NATIONAL AERONAUTICS AND SPACE …meridional (r-z) plane for stations that are lines into two...

AD-AL09 BOO NATIONAL AERONAUTICS AND SPACE ADMINISTRATION CLEVEL--ETC F/6 21/5COMPUTER PROGRAM FOR AERODYNAMIC AND BLADING DESIGN OF MULTISTA-EC(flDEC 81 J E CROUSE. W T GORRELL

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DesgnOf MultistageAxia-Flow Compressors.

Jamnes E. Crouseand Williamr T Gorrell

JAN 2 0 1982

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AVRADCOMTechnicalReport Computer Program for80-C-21

Aerodynamic and B ladingDesign of MultistageAxial-Flow Compressors

Jani's E. (:roisclen'i.s I?e. 'a ry (.'c'nh'r

Ci , ,elawl, Ohio

NVill Iam T GorrdllIPropll.iUo? Laboratory.. FRA,)COAI Resear h and Telmoloiy l.aboratorie.,I.ewn'A esearch (enter(.e'eland. Ohio

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Contents

PageSummary ............................................................................................

Introduction .......................................................................................... I

Compressor Design Procedures............................................................................ 2Input Initialization .................................................................................... 2General Information .................................................................................. 2Calculation Station Data Sets........................................................................... 3Initialization ........................................................................................... 5Iteration ........................................................................................... 6Aerodynamic Design ................................................................................ 6Blade Design ........................................................................................... 8Terminal Calculations and Output ...................................................................... 9

User Information..................................................................................... . 0

AppendixA-Symbols ........................................................................................ IS5B-Input Parameters for Compressor Design Program .................................................. 16C-Development of Equations of Motion into Form Used in Computer Program.......................27D-Conic Coordinates of Blade Centerline Path........................................................32

References............................................................................................. 38

Summary performance advantages over other compressor

types, but the good potential performance is notA code for computing the aerodynamic design of a easily attained. The problem and challenge to the

multistage axial-flow compressor and, if desired, the designer is to model the actual flows well enough toassociated blading geometry input for internal flow adequately predict aerodynamic performance.analysis codes is presented. The aerodynamic Progress is continually being made with codes forsolution gives velocity diagrams on selected computing the complex three-dimensional flows instreamlines of revolution at the blade row edges. turbomachinery. However, it is extremely difficult toBlading is defined from stacked blade elements design mechanically acceptable turbomachineryassociated with the selected streamlines. The blade blading by using the direct approach (i.e., specifyingelement inlet and outlet angles are established inviscid blade surface velocities and computing thethrough empirical incidence and deviation angle blade geometry). Consequently, the more detailedadjustments to the relative flow angles of the velocity codes are generally used in the analysis mode; that is,diagrams. The blade element centerline is composed the flow field is calculated for a fixed geometry. Theof two segments tangentially joined at a transition current procedure is to establish blading geometrypoint. The local blade angle variation of each with simpler design codes and then to use the moresegment can be specified with a fourth-degree detailed analysis codes in blade rows wherepolynomial function of path distance. Blade element troublesome flow conditions are likely to exist. Inthickness also can be specified with fourth-degree this way prototype designs can often be adjustedpolynomial functions of path distance from the before hardware is built and tested.maximum thickness point. The time and effort needed to get acceptable

Steady axisymmetric fw js assumed; so the configurations can be reduced if the design code canaerodynamic problem canj reduced to solving the be made to yield a good initial solution and if thetwo-dimensional flow field in the meridional plane. design and analysis codes can be made moreBecause the equations of motion as developed herein compatible with one another. This compatibility canare only applicable for calculation stations outside be achieved (1) if the output from a design code canthe blade rows, stations at the blade edges, but not be directly used by flow and mechanical analysisinside the blade rows. are used. The streamline codes and (2) if corrective adjustments indicated bycursature method is used for the iterative the analysis codes can effectively be made in theaerodynamic solution. If a blade design is desired, design code. With these objectives in mind athe blade elements are defined and stacked within the composite aerodynamic and blade design code foraerodynamic solution iteration. Thus the design axial-flow compressors has been developed. The codevelocity diagrams can be located at the blade edges. and its capabilities are the subjects of this report.

The program input includes the annulus profile, The aerodynamic solution assumes steady,the oxerall compiessor mass flow, the pressure ratio, axisymmetric flow and uses a streamline curvatureand the rotative speed. A number of parameters are method for calculation stations outside the bladeinput to specify and control the blade row rows. The program is structured so that the empiricalaerodynanics and geometry. There are numerous correlations (such as those for loss. deviation angle,options for controlling the way information is input and incidence angle) can readily be changed when theand for specifying the amount of output. The output need or desire exists. The method of describingfnom the aerodynamic solution has an overall blade blading is a compromise between the %ast amount ofrow arid conipressor performance summary followed input needed for completely general blade elementsby blade element parameters for the individual blade and the restrictions of simple shapes. A blader,)sw. It desired. lade coordinates in the streamwise element is defined on a conic surface with thicknesstlireit.n tot internal flow analysis codes and/or applied to a centerline that is composed of tsxo;ordmiate on plane sections through blades for segments tangentially joined at a transition point

!Ahf-ricalion Jiawings, can be printed and punched. The blade angle function of each segment can bedefined with a fourth-degree polynomial. Thicknessis prescribed by first specifying a ma\imum thickness

Introduction value and location. The distribution of thicknes,, ineach direction from the maxinum luckn|ess location

lhc axial-t1o conpressor i% used for aircraft is then prescribed with a fOurth-degree polynotnial.ezigme, heAuse it has distinct configuration and Finally each pol.\nomial coefficient is detined across

4(.

blade elements with a third-degree polynomial deviation angles, the layout and stacking of bladefunction of annulus height. elements, and the realignment of the elements.

The terminal calculation phase performs the finalcalculations and generates the output. Mass-averaged

Compressor Design Procedures parameters for the individual and cumulativecompressor blade rows are computed and printed

The discussion o~f the compressor design first. Then tabulated values of aerodynamic andprocedures is organized according to usage in the blading parameters along the station lines arecomputer program; so for better orientation an computed and printed. Finally blade sectionoperational overview of the program is given first coordinates and other section mechanical properties(table 1). The computer program can be divided into can be computed and printed if desired.three major phases of calculation: (1) the input and The program is discussed in greater detail in theinitialization phase, (2) the iteration phase, and (3) following subsections.the terminal calculation phase. In the input andinitialization phase the input data are read and Input and Initializationinterpreted, the calculation stations are located withestimated values for the blade edges, and streamlines The basic computational plane is the meriodionalare located on the basis of annulus area. Estimates of (r-z) plane of a cylindrical coordinate system. Astagnation temperature and pressure and axial and graphic view of an example compressortangential velocity components are also made for all configuration is shown in figure 1. The hub and tipcalculation points in the flow field. casing walls are fixed input. Calculation stations are

The iteration phase includes both the flow field located at the blade row leading and trailing edgesand the blade design iterations. In the flow field and at other annular locations for the purpose ofiteration the equations of motion are satisfied in the locating streamlines. The input data can be classifiedmeridional (r-z) plane for stations that are lines into two groups: general information and calculation

across the flow annulus. At the stations the equations station and blade row information. The inputof motion and overall flow continuity are satisfied parameters and options, along with the input dataAwith fixed values of streamline slope and curvature format, are described in appendix B. (Allfor a complete computational pass across the mathematical symbols are defined in appendix A.)annulus. After the overall flow continuity conditioni Additional advice on how to set up the input is givenat a calculation station is satisfied, the internal in the section User Information.streamline intersections with the station lines areupdated by solving for the locations that give General Informationspecified fractions of overall station weight flow. Atthe completion of a pass through all the calculation All the general information is read in first.stations in the annulus, the new streamline locations Included are the following:

are curve fit for new streamline slope and curvaturevalues. (1) Compressor rotational speedI

To insure proper location of the blade edge (2) Inlet flow ratestations, most of the blade design iteration is made (3) Desired compressor pressure ratioconcurrent with the flow field iteration. This (4) Gas molecular weight

operation includes the calculation of incidence and (5) Number of streamlines

Annuar tatln-~Blade edgeAnnular tation-,stations-7.Tip 5 tao

Rotorto

I I Rotor IIII I stator Roo

I Hub r

________ Axis of rotation ______ ___

Figure 1. -Calculation stations in compressor flow Path.

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(6) Number of blade rows polynomial function of annulus height in the radial(7) Number of annular stations direction. The pressure level is computed internally(8) Coefficients for cp as a fifth-degree to the program from the energy input level and the

polynomial function of temperature computed losses. With the other rotor options the(9) Far upstream values of total temperature, exit temperature profile is input instead of the energy

total pressure, and inlet tangential velocity for each addition fraction being specified. For either a rotorstreamline or stator, stagnation pressure profiles can be input

(10) Streamitube mass flow fractions between instead of the losses being computed internal to thestreamlines program. These options can be useful to users who

(11) Sets of points to define tip and hub casing have existing aerodynamic designs but want to usecontours this program for blade description and fabrication

(12) Sets of blade element profile loss parameters coordinates.that are tabulated as functions of blade element At a stator exit a tangential velocity profile is input

loading parameter and fraction of passage height as a polynomial function of radius. Unless specified,the stator outlet pressure profile is determined from

loss set used for a given blade row is designated in the upstream station.blade row input. Usually at least two loss sets are There are some input aerodynamic limits that theinput-one for rotors and another for stators. program will not allow to be exceeded. For a rotor

telimiting parameters are tip diffusion factor andCalculation Station Data Sets absolute flow angle at the hub. The stator

aerodynamic limits are diffusion factor and inletThe data sets that contain information about the Mach number at the hub. If an aerodynamic limit is

calculation stations and blade rows are read in order exceeded during iteration, the stage energy additionfrom annulus inlet to outlet. The first card of the is lowered by the amount needed to get thedata set identifies the type of station, the tip and hub aerodynamic limit within bounds. If any other stageaxial locations, the tip and hub boundary layer is not up to one of the aerodynamic limits, the energyblockage factors and the station mass flow bleed. For decrement is made up among such stages. If all theannular stations the single card is the whole data set. stages reach an aerodynamic limit, the input overallFor rotors and stators several cards are used to compressor pressure ratio is lowered.describe (1) the blade row inlet and outlet station The blade angles are related to fluid flow anglesinformation, (2) the blade row aerodynamic along streamlines by two key correctionparameter input and controls, and (3) the parameters parameters-incidence angle at the inlet anddefining blade geometry. A blade and the associated deviation angle at the outlet (fig. 2). There are severaledge calculation stations are located in the annulus options for specifying each. Two of the options forby using a reference blade element stacking line, both the incidence and deviation angles are the two-Stacking axis tip and hub axial locations and lean and three-dimensional methods of reference 1. Tlbangle in the circumferential direction are input, other incidence angle option is user-entered tabular

The locations of the calculation stations at theblade edges are at first approximated from some ofthe input blade geometry information. The stationlocations are moved during later iterations when the Flowblade elements are defined and stacked. However,6the input tip and hub boundary layer blockages andmass flow bleeds for the inlet and outlet stations areconstant.

Aerodynamic parameter input and controls. -Theblade aerodynamic design is controlled with severalparameters that impose the necessary and sufficientconditions for a solution. The options as to how suchconditions can be imposed are shown in table 11. Forrotors the most convenient option is to specify thestage energy addition as a cumulative fraction of tbeoverall compressor energy addition. With this option Fothe radial distribution of energy addition is not input /Fo

directly but is imposed through a normalized rotorexit stagnation pressure profile that is expressed as a Figure 2. -Blade element incidence and deviation angles.

