Gas Compression

k 

k 

 V 1 = P 2 V  V 2 = C   P 1 V

(5.2)

where; k = the isoentropic exponent, which is equal to the ratio of the specific heat at constant  pressure  pressure (C p) and the specific heat at constant volume (Cv) for the gas: k=

C  p

(5.3)

C v

For ideal gases, C  p - C v = R

(5.4)

where; R = universal gas constant. C p and Cv are functions of temperature only for ideal gases. For real gases, however, C p and Cv are functions of both pressure and temperature.

∂ P  2 (   ) ∂T  v C  p - C v = -T  ∂ P  (   ) ∂V  T 

(5.5)

5.1.3 Polytropic Compression For "real" gases under actual conditions (with friction and heat transfer), the compression process is  polytrop  polytropic, ic, where where the the poly polytro tropic pic expone exponent nt n applie appliess inste instead ad of of the the adia adiabat batic ic expo exponen nentt k: n

n

 V 1 = P 2 V  V 2 = C   P 1 V

(5.6)

5.1.4 Evaluation of Work Required in Compression According to the general energy equation the theoretical work required to compress a unit mass of  gas from pressure P1 at state 1 to P2 at state 2 is given by:  P 2

W = ∫  P 1 VdP +

∆ v2

 g  + ∆z + l w 2 g c  g c

where; W = work done by the compressor on the gas, P = pressure, V = volume of a unit mass of gas, z = elevation above a datum plane, lw = lost work due to friction and irreversibility, g = gravitational acceleration, gc = conversion constant relating mass and weight, 59

(5.7)

 Neglecting frictional losses, and the changes in kinetic and potential energies, the energy balance of  Equation 5.7 can be written as:  P 2

W = ∫  P 1 Vdp

(5.8)

Substituting for V from Equation 19, we get: 1

 P 2

-

1

W = C k  ∫  P 1 p k  dP 

(5.9)

where C is a constant. Upon integration, Equation 5.9 becomes: W=

k  k -1

1

(k-1)

(k-1)

C k  [  P 2 k  - P 1 k  ]

1

(5.10)

(k-1)

k 

C  k   P 2 W=  P 1 (   ) [ (   ) k -1  P 1  P 1

k 

- 1]

(5.11)

Substituting for C/P1 from Equation 5.2, we get: k -1

W=

k 

 P 2 k   P1V 1 [ (   ) - 1] k -1  P 1

(5.12)

From the ideal gas law, for a unit mass of gas:  P1V 1 = R T 1

(5.13)

Substituting for P1V1 from Equation 5.13 into Equation 5.12:

W = R T 1

k  k -1

k -1

( r  k  - 1)

(5.14)

where; r = compression ratio (P2/P1) This analysis assumed an ideal gas. Where deviation from ideal gas behavior is significant, Equation 5.14 empirically modified in several different ways. One such modification is: k -1

k 

 P 2 k   z 1 + z 2 W = R T 1 [ (   )   -1](   ) k - 1  P 1 2 z 1

60

(5.15)

where; z1 and z2 = compressibility factors of the gas at inlet and outlet conditions. During compression the discharge temperature of the gas changes and can be estimated by the following formula. k -1

 P 2 = (   )  P 1 T 1

T 2

k 

(5.16)

where; T1 and T2 = temperature of the gas at the inlet and outlet conditions. 5.1.5 Multi-staging There are practical limits to the permissible amount of compression for a single compression stage. The limitations vary with the type of compressor, and include the following: - Discharge temperature. - Compression efficiency. - Mechanical stress problems. - Compression ratio. Whenever any limitation is involved, it becomes necessary to use multiple compression stages (in series). Furthermore, multi-staging may be required from a purely optimization standpoint. For  example, with increasing compression ratio r, compression efficiency decreases and mechanical stress and temperature problems become more severe. Inter-coolers are generally used between the stages to increase compression efficiency as well as to lower the gas temperature that may become undesirably high. Theoretically minimum power requirement is obtained with perfect inter-cooling and no pressure loss between stages by making the ratio of compression the same in all stages. The following formula uses the overall compression ratio, 1

