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iRT SECTION 

NPS-57MA70081A 

ft 93940 



// 



United States 
Naval Postgraduate School 




# 



COMPUTER PROGRAM FOR PREDICTION 
OF AXIAL FLOW TURBINE PERFORMANCE 

by 
Ennio Macchi 
August 1970 



This document has been approved for public release 
or sale, its distribution is unlimited. 



FEDDOCS 

D 208.14/2:NPS-57MA70081A 



NAVAL POSTGRADUATE SCHOOL 
Monterey, California 

Rear Admiral R. W. McNitt R. F. Rinehart 

Superintendent Academic Dean 



ABSTRACT: 

The report presents a computer program for prediction of performance 
of single-stage axial turbines of given geometry. The three-dimensional 
method developed by Vavra is applied, taking account of streamline 
curvatures and slopes, as well as enthalpy and entropy gradients in 
the solutions of the equation of motion, and of boundary layer thick- 
nesses in the continuity equation. 

A choice among five different loss correlation methods and two 
flow angles correlations is offered. Loss coefficients and flow 
angles are automatically calculated from blading geometry and actual 
flow conditions for every streamline, according to the selected 
correlation method. 

A fair agreement of predictions with several actual turbines 
experimental results was found in ref. ihl , where also the applicability 
of different available correlations is discussed. 



TABIE OF CONTENTS 





Page 


Introduction 


9 


Method of Analysis 


10 


Method of Solution 


19 


Flow Angles and Loss Coefficients Correlation 


21 


Computer Program 


25 


Acknowledgments 


29 


Appendix 


51 


Bibliography 


101 



LIST OF ILLUSTRATIONS 

Page 

Fig. 1 Coordinates, Stations and Streamline Nomenclature 30 

Fig. 2 Turbine Thermodynamic Process 31 

Fig. 3 Symbols in Continuity Equation 32 

Fig. k Computer Program Flow Diagram 33 

Fig. 5 Velocity Diagrams 3I+ 

Fig. 6 Blade Geometry Nomenclature 35 



LIST OF TABLES 

Page 

TABLE I Fixed Imput Data 36 

TABLE II Variable Imput Data 38 

TABLE III Example of Variable Imput Data kl 

TABLE IV Instructions Relative to Print Indices 43 

TABLE V Instructions Relative to Correlation Indices 44 

TABLE VI Example of Normal Output Data 46 



LIST OF SYMBOLS 



SYMBOLS 

A 

ACL 
a 

a 

C 

ci 

C P 

dA n 

E 

h 

H 

H 

K 

KCL 
k 

— » 

i 
i 
L 
M 
m 
m 

mCL 

NSETS 



P T 
P 

q 

% 
Rh 
Rt 

r 

S 
S* 



Area (in ) 

Tip clearance area (in^) 

Throat opening of blade channel (in) 

Non-dimensional throat opening, a/ a 

Blade chord, see Fig. 6 (in) 

Rotor tip clearance (in) 

Specific heat at constant pressure ( BTU/ lb m - °R ) 

Infinitesimal area, perpendicular to the flow (in^) 

Blade radius of curvature, see Fig. 6 (in) 

Frictional force per unit mass (lb/ slug) 

Total enthalpy (BTU/lb m ) 

Boundary layer form factor 

Boundary layer energy factor 

Coefficient, see eq. (31) 

Curvature coefficient, see eq. (l6) 

Unit vector 

Incidence angle 

Mean distance between the stations and 1 and 1 and 2 (in) 

Mach number 

Boundary layer exponent 

Mass flow rate (slug/ sec) 

Mass flow rate passing thru the rotor tip clearance (slug/ sec) 

Number of conditions for which performance values have to be 
found 

Blade opening (in) 

Total pressure (psi) 

Static pressure (psi) 

Heat exchanged with external sources per unit mass (BTU/lb) 

Gas constant (ft-lb/lb m -°R) 

Radius at blade hub (in) 

Radius at blade tip (in) 

Radius (in) 

Entropy (BTU/lb-°R) 

Non-dimensional entropy, S/ Cp 



T Blade maximum thickness, see Fig. 6 (in) 

T Static temperature (°R) 

Tip Total temperature ( R) 

TE Thickness of "blade trailing edge, in peripheral direction, 
see Fig. 6 (in) 

TN Thickness of blade trailing edge, in normal direction, 
see Fig. 6 (in) 

t Time (sec. ) 

u Peripheral velocity (ft/ sec) 

Y Absolute velocity (ft/ sec) 
W Relative velocity (ft/ sec) 

x Non-dimensional radius, r/r m 

Xg See eq. (30) 

Y Non-dimensional axial velocity, VpfVpm 

Y Total pressure loss coefficient, see eq. (h-0) 
z Number of blades 



GREEK SYMBOLS 

oi Absolute gas flow angle, see Fig. 5 (deg or radians) 

P Relative gas flow angle, see Fig. 5 (deg or radians) 

g* Pressure ratio, see eq. (hk) 

P Q Blade inlet angle, see Fig. 5 (deg or radians) 

Y Specific heat ratio 

Atw Decrease in efficiency, due to rotor tip leakage, see eq. (V?) 

AR Streamline displacement, see Fig. 1 

<5r Streamline displacement, see Fig. 1 

Q Kinetic energy loss coefficient 

C* Kinetic energy loss coefficient used in the continuity 

equation, see eq. (28) 

£ Kinetic energy loss coefficient for low Mach numbers, 
see eq. (k2) 

Cql Kinetic energy loss coefficient due to rotor tip leakage, 
see eq. (hf) 

TV™ Total/ static efficiency 

^ TT Total/total efficiency 

9 Angle between two radial planes (deg. or radians) 

K Angle between the flow and the axis of the turbine in a 

meridional plane (deg. or radians) 

\i Gas viscosity (lb/ sec- ft) 

§ Area restriction factor, see eq. (27) 

p Gas density (lbjj/ft 3 ) 

$ Non-dimensional flow function 

Xp Coefficient in Traupel's method of loss correlation, 
see eq. (kG) 



O) 



Angular velocity (radians/sec) 



SUBSCRIPTS 

A Axial 

E Equivalent 

is Isentropic expansion from total inlet conditions 

m Mean streamline 

m Meridional 

R Rotor 

r Radial direction, cylindrical coordinates 

S St at or 

z Axial direction, cylindrical coordinates 

9 Peripheral direction, cylindrical coordinates 

Station ahead the stator 

1 Station between stator and rotor 

2 Station after the rotor 



INTRODUCTION 

An accurate knowledge of performance achievable with turbines 
of fixed geometry, both at design and off-design conditions, is 
important in advanced propulsion and power systems. The more accurate 
the prevision is, the more meaningful the optimization of the turbine 
itself and of the overall system will be. 

