A Study on Design Criteria and Matching of Turbomachines ...

0. E. BALJE Engineering Consultant,

Hollywood, Calif. Mem. A S M E

Study on Design Criteria and Matching of

Fart A—Similarity Relations and Desk

Only four parameters are needed to describe the characteristics of turbomachines com-pletely. This concept is used to present maximum obtainable efficiencies and the optimum design geometry of turbines as function of specific speed, specific diameter, Mach number, and Reynolds number, based on the state-of-the-art knowledge. Addi-tionally, other aspects such as unit weight and rotor stresses are discussed. Some of the information is preliminary because of limited information on loss relations for tur-bines, particularly at low Reynolds number and high Mach numbers.

Introduction E V E R A L characteristic values are commonly used

for defining significant performance criteria of turbomachines, such as turbine velocity ratio u/co, pressure coefficient qad, flow factor ip, specific speed Ars, Reynolds number Re, Mach number M, and so on. Each of these parameters represents important aspects and is used to express, in a dimensionless form, significant criteria. Similarity considerations [l]1 show that actually only four parameters are needed to describe completely the charac-teristics of turbomachines handling compressible medii; namely, the Mach number, the Reynolds number, and two characteristic velocity ratios. These velocity ratios can either be the flow factor and turbine-velocity ratio or equivalent values. Practical con-siderations, dealing with the design aspect of turbomachines, would indicate that parameters which contain the rotative speed and rotor diameter would be desirable terms for the equivalent values. Such values can be provided bj' the similarity concept in

1 Numbers in brackets designate References at end of paper. Contributed by the Gas Turbine Power Division and presented

at the Winter Annual Meeting, New York , N. Y . , November 2 7 - D e -e e m b e r 2 , 1 9 6 0 , o f T H E A M E R I C A N S O C I E T Y OF M E C H A N I C A L E N -GINEERS. Manuscript received at A S M E Headquarters, August 29, 1900. Paper N o . 60—WA-230 .

the form of specific speed N, and its correlate, specific diameter D, [2],

An interesting aspect of the specific diameter Ds is that this value represents a critical dimension of the turbomachine, the rotor diameter; i.e., introduces a geometry-value into the similarity concept. This is a slight departure from the "classical" concept which uses velocity ratios (such as M/C0 and if>*) as parame-ters. The fact that now the geometry of the turbomachine is directly represented in the similarity concept is a distinct ad-vantage since it offers the opportunity to recognize relations for the optimum channel geometry rather readily, by expressing op-timum geometrical values in terms of Ds. This concept is used to present the available information on turbine-performance data in convenient design diagrams which show the maximum ob-tainable efficiency together with optimum design geometry.

Such an undertaking is necessarily presumptuous since the knowledge of the interrelation between geometry and losses is still incomplete and in many cases very little explored; e.g., Reynolds-number and Mach-numbcr influences. Thus many of the presented diagrams have to be labeled preliminary. The systematic study of the available information on losses, how-ever, did indicate some interesting trends and contributed con-siderably to improve the state of the art in the low-specific-speed regime. Hence this study may be considered a first attempt to

'Nomenclature A = area, ft2 II = head, ft T = temperature, deg R, torque, ft-lb a = arc of admission, ft h = blade heights, ft t = blade pitch, ft; time, sec

a* — cutter diameter, ft HP = horsepower, hp u = peripheral speed, fps B = length to diameter ratio of rotor ./ = mechanical equivalent of heat, V = volume flow, ft3/sec

C chord length, ft Btu/ft-lb w = weight flow, lb/sec chord length, ft K, k = coefficient w — relative velocity, ft/sec

r = absolute velocity, ft/sec I = rotor length, ft X = ratio of free area to swept area Co = spouting velocity M = Mach number X = Reynolds number exponent cv = specific heat at constant pressure, N = rotative speed, rpm Y = pressure-ratio function

Btu/lb deg F n = number of stages y = percentage of total head ex-D • rotor diameter, ft V = pressure, lb/ft2 panded in first stage

D* = equivalent hydraulic diameter, ft r R

Re

= radius z = coefficient as defined in equation

F = force, lb

r R

Re _ gas constant, ft /deg F Reynolds number a =

(16) absolute angle

G = weight, lb S = stress factor P = relative angle (I = gravitational constant, ft/sec2 s = tip clearance, ft (Continued on next page)

Journal of Engineering for Power J A N U A R Y 1 96 2 / 8 3

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present a somewhat simplified but rigorous picture of the design and performance aspect of the different turbine types.

It is to be noted that only the design-point operation is con-sidered in this paper, since a treatise of the "off-design" per-formance of the different turbine tj'pes would exceed the scope of a technical paper. Due to space limitations, the analytical treat-ment is confined to the discussion of highlights and major design trends.

Similarity Relations for Turbine Designs It is the purpose of the similarity parameters to define the

operating and design conditions for which turbines of similar de-sign geometry experience similar fluid dynamic conditions. For the derivation of these parameters the interrelation between the main criteria of turbine designs, namely, the relation between the turbine exhaust flow F3, the supply head Had, and the rotor dimensions, is investigated. The following sequence can be written: The flow Vs passing through the turbine exhaust is pro-portional to a characteristic velocity c and the through-flow area A. The area A is proportional to the square of the rotor diameter D, and the characteristic velocity c is proportional to the rotor tip speed u. Hence the volume flow passing through the turbine becomes proportional to the product of rotational speed N and the cube of the rotor diameter D

cA ~ cD2 uD2 ~ ND3 (1)

H ad N2D2 (2)

Z L F3 - ,

NIP N.D-

and from equation (2)

gad = NW* ifad-» ~ N*D*

(3)

(4)

Solving equations (3) and (4) for the rotor diameter D, assuming that the characteristic values of the standard turbine are unity (Fa-, = fl„d-, = 1) it follows

I) = Vz'/'N.'/'D,

AT'/."" H^'W.D,

N

N. = NVi'"-

(TO

(6)

By solving equations (3) and (4) for the rotative speed N, again assuming the characteristic values of the standard turbine are unity, the relation

N = V,NSD,3 Hj/'DJi,

D3 D

is obtained, which can be rewritten in the form

DHj/< D,

F , *

(7)

(8)

The head expanded in the turbine is proportional to the square of the characteristic velocity c or tip speed u2 and, conse-quently, proportional to the product of the square of the rotative speed and rotor diameter

Comparing, now, the volume flow and the head of a turbine with the volume flow and head of a standard turbine (subscript „), it follows from equation (1)

The term Ns as defined in equation (6) usually is referred to as specific speed so that the expression D, as defined in equation (S) can be termed as specific diameter [2]. These terms represent the rotative speed and rotor diameter, respectively, of a turbine which handles a volume flow of unity and expands a head of unity. Since turbines of similar design geometry having the same specific speed and specific diameter are similar in geometry and flow mechanism, it follows that turbines of similar design geometry which have the same specific speed and specific diameter have the same efficiency as long as Reynolds-number and Mach-number effects are neglected.

It also may be noted that the specific speed and specific diame-ter are truly dimensionless. This becomes evident when equa-tions (1) and (2) are written in the form Hadg ~ to2!)2 and V ~ coD3 with co denoting the angular velocity. Then it results for the specific speed

Fa'/!a. F S ' / W T T

(gH, d)V< (!/ff„d)V! 30

and for the specific diameter

d. = Fa'/'

(6a)

(8a)

For convenience, and in keeping with specific speed data com-monly found in the literature, the definitions given in equations (6) and (8) are used in this paper.

Further analysis makes it evident that the terms specific speed

— Nomenclature

fi* = coefficient 7 = specific weight, lb/ft3

A/3 = deflection angle 5 = degree of admission

<5* = percentage of viscous losses £ = loss coefficient 6 = diameter ratio i/ — efficiency X = leakage rate /u = dynamic viscosity, lb / f t sec a = stress, lb/ft2

k = ratio of specific heats p = degree of reaction ip = flow factor

<p* = gulp factor — velocity coefficient — nozzle flow coefficient

co = angular velocity

Subscripts

ad = adiabatic D = diffuser fr = friction h = hydraulic, hub

id = ideal I = leakage

M = material m = meridional N — nozzle

opt = optimum P — partial admission R — rotor S = cavitation suppression s = specific, sound

st = static T = turbine t = total

tk = tank u = in peripheral direction w = wheel disk 1 = inlet of machine 2 = before turbine rotor 3 = outlet of machine I = first stage

II = second stage

8 4 / J A N U A R Y 1 9 6 2 Transactions of f fie A S IE


iiiid .specific diameter are interrelated with other frequently used turbine parameters, velocity ratio and flow factor. Their numerical interrelations read when quoted in l.erms which also cover radial machines

N, =

Co

D. =

387(e-

0.4

'A

H z r ( e - 6A-2)' / !

\/2g 60

(9)

(10)

(11)

when eh denotes the hub ratio, e the rotor-diameter ratio as de-fined in Fig. 1, Co the spouting velocity, and <p the flow factor, defined as the ratio of axial leaving velocity and wheel speed: i.e. (see Fig. 1)

c m - 2

AX IAL TURB INES RADIAL TURB INES

dc

dh t . u th T

Fig. 1 Turbine-velocity triangles and geometries

<P = Cm-3 Ui

(12)

It is interesting to note that the frequently used term "gulp factor" [3] is also interrelated with specific speed and specific diameter. This relation reads for full-admission turbines , = _y_ =

V ND3 V

(6-2 - eh

240 1

(1.3)

These comments indicate that the terms Ns and Da are not really "new" parameters, but merely a restatement of well-established characteristic values in terms of rotative speed and rotor diameter. Actually, the specific-speed criterion has been used widely in pumps and water turbines; i.e., machines handling

incompressible media. Its application for units handling com-pressible media was somewhat inhibited, apparently due to the fact that the volume flow changes within the machines, thus raising the question which volume flow, inlet or exhaust or an average volume flow, to use for defining N, and Da. A closer examination of the flow mechanism makes it apparent that for impulse turbines the volume flow at the exhaust is the most significant volume flow, as this value determines the rotor exit-velocity triangle as well as the rotor inlet-velocitj' triangle, since no density change occurs between these two points in an ideal impulse turbine. Even for reaction-tj'pe turbines the exhaust-volume flow is more significant than the inlet-volume flow, since it controls the rotor exit-velocity triangle, whereas the inlet-volume flow does not even control the rotor-inlet triangle except for the unusual case of "100 per cent reaction" turbines (p = 1). Thus it becomes evident that for turbines the exhaust-volume flow V3 should be taken for the definition of Ns and Da. Like-wise, it is apparent that for compressors the inlet-volume flow Vi is the more significant value, since it controls the inlet-velocity triangle (whereas the exhaust-volume flow is only significant for p = 1), so that Na and Ds for compressors and pumps should be de-fined by Vi.

