Compressible flow implies that variations exist in the density of a fluid. The vari-ations are caused by pressure and temperature changes from one point to another.
The rate of change is important in the analysis of compressible flow and is con-nected closely with the velocity of sound. When dealing with compressible flu-ids, when the density change is gradual and not more than a few percent, the flow can be treated as incompressible by using an average density. If the change in pressure divided by the initial pressure is greater than 0.05, however, the effects of compressibility must be considered. In plastic piping systems, compressible flow is encountered most often in gases, such as natural gas. The following section pro-vides many of the frequently used formulas in the design of plastic piping for nat-ural gas applications. The following variables will apply to each equation described in the following pages:
Cw⫽ Hazen-Williams Coefficient d⫽ Inside pipe diameter, inches E⫽ Pipeline efficiency factor
e⫽ 2.71828, natural logarithm base F⫽ Transmission factor, dimensionless
f⫽ Friction factor, dimensionless G⫽ Specific gravity of gas, dimensionless H⫽ Elevation, ft
⌬H ⫽ Change in elevation
PLASTIC PIPING HANDBOOK 3.12
h⫽ H2O pressure, in.
k⫽ Internal pipe wall roughness, in.
Lf⫽ Pipe length, ft Lm⫽ Pipe length, miles
P⫽ Pressure, lb/in2(psia)
⌬P ⫽ Change in pressure, psia P1⫽ Inlet or upstream pressure, psia P2⫽ Outlet or downstream pressure, psia Patm⫽ Average atmospheric pressure, psia
Pavg⫽ Average pressure along the pipeline segment, psia Pb⫽ Base pressure, psia
Qh⫽ Volumetric flow rate, ft3/hr (cfh) Qd⫽ Volumetric flow rate, ft3/day (cfd)
Re⫽ Reynolds Number, dimensionless
⌬T ⫽ Change in temperature
Tavg⫽ Average gas flowing temperature, Rankine Tb⫽ Base temperature, Rankine
T1⫽ Initial temperature of the gas, Rankine
T2⫽ Temperature of the gas under the second conditions, Rankine V1⫽ Volume of the gas in original condition, ft3
V2⫽ Volume of the gas in second set of conditions, ft3 v⫽ Gas velocity, ft/sec
W⫽ Weight of the gas, lb
Z⫽ Gas compressibility factor, dimensionless
⫽ Kinematic viscosity, ft3/sec
⫽ Absolute (dynamic) viscosity, lbm/foot-second
⫽ Density of fluid, lb/ft3
Table 3.6 lists some general formulas used for gas hydraulics:
Gas Laws
Boyle’s Law. If the temperature of the gas remains constant, the volume of a quantity of gas will vary inversely as the absolute pressure. This is expressed mathematically by Boyle’s Law as:
When using Boyle’s Law, we are usually interested in the volume at the sec-ond set of csec-onditions. For this purpose, the equation often is rewritten as:
V2⫽ V1
EXAMPLE 1
A quantity of gas at 70 psia has a volume of 1000 cubic feet. If the gas is com-pressed to 150 psia, what volume would it occupy? The barometric pressure is 14.7 psia and the temperature remains constant.
V2⫽ 1000 ⫽ 514.3 ft3
Charles’ Law. Charles’ Law states that the volume occupied by a fixed amount of gas is directly proportional to its absolute temperature, if the pressure remains constant. This empirical relation was formulated by the French physicist J.A.
Charles about 1787 and reaffirmed later by Joseph Gay-Lussac. Charles’ Law is expressed as:
⫽
Like the example above, we usually are interested in the volume at a second set of temperature conditions, so this equation often is rewritten as:
V2⫽ V1
Charles’ Law also states that if the volume of a quantity of gas does not change, the absolute pressure will vary directly as the absolute temperature. This is expressed as:
⫽
If we are interested in the pressure at a second temperature condition, the equa-tion can be expressed as:
P2⫽ P1
EXAMPLE 2
A gas has a volume of 500 cubic feet when the temperature is 45°F and the pressure is 20 pounds per square inch gauge (psig). If the temperature is changed to 90°F and the pressure stays the same, what will be the gas volume?