A first way to solve this problem consists in the testing of a 
prototype, but it presents several disadvantages, namely it is 
expensive and time consuming. Furthermore, if effects of turbine 
geometry changes have to be investigated, different prototypes 
will be required. 

An alternative solution to the problem consists in theoretical 
methods of prediction. The present availability of high speed 
computers allows one to solve systems of differential equations and 
to perform a large number of iterations in a short time. Moreover 
computer programs give the possibility of changing imput data, so that 
a large variety of different turbine configurations and of working 
conditions can be investigated. 

This report presents a computer program, derived by the 
three-dimensional calculation method given first by Vavra [l] and 
then developed by Eckert L2J and Harrison [3J. This method takes 
account of enthalpy and entropy gradients and of streamline curvatures 
and slopes in equations of motion, and of boundary layer thicknesses 
in continuity equation. It is applicable to single-stage axial 
turbines with unchocked flow. 

A critical point in all theoretical performance calculations 
is the prediction of the losses that occur in the blade rows. In 
the present program, one can choose among five different loss correla- 
tion methods. For each method pertinent subroutines compute loss 
coefficients from blading geometry and flow characteristics. A 
comparative study of these methods can be found in [U], where they 
were applied to actual turbines of different design practices, in a 
wide range of all significant flow and geometrical properties. 



The comparisons presented in [4] with experimental performance 

showed a fair agreement of predictions with all turbine stages 
that were examined. 

The present report is divided into four sections: in the first 
the used equations are set up and necessary assumptions are discussed; 
in the second the method of solution is described; in the third 
different losses and discharge flow angles available prediction methods 
are described; and in the fourth the instructions for the use of 
computer program are given, with an example of imput and output data. 
Eventually in the appendix the FORTRAN list of the MAIN routine and 
of the 28 subroutines is given. 

METHOD OF ANALYSIS 

The flow in turbomachines is ruled by the three fundamental 
laws of fluid dynamics, namely: 

Equation of Motion 

Energy Equation 

Equation of Continuity. 

All proper three-dimensional calculation methods proposed 
in the technical literature solve the differential equations derived 
by these laws. Differences among these methods are due to the three 
following reasons: 

different stations are chosen where equations are solved; 
different hypotheses are formulated to describe the real 
processes occurring in the machine; 

different numerical methods are applied in the solutions of 
the equations. 

As far as the first problem is concerned in the present method, 
the equation of motion is applied at three stations, namely ahead of 
the stator, after the stator, and after the rotor, indicated respectively 
with subscripts 0, 1, 2 in Fig. 1. On the contrary, since blade geometry 
is given, the continuity equation is applied at blade throat sections 
1* and 2* in Fig. 1, as suggested in [5] . 



10 



those occurring in the "boundary layers along the blade surfaces, and 
flow frictional losses between blades can therefore be neglected. 

Assumption h) is reasonable for uncooled blades, where energy 
changes due to heat transfer are negligible with respect to those 
related to velocity changes. If blades are cooled, the additional 
term q in the energy equation must be specified. Assumption 5) is 
valid in most cases. If the thermodynamic processes occur in ranges 
where real gas effects are not too strong, average values of gas 
properties give good approximations. 

Some additional assumptions, regarding streamlines slope and 
curvature, are discussed in the section pertaining to the equation 
of motion. 

As far as the third problem is concerned, the numerical 
small difference method was found to give a good convergence, and 
was therefore applied for the solution of the differential equations, 

The following sections give the derivations of the equations 
with the above- listed assumptions. 
Equation of motion 

The general equation of motion for relative flow in vectorial form, 
is: 

H + V H R = Wx (V x W + 2uu) + T V S + f f (l) 

(2) 
For absolute flow, where \ v ■■■■ W (3) , equation (l) 




ih) 



becomes: 



|¥ +VH= V x (V x V) + T V S + ff (la) 



"35 



With hypothesis l) iH = o, S = 

ot i3t 



and hypothesis 2) "? = 



11 



This allows one to more accurately evaluate the pressure drops 
occurring in blade channels, since in this way it is possible to 
account for boundary layer thicknesses. This problem is particularly 
important at high Mach numbers where a small change in flow area causes 
large differences in pressure ratios. 

As far as the second problem is concerned, the following 
assumptions are made: 

1) Steady flow 

2) Frictionless flow at the stations where the equations of 
motion are solved. 

3) Axisymmetric flow at the stations where the equations of 
motion are solved. 

k) Adiabatic flow. 

5) The working fluid is a perfect gas, namely a gas having 
the equation of state: 

p =P% T 

and specific heat at constant pressure is independent of temperature 
and pressure. 

Hypotheses l) and 3) are interrelated since the flow can be steady 
in both rotor and stator only if it is axisymmetric. Although they 
are correct only for cascades with an infinite number of blades, 
axisymmetry and steady conditions are necessary assumptions since any 
other hypothesis would require unknown quantitative information on 
flow inside the blade rows. One problem connected with these assumptions 
is that downstream conditions cannot affect the flow upstream. There- 
fore pressure distributions at rotor exit can deviate from those imposed 
by the discharge conditions. 

