# Predicting the Onset of Cavitation in Automotive Torque Converters—Part II: A Generalized Model

Authors: D. L. Robinette; J. M. Schweitzer; D. G. Maddock; C. L. Anderson; J. R. Blough; M. A. Johnson
Journal: International Journal of Rotating Machinery
 (2008)

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## Abstract

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


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## Body

## 1. Introduction

Part I of this two-part
paper discussed the principles of cavitation inception in the torque converter
at stall,  acoustical detection with a dynamometer test cell, and dimensional
analysis for the strict case of exact geometric scaling. The focus of Part II is
on the application of dimensional analysis for development of empirical models
capable of predicting stator torque cavitation thresholds at stall for a wide
range of torque converter designs.The potential for
cavitation exists during stall operation (converter output speed held at zero)
for any torque converter above some critical torque and charge pressure. A
cavitation model could be used to optimize converter designs to achieve higher
cavitation torque thresholds at stall and minimize cavitation under normal
driving conditions. The objective then is to increase stator torque at
cavitation as a function of design parameters and flow properties, for example,
pressures and cooling flow temperature, for any general automotive torque
converter design.Although the
results of Part I are fundamental to the principles of dimensional analysis,
they provide little practical use in predicting cavitation thresholds for a
diverse population of torque converters needed to meet performance requirements
in a wide range of powertrain applications. To date, a comprehensive empirical
model, spanning the design space of torque converters or other turbomachines,
for predicting cavitation thresholds has not been developed. Dimensional
analysis using a power product model and using response surface methodology
(RSM) was applied to the torque converter data to develop models for cavitation.
However, it was found that the power product model did not adequately satisfy
the assumption of linear regression analysis, namely, normality and constancy
of error variance for model residuals. Instead, RSM was used to statistically
determine the functional form of the model (dimensionless regressors) resulting
in a model with better normality and constancy of error variance as well as
increased prediction accuracy. RSM is a technique that is particularly well
suited for the development of empirical models in which an optimal solution is
sought. By definition, RSM is a mathematical and statistical approach to
modeling and analysis of data to which the response of interest is influenced
by several variables and the objective is to optimize the response [1]. Application
of RSM and regression techniques has previously been implemented in the design optimization
of a compressor impeller  [2] and turbine blades [3–5], diffusers [1],
scroll fans [6], and torpedoes [7] for maximum component performance.The formulation of
a generalized torque converter cavitation prediction model requires knowledge
of operating thresholds at cavitation. For this investigation, these were
obtained by relaxing geometric similitude and testing a matrix of converters designs
on a dynamometer setup for determination ofTs,i, stator
torque at the onset of cavitation. Even with geometric similitude relaxed,
similarity was preserved by maintaining consistent design philosophies for each
element (i.e., pump, stator, or turbine) regardless of torus geometry. Blades
for turbines and pumps are constant thickness sheet metal designs defined by
inlet and outlet angles and with a ruled surface from hub to shroud. Stator
blades are defined by inlet and outlet angles, and also by airfoil shapes that
are similar between different angle designs. The stall hydrodynamic behavior of
any one design tested is also a source of similarity. Like all turbomachines,
changes to the geometry of the torque converter only modify its unique
performance metrics, and not its overall function. Every converter tested exhibits
a characteristic K-factor (see Part I) and stall torque ratio,  the ratio of
turbine to pump torque, as a function of geometry changes to one or multiple
elements. Previous experimental work by [8] found that the onset of cavitation
consistently correlated with the torus and element geometries of the torque converter.
Given these similarities, it is expected that the use of dimensional analysis
will not be hindered due to relaxed geometric similitude among the tested parts.Nondimensional
terms were determined from the important variables that describe cavitation
within a torque converter at stall and were used as inputs into a RSM to form a
generalized model. The resulting model is dimensionally homogenous and contains
dimensionless quantities that represent meaningful relationships among the
initial list of fundamental variables. A
similar approach of combining RSM and dimensional analysis was demonstrated by
[9] for the optimization of stresses and buckling loads of an isotropic plate
with an abrupt change in thickness.


## 2. General Considerations

### 2.1. Element Design Modifications and Cavitation

For any combination
of internal combustion engine and automatic transmission, the performance of
the torque converter must be optimized over all possible speed ratios for
efficiency, torque ratio, tractive effort, vehicle acceleration, and fuel
consumption. Torus dimensions and shape are usually determined first by
powertrain packaging constraints,K-factor performance targets, and
consideration for rotating inertia. Once the dimensions of the torus are
selected, element designs are explored to refine performance over all possible
speed ratios. For a given set of torus dimensions, a single turbine design is
developed for use with any combination of pumps or stators to be developed. The
stamped sheet metal blades of the pump lend themselves to adjustments in inlet
and outlet angles,
with blade count fixed for a given diameter. The stator can be modified by
changing blade angles, both inlet and outlet simultaneous, and the airfoil
shape of the blade.  Any change in
converter diameter will scale K-factor by the ratio of the diameters to the 2.5
power, while axial length has a marginal effect. Increasing pump outlet angle
will lower K-factor and torque ratio at stall, while improving efficiency at
high speed ratio. Stator outlet angles of increased magnitude produce higher
K-factors and lower torque ratios at stall but improve peak efficiency at high
speed ratio. The airfoil shape of the stator blade influences stall torque
ratio and K-factor at high-speed ratio. These design
parameters that are crucial to tuning torque converter performance are also
influential to cavitation characteristics and were evaluated in the dimensional
analysis. Torque converter designs, with increased diameter and axial length
that contain pumps and stators of low magnitude blade angles, will result in increased
cavitation thresholds as described in [8].  The reader is referred to [8] for a more
thorough discussion on the influence of design parameters on performance and
cavitation.


