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Pipe Network Analysis

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Pipe Network Analysis determine the flow rates and pressure drops in the individual sections of a hydraulic network. Lyrica online without prescription is the oldest and probably best known solution method for pipe networks. In this method, each loop correction is determined independently of other loops. Epp and Fowler (1970) developed a more efficient approach by simultaneously computing corrections for all loops. This article illustrates use of Simultaneous Loop Flow Adjustment Algorithm in modelling pipe network analysis in excel spreadsheet.

A water supply distribution system is shown in the figure below. All pipes are cast iron with lengths and diameters as provided in table below. Perform pipe network analysis and calculate water flow in all branches.

Pipe Network Analysis Example 1

An initial flow estimate is made across all pipe branches that satisfies continuity at all nodes. Head loss across a pipe is determined using Darcy – Weisbach equation.

 HLoss = K Q²
 K    = 8fL/ π²g D 5

Head loss across each loop is made.

F(I)  : K1.Q1² + K3.Q3² - K8.Q8² - K4.Q4² - K2.Q2² = 0
F(II) : K5.Q5² + K7.Q7² - K6.Q6² - K3.Q3² = 0 
F(III): K6.Q6² + K10.Q10² - K9.Q9² + K8.Q8² = 0

Newton Raphson method is used to solve above equations for change in flow ΔQ (ΔQI, ΔQII, ΔQIII) across each loop. In vector form all loops can be written as :

 JL.ΔQ = - F(Qm-1)
 JL    = δF / δ(ΔQ)

where Qm-1 is vector of pipe flow, ΔQ is vector of loop flow corrections and F(Qm-1) is the vector of residuals of loop conservation of energy equations evaluated at Qm-1. JL is first derivatives of the loop equations evaluated at Qm-1. Once above matrices are formed, it is solved linearly for ΔQ and pipe flows are updated by the loop corrections as Qm = Qm-1 +/- ΔQ.

Coefficient Matrix, JL

Pipe Network Jacobian Matrix

Derivative for single pipe is obtained as:

 δ(K1.Q1²)/δ(ΔQI) = 2.K1.Q1 = 2.HL1/Q1

Diagonal terms are obtained by adding derivatives of all pipes in a loop and always have a positive sign. eg.

 δFI/δ(ΔQI) = n( |HL1/Q1| + |HL3/Q3| + |HL8/Q8| + |HL4/Q4| + |HL2/Q2| )
            = 979.03

Off-diagonal terms are gradients for pipes that appear in different loops and always have a negative sign.

 δFI/δ(ΔQII) = -n( |HL3/Q3| ) = δFII/δ(ΔQI)
            = -165.60

In above derivative change in Loop I due to flow change in Loop II will be due to common pipe 3. Above gradient is also for δFII/δ(ΔQI). The JL matrix is obtained as following.

Pipe Network Jacobian Matrix

F(Qm-1) is evaluated based on head loss across each loop.

 FI = HL1 + HL3 - HL8 - HL4 - HL2
    = -10.94

F matrix is obtained as

 F(Qm-1)T = [-10.94 -6.40 43.62 ]

Below matrix is solved in excel to obtain ΔQ.

 JL.ΔQ = - F

JL-1 is determined using Minverse function. Select 3×3 cells in excel and type MINVERSE(Input Array) and press CTRL+SHIFT+ENTER to evaluate inverse of matrix.

Pipe Network Jacobian Matrix

Do multiplication of JL-1 and -F vector using MMULT function. Select 3×1 cells in excel and type MMULT(Inverse Array, F array) and press CTRL+SHIFT+ENTER to evaluate multiplication. ΔQ array is obtained as following.

 ΔQ = [ 0.00066 -0.00261 -0.03133]

As flow changes are larger another iteration is done, with flows adjusted based on ΔQ values. For example flow rate in Pipe 3 will become.

 Q3 = Q30 + ΔQI - ΔQII
    = 0.12 + (0.00066) - (-0.00261)
    = 0.123

New flowrates are calculated and above steps are repeated. To do more iterations copy entire rows and paste them below the above cells to carry out further calculation till change in flowrates become negligible. For this example ΔQ becomes negligible in 5 iterations. Final flows in m³/s are as following.

Pipe Network Solution

If a pipe’s flow direction changes from the assumed value, the signs for that pipe head loss terms are switched for all loops containing the pipe during the next iteration in loop equations. JL matrix signs will remain same as above.

