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Assessment of the blade skew and the through flow velocity gradients in hydrodynamic torque converters
lnt. J. Mech. Sci. Vol. 29, No. 10/11, pp, 695-712, 1987
0020-7403/87 $3.00+ .00 ~t~1987 Pergamon Journals Ltd.
Printed in Great Britain.
ASSESSMENT OF THE BLADE SKEW A N D THE THROUGH FLOW VELOCITY GRADIENTS IN HYDRODYNAMIC TORQUE CONVERTERS SVEN ANDERSSON Machine Elements Division, Lund Technical University, Box 118, S-221 00 Lund, Sweden
(Received 24 November 1986; and in revised form 30 May 1987) A b s t r a c t - - T h e well established one-dimensional mean flow approach in the analysis of hydrodynamic torque converters has been supplemented by equations regarding the balance in the transverse direction of the flow path. By these equations the validity of the design mean flow path can be considered as well as the impact of the skew of the blades. Also additional information about the flow is obtained in the form of through flow velocity gradients, which allows a refined calculation of the performance to be undertaken.
NOTATION
jo,
A cross-sectional area of flow path A , B , C flow equation coefficients C absolute fluid velocity Es, Ef power loss due to incidence and friction, respectively f friction factor H rothalpy h distance between shell and core I energy per unit mass (enthalpy) jo, j~, j~ values of specific integrals involving Hermite polynomials K curvature of flow path k blade force per unit mass L moment of momentum 1 length of design flow path within an element M torque rh mass flow rate m , S curvilinear co-ordinates n vector perpendicular to blade surface O,P,Q points used when deriving cpm P pressure R radius of curvature of flow path r, z, 0 cylinder co-ordinates t time u tangential velocity of rotating element v s shock velocity w fluid velocity relative element blade angle coefficient 7, e, 2 angles regarding the orientation of curvilinear co-ordinates F moment of the tangential component of the absolute fluid velocity /~ incremental change of . . . . vorticity relative position along the design flow path within an element ® angle describing the blade surface l arbitrary quantity for demonstration of formulas A, E geometrical coefficients of flow path P density of fluid Y parameter for blade angle derivative (p blade angle (relative the meridional plane) X coefficient for the blade force in the transverse direction of the flow path Stokesian stream function f~ denotes the flow equation 6O rotational speed .
.
.
695
696
SVEN ANDERSSON
subscripts 1,2 A,B
entry, exit of turbine just in front of, just behind turbine entry C, S at core and shell, respectively J element number r, O, Z, S, m with respect to the specific co-ordinate direction
superscripts derivative with respect to q~, i.e. d/dq ~ refined coefficient value N.B. Vectors are shown in bold type.
1. INTRODUCTION The established method to analyse hydrodynamic torque converters is based upon the assumption that the flow in it has a one-dimensional character. The flow is represented by a mean flow path. The Euler turbomachinery equation and a power balance for the converter then solves the task. The one-dimensional approach implies a neglect of the state of things in the direction transverse to the flow path. However, the validity of the flow path is due to an equilibrium in this direction. When executing the one-dimensional method the result may be significantly inconsistent with such an equilibrium, resulting in a discrepancy between real and assumed flow path location and between actual and predicted performance.. These problems are overcome by employing the two-dimensional approach, frequently used when analysing turbomachines, but the increase of the numerical effort will be substantial. Such an analysis was carried out in Ref. [1]. In this paper results from the two-dimensional theory are applied to the one-dimensional mean flow path in order to obtain the conditions for its validity. Some new properties then enter the ordinary one-dimensional theory: the blade angles in the transverse direction, i.e. the skew of the blades and the gradient of the through flow velocity. When both the blade angles in flow direction and the skew are known, the shape of the blades are uniquely determined. The flow gradients provide additional information about the flow field in the converter; an estimate of the flow velocities at the shell and the core can be made. These velocities are often considerably different. This paper also supplies a numerical example concerning a three-element converter of the type commonly used in automatic transmissions. 2. B A S I C E Q U A T I O N S
ONE-DIMENSIONAL
APPROACH
Analysis of hydrodynamic torque converters employing a one-dimensional mean flow path is well established and will be briefly related below. More thorough treatises are provided by Refs [2-5]. A meridional cross-section of a converter is shown in Fig. 1. Converters with three elements, pump, turbine and stator, arranged as shown in this figure is the most common type; other types of converters may be found in Ref. [6]. The mean flow path is indicated in Fig. 1. The points where the fluid enters and exits the various elements are denoted by indices of which the first refers to the number of the actual element and the second has the value 1 at entries and 2 at exits. At entries a letter A or B is added to the second index indicating that the point is just in front of or just behind the entrance, respectively. A sample of the velocity triangles is also shown in Fig. 1. The absolute fluid velocity c is made up of the relative fluid velocity w and the rotational velocity u. The velocity c is resolved into two components: Co in the tangential direction and c= in the mean flow path direction. The geometrical relations between the velocities are c,, = w cos q~,, Co = u + w sin ~0,, = r(o + c,, tan q~,,,
( 1)
Blade skew and flow gradients in torque converters
697
where ~o,~is the blade angle relative to the meridional plane. The cross-sectional area of the mean flow path is denoted A allowing the mass flow rate to be written as fia = pcmA. (2) Usually A may be regarded as constant which implies that cm has the same value at every point along the mean flow path. 2.1 Torque on an element Define the moment of the tangential flow velocity as F
=
(3)
rco.
Now the Euler turbomachinery equation for the j-th element yields Mj
(4)
= r i l ( F j . 2 - r j , la ).
If no torque is exerted on the non-bladed space in front of this element, the Euler equation gives gj,l~ = r~_ ~,2.
(5)
Insertion of equation (1) in equation (3) yields r tan ~0m
(6)
which together with equations (4) and (5) gives the torque for the j-th element as M j = fi'l[(~j. 2 - ~j_ 1,2)c,, + r~2zo~j - r 2 1.2 ~oj_ 1].
(7)
A peculiarity regarding the nomenclature ought to be commented on. Asj can take the values 1, 2, 3 the quantityj - 1 is 0, 1, 2, respectively, but no element has the index 0. However, the index j - 1 should be interpreted as 'preceding element' and as element 1 is preceded by element 3, index 0 refers to element 3. Analogouslyj + 1 should be interpreted as 'succeeding element', that is the indices 4 and 1 are synonymous. This state of things bring forth the useful summation formula, t an arbitrary quantity, 3
3
Z j
3
= Y, ' J - , = 1
Z 'J+,.
1-1
j
,o1,
[-I
Turb~ ir
2,1 B 2,1A 1,2 /
(8)
1
Meonflopwofh
co
!
i i
I-I
I 1 ii
I vs,~
C1,
~
f FIG. 1. Meridional cross-section o f a three-element torque converter; at line I-1 a section perpendicular to the meridional plane, showing the velocity triangle and the shock velocity, v s.
698
SVEN ANDERSSON
With this formula and equations ~4) and (5) the torque equilibrium of the entire converter is readily proved and shown to be an inherent quality of these equations
2
Mj ~- 1TI
i =I
2 i
l-'J. 2
Fi
1
1,2
=
0.
{9)
1
2.2 Power balance jbr a converter When the fluid is on the point of entry into an element, its velocity is usually not consistent with the blades in the entrance. The situation is illustrated in Fig. 1. The fluid experiences an abrupt change of the velocity at this point; the change is often called shock velocity and is defined as US, = C#,],IA-- COj, IB" (|(}) This abrupt change of the velocity also implies a loss of power, often referred to as incidence loss or shock loss, and is shown to be, Ref. [1] among others, E~ i - rhv~ " 2
(111
rh ('I'i"B-- lS"'A~ 2 Es'j = 5 \ ri, l /
(12)
or if equations (3) and (10) are used
Power losses due to friction, often called circulation losses, are expressed as .2
•
,~m
Et= r h J ~ ,
t13)
where f is a loss parameter• Now the power balance for the entire converter states that the net power supplied by the shafts of the machine equals the total power dissipated, that is 3
3
~ Es,j+E f.
My)j=
(14j
j=t
j:i
Rewriting this with the aid of equations {13), I 11), (4) and (5) and introduction of the quantity gives 3
3
' + Y n = Jc;,, i
v 2~,; - 2 ~ (Fj, 2 - V j t
1,2)(Dj=O.
{15j
/:= 1
Insertion of equations (10h (3) and (6) yields a quadratic equation for the velocity % ~)= Ac 2 , - 2 B c m - C = 0
A
/]e 3
B = V B/(t)j j i Bj : {~i.2--,3(j,1)+{O~j.1. | --O~j,2)( rj''2 ~2 k,rj+ 1.1 / 3
C=
V -- Cj~,)~ i :1 4-
Cj = 2r22
rJ.a r2 t.l
z rj.~.
(16)
This equation may be considered as the governing equation for the converter. For a specific geometry and operational mode, i.e. rotational speeds, c,, is solved from this equation and thereupon torques and powers are readily calculated.
Blade skew and flow gradients in torque converters
699
As c= is solved from a quadratic equation, two roots are possible but only the upper one is stable as shown in Ref. [1]. Often the lower root is negative, that is contradictory to the assumed direction of c=. Equation (15) is evolved for a three-element converter but the generalization of the equation is obvious--the number of terms in the sums equals the number of elements of the converter, cf. Ref. [6]. 2.3 Design procedure When converters are designed usually a specific operational mode is fixed, at which no incidence losses should occur, in practice the point of maximum efficiency. Seven quantities now have to be settled, c= and six blade angles ~0m.~.l. . . . ~om.3.2. Four relations are given by equations (15), (10), (5) and (3) 3
f~=fc2..-2 ~ (rj,2-rj-1,2la)j=O
f
j=l
j = 1, 2, 3.
Fj_ 1,2 = Fj.1B
(17)
As there are seven unknowns and four relations three quantities are for choice, resulting in converters with different characteristics. A design process for converters is presented by Ref. [7].
