MEB and Wellbore Performance
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This document discusses wellbore performance and flow modeling. It covers: 1) Single phase liquid, gas, and two phase flow models based on mechanical energy balance equations. Pressure drops are calculated considering elevation change, kinetic energy, and friction. 2) Methods for calculating friction factors including Fanning, Darcy, and Moody charts. Correlations for gas properties like viscosity and deviation factor are also presented. 3) Examples of calculating pressure drops in single phase liquid and gas flows. Numerical methods for solving gas flow equations are described. 4) Multiphase flow is more complex due to different flow regimes affecting pressure gradients. Models include homogeneous and separated flow approaches.
MEB and Wellbore Performance
- 1. WELLBORE PERFORMANCE Single phase Liquid flow, Gas flow, Two phase flow, MEB Notes from Petroleum Production Engineering by Boyun Guo; Petroleum Production Systems by Economides et al.; & Morrison, F. Dr. Ajay Suri, Associate Professor IIT (ISM) Dhanbad July 31, 2017 1
- 2. Application to Oil/Gas Wells • Wellbore performance establishes a relationship between well tubular sizes, wellhead and bottom-hole pressures, fluid properties and production rates • The production rate is determined by • Wellhead pressure • Geometry of production string and its components (tubing, casing or both) • Properties of produced fluids (oil, water, gas and sand) • Constraint on production rate or flowing pressures, to avoid coning/sanding July 31, 2017 2
- 3. Objectives • Understanding wellbore performance is important for production engineers • Designing well equipment • Optimizing production conditions • Oil can be produced from tubing, casing or both depending upon which flow path has better performance • Tubing is a better option in most cases to take advantage of gas lift effect July 31, 2017 3
- 4. Names for Wellbore Performance • Traditional terms – “tubing performance relationship” (TPR) and “vertical lift performance” (VLP) • Math models are valid for casing-tubing annular flow as long as hydraulic diameter is used • Excel sheets of the theory presented in this presentation is available at • http://books.elsevier.com/companions/9780750682701 July 31, 2017 4
- 5. Single-Phase Liquid Flow • Water or water based fluids (ex. polymer solutions) is being injected or produced • When oil is produced with wellhead pressure above the bubble-point pressure July 31, 2017 5
- 6. Flow Along a Tubing String July 31, 2017 6 Consider fluid flowing from point 1 to point 2 in a tubing of length L and height Dz P1, v1, h1 P2, v2, h2 = h2 - h1
- 7. From Mechanical Energy Balance • MEB in U.S. field units results in the following equation July 31, 2017 7 Pressure reqd. to lift the fluid Pr. reqd. to increase the velocity Pr. Reqd. to overcome friction/viscous effects Excess pr. at bottom of the string/hole compared to the top of the string
- 8. Pressure Drop in Tubing – Single-Phase Liquid • Pressure drops due to increase in elevation, kinetic energy, and friction loss • fF (Fanning friction factor) is based on Reynolds number and relative roughness • Pressure was already in lbf, hence it wasn’t divided by gc July 31, 2017 8 Du = velocity increase, ft/s
- 9. Fanning Friction Factor (Wiki, 2018) July 31, 2017 9 • Fanning friction factor is a local parameter defined as the ratio of local shear stress (wall force per unit wall area) to local flow kinetic energy density • f = local Fanning friction factor, dimensionless • t = local shear stress, Pa • u = bulk flow velocity, volumetric rate / area, m/s • r = density of the fluid, kg/m3 f = t ru2 2 Eq. 1
- 10. • Shear stress at wall is related to pressure loss • Dp = p1 - p2, pressure loss • L = length of pipe • r = radius of pipe Shear Stress and Pressure Loss Relationship (Wiki, 2018) July 31, 2017 10 Dppr2 =t2prL p1 p2 L t Two forces are equated that act on the fluid in the pipe 1. Pressure force by outside fluid at inlet and outlet 2. Friction force by pipe walls t = Dpr 2L Eq. 2Friction force Pressure force Shear stress and velocity profiles
