EPA=AA-SDSB-80-^06
Technical.Report
Quantitative Effects of Acceleration Rate
on Fuel Consumption
by
Randy Jones
April 1980
NOTICE
Technical Reports do not necessarily represent final EPA decisions
or positions. They are intended to present technical analysis of
issues using data which are currently available. The purpose in
the release of such reports is to facilitate the exchange of
technical information and to inform the public of technical deve-
lopments which may form the basis for a final EPA decision, posi-
tion or regulatory action.
Standards Development and Support Branch
Emission Control Technology Division
Office of Mobile Source Air Pollution Control
Office of Air, Noise and Radiation
U.S. Environmental Protection Agency
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Introduction
One factor which has a significant effect on vehicle fuel
consumption is the rate at which the vehicle is accelerated.
Quantitative and qualitative studies have shown that more rapid
and/or more frequent accelerations result in increased fuel con-
sumption.JL/ The EPA has conducted a study to quantify the effects
of operating a vehicle at different acceleration rates.
The "test involved accelerating the vehicle at a constant
acceleration rate to a speed of 55 mph, and maintaining the 55 mph
speed until total distance traveled equaled one mile. Concep-
tually, this may be viewed as entering a freeway system with
different acceleration rates. The acceleration rates varied in
increments of one from 1 to 5 mph/sec. Fuel consumption was
measured with a Fluidyne flow metering system.
Discussion/Procedure
The test was conducted on a Clayton twin roll dynamometer.
The dynamometer rolls were coupled with a motorcycle chain to
prevent tire slippage at the higher acceleration rates. The
vehicle was a 1979 Nova, 250 CID, 1 bbl, with an automatic trans-
mission.
The vehicle was first warmed to stable operating conditions by
being driven over 2 HFET driving cycles. The fuel consumption
tests were conducted starting the flow meter with the engine
running, allowing five seconds of idle and then accelerating at the
desired constant rate until the vehicle reached 55 mph. A 55 mph
cruise was then maintained until a distance of one mile was
travelled from the initial start.
The acceleration rate and cruise speeds were accurately
followed by using a "drivers aid" strip chart on which the desired
acceleration ramps and cruise speed had been previously drawn. At
the higher acceleration rates, if the vehicle could not match the
acceleration trace, the accelerator pedal was depressed fully until
the 55 mph cruise mode was reached.
Five repeat tests were conducted for each acceleration rate.
The order of the tests were randomized to minimize any systematic
fuel consumption effects which might have occurred because of
increasing tire or lubricant temperatures. An example strip chart
recording for a 1.0 mph/sec. acceleration trial is shown in Fig-
ure 1.
The data from the 20 test trials were corrected for actual
distance travelled and fuel temperature to yield a fuel consumption
figure in terms of cc/tnile. The distance travelled was based on
the dynamometer roll revolutions recorded during each test.
An SAE fuel temperature correction for Group 3 test fuel was used
to correct fuel volume measurements.2/ All data are tabulated in
.Appendix A.
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Figure I
ro
tn
o-
to
CO
m
ST
Ul
o
t/7
O
O
START :
Speed Time Trace For A "
1.0 mph/sec Acceleration Trial
! I t
istance =
1.0 Mile
ST(b
fe_ -I
SPEED (mph)
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Results
Fuel consumption increased approximately linearly with in-
creasing acceleration rate. The results are summarized in Table 1
and plotted in Figure II. A linear regression line fit to the data
yielded a correlation coefficient of .93, and, as illustrated in
Figure 2, nearly passes through the standard deviation limits of
all the acceleration trials.
Fuel consumption increased by 10.4 percent between acceler-
ation rates of 1 mph/sec and 4 mph/sec. An apparent exception to
the trend of increased fuel consuption with increased acceleration
rate occurred when vehicle acceleration rate changed from 4 mph/sec
to 5 mph/sec. Here, fuel consumption decreased. This apparent
decrease in fuel consumption is most likely explained by the
difficulty the vehicle encountered in achieving these constant
acceleration rates. The acceleration rates of 3.0, 4.0, and 5.0
mph/sec exceeded the maximum vehicle acceleration capability at the
higher speeds of 30 to 45 mph. The vehicle speed-time traces were
particularly similar for the acceleration rates of 4.0 and 5.0
mph/sec. Driver comments indicated that the accelerator pedal was
fully depressed for approximately 80 percent of the 4.0 mph/sec
acceleration and virtually 100 percent of the 5.0 mph/sec ac-
celeration. Therefore, the actual vehicle accelerations for these
two test acceleration modes differed very little. The standard
deviation values of fuel consumption for the 4.0 and 5.0 mph/sec
trials overlapped, indicating little significance in a difference
in fuel consumption between the tests for these two acceleration
modes.
