Annals of Occupational Hygiene Advance Access originally published online on March 2, 2007
Annals of Occupational Hygiene 2007 51(3):311-325; doi:10.1093/annhyg/mem001
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A Numerical Method of Reconstructing the Pollutant Concentration Field in a Ventilated Room
Process Engineering Department, Institut National de Recherche et de Sécurité Vandoeuvre les Nancy, France
*Author to whom correspondence should be addressed. E-mail: robert.braconnier{at}inrs.fr
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ABSTRACT |
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TOP
ABSTRACT
INTRODUCTIONPollutant source emission flow rates in the workplace are typically unknown in occupational hygiene. Similarly, a restricted number of concentration measurements can provide only spatial limited information on the pollutant distribution in the room. This paper presents a numerical method to evaluate the intensities of pollutant sources and to reconstruct the associated concentration field at every point of a ventilated enclosure containing one or several pollutant sources of unknown emission rate. This reconstructed concentration field is obtained both from the geometric and ventilation characteristics of the enclosure and from a limited number of fixed-station concentration measurements. The method is currently applicable to steady situations. The predictions obtained are then compared with concentration measurements in a laboratory closed cabin under controlled ventilation. Pollutant sources generated tracer gas emissions at known flow rates. Comparisons were performed successively for three different physical configurations.
Keywords: emission rates • inverse method • predictive ventilation • pollutant sources • pollution mapping • reconstruction
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INTRODUCTION |
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Pollutant concentration levels, to which personnel in the workplace may be exposed, depend on several factors and, in particular, the time spent at different workstations in the room, ventilation conditions within the enclosure and values of the emission rates of pollutant sources in the room. However, direct on-site measurement of the emission rates of these sources is a difficult and very often impossible operation.
In addition, the spatial distribution of the pollutants in the enclosure is generally heterogeneous. A map of the concentrations in the enclosure can highlight differences between occupied zones polluted to differing degrees. Knowing the pollutant spatial distribution in the room facilitates development and implementation of prevention solutions.
The authors have developed a numerical method, which allows the emission rates of contaminant sources contained in a ventilated room to be estimated and the associated concentration field to be reconstructed. Required data include the geometric and ventilation characteristics of the room and a limited number of fixed-station concentration measurements. The method is based on computational fluid dynamics and a measurement inversion technique, which allows working back to the source flow rates from the data provided by the fixed-station sensors (this technique belongs to the class of inverse convection problems). It is presently limited to situations for which:
- All the phenomena are stationary: the flows, pollutant emission conditions and boundary conditions (in the openings, thermal disturbances, etc.) are invariable in time;
- The pollutant can be considered as passive; this is generally the case in occupational hygiene, in which concentrations are sufficiently low to influence neither fluid physical properties (especially density) nor flows.
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INTRODUCTION TO THE RECONSTITUTION METHOD |
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In general, the concentration field in an enclosure results not only from the emission rates of the internal pollution sources, but also from the pollutant concentration levels in the air inlets. To simplify this presentation, only the pollutant source contribution is considered in the ensuing description, the presented method applying unchanged to determining possible unknown concentration levels at the room entrances (Braconnier et al., 2002). Each source can be of any shape (volume, point, surface, etc.) and is characterized by its emission rate h.
Let us consider the case of a ventilated enclosure for which the following data are available:
- All geometric and ventilation parameters allowing the flow to be calculated inside the enclosure using a predictive ventilation software package;
- Geometric characteristics of the pollutant sources located in the room;
- A set of M fixed-station concentration measurements,
1, 
2, . . . , M obtained at P distinct sampling points;
- Estimated standard deviations
1, 2, . . . , M corresponding to these M measurements: an arbitrary scale-based estimate in relative values is sufficient (e.g. 1 = 2 = ··· = M = 1).
- Emission rates, whose values are unknown during reconstruction of the concentration field; their number is termed N and their values h1, h2, ..., hN;
- Emission rates, whose values are known.
Reconstruction of the concentration field at all points in the enclosure involves estimating the N unknown emission rates using the information provided by both the flow and the M fixed-station measurements. This operation is possible if:
 | (1) |
Step 1: calculation of flow
A grid is generated and numerical simulation is performed to compute the flow inside the enclosure.
Step 2: calculation of background concentration field
A background concentration field is calculated by setting the N unknown emission rates to 0 and all the known emission rates to their nominal value. From this field, M background concentrations are deduced at the nodes corresponding to the measurements: f1, f2, . . . , fM.
Step 3: calculation of transfer coefficients
This step involves carrying out a series of N simulations. The following boundary conditions are used for the simulation j (1 
j N): all the known and unknown emission rates are set to 0, except for the unknown rate hj, which is set to an arbitrary, positive level 
j.