3. B A S I C E Q U A T I O N S - - T W O - D I M E N S I O N A L
APPROACH
Turbomachines often have so many blades so that it is justifiable to replace the discrete forces they bring about by a continuous field of forces. This is equivalent to considering a turbine with an infinite number of thin blades. Now the flow will be independent of the circumferential co-ordinate 0, that is the real three-dimensional flow is replaced by a twodimensional one. The problem is reduced to study the flow in the meridional plane, i.e. the r-z plane. This concept is widely used; Refs [8-10-1 are given as examples. As axial flow machines as gas and steam turbines are of particular interest much of the literature is devoted to these. Here the flow is almost parallel with the z axis, and important feature when developing the equations. The turbines in torque converters have such a shape that a curvilinear co-ordinate system is preferable, see Fig, 2. It is assumed that a function exists which transforms an orthogonal net to a deformed quasi-orthogonal one, where some specific lines in the net fit the boundaries of the turbine. The deformed net is sketched in Fig. 2.
Cm I
I ~
Quasi-oH'hogonot net
9s ...~
I
I I
u=r6o
I I
•
/0
Z
FIG. 2. Meridional cross-section of a turbine; r-z plane. The lines I and 11 show sections perpendicular to the meridional plane; the m-O and s-O planes, respectively.
700
SVEN ANDERSSON
The formulation of the flow equations for this type of co-ordinates is analysed in Ref. [1], The parts of that treatise that is used here will be related below. 3.1 Derivatives in the curvilinear system The transform that produces the quasi-orthogonal net shown in Fig. 2 does not have to be expressed explicitly. Define the quantities m and s as the lengths measured along the lines of the net, see Fig. 2. N o w the partial derivative of some quantity t with respect to the variable m is given by the chain rule
Ot ?;l Or Ot Oz + ~m - Or Om c3z Om"
(18)
The derivatives 6r/~3m and dz/~m may be expressed by the angle 7 shown in Fig. 2 giving & & & 3m = ~rr sin 7 + ~zz cos 7.
i 19)
In a similar Way the derivative &lOs can be expressed with the aid of the angle ~:. These two expressions solved with respect to the r and z derivatives yield
Or
sin (e - 7) ~s cos y - ~m cos
31 _ 1 { 8l . ~t ) ~?z sin(e-7)~0mmSme-s sin7 "
(20)
3.2 Blade 9eometry The blades may, as any surface in space, be described by a function ®(r, z) which for a specific r and z gives the angle (9 measured from a reference plane to the blade surface, see Fig. 3. The blade angle, ~o,, in the figure, is the angle between the meridional plane and the blade surface. Using the notation shown in the figure this angle is [QO I (r + Ar) A(9 0(9 tan q~.. = lim - lim - r--. Q~p IPQI am~0 Am c3m
(21 )
This formula is valid for any direction; for instance 0(9 tan q~s = r c~s "
(22)
_J
'~ /BI(lde surfoce
R~erence ptane FIG. 3. Blade angles; the points P and Q contained in the meridional plane, P and O lies on the blade surface.
Blade skew and flow gradients in torque converters
701
In Fig. 3 is also shown how a vector n perpendicular to the blade surface may be constructed. F o r cylinder co-ordinates this vector is
n=(-tancp,, 1, -tan~oz)=
( -r~rr, do
1,
gz/"
(23)
3.3 S t r e a m f u n c t i o n It is of especial convenience to assume that the 'm-lines' of the quasi-orthogonal net coincides with the flow paths in the meridional plane. This implies that the fluid velocity cm is tangent to the 'm-lines', see Fig. 2. The velocities in the r and z directions may now be written
{
c: = c,. cos 7 c m sin 7.
(24)
cr
The continuity equation in cylinder co-ordinates reads O(rcr)
dr
+
t~(rc~)
dz
= o.
(25)
This equation is always satisfied if c, and cz are expressed as 1 d~ Cr =
r
dZ
I d~
(26)
Cz ~ - - - - ~
r dr
where qJ is the Stokesian stream function. Using equations (19), (20), (26) and (24) it is found that = rcm sin (e - 7)
d~p Omm
(27)
O.
The latter expression is a consequence of the coincidence of the 'm-lines' and the flow paths. It is convenient to define an angle 2 according to, see Fig. 2, 2 = } , - (e - n/2)
(28)
which yields dq j ds
= r c,, cos 2.
(29)
3.4 E q u a t i o n s o f m o t i o n The flow is governed by Euler's equation which reads c3c ~ - + (c" V) c -
Vp
+ k.
(30)
P The field of forces accounting for the blades are represented by k, force per unit mass. The fluid is non-compressible and the flow is non-dissipative. Omission of the non-stationary term and introduction of the vorticity = V x c - curlc
(31)
and the energy per unit mass C2
I = p- + ~P MS 29:10/II--C
(32)
702
SVENANDERSSON
gives ~xc=
(33)
-Vl+k.
Carrying out the derivatives in equation (31) yields 1 O(rco) r
~z
?Jc, dz
~o-
Ocz dr
1 O(rco)
r
~34)
dr
Developing equation (33) and insertion of equations (34) and (3) gives F ?F r 2 Or
c, dF 4 c = d F
7~r
r i?z
-- Cr~ 0 --
0I +kr dr --
ko
F d[ OI r i ~-z = - O-z+ k~.
(35)
If the second of these equations is combined with equations (24) and (19) 0F
(36)
rk o = c,, ~m m .
This equation is a differential form of the Euler turbomachinery equation, see Section 2.1. As the flow is non-dissipative the force k is perpendicular to the blade surface, that is parallel to the vector n, equation (23) k = (kr, ko, kz) = kon = ( - k 0 tan ~o,, ko, - k0 tan ~o:).
(37)
Elimination of ~0 from the first and third of equations (35) and with the aid of equations (24), (27), (23), (19), (6)and (36)yields (31
0F
d ~ - ~ ~mm = o.
i381
If the rothalpy H is defined as H =
I -
coF
(39)
equation (38) states that H is constant along a flow path. Equation (38) is a differential form of the energy conservation theorem; it states that AI = u)AF = Me) for an element. Multiplying the first and third parts of equation (35) by c. and cr, respectively, and subtracting them and using the second part of equation (35), equations (39), (6), (19}-(22), (37), (23), (24), (29) and (38) yields dF 0 0 ( , o - dr dz
dF (3® dz dr
1
~H
cm cos2 ds
140)
Further development and remarking that d H / d m = 0 implies that H may be considered as a function of q~ only gives 1 (~__~ 0F ) dH G0 - r cos~ tan q),, - ~mmtan ~o= - r dq j .
(41)
In analogy to equation (37) the projection of the force k in the s direction is k= =
-
ko t a n ~p=
(42)
Blade skew and flow gradients in torque converters
703
which together with equation (36) turns equation (41) to ~0 r
tan ~0m 01~ -- r 2 c o s ) ~ t3s
I-
k,
dH
rcmcos 2
dW"
(43)
An expression for the vorticity is found by evolving the second part of equations (34). Insertion of equations (24) and (20) gives 1 (o = CmK
~c,.
cos )~ ds '
(44)
where K is the curvature of the flow path. K-
~37 (45)
Ore"
The curvature is the reciprocal of the radius of curvature; thus K - 0 for a straight line. Equation (44) holds under the condition that Oct.~din = 0, that is the flow velocity is constant along the flow path. This restrictive assumption is justifiable as the aim of this treatise is to prescribe a flow and to find the appropriate shape of the blades for a converter with constant through flow area; A in equation (2). Equation (44) is rewritten with the aid of equation (29) dc,.
~o = c,,K - re,. d ~ "
(46)
Henceforth the flow velocity will be characterized by two parameters, c,, and dcm/dq2, that is, the function is replaced by the two first terms of its Taylor series. The parameter c,, is constant at every position along the flow path while the parameter d c , , / d W may vary.
4. B A S I C E Q U A T I O N S
IN DIRECTION
TRANSVERSE
TO FLOW
PATH
The one-dimensional theory related in Section 2 utilizes two theorems: Euler's equation for turbo machinery, i.e. a relation between torque and change of moment of momentum, and the energy conservation theorem, equations (4) and (14), respectively, When using a twodimensional approach these two turn up again, equations (36) and (38), but also a third equation is found, (43), essentially a force equilibrium in the direction perpendicular to the flow path. The essence of this treatise is to study the middle flow path just as for the one-dimensional approach but to consider the state of things transverse to the path too. In this way the implications of the blade angle q~, the skew of the blades, can be considered without solving the comparatively vast two-dimensional problem. 4.1 Within an element At every 's-line' any quantity may be described as a function of q~ as well as ofs. Therefore, according to equation (29) t~l
0z
O--s = rc,. cos 2 Oqj ,
(47)
where t can be any quantity. Now equation (43) can be written as ~o r
_
c., tan qo,"(__ + ac~,) +Z - H'
(48)
r
_= = (2rco + c., tan ~0,.)sin e -~
rcm cos 2
Z-
ks rc m COS )~
1 ~ tan ~0," cos 2 t~s
(49) (50)
704
SVEN ANDERSSON
F' = E+~c~..
(51)
For convenience derivatives with respect to ~P are denoted by a prime sign. Often converters have m i n i m u m clearance between the elements why it is justifiable to put
r-i+ 1.1
= rj,2.
(52)
Such a property also applies for: K, c~., 7, e and thereby also for ~0 and ).. Insertion of equation (46) in equation (48) and employing the constancy of H' within an element; H~.~ = H~.2 yields Xj.2 - ~j.1 = - cm(c'~,-i,2 sec 2 ~0,..j,2 - c~..j_ 1,2 sec z ~o.,.j, 1) A-i,1)
153)
A = (E tan tp,. - K)/r.
(54)
-- c,n(Aj.2 -
Equation (53) constitutes a specific relationship between the forces ks and derivatives c',,; the other quantities in the expression, c,, and geometrical properties as r, ~ o , , , . . . , may be regarded as k n o w n constants after the design of the converter according to the onedimensional a p p r o a c h has been executed. 4.2 Transition between elements When entering an element the fluid experiences a sudden change of the flow velocity; equation (10). Here power is lost, see equation ill), but as F a ~ F g also an exchange of mechanical power takes place. Study the power balance for the entrance region; this region is a thin disc which A is just in front of and B just behind. r h ( i a _ l A ) = - ~1 m . v 2s + m" ( V a - FA) oga.
