# Predicting the Onset of Cavitation in Automotive Torque Converters—Part I: Designs with Geometric Similitude

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)

---

## Abstract

Dimensional analysis has been applied to automotive torque converters to understand the response of performance to changes in torque, size, working fluid, or operating temperature. The objective of this investigation was to develop a suitable dimensional analysis for estimating the effect of exact geometric scaling of a particular torque converter design on the onset of cavitation. Torque converter operating thresholds for cavitation were determined experimentally with a dynamometer test cell at the stall operating condition using nearfield acoustical measurements. Dimensionless quantities based upon either speed or torque at the onset of cavitation and flow properties (e.g., pressures and temperature dependent fluid properties) were developed and compared. The proposed dimensionless stator torque quantity was found to be the most appropriate scaling law for extrapolating cavitation thresholds to multiple diameters. A power product model was fit on dimensionless stator torque data to create a model capable of predicting cavitation thresholds. Comparison of the model to test data taken over a range of operating points showed an error of 3.7%. This is the first paper of a two-part paper. In Part II, application of dimensional analysis will be expanded from torque converters with exact geometric similitude to those of more general design.


---

## Body

## 1. Introduction

The automotive
torque converter is  the powertrain component that transfers power from the
engine to the gear system of the automatic transmission. It multiplies torque
at low speed to overcome the poor low speed torque characteristics of an internal
combustion engine. Torque converters are classified as a turbomachine as most modern
designs typically contain multibladed elements, including a mixed flow pump and
turbine and an axial flow stator. A typical cross-section diagram of a three-element converter and closed-loop toroidal flow path is shown in
Figure1. The flow is highly three dimensional with secondary
flow structures as a result of the close proximity of each element and the closed-loop
flow. Torque converters produce maximum torque multiplication at low speed
ratio and high efficiency at high speed ratio. Speed ratio is defined as the
rotation of turbine speed divided by pump speed.Figure 1
Cross-section schematic of a modern
three-element torque converter; arrows indicate
direction of toroidal flow.The pump rotates
at engine speed and torque, imparting an increase in angular momentum on the
fluid from inner to outer radius of the torus, guiding the fluid into the
turbine. The turbine extracts angular momentum from the fluid, transmitting a
torque to the automatic transmission via the turbine shaft. The fluid exits the
turbine and enters the stator, which redirects the fluid at a favorable angle into
the pump providing torque multiplication.The change in
angular momentum flux across each element produces a torque equal to the
product of the local static pressure and radius integrated over the entire
surface of each blade. An increase in element torque requires the pressure
differential across a blade to increase proportionately. At some critical value
of element torque, the pressure on the low pressure side of a blade drops below
the vapor pressure of the liquid, causing the nucleation of cavitation bubbles.The effect of
incipient cavitation is negligible on torque converter performance due to a minimal
amount of liquid displaced by vapor in the toroidal flow circuit and
confinement of bubbles to a relatively small region. Sustained operation at
conditions of heavy cavitation alters converter performance due to larger vapor
regions. Broadband noise that may become objectionable to vehicle occupants is
produced by the collapse of cavitation bubbles as they travel to regions of
higher pressure. Advanced stages of cavitation have been observed to cause a
decrease in element torques that alters the characteristic relationship between
speed and torque; see [1]. This can be a problem for vehicle operation in that
the converter no longer controls the engine speed as expected. Cavitation is
suppressed by superimposing increased pressures upon the torque converter. The pressure
imposed by the transmission control system is referred to as charge pressure
and is limited by the structural integrity of the torque converter housing and
capacity of the transmission pump.The purpose of
parts I and II of the study is to quantify the effects of torque converter
design parameters and operating conditions on cavitation through dimensional
analysis of test results. The intention is to develop a tool for designing
converters with minimal cavitation by fitting a power product model to the
dimensionless data. In Part I,
converters following strict rules of geometric similitude were analyzed. For Part II, the rules of similitude were
relaxed to allow application of the dimensional analysis tool to a broader
range of converter designs.


