A METHOD TO CALCULATE THE FUEL MASS FLOW RATE …tem2/Proceedings_TEMM2018/data/papers/7429.pdf ·...

Proceedings of the 1st Iberic Conference on Theoretical and Experimental Mechanics and Materials /

11th National Congress on Experimental Mechanics. Porto/Portugal 4-7 November 2018.

Ed. J.F. Silva Gomes. INEGI/FEUP (2018); ISBN: 978-989-20-8771-9; pp. 381-396.

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PAPER REF: 7429

A METHOD TO CALCULATE THE FUEL MASS FLOW RATE

CONSUMED BY A DIESEL ENGINE IN DRIVING CYCLES

Pedro de F.V. Carvalheira(*)

Departamento de Engenharia Mecânica, Faculdade de Ciências e Tecnologia da Universidade de Coimbra,

Coimbra, Portugal (*)

Email: [email protected]

ABSTRACT

The present work aims to present a method to calculate the fuel mass flow consumed by a

Diesel engine for any specified engine brake torque, engine rotational speed and lubricating

oil temperature to be used in a computer program to predict the fuel consumption and CO2

emissions of passenger car vehicles in standard driving cycles. The method uses the

information contained in the engine fuel consumption map and the application of the Willans

line method to predict the mass flow consumed by the engine for any working point within

the engine operation range specified by a pair of engine brake torque and engine rotational

speed for normal engine operating temperature. The Willans line method is generally used to

determine friction mean effective pressure of Diesel engines. The effect of lubricating oil

temperature on the fuel mass flow consumed by the engine is taken into account by the

evaluation of the effect of lubricating oil temperature on the evolution of engine friction mean

effective pressure with engine rotational speed. For the envisaged application this method has

the advantages of allowing a rather simple mathematical formulation for calculating with

precision the fuel mass flow consumed by the engine and allowing the calculation of the fuel

mass flow consumed by the engine even when the torque developed by the engine is negative

as it happens in real engines for a certain range of low negative torques, in order to maintain

the smoothness and linearity response of the engine to the throttle control.

Keywords: Fuel mass flow rate, driving cycle, Willans line method, fuel consumption map.

INTRODUCTION

In modelling internal combustion engine driven passenger car standard driving cycles the

calculation of the mass of fuel consumed in the cycle and the CO2 emissions in the cycle are a

main objective. To perform these calculations the fuel mass flow rate consumed by the

vehicle’s engine must be evaluated at any working point specified by any triplet of engine

brake torque, engine rotational speed and lubricating oil temperature within the engine

operating range. Standard driving cycles include working points where the brake torque

developed by the engine can be positive, null or negative. Data presented in engine fuel

consumption maps allow the calculation of fuel mass flow rate consumed by the engine for a

given pair of brake torque and engine rotational speed at normal engine operating temperature

but only if engine brake torque is positive and are generally inaccurate for low positive

torque. To overcome the disadvantages of being inaccurate for low positive torque and not

giving results for null or negative torque this work presents a method to determine accurately

the fuel mass flow rate consumed by the engine at any working point specified by the pair of

engine brake torque and engine rotational speed even when the engine brake torque is

Track-B: Computational Mechanics

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negative or null within the operating range of engine brake torque and engine rotational

speed. This method is based on the construction of Willans lines for several engine rotational

speeds (Millington and Hartles, 1968) from the data presented in the engine fuel consumption

contour map. Moreover in standard driving cycles the engine lubricating oil temperature

changes during the cycle and the fuel mass flow rate for a given engine brake torque and

engine rotational speed change with the engine temperature. In this work a method is also

presented to determine the mass fuel flow rate consumed by the engine at a given engine

brake torque, engine rotational speed and engine lubricating oil temperature.

METHODOLOGY

Following we present the methodology used to calculate the fuel mass flow rate consumed by

the engine at any condition specified by the pair of engine brake torque and engine rotational

speed within the engine operating range. This method is based on the construction of Willans

lines for several engine rotational speeds from the data presented in the engine fuel

consumption contour map. The engine chosen to make this study is the Volkswagen 2.0 TDI

engine code CBDB (Volkwagen, 2015) for which an engine fuel consumption contour map is

published (Ecomodder, 2018).

