The advection diffusion equation plays an important role in describing solute transport due to the combined effect of diffusion and advection in a medium. These equations have drawn significant attention from hydrologists, civil engineers, and mathematical modelers. Many methods have been used to obtain the analytical, semi-analytical, and numerical solutions of the one-dimensional, two-dimensional, and three-dimensional advection diffusion equations with different parameters, boundary conditions, and techniques. Kumar et al and Kumar et al solved analytically the one-dimensional advection diffusion equation with variable coefficients for a uniform input point source and of increasing nature by the Laplace transformation method in finite and semi-infinite media, respectively. Jaiswal et al derived the solution of the one-dimensional advection diffusion equation with temporally dependent coefficients for a uniform input point source and input point of increasing nature by the Laplace transformation technique. Jaiswal and Kumar obtained an analytical solution of the one-dimensional advection-dispersion equation with variable coefficients for a varying pulse-type input point source in a longitudinal domain for two problems: temporally dependent dispersion along uniform flow and spatially dependent dispersion along non-uniform flow by the Laplace transformation method. Kumar et al developed an analytical solution for one-dimensional solute dispersion by the Laplace transformation method along unsteady flow through a heterogeneous medium; the dispersion parameter was considered proportional to the square of the velocity. Kumar et al obtained an analytical solution for the one-dimensional advection diffusion equation for uniform flow in a longitudinal semi-infinite homogeneous porous medium by the Laplace method. The first-order decay, retardation factor, and zero-order production terms were also considered in this study. The first-order decay and zero-order production were considered inversely proportional to the dispersion coefficient. Chen and Liu obtained a generalized analytical solution to the one-dimensional advection dispersion equation. The generalized analytical solution was obtained by using the Laplace transform with respect to time and the generalized integral transform technique with respect to the spatial coordinate. The solution was obtained in a finite spatial domain subject to an arbitrary time-dependent inlet boundary condition. Daga and Pradhan obtained an analytical solution to the one-dimensional advection diffusion equation based on the mathematical combination between the variational iteration method and the homotopy perturbation method. Mazaheri obtained an analytical solution for the one-dimensional advection diffusion equation with several point sources through arbitrary time-dependent emission rate patterns by the Laplace transform. Kumar and Yadav obtained an analytical solution for conservative solute transport in homogeneous porous media for uniform and varying pulse-type input point sources in one dimension. Wadi et al obtained an analytical solution of the one-dimensional advection dispersion equation by using the Laplace transformation technique. Sanskrityayn and Kumar obtained an analytical solution of the advection-diffusion equation in a heterogeneous infinite medium by using Green's function method for two problems: continuous injection and instantaneous point injection. Kumar obtained an analytical approach for the one-dimensional advection diffusion equation with temporally dependent variable coefficients of hyperbolic function in a semi-infinite porous domain by the Laplace transformation; the solute dispersion parameter and seepage velocity were considered as hyperbolically decreasing functions of time. Yadav and Kumar derived an analytical solution for one-dimensional spatially dependent solute transport in semi-infinite porous media through Laplace transformation. Chaudhary et al developed research titled “Analysis of 1-D pollutant transport in semi-infinite groundwater reservoir.” Yadav and Roy . Jaiswal et al obtained an analytical solution for the transport of pollutant from time-dependent locations along groundwater. The Laplace integral transformation technique was applied to obtain the final solution. Yadav et al obtained an analytical solution for conservative solute transport in time- and space-dependent flow in heterogeneously adsorbing porous media. The solute transport also involved first-order decay and zero-order production. Kumar et al discussed the effect of sink/source on the transport of solutes for the 1-D ADE in porous media to derive a closed-form solution by the Laplace transform. Paudel et al obtained an analytical solution for the advection diffusion equation of pollutant concentration by using the Laplace transformation technique. Sanskrityayn et al proposed an analytical solution for the advection diffusion equation with variable flow and transport coefficients. Yadav et al obtained an analytical solution to the one-dimensional advection diffusion equation in a heterogeneous aquifer system with space- and time-dependent groundwater velocity and dispersion coefficient in a semi-infinite medium by the Laplace transformation method. Yadav et al obtained an analytical solution for the two-dimensional advection diffusion equation in semi-infinite homogeneous porous media by the Laplace transformation method. First-order decay proportional to the dispersion coefficient was also considered in the study. Mckaa et al obtained an analytical solution of a simple advection-diffusion model of an oxygen transfer device. To get the analytical solution, the Laplace transformation and convolution were applied. Chatterjee and Singh obtained a solution for the two-dimensional advection dispersion equation with depth-dependent variable source concentration in a semi-infinite medium. Thakur conducted research titled “Two-dimensional solute transport with exponential concentration distribution and varying flow velocity.” Yadav and Kumar obtained an analytical solution of the advection diffusion equation for conservative solute transport in a semi-infinite heterogeneous porous medium with a pulse-type input point source of uniform nature in two dimensions by using the Laplace transformation method. First-order decay and zero-order production were also considered. Essa et al developed research on the evaluation of the analytical solution of the advection diffusion equation in three dimensions. Ibrahim et al used the Laplace transformation technique and explicit finite difference method to obtain the analytical and numerical solution of the two-dimensional advection-diffusion equation for pollutant concentration in a river with time-dependent dispersion coefficients. Chaudhary and Singh . Chen et al . Buske et al carried out research on the solution of the coupled advection-diffusion and Navier-Stokes equations in the two-dimensional advection-diffusion equation. De Athayde et al carried out research titled “Analytical solution of the coupled advection-diffusion and Navier-Stokes equation for air pollutant emission simulation.”

