How to Use the Perf Curves to Evaluate Behavior of Cent Comp

8/12/2019 How to Use the Perf Curves to Evaluate Behavior of Cent Comp

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How to use the performancecurves to evaluate behaviorof centrifugal compressorsPressure, temperature, compressibility, molecular weight andspecific-heat ratio of the gas or gas mixture at the compressorinlet, and the machine s rotational speed affect the operatingperformance of single-stage compressors. Here is how tocalculate the effects of changes in these factors.

Ronald Lufnha Procon Inc.

Fluctuations of inlet conditions for a gas affect theperformance of centrifugal compressors. For example, acompressor receiving inlet air at atmospheric conditionswill develop higher discharge pressure on cold daysthan on hot days at a given rotational speed and inlet-volume flow. The power requirement also will behigher.Changes in atmospheric conditions such as relativehumidity and barometric pressure will affect perform-ance, although these factors are usually less significantthan inlet temperature.We can account for these changes and others encoun-tered during operation by modifying the performancecurve of the compressor. Manufacturers of centrifugalcompressors often supply curves that define the ma-chines aerodynamic performance. These curves takemany forms, some of which are:

Polytropic or adiabatic head and horsepower vs.inlet volume flow.Discharge pressure (psia) and horsepower vs. inletvolume flow.Discharge pressure (in. gage, water column) andhorsepower vs. inlet volume flow.

The data for the ~erformance urve are the name-plate rating conditions, i.e., inlet pressure, inlet temper-ature, molecular weight, ratio of specific heats and inletcompressibility. The manufacturer will not normallysupply performance curves for other than rated inletconditions unless specifically requested.We will develop application procedures for modifica-tions to the performance curve for a single-stage centrif-ugal air compressor. However, these procedures arevalid for any gases, and for multistage centrifugal com-preqsors, with somewhat reduced accuracy.

T h is article was written by R. P. Lapina while an employee of Elliott, aUnited Technologies company, and adapted from a prize-winning article n theCAGI (Compressed ir and Gas Institute) technical article program.

Compressor stage characteristicsFirst, let us consider the following equations to illus-

trate the procedure accounting for variations in inletconditions:(Had)s pu2/g I )

where: u N?rd/720

Eq. (1) shows that the head produced by an impelleris a function only of its mechanical tip speed, u andhead coefficient, p that, in turn, is a function of theinlet volume flow. Therefore, the head produced by animpeller at a fixed speed and inlet volume is a con-stant.+ This statement forms the basis uDon which wecan derive the procedures accounting for change ininlet conditions.

If we compare Eq. (2) and (3) at a fixed flow for theinlet volume, we find that variations in inlet conditionswill affect the power requirements. Here, an increase ininlet temperature will decrease the power requirements,and a n increase in inlet pressure will increasd the powerrequirement. These power effects arise from changes inthe inlet density and, hence, in the weight flow.Performance curves

Fig. 1is a typical performance curve for a single-stagecentrifugal compressor at rated inlet conditions, anddischarge expressed as adiabatic head. Fig. 2 is a similarcurve with the compressor s discharge expressed as pres-sure, psia.

Compressor manufacturers provide such curves toThisstatement is not rigorously true, due to volume ratio effects. Variationsin inlet Sonditions will affect the v alue of p However such deviations arenormally small, and we can safely ignore them for our purposes.

Originally published January 25 982


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8 C LCUL TIONS ND EV LU TIONS


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HOW TO USE THE PERFORMANCE CURVES 49

1 2 3 4 5 6lnlet flow, thousand icfmRated conditions

Gas: a ~ r Relative humidity: 50Volume flow, 0 : 42.200 icfm Ratio of specific heats. k: 1.4Barome tric pressure: 14.7 psia lnlet compressibility,ZI 1.0lnlet pressure,P 14.5 psia Rot ation al speed. N : 4,350 rpmlnlet tem perature,T 0F D~scharge ressure,P2: 20.6 psia

Adiabatic-head performance curve forsingle-stage centrifugal compressor Fia. 1

define the flange-to-flange performance of their ma-chines. .External hardware such as inlet an d dischargepiping, inlet filters, and inlet and discharge valves arenot normally considered in establishing the perform-ance curve. Hence, we must account for pressure dropdue to external hardware when using the performancecurve.

The term inlet volume flow will be used extensivelyin the developments that follow. Inlet volume flow isthe volume flow that exists at the compressor's inletflange.