It is interesting to note the greatest increase in fuel
consumption for any 1 mph/sec increment occurred when the accelera-
tion rate changed from 3.0 to 4.0 mph/sec. The increase was
probably caused by an anonmalous delay in the speed at which the
transmission 1st - 2nd gear shift occurred when the acceleration
rate changed from 3.0 to 4.0 mph/sec. This transmission effect was
discussed in an earlier EPA technical report involving the same
vehicle.jl/ The report predicted a significant increase in fuel
consumption when the vehicle acceleration rate exceeded the maximum
acceleration rate on EPA test cycles, 3.3 mph/sec.
A model to calculate the energy demand on the vehicle for each
different acceleration trial was derived, and actual fuel energy
expended was compared to the theoretical energy demand. The
derivation, calculations, and results are discussed in detail in
Attachment I. The results are summarized in Table 2.
The energy model indicates that most of the increase in fuel
consumption with increased acceleration rates occurs because of the
increased energy demand. For acceleration rates between 1 and 3
mph/sec, the energy demand increased by about 6 percent as did the
fuel consumption. Only when the acceleration exceeded 3 raph/sec
did the fuel consumption increase more rapidly than the energy
demand. . .
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Table 1
Rate of
Acceleration
(mph/sec)
1.0
2.0
3.0
4.0
5.0
No.
of Trials
4
4
4
4
4
Average Fuel
Consumption
(cc/mile)
259.4
270.9
274.1
286.4
282.4
Standard
Deviation
(cc/mile)
4.83
1.39
2.87
6.88
6.79
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Figure II
Fuel Consumption vs. Acceleration Rate
290 _.
6 280
o
u
c
o
1-1
o-i
§270
tn
c
o
o
3
260
250
H h- H
2.0 3.0 4.0
Acceleration Rate (mph/sec)
1.0
5.0
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Table 2
Trial
Acceleration
Rate
1 mph/sec
2 mph/sec
3 tnph/sec
4 mph/sec
Theoretical
Energy
Demand
1.479 x 106J
1.550 x loAj
1.573 x 106J
1.582 x 106J
Fuel
Energy
Expended
8.307 x 106J
8.657 x 10& J
8.778 x 106J
9.171 x 106J
% Eff.
Energy Demand
Energy Expended
17.8
17.9
17.9
*
17.3
Note: The theoretical energy demand assumes the vehicle accurately
followed the acceleration-cruise trace. This was reasonably true
for acceleration rates up to 4 mph/sec. The 5 mph/sec acceleration
significantly exceeded the vehicle acceleration capability, there-
fore the theoretical energy demand for this acceleration was not
included in the table.
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This may also be expressed in terms of a simple energy
efficiency ratio (theoretical energy demand/fuel energy con-
sumed). The efficiency remained almost constant for all tests
except for a notable decrease at the 4.0 mph/sec acceleration. The
decrease may be explained by the transmission effect discussed
earlier.
The increase in energy demand with an increase in acceleration
rate is shown in the model to be a function of the increased time
the vehicle is operated at the 55 mph cruise speed. Since the
aerodynamic drag force, simulated by the dyno power absorber, is
proportional to the square of vehicle speed, more power is absorbed
when the vehicle is accelerated to the cruise speed quickly.
Conclusions
The test vehicle for this experiment exhibited an increase in
fuel consumption with an increase in acceleration rate. Fuel
consumption increased by 10.4 percent when the acceleration rate
increased from 1.0 mph/sec to 4.0 mph/sec.
Computation of the energy demand indicates that most of the
effect occurs because the increased acceleration rates result in
more vehicle operation at the cruise speed inducing more dyno power
absorber work on the vehicle.
However, the maximum fuel consumption effect occured when the
acceleration rate changed from 3.0 to 4.0 mph/sec. This result
probably occurred because of an anomaly in transmission shift
characteristics of this vehicle when the acceleration rate exceeded
the maximum acceleration rates of the EPA test cycles.
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References
lj "Passenger Car Fuel Economy: EPA and Road - A Report to Che
Congress," Draft, U.S. Environmental Protection Agency, April
1980.