From the concentration field obtained after simulation j, the M transfer coefficient values are deduced for emission rate hj at the nodes corresponding to measurements: 
1j, 2j, . . . , Mj. Coefficient ij is calculated by dividing the concentration obtained at the node of measurement i subsequent to simulation j by the number j. This represents the value at the measurement node i of 
j, the field of transfer coefficients for the unknown emission rate hj.
Step 4: calculation of emission rates
After these N + 2 simulations, the predicted concentration ci at the node of measurement i can be expressed into the following linear relationship form, in which the values fi and ij are known and the emission rates hj are still unknown:
 | (2) |
i – fi, and N explanatory variables j. The purpose of this regression is to determine the unknown coefficients hj by applying the method of least squares. This regression has two special characteristics: first, there is no constant term in the model and, second, the hj coefficients are subject to a non-negativity constraint: negative values are excluded for the emission rates.
In the remainder of this section, we consider the case in which uncertainties in explanatory variables j (obtained from a numerical model requiring geometric and ventilation data) are low compared with uncertainties in concentration measurements i. Calculation of unknowns hj is therefore reduced to the following linear algebraic problem:
 | (3) |
 | (4) |
 | (5) |
Solving equation (3) provides values of the emission rates hj and of the residual variance of the regression s2:
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 | (7) |
 | (8) |
i.
Step 5: calculation of reconstructed concentration field
The emission rates are now all known. The pollutant concentration map can be reconstructed for the entire volume of the enclosure by application at each node of a linear relationship analogous with equation (2), in which the background concentration was obtained in step 2 and the transfer coefficients in step 4. The regression model is used here to predict new values.
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INTRODUCTION TO THE SAMPLING POINT POSITIONING ASSISTANCE METHOD |
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Choice of concentration measurement point positions in the enclosure influences reconstructed map quality. These sampling positions must be decided before measurements are actually taken and this decision can be made with a view to optimizing pollutant map reconstruction in specific, predetermined areas of interest, for example the room occupied areas. The optimization criterion implemented below minimises the variance of future reconstructed concentrations.
The number of sampling nodes P to be used in the future (P 
N) is firstly determined based on the resources available for the reconstruction operation. A set of R distinct prospective nodes in the enclosure is then identified (R P) and pre-selected as sampling nodes. These prospective nodes can be selected by considering the practical measurement constraints, the pollutant source and operator locations, the flow pattern provided by numerical simulation, etc.
Arbitrary scale-based, a priori estimated standard deviations to be assigned to the potential concentration measurement taken at each prospective node are given in relative values. These estimates are termed 
, 
, etc. In the absence of more detailed information, a uniform value = = · · · = 1 can usually be adopted for all these a priori, estimated standard deviations.
The user also defines the positions of 
criteria nodes ( 1), indicating pre-established areas of interest within the room. These nodes can, for example, correspond to room occupied areas. A weighting coefficient 
u is provided with each criterion node: this expresses the relative importance of each criterion node and can, for example, be chosen in relation to the workforce, presence time, the floor space or areas concerned.
The different steps involved in selecting sampling nodes are summarized below.
Step A: calculation of flow
The flow inside the room is calculated by numerical simulation.
Step B: calculation of transfer coefficients
This step involves performing a series of N simulations identical to those described in step 2 above. The transfer coefficient values can thereby be derived for each unknown emission rate hj (j = 1 to N). This includes:
- R-values µj at pre-selected nodes,
- -values Luj at criterion nodes.
Step C: calculation of emission rate variances and covariances
The number of ways of forming a set of P sampling nodes extracted from R pre-selected nodes is equal to the number of combinations 
:
 | (9) |
of this type ( = 1 to ), a matrix 
is formed, by analogy with equation (4), with P lines and N columns composed of the following elements:
 | (10) |
E (P-values).
This matrix can be decomposed as stated in step 4 above. All matrices can be calculated before concentration measurements i are taken, but the residual variance 
cannot be evaluated before obtaining these measurements. As an initial approximation, however, the latter variance can be considered independent of the set of sampling nodes E. All variances 
are therefore equal to the same constant, but unknown, value 
:
 | (11) |
can be estimated in relative values. These estimates are obtained without calculating the emission rates themselves:
 | (12) |
 | (13) |
Step D: calculation of optimization criterion
In the following section, equations (14)–(16) provide the derivation of the optimization criterion with respect to the choice of sampling positions, but do not correspond to calculations possible during this step D because values cu, Fu and hj are unknown at this stage.
Weighted concentration 
obtained from the reconstructed concentrations at the criteria nodes is given by the equation:
 | (14) |
 | (15) |
 | (16) |
 | (17) |
. This is a transfer coefficient of the same dimension as j. Before taking measurements, this transfer coefficient can be calculated from the given weighting coefficients u and the transfer coefficients at criteria nodes Luj obtained in step B.
