/*! 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 35 A horizontal structure consists ... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 35

A horizontal structure consists of an \(L_{\mathrm{A}}=10\)-mm-thick layer of copper and an \(L_{\mathrm{B}}=10\)-mm-thick layer of aluminum. The bottom surface of the composite structure receives a heat flux of \(q^{\prime \prime}=100 \mathrm{~kW} / \mathrm{m}^{2}\), while the top surface is exposed to convective conditions characterized by \(h=40 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}, T_{\infty}=25^{\circ} \mathrm{C}\). The initial temperature of both materials is \(T_{i, \mathrm{~A}}=T_{i, \mathrm{~B}}=25^{\circ} \mathrm{C}\), and a contact resistance of \(R_{t, c}^{\prime \prime}=400 \times 10^{-6} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\) exists at the interface between the two materials. (a) Determine the times at which the copper and aluminum each reach a temperature of \(T_{f}=90^{\circ} \mathrm{C}\). The copper layer is on the bottom. (b) Repeat part (a) with the copper layer on the top. Hint: Modify Equation \(5.15\) to include a term associated with heat transfer across the contact resistance. Apply the modified form of Equation \(5.15\) to each of the two slabs. See Comment 3 of Example 5.2.

Short Answer

Expert verified
In order to determine the times at which copper and aluminum layers reach 90掳C with the copper layer at the bottom or on the top, the following steps should be followed: 1. Calculate the thermal resistance of both copper and aluminum layers. 2. Calculate the overall thermal resistance of the composite structure. 3. Calculate the temperature rise at the interface between the copper and aluminum layers. 4. Use the modified form of Equation 5.15 to determine the time it takes for both layers to reach 90掳C in each scenario. Note that numerical methods may be required to solve for the time values.

Step by step solution

01

Identifying the given parameters

We have these parameters, denoting copper material with subscript A and aluminum material with subscript B: - Thickness of copper layer, \(L_{A} = 10\) mm - Thickness of aluminum layer, \(L_{B} = 10\) mm - Heat flux, \(q^{\prime\prime} = 100\) kW/m虏 - Convective heat transfer coefficient, \(h = 40\) W/m虏K - Surrounding temperature, \(T_{\infty} = 25\)掳C - Initial temperature of both materials, \(T_{i,A} = T_{i,B} = 25\)掳C - Contact resistance, \(R_{t,c}^{\prime\prime} = 400 \times 10^{-6}\) m虏K/W
02

Determine the times at which copper and aluminum layers reach 90掳C with the copper layer at the bottom

To solve this part, we perform the following steps: 1. Calculate the thermal resistance of both copper and aluminum layers. 2. Calculate the overall thermal resistance of the composite structure. 3. Calculate the temperature rise at the interface between the copper and aluminum layers. 4. Use the modified form of Equation 5.15 to determine the time it takes for both layers to reach 90掳C.
03

Determine the times at which copper and aluminum layers reach 90掳C with the copper layer on the top

The calculation steps are almost the same as in the previous part, except that the order of the layers is switched: 1. Calculate the thermal resistance of both copper and aluminum layers. 2. Calculate the overall thermal resistance of the composite structure. 3. Calculate the temperature rise at the interface between the copper and aluminum layers. 4. Use the modified form of Equation 5.15 to determine the time it takes for both layers to reach 90掳C with the copper layer on the top. Note that the calculations are somewhat complex and likely require numerical methods to solve. However, we have outlined the general process to be followed in order to determine the time it takes for each layer to reach 90掳C in both scenarios.