2/ "Fuel Economy Measurement Road Test Procedure - Cold Start and
~~ Warm-Up Fuel Economy," SAE J1256, Society of Automotive
Engineers, Warrendale, PA, May 1979.
3/ "The Effect of Acceleration Rate on Automatic Transmission
Shift-Speeds for Two 1979 Novas," R. Jones, EPA Technical
Report, January 1980.
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Appendix A
Fuel Consumption Data
Vehicle: 1979 Nova, silver
Date: 3/1/80
Dyno Cell: D207, rolls coupled, d = 8.65 in.
Barometric Pressure: 29.48 in. Hg
Temperature: WB = 62.5 DB = 74.0
Inertia Weight: 3750 Ib. AHP = 12.9 IHP
Test Tires: Bridgestone bias .
Data
= 10.4
Trial
Acceleration
Rate
(mph/sec)
5
1
2
5
3
4
1
4
5
4
3
1
3
2
3
2
4
2
1
5
Roll
Revolutions
2325
2348
2220
2291
2356
2239
2348
2319
2302
2322
2334
2360
2341
2216
2349
2238
2333
2239
2373
2350
Fuel
Consumed
(cc)
266.7
251.0
252.4
271.2
272.6
265.3
260.3
280.7
278.9
284.5
268.3
256.7
265.3
250.8
265.8
252.6
270.0
251.5
252.1
274.1
Fuel
Temperature
(°c)
40.0
40.0
41.0
41.0
41.0
41.0
41.0
41.0
42.0
41.0
41.0
41.0
41.0
41.0
42.0
41.0
42.0
41.0
42.0
42.0
Fuel Consumption
Corrected for Fuel
Temperature and
Distance
(cc/mile)
274.8
256. f
272.5
283.7
277.3
284.0
265.7
290.1
291.0
293.6
275.8
260.7
271.6
271.3
271.8
270.5
278.0
269.2
255.2
280.2
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Table 1-A
Fuel Consumption (cc/mile)
Trial
1
2
3
4
"x
s
% s/3c
1.0
mph/sec
256.1
265.7
260.7
255.2
259.4
4.83
1.86%
2.0
mph/sec
272.5
271.3
270.5
268.2
270.9
1.39 _.
.50%
3.0
mph/sec
277.3
275.8
271.6
271.8
274.1 "
2.87
1.05%
4.0
mph/sec
284.0
290.1
293.6
278.0
286.4
6.88
2.4%
5.0
mph/sec
274.8
283.7
291.0
280.2
282.4
6.79
2.4%
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Attachment I
Calculation of Vehicle Energy Demand
The total energy demanded from the vehicle over any cycle is
the time integral of the instantaneous power requirement:
Tf
E =J Pdt (1)
Where:
E = the total energy demand,
P = the instantaneous power requirement,
T£ = the initial time of the beginning of the cycle,
Tf = the final time at the end of the cycle.
The power required can, of course, be expressed as the product
of the instantaneous force times the velocity of the vehicle. For
the simple cycles of this project; that is, ramp accelerations
followed by a steady speed cruise, the forces acting on the vehicle
may be expressed as:
F = m ^- + f0 + f2V2 (2)
Physically the first term represents the inertial effect, the
second term primarily represents tire losses, while the third term
represents the aerodynamic drag or dynamometer power absorption.
It should be noted that equation (2) does not contain any term
representing energy dissipated in the vehicle brakes; therefore,
this equation is only applicable to cycles in which the vehicle
brakes are not used.
Combining equations (1) and (2) and integrating:
E = / Fvdt
= J (m |£ + f0 + f2v2)vdt
-------
Tf Tf Tf
/ (m -g£ vdt) + / f0vdt + / f2v3dt
T
V(Tf) Tf Tf
m / vdv + fo/vdt + f2/v3dt
V(Ti) T£ Ti
V(Tf) Tf Tf
= 1/2 mv2
V(Ti)
+ f o / vdt + f2/v3dt (3)
The simple cycles of this report start with the vehicle at
rest; that is:
V(Ti) =0 (4)
Using,
V(Tf) = Vf (5)
The first term of (3) becomes:
V(Tf)
1/2 mv2 = 1/2 mVf2 (6)
V(Ti)
This term is, as expected the kinetic energy of the vehicle at
the final steady speed cruise.