 P  final   s  ) r  s = (   P initial 

 

(5.17)

where; s = number of stages r s = theoretically best compression ratio per stage 5.1.6 The reciprocating compressor  The basic reciprocating compression element is a single cylinder compressing only on side of  the piston (single acting). A unit compressing on both sides of the piston (double acting) consists of two basic single acting elements operating in parallel in one casting. The reciprocating compressor uses automatic spring-loaded valves that open only when the  proper differential pressure exists across the valve. Inlet valves open when the pressure in the cylinder is slightly below the intake pressure. Discharge valves open when the pressure in the cylinder is slightly above the discharge pressure. Figure 5.1, diagram A, shows the basic element with the cylinder full of atmospheric air. On the theoretical PV diagram (indicator card), point 1 is the start of compression. Both valves are closed. 61

Diagram B shows the compression stroke, the piston having moved to the left, reducing the original volume of air with an accompanying rise in pressure. Valves remain closed. The PV diagram shows compression from point 1 to point 2, and that the pressure inside the cylinder  has reached that in the receiver . Diagram C shows the piston completing the delivery stroke. The discharge valves opened just  beyond point 2. Compressed air is flow out through the discharge valves to the receiver. After the piston reaches point 3, the discharge valves will close leaving the clearance space filled with air at discharge pressure. During the expansion stroke, diagram D, both the inlet and discharge valves remain closed and air trapped in the clearance space increases in volume causing a reduction in pressure. This continues, as the piston moves to the right, until the cylinder pressure drops below the inlet pressure at point 4 .The inlet valves now will open and air will flow into the cylinder until the end of the reverse stroke at point 1. This is the intake or  suction stroke, illustrated by diagram E. At point 1 on the PV diagram, the inlet valves will close and the cycle will repeat on the next revolution of the crank. In a simple two-stage reciprocating compressor, the cylinders are proportioned according to the total compression ratio, the second stage being smaller because the gas, having already  being partially compressed and cooled, occupies less volume than at the first stage inlet. Looking at the PV diagram (Figure 5.2), the conditions before starting compression are points 1 and 5 for the first and second stages, respectively; after compression, points 2 and 6, and, after delivery, 3 and 7. Expansion of air trapped in the clearance spaces as the pistons reverse  brings points 4 and 8, and on the intake stroke the cylinders are again filled at points 1 and 5 and the cycle is set for repetition. Multiple staging of any positive displacement compressor  follows the above pattern. Compression Cycles Two basic compression cycles that are applicable to all compressors are isothermal  compression and adiabatic compression. A third process, polytropic compression, is widely used, but, since it is a modification involving an efficiency to more nearly represent actual conditions, it is not a true basic cycle. Figure 5.3 shows the theoretical zero clearance isothermal and adiabatic cycles on a PV basis. The area ADEF represents the work required when operating on the isothermal basis; and ABEF, the work required on the adiabatic basis. Obviously the isothermal area is considerably less than the adiabatic and would be the cycle for greatest compression economy. However, it is never commercially possible to remove the heat of compression as rapidly as it is generated. Therefore, this cycle is not as logical a working base as the adiabatic although it was used for  many years. Compressors are designed, however, for as much heat removal as possible. Adiabatic compression is likewise never exactly obtained, since with some types of units there may be heat losses during part of the cycle and a gain in heat during another part. For this reason, polytropic compression cycle is generally used. The polytropic exponent, n, is experimentally determined for a given type of machine and may  be lower or higher than the adiabatic exponent, k. In positive displacement and internally cooled dynamic compressors n is usually less than k. In uncooled dynamic units it is usually higher than k due to internal gas friction. Although n is actually a changing value during compression, an average or effective value, as calculated from experimental information is used. 62

In addition to the isothermal and adiabatic compression curves shown in Figure 5.3, the dotted lines show typical polytropic curves for a water-cooled reciprocating cylinder (AC) and for a non-cooled dynamic unit (AC'). Although the exponent n is seldom required, the quantity n-l/n is frequently needed. This can  be obtained from the following equation, although it is necessary that the polytropic efficiency ηP be known from prior test. The k value of any gas or gas mixture is either calculable or  known. n −1 n

