Chemical Reactor Design Youn-Woo Lee School of Chemical and Biological Engineering Seoul National University 155-741, 599 Gwanangro, Gwanak-gu, Seoul, Korea  ywlee@snu.ac.kr  http://sfpl.snu.ac.kr

第7章 Collection and Analysis of Rate Data Seoul National University

개요 제5장과 6장에서 우리는 일단 속도식을 알면, 이를 적절한 몰수지에 대입하고, 적절한 화학양론관계를 사용함으로써, 어떤 등온 반응계를 분석하기 위한 화학반응공학 알고리즘을 활용할 수 있다는 것을 보여주었다. 이 장에서는 특정반응에 대한 속도식을 얻기 위하여 반응속도자료를 수집하고 해석하는 방법에 대해 중점을 두고자 한다. 특히 속도자료를 얻기 위한 두 가지의 일반적인 형태의 반응기를 다룬다: 균일반응에 주로 사용되는 회분식 반응기 고체-유체의 불균일반응에 사용되는 미분반응기 Seoul National University

수집된 자료를 분석하기 위해서 3가지 방법이 사용된다. 미분법 적분법 비선형 회귀분석법 (최소자승법) 회분식 반응기 실험에서는 농도, 압력, 또는 부피 등을 반응이 진행되는 시간에 따라 측정하여 기록하는 것이 보통이다. 자료는 회분식반응기에서는 반응진행 중에서 측정하여 얻는 반면에, 미분반응기에서는 정상상태조작 중에서 측정한다. 미분반응기에 의한 실험에서는, 일반적으로 반응물 공급조건을 변화시켜가면서 생성물의 농도를 측정하여 데이터를 얻는다. 수집된 자료를 분석하기 위해서 3가지 방법이 사용된다. 미분법 적분법 비선형 회귀분석법 (최소자승법) 미분법과 적분법은 일반적으로 회분식반응기의 자료를 해석하는 데 사용된다. 근래에는 자료를 분석하기 위해 여러 가지의 소프트웨어 (폴리매스 등)를 사용할 수 있기 때문에 비선형 회귀분석법에 의한 광범위한 자료처리 내용까지 포함시켰다. Seoul National University

Objective ★ Determine the reaction order and specific reaction rate from experimental data obtained from either batch or flow reactors. ★ Describe how to use equal-area differentiation, polynomial fitting, numerical difference formulas and regression to analyze experimental data to determine the rate law. ★ Describe how the methods of half lives, and of initial rate, are used to analyze rate data. ★ Describe two or more types of laboratory reactors used to obtain rate law data along with their advantages and disadvantages. Seoul National University

Collection and Analysis of Rate Data ★ Two common type reactors for obtaining rate data: (1) Batch reactor : Conc. vs. time - homogeneous reaction during transient operation - Concentration (or pressure) are usually measured and recorded at different times during the course of reaction. (2) Differential reactor : Conc. @ steady state - Solid-fluid heterogeneous reactions - Product concentration is usually monitored for different sets of feed conditions. Seoul National University

Collection and Analysis of Rate Data ★ Six different methods of analyzing the data collected (1) the differential method (2) the integral method (3) the method of half-lives (4) method of initial rates (5) linear regression (6) nonlinear regression (least squares analysis) primarily in analyzing batch reactor data Seoul National University

7.1 Algorithm for Data Analysis Steps in Analyzing Rate Data Seoul National University

(TE7-1.1) (TE7-1.2) Seoul National University

A. Differential Analysis (7.4) (TE7-1.3) (TE7-1.4) (TE7-1.5) (TE7-1.1) Seoul National University

B. Integral Analysis (7.3) (TE7-1.4) (TE7-1.6) (TE7-1.6). 7 Seoul National University

(7.6) t Seoul National University

Measuring concentration or nonlinear regression method 7. 2 Method of Excess Measuring concentration as a function of time t CA 0 50.0 50 38.0 100 30.6 . . Differential, integral or nonlinear regression method Data analysis Determining a and k in -rA = kCA Seoul National University

7. 2 Method of Excess  Irreversible reaction -Determine a and k by either nonlinear regression or by numerically differentiating concentration versus time data For example -for decomposition reaction (only one reactant) A  products -rA = kACA then differential method may be used. Assuming that the rate law is of the form -rA = kACA (7-1) Seoul National University

7. 2 Method of Excess Excess Experiments Excess B experiments: Consider the irreversible reaction : A + B  products (7-2) Excess B experiments: CB remains unchanged during the reaction (CBo) (7-3) Excess A experiments: CA remains unchanged during the reaction (CAo) (7-4) (7-5) Once  and  are determined, kA can be calculated from the measurement of -rA at known concentration of A and B Seoul National University

7.3 Integral Method ★ The integral method uses a trial-and-error procedure to find rxn order: We guess the reaction order and integrate the differential equation used to model the batch system. If the order we assume is correct, the appropriate plot of the concentration-time data should be linear. ★ It is important to know how to generate linear plots of functions of CA versus t for zero-, first-, and second-order reactions. ★ The integral method is used most often when the reaction order is known and it is desired to evaluate the specific reaction rate constants at different temperatures to determine the activation energy. ★ Finally we should also use the formula to plot reaction rate data in terms of concentration vs. time for 0, 1st, and 2nd order reactions. Seoul National University

zero order first order second order t t t slope = -k zero order t slope = k first order t slope = k second order The rxns are zero, first, and second order respectively since the plots are linear. Seoul National University

second order t non linear If the plots of concentration data versus time had turned out not to be linear, we would say that the proposed reaction order did not second order. Seoul National University

