Free jet analysis from nozzles - WIT Press · 2014-05-13 · Free jet analysis from nozzles of cross-flow turbine H. Olgun, A. Ulkii, E. Demirci Department of Mechanical Engineering,

Free jet analysis from nozzles

of cross-flow turbine

H. Olgun, A. Ulkii, E. Demirci

Department of Mechanical Engineering, Karadeniz

Technical University, 61080 Trabzon, Turkey

ABSTRACT

The flow from nozzles is calculated numerically byusing the boundary element method. The solutions fromthis method have been compared with the results obta-ined from electrical analogy method for two differentshapes of nozzles. Flow visulation studies for thesenozzles have been carried out and the trajectories ofthe flow have been observed and photographed. Thesetrajectories have been compared with the numericalsolutions. It has been found out that the numericallyobtained free jets from these nozzles are good agree-ment with those of the electrical analogy method,viewing and photography techniques. In addition, theeffects of three different types of nozzles on theefficiency of the cross-flow turbine have been expe-rimentally investigated.

NOMENCLATURE

Og,au,ciL : flow anglesR!q6SQSTVfs

outer radius of runner Pflow rate *nozzle entry arc anozzle throat width kfree jet width Trfree jet velocity d

contour velocitypressurepotential functionfree jet angleSo/2Ri

position of free jet-

1. INTRODUCTION

The nozzle of a cross-flow turbine has to give acertain circumferantial velocity and an optimum angleto the flow at the exit of the nozzle. Therefore,

Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

94 Boundary Elements

the nozzle shape has an important influence upon theturbine performance. Generally, the nozzle exit flowhas no solid boundary and the boundary condition ofthe nozzle exit flow is such that the velocity on afreestreamline is constant and its pressure isequalto atmospheric pressure. The objective of thisstudy is to determine the outline conditions of thecross-flow turbine nozzles. For this purpose, theboundaryelement method has been used as a tool ofinvestigation.

2. MATHEMATICAL MODEL

Mathematical modelling of the flow inside and outsideof a cross-flow turbine nozzle is assumed as a poten-tial flow and its flow model is given in Fig.l. It isassumed that the flow is uniform in the direction ofn at outlet and inlet section. The flow leaving thesolid walls of the nozzle is defined as a free jet.The streamlines of the free jet are parallel accor-ding to the potential theory. In Fig. 1; G is the in-let section, F is the free jet, C is the outlet sec-tion and S is the solid wall. J and LL are corner po-ints of outlet section, M and N are corner points ofinlet section, LF1 and LF2 are the starting points offree jet. 0 is the center of nozzle entry arc.

Fig. 1 A flow model of cross-flow turbine nozzle.

2.1. Equation of MotionThe governing equation for potential flows is theLaplace equation which is given by,

92*

3X2(1)

Where $ is the potential function (Brebbia[l]). Theaim is to find a solution satisfying the Laplace equ-ation for two dimensional, irrotational, incompress-ible flows. Assuming that a concentrated charge isacting at a point i, the governing equation becomes,


Boundary Elements 95

72$* + vi = 0 (2)

Where v^ is a Dirac delta function. Solution of thisequation is called the fundamental solution. For anisotropic two dimensional medium, the fundamentalsolution of equations ( 2 ) is

** = (1/2%) ln(l/r) (3)

Where f is the distance from the point of applicationof the unit potential to the point under considera-tion. The solution of the fundamental equation,applying the Green's theorem for any 4>i harmonicfunction is given by Schwank [2].

*i = — 6 [* -- In -- ] ds (4)r 3n

Where 9* / 3n = Vn is the normal velocity of the con-tour and is known all over the closed surface of thenozzle (Fig.l). Equation(4) is now applied on the bo-undary of the domain under consideration. For simpli-city only the two dimensional case is studied and itsboundary is divided into n segments. The points wherethe unknown values of 4>i are considered and taken tobe in the middle of each segment. The values of $ and3* /3n are assumed to be constant on each element.From the vector analysis the term " (3 (ln( 1/f ) )/3n)ds"in equation (4) can be rewritten as,

