Mathematical Modeling of the Normal Fuel Consumption of Certain Vehicles

The scientific paper introduces original research and unique perspectives on the technical and economic aspects of fuel consumption in motor vehicles. Numerous experts argue that conducting practical experiments, specifically through active determinations, can yield noteworthy outcomes in this domain. To substantiate this viewpoint, the paper includes practical examples of experiments and considerations related to the mathematical model used for validating objective functions. These functions encompass the average fuel consumption in liters per 100 kilometers and the quantity of CO2 emitted per kilometer by the thermal engine of motor vehicles. The research also articulates the authors’ stance, citing the World Health Organization’s recommendation to decrease urban traffic speed from 50 km/h to 30 km/h. This reduction aims to mitigate traffic accidents and curb chemical noise and vibration pollution. The study amalgamates results from tests conducted on both urban and extra-urban routes, incorporating eco-friendly driving practices for comparative purposes. The primary focus of the mathematical model is to validate fuel consumption under real driving conditions. The paper concludes with insights and future research directions in modeling.


Introduction
The motivation for investigating motor vehicle fuel consumption stems from the World Health Organization's recommendations to decrease urban speed limits from 50 km/h to 30 km/h.This suggestion is rooted in the high number of casualties resulting from road accidents and the escalating chemical pollution in the environment.We advocate for specific life-saving measures, including lower speed limits, particularly establishing an implicit 30 km/h limit for cities, expeditious transition to alternative transportation modes, and the establishment of a new European Road Transport Agency [1].
Analyzing the gathered data reveals a persistent global increase in environmental pollution, particularly carbon dioxide and other harmful gases.Concurrently, road accidents continue to rise.In Romania, this trend is attributed to an outdated national vehicle fleet, inadequate road infrastructure for high traffic volumes (limited high-speed roads and poorly connected streets), absence of bypasses, and major traffic arteries passing through city centers (dating back to the '50s-'60s infrastructure).This situation contributes to a significant number of casualties, with children and young adults emerging as the primary victims [2].1303 (2024) 012051 IOP Publishing doi:10.1088/1757-899X/1303/1/012051 2 Fuel consumption represents the maximum allowable quantity of gasoline, diesel, liquefied petroleum gas (LPG), or biofuel a heat engine vehicle can utilize over a specified distance (typically 1000 meters).This evaluation considers prevailing operating conditions, road category, and environmental factors.Stringent emission requirements for vehicle type approval align with evolving European standards.The Worldwide Harmonized Light Vehicle Test Procedure (WLTP), introduced by the EU to standardize global fuel consumption and CO2 emissions, replaced the previous New European Driving Cycle (NEDC) method from September 1, 2017 [5].The WLTP incorporates dynamic parameters to provide more realistic information on fuel consumption and pollutant emissions [6].
Vehicle fuel consumption is influenced by technical characteristics, equipment, and environmental conditions.Factors such as operating conditions, passenger and cargo weight, driving style, ambient temperature, tire type, vehicle aerodynamics (Cx), road category, traffic conditions, and driver comfort all contribute to fuel consumption variability [7, 103-118 p.]. Fuel consumption measurement serves as a performance examination, offering insights into design decisions that contribute to economical operation and reduced operating costs associated with fuel provision [8, p. 96].Considering these aspects, conducting experimental studies is recommended to practically demonstrate whether reducing urban traffic speed leads to a reduction in fuel consumption.

Organizational measures
The steps of the process (procedure) for determining the fuel consumption of motor vehicles are as follows:  Check the tire pressure [9], refuel the tank of the vehicle to full and reset the mileage in the dash to zero;  The engine starts and the car starts, accelerates, and shifts until the set speed is reached (30 km/h, 50 km/h, 130 km/h).This speed is maintained as uniform as possible along the entire route (prescribed route of travel), taking into account traffic conditions or requirements dictated by the road infrastructure.;  During the experimental research, the rules of defensive driving of the vehicle shall be observed as much as possible;  In urban areas, measurements are carried out both for speeds on one route (30 km/h and 50 km/h), and on a route outside the city (motorway) at a speed of 130 km/h, covering the same distance (10 km);  After completing the routes established for carrying out the experimental research, the vehicle and the engine are stopped, the tank is filled in full and calculations on the fuel consumption will be made according to the relation (1).The calculation relationship through wich the fuel consumptions is determinated is the following: [liter/100 km] (1) where, FC represents fuel consumption; AFF -the amount of fuel feed, in litres, completed after completion route for filling the tank, full; d -the distance in kilometers of the route traveled (it is read from the dashboard of the vehicle).