Example 7-1 Use the integral method to confirm that the reaction is second order with respect to trityl(A) as described in example 5-1 and to calculate the specific reaction rate k . ' Trityl(A) + Methanol (B)  Products Integrating with Time (min) 0 50 100 150 200 250 300 CA (mol/dm3) 0.05 0.038 0.0306 0.0256 0.0222 0.0195 0.0174 1/CA (dm3/mol) 20 26.3 32.7 39.1 45 51.3 57.5 Seoul National University

Example 7-1 second order Seoul National University

Example 7-1 (E7-1.7) Note that differential method gives Seoul National University

Linear Regression Report Example 7-1 Linear Regression Report by SigmaPlot Function values: x f(x) 0 20.1142857143 6 20.8634285714 12 21.6125714286 18 22.3617142857 24 23.1108571429 30 23.86 36 24.6091428571 42 25.3582857143 48 26.1074285714 54 26.8565714286 60 27.6057142857 66 28.3548571429 72 29.104 78 29.8531428571 84 30.6022857143 90 31.3514285714 96 32.1005714286 102 32.8497142857 108 33.5988571429 114 34.348 120 35.0971428571 126 35.8462857143 132 36.5954285714 138 37.3445714286 144 38.0937142857 150 38.8428571429 Function values: x f(x) 156 39.592 162 40.3411428571 168 41.0902857143 174 41.8394285714 180 42.5885714286 186 43.3377142857 192 44.0868571429 198 44.836 204 45.5851428571 210 46.3342857143 216 47.0834285714 222 47.8325714286 228 48.5817142857 234 49.3308571429 240 50.08 246 50.8291428571 252 51.5782857143 258 52.3274285714 264 53.0765714286 270 53.8257142857 276 54.5748571429 282 55.324 288 56.0731428571 294 56.8222857143 300 57.5714285714 Coefficients: a0 20.1142857143 k’ 0.1248571429 r ² 0.999903135 k=0.25 dm3/mol/min Seoul National University

Example 7-1: Determining the Rate Law zero order first order Seoul National University

Example 7-1: Determining the Rate Law 분석: 이 예제에서 반응차수를 알고 있고 따라서 적분법을 사용하여 반응차수가 trityl에 대하여 2차인 것을 확인하고, 메탄올 (B)이 과량으로 사용한 경우에 대해서, 비유사반응속도상수 (specific pseudo reaction rate) k' = kCB0를 찾을 수 있었다. k'과 CB0를 알게되면, 실제 속도상수 k를 구할 수 있게 된다. Seoul National University

7.4 Differential Method of Analysis Consider a reaction carried out isothermally in a constant-volume batch reactor and the concentration recorded as a function of time. By combining the mole balance with the rate law given by Equation (7-1), we obtain (7-6) After taking the logarithm of both sides of Eq. (7-6) (7-11) Seoul National University

Differential Method to determine α and k ln CA slope=a ln CAp P (CAp)a kA= Seoul National University

To obtain the derivative –dCA/dt used in this plot, How to get –dCA/dt ? To obtain the derivative –dCA/dt used in this plot, we must differentiate the concentration-time data either numerically and graphically. These methods are: ★ Graphical differentiation ★ Numerical differentiation formulas ★ Differentiation of a polynomial fit to the data Seoul National University

How to get –dCA/dt ? (Graphical Method) Time (min) t0 t1 t2 t3 t4 t5 Concentration (mol/dm3) C0 C1 C2 C3 C4 C5 1. Tabulate the (ti, Ci) 2. For each interval, calculate Dt and DC 3. Calculate DC/Dt as an estimate of the average slope in an interval. 4. Plot these values(DC/Dt) as a histogram versus ti. 5. Next draw in the smooth curve that best approximates the area under the histogram. 6. Read estimates of the dC/dt from this curve at the data points t1, t2, … and complete the table. ti t0 t1 t2 t3 t4 t5 Ci C0 C1 C2 C3 C4 C5 Dt t1-t0 t2-t1 t3-t2 t4-t3 t5-t4 DC C1-C0 C2-C1 C3-C2 C4-C3 C5-C4 DC/Dt (DC/Dt)1 (DC/Dt)2 (DC/Dt)3 (DC/Dt)4 (DC/Dt)5 dC/dt (dC/dt)0 (dC/dt)1 (dC/dt)2 (dC/dt)3 (dC/dt)4 (dC/dt)5 Seoul National University

How to Get –dCA/dt ? (Graphical Method) ★ Graphical Method (Equal-Area Graphical Differentiation) Draw smooth curve that best approximates the area under histogram A B 0 1 2 3 4 5 Seoul National University

How to Get –dCA/dt (Numerical Method) Time (min) t0 t1 t2 t3 t4 t5 Concentration (mol/dm3) CA0 CA1 CA2 CA3 CA4 CA5 ★ Numerical Method (Independent variables are equally spaced) The three-point differentiation formulas Seoul National University

How to Get –dCA/dt ? (Polynomial Fit) Time (min) t0 t1 t2 t3 t4 t5 Concentration (mol/dm3) CA0 CA1 CA2 CA3 CA4 CA5 Polynomial fit with software program to get best value of ai n-th order polynomial Differential equation Seoul National University

How to Get –dCA/dt (Polynomial Fit) 3rd-order polynomial 5th-order polynomial Negative derivative t5 Care must be taken in choosing the order of the polynomial. If the order is too low, the polynomial fit will not capture the trends in the data. If too large an order is chosen, the fitted curve can have peaks and valleys as it goes through most all of the data points, thereby producing significant errors when the derivatives, dCA/dt, are generated at the various points. Seoul National University

