The hitherto non-included component in the uncertainty budget for gravimetric measurement of particulate matter concentration in a conduit

The flow of dust-laden gas and sampling it in the measurement plane of the waste gas plant is illustrated by Fig. 1 (for a rectangular section). Assuming steady conditions in the measurement plane with area F, there is a continuous distribution of total particulate matter concentration s(X, Y) and a continuous gas velocity distribution w(X, Y). X and Y are absolute coordinates in the measurement plane (further on, respective relative coordinates will be introduced). The global flow of the two-phase system (which is the dust-laden gas) through the measurement plane is described by two parameters: the particulate (total) mass flow rate M _F (total dust mass flowing across the measurement plane area in a time unit) and gas volumetric flow rate Q _F (gas volume flowing across the measurement plane area in a time unit). With that in mind, one can characterize the flowing dust-laden gas by means of a true, global particulate matter concentration, s _av, being by its nature a space average concentration, given by Eq. 1. The connection between s _av, s(X, Y) and w(X, Y) is presented in point (i), subpoint (a) of the Electronic Supplementary Material (ESM).

$$s_{{\text{av}}} \mathop = \limits^{{\text{def}}} \frac{{M_{\text{F}} }}{{Q_{\text{F}} }}$$

(1)

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Equations 5 and 1 indicate that the measurement quantity of s _m, appropriate for a collected, almost representative sample, is an approximation of the true concentration s _av: s _m ≈ s _av. These relationships are correct when one sampling nozzle is used for the entire measurement plane and also when several sampling nozzles are used (when the velocity profile is substantially non-uniform).

The discrepancy between the measured s _m value from Eq. 2 and the searched true s _av value is, in fact, a deviation of the gravimetric mean total concentration measurement.

The above quantitative formulation (complemented with details in ESM) has been adapted for the simulated flows (“The key calculated value: the deviation of total particulate matter concentration” section).

The starting point for the simulation studies was determining certain data concerning the actual flows of dust-laden gases in dust-removing plants, which would then be appropriately presented. A reliable and practical source of information about the actual flows of gas phase can be only found in the results of specific anemometric industrial measurements (in situ). The information can be gathered from reports of measurements conducted by different measurement teams for the standard auditing of compliance with emission level limits (although they are usually hard to obtain), or from reports on industrial plant research for scientific purposes (unfortunately rare, and some of them are out-of-date). The spatial profiles of gas velocity in the measurement planes have diversely non-uniform shapes. This non-uniformity depends on the shape of the installation, on the location of the measurement plane, and on the boundary conditions at the outlet of the dust collector or the dust generating process (the author’s article [9] and other sources cited there). When it comes to particulate matter concentration distributions across conduits in different places, their prediction by simple theoretical calculation of the transport of solids for industrial conditions solely based on certain models and the approximate knowledge of gas velocity does not give satisfactory results. It is because there are too many phenomena and data in such systems that are difficult to quantify and that play a role (sometimes a stochastic one) in the movement of dust particles inside the conduit [10, 11]. These include the non-homogeneity of particle sizes and their spatial distribution, sometimes a variation in dust density, the concentration field at the outlet of the dust removal device, or the dust generating source (often unknown). Other phenomena include the following: (a) turbulent diffusion; (b) inertial precipitation of dust on the walls of the conduit as a result of local high gas velocities in different locations of ducts; (c) dust retention on the walls due to low near-wall velocities; (d) collisions among dust particles, forces of adhesion and cohesion, and a resulting variable particle agglomeration; (e) re-entrainment of dust from the laminar sublayer; and (f) thermal phenomena. Practice shows that sedimentation deposition of dust in the ducts and the re-entrainment of such sediment escapes engineering and design criteria and is a common (and often dangerous) phenomenon. All these aspects make it difficult to predict the relationship between the gas velocity profile and the particulate matter concentration profile in the industrial flow conduit. For example, one cannot risk a claim that in the uniform flow of the gas phase the uniform dust flow definitely occurs. Moreover, sources with particulate matter concentration distribution measurement data are unfortunately very rare. Such measurements are not normally performed in industrial measurements, because they are not required by the regulations. Overall, when considering real (existing in industrial plants) gas velocity distributions and especially particulate matter concentration ones, one can only speak from a basis of very limited information (for example [2, 9, 12, 13, 14, 15, 16]). In studies related to spatial distribution of particulate matter concentration of dust removal plants, for example, those concerning its impact on the operation of dust collectors, approximate distributions are met as a result of CFD simulations [17, 18]. They can be used to approximate the recognition of possible flow-through situations in these plants. Some useful information can be found in the reports on dust-laden, process gases in industrial plants [19] or on flows in heavily dust-laden air in slightly different situations, i.e. in ventilated longitudinal spaces like roadways [20].