3

data referenced to either the centerline or the suction- Blade geometry parameters.-A number of bladesurface blade angles at the inlet. Other deviation geometry parameters are input for the pupose ofoptions are user-entered tabular data and a version of defining a blade. Blade chord is defined along flowCarter's rule, which was modified to account for streamlines, but for the purpose of this bladecenterline shapes other than a circular arc. The definition a radial projection of streamline chord ismodification is shown in figure 3. specified because it is more meaningful for defining a

Another input aerodynamic parameter is the structurally sound configuration. The radiallyminimum blade choke margin (A/A ) - 1, where A is projected chord is defined from the number ofthe local streamtube cascade channel area and A* is blades, the tip solidity, and a normalized polynomialthe corresponding area needed for choked flow. The for the radial variation of chord. The blade isA * value is the area needed to pass the streamtube basically defined from a stacked series of graduallyflow at a relative Mach number of 1.0. The effects of changing airfoil shapes or "blade elements" in thelosses in all blade rows and energy addition in rotors radial direction.are included in the computation of A*. Choke Each blade element, as shown in figure 4, ismargin depends on the flow conditions and geometry defined from a thickness distribution applied to adefining the channel area. If insufficient choke two-segment centerline. The variation of the localmargin exists in a prototype design, some centerline angles K with path distance can be specifiedcompromise must be made in either the aerodynamic by option through the parameter IDEF(IROW). Ifrequirements or the geometry. Minor choke margin IDEF(IROW) equals zero, the K for each segmentdeficiencies can usually be accommodated with varies linearly with path distance (as a circular arc).adjustments in geometry. Logical procedures for When IDEF(IROW) does not equal zero, the K forgeometry adjustments are not obvious; however, if each segment is expressed as a fourth-degreethe minimum margin occurs at the channel entrance, polynomial function of path distance. The bladeincreased incidence is an effective method of relief. If angle is continuous at the transition point, but thea minimum desired choke margin is input, the rate at which the angle changes with distanceprogram will adjust incidence angle up to + 2* to the (curvature) can be discontinuous. The ratio ofleading-edge suction surface in order to attain the curvature for the first segment to that for the secondspecified choke margin if the channel entrance is the segment is defined as the turning rate ratio. When theproblem. When the minimum margin occurs at other blade local centerline angle K is specified bylocations in the channel, the minimum value and its polynomial coefficients, the turning rate ratio islocation are printed in the output and it is up to the controlled by the relative magnitudes of the linearuser to decide if he wants to make compromises to term coefficients of the polynomials for eachimprove the choke margin, segment. However, when the segments are treated

c Rte

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Bld c od nle ' (e a//'

x(Zac.62

cP' .0 5 2 " 0 0 1 1 ?1 )c n e l n n thickness p l n m a s

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0 10 20 30 40 50 60 70 1Blade chord angle, y, deg Ii

Figure 3. .Variation of coefficient for Carter's deviation equation rlewith location of blade element maximum camber point.m - (. 219 -0. NU916 y 0. 0002705 ? l Figure 4. -Reference and direction nomenclature for prescribed blade element

itxf2alcltZ 175-1035525y +0. 0019167 y2 ) centerl ine and thickness polynomials.

4

simply as circular arcs, the turning rate ratio is a surfaces for each segment are defined by polynomialblade element input parameter. distributions of the normal -to-centerline distance.

When IDEF(IROW) equals zero, there are some The functional relation for this distance isoptions for specifying the turning rate ratio at thetransition point. With the CIRCULAR option~ the t__ aS S S Svalue is set at 1.0, as for a circular arc blade element, t= '-- -avS 0 + aV S- S + - -bS-S 3 dSfor all blade elements in the blade row. With the 2v§s 0TABULAR option a table of values for the elementsis read. With the OPTIMUM option a value will be where S is the centerline distance (normalized byset by an empirical function of inlet relative Mach chord) from the maximum thickness point. Values ofnumber. For this option the blade element will be a S are positive in either direction from the maximumcircular arc below a relative Mach number of 0.8. As thickness point; and S, is the maximum S, which isrelative Mach number increases, the ratio of first- to the distance to the point where the end ellipsesecond-segment turning rate at the transition point is intersects the centerline (fig. 4).reduced. A limit of zero camber on the suction There are two other input parameters for bladesurface of the first segment is approached at an inlet rows. One is a material density for rotors. If arelative Mach number of about 1.60. nonzero value is input for a rotor, the stacked blade

The coefficients for the centerline polynomial (i.e., will lean in both the meridional and the r-O planes sowhen IDEF(IROW) 00) are input as a cubic function that the centrifugal force on a blade with the inputof blade span. There are two reasons for this method material density will balance the aerodynamic forcesof specification. First, the user is mogre confident of at the design point. The objective is to minimize thespecifying a relatively smooth blade surface; and blade root stress. With atmospheric air as thesecond, the amount of input is reduced over that working fluid, the lean is normally only a fraction of

required by individual coefficients for as many as 11. a degree.Iblade elements. The final input parameter, NXCUT, controls the

Blade element surface definition begins with three number and location of planes through a blade rowanchor points from the centerline. These points are a for which fabrication coordinates are desired. If themaximum thickness point and the two end points. A parameter is zero, the program will set the number ofmaximum thickness value normalized by chord and XCUT's on the basis of aspect ratio, which is theits location as a fraction of chord are input. At the ratio of overall radial to axial blade lengths. Formaximum thickness point the normal-to-centerline positive parameter input values the program willdistance to each surface is one-half the maximum determine appropriate locations for that number ofthickness, and the surface K angles are equal to the planes to represent the blade. Negative parametercenterline K. values trigger an option to read cards for the XCUT

At the blade element ends the leading- and trailing- plane values. The number of input values expectededge end circle radii normalized to chord are input. If for a blade row is the absolute value of the negativeIDEF(IROW) does not equal zero, the end parameter.configurations are ellipses with semnimajor axestangent to the local centerline. For this case the input Iiializationend circle radius is used as the minimum radius valueof the ellipse. For each ellipse one other parameter is Once the input is read, a number of initializationinput to specify elongation. The parameter is calculations are made in subroutine START ine =(b/a) - 1, where b and a are the semnimajor and preparation for the iterative phase of computation.semiminor axes, respectively. Note that as e The axial locations of the blade edges are

* papproaches zero, the ellipse approaches a circle with approximated and the intersections of all station linesthe input radius. with the casing walls are determined. Checks are

A surface definition criterion is that the surface made to be certain that the spacing of calculationcurve join the end circles or ellipses at a point of stations is appropriate. Annular stations will betangency. When IDEF(IROW) equals zero, the shifted by the program if calculation stations crosssurface curves are defined with K being a linear one another or if adjacent spacing is less than 30function of path distance for each segment. As percent of the spacing of neighboring stations.explained in reference 2, necessary and sufficient Streamlines are initially positioned by applyingconditions exist to completely define the surfaces the input stream-tube weight flow fractions to thewhen the computation is begun on the segment where annulus area. From the input data the cir-the maximum thickness occurs. cumferential component of velocity and the

When IDEF(IROW) does not equal zero, the blade stagnation temperature and pressure are

hA.~La I_~ *-

approximated for all streamlines at all calculation conserved along streamlines outside the blade rowsstations throughout the flow field. Finally an axial except for mixing effects from turbulence andvelocity is computed for each station by using secondary flows. The stagnation pressuremeanline values in a continuity calculation at the distribution is input behind the rotors; so pressurestation. gradients are reasonably well controlled in the design

process without using empirical mixing terms.

Iteration In the design process the rotor energy additionmust cover nonproductive losses in addition to

The general objective of the program is to obtain producing a desired pressure. With the usual inputboth an aerodynamic solution and a blade design. options, losses are computed internal to the program.Both are achieved with iterative procedures. The Normally there is a significant radial gradient of loss;aerodynamic design has the greater sensitivity, and it so there is also a radial gradient of work. Therequires more iterations. The program is set up to do stagnation temperature increase along a streamline isthe aerodynamic and blade design iterations in almost direct proportion to the blade elementconcurrently. However, the blade design is done less work; so temperature gradients are generated.frequently and lags the aerodynamic iteration. The Because these gradients through compressor stagesfirst blade design iteration occurs on the fourth are basically additive, theoretically the gradients canaerodynamic iteration, and the final blade design grow very large. The real flows in compressorspass is made after the aerodynamic solution is reduce this effect somewhat with fluid mixing. To atprinted, least partially account for fluid mixing in an

empirical manner, a mixing term for temperature isused in the program. The mass average temperature

Aerodynamic Design is held constant at a station, but specific streamlinevalues outside the blade rows are modified from the

The aerodynamic design solution establishes previous station values by equation (1).complete velocity diagrams and fluid state propertieson streamlines at the blade row inlet and exit. Abilevel iteration is used to arrive at the solution. In (dT I dT), IT0.002( )( zthe outer loop the variables are stagnation kdr} = dr exp [_0 2(1)temperature and pressure; the tangential componentof velocity; and the streamline location, slope, andcurvature. The inner loop is the station flow

continuity calculation in which the axial component where AZ is the axial distance between the adjacentof velocity is the variable and the outer loop stations. Future adjustments in this functional

parameters are held fixed. An example flow field relation are probable as data from multistagewith typical placement of calculation stations is compressors become available.shown in figure 1. Stagnation temperature and pressure values are the

Outer Ioop.-In the program the control routine most difficult to set at blade row exits. This is mainlyfor the outer loop is VDIAG. The basic procedure is because of the complex real flow effects through astation marching from inlet to outlet with streamline blade row that must be represented either throughparameters fixed. Only after a pass through all the theoretical models of loss or by empiricalstations are the streamlines relocated from the correlations. Representation of losses is, of course,current flow solution. Normally between 10 and 20 one of the major problems for an aerodynamicof the cycles are needed to converge to a solution. solution. In this program the losses are represented

The major part of the blade design is also by two additive components: shock losses, and allcontrolled in the outer loop. When a blade design other losses.iteration is made, the blade edge station locations are The shock losses are a modification of those givenmoved to the new blade edge locations, in reference 3. This reference, in essence, gives the

The tangential velocity and the stagnation shock loss associated with a normal shock with antemperature and pressure at a station are determined approach Mach number equal to the average relativeas changes from values of the preceding station on Mach number at the suction and pressure surfaces ofthe particular streamline. For annular stations and the blade at the normal shock. The suction-surfaceblade row inlets the tangential velocity is determined Mach number at the shock is determined by Prandtl-from the conservation of angular momentum; that is, Meyer turning from the inlet.the product of radius and tangential velocity remains Unless the flow is in the low transonic range, athe same along streamlines outside the blade rows. normal shock cannot be maintained in a bladeStagnation temperature and pressure should also be channel. Either the shock is oblique or it develops a

6

foot at the blade surface because the boundary layer a Vm v 2cannot sustain the sudden static pressure rise. In + V sin(a + X) + I cos(a + X) (6)

either case the shock losses are less than those m Rm

predicted by a normal shock. To empirically accountfor these effects, the computed normal shock loss is withreduced by dividing by the average inlet relativeMach number squared.

All the other blade row losses-profile, secondary, aVm Vm

etc.-are represented by a correlation with fraction am M 2 -1of passage height and aerodynamic blade loading. rThe values for such a correlation are input in tabularform. The aerodynamic blade loading parameter inthe table is the diffusion factor of reference 1. In M2+ 1 )equation form it is [M sin a+d sec(a+ X) tan(of (7)

Vi A(rVo)D=l- F c(r+r 2 )Vj (2)

A velocity gradient procedure is used to constructthe axial velocity profile from the tip to the hub withthe stagnation state values, the streamlinecharacteristics, and the tangential component of

Wcos3 (3) velocity held fixed. Since this inner loop of the2a program is used many times, some effort was made

to evaluate its accuracy and efficiency for typicalwhere w is the loss coefficient. streamline spacing. Reasonably good accuracy and

stability were found to result from a rather simplePi-P (4) procedure. Let

Pi -P) dVm a +bVm(8

The rotor exit tangential velocity is calculated + b V-8directly from the Euler equation

H2 -H ! T z where

H 2 -HI= T2Cp dt = U2 V 2 - U VO, (5)T,

Note that the enthalpy change is evaluated by using a - I dH + tRd In P d(rV (9)an integral for the calorically nonperfect gas; that is, \1 T dl dl dlc is a function of temperature. All state processes intWe program use thermally perfect, but calorically andnonperfect, gas relations; so integrations and in somecases iterations are used in several small functionroutines. b= cos(e + X) + sin(a + X)

Inner loop.-The basis function of the inner loop Rm M2 -1is to determine the axial velocity profile at the

calculation station. The axial velocity level is set byflow continuity, and the distribution is controlled by [M0 I di tan(a+ Xthe radial equation of motion. The differential x sina+ - sec(c+X) - (10)equation is developed in appendix C. The form used mr ]in the program is

dVm (T- tdH With a and b constants for the I interval along the

mdl - T / dl station path, the solution for Vm is

dlnP d(rVo) V2 / . \! +V2.2b(l-,10 ) a-ldl 4+ (

&7

A two-step procedure is used in the program. First reference 2. Only a summary description is givena, b, and Vm values on the streamline j are used to herein. A blade element is laid out on a cone with adetermine a temporary Vmj , I. The a and b values center axis coincident with the turbomachine axis ofare slightly dependent on Vm so Vmj +I is used to rotation. The angle and location of the cone are fixeddetermine new a and b values. The second step uses by the intersection of the streamline with the leading-the average of the old and new respective values of a and trailing-edge station lines of the blade (fig. 6).and b to compute a final Vmj+ value. This V,j +I The leading- and trailing-edge blade angles arevalue will then be used as the current Vmj value for related to aerodynamic flow angles primarily throughthe next I interval. two key correlation parameters -incidence angle and

When Vm values are set on all streamlines, flow deviation angle. The user has some options for thecontinuity is checked by using dr integration of a specification of these correlation parameters, aspiecewise cubic curve fit of pVmr values at the already discussed in the section on data input.streamlines. If the integrated weight flow is not Application of incidence and deviation angles to thewithin 0.01 percent of its specified value, the tip flow angles at the blade edges gives blade angles inreference Vm is adjusted and the Vm profile is the local streamwise direction. Corrections toreconstructed. The method of adjusting the reference "cascade" deviation angle for a change in radius andvalue of Vm is shown graphically in figure 5. There axial velocity are made internally to the program.are two solutions to the continuity equation in These corrections are presented in reference 4 tocompressible flow-the subsonic and supersonic relate deviation angle to a cascade section withsolutions. When a parabolic fit of trial solutions is equivalent circulation rather than with the sameused to get a new trial value of Vm, the lower or camber angle.subsonic solution is always sought. The Vm Because the cone angle of the associated bladeadjustment between iterations usually is small; so element is usually a little different from these localconvergence normally is achieved in three or four streamwise blade angles, corrections are made withpasses. current streamwise and radial direction derivatives.

