Turbocharger Modeling For Automotive Control Applications

SAE TECHNICAL PAPER SERIES

1999-01-0908

Turbocharger Modeling for Automotive Control Applications Paul Moraal Ford Forschungszentrum Aachen

Ilya Kolmanovsky Ford Research Laboratories

Reprinted From: SI Engine Modeling (SP-1451)

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1999-01-0908

Turbocharger Modeling for Automotive Control Applications Paul Moraal Ford Forschungszentrum Aachen

Ilya Kolmanovsky Ford Research Laboratories Copyright © 1999 Society of Automotive Engineers, Inc.

ABSTRACT

engine models because the standard linear interpolation routine is not continuously differentiable, sometimes leading to apparent discontinuities in simulations. Furthermore, and more seriously, this type of model does not adequately handle operating conditions outside of the mapped data range, for example at very low turbocharger rotational speeds.

Dynamic simulation models of turbocharged Diesel and gasoline engines are increasingly being used for design and initial testing of engine control strategies. The turbocharger submodel is a critical part of the overall model, but its control-oriented modeling has received limited attention thus far. Turbocharger performance maps are typically supplied in table form, however, for inclusion into engine simulation models this form is not well suited. Standard table interpolation routines are not continuously differentiable, extrapolation is unreliable and the table representation is not compact. This paper presents an overview of curve fitting methods for compressor and turbine characteristics overcoming these problems. We include some background on compressor and turbine modeling, limitations to experimental mapping of turbochargers, as well as the implications of the compressor model choice on the overall engine model stiffness and simulation times.

While engine mapping usually covers the entire operating range, the situation for the turbocharger unit is different. Generally, it is possible to obtain the performance characteristics from the supplier. However, the turbocharger characteristics are typically only mapped for higher turbo speeds (typically 90000 RPM and up) and pressure ratios, whereas the operating range on the engine ex2.8 2.6

Compressor map Measured engine data

2.4 2.2

pressure ratio

The emphasis in this paper is on compressor flow rate modeling, since this is both a very challenging problem as well as a crucial part of the overall engine model. For the compressor, four different methods, including neural networks, are presented and tested on three different compressors in terms of curve fitting accuracy, model complexity, genericity and extrapolation capabilities. Curve fitting methods for turbine characteristics are presented for both a wastegated and a variable geometry turbine.

2 180000

1.8 1.6

160000

1.4

140000 120000

1.2 90000 1 0

INTRODUCTION AND BACKGROUND

0.05

0.1

0.15 0.2 0.25 0.3 Scaled compressor flow parameter

0.35

Figure 1. Typical compressor map. Usually, such a compressor map shows constant speedlines and constant isentropic efficiency lines. Here, we have omitted the efficiency lines and instead superimposed the compressor flow rate determined from engine mapping data. It is readily apparent that the compressor mapping data does not cover the operating range of the compressor on the engine. In particular, low speed and low pressure ratio data are lacking.

Dynamic simulation models of turbocharged Diesel and gasoline engines are increasingly being used for design and initial testing of engine control strategies. The turbocharger submodel is a critical part of the overall model, but its control-oriented modeling has received limited attention thus far. A standard approach still appears to be to include the turbocharger performance data in the form of lookup tables directly into the model [6], [12]. However, this form is not ideally suited for use in control-oriented 1

tends down to 10000 RPM shaft speed and pressure ratios close to unity. This is illustrated in Figure 1. As a result, standard interpolation methods (polynomial regression, look-up tables) generally fail to produce reasonable results outside of the region where experimental data is available. In fact, because of the nonlinear nature of the compressor and turbine characteristics even interpolation through lookup tables has been found to cause unacceptable performance in simulations.

HEAT TRANSFER EFFECTS A second problem associated with the low speed region of the maps is caused by the heat conduction from the hot lubricating oil to the cool compressor end. At higher speeds, the conduction can reverse and go from the hot air to the lubricating oil. This heat added to the air is measured by the temperature sensors measuring the temperature rise of the air and used to calculate the work input and hence the efficiency of the compressor and the efficiency of the turbine. It makes the compressor look worse because apparently more turbine work is required than is really used. The same effect makes the turbine look better. This problem is worst at low speeds because of the higher temperature differences. With relatively large areas, higher temperature drops and low flow rates, the backplate and compressor housing are relatively efficient heat exchangers. This effect can be seen on the compressor maps which all show efficiency collapsing at low pressure ratios. This is not true aero performance. On the turbine maps, the opposite effect of the turbine looking better than it really is is apparent. Again this is not true aerodynamic performance. Lower speeds would show the heat transfer effect as more extreme. This means that the predicted transient performance would be in error if the maps were used without correction because the turbine work to drive the compressor is overestimated and more work is actually available to accelerate the rotor. The heat transfer problem is not easily overcome without separate turbine, bearing and compressor dynamometers. When these are in use, the compressor mapping method could be adjusted to control the temperature differences driving the heat transfer so that it could be largely eliminated. However, different oil temperatures may require different bearings to be used and either of these two changes would affect the turbine maps. This is a problem the turbocharger manufacturers are well aware of, but simple solutions presently don't appear to be available.

In this paper we will present a number of different curve fitting methods for turbochargers which allow a more compact implementation of the turbocharger submodel. Special emphasis will be on the abilities of the different methods to extrapolate the performance characteristics into regions which are usually not mapped, but which are encountered during normal operation of the engine. First we'll provide some insight into the limitations of experimental turbocharger mapping.

