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Pressure, Volume, Temperature (PVT) Behaviour

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Pressure, Volume, Temperature (PVT) Behaviour

1998 AEA Technology plc - All Rights ReservedNat Gas Lec 1.pdf

2 Lecture

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LectureThe Pressure, Volume, Temperature (PVT) Behaviour moduleintroduces you to some of the basic concepts needed in the GasProcessing industry. You will learn to use the GPSA databooks todetermine PVT behaviour and properties for Gas Processing Streams.

Learning ObjectivesOnce you have completed this module, you will understand thefollowing concepts:

Concept of the Mole Ideal and Real Gas Laws PVT behaviour Equations of State Calculation of heating value for natural gas

TerminologyCalorimeter

An apparatus which is used to determine the heating value of acombustible material.

Compressibility Factor

A factor, usually expressed as "Z", which gives the ratio of the actualvolume of gas at a given temperature and pressure to the volume of gaswhen calculated by the ideal gas law.

Lecture 3

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Critical Pressure

The vapour pressure of a substance at its critical temperature.

Critical Temperature

For a pure component, the maximum temperature at which thecomponent can exist as a liquid.

Gas Constant (R)

The constant multiplier in the Ideal Gas Law. Numerically, R=PV/T, if V is the volume of one mole of an ideal gas at temperature T and pressure P.

Gross Heating Value

Gross heating value, MJ/m3, is the amount of energy transferred asheat, upon the combustion of one standard cubic metre of gas in anideal combustion reaction, at standard pressure and temperature, withall combustion products cooled to the standard temperature and allwater formed during the combustion process being condensed.

Latent Heat of Condensation

Amount of enthalpy (energy) per unit mass, required to change phasefrom liquid to vapour. Energy must be supplied to vaporize; it isreleased on condensation.

Net Heating Value

Net heating value, MJ/m3, is the amount of energy transferred as heatin an ideal combustion reaction, but with the water formed in thecombustion remaining in the vapour phase.

Relative Density

The Relative Density of a gas is the ratio of the densities of the gas andair both at standard conditions of 15C and 101.325 kPa (abs).

4 Lecture

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Theoretical Foundations

Concept of the Mole

Natural gas is a fluid that will fully occupy the space in the vessel or pipein which it is contained. At low pressures, in the order of one or twoatmospheres, the volume of an amount of gas will vary inversely withabsolute pressure at constant temperature (Boyles Law), and inproportion to absolute temperature at constant pressure (CharlesLaw). It is also accepted that at the same pressure and temperature, agiven volume of any gas will contain the same number of molecules asthe same volume of any other gas (Avogadros hypothesis). This leadsto the concept of the "mole", with the symbol "mol". For engineeringpurposes, a mole is simply an amount of mass of any substance (solid,liquid or gas), which has the same number of molecules as a mole ofany other substance. By assuming the molar mass of one element(0.012 kg of Carbon for example), the relative mass of a mole of all otherelements and compounds (having the same number of molecules as0.012 kg of Carbon) can be determined. The molar mass of compoundsencountered in the gas industry is given in Fig. 23-2, Column A in theData Book, in kilograms per kilomole (kg/kmol).

The equivalent molar mass of natural gas mixtures is calculated on aweighted mole fraction basis, using the molar mass of each individualcompound and multiplying by the mole fraction of that compound inthe gas mixture. The relative molecular mass of gas is designated by theletter M. The relative molecular mass of air is 28.9625, making themolar mass of air 0.0289625 kg per mole. In natural gas engineering,the ratio of the molar mass of gas to that of air is frequently used. Thisratio is called "gas gravity" in British units and "relative density" in SIunits. It is usually designated by the letter G, and since the ratio ofmolecular masses is the same as the ratio of molar masses, then:

Molar mass of gas industry compounds is given in Fig. 23-2, Column A.