## 2. General Considerations

### 2.1. Torque Converter Cavitation

The operating
point most susceptible to the occurrence of cavitation in an automotive torque
converter is stall, or zero turbine speed. Converter stall is produced in a
vehicle by applying the brake and throttle simultaneously. This extends the
instantaneous stall condition a vehicle undergoes when launching from a stopped
position. Stall tests are useful for creating extreme conditions to study
phenomena such as cavitation. Also, a vehicle can undergo sustained operation
near stall when climbing a steep grade. The toroidal flow is highly three
dimensional with high element flow incidence angles. The stator has previously
been identified as the critical element in which incipient cavitation bubbles
form as indicated by computational results
of [2, 3]. Governed by
hydrodynamic theory, pressures on the pressure surface of the stator blade are
higher than the suction surface, with the greatest differential located at the
leading edge of the blade. The incidence angle of flow into a stator at stall
can be as high as 70 degrees for some designs, resulting in relatively large
areas of flow separation and regions of low pressure on the suction surface of
the blade. Experimentally, these flow phenomena have been measured using dynamometer
test setups by [4, 5].Fluctuating
pressure measurements were made within the pump passage by [1] which recorded
the pressure disturbances caused by the collapse of the cavitation bubbles as
they traveled into regions of higher pressure within the pump. Mekkes et al.
[6] performed measurements of the fluctuating component of pressure at the
leading edge of the stator, which showed that the fluctuating pressure signal became
damped as cavitation developed at the nose of the stator. The computational investigation
performed by [2] simulated the formation of cavitation bubbles within the
toroidal flow circuit at stall. The results showed that at a particular pump
speed, cavitation bubbles begin to develop at the leading edge of the stator. A
subsequent increase in pump speed causes the pressure differential at the
stator blade leading edge to increase. This
produces an expansion of cavitation and greater displacement of liquid with
vapor.


### 2.2. Acoustical Detection of Cavitation

Nearfield
acoustical measurements were utilized to facilitate testing of multiple
automotive torque converters without the need for specially instrumented
converters. Similar measurement and post processing techniques were developed by
[7] to identify cavitation via the noise produced by the collapse of cavitation
bubbles within the torque converter. It was observed from the sound measurements
that a particular frequency characteristic correlated to the noise produced by
the collapse of incipient cavitation bubbles. Cavitation related noise
manifested itself in the frequency range of 6 to 20 kHz, the maximum measurable
frequency. To remove noise not associated with cavitation, a Kaiser window high-pass
filter with a cutoff frequency of 6 kHz was applied to all acoustical
measurements. Time averaging of filtered nearfield acoustical data at discrete,
equally spaced pump speeds, was performed to further enhance the ability to
identify cavitation inception. Post processing
the nearfield acoustical data in this manner revealed that an abrupt increase
in filtered sound pressure level (SPL) occurred at a particular pump speed. This
pump speed was identified as the onset of cavitation in this investigation. The
specific procedure used to post process the data and identify the exact pump
speed at the onset of cavitation is detailed in [8].An example of a
filtered SPL versus pump speed plot from this study is shown in
Figure2. At the transition to cavitating operation, an abrupt
increase in filtered SPL is evident along with a drastic change in slope.
Experimental investigations,
[9, 10], on centrifugal pumps have demonstrated
the utility in acoustical measurements for detecting cavitation in
turbomachinery. In both investigations, the cavitation threshold was noted by an
abrupt increase in SPL as the turbomachine transitioned from a noncavitating to
cavitating operating regime. Consistent with these investigations, the first
abrupt change in slope of the filtered SPL curve was recognized as the onset of
cavitation.Figure 2
Filtered SPL versus pump speed for
an automotive torque converter operating at stall; vertical dashed line
indicates pump speed at onset of cavitation.


## 2.1. Torque Converter Cavitation

The operating
point most susceptible to the occurrence of cavitation in an automotive torque
converter is stall, or zero turbine speed. Converter stall is produced in a
vehicle by applying the brake and throttle simultaneously. This extends the
instantaneous stall condition a vehicle undergoes when launching from a stopped
position. Stall tests are useful for creating extreme conditions to study
phenomena such as cavitation. Also, a vehicle can undergo sustained operation
near stall when climbing a steep grade. The toroidal flow is highly three
dimensional with high element flow incidence angles. The stator has previously
been identified as the critical element in which incipient cavitation bubbles
form as indicated by computational results
of [2, 3]. Governed by
hydrodynamic theory, pressures on the pressure surface of the stator blade are
higher than the suction surface, with the greatest differential located at the
leading edge of the blade. The incidence angle of flow into a stator at stall
can be as high as 70 degrees for some designs, resulting in relatively large
areas of flow separation and regions of low pressure on the suction surface of
the blade. Experimentally, these flow phenomena have been measured using dynamometer
test setups by [4, 5].Fluctuating
pressure measurements were made within the pump passage by [1] which recorded
the pressure disturbances caused by the collapse of the cavitation bubbles as
they traveled into regions of higher pressure within the pump. Mekkes et al.
[6] performed measurements of the fluctuating component of pressure at the
leading edge of the stator, which showed that the fluctuating pressure signal became
damped as cavitation developed at the nose of the stator. The computational investigation
performed by [2] simulated the formation of cavitation bubbles within the
toroidal flow circuit at stall. The results showed that at a particular pump
speed, cavitation bubbles begin to develop at the leading edge of the stator. A
subsequent increase in pump speed causes the pressure differential at the
stator blade leading edge to increase. This
produces an expansion of cavitation and greater displacement of liquid with
vapor.


