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Data-based mechanistic modelling approach to determine the age of air in a ventilated space
ARTICLE IN PRESS
Building and Environment 41 (2006) 557–567 www.elsevier.com/locate/buildenv
Data-based mechanistic modelling approach to determine the age of air in a ventilated space S. Van Buggenhouta, T. Zerihun Destaa, A. Van Brechta, E. Vrankena, S. Quantena, W. Van Malcotb, D. Berckmansa, a
Laboratory for Agricultural Buildings Research, Department of Agro-engineering and -Economics, Catholic University of Leuven, Kasteelpark Arenberg 30, Leuven B-3001, Belgium b Katholieke Hogeschool Kempen, Kleinhoefstraat 4, 2440 Geel Received 27 January 2005; received in revised form 21 February 2005; accepted 23 February 2005
Abstract In literature, local mean age of air is used as an important index to evaluate indoor air quality in ventilated rooms. In this research, a data-based mechanistic approach is used to model the spatial–temporal mass distribution in an imperfectly mixed forced ventilated installation. A first-order transfer function model has proved to be sufficiently good in describing the mass transfer dynamics ðR2t ¼ 0:987Þ of the system. Furthermore, it was possible to fully understand the physical meaning of the model parameter. The parameter is found to be an inverse of the age of air. This Data-Based Modelling approach proved to be more robust when dealing with measurement noise. Finally, the modelled age of air was validated with a classical step up determination of the age of air for experimental data. Good correlation ðR2t ¼ 0:77Þ was found between both results, which proved the physical background of the model parameter. r 2005 Elsevier Ltd. All rights reserved. Keywords: Imperfectly mixed fluids; Data-based mechanistic modelling; Age of air; Mass transfer; Ventilation effectiveness
1. Introduction In everyday life, we are always confronted with imperfectly mixed fluids. Every fluid in nature, whether it is a gas or a liquid, is imperfectly mixed and characterised by spatial–temporal gradients of heat and mass transfer variables. In ventilated airspaces (domestic buildings, office rooms, supermarkets, transport systems, medical facilities, etc.) and in agricultural and industrial process rooms (greenhouses, animal houses, chemical vessels, bio-reactors, etc.), it is desirable to control the spatial–temporal heat and mass distribution in the imperfectly mixed fluid in order to achieve optimum process quality with a minimum use of energy [1]. Corresponding author. Tel.: +32 16 321726; fax: +32 16 321480.
E-mail address: [email protected] (D. Berckmans). 0360-1323/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.buildenv.2005.02.029
Local mean age of air is an important index for the evaluation of ventilation effectiveness and air distribution in buildings. Recent researches have revealed the importance of providing fresh air to the breathing zone of the living organism. On humans, a close link was observed between the micro-environment and climaterelated diseases: e.g. Sick Building Syndrome (SBS), asthma and allergic reactions [2–5]. Distribution of ventilated air can be visualised by measuring the ‘local mean age of air’ [6,7] at different positions. The local mean age of air at a point in a room is the mean time period required for outdoor air to reach that particulate location since entering the room. This parameter is used to assess ventilation effectiveness (e.g. [7]). Quantification is done by injecting tracer gases (CO2, SF6, etc.) at the inlet and recording gas concentration at the position of interest. The tracer gas can be injected by pulse, step-up or step-down methods. However, due to integration errors [8–10], the
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determination of this local mean age of air is not straightforward. This paper presents a new approach for the determination of this ventilation effectiveness parameter. A Data-Based Mechanistic (DBM) modelling technique is used for modelling the mass transfer phenomena in a forced ventilated room. Most existing models are or mechanistic (white box) models or data-based (black box) models. Mechanistic models describe the system based on a priori defined physical, mechanical, chemical and/or biological mechanisms underlying the physics of the system [11]. These mechanistic models (e.g. Computational Fluid Dynamics (CFD)) are restrictive due to their exceptional complexity. One must be aware of the fact that mechanistic models constitute a large number of assumptions and approximations [12] resulting in models that lack the necessary accuracy to be appropriate for control purposes. In the data-based modelling approach, the model structure is inferred and the model parameters are identified using informative experimental data and applying more objective, statistically based methods (e.g. [13]). These techniques can deliver accurate models but the complete lack of physical insight is a major drawback in their formulation, making them application specific models. A DBM model is an intermediate type of model, which exploits the availability of time-series data in statistical terms and also attempts to produce models which have a physical (mechanistic) meaning. These data-based mechanistic (or grey box) models provide a physically meaningful description of the dominant internal dynamics of heat and mass transfer in the imperfectly mixed fluid, while providing low order models [14,15]. The strength of these models is that they combine the advantages of both mechanistic or white box (generality, knowledge based) and data-based or black box (compact, accurate) models and are, therefore, an ideal basis for model-based control system design [16]. In the present paper, the DBM approach is applied in order to assess ventilation performance in a forced ventilation type of building. More precisely, the age of air of a mechanically ventilated structure is determined by using a data-based model with physically meaningful parameters. The values of the modelled age of air are validated with a ‘classical’ numerical (step inlet injection method, [7]) determination of the age of air.
2. Materials and methods 2.1. Test installation The laboratory test room, represented in Fig. 1, is a mechanically ventilated room with a length of 3 m, a height of 2 m and a width of 1.5 m. It has a slot inlet (1 in
Fig. 1) in the left sidewall just beneath the ceiling and an asymmetrically positioned, circular air outlet (2 in Fig. 1) in the right sidewall just above the floor. The volume of the room is 9 m3. An enveloping chamber of length 4 m, width 2.5 m and height 3 m (6 in Fig. 1) is built around the test room to reduce disturbing effects of varying laboratory conditions (fluctuating temperature, opening doors, etc.). The test room and the enveloping chamber are both constructed of transparent Plexiglas through which the air flow pattern can be observed during flow visualisation experiments. A mechanical ventilation system enables an accurate control (accuracy of 76 m3/h) of the ventilation rate in the range 70–420 m3/h. A heat exchanger is provided in the supply air duct to regulate the temperature of the inflowing air. A series of five aluminium heat sources (4 in Fig. 1), with a semi-conductor heat source is used to physically simulate the heat production of the occupant(s). To measure the dynamic spatial gas concentration distribution in the test chamber, 36 air sampling tubes are located in a 3-D measuring grid (7 in Fig. 1) covering a large part of the room. The tubes are located in two vertical xy-planes: a ‘front sensor plane’ (0.375 m from the front wall) and a ‘rear sensor plane’ (0.375 m from the back wall). The lower sensors in both sensor planes are positioned at a height of 0.8 m above the floor, the upper sensors are 0.4 m beneath the ceiling and the middle sensors are at a height of 1.2 m above the floor. The left sensors are positioned 0.4 m from the inlet wall and the right sensors are positioned 0.6 m from the outlet wall (see Fig. 1). A pneumatic system (9 in Fig. 1) with two-way solenoid valves is used to direct the gas sample at the selected position in the room to the gas analyser. The measurement accuracy of the gas analyser is 710 ppm for CO2. The sampling rate used is 3.3 s. Moreover, a data logger is provided that monitors ventilation rate and temperature inside the room. All the recorded data are stored on a central computer. A pressurised CO2 gas bottle (N95) in combination with a computerised gas flow rate controller is used to inject the tracer gas into the air inlet. 2.2. Data-based mechanistic modelling In DBM modelling techniques, the model structure is first identified using objective methods of time series analysis based on a given, general class of time series model (here linear, continuous-time transfer functions (TF) or the equivalent ordinary differential equations are used). A continuous-time TF model for a singleinput single output (SISO) system has the following general form: yðtÞ ¼
BðsÞ uðt dÞ þ xðtÞ, AðsÞ
(1)
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Fig. 1. Laboratory test chamber with sampling system to measure spatial gas concentration distributions. 1. Pressurised CO2 gas bottle, 2. gas flow rate controller, 3. air inlet, 4. air outlet, 5. shallow hot water reservoir, 6. aluminium conductor heat sink, 7. 3-D measurement grid consisting of 36 sampling tubes, 8. envelope chamber or buffer zone, 9. multipoint sampler consisting of pc-controlled solenoid valves, 10. gas analyser, 11. datalogging system.
