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Design of three element hydrokinetic torque converters
IaZ. J. Mech. Sci. Vol. 25, No. 7, pp. 485--497,1983 Printed in Great Britain.
0020-7403/83 $3.00+ .00 Pergamon Press Ltd.
DESIGN OF THREE ELEMENT HYDROKINETIC TORQUE CONVERTERS A. WHITFIELD,F. J. WALLACEand A. PATEL University of Bath, School of Engineering, Claverton Down, Bath, BA2 7AY, England
(Received 23 May 1982; in revised form 20 September 1982) Summary--This paper describes a design procedure for three element torque converters. The torque converter, consisting of closely coupled hydraulic turbines and pumps, forms an essential part in the transmission systems of many power lines. Efficient torque converters are therefore essential if maximum energy is to be transmitted to the output shaft. The design procedure consists of two components, the first establishes the overall dimensions and mean streamline blade angles using a one-dimensional fluid dynamic analysis, whilst the second uses a detailed integrated fluid dynamic procedure to predict the performance of the proposed torque converter design. The design procedure is developed in full whilst the prediction procedure which is published elsewhere, is used along with published data to assess the quality of the torque converter designs.
A
AK a C CR2 g rh N PL PI
Pc R
Ro Rc R, r T U W a /3 p •R co l~ ~
NOTATION flow area empirical loss parameter non-dimensional area AIA2 with appropriate subscript absolute fluid velocity non-dimensional velocity Cm21U2p gravitational acceleration mass flow rate of fluid rotational speed rev/min total power dissipated power dissipated due to incidence power dissipated due to fluid circulation radius to mean streamline ~/(Rs2/2 + Rc2/2) outer radius of torque converter radius to core radius to shell non-dimensional radius R/R2 non-dimensional torque linear velocity of rotational relative velocity of fluid direction of absolute velocity form the meridional direction direction of relative velocity from the meridionai direction efficiency fluid viscosity fluid density torque torque ratio rotational velocity rad/min speed ratio ~o~/~op speed ratio co, coo
Subscripts 1 2 3 P T S 0 m " '
pump inlet, stator exit turbine inlet, pump exit stator inlet, turbine exit pump impeller turbine rotor stator tangential component of velocity meridional component of velocity stall condition zero incidence design condition. 1. I N T R O D U C T I O N
Hydrokinetic devices, in the form of fluid couplings and torque converters, are widely used in the transmission systems of many road vehicles, cranes, lift trucks, earth moving vehicles etc. They provide the advantages of shock and vibration free power 485
486
A. WHITFIELDet al.
t r a n s m i s s i o n , a n d a u t o m a t i c t r a n s m i s s i o n s w h i c h do n o t r e q u i r e m a n u a l g e a r c h a r g e s . These hydrokinetic devices consist of a hydraulic pump and turbine closely coupled, w i t h o u t a n y m e c h a n i c a l link, so t h a t t h e fluid flows in a c l o s e d circuit. T h e fluid d i s c h a r g e d f r o m the c e n t r i f u g a l p u m p is d e l i v e r e d to a r a d i a l flow t u r b i n e a n d then r e t u r n e d to the p u m p inlet, t h r o u g h a s a t i o n a r y g u i d e v a n e in the c a s e o f a t o r q u e converter. T o r q u e c o n v e r t e r d e s i g n is b a s e d l a r g e l y on t h e u s e o f d i m e n s i o n l e s s a n a l y s i s , to b u i l d s i m i l a r units to o n e of k n o w n s a t i s f a c t o r y p e r f o r m a n c e , s e e Ref. [1], a n d u p o n o n e - d i m e n s i o n a l single m e a n s t r e a m a n a l y s e s to p r e d i c t the p e r f o r m a n c e , R e f s . [2-4]. T h e m o s t c o m p r e h e n s i v e p r e s e n t a t i o n o f t o r q u e c o n v e r t e r p e r f o r m a n c e a n d d e s i g n is t h a t b y J a n d a s e k [ l ] . T h e d e s i g n a n d p e r f o r m a n c e p a r a m e t e r s w h i c h he p r e s e n t s in d e t a i l h a v e b e e n e x t e n s i v e l y u s e d in t h e d e v e l o p m e n t o f the d e s i g n a n d p e r f o r m a n c e p r e d i c t i o n p r o c e d u r e s p r e s e n t e d here. T h e d e s i g n p r o c e d u r e a d o p t e d is to u s e a o n e - d i m e n s i o n a l single m e a n s t r e a m a n a l y s i s to e s t a b l i s h the o v e r a l l d i m e n s i o n s a n d b l a d e a n g l e s for t h e t o r q u e c o n v e r t e r . H o w e v e r , it h a s b e e n s h o w n , R e f s . [6] a n d [7] t h a t this a p p r o a c h is n o t satisfactory for accurate detailed performance prediction and the more thorough a n a l y s i s o f Ref. [7] is u s e d . T h i s a p p r o a c h a s s u m e s t h a t the fluid c i r c u l a t e s as a f o r c e d v o r t e x a n d u s e s a n i n t e g r a t i o n p r o c e d u r e b e t w e e n t h e o u t e r shell a n d i n n e r c o r e in o r d e r to c a l c u l a t e t h e t o r q u e d e v e l o p e d a n d p o w e r d i s s i p a t e d . T h i s v e r y d e t a i l e d flow m o d e l , d e s c r i b e d f u l l y in R e f . [7] is, t h e r e f o r e , the m a i n p r e d i c t i v e t o o l ; the m e a n s t r e a m l i n e a n a l y s i s , d e s c r i b e d h e r e , p r o v i d i n g t h e m a i n d e s i g n p r o c e d u r e , a n d an initial e s t i m a t e o f p e r f o r m a n c e . 3. T H E O R E T I C A L D E S I G N P R O C E D U R E F O R T H R E E E L E M E N T TORQUE C O N V E R T E R S The objective of the design procedure is to calculate the basic geometric parameters of a three element torque converter for specified operating conditions. The main operating requirements of a torque converter are the stall torque capacity and torque ratio, a broad operating range up to the coupling point, and high peak efficiency at the designed operating condition. The specified parameters are, therefore, the input stall torque, the output stall torque, or torque ratio, and the speed ratio ll' at which the circulating fluid moves smoothly from one component to the next, i.e. the condition at which no incidence loss will occur. The required stall torques must be specified from the performance requirements projected for the vehicle under consideration. Tractive effort requirements for start up determines the stall torque ratio, if a gearbox is used to supplement this ratio its effect must be included in the determination of the required torque converter output. The basic geometric parameters to be calculated are the outer diameter, the size and position of the central core, and the blade angles for each element. The outer diameter is calculated by systematically increasing the size of the torque converter until a size is found which is able to meet the specified torque capacity, this illustrated in Section 3. The position and size of the central core is calculated using the procedure given by Jandasek[l]. This fixes the position and size of the core to give a constant flow area throughout the circuit equal to 23% of the cross sectional area, i.e. 0.23 ~rR2o. This leaves the calculation of the blade angles of each element as the main objective of the design procedure. Figure 1 shows the notation and typical arrangement of a three element torque converter considered. The design procedure is based upon a one-dimensional single mean streamline analysis approach which is well documented in the literature for performance prediction only. 2.1 Fundamental turbomachinery theory
Central to all analyses of turbomachines is the Euler turbomachinery equation which relates torque to the rate of change of angular momentum. This can be written for each element of the torque converter, as Tp = m ( G 2 P R 2 - G I s R , )
(1)
rr = m(G3TR3 -- Co2pR2)
(2)
rs = ra( ColsRi - G3rR3).
(3)
and
As fluid which completes one complete circuit from pump inlet through each element and back to pump inlet, must have no net change in moment of momentum, there can be no net torque exerted, i.e. "rp + "rr + ~'S= 0.
(4)
This is clear from the summation of equations (1), (2) and (3). It should be noted that the input torque % is defined as a positive torque and consequently the output torque, ~'r is negative. As it is conventional to
487
Design of three element hydrokinetic torque converters
Meon Streomline
\ //: / Pump
Turbine
j'
\,
/ /
\
StQtor i
FIG. 1. Cross section of torque converter showing mean streamline. have a positive torque ratio this is defined as: ~-R =
- rr
(5)
rp
The tangential components of velocity are calculated from the velocity triangles at each station, Fig. 2, which are constructed by assuming that the fluid is perfectly guided by the blades. This leads to: Co2e = UEp + C=2 tan fl2p
(6)
Co3-: = U3r + C,,3tan/33r
(7)
Cols = U1s + C=l tan/31s.
(8)
Emls
E m;
Cm2 [m2T
........
_ _[m_2~. . . . t
~,.Wezp',
~--~ E/ . . . . -, E O 2 T ~ rW021[ W~
[ " ~ W aT
~
/- ~3r
[02P
~T
¢ m3
[83T~ FiG. 2. Velocity triangles.