## 2.2. Acoustical Detection of Cavitation

Nearfield
acoustical measurements were utilized to facilitate testing of multiple
automotive torque converters without the need for specially instrumented
converters. Similar measurement and post processing techniques were developed by
[7] to identify cavitation via the noise produced by the collapse of cavitation
bubbles within the torque converter. It was observed from the sound measurements
that a particular frequency characteristic correlated to the noise produced by
the collapse of incipient cavitation bubbles. Cavitation related noise
manifested itself in the frequency range of 6 to 20 kHz, the maximum measurable
frequency. To remove noise not associated with cavitation, a Kaiser window high-pass
filter with a cutoff frequency of 6 kHz was applied to all acoustical
measurements. Time averaging of filtered nearfield acoustical data at discrete,
equally spaced pump speeds, was performed to further enhance the ability to
identify cavitation inception. Post processing
the nearfield acoustical data in this manner revealed that an abrupt increase
in filtered sound pressure level (SPL) occurred at a particular pump speed. This
pump speed was identified as the onset of cavitation in this investigation. The
specific procedure used to post process the data and identify the exact pump
speed at the onset of cavitation is detailed in [8].An example of a
filtered SPL versus pump speed plot from this study is shown in
Figure2. At the transition to cavitating operation, an abrupt
increase in filtered SPL is evident along with a drastic change in slope.
Experimental investigations,
[9, 10], on centrifugal pumps have demonstrated
the utility in acoustical measurements for detecting cavitation in
turbomachinery. In both investigations, the cavitation threshold was noted by an
abrupt increase in SPL as the turbomachine transitioned from a noncavitating to
cavitating operating regime. Consistent with these investigations, the first
abrupt change in slope of the filtered SPL curve was recognized as the onset of
cavitation.Figure 2
Filtered SPL versus pump speed for
an automotive torque converter operating at stall; vertical dashed line
indicates pump speed at onset of cavitation.


## 3. Experimental Setup and Test Procedure

### 3.1. Dynamometer Test Cell

A torque converter
dynamometer test cell was constructed to acquire nearfield acoustical measurements
for a variety of torque converter designs. The test cell consists of four major
subsystems; dynamometers, a hydraulic supply system, acoustically treated test
fixture, and data acquisition systems, see
Figure3. A 213 kW direct current
(DC) dynamometer was used to simulate engine speeds and torques at stall. A
second DC dynamometer, used to simulate automatic transmission speeds and
torques, was prevented from rotating to simulate stall operation. The hydraulic
system allows setpoint control of the fluid pressures and temperature
superimposed upon the torque converter to measure their impact on cavitation
thresholds. A test fixture with acoustical treatment was necessary to create a
near anechoic environment to accurately measure the cavitation noise produced
within the torque converter. A minimum of 6 inches of acoustical treatment
surrounded the torque converter, providing approximately 20 dB attenuation of standing
waves and background noise. The configuration of torque converter, acoustical
foam, and nearfield microphone is depicted in Figure 4. Two data acquisition
(DAQ) systems were used to acquire data during experimental testing. One DAQ system
acquired nearfield acoustical data and pump speed at 51.2 kHz for determining
the onset of cavitation. The other DAQ system acquired operational speeds,
torques, pressures, temperatures, and flows for use in dimensional analysis.Figure 3
Torque converter dynamometer test
cell, from left to right, hydraulic pump, input dynamometer, test fixture, and
output dynamometer.Figure 4
Torque converter, acoustical
foam, and nearfield micro- phone configuration in test fixture.A standardized
test procedure was developed to transition each torque converter from a
noncavitating to a cavitating condition. A series of stall speed sweep tests
were completed to characterize cavitation thresholds for each torque converter
at various combinations of pressures and input temperature. Each stall speed
sweep test began at a pump speed of 500 RPM and increased at 40 RPM/s until the
test was stopped at pump speeds above the onset of cavitation (~1200 to
2300 RPM). Fourteen stall speed sweeps were conducted for each design with charge
pressure ranging from 483 to 896 kPa, back pressure from 276 to 827 kPa, and
input temperatures of 60, 70, and 80°C.


