.
ComDressor Design for the Process Industries Redd q 4 d , ]a* Carnegie Institute of Technology, Pittsburgh, Penna.
In the application of reciprocating compressors to the process industries, power consumption, volumetric capacity, methods of control, and flexibility of operation are important. This paper presents quantitative design methods applicable to the problems of applying compressors to processing. The cases considered are perfect or theoretical compressors operating on perfect gases, actual compressors operating on perfect gases, actual compressors operating on imperfect gases, and actual compressors operating on mixtures of imperfect gases. Criteria for predicting and alleviating liquefaction in compressors are discussed. A numerical example on the performance of a compressor operating on an imperfect gas mixture is included.
equipped with as many as eight unloader bottles or pockets; a snap-acting pilot valve is used to open (or close) progressively the valves within the bottles. Frequently this control may be operated by a process variable, especially suction or discharge pressure and flow rate, which thus ties in the performance of the compressor directly with the process. Control of the gas flow by speed regulation depends largely upon the driving unit. Gas compressors may be driven by electric motors, steam engines, or internal combustion engines. Where close regulation of gas flow is of primary importance, the electric motor is sometimes operated on direct current so that wide variation in speed regulation is practical. Alternating current motors operate at a substantially constant speed. Steam engines are capable of some speed regulation, whereas internal combustion engines may be varied in speed by as much as 70 per cent. Where speed regulation by the driving unit does not give sufficient flexibility, cylinder unloading must be used.
T H E process industries gases may be compressed to overcome pressure drop due to friction in transmission lines, to separate components in a mixture by flash vaporization, to liquefy by condensation, or to increase the yield of products in a high-pressure synthesis. In many processes the gas compressor is one unit, usually connected with reactors, heat exchangers, condensers, or absorption towers, and thereby functions as an integral part of the process equipment rather than as a single unit. I n this respect the performance of the compressor is important in supplying gas at the correct pressure and flow rate; otherwise the interconnected equipment may be overloaded, underloaded, or out of balance. Frequently it is desirable to run a plant at fractional loads. This may be accomplished with the compressor by changing speed or by unloading the cylinders, manually or automatically. Control of the gas flow by cylinder unloading is accomplished by lifting the suction valves and by varying the clearance volume. In practice, two methods are utilized to vary the clearance. The first method, applicable only to single-acting cylinders, is to increase the clearance of the head end by screwing the piston rod into the compressor crosshead. This arrangement changes the location of the piston within the cylinder a t the extremity of its stroke and thereby alters the clearance volume. The second method is to equip the cylinder with additional clearance volume, known as a clearance pocket or as an unloader bottle. The bottle is usually mounted upside down so that condensable gases will not collect in it. The opening is controlled by means of a valve. The pocket or bottle arrangement has the advantage, particularly in the process industries, that the clearance may be varied by automatic control. In practice the cylinder is
In the design and performance of reciprocating gas compressors, the important items are capacity and power requirement. Since the power requirement is directly proportional to the capacity, the equations relating to both these will be discussed together. It is convenient to write an over-all energy balance (11; 17, page 17) on the gas as the thermodynamic system. For all practical purposes the kinetic energy and potential energy terms in this balance may be neglected in applications to gas compressors. The siqplified energy balance is:
Power Requirements
IN
Q - W = A H
(1)
A large number of tests have established that compression of the gas within the cylinder is substantially reversible and adiabatic. This is substantiated by a rough calculation to show that the heat transferred by natural convection (but with disturbances from the moving piston) is negligible in a time interval of something like 0.1 second. Hence the Q term of Equation 1 may be disregarded:
-w
= (Am8
(2)
where ( A H ) s = enthalpy change along an isentropic path of compression The ( A H ) , of Equation 2 is a property of the gaa, and therefore the work required to compress a gas at the gas-piston interface may be evaluated from this property. The evaluation of (M),is greatly simplified for a perfect gas and will therefore be discussed for this case. THEORETICAL POWERREQUIREMENTSWITH PERFECT GASES. For perfect gases of constant heat capacity, the work of reversible adiabatic compression without clearance is given by the familiar expression, 535
INDUSTRIAL AND ENGINEERING CHEMISTRY
536
(3)
where k = cp/cuand-subscripts for pressure P and volume V correspond to the numbers of cycle 1-2-5-6 of Figure 1. Equation 3 does not allow for the effect of clearance and must therefore be modified. With clearance in a compressor cylinder, a the network required is represented by the d i f f er en c e b e t w e e n the areas, taken toward the P axis, of V the compression and expansion curves, Figure I. Ideal Pressureas shown by Figure Volume Diagram 1. If the Dath of expansion is‘ expressed as PV“ = constant, then this network is:
Vol. 34, No. 5
or in terms of the compression ratio, R., of measuring pressure, PO,and suction temperature, theoretical gas horsepower
=
where 7 0 , the capacity of gas in cfm. a t suction temperature, is corrected by the gas laws from suction pressure PI to measuring pressure Po. ACTUALPOWER REQUIREMENTS.Equations 8 and 9 are theoretical expressions for gas horsepower under ideal conditions of reversible adiabatic compression and limited to perfect gases of constant heat capacity. To make these equations represent actual power requirements, it is necessary to introduce a correction factor. One such factor heretofore used is the compression efficiency, defined by Kiefer and Stuart (11) as the “work for completely isentropic compression divided by actual work input”. ’When such a compression efficiency is used, values have been found t o vary from 0.50 to 0.95. An examination of losses occurring for gas compression within the cylinder reveals that the major deviation from ideal power requirements is caused by pressure drop through the suction (46 and discharge valves and ports. Thus gas being drawn into the cylinder will be a t a pressure less than suction pressure, PI, because of pressure drop through suction valves APl. Likewise, gas being discharged from the cylinder will be a t a pressure greater than discharge pressure Pz, because of pressure drop through discharge valves AP2. I n over-all effect this means that, t o compress a gas from P1 to Pz,the gas must be compressed between greater pressure ranges-namely, (Pl - AP1) and (Pz Ape). Let the ratio (P2 -/- A P J / ( P 1 - API) be termed the “cylinder compression ratio”, denoted by Rl, as separate and distinct from the line compression ratio P2/PI, denoted as R,. The valve losses caused by pressure drop are the major Posses, even though small in magnitude, in compressing a gas. Minor losses result from gas leakage through valves and piston rings and from heat radiation during re-expansion. All these losses increase the power requirement as computed by Equations 8 and 9. On the other hand, as a study of Equation 1 shows, the power requirement is slightly diminished by heat transferred during compression. The combined effect of all these factors may be allowed for in the value of R:. From several field tests the writer has found that RL is from 3 t o 20 per cent greater than R, and, even more significant, that for several types of cylinders R: may be correlated as a function of R,. For design purposes the relation of R: to R, will need to be determined from tests on each manufacturer’s machines. The gas horsepower may now be computed by
1
If the gas in the clearance space expands reversibly and adiabatically, exponents n and k become identical. Experiment has shown that for cylinders with normal clearances this relation is approximately true. At least, according to Gill (7) the values of n and k are so nearly the same that the use of the more complicated Equation 4 is not justified in computing power requirements. Because PI and P4 as well as P2 and Pa are numerically the same, the network becomes:
If the clearance is expressed as a fraction of the piston displacement, D, and as such is denoted by c, then under ideal conditions, r
.l
The bracketed term in Equation 6 is known as the ideal volumetric efficiency. Whether ideal or not, this efficiency is denoted by Ey. By these substitutions the network becomes:
(7)
+
actual gas horsepower
= P
