Simulation of fast transients in fluid transport equipments and utility networks

SIMULATION OF FAST TRANSIENTS IN FLUID TRANSPORT EQUIPMENTS AND UTILITY NETWORKS G. Heyen 1,B. Kalitventzeff 1,P. Hutchinson 2, Hoi Yeung 2, M. Gill3 1 : L.A.S.S.C., Universite deLiege, Sart Tilman B6, B4000 LIEGE (Belgium) 2 : School ofMechanical Engineering, Cranfield Institute ofTechnology, Bedford MK43 OAL (UK) 3 : W.S. Atkins, Woodcote Grove, Ashley Road, Epsom Surrey KT18 SBW (UK)

ABSTRACT

A model allowing dynamic simulation of processes involving compression, expansion, heat generation and transfer in fluids is presented. Equations are developed for some equipment models, based on onedimensional flow approximation. The architecture of a simulation package basedon the objectparadigm is described. The programme has been validated withexperimental data: surge onsetin compression systems has been satisfactorily predicted. KEYWORDS

Dynamic simulation, Integration of DAEsystems, Fluiddynamics, Compressors, Object programming INTRODUCTION

The following text describes part of a research programme where Cranfield Institute of Technology (UK), University of Liege (B) and Ecole des Mines de Paris (F) join efforts to developmodelling and control design tools. A first goal of the project is modelling of fast transients (time scale from 10-3 to 10 s) in rotary machines andutilitydistribution networks. Improving energyefficiency is the incentive to study the behaviourof such systems in operation rangethat are closer to instability than usual operation mode. For instance, maximum of polytropic efficiency for compressors is usually located closeto the surge limit. Precise modelling of the process behaviour in the domain of interestshould allowto reproduce occurrence of instabilities, or even predictthem at the design stage. Next control algorithms should be developed in order to allow safe operation at improved energy efficiency. This paper will focus on the work that has beencarried out on dynamic simulation. MATHEMATICAL MODELS

The mathematical modelconsiders the dynamics of the fluid flowin the process system by the application of time dependent conservation equations to elements of the system. Typical elements might be : a compressor or compressor stage, a turbine, a length of pipe, a valve, or the process stream in a heat exchanger. The generalform of the conservation equations is simplified to keep the essential terms. Each type of flow element is then described by appropriate input data and characteristics. Such data may be assumed known in steady state operation of the equipment from either design values or from precommissioning testsin isolation. The model derived satisfies two requirements. First, it contains the fluid dynamic representation to allow to simulate faithfully fast transients in the system, for example compressor surge. Second, the use of nonideal gas relations gives it sufficient thermodynamic flexibility to simulate non-ideal gases and gas mixtures, and allows it to incorporate physical property data generated by a separate physical properties package.

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Derlyatlon of the equations

To achieve a concise model, the flow in the system is assumed to be one-dimensional, i.e. variations across the direction normal to the mainflow direction aresmall. Using this assumption the conservation equations maybe written for a general system element (seeFig. 1),as follows:

Conservation of mass

JP A dx = M1 - M2

d~

(1)

x

Conservation of linear momentum d

at fM dx = (pA+Muh - (pA+Muh +Fnet

(2)

x

Conservation of energy d at

J (U

2

p

. e+T) Adx=(Mho)1-(Mho)2+ Eret

(3)

x Fnet and &tet represent the net force actingwithin the element, and the rate of energyinput, respectively. Equations to evaluate them are required for each type of element.

~

(0 P1

i i I

M1---t-

T1

i i

, I

Fnet

~

....

.A i

CY i i i

P2

12

_----~o::::::

L y

/

T25000

R

~2oo00

Fig. 1 Generalized one-dimensional flow element

. I'~'.

