Question

A Venturi tube has a pipe diameter of \(25.4 \mathrm{~cm}\) and a throat diameter of \(11.3 \mathrm{~cm}\). The water pressure in the pipe is \(57.1 \mathrm{kPa}\) and in the throat is \(32.6 \mathrm{kPa}\). Calculate the volume flux of water through the tube.

Short answer

To find the volume flux, first calculate the speed of water in pipe and in the throat using Bernoulli equation, considering energy conservation. After getting the two speeds, calculate the volume flux of the water using the continuity equation, considering the constancy of the mass flow rate in the pipe.

Step-by-step solution

Decoding the problem statement

In this problem, diameters of the pipe and throat are given as \(25.4 \mathrm{cm}\) and \(11.3\mathrm{cm}\) respectively. The corresponding pressures in the pipe and throat are given as \(57.1\mathrm{kPa}\) and \(32.6\mathrm{kPa}\) respectively. Convert these given diameters and pressure to SI units before attempting to solve for the problem. Diameter in SI units is meter (m) and pressure in SI units is Pascal (Pa).

Calculate the speed of water in the pipe and in the throat

Use the Bernoulli equation to find the speed in the pipe and at the throat. Bernoulli equation states that the total mechanical energy in a fluid system is conserved (i.e. energy cannot be created or lost), hence, the sum of pressure energy, kinetic energy and potential energy per unit volume remains constant. Thus, applying Bernoulli equation, equate the total mechanical energy in the pipe to that at the throat to determine the speed at both ends of the venturi tube.

Calculate the volume flux of water

Use the continuity equation, which states that the mass flow rate must remain constant in the pipe in absence of any sinks or sources. Thus, the product of cross-sectional area of the pipe or throat and speed at these respective locations will remain constant. Calculate the cross-sectional areas using the given diameters (Area = \(πd^2/4\)). Hence, equate mass flow rate at the pipe to that at the throat to calculate the volume flux.

Key concepts

Bernoulli's Principle

Bernoulli's Principle is a fundamental concept in fluid dynamics. It describes the behavior of a moving fluid and is essential for understanding how pressure, velocity, and height interrelate in fluid systems. According to Bernoulli's Principle, an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. This principle can be summed up in the equation: \[ P + \frac{1}{2} \rho v^2 + \rho gh = ext{constant} \]where:

• \(P\) is the fluid pressure,
• \(\rho\) is the fluid density,
• \(v\) is the fluid velocity,
• \(g\) is the acceleration due to gravity,
• \(h\) is the height above a reference point.

In the Venturi tube problem, Bernoulli's equation helps us find the velocity of the water both in the wider section and in the narrow section (the throat). By applying this principle, you can equate the sum of pressure energy, kinetic energy, and potential energy at two different points in the flow system. This allows the calculation of changes in speed or pressure when a fluid is passing through different diameters.

Continuity Equation

The Continuity Equation is a crucial concept in fluid dynamics. It ensures that the mass of fluid remains conserved as it flows through a system, even when the system varies in diameter. This principle is especially important for understanding flow in systems that include changing cross-sections, like pipes and Venturi tubes.
The equation is expressed as: \[ A_1 v_1 = A_2 v_2 \]where:

• \(A_1\) and \(A_2\) are the cross-sectional areas at two points in the system,
• \(v_1\) and \(v_2\) are the fluid velocities at these points.

In simpler terms, the same amount of fluid must pass through any two points in the system over a given time, so the product of area and velocity at one point must equal that at another point.
For the Venturi tube, this continuity ensures that any change in diameter results in a compensating change in velocity, which allows us to compute the volume flux effectively.

Venturi Effect

The Venturi Effect is a phenomenon that occurs when a fluid flows through a constricted section of a pipe. As the fluid enters a narrower section, it speeds up, and the pressure decreases. This effect is a direct consequence of Bernoulli's Principle and the Continuity Equation in action. The change in velocity and pressure in the Venturi tube is used in various applications, from measuring flow rates to aircraft instrumentation. In the Venturi tube problem, the Venturi Effect allows you to solve for differences in speed and pressure between the wider pipe and the narrow throat. This effect results from carefully designed changes in the tube's cross-sectional area, guiding the fluid dynamics principles.
Using these principles, the Venturi tube provides empirical data that demonstrates how fluid speed and pressure vary inversely with changes in diameter. The Venturi tube is thus a practical application of both Bernoulli's Principle and the Continuity Equation, illustrating foundational fluid dynamic concepts in a tangible form.