### 3.2. Experimental Torque Converters

Two automotive
torque converters were tested which maintained near exact geometric scaling. Every
size dimension, such as thicknesses, torus profiles, and clearances was scaled
by the ratio of the two converter diameters. Inlet and outlet blade angles were
the same for both converters. Table1 contains the design information for each
element of the two torque converters tested. The only major difference between
the small diameter, D1, and large diameter, D2, torque
converters was the number of pump blades, 32 and 29, respectively. A difference
in pump blade count was necessary to maintain identical performance metrics as
discussed in the next section. The difference in blade count is not expected to
cause a drastic departure from geometric similitude or to affect the analysis
and conclusion of this investigation.Table 1
Torque converter element design information.
D1D2ElementInlet/outlet
anglesBlade
countInlet/outlet
anglesBlade
countPump−30°/+30°32−30°/+30°29Stator+27°/+70°17+27°/+70°17Turbine+61°/−63°36+61°/−63°36


## 3.1. Dynamometer Test Cell

A torque converter
dynamometer test cell was constructed to acquire nearfield acoustical measurements
for a variety of torque converter designs. The test cell consists of four major
subsystems; dynamometers, a hydraulic supply system, acoustically treated test
fixture, and data acquisition systems, see
Figure3. A 213 kW direct current
(DC) dynamometer was used to simulate engine speeds and torques at stall. A
second DC dynamometer, used to simulate automatic transmission speeds and
torques, was prevented from rotating to simulate stall operation. The hydraulic
system allows setpoint control of the fluid pressures and temperature
superimposed upon the torque converter to measure their impact on cavitation
thresholds. A test fixture with acoustical treatment was necessary to create a
near anechoic environment to accurately measure the cavitation noise produced
within the torque converter. A minimum of 6 inches of acoustical treatment
surrounded the torque converter, providing approximately 20 dB attenuation of standing
waves and background noise. The configuration of torque converter, acoustical
foam, and nearfield microphone is depicted in Figure 4. Two data acquisition
(DAQ) systems were used to acquire data during experimental testing. One DAQ system
acquired nearfield acoustical data and pump speed at 51.2 kHz for determining
the onset of cavitation. The other DAQ system acquired operational speeds,
torques, pressures, temperatures, and flows for use in dimensional analysis.Figure 3
Torque converter dynamometer test
cell, from left to right, hydraulic pump, input dynamometer, test fixture, and
output dynamometer.Figure 4
Torque converter, acoustical
foam, and nearfield micro- phone configuration in test fixture.A standardized
test procedure was developed to transition each torque converter from a
noncavitating to a cavitating condition. A series of stall speed sweep tests
were completed to characterize cavitation thresholds for each torque converter
at various combinations of pressures and input temperature. Each stall speed
sweep test began at a pump speed of 500 RPM and increased at 40 RPM/s until the
test was stopped at pump speeds above the onset of cavitation (~1200 to
2300 RPM). Fourteen stall speed sweeps were conducted for each design with charge
pressure ranging from 483 to 896 kPa, back pressure from 276 to 827 kPa, and
input temperatures of 60, 70, and 80°C.


## 3.2. Experimental Torque Converters

Two automotive
torque converters were tested which maintained near exact geometric scaling. Every
size dimension, such as thicknesses, torus profiles, and clearances was scaled
by the ratio of the two converter diameters. Inlet and outlet blade angles were
the same for both converters. Table1 contains the design information for each
element of the two torque converters tested. The only major difference between
the small diameter, D1, and large diameter, D2, torque
converters was the number of pump blades, 32 and 29, respectively. A difference
in pump blade count was necessary to maintain identical performance metrics as
discussed in the next section. The difference in blade count is not expected to
cause a drastic departure from geometric similitude or to affect the analysis
and conclusion of this investigation.Table 1
Torque converter element design information.
D1D2ElementInlet/outlet
anglesBlade
countInlet/outlet
anglesBlade
countPump−30°/+30°32−30°/+30°29Stator+27°/+70°17+27°/+70°17Turbine+61°/−63°36+61°/−63°36


