Mathematical Model For Delayed Particle Flow In Cascading Sub-Surface Water Reservoirs With Validation On River Nyakomisaro In Kisii County

Abstract

Pollution of sub-surface water reservoirs mainly rivers and streams, through contaminated water point sources was studied. Possible potential sources of pollution to these point sources include municipal wastes, septic systems, landfills, uncontrolled hazardous wastes and sewage storage tanks. These present a unique dynamic system of water reservoir pollution which was described using mixing problem processes. A conceptual perspective of mixing problem process in water tanks was applied to model delayed particle flow in water reservoir systems. The concentration ( ) of pollutants was expressed as a function of the inflow and outflow rates using the principle for the conservation of mass. Systems of Ordinary Differential Equations (ODE) were then generated to describe contaminant flow and transport through cascading water reservoirs in terms of the concentration of pollutant over time. The major assumption made in modeling of mixing problems using tanks is that mixing is instantaneous. Practical realities dictate that mixing cannot occur instantaneously throughout the tank. So as to accommodate these realities, the study refined the ODEs generated from principles of mixing problems, into a system of Delayed Differential Equations (DDEs) so that the concentration of pollutant leaving the reservoir at time would be equal to the average concentration at some earlier instant, for the delay . The formulated model is a mathematical discrete time delay model which was used to describe the dynamics of sub-surface water reservoir pollution in a river. The formulated model was validated on river Nyakomisaro in Kisii County. The system of DDEs from this validation was solved numerically on MATLAB using dde23 solver. From the results obtained, DDEs generated longer time series solutions (characteristic curves) than the corresponding ODEs in the same reservoir indicating that time necessary for particle flow through water reservoir is underestimated if ordinary differential equations are used to describe particle flow in SWR.