compressor performance characteristics

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Note: The source of the technical material in this volume is the ProfessionalEngineering Development Program (PEDP) of Engineering Services.

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Engineering Encyclopedia

Saudi Aramco DeskTop Standards

COMPRESSORPERFORMANCE CHARACTERISTICS


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Engineering Encyclopedia Compressors

Compressor Performance Characteristics

Saudi Aramco DeskTop Standards i

Section Page

INFORMATION............................................................................................................... 3INTRODUCTION............................................................................................................. 3

DETERMINING DYNAMIC COMPRESSOR PERFORMANCE CHARACTERISTICS ... 4

Thermodynamics of Compression........................................................................ 4

Isothermal Process............................................................................................... 8

Isentropic Process................................................................................................ 8

Polytropic Process.............................................................................................. 12

Compressibility ................................................................................................... 15

DETERMINING POSITIVE-DISPLACEMENT COMPRESSOR PERFORMANCECHARACTERISTICS .................................................................................................... 58

WORK AIDS.................................................................................................................. 62

WORK AID 1A: CALCULATION PROCEDURES AND CHARTS FORDETERMINING DYNAMIC COMPRESSOR PERFORMANCECHARACTERISTICS.......................................................................... 62

WORK AID 1B: CHARTS FOR DETERMINING COMPRESSORPERFORMANCE CHARACTERISTICS............................................. 65

WORK AID 2: CALCULATION PROCEDURES FOR DETERMINING POSITIVE-

DISPLACEMENT COMPRESSOR PERFORMANCECHARACTERISTICS.......................................................................... 69

GLOSSARY .................................................................................................................. 71


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Saudi Aramco DeskTop Standards ii

LIST OF FIGURES

Figure 1. Relationship Between Isentropic Efficiency and Polytropic Efficiency Basedon an Ideal Gas...................................................................................................... 15

Figure 2. Compressibility Factors at Low Reduced Pressures..................................... 20

Figure 3. Psychometric Chart for Normal Temperatures .............................................. 31

Figure 4. Mollier Diagram for Propane ......................................................................... 41

Figure 5. Basic Head Versus Flow Performance Curve ............................................... 43

Figure 6. Centrifugal Compressor Impeller and Vector Diagram.................................. 44

Figure 7. Effect of Impeller Blade Angle on Head and Efficiency ................................. 46

Figure 8. Dynamic Compressor Surge Line ................................................................. 49

Figure 9. Dynamic Compressor Stonewall ................................................................... 50

Figure 10. Graphical Representation of the Effect of Molecular Weight on CompressorHead Versus Flow Curves...................................................................................... 51

Figure 11. Typical Head Curve..................................................................................... 56

Figure 12. Typical Horsepower Curve.......................................................................... 57

Figure 13. Pressure Volume Cycle............................................................................... 60

LIST OF TABLES

Table 1. Critical Constants of Gases............................................................................ 18

Table 1. Critical Constants of Gases (Contd) .............................................................. 19

Table 2. Computation of the Physical Characteristics of a Sales Gas/Fuel Gas Mixture23

Table 3. Water Content of Air in Gallons Per 1000 ft

3

at Various Relative Humidities. 32Table 4. Water Content of Saturated Air in Gallons per 1000 ft

3at Various

Temperatures and Pressures with 100% Relative Humidity........................... 34

Table 5. Critical Constants of Gases............................................................................ 65


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Saudi Aramco DeskTop Standards 3

INFORMATION

INTRODUCTION

Compressor performance characteristics can be described asthe operating characteristics that define the ratings of acompressor. An understanding of compressor performancecharacteristics is important when determining compressorrequirements for a system and when evaluating compressoroperation. This module describes compressor performancecharacteristics and the methods of determining the compressorperformance characteristics for dynamic and positive-displacement compressors.


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DETERMINING DYNAMIC COMPRESSOR PERFORMANCECHARACTERISTICS

The major performance characteristics of a dynamiccompressor are flow, head, and efficiency. To determine theseperformance characteristics, the Mechanical Engineer mustunderstand the following subjects:

Thermodynamics of Compression

Properties of Gas Mixtures

Volumetric Flow

Mollier Diagrams

Dynamic Compressor Characteristics

Thermodynamics of Compression

No gas exactly conforms to the Ideal Gas Law, which show therelationship between the volume, the absolute pressure, and theabsolute temperature of an ideal gas. Most gases, however,conform to these laws with sufficient accuracy to yield soundengineering answers relevant to engineering problems. Tounderstand and to calculate the thermodynamics ofcompression, the Mechanical Engineer uses the followingfundamental laws:

Boyles Law

Charles Law

Daltons Law

Avogadros Law

As explained below, these gas laws combine to form the IdealGas Law.

Boyles Law states that when the temperature of a gas is kept

constant, the volume of an enclosed mass of gas is inverselyproportional to varying pressure upon the gas. Another way tostate Boyles Law is that the product of the pressure multipliedby the volume remains constant at a constant temperature. Therelationship between pressure and volume can be convenientlyexpressed as the following equation:

ttanconsisetemperaturwhen;VPVP 2211 =


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Where: V = Volume

P = Pressure

Although Boyles Law assumes the condition of constanttemperature; constant temperature is seldom the case in actualindustrial situations. Temperature continually changes, andsuch changes in temperature affect the volume of a given massof gas.

Charles Law states that, if constant pressure is maintained, thevolume of gas is directly proportional to its absolutetemperature. The relationship between volume and absolutetemperature can be conveniently expressed as the followingequation:

1

2

1

2

T

T

V

V= when pressure is constant

Where: V = Volume

T = Temperature

Daltons Law states that, in a mixture of gases, the summationof partial pressures is equal to the total pressure of the mixture.

A partial pressure is defined as the pressure that a specific gas

in a gas mixture would exert if the gas alone occupied the totalvolume occupied by the mixture at the mixture temperature.The relationship between partial pressures can be convenientlyexpressed as the following equation:

PnP3P2P1PT .P...PPPPP ++==

Where: PT = Total pressure

PP = Partial pressure

= Summation

n = Number of component gases

Avogadros Law states that all gases have the same number ofmoles in the same volume and at the same pressure andtemperature. This relationship can be stated through thefollowing equation:


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ttanConsT

PV=

Where: P = Pressure

V = Volume

T = Temperature

As pointed out above, Boyles Law, Charles Law, Daltons Law,and Avogadros Law combine to form the Ideal Gas Law, whichshows the relationship between the volume, the absolutepressure, and the absolute temperature of an ideal gas. TheIdeal Gas Law can be expressed with either of the two followingformulas:

=P

RT

or

=RT

P

Where: = Specific volume (ft3/lbm)

= Density (lbm/ ft3)

R = Gas constant =Runiv/MW

= 1545.32 ft-lbf/lbm-Mol-R/MW

Runiv = Universal Gas constant

= 1.98587 Btu/lbm-Mol-R

= 1545.32 ft-lbf/lbm-Mol-R

= 8.3143 Joules/gm-Mol-R

= 10.73 psia-ft3/ lbm-Mol-R

MW = Gas molecular weight (lbm/mole)

P = Absolute pressure (lbf/in2)

T = Absolute temperature (R, where R =F + 460


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The General Gas Law derives from simplification of the IdealGas Law applied to a fixed mass. The General Gas Law relatesthe properties of an ideal gas in response to varyingtemperatures and volumes, with pressure held constant:

2

22

1

11

T

VP

T

VP=

Where: V = Volume

P = Pressure

T = Temperature

Variation in temperature is a function of the specific heat (C) ofa gas, or the amount of energy that is required to raise the

temperature of one pound of gas one degree Fahrenheit. If thevolume of the gas is kept constant while the heat is added, all ofthe heat is used to increase the temperature of the gas. Thespecific heat at a constant volume is denoted CV. If thepressure is kept constant and if the volume is allowed to varywhile the heat is added, an increased amount of heat will berequired. The increased amount of heat is required because, inaddition to increasing the temperature, the gas expands andthus performs external work. The specific heat at constantpressure is denoted CP.