C 15000

~ 10000

L

.... ~~~

s~ •• ~ •

I

~x ------....

j;

0.8

o

~M2 I i

6

p

"'"---

10700 rpm 9600

rpm

8550 rpm 7500 rpm

A

D 5000""+' <4000

_ 6000

8000

--'-_ 10000

INLET FLOW

_

'2000

Fig. 2 Compressor performance map The equations (1) to (3) can be simplified by use of some assumptions which are discussed in somedetail in Elder et al. (1986,1987), and are summarized here.These assumptions are generally valid in process and utility plants, where the Mach number of the flows will remain low «0.3). The contribution of the momentum term (Mu) in equation (2) and the dynamic head term in equation (3) are small relativeto Ire pressure and internalenergy terms. They can therefore be neglected for simplification of the model and computational speed. The left hand side of the equations also require the integration of state variables along the length of the element. By appropriate selectionof the elements of the system, this term may be replaced by average values of the variables in theelement. The equations can thus be simplified to : dp VdT=M1- M2

(4)

dM

(5)

L\X d"t = (PA)l - (PA)z + Fnet V

d( p e) d t

(Mho)l - (Mhon +

Em

(6)

dN dN where dt dx =L\ x
fa

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Slll

The above equations provide the description to be used for a general process element, e.g. a heat exchanger, boiler, compressor or turbine. The equations maybe usedin thisform, provided definitions for the energy and force inputterms may be derived from an appropriate model of the physical processes. The example process which is discussed below consists of a compressor and ducting system, in which the dynamics of the energy equation is found to addlittle to the accuracy of the model (Elderet al, 1986). The energy equation is therefore replaced by the steady-state form appropriate to eachcomponent. Compressor model

The simulation of a compressor element requires definition of the force and energy input term, Fnet and Enetin equations (5) and (6) above. Compressor steady stateperformance is described by the compressor map, whichgives, for example, compressor polytropic headand efficiency as a function of inlet flow rate and compressor rotational speed (Fig. 2). It is assumed that the compressor performance may be represented by these characteristics during transient operation. Analysis proposed by Schultz (1962) is employed, to givethe following relationships for thecompressor model. Wp= where

~:T~ {(~~)m

-1 }

(7)

11

m=ZR(.!.+ Cp 11

x)

Enet =_W....p,-M--",-l 11 Fnet=APl {

(8)

(~tl (~+

XJIJ/m_ l}

(9)

The gasequation of state : P=ZRTp is employed together withthe polytropic relation:

(10)

(~Dm=(~~)

(11)

to reduce equation (4)to equation (12) : dP Z R TI Vat (1-m) (Ml-M2>

(12)

The dynamics of the energy equation are to be neglected, andequation (6) may be replaced by the steadystate energy inputrelation: W

T2 =TI + ~ 11 Cp

(13)

Equations 5, 12, 7, 9, 13 provide the basisfor the compressor model. Pipe model

The complexity of the pipemodel usedin the simulation depends on the nature of the flow andtransients to which the system is expected to be subjected. The simplest pipemodelassumes a constant friction factor, isothermal flow and an isentropic relation between pressure anddensity transients. Theseassumptions lead to the following equations forthe simple transient pipemodel: dP VdT= yZR TI (Ml -M2) (14) dM

~x CIT = (P Ah

- (PA)z + Fnet ~x

I

(15) 2

where Fnet = Al CF If (2 PI u1) Assuming no heatloss out of the pipeelements, the energy equation (6)is replaced by :

Sl12

TZ

European Symposium on Computer Aided Process Engineering-I

= TI

(16)

It should be notedthat, by retaining the dynamic momentum and continuity equations, the propagation of wave motion can be simulated, and the equations are therefore accurate in simulating fast transients. Pressure waves will be propagated through the pipe at the speedof sound. Where the response of a pipe network is important, Elder at al (1986) have shownthat the frequency f of expected transients and the length of pipeelements should satisfy the criterion:

21tf~< {;RT-~

(17)

Ap

Other process elements

From the above examples, it is easy to see how models for other plant elements, such as valves, heat exchangers andturbines, may be builtup. A detailed model of a gasturbine is under development; it takes account of the transient flow of gas through the gas turbine components - compressors, combustor and turbines - together withshaftspeedchanges and fuel control systems. Such a model will be used to investigate changes in output from the gas turbine in terms of shaftpoweroutputand exhaust gas flow temperature, for wasteheat recovery purposes. INTEGRATION OF MODEL EQUATIONS

