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Chapter 5: Problem 1

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

Short Answer

Expert verified
(a) The temperature distributions on T鈫恱 coordinates are as follows: 1. Initial state: Horizontal line at \(T_{\infty}\). 2. Steady-state: Linear temperature distribution with highest temperature at the inner surface and lowest at the outer surface. 3. Intermediate time 1: Exponential temperature distribution with most significant difference between initial and steady-state distributions near the heater's surface. 4. Intermediate time 2: Exponential temperature distribution closer to the steady-state distribution, with the largest differences near the heater's surface. (b) The heat flux at the outer surface \(q_{x}^{\prime \prime}(L, t)\) as a function of time will show an exponential increase until reaching a steady constant value, indicating a constant heat conduction rate through the plate at the steady-state.

Step by step solution

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a) Sketching the temperature distributions on T鈫恱 coordinates

We need to sketch the temperature distributions at four different stages: initial, steady-state, and two intermediate times. 1. Initial state: The heater and plate are at the ambient temperature, \(T_{\infty}\), which means that the whole plate will be at the same temperature, so the graph will be a horizontal line at \(T_{\infty}\). 2. Steady-state: After some time, the system reaches a steady state where the temperature distribution within the plate becomes linear, with the highest temperature at the inner surface (where the heater is located) and lowest at the outer surface (backed by insulation. 3. Intermediate time 1: At the first intermediate time, when the heater is activated, we will see an exponential temperature distribution with the most significant difference between the initial and steady-state distributions near the heater's surface. 4. Intermediate time 2: At the second intermediate time, the temperature distribution is still exponential but closer to the steady-state distribution, meaning the plate has started to reach equilibrium, but still, the largest differences will be seen near the heater's surface.
02

b) Sketching the heat flux at the outer surface \(q_{x}^{\prime \prime}(L, t)\) as a function of time.

Initially, the heat flux at the outer surface is zero because the heater is not yet activated. After the heater is activated, heat will be conducted through the plate. As time progresses, we will see an exponential increase in the heat flux at the outer surface of the plate, indicating that more heat is being conducted through the plate until it reaches a maximum value. Once the maximum value of the heat flux at the outer surface is reached, the system reaches a steady-state, and the heat flux remains constant. The heat flux will be constant as a function of time, indicating that the heat is being conducted through the plate at a steady rate. So, the graph of the heat flux at the outer surface \(q_{x}^{\prime \prime}(L, t)\) as a function of time will show an exponential increase until reaching a steady constant value.

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Key Concepts

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

Temperature Distribution
Temperature distribution describes how temperature varies across different points on a material or object. Consider a thin plate with a heater attached to one side. Initially, the entire plate is at the ambient temperature, denoted as \( T_{\infty} \), so the temperature distribution is a flat line on a temperature vs. position graph.

Once the heater is turned on, the inner surface starts to warm up. At this stage, the temperature distribution becomes exponential. It has a higher temperature at the heater's surface and gradually decreases as you move away from the heater. As time passes, this distribution approaches a steady-state.
  • Initial State: Uniform temperature \( T_{\infty} \).
  • Intermediate Times: Exponential curve starting to form.
  • Steady-State: Linear temperature drop from inside to outside.
This progression is crucial for understanding how materials respond to thermal inputs over time. The intermediate stages show how heat is conducted through the plate until it achieves a homogeneous temperature gradient in the steady-state.
Steady-State Heat Transfer
Steady-state heat transfer occurs when the temperatures in a system no longer change over time. Once the plate reaches this state, its temperature distribution becomes linear. This linearity results from a consistent flow of heat across the plate, maintaining a balance between heat entering at the heater and heat leaving at the outer insulated surface.

In the context of our plate:
  • The surface closest to the heater will have the highest temperature.
  • There will be a gradual, linear temperature drop across the thickness of the plate.
  • The outer surface will maintain the lowest temperature, determined by the insulation and ambient conditions.
Reaching steady-state is essential in thermal system design, as it allows for the prediction of system behavior and performance. It ensures that temperatures remain consistent, avoiding fluctuations that could lead to inefficiencies or damage.
Heat Flux
The term 'heat flux' refers to the rate of heat flow through a surface per unit area, measured typically in \( \mathrm{W/m}^2 \). In our heater-plate scenario, the heat flux has dynamic behaviors over time.

Initially, before the heater is activated, no heat is conducted through the plate, giving a zero heat flux at the outer surface.
  • Initial: Zero heat flux.
  • Post Activation: As the heater turns on, heat starts conducting through the plate, and so the heat flux increases.
  • Exponential Increase: Heat flux rises steeply over time until the system reaches equilibrium.
  • Steady-state: When equilibrium is achieved, the heat flux becomes constant, reflecting a persistent heat transfer across the plate from the heater to the outer surface.
Understanding the behavior of heat flux is vital in informing how quickly a system can respond to changes and how much energy is required to maintain desired conditions. It aids in the effective design and management of thermal systems.