Table 1 presents some of the main characteristics of the Volkswagen 2.0 TDI engine code

CBDB, which is very similar to the Volkswagen 2.0 TDI engine code CBAB (Volkwagen,

2007), which is a four-stroke turbocharged Diesel engine with intercooler, with four cylinders

in-line, with four valves per cylinder, with common rail direct fuel injection, exhaust gas

recirculation and Diesel particulate filter for exhaust gas treatment and satisfying the Euro 4

emissions standard for passenger car vehicles. Figure 1 presents the Volkswagen 2.0 TDI

engine code CBDB fuel consumption map.

Table 1 - Main characteristics of Volkswagen 2.0 TDI engine code CBDB used in this study.

Type 4 cylinder in-line engine

Displacement volume [cm3] 1968

Bore [mm] 81.0

Stroke [mm] 95.5

Valves per cylinder 4

Compression ratio 16.5

Maximum brake power, kW/rpm 103/4200

Maximum brake torque, N·m/rpm 320/1750-2500

Engine management EDC 17 (Common Rail Control Unit)

Fuel Diesel fuel in accordance with DIN EN 590

Exhaust gas treatment Exhaust gas recirculation, Diesel particulate filter

Emissions standards Euro 4

Proceedings TEMM2018 / CNME2018

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Fig. 1 - Volkswagen 2.0 TDI engine code CBDB fuel consumption map [4].

From the map in Figure 1 were taken data of engine working points specified by pairs of

brake mean effective pressure (bmep) and engine rotational speed for selected engine

rotational speeds. The selected engine rotational speeds were 1000 rpm, 1500 rpm, 2000 rpm,

2500 rpm, 3000 rpm, 3500 rpm, 4000 rpm and 4500 rpm.

As this is a four-stroke cycle engine, the brake torque developed by the engine, in a specified

working point, is given by Eq. (1) where bmep is the brake mean effective pressure of the

working point and �� is the engine displacement volume.

��N ∙ m

bmep�kPa � ��dm�4�

(1)

The fuel mass flow rate consumed by the engine, �� , for each point of the fuel consumption

map specified by a given brake specific fuel consumption, bsfc, for a pair of brake mean

effective pressure, bmep, and engine rotational speed, �, was calculated using Eq. (2).

�� kg/s

bsfc�g/kW ∙ h � bmep�kPa � ��dm� � ��rpm2 � 60 � 3.6 � 10(

(2)

Then for each engine rotational speed the data of bmep and bsfc was collected from the

engine fuel consumption map and �� and �� were calculated respectively using Eq. (1) and

Eq. (2) as presented in Table 2 for � = 1000 rpm. Tables 3 to 9 present the data of bmep and

bsfc that was collected from the engine fuel consumption map and �� and �� calculated respectively for � = 1500 rpm, � = 2000 rpm,� = 2500 rpm,� = 3000 rpm, � = 3500 rpm, � = 4000 rpm and � = 4500 rpm. Then for each engine rotational speed a plot was made of ��


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as a function of bmep, a straight line was fit to the data and the values of the slope � and of

the y-intercept * of the straight line and the square of the correlation coefficient +, of the

straight line fit were recorded for each engine rotational speed as presented in Table 10.

Figures 2 to 5 present a plot that was made of �� as a function of bmep, a straight line fit to

the data, the values of the slope and of the y-intercept of the straight line and the square of the

correlation coefficient of the straight line fit, respectively for� = 1000 rpm, � = 1500 rpm, � = 2000 rpm,� = 2500 rpm,� = 3000 rpm, � = 3500 rpm, � = 4000 rpm and � = 4500 rpm.

