CEN/TR 16988:2016

Estimation of uncertainty in the single burning item test

The measuring technique of the SBI (single burning item) test instrument is based on the observation that, in general, the heats of combustion per unit mass of oxygen consumed are approximately the same for most fuels commonly encountered in fires (Huggett [12]). The mass flow, together with the oxygen concentration in the extraction system, suffices to continuously calculate the amount of heat released. Some corrections can be introduced if CO2, CO and/or H2O are additionally measured.

Messunsicherheit - Thermische Beanspruchung durch einen einzelnen brennenden Gegenstand (SBI)

Ocena negotovosti s preskusom enega samega gorečega predmeta

Merilna tehnika instrumenta za preskušanje gorljivosti posameznega predmeta (SBI) temelji na ugotovitvi, da je toplota zgorevanja na enoto mase porabljenega kisika na splošno približno enaka za večino goriv, ki so običajna pri požarih (Huggett [12]). Masni pretok skupaj s koncentracijo kisika v sistemu za odvajanje zadostuje za neprekinjeno izračunavanje količine sproščene toplote. Nekateri popravki se lahko uporabijo, če se dodatno izmerijo vrednosti CO2, CO in/ali H2O.

General Information

• Status: Published
• Publication Date: 26-Jul-2016
• ICS: 17.200.01 - Thermodynamics in general
• Technical Committee: CEN/TC 127 - Fire safety in buildings
• Drafting Committee: CEN/TC 127/WG 4 - Reaction to fire
• Current Stage: 6060 - Definitive text made available (DAV) - Publishing
• Start Date: 27-Jul-2016
• Due Date: 30-Jan-2017
• Completion Date: 27-Jul-2016
• Directive: 305/2011 - Regulation (eu) No 305/2011 of the European Parliament and of the Council of 9 march 2011 laying down harmonised conditions for the marketing of construction products and repealing council directive 89/106/eec
• Mandate: M/117 - Horizontal complement to the mandates to CEN/CENELEC concerning the execution of standardization work for the valuation of construction products and elements in respect of their resistance to fire
• Ref Project: SIST-TP CEN/TR 16988:2016 - Estimation of uncertainty in the single burning item test

Overview

CEN/TR 16988:2016 is a CEN technical report that describes methods for the estimation of measurement uncertainty in the single burning item (SBI) test. The SBI test uses the oxygen-consumption principle (Huggett method) to derive heat release from measured exhaust volume flow and oxygen concentration, with optional corrections when CO2, CO and H2O are also measured. This report-prepared by CEN/TC 127 and published July 2016-documents calculation procedures, uncertainty propagation, and classification of confidence intervals for SBI-derived fire performance parameters.

Key topics and technical requirements

• Measurement principle: Oxygen-consumption (Huggett) method with a reference heat per mole O2 (~17 200 kJ/m3 at 298 K).
• Calculation procedures:
  • Synchronization of analyzer and flow data (due to transport delays and analyser response).
  • Computation of total and specimen heat release rate (HRR), subtraction of burner baseline, and 30 s flat running-average smoothing for HRR (used for FIGRA).
  • Derived metrics: THR (total heat release, e.g., THR600s), FIGRA (fire growth rate indices for 0.2 MJ / 0.4 MJ thresholds), SPR/SMOGRA/TSP (smoke metrics).
• Uncertainty framework:
  • Definitions of mean, variance, confidence intervals, combined standard uncertainty and expanded uncertainty.
  • Propagation formulas for sums, averages, products/divisions and specific SBI outputs (HRR, φ oxygen depletion factor, V(t), air density, specimen HRR, FIGRA, THR600s, SPR, SMOGRA, TSP600s).
• Sources of measurement uncertainty:
  • Data acquisition (DAQ) resolution and timing, transient and aliasing errors, data synchronization, probe pressure difference (Δp), temperature (Tms), flow profile correction (kt), duct area (A), analyzer uncertainties (O2, CO2, CO), and others.
• Components & symbols: Annex with list of symbols and abbreviations for reproducibility.

Practical applications and users

• Fire test laboratories performing SBI (EN 13823) tests and reporting HRR, THR, FIGRA and smoke metrics.
• Product manufacturers (building products, wall/floor coverings, furniture) validating reaction-to-fire performance.
• Building engineers, fire safety consultants and regulators who evaluate conformity with reaction-to-fire criteria.
• Standards developers and accreditation bodies assessing measurement uncertainty, test repeatability and laboratory competence.

Related standards

• SBI test normative procedures (EN 13823) - CEN/TR 16988 provides uncertainty estimation guidance specifically for SBI outputs.
• Standards on uncertainty and metrology (e.g., ISO/IEC Guide 98 / GUM) - referenced for uncertainty calculation methods.

Keywords: CEN/TR 16988:2016, single burning item test, SBI, uncertainty estimation, heat release rate (HRR), FIGRA, THR600s, oxygen consumption method, fire testing, measurement uncertainty.