Once convergence is achieved, the profile is back The blade element leading- and trailing-edge anglesintegrated to find the fraction of weight flow points are calculated from aerodynamic flow angles inrepresented by the streamlines. These points are subroutine BLADE.saved until the outer loop pass through all the Blade element layout.-There are several optionsstations is completed for the purpose of relocating for controlling the blade element layout (see thestreamlines. IDEF (IROW) parameter description in appendix B).

With all but one of these options a blade element is

Blade Design described by a prescribed thickness applied to aprescribed centerline (fig. 4). The centerline is treated

A blade is defined from stacked blade elements. as two segments that are joined at the referenceThe procedure for laying out blade elements and transition point. The rate of change of the local bladestacking them for blade definition is given in detail in angle with path distance, K=f(S) (fig. 4), is controlled

by a fourth-degree polynomial for each segment. Thecoefficients for the polynomials are input, but theyare scaled in the program to match blade element

Subsonic S upersonic inlet and outlet angles. The fourth-degree polynomialS o l u t iot i o1.0o . .

Third V, iinear/ IUnrapped coniiao " surae-•th ro u g h first u l e _and second

g. poiltsi -Second Vm ,,near R

rou , h p nt " tar ing re

Meridional velocity, Vrq - - - -i

Figure 5. - M ,eridional velocity adjustrnent for flow conf n uiiy I yore 5. - Len cdl coordinate s,,ste.-iteration, for blade element layout.

8i

O -

representation of segment blade angle representsgreater specification freedom than does the linearspecification of reference 2, where the ratio of inlet-to-outlet segment curvature at the transition point isinput rather than any polynomial coefficients. A - stacKnisummary derivation of the equations for the - line

centerline coordinates is given in appendix D.Blade element thickness is defined along a path

that is locally normal to the centerline. The pressureand suction surfaces are equidistant from the -ncenterline. Thickness is specified in both the forwardand rearward directions from the maximum

thickness point by polynomials of the form

t' -aVS o +avSo -S + as2 0

-bS 2 - cS 3 -dS 4 (12)

The input coefficients are scaled to meet the leading-and trailing-edge ellipses at the appropriate tangency Xis of rolation

points. The control routine for the blade element fqure 7. :toof Dlare sections for olade elementlayout in the program is CONIC. stack tit$

Blade element stacking.-The rotating parts ofturbomachinery normally operate at high stress levelsbecause of high centrifugal force. The high recomputed and adjusted if the stacking axis leancentrifugal acceleration also causes stress from option is activated through the input data.bending moments to be very sensitive to bladeelement location. Thus it behooves the designer, Terminal Calculations and Outputfirst, to be reasonably accurate in the stackingcomputation and, second, to try to minimize stresses The program output of an example two-stagethat can be easily reduced-namely, those from the compressor is shown in table Ill. In general thesteady-state bending. The blade bending moments output is printed shortly after its computation so thatfrom aerodynamic forces can be counterbalanced by large arrays of data are not stored. Data are printedcentrifugal force moments with slight blade lean in from each of the major phases ofboth the (r-z) and (r-6) planes. computation-input, iteration, and terminal. The

The reference line for stacking purposes is a radial first information (table Ill (a)) is the input data,line through the hub stacking reference point (fig. 7). which are printed directly from input routines in veryThe sections used for stacking alignment are planes nearly the order in which the input was read.normal to this reference line in space. Such planes are The second major part of output (table Ill (b)),used because their centers of area are essentially the from the iterative phase of computation, is printed tocenters of centrifugal force also. The stacking line is help the user monitor the solution. Although thesea line that can be leaned from the reference line at the data have little value once the solution is converged,hub reference point. For alignment purposes the they are quite helpful in disclosing bad input and inplanes pass through the stacking line intersection of finding sources of problems when solutions are notblade elements (fig. 7). Blade sections are defined by achieved.interpolation across blade elements. When the For computational stability a station aspect ratio,section center of area does not match the stacking defined as (r t -rh)/(Zl+ I -zl), is limited to 7 forline, the corresponding blade element is translated streamline fits. When the limit is exceeded, particularand rotated on its cone for the stacking adjustment. stations (according to the priorities set forth in theNormally the adjustments decrease by about an order section User Information) are eliminated from theof magnitude for successive passes through the curve fits used to locate streamlines. The first datastacking procedure. For each pass the stacking axis shown from the iteration phase are a table of suchlean angles in both the (r-z) and (r-0) planes are calculation station information (table 11 (b)). On the

9

left is a list of calculation station locations used to Stackingcompute streamlines, along with the associated pointaspect ratios. On the right is the input list of station Clocations and aspect ratios. When blade rows are o

stacked, the blade edge stations are relocated, and a

thus the station aspect ratios change. After the first 'Fi Zstacking on iteration 4, the station aspect ratios are Hs Hrechecked and changes in the station list are made if Lnecessary. ca Lsp

Arrays of axial velocities throughout the flow fieldfor each iteration are the bulk of the output printed N".

from the iterative phase of computation. These dataare useful for observing solution stability since the Figure 8. - Coordinate system for lade sectionsolution convergence criterion is based on changes of output data.

axial velocity between successive iterations. Somecompressor overall parameters are shown above thevelocity arrays. Parameters included are the overall If the input options call for fabricationvalues of input pressure (PR), current computed coordinates, they are printed after all thepressure ratio (CPR), enthalpy increase (DHC), and aerodynamic output. The coordinates are printed inideal enthalpy increase (DHI). tabular form with four sections on a page, as shown

When the aerodynamic solution is converged, the in table 111(c). The length coordinate L is a distanceoverall parameters for individual blide rows and the along the chord line, with the most forward pointoverall cumulative values in the compressor are being zero (fig. 8). The pressure- and suction-surfacecomputed and printed. Overall temperature and height values Hp and H,, respectively, are referencedpressure values are calculated by mass averaging their from the chord line. Surface height values are givenequivalent enthalpy values. The cumulative forward for at least 20 round-value increments of L; alsoaxial thrust is the axial force exerted on the rotating surface coordinates are given for three specific valuesshaft by aerodynamic forces from the hub inlet of L-the blade trailing edge and the leading- andstation of the first blade row to the local point. The trailing-edge ellipse tangency points with thethrust force shown for individual blade rows is the surfaces.axial force on the shaft from the trailing edge of the A blade section's properties are shown above itsupstream blade to the trailing edge of this blade row. table of coordinates (table 111(c)). The blade sectionSince the blade forces on stationary blade rows act on radial location, the L and H stacking point values,the casing, the thrust value on the rotating shaft is and the section setting angle are given to locate andsimply the static pressure force on the tapered shaft orient the blade section. The blade section center-of-in the forward axial direction. Effects of cavities area coordinates, section area, minimum andbelow the hub flow path are not included since maximum moments of inertia through the centerundetermined information about seal locations and area, orientation angle of the maximum moment ofpressure differences would be needed. The gas inertia with respect to the axial direction, sectionbending moments are values for a single blade. The torsion constant, and twist stiffness are all usefulbending moments are referenced to the stacking axis information for design and stress analysis.intersection with the flow path wall from which the After all the fabrication coordinates for a givenblades are attached. blade row are printed, the blade section coordinates

Sets of calculation station data for streamlines are presented in another orientation that may beacross the channel follow the overall data. For all more useful for further flow analysis. With astations, velocity components, streamline slope and stacking axis reference, coordinates for the samecurvature, and both stagnation and static values of blade sections are given in the axial and tangentialtemperature and pressure are given. For stations at directions.blade row edges, additional information is computedand printed. These parameters are (1) a completedescription of velocity triangles, (2) definition of User Informationblade elements, (3) relations between aerodynamicand blade angles, (4) aerodynamic performance Since earlier sections of the report discuss theparameters, (5) streamline choke area margin, (6) input, output, and main centers of program control,local blade force intensity in pounds per radial inch this discussion is directed at the user who is trying toon a blade, and (7) blade edge direction derivatives get the program on his computer and to make it runr dO/dr. efficiently. Some facts about the program as well as

10

IILI, 2-- - -'-

some advice about the input are given, curve fits of these intersections. The consequence ofThe code, which is written in FORTRAN, takes this procedure is that the number of iterations and

about 80,000 decimal words of computer storage, the program convergence characteristics areThe call relation among the subroutines is shown dependent on the calculation station locationgraphically in figure 9. Note that the tickmarks on although the final solution, in general, is not verythe routine boxes in the figure mean that there are dependent on the location of the calculation stations.other call lines to the routine. These lines are shown The user can reduce the number of iterations andon the other part of the figure where the routine hence the program running time with good placementname is repeated. The program running time on of calculation stations. The first calculation stationeither a Univac 1 110 or an IBM 360-67 is about 2 should be placed upstream of the first blade row aminutes for a single-stage compressor and about 5 distance at least equal to two or three annulusminutes for a five-stage compressor. Several of the heights. The best far-upstream inlet condition iskey indices in COMMON/SCALAR/ are described straight axial flow with no wall curvature. Lessin the following tabulation. iterations are usually needed for more widely spaced

calculation stations; however, enough iterationsInde Decripionshould be used to properly locate the streamlines.Inde DecripionCalculation station spacing can vary somewhat along

indexafter the annulus but, as a general guideline, successiveIcalculation station idxatr station increments should not be changed more than

preliminary calculations are 35pretcompeted Th proram Is When calculation stations are input close together,dtmensioned for 50 calculation stations only some of them will be used for locating the

and 20 blade rows, of which only 10 streamlines if the station aspect ratio is above 7.0.can be rotors. Each blade row accounts This is done for program stability and convergencefor two calculation stations-one at toward a solution. If the user does not specify whichthe leading edge of the blade and the stations to eliminate from the streamline locationstators and teranlnula calculaotons procedure, the program has logic to do so when thestatrs, nd anulr caculaion station aspect ratio exceeds 7.0. The priority ofstations can be put together in any sain etfrsraln oaini sflos Icombination with the following bladeiow kext fstranloations arlas sed (2)oladow(1constraints: The number of stations ilaerext stations are epty ifed the blade rowrioscannot exceed 50. There must be at iltsain r eti h ld o setrtoileast four annular stations ahead of the less than 7.0, and (3) an annular station is kept iffirst blade row and at least three neither adjacent station is closer than the aspect ratioannular stations behind the last blade tolerance.

row. The user can also specify that particular annularrow. stations not be used for streamline definition through

IROTR roor idexthe alphanumeric station designation. The programIROTR roor idexlooks for ROTO for rotor, STAT for stator, or

IROW laderow ndexANNU for regular annular. Any other combinationsIROW laderow ndexof letters, numbers, or symbols designates the station

J steamine nde. Steaminesare as the extra-annular type. All the computations thatSstmreline ine Sttreminesar are done for regular annular stations also are done

numbredfro oneat he ipfor the extra-annular stations. The only difference is

K los st idexfor ubrutie IPUT that the new streamline locations at that station areK los st idexfor ubrutie IPUT not used for the curve fit for streamline parameters.