Weighted concentration cannot be calculated before taking measurements. On the other hand, its variance can be estimated in relative values for each combination E of P sampling nodes defined in step C using the equation:
 | (18) |
in relative values for the combination of sampling nodes E now being considered. Therefore:
 | (19) |
combination to be retained is the one that minimises this criterion when embraces all the possible combinations and therefore minimises the predictable uncertainty of the reconstructed concentration . |
EXPERIMENTAL TESTING CONDITIONS |
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Testing cabin
Tests were conducted in a parallelepiped-shaped closed cabin 5.69 m long, 4.0 m wide and 2.97 m high (Figs 1 and 2). A 0.42 m-sided square air inlet is incorporated in the lower corner of one of the vertical walls perpendicular to the cabin length.
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It was agreed to assign an OXYZ orthonormal reference frame (Fig. 1) to the testing cabin, defined as follows:
- Origin O is located on the floor, on the same vertical wall as the air inlet and at the cabin corner opposite the inlet;
- Axis OX is marked on the floor, parallel to the cabin long sides;
- Axis OY is vertical and ascending;
- Axis OZ is marked on the floor, from the origin to the air inlet.
Air is extracted from the cabin at the upper end of a 0.2 m diameter vertical pipe located at a height of Y = 2.33 m above the floor; the centre of the opening is located at abscissa X = 5.19 m and at offset Z = 3.09 m. Figure 2 shows the air inlet (perpendicular to wall X = 0) and outlet (near the point marked 12) locations.
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Some tests were conducted with the cabin empty, whilst for others, the cabin was partially divided by a vertical, rectangular thin panel positioned perpendicular to the cabin longitudinal X-axis (Fig. 1). This panel is placed on the floor with one end against the rear cabin wall at offset Z = 4 m. The flat obstruction thereby created is 1.5 m high and 2.0 m wide, and therefore obstructs approximately one quarter of the cabin cross-section. The panel is placed facing the air inlet opening at abscissa X = 3.16 m.
Pollutant emission
Helium tracer gas is used to simulate pollutant releases inside the cabin. For each pollutant source, emission passes through an 80 mm diameter sphere of filter material. Pollutant sources can therefore be considered as point sources at cabin scale. Pollutant is released at very low initial velocity: emission velocities at the periphery of the source are 6 mm/s.
In some tests, the tracer gas is emitted by a single pollutant source located at point S1 of Fig. 2. In other tests, the tracer gas is simultaneously emitted by two sources located at points S1 and S2 of Fig. 2. Points S1 and S2 are at the same height above the floor, namely 1.04 m. Their coordinates, expressed in meters, are as follows:
- For S1: X = 4.84; Y = 1.04; Z = 1.40;
- For S2: X = 1.95; Y = 1.04; Z = 1.40.
Measuring equipment
Measuring equipment used for the tests is briefly introduced below. Tracer gas measurements were taken with measuring apparatus and software described in detail in reference (Dessagne et al., 1993).
The airflow introduced into the cabin is measured inside the inlet pipe by the tracer gas dilution method (AFNOR, 1982; ASTM, 2004). Air velocity distribution within the inlet opening is determined using a thermal anemometer fitted with an 80 mm long cylindrical directional barrel.
Air velocities inside the cabin are measured using a 3D ultrasonic anemometer. Velocities are determined at the centre of a 50 mm-sided parallelepiped. The anemometer is oriented such that the three components U, V and W of the air velocity are measured parallel to the X, Y and Z coordinate axes. Each measurement results from calculating the average of three unrelated determinations, each lasting 30 s. Measured components vary between 0.003 and 0.790 m/s, the average value being close to 0.080 m/s: these values therefore fall within the low-speed range.
The graphite piston flowmeter used to calibrate the mass flow controllers is traceable to French national accredited laboratories; its measurement uncertainty is of 0.4%.
A helium-calibrated mass spectrometer is used to measure pollutant concentrations inside the cabin (Dessagne et al., 1993). This spectrometer emits a continuous electrical signal proportional to the tracer concentration in an internal measuring cell. This cell has a 3 s time constant. The measuring line uncertainty (including calibration operations and drift phenomena) is 3%. A 1 Hz frequency data acquisition board is used to sample the continuous signal emitted by the spectrometer. A specific programme developed in C language is implemented for the flowmeters control, data acquisition and digital processing functions.
Measuring positions
Measurements of air velocity and tracer gas concentration inside the cabin are taken at T = 48 points distributed between the following three horizontal planes: Y = 0.59 m (plane 1), Y = 1.48 m (plane 2) and Y = 2.38 m (plane 3). In each plane, measuring positions are numbered from 1 to 16 and are distributed as shown in Fig. 2. They correspond to the following X-axis abscissas and Z-axis offsets (in meters): X = 1.14, 2.85, 4.55 and 5.12; Z = 0.8, 2.0, 3.2 and 3.7.
