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
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
