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Chapter 5: Problem 1

Consider a thin electrical heater artached to a plate and hacked by insulation. Initially, the heater and plate are at the temperafure of the ambient air, \(T\) - Suddenly, the power to the heater is activated, yielding a constant heat fux \(q_{e}^{*}\left(\mathrm{~W} / \mathrm{m}^{2}\right)\) at the inner surface of the plate. (a) Sketch and label, on \(T-x\) cocrdinates, the temperature distributions: initial, steady-state, and at two intermediate times. (b) Sketch the heat flux at the outer surface \(q_{z}^{\prime \prime}(L, t)\) as a function of time.

Short Answer

Expert verified
The solution involves sketching: (a) temperature profiles at initial, intermediate, and steady-state times showing curving to linear; (b) a heat flux curve rising from zero to a constant value.

Step by step solution

01

Understand the Problem

The problem involves a thin electrical heater attached to a plate with insulation at the back. Initially, the system is at ambient temperature \( T \). Then, the heater is turned on, providing a constant heat flux \( q_e^{*} \) at the plate's inner surface. We need to sketch the temperature distributions at various times and the heat flux at the outer surface as a function of time.
02

Initial Temperature Distribution

Since the system starts at a uniform ambient temperature \( T \), the initial temperature distribution on a \( T-x \) graph is simply a horizontal line at temperature \( T \) across the entire plate's thickness.
03

Steady-State Temperature Distribution

Once the system reaches steady-state, the temperature will vary linearly from the inner surface, where the constant heat flux is applied, to the outer surface. On the \( T-x \) graph, this is depicted as a linear line starting at a higher temperature at the inner surface and sloping down to a lower temperature at the outer surface.
04

Temperature Distribution at Intermediate Times

At intermediate times, the temperature distribution will not be linear. Initially, the temperature near the heater rises while the rest of the plate remains close to the ambient temperature. As time progresses, the heat spreads through the plate, creating curves on the \( T-x \) graph that gradually transition from the initial horizontal line to the final straight line of steady-state.
05

Heat Flux at the Outer Surface Over Time

Initially, the heat flux at the outer surface \( q_z''(L, t) \) is approximately zero since the entire plate is at ambient temperature. As time progresses and the heat moves through the plate, the heat flux will increase. Eventually, in the steady-state condition, it will stabilize at a constant value, equal to the input heat flux \( q_e^{*} \). On a \( q_z''(L, t)-t \) graph, this appears as a curve starting from zero and leveling off to a constant value.

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Key Concepts

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

Temperature Distribution
In heat transfer analysis, understanding how temperature varies across a material is essential. Initially, the temperature distribution in the plate is uniform, matching the ambient temperature around it. This can be visualized as a flat horizontal line on a graph plotting temperature ( T ) against the position within the plate ( x ).
As the electrical heater begins to emit heat, the temperature closest to the heater increases first.
This creates a curved profile as the heat permeates through the plate. The section nearest the heater will be warmer, while the rest of the plate slowly catches up.
Over time, the temperature profile shifts from a curve to a straight, linear line. This occurs once the plate reaches a point where the temperature is constant over time, known as steady-state.
At this point, the temperature seamlessly decreases from inside the plate, where it is warmed, to the cooler outer surface. This linear distribution reflects how heat continuously distributes via conduction from the heater across the thickness of the plate until it balances out toward steady state.
Steady-State Conditions
Steady-state conditions occur when the temperature no longer changes with time at any point in the system.
In the context of our heater and plate setup, this means the temperature distribution becomes time-invariant across the plate's material.
Physically, this implies that the heat entering at the inner surface exactly matches the heat leaving the outer surface of the plate.
With a constant heat flux being applied by the heater, the system steadily loses heat to the surroundings at a matching rate, achieving equilibrium.
  • Heat transfer within the plate reaches equilibrium between input and output.
  • Temperature difference between the surfaces establishes a gradient sufficient to conduct all incoming heat through the material.
This is represented graphically by a straight, descending line from the inner surface to the outer surface, marking a constant rate of temperature decline.
No more temperature change means the steady-state is achieved, maintaining a consistent energy balance across the entire system.
Heat Flux
Heat flux describes the rate of heat transfer per unit area across a surface, influencing how energy moves through different materials.
At the inner surface of the plate, a constant heat flux is initiated by the heater, starting the process of warming the material.
Initially, at the outer surface, heat flux is minimal because the temperature of the entire plate has yet to rise significantly.
  • As time continues, heat flux at the outer surface increases as the energy transfers through the plate.
  • This increase continues until steady-state conditions are achieved, whereby the heat flux stabilizes.
During steady-state, the heat flux at the outer surface maintains a constant value equivalent to the heat flux introduced by the heater.
This constancy in heat flux ensures that any heat applied to one side exits through the other without accumulation or depletion, crucial for maintaining thermal stability and efficiency in heating applications.
The graph representing heat flux over time at the outer surface reflects a rising curve that eventually levels out, highlighting the transition from initial heating to a stable state.