It was agreed to refer to a measuring point inside the cabin by its number in the plane followed by the number of the plane: e.g. 7-2 for point 7 in horizontal plane Y = 1.48 m.
The presence of the extract pipe means that:
- Velocity measurements could not be taken at points 12 and 16 in all three planes;
- Concentration measurements are slightly offset at points 12-1 and 16-1.
Pollutant emission points S1 and S2 are located approximately mid-distance along horizontal planes 1 and 2.
Measuring conditions
For internal concentration measurements, the testing cabin is equipped with a fixed network of 48 polyurethane tubes of the same length and internal diameter (5.5 mm). These tubes terminate at the different measuring positions and are connected to an external terminal board. Helium concentration is determined point by point by manually connecting the corresponding sampling tube to an external pump extracting a 15 l/min flow. The velocity field within the cabin is not influenced by this low extraction airflow, part of which is continuously drawn off and directed to the spectrometer for analysis.
Each concentration measurement is taken for an average time of 200 s. This measuring time is increased to 400 s for points with high, turbulent fluctuations in instantaneous concentration and is reduced to 100 s for concentrations fluctuating little. These measuring times exclude transient signals recorded during measuring point changeovers. Acquisition of the 48 concentrations takes one day. Steadiness in ventilation conditions and possible drift in the measuring system are monitored by additional samples taken cyclically at four points: outside the building, at the cabin air inlet and outlet and at a fixed point inside the cabin. These monitoring measurements correspond to 40% of the total measuring time. Relative standard deviations of the monitoring measurements taken at the air outlet and the internal point are 2 and 3%, respectively.
Concentrations inside the cabin are shown below in the form of net pollutant concentrations (measured values minus natural atmospheric helium content of air entering). Values given are volume-based helium fractions expressed in parts per million (ppm).
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NUMERICAL FLOW SIMULATION TECHNIQUES |
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The method described above for evaluating source emission flows and reconstructing the pollution map in a room is independent of any specific flow simulation software. It can be applied with any CFD programme for general usage. In this study, flows inside the testing cabin and pollutant transport and diffusion have been computed using EOL-3D, version 3.1, ventilation forecasting software (Fontaine et al., 1993, 1996). This application solves fluid mechanics partial differential equations by the finite volume method, using the ‘SIMPLEC’ algorithm. It uses Cartesian grids at variable spacing and takes into account turbulence phenomena by means of the standard k-
model (Fontaine et al., 1996). Simulations performed correspond to turbulent, isothermal, incompressible and steady flows and to a passive pollutant. Nodes were located in each case at the 48 measuring positions as well as at pollutant point source positions S1 and S2. To facilitate setting up the grid, we used a 0.177 m-sided pipe of the same normal cross-section to model the 0.200 m diameter circular extract pipe. Possible flow modifications resulting from this simplification remain limited to the extract pipe surroundings because of the small dimensions of the pipe compared with those of the testing cabin.
A 508 810-node grid was used in the ‘empty cabin’ case (85, 73 and 82 nodes on X, Y and Z coordinate axes, respectively). Flow convergence was reached after 6177 iterations, based on the software default convergence threshold (<0.5% residues). A 603 064-node grid was applied in the ‘cabin with flat obstruction’ case (89, 77 and 88 nodes on X, Y and Z coordinate axes, respectively) and convergence was reached after 2957 iterations, based on the same convergence threshold.
Complementing these reference simulations, we performed additional simulations for each considered geometry to ensure independence of results with respect to both grid density and convergence tolerance. These additional simulations implemented grids comprised, on the one hand, 1 125 300 nodes and, on the other hand, a convergence threshold divided by 10 (<0.05% residues). In every case, the velocity and concentration field plots reveal no notable differences compared with the reference simulations described above. More specifically, variations in the 48-value series of transfer coefficients, calculated at the nodes corresponding to the measuring points, can be investigated for each pollutant source.
In the ‘empty cabin’ case and for source S1, lowering of the convergence threshold induces a transfer coefficient variation of 0.39% on average and a maximum variation of 0.87% at one of the 48 points. Corresponding average and maximum variations are 0.45 and 1.79%, respectively, when the grid density is increased. In the ‘cabin with flat obstruction’ case, lowering of the convergence threshold leads to average transfer coefficient variations of 1.01% for source S1 and 0.60% for source S2, with maximum variations of 3.15% at one of the 48 points for source S1 and 2.93% for source S2, respectively. When the grid density is increased, these average and maximum transfer coefficient variations are 3.71 and 8.89%, respectively for source S1 and 0.62 and 3.92%, respectively for source S2.
