0., 0.0 00 00 00 0.0 +0 0. . . . . . . . . . . .0 N.0 ..0 00 . 0 0 <9. 20 o 000 0 0 0 <9. 10 n o <9. 30 a a o a\os 3.0 o . . . <9. 00.0 o 4 00:93:. 05 1. 3.0 a an 3.0.. DOE O C 0 O O u 0 [~00 D C ... . E «0.0 I .05 0.00:0: . u 1000 E 00000.0 u .05 3E5 . . nofintfluoéco 0.87.0: . u o I... . 0.0 I c . 30 0 Ca O.F - v— 0 330.5008on 20.... O no.0 (s/fix) 9103 mou ssow 78 0.8.008 05 0. 0030.020 motto “00.0.0.0 00.. 00.30520 30... 0.259: 030.0005 .00 0E AEV 505.. 28.8.2 0.. 0.0 0.0 0.0 0.0 0.0 to n0 «.0 F. . _ _ F _ _ r _ _ 00 00 . q u a E 0.0000 o 0000 0 o . E 09000 a o u E 0900.0 0 o E 0800.0 0 . 9 o o E 00.0000 . I080 o 0 0 m o . 6.0 8.0.0 m o [03.0 0 D O . a T0000 0 {9. 0.0 . 6000.05 05 go 30¢ 30.... n I. 9 0. 0 d E «0.0 u .05 0.8.002 . 0000.00.60.05 0.8.5.2 . 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The same pattern is found in Figure 35 where the fluid consistency coefficient took different values. When the fluid consistency coefficient is 0.3 Pa 5n the difference between the flow rate in the first orifice with the last orifice is less than that found when the fluid consistency coefficient is 3.0 Pa s“. This behavior is present because the more viscous fluid produces higher energy loss due to friction causing the pressure along the manifold decrease rapidly. We can conclude that using water (less viscous fluid) it would be possible to obtain almost a uniform flow distribution in a manifold with these characteristics. Figure 36 shows the simulated orifice flow rates when the manifold diameter was varied and everything was kept constant. When the manifold diameter is increased the difference between the flow rate in the first orifice and the last one is decreased. This means that manifold flow distribution tends to be uniform when the manifold diameter is increased. This is because the energy loss due to the friction is related to the surface area (diameter) of the manifold and when the manifold diameter is increased the surface area increases and, therefore, the energy loss due to the friction is decreased. 82 5.8 Strategies for Ashieving Uniform Flow. The strategies consist of developing the necessary conditions to insure that static pressure remains constant along the entire length of the manifold which will insure a uniform flow distribution from the manifold. It must be noted that the pressure drop is due to friction losses from flow through the pipe and the orifice; therefore, pressure drop is related to the orifice diameter, pipe diameter, space between orifices, and fluid properties (consistency coefficient and flow behavior index). Uniform flow distribution can be accomplished by several means, such as increasing the pipe diameter to have less pressure drop due to friction, decreasing the orifice diameter or both. The fluid properties play a very important role in the uniformity of the flow from a manifold. When the fluid has a high consistency coefficient, the pressure drop in the pipe and in the orifice is larger and the static pressure is significantly decreased. In some cases this static pressure can be zero (no flow in the orifice). If one have a very viscous fluid (power-law model) and the orifices are the same size, uniform flow distribution can be accomplish by increasing the pipe diameter and decreasing the flow rate at the entrance. According to Dow (1950), uniform distribution in the manifold is achieved when the necessary conditions to insure that the pressure drop due to friction losses from flow through the pipe and the orifice are exactly balanced by the pressure due to the deceleration of the flow in the pipe which necessarily occurs when part of the fluid escapes through the orifices. Wu and Gitlin (1974) said that if the pressure distribution along the pipe can be determined, uniform flow can be achieved by adjusting size of the orifices, length and size of 83 the microtube (a special type of emitter) and slightly adjusting the spacing between orifices. The microtube idea may have excellent potential for fluid foods. 6. SUHHARI'AND OOHCEUSIONS. . A laboratory manifold system was successfully designed and tested in the collection of the fluid flow rate at each orifice for different orifice diameters, pipe diameters and fluid properties (consistency coefficient, flow behavior index and density). . The rheological properties of the non-Newtonian fluid (consistency coefficient and flow behavior index) and the orifice diameter affect the orifice discharge coefficient. The orifice discharge coefficient for a non-Newtonian fluid is in the range of O - 0.5. . A mathematical expression that correlate the orifice discharge coefficient with generalized Reynolds number of the fluid in the orifice was obtained. . The pressure calculated by means of the mechanical energy balance is higher than the experimental pressure, therefore it is necessary to include a parameter that accounts for energy loss due to turbulence at the orifice. Calculated values indicate that the energy loss coefficients due to turbulence increase significantly for decreasing values of the generalized Reynolds number. . Using the theoretical model developed, it is possible to determine a correction factor for the orifice discharge coefficient and also the corrected orifice discharge coefficient for the flow distribution from a manifold. This corrected orifice discharge coefficient (C') can be expressed as a function of the generalized Reynolds number 84 6. 7. 