(55)
Insertion of equation (39)
Ha_H
A = _½ Vs_2
(COB__ ( D A ) F A
{56)
or using equation (51) HS+ 1,1 - H),2 = - Vs0 + 1V's0+ 1- (coj+ 1 - ~oj) (Zj.2 + o~j,2Cm,j.2 ).
(57)
This equation holds for the transition between elements j and j + 1. For the design mode, Section 2.3, no incidence losses take place, that is v s = 0. Elimination of H' and (0 by equations (48) and (46) and observing that {0, e~,, F and K are not changed between the positions (j, 2) and (j + 1, 1) gives Z j + 1,1 - - Zj,2 •
- - Cm( s e c 2 -
~Om,j+1, 1 - -
c,~(Aj+ 1.1 -
Aj,2) -
sec2
(tOm,j.2)Cm.j,2
(foj+ 1 -
ogj)(•j.2 + o~j.zC',n.j.2).
(58)
This equation is similar to equation (53). In the case of a three-element converter equations (53) and (58) constitute a set of six equations. Summing these six equations yields 3
3
~, (ct-i.a(coj +1 - o i)) c',,.-i.2 = - ~, (co-i+1 - eJ j) Ej.2 j=l
(591
j=l
which expression is equivalent to 1
df~
2dq /
-
0.
{60)
Here the loss parameter f = 0, that is non-dissipative flow. The equivalence between equations (59) and (60) is shown with the aid of equations (17), (51) and (8), and is an obvious consequence of the energy conservation theorem; the function f~ = 0. When designing a converter according to the one-dimensional method the flow in it is described by one parameter, c,,, but now additional information a b o u t the flow is supplied by the gradients c~,. Equation (59) constitutes a specific relationship between them.
Blade skew and flow gradients in torque converters
705
If the c~.s are chosen, in compliance with equation (59), and one of the Xs then the remaining Xs are given by equations (53) and (58). In the case of a three-element converter the p r o b l e m has nine unknowns, three c~,s and six Ks, but six equations. Thus, a variety of solutions are possible; the designer has to formulate additional criteria in order to select a suitable one. If the condition ~O.j,2 ~- ~O,j + 1.1 was not valid, i.e. equation (58) not satisfied, the flow path will have a discontinuity in the curvature K when entering an element. This implies that the fluid experiences a jerk here despite continuous flow velocities c" and Co.
5. BLADE ANGLES AND A NUMERICAL EXAMPLE Assume that a circular flow path has been chosen for a three-element converter; Fig. 4. According to equation (21) the angle ®, which describes the position of the blade surface, Fig. 3, m a y be calculated as ® = ®1 + fo tan tO" l dr/ r
(61)
where 1 is the length of the flow path within an element and r / a parameter describing the relative position along the flow path; r / = 0 at inlet, r / = 1 at outlet. Moving the curve a distance As inwards at every 's-line' changes the angle ® according to t~s
-
t3~- +
tan tO,.
ds
r2 sin e dr/.
(62)
It has been assumed that O tan to,, ~
-
0.
(63)
This is in good agreement with measurements reported in Ref. [1], but also other types of blade constructions exist, Refs [2-1 and [4-1. F o r this particular circular flow path is
{ --1-R(l+cos,) ~ dl
(72 --Tl)R - (72 - 70 + (tan 22 - tan 21)
I/R.
(64)
5.1 Formulas for co,, and to~ The function tan to,.(r/) is described as tan tO,. = (1 - 3r/2 + 2r/3) tan tO,.. 1 + (3r/2 - 2r/3) tan tOm.2 + (q _ 2q2 + r/3)y1 + (q3 _ q2)y2,
(65)
Turbine ./i~-~.~... Pump !
/~
\ i~Fto~pa~h
T , FIG. 4. Skeleton sketch of a three-element converter.
706
SVEN ANDERSSON
that is by four parameters: ~o,..~, Y~, q~,,.2, Y2 and the adequate Hermite polynomials. The quantity r is defined by c3tan ~p,, 1 d tan ~,. . . . . (Jm 1 dr/
Y 1
t66~
When constructing the blades the angles q~= are given but the derivatives "fl and Y2 are for choice. Define the coefficients J:
jo Jl
1 -- 3r/2 + 2r/31 3r/2 --2r/3 / r / - 2r/2 + r/3 | ( ! d l ds
-1 -- jo
Lr/3 __712
l sin e r2 ) dr/
~67t
J
which allows the angle ~p~to be written as, equations (22), (62), (65) and (67) tan ~os.2 r2
tan ~Os.~ + (J1° tan ~0m.l + jo tan q~m.2)+ (Jl r~ + J~ Y2). rl
(68)
With equation (66)
~?F
(2rco+ cm tan
'
rcm `1.
(69)
which combined with equations (42) and (36) gives ks-
cm tan r %
{
r'i
(2to) + Cmtan ~o,,)sin 7 + ~ - Y .
(70t
Assume that k~.: is known. If q)~,l is chosen YI is given by equation (70). Elimination of Y~ between equations (68) and (70) gives a second-order equation for tan ~os,2; tan2 ~%2 - tan ~o~.2
tan q~,l +r2 (J° tan ~o,,,l + J2° tan q?m.2)
+r2J~`1.1 _ j ~ l(2r2u)+c,,tan~o,,.2)sinTz
4 J12ks.21r2 .2 - 0.
Cm
{71!
(- m
Insertion of k~,2 and other known values yields ~os,2and thereupon "i"2,solved from equation (70). This problem has four unknowns: ~0s.~, Y l, ~0s.z,Y2 but three equations, equations (68) and (70) at inlet and outlet. Hence the designer has to settle which one of the possible solutions that is best fitted. 5.2 Numerical values for a.How path The geometrical properties of the flow path are presented in Table 1. The radius r and other lengths are given by relative numbers; the outer radius of the flow path is used as a norm. The J-coefficients are given in Table 2. The angles y and ). vary linearly along the flow path and the integrals, equation (67), are evaluated by employing the trapezoidal formula.
TABLE 1. GEOMETRICALPROPERTIESFOR FLOWPATH
Position
"; C)
~: (')
2 C)
1, 1 l, 2
45 180
117 270
2, 2, 3, 3,
180 315 315 405
270 423 423 477
1 2 1 2
r
K
+ 18 0
0.500 1.000
3.414 3.414
0 - 18 - 18 + 18
1.000 0.500 0.50{) 0.500
3.414 3.414 3.414 3.414
B l a d e s k e w a n d flow g r a d i e n t s in t o r q u e c o n v e r t e r s
707
TABLE 2. J-COEFFICIENTS Element
jo
jo
j~
j 2~
1 2
-2.3928 - 1.2860
- 1.2860 -2.3928
-0.3249 -0.2390
+0.2390 +0.3249
3
-2.1822
-2.1822
-0.3785
+0.3785
5.3 Numerical example--blade angles The presented theory will now be d e m o n s t r a t e d by a complete numerical example. First the blade angles ~o,, are chosen according to the procedure mentioned in Section 2.3. Assume that the rotational velocities: ~Ol = I, to 2 = 0.65, ~o3 = 0 are settled for the point of no incidence losses. The rotational velocities are given by non-dimensional values; the velocity of the pump, element 1, is used as a norm. The process of selecting the q~,,s is connected to the characteristics of the converter needed and is overlooked here; the choice cm = 0.25, F3.~ = 0 and F~,2 = 0.75 will do as an adequate example. In Table 3 the blade angles and the ct, E and A coefficients are shown. 5.4 Numerical example--d,, and X N o w the properties c~, and X may be settled. Assume that tps.3.~ = 0 is demanded, i.e. X3.1 = 0. Stators in existing converters are often constituted in this way. The gradients c~, have to fulfil equation (59); if one X is k n o w n the remaining are given by equations (53) and (58). T w o alternatives are examined: in the first all the c;,s are equal but in the second these are varied in order to obtain low values for the Xs. The result is shown in Table 4. The second alternative is chosen as low values for the Xs counteract excessive values for the angles ~0,. However, the effect of this step is yet to be seen at this point in the numerical evaluation. 5.5 Numerical example--qg~ and Y When c~ and X, i.e. k~, have been settled the properties ~0~ and Y remain. The procedure described in Section 5.1, equations (68), (70), (71) is used. If Y1 is used as a parameter Fig. 5 shows an example of how things turns out. The designer has to select a solution which implies a reasonable shape of the blades. The choice in this example is given in Table 5. TABLE 3. BLADE ANGLES AND ASSOCIATED COEFFICIENTS cm = 0.25 Position
u9
F
1, 1 1, 2
1.00
0.2625 0.7500
0.100 - 1.000
0.050 - 1.000
7.682 - 7.000
- 5.292 3.586
2, 1 2, 2
0.65
0.7500 0.0000
0.400 - 1.300
0.400 -0.650
- 5.600 2.436
- 5.654 - 13.161
3, 1 3, 2
0.00
0.04300 0.2625
0.000 2.100
0.000 1.050
0.000 3.935
- 6.828 9.698
TABLE 4.
t a n qg,,
c',,,X
~
_=
A
AND k s - - T W O ALTERNATIVES
c"
Alternative 1 X
ks
c~,
Alternative 2 )~
k5
-
1, 2
2.6346 -2.6346
- 1.546 -3.113
- 0.1838 - 0.7784
- 3.6420 -4.0000
0.6747 -0.4643
0.0802 -0.1161
2, 1 2, 2
-2.6346 - 2.6346
-2.885 0.0430
-0.7212 0.0004)
-4.0000 1.0000
-0.0443 0.00430
-0.0111 0.00(~
3, 1 3, 2
- 2.6346 - 2.6346
0.000 - 1.227
0.04300 - O. 1459
1.004)0 - 3.6420
0.004)0 1.0442
0.00430 O. 1241
Position 1, 1
708
SVEN ANDERSSON +1.2 tangs
~
+0.8
ton~sAupper
root
i
+0./~
~ 7~upper root
÷2 fi
o.o
.