## 4. Results and Discussion

### 4.1. Nondimensionalizing the Onset of Cavitation

Analysis of test
results from the similar converters was done to determine if cavitation
threshold measurements can be manipulated into dimensionless parameters that
include converter diameter. This would allow speed or torque thresholds determined
experimentally for a given diameter, design and operating point to be scaled to
multiple diameters when exact dimensional scaling and operating points are
preserved.Parameters
commonly used to characterize torque converter performance areK-factor and
unit input speed, U, shown in
(1)K=NpTp,(2)U=KρD5.K-factor is a semidimensionless
parameter relating input torque, and speed is used to compare performances of
converters with constant diameter and constant fluid properties. Test results can be scaled to other diameters
using the dimensionless version of K-factor, unit input speed, U, which
incorporates terms for diameter and fluid density. The inverse square of unit
input speed is a more common presentation of the dimensionless turbomachinery
parameter known as the torque capacity coefficient. The two converters used in
this study have different values for K-factor but the same U because of their
similar torus shape and blade angles. During stall operation, U for
similar converter designs remains fixed, so it is not suitable for scaling
onset of cavitation thresholds. A new dimensionless parameter must be
developed.Due to the
relationship between speed and torque, either could be used to describe
critical operating thresholds at the onset of cavitation. Torque was selected
since torque capacity is an important factor when matching a converter to a
particular engine. Further, stator torque at the onset of cavitation,Ts,i,
was selected over pump torque for comparing different converter designs since
it has been shown that cavitation usually starts in the stator; see [8]. In
general, small diameter torque converters will have higher pump speeds, but
lower torque thresholds at the onset of cavitation than larger diameter units
of similar element designs. This is readily seen in the plots of (a) filtered
SPL versus pump speed and (b) stator torque in
Figure 5. Figure 6 is a plot of Ts,i versus
charge pressure for all fourteen operating conditions. The stator torque
cavitation threshold is greater for the larger diameter torque converter by 95
to 118 Nm with 95% confidence.(a) Filtered SPL versus pump speed
and (b) stator torque for stall speed sweep of 40 RPM/s atpc=791 kPa,
pb=584 kPa, and θin = 70°C.
Vertical dashed lines indicate onset of cavitation.
(a)(b)Figure 6
Stator torque at the onset of
cavitation threshold versus average absolute pressure superimposed upon the
torque converter.Stator torque at incipient
cavitation is a function of those variables that determine the boundary value
problem of the toroidal flow, primarily the size and shape of the torque
converter, blade geometries, and the charge pressure,pc, and
back pressure, pb, superimposed upon the unit. Charge and
back pressures can be transformed into average pressure, pave,
and pressure drop, Δp, which are more suitable for the dimensional
analysis. Secondary variables include cooling through-flow rate, Q, and the
relevant mean fluid properties at a representative temperature. Through-flow
cooling is a function of the controlled operating points of charge pressure and
pressure drop across the unit and was not explicitly included in the list of
variables that affect stator torque at the onset of cavitation.
Equation (3)
represents Ts,i as a
function of both the primary and secondary variables, with all temperature
dependent fluid properties determined at the input temperature of the cooling
flow supplied to the torque converter:
(3)Ts,i=f(D,pave,Δp,ρ,μ,Cp,k).The dimensionless
form of (3)
can be resolved into a function of two dimensionless operating
point parameters using the repeating variables of D, ρ, pave and Cp. The result,
(4), is a dimensionless quantity relating Ts,i to size of the torque
converter and average pressure superimposed upon the unit and will be referred
to as dimensionless stator torque. This quantity is then a function of two
dimensionless operating point parameters, dimensionless operating pressure and the
Prandtl number:
(4)Ts,iD3pave=f(Δppave,Pr).Dimensionless
stator torque represents a similarity condition when geometric similitude is
observed. Thus, the stator torque cavitation threshold for a particular stall
operating point can be scaled to other diameters. The similarity condition for
dimensionless stator torque is expressed in (5).
The subscript 1 represents the
known cavitation threshold and diameter, while subscript 2 represents the cavitation
threshold and diameter that are to be scaled. Figure 7 illustrates this point
more clearly for the converters tested in this investigation. The onset of
cavitation in Figure 7 is indicated by the vertical dashed lines, which differ
by 1.69%. This observation was found
to be valid for all fourteen stall operating
points tested:(5)(Ts,iD3pave)1=(Ts,iD3pave)2.It
should be noted that pave in (5)
must be equivalent, that is, same combination of charge and back
pressures, to properly scale Ts,i from one diameter to another. The ratio of average pressures for each diameter
simply becomes unity, and can be subsequently dropped
from (5).Figure 7
Filtered SPL
versus dimensionless stator torque for stall speed sweep of 40 RPM/s atpc=791 kPa, pb=584 kPa, and θin = 70°C. Vertical dashed
lines indicate onset of cavitation.