The external work that is done when a unit mass of gas isheated at constant pressure is equal to the gas constant (R).The external work can be shown by the following formula:

J

RCC VP =

Where: CP = Specific heat at constant pressure (Btu/F/lb)

CV = Specific heat at constant volume (Btu/F/lb)

R = Specific gas constant (ftlb/R)

J = Joules constant, a ratio of the mechanicalwork done to the heat that is produced (equalto 778 ft-lb/Btu)

The following reversible (Ideal) compression processes can beapplied to compressors:


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Isothermal Process

Isentropic Process

Polytropic Process

Although they describe ideal gases and are not commerciallyattainable, these processes are used as a basis for calculationsand comparisons. The variance of a gas from laws andprocesses for an ideal gas is referred to as compressibility. Thereversible (ideal) compression processes and compressibilityare discussed below. The discussions will focus on head andefficiency in the ideal compression processes and in the actualcompression process.

Isothermal Process

The isothermal compression process is compression that takesplace at a constant temperature. Because large amounts ofheat transfer area must be supplied to keep the temperatureconstant, isothermal compression is not common in the actualoperation of machinery.

The equation for isothermal efficiency is as follows:

ttanConsVPVP 2211 ==

Where: P = PressureV = Volume

Isentropic Process

The isentropic compression process follows a path of constantentropy. In the isentropic process, heat is neither added to norremoved from the gas during compression. The fact that heat isneither added nor removed does not mean that the temperatureis constant. Because of the work of compression that isperformed on the gas, temperature increases as the pressure

increases. In compressor theory, the terms isentropic(constant entropy) and adiabatic (no heat transfer) areinterchangeably used. This interchangeability is valid for thecontext in which the terms are used. The actual definition of anisentropic process is an adiabatic, reversible process.

The following equation shows the relationship between pressureand volume for isentropic compression:


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ttanConsPVk =

Where: P = Pressure

V = Volumek = Isentropic exponent

The isentropic exponent (k) is the ratio of the specific heat at aconstant pressure (CP) to the specific heat at constant volume(CV). The isentropic exponent is equal to CP/CV.

The following equations are used to calculate the total work(Workisen) that is done on a unit mass of gas in the isentropiccompression process:

Workisen:

=

=

11

/1

1

1

kk

r

P

P

k

kP1

1rp

V

)T(TC

or

Workisen:

( )

=

1PP

1kkRT

1/kk

i

fi

Where: CP = Specific heat at constant temperature

Tf = Final temperature (R)

Ti = Initial temperature (R)

Vi = Initial volume (ft3)

Pf = Final pressure (psia)

Pi = Initial pressure (psia)

k = Isentropic exponent

R = Gas constant


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As listed above, for all gases, the gas constant (R) is equal tothe universal gas constant (Runiv) divided by the gas molecularweight (MW), or 1545.32/MW. By definition, the air has auniform molecular weight of 28.966; therefore, the specific

gravity (sg) of any gas, relative to air, of molecular weight MW isequal to MW/28.966. The gas constant (R) for any gas can nowbe defined as 1545.32/MW, or 53.34/sg.

The energy (lbf) of the compression of a gas can be thought ofas lifting a given weight of gas (lbm) at inlet pressure andtemperature to a height (feet) at which the gas is discharged atthe same pressure and temperature. The unit for head is asfollows:

lbm

lbfft

masspoundperforcepoundFoot

Head (Hp) is frequently expressed as feet, which relates to theheight of the gas column at which the gas is discharged at thesame pressure and temperature as the inlet gas.

Head (Hp) is a fundamental property of a compressor. Head is afunction of the compressor design and of the compressorspeed. Head is not affected by the nature of the compressedgas, the thermodynamic properties of the gas, or the addition orsubtraction of heat as the gas flows through the compressor.

The following equation for head is usually stated in terms ofmolecular weight:

( )

=

1P

P

1k

k

MW

1545.32THead

1/kk

i

fiisen

Where: Head = ft-lbf/lbm

T = R

P = psia

The efficiency of a compressor is the ratio of the theoreticalenergy output of the system to the actual energy input of thesystem. For an isentropic process, the theoretical energy outputis the isentropic work output. To determine the efficiency in theisentropic process ( ), the isentropic process must beunderstood. Because of the second law of thermodynamics, theideal adiabatic compression occurs at constant entropy.


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Efficiency in isentropic compression, as shown in the followingequation, can then be defined as the ratio of isentropic work toactual work:

WorkActual

WorkIsentropic=

The overall efficiency in isentropic compression (also referred toas isentropic efficiency) is used as a measure of the overallperformance of a compressor.

A variation of the isentropic process occurs when compressionwith intercooling is used. Multi-stage compressors may useintercoolers between stages to lower the gas temperature.Compression with intercooling results in an isothermal

approximation of an isentropic process. When intercooling isused, the compressor head can be approximated through use ofthe following isothermal head equation:

( )12i

iso /PPlnMW

RTH =

Where: Hiso = Isothermal head

R = Gas constant (1545.32 ft-lbf/lbm - Mol - R)

Ti = Initial temperature in R

ln = Log to base e

MW = Gas molecular weight

P1 = Initial pressure in psia

P2 = Final pressure in psia


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Polytropic Process

Polytropic compression is the compression path that closelyfollows the compression path in a real centrifugal compressor.Centrifugal compression is not an ideal thermodynamic process.The inefficiency of the compression process results in excessheat input to the process gas, which causes the temperature toincrease faster than it would in isentropic compression.Because of the temperature increase, the volume at the end ofpolytropic compression is larger than the volume at the end ofisentropic compression.

The following equation shows the relationship between pressureand volume for polytropic compression:

ttanConsPVn =

Where: P = Pressure

V = Volume

n = Polytropic exponent

In terms of required energy, all compressors operate closest tothe polytropic process. In any gas compression, the actual workinput is greater than the polytropic work input. In a polytropicprocess, the temperature rise occurs at a faster rate than it does

in an isentropic process. The faster rise in temperature isaccounted for mathematically by the substitution of thepolytropic exponent (n) for the isentropic exponent (k) in thefollowing polytropic head equation:

( )

=

1P

P

1n

nRTHead

1/nn

i

fipoly

The following equation for head is usually stated in terms ofmolecular weight:

( )

=

1P

P

1n

n

MW

1545.32THead

1/nn

i

fipoly


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The following equation shows the relationship between thepolytropic exponent (n) and the isentropic exponent (k):

n 1

n

k 1

k

1

p

=

Where: p = Polytropic efficiency

The equation that shows the relationship between the polytropicexponent (n) and the isentropic exponent (k) indicates that when

p is equal to 100%,k

1k

n

1n =

and the process becomes

isentropic (adiabatic). As mentioned in compressor theory, theterms isentropic (no heat transfer) and adiabatic (no entropy

change) are used interchangeably.

If the proper mathematical substitution is performed, thefollowing equation for polytropic efficiency results:

1k

k1n

n

p

=

The polytropic exponent for Ideal Gases can be obtained

independent of polytropic efficiency by the following equation,which relates suction and discharge temperature and pressure:

=

1

2n

1

2n

P

PL

T

TL

n

1n

Polytropic efficiency is a characteristic of each compressor.Polytropic efficiency is equal to the reversible work divided by

the total work applied to the gas. Because of the various lossesthat are caused by the gas as it passes through the impellersand diffusers at high velocity, reversible work and total work aredifferent. For centrifugal compressors, the polytropic efficiencyis usually between 60% and 85%. For axial compressors,efficiencies can be as high as 92%.


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The polytropic exponent (n) is always larger than the isentropicexponent (k). For the same actual performance, the value ofthe polytropic efficiency will be higher than the value of theisentropic efficiency.

The ratio of any reversible (Ideal) process, isothermal,isentropic, or polytropic, is equal to the actual work (energy) asillustrated in the following equation:

poly

poly

isen

isen

iso

iso

Head

Head

HeadWorkActual ===

The relationship between isentropic efficiency and polytropicefficiency (based on a perfect gas) is shown in Figure 1. If the

inlet and outlet pressure of the compressor are known, Figure 1can be used to convert isentropic efficiency to polytropicefficiency, or polytropic efficiency to isentropic efficiency. Toconvert known efficiency to the unknown efficiency, atemperature rise factor (X) must be calculated. The followingequation is used to calculate the temperature rise factor (X):

( )

=

1P

PX

1/kk

1

2

Where: X = Temperature rise factor

P2 = Discharge pressure

P1 = Inlet pressure

k = Isentropic exponent

Once the temperature rise factor (X) has been calculated, theline that corresponds to the temperature rise factor (X) is usedto convert the known efficiency to unknown efficiency. Thepoint on the unknown efficiency axis that corresponds to theintersection of the known efficiency and the temperature rise

factor (X) is the unknown efficiency. This point is calledequivalent efficiency.