=

k − 1 k 

×

1

η P 

(5.18)

where ηP is the polytropic compression efficiency. Figure 5.4 solves this equation in curve form. If, either n or n-l/n is known, the discharge temperature can be estimated from the following equation. Figure 5.5 will be useful for this purpose. n −1

T 2 T 1

n −1   P 2   n =    = r  n     P 1  

(5.19)

In adiabatic cycle, it is customary to use the theoretical discharge temperature in calculations. In an actual compressor, there are many factors acting to cause variation from the theoretical  but, on an average, the theoretical temperature is closely approached and any error introduced is slight. The power requirement of isothermal compression cycle is the absolutely minimum power  necessary to compress the gas. An actual compressor with an infinite number of intercoolers and stages of compression would approach Isothermal conditions if the gas were cooled to the initial temperature in the intercoolers. Qualitatively, Figure 5.6 shows horsepower  requirements versus number of compression stages. The horsepower approaches the isothermal value as the number of stage increases. Mollier Charts Mollier charts for a gas are plots of enthalpy versus entropy as a function of pressure and temperature. Mollier charts for natural gases with specific gravities in the range of 0.6 to 1.0 are shown in Figures 5.7 through 5.12. In developing the use of an enthalpy-entropy (H-S) diagram for calculating the power required compressing a gas, the following equation best describes the process.

ν2 ∆ H  + ∆ 2 g c

+

 g   g c

∆ X  = q − w

(5.20)

where AH is the increase in enthalpy between initial and final states, 2nd term of the left hand side is the difference in kinetic energy and 3rd term is the change in potential energy . Q is the heat absorbed by system from surroundings and w is work done by the fluid while in flow. 63

Kinetic energy and potential energy due to differences in elevation are usually neglected, leaving

∆ H  = q − w

(5.21)

For isothermal and frictionless conditions

∆ H  = T ∆S − w

(5.22)

or 

− w = ∆ H  − T ∆S 

(5.23)

where T is the absolute temperature and ∆S is the difference in entropy . To find the theoretical horsepower required compressing isothermally 1 MMscf/day at 60 °F and 14.7 psia, equation 5.23 is written

− w = 0.0432(∆ H  − T ∆S )

(5.24)

where ∆H is in Btu per pound mole and ∆S is in Btu peer pound mole per degree Rankine. By following a line of constant temperature on an enthalpy-entropy diagram between initial and final pressure conditions, ∆H and ∆S can be determined. In the case of adiabatic compression, q of Equation 5.21 is also zero; so

− w = ∆ H 

(5.25)

That is, at adiabatic or constant-entropy conditions for a single stage of compression, the work  necessary to compress the gas is equal to the difference in enthalpy between the initial and final stages of compression. Expressing the adiabatic theoretical work necessary to compress 1 MMscf/day at 60 °F and 14. 7 psia results in

− w = 0.0432∆H 

(5.26)

where ∆H is expressed in Btu per pound mole. For multistage compression the ∆H must be calculated separately for each stage and totaled. In addition to the horsepower, the final temperature of compression and the heat removed in the intercoolers can be obtained from enthalpy-entropy diagrams. The procedure for calculating horsepower from an enthalpy-entropy diagram can be best shown diagrammatically. Point 1 in Figure 5.13 is the initial state of the gas as it enters the compressor. Path l-2 shows the first stage of compression (constant entropy). The gas is then cooled in the intercoolers at constant pressure (path 2-3); the difference in enthalpy along this  path is equal to the heat removed in the intercooler. Path 3-4 shows the second stage of  compression. The temperatures at points 2 and 4 are the temperatures of the gas at the end of  64

the first and second stages of compression. The temperature at point 3 is the temperature to which the gas is cooled in the intercooler. 5.2 GAS FLOW MEASUREMENTS 

