Predicting the Onset of Cavitation in Automotive Torque

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2008, Article ID 312753, 8 pagesdoi:10.1155/2008/312753

Research ArticlePredicting the Onset of Cavitation in Automotive TorqueConverters—Part II: A Generalized Model

D. L. Robinette,1 J. M. Schweitzer,1 D. G. Maddock,1 C. L. Anderson,2 J. R. Blough,2 and M. A. Johnson2

1 General Motors Powertrain Group, General Motors Corporation, Pontiac, MI 48340, USA2 Department of Mechanical Engineering—Engineering Mechanics, Michigan Technological University, Houghton, MI 49931, USA

Correspondence should be addressed to D. L. Robinette, [email protected]

Received 10 March 2008; Accepted 9 June 2008

Recommended by Yoshinobu Tsujimoto

The objective of this investigation was to develop a dimensionless model for predicting the onset of cavitation in torque convertersapplicable to general converter designs. Dimensional analysis was applied to test results from a matrix of torque converters thatranged from populations comprised of strict geometric similitude to those with more relaxed similarities onto inclusion of all thetorque converters tested. Stator torque thresholds at the onset of cavitation for the stall operating condition were experimentallydetermined with a dynamometer test cell using nearfield acoustical measurements. Cavitation torques, design parameters, andoperating conditions were resolved into a set of dimensionless quantities for use in the development of dimensionless empiricalmodels. A systematic relaxation of the fundamental principle of dimensional analysis, geometric similitude, was undertaken topresent empirical models applicable to torque converter designs of increasingly diverse design parameters. A stepwise linearregression technique coupled with response surface methodology was utilized to produce an empirical model capable of predictingstator torque at the onset of cavitation with less than 7% error for general automotive torque converter designs.

Copyright © 2008 D. L. Robinette et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

Part I of this two-part paper discussed the principles of cav-itation inception in the torque converter at stall, acousticaldetection with a dynamometer test cell, and dimensionalanalysis for the strict case of exact geometric scaling. Thefocus of Part II is on the application of dimensional analysisfor development of empirical models capable of predictingstator torque cavitation thresholds at stall for a wide range oftorque converter designs.

The potential for cavitation exists during stall operation(converter output speed held at zero) for any torqueconverter above some critical torque and charge pressure. Acavitation model could be used to optimize converter designsto achieve higher cavitation torque thresholds at stall andminimize cavitation under normal driving conditions. Theobjective then is to increase stator torque at cavitation asa function of design parameters and flow properties, forexample, pressures and cooling flow temperature, for anygeneral automotive torque converter design.

Although the results of Part I are fundamental tothe principles of dimensional analysis, they provide little

practical use in predicting cavitation thresholds for a diversepopulation of torque converters needed to meet performancerequirements in a wide range of powertrain applications.To date, a comprehensive empirical model, spanning thedesign space of torque converters or other turbomachines,for predicting cavitation thresholds has not been developed.Dimensional analysis using a power product model andusing response surface methodology (RSM) was applied tothe torque converter data to develop models for cavitation.However, it was found that the power product model didnot adequately satisfy the assumption of linear regressionanalysis, namely, normality and constancy of error variancefor model residuals. Instead, RSM was used to statisticallydetermine the functional form of the model (dimensionlessregressors) resulting in a model with better normality andconstancy of error variance as well as increased predictionaccuracy. RSM is a technique that is particularly well suitedfor the development of empirical models in which an optimalsolution is sought. By definition, RSM is a mathematicaland statistical approach to modeling and analysis of datato which the response of interest is influenced by severalvariables and the objective is to optimize the response

mailto:[email protected]

2 International Journal of Rotating Machinery

[1]. Application of RSM and regression techniques haspreviously been implemented in the design optimizationof a compressor impeller [2] and turbine blades [3–5],diffusers [1], scroll fans [6], and torpedoes [7] for maximumcomponent performance.

