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Chapter 2: Problem 14

Consider a cold canned drink left on a dinner table. Would you model the heat transfer to the drink as one-, two-, or three-dimensional? Would the heat transfer be steady or transient? Also, which coordinate system would you use to analyze this heat transfer problem, and where would you place the origin? Explain.

Short Answer

Expert verified
Answer: For the cold canned drink problem, the heat transfer process is one-dimensional and transient. A cylindrical coordinate system should be used for analysis, with the origin placed at the center of the can's bottom surface.

Step by step solution

01

Determine if the heat transfer is one-, two-, or three-dimensional

First, let's examine the geometry of the canned drink and how heat transfers. Heat transfers from the surrounding air to the canned drink in all directions. However, the can is cylindrical and its height or length is much greater than its radius or diameter. Due to the predominant heat transfer occurring across the height of the can, we can assume that the heat transfer process is one-dimensional along its height or length.
02

Determine if the heat transfer is steady or transient

The temperature of the canned drink will change over time as it gains heat from the surrounding air, and that change will eventually slow down when it reaches the equilibrium with the surrounding temperature. Since the temperature of the drink changes over time, the heat transfer process is transient.
03

Choose the appropriate coordinate system

Since we have established that the heat transfer is one-dimensional along the height of the can, it is best to use a cylindrical coordinate system. In this coordinate system, we have three variables: radial distance (r), azimuthal angle (θ), and axial distance (z). Since the heat transfer is mainly along the height of the can (one-dimensional), we will focus on the z-coordinate.
04

Placement of the origin

To analyze the heat transfer problem, we can place the origin at the center of the can's bottom surface. In the cylindrical coordinate system, the origin placement would be (r, θ, z) = (0, 0, 0). This position helps to simplify the mathematical analysis and allows us to focus on the axial distance (z) for understanding the heat transfer along the height of the can. In conclusion, we have established the following for this heat transfer problem: The heat transfer to the cold canned drink is one-dimensional and transient. A cylindrical coordinate system is suitable for analyzing this problem, with the origin placed at the center of the can's bottom surface.

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Most popular questions from this chapter

A flat-plate solar collector is used to heat water by having water flow through tubes attached at the back of the thin solar absorber plate. The absorber plate has an emissivity and an absorptivity of \(0.9\). The top surface \((x=0)\) temperature of the absorber is \(T_{0}=35^{\circ} \mathrm{C}\), and solar radiation is incident on the absorber at $500 \mathrm{~W} / \mathrm{m}^{2}\( with a surrounding temperature of \)0^{\circ} \mathrm{C}$. The convection heat transfer coefficient at the absorber surface is $5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, while the ambient temperature is \(25^{\circ} \mathrm{C}\). Show that the variation of temperature in the absorber plate can be expressed as $T(x)=-\left(\dot{q}_{0} / k\right) x+T_{0}\(, and determine net heat flux \)\dot{q}_{0}$ absorbed by the solar collector.

Consider a large plate of thickness \(L\) and thermal conductivity \(k\) in which heat is generated uniformly at a rate of \(\dot{e}_{\text {gen }}\). One side of the plate is insulated, while the other side is exposed to an environment at \(T_{\infty}\) with a heat transfer coefficient of \(h\). (a) Express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the plate, (b) determine the variation of temperature in the plate, and (c) obtain relations for the temperatures on both surfaces and the maximum temperature rise in the plate in terms of given parameters.

A 6-m-long, 2-kW electrical resistance wire is made of \(0.2-\mathrm{cm}\)-diameter stainless steel $(k=15.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$. The resistance wire operates in an environment at \(20^{\circ} \mathrm{C}\) with a heat transfer coefficient of $175 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ at the outer surface. Determine the surface temperature of the wire \((a)\) by using the applicable relation and \((b)\) by setting up the proper differential equation and solving it.

In metal processing plants, workers often operate near hot metal surfaces. Exposed hot surfaces are hazards that can potentially cause thermal burns on human skin. Metallic surfaces above \(70^{\circ} \mathrm{C}\) are considered extremely hot. Damage to skin can occur instantaneously upon contact with metallic surfaces at that temperature. In a plant that processes metal plates, a plate is conveyed through a series of fans to cool its surface in an ambient temperature of \(30^{\circ} \mathrm{C}\). The plate is \(25 \mathrm{~mm}\) thick and has a thermal conductivity of $13.5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The temperature at the bottom surface of the plate is monitored by an infrared (IR) thermometer. Obtain an expression for the variation of temperature in the metal plate. The IR thermometer measures the bottom surface of the plate to be \(60^{\circ} \mathrm{C}\). Determine the minimum value of the convection heat transfer coefficient needed to keep the top surface below \(47^{\circ} \mathrm{C}\).

A spherical vessel is filled with chemicals undergoing an exothermic reaction. The reaction provides a uniform heat flux on the inner surface of the vessel. The inner diameter of the vessel is \(5 \mathrm{~m}\) and its inner surface temperature is at \(120^{\circ} \mathrm{C}\). The wall of the vessel has a variable thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where $k_{0}=1.01 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \beta=0.0018 \mathrm{~K}^{-1}\(, and \)T\( is in \)\mathrm{K}$. The vessel is situated in a surrounding with an ambient temperature of \(15^{\circ} \mathrm{C}\), and the vessel's outer surface experiences convection heat transfer with a coefficient of \(80 \mathrm{~W} / \mathrm{m}^{2}\). K. To prevent thermal burns to workers who touch the vessel, the outer surface temperature of the vessel should be kept below \(50^{\circ} \mathrm{C}\). Determine the minimum wall thickness of the vessel so that the outer surface temperature is \(50^{\circ} \mathrm{C}\) or lower.
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