In previous work at Cranfield on compressor systems modelling (Elderet al (1986,1987)), explicit ODE solvers, such as Runge-Kutta method, were used to solve the model equations described in the previous section afterelimination of algebraic equations. However sucha problem reformulation maynot always be practical for other plant elements models, such as boilers. A solver capable of handling differential algebraic equation systems suchas DASSL, is required under thiscircumstance. A comparison between both methods has been madeby using a simplevalve - duct - compressor - duct nozzle plant. For this system (see Fig. 3), the inlet and outletductsare further divided into threeelements each. The characteristics of the compressor can be found in Fig. 2. The dynamics of the energy equation has been neglected; therefore eachelement of the system is modelled by two ODE's, an algebraic equation (AE) representing the energy equation, and other AE's describing the element performance, the Fnet and Enetterms. A system of23 equations represents the plant: 14 ODEs (2 for eachpipeelement and 2 for the compressor) and 9 AEs. RKMwas used to solvethe set of ODE's obtained afterexplicit elimination of the AE's, while DASSL was ableto handle theoriginal problem directly. Since the twomethods gavecomparable results in term of accuracy and computing time, DASSL, beingmoregeneral and flexible, has been selected as the basic integration algorithm forthe development of FASTran package described below.

~I , I

2

I

I'

~'-I......-'-1--.2,.......,1---,3,......,1.

3

j

Fig 3. Compressor model test case

FAST~n

PROGRAMME

FASTran is a general dynamic simulation package whose development is under way, based on the modelling approach described hereabove. Its mainfeatures willbe discussed. Problem set yp

Problems are defined in free-format ASCII datafile. The inputdatafilehas beendesigned to be botheasily readable and easilydecoded. It is thus quiteportable from one computer system to another. Data transfer using this type of file should allow connection to almost any type of data base management system or graphical userinterface. Data storage

The description of all items(equipments, streams, components, etc) making up the process has to be stored in computercore in an orderly manner. However the type and amount of information neededdiffers from

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one item to the next. Thus some kind of memory management subsystem has to be designed. In the past, use of PLEX data structures has been proposed (Evans (1977» : these are large linked lists containing the whole process structure (in the form of pointers) and technical information (in the form of arrays of numbers allocated from a memory pool). Information about connected plant items can be retrieved by knowing how the list of pointers is organized, and by "walking through" the information tree. This architecture is memory efficient, but proneto error,sinceany subroutine has access to the whole data base. Furthermore, any modification in the data structure of one item (e.g. modifying the pure component physical property list) may require manymodifications in other parts of the programme. Thus it may lead to difficulty in debugging and maintenance of the code. This is why concepts from object oriented programming methodology have been adopted to design the structure of the FASTran programme. Dataindependence of the modules has been secured by developing a memorymanagement system and an information exchange protocolthat avoid sharing global variables: each item corresponds to an independent object, that has to manage its own data and respond to processing actions or data exchange requests that aretransmitted by an objectmanager. Two types of information must be organized and stored: processinformation and problem information. Process information is managed by the objects: they request some part of memory from a memory manager and use it in the way that is the most appropriate. Processinformation contains all information needed to describe the process from the engineers pointof view. It is not related to the type of problem that has to be solved. One could imagineto use the same type of processinformation for transientsimulation, for steady state models or for plant data reconciliation. Process information contains both constants and variables. It may contain redundant data (e.g. stream composition, temperature, pressure, enthalpy, density, viscosity, etc) even if it is knownthat someproperties may be calculated from the valuesof other properties. Problem information is specific to the problem to be solved. In the case of transient simulation, it refers to the array of integration variables, to the array of equation right hand side, and to the Jacobianmatrix of these equations. Normally, no redundant information is storedin the problem arrays. Problem information is generated by the process objects, but used and updated by the mathematical package (integration procedures, such as DASSL). When a problem solution has been found, objects can use this result to update process information accordingly. Design using object oriented approach.

Objects are defined as a set of variables and a pieceof computer code that has to manipulate thosevariables. The same piece of code can be reused and associated with several data sets of the same type. In the FASTranprogramme, objectscorrespond to actualplant items. They containcode to model the behaviour of the corresponding plant itemsand to communicate withthe environment. They respond to requests from otherpiecesof code or objects, that will be transmitted by argument passing. Five main objects classes have been defined: equipments, streams,components, mixtures and reactions. Equipments correspond to basic processing operations (compressor, piping section, valve, expander, heat exchanger, combustor, etc). Streams correspond to the stateof anymaterial, energyor information transfer between equipments; shaft power, electricity or heat are treated as stream objects, as well as standard material flows. Components correspond to the chemical species, or to identifiable materials whose properties can be characterized (e.g. a petroleum fraction). Mixture objects handle the physicalproperty modelsthat allow to estimate thermodynamic properties and phase equilibria for any stream containing a givenlist of components. Reactions objects are ableto handle massand energy balance in reacting systems, to evaluate chemical equilibria or kinetic expressions. Advantages of object-based architecture

