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will be obtained, and a correlation between C and n will be found: the larger C
values correspond to the smaller n and vice versa. A good identification
technique should give the most likely couple of coefficients together with the
probability density fðC; nÞ. Such a technique exists (Tarantola, 1987) and is
summarized below.
Identification of the model parameters
Let us put in a vector, z, both measured data and model parameters that have to
be determ ined and assume that this vector is a random variable with a normal
distribution in k-fold space (k ¼ number of parameters and data):
fðzÞ¼A exp ½
1
2
ðz z
p
Þ
T
C
1
z
ðz z
p
Þ ð7:35Þ
where:
z
p
is the a priori vector, z, containing the measured values and reasonable
estimates of the parameters to be identified,
C
z
is the covariance matrix between the elements of z. Its diagonal elements
are the variances of the measured quantities and a priori estimated
variances of the parameters. These latter varian ces are generally large,
since the parameters are generally not known before the me asurements.
The components of the vector, z, are linked by a mathematical model or a set of
equations that can be written:
ðzÞ¼0 ð7:36Þ
For example, if a linear relationship is assumed between two measured vari-
ables, x and y, the set of equations:
y
i
¼ a þ bx
i
ð7:37Þ
can be written in a matrix form:
1 x
1
y
1
1 x
2
y
2
1 x
n
y
n
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
a
b
1
0
B
@
1
C
A
¼ 0 ð7:38Þ
Generally, the proposed model is not exact and it may be assumed that it has a
normal distribution:
gðzÞ¼B exp½
1
2
ðzÞ
T
C
1
T
ðzÞ ð7:39Þ
where C
T
is the covariance matrix of the model. If the model is exact, this
distribution is a Dirac distribution:
gðzÞ¼½ ðzÞ ð7:40Þ
Combining the prior knowledge contained in the distribution fðzÞ with the
model described with the distribution gðzÞ gives a new distribution containing
152 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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