A unified notation for turbo machinery

A UNIFIED NOTATIO FOR T TR B O MACHINERY

by

G. K. KORBACHER

BibliotheekTUDelft

Faculteit der Luchtvaart· en Ruimlevaartlechllielc

Kluyverweg 1 2629 HS Delft

AUGUST 1954

TURBO MACHINERY

by

G.K. KORBACHER

'

.

.,

ACK~OWLEDGEMEN TS

The author wishesto express his sincere

thanks to Dr. G. N. Pa t t e r son, Di r ec to r of the

Institut e of Aerop hysics, fo r his enco ura ge m e nt and his interest in the progr e s s of this work.

He is also grate f u l to Mr . R. J . Thomson, Section Engine er (T h er m o dy nami cs) of A. V. Roe, Canada, Lt d ,, for his very ca r e fu l review of the manuscr ipt.

This work was ca r ried out with th e fina n cia l assistance of th e Defenc e Res e a rch Board whic h is

SUMMARY

In gas turbine analysis work, cascade tunnel investigations, and textbook treatment of turbo machinery, it would be very convenient and useful if a skeleton of identically applicable basic equations , com m o n notations and definiÜons, and a common conven tion of signs for I angles, velocity veetor-s and fo r c es, cou ld be used for

both centrifugal and axia l flow compressors an d for turbine s. The purpose of th is note is to presen t suc h a scheme. A skeleton of basic an d identi cally applica ble equations and coefficients is recorded and a sy st e m of

ii T ABLE OF CONTENTS SUMMARY CONTENTS 1. 0 INTRODUCTION 2.0 BASIC NOTATION 2.1 Symbols 2.2 Greek Symbols 2. 3 Subsc ripts 2.4 Superscripts i ii 1 2 5 6 6 3.0 SOME EXAMPLES OF FUNDAMENTAL. IDENTICALLY

APPLICABLE EQUATIONS IN FLOW MACHINERY 7

4.0 NOTES ON THE PROPOSED SIGN CONVENTION 18

4. 1 Angle Sign Convention

4.2 Velocity Vector and Force Sign Convention REFERENCES

ILLUSTRATIONS

Fig. No. Tit le

1 Stations and the mid-blade height flow element in compressors and turbines.

2 Blade diagrams and velocity triangles of a com-pressor and turbine blade row.

3 Enthalpy - Entropy diagrams of a compressor and turbine stage.

4 Sign .c o n v e ntion in stator blade rows. 5 .Sign convention in rotor blade rows.

18 19

L 0 INTRODUC TION

Th e notati on in reports and text books on gas turbines

is often confu s ing to the reader, sin ce me thods of notation and sign

conventions are ch a nged even in th e prese ntation of fundamental

equations, ac c ording to wh e th er the dis c u ssi on dea ls with cent

ri-fugalor axi a l com press ors, compre s sors or turbines and so forth.

These discrepanc i e s in notation ar e a manifestation of the pre- gas

turbine age when, for exampl • a desi g n e r of turbines might never

have des Igneda com pressor.

Toda y a gas turbin e has to be desi gn ed as a unit, an d this fact adds to th e de s i r a bility of using common notation and de fi ni

-tions and a common sign conven tion w ich wiU result in identically

ap plicable equations for the principal components of a ga s turbine.

In case s wher e res e ar c h in two relat ed fie lds, e. g. • tha t of axi a l flow compr ess o rs and that of turbine s , is combin ed,

the need for a "comm on convention of sig ns for the st a ge velocity

triangles of bo th compress ors and turbine s" is already recognized (1).

A recent publica tio n (2) emphasizes this trend. However , in favouring the conven tio n for notation and si gns the sys t e m already established

for axia l flow compr ess o rs was rigidly adher e d to, and the turbine

sys tem was adj uste d to suit th e sa me sche me . Th i s doe s in fact

neithe r resu lt in a simple nor in a universa l sys te m . On th e contr-ary,

som e sig n de finitio ns are mor e or less ar b it r a r y and therefore not

very logical. "

Payin g no tribute to what is "usua lor established " fo r

just one of th e compone n ts of a ga s tu r bine. but conside rin g c

om-pressors. turbine s. nos zles, et c .• as wha t they ar e - as flow mac h

-inery - allo ws one to build up the th eory for the whole gas turbine field

on identica lly applicabl e ba s ic eq uat.ions , Besfdes , the resulting

common sig n con ve n tion becomes natur-a l ., s im ple and therefore easy

to remem ber.