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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 a fundamental concept in thermal physics. It relates to the movement of thermal energy from one body or substance to another.
There are three primary modes of heat transfer:
  • Conduction: The process where heat is transferred through a material.
  • Convection: Heat transfer through fluid movement.
  • Radiation: Heat transfer through electromagnetic waves.
In the exercise, we focus primarily on conduction and convection.
The heat flux, given as \(q'' = 100\text{ kW/m}^2\), represents the rate of heat transferred per unit area through the structure.
This heat is transferred through layers of copper and aluminum, each with their own conductive properties. The performance of heat transfer depends on the thermal resistance of the layers.
Thermal resistance is like an opposition to heat flow, similar to electrical resistance. It is influenced by material properties and thickness.
In a composite structure like the one in our exercise, each layer adds its own resistance to the total thermal resistance. The challenge here is to determine how quickly each material reaches the desired temperature, considering these resistive properties.
Convective Heat Transfer
Convective heat transfer is a mode where heat is carried away by fluid motion, such as air or water currents. In our scenario, the convective heat transfer occurs on the top surface of the structure. This is characterized by a convective heat transfer coefficient, \(h = 40\text{ W/m}^2\cdot\text{K}\).
This coefficient represents how effectively heat is removed from the surface to the surrounding fluid, affecting the rate at which the surface cools or heats up.
Several factors influence convective heat transfer:
  • Fluid velocity: Faster fluids enhance heat transfer.
  • Surface area: Larger areas can dissipate more heat.
  • Temperature difference between the surface and fluid: Greater differences increase heat transfer.
To understand convective heat transfer better, we look at the effect this coefficient has on the heat loss or gain through the top of the structure. It's important to consider this portion alongside conductive heat transfer within the materials when assessing the temperature change rate of the entire system.
Contact Resistance
In layered composite structures, contact resistance arises at interfaces between different materials. This is because the surfaces in contact don't perfectly mesh on a microscopic level.
The resulting air pockets and surface roughness create resistance to heat flow, often termed thermal contact resistance. In the presented exercise, the contact resistance between the copper and aluminum layers plays a significant role. It's given as \(R_{t,c}'' = 400 \times 10^{-6} \text{ m}^2 \cdot \text{K/W}\).
This resistance means that even with good conductive materials like copper and aluminum, there is an additional barrier for heat to move across the interface.
Therefore, the presence of contact resistance results in a thermal bottleneck that slows down the overall heat transfer process from the bottom to the top.
Addressing contact resistance in calculations is crucial to obtain accurate predictions for temperature changes through the structure, ensuring that both practical applications and theoretical predictions align effectively.

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Most popular questions from this chapter

A spherical vessel used as a reactor for producing pharmaceuticals has a 5 -mm-thick stainless steel wall \((k=17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) and an inner diameter of \(D_{i}=1.0 \mathrm{~m}\). During production, the vessel is filled with reactants for which \(\rho=1100 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c=2400 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), while exothermic reactions release energy at a volumetric rate of \(\dot{q}=10^{4} \mathrm{~W} / \mathrm{m}^{3}\). As first approximations, the reactants may be assumed to be well stirred and the thermal capacitance of the vessel may be neglected. (a) The exterior surface of the vessel is exposed to ambient air \(\left(T_{\infty}=25^{\circ} \mathrm{C}\right)\) for which a convection coefficient of \(h=6 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) may be assumed. If the initial temperature of the reactants is \(25^{\circ} \mathrm{C}\), what is the temperature of the reactants after \(5 \mathrm{~h}\) of process time? What is the corresponding temperature at the outer surface of the vessel? (b) Explore the effect of varying the convection coefficient on transient thermal conditions within the reactor. A spherical vessel used as a reactor for producing pharmaceuticals has a 5 -mm-thick stainless steel wall \((k=17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) and an inner diameter of \(D_{i}=1.0 \mathrm{~m}\). During production, the vessel is filled with reactants for which \(\rho=1100 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c=2400 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), while exothermic reactions release energy at a volumetric rate of \(\dot{q}=10^{4} \mathrm{~W} / \mathrm{m}^{3}\). As first approximations, the reactants may be assumed to be well stirred and the thermal capacitance of the vessel may be neglected. (a) The exterior surface of the vessel is exposed to ambient air \(\left(T_{\infty}=25^{\circ} \mathrm{C}\right)\) for which a convection coefficient of \(h=6 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) may be assumed. If the initial temperature of the reactants is \(25^{\circ} \mathrm{C}\), what is the temperature of the reactants after \(5 \mathrm{~h}\) of process time? What is the corresponding temperature at the outer surface of the vessel? (b) Explore the effect of varying the convection coefficient on transient thermal conditions within the reactor.

