The transport of contaminant species by the moving air is of great importance in 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 (Com- putational 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.

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] :

∫ A i n q a [ t − Δ t ( A ) ] d Α + ∫ A o u t q a ( t ) d Α + ∫ A q d + c ( t ) d A + ∫ V − V a ∂ ρ e ( t ) ∂ t d V = ∫ V ρ f e d V (1)

In (1), q a is advective mass flux density of species e; Δ t ( A ) is the average travel time of the advective fluidoriginating from A i n ; q d + c is the sum of diffusion, dispersion and convection of e across area, A; ρ e and ρ are the densities of e and the bulk flow, respectively, and f e is the source term for species e.

Integral Equation (1) may be written in a finite difference form using finite volume and surface elements. With the introduction of the mass fraction variable, ω e = ρ e / ρ for species e, a State-Flux (SF) model is formulated in which the state, ω e 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 = Δ x Δ y Δ z ; stagnant volume V s = S Δ x Δ y Δ z ; and advection channel cross section A a = ( 1 − S ) Δ y Δ z surface. Note that the advection travel time, the spatial division, and flow velocity must obey the relationship Δ t = Δ x / v , that is,

C u = v Δ t Δ x = 1 , 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, ω e i n − ω e i − 1 n − 1 , where i and i + 1 denotes the input and the output points, respectively [5] :

∫ A i n q a [ t − Δ t ( A ) ] d Α + ∫ A o u t q a ( t ) d Α = ( 1 − S ) ρ v Δ y Δ z ( ω e i n − ω e i − 1 n − 1 ) (2)

The third term in (1) gives the difference of the flux by diffusion and convection (if applicable) driven by mass fraction differences, ( ω e i + 1 n − ω e i n ) and ( ω e i n − ω e i − 1 n ) :

∫ A q d + c ( t ) d A     i R D = − ρ D e Δ y Δ z / Δ x [ ( ω e i + 1 n − ω e i n ) − ( ω e i n − ω e i − 1 n ) ] (3)

The fourth term in (1) is the accumulation of the substance flux, proportional to the mass fraction change ( ω e i n − ω e i n − 1 ) over the Δ t time period:

∫ V − V a ∂ ρ e ( t ) ∂ t d V = S ρ Δ x Δ y Δ z / Δ t ( ω e i n − ω e i n − 1 ) (4)

Note that the accumulation term is reduced to the stagnant volume only, V − V a

The right side of (1) expresses the source term of species e in control volume V:

∫ V ρ f e d V = ρ f e i Δ x Δ y Δ z (5)

Choosing a unit advective admittance, i R A = ( 1 − S ) ρ v Δ y Δ z = 1 , the three transport admittances and the source term in (2)-(5) can be reduced to two, non-dimensional parameters, i R D and i R S , and a cell source term, F c i , following [5] :

i R A = ( 1 − S ) ρ v Δ y Δ z = 1 i R D = D e v Δ x ( 1 − S ) = 1 R e ( 1 − S ) S c i R S = S / ( 1 − S ) F c i = F i / i R A = f e i Δ t / ( 1 − S ) } (6)

The reciprocal of the D e / ( v Δ x ) term in Equations (6) is recognized as the multiple of the Reynolds, Re, and Schmidt, Sc, numbers, two basic, non-dimen- sional parameters of transport processes [1] :

R e = v Δ x / ν S c = ν / D e R e S c = ( v Δ x / ν ) ( ν / D e ) = v Δ x / D e } (7)

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] :

( ω e i n − ω e i − 1 n − 1 ) − i R D [ ( ω e i + 1 n − ω e i n ) − ( ω e i n − ω e i − 1 n ) ] + i R S ( ω e i n − ω e i n − 1 ) = F c i (8)

Note that the condition of validity of (8) is the unity of the Courant number, C u = v Δ t / Δ x = 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 Δ t = Δ 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 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.