G M28.9625-------------------=

M 28.9625 G=

Lecture 5

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Ideal / Real Gas Laws

The relationships among pressure, volume, temperature (PVT) andnumber of moles can be combined and expressed by the Ideal Gas Lawin terms commonly used in the gas industry:

,

where: P = absolute pressure, kPa

V = volume, m3

n = number of kilomoles

R = Gas Constant,

T = absolute temperature, K (= 273.15 + C)

When dealing with gas systems, the mole fraction of a gas mixture isequal to the volume fraction, as it is assumed that any mole of gas willoccupy the same volume as any other mole of gas at the same pressureand temperature. This is not the case for liquid mixtures, that is to say,when dealing with liquids, the mole fraction differs from the volumefraction.

At low pressures, the Ideal Gas Law is reasonably accurate in describingthe PVT behaviour of gases. When gases are compressed, theirmolecular volumes and intermolecular forces cause deviations fromideal behaviour. To account for this deviation from ideal behaviour forreal gases, the concept of a dimensionless "deviation factor", normallyreferred to as the "compressibility factor", has been introduced into theIdeal Gas Law. This factor has been given the designation "Z", and iscommonly referred to as the Z factor in natural gas calculations. TheIdeal Gas Law is transformed to the Real Gas Law by adding the Zfactor:

, with all terms as defined above.

PV nRT=

8.3145 kPa m3kmol K

----------------------------------------

See Fig. 23-3, Column E for data on liquid volume per kmol.

PV ZnRT=

6 Lecture

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Equations of State

Other, more complex equations have been developed in the last 50years or more, to account for molecular volumes and molecularattractive forces. These are generally called "Equations of State". Oneof the most popular and accurate equation of state for describing thebehaviour of natural gases is the Peng-Robinson equation of statewhich was developed at the University of Alberta in 1975. Sinceequations of state are difficult to handle without computers, no furtherdiscussion on equations of state will be provided in this section.Equations of state and their use in HYSYS are discussed in more detail inthe Workshop section of this Module.

PVT Behaviour

Since gas is a compressible fluid, it is necessary to agree on standardconditions of pressure and temperature, for expressing any amounts ofgas in terms of volumes at the standard or reference conditions. Thestandard conditions of pressure and temperature for determiningstandard volumes of gas are the following:

Standard pressure: 101.325 kPa (abs)

Standard temperature: 15C or 288.15 K

Standard volume : Sm3 (cubic metre of gas at a pressure of 101.325 kPa

(abs) and 15C

In all natural gas engineering calculations, the estimation of the PVTbehaviour is a requirement. Without the use of a computer, chartmethods are available to perform these tasks. Before the advent of thecomputer and the development of the powerful software programs thatare nowadays available to virtually every engineer, all of the designcalculations were made using chart approximations for determiningthe properties and PVT relationships of natural gas. These notes willexplain what calculations are necessary, and how to perform themwithout computers.

Equations of State are available in Section 25, page 25-7.

In this course standard conditions are 101.325 kPa and 15C.

Lecture 7

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Z Factor

The Z factor is one of the most frequently calculated properties in anyengineering calculation dealing with natural gas. The determination ofthe Z factor by chart methods is based on the concept of the "Law ofCorresponding States". This law postulates that at the same ratio ofpressure and temperature to the critical pressure and temperature, allgases have the same volumetric deviation from ideal volume. The ratioof pressure to critical pressure is called the "reduced pressure". The"reduced temperature" is arrived at in a similar manner.

where: P = pressure at which Z is to be determined, kPa (abs)

T = temperature at which Z is to be determined, K

Pc = critical pressure, kPa (abs)

Tc = critical temperature, K

Pr = reduced pressure, ratio

Tr = reduced temperature, ratio

Once the reduced values have been determined, the Z factor isobtained from the Z factor chart for natural gas, which is given inFig. 23-4 of the Data Book. The critical properties of pressure andtemperature for each compound are given in Fig. 23-2, page 23-2, ofthe Data Book.

