Fundamentals of Compressible Flows in Pipelines

Fundamentals of Compressible Flows in pipelines

Dr. Ahmed Elmekawy Fall 2018

1 Compressible Flow

Flow in a Single Pipe – Branched Pipes - Network of Pipes – Unsteady Flow

( Density is Constant W.R.T Pressure)

• Compressible Flow Study Density is VARIABLE W.R.T Pressure or Compressibility of the Fluid Egyptian Liquefied Natural Gas Plant (ELNG)

Ov er v iew Transportation of natural gas

• Pipeline

• By ship as liquefied natural gas (LNG)

• By ship as compressed natural gas (CNG)

Compressible Flow

• Incompressible Flow Study

(1) Mathematical Formulations f l Q2 Darcy-Weisbach Equation h = 0.8 l g d 5 (2) Empirical Formulae Hazen William Equation Colbroack Equation Compressible Flow

• Flowrate Analysis m=  Q =   AV

Incompressible Flow: Density is Constant… Thus, m/  = Q = AV

Constant Constant Compressible Flow

• Flowrate Analysis m=  Q =   AV Constant

Variable Variable Volume Flowrate

We Can NOT Use Darcy-Weisbach Equation Directly in Compressible Flow Analysis… ( Variable HEAD LOSS !!! ) Compressible Flow

• Compressible Flow Study

1 2 N

P 

Assume a Pipeline 1-N is divided into sections1-2, 2-3, 3-4, …etc. Pressure Loss or Pressure Change over any section is decreased. Thus, Density Change is decreased Too

Density can be considered as a CONSTANT Value for each pipe segment only. Compressible Flow

• Compressible Flow Study

1 2 N

P      Const.

Darcy-Weisbach Equation Could Be Used Now on Every Section of The Pipe Compressible Flow

1 2 N

P      Const. f  x (m /  ) 2 h = 0.8 1 l1−2 g d 5 Compressible Flow

• Compressible Flow Study

f  x (m /  ) 2 h = 0.8 1 l1−2 g d 5 P1 / 1 = RT1

P1 Assuming Constant Density 1 = RT1 Compressible Flow

• Compressible Flow Study

f  x (m /  ) 2 h = 0.8 1 l1−2 g d 5 P1 − P2 = P

P1− P2=   g hl1−2

P2= P1−   g hl1−2 Compressible Flow

• Compressible Flow Study Pressure Values at 1 and 2 are known With The Same Procedure We Could get the Value of the end pressure at the pipeline outlet and overall head loss.

f  x (m /  )2 h = 0.8 2 l2−3 g d 5 Compressible Flow

• Example Compressible Flow

• Compressible Flow Study

Values obtained from the previous procedures have some error because of a lot of criterion:

1. Ideal Gas Assumption

2. Assuming constant Density in each section.

3. Accuracy depends on Section Numbers (directly) Compressible Flow

• Compressible Flow Study

Compressible Flow Analysis Depends on Supplied Pressure or Delivered Pressure

Compressible Flow Gas Well Storage

P1 = Known P2 = ????

Compressor Industry Station Compressible Flow P1 = ???? P2 = Known Compressible Flow

Compressibility Factor (Z)

Low or Moderate Pressure-Temperature Conditions P = R T At High or Very Low Pressure-Temperature Conditions P = Z  R T

Where Z is a dimensionless factor represents the fluid behavior deviation of ideal gas to account for higher pressure and temperature. At low pressures and temperatures Z is nearly equal to 1.00 whereas at higher pressures and temperatures it may range between 0.75 and 0.90 Compressible Flow

Compressibility Factor (Z)

At High Pressure-Temperature Conditions P = Z  R T

Z = Fn ( P , T ) Compressibility Factor could be obtained through Engineering Tables or Charts as follows Compressible Flow

• Compressible Factor (Z) The critical temperature of a pure gas is that temperature above which the gas cannot be compressed into a liquid, however much the pressure. The critical pressure is the minimum pressure required at the critical temperature of the gas to compress it into a liquid. As an example, consider pure methane gas with a critical temperature of 343 R and critical pressure of 666 psia. The reduced temperature of a gas is defined as the ratio of the gas temperature to its critical temperature, both being expressed in absolute units (R or K). It is therefore a dimensionless number. Similarly, the reduced pressure is a dimensionless number defined as the ratio of the absolute pressure of gas to its critical pressure. Therefore we can state the following: Compressible Flow