85 for the fluid in the orifice. The theoretical model developed for the manifold in conjunction with the mathematical model for corrected orifice discharge coefficient simulate the fluid flow rate distribution from a manifold under the conditions studied. The use of the simulation model for less viscous fluids (5% starch solution) caused significant errors in the flow distribution from the manifold due to the experimental error in the determination of the flow rate in the orifice and the pressure, the effect of the consistency coefficient in the correct orifice discharge coefficient and the system complexity due to large number of interactive variables presented. 7. SUGGESTIONS FOR FUTURE RESEARCH. . Investigate the importance of energy loss due to turbulence at the orifice by measuring the pressure drop at each orifice and comparing this with the pressure calculated by means of the mechanical energy balance equation. . Validate the theoretical model developed in this study for a non-Newtonian fluid having a yield stress, i.e. Herschel-Bulkley or Bingham plastic materials. . Investigate the effect of the flow behavior index on the orifice discharge coefficient for a non-Newtonian fluid. . Evaluate the theoretical model developed in this research to network systems using non-Newtonian fluids. 86 8. REFERENCES Bird, R. 8., Armtrong, R. C. and Hassanger O. 1987. ”Dynamics of Polymeric Liquids.". Vol 1. Wiley-Interscience Publication. John Wiley and Sons, New York, NY. Bralts, V. 1983. Hydraulic design and field evaluation of drip irrigation submain units. Ph. D. Thesis, Michigan State University, East Lansing, MI. Dow, W. M. 1950. The uniform distribution of a fluid flowing through a perforated pipe. J. of Appl. Mech. 431 -438. Eskinazi, S. 1962. "Principles of Fluid Mechanics." Allyn and Bacon, Inc., Boston. Garcia, E. J. and Steffe, J. F. 1986. Optimum economic pipe diameter for pumping Herschel—Bulkley fluids in a laminar flow. J. of Food Process Eng. 8, 117-136. Garcia, E. J. 1985. Optimum economic tube diameter for pumping Herschel-Bulkley fluids, M.S. Thesis. Michigan State University, East Lansing, MI. Govier, G. W. and Aziz, K. 1972. "The Flow of Complex Mixtures in Pipe." Van Nostrannd Reinhold Co., New York, NY. 87 88 Hanks, R. W. 1978. Low Reynolds number turbulent pipeline flow of pseudohomogeneous slurries. In Proceedings of the Fifth International Conference on the Hydraulic Transport of Solids in Pipe. (Hydrotransport) May 8-11 Paper CZ p. C2-23 to C2-34. Hannover, Federal Republic of Germany. BHRA Fluid Engineering, Cranfild, Bedford, England. Hanks, R. W. and Ricks, B. L. 1974. Laminar turbulent transition in flow of pseudoplastics fluid with yield stress. J. Hydronautic 8(4):l63-l66. Hanks, R. W. 1969. A theory of laminar flow stability. AIChE Journal 15(1):25-27. Keller, J. D. 1949. The manifold problem. J. of Appl. Mech. 16, 77-85. Krieger, I. M. 1968. Shear rate in the Couette viscometer. Trans. of the Soc. of Rheology 12(1):S-11. Osorio, F. and Steffe, J. F. 1984. Kinetic energy calculation for non- Newtonian fluids in circular tubes. J. of Food Sci. 49(5):1295-1296, 1315. Perry, R. H. and Chilton, C. H. 1963. "Chemical Engineers' Handbook." MacGraw-Hill Book Co., New York, NY Ramirez-Guzman, H. and Manges, H. 1971. Uniform flow from orifices in irrigation pipe. Transactions of the ASAE. 14(1):127-129. Segerlind, L., Steffe, J. F. and Bralts, V. 1985. Network analysis for 89 non-Newtonian fluid foods. ASAE paper No 83-6005. Am. Soc. Agr. Eng., St. Joseph, Michigan. Skelland, A. H. P. 1967. "Non-Newtonian Flow and Heat Transfer". John Wiley and Sons, Inc., New York, NY. Steffe, J. F., Mohamed, I. 0. and Ford, E. W. 1984. Pressure drop acrooss valves and fittings for pseudoplastic fluids in laminar flow. Transactions of the ASAE. 27(2):6l6-619. Steffe, J. F. and Ford, E. 1985. Rheological techniques to evaluate the shelf-stability of starch-thickened strained apricots. J. of Texture Studies 16, 179-192. Steffe, J. F. and Morgan, R. C. 1986. Pipeline design and pump selection for non-Newtonian fluid foods. Food Technology 40(12):78-8S. Whorlow, R. W. 1980. "Rheological Techniques," Halstead Press, New York, NY. Wood, I. and Charles, C. 1972. Hydraulic network analysis using linear theory. J. Hydraulic Div. Am. Soc. Civ. Eng. 98, 1157-1170. Wu, I. and Gitlin, H. M. 1974. Drip irrigation design based on uniformity. Transactions of the ASAE l7(3):429-432. APPENDICES 91 APPENDIX A Pressure and Fluid Flow Rate in the Orifice for 5, 7.5 and 10% Corn Starch Solutions and Different Orifice Diameters 92 Table A1. Pressure and fluid flow rate in the orifice for a 5% starch solution and different orifice diameters Experiment: Starch solution at 23°C Density: 1010 kg/m8 Flow behavior index: 0.8 Consistency Coefficient: 0.105 Pa sn Orifice Diameter: 0.00318 m. Pressure Flow Rate Velocity Orifice Generalized Pa kg/s m/s Discharge Reynolds Coefficient Number 196.2 0.0005 0.07 0.112 23.8 264.9 0.0087 0.08 0.116 29.7 421.8 0.0010 0.12 0.137 48.2 598.5 0.0017 0.21 0.198 92.5 784.9 0.0005 0.07 0.057 24.4 843.7 0.0008 0.11 0.086 42.1 5141.2 0.0103 1.29 0.406 792.5 7260.5 0.0161 2.02 0.533 1351.0 9026.5 '0.0186 2.33 0.552 1605.8 Orifice Diameter: 0.00476 m. Pressure Flow Rate Velocity Orifice Generalized Pa kg/s m/s Discharge Reynolds Coefficient Number 49.0 0.0006 0.03 0.111 14.1 245.2 0.0028 0.15 0.225 87.3 3924.6 0.0282 1.56 0.562 1378.3 5003.8 0.0323 1.79 