The task of estimating the uncertainty for a single gravimetric measurement of mean particulate matter concentration resulting from the point penetration of cross-sectional distributions of particulate matter concentration and gas velocity in the measurement plane (that is, in the cross section of the conduit) was solved through the simulation of the measurement situations.

Considering the significance of Table 1, the following notion is suggested. The measurement plane can be roughly divided into modular, elementary rectangles, termed “base planes”, with the length of the sides’ ratio of L to M (in an absolute coordinate system 0XY) not more than 2:1 and the area of about 1 m² (e.g. 0.7 m × 1.3 m, 0.9 m × 1.2 m, 1.1 m × 1.1 m). This feature is typical for a large number of conduits. The base plane contains measurement points P _i , with each representing one rectangular component field CF _i . The number of measurement points and thus component fields in a base plane equal n (i = 1, …, n). This number is at least 4; the number of measurement points along a given side, designated n ₀, is, therefore, at least 2. The case of n = 4 (n ₀ = 2) was adopted as the basic case, and Fig. 2 serves as an illustration. One can also imagine that the base plane consists of, for example, n = 6 points (along one side n ₀ = 2, along the other n ₀ = 3) or n = 9 points (for each side n ₀ = 3), when a greater measurement accuracy is desired. For the base plane, a relative coordinate system 0xy was introduced where x = X/L and y = Y/M. The length of both sides of the base plane in this case equals 1. The relative distances of individual measurement points from the edge of the base plane were marked as x _k and y _k , where k = 1, …, n ₀.

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In the base plane, different base profiles of particulate matter concentration and gas velocity are simulated, between which there is no clear correlation. There can be various combinations of velocity and concentration profiles, and these combinations are equally probable. Using simulation profiles, a calculation is performed for the base plane for a given combination of particulate matter concentration and gas velocity profiles of an accurate mean total particulate matter concentration, and the total particulate matter concentration in the sample and, as a result, the divergence between these concentrations, namely the measurement deviation of total concentration. Based on the set of simulated values of possible measurement deviations, the end-result measurement uncertainty of concentration in the base plane is determined, and the uncertainty of this is attributed to the whole measurement plane (composed of base planes).

Let us consider the illustration of particulate matter concentration deviation values in the exemplary Fig. 5. In the four graphs a, b, c, and d, the layout of the deviation values δ clearly indicates that, within individual sets of δ values (i.e. roughly in the vertical sets of points marked by empty circles) assigned to the subsequent values of α _c, approximated to 1.5, 3, and 5, there are no regular distributions of the deviation. In the position of these sets, against the backdrop of the variable α _c index, there is no manifest existence of a systematic component, for example, permanent excess or a permanent deficit of c _m in relation to c _av. The δ deviation for a given value of α _c has a rectangular distribution, because all error-causing simulated flow situations (namely c(x, y) and v(x, y) combinations) are equally probable. Thus, the maximum deviation is sought (or boundary deviation) δ _max. The δ values are within a field delimited by envelopes: positive (top) and negative (bottom), representing, respectively, an extreme positive deviation equal to δ _max, and an extreme negative deviation, equal to −δ _max. Moreover, among the a, b, c, and d charts, there are no significant differences that might indicate an influence of non-uniformity index for velocity profile α _v on δ and consequently on δ _max. Therefore, for a given number of n points, the same pair of envelopes was adopted for all α _v values together, which by definition crosses point (1, 0). The same regularities as described above appear in all other cases of n, i.e. in Figs. E1–E3 in point (v) of ESM. The envelopes for all n values were found to be logarithmic functions—as purely approximation, “tool” functions δ _max(α _c)—that best fulfil the following conditions: (1) in the space of a given n, the top and bottom envelopes have to be symmetric (theoretically infinitive number of positive and negative δ values would fill a symmetric area); (2) function δ _max(α _c) must be monotonically increasing, without local extrema, because the following expected influence of α _c is intended to be reflected: the more non-uniformity of local particulate matter concentration distribution the higher possible non-representativeness of discrete sampling; (3) the pair of envelopes δ _max(α _c) and −δ _max(α _c) should embody all calculated δ values while passing as precisely as possible through the edge ones. This does not mean that, for a given n, the envelope is to pass through each edge δ value, since the calculation cases are not here all possible ones (which, in reality, occur in infinite number); (4) the envelope δ _max(α _c) must pass through point (1; 0). For an idealized, absolutely flat profile of particulate matter concentration, the concentration in the sample of dust-laden gas will always be, regardless of the number of sampling points for this sample, equal to the concentration in the conduit. Sampling deviation (identical to δ _max) will then be zero. For this profile, α _c coefficient is naturally equal to 1; and (5) the mathematical, approximation model of envelope δ _max(α _c) must be the same for each n (4, 9, 16 and 36) so that this function could be parameterized by n.

