Fluid Flow & Continuity Equation Practice Problems

Reducers are devices that connect pipes of different cross-sections. A pipeline has a reducer that connects a 3.0 inch diameter pipe to a 2.0 inch diameter pipe. If water in the large pipe flows at 10 cm/s, how quickly does water flow in the narrow pipe?

A pipeline consists of sections of different cross-sections.  Oil flowing in the pipeline fills all cross-sections completely. A section Y of the pipeline has a cross-sectional area of 1.15 m² and the velocity magnitude at that point is 1.8 m/s. Determine the volume delivered to a tank every hour at the pipeline's terminal.

A water bottling plant uses a pump that fills 900 0.500L bottles every 5 minutes. A gauge connected to the pipe originating from the pump at point A reads 135 kPa and the pipeline cross-section at the point is 20.0 cm². Point B is elevated by 2.0 m from point A and the pipe cross-section is 5.0 cm². Determine the volume (in L/s) passing through a fixed point in the pipeline, and the speed of the water at points A and B.

A sprinkler head has 41 openings used to sprinkle water. The openings are 1.5 mm wide. The sprinkler head is supplied with water from a 1.0-inch diameter hose pipe. If water flows at 2.0 m/s in the hose, calculate the speed of the water jets leaving the sprinkler head.

A water tank contains a large volume of water. Three pipes, each with a radius of 2.54 m, drain water from the tank at a combined rate of 8.0 × 10⁷ gallons/h. Assuming the pipes carry equal volumes, how fast does water move in the pipes?

Capillary action causes water to rise in a tube due to adhesive forces between water and the walls of the tube. Suppose a capillary action pump moves water through a compound pipe with a net flow rate of 50 g/h. If the compound pipe has 3500 bores each of diameter 600 μm, find the water speed (in mm/s) in the bores. Assume the water is salty with a density of 1010 kg/m³.

A hydraulic piston has a diameter of 4.45 cm and a volume of 0.398 L. It is supplied/drained of hydraulic fluid using a pipe of diameter 1.27 cm.  Suppose fluid from the piston is drained into a reservoir whose pressure is 120 mmHg, and if all the fluid is sent back to the reservoir in 2.5 s, determine the speed of the hydraulic fluid as it flows through the pipe.

A swimming pool has dimensions of 38 cm wide by 2.0 m long by 0.70 m high. What should the flow speed be in the 4.0-cm-diameter pipe for the pump if the pool pump should circulate all the water in the pool once every 3.5 hours?

Suppose water flows through a horizontal hose that narrows from an 8.0 cm diameter to a 4.0 cm diameter. What is the volume rate of flow if the gauge pressures in these sections are 33.0 kPa and 25.0 kPa, respectively?

A tube of 2.5 cm diameter has three nozzles connected to its end, each with a diameter of 0.2 cm. Water flows through the tube and comes out of the nozzles at an angle of 30° with the horizontal. The nozzles sprinkle water over a circular area with a radius of 3.0 m. Determine the speed at which water flows inside the 2.5 cm diameter tube. Assume that the gravitational acceleration is g = 9.81 m/s².

Determine the average speed of water flow in a network of pipes with a combined cross-sectional area of 3.5 cm². The main supply pipe has a radius of 1.5 cm and a water speed of 30 cm/s.

A garden hose nozzle sprays water to a height of 25 cm. The nozzle has a diameter of 1.0 cm, while the supply pipe feeding the nozzle has a diameter of 2.0 cm. The water is pumped from a ground-level reservoir 0.75 m below the nozzle. Determine the gauge pressure the pump must generate. Ignore the effects of viscosity. Use $v=\sqrt{2 g h}$ for the speed of the water at height h₂ (at the nozzle), $A_{1} v_{1}=A_{2} v_{2}$ for the continuity equation, and Bernoulli's equation: $P_1+\frac{1}{2}\rho v_1^2+\rho gh_1=P_2+\frac{1}{2}\rho v_2^2+\rho gh_2$ .

In a skyscraper, water at the ground floor flows through a pipe with a gauge pressure of 5.0 atm and a speed of 1.1 m/s. The pipe, which has a diameter of 7.0 cm at the ground floor, narrows to 4.5 cm in diameter by the time it reaches the 25th floor, 50 m above. The faucet at the 25th floor has been left open. Determine the flow speed in the pipe at the 25th floor. Assume no branch pipes and ignore the effects of viscosity.

Water flows at a gauge pressure of 5.0 atm and a speed of 1.10 m/s through a pipe in an industrial plant. In another section of the plant, 50 m higher in elevation, the pipe narrows, and the water flows at a speed of 2.66 m/s. Calculate the gauge pressure in this higher section. Ignore branch pipes and viscosity. Use density of water = 1000 kg/m³, 1 atm = 101325 Pa, and g=9.8 m/s².