The main advantage lies in the better structuring of data. Each module has the capability to access and structure its own data in the most convenient way. For instance, reading data from the input file can be done differently for each object, if this looks easier for the user (or the programmer). This allows easier interfacing with other data structures. To give an example, the physical property modelto evaluate enthalpy has to integrate a Cp equation from a reference temperature to the targettemperature. In the "classical" way of programming, it would retrieve the Cp parameters (usually coefficients of a polynomial), weightthem with the mixture mole fractions to obtainthe mixture Cp coefficients, and evaluatethe integral. Usingthe "object" approach, the property model will call eachcomponent in turn, and ask it to return its evaluation of the pure component Cp integral. This allows to mix in the same stream components of differing types, and to use freely data from differentdata bases.The enthalpyfunction will be written without knowing how pure component specific heat willbe coded, andneed not to be updated if a betterexpression is laterused.

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European Symposium on Computer Aided Process Engineering-I

Most of the objecttypesdeal with a specific case of a moregeneral family of objects. Thus the subroutines implementing an object behaviourtake care of the peculiarities of the given object. and pass the job to a more general object when this is possible. For instance. whenreading the inputdata file. some equipment objects may need to read data in a specific format. and thus contain code to deal with it; most of the equipment objects acceptdata according to a standard layout, andpass thejob of reading incoming data to a moregeneral equipment objectthat handles default classbehaviour. As a second example, we can consider the modelling of heat exchange. In FASTran. a heat exchanger model is build by combining a hot cell, a cold cell and a wall object. all connected by energy streams. Different cell typescorrespond to differentflow patterns (e.g. flow in a tube bundlecan be represented by a tank-in-series model, whilea combustion chambercouldbe represented by a single well-mixed cell). Wall objectshandle the calculation of heat transfer; they call connected cells to obtain information about flow turbulence andphysical properties. The wallequipment classcontains several objects. allowing to ignore or to take into accountheat capacity of the wall, and to calculateinternal temperature profiles with different degrees of detail. By decoupling the various phenomena. the samepieceof code can be reused in the model of several equipment types. APPLICATION OF THE DYNAMIC MODELS

The dynamic models have been used for many applications, for compressors and gas turbine systems, ranging from aero-engine multistage compressors, to industrial centrifugal compressors and gas turbine power plants for aircraft and power generation, In all cases. the objective was to provide a detailed simulation of the system, to allow the effectof fast actingfluid dynamictransients on the performance and stability of the system (compressor surge) to be investigated. Two examples of compressor system simulations in process andutilityplantare presented. Compressor system

A simple stagecompressor andits associated test system was simulated (Fig. 4).

L.......-1 D

Fig. 4. Compressor test rig

IJ

.

~: 6 -

r;

,

._

•

( :rP(QI1 (N ' .&.l

-

S",UJON

"

•••

••••

L-.-I- L - - l_ _ L.-.-J__ -.J ~

a

02

Ql

M

ee

10

11

U,

'

I

(K

1l'6

t

ee

I

I

to

U

1

" rIME ,""

Fig. Sb : Simulation closingblow-offvalve

~.,.:!J- ...;,~ • •~ .....

~.

0 -

1>1

I

I

t-I

II

..

[ )'p( RIHf NUl

-

SMJ.....llOK

!O
18:)

1'0

I'll

I"

.'"

tl loil l ,. " .

Fig, 5a : Simulation openingblow-offvalve

tr-tr-rt;;-!t;;--rr---ro--tr -rr--ii------trTI"€ ? "' t'

Fig. Sc : Simulation closingoutlet valve(surge)

Transientswere introduced into the compression system by opening and closing the blow-offvalve, thus changing the compressor operating point. Finally. the downstream gate valve was closed pushing the compressor into surge.The results of thesesimulations are shown in Fig. Sa-c.

