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Chapter 4: Problem 92

A thick wall made of refractory bricks \((k=1.0 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\alpha=5.08 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\) ) has a uniform initial temperature of \(15^{\circ} \mathrm{C}\). The wall surface is subjected to uniform heat flux of \(20 \mathrm{~kW} / \mathrm{m}^{2}\). Using EES (or other) software, investigate the effect of heating time on the temperature at the wall surface and at \(x=1 \mathrm{~cm}\) and \(x=5 \mathrm{~cm}\) from the surface. Let the heating time vary from 10 to \(3600 \mathrm{~s}\), and plot the temperatures at \(x=0,1\), and \(5 \mathrm{~cm}\) from the wall surface as a function of heating time.

Short Answer

Expert verified
Answer: The key steps to analyze the effect of heating time on the temperature at different points within a refractory brick wall are: 1. Prepare for calculation in a compatible software. 2. Input given values and create variables for unknowns. 3. Calculate the temperature at different distances. 4. Vary the heating time and calculate temperatures. 5. Plot the temperatures as a function of heating time.

Step by step solution

01

Prepare for calculation in the software

Open your chosen software and ensure you are familiar with entering data and equations, as well as plotting data.
02

Input given values and create variables for unknowns

Enter the given values of thermal conductivity (k), thermal diffusivity (α), initial temperature (T_initial), and heat flux (q) into the software. Also, create variables for the temperature at different distances (T0, T1, T5) and heating times (t).
03

Calculate the temperature at different distances

Using the software's equation solver, apply the appropriate heat conduction equation to calculate the temperature at each distance (x = 0, 1, and 5 cm). You will need to modify the equation depending on the distance from the surface.
04

Vary the heating time and calculate temperatures

Now, create a loop (or similar function) in the software to vary the heating time from 10 to 3600 seconds, with a reasonable step size (such as 10 seconds). In each iteration of the loop, calculate the temperature at the given distances for the current heating time and store the results in a table or array.
05

Plot the temperatures as a function of heating time

Once the loop finishes and you have the calculated temperatures at each distance for every heating time, plot the results. Create a plot with heating time on the x-axis and temperature on the y-axis. Plot T0, T1, and T5 as separate curves to compare their temperature evolution over time. With these steps, you can study the effect of heating time on the temperature at the wall surface and at different distances from the surface. The plot will show how the temperature changes with heating time at each location.

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Most popular questions from this chapter

Aluminum wires, \(3 \mathrm{~mm}\) in diameter, are produced by extrusion. The wires leave the extruder at an average temperature of \(350^{\circ} \mathrm{C}\) and at a linear rate of \(10 \mathrm{~m} / \mathrm{min}\). Before leaving the extrusion room, the wires are cooled to an average temperature of \(50^{\circ} \mathrm{C}\) by transferring heat to the surrounding air at \(25^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Calculate the necessary length of the wire cooling section in the extrusion room.

The soil temperature in the upper layers of the earth varies with the variations in the atmospheric conditions. Before a cold front moves in, the earth at a location is initially at a uniform temperature of \(10^{\circ} \mathrm{C}\). Then the area is subjected to a temperature of \(-10^{\circ} \mathrm{C}\) and high winds that resulted in a convection heat transfer coefficient of \(40 \mathrm{~W} / \mathrm{m}^{2}\). \(\mathrm{K}\) on the earth's surface for a period of \(10 \mathrm{~h}\). Taking the properties of the soil at that location to be \(k=0.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\alpha=1.6 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\), determine the soil temperature at distances \(0,10,20\), and \(50 \mathrm{~cm}\) from the earth's surface at the end of this \(10-\mathrm{h}\) period.

In Betty Crocker's Cookbook, it is stated that it takes \(2 \mathrm{~h} \mathrm{} 45 \mathrm{~min}\) to roast a \(3.2-\mathrm{kg}\) rib initially at \(4.5^{\circ} \mathrm{C}\) "rare" in an oven maintained at \(163^{\circ} \mathrm{C}\). It is recommended that a meat thermometer be used to monitor the cooking, and the rib is considered rare done when the thermometer inserted into the center of the thickest part of the meat registers \(60^{\circ} \mathrm{C}\). The rib can be treated as a homogeneous spherical object with the properties \(\rho=1200 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=4.1 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=0.45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(\alpha=\) \(0.91 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\). Determine \((a)\) the heat transfer coefficient at the surface of the rib; \((b)\) the temperature of the outer surface of the rib when it is done; and \((c)\) the amount of heat transferred to the rib. \((d)\) Using the values obtained, predict how long it will take to roast this rib to "medium" level, which occurs when the innermost temperature of the rib reaches \(71^{\circ} \mathrm{C}\). Compare your result to the listed value of \(3 \mathrm{~h} \mathrm{} 20 \mathrm{~min}\). If the roast rib is to be set on the counter for about \(15 \mathrm{~min}\) before it is sliced, it is recommended that the rib be taken out of the oven when the thermometer registers about \(4^{\circ} \mathrm{C}\) below the indicated value because the rib will continue cooking even after it is taken out of the oven. Do you agree with this recommendation? Solve this problem using analytical one-term approximation method (not the Heisler charts).

A 6-mm-thick stainless steel strip \((k=21 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(\rho=8000 \mathrm{~kg} / \mathrm{m}^{3}\), and \(\left.c_{p}=570 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) exiting an oven at a temperature of \(500^{\circ} \mathrm{C}\) is allowed to cool within a buffer zone distance of \(5 \mathrm{~m}\). To prevent thermal burn to workers who are handling the strip at the end of the buffer zone, the surface temperature of the strip should be cooled to \(45^{\circ} \mathrm{C}\). If the air temperature in the buffer zone is \(15^{\circ} \mathrm{C}\) and the convection heat transfer coefficient is \(120 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the maximum speed of the stainless steel strip.

Conduct the following experiment at home to determine the combined convection and radiation heat transfer coefficient at the surface of an apple exposed to the room air. You will need two thermometers and a clock. First, weigh the apple and measure its diameter. You can measure its volume by placing it in a large measuring cup halfway filled with water, and measuring the change in volume when it is completely immersed in the water. Refrigerate the apple overnight so that it is at a uniform temperature in the morning and measure the air temperature in the kitchen. Then take the apple out and stick one of the thermometers to its middle and the other just under the skin. Record both temperatures every \(5 \mathrm{~min}\) for an hour. Using these two temperatures, calculate the heat transfer coefficient for each interval and take their average. The result is the combined convection and radiation heat transfer coefficient for this heat transfer process. Using your experimental data, also calculate the thermal conductivity and thermal diffusivity of the apple and compare them to the values given above.
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