The units in Equation 7 have not been discussed. In fact, all terms are dimensionless except P1 and D. It is customary to express the suction pressure, P I , in pounds per square inch absolute (abbreviated psia.) and the piston displacement D in cubic feet per minute (cfm.). This unit of time being introduced, Equation 7 becomes one for power. Let the term “gas horsepower” be used to denote the power required a t the gas-piston interface as evaluated from (AH)& Then, theoretical gas horsepower
=
r
k1i
9
--
33,000 k - 1
I n practice it is customary to choose a measuring pressure,
PO,as 14.70 psia., and a volumetric capacity, VO(corrected to PO),of 100 cfm. Under such conditions the solution of Equation 10 will give the gas horsepower per hundred cfm. a t 14.7 psia. and suction temperature, here denoted by F . Since F depends upon k , the ratio of heat capacities, it is convenient to present values as a plot of F against k for constant R,. Such a plot is given in Figure 2. The shaft or brake horsepower may be computed by adding frictional losses to the gas horsepower. These losses vary
INDUSTRIAL AND ENGINEERING CHEMISTRY
May, 1942
531
Courtesy, Cooper-Bessemer Corporatlon
Electrically Driven
Gas Compressor with Automatic
from 3 to 10 per cent of the gas horsepower, depending upon the type of drive-i. e., electric motor, steam, gas, or Diesel engine. The greater part of these frictional losses is due to sliding friction in the piston rings. But here again the magnitude varies with the machines of different builders. ACTUALVOLUMETRIC EFFICIENCY. The volumetric efficiency, E,, appears in the expression for gas horsepower (Equation 8). Thus the power requirement is directly proportional to the volumetric efficiency of a compressor cylinder. Previously the volumetric efficiency was discussed in connection with Equation 6 for ideal conditions. Because of the
Controls
importance of EV in calculating the power requirement, it is necessary to modify the idealized expression, 1 C - C X (P2/Pl)1/’,in order to make allowances for actual conditions. Thus wire drawing and valve leakage must be taken into account. I n addition, the P-V path may vary from PVk = constant because of heat transfer from the gas in the clearance volume to the cylinder walls and head. If the actual P-V path of re-expansion is expressed by PV“ = constant, the following empirical expression for the volumetric efficiency is obtained :
+
Ev
= 1
- 0.01RC- C
-
P1 (R,)
-1
(11)
where the term 1 0.01R,allows for wire drawing and valve leakage and where the exponent n fits the actual P-V path during re-expansion. While the variation between n and k is slight, the effect upon volumetric efficiency is much more pronounced than upon power requirements. After trying several methods of relating n to k , from the results of field tests the writer has found that n/k may be plotted against R. with curves of constant percentage clearance. Such a correlation will depend upon shape factors and therefore will vary from that of one compressor to that of another. It is customary among compressor designers to present values as a plot of volumetric efficiency against R, with curves of constant clearance and for a gas of fhed k. Figure 3 is such a plot for a gas with k = 1.25. Design and Performance with Perfect Gases
k,RATIO
Figure
2.
OF
H E A T CAPACITIES, J ‘& ’
Gas H o r s e p o w e r Curves
The general problems of design and performance fall-into three categories: total power requirement to handle a given gas flow; the number of machines, stages, and cylinders, and cylinder sizes for each stage; the gas flow that can be handled with a given power input and between fixed pressure conditions. POWER REQUIREMENT.For compressor problems where gas law deviations are negligible, the design procedure will
INDUSTRIAL AND ENGINEERING CHEMISTRY
538
gas horsepower =
TABLEI. COMPRESSOR CYLINDERDATA^ Max. Working Pressure Lb./Sq. 1;.
Clearance without Unloaders,
Cylinder Diam., In.
Stroke, In.
Speed, R. P. M.
Piston Displacement, Cu. Ft./Min.
Rod Diam., In.
6 9
9 9
500 500
114.1 261.5
1.50 1.50
600 400
14.8 8.0
3 6 9 12 18 24
14 14 14 14 14 14
300 300 300 300 300 300
17.18 68.7 297.1 538.0 1226.0 2187.0
2.50 2.50 2.50 2 50 2.50 2.50
6000 2500 650 525 350 250
10.2-24.5 10.1-24.4 13.3 14.8 7.9 5.7
9 16 9
16 16 20
250 250 200
286.0 810.0 286.0
2.25 2.25 2.25
300 125 300
18.9 15.1
15 4 8 12 16
20 24 24 24 24
200 180 180 180 180
810.0 45.2 233.7 547.9 987.7
2.25 3.00 3.00 3.00 3.00
175 4500 1500 525 350
7.9 15.0 2.9 11.9 6.7
9 12 16
36 36 36
125 125 125
229.0 556.0 1013.0
4.00 4.00 4.00
1500 500 500
4.4 7.7 3.0
a
DEvPlF
1470
(capacity in cfm. at 14.7 psia. and S. T.)