_ _ [mas__ _
~ / ~ ' C 3S
488
A. WHITFIELD et al.
The blade angles are defined in Fig. 2, the sign convention is such that a positive blade angle directs the fluid in the direction of rotation of the blades. Substituting equations (6), (7) and (8) into equations (1), (2) and (3) and noting that continuity yields
C~tAI = C~2A2= C,n3A3 the following expressions for torque can be derived: Tp = CR2(l+CR2(tan/3n, - r-L' a, tan/3,s) - rl2[~s)
TT=CR2(flr23+CR2(r3tan$3r-tan$2") \a3 -I Ts = CR2(II,r~2+ CB2(~ tan fl~s- r3/a3 tan fl3r) - r3ZIl)
(9)
(10) (ll)
where torque d(pA2R2U2e) velocity CmJ Uw radius R/R2 area AIA2.
T is a non-dimensional CR2 . . . . . . . . r ........ a " " " "
For a design calculation it is desired to compute the blade angles for known torques from equations (9), (10) and (11), whilst for a prediction procedure the torques are to be calculated for known blade angles. In either case the major difficulty is the calculation of the rate of circulation of the fluid through the torpus, i.e. velocity CR2 in equations (9), (10) and (11). This is done by considering the total energy flow through the system. 2.1.1 Torque converter power balance. The total energy input to the torque converter must be equal to the total energy output. The total input power is supplied by the prime-mover driving the torque converter, the output power is composed of that available at that output shaft and that dissipated as heat from the casing. The difference between the shaft input and output power is equal to that dissipated, as heat, by the fluid circulating through the toffs, i.e. PL = ¢ p % + ¢r~or
(12)
PL
(13)
and non-dimensionally
pA2 U~pR2~op
= Tp + Tr[l.
The power loss is considered to be absorbed by the circulating fluid in the form of incidence and circulation losses. These losses cannot be calculated theoretically and it is necessary to restort to empirical models and equations for their derivation. These are given in Refs. [2--4] and are only briefly summaffsed here. The incidence loss at pump inlet is given by rn
P~ = :~ w~,, - w,,) ~
(14)
where Wetp and Wot are derived from the velocity triangles Fig. 3, as
Wo,e = Cruztan ~lP
(15)
W01 = C,~, tan//,~ - U,p + U,s.
(16)
Substituting equations (15) and (16) into equation (14) yields P/e
= C]u/tan/3w - tan 181~
pA2U]e -'-2"\
-~1
rl
~'~srl~ 2
~ CR2 CR2/ "
(17)
Similarly the incidence loss at turbine and stator inlet is given by
Prr
C~2/
= -Titan
f~ ~---~2f
#~r - tan t3~, + G - S -
(18)
and
Pts pA2 U]o
tan//3~ - tan/3~r 4
a3
(19)
CR2 CR2]"
Design of three element hydrokinetic torque converters
489
m cos L \R
t/.I
2p i
,?
Prewhirt /
o
-7;
-610 -50 - ~0 -3'0 lmpetLer Inlet Angte ~
-2J0
-ITO-----'-0
FIG. 3. Graphical representation of equation (37).
The circulation loss is simply considered to be given by
Pc = FaAK-~-
(20)
where AK is the circulation loss parameter. If the central core is located so that the flow area around the torus is not constant the circulating velocity Cm varies. In this case an average circulation loss is considered as:
Pc = ----~ ( C2' + C2m:+ C:m3).
(21)
The total power dissipated, equation (13) is then equated to the sum of the losses, i.e. equations (17), (18), (19) and (21) to yield:
Tp + Trfl=_.~[AK. Ac +(tan~to~tan~l, + rl [~srl ~2 CR2 CR21 + (tan/32r - tan/32p + ~e21 f + (tanff3s-tan/i3r+flsr3
a3
CR2
[~r3~2] CRT/ ]
(22)
where A c = a l 2 ..[- d12a3 2 Jr a3 2 3a12a3 2
Substituting from equations (9) and (10), for Tp and TT into equation (22) yields a quadratic equation to be solved for Cez. Up to the coupling point fl, is zero; when the torque ratio becomes less than unity the rotational speed of the stator is calculated from equation (11) by setting. Ts=0,
490
A. WHITFIELD et al,
than
l~ = f~ < + CR$( r~ tan t33r - rl tan t3,s) . •
rc \a3
rl"
(23)
a~
Once the circulating velocity CRy. has been computed the input and output torques, power losses, and complete torque converter performance can be obtained. 2.2 Application to torque converter design For the design procedure the main specified parameters are the input and output torques at stall, and the speed ratio at which no incidence losses will occur. The basic equations developed in the previous section must itherefore be applied at stall, where [1 = o, and at the zero incidence condition, where fl = II'. Application of the power balance at these operating conditions enables the fluid circulation velocities C~ and Ckz, at stall and zero incidence respectively, to be calculated. At the stall condition the input and output torques are derived from equations (9) and (10) as
tan.is))
(24)
and
{ 1 C~2 (tan ~2P - r 3a3t a n / 3 3 r ) ) . T~= C"R2~--
(25)
- -
The power balance, equation (22), becomes
T"P=(CR2) ,, 2 [ A K ' A " + ( tanfl~e-tanfl'sa~
(~):
+__
+
+(tan~2r-tanfl2°--~'~)2+( tan[33s-tanf13r~2]'a3
/ 3
(26)
At the zero incidence condition the input and output torques are given by
T'p = C'n2(l + C'R2(tan [32~-~ tan [31~))
(27)
and
T~= C'ra(-1-C~(tanf12p- r3 tan~r)+r32[~'). a3
(28)
The power balance reduces to
T'p + T'rfl' = (C'~)3AK. At~2.
(29)
Also at the zero incidence condition equations (17), (18) and (19) yield C~2(tan/3~p -tan/~1,) + rlal = 0
(30)
Ck2(tan/3zr - tan/32p) + ll' - 1 = 0
(31)
Ckz(tan//3, - tan//3r) - fl'r3a3 = O.
(32)
and
Equations (24) to (32) provide 9 equations which must be solved for the unknowns T~, T~-, Ck2, C~ and the six blade angles, i.e. ten unknowns. Consequently the blade angle at pump inlet is specified using the theoretical procedure described in the next section. By combining equations (24) and (32) two equations can be derived with Ck2 and C~ as the unknowns. Substituting equations (30), (31) and (32) into equation (26) and rearranging leads to:
[1~. %3
+ ~--~-L[ rl2 + (l - fI') 2+ ([1'r3)2] = 0.
(33)
Bu substituting equations (24) and (25) into equations (27) and (28) the blade angles, flEp, ills and fin can be eliminated and the torques at the zero incidence condition can be expressed as a function of the specified stall torques. Substituting the resulting expressions into equation (29) yields:
( C~2)2AK. Ac/2 + Cb[( - T'~' - C~fI' - T~ + C~)/( C'~)2] + D' - ([~'r3)2- 1 = 0.
(34)
Design of three element hydrokinetic torque converters
491
Equations (33) and (34) provide two equations for the unknown circulating velocities C ~ and C~. It is interesting to note that the blade angles do not appear in these equations. After solving for C~ and C~ the blade angles, with//ip specified, can be found from the systematic application of equations (30), (24), (25), (31) and (32). With the blade angles known the full performance characteristics of the torque converter, in the form of input torque and torque ratio as a function of speed ratio, can be calculated from the equations developed in Section 2.1. 2.2.1 Design of pump impeller blade inlet angle. The above blade angle design procedure leads to a range of possible solutions as it is necessary to specify one blade angle. In centrifugal compressor design the inlet blade angle at the inducer tip is always approx. - 6 0 °. This is based upon a design procedure by Stanitz[5] which designs for the maximum flow rate per unit frontal area. The procedure designs the blade angle to ensure maximum flow rate for any specified limiting Mach number. The effects of high relative velocity at inlet are apparent as Mach number effects in compressors and as cavitation effects in pumps. In torque converters undesirable cavitation can usually be eliminated by increasing the fluid charge pressure; however, as the friction and circulation losses are a function of the velocity relative to the impeller it is important to ensure that the relative velocity is the minimum necessary for any required flow rate. With a uniform axial component of velocity at the impeller inlet, the critical region is at the inducer blade tip, where the blades speed and hence the relative velocity is highest. Usually the impeller inlet blade root diameter is fixed by design requirements of the shaft bearings, and stator. For a given flow Q the area of the impeller inlet may be large, giving a low velocity Cst and a high tip speed U, or it may be small giving a high inlet velocity Cs~ but a low tip speed U. Either case in the extreme gives a high value of relative velocity Wt, and it is necessary to find the condition in between which yields the minimum relative velocity. In compressor and pump design it is usual to assume zero prewhirl at inlet to the impeller; this is not the case for a torque converter where the flow from the stationary guide vane will have a high degree of swirl. Assuming a uniform meridional component of velocity C,.,t at inlet the mass flow rate is given by
= AtpCst
(35)
where the subscript "1" refers to the pump inlet radius at the core where the relative velocity will be maximum. The objective of the analysis is to ensure that the maximum flow circulates through any given torque converter size, i.e. the torque converter is not oversize for the required capacity. The mass flow rate is, therefore, non-dimensionalised using the outer radius of the torque converter to yield
th
R~,- R~, Cm,
7rR22pU2p - cos ~bR2: U2p"
(36)
Substituting 2
2
u_U_~_ R,__~c U~p- R22' C,, = W, cos//,, and
Ulp = Col- Wt sin//i = (Wi cos//1 tan al - sin//d into equation (36) yields COS ~b
,
[
¢rR2"pU2p 1 -
3
( R , ~ ' ] = ~U,p cos BI[cos//I tan al - sin//i]:.