With assumption 2), all entropy changes are supposed to occur in 
the blade rows ahead of the considered station. This assumption can 
be justified by the fact that velocity gradients in the regions, where 
the equation of motion is applied, are in all probability smaller than 



12 



equations (l) and (la) become respectively: 

VH R = Wx(vxW+2ao) + TVS (5) 

VH = Vx(^7xV) + TVS (5a) 

The following derivations for equation (5) are valid also for 
equation (5a), if substitutions (2), (3), (h) are performed. 

It is convenient to express equation (5) in cylindrical coordinates 
(see Fig. 1 for symbols). Equating the three components of equation (5), 
the following equations are obtained: 



a. tangential component: 

.)] w r i d(rW u ) 

—J - *» w r - - -§q 



1 BHr W A '% 3(rW u ) ] W r ," a (rW u ) aw r "j T as ,,, 



b. meridional component: 

3H R T7 \ d W r Sw A 1 Wu f ^W A Q(rW u ) ] dS 

c. radial component: 

dH w u ,"a(rw u ) sw r l f aw aw A -, d 

__£ = _ L — s ^J - w A L -s=- - -3- + 2uu w u + t ^ 

ar r or aG A dz Sr J u or 



(7) 



with hypothesis 3) To ( ) = ° 



and hypothesis ^) w d t • v Hj> = 

the system of equations (6), (7), (8) reduces to equation (9): 

sh r w. a(rw ) aw aw A ^ s 

-Jl = -S .^ - W A — £ + W A -4 + 2uu W u + T ^ (9) 

<5r r or A az A or u or 



13 



2 
Using substitution: H E = H R + ^ (l°) 

with hypotheses 5) H = c p T and k) H^ = const (see Fig. 2) 

T 2 = -T " "2^ (U) 

Up ^>-p 

and introducing the non-dimensional quantities: 



(12) 



Y = 


w Am 


X = 


r 

m 


S* 


S 
= C P 



(13) 



(1*0 



Equation (9) becomes: 

1 SY 2 2 . , 2r m 6W r 1 d S *s + dP 

2 . 2ft 4 sin p cos g 2 U m U cos 2 p 2 cos 2 p 3H E 

+ — sin p + TT 1 ~ + 5 — 5 o — o~ —^ + 

x W Am Y W A 2 Y 2 W M 2 Y 2 ° X 

r 2 h e cos2p . 2r 1 as* n ,^ 

+ L w 2 y 2 - Sin ^J -^ = ° (15) 

Am 

aw r 

In this equation, terms containing cos \ and -r — depend respectively 

on streamlines slope and curvature. Vavra [lJ showed that the most 
reasonable assumptions, when only three points of a streamline are known, 
are the following (see Fig. 1 for symbols): 

W A Sz L 2 



2 A 
cos A = 



L 2 + (f) 2 



(17) 



14 



In equation (l6) the positive sign holds for station (2) and the negative 
one for station (l), for 6r positive as shown in Fig. 1. Suggested values 
for k are between h and 6. Finally with equations (l6) and (17) into (15) 
the following equation is obtained: 



dS* 



* (^ y2 > cos% r £kr a r * + if f 

= - COS p ■ 2 k r — y? - 75 

dx m ]? l 2 dx 

n d6 2 ? ^ u m sing cosP 

- 2 tan 6 — - — sin p 



dx - x ^ Xii ^ " W^Y 

2 U m U cos 2 p 2. difc -2 H E cos 2 ? . 

" 2—5 + g°V — " I —3 3 ' Sin P T^ ( l8 ) 



dS* 



For the station after the stator row, the following equation can be 
derived with (2), (3) and (k) in equation (l8): 

&±£l . - co, 2 a [(-2 K r m i§ ) - lU^l^f -#|1 J 



dcv 2.2 
-2 tan *,. -g; - 5 sin ^ + 



-"-m 2 y ~ 




T 2 dx 

Li 


2 cos 2 a 

Y2 V, 2 
Am 


dH 
dx 


j 2 H cos 2 Oi 
Y 2 V M 2 



. 2 
sin a 1 



dS* 
dx 



(18a) 



Equation (l8a) could also have been derived directly from equation (5a) 
In equations (l8) and (l8a) the last term is depending on flow 

irreversibilities due to frictional losses. The specific entropy S* will 

be calculated, after LlJ, by: 

Y ' W Am 2 



r 2 Hp cos 2 fi 
S* = lnl 2— 1 1 + S v * (19) 

Y W M 
1- 



2 % cos 2 ? (l-;) 



l E 



15 



and by: 



S* = In 



2 2 

- Y V Am 




2 


H Q cos 2 ^ 




Y 


VAm 2 




2 


H cos 2 p 


'1-0 



+ s. 



(19a) 



where £ represents the kinetic energy loss coefficient which is 
established by the chosen loss correlation method. 
Energy equation 

The energy equation in a relative flow is given by: 

a w 2 
fldt-V (H R ) = Wdt-V(q) - ^ (iL) dt 



(20) 



and in an absolute flow, with (2), (3) and (h) in (20), : 



o ,V C 



Vdt-V (H R ) = Vdt'V(q) _ ^- (^-) dt 



(20; 



where q is the heat added to a unit mass particle from external 
sources. With assumptions l) d/dt ( ) = and h) q = 0, equations 
(20) and (20a) become: 



Wdt-V (H R ) = 
Vdr-V (H) = 



(21) 
(21a) 



Equations (21) and (21a) indicate that for steady, adiabatic flow 
the relative total enthalpy is constant along any streamline. This 
does not mean that V (H R ) = 0, since relative total enthalpy can assume 
different values for different streamlines. 
Equation of continuity 

Pressure ratios across a stage necessary to discharge a given 
flow rate are function of inlet Mach number, blade openings and the 
losses occurring in the blades. For an isentropic flow the mass flow 
rate passing thru a section dA n perpendicular to the flow is given by: 



16 



a * is = Pis V is dAn (22) 

where Pj_ s and V\ g are respectively the density and the velocity 
corresponding to an isentropic process. It can be shown that 
equation (22) can be expressed for a perfect gas (see assumption 5) 
by: 

dm is = T $ dA n (23) 

v'R G T T 

where Pp, Tip are, respectively, total pressure and total temperature 
(constant for isoentropic and adiabatic flows) and * is a function of 
pressure ratio and gas isentropic exponent, namely: 



/p v i~p 2/y p Y+l l 

• ■ / Hi !J%> - <%> Y J (2k) 

With the symbols in Fig. 3? assuming the pressure to be constant along 
the throat "a" equation (23) becomes: 

P T 
d mj_ s = — — - — — _ $ a cos X d r (25) 

/ Rq Tip 

Integrating equation (25) along the radius and for z blades the 

flow rate passing thru the stage is: 

Rp 

P T 
m. = z f ' $ a cos h dr (26) 

4 ' * t ? 