As we have seen in the introduction, natural gas is a mixture of manycompounds, so to apply the Law of Corresponding States to natural gasmixtures, it is necessary to determine a substitute for the criticalpressure and critical temperature of a pure gas. A method wasproposed by Kay, which is based on the mole fraction weighted averagepressure and temperature of the gas mixture.

Reduced values are always based on absolute pressure and temperature.

PrPPc-----=

TrTTc-----=

Fig. 23-4: Z Factors.

Fig. 23-2: Critical temperature and pressure.

8 Lecture

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The result is a pseudo-critical pressure and pseudo-criticaltemperature for a particular gas mixture. These pseudo-values are thenused as if they were the true critical values of pressure and temperaturefor the mixture which, strictly, they are not.

where: Ppc = pseudo-critical pressure of gas mixture, kPa (abs)

Pci = critical pressure of compound "i" in the gas mixture, kPa(abs)

Tpc = pseudo-critical temperature of gas mixture, K

Tci = critical temperature of compound "i" in the gas mixture, K

Yi = concentration of compound "i" in the gas mixture, mole fraction.

When natural gas contains significant amounts (more than onepercent) of H2S or CO2, then the Z factor obtained from Fig. 23-4 willsuffer in accuracy. This was experienced in western Canada many yearsago, when the sour gas pools were discovered. Companies tooksamples of gas from sour wells and had laboratories determine a matrixof Z factors for such sour gases. Various attempts were made to developnew correlations for estimating the Z factor for sour gases in the 1950sand 1960s. In 1970, Wichert and Aziz published their research work,which was done at the University of Calgary, on new computermethods for estimating the Z factor for sour gases.

While three methods were developed by Wichert-Aziz, two have foundgeneral acceptance. The third method, a modification of a method forZ factor calculation developed by Pitzer et al., has the drawback ofrequiring the storage of two matrices of factors used in thedetermination of the Z factor. This uses up computer storage, andrequires interpolation to extract the appropriate factors at the desiredconditions of pseudo-reduced pressure and temperature. As a result,the Wichert-Aziz modification of the Pitzer et al. Z factor calculationmethod has not proven as popular. The other two methods are virtuallyas accurate, and are described in the following pages.

Ppc Yi Pci=

Tpc Yi Tci=

The Z factor chart in Fig. 23-4 of the Data Book is only valid for sweet natural gas.

Lecture 9

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Wichert-Aziz Modification of the Redlich-Kwong Equation of State

Redlich and Kwong (R-K) proposed an equation of state in 1949 whichhas gained popularity, as it is relatively simple yet accurate whencompared to more complex equations. It has also been modified byothers over the years to further enhance its accuracy. Equations of stateare suited for computer use as the need for storing data is low, and isusually limited to very basic data, such as critical pressures andtemperatures of the components making up the gas mixture.

The Wichert-Aziz modification of the R-K equation of state requires thedetermination of a parameter, , as follows:

, with units of C

where: A = mole fraction (H2S + CO2) in sour gas mixture,

B = mole fraction H2S.

The critical temperature (in Kelvin)of each component in the gasmixture is then adjusted by subtracting the value of , and eachcomponent critical pressure (in kPa) is adjusted as follows:

where: Tci and Pci are critical temperature, in K, and critical pressure, in kPa (abs), of component "i" in the gas mixture, and and are the adjusted critical conditions for each compound.

The adjusted critical values are used to calculate the constants in theR-K equation with the mixing rules as proposed by Redlich and Kwong.An adjustment to the temperature, T, (in K), at which Z is to bedetermined is necessary in the pressure range 0 to 17.24 MPa, asfollows:

2

2 15 A( A2 ) 4.167 B( 0.5 B2 )+=

2

Tci Tci 2=

Pci Pci Tci( ) Tci=

Tci Pci

T T 1.94 P( 2760 2.1 10 8 P2 ) +=

10 Lecture

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Modification of the Standing-Katz (S-K) Chart Method

The S-K chart method for estimating the compressibility factor forsweet natural gases was the only method that was in widespread useuntil the advent of the computer. It seemed appropriate to develop amodification for this method to make the chart applicable for sourgases also. The method that was developed involves adjusting thepseudo-critical properties of the gas mixture, so that the resultingpseudo-reduced pressure and temperature can be used to look up the Zfactor on the S-K chart.