LIMITATIONS OF EXPERIMENTAL TURBOCHARGER MAPPING Whenever the choice is given, one should obviously aim for extending the range of available experimental data, rather than trying to predict/extrapolate the behavior outside of the given range (unless good models of the underlying physics are available, in which case a few experimentally measured data points may suffice to characterize a large operating region and experimental development time can thus be reduced). With regard to extending the range of experimental mapping of turbocharger units on a flow bench (in particular, lower turbine speeds and lower mass flow rates), two problems arise, which would require significant effort to work around. The following is an explanation provided by David Flaxington [2], at the time working at Allied Signal - Garrett.

FLOW SENSOR ACCURACY The flow through the compressor (the same arguments hold for the turbine) is usually measured by determining the differential pressure across a properly sized orifice placed in the flow path, and using Bernoulli's law, assuming incompressible flow (constant density), to determine the flow rate [3]. This gives a smooth variation of compressor characteristics as the flow rate is varied. One can map at lower speeds if smaller orifice plates and nozzles are used to measure the lower flows, but this produces a problem. As the flow rate is reduced, the accuracy of the readings with any one measuring device reduces. Changing at some point to another measuring device sized for lower flow rates then causes a step change in the mass flow readings as the accuracy of the measurement is again improved.

COMPRESSOR BACKGROUND – The rotating impeller of a centrifugal compressor imparts a high velocity to the air. The air is then decelerated in a diffuser with a consequent rise in static pressure. Neglecting heat losses, the power, P, required to drive the compressor can be related to the mass flow rate through the compressor, W, and the total enthalpy change across the compressor from the first law of thermodynamics as (1) Assuming constant specific heat coefficients, the power is given by

With a vast set of orifice plates and nozzles this problem could be circumvented. However, this is not standard practice and improvements are largely dependant upon the goodwill of the supplier.

(2) The subscripts 01 and 02 refer to the stagnation conditions at the compressor inlet and outlet, respectively. If cp 2

is specified in kJ/kg/K, W in kg/sec and T01,T02 are in K then units of P are kW. For an isentropic process the temperature ratio can be related to the pressure ratio using the relation,

through the compressor [13]. Thus, from equation (10) we obtain (11) C2 Cθ2

(3) To account for the fact that the compression process is not isentropic, the compressor isentropic efficiency, 0 ≤ ηc,is ≤ 1, is introduced and defined as the ratio of theoretical (isentropic) temperature rise and actual temperature rise:

U2

ω

Figure 2. Generic compressor flow velocity diagram at impeller tip, indicating blade tip velocity U2, air flow absolute velocity C2 , and the tangential component of the air leaving the compressor, Cθ2.

(4) Combining equations (3) and (4),

If the dependence of the slip factor on the mass flow rate is neglected, equations (7) and (11) suggest that, ideally (i.e. if ∆h = ∆hideal), the pressure increase across the compressor should only be a function of the turbocharger speed Ntc. However, this is not the case because of the losses that do depend on the mass flow rate, W. The actual enthalpy change across the compressor is larger than ∆hideal and the ratio is precisely the compressor efficiency:

(5) hence,

(6) An insight into various terms comprising equation (6) can be obtained by considering the enthalpy change across the compressor, ∆h. From equations (1) and (6), we obtain the following relation between enthalpy change and pressure risse across the compressor:

(12) There are several sources of losses that can be broadly categorized as (1) incidence losses in the impeller and diffuser caused by flow instantaneously changing direction to comply with geometry, (2) friction losses in impeller, diffuser and collector (e.g. due to viscous drag on the walls), (3) clearance losses in the impeller, (4) losses in the impeller due to the backflow and several others, see [13]. Accounting for these losses individually with physics-based sub-models may be very difficult. Furthermore, it is not clear how to validate the individual submodels from the experimental data. Consequently, one can view their use as simply a way of introducing a particular parametrization with multiple parameters to be regressed from compressor performance maps. Simpler expressions can potentially be generated by only partially restricting ourselves to the structure suggested by the physics based models. This is, basically, the curve fitting approach pursued in this paper.

(7) Assuming that there are no losses (i.e. in the ideal case), ∆h can be estimated from Euler's equation for turbomachinery. Consider a compressor with a radial vaned impeller, no backsweep and no inlet pre-whirl. Then [13], (8) where U2 is the velocity of the impeller at the impeller tip, Cθ2 is the tangential component of the air velocity leaving the impeller, while U1, is the velocity of the impeller at the impeller entry (where air enters the impeller), and Cθ1 the tangential component of the air velocity entering the impeller (see Figure 2 for a velocity diagram). The no inlet pre-whirl assumption implies that Cθ2 = 0 and

The turbocharger manufacturer specifies the performance characteristics in terms of the mass flow rate and isentropic efficiency for varying compressor speeds and pressure ratios on a flow stand and supplies the information as performance maps in table form. It is conventional to specify the performance maps in terms of scaled mass flow rate parameter, φ, and compressor rotational speed parameter, , that are defined as

(9) The ratio (10) is known as the slip factor and depends, for example, on blade spacing, backsweep angle, and mass flow rate 3

or the pressure ratio across the compressor can be modeled as a function of compressor flow Wc and turbine speed Ntc:

(13) where Ntc denotes the compressor rotational speed (rpm). The use of the scaled parameters eliminates the dependence of the performance maps on inlet conditions (Tin and pin). For the remainder of this section, we will assume that the inlet and outlet velocities for the compressor are small enough to ignore the difference between static and stagnation pressure and temperature.

(17) When viewing the compressor model in isolation, Model II appears to be the better choice: • In the areas where the speedlines are almost horizontal,coinciding with a large part of the engine operating regime, as illustrated in Figure 1, this model is less sensitive to input or modeling errors. In Model I, lines of constant pressure ratio intersect the speedlines with a very small angle, resulting in high sensitivity of Wc for small changes in pressure ratio pout/ pin.

The dependencies are

(14) (15)

• Unlike Model II, Model I cannot be easily incorporated into a dynamic model that exhibits surge behavior (as will be made clear below).