Perform pipe network analysis and calculate water flow in all branches.

Pipe Network Analysis Example 2

This problem is solved based on methodology developed above, refer attached excel spreadsheet for solution. In this case, 3 loop equations will be made and it takes 4 iterations to converge.

Perform pipe network analysis and calculate water flow in all branches. Hazen Williams coefficient for each pipe is provided in table below.

Pipe Network Analysis Example 3
Pipe Network Analysis Example 3

Head loss is calculated using Hazen-Williams equation.

 HL = 10.67LQ1.85 / C1.85 D4.87

Based on method discussed above 7 loop equations are formed and it takes 5 iterations to converge to final flow values. Refer attached excel spreadsheet for solution.

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Centrifugal Pump Curve Fitting

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Pump manufacturers provide Head versus Flow curve for various pump models. This article shows how to convert it into a 2nd order polynomial using Excel’s LINEST function. This curve can then be used to predict pump head curve at different impeller diameter and pump speed based on Affinity Laws.

Pump head versus flow curve is available for impeller diameter 210 mm. Estimate modified head vs flow curve at impeller diameter 250 mm.

Flow (m³/h) Head (m)
4 15.00
6 14.50
8 13.75
10 12.70
12 11.25
14 9.50

LINEST function in Excel is used to do 2nd order polynomial curve fitting to get constants a0,a1 and a2.

 Head(x) = a2.x² + a1.x + a0

centrifugal pump curve fitting

LINEST function formula is copied in an empty cell e.g G8.

 = LINEST(C10:C15,B10:B15^{1,2},TRUE,TRUE)

After copying the formula, select the range G8:I10 starting with formula cell. Press F2, and then press CTRL+SHIFT+ENTER. Resulting array G8, H8 & I8 provides value for a2,a1 & a0.

Affinity Laws

The Affinity Laws for centrifugal pumps are used to determine the performance of a pump at different impeller diameters or speeds. Head vs flow curve are related by Affinity laws for impeller diameter change as follows:

 Q2/Q1 = D2/D1 = r
 H2/H1 = (D2 / D1)² = r²

Head vs flow curve for speed change are related as follows:

 Q2/Q1 = N2/N1 = R
 H2/H1 = (N2 / N1)² = R²

Modified head vs flow curve is obtained as

 Head'(x) = a2.x² + a1.r.x + a0.r²
 Head'(x) = a2.x² + a1.R.x + a0.R²

modified pump head curve

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Pipe Fitting Losses

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Head loss in a pipe is sum of following –

  • Elevation difference, hZ
  • Fitting losses, hL
  • Friction losses, hF

Fitting losses hL is calculated as

hL = K(V²/2g)

where, K is resistance coefficient due to fittings, V is fluid velocity and g is acceleration due to gravity.

Friction losses hF is calculated as

hF = f(L/D)(V²/2g)

where, f is Darcy’s pipe friction factor, L is pipe length and D is pipe inside diameter.

Total head loss in a pipe –

hTotal = hZ + hL + hF

Pressure drop due to head loss in pipe is calculated as

ΔP = hTotal.ρ.g

where, ρ is fluid density.

There are several methods for estimating pipe fitting losses like equivalent length method, K method, 2-K (Hooper) method and 3-K (Darby) method. 3-K method is most accurate followed by 2-K method.

2-K (Hooper) Method

K = K1/Re + K (1 + 1/ID )

where, Re is Reynold’s number, K1, K are constants and ID is inside diameter in inches.

3-K (Darby) Method

K = K1/Re + K (1 + Kd/Dn0.3 )

where, K1, K, Kd are constants and Dn is nominal pipe diameter in inches.

Constants for 3K and 2K method for some common fittings.