.
.
.
.
.
.
.
.
!o
.
_0./~'
tan~s.~SJ
-O.B
J
-1.2 . . . . . . . .
-6
-L
-2
~,fQn~.2tower roof -2
~'.......t J-. . . . ., . , ,
o
.2
,~
i-4
J
)i+6
6
FIG. 5. Y2 and tan (p~ vs Y~ for element 1 - t w o roots due to equation (71). Y~ = 0 and the lower root case was chosen, see Table 5.
TABLE 5. • AND (Ps
Position
Y
tan q),
1, 1 1, 2
0 - 3.902
-0.221 - 0.329
2, 1 2, 2
-0.2 1.269
-0.613 1.222
3,1 3, 2
0 - 0.998
0 - 2.480
5.6 Numerical example--closure Now when all parameters have been settled any quantity can be evaluated at any position, t/-value, along the flow path. In Fig. 6 the blade angles and Figs 7 and 8 some flow properties are shown. When evaluating q~sand ® numerical integration of equations (62)and (61) with equation (65) inserted was done. Now the upper integration limit in equation (67), "1",takes the t/-value for the actual position.
+8oI
..........................................
.60~
(deg,
~
'°k +2ok
4 / / / / /
\
-60 Pump -80
~ , Sfo~or
Turbine '
/'
L
FIG. 6. The angles ~Ps, q~m a n d ® along the flow path.
"~f I
Blade skew and flow gradients in torque converters
709
-12 c~ -8
-4 0
Pump
+L,
Turbine
Stutor
FIG. 7. The gradient c~. along the flow path. According to equation (29) A% ° = rcmAs. Assuming As = 0.14 for r = 1 yields AW = 0.035. Then, for instance, c~. = - 7 implies that cm = 0.37 at the shell and 0.13 at the core, respectively.
1.0
ko,ks 0.8
E
0.6 F
0.t~ 0.2 0
ks
-0.2 -0.t. -0.6
Pump
Turbine
Sfnfor
FiG. 8. The quantity F and the forces ko and ks along the flow path.
Sections 5.2-5.5 describe a design a n d decision hierarchy in four levels namely: (1) a r r a n g e m e n t o f converter; (2) p e r f o r m a n c e characteristics; (3) details regarding the flow and blade forces and; (4) blade design. H e r e b y the p a r a m e t e r s will be settled in the o r d e r o f decreasing prominency. However, p a r a m e t e r s settled at a specific level become prerequisites for the lower levels a n d m a y then turn out to have d i s a d v a n t a g e o u s values. Eventually these high level p a r a m e t e r s have to be reconsidered. This feed-back process has to be executed by the designer. The four level a p p r o a c h is useful when creating a c o m p u t e r - a i d for the designer. The levels are i m p l e m e n t e d as separate p r o g r a m blocks a n d the p a r a m e t e r s as a d a t a base accessible for all the blocks.
6. I N F L U E N C E OF THE FLOW VELOCITY GRADIENTS ON THE P E R F O R M A N C E W h e n the flow velocity gradients c~, are k n o w n the calculation o f the torques m a y be refined. The m o m e n t o f m o m e n t u m , L, at a specific p o s i t i o n along the flow p a t h is, e m p l o y i n g e q u a t i o n s ( 3 ) a n d (1) L = Srco drh = S F dria = ~ (r2o~ + rc,. tan q~,,) drh.
(72)
Hence the t o r q u e is, see e q u a t i o n s (4) a n d (5) M j = Li, 2 - L~_ 1,2.
(73)
710
SVEN ANDERSSON
The differential drh is
dria = 2nprc,.cos ).ds = 2 n p ~ - ds = 2 n p d q j,
(74)
where equation (29t has been inserted. The differential ds = dr/sin c, which inserted in equation (74) gives the mass flow rate COS ).
rh = npc., _
s i n ~;
(r~-r~) = 2np(qJc-- qJs).
(75)
The radii to the shell and the core are denoted by rsand rc, respectively. Denoting the distance between the shell and core along a 's-line' by h, h = (r,, - r~)/sin e
(76)
and comparing equations (75) and (2) yields
ria = npc.,h cos ). (r e + rs)
i 77)
A = nhlr~+r~)cos)~.
(78)
As the influence of c~. on the quantity L is moderate a series expansion is convenient.
+c'm( dL ~ =~(r~o+(~c,,). \de',. ],.;. = o
L=(L),;=o
(79)
Appropriate values for the coefficients r 2 and ~ in equation (6) are also defined by equation (79). Evaluation of L when c~ = 0 yields, employing equations (74) and (75)
rc+ r~ 2(r2c+ r cr s + r s (L),,, =o = r h l e ) ' 2 , +c,.tanq)m 3 ( r c + r s ) 2} .
180)
The derivative of L is given by equation (72) with equation (74) inserted dL ..... dc'.,
.
cos ). i'r~ dc,. ~zt~ " - I (r2")+2c"rtan~°~)rdc'dr" sin c j r,
{811
According to equation (75) the stream function may be written as COS ,,:~ r 2
q~ (r) = c , . .
sin g
(821
2
Recalling the assumptions in the end of section 3 and assigning the stream function value (q'~+ q~)/2 to the mean flow path allows the flow velocity to be written as
c.,(r)=c,.+c'{qqr)
q"c+"~s }2.
=c,.+c',.c.o..~\~c°s2{rZ
r2c+rZs)4'
183)
Insertion of this in equation (74) and recalculation ofria confirms that rh is independent of the parameter c;,. Equation (83) gives arm cos2 ( 2 ~ (84)
dc"
-
Cm
sin-/:
which inserted in equation (81) yields dL
_
{ ,r +rs,2
= rh cos). ( r e - rs)2 o~ c:. = 0 sin e
24
_
r~+3rcrs+rZs ) , ?. outr c-t-rs) )
I- 2c,. tan ~o., ~
(85)
Comparison between the expressions within the square brackets in equations (80) and (85) shows that the latter, with sufficient accuracy, may be regarded as one sixth of the former. hence, by equation (76) (dL) ~ h2 cos2 sin~ (L)c; = 0" [86) if',,, ,;, 6 =
,,
Blade skew and flow gradients in torque converters
711
TABLE6. CORRECTIVECOEFFICIENTS;EQUATIONS(87}AND(88) 1,2 2, 1
Position
I1 + ¢~h2 COS.~sine 6
2,2 3, 1
1.0131
]
3,2 1, 1
1.0122 0.9554
Equations (79) and (80) now give the coefficients +
2(r2+rcrs+r])[ & = t a n q~,.
3 (% + rs)
1
g
(87)
c~'hZc°s)~ siri e ] 1+
6
(88) "
These equations imply that the influences due to the through flow velocity gradients are regarded by multiplying the ordinary r 2 and :t by a specific corrective coefficient. Table 6 shows the values of these for the numerical example in Section 5. The through flow area A = 2n × 0.14 was assumed, which when inserted in equation (78) gave h. Using these corrections and the values in Tables 2 and 3 together with equations (16), (79) and (73) showed that the value of c,, changed from 0.250 to 0.258 (3.3 ~o),and that the capacity of the torque converter, i.e. M1, increased by 4.3 'Yo. Thus the flow velocity gradients cause a discernible increase of the capacity, this effect is also reported by Ref. [4]. There the flow gradients are obtained in quite another way by a geometrical reasoning. The flow in the meridional plane is assumed to be a type of solid body rotation about the kernel of the core. However, the main gain of more knowledge of the flow field is that this provides a mean for improvement of the efficiency of the machine. 7. CONCLUSIONS The well established one-dimensional mean flow approach in the analysis of hydrodynamic torque converters has been supplemented by equations regarding the balance in the transverse direction of the flow path. These equations are derived from the two-dimensional theory for turbomachines. By these equations the validity of the design mean flow path can be considered as well as the impact of the skew of the blades. Further parameters regarding the blade shape now enter the analysis: the skew angle and the derivative of the blade angle in the through flow direction. Also additional information about the flow is obtained in form of through flow velocity gradients allowing the velocities at the shell and the core to be estimated. Furthermore the calculation of the performance of the converter can be refined. REFERENCES 1. S. ANDERSSON,On hydrodynamic torque converters. Transactions of Machine Elements Division, Lund Technical University, Lund (1982). 2. M. WOLF, Str6mungskupplungen und Str6mungswandler. Springer-Verlag, Berlin (1962). 3. R. HERBERTZ,Untersuchung des dynamischen Verhaltens von F6ttinger-Getrieben. Diss. T. H. Hannover (1973). 4. A. WHITEFIELD,F. J. WALLACEand R. SIVALINGHAM,A performanceprediction procedure for three-element torque converters. Int. J. Mech. Sci. 20, 801-814 (1978). 5. A. WHITFIELD,F. J. WALLACEandA. PATEL,Performanceprediction of multi-elementtorque converters. Int. J. Mech. Sci. 25, 77-85 (1983). 6. S. ANDERSSON,Analysisof multi-elementtorque converter transmissions. Int. J. Mech. Sci. 28, 431~,41 (1986).
712
SVEN ANDERSSON
7. A. WHITFIELD,F. J. WALLACEand A. PATEL,Design of three-element hydrokinetic torque converters. Int. J. Mech. Sci. 25, 485--497 (1983). 8. R. A. NOVAK, Streamline curvature computing procedures for fluid-flow problems. J. Enono Power, Trans. A S M E 478490 11967). 9. J. J. OTTE,,A method of analysis of the axi-symmetrical flow in the blade channels of turbines. Third Scient!lic ConJl on Steam Turbines of Great Output, Gdansk, 24-27 Sept 1974. Prace instytutu maszyn przplywowych, Warszawa-Poznan, Zeszyt 7(~72, pp. 597-614 (1976). 10. A. J. WENNERSTROM,On the treatment of body forces in the radial equilibrium equation of turbomachinery. TraupeI-Festschrifl, pp. 35l 367. Juris-Verlag, Z~rich (1974).