where s is the Laplace operator, i.e. s ¼ d=dt; yðtÞ is the noisy measured output; uðtÞ is the model input; d is the time delay; xðtÞ is additive noise, assumed to be a zero mean, serially uncorrelated sequence of random variables with variance s2 accounting for measurement noise, modelling errors and effects of unmeasured inputs to the process; and finally, AðsÞ and BðsÞ are polynomials in the s operator of the following form: AðsÞ ¼ sn þ a1 sn1 þ . . . þ an ,
(2)
BðsÞ ¼ b0 sm þ b1 sm1 þ . . . þ bm ,
(3)
where mpn; a1, a2, y, an and b0, b1, y, bm are the TF denominator and numerator parameters, respectively. Once the input–output data are available, model parameters (Eqs. (2) and (3)) can be identified by using statistical procedures. But the resulting model is only considered fully acceptable if, in addition to explaining the data well, it also provides a description that has relevance to the physical reality of the system under consideration. The ability to estimate parameters represents only one side of the model identification problem. Equally important is the problem of objective model order identification. This involves the identification of the best choice of orders of the numerator and denominator polynomials, together with the time delay. The parameters of a TF model may be estimated using
various methods of identification and estimation procedures [17,18]. Although Least Squares (LS) is one of the most commonly used model estimation algorithm, the estimated model parameters become asymptotically biased away from their true values in the presence of measurement or disturbance noise signal [17]. Here we use the more complex Simplified Refined Instrumental Variable (SRIV) algorithm [17], that uses the Instrumental Variable (IV) approach coupled with special adaptive prefiltering to avoid this bias and to achieve good estimation performance. A reasonably successful identification is based on minimisation of the Young Identification Criterion YIC: ! np 1 X s^ 2 p^ ii s^ 2 YIC ¼ ln 2 þ ln , np i¼1 a^ 2i sy
(4)
where s^ 2 is the sample variance of the model residuals, s2y is the sample variance of the measured system output about its mean value, np is the total number of model parameters, i.e. np ¼ n þ m þ 1, a^ 2i is the square of the ^ p^ ii is the ith ith element in the parameter vector a, diagonal element of the inverse cross product matrix PðNÞ, s^ 2 p^ ii can be considered as an approximate estimate of the variance of the estimated uncertainty on the ith parameter estimate.
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YIC is a heuristic statistical criterion which consists of two terms, as shown in Eq. (2). The first term provides a normalised measure of how well the model fits the data: the smaller the variance of the model residuals in relation to the variance of the measured output, the smaller this term becomes. The second term is a normalised measure of how well the model parameter estimates are defined. This term tends to become bigger when the model is over-parameterised and the parameter estimates are poorly defined. Consequently, the model which minimises the YIC provides a good compromise between goodness of fit and parametric efficiency. While the YIC can be a great help in ensuring that the model is not over-parameterised, it is not always good at discriminating models that have a lower order than the ‘best’ model [19]. Hence, the YIC will often, if applied strictly, identify a model that is under-parameterised. Therefore, it is used
together with the coefficient of determination R2T . The coefficient of determination is a statistical measure of how well the model fits the experimental data. If the sample variance s^ 2i of the model residuals is low compared with the sample variance sy2 of the measured system output about its mean value, then R2 tends towards unity. If s^ 2 is of similar magnitude to s2y , then it tends towards zero. R2 ¼ 1
s^ 2 . s2y
(5)
If the YIC identified model has an adequate R2T which is not significantly lower than the R2 of the higher order models, it may be fully accepted as the best model in statistical terms. The SRIV structure identification criterion has been proven very successful in practical applications [19].
Table 1 Overview of the identification experiments Experiment
Ventilation rate (m3/h)
Refreshment rate (1/h)
Supply air temperature (1C)
Air flow pattern
1 2 3 4 5 6
80 120 160 200 240 300
9 13 18 22 27 33
11.5 11.5 11.5 11.5 11.5 11.5
Anti-clockwise Anti-clockwise Anti-clockwise Clockwise Clockwise Clockwise
Fig. 2. The output of a first-, second- and third-order TF models compared with the measured CO2 concentration response at sensor position inside the measuring grid (position 15) for experiment 2.
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2.3. Experiments for identification of a reduced-order linear model When a set of informative input–output data is generated, a reduced-order, linear model can be identified to describe the data in a sufficiently accurate way. To identify the parameters of the TF model for each of the 36 sampling positions in the test chamber, 6 identification experiments were carried out as presented in Table 1. For each experiment, the CO2 concentration at the air supply was raised to 1905 ppm during 47 min (time needed to obtain steady-state conditions in test installation), whilst maintaining a constant ventilation rate. The experiments were carried out over a wide range of low (80, 120 and 160 m3/h) and high (200, 240 and 300 m3/h) ventilation rates. The total internal heat production of the 5 heating elements was maintained constant at 300 J/ s and the temperature of the incoming air is kept constant at 11.5 1C. In each of the 6 experiments, the ventilation rate, supply air CO2 concentration and the CO2 concentration response at each of the 36 spatially distributed sampling positions in the test chamber were recorded every 3.3 s. A typical identification experiment at a position inside the measuring grid is presented in Fig. 2. Further, in each experiment, non-toxic smoke is injected in the room and flow patterns are recorded by a digital camera.
3. Results and discussion 3.1. Parameter identification The continuous-time SRIV algorithm [19] is used to identify the linear TF model between the supply air CO2
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concentration and the CO2 concentration at a certain spatial point. Table 1 shows that a first-order model (Eq. (5)) yields an excellent explanation of the convective mass transfer of the system according to the three basic principles of the SRIV algorithm: minimal number of parameters, high reliability of the parameters estimation and high accuracy of the model to describe the CO2 concentration dynamics (Table 2 and Fig. 3). ci ðtÞ ¼
b0 c0 ðtÞ, s þ a1
(6)
where ci is the dynamic CO2 concentration at position i (ppm); c0 is the dynamic supply air CO2 concentration (ppm). The R2T values are similar for first-, second- and third-order models. However, when the YIC-values are considered the first-order system is more parametrical efficient than the higher order models. This demonstrates that the mass transfer phenomenon obeys firstorder dynamics. When applied to all 36 sensor positions in all experiments, the first-order model describes the data with an average R2 of 0.987 and an average YIC-value of 14.91 (Table 3). 3.2. Physical interpretation of model parameter—WellMixed Zone Concept The DBM approach represents the imperfectly mixed fluid in a process room by a number of well-mixed zones (WMZs) which are defined around the nodes of a sensor grid. A WMZ is a 3-D-zone of improved mixing with a certain volume wherein acceptably low spatial gradients occur. These WMZs exist in every imperfectly mixed fluid and for each considered variable such as, for example, CO2-concentration, temperature, humidity, etc. A schematic representation of a WMZ in a process room with inlet fluid flow rate V (m3/s) is given in Fig. 3.
Table 2 The model parameter estimates with associated relative standard error Order of TF
Parameter estimates
Relative standard error (%)
R2T (dimensionless)
YIC value (dimensionless)
[1 1 9]
a1 ¼ 0:013 b0 ¼ 22; 094
x ða1 Þ ¼ 0:32 x ðb0 Þ ¼ 0:31
0.998
17,146
[2 2 11]
a1 a2 b0 b1
¼ 0:022 ¼ 0:000 ¼ 24; 190 ¼ 0:157
x x x x
ða1 Þ ¼ 2:89 ða2 Þ ¼ 8:21 ðb0 Þ ¼ 0:42 ðb1 Þ ¼ 8:16
0.999
10,801
[3 3 11]
a1 a2 a3 b0 b1 b2
¼ 0:050 ¼ 0:001 ¼ 0:000 ¼ 26; 697 ¼ 0:809 ¼ 0:002
x x x x x x
ða1 Þ ¼ 7:04 ða2 Þ ¼ 22:67 ða3 Þ ¼ 141:22 ðb0 Þ ¼ 0:68 ðb1 Þ ¼ 9:92 ðb2 Þ ¼ 136:29
0.999
5297
The YIC value and the coefficient of determination R2 of a first-, second- and third-order continuous-time TF model for sensor position 1.
