Question

Designing a Drip Dispenser for a Hydrology Experiment

In order to make laboratory measurements of water filtration and saturation rates in various types of soils under the condition of steady rainfall, a hydrologist wishes to design drip dispensing containers in such a way that the water drips out at a nearly constant rate. The containers are supported above glass cylinders that contain the soil samples (Figure 2.P.1). The hydrologist elects to use the following differential equation, based on Torricelli's principle to help solve the design problem,

A(h) dh/dt=-α a √(2gh)

In Eq. (1), h(t) is the height of the liquid surface above the dispenser outlet at time t, A(h) is the cross-sectional area of the dispenser at height h, a is the area of the outlet, and α is a measured contraction coefficient that accounts for the observed fact that the cross section of the (smooth) outflow stream is smaller than a. Note that the hydrologist is using a laminar flow model as a guide in designing the shape of the container. Forces due to surface tension at the tiny outlet are ignored in the design problem. Once the shape

1. Assume that the shape of the dispensers are surfaces of revolution so that A(h)=π[r(h)]², where r(h) is the radius of the container at height h. For each of the h-dependent cross-sectional radii prescribed below 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 ≤ t ≤ 60 :

i. r(h)=r₁,    0 ≤ h ≤ H

ii. r(h)=r₀+(r₁-r₀) h / H,    0 ≤ h ≤ H

iii. r(h)=r₀+(r₁-r₀) \sqrt{h / H}, 0 ≤ h ≤ H

iv. r(h)=r₀ exp [(h / H) ln (r₁ / r₀)],

Answer 1