90° Elbow K1 K Kd
Threaded, r/D = 1 800 0.14 4.0
Threaded, Long Radius, r/D = 1.5 800 0.071 4.2
Flanged, Welded, Bend, r/D = 1 800 0.091 4.0
Flanged, Welded, Bend, r/D = 2 800 0.056 3.9
Flanged, Welded, Bend, r/D = 4 800 0.066 3.9
Flanged, Welded, Bend, r/D = 6 800 0.075 4.2
Mitered, 1 Weld, 90° 1000 0.270 4.0
Mitered, 2 Weld, 45° 800 0.068 4.1
Mitered, 3 Weld, 30° 800 0.035 4.2
2K Method
Mitered, 4 Weld, 22.5° 800 0.27
Mitered, 5 Weld, 18° 800 0.25
45° Elbow K1 K Kd
Standard, r/D = 1 500 0.071 4.2
Long Radius, r/D = 1.5 500 0.052 4.0
Mitered, 1 Weld, 45° 500 0.086 4.0
Mitered, 2 Weld, 22.5° 500 0.052 4.0
180° Bend K1 K Kd
Threaded, r/D = 1 1000 0.230 4.0
Flanged/ Welded, r/D = 1 1000 0.120 4.0
Long Radius, r/D = 1.5 1000 0.100 4.0
Tees K1 K Kd
Standard, Threaded, r/D = 1 500 0.274 4.0
Long Radius, Threaded, r/D = 1.5 800 0.140 4.0
Standard, Flanged/ Welded, r/D = 1 800 0.280 4.0
Stub-in Branch 1000 0.340 4.0
Run Through, Threaded, r/D = 1 200 0.091 4.0
Run Through, Flanged/ Welded, r/D = 1 150 0.050 4.0
Run Through Stub in Branch 100 0 0
Valves K1 K Kd
Angle Valve = 45°, β = 1 950 0.250 4.0
Angle Valve = 90°, β = 1 1000 0.690 4.0
Globe Valve, β = 1 1500 1.700 3.6
Plug Valve, Branch Flow 500 0.410 4.0
Plug Valve, Straight Through 300 0.084 3.9
Plug Valve, 3-way, Flow Through 300 0.140 4.0
Gate Valve, β = 1 300 0.037 3.9
Ball Valve, β = 1 300 0.017 3.5
Butterfly Valve 1000 0.690 4.9
Swing Check Valve 1500 0.460 4.0
Lift Check Valve 2000 2.850 3.8
2K Method
Diaphragm Valve, Dam Type 1000 2.0
Tilting Disk Check Valve 1000 0.5

Square Reduction

Square Reduction

For Re1 < 2500

K = (1.2 + 160/Re1)[(D1/D2)4 - 1]

For Re1 > 2500

K = (0.6 + 0.48f1)(D1/D2)²[(D1/D2)² - 1]

Re1 is upstream Reynold’s number at D1 and f1 is friction factor at this Reynold’s number.

Tapered Reduction

Tapered Reduction

For θ < 45°, multiply K from square reduction by 1.6 sin(θ/2).
For θ > 45°, multiply K from square reduction by sin(θ/2)0.5.

Rounded Pipe Reduction

Rounded Pipe Reduction

K = (0.1 + 50/Re1)[(D1/D2)4 - 1]

Square Expansion

Square Expansion

For Re1 < 4000

K = 2[1 - (D1/D2)4]

For Re1 > 4000

K = (1 + 0.8f1)[1 - (D1/D2)²]²

Re1 is upstream Reynold’s number at D1 and f1 is friction factor at this Reynold’s number.

Tapered Expansion

Tapered Expansion

For θ < 45° multiply K for square expansion by 2.6 sin(θ/2).
For θ > 45° use K for square expansion.

Rounded Pipe Expansion

Rounded Pipe Expansion

Use K for square expansion.

Thin Sharp Orifice

Thin Sharp Orifice

For Re1 < 2500
K Value Thin Sharp Orifice Re < 2500

For Re1 > 2500
K Value Thin Sharp Orifice Re > 2500

Thick Orifice

Thick Orifice
For L/D2 > 5, use equations for square reduction and a square expansion.
For L/D2 < 5, multiply K for a thin sharp orifice by

0.584 + (0.0936 / ( (L/D2)1.5 + 0.225))

Pipe Entrances

Flush/ Square Edged

Flush Square Edged

K = 0.5


Rounded Pipe Entrances

r/D K
0.02 0.28
0.04 0.24
0.06 0.15
0.10 0.09
0.15+ 0.04

Inward Projecting (Borda)

Inward Projecting (Borda)

K = 0.78


Chamfered Pipe Entrances

K = 0.25

Pipe Exits

K = 1.0 for all geometries

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Two Phase Flow – Horizontal Pipe

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Criterion for Line size selection is generally pressure drop expressed as ΔP per 100 feet or meters of pipe. There are different type of two phase pressure drop correlations determined by viscosity ratio and mass flux.