The boundary nodes with known driving potentials at n = 1 , ⋯ , M time steps are separated from the active nodes of unknown potentials along i = 1 , ⋯ , N spatial divisions. The advective transport connections between consecutive time levels are separated by potential followers, in order to provide the correct initial state values from time n − 1 to time n, affecting the substance flux density balance only at time n, but not at time n − 1. The initial potentials are established from time-step to time-step with no feedback flux effect from future time. The size of the network model to be solved simultaneously is (N ´ M)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 A ( i , i − N − 1 ) = − 1 , i = N + 2 , ⋯ , N × M . The other off-diagonal line includes the accumulation connections for the stagnant volume, A ( i , i − N ) = − i R S , i = N + 1 , ⋯ , N × M for each internal time step. The five-diagonal matrix,#Math_43# for a positive, left-to-right, v > 0 velocity is [5] :

A ( i , j ) = { − i R D   if   j = i + 1 , i = 1 , ⋯ , N × M − 1 1 + 2 i R D + i R S   if   j = i , i = 1 , ⋯ , N × M − i R D     if   j = i − 1 , i = 2 , ⋯ , N × M − 1     if   j = i − N − 1 , i = N + 2 , ⋯ , N × M − i R S     if   j = i − N , i = N + 1 , ⋯ , N × M 0                 otherwise (9)

For a right-to-left, v_a < 0 velocity, the advection connections are transposed:

A ( i , j ) = { − i R D   if   j = i + 1 , i = 1 , ⋯ , N × M − 1 1 + 2 i R D + i R S   if   j = i , i = 1 , ⋯ , N × M − i R D     if   j = i − 1 , i = 2 , ⋯ , N × M − 1     if   j = i − N − 1 , i = N + 2 , ⋯ , N × M − i R S     if   j = i − N , i = N + 1 , ⋯ , N × M 0                 otherwise (10)

Matrix A may be viewed as a composite array of M ´ M sub-matrices, a_i,j, i.e., A ( i , j ) = { a i , j } , i = 1 , ⋯ , M ,   j = 1 , ⋯ , M :

A ( i , j ) = [ a 1 , 1 ⋯ a 1 , M ⋮ ⋱ ⋮ a M , 1 ⋯ a M , M ] (11)

In (11), the diagonal sub-matrices, a i , i ( k , l ) are N × N in size, a i , i ( k , l ) ∈ R 2 , i = 1 , ⋯ , M , k = 1 , ⋯ , N , l = 1 , ⋯ , N . The a_i,i elements constitute triple-diagonal sub-matrices:

a i , i ( k , l ) = { − i R D                               if   l = k + 1 ,   k = 1 , ⋯ , N − 1 1 + 2 R D + i R S         if   l = k ,   k = 1 , ⋯ , N − R D                                 if   l = k − 1 ,   k = 2 , ⋯ N 0                                           otherwise (12)

The off-diagonal sub-matrices, a i , j ( k , l ) are mainly zeros. For left-to-right, v > 0 velocity, the non-zero off-diagonal sub-matrices for i = 2, ⋯ , M are:

a i , i − 1 ( k , l ) = { − i R S if   l = k ,   k = 2 , ⋯ , N − 1 if   l = k − 1 ,   k = 2 , ⋯ , N 0 otherwise (13)

The advection connections are transposed for the right-to-left, v < 0 velocity in the off-diagonal elements, a i , j ( k , l ) as follows for j = 2, ⋯ , M :

a j − 1 , j ( k , l ) = { − i R S if   l = k ,   k = 2 , ⋯ , N − 1 if   k = l − 1 ,   l = 2 , ⋯ , N 0 otherwise (14)

For all other sub-matrixes, not defined by (12) through (14), are null matrices, a_i,j = {0}. Consequently, A(i,j) is dominantly a sparse matrix.

Although the structure of the matrix operators are different for v > 0 from v < 0 velocity directions, the notations are simplified and applied for left-to-right, v > 0 velocity, shown in Figure 2.