9 (In 1/f) 1 (r - ri)- ds = --- dn = EGYH, (5)

and using the relationship Vnds = dq, and integra-tion from k=l to kmax ,

kmax (rk-ri) kmaxE (*k-*i) — - dnk - E ln( rk-ri )dqk-A*dqi= 0(6)

k=l (rk-ri) ̂ k=lkfi

Where k denotes any point over the boundary andis equal to n.JThe values of 1/r and ln(?k-?i) becomeinfinite when ?k tends to ?±. Therefore A* can notbe calculated analytically. However, it can be calcu-lated by assuming uniform and horizantal flow condi-tions from hydrodynamics. Where A=2A* and

*k-*i

" (7)



kmax _^ . . . . _^E 2EGYH(rk-fi)ni - E ln(rk-r±)2ni

k=l k=l

(8)

kmax _^ ^E EGYH(*k-*i) - E ln(rk-r±)2dqk = Adq± (9)

k=l k=l

Then, Equation (6) becomes

This is the main equation for the boundary elementmethod presented in this work. If equation (9) iswritten for each ith node, it results in n equationsfor *i unknowns. The whole set of equations for the nnodes can be expressed in the following manner.

A(i,k) *k + A(i,i) *i = Vi, i=l,.n, k=l,.n (10)

Where, A(i,k) = EEGYH, A(i,i) = EEGYH,

V(i) = Eln(rk - r±)2dqk + Adq±

This is a set of linear equations with the unknowns*!/ *2/ *n- These equations are solved iterativelyand then the velocities of Ci around the contour arecalculated by using the finite diference method.

2.2 Calculation of EGYHIn the present program for fixed "i" (Fig. 2),

Pk+l(Xk+l,Yk+l

PkUk,Yk)

Pk-i A<+i,i *

*EGYH = E 4>da (11)Ok+l,i ) k=1 J

Pk

X

Fig.2 Calculation of EGYH.

Where Pk is the point (Xk, Yk), Pi is the point ( XJL, YI ),

Pk+1

Uda = (*k+i + *k)(ak+l,i-(%k,i)/2 (12)

Pk



The first term is the average potential flow, thesecond term is the length of interval a. Thisapproximation is applied for each section. In thecase of PI = Pk or P± = Pk+l , a does not change withits integration along Pkfk+1 , so the integral overPkPk+1 is zero. In order to calculate the EGYH onepossible approximation is the following:It is considered that * changes linearly over PkPk+1-In this case with the geometrical configuration inFig.3, X coordinate varies over PkPk+1, and X = 0 atmiddle point. The relation between * and X becomes.

Pk+l(Xk+l,Yk+l)

-c<x<+c

Fig. 3 Calculation of Equation( 12) .

Pk+l

*k+l)/2)da + ((*k+l - 4>k)x/2c)da (13)

Pk

This is the main equation of EGYH. When this integralequation is solved,

EGYH=-[(ok+i,i-ak_i,i)+(bk-l/ck-l)(ak,i-ak-l,i)-(bk/ck)(ak+l,i-ak,i)-(ak-l/2ck-l)ln(Sqk-l/Sqk)

] (14)

Where Sqk=Pi?k

2.3. Calculation of Velocities Along to Nozzle EntryArc

In order to calculate the velocities along the nozzleentry arc, Green's theorem is applied for point TL,where it's located inside the flow region. This isgiven by Schwank [2].

H

3(ln(l/r)) 1 9**L = — <j> [ * In ] ds (15)

3 n r 3 n

Above equation represents the integral relationshipobtained between an integral point TL and the boun-dary values of *± and 9*/9n as shown in Fig.4. Equa-tion (15) can be rewritten in discreatized form as



Lmax Lmax , _.E EGYHM (*k-*L) + Z In(rk-rL)

L=l L=l(16)

Where EGYHM is a new coefficient and is calculatedas,

(Xk-Xoi)(Yk-l-Yk+l)+(Yk-YOL)(Xk+l-Xk_l)EGYHM = (17)

(Xk - XOL)^ + (Yk-YOL,)*

Then all of the $L internal potentials are calculat-ed. The velocities along the circular arc of thenozzle can be obtained by using the finite differencemethod as explained above (Fig.4).

Fig. 4 The flow model along the circular arc.

2.4. The Velocity Calculations Around the Guide VaneThe basic equation (4) will be valid even if a guidevane is mounted to the flow channel of the nozzle. Inthis case the equation must be integrated twice alongthe dotted line but in the reverse direction. Thedotted line can be placed anywhere between the guidevane and the nozzle wall as shown in Fig. 5.