Standards Content (Sample)

SLOVENSKI STANDARD
01-november-2016
2FHQDQHJRWRYRVWLVSUHVNXVRPHQHJDVDPHJDJRUHþHJDSUHGPHWD
Estimation of uncertainty in the single burning item test
Messunsicherheit - Thermische Beanspruchung durch einen einzelnen brennenden
Gegenstand (SBI)
Ta slovenski standard je istoveten z: CEN/TR 16988:2016
ICS:
13.220.40 Sposobnost vžiga in Ignitability and burning
obnašanje materialov in behaviour of materials and
proizvodov pri gorenju products
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

CEN/TR 16988
TECHNICAL REPORT
RAPPORT TECHNIQUE
July 2016
TECHNISCHER BERICHT
ICS 17.200.01
English Version
Estimation of uncertainty in the single burning item test
Messunsicherheit - Thermische Beanspruchung durch
einen einzelnen brennenden Gegenstand (SBI)

This Technical Report was approved by CEN on 4 July 2016. It has been drawn up by the Technical Committee CEN/TC 127.

CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia,
Finland, Former Yugoslav Republic of Macedonia, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania,
Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and
United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION

EUROPÄISCHES KOMITEE FÜR NORMUNG

CEN-CENELEC Management Centre: Avenue Marnix 17, B-1000 Brussels
© 2016 CEN All rights of exploitation in any form and by any means reserved Ref. No. CEN/TR 16988:2016 E
worldwide for CEN national Members.

Contents Page
European foreword . 4
1 Scope . 5
1.1 General . 5
1.2 Calculation procedure . 5
1.2.1 Introduction . 5
1.2.2 Synchronization of data . 5
1.2.3 Heat output . 5
2 Uncertainty . 9
2.1 Introduction . 9
2.2 Elaboration of terms and concepts . 11
2.2.1 Mean and variance . 11
2.2.2 Estimation of the confidence interval for the population mean . 12
2.2.3 Sources of uncertainty . 12
2.2.4 Standard uncertainties for different distributions . 12
2.2.5 Combined uncertainty . 15
2.2.6 Expanded uncertainty . 16
2.2.7 Uncorrected bias . 16
2.3 Combined standard uncertainties . 17
2.3.1 Combined standard uncertainty on sums . 17
2.3.2 Combined standard uncertainty on averages . 18
2.3.3 Combined standard uncertainty of a product and a division . 19
2.3.4 Combined standard uncertainty on the heat release rate (Q) . 20
2.3.5 Combined standard uncertainty on the depletion factor (ϕ) . 22
D°
2.3.6 Combined standard uncertainty on the initial O -concentration (X ) . 22
2 O2
2.3.7 Combined standard uncertainty on the volume flow rate (V ) . 23
D298
2.3.8 Combined standard uncertainty on the air density (ρ ) . 24
air
2.3.9 Combined standard uncertainty on specimen heat release rate (Qspecimen) . 24
2.3.10 Combined standard uncertainty on the average heat release rate (Q ) . 24
av
2.3.11 Combined standard uncertainty on FIGRA . 25
2.3.12 Combined standard uncertainty on THR600s . 25
2.3.13 Combined standard uncertainty on the volume flow (V(t)) . 25
2.3.14 Combined standard uncertainty on the smoke production rate (SPR) . 25
2.3.15 Combined standard uncertainty on specimen smoke production rate (SPR) . 26
2.3.16 Combined standard uncertainty on the average smoke production rate (SPR ) . 26
av
2.3.17 Combined standard uncertainty on SMOGRA . 26
2.3.18 Combined standard uncertainty on TSP600s . 27
2.4 Confidence interval classification parameters . 27
2.5 Standard uncertainty on the different components . 28
2.5.1 Uncertainty on the data acquisition (DAQ). 28
2.5.2 Transient error . 28
2.5.3 Aliasing error . 28
2.5.4 Uncertainty on data synchronicity . 29
2.5.5 Uncertainty on the component E and E’ . 30
2.5.6 Uncertainty on the component φ . 36
2.5.7 Uncertainty on the component p . 36
atm
2.5.8 Uncertainty on the component T . 36
room
2.5.9 Uncertainty on the component α . 38
2.5.10 Uncertainty on the component c . 38
2.5.11 Uncertainty on the component A and L . 39
2.5.12 Uncertainty on the component q . 40
gas
2.5.13 Uncertainty on the component k . 40
t
2.5.14 Uncertainty on the component k . 43
p
2.5.15 Uncertainty on the component Δp . 44
2.5.16 Uncertainty on the component T . 44
ms
2.5.17 Uncertainty on the component I . 46
Annex A (informative) List of symbols and abbreviations . 48

European foreword
This document (CEN/TR 16988:2016) has been prepared by Technical Committee CEN/TC 127 “Fire
Safety in Buildings”, the secretariat of which is held by BSI.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. CEN shall not be held responsible for identifying any or all such patent rights.
This document has been prepared under a mandate given to CEN by the European Commission and the
European Free Trade Association.
1 Scope
1.1 General
The measuring technique of the SBI (single burning item) test instrument is based on the observation
that, in general, the heats of combustion per unit mass of oxygen consumed are approximately the same
for most fuels commonly encountered in fires (Huggett [12]). The mass flow, together with the oxygen
concentration in the extraction system, suffices to continuously calculate the amount of heat released.
Some corrections can be introduced if CO , CO and/or H O are additionally measured.
2 2
1.2 Calculation procedure
1.2.1 Introduction
The main calculation procedures for obtaining the HRR and its derived parameters are summarized
here for convenience. The formulas will be used in the following clauses and especially in the clause on
uncertainty.
The calculations and procedures can be found in full detail in the SBI standard [1].
1.2.2 Synchronization of data
The measured data are synchronized making use of the dips and peaks that occur in the data due to the
switch from ‘primary’ to ‘main’ burner around t = 300 s, i.e. at the start of the thermal attack to the test
specimen. Synchronization is necessary due to the delayed response of the oxygen and carbon dioxide
analysers. The filters, long transport lines, the cooler, etc. in between the gas sample probe and the
analyser unit, cause this shift in time.
After synchronization, all data are shifted so that the ‘main’ burner ignites – by definition – at time
t = 300 s.
1.2.3 Heat output
1.2.3.1 Average heat release rate of the specimen (HRR )
30s
A first step in the calculation of the HRR contribution of the specimen is the calculation of the global
HRR. The global HRR is constituted of the HRR contribution of both the specimen and the burner and is
defined as
 φ(t) 

