/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 122 A rod of diameter \(D=25 \mathrm... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 122

A rod of diameter \(D=25 \mathrm{~mm}\) and thermal conductivity \(k=60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) protrudes normally from a furnace wall that is at \(T_{w}=200^{\circ} \mathrm{C}\) and is covered by insulation of thickness \(L_{\text {ins }}=200 \mathrm{~mm}\). The rod is welded to the furnace wall and is used as a hanger for supporting instrumentation cables. To avoid damaging the cables, the temperature of the rod at its exposed surface, \(T_{o}\), must be maintained below a specified operating limit of \(T_{\max }=100^{\circ} \mathrm{C}\). The ambient air temperature is \(T_{\infty}=\) \(25^{\circ} \mathrm{C}\), and the convection coefficient is \(h=15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Derive an expression for the exposed surface temperature \(T_{o}\) as a function of the prescribed thermal and geometrical parameters. The rod has an exposed length \(L_{o}\), and its tip is well insulated. (b) Will a rod with \(L_{o}=200 \mathrm{~mm}\) meet the specified operating limit? If not, what design parameters would you change? Consider another material, increasing the thickness of the insulation, and increasing the rod length. Also, consider how you might attach the base of the rod to the furnace wall as a means to reduce \(T_{o}\).

Short Answer

Expert verified
In this problem, we derived an expression for the exposed surface temperature, \(T_o\), for a rod protruding from a furnace wall as a function of given thermal and geometrical parameters. The expression was found to be \(T_o = T_\infty + Q \cdot R_{conv}\), where \(Q\) is the heat transfer rate, and \(R_{conv}\) is the resistance due to convection. By substituting the given values, we can evaluate if the rod with an exposed length of \(L_o = 200mm\) meets the specified operating limit of \(T_{max} = 100^{\circ}C\). If it does not, alternative design parameters or changes to the current design need to be considered, such as using a different material, increasing insulation thickness, increasing rod length, or attaching the base of the rod differently to the furnace wall.

Step by step solution

01

Calculate the Thermal Resistances

First, we need to determine the resistance due to conduction through the insulation and convection from the exposed surface. Resistance due to conduction, \(R_{cond} = \frac{L_{ins}}{k \cdot A}\), where \(A\) is the cross-sectional area of the rod, which can be found by \(A = \pi \frac{D^2}{4}\). Resistance due to convection, \(R_{conv} = \frac{1}{h \cdot A}\).
02

Calculate the Heat Transfer Rate

We'll use the concept of thermal resistance to calculate the heat transfer rate through the system. The total resistance of the system, \(R_{total} = R_{cond} + R_{conv}\). The heat transfer rate, \(Q = \frac{T_w - T_\infty}{R_{total}}\).
03

Derive Expression for Exposed Surface Temperature

Now, we'll derive an expression for the exposed surface temperature, \(T_o\). Using the heat transfer rate and resistance due to convection, \(Q = \frac{T_o - T_\infty}{R_{conv}}\). Rearranging the equation to make \(T_o\) the subject gives us: \(T_o = T_\infty + Q \cdot R_{conv}\).
04

