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Chapter 8: Problem 11

Consider the flow of oil in a tube. How will the hydrodynamic and thermal entry lengths compare if the flow is laminar? How would they compare if the flow were turbulent?

Short Answer

Expert verified
Question: Compare the hydrodynamic and thermal entry lengths in laminar and turbulent flows within a tube. Answer: In laminar flow, the hydrodynamic and thermal entry lengths have a similar magnitude due to their dependence on the Reynolds number and Prandtl number. In turbulent flow, the hydrodynamic entry length is approximately constant and typically shorter than the thermal entry length, as it takes longer for a stable temperature profile to establish in turbulent flows due to increased mixing.

Step by step solution

01

Define hydrodynamic and thermal entry lengths

The hydrodynamic entry length is the distance it takes for the flow to become fully developed from the point where the fluid enters a tube. The thermal entry length, on the other hand, is the distance required for the temperature profile to become established. These lengths depend on the Reynolds number (Re) and Prandtl number (Pr), respectively, for laminar flow, and on the tube diameter, fluid properties, and flow rate for turbulent flow.
02

Compare hydrodynamic and thermal entry lengths in laminar flow

For laminar flow, the relationship between the hydrodynamic entry length (Lh) and the Reynolds number (Re) is given by: Lh = 0.05 * Re * D where D is the diameter of the tube The relationship between the thermal entry length (Lt) and the Prandtl number (Pr) is given by: Lt = 0.05 * Re * Pr * D In laminar flow, the Reynolds number is small, and the Prandtl number is typically close to 1. Therefore, the thermal entry length and hydrodynamic entry length have a similar magnitude in a laminar flow.
03

Compare hydrodynamic and thermal entry lengths in turbulent flow

In turbulent flow, the hydrodynamic entry length is approximately constant and not dependent on the Reynolds number, with a value around 10 times the tube diameter: Lh_turbulent ≈ 10 * D The thermal entry length in turbulent flow is more difficult to predict and depends on various factors, but it typically tends to be larger than the hydrodynamic entry length. This is because, in turbulent flow, there is a higher degree of mixing making it harder for a stable temperature profile to be established quickly.
04

Summarize the comparison

In laminar flow, the hydrodynamic and thermal entry lengths are of a similar magnitude due to their dependence on the Reynolds number and Prandtl number. In turbulent flow, the hydrodynamic entry length is approximately constant and is typically shorter than the thermal entry length, as it takes longer for a stable temperature profile to establish in turbulent flows due to increased mixing.

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Most popular questions from this chapter

Combustion gases passing through a 3-cm-internaldiameter circular tube are used to vaporize waste water at atmospheric pressure. Hot gases enter the tube at \(115 \mathrm{kPa}\) and \(250^{\circ} \mathrm{C}\) at a mean velocity of \(5 \mathrm{~m} / \mathrm{s}\), and leave at \(150^{\circ} \mathrm{C}\). If the average heat transfer coefficient is \(120 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and the inner surface temperature of the tube is \(110^{\circ} \mathrm{C}\), determine \((a)\) the tube length and (b) the rate of evaporation of water.

In a food processing plant, hot liquid water is being transported in a pipe \(\left(k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, D_{i}=2.5 \mathrm{~cm}\right.\), \(D_{o}=3 \mathrm{~cm}\), and \(\left.L=10 \mathrm{~m}\right)\). The hot water flowing with a mass flow rate of \(0.15 \mathrm{~kg} / \mathrm{s}\) enters the pipe at \(100^{\circ} \mathrm{C}\) and exits at \(60^{\circ} \mathrm{C}\). The plant supervisor thinks that since the hot water exits the pipe at \(60^{\circ} \mathrm{C}\), the pipe's outer surface temperature should be safe from thermal burn hazards. In order to prevent thermal burn upon accidental contact with skin tissue for individuals working in the vicinity of the pipe, the pipe's outer surface temperature should be kept below \(45^{\circ} \mathrm{C}\). Determine whether or not there is a risk of thermal burn on the pipe's outer surface. Assume the pipe outer surface temperature remains constant.

Air at \(10^{\circ} \mathrm{C}\) enters an \(18-\mathrm{m}\)-long rectangular duct of cross section \(0.15 \mathrm{~m} \times 0.20 \mathrm{~m}\) at a velocity of \(4.5 \mathrm{~m} / \mathrm{s}\). The duct is subjected to uniform radiation heating throughout the surface at a rate of \(400 \mathrm{~W} / \mathrm{m}^{3}\). The wall temperature at the exit of the duct is (a) \(58.8^{\circ} \mathrm{C}\) (b) \(61.9^{\circ} \mathrm{C}\) (c) \(64.6^{\circ} \mathrm{C}\) (d) \(69.1^{\circ} \mathrm{C}\) (e) \(75.5^{\circ} \mathrm{C}\) (For air, use \(k=0.02551 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=0.7296, v=1.562 \times\) \(10^{-5} \mathrm{~m}^{2} / \mathrm{s}, c_{p}=1007 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \rho=1.184 \mathrm{~kg} / \mathrm{m}^{3}\).)

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 \(k=0.607 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=6.14, v=0.894 \times\) \(10^{-6} \mathrm{~m}^{2} / \mathrm{s}, c_{p}=4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \rho=997 \mathrm{~kg} / \mathrm{m}^{3}\) )

The velocity profile in fully developed laminar flow in a circular pipe of inner radius \(R=10 \mathrm{~cm}\), in \(\mathrm{m} / \mathrm{s}\), is given by \(u(r)=4\left(1-r^{2} / R^{2}\right)\). Determine the mean and maximum velocities in the pipe, and the volume flow rate.
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