- 11. Pressure Loss from Fanning Friction Factor (Wiki, 2018) July 31, 2017 11 • Using t from eq. 2 in eq. 1, we can calculate frictional pressure loss as • Fanning friction factor is 1/4th of Darcy friction factor Dp = 2 fLru2 D f = t ru2 2 = Dpr 2L 1 ru2 2
- 12. Pressure Loss in a Pipe (Wiki, 2018) July 31, 2017 12 • Cylindrical pipe of diameter D, flowing full, pressure loss due to viscous effects is given by Darcy-Weisbach eq. • Dp/L = frictional pressure loss per unit length, Pa/m • r = density of fluid, kg/m3 • <v> = mean flow velocity (volumetric rate / cross-sectional area), m/s • fD = Darcy friction factor, dimensionless
- 13. MEB with Friction • Its relationship to pressure drops, flow rates, and geometric factors may be understood/calculated using MEB July 31, 2017 13
- 14. Ex – Newtonian Fluid Steady State Flow F July 31, 2017 14
- 15. F for Newtonian Fluid Steady State Flow July 31, 2017 15 Experimentally we can measure frictional pressure loss using the above method
- 16. F for Newtonian Fluid in Steady State Flow • Data for various flow rates, tube lengths and diameters, fluid densities and viscosities could be tabulated and published • Dimensional analysis makes the collection and reporting of pressure drop and flow rate data more accessible and rational • Darcy / Fanning friction factors, dimensionless wall force may be used to correlate friction in pipes with Re (dimensionless flow rate) July 31, 2017 16
- 17. f for Newtonian Fluid in Steady State Laminar Flow • Dimensional analysis tells that f is a function of Re only • F can be determined for any fluid in any tube July 31, 2017 17 Ratio of inertial forces to viscous forces
- 18. ff and NRe for Steady State Laminar Flow July 31, 2017 18
- 19. Darcy/Moody Friction Factor, fM or fD July 31, 2017 19 Moody friction factor, fM is also referred to as Darcy- Weisbach friction factor, fD
- 20. Fanning Friction Factor, ff • For Re > 2,100 • Chen’s (1979) corelation has explicit form • Similar accuracy as Cole-brook-White equation used to generate friction factor chart in petroleum industry • d is the absolute roughness of the pipe wall, inch with d in inches too July 31, 2017 20
- 21. Example Problem – Single Phase Liquid Flow in a Pipe July 31, 2017 21 Tubing I.D. = 2.259 inches for the given tubing
- 22. Example Solution – Single Phase Liquid Flow in a Pipe July 31, 2017 22
- 23. Example Solution – Single Phase Liquid Flow in a Pipe July 31, 2017 23
- 24. Example Solution – Single Phase Liquid Flow in a Pipe July 31, 2017 24
- 25. Example Solution – Single Phase Liquid Flow in a Pipe July 31, 2017 25
- 26. Example Solution – Single Phase Liquid Flow in a Pipe July 31, 2017 26 Elevation component = 49816.6 lbf/ft2, 345.9 psi, 98.8 % Friction component = 618.64 lbf/ft2, 4.3 psi, 1.2 %
- 27. Single-Phase Gas Flow • Same MEB governs gas flow in tubing • Kinetic energy change is taken negligible because tubing dia. is almost constant July 31, 2017 27 fM = Moody friction factor = 4*fanning friction factor d(v2 ) 2gc +
- 28. Single-Phase Gas Flow • Ordinary differential equation • With z, T, P varying with tubing length • T can be approximated from linear geothermal gradient • z is a function of both P and T • Hence analytical solution difficult • P is not a strong function of T and z • Approximate solutions sought and used in gas industry July 31, 2017 28
- 29. Single-Phase Gas Flow – Avg. T and z • If single avg. T and z is used over entire tubing length, • By separation of variables, eq. is integrated over L July 31, 2017 29
- 30. Single-Phase Gas Flow – Avg. T and z July 31, 2017 30 • Avg. z is function of P (dependent variable) itself, numerical iterative method, like trial and error or Newton- Raphson method is required. • Example program - AverageTZ.xls
- 31. NRe, Reynolds Number for Gas Flow July 31, 2017 31 NRe = 20.09gg qsc Dm • gg = gas specific gravity, gas M.W. / 28.97 (air M.W.) • qsc = gas rate at standard conditions, Mscf/D • D = diameter of pipe, inch • m = viscosity of gas, cp
- 32. Friction factor for Gas Flow • Moody (Darcy-Wiesbach) friction factor calculated conventionally (ex. chen’s correlation) • For fully turbulent flow (in most gas wells), simpler relation by Katz and Lee (1990) can be used July 31, 2017 32
- 33. Simplified Friction factors for Gas Flow • Guo (2001) used Nikuradse friction factor for fully turbulent flow in rough pipes July 31, 2017 33