It is evident that the terms N, and D, are parameters which represent significant fluid-dynamic aspects of turbomachines. Of particular significance in this respect is equation (11) which reveals that the turbine-velocity ratio is directly related to Na

and Da. Since the turbine efficiency is predominantly a function of the turbine-velocity ratio, it follows that Ars and Da determine also, to a large degree, the turbine efficiency.

Additional significant interrelations can be recognized bj' dis-cussing the term "turbines of similar design geometry" in more detail. This definition is used to mean that certain geometric parameters, such as the ratio of blade heights h to rotor diameter, the nozzle angle a2, the blade angles ft and /S3, the degree of ad-mission 5, the ratio of rotor clearance s to blade heights, the ratio of blade spacing t to rotor diameter, the ratio of chord length C to blade heights, the ratio of trailing-edge thickness te to blade spacing and the relative roughness are equal for the designs under consideration.

Observing now that the losses in turbines are functions of these geometric parameters, it becomes apparent that turbines with equal N, and Devalues can have equal efficiencies only (still neglecting Re and M influences) when also the quoted geometric parameters are equal; i.e., for "geometric similarity." Further, analysis indicates that certain geometries are optimum; i.e., yield highest efficiencies for certain specific speeds and specific diame-ters. This means that the optimum geometry, e.g., optimum h/D, oil, and so on values, can be quoted as function of Na and D„ in all cases where the interrelation between the losses and the turbine geometry, particularly the blade geometry, is known. Thus diagrams can be computed which show the maximum ob-tainable efficiency and the optimum design geometry as function of N, and D, for constant Re and M values. The mathematical concept for the computation of these diagrams reads: The efficiency is a function of Na, Da, and several loss coefficients, which, in turn, are a function of channel geometry, Reynolds number Re and Mach number M. By partial differentiation of the relation, which describes the efficienc}' in the foregoing terms, the optimum geometrj' and, consequently, the maximum ob-tainable efficiency is found as function of Ars and Da for constant values of Re and M. This information can then be presented in a two-dimensional AfsDs-diagram, showing lines of constant ef-ficiency and lines for the optimum geometry. Additionally, such a diagram can show lines of constant turbine-velocity ratio which, according to equation (11) have a slope of one-to-one in a logarithmic ArsDs-diagram, and lines of constant gulp factor which have a slope of tliree-to-one, according to equation (13).



It is important to realize that the accuracy of the loss-coelHcicnt relations, which necessarily are based on the presently available experimental evidence, determines the validity of the "dia-gram and that these relations make the diagram dependent on the present state of the art. New information on loss coefficients, therefore, may change the location of the lines in the presently computed diagrams but it does not change the validity of the concept; i.e., it does not invalidate the meaning of the ArsDs-dia-grams, which is: To present the maximum efficiency and the op-timum geometry of turbomaehines in a rigorous, practical, and theoretically correct form.

One of the major advantages of this technique is that the maxi-mum obtainable efficiency is shown as a function of parameters which are of immediate concern to the designer; namely, the rotative speed and the rotor diameter. Another interesting ad-vantage is realized when it is considered that the unit weight G for designs with rotor diameters between 3 and 10 in. is proportional to the square of the rotor diameter; i.e.,

G = D'-K (14) with /,'„-values as shown in Fig. 2 for single-disk machines. Hence, approximate values for the turbine weight are directly obtainable from the specific diameter.

It is also interesting to note that the specific diameter gives

Z>, = VAN/D2

(15)

whereby the factor Z is a function of pressure ratio, efficiency, de-gree of reaction and ratio of specific heats

-Vr* 1 1 - r

K - 1

(1 - -OYM*(VI/V*) with YR representing the turbine-expansion ratio

YT = 1 -

(10)

(17)

and ij/* denoting a nozzle flow coefficient which, for a choked nozzle, becomes a constant (depending on k exclusively) and which for subsonic-nozzle velocities follows the relation

i * = % 2 f / v ; K - 1 (18)

The factor Z is graphically represented in Fig. 3 as a function of pressure ratio for different ratios of specific heats and different p-values. It is apparent that Z depends mainly on pressure ratio, and only to a very small degree on k and p. For incom-pressible media the factor Z becomes a function of the degree of reaction; i.e., independent of turbine pressure ratio

2 = (2f/)- ' /<(l - p ) - ' / J

- It is also worth mentioning that the specific speed interrelates the net output HP and rotative speed of the turbine as indicated by

N _ N 1 5 5 0 H P / r / V

H|5/< T Ws-St (10)

Equation (10) makes it evident that the turbine back pressure pt-at is an important criterion for the specific speed, indicating that for given power requirements and rotative speeds the specific speed increases with decreasing turbine back pressures.

The interrelations outlined in the foregoing indicate that NSD,-diagrams can be devised which present the performance and de-sign criteria in considerable detail. Since the -diagram also shows lines of constant turbine-velocity ratio, the design limita-tion as dictated by the maximum allowable tip speed becomes also immediately evident since for given gas temperatures and disk materials certain turbine-velocity ratios (i( /ci)„„ s cannot be exceeded. Hence, in these cases efficiencies, diameters, and rotative speeds occurring at turbine-velocity ratios smaller than (w/co) max must be considered exclusively. A more detailed dis-cussion of these interrelations is presented in subsequent sections.

Optimization Analysis for Axial Impulse Turbines Full-Admission Designs. Fig. 1 shows a typical velocity triangle

where the following notations are used: c denotes absolute veloci-ties, w denotes the relative velocities, u denotes wheel speed, subscript 2 refers to the inlet of the rotor and subscript 3 to the exit of the rotor. A typical approach to the performance analysis is to consider the force F which is exerted by the velocity vectors on the rotor. This relation reads for axial turbines:

IF F = - (cti-2 + c„-3)

(I (20)

In equation (20) c„-2 denotes the peripheral component of the inlet velocity c2, and cu-3 denotes the peripheral component of the

C % A O M I S S I O N

3% ADMISSION

Fig. 2 Approximate values of weight factor for different machine types Fig. 3 Z-values as function of pressure ratio

8 6 / J A N U A R Y 1 9 6 2 Transactions of the A S M E


absolute leaving velocity c3. Introducing, now, the term spouting velocity Co (which denotes the velocity obtained when the total turbine pressure ratio is ideally expanded into kinetic energy) and the term degree of reaction p (which denotes the ratio of the head expanded in the rotor and the head expanded in the tur-bine), the peripheral components in equation (20) can be ex-pressed by

It is to be expected that ipR is also affected by the relative ap-proach Mach number M . ^ . Adequate relations for this effect do not yet appear to be established. For conventional (subsonic) impulse blades the aspect ratio can be expressed by

C_ h

4 cos 10?.a*/D h/D

(28)

c,t—2 = c-2 cos a2 = t/'.y "\/1 p Co cos a'2 (21)

and

with a* denoting the "cutter diameter," Fig. 1. Observing now that equation (23) holds for incidence-free inlet (maximum ef-ficiency condition), equation (27) can be rewritten in the form

Cu-A = wu-3 — u = CoV'B COS ft X

\ p + (1 - p M v 2 - 2 ^ v cos a 2 ( l - p) ' / 2 — + ( — Y -N Co \ Co /

when incidence free inlet is assumed, i.e.,

cot ft = cot a-2 — u/co

sm oc2

and when the relative leaving velocity is defined by

w3 = iPa's/c0-p + W22

u

(22)

(23)

(24)

in =

cot a2

30 ( 1 ) I ' l

sin a-,\pN4620 )

1 0.24 cos ft a*/D~

h/D (29)

after approximating 0.228 (1 - /32/90)3 by cot ft/30 and by de fining u as the mean blade speed. Observing additionally that for full-admission impulse turbines, with 72/73 denoting the density ratio across the rotor

sm «2 = Ya/72

The terms \pN and ipit denote velocity coefficients expressing the ratio of the actual velocity and the theoretical velocity, when the theoretical velocity refers to the velocity which would be ob-tained when no losses have to be considered in the flow passage. Introducing equations (21) and (22) into equations (20), the head H R which is transferred into rotative power becomes

L V ^ v V 2 g

(30)

equation (26) can be written in terms of NSDS, h/D, a*/D for p = 0 (impulse). It is evident from the structure of this equation that an optimum /i/D-value (for 7jmas) exists which can be found

I II,, =

Fu W

VCo 1 l/'.Y "vA - P COS C*2 + in cos ft -y \ /p + (1 - 2\pN cos a 2 ( l — pY'" u Co

+ - -Co 4 Co ) 'J

(25)

The hydraulic efficiency T),, being the ratio of the head transferred into rotative power and the head supplied by the gas (co2/2g) then becomes

by differentiating in terms of h/D. This yields, after some simplification, an equation of the seventh degree which can be ap-proximated by

?,„ = 2 — Co

2(1 - p) , / 2 + \pR cos ft ^p + (1 - p ) i / V - cos a,Vl -- u P - +

Co ^ (26) Co .

This relation describes the hydraulic-turbine efficienc}' (neglecting wheel-disk friction) of axial turbines and reveals that the hy-draulic efficiency depends predominantly on the ratio of wheel-tip speed to spouting velocity M/CO, the degree of reaction p, and the blading geometry as expressed bj' the nozzle angle a2, the rotor-blade exit angle ft, the velocity coefficient of the nozzle 1pN, and the velocity coefficient of the rotor xpR.

