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Chapter 4: Problem 41

What is the physical significance of the Fourier number? Will the Fourier number for a specified heat transfer problem double when the time is doubled?

Short Answer

Expert verified
Answer: Yes, doubling the time in a heat transfer problem will double the Fourier number. This indicates that for the same system and material properties, an increase in the observation time will result in conduction becoming more dominant, relative to heat storage, for the same fractional increase in time.

Step by step solution

01

Understand the Fourier number

The Fourier number (Fo) is a dimensionless parameter that represents the balance between heat conduction and heat storage in a system experiencing unsteady-state heat transfer. It can be defined as follows: Fo = (α * t) / L² Where: - α (alpha): Thermal diffusivity of the material (m²/s) - t: Time (s) - L: Characteristic length of the system (m) In simple terms, a high Fourier number means that heat conduction is dominating the process, and a low Fourier number means that heat storage is dominating. This parameter helps in predicting the behavior of heat transfer systems over time and finding solutions to transient heat conduction problems.
02

Determine the impact of doubling the time on the Fourier number

Now, let's analyze the effect of doubling the time on the Fourier number (Fo). Since Fo is directly proportional to t: Fo' = (α * 2t) / L² We can express the new Fourier number (Fo') in terms of the initial Fourier number (Fo): Fo' = 2 * (α * t) / L² = 2 * Fo So, if the time is doubled, the Fourier number will also double. This means that for the same system and material properties, an increase in the observation time will result in conduction becoming more dominant, relative to heat storage, for the same fractional increase in time.

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Most popular questions from this chapter

A hot brass plate is having its upper surface cooled by impinging jet of air at temperature of \(15^{\circ} \mathrm{C}\) and convection heat transfer coefficient of \(220 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The \(10-\mathrm{cm}\) thick brass plate \(\left(\rho=8530 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=380 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=110 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.\), and \(\alpha=33.9 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) ) has a uniform initial temperature of \(650^{\circ} \mathrm{C}\), and the bottom surface of the plate is insulated. Determine the temperature at the center plane of the brass plate after 3 minutes of cooling. Solve this problem using analytical oneterm approximation method (not the Heisler charts).

An ordinary egg can be approximated as a \(5.5-\mathrm{cm}-\) diameter sphere whose properties are roughly \(k=0.6 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\alpha=0.14 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\). The egg is initially at a uniform temperature of \(8^{\circ} \mathrm{C}\) and is dropped into boiling water at \(97^{\circ} \mathrm{C}\). Taking the convection heat transfer coefficient to be \(h=\) \(1400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine how long it will take for the center of the egg to reach \(70^{\circ} \mathrm{C}\). Solve this problem using analytical one-term approximation method (not the Heisler charts).

The walls of a furnace are made of \(1.2\)-ft-thick concrete \(\left(k=0.64 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right.\) and \(\left.\alpha=0.023 \mathrm{ft}^{2} / \mathrm{h}\right)\). Initially, the furnace and the surrounding air are in thermal equilibrium at \(70^{\circ} \mathrm{F}\). The furnace is then fired, and the inner surfaces of the furnace are subjected to hot gases at \(1800^{\circ} \mathrm{F}\) with a very large heat transfer coefficient. Determine how long it will take for the temperature of the outer surface of the furnace walls to rise to \(70.1^{\circ} \mathrm{F}\). Answer: \(116 \mathrm{~min}\)

An experiment is to be conducted to determine heat transfer coefficient on the surfaces of tomatoes that are placed in cold water at \(7^{\circ} \mathrm{C}\). The tomatoes \((k=0.59 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=\) \(\left.0.141 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}, \rho=999 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=3.99 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)\) with an initial uniform temperature of \(30^{\circ} \mathrm{C}\) are spherical in shape with a diameter of \(8 \mathrm{~cm}\). After a period of 2 hours, the temperatures at the center and the surface of the tomatoes are measured to be \(10.0^{\circ} \mathrm{C}\) and \(7.1^{\circ} \mathrm{C}\), respectively. Using analytical one-term approximation method (not the Heisler charts), determine the heat transfer coefficient and the amount of heat transfer during this period if there are eight such tomatoes in water.

Consider a 1000-W iron whose base plate is made of \(0.5-\mathrm{cm}\)-thick aluminum alloy \(2024-\mathrm{T} 6\left(\rho=2770 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=\right.\) \(\left.875 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \alpha=7.3 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\right)\). The base plate has a surface area of \(0.03 \mathrm{~m}^{2}\). Initially, the iron is in thermal equilibrium with the ambient air at \(22^{\circ} \mathrm{C}\). Taking the heat transfer coefficient at the surface of the base plate to be \(12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and assuming 85 percent of the heat generated in the resistance wires is transferred to the plate, determine how long it will take for the plate temperature to reach \(140^{\circ} \mathrm{C}\). Is it realistic to assume the plate temperature to be uniform at all times?
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