The advantage s of introducing a un ivers a l sche m e for

the who le ga s tur bine unit are: no furth er confus ion fr om a represen

-tation of one and the same equa tion in differen t fo r m s fo r rela ted flo w

machine s or engine s; simplifie d mathema tic s ; ea sier to memo r iz e

equations; less liability to errors on the part of inex p e r i e n c ed ca

2 -2.0 BASIC NüTATIüN 2.1 Symbols A a b C c d e F G g area throat area

frontal annulus area

distance of point of maximum camber.frorn leading edge maximum camber

constant

blade chord

specific heat at constant pressure (BTUjlb, oR)

o specific heat at constant volume (BTUjlb, R)

o specific heat at constant pressure (BTUjlb mol, R)

o specific heat at constant volume (BTUjlb mol, R) lift coefficient

drag coefficient

skin friction coefficient drag force

diameter

blade row width air forces

weight flow rate of gas (lb. jsec.) acceleration due to gravity

LlJ' ho rs e po w e r

,1.;, specifi c enthalpy (BTU

I

lb. )

h_T -, tota.l.head en th a l py :( h

+

(v 212gJ ) ) J: .sm ech. ' equivalent of he a t KW kil ow a t t k - cp/cv ra ti o of specific heats k 0;.C pi Cv ra ti o of s pecific heats L lift force

J. aer of o il length (span) or blade height

M .mass flow rate of gas (Gig ) in (slu gsj sec) .Ma Mac h number

ID. em pir i ca l factor Iór- devi ation

N rota ti ona l speed (r. p. m ,)

.n numbe r of stages

o b lade opening (turbine)

P abso l u te static pr-ess ur-e

'P_T abs olu t e tot a 1 head pressure

t .

.

Q am o un t of heat added (+) or ex t r a c t e d ( - ) q stag natio n pressure (

t

12g • v2)

R ga s.constant

R e Rey nolds number

RH rehea t factor

4

-S entropy

s blade pitch

T absolute statie temperature

TT absolute total head temperature ( T

+

v2 j2g.~ .opm) t profile thickness

U _{internal energygc v• dT)} u peripheral blade velocity

V volume

V absolute or relative gas.velocity

V_s sonic velocity

W e x t e r h al'. work done p. unit weight (ft. lb.'jlb.) W' exterhal' work done p. unit mass (ft. lb. tsi~) w los s in total head pressure

wp profile loss

X los s coefficient (related to inlet velocity) Y 1055 coefficient (related to outlet velocity) A.R. aspect ratio (lj c)

L. E. 1eading edge T. E. trailing edge T.F. turbulence factor

e]

c pitch chord ratio

·2 . 2 Gr-eek.Svm bols . .~

(

:!

.

cf

cf*

~

.

r

angles of the absolute or rela tive air

ve lo city vectors blade angles deviation

nom ina1 devi ation

,-de flec tion nomina1 defle.ction 5talling.deflec tion blade sta g ge r angle effi c i e nc y blade camber angle vis c osity

kinematic viscos ity density

blade cam be r inlet angle blade camber outlet an g 1e angu1ar veloc ity

Ios s facto r

work done fact or circu1ati on

CP)

flow coefficient (Va/ u)

bladel ga s velocity ratio (u/ v)

=

\Io.~

"0.1

- 6 -. 2.3 Subscripts . 1, 2.. stations a axial component u circumferential component t tip value

m mid b lade height value

h hub value i isentropic -th -theoretical max. maximum D diffuser T total head S stator R rotor ST stage P profile

00 vector mean, or relating cascade

conditions to infinity pitch

opt, optimum 2.4 SupeI'scripts

a mean or average value gained fr om integration

+

correct only under limiting conditions

*

nominal values

~.

3-.GA.SKELETON OF FUNDAMENTAL, IDENTICALLY APPLICABLE

;;

EQUATIONS FOR FLOW MACBINERY

There are two limitations which should be clearly und er-st o od and fully realized in conneefion with the equafions r-e p r e sen t edbélow:

(1) these equations are derived for the flow of a gas at low Mach

numbers through a compressor or turbine, that is for a flow,

in whic h the density

f

is conside re d to be c ons tant, unle s s otherwise indicated.

(2) these equations are derived and therefore co r r ect only for a ,s ingle radial flow element (see Fig. 1) i. e. , for tw o d im e n -sional flow only. As the flow through actual flow machinery is three- dim ens iona l, the parameters used in the e quations have to be mean'para m e t e r values taken ove r the flow cross section.

Note further the foUowing rules for th e application and for th e mem o riz-atio n of the equations th a t follow .

(1) A 11 basic eq uati ons ar e applicable as they stand to centrifuga l and axial flow compressors and turbines.

(2) The!" signs in front of a parameter (e. g .• for the work done parameter: ±W) means two things ;

(a) the equat ion is applicable to compressors and turbines. (b) if the actual values are inserted into such an equa tio n ,

the paramete r m ar-k e d with th e

±

sign come s ou t with the positive sign (+) in case of a turbine (work gained, enthalpy , pressure ar temperature drop) or with the (- ) negative si g n in case of a compressor (work input, en -tha lpy, pressure or temperatu r e rise).

(3) Ind e p e n d e nt of the type of flow mach i n e r y an equation is applie d to, a drop res u lts always in a positive and a rise in a nega ti ve resultant va lue .