′
HRR (t)= EV (t)x   (1)
total D298 a_O2
 
1+ 0,105φ(t)
 
where
is the total heat release rate of the specimen and burner (kW);
HRR (t)
total
′ is the heat release per unit volume of oxygen consumed at 298 K, = 17 200 (kJ/m3);
E
 is the volume flow rate of the exhaust system, normalized at 298 K (m3/s);
Vt()
D298
is the mole fraction of oxygen in the ambient air including water vapour;
x
a_O2
is the oxygen depletion factor.
ϕ()t
 φ(t) 
The last two terms x and   express the amount of moles of oxygen, per unit volume,
a_O2
 
1+ 0,105φ(t)
 
that have chemically reacted into some combustion gases. Multiplication with the volume flow gives the
amount of moles of oxygen that have reacted away. Finally this value is multiplied with the ‘Huggett’
factor. Huggett stated that regardless of the fuel burnt roughly a same amount of heat is released.
The volume flow of the exhaust system, normalized at 298 K, V (t) is given by
D298
k Dp(t)
t
V (t)=cA (2)
D298
k T (t)
ρ ms
where
c
0,5 1,5 −0,5
(2T /ρ ) 22, 4 [K⋅ m ⋅ kg ]
A is the area of the exhaust duct at the general measurement section (m2);
is the flow profile correction factor; converts the velocity at the height of the bi-directional
k
t
probe in the axis of the duct to the mean velocity over the cross section of the duct;
is the Reynolds number correction for the bidirectional probe, taken as 1,08;
k
ρ
Dpt() is the pressure difference over the bi-directional probe (Pa);

is the temperature in the measurement section (K).
Tt()
ms
The oxygen depletion factor ϕ()t is defined as
xO (30s.90s){1−xCO (t)}−xO (t){1−xCO (30s.90s)}
2 2 2 2
φ(t)= (3)
xO (30s.90s){1−xCO (t)−xO (t)}
2 2 2
where
is the oxygen concentration in mole fraction;
xtO ()
is the carbon dioxide concentration in mole fraction;
xtCO ( )
Ys.Zs mean taken over interval Y s to Z s.
The mole fraction of oxygen in ambient air, taking into account the moisture content, is given by
 
 
H 3816
x = xO (30s.90s) 1− exp 23,2− (4)
  