Figure 1 shows several speedlines (lines of stable operating points of pressure ratio versus mass flow parameter for constant speeds) for a typical compressor. For each speedline, there are two limits to the flow range. The upper limit is due to choking, when the flow reaches the velocity of sound at some cross-section. In this regime no further flow increase can be obtained by reducing the compressor outlet pressure and the speedline slope becomes infinite. The lower limit is due to a dangerous instability known as surge [13]. During surging a noisy and often violent flow process can occur causing cycle periods of backflow through the whole compressor and the installation downstream the compressor. The specific value of φ at which surge occurs depends not only on the compressor characteristics but also on the properties of the installation downstream of the compressor. Typically, this value is where the slope of the speedline is zero or slightly positive. The left-hand extremities of the speedlines may be joined up to form what is known as the surge line. When the turbocharger compressor is connected to an engine intake manifold, the volume of the manifold is often not sufficient to damp out the pressure fluctuations arising from periodic suction strokes of the pistons. As a result, even though the mean value of φ may lie to the right of a surge line obtained in a steadystate flow stand, the minimum mass flow rate (at the peak of the pulse) may cause surge to develop. Other instabilities that can develop during operation of a centrifugal compressor include stalls. See [13] for a discussion of inducer stalls, impeller stalls and rotating stalls.

However, when the compressor model is implemented as part of an engine model, other aspects need to be taken into account. SENSITIVITY – When the compressor model is connected to the engine model, the equilibrium compressor flow and boost pressure levels are determined from an equilibrium of the engine pumping rate and the compressor flow map. The engine pumping is given by the following equation:

(18) where, for fixed fueling rate, the volumetric efficiency ηvol and engine speed N are only weak functions of boost pressure pi and compressor flow Wc. The subscript i refers to intake manifold conditions. For fixed engine speed, the term ηvol Vd N/(120 R Ti) is sometimes referred to as engine pumping constant, and the pumping map is a linear function of intake manifold pressure. Superimposing the engine pumping map for various values of the engine pumping constant onto the compressor map (Figure 3) illustrates that the sensitivity issue has changed significantly: the engine operating points are determined by the intersection of the two sets of lines. The angle of intersection is never close to zero, but rather around 60-90 degrees, indicating that the sensitivity to compressor modeling error is almost the same for Models I and II. Of course, this analysis is only correct for steady-state operating points. During transients the engine operating trajectory behaves differently with respect to the compressor speed lines due to the fact that the pressure ratio changes much faster than the compressor flow rate.

CURVE FITTING OF COMPRESSOR MASS FLOW For the mathematical representation of the compressor flow characteristics, there are two options. The flow through the compressor can be expressed as a function of pressure ratio pout/pin and turbine shaft speed Ntc

(16) 4

earized system was found to be between -0.5 and -280. The additional state introduced an eigenvalue of -4100 to the system. The resulting increase in simulation time by a factor of 3 - 4 may be unacceptable. However, if the connecting tube between compressor and intake manifold is longer (this may be required for packaging of the intercooler, for example), say, l=100cm, the additional state introduces an eigenvalue of -450 to the system. In this latter case, the simulation time will not be significantly affected. In the remainder of the section we'll describe four different curve fitting techniques for the flow characteristics of a radial compressor. Alternative approaches have been reported in [10].

Figure 3. Generic compressor speed lines and engine pumping map (for fixed engine speeds). The intersections of the two sets of lines determine engine operating equilibria

Jensen & Kristensen Method (5) – In [5], Jensen & Kristensen present a simplification of a model due to Winkler [14]. The model uses the dimensionless head parameter Ψ, equivalent to the slip factor defined in equation [10]:

MODEL STIFFNESS – Model I can easily be incorporated into a mean value engine model with ambient pressure pamb as (constant) input and intake manifold pressure pi as state variable. The intake manifold dynamics are modeled by

(19)

(23)

where the implicit assumptions of constant temperature and absence of EGR are without loss of generality. Using Model I, the compressor flow Wc is simply given by

where Uc is the compressor blade tip speed (24) The normalized compressor flow rate Φ is defined by

(20) Using Model II, an additional state variable must be introduced based on the momentum equation for the air mass mc in the tube connecting the compressor outlet and the intake manifold. With pc denoting the compressor outlet pressure, and A and l representing the cross sectional area and length respectively of the connecting tube, the momentum equation is given by

(25) and the inlet Mach number M introduced by

(26) (21)

The head parameter Ψ and compressor efficiency ηc are then expressed as functions of Φ and M in the following way:

(22)

(27)

hence

Depending on the geometry of the intake assembly, the addition of this state variable can increase the model stiffness considerably. Model stiffness refers to the ratio of smallest to largest eigenvalue of the linearization around a given operating point, or, equivalently, to the difference of time scales in the system’s dynamics. Dynamic models for the complete engine dynamics can be found in, e.g., [5]-[8]. Using typical values for A and l, say A=50cm2 and l=20cm, at a nominal operating point (fuel=1.5 kg/hr, load=40Nm, no EGR), the range of eigenvalues of the lin-

(28) The coefficients k and a are determined through a least squares fit on experimental data. Since equation (27) is invertible, this compressor model is also capable of describing the compressor flow as a function of pressure ratio (Model I) by

5

(29) Depending on the model chosen, the output is then given by (30) (33)

or by

This model is expressed in the form of Model II and cannot easily be inverted to a Model I form. Whatever finite period of time is needed for determining the sign of the square root term in the inversion (during which the sign may be incorrect, and if a globally stable sign detection algorithm can be found at all) will more than likely be unacceptable in dynamic simulations.