μL/ μG Mass Flux, (kg/m².sec) Correlation
< 1000 All Friedel
> 1000 > 100 Chisolm – Baroczy
> 1000 < 100 Lockhart – Martinelli

Select a pipe and estimate inner diameter (D) based on pipe schedule. Obtain important properties for both Gas and Liquid like flowrate, density (ρ), viscosity (μ) and surface tension (σ).

Density, velocity and viscosity are averaged for the combined phases in fluid flow and two-phase Reynolds number is calculated.

ρave = (QG + QL)/(QGG + QLL)
Vave = (QG + QL)/(ρave(πD²/4))
μave = (QG + QL)/(QGG + QLL)
Reave = 4(QG + QL)/(πDμave)

where, QG, QL are gas and liquid mass flowrate. ρG and ρL are gas and liquid density. μG and μL are gas and liquid viscosity.

Chisolm – Baroczy method

Pressure drop for each of the phases are calculated assuming that the total mixture flows as either liquid or gas based on procedure provided for purchase Lyrica.

G = QG + QL
(ΔP/L)Go = function(G, D, μG, ρG, ε)
(ΔP/L)Lo = function(G, D, μL, ρL, ε)

A pressure ratio, Y is calculated –

Y² = (ΔP/L)Go / (ΔP/L)Lo

Based on pressure ratio, a constant is calculated:

B = 55 / G 0.5, 0 < Y < 9.5
  = 520 / YG 0.5, 9.5 < Y < 28
  = 15,000 / Y 2G 0.5, 28 < Y

Two phase correction factor is calculated as following:

φ²Lo = 1+(Y²-1)[BX(2-n)/2(1-X)(2-n)/2+X2-n]

where n is 0.25 and X is gas mass fraction. Two phase pressure drop is calculated as following:

(ΔP/L)TP = φ²Lo (ΔP/L)Lo

Lockhart – Martinelli method

Pressure drop for each of the phases are calculated explicitly assuming that either liquid or gas is flowing through the pipe based on procedure provided for purchase Lyrica.

(ΔP/L)G = function(QG, D, μG, ρG, ε)
(ΔP/L)L = function(QL, D, μL, ρL, ε)

A pressure ratio is calculated.

X² = (ΔP/L)L / (ΔP/L)G

Lockhart - Martinelli Correlation

A separate pressure drop is calculated for each phase.

(ΔP/L)L1 = φ²L (ΔP/L)L
(ΔP/L)G1 = φ²G (ΔP/L)G

Estimated two phase pressure drop is maximum of these

(ΔP/L)TP = Max((ΔP/L)L1,(ΔP/L)G1)

Friedel method

Calculate Froude and Weber number using mass flux of the total mass flowing in pipe.

Fr = G² / ( gcDρ²ave )
We = G²D / ( ρave σ )

where gc is gravitational constant. Friction factor for each of the phases are calculated assuming that the total mixture flows as either liquid or gas based on procedure provided for purchase Lyrica.

fGo = function(G, D, μG, ρG, ε)
fLo = function(G, D, μL, ρL, ε)

Following constants are calculated –

E = (1 - X)² + X² ρLfGo / (ρGfLo)
H = (ρLG)0.91GL)0.19(1-μGL)0.7
F = X 0.78 (1 -X) 0.24

where, X is gas mass fraction. Two phase correction factor is calculated as following –

φ²Lo = E + 3.24 FH / (Fr 0.045 We 0.035)

Two phase pressure drop is calculated as following –

(ΔP/L)TP = φ²Lo (ΔP/L)Lo

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Beggs & Brill Method

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Beggs and Brill (1973) correlation, is one of the few correlations capable of handling all flow directions encountered in oil and gas operations, namely uphill, downhill, horizontal, inclined and vertical flow for two phase fluid.

Total pressure gradient is described by following relation.

dP/dZ = [(dP/dZ)Fric. +(dP/dZ)Ele.]/(1-Ek)

where, (dP/dZ)Fric. is pressure gradient due to friction, (dP/dZ)Ele. is hydrostatic pressure difference and Ek estimates pressure loss due to acceleration.

Flow Pattern Map

A flow regime is identified based on the Froude number of the mixture (Frm) and input liquid content (no slip liquid holdup CL).

Frm = vm²/ g.D

where, vm is mixture velocity, D is pipe inside diameter and g is gravitational constant.

CL = QL/ (QL + QG)

where, QL is liquid volumetric flow and QG is gas volumetric flow.