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To describe the dynamic behaviour of mass transfer in the considered WMZs, standard mass transfer theory can be applied. In case of a constant ventilation rate, a linear, first-order differential equation (as previously identified from experimental data) can be formulated as
Vc,i.Co(t−i).o V Co (t)
Ci(t) well mixed zone
dC i ðtÞvoli ri ¼ V c;i C o ðtÞro V c;i C i ðtÞri , dt
Vc,i.Ci(t).i
Fig. 3. Schematic representation of the WMZ concept.
Table 3 The mean value over the 6 experiments of the relative standard errors xðbÞ, xðaÞ on the model parameter estimates Sensor
x (b)
x (a)
R2T
YIC
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
1.66 0.69 0.55 1.02 0.58 0.79 1.51 0.72 0.65 0.55 0.65 1.04 0.55 0.57 0.82 0.68 0.95 1.29 0.85 0.37 0.75 0.78 0.61 0.68 0.68 0.76 0.68 0.66 0.64 0.81 0.38 0.57 0.60 1.76 0.64 1.19
1.70 0.72 0.57 1.04 0.60 0.82 1.55 0.78 0.66 0.57 0.68 1.07 0.57 0.59 0.80 0.71 1.00 1.32 0.88 0.35 0.76 0.79 0.63 0.70 0.70 0.79 0.70 0.68 0.66 0.83 0.40 0.59 0.62 1.80 0.67 1.23
0.985 0.989 0.995 0.990 0.994 0.982 0.987 0.962 0.994 0.994 0.982 0.992 0.994 0.994 0.983 0.987 0.949 0.984 0.959 0.988 0.991 0.987 0.992 0.991 0.992 0.989 0.992 0.989 0.992 0.989 0.987 0.994 0.993 0.989 0.991 0.988
13.55 14.12 15.19 14.00 15.07 13.49 13.82 13.97 14.96 14.91 14.21 14.17 14.88 14.79 13.61 14.47 12.99 14.00 13.88 16.67 14.38 13.86 14.95 14.27 14.61 13.71 14.18 14.05 14.23 13.75 38.12 14.70 14.46 13.42 14.19 13.28
Mean
0.80
0.82
0.987
14.91
The coefficient of determination and the YIC-value for each of the 36 sensor positions.
(7)
where t is the time (s); Ci is the concentration of CO2 in the WMZ (ppm); C0 is the concentration of CO2 in the supply air (ppm); voli is the volume of the WMZ (m3); Vc,i is the part of the total ventilation rate that enters the considered WMZ i (m3/h); ri is the density (kg/m3) of the air in WMZ; r0 is the density (kg/m3) of the supply air and subscript i is the position index. But because of small refreshment rates used and since indoor air temperature is kept constant, the density difference of the air can be neglected in these experiments. The average concentration of CO2 in the air before injection is equal to 500 ppm (750 ppm). The average temperature of the surrounding laboratory is equal to an average value of 21 1C. In contrast to the zonal and nodal models in literature [20,21], the different WMZs are considered as decoupled or non-interactive zones, because the objective was to control the conditions in a WMZ by using the fresh air at the inlet conditions. To do so the influence of the inlet conditions on a specific WMZ must be modelled. Since the air inlet conditions are responsible for the mass concentration characteristics of the zone, the model only creates a link between the inlet and individual zones. Since every living organism in a bio-process needs an adequate amount of oxygen rich fresh air, the focus on the WMZ model concept is to model the 3-D- transport of fresh air to a particular zone. This is established by using local fresh air flow rate VCi. The local fresh air flow rate is the air flow rate with the same gas concentrations like the incoming air at the inlet that would have created the aggregate effect of the convective flux interaction with the neighbouring zones on the WMZ in consideration. This is a very useful assumption, because it creates a direct relationship between individual zones without the need for modelling zonal interactions. When this concept is applied in an on-line way, the influence of zonal interactions on the relationship inlet-specific zone is taken into account. When considering non-interactive WMZs, the spatial–temporal model is also less complex and it is an excellent basis for control purposes. The response of each WMZ to changes in the inlet conditions is described by a single model, resulting in n first-order models for n WMZs that can be used for controlling the conditions in the n WMZs each time only using the air inlet conditions.
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Under constant ventilation rate, Eq. (6) can be rewritten as dC i ðtÞ ¼ bi C o ðtÞ bi C i ðtÞ, dt
(8)
where bi is the local refreshment frequency, bi ¼
V c;i . voli
Integrating Eq. (7) gives rise to following equation: Z 0
Ci
dC i ðtÞ ¼ C o ðtÞ C i ðtÞ
Z
t
bi dt.
(9)
0
Solving the integral in Eq. (8) results in C i ðtÞ ¼ C 0 ðtÞð1 ebi t Þ.
(10)
This function represents the exponential accumulation of the tracer gas inside the ventilated room. At infinity, C i ðtÞ will be equal to C o ðtÞ, the constant concentration at the end of the experiments. 3.3. Local mean age of air The local mean age of air is an index used to quantify ventilation effectiveness and to visualise air flow patterns in a ventilation system [22]. Local mean age of air is the mean time required for all fluid particles to arrive at a position of interest from the inlet. It is the mean time required to refresh a certain position in a room once. There are three methods of quantifying this index [7]. All are based on tracer injection at the inlet and gas concentration readings at sampling positions in the room. These methods use advanced statistical distribution analysis [6]. Local mean age of air for step inlet injection is defined as [6] Z 1 C i ðtÞ ti ¼ 1 dt, (11) C i ð1Þ 0 where ti is the local mean age of air at the position i, C i ðtÞ is the contaminant concentration at position i at time t, C i ð1Þ is the contaminant concentration at position i after a long time. Numerical procedures can be applied to solve the above equation on condition that concentration data are available. In this research, the numerical integral is calculated using the Simpson rule [23]. Plugging Eq. (9) into Eq. (10) and taking C i ð1Þ is equal to C 0 yields Z 1 C 0 ðtÞð1 ebi t Þ 1 ti ¼ dt. (12) C0 0
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Solving gives Z
1
ebi t dt ¼
ti ¼ 0
ebi t 1 , b 0
1 ti ¼ 0 , bi 1 ti ¼ . bi
ð13Þ
For first-order system with zero time delay the local age of air, ti ðsÞ, is the inverse of the local refreshment frequency, bi (s1). Zerihun Desta et al. [24] proved the potential of this equation based on numerical CFDdata. The WMZ concept, which results in the formulation of the DBM model, and the associated mass balance differential equation, can be applied to each of the spatially distributed monitoring positions in the test room. 3.4. Parameter contour plots Fig. 4 shows an example of the spatial contours of the parameter bi (s1) in the front and the rear sensor plane of the test installation at a ventilation rate of 240 m3/h (identification experiment 5). Further, the contour plots relate well to the air flow pattern, which is presented in Fig. 5 and was visualised through smoke experiments [25]. At high ventilation rates, the incoming fresh air rapidly moves across the top of the ventilated chamber, hits the right sidewall and then descends towards the exit at the lower right, where some of the air flow recirculates in a clockwise direction. An increase in the air freshness in the direction of the air flow is quite noticeable from the contours of the local refreshment frequency bi. At low ventilation rates, the incoming air moves downwards, hits the floor and moves to the outlet, some air makes a counter clock wise circulation. 3.5. Sensitivity analysis to measurement noise To demonstrate the usefulness of the proposed methodology for the determination of the local mean age of air, a sensitivity analysis was performed. The consistency of both methods was checked when dealing with different levels of measurement noise. To show the effects of noise on both methods, a tracer step injection was simulated (time constant ¼ 20 s, gain ¼ 1) and the local mean age of air was calculated through both methods. This simulation made it possible to demonstrate the effects of introducing random noise. In Fig. 6, the relative error (RE) on the age of air determination is plotted as a function of relative sampling rate (%) without the presence of noise for
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Fig. 4. Spatial contour plots of parameter bi (s1) at 240 m3/h in the 2 vertical planes.
34
28
29
36
23
30 33
7
14
21
1
8
15
4
5
11 2
6
12
18
24 27
13
10
17 20
26
32
16 19
25
31
35
22
9
3
Fig. 5. The visualised air flow pattern at 240 m3/h.
both methods. This relative sampling rate is defined as the ratio of the sampling rate over the time constant. In this case, both methods give reliable results (RE o0.5%) for all sampling rates. When dealing with noise free data, it can be concluded that the numerical integration method for the determination of the age of air is accurate enough. The tendency can be observed that the measurement error augments with higher sampling rates.