Finding the Rate Law Parameter (7-7) Seoul National University

Finding the Rate Law Parameter Seoul National University

Example 7-2: Determining the Rate Law The reaction of triphenyl methyl chloride (trityl) (A) and methanol (B) was carried out in a solution of benzene and pyridine at 25oC. Pyridine reacts with HCl that then precipitates as pyridine hydrochloride thereby making the reaction irreversible. The concentration-time data was obtained in a batch reactor. The initial concentration of methanol was 0.5 mol/dm3. (C6H5)3CCl (A) + CH3OH (B)  (C6H5)3COCH3 (C) + HCl (D) Part (1) Determine the reaction order with respect to trityl (A) Part (2) In a separate set of experiments, the reaction order with respect to methanol was found to be first order. Determine the specific reaction rate constant. Time (min) 0 50 100 150 200 250 300 CA (mol/dm3) x 103 50 38 30.6 25.6 22.2 19.5 17.4 CAo=0.05 M CBo=0.5 M Seoul National University

Example 7-2: Determining the Rate Law Concentration of triphenyl methyl chloride as a function of time Seoul National University

Example 7-2: Determining the Rate Law Solution Part (1) Determine the reaction order with respect to trityl (A) Step 1 Postulate a rate law (E7-2.1) Step 2 Process your data in terms of the measured variable, which is this case is CA. Step 3 Look for simplifications. Because concentration of methanol is 10 times the initial concentration of triphenyl methyl chloride, its concentration is essentially constant (E7-2.2) Substituting for CB in Equation (E5-1.1) (E7-2.3) Seoul National University

Example 7-2: Determining the Rate Law Step 4 Apply the CRE algorithm Mole Balance (E7-2.4) Rate law (E7-2.3) Stoichiometry: Liquid Combine (E7-2.5) Seoul National University

Example 7-2: Determining the Rate Law Taking the natural log of both sides of Equation (E5-1.5) (E7-2.5) (E7-2.6) The slope of plot of versus will yield the reaction order a with respective to triphenyl methyl chloride (A). Step 5 Find as a function of CA from concentration-time data. We will find by each of the three methods just discussed, (1) the graphical (2) finite difference, and (3) polynomial methods. Seoul National University

Example 7-2: Determining the Rate Law Step 5A.1a Graphical method 2.40 1.48 1.00 0.68 0.54 0.42 3.0 1.86 1.2 0.8 0.5 0.47 50 100 150 200 250 300 50 38 30.6 25.6 22.2 19.5 17.4 Seoul National University

Example 7-2: Determining the Rate Law Step 5A.1a Graphical method 3.0 1.86 1.2 0.8 0.5 0.47 3.0 2.4 1.48 1.00 0.68 0.54 0.42 2.0 1.0 0.0 50 100 150 200 250 300 Seoul National University

Example 7-2: Determining the Rate Law Step 5A.1b Finite Differential Method Seoul National University

Example 7-2: Determining the Rate Law Step 5A.1c Polynomial method t CA dCA/dt 50 -0.299 38 -0.189 100 30.6 -0.120 150 25.6 -0.081 200 22.2 -0.061 250 19.5 -0.049 300 17.4 -0.034 Another method to determine (dCA/dt) is to fit the concentration of A to a polynomial in time and then to differentiate the resulting polynomial. We first choose the fourth polynomial degree Seoul National University

Example 7-2: Determining the Rate Law Step 5A.1c Polynomial method Seoul National University

Example 7-2: Determining the Rate Law SUMMARY: There is quite a close agreement between the graphical technique, finite difference, and polynomial methods 　min mol/dm3 Graphical Finite Difference Polynominal t CAx103 -dCA/dt 50.0 3.00 2.86 2.98 50 38.0 1.86 1.94 1.88 100 30.6 1.20 1.24 1.19 150 25.6 0.80 0.84 200 22.2 0.68 0.61 0.60 250 19.5 0.54 0.48 300 17.4 0.42 0.36 0.33 Seoul National University

Example 7-2: Determining the Rate Law From Figure E5-1.3, we found the slope to be 1.99 so that the reaction is said to be second order w.r.t. triphenyl methyl chloride. To evaluate k’, we can evaluate the derivative and CAp=20x10-3 mol/dm3, which is then Seoul National University

Example 7-2: Determining the Rate Law Excel plot to determine a and k Coefficients: b[0] -6.6685984257 b[1] 1.9958592737 r ² 0.9942392165 0.5 Regression again 20 Seoul National University

Example 7-2: Determining the Rate Law Part (2) The reaction was said to be first order wrt methanol, b=1, Assuming CB0 is constant at 0.5 mol/dm3 and solving for k yields The rate law is Seoul National University

Example 7-2: Determining the Rate Law 분석: 이 예제에서 트리페닐메틸클로라이드(trityl) 농도에 대한 반응차수 (α = 1.99)와 반응상수(k' = 0.125 (dm3/mol)/min)을 구하기 위하여 미분법을 이용한 자료분석을 수행하였다. 반응차수는 반올림하여 α = 2였고 자료는 다시 회기 분석 하여 k' = 0.122 (dm3/mol)/min)을 얻었으며, 다시 k'값과 CB0 값을 이용하여 실제 반응상수 k = 0.244 (dm3/mol)/min) 임을 알 수 있었다. Seoul National University