- 34. Single-Phase Gas Flow – Avg. T and z July 31, 2017 34
- 35. Carr et al. Corelation for Gas Viscosity July 31, 2017 35
- 36. Carr et al. Corelation for Gas Viscosity July 31, 2017 36
- 37. Carr et al. Corelation for Gas Viscosity July 31, 2017 37
- 38. Brill and Beggs Corelation for Gas Deviation factor (Z factor) July 31, 2017 38
- 39. More Accurate Corelation for Gas Deviation factor (Z factor) July 31, 2017 39
- 40. More Accurate Corelation for Gas Deviation factor (Z factor) July 31, 2017 40
- 41. Single-Phase Gas Flow – Avg. T and z July 31, 2017 41 Pressure gradient = 0.21 psi/ft
- 42. Single-Phase Gas Flow – Avg. T and z July 31, 2017 42 m = 0.013 cp (Carr et al. corelation) Re ~ 106 fM ~ 0.018 (from Moody chart)
- 43. Single-Phase Gas Flow – Kinetic pr. drop July 31, 2017 43
- 44. • The original equation can be solved by a fast numerical algorithm by Cullender and Smith • Rearranging the above eq. Single-Phase Gas Flow – Cullender and Smith Method July 31, 2017 44
- 45. • If qsc is in MMscf/d (U.S. field units) Single-Phase Gas Flow – Cullender and Smith Method July 31, 2017 45
- 46. Single-Phase Gas Flow – Cullender and Smith Method July 31, 2017 46 • Integrant is denoted with symbol I • Integrating numerically, with pmf is pressure at mid-depth and Imf is integrant evaluated at it
- 47. Single-Phase Gas Flow – Cullender and Smith Method July 31, 2017 47 • Assuming both terms are half of the right hand side Imf is a f(pmf), hence numerical technique such as N-R iteration is required for pmf Iwf is a f(pwf), hence N-R iteration required for pwf
- 48. Single-Phase Gas Flow – Cullender and Smith Method July 31, 2017 48
- 49. Single-Phase Gas Flow – Cullender and Smith Method July 31, 2017 49
- 50. Multiphase Flow in Oil Wells July 31, 2017 50 • Almost all oil wells produce certain amount of water, gas, and sometimes sand • These are called multiphase-oil wells • TPR for single-phase flow isn’t valid for multi-phase oil wells, rigorously, a multiphase flow model is required • Complicated flow regime / pattern in the well in multiphase flow • Fluid distribution changes greatly in different flow regimes, significantly affecting the pressure gradient
- 51. Tubing Performance Relationship (TPR) Models • Numerous TPR models have been developed for vertical pipes • Brown (1977) presented a review of the models • Two categories • Homogeneous-flow models • Separated-flow models • Homogeneous treats multiphase as a homogeneous mixture and do not consider liquid holdup (no slip between flowing phases) July 31, 2017 51
- 52. TPR – Homogeneous-Flow Models • Less accurate and usually calibrated with local operating conditions in field applications • Can handle gas-oil-water 3-phase and gas-oil-water-sand 4 phase systems. • Easy to code July 31, 2017 52
- 53. TPR – Separated-Flow Models • More realistic • Empirical corelations • Effect of liquid holdup (slip) and flow regime considered • Difficult to code because corelations are graphs July 31, 2017 53
- 54. TPR – Homogeneous-Flow Models • Pioneers were Poettmann and Carpenter (1952). Used two-phase friction factor without considering the effect of liquid viscosity • Cicchitti (1960) and Dukler et al. (1964) considered liquid viscosity • Hasan and Kabir (2002) reviewed the above models • Guo and Ghalambor (2005) presented work on gas-oil- water-sand 4 phase flow July 31, 2017 54
- 55. TPR – Homogeneous-Flow Models • With no slip, P&C presented a simplified gas-oil-water 3 phase flow model to compute pressure losses by estimating mixture density and friction factor • Acceleration term is neglected July 31, 2017 55
- 56. July 31, 2017 56 lb 350.17 lb is the mass of 1 bbl of water (S.G. = 1) 0.0765 lb/ft3 is the air density at 14.7 psi and 60 F Mass of the MixtureAssociated with 1 STB of Oil scf/stbstb of water / stb of oil
- 57. Volume of MixtureAssociated with 1 STB of Oil July 31, 2017 57 Oil Volume = 1 STB Associated water Volume = WOR (bbl/STB) Associated Gas Volume = GOR (scf/stb) Stock Tank Conditions At higher PT conditions (anywhere in the wellbore) Oil Volume = Bo bbls Associated water Volume = 5.615*WOR*Bw, cf Associated Gas Volume = (GOR-Rs)*Bg, cf/stb Same Mass of fluids
- 58. P&C Homogeneous Flow Model for TPR July 31, 2017 58 lb/bbl of water lbm cuft.
