Petrowiki Pressure Drop Equations

Pressure drop evaluation along pipelinesThe simplest way to convey a uid, in a contained system from Point A to Point B, is by means of a conduit or pipe (Fig. 1).

(/File%3AVol3_Page_319_Image_0001.png)

Fig. 1Fluid-ow system (courtesy ofAMEC Paragon).

Contents1 Piping design2 Bernoulli equation3 Reynolds number and Moody friction factor4 Pressure drop for liquid ow

4.1 General equation4.2 Hazen Williams equation

5 Pressure drop for gas ow5.1 General equation5.2 Simplied equation

5.2.1 Weymouth equation5.2.2 Panhandle equation5.2.3 Spitzglass equation

5.3 Application of the formulas5.3.1 Simplied gas formula5.3.2 Weymouth equation5.3.3 Panhandle equation5.3.4 Spitzglass equation

6 Multiphase ow6.1 Flow regimes

6.1.1 Bubble6.1.2 Slug ow6.1.3 Transition ow6.1.4 Annular mist ow

6.2 Two phase pressure drop6.3 Simplied friction pressure drop approximation for two phase ow6.4 Pressure Drop Because of Changes in Elevation

7 Pressure drop caused by valves and ttings7.1 Resistance coecients7.2 Flow coecients7.3 Equivalent lengths

8 Nomenclature9 References10 Noteworthy papers in OnePetro11 External links12 See also

Piping designThe minimum basic parameters that are required to design the piping system include, but are not limited to, the following.

The characteristics and physical properties of the uid.The desired mass-ow rate (or volume) of the uid to be transported.The pressure, temperature, and elevation at Point A.The pressure, temperature, and elevation at Point B.The distance between Point A and Point B (or length the uid must travel) and equivalent length (pressure losses) introduced by

Pressure drop evaluation along pipelines - http://petrowiki.org/Pressure_drop_evaluation_al...

1 of 14 03/19/2014 09:39 PM

valves and ttings.

These basic parameters are needed to design a piping system. Assuming steady-state ow, there are a number of equations, which arebased upon the general energy equation, that can be employed to design the piping system. The variables associated with the uid (i.e.,liquid, gas, or multiphase) aect the ow. This leads to the derivation and development of equations that are applicable to a particularuid. Although piping systems and pipeline design can get complex, the vast majority of the design problems encountered by theengineer can be solved by the standard ow equations.

Bernoulli equationThe basic equation developed to represent steady-state uid ow is the Bernoulli equation which assumes that total mechanical energyis conserved for steady, incompressible, inviscid, isothermal ow with no heat transfer or work done. These restrictive conditions canactually be representative of many physical systems.The equation is stated as

(/File%3AVol3_page_319_eq_001.PNG) (Eq. 1)whereZ = elevation head, ft,P = pressure, psi, = density, lbm/ft3,V = velocity, ft/sec,g = gravitational constant, ft/sec2,andHL = head loss, ft.

Fig. 2 presents a simplied graphic illustration of the Bernoulli equation.


Fig. 2Sketch four Bernoulli equation(courtesy of AMEC Paragon).

Darcys equation further expresses head loss as (/File%3AVol3_page_319_eq_002.PNG) (Eq. 2)

and (/File%3AVol3_page_319_eq_003.PNG) (Eq. 3)

whereHL = head loss, ft,f = Moody friction factor, dimensionless,L = pipe length, ft,D = pipe diameter, ft,V = velocity, ft/sec,g = gravitational constant ft/sec2,P = pressure drop, psi, = density, lbm/ft3,and


2 of 14 03/19/2014 09:39 PM

d = pipe inside diameter, in.

Reynolds number and Moody friction factorThe Reynolds number is a dimensionless parameter that is useful in characterizing the degree of turbulence in the ow regime and isneeded to determine the Moody friction factor. It is expressed as

(/File%3AVol3_page_320_eq_001.PNG) (Eq. 4)where = density, lbm/ft3,D = pipe internal diameter, ft,V = ow velocity, ft/sec,and = viscosity, lbm/ft-sec.