In actuality, the flow rate will be less than m is , because of the losses 
occurring in the blade channels. 

A so-called restriction factor: 

S«-A£- (27) 

d m is 

is now introduced. 

17 



In [ 5] the following relationship between § and the kinetic 
energy loss coefficient C* is derived: 



1 



1 + '* - 

± + a K 



(28) 



where H and K are respectively the boundary layer wake form factor 
and the energy factor. For boundary layer profiles following the 

TJ 

power law, the ratio— can be expressed in power series: 



H 
K 



"»i-l)ml 



1 - (1-Xe) LXe 1_ 7l+(2i-l)] 



1 • -1 . 

(1-Xe) Si [Xe 1- /l+(2i-l)mj- (l-Xe) Si Lxe 1_1 /l+(2i+l)m J 



(29) 



where : 

7-1/ y 
Xe = 1 - (P/P T ) (30) 

and m is the exponent of the power law for the boundary layer profile 
(for incompressible flow, m depends on Reynolds number and varies between 
l/7 and l/lO). Experience has shown that the loss coefficient C* has 
to be taken less than the value C^OT that corresponds to the overall 
energy losses that are given by the correlation methods. In the present 
program one of the three following different assumptions for C* can be 
chosen: l) C* = ^ TO t > 2 ) C* = JCtoT > 3) £* = -^PROFILE • 

While assumption l) is certainly conservative, it is difficult to state 
which of assumptions 2) or 3) is the best one. Good agreement between 
experimental values and predicted flow rates for all considered turbines 
at low tip clearance values was found in L4J, where assumption 2) was 
adopted. Assumption 3) was however found in closer agreement with 
test data for large values of rotor tip clearances. 



18 



Tip leakage flow rate will be calculated by using the relation: 

1 P T 
A CL = — L (5 ^=-^ ^s X) TIp A CL (31) 



where: A CL = 2n R H d (32) 

Since the flow is not perpendicular to the leakage area, and since the 
restriction factor § TI p does not account for boundary layer blockage 
occurring on shroud, the leakage flow rate must be divided by a 
coefficient K CL (larger than l) } assumed equal to 2 in the present 
program. 

With equations (29) and (33), the overall flow rate passing thru 
a blade row is given by: 



? T P T P T (33) 

A TOT = A + m CL = z J § =L=, $ a cos X dr + -L (§ — Lr $ cos\) TIp A^ 

K H G T T 



Equation (33) can be expressed in the following dimensionless form: 



T 

>' , - P T t _ , v 2tt _ X T c^ / P T 



(34; 



1=2%^ (? , $ a cos \) x dx + -rr— S TIp — — ( i cos^) 

x H /lfcT T K CL iJ ^ \za yTTj TIP 



where a = — (35) 



a 
m 



METHOD OF SOLUTION 



Since no direct solution of governing equations is possible, an 
iterative method is followed. At first, solutions are found without 
considering the influence of streamline curvature and slope. The 
derived radial shifts are then used for the second and final cycles of 
calculation. In Fig. k the scheme followed in the computer program is 
indicated. The names in brackets correspond to the names of the 
subroutines which perform the operations described next to the brackets. 



19 



The flow ahead of the stator is supposed to be uniform, so that the radii of 
streamlines dividing the flow rate in equal parts can be easily computed. 
For stator and rotor rows the following functions are supposed to be 
known: 



STATOR 

a - ot (r, flow condition) 



a x = a 1 (r) 



£ = Co (r, flow condition) 

S " 



ROTOR 

^2 = ^2 ( r > fl° w condition) 



a 2 = a 2 (r) 



^R = *r ( r > flow condition) 



All derivatives used in the following analysis are computed by 
finite difference methods. The computer program calculates flow 
conditions along 5 streamlines. However the same scheme of calcu- 
lation can be applied to any number of streamlines. By assuming the 
inlet Mach number M , the flow rate passing thru section can easily be 
determined. Radial position of streamlines at station 1, Kj_, and the axial 
velocity at mean streamline? V^-^, are also assumed a priori, and will 
be checked later. 

For any given values of V., , AR, and $r, equation (l8a) can be 
solved by iteration. In fact it can be reduced to: 



l(ln Y 2 ) 
dx 



I (x) 



X = 



Equation (36) integrated, with boundary condition \ y _ 



gives 



x 

lnY 2 = 'i dx 



(36) 
(37) 
(38) 



and 



x 



_ A Ji Idx 



Y = e 



(39) 



20 



Solutions are found by trial and error, assuming approximate values 

of Y(x) and verifying equation (39) • Once equation (39) has been 

solved, all flow conditions at station (l) are known and the overall 

continuity equation (3*0 can be solved. If the calculated flow rate 

is different from the assumed one, V" am 2 will be changed and the 

iteration shown in Fig. h is performed. When overall continuity 

equation is satisfied, the assumed radial position of streamlines 

is checked, and necessary iterations are performed. If streamlines 

continuity is met also, the flow at the rotor inlet is known completely. 

A similar iterative method is then followed for station 2 after the 

rotor as indicated in Fig. k. If the flow after the rotor is known 

also, the radial shift of the streamlines can be determined, and the 

whole calculating procedure is repeated by taking account of the 

streamlines curvatures and the slopes in solving the equation of 

motion. 