- 59. Rs and Bo Corelations July 31, 2017 59
- 60. Gas Density July 31, 2017 60 In field units Note density of air at 14.7 psi and 60 F is 0.0763 lbm/cu.ft.(1.23 kg/m3)
- 61. Nomenclature July 31, 2017 61
- 62. Friction Factor Term in P&C Model July 31, 2017 62
- 63. Poettman-Carpenter Model - Comments • Accurate for short depth increments • For deep wells, well length should be broken in segments • Excel spreadsheet Poettman-CarpenterBHP.xls is available July 31, 2017 63
- 64. Poettman-Carpenter Example Problem July 31, 2017 64
- 65. Poettman-Carpenter Example Solution July 31, 2017 65
- 66. Poettman-Carpenter Example Solution July 31, 2017 66
- 67. Guo-Ghalambor (2005) 4 Phase Model • Gas-oil-water-sand 4 phase model is similar to gas-oil- water 3 phase flow model by P-C with no slip July 31, 2017 67
- 68. Guo-Ghalambor (2005) 4 Phase Model July 31, 2017 68 Excel program – Guo-GhalamborBHP.xls
- 69. Guo-Ghalambor (2005) 4 Phase Model July 31, 2017 69
- 70. July 31, 2017 70
- 71. July 31, 2017 71
- 72. Flow Regimes in Gas-Liquid Two Phase Flow July 31, 2017 72 At least 4 flow regimes are identified in vertical flow 1. Bubble 2. Slug 3. Froth/Churn 4. Annular/Mist Occurs in progression with increasing gas flow rate at a given liquid flow rate
- 73. Flow Regimes Described • Bubble flow – gas is dispersed in the form of small bubbles in a continuous liquid phase • Slug flow – gas bubbles coalesce into large bubbles that eventually fill the entire pipe cross-section. Between the large bubbles, are slugs of liquid that contain smaller bubbles of entrained gas • Churn flow – larger gas bubbles become unstable and collapse, resulting in a turbulent flow pattern with both phases dispersed, & oscillatory up and down motions of the liquid • Annular flow – gas becomes the continuous phase, with liquid flowing in an annulus, coating the surface of the pipe with droplets entrained in the gas phase in the middle July 31, 2017 73
- 74. Liquid Holdup • Amount of the pipe occupied by a phase is often different from its proportion of the total volumetric flow rate • This is due to density difference between phases • Dense phase slips down in an upward flow (lighter phase moves faster than denser phase) • In situ volume fraction of denser phase will be greater than its input volume fraction (i.e. it is held up in the pipe relative to the lighter phase) July 31, 2017 74
- 75. Liquid Holdup • Liquid holdup is defined as • Liquid holdup, yL, depends on flow regime, fluid properties, pipe size and configuration • Value can be determined only through experiments July 31, 2017 75
- 76. Number of Separated-Flow Models for TPR • Lockhart and Martinelli correlation (1949) • Duns and Ros correlation (1963) • Hagedorn and Brown (H-B) method (1965) • Ansari et al. (1994) and Hasan and Kabir (2002) compared the above models and recommended the H-B method with modifications (mH-B) for near-vertical flow • Modifications are • Use no slip holdup when calculated holdup is less than no slip holdup • Using Griffith and Wallis (1961) correlation in bubble flow regime July 31, 2017 76
- 77. Original H-B Correlation (Near Vertical Wells) July 31, 2017 77 Note mostly gas density and rate change in the well due to its compressibility Q: At steady state, would the gas rate increase or decrease coming up the well
- 78. Gas Volumetric Rate Up the Well July 31, 2017 78 Q: At steady state, would the gas flow rate increase or decrease coming up the well Ans: At steady state, the total mass rate in and out of the well should be constant. If we assume that the two phases remain intact with no mass transfer between the phases, then Since the pressure decreases up the well, the density of the gas should decrease. If the density of the gas decreases, its volumetric flow rate should increase and so its superficial gas velocity. rG ¯ qG -
- 79. Original H-B Correlation (Vertical Wells) July 31, 2017 79 dp/dz = psi/ft, D = ft, z = ft
- 80. Original H-B Correlation July 31, 2017 80 usL = qL A usG = qG A Note the liquid density is assumed to be constant
- 81. July 31, 2017 81 Flow Regime Map Air-Water System 2 inch Pipe
- 82. July 31, 2017 82 Taitel and Dukler Flow Regime Described
- 83. Dimensionless nos. for Liquid Hold-up Calculation • Liquid holdup, yL is calculated from 3 charts that depends on following dimensionless numbers. July 31, 2017 83
- 84. July 31, 2017 84 Nomenclature for Dimensionless Nos.