(37)
\R-~lc) J
To find the flow angles al and /31 which yields the maximum flow rate for any non-dimensional relative velocity WJU2p, equation (37) is presented graphically in Fig. 3. From this it can be seen that there is a maximum value for each curve. The effect of increasing positive prewhirl a] is to increase the peak value of the curve and to move it to lower flow angles,//i, i.e. for a given//~ more flow can be handled for the same relative velocity with increased prewhirl. The flow angle//i at which the peak occurs can also be calculated by differentiating equation (37) with respect to/3~ and equating to zero to yield 3 tan 2al (sin//i - sin 3/3t) + 2 tan at cos//1(1 - 3 sin 2//0 + sin Bl(3 sin 2//1 - 2) = 0.
(38)
This must be solved for/31 for any prewhirl angle al. This is done most easily by specifying/31 and solving for tan at as equation (38) then reduces to a quadratic. The result is shown graphically in Fig. 4. This procedure, therefore, yields a series of possible pump inlet and stator discharge blade angles; for any specified stator discharge blade angle Fig. 4 yields the appropriate pump inlet angle. It should be noted that the prewhiri angle a~ in the above analysis is equal to the stator blade angle//ts, provided the coupling point has not been reached causing the stator to rotate. From the design of the previous section the zero incidence condition at pump inlet, equation (30) gives tan/3~p -tan/3~s = -rta~/C'~ which can be simplified to tan $1, - t a n $~ = UllCmt.
(39)
492
A. WHITF1ELD et al. Condition
for Maximum Flow
Rote ~ Equohon 38
......
CondiHon
for Zero
nc dence
Equation /,0
100
_•___
80
2O
i
-60
---
-50
-40 Impelter
-30
-20
Inter Angle
~
~_
0
-10
0
p
FIG. 4. Graphical representation of equations (38) and (40).
At the zero incidence condition the flow angle/3~ of the above analysis will be equal to the blade angle/31p. By combining equations (38) and (39) a solution is obtained for/~lp and/3~,. The solution of equation (39) depends upon the calculation of the circulating velocity C,,~, through the empirical losses, i.e. the solution of equations (33) and (34). This calculation was performed for the Jandasek torque converter, assuming A K = 3.3 and fl' = 0.7, to yield tan/3,s - tan/31p = 2.35
(40)
where/3t, and a, are identical if the coupling point has not been reached. Equation (40) is also plotted on Fig. 4, broken line, and the point of intersection of the two curves yields the required blade angles. The actual blade angles of the Jandasek torque converter are /31p = - 15" and /31, = 68*. These are remarkably close to those given by the above design procedure; substitution of /3, = - 15" into equation (57) yields a~ and hence/~,s, equal to 67.37 °. These results are discussed more fully in Section 2. 4. P R E S E N T A T I O N A N D D I S C U S S I O N OF R E S U L T S The design and analysis procedure has been applied extensively to the torque converter described in detail by Jandasek. The basic torque converter is 305 mm (12 in.) in diameter with mean streamline blade angles as given in Table 1. In addition detailed performance characteristics were presented for six alternative impeller blade exit angles. This data has been used to asses the design and performance preddiction procedure.
Table !. Designed blade angles for the Jandasek torque converter, IMPELLER IMPELLER INLET ANGLE EXIT ANGLE
ROTOR ROTOR INLET ANGLE EXIT ANGLE
AK
= 3 . 5 , fl' =
STATOR STATOR INLET ANGLE EXIT ANGLE
82~
82T
63T
133s
75
66.59
74.72
68.27
76.46
80.66
5O
44.53
66.81
-1.88
58.19
74 •24
25
31.17
62.92
-37.18
41.56
70.46
-15
12.49
57.54
-56.19
8.64
64 •36 62.05
gts
-25
6.73
55.75
-59.41
-2.66
-50
-14.62
47.46
-67.53
-37.68
49.22
-75
-57.79
i -13.31
-78.60
-73.21
-54.09
Actual angles of Jandasek torque c o n v e r t e r -15
15
58
-60
0
68
0.7
Design of three element hydrokinetic torque converters
493
The full design procedure consists of two components, the first to design the torque converter geometry and the second to predict the full performance characteristics. In the light of the predicted performance the designed geometry may be systematically varied in order to achieve the desired optimum design. The performance curves are presented as torque ratio (¢dep) and non-dimensional torque (~-~p%2RoS))as functions of speed ratio (to./top). The efficiency (¢rtod(¢p%) can be readily derived from the torque ratio-speed ratio results. 3.1
Assessment of the design procedure
The key to the blade design procedure lies in the simultaneous solution of equations (33) and (34) which, as already stated in Section 2.2, are independent of the blade angles. This is due to the method used to calculate the circulation losses, equation (20). Strictly the circulation loss should be considered to be a function of relative velocity, IV, rather than the meridional component of veloicty Cm; however this leads to added complexity without any obvious benefits. (The philosophy used has been to develop a satisfactory design procedure without unnecessary complications to complement the detailed performance prediction procedure developed in Ref. 7.) Equations (33) and (34) are presented graphically in Fig. 5(a-c). A solution exists at the point of intersection of the two curves; in cases where the curves do not intersect it is not possible to obtain a solution for the particular conditions specified. Figures 5(a-c) show how the specified conditions can be modified in order to obtain a solution. The circulation loss coefficient, AK, can be reduced, the design speed ratio ~ ' can be reduced, and the outer radius of the torque converter Ro can be increased. Whilst these are mathematical modifications which can be applied to obtain a solution to equations (33) and (34), it may not always be satisfactory or even possible to make these modifications. The physical reason for the non-existence of a solution is that it is not possible for the fluid to circulate through the torus at a rate sufficient to meet the specified stall torque capacity. Modifying any of the three parameters, AK, ~' and Ro leads to an increase in the rate of fluid circulation. However, in practice the designer will not be able to modify AK, and will wish to specify D' to give peak efficiency at a specified speed ratio and to provide a satisfactory coupling range; this leaves little option but to increase the size of the torque converter if the specified torque capacity is to be achieved. Table 1 shows the blade angles calculated for the Jandasek torque converter. As one angle has to be specified the results are shown for a series of impeller inlet angles/3zp. From these results it can be seen that t a n / ~ - tan flip = 2.35. Plotting this result onto Fig. 4 yields a point of intersection with the curve for maximum flow rate at/3~p = - 17; this is remarkable close to the actual angle of - 15° given by Jandasek. A point which should not be overlooked, however, is the fact that the angles considered in the analysis of Section 2.21 are considered to be at the outer radius of the impeller inlet, i.e. at the core, whereas those for the blade angle design are considered at the radius of the mean streamline. The two curves of Fig. 4 are, therefore, not strictly comparable; however, the difference between the blade angle at the mean radius and that at the core radius is sufficiently small ( - 15 compared to - 17.3) as to make any error insignificant
Equation 33 1 Equation 3/. I AK :/.. S o
o
• ....
•
Equation 33[ AK : 1 '5 Equation 3~ J I
I "ca
/
0-3
i
I I
fi /P
o N
U-
/
I
0.2
0.1
"O 5
*
L
I
1
0.1
0.2
03
0.4
Non- dimensionol Fluid VeLocity o;" ShOL[ CR2 j'
FIG. 5(a). Effect of loss coefficient on convergence.
/
A. WHITFIELD et al.
494
Imaginary o
~utions
o Equation 3 3 [
H
Equo.tion 36 I caR'= 0 9
o----o ° - - ---.,
Equation '3'] l Equation 3t, j ~R' = 0 6 I Imaginary Sotutions
- ,.,,, 0 3
,._,=
I I ,/0~
~u
Y=
.~_
/,,'° 7 ,.~ 0.2 .-'e
,o'/
. / ~
5
04 o
i
g z
0
±
/
01
0'2
0"3
0"/,
Non-dimenstonat Fluid Velocity a}" ~olt CR2 s'
FIG. 5(b). Effect of varying ~ok on c o n v e r g e n c e A K = 4.5.
O
Equation
o....
o
Equation 3 3 ]
°....
•
Equation 3l,
D
33},] Ro = 0 1 6 m Equation 3&
J
a
R o =019m
-,-,. 0 t~
o.,
[magi nory Solu}'ions o
,~0
Y
l Imaginary
~olution~/
-'e >-,
I
=
J ,,~',,'°
0.1
,o
J/
,S ? ~ ~ i
0
o
011
i
i
02
03
Non-dimens.ior~l Fluid Velocity at Sl'olt £R2 sr FIG. 5(c). Effect of varying outer radius Ro on convergence. A K = 4-5, o~k = 0.8.
