Time-Dependent Contaminant Transport in Ventilating Air from a Moving Source

A new method which combines the Eulerian, fixed control volume with a moving, Lagrangean flow channel is described for the solution of the conjugate, advection-diffusion problem for modeling transport processes of contaminant species. The transport model is presented as a conservative mass balance equation in a state-flux, species transport form in the space-time domain. A fully-implicit, general solution scheme is formulated with matrix operators in the space-time domain. The particular solutions for specific initial and boundary conditions and source term are constructed with the help of a single, inverse matrix operator, A, which has to be calculated only once for all possible particular problems. Although A involves a large number constants, all are independent from the initial, boundary, and source term input vectors. The multi-level, state-flux, space-time (SFST) scheme brings a significant computational acceleration since A−1 has to be calculated only once, such as in mine ventilation cases involving long drifts with constant air flow velocities. Such application is shown in an example for analyzing the transport and concentration distributions of diesel particulate matter (DPM) in the ventilation air at the working area with the interactions between ventilation and a moving diesel loading machine. Comparison between simulation and in situ DPM monitoring results suggests that reliable evaluation of average exposure of DPM to mine workers may be accomplished directly from tailpipe DPM emission data, ventilation air velocity, and mine geometry with the use of the SFST model even in a highly dynamic working area, potentially reducing the need for real-time DPM monitoring.

the natural atmosphere as well as in the built environment. Concentration of harmful chemical components or dust particles negatively affects health and safety of humans as well as the biosphere and must be controlled. Specific interest is of concentration distribution spread in space and with time originating from contaminant sources. Analysis, numerical model simulation and measurement are often needed to understand the transport of contaminant species for checking the concentration against regulatory compliance.
The transport mechanism of contaminants is complex involving advection, diffusion, dispersion, convection and accumulation in the moving air. Further complexity is added if the contaminant source is moving relative to the flow of air. Such problems are encountered in transportation tunnels, congested cities, and underground mines. Fick's second law may serve as the fundamental component of the governing equations [1] [2], but the formulation is based on the Eulerian, fixed control volume approach which requires fine spatial discretization. In the case of advection and diffusion, the Courant number, Cu, must be kept at Cu = 1 for accurate prediction from time-dependent simulation [3] [4] [5]. In case of moving source and independently moving air, the grid selection must be further constrained. The resulting solution in the form of a CFD (Computational Fluid Dynamics) model [6] [7] often require millions of spatial grid elements in sub-meter, or even micrometer size and of corresponding time steps in seconds, or down to even shorter temporal divisions.
A new method for modeling transport processes has been introduced that combines the Eulerian, fixed control volume with a moving, Lagrangean flow channel for a solution scheme for advection-diffusion problems [5]. The method promises a reduced grid sensitivity due to the intrinsic Cu = 1 condition built into the solution. The method, which can be used for the simulation of macroscopic flow, heat, and momentum transport, is briefly described here for contaminant transport studies. An example is provided for studying diesel emission variation in a ventilated underground working area with a moving loader machine. The numerical simulation results are compared with measurement data collected with stationary as well as moving sensors. The results are discussed and conclusions are drawn from the exercise.

Eulerian Balance Equation with Lagrangean Internal Transport
The general balance equation for advection, diffusion, dispersion, convection and accumulation of species e in the moving air is described in [5] in a stationary Eulerian volume V, swept with a Lagrangean flow channel of advection volume V a . The notations are shown in Figure 1 with a stagnant volume space as V-V a , filled by eddies, but with no net advection transport. The advection flux density component, q a flows as a Lagrangean wave front traveling at velocity v from A in through A out . The transport balance equation for species e in the advection volume is [5]: Figure 1. Control volume and surface for advection, convection, diffusion and accumulation (Reprinted from [5] with permission from Springer).
In (1) ω is the driving force for the transported mass flux density of species e [5]. The SF model is derived directly from (1), substituting volume V x y z =∆ ∆ ∆ ; stagnant volume s V S x y z = ∆ ∆ ∆ ; and advection channel cross sec- Note that the advection travel time, the spatial division, and flow velocity must obey the relationship t x v ∆ =∆ , that is, , for the validity of (1), where Cu is the Courant number [5].
The sum of the first and second flux terms in (1) gives the advective flux driven by mass fraction difference, , where i and i + 1 denotes the input and the output points, respectively [5]: The third term in (1) gives the difference of the flux by diffusion and convection (if applicable) driven by mass fraction differences, ( ) The fourth term in (1) Note that the accumulation term is reduced to the stagnant volume only, The right side of (1) expresses the source term of species e in control volume V: Choosing a unit advective admittance, , the three transport admittances and the source term in (2) The reciprocal of the (6) is recognized as the multiple of the Reynolds, Re, and Schmidt, Sc, numbers, two basic, non-dimensional parameters of transport processes [1]: Re v x Sc D The normalized, finite difference form of (2) with the use of (6) and (7) constitutes the SF network equation for a network branch between nodes i − 1 and i, however, with the connection to node i + 1 also due to diffusion, dispersion, and convection [5]: Note that the condition of validity of (8) is the unity of the Courant number, 1 Assuming a homogeneous flow and transport field with constant material properties and transport coefficients, the SF network Equation (8) can be applied to a series of finite volume cells connected together. A fully implicit in space, time-marching in time solution scheme may be used for solving a set of network equations [5]. The time-marching step must be selected as t x v ∆ =∆ . A multiple time step, fully implicit, State-flux, Space-Time (SFST) solution scheme can also be constructed offering a closed-form, operator solution for the conjugate advection, convection and diffusion-dispersion problem [5].