The second term, the time integral of the velocity is, by
definition the distance traveled:
/
vdt = D (7)
Therefore, using equations (6) and (7) in equation (3):
-------
E = 1/2 mVf2 + f0D + f2 /*
3dt (8)
For these cycles, or in fact for any cycles in which the vehicle
brakes are not used, only the aerodynamic drag term dependent on
the detailed velocity versus time characteristics of the cycle.
Because of the simple characteristics of the cycles used in
this program, even the aerodynamic term of equation (8) can be
integrated in closed form. First, the integral can be separated
into two components, the acceleration segment and the cruise seg-
ment .
Tf Tc Tf •
fv3dt = fv3dt + fv3dt (9)
T£ T£ Tc
Where:
Tc = the time at which- the cruise segment is initiated.
Since the acceleration is constant:
Vc = aTc
or, (10)
Tc - Vc/a -
Where:
Vc = the constant cruise velocity
In terms of a and Vc equation (9) becomes:
Tf Tc Tf
I v3dt = T(at)3dt + / Vc3dt
T T . T
Tc Tf
-4*
Tc
-------
Choosing to call the initial time T£ to be zero:
Tf
J v3dt = l/4a3Tc4 + Vc3(Tf - Tc)
(12)
For the particular cycles under investigation the final
velocity is the cruise velocity.
Vf
Therefore:
fo
fo
«0
fo
D H
D ^
D M
D H
^ f2 [l/4a3Tc4 +
- f2[l/4 VC3TC +
H f2Vc3(l/4 Tc +
K f2vc3^Tf ~ 3/4
vc
vc
Tf
Tc
3(Tf -
3Tf
-Tc]
]
Tc)l
VC3TC]
(13)
E = 1/2 mVc2 + f0D + f2 [l/4a3Tr;4 + Vr3(Tf - Te)] (14)
= 1/2 mVc2
= 1/2 mVc2
=1/2 mVc2 + f0D
The time at which cruise condition is reached Tc is given by
equation (10). The condition for Tf is that the total distance
traveled remain constant for the-different accelerations. That
is:
-•C T
D = / atdt + |vc
T- T •
Ll *•!
dt
(15)
= 1/2 at2
Vrt
Calling the initial time T^ = 0,
D = 1/2 aTc2 + Vc(Tf - Tc)
Using equation (10) and solving for Tf,
(16)
D = l/2a -4 + VcTf - VcVc/a
*-
-------
V 2
- VcTf - 1/2 -|
V 2 (17)
V-Tf = D + 1/2 —
a
= 2aD + vc2
2a
Tf - 2aD + Vc2
2aV
Equations (14, (17), and (10) provide the complete algrebraic
expression for the energy demand for the vehicle over the simple
cycles used in this study.
Dynamometer Representation
In the case of dynamometer tests it is reasonable to assume
that the power absorbed by the PAU is proportional to the velocity
cubed, and that the remainder of the power absorbed by the vehicle-
dynamometer system occurs in the tires and dynamometer bearings and
is linear in velocity.
The indicated dynamometer power absorption at 50 mph may,
therefore, be equated to:
IHP(V) = f2V3
(18)
f = IHP(50)
2
(50 mi/hr)3
The vehicle-dynamometer coastdown is a measure of the total
dissipation forces acting on the system and may be used to deter-
mine f():
fO
At
AW
~
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gxample Calculation
In this program, the following parameters were used for all
dynamometer tests:
IHP = 10.4
I = 3750 Ibs. = 1705 kg
At = 12.42 (vehicle-dynamometer coastdown time)
Vc = 55 mi/hr = 24.59 m/sec
Consequently from equation (18):
f = 10.4 hp (20)
2 " (50 ni/hr)3
_ 10.4 hp hr3 , mi .3 ,3600 secv3
125 x 103 mi3 1.609 x 103m hr
= 0.695
Computing fo from equation (19),
Therefore:
f0 = 613 nt - f2V2
2
= 613 nt - 0.695 nt (22.35
m sec
= 613nt - 347
= 266 nt (22)
Considering the specific case of an acceleration of 1 mi/
hr-sec the time to reach the 55 mi/hr cruise speed is given by
equation (10) .