Then a plot was made of the evolution of the slope of the straight line fit of �� as a function of bmep for each engine rotational speed with the engine rotational speed as presented in

Figure 6 A polynomial of second degree was fit to the plot of the evolution of the slope of the

straight line fit to the evolution of �� as a function of bmep for each engine rotational speed

with the engine rotational speed. The coefficients of the polynomial of the evolution of the

slope of the straight line fit with the engine rotational speed are presented in Table 11. Then a

plot was made of the evolution of the y-intercept of the straight line fit of �� as a function of bmep for each engine rotational speed with the engine rotational speed as presented in Figure

7. The coefficients of the polynomial of the evolution of the y-intercept of the straight line fit

with the engine rotational speed are presented in Table 11. The value of �� for a given value of engine brake torque �� and engine rotational speed � can be calculated in the following

way. As this engine is a four-stroke cycle the engine bmep for a given engine brake torque �� is calculated using Eq. (3).

bmep�kPa

��N ∙ m � 4��dm�

(3)

The slope of the straight line fit of �� as a function of bmep for the engine rotational speed �, �-�. is given by Eq. (4).

�-�. /,0 � �, + /20 � � + /30 (4)

The y-intercept of the straight line fit of �� as a function of bmep for the engine rotational

speed �, *-�. is given by Eq. (5).

*-�. /,� � �, + /2� � � + /3� (5)

The value of �� for a given value of engine brake torque �� and engine rotational speed � can be calculated by Eq. (6).

�� -�, ��.�kg/s �-�. �

��N ∙ m � 4��dm�

+ *-�. (6)

Table 2 - bmep, bsfc, �� and �� at 1000 rpm.

bmep [kPa] bsfc [g/kW·h] �� [N·m] �� [kg/s]

1100 232 172.31 1.163E-03

975 230 152.73 1.022E-03

615 230 96.34 6.445E-04

470 240 73.62 5.140E-04

341 260 53.42 4.040E-04

118 360 18.48 1.936E-04


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1820 213 285.09 2.650E-03

1725 210 270.21 2.476E-03

1431 210 224.16 2.054E-03

759 220 118.89 1.141E-03

566 230 88.66 8.898E-04

441 240 69.08 7.234E-04

328 260 51.38 5.829E-04

98 360 15.35 2.411E-04

(a) (b)

Fig. 2 - (a) Evolution of �� as a function of bmep, straight line fit to the data, equation of the straight line fit and

square of the correlation coefficient for n = 1000 rpm. (b) Evolution of �� as a function of bmep, straight line fit

to the data, equation of the straight line fit and square of the correlation coefficient for n = 1500 rpm.



2040 201 319.55 3.737E-03

1672 200 261.91 3.047E-03

1266 210 198.31 2.423E-03

934 220 146.31 1.873E-03

660 230 103.38 1.383E-03

484 240 75.82 1.059E-03

336 260 52.63 7.961E-04

105 360 16.45 3.445E-04


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2040 200 319.55 4.648E-03

1630 200 255.33 3.714E-03

943 210 147.72 2.256E-03

784 220 122.81 1.965E-03

702 230 109.96 1.839E-03

616 240 96.49 1.684E-03

410 260 64.22 1.214E-03

108 360 16.92 4.429E-04

(a) (b)






1910 206 299.19 5.379E-03

1426 201 223.37 3.918E-03

875 210 137.06 2.512E-03

697 220 109.18 2.096E-03

579 230 90.70 1.820E-03

505 240 79.11 1.657E-03

400 260 62.66 1.422E-03

236 360 36.97 1.161E-03


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1749 215 273.97 5.997E-03

1630 210 255.33 5.459E-03

1056 210 165.42 3.537E-03

775 220 121.40 2.719E-03

616 230 96.49 2.260E-03

530 240 83.02 2.029E-03

418 260 65.48 1.733E-03

208 360 32.58 1.194E-03

(a) (b)






1590 227 249.06 6.578E-03

1360 220 213.04 5.453E-03

1118 220 175.13 4.483E-03

774 230 121.24 3.245E-03

633 240 99.16 2.769E-03

482 260 75.50 2.284E-03

225 360 35.24 1.476E-03



1348 238 211.16 6.578E-03

869 240 136.12 4.276E-03

570 260 89.29 3.039E-03

238 360 37.28 1.757E-03


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(a) (b)




Table 10 - Slope, y-intercept, square of the correlation coefficient of the straight line fit of �� as a function of bmep and calculated fmep for selected engine rotational speeds.

n [rpm] m [kg/s·kPa] b [kg/s] +, fmep [kPa]

1000 9.8578E-07 6.2195E-05 0.99821 -63.09

1500 1.3766E-06 1.1127E-04 0.99956 -80.83

2000 1.7258E-06 2.1547E-04 0.99905 -124.85

2500 2.1224E-06 3.0137E-04 0.99826 -141.99

3000 2.5411E-06 3.9027E-04 0.99496 -153.58

3500 3.1192E-06 3.9365E-04 0.99625 -126.20

4000 3.7006E-06 4.8737E-04 0.99463 -131.70

4500 4.3511E-06 6.2212E-04 0.99698 -142.98

Fig. 6 - Evolution of the slope of the straight line fit of �� as a function of bmep for each engine rotational speed

with the engine rotational speed.