The transition lines for correlation are defined as follows:

L1 = 316 CL0.302
L2 = 0.0009252 CL-2.4684
L3 = 0.1 CL-1.4516
L4 = 0.5 CL-6.738

Segregated Flow

CL < 0.01 and Frm < L1
OR CL >= 0.01 and Frm < L2

Intermittent Flow

0.01 <= CL < 0.4 and L3 < Frm <= L1
OR CL >= 0.4 and L3 < Frm <= L4

Distributed Flow

CL < 0.4 and Frm >= L4
OR CL >= 0.4 and Frm > L4

Transition Flow

L2 < Frm < L3

Liquid Holdup, EL(θ)

Once flow type has been determined, liquid holdup for horizontal flow EL(0) is calculated.

EL(0) = a CLb / Frmc
Flow Regime a b c
Segregated 0.98 0.4846 0.0868
Intermittent 0.845 0.5351 0.0173
Distributed 1.065 0.5824 0.0609

EL(0) must be greater than CL, if EL(0) is smaller than CL, then EL(0) is assigned a value of CL. Actual liquid volume fraction is obtained by multiplying EL(0) by a correction factor, B(θ).

EL(θ) = B(θ) x EL(0)

B(θ) is obtained as –

B(θ) = 1 + β(sin(1.8θ) - (1/3)sin³(1.8θ))

where θ is the angle of inclination of pipe with horizontal.
Correction factor β is calculated as following –

β = (1 - CL)ln( d.CLe.NLVf.Frmg )
Uphill d e f g
Segregated 0.011 -3.768 3.539 -1.614
Intermittent 2.96 0.305 -0.4473 0.0978
Distributed β = 0
Downhill d e f g
All 4.7 -0.3692 0.1244 -0.5056

Liquid velocity number, NLV is given by:

NLV = 1.938 VslL/ (g.σ))1/4

Vsl is no slip liquid velocity, ρL is liquid density, g is gravitational constant and σ is surface tension.

For transition flow,

EL(θ)transition = AEL(θ)segregated + BEL(θ)intermittent

where A and B are as following –

A = ( L3 - Frm)/(L3 - L2)
B = 1- A

Liquid holdup, EL(θ) is used to calculate mixture density ρm.

ρm = ρL.EL(θ) + ρG.(1-EL(θ))


Pressure change due to the hydrostatic head of the vertical component of the pipe is given by:

(dP/dZ)Ele. = ρm.g.sin(θ)/(144.gc)


Calculate no slip Reynold’s number using no slip mixture density and viscosity.

ReNS = ρNS.Vm.D/μNS

No slip friction factor, fNS is then calculated using Colebrook-White equation.

Ratio of friction factor is defined as

fTP/ fNS = eS

Value of S is governed by following conditions –

S = ln(2.2y - 1.2)

if 1 < y < 1.2, otherwise –

S = ln(y)/(-0.0523 + 3.182.ln(y) - 0.8725.(ln(y))2 + 0.01853.(ln(y))4 )

where y is defined as

y = CL / EL(θ)²

Pressure loss due to friction is:

(dP/dZ)Fric. = 2.fTP.Vm².ρNS /(144.gc.D)

Pressure loss due to acceleration, factor Ek is given by:

Ek = ρm.Vm.Vsg/(gc.P)

where, Vsg is no slip gas velocity and P is gas pressure.

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Bingham Plastic Fluid

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Bingham plastic is a material that behaves as rigid body at low stresses but flows as a viscous fluid at high stress. This behaviour is exhibited by slurries, suspensions of solids in liquids, paints, emulsions, foams, etc.

Bingham model is described by following relation.

τ = τo + μp γ

where, τ is shear stress, γ is shear rate, τo is called minimum yield stress and μp is called plastic viscosity.

Reynolds number for Bingham plastic fluid is defined as

Re = D V ρ / μp

where, D is pipe inside diameter, V is fluid velocity and ρ is fluid density.

For Laminar flow, friction factor is provided by Buckingham Reiner equation.

Friction factor laminar flow Buckingham Reiner

where, He is Hedstrom number and is calculated as

He = D²ρτo / μp²

For turbulent flow, an empirical relationship was developed by Darby and Melson.