• Compressible Factor (Z) Using the preceding equations, the reduced temperature and reduced pressure of a sample of methane gas at 70 F and 1200 psia pressure can be calculated as follows

The Standing-Katz chart, Fig. can be used to determine the compressibility factor of a gas at any temperature and pressure, once the reduced pressure and temperature are calculated knowing the critical properties Compressible Flow Compressible Flow

• Compressible Factor (Z) Another analytical method of calculating the compressibility factor of a gas is using the CNGA equation as follows:

The CNGA equation for compressibility factor is valid when the average gas pressure Pavg is greater than 100 psig. For pressures less than 100 psig, compressibility factor is taken as 1.00. It must be noted that the pressure used in the CNGA equation is the gauge pressure, not the absolute pressure. Compressible Flow

• Compressible Factor (Z) Compressible Flow

• Compressible Factor (Z) Fundamentals of Gas Transmission Governing Equations

Real Gas Law

P v = Z R T P: Pressure α v: Specific Volume R: Gas Constant T: Temperature ρ: Gas Density Z: Compressibility Factor

Continuity Equation o m =.A .u = Mass Flow Rate A: Cross SectionalArea u: Gas Velocity

Bernoulli’s Equation

D : Diameter u : Gas Velocity A : Cross SectionalArea τ : Shear Stress v : Specific Volume u du : Kinetic Energy dP : Pressure Differential v dP : Pressure Energy dH : Elevation Differential dH : Potential Energy (π D dY v/A) . τ : Friction Energy Loss The general flow Equation It can be used instead of dividing the pipe line into segments The general flow Equation

Widely used Steady State Flow Equations

Transmission Factor Flow Description Equation Formula* (F)

0.538 High Pressure T (P 2 −P 2 )D 5  1  0.462 b 1 2 0 .071 High Flow Rate Fritzsche Q b = 1.72     5 . 145 ( Re .D ) Pb  Tf L  G  Large Diameter

2 2 0.5  3.7D High Pressure T P − P  3.7 D 4 log AGA Fully b 1 2 2.5   High Flow Rate Turbulent Q b = 0.4696   log .D     K e  Pb GTf Zav  K e Medium to Large Diameter Medium to High Pressure 1.02 2 2 0.51  T   P − P  0 . 01961 b 1 2 2.53 16 . 49 (R e ) High Flow Rate Panhandle B Q b = 2.431    D   0.96 Large Diameter  Pb  L Tf G Z av  Used when Re < 40 million 0.5 T  P 2 − P 2  K 1 .4126F   2 .5 Colebrook- Q = 0 .4696 b 1 2 log e + D k e 2 .5 F Used when the flow is near the b   −2 L o g  +  White Pb L G Tf Zav  3.7 D Re  3.7 D Re  transition zone (border line)

5 /9 T P 2 − P 2   D 8/ 3  IGT b 1 2 Used in Natural Gas Distribution Q b = 0.6643     0 . 1 Pb  LTf  G4 / 9  1 / 9 4 . 619 ( Re ) Distribution   Networks.

 2 2 0.575  2.275  T b P1 − P 2   D Q b = 0.4973   3 . 35 ( Re ) 0 .13 Mueller P LT G 0.425 0.15  b  f   

0.539  1.078  2 2   2.618  Medium to High Pressure T b P1 − P 2 D 0 .073 Panhandle A Q b = 2.45     0.461  6 . 872 (Re ) Moderate Flow Rate Pb  L Tf Zav  G   Medium Diameter

0.5 T  (P 2 −P 2 )D16 / 3  High Pressure b 1 2 1 /6 Weymouth Q b = 1.3124   11 . 19 D High Flow Rate Pb  LGTf  Large Diameter Summary of Pressure Drop Equations Equation Application

Fundamental flow equation using friction or transmission factor; used General Flow with Colebrook-White friction factor or AGA transmission factor

Friction factor calculated for pipe roughness and Reynolds number; Colebrook-White most popular equation for general gas transmission pipelines

Modified Modified equation based on U.S. Bureau of Mines experiments; gives Colebrook-White higher pressure drop compared to original Colebrook equation

Transmission factor calculated for partially turbulent and fully AGA turbulent flow considering roughness, bend index, and Reynolds number

Panhandle A Panhandle equations do not consider pipe roughness; instead, an Panhandle B efficiency factor is used; less conservative than Colebrook or AGA