For simplicity, sub-matrix notations are used for the upward and downward boundary condition vectors, BU^d, BU^a and BD^d, as well as the initial condition vector, IC. The boundary and initial condition sub-vectors in the matrix-vector balance equation have to be all in the length of N × M, filling them with zeros 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 b u i d ( j ) = B U d ( n ) , b u i a ( j ) = B U a ( n ) , b d i d ( j ) = B D d ( n ) and i c i ( j ) = I C ( j ) , respectively, the boundary vector elements are:

b u i d ( j ) = { B U d ( i ) if   j = i , i = 1 , ⋯ , M , j = 1 , ⋯ , N 0 otherwise (15)

b d i d ( j ) = { B D d ( i ) if   j − N + 1 = i , i = 1 , ⋯ , M , j = 1 , ⋯ , N 0 otherwise (16)

b u i a ( j ) = { B U a ( i ) if   j = i , i = 1 , ⋯ , M , j = 1 , ⋯ , N 0 otherwise (17)

i c i ( j ) = { I C ( j ) if   i = 1 , j = 1 , ⋯ , N 0 otherwise (18)

The unknown mass fraction vector, ω e , is also used in sub-vectors form of ω i = ω e [ ( i − 1 ) M + j ] :

ω i ( j ) = ω e [ ( i − 1 ) M + j ] , i = 1 , ⋯ , M , j = 1 , ⋯ , N (19)

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:

F i ( j ) = F c i [ ( i − 1 ) M + j ] , i = 1 , ⋯ , M , j = 1 , ⋯ , N (20)

The balance equation of the SF network of Figure 2 is now written in sub-matrix notation:

[ a 1 , 1 ⋯ a 1 , M ⋮ ⋱ ⋮ a M , 1 ⋯ a M , M ] [ ω 1 ⋮ ω M ] = [ i c 1 ⋮ 0 ] + i R D [ b u 1 d ⋮ b u M d ] + [ b u 1 a ⋮ b u M a ] + i R D [ b d 1 d ⋮ b d M d ] + [ F 1 ⋮ F M ] (21)

The simultaneous solution for the entire mass fraction field with space and time is:

[ ω 1 ⋮ ω M ] = A − 1 [ i c 1 ⋮ 0 ] + A − 1 i R D [ b u 1 d ⋮ b u M d ] + A − 1 [ b u 1 a ⋮ b u M a ] + A − 1 i R D [ b d 1 d ⋮ b d M d ] + A − 1 [ F 1 ⋮ F M ] (22)

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 i = 1, ⋯ , N line, or sampling the values of the F_i(j) vector along a desired trajectory, e.g., along a moving source, in the x-t space within the cell domain. With these simplifications by algebra, a new matrix-vector equation is obtained with matrices of MN rows and M columns, with still full expression for all MN mass fraction values, but in need of M-vectors only on the right side [5] :

[ ω 1 ⋮ ω M ] = i A I C I C + i A B U d B U d + i A B U a B U a + i A B D d B D d + i A F F s (23)

In (23), five different coefficient matrices emerged with the definitions as follow:

i A I C ( i , j ) = A − 1 ( i , j )         for   i = 1 , ⋯ , M ⋅ N ,   j = 1 , ⋯ N i A B U d ( i , j ) = A − 1 ( i , j ) i R D       for   i = 1 , ⋯ , M ⋅ N ,   j = 1 , N + 1 , 2 N + 1 , ⋯ , ( M − 1 ) N + 1 i A B U a ( i , j ) = A − 1 ( i , j )         for   i = 1 , ⋯ , M ⋅ N ,   j = 1 , N + 1 , 2 N + 1 , ⋯ , ( M − 1 ) N + 1 i A B D d ( i , j ) = A − 1 ( i , j ) i R D         for   i = 1 , ⋯ , M ⋅ N ,   j = N , 2 N , ⋯ , M ⋅ N i A F ( i , j ) = A − 1 ( i , j ) A s ( i , k )         for   i = 1 , ⋯ , M ⋅ N ,   k = 1 , ⋯ , M ; A s : samplesof   F s } (24)

The coefficient matrices in (23) and (24) all have M ⋅ N rows, but the number of columns is N in i A I C and M in all the other four matrices. Therefore, the full matrices in (23) are different from each other; and all have much smaller size that A⁻¹ in (22).