### 2.2. Parameters for Dimensional Analysis

Stator torque at
the onset of cavitation for any torque converter design is a function of the
specific design and operating point parameters that determine the boundary
value problem of the toroidal flow. The stator torque cavitation threshold,Ts,i,
for any torque converter design is primarily a function of torus dimensions D and Lt, and element blade designs comprising of blade angles
for pump, turbine, and stator, and blade airfoil shape for the stator. Element design
parameters manifest themselves as the integrated quantities of K-factor (see
Part I), K, and the ratio of turbine to pump torque, known as torque
ratio, TR. Both quantities at stall correlate to cavitation behavior in
the converter, see [8], and their simplicity is preferred over a lengthy and
detailed list of individual blade design parameters for each element.  However, the number of blades and airfoil
shape of the stator can strongly influence performance and cavitation behavior
and was also included.
Airfoil shape is specified by the maximum blade thickness, tmax,
and chord length, lc. Stator blade count and shape have been
shown to affect Ts,i as described by [8]. Pressures
superimposed upon the torque converter can alter Ts,i by as
much as 150 Nm as determined by [8] and are considered primary variables. As
in Part I, average pressure, pave, and pressure drop, Δp,
are more suitable to the dimensional analysis than charge and back pressures.Secondary variables
that influenceTs,i include temperature dependent fluid
properties and cooling through flow rate, Q. Since cooling through flow
rate is determined by the charge and back pressures imposed upon the converter,
it is not explicitly included in the variable list. The temperature dependent
properties of density, ρ, viscosity, μ, specific heat, Cp,
and thermal conductivity, k, were all taken at the input temperature of
the cooling flow supplied to the torque converter. Equation (1) gives a
functional list of the primary and secondary variables that determine Ts,i in an automotive torque converter:
(1)Ts,i=f(D,Lt,K,TR,nsb,tmax,lc,pave,Δp,ρ,μ,Cp,k).The dimensionless
form of (1) was resolved into dimensionless stator torque as a function of five
dimensionless design parameters and two dimensionless operating point
parameters using the repeating variables of D, ρ, pave, and Cp:
(2)Ts,iD3pave=f(LtD,U,TR,nsb,tmaxlc,Δppave,Pr).The dimensionless design parameters
include torus aspect ratio, Lt/D, unit input speed
(see Part I), U, stall torque ratio, TR, number of stator blades, nsb, and stator blade thickness ratio, tmax/lc.
The dimensionless operating point parameters are dimensionless operating
pressure, Δp/pave, and the Prandtl number, Pr. These
seven dimensionless parameters were used as dimensionless regressors to fit empirical
model(s) to dimensionless stator torque.Not all of the dimensionless
design parameters of (2) are necessary when considering a particular population
of torque converters. Those dimensionless design parameters that remain
constant are removed from (2) before fitting a model to the data. This was the
case in Part I, where the dimensions of both torque converters were exactly scaled
by the ratio of their diameters. This resulted in U and TR being
constant for each diameter along with the design parameters of nsb,
and tmax/lc. These dimensionless regressors
were subsequently removed from the functional list of (2) before proceeding to
fit a model to the data.


### 2.3. Response Surface Methodology

Response surface
methodology (RSM) and a stepwise regression procedure were used to model the
relationship of dimensionless stator torque with the dimensionless design and
operating point parameters of (2). The general form of the second-order response
surface (RS) assumed for this investigation is given by (3), which includes
linear, quadratic, and two-factor interactions:
(3)y^=b0+∑i=1kbixi+∑i=1kbiixi2+∑i=1k−1∑j=i+1kbijxixj+ε.The response, y^,
is dimensionless stator torque at cavitation and the linear, quadratic, and two-factor
interactions, xi, xi2, and xixj,
respectively, are the dimensionless regressors of (2). The estimated regression
coefficients, b0, bi, bii, and bij were determined using least squares method. The term, ε, is the residual or error of the model, the difference between the
experimentally measured and predicted value of dimensionless stator torque.A stepwise
regression technique was utilized to produce a RS model with the minimum number
of dimensionless regressors that are statistically significant with 95%
confidence. Equation (4) was the initial form of the RS model assumed, from which
dimensionless regressors are sequentially added based upon meeting the statistical
criterion of a 95% joint Bonferroni confidence interval and t-test; see
[10]:
(4)y^=b0+ε.
During the stepwise regression
process, regressors can be added or removed from the model as dictated by the statistical
criterion. The overall reduction in error realized by adding any one regressor is
dependent upon the regressors already in the model. For a more detailed
description of RSM and the stepwise linear regression procedure, the reader is
referred to [10, 11].The accuracy and
goodness of fit for any RS model developed were checked by computing the root mean
square error (RMSE) and determining the linear association between the
dimensionless response and regressors. RMSE is an estimator of a model's
standard deviation and for this investigation was computed as a percentage, denoted
as %RMSE. Equation (5) defines %RMSE, where yi
is
the measured response, n is the number of data points, and p is the
number of dimensionless regressors in the model:
(5)%RMSE=∑i=1n(((yi−y^i)/yi)*100)2n−p.A measure of the
proportionate amount of variation in the response explained by a particular set
of regressors in the model is the adjusted coefficient of multiple
determinationRa2:
(6)Ra2=1−∑i=1n(yi−y^i)2/n−p∑i=1n(yi−y¯)2/n−1.
This metric calculates to a value
from 0 to 1. Results above 0.85 are generally needed to signify an accurate model
of the data set. Ra2 is typically preferred over R2,
the coefficient of multiple determination, as it is a better evaluator of the
number of regressors included in the model. As the number of regressors increases, R2 will always increase, whereas Ra2
may increase or decrease depending on whether or not the additional explanatory
regressors actually reduce variation in the response. Computationally, the difference
between these metrics is that the factors of n−p and n−1 are removed
from the numerator and denominator of (6), respectively, to compute R2.