Fig. 5 Flow model of nozzle with the guide vane

At the trailing point of the guide vane, the KuttaJoukowski condition should be satisfied. That is,

C(Jl+3) = C(Ll-2) (18)



Also the following conditions must be provided.

*(L2+1) - *(J2) = *(L2) - *(J2+1)$(J2) - *(J2-1) = $(L2+2) - *(L2+1)

*(J2+1) - *(J2+2) = *(L2-1) - *(L2) (19)

2.5. Boundary ConditionsAt a solid boundary, the velocity component normal tothe boundary must^be zero at every point on the boun-dary, that is, n.V = 0. Where n is the normal vectorto the boundary, V is the velocity vector on theboundaries. Assuming uniform flow in the direction ofn at the inlet and outlet section (Fig.l), the boun-dary condition becomes Vxny-Vyn% = 0 .The nozzle exitflow does not have any solid boundary. The velocitiesof the free jet are constant and its pressure is equ-al to atmospheric pressure. By applying the Bernoulliequation for two points on a streamline gives

V=i/2 + Pi/p = V=fs/2 + Pat/P (20)

Where Vi is the velocity of inlet section, Vfs isconstant velocity of the free jet.

2.6 Numerical CalculationsA computer program is written for the solution ofequation (10). The procedure used in the computerprogram for the calculation of velocities along theclosed surface of nozzle (Fig.l) is given as follows:The nozzle walls and free jet are assumed as a solidboundary such that the velocities of the contour arecalculated. Thus the calculations are carried outuntil the contour velocities of the free jet areconstant. This condition is achieved by varying thetrajectory of free jet in each iteration (Olgun[5]).

2.7. Control of the Trajectory of Free JetThe velocity of the middle section of free jet can becalculated by using the momentum theorem for acontrol volume as shown in Fig. 6. The momentumtheorem is given as follows:

P dn + V(V dn) = 0 (21)

This integration is made along the^cqntour (excluding'the free jet) and the term of "dq=V.dn" is calculatedonly at the inlet and outlet sections. When theBernoilli equation is used for P,

Q(V*fs- V%) dn ] - V2 (22)



r*(xo yo)

LL

Fig. 6 Control of the free jet.

The angle between the middle freestreamline of freejet and X-axis is

a=arctan(Viy/Vix) (23)

and the thickness of the free jet is calculated as

ST = Q/Vi (24)

The momentum theorem is^valid for a given TO, then itis valid for any other ro\

i

(r-ro)x ?dn + <J>(r-ro)xpV(Vdn) =0 (25)

This equation is known as a moment of momentum equa-tion. When the Bernoulli equation is written,

r %(r-ro)xdn(vfs-V)/2+(rc2-ro)xV2Q+(rci-ro)xViQ=0 (26)

is written as a "d". The numerical solutionof Equation (10) is compared to these control valuesof Vi, a, ST, d whose values are taken to be measureof convergence in the iterative method.

3. CALCULATION RESULTS

The technical features of the nozzles A and B, andthe results obtained from electrical analogy given byChrist [3] are shown in Fig. 7. Both of the nozzleshave a rectangular cross-sectional channel. Numericalsolutions of these nozzles are given in Figs. 8 and 9.The comparison of these methods shows that the tra-jectories of the free jets are almost the same. Thecomparison values of flow angle and width of the freejet for the nozzles A and B are given in Table 1.



NOZZLE B

Fig. 7 Results from electrical analogy method [3].

In addition, the control values found from numericalcomputations are also given in Table 1. The resultsindicate close agreement between boundary element andelectrical analogy methods. It should be noted that,the electrical analogy method gives only the trajec-tory of the free jet.

Table 1. The control values

Christ [3]Appel [4]Numerical [5]Control [5]Error(%)

a(o)

8.509.309.309.380.80

Noz

ST(mm

40.40.39.40.1.

z

)

05820

le A

V(m/s)

10.010.11.0

d(mm)

44.544.70.45

Noz

a(o)

-29.0-28.8-29.7-29.90.7

zle

ST(mm)

35.37.35.34.1.

B

52478

V(m/s)

10.010.181.8

d(m

46451

m)

. 5

.59

.90

Photograph recordings of the nozzle exit flows aregiven in Figs. 10 and 11. In order to compare nume-rical solutions to the observed flows, the Figs. 12and 13 are given. Those figures show that, the ob-served results are slightly different from computa-tional results. These differences are almost insig-nificiant at the nozzle exit but are more signifi-ciant as the distance from the nozzle exit increa-ses because of the weight force effects.