The Reynolds number for liquids can be expressed as (/File%3AVol3_page_320_eq_002.PNG) (Eq. 5)

where = viscosity, cp,d = pipe inside diameter, in.,SG = specic gravity of liquid relative to water (water = 1),Ql = liquid-ow rate, B/D,andV = velocity, ft/sec.

The Reynolds number for gases can be expressed as (/File%3AVol3_page_321_eq_001.PNG) (Eq. 6)

where = viscosity, cp,d = pipe inside diameter, in.,S = specic gravity of gas at standard conditions relative to air (molecular weight divided by 29),andQg = gas-ow rate, MMscf/D.

The Moody friction factor, f, expressed in the previous equations, is a function of the Reynolds number and the roughness of the internalsurface of the pipe and is given by Fig. 3. The Moody friction factor is impacted by the characteristic of the ow in the pipe. Forlaminar ow, where Re is < 2,000, there is little mixing of the owing uid, and the ow velocity is parabolic; the Moody friction factoris expressed as f = 64/Re. For turbulent ow, where Re > 4,000, there is complete mixing of the ow, and the ow velocity has auniform prole; f depends on Re and the relative roughness (/D). The relative roughness is the ratio of absolute roughness, , ameasure of surface imperfections to the pipe internal diameter, D. Table 9.1 lists the absolute roughness for several types of pipematerials.


Fig. 3Friction-factor chart (courtesyof AMEC Paragon).


Table 1


3 of 14 03/19/2014 09:39 PM

If the viscosity of the liquid is unknown, Fig. 4 can be used for the viscosity of crude oil, Fig. 5 for eective viscosity of crude-oil/watermixtures, and Fig. 6 for the viscosity of natural gas. In using some of these gures, the relationship between viscosity in centistokesand viscosity in centipoise must be used

(/File%3AVol3_page_321_eq_002.PNG) (Eq. 7)where = kinematic viscosity, centistokes, = absolute viscosity, cp,andSG = specic gravity.


Fig. 4Standardviscosity/temperature charts for liquidpetroleum products (courtesy ofASTM).


Fig. 5Eective viscosity of anoil/water mixture (courtesy of AMECParagon).


Fig. 6Hydrocarbon-gas viscosity vs.temperature (courtesy Western SupplyCo.).

Pressure drop for liquid owGeneral equationEq. 3 can be expressed in terms of pipe inside diameter (ID) as stated next.

(/File%3AVol3_page_323_eq_001.PNG) (Eq. 8)whered = pipe inside diameter, in.,f = Moody friction factor, dimensionless,L = length of pipe, ft,Ql = liquid ow rate, B/D,SG = specic gravity of liquid relative to water,andP = pressure drop, psi (total pressure drop).

Hazen Williams equationThe Hazen-Williams equation, which is applicable only for water in turbulent ow at 60F, expresses head loss as

(/File%3AVol3_page_323_eq_002.PNG) (Eq. 9)whereHL = head loss because of friction, ft,L = pipe length, ft,C = friction factor constant, dimensionless (Table 2),d = pipe inside diameter, in.,Ql = liquid ow rate, B/D,and


4 of 14 03/19/2014 09:39 PM

gpm = liquid ow rate, gal/min.


Table 2

Pressure drop can be calculated from (/File%3AVol3_page_323_eq_003.PNG) (Eq. 10)

Pressure drop for gas owGeneral equationThe general equation for calculating gas ow is stated as

(/File%3AVol3_page_323_eq_004.PNG) (Eq. 11)

wherew = rate of ow, lbm/sec,g = acceleration of gravity, 32.2 ft/sec2,A = cross-sectional area of pipe, ft2,V1 = specic volume of gas at upstream conditions, ft3/lbm,f = friction factor, dimensionless,L = length, ft,D = diameter of the pipe, ft,P1 = upstream pressure, psia,andP2 = downstream pressure, psia.