a_O2 2
100p T (30s.90s)− 46
 ms 
 
where
is the oxygen concentration in mole fraction;
xtO ()
H is the relative humidity (%);
p is the ambient pressure (Pa);
Tms(t) is the temperature in the general measurement section (K).
Since we are interested in the HRR contribution of the specimen only, the HRR contribution of the
burner should be subtracted. An estimate of the burner contribution HRR (t) is taken as the
burner
HRR (t) during the base line period preceding the thermal attack to the specimen. A mass flow
total
controller ensures an identical HRR through the burners before and after switching from primary to the
main burner. The average HRR of the burner is calculated as the average HRR (t) during the base line
total
period with the primary burner on (210 s ≤ t ≤ 270 s):
=
HRR = HRR total (210s.270s) (5)
av_burner
where
HRRav_burner is the average heat release rate of the burner (kW);
HRRtotal(t) is the total heat release rate of specimen and burner (kW).
HRR of the specimen
In general, the HRR of the specimen is taken as the global HRR, HRR (t), minus the average HRR of the
total
burner, HRR :
av_burner
For t > 312 s:
HRR(t)= HRR (t)− HRR (6)
total av_burner
where:
HRR(t) is the heat release rate of the specimen (kW);
HRRtotal(t) is the global heat release rate of specimen and burner (kW);
HRRav_burner is the average heat release rate of the burner (kW).
During the switch from the primary to the main burner at the start of the exposure period, the total heat
output of the two burners is less than HRR (it takes some time for the gas to be directed from one
av_burner
burner to the other). Formula (24) gives negative values for HRR(t) for at most 12 s (burner switch
response time). Such negative values and the value for t = 300 s are set to zero, as follows:
For t = 300 s:
HRR(300)= 0 kW (7)
For 300 s < t ≤ 312 s:
HRR(t)= max.{0 kW, HRR (t)− HRR } (8)
total av_burner
where
max.[a, b] is the maximum of two values a and b.
Calculation of HRR
30s
In view of the calculation of the FIGRA index, the HRR data are smoothened with a ‘flat’ 30 s running
average filter using 11 consecutive measurements:
0,5HRR(t−15)+ HRR(t−12 )+ .+ HRR(t+12)+ 0,5HRR(t+15)
(9)
HRR (t)=
30s
where
HRR (t) is the average of HRR(t) over 30 s (kW);
30s
HRR(t) is the heat release rate at time t (kW).
1.2.3.2 Calculation of THR(t) and THR
600s
The total heat release of the specimen THR(t) and the total heat release of the specimen in the first
600 s of the exposure period (300 s ≤ t ≤ 900 s), THR , are calculated as follows:
600s
t
a
THR(t )= HRR(t)×3 (10)
a ∑
300s
900s
THR = HRR(t)×3 (11)
600s ∑
300s
whereby the factor 1 000 is introduced to convert the result from kJ into MJ and the factor 3 stands for
the time interval in-between 2 consecutive measurements,
and where
THR(t ) is the total heat release of the specimen during the period 300 s ≤ t ≤ t (MJ);
a a
HRR(t) is the heat release rate of the specimen (kW);
THR is the total heat release of the specimen during the period 300 s ≤ t ≤ 900 s (MJ);
600s
(equal to THR(900)).
1.2.3.3 Calculation of FIGRA and FIGRA (Fire growth rate indices)
0.2MJ 0.4MJ
The FIGRA is defined as the maximum of the ratio HRR (t)/(t − 300), multiplied by 1 000. The ratio is
av
calculated only for that part of the exposure period in which the threshold levels for HRR and THR
av
have been exceeded. If one or both threshold values are not exceeded during the exposure period,
FIGRA is equal to zero. Two combinations of threshold values are used, resulting in FIGRA and
0,2MJ
FIGRA .
0,4MJ
a) The average of HRR, HRR , used to calculate the FIGRA is equal to HRR , with the exception of the
av 30s
first 12 s of the exposure period. For data points in the first 12 s, the average is taken only over the
widest possible symmetrical range of data points within the exposure period:
For t 300 s: HRR (300 s) 0 (12)
av
For t 303 s: HRR (303 s) HRR(300 s306 s) (13)
av
For t 306 s: HRR (306 s) HRR(300 s312 s) (14)
av
For t 309 s: HRR (309 s) HRR(300 s318 s) (15)
av
For t 312 s: HRR (312 s) HRR(300 s324 s) (16)
av
For t≥=315 s: HRR (tt) HRR ( ) (17)
av 30s
b) Calculate FIGRA for all t where:
0,2MJ
(HRR (t) > 3 kW) and (THR(t) > 0,2 MJ) and (300 s < t ≤ 1 500 s);
av
and calculate FIGRA for all t where:
0,4MJ
(HRRav(t) > 3 kW) and (THR(t) > 0,4 MJ) and (300 s < t ≤ 1 500 s);
both using:
==
==
==
==
==
HRR (t)
 
av
FIGRA=1000× max. (18)
 