### 4.2. Power Product Model

At stall,K-factor
(1)
is a constant value for a given design so long as heavy and sustained
cavitation does not disrupt normal toroidal flow conditions. For a particular
operating point, the onset of cavitation will occur at some pump speed and
corresponding value of pump torque and lay on the K-factor curve for a given
design. Figure 8 contains all Ts,i versus Np,i data, indirectly demonstrating the K-factor relationship for either diameter,
since stator torque can be determined from pump and turbine torques. The results
of Figure 8, however, are not readily applicable to other diameter converters. A
power product model (6)
was fit to dimensionless stator torque values at the onset
of cavitation for the range of stall operating points tested, as noted in
Table 2, to determine the functional relationship for (4)
and develop a predictive tool:
(6)Ts,iD3pave=0.006[(Δppave)0.042Pr0.11].For (6), dimensionless stator
torque is the response variable and the dimensionless quantities of operating pressures,
and Prandtl number are the dimensionless regressors. The coefficients and
exponents of (6)
were determined using a nonlinear least squares fit using a Gaussian
Newton numerical approximation technique. The power product model shown in
Figure 9 had a computed percent root
mean square error (%RMSE) of 3.7%:
(7)%RMSE=∑i=1n(((yi−y^i)/yi)*100)2n−p.From studentized residual and
leverage rule statistical tests, see [11], it was determined that the power
product model did not contain outliers.Table 2
Range of operating point parameters
tested.
Operating pointLowHighAverage pressure481 kPa963 kPaPressure drop69 kPa345 kPaInput temperature60°C80°CFigure 8
Stator torque at the onset of
cavitation plotted against pump speed at the onset of cavitation over all stall
operating points.Figure 9
Dimensionless power product model
fit on dimensionless stator torque for torque converters of exact geometric
scaling, over range of operating points in Table2.Model goodness of fit
was checked by computing the adjusted coefficient of multiple determination,Ra2,
using (8). The value of Ra2 can be interpreted as
the proportionate reduction in the variation in the response given the set of
regressors in the model:
(8)Ra2=1−∑i=1n(yi−y^i)2/n−p∑i=1n(yi−y¯)2/n−1.The computed value was relatively
low, at 0.434, compared to values of around 0.85 that are typical of good fit models.
The low Ra2 value indicates that the model and
regressors did not account for approximately 60% of the variation in
dimensionless stator torque. This is caused mainly by a limited linear
association between the dimensionless response and model parameters. More
specifically, a multitude of least squares solutions exists in terms of
estimated slopes and exponents. Thus, a variety of least squared determined
power product models may be fit to the data set resulting in the same overall
minimized error. However, the low value Ra2 does
not detract from the predictive capability of the model, but its overall
utility is minimal as predictions are limited to scaling cavitation thresholds
for a single set of element designs and torus dimensions.Use of
dimensionless stator torque results in a nearly constant term for incipient
cavitation over the range of operating points tested for a given diameter
converter. It can be noted from Figure10,
a plot of mean and predicted values for each
diameter, that there is little variation between either diameter for all average
pressures (see Figure 10(a))
and input temperatures (see Figure 10(b)) tested. Dimensionless
stator torque does not vary greatly with average pressure (see Figure 10(a)), a
consequence of nondimensionalizing stator torque by average pressure. Some
degree of variation with input temperature is evident (see Figure 10(b)), as it
is not used explicitly or implicitly through fluid properties to
nondimensionalize stator torque at the onset of cavitation. The small variation
in dimensionless stator torque does not particularly lend itself to empirical
modeling. The results are not surprising since designs with near exact
geometric similitude have identical dimensionless quantities that characterize
their design, causing potential regressor terms to fall out of the power
product model equation. For an adequate model to be developed, the variation in
dimensionless stator torque would have to be increased by including test
results from converters that do not follow exact geometric similitude, but are
defined by ranges of design parameters. Additional testing and analysis were
completed and are detailed in Part II of the paper. Results demonstrated a
stronger linear association between dimensionless response and regressors and
increased value of Ra2.Predicted value and mean measured
value of dimensionless stator torque as (a) a function of average pressure and (b) input temperature from PP model.
(a)(b)