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Fig. 7 - Evolution of the y-intercept of the straight line fit of �� as a function of bmep for each engine rotational

speed with the engine rotational speed.

Table 11 – Coefficients of the polynomial fit to the evolution of the slope of the straight line fit with the engine

rotational speed and coefficients of the polynomial fit to the evolution of the y-intercept of the straight line fit

with the engine rotational speed.

/,0 /20 /30 /,� /2� /3�

1.0912E-13 3.4691E-10 5.6791E-07 2.4592E-12 1.3941E-7 -8.2247E-5

The Willans line method is generally used to determine friction mean effective pressure of

Diesel engines. This was also made in this work. The friction mean effective pressure for a

given engine rotational speed �, fmep-�., is equal to the x-intercept of the straight line fit made to the plot of �� as a function of bmep for the same engine rotational speed �. The x-intercept of the straight line fit of �� as a function of bmep for a given engine rotational

speed � is given by Eq. (7) where �-�. is the slope of the straight line fit of �� as a function of bmep for the engine rotational speed � and *-�. is the y-intercept of the straight line fit of �� as a function of bmep for the engine rotational speed n.

fmep-�. 5

*-�.�-�.

(7)

Table 10 presents fmep calculated by Eq. (7) for selected engine rotational speeds. Figure (8)

presents the evolution of fmep as a function of the engine rotational speed and a second order

polynomial fit to the plot of fmep as a function of the engine rotational speed. The evolution

of fmep as a function of the engine rotational speed generally follows a second order

polynomial fit according to (Heywood, 1988). Here the evolution of fmep as a function of the

engine rotational speed is very irregular, the second order polynomial does not fit very well to

the data as indicated by the value of the square of the correlation coefficient, and the

concavity of the second order curve is upwards, as shown in Figure (8), while in general the

concavity of the curve fit is to the down side, the opposite side of the curve presented here.


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Fig. 8 - Evolution of fmep as a function of engine rotational speed and a second order polynomial fit to the plot

of fmep as a function of the engine rotational speed.

To get a curve of fmep as a function of engine rotational speed with a shape similar to the

shape expected we decided to eliminate the points of fmep corresponding to engine speeds of

n = 2000 rpm, n = 2500 rpm and n = 3000 rpm. Figure 9 presents this new curve of evolution

of fmep as a function of the engine rotational speed which is very smooth, the second order

polynomial fits very well to the data, as indicated by the value of the square of the correlation

coefficient, and the concavity of the second order curve is still upwards as opposite to the side

expected but is rather smaller. This curve will be compared to the curves of calculated fmep

as a function of engine rotational speed based on a given engine lubricating oil, engine

geometric and operating parameters for selected engine lubricating oil temperatures.

Fig. 9 - Evolution of fmep as a function of engine rotational speed with the points correspondent to engine

rotational speeds of n = 2000 rpm, n = 2500 rpm and n = 3000 rpm removed and a second order polynomial fit to

the plot of fmep as a function of the engine rotational speed.


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FUEL MASS FLOW RATE

The data processed until now was obtained with the engine operating at normal engine

temperature. To obtain the fuel mass flow rate consumed by the engine for a given engine

bmep, engine rotational speed and engine lubricating oil temperature the following method is

used. The engine fmep is calculated as a function of engine speed for selected temperatures of

a selected engine lubricating oil based on the engine geometric and operating parameters. The

fuel mass flow rate consumed by the engine for a given bmep, engine rotational speed and

engine lubricating oil temperature is given by Eq. (8) where �� is in [kg/s],� is in [rpm],

bmep is in [kPa], 6789 is in [ºC], �-�. is in [kg/(s�kPa)] and fmep is in [kPa].