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CONFIGURATIONS STUDIED |
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Main characteristics
Three test configurations, namely A, E and J, were successively studied:
- Configuration A: empty cabin with pollutant source positioned at point S1;
- Configuration E: cabin partially divided by flat obstruction and enclosing a pollutant source at point S1;
- Configuration J: cabin with obstruction enclosing two pollutant sources at points S1 and S2.
The 25 air velocity measurements taken at the inlet show that the velocity distribution within this opening is not particularly irregular. Thus, a uniform air velocity condition at the inlet was applied in the different simulations. The turbulence level value at the inlet was estimated at 4.7%, based on time-related recordings of velocity in front of this opening. Fixed values were assigned to the turbulent kinetic energy and the turbulent dissipation at the inlet, based on this estimate and the opening dimensions.
The main characteristics of the three configurations are summarized in Table 1. In particular, this table shows the measured values of pollutant source emission rate, expressed in litres of helium per minute reduced to 0°C and 1 atmosphere.
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Air velocities inside the cabin
The specific flow and transfer coefficient computation software, EOL-3D, used in this study has been the object of several experimental validation campaigns (Braconnier et al., 1991; Fontaine, Rapp et al., 1994; Braconnier, 1994; Fontaine, Biolley et al., 1994; Collineau et al., 1997). An example has also demonstrated that the velocity and pollutant concentration fields given by both this programme and FLUENT software for general usage are very similar (Braconnier, 2006). The scope of this paper does not embrace detailed comparison of measured and computed flows inside the testing cabin. Flows measured and computed for the two geometries studied (with or without a flat obstruction) are in fact very similar, although significant differences in velocity magnitude or direction can be observed at some points.
The local flow direction near to pollutant emission points determines the sampling point locations closest to the initial plume emerging from each source; these locations are discussed below. Flow simulations show that, for the configuration J featuring an obstruction, the simulation-computed air velocity at point S2 is low: in fact only 0.05 m/s, whilst the computed velocity at point S1 is 0.33 m/s in configurations J or E and 0.12 m/s for configuration A featuring no obstruction. The velocity field plot confirms that, for configuration J, the emission point S2 is located in a dead zone near the centre of an eddy.
Tracer gas concentrations
Results of measurements taken at 48 points within the testing cabin reveal significant spatial heterogeneousness of pollutant concentrations. The ratio between the highest (observed near a source) and the lowest (observed vertically above the point of impact of fresh air jet from the inlet on the obstruction or on the vertical wall opposite the inlet) local concentrations is 11.9 for configuration A, 13.8 for configuration E and 7.5 for 2-source configuration J.
Moreover, continuous signal recording (at a frequency of 1 Hz) throughout the duration of each measurement 
k allows us to determine the standard deviation 
k of turbulent fluctuations in instantaneous concentration and the relative standard deviation of these fluctuations: 
k = k / k. This fluctuation level is 13% on average, but varies widely according to the local measurement considered: between 2.5 and 97%. The highest values of the turbulent fluctuation level are only measured at the highest concentrations, obtained at just a few measuring points near to the pollutant sources. Thus, for configuration A, the turbulent fluctuation level lies between 54 and 69% for the six highest concentration measurements and is <21% for the other 42 measurements. In configuration E, only the maximum concentration point (point 7-1) has a high -value (97%), the turbulent fluctuation level of all the other measurements being <24%. In configuration J, reaches 52 and 64% at two measuring points and is <29% at other positions.
This phenomenon is probably due to turbulent oscillations in the trajectories of the highly polluted air streams emitted from the pollutant sources. The effects of these random oscillations on the instantaneous concentrations received by the sensors located downstream become less and less marked as we move away from the pollutant sources. This is because pollutant diffusion phenomena cause the polluted air streams to widen.
In the specific case of the three laboratory configurations considered here, concentrations inside the cabin can be numerically determined by direct simulation, i.e. by entering the experimental values of pollutant source emission rate into the predictive ventilation software program. This calculation is obviously impossible in the general case involving reconstruction in which the emission rates are unknown. In particular, the T = 48 pollutant concentrations g can be obtained at the same points as the experimental concentrations and the two sets of values can be compared for each configuration. For configurations A and E, which feature only one pollutant source S1, this comparison is equivalent to comparing numerical and experimental transfer coefficients.
For each configuration, the relative difference ek at measuring point k (k varies from 1 to T) between the measured concentration k and the concentration obtained by direct simulation gk is given by the following formula:
 | (20) |
ave given by:
 | (21) |
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Correlation between measured concentrations and concentrations calculated by direct simulation is highest (r2 = 0.880) in the configuration E case (obstruction and single source) (Fig. 4). In this case, only one point deviates widely from the diagonal. This point corresponds to measuring position 7-1 located, in this configuration, in the initial plume from source S1; the value of the concentration turbulent fluctuation level is high at this position (
= 97%). Scatter is slightly greater in Fig. 5 (case J: obstruction and two sources), but the average relative difference remains low: ave = 12.8%. For the no obstruction configuration A (Fig. 3), at several measuring points the transfer coefficients obtained by simulation are significantly lower than the experimental coefficients, the relative difference even reaching –119.5% locally. The average relative difference rises to 25%.