## 2.1. Element Design Modifications and Cavitation

For any combination
of internal combustion engine and automatic transmission, the performance of
the torque converter must be optimized over all possible speed ratios for
efficiency, torque ratio, tractive effort, vehicle acceleration, and fuel
consumption. Torus dimensions and shape are usually determined first by
powertrain packaging constraints,K-factor performance targets, and
consideration for rotating inertia. Once the dimensions of the torus are
selected, element designs are explored to refine performance over all possible
speed ratios. For a given set of torus dimensions, a single turbine design is
developed for use with any combination of pumps or stators to be developed. The
stamped sheet metal blades of the pump lend themselves to adjustments in inlet
and outlet angles,
with blade count fixed for a given diameter. The stator can be modified by
changing blade angles, both inlet and outlet simultaneous, and the airfoil
shape of the blade.  Any change in
converter diameter will scale K-factor by the ratio of the diameters to the 2.5
power, while axial length has a marginal effect. Increasing pump outlet angle
will lower K-factor and torque ratio at stall, while improving efficiency at
high speed ratio. Stator outlet angles of increased magnitude produce higher
K-factors and lower torque ratios at stall but improve peak efficiency at high
speed ratio. The airfoil shape of the stator blade influences stall torque
ratio and K-factor at high-speed ratio. These design
parameters that are crucial to tuning torque converter performance are also
influential to cavitation characteristics and were evaluated in the dimensional
analysis. Torque converter designs, with increased diameter and axial length
that contain pumps and stators of low magnitude blade angles, will result in increased
cavitation thresholds as described in [8].  The reader is referred to [8] for a more
thorough discussion on the influence of design parameters on performance and
cavitation.


## 2.2. Parameters for Dimensional Analysis

Stator torque at
the onset of cavitation for any torque converter design is a function of the
specific design and operating point parameters that determine the boundary
value problem of the toroidal flow. The stator torque cavitation threshold,Ts,i,
for any torque converter design is primarily a function of torus dimensions D and Lt, and element blade designs comprising of blade angles
for pump, turbine, and stator, and blade airfoil shape for the stator. Element design
parameters manifest themselves as the integrated quantities of K-factor (see
Part I), K, and the ratio of turbine to pump torque, known as torque
ratio, TR. Both quantities at stall correlate to cavitation behavior in
the converter, see [8], and their simplicity is preferred over a lengthy and
detailed list of individual blade design parameters for each element.  However, the number of blades and airfoil
shape of the stator can strongly influence performance and cavitation behavior
and was also included.
Airfoil shape is specified by the maximum blade thickness, tmax,
and chord length, lc. Stator blade count and shape have been
shown to affect Ts,i as described by [8]. Pressures
superimposed upon the torque converter can alter Ts,i by as
much as 150 Nm as determined by [8] and are considered primary variables. As
in Part I, average pressure, pave, and pressure drop, Δp,
are more suitable to the dimensional analysis than charge and back pressures.Secondary variables
that influenceTs,i include temperature dependent fluid
properties and cooling through flow rate, Q. Since cooling through flow
rate is determined by the charge and back pressures imposed upon the converter,
it is not explicitly included in the variable list. The temperature dependent
properties of density, ρ, viscosity, μ, specific heat, Cp,
and thermal conductivity, k, were all taken at the input temperature of
the cooling flow supplied to the torque converter. Equation (1) gives a
functional list of the primary and secondary variables that determine Ts,i in an automotive torque converter:
(1)Ts,i=f(D,Lt,K,TR,nsb,tmax,lc,pave,Δp,ρ,μ,Cp,k).The dimensionless
form of (1) was resolved into dimensionless stator torque as a function of five
dimensionless design parameters and two dimensionless operating point
parameters using the repeating variables of D, ρ, pave, and Cp:
(2)Ts,iD3pave=f(LtD,U,TR,nsb,tmaxlc,Δppave,Pr).The dimensionless design parameters
include torus aspect ratio, Lt/D, unit input speed
(see Part I), U, stall torque ratio, TR, number of stator blades, nsb, and stator blade thickness ratio, tmax/lc.
The dimensionless operating point parameters are dimensionless operating
pressure, Δp/pave, and the Prandtl number, Pr. These
seven dimensionless parameters were used as dimensionless regressors to fit empirical
model(s) to dimensionless stator torque.Not all of the dimensionless
design parameters of (2) are necessary when considering a particular population
of torque converters. Those dimensionless design parameters that remain
constant are removed from (2) before fitting a model to the data. This was the
case in Part I, where the dimensions of both torque converters were exactly scaled
by the ratio of their diameters. This resulted in U and TR being
constant for each diameter along with the design parameters of nsb,
and tmax/lc. These dimensionless regressors
were subsequently removed from the functional list of (2) before proceeding to
fit a model to the data.