V2⫽ 50090⫹ 460⫽ 544.6 ft3
What would the pressure be for the gas above if the volume remains constant and the temperature changes from 45°F to 80°F? Atmospheric pressure is 14.7 psia.
P2⫽ (20 ⫹ 14.7) ⫽ 37.8 psia Boyle’s and Charles’ laws can be combined and expressed as:
⫽
We can substitute known values in the above equation and solve for any one unknown.
Avogadro’s Law. Avogadro’s Law states that equal volumes of gases at the same temperature and pressure contain an equal number of molecules. From this law, we see that the weight of a volume of gas is a function of the weight of the molecules. In addition, at a certain volume, the gases weigh in pounds the numer-ical value of its molecular weight. This is known as the volume. The mol-volume is 378.9 cubic feet for gases at 60°F and 14.73 psia. Table 3.7 lists the molecular weights for some of the compounds often associated with natural gas.
The molecular weight for methane is 16.043. From the mol-volume explanation above it can be determined that 378.9 cubic feet of methane at 60°F and 14.73 psia weighs 16.043 pounds.
Ideal Gas Law. The Ideal Gas Law is the basic law for gas equations. It is used in many arrangements but often is written as:
PV⫽nRT where P⫽ Pressure of the gas
V⫽ Volume of the gas
n⫽ Number of pound-mols of the gas
R⫽ Universal gas constant which varies depending on the pressure, volume, and temperature of the gas.
The number of pound-mols is equal to the weight of the gas divided by the molecular weight of the gas. Therefore, we write the Ideal Gas Law as:
PV⫽ 10.722 T where P⫽ Pressure of the gas
V⫽ Volume of the gas W⫽ Weight of the gas, pounds M⫽ Molecular weight of the gas
T⫽ Temperature of the gas, Rankine.
The constant 10.722 is from the generally used value for the universal gas constant of 1544 when the pressure is in lb/ft2absolute.
ᎏW
The above equation also can be written in many forms to find an unknown.
An often-used equation to find the weight of a quantity of gas is:
W⫽ 0.0933
EXAMPLE 3
Find the weight of a gas in a 2000-cubic foot tank. The gas pressure is 200 psig at 90°F. The molecular weight of the gas is 16.535 and the barometric pressure
ᎏMVP T
PLASTIC PIPING HANDBOOK 3.16
TABLE 3.6 General Formulas for Gas
Formulas Symbols
Gas velocity: V⫽ Velocity in feet per second
V⫽ D⫽ Inside pipe diameter inches
Alternate method with gas temperature as
QMcfh⫽ Flow rate in Mcfh a factor:
Qcfh⫽ Flow rate in cfh
P2⫽ Downstream pressure in psia
V⫽ T⫽ Gas temperature in degrees Rankine
Pipeline pack for gas pipelines: V⫽ Pipeline volume in scf per 1000 feet v⫽ d2P1.378 For Pavg⬍ 100 D⫽ Inside pipe diameter
v⫽ For Pavg⬎ 100 P1⫽ Upstream pressure in psia P2⫽ Downstream pressure in psia Pavg⫽ 冢P1⫹ P2⫺ 冣 Pavg⫽ Average pressure in the pipeline
Z⫽ Gas compressibility factor
Pipeline blowdown time: BTm⫽ Blowdown time in minutes
BTm⫽ .0588 D⫽ Inside pipe diameter
Dblowdown⫽ Inside diameter of blowdown pipe inches
P1⫽ Upstream pressure in psia P2⫽ Downstream pressure in psia Sg⫽ Gas specific gravity L⫽ Length of pipe section in miles Fc⫽ Blowdown valve choke factor:
Ideal nozzle: 1.0
is 14.7 psia. This problem can be written as:
W⫽ ⫽ 1204 pounds
Gas Pipeline Hydraulics
Several equations are used in gas pipeline hydraulics and selecting the proper equation is as important as doing the calculations correctly. Selecting the correct equation requires an understanding of the equations and parameters for the sys-tem being designed. Because of the assumptions made in each equation, a slight difference in the calculation can result when the equations are compared. The fol-lowing section provides some of the frequently used equations and when they should be used.
Colebrook-White Equation. The Colebrook-White Equation is recommended for those who are not familiar with pipeline flow equations because it produces the greatest consistency of accuracy over a wide range of variables.