(31) Due to the particular choice of basis functions, namely rational polynomials, this method is effective in describing both the flat speedlines at low flow rates, as well as the almost vertical speedlines at high compressor speeds. The functional form (27) has a singularity at Φ=k3. Only the curve to the left of this asymptote is used. During simulations, care must be taken to avoid crossing this singularity during the numerical integration.

Zero Slope Line method (ZSLM) – Another model for the compressor flow map was developed in an internal publication [7]. It describes the compressor flow parameter, φ, as a function of pressure ratio, r, and speed parameter, (Model I). First, the curve connecting the maximum mass flows on each speedline (also referred to as the zero-slope line) is characterized by a quadratic in :

Even though the model is not entirely physics-based, equation (31) shows that extrapolation to low compressor speeds and low mass flow rates causes the resulting pressure ratio to decrease continuously to unity for zero mass flow, which is exactly what one would expect to happen physically. The only constraint on the curve fitting parameters is that k3>0 for all compressor speeds.

(34) where rp,top are the values of the pressure ratio corresponding to φtop. Then, in order to capture the steep slope of the speedlines near the choke limit, the speedlines to the right of the zero-slope line are modeled as exponentials:

Mueller method – In his Master's Thesis [8], Martin Mueller from DTU derives a compressor model from first principles, incorporating the underlying physical principles as well as the compressor assembly geometry. The models thus derived can predict the compressor characteristics surprisingly well. However, once experimental data is available, still better accuracy can always be achieved by appropriate curve fitting.

(35) for rp,< rp,top and are linearly extended to the left of the zero-slope line:

Based on physical considerations, Mueller proposes to model Ψ as a quadratic function in Φ: (32)

(36)

This model is claimed to be generic, however, the parameters A, B, and C are known to be speed dependent and the way in which this speed dependence is modeled is again a design choice. The obvious choice is to model A, B, and C as linear or quadratic in Uc. However, Mueller observes that this choice does not lead to a generic model because it fails to give satisfactory results for some of the compressor types under investigation. It turns out that, rather than fitting A, B, and C independently, it is advantageous to exploit one more observation, namely that the curve connecting the maximum mass flows Wc,top on each speedline is typically a quadratic function in Uc. This then leads to the following parametrization:

where the parameter α is modeled as a constant, or as a function of : (37) Neural networks – Neural networks are becoming increasingly popular for a wide range of applications, including curve fitting, system identification, etc. The use of neural networks for representing turbocharger characteristics is reported in [4], among others. However, even though neural networks are claimed to be universal function approximators, which they are in even a quite general sense, finding the right structure and coefficients requires some amount of trial and error. 6

zation method (also referred to as network training) needs to be computationally very efficient and have builtin mechanisms for escaping from local minima. Over the past couple of years, the issue of computational speed has improved significantly compared to the initial backpropagation algorithms, which are a type of gradient descent method. Commercially available software, such as The Mathworks' Neural Network Toolbox [1], implement a number of different second order methods, such as Levenberg-Marquardt, which are orders of magnitude faster than gradient descent based backpropagation methods.

In [4], a network with one hidden layer and five neurons is used to represent a compressor flow map in the form of model II, i.e., pressure ratio is fitted as a function of mass flow and compressor rotational speed. A network with nu inputs (nu=2 in this case), one hidden layer containing nn neurons and ny outputs (ny=1 for now) would mathematically be represented by

(38) where b and W are coefficient vectors and matrices respectively, and the function f is the neuron transfer function (basis function) typically of the form

The issue of global convergence on the other hand, has not been solved with the same level of success. The crude method is simply to start the optimization many times from different, semi-randomly generated initial conditions (knowledge about the output range allows one to restrict some coefficients to a sensible range). However, this can be quite tedious. A large number of trials with 4 and 5-node networks for the model I compressor map revealed that about one out of ten times the resulting network starts to look acceptable, but a sufficiently accurate fit was not obtained. A more systematic way of dealing with local minima was developed by Puskorias and Feldkamp [9] and, undoubtedly, more exist; however, a review of the neural network literature is beyond the scope of this paper.

(39) In general, such a network requires a total of (nu+ny+1)*nn+ ny coefficients to be fitted. A network with 2 inputs, 5 hidden neurons and 1 output would therefore have 21 parameters to be fitted, the same network with two outputs would have 27 coefficients. Considering the fact that a typical compressor map is specified by 25-40 points, this is a large number of coefficients. Of course, if one given network is used to fit a family of compressor maps, the ratio of data points to coefficients improves substantially.

The conclusion is that for a generic compressor characteristics curve fitting method, neural networks are very well suited for models of type II, but appear not to be suited for models of type I (or, at least, it is not straightforward to find the right structure and training procedure, and significant manual modifications of the mapping data are required).

Here we have used a smaller network to fit the compressor maps; in fact, both mass flow map and efficiency are represented in one single network with two inputs and two outputs. For a compressor map in the form of model II, i.e., pressure ratio is fitted as a function of mass flow and compressor rotational speed, 3 and 4-node networks with 17 and 22 coefficients respectively gave excellent results for the compressor pressure ratio (see Figure 7). The problem of fitting a compressor model of type I, i.e., compressor mass flow as a function of pressure ratio and compressor speed, proved much more difficult - practically impossible actually. Any combination of 3-5 neurons and one or two hidden layers was tried up to 10 times, with initial coefficients generated in a partially random way without success. The problem here is that the fit through the mapped data points may have to be less than optimal in order to get sensible extrapolation results. This type of behavior is difficult to enforce in a generic structure such as a neural network. Even the addition of "artificial" mapping points in the surge region and at low compressor speeds, or the deletion of the positive slope speedline segments was without success.