Experimental data however is always biased with different levels of noise. Fig. 7 shows the results of age of air determination when noise level is increasing for a relative sampling rate of 1.6% ( ¼ sampling rate of 3 s) of both methods. A Monte Carlo Analysis was carried out to determine the confidence intervals of both methods. One thousand simulations were performed. The mean RE of the numerical integration method was below 3%, but this method becomes biased in the
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Fig. 6. Relative error (%) as a function of relative sampling rate, on the determination of the local mean age of air based on the trapezoidal integration rule ([6]) and the DBM method.
presence of noise. The confidence interval of the RE becomes larger for higher noise levels, more precisely ranging from 0.6 to 0.4% for 1% noise; from 3.6 to 1.8% for 5% noise and from 6% to 1.4% for 7% noise. The confidence intervals of the DBM method remain constant for different levels of noise (1% noise: 0.01–0.01%; 5% noise: 0.04–0.03% and 7% noise: 0.03–0.07%). When collecting experimental data, repetitions are a necessity, but the accuracy of the numerical integration method will be far less than the accuracy of the DBM method. This indicates that when experimental data (with inevitable measurement noise) is used the actual determination of the age of air with the numerical integration method becomes biased away from the actual value depending on the noise level of the data. Besides being suitable for controlling purposes, due to the compact model structure, the DBM method for the determination of the local mean age of air proved to be more robust for measurement noise and gave more reliable results. 3.6. Validation of the physical meaning of the modelled age of air In Fig. 8, the results for all ventilation rates and all 36 positions in the room are presented in a graphical way, more precisely; the numerically calculated age (integra-
tion method) is plotted against the modelled version (model parameter ti ¼ 1=bi ). Fig. 8 clearly demonstrates that a linear relationship exists between the modelled and the numerical integration values of the age of air. A linear curve was fitted (R2T ¼ 0:77) through the data. The regression error can be explained by the fact that this formula (Eq. (10)) is based on the calculation of an integral, where the accuracy is directly dependent with the sampling rate. The larger the sampling rate is, the larger the error will be. In this research, the minimum sampling rate was equal to 3.3 s. When the linear fit is forced without intercept, an R2T value of 0.62 is obtained. This supports the physical background of the model parameter bi, as being an unbiased estimator of the actual local age of air in ventilated rooms.
4. Conclusions This paper has proved that a Data-Based Mechanistic (DBM) approach could be applied to determine the local mean age of air in a forced ventilated space. Here a minimally parameterised transfer function model is first identified and estimated from the experimental data without any prior assumptions about the physical nature of the system. Having objectively identified the
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mean age of air (Sandberg) + confidence intervals mean age of air (DBM) + confidence intervals
6
Relative Error [%]
4
2
0
-2
-4
-6
0
1
2
3
4 5 Noise Level [%]
6
7
8
9
Fig. 7. Mean value (1000 repetitions) of the relative error in function of different levels of noise and corresponding confidence intervals for a relative sampling rate of 1.6%.
500 450
y = 0.739*x + 80.9 Rt2 = 0.77
400
300 250 200
βi
1
age modelled [s]
350
150 y = 1.03 * x
100
Rt2 = 0.62
50 0
0
50
100
150
200
250
300
350
400
450
500
τi age numerical [s]
Fig. 8. X–Y scatter plot of the numerical versus the modelled age of air for all 6 ventilation rates and 36 positions in the room. The full line represents a regression curve through the data; the dotted line is the bisector where the modelled value equals the numerical values (integration method).
dominant modes of dynamic behaviour in this manner, the model is then interpreted in physically meaningful terms; more precisely, the inverse of the model parameter is equal to the local mean age of air. This
model explains the data very well (average R2T ¼ 0:987), with the minimum number of identifiable parameters. The local air refreshment parameter bi has proven to be the inverse of the age of air and good agreement ðR2T ¼
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0:77Þ was found with the calculation of the age of air by means of the numerical integral method. Besides being suitable for controlling purposes, the DBM method for the determination of the local age of air proved to be more robust for measurement noise. References [1] Roulet C-A, Vandaele L. Airflow patterns within buildings: measurements techniques. Technical Note AIVC 1991;34. [2] Wargocki P, Wyon DP, Sundell J, Clausen G, Fanger PO. The effects of outdoor air supply rate in an office on perceived air quality, Sick Building Syndrome (SBS) symptoms and productivity. Indoor Air 2000;10:222–36. [3] Xing H, Hatton A, Awbi HB. A study of the air quality in the breathing zone in a room with displacement ventilation. Buildings and Environment 2001;36(7):809–20. [4] Cheong KW, Chong KY. Development and application of an indoor air quality audit to an air-conditioned building in Singapore. Buildings and Environment 2001;36(2):181–8. [5] Cheong KWD, Lau HYT. Development and application of an indoor air quality audit to an air-conditioned tertiary institutional building in the tropics. Buildings and Environment 2003;38(4):605–16. [6] Sandberg M, Sjo¨berg M. The use of moments for assessing air quality. Buildings and Environment 1983;18:181–97. [7] Etheridge DW, Sandberg M. Building Ventilation: Theory and Measurement. West Sussex: Wiley; 1996. [8] Sandberg M, Blomqvist C. A quantitative estimate of the accuracy of tracer gas methods for the determination of the ventilation flow rate in buildings. Buildings and Environment 1985;20(3):139–50. [9] Sherman MH. Analysis of errors associated with passive ventilation measurement techniques. Buildings and Environment 1989;24(2):131–9. [10] Sherman MH. Uncertainty in air flow calculations using tracer gas measurements. Buildings and Environment 1989;24(4): 347–54. [11] Baldwin RL. Modeling Ruminant Digestion and Metabolism. London: Chapman & Hall; 1995. [12] Oltjen JW, Owens FN. Beef cattle feed intake and growth: empirical Bayes derivation of the Kalman filter applied to a
[13]
[14]
[15]
[16] [17] [18] [19]
[20]
[21]
[22] [23] [24]
[25]
567
nonlinear dynamic model. Journal of Animal Science 1987;65: 1362–70. Aerts J- M, Berckmans D, Saevels P, Decuypere E, Buyse J. Modelling the static and dynamic responses of total heat production of broiler chickens to step changes in air temperature and light intensity. British Poultry Science 2000;41: 651–9. Berckmans D, De Moor M, De Moor B. New model concept to control the energy and mass transfer in a three-dimensional imperfectly mixed ventilated space. In: Proceedings of Roomvent 1992, Aalborg, Denmark, pp. 151–168. Janssens K, Van Brecht A, Zerihun Desta T, Boonen C, Berckmans D. Modelling the internal dynamics of energy and mass transfer in an imperfectly mixed ventilated airspace. Indoor Air 2004;14:146–53. Maciejowski JM. Predictive Control with Constraints. UK: Prentice-Hall; 2002 331pp. Young PC. Recursive Estimation and Time-Series Analysis. Berlin: Springer; 1984. Ljung L. System Identification: Theory for the User. Englewood Cliffs, NJ: Prentice-Hall; 1987. Young PC, Jakeman AJ. Refined instrumental variable methods of recursive time-series analysis: part II, single input output systems. International Journal of Control 1979;29:1–30. Dalicieux P, Bouia H, Blay D. Simplified modelling of air movements in room and its first validation with experiments. In: Proceedings of roomvent 1992, Aalborg, Denmark, p. 383–97. Li Y, Sandberg M, Fuchs L. Vertical temperature profiles in rooms ventilated by displacement: full-scale measurement and nodal modelling. In: Proceedings of indoor air 1992, vol. 2, p. 225–43. Federspiel CC. Air-change effectiveness: theory and calculation methods. Indoor Air 1999;9(1):47–56. Kreyszig E. Advanced engineering mathematics. West Sussex: Wiley; 1993. Zerihun Desta T, Van Buggenhout S, Van Brecht A, Meyers J, Aerts J-M, Baelmans M, Berckmans D. Modelling mass transfer phenomena and quantification of ventilation performance in a full scale installation. Building and Environment 2005; Accepted for publication. Van Brecht A, Janssens K, Berckmans D, Vranken E. Image processing to quantify the trajectory of a visualized air jet. Journal of Agricultural Engineering Research 2000;76(1):91–100.