- 85. July 31, 2017 85
- 86. July 31, 2017 86
- 87. July 31, 2017 87
- 88. July 31, 2017 88
- 89. July 31, 2017 89 Liquid Hold-up Charts / Corelations based on Dimensionless Nos.Hold up Chart 1
- 90. • First chart gives CNL based on NL July 31, 2017 90 Hold up Chart 1 Corelation
- 91. July 31, 2017 91 Hold up Chart 2
- 92. Holdup Chart 2 Corelation July 31, 2017 92 Above group is used in the second chart to determine From chart 1
- 93. July 31, 2017 93 Hold up Chart 3
- 94. Chart 3 Corelation July 31, 2017 94
- 95. Hold up July 31, 2017 95 From chart 2 and chart 3 values
- 96. Friction Factor July 31, 2017 96 Fanning friction factor can be determined by Moody plot or Chen’s correlation where Re for the mixture is calc. as followed In field units
- 97. Bubble Flow Regime for mH-B July 31, 2017 97 Bubble-flow regime is when input gas fraction is less than LB Where lG = Input Gas Fraction if LB < 0.13, LB = 0.13 Bubble Flow Regime
- 98. Griffith Corelation during Bubble Flow Regime July 31, 2017 98 • Different Holdup corelation • Frictional pressure drop based on in-situ avg. liq. velocity • Neglects kinetic energy pressure gradient
- 99. Liquid hold is given as mH-B Correlation – Griffith Corelation Hold up July 31, 2017 99
- 100. Griffith Corelation Friction Factor July 31, 2017 100 Friction factor is based in-situ avg. liquid velocity Re is based on in situ average liquid velocity, i.e. Excel pgm – HagedornBrownCorrelation.xls has the code
- 101. July 31, 2017 101 Surface tubing pressure = 800 psia Surface temperature = 175 oF Liquid rate = Oil rate = 2000 bpd Density of oil = 0.8 g/cc Viscosity of oil = 2 cp Gas rate = 1 MMSCF/d Gas specific gravity = 0.709 Compressibility factor = 0.935 at surface P & T Gas viscosity = 0.0131 cp at surface P & T Surface tension = 30 dynes/cm Relative roughness = 0.0006 From corelations
- 102. July 31, 2017 102
- 103. July 31, 2017 103
- 104. July 31, 2017 104
- 105. July 31, 2017 105
- 106. July 31, 2017 106
- 107. July 31, 2017 107 Fig 4.4
- 108. July 31, 2017 108 Table 4.3
- 109. Mist Flow in Gas Wells • Almost all gas wells produce certain amount of liquids • Water and/or gas condensate (light oil) • In some gas wells, gas condensate is in the well and not at the surface depending upon P, T • Sand and coal particles also produced • Multi-phase-gas wells • Homogeneous 4-phase flow model (Guo-Ghalambor) can be applied to mist flow in gas wells July 31, 2017 109
- 110. Summary • Illustrated different math models for wellbore/tubing performance • mH-B has been found to give results with good accuracy • Industry practice is to conduct a flow gradient (FG) survey to measure the flowing pressures along the tubing string • FG data are employed to validate and tune one of the models to use in on a large scale July 31, 2017 110
- 111. Energy Conservation July 31, 2017 111 DETot =Qin +Won +DEconvection DEconvection (Net energy added to the system due to entry of mass) Won (Work done on the system by compressing the system) System Qin
- 112. Kinetic Energy of the System (assume ball shape instead of the piston compartment) July 31, 2017 112
- 113. Potential Energy of the System July 31, 2017 113 Accounting only gravitational potential energy Neglecting electromagnetic potential energy
- 114. Internal Energy of the System July 31, 2017 114 Typically a function of • Temperature, Pressure • Phase • Chemical composition
- 115. Case1: Closed System – No Convection (No mass leaving/entering the system) July 31, 2017 115
- 116. Case 2: Open Systems – with Convection July 31, 2017 116
- 117. Open Systems – Steady State July 31, 2017 117 0
- 118. Open Systems – Steady State July 31, 2017 118 From now on D refers to Out - In
- 119. Open Systems – Steady State July 31, 2017 119
- 120. Open Systems – Steady State July 31, 2017 120
- 121. Open Systems – Steady State July 31, 2017 121
- 122. Open Systems – Steady State July 31, 2017 122
