Collection and Analysis of Rate Data

1. Collection and Analysis of Rate Data 授課教師：林佳璋

2. Algorithm for Data Analysis For batch system, the usual procedure is to collect concentration time data which we then use to determine the rate law. The following procedure is what we will emphasize in analyzing reaction engineering data. 1.Postulate a rate law 3.Process your data in terms of measured variable (e.g., NA, CA, or PA). If necessary rewrite your mole balance in terms of the measured variable (e.g., PA). 4.Look for simplifications For example, if one of the reactants in excess, assume its concentration is constant. If the gas phase mole fraction of reactant is small, set =0. 2.Select type and corresponding mole balance

3. 5.For a batch reactor, calculate –rA as a function of concentration CA to determine reaction order reaction order specific reaction rate constant

4. 6.For differential PBR calculate –rA’ as a function of CA or PA 7.Analyze your rate law model for “goodness fit” Calculate a correlation coefficient One will be able to postulate different rate laws and then use polymath nonlinear regression to choose the “best” rate law and reaction rate parameters.

5. Batch Reactor Data Batch reactors are used primarily to determine rate law parameters for homogeneous reactions. This determination is usually achieved by measuring concentration as a function of time and then using either the differential, integral or nonlinear regression method of data analysis to determine the reaction order, , and specific reaction rate constant, k.  excess of B  excess of A

6. 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. Figure 5-1(a) shows a plot of [-(dCA/dt)] versus [CA] on log-log paper where the slope is equal to the reaction order . The specific reaction rate, kA, can be found by first choosing a concentration in the plot, say CAp, and then finding the corresponding value of [-(dCA/dt)] as shown in Figure 5-1(b). After raising CAp to the  power, we divide it into [- (dCA/dt)p] to determine kA:

7. We describe three methods to determine the derivate (-dCA/dt) from the data giving the concentration as a function of time. These methods are: ~Graphical differentiation ~Numerical differentiation formulas ~Differentiation of a polynomial fit to the data Numerical Method Initial point Interior point Last point

8. Polynomial Fit If the order is too low, the polynomial fit will not capture the trends in the data and not go through many of the points. 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 point. third-order fifth-order negative slope

9. Finding the Rate Law Parameters Using either the graphical method, differentiation formulas, or the polynomial derivative, the following table can be set up:

10. Example 5-1 The reaction of triphenyl methyl chloride (trityl) (A) and methanol (B) was carried out in a solution of benzene and pyridine at 25 C. Pyridine reacts with HCl that then precipitates as pyridine hydrochloride thereby making the reaction irreversible. The concentration-time data in Table E5-1.1 was obtained in a batch reactor The initial concentration of methanol was 0.5 mol/dm3. Part(1) Determine the reaction order with respect to triphenyl methyl chloride. Part(2) In a separate set of experiments, the reaction order wrt methanol was found to be first order. Determine the specific reaction rate constant.

11. Solution Part(1) Find reaction order wrt trityl Step 1 Postulate a rate law Step 2 Process your data in terms of measured variable, which in this case is CA. Step 3 Look for simplifications concentration of methanol is essentially constant due to that concentration of methanol is 10 times the initial concentration of triphenyl methyl chloride. Step 4 Apply the CRE algorithm Mole Balance Rate Law Stoichiometry: Liquid

12. Step 5 Find (-dCA/dt) as a function CA from concentration-time data Graphical Method

13. Finite Difference Method

14. Polynomial Method

16. Part(2) The reaction was said to be first order wrt methanol, =1, The rate law is

17. Integral Method ~To determine the reaction order by the integral method, 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 (determined from this integration) of the concentration-time data should be linear. ~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. ~In the integral method of analysis of rate data, we are looking for the appropriate function of concentration corresponding to a particular rate law that is linear with time.

18. Considering the reaction AProducts carried out in a constant-volume batch reactor, the mole balance is

19. Figure 5-6 shows that the plots of concentration data versus time had turned out not to be linear. We would say that the proposed reaction order did not fit the data. In the case of Figure 5-6, we would conclude the reaction is not second order. The idea is to arrange the data so that a linear relationship is obtained.

20. Example 5-2 Use the integral method to confirm that the reaction is second order wrt triphenyl methyl chloride as described in Example 5-1 and to calculate the specific reaction rate k’ Trityl (A) + Methanol (B)  Products Solution The rate law is Polymath ?

21. Nonlinear Regression Not only can nonlinear regression find the best estimates of parameter values, it can also be used to discriminate between different rate law models. If we carried out N experiments, we would want to find the parameter values (e.g., reaction order, specific rate constant) that would minimize the quantity AProducts initial guesses is important

22. Concentration-Time Data For a constant-volume batch reactor integrating For N data points

23. Example 5-3 We shall use the reaction and data in Example 5-1 to illustrate how to use regression to find  and k’. Solution

25. Method of Initial Rates ~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 case, 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 concentration, CA0, and the initial rate of reaction, -rA0, is determined for each run. The initial rate, -rA0, can be found by differentiating the data and extrapolating to zero time. ~By various plotting or numerical analysis techniques relating –rA0 to CA0, we can obtain the appropriate rate law. If the rate law is the form the slope of the plot of ln(-rA0) versus lnCA0 will give the reaction order .