A water pipe equipped with a flow meter shows the following setup: the pipe's main section has a diameter of 4.0 cm, which narrows down to a diameter of 1.5 cm at the meter. The pressure difference between these sections is recorded as 25 mm-Hg. Using the relation $v_{1}=A_{2} \sqrt{\frac{2\left(P_{1}-P_{2}\right)}{\rho\left(A_{1}^{2}-A_{2}^{2}\right)}}$, calculate the flow speed of water entering the narrower section of the pipe. Use $\rho _{water}=1000\ \frac{kg}{m^3}$

In a fluid dynamics experiment, a container with oil is subjected to an external pressure P_B along with atmospheric pressure P_o. If this external pressure, P_B = 0.75 atm, and the distance between the two heights y_B and y_A within the fluid, y_B - y_A = 0.5 m, calculate the velocity v_A at height y_A with which the oil comes out of the container through the opening.

[Assume that atmospheric pressure 1 atm = 1.0 × 10⁵ N/m² and the density of oil is 7.5 × 10² kg/m³.]

A pool maintenance worker finds it takes too long to drain a swimming pool using a hose with a 1/2-inch diameter. By what factor will the time be reduced if they switch to using a hose with a 1-inch diameter? Assume all other conditions remain constant.

While playing with a water gun, you partially cover the nozzle to increase the range of the water stream. Initially, the water stream reaches a certain distance. By partially covering the nozzle, the range increases by a factor of 5. If you are holding the water gun horizontally, what fraction of the nozzle opening did you block?

A physics student is experimenting with a faucet. At first, she turns on the faucet, and the water flows out. Afterward, she reduces the diameter of the nozzle by 35%. If the pressure difference is kept constant, determine by what percent the flow rate decreases.

Engineers are experimenting with a fire-extinguishing system that consists of a nozzle of diameter 0.86 cm at the end of a 0.80 m long vertical pipe installed on the floor. The water comes out of the nozzles at an angle of 15° and lands 2.5 m away from the base of the pipe in 0.5 seconds. Given that the inner diameter of the pipe is 1.4 cm, determine the pressure of water inside the pipe at its base.

Determine the atmospheric pressure within a formidable tornado, with wind speeds reaching 250 km/h, employing Bernoulli's equation. The density of air is 1.29 kg/m³.

A viscometer's inner and outer cylinders have diameters of 12.30 cm and 12.70 cm. The fluid depth is 10.0 cm. A torque of 0.030 N·m rotates the inner cylinder at 60 rev/min. Find the fluid's viscosity.

Glycerin passes through a small pipe with a diameter of 2.2 mm and a length of 15 cm. Calculate the pressure difference needed to maintain a flow rate of 8 mL/min. Assume glycerin has a dynamic viscosity of 1.5 Pa·s.

Determine if water flow through a cylindrical pipe is laminar or turbulent when the average speed of water in the pipe (with a radius of 1.5 cm) is 50 cm/s. The density of water is 1000 kg/m³ and its viscosity is $1.0 \times 10^{-3} \mathrm{~Pa} \cdot \mathrm{s}$ . The onset of turbulence occurs when the Reynolds number, Re, exceeds approximately 2000. The Reynolds number is defined as:

$R e=\frac{2 \bar{v} r \rho}{\eta}$

where $\bar{v}$ is the average speed of the fluid, 𝜌 is its density, 𝜂 is its viscosity, and 𝑟 is the radius of the pipe.

An elongated cylindrical container has an internal diameter of 0.80 cm and is filled with glycerin. A cylindrical rod that has a length of 25.0 cm and a diameter of 0.70 cm is dropped vertically through the glycerin. The rod has a mass of 100 g. Determine the maximum velocity the rod will achieve as it falls through the glycerin, given that the viscosity of glycerin is 1.49 N$\cdot$s/m².

An air conditioning system is designed to replace the air in a storage room that measures 10 m by 6 m by 3 m every 10 minutes. If the air flows through a duct that has a radius of 15 cm, what is the speed of the air inside of the duct?

A garden hose that has a diameter of 2.5 cm is used to fill a swimming pool located at ground level. The water pressure at the source is equivalent to a pressure head of 8.0 meters. Calculate the volume rate of the flow of water from the hose.

A garden water sprinkler rotates and ejects water through a small nozzle. The sprinkler is connected to a hose that has a diameter of 2.0 cm and is designed to deliver water at a rate of 300 L/h. The water exits the sprinkler through a nozzle that has a diameter of 0.50 cm. How much force is exerted on the sprinkler due to the water flowing from the hose through the nozzle?

At a water storage facility, you see a 6.3-meter-deep tank with a hole at the bottom. The manager is concerned about the flow rate of the water draining out of the tank. Ignoring viscosity, calculate the speed of the water that is draining out of the tank?

Determine the pressure difference between the two ends of a section of a tube that is used for transporting Ethanol which has a density of ρ = 789 kg/m³ and a viscosity of η = 0.0012 Pa⋅s. The section of the tube being examined has a length and diameter of 1.8 km and 15 cm respectively, and the rate of transportation is 1000 cm³/s.

A fireman uses a firehose with an internal diameter of 1/2-inch to fill up an empty cylindrical water tank that has a diameter of 4.5 m and a height of 2.0 m. If water flows from the hose at a speed of 0.35 m/s, determine the time it will take for the tank to fill up completely.