Upload
others
View
18
Download
1
Embed Size (px)
Citation preview
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.
-381-
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
-382-
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
-383-
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 �� �
Track-B: Computational Mechanics
-384-
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
Proceedings TEMM2018 / CNME2018
-385-
Table 3 - bmep, bsfc, �� and �� � at 1500 rpm.
bmep [kPa] bsfc [g/kW·h] �� [N·m] �� � [kg/s]
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.
Table 4 - bmep, bsfc, �� and �� � at 2000 rpm.
bmep [kPa] bsfc [g/kW·h] �� [N·m] �� � [kg/s]
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
Track-B: Computational Mechanics
-386-
Table 5 - bmep, bsfc, �� and �� � at 2500 rpm.
bmep [kPa] bsfc [g/kW·h] �� [N·m] �� � [kg/s]
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)
Fig. 3 - (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 = 2000 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 = 2500 rpm.
Table 6 - bmep, bsfc, �� and �� � at 3000 rpm.
bmep [kPa] bsfc [g/kW·h] �� [N·m] �� � [kg/s]
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
Proceedings TEMM2018 / CNME2018
-387-
Table 7 - bmep, bsfc, �� and �� � at 3500 rpm.
bmep [kPa] bsfc [g/kW·h] �� [N·m] �� � [kg/s]
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)
Fig. 4 - (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 = 3000 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 = 3500 rpm.
Table 8 - bmep, bsfc, �� and �� � at 4000 rpm.
bmep [kPa] bsfc [g/kW·h] �� [N·m] �� � [kg/s]
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
Table 9 - bmep, bsfc, �� and �� � at 4500 rpm.
bmep [kPa] bsfc [g/kW·h] �� [N·m] �� � [kg/s]
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
Track-B: Computational Mechanics
-388-
(a) (b)
Fig. 5 - (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 = 4000 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 = 4500 rpm.
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.
Proceedings TEMM2018 / CNME2018
-389-
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.
Track-B: Computational Mechanics
-390-
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.
Proceedings TEMM2018 / CNME2018
-391-
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.
Track-B: Computational Mechanics
-392-
>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
Proceedings TEMM2018 / CNME2018
-393-
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.
Track-B: Computational Mechanics
-394-
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
Proceedings TEMM2018 / CNME2018
-395-
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
Track-B: Computational Mechanics
-396-
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.