CURVE FITTING RESULTS FOR THREE DIFFERENT COMPRESSORS In this section we present the curve fitting results on three compressors from two different manufacturers for the methods discussed sofar. The compressors are identified here only as compressors 1,2, and 3. In Figures 47, the curve fitting methods are applied to the compressor flow map for three compressors used for engine displacements in the range from 1.1 to 2.4 liter. The data supplied by the manufacturer is indicated by the solid lines - the speedlines cover speeds in the range of 90 230 kRPM. Superimposed onto the manufacturer's data are the results of the curve fits (dashed lines), extended down to very low turbine speeds (10kRPM, 30kRPM, and 60kRPM) and pressure ratios or mass flows. The extension of the maps gives an indication as to whether the curve fits produce sensible results in extrapolation. Especially for lower turbine speeds, this is important, because they represent operating conditions which are frequently encountered.

Of course, with an increasing number of neurons and layers, the optimization problem to determine the best coefficients increases in dimensionality and number of local extrema. This means that the optimization process will take longer, and at the same time, it may become more difficult to find a sufficiently good initialization because of the larger number of local minima. Therefore, the optimi-

7

It can be noted that all methods yield quite similar results for the lower speed lines of the supplied compressor maps. Also, except for the neural network approach, each method has difficulties describing the highest speed lines for at least one of the given compressors. The extensions into the surge region, i.e., extending the speed lines to the left, yield very different results depending on the curve fitting method chosen. This was to be expected since the methods of Jensen & Kristensen and ZSLM require the speed lines to be strictly monotonic. However, for use in mean value control oriented engine models, any of the proposed extensions will work since they all provide bounded and continuously differentiable extensions, and the surge region is expected to be entered only for very short periods during transients, if at all. Finally, Mueller's method applied to compressors 2 and 3 reveals some slight difficulties with the extension to very low speed lines: the extrapolated speed lines for 30 kRPM in the second and third fit in Figure 5 are obviously incorrect.

Compressor 1 2.8

18

2.6

pressure ratio

2.4

16

2.2 2

14 180000

1.8 12 1.6 1.4 1.2 1 0

160000 9

140000 120000

6 3 1

90000 0.05 0.1 0.15 0.2 scaled mass flow parameter mean eror 0.021426%, std 0.059537

0.25

0.3

Compressor 2 20 3

In order to validate the extrapolation results to the lower speed lines, all the compressor fits were repeated and supplied with the manufacturer's data excluding the lowest speed line. All methods showed acceptable results for the extrapolated lowest speedline. In fact, the methods by Jensen & Kristensen and Mueller yielded almost identical results compared to the fits where all the original data were used for curve fitting.

pressure ratio

18 2.5 16 200000 2

14 180000 12 160000

1.5

CURVE FITTING OF COMPRESSOR EFFICIENCY 1 0

One method for describing the compressor isentropic efficiency was already given in equation (28). It models the efficiency as a quadratic function of the compressor flow rate, with coefficients depending on the speed parameter. For compressor 1, the corresponding efficiency fit is shown in Figure (8)

10

140000 120000

6 3 1

100000 0.05 0.1 0.15 0.2 scaled mass flow parameter mean eror -0.217840%, std 0.094753

0.25

0.3

Compressor 3 3.5

23

21

Alternatively, if the compressor flow is represented using a neural network, the network can be augmented with a second output (and additional nodes if necessary) and be retrained to provide an approximation for the isentropic efficiency as well as for the mass flow rate. This was actually already done for the neural network curvefits shown in Figure (7). For compressor 1, the corresponding efficiency fit is shown in Figure (9)

pressure ratio

3 19 2.5

230000 17 210000

2

15 190000 13

1.5

11 9

1 0

6 3 1

170000 150000 130000 110000 90000 0.05 0.1 0.15 scaled mass flow parameter mean eror 1.664988%, std 0.097928

0.2

0.25

Figure 4. Curve fits for three different compressors using method proposed by Jensen & Kristensen [5].

8

Compressor 1 2.8

compressor data curve fit

2.6

2.6

2.4

2.4 pressure ratio

pressure ratio

2.8

Compressor 1

2.2 2 180000 18

1.8 1.6 1.4

9

1.2

6

1 0

2

160000 140000

16

120000

120000

1.2

90000

14

0.05 0.1 0.15 0.2 scaled mass flow parameter mean eror 0.053687%, std 0.058212

180000

1.8

1.4

140000

3 1

2.2

1.6

160000

12

compressor data curve fit zero slope line

0.25

1 0

0.3

1

900009 3 6 12 14 0.05 0.1 0.15 0.2 0.25 scaled mass flow parameter mean error = -4.73764 % , std = 0.0023737

16 18 0.3

Compressor 2 Compressor 2 compressor data curve fit

3

pressure ratio

pressure ratio

3

compressor data curve fit zero slope line

2.5

200000

20

2.5

200000 2

2

180000 180000 160000 12 10

1.5

160000 140000

120000

100000

1 0

16 14

0.05 0.1 0.15 0.2 0.25 scaled mass flow parameter mean eror 0.041969%, std 0.060226

0.3

3.5

compressor data curve fit

3.5

2.5

pressure ratio

pressure ratio

18 0.3

20

compressor data curve fit zero slope line

3

3

230000 210000

23

190000

190000 170000

90000

0.04 0.08 0.12 scaled mass flow parameter mean eror 0.189370%, std 0.055547

1.5

19

150000 130000 110000

17 0.16

230000 210000

21

170000 11 9 6 3

2.5

2

2

1 0

1

100000 3 6 10 12 14 16 0.05 0.1 0.15 0.2 0.25 scaled mass flow parameter mean error = -5.47799 % , std = 0.0031010

Compressor 3

Compressor 3

1.5

140000

120000

6 3 1 0

1.5

18

1 1 0

15 13 0.2

150000 130000 110000 90000 3 6 9 11 13 0.05 0.1 0.15 scaled mass flow parameter mean error = 0.114804 % , std = 0.00070898

15 23 17 21 19 0.2

0.25

Figure 6. Curve fits for three different comprressors compressors using curve fitting method proposed in [7].