## 4.1. Nondimensionalizing the Onset of Cavitation

Analysis of test
results from the similar converters was done to determine if cavitation
threshold measurements can be manipulated into dimensionless parameters that
include converter diameter. This would allow speed or torque thresholds determined
experimentally for a given diameter, design and operating point to be scaled to
multiple diameters when exact dimensional scaling and operating points are
preserved.Parameters
commonly used to characterize torque converter performance areK-factor and
unit input speed, U, shown in
(1)K=NpTp,(2)U=KρD5.K-factor is a semidimensionless
parameter relating input torque, and speed is used to compare performances of
converters with constant diameter and constant fluid properties. Test results can be scaled to other diameters
using the dimensionless version of K-factor, unit input speed, U, which
incorporates terms for diameter and fluid density. The inverse square of unit
input speed is a more common presentation of the dimensionless turbomachinery
parameter known as the torque capacity coefficient. The two converters used in
this study have different values for K-factor but the same U because of their
similar torus shape and blade angles. During stall operation, U for
similar converter designs remains fixed, so it is not suitable for scaling
onset of cavitation thresholds. A new dimensionless parameter must be
developed.Due to the
relationship between speed and torque, either could be used to describe
critical operating thresholds at the onset of cavitation. Torque was selected
since torque capacity is an important factor when matching a converter to a
particular engine. Further, stator torque at the onset of cavitation,Ts,i,
was selected over pump torque for comparing different converter designs since
it has been shown that cavitation usually starts in the stator; see [8]. In
general, small diameter torque converters will have higher pump speeds, but
lower torque thresholds at the onset of cavitation than larger diameter units
of similar element designs. This is readily seen in the plots of (a) filtered
SPL versus pump speed and (b) stator torque in
Figure 5. Figure 6 is a plot of Ts,i versus
charge pressure for all fourteen operating conditions. The stator torque
cavitation threshold is greater for the larger diameter torque converter by 95
to 118 Nm with 95% confidence.(a) Filtered SPL versus pump speed
and (b) stator torque for stall speed sweep of 40 RPM/s atpc=791 kPa,
pb=584 kPa, and θin = 70°C.
Vertical dashed lines indicate onset of cavitation.
(a)(b)Figure 6
Stator torque at the onset of
cavitation threshold versus average absolute pressure superimposed upon the
torque converter.Stator torque at incipient
cavitation is a function of those variables that determine the boundary value
problem of the toroidal flow, primarily the size and shape of the torque
converter, blade geometries, and the charge pressure,pc, and
back pressure, pb, superimposed upon the unit. Charge and
back pressures can be transformed into average pressure, pave,
and pressure drop, Δp, which are more suitable for the dimensional
analysis. Secondary variables include cooling through-flow rate, Q, and the
relevant mean fluid properties at a representative temperature. Through-flow
cooling is a function of the controlled operating points of charge pressure and
pressure drop across the unit and was not explicitly included in the list of
variables that affect stator torque at the onset of cavitation.
Equation (3)
represents Ts,i as a
function of both the primary and secondary variables, with all temperature
dependent fluid properties determined at the input temperature of the cooling
flow supplied to the torque converter:
(3)Ts,i=f(D,pave,Δp,ρ,μ,Cp,k).The dimensionless
form of (3)
can be resolved into a function of two dimensionless operating
point parameters using the repeating variables of D, ρ, pave and Cp. The result,
(4), is a dimensionless quantity relating Ts,i to size of the torque
converter and average pressure superimposed upon the unit and will be referred
to as dimensionless stator torque. This quantity is then a function of two
dimensionless operating point parameters, dimensionless operating pressure and the
Prandtl number:
(4)Ts,iD3pave=f(Δppave,Pr).Dimensionless
stator torque represents a similarity condition when geometric similitude is
observed. Thus, the stator torque cavitation threshold for a particular stall
operating point can be scaled to other diameters. The similarity condition for
dimensionless stator torque is expressed in (5).
The subscript 1 represents the
known cavitation threshold and diameter, while subscript 2 represents the cavitation
threshold and diameter that are to be scaled. Figure 7 illustrates this point
more clearly for the converters tested in this investigation. The onset of
cavitation in Figure 7 is indicated by the vertical dashed lines, which differ
by 1.69%. This observation was found
to be valid for all fourteen stall operating
points tested:(5)(Ts,iD3pave)1=(Ts,iD3pave)2.It
should be noted that pave in (5)
must be equivalent, that is, same combination of charge and back
pressures, to properly scale Ts,i from one diameter to another. The ratio of average pressures for each diameter
simply becomes unity, and can be subsequently dropped
from (5).Figure 7
Filtered SPL
versus dimensionless stator torque for stall speed sweep of 40 RPM/s atpc=791 kPa, pb=584 kPa, and θin = 70°C. Vertical dashed
lines indicate onset of cavitation.