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The concentration fields computed by direct numerical simulation can be used to locate the sampling positions nearest to the initial plume emerging from each pollutant source. These positions correspond to the sampling points with the highest calculated concentrations amongst the 48 concentrations gk. This operation was performed for single source configurations A and E and for an additional simulation with the obstruction and only source S2. The first sampling points within the S1 plume are point 4-1 in configuration A and point 7-1 in configurations E and J. For source S2 in configuration J, point 6-3 is concerned. These results are fully compatible with the signs of the air velocity components calculated at the emission points.
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RESULTS GIVEN BY SAMPLING POINT POSITIONING METHOD |
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The sampling point positioning assistance method was successively applied to each study configuration for different numbers of future sampling points. P varied from 1 to 10 in configurations A and E, which feature only one pollutant source, and varied from 2 to 10 in configuration J, which features two pollutant sources. Prospective nodes suitable for sampling point use correspond to all experimental positions (R = 48).
Configurations with a single pollutant source (A and E)
In the case of a single pollutant source, the theory (Braconnier et al., 2002) predicts that the choice of recommended sampling nodes is independent of the location of the criterion node(s) used. Several calculations were performed for configurations A and E by varying the position of a criterion node. The results obtained are effectively constant for all the attempted positions. Sets of recommended sampling points for P varying from 1 to 10 confirm another theoretical prediction in the case of a single pollutant source, namely that, for each P-value, the recommended points correspond to the prospective nodes possessing the P highest transfer coefficient values obtained by numerical simulation for source S1.
In configuration A, for example, the recommended sampling point, when P = 1, is point 4-1, the point closest to the initial plume from source S1 (as stated previously). For P = 2, point 4-1 is complemented by point 3-1, which the pollutant then reaches. The recommended sets for configuration A and E differ significantly due to flow modifications resulting from the inclusion of a flat obstruction, especially near source S1. Recommended sampling positions in configuration E include, for example, 7-1, if we take P = 1 or 7-1 and 7-3, if we take P = 2.
Configuration with two pollutant sources (J)
In the presence of several pollutant sources, the sampling point positioning assistance method provides results, which depend on the location of the criterion node(s) used. Two sets of calculations were performed for configuration J with one ( = 1) criterion node located in succession at two intentionally dissimilar positions, namely:
- At point 6-2 located at mid-height, near the cabin centre (Fig. 2);
- At point 9-3 nearer to the ceiling and one corner of the cabin.
The testing cabin can be artificially divided into two adjacent volumes by imagining that the flat obstruction extends as far as the front wall (Z = 0) and ceiling. This process defines:
- An upstream zone, in which abscissa X < 3.16 m, occupies 61% of the total volume, communicates with the air inlet and encloses source S2;
- A downstream zone, in which abscissa X > 3.16 m, occupies 39% of the total volume, communicates with the air outlet and encloses source S1 (Fig. 2).
- The criterion node is located at 6-2, the recommended sets comprise a point in the downstream zone (7-1) and between 1 and 9 points in the upstream zone;
- The criterion node is located at 9-3, the recommended sets comprise points exclusively in the upstream zone.
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RESULTS OF RECONSTRUCTION CALCULATIONS AND DISCUSSION |
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Reconstruction operations performed
Four series of pollutant concentration map reconstruction operations were performed. In each series, calculations were successively performed for various values of P, the number of sampling points used as reconstruction data. This number varied from 1 to 10 for configurations with only one pollutant source (A and E) and from 2 to 10 for configurations with two pollutant sources (J). For each P-value, the measurements used as reconstruction data were those obtained at positions recommended by the positioning assistance method for this number of points P. Just one concentration value is available for each sampling position, so the number of measurements M used for each reconstruction is always equal to the number of sampling points P (M = P).
A value proportional to the standard deviation i in turbulent fluctuations recorded during the measurement period was retained for the standard deviation i in each concentration measurement i used as data in a reconstruction. Since the values i can be expressed on an arbitrary scale, i was simply fixed equal to i.
The four series of reconstruction operations correspond to the following calculation conditions:
- Configuration A using recommended sets of sampling points, independent of criterion node location;
- Configuration E using recommended sets of sampling points, also independent of criterion node location;
- Configuration J using recommended sets of sampling points for a criterion node located at point 6-2 ( = 1);
- Configuration J using recommended sets of sampling points for a criterion node located at point 9-3 ( = 1).
j in the remainder of this paper.
 | (22) |
can also be compared. As stated previously, the relative difference ek at measuring point k (k varying from 1 to T) between the measured concentration k and the reconstructed concentration ck is given by the following formula.
 | (23) |
ave, given by formula (2), can be derived from the above formula. The value of the square of the correlation coefficient, r2, between the two sets of concentrations (reconstructed and experimental) was also calculated after each reconstruction operation.