t− 300
 
where:
FIGRA is the fire growth rate index
HRR (t) is the average of HRR(t) as specified in a) (kW);
av
As a consequence, specimens with a HRR not exceeding 3 kW during the total test have FIGRA values
av
FIGRA and FIGRA equal to zero. Specimens with a THR not exceeding 0,2 MJ over the total test
0,2MJ 0,4MJ
period have a FIGRA equal to zero and specimen with a THR not exceeding 0,4 MJ over the total test
0,2MJ
period have a FIGRA equal to zero.
0,4MJ
2 Uncertainty
2.1 Introduction
According to EN ISO/IEC 17025 [3], which sets out the general requirements for the competence of
testing and calibration laboratories, and EN ISO 10012 [7], which sets out the requirements for assuring
the quality of measuring equipment, uncertainties shall be reported in both testing and calibration
reports.
The general principles for evaluating and reporting uncertainties are given in the ISO Guide to the
Expression of Uncertainty in Measurement (GUM) [6], but need to be applied to the specific case of fire
testing. Due to the harmonization of fire testing in the European Community (EUROCLASSES;
EN 13501-1 [21]) and the pressure on testing laboratories to operate under accreditation, this is
becoming even more important.
It is of common knowledge that measurement results are never perfectly accurate. In practice the
sources of systematic and random errors which can affect the results of measurement are numerous,
even for the most careful operators. To describe this lack of perfection, the term 'uncertainty' is used.
Although the concept of uncertainty may be related to a 'doubt', in the real sense the knowledge of
uncertainty implies increased confidence in the validity of results.
The qualitative concept of accuracy is quantified by the uncertainty which varies inversely
‘proportioned’ to it. Accuracy consists of both trueness and precision as shown in Figure 1. A numerical
measure for precision is the standard deviation, while trueness is expressed numerically by the
systematic error or the bias.
It is considered good practice to eliminate any systematic errors. However, if the value of a systematic
error is unknown it may be regarded as a random error. Random errors result in a spread of the values
and can usually be reduced by increasing the number of observations. Its expectation or expected value
is zero.
high precision low precision
high
trueness
(high accuracy)
low
trueness
Figure 1 — Concepts of accuracy (uncertainty), precision (standard deviation) and trueness
(bias)
In general, the result of a measurement is only an approximation or estimate of the value of the specific
quantity subject to measurement, that is, the measurand, and so the result is complete only when
accompanied by a quantitative statement of its uncertainty.
Without knowledge of the accuracy (trueness and precision) of measurement methods and/or the
uncertainty of measurement results, it can appear very easy to make decisions. But, in practice, these
decisions might be incorrect and sometimes lead to serious consequences, if the measurement
uncertainty is not taken into account.
For example, in fire testing, when rejecting instead of accepting a good product during a certification
process or, conversely, when accepting a bad product by error. So, it is vital to quantify the reliability of
the measurement results to greatly reduce any disputes and adverse consequences of legal proceedings.
This is of particular importance if the growing number of cases of litigation in Europe and the liability
problems of manufacturers in case of accidents are considered.
The difference between error and uncertainty should always be borne in mind. For example, the result
of a measurement after correction can unknowably be very close to the unknown value of the
measurand, and thus have negligible error, even though it might have a large uncertainty.
Key
X value
Y frequency
1 bias
2 repeated measurements would give values with this frequency curve
3 standard deviation (σ)
4 true value
5 expected value
Figure 2 — Concepts of accuracy (uncertainty), precision (standard deviation) and trueness
(bias)
2.2 Elaboration of terms and concepts
2.2.1 Mean and variance
A population with a ‘normal’ probability density function is characterized by its mean value μ and its
2 2 2
variance σ : N(μ,σ ). When both μ and σ are unknown, they can be estimated by taking a number n of
samples and by calculating the estimated mean x , the estimated variance s and the estimated
standard deviation s.
n
(19)
x= x
∑
i
n
i=1
n
( ) (20)
s = x−x
∑
i
n−1
i=1
If a covariance exists between two variables x and y, it is given by
n
s = (x−x)(y− y) (21)
ij ∑ i i
n−1
i=1
2.2.2 Estimation of the confidence interval for the population mean
Often the standard deviation σ is unknown. To evaluate the confidence interval, some estimate of σ shall
be made. The most obvious candidate is the sample standard deviation s. But the use of s introduces an
additional source of unreliability, especially if the sample is small. To retain the confidence interval, the
interval shall therefore be widened. This is done by using the t distribution instead of the standard
normal distribution. For a sample size larger than 100, the t-distribution approaches the normal
distribution. For a 95 % (two tails of 2,5 %) confidence interval – which we strive for – the uncertainty
is estimated by
s
t (22)
0.025
n
The value t depends on the amount of information used in calculating s , i.e. on the degrees of
0.025
freedom. For large sample sizes, t approaches 1,96 which is the value for a normal distribution. For a
0.025
normal distribution, a coverage factor 2 (1,96) corresponds to a 95 % confidence interval (see 2.2.6).
2.2.3 Sources of uncertainty
According to GUM [6] any detailed report of the uncertainty should consist of a complete list of the
components, specifying for each the method used to obtain its numerical value. The components may be
grouped into two categories based on their method of evaluation:
Type A The components in category A are characterized by the estimated variances s
i
or by the estimated standard deviation s derived from data by statistical
i
methods. Where appropriate the covariance s should be given.
ij
For such a component, the standard uncertainty is ui = si.
Type B The standard uncertainty of a Type B evaluation is approximated based on
specifications, calibrations, handbooks, experience, judgements etc. and is
represented by a quantity uj. It is obtained from an assumed probability
distribution based on all the available information.
Where appropriate the covariance should be given and should be treated in a
similar way.
The ‘type’ classification does not indicate any difference in the nature of the components resulting from
the two types of evaluation. Both are based on probability distributions, and the uncertainty
components resulting from either type are quantified by standard deviations. It should be recognized
that a Type B evaluation of standard uncertainty can be as reliable as a Type A evaluation.
The standard deviation of a Type B evaluation is based on the shape of the distribution. Distributions
used in this dcoument are the rectangular, the triangular, the trapezoidal and the normal distribution.
For the rectangular and triangular also asymmetric distributions are discussed.
2.2.4 Standard uncertainties for different distributions
Normal distribution
Often calibration certificates, handbooks, manufacturer’s specifications, etc. state a particular multiple
of a standard deviation. In this case, a normal distribution is assumed to obtain the standard
uncertainty.
Rectangular distribution
In other cases the probability that the value of X lies within the interval a- to a+ for all practical
i
purposes is equal to one and the probability that X lies outside this interval is essentially zero. If there
i
is no specific knowledge about the possible values of X within the interval, a uniform or rectangular
i
distribution of values is assumed. The associated standard deviation is function of the width of the
distribution as:
a
(23)
u =
rect
Indeed, for a rectangular distribution, the variance is obtained as in 24. Given the probability function of
the rectangular distribution
0 x 