(4) In all e quat ions , the inlet or initial condition parameters ar e quoted first (e. g.,

zs

h

=

(hl- h2»'

- 8

-(3 - 2) (ft lb/ lb)

)

From th e flow en!~r~y eg~ation:~~:l~ea~y:8-im p lifi e d form:

"

Q

"

'

-

i

"

-"

ei

.

[

.t~

.:

~l

"

+

V~

- VI

'

+

~]

(B TU/ sec) (3 -1)

d..~.'J .J

follows th e external, wo r k done W in a rotor blade row (compres sor or

tur bin e notation s of Fig. l a nd 2), if Ql-2 is assumed to be zero an d

G (lb. /sec) to be equal to un ity, as: ~ ~ ( ·0 V_o - V3

+W

=

J

~

,-4\.~

+

:Lr

J

where

+

W me a ns wo r k gained from the gas (turbine) and -W me a ns work done on th e gas flow (com p r ess o r).

From the momentum considerations follows another ro t or b1ade row wo r k done equatio n as:

+

W

=

t (

(J.".Vu,o - u,;.'

VIA.~

) (ft lb/lb) (3 - 3

and by the la ws of th erm odynam i cs, the work done by or on a unit weight of ga s is:

(ft lb/ lb ) (3 - 4)

Combin i ng Eqns. (3-3) and (3-4), we get the work done equation for a unit mass of gas as :

A11 these an d the fo11owing equations representing work done fo r a rotor bIade row also represent stage work done if ze ro 1055 is as s um e d, si nce no wor k is done in a stator or fixed row.

(3 -6) (3-7) (ft lb/lb) (ft lb/ lb)

)

+W'

=

J

(iv..

-.Iv . +

- l. o , - ; ; l . l .

In case of no los ses, Eqn. (3-2) written in sta ge nota tion

(see Fig. l an d 3) becomes :

and Eqn. (3 - 4) ch anges to

In case of los ses

W

s

in the stator and wR in the rotor b1ade row, .w e get for Eqns. (3-6) and (3-7):

and

±

W

=

J (

Á\,o~

-

k~

+

J.~.

v;).. )

J

3 (ft 1b/1b) (3-8)

+

W

=

J.

ty>"W\.

.

(T

oT ,- -

T~

,.

)

(ft 1bj lb) (3- 9) (3-11) (3-10) an d for-the stage:

. 'Ihe .entha1py changes, e.g. (hl -_{h 2)} i_{n Eqn. (3-2) or (hoi -h2 )}

..in Eqn. (3-8). represent the change in thermal energy of the gas flow on its

way betw e e n.tho a e stations consi dere d. Thus we get for th e rotor blade row:

~ ~

r.

a,

;;L.)

+

A~

=

(~

, _ ""~)

.

Ik, -

lA.~

\v,

-V~

(BTUjlb)

J.

~'J

d.j

.J

(V~-VO~)

(

v,J._ V:J.,.O-)

l~

.J

J.~.'J

These equations are applicable to cen trifuga l compressors

as we U. Substituting Eqn s . (3-10 ) and (3-11) into Eqns. (3-2) and (3-8 )

r-espective ly, we .g et.the work done equation in terms of ve locitie s , and this equation is agai n id e n tica11y applicable to centrifugal, axial flow

compressors and turbines.

~ :l.. ~ Jo. a, ~

+

W

= Vo - V;?,

+

u" - l.t.~ VI - v~ (ft lb/ lb) (3-J.2)

~~

.

~

~

~

~

In gener-al, for axial flow compr e s s ors and turbines Ut is

.a ssumed to be equal to u2 = u, thus changin g Eqn. (3 - 12) into the familia r

form:

(ft 1b/1b) (3- 13}

The actual temperature change À TT in a flow machinery

stage follows from Eqn. (3-9) as:

+~1;-

.

(T

OT~-T~T'=

j

'(lk,.VlAO-

u,~'

V~~)

(oR) (3-14

10

-or the actua l tota l head pr ess u r e change ~PT is given by:

+

Àï>T

=('POT~

-

~T)-

+

W'r

+

W

s

+

Wl\

=f(vo~-v?,~)

+

I (

u,,;L -

u,~)

-

f-(

v,~_v:)+

Ws

+W"p.

,

~

t

(lbj ft 2)

whereas the id eal or adiab a tic isentropic total head pressure change

A PTi is :

-+

+(~"Pr~)

=

(POT~ --P~Ti.)=

+

W·

r

(lbj ft2)

(3- 15 )

(3-16)

whe r e.th e.au p er-sc'ript H-)me ans that this equation is only correct under the

limiti n g condit ion of small pr-e ss ure ratios ( b.PTj PoTi) and under the

assum ption of con stant density.