- 123. Open Systems – Steady State July 31, 2017 123 Won, work done on the system has two contributions: 1. Shaft work from moving parts like shafts, turbines, and pumps 2. Flow work by the fluid itself as it enters and leaves the system
- 124. Shaft Work July 31, 2017 124
- 125. Flow Work - Open Systems July 31, 2017 125 Volumetric rate Rate of Rate of
- 126. Inlet Flow Work Rate July 31, 2017 126
- 127. Exit Flow Work Rate July 31, 2017 127
- 128. Open Systems – Steady State July 31, 2017 128 Will show the enthalpy equivalence
- 129. Open Systems – Steady State July 31, 2017 129
- 130. Open Systems – Steady State July 31, 2017 130
- 131. Open Systems – Steady State July 31, 2017 131
- 132. Heat Transfer Open Systems – Steady State July 31, 2017 132 and the dot is the rate
- 133. Mechanical Energy Balance (MEB) July 31, 2017 133 • Open-system macroscopic energy balance is quite common in heat exchangers and reactors • Flow of liquids and gases in conduits, kinetic, potential and shaft work dominates. • Bernoulli equation is an example of simple MEB
- 134. Bernoulli’s equation as MEB July 31, 2017 134 • Special case of single-input and output system of liquid pushed thru a pipe by pump Open-system Energy Balance Assumptions
- 135. Kinetic Energy Difference (out, 2 – in, 1) July 31, 2017 135 where
- 136. Potential Energy Difference (out, 2 – in, 1) July 31, 2017 136 where
- 137. Enthalpy Difference (out, 2 – in, 1) July 31, 2017 137 where U with a pointed cap is the internal energy per unit mass and V with a pointed cap is the volume per unit mass j is the outlet, denoted as 2 i is the inlet, denoted as 1
- 138. Enthalpy Difference (out, 2 – in, 1) continued July 31, 2017 138 where For incompressible system, r1 = r2
- 139. MEB Simplified July 31, 2017 139 Square bracket terms are small for incompressible fluids in pipe
- 140. MEB Simplified • Internal energy term is small since temperature is almost constant and assumingly no phase change or chemical reaction occurred • Heat loss or gain term is small – • We group these terms as Friction factor, F July 31, 2017 140
- 141. MEB Simplified July 31, 2017 141
- 142. MEB Simplified July 31, 2017 142 • a in the denominator of the kinetic energy term accounts for the variation in the velocity of fluid at different radii of the pipe • Approx. 1 for turbulent, and exactly 0.5 for laminar flow • Can be deduced from momentum balance (Geankoplis)
- 143. MEB Simplified to Bernoulli’s Equation July 31, 2017 143 • When friction term, F and shaft work are neglected or not there, the MEB simplifies to Bernoulli’s equation
- 144. MEB – No Friction July 31, 2017 144
- 145. MEB – No Friction July 31, 2017 145
- 146. MEB – No Friction July 31, 2017 146
- 147. MEB – No Friction - Flow in 3 inch Pipe July 31, 2017 147
- 148. MEB – No Friction – Laminar/Turbulent July 31, 2017 148
- 149. MEB – No Friction – Turbulent July 31, 2017 149
- 150. MEB – No Friction – Turbulent July 31, 2017 150 Kinetic energy contribution Potential energy contribution
- 151. Example – Venturi Flow Rate & Pressure Drop July 31, 2017 151 • A Venturi is a tapered tube with a throat that allows us to measure the flow rate of an incompressible fluid in a pipe • Takes lot of space but is accurate & doesn’t disturb flow much
- 152. Example – Venturi Flow Rate & Pressure Drop July 31, 2017 152 • MEB without friction can be used to deduce the relationship between flow rate and pressure drop • Calibrated device can be used to take into account the friction effects • Converging and then diverging
- 153. Example – Venturi Flow Rate & Pressure Drop July 31, 2017 153 • Tapering is gradual to minimize friction losses
- 154. MEB with Friction • Friction term is important when there are changes in pipe diameter, twists, turns, flow obstructions such as orifice plate or when there are very long runs of piping • F must be determined experimentally, as Cv for Ventur • MEB is not very useful then; however MEB is applied from one apparatus (variables) to another system of variables. • Ex. to calculate shaft work of a pump in a loop July 31, 2017 154