In order to recognize additional interrelations the velocity co-efficients 1pH and \pN are investigated more closely. Experimental investigations [4, 5] reveal that 1pN can for most cases be con-sidered a constant with a numerical value of = 0.96. The rotor loss coefficient \pR depends mainly on the turning angle A ft blade spacing t, chord length C, and blade height h, Fig. 1. Ref-erence [6] indicates that t/C-values between 0.5 and 0.9 yield the smallest losses for impulse blades. Reference [7] indicates that the aspect ratio is of significant influence, suggesting that the losses increase with increasing C/fo-values. It appears that the influence of the blade geometry on the rotor coefficient can for im-pulse blades, i.e., ft = ft, be expressed by a relation of the form [8]

D,

/ 1

j Ds 460 + (NSDA

K 10 j

2~ (31)

Considering now that the turbine-shaft efficiency is smaller than the hydraulic efficiency by the wheel-disk friction losses, which again can be expressed by N s and Ds, it results that the maximum turbine efficiency becomes a unique function of N s and D, which reads for full-admission, axial-impulse turbines

N.D,

V = 1 1 - 2 + ( 0

77

[1 + is) ATSD,

_V COS C(2 — Ml) • h

154

- N.'D'lCiPv, (32)

in 1 - 0.228 1 90 - 006 f ]

1543

(27) with h / D according to equation (31), 1pn according to equation (29), a2 according to equation (30) and



A\D. cot 13? - cot a» —

sin otiipN 154 (33)

The term /3,„* in equation (32) denotes a wheel-disk friction co-efficient which for large Reynolds number has a numerical value of 3 X 10~3.

Equation (32) is graphically presented in Fig. 4 in the form of an iVjDj-diagram for full-admission, axial-impulse turbines. This diagram shows that maximum efficiencies are obtained at specific speeds of 40 to 80 and specific diameters of 1.8 to 1.1; i.e., at tur-bine-velocity ratios of n/co = 0.45 to 0.55 as indicated by the lines of w/co = const. The optimum gulp factor varies from ip* = 0.044 to 0.001. The lines of constant h/D and a« are also indi-cated in Fig. 4 and reveal that optimum values are h/D = 0.05 to 0.11 and a2 = 14 to 21 deg. For specific-speed values which are smaller than the optimum values, the specific diameter has to increase and h/D and a> have to decrease when the maximum obtainable efficiencies are desired.

Another interesting aspect is revealed when the ratio of exhaust energy c32/2j7 to input energy (c02/2ry) is considered which is repre-sented in Fig. 4 by lines of constant (C3/C0)2 values. This ratio again is a function of N„ D, and turbine geometry since for axial impulse turbines

(!) — = sill2 a«

y sin cot ft AJSDS - y i - 2 — + 2 D

154

These lines show that in the optimum-efficiency regime (c'3/co)2 =

0.05 to 0.1 and that the exhaust energy increases for specific diameters which are different from the optimum Devalues. The significance of the (c3/c0)2-value is demonstrated when the dif-ferent definitions of the turbine efficiency arc discussed.

It is evident from equation (26) that the supply head H„d = cl>2/2(/ refers to the ratio of total inlet pressure pi_( to static ex-haust pressure p3-at, i.e.,

H ad K ~ 1

1 P 3 - s t

Pl-I (35)

Another frequently used definition of turbine efficiency (r;,) refers the supply head to the ratio of total inlet pressure and total exit pressure pt-i', i.e., assumes that the absolute leaving velocity cz can be recovered full}' in a diffuser. This definition is particularly advantageous when multistage designs are considered, since in this case the leaving velocity, particularly in counterrotating-turbine arrangements, is utilized in the next stage and not en-tirely lost. For single-stage designs only a part of the leaving velocity can usually be recovered, depending on the diffuser ef-ficiency i)Dso that the "total" efficiency becomes

Vt = V

VD (36)

It is, therefore, possible to convert the efficiency ?j quoted in Fig. 4 to the efficiency ? b y considering the lines of cs/co = const. In applying equation (36) it is found that considerably higher turbine efficiencies rresult and that lines of r]t = const have a maximum at larger specific speeds and smaller specific diameters than lines of i] = const (see Fig. 14). It is to be noted that the 17,-values re-sulting from equation (36) and Fig. 4 are not necessarily maxi-mum possible values since the optimization was performed for

3 .6 I 3 6 10 "S

Fig. 4 NsDs-diagrums for single-stage, full-admission, axial-impulse turbines

V DENOTES EFFICIENCY RELATED TO STATIC EXHAUST P R E S S U R E ANO TOTAL INLET PRESSURE

8 8 / J A N U A R Y 1 9 6 2 Transactions of f fie A S IE


ijmax- This means also that equation (31) is not necessarilj' valid for i ) i - » , so that the h/D-values quoted in Fig. -4 may not be optimum values for 17(-max.

Since the similarity parameters together with the optimum geometry relations describe the velocity diagram completely, it is also possible to show lines of constant leaving angle in the jVsDs-diagram. This angle is of importance for the diffuser de-sign in single-stage turbines and for the consecutive stage in multi-stage designs. Fig. 4 reveals that a3 = 90, i.e., swirl-free ex-haust, for the maximum efficiency conditions. For specific diameters which are different from D„-opt, a certain amount of exhaust swirl exists (a3 ± 90°).

It is evident from Fig. 4 that extremely small blade heights are required for obtaining maximum efficiencies at low specific speeds. This also means that small rim clearances s are required. For large clearances the efficiencies have to be corrected by

1 -1.5 s / 2 sin 2ft

1 + h/s

(references [8, 9]), i.e., can be considerably smaller than quoted in Fig. 4 (see also Fig. 11). In these cases partial-admission tur-bines promise a better solution. This turbine type shows the additional advantage that even higher efficiencies are obtainable at low specific speeds than quoted in Fig. 4.

Partial-Admission Designs. In order to optimize the design geome-try of partial-admission turbines a procedure is applied which is similar to the technique described before. The main difference is that the rotor velocity coefficient 1pR~P for partial-admission turbines is smaller than for full-admission turbines [10] and can be expressed by the approximate relation

'Pit-p = in ( 1 -2 a

(37)

when t denotes the blade pitch and a the extension of the nozzle arc, Fig. 1. The ratio t/2a in equation (37) can again be ex-pressed by the similarity parameters yielding

J_ 2 a

D,2 sin a-2 # , v V 2 | a • ~D 7-2

(38) 2 sin (3273

when 72/73 denotes the densitj' ratio across the rotor. Introduc-

ing now equations (11), (23), (27), (37), and (38) into equation (26), and observing that equation (30) is not applicable for partial-admission designs, its counterpart being a relation for the optimum degree of admission, a relation for the optimum h/D-value results [8] which reads

h_ D / opt-p V; 0.24 sin 2 ft

ypND2 sin a2 V / 2 (39)

It is interesting to observe that the optimum h/D-value depends mainly on the specific diameter, as it was the case for full-admis-sion impulse turbines, equation (31). It is to be noted that equation (39) is only rigorous for cases where the density ratio 7 2 / 7 3 is unity. Actually, this ratio depends 011 the nozzle effi-ciency, the bucket efficienc}', the turbine efficiency, and the y-fac-tor, yielding 72/73-values which differ somewhat from unity. For most cases, however, this difference is sufficiently small and can be neglected. Even in cases where the density ratio dif-fers by as much as 30 per cent from unity, a correction of equation (39) appears to be unwarranted, clue to the approximate nature of equation (37). With equation (39) the relation for the maxi-mum hydraulic efficiency of partial-admission turbines reads

Vh-p = 77 1 + 1 - 0.228 1 A

90

1 — V si sm p3

sin a-2 si 1 1 2 f t V2g

NSDS

154 \pN cos a'-> (40)

i.e., i)h is again a unique function of the similarity parameters. Equation (40) can now be expanded to express the shaft efficiency by accounting for the wheel-disk friction, pumping and scavenging losses, so that A'SD,-diagrams can be calculated for optimized designs [8]. A typical diagram is shown in Fig. 5, which again shows lines of the optimum /i/D-values, constant (c3/c0)2-values, constant a3-values, and additionally lines of constant degree of ad-mission 5. The trends are similar as in full-admission designs; namely, increasing specific diameters and decreasing blade heights are required for obtaining maximum efficiencies with decreasing specific speeds. Additionally, decreasing degrees of admission are desired with decreasing specific speeds. Fig. 5 is calculated

I 3 6 1 3 R e > 1 0 5

dj'ie* Fig. 5 N.Di-diagram for single-stage, partial-admission, axial turbines with a*/D = 0.01



for a nozzle angle of a2 = 16 deg but the efficiency lines are equally valid for a? = 15 to 22 deg. In contrast the lines for h/D, (C3/C0), «3, and degree of admission are valid only for = 16 deg.

A significant criterion for the validity of Fig. 5 is the ratio a */D or its equivalent, the blade number

it sin jS2

2 = T v F -

It is evident from equation (40) that the efficiency decreases with increasing a*/D-values, particularly at large specific diameters. This means that low a*/D-values are essential in low-specific-speed designs, but of little consequence in large specific-speed units. Fig. 5 is calculated on the basis of a*/D = 0.01; i.e., by assuming that about 100 blades can be incorporated in the rotor geometry. The importance of this parameter for low-specific-speed designs is demonstrated in Fig. 11 by showing the maximum obtainable efficiencies (for D5-opt) as function of specific speed for different o*/D-values. This diagram indicates that the efficiency drops by about 9 per cent when a*/D is increased from 0.01 to 0.015 at N , = 3.

It is to be noted that the information presented in Figs. 4, 5, and 10 is calculated on the basis of the available information and that simplifying approximations had to be made for the in-terrelation between the blade losses and the blade geometry. While these approximations do not invalidate the similarity con-cept, it has to be kept in mind that the numerical values reflect the presently available information 011 loss coefficients. Hence the ArsDt-diagrams necessarily reflect the present state of the art. Good conformance of Figs. 4 and 5 with the available but limited test information has been found.

