化工應用數學

1. 化工應用數學 授課教師： 郭修伯 Lecture 1

2. 實驗數據處理與分析 應用數學表達物理現象 常微分方程式 傅立葉分析 偏微分方程式 數值解析 最佳化設計 給分標準 期中考 30% 期末考 40% 作業 30% 授課內容

3. 教科書 及 參考書 • 教科書： • Mathematical Methods in Chemical Engineering (Author: V.G. Jenson and G.V. Jeffreys; Second Edition; Publisher: Academic Press) • 參考書： • Partial Differential Equations and Boundary-Value Problems with Applications (Author: M.A. Pinsky; Publisher: McGraw-Hill) • Applied Mathematics in Chemical Engineering (Author: H. S. Mickley, T. K. Sherwood, and C. E. Reed) • Advanced Engineering Mathematics (Author: P.V. O’Neil; Publisher: Wadsworth)

4. Identify the main components of the system Attempt to relate the variables by means of one or more equations Observe the behaviour of the system 應用數學 • Constructing a mathematical model to describe, understand, and predict behaviour of a physical process or system.

5. 應用數學 • Experiment • How to analyse experimental results? • Is it accurate? • Mathematics • How to build mathemitical models to describe physical problems? • How to solve the mathematical equations?

6. 實驗 • Why do experiments? • test theoretical predictions by comparison with experiment • develop empirical formulae • Data analysis • Some of the data points are often inaccurate and methods must be found for eliciting reliable information with reasonable certainty. • Statistics • Presenting the results graphically Purpose: to extract maximum information

7. Chapter 10 Treatment of experimental results

8. 實驗結果繪圖 • 選用何種座標？ • 有因次式 • 無因次式 • 選用何種繪圖紙？ • Linear graph paper • Semi-logarithmic graph paper • Logarithmic graph paper • Triangular graph paer

9. 無因次群 • If a particular piece of equipment compares with other work on similar equipment, the appropriate dimensionless groups should be used along the axes. • If the dimensionless groups are chosen correctly, each group should only contain one principal variable and the rest of its constituent parts should be parameters. 目的在比較scale不同，但進行相似實驗的結果

10. 繪圖 • Axes: avoid plotting a dimensionless group against a dimensional variable. • Many types of graph paper can be used to present experimental data. The most desirable shape for a curve is a straight line.

11. 繪圖紙 • Linear graph paper • y = a + bx • Semi-logarithmic graph paper • useful for the cases involving the approach to steady-state conditions. The dependent variable is a decaying exponential function of the independent variable. • y = ke bxln y = ln k + bx • The gradient, b,must be determined from the linear measurement of lny and not from reading the scales which gives values of y

12. 繪圖紙 • Logarithmic graph paper • covers a wide range • Dimensional analysis frequently indicates an empirical equation of the form y and x are dimensionless groups 1 1 y ln y 2 3 2 3 x ln x

13. 繪圖紙 • Triangular graph paper • In the study of liquid-liquid extraction systems where three components are present in two phases, a convenient graphical representation of the composition of a phase is needed. • The data can be presented on a triangular diagram by using the geometrical properties of a triangle. A F E P C B D

14. 誤差擴大(Propagation of errors) • The absolute error in the result is the sum of the absolute errors in the constituent parts. • z = x + y • z = x + y • The absolute error in the difference of two quantities is the sum of the absolute errors in those quantities. • z = x - y • z = x + y • The relative error in a production is the sum of the relative errors in the constituent parts. • z = x y • zr = xr + yr • The relative error in a quotient is the sum of the relative errors in the constituent parts. • z = x / y • zr = xr + yr derivation

15. 誤差範例 • Propagation of errors through a general functional relationship • z = f (x,y) • Example: • If z = x y n with n known and x and y determined experimentally, determine the relative error in z in terms of the relative errors in x and y.

16. z = x y n Z 相對於 x 和 y 的誤差為何？ Ans: zr = xr + n yr

17. 誤差範例 If a chemical reaction A B has a first order reaction rate constant k (s-1), the concentration of A leaving a tubular reactor of length L (m) with velocity u (m/s) is: where C0 is the initial concentration of A, diffusion has been neglect and plug flow has been assumed. How accurately must k be known and the flow rate be steady for it to be possible to design a reactor to give 94.5 - 95.5% completion? 0.95 0.05

18. C0 C L f = C/C0 假設誤差的來源為 k 及 u

19. f = 0.95 f = 0.005 兩者的相對誤差和不能大於10％

20. 實驗式 (Empirical equation) • The equation must be truely representative of the experimental data and it should be simple in form. • The form of the equation is frequently suggested by a theoretical analysis and it is necessary only to evaluate certain contants. • The general problem of fitting data by an empirical equation may be divided into two parts: • the determination of a suitable form of equation • the evaluation of the constants

21. 常見的實驗式型態 • Equations involving more than two constants should be avoided.