0.570 1620.5 6475.5 0.0405 2.25 0.629 2129.3 Orifice Diameter: 0.0079375 m. Pressure Flow Rate Velocity Orifice Generalized Pa kg/s m/s Discharge Reynolds Coefficient Number 1667.9 0.0606 1.21 0.668 1524.3 1815.1 0.0632 1.26 0.667 1600.9 2354.7 0.0754 1.50 0.699 1980.4 4120.8 0.0943 1.68 0.690 2262.7 93 Table A2. Pressure and fluid flow rate in the orifice for a 5% starch solution and different orifice diameters Experiment: Starch solution at 23°C Density: 1021 kg/m3 Flow behavior index: 0.77 Consistency Coefficient: 1.3 Pa sn Orifice Diameter: 0.00318 m. Pressure Flow Rate Velocity Orifice Generalized Pa kg/s m/s Discharge Reynolds Coefficient Number 2256.6 0.0042 0.52 0.249 24.9 2943.4 0.0083 1.03 0.431 57.8 3139.6 0 0104 1.29 0.523 76.2 3532.1 0.0078 0.96 0.367 52.9 5494.4 0.0114 1.41 0.431 84.7 7113.3 0 0151 1.87 0.500 119.2 8192.6 0.0134 1.66 0.416 103.5 8506.2 0.0134 1.66 0.408 103.5 18221.4 0.0139 1.72 0.289 108.1 20047.6 0.0180 2.23 0.356 148.2 20412.8 0.0192 2.38 0.376 160.4 Orifice Diameter: 0.00476 m. Pressure Flow Rate Velocity Orifice Generalized Pa kg/s m/s Discharge Reynolds Coefficient Number 49.0 0.0018 0.09 0.319 4.3 147.1 0.0006 0.03 0.064 1.2 690.2 0.0023 0.13 0.112 6.1 1079.2 0.0114 0.62 0.432 42.6 1962.3 0.0182 1.00 0.511 75.7 2747.2 0.0149 0.82 0.354 59.3 6533.9 0.0262 1.44 0.403 118.6 7264.4 0.0350 1.92 0.510 169.0 9455.8 0.0272 1.50 0.348 124.3 9821.0 0.0430 2.36 0.539 217.8 94 Table A2. (Cont'd.) Orifice Diameter: 0.00794 m. Pressure Flow Rate Velocity Orifice Generalized Pa kg/s m/s Discharge Reynolds Coefficient Number 44.1 0.0020 0.04 0.137 2.2 93.2 0.0055 0.10 0.256 7.4 1569.8 0.0435 0.85 0.486 92.1 1717.0 0.0424 0.84 0.458 90.4 3237.7 0.0789 1.56 0.620 193.7 3685.1 0.0697 1.38 0.513 166.2 95 Table A3. Pressure and fluid flow rate in the orifice for a 10 % starch solution and different orifice diameters Experiment: Starch solution at 23°C Density: 1034 kg/m3 F1ow behavior index: 0.68 Consistency Coefficient: 4.5 Pa sn Orifice Diameter: 0.00318 m. Pressure Flow Rate Velocity Orifice Generalized Discharge Reynolds Pa -kg/s m/s Coefficient Number 1344.1 0.0004 0.05 0.035 0.6 10916.7 0.0079 0.96 0.211 24.8 11647.2 0.0100 1.23 0.259 34.0 15153.4 0.0122 1.49 0.275 43.7 20412.8 0.0175 2.14 0.341 70.7 Orifice Diameter: 0.00476 m. Pressure Flow Rate Velocity Orifice Generalized Pa kg/s m/s Discharge Reynolds Coefficient Number 490.5 0.0003 0.019 0.019 0.2 883.0 0.0007 0.041 0.031 0.5 4169.8 0.0067 0.365 0.128 9.0 7064.2 0.0197 1.074 0.290 37.4 9455.8 0.0280 1.522 0.356 59.3 11647.2 0.0353 1.919 0.404 80.4 16030.0 0.0416 2.260 0.405 99.8 Orifice Diameter: 0.00794 m. Pressure Flow Rate Velocity Orifice Generalized Pa kg/s m/s Discharge Reynolds Coefficient Number 461.1 0.003 0.06 0.069 1.3 735.8 0.003 0.06 0.056 1.3 981.1 0.004 0.08 0.063 1.9 1226.4 0.006 0.11 0.077 2.8 2256.6 0.036 0.71 0.342 30.9 3090.6 0.047 0.92 0.379 43.5 4807.6 0.060 1.19 0.390 60.6 6426.5 0.085 1.66 0.472 94.5 APPENDIX B Manifold Flow Distribution: Experimental Data 97 Table Bl. Manifold flow distribution for a 5% starch solution Experiment: Starch solution at 23°C Density: 1010 kg/m3 Flow behavior index: 0.8 Consistency Coefficient: 0.105 Pa sn Orifice Diameter: 0.00318 m. Pipe Diameter: 0.0158 m. Orifice Diameter: 0.00318 m. Orifice Mass Flow Pressure Mass Flow Pressure Number Rate Pa Rate Pa kg/s ks/s 1 0.00805 1363.8 0.01041 2256.6 2 0.00715 0.00910 3 0.00512 0.00779 4 0.00501 0.00680 5 0.00298 0.00585 6 0.00330 0.00472 7 0.00246 0.00441 8 0.00201 0.00244 9 0.00154 0.00215 10 0.00056 196.2 0.00089 843.8 Total 0.03820 0.05014 Pipe Diameter: 0.0158 m. Orifice Diameter: 0.00476 m. Orifice Mass Flow Pressure Mass Flow Pressure Number Rate Pa Rate Pa kg/s R8/8 1 0.03333 2904.2 0.02374 1775.0 2 0.02860 0.02009 3 0.02533 0.01665 4 0.02113 0.01346 5 0.01666 0.01055 6 0.01293 0.00587 7 0.00986 0.00499 8 0.00697 0.00296 9 0.00442 0.00159 10 0.00283 245.2 0.00632 49.0 Total 0.16210 0.10276 Table 81. Cont'd. Pipe Diameter: Orifice Diameter: 98 0.0158 m. 0.00794 m. Orifice Mass Flow Pressure Number H 000NOUIJ-‘WNH Total Rate kg/s .06067 .05196 .04410 .04456 .03276 .02660 .01454 00000000 Pa 1667.9 99 Table B2. Manifold flow Distribution for a 7.5 % starch solution Experiment: Starch solution at 23°C Density: 1021 kg/m3 Flow behavior index: 0.77 Consistency Coefficient: 1.3 Pa sn 0.0158 m. 0.00318 m. Pipe Diameter: Orifice Diameter: Orifice Mass Flow Pressure Mass Flow Pressure Number Rate Pa Rate Pa kg/s kg/s 1 0.02443 13838.8 0.02282 18586. 2 0.02239 0.02342 3 0.02170 '0.02278 4 0.01518 0.01953 5 0.01840 0.01786 6 0.01242 0.01584 7 0.01091 0.01336 8 0.00864 0.01228 9 0.00692 0.01011 10 0.00424 2256.7 0.00781 3532. Total 0.14526 0.16581 Pipe Diameter: 0.0158 m. Orifice Diameter: 0.00476 m. Orifice Mass Flow Pressure Mass Flow Pressure Number Rate Pa Rate Pa kg/s ks/s. 1 0.04181 6279.5 0.05472 11282. 