Chapter 8.P1, Problem 1P

Assume that the shape of the dispensers are surfaces of revolution so that $A\left(h\right)=\pi {\left[r\left(h\right)\right]}^2$, where $r\left(h\right)$ is the radius of the container at height $h$. For each of the $h$-dependent cross-sectional radii prescribed in (i)-(v):

(a) Create a surface plot of the surface of revolution, and

(b) Find numerical approximations of solutions of Eq. (1) for $0\le t\le 60$:

i     $r\left(h\right)=r_1,0\le h\le H$

ii    $r\left(h\right)=r_0+\left(r_1-r_0\right)\frac{h}{H},0\le h\le H$

iii   $r\left(h\right)=r_0+\left(r_1-r_0\right)\sqrt{\frac{h}{H}},0\le h\le H$

iv   $r\left(h\right)=r_0\exp \left[\left(\frac{h}{H}\right)\ln \left(\frac{r_1}{r_0}\right)\right],0\le h\le H$

v     $r\left(h\right)=\frac{r_0{r}_{1}H}{r_{1}H-\left(r_1-r_0\right)h},0\le h\le H$

Use the parameter values specified as follows:

$\begin{array}{l}{r}_0=0.1\text{ft}\\ {\text{r}}_1=1\text{ft}\\ \alpha =0.6\\ a=0.1{\text{ft}}^2\end{array}$

In addition, use the initial conditions $h\left(0\right)=H$, where the initial height $H$ of water in each of the containers is determined by requiring that the initial volume of water satisfies

$V\left(0\right)={\displaystyle \underset{0}{\overset{H}{\int }}\pi r^2\left(h\right)}dh=1{\text{ft}}^3$

Determine the qualitative shape of the container such that the output flow rate given by the right hand side of Eq. (1), $F_R\left(t\right)=\alpha a\sqrt{2gh\left(t\right)}$, varies slowly during the early stages of the experiment. As a design criterion, consider plotting the ratio $R=\frac{F_R\left(t\right)}{F_R\left(0\right)}$ for $0\le t\le 60$, where values of $R$ near $1$ are most desirable. Based on the results of your computer experiments, sketch the shape of what a suitable container should look like.