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where
1
and
n
are respectively the largest and the smallest eigenvalues of
A
H
A. This number is the spectral condition number.
Constant absolute error
If the absolute error is constant:
A ¼ e1 and y ¼ y1 ð7:93Þ
where 1 and 1 are respectively a matrix of order N and an N-component vector
with all elements equal to 1 (they are not the identity matrix and the unit
vector).
It is easy to see that:
j1j
2
¼ N and j1j
2
¼
ffiffiffiffiffi
N
p
ð7:94Þ
because the eigenvalues of 1 are N and 0
N 1
, and those of 1
H
1 ¼ 1
2
are N
2
and
0
N 1
. It follows that:
jAj
2
¼ eN and jyj
2
¼
ffiffiffiffiffi
N
p
y ð7:95Þ
and
jxj
jxj
cond
2
ðAÞ
1 eNjA
1
j
ffiffiffiffiffi
N
p
y
jyj
þ
Ne
jAj
ð7:96Þ
Constant relative error
In this case, if e and " are the constant relative errors on A and y:
A ¼ eA and y ¼ "y ð7:97Þ
and, from the definitions of the norms:
jAj¼ejAj and jyj¼"y ð7:98Þ
and we get, for any norm sa tisfying j Ij¼1:
jxj
jxj
condðAÞ
1 cond ðAÞ
ð" þ eÞð7:99Þ
assuming that e condðAÞ < 1, that is that A þ dA is regular.
Example: A measurement with two tracer gases at constant concentration in
two zones gives the results in Table 7.7.
Table 7.7 Data measured during a tracer gas experiment in two connected rooms
Zone 1 Zone 2
Tracer concentration 1 10 ppm 2.29 ppm
Tracer concentration 2 6.46 ppm 10 ppm
Injection rate, tracer 1 2:65 10
4
m
3
/h 0 m
3
/h
Injection rate, tracer 2 0 m
3
/h 3:6 10
4
m
3
/h
164 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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