Because of the evident importance of accurate measurement of the quantity of gas delivered, various technical societies have combined their efforts to arrive at standards and procedures for  quantitative measurement of gas that are mutually acceptable to both purchasers and sellers of gas  products. Also, there is interest in the orifice meter throughout industry in general. Two general classes of metering devices are available for measuring gas rates: dynamic and volumetric meters (Table 5.1). For measuring large volumes, the orifice flow meter is the primary type. In these meters, the pressure drop occurring at a restriction indicates the flow rate. With the exception of domestic or space heating sales, essentially all measurements are based on orifice meter  The second class of meter, which is used for domestic sales, is the volumetric meter; this meter  receives mechanically a definite volume of gas from an upstream source, counts the volume, and discharges it downstream. However, to make this volumetric measurement more meaningful, base or standard pressure and temperature conditions are defined that yield measurement in standard cubic feet. The most common basis is the AGA and API recommended pressure of 14.73 psia and o temperature of 60 F. Table5.1 Principal Types of Gas Flow Meters

   

Dynamic measurement Orifice meter Venturi meter Flow nozzle Critical-flow prover  Pitot tube Rotameter  Choke

Volumetric Measurement Diaphragm meter  Laboratory wet-test meter

Gas is most commonly measured in terms of volume because of the simplicity of such a procedure. However, to make this volumetric measurement more meaningful, base or standard pressure and temperature conditions are defined that yield measurement in standard cubic feet. This volumetric rate can be converted to mass flow rate by multiplying with the gas density at the standard pressure and temperature conditions. Since the gas density at the standard pressure and temperature is a constant for the particular gas under consideration, measurements in standard cubic feet are synonymous to mass flow rate measurements. Flow is one of the most difficult variable to measure, because it cannot be measured directly like  pressure and temperature. It must be inferred by indirect means, such as pressure differential over a specified distance, speed of rotation of a rotating element, displacement rate in a measurement chamber, etc. 5.2.1 Attributes of Flow Devices A flow-meter or measurement device is characterized using the following parameters: - Accuracy - Rangeability - Repeatability - Linearity Accuracy: This is a measure of a flow-meter's ability to indicate the actual flow rate within a 65

specified flow-rate range. It is defined as the ratio of the difference between the actual and measured rates to the actual rate. ABS [Actual rate - Measured rate] Accuracy =   × 100 (%) Actual rate Accuracy is reported in either of two ways: percent of full scale, or percent of reading. For example, for a 100 MMscfd flow-meter, a ± 1% of full scale accuracy means that the measured flow rate is within ± 1 MMscfd of the actual flow rate, regardless of the value of the flow rate. Thus, for a measured flow rate of 10 MMscfd, the actual flow rate is between 9 and 11 MMscfd, and for a measured rate of 100 MMscfd, it is between 99 and 101 MMscfd. An accuracy of ± 1 % of reading, however implies that the measured flow rate is within 9.9 to 10.1 MMscfd for a measured rate of 10 MMscfd, 49.5 to 50.5 for a measured rate of 50 MMscfd, 99 to 101 MMscfd for a measured rate of 100 MMscfd, etc. Thus, the percent reading results in a better overall performance because the error is proportional to the magnitude of rate. Orifice meters and rotameters have a percent of fullscale accuracy in their specifications.  Rangeability: A flow-meter's rangeability is the ratio of the maximum flow rate to the minimum flow rate at the specified accuracy. Maximum flow rate that can be measured Rangeability =   Minimum flow rate that can be measured Rangeability is usually reported as a ratio X:1. For example, a meter with maximum and minimum rates of 10 and 50 MMscfd, respectively, for a specified accuracy of ± 1%, has a rangeability of 5:1. Thus, it is important to know the flow rate range over which a quoted rangeability applies. Repeatability: Also known as reproducibility or precision, repeatability is the ability of a meter to reproduce the same measured readings for identical flow conditions over a period of time. It is computed as the maximum difference between measured readings, sometimes expressed as a  percent of full scale. Note that repeatability does not imply accuracy; a flow-meter may have a good repeatability, but a lower overall accuracy. Linearity: This is a measure of the deviation of the calibration curve of a meter from a straight line. It can be specified over a given flow rate range, or at a given flow rate. A calibration curve is desirable because it leads to a constant metering accuracy, with no portion of the scale being relatively more or less sensitive than the other. Note that a flow-meter could have a good linearity,  but poor accuracy if its calibration curve is offset (shifted). 5.2.2 Measurement by Orifice Meters This is by far the most commonly used device for metering natural gas. It consists of a flat metal  plate with a circular hole, centered in a pair of flanges in a straight pipe section (Figure 5.14). The  pressure differential is measured across this plate to yield the flow rate. The relationship for orifice meters can be derived from the general energy equation, written between two points in the flowing stream- Point 1 being some point upstream of the orifice plate, and point 2 representing the orifice throat: 1

