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Chapter 8: Problem 4

What is hydraulic diameter? How is it defined? What is it equal to for a circular tube of diameter \(D\) ?

Short Answer

Expert verified
Answer: The hydraulic diameter is a parameter used to characterize the efficiency of fluid flow through a channel or conduit. For a circular tube with diameter D, the hydraulic diameter can be calculated using the formula: Hydraulic Diameter = D. This is obtained by dividing the flow area (A) by the wetted perimeter (P) and multiplying by 4, as shown in the formula: Hydraulic Diameter = \(\frac{4 \times A}{P}\).

Step by step solution

01

Definition of Hydraulic Diameter

The hydraulic diameter is a parameter that is used to characterize the efficiency of fluid flow through a channel or conduit, especially in situations related to internal flow and heat transfer. It is calculated by dividing the cross-sectional area of the flow by the wetted perimeter. The hydraulic diameter is a useful parameter for comparing the performance of different shapes of conduits with similar cross-sectional areas and wetted perimeters. Here is the formula for hydraulic diameter: Hydraulic Diameter = \(\frac{4 \times \text{Flow Area}}{\text{Wetted Perimeter}}\)
02

Hydraulic Diameter of a Circular Tube

For a circular tube with diameter D, the flow area (A) and the wetted perimeter (P) can be calculated using the following formulas: Flow Area (A) = \(\pi \Big(\frac{D}{2}\Big)^2\) Wetted Perimeter (P) = \(\pi D\) Now, by plugging these values into the hydraulic diameter formula, you get: Hydraulic Diameter = \(\frac{4 \times A}{P}\) = \(\frac{4 \times [\pi \Big(\frac{D}{2}\Big)^2]}{\pi D}\) Simplifying this expression, we find that the hydraulic diameter for a circular tube with diameter D is equal to: Hydraulic Diameter = \(D\)

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Most popular questions from this chapter

Someone claims that the volume flow rate in a circular pipe with laminar flow can be determined by measuring the velocity at the centerline in the fully developed region, multiplying it by the cross-sectional area, and dividing the result by 2 . Do you agree? Explain.

Oil at \(15^{\circ} \mathrm{C}\) is to be heated by saturated steam at 1 atm in a double-pipe heat exchanger to a temperature of \(25^{\circ} \mathrm{C}\). The inner and outer diameters of the annular space are \(3 \mathrm{~cm}\) and \(5 \mathrm{~cm}\), respectively, and oil enters with a mean velocity of \(0.8 \mathrm{~m} / \mathrm{s}\). The inner tube may be assumed to be isothermal at \(100^{\circ} \mathrm{C}\), and the outer tube is well insulated. Assuming fully developed flow for oil, determine the tube length required to heat the oil to the indicated temperature. In reality, will you need a shorter or longer tube? Explain.

An engineer is to design an experimental apparatus that consists of a \(25-\mathrm{mm}\)-diameter smooth tube, where different fluids at \(100^{\circ} \mathrm{C}\) are to flow through in fully developed laminar flow conditions. For hydrodynamically and thermally fully developed laminar flow of water, engine oil, and liquid mercury, determine \((a)\) the minimum tube length and \((b)\) the required pumping power to overcome the pressure loss in the tube at largest allowable flow rate.

Water enters a 5-mm-diameter and 13-m-long tube at \(45^{\circ} \mathrm{C}\) with a velocity of \(0.3 \mathrm{~m} / \mathrm{s}\). The tube is maintained at a constant temperature of \(8^{\circ} \mathrm{C}\). The exit temperature of water is (a) \(4.4^{\circ} \mathrm{C}\) (b) \(8.9^{\circ} \mathrm{C}\) (c) \(10.6^{\circ} \mathrm{C}\) (d) \(12.0^{\circ} \mathrm{C}\) (e) \(14.1^{\circ} \mathrm{C}\) (For water, use \(k=0.607 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=6.14, v=0.894 \times\) \(10^{-6} \mathrm{~m}^{2} / \mathrm{s}, c_{p}=4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \rho=997 \mathrm{~kg} / \mathrm{m}^{3}\) )

A geothermal district heating system involves the transport of geothermal water at \(110^{\circ} \mathrm{C}\) from a geothermal well to a city at about the same elevation for a distance of \(12 \mathrm{~km}\) at a rate of \(1.5 \mathrm{~m}^{3} / \mathrm{s}\) in \(60-\mathrm{cm}\)-diameter stainless steel pipes. The fluid pressures at the wellhead and the arrival point in the city are to be the same. The minor losses are negligible because of the large length-to-diameter ratio and the relatively small number of components that cause minor losses. (a) Assuming the pump-motor efficiency to be 65 percent, determine the electric power consumption of the system for pumping. \((b)\) Determine the daily cost of power consumption of the system if the unit cost of electricity is $$\$ 0.06 / \mathrm{kWh}$$. (c) The temperature of geothermal water is estimated to drop \(0.5^{\circ} \mathrm{C}\) during this long flow. Determine if the frictional heating during flow can make up for this drop in temperature.
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