22. 繪曲線(Curve fitting) • How do you fit a set of experimental data for x and y? • graph fitting • method of averages • method of least squares

23. 平均法 (Method of averages) • The best curve is the one passing through the average points. • Procedure • determine the type of curve • arrange the value of x in ascending order • divide the experimental results into groups and the number of groups must equal the number of unknown parameters • groups contain approximately equal numbers of points • substitute the “average point” from the grouped data into the equation of the chosen curve • determine the equation of the curve

24. 平均法範例 The thermal conductivity of graphite varies with temperature according to the equation Experimentally, it is only possible to obtain a mean conductivity over a temperature range. It is required to find the point conductivity from the mean conductivity given below. km is determined between 25°C and T

25. T1 = 25 ºC T2 L An elementary heat balance B.C. x=0, T=T1 x=L, T=T2 對兩邊積分

26. 續上頁... 平均法： 將數據分為兩組

27. 最小平方法 (Method of least squares) • Most frequenly used method to fit the best straight line to a set of data. • This method defines the best straight line as the one for which the sum of the squares of the error terms is a minimum. • Determine the values of m and c which gives the least mean squares fit of the equation: y = m x + c derivation

28. 最小平方法範例 It has been proposed that the second order chemical reaction proceeds on the surface of an activated carbon catalyst after adsorption of the two reactants. Each of the three substances is adsorbed to a different, but the number of sites occupied by carbon monoxide is small compared with sites otherwise occupied. Assuming that the process is controlled by the surface reaction, which is irreversible, find the best values of the adsorption coefficients from the following experimental results. CO + Cl2 COCl2

29. CO + Cl2 COCl2 a + b  c 光氣 Let C = number of sites in the stage specified K = adsorption coefficient k = specific reaction rate constant p = partial pressure r = rate of reaction t = total number of sites v = vacant site The equilibrium of the three components between the catalyst and the vapour phase can be expressed as: The rate law of the elementary reaction 反應速率方程式 吸附方程式 Number of total active sites

30. We measured pa, pb, pc, and rc 消去 Ca, Cb, Cc 消去 Cv Experimental error

31. Square the left-hand of the equation and sum over the N experimental points: Differentiate the equation partially with respect to , , and  equal to zero, which gives the minimum value of sum (Rn2)

32. 數值積分 (Numerical integration) • It is sometimes necessary to perform a calculation which involves integration. For example, the volumetric flow rate of a gas through a duct can be determined from the linear velocity distribution by evaluating a suitable integral. • One way of integrating a set of data is by fitting an empirical equation to the points and then integrating the equation analytically. • Polynomials are used to fit data for integration purposes.

33. The trapezium rule • The linear equation I1 I = average height x width

34. Simpson’s rule • The cubic equation • It passes through any three chosen points at equally spaced values of x. • If the range of integration is subdivided into equal intervals by using any odd number of ordinates (say 7), Simpson’s rule can be applied to each group of three points and the result of all integrations added together. I = average height x width

35. -1 1 Gauss’s method • It enables a polynomial of degree (2n-1) to be fitted to n points. • If the values of y at the three positions indicated by the green arrows can be determined, the shape of the quintic curve can also be determined. (n = 3) • Two-points method: y1 and y2 are evaluated at x =  0.5773 • Four-points method: y1, y2, y3 and y4 are evaluated at x = -0.8611, -0.3400, 0.3400, 0.8611 Homework

36. Summary of today’s course • 無因次群的目的在於比較「相似」系統的實驗數據。 • 避免無因次群對有因次群作圖。 • 應用不同的繪圖紙，以利繪出直線關係式。 • 三角座標圖可用於顯示溶劑中三種不同溶質的濃度。 • 實驗誤差隨著之後的運算而擴大。 • 實驗值的 fitting可用平均法及最小平方法。後者較常使用。 • 數值積分可用 trapezium, Simpson’s, Gauss’s 等三個方法。 • 似線性關係時，多用 trapezium 方法。