Note that the last row in each matrix in Equation (23) is not sparse and may involve a large number constants, carrying all of information necessary for expressing the solution the conjugate advective and diffusive transport in the ST domain with matrix operators, all independent from the IC, BU, and F^S input vectors. At the same time, the multi-level scheme brings a significant computational acceleration if A⁻¹ has to be calculated only once, such as in mine ventilation for each long drift with a constant, advection velocity. Such application is shown in an example as follows.

Based on an underground mine experiment [9] , an example of the method is described for contaminant concentration distribution in the ventilating air in a mine drift. A mine haulage route for a diesel load-haul-dump (LHD) machine in the experiment is shown in Figure 3. Diesel particulate matter (DPM), a contaminant is modeled as it is transported in the ventilating air by advection and dispersion, emitted from the tail pipe as a moving source of the engine. The machine travels back and forth along the specified route, continuously repeating a loading cycle.

In order to capture the tailpipe DPM contaminants from the exhaust fumes directly, two sampling sensors were placed 2.1 m away from the tailpipe. Three pairs of stationary sensors were placed, two at each location at three different points on the wall, marked as m₁, m₂, and m₃ at 3.1 m, 41 m, and 78 m from the loading area to measure the DPM concentration in the ventilating air.

The ventilating air enters the mine drift at the mid-section and splits into two directions. The left side (LS) drift segment, 78 m long, leads to the loading zone. The right side (RS) segment, 241 m long, stretches to the dumpling zone. The two drift segments require two separate DPM transport models. The connection between the two model sections is the loading machine crossing from one to the other in a cyclic manner. The diesel exhaust source of the loading machine moves with a travel velocity of v T = 2.18   m / s in either the LS or RS drift section. The airflow velocity of v A = 1.3   m / s is also the same in both the LS and the RS segments.

Figure 4 depicts the cycle time diagram for the loading machine and the airflow directions. The moving sensor measuring concentration is placed 2.1 m behind the moving point source and modeled independently in the two separate sections. The point where the air flow splits into two opposite directions to the left and right sections is used as a reference point for background concentration. The DPM concentration of 90 μg/m³, measured as an incoming value at the intake air from the other areas of the mine, is used in the example as a constant additive to the zero concentration level in the numerical model. The concentration from the moving sensors on the loading machine is modeled by sampling the spatial concentration from the SFST solution at an offset of 2.1 m distance from the exhaust point of the tailpipe.

The airflow directions and the spread of the tailpipe exhaust fumes are illustrated in Figure 5 as the machine travels in the haulage drift in and out of the LS and the RS drift segments. The efficiency of catching the DPM from the exhaust plumes by the moving sensor depends on the direction of the movement of the machine as well as the air flow direction relative to the machine.

A fine spacial mesh of 1.3 m and 1 second of temporal discretization is used in both the LS and RS for the numerical models, satisfying the condition for the Courant number of Cu = 1.3/1.3 = 1. The model is setup using the geometry of

the mine experiment. Accordingly, the N and M divisions in the spatial and temporal meshes are N = 60, M = 162 for the LS and N = 186, M = 258 for the RS model segments.

Figure 6 and Figure 7 show the machine velocity versus distance diagram per cycle and machine velocity versus time diagram, respectively. The source term is determined from measured diesel fuel use for the haulage and the tailpie emis- sion data [9] . The DPM concentration source term entering the air flow from the tailpipein the model is 1000 μg/kg/s, determined based on LHD machine data. Note that the moving point source is a cumulative transport term that cannot be converted directly into a mass fraction (or volumetric concentration) in the moving air stream without considering the advetive, dispersive, and accumulation components. Assuming no stagnant volume in the drift cross section (S = 0), no dispersion, (D = 0), and near-zero advective air flow velocity, (such as that

lim v → 0 ( D v ) = 0 ), the source term would continuously increase the contaminant mass fraction according to ω i n − ω i − 1 n − 1 = Δ ω = F c Δ t as seen from Equation (8).

Therefore, the mass fraction (and the corresponding volumetric concentration) would both increase linearly with time to infinity.

The transient mass concentration distributions are of interest at various moving and fixed points: at the moving source i.e., at the exhaust tailpipe; at a sensor fastened on the machine with an offset of 2.1 m behind the tailpipe and at three fixed locations in the airway. The fixed locations are: at x = L/25, close to the en-

trance to the tunnel; at x = L/2; and at x = L.

