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Chapter 1: Problem 15

The 5 -mm-thick bottom of a \(200-\mathrm{mm}\)-diameter pan may be made from aluminum \((k=240 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) or copper \((k=390 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). When used to boil water, the surface of the bottom exposed to the water is nominally at \(110^{\circ} \mathrm{C}\). If heat is transferred from the stove to the pan at a rate of \(600 \mathrm{~W}\), what is the temperature of the surface in contact with the stove for each of the two materials?

Short Answer

Expert verified
The temperature of the surface in contact with the stove for the aluminum pan is \(129.90^{\circ}C\) and for the copper pan is \(122.42^{\circ}C\).

Step by step solution

01

Convert given values to SI Units.

First, we need to convert the measurements given in millimeters to meters, so that we use the right units in our calculations: Thickness of the pan: \(5~mm = 0.005~m\) Diameter of the pan: \(200~mm = 0.2~m\)
02

Calculate the surface area of the pan.

Next, we'll calculate the surface area in contact with the stove. Since the pan is circular, the surface area formula is given by: \[ A = \pi r^2 \] where \(r\) = radius of the pan. The radius is half the diameter, so that's \(0.1~m\). Thus: \[ A = \pi (0.1)^2 = 0.0314~m^2 \]
03

Reorganize the conduction equation for temperature difference.

We need to find the temperature difference \(\Delta T\) across the material. Rearrange the conduction equation to solve for \(\Delta T\): \[ \Delta T = -\frac{q \Delta x}{kA} \]
04

Calculate temperature difference for each material.

Now, we will plug in the given values for heat transfer rate, thickness, thermal conductivity, and surface area to find the temperature difference for each material. For aluminum: \[ \Delta T_{Al} = -\frac{(600)(0.005)}{(240)(0.0314)} = -19.90~K \] For copper: \[ \Delta T_{Cu} = -\frac{(600)(0.005)}{(390)(0.0314)} = -12.42~K \]
05

Calculate the temperature of the surface in contact with the stove.

We are now able to find the temperature of the surface in contact with the stove. Since we have the temperature of the surface in contact with the water (that is, 110掳C), we can subtract the temperature difference to find the required temperature surface. Remember that the temperature difference is negative. For aluminum: \[ T_{Al} = (110) - (-19.90) = 129.90^{\circ}C \] For copper: \[ T_{Cu} = (110) - (-12.42) = 122.42^{\circ}C \] Thus, the temperature of the surface in contact with the stove is \(129.90^{\circ}C\) for aluminum and \(122.42^{\circ}C\) for copper.

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Key Concepts

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

Thermal Conductivity
Thermal conductivity, represented by the symbol \(k\), is a measure of a material's ability to conduct heat. It quantifies the rate at which heat energy is transferred through a material when there is a temperature difference across it. The higher the thermal conductivity of a material, the more effectively it can transport heat from one side to the other.

In the context of our problem, different materials for the pan鈥攁luminum and copper鈥攈ave different thermal conductivities. Aluminum has a thermal conductivity of \(240 \mathrm{~W} / \mathrm{m} \textrm{K}\), while copper's thermal conductivity is \(390 \mathrm{~W} / \mathrm{m} \textrm{K}\). These values indicate that copper, having higher \(k\), can transfer heat more efficiently than aluminum. This implies that if both materials are used to fabricate pans with equivalent dimensions and under identical conditions, the pan made of copper will have a lower temperature gradient compared to the one made of aluminum, suggesting a more uniform heat distribution across the pan.
Conduction Equation
The conduction equation relates the rate of heat transfer through a material to the material's properties and temperature difference. It is generally expressed in the form:

\[\begin{equation} q = \frac{kA\Delta T}{\Delta x} \end{equation}\]
where:
  • \(q\) is the rate of heat transfer in watts (W),
  • \(k\) is the thermal conductivity of the material,
  • \(A\) is the cross-sectional area through which heat is being transferred,
  • \(\Delta T\) is the temperature difference across the material, and
  • \(\Delta x\) is the thickness of the material.

In our exercise, we're interested in solving for the temperature difference \(\Delta T\) using the rearranged conduction equation:

\[\begin{equation} \Delta T = -\frac{q \Delta x}{kA} \end{equation}\]
This equation allows us to calculate how much hotter or cooler one side of a material will be in comparison to the other when heat is applied at a known rate. It's pivotal for engineers to calculate the appropriate thickness and materials for effective heat distribution in objects such as cooking pans.
Temperature Difference Calculation
Calculating temperature difference is an essential step in understanding heat conduction in materials. The temperature difference, \(\Delta T\), is simply the difference between the temperature on one side of a material and the temperature on the opposite side.

If heat is being added or removed from a system, the temperature difference becomes dynamic and can change over time. However, in many practical steady-state cases, such as in our pan example, the rate of heat transfer is constant, and so is the temperature difference.

By using the conduction equation, we can find the temperature difference caused by a certain amount of heat transfer for any given material. As we saw in our exercise, the calculation helps us predict the temperature at the surface in contact with the stove which is critical for safety and cooking performance considerations. Knowing the temperature difference and thermal properties of materials also aids in the design of thermal systems, such as in insulation installation, HVAC system planning, and even in the culinary design of kitchenware.