Design of three element hydrokinetic torque converters
495
when compared with the uncertainties associated with a one-dimensional mean streamlines design procedure. In order to assess these uncertainties the impeller inlet angle was assumed to be - 1 5 °, as given by Jandasek, and all other angles calculated for a number of assumed loss coefficients, AK, and design speed ratios D'; the result is given in Table 2. The calculated blade angles are in close agreement with the actual angles given by Jandasek when a loss coefficient of 3.5 is used at a design speed ratio of 0.7. From Table 2 it can be seen that the calculated impeller exit angle could be forced to a value of 15°, as specified by Jandasek, by increasing the loss coefficient to a value between 3.5 and 4.5. If this were done all the other calculated angles would then be brought into even closer agreement with those specified by Jandasek. 3.2 Application of the design procedure In Ref. [1] Jandasek presents experimental data, obtained over 15 years of experimental development, to be used as design data, to meet any specified requirements. This data and the procedure developed here have been applied to the design of a torque converter with a 3-1 stall torque ratio and an input stall torqtle of 270 Nm (200 Ibft) at a rotational speed of 1480 revs/min, which is the stall torque and speed of the basic Jandasek design, and the resultant designs compared. The Jandasek design employs the blade angles of the basic design, Table 1, with the impeller exit angle modified to - 5 5 ° in order to give the 3-1 stall torque ratio. With this design the non-dimensional stall torque is 0.042, and for an input torque of 270 Nm the outer radius of the torque converter is given by
Ro5 = ~'/(0.042. p. to2p), leading to an outer radius of 198 ram. For the design procedure developed here the parameters used in the design assessment were initially used, i.e. AK = 3.5 and (oR,= 0.7. With an assumed impeller inlet angle of - 1 5 ° the outer radius of the torque converter was progressively increased from 152 mm until a converter size was reached which was
Table 2. Effect of loss coefficient and speed ratio on designed blade angles IMPELLER EXIT ANGLE
ROTOR INLET ANGLE
ROTOR EXIT ANGLE
B2~
B2T
46 •88
69.69
-66.79
24.34
63.13
-60.72
12.49
57.54
-16.25
83T
STATOR INLET ANGLE B3s
STATOR EXIT ANGLE
DES I GN SPEED RATIO
LOS S COEFFICIENT
~is
J~
12.67
72.35
0.75
3.95
67.21
0.7
!
4.5
-56.19
8.64
64.36
0.7
3.5
50.08
-47.17
-16.84
52.1
0.5
i i !
-I0.23
55.82
-52.51
-23.79
55.56
0.5
1
4.5
-4.13
59.91
-56.71
-30.3
58.15
0.5
]
5.5
15
58
-60
68
AK 4.5
3.5
ANGLES FOR THE FANDASEK TORQUE ?ONVERTER
Impeller inlet angle -15
Table 3. Blade angle design for 3 : 1 stall torque ratio converter with Ft' = 0.7 IM~aELLER INLET ANGLE
-15
-5
-20
IMPELLER EXIT ANGLE
ROTOR INLET ANGLE
B2~
B2T
ROTOR EXIT ANGLE
B3T
STATOR INLET ANGLE B3s,
STATOR EXIT ANGLE
Bls
OUTER RADIUS
E0
7.97
77.47
-82.96
-72.8
81.49
192
-40.83
71.68
-81.21
-613.38
80.3
199
-60.41
66.38
-80.86
-61.37
80.54
210
12.75
77.7
-82.8
-71.85
81.71
192
-37.91
72.51
-80.96
-63.45
80.59
199
-59.2
67.13
-80.6
-58.8
80.81
210
5.37
77.34
-83.04
-73.27
81.37
192
-42.28
71.42
-81.33
-66.3
80.14
199
-61.O2
65.96
-81
-62.58
80.39
210
~
A. WHITFIELD et al.
496
Table 4. Blade an le design for IMPELLER INLET ANGLE
Blp -15
-5
ROTOR INLET ANGLE
IMPELLER EXIT ANGLE
B2T
g2p
: 1 stall torque ratio converter with fF : 0.6 ROTOR EXIT ANGLE
STATOR INLET ANGLE
fl3s
83T
STATOR EXIT ANGLE
g[s
OUTER RADIUS R m~m o
-48.79
48.4
-69.58
-46.14
68.02
-55.26
42.46
-69.13
-42.99
68.59
[92
-60.5
35.95
-68.93
-39.67
69.52
i99
185
-46.52
50.52
-68.25
-40.69
69.38
185
-53.58
45.04
-67.73
-36.93
69.89
192
-59.27
39.O2
-64.51
-32.97
70.72
199
sufficient to meet the torque capacity requirements. Table 3 presents the calculated blade angles for three outer radii Re. In addition the results for alternative impeller inlet angles are presented; an inlet angle of - 5 ° was selected based upon the curve of Fig. 4, and the angle of -200 is presented to provide further comparison. The high turbine and stator exit angles shown in Table 3 are not considered desirable as any blade blockage effect may become significant; blade thickness has generally been ignored in both the design and prediction procedure. Attempts were therefore made to reduce these angles. As can be seen from Table 3 increasing the outer radius leads to a reduction in all angles, but the smallest changes occur with the largest angles, and the starer exit angle shows a minimum value after which it increases again as the radius increases. From Table 2 it can be seen that reducing the design point speed ratio leads to a reduction in the designed blade angles. The design exercise was therefore repeated for a specified speed ratio of 0.6. Table 4 presents the results for two impeller inlet blade angles. It can be seen that the blade angles have been substantially reduced, and that with an outer radius of 0.192 m and an impeller inlet angle of - 15° both the impeller and stator exit angles are approximately the
o
o Designed using Jandosek Procedure
D----o
Designed using Present Procedure with uoR' = 0 7 end ~Sip = -15
x----~x
Designed using Present" Procedure wiih ~ R '= 0 6 end )Sip : - 1 5
2,8
24
2.C
18
°
l
1'6,
,\
12
1'C
~
011
0'.2
03
0~,
0'$
0.6
0'7
0'-8
0.9
10
Speed I~'io
FIG. 6(a). Predicted torque characteristic for proposed 3 : 1 stall torque radio design.
Design of three element hydrokinetic torque converters o
,. o
Designed
using
Jondosek
~
m -- --~
Designed
using
Present
x-----~x
497
Procedure Procedure
with UJR'= 0 7 and j61p=-15 Designed using Present Procedure with ("JR: 06 (and /Sip:-15
006
tT" 0.06
o
i
0-02
g
z
.
0.1
.
0.2
.
.
03
.
.
.
.
.
0t,
05 06 07 08 09 10 £pe~ Rotio FIG. 6(b). Predicted torque capacity characteristic for proposed 3 : 1 stall torque ratio design.
same as those specified by Janasek. Also the impeller inlet angle of - 1 5 ° and stator exit angle of 68° satisfies the maximum flow rate requirement of equation (57), Fig. 4. The predicted performance, obtained with the forced vortex procedure of Ref. [7], for the three designs with an impeller inlet angle of - 150 and an outer radius of 0.199 m, i.e. (a) the Jandasek design, (b) the blade design with II' = 0.7, and (c) the blade design with 1)' = 0.6, are compared in Fig. 6(a, b). From this it can be seen that it is predicted that none of the designs will achieve the required 3 : 1 stall torque ratio. As the torque converter efficiency is given by the product of torque ratio and speed ratio it can be seen that the Jandasek design shows the lowest predicted efficiency, a peak of 90% compared to 96% for the blade design with f~' = 0.7. For the blade design with ~ ' = 0.6 a stall torque ratio greater than 2.7 is predicted and clearly further modifications are needed before the proposed 3 : I stall torque can be achieved. The capacity curves of Fig. 6(b) shows that the stall torque capacity requirement of 0.042 is approximately achieved, although the Jandasek design is a little high. The major difference between the torque capacity results lies in the continuously falling characteristic achieved for the blade design with f l ' = 0.7. The rising characteristics of the other two designs is due to the high backward swept impeller exit blade, - 55o.
4. CONCLUSIONS A detailed theoretical design procedure for three element torque converters has been developed to complement an extensive prediction procedure presented in Ref. [7]. It has been shown that the theoretical procedure leads to a torque converter design similar to that developed experimentally by Jandasek. Whilst the work presented here has been confined to the simple three element torque converter it provides the basis for development to multi-ilement designs. The mean streamline prediction procedure has been extended to accommodate any multielement design and is currently being tested. A design procedure for multi-element torque converters has been formulated but has yet to be developed into a computer program. REFERENCES 1. V. J. JANDASEK, Design of a single stage three element torque converter. Passenger Car Automatic Transmission, S A E Transm. Workshop Meeting, 2nd Edn, Vol. 5, p. 201 (1963). 2. R. EKSER61AN,The fluid torque converter and coupling. Z Franklin Inst. 235(5), 441 (1943). 3. J. W. QUALMAN and E. L. EeBERT, Fluid couplings. Passenger Car Automatic Transmission, SAE. Transm. Workshop Meeting, 2nd Edn, Vol. 5, p. 137, 1963. 4. G. G. LUCASand A. RAYNER,Torque converter design calculations. Automobile Engng 56 (1970). 5. J. D. STANEZ, Design considerations for mixed flow centrifugal compressors with high flow rates per unit frontal area. N A C A R M E 53 AI5 0953). 6. F. J. WALLCE, A. WrUTFIELD and S1VALINGAM,A theoretical model for the performance prediction of fully filled fluid couplings. Int. Z Mech. Sci. 20, 335 (1978). 7. A. WHITF1ELD, F. J. WALLACE a n d SIVALINGAM, A performance prediction procedure for three element torque converters. Int. Z Mech. Sci. (1978).