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Figure 1. Relationship Between Isentropic Efficiency andPolytropic Efficiency Based on an Ideal Gas

Compressibility

The relationship of specific volume to pressure and temperaturefor an ideal gas can be defined by the equation (P)() = (R)(T).However, most gases that are encountered in industrialcompression do not exactly obey the Ideal Gas Law equation.Deviation from the Ideal Gas Law is referred to ascompressibility. Compressibility is specifically defined as thedegree to which any given gas varies from the Ideal Gas Law.


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Compressibility is stated as a ratio of actual gas volume at agiven pressure and temperature to the volume that is calculatedby theoretical law. The compressibility modifies the equation forrelationship of specific volume to pressure and temperature for

an ideal gas from P= RT to:

( )( ) ( )( )( )TRZP =

Where: Z = Compressibility factor

P = Pressure

= Specific volume

R = Gas constant

T = Temperature

The compressibility factor (Z) is a dimensionless factor that isindependent of the quantity of gas. The compressibility factor(Z) is determined by the type, the temperature, and the pressureof the gas. The compressibility factor (Z) can be derived fromthe rule of corresponding states through the use of reducedtemperature and pressure. The reduced values of temperatureand pressure are ratios of actual conditions to critical constantsas shown in the following formulas:

cr T

T

T =

Where: Tr = Reduced temperature in R

Tc = Critical temperature in R

T = Temperature actual in R

c

rP

PP =

Where: Pr = Reduced pressure in lbf/in2

Pc = Critical pressure in lbf/in2

P = Actual pressure in lbf/in2


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Values of the critical constants, Tcand Pc, for individual gasesare given in Table 1. The following example shows how todetermine the compressibility factor of propane gas with

pressure (P) of 300 psia and temperature (T) of 140F. Table 1is used to determine the following critical constants of propane:

R666.2=T

psia617.4=P

c

c

To calculate the reduced temperature (Tr), the temperature of

the propane (140F) must be converted to degrees Rankine asfollows:

R600460140

R460TT

=+=

+=

The reduced temperature (Tr) is calculated by dividing the

temperature (T) of the propane (600R) by the criticaltemperature constant (Tc) for propane (666.2R).

0.9006666.2

600

T

TT

c

r

=

=

=

The reduced pressure (Pr) is calculated by dividing the pressure(P) of the propane (300 psia) by the critical pressure constant(Pc) for the propane (617.4 psia).

0.4859617.4

300

P

PP

c

r

=

=

=


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Table 1. Critical Constants of Gases

Cpand Cp/Cr

at 14.7 psia and

60FCritical Constants

Compound Formula

Mol. Wt.

M Cp Cp/Cr

Pressure

psia Pc

Temp. RTc

Mcp

at 60F

Mcp

at

100F

Mcp

at

200F

Acetylene C2H2 26.036 0.3966 1.238 905.0 557.4 10.33 10.69 11.53

Air N+O2 28.966 0.2470 1.395 547.0 238.7 6.96 6.96 6.99

Ammonia NH3 17.032 0.5232 1.310 1,657.0 731.4 8.91 8.57 9.02

Benzene C6H6 78.108 0.2404 1.118 714.0 1,013.0 18.78 20.47 24.46

1,2-Butadiene C4H6 54.088 (0.3458) (1.120) 653.0 799.0 18.70

1,3-Butadiene C4H6 54.088 (0.3412) 1.120 628.0 766.0 18.45

N-Butane C4H10 58.120 0.3970 1.094 550.7 765.6 23.07 24.51 26.16

Isobutane C4H10 58.120 0.3872 1.097 529.1 734.9 22.50 23.96 27.62

N-Butene C4H6 56.104 0.3703 1.105 583.0 755.6 20.77 22.09 25.18

Isobutene C4H6 56.104 0.3701 1.106 579.8 752.5 20.76

Butylene C4H6 56.104 0.3703 1.105 583.0 755.6 20.78 21.94 24.86

Carbon dioxide CO2 44.010 0.1991 1.300 1,073.0 548.0 8.76 9.00 9.35

Carbonmonoxide

CO 28.010 0.2484 1.403 510.0 242.0 6.96 6.96 6.98

Chlorine Cl2 70.914 0.1149 1.366 1,120.0 751.0 8.15

Ethane C2H4 30.068 0.4097 1.193 708.3 550.1 12.32 12.96 14.68

Ethyl alcohol C2H5OH 46.069 0.3070 1.130 927.0 629.6 14.14

Ethylene C2H4 28.052 0.3622 1.243 742.1 509.8 10.16 10.68 12.08

N-Hexane C6H14 86.172 0.3984 (1.062) 439.7 914.5 34.33 36.23 41.08

Helium He 4.003 1.2480 1.6598 480.0 510.0 5.00

Hydrogen H2 2.016 3.408 1.408 188.0 60.2 6.87 6.90 6.95

Hydrogensulfide H2S 34.076 0.254 1.323 1,306 672.7 8.66 8.18 8.36

Methane CH4 16.042 0.5271 1.311 673.1 343.5 8.46 8.65 9.30

Methyl alcohol CH3OH 32.042 0.2700 1.203 1,157.0 924.0 8.65

Nitrogen N2 28.016 0.2482 1.402 492.0 227.2 6.95 6.96 6.963

N-Octane C8H18 114.224 0.3998 (1.046) 362.1 1,025.2 45.67

Oxygen O2 32.00 0.2188 1.401 730 278.2 7.00 7.03 7.120


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Table 1. Critical Constants of Gases (Contd)

Cpand Cp/Cr

at 14.7 psia and60F Critical Constants

Compound FormulaMol. Wt.

MCp Cp/Cr

Pressure

psia Pc

Temp. RTc

Mcp

at 60F

Mcp

at 100F

Mcp

at 200F

N-Pentane C5H12 72.146 0.3972 1.074 489.5 845.9 28.66 30.30 34.41

Isopentane C5H12 72.146 0.3880 1.075 483.0 830.0 27.99 29.90 34.44

Propane C3H8 44.094 0.3885 1.136 617.4 666.2 17.13 18.21 20.90

Propylene C3H6 42.078 0.3541 1.154 667 657.4 14.90 15.77 17.88

Sulfur dioxide SO2 64.060 0.1470 1.246 1.142 775.0 9.42

Toluene C7H8 92.134 0.2599 1.091 611 1,069.5 23.95

Water H2O 18.016 0.4446 1.335 3,206 1,165.4 8.01 8.03 8.12

Hydrogenchloride

HCl 36.465 0.1939 1.410 1,199.2 584.5 7.07

The compressibility factor curves are graphs of reducedpressure (Pr) versus compressibility factor (Z) for variousreduced temperatures (Tr). The compressibility factor (Z) shownin Figure 3 is for low reduced pressure. As shown in Figure 2, a

compressibility factor curve is used in conjunction with thecalculated reduced temperature (Tr) and reduced pressure (Pr)to determine the compressibility factor (2) determined bylocating the point at which the reduced temperature (Tr=0.9006) and the reduced pressure (Pr= 0.4859) intersect and,then, by reading horizontally to find the compressibility factor (Z= 0.675).


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Figure 2. Compressibility Factors at Low Reduced Pressures


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In our previous discussion of the isentropic process and thepolytropic process, the equations for work or head were onlytrue for the Ideal Gas Law equation. The compressibility factor

is used to account for the deviation of a gas from the Ideal GasLaw equations. To correct for deviation from Ideal Gas Law, thecompressibility factor must be used in the work or headequations.