Does not consider pipe roughness; uses an efficiency factor Weymouth used for high-pressure gas gathering systems; most conservative equation that gives highest pressure drop for given flow rate

Does not considerPowerpoint pipeTemplates roughness; uses an efficiency factor used on IGT Page 227 gas distribution piping Determining the Flow Regime

Flow regimes experienced in gas transmission:

Partially Turbulent Flow Fully Turbulent Flow

Reynold’s Number Q G The units used are: Re =45. b Qb : ft³/hr D G : Dimensionless D : Inches

The Prandtl-Von Karman Equation

    If Reynold’s Number is larger that the 1 = 4.log Re − 0.6 f  1  Prandtl-Von Karman’s Reynold’s Number, the flow is Fully Turbulent.  f  Moody Chart Hydraulic Analysis Parameters

Gas Gravity:

The ratio of gas molecular weight to air molecular weight.

Mg G = Mg depends on the GasComposition. Ma

Compressibility Factor:

Two methods were used:

Van Der Waals Equation – Long iterative solution

3 2 Z −a 1 Z +a 2 Z −a 3 =0 .0 a1, a2, a3 are function of pseudo-reduced properties.

CNGA Equation – Direct solution 1 Z = Where,   P 344400(10)1.785G   avg  Pavg : Average Gauge Gas Pressure, psig 1 +  Tf : Fluid Temperature, R   T3.825    f  G : Gas Gravity

After comparing both equations, the results of the CNGA Equation were very accurate to Van Der Waals Equation. The comparison was done at constant temperature and for the same gascomposition. Temperature Profile Temperature has a significant effect on the pressuredrop. As temperature decreases Gas Viscosity decreases Pressure Drop decreases

Temperature Profile Calculation

− UA o m CP Ti+1 =(Ti −Tg ).e

Where: Pressure

Ti+1 : Downstream Temperature Ti : Upstream Temperature U : Overall Heat Transfer Coefficient A : Heat Transfer Area (Lateral) m : Mass Flow Rate Pipeline Length Cp : Gas Specific Heat Tg : Ground/Surrounding Temperature Temperature Profile Temperature has a significant effect on the pressuredrop. As temperature decreases Gas Viscosity decreases Pressure Drop decreases

Temperature Profile Calculation

− UA o m CP Ti+1 =(Ti −Tg ).e

Where: Pressure

Studies have proven that a pressure drop of (15 -25 Kpa/Km) or (3.5 -5.85 psi/mile) is optimal. Pressure drops below 15 Kpa/Km are an indication that too many facilities have been installed) Compressible Flow

• General Flow Equation for Compressible Flow (Empirical)

T P2 −P2 Q =1.149410−3  b  D 2.5 1 2 Pb G Tf l  z  f Compressible Flow

T P2 − P2 Q =1.149410−3  b  D 2.5 1 2 Pb G Tf l  z  f

Q Gas Flow rate (m3/day) L Pipe Length (m) D Diameter (mm) G Gas Gravity (Specific Gravity) f Friction Coefficient (Dimensionless)

P1 Upstream Pressure or Supplied (kPa)

P2 Downstream Pressure or Delivered (kPa) Compressible Flow

T P2 − P2 Q =1.149410−3  b  D 2.5 1 2 Pb G Tf l  z  f

Z Compressibility Factor

Pb Base Pressure (kPa) - Reference Value - 0 Tb Base Temperature (K ) - Reference Value -

Tf Flow Average Temperature (K0) Compressible Flow

Base Parameters (P,T)

For constant flow rate (m=const.) m = 1  Q1 = 2  Q2  = P / RT

Q1(P1 /T1 )= Q2 (P2 /T2 )

Thus, the flowrate could be obtained W.R.T standard flowrate at standard atmospheric pressure and temperature as a reference… Compressible Flow

Base Parameters (P,T)

 P  P  Q  = Q b   T   Tb 

And the general flow equation could be… Compressible Flow

Base Parameters (P,T)

2 2 Pb −3 2.5 P1 −P2 Qb  =1.149410  D Tb G Tf l  z  f

 P  P2 −P2 Q   =1.149410 −3  D 2.5 1 2  T  G Tf l  z  f

Compressible Flow

Erosional Velocity of gas in pipe flow Gas Composition Dahshour –Assiut- Aswan Gas pipe Line Compressible Flow

Example Compressible Flow

Example