Comparison The differential method tends to accentuate the uncertainties in the data, while the integral method tends to smooth the data, thereby disguising the uncertainties in it. In most analyses, it is imperative that the engineer know the limits and uncertainties in the data. This prior knowledge is necessary to provide for a safety factor when scaling up a process from laboratory experiments to design either a pilot plant or full-scale industrial plant Accentuate : 강조하다 . 두드러지게하다. Disguise: 위장하다. 변장하다. Seoul National University

Comparison 예제 7-1과 예제 7-2에 나타낸 속도자료의 해석방법들을 비교해보면, 미분법은 자료에 불확실성이 증폭되는 반면에, 적분법은 자료를 부드럽게 하는 경향이 있어서 불확실성을 어느 정도 감쇠시키는 경향이 있음을 알 수 있다. 대부분의 분석에 있어서, 공학도는 자료 속에 존재하는 한계와 불확실성을 아는 것이 꼭 필요하게 된다. 이러한 사전 지식은 실험실의 실험으로부터 파이롯트 공장이나 대규모 공업용 공장을 설계하기 위한 공정의 스케일업을 할 때 안전지수(safety factor)를 결정하는 데 꼭 필요하다. Seoul National University

5.2.3 Nonlinear Regression ★ In nonlinear regression analysis, we search for those parameter values that minimize the sum of squares of the differences between the measured values and the calculated values. Not only can nonlinear regression find the best estimates of parameter values, it can also be used to discriminate between different rate law models, such as Langmuir-Hinshelwood models. ★ Many software programs are available to find these parameter values so that all one has to do is enter the data. In order to carry out the search efficiently, in some cases one has to enter initial estimates of the parameter values close to the actual values. These estimates can be obtained using the linear-least-squares technique. Seoul National University

5.2.3 Nonlinear Regression ★ We will now apply nonlinear least-squares analysis to reaction rate data to determine the rate law parameters (k, a, b..). We then search for those values that will minimize the sum of the squared differences of the measured reaction rates, rm, and the calculated reaction rates, rc. That is, we want the sum of (rm- rc)2 for all data points to be minimum. If we carried out N experiments, we would want find the parameter values that minimize the quantity N = number of runs K = number of parameters to be determined rim = measured reaction rate for run i (i.e., -rAim) ric = calculated reaction rate for run i (i.e., -rAic) (5-34) Seoul National University

Nonlinear Regression A Product To illustrate this technique, let’s consider the first-order reaction A Product for which we want to learn the reaction order, a, and the specific reaction rate, k, The reaction rate will be measured at a number of different concentrations. We now choose values of k and a and calculate the rate of reaction (ric) at each concentration at which an experimental point was taken. We then subtract the calculated value (ric) from the measured value (rim), square the result, and sum the squares for all the runs for the values of k and a we have chosen. Seoul National University

Nonlinear Regression This procedure is continued by further varying a and k until we find their best values, that is, those values that minimize the sum of the squares. Many well-known searching techniques are available to obtain the minimum value . Figure 5-7 shows a hypothetical plot of sum of the squares as a function of the parameters a and k: 2 5 Seoul National University

Nonlinear Regression In searching to find the parameter values that give minimum of the sum of squares s2, one can use a number of optimization techniques or software packages. A number of software packages are available to carry out the procedure to determine the best estimates of the parameter values and corresponding confidence limits. All on has to do is to type the experimental values in the computer, specify the model, enter the initial guesses of the parameter values along with 95% confidence limit appear. 최적의 매개변수 값의 추정과 이에 상당하는 신뢰한계를 구하는 절차를 수행하는 많은 소프트웨어 패키지가 있다. 여러분은 단지 컴퓨터에 실험치를 타자치고, 모델을 정하고, 매개변수의 초기 추정치를 적어 놓고는 “계산” 버튼을 누르기만 하면 된다. 그러면 최적의 매개변수 값과 95% 신뢰한계가 나타날 것이다. 만약에 주어진 매개변수의 신뢰한계가 매개변수 값 자체보다 크다면, 그 매개변수는 별 의미가 없는 것이며, 그 매개변수는 버려야 할 것이다. 적절한 모델 매개변수를 제거한 후에, 소프트웨어를 다시 돌려 새로운 모델식에 가장 잘 맞는 것을 결정해야 한다. Seoul National University

Nonlinear Regression Concentration-time data. We will now use nonlinear regression to determine the rate law parameters from concentration-time data obtained in batch experiments. We recall that the combined rate law-stoichiometry-mole balance for a constant-volume batch reactor is (7-6) We now integrate Eq. (5-6) to give (7-15) We want the value of a and k that will make s2 a minimum. (7-16) Seoul National University

Nonlinear Regression (Nonlinear Least-Squares Analysis) (7-18) (7-19) Seoul National University

Example 5-3 Use of Regression to find the Rate Law Parameter We shall use the reaction and data in Ex. 5-1 to illustrate how to use regression to find a and k’ (C6H5)3CCl (A) + CH3OH (B)  (C6H5)3COCH3 (C) + HCl (D) (E7-2.5) Integrating with the initial condition when t=0 and CA=CA0 for a ≠1.0 (E7-3.1) Substituting for the initial concentration CA0=0.05 mol/dm3 (E7-3.2) Let’s do a few calculations by hand to illustrate regression. Seoul National University

Example 7-3 Use of Regression to find the Rate Law Parameter We now first assume a value of a and k and then calculate t for the concentrations of A given in Table E5-1.1 (pp 261). We will then calculate the sum of the squares of the difference between the measured time, tm and the calculated times (i.e., s2). For N measurements, (7-19) Our first guess is going to be a = 3 and k’ = 5, with CA0 = 0.05. Equation (E7-3.1) becomes We now make the calculations for each measurement of concentration and fill in column 3 and 4 of Table E5-3.1. For example, when CA=0.038 mol/dm3 then Seoul National University