The compressibility factor will vary from compressor inletconditions to compressor outlet conditions. In most cases, thecompressibility factor remains fairly constant over the range ofcompression, and an average value for the compressibilityfactor can be used. The average compressibility factor can bedetermined through use of the following calculation:

2

ZZZ 12avg

=

Where: Zavg = Average compressibility factor

Z2 = Compressibility factor at dischargeconditions

Z1 = Compressibility factor at inlet conditions

If the proper mathematical substitutions are made to theisentropic and polytropic head equations, the followingisentropic and polytropic head equations would result:

Isentropic:

( )

( )

=

=

1P

P

1k

k

MW

T1545.32Z

1P

P

1k

kRTZHead

1/kk

i

fiavg

1/kk

i

fiavgisen


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Polytropic:

( )

( )

=

=

1P

P

1n

n

MW

T1545.32Z

1P

P

1n

n/TRZHead

1/nn

i

fiavg

1/nn

i

fiavgpoly

Properties of Gas Mixtures

Many of the gases that are involved in engineering systems arephysical mixtures of either the permanent gases or one or moreof these gases with superheated or saturated vapors. Forexample, atmospheric air is a mixture of oxygen and nitrogenwith traces of other gases, with superheated or saturated watervapor or, at times, with saturated vapor and liquid. This sectiondiscusses the properties of the following gas mixtures:

Dry Gas Mixtures

Wet Gas Mixtures

Dry Gas Mixtures

The procedures that are required to individually consider theproperties of each constituent of a dry gas mixture are verycomplex. Experience has demonstrated that a mixture of drygases may be regarded as an equivalent gas. The properties ofthe equivalent gas depend upon the types of gases and theproportion of each of the gases that make up the equivalentgas.

If the chemical composition of a dry gas mixture is known, it ispossible to determine the gas characteristics that are necessaryto perform compressor calculations. The following are theproperties of a dry gas mixture that are required for adiabaticcompressor calculations:

Gas constant (dependent on molecular mass MW)

k, specific heat ratio and adiabatic exponent


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P1, T1, V1, and P2

Critical pressure , PC

Critical temperature, TC

Compressibility factor, Z

Of the above properties, MW, CP, CV, PC, and TCare calculatedby addition of the products of the individual mol fraction of eachof the constituents of the gas mixture multiplied by the specificproperties of the individual gas. An application of the individualmol fraction calculations is shown in Table 2, which presents thecomputation of the physical characteristics of a typical salesgas/fuel gas mixture. The composition is known on the

volumetric basis.

Table 2. Computation of the Physical Characteristics of aSales Gas/Fuel Gas Mixture

GasComponent

MolFraction

(y)

MolecularWeight(MW)

(y)x(MW)

MCPat100F

(y) x MCPat 100F

CriticalPressure

Pc (y) x Pc

CriticalTemperature

Tc (y) x Tc

Methane

Ethane

Propane

i-Butane

N-butane

i-pentane

0.922

0.048

0.019

0.004

0.006

0.001

16.04

30.07

44.09

58.12

58.12

72.15

14.78

1.44

0.84

0.23

0.35

0.07

8.65

12.96

18.21

23.96

24.51

29.90

7.975

0.622

0.346

0.096

0.147

0.030

673.1

708.3

617.4

529.1

550.7

483.0

620.6

34.0

11.7

2.1

3.3

0.5

343.5

550.1

666.2

734.9

765.6

830.0

316.7

26.4

12.7

2.9

4.6

0.8

Total 1.00 MW = 17.71 Mcp = 9.216 Pcmix = 672.2 Tcmix = 364.1

MCv = MCP - 1.986 = 7.230 k = MCP/MC


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Several individual equations are used to calculate the individualproperties of a dry gas mixture. The molecular weight of a gasmixture is determined from the following equation:

))(MW(XMW ii=

Where: Xi = Mol fraction of the individual component ofthe mixture

MWi = Molecular weight of the individualcomponent of the mixture

MW = Molecular weight of the mixture

= Sum

The premise for the calculation of MW is the following equation:

MW

RR univ=

Where: Runiv = Universal gas constant

R = Gas constant of mixture

Given that MW X MWi i

= ( )( ) , it follows that:

))(MW(X

RR

ii

univ

=

The k value of a gas mixture is determined from the followingequation:

( )( )( )( ) 1.986cM

cMk

pii

pii

=

For metric values (Cpiin kJ/KmolK), the k value of a mixture isdetermined from the following equation:

( )( )( )( ) 8.32cM

cMk

pii

pii

=

Where: Mi = Molecular weight of the individual


26/72




component of the mixture

Cpi = Specific heat at constant pressure of theindividual component of the mixture

The pressure of a gas mixture can be calculated from thefollowing equation:

))(P(XP ii=

Where: P = Pressure of the mixture

Xi = Mol fraction of the individual component

Pi = Pressure of the individual component

The temperature of a gas mixture can be calculated from thefollowing equation:

( )( )ii TXT =

Where: T = Temperature of the mixture


Ti = Temperature of the individual component

The specific volume of a gas mixture can be calculated from the

following equation:

mix = (Xi)(i)

Where: mix = Specific volume of the mixture


i = Specific volume of the individual component

The critical pressure of a gas mixture can be calculated from thefollowing equation:

)ciimixc )(P(XP =


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Where: Pcmix = Critical pressure of the mixture


Pci = Critical pressure of the individualcomponent

The critical temperature of a mixture can be calculated from thefollowing equation:

))(TX(T imixc ci=

Where: Tcmix = Critical temperature of the mixture


Tci = Critical temperature of the individualcomponent

The compressibility factor (Z) of the mixture is determined bythe calculation of the reduced temperature (Tr) and the reducedpressure (Pr) through the use of the following equations:

cmix

rT

TT =

cmix

rP

PP =

Where: Tr = Reduced temperature of the gas mixture

Tcmix = Critical temperature of the mixture

Pr = Reduced pressure of the gas mixture

Pcmix = Critical pressure of the mixture

T = Temperature of the gas in R

P = Pressure of the gas in psia

The compressibility factor for the inlet condition (Z1) isdetermined through the use of inlet pressure (P1) andtemperature (T1). The compressibility factor for the outletcondition (Z2) is determined through the use of outlet pressure


28/72




(P2) and temperature (T2).

The calculated values of reduced pressure and reducedtemperature for the gas mixture are then used with the curves of

compressibility factors at low reduced pressure that werepreviously shown in Figure 2 to determine the compressibilityfactor (Z) of the gas mixture.

Wet Gas Mixtures

Compressor performance is affected by compressing wet gas(gases that contain water vapor). As the gas pressure isincreased during compression, the gas reaches the water vaporsaturation point. The weight of a cubic foot of gas at standardtemperature and pressure and when entering the compressor

will be more than the weight of a cubic foot of gas at standardtemperature and pressure and when leaving the compressor.Compressor inlet flow rate is typically rated for dry airconditions. The compressor inlet flow rate must be corrected toreflect the capacity at wet gas conditions. Gas density and thepolytropic exponent (n) must also be adjusted for the effect ofwater vapor.

The amount of water vapor that is contained in the air ismeasured in two ways: specific humidity and relative humidity.Specific humidity is the ratio of the mass of water vapor present

in a gas to mass of dry gas. Relative humidity is the ratio of theamount of water vapor that is actually present in the gas to theamount of water vapor that would be present if the air weresaturated.

Specific humidity is also known as absolute humidity, or thehumidity ratio. Specific humidity can be expressed by thefollowing equation:

a

vs

m

m=

Where: s = Specific humidity

mv = Mass of water vapor in kg or lbm

ma = Mass of dry air in kg or lbm


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The specific humidity can also be expressed by the followingequations:

av

va

a

a

v

v

a

a

v

v

a

vs

PR

PR

RP

RP

TRVP

TRVP

m

m ====

or

( )vvva

sPPR

PR

=

Where: s = Specific humidity

Pv = Partial pressure of the water vapor

Pa = Partial pressure of the dry gas

P = Total pressure of the gas mixture

Rv = Gas constant of the gas-water vapormixture

Ra = Gas constant for the dry gas

T = Absolute temperature

Relative humidity can be determined from the followingequation:

g

v

v

g

v

v

g

v

P

P

TRVP

TRVP

m

m ===

Where: = Relative humidity as a decimal fraction

Pv = Partial pressure of the water vapor

Pg = The saturation pressure at the gastemperature

Rv = Gas constant of the gas-water vapor mixture


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T = Absolute temperature

The effect of humidity on compressor flow rate can be shown bythe following scenario for an air compressor:

Compressor design conditions:

Dry air inlet capacity at 60F: 62,000 cfm

Molecular weight of air: 28.95

Inlet temperature: 80F

Inlet pressure: 14.7 psia

Relative humidity: 48%

Discharge pressure: 250 psig

Saturation pressure at 80F: 0.507 psia

Partial pressure of the watervapor at 48% RH: 0.48 x 0.507 = 0.243 psia

Partial pressure of air: 14.7 - 0.243 = 14.56 psia

The weight flow of dry air at the inlet can be calculated by thefollowing equation:

lbm/min4735

52053.3

14414.762,000

TR

144QP(W)AirDryofFlowWeight

a

1

=

=

=

The volume flow rate of air and water vapor at the inletconditions can be calculated by the following equation:


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cfm62,59314414.56

52053.34735

144P

WRTQi

a

i

=

=

=

The example shows the effect of humidity on the air compressorflow rate.