We will develop our techniques by using the adia-batic-head curve because adiabatic head lends itselfmore readily to our developments. The final equations,however, are applicable to the performance curves ex-pressed as adiabatic head or discharge pressure.nlet pressures

Let us begin by considering the effects of a variationin inlet pressure. For our discussion, consider that thecompressor draws atmospheric air through an inlet fil-ter, s shown in Fig. 3. The rated compressor inlet pres-sure is 14.5 psia. When the filter becomes dirty, the inletpressure at the compressor flange drops to 14.2 psia.What is the effect on discharge pressure and shaft horse-power at the rated inlet volume flow?

lnlet flow, thousand icfmRated conditions

Gas: air Relative humidity: 50Volume flow. : 42 ,20 0 icfm Ratio of specific heats,k: 1.4Barom etr~c ressure: 14.7 psia lnlet compressibility,2 1.0Inlet pressure, P 14.5 psia Rotational speed,N : 4,350 rpmlnlet temperature. T 0F Discharg e pressure. P : 20 .6 psia

Discharge-pressure performance curvefor single-staqe centrifuaal compressor Fia.

Discharge pressure is related to the adiabatic headby:

For a given inlet volume flow at a given rotationalspeed, the head output is constant, and because there isno change in the other inlet conditions, the pressureratio does not vary. Therefore:

rp = rp)7= 20 . 6 /14 . 5 = 1.42P, = P1 rp) , 14 .2 1 .42)= 20.2 psia (5)

Rearranging Eq. 2 ) yields:

On comparing Eq. 3) and 6 ) , we find that shafthorsepower is proportional to weight flow or inlet pres-sure, or:

Eq.. 7 ) is not strictly valid, because shaft horsepoweris composed of gas horsep~wer nd mechanical losses.Th e mechanical losses are approximately constant for agiven speed but are generally a small part of the total


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50 CALCULATIONS AND EVALUATIONS

Nomenclature r;C ConstantC Constantd Impeller tip diameter, in.

Gravitational constant, 32 2 ft-lbf/(lb,)(s2)Ha, Adiabatic head, ft-lbf/lbm(Had), Adiabatic head developed by a centrifugal com-pressor stage, ft-lb,/lb,k Ratio of specific heats, cp/cVL Mechanical lossesMW Molecular weightN Rotational speed, rpmP Pressure, psiaQ Volume flow, icfm (inlet ft3/min)R Gas constant, ft-lb,/(lb,)( R)rp Pressure ratio, P J 4(SHP) Shaft horsepower, hpT Temperature, R

Mechanical tip speed, ft /sSpecific volume, ft3/lbm

W Weight flow, lb,/minZ Compressibilityqaa Adiabatic efficiency

Head coefficientSubscriptsP Fan lawi c Inlet conditions

Ratedreg Requireds Stage1 Inlet2 Discharge

horsepower. Therefore, ignoring mechanical losses willusually yield a useful approximation in our procedures.

Th e performance curve (such as Fig. 1) indicates thatthe power requirement for the rated conditions is1,315 hp. Substituting in Eq. 7), we find:

(SHP) = (14.2/14.5)(1,315) 1,290 hpIn this example, we have ignored the effects of system

resistance downstream of the compressor dischargeflange. For many applications, system resistance is smallwhen compared to the total pressure requirements ofthe compressor and will therefore have minimal effecton the analysis.In some applications, however, system resistance ef-fects are large and these will actually define the com-lpressor operation.

We can view system resistance as the sum of pipingand system losses and utility pressure drops. Theseshould not be thought of as losses. As the volume flowthrough the system increases, frictional losses increaseand require a higher pressure at the compressor's dis-charge flange to be overcome.

Fig. 4 shows a typical system resistance line superim-posed on two compressor performance curves. The solidcurve represents rated inlet conditions. The dashedcurve shows the effects of a reduction in inlet pressure

l l Atmospheric pressure = 14.7 psiaair filter

Inlet pressure =

CompressorBDi rt y fi lter decreases inlet pressure Fig

~nly. oint A on the solid curve is the rated operating)oint.

In this example, we have assumed constant inlet vol-lme flow and, therefore, calculated the discharge pres-ure at Point C. The power requirement calculated as,290 hp was for operation at Point C.If the compressor in this example operated with sys-

em resistance, it would search its new performance:urve (the dashed line in Fig. 4 until it intersected sys-em requirements. The result would be operation at'oint B Therefore, the inlet volume flow would beomewhat less than rated and the discharge pressurevould be slightly higher than calculated. Referring tohe horsepower curve of Fig. 1 or 2 would show that theewer requirement would be less than calculated.

et temperaturesLet us assume that the inlet temperature drops to

10F while other inlet conditions remain at rated val-les. Wha t is the effect on discharge pressure a nd shaftlorsepower for the compressor defined by Fig. 1 at the.ated inlet volume flow?

