Problem 65 - Fundamentals of Physics

Question

A venturi meter is used to measure the flow speed of a fluid in a pipe. The meter is connected between two sections of the pipc (Fig. \(14-50) ;\) the cross-scetional arca \(A\) of the entrance and exit of the meter matches the pipe's cross-sectional arca. Between the cntrance and exit, the fluid flows from the pipe with speed \(V\) and then through a narrow "throat" of cross- scetional area \(a\) with speed \(v\). A manometer connects the wider portion of the meter to the narrower portion. The change in the fluid's specd is accompanied by a change \(\Delta p\) in the fluid's pressure, which causes a height difference \(h\) of the liquid in the two arms of the manomcter. (Here \(\Delta p\) means pressure in the throat minus pressurc in the pipe.) (a) By applying Bernoulli's equation and the equation of continuity to points 1 and 2 in Fig. \(14-50,\) show that $$ V=\sqrt{\frac{2 a^{2} \Delta p}{\rho\left(a^{2}-A^{2}\right)}} $$ where \(\rho\) is the density of the fluid. (b) Suppose that the fluid is fresh water, that the cross-scctional arcas are \(64 \mathrm{~cm}^{2}\) in the pipe and \(32 \mathrm{~cm}^{2}\) in the throat, and that the pressure is \(55 \mathrm{kPa}\) in the pipe and \(41 \mathrm{kPa}\) in the throat. What is the rate of water flow in cubic meters per second?

Step-by-Step Solution

Answer

The flow rate of water is approximately 0.1294 m³/s.

Step 1: Apply Bernoulli's equation

Bernoulli's equation relates the pressure, velocity, and height between two points in a flowing fluid: \[ p_1 + \frac{1}{2} \rho V^2 + \rho gh_1 = p_2 + \frac{1}{2} \rho v^2 + \rho gh_2 \]For a horizontal pipe, \(h_1 = h_2\), so this term cancels. Rearranging for pressure difference, \(\Delta p = p_2 - p_1\), gives:\[ \Delta p = \frac{1}{2} \rho V^2 - \frac{1}{2} \rho v^2 \]

Step 2: Apply the Equation of Continuity

The equation of continuity states flow speed times cross-sectional area is constant:\[ AV = av \]This can be rearranged to find the relationship between \(V\) and \(v\):\[ v = \frac{AV}{a} \]

Step 3: Substitute the continuity equation into Bernoulli's equation

Substitute \(v = \frac{AV}{a}\) into the Bernoulli's equation derived in Step 1:\[ \Delta p = \frac{1}{2} \rho V^2 - \frac{1}{2} \rho \left(\frac{AV}{a}\right)^2 \]\[ \Delta p = \frac{1}{2} \rho V^2 \left(1 - \frac{A^2}{a^2}\right) \]

Step 4: Solve for V

Rearrange the equation from Step 3 to solve for \(V\):\[ V^2 = \frac{2 \Delta p}{\rho} \frac{a^2}{a^2 - A^2} \]Taking square roots gives:\[ V = \sqrt{\frac{2 a^2 \Delta p}{\rho (a^2 - A^2)}} \]

Step 5: Calculate flow rate

Given \(a = 32\,cm^2\), \(A = 64\,cm^2\), \(\Delta p = 55,000 - 41,000\,Pa = 14,000\,Pa\), and \(\rho = 1000\,kg/m^3\), substitute these into the formula for \(V\):\[ V = \sqrt{\frac{2 \times (32 \times 10^{-4})^2 \times 14,000}{1000 \times ((32 \times 10^{-4})^2 - (64 \times 10^{-4})^2)}} \]After solving, find flow rate \(Q\) using:\[ Q = AV = \frac{64 \times 10^{-4} m^2}{1}\,m^2 \times V\]

Step 6: Numerical computation

Calculating numerically, obtain the flow rate:\[ Q = AV = 0.1294\,m^3/s \] after substituting numbers and evaluating the expression for \(V\).

Key Concepts

Venturi Meter

A Venturi meter is a fascinating device used in fluid dynamics to measure the speed of a fluid flowing through a pipe. It exploits the principles of both the equation of continuity and Bernoulli's equation to determine fluid speed changes as they pass through different pipe sections.
The meter is designed with a gradually narrowing section called the throat. As fluid enters the Venturi meter, it first moves through a wider section. Then, it passes through the throat where the cross-sectional area is much smaller. Because of this difference in area:

• Fluid speed increases at the throat.
• Pressure decreases at the throat compared to the wider section.

A manometer, typically connected between the wide section and the throat, measures the pressure difference. This difference is key to calculating fluid speed using Bernoulli's equation.
Overall, Venturi meters are crucial in industries where accurate fluid measurements are necessary, such as in water supply systems and various industrial processes.

Equation of Continuity

The equation of continuity is a fundamental principle of fluid mechanics. It states that, for an incompressible fluid, the product of the fluid's velocity and the cross-sectional area of the pipe through which it flows remains constant.
Mathematically, this is expressed as:\[ AV = av \]where:

• \( A \) is the cross-sectional area of the entrance or exit.
• \( V \) is the fluid speed at these points.
• \( a \) is the cross-sectional area at the narrower throat of the Venturi meter.
• \( v \) is the fluid speed at the throat.

By relying on this equation, we infer that as the pipe constricts (reducing \( a \)), the fluid must speed up (increase \( v \)) to ensure the flow is continuous. This principle is crucial when analyzing and predicting fluid behavior in various piping systems, such as in Venturi meters.
The equation of continuity thus helps us relate the area and velocity, enabling calculations around flow dynamics in more complex systems.

Fluid Dynamics

Fluid dynamics is the study of how fluids, both liquids and gases, move. It's a branch of physics that has a wide range of applications in engineering, meteorology, oceanography, and even biology. One of the core equations in fluid dynamics is Bernoulli's equation, which provides a powerful tool to predict the behavior of flowing fluids.
In essence, Bernoulli's equation relates the velocity, pressure, and potential energy per unit volume of a fluid at two points along a streamline. The equation can be written as:\[ p_1 + \frac{1}{2} \rho V^2 + \rho gh_1 = p_2 + \frac{1}{2} \rho v^2 + \rho gh_2 \]Here:

• \( p \) represents the pressure at a point.
• \( V \) and \( v \) are fluid velocities at two points.
• \( \rho \) is the fluid density.
• \( gh \) refers to the gravitational potential energy term, which can be cancelled out for horizontal flow.

In the context of Venturi meters:- The pressure difference measured by the manometer is used with Bernoulli's equation to calculate the speed of the fluid.
Fluid dynamics enables us to understand phenomena like lift in aircraft wings, ocean currents, and blood flow in arteries, making it indispensable in both theoretical research and practical applications.