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Chapter 4: Problem 56

A long cylindrical wood \(\log (k=0.17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\left.\alpha=1.28 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)\) is \(10 \mathrm{~cm}\) in diameter and is initially at a uniform temperature of \(15^{\circ} \mathrm{C}\). It is exposed to hot gases at \(550^{\circ} \mathrm{C}\) in a fireplace with a heat transfer coefficient of \(13.6 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the surface. If the ignition temperature of the wood is \(420^{\circ} \mathrm{C}\), determine how long it will be before the log ignites. Solve this problem using analytical one-term approximation method (not the Heisler charts).

Short Answer

Expert verified
Answer: First, calculate the Biot number and then use the one-term approximation method formula to find the time needed for ignition, as specified in the provided step-by-step solution.

Step by step solution

01

Calculate the Biot Number

Calculate the Biot number using the formula: Bi = (h * L)/k where h is the heat transfer coefficient, L is the log radius(diameter/2), and k is the thermal conductivity. We can assume the one-term approximation method is appropriate for this exercise if the Biot number is less than or equal to 0.1. Bi = (13.6 (W/m²⋅K) * 0.05 (m)) / 0.17 (W/m⋅K)
02

Calculate the time needed for ignition

We can use the following formula to find the time needed for ignition using the one-term approximation method: t = [ (h * L * (T_s - T_i) * sqrt(pi))/(3 * k * (T_i - T_o)) ]² / alpha where T_i= Initial temperature of the wood T_o= Temperature of the hot gases T_s= Ignition temperature of the wood L= Radius of the log k= Thermal conductivity of the wood h= Heat transfer coefficient alpha= Thermal diffusivity of the wood t = [ (13.6 (W/m²⋅K) * 0.05 (m) * (420 (°C) - 15 (°C)) * sqrt(pi))/(3 * 0.17 (W/m⋅K) * (15 (°C) - 550 (°C))) ]² / (1.28 * 10^(-7) (m^2/s)) Calculate the value of t to find the time it takes for the log to ignite.

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Most popular questions from this chapter

Consider the freezing of packaged meat in boxes with refrigerated air. How do \((a)\) the temperature of air, (b) the velocity of air, \((c)\) the capacity of the refrigeration system, and \((d)\) the size of the meat boxes affect the freezing time?

A large heated steel block \(\left(\rho=7832 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=\right.\) \(434 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=63.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(\left.\alpha=18.8 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)\) is allowed to cool in a room at \(25^{\circ} \mathrm{C}\). The steel block has an initial temperature of \(450^{\circ} \mathrm{C}\) and the convection heat transfer coefficient is \(25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming that the steel block can be treated as a quarter-infinite medium, determine the temperature at the edge of the steel block after 10 minutes of cooling.

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.

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).

Chickens with an average mass of \(2.2 \mathrm{~kg}\) and average specific heat of \(3.54 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}\) are to be cooled by chilled water that enters a continuous-flow-type immersion chiller at \(0.5^{\circ} \mathrm{C}\). Chickens are dropped into the chiller at a uniform temperature of \(15^{\circ} \mathrm{C}\) at a rate of 500 chickens per hour and are cooled to an average temperature of \(3^{\circ} \mathrm{C}\) before they are taken out. The chiller gains heat from the surroundings at a rate of \(210 \mathrm{~kJ} / \mathrm{min}\). Determine \((a)\) the rate of heat removal from the chicken, in \(\mathrm{kW}\), and \((b)\) the mass flow rate of water, in \(\mathrm{kg} / \mathrm{s}\), if the temperature rise of water is not to exceed \(2^{\circ} \mathrm{C}\).
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