The adjustment is applied to the pseudo-critical pressure andtemperature of the gas mixture, rather than to the critical properties ofthe individual components, as is the case in the modification of the R-Kequation method. In the modification of the S-K chart method, theadjustment parameter, , is calculated as follows:

, with units of C

where: A = mole fraction (H2S + CO2) in sour gas mixture

B = mole fraction H2S

The pseudo-critical pressure and temperature, which are calculated byKays molar average rules, are then adjusted as follows:

where: Tpc and Ppc are the pseudo-critical temperature and pressure of the gas mixture, and and are the adjusted pseudo-critical values.

The adjusted pseudo-critical temperature and pressure are then usedto determine the pseudo-reduced temperatures and pressures for theconditions at which the Z factor is to be determined from the S-K chart.

A chart has been developed for , which is shown in

Fig. 23-8.3

3

3 66.667 A(0.9

A1.6 ) 8.333 B0.5( B4 )+=

Tpc Tpc 3=

PpcPpc Tpc

Tpc B 1 B( ) 3 +--------------------------------------------------=

Tpc Ppc

Lecture 11

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Application of the method should be limited to sour natural gases thathave methane as the major constituent of the hydrocarboncomponents of the gas mixture. The method can be used for sournatural gases containing up to 80 vol % total acid gas.

Example

What is the compressibility factor of the sour gas mixture of thecomposition shown below at a pressure of 6895 kPa (abs) (1000 psia)and 37.8 C (100 F)?

1. Calculate the pseudo-critical properties by the molar averagemethod:

Comp.* Mole Fr. Tc,K Pc,kPa

0.0304 126.21 3.84 3398 103.3

0.0287 304.11 8.73 7374 211.6

0.2327 373.37 86.88 8963 2085.7

0.5601 190.56 106.73 4599 2575.9

0.0820 305.41 25.04 4880 400.2

0.0345 369.77 12.76 4240 146.3

0.0085 407.82 3.47 3640 30.9

0.0110 425.10 4.68 3784 41.6

0.0000 460.35 0.00 3381 0.0

0.0071 469.65 3.33 3365 23.9

0.0028 506.4 1.42 3030 8.5

0.0022 539.2 1.19 2740 6.0

Totals 1.0000 258.07 5633.9

0L[WXUH5/5RELQVRQ-UDQG5+-DFRE\%HWWHU&RPSUHVVLELOLW\)DFWRUV

+\GURFDUERQ3URFHVVLQJ$SULOSS

Pc,kPa yiPciN2

CO2H2S

C1C2C3iC4nC4iC5nC5C6C7

12 Lecture

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2. From Fig. 23-8 of the Data Book, obtain the pseudo-criticaltemperature adjustment factor, :

: for % and %, C

3. Calculate the adjusted pseudo-critical temperature, , where:

4. Calculate the adjusted pseudo-critical pressure, , where

and B= mole fraction H2S

= 5224.2 kPa (abs)

5. Calculate the pseudo-reduced temperature, , and the pseudo-reduced pressure, , using the adjusted pseudo-critical values:

6. From Fig. 23-4 of the Data Book, at and ,

*Experimental Z = 0.774.