Qh⫽ 234.8 冢 冣.5冢 冣.5D2.5E
Panhandle A Equation. This equation is best used for pipelines with Reynolds numbers in the range of 5⫻106to 11⫻106. The average pipeline efficiency fac-tor used in this equation is 92 percent. This number is based on actual empirical experience with the metered gas flow rates corrected to standard conditions. For larger diameter pipelines, the pipeline efficiency factor can be as high as 98 per-cent. With this equation, pipeline efficiency factors should be reduced for smaller pipe diameters. The Panhandle A Equation provides a reasonable approximation for partially turbulent flow. For fully turbulent flow, this equation does not pro-duce accurate results. In the fully turbulent flow region, the Panhandle B Equation is recommended.
where ⌬P ⫽ P12⫺ P22
and Pavg⫽
[
P1⫹ P2⫺]
E⫽ 1 for new straight pipes without fittings or pipe diameter changes.
E⫽ 0.95 for good pipe typically during the first 1-2 years of operation.
E⫽ 0.92 for average operating conditions.
E⫽ 0.85 for poor operating conditions.
Panhandle B Equation. This equation is best used for pipelines with Reynolds numbers in the range of 4⫻106to 40⫻106. The Panhandle B Equation is used in the design of large diameter, high-pressure, long pipelines. The average pipe-line efficiency factors are the same as the Panhandle A Equation. The Panhandle A Equation provides a reasonable approximation for turbulent flow conditions.
where ⌬P ⫽ P12⫺ P22
and Pavg⫽
[
P1⫹ P2⫺]
Weymouth Equation. This is one of the older equations, but it still is used widely for natural gas distribution and gathering systems. The Weymouth Equation
orig-(P1P2)
inally was developed from data taken from low- to medium-pressure pipelines. The results obtained are conservative when the equation is used for higher-pressure pipe-lines. The Weymouth Equation typically is used for pipe diameters less than 6 inches in pressure ranges greater than 1.5 psig and less than 300 psig. This equation is not recommended for gas transmission through long pipelines.
where Pavg⫽
[
P1⫹ P2⫺]
IGT-Improved Equation. This equation is used widely for natural gas distribution systems. When used for higher-pressure pipelines, the results obtained are con-servative. The IGT Equation typically is used for pressure ranges between 1.5 and 100 psig. This equation is not recommended for gas transmission through long pipelines.
Qh⫽ 664 冢 冣.556D2.667E
Mueller-High-Pressure Equation. This equation is used in natural gas distribu-tion systems with pressures above 1 psig.
Qh⫽ 2826冢 冣.575D2.275E
where ⌬P ⫽ P12⫺ P22
Mueller-Low-Pressure Equation. This equation is used in natural gas distribu-tion systems with pressures below 1 psig.
Qh⫽ 735.4 冢 冣.575D2.725E
where ⌬h ⫽ h1⫺ h2
Spitzglass Equation. This equation is best used for pipe diameters of 10 inches or less with pressures between 1.5 and 50 psig.
Qh⫽ 2209ᎏTPbb冢 冣ᎏ1f .5冢ᎏPGT12⫺ PavgL2f2冣.5D2.5E
where
Another popular version of the Spitzglass Equation often is used for natural gas distribution systems operating at 1 psig or less:
where Qh⫽ Flow rate, cubic feet per hour at 14.7 psia and 60°F h⫽ Static pressure head, inches of water.
REFERENCES
1. “Charles’s law,” Encyclopædia Britannica Online. http://subscribe.eb.com [Accessed February 9, 2000].
2. “Fluid,” Encyclopædia Britannica Online. [Accessed February 12, 2000].
3. ASME B31, Code for Pressure Piping, Section B31.8, Gas Transmission and Distri-bution Piping Systems, American Society of Mechanical Engineers, New York, 1989 ed.
4. Nayyar, M.L. 1992. Piping Handbook. New York: McGraw-Hill, Inc.
5. Chevron, Plexco/Spirolite Engineering Manual. 1998. Vol 2, 2d Edition.
6. Irving Granet, 1996. Fluid Mechanics. New Jersey: Prentice Hall.
PLASTIC PIPING HANDBOOK 3.20
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