 g 

∫ 12 VdP+  g   ∫ 12 vdv+  g  ∫ 12 dz = w s - l w c

c

where; 66

(5.27)

V = specific volume of the fluid, P = pressure, v = fluid velocity, z = elevation above a given datum plane, ws = shaft work done by the fluid on the surroundings, lw = work energy lost due to friction, g = gravitational acceleration, gc = conversion factor relating mass and weight. For most meters, the elevation change between points 1 and 2 is zero, and no work is done by the flowing fluid stream. Therefore, Equation 5.27 can be written as: 1

∫ 12 Vdp+ g   ∫ 12 vdv+ l w = 0

(5.28)

c

Incorporating the friction loss term lw in the compression-expansion term to avoid the complexity of  referring to the friction factor, and multiplying both sides by fluid densityρ, we get : 2

C 2 ∫ 1 dP +

1 2 ∫   ρvdv=0  g c 1

(5.29)

where C is an empirical constant. Since the pressure differential is small as compared with the pressure, an average density ρav is used and this variable is removed from under the integral as if it were constant.

C 2 (  P 2 - P 1 )+

ρ av 2 g c

( v 22 - v12 ) = 0

(5.30)

or, as more commonly written, v 22 - v 21 = C 2 g c

P 1 - P 2

ρ av

(5.31)

The mass flow rate, m , is given by: m = ρvA

(5.32)

Substituting for v in Equation 5.31, recognizing that v1 is measured at area A1, the pipe, and v2 at A2, the orifice; and defining b = Dorifice / D pipe, one obtains Equation 5.33.

m = C  A2

2 g c ρ av (  P 1 - P 2 ) 1- β

4

(5.33)

The value of C in Equations 5.31 and 5.33 varies with the area ratio b as well as with the flow rate. Figure 5.15 shows values of C, based on data with liquids, with the variation in flow rate given as a 67

Reynolds number through the orifice. For liquids at appreciable velocities, the orifice coefficient  becomes 0.610. Equation 5.31 may be reduced to a single velocity v1 or v2 by inserting a function of b as follows. 4 2 g c β (  P 1 - P 2 )

v1 = v pipe = C 

4

(1- β ) ρ av 2 g c (  P 1 - P 2 )

v 2 = vorifice = C 

4

(1- β ) ρ av

(5.34) (5.35)

These formulas apply for single-phase fluids, either liquids or gases. The orifice meter formulas, Equations 5.31 and 5.33 have included the average density ρav of the gas. For gases, the average density may be expressed in terms of pressure, temperature, and compressibility factor.

ρ av =

29G P 

(5.36)

 zRT 

The average values of pressure, temperature, and compressibility factor from points 1 to 2 in the meter should be used. It is customary to use either the upstream pressure or the downstream  pressure, the flowing temperature, and the compressibility factor at the temperature and the  pressure selected. In using an orifice formula for thousands of measurements each day, it is necessary to devise calculation procedures that can be followed easily in a routine manner. This has been done for the orifice formula, with the development of a series of factors or corrections that can be taken from tables. To convert Equation 5.33 into units and form similar to those used for natural gas, the following conversions will be made. Q=

(379)(3600)m

(5.37)

29G

where; Q = gas flow at standard conditions, cuft/hr  G = gas gravity m = mass velocity, lb/sec Combining Equations 5.33, 5.36, and 5.37,

Q=

(3600)(379) C  A2 29G

1- β

4

68

2 g c

(  P 1 - P 2  )29GP   zRT 

(5.38)

Q=

(3600)(379) 64.34C  A2 29 R G T z 1 - β

4

(5.39)

(  P 1 - P 2 )P 

Since the differential pressure h is measured in inches of water and (P1 - P2) is in pounds per square foot,

h=

(  P 1 - P 2 )12

 P 1 - P 2 =

Q=

62.43h

(5.41)

12

(3600)(379) 62.43 29 10.73

(5.40)

62.43

64.34C  A2

12

GTz 1- β

(5.42)

hP 

4

o

This equation gives the flow volumes computed at base conditions of 60 F and 14.7 psia. Since other base conditions may be used on occasion, factors are used to convert base conditions.