When the new curve fit streamlines are established,As indicated in the table at least four annular stations their intersections with the station line are found andare expected upstream of the first blade row and at the streamline parameters at that point are used inleast three downstream of the last blade row. the equation -o f-mot ion calculations.Additional annular stations can be located between The arrays of points that describe the hub and tipblade rows but not within blade rows; that is, not casing contours should extend at least from thebetween the inlet and outlet stations of a given blade furthest upstream calculation station to the furthestrow. downstream one. There should be enough data

Streamline intersections of station lines are points to adequately define the desired casingdetermined by integrating velocity profiles at station contours with a spline curve fit.lines to the specified mass flow fractions. Streamline The input boundary layer blockage factors have an

Ifslope and curvature are determined from streamwise option. A displacement thickness from the wall can

be specified instead of blockage as a fraction of whereannulus height. This is done by using a negativenumber the magnitude of which is the value ofdisplacement thickness. R= r , -r

A total pressure profile can be input in place of rt-rhlosses. Although the way to activate this option hasbeen discussed earlier, its full effects need to be or the fraction of passage height at the blade rowunderstood. This option is activated for a particular exit. Because these coefficients are stored into theblade row by using zero or a negative number in locations of loss sets 4 and 5, those loss sets areILOSS (IROW). When the option is activated, an destroyed for the run even if read in.additional data card is required for that blade row When the pressure level is specified instead of(fig. 12(a)). The first parameter PTT(IROW), or P, losses for the last blade row of the compressor, therein the equation, is the blade row tip (larger radius) is an overspecification of data because the inlettotal pressure in psia. The five other parameters are pressure and compressor pressure ratio are input too.polynomial constants PI to P 5 ; therefore a total In computation the pressure ratio predominates; sopressure at some other radial location is the pressure levels will be adjusted as necessary. Also

note that when the pressure level is input, the totalP=P,(1.O + PI R + P 2 R 2

+P 3 R 3 +P 4 R4 +P 5 R 5) temperature profile must also be input (table 11).

MAIN

IN PUT START DA

WaI Subroutines used in input and iteration phases.

#. Figure 9. - Line representation ot subroutine calls.

2

L ~~ ~IPT CS.TN_ DER.i, ,, . .. . .I.,"...... .,> .. ....- ,-

STEA LOSS SBE-"T A

fbi Subroutines used for terminal calculations.

Figure L . - Concluded.

At a rotor exit the total temperature level can be for rotor exit total temperature distribution wheninput in place of the cumulative energy addition IPRA(IROW)I->i100.0. The polynomial coefficientfraction. If the input CRENGY (IROTOR) is greater represented by PRA(IROW) is found by adding orthan 2.0, the value is interpeted to be the rotor exit subtracting the number of 100's needed to give atip temperature in degrees Rankine. In the remainder in the range - 100.0 to 100.0.preexecution phase of computation the temperature When the total temperature level is input, the totalis converted and used as an appropriate energy pressure level can be set in two ways. It can either beaddition value. The polynomial coefficients for the determined from losses or input directly by aradial distribution of total temperature are input in polynomial, as discussed earlier in this section.the former pressure polynomialicoefficient locations, The description of parameter variations withPARA(IROW). ..PRE(IROW). During regular polynomials assures smoothness, but the

~iteration the program will use the polynomial form specification of polynomial coefficients is not alwaysTA3

LC

L ~ ~ ~ I SnP-N TE- A NC MRAITR

easy. In most cases the range of applicability for the generated by this input for several fractions ofpolynomial independent variable is 0 to 1.0. This annulus height is very helpful in avoiding obviouslyconsiderably eases the burden on the user since unacceptable configurations.computation is normally not needed to choose and The computer peripheral equipment also can beset the polynomial coefficients. When the higher used by some other subroutines when options aredegree terms are used to define distributions, the end activated with the parameter OPO. When the punchconditions are relatively easy to meet. However, option is activated, the tables of fabricationsome simple computations are needed to check the coordinates shown on the listing are punched ondistribution. cards in subroutine COORD. When the plot option

Another caution is that combinations of of OPO is activated, subroutine BLUEPT plotsreasonable-looking numbers often give blade tables of fabrication coordinates on a blueprintelements that one can judge to be poor by visual format. If a plot option is activated by either OPM orobservation. The capability to make machine graphic OPO, subroutine MERID is also called. It produces aplots of blade elements and the channel formed by meridional plane plot of the annulus flow path withadjacent blades is very useful. Such plots are made in the calculation stations and streamlines included.subroutine EPLOT, which is activated by the input This code is interfaced with three other NASAparameter OPM. Since graphics packages differ with codes through punched card output. Input for thecomputer systems, the program presented will not TSONIC code (ref. 5), which is a blade-to-bladenecessarily work directly on a user's computer. channel flow analysis code, is obtained with the THowever, it is suggested that the user make the option of OPM. Input for the MERIDL code (ref. 6),conversions necessary to plot the blade element which is a more detailed hub-to-shroud flow analysissurface arrays generated in EPLOT. code within a blade row, is obtained with the M

The determination of acceptable polynomial option of OPO. Input for an off-design performancecoefficients for the centerline and thickness of an prediction code that is being developed at NASAentire blade row can be difficult when high-degree Lewis is obtained with the 0 option of OPM.terms are used. This task was eased considerably at The computer program can be obtained fromNASA Lewis with an interactive graphics capability. COSMIC, 112 Barrow Hall, University of Georgia,A series of computer programs were developed to 30601. The COSMIC program number isdesign particular blade elements from actual LEW-13505.centerline angle and thickness distributions. Thesedata were then curve fit by least-squares methods to Lewis Research Centerproduce the input required by the program described National Aeronautics and Space Administrationin this report. Visual observaton of blade elements Cleveland, Ohio, December 29, 1980

14

Appendix A

Symbols

A annulus area; also streamtube channel area U local blade velocity, ft/secA i polynomial constants for as a function of S u generalized variable in a differentiala sonic velocity, ft/sec; also a coefficient in equation

velocity gradient equation; also a V velocity, ft/secpolynomial coefficient v generalized variable in a differential

b coefficient in velocity gradient equation; equationalso a polyr. -nial coefficient w weight flow, lb/sec

C constant z axial distance, in.Ci polynomial constants for conic radius as a ai angle of streamline with reference to axial

function of S direction, degc blade chord, in.; also a polynomial j flow angle relative to meridional direction,

coefficient degcP(t) specific heat function for constant 'Y blade chord angle, deg

pressure, ft/sec2 'R 6 deviation angle, degD blade element diffusion factor E angular coordinate on blade element layoutDi,= 1.* simplified nomenclature, D i = -(Ci)/(i)R t cone, rad

d polynomial coefficient 0 circumferential direction, radf friction force, ft/sec 2 K blade angle relative to local conic ray, degH stagnation enthalpy, ft/sec2 X local angle of calculation station line withHp pressure-surface height, in. reference to radial direction, degH s suction-surface height, in. p static density, slug/ft 3

h static enthalpy, ft/sec2 a blade element solidity, chord/tangentiali integer index; also incidence angle, deg spacing

j integer index r time, sec

k curvature in curvilinear coordinate system, C loss coefficientft - I; also an integer index Subscripts:

L distance along chord line, in. ca center of areaI distance along calculation station line, in. I calculation station indexM Mach number i ideal value, as by an isentropic processm streamline direction in meridional plane, j streamline index

in.; also an integer indexn streamline normal direction in meridional le leading edge

plane, in. m streamline direction in meridional plane;

P stagnation pressure, lb/ft2 also maximum thickness

p static pressure, lb/ft2 n streamline normal direction in meridionalplane

R conic coordinate radius, in. o initial valueRi~i=l,o, series coefficients for polynomial, sp stacking point

R t /R= I +RI S+R 2S2 +R 3

3 + t transition pointRm radius of curvature in meridional plane, ftMte trailing edge(R gas constant, ft lb/slug *R

0 circumferential directionr radius from axis of rotation, in. I blade row inletS blade element path distance, in. 2 blade row outlets entropy, ft/sec 2 *RT stagnation temperature, "R Superscript:

I static temperature, *R; also blade element ( )' relative to rotorthickness, in. ( ) flow at sonic condition (M'= 1.0)

15

Appendix B

Input Parameters for Compressor Design Program

The input variables for the compressor design program and the associated options are described in thisappendix. The format for the input data is given in figures 10 to 12. The calculation station and blade row datasets are input in the order in which they occur in the compressor flow. If any of the sets of option cards forblade rows are needed, they are considered part of the blade row set and they follow the particular basic bladerow data set in the order shown in figure 12. The only exception is any XCUT cards that are read in the outputroutines. These cards are at the end of the input data, but of course the sets of XCUT values must be placed inthe same order as the stations specifying them.

In the following list of parameters the independent variable S appears frequently. Since it is an importantblade element definition variable, this preliminary explanation of its definition and usage is given. The variableSin equations for the blade element centerline is the distance in either direction from the transition point as a

reference. The variable S in equations for the thickness distribution is the distance in either direction from themaximum thickness point as a reference. All four of these usages of S are shown in figure 4. In all cases, Svalues are positive away from their reference point. The S values for thickness definition are normalized byblade element chord. The S values for centerline definition are also normalized by blade element chord whenIDEF(IROW) is less than zero; however, when IDEF(IROW) is greater than zero, S is normalized to 1.0; thatis, the maximum segment S is 1.0.

NS r Nt AF 11 1.( N Y I' P N 11F If -I IF1 0 1F F t IF II IF

C ,C' 1FF ( I I C F1 ( 2 ,CF1(:

I I I, Ft1 A ( 1 F F Ft Ft A 12 1 10 it t A 1 3 1 1OfF IAN1 Fti"f-

r I I -(I IF1 2 T F 1 1 NIF N IFF

I, I' If I % T H I f I II 1 t Nm

it T I F' I I I it I F PF f 1 1 1 1i I P 1 o1 1

X~F Itt 1 31 IIFi12F t 1It I T f NF11111 N Il1l1

'fill I I+I I III I t tI It 1 1\ f 1 1 FF NF 1 FF 1

(2,L% 1) 0 (E . 2 2) 1) r-T FF (1.F 1)F ' ]- 'OF 12F 21, 21F -.- A 5.21

F ~ - 7FF tFF IF F

Figure 10. -input data format of general information.

16

A JA z I , A l ,

1 ,

1 ,III I % NA I EI Ii I I I ,,E l), I

(al Annular stations.

DL I z ; V IM1 IIIWOl-r-'-"-'----.J J------)-[-B-EE - I EEN( IRl)H) LM IE) 1)

I L ) 1, ) 1 l. .- I I t .I ( I I T F

Wc Statior os.

A ___ ZT 1 1'1 1NA 1 . 1 ItL)Nf l J l-) I N AI

IlL! ~ ~ ~ ~ ~ t it I (W I.51 111) l II Iif I I HE 1E)5f) 13 PSI

DLAD M IIROW)A) 0LA M())R~l IIT EIPE I ROW~l 13'T,( I HOl IEI I I H LE E 1) ()H I I I I lw,

(c) Stationary blade rows.

Figure 11. - Input data format of calculation stations and basic blade row information.

Parameter Description Format

AA This parameter is used twice to indicate options in alphanumeric form. As the A4first term of a data set it indicates the type of calculation station or blade row(ANNULAR, ROTOR, or STATOR). Any station description other thanANNU, ROTO, or STAT will be treated as an extra-annular station, that is, thestreamlines will not be forced to pass through the streamtube-fraction-ofweight-flow point as determined by continuity at the station. The second use ofAA later in the data set is the incidence angle option for blade design purposes.Interpretable options are 2-D, 3-D, SUCTION, and TABLE. Anoninterpretable incidence option word is set to the 2-D option. The 2-D and3-D options mean incidence angles are determined by procedures in reference Ifor the respective option. The suction option gives zero incidence to the suctionsurface of the blade at the leading edge. The TABLE option means the bladeincidence angles for the blade element will be input in tabular form,INC(IROW,J), at the end of the data set.

AB This parameter completes the incidence TABLE option discussed above. To A4reference incidence to the suction surface at the leading edge, the eight spaces of

the card for AA and AB must read

TABLE SS

AA AB

(If AB is anything other than E SS, the incidence angles will be referenced to theleading-edge centerline.)

ACF(I,IROW), polynomial coefficients for linear coefficient of blade element centerline angle FO.4

ACF(2,IROW), equation for front segment, K=K +aS+bS 2 +cS 3 +dS 4 with

ACF(3,IROW), a= ACFI + ACF2.R + ACF3°R2 + ACF4*R, where R =(r, -r)/(rt -rh)-ACF(4,IROW) fraction of passage height at blade leading edge

ACR(I,IROW), same as above for rear segment with same R FIO.4

ACR(2,IROW),ACR(3,IROW),ACR(4,IROW)

17

• 0\ bi .

fI

' , 1.. .. 1 it I, , r -" ti" ...... 1 c I it. o " .... I -C 4 j ' I"t' (M' , , F. .. .. . .