3

3 CO2 2.87= H S2 23.27= 3 16.1=

TpcTpc Tpc 3=

Tpc 258.07 16.1 241.97K= =

Ppc

PpcPpc Tpc

Tpc B+ 1( B ) 3--------------------------------------------------=

Ppc(5633.9 241.97 )

(258.07 0.2327 1( 0.2327 ) 16.1 ) +-----------------------------------------------------------------------------------------------=

TprPpr

TprT

Tpc----------

37.8( 273.15 )+241.97

--------------------------------------- 1.29= = =

PprP

Ppc----------

68955224.2---------------- 1.32= = =

Ppr 1.32= Tpr 1.29=Z 0.775=

Lecture 13

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AGA # 8 (1992) Method

The American Gas Association (AGA) has conducted research todevelop a method for accurately calculating the Z factor for pipelinequality natural gas. In 1985, AGA published report AGA # 8, whichdeveloped a highly accurate method for estimating the Z factor fornatural gas. However, this method limited the H2S content to about1 vol %. Upon further research, AGA released the second edition ofAGA # 8 in 1992.

AGA # 8 (1992) presents a very complex calculation method, which canonly be done by computer. The list of the computer source program inFORTRAN 77 is included in the AGA # 8 (1992) report, which can beobtained from AGA. A diskette containing the program is also availableat a nominal additional cost.

Table 1.1 provides a comparison of experimental and calculated Zfactors for a sour gas mixture. The calculations include a comparisonwith results obtained by HYSYS, using the Peng Robinson and theSoave-Redlich-Kwong equations of state.

The method contained in the 1992 edition of AGA #8 is highly accurate for sweet as well as for sour natural gases.

14 Lecture

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Derivations of the Real Gas Law

On the basis of the Real Gas Law, PV = ZnRT, many useful formulas canreadily be derived that find application in natural gas engineering. Oneof the most basic relationships is the relative volume of a certainamount of gas at one set of conditions of pressure and temperature ascompared to another set:

Other useful formulas that can be derived from the Real Gas Law arethe following:

TABLE 1.1

Comparison of Experimental* and Calculated Z Factors

Exp. W-A W-A AGA#8

HYSYS HYSYS

Pressurepsia/kPa (abs)

Temp.F / C

ZS-K Chart

R-K (1992) PR SRK

258/1779 100/37.8 .962 .961 .961 .963 .955 .963

377/2599 100/37.8 .945 .943 .943 .946 .935 .947

547/3772 100/37.8 .921 .920 .919 .922 .907 .924

785/5413 100/37.8 .889 .887 .886 .889 .870 .892

1111/7660 100/37.8 .847 .847 .845 .846 .824 .853

1558/10742 100/37.8 .798 .801 .798 .796 .774 .810

* Mixture 14, R.L. Robinson Jr. and R. H. Jacoby, Better Compressibility Factors, Hydrocarbon Processing, April 1965, pp. 141-145.

Composition (mole fr.): 0.0, 0.0077, 0.0144, 0.1630, 0.7948, 0.0153, 0.0048, 0.0

He N2 CO2 H2S C1 C2C3 C4+

P1 V1Z1 T1----------------

P2 V2Z2 T2----------------=

Lecture 15

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1. Volume of one kilomole of gas at Standard Conditions of P and T:

= 23.64 m3/kmol

2. Actual volume when volume at Standard Conditions is known:

, m3

3. Standard volume of gas, when volume at P and T is known:

, m3

4. Gas density at P and T:

, kg/m3

, kg/m3

5. Superficial gas velocity in pipes or vessels:

,m/s

Molar volume Z1 n R T1 P1 1 1 8.3145 288.15 101.325 = =

V20.352 V1 Z T

P---------------------------------------=

V12.84 V P

Z T--------------------------=

v0.120 P M

Z T-------------------------------=

v3.48 P G

Z T---------------------------=

v5.187 Q Z T

P D2--------------------------------------=

16 Lecture

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where: P = pressure, kPa (abs.)