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Most popular questions from this chapter

Thermal energy storage systems commonly involve a packed bed of solid spheres, through which a hot gas flows if the system is being charged, or a cold gas if it is being discharged. In a charging process, heat transfer from the hot gas increases thermal energy stored within the colder spheres; during discharge, the stored energy decreases as heat is transferred from the warmer spheres to the cooler gas. Consider a packed bed of \(75-\mathrm{mm}\)-diameter aluminum spheres \(\left(\rho=2700 \mathrm{~kg} / \mathrm{m}^{3}, c=950 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=\right.\) \(240 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) ) and a charging process for which gas enters the storage unit at a temperature of \(T_{g, i}=300^{\circ} \mathrm{C}\). If the initial temperature of the spheres is \(T_{i}=25^{\circ} \mathrm{C}\) and the convection coefficient is \(h=75 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), how long does it take a sphere near the inlet of the system to accumulate \(90 \%\) of the maximum possible thermal energy? What is the corresponding temperature at the center of the sphere? Is there any advantage to using copper instead of aluminum?

A plane wall of a furnace is fabricated from plain carbon steel \(\left(k=60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=7850 \mathrm{~kg} / \mathrm{m}^{3}, c=430 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) and is of thickness \(L=10 \mathrm{~mm}\). To protect it from the corrosive effects of the furnace combustion gases, one surface of the wall is coated with a thin ceramic film that, for a unit surface area, has a thermal resistance of \(R_{t, f}^{\prime \prime}=0.01 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\). The opposite surface is well insulated from the surroundings. At furnace start-up the wall is at an initial temperature of \(T_{i}=300 \mathrm{~K}\), and combustion gases at \(T_{\infty}=1300 \mathrm{~K}\) enter the furnace, providing a convection coefficient of \(h=25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) at the ceramic film. Assuming the film to have negligible thermal capacitance, how long will it take for the inner surface of the steel to achieve a temperature of \(T_{s, i}=1200 \mathrm{~K}\) ? What is the temperature \(T_{s, o}\) of the exposed surface of the ceramic film at this time?

One end of a stainless steel (AISI 316) rod of diameter \(10 \mathrm{~mm}\) and length \(0.16 \mathrm{~m}\) is inserted into a fixture maintained at \(200^{\circ} \mathrm{C}\). The rod, covered with an insulating sleeve, reaches a uniform temperature throughout its length. When the sleeve is removed, the rod is subjected to ambient air at \(25^{\circ} \mathrm{C}\) such that the convection heat transfer coefficient is \(30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Using the explicit finite-difference technique with a space increment of \(\Delta x=0.016 \mathrm{~m}\), estimate the time required for the midlength of the rod to reach \(100^{\circ} \mathrm{C}\). (b) With \(\Delta x=0.016 \mathrm{~m}\) and \(\Delta t=10 \mathrm{~s}\), compute \(T(x, t)\) for \(0 \leq t \leq t_{1}\), where \(t_{1}\) is the time required for the midlength of the rod to reach \(50^{\circ} \mathrm{C}\). Plot the temperature distribution for \(t=0,200 \mathrm{~s}, 400 \mathrm{~s}\), and \(t_{1}\).

The density and specific heat of a particular material are known \(\left(\rho=1200 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=1250 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\), but its thermal conductivity is unknown. To determine the thermal conductivity, a long cylindrical specimen of diameter \(D=40 \mathrm{~mm}\) is machined, and a thermocouple is inserted through a small hole drilled along the centerline. The thermal conductivity is determined by performing an experiment in which the specimen is heated to a uniform temperature of \(T_{i}=100^{\circ} \mathrm{C}\) and then cooled by passing air at \(T_{\infty}=25^{\circ} \mathrm{C}\) in cross flow over the cylinder. For the prescribed air velocity, the convection coefficient is \(h=55 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) If a centerline temperature of \(T(0, t)=40^{\circ} \mathrm{C}\) is recorded after \(t=1136 \mathrm{~s}\) of cooling, verify that the material has a thermal conductivity of \(k=0.30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). (b) For air in cross flow over the cylinder, the prescribed value of \(h=55 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) corresponds to a velocity of \(V=6.8 \mathrm{~m} / \mathrm{s}\). If \(h=C V^{0.618}\), where the constant \(C\) has units of \(\mathrm{W} \cdot \mathrm{s}^{0.618} / \mathrm{m}^{2.618} \cdot \mathrm{K}\), how does the centerline temperature at \(t=1136 \mathrm{~s}\) vary with velocity for \(3 \leq V \leq 20 \mathrm{~m} / \mathrm{s}\) ? Determine the centerline temperature histories for \(0 \leq t \leq 1500 \mathrm{~s}\) and velocities of 3,10 , and \(20 \mathrm{~m} / \mathrm{s}\).

When a molten metal is cast in a mold that is a poor conductor, the dominant resistance to heat flow is within the mold wall. Consider conditions for which a liquid metal is solidifying in a thick-walled mold of thermal conductivity \(k_{v}\) and thermal diffusivity \(\alpha_{w}\). The density and latent heat of fusion of the metal are designated as \(\rho\) and \(h_{s f}\), respectively, and in both its molten and solid states, the thermal conductivity of the metal is very much larger than that of the mold. Just before the start of solidification \((S=0)\), the mold wall is everywhere at an initial uniform temperature \(T_{i}\) and the molten metal is everywhere at its fusion (melting point) temperature of \(T_{f}\). Following the start of solidification, there is conduction heat transfer into the mold wall and the thickness of the solidified metal \(S\) increases with time \(t\). (a) Sketch the one-dimensional temperature distribution, \(T(x)\), in the mold wall and the metal at \(t=0\) and at two subsequent times during the solidification. Clearly indicate any underlying assumptions. (b) Obtain a relation for the variation of the solid layer thickness \(S\) with time \(t\), expressing your result in terms of appropriate parameters of the system.
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