Continuous System Simulation $($Water Reservoir System$)$

Consider the water reservoir system in which two inputs are river infl ow and direct rainfall and one output is demand of water for irrigation and for other purposes, as shown in Figure $1.$ Some of the water is lost in the form of seepage and evaporation. It is known that the evaporation loss depends upon the surface area and the atmospheric temperature and the seepage loss depends upon the volume water stored in the reservoir. Develop a model to determine the volume of water stored in the reservoir so that it does not spill over and there is no shortage in summer.

FIGURE $1$

Water reservoir system.

Data:

Coefficient of evaporation: $0\mathrm{.}1$

Water from Rain: $[100400400200200100505050100150200]$

Initial Demand: $\{00000050300500000]$

Strategy

As given in the problem, seepage loss is function of volume.

Seepage loss $=f(volume)$

Evaporation loss is function of area of exposed surface and coefficient of evaporation.

Evaporation loss $=f(surface$ area, $C_{\text{e}\text{r}\text{a}\text{p}})$

The state variable is the volume which changes with time.

Model development

Total input $V_{in}=$ Rain $+$ river flow

Gross volume $=V_{in}+$ earlier volume

Total loss $=$ Seepage loss $+$ evaporation loss

Net volume $=$ Gross volume $-$ total loss $-$ Demand

If Demand $$ Net volume then reservoir is dry and shortage of water $(=$ Dem$-($:$V_{\text{n}\text{e}\text{t}}\}$

Otherwise Diff $=$ Demand $-V_{\text{n}\text{e}\text{t}}$

If Diff $>$ Cap then Spill over