Skip to main content
Ch 14: Fluids and Elasticity
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 14: Fluids and ElasticityProblema 65a
Capitolo 14, Problema 65a

A cylindrical tank of radius 𝑅, filled to the top with a liquid, has a small hole in the side, of radius 𝓇, at distance d below the surface. Find an expression for the volume flow rate through the hole.

Guida verificata passo dopo passo
1
Start by identifying the principle governing the flow of liquid through the hole: Torricelli's Law. This law states that the speed of efflux, 𝑣, of a liquid under gravity through a small hole is given by 𝑣 = √(2𝑔ℎ), where ℎ is the height of the liquid above the hole and 𝑔 is the acceleration due to gravity.
Relate the height ℎ to the given distance 𝑑. Since the hole is located at a distance 𝑑 below the surface of the liquid, we can write ℎ = 𝑑.
The volume flow rate, 𝑄, is defined as the product of the cross-sectional area of the hole and the velocity of the liquid through the hole. The cross-sectional area of the hole is 𝐴 = π𝓇², where 𝓇 is the radius of the hole. Thus, 𝑄 = 𝐴 × 𝑣 = π𝓇² × √(2𝑔𝑑).
Combine the expressions to write the final formula for the volume flow rate: 𝑄 = π𝓇²√(2𝑔𝑑). This expression shows that the flow rate depends on the size of the hole (𝓇), the distance below the surface (𝑑), and the gravitational acceleration (𝑔).
Verify the assumptions: This derivation assumes that the hole is small enough for the liquid's velocity profile to remain uniform and that the liquid is incompressible and non-viscous. Additionally, the tank is assumed to be large enough that the liquid's surface area does not significantly change as the liquid flows out.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
13m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Bernoulli's Principle

Bernoulli's Principle states that in a fluid flow, an increase in the fluid's speed occurs simultaneously with a decrease in pressure or potential energy. This principle is crucial for understanding how fluids behave in motion, particularly in scenarios involving openings or holes, as it helps relate the pressure difference to the velocity of the fluid exiting the hole.
Video consigliato:
Percorso guidato
14:47
Diffraction with Huygen's Principle

Continuity Equation

The Continuity Equation is a fundamental principle in fluid dynamics that asserts that the mass flow rate must remain constant from one cross-section of a pipe to another. For incompressible fluids, this means that the product of the cross-sectional area and the fluid velocity is constant, which is essential for determining the flow rate through the hole in the tank.
Video consigliato:
Percorso guidato
11:08
Flow Continuity

Hydrostatic Pressure

Hydrostatic Pressure refers to the pressure exerted by a fluid at rest due to the weight of the fluid above it. In the context of the cylindrical tank, the pressure at the hole is determined by the height of the liquid column above it, which influences the velocity of the fluid exiting the hole according to Bernoulli's Principle.
Video consigliato:
Percorso guidato
17:04
Pressure and Atmospheric Pressure
Pratica correlata
Domanda del libro di testo

The 30-cm-long left coronary artery is 4.6 mm in diameter. Blood pressure drops by 3.0 mm of mercury over this distance. What are the (a) average blood speed and (b) volume flow rate in L/min through this artery?

1697
views
Domanda del libro di testo

Air flows through the tube shown in FIGURE P14.62 at a rate of 1200 cm³/s. Assume that air is an ideal fluid. What is the height h of mercury in the right side of the U-tube?

2523
views
1
rank
Domanda del libro di testo

A water tank of height h has a small hole at height y. The water is replenished to keep h from changing. The water squirting from the hole has range 𝓍. The range approaches zero as y → 0 because the water squirts right onto the ground. The range also approaches zero as y → h because the horizontal velocity becomes zero. Thus there must be some height y between 0 and h for which the range is a maximum. Find an algebraic expression for the flow speed v with which the water exits the hole at height y.

2324
views
Domanda del libro di testo

Air flows through the tube shown in FIGURE P14.63. Assume that air is an ideal fluid. What is the volume flow rate?

3093
views
1
rank
Domanda del libro di testo

20°C water flows through a 2.0-m-long, 6.0-mm-diameter pipe. What is the maximum flow rate in L/min for which the flow is laminar?

1689
views
Domanda del libro di testo

20°C water flows at 1.5 m/s through a 10-m-long, 1.0-mm-diameter horizontal tube and then exits into the air. What is the gauge pressure in kPa at the point where the water enters the tube?

1908
views