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 pipe (Fig. \(14-50\) ); the cross-sectional area \(A\) of the entrance and exit of the meter matches the pipe's cross-sectional area. Between the entrance and exit, the fluid flows from the pipe with speed \(V\) and then through a narrow "throat" of crosssectional area \(a\) with speed \(v .\) A manometer connects the wider portion of the meter to the narrower portion. The change in the fluid's speed 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 manometer. (Here \(\Delta p\) means pressure in the throat minus pressure 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-sectional areas 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

\(V = \sqrt{\frac{2a^2(-14,000)}{1000(a^2 - A^2)}}\), and flow rate is \(A \times V\).

Step 1: Understanding Bernoulli’s Equation

Bernoulli’s equation relates the pressure, velocity, and height of a flow in two points along a streamline. The equation is given by: \[ p_1 + \frac{1}{2} \rho V_1^2 + \rho g h_1 = p_2 + \frac{1}{2} \rho V_2^2 + \rho g h_2 \] Since we assume horizontal flow (no change in height), terms with \( \rho gh \) cancel out.

Step 2: Applying Bernoulli's Principle

According to Bernoulli's principle, the velocities \(V\) and \(v\) at points with pressures \(p_1\) and \(p_2\) relate as: \[ p_1 + \frac{1}{2} \rho V^2 = p_2 + \frac{1}{2} \rho v^2 \] Rearranging for pressure difference, we get: \[ \Delta p = \frac{1}{2}\rho(v^2 - V^2) \] where \( \Delta p = p_2 - p_1 \).

Step 3: Using the Continuity Equation

The continuity equation for fluid flow through the meter is given by: \[ A V = a v \] Express \( v \) in terms of \( V \): \[ v = \frac{A}{a} V \]

Step 4: Substituting Continuity into Bernoulli’s

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

Step 5: Solving for Velocity \( V \)

Rearrange to solve for \( V \): \[ 2 \Delta p = \rho V^2 \left( \frac{A^2}{a^2} - 1 \right) \] \[ V^2 = \frac{2 \Delta p}{\rho \left( \frac{A^2}{a^2} - 1 \right)} \] Factor out \( \frac{a^2}{a^2} \): \[ V^2 = \frac{2 a^2 \Delta p}{\rho (a^2 - A^2)} \] Finally: \[ V = \sqrt{\frac{2 a^2 \Delta p}{\rho (a^2 - A^2)}} \] which matches the given expression.

Step 6: Calculating the Rate of Flow

Given: - \( A = 64 \text{ cm}^2 = 0.0064 \text{ m}^2 \) - \( a = 32 \text{ cm}^2 = 0.0032 \text{ m}^2 \) - \( \Delta p = 41 \text{ kPa} - 55 \text{ kPa} = -14 \text{ kPa} = -14,000 \text{ Pa} \) - \( \rho = 1000 \text{ kg/m}^3 \) (density of water)Plug these values into the velocity formula: \[ V = \sqrt{\frac{2 \cdot 0.0032^2 \cdot -14,000}{1000 \cdot (0.0032^2 - 0.0064^2)}} \]Calculate this to find \( V \) and then multiply by the cross-sectional area to find the flow rate.

Key Concepts

Bernoulli's equation

Bernoulli's equation is a cornerstone of fluid dynamics that describes how the pressure of a fluid varies with its speed and height. The equation is formulated as: \[ p_1 + \frac{1}{2} \rho V_1^2 + \rho g h_1 = p_2 + \frac{1}{2} \rho V_2^2 + \rho g h_2 \] where \( p \) represents pressure, \( \rho \) is the fluid's density, \( V \) represents velocity, and \( h \) represents height. In many real-world applications, such as in a venturi meter, the fluid is assumed to be flowing horizontally, removing the height terms from the equation. This assumption simplifies the equation, making it easier to relate pressure changes directly to velocity changes. This is particularly useful when determining flow speeds in pipes where vertical movement is minimal.

continuity equation

The continuity equation in fluid dynamics ensures that the mass of a fluid remains constant as it flows through a pipe. Given by the formula: \[ A_1 V_1 = A_2 V_2 \] it expresses that the product of the cross-sectional area of a pipe and the velocity of the fluid at any point is constant. This is particularly helpful in determining flow characteristics when the pipe width changes, as seen in the usage of venturi meters. By understanding how fast the fluid moves at one cross-section, you can determine its speed at another. This relationship is especially vital when using various cross-sectional areas, such as in venturi meters.

venturi meter

A venturi meter is a device used to measure the flow rate or speed of a fluid in a pipeline. This device exploits the principles of Bernoulli's equation and the continuity equation by having sections of varying cross-sectional areas. It comprises a larger entry section, a narrow throat, and a wide exit section. By measuring the pressure difference between the wide section and the throat via a manometer, it becomes possible to calculate the fluid's flow speed. The venturi meter is extensively used in engineering because it provides an effective means of measuring fluid flow with a simple mechanical design.

pressure difference

The pressure difference in fluid dynamics, commonly denoted as \( \Delta p \), is the change in pressure a fluid experiences between two points in its flow path. In a venturi meter, this difference is created when a fluid flows from a wider section to a narrower throat. The speed increase at the throat reduces pressure, creating a noticeable pressure drop. This difference is recorded using a manometer, which displays the variation as a height difference in the fluid levels in its arms. The pressure difference is crucial for calculating flow speeds, as it directly shows how energy is distributed between kinetic and potential forms.

flow rate calculation

Flow rate calculation is an essential aspect of fluid dynamics for determining how much fluid passes through a point in the system per unit time. Utilizing both the continuity equation and Bernoulli's principle, one calculates the flow rate by determining the fluid velocity and multiplying it by the cross-sectional area. For example, using the formula for velocity obtained from the Bernoulli's and continuity equations: \[ V = \sqrt{\frac{2 a^2 \Delta p}{\rho (a^2 - A^2)}} \] Subsequently, the flow rate \( Q \) can be found using: \[ Q = A \cdot V \] This calculation allows engineers to design systems that accommodate required fluid volumes and pressures. It demonstrates how theoretical principles translate into practical applications, ensuring optimal fluid management.