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 Pl4.70 (page 416$)$ . 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 cross- sectional area of the siphon tube is $A^{\prime} .$ Model the water as flowing without friction. (a) Show that the time interval required to empty the tank is $$ \Delta t=\frac{A h}{A^{\prime} \sqrt{2 g d}} $$ (b) Evaluate the time interval required to empty the tank if it is a cube 0.500 $\mathrm{m}$ on each edge, taking $A^{\prime}=2.00 \mathrm{cm}^{2}$ and $d=10.0 \mathrm{m}$

A woman is draining her fish tank by siphoning the water into an outdoor drain as shown in Figure Pl4.70 (page 416$)$ . 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 cross- sectional area of the siphon tube is $A^{\prime} .$ Model the water as flowing without friction. (a) Show that the time interval required to empty the tank is
$$
\Delta t=\frac{A h}{A^{\prime} \sqrt{2 g d}}
$$
(b) Evaluate the time interval required to empty the tank if it is a cube 0.500 $\mathrm{m}$ on each edge, taking $A^{\prime}=2.00 \mathrm{cm}^{2}$ and $d=10.0 \mathrm{m}$

Key Concepts

Bernoulli's Principle

This principle states that for an ideal fluid flowing along a streamline, the sum of its pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant. In problems like siphoning from a tank, Bernoulli’s principle is used to relate the height (and therefore potential energy) of the water to the speed of the water exiting the system, leading to the derivation of the outflow speed.

Torricelli's Law

Derived from Bernoulli’s principle, Torricelli’s law gives the speed of efflux of a fluid under gravity from an orifice as the square root of twice the gravitational acceleration times the height of the fluid above the orifice. In this context, it is pivotal in determining the velocity of water exiting the siphon, which is then used to calculate the flow rate.

Continuity Equation

The continuity equation in fluid dynamics expresses the conservation of mass for incompressible fluids, stating that the product of cross-sectional area and velocity remains constant along a streamline. This concept allows one to relate the large cross-sectional area of the tank to the much smaller area of the siphon tube, thereby linking the decrease in water level to the flow rate through the tube.

Flow Rate and Volume Discharge

The flow rate is defined as the volume of fluid passing a given surface per unit time and is calculated as the product of the cross-sectional area and the velocity of the fluid. In draining problems, integrating the flow rate over time provides the total time required for emptying the container. This concept underpins the derivation of the formula for the emptying time of the tank.

Ideal (Frictionless) Flow Assumption

Assuming that the fluid flows without friction simplifies the analysis by eliminating viscous losses and turbulence effects. This assumption allows the direct application of Bernoulli's equation and Torricelli’s law without additional correction factors, leading to an exact analytical expression for the time required to empty the tank.

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