0020-7403/83 $3.00+ .00 Pergamon Press Ltd.
DESIGN OF THREE ELEMENT HYDROKINETIC TORQUE CONVERTERS A. WHITFIELD,F. J. WALLACEand A. PATEL University of Bath, School of Engineering, Claverton Down, Bath, BA2 7AY, England
(Received 23 May 1982; in revised form 20 September 1982) Summary--This paper describes a design procedure for three element torque converters. The torque converter, consisting of closely coupled hydraulic turbines and pumps, forms an essential part in the transmission systems of many power lines. Efficient torque converters are therefore essential if maximum energy is to be transmitted to the output shaft. The design procedure consists of two components, the first establishes the overall dimensions and mean streamline blade angles using a one-dimensional fluid dynamic analysis, whilst the second uses a detailed integrated fluid dynamic procedure to predict the performance of the proposed torque converter design. The design procedure is developed in full whilst the prediction procedure which is published elsewhere, is used along with published data to assess the quality of the torque converter designs.
A
AK a C CR2 g rh N PL PI
Pc R
Ro Rc R, r T U W a /3 p •R co l~ ~
NOTATION flow area empirical loss parameter non-dimensional area AIA2 with appropriate subscript absolute fluid velocity non-dimensional velocity Cm21U2p gravitational acceleration mass flow rate of fluid rotational speed rev/min total power dissipated power dissipated due to incidence power dissipated due to fluid circulation radius to mean streamline ~/(Rs2/2 + Rc2/2) outer radius of torque converter radius to core radius to shell non-dimensional radius R/R2 non-dimensional torque linear velocity of rotational relative velocity of fluid direction of absolute velocity form the meridional direction direction of relative velocity from the meridionai direction efficiency fluid viscosity fluid density torque torque ratio rotational velocity rad/min speed ratio ~o~/~op speed ratio co, coo
Subscripts 1 2 3 P T S 0 m " '
pump inlet, stator exit turbine inlet, pump exit stator inlet, turbine exit pump impeller turbine rotor stator tangential component of velocity meridional component of velocity stall condition zero incidence design condition. 1. I N T R O D U C T I O N
Hydrokinetic devices, in the form of fluid couplings and torque converters, are widely used in the transmission systems of many road vehicles, cranes, lift trucks, earth moving vehicles etc. They provide the advantages of shock and vibration free power 485
486
A. WHITFIELDet al.
t r a n s m i s s i o n , a n d a u t o m a t i c t r a n s m i s s i o n s w h i c h do n o t r e q u i r e m a n u a l g e a r c h a r g e s . These hydrokinetic devices consist of a hydraulic pump and turbine closely coupled, w i t h o u t a n y m e c h a n i c a l link, so t h a t t h e fluid flows in a c l o s e d circuit. T h e fluid d i s c h a r g e d f r o m the c e n t r i f u g a l p u m p is d e l i v e r e d to a r a d i a l flow t u r b i n e a n d then r e t u r n e d to the p u m p inlet, t h r o u g h a s a t i o n a r y g u i d e v a n e in the c a s e o f a t o r q u e converter. T o r q u e c o n v e r t e r d e s i g n is b a s e d l a r g e l y on t h e u s e o f d i m e n s i o n l e s s a n a l y s i s , to b u i l d s i m i l a r units to o n e of k n o w n s a t i s f a c t o r y p e r f o r m a n c e , s e e Ref. [1], a n d u p o n o n e - d i m e n s i o n a l single m e a n s t r e a m a n a l y s e s to p r e d i c t the p e r f o r m a n c e , R e f s . [2-4]. T h e m o s t c o m p r e h e n s i v e p r e s e n t a t i o n o f t o r q u e c o n v e r t e r p e r f o r m a n c e a n d d e s i g n is t h a t b y J a n d a s e k [ l ] . T h e d e s i g n a n d p e r f o r m a n c e p a r a m e t e r s w h i c h he p r e s e n t s in d e t a i l h a v e b e e n e x t e n s i v e l y u s e d in t h e d e v e l o p m e n t o f the d e s i g n a n d p e r f o r m a n c e p r e d i c t i o n p r o c e d u r e s p r e s e n t e d here. T h e d e s i g n p r o c e d u r e a d o p t e d is to u s e a o n e - d i m e n s i o n a l single m e a n s t r e a m a n a l y s i s to e s t a b l i s h the o v e r a l l d i m e n s i o n s a n d b l a d e a n g l e s for t h e t o r q u e c o n v e r t e r . H o w e v e r , it h a s b e e n s h o w n , R e f s . [6] a n d [7] t h a t this a p p r o a c h is n o t satisfactory for accurate detailed performance prediction and the more thorough a n a l y s i s o f Ref. [7] is u s e d . T h i s a p p r o a c h a s s u m e s t h a t the fluid c i r c u l a t e s as a f o r c e d v o r t e x a n d u s e s a n i n t e g r a t i o n p r o c e d u r e b e t w e e n t h e o u t e r shell a n d i n n e r c o r e in o r d e r to c a l c u l a t e t h e t o r q u e d e v e l o p e d a n d p o w e r d i s s i p a t e d . T h i s v e r y d e t a i l e d flow m o d e l , d e s c r i b e d f u l l y in R e f . [7] is, t h e r e f o r e , the m a i n p r e d i c t i v e t o o l ; the m e a n s t r e a m l i n e a n a l y s i s , d e s c r i b e d h e r e , p r o v i d i n g t h e m a i n d e s i g n p r o c e d u r e , a n d an initial e s t i m a t e o f p e r f o r m a n c e . 3. T H E O R E T I C A L D E S I G N P R O C E D U R E F O R T H R E E E L E M E N T TORQUE C O N V E R T E R S The objective of the design procedure is to calculate the basic geometric parameters of a three element torque converter for specified operating conditions. The main operating requirements of a torque converter are the stall torque capacity and torque ratio, a broad operating range up to the coupling point, and high peak efficiency at the designed operating condition. The specified parameters are, therefore, the input stall torque, the output stall torque, or torque ratio, and the speed ratio ll' at which the circulating fluid moves smoothly from one component to the next, i.e. the condition at which no incidence loss will occur. The required stall torques must be specified from the performance requirements projected for the vehicle under consideration. Tractive effort requirements for start up determines the stall torque ratio, if a gearbox is used to supplement this ratio its effect must be included in the determination of the required torque converter output. The basic geometric parameters to be calculated are the outer diameter, the size and position of the central core, and the blade angles for each element. The outer diameter is calculated by systematically increasing the size of the torque converter until a size is found which is able to meet the specified torque capacity, this illustrated in Section 3. The position and size of the central core is calculated using the procedure given by Jandasek[l]. This fixes the position and size of the core to give a constant flow area throughout the circuit equal to 23% of the cross sectional area, i.e. 0.23 ~rR2o. This leaves the calculation of the blade angles of each element as the main objective of the design procedure. Figure 1 shows the notation and typical arrangement of a three element torque converter considered. The design procedure is based upon a one-dimensional single mean streamline analysis approach which is well documented in the literature for performance prediction only. 2.1 Fundamental turbomachinery theory
Central to all analyses of turbomachines is the Euler turbomachinery equation which relates torque to the rate of change of angular momentum. This can be written for each element of the torque converter, as Tp = m ( G 2 P R 2 - G I s R , )
(1)
rr = m(G3TR3 -- Co2pR2)
(2)
rs = ra( ColsRi - G3rR3).
(3)
and
As fluid which completes one complete circuit from pump inlet through each element and back to pump inlet, must have no net change in moment of momentum, there can be no net torque exerted, i.e. "rp + "rr + ~'S= 0.
(4)
This is clear from the summation of equations (1), (2) and (3). It should be noted that the input torque % is defined as a positive torque and consequently the output torque, ~'r is negative. As it is conventional to
487
Design of three element hydrokinetic torque converters
Meon Streomline
\ //: / Pump
Turbine
j'
\,
/ /
\
StQtor i
FIG. 1. Cross section of torque converter showing mean streamline. have a positive torque ratio this is defined as: ~-R =
- rr
(5)
rp
The tangential components of velocity are calculated from the velocity triangles at each station, Fig. 2, which are constructed by assuming that the fluid is perfectly guided by the blades. This leads to: Co2e = UEp + C=2 tan fl2p
(6)
Co3-: = U3r + C,,3tan/33r
(7)
Cols = U1s + C=l tan/31s.
(8)
Emls
E m;
Cm2 [m2T
........
_ _[m_2~. . . . t
~,.Wezp',
~--~ E/ . . . . -, E O 2 T ~ rW021[ W~
[ " ~ W aT
~
/- ~3r
[02P
~T
¢ m3
[83T~ FiG. 2. Velocity triangles.