Building and Environment 41 (2006) 557–567 www.elsevier.com/locate/buildenv
Data-based mechanistic modelling approach to determine the age of air in a ventilated space S. Van Buggenhouta, T. Zerihun Destaa, A. Van Brechta, E. Vrankena, S. Quantena, W. Van Malcotb, D. Berckmansa, a
Laboratory for Agricultural Buildings Research, Department of Agro-engineering and -Economics, Catholic University of Leuven, Kasteelpark Arenberg 30, Leuven B-3001, Belgium b Katholieke Hogeschool Kempen, Kleinhoefstraat 4, 2440 Geel Received 27 January 2005; received in revised form 21 February 2005; accepted 23 February 2005
Abstract In literature, local mean age of air is used as an important index to evaluate indoor air quality in ventilated rooms. In this research, a data-based mechanistic approach is used to model the spatial–temporal mass distribution in an imperfectly mixed forced ventilated installation. A first-order transfer function model has proved to be sufficiently good in describing the mass transfer dynamics ðR2t ¼ 0:987Þ of the system. Furthermore, it was possible to fully understand the physical meaning of the model parameter. The parameter is found to be an inverse of the age of air. This Data-Based Modelling approach proved to be more robust when dealing with measurement noise. Finally, the modelled age of air was validated with a classical step up determination of the age of air for experimental data. Good correlation ðR2t ¼ 0:77Þ was found between both results, which proved the physical background of the model parameter. r 2005 Elsevier Ltd. All rights reserved. Keywords: Imperfectly mixed fluids; Data-based mechanistic modelling; Age of air; Mass transfer; Ventilation effectiveness
1. Introduction In everyday life, we are always confronted with imperfectly mixed fluids. Every fluid in nature, whether it is a gas or a liquid, is imperfectly mixed and characterised by spatial–temporal gradients of heat and mass transfer variables. In ventilated airspaces (domestic buildings, office rooms, supermarkets, transport systems, medical facilities, etc.) and in agricultural and industrial process rooms (greenhouses, animal houses, chemical vessels, bio-reactors, etc.), it is desirable to control the spatial–temporal heat and mass distribution in the imperfectly mixed fluid in order to achieve optimum process quality with a minimum use of energy [1]. Corresponding author. Tel.: +32 16 321726; fax: +32 16 321480.
E-mail address: [email protected] (D. Berckmans). 0360-1323/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.buildenv.2005.02.029
Local mean age of air is an important index for the evaluation of ventilation effectiveness and air distribution in buildings. Recent researches have revealed the importance of providing fresh air to the breathing zone of the living organism. On humans, a close link was observed between the micro-environment and climaterelated diseases: e.g. Sick Building Syndrome (SBS), asthma and allergic reactions [2–5]. Distribution of ventilated air can be visualised by measuring the ‘local mean age of air’ [6,7] at different positions. The local mean age of air at a point in a room is the mean time period required for outdoor air to reach that particulate location since entering the room. This parameter is used to assess ventilation effectiveness (e.g. [7]). Quantification is done by injecting tracer gases (CO2, SF6, etc.) at the inlet and recording gas concentration at the position of interest. The tracer gas can be injected by pulse, step-up or step-down methods. However, due to integration errors [8–10], the
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determination of this local mean age of air is not straightforward. This paper presents a new approach for the determination of this ventilation effectiveness parameter. A Data-Based Mechanistic (DBM) modelling technique is used for modelling the mass transfer phenomena in a forced ventilated room. Most existing models are or mechanistic (white box) models or data-based (black box) models. Mechanistic models describe the system based on a priori defined physical, mechanical, chemical and/or biological mechanisms underlying the physics of the system [11]. These mechanistic models (e.g. Computational Fluid Dynamics (CFD)) are restrictive due to their exceptional complexity. One must be aware of the fact that mechanistic models constitute a large number of assumptions and approximations [12] resulting in models that lack the necessary accuracy to be appropriate for control purposes. In the data-based modelling approach, the model structure is inferred and the model parameters are identified using informative experimental data and applying more objective, statistically based methods (e.g. [13]). These techniques can deliver accurate models but the complete lack of physical insight is a major drawback in their formulation, making them application specific models. A DBM model is an intermediate type of model, which exploits the availability of time-series data in statistical terms and also attempts to produce models which have a physical (mechanistic) meaning. These data-based mechanistic (or grey box) models provide a physically meaningful description of the dominant internal dynamics of heat and mass transfer in the imperfectly mixed fluid, while providing low order models [14,15]. The strength of these models is that they combine the advantages of both mechanistic or white box (generality, knowledge based) and data-based or black box (compact, accurate) models and are, therefore, an ideal basis for model-based control system design [16]. In the present paper, the DBM approach is applied in order to assess ventilation performance in a forced ventilation type of building. More precisely, the age of air of a mechanically ventilated structure is determined by using a data-based model with physically meaningful parameters. The values of the modelled age of air are validated with a ‘classical’ numerical (step inlet injection method, [7]) determination of the age of air.
2. Materials and methods 2.1. Test installation The laboratory test room, represented in Fig. 1, is a mechanically ventilated room with a length of 3 m, a height of 2 m and a width of 1.5 m. It has a slot inlet (1 in
Fig. 1) in the left sidewall just beneath the ceiling and an asymmetrically positioned, circular air outlet (2 in Fig. 1) in the right sidewall just above the floor. The volume of the room is 9 m3. An enveloping chamber of length 4 m, width 2.5 m and height 3 m (6 in Fig. 1) is built around the test room to reduce disturbing effects of varying laboratory conditions (fluctuating temperature, opening doors, etc.). The test room and the enveloping chamber are both constructed of transparent Plexiglas through which the air flow pattern can be observed during flow visualisation experiments. A mechanical ventilation system enables an accurate control (accuracy of 76 m3/h) of the ventilation rate in the range 70–420 m3/h. A heat exchanger is provided in the supply air duct to regulate the temperature of the inflowing air. A series of five aluminium heat sources (4 in Fig. 1), with a semi-conductor heat source is used to physically simulate the heat production of the occupant(s). To measure the dynamic spatial gas concentration distribution in the test chamber, 36 air sampling tubes are located in a 3-D measuring grid (7 in Fig. 1) covering a large part of the room. The tubes are located in two vertical xy-planes: a ‘front sensor plane’ (0.375 m from the front wall) and a ‘rear sensor plane’ (0.375 m from the back wall). The lower sensors in both sensor planes are positioned at a height of 0.8 m above the floor, the upper sensors are 0.4 m beneath the ceiling and the middle sensors are at a height of 1.2 m above the floor. The left sensors are positioned 0.4 m from the inlet wall and the right sensors are positioned 0.6 m from the outlet wall (see Fig. 1). A pneumatic system (9 in Fig. 1) with two-way solenoid valves is used to direct the gas sample at the selected position in the room to the gas analyser. The measurement accuracy of the gas analyser is 710 ppm for CO2. The sampling rate used is 3.3 s. Moreover, a data logger is provided that monitors ventilation rate and temperature inside the room. All the recorded data are stored on a central computer. A pressurised CO2 gas bottle (N95) in combination with a computerised gas flow rate controller is used to inject the tracer gas into the air inlet. 2.2. Data-based mechanistic modelling In DBM modelling techniques, the model structure is first identified using objective methods of time series analysis based on a given, general class of time series model (here linear, continuous-time transfer functions (TF) or the equivalent ordinary differential equations are used). A continuous-time TF model for a singleinput single output (SISO) system has the following general form: yðtÞ ¼
BðsÞ uðt dÞ þ xðtÞ, AðsÞ
(1)
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Fig. 1. Laboratory test chamber with sampling system to measure spatial gas concentration distributions. 1. Pressurised CO2 gas bottle, 2. gas flow rate controller, 3. air inlet, 4. air outlet, 5. shallow hot water reservoir, 6. aluminium conductor heat sink, 7. 3-D measurement grid consisting of 36 sampling tubes, 8. envelope chamber or buffer zone, 9. multipoint sampler consisting of pc-controlled solenoid valves, 10. gas analyser, 11. datalogging system.