Vitryalice1P(x)= a< x 
b−a

0 x>b

This can be written in terms of the Heaviside step function H(x) as
H (x−a)−H (x−b)
P(x)= (25)
b−a
This makes that the variance σ with population mean μ for an asymmetric distribution becomes
∞ b
x b+a
µ= P(x)xdx= dx= (26)
∫ ∫
(b−a) 2
−∞ a
∞
2 2
σ = P(x)(x−µ) dx (27)
∫
−∞
∞
H (x−a)−H (x−b) a+b
2 2
(28)
σ = (x− ) dx
∫
b−a 2
−∞
a+b
b (x− ) 2
(a+b)
σ = dx= (29)
∫
b−a 12
a
So for a symmetric rectangular interval a- to a+, the variance reduces to
a
σ = . (30)
The sample estimate of the standard deviation thus is:
a
u = . (31)
rect
The rectangular distribution is a reasonable default model in the absence of any other information. But
if it is known that values of the quantity in question near the centre of the limits are more likely than
values close to the limits, a triangular or a normal distribution migth be a better model.
Triangular and trapezoidal distribution
In many cases it is more realistic to expect that values near the bounds are less likely than those near
the midpoint. It is then reasonable to replace the symmetric rectangular distribution by a symmetric
trapezoidal distribution having equal sloping sides, a base of width 2a and a top of width 2aβ where
0 ≤ β ≤ 1. Similar as for a rectangular distribution, for a trapezoidal distribution the standard deviation
becomes:
a (1+β )
u = (32)
trap
As β goes to 1 this trapezoidal distribution approaches the rectangular distribution while for β = 0 it is a
triangular distribution.
a
(33)
u =
trian
Asymmetric distributions
For an asymmetric triangular distribution the mean value and standard deviation become

µ= (a+b+c) (34)
2 2 2
u = (a +b +c −ab−ac−bc) (35)
asymmetric _triangular
For the limit case of a ‘one sided triangular distribution (a = c = 0; b = b) this reduces to
b
µ= (36)
b
u = (37)
asymmetric _triangular
3 2
The limit case of a ‘one sided rectangular distribution (a = 0; b = b) reduces to
b
µ= (38)
b
u = (39)
asymmetric _rec tangular
For asymmetric distributions the difference between the mean value μ and c will be considered as a bias
and, as recommended by the GUM [6], it will be corrected for.
2.2.5 Combined uncertainty
2.2.5.1 Uncorrelated input quantities
The standard uncertainty of y, where y is the estimate of the measurand Y and thus the measurement, is
obtained by appropriately combining the standard uncertainties of the input estimates x. In case all
i
input quantities are independent, this combined standard uncertainty u (y) is the positive square root
c
of the combined variance u (y) and is given by
c
N
u = [cu(x )] (40)
c ∑ i i
i=1
where the sensitivity coefficient is given by
∂y
c= (41)
i
∂x
i
2.2.5.2 Correlated input quantities
Formula (41) is only valid when the input quantities X are independent or uncorrelated. When the
i
input quantities are correlated, the appropriate expression for the combined variance u (y) associated
c
with the result of the measurement is
N N−1 N
u = [cu(x )] + 2 cc u(x )u(x )r(x ,x ) (42)
c ∑ i i ∑∑ i j i j i j
i=1 i=1 j=i+1
whereby the degree of correlation between x and x is characterized by the estimated correlation
i j
coefficient (Pearson)
u(x ,x )
i j
r(x ,x )= −1≤rx( ,x )≤+1  (43)
i j i j
u(x )u(x )
i j
If the estimates x and x are independent, r(x ,x ) = 0, and a change in one does not imply an expected
i j i j
change in the other. Pearson’s coefficient reflects the degree of linear relationship between two data
sets. Its value is between −1 and +1. A value of +1 means that there is a perfect positive linear
relationship between the two data sets. A value of −1 means that there is a perfect negative linear
relationship, and a value of 0 means there is no linear relationship at all between the data sets.
The measurands E’, p , T , φ, α, c, A, k and k , are constant or can be considered constant
atm room t p
throughout the test. They are treated as independent input quantities.
The oxygen concentration (XO ), the carbon dioxide concentration (XCO ), the exhaust gas temperature
2 2
in the measurement section (T ) and the differential pressure over the velocity probe (Δp) are
ms
significantly correlated to each other and the correlation coefficients will be chosen as indicated in
Table 1.
Table 1 — Assumed correlation coefficients between XO , XCO , T and Δp
2 2 ms
O CO Δp T
2 2 ms
O 1 −1 −1 −1
CO −1 1 1 1
Δp −1 1 1 1
T −1 1 1 1
ms
2.2.6 Expanded uncertainty
Although the combined standard uncertainty u is used to express the uncertainty of a wide variety of
c
applications, what is often required is a measure of uncertainty that defines an interval about the
measurement result y within which the value of the measurand Y can be confidently asserted to lie. The
measure of uncertainty intended to meet this requirement is termed expanded uncertainty, U, and is
obtained by multiplying u (y) by a coverage factor k. So U = ku (y) and it is confidently believed that Y is
c c
greater than or equal to y - U, and is less than or equal to y + U, which is commonly written as Y = y ± U.
In general, the value of the coverage factor k is chosen on the basis of the desired level of confidence to
be associated with the interval defined by U = ku . Typically, k is in the range 2 to 3. When the normal
c
distribution applies and u is a reliable estimate of the standard deviation of y, U = 2 u (i.e. k = 2)
c c
defines an interval having a level of confidence of approximately 95 % (95,44 %), and U = 3 u (i.e. k = 3)
c
defines an interval having a level of confidence greater than 99 % (99,73 %).
In this document, we will work with standard deviations to express the uncertainty of individual
measurands and combined standard uncertainties, and with a coverage factor of 2 to express the
confidence interval on the estimate of the overall uncertainty.
On occasion, a self-defined apparent standard deviation defined as the uncertainty for a 95 % interval
divided by 1,96 will be used. This is useful when for example using t-distributions since for this
distribution the expanded uncertainty U(k = 1,96) ≠ 1,96*U(k = 1). Since we know that we finally want
to end up with a 95 % confidence interval but we are working with standard deviations, t-distributions
will be calculated based on a 95 % confidence interval divided by 1,96.
This allows us to work with standard deviations all the time and to, at the end, multiply with the
coverage factor of k = 1,96.
 Ux  Ux 
( ) ( )
Ux( ) 1,96 c++c . (44)
  