The formulation of a generalized torque converter cav-itation prediction model requires knowledge of operatingthresholds at cavitation. For this investigation, these wereobtained by relaxing geometric similitude and testing amatrix of converters designs on a dynamometer setupfor determination of Ts,i, stator torque at the onset ofcavitation. Even with geometric similitude relaxed, similaritywas preserved by maintaining consistent design philosophiesfor each element (i.e., pump, stator, or turbine) regardless oftorus geometry. Blades for turbines and pumps are constantthickness sheet metal designs defined by inlet and outletangles and with a ruled surface from hub to shroud. Statorblades are defined by inlet and outlet angles, and also byairfoil shapes that are similar between different angle designs.The stall hydrodynamic behavior of any one design tested isalso a source of similarity. Like all turbomachines, changes tothe geometry of the torque converter only modify its uniqueperformance metrics, and not its overall function. Everyconverter tested exhibits a characteristic K -factor (see Part I)and stall torque ratio, the ratio of turbine to pump torque, asa function of geometry changes to one or multiple elements.Previous experimental work by [8] found that the onset ofcavitation consistently correlated with the torus and elementgeometries of the torque converter. Given these similarities,it is expected that the use of dimensional analysis will notbe hindered due to relaxed geometric similitude among thetested parts.

Nondimensional terms were determined from theimportant variables that describe cavitation within a torqueconverter at stall and were used as inputs into a RSM to forma generalized model. The resulting model is dimensionallyhomogenous and contains dimensionless quantities thatrepresent meaningful relationships among the initial list offundamental variables. A similar approach of combiningRSM and dimensional analysis was demonstrated by [9]for the optimization of stresses and buckling loads of anisotropic plate with an abrupt change in thickness.

2. GENERAL CONSIDERATIONS

2.1. Element design modifications and cavitation

For any combination of internal combustion engine andautomatic transmission, the performance of the torqueconverter must be optimized over all possible speed ratios forefficiency, torque ratio, tractive effort, vehicle acceleration,and fuel consumption. Torus dimensions and shape are usu-ally determined first by powertrain packaging constraints, K-factor performance targets, and consideration for rotatinginertia. Once the dimensions of the torus are selected,element designs are explored to refine performance over allpossible speed ratios. For a given set of torus dimensions,a single turbine design is developed for use with anycombination of pumps or stators to be developed. The

stamped sheet metal blades of the pump lend themselvesto adjustments in inlet and outlet angles, with blade countfixed for a given diameter. The stator can be modified bychanging blade angles, both inlet and outlet simultaneous,and the airfoil shape of the blade. Any change in converterdiameter will scale K-factor by the ratio of the diametersto the 2.5 power, while axial length has a marginal effect.Increasing pump outlet angle will lower K-factor and torqueratio at stall, while improving efficiency at high speed ratio.Stator outlet angles of increased magnitude produce higherK-factors and lower torque ratios at stall but improve peakefficiency at high speed ratio. The airfoil shape of the statorblade influences stall torque ratio and K-factor at high-speedratio. These design parameters that are crucial to tuningtorque converter performance are also influential to cavita-tion characteristics and were evaluated in the dimensionalanalysis. Torque converter designs, with increased diameterand axial length that contain pumps and stators of lowmagnitude blade angles, will result in increased cavitationthresholds as described in [8]. The reader is referred to [8]for a more thorough discussion on the influence of designparameters on performance and cavitation.

2.2. Parameters for dimensional analysis

Stator torque at the onset of cavitation for any torqueconverter design is a function of the specific design and oper-ating point parameters that determine the boundary valueproblem of the toroidal flow. The stator torque cavitationthreshold, Ts,i, for any torque converter design is primarilya function of torus dimensions D and Lt, and element bladedesigns comprising of blade angles for pump, turbine, andstator, and blade airfoil shape for the stator. Element designparameters manifest themselves as the integrated quantitiesof K-factor (see Part I), K , and the ratio of turbine topump torque, known as torque ratio, TR. Both quantitiesat stall correlate to cavitation behavior in the converter, see[8], and their simplicity is preferred over a lengthy anddetailed list of individual blade design parameters for eachelement. However, the number of blades and airfoil shape ofthe stator can strongly influence performance and cavitationbehavior and was also included. Airfoil shape is specifiedby the maximum blade thickness, tmax, and chord length,lc. Stator blade count and shape have been shown to affectTs,i as described by [8]. Pressures superimposed upon thetorque converter can alter Ts,i by as much as 150 Nm asdetermined by [8] and are considered primary variables. Asin Part I, average pressure, pave, and pressure drop, Δp, aremore suitable to the dimensional analysis than charge andback pressures.