The idea l st a tic pressure change in a flow machinery stage

de r i ved from the'Ber- no ul .II Equation is:

(3-17)

which is the mos t ge nera l farm, (se e Fig. 1) also applicable to the

centri-fuga l compres sor . For the axial flow compressor and turbine , where

uj = u2 an d V3

=

VD, we get:

(3 -18)

an d similarly fo r the ac t ua l stat i c pressure change AP:

wh ere the stat o r and rotor row losses

Ws

and wR are defined as total head

pre s sure losses, related to the inlet (compressor) or outlet (t u r b i ne )

or

. ~

w~=

Y'R.

.i:»;

~a

(3 - 2 0)

With re spect to blade cascade tunne l results it is conventent

if at le a st all basi c equa tions are rela t e d to geometri e par a m e te rs of the

velocity triangles (see Fig . 2j. The principle corre la ting re la tio n s h i ps are:

and (3-21)

Using these and other simp1e triang le re1ationships, 'Ne can

re - write our eq ua tio ns in terms of veloc ity triang1e parameters.

Eqn. (3-3) bec ome s:

+

W

=

t

_C

lA.,'V""'

~

<>G. - 1A.;a,'V,,-;..,

-l.o-n.

oC..)

(ft lb/lb)

and comhinin g Eqn. (3-9) with Eq n . (3-22) we get :

(3-22)

+w'=

J

. ~

'

cr~' (+ ÀTT)

=(

~,'

''Q.''

ioM,Clt

o-

u.~'''Q,a...~~~)

(ft lb/s lug) (3-2 3)

(3- 21)give s:

Introducing

À

= ~a. and replacing tan QC,3 fr om Eqn.

VI1.I

J

·

t

.

Cf-m.

·

(

+

À

T,.)

=

YQ.., ( \A, • •

~

a(.o -

u,~.

À

.

~ ~~).- \A..~

(ft lb/ s lug) (3-24)

- 12

-which for axia 1 flow co mprèsso rs an d turb i nes with u

1

=

u2

=

u

reduces to:

(3-26)

The ac t ual tota1head tem pe r a t u r e ch ange b. TT in a.stage.

Eqn. (3-14), can be de r i ved from E~n . (3 -25) as : ]

(±AT

T

)=

(~T\.-T~T)== cL·I.(,~·VQ,.1 (~·~~o-~~~)- ~

\:

'J .

J.o.

c.o"""

a, I

-4

T (OR) (3 - 2 7)

In the same way Eqn. (3-15) for th e actual tota l head pressure change can be written as :

+

ll.P

T

=("POTl-iiT)=J·t·~(±t:.TT)

+

~

.

,,:t~~<Lo

if the los s is re lated to the blade row outle t gas ve Ioci.tie s , Another farm for ÀP_{T i}s :

+

_

a

v.

4[ ,

_

À.~

_( , _

- A

~ ~t

11.1

~.1.~o ~~cl~

cm~~1

(3-29) =

.

+

'1'R..

]

+

1:.. (

\A,1~

-

u,~)

(1'"'/ft2) ~;l..oL~

d..2

,...

(3-28)

If we di v ide AP_{T by} Cr/2g). u22 again (see tempera tur e coefficient Eqns,

(3-25) an d (3-26), we get the so-c alled stage pres sure coeffic i e nt as:

+

±

Ä"P

T

..:L.

~~'J...

J.t

whe re

l'f.

is the tota1 h ad adi a ba tic isentro pi c sta ge effic i ency of either

a turbine or a com pr es so r . The efficiencies defined as:

1+

('1lti

w

an d

are in analysis work often as we ll exp r e s s e d by:

1Turb. =

J..

.~.

r.

Vs

+

YB.

·À~

]

, +

Jo. \A, l~~eto ~5:at~

(~O(,o- ~~<ia..)- ~

an d for the centrifugal compressor, including the diffuser behind the impeller by:

'Y},cc.

=

,+

(3-30) (3 - 31) (3-32) (3-34)

wh i c h reduces for a centrifugal com p r e s s or stage (without diffuser) to:

VQ.,I

[X

s

+ )(."

>t

1

'Yf.

=

I

+

J..

~

i: Gcx\;:0(, CAX\

~a("

(3 - 3 3)

CC

r~. ~a(,o

_

~~ci~

-

lA.~

1

llA.~ V~I

fr om which the stage efficien cy for an axial flow c ompr-es so r follows dir-e ctly, when u is substituted for UI and Uz again.

It should be clearly understood that all these equations only apply st r i c tly to a single radial flow element (see Fig. 1) through a stage.

It is cu st om a r y in practice , how e ver , to consider that radial flow element

passing through the mid-blade height of a blade row as representing the

mean blade row or mean stage va lues, which are defined as, e. g.•

",I;

1

~It

X .

d.A .

"Q. .

O'

rAl;.

/tI..

J

.\.t

~It

'Jt .

riA .

VQ..'

r

M

for the mean total head tempe r at ure . Unfortunately this simple assumption doe s no t ho l d in practice. Tes ts on axial flow compressors have shown

- L4,-

-calculated for a flow element pas sing thr o u gh the mid blade he igh t ,

'There for-e, a factor ca lled the "work done factor

n

" had to be introduced to correct for th e error involve d when mid-blade hei gh t values are taken instead of me an va lu e s.