Evaluate the Design Parameters

We'll substitute the given values of \(D\), \(k\), \(T_w\), \(T_\infty\), \(L_{ins}\), \(h\), and \(L_o = 200mm\) into the expression for \(T_o\) to see if it meets the specified operating limit. If the calculated \(T_o > T_{max}\), then we'll consider alternative design parameters or changes to the current design, such as using a different material, increasing the insulation thickness, increasing the rod length, or attaching the base of the rod differently to the furnace wall. #Phase 1: Conclusion# In this exercise, we derived an expression for the exposed surface temperature of a rod as a function of the given thermal and geometrical parameters. We also evaluated if the given rod with a specific exposed length fulfills the boundary conditions regarding the maximum allowable temperature (\(T_{max}\)). If the given design does not meet the requirements, alternative design parameters should be considered.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Heat Transfer
Heat transfer is the process by which heat energy is exchanged between materials or systems. In this example, the rod connected to the furnace wall transfers heat from the furnace to the surrounding air.
The process can occur through conduction, convection, and radiation. Here, we'll focus on conduction and convection, which are the main forms of heat transfer in the rod scenario.
  • Conduction: This happens when heat moves through a solid, such as the rod itself and the insulation. The heat flows from areas of higher temperature (the furnace side) to areas of lower temperature (the exposed surface).
  • Convection: It comes into play when heat is transferred from the rod's surface to the air. The surrounding air absorbs heat and carries it away from the surface of the rod.
Understanding these two forms of heat transfer helps calculate the temperature at the rod's exposed surface and ensures the temperature is kept below a safe limit to prevent cable damage.
Thermal Resistance
Thermal resistance is a measure of how well a material resists the flow of heat. It acts much like electrical resistance, helping us understand how heat flows through different layers in the system.
  • Conduction Resistance: This is due to the insulation around the rod. The thermal resistance due to conduction, given by \( R_{cond} = \frac{L_{ins}}{k \cdot A} \), shows how much the material opposes the heat flow due to its thermal conductivity \( k \), insulation thickness \( L_{ins} \), and area \( A \).
  • Convection Resistance: Refers to the resistance on the surface of the rod, expressed as \( R_{conv} = \frac{1}{h \cdot A} \). It depends on the convection coefficient \( h \) and the surface area \( A \).
By summing these resistances, we get the total thermal resistance. It helps predict the heat transfer rate, which is crucial for finding the exposed surface temperature \( T_o \). Proper management of thermal resistance ensures the temperature limits are not exceeded.
Convection Coefficient
The convection coefficient \( h \) is a key factor in the rate of convective heat transfer, which describes how efficiently heat is transferred from a surface to a fluid. A higher \( h \) implies better heat transfer as the air absorbs the heat more effectively.
For a rod exposed to air, which in this case refers to the ambient air surrounding the rod, the value \( h = 15 \text{ W/m}^2 \cdot \text{K} \) was given, indicating how heat flows to the air from the rod.

This coefficient is essential in tailoring the design to control temperature effectively. If the calculated exposed surface temperature \( T_o \) exceeds \( T_{max} \), altering \( h \) by changing environmental conditions or surface treatments can often help achieve desired heat transfer.
Thermal Insulation
Thermal insulation plays a crucial role in reducing heat flow from the furnace wall to the environment. By adding an insulating layer, you lower the rate at which heat is conducted along the rod, keeping the exposed surface temperature in check.
Insulation is quantified by its thickness \( L_{ins} \) and material thermal conductivity \( k \). In this exercise, a thickness of 200 mm is specified. A thicker or more conductive insulating material can help keep the temperature lower.
  • Using thicker insulation can improve thermal resistance, thereby lowering the heat transfer rate and protecting against excess heating.
  • Choosing a material with low thermal conductivity enhances insulation capabilities, effectively acting as a barrier to heat flow.
By optimizing insulation properties, you can maintain safe temperature conditions for the rod, ensuring it doesn't exceed operating limits.

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Most popular questions from this chapter

An electrical current of 700 A flows through a stainless steel cable having a diameter of \(5 \mathrm{~mm}\) and an electrical resistance of \(6 \times 10^{-4} \mathrm{\Omega} / \mathrm{m}\) (i.e., per meter of cable length). The cable is in an environment having a temperature of \(30^{\circ} \mathrm{C}\), and the total coefficient associated with convection and radiation between the cable and the environment is approximately \(25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) If the cable is bare, what is its surface temperature? (b) If a very thin coating of electrical insulation is applied to the cable, with a contact resistance of \(0.02 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\), what are the insulation and cable surface temperatures? (c) There is some concern about the ability of the insulation to withstand elevated temperatures. What thickness of this insulation \((k=0.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) will yield the lowest value of the maximum insulation temperature? What is the value of the maximum temperature when this thickness is used?