Secondary variables that influence Ts,i include temper-ature dependent fluid properties and cooling through flowrate, Q. Since cooling through flow rate is determined by thecharge and back pressures imposed upon the converter, it isnot explicitly included in the variable list. The temperaturedependent properties of density, ρ, viscosity, μ, specific heat,Cp, and thermal conductivity, k, were all taken at the inputtemperature of the cooling flow supplied to the torqueconverter. Equation (1) gives a functional list of the primary

D. L. Robinette et al. 3

and secondary variables that determine Ts,i in an automotivetorque converter:

Ts,i = f(D,Lt,K , TR,nsb, tmax, lc, pave,Δp, ρ,μ,Cp, k

).

(1)

The dimensionless form of (1) was resolved into dimen-sionless stator torque as a function of five dimensionlessdesign parameters and two dimensionless operating pointparameters using the repeating variables of D, ρ, pave, andCp:

Ts,iD3pave

= f(LtD

,U , TR,nsb,tmax

lc,Δp

pave, Pr

). (2)

The dimensionless design parameters include torus aspectratio, Lt/D, unit input speed (see Part I), U, stall torqueratio, TR, number of stator blades, nsb, and stator bladethickness ratio, tmax/lc. The dimensionless operating pointparameters are dimensionless operating pressure, Δp/pave,and the Prandtl number, Pr. These seven dimensionlessparameters were used as dimensionless regressors to fitempirical model(s) to dimensionless stator torque.

Not all of the dimensionless design parameters of (2)are necessary when considering a particular population oftorque converters. Those dimensionless design parametersthat remain constant are removed from (2) before fitting amodel to the data. This was the case in Part I, where thedimensions of both torque converters were exactly scaled bythe ratio of their diameters. This resulted in U and TR beingconstant for each diameter along with the design parametersof nsb, and tmax/lc. These dimensionless regressors weresubsequently removed from the functional list of (2) beforeproceeding to fit a model to the data.

2.3. Response surface methodology

Response surface methodology (RSM) and a stepwise regres-sion procedure were used to model the relationship ofdimensionless stator torque with the dimensionless designand operating point parameters of (2). The general formof the second-order response surface (RS) assumed for thisinvestigation is given by (3), which includes linear, quadratic,and two-factor interactions:

y = b0 +k∑

i=1

bixi +k∑

i=1

biix2i +

k−1∑

i=1

k∑

j=i+1

bi jxix j + ε. (3)

The response, y, is dimensionless stator torque at cavitationand the linear, quadratic, and two-factor interactions, xi,x2i , and xixj , respectively, are the dimensionless regressors

of (2). The estimated regression coefficients, b0, bi, bii,and bi j were determined using least squares method. Theterm, ε, is the residual or error of the model, the differencebetween the experimentally measured and predicted value ofdimensionless stator torque.

A stepwise regression technique was utilized to producea RS model with the minimum number of dimensionlessregressors that are statistically significant with 95% confi-dence. Equation (4) was the initial form of the RS model

assumed, from which dimensionless regressors are sequen-tially added based upon meeting the statistical criterion of a95% joint Bonferroni confidence interval and t-test; see [10]:

y = b0 + ε. (4)

During the stepwise regression process, regressors can beadded or removed from the model as dictated by thestatistical criterion. The overall reduction in error realized byadding any one regressor is dependent upon the regressorsalready in the model. For a more detailed description of RSMand the stepwise linear regression procedure, the reader isreferred to [10, 11].

The accuracy and goodness of fit for any RS modeldeveloped were checked by computing the root meansquare error (RMSE) and determining the linear associationbetween the dimensionless response and regressors. RMSEis an estimator of a model’s standard deviation and forthis investigation was computed as a percentage, denotedas %RMSE. Equation (5) defines %RMSE, where yi is themeasured response, n is the number of data points, and p isthe number of dimensionless regressors in the model:

%RMSE =√√√√∑n

i=1

(((yi − yi

)/yi)∗100

)2

n− p. (5)

A measure of the proportionate amount of variation inthe response explained by a particular set of regressors in themodel is the adjusted coefficient of multiple determinationR2a:

R2a = 1−

∑ni=1

(yi − yi

)2/n− p

∑ni=1

(yi − y

)2/n− 1

. (6)

This metric calculates to a value from 0 to 1. Results above0.85 are generally needed to signify an accurate model ofthe data set. R2

a is typically preferred over R2, the coefficientof multiple determination, as it is a better evaluator of thenumber of regressors included in the model. As the numberof regressors increases, R2 will always increase, whereas R2

a

may increase or decrease depending on whether or not theadditional explanatory regressors actually reduce variation inthe response. Computationally, the difference between thesemetrics is that the factors of n − p and n − 1 are removedfrom the numerator and denominator of (6), respectively, tocompute R2.