This work do ne fac to r may be introduced now into, e. g .•

Eqn. (3 - 3) or Eqn. (3-22) giving:

or:

+w

=

.n.. .(

'A",

'1~o

- u,)..'1\A,

~)

m

t

.

(f t lb/ lb ) (3-35)

(3-36)

where m means th a t in these equations , mid-blade height va lue s are to be used . ..Q. is also us ed in the tem pe ra t u r e coefficient equations (3-25) an d (3 - 26). the te mpera ture change equation (3 - 2 7). the pres sure change equation (3- 2 8) an d the pre s s u r e coe ffi cient equa ti on (3 - 2 9) . But it is not us ed in efficiency equa tions like Eqn. (3 -3 0) or Eqns, (3 - 31) and (3-32). as it is as sumed that the mean stage pressure losses are affected in the sa m e proportion as the mean stage isentropic pressure change (s e e Fig. 3. h-S diagram). For th e centrifuga l com p r e s s or s .Q. is not used so far; and for turbines th e wor k done fa c t or is assumed to be unity due to lack of better knowledge.

Th e r e are numer-ous other equations wh i ch could be quoted he re, as. e.g.• th e to tal hea d or sta ti c pre s sure rati os:

(3-37)

for a centrifuga l compre s so r. including the diffusing se c tion adj a c e n t to the impe l ler-, or

= ( \ -

(~o~

-

..t~)

tr~

.1öi.

A

] -!I-I

_(3-38)

for the static pressure ratio in a compressor or turbine . But the r e is still an othe r group of equati ons whic h sho uld be mentioned. Sta rting off

with the well known relations h i p between los s wand drag coefficient

Co

i

r

_

S'

w·

coe

o(,co$ S

X

~

-

--I

.

~J,~s

c.L

.

v~s

c

~,

where oL_oos can be calculated from (see Fig. 4):

(3-39)

(3-40)

which reduces for axial flow compressors an d turbines in genera l, assuming

v" -

VQ.o - VQ..~ S ' to:

(3-41)

Combining Eqn . (3 - 3 9j with th e loss equation (3-20), related

to the blade row inlet ve loc ity as usua l for com p r e s s ors (for turbines to v 2)'

we get for the stator blade row:

X

s

an d for the rotor blade row:

S

?>

- . ~ c(.aos

C

(3-42)

(3-43)

Introducing these ter ms into Eqn. (3-33), a,.ssuming

ul :;; u2 = u and À = 1, we get the axial flo w compressor stage efficiency as:

- 16

-A pplying this equation to a 50% re act i on axial flow com

-pres sor sta ge (s ee Fig. 2) in whi ch: ~ ~ 1.0, d,o :: - t/..₂; CDs ~ CDR or _{X s} ~ XR; d, =: - ~l ;r. ~3 and 0(,00S - - d.co ~ we ge t.: (3-4 5) VG/

X

~. ~~c(..

~

(J,o -

iwn..

~

-

I

+

~~--...;;;~-=---If finally the Uft coefficie nt is int roduced, which in its general form is given by:

(3- 4 6)

and which reduces to the theoretical lift coefficient CL if the drag term

C!)oo ' tan c:J.,oos is neglec ted, Eqn. (3- 45) bec omes : CD

_ I -

__

..;..,;J... • _C_D_ =

~

J..cL

ao S CL~

,+

CLt.~ (3 - 4 7)

whe r e Va /u ~

cp

is the so -c aUed flow coeffic ient. Now compare Eqn.

(3-31) with Eqn. (3- 44). and it is ob vious th at from Eqn.. (3- 4 7) follows

immediate ly the turbine e

,

ffic i e ncy as:

(3 - 4 8)

As a last examp le for th e advantages con n ec ted with tdentica. Ily applic a ble

equati ons for comp r es sor and turbine s , the "s tage re ac t ton" may be

considered. Th e rea ction is defineda.s ::

~I

-..ft.

Re ac t.

=

(3- 4 9)

Usin g Eqns. (3-2) and (3-4 ) to eliminate (h l -h₂₎ and Eqns . (3-8) and (3-9)

to rep1ace (hoi -h2) we get :

React .

_

J,

g..J

.

Cf

-m.

.

a

TT

(V

o~

-

Vö"J-)

~

.

J

.

'fm

.

A

TT -

(v

~

-

v./

-)

(3-50)

In pr-actice , as À is not far off unity fo r compressors and turbines.

v

may be conside r ed equal to

v

3' thus sim plify in g Eqns. (3-50) and (3-5 1) to:

Rea ct.

=

-

~:

. (

~ ~I

+

I\'

-kun.

~

-cl

J&...

.