T = absolute temperature, K

V = volume, m3

V1 = volume @ standard condition of 101.325 kPa and 288.15 K, Sm3

V2 = actual volume at P and T, m3

D = internal diameter, mm

Q = flow rate at standard conditions, Sm3/day

n = number of kilomoles

Z = compressibility factor for gas at P and T

R =

M = relative molecular mass of gas, kg

G = relative density = M / 28.9625

Problem 1-1

The Z factor for the Sales Gas in Table A of these Notes is 0.843 at 10Cand the relative density of the gas is 0.628. The composition for SalesGas from Table A is included here:

8.3145 kPa m3kmol K

----------------------------------------

Lecture 17

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Determine the following:

a) The volume of gas at standard conditions stored in a pipeline 10 kmlong and with an internal diameter of 260 mm, at a pressure of 6000 kPaand a temperature of 10C.

b) The density of the gas in the pipeline, in kg/m3.

c) The velocity of the gas flowing through the line, in km/h, at a salesgas rate of Sm3/d (3 000 000 Sm3/d).

TABLE A

Sales Gas Compositon

Component Mole Percent

H2 0.01

He 0.00

N2 0.55

CO2 0.88

H2S 0.00

C1 89.30

C2 6.25

C3 2.35

iC4 0.21

nC4 0.32

iC5 0.05

nC5 0.08

C6 0.00

C7+ 0.00

TOTAL 100.00

3 106

18 Lecture

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Heating Value of Natural Gas

Throughout the world, natural gas is mostly burned to generate heat forindustrial, commercial and domestic applications. Other uses, such asdriving electric generators or automobiles, also make use of the heatenergy of natural gas. Thus it is very important for the consumer toknow what the energy content is of the natural gas that is being bought.

The heating value of natural gas has been defined in several ways, and itis important to understand the differences between the variousdefinitions of heating value, namely:

gross heating value, dry,

gross heating value, wet,

net heating value, dry,

net heating value, wet.

Additionally, the terms "ideal gas" or "real gas" have to be used tofurther classify the heating value, making it necessary to define theseterms.

Gross Heating Value

Gross heating value, MJ/m3, is the amount of energy transferred asheat, upon the combustion of one standard cubic metre of gas in anideal combustion reaction, at standard pressure and temperature, withall combustion products cooled to the standard temperature, and allwater formed during the combustion process being condensed. If thegas does not contain any water vapour, the heating value is on a drybasis. If the gas is fully saturated with water vapour at standardconditions, the heating value is on a wet or saturated basis. The termideal assumes that the gas behaves as an ideal gas, and therefore thecompressibility factor is 1. Real gas has a compressibility factor lessthan 1.

Lecture 19

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The relationship between compressibility factor, Z, ideal gas and realgas heating values (HV) is:

Net Heating Value

Net heating value, MJ/m3, is the amount of energy transferred as heatin an ideal combustion reaction, as defined above, but with the waterformed in the combustion remaining in the vapour phase. Thedifference between gross and net values is the Latent Heat ofCondensation of the water produced in the combustion process.

The heating value of natural gas can readily be calculated from the gasanalysis. The heating value can then be expressed in gross, net, dry,wet, ideal, or real terms, as described above.

The gross heating values and respective net values for pure methane,ethane and propane for an ideal gas at standard pressure 101.325 kPa(abs) and a temperature of 15C, dry basis, are shown in the followingtable:

HVreal HVideal Z=

Many years ago instrumentation such as the "calorimeter" were used for actually determining the heating value.

C1 C2 C3

Gross 37.708 MJ/m3 66.065 MJ/m3 93.963 MJ/m3

Net 33.949 MJ/m3 60.429 MJ/m3 86.418 MJ/m3

20 Lecture

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To obtain the heating value of a natural gas, it is necessary to determinethe mole fraction weighted heating value, using the heating value ofeach component from the Data Book. When determining the heatingvalue of gas on a "wet" basis, the calculation procedure of GPA 2172should be followed.

The Gas Processors Association (GPA) of the United States has publishedseveral standards which are available at a nominal cost. A list of thestandards and various research reports is provided on pages 1-13 to 1-32of the Data Book.

Problem 1-2

Determine the Gross Heating Value per Sm3 of the Sales Gas in Problem1-1, in terms of ideal gas, dry basis.