New Leiderman–Khlystov Coefficients for Estimating Engine Full Load Characteristics and Performance
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Abstract
The paper presents a method of calculating the full load engine characteristics based on the Leiderman–Khlystov relation. Because the values of the coefficients of the discussed function available in literature were determined for obsolete engine designs, an attempt was made to update them. To this end, a chassis dynamometer was used where a database of results had been built for a variety of vehicles. Following the data collection, the coefficients for variety of fueling system (six groups: fuel injected gasoline and turbocharged gasoline, spark ignition LPG I–II and IV generation, naturally aspirated diesel and turbocharged diesel) were determined. The identification of the coefficients was carried out in Matlab-Simulink indicating the applicability of the said function for most of the engines, yet the recent popularity of turbocharged gasoline engines requires an additional analysis of the possibility of use of a different functional description. The full load engine characteristics is a basis for the vehicle performance characteristics and, further, for modeling of traffic in a variety of aspects of the vehicle operation.
Keywords
Passenger car Combustion engines Full load engine characteristics Numerical identificationAbbreviations
- n
Engine speed (rad/s) or (r/min)
- T
Maximum value of torque (Nm)
- P
Power (kW)
- m
Number of points of the identified curve
- l
Number of significant model coefficients
- A, B, C
Coefficients of the curve
- max
Maximum
- P_{max}
Corresponding to the maximum power
- T_{max}
Corresponding maximum torque
- e
Experimental
- m
Model
- –
Average
- FPE
Final prediction error
- LPG
Liquefied petroleum gas
- V.A.G.
Volkswagen Audi Group
1 Introduction
Despite the fact that hybrid vehicles are offered increasingly, the prevailing source of propulsion are still gasoline and diesel internal combustion engines. An appropriate combination of the elements of the drivetrain must always be made with respect to the vehicle performance. The selection of the engine is usually a combination of technical, economical and ecological aspects. A task of a designer is to make the right choice while facing contradictory factors. The selection of power output on the wheels must allow for the most important criteria for a given vehicle type. For a passenger vehicle, it is the maximum speed and acceleration.
During the design of the drivetrain it is necessary to determine the engine power output and torque as a function of engine speed (full load characteristics) assuming: maximum power output, engine speed and type of fueling system as the input data.
The calculations may be related to the vehicle maintenance/operation and vehicle performance. The differences in the values of the engine indexes at (100, 75, 50, 30)% of the power have been presented because the engine does not always operate at full load [1]. The full load engine characteristics and, thus the performance characteristics constitute bases for the calculations related to the traffic flow on the roads or the automation of the traffic lights operation.
The full load engine characteristics can be used in many aspects, such as modeling of the vehicle in motion—performance with automatic transmission [2, 3], vehicle body behavior [4, 5], performance under different soil conditions [6], when changing lanes [7] or the assessment of motion stability—vehicle-driver in the ADA method [8].
Additionally, the introduction of the said full load engine characteristics as additional information allows a virtual diagnostic assessment in real time (as has been proposed by An et al. [9] or a vehicle acceleration simulation by Stonys et al. [10]).
Full load engine characteristics is also needed for safety issues such as the influence of the driver and vehicle characteristics on the speed-related decisions of the driver on the roads as well as road safety by Rothengatter and Debruin [11].
The simplest method of calculating the full load engine characteristics is the application of the Leiderman–Khlystov relation whose coefficients are obsolete for modern vehicles. On the other hand, based on commercial software (GT Suite) we can calculate the characteristics but the workload will be much higher [12]. In this case, the investigations are based on a particular engine type and in the previous case, generalizations are allowed.
2 Engine Characteristics
The values of flexibility of different groups of engines are given by Grishkevich [13], Myslowski and Koltun [14] wheras those of the 1.9 TDi engine by V.A.G by Szpica and Czaban [15], and also Szpica [16].
