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Chapter 5: Problem 140

What is a practical way of checking if the round-off error has been significant in calculations?

Short Answer

Expert verified
Answer: Round-off errors are small inaccuracies that occur when numbers are rounded or truncated to a fixed number of digits in numerical calculations. The significance of round-off errors lies in their potential to accumulate and propagate during calculations, leading to inaccuracies in the final result. Estimating their significance can be done by performing calculations with different levels of precision and comparing the results. If the results do not change significantly with varying precision levels, it can be concluded that the round-off error is not significant, and the results are reliable.

Step by step solution

01

Understanding Round-off Errors

In numerical calculations, especially when using computers, numbers are often rounded or truncated to a fixed number of digits due to limitations in memory and processing capacity. This process can lead to small inaccuracies called round-off errors. These errors can accumulate and propagate during calculations, leading to significant inaccuracies in the final result.
02

Importance of Checking Round-off Errors

It is important to check for the significance of round-off errors to ensure the accuracy and reliability of the numerical results. In some cases, if the round-off error is too large, it can lead to incorrect conclusions or decisions based on the output of the calculations. Therefore, understanding and managing round-off errors is a crucial aspect of numerical analysis.
03

The Simplest Way of Estimating Round-off Errors

A practical method for checking the significance of round-off errors is to perform the calculations with different levels of precision (i.e., change the number of significant digits or the number of decimal places) and compare the results. If the results of the calculations do not change significantly with different levels of precision, it can be concluded that the round-off error is not significant, and the results are reliable.
04

Example of Checking Round-off Error Significance

Let's consider a simple example of calculating the volume of a cylinder with a radius of 3.14159 meters and a height of 1.234567 meters. Both radius and height are given to six decimal places. Perform the calculation for the volume of the cylinder \(V = \pi r^2 h\) using different levels of precision for radius and height values: 1. Truncate numbers to four decimal places: \(V_1 = \pi (3.1415)^2 (1.2345)\) 2. Truncate numbers to five decimal places: \(V_2 = \pi (3.14159)^2 (1.23456)\) 3. Original values: \(V_3 = \pi (3.14159)^2 (1.234567)\) Compare the results: If the change in the calculated volume is insignificant between different levels of precision, then the round-off error can be assumed as not significant in this case.

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Most popular questions from this chapter

Consider a nuclear fuel element $(k=57 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\( that can be modeled as a plane wall with thickness of \)4 \mathrm{~cm}\(. The fuel element generates \)3 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}$ of heat uniformly. Both side surfaces of the fuel element are cooled by liquid with temperature of \(80^{\circ} \mathrm{C}\) and convection heat transfer coefficient of $8000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. Using a uniform nodal spacing of \)8 \mathrm{~mm}$, (a) obtain the finite difference equations, (b) determine the nodal temperatures by solving those equations, and (c) compare the surface temperatures of both sides of the fuel element with the analytical solution.

A hot surface at \(100^{\circ} \mathrm{C}\) is to be cooled by attaching \(3-\mathrm{cm}-\) long, \(0.25-\mathrm{cm}\)-diameter aluminum pin fins $(k=237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ with a center-to-center distance of \(0.6 \mathrm{~cm}\). The temperature of the surrounding medium is \(30^{\circ} \mathrm{C}\), and the combined heat transfer coefficient on the surfaces is \(35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming steady one-dimensional heat transfer along the fin and taking the nodal spacing to be \(0.5 \mathrm{~cm}\), determine \((a)\) the finite difference formulation of this problem, \((b)\) the nodal temperatures along the fin by solving these equations, \((c)\) the rate of heat transfer from a single fin, and \((d)\) the rate of heat transfer from a \(1-\mathrm{m} \times 1-\mathrm{m}\) section of the plate.

Hot combustion gases of a furnace are flowing through a concrete chimney \((k=1.4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) of rectangular cross section. The flow section of the chimney is $20 \mathrm{~cm} \times 40 \mathrm{~cm}\(, and the thickness of the wall is \)10 \mathrm{~cm}$. The average temperature of the hot gases in the chimney is \(T_{i}=280^{\circ} \mathrm{C}\), and the average convection heat transfer coefficient inside the chimney is \(h_{l}=75 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The chimney is losing heat from its outer surface to the ambient air at $T_{0}=15^{\circ} \mathrm{C}\( by convection with a heat transfer coefficient of \)h_{o}=18 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and to the sky by radiation. The emissivity of the outer surface of the wall is \(\varepsilon=0.9\), and the effective sky temperature is estimated to be \(250 \mathrm{~K}\). Using the finite difference method with \(\Delta x=\Delta y=10 \mathrm{~cm}\) and taking full advantage of symmetry, \((a)\) obtain the finite difference formulation of this problem for steady two-dimensional heat transfer, (b) determine the temperatures at the nodal points of a cross section, and \((c)\) evaluate the rate of heat loss for a \(1-m\)-long section of the chimney.

A nonmetal plate \((k=0.05 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) is attached on the upper surface of an ASME SB-96 copper-silicon plate $(k=36 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$. The nonmetal plate and the ASME SB-96 plate have thicknesses of \(20 \mathrm{~mm}\) and \(30 \mathrm{~mm}\), respectively. The bottom surface of the ASME SB-96 plate (surface 1) is subjected to a uniform heat flux of \(150 \mathrm{~W} / \mathrm{m}^{2}\). The top nonmetal plate surface (surface 2) is exposed to convection at an air temperature of \(15^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of \(5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Copper- silicon alloys are not always suitable for applications where they are exposed to certain median and high temperatures. Therefore, the ASME Boiler and Pressure Vessel Code (ASME BPVC.IV-2015, HF-300) limits equipment constructed with ASME SB-96 plate to be operated at a temperature not exceeding \(93^{\circ} \mathrm{C}\). Using the finite difference method with a uniform nodal spacing of \(\Delta x=5 \mathrm{~mm}\) along the the temperature distribution as a function of \(x\). Would the use of the ASME SB- 96 plate under these conditions be in compliance with the ASME Boiler and Pressure Vessel Code? What is the lowest value of the convection heat transfer coefficient for the air so that the ASME SB-96 plate is below $93^{\circ} \mathrm{C}$ ?

A stainless steel plate is connected to a copper plate by long ASTM B98 copper-silicon bolts \(9.5 \mathrm{~mm}\) in diameter. The portion of the bolts exposed to convection heat transfer with hot gas is \(5 \mathrm{~cm}\) long. The gas temperature for convection is at \(500^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of $50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. The thermal conductivity of the bolt is known to be \)36 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The stainless steel plate has a uniform temperature of \(100^{\circ} \mathrm{C}\), and the copper plate has a uniform temperature of \(80^{\circ} \mathrm{C}\). According to the ASME Code for Process Piping (ASME B31.3-2014, Table A-2M), the maximum use temperature for an ASTM B98 copper-silicon bolt is \(149^{\circ} \mathrm{C}\). Using the finite difference method with a uniform nodal spacing of \(\Delta x=5 \mathrm{~mm}\) along the bolt, determine the temperature at each node. Plot the temperature distribution along the bolt. Compare the numerical results with the analytical solution. Would any part of the ASTM B 98 bolts be above the maximum use temperature of \(149^{\circ} \mathrm{C}\) ?
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