14.7
D=
(gas handled in cfm. at 14.7 psia. and suction temp.) (14.7)
Properties of Actual Gases
(16)
Reduced quantities are ratios of the value to that at the critical state; thus P R = P/P, and T R = T/T,. This method of correlating P-V-T relations was proposed by Cope, Lewis, and Weber (S), was later extended by Brown, Souders, and Smith (9) and Dodge ( I ) ,and is explained in detail in such texts as Weber (17,page 108) and Hougen and Watson (8). Weber presents a plot of the average compressibility factor for several gases against reduced pressure with curves of constant reduced temperatures. Unfortunately the P-V-T relations of methane do not agree well with this average plot. For methane or gas mixtures of high methane content frequently encountered in the process industries, it is preferable to use a compressibility factor chart based upon the P-V-T relations of methane, as given by Figure 4. I
I
I
( E T )(Pl)
(12)
where Evis a decimal fraction. With the piston displacement known, the cylinder size may be chosen from the displacement tables of a manufacturer. Table I is an abbreviated table of this type. Actually, EY in Equation 12 is unknown, so that it is usually necessary to determine D by trial and error, because EY depends upon the property, k , of the gas compressed. ThiR procedure, however, offers little difficulty. For multiple-stage machines, the criterion of equal horsepower per stage is usually followed. This criterion gives interstage pressures as the geometric means between suction snd discharge. Hence with m stages the gas horsepower available per stage will be the total gas horsepower divided by m, so that (gas handled per stage in 100 cfm. at 14.7 psia. and S. T.) = gas horsepower available - (brake horsepower available) (Em) mF
mF
(13)
The number of cylinders and size of each cylinder will be calculated just the same as for single-stage machines. PERFORMANCE. The previous equations used in the design method will now be modified so as to predict the performance for a given compressor cylinder when handling a perfect gas. Briefly the equations for gas horsepower and capacity for a given compressor cylinder are:
(15)
The preceding discussion concerning the design and performance of gas compressors has been limited to perfect gases. Consequently the equations can be used only for gases at relatively low pressures or for a gas which is difficult to liquefy. Such a restriction would seriously handicap the utility of gas compressors in the process industries. It is therefore desirable to develop suitable methods to allow for deviations from the perfect gas laws. One useful method has been suggested to allow for deviations of the volume in terms of the compressibility factor ,u defined by the equation, PV = pNRT
consist of calculating k , the ratio of Mc, t o Mc,, and the value of the line compression ratio, R.. With these values of k and R, the gas horsepower may be readily computed from Figure 2. This gas horsepower can be converted to shaft or brake horsepower by dividing by the mechanical efficiency, E,,,, which varies from 0.90 to 0.97. CYLINDER S I Z ~ SAfter . the total horsepower is known, the number and type of prime movers may be chosen. This choice varies, of course, with the individual builders, who will Supply the information on their separate machines. Compres8or builders usually prefer to stage their machines when the compression ratio exceeds 4; this practice therefore fkes the number of stages. With a given gas flow, the piston displacement for a single stage machine may be calculated from a convenient modification of Equation 6:
=
%
9.9
Courtesy of Cooper-Bessemer Corporation.
Vol. 34, No. 5
R,,LINE
Figure 3.
COMPRESSION
RATIO
Volumetric Efficiency Curve
I
0.2”JI
! I l l
I
I l l /
/ / / I
I l l 1
j j j j
~
j
j
j
~ilr
(usually 14.4 or 14.7 psia.) and at suction temperature. The state-
540
INDUSTRIAL AND ENGINEERING CHEMISTRY
The factor may be evaluated by two methods, according to whether or not the thermodynamic properties of the gas are known. The first method to be discussed for evaluating 4 is applicable to those gases whose thermodynamic properties are known. Thermodynamic properties are known and graphed as Mollier diagrams for several gases (particularly refrigerants and hydrocarbons), including steam, oxygen, nitrogen, air, hydrogen, helium, carbon dioxide, sulfur dioxide, ammonia, methane, ethane, propane, butane, and some halogen derivatives of methane and ethane. For such gases values for enthalpy may be read from the diagram along a constant entropy path, and AHs computed as the difference between the enthalpies a t initial and final pressures. Values of c,T1 X k-
[(.P2/PI)"
1
11