Eventually the resulting average pressure ratio across the turbine 

is compared with the desired one , and if the difference is larger than 

a smallprefixed error, the complete calculating scheme is repeated 

again with a new value of inlet Mach number. After the specified 

pressure ratio has been obtained, all output information is calculated 

and printed. 

FLOW ANGLES AND LOSS COEFFICIENTS C0REELATI0NS 

Evidently, even the most accurate method of performance prediction 
will give results in disagreement with actuality, if the flow angles 
and/ or the losses in the blade rows are not predicted with the necessary 
accuracy. 

Several correlations, either empirical or theoretical, have been 
proposed in the technical literature. A comparative study of five 

correlations methods available in the present program can be found in 
Reference I kl . The purpose of the present chapter is not to discuss 
the applicability of these different correlations, but only to indicate 
the problems which arose, in applying the different correlation methods. 
In Fig. 5 the nomenclature and sign convection of the velocity triangles 
is indicated. Unless otherwise stated, the variables are positive if 
directed as in Fig. 5. In Fig. 6 the blading geometry characteristics 
used in the different correlation methods are indicated. The names 

21 



given in Fig. 6 are the FORTRAN names used in the program for stator 
quantities at the mean radius. 

The program is organized in such a way that other than considered 
flow angles and loss coefficients correlations can be introduced by- 
adding a few cards and pertinent subroutines. Since the solution method 
is iterative, added subroutines can use blading and flow characteristics 
that are not used in the present program. 

a) Flow angles correlations 

Two methods, one given by Ainley and Mathieson [6J and the other 
given by Traupel [ 10J are available in subroutines. In both these 
methods, exit flow angles ( o^, &2 ) are considered to be independent 
on inlet angles ( Oq, 3}_). While in Traupel's method the angles are 
a function of blading geometry only, in A&M method also the exit Mach 
number is supposed to affect the angles. Although both these methods 
are given for a "reference diameter approach", they will be applied 
in three-dimensional calculations with the assumption that at any 
radius the angle is related only to the corresponding characteristics. 
Since during the calculations both Mach numbers and streamlines 
radii will be changed, flow angles corrections are automatically 
performed during the solution of the equation of motion. 

b) Loss coefficients correlations 

The assumption is made that at any particular radius the loss 
coefficient is a function of the characteristics at that radius only. 
It is realized that this assumption is not realistic, since secondary 
and leakage losses are certainly not equally distributed along the 
radius. On the other hand, secondary and tip leakage flows are not 
understood well enough at the present time to formulate any other 
reasonable assumption. 

Five loss correlation methods given in Ref . [6-10j are available 
in subroutines. They calculate loss coefficients from given blade 
geometry and flow angles and velocities. 

The first considered method is the Ainley and Mathieson L6] 
method. Recently an improvement to the original method was given 
by Dunham and Came |_7]. Both methods are available in subroutines. 
These methods express losses by total pressure loss coefficient Y, 



22 



defined by: 



P - P 

y = -S * (Uo 

p t " p 



and assume Y to be independent on pressure ratio P^q/P . The 
relationship between the kinetic energy loss coefficient C and Y is 
given by: 

Y-l 

r i + y v . 

, ^l + YP/P To -' _ m 

Y-l 

The assumption that Y is constant implies that £ decreases with increasing 
Mach number. Since this behavior is not experimentally proved, two 
alternatives to equation (4l) are available in the program. In both 
these alternatives Q is supposed to be constant with pressure ratio, 
and assumes respectively the value corresponding to very low Mach 
numbers (eq. k-2) or to an exit Mach number of 0.8 (eq. k-3) . 

C = C = lim C =-j~ (1*2) 

P /PTo-l 



Y-l 



c = c m=.8 = — *~n — ( ^ 3) 

(l/p*) V -1 



where p* i s the pressure ratio corresponding to an isentropic Mach 
number of 0.8, given by: 

Y 

p * = [i + ^ x .8 2 ] " Y - x w 



23 



The same alternatives are given also for the simple method of 
Lenherr and Carter [9J • 

For the method of Balje ' L 8J , the boundary layer wake form 
factor H must be introduced. 

With the same nomenclature and assumptions as in eq. (29), H can 
be expressed by: 



1 - (1-X ^ 



i-1 



co i_l co 1-X 

- (1-X e ) S n -( X e ) + (1-Xj El ( X e 

1 v l+m(2i) 1 l+m(2i-l) 



In this program the assumption m = 0.15 is made. If another exponent 
has to be introduced, card #17 must be changed in the MAIN routine. 

In Traupel's method, the loss coefficients depend on the 
coefficient: 

X = X (Re, relative surface roughness, turbulence factor). (k6) 

In the present program, the standard value of X =1 is assumed. If 
the considered turbine has peculiar values of surface roughness or 
Reynolds numbers, it will be necessary to change card #15 in the MAIN 
routine. 

For conformity with the other methods, tip clearance losses are 
expressed by the kinetic energy loss coefficient Qqt, instead of by 
the overall efficiency decrease AT]^ as in [ 10] . The relationship 
between AT]cl and £cL is given by: 



Ah. AT] 2 "I 

C CL =(i-C r )x Li- U-^fw- 2 -") J <W 

^ w2 



where Ah is is the isentropic enthalpy drop through the turbine and 
C r is the rotor kinetic energy loss coefficient for zero tip clearance, 



2k 



COMPUTER PROGRAM 

The present computer program is composed of a MAIN routine 
and 28 subroutines, eight of which are concerned with flow calcula- 
tions, two with flow angle correlations, 15 with blade loss 
correlations, and three with numerical interpolations. A brief 
description of the purpose of each routine is given below. 

a) MAIN routine 

1. The primary function of the MAIN routine is to control 
the overall logic of the computer program which is indicated in 
Fig. k. Furthermore the imput data are read, numerical constants 
are introduced, interpolations concerning stall and blade angles and 
flow areas are performed and the flow angles are computed. Eventually, 
also the output data are printed. 

b) Flow calculation subroutines 

2. In subroutine CHAN the flow rate passing through the section 
is calculated from the conditions ahead of the stator. 

3. In subroutine STATER the equation of motion (l8a) is solved, 
and the flow conditions after the stator are calculated. If necessary, 
new values of the flow angles are also calculated. 

k. In subroutine RfbTjbKL the absolute flow properties calculated 
in STATER are converted into relative properties at rotor inlet. 