0020-7403/87 $3.00+ .00 ~t~1987 Pergamon Journals Ltd.
Printed in Great Britain.
ASSESSMENT OF THE BLADE SKEW A N D THE THROUGH FLOW VELOCITY GRADIENTS IN HYDRODYNAMIC TORQUE CONVERTERS SVEN ANDERSSON Machine Elements Division, Lund Technical University, Box 118, S-221 00 Lund, Sweden
(Received 24 November 1986; and in revised form 30 May 1987) A b s t r a c t - - T h e well established one-dimensional mean flow approach in the analysis of hydrodynamic torque converters has been supplemented by equations regarding the balance in the transverse direction of the flow path. By these equations the validity of the design mean flow path can be considered as well as the impact of the skew of the blades. Also additional information about the flow is obtained in the form of through flow velocity gradients, which allows a refined calculation of the performance to be undertaken.
NOTATION
jo,
A cross-sectional area of flow path A , B , C flow equation coefficients C absolute fluid velocity Es, Ef power loss due to incidence and friction, respectively f friction factor H rothalpy h distance between shell and core I energy per unit mass (enthalpy) jo, j~, j~ values of specific integrals involving Hermite polynomials K curvature of flow path k blade force per unit mass L moment of momentum 1 length of design flow path within an element M torque rh mass flow rate m , S curvilinear co-ordinates n vector perpendicular to blade surface O,P,Q points used when deriving cpm P pressure R radius of curvature of flow path r, z, 0 cylinder co-ordinates t time u tangential velocity of rotating element v s shock velocity w fluid velocity relative element blade angle coefficient 7, e, 2 angles regarding the orientation of curvilinear co-ordinates F moment of the tangential component of the absolute fluid velocity /~ incremental change of . . . . vorticity relative position along the design flow path within an element ® angle describing the blade surface l arbitrary quantity for demonstration of formulas A, E geometrical coefficients of flow path P density of fluid Y parameter for blade angle derivative (p blade angle (relative the meridional plane) X coefficient for the blade force in the transverse direction of the flow path Stokesian stream function f~ denotes the flow equation 6O rotational speed .
.
.
695
696
SVEN ANDERSSON
subscripts 1,2 A,B
entry, exit of turbine just in front of, just behind turbine entry C, S at core and shell, respectively J element number r, O, Z, S, m with respect to the specific co-ordinate direction
superscripts derivative with respect to q~, i.e. d/dq ~ refined coefficient value N.B. Vectors are shown in bold type.
1. INTRODUCTION The established method to analyse hydrodynamic torque converters is based upon the assumption that the flow in it has a one-dimensional character. The flow is represented by a mean flow path. The Euler turbomachinery equation and a power balance for the converter then solves the task. The one-dimensional approach implies a neglect of the state of things in the direction transverse to the flow path. However, the validity of the flow path is due to an equilibrium in this direction. When executing the one-dimensional method the result may be significantly inconsistent with such an equilibrium, resulting in a discrepancy between real and assumed flow path location and between actual and predicted performance.. These problems are overcome by employing the two-dimensional approach, frequently used when analysing turbomachines, but the increase of the numerical effort will be substantial. Such an analysis was carried out in Ref. [1]. In this paper results from the two-dimensional theory are applied to the one-dimensional mean flow path in order to obtain the conditions for its validity. Some new properties then enter the ordinary one-dimensional theory: the blade angles in the transverse direction, i.e. the skew of the blades and the gradient of the through flow velocity. When both the blade angles in flow direction and the skew are known, the shape of the blades are uniquely determined. The flow gradients provide additional information about the flow field in the converter; an estimate of the flow velocities at the shell and the core can be made. These velocities are often considerably different. This paper also supplies a numerical example concerning a three-element converter of the type commonly used in automatic transmissions. 2. B A S I C E Q U A T I O N S
ONE-DIMENSIONAL
APPROACH
Analysis of hydrodynamic torque converters employing a one-dimensional mean flow path is well established and will be briefly related below. More thorough treatises are provided by Refs [2-5]. A meridional cross-section of a converter is shown in Fig. 1. Converters with three elements, pump, turbine and stator, arranged as shown in this figure is the most common type; other types of converters may be found in Ref. [6]. The mean flow path is indicated in Fig. 1. The points where the fluid enters and exits the various elements are denoted by indices of which the first refers to the number of the actual element and the second has the value 1 at entries and 2 at exits. At entries a letter A or B is added to the second index indicating that the point is just in front of or just behind the entrance, respectively. A sample of the velocity triangles is also shown in Fig. 1. The absolute fluid velocity c is made up of the relative fluid velocity w and the rotational velocity u. The velocity c is resolved into two components: Co in the tangential direction and c= in the mean flow path direction. The geometrical relations between the velocities are c,, = w cos q~,, Co = u + w sin ~0,, = r(o + c,, tan q~,,,
( 1)
Blade skew and flow gradients in torque converters
697
where ~o,~is the blade angle relative to the meridional plane. The cross-sectional area of the mean flow path is denoted A allowing the mass flow rate to be written as fia = pcmA. (2) Usually A may be regarded as constant which implies that cm has the same value at every point along the mean flow path. 2.1 Torque on an element Define the moment of the tangential flow velocity as F
=
(3)
rco.
Now the Euler turbomachinery equation for the j-th element yields Mj
(4)
= r i l ( F j . 2 - r j , la ).
If no torque is exerted on the non-bladed space in front of this element, the Euler equation gives gj,l~ = r~_ ~,2.
(5)
Insertion of equation (1) in equation (3) yields r tan ~0m
(6)
which together with equations (4) and (5) gives the torque for the j-th element as M j = fi'l[(~j. 2 - ~j_ 1,2)c,, + r~2zo~j - r 2 1.2 ~oj_ 1].
(7)
A peculiarity regarding the nomenclature ought to be commented on. Asj can take the values 1, 2, 3 the quantityj - 1 is 0, 1, 2, respectively, but no element has the index 0. However, the index j - 1 should be interpreted as 'preceding element' and as element 1 is preceded by element 3, index 0 refers to element 3. Analogouslyj + 1 should be interpreted as 'succeeding element', that is the indices 4 and 1 are synonymous. This state of things bring forth the useful summation formula, t an arbitrary quantity, 3
3
Z j
3
= Y, ' J - , = 1
Z 'J+,.
1-1
j
,o1,
[-I
Turb~ ir
2,1 B 2,1A 1,2 /
(8)
1
Meonflopwofh
co
!
i i
I-I
I 1 ii
I vs,~
C1,
~
f FIG. 1. Meridional cross-section o f a three-element torque converter; at line I-1 a section perpendicular to the meridional plane, showing the velocity triangle and the shock velocity, v s.
698
SVEN ANDERSSON
With this formula and equations ~4) and (5) the torque equilibrium of the entire converter is readily proved and shown to be an inherent quality of these equations
2
Mj ~- 1TI
i =I
2 i
l-'J. 2
Fi
1
1,2
=
0.
{9)
1
2.2 Power balance jbr a converter When the fluid is on the point of entry into an element, its velocity is usually not consistent with the blades in the entrance. The situation is illustrated in Fig. 1. The fluid experiences an abrupt change of the velocity at this point; the change is often called shock velocity and is defined as US, = C#,],IA-- COj, IB" (|(}) This abrupt change of the velocity also implies a loss of power, often referred to as incidence loss or shock loss, and is shown to be, Ref. [1] among others, E~ i - rhv~ " 2
(111
rh ('I'i"B-- lS"'A~ 2 Es'j = 5 \ ri, l /
(12)
or if equations (3) and (10) are used
Power losses due to friction, often called circulation losses, are expressed as .2
•
,~m
Et= r h J ~ ,
t13)
where f is a loss parameter• Now the power balance for the entire converter states that the net power supplied by the shafts of the machine equals the total power dissipated, that is 3
3
~ Es,j+E f.
My)j=
(14j
j=t
j:i
Rewriting this with the aid of equations {13), I 11), (4) and (5) and introduction of the quantity gives 3
3
' + Y n = Jc;,, i
v 2~,; - 2 ~ (Fj, 2 - V j t
1,2)(Dj=O.
{15j
/:= 1
Insertion of equations (10h (3) and (6) yields a quadratic equation for the velocity % ~)= Ac 2 , - 2 B c m - C = 0
A
/]e 3
B = V B/(t)j j i Bj : {~i.2--,3(j,1)+{O~j.1. | --O~j,2)( rj''2 ~2 k,rj+ 1.1 / 3
C=
V -- Cj~,)~ i :1 4-
Cj = 2r22
rJ.a r2 t.l
z rj.~.
(16)
This equation may be considered as the governing equation for the converter. For a specific geometry and operational mode, i.e. rotational speeds, c,, is solved from this equation and thereupon torques and powers are readily calculated.
Blade skew and flow gradients in torque converters
699
As c= is solved from a quadratic equation, two roots are possible but only the upper one is stable as shown in Ref. [1]. Often the lower root is negative, that is contradictory to the assumed direction of c=. Equation (15) is evolved for a three-element converter but the generalization of the equation is obvious--the number of terms in the sums equals the number of elements of the converter, cf. Ref. [6]. 2.3 Design procedure When converters are designed usually a specific operational mode is fixed, at which no incidence losses should occur, in practice the point of maximum efficiency. Seven quantities now have to be settled, c= and six blade angles ~0m.~.l. . . . ~om.3.2. Four relations are given by equations (15), (10), (5) and (3) 3
f~=fc2..-2 ~ (rj,2-rj-1,2la)j=O
f
j=l
j = 1, 2, 3.
Fj_ 1,2 = Fj.1B
(17)
As there are seven unknowns and four relations three quantities are for choice, resulting in converters with different characteristics. A design process for converters is presented by Ref. [7].