»1JOLF1

1.0 1.5

OS

1JO

2.0 15

•"" X(dm)

LF2

B point of tho *ali4 wellO point of tho fro* jot4 voloeity of th* moll* vailO voAoeity of tho froo jot

LL

Fig. 8 Numerical solution of nozzle A.

1 2 3

05 10

05

10

2.0 2.5

X(dm>

m polfrt of tho 0*114 wollO point of tho fro* jotM velocity of th* *oll« woliO velocity of t*o froo jot

IF?II

Fig. 9 Numerical solution of nozzle B.



Fig.10 Flow from nozzle A.

Fig.11 Flow from nozzle B.

NOZZLE Aci(cuUtrd\riptrimrntatv

Fig.12 Results of nozzle A,

NOZZLE 8calculated /

—eipenmentat,'

Fig.13 Results of nozzle B.

Numerical computations have been carried outfor three different types of cross-flow turbinenozzles. The technical feautures of the nozzlesused for the computational process are given inTable 2.

Table 2. The technical features of the nozzles

Nozzle

C

D

E

ri(mm)

86

86

86

So(mm)

17

42

21

k

0.100

0.247

0.143

eC)

46

104

65

Tr

0.249

0.544

0.259

Both of the nozzles have a rectangular cross sec-tional channel. Nozzle outlet angles of both solidwalls measured from the circumferantial direction



are designed to be 16 degrees as shown in Fig. 14.for nozzle C,D and E. Nozzle C has a doubly evol-vent walls at the exit so that one of them servesas a guide vane. Nozzle D, also has doubly evolventwalls at the exit and a guide vane is located in-side the flow region where the trailing edge of theguide vane is arranged at 16 degrees from the pres-sure surface. Nozzle E has a bigger evolvent curv-ed walls.

Fig. Nozzles.

The computed trajectory of the free jet and thevelocity of contours for nozzles C,D and E areshown in Figs. 15,16 and 17 respectively. It can beseen that the velocities on the free jet are const-ant. The exit wall angles obtained from these figu-res are given in Table 3.

Table 3. The exit walls angles.

Nozzles

C

D

E

*g16

16

16

<%u

1.0

0.0

15.0

<%1

16.5

23.0

22.0

Where Og is the design angle,au is upper wall angleand <%]_ is lower wall angle of these nozzles. Upperwall angles <%u have been found to be smaller thanthe design values for nozzle C and D,however it hasbeen found nearly to be equal to the design valuefor nozzle E. Lower wall angles have been foundto be greater than the design values for nozzles Dand E, on the other hand it has been found to benearly equal to the design value for nozzle C.



In this case, if the runner was presented for nozz-les C and D, the flow from the upper part would notflow through the center opening space but wouldfollow directly the outside of the runner. Nozzle Egives a better velocity distributions on the con-tour as shown in Fig.17. The velocity distributionsaround the guide vane are found to be uniform asshown in Fig.16.

*' ***** ** *"* Mli« Mil VO> *»•»& mf ll* #,.. )•« \0< »•>•«.IT *r ik« •*!«< MII Vo> ••»•«»•, .* ih. i,.. ).& v i

a

ST

vrs

0

Ol*«n

23.4

16.3

10.0

31.0

Control

22. $2

16.27

10.02

30.59

Error

2.09

0.1M

0.16

1.322

Fig.15 The calculated free jet for nozzle C,

m t point *f th« »oJH walO : p**n& el tk* fr«« )*tty #f tb« »e)i4#( th« fr.« )

a

#T

VF9

0

Olvom

25.0

34.15

10.0

32 5

Control

24.96

33.96

10.06

30.65

Error

0.17

0.55

0.57

1.54m u

Fig. 16 The calculated free jet for nozzle D.

I7̂ rrrz-T\O i **&** ** *k* fr«« )«t ILZ : v«J*caty *f tk* *#1W wall %^

irj

a

ST

VFS

D

Gl»*n

37.2

23.0

10.0

36.2

Control

37.16

22.83

10.07

36.43

Error

0.12

0.74

0.73

3.50

Fig. 17 The calculated free jet for nozzle E.