where s is the Laplace operator, i.e. s ¼ d=dt; yðtÞ is the noisy measured output; uðtÞ is the model input; d is the time delay; xðtÞ is additive noise, assumed to be a zero mean, serially uncorrelated sequence of random variables with variance s2 accounting for measurement noise, modelling errors and effects of unmeasured inputs to the process; and finally, AðsÞ and BðsÞ are polynomials in the s operator of the following form: AðsÞ ¼ sn þ a1 sn1 þ . . . þ an ,
(2)
BðsÞ ¼ b0 sm þ b1 sm1 þ . . . þ bm ,
(3)
where mpn; a1, a2, y, an and b0, b1, y, bm are the TF denominator and numerator parameters, respectively. Once the input–output data are available, model parameters (Eqs. (2) and (3)) can be identified by using statistical procedures. But the resulting model is only considered fully acceptable if, in addition to explaining the data well, it also provides a description that has relevance to the physical reality of the system under consideration. The ability to estimate parameters represents only one side of the model identification problem. Equally important is the problem of objective model order identification. This involves the identification of the best choice of orders of the numerator and denominator polynomials, together with the time delay. The parameters of a TF model may be estimated using
various methods of identification and estimation procedures [17,18]. Although Least Squares (LS) is one of the most commonly used model estimation algorithm, the estimated model parameters become asymptotically biased away from their true values in the presence of measurement or disturbance noise signal [17]. Here we use the more complex Simplified Refined Instrumental Variable (SRIV) algorithm [17], that uses the Instrumental Variable (IV) approach coupled with special adaptive prefiltering to avoid this bias and to achieve good estimation performance. A reasonably successful identification is based on minimisation of the Young Identification Criterion YIC: ! np 1 X s^ 2 p^ ii s^ 2 YIC ¼ ln 2 þ ln , np i¼1 a^ 2i sy
(4)
where s^ 2 is the sample variance of the model residuals, s2y is the sample variance of the measured system output about its mean value, np is the total number of model parameters, i.e. np ¼ n þ m þ 1, a^ 2i is the square of the ^ p^ ii is the ith ith element in the parameter vector a, diagonal element of the inverse cross product matrix PðNÞ, s^ 2 p^ ii can be considered as an approximate estimate of the variance of the estimated uncertainty on the ith parameter estimate.
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YIC is a heuristic statistical criterion which consists of two terms, as shown in Eq. (2). The first term provides a normalised measure of how well the model fits the data: the smaller the variance of the model residuals in relation to the variance of the measured output, the smaller this term becomes. The second term is a normalised measure of how well the model parameter estimates are defined. This term tends to become bigger when the model is over-parameterised and the parameter estimates are poorly defined. Consequently, the model which minimises the YIC provides a good compromise between goodness of fit and parametric efficiency. While the YIC can be a great help in ensuring that the model is not over-parameterised, it is not always good at discriminating models that have a lower order than the ‘best’ model [19]. Hence, the YIC will often, if applied strictly, identify a model that is under-parameterised. Therefore, it is used
together with the coefficient of determination R2T . The coefficient of determination is a statistical measure of how well the model fits the experimental data. If the sample variance s^ 2i of the model residuals is low compared with the sample variance sy2 of the measured system output about its mean value, then R2 tends towards unity. If s^ 2 is of similar magnitude to s2y , then it tends towards zero. R2 ¼ 1
s^ 2 . s2y
(5)
If the YIC identified model has an adequate R2T which is not significantly lower than the R2 of the higher order models, it may be fully accepted as the best model in statistical terms. The SRIV structure identification criterion has been proven very successful in practical applications [19].
Table 1 Overview of the identification experiments Experiment
Ventilation rate (m3/h)
Refreshment rate (1/h)
Supply air temperature (1C)
Air flow pattern
1 2 3 4 5 6
80 120 160 200 240 300
9 13 18 22 27 33
11.5 11.5 11.5 11.5 11.5 11.5
Anti-clockwise Anti-clockwise Anti-clockwise Clockwise Clockwise Clockwise
Fig. 2. The output of a first-, second- and third-order TF models compared with the measured CO2 concentration response at sensor position inside the measuring grid (position 15) for experiment 2.
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2.3. Experiments for identification of a reduced-order linear model When a set of informative input–output data is generated, a reduced-order, linear model can be identified to describe the data in a sufficiently accurate way. To identify the parameters of the TF model for each of the 36 sampling positions in the test chamber, 6 identification experiments were carried out as presented in Table 1. For each experiment, the CO2 concentration at the air supply was raised to 1905 ppm during 47 min (time needed to obtain steady-state conditions in test installation), whilst maintaining a constant ventilation rate. The experiments were carried out over a wide range of low (80, 120 and 160 m3/h) and high (200, 240 and 300 m3/h) ventilation rates. The total internal heat production of the 5 heating elements was maintained constant at 300 J/ s and the temperature of the incoming air is kept constant at 11.5 1C. In each of the 6 experiments, the ventilation rate, supply air CO2 concentration and the CO2 concentration response at each of the 36 spatially distributed sampling positions in the test chamber were recorded every 3.3 s. A typical identification experiment at a position inside the measuring grid is presented in Fig. 2. Further, in each experiment, non-toxic smoke is injected in the room and flow patterns are recorded by a digital camera.
3. Results and discussion 3.1. Parameter identification The continuous-time SRIV algorithm [19] is used to identify the linear TF model between the supply air CO2
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concentration and the CO2 concentration at a certain spatial point. Table 1 shows that a first-order model (Eq. (5)) yields an excellent explanation of the convective mass transfer of the system according to the three basic principles of the SRIV algorithm: minimal number of parameters, high reliability of the parameters estimation and high accuracy of the model to describe the CO2 concentration dynamics (Table 2 and Fig. 3). ci ðtÞ ¼
b0 c0 ðtÞ, s þ a1
(6)
where ci is the dynamic CO2 concentration at position i (ppm); c0 is the dynamic supply air CO2 concentration (ppm). The R2T values are similar for first-, second- and third-order models. However, when the YIC-values are considered the first-order system is more parametrical efficient than the higher order models. This demonstrates that the mass transfer phenomenon obeys firstorder dynamics. When applied to all 36 sensor positions in all experiments, the first-order model describes the data with an average R2 of 0.987 and an average YIC-value of 14.91 (Table 3). 3.2. Physical interpretation of model parameter—WellMixed Zone Concept The DBM approach represents the imperfectly mixed fluid in a process room by a number of well-mixed zones (WMZs) which are defined around the nodes of a sensor grid. A WMZ is a 3-D-zone of improved mixing with a certain volume wherein acceptably low spatial gradients occur. These WMZs exist in every imperfectly mixed fluid and for each considered variable such as, for example, CO2-concentration, temperature, humidity, etc. A schematic representation of a WMZ in a process room with inlet fluid flow rate V (m3/s) is given in Fig. 3.
Table 2 The model parameter estimates with associated relative standard error Order of TF
Parameter estimates
Relative standard error (%)
R2T (dimensionless)
YIC value (dimensionless)
[1 1 9]
a1 ¼ 0:013 b0 ¼ 22; 094
x ða1 Þ ¼ 0:32 x ðb0 Þ ¼ 0:31
0.998
17,146
[2 2 11]
a1 a2 b0 b1
¼ 0:022 ¼ 0:000 ¼ 24; 190 ¼ 0:157
x x x x
ða1 Þ ¼ 2:89 ða2 Þ ¼ 8:21 ðb0 Þ ¼ 0:42 ðb1 Þ ¼ 8:16
0.999
10,801
[3 3 11]
a1 a2 a3 b0 b1 b2
¼ 0:050 ¼ 0:001 ¼ 0:000 ¼ 26; 697 ¼ 0:809 ¼ 0:002
x x x x x x
ða1 Þ ¼ 7:04 ða2 Þ ¼ 22:67 ða3 Þ ¼ 141:22 ðb0 Þ ¼ 0:68 ðb1 Þ ¼ 9:92 ðb2 Þ ¼ 136:29
0.999
5297
The YIC value and the coefficient of determination R2 of a first-, second- and third-order continuous-time TF model for sensor position 1.
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To describe the dynamic behaviour of mass transfer in the considered WMZs, standard mass transfer theory can be applied. In case of a constant ventilation rate, a linear, first-order differential equation (as previously identified from experimental data) can be formulated as
Vc,i.Co(t−i).o V Co (t)
Ci(t) well mixed zone
dC i ðtÞvoli ri ¼ V c;i C o ðtÞro V c;i C i ðtÞri , dt
Vc,i.Ci(t).i
Fig. 3. Schematic representation of the WMZ concept.