The lack of an advection diffusion equation coupled with a linear partial differential equation is the motivation of this work to obtain a semi-analytical solution for the one-dimensional advection diffusion equation coupled with a linear partial differential equation with constant coefficients. The Laplace transformation technique and Stehfest algorithm are applied to obtain the final solution to the system. The solution to the system is obtained in homogeneous and semi-infinite media. The computations are done with the help of MATLAB software. The solution to the coupled system is illustrated through graphs, presenting the effect of concentration and velocity.

The advection diffusion equation coupled with a linear partial differential equation is modeled from an application of grain fumigation in a grain store, which is the grain fumigant air system. During fumigation, fumigant gas is pumped into the storage with constant velocity. Then, the gas mixes with the air in the silo. The flow of the gas into the silo can be modeled as a transport process where the processes of diffusion and advection occur. The fumigant transfers from the inlet into the storage and accumulates on the grain solid. Work by Darby has proven that some amount of the fumigant will absorb into the grain, and at the same time some amount also desorbs back to the air. Therefore, Darby introduced a coupled advection diffusion equation and linear partial differential equation as follows:

Where x is the direction of flow (m), t is time(s), c(x,t) the concentraion of the fumigant in air, q(x,t) is the concentration of fumigant in grain, V is the velocity that the quantity is moving, and D is the diffusivity and the additional parameter

Based on James Darby (2011) which is about the mathematical model defining the kinetics of the sorption and application to predict concentrations and grain residues for practical fumigation scenarios, the formula of the constant are given by:

: The mass transfer coefficient of gaseous fumigant from air to grain.

: The mass transfer coefficient of adsorbed fumigant in grain.

: The rate of the first-order reaction of adsorbed fumigant in the grain.

: Rate of first order “reaction” of gaseous fumigant in air.

: Specific surface area for sorption.

: partition factor.

: The true density of the grain kernels.

: porosity of the bulk grain.

: The diffusivity of fumigant into grain.

With initial and boundary conditions:

Initial Condition: $c(x,0)=0,\mathrm{ }q(x,0)=0$

Boundary condition: , $c(\infty ,t)=0$,

Applying the Laplace transform:

Then (3) becomes:

and (4) becomes:

Appliying the initial condition, for equations (5) and (6) become:

Substitute (8) into equation (7), we get.