95% xx1 2
1,96 1,96
  
2.2.7 Uncorrected bias
Although it is recommended (and strongly preferred) practice of correcting for all known bias effects, in
view of backwards compatibility of test results for example, an increased uncertainty interval may be
the preferred option.
Unfortunately, the GUM [6] does not deal directly with the situation where a known measurement bias
is present but is uncorrected.
Several proposed methods of treating uncorrected bias are available. We propose to follow the
guidelines of Phillips et al. [14] because of its conservative approach. This method algebraically sums
the signed bias δ with the expanded uncertainty, unless the bias is larger:
+
+U
Y= y (45)
−
−U
Where
=
ku −δ ku −δ> 0

+ c c
U = if  (46)

0 ku −δ≤ 0
 c
And
ku +δ ku +δ> 0

− c c
U = if  (47)

0 ku +δ≤ 0
 c
Note that the expanded uncertainty shall be re-computed if the coverage factor is changed, and in
particular, that U ± (k = 2) ≠ 2*U ± (k = 1).
The combined standard uncertainty u is calculated out of the standard uncertainty associated with the
c
bias u and the standard uncertainty u that accounts for the combination of all other uncertainty
b
sources not directly associated with the bias.
2 2
u =(u +u ) (48)
c b
The proposed approach can somewhat overestimate the uncertainty. In the case of a coverage factor
k = 2, the method maintains the 95 % confidence interval until the ratio of the bias to the combined
standard uncertainty becomes larger than the coverage factor. For such large bias values, the method
produces uncertainty intervals that are slightly conservative.
Note that the sign of the sensitivity coefficient is important to know the effect on the global uncertainty.
As an example, suppose x and x both have uncorrected bias and the expanded uncertainty is given by
1 2
1+
+U
x 1− (49)
−U
2+
+U
x 2− (50)
2−U
The uncertainty interval on x defined as
x
x= (51)
x
then becomes
2 2
 1+ 2−
   
U U

+   + 
   
x x

u(x)
 1   2 
= (52)

2 2
1− 2+
x
 U  U 
   
− +

   
x x
 1   2 

assuming x ≠ 0 and x ≠ 0.
1 2
An underestimation of x leads to an underestimation of x, while an underestimation of x leads to an
1 2
overestimation of x.
2.3 Combined standard uncertainties
2.3.1 Combined standard uncertainty on sums
Since the discussion on the uncertainty of a data acquisition system often requires the standard
uncertainty on the sum of N independent variables/measurements, a short review is as follows.
Assume the sum
N
y=a x (53)
∑ i
i
Taking the partial derivative to the different components x results in the corresponding sensitivity
i
coefficients c = a.
i
The standard uncertainty of y is obtained by appropriately combining the standard uncertainties of the
input estimates x . This combined standard uncertainty u (y) is the positive square root of the combined
i c
variance u (y) and is given by
c
N N
2 2
u (y)= [cu(x )] =a u (x ) (54)
∑ ∑
c i i i
i=1 i=1
Note that for correlated measurements this is no longer true as will be discussed in the next clause.
2.3.2 Combined standard uncertainty on averages
If x is a repetitive independent measurement of a measurand X, the uncertainty on the average is given
i
by
N
u (x )
∑
i 2
Nu (x )
u(x )
i=1 i
i
(55)
u (x)= = =
c
N N N
If however the uncertainty of a component is related to an effect with periodicity exceeding the
weighing interval (t – t < < T ), the uncertainty on the average is more likely to be
N 1 effect
u (x)=u(x ) (56)
c i
One could say that the measurement results are highly correlated (r = 1) such that Formula (42)
becomes (c = 1/N)
i
N
u(x )
∑ i
N N−1 N
i=1
u = [cu(x )] + 2 cc u(x )u(x )= =u(x ) (57)
c ∑ i i ∑∑ i j i j i
N
i=1 i=1 j=i+1
When for example calculating the 30 s running average of the HRR (Formula (27) this should be kept in
mind. The running average will reduce the uncertainty associated with noise, but will not, for example,
eliminate the uncertainty related to daily cycle temperature variations.
Figure 3 illustrates this by way of example. Suppose a cyclic phenomenon with periodicity t = 360 s
s
introduces an uncertainty of 500 ppm. Noise on the signal introduces an uncertainty estimated at 100
ppm.
Key
X time (s)
running average over 30 s
signal with noise
Figure 3 — Effect of averaging on uncertainty (function of periodicity)
From this example it is clear that the running average over 10 samples (30 s) may reduce the
uncertainty associated with noise with a factor 10 , but does not affect the uncertainty related to
longer term variations (t > > 30 s).
s
A similar statement is true for the calculation of the total heat release in the first 10 min of a test
(600 s). Suppose the measurements are perfectly correlated (r = 1) the uncertainty on the sum becomes
(c = 1):
i
N N−1 N N
u = [cu(x )] + 2 cc u(x )u(x )= u(x ) (58)
c ∑ i i ∑∑ i j i j ∑ i
i=1 i=1 j=i+1 i=1
which is higher than Formula (55). In this case, events with a periodicity of approximately 10 minutes
or less will be dampened out while events with a longer periodicity will not be dampened out (r goes to
1).
For parameters like MARHE, which is also based on total heat release, the behaviour with respect to
uncertainty will depend upon the integration time which is variable.
On the other hand however, uncertainties related to very slow processes (τ > 10 times test run) hardly
contribute to the uncertainty on the oxygen depletion since it is a relative measurement, i.e. the actual
status is compared with the initial status at the start of the test.
This document therefore considers measurements as being independent, include the uncertainty
related to slow processes (τ > 10 times test run) in the zero calibration (= daily calibration of zero
points), include the uncertainty related to drift over one test run in the uncertainty related to the actual
measurement point.
2.3.3 Combined standard uncertainty of a product and a division
Throughout the document, often the uncertainty has to be calculated for a product and for a division.
Starting from the general form
y=axx (59)
1 2
The partial derivative to the variables x and x is given by
1 2
∂y y
=ax = (if x1 ≠ 0) (60)
∂x x
1 1
∂y y
=ax = (if x ≠ 0) (61)
∂x x
2 2
Thus the combined standard uncertainty on y is estimated out of
2 2
   