Assumptions: no work performed, steady-state ow, and f = constant as a function of the length.

Simplied equationFor practical pipeline purposes, Eq. 11 can be simplied to

(/File%3AVol3_page_327_eq_001.PNG) (Eq. 12)whereP1 = upstream pressure, psia,P2 = downstream pressure, psia,S = specic gravity of gas,Qg = gas ow rate, MMscf/D,Z = compressibility factor for gas, dimensionless,T = owing temperature, R,f = Moody friction factor, dimensionless,


5 of 14 03/19/2014 09:39 PM

d = pipe ID, in.,andL = length, ft.

The compressibility factor, Z, for natural gas can be found in Fig. 7.


Fig. 7Compressibility oflow-molecular-weight natural gases(courtesy of Natl. Gas ProcessorsSuppliers Assn.).

Three simplied derivative equations can be used to calculate gas ow in pipelines:The Weymouth equationThe Panhandle equationThe Spitzglass equation

All three are eective, but the accuracy and applicability of each equation falls within certain ranges of ow and pipe diameter. Theequations are stated next.

Weymouth equationThis equation is used for high-Reynolds-number ows where the Moody friction factor is merely a function of relative roughness.

(/File%3AVol3_page_327_eq_002.png) (Eq. 13)whereQg = gas-ow rate, MMscf/D,d = pipe inside diameter, in.,P1 = upstream pressure, psia,P2 = downstream pressure, psia,L = length, ft,T1 = temperature of gas at inlet, R,S = specic gravity of gas,andZ = compressibility factor for gas, dimensionless.

Panhandle equationThis equation is used for moderate-Reynolds-number ows where the Moody friction factor is independent of relative roughness and isa function of Reynolds number to a negative power.

(/File%3AVol3_page_330_eq_001.PNG) (Eq. 14)whereE = eciency factor (new pipe: 1.0; good operating conditions: 0.95; average operating conditions: 0.85),Qg = gas-ow rate, MMscf/D,d = pipe ID, in.,


6 of 14 03/19/2014 09:39 PM

P1 = upstream pressure, psia,P2 = downstream pressure, psia,Lm = length, miles,T1 = temperature of gas at inlet, R,S = specic gravity of gas,andZ = compressibility factor for gas, dimensionless.

Spitzglass equation


whereQg = gas-ow rate, MMscf/D,hW = pressure loss, inches of water,andd = pipe ID, in.

Assumptions:f = (1+ 3.6/ d + 0.03 d ) (1/100),T = 520R,P1 = 15 psia,Z = 1.0,andP = < 10% of P 1 .

Application of the formulasAs previously discussed, there are certain conditions under which the various formulas are more applicable. A general guideline forapplication of the formulas is given next.

Simplied gas formulaThis formula is recommended for most general-use ow applications.

Weymouth equationThe Weymouth equation is recommended for smaller-diameter pipe (generally, 12 in. and less). It is also recommended for shorterlengths of segments ( < 20 miles) within production batteries and for branch gathering lines, medium- to high-pressure (+/100 psig to> 1,000 psig) applications, and a high Reynolds number.

Panhandle equationThis equation is recommended for larger-diameter pipe (12-in. diameter and greater). It is also recommended for long runs of pipe ( >20 miles) such as cross-country transmission pipelines and for moderate Reynolds numbers.

Spitzglass equationThe Spitzglass equation is recommended for low-pressure vent lines < 12 in. in diameter (P < 10% of P1).The petroleum engineer will nd that the general gas equation and the Weymouth equation are very useful. The Weymouth equation isideal for designing branch laterals and trunk lines in eld gas-gathering systems.

