Mechanical Energy Balance and Wellbore Performance of Oil and Gas Wells

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
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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
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34. Single-Phase Gas Flow – Avg. T and z
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35. Carr et al. Corelation for Gas Viscosity
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36. Carr et al. Corelation for Gas Viscosity
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37. Carr et al. Corelation for Gas Viscosity
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38. Brill and Beggs Corelation for Gas Deviation
factor (Z factor)
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39. More Accurate Corelation for Gas Deviation
factor (Z factor)
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40. More Accurate Corelation for Gas Deviation
factor (Z factor)
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41. Single-Phase Gas Flow – Avg. T and z
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Pressure gradient = 0.21 psi/ft

42. Single-Phase Gas Flow – Avg. T and z
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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
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46. Single-Phase Gas Flow – Cullender and Smith
Method
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• 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
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65. Poettman-Carpenter Example Solution
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66. Poettman-Carpenter Example Solution
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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
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Excel program – Guo-GhalamborBHP.xls

69. Guo-Ghalambor (2005) 4 Phase Model
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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
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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
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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
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161. Fanning Friction Factor, f, for Steady Laminar Flow
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162. Fanning Friction Factor, f, for Fully Turbulent Flow
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Colebrook Equation

163. Fanning Friction Factor, f, for Turbulent Flow
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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
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165. Friction loss Factors, Ki, for Valves, Fittings (Laminar Flow)
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166. Friction loss
Factors, Ki, for
Valves, Fittings
(Turbulent Flow)
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167. Friction Term, F, for Complete Piping System
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• 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
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2 contractions (tank to inlet, inlet
pipe to pump outlet pipe)
2 90o elbows

169. Previous Example with Friction Now
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• 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
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• Fanning friction factor, f, from Colebrook formula for the
straight pipes give
• From Table of ki for turbulent flow

171. Friction Term, F
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172. Friction Term, F Included for Shaft Work
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• 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)
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• 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
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