WEIGHTING ERROR –a Potential Source of Systematic Measurement Errors in Process Analysis

Abstract

Introduction: What is Weighting Error (WE)?
•Questions:
–Can WE be eliminated/minimized by increasing the number of samples?
–Can WE be eliminated/minimized by sampling at fixed volume intervals?
•Simulation studies
•Real case examples

Full text

WEIGHTING ERROR –a Potential
Source of Systematic Measurement
Errors in Process Analysis
Pentti Minkkinen
Lappeenranta University of Technology
e-mail: Pentti.Minkkinen@lut.fi
WSC-9, Tomsk,17-21 February, 2014

Outline
•Introduction: What is Weighting Error (WE)?
•Questions:
–Can WE be eliminated/minimized by increasing
the number of samples?
–Can WE be eliminated/minimized by sampling
at fixed volume intervals?
•Simulation studies
•Real case examples

Error components of analytical determination according to P.Gy
Global Estimation Error
GEE
Total Sampling Error
TSE
Point Selection Error
PSE
Total Analytical Error
TAE
Point Materialization Error
PME Weighting Error
SWE
Increment Delimi-
tation Error
IDE
Long Range
Point Selection Error
PSE1
Periodic
Point Selection Error
PSE2
Fundamental
Sampling Error
FSE
Grouping and
Segregation Error
GSE
Increment Extraction
Error
IXE
Increment and Sample
Preparation Error
IPE
GEE=TSE +TAE
TSE= (PSE+FSE+GSE)+(IDE+IXE+IPE)+SWE

Weighting error
Weighting error is made if the lot mean aLis
estimated as simple arithmetic mean,
, when
•the sampling target consists of several sub-
strata of different sizes (or densities) and
different sub-lot average concentrations
•in process analysis, when the flow-rate varies
during the sampling campaign
n
a
Li
a


Lot consisting of strata (sublots) of
different sizes, MLi and concentrations, aLi
LOT
N1
n1
s
1
N2
n2
s
2
Nk
nk
s
k
aL1, ML1
LOT
aL2, ML2
aLk, MLk
k
a
Li
L
a

Mean is biased because of WE



i
L
i
L
i
L
M
aM
L
a
Weighted mean is unbiased

Effect of density
•If the density of the material varies within
the lot and equal volumes are sampled
simple mean is erroneous

Total mass of the drill core: Mtot=152.8 kg
Mass of the valuable mineral = 47.5 kg
Density of the valuable mineral = 5 kg/dm3
Density of the gangue = 2.6 kg/dm3
Average density = 3.056 kg/dm3
True mass fraction of the mineral = 47.5kg/(152.8 kg) = 0.3109
= 31.09 %
Example on weighting error: Drill core of
stratified rock type (density effect)
SAMPLING PLAN:
The drill core is divided into 100 slices of equal sizes,
volume = 0,5 dm3and average mass, Ms=1.528 kg

Example on weighting error(cont.)
Each sample is analyzed separately. Without weighting the mean
concentration as mass fraction, cm= 0.190
Based on this result the average mass of the valuable mineral in the
core is = cm ·Ms·100=29.03 kg
If every sample is weighed (mass Mi) and the weighted mean of the
mass fraction is estimated the correct mean concentration is
obtained:
and the total mass of the valuable mineral
Relative weighting error is thus:
(0.19-0,3109)/0.3109 = -0.389 = -39,9 %
3109,0



i
ii
wM
Mc
c
kg5,47100
min  McM w
The drill core is divided into 100 slices of equal sizes,
volume = 0,5 dm3and average mass, Ms=1.528 kg

Sample
#
Mass
fraction
Sample mass
kg
Mineral content
kg
1
1
5,0
5,0
2
0
2,6
0
3
0
2,6
0
4
0
2,6
0
5
0,6579
3,8
2,5
6
0
2,6
0
7
0
2,6
0
8
0
2,6
0
9
0
2,6
0
10
0,6579
3,8
2,5
11
0
2,6
0
12
0
2,6
0
13
0
2,6
0
14
0
2,6
0
15
0
2,6
0
16
0
2,6
0
17
0,6579
3,8
2,5
18
0,6579
3,8
2,5
19
0
2,6
0
20
0
2,6
0
MEAN
0,1816
2,96
SUM
59,2
15,0
Weighted mean= 15 kg/(59,2 kg) = 0,2534
20 samples, 1 dm3 by volume analysed
Correct mean =
0.3109 = 31,09 %

Weighting error in process
analysis
•In process analysis the fluctuation of the flow-rate
should be taken into account in estimating the mean
over a time interval, or in designing the sampling plan
•Sample weight can be used as the stratum size in eq.
1 below to estimate the mean, provided that the
sample is cut proportional to the process flow-rate.
•Alternatively, flow-rate at sampling time could be
used as weight, if reliable measurement is available
(especially gaseous and liquid streams)
•Sampling systematically, when a fixed volume has
passed the sampling point, also eliminates the
weighting error

Sampling error in process analysis
•In process analysis the fluctuation of the
flow-rate should be taken into account in
estimating the mean over a time

Proportional sampling
•Correctly executed proportional sampling
eliminates the sampling error, if the
samples are weighed and the lot mean is
calculated as weighted mean by using the
sample masses as weights (Gy).
•Sub-samples have to be sampled
proportionally, if they are combined into a
composite sample.