(a) If ILOSS(IROW) : 0.

II N 1 ,11 I, I I I It11 Ic 1. 11 1 it 1%, 1.1 r 1)1 1 1 I A U t . 1 4 I .. H1( II 1 41 It 0 kk11 C I'I I It k11 T 1 T il . I I 1

l. I x I , I )11 I Mx x 1 OIi % r, T x, I it%1-1 N. M X, II 0 1 CH0 I t ) Al I It 1(1(11o1 fit I II 11 II it('

(b) If OP is DESIGN, COORD, PUNCH, or ALL.

( F I 1 It O1 I A 1 I I2. 0 %k I I ( I t O , I C I . 1 1 1 I.(I4. ( ,O\ I 1.l I t I I It ( I, lIItO\2 . I C" I.: 11 , 1 11 1 %k 11 (1 (4, I 0 1 %1

C," f 1, 1 H 0 1it \ it 0 %k .O \

. . VL I it , I l\1 t R I , 1ito \k 1 1) C 1-( 1,. It ON\ 1 1)(C I-. O. I i )% )11 ( 1 , I kk. 1 ) B ( F (4, 1 It 1) L k I

A It, I I 1 :(1 It 12 t1111 13, 1 ;l It 11) % C it 1(4, It I I t1 1 1 It 0OXl IB C11 2., It 0 %k ll ( it11 1. I it 0 (1 I it (4, I PIt01c it, I . I 1M% 1' 1CIt 2. It(11\ l (1., 11( ' It(4, 0I %N,\) I)1' I1). I It [) ' , N 0C(R D (3'(. 1 I(0\%1 DT I 14, IR O101

....I.IDEC'RO I>40

01 " 1 I I I lOl, 1 1 2 1 It 0 1 AT 1 :1, 1 i(0 \%I V LL.' 14 R1 O\k1 IF T , I11 1 I " I I 1 1 01 , o f(f :, I it1( E I t 0

I'll i 111 , 1\1 IC' r ft I It 01,1 IC I 1F 1 1,11 C T 14 , I -%%-)1 -r 1t \\ 1) 'I" F I I t ( l. [-llI:, IIN 1O T I Ii t

r' ( i 1 , I lit o % I " 11, I i I I .II iA (I 01\1 T1, I , I R1 I1\\I l l f l l Ii11 \ r II i ' J . J 0Ii B .I l ( 4,i 1 it

it 11 l it " I 1 1 0 , 7 M I i1 T ( .1 , M. L I I (4,., 1t %V ) D'-Ii 11, 1 It 10%% I (l ,riI I It 0 II .I 1 \ I .

c) If IDEF(ROW) > 0.

rD F V( it o tk,)I)F . v k ,t % ,70 iO \ 2 [ ' 1) 1. 'I, 1" II1ol\\, 4 1 1) 1 \ I I\\N:"]'If. if l t - T %Bl~f'2

7, M a '<( RJO \%,I 1 1 Xi t0 ,2 IjZ II A X I( I If (, 't .. Z Mt A X IRIO\\,41 \[Vv r B F

d) If indicated parameters are TABLE.

,%- r 0 1 1 1 U T 1 2 N TV T HI - 1 I ' V T1 (- , 1 1 1O0 ( I I ,' P I I 3 I I I - 1- 4 O I

T IfI 1 1, 11) IV TIf ( I , 2 1 \'TIf I , .1 TtI ( 1 4 1 J, TH I , I Pq(I ,1,1 PO(I 1 2 '' 1,O l , 3 PO(I . 4 E

(e) If OP is VEL. DIA.

X I I X C 'U T 2 X VV 1 3 1 X U (N I I C I T .,I, X C 1 6 V T ' 1 I T I I

.. .i .... • ...... ..

f) If NXCUTCIROW) < 0.

Figure 12. - Input data format of additional blade row information if needed by the options.


ALIM(IROW) For a data set designated ROTOR, ALIM(IROW) is the minimum allowablerelative flow angle (deg) leaving the rotor hub. For a data set designatedSTATOR, ALIM(IROW) is the maximum Mach number entering the stator atthe hub. The program will reduce the stage energy addition to satisfy theseconditions if a limit criterion has been reached during computation. If noaerodynamic limits have been reached in some other stages of a multistagecompressor, the program will try to pick up the energy loss of the limiting stagein the stages free of aerodynamic limits. If all stages have reached someaerodynamic limit, the overall compressor pressure ratio is degraded to get allstages within the specified aerodynamic limits. The most efficient way to run theprogram is to specify the stage energy addition levels so than aerodynamic limits

Pare not reached or at least not reached in a drastic fashion.

18

-zi


ATF(I,IROW), polynomial coefficients for first coefficient a of blade element thickness equation FIO.4ATF(2,IROW), forward of maximum thickness pointATF(3,IROW), t tin _ S ) -cS 3 -dS 4

ATF(4,IROW) 2c 2c +S, 2v- /bS2-/S-dS

with a=ATFI+ATF2*R+ATF3*R 2 +ATF4,R 3 , where R is fraction ofpassage height at blade leading edge and S, is distance from maximumthickness point to centerline intersection of edge ellipse (fig. 4)

ATR(I,IROW), same as above for rearward thickness with same R FlO.4ATR(2,IROW),ATR(3,IROW),ATR(4,IROW)

BB deviation angle option for blade design purposes. Interpretable options are 2-D, A43-D, TABLE, CARTER, and MODIFY. Noninterpretable input is set to the2-D option. For the 2-D and 3-D options, deviation angles are determined byprocedures of reference 1 for the corresponding option. The CARTER andMODIFY options are now the same in the program. They indicate the use of aCarter's rule with a modification when the front and rear segments of a bladeelement have different camber rates. The TABLE option means that the bladedeviation angles for the blade elements will be input in tabular form,DEV(IROW,J), at the end of the data set.

BCF(I,IROW), polynomial coefficients for quadratic coefficient of blade element centerline FIO.4BCF(2,IROW), angle equation for front segment, K= K, +aS+bS2 +cS 3 +dS 4

BCF(3,IROW), with b = BCFI + BCF2*R + BCF3°R 2 + BCF4*R 3 , whereBCF(4,IROW) R = (r, - r)/(r, - rh)-fraction of passage height at blade leading edge

BCR(I,IROW), same as above for rear segment with same R F10.4BCR(2,IROW),BCR(3,IROW),BCR(4,IROW)

BH(l) hub blockage factor for each calculation station; fraction of the station annular FIO.4area to be allowed for hub annular surface boundary layer blockage. The hubstreamline will be displaced away from the physical wall a distance that gives thespecified annular fraction. Negative input values are used as the magnitude ofboundary layer displacement in inches.

BMATL(IROTOR) rotor material density (lb/in 3). If a positive nonzero number is input, the blade FIO.4will be stacked so as to balance out gas bending moments with the centrifugalforce moment for the material density. Because the hub stacking point staysfixed, the tip location is moved if necessary.

BLADES(IROW) number of blades in each rotor or stator blade row FIO.4

BLEED(I) fraction of weight flow bled off at particular calculation station FI0.4

BT(l) same as BH(l) except applicable at tip Fi0.4

19

" •. "- . . . . -' "-. " *i~


BTF(I,IROW) polynomial coefficients for quadratic coefficient of blade element thickness FIO.4equation forward of maximum thickness point

t -tm t S+a V.S;--S-vS o + - -bS 2 -cS3 -dS4

2-c 2c 2VS 0, )/with b=BTFI+BTF2*R+BTF3*R 2 +BTF4*R 3 , where R is fraction ofpassage height at blade leading edge.

BTR(I,IROW), same as above for rearward thickness with same R Fl0.4BTR(2,IROW),BTR(3,IROW),BTR(4,IROW)

CC blade element geometry option for blade design purposes. Interpretable options A4are CIRCULAR, OPTIMUM, and TABLE. The CIRCULAR option givescircular arc blade elements. Noninterpretable input will be set to theCIRCULAR option. The OPTIMUM option means that the ratio of bladeelement segment turning rates will be set by an empirical function of inletrelative Mach number. Below an Mi of 0.8 the blade element will be a circulararc. As M i is increased, the ratio of front segment turning rate to rear segmentturning rate is reduced. A limit of zero camber on the suction surface of thefront segment is approached at an Mi of about 1.60. The TABLE option meansthe ratio of blade segment turning rates will be input in tabular form,PHI(IROW,J), at the end of the data set.

CCF(I,IROW), polynomial coefficients for cubic coefficient of blade element centerline angle FIO.4CCF(2,IROW), equation for front segment, K = K, +aS+bS2 +cS 3 4- dS4

CCF(3,IROW), with c = CCFI + CCF2*R + CCF3*R 2 + CCF4*R 3, whereCCF(4,IROW) R = (r, - r)/(r, - rh)-fraction of passage height at blade leading edge

CCR(I,IROW), same as above for rear segment with same R FI0.4CCR(2,IROW),CCR(3,IROW),CCR(4,IROW)

CHORDA(IROW), constants to define ratio of blade element chord to tip chord on projected FI0.4CHORDB(IROW), planeCHORDC(IROW) c- = I + R*CHORDA(IROW) + R 2 *CHORDB(IROW)

ctip

+ R 3 *CHORDC(IROW)

where R = (r, -r)/(r -rh)-fraction of annulus height at blade stacking line

CHOKE(IROW) desired minimum value of (A/A *)- 1.0, where A/A* is the ratio of localstreamtube area in the channel to the area required when M' = 1.0 within ablade passage. If zero is input, no adjustment will be attempted within theprogram. For input values greater than zero,incidence angle will be increased asnecessary up to a maximum of + 2.0* on the leading edge of the suction surfacein an attempt to give the specified choke margin at the covered channel entranceif the minimum occurs at the channel inlet.

CPCO(I) constants for specific heat polynomial function of temperature E20.8for I =1,6 C= CPCO(I) + CPCO(2)* T+ CPCO(3)* T2 + CPCO(4)* T3

+ CPCO(5). T4 + CPCO(6) •T5

20


CRENGY desired cumulative energy addition fraction through particular rotor to total FIO.4(IROTOR) energy addition of compressor. Thus the fractions are progressively larger

positive numbers through successive rotors. The last rotor must haveCRENGY = 1.0 to meet the input pressure ratio. If a value greater than 2.0 isinput, the value is interpreted as a rotor exit total temperature level in degreesRankine instead of the cumulative energy addition fraction. In the preexecutionphase of computation the input temperature is converted and used as anappropriate energy addition value.

CTF(I,IROW), polynomial coefficients for cubic coefficient of blade element thickness FIO.4CTF(2,IROW), equation forward of maximum thickness pointCTF(3,IROW), t _tin / S )CTF(4,IROW) 2c 2c +a x/S-s + 2 b S 2

-cS3 -dS4

with c=CTFI+CTF2*R+CTF3*R 2 +CTF4*R 3 , where R is fraction ofpassage height at blade leading edge

CTR(I,IROW), same as above for rearward thickness with same R F0.4CTR(2,IROW),CTR(3,IROW),CTR(4,IROW)

DCF(I,IROW), polynomial coefficients for fourth degree coefficient of blade element centerline FIO.4DCF(2,IROW), angle equation for front segment, K = K, +aS + bS2 + cS3 t dS4

DCF(3,IROW), with d= DCFI + DCF2*R + DCF3*R 2 + DCF4*R 3 , whereDCF(4,IROW) R = (rt -r)/(r t -rh)-fraction of passage height at blade leading edge

DCR(I,IROW), same as above for rear segment with same R F10.4DCR(2,IROW),DCR(3,IROW),DCR(4,IROW)

DD option control of location of transition point between segments of a blade A4element. The interpretable options are CIRCULAR, SHOCK, and TABLE.The SHOCK option locates the transition point on the suction surface at thenormal shock impingement point from the leading edge of the adjacent blade.The TABLE option means the location of the transition point will be input intabular form, TRANS (IROW,J), at the end of the data set. The CIRCULARoption and noninterpretable data put the transition point at midchord.

DEV(IROW,J) deviation angle (deg) that can be specified by option. If the tabular option is FIO.4used, a value is expected for each streamline starting from the tip.