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The simulations of the blow-off valve opening, Fig. Sa, shows good agreement on mass flow, but a slightly highpressure history. This is thought to be due to an inaccuracy in the representation of the blowoff valvecharacteristics and final opening, leading to a different finalcondition of the compressor between the experiment and the simulation. It is worthnotingthat the 'kink' appearing in both pressure and mass flow curves are reproduced in the simulation. In the simulation of the blow-offvalve closing, Fig. Sb, a slightdiscrepancy in final mass flow is seen. Mass flow measurements are taken from the pressuredrop across the orifice plateat compressor suction, an are subject to a degree of experimental error. Of particular interest are the post-transient oscillations which can be seenin the prediction, and alsoin the experimental data, although thereis somescatterin this data. The frequencies of the oscillations agreewellbut the phase differs by a smallamount, probably due to a misalignment of the transient start. The ability to simulate this wave structure is a feature of the model, andhas beennoted in workcarried out since. Simulation of pre-surge oscillations, Fig. Sc, occurs at about correct point in time and at the correct compressor operating point. Frequency response is about the rightorderof magnitude. Simulation of a Gas-pumping Station

A second example (Fig. 6) is the simulation of part of a natural gas pumping station where compressor instability and control problems had beenencountered whenreducing the compressor speed. This example is discussed in moredetail in Elderet al. (1987). 35 )411.

JJ

Jl

11

17 J

• ~50

", 47

52

" J

s

20000 22

ss

'8

•

l' " "

L2

,8

57~5

,,-

sz

"

22

,

~

18000

POINT

TIME lSI

POINT

TIME ('

I 2 3 4 5 6 7 8 9 10 11 12 11 ,I, 15 16 17 18

00 0·1 02 03 01, 05 0·6 07 08 09 10 1·1 12 13 11, I5 16 17

19 20 21

'8 19 20 21 22 2J 21, 25 26 27 28 29 30 31 J2 B 3.1, 35 36

~

8

" " 5

~

16000

,

2

1

,

,.., 5

11611.17

8

,

23 21, 25 26 27 28 29 JO 31 32 JJ 31, 35 J6 37 38 39 40

].7

],8 39

It' ~ 11,000 0

ex

>:

is 12000 ': a. 10000

1

22

65DO

~

8000

/Ill 10 11

LJ

Fig. 6. Gas pumping station Decomposition into subsystems

./

II, 18 12 16 10 VOLUME FLOW RATE m)/m

Fig. 7. Gas pumping station Model response to speed reduction

No experimental data was available to check out the simulation but the results produced were able to explain qualitatively the observations of the operators on the station. The system consisted of a single operational compressor with a recycle valve providing anti-surge protection. Downstream of the system illustrated in Fig.6was a pipe network system including non-return valves and a choiceof gas delivery via single or double discharge pipes. The simulation of this system involved the solution of 300ODE's. The simulation of interest was that of reducing speed. The operators of the plant had observed that, on shutting downthe compressor, therewas a tendency for compressor surgeto occur,andthatthe non-return valves in the compressor discharge lines could be hard to be closing during the surge. The compressor modelwas set up, withthe compressor at full speedandthe anti-surge recycle valveclosed. An anti-surge control was simulated such that at a predefined compressor discharge pressure the recycle valvewillopen fully. The compressor speed was reduced and the results are shown in Fig. 7, superimposed on the compressor performance map. The compressor operating pointis seento movetowards the surge line,on a trajectory whichis determined by the match between the compressor andthe restof the system. Dueto the largelengthof pipe work downstream of the compressor, the pressure can vary only slowly, and the compressor transient line has a shallow gradient. At point 26 on Fig. 7, the anti-surge control causes the

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recycle valve to open, and the compressor operating point moves back in the stable region. As the speed continues to reduce, the compressor moves again towards surge until surgeoccurs again at point37. Sometest examples showed that the stability of the system depended on the delivery arrangements being used. This is because the friction losses in the delivery pipe system influence the gradient of the compressor transienttrajectory, points 1 to 26 of Fig. 7, and the steeperthe gradientthe more stable Ire system. This effecthadbeennoted in the plantoperation. CONCLUSIONS

A general modelfor the simulation of fluid flow systems has been presented. Validation of the modelhas beenprovided by various examples. The model has beenshown to be able to simulate the surgeonsetpoint of a multistage compressor. Simulation of compressor transients have agreed well with test results and oscillatory features in the system are well reproduced. A general dynamic simulation programme is builtto allow easyuse of suchmodels. The DASSL package has beenselected in the firstversion of the programme to solve the DAEsystem. Programme structure has been designed using lessons from object oriented programming. Reduction in programme maintenance effortandincreased flexibility in laterdevelopment is expected. The simulation of a plant operation showshow the model may be used to study the system response to control actions, andhence provide a design aidfor control system specification. ACKNOWLEDGEMENT