Usually for carburetor engines without the engine speed limit λ_{max} = 1.15,…, 1.30; for other engines with the engine speed limit λ_{max} = 0.9,…, 1.15; for diesel engines λ_{max} = 0.9,…, 1.0. The calculations are usually made from values λ = 0.2 to λ_{max} = n_{vmax}/n_{Pmax} [17]. Where n_{vmax} is the engine speed corresponding to the maximum vehicle speed.
Leiderman–Khlystov relation coefficient values [18]
Coefficients | Petrol | Diesel | ||
---|---|---|---|---|
Carburetor | Direct injection | Pre-chamber | Swirl chamber | |
A | 1.0 | 0.5 | 0.7 | 0.6 |
B | 1.0 | 1.5 | 1.3 | 1.4 |
C | 1.0 | 1.0 | 1.0 | 1.0 |
n/n_{Pmax} | \(A\lambda + B\lambda^{2} - C\lambda^{3}\) | |||
0.2 | 0.232 | 0.125 | 0.184 | 0.168 |
0.3 | 0.363 | 0.258 | 0.300 | 0.279 |
0.4 | 0.496 | 0.376 | 0.424 | 0.400 |
0.5 | 0.625 | 0.500 | 0.550 | 0.525 |
0.6 | 0.744 | 0.624 | 0.672 | 0.646 |
0.7 | 0.847 | 0.742 | 0.784 | 0.763 |
0.8 | 0.928 | 0.818 | 0.880 | 0.864 |
0.9 | 0.981 | 0.936 | 0.954 | 0.945 |
1.0 | 1.000 | 1.000 | 1.000 | 1.000 |
1.1 | 0.980 | ‒ | ‒ | ‒ |
Depending on the engine type (gasoline carburetor, direct injected diesel, with a pre-chamber or swirl chamber), we can determine the characteristics having P_{max} at n_{Pmax}.
Lenin relation coefficient values [18]
Engine type | n/n_{max} (%) | |||||
---|---|---|---|---|---|---|
20 | 40 | 60 | 80 | 100 | 120 | |
P/P_{max} (%) | ||||||
Petrol-carburetor | 20 | 50 | 73 | 92 | 100 | 92 |
Diesel-direct injection | 15 | 38 | 62 | 85 | 100 | ‒ |
Diesel-pre-chamber | 17 | 40 | 65 | 84 | 100 | ‒ |
Modern vehicle engines are mostly fuel injected gasoline engines and, turbocharged. In diesel engines, the turbocharged versions are the dominant ones. The literature omits LPG (Liquefied petroleum gas) fueled engines. Only Szpica and Czaban [20] was the attention to this problem drawn and attempts were made to initially estimate the values of the characteristic parameters of the power curve. The results however were based on a small representative group and the publication was only demonstrative.
Partial data for selected engine models are given by Myslowski and Myslowski [21], Prajwowski and Tarczynski [22], Prajwowski and Golebiewski [23].
Hence, the proposal for a more in-depth analysis of the topic along with a comparison against engines that were not directly included in the research.
In the attempts to describe characteristics of a diesel engine fueled with different fuels polynomials are also used, as presented by Stoeck [24], yet the analysis was based on several variants without the statistical analysis on a larger number of samples.
3 Material and Methods
3.1 Subject of the Research
fuel injected gasoline engines—237 units
fuel injected, turbocharged gasoline engines—9 units
spark ignition LPG, I and II generation engine—64 units
spark ignition LPG, IV generation—23 units
naturally aspirated diesel—11 units
turbocharged diesel—175 units
The authors also had the characteristics of the carburetor engines and they were only used to validate information found in the literature.
3.2 Research Methodology
On the LPS 3000 dynamometer operating in the load-applying mode, at a continuous test, the cycle during the measurement is realized assuming a constant acceleration of the roller.