2 0.02541 0.04255 3 0.02195 0.03401 4 0.01335 0.02368 5 0.00737 0.01700 6 0.00327 0.00800 7 0.00136 0.00418 8 0.00074 0.00196 9 0.00044 0.00102 10 0.00018 49.0 0.00063 147 1 Total 0.11589 0.19342 100 Table 82. Cont'd. 0.0158 m. 0.00794 m. Pipe Diameter: Orifice Diameter: Orifice Mass Flow Pressure Number Rate Pa kg/s 1 0.10478 3385.0 2 0.06844 3 0.03450 4 0.00853 5 0.00172 6 no flow 7 8 9 10 0 Total 0.21797 Pipe Diameter: 0.0409 m. Orifice Diameter: 0.00318 m. Orifice Mass Flow Pressure Mass Flow Pressure Number Rate Pa Rate Pa ks/s ks/s 1 0.01572 6534.0 0.01038 3612.1 2 0.01456 0.01048 3 0.01592 0.01065 4 0.01619 0.01059 5 0.01412 0.01007 6 0.01605 0.01032 7 0.01661 0.01001 8 0.01445 0.01001 9 0.01394 0.01001 10 0.01510 7113.4 0.01004 3139.7 0.15267 0.10259 101 Table 82. Cont'd. 0.0409 m. 0.00476 m. Pipe Diameter: Orifice Diameter: Orifice Mass Flow Pressure Mass Flow Pressure Number Rate Pa Rate Pa kg/s kg/s 1 0.01615 1471.7 0.02632 3335.9 2 0.01477 0.02420 3 0.01190 0.01970 4 0.00916 0.01696 5 0.01101 0.02009 6 0.00858 0.01643 7 0.01041 0.01861 8 0.00977 0.01569 9 0.00875 0.01626 10 0.00237 1470.0 0.01494 2747.2 0.10290 0.18920 Pipe Diameter: 0.0409 m. Orifice Diameter: 0.0079375 m. Orifice Mass Flow Pressure Mass Flow Pressure Number Rate Pa Rate Pa ks/s kg/s 1 0.03645 461.1 0.02580 166.8 2 0.03309 0.02341 3 0.02916 0.01974 4 0.01986 0.01326 5 0.01947 0.01302 6 0.01500 0.00741 7 0.01152 0.00825 8 0.00967 0.00606 9 0.00710 0.00406 10 0.00554 93.2 0.00204 44.1 Total 0.18686 0.123088 102 Table 82. Cont'd. Pipe Diameter: 0.0525 m. Orifice Diameter: 0.00318 m. Orifice Mass Flow Pressure Mass Flow Pressure Number Rate Pa Rate Pa kg/s ks/s 1 0.01332 5438.4 0.00865 3246.9 2 0.01139 0.00859 3 0.00912 0.00687 4 0.01237 0.00584 5 0.01188 0.00644 6 0.01014 0.00787 7 0.01012 0.00794 8 0.01015 0.00807 9 0.00831 0.00625 10 0.01144 5494.5 0.00838 2943 4 0.10828 0.07491 Pipe Diameter: 0.0525 m. Orifice Diameter: 0.00476 m. Orifice Mass Flow Pressure Mass Flow Pressure Number Rate Pa Rate Pa 1‘3/8 kg/s 1 0.01437 1049.8 0.02077 1560.0 2 0.01382 0.01971 3 0.01341 0.01956 4 0.01337 0.02084 5 0.01243 0.02078 6 0.01320 0.02012 7 0.01242 0.01989 8 0.01153 0.01812 9 0.01192 0.01846 10 0.01142 1079 2 0.01823 1962.3 Total 0.12793 0.19648 103 Table 83. Manifold flow distribution for a 10% starch solution: Experimental results. Experiment: Starch solution at 23°C * no data. Density: 1034 kg/m3 Flow behavior index: 0.68 Consistency Coefficient: 4.5 Pa sn Pipe Diameter: 0.0158 m. Orifice Diameter: 0.00318 m. Orifice Mass Flow Pressure Mass Flow Pressure Number Rate Pa Rate Pa kg/s kg/s 1 0.09575 8275.5 0.01681 15229.8 2 0.00563 0.00709 3 0.00365 0.00728 4 0.00320 0.00543 5 0.03165 0.00359 6 0.00190 0.00189 7 0.00075 0.00304 8 0.00135 0.00272 9 0.00070 0.00100 10 0.00071 1344.2 0.00214 n d.* Total 0.05911 0.05099 ' Pipe Diameter: 0.0158 m. Orifice Diameter: 0.00476 m. Orifice Mass Flow Pressure Mass Flow Pressure Number Rate Pa Rate Pa kg/s kg/8 1 0.04380 31954.6 0.03139 24796.0 2 0.02596 0.03581 3 0.02149 0.02868 4 0.01091 0.01856 5 0.00523 0.00734 6 0.00608 0.00718 7 0.00368 0.00387 8 0.00240 0.00276 9 0.00153 0.00980 10 0.00035 490.5 0.00076 883.0 Total 0.12144 0.14615 104 Table B3. Cont'd. 0.0158 m. 0.0079375 m. Pipe Diameter: Orifice Diameter: Orifice Mass Flow Pressure Number Rate Pa kg/s 1 0.10834 8725.5 2 0.03924 3 0.01732 4 0.00623 5 0.00263 6 0.00040 7 no flow 8 9 10 0 Total 0.17417 Pipe Diameter: 0.0409 m. Orifice Diameter: 0.00318 m. Orifice Mass Flow Pressure Mass Flow Pressure Number Rate Pa Rate Pa ks/s ks/s 1 0.01137 10186.5 0.01945 16030.3 2 0.00875 0.01498 3 0.00946 0.01480 4 0.00543 0.01386 5 0.00918 0.00974 6 0.00526 0.00856 7 0.00694 0.01296 8 0.00724 0.00936 9 0.00550 0.00595 10 0.00080 n d.* 0.00297 n d.* Total 0.06995 0.11265 * no data 105 Table B3. Cont'd. * no data. Pipe Diameter: 0.0409 m. Orifice Diameter: 0.00476 m. Orifice Masc.Plow Pressure Mass Flow Pressure Number Rate Pa Rate Pa kg/s ks/s - 1 0.01971 7264.5 0.02326 10040.4 2 0.01855 0.02789 3 0.01782 0.02397 4 0.01312 0.01463 5 0.01432 0.01863 6 0.00890 0.01363 7 0.01262 0.02026 8 0.00740 0.01674 9 0.00269 0.01204 10 0.00017 6573.8 0.00721 n. d.* Total 0.11533 0.19888 Pipe Diameter: 0.0409 m. Orifice Diameter: 0.00794 m. Orifice Mass Flow Pressure Mass Flow Pressure Number Rate Pa Rate Pa ks/s ks/s 1 0.02531 470.9 0.03362 1942.7 2 0.01986 0.02630 3 0.01300 0.01504 4 0.00874 0.01344 5 0.00734 0.01273 6 0.00676 0.01092 7 0.00644 0.00864 8 0.00486 0.00808 9 0.00450 0.00459 10 0.00337 46.1 0.00470 981.2 Total 0.10022 0.13806 106 Table B3. Cont'd. * no data Pipe Diameter: 0.0525 m. Orifice Diameter: 0.00318 m. Orifice Mass Flow Pressure Mass Flow Pressure Number Rate Pa Rate Pa kg/s kg/s 1 0.00758 4611.5 0.01256 11647.4 2 0.00709 0.01192 3 0.00863 0.01042 4 0.00646 0.01018 5 0.00652 0.01148 6 0.00709 0.01138 7 0.00547 0.00966 8 0.00521 0.00924 9 0.00498 0.00938 10 0.00150 n d.* 0.00363 n d.* Total 0.06055 0.09988 Pipe Diameter: 0.0525 m. Orifice Diameter: 0.00476 m. Orifice Mass Flow Pressure Mass Flow Pressure Number Rate Pa Rate Pa kg/s kg/s 1 0.01298 3532.2 0.01908 3612.2 2 0.01238 0.00880 3 0.00878 0.01610 4 0.00800 0.01400 5 0.00770 0.01402 6 0.00840 0.01528 7 0.00859 0.01243 8 0.00507 0.01250 9 0.00791 0.00446 10 0.00672 4169.9 0.00301- 7456.8 Total 0.08657 0.11968 Table BS. Cont'd. Pipe Diameter: Orifice Diameter: Orifice Mass Flow Number H COQNO‘UMFWNH 00000000A00 Rate kg/s .01923 -01814 .01171 .00988 .00798 .00705 .00933 .00664 .00669 .00344 107 0.0525 m. 0.00794 m. Pressure Mass Flow Pressure Pa Rate Pa kg/s 7358.4 0.02535 1226.4 0.02296 0.0185 0.01206 0.01036 0.0199 0.01106 0.00957 0.00909 735.8 0.00608 245.2 APPENDIX C Energy Loss Coefficient Due to Turbulence and the Corrected Orifice Discharge Coefficient for 5, 7.5 and 10% Starch Solutions 109 Table C1. Results of the energy loss coefficient due to the turbulence and the corrected orifice discharge coefficient for a 5% starch solution. Manifold Diameter: 0.0518 m. Orifice Diameter: 0.00318 m. Orif. Num. Press. k