Figure 5. Curve fits for three different compressors using curve fitting method proposed by Mueller [8]

9

Compressor 1 2.8 2.6

compressor data curve fit

0.75

18 Compressor efficiency

pressure ratio

2.4 2.2

16

2 14

180000

1.8 1.6

12

0.7 0.65 0.6 120000 0.55

9

1.2

6 3 1

180000 140000160000

90000

160000

1.4

1 0

Compressor 1

0.8

0.5 140000 120000

0.45

90000

0.05 0.1 0.15 scaled mass flow parameter mean eror 0.002116%, std 0.016118

0.2

0.25

0.4 0

0.3

0.05

0.1

0.15 0.2 0.25 Scaled mass flow parameter

0.3

0.35

Figure 8. Compressor efficienc curve fit for compressor 1 using curve fitting method proposed by Jensen & Kristensen [5].

Compressor 2 compressor data curve fit

3

Compressor 1

0.8

20

18

16

Compressor efficiency

pressure ratio

0.7 2.5

200000

2 14

180000

12 1.5

160000

10

140000

0.2 0.025

0.03 0.1 0

pressure ratio

0.05 0.1 0.15 Scaled mass flow parameter

0.2

0.25

0.3

0.35

Figure 9. Compressor efficiency curve fit for compressor 1 using the same 3 node neural network as was used for the compressor flow rate in the top graph of Figure (7).

compressor data curve fit

TURBINE

23 21

BACKGROUND – The turbine is powered by the energy of the exhaust gas. The power input to the turbine, P, can be obtained from the first law of thermodynamics, neglecting heat transfer, as

19 230000

17

210000 2

0.4

100000

0.005 0.01 0.015 0.02 scaled mass flow parameter mean eror -0.004643%, std 0.014558

3.5

2.5

90000

0.5

0.3

Compressor 3

3

180000 120000 140000 160000

120000

6 3 1 1 0

0.6

15 190000 13

1.5

1 0

11 9 6 3 1

(40)

170000 150000

where W is the mass flow rate of the exhaust through the turbine. As previously, the subscripts 01 02 refer to stagnation conditions at the turbine inlet the turbine outlet, respectively. Treating the exhaust as an ideal gas, we obtain

130000 90000

110000

0.04 0.08 scaled mass flow parameter mean eror 0.312992%, std 0.02571

0.12

0.16

Figure 7. Curve fits for three different compressors using a neural network with 3 neurons in one hidden layer.

gas and and gas

(41)

10

If W is specified in kg/sec, cp in kJ/kg/K, and T in K then P is in kW. For a given pressure ratio across the turbine, the outlet temperature can be computed assuming isentropic expansion,

Typically, the turbine model is used during simulations to calculate turbine power and mass flow rate given inlet and outlet pressure values and turbocharger speed. The turbocharger speed and inlet pressure are usually state variables whose behavior is determined by differential equations based on compressor-turbine power balance and ideal gas law respectively. The turbine outlet pressure is a function of the flow restriction of the exhaust system assembly downstream of the turbine. Ideally, it is equal to atmospheric. As a consequence of this use of the model, it is convenient to express the turbine characteristics in the same form as the compressor Model I, i.e., the turbine mass flow is expressed as a function of turbine pressure ratio and rotational speed.

(42) In order to account for the fact that the expansion through the turbine is not isentropic, the turbine (total-to-static) isentropic efficiency is introduced and defined as

(43)

Using equation (42) and the above defined isentropic efficiency, we obtain the following expression for the turbine power:

The turbocharger manufacturer characterizes the mass flow rate and isentropic efficiency over a certain operating range (typically for turbine speeds between 100 kRPM and 180 kRPM and pressure ratio between 0.3 and 0.8) on a flow stand and supplies the information in table form. For turbines with variable inlet geometry (generically abbreviated as VGT - variable geometry turbochargers), an additional input for these maps is the inlet geometry setting ϑvgt

(44)

Again, we'll use the scaled mass flow parameter, φ, and turbine speed parameter, , and neglect differences between static and stagnation pressures and temperatures:

where T2,is is the temperature of the exhaust gas leaving the turbine if the expansion were isentropic. Note that the turbine outlet temperature is evaluated as static, because no use can be made of the kinetic energy left in the exhaust gas at the turbine outlet.

The turbine outlet temperature T02 is given by (48) (45)

The use of these parameters eliminates the dependence of the performance maps on inlet conditions (Tin and pin). These maps are now only a function of speed parameter, pressure ratio across the turbine, and, if applicable, VGT setting ϑvgt:

Similar to the analysis done for the compressor, it is possible to obtain more insight into various terms that comprise the above equation. From Euler's equation for the turbine rotor we obtain (46)

(49)

where ∆hideal is the ideal (or isentropic) enthalpy drop across the turbine, Cθ1 is the tangential velocity component of the flow at the entry to the rotor and U1 is the velocity of the turbine rotor at the point where the flow enters. Assuming that there is no swirl at the turbine outlet we obtain Cθ2 = 0, and ∆hideal = U1Cθ1. Similar to the compressor slip factor, the ratio Cθ1/U1 is a function of several variables, e.g. the number of rotor blades. Consequently, in the ideal case

and

(50) where U/C is the blade-speed ratio [13], defined as

(51) (47)

and D denotes the turbine blade diameter. Note that the blade speed ratio is a function of pressure ratio and speed parameter, and hence no new independent variables are introduced.

is proportional to the square of the turbo speed. Of course, various sources of energy losses (accounted for by the turbine efficiency) do introduce the dependence on the turbine mass flow rate. See [13] for more details.