Table 3 The mean value over the 6 experiments of the relative standard errors xðbÞ, xðaÞ on the model parameter estimates Sensor
x (b)
x (a)
R2T
YIC
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
1.66 0.69 0.55 1.02 0.58 0.79 1.51 0.72 0.65 0.55 0.65 1.04 0.55 0.57 0.82 0.68 0.95 1.29 0.85 0.37 0.75 0.78 0.61 0.68 0.68 0.76 0.68 0.66 0.64 0.81 0.38 0.57 0.60 1.76 0.64 1.19
1.70 0.72 0.57 1.04 0.60 0.82 1.55 0.78 0.66 0.57 0.68 1.07 0.57 0.59 0.80 0.71 1.00 1.32 0.88 0.35 0.76 0.79 0.63 0.70 0.70 0.79 0.70 0.68 0.66 0.83 0.40 0.59 0.62 1.80 0.67 1.23
0.985 0.989 0.995 0.990 0.994 0.982 0.987 0.962 0.994 0.994 0.982 0.992 0.994 0.994 0.983 0.987 0.949 0.984 0.959 0.988 0.991 0.987 0.992 0.991 0.992 0.989 0.992 0.989 0.992 0.989 0.987 0.994 0.993 0.989 0.991 0.988
13.55 14.12 15.19 14.00 15.07 13.49 13.82 13.97 14.96 14.91 14.21 14.17 14.88 14.79 13.61 14.47 12.99 14.00 13.88 16.67 14.38 13.86 14.95 14.27 14.61 13.71 14.18 14.05 14.23 13.75 38.12 14.70 14.46 13.42 14.19 13.28
Mean
0.80
0.82
0.987
14.91
The coefficient of determination and the YIC-value for each of the 36 sensor positions.
(7)
where t is the time (s); Ci is the concentration of CO2 in the WMZ (ppm); C0 is the concentration of CO2 in the supply air (ppm); voli is the volume of the WMZ (m3); Vc,i is the part of the total ventilation rate that enters the considered WMZ i (m3/h); ri is the density (kg/m3) of the air in WMZ; r0 is the density (kg/m3) of the supply air and subscript i is the position index. But because of small refreshment rates used and since indoor air temperature is kept constant, the density difference of the air can be neglected in these experiments. The average concentration of CO2 in the air before injection is equal to 500 ppm (750 ppm). The average temperature of the surrounding laboratory is equal to an average value of 21 1C. In contrast to the zonal and nodal models in literature [20,21], the different WMZs are considered as decoupled or non-interactive zones, because the objective was to control the conditions in a WMZ by using the fresh air at the inlet conditions. To do so the influence of the inlet conditions on a specific WMZ must be modelled. Since the air inlet conditions are responsible for the mass concentration characteristics of the zone, the model only creates a link between the inlet and individual zones. Since every living organism in a bio-process needs an adequate amount of oxygen rich fresh air, the focus on the WMZ model concept is to model the 3-D- transport of fresh air to a particular zone. This is established by using local fresh air flow rate VCi. The local fresh air flow rate is the air flow rate with the same gas concentrations like the incoming air at the inlet that would have created the aggregate effect of the convective flux interaction with the neighbouring zones on the WMZ in consideration. This is a very useful assumption, because it creates a direct relationship between individual zones without the need for modelling zonal interactions. When this concept is applied in an on-line way, the influence of zonal interactions on the relationship inlet-specific zone is taken into account. When considering non-interactive WMZs, the spatial–temporal model is also less complex and it is an excellent basis for control purposes. The response of each WMZ to changes in the inlet conditions is described by a single model, resulting in n first-order models for n WMZs that can be used for controlling the conditions in the n WMZs each time only using the air inlet conditions.
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Under constant ventilation rate, Eq. (6) can be rewritten as dC i ðtÞ ¼ bi C o ðtÞ bi C i ðtÞ, dt
(8)
where bi is the local refreshment frequency, bi ¼
V c;i . voli
Integrating Eq. (7) gives rise to following equation: Z 0
Ci
dC i ðtÞ ¼ C o ðtÞ C i ðtÞ
Z
t
bi dt.
(9)
0
Solving the integral in Eq. (8) results in C i ðtÞ ¼ C 0 ðtÞð1 ebi t Þ.
(10)
This function represents the exponential accumulation of the tracer gas inside the ventilated room. At infinity, C i ðtÞ will be equal to C o ðtÞ, the constant concentration at the end of the experiments. 3.3. Local mean age of air The local mean age of air is an index used to quantify ventilation effectiveness and to visualise air flow patterns in a ventilation system [22]. Local mean age of air is the mean time required for all fluid particles to arrive at a position of interest from the inlet. It is the mean time required to refresh a certain position in a room once. There are three methods of quantifying this index [7]. All are based on tracer injection at the inlet and gas concentration readings at sampling positions in the room. These methods use advanced statistical distribution analysis [6]. Local mean age of air for step inlet injection is defined as [6] Z 1 C i ðtÞ ti ¼ 1 dt, (11) C i ð1Þ 0 where ti is the local mean age of air at the position i, C i ðtÞ is the contaminant concentration at position i at time t, C i ð1Þ is the contaminant concentration at position i after a long time. Numerical procedures can be applied to solve the above equation on condition that concentration data are available. In this research, the numerical integral is calculated using the Simpson rule [23]. Plugging Eq. (9) into Eq. (10) and taking C i ð1Þ is equal to C 0 yields Z 1 C 0 ðtÞð1 ebi t Þ 1 ti ¼ dt. (12) C0 0
563
Solving gives Z
1
ebi t dt ¼
ti ¼ 0
ebi t 1 , b 0
1 ti ¼ 0 , bi 1 ti ¼ . bi
ð13Þ
For first-order system with zero time delay the local age of air, ti ðsÞ, is the inverse of the local refreshment frequency, bi (s1). Zerihun Desta et al. [24] proved the potential of this equation based on numerical CFDdata. The WMZ concept, which results in the formulation of the DBM model, and the associated mass balance differential equation, can be applied to each of the spatially distributed monitoring positions in the test room. 3.4. Parameter contour plots Fig. 4 shows an example of the spatial contours of the parameter bi (s1) in the front and the rear sensor plane of the test installation at a ventilation rate of 240 m3/h (identification experiment 5). Further, the contour plots relate well to the air flow pattern, which is presented in Fig. 5 and was visualised through smoke experiments [25]. At high ventilation rates, the incoming fresh air rapidly moves across the top of the ventilated chamber, hits the right sidewall and then descends towards the exit at the lower right, where some of the air flow recirculates in a clockwise direction. An increase in the air freshness in the direction of the air flow is quite noticeable from the contours of the local refreshment frequency bi. At low ventilation rates, the incoming air moves downwards, hits the floor and moves to the outlet, some air makes a counter clock wise circulation. 3.5. Sensitivity analysis to measurement noise To demonstrate the usefulness of the proposed methodology for the determination of the local mean age of air, a sensitivity analysis was performed. The consistency of both methods was checked when dealing with different levels of measurement noise. To show the effects of noise on both methods, a tracer step injection was simulated (time constant ¼ 20 s, gain ¼ 1) and the local mean age of air was calculated through both methods. This simulation made it possible to demonstrate the effects of introducing random noise. In Fig. 6, the relative error (RE) on the age of air determination is plotted as a function of relative sampling rate (%) without the presence of noise for
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Fig. 4. Spatial contour plots of parameter bi (s1) at 240 m3/h in the 2 vertical planes.
34
28
29
36
23
30 33
7
14
21
1
8
15
4
5
11 2
6
12
18
24 27
13
10
17 20
26
32
16 19
25
31
35
22
9
3
Fig. 5. The visualised air flow pattern at 240 m3/h.
both methods. This relative sampling rate is defined as the ratio of the sampling rate over the time constant. In this case, both methods give reliable results (RE o0.5%) for all sampling rates. When dealing with noise free data, it can be concluded that the numerical integration method for the determination of the age of air is accurate enough. The tendency can be observed that the measurement error augments with higher sampling rates.
