A woman is draining her fish tank by siphoning the water...

A woman is draining her fish tank by siphoning the water into an outdoor drain as shown in Figure $\mathrm{P} 14.42$. The rectangular tank has footprint area $A$ and depth $h .$ The drain is located a distance $d$ below the surface of the water in the tank, where $d>>h .$ The crosssectional area of the siphon tube is $A^{\prime}$. Model the water as flowing without friction. Show that the time interval required to empty the tank is given by $$ \Delta t=\frac{A h}{A^{\prime} \sqrt{2 g d}} $$

A woman is draining her fish tank by siphoning the water into an outdoor drain as shown in Figure $\mathrm{P} 14.42$. The rectangular tank has footprint area $A$ and depth $h .$ The drain is located a distance $d$ below the surface of the water in the tank, where $d>>h .$ The crosssectional area of the siphon tube is $A^{\prime}$. Model the water as flowing without friction. Show that the time interval required to empty the tank is given by
$$
\Delta t=\frac{A h}{A^{\prime} \sqrt{2 g d}}
$$

Key Concepts

Differential Equations in Fluid Drainage

Modeling the emptying of a tank involves setting up and solving a differential equation that captures the relationship between the fluid’s outflow rate and the decreasing water height. By applying separation of variables and integrating over the appropriate limits, one can derive an expression for the time required to drain the tank, incorporating the geometry of the tank and the gravitational driving force.

Conservation of Mass

The conservation of mass in fluid dynamics is expressed through the continuity equation, which relates the flow rate through any two cross sections of a stream tube. This ensures that the rate at which fluid leaves the tank (product of the tube’s cross-sectional area and the exit velocity) is equal to the rate at which the water level in the tank decreases (product of the tank’s area and the rate of change of the height).

Torricelli's Law

Torricelli's law is a specific application of Bernoulli's principle, which shows that the speed of fluid exiting an opening under the influence of gravity is equivalent to the speed that an object would gain by free-falling from the same vertical distance. This law provides the idealized velocity expression, ?(2g?h), which is central in calculating outflow speeds in drainage problems.

Bernoulli's Principle

Bernoulli's principle is a fundamental concept in fluid dynamics that states that the total mechanical energy of the flowing fluid (including kinetic, potential, and pressure energy) remains constant along a streamline for an inviscid, incompressible fluid. This principle is used to relate the pressure and velocity of the fluid at different points, allowing for the derivation of expressions for fluid exit velocity from an orifice or tube under gravity.

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