Veloc. C e C' Reo Pa f m/s 1 1363.8 635 1.00 0.455 1.329 0.605 140.33 2 855.6 74 0.89 0.437 1.544 0.675 121.60 3 758.0 0 0.64 0.381 1.341 0.511 81.45 4 698.6 0 0.63 0.377 1.381 0.521 79.42 5 652.9 0 0.37 0.279 1.132 0.316 42.54 6 616.0 0 0.41 0.298 1.212 0.361 48.09 7 589.9 0 0.31 0.245 1.116 0.273 33.91 8 570.9 0 0.25 0.210 1.064 0.223 26.52 9 560.2 0 0.19 0.171 1.001 0.171 19.37 10 556.4 0 0.07 0.070 1.000 0.070 5.72 1 2256.7 1042 1.29 0.484 1.257 0.608 190.86 2 1023.0 26 0.94 0.470 1.679 0.803 129.74 3 900.9 0 0.97 0.450 1.596 0.718 134.78 4 817.6 0 0.53 0.430 1.528 0.657 65.14 5 751.4 0 0.73 0.406 1.453 0.589 95.70 6 700.8 0 0.59 0.367 1.337 0.491 73.95 7 663.8 0 0.55 0.353 1.328 0.469 68.15 8 640.7 0 0.30 0.244 1.067 0.260 33.52 9 626.3 0 0.27 0.221 1.037 0.229 28.75 10 620.8 0 0.11 0.108 1.000 0.108 10.09 Manifold Diameter: 0.0518 m. Orifice Diameter: 0.00476 m. Orif. Num. Press. k Veloc. C e C' Reo Pa f m/s 1 2904.2 0 1.86 0.588 1.312 0.771 404.52 2 2280.4 33 1.59 0.584 1.263 0.738 336.61 3 1969.4 9 1.41 0.581 1.224 0.711 291.02 4 1528.6 44 1.18 0.573 1.175 0.673 234.13 5 1109.0 77 0.93 0.554 1.123 0.622 176.08 6 794.7 96 0.72 0.523 1.089 0.569 129.88 7 576.5 120 0.55 0.478 1.074 0.513 93.86 8 381.9 215 0.39 0.407 1.080 0.439 61.89 9 246.8 414 0.25 0.310 1.111 0.344 35.87 10 178.6 509 0.16 0.224 1.142 0.256 21.00 Table C1. Cont'd. Manifold Diameter: Orifice Diameter: Orif. Num. Press. kf Pa 1 1775.9 0 2 1429.3 30 3 1108.3 48 4 836.4 68 5 615.6 97 6 319.7 544 7 247.5 23 8 183.6 607 9 136.9 1085 10 113.7 3714 Manifold Diameter: Orifice Diameter: Orif. Num. Press. kf Pa. 1 1688.0 0 2 1226.6 0 3 871.2 0 4 592.9 0 5 398.8 0 6 272.0 0 7 209.0 0 8 134.2 185 110 0.0518 m. 0.00476 m. Veloc. 000000000-‘H 0.0518 m. 0.0079375 m. Veloc. 0000000-‘H m/s 0000000000 00000000 C 00HD-‘I-‘000 rdrdrahihuarahahud .980 .982 .990 .208 .092 .087 .770 .572 0000000000 00000000 Cl .701 .668 .622 .577 .527 .403 .370 .265 .159 .063 CI .667 .667 .673 .821 .736 .721 .456 .189 Re 0 269. 220. 175. 136. 101. 50. 41. 22. 10. 5. Re 0 1366. 1266. 1040. 1053. 728. 567. 274. 72. 03 111 Table C2. Results of the energy loss coefficient due to turbulence and the corrected orifice discharge coefficient for a 7.5% starch solution. Manifold Diameter: 0.0158 m. Orifice Diameter: 0.00318 m. Orif. Num. Press. kf Veloc. C e C' Reo Pa. m/s 1 13838.8 0 3.01 0.461 1.256 0.577 55.22 2 12443.9 0 2.76 0.459 1.228 0.563 50.23 3 8155.7 490 2.20 0.453 1.220 0.552 37.91 4 6221.8 11 1.87 0.445 1.199 0.533 30.75 5 5042.0 0 1.59 0.434 1.168 0.501 23.34 6 4771.2 0 1.52 0.430 1.158 0.498 23.93 7 4125.1 ' 0 1.33 0.417 1.125 0.469 20.36 8 3315.0 104 1.04 0.389 1.064 0.413 15.17 9 2809.6 77 0.85 0.356 1.008 0.359 11.52 10 2171.2 1829 0.53 0.274 0.947 0.259 6.56 Manifold Diameter: 0.0525 m. Orifice Diameter: 0.00318 m. Orif. Num. Press. kf Veloc. C e, C' Reo Pa. m/s 1 3246.9 0 1.06 0.389 1.077 0.418 15 39 2 3246.9 0 1.05 0.308 1.077 0.418 15 17 3 3246.9 0 0.86 0.355 0.961 0.331 11.94 4 3246.9 0 0.72 0.328 0.884 0.289 9.68 5 3246.9 0 0.80 0.344 0.928 0.319 10 90 6 3246.9 0 0.96 0.375 1.013 0.379 13 64 7 3246.9 0 0.96 0.377 1.019 0.384 13 64 8 3246.9 0 0.98 0.379 1.029 0.389 14 08 9 3246.9 ‘0 0.79 0.340 0.915 0.311 10 70 10 3246.9 0 1.02 0.384 1.054 0.405 14 73 Table C2. Cont'd Manifold Diameter: Orifice Diameter: Orif. Num. Press. kf Pa. 1 3336.0 0 2 3252.2 0 3 3101.0 90 4 2993.4 0 5 2817.3 0 6 2726.6 50 7 2650.2 0 8 2591.1 171 9 2548.0 0 10 2523.5 970 Manifold Diameter: Orifice Diameter: Orif. Num. Press. kf Pa. 000N00§WNH w 0‘ 0 0000000000 H 112 0.0409 m. 0.00476 m. Veloc. m/s 000HOHOHHH 0.0158 m. 0.00476 m. Veloc. m/s 0000000r-‘HN 0000000000 0000000000 C .486 .485 .484 .482 .479 .482 .477 .479 .476 .486 .486 .485 .470 .449 .275 .143 .085 .022 00000000HH F‘PHHPHPJP‘F‘hHHr‘ .163 .092 .973 .900 .863 .805 .885 .787 .799 .750 .251 -381 .420 .300 .092 .092 .000 .000 .000 0000000000 0000000000 CI .565 .530 .471 .435 .415 .386 .427 .371 .383 .357 .608 .671 .675 .642 .358 .156 .085 .022 Re 0 Re 0 Table C2. Cont'd. Manifold Diameter: Orifice Diameter: Orif. Num. Press. kf Pa. 1 1079.8 0 2 1129.9 0 3 1103.6 0 4 1103.6 0 5 1050.0 830 6 1093.5 0 7 1050.0 900 8 1002.5 1720 9 1023.2 0 10 997.1 1060 Manifold Diameter: Orifice Diameter: Orif. Num. Press. kf Pa. 1 461.1 0 2 710.9 0 3 530.0 240 4 347.2 520 5 257.0 90 6 177.3 280 7 141.9 0 8 124.4 0 9 102.0 0 10 90.8 0 113 0.0525 m. 0.00476 m. Veloc. 0000000000 m/s .79 .76 .74 .74 .68 .73 .68 .64 .66 .63 0.0409 m. 0.0079375 m. Veloc. m/s 0000000000 .72 .66 0000000000 0000000000 C .474 .472 .470 .470 .466 .470 .466 .461 .460 .517 .514 .508 .494 .476 .437 .366 .307 .261 hlhudrdhdhud rahlrndrdhahuaraha .467 .084 .111 .137 .135 .108 .053 .009 .000 0000000000 0000000000 CI .541 .515 .499 .497 .473 .494 .473 .457 .463 .454 C' .616 .628 .626 .578 .548 .470 .401 .335 .280 114 Table C2. Cont'd. Manifold Diameter: 0.0158 m. Orifice Diameter: 0.0079375 m. Orif. Num. Press. kf Veloc. C e C' Reo Pa. m/s 1 3385.0 0 2.07 0.522 1.544 0.806 71.04 2 2211.0 0 1.36 0.522 1.250 0.653 42.19 3 1114.6 0 0.68 0.515 1.000 0.515 18.24 4 275.6 441 0.17 0.343 0.679 0.235 3.37 5 55.6 2890 0.03 0.101 1.000 0.101 0.44 115 Table C3. Results of the corrected orifice discharge coefficient for a a 10% starch solution. Manifold Diameter: 0.0409 m. Orifice Diameter: 0.00318 m. Orif. Num. Press. Flow Rate Veloc. C e C' Reo Pa. kg/s m/s 1 10186.5 0.0114 1.39 0.270 1.051 0.284 12.76 2 9984.1 0.0088 1.07 0.230 1.051 