(-kwn. ~, -À~~\

V"' I 0) (3-52)

This equation reduc es fo r À ::: 1to:

(3-53)

In case of a 50% reaction sta ge (ax i a 1flow com.pr-easor- or turbine). substituting

we get:

- 18

-4.0 NOTES ON THE PROPOSED SIGN CONVENTION

As, of cou rse, definitions of factors, ïor ces, coeff'icierits

etc, , are not ch an ged, they are omi tt ed here. An exhaustive classification of all definitions in use is gi ven in (2). as we ll as a complete lis t of pro-pos ed nota tio n , w ich is us ed in this repo rt in principle.

Th e pr op osed sign convent ion for angle s , forces and

veloc ity vec t o rs is shown in Fig. 4 for sta tor (fixed) blade rows and in Fig. 5 for rotor (m ovi ng) blade rows .

4. 1 ANGLE SIGN CONVENTION

All flow and blad e an gles ar e measured from the axial

.

.

direction. Only th e cam b .r angles Xl ' an d :x.₂ are measured from the cho rd Iine, whic h takes over th e fu nc tions of the axial direction in de

fin-ing the cambe r.angle si gns.

• I.. ..

One flow veloc ity vector on ly, that of the abso lute inlet

ve locity into a row, define s th e si gns for all blade an d flow an g les of that

row. In a stator blade row (see Fi g . 4), 0(.. is always positive, whe r e a s in a rotor blade row (s e e Fi g . 5), 0(.0 is defined as positive on ly if its velocity vecto r has a component in th e u-'direction, the direction of

rotat~on. · But 0(,0 bec ome s negative if its ve l oc ity vector v0 has a

com-ponent pointing aga i ns t u. This si m ple an d easy to remember sign

con-ve n tion si m plifies som e userulangle relationships to:

(3 -

~

+

JC with

(3, ..

~

+

:X,

an d ~~

=

~

+

X;».

or

e

= Je, -

X

~

=

(JJ,-

~~

wh erea s the following relationshi~s remain unchanged:

E -

~,

-~~

=

a

+

À,-d

tan do

=

4.2 VE L O C I T Y VE CT OR AND FOR C E SIGN CONVENTION

Ve locity vector signs follow directly fro~ the ve Io c ity

ang1es, as:

"lAl

_va.,

i. e. , the circ umferentia1 velocity vector v u has always the same sign as

the angle oe. • Or ind ependantly of flow angles, a vetocity vector v u

(whirl velocity) pointing in or against the direction of rotor rotation is

positive or nega ti ve res p ec tive ly.

Th e rota tiona 1 spee d u is always positive. '{he velocity

vect or in th e ax ial di rection is pos itive, if pointing downstream and

negative if po i nting up s t r e am.

The lift an d dr ag Ier-ces the gas flow exerts on the blades

are positive on 1y , and the same holds fo r F u. The pressure forèe<F'a~anbe

positive or negative, deperiding on whethe r acting down or upstream,

respectively. Since

(lb/ft)

assum i n g val

=

_{v a 2 an}d a blade length of unity, it is on1y logical to define

Fa pointing upst ream (- A P) as negative.

If F u is used to calcu la t e the rotor power output or input

(com pr es s o r ) from:

+

W

=

+

1=u. .

11.' ( i j

~

(ft. lb/ lb)

in ca s e of th e compr e s so r, the reaction force to F u in the direction of

-1. 2. Ainley, D.G. Gray. S. REFERENCES

A Common Convention of Signs for th e

Stage Velocity .T r iangle s of Both Corn

pressors and Turbines.

N.G .T.E. Aerodyna.mic Note No. 254

(1949)

F1uid Dynamic Notation in Current Use

atN.G.T .E .

A.R. C. Technica1Report. C.P. No. 97

CJ

I

~ ~ • I

---t-~-o

COMPRESSOR STATIONS "4 • 3 Jo _..:r...,_ ..~.'f"~-;;

o

_-lJ

-OR TURBINE

FIG. 1 STATIONS AND THE MlD BLADE HEIGHT FLOW ELEMENT IN COMFRESSORS AND TURBINES

TURBINE BLADE DIAGRAMS \A,~ COMPRESSOR

u..

~ ROTOR :.;': ~ ,( ,( ( 0 STATION: 0 STATOR l.C.,), VELOCITY TRIANGLES

Vlt~

~

Vu.,..o- - - -·

u...

~. Lt.,),

FIG. 2 BLADE DIAGRAMS AND VELOCITY TRIANGLES OF A COMFRESSOR AND TURBINE STAGE

t

.

-i1--~

+

~~

. , J

...

~-~-~-~

:J

.J

~~

...

J

"'r~

,

o

STATION:

o

STATION: 0

ANGLE SIGN CONVENTION

~ÄlI

/ : ®-'-.l--B

~

---STATOR BLADE ROW FORCE DEFINITION

G-"

~ / ~I - t .- ..

y5r

\

TURBINE I I _I!l~:" ~~

<:

//1 I COMPRESSOR

u.