3. EXPERIMENTAL SETUP AND TEST PROCEDURE

3.1. Dynamometer test cell

The torque converter dynamometer test cell and test proce-dure used to acquire stator torque cavitation thresholds atstall using nearfield acoustical measurements was describedin Part I. A 40 RPM/s stall speed sweep procedure wasagain utilized to transition all torque converters from anoncavitating to a cavitating condition. Fourteen operatingpoints were applied to each design to characterize their effecton Ts,i. Table 1 lists the ranges of the three operating points


Table 1: Range of stall operating point parameters tested.

Operating point Low High

Average pressure 481 kPa 963 kPa

Pressure drop 69 kPa 345 kPa

Input temperature 60◦C 80◦C

varied in this investigation. The signal processing techniqueutilized to determine the exact threshold of cavitation wasbriefly discussed in Part I and further detailed by [8].

3.2. Matrix of experimental torque converters

Four major populations of torque converters were tested withprogressively decreasing levels of geometric similitude, asshown in Figure 1 from top to bottom. The first population,referred to as geometric scaling, contains the same torqueconverters as presented in Part I. The converters maintainexact geometric similitude between diameters. This popula-tion served as a baseline comparison for RS models in whichgeometric similitude was relaxed. The second population,multiple torus geometry, includes three torque converters ofsimilar element design, but varying torus dimensions (D andLt) to create converters with the same unit input speed, U.The next population consists of multiple pump and statorblade designs, with torus dimensions and turbine design heldconstant, for 16 total configurations that achieve a rangein K-factor performance. The final population includes allconverters tested for a total of 51 widely varying designs thatencompass a typical design space for powertrain componentmatching. Within this population, referred to as generaltorque converter design, geometric similitude was greatlyrelaxed, allowing the development of a comprehensive RSmodel capable of predicting Ts,i. Because of the loosenedrestrictions on similitude, the resulting model is applicableto general torque converter designs. The details for eachpopulation are provided with the discussion of each RSmodel. The six diameters tested in this investigation willbe denoted D1 through D6, with D1 corresponding to thesmallest diameter and D6 to the largest diameter.

4. RESULTS AND DISCUSSION

The RS models developed using the RSM and stepwiseregression techniques are presented in the order depictedin Figure 1, from strictest geometric similitude to themost relaxed. The full RS model equations for all of thepopulations are listed in Table 4, at the end of the paper. It isexpected that the RS models presented will be integrated intoinitial converter design and powertrain component matchingto maximize Ts,i within a given set of packaging, perfor-mance, and stall operating point specifications. Predictionof Ts,i during this phase will enable optimization of torusand/or element geometries along with operating pressuresand temperature to minimize the effects of cavitation relatednoise.

Geometric scaling

Multiple torusdesign

Multiple pumpand stator design

General torqueconverter design

All automotivetorque

converters

Dec

reas

ing

geom

etri

csi

mili

tude

Diameters:D2

D6

Diameters:D2

D3

D5

Diameter:D3

K-factors:125 to 175 K

Diameters:D1 to D6

K-factors:90 to 200 K

Figure 1: Organizational chart for torque converter populationstested; range of converters tested is denoted by diameters and K-factors.

0.008

0.009

0.01

0.011

0.012

Mea

sure

ddi

men

sion

less

stat

orto

rqu

e

0.008 0.009 0.01 0.011 0.012

Predicted dimensionless stator torque

Geometric scaling

%RMSE = 3.44% R2a = 0.566

D2

D6

pave

Δp

Pr

Low481 KPa69 KPa

116

High963 KPa345 KPa

300

Figure 2: Geometric scaling RS model for predicting stator torqueat onset of cavitation (28 points).

4.1. Geometric scaling

The cavitation threshold data used to fit the RS model wasthe same as that used to fit the power product (PP) modelof Part I. Equation (7) gives the functional form of the RSmodel for this population, and the RS model versus test datagraph is shown in Figure 2:

Ts,iD3pave

= f(Δp

pave, Pr

). (7)

The model diagnostics (%RMSE and R2a) provided in

Table 2 show the RS model to have a 23% increase in R2a


and a 9% decrease in %RMSE over the corresponding PPmodel for the same number of dimensionless regressors.Although both models have %RMSE values below 4%, thesmall R2

a values suggest that both models have a low linearassociation between dimensionless response and regressors.As with the PP model of Part I, a multitude of least squaressolutions exists for the geometric scaling data set. Use ofdimensionless stator torque results in a nearly constant termfor the onset of cavitation over the range of operating pointstested. The small variation in dimensionless quantities doesnot particularly lend itself to empirical modeling, requiringincreased variation in dimensionless quantities to adequatelydefine a curve. However, even with relatively low values of R2

a,the prediction accuracy for scaling stator torque cavitationthresholds for a given torus and set of element designs is high.