Which is the homogenous second order linear ODE.

Applying the boundary conditions, we find: A=0 Hence B . Thus:

After simplification,

Figure 1 shows the concentration of the phosphine gas flow in air versus the vertical height of the silo. The initial concentration of the phosphine gas is 10 mg/L with constant velocity at V=0.002 m/s. The concentration of phosphine gas is plotted for four intervals of time to illustrate the gas movement behavior until the gas completely fills the silo.

Based on the graph, the fumigation gas process needs more than 120 minutes to ensure the phosphine gas completely fills the silo of 6 m height. At t=30 minutes, fumigation gas of 10 mg/L has reached a height of more than 1 m and continues moving until the entire silo space is occupied. Therefore, it can be concluded that fumigation with constant velocity V=0.002 m/s takes at least 120 minutes to fill a 6 m silo, and the gas will be absorbed by the grain after 120 minutes.

Figure 2 shows that at t=150 minutes, the concentration of the fumigation gas is zero because the gas is still moving to fill the entire space of the silo. The grain at a height of more than 1 m starts to absorb the gas after the gas has been flowing for 180 minutes through the silo. The gas continues to diffuse into the grain at higher parts until the grain at all heights contains 1 mg/L of gas concentration after 240 minutes of gas flow.

Figure 3 represents the concentration of the phosphine gas in air when the constant velocity is increased to 0.004 m/s. At t=30 minutes, the fumigation gas of 10 mg/L has reached a height of more than 4 m, which is better than the result in Figure 2.

Moreover, the maximum time for the gas to completely fill the silo is t=60 minutes, whereas in Figure 1 it is t=120 minutes. It can be concluded that changing the velocity from 0.002 m/s0 to 0.004 m/s affects the fumigation process: less time is needed for the gas to completely fill the 6 m silo when the velocity is increased.

Figure 4 shows that when the velocity is increased, the gas absorption by the grain becomes faster; the grain in all parts takes only 120 minutes to absorb 1 mg/L of phosphine gas, compared with Figure 2, which required 240 minutes.

Figure 5 illustrates the behavior of phosphine gas concentration when the initial concentration of fumigation gas is decreased from 10 mg/L to 5 mg/L at constant velocity V=0.002 m/s. The gas with an initial concentration of 5 mg/L needs at least 90 minutes to reach the maximum height of the silo, which is faster than the gas with an initial concentration of 10 mg/L.

Since the value of constant velocity is the same in both observations, it is clear that the initial concentration of the fumigation gas also affects the rate at which the gas moves.

Figure 6 shows that the concentration of the gas in the grain remains zero as time increases when the applied initial concentration is 5 mg/L, meaning that there is no gas absorbed by the grain.

Figure 7 represents the behavior of phosphine gas concentration when the initial concentration is 5 mg/L and the constant velocity is increased from 0.002 m/s to V=0.004 m/s. Based on the result in Figure 7, the gas moves faster to reach the top of the silo at 6 m.

At t=30 minutest=30 minutes, the concentration of 5 mg/L has achieved a height of more than 3 m, which is better than the result with constant velocity 0.002 m/s However, as time increases, the fumigation gas still does not appear in the grain.

An analytical solution to the one dimensional advection diffusion equation coupled with a linear partial differential equation with constant coefficients is obtained by the Laplace transformation method. The Stehfest algorithm is used to obtain the inverse Laplace transformation. Based on the results, it can be concluded that the initial concentration and velocity of the fumigation gas affect the rate of gas absorption. When the initial concentration is lower, no gas is absorbed by the grain. Higher values of initial concentration and velocity lead to a faster gas-absorption process, since less time is needed for the gas to fill the top surface of the silo and reach the grain.

Mohammad Jawad Qasimi: Conceptualization, Formal Analysis, investigation, Methodology, Software, visualization, Writing – original draft, Writing – review & editing

Norma Alias: Supervision, Writing – review & editing

The authors declare no conflicts of interest.