## 2.3. Response Surface Methodology

Response surface
methodology (RSM) and a stepwise regression procedure were used to model the
relationship of dimensionless stator torque with the dimensionless design and
operating point parameters of (2). The general form of the second-order response
surface (RS) assumed for this investigation is given by (3), which includes
linear, quadratic, and two-factor interactions:
(3)y^=b0+∑i=1kbixi+∑i=1kbiixi2+∑i=1k−1∑j=i+1kbijxixj+ε.The response, y^,
is dimensionless stator torque at cavitation and the linear, quadratic, and two-factor
interactions, xi, xi2, and xixj,
respectively, are the dimensionless regressors of (2). The estimated regression
coefficients, b0, bi, bii, and bij were determined using least squares method. The term, ε, is the residual or error of the model, the difference between the
experimentally measured and predicted value of dimensionless stator torque.A stepwise
regression technique was utilized to produce a RS model with the minimum number
of dimensionless regressors that are statistically significant with 95%
confidence. Equation (4) was the initial form of the RS model assumed, from which
dimensionless regressors are sequentially added based upon meeting the statistical
criterion of a 95% joint Bonferroni confidence interval and t-test; see
[10]:
(4)y^=b0+ε.
During the stepwise regression
process, regressors can be added or removed from the model as dictated by the statistical
criterion. The overall reduction in error realized by adding any one regressor is
dependent upon the regressors already in the model. For a more detailed
description of RSM and the stepwise linear regression procedure, the reader is
referred to [10, 11].The accuracy and
goodness of fit for any RS model developed were checked by computing the root mean
square error (RMSE) and determining the linear association between the
dimensionless response and regressors. RMSE is an estimator of a model's
standard deviation and for this investigation was computed as a percentage, denoted
as %RMSE. Equation (5) defines %RMSE, where yi
is
the measured response, n is the number of data points, and p is the
number of dimensionless regressors in the model:
(5)%RMSE=∑i=1n(((yi−y^i)/yi)*100)2n−p.A measure of the
proportionate amount of variation in the response explained by a particular set
of regressors in the model is the adjusted coefficient of multiple
determinationRa2:
(6)Ra2=1−∑i=1n(yi−y^i)2/n−p∑i=1n(yi−y¯)2/n−1.
This metric calculates to a value
from 0 to 1. Results above 0.85 are generally needed to signify an accurate model
of the data set. Ra2 is typically preferred over R2,
the coefficient of multiple determination, as it is a better evaluator of the
number of regressors included in the model. As the number of regressors increases, R2 will always increase, whereas Ra2
may increase or decrease depending on whether or not the additional explanatory
regressors actually reduce variation in the response. Computationally, the difference
between these metrics is that the factors of n−p and n−1 are removed
from the numerator and denominator of (6), respectively, to compute R2.


## 3. Experimental Setup and Test Procedure

### 3.1. Dynamometer Test Cell

The torque
converter dynamometer test cell and test procedure used to acquire stator
torque cavitation thresholds at stall using nearfield acoustical measurements was
described in Part I. A 40 RPM/s stall speed sweep procedure was again utilized to
transition all torque converters from a noncavitating to a cavitating condition.
Fourteen operating points were applied to each design to characterize their
effect onTs,i. Table 1 lists the ranges of the three
operating points varied in this investigation. The signal processing technique
utilized to determine the exact threshold of cavitation was briefly discussed
in Part I and further detailed by [8].Table 1
Range of stall operating
point parameters tested.
Operating pointLowHighAverage pressure481 kPa963 kPaPressure drop69 kPa345 kPaInput temperature60°C80°C


### 3.2. Matrix of Experimental Torque Converters

Four major
populations of torque converters were tested with progressively decreasing
levels of geometric similitude, as shown in Figure1 from top to bottom. The first population, referred to
as geometric scaling, contains the same torque converters as presented in Part
I. The converters maintain exact geometric
similitude between diameters. This population served as a baseline comparison
for RS models in which geometric similitude was relaxed. The second population,
multiple torus geometry, includes three torque converters of similar element
design, but varying torus dimensions (D and Lt) to
create converters with the same unit input speed, U. The next population
consists of multiple pump and stator blade designs, with torus dimensions and
turbine design held constant, for 16 total configurations that achieve a range
in K-factor performance. The final population includes all converters tested
for a total of 51 widely varying designs that encompass a typical design space
for powertrain component matching. Within this population, referred to as
general torque converter design, geometric similitude was greatly relaxed,
allowing the development of a comprehensive RS model capable of predicting Ts,i.
Because of the loosened restrictions on similitude, the resulting model is
applicable to general torque converter designs. The details for each population
are provided with the discussion of each RS model. The six diameters tested in
this investigation will be denoted D1 through D6, with D1 corresponding to the smallest diameter and D6 to the largest
diameter.Figure 1
Organizational chart for torque
converter populations tested; range of converters tested is denoted by
diameters andK-factors.