A Fully Implicit, SFST Numerical Solution
A fully-implicit transport network solution is given in [5] using Equation (8) at all the nodes and Equations (6) everywhere at the branches of a high-density internal grid. The transport network for such a multiple-level implicit scheme with inter-connected spatial and temporal grids is shown in Figure 2. It is shown that the solution can be expressed in a matrix-vector form with a five-diagonal admittance matrix, A, [5]. Out of the five diagonals, there is a triple-diagonal strip matrix symmetric around the main diagonal, stretching to the size of (N × M) 2 . In addition, there are two off-diagonal lines to include transport connections from the previous time interval. One off-diagonal line models the advection connections for each time step with the time-shifted potentials in the new A matrix, with ( ) For a right-to-left, v a < 0 velocity, the advection connections are transposed: Matrix A may be viewed as a composite array of M × M sub-matrices, a i,j , i.e., In (11), the diagonal sub-matrices, . The a i,i elements constitute triple-diagonal sub-matrices: The off-diagonal sub-matrices, ( ) , , i j a k l are mainly zeros. For left-to-right, v > 0 velocity, the non-zero off-diagonal sub-matrices for The advection connections are transposed for the right-to-left, v < 0 velocity in the off-diagonal elements, For all other sub-matrixes, not defined by (12)  where no connections are defined in the network of Figure 2, and keeping their non-zero value at the active connections. Using the notation of The unknown mass fraction vector, e ω , is also used in sub-vectors form of ( ) Distributed substance source for each node in Figure 2 may be included, varying with space and time. The source term in sub-vector form is: The balance equation of the SF network of Figure 2 is now written in sub-matrix notation: The simultaneous solution for the entire mass fraction field with space and time is: All sub-vectors except for the distributed source term vector are substantially sparse in (22). It is possible to eliminate the zero elements from the terms on the left side of (22) and to return to the full initial and boundary condition vectors [5]. The source term is also reduced to an M-element F s vector by either accepting the average of the nodal sources along each 1, , In (23), five different coefficient matrices emerged with the definitions as follow: The coefficient matrices in (23) and (24)

Application Example of the SFST Model with Moving Source Term
Based on an underground mine experiment [9], an example of the method is de-         Figure 12 and Figure 13 for the LS and RH segments, respectively. The concentration for the fixed location sensors are shown in Figure 14 for D = 0.5 m 2 /s. Figure 15 and Figure 16 depict the DPM concentration for the LHD tailpipe and the DPM moving sensors for the LS and RH segments, respectively, assuming a dispersion coefficient of D = 2.5 m 2 /s; the corresponding concentrations for the stationary sensors are shown in Figure 17.

Discussion of the SFST Method and the Results in the Example
The SFST solution for a moving substance source is given in a The results for the three independent simulations performed at two separate sections such as LS and RS of the mine drift using the three different dispersion coefficient are delineated as follows.

