Consider a tlywheel storage system working on a daily cycle. The power supply received by the storage system is given by the function:

$W_{in}^{}(t)=\{\begin{array}{ccc}0& \text{f}\text{o}\text{r}t=0\text{c}\text{d}\text{o}\text{t}\text{s}7\text{a}\text{n}\text{d}17\text{d}\text{o}\text{t}\text{s}& 24\\ 2(t-7)kW& \text{f}\text{o}\text{r}t=7\text{c}\text{d}\text{o}\text{t}\text{s}& 12\\ 2(17-t)& \text{f}\text{o}\text{r}t=12\text{c}\text{d}\text{o}\text{t}\text{s}& 17\end{array}$

where the time, $t,$ is in hours $(t=0$ and $t=24$ both represent midnight, $t=12$ represents noon$).$

The power usage from the storage system is given by the function

$W_{\text{o}\text{u}\text{t}}^{}(t)=\{\begin{array}{ccc}0& \text{f}\text{o}\text{r}t=10\text{c}\text{d}\text{o}\text{t}\text{s}& 24\\ C=& \text{c}\text{o}\text{n}\text{s}\text{t}\text{f}\text{o}\text{r}t=0\text{c}\text{d}\text{o}\text{t}\text{s}& 10\end{array}$

Where C is such that the energy balance is maintained over any $24$ hour period. All efficiencies are assumed to be $100\%.$

a$)$ Calculate C and show on the same graph $($printed$,$ generated using software$)$ the following $3$ curves: power supply, power usage, and net power to the storage system.

b$)$ The ratio of the maximum to the minimum energy stored in the flywheel is assumed to be $\frac{E_{\text{m}\text{a}\text{x}}}{E_{\text{m}\text{i}\text{n}}}=6($this is a design constraint$).$ Calculate the values for the maximum and the minimum energy stored in the flywheel $E_{\text{m}\text{a}\text{x}}=?kWh,E_{\text{m}\text{i}\text{n}}=?$ kWh $.$ Show on a graph the energy stored in the flywheel as a function of time for $t=0$cdots$24$ hours.

c$)$ The flywheel is a solid disc $($flat unpierced disc$)$ with a diameter of $2$ meters and is made of steel with a density of $7800k\frac{g}{m^3}$ and tensile strength of $550$ MPa $.$ For the flywheel, calculate the specific energy density $(\frac{E_{\text{r}\text{o}\text{t}\text{a}\text{t}\text{i}\text{o}\text{n}\text{a}\text{l}\text{,}\text{m}\text{a}\text{x}}}{m}),$ mass $(m),$ thickness $(b),$ maximum rotational velocity $({}_{\text{m}\text{a}\text{x}})$ and $2$ minimum rotational velocity, $({}_{\text{m}\text{i}\text{n}}).$