It is appropriate to note at this point that the iY ̂ - t e chnique and, in particular, its reference to the optimum geometry did contribute materially to improve the state of the art of designing low-specific-speed turbines. This is demonstrated in Fig. 6 where the shaded regime represents the efficiencies of low-specific-speed designs, reported in the literature up to 1957, as function of specific speed. According to this survey, Terry turbines ap-peared to be better suited for low-specific-speed application than axial turbines. The solid lines represent the calculated maximum efficiency values (for Ds~opt) assuming that the optimum geometry is incorporated in the design. The test data, shown in Fig. 6, represent the test information obtained at Sundstrand-Turbo for partial admission single-stage, axial-impulse turbines which did exhibit the optimum geometry, i.e., the h/D, a*/D, and De-values which yield rim;iS, Fig. 5, and confirmed the calculated data, thus evidencing that higher efficiencies can be obtained with

low-specific-speed turbines than reported previously in the litera-ture [8].

Comments on Reynolds-Number and Mach-Number Effects It will now be necessary to discuss the limitations imposed on

the validity of ArsD5-diagrams by Mach-number and Reynolds-number effects.

The main influence of the Mach number is felt in the rotor-velocity coefficient \pR. It is customary to define the Mach num-ber of impulse turbines in terms of rotor-blade approach velocity,

i.e., M,t-2 = vh c.

(41)

Expressing the Mach number M ^ ; in terms of turbine pressure ratio, it results for impulse turbines (design point)

M r ! =

I 2 cos ao(t(/co) , / w / c o V ' V 'I'n )

- 1 (42)

This interrelation is graphically represented in Fig. 7 for a? = 16 deg and several k and w/co-values. This diagram indicates that the turbine pressure ratio increases rapidly with Mach number as well as turbine-velocity ratio, and indicates the operating regimes, where supersonic blade designs should be applied (M.,-2 > 1) in order to obtain maximum efficiencies. In cases where sub-sonic blade designs are applied for M„,-2 > 1, performance penal-ties are to be anticipated since then 1pR is smaller than indicated by equation (27). Hence the turbine efficiency will be smaller than quoted in Figs. 4, 5, and 11. By using the information pro-vided in [11] a correction coefficient 7)3Ub/i;o can be devised, when ijsuo denotes the efficiency obtainable with subsonic blade designs and % denotes the efficiency quoted in the ;YsDs-diagram. Ap-proximate values for this coefficient are shown in Fig. 8, indicating a substantial decrease in efficiency at large relative Mach num-bers when subsonic blade designs are applied.

The Reynolds number in impulse turbines can be defined by

Re = W2D*y3

M (43)

when n denotes the dynamic viscosity and D* the equivalent hy-draulic diameter of the blade passage D* = 2hi sin /32/(£ sin /3? + h) with t denoting the blade distance, Fig. 1. By replacing D * and vh with previously defined values, equation (43) can be re-written in the form

^ man ..oi"1

/ f

4 .4 .6 I 4 6 10 20 40 60 IOO

N S

g. 6 Comparison of turbine efficiencies

90 / J A N U A R Y 1 9 6 2 Transactions of f fie A S IE


R e =

120gKYT ( f o ) ^ - 2 C ° 7 f / C 0 ) + ( ^ ° ) ;

Fig. 9 reveal that the Reynolds number increases with increas-ing pressure ratios, increasing turbine back pressure p3, increasing turbine-velocity ratios, and decreasing rotative speeds.

The efficiencies shown in Figs. 4, 5, and 11 are calculated for Re > 2 X 106. Hence, for conditions where equation (44) indi-cates smaller Reynolds number, an efficiency correction is re-quired. This correction can be provided by the argument that the Reynolds number only affects the viscous losses. For full-admission turbines all losses (at design-point operation) with the exception of the loss inherent in the absolute leaving velocity are of viscous nature, whereas for partial-admission turbines a smaller percentage is of the viscous type. This is shown in Fig. 10 where the percentage d* of the viscous to over-all losses (for design-point operation) is plotted against specific speed for the optimum specific diameter. It is conventionally assumed that the viscous losses are inversely proportional to the Reynolds number, al-though the exact function is not too well established. With this presumption the efficiency actually obtainable for turbine designs operating at Reynolds numbers smaller than 2 X 105 can be ex-pressed by the relation

7) = r)0 (1 - 5*) + 5* Re0

Re - 5*

R e 0

R e 1

(45)

Fig. 7 Pressure ratio as function of Mach number for single-stage, axial-impulse turbines

(44)

which is graphically represented in Fig. 9. Equation (44) and

when r/o denotes the (diagram) efficiency calculated for Rep = 2 X 10s and whereby the exponent x appears to have numerical values between 0.02 and 0.4, depending on the Reynolds number itself and increasing with decreasing Reynolds number [12].

Examining Fig. 9 more closely, it is observed that the Reynolds number is in the vicinity of 105 to 2 X 105 for cases where the back pressure p3 is about 15 psia; i.e., for typical sea-level designs. For smaller back pressure, the Reynolds number most likely is smaller than 105 so that for designs operating at back pressures below 15 psia, an efficiency correction as quoted in equation (45) may be necessary.

Comparison oi Full-Admission and Partial-Admission, Axial, Impulse Turbines

A comparison of the performance characteristic of full-admis-sion and partial-admission turbines is now of interest in order to establish the respective operating regimes for these two turbine types. A pertinent diagram is obtained when the maximum ob-tainable efficiencies are shown as function of specific speed (for

liub 1o

RJ IC

^ ^ r / — "L

K = 1.3 <*2 = 16'

Fig. 8 Approximately efficiency correction due to M a c h number influence for single-stage, axial-impulse turbines, using subsonic blade designs Fig. 9 Reynolds-number interrelation for single-stage, axial-impulse turbines



.1 1 10 100 KX» MS

Fig. 11 Comparison of performance of full-admission and partial-admission, axial-impulse turbines

Ds-opt), Fig. 11. This diagram also shows the required h/D-values which are of particular significance for this comparison. Fig. 11 indicates that at specific speeds below 25 the partial-ad-mission turbine obtains higher maximum efficiencies than the full-admission turbine when «*/jD-values of 0.01 can be realized. For larger a*/£)-values the crossover occurs at lower specific speeds. Another aspect is revealed when the h/D-values are reviewed for the different turbine types. It is apparent from Fig. 11 that a comparatively small blade length results, amounting to 3 per cent of the turbine diameter at a specific speed of 15 and to about 1 per cent of the turbine diameter at a specific speed of 10 for the full-admission turbine. In contrast the partial-admission turbine has an optimum blade length of 14 per cent of the rotor diameter at a specific speed of 15 and (3 per cent of the rotor diameter at a specific speed of 10. This difference is of importance for un-shrouded rotor designs, since the tip leakage loss is a function of the /i/D-value for given clearance ratios s/D as indicated in Fig. 12 (computed from [8]), meaning that extremely small clearances of s/D = 5/10,000 will be required for efficient full-admission turbines in the specific-speed range below 15, whereas partial-admission turbines in this operating regime can afford about 3 to 4

times the clearance for the same leakage loss. Hence the choice of the optimum turbine tj'pe in the specific-speed regime be-tween 8 and 15 depends on the a*/D-value; i.e., blade number as well as the allowable radial clearance. Thus turbines operating with high-temperature gases and consequently requiring large radial clearances will preferably be designed for partial admission in this regime.

Axial, Reaction-Type Turbines The ArsDs-diagrams presented so far have been computed on

the basis of a loss analysis. This procedure presumes that suf-ficiently exact relations for the component losses can be estab-lished. This is generally true for subsonic impulse-type turbines, but not necessarily for reaction-type turbines, since for this tur-bine type, particularly in designs which require comparatively long blades, a three-dimensional flow exists which makes the op-timization technique cumbersome. It may be expected that the single-streamline concept, used so far, will not yield data of suf-ficient accuracy for reaction-type turbines, i.e., for optimized de-signs in the large specific-speed regime. Actually, this reservation

9 2 / J A N U A R Y 1 9 6 2 Transactions of f fie AS IE


Fig. 13 Efficiency border lines in NSDS diagram

is also valid for impulse turbines with large /i/D-values; i.e., for full-admission impulse turbines when Ds < 0.9 and partial-admis-sion turbines with Ds < 2. It might, therefore, be attempted to use a somewhat different procedure for obtaining ATSDS-diagrams for axial, reaction turbines. This approach is to use available test data on efficiencies of turbines and to convert this informa-tion to jVsDs-values and thus to arrive at lines of constant ef-ficiencies. This process is only feasible when the available test data cover a sufficient spread of specific speeds and by considering the peak efficiencies of the different turbines exclusively. These data can then be extrapolated by the following argument:

Equation (26) indicates that for a given design geometry, i.e., given ft, and p-values (since p determines also the incidence-free inlet angle ft) the hydraulic efficiency is a function of the turbine-velocity ratio. By writing equation (12) in the form

By combining Figs. 4 and 5 and extending Fig. 4 to include reaction turbines, the maximum obtainable efficiencies for axial, single-stage turbines can be represented in one diagram as shown in Fig. 14, which now covers the full operating regime of single-stage turbines. The shaded line in this diagram indicates the regime where partial-admission and full-admission impulse tur-bines yield about the same efficiency, thus revealing that the operating regime to the left of the shaded line is covered best by partial-admission turbines, whereas the operating regime on the right of the shaded area requires full-admission designs. The numerical values quoted for the efficiency are only valid for the Reynolds numbers quoted in Fig. 14 and for M„-o < 1.0 when subsonic rotor-blade designs are used; i.e., for cor-responding pressure ratios and turbine back pressures. Correc-tions, Figs. 8 and 9, are required when operating beyond the quoted limits. It is important to note that the data for high-specific-speed designs are considerably more tentative than the data for low-specific-speed designs since they are extrapolated by con-sidering generalized trends instead of being calculated by a more rigorous optimization procedure.

