The application of Integration to measure volumes
Introduction:
For my year in industry, I am placed with Delphi Diesel systems.
Delphi design, develop and manufacture fuel injectors for many
of the world’s leading truck engine manufacturers.
Fuel injectors expel fuel into the cylinder at high pressures.
When compressed, the fuel and air combust, forcing the piston
down.
This product delivers multiple volumes of fuel every second.
Set Rail Pressure = 800 bar
PE1750-03
Replt5P3v2 case 3
Replt5P3v2 case 3
20
10
Volts
0
To (V)
measure
the volume of fuel expelled, the injector injects90into a cylinder of known diameter,
-10
80
-20
displacing
a piston. Due to a restriction imposed
by the custom
setup of the measurement system,
DFI5.1 Gain Curves
70
-30
we are
the final
-40 unable to measure the injected volume from
60 position of the piston. Instead, we
Poster
Set Rail Pressure = 800 bar
50injected.
have140
to use other measurements to calculate the total Fuel
fuel810
Rail
3
40
40
(mm
Pres) 800
120
30
30injection. This is the amount of fuel
(bar)of
From100
the
piston’s
speed
we
are
able
to
measure
the
rate
790
20
20
100
10
delivered
by the product, per unit time- i.e. mm³/s. We are10
also able to measure the time period for
Volts 800
Rate
90
(V) 60
[mm³/ms]
-10
which the injector is injecting. Using this data, we can plot a800rate trace graph.
-20
40
3*SD703
-30
20
Fuel602
-40
1
3
(mm ) 0
0
50
Fuel 600
140
0.1
3 40
(mmThis
) 550 makes it clear to see that the area
120
0.08
SOrate 500
30
100
450 the graph (for the time period) is
(us)
under
0.06
20
400
Rate 80
Dvolts
650
equal
10 to the total volume of fuel injected. In
0.04
[mm³/ms]
(V/us) 60
6000
its most crude form, we could estimate the
0.02
40
5503
3*SD
2 by counting the squares.
area
200
Fuel
5001
3
(mm ) 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6 EOrate 450
0.10
600
Length
Time from Start of Cur. (ms)
550 of
(us) 400
Time (ms)
0.08
500
SOrate
injection
350
Test Trc Inj
Stat
NCV
PSG
PSGd NdlNoz
VL FG Comment
(us) 450
16810 2220.06
5831CMaster Master Master Master Master
Master
Analyser 3 5 Trial
300
400
Dvolts
650
250
0.04
(V/us)
600
200
0.02
350
550
0
300
Total
number of squares ≈ 8
500
250
T4glt Volume
per square= 20 x 0.2= 4mm³
450
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6 EOrate
us 200
However, we requireTime
a much
more
accurate
answer.
from Start of Cur. (ms)
(us) 400
0.2
0.3
0.5
0.6
0.7
0.8
0.9
3500.1 volume=
Test Trc Inj
Stat
NCV
PSG
PSGd NdlNoz
VL FG Comment
Total
80.4
x 4=
32mm³
Logic (ms)
16810 222 5831CMaster Master Master Master Master
Master
Analyser 3 5 Trial
300
18-May-2012
Rail 810
Pres 800
(bar) 790
100
Z:\PE1750-03\MATLAB\5P3v2
40
Calculating
the fuel delivered per injection:
30
18-May-2012
Fuel injector
Consequently, it is essential to know how much fuel is being
expelled per injection. This information is needed
inCurves
order to
DFI5.1 Gain
meet the energy and emissions requirements of Poster
the engine.
Time (ms)
200
350
300
T4glt 250
us 200
0.1
William Springthorpe
0.2
0.3
0.4
0.5
0.6
Logic (ms)
0.7
0.8
0.9
MEI Mathematics in Work Competition 2012
Z:\PE1750-03\MATLAB\5P3v2
PE1750-03
250
Remembering that the rate is the change in volume, per unit time, we can use calculus to find our
answer.
d (V )
d (t )
Volume
(mm³)
Rate
(mm³/s)
 Vt
1
dt
As shown, integration is the method to use to find the volume from the rate. It is, however, not
possible to simply integrate, as this curve is a series of connected data points - it has no equation.
Instead we must use one of the rules of integration to find the area under the curve - The Trapezium
Rule, given as:
Time (ms)
The area under the curve can be thought of much like a bar chart. It is divided into lots of individual
rectangles of the same width but differing length. The combined area of these rectangles is the total
volume of fuel ejected by this particular injection.
In this particular example, there are 215 invisible trapeziums, in the thresholds of 1.3ms and 0.44ms,
each 4µs wide:
∫
≈
x 215 {(y0 + y215) + (y1 + y2 +…+ y214)}
h=
= 215
This equation is written and used on computer software. The software, Matlab, has calculated the
volume of fluid injected in this injection to be 32.02mm³.
This is very close to our initial guess of 32mm³.
Note: For a typical injector test, the Matlab software integrates 1675 rate traces. Each rate trace
containing 1000 data points at 4µs intervals, a total 1,675,000 points per test are evaluated!
Integration is an essential mathematical tool in our test work.
William Springthorpe
MEI Mathematics in Work Competition 2012