Example 7-3 Use of Regression to find the Rate Law Parameter Which is shown in Table E5-3.1 on line 2 for guess 1. We next calculate the squares of difference (tm1-tc1)2=(50-29.2)2=433. We continue in this manner for points 2, 3, and 4 to calculate the sum s2=2916. After calculating s2 for a = 3 and k = 5, we make a second guess for a and k’. For our second guess we choose a = 2 and k = 5; Equation (E7-3.2) becomes (E7-3.2) We now proceed with our second guess to find the sum of (tm1-tc1)2 to be s2=49,895, which is far worse than our first guess. So we continue to make more guesses of a and k and find s2. Let’s stop and take a look at tc for guesses 3 and 4. Seoul National University

Example 7-3 Use of Regression to find the Rate Law Parameter Table E5-3.1. Regression of data Original Data t CA x 103 (min) (mol/L) Guess 1 Guess 2 Guess 3 Guess 4 a=3 k’=5 a=2 k’=5 a=2 k’=0.2 a=2 k’=0.1 tc (tm-tc)2 tc (tm-tc)2 tc (tm-tc)2 tc (tm-tc)2 0 50 0 0 0 0 0 0 0 0 1 2 3 4 50 38 29.2 433 1.26 2,375 31.6 339 63.2 174 100 30.6 66.7 1,109 2.5 9,499 63.4 1,340 126.8 718 200 22.2 163 1,375 5.0 38,622 125.2 5,591 250 2,540 s2 = 2,916 s2 = 49,895 s2 = 7,270 s2 = 3,432 We see that (k’=0.2 dm3/mol·min) underpredicts the time (e.g., 31.6 min versus 50 minutes), while (k’=0.1 dm3/molmin) overpredicts the time (e.g., 63 min versus 50 minute) Seoul National University

Example 7-3 Use of Regression to find the Rate Law Parameter Regression again Seoul National University

Example 7-3 Use of Regression to find the Rate Law Parameter 분석: 이 예제를 통해 우리는 k'와 α값을 구하기 위하여 어떻게 비선형회귀분석을 이용하는 가하는 방법을 보여 주었다. 첫 번째 회귀분석을 통해 α값은 2.00 근처인 α = 2.04임을 알 수 있었고, 회귀분석을 한 번 더 실시하여 α = 2.0에서의 최적의 k'값은 k' = 0.125 dm3/mol · min이 되므로, k = 0.25 (mol/dm3)2· min이 됨을 알 수 있었다. 예제 7-1, 7-2에서 반응차수는 같은 값을 갖지만, k 값은 약 8%가 큰 것에 주목하여라. r 2과 다른 통계는 폴리매스의 결과이다. Seoul National University

Method of Initial Rates 가역반응의 속도식 The use of the differential method of data analysis to determine reaction orders and specific reaction rates is clearly one of the easiest, since it requires only one experiment. However, other effects, such as the presence of a significant reverse reaction, could render the differential method ineffective. In these cases, the method of initial rates could be used to determine the reaction order and the specific rate constant. Here, a series of experiments is carried out at different initial concentrations, CA0, and the initial rate of reaction, -rA0, is determined for each run. The initial rate of reaction –rA0 can be found by differentiating the data and extrapolating to zero time. Seoul National University

Method of Initial Rates By various plotting or numerical analysis techniques relating –rA0 to CA0, we can obtain the appropriate rate law. the slope of the plot of ln(-rA0) versus lnCA0 is the reaction order a. Seoul National University

Method of Initial Rates Seoul National University

Method of Initial Rates Seoul National University

Method of Initial Rates Solution Seoul National University

Method of Initial Rates Seoul National University

Method of Initial Rates Seoul National University

Method of Initial Rates Seoul National University

A  products (irreversible) Method of Half-Lives A  products (irreversible)  The half-life of a reaction = the time it takes for the concentration of the reactant to fall to half of its initial value  By determining the half-life of a reaction as a function of the initial concentration, the reaction order and specific reaction rate can be determined. Slope=1-a Seoul National University

7.6 Differential Reactors 미분반응기는 농도 또는 분압의 함수로서 반응속도를 구하는 데 사용된다. 이러한 미분반응기는 관형 반응기에 매우 소량의 촉매를 얇은 웨이퍼로 배치하여 사용한다. 미분반응기가 되기 위한 기준은 촉매층에서의 반응물의 전화율을 아주 낮고, 촉매층 전반에 걸쳐서 온도의 변화나 반응물 농도의 변화가 극히 작은 경우이다. 그 결과 반응기 전반에 걸친 반응물의 농도는 거의 일정하며 근사적으로 유입 농도와 같은 값을 갖는다. 즉 반응기의 농도구배는 없다고 가정할 수 있으면, 반응속도는 촉매층 내에서 공간적으로 일정하다고 볼 수 있다. 미분반응기는 상대적으로 저렴한 가격으로 제조할 수 있다. 반응기에서의 낮은 전화율 때문에 단위 부피당 열 발생률이 극히 낮아 (혹은 불활성 고체로 촉매를 희석하여 열 발생률을 작게 유도) 반응기가 기본적으로 등온 상태로 조작된다. 미분반응기를 사용하는 경우에는 반응물 기체 또는 액체가 우회(by-pass)나 편류(channeling) 현상이 일어나지 않고 촉매층을 균일하게 통과할 수 있도록 세심한 주의를 필요로 한다. 연구 중에 있는 촉매의 활성이 급격하게 줄어드는 경우에는 미분반응기가 적절치 않다. 왜냐하면 반응 초기와 반응 후기에서의 반응속도 파라미터가 변하기 때문이다. 다성분계에서 전화율의 변화가 미세할 경우에는 미분반응기상에서는 생성물의 시료 채취와 분석이 쉽지 않다. Seoul National University