DFTAB(K,J,I) blade element diffusion factor (D factor) for which profile losses are tabulated. F8.4Five values are input for each streamline; that is, K always has values from I to5, J is the streamline index, and I is the loss set index. The maximum number ofsets is 5. Because D-factor values normally fall between 0.3 and 0.7, values of0.3, 0.4, 0.5, 0.6, and 0.7 for DFTAB on a streamline can be implied by leavingthe DFTAB values blank. As a consequence of this option the DFTAB cannotbe exactly 0.0 when K = I if you do not want the implied values of DFTAB.

DLIM(IROW) aerodynamic D-factor limit. In a data set designated ROTOR this limit applies at FIO.4the tip streamline. For a STATOR data set the limit applies at the hub. Theprogram operates with this limit criterion in the same way as it did withALIM(IROW).

21


DLOS(K,J,I) profile loss parameter w cos j3 /2o corresponding to DFTAB(K,J,I) reference F8.4arrays

DTF(1,IROW), polynomial coefficient for fourth coefficient of blade element thickness equation F10.4DTF(2,IROW), forward of maximum thickness pointDTF(3,IROW), t = ti + S )-bS 2DTF(4,1ROW) 2 2c + a 'VS - S - ' + - -b 2 CS 3 - d S 4

2c 2c 2yS 2vS 0 }-with d=DTFI+DTF2°R+DTF3°R 2 +DTF4°R 3 , where R is fraction ofpassage height at blade leading edge

DTR(1,IROW), same as above for rear segment with same R FIO.4DTR(2,IROW),DTR(3,IROW),DTR(4,IROW)

EB EB completes CABLE option of maximum thickness location. If the eight spaces A4controlling the option appear as

TABLE LE

EE EB

the input values of ZMAX(IROW.J) will be used as the fraction of chorddistance from the leading edge. If EB is not as shown, the values ofZMAX(IROW,J) will be used as the fraction of chord distance behind thetransition point.

EE option control of location of maximum thickness point of a blade element. The A4interpretable options are TRAN and TABLE. The TRAN option andnoninterpretable options will set the maximum thickness point at the transitionpoint. The TABLE option means the maximum thickness point location will beinput in tabular form, ZMAX(IROW,J), at the end of the data set.

ELE(I,IROW), coefficients for leading-edge ellipse FlO1.4ELE(2,IROW), ratio of semimajor to semiminor axesELE(3,IROW), minus 1 bELE(4,1ROW) /b

e= - -e = ELEI +ELE2*,R +ELE3*R 2 +ELE4*R 3

awhere R is fraction of passage height at blade leading edge

ETE(I,IROW), coefficients for trailing-edge ellipse ratio of semimajor to semiminor axes minus I*1 ETE(2,IROW), bETE(3,IROW), e=- - I = ETEI + ETE2*R + ETE3*R 2 + ETE4*R 3

ETE(4,IROW) awhere R is fraction of passage height at blade trailing edge

FLOFRA(I) cumulative weight-flow split between streamlines starting from tip. NTUBES, FIO.4which is NSTRM-I, values are read. Thus the first value is greater than zero andsucceeding values must increase to 1.0 in order for the last value to account forthe accumulation of flow for all streamtubes.

FLOW(l) mass flow (ib/sec) entering the first calculation station FIO.4

22


IDEF(IROW) blade definition index. When the index is zero, the blade segment centerline andsurfaces are defined by dK/dS=constant. When the index is not zero, thesegment centerline and thickness are defined with fourth-degree functions ofpath distance from the transition and maximum thickness points, respectively.The specification of the coefficients for these functions is extra input, for whichthe format is shown in figure 12(c). If IDEF(IROW) is positive, the coefficientsfor the definition polynomials are interpreted to be functions of segment lengthnormalized to 1.0; but if IDEF(IROW) is negative, the coefficients areinterpreted to be functions of segment length normalized by chord. Thereference point for the centerline polynomials can be either the transition pointor the segment ends. The possible combinations are shown in the IDEF(IROW)summary in table IV.

ILOSS(IROW) designation of which profile loss set (I variable in DLOS(K,J,I)) to use with 15particular blade row. If the input value of ILOSS(IROW) is less than or equal tozero, a total pressure level is input in place of losses. The pressure is input withthe parameters shown in the first option of figure 12. These parameters arestored into the locations of loss sets 4 and 5; so those loss sets are not availablefor use with any blade row.

INC(IROW,J) incidence angle (deg) that can be input by option. If the tabular option is used, a FI0.4value is expected for each streamline starting from the tip.

MOLE molecular weight of gas (28.97 for dry air) I5

NA number of annular stations at which radial velocity profiles are constructed 15during computation

NBROWS number of blade rows (maximum of 20) 15

NHUB number of points input to describe hub geometric boundary (maximum of 40)

NLOSS number of loss sets input (maximum of 5) 15

NTIP number of points inpit to describe tip geometric boundary (maximum of 40).

NXCUT number of sections across blade for which fabrication coordinates are desired. If 110zero, the program will set the number of XCUT's on the basis of aspect ratio.For all positive values the program will set appropriate locations to represent theblade. Negative values of NXCUT(IROW) trigger an option to read cards forthe XCUT values. The number of values expected for a blade row is the absolutevalue of NXCUT(IROW).

NSTRM number of streamlines (maximum of il) 15

OP option controlling amount of output information desired. Interpretable options A4are APPROX, VEL. DIA., DESIGN, and COORD. If the first four charactersinput in OP match none of the above, the program will try to proceed with theVEL. DIA. option. The program completes only velocity diagram informationwhen run with the APPROX and VEL. DIA. options. With the APPROXoption the locations of blade edges are estimated from the stacking line, butwith the VEL. DIA. option the blade edge locations are input. The blade edgedata are read from extra cards at the end of the data set for a particular bladetype. The axial coordinates are temporarily read into VTH(I,J), and the radialcoordinates are temporarily read into PO(IJ). When run with the DESIGN and4.

23

i. . . .. . . . . . . . ..+ + + , "+,..A X . . . .--+, . . .. .


COORD options, the program designs and stacks that particular blade row.With the DESIGN option only velocity diagram information is printed, but theblade leading- and trailing-edge locations are for the stacked blade. TheCOORD option includes the printout of blade section properties andcoordinates for fabrication.

OPM additional output options in effect if OP is DESIGN or COORD A4

Card column Additional output

7 8 9 10

0 Off-design punchT TSONIC punchM Blade element channel microfilm

IM 0 M and 0 optionsM, TM and T options

OPO Additional output options in effect when OP is COORD A4

Card column Additional output

17 18 19 20

N1 Fabrication coordinate on microfilmP Fabrication coordinate punchC MERIDL punchM P

M and P options%M M and C options

PHI(IROW,J) ratio of inlet segment turning to outlet segment turning (ratio of FIO.4j(dK/dS)1 /I(dK/dS) 2 I) for a blade element. If input values are expected by use ofthe tabular option, the data cards go with the optional cards at the end of thedata set for each blade row. A value is expected for each streamline beginningfrom the tip.

PRA(IROW), coefficients for polynomial equation to define profile behind 1' ~ row. P-,.,.d FI0.4PRB(IROW), a rotor the pressure ratio profile is specified asPRC(IROW),PRD(IROW), - =1.0 + PRA*R + PRB*R 2 + PRC*R3 + PRD*R 4 + PRE*R 5

PRE(IROW) P,where P, is the stagnation pressure at the rotor exit tip and R=(r, -r)/(r, -rh)-a fraction of passage height. When PRA(IROW)I 2 100.0,another option is activated. The input profile is for a temperature profile T/T,instead of a pressure profile PIP,. The data value of PRA(IROW) is extracted

* "from the input value by adding or subtracting 100's until the remainder is in therange of - 100.0 to 100.0. At a stationary blade row the polynomial is for theblade row exit tangential velocity profile in ft/sec.Vo = PRA/R 2 + PRB/R + PRC + PRD*R + PREoR 2 where R =r/rt

PO(I,J) general stagnation pressure array in lb/ft 2 within program. The I index is the FIO.4station index and J is the streamline index. Only (PO(I,J), J = 1, NSTRM)values are input; that is, the streamline value for the first calculation station.The input values are read in units of psia.

When blade edge coordinates are input, some of the other PO(I,J) locations are F8.4

used for temporary storage of the input values of radius.

PR desired overall compressor pressure ratio FIO.44?

24

I L . . . L : . . . " F - "" :- '

. . "r '. .... *i L 2_. , . ,-


PTT(IROW), coefficients that describe blade row exit profile when it is input as an option. FIO.4PTC(I,IROW), PTT is the blade row exit pressure in psia at the tip (highest radius). The otherPTC(2,IROW), five values are polynomial coefficients forPTC(3,IROW),PTC(4,IROW), P= PTT*(1.0 + PTCI *R + PTC2.R 2 + PTC3.R 3 + PTC4.R 4 + PTC5.R 5

PTC(5,IROW) where R = (r, - r)/(r, - rh)-fraction of passage height at blade row exit

RHUB(I) radius coordinates of a set of points that define geometric hub boundary F10.4(maximum of 40)

ROT compressor rotational speed, rpm FIO.4

RTIP(1) radius coordinates of set of points that define geometric tip boundary (maximim FlO.4of 40)

SOLID(IROW) tip solidity of a blade row (ratio of chord to circumferential spacing) FIO.4

TALE(IROW), polynomial coefficients of ratio of blade element leading-edge radius to chord. FIO.4TBLE(IROW), where tie/c =TALE + TBLE'R +TCLE*R 2 + TDLE.R 3

TCLE(IROW), where R (r, -r)/(r, -rh)-fraction of passage height at blade leading edgeTDLE(IROW)

TAMAX(IROW), polynomial coefficients of ratio of blade element maximum thickness to chord, FIO.4TBMAX(IROW), where tm/c=TAMAX +TBMAX*R +TCMAX.R 2 +TDMAX.R 3

TCMAX(IROW),TDMAX(IROW)

TATE(IROW), polynomial coefficients of ratio of blade element trailing-edge radius to chord, F10.4TBTE(IROW), where tte/c-TATE + TBTE*R + TCTE.R 2 + TCTE.R 3

TCTE(IROW), where R (r, -r)/(r t -rh)-fraction of passage height at blade trailing edgeTDTE(IROW)

TILT(IROW) angle of stacking axis tilt (deg) in circumferential direction (r-O plane). The angleis positive in the direction of rotor rotation. If jTlLT(IROW)j > 100.0, a curvedstacking line is specified according to r-rref = C(sin -y- sin -Yef), and the codeof the TILT(IROW) is-

......-- tilt angle at hub in degrees with sign ofoverall TILT(IROW) number

""---tilt angle at tip in degrees. Circled digitcontrols sign of tip tilt angle. Even digitgives tip tilt angle same sign as hub tiltangle. Odd digit gives tip tilt angleopposite sign of hub tilt angle.

For example: 12332.65 gives a hub angle of 23* and a tip angle of -32.65*.

TITLE(I) description of compressor for printout and later identification 18A4

TO(IJ) general stagnation temperature array in program. Only (TO(IJ), J = 1, FIO.4NSTRM) values are input; that is the streamline value for the first calculationstation. The input values are in units of *R.

25

&t


TRANS(IROW,J) location of transition point on blade element centerline as fraction of blade FIO.4element chord. If input values are expected by use of the tabular option, thedata cards go with the optional cards at the end of the data set for each bladerow. A value is expected for each streamline beginning from the tip.

VTH(I,J) general tangential component of velocity array in program. Only (VTH(l,J), FIO.4J=1, NSTRM) values are input; that is, the streamline value for the firstcalculation station. The input values have units of ft/sec.

When blade edge coordinates are input, some of the other VTH(I,J) locations F8.4are used for temporary storage of the axial coordinates of the points.

XCUT(IC) radial location of blade section planes. Whether or not data cards are read for FIO.4values of XCUT(IC) for a blade row is controlled by the value of NXCUT (IC).Any XCUT(IC) cards are read in an output routine. Therefore they must followall cards read in subroutine INPUT; that is, they follow the ANNULAR cardfor the last calculation station. There is no index identifying the data with aparticular blade row, so the data sets for the blade rows are expected in theorder that one would see the blade rows in moving through the compressor fromthe inlet. Start the set of points for each blade row on a new card. It ispreferable, but not necessary, to list the XCUT(iC) for a blade row in orderstarting from the tip.

XHUB(I) axial coordinates of set of points that define geometric hub boundary. The axial F10.4extent of the coordinates must at least reach the first and last calculationstations. The hub coordinates must have the same reference origin as otherinput axial coordinates, that is, casing, blade edge, and stacking linecoordinates. The number of points input should be 4 <n s40.

