Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 9: Problem 17

Under what conditions can the outer surface of a vertical cylinder be treated as a vertical plate in natural convection calculations?

Short Answer

Expert verified
Answer: The outer surface of a vertical cylinder can be treated as a vertical plate in natural convection calculations when the height of the cylinder is much larger than its diameter (h >> 2r). This condition ensures that the fluid flow around the cylinder is dominated by vertical motion and the curved geometry's effect on the flow pattern is negligible.

Step by step solution

01

Understand the concepts of natural convection and vertical plate approximation

Natural convection is the process in which heat transfer occurs due to the movement of fluid caused by the temperature difference between the fluid and the surface. When the fluid is in contact with a heated (or cooled) plate, buoyancy forces cause the fluid to rise (or fall), transferring heat from the hotter surface to the cooler fluid. The vertical plate approximation simplifies the analysis by treating the surface as an infinitely long, vertical surface; this is valid when the height of the plate is much larger than its thickness.
02

Review the vertical cylinder geometry

A vertical cylinder is a 3D geometric shape with a circular base and a height, h. Its outer surface extends around the entire circumference, which is 2π times the radius, r. The vertical plate geometry, on the other hand, considers only vertically-oriented, flat surfaces with a length (or height) and width.
03

Assess the fluid flow around the vertical cylinder

When a vertical cylinder is subjected to a temperature difference, the fluid flow is influenced by the combination of the cylinder's diameter (2r) and its height (h). If the diameter is significantly smaller than the height, the fluid flow in the vertical direction will dominate, making it similar to the flow pattern around a vertical plate.
04

Establish the condition for treating a vertical cylinder as a vertical plate

To treat the outer surface of a vertical cylinder as a vertical plate in natural convection calculations, the height of the cylinder must be much larger than its diameter (i.e., h >> 2r). This implies that the fluid flow around the cylinder will mostly be in the vertical direction, and the curved geometry will have a minimal impact on the fluid flow pattern.
05

Conclusion

The outer surface of a vertical cylinder can be treated as a vertical plate in natural convection calculations under the condition that the height of the cylinder is much larger than its diameter (h >> 2r). In this scenario, the fluid flow around the cylinder will be dominated by vertical motion, and the curved geometry's effect on the flow pattern will be negligible.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Flat-plate solar collectors are often tilted up toward the sun in order to intercept a greater amount of direct solar radiation. The tilt angle from the horizontal also affects the rate of heat loss from the collector. Consider a \(1.5-\mathrm{m}\)-high and 3-m-wide solar collector that is tilted at an angle \(\theta\) from the horizontal. The back side of the absorber is heavily insulated. The absorber plate and the glass cover, which are spaced \(2.5 \mathrm{~cm}\) from each other, are maintained at temperatures of \(80^{\circ} \mathrm{C}\) and \(40^{\circ} \mathrm{C}\), respectively. Determine the rate of heat loss from the absorber plate by natural convection for \(\theta=0^{\circ}, 30^{\circ}\), and \(90^{\circ}\).

Skylights or "roof windows" are commonly used in homes and manufacturing facilities since they let natural light in during day time and thus reduce the lighting costs. However, they offer little resistance to heat transfer, and large amounts of energy are lost through them in winter unless they are equipped with a motorized insulating cover that can be used in cold weather and at nights to reduce heat losses. Consider a 1 -m-wide and \(2.5\)-m-long horizontal skylight on the roof of a house that is kept at \(20^{\circ} \mathrm{C}\). The glazing of the skylight is made of a single layer of \(0.5\)-cm-thick glass \((k=0.78 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\varepsilon=0.9)\). Determine the rate of heat loss through the skylight when the air temperature outside is \(-10^{\circ} \mathrm{C}\) and the effective sky temperature is \(-30^{\circ} \mathrm{C}\). Compare your result with the rate of heat loss through an equivalent surface area of the roof that has a common \(R-5.34\) construction in SI units (i.e., a thickness-to-effective-thermal- conductivity ratio of \(\left.5.34 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\right)\). Evaluate air properties at a film temperature of \(-7^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) pressure. Is this a good assumption?

A \(1.5\)-m-diameter, 4-m-long cylindrical propane tank is initially filled with liquid propane, whose density is \(581 \mathrm{~kg} / \mathrm{m}^{3}\). The tank is exposed to the ambient air at \(25^{\circ} \mathrm{C}\) in calm weather. The outer surface of the tank is polished so that the radiation heat transfer is negligible. Now a crack develops at the top of the tank, and the pressure inside drops to \(1 \mathrm{~atm}\) while the temperature drops to \(-42^{\circ} \mathrm{C}\), which is the boiling temperature of propane at \(1 \mathrm{~atm}\). The heat of vaporization of propane at \(1 \mathrm{~atm}\) is \(425 \mathrm{~kJ} / \mathrm{kg}\). The propane is slowly vaporized as a result of the heat transfer from the ambient air into the tank, and the propane vapor escapes the tank at \(-42^{\circ} \mathrm{C}\) through the crack. Assuming the propane tank to be at about the same temperature as the propane inside at all times, determine how long it will take for the tank to empty if it is not insulated.

A 3 -mm-diameter and 12-m-long electric wire is tightly wrapped with a \(1.5-\mathrm{mm}\)-thick plastic cover whose thermal conductivity and emissivity are \(k=0.20 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\varepsilon=0.9\). Electrical measurements indicate that a current of \(10 \mathrm{~A}\) passes through the wire and there is a voltage drop of \(7 \mathrm{~V}\) along the wire. If the insulated wire is exposed to calm atmospheric air at \(T_{\infty}=30^{\circ} \mathrm{C}\), determine the temperature at the interface of the wire and the plastic cover in steady operation. Take the surrounding surfaces to be at about the same temperature as the air. Evaluate air properties at a film temperature of \(40^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) pressure. Is this a good assumption?

Hot water is flowing at an average velocity of \(4 \mathrm{ft} / \mathrm{s}\) through a cast iron pipe \(\left(k=30 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right)\) whose inner and outer diameters are \(1.0\) in and \(1.2\) in, respectively. The pipe passes through a 50 -ft-long section of a basement whose temperature is \(60^{\circ} \mathrm{F}\). The emissivity of the outer surface of the pipe is \(0.5\), and the walls of the basement are also at about \(60^{\circ} \mathrm{F}\). If the inlet temperature of the water is \(150^{\circ} \mathrm{F}\) and the heat transfer coefficient on the inner surface of the pipe is \(30 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}\), determine the temperature drop of water as it passes through the basement. Evaluate air properties at a film temperature of \(105^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) pressure. Is this a good assumption?
See all solutions

Recommended explanations on Physics Textbooks

Particle Model of Matter

Read Explanation

Magnetism

Read Explanation

Space Physics

Read Explanation

Fields in Physics

Read Explanation

Kinematics Physics

Read Explanation

Measurements

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free