7.6 Differential Reactors • Most commonly used catalytic reactor to obtain experimental data - use to determine the rate of reaction as a function of either concentration or partial pressure - the conversion of the reactants in the bed is very small - reactant concentration is constant: gradientless (~CSTR) - reaction rate is spatially uniform (~CSTR) A  P FA0 FAe DL Inert filling Catalyst wafer (Const. Temp.) Seoul National University

7.6 Differential Reactors Steady state mole balance on reactant A (CSTR) DL FA0 FAe CA0 FP CP DW A  P (7-20) Seoul National University

7.6 Differential Reactors Steady state mole balance on reactant A (CSTR) (7-21) Reactor Design Equation (7-22) Seoul National University

7.6 Differential Reactors Measuring the product concentration For constant volumetric flow known Can be determined (7-23)    known - using very little catalyst and large volumetric flow rates: where CAb the concentration of A within the catalyst bed the arithmetic mean of the inlet and outlet concentration: - very little reaction takes place within bed: (7-24) (7-25) (7-26) Seoul National University

Example 7-4 Differential Reactors The synthesis of CH4 from CO and H2 using a nickel catalyst was carried out at 500oF in a differential reactor where the effluent concentration of CH4 was measured. Relate the rate of reaction to the exit methane concentration. The reaction rate law is assumed to be the product of a function of the partial pressure of CO and a function of the partial pressure of H2: (E7-4.1) Determine the reaction order w.r.t. CO, using the data in Table E5-5.1. Assume that the functional dependence of on is of the form (E7-4.2) Seoul National University

Example 7-4 Differential Reactors Table E7-4.1. Raw Data Run PCO (atm) PH2 (atm) CCH4 (mol/dm3) 1 1.0 1.0 1.73x10-4 2 1.8 1.0 4.40x10-4 3 4.08 1.0 10.0x10-4 4 1.0 0.1 1.65x10-4 5 1.0 0.5 2.47x10-4 6 1.0 4.0 1.75x10-4 The exit volumetric flow rate a differential packed bed containing 10 g of catalyst was maintained at 300 dm3/min for each run. The partial pressure of H2 and CO were determined at the entrance to the reactor, and the methane concentration was measured at the reactor exit. Seoul National University

Example 7-4 Differential Reactors (a) In this example the product composition, rather than the reactant concentration, is being monitored. r´CH4 can be written in terms of flow rate of methane from the reaction, (E7-4.3) Table E7-4.1. Raw and Calculated Data Run PCO (atm) PH2 (atm) CCH4 (mol/dm3) r´CH4 (mol CH4/cat·min) 1 1.0 1.0 1.73x10-4 5.2x10-3 2 1.8 1.0 4.40x10-4 13.2x10-3 3 4.08 1.0 10.0x10-4 30.0x10-3 4 1.0 0.1 1.65x10-4 4.95x10-3 5 1.0 0.5 2.47x10-4 7.42x10-3 6 1.0 4.0 1.75x10-4 5.25x10-3

Example 7-4 Differential Reactors (b) Determining the rate law dependence in CO For constant hydrogen concentration (run 1,2,3), the rate law can be written as constant (E7-4.4) Taking the log Seoul National University

Example 7-4 Differential Reactors We now plot versus for runs 1, 2, and 3. Table E5-5.1. Raw and Calculated Data Run PCO (atm) PH2 (atm) CCH4 (mol/dm3) r´CH4 (mol CH4/cat·min) 1 1.0 1.0 1.73x10-4 5.2x10-3 2 1.8 1.0 4.40x10-4 13.2x10-3 3 4.08 1.0 10.0x10-4 30.0x10-3 4 1.0 0.1 1.65x10-4 4.95x10-3 5 1.0 0.5 2.47x10-4 7.42x10-3 6 1.0 4.0 1.75x10-4 5.25x10-3 Seoul National University

Example 7-4 Differential Reactors Seoul National University

Example 5-5 Differential Reactors (b) Determining the rate law dependence in CO Had we include more experiment points, we would have found the reaction rate is essentially first order with a=1, that is, a=1.22 → 1 (E5-5.5) H2 분압이 일정한 처음 3개의 자료로부터, 반응속도가 CO 분압에 선형적임을 알 수 있다. Seoul National University

Example 5-5 Differential Reactors (c) Determining the rate law dependence in H2 From Table E7-4.2, it appears that the dependence of r’CH4 on PH2 cannot be represented by a power law. The reaction rate first increases with increasing partial pressure of hydrogen, and subsequently decreases with increasing partial pressure of hydrogen. That is, there appears to be a concentration of hydrogen at which the rate is maximum. Run PCO (atm) PH2 (atm) CCH4 (mol/dm3) r´CH4 (mol CH4/cat·min) 1 1.0 1.0 1.73x10-4 5.2x10-3 2 1.8 1.0 4.40x10-4 13.2x10-3 3 4.08 1.0 10.0x10-4 30.0x10-3 4 1.0 0.1 1.65x10-4 4.95x10-3 5 1.0 0.5 2.47x10-4 7.42x10-3 6 1.0 4.0 1.75x10-4 5.25x10-3 Seoul National University