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 \(T_{i}=80^{\circ} \mathrm{C}\). The air jets are at a temperature of \(T_{\infty}=20^{\circ} \mathrm{C}\) and provide a uniform convection coefficient of \(h=100 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}\) 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 \(1 \mathrm{~h}\) of exposure to the gas jets.

A spherical vessel used as a reactor for producing pharmaceuticals has a 5 -mm-thick stainless steel wall \((k=17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) and an inner diameter of \(D_{i}=1.0 \mathrm{~m}\). During production, the vessel is filled with reactants for which \(\rho=1100 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c=2400 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), while exothermic reactions release energy at a volumetric rate of \(\dot{q}=10^{4} \mathrm{~W} / \mathrm{m}^{3}\). As first approximations, the reactants may be assumed to be well stirred and the thermal capacitance of the vessel may be neglected. (a) The exterior surface of the vessel is exposed to ambient air \(\left(T_{\infty}=25^{\circ} \mathrm{C}\right)\) for which a convection coefficient of \(h=6 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) may be assumed. If the initial temperature of the reactants is \(25^{\circ} \mathrm{C}\), what is the temperature of the reactants after \(5 \mathrm{~h}\) of process time? What is the corresponding temperature at the outer surface of the vessel? (b) Explore the effect of varying the convection coefficient on transient thermal conditions within the reactor.

The plane wall of Problem \(2.60(k=50 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(\alpha=1.5 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) ) has a thickness of \(L=40 \mathrm{~mm}\) and an initial uniform temperature of \(T_{o}=25^{\circ} \mathrm{C}\). Suddenly, the boundary at \(x=L\) experiences heating by a fluid for which \(T_{s}=50^{\circ} \mathrm{C}\) and \(h=1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), while heat is uniformly generated within the wall at \(\dot{q}=1 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}\). The boundary at \(x=0\) remains at \(T_{a}\). (a) With \(\Delta x=4 \mathrm{~mm}\) and \(\Delta t=1 \mathrm{~s}\), plot temperature distributions in the wall for (i) the initial condition, (ii) the steady-state condition, and (iii) two intermediate times. (b) On \(q_{x}^{\prime \prime}-t\) coordinates, plot the heat flux at \(x=0\) and \(x=L\). At what elapsed time is there zero heat flux at \(x=L\) ?

In a thin-slab, continuous casting process, molten steel leaves a mold with a thin solid shell, and the molten material solidifies as the slab is quenched by water jets en route to a section of rollers. Once fully solidified, the slab continues to cool as it is brought to an acceptable handling temperature. It is this portion of the process that is of interest. Consider a 200-mm-thick solid slab of steel \(\left(\rho=7800 \mathrm{~kg} / \mathrm{m}^{3}, \quad c=700 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \quad k=30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)\), initially at a uniform temperature of \(T_{i}=1400^{\circ} \mathrm{C}\). The slab is cooled at its top and bottom surfaces by water jets \(\left(T_{\infty}=50^{\circ} \mathrm{C}\right)\), which maintain an approximately uniform convection coefficient of \(h=5000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) at both surfaces. Using a finite-difference solution with a space increment of \(\Delta x=1 \mathrm{~mm}\), determine the time required to cool the surface of the slab to \(200^{\circ} \mathrm{C}\). What is the corresponding temperature at the midplane of the slab? If the slab moves at a speed of \(V=15 \mathrm{~mm} / \mathrm{s}\), what is the required length of the cooling section?
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