3. B A S I C E Q U A T I O N S - - T W O - D I M E N S I O N A L
APPROACH
Turbomachines often have so many blades so that it is justifiable to replace the discrete forces they bring about by a continuous field of forces. This is equivalent to considering a turbine with an infinite number of thin blades. Now the flow will be independent of the circumferential co-ordinate 0, that is the real three-dimensional flow is replaced by a twodimensional one. The problem is reduced to study the flow in the meridional plane, i.e. the r-z plane. This concept is widely used; Refs [8-10-1 are given as examples. As axial flow machines as gas and steam turbines are of particular interest much of the literature is devoted to these. Here the flow is almost parallel with the z axis, and important feature when developing the equations. The turbines in torque converters have such a shape that a curvilinear co-ordinate system is preferable, see Fig, 2. It is assumed that a function exists which transforms an orthogonal net to a deformed quasi-orthogonal one, where some specific lines in the net fit the boundaries of the turbine. The deformed net is sketched in Fig. 2.
Cm I
I ~
Quasi-oH'hogonot net
9s ...~
I
I I
u=r6o
I I
•
/0
Z
FIG. 2. Meridional cross-section of a turbine; r-z plane. The lines I and 11 show sections perpendicular to the meridional plane; the m-O and s-O planes, respectively.
700
SVEN ANDERSSON
The formulation of the flow equations for this type of co-ordinates is analysed in Ref. [1], The parts of that treatise that is used here will be related below. 3.1 Derivatives in the curvilinear system The transform that produces the quasi-orthogonal net shown in Fig. 2 does not have to be expressed explicitly. Define the quantities m and s as the lengths measured along the lines of the net, see Fig. 2. N o w the partial derivative of some quantity t with respect to the variable m is given by the chain rule
Ot ?;l Or Ot Oz + ~m - Or Om c3z Om"
(18)
The derivatives 6r/~3m and dz/~m may be expressed by the angle 7 shown in Fig. 2 giving & & & 3m = ~rr sin 7 + ~zz cos 7.
i 19)
In a similar Way the derivative &lOs can be expressed with the aid of the angle ~:. These two expressions solved with respect to the r and z derivatives yield
Or
sin (e - 7) ~s cos y - ~m cos
31 _ 1 { 8l . ~t ) ~?z sin(e-7)~0mmSme-s sin7 "
(20)
3.2 Blade 9eometry The blades may, as any surface in space, be described by a function ®(r, z) which for a specific r and z gives the angle (9 measured from a reference plane to the blade surface, see Fig. 3. The blade angle, ~o,, in the figure, is the angle between the meridional plane and the blade surface. Using the notation shown in the figure this angle is [QO I (r + Ar) A(9 0(9 tan q~.. = lim - lim - r--. Q~p IPQI am~0 Am c3m
(21 )
This formula is valid for any direction; for instance 0(9 tan q~s = r c~s "
(22)
_J
'~ /BI(lde surfoce
R~erence ptane FIG. 3. Blade angles; the points P and Q contained in the meridional plane, P and O lies on the blade surface.
Blade skew and flow gradients in torque converters
701
In Fig. 3 is also shown how a vector n perpendicular to the blade surface may be constructed. F o r cylinder co-ordinates this vector is
n=(-tancp,, 1, -tan~oz)=
( -r~rr, do
1,
gz/"
(23)
3.3 S t r e a m f u n c t i o n It is of especial convenience to assume that the 'm-lines' of the quasi-orthogonal net coincides with the flow paths in the meridional plane. This implies that the fluid velocity cm is tangent to the 'm-lines', see Fig. 2. The velocities in the r and z directions may now be written
{
c: = c,. cos 7 c m sin 7.
(24)
cr
The continuity equation in cylinder co-ordinates reads O(rcr)
dr
+
t~(rc~)
dz
= o.
(25)
This equation is always satisfied if c, and cz are expressed as 1 d~ Cr =
r
dZ
I d~
(26)
Cz ~ - - - - ~
r dr
where qJ is the Stokesian stream function. Using equations (19), (20), (26) and (24) it is found that = rcm sin (e - 7)
d~p Omm
(27)
O.
The latter expression is a consequence of the coincidence of the 'm-lines' and the flow paths. It is convenient to define an angle 2 according to, see Fig. 2, 2 = } , - (e - n/2)
(28)
which yields dq j ds
= r c,, cos 2.
(29)
3.4 E q u a t i o n s o f m o t i o n The flow is governed by Euler's equation which reads c3c ~ - + (c" V) c -
Vp
+ k.
(30)
P The field of forces accounting for the blades are represented by k, force per unit mass. The fluid is non-compressible and the flow is non-dissipative. Omission of the non-stationary term and introduction of the vorticity = V x c - curlc
(31)
and the energy per unit mass C2
I = p- + ~P MS 29:10/II--C
(32)
702
SVENANDERSSON
gives ~xc=
(33)
-Vl+k.
Carrying out the derivatives in equation (31) yields 1 O(rco) r
~z
?Jc, dz
~o-
Ocz dr
1 O(rco)
r
~34)
dr
Developing equation (33) and insertion of equations (34) and (3) gives F ?F r 2 Or
c, dF 4 c = d F
7~r
r i?z
-- Cr~ 0 --
0I +kr dr --
ko
F d[ OI r i ~-z = - O-z+ k~.
(35)
If the second of these equations is combined with equations (24) and (19) 0F
(36)
rk o = c,, ~m m .
This equation is a differential form of the Euler turbomachinery equation, see Section 2.1. As the flow is non-dissipative the force k is perpendicular to the blade surface, that is parallel to the vector n, equation (23) k = (kr, ko, kz) = kon = ( - k 0 tan ~o,, ko, - k0 tan ~o:).
(37)
Elimination of ~0 from the first and third of equations (35) and with the aid of equations (24), (27), (23), (19), (6)and (36)yields (31
0F
d ~ - ~ ~mm = o.
i381
If the rothalpy H is defined as H =
I -
coF
(39)
equation (38) states that H is constant along a flow path. Equation (38) is a differential form of the energy conservation theorem; it states that AI = u)AF = Me) for an element. Multiplying the first and third parts of equation (35) by c. and cr, respectively, and subtracting them and using the second part of equation (35), equations (39), (6), (19}-(22), (37), (23), (24), (29) and (38) yields dF 0 0 ( , o - dr dz
dF (3® dz dr
1
~H
cm cos2 ds
140)
Further development and remarking that d H / d m = 0 implies that H may be considered as a function of q~ only gives 1 (~__~ 0F ) dH G0 - r cos~ tan q),, - ~mmtan ~o= - r dq j .
(41)
In analogy to equation (37) the projection of the force k in the s direction is k= =
-
ko t a n ~p=
(42)
Blade skew and flow gradients in torque converters
703
which together with equation (36) turns equation (41) to ~0 r
tan ~0m 01~ -- r 2 c o s ) ~ t3s
I-
k,
dH
rcmcos 2
dW"
(43)
An expression for the vorticity is found by evolving the second part of equations (34). Insertion of equations (24) and (20) gives 1 (o = CmK
~c,.
cos )~ ds '
(44)
where K is the curvature of the flow path. K-
~37 (45)
Ore"
The curvature is the reciprocal of the radius of curvature; thus K - 0 for a straight line. Equation (44) holds under the condition that Oct.~din = 0, that is the flow velocity is constant along the flow path. This restrictive assumption is justifiable as the aim of this treatise is to prescribe a flow and to find the appropriate shape of the blades for a converter with constant through flow area; A in equation (2). Equation (44) is rewritten with the aid of equation (29) dc,.
~o = c,,K - re,. d ~ "
(46)
Henceforth the flow velocity will be characterized by two parameters, c,, and dcm/dq2, that is, the function is replaced by the two first terms of its Taylor series. The parameter c,, is constant at every position along the flow path while the parameter d c , , / d W may vary.
4. B A S I C E Q U A T I O N S
IN DIRECTION
TRANSVERSE
TO FLOW
PATH
The one-dimensional theory related in Section 2 utilizes two theorems: Euler's equation for turbo machinery, i.e. a relation between torque and change of moment of momentum, and the energy conservation theorem, equations (4) and (14), respectively, When using a twodimensional approach these two turn up again, equations (36) and (38), but also a third equation is found, (43), essentially a force equilibrium in the direction perpendicular to the flow path. The essence of this treatise is to study the middle flow path just as for the one-dimensional approach but to consider the state of things transverse to the path too. In this way the implications of the blade angle q~, the skew of the blades, can be considered without solving the comparatively vast two-dimensional problem. 4.1 Within an element At every 's-line' any quantity may be described as a function of q~ as well as ofs. Therefore, according to equation (29) t~l
0z
O--s = rc,. cos 2 Oqj ,
(47)
where t can be any quantity. Now equation (43) can be written as ~o r
_
c., tan qo,"(__ + ac~,) +Z - H'
(48)
r
_= = (2rco + c., tan ~0,.)sin e -~
rcm cos 2
Z-
ks rc m COS )~
1 ~ tan ~0," cos 2 t~s
(49) (50)
704
SVEN ANDERSSON
F' = E+~c~..
(51)
For convenience derivatives with respect to ~P are denoted by a prime sign. Often converters have m i n i m u m clearance between the elements why it is justifiable to put
r-i+ 1.1
= rj,2.
(52)
Such a property also applies for: K, c~., 7, e and thereby also for ~0 and ).. Insertion of equation (46) in equation (48) and employing the constancy of H' within an element; H~.~ = H~.2 yields Xj.2 - ~j.1 = - cm(c'~,-i,2 sec 2 ~0,..j,2 - c~..j_ 1,2 sec z ~o.,.j, 1) A-i,1)
153)
A = (E tan tp,. - K)/r.
(54)
-- c,n(Aj.2 -
Equation (53) constitutes a specific relationship between the forces ks and derivatives c',,; the other quantities in the expression, c,, and geometrical properties as r, ~ o , , , . . . , may be regarded as k n o w n constants after the design of the converter according to the onedimensional a p p r o a c h has been executed. 4.2 Transition between elements When entering an element the fluid experiences a sudden change of the flow velocity; equation (10). Here power is lost, see equation ill), but as F a ~ F g also an exchange of mechanical power takes place. Study the power balance for the entrance region; this region is a thin disc which A is just in front of and B just behind. r h ( i a _ l A ) = - ~1 m . v 2s + m" ( V a - FA) oga.
