1 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. CHAPTER 20 20.1 A plot of log10k versus log10f can be developed as y = 0.4224x - 0.83R2 = 0.9532-0.6-0.4-0.200.20.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 As shown, the best fit line is 10 10log 0.422363log 0.83fk Therefore, 2 = 100.83 = 0.147913 and 2 = 0.422363, and the power model is 0.4223630.147913yx The model and the data can be plotted as y = 0.1479x0.4224R2 = 0.953200.511.50 20 40 60 80 100 120 140 160 20.2 We can first try a linear fit y = 2.5826x + 1365.9R2 = 0.96081200140016001800-60-30 0 30 60 90120 As shown, the fit line is somewhat lacking. Therefore, we can use polynomial regression to fit a parabola2 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. y = 0.0128x2 + 1.8164x + 1331R2 = 0.99341200140016001800-60 -30 0 30 60 90 120 This fit seems adequate in that it captures the general trend of the data. Note that a slightly better fit can be attained with a cubic polynomial, but the improvement is marginal. 20.3 (a) The linear fit is y = 1.05897x + 0.81793R2 = 0.90583020406080020406080 The tensile strength at t = 32 can be computed as 1.05897(32) 0.81793 34.7048913y (b) A straight line with zero intercept can be fit as y = 1.07514xR2 = 0.905560204060800 20406080 For this case, the tensile strength at t = 32 can be computed as 1.07514(32) 34.40452y 20.4 Linear regression with a zero intercept gives [note that T(K) = T(oC) + 273.15].3 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. y = 29.728xR2 = 0.99990500010000150000 100 200 300 400 500 Thus, the fit is 29.728pT Using the ideal gas law pVRTn For our fit 29.728pT For nitrogen, 1 kg28 g/molen Therefore, 31029.728 8.32410 / 28R This is close to the standard value of 8.314 J/gmole. 20.5 This problem is ideally suited for Newton interpolation. First, order the points so that they are as close to and as centered about the unknown as possible. x0 = 740 f(x0) = 0.1406 x1 = 760 f(x1) = 0.15509 x2 = 720 f(x2) = 0.12184 x3 = 780 f(x3) = 0.16643 x4 = 700 f(x4) = 0.0977 The results of applying Newton’s polynomial at T = 750 are Order f(x) Error 0 0.14060 0.007245 1 0.14785 0.0005344 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. 2 0.14838 -7E-05 3 0.14831 0.00000 4 0.14831 Note that the third-order polynomial yields an exact result, and so we conclude that the interpolation is 0.14831. 20.6 A program can be written to fit a natural cubic spline to this data and also generate the first and second derivatives at each knot. Option Explicit Sub Splines() Dim i As Integer, n As Integer Dim x(100) As Double, y(100) As Double, xu As Double, yu As Double Dim dy As Double, d2y As Double Dim resp As Variant Range("a5").Select n = ActiveCell.Row Selection.End(xlDown).Select n = ActiveCell.Row - n Range("a5").Select For i = 0 To n x(i) = ActiveCell.Value ActiveCell.Offset(0, 1).Select y(i) = ActiveCell.Value ActiveCell.Offset(1, -1).Select Next i Range("c5").Select Range("c5:d1005").ClearContents For i = 0 To n Call Spline(x(), y(), n, x(i), yu, dy, d2y) ActiveCell.Value = dy ActiveCell.Offset(0, 1).Select ActiveCell.Value = d2y ActiveCell.Offset(1, -1).Select Next i Do resp = MsgBox("Do you want to interpolate?", vbYesNo) If resp = vbNo Then Exit Do xu = InputBox("z = ") Call Spline(x(), y(), n, xu, yu, dy, d2y) MsgBox "For z = " & xu & Chr(13) & "T = " & yu & Chr(13) & _ "dT/dz = " & dy & Chr(13) & "d2T/dz2 = " & d2y Loop End Sub Sub Spline(x, y, n, xu, yu, dy, d2y) Dim e(100) As Double, f(100) As Double, g(100) As Double, r(100) As Double, d2x(100) As Double Call Tridiag(x, y, n, e, f, g, r) Call Decomp(e(), f(), g(), n - 1) Call Substit(e(), f(), g(), r(), n - 1, d2x()) Call Interpol(x, y, n, d2x(), xu, yu, dy, d2y) End Sub Sub Tridiag(x, y, n, e, f, g, r) Dim i As Integer f(1) = 2 * (x(2) - x(0)) g(1) = x(2) - x(1)5 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. r(1) = 6 / (x(2) - x(1)) * (y(2) - y(1)) r(1) = r(1) + 6 / (x(1) - x(0)) * (y(0) - y(1)) For i = 2 To n - 2 e(i) = x(i) - x(i - 1) f(i) = 2 * (x(i + 1) - x(i - 1)) g(i) = x(i + 1) - x(i) r(i) = 6 / (x(i + 1) - x(i)) * (y(i + 1) - y(i)) r(i) = r(i) + 6 / (x(i) - x(i - 1)) * (y(i - 1) - y(i)) Next i e(n - 1) = x(n - 1) - x(n - 2) f(n - 1) = 2 * (x(n) - x(n - 2)) r(n - 1) = 6 / (x(n) - x(n - 1)) * (y(n) - y(n - 1)) r(n - 1) = r(n - 1) + 6 / (x(n - 1) - x(n - 2)) * (y(n - 2) - y(n - 1))
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