The specific humidity of process gases can be determined bysampling and analyzing the gas stream at the compressorsuction. The relative humidity for air compressors can be

determined by using a sling psychrometer.

The sling psychrometer consists of two identical thermometersthat are mounted on a light frame. One thermometer, which iscalled the wet bulb (WB), is covered with a wick that is saturatedwith water before a reading is taken. The other thermometer,which is called the dry bulb (DB), has no wick. The slingpsychrometer is whirled or slung through the air. As the slingpsychrometer is whirled through the air, the water evaporatesfrom the wick. The amount of evaporation depends on thedegree of saturation of the surrounding air with water vapor.

The evaporation cools the bulb of the wet-bulb thermometer andcauses its temperature reading to fall below the temperaturereading of the dry-bulb thermometer. The difference betweenthe two temperature readings is called the wet-bulb depression.The wet-bulb depression is a measure of the relative humidity.The cooling effect of the wet bulb depends on the evaporationrate from the wick, which depends on the degree of saturation inthe surrounding air.

The properties of air are normally presented in a graphical formthat is called a psychometric chart. Figure 3 shows a portion of

a psychometric chart for normal temperatures. To determinethe relative humidity of the atmosphere with a slingpsychrometer, the wet-bulb temperature and the dry-bulbtemperature are determined. For example, the dry-bulb

temperature is found to equal 85F, and the wet-bulbtemperature is found to equal 77F. The dry-bulb temperature(85F) is found on the psychometric chart. A vertical line isfollowed upward until the line intersects with the 77F axis for


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wet-bulb temperature. This intersection is located on the 70%relative humidity curve (Point A).

Figure 3. Psychometric Chart for Normal Temperatures

Relative humidity is expressed as a percentage of saturation.

Air is said to be saturated with water vapor when the aircontains as much water as it can possibly hold at a specifictemperature. At saturation, the relative humidity is 100%, whileabsolutely dry air has a relative humidity of 0%. Table 3 lists thewater content of air (in gallons per 1000 ft3) at various

temperatures (F) and relative humidities (%RH)


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The temperature has a major effect on the ability of air at agiven pressure to hold vapor content. As the temperatureincreases, the amount of water vapor that can be mixed with the

air before saturation occurs also increases. For example, at80F and with a relative humidity of 100%, 1000 ft3of air wouldcontain 0.2046 gallons of water. At 120F and with a relativehumidity of 100%, 1000 ft3of air would contain 0.7460 gallons ofwater. Conversely, as the air temperature decreases, the airscapacity to hold water vapor also decreases.

Table 3. Water Content of Air in Gallons Per 1000 ft3 at Various

Relative Humidities

Temperature, F

%RH 35 40 50 60 70 80 90 100 110 120

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

.0019

.0039

.0058

.0078

.0098

.0117

.0137

.0156

.0176

.0195

.0215

.0235

.0254

.0274

.0294

.0313

.0333

.0353

.0372

.0392

.0024

.0047

.0071

.0095

.0119

.0143

.0166

.0190

.0214

.0238

.0262

.0286

.0310

.0334

.0358

.0382

.0406

.0430

.0454

.0478

.0035

.0069

.0104

.0139

.0174

.0209

.0244

.0279

.0314

.0349

.0384

.0419

.0454

.0490

.0525

.0560

.0596

.0631

.0666

.0702

.0050

.0100

.0150

.0200

.0251

.0301

.0351

.0402

.0453

.0503

.0554

.0605

.0656

0.707

.0758

.0810

.0861

.0913

.0964

.1016

.0071

.0142

.0213

.0284

.0356

.0427

.0499

.0571

.0644

.0716

.0789

.0861

.0934

.1007

.1081

.1154

.1228

.1302

.1376

.1450

.0099

.0198

.0298

.0398

.0498

.0599

.0700

.0801

.0903

.1005

.1107

.1210

.1313

.1417

.1521

.1625

.1730

.1835

.1940

.2046

.0136

.0273

.0411

.0549

.0689

.0828

.0969

.1110

.1251

.1394

.1537

.1681

.1825

.1970

.2116

.2263

.2410

.2559

.2707

.2857

.0186

.0372

.0561

.0750

.0940

.1132

.1325

.1519

.1715

.1912

.2110

.2310

.2511

.2713

.2917

.3122

.3328

.3536

.3745

.3956

.0250

.0501

.0755

.1012

.1270

.1531

.1794

.2060

.2328

.2598

.2871

.3146

.3424

.3705

.3988

.4273

.4562

.4853

.5147

.5443

.0332

.0668

.1007

.1351

.1699

.2051

.2407

.2768

.3133

.3502

.3876

.4254

.4637

.5025

.5418

.5816

.6219

.6627

.7041

.7460


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Pressure also has a major effect on the ability of air to holdvapor content. The capacity of air at a given temperature tohold moisture in vapor form decreases as the air pressureincreases. Table 4 lists the water content of saturated air

(relative humidity of 100 percent) at given temperatures andpressures. For example, if 1000 ft3of saturated air iscompressed from 0 to 200 psig while the temperature ismaintained constant at 100F, the ability of the air to holdmoisture in vapor form decreases. The moisture wouldcondense. The amount of moisture that will condense is thedifference between the amount of moisture that air can hold atthe two pressures, 0.3956 gallons at 0 psig minus 0.0254gallons at 200 psig, or 0.3701 gallons.


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Table 4. Water Content of Saturated Air in Gallons per 1000 ft3at VariousTemperatures and Pressures with 100% Relative Humidity

Temperature, F

psig 35 40 50 60 70 80 90 100 110 120

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

200

.0392

.0233

.0165

.0128

.0165

.0089

.0077

.0068

.0060

.0055

.0050

.0046

.0043

.0040

.0037

.0035

.0033

.0031

.0029

.0028

.0027

.0479

.0283

.0201

.0156

.0128

.0108

.0093

.0082

.0074

.0067

.0061

.0056

.0052

.0048

.0045

.0042

.0040

.0038

.0036

.0034

.0032

.0702

.0416

.0295

.0229

.0187

.0158

.0137

.0121

.0108

.0098

.0089

.0082

.0076

.0071

.0066

.0062

.0058

.0055

.0052

.0050

.0048

.1016

.0600

.0426

.0330

.0269

.0228

.0197

.0174

.0155

.0140

.0128

.0118

.0109

.0102

.0095

.0089

.0084

.0080

.0075

.0072

.0068

.1450

.0854

.0605

.0469

.0383

.0323

.0280

.0246

.0220

.0199

.0182

.0167

.0155

.0144

.0135

.0126

.0119

.0113

.0107

.0102

.0097

.2046

.1200

.0849

.0657

.0536

.0452

.0391

.0345

.0308

.0279

.0254

.0234

.0216

.0201

.0188

.0177

.0167

.0158

.0149

.0142

.0136

.2857

.1667

.1176

.0909

.0741

.0625

.0540

.0476

.0425

.0385

.0351

.0323

.0298

.0278

.0260

.0244

.0230

.0217

.0206

.0196

.0187

.3956

.2290

.1612

.1213

.1012

.0853

.0737

.0649

.0580

.0524

.0478

.0439

.0407

.0378

.0354

.0332

.0313

.0296

.0281

.0267

.0254

.5443

.3119

.2186

.1682

.1367

.1152

.0995

.087

.078

.0706

.0644

.0592

.0548

.0509

.0476

.0447

.0421

.0398

.0378

.0359

.0342

.7460

.4217

.2939

.2256

.1830

.1540

.1329

.1169

.1043

.0942

.0858

.0789

.0729

.0678

.0634

.0595

.0561

.0530

.0503

.0478

.0455


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Volumetric Flow

A compressor is typically specified by the required volumetricflow. Volumetric flow is the gas flow rate through the

