Heavy metal transport with adsorption for instantaneous and exponential attenuation of concentration

Heavy metal pollutant is a serious problem in environmental pollution, and it is very difficult to eradicate once it enters the soil. As heavy metal adsorption has been proven to occur, the heavy metal's behaviour can be modeled as a transport equation with adsorption. Previous adsorption term mostly due to the concentration alone, while in here, the desorption effect given by the rate of change of the concentration is also included. Also, the heavy metals are frequently considered to enter the soil after being dumped into the soil for a certain period of time. But, quick dumping onto the soil can introduce heavy metal instantaneously. Heavy metals entering the soil through leaching or when their concentration in the soil is influenced by chemical reactions, can all lead to the exponential decay of heavy metals entering the soil. Based on two-dimensional advection diffusion equation (ADE) with the new adsorption term, analytical solutions are obtained for the cases of instantaneous and exponential attenuation of heavy metals emission to soil by the method of Laplace transform. The results highlight the significant influence of emission type on the peak concentrations. If heavy metals are instantaneously enter the soil, the peak occurs in the range of 1-3 m radius from the point of emission on the first day, while for exponential attenuation the peak occurs close to the point of emission. Furthermore, there exists a correlation between retardation factors and heavy metal concentrations, where a decrease in retardation factors leads to an increase in heavy metal concentration. It is essential to investigate both types of heavy metals emission to provide valuable information for proper pollution management, effective environmental regulations and enforcement.