�� -�, bmep, 6789. �-�. � :bmep 5 fmep-�, 6789.; (8)

�-�. in Eq. (8) is calculated using Eq. (4) with the values of /,0, /20 and /30 presented in

Table 11.

ENGINE LUBRICATING OIL

Table 12 presents physical properties of the engine lubricating oil used in the VW 2.0 TDI

engine code CBDB. The engine lubricating oil is the Castrol Edge 5W30 (Castrol, 2013). This

engine lubricating oil satisfies the VW 507.00 standard which the lubricating oil must comply

to be recommended for use in this engine. This engine lubricating oil also satisfies the ACEA

C3 standard. These physical properties allow the calculation of the evolution of the dynamic

viscosity with temperature in a procedure explained in the following text. The evolution of the

engine friction mean effective pressure with the engine temperature is calculated based on the

evolution of the lubricating oil dynamic viscosity with temperature and on engine geometric

data and operating parameters.

Table 12 - Physical properties of Castrol EDGE 5W30 engine lubricating oil

recommended for use in VW 2.0 TDI engine code CBDB [6].

Property Method Castrol EDGE 5W30

SAE Viscosity Grade 5W-30

µ @-30°C [mPa·s] ASTM D5293 5800

ν @40°C [mm2/s] ASTM D445 70.0

ν @100°C [mm2/s] ASTM D445 12.0

HTHS viscosity @150°C [mPa·s] 3.50

Viscosity index ASTM 2270 169

Density @15°C [kg/m3] ASTM D4052 851

The dynamic viscosity <789-�789. of a given engine lubricating oil, in Pa·s, at a given oil temperature �789 is calculated by (Eq. 9), based on the oil kinematic viscosity =789-�789., in m

2/s, and oil density >789-�789., in kg/m

3, at the same temperature.

<789-�789. =789-�789. � >789-�789. (9)

The oil density >789-�789., in kg/m3, at a given oil temperature �789, in K, is given by Eq. (10)

(Maciel, 2000), where �3, the reference temperature is 288.15 K.


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>789-�789.

>789-�3.

exp:@-�789 5 �3.; (10)

@ in (Eq. 10) is given by (Eq. 11) and @A is given by Eq. (12) where for engine lubricating

oils B3 = 0 and B2 = 0.6278 (Maciel, 2000).

@ @A + 0.8@A,-�789 5 �3. (11)

@A

B3 + B2>789-�3.->789-�3..,

(12)

The viscosity data presented in Table 12 for the engine lubricating oil are the kinematic

viscosity for temperatures of 313.15 K (40 ºC) and 373.15 K (100 ºC) and the dynamic

viscosity for temperatures of 243.15 K (-30 ºC) and 423.15 K (150 ºC). Using Eq. (9) to Eq.

(12) the dynamic viscosities at 313.15 K and 373.15 K were calculated for the lubricating oil

based on the kinematic viscosity presented in Table 12 and the calculated density of the

lubricating oil at 313.15 K and 373.15 K respectively. Once we have the data of the dynamic

viscosity for several temperatures for the engine lubricating oil we made a plot of

log-<789-�789./<789-313.15K.. as a function of �789, in K, and fitted a polynomial to the data.

For the engine lubricating oil 5W30 we had the dynamic viscosity data for four temperatures,

243.15 K, 313.15 K, 373.15 K and 423.15 K, so we fitted to the data a polynomial of third

degree. With the coefficients of the polynomials calculated in this way for the lubricating oil,

the dynamic viscosity of the lubricating oil at any lubricating oil temperature, in K, inside the

temperature range where the fit was made can be calculated using Eq. (13).

<789-�789. <789-313.15K. � 10HIAJKLI MHNAJKL

N MHOAJKLMHP (13)

Table 13 presents the calculated dynamic viscosity at 313.15 K and coefficients of the

polynomial to calculate the dynamic viscosity of Castrol EDGE 5W30 engine lubricating oil

as a function of temperature.

Table 13 - Dynamic viscosity at 313.15 K and coefficients of the polynomial to calculate

the dynamic viscosity as a function of the oil temperature, in K, for Castrol EDGE 5W30

engine lubricating oil, recommended for the VW 2.0 TDI engine code CBDB.

