“…It is clear that near the source ( ), as increases the value of increases and the value of decreases. This result agrees with that obtained by 11 , 13 . Figure 5 The variations of and with for the values for the steady state analytical solutions given by Eqs.…”
Section: Resultssupporting
confidence: 94%
“…The coupled non-linear PDEs modelling pollutant and DO concentrations in a river can be written as 4 , 10 , 11 : where is the retardation factor, is the cross-section area of the river which is taken to be constant, is the pollutant concentration, is the DO concentration, is the diffusion coefficient of the pollutant concentration, is the diffusion coefficient of DO, is the river’s velocity, is the degradation rate coefficient for pollution at a constant temperature, is the degradation rate coefficient for DO at a constant temperature, is the half-saturated oxygen demand concentration for pollution decay, is the rate of the injected pollutant discharge, is an arbitrary constant of pollution source terms, is the mass transfer of oxygen from air to water and is the saturated oxygen concentration.…”
Section: The Governing Equationsmentioning
confidence: 99%
“…The flow domain parameters are considered to be spatially and temporally dependent so we have: where a is a heterogeneity parameter and b is an arbitrary dimensionless constant 11 , is the initial retardation factor, is the initial pollution diffusion coefficient, is the initial DO diffusion coefficient, is the initial velocity component, is the initial rate of the injected pollutant discharge, is the initial degradation rate coefficient for pollution at a constant temperature, is the initial degradation rate coefficient for DO at a constant temperature, is the initial oxygen mass transfer, (day -1 ) is a constant. We choose such that for or we have which ensures that the nature of the initial condition doesn’t change in the new time domain (day) which is defined by 24 : …”
Section: The Governing Equationsmentioning
confidence: 99%
“…Manitcharoen and Pimpunchat 10 acquired analytical and numerical solutions of pollution concentration with uniformly and exponentially increasing forms of sources. Yadav and Kumar 11 solved the space and time-dependent ADE and obtained an analytical solution of two‑dimensional conservative solute transport in a heterogeneous porous medium for varying input point sources. Saleh et al 12 solved the one-dimensional ADE describing exponential variations in pollutant concentration numerically and analytically and captured results regarding the remediation of pollution by releasing clean water.…”
The coupled equations of pollution and aeration for flow in a river were studied under generalized assumptions in terms of parameter dependency on space and time, as well as general boundary constraints. An analytical solution was obtained in the steady-state case. Also, the system was solved in its unsteady state numerically in a dimensionless form using the finite difference scheme. The effect of different parameters controlling the flow (such as the velocity, Peclet number, injected pollutants, and so on…) was studied. Investigations indicate that the special cases of the proposed model (i.e., uniform distribution of pollutant and Dissolved Oxygen concentrations, and zero injected pollutants along the river) give results that agree with the previous studies. This simple model helps in understanding the behavior of the pollution-aeration process and its relation to the injected pollution along a river and its effect on fish survival. A simple procedure was discussed in this study to help in regulating farming, industrial, and urban practices and impose restrictions if necessary. This study determines with accuracy the intervals of the river at which fish can survive at a given time, as well as the maximum amount of pollutants allowed to be injected along the river for fish survival.
“…It is clear that near the source ( ), as increases the value of increases and the value of decreases. This result agrees with that obtained by 11 , 13 . Figure 5 The variations of and with for the values for the steady state analytical solutions given by Eqs.…”
Section: Resultssupporting
confidence: 94%
“…The coupled non-linear PDEs modelling pollutant and DO concentrations in a river can be written as 4 , 10 , 11 : where is the retardation factor, is the cross-section area of the river which is taken to be constant, is the pollutant concentration, is the DO concentration, is the diffusion coefficient of the pollutant concentration, is the diffusion coefficient of DO, is the river’s velocity, is the degradation rate coefficient for pollution at a constant temperature, is the degradation rate coefficient for DO at a constant temperature, is the half-saturated oxygen demand concentration for pollution decay, is the rate of the injected pollutant discharge, is an arbitrary constant of pollution source terms, is the mass transfer of oxygen from air to water and is the saturated oxygen concentration.…”
Section: The Governing Equationsmentioning
confidence: 99%
“…The flow domain parameters are considered to be spatially and temporally dependent so we have: where a is a heterogeneity parameter and b is an arbitrary dimensionless constant 11 , is the initial retardation factor, is the initial pollution diffusion coefficient, is the initial DO diffusion coefficient, is the initial velocity component, is the initial rate of the injected pollutant discharge, is the initial degradation rate coefficient for pollution at a constant temperature, is the initial degradation rate coefficient for DO at a constant temperature, is the initial oxygen mass transfer, (day -1 ) is a constant. We choose such that for or we have which ensures that the nature of the initial condition doesn’t change in the new time domain (day) which is defined by 24 : …”
Section: The Governing Equationsmentioning
confidence: 99%
“…Manitcharoen and Pimpunchat 10 acquired analytical and numerical solutions of pollution concentration with uniformly and exponentially increasing forms of sources. Yadav and Kumar 11 solved the space and time-dependent ADE and obtained an analytical solution of two‑dimensional conservative solute transport in a heterogeneous porous medium for varying input point sources. Saleh et al 12 solved the one-dimensional ADE describing exponential variations in pollutant concentration numerically and analytically and captured results regarding the remediation of pollution by releasing clean water.…”
The coupled equations of pollution and aeration for flow in a river were studied under generalized assumptions in terms of parameter dependency on space and time, as well as general boundary constraints. An analytical solution was obtained in the steady-state case. Also, the system was solved in its unsteady state numerically in a dimensionless form using the finite difference scheme. The effect of different parameters controlling the flow (such as the velocity, Peclet number, injected pollutants, and so on…) was studied. Investigations indicate that the special cases of the proposed model (i.e., uniform distribution of pollutant and Dissolved Oxygen concentrations, and zero injected pollutants along the river) give results that agree with the previous studies. This simple model helps in understanding the behavior of the pollution-aeration process and its relation to the injected pollution along a river and its effect on fish survival. A simple procedure was discussed in this study to help in regulating farming, industrial, and urban practices and impose restrictions if necessary. This study determines with accuracy the intervals of the river at which fish can survive at a given time, as well as the maximum amount of pollutants allowed to be injected along the river for fish survival.
“…In addition, Derya and Yetim [16] focused on the advection-diffusion equation with the Atangana-Baleanu derivative, which is a fractional derivative. Furthermore, Yadav and Kumar [17] found that the concentration patterns and levels are influenced by the varying nature of the input concentration.…”
This paper studies a two-dimensional transport model to investigate the behavior of heavy metal migration in porous media, specifically considering their transport in soil. The model takes into consideration the combination of adsorption term as well as instantaneous injection at the boundary. Also, the model accounts for both longitudinal and transverse movements, providing a comprehensive understanding of heavy metal transport phenomena. In order to obtain the analytical solution Laplace transform has been implemented. It is found that the peak concentration of heavy metals is found to be greatly affected by the changes in the instantaneous injection value. Additionally, a correlation is observed between retardation factors and heavy metal concentrations, with a decrease in retardation factors resulting in an increase in heavy metal concentration.
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