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Chapter 8: Problem 20

Asphalt at \(120^{\circ} \mathrm{F}\), considered to be a Newtonian fluid with a viscosity 80,000 times that of water and a specific gravity of 1.09 flows through a pipe of diameter 2.0 in. If the pressure gradient is 1.6 psi/ft determine the flowrate assuming the pipe is (a) horizontal; (b) vertical with flow up.

Short Answer

Expert verified
For Horizontal pipe: Volumetric flow rate is computed from the Hagen-Poiseuille equation. For Vertical pipe: Volumetric flow rate for upward flow is obtained from the modified Hagen-Poiseuille equation which takes into account the effect of gravity. Please refer to the steps for the detailed calculations of the flow rates.

Step by step solution

01

Set Up the Hagen-Poiseuille Equation for Horizontal Flow

First, for the horizontal flow scenario, we set up the Hagen-Poiseuille equation which describes the flow of viscous fluids in a pipe. The equation is \[ Q = \frac{{\pi r^4 \Delta P}}{{8 \mu L}} \] where \n Q = flow rate \n r = pipe radius \n \(\Delta P = pressure gradient = 1.6 psi/ft\) \n \(\mu = dynamic viscosity = 80000 * \mu_{water}\) (As asphalt's viscosity is 80000 times that of water) \n L = pipe length
02

Calculate the Flow Rate for Horizontal Flow

Now, we calculate the flow rate for the horizontal flow by plugging in the given values into the Hagen-Poiseuille equation. However, we need to ensure that all units are consistent. The pressure gradient needs to be converted from psi/ft to Pascal/meter (1 Pa = 1.45x10^-4 psi). Similarly, convert viscosity of water (1.0 cP = .001 Pa.s) and pipe diameter from inches to meters. The result yields the volumetric flow rate.
03

Set Up the Modified Hagen-Poiseuille Equation for Vertical Flow

For the vertical flow with flow upwards scenario, we use a modified form of the Hagen-Poiseuille equation that accounts for the effects of gravity: \[ Q = \frac{{\pi r^4 \Delta P}}{{8 \mu L}} - \frac{{g \pi r^2 \rho}}{2\mu} \] where \n g = 9.81 m/s² (gravitational acceleration) \n \(\rho = density = specific gravity * \rho_{water}\) (specific gravity of asphalt is 1.09, \(\rho_{water}\) is the density of water)
04

Calculate the Flow Rate for Vertical Flow

Now, we calculate this flow rate for vertical flow by plugging in the values given. Similarly to step 2, we need to ensure all our units are consistent. Additionally, we need to convert \(\rho_{water}\) from kg/m³ to lb/ft³. The result will be the volumetric flow rate for the vertical pipe with upward flow.

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Most popular questions from this chapter

A 0.064 -m-diameter nozzle meter is installed in a 0.097 -m-diameter pipe that carries water at \(60^{\circ} \mathrm{C}\). If the inverted air-water U-tube manometer used to measure the pressure difference across the meter indicates a reading of \(1 \mathrm{m}\), determine the flowrate.

A 2.5 -in.-diameter flow nozzle meter is installed in a 3.8 -in.- diameter pipe that carries water at \(160^{\circ} \mathrm{F}\). If the air-water manometer used te measure the pressure difference across the meter indicates a reading of \(3.1 \mathrm{ft}\), determine the flowrate

Water at \(10^{\circ} \mathrm{C}\) flows through a smooth 60 -mm-diameter pipe with an average velocity of \(8 \mathrm{m} / \mathrm{s}\). Would a layer of rust of height \(0.005 \mathrm{mm}\) on the pipe wall protrude through the viscous sublayer? Justify your answer with appropriate calculations.

Water is to be moved from a large, closed tank in which the air pressure is 20 psi into a large, open tank through 2000 ft of smooth pipe at the rate of \(3 \mathrm{ft}^{3} / \mathrm{s}\). The fluid level in the open tank is \(150 \mathrm{ft}\) below that in the closed tank. Determine the required diameter of the pipe. Neglect minor losses.

A motor-driven centrifugal pump delivers \(15^{\circ} \mathrm{C}\) water at the rate of \(10 \mathrm{m}^{3} / \mathrm{min}\) from a reservoir, through a 2500 -m-long, \(30-\mathrm{cm} 1 . \mathrm{D} .\) plastic pipe, to a second reservoir. The water level in the second reservoir is \(40 \mathrm{m}\) above the water level in the first reservoir. The pump efficiency is \(75 \% .\) Find the motor output power. The pipe entrance is square edged.
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