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Chapter 3: Problem 6

What pressure gradient along the streamline, \(d p / d s\), is required to accelerate water upward in a vertical pipe at a rate of \(30 \mathrm{ft} / \mathrm{s}^{2} ?\) What is the the if the flow is downward?

Short Answer

Expert verified
The pressure gradient for upward flow is 3872.64 lb/ft²/ft, and for the downward flow, it is 138.24 lb/ft²/ft.

Step by step solution

01

Find Pressure Gradient for Upward Flow

First, let's consider the upward flow. Let's denote the absolute pressure in the fluid as \(P\), density as \(ρ\), acceleration due to gravity as \(g\), additional acceleration as \(a\), height as \(h\) and the pressure gradient as \(dp/ds\). For upward flow, the acceleration of gravity and the flow acceleration are in opposite directions. Using Bernoulli's equation, we get \(dp/ds = ρ(g + a)\).
02

Substitute Known Values for Upward Flow

Next, let's plug the given values into our equation. Given that \(a = 30 ft/s^2\), \(g = 32.2 ft/s^2\), and \(ρ (for water) = 62.4 lb/ft^3\), so the pressure gradient becomes: \(dp/ds = 62.4 (32.2 + 30) lb/ft^2/ft\).
03

Calculate Pressure Gradient for Upward Flow

After doing the calculation, we obtain \(dp/ds = 3872.64 lb/ft^2/ft\) for the upward flow.
04

Find Pressure Gradient for Downward Flow

Now let's find the pressure gradient for the downward flow. Here, the acceleration of gravity and the flow acceleration are in the same direction. Using Bernoulli's equation, we get \(dp/ds = ρ(g - a)\).
05

Substitute Known Values for Downward Flow

Substituting the values into our equation, we get \(dp/ds = 62.4 (32.2 - 30) lb/ft^2/ft\).
06

Calculate Pressure Gradient for Downward Flow

Doing the arithmetic, we obtain \(dp/ds = 138.24 lb/ft^2/ft\) for the downward flow.

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Most popular questions from this chapter

Carbon dioxide flows at a rate of \(1.5 \mathrm{ft}^{3} / \mathrm{s}\) from a 3 -in. pipe in which the pressure and temperature are 20 psi (gage) and \(120 \%\) into a 1.5 -in. pipe. If viscous effects are neglected and incompressible conditions are assumed, determine the pressure in the smaller pipe.

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