In a wastewater facility, the effluent is treated both chemically and biologically. For biological treatment, water needs to be passed through a tank loaded with microorganisms. The tank has a maximum volume of $1000L.$ When the tank reaches that point, the feeds are diverted to another identical $1000$L tank to allow the first to process and drain. Water enters the tank at a flowrate of $30\frac{L}{h}$ with a substrate concentration of $1\frac{g}{L}.$ The tank is also taking runoff from a nearby farm with a much higher substrate concentration of $300\frac{g}{L}$ at a flowrate of $3\frac{L}{h}.$ The organisms can consume an estimated $100\frac{g}{h}$ of substrate. It is estimated that a substrate concentration higher than $18\frac{g}{L}$ is unsafe for microbes. At this point, the flows must be diverted to the new tank, even if the level has not reached $1000L.$ If the tank is initial filled with $100L$ of fresh water before use, do the following:

a$.$ Sketch the system

b$.$ Find the equations governing the tank behavior for substrate concentration and volume in the tank

c$.$ Develop an equation for estimating tank switch time for the system described