u(y) u(x ) u(x )
1 2
=   +  (if x ≠ 0 and x ≠ 0) (62)
1 2
   
y x x
 1   2 
This formula holds for a division. Indeed, taking the partial derivatives of the general form
x
y=a (63)
x
∂y a y
= = (if x ≠ 0) (64)
∂x x x
1 2 1
∂y ax y
=− =− (if x ≠ 0) (65)
∂x x x
2 2 2
NOTE The statement (if x ≠ 0) will be omitted from now on if it can reasonably be assumed that x will always
i i
fulfil this requirement in a practical sense.
2.3.4 Combined standard uncertainty on the heat release rate (Q)
The heat release rate (Q = HRR ) is given by Formula (19)
total
φ
D°


Q= E′X V (66)
O2 D298
{ ( )}
1+α−1φ
Substituting the volume flow V (2.3.7) this becomes
D298
φ k Dp(t)
D°
t

′
Q= EX cA (67)
O2
{1+(α−1)φ} k T (t)
p ms
Taking the partial derivative of Q to the different components x results in the corresponding
i
sensitivity coefficients :
ci
D°
  
′ ( )
∂Q EX V −α−1Q
Q
O2 D298
c ≡ = (68)
φ
∂φ [1+(α−1)φ]
Q Q φ
c =c c see also 2.3.5 (69)
A A
φ
XO XO
2 2
Q Q φ
see also 2.3.5 (70)
c =c c
A A
φ
XCO XCO
2 2
 
∂Q Q
Q
(71)
c ≡ =
E'
′ ′
∂E E

∂Q  −φ 
Q

c ≡ =Q (72)
α
 
∂α 1+(α−1)φ
 
 
∂Q Q
Q
c ≡ = (73)
D
D° D°
XO
∂X X
O2 O2
 
∂Q Q
Q
c ≡ = (74)
V
D
 
∂V V
D D
Q Q V
D
c =c c see also 2.3.7 (75)
c V c
D
Q Q V
D
c =c c see also 2.3.7 (76)
A V A
D
Q Q V
D
c =c c see also 2.3.7 (77)
k V k
t D t
Q Q V
D
c =c c see also 2.3.7 (78)
k V k
p D p
Q Q V
D
c =c c see also 2.3.7 (79)
Dp V Dp
D
Q Q V
D
c =c c see also 2.3.7 (80)
T V T
ms D ms
D
Q Q φ Q XO
c =c c +c c see also 2.3.5 and 2.3.6 (81)
A° A° D A°
φ
XO XO XO XO
2 2 2 2
Q Q φ
c =c c see also 2.3.5 (82)
A° A°
φ
XCO XCO
2 2
and to the combined standard uncertainty
2 2 2 2 2
Q A Q A Q Q
 
(c u(XO ))+(c u(XCO ))+(c u(E')) +(c u(α))
A A
2 2 E' α
XO XCO
2 2
 
2 2 2 2
 Q D° Q Q Q 
+(c u(XO ))+(c u(c)) +(c u(A)) +(c u(k ))
D
2 c A k t
XO t
 
 2 2 2 
Q Q Q
( )
+ c u(k ) +(c u(Dp)) +(c u(T ))
 k p Dp T ms 
p ms
 
2 2
Q A° Q A°
( ) ( )
+ c u(XO ) + c u(XCO )
 A° A° 
2 2
XO XCO
2 2
 
Q Q A A
− 2c c u(XO )u(XCO )
 A A 
2 2
XO XCO

2 2 (83)
u(Q)=
 
Q Q A
− 2c c u(XO )u(T )
 A 
T 2 ms
XO ms
 
Q Q A
− 2c c u(XO )u(Dp)
 A 
Dp 2
XO


Q Q A
+ 2c c u(XCO )u(T )
 A 
T 2 ms
XCO ms


Q Q A
+ 2c
...