compressor at specified conditions. Standard cubic feet perminute (SCFM) is a common method of describing the capacityof a compressor; however, the specified standard conditionsthat define volumetric flow may vary. For example, one source

defines SCFM conditions as 14.7 psia at 60F (15.5C) and 0%relative humidity, while another source defines SCFM conditions

as 14.7 psia at 68F (20C) and 36% relative humidity. Themetric standard for volumetric flow conditions, standard cubic

meter per hour (SCMH), is defined as 1 atmosphere at 0C(32F) and 0% relative humidity. Compressor manufacturersfrequently define the volumetric flow of a compressor by theactual volume used to obtain the actual gas velocity. Inlet cubicfeet per minute (ICFM) or inlet cubic meter per hour (ICMH)indicates the actual volumetric flow of gas entering thecompressor at the expected operating conditions. The inletcubic feet per minute is also referred to as the actual cubic feetper minute (ACFM). Likewise, the inlet cubic meter per hour isalso referred to as the actual cubic meter per hour (ACMH).

The manufacturers curves for the performance of a compressorare based on the actual volumetric flow at the inlet of thecompressor (ACFM). As the following equation shows,

calculations of the value of actual volumetric flow can bedetermined from the standard flow (SCFM).

Actual Volumetric Flow = Standard Volumetric

Flow 1std

1

1

std ZT

T

P

P

Where:

Actual Volumetric Flow = Volumetric flow in actual cubicfeet per minute for English units or actual cubic meter per

hour for metric units.

Standard Volumetric Flow = Volumetric flow in standardcubic feet per minute for English units or standard cubicmeter per hour for metric units.


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P1 = Inlet pressure, psia

Pstd = 14.7 psia in English units, 1 atmosphere in metric

units

T1 = Inlet temperature, R

Tstd = Standard temperature, 520R

Z1 = Inlet compressibility factor

The following example shows how to use this equation tocalculate the inlet flow of a centrifugal compressor that delivers5000 SCFM of natural gas. The inlet pressure (P1 ) is 25 psia,

the inlet temperature (T1 ) is 560R, and the inlet compressibilityfactor (Z1 ) is 0.95. The inlet compressibility factor wouldnormally need to be calculated as previously discussed in thismodule.

ACFM30080.95520

560

25

14.7

SCFM5000ACFMin,InletatFlowVolumetricActual

=

=

Mechanical Engineers should note that, for this examplecalculation, the inlet temperature was given in degrees Rankine

(R) and that the inlet pressure was given in pounds per squareinch absolute (psia). In actual field calculations, these valuesmust be obtained from installed instrumentation, which normallyindicates the inlet temperature in degrees Fahrenheit (F) andthe inlet pressure in pounds per square inch gauge (psig). Thesuction temperature (in F) and the inlet pressure pounds persquare inch gauge (psig) must be converted to R and psiabefore they can be used in the ACFM calculation of actualvolumetric flow at the inlet. The following equation is used toconvert temperatures in F to temperature in R:

R = F + 460

The following equation is used to convert pressures in psig topressure in psia:

PSIA = PSIG + 14.7


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Actual volumetric flow can also be determined from mass flowthrough the use of the following equation:

ACFM = W x V

Where: W = Weight flow in lb/min (kg/min)

V = Inlet specific volume in ft3/lb (m3/lb)

Inlet specific volume may be determined through the use of thefollowing equation:

1

11

P144

RTZV=

Where: V = Inlet specific volume in ft3

/lb

Z1 = Compressibility factor at inlet conditions

R = Gas constant fromWeightMolecular

1545.32

MW

Runiv =

T1 = Temperature at inlet conditions, R

P1 = Pressure at inlet conditions, psia

For metric calculations:

R = 8.3143Rmolgm

Joules

and 144 is replaced

with 18.129

Mollier Diagrams

Compressor performance cannot be accurately predicted

without detailed knowledge of how a gas or gases will behavewhen compressed. The behavior of a wide variety of gases inany conceivable mixture can be accurately computed, plotted,and offered to the process engineer in the form of a pressure-enthalpy diagram, which is called a Mollier diagram. A Mollierdiagram is a graphical representation of the relationshipbetween the pressure, the temperature, the volume, theenthalpy, and the entropy of a gas. Mollier diagrams are readily


39/72




available for most pure gases at conventional pressures andtemperatures, but Mollier diagrams are not readily available forgas mixtures. Because most of the gases that are used atSaudi Aramco are gas mixtures, Mollier diagrams are not that

widely used in Saudi Aramco. The Saudi Aramco applicationsfor which Mollier diagrams are useful are the refrigerant gases,namely, propane and freon.

From the Mollier diagram, enthalpy and specific volume canthen be directly determined. The use of a Mollier diagramenables calculation of head, efficiency, and specific volume(ft3/lbm).

Mollier diagrams display gas properties. The process of gascompression is easy to visualize when plotted on a Mollier

diagram. The phase change, the expansion, and thecompression process can be seen, and it is easier tocomprehend the overall process and the effect of processchanges.

On a Mollier diagram, as shown in Figure 4, the pressure istaken as the ordinate, and enthalpy is taken as the abscissa.Lines of constant entropy and constant volume slope upwardfrom left to right. Lines of constant temperature slopedownward from left to right. The area on the diagram that isenclosed by the saturated vapor line represents the liquid-vapor

region of the gas. The critical point represents the top-most partof the saturated vapor line. Above the critical point, a gascannot be liquefied.

The following equation is used to calculate isentropic head:

12isis hhH =

Where: His = Isentropic head

h2is = Isentropic discharge enthalpy

hi = Inlet enthalpy


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The following equation is used to calculate isentropic efficiency:

12

12is

is hh

hh

=

Where: is = Isentropic efficiency

h2 = Discharge enthalpy

The following example illustrates how to use the Mollier diagramthat is shown in Figure 4 to find the inlet and discharge specificvolume, the enthalpy, and the isentropic discharge enthalpy.Figure 4 is a section of the Mollier diagram for propane. Thecompressor gas inlet pressure is 14.7 psia at a temperature of

40F. The compressor gas discharge pressure is 310 psia at atemperature of 315F.

The inlet pressure, 14.7 psia (P1), is located on the ordinate. Aline is horizontally followed from P1until it intersects with thetemperature line that corresponds to the given inlet temperature

of 40F (T1). This intersection is labeled point 1.

The specific volume for point 1 (v1) is estimated from the twoadjacent constant volume lines. For this example, v1isapproximately 8.25 cubic feet per pound.

A line is vertically followed from point 1 down to the abscissa.This point on the abscissa is the inlet enthalpy (hi). For thisexample, inlet enthalpy is approximately 128 BTUs per pound.

The given discharge pressure, 310 psia (P2), is located on theordinate. A line is horizontally followed from P2until it intersectswith the temperature line that corresponds to the discharge

temperature, 315F (T2). This intersection point is labeled point2.

The specific volume for point 2 (v2) is estimated at 0.57 cubicfeet per pound from the two adjacent constant volume lines.

A line is vertically followed from point 2 up to the abscissa. Thispoint on the abscissa is the actual discharge enthalpy (h2).


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The isentropic discharge enthalpy (h2is) is located by following aconstant entropy line from point 1 to point 2 until the dischargepressure line (P2) is intersected. This intersection is point 2is. A

line is vertically followed from point 2isdown to the abscissa.This point on the abscissa is the isentropic discharge enthalpy(h2is). For this example, h2isis approximately 206 BTUs perpound.

The isentropic efficiency is the ratio of ideal (isentropic) energyto actual energy.


