Ch 12: Fluid Mechanics

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 12: Fluid Mechanics

Problem 44bc

Chapter 12, Problem 44bc

A soft drink (mostly water) flows in a pipe at a beverage plant with a mass flow rate that would fill 220 0.355-L cans per minute. At point 2 in the pipe, the gauge pressure is 152 kPa and the cross-sectional area is 8.00 cm². At point 1, 1.35 m above point 2, the cross-sectional area is 2.00 cm². Find the (b) volume flow rate. (c) flow speeds at points 1 and 2.

Verified step by step guidance

To find the volume flow rate, first calculate the total volume of soft drink flowing per minute. Since 220 cans are filled per minute and each can has a volume of 0.355 L, multiply these values to get the total volume per minute: \( V = 220 \times 0.355 \text{ L} \).

Convert the volume flow rate from liters per minute to cubic meters per second. Use the conversion factor \( 1 \text{ L} = 0.001 \text{ m}^3 \) and \( 1 \text{ min} = 60 \text{ s} \).

To find the flow speeds at points 1 and 2, use the continuity equation, which states that the volume flow rate \( Q \) is constant throughout the pipe: \( Q = A_1 v_1 = A_2 v_2 \). Here, \( A_1 \) and \( A_2 \) are the cross-sectional areas at points 1 and 2, respectively, and \( v_1 \) and \( v_2 \) are the flow speeds.

Calculate the flow speed at point 2 using the formula \( v_2 = \frac{Q}{A_2} \), where \( A_2 = 8.00 \text{ cm}^2 \). Convert \( A_2 \) to square meters by using the conversion factor \( 1 \text{ cm}^2 = 0.0001 \text{ m}^2 \).

Calculate the flow speed at point 1 using the formula \( v_1 = \frac{Q}{A_1} \), where \( A_1 = 2.00 \text{ cm}^2 \). Again, convert \( A_1 \) to square meters using the same conversion factor.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Continuity Equation

The continuity equation in fluid dynamics states that the mass flow rate must remain constant from one cross-section of a pipe to another, assuming incompressible flow. It is expressed as A1V1 = A2V2, where A is the cross-sectional area and V is the flow speed. This principle helps determine the flow speeds at different points in the pipe.

Bernoulli's Principle

Bernoulli's principle relates the pressure, velocity, and height in a flowing fluid, stating that an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or potential energy. It is crucial for understanding how pressure and velocity change between two points in the pipe, especially when elevation changes are involved.

Volume Flow Rate

Volume flow rate is the volume of fluid that passes through a given surface per unit time, typically measured in cubic meters per second (m³/s). It can be calculated using the mass flow rate and the density of the fluid, or directly from the product of cross-sectional area and flow speed. This concept is essential for determining how much fluid is moving through the pipe.

Recommended video:

Guided course

6:34

Fluid Speed & Volume Flow Rate

Related Practice

Textbook Question

A small circular hole 6.00 mm in diameter is cut in the side of a large water tank, 14.0 m below the water level in the tank. The top of the tank is open to the air. Find (a) the speed of efflux of the water and (b) the volume discharged per second.

views

Textbook Question

A shower head has 20 circular openings, each with radius 1.0 mm. The shower head is connected to a pipe with radius 0.80 cm. If the speed of water in the pipe is 3.0 m/s, what is its speed as it exits the shower-head openings?

views

Textbook Question

Water is flowing in a pipe with a varying cross-sectional area, and at all points the water completely fills the pipe. At point 1 the cross-sectional area of the pipe is 0.070 m², and the magnitude of the fluid velocity is 3.50 m/s. (c) Calculate the volume of water discharged from the open end of the pipe in 1.00 hour.

views

Textbook Question

At one point in a pipeline the water's speed is 3.00 m/s and the gauge pressure is 5.00×10⁴ Pa. Find the gauge pressure at a second point in the line, 11.0 m lower than the first, if the pipe diameter at the second point is twice that at the first.

views

Textbook Question

A cubical block of wood, 10.0 cm on a side, floats at the interface between oil and water with its lower surface 1.50 cm below the interface (Fig. E12.33). The density of the oil is 790 kg/m³. (a) What is the gauge pressure at the upper face of the block? (b) What is the gauge pressure at the lower face of the block?

views

Textbook Question

BIO. Artery Blockage. A medical technician is trying to determine what percentage of a patient's artery is blocked by plaque. To do this, she measures the blood pressure just before the region of blockage and finds that it is 1.20×10⁴ Pa, while in the region of blockage it is 1.15×10⁴ Pa. Furthermore, she knows that blood flowing through the normal artery just before the point of blockage is traveling at 30.0 cm/s, and the specific gravity of this patient's blood is 1.06. What percentage of the cross-sectional area of the patient's artery is blocked by the plaque?