## 3.1. Dynamometer Test Cell

The torque
converter dynamometer test cell and test procedure used to acquire stator
torque cavitation thresholds at stall using nearfield acoustical measurements was
described in Part I. A 40 RPM/s stall speed sweep procedure was again utilized to
transition all torque converters from a noncavitating to a cavitating condition.
Fourteen operating points were applied to each design to characterize their
effect onTs,i. Table 1 lists the ranges of the three
operating points varied in this investigation. The signal processing technique
utilized to determine the exact threshold of cavitation was briefly discussed
in Part I and further detailed by [8].Table 1
Range of stall operating
point parameters tested.
Operating pointLowHighAverage pressure481 kPa963 kPaPressure drop69 kPa345 kPaInput temperature60°C80°C


## 3.2. Matrix of Experimental Torque Converters

Four major
populations of torque converters were tested with progressively decreasing
levels of geometric similitude, as shown in Figure1 from top to bottom. The first population, referred to
as geometric scaling, contains the same torque converters as presented in Part
I. The converters maintain exact geometric
similitude between diameters. This population served as a baseline comparison
for RS models in which geometric similitude was relaxed. The second population,
multiple torus geometry, includes three torque converters of similar element
design, but varying torus dimensions (D and Lt) to
create converters with the same unit input speed, U. The next population
consists of multiple pump and stator blade designs, with torus dimensions and
turbine design held constant, for 16 total configurations that achieve a range
in K-factor performance. The final population includes all converters tested
for a total of 51 widely varying designs that encompass a typical design space
for powertrain component matching. Within this population, referred to as
general torque converter design, geometric similitude was greatly relaxed,
allowing the development of a comprehensive RS model capable of predicting Ts,i.
Because of the loosened restrictions on similitude, the resulting model is
applicable to general torque converter designs. The details for each population
are provided with the discussion of each RS model. The six diameters tested in
this investigation will be denoted D1 through D6, with D1 corresponding to the smallest diameter and D6 to the largest
diameter.Figure 1
Organizational chart for torque
converter populations tested; range of converters tested is denoted by
diameters andK-factors.


## 4. Results and Discussion

The RS models
developed using the RSM  and stepwise regression techniques are presented in the
order depicted in Figure1, from strictest geometric similitude to the most
relaxed. The full RS model equations for all of the populations are listed in
Table 4, at the end of the paper. It is expected that the RS models presented will
be integrated into initial converter design and powertrain component matching
to maximize Ts,i within a given set of packaging, performance,
and stall operating point specifications. Prediction of Ts,i during
this phase will enable optimization of torus and/or element geometries along
with operating pressures and temperature to minimize the effects of cavitation
related noise.


### 4.1. Geometric Scaling

The cavitation
threshold data used to fit the RS model was the same as that used to fit the
power product (PP) model of Part I. Equation (7) gives the functional form of the
RS model for this population, and the RS model versus test data graph is shown
in Figure 2:
(7)Ts,iD3pave=f(Δppave,Pr).Figure 2
Geometric scaling RS model for
predicting stator torque at onset of cavitation (28 points).The model
diagnostics (%RMSE andRa2) provided in Table 2 show the RS model to have a 23% increase in Ra2 and a 9% decrease in %RMSE over the corresponding PP model for the same
number of dimensionless regressors. Although both models have %RMSE values below 4%, the small Ra2
values suggest that
both models have a low linear association between dimensionless response and
regressors. As with the PP model of Part I, a multitude of least squares
solutions exists for the geometric scaling data set. Use of dimensionless
stator torque results in a nearly constant term for the onset of cavitation over
the range of operating points tested. The small variation in dimensionless
quantities does not particularly lend itself to empirical modeling, requiring
increased variation in dimensionless quantities to adequately define a curve.
However, even with relatively low values of Ra2,
the prediction accuracy for scaling stator torque cavitation thresholds for a
given torus and set of element designs is high.Table 2
Comparison of geometric scaling models.
Model typeData
pointsRegressorsRa2%RMSERS2820.5663.340PP2820.4343.733


### 4.2. Multiple Torus Design

For the multiple
torus design population, element blade designs were held constant, while the
dimensions of the torus were varied for three ratios ofLt/D.
Performances for the 3 designs result in equivalent unit input speeds, U. The
functional form for this RS model is given by (8) and includes torus aspect ratio
and number of stator blades in addition to the terms included in (7):
(8)Ts,iD3pave=f(LtD,nsb,Δppave,Pr).
The addition of the dimensionless design
parameters was required as torus dimensions changed along with some modifications
to the stator. The number of stator blades is different between the 3
converters due to differences in axial length, but blade angles are the same. Figure 3 contains the RS model, showing an increase in the
range of dimensionless response and regressors compared to the geometric
scaling RS model shown in Figure 2. A decrease in model %RMSE from 3.40%
to 2.87% and an increase in Ra2
from 0.566 to
0.925 were realized compared to the geometric scaling model. The multiple torus
design RS model includes five regressors, consisting primarily of two factor
interactions between dimensionless design and operating point parameters. The
utility of this particular model allows prediction of Ts,i
for
a single set of element designs spanning the range of torus dimensions provided
in Figure 3.Figure 3
Multiple torus design RS model for
predicting stator torque at onset of cavitation (42 points).