Multistage Axial Turbines The iVs-Ds-diagrams can also be used for calculating the per-

formance and design criteria for multistage arrangements. This becomes evident if it is considered that the specific speed and specific-diameter of the different stages (I, II) are directly in-terrelated, namely,

AVi Ni ( Yn V/4(l - i?iFi)v

JV,-II A rh \ F , / V l - I?iFI

and

D,-1 fli ( Yi \ ' A V l - VuYu Dr D i YuJ (l-7)i F,)'/4

(1 - Fu) 2 ** - 1 ' (47)

(48) (1 - F I T ) 2 ' * - 1 '

<P =

sin fti/^ -yp + (1 - p)'./V - 2 c o s oc-2 (1 - p) v . - + ( 2 L V Co \ Co /

u/co (46)

it becomes apparent that the flow factor is also a function of the parameters which determine the efficiency (see also [13]). Thus a certain combination of flow factors and turbine-velocity ratios is required to obtain a desired efficiency, whereby the optimum com-bination is largely a function of the degree of reaction. Consider-ing now the design parameters which are required to obtain a certain efficiency, for example, rj = 0.5, it results from a numerical evaluation of equations (26) and (46) that a certain minimum and maximum flow factor exist for this condition, and that also a cer-tain minimum and maximum turbine-velocity ratio exists. Hence, for given degrees of reaction, the borderlines for t] = 0.5 are limiting flow factors and limiting turbine-velocity ratios as indicated by the dashed lines in Fig. 13. Since lines of constant flow factors (or gulp factors) have a slope of 3:1 and lines of constant-velocity ratios have a slope of 1:1 in a logarithmic ArsZ)3-diagram, it results that the borderlines for rj = 0.5 form a parallelogram. This consideration gives only a rough approxima-tion of the constant 7)-lines since the interrelation of the loss co-efficients with Ars and Ds imposes some additional limits which tend to cut off some extreme sections of the parallelogram as schematically shown by the solid line in Fig. 13. This considera-tion, however, is sufficient to recognize major trends and to serve as a fair basis for the extrapolation of the available test data. It indicates, for example, that the degree of reaction must increase with increasing specific speeds for optimized designs and that lines of constant degree of reaction and, consequently, lines of constant ft will follow lines of constant turbine-velocity ratio.

meaning that the operating points of the different stages can be identified in the Ar„Ds-diagram and that the efficiency and geometry of the different stages can be read directly from the diagram. With the proper selection of the (optimum) stage pres-sure ratios, the most efficient stage arrangements can be found readily. Such calculations can then be used to compute NSDS-diagrams for multistage designs, which, after proper optimization, can show the optimum pressure split and optimum geometry as function of N, and Ds. Instead of plotting ArsDs-diagrams for two-stage, three-stage, and so on, turbines, it is possible to con-dense the presentation to a single .2V,D,-diagram for multistage designs by showing the performance of the two-stage turbine only in that specific-spced regime where it is superior to the single-stage and three-stage designs, and by showing the performance of the three-stage design only for the specific-speed regime where it is superior to the two-stage and four-stage designs, and so on. In this way lines of optimum stage numbers are obtained (assuming that all disks rotate with the same speed, and that Ds is referenced to a particular stage and that disks, which have a different diameter from the reference disk, are properly identified).

As an example, the over-all efficiency of multistage (pressure-staged) single-disk turbines (according to a patent from Perrigault and Farcot, described in Stodola's third edition, 1905) may be quoted based on the calculations presented in [8]. These data are presented in Fig. 14 as short dashed lines in the form of lines of constant efficiency, whereby these lines are only shown in the regime where this turbine type gives better performance than the

Journal of Engineering for Power J A N U A R Y 1 9 6 2 / 9 3


Fig. 14 N sD s -d iagram for single-disk turbines

Fig. 15 N.SD,-diagram for single-disk turbines showing total efficiency



single-stage arrangement. The long clashed lines indicate the optimum number of stages. It is evident that this turbine type offers better performance in the low-specific-speed regime, par-ticularly in the specific-speed range from 0.5 to 10. It is to be noted that the optimum number of stages increases with de-creasing specific speeds but that at specific speeds below 0.5 the performance of the multistaged single disks approaches rapidly the performance of the partial-admission single stage.

The data quoted for the multistage single disk are calculated on the basis that the axial clearance between rotor blade and nozzle is one thousandth of the turbine diameter. For larger axial clearances the interstage leakage begins to offset the ad-vantage achieved by multistaging, particularly in the low-specific-speed regime. The performance advantage obtained by pressure staging on a single wheel is comparatively small at stage numbers above 4 so that most likely for all practical purposes more than four stages appear unwarranted for a single-disk design. Even then a stage number of 4 is only feasible for limited pressure ratios, not exceeding 100:1 per disk. This becomes evident when the operating points of the different stages are located in the stage diagram, Fig. 4, revealing that the first stage operates at lower specific speeds than the over-all specific speed, i.e., largo specific diameters and, consequently, small relative blade length h/D. The subsequent stages operate at larger specific speeds and, con-sequently, larger relative blade lengths. The difference in opti-mum blade lengths increases with increasing over-all pressure ratios. The actual blade length, incorporated in the design, will have to be a compromise, whereby a small blade length will have to be favored when a good first-stage efficiency is essential. This means that the blade length of the last stage is smaller than optimum. This is acceptable from an efficiency point of view but increases the required arc of admission so that the number of stages which can be accommodated on a single wheel is limited. By favoring a comparatively long blade, the efficiency of the first stage is severely affected, so that performance sacrifices result, which tend to offset the advantage of multistaging. This then means that even smaller stage numbers than 4 become optimum in eases where the over-all pressure ratios exceed 100:1.

There is, however, one remaining advantage for a large number of stages; namely, that large stage numbers decrease the opti-mum specific diameter and, consequently, the turbine-velocity ratio and, therefore, the required tip speed. Hence, for cases where the tip speed becomes the governing criterion, due to stress limitations, a comparatively large number of stages may present the most feasible solution.

Special Turbines A turbine type which has some of the characteristics of a multi-

stage single-disk turbine is the drag turbine [8] which exhibits good performance at low specific speeds. Lines of constant ef-ficiency for this turbine type arc shown in Fig. 14 by clotted lines. It is evident that this turbine type has better efficiencies at low turbine-velocity ratios than the single-stage turbine, but is in-ferior to the performance of multistaged single-disk turbines. Due to the simplicity of its rotor and the comparatively low tip speed required in the low-specific-speed regime, this turbine t}'pe will offer a preferable solution for applications where low manufacturing costs and small rotor diameters (low weight) are the prime criteria. The performance quoted in Fig. 14 for drag turbines so far is valid only for low pressure ratios. Further research, however, may indicate design geometries which enable the same performance at high pressure ratios or improved per-formance at low pressure ratios.

It is interesting to note that the Terry turbine offers also good performance in the low-specific-speed regime with, however, lower peak efficiencies than the corresponding axial design. The main reason for the lower efficiency is a somewhat smaller rotor-velocity

coefficient iplt which causes the turbine peak efficiencies to be about 20 per cent lower than the peak efficiencies of axial turbines having a*/D-values of 0.025, reference [8].

"Tota l " Efficiency It is to lie emphasized that the efficiency values, shown so far

in the ArsDs-diagrams, with the exception for the drag-turbine data, are calculated with the assumption that the kinetic leaving energy (C3-/2IJ) is considered lost. Thus the efficiency values are conservative since often at least a part of this energy can be re-covered, equation (36). The total efficiency defined in equation (36) is higher. The difference between the two definitions is particularly noticeable for designs with small specific diameters; i.e., large specific speeds. Actually, the term specific diameter indicates the minimum value of the ratio (c3/c0)2, the decisive criterion between ?/, and r\. This becomes evident by the follow-ing consideration:

The minimum possible c3-value occurs when a3 = 00 cleg, i.e., for zero swirl exhaust and when h/D = 0.5, i.e., when it is assumed that the displacement clue to the rotor hub is negligible. For this case

Co / ni i n 0.025

D,4 (49)

revealing that for a specific diameter of Ds = 1, the minimum leaving energy is 2.5 per cent of the supply energy; whereas for Ds = 0.5, the minimum leaving energy is 40 per cent of the supply energy. Actually, the (c3/co)2-values are larger in many cases since h/D < 0.5.

ArsDs-diagrams, showing total efficiencies, therefore, differ considerably from ArsDs-diagrams showing "static" efficiencies, particularly in the low-specific-diameter regime. Fig. 15 shows the total efficiencies for single disk, single-stage designs revealing peak efficiencies in excess of 85 per cent for full-admission designs and also "improved" performance of partial-admission turbines, thus indicating the benefits which can be obtained by effective exhaust diffusers.

Radial Turbines As another example for a calculated ArsD„-diagram, Fig. 10 is

shown which represents the optimized performance for radial turbines with blade angles of (S2 = 90°; i.e., a turbine type which is frequently considered the preferred arrangement for small gas turbines and turbosuperchargers. This turbine has a certain natural degree of reaction clue to the radius ratio of the impeller; i.e., the centrifugal field which causes p-values between 0.5 and 0.7 for the optimum Ns and Devalues. The comparatively high degree of reaction causes the Mach number to exert a somewhat larger influence on the performance than is usually encountered in impulse turbines. This becomes evident when the wheel-disk friction losses are examined in detail. The wheel-disk friction is proportional to the density surrounding the turbine disk. For impulse turbines, this density is equal to the exhaust density and, therefore, independent of turbine pressure ratio. For reac-tion-type turbines, the density of the gas surrounding the disk is larger than the exhaust density and increases with increasing tur-bine pressure ratios. This effect occurs only on the upstream side of the disk in axial, reaction turbines but is felt on both sides of the radial-turbine wheel for conventional designs and, there-fore, is of larger influence. With some simplifying assumptions the change in efficiency can be expressed bj' the relation

•q = t)o -NS3D*5 X 10-9 L\ P3

(50)

Journal of Engineering for Power J A N U A R Y 1 96 2 / 95


when subscript 0 denotes the reference pressure ratio denoted in Fig. 1G with (pi/p3)mai = 6. Sample calculations indicated that increase in pressure ratio to 20 :1 decreases the turbine efficiency by 2 to 4 per cent for the optimum A*s and Devalues.