5. In subroutine R$t/)R2 the equation of motion (l8) is solved, 
and the flow conditions after the stator are calculated. If necessary, 
also new values of flow angles are calculated. 

6. In subroutine Fl/)WR the flow rate passing through the blade 
sections, either at station 1 or 2, is computed by equation (3^) 5 and 
compared with the value calculated in CHAN. If the difference is larger 
than a small prefixed error, a new value of the axial velocity at the 
mean stremaline is computed. The fractional values of the flow rate 
passing between the hub and each streamline are calculated also. 

7. In subroutine SLINE the fractional values of the flow rate 
computed in FL0WR are compared with prefixed ones. If the differences 
are larger than a small specified error, new values of the streamline 
radii, either at station 1 or 2, are computed. 



25 



8. In subroutine ALj6si the stator loss coefficients are 
calculated from the chosen loss correlation, as a function of 
blading geometry and flow characteristics. Coefficients C* of (28) 
are computed for the stator blades. 

9. In subroutine AL6S2 the rotor loss coefficients are calcu- 
lated from the chosen loss correlation as function of blading 
geometry and flow characteristics. Coefficients £* of equation (28) 
are computed for the rotor blades. 

c) Flow Angles Subroutines 

10. In subroutine ANGAIN the flow angles are computed from 
blading geometry and exit Mach numbers with the method of Ainley 
and Mathieson [6J. 

11. In subroutine ANGTRA the flow angles are computed from 
blading geometry with Traupel's method L10]. 

d) Loss Correlation Subroutines 

The six subroutines given below are concerned with the methods L6J and 

[7]. 

11. In subroutine Cj&EFFI the imput cards concerning the profile 
losses of nozzle blades given in [6] are read, and the necessary 
interpolations are performed. 

12. In subroutine C/)EFF2 a similar process is followed for the 
profile losses of impulse blades. 

13. In subroutine CALYAjft the profile losses of nozzle blades are 
calculated with the coefficients computed in CyCEFFI. 

1^. In subroutine CALYA2 the profile losses of impulse blades 
are calculated with the coefficients computed in Q0EFF2 . 

15. In subroutine FIG6 the necessary interpolations for calcu- 
lating the secondary losses are performed. 

16. In subroutine AINL/)S the overall loss coefficients with 
either method 1 6J or L 7] are calculated from blading geometry and 
flow angles. Subroutine AINL/)S must be called after subroutines 
C/)EFFI and c/)EFF2, and calls subroutines CALYA0, CALYA2 and FIG6. 



26 



The eight subroutines given below are concerned with method (ll) 
of the loss correlation. 

17. In subroutine TRAUP1 the imput cards concerning the 
profile losses given by Traupel's method are read, and the necessary- 
interpolations are performed. 

18. In subroutine XPfi the basic profile loss coefficients are 
computed with the coefficients calculated in TRAUP1. 

19. In subroutine CSIM Mach number effects on the loss 
coefficients are computed. 

20. In subroutine CID the effects due to the trailing edge 
thichness and nontwisted blades on the loss coefficients are computed, 

21. In subroutine CSIW the end wall loss coefficients are 
computed. 

22. In subroutine CSIR the secondary loss coefficients are 
computed . 

23. In subroutine ALEAK the tip leakage efficiency losses are 
computed. 

2k. In subroutine TRAUP2 the overall loss coefficients are 
computed with Traupel's method LlOJ from both blade geometry and flow 
characteristics. Subroutine TRAUP2 has to be called after TRAUP1, 
and calls subroutines XP0, CSIM, CID, CSIW, CSIR and ALEAK. 

25. In subroutine ZA1L0S the overall loss coefficients are 
computed with the method of Lenherr and Carter L9J from the flow 
angles and the relative velocities. 

26. In subroutine BALJE the overall loss coefficients are 
computed with Balje's method [8j from blade geometry and flow 
conditions. 

e) Numerical Interpolation Subroutines 

27. In subroutine PARAB the coefficients of the parabolic 

interpolations used to approximate curves are calculated. 

28. In subroutine CHBFT the coefficients of the polinomial of 
arbitrary degree, which is the best fit of a given array of points 
in the Chebyschev sense, are calculated. 



27 



29. In subroutine YC the value of the polynomial determined 
in CHBFT, corresponding to a given value of the independent 
variable, is calculated. 
Description of Imput Data 

The imput data of the present computer program consist of 63 + 
NSETS cards, NSETS being the number of conditions for which 

performance values are to be obtained. They can be divided in three 
sections. In the first, the constant values used in the methods 
of Ainley and Mathieson and of Traupel are introduced (these 36 
cards are read respectively in subroutines Cj^EFFI, Cj$EFF2 and TRAUPl). 
In the second, the specifications about the detail of output desired, 
the tolerances of the numerical solutions, the systems of correlation 
of the flow angles and the loss coefficients, the coefficients used in 
the equation of continuity, the gas properties and the turbine geometry 
are stated. In the third, the conditions for which performance values 
are to be found are introduced. 

The 36 input cards for the first section are given in Table I. 
They will remain unchanged for all turbines. Detailed description 
of the 27 cards composing the second section of the imput are given 
in Table II, where also a description of the third imput section can be 
found in cards #28 and #29- An example of the imput cards described 
in Table II is given in Table III. 
Description of the normal output 

In Table IV the output corresponding to the imput of Tables I 
and III is given. It can be seen that it consists of 1 + 3 NSETS 
sheets. In the first sheet the turbine geometrical dimensions and 
the choice among the different correlation methods are described. 
Then three sheets for every set of operating conditions are printed. 
The first of these sheets describes the flow conditions at station 1 
after the stator row; the second one the flow conditions after the 
rotor row; and the third one gives the main overall turbine characteristics 
and the mass averaged performance values. 