### 4.3. Multiple Pump and Stator Design

The dimensionless
functional form of the RS model for the population with multiple pump and
stator designs is given by(9)Ts,iD3pave=f(U,TR,nsb,tmaxlc,Δppave,Pr).
The list of dimensionless
regressors includes those that characterize changes to the pump and stator and
stall operating point. The converters in this population all have the same
torus design, so the Lt/D term included in (8) drops out of (9).Figure4
summarizes results from the stepwise regression procedure used to determine statistically
significant regressors for the RS model. The figure shows the reduction in
model error and an increase in linear association between response and
regressors as regressors are added to the model. Only regressors that have a
statistically significant effect are retained in the final equation. In this
way, the number of regressors was reduced to 10, from a possible 27, including
the constant term, b0. It can be noted in Figure 4 that the
addition of the first three regressors, nsb2, TR2,
and the interaction U*TR accounts for 88% of the variation in
dimensionless stator torque and reduces %RMSE from 27.6% to 7.78%. Building
the RS model by sequentially adding dimensionless regressors other than in the
order shown in Figure 4 would produce different characteristic values of %RMSE and Ra2. During the stepwise regression process, models %RMSE and Ra2 are dependent upon the statistical
significance of the individual regressor to be added and those regressors
already present in the model. For this investigation, regressors were added to
the RS model according to their influence on %RMSE and Ra2.
Those regressors, that were deemed statistically significant and generated the
highest reduction in %RMSE and increase in Ra2,
were added first. It was found that the regressors based on design parameters
produced more statistically significant contributions to reducing model %RSME
and increasing Ra2 than those based on fluid properties,
even though cavitation is a fluid dependent phenomenon influenced by pressures
and temperatures.Figure 4
Sequential reduction in %RMSE and increase inRa2
during stepwise regression procedure
for multiple pump and stator design RS model.The RS model for the
multiple pump and stator design population resulting from the stepwise
regression is provided in Figure5. For the 16 different pump and stator
combinations included, the RS model has a %RMSE of 5.16% and Ra2
of 0.952. Compared with the multiple torus geometry RS model, %RMSE increased by 2.29%. This increase is expected as more data points are included,
introducing more variation and departing further from geometric similitude. The
predicted value of dimensionless stator torque from this RS model can be scaled
to other diameters if exact dimensional scaling is observed, and design
modifications to pump and stator fall within the scope of the model.Figure 5
RS model for multiple pump and
stator design population, data is for 6 pump and 11 stator designs with fixed
torus geometry.


### 4.4. Generalized Torque Converter Design

A population of
converters with variations in both torus and element geometries was utilized to
develop a generalized RS model for predictingTs,i
at stall.
The experimental data, shown in Figure 6 as Ts,i versus Np,i,
pump speed at onset of cavitation, spans a broad range of torques and speeds
for the matrix of converters and stall operating points tested.Figure 6
Stator torque versus pump speed at
onset of cavitation data for 51 widely varying
torque converter designs (714 points).All of the dimensionless
regressors from (2) were used for developing the generalized RS model, as all
varied for the matrix of designs tested. The stepwise regression procedure
resulted in a RS model with 18 dimensionless regressors and the constant term, b0.
Dimensionless design parameters dominated the reduction in %RMSE and an increase
in Ra2, most notably U, TR, and
their interaction. These two regressors along with the constant term reduce %RMSE from 27.76%, for the initial model (4), to 10.36%, and account for 81% of the
variation in the dimensionless response. The generalized torque converter RS
model for predicting Ts,i
is shown in Figure 7, differentiated by diameter. The range of design
parameters tested are provided, showing the range of torus, stator design
specifics, and overall stall performance metrics as determined by element
geometries.Figure 7
RS model for prediction of stator
torque at onset of cavitation for general torque converter design.The generalized RS
model’s %RMSE of 6.52% andRa2
of 0.936 demonstrates
that even with greatly relaxed geometric similitude, an accurate prediction model
was developed. Figure 8 is a histogram of the model residuals; the error term
defined in (3). It shows that no significant deviations from normalcy and
roughly follow that of a normal distribution.Figure 8
Histogram of residuals for
general torque converter design population RS model.Table3  compares
each RS model developed from a population of torque converters with complete
geometric similitude, to populations with diminishing restrictions on geometric
similitude. It can be noted that as geometric similitude is relaxed, the number
of dimensionless regressors required to realize a highly accurate RS model
increases. This is expected as the numerous interactions between design
parameters become increasingly important in determining Ts,i
at stall. The increase in %RMSE occurring sequentially from the
geometric scaling to the general torque converter design RS model is not
substantial enough to consider RS models lacking geometric similitude to be inaccurate.Table 3
Comparison of RS model performance
by population.
PopulationGeometric
similitudeData
pointsRegressorsRa2%RMSEGeometric scalingExact2830.5663.34Multiple torus designElements
only4260.9262.87Multiple pump and stator designTorus
only224100.9525.16General torque converter designNone714190.9366.52Table 4
Summary of RS models by
torque converter population excluding regression coefficients.
PopulationResponse
surface modelGeometric
scalingTs,iD3pave=b0+b1(Δppave)2+b2(Pr)2.Multiple
torus designTs,iD3pave=b0+b1(Δppave)+b2(LtD)2+b3(LtD)(Δppave)+b4(LtD)(Pr)+b5(nsb)(Δppave).Multiple
pump and stator designTs,iD3pave=b0+b1(tmaxlc)+b2(U)2+b3(TR)2+b4(nsb)2+b5(Δppave)2+b6(Pr)2+b7(U)(TR)+b8(U)(tmaxlc)+b9(nsb)(tmaxlc).General
torque converter designTs,iD3pave=b0+b1(LtD)+b2(U)+b3(TR)+b4(nsb)+b5(tmaxlc)+b6(LtD)2+b7(nsb)2+b8(tmaxlc)2+b9(Δppave)2+b10(Pr)2+b11(LtD)(nsb)+b12(LtD)(tmaxlc)+b13(U)(TR)+b14(U)(nsb)+b15(U)(tmaxlc)+b16(TR)(nsb)+b17(TR)(tmaxlc)+b18(nsb)(tmaxlc).Table4 summarizes
the dimensionless regressors that comprise each RS model presented in this
paper. In general, the dimensionless regressors found to be most influential in
reduction of RS model error reduction are those most closely related to the
design and performance characteristics of the turbomachine rather than the
dimensionless operating point parameters. Depending on the data set being
modeled, the dimensionless design parameter regressors that result in the
largest error reduction varied. In the case of the generalized model, 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, and dimensionless operating point serve to
explain additional but minor variations in the dimensionless response. The
regression coefficients contained in each RS model were determined from proprietary
geometries and are withheld from Table 4.