CORRECT SAMPLE MATERIALIZATION:
Proportional sample delimitation
Correct sample profilesCutter movement

Incorrect sample delimitation
Incorrect sample profile
Cutter movement

Incorrect: Arithmetic mean
Correct: Weighted mean
i
s
M
i
a
where = sample mass (proportional sampling),
or flow-rate
= concentration of analyte in sample i





Process analyzers
Dp
Probe/ analyzer
Sampling errors, especially IDE and WE are difficult to control
when modern process analyzers and sensors are used

Process sampling: Simulation study
Three processes with 1000 data points were
generated with
low
medium
and high
correlation between concentration and flow-rate

Weighted mean Simple mean Relat. Error (%)
All sampled 45.3669 45.3429 -0.053
Every tenth sampled 45.5452 0.3929
Every tenth sampled 45.4100 0.0948
25 30 35 40 45 50 55 60 65
0
2
4
6
8
Flow-rate
ai
r = 0.011
0100 200 300 400 500 600 700 800 900 1000
-4
-2
0
2
4
Sample No.
ai, Vi

Weighted mean Simple mean Relat. Error (%)
All sampled 43.3210 45.3429 4.67
Every tenth sampled 43.4187 0.225
Every tenth sampled 45.4100 4.82
25 30 35 40 45 50 55 60 65
0
5
10
15
20
Flow-rate
ai
r = -0.558
0100 200 300 400 500 600 700 800 900 1000
-4
-2
0
2
Sample No.
ai,Vi

Weighted mean Simple mean Relat. Error (%)
All sampled 47.1692 45.3429 -3.87
Every tenth sampled 47.2751 0.225
Every tenth sampled 45.4100 -3.73
25 30 35 40 45 50 55 60 65
20
40
60
80
Flow-rate
ai
r = 0.972
0100 200 300 400 500 600 700 800 900 1000
-4
-2
0
2
4
Sample No.
ai,Vi

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-10
-5
0
5
10
CORRELATION COEFFICIENT
RELATIVE ERROR (%)
Weighting error is systematic with structurally
(red) and circumstantially (blue) correlated data,
but is random for uncorrelated (black) data. 10 %
of data sampled.
Simulation results

Comparison of two different systematic sampling
modes; results calculated as weighted mean.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-0.015
-0.01
-0.005
0
0.005
0.01
CORRELATION COEFFICIENT
RELATIVE ERROR
Systematic sampling at constant volume and time intervals
= constant time interval = constant volume interval

EXAMPLES ON TRUE
PROCESS DATA
Atmospheric emissions
Wastewater
Solid wastes

Weighting error NOxmeasurement
0100 200 300 400 500 600 700 800
100
150
200
250
300
350
NOx (mg/m3)
0100 200 300 400 500 600 700 800
1
1.5
2
2.5
3
3.5 x 105
Time (h)
FLOWRATE (m3/h)
NOxconcentrations and total gas flow-rate measured as one-
hour averages from a power plant during one month

Weighting error of simple arithmetic mean:
in mean concentration = -7.97 mg/m3
in total monthly emission = -1400 kg
Mean of NOxconcentrations: 229.5 mg/m3
Mean of gas flow-rate : 2.327 ·105m3/h
Total gas flow: 1.718·108m3
Total NOxemitted (unweighted): 39400 kg
Weighted mean of NOxconcentration:= 237.5 mg/m3
Vi
ciVi
Total NOxemitted (weighted): 40800 kg
CALCULATION OF TOTAL MONTHLY NOxEMISSION

050 100 150 200 250 300 350 400
-5
0
5
10 COD in WASTEWATER; r = -0.383
SAMPLING DAY
ai, FLOW-RATE
Weighting error of COD measurement
Daily COD concentrations and flow-rates measured from a
wastewater treatment plant

Annual COD discharge from a
wastewater treatment plant
•Mean of COD = 159.2 mg O2/dm3
•Mean of flow-rate = 8.21 ·104m3/d
•Total volume of water = 2.995·107m3
•Annual COD discharge (not weighted) = 4.768·106kg O2
•Weighted mean of COD: = 155.1 mg O2/dm3
•Annual COD discharge (weighted) = 4.646·106kg O2
•Weighting error of simple mean:
–as mean COD = 4.06 mg O2/dm3
–as total annual discharge = 0.122·106 kg O2
In this case the simple mean is 2.6 %
higher than the weighted mean.

Minimization of weighting error in
process analysis, when proportional
cross-stream sampling cannot be used
•Flow-rate is measured simultaneously with
sampling and used as weight in calculating the
mean.
•Sampling system is coupled to a flow meter so
that a fixed sample volume is taken when the
preset total volume has passed the sampling
point. In this case the simple arithmetic average
can be used as the mean concentration.

Conclusions on weighting errors
Weighting error is often a significant component of
sampling errors and has to taken into account when the
average value, mean concentration or total mass of analyte
in the sampling target is estimated.
Increasing the No. of samples does not necessarily
reduce the sampling error, if the flow-rate and
concentration are correlated.
Sampling at constant volume intervals eliminates WE in
process analysis; important also in composite sampling.
WE is an important reason why composite samples
analyzed in laboratory show significant differences
compared to average calculated from process
analyzers for the same period.

THANK YOU KIITOS
Спасибо