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Most popular questions from this chapter

Copper-coated, epoxy-filled fiberglass circuit boards are treated by heating a stack of them under high pressare as sown in the sketch. The perpose of the pressing-heuting operation is to cure the eposy that bonds the fiberglass sheets, imparting siffness to the boards. The stack. riferted to as a book, is comprised of 10 boards and 11 pressing plates, which prevent epoxy from flowing between the boards and impar a-smooth finish to the cured boards. In order to perform simplified thermal analyses, it is reasonable to approximule the book as having an effective themal conductivity \((k)\) and an effective thermal capacitance \(\left(\rho c_{p}\right)\). Calculate the effective propertics if each of the boards and plates has a tickness of \(2.36 \mathrm{~mm}\) and the following thermophysical properties: board (b) \(\rho_{b}=1000 \mathrm{~kg} / \mathrm{m}^{\prime}, c_{p,}=1500\) \(\mathrm{W} / \mathrm{Kg} \cdot \mathrm{K}, k_{p}=0.30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}_{;}\)plate \((p) \rho_{p}=8\left(000 \mathrm{~kg} / \mathrm{m}^{3}\right.\). \(c_{v}=480 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k_{p}=12 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\).

It is well known that, although two materials are at the same temperature, one may feel cooler to the touch than the other. Consider thick plates of copper and glass, each at an initial temperature of \(300 \mathrm{~K}\). Assuming your finger to be at an initial temperature of \(310 \mathrm{~K}\) and to have thermophysical properties of \(\rho=1000 \mathrm{~kg} / \mathrm{m}^{3}, c=4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k=0.625 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), determine whether the copper or the glass will feel cooler to the touch.

A chip that is of length \(L=5 \mathrm{~mm}\) on a side and thick. ness \(t=I \mathrm{~mm}\) is encased in a ceramic substrate, and its exposed surface is convectively cooled by a diefectric liquid for which \(h=150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(T_{\mathrm{z}}=20^{\circ} \mathrm{C}\). In the off-mode the chip is in thermal equilibrium with the coolant \(\left(T_{i}=T_{*}\right)\). When the chip is energized, however, its Icmperafure increases until a new steady-state is established. For purpeises of analysis, the cnergized chip is characterized by uniform volumetric heating with \(q=9 \times 10^{6} \mathrm{~W} / \mathrm{m}^{3}\). Assuming an infinite contacl resistance between the chip and substrate and negligible conduction resistance within the chip. determine the steady-state chip temperature \(T_{f}\). Following activation of the chip, how long does it take to come within \(1^{-1} C\) of this temperature? The chip density and specific heat are \(\rho=2000 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c=700 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), respectively.

A solid steel sphere (AISI 1010), \(300 \mathrm{~mm}\) in diameter, is couted with a dielectric material layer of thickness \(2 \mathrm{~mm}\) and thermal conductivity \(0.04 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The couted sphere is initially at a uniform temperabure of \(500^{7} \mathrm{C}\) and is suddenly quenched in a large oil bath for which \(T_{x}=100^{\circ} \mathrm{C}\) and \(h=3300 \mathrm{~W} / \mathrm{m}^{2}-\mathrm{K}\). Estimate the time required for the coated sphere temperature to reach \(140^{\circ} \mathrm{C}\). Hint: Neglect the effect of energy storage in the dielectric material, since its thermal capocitance \((\rho \in V)\) is small compared to that of the steel sphere.

A molded plastic product \(\left(\rho=1200 \mathrm{~kg} / \mathrm{m}^{3} . c=\right.\) \(1500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=0.30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) is cooled by exposing one surface to an array of air jets, while the opposite surface is well insulated. The product may be approximated as a slab of thickness \(L=60 \mathrm{~mm}\). which is initially at a uniform temperature of \(T_{i}=\) \(80 \% \mathrm{C}\). The air jets are at a temperature of \(T_{2}=20^{\circ} \mathrm{C}\) and provide a uniform convection coefficient of \(h=\) \(100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) at the cooled surface. Using a finite-difference solution with a space increment of \(\Delta x=6 \mathrm{~mm}\), determine temperatures at the cooled and insulated surfaces after 1 hour of exposure to the gas jets.
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