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turbine flow, not only do we know that the expansion is not isentropic, but we also know the isentropic efficiency. Hence, we could use a more general form of the equations presented here, taking into account the isentropic efficiency. However, at least for the turbines considered here, this modification did not result in noticeably better fits, hence we'll stick with the standard equations. Therefore, turbine flow equations are given by

It should be noted that the manufacturer supplied data on turbine characteristics is not as representative of actual turbine behavior on the engine as is the case for compressor data. The turbine is characterized on a flow bench with steady flow, whereas the turbine on the engine experiences strong pressure pulsations, which will influence its performance. For that reason, it may be preferable to model the turbine using data measured directly on the engine. However, this does require the turbine speed measurement, which may not always be available. Figure (10) illustrates this to some extent: the flow measured on the engine is slightly higher than the mapped flow for a given mean value of the pressure ratio across the turbine. The figure also clearly illustrates the effect of the wastegate: for expansion ratios close to 2 and higher, an increasing portion of the exhaust flow bypasses the turbine through the wastegate. As a result, the total flow through the exhaust increases without building up additional boost pressure or exhaust manifold pressure. A final observation on Figure (10) is that here, again, we see that the turbine map does not cover the entire region of turbine operation on the engine. Just as with the compressor, lower flows and lower expansion ratios were not mapped. In contrast to the compressor, however, the turbine characteristics can be extrapolated into those regions without real difficulty.

Turbine mass flow parameter φ

where the effective turbine area At . is modeled as a function of turbine pressure ratio and speed parameter. For a fixed geometry turbocharger, a simple fit for the effective turbine flow area At was proposed by Jensen & Kristensen:

(53) where the parameters kti are functions of the speed parameter:

Turbine flow data

0.18 0.16

(52)

This representation is only approximate because the pressure ratio across the turbine at which choked flow occurs is actually lower than the expected value of approximately 0.55 (depending on exhaust gas composition and temperature). The reason is that the turbine effectively behaves as a series of two nozzles (inlet nozzle vanes and rotor passages), which each individually experience higher pressure ratios than the total across the turbine. However, by including the pressure ratio as a parameter in the effective area fit (53), the model can still represent an increasing mass flow parameter at pressure ratios beyond the critical pressure ratio. A second restriction on the validity of the adiabatic nozzle flow model is the fact that, in the limit, the turbine can actually act as a compressor with flow against a positive pressure gradient, whereas in an adiabatic nozzle flow reversal would occur under those conditions.

Turbine map Measured engine data

0.14 0.12 0.1

0.08 0.06 0.04 0.02 0 1

1.5

2 2.5 Turbine expansion ratio

3

3.5

Figure 10. Comparison of turbine flow characteristics provided by the turbine manufacturer and measured on the engine. At higher boost pressures, the wastegate opens and part of the exhaust flow bypasses the turbine. Since the wastegate flow was not measured separately, the turbine flow parameter includes the wastegate flow.

Figure (11) illustrates the results of the turbine flow curve fit. The accuracy is more than adequate. VARIABLE GEOMETRY TURBINE – For variable geometry turbochargers, the turbine area is modified by changing the inlet geometry. For example, in the case of variable nozzle geometry, the throat area of the nozzles is modified, resulting in a change in expansion ratio for the same mass flow rate. One way to model the turbine flow in the presence of a variable inlet geometry is to use equation (51) where the effective area is also a function of the (normalized) geometry setting. For turbine 2 a reasonably good fit could be obtained this way without a turbine speed dependence, i.e., the mass flow parameter is

CURVE FITTING OF TURBINE MASS FLOW FIXED GEOMETRY TURBINE – The mass flow rate through the turbine can be modeled as an adiabatic nozzle flow, where the effective flow area is a function of the turbine speed parameter, and pressure ratio. In [11] it is pointed out that the standard orifice flow equations are derived assuming isentropic expansion. In the case of the

12

sion ratios. However, in light of the comments on turbocharger mapping limitations described earlier, the observed decreased efficiency at lower expansion ratios may be an artifact of heat transfer effects occurring during the experimental procedure.

expressed as a function only of the pressure ratio and the inlet geometry setting. A slight variation of equation (51) is given by

Turbine 2 0.4

Scaled turbine mass flow parameter

φ

1

where 1-g is the theoretical zero flow pressure ratio (the intersection of the curves with the abscissa in Figure 12), and g is fitted as a quadratic function in vane position. The result of this fit is shown in Figure 12. Turbine data for turbine 1

0.24

0.5

0.2 0.333

0.167 0

0

φ Scaled mass flow parameter

0.667

1

1.5

6092

3.0

Figure 12. Curve fit of turbine flow for turbine 2, using modified version of adiabatic nozzle flow. VGT setting, normalized between 0 and 1 is indicated in the graph.

5415

0.16

2.5 2.0 Turbine expansion ratio

4738 4061 3045

0.08 0.4

Turbine data for turbine 2

0 1

1.5

2.0 2.5 Turbine expansion ratio

Scaled turbine mass flow parameter

φ

1

3.0

Figure 11. Curve fit of turbine flow for turbine 1 for different values of the speed parameter. An alternative is to use a neural network with all inputs (speed parameter, pressure ratio, and vane setting) and one hidden layer with three neurons. The result of that fit is shown in Figure 13. An interesting note on extrapolation: the asterisks in Figure 13 mark the extrapolated turbine flows at 10kRPM. It is apparent that these extrapolation results do not make physical sense. Hence, in order to get sensible extrapolation results for low turbine speeds, the neural network needs to be supplied with additional, artificial mapping points forcing the network to provide much lower turbine flows at these low speeds. For the previous fit based on the orifice flow equations the extrapolation did give sensible results.

