化工應用數學

1. 化工應用數學 Treatment of Experimental Results 授課教師： 林佳璋

2. 應用數學與實驗數據 • Experiment -How to analyse experimental results?(如何分析實驗數據) -Is it accurate?(實驗誤差) • Mathematics -How to build mathematical models to describe physical problems? (建立數學模式描述問題) -How to solve the mathematical equations?(數學模式求解) • Why do experiments? -Test theoretical predictions by comparison with experiment (理論值與實驗值比較；驗證理論的合理性與準確性) -Develop empirical formulae(求得實驗經驗式) • Data analysis -Some of the data points are often inaccurate and methods must be foundfor eliciting reliable information with reasonable certainty. (減少實驗誤差) -Statistics(統計問題) -Presenting the results graphically(實驗數據繪圖)

3. 實驗數據分析 列表 圖示 缺點：無法明確比較操作變數對系統的影響 優點：可以比較實驗值與理論值之差異

4. 實驗數據繪圖 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. • 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 paper 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.

5. 實驗數據繪圖

6. 1 3 2 2 3 實驗數據繪圖模式 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 independentvariable. y = kebx ln y = ln k + bx The gradient, b, must be determined from the linear measurement of lny and not from reading the scales which givesvalues of y logarithmic graph paper covers a wide range Dimensional analysis frequently indicates an empirical equation of the form y and x are dimensionless groups y ln y 1 x ln x

7. 常見的實驗式型態 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 constants. The general problem of fitting data by an empirical equation may be divided into two parts: -the determination of a suitable form of equation (合適的實驗經驗式) -the evaluation of the constants (常數求取) Equations involving more than two constants should be avoided.

8. Curve Fitting How do you fit a set of experimental data for x and y? • graph fitting • method of averages • method of least squares

9. Curve Fitting~平均法 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

10. Curve Fitting~平均法 Suppose that eight experimental values of a variable y are available at eight different known values of x, and the best curve of the type …. …. where Rn are the error terms Assuming that R1+R2+R3=0, R4+R5=0, R6+R7+R8=0

11. Curve Fitting~平均法 平均數據降低誤差

12. 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

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

14. 平均法範例 平均法：將數據分為兩組

15. Curve Fitting~最小平方法 Most frequently used method to fit the best straight line to a set of data.(直線迴歸實驗數據) This method defines the best straight line as the one for which the sum of the squares of the error terms is a minimum. (誤差平方總和為最小) Determine the values of m and c which gives the least mean squares fit of the equation: y = m x + c

16. Curve Fitting~最小平方法 誤差值 N個實驗數據 對m及c微分

17. 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 extent, 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 adsorptioncoefficients from the following experimental results. CO + Cl2 COCl2

18. 最小平方法範例 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 rate law of the elementary reaction The equilibrium of the three components between the catalyst and the vapour phase can be expressed as: 反應速率方程式 Number of total active sites 吸附方程式

19. 最小平方法範例 We measured pa, pb, pc, and rc 消去 Ca, Cb, Cc 消去 Cv

20. 最小平方法範例 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)

21. 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.

22. 梯形法(Trapezium Rule) two points I1 three points four points five points six points seven points eight points nine points I = average height  width

23. y h h y2 y3 y1 x1 x2 x3 x 辛普森法(Simpson’s Rule) The cubic equation It passes through any three chosen points at equally spaced values of x. 定義新的參數 z=-1, y=y1 z=0, y=y2 z=1, y=y3

24. 辛普森法(Simpson’s Rule) I = average height width

25. 辛普森法(Simpson’s Rule) 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. five points nine points

26. a b 高斯法(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. x=a, u=-1 x=b, u=1

27. 高斯法(Gauss’s Method)

28. 高斯法(Gauss’s Method) n=2 n=5 n=4 n=3

29. 數值積分範例 Evaluate by using the following methods (a) Analytical (b) Trapezium rule (three points) (c) Trapezium rule (nine points) (d) Simpson’s rule (three points) (e) Simpson’s rule (nine points) (f) Gauss three-point (g) Gauss four-point (a)x=sinhx dx=coshxdz

30. 數值積分範例 (b) Trapezium rule (three points) I=y1+2y2+y3=1.00000+2(0.44722)+0.24254=2.1369 (c) Trapezium rule (nine points) I=1/4(y1+2y2+2y3+2y4+2y5+2y6+2y7+2y8+y9) =1/4(1.00000+0.24254)+ 1/2(0.89445+0.70711+0.55475+0.44722+0.37138+0.31623+0.27473) =2.0936

31. 數值積分範例 (d) Simpson’s rule (three points) I=1/6(y1+4y2+y3)4=2/3(1.00000+1.78888+0.24254)=2.0209 (e) Simpson’s rule (nine points) I=1/24(y1+4y2+2y3+4y4+2y5+4y6+2y7+4y8+y9)4 =1/6(1.00000+0.24254)+1/3(0.70711+0.44722+0.31623) +2/3(0.8945+0.55475+0.37138+0.27473) =2.0941 (f) Gauss three-point x=0.4508, 2, 3.5492 I=4/18[5(0.91165+0.27118)+8(0.44722)]=2.1093 (g) Gauss four-point I=4[0.1739(0.96351+0.25945)+0.3261(0.60386+0.34959)]=2.0944 x=0.2778, 1.3200, 2.6800, 3.7222

32. 誤差來源 Accidental errors of measurement Such errors are inevitable in all measurements and that they result from small unavoidable errors of observation due to more or less fortuitous variation in the sensitivity of measuring instrument and the keenness of the senses of perception. (例如採用不準確的A來校正B，用B的錯誤校 正曲線進行量測） Precision and constant errors A result may be extremely precise and at the same time highly inaccurate. Constant errors can be detected only by performing the measurement with a number of different instruments and , if possible, by several independent methods and observers. (例如採用不準確的儀器或樣品取 樣在不具代表性的地方） Errors of methods These arises as a result of approximations and assumptions made in the theoretical development of an equation used to calculate the desired result. (例如在計算時，使用錯誤的假設）

33. 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=xy 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

34. 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. Ans: zr = xr + n yr x, y符號相反 Ans: zr = xr -n yr

35. 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? f = C/C0 0.95 0.005