0.242 9.35 3 9802.8 0.0095 1.15 0.234 1.064 0.249 10.01 4 9644.5 0.0054 0.66 0.162 0.962 0.156 4.81 5 9500.9 0.0092 1.12 0.237 1.088 0.258 9.62 6 9383.8 0.0053 0.64 0.158 0.969 0.153 4.61 7 9283.2 0.0069 0.85 0.195 1.007 0.196 6.65 8 9206.7 0.0072 0.88 0.201 1.024 0.206 7.04 9 9160.1 0.0055 0.67 0.164 0.989 0.162 4.89 10 9145.2 0.0008 0.10 0.029 0.901 0.026 0.38 Manifold Diameter: 0.0525 m. Orifice Diameter: 0.00318 m. Orif. Num. Press. Flow rate Veloc. C e C' Reo Pa. kg/s m/s 1 11647.4 0.0126 1.53 0.287 1.123 0.322 14.56 2 11522.9 0.0119 1.45 0.278 1.105 0.307 13.58 3 11410.1 0.0104 1.27 0.257 1.050 0.270 11.37 4 11308.1 0.0102 1.24 0.253 1.047 0.265 11.04 5 11217.3 0.0115 1.40 0.251 1 044 0.262 12.93 6 11138.0 0.0114 1.39 0.247 1 040 0.257 12.78 7 11070.9 0.0097 1.18 0.245 1.038 0.254 10.29 8 11016.9 0.0092 1.13 0.240 1.028 0.247 9.70 9 10977.3 0.0094 1.14 0.239 1.028 0.246 9.90 10 10954.9 0.0036 0.44 0.196 0.943 0.185 2.83 116 Table C3. Cont'd. Manifold Diameter: 0.0409 m. Orifice Diameter: 0.00476 m. Orif. . Num. Press. Flow Rate Veloc. C e C' Pa. kg/s m/s 1 7264.5 0.0197 1.07 0.263 1.010 0.266 2 6863.4 0.0186 1.01 0.255 1.090 0.278 3 6494.5 0.0178 0.97 0.247 1.100 0.272 4 6157.8 0.0131 0.71 0.221 1.070 0.236 5 5849.3 0.0143 0.78 0.188 1.030 0.194 6 5564.5 0.0089 0.48 0.176 1.030 0.181 7 5302.0 0.0126 0.69 0.151 1.200 0.181 8 5058.6 0.0074 0.40 0.129 1.100 0.142 9 4831.4 0.0027 0.15 0.052 0.950 0.049 10 4610.2 0.0002 0.01 0.004 0.950 0.004 Manifold Diameter: 0.0525 m. Orifice Diameter: 0.00476 m. Orif. Num. Press. Flow Rate Veloc. C e C’ Pa. kg/s m/s 1 3612.2 0.0191 1.04 0.258 1.520 0.392 2 3475.3 0.0880 4.78 0.148 1.246 0.184 3 3346.8 0.0161 0.88 0.232 1.485 0.345 4 3234.0 0.0140 0.76 0.217 1.465 0.318 5 3136.5 0.0140 0.76 0.211 1.468 0.310 6 3054.9 0 0153 0.83 0.224 1.528 0.342 7 2992.3 0.0124 0.68 0.193 1.554 0.300 8 2947.7 0.0125 0.68 0.194 1.466 0.284 9 2925.7 0.0045 0.24 0.083 1.232 0.102 10 2914.8 0.0030 0.16 0.058 1.195 0.069 Fwd Gnocnaxbwn~uc5ha HHO‘O‘QVVQ Table C3. Cont'd. Manifold Diameter: Orifice Diameter: Orif. Num. Press. Pa. 1942. 1639. 1388. 1174. 995. 847. 730. 642. 584. 547. COQNOUIJ-‘WNH OWHNNOONHN 0" Flow Rate kg/s .0336 .0263 .0150 .0134 .0127 .0109 .0086 .0081 .0046 .0047 0000000000 Manifold Diameter: Orifice Diameter:- Orif. Num. Press. Pa. 1 8725.5 2 4772.4 3 2640.0 4 1598.3 5 1110.0 6 985.2 Flow rate kg/s 0.1083 0.0392 0.0173 0.0062 0.0026 0.0004 m/s .66 .51 .29 .26 .25 .21 .17 .16 .09 .09 0000000000 m/s 2.12 0.77 0.34 0.12 0.05 0.01 117 0.0409 m. 0.0079375 m. Veloc. 0.0158 m. 0.0079375 m. Veloc. 0000000000 000000 C .323 .284 .215 .181 .174 .154 .127 .120 .073 .074 .434 .346 .218 .096 .043 .007 nardhdhndraCDCHdrd 00000H .010 .020 .940 .960 .030 .080 .110 .170 .170 .180 .188 :691 .711 .779 0000000000 000000 Cl .326 .290 .202 .174 .179 .166 .141 .140 .085 .087 .515 .253 .151 .068 .033 .005 Re 0 28.57 25.14 12.64 11.01 H WNUmWC U) \0 Re 129.78 33.96 11.53 2.99 0.96 0.08 APPENDIX D Listing of the Computer Program Used to Simulate the Manifold Flow Distribution 119 10'******************************************************************** ' MANIFOLD SYSTEM PROGRAM 20 ' VERSION Sept 1988 By Walter F. Salas Valerio Michigan State University Deparment of Agricultural Egineering ' Language : Basic 'THIS PROGRAM COMPUTES TTHE FLOW RATE AT EACH ORIFICE FROM A 'MANIFOLD SYSTEM BASED ON THE CHARACTERISTIC OF THE FLUID (NON- 'NEWTONIAN FLUID), MAIN PIPE 30 'AND ORIFICE DIAMETER. 'THE INPUT VARIABLES REQUIRED ARE: PIPE INSIDE DIAMETER, ORIFICE DIA., LENGTH OF THE PIPE, NUMBER OF ORIFICES, FLOW RATE AT THE ENTRANCE 50 'CHARATERISTIC OF THE FLUID (CONSISTENCE COEFFICIENT, FLOW BEHAVIORAL INDEX AND DENSITY). Q‘. 60 ' mmmm**m**********************************~k**********~k* 70 CLs 80 PRINT 9O AH$- -###############" 100 LOCATE 2, 1: PRINT AH$: PRINT 110 PRINT " DETERMINATION OF THE DISCHARGE DISTRIBUTION": PRINT 120 PRINT " IN A MANIFOLD SYSTEM": PRINT : PRINT 130 PRINT " BY WALTER F. SALAS VALERIO": PRINT 140 LOCATE 24, 1: INPUT "(PREES RETURN TO CONTINUE)", ZS: PRINT 150 CLS 160 ' MAIN MENU 170 PRINT ” MAIN MENU ": PRINT : PRINT 180 PRINT ” 1) INPUT PIPE CHARACTERISTICS": PRINT 190 PRINT ” 2) INPUT FLUID CHARACTERISTICS": PRINT 200 PRINT " 3) INPUT EXPERIMENTAL DATA": PRINT 201 PRINT ” 4) EXIT OF THIS PROGRAM": PRINT 210 INPUT "(CHOOSE 1, 2, 3.0R 4)", IMM: PRINT 211 IF IMM - 1 THEN GOTO 220 212 IF IMM - 2 THEN GOTO 220 213 IF IMM 3 THEN GOTO 220 214 IF IMM - 4 THEN GOTO 5000 220 UPR$ - ”N / m22': ULE$ - "m": UVE$ - "m/s” : UMF$ - "kg/s": UTE$ - ”C”: UCC$ a ”N San / mA2": UDE$ - "kg / m 3": CC - l: URS$ - "l/s": ULG$ - ”m": UDIS - m": UEE$ - "m” : DI$ - "m": ULK$ - "m": ULCHS - "m": RH$ - ”m” 230 UMFQ$ - "kg/s”: UEE$ - ”m": PR$ - "Pa": Q$ - "kg/s" 240 CLS 250 INPUT " FLUID NAME : ”, FLNAS: LOCATE 2, 58 270 PRINT ”DATE : "; DATE$ 290 PRINT ' TEMPERATURE ("; UTE$; ") z”; 310 INPUT ” ', TEMP$: LOCATE 3, 58: PRINT "TIME : "; TIMES 320 PRINT 340 PRINT " Manifold characteristic (manifold pipe)" 360 PRINT ' Length of the pipe ("; ULGS; ") z", 380 INPUT ULC: LOCATE 6, 43: PRINT USING "######. ###"; ULG 400 PRINT " Manifold diameter("; UDI$; z", , 420 INPUT UDI: LOCATE 7, 43: PRINT USING) " .####"; UDI 440 PRINT " Number of orifices z", , 460 INPUT UNT: LOCATE 8, 43: PRINT USING ”######"; UNT 120 470 PRINT ” Distance b. orifice ("; UEE$; z", 490 INPUT UEE: LOCATE 9,43: PRINT USING "######. ###"; UEE 500 PRINT ' 510 PRINT ' 530 PRINT “ Fluid characteristic " 550 PRINT " Flow behavior index z", 570 INPUT N: LOCATE 13,43: PRINT USING "######. ###"; N 590 PRINT ' Consistency coefficient ("; UCC$; ") :”; 610 INPUT K: LOCATE 14, 43: PRINT USING ”######.