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Most popular questions from this chapter

If \(T_{s} \approx T_{\text {sur }}\) in Equation \(1.9\), the radiation heat transfer coefficient may be approximated as $$ h_{r, a}=4 \varepsilon \sigma \bar{T}^{3} $$ where \(\bar{T} \equiv\left(T_{s}+T_{\text {sur }}\right) / 2\). We wish to assess the validity of this approximation by comparing values of \(h_{r}\) and \(h_{r, a}\) for the following conditions. In each case, represent your results graphically and comment on the validity of the approximation. (a) Consider a surface of either polished aluminum ( \(\varepsilon=\) \(0.05)\) or black paint \((\varepsilon=0.9)\), whose temperature may exceed that of the surroundings \(\left(T_{\text {sur }}=25^{\circ} \mathrm{C}\right)\) by 10 to \(100^{\circ} \mathrm{C}\). Also compare your results with values of the coefficient associated with free convection in air \(\left(T_{\infty}=T_{\text {sur }}\right)\), where \(h\left(\mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)=0.98 \Delta T^{1 / 3}\). (b) Consider initial conditions associated with placing a workpiece at \(T_{s}=25^{\circ} \mathrm{C}\) in a large furnace whose wall temperature may be varied over the range \(100 \leq\) \(T_{\text {sur }} \leq 1000^{\circ} \mathrm{C}\). According to the surface finish or coating, its emissivity may assume values of \(0.05\), \(0.2\), and \(0.9\). For each emissivity, plot the relative error, \(\left(h_{r}-h_{r, a}\right) / h_{r}\), as a function of the furnace temperature.

Single fuel cells such as the one of Example \(1.5\) can be scaled up by arranging them into a fuel cell stack. A stack consists of multiple electrolytic membranes that are sandwiched between electrically conducting bipolar plates. Air and hydrogen are fed to each membrane through fiw channels within each bipolar plate, as shown in the sketch. With this stack arrangement, the individual fuel cells are connected in series, electrically, producing a stack voltage of \(E_{\text {stack }}=N \times E_{c}\), where \(E_{c}\) is the voltage produced across each membrane and \(N\) is the number of membranes in the stack. The electrical current is the same for each membrane. The cell voltage, \(E_{c}\), as well as the cell efficiency, increases with temperature (the air and hydrogen fed to the stack are humidified to allow operation at temperatures greater than in Example 1.5), but the membranes will fail at temperatures exceeding \(T \approx 85^{\circ} \mathrm{C}\). Consider \(L \times w\) membranes, where \(L=w=100 \mathrm{~mm}\), of thickness \(t_{m}=0.43 \mathrm{~mm}\), that each produce \(E_{c}=0.6 \mathrm{~V}\) at \(I=60 \mathrm{~A}\), and \(\dot{E}_{c g}=45 \mathrm{~W}\) of thermal energy when operating at \(T=80^{\circ} \mathrm{C}\). The external surfaces of the stack are exposed to air at \(T_{\infty}=25^{\circ} \mathrm{C}\) and surroundings at \(T_{\text {sur }}=30^{\circ} \mathrm{C}\), with \(\varepsilon=0.88\) and \(h=150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Find the electrical power produced by a stack that is \(L_{\text {stack }}=200 \mathrm{~mm}\) long, for bipolar plate thickness in the range \(1 \mathrm{~mm}

The diameter and surface emissivity of an electrically heated plate are \(D=300 \mathrm{~mm}\) and \(\varepsilon=0.80\), respectively. (a) Estimate the power needed to maintain a surface temperature of \(200^{\circ} \mathrm{C}\) in a room for which the air and the walls are at \(25^{\circ} \mathrm{C}\). The coefficient characterizing heat transfer by natural convection depends on the surface temperature and, in units of \(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\), may be approximated by an expression of the form \(h=0.80\left(T_{s}-T_{\infty}\right)^{1 / 3}\). (b) Assess the effect of surface temperature on the power requirement, as well as on the relative contributions of convection and radiation to heat transfer from the surface.

A rectangular forced air heating duct is suspended from the ceiling of a basement whose air and walls are at a temperature of \(T_{\infty}=T_{\text {sur }}=5^{\circ} \mathrm{C}\). The duct is \(15 \mathrm{~m}\) long, and its cross section is \(350 \mathrm{~mm} \times 200 \mathrm{~mm}\). (a) For an uninsulated duct whose average surface temperature is \(50^{\circ} \mathrm{C}\), estimate the rate of heat loss from the duct. The surface emissivity and convection coefficient are approximately \(0.5\) and \(4 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. (b) If heated air enters the duct at \(58^{\circ} \mathrm{C}\) and a velocity of \(4 \mathrm{~m} / \mathrm{s}\) and the heat loss corresponds to the result of part (a), what is the outlet temperature? The density and specific heat of the air may be assumed to be \(\rho=1.10 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c_{p}=1008 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), respectively.

An instrumentation package has a spherical outer surface of diameter \(D=100 \mathrm{~mm}\) and emissivity \(\varepsilon=0.25\). The package is placed in a large space simulation chamber whose walls are maintained at \(77 \mathrm{~K}\). If operation of the electronic components is restricted to the temperature range \(40 \leq T \leq 85^{\circ} \mathrm{C}\), what is the range of acceptable power dissipation for the package? Display your results graphically, showing also the effect of variations in the emissivity by considering values of \(0.20\) and \(0.30\).
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