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represented by a vector x and a matrix A. The question is: which is the
resulting error y on the vector y, which is the vector containing the final results?
If the matrix A and the vector x were known, we could write:
ðA þ AÞðy þ yÞ¼x þ x ð7:72Þ
and, taking Equation 7.71 into account, we could solve:
y ¼ðA þ AÞ
1
ðx AyÞð7:73Þ
This equ ation can be used many times in a Monte-Carlo process, varying each
time all the components of A and x at random, according to their probability
density function. This provides a series of vectors y from which an estimate
of the probability density functions of the components can be calculated.
However, this procedure is time consuming and, assuming a normal distribu-
tion of the measurement methods, simpler methods are available, which are
described next.
Complete error analysis
The requested final result is calculated using:
y ¼ A
1
x hence y
i
¼
X
j
ij
x
j
ð7:74Þ
where the coefficients
ij
are those of the inverse matrix A
1
. The error
calculated with the most simple (or the differential) method will then be:
y
i
¼
X
k
@y
i
@x
k
x
k
þ
X
k;l
@y
i
@a
kl
a
kl
¼
X
k
j
ik
x
k
jþ
X
k;l
@
ij
@a
kl
x
j
a
kl
ð7:75Þ
But since
@
ij
@a
kl
¼
ik
lj
ð7:76Þ
we get finally:
y
i
¼
X
k
j
ik
x
k
jþ
X
kl
X
j
ik
jl
x
j
kl
ð7:77Þ
If the variances and covariances s
2
xk;xl
, s
2
aki;amn
and s
2
aki;x
m
are known, the
covariance of the results s
2
yi; yj
is well estimated using a first order Taylor’s
expansion (Bevinton, 1969). We get:
s
2
y
i
; y
j
¼
X
klmn
@y
i
@a
kl
@y
j
@a
mn
s
2
a
kl
a
mn
þ
X
kl
@y
i
@x
k
@y
j
@x
j
s
2
x
k
x
l
þ
X
klmn
@y
i
@a
kl
@y
j
@x
m
þ
@y
j
@a
kl
@y
i
@x
m
s
2
a
kl
x
m
ð7:78Þ
Common Methods and Techniques 161
- Claude-Alain Roulet 4
- Contents 6
- List of Figures and Tables 8
- List of Figures and Tables ix 10
- List of Figures and Tables xi 12
- Preamble 13
- Introduction 14
- Outdoor airflow rate 15
- Recirculation rates 16
- Exfiltration 17
- Ventilation efficiency 18
- Airtightness 20
- Energy efficiency 20
- Common techniques 21
- Airflow Rates in Buildings 22
- Tracer decay, no injection 24
- Constant injection rate 24
- Constant concentration 25
- Pulse injection 25
- Airflow Rates in Buildings 5 26
- Airflow Rates in Buildings 7 28
- Global system of equations 29
- Airflow Rates in Buildings 9 30
- Properties of the flow matrix 31
- Airflow Rates in Buildings 11 32
- Note: y Under condition 34
- Source: Sherman, 1990 34
- Airflow Rates in Buildings 13 34
- Airflow Rates 36
- Source: Roulet et al., 2000a 37
- Velocity traverse 38
- Test procedure 39
- Tracer gas dilution 39
- Source: Awbi, 2007 40
- Inflatable bag 41
- Flowmeter 41
- Compensated flowmeter 41
- Tracer gas injection ports 41
- Source: Roulet et al., 1999 42
- Node by node method 44
- Least square solution 48
- Eliminating some equations 48
- Simplest way 49
- Less than four tracer gases 53
- Planning tool 54
- Simple measurement using CO 55
- Age of Air and 60
- Ventilation Efficiency 60
- Nominal time constant 61
- Air exchange efficiency 61
- Source: Roulet, 2004 62
- Measurement method 63
- Concentration [ppm] 65
- Source: Roulet et al., 1998 69
- The model matrix M 73
- Condition of the model matrix 74
- Factorial designs 75
- 3-D designs 76
- Measurement methods 80
- Reductive sealing 82
- Density corrections 84
- Two measurement points 85
- Airtightness 65 86
- Airtightness of buildings 88
- External fan 89
- Internal fan 90
- Leakage visualization 90
- The stack effect method 91
- Airtightness 71 92
- Neutral height method 93
- Airtightness 73 94
- Pressurization method 95
- Flow rate difference method 96
- Conditioned space 97
- SealSeal 97
- Measurements and Measures 98
- Related to Energy Efficiency 98
- Energy in air handling units 100
- Heat exchangers 104
- Types of heat exchangers 105
- Rotating heat exchangers 106
- Glycol heat exchanger 107
- Heat pump 107
- Heat exchange efficiency 107
- Ventilated 110
- Global heat recovery efficiency 111
- Source: Roulet et al., 2001 114
- Examples of application 115
- Energy for ventilation 118
- Measurement of airflow rate 120
- Measurement of electric power 121
- Simulation results 126
- Recirculation 126
- Humidification 127
- Heat recovery 127
- Ductwork 128
- Contaminants in 129
- Air Handling Units 129
- Source: Bjo 130
- Humidifiers 131
- Source: Roulet et al., 2000 133
- Measurement protocols 134
- Selecting the panel 135
- Source: Bluyssen, 1990 136
- Training procedure 137
- Experimental day 138
- Principle of the method 138
- Injection technique 139
- Hot air blower (200 °C) 140
- Stainless steel 140
- Ø 6 mm copper tube 140
- Example of application 141
- Evidence for adsorption 144
- HVAC systems 146
- Common Methods 153
- Properties of tracer gases 155
- Source: Dietz et al., 1983 157
- Mixing tracer gases 158
- Tracer density problem 159
- Sampling methods 160
- Grab sampling 160
- Passive sampling 160
- Networks, pumps and pipes 161
- Tracer gas analysers 163
- Objective of the analysis 163
- Photo-acoustic detector 165
- Mass spectrometry 165
- Gas chromatography 167
- Chemical indicator tubes 167
- Identification methods 168
- Confidence in the coefficients 170
- Regression of the second kind 171
- Orthogonal regression 171
- Bayesian identification 172
- Error analysis 174
- Definitions 175
- A few statistics 175
- Covariance 177
- Statistical distributions 177
- What is the problem? 180
- Most simple error analysis 180
- Estimate of the variance 181
- Linear equations systems 181
- Upper bound of the errors 183
- References 187
- Unit Conversion Tables 192
- Pressure 194
- Volume flow rate 194
- Mass flow rate 194
- Glossary 195
- Glossary 175 196
- Glossary 177 198
- Glossary 179 200
- Glossary 181 202
- Glossary 183 204
- Glossary 185 206
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