4.2. Multiple torus design

For the multiple torus design population, element bladedesigns were held constant, while the dimensions of thetorus were varied for three ratios of Lt/D. Performances forthe 3 designs result in equivalent unit input speeds, U. Thefunctional form for this RS model is given by (8) and includestorus aspect ratio and number of stator blades in addition tothe terms included in (7):

Ts,iD3pave

= f(LtD

,nsb,Δp

pave, Pr

). (8)

The addition of the dimensionless design parameters wasrequired as torus dimensions changed along with somemodifications to the stator. The number of stator bladesis different between the 3 converters due to differencesin axial length, but blade angles are the same. Figure 3contains the RS model, showing an increase in the rangeof dimensionless response and regressors compared to thegeometric scaling RS model shown in Figure 2. A decreasein model %RMSE from 3.40% to 2.87% and an increasein R2

a from 0.566 to 0.925 were realized compared tothe geometric scaling model. The multiple torus designRS model includes five regressors, consisting primarily oftwo factor interactions between dimensionless design andoperating point parameters. The utility of this particularmodel allows prediction of Ts,i for a single set of elementdesigns spanning the range of torus dimensions provided inFigure 3.

4.3. Multiple pump and stator design

The dimensionless functional form of the RS model for thepopulation with multiple pump and stator designs is givenby

Ts,iD3pave

= f(U , TR,nsb,

tmax

lc,Δp

pave, Pr

). (9)

The list of dimensionless regressors includes those thatcharacterize changes to the pump and stator and stalloperating point. The converters in this population all have

0.008

0.01

0.012

0.014

0.016

Mea

sure

ddi

men

sion

less

stat

orto

rqu

e

0.008 0.01 0.012 0.014 0.016


Multiple torus design

% RMSE = 2.87 % R2a = 0.926

D2

D3

D5

Lt/Dnsb

Low0.2919

High0.317

21

Figure 3: Multiple torus design RS model for predicting statortorque at onset of cavitation (42 points).

0

5

10

15

20

25

30R

MSE

(%)

b0

0

0.2

0.4

0.6

0.8

1

R2 a

Regressors in the model

(tmax

lc

)2

nsb

(tmax

lc

)

U(tmax

lc

)(Δp

pave

)2

U (TR)

n2sb TR2

Pr2 U2

Figure 4: Sequential reduction in %RMSE and increase in R2a

during stepwise regression procedure for multiple pump and statordesign RS model.

Table 2: Comparison of geometric scaling models.

Model type Data points Regressors R2a %RMSE

RS 28 2 0.566 3.340

PP 28 2 0.434 3.733

the same torus design, so the Lt/D term included in (8) dropsout of (9).

Figure 4 summarizes results from the stepwise regressionprocedure used to determine statistically significant regres-sors for the RS model. The figure shows the reduction inmodel error and an increase in linear association betweenresponse and regressors as regressors are added to the model.Only regressors that have a statistically significant effect areretained in the final equation. In this way, the number of


0.004

0.008

0.012

0.016

0.02M

easu

red

dim

ensi

onle

ssst

ator

torq

ue

0.004 0.008 0.012 0.016 0.02


Multiple pump and stator design

%RMSE = 5.16% R2a = 0.952

Pump 1Pump 2Pump 3

Pump 4Pump 5Pump 6

Lt/DK(U)

TRnsb

tmax/lc

Low

126 (123)1.7613

0.158

High

174 (170)2.2122

0.224

0.3

Figure 5: RS model for multiple pump and stator design popu-lation, data is for 6 pump and 11 stator designs with fixed torusgeometry.