###"; K 650 INPUT YS: LOCATE 15, 43: PRINT USING "######.###"; YS 670 PRINT n Fluid density (a; UDES; ") :", 690 INPUT DE: LOCATE 16, 43: PRINT USING "#####,.###"; DE 710 PRINT ' Diameter of the orifice ("; ULES; ") :", 730 INPUT DI: LOCATE 17, 43: PRINT USING "######.####"; DI 760 PRINT " Flow rate at the entrance "; UMFQ$; ") z", 780 INPUT UMFQ: LOCATE 18, 43: PRINT USING "######.####"; UMFQ 800 PRINT " Pressure at the entrance ("; PRS; ") z", 820 INPUT PR: LOCATE 19, 43: PRINT USING "######.####"; PR 830 CLS 910 CLS 930 ' 940 PRINT " COMPUTING..." 960 PRINT "Orifice Ratio Press. Flow rate Total F.R 980 PRINT " No (x/L) (Pa ) Kg /s Kg/s " 990 ULC - 1 1260 MAX - 10: ER - .000001: NOROOT - 1: PI - 3.141592 1270 QQ - UMFQ 1280 '**** PRELIMINARIES COMPUTATIONS *** 1290 '*** AREA OF THE PIPE *** 1300 AT - PI * UDI A 2 / 4 1320 '*** AREA OF THE ORIFICE *** 1330 AP - PI * DI A 2 / 4 1340 ' 1345 ' **** FLUID VELOCITY IN THE ORIFICE *** 1348 ' 1350 GOTO 1390 1360 CLS 1440 ' PE is specific weight in N/m23 . 1450 ' PR is pressure drop in the orifice in N/m22. 1460 ' 1461 '**** FLUID VELOCITY IN THE PIPE ***** v - UMFQ / (AT * DE): '**** CALCULATION OF THE ENERGY LOSS DUE TO THE FRICTION *** ' GOSUB 2230 I465 EFP - 2 * FRI.FAC * V A 2 * UEE / UDI '**** CALCULATION OF THE ENERGY LOSS DUE TO TURBLULENCE ****** KF - 281.2 * RE A (-.97) + 148.4 EFK - RF * (QQ / (AT * 03)) ‘ 2 / 2 1466 '*** PRESSURE AT THE ORIFICE ***** 1467 PRPI - (PRP / DE - EFP - EFK) * DE 1468 '*** FLUID VELOCITY IN THE ORIFICE **** 1469 vv - QQ / (AP * DE) '** CALCULATION OF THE C' VALUE **** 1470 1480 1490 1540 1590 1610 1630 1650 1680 1700 1710 1730 1740 1750 1770 1820 1840 1860 1880 1900 1920 1950 1960 1970 1980 1995 2010 2040 2050 2060 2090 2100 2110 2120 2130 2140 2150 2170 2190 2220 2230 2250 2260 2270 2310 2320 121 GOSUB 4110 '** PRESSURE IN THE FIRST ORIFICE *** FR - (QQ / (AF * c * DE)) * 2 * DE / 2 LOCATE 23, 1: PRINT PR, PRPl, QQ, EFK IF PR - PRP1 < 5 THEN GOTO 1490 00 - QQ - .0001#: GOTO 1465 FRP - PRPl . '*** CALCULATE THE PRESSURE AT THE NEXT ORIFICES *** XX - UEE + .1 QQQ - QQ v0 - v FOR I - 2 TO UNT '*** MASS BALANCE IN THE ORIFICE *** v - v0 - QQ / (AT * DE) ' *** CALCULATION OF THE FRICTION FACTOR *** GOSUB 2230 '**** CALCULATION OF THE ENERGY LOSS COEFFICIENT **** *** DUE TO TURBULENCE **** - 281.2 * RE (- .97) + 148. 4 '*** CALCULATION OF THE ENERGY LOSSES DUE TO FRICTION *** EFF - 2 * FRI.FAC * UEE * v ‘ 2 / UDI '**** CALCULATION OF THE ENERGY LOSS DUE TO THE TURBULENCE *** EKF - KF * (QQ / (AT * DE)) ‘ 2 / 2 '*** PRESSURE IN THE ORIFICE *** FRF1 - (PRP / DE - EFF - EFR) * DE '*** FLOW RATE IN THE NEXT ORIFICE *** vv - QQ / (AP * DE) '*** CALCULATE THE CORRECTED ORIFICE DISCHARGE COEFFICIENT *** GOSUB 4110: ' '**** PRESSURE IN THE ORIFICE *** PR - (QQ / (DE * c * AP)) ‘ 2 * DE / 2 '*** COMPARE THE CALCULATED PRESSURES **** IF PR - FRF1 < 3 THEN GOTO 2090 00 - QQ - .00001 GOTO 1870 QQQ - QQQ + QQ FRF - PRPl 'LPRINT 1+1 ,:LPRINT USING "##.###“‘ ":XX/ULG ,:LPRINT USING "##.###"; PRP,: LPRINT USING "##.###"; QQ,:LPRINT USING '#.###";QQQ LOCATE I + 3, 2: PRINT 1: LOCATE I + 3, 10: PRINT USING "#.#"; xx / ULG, : LOCATE I + 3, 18: PRINT USING "######.#"; PRP, : LOCATE I + 3, 30: PRINT.USING "#.####"; QQ, LOCATE I + 3, 40: PRINT USING "#.###"; QQQ, : LOCATE I + 3, 50: PRINT USING "#.###"; c: LOCATE I + 3, 60: PRINT USING "####.#"; RF XX - XX + UEE v0 - v NEXT I FOR I - 1 TO 500: NEXT I END 'SUBROUTINE P1 - 1 + 3 * N. P2 - 1 + 2 * N: P3 - 1 + N: P4 = 2 + N PS - 2 / N - 1: P6 - 2 / N + 1: P7 - 2 / N - 2 P8 - 16800 * SQR(1 / 27) * P4 (P4 / F3) / N '*** COMPUTE THE GENERALIZED REYNOLDS NUMBER AT ANY POINT *** LOCATE 18, 3: PRINT RE, v RE - 8 * DE * (N / P1) ‘ N * (UDI / 2) “ N * v “ (2 - N) / (K * CC) 2330 2350 2360 2370 2380 2390 2400 2410 2420 2430 2440 2450 2460 2470 2480 2490 2510 2520 2540 2550 2560 2580 2590 2600 2620 2630 2640 2650 2660 2680 2700 2710 2720 2730 2740 2750 2760 2770 2780 2790 2800 2820 2830 2840 2860 2870 2880 122 '*** COMPUTE THE GENERALIZED HEDSTROM NUMBER *** UNI - 2 IF YS - 0 THEN HE - 0: UPR - 0: UPR.CR - 0 ELSE HE - DE * UDI A * (YS/K)“(2/N)/(YS*GC) GOTO 2390 '***COMPUTE THE CRITICAL UNSHEARED PLUG RADIUS THROUGH ITERATION X0 - 0: X1 - .999: m - 1 GOSUB 3230 IF NOROOT - 1 GOTO 2480 LOCATE 14, l PRINT ' THE CRITICAL UNSHEARED PLUG RADIUS WAS NOT FOUND IN THE RANGE"; X0; "TO"; X1 PRINT ' WHAT IS THE NEW RANGE (X0,X1): WARNING: 0 <- X0,X1 < 1 " INPUT " X0 - '; X0 INPUT ' X1 - '; X1 NOROOT - 1: GOTO 2400 UPR.CR - X '*** CALCULATON OF THE FRICTION FACTOR **** *** COMPUTE CRITICAL PSI *** GOSUB 3740 ‘ PSI.CR - LF *** COMPUTE THE CRITICAL REYNOLDS NUMBER *** RECl - 2 * P8 * N A 2 * PSI.CR ‘ P5 REC2 - P1 ‘ 2 * (1 - UPR.CR) ‘ P6 RE.CR - RECl / REC2 *** COMPUTE THE CRITICAL FRICTION FACTOR *** FF.CR - 16 / (RE.CR * PSI.CR) I IF RE <> RE.CR GOTO 2660 *** THE FLOW IS CRITICAL *** FLW.CON$ - ' CRITICAL" FRI.FAC - FF.CR UPR - UPR.CR GOTO 3030 , IF RE > RE.CR GOTO 2900 *** THE FLOW IS LAMINAR *** FLW.CON$ - " LAMINAR” *** COMPUTE THE UNSHEARED PLUG RADIUS TRHOUGH ITERATION *** IF HE - 0 THEN PSI - l: GOTO 2860 m - 2: X0 - UPR.CR: X1 - .999 GOSUB 3230 IF NOROOT - 1 GOTO 2800 LOCATE 14, 1 PRINT " THE UNSHEARED PLUG RADIUS WAS NOT FOUND IN THE RANGE"; X0; "TO"; X1 PRINT ” WHAT IS THE