COMPRESSOR

~---

...

~~

+

"Fu.

,

cl. \..,~ - •

ANGLE AND FORCE SIGN CONVENTION

®-

-

-I STATION I ---~"':w:K"""":'~. . ; . , " "

-- rU. ...--.---...;:

a

ROTOR BLADE ROW

+L

ANGLE AND VELOCITY

VECTOR SIGN CONVENTION . /

Va...T ,

v,t

~ ~ ~ !

.-+

'Fu.

w-w

cLt. -0(

...

.

I

Qo .

_

.

.

G)

STATION

rA

U,I

--~~~

11

~",

..

I..--VlL

--.l "

...

u

_

--~-,I - vU,3

10.-

--lA, TURBINE 1-4-- -vI.&. J.

UTIA REVIEW NO. 7

InstituteoF Aerophysics, University of Toronto

A Unified NotationFor TurboMa chin ery

G.K. Korbacher, August, 1954, 19 pp., 5 fi gs.

~Q

~~~_~

UTIA REVIEW NO. 7

Institute of Aerophysics, Univeristyof Toronto

A Unified Notation For TurboMachinery

G. K. Korbacher, August, 1954, 19 pp., 5 figs.

•

1. Compressors 2.

I Korbacher , G.K.

Engines, Aircra ft 3. Ga s Turb ines

II UTIARevie w No. 7

4. Jet Propulsion l. Compressors 2.

I Korbache r , G.K.

Engines, Aircraft 3. GasTurbines

11 UTIA Review No. 7

4. Je tPropulsion

In gas tur b i ne ana lys iswork, cascade tunnelinvestigations, andtextb o ok tr e a tme n t

ofturboma chi nery , it wo uld be ve r y convenientand us eful if a skelet on of identica ll y applicab le basic equations, commonnotations and definitio ns , an da

com monconvention of signs for angles, velocity ve c to r s an dforces , could be used

for both centrifugal andax ia l flow com p r e s s o r s and for turbines.

The purpose of this note is to present suc ha sch e m e . Askeletonof bas ic and

identically ap p licab le equatio ns and eoefficien ts is ree ordedand a system of

eommonno tation anda co mmon co n v ent ionof sig ns is de fined .

In gas tur bineana ly sis work, cas c a d e tu nn el investiga tions, an d te xtb ook tr-eatrnent

oftur b omachiner y, it wou ld be very co nv e nient and usefu lif a skele t o n of

identically applicable ba s i c equati o ns, commo n notationsand definitions, and a commonconventionof signs for angle s, veloc it y vectors and forces, could be used for both centrifug a land axial flow compressors and for turbines.

The purpose of this note is to present su eh a seheme. A skeleton of ba sicand

ide nlically appli eableequations and eoefficientsis re e orded an d a systemof

eom mon notalionand a com mç ncon ve n t ion of signs is defin ed.

Copies obtainable from: Instituteof Aer ophysics, Universityof Toronto,Toronto 5, Ontariq Copies obtainable from: Instttute of Aerophysics, University of Toronto, Toronto 5,Ontario UTIA REVIEW NO. 7

Insti tu te of Aerophysics, Uni ve rsity of Toronto

A Unified Notation For TurboMach inery G. K. Korbacher, August, 1954 , 19 pp., 5 figs.

•

UTIA REVIEW NO. 7

Inslitute of Aerophys ics, Unive rsityof Toro ntö

A UnifiedNotation For Tu rboMac hin e ry

G. K. Korbache r, August , 1954, 19 pp. , 5 figs.

•

l. Com pressors IKorbaeher , G. K.

2. Engines, Ai rc ra ft 3. 11 UTIA ReviewNo. 7

Gas Turbines 4. Jet Propulsion l. Compressors 2.

I Korbacher, G.K.

Engines , Aircra ft 3. Ga s Turbine s II UTIARe view No. 7

4. Jet Propulsion

In gas turbine analysis work, cascad e tunnelinvestigations, and tex tbooktreatme nt

oftu rbomachinery, it wouldbe ve r y convenient an d usefulifa ske le to n of

identicallyapplicable ba sic equ a tions, commonnotations anddefinit ions, an da

common convention of si g ns for angles, velocity vectorsand forces, could be used for both centrifugaland axial flow com p ress o r s an d for turbines.

The purpos e ofth is no t e is to pres ent sueh a scheme. A skeleton of basic and

identically applieab le equatio ns an dcoe fficien t s is recor d e d and a system of

common nota tion anda com mo nconvention ofsig ns is defined.

In gas turbineanalysis work, cascad e tunne Iin vestigations, an d textbook treatment

of turbo machiner y, it would be ve ry con ve n ie nt anduse fu l if a skeleton of

identicallyapplicable basic equations, commonno ta tio ns andde fi n itio ns , and a common convention of signs for angles, velocity vectors and forces, could be used for both centrifugal and axial flow compressors and for turbines.

The purpose of this note is to present such a scheme. A skeleton of basic and