Lubricating oil Castrol EDGE 5W30

<789-313.15K. [Pa·s] 5.84655E-2

/� -4.37686E-7

/, 5.25387E-4

/2 -2.18670E-1

/3 3.03963E+1

FRICTION MEAN EFFECTIVE PRESSURE

For the selected engine lubricating oil the engine fmep was calculated as a function of engine

speed for selected temperatures. To calculate fmep the geometric and operating parameters of

the engine were considered. It was calculated the fmep due to the crankshaft main bearings,

connecting rod big end bearings, piston pins bearings, piston-cylinder friction, piston rings-

cylinder friction, camshafts bearings, camshafts cams, crankshaft seals, camshaft seals, oil

pump, water pump, injection pump and alternator. For the calculation of friction of the piston-

cylinder and piston rings-cylinder the viscosity of the oil at normal engine temperature was


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considered, for the friction of the other components the viscosity of the oil at the engine

temperature was considered. Only one temperature was considered for the engine which was

considered equal to the engine lubricating oil temperature. Figures 10(a) and 10(b) present the

evolution of calculated fmep, based on engine geometrical and operating parameters, as a

function of engine rotational speed for Castrol Edge 5W30 engine lubricating oil, respectively

at 25 ºC and 50 ºC.

(a) (b)

Fig. 10 - (a) Evolution of calculated fmep, based on engine geometrical and operating parameters, as a function

of engine rotational speed for Castrol Edge 5W30 engine lubricating oil at 25 ºC. (b) Evolution of calculated

fmep, based on engine geometrical and operating parameters, as a function of engine rotational speed for Castrol

Edge 5W30 engine lubricating oil at 50 ºC.

Figures 11(a) and 11(b) present the evolution of calculated fmep, based on engine geometrical

and operating parameters, as a function of engine rotational speed for Castrol Edge 5W30

engine lubricating oil, respectively at 75 ºC and 100 ºC.

(a) (b)

Fig. 11 - (a) Evolution of calculated fmep, based on engine geometrical and operating parameters, as a function

of engine rotational speed for Castrol Edge 5W30 engine lubricating oil at 75 ºC. (b) Evolution of calculated

fmep, based on engine geometrical and operating parameters, as a function of engine rotational speed for Castrol

Edge 5W30 engine lubricating oil at 100 ºC.

Based on the data presented on Figures 10 and 11 for each engine lubricating oil temperature

fmep can be calculated as a function of � by Eq. (14) where the coefficients of the second

order polynomial are functions of the engine lubricating oil temperature 6789 in ºC.


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fmep-6789, �. /�-6789.�, + *�-6789.� + Q�-6789. (14)

Figures 12(a,b) and 13 present respectively the evolution of /�, *� and Q� with the engine

lubricating oil temperature 6789 in ºC. The evolution of /�, *� and Q� with the engine

lubricating oil temperature can be fitted with polynomials of third order according

respectively to Eq. (15), (16) and (17). The coefficients of the polynomials of third order

fitted to the evolution of /�, *� and Q� with the engine lubricating oil temperature 6789 in ºC are presented in Table 14.

/�-6789. /�H�6789� + /,H�6789, + /2H�6789 + /3H� (15)

*�-6789. /��6789� + /,��6789, + /2��6789 + /3�� (16)

Q�-6789. /�R�6789� + /,R�6789, + /2R�6789 + /3R� (17)

(a) (b)

Fig. 12 - (a) Evolution of /�with the engine lubricating oil temperature 6789 in ºC. (b) Evolution of *�with the

engine lubricating oil temperature 6789 in ºC.

Fig. 13 - Evolution of Q� with the engine lubricating oil temperature 6789 in ºC.

Table 14 - Coefficients of the polynomials of third order fitted to the evolution of

/�, *� and Q� with the engine lubricating oil temperature 6789 in ºC.

i /SH� /S�� /SR�

3 1.8133E-14 2.5580E-07 3.1950E-04

2 -4.4800E-12 -6.3704E-05 -7.9567E-02

1 3.2067E-10 5.4128E-03 6.7596E+00

0 -1.9639E-06 -1.7100E-01 -2.4335E+02


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Eqs. (8), (14), (15), (16) and (17) allow the calculation of the fuel mass flow rate consumed

by the engine for a given bmep, engine rotational speed and engine lubricating oil

temperature.