0.667 0.5

0.2 0.333

0.167 0

0

1.5

2.5 2.0 Turbine expansion ratio

3.0

Figure 13. Curve fit of turbine flow for turbine 2, using 3node neural network. VGT setting, normalized between 0 and 1 is indicated in the graph. The asterisks near the y-axis are the result of extrapolation of the turbine flow parameter to 10kRPM turbine speed for the given VGT settings. This network would have to be retrained and supplied with artificial mapping points at low turbine speeds to force it to give more sensible low speed turbine flows.

CURVE FITTING OF TURBINE EFFICIENCY For fixed turbine speed, the turbine efficiency typically has the shape of an inverted parabola, and can usually be modelled by a quadratic or cubic polynomial in blade speed ratio, with coefficients depending on the speed parameter [5], [7], [8]. In [10], a correction factor at low expansion ratios is used to account for the observed accelerated decrease in efficiency at those lower expan-

The curve fit illustrated in Figure 14 is a quadratic polynomial in blade speed ratio with coefficients linearly dependant on the speed parameter:

(54)

13

SUMMARY AND CONCLUSIONS

empirical model proposed by Jensen & Kristensen appeared to be best suited.

In this paper we have presented an overview of curve fitting techniques for automotive turbochargers, motivated by two facts: Turbocharger mapping data usually do not span the operating range experienced on the engine, hence a need for reliable extrapolation, and, even though it still appears to be commonly used, the representation in lookup tables is not well-suited for implementation in dynamic engine simulation models. .

For the representation of the turbine characteristics we made a similar experience as for the compressor Model I representation: a neural network can give an accurate fit of the mapping points, but a sensible extrapolation to lower expansion ratios and turbine speeds can only be obtained by augmenting the experimental mapping data with a suitable number of articial mapping points in those areas. On the other hand, a more physically based model using adiabatic nozzle flow equations extrapolated very well without manual intervention in the mapping data.

Turbine data for turbine 1

0.75 0.74

ACKNOWLEDGEMENT

Turbine efficiency

0.73 3046

0.72

4738

Thanks to Michiel van Nieuwstadt from Ford Research Laboratories and an anonymous reviewer for careful reading of the draft paper. Thanks also to David Flaxington from Allied Signal/Garrett for a number of fruitful discussions.

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REFERENCES

0.71 4061 0.7 0.69 0.68 0.67 6092 0.66

0.55

0.6

0.65 0.7 Blade speed ratio U/C

0.75

1. Demuth, H., Beale, M., "Neural Network Toolbox, version 3.0", The Mathworks, Inc., 1988.

0.8

Figure 14. Curve fit of turbine efficiency turbine 1 for different values of the speed parameter

2. Flaxington, D., Allied Signal/Garrett. Personal communication. July1996. 3. Fraden, J., "AIP Handbook of modern sensors", AIP Press, 1993.

The emphasis was on compressor flow rate curve fitting for two reasons: engine models are generally more sensitive to errors in mass flow than to errors in temperatures (within reason, of course), and the compressor characteristics are more difficult to capture based on first principles models than the turbine characteristics. In other words, the compressor flow rate representation is both the more critical and the more challenging task. Four different techniques were described and illustrated on three different compressors. Based on the choice of input and output, the methods can be classified into two categories: Model I uses pressure ratio and speed to compute mass flow, Model II uses mass flow and speed to compute pressure ratio. The particular choice has interesting implications on the overall engine model. For a compressor characterisation in the form of Model II, a neural network appeared to be the most accurate technique for all three compressors. It is also the most flexible in that its complexity is easily changed by changing the number of neurons or hidden layers. Furthermore, by combining compressor pressure ratio and isentropic efficiency in one network, the total number of coefficients compares not unfavorably to the other techniques described. For a compressor characterisation in the form of Model I however (typically used in control oriented mean value engine models), a neural network was not found to give an acceptable representation. Sensible extrapolation of the model to lower turbocharger speeds and compressor flow rates could not be obtained, even after manual modifications in the compressor mapping data. Instead, an

4. Nelson, S.A., Filipi, Z.S., Assanis, D.N., "The use of neural networks for matching compressors with diesel engines," Spring Technical Conference, volum ICE-26-3, pages 3542, 1996. 5. Jensen, J.P, Kristensen, A.F., Sorenson, S.C., Houbak, N. , Hendricks, E., "Mean value modeling of a small turbocharged diesel engine," SAE 910070. 6. Kao, M., Moskwa, J.J., "Turbocharged diesel engine modeling for nonlinear engine control and estimation", ASME Journal of Dynamic Systems, Measurement and Control, Vol 117, 1995. 7. Kolmanovsky, I.V., Moraal, P.E., van Nieuwstadt, M.J., Criddle, M., Wood, P., "Modeling and identification of a 2.0 l turbocharged DI diesel engine". Ford internal technical report SR-97-039, 1997. 8. Mueller, M., "Mean value modeling of turbocharged spark ignition engines", Master’s thesis, DTU, Denmark, 1997. 9. Puskorius, G.V., Feldkamp, L.A., "Decoupled extended Kalman filter training of feedforward layered networks", Proceedings IJCNN, 1991. 10. Sher, E., Rakib, S., Luria, D., "A practical model for the performance simulation of an automotive turbocharger", SAE 870295. 11. Sokolov, A.A., Glad, S.T., "Identifiability of turbocharged IC engine models", SAE 1999.

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12. Watson, N., "Dynamic turbocharged diesel engine simulator for electronic control system development", Journal of Dynamic Systems, Measurement, and Control 106, pp.2745, 1984. 13. Watson, N., Janota, M.S., "Turbocharging the internal combustion engine", John Wiley & Sons, 1982. 14. Winkler, G., "Steady state and dynamic modeling of engine turbomachinery systems", PhD Thesis, University of Bath, 1977.

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