0

100

200

300

400

500

600

Stat

orto

rqu

eat

onse

tof

cavi

tati

on,T

s,i(N

m)

1000 1500 2000 2500

Pump speed at onset of cavitation, Np,i (RPM)

General torque converter design

D1

D2

D3

D4

D5

D6

Figure 6: Stator torque versus pump speed at onset of cavitationdata for 51 widely varying torque converter designs (714 points).

regressors was reduced to 10, from a possible 27, includingthe constant term, b0. It can be noted in Figure 4 thatthe addition of the first three regressors, n2

sb, TR2, andthe interaction U∗TR accounts for 88% of the variationin dimensionless stator torque and reduces %RMSE from27.6% to 7.78%. Building the RS model by sequentiallyadding dimensionless regressors other than in the ordershown in Figure 4 would produce different characteristicvalues of %RMSE and R2

a. During the stepwise regressionprocess, models %RMSE and R2

a are dependent upon thestatistical significance of the individual regressor to be added

0.005

0.01

0.015

0.02

0.025

Mea

sure

ddi

men

sion

less

stat

orto

rqu

e

0.005 0.01 0.015 0.02 0.025



%RMSE = 6.52% R2a = 0.936

D1

D2

D3

D4

D5

D6

Lt/D

K(U)TRnsb

tmax/lc

Low0.208

93 (117)1.6813

0.158

High0.321

203 (182)2.4835

0.413

Figure 7: RS model for prediction of stator torque at onset ofcavitation for general torque converter design.

0

20

40

60

80

100

120

Nu

mbe

rof

resi

dual

s

−0.004 −0.002 0 0.002 0.004

Value of residual (error)

Figure 8: Histogram of residuals for general torque converterdesign population RS model.

and those regressors already present in the model. Forthis investigation, regressors were added to the RS modelaccording to their influence on %RMSE and R2

a. Thoseregressors, that were deemed statistically significant andgenerated the highest reduction in %RMSE and increase inR2a, were added first. It was found that the regressors based

on design parameters produced more statistically significantcontributions to reducing model %RSME and increasing R2

a

than those based on fluid properties, even though cavitationis a fluid dependent phenomenon influenced by pressuresand temperatures.


Table 3: Comparison of RS model performance by population.

Population Geometric similitude Data points Regressors R2a %RMSE

Geometric scaling Exact 28 3 0.566 3.34

Multiple torus design Elements only 42 6 0.926 2.87

Multiple pump and stator design Torus only 224 10 0.952 5.16

General torque converter design None 714 19 0.936 6.52

Table 4: Summary of RS models by torque converter population excluding regression coefficients.

Population Response surface model

Geometric scalingTs,i

D3pave= b0 + b1

(Δp

pave

)2

+ b2(Pr)2.

Multiple torus designTs,i

D3pave= b0 + b1

(Δp

pave

)+ b2

(LtD

)2

+ b3

(LtD

)(Δp

pave

)+ b4

(LtD

)(Pr) + b5(nsb)

(Δp

pave

).

Multiple pump and stator design

Ts,iD3pave

= b0 + b1

(tmax

lc

)+ b2(U)2 + b3(TR)2 + b4(nsb)2 + b5

(Δp

pave

)2

+b6(Pr)2 + b7(U)(TR) + b8(U)(tmax

lc

)+ b9(nsb)

(tmax

lc

).


Ts,iD3pave

= b0 + b1

(LtD

)+ b2(U) + b3(TR) + b4(nsb) + b5

(tmax

lc

)+ b6

(LtD

)2

+ b7(nsb)2 + b8

(tmax

lc

)2

+b9

(Δp

pave

)2

+ b10(Pr)2 + b11

(LtD

)(nsb) + b12

(LtD

)(tmax

lc

)+ b13(U)(TR) + b14(U)(nsb)

+b15(U)(tmax

lc

)+ b16(TR)(nsb) + b17(TR)

(tmax

lc

)+ b18(nsb)

(tmax

lc

).

The RS model for the multiple pump and stator designpopulation resulting from the stepwise regression is providedin Figure 5. For the 16 different pump and stator combina-tions included, the RS model has a %RMSE of 5.16% andR2a of 0.952. Compared with the multiple torus geometry

RS model, %RMSE increased by 2.29%. This increase isexpected as more data points are included, introducing morevariation and departing further from geometric similitude.The predicted value of dimensionless stator torque fromthis RS model can be scaled to other diameters if exactdimensional scaling is observed, and design modifications topump and stator fall within the scope of the model.

4.4. Generalized torque converter design

A population of converters with variations in both torusand element geometries was utilized to develop a generalizedRS model for predicting Ts,i at stall. The experimental data,shown in Figure 6 as Ts,i versus Np,i, pump speed at onset ofcavitation, spans a broad range of torques and speeds for thematrix of converters and stall operating points tested.