Friction factor turbulent flow Darby and Melson

The friction factor for a Bingham plastic can be calculated for any Reynolds number from the equation.

f = ( fLm + fTm ) 1/m

where, fL is laminar flow friction factor and fT is turbulent flow friction factor. Factor m is calculated from following equation.

m = 1.7 + 40,000 / Re

Pressure drop is calculated as

ΔP = 2fρV² (L'/D)
L' = L + Le

where, L is the pipe length and Le is equivalent length due to loss in pipe fittings and is calculated as

Le = kD / 4f

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Power Law Fluid

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Non-Newtonian fluids occur commonly in our world. Power law model is applicable for time independent non-Newtonian fluids and can be written as

τ = K γn

where, τ is shear stress, γ is shear rate, K is called flow consistency index and n is called flow behavior index.

For n < 1, the apparent viscosity decreases with increasing shear rate and fluid is called pseudoplastic or shear-thinning. A majority of non-Newtonian fluids like polymer solutions, pulp suspensions, pigments and food materials can be found in this category.

For n > 1, the apparent viscosity increases with shear rate increase and fluid is termed dilatant or shear-thickening. Examples are starch and clay suspensions in water.

For n = 1, Newtonian flow behavior is expected.

Reynolds number for the power law fluid is defined as

Reynolds number for power law non-newtonian fluid

where, D is pipe inside diameter, V is fluid velocity and ρ is fluid density.

For Re < 2100, flow is laminar and friction factor is calculated as

f = 16 / Re

For turbulent flow, following relationship was developed by Dodge and Metzner.

Friction factor turbulent flow Dodge Metzner

Pressure drop is calculated as

ΔP = 2fρV² (L'/D)
L' = L + Le

where, L is the pipe length and Le is equivalent length due to loss in pipe fittings and is calculated as

Le = kD / 4f

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Compressible Fluid Flow

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Flow of compressible fluid like vapors and gases through pipe is affected by changing conditions of pressure, temperature and physical properties. General relation for evaluating such flow is given by following equation in English units.

Compressible fluid general flow equation

where, f is Darcy’s friction factor, L is pipe length (ft), D is pipe diameter (ft), ρ is fluid density (lb/ft³), A is pipe cross-sectional area (ft²), g is constant 32.174 ft/sec², P is pressure in psi and Ws is gas flow (lbm/sec). Subscript 1 denotes conditions at pipe inlet and 2 denotes at pipe outlet.

Friction factor is determined based on Reynolds’s number (Re).

Re = D V ρ / μ

where, μ is fluid viscosity.

For Re < 2100, flow is laminar and friction factor is calculated as following.

f = 64 / Re

For Re > 4000, flow is turbulent and friction factor is obtained by solving Colebrook White equation.

Colebrook White Equation

Sonic Velocity

The maximum possible velocity of a compressible fluid in a pipe is called sonic velocity.

Vs = 68.1 [ (Cp/Cv) P/ρ]0.5

where, Cp/Cv is gas specific heat ratio, P is pressure in psi, ρ is density in lb/ft³ and Vs is sonic velocity in feet/sec.

Mach Number

The Mach number is the velocity of the gas divided by the sonic velocity in gas.

Ma = V/ Vs

where V is gas velocity in pipe and Vs is the sonic velocity.

Erosional Velocity

Erosional velocity is maximum allowable gas velocity in a pipeline, as gas velocity increases, vibration and noise increases. Erosional velocity can be estimated as following.

Vmax = 100 / ρ0.5

where ρ is gas density in lb/ft³ and Vmax is erosional velocity in ft/sec.

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Single Phase Fluid Flow Pressure Drop

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Pressure drop (ΔP) in a pipe is calculated using Darcy – Weisbach equation.

ΔP = f (L/D) (ρV²/2)

where, f is Darcy friction factor, L is pipe length, D is pipe diameter, ρ is fluid density and V is fluid velocity.

Friction factor is determined based on Reynolds’s number (Re).

Re = D V ρ / μ

where, μ is fluid viscosity.

For Re < 2100, flow is laminar and friction factor is calculated as following.

f = 64 / Re

For Re > 4000, flow is turbulent and friction factor is obtained by solving Colebrook White equation.

Colebrook White Equation

where, ε is pipe roughness.

For 2100 < Re < 4000, flow is in critical zone and there is no definite friction factor. Churchill equation (1977) predicts friction factor for entire flow regime from laminar to turbulent, it can be used to get an estimate for friction factor in critical zone.

Churchill Equation

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