_ _ [mas__ _
~ / ~ ' C 3S
488
A. WHITFIELD et al.
The blade angles are defined in Fig. 2, the sign convention is such that a positive blade angle directs the fluid in the direction of rotation of the blades. Substituting equations (6), (7) and (8) into equations (1), (2) and (3) and noting that continuity yields
C~tAI = C~2A2= C,n3A3 the following expressions for torque can be derived: Tp = CR2(l+CR2(tan/3n, - r-L' a, tan/3,s) - rl2[~s)
TT=CR2(flr23+CR2(r3tan$3r-tan$2") \a3 -I Ts = CR2(II,r~2+ CB2(~ tan fl~s- r3/a3 tan fl3r) - r3ZIl)
(9)
(10) (ll)
where torque d(pA2R2U2e) velocity CmJ Uw radius R/R2 area AIA2.
T is a non-dimensional CR2 . . . . . . . . r ........ a " " " "
For a design calculation it is desired to compute the blade angles for known torques from equations (9), (10) and (11), whilst for a prediction procedure the torques are to be calculated for known blade angles. In either case the major difficulty is the calculation of the rate of circulation of the fluid through the torpus, i.e. velocity CR2 in equations (9), (10) and (11). This is done by considering the total energy flow through the system. 2.1.1 Torque converter power balance. The total energy input to the torque converter must be equal to the total energy output. The total input power is supplied by the prime-mover driving the torque converter, the output power is composed of that available at that output shaft and that dissipated as heat from the casing. The difference between the shaft input and output power is equal to that dissipated, as heat, by the fluid circulating through the toffs, i.e. PL = ¢ p % + ¢r~or
(12)
PL
(13)
and non-dimensionally
pA2 U~pR2~op
= Tp + Tr[l.
The power loss is considered to be absorbed by the circulating fluid in the form of incidence and circulation losses. These losses cannot be calculated theoretically and it is necessary to restort to empirical models and equations for their derivation. These are given in Refs. [2--4] and are only briefly summaffsed here. The incidence loss at pump inlet is given by rn
P~ = :~ w~,, - w,,) ~
(14)
where Wetp and Wot are derived from the velocity triangles Fig. 3, as
Wo,e = Cruztan ~lP
(15)
W01 = C,~, tan//,~ - U,p + U,s.
(16)
Substituting equations (15) and (16) into equation (14) yields P/e
= C]u/tan/3w - tan 181~
pA2U]e -'-2"\
-~1
rl
~'~srl~ 2
~ CR2 CR2/ "
(17)
Similarly the incidence loss at turbine and stator inlet is given by
Prr
C~2/
= -Titan
f~ ~---~2f
#~r - tan t3~, + G - S -
(18)
and
Pts pA2 U]o
tan//3~ - tan/3~r 4
a3
(19)
CR2 CR2]"
Design of three element hydrokinetic torque converters
489
m cos L \R
t/.I
2p i
,?
Prewhirt /
o
-7;
-610 -50 - ~0 -3'0 lmpetLer Inlet Angte ~
-2J0
-ITO-----'-0
FIG. 3. Graphical representation of equation (37).
The circulation loss is simply considered to be given by
Pc = FaAK-~-
(20)
where AK is the circulation loss parameter. If the central core is located so that the flow area around the torus is not constant the circulating velocity Cm varies. In this case an average circulation loss is considered as:
Pc = ----~ ( C2' + C2m:+ C:m3).
(21)
The total power dissipated, equation (13) is then equated to the sum of the losses, i.e. equations (17), (18), (19) and (21) to yield:
Tp + Trfl=_.~[AK. Ac +(tan~to~tan~l, + rl [~srl ~2 CR2 CR21 + (tan/32r - tan/32p + ~e21 f + (tanff3s-tan/i3r+flsr3
a3
CR2
[~r3~2] CRT/ ]
(22)
where A c = a l 2 ..[- d12a3 2 Jr a3 2 3a12a3 2
Substituting from equations (9) and (10), for Tp and TT into equation (22) yields a quadratic equation to be solved for Cez. Up to the coupling point fl, is zero; when the torque ratio becomes less than unity the rotational speed of the stator is calculated from equation (11) by setting. Ts=0,
490
A. WHITFIELD et al,
than
l~ = f~ < + CR$( r~ tan t33r - rl tan t3,s) . •
rc \a3
rl"
(23)
a~
Once the circulating velocity CRy. has been computed the input and output torques, power losses, and complete torque converter performance can be obtained. 2.2 Application to torque converter design For the design procedure the main specified parameters are the input and output torques at stall, and the speed ratio at which no incidence losses will occur. The basic equations developed in the previous section must itherefore be applied at stall, where [1 = o, and at the zero incidence condition, where fl = II'. Application of the power balance at these operating conditions enables the fluid circulation velocities C~ and Ckz, at stall and zero incidence respectively, to be calculated. At the stall condition the input and output torques are derived from equations (9) and (10) as
tan.is))
(24)
and
{ 1 C~2 (tan ~2P - r 3a3t a n / 3 3 r ) ) . T~= C"R2~--
(25)
- -
The power balance, equation (22), becomes
T"P=(CR2) ,, 2 [ A K ' A " + ( tanfl~e-tanfl'sa~
(~):
+__
+
+(tan~2r-tanfl2°--~'~)2+( tan[33s-tanf13r~2]'a3
/ 3
(26)
At the zero incidence condition the input and output torques are given by
T'p = C'n2(l + C'R2(tan [32~-~ tan [31~))
(27)
and
T~= C'ra(-1-C~(tanf12p- r3 tan~r)+r32[~'). a3
(28)
The power balance reduces to
T'p + T'rfl' = (C'~)3AK. At~2.
(29)
Also at the zero incidence condition equations (17), (18) and (19) yield C~2(tan/3~p -tan/~1,) + rlal = 0
(30)
Ck2(tan/3zr - tan/32p) + ll' - 1 = 0
(31)
Ckz(tan//3, - tan//3r) - fl'r3a3 = O.
(32)
and
Equations (24) to (32) provide 9 equations which must be solved for the unknowns T~, T~-, Ck2, C~ and the six blade angles, i.e. ten unknowns. Consequently the blade angle at pump inlet is specified using the theoretical procedure described in the next section. By combining equations (24) and (32) two equations can be derived with Ck2 and C~ as the unknowns. Substituting equations (30), (31) and (32) into equation (26) and rearranging leads to:
[1~. %3
+ ~--~-L[ rl2 + (l - fI') 2+ ([1'r3)2] = 0.
(33)
Bu substituting equations (24) and (25) into equations (27) and (28) the blade angles, flEp, ills and fin can be eliminated and the torques at the zero incidence condition can be expressed as a function of the specified stall torques. Substituting the resulting expressions into equation (29) yields:
( C~2)2AK. Ac/2 + Cb[( - T'~' - C~fI' - T~ + C~)/( C'~)2] + D' - ([~'r3)2- 1 = 0.
(34)
Design of three element hydrokinetic torque converters
491
Equations (33) and (34) provide two equations for the unknown circulating velocities C ~ and C~. It is interesting to note that the blade angles do not appear in these equations. After solving for C~ and C~ the blade angles, with//ip specified, can be found from the systematic application of equations (30), (24), (25), (31) and (32). With the blade angles known the full performance characteristics of the torque converter, in the form of input torque and torque ratio as a function of speed ratio, can be calculated from the equations developed in Section 2.1. 2.2.1 Design of pump impeller blade inlet angle. The above blade angle design procedure leads to a range of possible solutions as it is necessary to specify one blade angle. In centrifugal compressor design the inlet blade angle at the inducer tip is always approx. - 6 0 °. This is based upon a design procedure by Stanitz[5] which designs for the maximum flow rate per unit frontal area. The procedure designs the blade angle to ensure maximum flow rate for any specified limiting Mach number. The effects of high relative velocity at inlet are apparent as Mach number effects in compressors and as cavitation effects in pumps. In torque converters undesirable cavitation can usually be eliminated by increasing the fluid charge pressure; however, as the friction and circulation losses are a function of the velocity relative to the impeller it is important to ensure that the relative velocity is the minimum necessary for any required flow rate. With a uniform axial component of velocity at the impeller inlet, the critical region is at the inducer blade tip, where the blades speed and hence the relative velocity is highest. Usually the impeller inlet blade root diameter is fixed by design requirements of the shaft bearings, and stator. For a given flow Q the area of the impeller inlet may be large, giving a low velocity Cst and a high tip speed U, or it may be small giving a high inlet velocity Cs~ but a low tip speed U. Either case in the extreme gives a high value of relative velocity Wt, and it is necessary to find the condition in between which yields the minimum relative velocity. In compressor and pump design it is usual to assume zero prewhirl at inlet to the impeller; this is not the case for a torque converter where the flow from the stationary guide vane will have a high degree of swirl. Assuming a uniform meridional component of velocity C,.,t at inlet the mass flow rate is given by
= AtpCst
(35)
where the subscript "1" refers to the pump inlet radius at the core where the relative velocity will be maximum. The objective of the analysis is to ensure that the maximum flow circulates through any given torque converter size, i.e. the torque converter is not oversize for the required capacity. The mass flow rate is, therefore, non-dimensionalised using the outer radius of the torque converter to yield
th
R~,- R~, Cm,
7rR22pU2p - cos ~bR2: U2p"
(36)
Substituting 2
2
u_U_~_ R,__~c U~p- R22' C,, = W, cos//,, and
Ulp = Col- Wt sin//i = (Wi cos//1 tan al - sin//d into equation (36) yields COS ~b
,
[
¢rR2"pU2p 1 -
3
( R , ~ ' ] = ~U,p cos BI[cos//I tan al - sin//i]:.
