Suppose water is leaking from a tank through a circular hole...

Suppose water is leaking from a tank through a circular hole of area $A_{h}$ at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to $c A_{h} \sqrt{2 g h},$ where $c(0 < c < 1)$ is an empirical constant. Determine a differential equation for the height $h$ of water at time $t$ for the cubical tank shown in Figure $1.3 .12 .$ The radius of the hole is 2 in., and $g=32 \mathrm{ft} / \mathrm{s}^{2}$.

Key Concepts

Formulation of Differential Equations

Differential equations are used to model the rate of change of physical quantities. In the context of a leaking tank, the change in water height over time is governed by a differential equation that arises from equating the rate of loss of water volume (due to outflow through the hole) with the rate of decrease in the tank’s water volume as a function of its height. This formulation involves techniques from calculus, particularly related rates and integration.

Volume-Height Relationship in Tanks

For tanks with a uniform or known geometric shape, there is a specific relationship between the volume of water and its height. Understanding this relationship is crucial for translating a change in volume (due to outflow) into a change in the height of water, thereby allowing the formulation of a differential equation that describes how the water level varies over time.

Empirical Discharge Coefficient

The discharge coefficient, typically denoted by c, accounts for real?world factors such as friction and the contraction of the streamline as water exits an orifice. Since these effects reduce the ideal flow rate, the coefficient (a value between 0 and 1) is introduced into the model to adjust the theoretical result to better match observed behavior.

Conservation of Mass

This principle states that the mass of water in the tank changes only by the water flowing out. In a closed system like a leaking tank, the rate of decrease in water volume inside the tank is equal to the outflow rate. This concept is fundamental in setting up differential equations to relate the changing water height to the volume lost over time.

Torricelli’s Law

Torricelli’s law provides that the speed of fluid flowing out of an orifice under gravity is proportional to the square root of the height of the fluid above the hole. This relationship is used to determine the outflow velocity and, when combined with the area of the hole, helps model the rate at which water exits the tank.

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