## 4.1. Geometric Scaling

The cavitation
threshold data used to fit the RS model was the same as that used to fit the
power product (PP) model of Part I. Equation (7) gives the functional form of the
RS model for this population, and the RS model versus test data graph is shown
in Figure 2:
(7)Ts,iD3pave=f(Δppave,Pr).Figure 2
Geometric scaling RS model for
predicting stator torque at onset of cavitation (28 points).The model
diagnostics (%RMSE andRa2) provided in Table 2 show the RS model to have a 23% increase in Ra2 and a 9% decrease in %RMSE over the corresponding PP model for the same
number of dimensionless regressors. Although both models have %RMSE values below 4%, the small Ra2
values suggest that
both models have a low linear association between dimensionless response and
regressors. As with the PP model of Part I, a multitude of least squares
solutions exists for the geometric scaling data set. Use of dimensionless
stator torque results in a nearly constant term for the onset of cavitation over
the range of operating points tested. The small variation in dimensionless
quantities does not particularly lend itself to empirical modeling, requiring
increased variation in dimensionless quantities to adequately define a curve.
However, even with relatively low values of Ra2,
the prediction accuracy for scaling stator torque cavitation thresholds for a
given torus and set of element designs is high.Table 2
Comparison of geometric scaling models.
Model typeData
pointsRegressorsRa2%RMSERS2820.5663.340PP2820.4343.733


## 4.2. Multiple Torus Design

For the multiple
torus design population, element blade designs were held constant, while the
dimensions of the torus were varied for three ratios ofLt/D.
Performances for the 3 designs result in equivalent unit input speeds, U. The
functional form for this RS model is given by (8) and includes torus aspect ratio
and number of stator blades in addition to the terms included in (7):
(8)Ts,iD3pave=f(LtD,nsb,Δppave,Pr).
The addition of the dimensionless design
parameters was required as torus dimensions changed along with some modifications
to the stator. The number of stator blades is different between the 3
converters due to differences in axial length, but blade angles are the same. Figure 3 contains the RS model, showing an increase in the
range of dimensionless response and regressors compared to the geometric
scaling RS model shown in Figure 2. A decrease in model %RMSE from 3.40%
to 2.87% and an increase in Ra2
from 0.566 to
0.925 were realized compared to the geometric scaling model. The multiple torus
design RS model includes five regressors, consisting primarily of two factor
interactions between dimensionless design and operating point parameters. The
utility of this particular model allows prediction of Ts,i
for
a single set of element designs spanning the range of torus dimensions provided
in Figure 3.Figure 3
Multiple torus design RS model for
predicting stator torque at onset of cavitation (42 points).


## 4.3. Multiple Pump and Stator Design

The dimensionless
functional form of the RS model for the population with multiple pump and
stator designs is given by(9)Ts,iD3pave=f(U,TR,nsb,tmaxlc,Δppave,Pr).
The list of dimensionless
regressors includes those that characterize changes to the pump and stator and
stall operating point. The converters in this population all have the same
torus design, so the Lt/D term included in (8) drops out of (9).Figure4
summarizes results from the stepwise regression procedure used to determine statistically
significant regressors for the RS model. The figure shows the reduction in
model error and an increase in linear association between response and
regressors as regressors are added to the model. Only regressors that have a
statistically significant effect are retained in the final equation. In this
way, the number of regressors was reduced to 10, from a possible 27, including
the constant term, b0. It can be noted in Figure 4 that the
addition of the first three regressors, nsb2, TR2,
and the interaction U*TR accounts for 88% of the variation in
dimensionless stator torque and reduces %RMSE from 27.6% to 7.78%. Building
the RS model by sequentially adding dimensionless regressors other than in the
order shown in Figure 4 would produce different characteristic values of %RMSE and Ra2. During the stepwise regression process, models %RMSE and Ra2 are dependent upon the statistical
significance of the individual regressor to be added and those regressors
already present in the model. For this investigation, regressors were added to
the RS model according to their influence on %RMSE and Ra2.
Those regressors, that were deemed statistically significant and generated the
highest reduction in %RMSE and increase in Ra2,
were added first. It was found that the regressors based on design parameters
produced more statistically significant contributions to reducing model %RSME
and increasing Ra2 than those based on fluid properties,
even though cavitation is a fluid dependent phenomenon influenced by pressures
and temperatures.Figure 4
Sequential reduction in %RMSE and increase inRa2
during stepwise regression procedure
for multiple pump and stator design RS model.The RS model for the
multiple pump and stator design population resulting from the stepwise
regression is provided in Figure5. For the 16 different pump and stator
combinations included, the RS model has a %RMSE of 5.16% and Ra2
of 0.952. Compared with the multiple torus geometry RS model, %RMSE increased by 2.29%. This increase is expected as more data points are included,
introducing more variation and departing further from geometric similitude. The
predicted value of dimensionless stator torque from this RS model can be scaled
to other diameters if exact dimensional scaling is observed, and design
modifications to pump and stator fall within the scope of the model.Figure 5
RS model for multiple pump and
stator design population, data is for 6 pump and 11 stator designs with fixed
torus geometry.