Another parameter which is of importance in radial turbines is the diameter ratio e defined as the ratio of the wheel-tip diameter D to exducer diameter d, Fig. 1. It is apparent from the previous considerations that optimum values for this parameter exist and that this optimum is a function of specific speed and specific diameter. Hence, lines of optimum diameter ratio can be shown in an ArsDs-diagram. No exact relations for the interrelation of the loss coefficients with the diameter ratio have been found in the literature so that only the main trends of this parameter can be indicated. The lines of constant diameter ratio shown in Fig. 16, therefore, are not to be considered as absolute values but rather as approximations demonstrating trends, resulting from the analysis presented in [13].

Another interesting fact is revealed when lines of constant exhaust energy expressed in percentage of the total supply energy (CJVOT) are presented. This is expressed by the daslied-dotted lines in Fig. 16, indicating that exhaust diffusers become of in-creasing importance for turbines showing small specific diameters, i.e., comparatively large specific speeds. In this respect it is im-portant to note that the static efficiency is shown in Fig. 16 and that higher total efficiencies result as indicated by equation (36). It should be realized that the absolute leaving velocity c3

depends on the diameter ratio e. Hence, the location of the (cs/co)1 lines is, to a large degree, dictated by the e-values. The (C3/C0)2 lines shown in Fig. 16 are in accordance with the selected e approximations.

In general, it can be said that this turbine type shows maximum static, efficiencies at specific speeds between 45 and 90; and specific diameters of about 2 to 1.2. The limited specific-speed range is caused mainly by the assumption that the blade-inlet angle is /32 = 90°. By selecting smaller blade-inlet angles, i.e., by designing this turbine type for a smaller degree of reaction, the optimum-performance regime shifts to lower specific speeds, whereas, by selecting blade angles larger than 90 deg, optimum efficiencies occur at larger specific speeds.

Actually, the radial turbine shows performance criteria which are very similar to axial turbines so that a good part of the per-formance regime covered by axial, single-stage turbines, Figs. 4 and 14, can be covered equally well by radial turbines. However, this might be doubtful for partial-admission, single-stage designs

REACTION BLADES (NOZZLE)

2 0 ° 4 0 ° 6 0 ° 8 0 ° 12 .0° 1 4 0 ° 1 6 0 ° 180°

A f b

Fig. 17 Velocity coefficients for nozz les a n d buckets

Re = -p Fig, 18 Wheel -d i sk friction coefficients

q DENOTES EFFICIENCY RELATED TO 8TATIC EX-HAUGT PRESSURE ABO TOTAL 1MLET PRE88URE.

fit " SO" Ro* > I 0°

n/VT »s • —T Hod %

D Hod ** / v j

N • rpm V « ft3/«oc Hod n ft Ib/H 0 . II

Fig.

I ! 3 6 10 30 60 100 Ng 300

16 Calculated N ,D 6 -d iag ram for s ing le-s tage radial turbines

YR

Fig. 19 Effect of velocity coefficients on turbine hydraul ic efficiency

96 / J A N U A R Y 1 9 6 2 Transactions of f fie AS IE


where the pumping losses in the unadmitted portion of the periphery most likely are larger, due to the radial pressure gradient, than in the corresponding axial design.

Comments on Peak Efficiencies It has to be realized that some generalizations had to be made

in order to present definite efficiency values in Figs. 4, 5, 11, 14, 15, and 16. This necessarily means that only average values have been presented in these diagrams, whereby the prime emphasis was to indicate trends, thus leaving some degree of uncertainty regarding the accuracy of the absolute values. This is primarily due to the difficulty in finding accurate data on the loss coefficients of the turbine elements. Different values are reported by dif-ferent investigators. The influence of these differences may be discussed on the example of a full-admission, axial, impulse tur-bine. It is evident from equation (32) that the velocity co-efficients tpjf, and the loss coefficient /?„,* determine the ef-ficiency besides the similarity parameters Ns and Ds. Fig. 17 shows the scattering zone of the and j-data, reported in the literature, by plotting these values as function of the deflection at angle A/3. Fig. IS shows the different values for the wheel-disk friction coefficient /3,„*, found in the literature. Calculating now the hydraulic efficiency of axial-impulse turbines for (w/co)0pt as function of ipK and \pv, Fig. 19 is obtained, which indicates that the hydraulic efficiency is particularly sensitive to \pN and that the difference between the average l/'-values, used in the NSD3-diagrams, and the maximum l/'-values can cause efficiency dif-ferences of as much as 5 to 7 per cent. Considering now addi-tionally the differences in /S^-values, it results that higher (as well as lower) efficiencies than quoted in the ArsDs-diagrams can be obtained and that this difference can amount to as much as 6 to 8 per cent. It appears reasonable to assume that the efficiencies quoted in Figs. 4, 5, 11, 14, 15, and 16 are obtained with good aerodynamic design features, whereas up to 8 per cent higher efficiencies should be obtainable with aerodynamically refined designs; i.e., designs with carefully considered flow-path details, such as surface roughness, avoiding sudden flow decelerations or sharp corners, and so on.

Stress Considerations Important considerations in the selection of the turbine-wheel

design are the disk stresses. These are for similar wheels a func-tion of the wheel-tip speed [14, 15] and the specific weight y M of the rotor material. Similarity considerations indicate that the stress tr follows the relation

<t = Sll2 Tit (I

(51)

whereby the factor S represents a characteristic value which depends mainly on the wheel type. Typical values are shown in Fig. 20. These values are valid only in cases where compara-tively small thermal stresses exist in the wheel, i.e., when it is assumed that the temperature gradient in the wheel disk is comparatively small, and that for axial wheels the blade length does not exceed 20 per cent of the disk diameter (h/D < 0.2). For larger /i/D-values the stresses in the blade base become larger than the disk stresses and, consequently, are the main criterion. It is to be noted that the <S-fac.tors presented in Fig. 20 represent average values for typical design geometries. Lower S-values, i.e., higher admissible tip speeds, may prove to be ob-tainable with rotor geometries which are carefully stress-balanced by a rigorous stress analysis.2

The maximum allowable stress tr depends on the material composition and on the disk temperature. Some typical values

2 Apparently the S-values can be reduced to of the values shown in Fig. 20 with reasonable disk and blade taper ratios.

[16] are quoted in Fig. 21 by plotted <r/Y.u against temperature. This diagram indicates that nickel and cobalt-based alloys are applicable up to 1500 to 2000 R and that for higher temperatures molybdenum alloys must be applied.

The relation quoted in equation (51) and Figs. 20 and 21, can also be interpreted as meaning that for certain temperature levels a certain maximum allowable turbine-velocity ratio exists. This means that in the A'sDs-diagram a limiting turbine-velocity ratio exists, equation (11), so that only operating points to the left of the limiting »/co-line can be selected since otherwise the maximum allowable stress will bo exceeded. This interrelation is an im-portant criterion for turbines using high-energy propellant gases as a power source. In order to give some indication of commonly experienced limits for these designs, the interrelation between disk temperature and gas temperature must be determined.

An approximate value for the rim disk temperature in axial turbines can be obtained by assuming that the rim disk tem-perature is equal to the blade temperature. For the average blade temperature the argument holds that about 85 per cent of the dynamic temperature rise is effective at the blade. This then means that the average blade temperature can be computed as a function of the gas temperature at the turbine inlet and the tur-bine-velocity ratio, yielding the relation

1

Fig. 20 Approximate values for characteristic stress factor

9 mox

Y

Fig. 21 M a x i m u m allowable stresses for different materials for 1000-hr rupture life

Journal of Engineering for Power J A N U A R Y 1 9 6 2 / 9 7


Tl\ = 'A Co2<Ml

2]gcp

_p) ( O.Soii.'r _ + 2}gcf

= r , < i - YTiN\ 1 - P) 0.S5 sin a'2

sin /3o (52)

Since the blade temperature decreases with decreasing degrees of reaction, it becomes apparent that impulse turbines are most suitable for operation at high gas temperatures. The approxi-mate temperature limits for this turbine type may now be dis-cussed in more detail.

With the approximation (sin a2/siii (S2)2 = (1 — u/coY the dif-ference between average blade temperature and gas temperature can be expressed by the relation [from equation (52) for p = 0]

A I = T, - T„ = O.loco2 + 1.7-»Co - 0.85»2

K 2 g R T K - 1 l/'.V"

(53)

Thus the temperature drop can be calculated as a function of wheel speed and spouting velocity for selected K and 7?-values. Using now equation (51), the required cr/yM ratio can be cal-culated as a function of wheel speed after selecting the proper stress factor S, so that the allowable blade temperature can be found from Fig. 21 for the different materials. Adding, now, the temperature drop A/, equation (53), to the blade temperature, the allowable gas temperature is found. This procedure yields the interrelation between turbine-velocity ratio and allowable gas temperature for different spouting velocities and wheel ma-terials as presented in Fig. 22 for S = 0.21, for a gas with k = 1.4 and R = 53.3. This diagram reveals the interesting fact that the allowable gas temperatures increase with increasing spouting velocities. The influence of the turbine-velocity ratio depends on the material, or more specifically, on the slope of <r/yM = f(Tb).

For a material with a steep slope (aluminum) the admissible gas temperature increases with increasing turbine-velocity ratios (up to the point where crmax is reached), whereas for materials with a more gradual slope (nickel and molybdenum alloys) the ad-missible gas temperature may increase, decrease or be almost in-dependent of the turbine-velocity ratio (up to the point where

3 0 0 0

Tgaa

Co

Vmu is reached). It is to be noted that this diagram is computed by assuming a constant "recovery factor" of 0.85 and using the average blade temperature as a basis; i.e., by using simplifying assumptions, which may not be entirely correct at supersonic speeds in the rotor blade passage; i.e., at high values of Co. This diagram is shown mainly to indicate trends rather than absolute values. It stands to reason that a more refined stress analysis which duly accounts for the thermal disk stresses and the interre-lation between disk shape and stress can yield values which differ from Fig. 22. Additionally, it is to be noted that higher gas temperatures than shown in Fig. 22 may be obtained by provid-ing external cooling, particularly in partial-admission designs.