Multiphase ow


7 of 14 03/19/2014 09:39 PM

Flow regimesFluid from the wellbore to the rst piece of production equipment (separator) is generally two-phase liquid/gas ow.The characteristics of horizontal, multiphase ow regimes are shown in Fig. 8. They can be described as follows:

Bubble: Occurs at very low gas/liquid ratios where the gas forms bubbles that rise to the top of the pipe.Plug: Occurs at higher gas/liquid ratios where the gas bubbles form moderate-sized plugs.Stratied: As the gas/liquid ratios increase, plugs become longer until the gas and liquid ow in separate layers.Wavy: As the gas/liquid ratios increase further, the energy of the owing gas stream causes waves in the owing liquid.Slug: As the gas/liquid ratios continue to increase, the wave heights of the liquid increase until the crests contact the top of thepipe, creating liquid slugs.Spray: At extremely high gas/liquid ratios, the liquid is dispersed into the owing-gas stream.


Fig. 8Two-phase-ow patterns inhorizontal ow (courtesy of AMECParagon).

Fig. 9[1] shows the various ow regimes that could be expected in horizontal ow as a function of the supercial velocities of gas andliquid ow. Supercial velocity is the velocity that would exist if the other phase was not present.


Fig. 9Horizontal multiphase-owmap (after Grith).[1]

The multiphase ow in vertical and inclined pipe behaves somewhat dierently from multiphase ow in horizontal pipe. Thecharacteristics of the vertical ow regimes are shown in Fig. 10 and are described next.


8 of 14 03/19/2014 09:39 PM


Fig. 10Two-phase-ow patterns invertical ow (courtesy of AMECParagon).

BubbleWhere the gas/liquid ratios are small, the gas is present in the liquid in small, variable-diameter, randomly distributed bubbles. Theliquid moves at a fairly uniform velocity while the bubbles move up through the liquid at diering velocities, which are dictated by thesize of the bubbles. Except for the total composite-uid density, the bubbles have little eect on the pressure gradient.

Slug owAs the gas/liquid ratios continue to increase, the wave heights of the liquid increase until the crests contact the top of the pipe, creatingliquid slugs.

Transition owThe uid changes from a continuous liquid phase to a continuous gas phase. The liquid slugs virtually disappear and are entrained inthe gas phase. The eects of the liquid are still signicant, but the eects of the gas phase are predominant.

Annular mist owThe gas phase is continuous, and the bulk of the liquid is entrained within the gas. The liquid wets the pipe wall, but the eects of theliquid are minimal as the gas phase becomes the controlling factor. Fig. 11[2] shows the various ow regimes that could be expected invertical ow as a function of the supercial velocities of gas and liquid ow.


Fig. 11Vertical-multiphase-ow map(after Yaitel et al.).[2]

Two phase pressure dropThe calculation of pressure drop in two-phase ow is very complex and is based on empirical relationships to take into account thephase changes that occur because of pressure and temperature changes along the ow, the relative velocities of the phases, andcomplex eects of elevation changes. Table 3 lists several commercial programs that are available to model pressure drop. Because allare based to some extent on empirical relations, they are limited in accuracy to the data sets from which the relations were designed. Itis not unusual for measured pressure drops in the eld to dier by 20% from those calculated by any of these models.


9 of 14 03/19/2014 09:39 PM


Table 3

Simplied friction pressure drop approximation for two phase owEq. 16 provides an approximate solution for friction pressure drop in two-phase-ow problems that meet the assumptions stated.

(/File%3AVol3_page_333_eq_001.PNG) (Eq. 16)whereP = friction pressure drop, psi,f = Moody friction factor, dimensionless,L = length, ft,W = rate of ow of mixture, lbm/hr,M = density of the mixture, lbm/ft3,andd = pipe ID, in.

The formula for rate of mixture ow is (/File%3AVol3_page_333_eq_002.PNG) (Eq. 17)

whereQg = gas-ow rate, MMscf/D,QL = liquid ow rate, B/D,S = specic gravity of gas at standard conditions, lbm/ft3 (air = 1),andSG = specic gravity of liquid, relative to water, lbm/ft3.