- 155. MEB with Friction • We draw on past experiments of prior researchers to estimate F for systems that interests us • We may be able to use the data for similar experiments where apparatus is not the same • Resolution is dimensional analysis based on correct observation that the laws of physics apply to all systems • Simple systems – engineering analysis; complex systems – start from laws of physics July 31, 2017 155
- 156. MEB with Friction • From dim. analysis on laws of physics, deduce interest quantities (wall friction or heat transfer coefficient) vary with certain identified system quantities • Targeted experiments are done to publish data corelations to be used by engineers to calculate quantities of interest on similar systems (Geankoplis) • Data corelations for F in straight pipes, valves, fittings, etc. • Liquid flow in straight pipes – Fanning friction factor, f as a function of Reynolds number, Re July 31, 2017 156
- 157. Example – Venturi Flow Rate & Pressure Drop July 31, 2017 157 • Incompressible flow
- 158. Example – Venturi Flow Rate & Pressure Drop July 31, 2017 158
- 159. Venturi Meters with Friction • When flow is sufficiently rapid (Re > 104), previous no friction relationship holds well • For slower flows, friction is important to total energy and calibration should be performed to determine an empirical friction correction factor Cv July 31, 2017 159
- 160. Fanning Friction Factor, f, for Steady Laminar Flow • Corelation for f and Re can be determined experimentally • Laminar flow is simple, microscopic momentum balance gives the relationship between pressure drop and Re • Newtonian fluid at steady state results in Hagen-Poiseuille equation July 31, 2017 160
- 161. Fanning Friction Factor, f, for Steady Laminar Flow July 31, 2017 161
- 162. Fanning Friction Factor, f, for Fully Turbulent Flow July 31, 2017 162 Colebrook Equation
- 163. Fanning Friction Factor, f, for Turbulent Flow July 31, 2017 163
- 164. Friction, F, in Other Devices • Valves, fittings, pumps, expansions, contractions, twists, turns etc. • Same procedure as earlier for estimating F, simplify MEB • Use dim. analysis to guide experiments for corelations • For valves, fittings, expansions and contractions, F is July 31, 2017 164
- 165. Friction loss Factors, Ki, for Valves, Fittings (Laminar Flow) July 31, 2017 165
- 166. Friction loss Factors, Ki, for Valves, Fittings (Turbulent Flow) July 31, 2017 166
- 167. Friction Term, F, for Complete Piping System July 31, 2017 167 • vj is the average flow velocity in the pipes of different dia. • vi is the faster average velocity in the fittings (downstream in case of contraction and upstream in case of expansion)
- 168. MEB with Friction July 31, 2017 168 2 contractions (tank to inlet, inlet pipe to pump outlet pipe) 2 90o elbows
- 169. Previous Example with Friction Now July 31, 2017 169 • The 3 inch pipe before the pump is 50 ft long • The 2 inch pipes after the pump are (40+8+75+20) =143 ft • Recall Re was > 4000; as a result it was turbulent flow
- 170. Previous Example with Friction Now July 31, 2017 170 • Fanning friction factor, f, from Colebrook formula for the straight pipes give • From Table of ki for turbulent flow
- 171. Friction Term, F July 31, 2017 171
- 172. Friction Term, F Included for Shaft Work July 31, 2017 172 • Shaft work without friction = 0.113 hp • Shaft work with friction = 0.114 hp • Not much difference, potential energy still dominates
- 173. Energy Terms in Elevation Head (ft of head) July 31, 2017 173 • All in energy terms in elevation head • Elevation head is therefore a convenient concept for comparison Kinetic head change Elevation head change Friction head
- 174. Acknowledgement • Notes taken from Faith A. Morrison, Associate Professor, Chemical Engineering, Michigan Technological University July 31, 2017 174