Another interesting feature regarding allowable gas tempera-tures is exhibited by the double-staged (pressure-staged), single-disk design. For many designs, particularly when operating at large pressure ratios, it can be assumed that the last stage occupies about 00 to 80 per cent of the periphery so that the disk tempera-ture is mainly determined by the blade temperature of the last stage. This means that the disk temperature is lower than ex-pressed by equation (52) and, consequently, the allowable gas temperature is higher than shown in Fig. 22. Actually, this can only be the case when the first-stage efficiency is reasonably high. This is demonstrated in Fig. 23 by comparing the blade tempera-ture of a single-stage design with the blade temperature of a two-staged, single-disk design having rji = 0.6 and r]i = 0.4 for dif-ferent over-all pressure ratios (expressed by F0), pressure splits between the stages (expressed by y), and different wheel speeds, f t is evident that in this case (hydrazine) the blade temperature of the two-stage design is smaller than the blade temperature of the single-stage design, particularly for high expansion ratios when iji = 0.6 (solid lines in Fig. 23). Thus in many cases where single-stage designs have to resort to molybdenum alloys, double-staged, single-disk designs can still use nickel alloys. When, how-ever, the first-stage efficiency of a double-staged, single-disk de-sign is only TJI = 0.4, the blade temperature is about equal to the blade temperature of a single-stage design, as evidenced by the dashed lines in Fig. 23, so that no temperature advantage results for the double-staged, single-disk design for these conditions.

Similarity Relations for Rotary-Displacement Machines The similarity laws for displacement machines differ to some

degree from the similarity laws for turbomachines. However, it can be demonstrated that the maximum obtainable efficiency of

= "

u = 1 6 0 0 f t / o o c _ _

^-"^mifcqcL ZL _ _ [ _

Fig. 24 Typical cross section through rotary-displacement machine (Roots type)

Yi

Fig. 22 Interrelation between al lowable turbine-velocity ratio and gas temperature for different spouting velocities and materials (1000-hr rupture life)

Fig. 23 Comparison of blade temperatures for double-stage, single-disk, axial turbines and single-stage axial turbines



displacement machines is a unique function of the specific speed and specific diameter as it was for turbomachines. Hence N,Da-diagrams can also be calculated for optimized designs of dis-placement machines. The validity of the foregoing argument may be demonstrated for the case of a Roots-type expander.

A typical cross section through a Roots expander is given in Fig. 24, indicating two lobe-type rotors with a diameter of D, ar-ranged in such a manner that they constantly intermesh, thus sealing the exit area from the inlet area at any position of the lobes. The weight flow W which passes through the machine can be expressed by the relation

TF = WK + W, (54)

V n = TTD'LXN

120 (55)

Yi + 73 Pi RTi

1 + Pi/Pi 1 - VY,

(50)

From equations (54), (55), (56), and (1) it results for the exhaust volume flow

Fs ND3

irBX Pl/P3 1 - i)Y

+ 1 (1 + X )

240 (57)

X = TF, TF®

XB — I 1 P-i/V i 1 - i)Y

K K - 1

RTi

W, =

3 ± - ( l + B + ± ) r

Vr'I\

(58)

(59)

when e denotes the ratio of the outer diameter of the lobe to the hub diameter of the lobe, Fig. 24. Comparing equation (57) with (1), it is found that the similarity relation for the volume flow of the rotarv-displacement machine and the turbomachine are equal as long as X is constant; i.e., as long as similar machines operating at equal w/co-values and pressure ratios are considered. The rela-tion for the expanded heads, however, are different. The head expanded in the Roots motor for the ideal case, i.e., without con-sidering friction and windage losses, follows the relation

H id = Taw TmttN

W 3 0 F k 7 „ (60)

when 2'id denotes the torque which is exerted by the airflow on the lobes. For T\i it holds

= I 2ApLr dr = V i P 3 D3B{ 1 - e~2) (61)

Journal of Engineering for Power

Introducing equations (55), (56), and (61) into equation (60), the relation for the ideal head expanded in the Roots motor reads

H id = (1 - p3/pi)(l - e~2)2RTi

X 1 + Pi/Pi - vYi

(62)

when TFB denotes the weight flow transported from inlet section to exhaust section due to the rotation, and when IF; denotes the leakage flow passing through the clearance s, Fig. 24. The volume flow VR transported by the rotor is

when L denotes the length of the rotor and X the ratio of the free area compared to the area of the full circle; i.e., the ratio of the actually swept volume to the volume swept in the ideal case for zero thickness of the lobes. This volume flow passes through the machine with a mean density y m which may be approximated by

i.e., is entirely independent of diameter and rotative speed of the machine, in contrast to the head relation for turbomachines, equation (2). Hence the basic similarity relations for positive-displacement machines do not allow formulation of such terms as specific speed and specific diameter. Equation (57), however, indicates that the similarity relation for the volume flow is de-scribed by the flow of the gulp factor as becomes evident by com-paring equations (57) and (13). Since the gulp factor is in-terrelated with specific speed and specific diameter, as indicated in equation (13), it becomes evident that the terms specific speed and specific diameter will have a definite meaning, also, for dis-placement machines. This becomes even more evident when it is considered that the efficiency relation for displacement machines is a function of the turbine-velocity ratio u/co, i.e., Na and D, ac-cording to equation (11).

The basic definition for efficiency states that the efficiency is the ratio of the work output to work input, whereby work input is de-fined by the product of the head Hid and flow TF delivered to the machine. The output is defined by the head H* transferred into rotative power and the weight flow IF® participating in the process, i.e.

V H* Wn Had TF

(63)

when the length L in equation (55) is expressed as percentage of the rotor diameter by writing L = BD, and when X denotes the leakage rate

The head ratio in equation (63) can also be written in the form

H* = Hid - Hw

H ad Hid + H lr

when Hit denotes the head loss due to wall friction

(64)

o,2 V2 Hfr = — ? (65)

with £ denoting a loss coefficient which depends mainly on the passage geometry', on the pressure ratio, and on the Reynolds number. The term Hw in equation (64) denotes the head lost due to wheel-disk friction and windage, i.e.,

H,„ = u22

gBX1 (66)

with (3W* denoting a loss coefficient which is mainly a function of the Reynolds number. The weight flow ratio in equation (63) can also be written in the form

1F« = 1F„ = 1 TF TFk + TF, 1 + X

(67)

Introducing, now, equations (58), (62), (64), (65), (66), and (67) into equation (63) the relation for the efficiency of rotary-displace-ment motors is obtained and reads

V =

2 ft/fc,

,c0/ B( 1 - e"

1 + 1 - , 1 Co J 16(1 - e"2).

12 D

1+B + - ) k .

{u/co)XB

(68)

J A N U A R Y 1 9 6 2 / 9 9


600 1000

Fig, 25 Preliminary N s D s -d iagram for rotary-displacement machines operating as motors (low-pressure ratio)

where ki and fa are factors which depend mainly on pressure ratio

fe. =

h = 2;,

k — 1 T \ ' 1 — f]YT

l - in/Pi

r ((59)

k - 1 YT I ( 1 Ps/Vi

1 - 7]Y

Equation (68) reveals that the efficiency depends on the velocity ratio it/ct, the pressure ratio, and the geometry as expressed by the X and B-factors. If it is now considered that the A* and B-factors are interrelated with specific diameter and specific speed as indi-cated by equation (57), and that the velocity ratio u/c0 is also a function of Nt and D, as indicated by equation (11), it becomes apparent that the efficiency of rotary-displacement machines is a unique function of the similarity parameters N, and D, and that the leakage relations are mainly responsible for this result. This then means that lines of constant efficiencies can be plotted in an iYjDj-diagram together with lines for the optimum geometry. Hence the similarity concept developed for turbomac-hincs is also directly applicable for rotary-displacement machines and has equal significance for both machine types. Tt is, however, to be noted that the pressure ratio is more critical in displacement machines than in turbines so that an jVsDs-diagram for displacement machines usually is valid only for a narrow range of pressure ratios. This is particularly true for compressible media, since the maximum obtainable efficiency depends to a major degree on the type of expansion process which can be utilized in the machine; i.e., to which degree the full expansion energy (in contrast to the displacement energy) is utilized. This can be expressed in equa-tion (68) by the loss coefficient J, meaning that J will be com-

L/o Fig. 26 Approximate values for weight factor of rotary-displacement machines

paratively large for large pressure ratios in "displacement" machines representing sudden expansion losses in addition to the friction losses, but smaller in true "expansion" machines.

Due to the lack of detailed information on the different loss co-efficients in rotary-displacement machines, no detailed per-formance diagrams of this machine type are presented. It ap-pears, however, feasible to indicate from the available per-formance data of this machine type in conjunction with equation (68) the approximate operating regime for rotary-displacement devices. These lines necessarily are preliminary and merely serve to indicate in a somewhat generalized form the most feasible operating regime of rotary-displacement machines. A typical diagram is shown in Fig. 25, indicating that displacement ma-chines cover the low-specific-speed regime extremely well and are efficiencywise superior in this regime to single-stage turbines. Considering only the maximum obtainable efficiency (D,-opt) as function of specific speed, it is found that the rotary-displacement machine exhibits about the same efficiency as the pressure-staged, single-disk turbine at, however, specific diameter and conse-

1 0 0 / J A N U A R Y 1 9 6 2 Transactions of f fie AS IE


quently tip speeds which are only '/a to ' / j of the values required for the single-disk turbine. This advantage is for some applica-tions offset by the fact that these efficiencies are obtained, at least for the true "displacement" type, only at comparatively low pres-sure ratios.

It may be pointed out also that for many designs the weight of the unit is again proportional to the square of the diameter. The weight factor, however, is considerably larger for this machine type than it was for turbines, ranging, for example, from 150 to 350 for Roots-type motors, as indicated in Fig. 26.