The velocity and flow angle distributionsalong the nozzle exits for nozzles C,D and E areshown in Figs. 18,19 and 20 respectively. The flowangle distributions have been found to be compara-tively uniform in the X direction, but they have afluctuation near the trailing edge of the guidevane for nozzle D. The velocity distributions havebeen found to be decreasing with the nozzle periph-eral position. The exit velocity has a smaller va-lue than the design value with an increase of thenozzle entry arc. Thus, the flow at the nozzle exitof a cross-flow turbine does not drop to the atmos-pheric pressure immediately. It is concluded thatthe nozzle exit velocity and angle distribution areinfluenced by the nozzle solid wall shapes (Olgun[5]).

LF2 LF1 LF2 LFI

.10

Fast01

07

Notitt c

i O%bqj

__, 6_3JD 11

1.0

QJ

M

i Noiiie E

ijQQTDOQQQQ

O:a.

0 OQO

QQQDQ

V/Vf,

VcPoo

11

i

k\

a»toa.302010n

Fig. 18 The velocity andflow angle distributionsfor nozzle C.

Fig.19 The velocity andflow angle distributionsfor nozzle E.

LF2 JU3 LF1

14

. O.I

* o.s

0.4

- <

1

> Nozzit

o o0 OQ

^ 0o

1n DO

io !ii

0° 1 ad°o

0 V/Vf, <D a,

.00°°°

°°°°oo •

>

01* •a*"

, ,

3020100

2.1 XO 12 U If XI 40*' X(dm)

Fig. 20 The velocity and flow angle distributionsFor nozzle D.

Photograph of the nozzle E with a flow are given inFigs. 21. The comparison between the observationand the numerical solutions are given in Figs. 22.These figures show that the observed trajectoryare slightly different from the computed result.



Fig. 21 Flow from nozzle Fig. 22 Results of nozzleE E

The effects of three different types ofnozzles on the efficiency of the cross-flow turbinehave been experimentally investigated. In thetests, single turbine runner was used. It has beenfound out that the maximum efficiency correspondedto using the nozzle E is 74%. By using the nozzleC and D the ef-ficiencies of these turbines are 72% and 71%(Olgun[8]> .

4. CONCLUSIONS

The boundary element method has been used incomputing the nozzle exit flow. It was shown thatthe numerical results for the nozzles agreed wellwith the electrical analogy methods and the observ-ed flows. In the numerical solution, the flow istaken as a potential flow. In fact the observedflow is the real flow. Howewer, the comparisons ofthe numerical and photographically recorded flowshave been found to have satisfactory agreement atthe nozzle exit. Also results obtained by the elec-trical analogy display a good agreement with thecomputational results. The theoretical results alsoshowed that the computed trajectory of the free jetis departed from the assumed trajectory. Also it ispresented that, the nozzle shape has an importanteffect on the out line conditions. The flow anglealong the nozzle entry arc is not constant.



REFERENCES

1. Brebbia, C.A. Boundary Element Methods,Springer Verlag, Germany, 1981.

2. Schwank, F. Randwertproblema, B.G. TeubnerVerlagesgesellschaff, Leipzig, 1951.

3. Christ, A. and Schreiber, V.'FreistrahlstromungNach Einer Beliebig Geformten Spaltblende'Escher Wysess Mitteilungen 2, 28-34, 1977.

4. Appel, D. W. 'Free Streamline Analysis ofNozzles' University of Cansas, Report Nr. 17,1963.

5. Olgun, H. 'Cross-Flow Turbininin Tasarim Para-metrelerinin Incelenmesi', Ph.D. Thesis, Kara-deniz Technical University, Trabzon, Turkey,1990.

6. Fukutomi, J., Nakase, Y. and Watanabe, T. A.'Numerical Method of Free Jet From A Cross-FlowTurbine Nozzle', 28,1436-1440, Bulletin ofJSME, 1985.

7. Olgun, H. , Ulku, A. and Fay, A.'An Investiga-tion of Free Jet From a Cross-Flow TurbineNozzle', 7th International Conference on Nume-rical Methods in Laminar and Turbulent Flow, 2,543-553, 1991.

8. Olgun, H. and Ulku,A. 'An Experimental Study ofCross-Flow Turbine', Vol.1, pp.71-76 Proceedingof the Engineering Systems Design and AnalysisConference, Istanbul, Turkey, 1992.


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Free jet analysis from nozzles - WIT Press · 2014-05-13 · Free jet analysis from nozzles of cross-flow turbine H. Olgun, A. Ulkii, E. Demirci Department of Mechanical Engineering,