The density of the mixture is given by (/File%3AVol3_page_334_eq_001.PNG) (Eq. 18)

whereP = operating pressure, psia,R = gas/liquid ratio, ft3/bbl,T = operating temperature, R,SG = specic gravity of liquid, relative to water, lbm/ft3,S = specic gravity of gas at standard conditions, lbm/ft3 (air = 1),andZ = gas compressibility factor, dimensionless.

The formula is applicable if the following conditions are met:P is less than 10% of the inlet pressure.Bubble or mist exists.There are no elevation changes.There is no irreversible energy transfer between phases.


10 of 14 03/19/2014 09:39 PM

Pressure Drop Because of Changes in ElevationThere are several notable characteristics associated with pressure drop because of elevation changes in two-phase ow. The owcharacteristics associated with the elevation changes include:

In downhill lines, ow becomes stratied as liquid ows faster than gas.The depth of the liquid layer adjusts to the static pressure head and is equal to the friction pressure drop.There is no pressure recovery in the downhill line.In low gas/liquid ow, the ow in uphill segments can be liquid "full" at low ow rates. Thus, at low ow rates, the total pressuredrop is the sum of the pressure drops for all of the uphill runs.With increased gas ow, the total pressure drop may decrease as liquid is removed from uphill segments.

The pressure drop at low ow rates associated with an uphill elevation change may be approximated with Eq. 19. (/File%3AVol3_page_335_eq_001.PNG) (Eq. 19)

wherePZ = pressure drop because of elevation increase in the segment, psi,SG = specic gravity of the liquid in the segment, relative to water,andZ = increase in elevation for segment, ft.

The total pressure drop can then be approximated by the sum of the pressure drops for each uphill segment.

Pressure drop caused by valves and ttingsOne of the most important parameters aecting pressure drop in piping systems is pressure loss in the ttings and valves, which isincorporated in the system. For piping systems within production facilities, the pressure drop through ttings and valves can be muchgreater than that through the straight run of pipe itself. In long pipeline systems, the pressure drop through ttings and valves canoften be ignored.Resistance coecientsThe head loss in valves and ttings can be calculated with resistance coecients as

(/File%3AVol3_page_336_eq_001.PNG) (Eq. 20)whereHL = head loss, ft,Kr = resistance coecient, dimensionless,D = pipe ID, ft,andV = velocity, ft/sec.

The total head loss is the sum of all Kr V2/2g.The resistance coecients Kr for individual valves and ttings are found in tabular form in a number of industry publications. Mostmanufacturers publish tabular data for all sizes and congurations of their products. One of the best sources of data is the Crane Flowof Fluids, technical paper No. 410. [3] The Natural Gas Processors Suppliers Assn. (NGPSA) Engineering Data Book[4] and Ingersoll-Rands Cameron Hydraulic Data Book[5] are also good sources of references for the information. Some examples of resistancecoecients are listed in Tables 4 and 5.


11 of 14 03/19/2014 09:39 PM


Table 4


Table 5


Table 5 (Cont'd)


Table 5 (Cont'd)


Table 5 (Cont'd)

Flow coecientsThe ow coecient for liquids, CV, is determined experimentally for each valve or tting as the ow of water, in gal/min at 60F for apressure drop of 1 psi through the tting. The relationship between ow and resistance coecients can be expressed as

(/File%3AVol3_page_336_eq_002.PNG) (Eq. 21)In any tting or valve with a known CV, the pressure drop can be calculated for dierent conditions of ow and liquid properties withEq. 22.

(/File%3AVol3_page_336_eq_003.PNG) (Eq. 22)whereQL = liquid-ow rate, B/D,andSG = liquid specic gravity relative to water.

Again, the CV is published for most valves and ttings and can be found in Crane Flow of Fluids,[3] Engineering Data Book,[4] CameronHydraulic Data Book,[5] as well as the manufacturers technical data.Equivalent lengthsThe head loss associated with valves and ttings can also be calculated by considering equivalent "lengths" of pipe segments for eachvalve and tting. In other words, the calculated head loss caused by uid passing through a gate valve is expressed as an additionallength of pipe that is added to the actual length of pipe in calculating pressure drop.All of the equivalent lengths caused by the valves and ttings within a pipe segment would be added together to compute the pressuredrop for the pipe segment. The equivalent length, Le, can be determined from the resistance coecient, Kr, and the ow coecient, CV,using the formulas given next.