(55)
Insertion of equation (39)
Ha_H
A = _½ Vs_2
(COB__ ( D A ) F A
{56)
or using equation (51) HS+ 1,1 - H),2 = - Vs0 + 1V's0+ 1- (coj+ 1 - ~oj) (Zj.2 + o~j,2Cm,j.2 ).
(57)
This equation holds for the transition between elements j and j + 1. For the design mode, Section 2.3, no incidence losses take place, that is v s = 0. Elimination of H' and (0 by equations (48) and (46) and observing that {0, e~,, F and K are not changed between the positions (j, 2) and (j + 1, 1) gives Z j + 1,1 - - Zj,2 •
- - Cm( s e c 2 -
~Om,j+1, 1 - -
c,~(Aj+ 1.1 -
Aj,2) -
sec2
(tOm,j.2)Cm.j,2
(foj+ 1 -
ogj)(•j.2 + o~j.zC',n.j.2).
(58)
This equation is similar to equation (53). In the case of a three-element converter equations (53) and (58) constitute a set of six equations. Summing these six equations yields 3
3
~, (ct-i.a(coj +1 - o i)) c',,.-i.2 = - ~, (co-i+1 - eJ j) Ej.2 j=l
(591
j=l
which expression is equivalent to 1
df~
2dq /
-
0.
{60)
Here the loss parameter f = 0, that is non-dissipative flow. The equivalence between equations (59) and (60) is shown with the aid of equations (17), (51) and (8), and is an obvious consequence of the energy conservation theorem; the function f~ = 0. When designing a converter according to the one-dimensional method the flow in it is described by one parameter, c,,, but now additional information a b o u t the flow is supplied by the gradients c~,. Equation (59) constitutes a specific relationship between them.
Blade skew and flow gradients in torque converters
705
If the c~.s are chosen, in compliance with equation (59), and one of the Xs then the remaining Xs are given by equations (53) and (58). In the case of a three-element converter the p r o b l e m has nine unknowns, three c~,s and six Ks, but six equations. Thus, a variety of solutions are possible; the designer has to formulate additional criteria in order to select a suitable one. If the condition ~O.j,2 ~- ~O,j + 1.1 was not valid, i.e. equation (58) not satisfied, the flow path will have a discontinuity in the curvature K when entering an element. This implies that the fluid experiences a jerk here despite continuous flow velocities c" and Co.
5. BLADE ANGLES AND A NUMERICAL EXAMPLE Assume that a circular flow path has been chosen for a three-element converter; Fig. 4. According to equation (21) the angle ®, which describes the position of the blade surface, Fig. 3, m a y be calculated as ® = ®1 + fo tan tO" l dr/ r
(61)
where 1 is the length of the flow path within an element and r / a parameter describing the relative position along the flow path; r / = 0 at inlet, r / = 1 at outlet. Moving the curve a distance As inwards at every 's-line' changes the angle ® according to t~s
-
t3~- +
tan tO,.
ds
r2 sin e dr/.
(62)
It has been assumed that O tan to,, ~
-
0.
(63)
This is in good agreement with measurements reported in Ref. [1], but also other types of blade constructions exist, Refs [2-1 and [4-1. F o r this particular circular flow path is
{ --1-R(l+cos,) ~ dl
(72 --Tl)R - (72 - 70 + (tan 22 - tan 21)
I/R.
(64)
5.1 Formulas for co,, and to~ The function tan to,.(r/) is described as tan tO,. = (1 - 3r/2 + 2r/3) tan tO,.. 1 + (3r/2 - 2r/3) tan tOm.2 + (q _ 2q2 + r/3)y1 + (q3 _ q2)y2,
(65)
Turbine ./i~-~.~... Pump !
/~
\ i~Fto~pa~h
T , FIG. 4. Skeleton sketch of a three-element converter.
706
SVEN ANDERSSON
that is by four parameters: ~o,..~, Y~, q~,,.2, Y2 and the adequate Hermite polynomials. The quantity r is defined by c3tan ~p,, 1 d tan ~,. . . . . (Jm 1 dr/
Y 1
t66~
When constructing the blades the angles q~= are given but the derivatives "fl and Y2 are for choice. Define the coefficients J:
jo Jl
1 -- 3r/2 + 2r/31 3r/2 --2r/3 / r / - 2r/2 + r/3 | ( ! d l ds
-1 -- jo
Lr/3 __712
l sin e r2 ) dr/
~67t
J
which allows the angle ~p~to be written as, equations (22), (62), (65) and (67) tan ~os.2 r2
tan ~Os.~ + (J1° tan ~0m.l + jo tan q~m.2)+ (Jl r~ + J~ Y2). rl
(68)
With equation (66)
~?F
(2rco+ cm tan
'
rcm `1.
(69)
which combined with equations (42) and (36) gives ks-
cm tan r %
{
r'i
(2to) + Cmtan ~o,,)sin 7 + ~ - Y .
(70t
Assume that k~.: is known. If q)~,l is chosen YI is given by equation (70). Elimination of Y~ between equations (68) and (70) gives a second-order equation for tan ~os,2; tan2 ~%2 - tan ~o~.2
tan q~,l +r2 (J° tan ~o,,,l + J2° tan q?m.2)
+r2J~`1.1 _ j ~ l(2r2u)+c,,tan~o,,.2)sinTz
4 J12ks.21r2 .2 - 0.
Cm
{71!
(- m
Insertion of k~,2 and other known values yields ~os,2and thereupon "i"2,solved from equation (70). This problem has four unknowns: ~0s.~, Y l, ~0s.z,Y2 but three equations, equations (68) and (70) at inlet and outlet. Hence the designer has to settle which one of the possible solutions that is best fitted. 5.2 Numerical values for a.How path The geometrical properties of the flow path are presented in Table 1. The radius r and other lengths are given by relative numbers; the outer radius of the flow path is used as a norm. The J-coefficients are given in Table 2. The angles y and ). vary linearly along the flow path and the integrals, equation (67), are evaluated by employing the trapezoidal formula.
TABLE 1. GEOMETRICALPROPERTIESFOR FLOWPATH
Position
"; C)
~: (')
2 C)
1, 1 l, 2
45 180
117 270
2, 2, 3, 3,
180 315 315 405
270 423 423 477
1 2 1 2
r
K
+ 18 0
0.500 1.000
3.414 3.414
0 - 18 - 18 + 18
1.000 0.500 0.50{) 0.500
3.414 3.414 3.414 3.414
B l a d e s k e w a n d flow g r a d i e n t s in t o r q u e c o n v e r t e r s
707
TABLE 2. J-COEFFICIENTS Element
jo
jo
j~
j 2~
1 2
-2.3928 - 1.2860
- 1.2860 -2.3928
-0.3249 -0.2390
+0.2390 +0.3249
3
-2.1822
-2.1822
-0.3785
+0.3785
5.3 Numerical example--blade angles The presented theory will now be d e m o n s t r a t e d by a complete numerical example. First the blade angles ~o,, are chosen according to the procedure mentioned in Section 2.3. Assume that the rotational velocities: ~Ol = I, to 2 = 0.65, ~o3 = 0 are settled for the point of no incidence losses. The rotational velocities are given by non-dimensional values; the velocity of the pump, element 1, is used as a norm. The process of selecting the q~,,s is connected to the characteristics of the converter needed and is overlooked here; the choice cm = 0.25, F3.~ = 0 and F~,2 = 0.75 will do as an adequate example. In Table 3 the blade angles and the ct, E and A coefficients are shown. 5.4 Numerical example--d,, and X N o w the properties c~, and X may be settled. Assume that tps.3.~ = 0 is demanded, i.e. X3.1 = 0. Stators in existing converters are often constituted in this way. The gradients c~, have to fulfil equation (59); if one X is k n o w n the remaining are given by equations (53) and (58). T w o alternatives are examined: in the first all the c;,s are equal but in the second these are varied in order to obtain low values for the Xs. The result is shown in Table 4. The second alternative is chosen as low values for the Xs counteract excessive values for the angles ~0,. However, the effect of this step is yet to be seen at this point in the numerical evaluation. 5.5 Numerical example--qg~ and Y When c~ and X, i.e. k~, have been settled the properties ~0~ and Y remain. The procedure described in Section 5.1, equations (68), (70), (71) is used. If Y1 is used as a parameter Fig. 5 shows an example of how things turns out. The designer has to select a solution which implies a reasonable shape of the blades. The choice in this example is given in Table 5. TABLE 3. BLADE ANGLES AND ASSOCIATED COEFFICIENTS cm = 0.25 Position
u9
F
1, 1 1, 2
1.00
0.2625 0.7500
0.100 - 1.000
0.050 - 1.000
7.682 - 7.000
- 5.292 3.586
2, 1 2, 2
0.65
0.7500 0.0000
0.400 - 1.300
0.400 -0.650
- 5.600 2.436
- 5.654 - 13.161
3, 1 3, 2
0.00
0.04300 0.2625
0.000 2.100
0.000 1.050
0.000 3.935
- 6.828 9.698
TABLE 4.
t a n qg,,
c',,,X
~
_=
A
AND k s - - T W O ALTERNATIVES
c"
Alternative 1 X
ks
c~,
Alternative 2 )~
k5
-
1, 2
2.6346 -2.6346
- 1.546 -3.113
- 0.1838 - 0.7784
- 3.6420 -4.0000
0.6747 -0.4643
0.0802 -0.1161
2, 1 2, 2
-2.6346 - 2.6346
-2.885 0.0430
-0.7212 0.0004)
-4.0000 1.0000
-0.0443 0.00430
-0.0111 0.00(~
3, 1 3, 2
- 2.6346 - 2.6346
0.000 - 1.227
0.04300 - O. 1459
1.004)0 - 3.6420
0.004)0 1.0442
0.00430 O. 1241
Position 1, 1
708
SVEN ANDERSSON +1.2 tangs
~
+0.8
ton~sAupper
root
i
+0./~
~ 7~upper root
÷2 fi
o.o
.