Configuration A
Table 3 and Fig. 6 give the reconstruction calculation results for configuration A (one source, no obstruction). The calculated emission flow deviation 1 is greatest when only one sampling point is used (P = 1); it then reaches 24.3%. For higher values of P, this deviation decreases significantly and remains between –2.9 and +9.5% (Table 3).
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Configuration A features only one pollutant source. Following each measurement inversion, the reconstructed concentration values are directly proportional to the corresponding calculated emission flow h1. The scatter of points in the figures representing the concentrations reconstructed according to the concentrations measured at the same positions is therefore analogous, for all values of P, apart from a stretching of the vertical axis. This scatter is also analogous with the above-described scatter for concentrations obtained by direct calculation (Fig. 3). This observation is indeed confirmed by the square of correlation coefficient r2, whose values have been found to be independent of P (Table 3) and identical to the directly calculated value (Table 2).
The values of the 48 reconstructed pollutant concentrations are shown in Fig. 6 with respect to the concentrations measured at the same positions in the P = 10 case. Most of the points are close to the diagonal, but the reconstructed concentrations are significantly lower than the experimental concentrations for several measuring points.
The average standard deviation ave between two series of concentrations varies little with the number of sampling points P used; it lies between 18.4 and 27.2%. When P varies, local extreme deviations emin and emax remain approximately –115 and +30%, respectively. Configuration A features only one pollutant source, so these three deviations approximate all the better to those computed by direct simulation, when the predicted source flow h1 is close to the experimental flow.
Configuration E
The results of the reconstruction calculations performed on configuration E (one source, presence of obstruction) are shown in Table 4 and Fig. 7. When a single sampling point is used (P = 1), the calculated emission rate deviation 1 reaches 38%. For higher values of P, this deviation remains below 6% and even below 1% in four cases (P = 3, 7, 8 and 9).
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The values of the 48 reconstructed pollutant concentrations are shown in Fig. 7 with respect to the concentrations measured at the same positions in the P = 10 case. As in the previous configuration, the scatter of points is analogous to other values of P (only one pollutant source). Agreement between both sets of concentrations appears closer than for configuration A, which confirms the values of the square of the correlation coefficient (independent of P) r2 = 0.880 and of the extreme local differences emin and emax, which remain close of –39 and +46%, respectively when P is >1.
The average relative difference ave between the measured and the reconstructed concentrations remains between 12.2 and 16.6% when P is >1.
Configuration J
Tables 5–7 and Fig. 8 give the results of the reconstruction calculations performed on configuration J for both criterion node locations 6-2 and 9-3 and for a number of sampling points P varying from 2 to 10. All reconstruction operations give a calculated value of the emission flow h1 at source S1 greater than the measured value and an emission flow h2 at source S2 less than the experimental flow.
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Results obtained when the criterion node is located at point 6-2 (Table 5) show that the calculated flow at source S1 is more sensitive to the number of sampling points P than the calculated flow at source S2. The deviation
1 is of the order of 18% at low P-values. It decreases to 
8% at intermediate P-values, then increases rapidly to reach a maximum of 96% with P = 9. The deviation 2 is 26% with P 5 and reaches a maximum, 48.3%, with P = 9.
Differences between reconstructed emission rates and experimental emission rates are much greater when the criterion node is located at point 9-3 (Table 6). Overestimation of emission flow h1 varies from 64 to 218% and underestimation of emission flow h2 varies from 32 to 74%. As stated previously, the greatest differences are obtained at high P-values (in this case P 8).
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Within the limits of the reconstruction operations performed on configuration J, it is observed that emission rate deviations pass through a minimum for an intermediate P-value. This minimum is reached when P approaches 4 in Table 5 and when P = 7 in Table 6.
The pollutant total flow in the cabin, ht, is available following each reconstruction operation by adding the calculated flows h1 and h2. The relative deviation t of this flow with respect to the experimental total flow is given by a formula similar to formula (3). Values obtained for configuration J are shown in Table 7. The relative deviation is a maximum at high P-values. On average, it is 11% when the criterion node is located at point 6-2 and is 47% when the criterion node is located at point 9-3. For both these positions, the predicted total emission flow is therefore more accurate than its distribution between the individual flows from each of the two pollutant sources.
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Emission flow results obtained in configuration J are perhaps linked to the position of source S2, located within a dead zone, and this could cause differences between the numerical and experimental transfer coefficients for this same source. However, this hypothesis cannot be checked without concentration measurements for source S2 alone. Furthermore, comparisons between experimental concentrations and those obtained by direct simulation (Table 2 and Fig. 5) did not reveal any particular degradation with respect to configurations A and E.