Experimental data however is always biased with different levels of noise. Fig. 7 shows the results of age of air determination when noise level is increasing for a relative sampling rate of 1.6% ( ¼ sampling rate of 3 s) of both methods. A Monte Carlo Analysis was carried out to determine the confidence intervals of both methods. One thousand simulations were performed. The mean RE of the numerical integration method was below 3%, but this method becomes biased in the
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Fig. 6. Relative error (%) as a function of relative sampling rate, on the determination of the local mean age of air based on the trapezoidal integration rule ([6]) and the DBM method.
presence of noise. The confidence interval of the RE becomes larger for higher noise levels, more precisely ranging from 0.6 to 0.4% for 1% noise; from 3.6 to 1.8% for 5% noise and from 6% to 1.4% for 7% noise. The confidence intervals of the DBM method remain constant for different levels of noise (1% noise: 0.01–0.01%; 5% noise: 0.04–0.03% and 7% noise: 0.03–0.07%). When collecting experimental data, repetitions are a necessity, but the accuracy of the numerical integration method will be far less than the accuracy of the DBM method. This indicates that when experimental data (with inevitable measurement noise) is used the actual determination of the age of air with the numerical integration method becomes biased away from the actual value depending on the noise level of the data. Besides being suitable for controlling purposes, due to the compact model structure, the DBM method for the determination of the local mean age of air proved to be more robust for measurement noise and gave more reliable results. 3.6. Validation of the physical meaning of the modelled age of air In Fig. 8, the results for all ventilation rates and all 36 positions in the room are presented in a graphical way, more precisely; the numerically calculated age (integra-
tion method) is plotted against the modelled version (model parameter ti ¼ 1=bi ). Fig. 8 clearly demonstrates that a linear relationship exists between the modelled and the numerical integration values of the age of air. A linear curve was fitted (R2T ¼ 0:77) through the data. The regression error can be explained by the fact that this formula (Eq. (10)) is based on the calculation of an integral, where the accuracy is directly dependent with the sampling rate. The larger the sampling rate is, the larger the error will be. In this research, the minimum sampling rate was equal to 3.3 s. When the linear fit is forced without intercept, an R2T value of 0.62 is obtained. This supports the physical background of the model parameter bi, as being an unbiased estimator of the actual local age of air in ventilated rooms.
4. Conclusions This paper has proved that a Data-Based Mechanistic (DBM) approach could be applied to determine the local mean age of air in a forced ventilated space. Here a minimally parameterised transfer function model is first identified and estimated from the experimental data without any prior assumptions about the physical nature of the system. Having objectively identified the
ARTICLE IN PRESS S. Van Buggenhout et al. / Building and Environment 41 (2006) 557–567
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mean age of air (Sandberg) + confidence intervals mean age of air (DBM) + confidence intervals
6
Relative Error [%]
4
2
0
-2
-4
-6
0
1
2
3
4 5 Noise Level [%]
6
7
8
9
Fig. 7. Mean value (1000 repetitions) of the relative error in function of different levels of noise and corresponding confidence intervals for a relative sampling rate of 1.6%.
500 450
y = 0.739*x + 80.9 Rt2 = 0.77
400
300 250 200
βi
1
age modelled [s]
350
150 y = 1.03 * x
100
Rt2 = 0.62
50 0
0
50
100
150
200
250
300
350
400
450
500
τi age numerical [s]
Fig. 8. X–Y scatter plot of the numerical versus the modelled age of air for all 6 ventilation rates and 36 positions in the room. The full line represents a regression curve through the data; the dotted line is the bisector where the modelled value equals the numerical values (integration method).
dominant modes of dynamic behaviour in this manner, the model is then interpreted in physically meaningful terms; more precisely, the inverse of the model parameter is equal to the local mean age of air. This
model explains the data very well (average R2T ¼ 0:987), with the minimum number of identifiable parameters. The local air refreshment parameter bi has proven to be the inverse of the age of air and good agreement ðR2T ¼
ARTICLE IN PRESS S. Van Buggenhout et al. / Building and Environment 41 (2006) 557–567
0:77Þ was found with the calculation of the age of air by means of the numerical integral method. Besides being suitable for controlling purposes, the DBM method for the determination of the local age of air proved to be more robust for measurement noise. References [1] Roulet C-A, Vandaele L. Airflow patterns within buildings: measurements techniques. Technical Note AIVC 1991;34. [2] Wargocki P, Wyon DP, Sundell J, Clausen G, Fanger PO. The effects of outdoor air supply rate in an office on perceived air quality, Sick Building Syndrome (SBS) symptoms and productivity. Indoor Air 2000;10:222–36. [3] Xing H, Hatton A, Awbi HB. A study of the air quality in the breathing zone in a room with displacement ventilation. Buildings and Environment 2001;36(7):809–20. [4] Cheong KW, Chong KY. Development and application of an indoor air quality audit to an air-conditioned building in Singapore. Buildings and Environment 2001;36(2):181–8. [5] Cheong KWD, Lau HYT. Development and application of an indoor air quality audit to an air-conditioned tertiary institutional building in the tropics. Buildings and Environment 2003;38(4):605–16. [6] Sandberg M, Sjo¨berg M. The use of moments for assessing air quality. Buildings and Environment 1983;18:181–97. [7] Etheridge DW, Sandberg M. Building Ventilation: Theory and Measurement. West Sussex: Wiley; 1996. [8] Sandberg M, Blomqvist C. A quantitative estimate of the accuracy of tracer gas methods for the determination of the ventilation flow rate in buildings. Buildings and Environment 1985;20(3):139–50. [9] Sherman MH. Analysis of errors associated with passive ventilation measurement techniques. Buildings and Environment 1989;24(2):131–9. [10] Sherman MH. Uncertainty in air flow calculations using tracer gas measurements. Buildings and Environment 1989;24(4): 347–54. [11] Baldwin RL. Modeling Ruminant Digestion and Metabolism. London: Chapman & Hall; 1995. [12] Oltjen JW, Owens FN. Beef cattle feed intake and growth: empirical Bayes derivation of the Kalman filter applied to a
[13]
[14]
[15]
[16] [17] [18] [19]
[20]
[21]
[22] [23] [24]
[25]
567
nonlinear dynamic model. Journal of Animal Science 1987;65: 1362–70. Aerts J- M, Berckmans D, Saevels P, Decuypere E, Buyse J. Modelling the static and dynamic responses of total heat production of broiler chickens to step changes in air temperature and light intensity. British Poultry Science 2000;41: 651–9. Berckmans D, De Moor M, De Moor B. New model concept to control the energy and mass transfer in a three-dimensional imperfectly mixed ventilated space. In: Proceedings of Roomvent 1992, Aalborg, Denmark, pp. 151–168. Janssens K, Van Brecht A, Zerihun Desta T, Boonen C, Berckmans D. Modelling the internal dynamics of energy and mass transfer in an imperfectly mixed ventilated airspace. Indoor Air 2004;14:146–53. Maciejowski JM. Predictive Control with Constraints. UK: Prentice-Hall; 2002 331pp. Young PC. Recursive Estimation and Time-Series Analysis. Berlin: Springer; 1984. Ljung L. System Identification: Theory for the User. Englewood Cliffs, NJ: Prentice-Hall; 1987. Young PC, Jakeman AJ. Refined instrumental variable methods of recursive time-series analysis: part II, single input output systems. International Journal of Control 1979;29:1–30. Dalicieux P, Bouia H, Blay D. Simplified modelling of air movements in room and its first validation with experiments. In: Proceedings of roomvent 1992, Aalborg, Denmark, p. 383–97. Li Y, Sandberg M, Fuchs L. Vertical temperature profiles in rooms ventilated by displacement: full-scale measurement and nodal modelling. In: Proceedings of indoor air 1992, vol. 2, p. 225–43. Federspiel CC. Air-change effectiveness: theory and calculation methods. Indoor Air 1999;9(1):47–56. Kreyszig E. Advanced engineering mathematics. West Sussex: Wiley; 1993. Zerihun Desta T, Van Buggenhout S, Van Brecht A, Meyers J, Aerts J-M, Baelmans M, Berckmans D. Modelling mass transfer phenomena and quantification of ventilation performance in a full scale installation. Building and Environment 2005; Accepted for publication. Van Brecht A, Janssens K, Berckmans D, Vranken E. Image processing to quantify the trajectory of a visualized air jet. Journal of Agricultural Engineering Research 2000;76(1):91–100.