NEW RANGE (X0,X1): WARNING: 0 <- X0,X1 < 1 " INPUT " X0 - ”; X0 INPUT ' X1 - "; X1 NOROOT - l: GOTO 2720 UPR - X '*** COMPUTE PSI *** GOSUB 3740 PSI - LF *** COMPUTE THE LAMINAR FRICTION FACTOR *** FRI.FAC - 16 / (RE * PSI) GOTO 3030 '*** THE FLOW IS TURBULENT *** 2 2900 2910 2920 2930 2940 2950 2960 2970 2980 2990 3000 3010 3020 3030 3040 3050 3190 3200 3210 3220 3230 3240 3250 3260 3270 3280 3290 3300 3310 3320 3330 3340 3350 3360 3380 3390 3400 3410 3420 3430 3440 3450 3460 3470 3480 3490 123 FLW.CON$ - " TURBULENT" E0 - 16 * (2 * HE) ‘ (N / (2 - N)) * (N / F1) “ (2 * N / (2 - N)) / F0 - 2 * YS / (DE * v A 2): m - 3: X0 - F0 + .00001: X1 - 1 GOSUB 3230 IF NOROOT - 1 GOTO 3010 LOCATE 14, 1 PRINT ' THE TURBULENT FRICTION FACTOR WAS NOT FOUND IN THE RANGE '; X0; ”TO”; X1 PRINT " WHAT IS THE NEW RANGE (X0,X1): WARNING: '; F0; " < X0,X1 " INPUT " X0 - '; X0 INPUT ' X1 - ”; X1 NOROOT - l: GOTO 2930 FRI.FAC - X UPR - E0 / FRI.FAC wss - FRI.FAC * DE * v “ 2 / (2 * GC) WSR - ((wss / R) * (1 - UPR)) ‘ (1 / N) '************************************************************ SUBROUTINE : ROOT FINDING-1. THIS SUBROUTINE IS A COMBINATION OF THE BISECTION AND NEWTON ROOT FINDING ITERAION. THE ' MIDPOINT OF THE INITAL INTERVAL IS USED TO START THE NEWTON ITERATION. THE PROGRAMS CONTINUES WITH THIS METHOD UNTIL THE SOLUTION IS FOUND OR ONE OF THE ' FOLLOWING SITUATIONS OCCURS: THE DERIVATIVE OF THE FUNCTION IS EQUAL TO ZERO; 2- X FALL OUTSIDE THE INTERVAL KNOWN TO CONTAIN THE SOLUTION; 3- THE ' DIFFERENCE IN SUCCESIVE APROXIMATION DOES NOT DECREASES: 47 THE NUMBER OF ITERATION EXCEEDS MAX. IF ANY OF THE ' ABOVE SITUATIONS HAPPEN, THE PROGRAM SWITCHS TO THE BISECTION METHOD TO OBTAIN A SMALLER INTERVAL. XA - X0: XB - X1 IF XA > XB THEN SWAP XA, XB X - XA ON m GOSUB 3640, 3800, 3950 FA - Y IF FA - 0 THEN RETURN' ROOT HAS BEEN FOUND X - XB ON m GOSUB 3640, 3800, 3950 FB - Y IF FB - 0 THEN RETURN' ROOT HAS BEEN FOUND IF FA * FB > 0 THEN NOROOT - 0: RETURN' ROOT WAS NOT FOUND XM - (XA + X3) / Z OLDDIF - ABS(XA34 XB) / 2 x - XM . *** NEWTON ITERATION *** FOR J - 1 TO MAX OLDX - x 0N m GOSUB 3680, 3850, 4020 YPRIME - Y IF YFRIME - 0 THEN GOTO 3510 ON m GOSUB 3640, 3800, 3950 x - x - Y / YFRIME DIFF - ABS(X - OLDX) IF DIFF <- ABS(X * ER) THEN RETURN' ROOT HAS BEEN FOUND IF DIFF >- OLDDIF THEN GOTO 3510 OLDDIFF - DIFF NEXT 3680 3690 3700 3710 3720 3730 3740 3750 3760 3770 3780 3790 3800 3810 3820 3830 124 *** BISECTION ITERATION *** X - XM ON m GOSUB 3640, 3800, 3950 FM -'Y IF FM - 0 THEN RETURN' ROOT HAS BEEN FOUND IF FA * FM <- 0 THEN GOTO 3590 (XA+XB)/2 IF ABS(XA - XB) > ABS(XM * ER) THEN GOTO 3350 ELSE X - XM: RETURN '************************************************************* ' FUNCTION SUBROUTINE : CRITICAL UNSHEARED PLUG RADIUS EQUATION WRITEN AS Y-FUNC.(X)-O Y - HE - F8 * X ‘ F5 / (1 - X) ‘ F6 RETURN '************************************************************* FUNCTION SUBROUTINE : DERIVATIVE OF THE CRITICAL UNSHEARED PLUG RADIUS EQUATION WITH RESPECT TO THE CRITICAL Y1 - (2 - N) * X ‘ F7 / (1 - X) ‘ P6 Y2 - F4 * X ‘ F5 / (1 - X) ‘ (2 / N + 2) Y - -P8 * (Y1 + Y2) / N RETURN '***************************************************************** ' FUNCTION SUBROUTINE : LAMINAR FUNCTION PSI LF1 - 1 - X LF2 - 2 * X * LF1 * P1 / F2 LF3 - X ‘ 2 *‘F1 / P3 LF - LF1 “ F3 *.(LF1 ‘ 2 + LF2 + LF3) “ N RETURN '************************************************************ ' FUNCTION SUBROUTINE : UNSHEARED PLUG RADIUS EQUATION WRITEN AS Y-FUNC.(X)-0 GOSUB 3740 Y - RE * X ‘ P5 - 2 * HE * LF ‘ P5 * (N / P1) ‘ 2 RETURN '****************************************************************** 3840 3850 3860 3870 3880 3890 3900 3910 3920 3930 3940 FUNCTION SUBROUTINE : DERIVATIVE OF THE UNSHERED PLUG RADIUS ' EQUATION WITH RESPECT TO THE UNSHEARED PLUG RADIUS Y1 - 1 - X Y2 - P1 * F3 * Y1 ‘ 2 + 2 * F2 * P1 * Y1 * X + P1 * P2 * P3 * X “ 2 Y3 - F2 * P3 * Y1 ‘ 3 + 2 * P1 * P3 * Y1 ‘ 2 * X + P1 * P2 * Y1 * X “ 2 SIGMMA - Y2 / Y3 GOSUB 3740 Y4 - 2 * HE * P5 * (N / P1) ‘ 2 * SIGMMA * LF ‘ P5 Y5 - P5 * RE * X ? P7 Y - Y4 + Y5 RETURN '****************************************************************** FUNCTION SUBROUTINE : FRICTION FACTOR EQUATION FOR TURBULENT FLOW WRITEN AS Y-FUNC.(X)=0 125 3950 Y1 - .45 - 2.75 / N + 1.97 * LOG(1 - EO / X) / N 3960 Y2 - (P1 / (4 * N)) ‘ N 3970 Y3 - 1.97 * LOG(RE * Y2 * X 2 (1 - N / 2)) / N 3980 Y - Y1 + Y3 - 1 / SQR(X) 3990 RETURN 4000 ' WWW*WW*WW**************** 4010 'FUNCTION SUBROUTINE : DERIVATIVE OF THE FRICTION FACTOR EQUATION FOR TURBULENT FLOW WITH RESPECT TO THE FRICTION FACTOR 4020 Y1 - 3.94 * E0 * SQR(X) / X + N * (1 - E0 / X) 4030 Y2 - 3.94 * (1 - N / 2) * (1 - E0 / X) * SQR(X) 4040 Y3 - 2 * N * (1 - E0 / X) * X “ 1.5 4050 Y - (Y1 + Y2) / Y3 4060 RETURN 4070 ' WW**W*MW*****WM******************************** 4080 IF (Q1 - QQQ) >- .002 THEN GOTO 1520 4090 v****************************************************************** 4100 ' 4110 ' FUNCTION SUBROUTINE : CALCULATE THE DISCHARGE COEFFICIENT AT 4120 ' THE ORIFICE 4130 REO - (DI ‘ N * vv ‘ (2 - N) * DE) / (8 ‘ (N - 1) * K) * (4 * N / (1 + 3 * N)) ‘ N 4140 ' CALCULATE THE GENERALIZED REYNOLDS NUMBER IN THE 4150 ' IN THE ORIFICE 4160 ' IF DI - .00318 THEN GOTO 4170 ' IF DI - .00476 THEN GOTO 4170 4170 c - .59 * (1 - EXP(-.071 * REO)) + .027: GOTO 4220 4220 RETURN , 4230 '****************************************************************** 5000 CLS 5001 FOR III - 1 TO 50 5002 PRINT " BYE!" 5003 NEXT III 5004 SYSTEM