Unlike the two previous configurations featuring a single pollutant source, the point scatter shown in the configuration J figures, representing 48 pollutant concentration values reconstructed according to the concentrations measured at the same positions, depends on both the position of the criterion node and the number of sampling points used. This scatter also differs from that obtained through direct simulation. As an example, Fig. 8 illustrates this plot for a criterion node located at point 6-2 and for P = 9, i.e. for a case in which the pollutant source flow deviations are relatively large.
The largest source flow deviations noted for configuration J compared with the two previous configurations cause no particular increase in the relative differences between the reconstructed and the measured concentrations (Tables 5 and 6). In fact, the average relative difference ave varies between 13 and 20% for a criterion node located at point 6-2 and between 12.9 and 28.7% for a criterion node located at point 9-3. These differences are comparable with those obtained for configuration A (18.4 to 27.2%). Similarly, the extreme local differences emin and emax are, on average, equal to –48 and +39%, respectively.
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CONCLUSION |
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A mathematical method has been developed, which allows pollutant source emission flows to be estimated and associated concentration field to be reconstructed for a ventilated room. The data required comprise both room geometrical and ventilation characteristics and a restricted number of fixed-station concentration measurements.
The predictions obtained were compared with laboratory concentration measurements in a ventilated cabin. Three configurations were studied: two with a single pollutant source and one with two pollutant sources. For configurations with a single source and when at least two sampling points are used, the mean difference between predictions and measurements is of the order of 4.4% in relation to the emission rate and 18% in relation to pollutant concentrations.
When a second pollutant source is added, the mean difference between total emission rate predictions and measurements remains <29%. On the other hand, difficulties arise in predicting the total emission rate distribution between two sources. The first source mean deviation then increases to 91% and the second source mean deviation increases to 44%. However, these high values do not affect the difference between the reconstructed and measured concentrations, which remains of the order of 18%. Difficulties identified are possibly linked to the position of the second source, which is located in a flow dead zone.
The reconstruction method introduced here is, in principle, not limited to isothermal situations encountered in the testing cabin. It applies to steady flows influenced by heat sources contained in the considered room, for example people present at static workstations. Similarly, as long as suitable turbulence models are used, the method can be applied to rooms with lower air renewal rates, for example office-type rooms.
It should be noted that methods of evaluating emission flows and reconstructing a concentration field do not necessarily require resorting to ventilation forecasting software. Simulation is effectively used only for obtaining the transfer coefficients needed for inverting the concentration measurements. These coefficients could also be determined experimentally, e.g. by gas tracing. Moreover, research is in progress in relation to extending the area of application to cases involving unsteady pollutant emission. Preliminary results have been presented by Girault et al. (2006).
Potential applications of inverse methods in occupational hygiene may involve mapping of pollutants in workplaces, estimating emission rates of pollutant sources, controlling ventilation systems on demand and detecting leaks.
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Appendix 1 |
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Nomenclature of main symbols used
c (ppm): reconstructed pollutant concentration;
e (%): relative difference between two concentration values;
emin (%): minimum relative difference between two sets of concentrations;
emax (%): maximum relative difference between two sets of concentrations;
f (ppm): background concentration at a single sampling point;
F (ppm): background concentration at any point;
g (ppm): concentration calculated by direct simulation;
hj (kg/s or L/min): emission rate of source j;
(kg/s or L/min): experimental emission rate of source j;
ht (kg/s or L/min): total emission rate;
L, (s/kg): transfer coefficient;
M (-): number of concentration measurements used for a reconstruction operation;
N (-): number of pollutant sources;
P (-): number of sampling points used for a reconstruction operation;
R (-): number of prospective nodes suitable for use as sampling points;
r2 (-): square of the correlation coefficient;
s2 (-): regression estimated residual variance;
T (-): number of measuring positions in cabin;
U, V, W (m/s): air velocity components with respect to X, Y and Z coordinate axes;
X, Y, Z (m): Cartesian coordinates;
k (ppm): fluctuation standard deviation during concentration measurement k;
ß (s/kg): transfer coefficient applied to weighted concentration ;
(ppm): measured pollutant concentration;
(ppm): weighted concentration;
j (%): relative deviation of source j emission rate;
t (%): relative deviation of total emission rate;
ave (%): average relative difference between two pollutant concentration sets;
(s/kg): transfer coefficient at a single sampling point;
i (ppm): relative value-based, estimated standard deviation of concentration measurement i;
k (-): fluctuation relative standard deviation during concentration measurement k (k = k/k);
(ppm): concentration measurement used for a reconstruction operation;
(-): weighting coefficient at a criterion node;
(-): number of criterion nodes.
Received January 13, 2006; in final form December 19, 2006
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