## 4.4. Generalized Torque Converter Design

A population of
converters with variations in both torus and element geometries was utilized to
develop a generalized RS model for predictingTs,i
at stall.
The experimental data, shown in Figure 6 as Ts,i versus Np,i,
pump speed at onset of cavitation, spans a broad range of torques and speeds
for the matrix of converters and stall operating points tested.Figure 6
Stator torque versus pump speed at
onset of cavitation data for 51 widely varying
torque converter designs (714 points).All of the dimensionless
regressors from (2) were used for developing the generalized RS model, as all
varied for the matrix of designs tested. The stepwise regression procedure
resulted in a RS model with 18 dimensionless regressors and the constant term, b0.
Dimensionless design parameters dominated the reduction in %RMSE and an increase
in Ra2, most notably U, TR, and
their interaction. These two regressors along with the constant term reduce %RMSE from 27.76%, for the initial model (4), to 10.36%, and account for 81% of the
variation in the dimensionless response. The generalized torque converter RS
model for predicting Ts,i
is shown in Figure 7, differentiated by diameter. The range of design
parameters tested are provided, showing the range of torus, stator design
specifics, and overall stall performance metrics as determined by element
geometries.Figure 7
RS model for prediction of stator
torque at onset of cavitation for general torque converter design.The generalized RS
model’s %RMSE of 6.52% andRa2
of 0.936 demonstrates
that even with greatly relaxed geometric similitude, an accurate prediction model
was developed. Figure 8 is a histogram of the model residuals; the error term
defined in (3). It shows that no significant deviations from normalcy and
roughly follow that of a normal distribution.Figure 8
Histogram of residuals for
general torque converter design population RS model.Table3  compares
each RS model developed from a population of torque converters with complete
geometric similitude, to populations with diminishing restrictions on geometric
similitude. It can be noted that as geometric similitude is relaxed, the number
of dimensionless regressors required to realize a highly accurate RS model
increases. This is expected as the numerous interactions between design
parameters become increasingly important in determining Ts,i
at stall. The increase in %RMSE occurring sequentially from the
geometric scaling to the general torque converter design RS model is not
substantial enough to consider RS models lacking geometric similitude to be inaccurate.Table 3
Comparison of RS model performance
by population.
PopulationGeometric
similitudeData
pointsRegressorsRa2%RMSEGeometric scalingExact2830.5663.34Multiple torus designElements
only4260.9262.87Multiple pump and stator designTorus
only224100.9525.16General torque converter designNone714190.9366.52Table 4
Summary of RS models by
torque converter population excluding regression coefficients.
PopulationResponse
surface modelGeometric
scalingTs,iD3pave=b0+b1(Δppave)2+b2(Pr)2.Multiple
torus designTs,iD3pave=b0+b1(Δppave)+b2(LtD)2+b3(LtD)(Δppave)+b4(LtD)(Pr)+b5(nsb)(Δppave).Multiple
pump and stator designTs,iD3pave=b0+b1(tmaxlc)+b2(U)2+b3(TR)2+b4(nsb)2+b5(Δppave)2+b6(Pr)2+b7(U)(TR)+b8(U)(tmaxlc)+b9(nsb)(tmaxlc).General
torque converter designTs,iD3pave=b0+b1(LtD)+b2(U)+b3(TR)+b4(nsb)+b5(tmaxlc)+b6(LtD)2+b7(nsb)2+b8(tmaxlc)2+b9(Δppave)2+b10(Pr)2+b11(LtD)(nsb)+b12(LtD)(tmaxlc)+b13(U)(TR)+b14(U)(nsb)+b15(U)(tmaxlc)+b16(TR)(nsb)+b17(TR)(tmaxlc)+b18(nsb)(tmaxlc).Table4 summarizes
the dimensionless regressors that comprise each RS model presented in this
paper. In general, the dimensionless regressors found to be most influential in
reduction of RS model error reduction are those most closely related to the
design and performance characteristics of the turbomachine rather than the
dimensionless operating point parameters. Depending on the data set being
modeled, the dimensionless design parameter regressors that result in the
largest error reduction varied. In the case of the generalized model, 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, and dimensionless operating point serve to
explain additional but minor variations in the dimensionless response. The
regression coefficients contained in each RS model were determined from proprietary
geometries and are withheld from Table 4.


## 5. Conclusions

Experimentally
obtained values of stator torque at the onset of cavitation for torque converters
with varying degrees of geometric similitude were nondimensionalized and used
as dimensionless response and regressors for developing response surface (RS)
models. A stepwise linear regression procedure reduced the complexity of each
RS model by only including statistically significant dimensionless regressors. RS
models created from data sets of decreasing geometric similitude showed thatRa2 values above 0.85 are achieved even with greatly relaxed geometric similitude. The %RMSE values resulting for each RS model of decreasing geometric
similitude were not substantial enough to indicate inadequacies in the
dimensional analysis or data modeling methodology. A RS model was presented
which is capable of predicting Ts,i
for general automotive
torque converter design with a %RMSE of 6.52%. This error is deemed
sufficiently low and its scope of prediction large enough to rate the RS model;
a valuable tool for optimizing torus and element geometries with respect to the
onset of cavitation at stall in three  element automotive torque converters.



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*Source: 312753-2008-07-28.xml*