XTIP(I) axial coordinates of set of points that define geometric tip boundary (See FIO.4XHUB(I) for additional comments.)

ZHUB(I) blade data set hub-axial coordinate. When the data set is a blade rather than an FIO.4ANNULAR station, ZHUB(I) is the axial location of the blade stacking line atthe hub.

ZMAX(IROW,J) location of maximum thickness point as fraction of blade element chord. If input FIO.4values are expected by use of the tabular options, the data cards go with theoptional cards at the end of the data set for each blade row. A value is expectedfor each streamline beginning from the tip with a leading-edge or transition-point reference according to option (see EB). With a transition point referencethe values input are (m-t)/c

t -MAXC -TRAN

ZTIP blade data set tip-axial coordinate. (See ZHUB(I) for similar additional FIO.4comments.)

26

-"

Appendix C

Development of Equations of Motion into Form

Used in Computer Program

In the computer program the equations of motion In terms of equations (C3) and (C6) the termare applied at calculation stations that are presumed V x (V x I) can be expressed asto be outside the blade rows; so the equations ofmotion are more conveniently developed in an fn habsolute, rather than a relative, coordinate system.The general equation of motion (eq. 3(21) of ref. 7) is Vx (V X V) = Vo Vm 0

av- + VH= Vx(V x V)+tVs+f (CI) aVm a(rVO) 0

an r an

When steady flow is assumed and the local friction

force is ignored, equation (CI) reduces to =6a0) + fn[0) + ,[ V, a(rVo )

VH= Vx(V x P)+tVs (C2) r an

In orthogonal curvilinear coordinates the velocity + Va-V n 1 (C7)vector can be expressed as an I

' Vo + fn Vm +)i Vn (C3) Now break equation (C2) into the three componentequations. In the 0 direction

where m is in the streamline direction in themeridional plane and n is in the normal direction in aH asthe meridional plane. Of course V, is zero 1 t- 0 (C8)everywhere for this application. The curl term ingeneral can be expressed as

The zero in equation (C8) recognizes circumferentialsymmetry of s. In the meridional plane streamline

-- O{ a Vn a Vm directionV× X 1= 0- + Vn k n - an + Vmk m

n[a(rV,) V]f aVm a(r Vo) H =tas =09)+ ar m __ (C4) a am

The zero in equation (C9) comes from thewhere km and k, are the curvature of the streamline assumption that entropy does not change along

I and the normal, respectively. All terms containing streamlines that are outside the blade rows. In theV, are zero for this application. The assumption of meridional plane normal directionsymmetric flow in the circumferential directionmakes n Vm / 80 equal to zero. Also, because angular an V m - _V

2 k asthemmblade rowdes not change on streamlines outside = m (CiO)

a(rV) =0 (CS) Equations (C8) to (CIO) apply to the three curvilinearam component directions. However, in the program

Thus equation (C4) reduces to velocity and state values are available along stationlines; so it is of computational convenience to apply a

S ) aV@)component equation along a station line. ToV ( -- !nVmkm + r (C6) accomplish this objective, the derivatives in the

meridional plane are converted from the orthogonal

27

hLI

streamline and normal directions to the generally Thereforenonorthogonal streamline and station line directions.The angle nomenclature for the conversion is shown a(rVq) = d(rVe) Iin figure 13. --- d o~ X C3

The enthalpy gradient in the station line direction On d coa+ )C13can be expressed as

dH d -, aV cos(a + X) + a -sin(at + X)

_ H dO OH dm OH dn Therefore+O dl m dl- O n

OH aV.mdVm oI a VM +X (C 14)IOli- OI+I101 sin( + X) + ~-cos(ci +X) an mlcsc+)---tnaX

dH _ H ds Os asdl- cos(a +X) (CII1) nj cos(a +X) + .sin(ce+ X)

OsIn general a station line derivative can be expressed as cn~x+X Osna+Xd _ Odn a0dm-=- - +--- Therefore

di an Os am dsla cos(a +X) + asin(c + X) (C12) as=- (CIS)On am andcos(a +X)

When equation (C02) is applied to the other normal The application of equations (C12) through (C15)derivatives of equation (CIO), the following relation to (CIO) givesdevelops:

d(r V6 ) O(rVO) O(rV) dH _V 6 d(r V) dV a±tt v ~!!snadl On am(~X) sin(ci+X) -dJl r 7 l dI mm

- rO cos(cv + X) + [Osin(a +X) - T1mosa ) C6

The streamline curvature k, is

ai Ikm (C17)

where Rm is the meridional plane streamline radius

A of curvature. Substituting equation (C17) into (C16)'Station line yields the following form for the meridional velocity

gradient:

StemiedVm = -H ~V - + V, Om sin(ci + X)V~t 'di -dl- r dl aV2 ds

+ m

AxialdiretionThe state properties appearing in equation (CIS8) areAxialdiretionH, t, and s. However, two state properties are

Figure 13. -Angle nomenclature for direction derivatives. sufficient to establish the others at a point. For a

28

thermally perfect gas (p=p{t) it is rather easy to ds 1 dh Icompute other state properties from two selected l =t dl pproperties; so it is desirable from a computer storagestandpoint to store only two properties throughout d P[ 1 T , (t)1tdtjjthe flow field. The two properties selected were .1 "stagnation temperature and pressure. These two × dlproperties, along with the velocity components, are

sufficient information for the calculation of the other ds I l I dP e I T dtstate properties. If these two properties can be used - t dl p d exp [p- Y tdirectly in the equations of motion, the need tocompute some state properties may not exist. Toexpress s in terms of Tand'P, start with the property _ P exp- IT cP(t)dtlrelations p1 11 t t

Ap =d- tds (C 19) - 1 dl/[JT -- d t

For the introduction of stagnation properties note I dh I dP , P d T c(t)d]that the thermodynamic process of moving between --- (-h + P t__the static and stagnation states is isentropic by t dl p t dt tdJ

definition. Thus equation (C19) for this processb e c o m e s I A d1 d P d " [ t ) (p=dh d Pd dt (C22)p

For a calorically nonperfect gas this becomesThe application of Liebnitz's rule to the last term

dp gives- = cp ([]dtP

dp I c.(t) dt-p +1 t + 7) dT cpCt)dt__ _ _ _

T dl dl

The variable (cp(t)/t) is not a direct function of pathP I I (t ) dt distance; it is a function of temperature alone.

InpI = dt Therefore the partial derivative with respect todistance must be zero. Thus the derivative of theintegral can be expressed in terms of gradients at the

exp[ IT C0 Q) dt] (C20) limits so that

d [ T E )] dt c c ) dT _ cpt) dtEquation (C19) used as a derivative with path 1- t I T d i ddistance can be written as

I dH I dh (C23)ds= dh dp T di t dldI t dI pt dl (C21)

Substituting equation (C20) gives Substituting (C23) into (C22) gives

29

L • • . .: _ ,...., .... -

ds I dh (R dP I dH I dh so equation (C26) can be written asdl Tdl P di T di t dl

I Dh-- 5- + VV=O (C27)

ds I dH (R dP = dH I dP (C24) a2

dl T dl P d T d poTdl

Equation (C27) expanded from its vector form is

Equation (C24) is essentially equation (C21)expressed in stagnation state variables. Equation(C24) would turn out to be the same for a calorically 1 (ah V6 ah ah ah)

perfect gas. Substituting equation (C24) into (CI8) a2 \a + 7 + Vm am + VT"gives

I a(r Vm) 1 V+ Ia(r V,)dVm _dH ~ d(rVe) a Vm (X Om +rO +r On-Vm rll m- d - Vordl am+ Vm Vmsin(a + X) r- am r -- r an

Vmdi dl- r dl mm+ Vmkm + Vnk n =0

V2 cos(, + t dH (Rt dP+ T +WT T -T

Outside the blade rows the flow is assumed to beaxisymmetric and steady. Also, because there is no

A rearrangement with all the state property terms velocity component normal to the streamline, thetogether gives equation reduces to

Vm Ohl I O(r Vm)V a + I am( T + Vmkm =0 (C28)V m=( t)dHi_+ (,,_d n d in P Vodrd OIa 2 am- r am

VVm )+ a2 Stagnation enthalpy is defined as+ Vm -5- sin( o + A) + mc sa+X ~+ I

C 9Om m +V m + V (C29)

(C25) 2 2dH Oh 0 Vm 0 Vo

All the terms on the right side of equation (C25) - = Vm + V 0can be computed quite accurately except aVm/ am, Om =m + V- Om omwhich is the gradient of Vm along a streamline in themeridional plane. The distance over which a Vm/ am But because allam = 0 outside the blade rows,changes sign are of the order of the calculationstation spacing so that representative values of Oh a m a V0OaVm/am cannot be obtained from a Vm curve fit - - m -V 0 (C30)along meridional streamlines. A better value of this m m m -mderivative probably can be obtained by means oflocal continuity. From equation 9(12) of reference 7 Outside the blade rows angular momentum isdifferential continuity can be expressed as conserved along streamlines; so

I dp + V. V= 0 (C26) 0 = (rV o) ar V VoPDt am Am +r am

However, Rearrangement gives

I Dp_ I Dh a Vo _ Vo ar _ Vsina (C31)p Dt a2 Dt 0m r am r r

30

Lr

Substituting equation (C31) into (C30) gives Substituting equation (C34) into (C33) gives

ah / ~ V aVm _Vm [ M~o+ ls.am VMm+ !1 sin a (C32) am - M -- +- -- m r -jm - I [q sml

Substituting equation (C32) into (C28) givesdci tan(a + h)1

VV V 2 ) V ar + dasec(a +) Rm + (C35)V, L--+ "Osi n aa +---m

a2 am r I rm

a Vm Calculation of aVm/am by using equation (C35)+ + Vmkm = should give a somewhat more accurate result than a

curve fit or a finite difference computation acrossincrements that span whole blade elements.However, a potential divide-by-zero complication(1_ Vr. +V__ + )1- ina Vmk,=0 has been introduced with the term M2,-I1. In

a 2 \ a a2 r equation (C35) the term in braces in essencerepresents the dAIA term of one-dimensional flowtheory. At a Mach number of 1.0, dAIA is zero,

a[ _v vk which is the throat of a nozzle. For compressor blade- Vr[(MO + 1) - 1- sin o + Vmk, (C33) rows the throat occurs within the blade passages.a m 1 Internal flows adjust around locally choked regions

so that the throughflow Mach number outside theblade only approaches 1. Computation of the

The curvature of the streamline normal k,, which is detailed nature of the flow is not available from onlyac/an, needs to be expressed in terms that can be stations outside the blade row; so a minimum value isevaluated. imposed on the denominator through an empirical

additive term to help stabilize the iterative procedure.

dbi ac act The additive center term iscos(ci + X) + - sin(ci + X)

aci dot I aci sin(a +X) f=0.l - exp! - IOM2 - I8n = dl cos(+ X) am cos(ot + X) Mm I OM

k, a z= d sec( + tan(ci + X) (C34) Its characteristics and effect on the denominator areT W/ c+ R , shown in table V.

i-

......

.-..-

Appendix D

Conic Coordinates of Blade Centerline Path

Local blade angle is defined with respect to the s <-'-t

local conic ray (fig. 14). Let the blade angle vary with T

path distance along the cone according to the Rte R -d -

polynomial R I dR -S'

K=Kt+aS+bS2 +CS 3 +dS 4 (DI),- " t

"

where K, is the blade angle at the transition point Rle _L

between segments in this application. The path Figure14.-Bladeelementcenterlinenomenclature.distance S is with respect to the transition pointreference but always positive in the direction frominlet to outlet.