28 



ACKNOWIEDGEMENTS 
The author would like to express his appreciation to Dr. M. H, 
Vavra for providing guidance and counsel during the investigation. 



29 



CM 




cV 



it-" 



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t- 



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O 

o 

c 
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E 
o 
a> 

CO 

"O 

c 
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CO 

c 
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a 

CO 



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9 



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30 




Fig. 2 — Turbine thermodynamic process, 



31 




Fi g. 3 Symbols in continuity equation 



3? 



READ: LOSS CORRELATION IMHJT 
TURBINE GEOMETRY IMPUT 



CALCULATE : FLOW ANGLES 
INTERPOLATE : FLOW AREA FUNCTIONS 

: INLET AND STALL FLOW ANGLES 



(ANGAIN) OR (ANGTRA) 
(CHBFT), (PARAB) 



INLET CONDITION. RPM, OVERALL PRESSURE RATIO, 



APPROXIMATE VALUES OF INLET MACH NUMBER, Mq 



READ: 

READ 

READ APPROXIMATE VALUES OF AXIAL VELOCITY V A1 (3) VW3' 

ASSUME APPROXIMATE VALUES OF RADII OF STREAMLINES Rl, R2 ' 

ASSUME APPROXIMATE VALUES OF LOSS COEFFICIENTS, £ S , L R 



CALCULATE FLOW RATE AT SECTION 0, CORRESPONDING TO M 



n. 



SOLVE MOTION AND ENERGY EQUATIONS AT STATOR EXIT, WITHOUT 
ACCOUNTING FOR STREAMLINE CURVATURE AND SLOPE 



CHANGE V 



Al(3) 



K 



CHECK OVERALL CONTINUITY 
AT STATOR EXIT 



CALCULATE STATOR LOSS COEFFICIENTS 



{CHANGE Rl — « — ^ CHECK CONTINUITY FOR EVERY STREAMLINE ^ 



j CALCULATE INLET ROTOR CONDITIONS 



SOLVE MOTION AND ENERGY EQUATIONS AT ROTOR EXIT, WITHOUT 
ACCOUNTING FOR STREAMLINE CURVATURE AND SLOPE 



CHANGE Va2(3' 




X 



CHECK OVERALL CONTINUITY 
AT ROTOR EXIT 



CALCULATE ROTOR LOSS COEFFICIENTS 



CHANGE R2 



CHECK CONTINUITY FOR EVERY STREAMLINE^ 



CALCULATION OF STREAMLINE 
CURVATURES AND SLOPES 



RECYCLE OF ALL THE CALCULATIONS 
FROM POINT 1, ACCOUNTING FOR STREAM- 
LINE CURVATURE AND SLOPE 



CHANGE M ("CHECK MASS AVERAGED TURBINE PRESSURE RATIO~^ 



WRITE OUTPUT 



STOP 



(chan; 



( STATOR ; 



(FLOWR) 

( AL0S1 ) 
(SLINE) 
(ROTORl) 



( R0T0R2 ) 



(FLOWR) 

( AL0S2 ) 
( SLINE [ 



Fig. 4 Flow diagram 



^ 



tangent to 
camber line 



V A1= W A1 




Fig. 5 Velocity diagrams 




AL = 
C = 
= 
E = 
S = 
T = 

TE = 
TN = 



CAMBER LINE LENGTH 

CHORD 

THROAT DIAMETER 

CURVATURE RADIUS 

SPACING 

MAXIMUM THICKNESS 

TRAILING EDGE THICKNESS PROJECTED IN 
PERIPHERAL DIRECTION 

TRAILING EDGE THICKNESS, NORMALTO FLOW 
DIRECTION 



Fig. 6 Blade geomety nomenclature 



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i+3 



CORRELATION 
INDEX 



IC0R 



IAI 



POSSIBLE 
i VALUES 



SPECIFICATIONS 



A&M [6] or D&C [7] loss correlation 
method is adopted, depending on IAI 
value . 

Balje [8] loss correlation method is 
adopted . 

L&C [9] loss correlation method is 
adopted. 

Traupel [lOJ loss correlation method 
is adopted. 



If IC0R=1, A&M [6] loss correlation 
method is adopted. 

If IC0R=1, D&C L7J loss correlation 
method is adopted. 





1 


A&M L"6J flow angles correlation is 
adopted. 


IAN 








2 


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is adopted. 


1 





Rotor is not shrouded. 


ICL 




: 




1 


Rotor is shrouded. 



TABLE V (1) 



kk 



CORRELATION 
INDEX 


POSSIBLE 
VALUES 


SPECIFICATIONS 


IINC 




1 


Loss coefficients axe function of 
flow angles 

Loss coefficients are function of 
blade inlet angles, for negative 
incidence 


icjte 


1 
6 

8 


Loss coefficient £ = £ (see eq. (42)) 

Loss coefficient £ = £ (Y, pressure 
ratio) (see eq. £n)) 

Loss coefficient £ = £ _ o (see eq. (43)) 


IC0N 


1 
2 

3 


Loss coefficient in continuity equation 
C* = .5 £tot 

Loss coefficient in continuity equation 
£* = ^PROF 

Loss coefficient in continuity equation 

C* = C 
b b TOT 



TABLE V (2) 



^5 



TABLE VI 



EXAMPLE OF NORMAL OUTPUT DATA 



1*6 



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BIBLIOGRAPHY 

1. Vavra, M. H. "Aero thermodynamic and Flow in Turbomachines", Wiley 
and Sons, N. Y., i960, pg. 439 - ^70. 

2. Eckert, H. E. "Performance Analysis and Initial Tests of a Transonic 
Turbine Test Rig," NPS Thesis, May 1966. 

3. Harrison, R. G. "An Analysis of Single Stage Axial- Flow Turbine 
Performance Using Three- Dimensional Calculating Methods," NPS Thesis, 
September 1967. 

k. Vavra, M. H. and Macchi, E. "Comparison of Performance Prediction 
Methods for Axial Turbine Stages," to be published. 