Seoul National University 높은 수소농도 낮은 수소농도 모든 수소농도를 만족시키는 식 Seoul National University

dissociate adsorption on the catalyst surface Hydrogen undergo dissociate adsorption on the catalyst surface → Dependence of H2 to the ½ power Polymath Regression again (E7-4.12) Polymath Regression (E7-4.11) Seoul National University

Rate law is indeed consistent with the rate law data rearranging (E7-4.13) This plot should be a straight line with an intercept of 1/a and a slope b/a. 1 2 3 4 400 300 200 100 Rate law is indeed consistent with the rate law data Seoul National University

7.7 Evaluation of Laboratory Reactor The successful design of industrial reactors lies primarily with the reliability of the experimentally determined parameters used in the scale-up. Consequently, it is imperative to design equipment and experiments that will generate accurate and meaningful data. Unfortunately, there is usually no single comprehensive laboratory reactor that could be used for all types of reactions and catalysts. In this section, we discuss the various types of reactors that can be chosen to obtain the kinetics parameters for a specific reaction system. We closely follow the excellent strategy presented in the article by V.W. Weekman of ExxonMobil. Seoul National University

7.7 Evaluation of Laboratory Reactor Criteria used to evaluate laboratory reactor 1. Ease of sampling and product analysis 2. Degree of isothermality 3. Effectiveness of contact between catalyst and reactant 4. Handling of catalyst decay 5. Reactor cost and ease of construction Seoul National University

Evaluation of Laboratory Reactor (Types of Reactors) Differential reactor (Fixed bed) Integral reactor (Fixed bed) Stirred-Batch Reactor catalyst slurry Continuous-Stirred Tank Reactor (CSTR) Stirred Contained Solids Reactor (SCSR) Straight-through transport reactor Recirculating transport reactor Figure 5-12 Type of Reactors Seoul National University

Evaluation of Laboratory Reactor (Reactor Ratings) Reactor type Sampling Isothermality F-S contact Decaying Catalyst Ease of construction Differential P-F F-G F P G Fixed bed G P-F F P G Stirred batch F G G P G Stirred-contained solids G G F-G P F-G Continuous-stirred tank F G F-G F-G P-F Straight-through transport F-G P-F F-G G F-G Recirculating transport F-G G G F-G P-F Pulse G F-G P F-G G G=Good; F=Fair; P=Poor CSTR and recirculating transport reactor appear to be the best choice, because they are satisfactory in every category except for construction. However, if the catalyst under study does not decay, the stirred batch and contained solid reactors appear to be best choices. If the system is not limited by internal diffusion in the catalyst pellet, larger pellets could be used, and the stirred-contained solids is the best choice. If the catalyst is nondecaying and heat effects are negligible, the fixed-bed (integral) reactor would be the top choice, owing to its ease of construction and operation. However, in practice, usually more than one reactor type is used in determining the reaction rate law parameters. Seoul National University

7.7 Experimental Planning Four to six weeks in the lab can save you an hour in the library G.J. Quarderer, Dow Chemical Co. So far, this chapter has presented various methods of analyzing rate data. It is just as important to know in which circumstances to use each method as it is to know the mechanics of these methods. In this section we discuss a heuristic to plan experiments to generate the data necessary for reactor design. However, only a thumbnail sketch will be presented; for a more thorough discussion the reader is referred to the books and articles by Box and Hunter.

Flowchart for experimental projects a heuristic to plan experiments to generate the data necessary for reactor design.

1. Do You Really Need the Experiments? When you are preparing to initiate an experimental program, be sure to question yourself and others to help guide your progress. The following questions will help you dig deeper into your project. Why perform the experiments? Can the information you are seeking be found elsewhere (such as literature journals, books, company reports, patent, etc.)? Can you do some calculations instead? Have sufficient time and money been budgeted for the program? Are you restricted to specific materials or equipment? Will the safety of the investigators be endangered to such a degree that the program should not be carried out?

2. Define the Objectives of the Experiment Prepare a list of all the things you want to accomplish. Next try to prioritize your list, keeping in mind the following: What questions regarding your problem would you most like to answer? Are you sure you are not losing sight of the overall objectives and other possible alternative solutions ("can't see the forest for the trees" syndrome)? How comprehensive does the program need to be? Are you looking at an exhaustive study or a cursory examination of a narrow set of conditions?

3. Choose the Responses You Want to Measure There are generally two different types of variables that are considered in an experimental program. The independent variables make things happen. Changes in the independent variables cause the system to respond. The responses are the dependent variables. Changing any one of the independent variables will change the system response (the dependent variable). As the experimental program is designed, the important dependent variables to be measured must be identified. What are the controlled or independent variables? What are the dependent variables? Are instruments or techniques available to make the measurements? Do they need to be calibrated? If so, have they been? Will the accuracy and precision of the expected results be sufficient to distinguish between different theories or possible outcomes?

4. Identify the Important Variables In any experimental program there will always be many quantities you can measure. However, you must decide which independent variables have the greatest influence on the dependent variable. What are the really important measurements to make? What are the ranges or levels of these variables to be examined? Instead of changing each independent variable separately, can dimensionless ratios or groups be formed (i.e., Schmidt or Sherwood numbers) and varied so as to produce the same end results with fewer measurements?

5. Design the Experiment To obtain the maximum benefit from a series of experiments, they must be properly designed. How can the experimental program be designed to achieve the experimental objectives in the simplest manner with the minimum number of measurements and the least expense? A successfully designed experiment is a series of organized trials which enables one to obtain the most experimental information with the least amount of effort. Three important questions to consider when designing experiments are: What are the types of errors to avoid? What is the minimum number of experiments that must be performed? When should we consider repeating experiments?