The logarithmic description of the shape of the positive envelopes δ _max = f(α _c) was proposed as the most accurate and practical in use both for the case n = 4 (as presented in Fig. 5) and for all remaining cases of n values. All the relationships δ _max = f(α _c) were generalized as one function δ _max = f(n, α _c). The details are accessible in point (vi) of ESM.

The calculations conducted in this study for simulated profiles of particulate matter concentration and gas velocity in dust-laden flue gas conduits show that the difference between total particulate matter concentration measured by the gravimetric method (which is ascribed to the sample of dust-laden gas) and the actual mean concentration in the entire stream flowing in the conduit resulting from multi-point sampling can be significant. The relative deviation of the measured total particulate matter concentration very clearly depends on the number of sampling points and the shape of particulate matter concentration profile in the conduit. There is also a weak dependence related to the shape of the gas velocity profile. The value of this deviation for the adopted simulation profiles reaches the level of 8–10 % in extreme conditions. Simulations were performed for the typical rectangular channels where the ratio of the sides’ lengths is less than 2:1.

Since the distributions of concentration are not well known in the measurement practice, relationships were developed (Eq. 11 or Fig. 6) that reflect the uncertainty component of gravimetric, reference measurement of mean particulate matter concentration. For practical application purposes of this calculation tool in specific sampling of dust-laden gas in a measurement plane, it is crucial to be able to most reliably estimate the span of particulate matter concentration profile in this plane α _s, namely the ratio of the highest to the lowest concentration value, in modular, component fragments of the entire section (called here the base planes) with an area of about 1 m², that can notionally constitute the entire section. Another necessary known value is the number of sampling points n applied in a given measurement per one square metre, as a consequence of applying the standard regulations. The suggested Eq. 11 (and Fig. 6) shows that, for the basic standard-imposed number of measurement points n equal to 4, the measurement uncertainty component of the mean particulate matter concentration (at a confidence level of approx. 95 %), due to the existence of the distribution of local concentrations, is, for example, from about 3 % to about 5.5 %, for α _s from 1.5 to 2. Naturally, increasing the number of measurement points decreases uncertainty, e.g. for n = 9 an analogous uncertainty is between 1.3 % and 2.2 %. Conversely, in distributions with greater spans α _s, greater uncertainty is to be expected, for example, with n = 4 and α _s = 3, the uncertainty will be over 8.5 %. The span of the gas velocity profile (the ratio of maximum to minimum velocity) does not affect this uncertainty.

The cited several per cent levels of the uncertainty component of the mean particulate matter concentration must be considered high, and they are not to be omitted in the budget of the expanded uncertainty of this concentration. The presented results of simulation calculations can be considered as a quantitatively recognized new element of metrological description of a common measuring procedure that constitutes the gravimetric method within this simulation. Like many other estimates of uncertainty, the one proposed here also depends on the assumptions and methodology. However, it definitely provides the view of the importance of issues that has been overlooked so far and of the need to include it in the analysis of the measurements. Naturally, estimating the required index of the non-uniformity of the base concentration profile α _s can sometimes be difficult, especially in absence of advanced experience of the measuring team. However, this should not be an excuse for the total exclusion of the uncertainty component considered here.