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Chapter 8: Problem 85

Water flows from a horizontal tube into a large tank. The tube is located \(2.5 \mathrm{m}\) below the free surface of water in the tank. The head loss is \(2 \mathrm{J} / \mathrm{kg}\). Compute the average flow speed in the tube.

Short Answer

Expert verified
After calculation, the average flow speed in the tube comes out to be approximately 6.65 m/s.

Step by step solution

01

Use Bernoulli's equation

Bernoulli's equation states that the sum of the pressure energy, kinetic energy, and potential energy per unit volume of a fluid is constant along a streamline, assuming no energy is added or removed by pumps or other means. The equation is formulated as \(P + \frac{1}{2}蟻v^2 + 蟻gh = constant\), where \(P\) is pressure, \(蟻\) is density, \(v\) is velocity, and \(h\) is height. But the pressure drops out because the start and end points are both at atmospheric pressure.
02

Set up the energy balance equation

We set the potential energy (in the tank) equal to the sum of the kinetic energy (in the pipe) and the energy loss: \(蟻gh_{tank} = \frac{1}{2}蟻v^2_{pipe} + h_{loss}\). Here, \(h_{tank}\) is the height of the liquid in the tank = 2.5m, \(v_{pipe}\) is the speed in the pipe that we want to find, and \(h_{loss}\) is the energy loss = \(2 \frac{J}{kg}\), which can be considered as another kind of effective height, \(h_{loss} = \frac{2}{g}\) where \(g\) is the gravitational acceleration.
03

Solve the energy balance equation for pipe speed

Rearranging the energy balance equation to solve for the average speed in the pipe \(v_{pipe}\) implies \(v_{pipe} = \sqrt{2 (gh_{tank}-h_{loss})}\). Plug in the given values into the equation: \(g = 9.81 m/s^2\), \(h_{tank} = 2.5m\), and \(h_{loss} = \frac{2}{9.81}\) m = 0.204m to calculate \(v_{pipe}\).

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Most popular questions from this chapter

\( \mathrm{A}\) smooth, 75 -mm-diameter pipe carries water \(\left(65^{\circ} \mathrm{C}\right)\) horizontally. When the mass flow rate is \(0.075 \mathrm{kg} / \mathrm{s}\), the pressure drop is measured to be 7.5 Pa per \(100 \mathrm{m}\) of pipe. Based on these measurements, what is the friction factor? What is the Reynolds number? Does this Reynolds number generally indicate laminar or turbulent flow? Is the flow actually laminar or turbulent?

A fire nozzle is supplied through \(100 \mathrm{m}\) of \(3.5-\mathrm{cm}\) diameter, smooth, rubber-lined hose. Water from a hydrant is supplied to a booster pump on board the pumper truck at \(350 \mathrm{kPa}(\text { gage }) .\) At design conditions, the pressure at the nozzle inlet is 700 kPa (gage), and the pressure drop along the hose is 750 kPa per \(100 \mathrm{m}\) of length. Determine (a) the design flow rate, (b) the nozzle exit velocity, assuming no losses in the nozzle, and (c) the power required to drive the booster pump, if its efficiency is 70 percent.

A water tank (open to the atmosphere) contains water to a depth of \(5 \mathrm{m} .\) A 25 -mm-diameter hole is punched in the bottom. Modeling the hole as square-edged, estimate the flow rate (L/s) exiting the tank. If you were to stick a short section of pipe into the hole, by how much would the flow rate change? If instead you were to machine the inside of the hole to give it a rounded edge \((r=5 \mathrm{mm})\), by how much would the flow rate change?

A fluid flows steadily between two parallel plates. The flow is fully developed and laminar. The distance between the plates is \(h\) (a) Derive an equation for the shear stress as a function of \(y\) Sketch this function. (b) For \(\mu=2.4 \times 10^{-5} \mathrm{lbf} \cdot \mathrm{s} / \mathrm{ft}^{2}, \quad \partial p / \partial x=-4.0 \mathrm{lbf} / \mathrm{ft}^{2} / \mathrm{n}\) and \(h=0.05\) in.., calculate the maximum shear stress, in \(\mathrm{lbf} / \mathrm{ft}^{2}\)

Two immiscible fluids are contained between infinite parallel plates. The plates are separated by distance \(2 h\), and the two fluid layers are of equal thickness \(h=5 \mathrm{mm}\). The dynamic viscosity of the upper fluid is four times that of the lower fluid, which is \(\mu_{10 \text { wer }}=0.1 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\). If the plates are stationary and the applied pressure gradient is \(-50 \mathrm{kPa} / \mathrm{m}\) find the velocity at the interface. What is the maximum velocity of the flow? Plot the velocity distribution.
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