Results and discussions 3.1 Proposed experiment
Experimental studies [10][11] were conducted on a vehicle equipped with a compression ignition (MAC) engine compliant with the European pollution standard Euro 3. The tests followed predefined sections and were conducted on three routes: an urban highway at a speed of 30 km/h, urban routes at a speed of 50 km/h, and highways outside the city at a speed of 130 km/h.The outcomes of these tests are summarized in Table 1.  1) shows the following data:  When driving a city route at a speed of 30 km/h, fuel consumption is 8.1 l/100 km. When driving a city route at a speed of 50 km/h, fuel consumption is 14.2 l/100 km. On the A1 -Sibiu highway, the consumption during the route is 3.5 l/100 km.
Chemically pure hexadecane [12], with a molecular formula n = 16, shares similar properties to diesel fuel, despite diesel fuel containing additional molecules and additives.A straightforward calculation enables the determination of hexadecane's molar mass: 12 x 6 + 1 x (2 x 16 + 2) = 226 g/mol.The CO2 produced by burning one mole of hexadecane is 44 x 16 = 704 g.Consequently, the diesel/CO2 emission ratio is 704 g/226 g = 3.11 g.The density of diesel fuel is 0.850 kg/l.Therefore, the emission of carbon dioxide per gram of burned diesel fuel is 3.11 g.A simple computation reveals that a car utilizing diesel in a heat engine emits 0.85 x 3.11 = 2.64 kg CO2/liter.Utilizing this information, we calculated the corresponding CO2 amounts associated with fuel consumption in columns 2, 4, and 6 of Table 1.
Analysis of the dynamics of fuel consumption (Table 1 and Fig. 1) shows the following data::  When driving a city route at a speed of 30 km/h, fuel consumption is 8.1 l/100 km. When driving a city route at a speed of 50 km/h, fuel consumption is 14.2 l/100 km. On the A1 -Sibiu highway, the consumption during the route is 3.5 l/100 km.The automaker initially endorsed the Worldwide Harmonized Light Vehicle Test Procedure (WLTP) method [5], [6] with the specified fuel consumption values: urban consumption = 5.3 l/100 km; extracity consumption = 3.5 l/100 km (refer to Table 2 and Figure 2).Nevertheless, experimental studies conducted under urban conditions (50 km/h) revealed that the actual fuel consumption was higher than the values declared by the manufacturer.Table 2 and Figure 2 present the manufacturer-approved fuel consumption figures for Renault Symbol 1.5 dCI, Euro-3 vehicles, both in urban and extracity settings.
Table 2. Urban and extra-urban consumption approved by the manufacturer, for the Renault Symbol 1.5 dCI Euro 3, car.

Mathematical model proposed in the framework of experimental research of active type
The proposed mathematical model faithfully reproduces the average fuel consumption obtained in the real case through experimental research.The determination of the parameters a,b,c, so that h (v; a, b, c) solves the problem of minimum, is done by using the method of least squares [13], which consists in determining the parameters a, b, c so that the sum S(a, b, c is minimal.The least squares define the solution that is obtained for minimizing the sum of squares of deviations from the values of the equations.The most important application is to determine the coefficients of a mathematical function that best approximates a set of data.[14].
The result is a Cramer system or a uniquely determined system, the values of the standard fuel consumption, necessary for the validation of the mathematical model, are obtained: After obtaining the values of θ, g, γ, β, α, e și f, according to the calculations in Table no.3, the will be replaced in the mathematical relations (2), ( 3) and ( 4

Conclusions
The dynamics of fuel consumption and CO2 emissions in Renault Symbol 1.5 dCI, Euro 3 vehicles reveal an increasing trend based on experimental studies conducted on two urban routes.The rise in fuel consumption and CO2 emissions is more pronounced.In suburban conditions on the highway, fuel consumption decreases to 3.5 l/100 km, accompanied by a corresponding reduction in CO2 emissions.Within urban settings, reducing the driving speed from 50 km/h to 30 km/h leads to a decrease in fuel consumption from 14.2 l/100 km to 8.1 l/100 km, representing a percentage reduction of 42.95%.Initially, the car manufacturer endorsed the following fuel consumption values using the WLTP method: urban consumption -5.3 l/100 km; extracity consumption -3.5 l/100 km (refer to Fig. 2 and Table 2).However, experimental studies in urban conditions (50 km/h) revealed higher actual fuel consumption than the manufacturer's claims.The increase in fuel consumption in urban conditions is 62.67% compared to the manufacturer's declared consumption, with a corresponding increase in CO2 emissions.In suburban conditions, the observed fuel consumption aligns with the manufacturer's approval.
The mathematical model demonstrates a high degree of correlation, as evidenced by a correlation coefficient -determination equal to 1 (refer to Fig. 3).Therefore, for a Renault Symbol 1.5 dCI, Euro 3 car, the applied mathematical model for average fuel consumption per liter/100 km is valid.Both mathematical formulas and calculations affirm the results obtained during experimental studies.

Figure 1 .
Figure 1.The fuel consumption obtained through experimental research in both urban and extra-urban settings for the Renault.Symbol 1.5 dCI car.

Figure 2 .
Figure 2. Urban and extra-urban consuption approved by the manufacturer, for the Renault Symbol 1,5 dCI car.

Figure 3 .
Figure 3.The graphical representation of the mathematical model for the fuel consumption of the Renault Symbol 1.5 dCI Euro 3 car at speeds of 30 km/h, 50 km/h, and 130 km/h.

Figure 4 .
Figure 4. Fuel consumption values obtained through experimental research in urban and extra-urban, for the speed range developed by the Renault Symbo1,5 dCI Euro 3 car.

Table 1 .
Fuel consumption data, derived from experimental research, for the Renault 1.5 dCI, Euro 3 car, under urban conditions at speeds of 30 km/h and 50 km/h, as well as extra-urban conditions at 130 km/h.

Table 3 .
Mathematical calculation model for obtaining the values of θ, g, γ, β, α, e și f.Note: For v = v0 (for example for the other speeds than those used in the research such as: 10 km/h, 20 km/h, 30,km/h, 40 km/h, 50 km/h, 60 km/h, 70 km/h, 80 km/h, etc, possible to obtain: y