(/File%3AVol3_page_341_eq_002.PNG) (Eq. 24)and


12 of 14 03/19/2014 09:39 PM

(/File%3AVol3_page_341_eq_003.PNG) (Eq. 25)whereKr = resistance coecient, dimensionless,D = diameter of the pipe, ft,f = Moody friction factor, dimensionless,d = pipe ID, in.,andCV = ow coecient for liquids, dimensionless.

Table 6 shows equivalent lengths of pipe for a variety of valves and ttings for a number of standard pipe sizes.


Table 6

NomenclatureZ = elevation head, ft,P = pressure, psi, = density, lbm/ft3,V = velocity, ft/sec,g = gravitational constant, ft/sec2,HL = head loss, ft.f = Moody friction factor, dimensionless,L = pipe length, ft,D = pipe diameter, ft,P = pressure drop, psi, = viscosity, lbm/ft-sec.SG = specic gravity of liquid relative to water (water = 1),Ql = liquid-ow rate, B/D,S = specic gravity of gas at standard conditions relative to air (molecular weight divided by 29),Qg = gas-ow rate, MMscf/D. = kinematic viscosity, centistokes, = absolute viscosity, cpQl = liquid ow rate, B/D,w = rate of ow, lbm/secP1 = upstream pressure, psiaP2 = downstream pressure, psia.hW = pressure loss, inches of water,W = rate of ow of mixture, lbm/hr,M = density of the mixture, lbm/ft3P = operating pressure, psia,R = gas/liquid ratio, ft3/bbl,T = operating temperature, R,PZ = pressure drop because of elevation increase in the segment, psi,Z = increase in elevation for segment, ft.


13 of 14 03/19/2014 09:39 PM

HL = head loss, ft,Kr = resistance coecient, dimensionlessCV = ow coecient for liquids, dimensionless.Kr = resistance coecient, dimensionless,

References 1.0 1.1 Grith, P. 1984. Multiphase Flow in Pipes. J Pet Technol 36 (3): 361-367. SPE-12895-PA. http://dx.doi.org/10.2118/12895-PA (http://dx.doi.org/10.2118/12895-PA).

1.

2.0 2.1 Taitel, Y., Bornea, D., and Dukler, A.E. 1980. Modelling ow pattern transitions for steady upward gas-liquid ow invertical tubes. AIChE J. 26 (3): 345-354. http://dx.doi.org/10.1002/aic.690260304 (http://dx.doi.org/10.1002/aic.690260304).

2.

3.0 3.1 Crane Flow of Fluids, Technical Paper No. 410. 1976. New York City: Crane Manufacturing Co.3. 4.0 4.1 Engineering Data Book, ninth edition. 1972. Tulsa, Oklahoma: Natural Gas Processors Suppliers Assn.4. 5.0 5.1 Westway, C.R. and Loomis,A.W. ed. 1979. Cameron Hydraulic Data Book, sixteenth edition. Woodcli Lake, New Jersey:Ingersoll-Rand.

5.

Noteworthy papers in OnePetroUse this section to list papers in OnePetro that a reader who wants to learn more should denitely read

External linksUse this section to provide links to relevant material on websites other than PetroWiki and OnePetro

See alsoPiping and pipeline systems (/Piping_and_pipeline_systems)Pipelines (/Pipelines)Pipeline pigging (/Pipeline_pigging)Pipeline design consideration and standards (/Pipeline_design_consideration_and_standards)PEH:Piping and Pipelines (/PEH%3APiping_and_Pipelines)


14 of 14 03/19/2014 09:39 PM

Documents

Petrowiki Pressure Drop Equations