Matrix A in (11) and matrices in (23) are calculated with N = 60 and M = 162 for the LS, and N = 186 and M = 258 for the RS drift segments, respectively. The size of A (and A⁻¹) in the operator solution are 9720 × 9720 in the LS and 47,988 × 47,988 in the RS models, both calculated only once in the examples for each dispersion coefficient. The dispersion coefficients is calculated as D = 2.5 m²/s for turbulent pipe flow in reference to Taylor’s results [8] , but two lower values of 0.05 m²/s and 0.5 m²/s are also used for sensitivity checks in the example. Zero initial and boundary values, i.e., IC = BU = BD = 0, are assumed in the Eulerian domain. The source points are defined along the traveling trajectory of the machine over the space-time plane intersecting or passing by minimum distance from M grid points which now represent the Lagrangean model domain. The mass concentration values are calculated from the full ST solution of (23), applied separately to the LS and RS drift segments. The spatial and temporal concentration values for the stationary and moving sensors; as well as the tailpipe of the LHD are determined from the full ST solutions by sampling the data sets obtained for the LS and RS drift segments.

The full ST solution is shown in Figure 8(a) and Figure 8(b). Both figures depict the mass concentration field with D = 2.5 m²/s. For graphical purposes, the large model size is reduced by sampling only every 10^th of the DPM mass fraction values, converted to the more convenient concentration units.

The desired solution for concentration both at fixed as well as at the moving points is expressed from the full solution for the entire space-time plane at the space-time grid points. Since both the result and source vectors have M elements, the solution equation is always in the form of:

[ ω 1 ⋮ ω M ] R = i A R F F s (25)

where i A R F is calculated from the common i A matrix operator by multiplication with individual sampling matrices defined in relations (24) for different model resultant solutions, R, according to the selection of the sampled points in the LS and RH segments along the characteristics lines of the traveling machine, shown in Figure 4.

Three different dispersion coefficients, 0.05 m²/s, 0.5 m²/s, and 2.5 m²/s are tested with the model for three independent simulations. The goal is to assess the effect of dispersion on the concentration predicted by the model. The simulated average concentrations are calculated for the moving sensor, as well as the stationary sensors. These values are used to calculate the cycle average concentrations.

Figure 9 and Figure 10 show the DPM concentration for the LHD tailpipe and the DPM moving sensors for the LS and RH segments, respectively. Figure 11 depicts the concentrations for the fixed locations sensors with D = 0.05 m²/s. Two higher dispersion coefficients are used in the simulations for sensitivity examination with D = 0.5 m²/s and D = 2.5 m²/s. For D = 0.5 m²/s, the DPM concentration for the LHD tailpipe and the DPM moving sensors are illustrated in 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 m²/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 m²/s; the corresponding concentrations for the stationary sensors are shown in Figure 17.

The SFST solution for a moving substance source is given in a Eulerian domain overlaid with a Lagrangean space. The particular solutions for the initial and boundary conditions and source term are constructed with the help of a matrix operator, A⁻¹, which has to be calculated only once for the LS and the RS drift segment for all particular, initial and boundary problems and even with the moving DPM source. The A⁻¹ matrix operators would remain valid for any other movement characteristics of the pollutant source term, however, the sampling matrices in (23) and (24) would have to change.

The method is advantageous to use where the air flow direction and velocity in the underground drifts are known, and the iA = A⁻¹ can be conveniently pre-calculated. Such a case is often found in underground mines with constant ventilation and long time periods of continuous loading operations such as in the application example for diesel emission measurement evaluation.

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.

・ A new, powerful, fully-implicit SFST solution is applied for interpreting mea- surement 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 m²/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.

A research support grant from the National Institute of Occupational Safety and Health, USA (NIOSH) is thankfully acknowledged.

Danko, G.L. and Asante, W. (2017) Time-Dependent Contaminant Transport in Ventilating Air from a Moving Source. Applied Mathematics, 8, 671-692. https://doi.org/10.4236/am.2017.85054