identicallyapplieable equationsand coefficients is recorded and a systemof

com m on notation and a com m on convention of signs is defined.

UTIA REVIEW NO. 7

Instituteof Aerophysics,Universityof Toronto

A Unified Not a tionForTu r b o Ma chin e ry

G.K. Korba cher, August, 195 4, 19pp., 5 figs .

•

~

~

'J

UTIA REVIEW NO. 7

Institute of Aerophysics, Univeristy of Toronto

AUnifiedNo tationFor Tu rboMa chin ery

G. K. Korbache r , August, 195 4, 19 pp., 5figs.

.,

I. Compre s s ors 2. Engi nes, Airc ra ft 3. Gas Turb i nes

I Korbache r, G. K. 11 UTIA ReviewNo. 7

4. Jet Propulsio n I. Compre ssors 2.

I Ko r b a che r, G.K.

Engines, Airc ra ft 3. Gas Turbine s

11 UT IA ReviewNo. 7

4. Je t Propulsion

Inga s turbine analy sis wor k, cascade tunnelinve stigations, and tex tb oo k tr eatmen t

of turbo ma chin er y, it wouldbe very convenie n t and us e ful ifa skele ton of identic ally app licab lebasicequa ti o ns , commonno ta ti o ns and definitio ns, anda

commonconve ntio nof signs ïo r- angles, velocityve c t o r s andforces, couldbe use d

Io r both centrifugal and axi alflow compress orsand fo r turbines.

The purpos e of this note is to pre sen tsuc ha sch e m e. Askeletonof basicand identi cally app lica b le equa ti ons an dcoefficien ts is re cordedand a syst emof

com mo n not ation anda co m monconve ntionof signs is defined.

Ingas turbine analy sis work, cascade tunn el investigations, andte x tboo ktrea tme nt of turboma chinery, it wo u ldbe very convenient andusefulif a skele ton of

identicallyapp licab le ba s ic equations, commonnotations and definitio ns , anda

commo nconve ntion of signs for angles , veloc it y ve c to r s and forces, couldbe used for bothcentrifu ga l and axialflowco m pressorsand fo r turbines .

The purposeof thi s note is to presen t such a scheme. A skeleton of ba s ic and identicallyapp li c ab le equations and coefflcientsis recordedand a system of

common notation and a com m o nco nve nti on of signs is defined.

Copies obtainabie from: Ins tit ut e of Aerophysics, UniversityofTo r o nto , Toronto 5, Ont a r to Copiesobtainablefrom: Instituteof Aerophysics, University of Toronto, Toronto 5,Ontario UTIA REVIEW NO. 7

InstituteofAerop hysics, Unive rsityof To ronlo

A Unified Not ationFor TurboMachinery

G.K. Korbacher, August, 1954 , 19 pp., 5 figs.

i

UTIA REVIEWNO. 7

Instituteof Aerophysics, University of Toronto

A Unifie d Notatio nForTurbo Machinery G.K. Korba c her , August, 1954 , 19pp,.; 5figs .

i

I. Compre sso r s

IKor bacher , G.K.

2. Engines, Aircraft 3. Ga s Tu r bine s

11 UTIAReview No. 7

4. Jet Pro puls io n I. Compre ss ors 2. Engines, Aircraft 3. GasTur b in es I Ko rbache r, G.K. 11 UTIA ReviewNo. 7

4. Jet Propulsion In gas turbineana lysis wor-k, cas ca de tunne linvestigations , and te x tb oo k treat me nt

of turbo machinery, it wou ld be very conve n ient and usefulIra skele ton of ide nt ic all yapp lica b le ba sic eq ua tio ns, com monnotations and definitions, and a commonco nve ntionof sig ns Ior- angles , velocity vec to r s and forces, co u ldbe used for bothcentrifugalandax ia l flow com p ress o r s and for turbines.

The purpose ofth is not e is topresent such a scheme. Po skeletonof basic and

identic ally applicable equa tio ns andcoeffic ien ts is recor de dand a system of

common not ationan da co m mo nconve nt ion of signs is defined.

Co pi e s obtainabie from: Institute of Aerophysics, UniversityofToronto, To r o nto 5, Ont a r io

In gas turb ine analy s is work, cascade tunne linvestigations. andtextbook treatment

of turboma c h ine ry , it wou ld be veryconve nientand useful if askele to n of identica llyapplicab le ba s i cequa tions, co m m onnotatio ns andde fin ition s, anda com mo nconventionofsigns for angles, velocity vee tor-s and for c e s, couldbe used

fo r bot h centrifuga l and axial flow co m p res so r s andfor tu rbi ne s.

The pur poseof this no t e is to pres e n tsucha scheme. A skeletonof ba s icand identica llyapplicab le equa ti o ns and coe fficients is re c orde d andasyste m of com m on not a ti on anda commonconve ntion of sig n s is de fi ned.