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the partial derivatives are computed as above and we get finally:
s
2
y
i
; y
j
¼
X
klmn
ik
y
l
jm
y
n
s
2
a
kl
a
mn
þ
X
kl
ik
jl
s
2
x
k
x
l
þ
X
klmn
ð
ik
y
l
jm
jk
y
l
im
Þs
2
a
kl
x
m
ð7:79Þ
which simplifies, if the variables are independent (that is, if the covariances are
zero, which is not always the case):
s
2
y
i
; y
j
¼
X
kl
ik
jk
y
2
l
n
s
2
a
ki
a
kl
þ
X
kl
ik
jk
s
2
x
k
x
k
ð7:80Þ
Upper bound of the errors
The vector y contains a large number of data, but it is helpful to represent the
error by a single value. To obtain such a single value, the following defini tions,
which can be found in the specific mathematical literature (for example, Deif,
1986) are used.
Vectorial norms and matrix norms
The norm jxj of a vector x is any operation of the n-fold real space R
n
in the
ensemble of real numbers R satisfying:
jxj0 and jxj¼0 if and only if x ¼ 0
jcxj¼jcjjxj for any c 2 R
jx þ yjjxjþjyj
ð7:81Þ
For example, the Euclidian norm that corresponds best to the standard
deviation:
jxj
2
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi
X
x
2
i
q
ð7:82Þ
complies with the relations in Equation 7.81, but there are many others, like
jxj
1
¼
P
x
i
or j xj
1
¼ maxðjx
i
jÞ.
The norm jAj of a matrix A is any application NðAÞ)jAj2R satisfying:
jAj0 and jAj¼0 if and only if A ¼ 0
jcAj¼jcjjAj for any c 2 R
jA þ BjjAjþjBj
jA BjjAj jBj
ð7:83Þ
The matrix norm jAj is consistent with the vectorial norm jxj if:
jAxjjAj jxj for any x ð7:84Þ
162 Ventilation and Airflow in Buildings
- 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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