Балкански 2024, Теория — Задача 4. Water purification by settling

Concours Général de Physique «Minko Balkanski» — 8 août 2024 · 8 август 2024 г.

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Условие

The aim of this problem is to study in a simplified way some stages of the treatment of river water in order to make it drinkable. The largest debris can easily be eliminated by filtration on a grid, but it seems more difficult to remove small or dissolved particles. In chronological order, some stages of the water's journey in the treatment plant: firstly, we are concerned with purification by decantation in a basin which eliminates particles larger than ten micrometers and, after coagulation, colloidal particles smaller than a few micrometers. Since some molecules cannot be eliminated by simple decantation, molecular adsorption by powdered activated carbon must be used. Finally, we deal with the problems of balancing the calcification of the water.

Decantation is a process that allows the elimination of particles suspended in water through the action of gravity. The particles settle at the bottom of a basin and form a sludge that can be evacuated later. It will be assumed that once deposited, the particles do not return to suspension. The water to be decanted fills a basin of depth $H$; the vertical axis, oriented upwards, is noted $Oz$ and its origin is placed at the bottom of the basin.

The kinematic viscosity of water at $10^{\circ}C$ is $\nu = 1.31 \times 10^{-6}SI$.

1. Give the expression for the force experienced by a solid of volume $V$ immersed in a liquid of density $\mu$. Under what conditions will a particle of density $\mu_{sol}$ be able to settle on the bottom of the settling basin, assuming that the water in the basin is immobile?

2. A spherical particle of radius $R$, moving at speed $\overrightarrow{v}$ in a stationary fluid, is subjected to a friction force, called the Stokes force, given by $\overrightarrow{F_S} = -6\pi\mu\nu R\overrightarrow{v}$, where $\mu$ and $\nu$ are the density and kinematic viscosity of the fluid, respectively. What is the dimension of $\nu$? Show that the particle reaches a limiting speed $v_d$, called the settling speed, which will be expressed as a function of $R$, $\nu$, $g$ and the density $d = \frac{\mu_{sol}}{\mu}$.

3. Calculate the settling speed of sand particles of radii $R = 50\mu m$, $5\mu m$ and $0.5\mu m$, and density $\mu_{sol} = 2.65gcm^{-3}$. How long do they take to travel one meter? What should we think of settling as a means of eliminating suspended particles?

Решение

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Оригинал в Архива: CGP_2024.pdf