All of the dimensionless regressors from (2) were usedfor developing the generalized RS model, as all varied for thematrix of designs tested. The stepwise regression procedureresulted in a RS model with 18 dimensionless regressorsand the constant term, b0. Dimensionless design parametersdominated the reduction in %RMSE and an increase in

R2a, most notably U, TR, and their interaction. These two

regressors along with the constant term reduce %RMSE from27.76%, for the initial model (4), to 10.36%, and accountfor 81% of the variation in the dimensionless response. Thegeneralized torque converter RS model for predicting Ts,i isshown in Figure 7, differentiated by diameter. The range ofdesign parameters tested are provided, showing the range oftorus, stator design specifics, and overall stall performancemetrics as determined by element geometries.

The generalized RS model’s %RMSE of 6.52% and R2a of

0.936 demonstrates that even with greatly relaxed geometricsimilitude, an accurate prediction model was developed.Figure 8 is a histogram of the model residuals; the error termdefined in (3). It shows that no significant deviations fromnormalcy and roughly follow that of a normal distribution.

Table 3 compares each RS model developed from apopulation of torque converters with complete geometricsimilitude, to populations with diminishing restrictions ongeometric similitude. It can be noted that as geometricsimilitude is relaxed, the number of dimensionless regressorsrequired to realize a highly accurate RS model increases. Thisis expected as the numerous interactions between designparameters become increasingly important in determiningTs,i at stall. The increase in %RMSE occurring sequentiallyfrom the geometric scaling to the general torque converterdesign RS model is not substantial enough to consider RSmodels lacking geometric similitude to be inaccurate.


Table 4 summarizes the dimensionless regressors thatcomprise each RS model presented in this paper. In general,the dimensionless regressors found to be most influential inreduction of RS model error reduction are those most closelyrelated to the design and performance characteristics of theturbomachine rather than the dimensionless operating pointparameters. Depending on the data set being modeled, thedimensionless design parameter regressors that result in thelargest error reduction varied. In the case of the generalizedmodel, the linear terms of unit input speed, U, torque ratio,TR, and their interaction, U∗TR, were most dominant.Other dimensionless quantities such as torus aspect ratio,number of stator blades, stator blade thickness ratio, anddimensionless operating point serve to explain additionalbut minor variations in the dimensionless response. Theregression coefficients contained in each RS model weredetermined from proprietary geometries and are withheldfrom Table 4.

5. CONCLUSIONS

Experimentally obtained values of stator torque at the onsetof cavitation for torque converters with varying degreesof geometric similitude were nondimensionalized and usedas dimensionless response and regressors for developingresponse surface (RS) models. A stepwise linear regressionprocedure reduced the complexity of each RS model by onlyincluding statistically significant dimensionless regressors.RS models created from data sets of decreasing geometricsimilitude showed that R2

a values above 0.85 are achievedeven with greatly relaxed geometric similitude. The %RMSEvalues resulting for each RS model of decreasing geo-metric similitude were not substantial enough to indicateinadequacies in the dimensional analysis or data modelingmethodology. A RS model was presented which is capableof predicting Ts,i for general automotive torque converterdesign with a %RMSE of 6.52%. This error is deemedsufficiently low and its scope of prediction large enough torate the RS model; a valuable tool for optimizing torus andelement geometries with respect to the onset of cavitation atstall in three element automotive torque converters.

NOMENCLATURE

Cp: Specific heat (kJ/kg·K)D: Diameter (m)K: K-factor (RPM/Nm0.5)Lt : Axial length (m)Np,i: Pump speed at onset of cavitation (RPM)Pr: Prandtl numberQ: Volumetric flow rate (m3/s)R2a: Adjusted coefficient of multiple determination

Ts,i: Stator torque at onset of cavitation (Nm)TR: Stall torque ratioU : Unit input speed%RMSE: Root mean square error (%)b: Regression coefficientsk: Thermal conductivity (W/m·K)lc: Chord length (m)

nsb: Number of stator bladespave: Average pressure (Pa)Δp: Pressure drop (Pa)tmax: Maximum blade thickness (m)xi: Dimensionless regressorsy: Predicted dimensionless responseε: Residual errorρ: Density (kg/m3)μ: Viscosity (N·s/m2).θin: Input temperature (◦C)

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Predicting the Onset of Cavitation in Automotive Torque