42/72




Figure 4. Mollier Diagram for Propane


43/72




Dynamic Compressor Characteristics

This section of the module examines the following areas that

Saudi Aramco Engineers must consider when determining theoperation of dynamic compressors:

Cause of the Performance Curve (Velocity Triangles)

Performance Curves

Performance Characteristics

Use of Fan Laws to Find Operating Points at DifferentSpeeds

Cause of the Performance Curve (Velocity Triangles)

A performance curve is a plot of the expected compressoroperating characteristics. For example, a performance curvecan be plotted as compressor head, volumetric flow rate, power,or efficiency. A performance curve usually sets the volumetricflow rate as the abscissa and either head, power, or efficiencyas the ordinate.

A compressor head versus volumetric flow performance curveprovides important compressor operating information. There

are three important aspects of a compressor head versusvolumetric flow performance curve: slope of the curve, surge,and stonewall (also called choke). Figure 5 illustrates a headversus volumetric flow diagram.

The change in compressor head for the change in gasvolumetric flow defines the slope of the performance curve. Theslope of the performance curve is defined by the gas velocitiesat the compressor impeller. A vector analysis of gas velocityand impeller blade tip speed can be graphically shown as acompressor velocity triangle.

The impeller design and the inlet design combine to greatlyaffect the gas velocity distribution in the impeller. The design ofthe impeller has a higher impact on the velocity triangle thandoes the design of the compressor inlet; therefore, the design ofthe impeller, such as blade angle, will be discussed.


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Figure 5. Basic Head Versus Flow Performance Curve

As shown in Figure 6, there are three blade profileconfigurations: forward leaning, radial, and backward leaning.The impeller blade profile influences the velocity of the gas as ittravels through the impeller and exits at the blade tip. Figure 6

illustrates the shape and impeller-exit velocity diagrams and theresulting head curves for the three conventional types of blades.The gas stream moves through the impeller blades with arelative velocity (Vr) while, at the same time, the impellerrotation imparts a tangential velocity (Vb) to the gas stream. Thegas stream possesses the resultant velocity (V) as it exits theimpeller. The resultant velocity is the vector sum of the relativevelocity (Vr) and the tangential velocity (Vb). The length of thevectors and the magnitude of the exit angle are determined bythe design of the impeller blades. The magnitude of the vectorsis determined by the tip speed of the impeller blade and by the

gas velocity relative to the blade.


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Figure 6. Centrifugal Compressor Impeller and Vector Diagram


46/72




Forward Leaning Blades- Forward leaning blades produce asignificant increase in the resultant velocity (V) when comparedto radial and backward leaning blades. The increase of the

resultant velocity is due to the coordinating vector sum of itscomponents, relative velocity (Vr), and tangential velocity (Vb).The direction of the relative velocity (Vr) allows all flow changesto dramatically affect the magnitude of the resultant velocity (V).Forward leaning blades produce a head versus flowperformance curve that does not continuously rise with adecrease in compressor flow. As a result of the saddle-shapedperformance curve, forward leaning blades produce inconsistenthead versus volumetric flow, which results in operationalinstability. The operational instability is the reason that forwardleaning blades are not used for centrifugal compressor

applications.Radial Blades- The increase in the resultant velocity (V) in theradial blades due to relative velocity (Vr) change is so small thatthe resultant velocity (V) is never appreciably different thantangential velocity (Vb), which results in nearly horizontalperformance curves. Any increase in head that is required bythe process will significantly reduce throughput and could easilysurge the compressor. Some older, open impellers weredesigned with radial blades because of the ease inmanufacturing.

Backward Leaning Blades- In contrast to forward leaningblades, backward leaning blades produce the lowest pressurerise for a given impeller tip speed. The direction of the relativevelocity (Vr) of backward leaning blades is such that itdecreases the magnitude of the resultant velocity (V).

The performance curve of a backward leaning blade impeller isa concave curve declining toward the right side of the plot.Because they produce the stable performance curves with thehighest efficiency, backward leaning bladed impellers are thepreferred choice for most compressor applications. The typical

standard for conventional closed impellers is 25 to 35 degreesof backward lean. A good design practice is to have abackward leaning impeller blades exit angle preferably between15 to 35 degrees. Typically, impellers that use a radial or nearradial blade design should not be used for process gascentrifugal compressors.


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Performance Curves

Figure 7 shows the effect of the impeller blade angle on head

and efficiency as compressor gas flow increases.

Figure 7. Effect of Impeller Blade Angle on Head and Efficiency

Because the magnitude of the resultant velocity that exits theimpeller produces the characteristics of the head curve, theforward leaning bladed impeller produces a greater head thanbackward lean or radial blade impellers when all other factorsare the same. The forward leaning blades provide a positivesloping head curve with the maximum head output. Althoughthe head profile is a positive attribute, the efficiency of the


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forward lean is the lowest of the three.

A radial bladed impeller has a theoretically constant head curvebecause tangential velocity does not effectively change with

flow. The fact that the head is reduced on increasing flow dueto a decrease in efficiency is attributable to higher frictionlosses. When going from design flow to minimum flow, theresulting basic slope normally shows a 2% to 3% head rise.

Overall stage efficiency is highest for backward leaningimpellers. The characteristics of the backward leaning bladeare such that, for a constant blade speed, the tangential velocityincreases as flow decreases, which is due to a lower relativevelocity. These factors result in an increased head output whenflow is decreased. When compared to forward leaning and

radial blades, a backward leaning blade has the greatest headrise, which results in the most stable performance curve of allblade profiles. The effect of the blade angle is not proportionalin regard to head, and the effect of backward lean on headoutput is minimized at low flow; therefore, a high backwardleaning impeller will produce almost as much head at minimumflow as a low backward leaning impeller running at the same tipspeed. As design flow is approached, however, the headdifference greatly changes. Because longer angles decreasethe slip factor, an increase in the backward lean angle to about45 degrees reduces the head that is produced, which partially

cancels out any positive effects of a greater backward lean. Slipis a consequence of the nonuniform velocity distribution acrossthe impeller channels, boundary-layer accumulation, and flowseparation.

Performance Curve Limits

Operation in some areas of a performance curve may bedetrimental to the operation of the compressor. The design of acompressor is controlled to minimize the likelihood of suchoccurrences; however, operation of a compressor outside the

design operating region may cause damage due to thephenomena that are known as surge and stonewall.

Surge- An important characteristic of a dynamic compressor isits surge point or surge limit. At some point on the operatingcurves for both centrifugal compressors and axial compressors,as shown in Figure 8, a condition of minimum flow exists inwhich the developed head is insufficient to overcome the head


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that is required by the system. This point or line is called thesurge point or surge line. When the compressor reaches thesurge point, flow separation (stall) occurs in the compressorblades and/or stationary passages and the gas in the discharge

piping backflows into the compressor. As the required headincreases, the flow decreases to produce enough head to matchthe system demand. When the highest point on the compressorcurve is reached, the compressor cannot increase the headfurther. At this point, the head that is required by the system ishigher than the maximum head that is produced by thecompressor. The flow in the impellers becomes unstable andreverses, which causes the discharge pressure to collapse. Thedischarge pressure will subsequently rise again, and the cyclerepeats. As many as six surge cycles can occur in one second.

Surge occurs at a predictable flow rate that is shown on themanufacturers curve as the surge point. The surge point on aperformance curve is specific to the speed of the compressor.

A surge point can be determined for various compressorspeeds. A plot of the surge points for each performance curveat a given speed provides a parabolic curve called the surgeline. A complete surge line, down to the origin of the plot, isneeded to assess the possibility of surge during compressorstartup and shutdown. A control system is used to keep theactual compressor flow rate above the minimum surge pointvalue.

The following are the most significant damaging effects ofsurge:

Rapid temperature rise

Increased thrust

Variable pressure

Variable flow

Variable speed

These effects can cause catastrophic compressor failure if theyare allowed to continue. The protection system (as specified inSAES-J-604) that is required by Saudi Aramco protects acompressor from extensive damage by tripping the unit beforesuch damage can occur.


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Figure 8. Dynamic Compressor Surge Line

Stonewall- Another important characteristic of dynamiccompressors is stonewall (or a choked flow condition), which isshown in Figure 9. As the flow rate through the compressorincreases beyond the design value, the amount of head that isproduced decreases because the tangential velocity of the gasdecreases. As the flow rate increases, the rate at which theproduced head decreases is accelerated. At a certain point, thehead that is produced drops rapidly to zero. This point is calledthe stonewall, or choked flow condition. The point at whichstonewall occurs is influenced by the Mach number.