Application of Design Diagrams A brief description may now be given for the application of the

design diagrams for finding the optimum design for given prob-lem statements.

It may be assumed that the problem statement calls for a tur-bine design delivering 60 hp at a rotative speed of 32,000 rpm operating on a gas with a ratio of specific heats of K = 1.3, a pres-sure ratio of 20:1, a gas temperature of 2000 R, a gas constant of R = 80 with a back pressure of p3 = 15 psia. Assuming, now, an efficiency, the specific speed can be calculated from equation (19)—or from equation (6), after calculating the adiabatic head with equation (35) and the required volume flow from the energy relation—and yields N, = 6.45 for t) = 0.605. From Fig. 14 it is found that an efficiencj' of rj = 0.605 can be obtained (thus confirming the initially assumed efficiency and eliminating the need for correcting the A^-value) for this condition if the optimum specific diameter is selected which has a numerical value of Ds = 7.3. This means that the optimum rotor diameter is D = 10.4 in., according to equation (8).

Fig. 14 also indicates that the partial-admission, axial turbine gives the best solution so that now the detailed geometry can be found from Fig. 5. For the specific speed and specific diameter quoted, it results that the optimum ratio of blade height to diame-ter is h/D = 0.042, meaning that the optimum nozzle height is h = 0.438 in. if a nozzle angle of a« = 16 deg is assumed. Equation (33) indicates that for these conditions the rotor angle becomes ft = 23.0 deg. It further results from Fig. 5 that a degree of ad-mission of 8 per cent is required, that the ratio of leaving energy to supply energy is (c3/c0)2 = 0.09. meaning that the exhaust loss is 9 per cent, and that the velocity vector has an angle of <23 = 50°. In order to obtain the quoted efficiency, a cutter diameter of a* = 0.104 is required, meaning that a blade number of 2 = tr sin ft/(a*/D) = 124 is desired. The maximum allowable radial clearance is s = 0.00876 in. since s/h = 0.02 and the maximum allowable trailing-edge thickness is te = 0.0053 since I j l = 0.02. From these values the wheel-tip speed is calculated to 1450 ft/sec which means, according to equation (51), a rim stress of 54,100 psi if a material density of 520 lb/ft3 is assumed. The blade temperature amounts to 1456 R according to equation (52), which means that a nickel alloy should be taken for the rotor material, as indicated by Fig. 21. The approximate weight of the turbine amounts to 28 lb according to equation (14) if a weight factor of ka = 38 is assumed, Fig. 2. A check of the Reynolds number indicates a value of Re = 4.45 X 105 if a /Li-value of 0.015 centipoises is assumed, meaning that the Reynolds number is sufficiently large to render Fig. 5 valid. A check of the Mach number reveals an approach Mach number of M = 1.7 which, according to Fig. 8, means that the efficiency quoted in Fig. 5 has to be corrected by a factor of 0.98, when a subsonic rotor blade design is selected, meaning that the obtainable ef-ficiency is 7} = 0.593. It is, however, possible to obtain the originally assumed value of 7) = 605 when an exhaust diffuser is provided, which yields an efficiency of = 0.222, according to equation (36). Hence the most significant values for the opti-mum design are readily found from the quoted diagram.

In cases that the resulting rotor diameter is too large or the calculated weight is too heavy for the intended application, Fig. 5 also indicates the design values for a lighter turbine. If it is, for example, assumed that the maximum allowable rotor diameter is D = 7 in., then it follows from equation (8) that the desired specific diameter is Ds = 4.9. For this specific diameter and the previously quoted specific speed, a maximum turbine efficiency of 7) = 0.51 results from Fig. 5, which also indicates that the blade length now has to be h = 0.058 X D = 0.406 in. and that 14 per cent admission is required. The leaving energy is 20 per-cent of the spouting velocity, and the cutter diameter has to be a* = 0.07 in., yielding a blade number of 2 = 109. The bucket angle now is 20.0 deg according to equation (33), i.e., somewhat smaller than the bucket angle required for the first design. It is to be observed that this design delivers only 50.5 hp due to the decreased efficiency. The design values for a 60-hp turbine with D = 7 in. and N = 32,000 rpm are found by correcting Ds and Ns

for the reduced efficiencj' by a trial-and-error calculation, yielding an efficiency value of 7? = 0.535, h/D = 0.062, (c3/c0)2 = 0.21, and 15 per cent admission.

It must be kept in mind that the i;-values quoted are average values as discussed in detail in the section "Comments 011 Peak Efficiencies."

It is evident, then, that fairly detailed design information can be read from the quoted diagrams, which, in most cases, is suf-ficient for judging the compatibility of the design with the re-quirement. It stands to reason that the values obtained this way in some cases may have to be refined by more exacting calcula-tions.

Conclusions The similarity parameters, specific speed N s and specific diame-

ter Ds, are adequate parameters for expressing the similarity rela-tions for turbomachines as well as rotary-displacement machines. Machines which have the same specific diameter and specific speed and similar geometries are similar in fluid-dynamic behavior and, consequently, have the same efficiency if Reynolds number and Mach number influences are neglected. This concept is used to compute the optimum geometry and maximum obtaina-ble efficiencies as function of Ns and D„. This information is presented in the form of AfsDs-diagranis, thus providing informa-tion on the maximum obtainable efficiencies and the desired ge-ometry of the machine types discussed. These diagrams are valid for a certain range of Mach numbers and Reynolds numbers and express the present state of the art. For Reynolds and Mach number different from those indicated in the diagrams, correction factors can be devised which account for the influence of Reynolds number and Mach number. By presenting the dif-ferent tj'pes of turbomachinery and rotary-expansion machines 011 the same diagram, the most feasible turbine type for given re-quirements is readily determined, together with its optimum geometry, based 011 present state-of-the-art knowledge, by merely determining two significant numbers. These numbers can be calculated readily from the design requirements.

The particular advantage of this technique is that the similarity parameters, specific speed and specific diameter, directly reflect values which are of immediate concern to the designer; namely, the rotative speed of the machine and the rotor diameter of the machine.

Although a conscientious effort has been made to collect all available information 011 turbine-performance data, it has to be realized that some of the ArsDs-diagrams presented in this paper are necessarily of preliminary nature, and are labeled as such, due to the still incomplete knowledge of the interrelation between losses and geometiy. This emphasizes the need for additional research in the field of turbomachines.

Journal of Engineering for Power J A N U A R Y 1 96 2 / 1 0 1


Acknowledgments The similarity concept described herein evolved while working

on an Office of Naval Research sponsored turbine research con-tract (Contract No . NONR-2292(00 ) Task N o . N R 094-343) with Sundstrand-Turbo, a division of the Sundstrand Corpora-tion. T h e diagrams for partial-admission axial turbines, pres-sure-staged single-disk axial turbines, and drag turbines, shown in this paper, were calculated under this contract. Acknowledg-ment is due to the Office of Naval Research and to the Sund-strand-Turbo Division for permission to use these diagrams.

T h e author is indebted to Mr . H. J. W o o d for his valuable c o m -ments in discussing the details of this paper.

References 1 D. G. Shepherd, "Principles of Turbomachinery," The Mac-

millan Company, New York, N. Y. , 1950. 2 0 . Cordier, " Aenlichkeitsbedingungen fUr Stromungsma-

schinen," VDI Berichte, vol. 3, 1955, pp. S5-8S. 3 H. J. Wood, "Parametric Data Presentation for Radial Flow

Turbines," A S M E Paper, 1952. 4 H. Kraft, "Reaction Tests of Turbine Nozzles for Subsonic

Velocities," TEAKS. ASME, vol. 71, 1949, pp. 781-787. 5 J. H. ICeenan, "Reaction Tests of Turbine Nozzles for Super-

sonic Velocities," TRANS. ASME, vol. 71, 1949, pp. 773-780. 6 D. G. Ainley and G. C. R. Mathieson,"An Examination of the

Flow and Pressure Losses in Blade Rows of Axial-Flow Turbines," A R C Technical Report, R and M No. 2891, Great Britain, 1955.

7 "Results of Systematic Investigations on Secondary Flow Losses in Cascades," Institute of Fluid Mechanics, University of Braunschweig, Germany, Report No. 54132a.

8 0 . E. Balje and D. II. Silvern, " A Study of High Energy Level, Low Power Output Turbines," A M F / T D No. 1190, Department of the Navy, Office of Naval Research, Contract No. NONR-2292(00), Task No. N R 094-343.

9 W. Traupel, "Neue allgemeine Theorie der mehrstufigen axialen Turbomachinen," Leeinan and Companv, Zurich,Switzerland, 1942.

10 A. H. Stenning, "Design of Turbines for High-Energy-Fuel, Low-Power-Output Applications," M I T Dynamic Analysis and Control Laboratory Report No. 79, 1953.

11 W. Traupel, "Thermische Turbomaschinen," Band 1, Springer Verlag, Berlin, Germany, 1958.

12 R. Rotzoll, " Untersuchungen an einer Langsamlaufigen Kreiselpumpe bei Verschiedenen Reynoldszahlen," Konstruction, v o l 10, 1958, p. 122.

13 O. E. Balje, " A Contribution to the Problem of Designing Ra-dial Turbomachines," TRANS. ASME, vol. 74, 1952, pp. 451-472.

14 K. T. Miiller, " D i e Festigkeit hoehbeanspruchter rein radial beschaufelter Kreiselverdichter-Laufrader," Jahrbuch dor Deutschen, Luftfahrtforschung, 1944.

15 R. E. Strong, "Axial Flow Gas Turbines," Summer Session 1955 on Gas Turbines and Free Piston Engines, Lecture 7, University of Michigan, College of Engineering, 1955.

16 J. W. Freeman, "High Temperature Materials," Summer Ses-sion 1955 on Gas Turbines and Free Piston Engines, Lecture 5, University of Michigan, College of Engineering, 1955.

1 0 2 / J A N U A R Y 1 9 6 2 Transactions of f fie AS IE


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A Study on Design Criteria and Matching of Turbomachines ...

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