5. Vavra, M. H., "Axial Flow Turbines," in Lecture Series 15, vcn Karman 
Institute for Fluid Dynamics, Brussels, April 1969. 

6. Ainley, D. G. and Mathieson, G. C. R. , "A Method of Performance 
Estimation for Axial- Flow Turbines," Aeronautical Research Council, 
R&M No. 2974, 1957. 

7. Dunham, J. and Came, P. M. , "Improvements to the Ainley-Mathieson 
Method of Turbine Performance Prediction," ASME paper 70-GT-2. 

8. Balje', 0. E. and Binsley, R. L. " "Axial Turbine Performance 
Evaluation, Part A - Loss Geometry Relationships," Journal of 
Engineering for Power, Trans. ASME, Oct. 1968. 

9. Lonherr, F. K. and Carter, A. F. , "Correlations of Turbine Blade 
Total Pressure Loss Coefficients Derived from Achievable Stage 
Efficiency Data," ASME paper 68-Fe-51, 1968. 

10. Traupel, W. , "Thermische Turbomaschinen, "Vol. 1, Springer 
Verlag, Berlin/Gottingen, Heidelberg, 1958, pg 269-298. 



101 



INITIAL DISTRIBUTION LIST 



No. Copies 



1. Defense Documentation Center 20 
Cameron Station 

Alexandria, Virginia 2231*+ 

2. Library, Code 0212 2 
Naval Postgraduate School 

Monterey, California 939^0 

3. Commander, Naval Air Systems Command 1 
Navy Department 

Washington, D.C. 20360 

h. RADM CO. Holmquist 1 

Chief of Naval Research ONR-100 
Navy Department 
Washington, D.C. 20360 

5. Professor M. H. Vavra 3 
Department of Aeronautics 

Naval Postgraduate School 
Monterey, California 939^0 

6. Chairman, Department of Aeronautics 1 
Naval Postgraduate School 

Monterey, California 939^0 

7. Professor Corrado Casci 3 
Director, Istituto di Macchine 

Politecnico di Milano 
Milano - Italy 

8. Dr. E. S. Lamar, Code 03C 1 
Chief Scientist, Research and Technology 

Naval Air Systems Command 
Navy Department 
Washington, D.C. 20360 

9. Dr. F. I. Tanczos, Code 033 1 
Technical Director, Research and Technology 

Naval Air Systems Command 
Navy Department 
Washington, D.C. 20360 



102 



10. Dr. M. J. Mueller, Code 310 
Naval Air Systems Command 
Navy Department 
Washington, D.C. 20360 

11. Mr. I. Silver, Code 330 
Propulsion Administrator 
Research and Technology 
Naval Air Systems Command 
Navy Department 
Washington, D. C. 2036O 

12. Office of Naval Research 
Air Programs Office 
Navy Department 
Washington, D.C. 20360 

13. Office of Naval Research (Power Branch) 
Attn: Mr. J. K. Patton, Jr. 

Navy Department 
Washington, D.C. 20360 

Ik. LCDR R. G. Harrison, Code 53622A 
Propulsion Division (JP-2) 
Naval Air Systems Command 
Navy Department 
Washington, D.C. 2036O 

15. Centro Nazionale di Ricerca 
Sulla Tecnologia della Propulsione 
e dei Materiali Relativi (CNPM) 
Milano - Italy 

16. Dr. E. Macchi 
Istituto di Macchine 
Politecnico 

Milano - Italy 

17. Dean of Research Administration, Code 023 
Naval Postgraduate School 

Monterey, California 93940 



103 



UNCLASSIFIED 



Security Classification 



DOCUMENT CONTROL DATA -R&D 

(Security classification ol title, body ol abstract and indexing annotation must be entered when the overall report is classified) 



originating activity (Corporate author) 

Naval Postgraduate School 
Monterey, California 



2a. REPORT SECURITY CLASSIFICATION 

UNCLASSIFIED 



26. GROUP 



3 REPORT TITLE 



Computer Program for Prediction of Axial Flow Turbine Performance 



4. descriptive NOTES (Type of report and.inclusive dates) 



Technical Report, 1970 



5 authoR(S) (First name, middle initial, last name) 



Macchi, Ennio 



6 REPOR T DATE 

August 1970 


7a. TOTAL NO. OF PAGES 

105 


lb. NO. OF REFS 

10 


8a. CONTRACT OR GRANT NO. 
b. PROJEC T NO 
C. 
d. 


9a. ORIGINATOR'S REPORT NUMBER(S) 

NPS-57MA70081A 


9b. OTHER REPORT NO(S) (Any other numbers that may be assigned 
this report) 



10 DISTRIBUTION STATEMENT 



This document has been approved for public release and sale; its distribution is 
unlimited. 



II SUPPLEMENTARY NOTES 



12. SPONSORING MILI TAR Y ACTIVITY 



I 3. ABSTR AC T 



The report presents a computer program for prediction of performance 
of single-stage axial turbines of given geometry. The three-dimensional 
method developed by Vavra is applied, taking account of streamline 
curvatures and slopes, as well as enthalpy and entropy gradients in 
the solutions of the equation of motion, and of boundary layer thick- 
nesses in the continuity equation. 

A choice among five different loss correlation methods and two 
flow angles correlations is offered. Loss coefficients and flow 
angles are automatically calculated from blading geometry and actual 
flow conditions for every streamline, according to the selected 
correlation method. 

A fair agreement of predictions with several actual turbines 
experimental results was found in ref . L^J , where also the applicability 
of different available correlations is discussed. 



DD , F °"»"..1473 



FORM 

i nov es 

S/N 0101 -807-681 1 



(PAGE 1) 



10U 



UNCLASSIFIED 



Security Classification 



A-31. 



UNCLASSIFIED 



Security Classification 



key wo RDS 



Analysis 

Single-Stage 

Axial-Flow 

Turbine 

Performance 

Three -Dimensional 

Calculating 

Methods 



DD, F N ° R V M 65 1473 (back) 



105 



UNCLASSIFIED 



I 



UUULbT WNUA Lionnni - ntot 



5 6853 01070494 3