The Mach number is the magnitude of the relative velocitycompared to the speed of sound (sonic velocity) of a particulargas. When the Mach number equals one, the point at whichstonewall occurs is reached. Stonewall occurs when sonicvelocity is reached at any point in the compressor, but it is

normally considered as stonewall when sonic velocity isreached at the compressor stage entrance. Once the sonicvelocity is reached, the flow through the compressor cannot beincreased. Because the system resistance is usually too greatto allow the compressor to reach this condition, stonewall orchoked flow is not usually reached in actual operation.


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The following are the most significant damaging effects ofstonewall:

Temperature rise due to low efficiency.

Gas turbulence that can excite blade natural frequencies(typically in axial compressors only).

Figure 9. Dynamic Compressor Stonewall

The molecular weight of the compressed gas also impacts thepoint of stonewall. The following is the equation for determiningthe Mach number:

sonic

rel

V

VM=

Where: M = Mach number

Vrel = The gas velocity relative to the blade

Vsonic = Sonic velocity


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The following equation is used for determining the sonic velocityof a gas:

( )MW

T1545KgVsonic =

Where: Vsonic = Sonic velocity

K = Ratio of specific heats

g = Gravitational constant, 32 ft-lbm/lbf-sec2

T = Temperature in R

MW = Molecular weight of the gas

Using the equations for Vsonicand the Mach number, highmolecular weight gases result in low Vsonicvalues and the Machnumber will quickly approach 1. Figure 10 shows a graphicalrepresentation of the effect of molecular weight on compressorhead versus flow curves.

Figure 10. Graphical Representation of the Effect of Molecular Weight onCompressor Head Versus Flow Curves


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Use of Fan Laws to Find Operating Points at Different Speeds

The general laws for speed characteristics (fan laws) are the

same for centrifugal compressors as for centrifugal fans andcentrifugal pumps. The three basic fan laws are as follows:

Equation 1:N

NQ=Q

1

2

12

Equation 2:

2

1

2

12 N

NH=H

Equation 3:

3

1

2

12 N

Nbhp=bhp

Where: Q1 = Initial flow rate, cfm

Q2 = Final flow rate, cfm

N1 = Initial speed, rpm

N2 = Final speed, rpm

H1 = Initial head, ft-lbs/lbm

H2 = Final head, ft-lbs/lbm

These equations show the relationship between the flow rate(Q), the head (H), the horsepower (bhp), and the compressorspeed (N). Basically, the performance of a centrifugalcompressor at speeds other than the speed for which the

compressor is designed is such that the capacity or flow rate willvary directly as the speed varies, as indicated in Equation 1.The head that is developed will vary as the square of the speed,as indicated in Equation 2. The horsepower will vary as thecube of the speed, as indicated in Equation 3.


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The fan laws can be used for estimation purposes; however, theaccuracy of the calculated results decreases with increasingspeed ratio. Because the change in energy in a fan is

significantly lower than in a compressor, the fan laws are moreaccurate for fans than for compressors. Other factors thatcontribute to the inaccuracies of the fan laws include thefollowing:

The higher the head, the greater the inaccuracy.

The heavier the gas, the greater the inaccuracy.

The greater the backward lean, the greater theinaccuracy.

Typically, the discrepancies will not be great until a speedchange of 30 to 40 percent is reached (except inmultistage compressors, where a change of 10 percentcan affect the fan laws). The fan laws only accuratelyapply to single-stage compressors with very lowcompression ratios.

The following examples illustrate the application of the fan laws:

Assume that a multistage compressor delivers 10,000 cfm at a

head of 30,000 lbm

lbfft

at an operational speed of 8000 rpm

with a required power input of 2200 bhp. The fan laws can beused to determine the speed, the head, and the power that arerequired from the same compressor system to deliver 11,000cfm.

The first fan law states that speed is proportional to flow rate.The required new speed can be found as follows:


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rpm8800800010,000

11,000

NNQ

Q

N

N

Q

Q

N

NQQ

21

1

2

1

2

1

2

1

212

=

=

=

=

The new speed that is required to obtain 11,000 cfm is 8800rpm.

The second fan law states that speed squared is proportional tohead. The required new head can be found as follows:

[ ]

lbm

lbfft36,300H

1.2130,000H

8000

8800

lbm

lbfft30,000H

N

NHH

2

2

2

2

2

1

212

=

=

=

=

The new head that is required to obtain 11,000 cfm at a speed

of 8800 rpm is 36,300lbm

lbfft .

The third fan law states that speed cubed is proportional tobrake horsepower. The new power required to obtain 11,000

cfm (at a speed of 8800 rpm and with a head of 36,300lbm

lbfft)

can be found as follows:


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[ ]

2928=bhp

331.12200=bhp

8000

88002200=bhp

N

Nbhp=bhp

2

2

3

2

3

1

2

12

The new required brake horsepower input is 2928 bhp.

These equations are used to draw the head curves at speed N2if the curve at speed N1is known, as shown in Figure 11.Starting with any point on the head curve at speed N1(point A1),both the head (H2) and the flow rate (Q2) are calculated byequations 1 and 2. Although the head is proportional to speedsquared, flow is proportional to speed; therefore, as point A2moves up to indicate the increase in head as speed increases,point A2also moves to the right to indicate increase in flow asspeed increases. These calculations give equivalent operatingpoints on the curve for speed N2(point A2). A series of thesepoints defines the head curves for the speed N2. Similarly, for

the horsepower curve that is shown in Figure 12, thehorsepower (bhp2) and the flow rate (Q2) for speed N2arecalculated from the horsepower (bhp1) and the flow rate (Q1) atspeed (N1point A1) to obtain the equivalent operating point A2.


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Figure 11. Typical Head Curve


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Figure 12. Typical Horsepower Curve

In the fan law example, a flow increase of only 10 percent

requires a driver horsepower increase of 33 percent. SaudiAramco specifications 31-SAMSS-001 and 31-SAMSS-006 onlyrequire that the compressor driver brake horsepower be rated10 percent greater than the compressor rated horsepower. As aresult, the driver and the coupling power ratings are typically thelimiting factors when considering a design flow increase of acompressor.


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DETERMINING POSITIVE-DISPLACEMENT COMPRESSORPERFORMANCE CHARACTERISTICS

This section discusses the following performance characteristicsfor positive-displacement compressors:

Isentropic Process

Pressure Volume Cycle

Clearance Volume

Pressure Effect on Volume

Isentropic Process

The isentropic process of a positive-displacement compressorvaries little from that of the dynamic compressor. The theory isthe same, but other factors are taken into account that affect theisentropic exponent.

The specific heat at constant pressure (Cp) and the specific heat

at constant volume (C) are affected by the variation intemperatures commonly occurring in reciprocating compressors.The temperature does not vary as much in centrifugalcompressors. These variations typically will increase thespecific heat constant.

The experimentally determined constant (n) in a polytropicequation is typically less than the ratio of specific heat constant(k) in the isentropic equation for a positive-displacementcompressor. The mechanical efficiency range also is slightlyhigher than for centrifugal compressors. It is approximately88% to 95% for positive-displacement compressors.

Pressure Volume Cycle

Figure 13 shows the pressure volume cycle of the reciprocatingtype of positive-displacement compressor. The positions of thepiston (a, b, c, d) correspond to the labeled points on thepressure volume diagram. As shown in Figure 13, the pressurevolume cycle includes:

Isentropic compression (the line between A and B). Thecylinder is filled with gas at the suction pressure with the


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piston at position A. As the piston starts to move, thesuction valve closes. As the piston continues to move fromposition A toward position B, the piston compresses thegas isentropically until the pressure within the cylinder

reaches the discharge pressure. At this point, thedischarge valve is closed.

Constant-pressure discharge (the line between B and C).At point B, the discharge valve opens and permits gas toflow from the cylinder into the discharge line at a constantpressure until the piston has reached the end of its stroke atpoint C.

Isentropic expansion (the line between C and D).Because it is impossible to build a compressor

compressor performance characteristics

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