5. Design the Experiment - The minimum number of experiments that must be performed is related to the number of important independent variables that can affect the experiment and to how precisely we can measure the results of the experiment. - One of the most important strategies to remember is to carry out first experiments at the extremes (maximum and minimum setting) of the range of the controlled variables. - For example, if the range of pressures that can be used to determine the rate law of a gas-phase reaction is from 1 to 100 atm, it is somewhat best to determine the rate at 1 atm and then at 100 atm.

5. Design the Experiment - If the independent variables have no effect on the dependent variables at the extremes, it is somewhat doubtful that there will be an effect in the intermediate range. Consequently, a lot of time, money, and energy would be lost if we progressed from a setting of 1 atm to 2 atm and found no effect, then to 10 atm and found no change, then to 50 atm and 80 atm with similar results. - In designing the experiments, we will first choose two levels (i.e., settings) for each independent variable. Because these levels are usually at the extremes of the variable range, we refer to these settings as high and low (e.g., on/off, red/green, 100 psi/14.7 psi, 100°C/0°C, etc.).

5. Design the Experiment - For example, consider an experimental program where the dependent variable is a function of three independent variables (A, B, and C), each of which can take on two possible values or levels. Independent Variable Names Possible Levels A High Low B High Low C High Low - If all possible variable combinations were to be tested, the number of experiments is equal to the number of levels, N, raised to the power of the number of independent variables, n. For the example for variables A, B, and C, the number of experiments necessary to test all combinations of independent variables is equal to Nn=23=8 experiments.

Placement (high/low) of controlled variables - These are detailed in Table A and Text Figure 5-13 [(+) indicates a high level, while (-) indicates a low level of a particular variable]. Experiment No. pH Temp. Conc. 1 - - - 2 + - - 3 - + - 4 - - + 5 + + + 6 - + + 7 + - + 8 + + - pH T C

urease + H2O + NH2CONH2 → CO2 + 2NH3 + urease Design experiments to determine the effect of pH and temperature on the rate of the enzyme-catalyzed decomposition of urea urease + H2O + NH2CONH2 → CO2 + 2NH3 + urease Enzyme degradation is believed to occur at temperatures above 50°C and pH values above 9.5 and below 3.0. The rate of reaction is negligible at temperatures below 6°C. For a urea concentration below 0.001 M, the reaction will not proceed at a measurable rate and the rate appears to be independent of concentration above 0.1 M. Consequently, the following high/low values of the parameters were chosen: A (-) pH 4 (+) pH 8 B (-) 10°C (+) 40°C C (-) 0.005 M (+) 0.1 M

If there is no interaction among the variables experiments 1-4 will yield all the necessary information Experiment No. pH Temp Comments 1 – – Base case 2 + – Reveals effect of high pH 3 – + Reveals effect of high temperature 4 – – Reveals effect of high concentration - How good are the measurements? - Is there any mathematical model or theory available that suggests how the data might be plotted or correlated? - What generalizations can be made from the data? - Should other experiments be run to extend the data into different regions? - Has an error analysis been performed? - Have all experimental objectives been satisfied?

9. Report 1. Abstract. This one-page summary of the report is usually written last. It defines the problem, tells how you approached the problem, and states the important results that were found. 2. Introduction. The introduction section defines the problem, tells why it is an important problem worthy of being studied, gives background information, describes the fundamental issues, and discusses and analyzes how they relate to published work in the area. 3. Experimental This section describes the equipment used to carry out the experiments, as well as instruments used to analyze the data. The purity of the raw materials is specified, as are the brand names of each piece of equipment. The accuracy of each measurement taken is discussed. The step-by-step procedure as to how a typical run is carried out is presented, and all sources of error are discussed. (If you developed a new model or theory, a theory section would come after section 3. The theory section would develop the governing equations that mathematically describe your phenomena and justify all assumptions in the development.) Designing Technical Reports by J. C. Mathes and D. W. Stephenson (Indianapolis, Ind.: Bobb-Merrill, 1976),

9. Report 4. Results. This section tells what you found. Make sure that figures and tables all have titles and the units of each variable are displayed. Discuss all sources of error and describe how they would affect your results. Put an error bar on your data where appropriate. 5. Discussion of results. This section tells why the results look the way they do. Discuss whether they are consistent with theory, either one you developed or that of others. You should describe where theory and experiment are in good agreement as well as those conditions where the theory would not apply. 6. Conclusion. The conclusion section lists all important information you learned from this work in numerical order; for example:(a) The reaction is insignificant below 0°C. (b) The results can be described by the Buckley-Leverette Theory. 7. References. List all resource material you referred to in this work in the proper bibliographical format Designing Technical Reports by J. C. Mathes and D. W. Stephenson (Indianapolis, Ind.: Bobb-Merrill, 1976),

마무리 이 장을 공부한 후 독자들은 도식법이나 수치 해석법은 물론 패키지 프로그램을 이용하여 속도식과 속도 파라미터를 결정하기 위하여 자료해석을 할 수 있어야 한다. 비선형 회귀분석법은 속도-농도 자료 해석으로 파라미터를 결정하는 가장 편리한 방법이다. 그러나 도식미분과 같은 다른 방법들은 자료의 격차에 대한 감을 갖는 데 도움을 준다. 독자들은 σ2에 대한 틀린 최소값에 다다르지 않았다는 것을 확인하기 위하여 비선형 회귀분석법을 사용할 수 있어야 하고 세심한 주의가 필요하다. 결론적으로 자료 분석을 위하여 한 가지 방법보다는 그 이상의 방법을 사용하는 것이 바람직하다.