In the preceding investigations, in the beginning of the research cycle preliminary tests were performed. They aimed at checking the reproducibility of the results on one hand (within 3 consecutively repeated tests no errors greater than 1% of the measurement range were recorded) and determining the full load engine characteristics at different settings of the transmission speeds on the other. It has been observed that only the gear ratio close to 1 allows an assessment of the full measurement range. For overdrives the risk of exceeding the maximum vehicle speed range declared by the manufacturer occurs. The test stand also has a security system against accidental vehicle takeoff from the rollers during the tests. It is rather important because the turbocharged diesel engines have a rapidly increasing torque at maximum charging, which may result in the vehicle uncontrolled takeoff from the stand. The chassis dynamometer systems calculate the slip between the front and the rear axles by monitoring the speeds of the front and rear rollers and can reduce the load in hazardous situations.
Basic technical data of the Maha LPS 3000 dynamometer (Maha)
Parameter | Unit | Values |
---|---|---|
Roller set R100/1 | ||
Axle load | t | 2.5 |
Length | mm | 3345 |
Width | mm | 1100 |
Height | mm | 625 |
Weight | kg | approx. 1200 |
Roller length | mm | 750 |
Track min. | mm | 800 |
Track max. | mm | 2300 |
Roller diameter | mm | 318 |
Roller axle separation | mm | 540 |
Running roller protrusion | mm | 45 |
Display range | ||
Test speed | kph | max. 250 |
Wheel power | kW | max. 260 |
Traction | kN | max. 6 |
Rotation speed | r/min | 0–10 000 |
Measurement accuracy of measurement value | % | ± 2 |
Aside from the chassis dynamometer, the integral part of the measurement track was the software made in Matlab-Simulink, Guide [25].
The minimization was performed numerically through a gradient less method of Nelder-Mead simplex [26]. The minimization was performed with the use of Matlab-Simulink, fminsearch procedure [27].
Through a proper selection of coefficients A, B and C we can obtain a high level of experiment to model conformity. The qualitative evaluation of the identification was done through determining of the average and maximum error and the coefficient of determination.
Because the values of the coefficients were identified based on a full range of determined full load engine characteristics, values A and B were sought for and on their basis C was calculated Eq. (9).
4 Results and Discussion
Identification results: average samples, standard deviation, asymmetry factor
Supply system | Number of samples | Average samples | Standard deviation | Asymmetry factor | |||
---|---|---|---|---|---|---|---|
A | B | A | B | A | B | ||
Fuel inj. gasoline | 237 | 0.511162 | 1.617855 | 0.313210 | 0.775373 | − 2.020228 | 1.899060 |
Fuel inj. turbocharged gasoline | 9 | 0.295132 | 2.120719 | 0.316841 | 0.818221 | − 0.413566 | − 0.108163 |
Spark ignition LPG, I and II gen. | 64 | 0.570199 | 1.574281 | 0.279793 | 0.693003 | − 0.279568 | 0.017163 |
Spark ignition LPG, IV gen. | 23 | 0.542335 | 1.544406 | 0.294183 | 0.652313 | 0.856567 | − 0.625658 |
Naturally aspirated diesel | 11 | 0.738505 | 1.194464 | 0.364956 | 0.840597 | − 1.105769 | 1.956045 |
Turbocharged diesel | 175 | 0.261225 | 2.568185 | 0.574244 | 1.348992 | − 0.452776 | 0.529523 |
The value of parameter A in all cases, except LPG IV shows a left-side skewness and B—only for turbocharged gasoline and LPG IV.
Identification results: kurtosis, confidence interval
Supply system | Number of samples | Kurtosis | Confidence interval | ||
---|---|---|---|---|---|
A | B | A | B | ||
Fuel inj. gasoline | 237 | 14.854857 | 15.803642 | 0.477565 | 1.534683 |
Fuel inj. turbocharged gasoline | 9 | 1.655907 | 2.266098 | 0.098739 | 1.613545 |
Spark ignition LPG, I and II gen. | 64 | 2.927065 | 3.067192 | 0.511813 | 1.429668 |
Spark ignition LPG, IV gen. | 23 | 5.101773 | 5.116707 | 0.437003 | 1.310846 |
Naturally aspirated diesel | 11 | 4.916271 | 6.344486 | 0.539064 | 0.735096 |
Turbocharged diesel | 175 | 2.737545 | 2.849502 | 0.189442 | 2.399555 |
The confidence interval, at which the hypothesis of regularity of the distribution with compliance of averages would not be rejected, remained (for all) below the critical values determined based on the standard deviation and the size the individual samples.