(37)
\R-~lc) J
To find the flow angles al and /31 which yields the maximum flow rate for any non-dimensional relative velocity WJU2p, equation (37) is presented graphically in Fig. 3. From this it can be seen that there is a maximum value for each curve. The effect of increasing positive prewhirl a] is to increase the peak value of the curve and to move it to lower flow angles,//i, i.e. for a given//~ more flow can be handled for the same relative velocity with increased prewhirl. The flow angle//i at which the peak occurs can also be calculated by differentiating equation (37) with respect to/3~ and equating to zero to yield 3 tan 2al (sin//i - sin 3/3t) + 2 tan at cos//1(1 - 3 sin 2//0 + sin Bl(3 sin 2//1 - 2) = 0.
(38)
This must be solved for/31 for any prewhirl angle al. This is done most easily by specifying/31 and solving for tan at as equation (38) then reduces to a quadratic. The result is shown graphically in Fig. 4. This procedure, therefore, yields a series of possible pump inlet and stator discharge blade angles; for any specified stator discharge blade angle Fig. 4 yields the appropriate pump inlet angle. It should be noted that the prewhiri angle a~ in the above analysis is equal to the stator blade angle//ts, provided the coupling point has not been reached causing the stator to rotate. From the design of the previous section the zero incidence condition at pump inlet, equation (30) gives tan/3~p -tan/3~s = -rta~/C'~ which can be simplified to tan $1, - t a n $~ = UllCmt.
(39)
492
A. WHITF1ELD et al. Condition
for Maximum Flow
Rote ~ Equohon 38
......
CondiHon
for Zero
nc dence
Equation /,0
100
_•___
80
2O
i
-60
---
-50
-40 Impelter
-30
-20
Inter Angle
~
~_
0
-10
0
p
FIG. 4. Graphical representation of equations (38) and (40).
At the zero incidence condition the flow angle/3~ of the above analysis will be equal to the blade angle/31p. By combining equations (38) and (39) a solution is obtained for/~lp and/3~,. The solution of equation (39) depends upon the calculation of the circulating velocity C,,~, through the empirical losses, i.e. the solution of equations (33) and (34). This calculation was performed for the Jandasek torque converter, assuming A K = 3.3 and fl' = 0.7, to yield tan/3,s - tan/31p = 2.35
(40)
where/3t, and a, are identical if the coupling point has not been reached. Equation (40) is also plotted on Fig. 4, broken line, and the point of intersection of the two curves yields the required blade angles. The actual blade angles of the Jandasek torque converter are /31p = - 15" and /31, = 68*. These are remarkably close to those given by the above design procedure; substitution of /3, = - 15" into equation (57) yields a~ and hence/~,s, equal to 67.37 °. These results are discussed more fully in Section 2. 4. P R E S E N T A T I O N A N D D I S C U S S I O N OF R E S U L T S The design and analysis procedure has been applied extensively to the torque converter described in detail by Jandasek. The basic torque converter is 305 mm (12 in.) in diameter with mean streamline blade angles as given in Table 1. In addition detailed performance characteristics were presented for six alternative impeller blade exit angles. This data has been used to asses the design and performance preddiction procedure.
Table !. Designed blade angles for the Jandasek torque converter, IMPELLER IMPELLER INLET ANGLE EXIT ANGLE
ROTOR ROTOR INLET ANGLE EXIT ANGLE
AK
= 3 . 5 , fl' =
STATOR STATOR INLET ANGLE EXIT ANGLE
82~
82T
63T
133s
75
66.59
74.72
68.27
76.46
80.66
5O
44.53
66.81
-1.88
58.19
74 •24
25
31.17
62.92
-37.18
41.56
70.46
-15
12.49
57.54
-56.19
8.64
64 •36 62.05
gts
-25
6.73
55.75
-59.41
-2.66
-50
-14.62
47.46
-67.53
-37.68
49.22
-75
-57.79
i -13.31
-78.60
-73.21
-54.09
Actual angles of Jandasek torque c o n v e r t e r -15
15
58
-60
0
68
0.7
Design of three element hydrokinetic torque converters
493
The full design procedure consists of two components, the first to design the torque converter geometry and the second to predict the full performance characteristics. In the light of the predicted performance the designed geometry may be systematically varied in order to achieve the desired optimum design. The performance curves are presented as torque ratio (¢dep) and non-dimensional torque (~-~p%2RoS))as functions of speed ratio (to./top). The efficiency (¢rtod(¢p%) can be readily derived from the torque ratio-speed ratio results. 3.1
Assessment of the design procedure
The key to the blade design procedure lies in the simultaneous solution of equations (33) and (34) which, as already stated in Section 2.2, are independent of the blade angles. This is due to the method used to calculate the circulation losses, equation (20). Strictly the circulation loss should be considered to be a function of relative velocity, IV, rather than the meridional component of veloicty Cm; however this leads to added complexity without any obvious benefits. (The philosophy used has been to develop a satisfactory design procedure without unnecessary complications to complement the detailed performance prediction procedure developed in Ref. 7.) Equations (33) and (34) are presented graphically in Fig. 5(a-c). A solution exists at the point of intersection of the two curves; in cases where the curves do not intersect it is not possible to obtain a solution for the particular conditions specified. Figures 5(a-c) show how the specified conditions can be modified in order to obtain a solution. The circulation loss coefficient, AK, can be reduced, the design speed ratio ~ ' can be reduced, and the outer radius of the torque converter Ro can be increased. Whilst these are mathematical modifications which can be applied to obtain a solution to equations (33) and (34), it may not always be satisfactory or even possible to make these modifications. The physical reason for the non-existence of a solution is that it is not possible for the fluid to circulate through the torus at a rate sufficient to meet the specified stall torque capacity. Modifying any of the three parameters, AK, ~' and Ro leads to an increase in the rate of fluid circulation. However, in practice the designer will not be able to modify AK, and will wish to specify D' to give peak efficiency at a specified speed ratio and to provide a satisfactory coupling range; this leaves little option but to increase the size of the torque converter if the specified torque capacity is to be achieved. Table 1 shows the blade angles calculated for the Jandasek torque converter. As one angle has to be specified the results are shown for a series of impeller inlet angles/3zp. From these results it can be seen that t a n / ~ - tan flip = 2.35. Plotting this result onto Fig. 4 yields a point of intersection with the curve for maximum flow rate at/3~p = - 17; this is remarkable close to the actual angle of - 15° given by Jandasek. A point which should not be overlooked, however, is the fact that the angles considered in the analysis of Section 2.21 are considered to be at the outer radius of the impeller inlet, i.e. at the core, whereas those for the blade angle design are considered at the radius of the mean streamline. The two curves of Fig. 4 are, therefore, not strictly comparable; however, the difference between the blade angle at the mean radius and that at the core radius is sufficiently small ( - 15 compared to - 17.3) as to make any error insignificant
Equation 33 1 Equation 3/. I AK :/.. S o
o
• ....
•
Equation 33[ AK : 1 '5 Equation 3~ J I
I "ca
/
0-3
i
I I
fi /P
o N
U-
/
I
0.2
0.1
"O 5
*
L
I
1
0.1
0.2
03
0.4
Non- dimensionol Fluid VeLocity o;" ShOL[ CR2 j'
FIG. 5(a). Effect of loss coefficient on convergence.
/
A. WHITFIELD et al.
494
Imaginary o
~utions
o Equation 3 3 [
H
Equo.tion 36 I caR'= 0 9
o----o ° - - ---.,
Equation '3'] l Equation 3t, j ~R' = 0 6 I Imaginary Sotutions
- ,.,,, 0 3
,._,=
I I ,/0~
~u
Y=
.~_
/,,'° 7 ,.~ 0.2 .-'e
,o'/
. / ~
5
04 o
i
g z
0
±
/
01
0'2
0"3
0"/,
Non-dimenstonat Fluid Velocity a}" ~olt CR2 s'
FIG. 5(b). Effect of varying ~ok on c o n v e r g e n c e A K = 4.5.
O
Equation
o....
o
Equation 3 3 ]
°....
•
Equation 3l,
D
33},] Ro = 0 1 6 m Equation 3&
J
a
R o =019m
-,-,. 0 t~
o.,
[magi nory Solu}'ions o
,~0
Y
l Imaginary
~olution~/
-'e >-,
I
=
J ,,~',,'°
0.1
,o
J/
,S ? ~ ~ i
0
o
011
i
i
02
03
Non-dimens.ior~l Fluid Velocity at Sl'olt £R2 sr FIG. 5(c). Effect of varying outer radius Ro on convergence. A K = 4-5, o~k = 0.8.