LS, LHD Tailpipe and Moving Sensor Concentration Variations
It is interesting to observe that the DPM concentration at the tailpipe exit point over the entire length of the inby section show a saw tooth-shape fluctuation.
With low dispersion (D = 0.05 m 2 /s, Figure 9), the amplitude of fluctuation does not change due to the fact that the machine always encounters a portion of fresh air at x ∆ with background concentration at the time of entering a new airway section of which is then charged by the source term resulting in a near-constant ω ∆ (and corresponding volumetric concentration) change. Since the speed of the LHD machine is higher than that of the air and the machine does not travel perfectly together with the same discrete air volume (however, part of the previous air section is still connected during a t ∆ time interval), the concentration at the next x ∆ section starts again at a lower concentration which is interestingly higher than the background concentration due to the near-instantaneous introduction of pollutant source to the control volume upon arrival at x ∆ at the tailpipe exit point. It is assuring to observe that when the transport connection is spread to a larger volume with increased dispersion (Figure 12, Figure   15), the DPM concentration shifts toward starting from the background concentration, and the amplitude of the saw tooth variation is decreasing.
The DPM concentration is shown to be sensed by the moving sensor very well over the entire length of the inby section in Figure 9, due to the favorable LHD movement direction for the exhaust plume in this section as illustrated in Figure   5, with the DPM sensor in the plume behind the tailpipe. It is assuring to see that the starting concentration of the moving sensor is at the background value and that the amplitude of the saw tooth variation is smaller, due to dispersion over a larger distance of the offset of 2.1 m that is larger than the value of 1.3 x ∆ = m. With increased dispersion shown in Figure 12 and Figure 15, the DPM concentration from the moving sensor is getting smoother following very well the concentration of the tailpipe emission curve.
At the loading point, the LHD and the moving sensor are stationary. As shown in Figure 9, the concentration gradually accumulates with time due to the continuous pollutant mass source as well as the arrival of the polluted air that is left behind the faster-moving machine. With increased dispersion shown in Figure 12 and Figure 15, the DPM concentration from the moving sensor at the loading point is getting lower and smoother, but always following very well the concentration of the tailpipe emission.
The DPM concentration at the tailpipe and the moving sensor in the LS drift section during the outby travel of the LHD are very different from that of the inby travel, as seen in Figure 9. The DPM concentration at the tailpipe appears to be smooth due to traveling against the air flow with low dispersion.  However, the difference is that, unlike the LS, the DPM concentration is shown not to be sensed by the moving sensor very well over the entire length of the inby section in Figure 10, Figure  Similar trends can be seen with increased dispersion coefficients in Figure 13 and Figure 16, albeit the higher dispersions appear to connect the moving sensor to the tailpipe better, elevating concentration levels.

RS, LHD Tailpipe and Moving Sensor Concentration Variations
At the dumping point, the LHD and the moving sensor are stationary. As shown in Figure 10, the concentration gradually accumulates with time due to the continuous pollutant mass source as well as the arrival of the polluted air left behind the moving machine. With increased dispersions in Figure 13 and Figure 16, the DPM concentration from the moving sensor at the dumping point is getting lower and smoother, but always following very well the concentration of the tailpipe emission.

LS, Stationary Drift Location Sensors
The stationary drift sensors at the LS drift section show increasing concentrations in the inby section in Figure 11 as the LHD moves from the reference point to the loading point. This is due to the lower velocity of the air than that of the tailpipe DPM source. The sensors show afluctuation in concentration with a reduced amplitude in the outby section in Figure

Cycle Weighted Averages (CWA) and Comparison of Model Results to Measured Data
Part of the process of evaluating the concentrations at the moving and stationary sensors is to calculate time-averaged values for the haulage cycles in both the LS and RS drift sections. These average values are depicted in Figure 9 through  The RS av used for the stationary sensors is 90 μg/m 3 , which is the background concentration entered into the model. Table 1 summarizes the CWA obtained from the SFST model for the different dispersion coefficients.
As seen in Table 1 The results of the SFST model are compared with in situ DPM in situ measurements performed at an underground mine [9]. The DPM measurement results are summarized in Table 2.
Comparison of DPM measurement data with the SFST model simulation results is shown in Table 3 for different dispersion coefficients. As seen, the dis-   can be predicted from the DPM mass transport model using the SFST simulation.
The DPM source term from the tailpipe DPM concentration and their fuel consumption data can be evaluated as an input to the SFST simulation model. The mass balance and transport network modeling method ensures a cost effective and accurate way of predicting average DPM concentrations in underground mines, reducing the need for real-time monitoring for appropriate ventilation design.

Conclusions
• A new, powerful, fully-implicit SFST solution is applied for interpreting measurement results for DPM contaminant concentration variations from a moving machine in an underground mine. • The mathematical model provides a link between the time-averaged and the peak DPM concentration values at the tailpipe.
• Very good match was obtained for all three drift stationary sensors as well as the moving sensor with D = 0.05, 0.5 and 2.5 m 2 /s between in situ measurement results and SFST model simulation.
• The DPM concentration variations with location and time in the air of the mine can be predicted from the known tailpipe DPM concentration from machine smog tests and the fuel consumption of the diesel machine.
• Therefore, the mathematical model may be used to evaluate the average concentration exposure value of the DPM for compliance analysis without real-time, complicated DPM measurements, relying basically on tailpipe smog test, fuel consumption and the SFST contaminant transport model, incorporated in the mine ventilation model. • With the simulation of total, accumulated DPM concentration at the working area, mining companies will be able to implement the right ventilation strategies to reduce or eliminate harmful DPM exposure to mine workers.