## 4.2. Power Product Model

At stall,K-factor
(1)
is a constant value for a given design so long as heavy and sustained
cavitation does not disrupt normal toroidal flow conditions. For a particular
operating point, the onset of cavitation will occur at some pump speed and
corresponding value of pump torque and lay on the K-factor curve for a given
design. Figure 8 contains all Ts,i versus Np,i data, indirectly demonstrating the K-factor relationship for either diameter,
since stator torque can be determined from pump and turbine torques. The results
of Figure 8, however, are not readily applicable to other diameter converters. A
power product model (6)
was fit to dimensionless stator torque values at the onset
of cavitation for the range of stall operating points tested, as noted in
Table 2, to determine the functional relationship for (4)
and develop a predictive tool:
(6)Ts,iD3pave=0.006[(Δppave)0.042Pr0.11].For (6), dimensionless stator
torque is the response variable and the dimensionless quantities of operating pressures,
and Prandtl number are the dimensionless regressors. The coefficients and
exponents of (6)
were determined using a nonlinear least squares fit using a Gaussian
Newton numerical approximation technique. The power product model shown in
Figure 9 had a computed percent root
mean square error (%RMSE) of 3.7%:
(7)%RMSE=∑i=1n(((yi−y^i)/yi)*100)2n−p.From studentized residual and
leverage rule statistical tests, see [11], it was determined that the power
product model did not contain outliers.Table 2
Range of operating point parameters
tested.
Operating pointLowHighAverage pressure481 kPa963 kPaPressure drop69 kPa345 kPaInput temperature60°C80°CFigure 8
Stator torque at the onset of
cavitation plotted against pump speed at the onset of cavitation over all stall
operating points.Figure 9
Dimensionless power product model
fit on dimensionless stator torque for torque converters of exact geometric
scaling, over range of operating points in Table2.Model goodness of fit
was checked by computing the adjusted coefficient of multiple determination,Ra2,
using (8). The value of Ra2 can be interpreted as
the proportionate reduction in the variation in the response given the set of
regressors in the model:
(8)Ra2=1−∑i=1n(yi−y^i)2/n−p∑i=1n(yi−y¯)2/n−1.The computed value was relatively
low, at 0.434, compared to values of around 0.85 that are typical of good fit models.
The low Ra2 value indicates that the model and
regressors did not account for approximately 60% of the variation in
dimensionless stator torque. This is caused mainly by a limited linear
association between the dimensionless response and model parameters. More
specifically, a multitude of least squares solutions exists in terms of
estimated slopes and exponents. Thus, a variety of least squared determined
power product models may be fit to the data set resulting in the same overall
minimized error. However, the low value Ra2 does
not detract from the predictive capability of the model, but its overall
utility is minimal as predictions are limited to scaling cavitation thresholds
for a single set of element designs and torus dimensions.Use of
dimensionless stator torque results in a nearly constant term for incipient
cavitation over the range of operating points tested for a given diameter
converter. It can be noted from Figure10,
a plot of mean and predicted values for each
diameter, that there is little variation between either diameter for all average
pressures (see Figure 10(a))
and input temperatures (see Figure 10(b)) tested. Dimensionless
stator torque does not vary greatly with average pressure (see Figure 10(a)), a
consequence of nondimensionalizing stator torque by average pressure. Some
degree of variation with input temperature is evident (see Figure 10(b)), as it
is not used explicitly or implicitly through fluid properties to
nondimensionalize stator torque at the onset of cavitation. The small variation
in dimensionless stator torque does not particularly lend itself to empirical
modeling. The results are not surprising since designs with near exact
geometric similitude have identical dimensionless quantities that characterize
their design, causing potential regressor terms to fall out of the power
product model equation. For an adequate model to be developed, the variation in
dimensionless stator torque would have to be increased by including test
results from converters that do not follow exact geometric similitude, but are
defined by ranges of design parameters. Additional testing and analysis were
completed and are detailed in Part II of the paper. Results demonstrated a
stronger linear association between dimensionless response and regressors and
increased value of Ra2.Predicted value and mean measured
value of dimensionless stator torque as (a) a function of average pressure and (b) input temperature from PP model.
(a)(b)


## 5. Conclusions

Dimensional
analysis was applied to experimentally obtained stall operating thresholds at
the onset of cavitation for two torque converters of exact geometric scaling. Stator
torque was determined to be the preferred operating threshold parameter at
cavitation as it properly conveys the torque capacity for a given design and is
most closely associated with the cavitation phenomenon. A scalable term for dimensionless stator
torque was developed by relating cavitation threshold stator torque to the size
of the torque converter and average pressure applied to the unit. Further, a
power product model based upon dimensionless stator torque was developed for
the range of operating points tested, resulting in a %RMSE of 3.7% andRa2 of 0.434. The low value of Ra2 computed for the
model indicates difficulties performing a least squares regression on a data
set of limited variation in response and regressor variables. The shortcomings
associated with the power product model and using converters only of geometric
similitude will be addressed in Part II of this two-part paper, along with
development of a more generalized model for predicting
Ts,i,
applicable to  three element torque converter designs with relaxed geometric
similitude.



---

*Source: 803940-2008-07-20.xml*