- are calculated directly but by using a constant value of c, (and hence k ) a t suction conditions. Strictly speaking, variation in heat capacity with temperature should be allowed for both in calculating the temperature after isentropic compression and in computing t h e enthalpy change. Actu80' F. S.T. ally this variPROPANE ation of heat capacity with temperature within t h e ranges involved in compressor problems is so slight for the nearly perfect gases-e. g . , helium, oxygen, 0 ATM. nitrogen, and air-that this IO ATM. refinement in calculation is beyond t h e 0 1 2 3 4 5 limits of design CR,; Y L IND E R c o M P RES s IO N RATIO tolerance or of experimental Figure 5. Isentropic Work Facerror in testing. tor for Propane with 80" F. SucFor imperfect tion Temperature gases the isentroDic w o r k factor has been so defined that corrections ark made for both variation in heat capacity with temperature and deviations from the gas laws. I n order that values of 4 may be presented in useful form, they should be plotted against the cylinder compression ratio R: for a fixed suction temperature and with curves for constant absolute suction pressure. A temperature of 80" F. was used as a practical average suction temperature. Values of AHs were determined from a Mollier diagram for propane
\\
,
unknown. For such gases these properties can be estimated only from generalized correlations. This procedure is not so simple as that oE reading values from a Mollier diagram. The first problem involved in evaluating 4 is that of estimating the temperature after constant entropy compression between any fixed pressures. This procedure involves a solution by successive approximation. Thus a final temperature is assumed and the change in entropy, AS, is calculated. When this change in entropy is zero, the proper final temperature has been found. Since the entropy is a point function, it is independent of path and may be calculated by choosing any convenient combination of paths. The change in entropy may be P evaluated by the following three-step path: (a) Re($HI, duce the pressure to one atmosphere (or less) a t constant temperature; (b) heat T the gas a t low pressure to Figure 6. Three-Step the final temperature; (6) raise the pressure from one Path 'Or Eva'uatin3 a t m o s p h e r e to the final and AH Dressure a t constant temperature. This three-step path is presented graphically as Figure 6. For convenience in presenting a generalized correlation for the effect of gas law deviation on the isothermal change in entropy above one atmosphere, let a residual entropy Q be defined as:
IjS,>
*'
-
( h s )= ~ -1.9871n P u (18) entropy change at constant temperature, B. t. u./(lb. mole) (" R.) P = absolute pressure, atmospheres u = residual entropy, B. t. u./(lb. mole) (" R.)
where ( AS)T
=
It is possible t o correlate residual entropy u in terms of reduced pressure and reduced temperature. Figure 7, based upon values published by Edmister (6),is such a plot for propane. The entropy change a t one atmosphere between T I and T2is : ( A ~ ) =P 1
atm.
[(g)
la-1
- 11 give These values divided by C,TI the desired isentropic work factor. The results are presented graphically as Figure 5 and are to be used later in an illustrative example. The second method to be discussed for evaluating 4 is applicable t o those gases whose thermodynamic properties are (18).
=
LTaiMcpdT-
T
. - approx. ( M C , ) ~In~ 2 TI
(19)
Within the temperature ranges involved in gas compression practice, this Mc, may be represented by a linear equation in temperature. Thus Mc, = A BT (20)
+
+
-
Vol. 34, No. 5
Itm.
= A In
T $ + B(Tz- T I )
(21)
After isentropic compression, the final temperature TZmust be such that AS = 0, or TI P A ln B ( T 2 - T I ) - 1.987 In $ U I - US = 0 (22) Ti
-+
+
where c1 and u2 are read as the ordinates of Figure 7, and A and B are summarized for hydrocarbons by Edmister (6). Attention is called to the fact that both pressure P and temperature T i n Equation 22 must be in absolute units. With temperature 2'2 known, the next step in evaluating cp is t o estimate AHs. The calculation for AHs is accomplished by the same three-step path as for AS. Isothermal changes in enthalpies may be estimated from the generalized correlations proposed by Edmister (6) or by York and Weber (19). In the correlation by York and Weber, the isothermal
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
May, 1942
enthalpy change denoted by (H* - H ) is based upon a datum of zero extrapolated pressure (H*) so that, to be consistent, values of Mc, should be used a t zero pressure. Actually the difference between Mc, at zero pressure and at one atmosphere is negligible for these calculations. The desired change in enthalpy between initial Pi and TIand final Ps;and Ts; is AH = (H*
- H)1 f
A(T2
B
- TI) + 2 (Ti - Tf) - (H* - H)a (23)
where (H*- H)l
=
- H)2 =
(H*
isothermal enthalpy change at TIfrom P1t o zero pressure (or 1 atm.) isothermal enthalpy change at Tz from Pz to zero pressure (or 1 atm.)
With AH now known, 6, is simply calculated according to Equation 17. This entire procedure for calculating 6, is illustrated by the following example: EXAMPLE. 4 will be calculated for gaseous propane between an initial pressure of 100 psia. and 80' F. and a final pressure of 400 Dsia. Then,
~
2
'= . 0 2
6
5
m'