.
.
.
.
.
.
.
.
!o
.
_0./~'
tan~s.~SJ
-O.B
J
-1.2 . . . . . . . .
-6
-L
-2
~,fQn~.2tower roof -2
~'.......t J-. . . . ., . , ,
o
.2
,~
i-4
J
)i+6
6
FIG. 5. Y2 and tan (p~ vs Y~ for element 1 - t w o roots due to equation (71). Y~ = 0 and the lower root case was chosen, see Table 5.
TABLE 5. • AND (Ps
Position
Y
tan q),
1, 1 1, 2
0 - 3.902
-0.221 - 0.329
2, 1 2, 2
-0.2 1.269
-0.613 1.222
3,1 3, 2
0 - 0.998
0 - 2.480
5.6 Numerical example--closure Now when all parameters have been settled any quantity can be evaluated at any position, t/-value, along the flow path. In Fig. 6 the blade angles and Figs 7 and 8 some flow properties are shown. When evaluating q~sand ® numerical integration of equations (62)and (61) with equation (65) inserted was done. Now the upper integration limit in equation (67), "1",takes the t/-value for the actual position.
+8oI
..........................................
.60~
(deg,
~
'°k +2ok
4 / / / / /
\
-60 Pump -80
~ , Sfo~or
Turbine '
/'
L
FIG. 6. The angles ~Ps, q~m a n d ® along the flow path.
"~f I
Blade skew and flow gradients in torque converters
709
-12 c~ -8
-4 0
Pump
+L,
Turbine
Stutor
FIG. 7. The gradient c~. along the flow path. According to equation (29) A% ° = rcmAs. Assuming As = 0.14 for r = 1 yields AW = 0.035. Then, for instance, c~. = - 7 implies that cm = 0.37 at the shell and 0.13 at the core, respectively.
1.0
ko,ks 0.8
E
0.6 F
0.t~ 0.2 0
ks
-0.2 -0.t. -0.6
Pump
Turbine
Sfnfor
FiG. 8. The quantity F and the forces ko and ks along the flow path.
Sections 5.2-5.5 describe a design a n d decision hierarchy in four levels namely: (1) a r r a n g e m e n t o f converter; (2) p e r f o r m a n c e characteristics; (3) details regarding the flow and blade forces and; (4) blade design. H e r e b y the p a r a m e t e r s will be settled in the o r d e r o f decreasing prominency. However, p a r a m e t e r s settled at a specific level become prerequisites for the lower levels a n d m a y then turn out to have d i s a d v a n t a g e o u s values. Eventually these high level p a r a m e t e r s have to be reconsidered. This feed-back process has to be executed by the designer. The four level a p p r o a c h is useful when creating a c o m p u t e r - a i d for the designer. The levels are i m p l e m e n t e d as separate p r o g r a m blocks a n d the p a r a m e t e r s as a d a t a base accessible for all the blocks.
6. I N F L U E N C E OF THE FLOW VELOCITY GRADIENTS ON THE P E R F O R M A N C E W h e n the flow velocity gradients c~, are k n o w n the calculation o f the torques m a y be refined. The m o m e n t o f m o m e n t u m , L, at a specific p o s i t i o n along the flow p a t h is, e m p l o y i n g e q u a t i o n s ( 3 ) a n d (1) L = Srco drh = S F dria = ~ (r2o~ + rc,. tan q~,,) drh.
(72)
Hence the t o r q u e is, see e q u a t i o n s (4) a n d (5) M j = Li, 2 - L~_ 1,2.
(73)
710
SVEN ANDERSSON
The differential drh is
dria = 2nprc,.cos ).ds = 2 n p ~ - ds = 2 n p d q j,
(74)
where equation (29t has been inserted. The differential ds = dr/sin c, which inserted in equation (74) gives the mass flow rate COS ).
rh = npc., _
s i n ~;
(r~-r~) = 2np(qJc-- qJs).
(75)
The radii to the shell and the core are denoted by rsand rc, respectively. Denoting the distance between the shell and core along a 's-line' by h, h = (r,, - r~)/sin e
(76)
and comparing equations (75) and (2) yields
ria = npc.,h cos ). (r e + rs)
i 77)
A = nhlr~+r~)cos)~.
(78)
As the influence of c~. on the quantity L is moderate a series expansion is convenient.
+c'm( dL ~ =~(r~o+(~c,,). \de',. ],.;. = o
L=(L),;=o
(79)
Appropriate values for the coefficients r 2 and ~ in equation (6) are also defined by equation (79). Evaluation of L when c~ = 0 yields, employing equations (74) and (75)
rc+ r~ 2(r2c+ r cr s + r s (L),,, =o = r h l e ) ' 2 , +c,.tanq)m 3 ( r c + r s ) 2} .
180)
The derivative of L is given by equation (72) with equation (74) inserted dL ..... dc'.,
.
cos ). i'r~ dc,. ~zt~ " - I (r2")+2c"rtan~°~)rdc'dr" sin c j r,
{811
According to equation (75) the stream function may be written as COS ,,:~ r 2
q~ (r) = c , . .
sin g
(821
2
Recalling the assumptions in the end of section 3 and assigning the stream function value (q'~+ q~)/2 to the mean flow path allows the flow velocity to be written as
c.,(r)=c,.+c'{qqr)
q"c+"~s }2.
=c,.+c',.c.o..~\~c°s2{rZ
r2c+rZs)4'
183)
Insertion of this in equation (74) and recalculation ofria confirms that rh is independent of the parameter c;,. Equation (83) gives arm cos2 ( 2 ~ (84)
dc"
-
Cm
sin-/:
which inserted in equation (81) yields dL
_
{ ,r +rs,2
= rh cos). ( r e - rs)2 o~ c:. = 0 sin e
24
_
r~+3rcrs+rZs ) , ?. outr c-t-rs) )
I- 2c,. tan ~o., ~
(85)
Comparison between the expressions within the square brackets in equations (80) and (85) shows that the latter, with sufficient accuracy, may be regarded as one sixth of the former. hence, by equation (76) (dL) ~ h2 cos2 sin~ (L)c; = 0" [86) if',,, ,;, 6 =
,,
Blade skew and flow gradients in torque converters
711
TABLE6. CORRECTIVECOEFFICIENTS;EQUATIONS(87}AND(88) 1,2 2, 1
Position
I1 + ¢~h2 COS.~sine 6
2,2 3, 1
1.0131
]
3,2 1, 1
1.0122 0.9554
Equations (79) and (80) now give the coefficients +
2(r2+rcrs+r])[ & = t a n q~,.
3 (% + rs)
1
g
(87)
c~'hZc°s)~ siri e ] 1+
6
(88) "
These equations imply that the influences due to the through flow velocity gradients are regarded by multiplying the ordinary r 2 and :t by a specific corrective coefficient. Table 6 shows the values of these for the numerical example in Section 5. The through flow area A = 2n × 0.14 was assumed, which when inserted in equation (78) gave h. Using these corrections and the values in Tables 2 and 3 together with equations (16), (79) and (73) showed that the value of c,, changed from 0.250 to 0.258 (3.3 ~o),and that the capacity of the torque converter, i.e. M1, increased by 4.3 'Yo. Thus the flow velocity gradients cause a discernible increase of the capacity, this effect is also reported by Ref. [4]. There the flow gradients are obtained in quite another way by a geometrical reasoning. The flow in the meridional plane is assumed to be a type of solid body rotation about the kernel of the core. However, the main gain of more knowledge of the flow field is that this provides a mean for improvement of the efficiency of the machine. 7. CONCLUSIONS The well established one-dimensional mean flow approach in the analysis of hydrodynamic torque converters has been supplemented by equations regarding the balance in the transverse direction of the flow path. These equations are derived from the two-dimensional theory for turbomachines. By these equations the validity of the design mean flow path can be considered as well as the impact of the skew of the blades. Further parameters regarding the blade shape now enter the analysis: the skew angle and the derivative of the blade angle in the through flow direction. Also additional information about the flow is obtained in form of through flow velocity gradients allowing the velocities at the shell and the core to be estimated. Furthermore the calculation of the performance of the converter can be refined. REFERENCES 1. S. ANDERSSON,On hydrodynamic torque converters. Transactions of Machine Elements Division, Lund Technical University, Lund (1982). 2. M. WOLF, Str6mungskupplungen und Str6mungswandler. Springer-Verlag, Berlin (1962). 3. R. HERBERTZ,Untersuchung des dynamischen Verhaltens von F6ttinger-Getrieben. Diss. T. H. Hannover (1973). 4. A. WHITEFIELD,F. J. WALLACEand R. SIVALINGHAM,A performanceprediction procedure for three-element torque converters. Int. J. Mech. Sci. 20, 801-814 (1978). 5. A. WHITFIELD,F. J. WALLACEandA. PATEL,Performanceprediction of multi-elementtorque converters. Int. J. Mech. Sci. 25, 77-85 (1983). 6. S. ANDERSSON,Analysisof multi-elementtorque converter transmissions. Int. J. Mech. Sci. 28, 431~,41 (1986).
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7. A. WHITFIELD,F. J. WALLACEand A. PATEL,Design of three-element hydrokinetic torque converters. Int. J. Mech. Sci. 25, 485--497 (1983). 8. R. A. NOVAK, Streamline curvature computing procedures for fluid-flow problems. J. Enono Power, Trans. A S M E 478490 11967). 9. J. J. OTTE,,A method of analysis of the axi-symmetrical flow in the blade channels of turbines. Third Scient!lic ConJl on Steam Turbines of Great Output, Gdansk, 24-27 Sept 1974. Prace instytutu maszyn przplywowych, Warszawa-Poznan, Zeszyt 7(~72, pp. 597-614 (1976). 10. A. J. WENNERSTROM,On the treatment of body forces in the radial equilibrium equation of turbomachinery. TraupeI-Festschrifl, pp. 35l 367. Juris-Verlag, Z~rich (1974).