The conic radial component of the centerline can K 2 K4 K6 K8

be found by integrating the differential equation for cos c= I - - + ! - +that component

When equation (Dl) is substituted, the terms of likedR=cos[dS=cs(xt. +aS+bS2 +cS 3 +dS4 )dS powers of Scan be summed to give in symbolic form

(D2) cos K= 12S+ 3S2+1 3 43+... (D4)

The problem is that a trigonometric function of apolynomial is not readily integratable in closed form.However, the function can be expanded in series R-R --cos ds=[ S+ 2

form and integrated term by term. Of course the 0series is infinite but it is convergent within the range ' $3 $4of our application. In the following presentation + S 3 -T-+11.4enough development is given to show the form of theseries. Upon application in the program a tolerance isused so that no more terms than necessary are When terms of similar coefficients are combined, thecalculated, following form evolves:

cos Kc dS = [cos K, sin(aS) + sin Kt cos(aS)] - sin Kt

( s 3 a 2 S 5 a 4 S 7 a 6 S 9

+ bsinKt -3+ 2- +47

+bost a-4 3! 6 5! 8"'

b2 ( 5a2 S7 a4 S9+ - -c s' - T T 7 4 ! V ..."

b2 ( 6 a3 S8 )+ -sint K(a S-36 8

32

b3(S7 a2 S 9

+ COS Kt (a---

3 ! CS K,~-

CO , 7 +~ S + ad2 2C ad74 So ad 16 C K -T T -2 -4+- -10

+[a(+ aC 2 9[ 4!C 05C2 _ 2 3 6S13

-- + T - -- + -! '-(2 (- -46! !(2) 1433!!5

+ b inSK ac S7+ (2 _ d S + ~ S[_ a2 d2 St3 1

a7 T S14 d ad32 11]

b25 Ia8 )~7! 14i K C !2)-f T+~ <

b2 S9 a~c 2 33

+ O fla i

b 2 s Kd9 a2d S1 a4d S'3 ad2 S14 a6 d SIS-T2in'41- 2 11 + 4! 13 2 14 6! 15 +

b2 C S 0 a3d S12 d2 S13 a5d S1 4 a2 d2 S15

2 st\a-] 3! 12 + 2 3- + 5! 14 2 (2)15 +

b3 K S

1 1 a2 d S 13 a

4 d S15 ad

2 S'6 1

+-!CosK' d--+ -- !-T- -i- +39. s~ \d 1- 2 13 + 4! 15 2 16 + "'

b3 , / S12 3d S14 d2 S1 5

+ sin c,(-ad - + 4 T 15 +

b4 / S

13 a2 S

15

+ T- sin Kt-d-iT- + -- d T--f .••

b4 t" S14 )CO Kt os t -ad--4- +..

b5 { S15 )+ T! COS Kt -d-Ti- +..

+ i , S 4 a 2 S 6 a 4 S 8 ++csn - T-4+ T w 4 ! T '

! ( S a 3 S a5 S 9 +

+-ad+

C2 S7 a2 S9+COSKt( -d + -

+

c2 S 6 a+C sinK, a +...

( S8- a2d s S7 a4d S J2 ad,2 S 3

2 2 13

+CCOSK-dj +r- F + T - + ~- -- +

"csiK(ads 9 a 3d S l d2 S' 2 a5 dS1 3 )si a - 3! +2 - - 5! 13 "

"C2 si ,( SII a2 dS13 +

34

C2 // S12+ f COS t (ad _ - . .

( S 5 a2 S 7 a4 S 9

+dsinKt- + 9 . .'

S6 a3 S 8

+dCOSKt -a- + -+...

+T-COS Kt - +.

iS11 a 1 ab S14 [a4 b0a I lS1+abcdcosKt {11 3! 3 2(2) 1 b.ac]SIS

+ [ akb be I S16 a+ a3c +a2b2 c2 S5173

+4!(2) - 6- + 4-..(2) + 3!)2 . T7

abcsin~ - T2 12 2 13 4! 2/ 14 3!(2) 15

+[ a5 ab2 a2C iS16 [_ a4b +abc + b3 1'-(3!) ++3-!(2) -i-6+ L +, -2 8 4! 7

d2 si S15 a2 S17 .+abc -sint -,1 + .1

+ OS t d2 15 3t'a S 17

"1

With these groupings shown, patterns of terms and S SMcoefficients can be observed. The whole equation was + [ +... + [ ]n -coded into three rather brief subroutines-one forterms with two coefficients, COEFI (two of the four Because in the following developments thesecoefficients a, b, c, and d); another for terms with coefficients appear frequently within parentheses,three coefficients, COEF2; and one for terms with all for simplicity the [ I's are replaced with c's; that is,four coefficients, COEF3. Finally the coefficients ofthe terms with the same powers of S are summed; so S2 S3 S4the [ I terms are known in R R, +C1S+C2 +C3 +C4 cn -

S2 S3 nR=Rr+I l1S+( 12 - +[ 13 (D6)

35

-. /

The conic angular coordinate can be expressed as The conversion from equation (D6) to (D8) beginsas

s dS (D7)

where both sin K and R can be expressed as infinite, R Rbut convergent for our purposes, polynomials of S. -R,+ Sc(S/)cS/3..Since a polynomial in the denominator is anundesirable form to integrate, the polynomial for Rwas converted to a polynomial in the numerator of-the form shown in equation (D8). 1 + (c I /R t )S + (C2/R)S 2

+ (c 3 /R t)S3

+.

_ sin K--- d S =

I -DIS-D2 $S

2 -D3S 3

-..

= R- -R sin K dS where

where

_CI 2 C

Rt 1 +RIS+R2S2 +R3S 3+.. (1) DI D 2 D 3 etc.R, 2R .. 3R,

R1 R 2 R3 R4

I +DS+(D2 +D2)S 2 +(D 3 +2DID 2 +D-)S 3 +(D4 +2DID 3 +D2 +3D1 D2 +D4)S4

I-DIS-D 2S2 -D 3S 3 -.. I

I -DIS-D 2 S2 -D3S 3 -D4S 4

DIS+D 2 S2 +D 3S3 +D4S 4

D1 S-D 2S2 -DD 2 S3 -D tD 3S

4

(D2 + Df)S2 + (D3 + D I D 2)S3 + (D4 + DI DOS4

(D2 + Dq)S2 D, D(D 2 + Df)S3 - D2(D2 + qS

(D 3 + 2DID 2 +D3)S 3 +(D 4 2+D4D3 +D+D 2 D )S

(D3 +2DID 2 +D3)S 3 -D I (D 3 + 2D3 + 2DI D 2 + D3)S4

(D4 + 2DI D 3 + D 2 + 3DjD 2 + D4)S4

~ 4

36

bt

Table VI summarizes the preceding division. S IS R sThe coefficients for equation (D8) are generated in ,= Rio R sin K

subroutine RCOEF. The coding for the procedure issomewhat complex, but in general not much l scomputation is required to satisfy a tolerance -t o(I+RIS+R 2 S2 +R 3 S

3 +...)

criterion of 1.OE-08.The conversion of sin P, where x(A +A 2 S+A 3 S2 +A 4 S 3 +.

K= K t -raS+bS2 +cS3 +dS

4

to the polynomial form I A, +(A 2 +RA 1 )S

sin K=A 1 +A 2 S+A 3S 2 +A 4S 3 +A 5S4... (D9) +(A 3 +R1 A 2 +R 2A 1 )S 2

is accomplished in the same way as it was for thecosine series (eqs. (DI) to (D5)). In fact, the cosine +(A 4 +R 1A 3 +R 2A 2 +R 3AI)S 3 +.

series can be converted to the sine series with thefollowing substitutions:

I -A I{S+A2+RIAI S2

Cosine series Sine series

+A 3 +R 1A2+R2AI S3

-sin Kt COS Kt 3-COS , --sin K t A 4 +RIA 3 +R 2 A 2 +R 3 A I S 1sin K1 -COS Kr +4cos Kt sin Kt 4

The general routine for establishing the

Consequently the same routines that are used to polynomial coefficients for the conic coordinates iscompute the cosine series can easily be modified to EPSL2. The end result is constant polynomialcompute the sine series coefficients also. coefficients for the conic coordinates (R and E) as a

When the polynomial series coefficients in function of S. These coefficients are saved so that the

equations (D8) and (D9) are known, the integration conic coordinate at any S of interest can be computedfor E is straightforward. easily with subroutine CONE.

37

4 -..

References

1. Johnsen, Irving A.; and Bullock, Robert O. eds.: Aerodynamic (GE-R66FPD321-PT-I, General Electric Co.; NASADesign of Axial-Flow Compressors. NASA SP-36, 1965. Contract NAS3-7617.) NASA CR-54581, 1967.

2. Crouse, James E.: Computer Program for Definition of 5. Katsanis, Theodore: FORTRAN Program for CalculatingTransonic Axial-Flow Compressor Blade Rows. NASA TN Transonic Velocities on a Blade-to-Blade Stream Surface of aD-7345, 1974. Turbomachine. NASA TN D-5427, 1969.

3. Schwenk, Francis C.; Lewis, George W.; and Hartmann, 6. Katsanis, Theodore; and McNally, William D.: RevisedMelvin J.: A Preliminary Analysis of the Magnitude of FORTRAN Program for Calculating Velocities andShock Losses in Transonic Compressors. NACA RM Streamlines on Hub-Shroud Midchannel Stream Surface onE57A30, 1957. an Axial-, Radial-, or Mixed-Flow Turbomachine or Annular

4. Seyler, D.R.; and Smith, L. H., Jr.: Single State Experimental Duct. -User's Manual. NASA TN D-8430, 1977.

Evaluation of High Mach Number Compressor Rotor 7. Vavra, Michael H.: Aero-Thermodynamics and Flow in Turbo-Blading. Part I-Design of Rotor Blading. machines. John Wiley & Sons, Inc., 1960.

TABLE I. - OVERVIEW OF COMPUTER PROGRAM

Program control

Input and initialization Iteration Terminal calculations

Read and interpret data Outer loop: Overall blade row performanceAt calculation stations on streamlines at calculation

Set coefficients of equation station:

At each station for each streamline, of motion General

estimate stagnation temperature t blade design option, set State properties (temper-

and pressure and axial and tan- incidence and deviation ature and pressure)

gential velocities angles, compute new blade Velocity diagramsedge location, and reset Streamline information

calculation station location Blade rows

Element definition param -Inner loop:etr

At each calculation station ee a t

Solve for meridional veloc- nges

ity distribution to satisfy Aerodynamic perforance

equations of motion and Aram per

contiuitvparameterscontinuity Streamline choke margin

Reset streamline location

Blade section parameters:

Surface coordinates

Area, moments, etc.

* 38

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CAO-A09 8 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION CLEVEL-ETC FIG 21/5COMPUTER PROGRAM FOR AERODYNAMIC AND RLADING DESIGN OF MULTISTA--ETC(U)DEC 81 J E CROUSE- W T GORRELL

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I e t o2 Government Accession No 3 Reipent's (tlog N,]NASA TP-1946 AVRADCOM~lTR 80-C-21],-l\ ij '

4 Title and Subttle 5 Report DJt

December 1981COMPUTER PROGRAM FOR AERODYNAMIC AND BLADING

6 PeCorring Orqani/ation QOJu

DESIGN OF MULTISTAGE AXIAL-FLOW COMPRESSORS 505-32-2A

7 Author(s) 8. Performingy Orgar at~or Rrp'j Ir

James E. Crouse and William T. Gorrell _-810 Work Uwt N ,

9 Performing Organization Name and Address

NASA Lewis Research Centerand

Propulsion LaboratoryAVRADCOM Research and Technology LaboratoriesCleveland, OH 44135 13. Type of Report and Pood Coered

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration Technical Paper

Washington, DC 20546 14, Sponsoring Agency Code

andU.S. Army Aviation Research and Development CommandSt. Louis. MO 63166

15. Supplementary Notes

James E. Crouse: Lewis Research Center.

William T. Gorrell: Propulsion Laboratory, AVRADCOM Research and Technology Laboratories.

16, Abstract

A code for computing the aerodynamic design of a multistage axial-flow compressor and, if

desired, the associated blading geometry input for internal flow analysis codes is presented.

Compressible flow, which is assumed to be steady and axisymmetric, is the basis for a two-

dimensional solution in the meridional plane with viscous effects modeled by pressure loss

coefficients and boundary layer blockage. The radial equation of motion and the continuity

equation are solved with the streamline curvature method on calculation stations outside the

blade rows. The annulus profile, mass flow, pressure ratio, and rotative speed are input.

A number of other input parameters specify and control the blade row aerodynamics and

geometry. In particular, blade element centerlines and thicknesses can be specified with

fourth-degree polynomials for two segments. The output includes a detailed aerodynamic

solution and, if desired, blading coordinates that can be used for internal flow analysis codes.

7 Key Words (Suggested by Avthor(sl I 1H [),strltsitrr Staterr t

Compressor design Unclassified - unlimited

Axial-flow compressor STAR Category 07Multistage compressor

19 Sei>,rty Classrf (of this reporl 20 Se(rtv C-0 "A tt,- p., 1 . Pr" 2? P' "

Unclassified Unclassifie 101 A06

i f , ' i't, hV tli,, N itr;mir ' ]T, ',, I '1 ' ,i ,' , ; i :

NATIONAL AERONAUTICS AND SPACE …meridional (r-z) plane for stations that are lines into two...

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