Fig. 14 - Evolution of fmep as a function of engine rotational speed calculated using the Willans line method and

calculated based on engine geometrical and operating parameters for Castrol Edge 5W30 engine lubricating oil

at temperatures of 50 ºC, 75 ºC, 88 ºC and 100 ºC.

Figure 14 presents the evolution of fmep with engine rotational speed calculated using the

Willans line method as presented in Figure 8 and the evolution of calculated fmep, based on

engine geometrical and operating parameters, as a function of engine rotational speed for

Castrol Edge 5W30 engine lubricating oil at 50 ºC, 75 ºC, 100 ºC, as presented respectively in

Figures 10(b), 11(a) and 11(b) and at 88 ºC. The temperature where the thermostat of the

engine cooling system starts to open is 88 ºC. What was expected to be observe in Figure 14

was that the fmep obtained from Willans line method would correspond to fmep calculated

for lubricating oil temperature close to the temperature where the thermostat of the engine

cooling system starts to open. Figure 14 indicates that points for n = 2000 rpm, n = 2500 rpm

and n = 3000 rpm where obtained for lubricating oil temperatures slightly above 50 ºC, points

for n = 1500 rpm and n = 3500 rpm where obtained for lubricating oil temperatures close to

75 ºC, points for n = 1000 rpm, n = 4000 rpm and n = 4500 rpm where obtained for

lubricating oil temperatures close to 88 ºC.

CONCLUSIONS

A method was developed to calculate the instantaneous fuel mass flow rate consumed by the

engine in any point of a driving cycle. This method allows the calculation of the engine fuel

mass flow rate in any working point in the driving cycle specified by a given bmep, engine

rotational speed and engine lubricating oil temperature. This method is based on the

construction of Willans lines for several engine rotational speeds from the data presented in

the engine fuel consumption contour map. The engine chosen to make this study is the

Volkswagen 2.0 TDI engine code CBDB for which an engine fuel consumption contour map

is published. To calculate the influence of the engine lubricating oil temperature on the engine


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fuel mass flow rate the engine fmep was calculated as a function of engine rotational speed

for a selected engine lubricating oil at selected engine lubricating oil temperatures based on

some detailed information about the engine geometrical and operating parameters.

This method allows the calculation of engine fuel mass flow rate in any working point in a

driving cycle specified by a given bmep, engine rotational speed and engine lubricating oil

temperature from the knowledge of the evolution of bsfc with engine rotational speed at

maximum brake torque, the knowledge of the evolution of engine fmep with engine rotational

speed at normal engine temperature and the availability of some detailed information about

the engine geometrical and operating parameters that allows the calculation of the engine

fmep as a function of engine rotational speed for a given engine lubricating oil temperature.

REFERENCES

[1]-Millington, B.W., and Hartles, E.R., Frictional Losses in Diesel Engines, paper 680590,

SAE Trans., Vol 77, 1968.

[2]-Volkswagen, ETKA - Engine Code, August 2015.

[3]-Volkswagen, Self-study Program 403 - 2.0l TDI Engine with Common Rail Fuel Injection

System - Design and Function, Volkswagen AG, Wolfsburg, October 2007.

[4]-Ecomodder, “Volkswagen Jetta TDI 2.0L 2009 Brake Specific Fuel Consumption Map,”

http://www.ecomodder.com/wiki/index/php/Brake_Specific_Fuel_Consumption_%28BSFC%

29_Maps#Volkswagen_Jetta_TDI_2.0L_2009. Accessed 28/04/2018.

[5]-Heywood, Internal Combustion Engine Fundamentals, McGraw-Hill, 1988.

[6]-Castrol, Castrol EDGE 5W30 Product Data Sheet, 26 September 2013.

[7]-Maciel, I. de F. 2000. Correção de Densidade e Volume, Tabelas API 2540 e ASTM D-

1250 de 1980. Bol. Téc. PETROBRAS, Rio de Janeiro, 43 (1): 11-18, jan./mar. 2000.

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