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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: Hydraulic diameter is a parameter used to characterize the flow complexity in ducts, channels, and pipes. It is defined as four times the ratio of the cross-sectional area (A) to the wetted perimeter of the conduit (P). The formula is given as: \(D_h = \frac{4 \times A}{P}\). For a circular tube with diameter D, the hydraulic diameter is equal to the diameter of the tube, so \(D_h = D\).

Step by step solution

01

Definition of Hydraulic Diameter

Hydraulic diameter is a calculated value used to characterize the flow complexity in ducts, channels, and pipes. It can be thought of as the diameter of a circular conduit that has the same flow characteristics as the given non-circular conduit. It's an important parameter when dealing with fluid flow problems, particularly in cases of non-circular cross-sections.
02

Formula for Hydraulic Diameter

The hydraulic diameter (\(D_h\)) is defined as four times the ratio of the cross-sectional area (A) to the wetted perimeter of the conduit (P). The formula is given as: \[ D_h = \frac{4 \times A}{P} \] Where: - \(D_h\) is the hydraulic diameter - A is the cross-sectional area of the conduit - P is the wetted perimeter (the perimeter in contact with the fluid)
03

Hydraulic Diameter for a Circular Tube

For a circular tube with diameter D, we can calculate the hydraulic diameter using the relationship established above. The cross-sectional area (A) and the wetted perimeter (P) for a circular tube with diameter D can be calculated as follows: - A = \(\frac{\pi \times D^2}{4}\) - P = \(\pi \times D\) Now, we can plug these values into the formula for hydraulic diameter: \[ D_h = \frac{4 \times (\frac{\pi \times D^2}{4})}{\pi \times D} \] Simplify the equation by cancelling out terms: \[ D_h = \frac{\pi \times D^2}{\pi \times D} \] \[ D_h = D \] In the case of a circular tube, the hydraulic diameter is equal to the diameter of the tube.

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Most popular questions from this chapter

Water is to be heated from \(10^{\circ} \mathrm{C}\) to \(80^{\circ} \mathrm{C}\) as it flows through a \(2-\mathrm{cm}\)-internal-diameter, \(13-\mathrm{m}\)-long tube. The tube is equipped with an electric resistance heater, which provides uniform heating throughout the surface of the tube. The outer surface of the heater is well insulated, so in steady operation all the heat generated in the heater is transferred to the water in the tube. If the system is to provide hot water at a rate of \(5 \mathrm{~L} / \mathrm{min}\), determine the power rating of the resistance heater. Also, estimate the inner surface temperature of the pipe at the exit.

The hot water needs of a household are to be met by heating water at \(55^{\circ} \mathrm{F}\) to \(180^{\circ} \mathrm{F}\) by a parabolic solar collector at a rate of \(5 \mathrm{lbm} / \mathrm{s}\). Water flows through a \(1.25\)-in-diameter thin aluminum tube whose outer surface is anodized black in order to maximize its solar absorption ability. The centerline of the tube coincides with the focal line of the collector, and a glass sleeve is placed outside the tube to minimize the heat losses. If solar energy is transferred to water at a net rate of \(350 \mathrm{Btu} / \mathrm{h}\) per \(\mathrm{ft}\) length of the tube, determine the required length of the parabolic collector to meet the hot water requirements of this house. Also, determine the surface temperature of the tube at the exit.

Air \(\left(c_{p}=1000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) enters a \(16-\mathrm{cm}\)-diameter and \(19-\mathrm{m}\)-long underwater duct at \(50^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) at an average velocity of $7 \mathrm{~m} / \mathrm{s}$ and is cooled by the water outside. If the average heat transfer coefficient is $35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and the tube temperature is nearly equal to the water temperature of \(5^{\circ} \mathrm{C}\), the exit temperature of the air is (a) \(6^{\circ} \mathrm{C}\) (b) \(10^{\circ} \mathrm{C}\) (c) \(18^{\circ} \mathrm{C}\) (d) \(25^{\circ} \mathrm{C}\) (e) \(36^{\circ} \mathrm{C}\)

The exhaust gases of an automotive engine leave the combustion chamber and enter an 8 -ft-long and 3.5-in-diameter thin-walled steel exhaust pipe at \(800^{\circ} \mathrm{F}\) and \(15.5 \mathrm{psia}\) at a rate of $0.05 \mathrm{lbm} / \mathrm{s}$. The surrounding ambient air is at a temperature of \(80^{\circ} \mathrm{F}\), and the heat transfer coefficient on the outer surface of the exhaust pipe is $3 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2},{ }^{\circ} \mathrm{F}$. Assuming the exhaust gases to have the properties of air, determine \((a)\) the velocity of the exhaust gases at the inlet of the exhaust pipe and \((b)\) the temperature at which the exhaust gases will leave the pipe and enter the air.

Hot air at atmospheric pressure and \(75^{\circ} \mathrm{C}\) enters a 10 -m-long uninsulated square duct of cross section $0.15 \mathrm{~m} \times 0.15 \mathrm{~m}\( that passes through the attic of a house at a rate of \)0.2 \mathrm{~m}^{3} / \mathrm{s}$. The duct is observed to be nearly isothermal at \(70^{\circ} \mathrm{C}\). Determine the exit temperature of the air and the rate of heat loss from the duct to the airspace in the attic. Evaluate air properties at a bulk mean temperature of \(75^{\circ} \mathrm{C}\). Is this a good assumption?
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