In order to provide the qualitative analysis of the identification, histograms indicating the distribution of errors and the value of the maximum error were developed from each of the measurement (with engine speed indicated).
In the further stages, the authors presented the courses of the reference characteristics calculated based on the determined coefficients and Eq. (5) against all recorded measurements in the analyzed groups.
The comparison of all with the model lines marked indicated the necessity of modification of the functional description for both diesel and gasoline turbocharged engines. Supercharging aims at increasing the torque at low engine speeds, thus making the function more convex. Later, the authors are planning to introduce their own function for the description of this type of engines.
5 Assessment of Application
Model to experiment comparison of the maximum power and torque
Engine | BMW 1.8MPi | Audi 1.8T | Audi 1.8 LPG I | Seat 1.8 LPG IV | Renault 1.8D | Toyota 2.0TD | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
P (kW) n (r/min) | T (Nm) n (r/min) | P (kW) n (r/min) | T (Nm) n (r/min) | P (kW) n (r/min) | T (Nm) n (r/min) | P (kW) n (r/min) | T (Nm) n (r/min) | P (kW) n (r/min) | T (Nm) n (r/min) | P (kW) n (r/min) | T (Nm) n (r/min) | |
Exp. | 84.58 | 154.30 | 125.11 | 281.60 | 60.09 | 146.10 | 63.03 | 121.60 | 63.98 | 175.80 | 102.75 | 331.20 |
5900 | 4200 | 5300 | 3400 | 4400 | 3300 | 5500 | 4100 | 4100 | 2200 | 3600 | 2300 | |
Mod. | 85.44 | 149.33 | 126.08 | 245.54 | 60.09 | 144.97 | 63.03 | 119.41 | 64.43 | 167.05 | 102.78 | 316.85 |
6100 | 4200 | 5500 | 4000 | 4400 | 3000 | 5500 | 3900 | 4200 | 2600 | 3500 | 2500 |
Having appropriate software for the identification, A = 0.831731, B = 1.205236 and C = 1.036967 were determined. The maximum difference of 6% was obtained in this way (Figure 25b).
6 Conclusions
- (1)
A method of calculation of full load engine characteristics has been presented in the paper based on hyperbolic equation where, at a correct selection, we can obtain compliance of the model with the experiment (these differences do not exceed 5% in most of the treated cases).
- (2)
The values of the characteristic coefficients of the power equation have been determined for engines that have not yet been analyzed in the literature (LPG).
- (3)
The correctness of the values presented for older variants of fuel systems has been confirmed.
- (4)
An applicability of the determined values was confirmed by the comparison with example engines, which were not part of the main research.
- (5)
The number of the tested vehicles with gasoline fuel injected and turbocharged engines and naturally aspirated diesel was insufficient for the statistical evaluation.
- (6)
The recently very popular turbocharging, particularly in spark ignition engines, forces the application of other functions (such a polynomials) for the description of the engine characteristics.
- (7)
It should be noted that the number of the tested vehicles was rather low compared to the number of available makes, models of the distinguished vehicle groups. That is why, some of the results were supported with vehicle names, which should not constitute a basis for the evaluation of the entire model group of a given make.
It is noteworthy that the test objects were random products of a given brand and model, which is why the results were supplemented by the manufacturers names. This however cannot be a basis for the assessment of the entire model group of a given make.
Notes
Authors’ Contributions
DS was in charge of the whole manuscript. The author read and approved the final manuscript.
Authors’ Information
Dariusz Szpica, born in 1971, is currently a PhD MEng at Faculty of Mechanical Engineering, Bialystok University of Technology, Poland. His research interests include fuel supply in combustion engines.
Competing Interests
The author declares no competing interests.
Funding
The research has been carried out within work no. S/WM/1/2018 realized at Bialystok University of Technology and financed from the funding allocated for science by the Ministry of Science and Higher Education—Poland.
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