Design of three element hydrokinetic torque converters
495
when compared with the uncertainties associated with a one-dimensional mean streamlines design procedure. In order to assess these uncertainties the impeller inlet angle was assumed to be - 1 5 °, as given by Jandasek, and all other angles calculated for a number of assumed loss coefficients, AK, and design speed ratios D'; the result is given in Table 2. The calculated blade angles are in close agreement with the actual angles given by Jandasek when a loss coefficient of 3.5 is used at a design speed ratio of 0.7. From Table 2 it can be seen that the calculated impeller exit angle could be forced to a value of 15°, as specified by Jandasek, by increasing the loss coefficient to a value between 3.5 and 4.5. If this were done all the other calculated angles would then be brought into even closer agreement with those specified by Jandasek. 3.2 Application of the design procedure In Ref. [1] Jandasek presents experimental data, obtained over 15 years of experimental development, to be used as design data, to meet any specified requirements. This data and the procedure developed here have been applied to the design of a torque converter with a 3-1 stall torque ratio and an input stall torqtle of 270 Nm (200 Ibft) at a rotational speed of 1480 revs/min, which is the stall torque and speed of the basic Jandasek design, and the resultant designs compared. The Jandasek design employs the blade angles of the basic design, Table 1, with the impeller exit angle modified to - 5 5 ° in order to give the 3-1 stall torque ratio. With this design the non-dimensional stall torque is 0.042, and for an input torque of 270 Nm the outer radius of the torque converter is given by
Ro5 = ~'/(0.042. p. to2p), leading to an outer radius of 198 ram. For the design procedure developed here the parameters used in the design assessment were initially used, i.e. AK = 3.5 and (oR,= 0.7. With an assumed impeller inlet angle of - 1 5 ° the outer radius of the torque converter was progressively increased from 152 mm until a converter size was reached which was
Table 2. Effect of loss coefficient and speed ratio on designed blade angles IMPELLER EXIT ANGLE
ROTOR INLET ANGLE
ROTOR EXIT ANGLE
B2~
B2T
46 •88
69.69
-66.79
24.34
63.13
-60.72
12.49
57.54
-16.25
83T
STATOR INLET ANGLE B3s
STATOR EXIT ANGLE
DES I GN SPEED RATIO
LOS S COEFFICIENT
~is
J~
12.67
72.35
0.75
3.95
67.21
0.7
!
4.5
-56.19
8.64
64.36
0.7
3.5
50.08
-47.17
-16.84
52.1
0.5
i i !
-I0.23
55.82
-52.51
-23.79
55.56
0.5
1
4.5
-4.13
59.91
-56.71
-30.3
58.15
0.5
]
5.5
15
58
-60
68
AK 4.5
3.5
ANGLES FOR THE FANDASEK TORQUE ?ONVERTER
Impeller inlet angle -15
Table 3. Blade angle design for 3 : 1 stall torque ratio converter with Ft' = 0.7 IM~aELLER INLET ANGLE
-15
-5
-20
IMPELLER EXIT ANGLE
ROTOR INLET ANGLE
B2~
B2T
ROTOR EXIT ANGLE
B3T
STATOR INLET ANGLE B3s,
STATOR EXIT ANGLE
Bls
OUTER RADIUS
E0
7.97
77.47
-82.96
-72.8
81.49
192
-40.83
71.68
-81.21
-613.38
80.3
199
-60.41
66.38
-80.86
-61.37
80.54
210
12.75
77.7
-82.8
-71.85
81.71
192
-37.91
72.51
-80.96
-63.45
80.59
199
-59.2
67.13
-80.6
-58.8
80.81
210
5.37
77.34
-83.04
-73.27
81.37
192
-42.28
71.42
-81.33
-66.3
80.14
199
-61.O2
65.96
-81
-62.58
80.39
210
~
A. WHITFIELD et al.
496
Table 4. Blade an le design for IMPELLER INLET ANGLE
Blp -15
-5
ROTOR INLET ANGLE
IMPELLER EXIT ANGLE
B2T
g2p
: 1 stall torque ratio converter with fF : 0.6 ROTOR EXIT ANGLE
STATOR INLET ANGLE
fl3s
83T
STATOR EXIT ANGLE
g[s
OUTER RADIUS R m~m o
-48.79
48.4
-69.58
-46.14
68.02
-55.26
42.46
-69.13
-42.99
68.59
[92
-60.5
35.95
-68.93
-39.67
69.52
i99
185
-46.52
50.52
-68.25
-40.69
69.38
185
-53.58
45.04
-67.73
-36.93
69.89
192
-59.27
39.O2
-64.51
-32.97
70.72
199
sufficient to meet the torque capacity requirements. Table 3 presents the calculated blade angles for three outer radii Re. In addition the results for alternative impeller inlet angles are presented; an inlet angle of - 5 ° was selected based upon the curve of Fig. 4, and the angle of -200 is presented to provide further comparison. The high turbine and stator exit angles shown in Table 3 are not considered desirable as any blade blockage effect may become significant; blade thickness has generally been ignored in both the design and prediction procedure. Attempts were therefore made to reduce these angles. As can be seen from Table 3 increasing the outer radius leads to a reduction in all angles, but the smallest changes occur with the largest angles, and the starer exit angle shows a minimum value after which it increases again as the radius increases. From Table 2 it can be seen that reducing the design point speed ratio leads to a reduction in the designed blade angles. The design exercise was therefore repeated for a specified speed ratio of 0.6. Table 4 presents the results for two impeller inlet blade angles. It can be seen that the blade angles have been substantially reduced, and that with an outer radius of 0.192 m and an impeller inlet angle of - 15° both the impeller and stator exit angles are approximately the
o
o Designed using Jandosek Procedure
D----o
Designed using Present Procedure with uoR' = 0 7 end ~Sip = -15
x----~x
Designed using Present" Procedure wiih ~ R '= 0 6 end )Sip : - 1 5
2,8
24
2.C
18
°
l
1'6,
,\
12
1'C
~
011
0'.2
03
0~,
0'$
0.6
0'7
0'-8
0.9
10
Speed I~'io
FIG. 6(a). Predicted torque characteristic for proposed 3 : 1 stall torque radio design.
Design of three element hydrokinetic torque converters o
,. o
Designed
using
Jondosek
~
m -- --~
Designed
using
Present
x-----~x
497
Procedure Procedure
with UJR'= 0 7 and j61p=-15 Designed using Present Procedure with ("JR: 06 (and /Sip:-15
006
tT" 0.06
o
i
0-02
g
z
.
0.1
.
0.2
.
.
03
.
.
.
.
.
0t,
05 06 07 08 09 10 £pe~ Rotio FIG. 6(b). Predicted torque capacity characteristic for proposed 3 : 1 stall torque ratio design.
same as those specified by Janasek. Also the impeller inlet angle of - 1 5 ° and stator exit angle of 68° satisfies the maximum flow rate requirement of equation (57), Fig. 4. The predicted performance, obtained with the forced vortex procedure of Ref. [7], for the three designs with an impeller inlet angle of - 150 and an outer radius of 0.199 m, i.e. (a) the Jandasek design, (b) the blade design with II' = 0.7, and (c) the blade design with 1)' = 0.6, are compared in Fig. 6(a, b). From this it can be seen that it is predicted that none of the designs will achieve the required 3 : 1 stall torque ratio. As the torque converter efficiency is given by the product of torque ratio and speed ratio it can be seen that the Jandasek design shows the lowest predicted efficiency, a peak of 90% compared to 96% for the blade design with f~' = 0.7. For the blade design with ~ ' = 0.6 a stall torque ratio greater than 2.7 is predicted and clearly further modifications are needed before the proposed 3 : I stall torque can be achieved. The capacity curves of Fig. 6(b) shows that the stall torque capacity requirement of 0.042 is approximately achieved, although the Jandasek design is a little high. The major difference between the torque capacity results lies in the continuously falling characteristic achieved for the blade design with f l ' = 0.7. The rising characteristics of the other two designs is due to the high backward swept impeller exit blade, - 55o.
4. CONCLUSIONS A detailed theoretical design procedure for three element torque converters has been developed to complement an extensive prediction procedure presented in Ref. [7]. It has been shown that the theoretical procedure leads to a torque converter design similar to that developed experimentally by Jandasek. Whilst the work presented here has been confined to the simple three element torque converter it provides the basis for development to multi-ilement designs. The mean streamline prediction procedure has been extended to accommodate any multielement design and is currently being tested. A design procedure for multi-element torque converters has been formulated but has yet to be developed into a computer program. REFERENCES 1. V. J. JANDASEK, Design of a single stage three element torque converter. Passenger Car Automatic Transmission, S A E Transm. Workshop Meeting, 2nd Edn, Vol. 5, p. 201 (1963). 2. R. EKSER61AN,The fluid torque converter and coupling. Z Franklin Inst. 235(5), 441 (1943). 3. J. W. QUALMAN and E. L. EeBERT, Fluid couplings. Passenger Car Automatic Transmission, SAE. Transm. Workshop Meeting, 2nd Edn, Vol. 5, p. 137, 1963. 4. G. G. LUCASand A. RAYNER,Torque converter design calculations. Automobile Engng 56 (1970). 5. J. D. STANEZ, Design considerations for mixed flow centrifugal compressors with high flow rates per unit frontal area. N A C A R M E 53 AI5 0953). 6. F. J. WALLCE, A. WrUTFIELD and S1VALINGAM,A theoretical model for the performance prediction of fully filled fluid couplings. Int. Z Mech. Sci. 20, 335 (1978). 7. A. WHITF1ELD, F. J. WALLACE a n d SIVALINGAM, A performance prediction procedure for three element torque converters. Int. Z Mech. Sci. (1978).