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 $\mathrm{P} 14.74$ . 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. 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

Bernoulli’s Principle

This principle states that for an ideal, incompressible, non?viscous fluid in steady flow, the total mechanical energy along a streamline is constant. In the context of draining a tank, it relates the pressure energy and gravitational potential energy in the fluid to the kinetic energy of the exiting water, allowing us to calculate the velocity at which the fluid exits the siphon.

Torricelli’s Law

Derived from Bernoulli’s principle, Torricelli’s law gives the exit speed of a fluid flowing under the force of gravity from an orifice. It states that the speed of efflux is proportional to the square root of twice the gravitational acceleration times the height of the fluid above the exit point. This law is crucial for determining the flow rate in the draining process.

Continuity Equation

The continuity equation is a statement of the conservation of mass in fluid mechanics, asserting that the mass flow rate must be constant along a streamline for an incompressible fluid. It links the cross?sectional areas and the velocities of the fluid in different regions of the flow, permitting the relationship between the velocity in the tank and the velocity in the siphon tube.

Differential Equations in Draining Problems

When analyzing the draining of a tank, the changing height of the water over time can be modeled with a differential equation derived from the relationship between the flow rate and the volume of fluid in the tank. Solving this differential equation, often through separation of variables, allows one to determine the time required for the fluid level to drop to a specific level or to be completely drained.

Assumptions of Ideal Fluid Flow

To simplify the analysis of fluid dynamics problems such as the draining of a tank, it is common to assume that the fluid is ideal—meaning it is incompressible and non-viscous—and that the flow is frictionless and steady. These assumptions enable the direct application of Bernoulli’s principle and the continuity equation without having to account for energy losses due to viscosity or turbulence.

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