 F  pb =

 F tb =

14.7 

(5.43)

 P b

T b

(5.44)

520

o

for T b in R. o

It is customary to establish the value to give Q, for a flowing temperature of 60 F, a gas gravity of  1.0 and a compressibility factor of 1.0. To do this, the standard values are inserted. 1 GTz 

=

520 T 

1

1

1

(5.45)

G  z  520

Therefore;

Q=

(3600)(379) 62.43 64.34C  A 2 4 29 10.73 12 520 1- β

520 T 

1

1

(  F pb F tb ) hP  G  z 

(5.46)

The numerical constants C, b, A2 are grouped as a single constant F b. The other factors are defined:

69

520

 F tf  =

T 

 F  g =

 F  pv =

1 G

1  z 

(5.47)

(5.48)

(5.49)

where; Ftf  = The flowing temperature factor  Fg = The gas gravity factor  F pv = The supercompressibility factor  Equation 5.37 thus becomes, Q = F b F tb F pb F tf F g F pv   hP 

(5.50)

where; Q = gas flow rate, cuft/hr  h = orifice differential, in. of water  P = pressure of gas at orifice, psia. The flowing pressure and temperature are taken normally as upstream values. The expansion of the gas through the orifice is essentially adiabatic. Under these conditions, the density of the stream changes because of the pressure drop and the adiabatic temperature change. An expansion factor Y, computed for the adiabatic and reversible case, is included in the formula to correct for this variation in density. The values of C, and hence of F b, vary with the Reynolds number and the diameter ratio b. Correction factors for variations in Reynolds numbers are given as Fr , with F b considered to be a constant for a given meter installation.  The density of the gas opposite the mercury influences the differential indicated. Since the calibration is normally made at or near atmospheric pressure, corrections should be used for the gas density at high pressures. This correction factor is Fm is

 F m =

ρ Hg - ρ f  ρ Hg 

where;

ρHg= the density of mercury ρf = the density of the fluid opposite the mercury column in the meter. The complete orifice equation recommended by AGA is 70

(5.51)

(5.52)

Q = F b F r Y F pb F tb F tf F g F pv F m  hP  1

Tables of the above factors are provided by the AGA and are given by Katz et. al . To obtain the quantity of gas flowing through an orifice meter from the above equation and the AGA tables, the following data are required: - Diameter of pipe - Diameter of orifice - Differential across meter  - Pressure on meter  - Flowing temperature - Gas gravity - Carbon dioxide and nitrogen content of gas - Type of meter, pipe or flange taps. REFERENCE

1. Katz, D,L., D.Cornell, R. Kobayashì , F.H. Poettmann, J.A. Vary, J.R. Elenbaas, and C.F. Weinaug, Handbook of Natural Gas Engineering, McGraw-Hill Publishing Co., New York, pp. 751-763 (1959).

71

Figure 5.1 The various steps in a reciprocating compression cycle.

72

Figure 5.2 Combined theoretical indicator card for a two-stage compressor.

Figure 5.3 Theoretical indicator card showing various gas compression processes.

73

Figure 5.4 Ratio (n-1)/n versus adiabatic exponent k .

Figure 5.5 Polytropic temperature ratio versus ratio (n-1)/n.

74

Figure 5.6 Variation of compression horsepower with number of stages.

75

76

77

78

Figure 5.13 Example of use of enthalpy-entropy diagram to find adiabatic work of compression.

Figure 5.14 Orifice meters; Flange taps, pipe taps

79

Figure 5.15 Discharge coefficient for liquids.

80