Consider a tube wall of inner and outer radii \(r_{i}\) and \(r_{o}\), whose temperatures are maintained at \(T_{i}\) and \(T_{o}\), respectively. The thermal conductivity of the cylinder is temperature dependent and may be represented by an expression of the form \(k=k_{o}(1+a T)\), where \(k_{o}\) and \(a\) are constants. Obtain an expression for the heat transfer per unit length of the tube. What is the thermal resistance of the tube wall?

A bonding operation utilizes a laser to provide a constant heat flux, \(q_{o}^{\prime \prime}\), across the top surface of a thin adhesivebacked, plastic film to be affixed to a metal strip as shown in the sketch. The metal strip has a thickness \(d=1.25 \mathrm{~mm}\), and its width is large relative to that of the film. The thermophysical properties of the strip are \(\rho=7850 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=435 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k=60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The thermal resistance of the plastic film of width \(w_{1}=40 \mathrm{~mm}\) is negligible. The upper and lower surfaces of the strip (including the plastic film) experience convection with air at \(25^{\circ} \mathrm{C}\) and a convection coefficient of \(10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The strip and film are very long in the direction normal to the page. Assume the edges of the metal strip are at the air temperature \(\left(T_{\infty}\right)\). (a) Derive an expression for the temperature distribution in the portion of the steel strip with the plastic film \(\left(-w_{1} / 2 \leq x \leq+w_{1} / 2\right)\). (b) If the heat flux provided by the laser is 10,000 \(\mathrm{W} / \mathrm{m}^{2}\), determine the temperature of the plastic film at the center \((x=0)\) and its edges \(\left(x=\pm w_{1} / 2\right)\). (c) Plot the temperature distribution for the entire strip and point out its special features.

A thermopane window consists of two pieces of glass \(7 \mathrm{~mm}\) thick that enclose an air space \(7 \mathrm{~mm}\) thick. The window separates room air at \(20^{\circ} \mathrm{C}\) from outside ambient air at \(-10^{\circ} \mathrm{C}\). The convection coefficient associated with the inner (room-side) surface is \(10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) If the convection coefficient associated with the outer (ambient) air is \(h_{o}=80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), what is the heat loss through a window that is \(0.8 \mathrm{~m}\) long by \(0.5 \mathrm{~m}\) wide? Neglect radiation, and assume the air enclosed between the panes to be stagnant. (b) Compute and plot the effect of \(h_{o}\) on the heat loss for \(10 \leq h_{o} \leq 100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Repeat this calculation for a triple-pane construction in which a third pane and a second air space of equivalent thickness are added.

In a test to determine the friction coefficient \(\mu\) associated with a disk brake, one disk and its shaft are rotated at a constant angular velocity \(\omega\), while an equivalent disk/shaft assembly is stationary. Each disk has an outer radius of \(r_{2}=180 \mathrm{~mm}\), a shaft radius of \(r_{1}=20 \mathrm{~mm}\), a thickness of \(t=12 \mathrm{~mm}\), and a thermal conductivity of \(k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). A known force \(F\) is applied to the system, and the corresponding torque \(\tau\) required to maintain rotation is measured. The disk contact pressure may be assumed to be uniform (i.e., independent of location on the interface), and the disks may be assumed to be well insulated from the surroundings. (a) Obtain an expression that may be used to evaluate \(\mu\) from known quantities. (b) For the region \(r_{1} \leq r \leq r_{2}\), determine the radial temperature distribution \(T(r)\) in the disk, where \(T\left(r_{1}\right)=T_{1}\) is presumed to be known. (c) Consider test conditions for which \(F=200 \mathrm{~N}\), \(\omega=40 \mathrm{rad} / \mathrm{s}, \tau=8 \mathrm{~N} \cdot \mathrm{m}\), and \(T_{1}=80^{\circ} \mathrm{C}\). Evaluate the friction coefficient and the maximum disk temperature.
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