The authors thank. the Commission of European Communities (JouleProgramme) for supporting part of the work described in the present paper. NOMENCLATURE

A

CF

~

e E

F f h M

m P R

section area Moody coefficient offriction specific heat pipediameter specific internal energy energy force frequency specific enthalpy massflowrate polytropic index pressure idealgasconstant

t

u V

Wp x

X

z

time axial velocity volume polytropic compression work axial coordinate, length compressibility function X=TN (av/iIT)p-l (see Schultz (1962»

y

compressibility factor Cp/Cv

11 P

density

polytropic efficiency

REFERENCES

ElderR.L, Gill M.E.,RazakA.M.Y. (1986). Validation of a compressor model. Trans. Inst. Measurement and Control, ,8.(4), 171-181 ElderR.L, Gill M.E., RazakA.M.Y. (1987). Simulation of the transient performance of a compressor in a natural gas pumping station. I. Mech. E; Third European Congress on Fluid machineryfor the oil, petrochemical and related industries, paper C116/87 EvansL.B., JosephB., SeiderW.D. (1977). System Structures for Process Simulation.A.I.Ch.E. Journal ,2.3.(5),658-666 Petzold L.R.. (1982). A description of DASSL, a differential/algebraic system solver Report SAND828637,SandiaNationalLab.•Livermore, Schultz, J.M.(1962).The Polytropic Analysis of Centrifugal Compressors J. Eng.for Power, 84,62-82

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recycle valve to open. and the compressoroperating poim moves back in the stable region. As the speed continues to reduce. the compressor moves againtowards surge until surgeoccurs again at point 37. Some test examples showed that the stability of the system depended on the delivery arrangements being used. This is because the friction losses in the delivery pipe system influence the gradient of the compressor transient trajectory. points 1 to 26 of Fig. 7. and the steeper the gradient the more stable Ire system. This effect had been noted in the plantoperation. CONCLUSIONS

A general model for the simulationof fluid flow systemshas been presented. Validation of the model has been provided by variousexamples. The modelhas been shown to be ableto simulatethe surge onset point of a multistage compressor. Simulation of compressor transients have agreed well with test results and oscillatory features in the systemare well reproduced. A general dynamic simulation programme is built to allow easy use of such models. The DASSL package has been selected in the first versionof the programme to solve the DAE system.Programme structure has been designed using lessons from object oriented programming. Reduction in programme maintenance effortand increased flexibility in later development is expected. The simulation of a plant operation shows how the model may be used to study the system response to control actions. and henceprovide a design aid for control systemspecification. ACKNOWLEDGEMENT

The authors thank the Commission of European Communities (Joule Programme) for supporting part of the workdescribed in the presentpaper. NOMENCLATURE

section area Moodycoefficient of friction specific heat pipediameter specific internal energy energy force frequency specific enthalpy mass flow rate polytropic index pressure ideal gas constant

A

CF

~

e E F f h

M

m P R

t

u V

Wp

x

X

z y

11 p

time axial velocity volume polytropic compression work axial coordinate, length compressibility function X=T/Y (ilV/aT)p-l (seeSchultz (1962»

compressibility factor Cp/Cv polytropic efficiency density

REFERENCES

ElderR.L. Gill M.E., Razak A.M.Y. (1986). Validation of a compressor model. Trans.Tnst. Measurement and Control. ~(4), 171-181 Elder R.L. Gill M.E., Razak A.M.Y. (1987). Simulation of the transientperformance of a compressor in a natural gas pumping station. I. Mech. E; Third European Congress on Fluid machineryfor the oil, petrochemical andrelated industries. paperC116/87 Evans L.B.. Joseph B., Seider W.D. (1977). System Structures for Process SimulationA1.Ch.E. Journal (5).658-666 Petzold L.R.. (1982). A description of DASSL. a differential/algebraic system solver Report SAND828637. Sandia National Lab .. Livermore, Schultz. J.M.(l962).The Polytropic Analysis of Centrifugal Compressors J. Eng.for Power, 84.62-82

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