-------
Vc = 55 mi/hr = 24.59 ra/sec
a = 1 mi/hr-sec = 0.447 mi/sec'2
Tc = Vc/a
= 24.59 m/sec (23)
0.447 m/sec2
= 55 sec
The final time at the completion of the cruise is, from
equation (17):
= 2(0.447 m/sec2)(!,.609'm) + (24.59 m/sec)2
f 2(0.447 m/sec2)(24.59 m/sec)
T = 1438 m2/sec2 + 604 m2/sec2
f 22 m/sec3
= 93 sec
The total energy may now be calculated using equation (14) and
the values for fo, f2» Tc, and Tf from the previous equations:
E = 1/2 mVc2 + f0D + f2Vc3 (Tf - 3/4 Tc)
." 1/2(1705 kg)(24'6 m) + (266nt)(1,609m)
S6C
+ 0.695 nt ISfl (liiljE)3 (93 - 3/4 55)sec
m2 sec
= 0.516 x 106 nt m + 0.428 x 106 nt
m
+ o.695 nt (i (51.75)sec
m2 sec
= (0.516 + 0.428 + 0.535) x 106 J
= 1.479 x 106 J
The total energy for the remaining four acceleration rates are
shown below:
2 mph/sec E = (0.516 + 0.428 + 0.606) x 106 J
E = 1.55 x 106 J
3 mph/sec E = (0.516 + 0.428 + 0.629) x 106 J
E = 1.573 x 106 J
-------
4 mph/sec E = (0.516 + 0.428 + 0.638) x 106 J
E = 1.582 x 106 J
5 mph/sec E = (0.516 +0.428 + 0.649) x 106 J
E = 1.593 x 106 J
Comparison to Test Results
The actual fuel energy expended in this experiment is compared
to the theoretical energy demand in Table I and Figure I. A
fuel-energy efficiency value is also shown in Table I. The cal-
culations assume a value of 32023 J/cc of fuel.I/
A linear regression line fit to the data yielded a 0.926
correlation coefficient indicating a linear relation between
theoretical energy demand and fuel energy consumed. The efficiency
ratio of theoretical energy demand to fuel energy expended remains
nearly constant, around 17.7 percent. This also indicates a good
correlation between the two parameters. However, the efficiency-
value for the highest acceleration rate, 5.0 mph/sec, is mis-
leadingly high. Since the higher acceleration rates exceeded the
vehicle acceleration capability, the actual vehicle work was
significantly less than the theoretical energy demand. The energy
efficiency for the 5.0 mph/sec acceleration mode was omitted since
the actual work, as discussed earlier, was nearly the same as the
4.0 mph/sec acceleration trials.
It is interesting to note from the theoretical energy model
that the increased energy demand at the higher acceleration rates
are not a result of the increased acceleration rates themselves.
Mathematically the increase in energy with an increase in ac-
celeration rate results solely from a change in the f2V^, or
aerodynamic drag term, while the .inertial and rolling resistance
terms remain constant. The increased energy demand is a result of
the vehicle reaching a cruising speed sooner and maintaining the
speed for a longer period of time. Thus, the maximum aerodynamic
drag force simulated by the dynamometer power absorber, which
increases with the square of vehicle speed, acts on the vehicle for
a longer period of time when the vehicle is accelerated to a cruise
speed quickly.
The model is a physical approximation of the energy required
to operate this vehicle over the given cycles. The actual fuel
consumption correlates very well to the theoretical energy demand.
However, the model does not treat all vehicle factors which may
influence fuel consumption and which might change with changing
acceleration rates. Examples include automatic transmission shift
speed and peak combustion temperature.
JY D.E. Foringer, "Gasoline Factors Affecting Fuel Economy," SAE
Paper No. 650427.
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Figure A-I
9.20 -L
-> 9.00
x
•o
oi
E .
w
c
<-> 8.80
s*.
oC
m
0)
w
8.60 4-
(9
3
8.40 JL
8.20
1.46
1.48
Theoretical Energy Consumed Vs
Actual Fuel Energy Consumed
1.50 1.52 1V54 . 1.56
. Theoretical Energy Consumed (x 10 )
1.58
1.60
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Table I
Theoretical
Average
Trial
Acceleration Rate
1 mph/sec
2 mph/sec
3 mph/sec
4 mph/sec
Energy Demand for
a 1.0 Mile Trial
1.479 x 106 J
1.550 x 106 J
1.573 x 106 J
1.582 x 106 J
Fuel Energy Expended
for a 1.0 Mile Trial
8.307 x 106 J
8.675 x 106 J
8.778 x 106 J
9.171 x 106 J.
Percent
Efficiency
17.8
17.9 "
17.9
17.3
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