Rearranging Eq. (4) yields:

Eq. (8) indicates that a change in inlet temperature,TI, inversely affects the pressure ratio. For a change inmlet temperature only, we can derive:

. (r p) y) /k 1 Cl/(Tl),where: C1 Had(k )/Z,Rk.

f- /k 1 C T

Solving this equation for rp, we get:


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erformance at ratedinlet co:ditions

A

~erf orhan ce t ratedinlet conditions exceptPl < PlI,

lnlet volume flowow inlet pressure affects centrifugalcompressor performance Fig

Substituting the new inlet temperature of 40F andthe rated inlet temperature of 90F into Eq. (9) yieldsthe discharge pressure:

By comparing Eq. (3) and 6), we find that shafthorsepower is inversely proportional to inlet tempera-ture, or:

SHP = [(Tl),/TlI(sm, (10)SHP [550/500](1,315) 1,450 hp

Again, we have neglected the effects of system resist-ance. Where system resistance prevails, we may refer toFig. 5. The solid curve represents the compressor per-formance for the rated inlet conditions. The dashedcurve represents a drop in inlet temperature with allother inlet conditions remaining at the rated values.Point A is the compressor rated point.

In our last example, we considered a constant inletvolume flow. Therefore, we calculated the performanceat Point C. The compressor will search its new perform-ance curve until it reaches Point B at its intersection'with the system resistance line. As shown in Fig. 5, inletflow will be somewhat higher than rated, a nd dischargepressure slightly lower than calculated. Reference toFig. or 2 will show that the power requirement will behigher than calculated.

If we compare Fig. 4 and 5, we see that a drop in inletpressure has the net result ~f lowering the dischargepressure curve; while a drop in inlet temperature resultsin raising the curve. From this, we can deduce that wecan obtain rated performance on cold days by suction

/ at erformanceated inletcond~tion s xcept

T < Ti),\

Performance at ratedinlet conditions

2lnlet volume flow

ow inlet temperature affects centrifugalcompressor performance Fig 5

hrottling of the inlet pressure. (It is also possible to)btain the same result by lowering the speed on varia-)le-speed drives. More on this later.)

Suction thrott ling also lowers the power requirement,Iecause horsepower is directly proportional to inletlressure.

In this discussion, we are trying to analyze each vari-lble independently. Hence, we have not taken into ac-:ount the change in water-vapor content of the air dueo the change in inlet temperature. For the rated condi-ions, the molecular weight of the air is 28.7. When thenlet temperature changes to 40F, the molecularveight becomes 28.9 (assuming relative humidity re-nains at 50 ). We will consider the effect's of changes invater content in the next example.Molecular weights

The molecular weight of an air/water-vapor mixturelraries with composition. The effect of this on air com-pressors is usually small, and can generally be ignored.However, gas compressors can operate over a widerange of molecular weights, making this variable signif-[cant. As an example of how to account for variation inmolecular weight, we will consider a change in relative~umidity t rated inlet temperature.

On a given day, the atmospheric temperature is 90Fand the relative humidity is 100 . At a barometricpressure of 14.7 psia, the molecular weight is approxi-mately 28.4; while at the rated conditions it is 28.7.

Let us investigate how the molecular weight entersthe head equation. The term R in Eq. (4) is given by:


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52 CALCULATIONS AND EVALUATIONS

Performance at ratedinlet conditions exceptMW


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Inlet volum flowotational speed of the compressoraffects its performance Fig. 8

If the specific heats vary, the discharge pressure is:

In either case, the equation for shaft horsepower is:

Since k does not vary appreciably for differentair/water-vapor mixtures, we can consider it constant,with little or no error introduced. The change in k canbe significant for gas compressors. Thus, we will have toconsider the effects of changing values of k.Let us consider the single-stage compressor as definedin Fig. 1. Ou r problem is to determine the dischargepressure and shaft horsepower at the rated inlet volumeflow for a change in inlet conditions to:

P 14.2 psiaT 40F = 500RW 28.4

Since the ratio of specific heats is constant, we useEq (14) to solve for the discharge pressure, and Eq. (16)for shaft horsepower.

= 20.8 psia

When necessary, we will use the system-resistancecurve t o make the proper ad.justments.Constant weight flow

Sometimes, it is necessary to consider constant-weightflow. We can use the performance curve to predict dis-charge pressure and shaft horsepower for this case.Changing the inlet conditions will affect the inlet vol-ume [see Eq. 2)]. Because we are dealing with variableinlet volume flow, the head produced by the impellerwill also vary.We will use the compressor defined by Fig. 1 andpredict the discharge pressure for a change in both theinlet temperature and inlet pressure. This procedurewill also be applicable to the conditions shown in Fig. 2.Let us consider a change in the inlet temperature to100F and inlet pressure to 14.0 psia. Molecular weight,ratio of specific heats and inlet compressibility will re-main at rated values, as given in Fig. 1.Eq. 2) implies that for a constant-weight-flow proc-ess, the inlet-volume flow is directly proportional to theinlet temperature, and inversely proportional to theinlet pressure: Hence:

560 45 (42,200) = 44,500 icfmA 4.0)where icfm inlet ft3/min.From Fig. 1 we find that at the rated inlet volumeflow of 42,200 icfm, the head produced is 11,000 ft-Ibf/lb,. Th e head produced at the new flow of44,500 icfm is 10,900 ft-lb,/lb,. From Eq. (4), we estab-lish that the expression containing the rated pressureratio is directly proportional to the adiabatic head andinversely proportional to the inlet temperature. Hence:

19.7 psiaShaft horsepower for rated inlet conditions at a flowof 44,500 icfm can be determined from the performancecurve (Fig. 1) as 1,380 hp. From Eq (16), we find:

(SHP) (=)(=)(1,380) = 1,310 hp14.5 560The procedure for using Fig. 2 follows from the pre-ceding. In this case, the discharge pressure at the newinlet volume flow for rated inlet temperature is ob-tained directly from the performance curve. For exam-ple, the discharge pressure at 44,500 icfm and 90F is20.5 >iii from Fig. 2. W&will consider this new dis-charge pressure rated and use the previously estab-


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54 CALCULATIONS AND EVALUATIONSk hed relationship [Eq. (9)] to correct for the new tem-perature.

otational speedCentrifugal-compressor performance varies with the

rotational speed. If the variation in speed is not toolarge-say 90 to 105 of rated speed-we can predictthe compressor's performance from the fan-law rela-tionships. These state that the adiabatic head, Had, ar-ies as the square of the speed, N2, and the inlet-volumeflow, Q, varies directly as the speed, N. If we use therated points as reference, we can write:

From Eq. (19) and (20), we can predict the speedrequired to overcome the effects of changes in inlet con-ditions. The procedure that will be outlined can be usedfor any variations in inlet conditions and for any type ofperformance curve.

Let us consider the earlier example involving achange in the inlet pressure only. Th e drop in inlet pres-sure caused the discharge pressure to decrease to20.2 psia. What speed is required to raise the dischargepressure to the rated value of 20.6 psia a t rated inletvolume conditions?

The pressure ratio at rated speed for the inlet condi-tions under consideration is determined from the proce-dures previously developed. For this example, it is:

The required pressure ratio is:

For any given set of inlet conditions, we have:

Hence, we can derive the following rotational-speedequation:

= 4,490 rpmAt 4,490 rpm, the compressor will produce the re-quired head. However, tbe increase in speed will also

affect the volume flow. Here, the new flow will become:Q = 42,200(4,490/4,350) = 43,560 icfm

Reduction of the flow to rated conditions at the newspeed of 4,490 rpm would result in too much head.

Fig. 8 shows the effect on inlet volume flow and headfor an increase in rotational speed. The solid curve rep-resents rated conditions. Point A is the rated operationpoint. Also plotted in Fig. 8 is the line for the fan law.Following this line to 4,490 rpm shows a shift in per-formance to Point B which represents the desired com-pressor head. Reducing the inlet volume flow to ratedconditions results in performance at Point C.

The volume-flow effect can best be handled by deter-mining the percentage change in head tha t occurs froma change in relative loading, as follows:

By referring to Fig. 1, the percentage change in headas a result of going from 100 Q o 96.9 Qmay be de-termined (i.e., flow is 42,200 icfm at 100 Qr, an d40,900 icfm a t 96.9 Qr).

At 42,900 icfm, Had 11,000 ft-lbf/lb,. At40,900 icfm, Had= 11,075 ft-lbf/lb,. Then:

We can now revise Eq. (21) to:

= 4,470 rpmThe result is operation at Point D in Fig. 8, which is

the desired operating point.The last step in our analysis of speed variation may

seem a little difficult for a performance curve plottedagainst discharge pressure. In using a curve such as thatin Fig. 2, we should remember th at for any given set ofinlet conditions:

SummaryWith these procedures, we should be able to accu-rately predict the performance for any single-stage cen-trifugal compressor. Furthermore, we should be able toanalyze plant-air centrifugal compressors by applyingthese techniques to each stage between coolers.The use of these techniques on centrifugal units hav-ing several impellers between cooling points will yielduseful qualitative results. However, accuracy of the re-sults will deteriorate in proportion to the number ofimpellers between coolers and differences in the molecu-lar weights of the gases involved.

The authorRonald P Lapina is a principalmecha nical engineer with Procon Inc. asubsidiary of Pro.con Interna tional Inc.16340 Park 10 Place Drive HoustonTX 77218. He has been with Procon forone year and has responsibility forspecifying and evaluating machinery.Previously he spent nine years withElliott Co. where his duties included thereratin of centrifugal compressors. Hehas a . in aerospace engineering andan M.S. in mechanical engineering bothfrom the University of Pittsburgh.

How to Use the Perf Curves to Evaluate Behavior of Cent Comp

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