26. Example 5-4 The dissolution of dolomite, calcium magnesium carbonate, in hydrochloric acid is a reaction of particular importance in the acid stimulation of dolomite oil reservoirs. The oil is contained in pore space of the carbonate material and must flow through the small pores to reach the well bore. In matrix stimulation, HCl is injected into a well bore to dissolve the porous carbonate matrix. By dissolving the solid carbonate, the pores will increase in size, and the oil and gas will be able to flow out at faster rates, thereby increasing the productivity of the well. The dissolution reaction is The concentration of HCl at various times was determined from atomic absorption spectrophotometer measurements of the calcium and magnesium ions. Determine the reaction order with respect to HCl from data presented in Figure E5-4.1 for this batch reaction

27. Solution For a constant-volume batch reactor 30 cm2/solid per liter of solution slope=0.44 The rate law is

28. Method of Half-Lives The half-life of a reaction, t1/2, is defined as 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. =1

29. Differential Reactors Data acquisition using the method of initial rates and a differential reactor are similar in that the rate of reaction is determined for a specified number of predetermined initial or entering reactant concentrations. A differential reactor is normally used to determine the rate of reaction as a function of either concentration or partial pressure. If consists of a tube containing a very small amount of catalyst usually arranged in the form of a thin wafer or disk. A typical arrangement is shown schematically in Figure 5-10. The criterion for a reactor being differential is that the conversion of the reactants in the fed is extremely small, as is the change in temperature and reactant concentration through the bed. As a result, the reactant concentration through the reactor is essentially constant and approximately equal to the inlet concentration. That is, the reactor is considered to be gradientless, and the reaction rate is considered spatially uniform within the bed. ~low cost ~small heat release ~operating in an isothermal manner ~not a good choice for the case that catalyst decays rapidly ~sampling and analysis may be difficult for small conversions in multicomponent system

30. The volumetric flow rate through the catalyst bed is monitored, as are the entering and exiting concentrations (Figure 5-11). Therefore, if the weight of catalyst, W, is known, the rate of reaction per unit mass of catalyst, -rA’, can be calculated. Since the differential reactor is assumed to be gradientless, the design equation will be similar to the CSTR design equation. A steady-state mole balance on reactant A gives when the stoichiometric coefficients of A and P are identical For constant volumetric flow using very little catalyst and large volumetric flow rates Using high flow rates through the differential reactor and small catalyst particle sizes is to avoid mass transfer limitations.

31. Example 5-5 The formation of methane from carbon monoxide and hydrogen using a nickel catalyst was studied by Pursley. The reaction was carried out at 500F in differential reactor where the effluent concentration of methane was measured. (a)Relate the rate of reaction to the exit methane concentration. (b)The reaction rate law is assumed to be the product of a function of the partial pressure of CO, f(CO), and a function of the partial pressure H2, g(H2): Determine the reaction order with respect to carbon monoxide, using the data in Table E5- 5.1. Assume that the functional dependence of rCH4’ on PCO is the form The exit volumetric flow rate from 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.

32. Solution (a) (b) =1.22 Had we included more points we would have found that the reaction is essentially first order with =1

33. At low H2 concentration where rCH4’ increases as PH2 increase, the rate law may be of the form At high H2 concentration where rCH4’ decreases as PH2 increase, the rate law may be of the form Polymath If we assume hydrogen undergoes dissociative adsorption on the catalyst surface one would expect a dependence of hydrogen to the 0.5 power Polymath

34. Evaluation of Laboratory Reactors ~The successful design of industrial reactors lies primarily with the reliability of the experimentally determined parameters used in the scale-up. ~It is imperative to design equipment and experiments that will generate accurate and meaningful data. ~There is usually no single comprehensive laboratory reactor that could be used for all types of reactions and catalyst. ~In this section, we discuss the various types of reactors that can be chosen to obtain the kinetic parameters for a specific reaction system.

35. Criteria The criteria used to evaluate various types of laboratory reactors are listed in Table 5-2. Each type of reactor is examined with respect to these criteria and given a rating of good (G), fair (F), or poor (P).

36. Integral (Fixed-Bed) Reactor One advantage of the integral reactor is its ease of construction (see Figure 5-14). On the other hand, while the channeling or bypassing of some of the catalyst by the reactant stream may not be as fatal to data interpretation in the case of this reactor as in that of the differential reactor, it may still be a problem. There is more contact between the reactant and catalyst in the integral reactor than in the differential reactor, owing to its greater length. Consequently, more product will be formed, and the problems encountered in the differential reactor in analyzing small or trace amounts of product in the effluent stream are eliminated. However, if a reaction is highly endothermic or exothermic, significant axial and radial temperature gradients can result, and this reactor will receive a poor-to-fair rating on its degree of isothermality. If a reaction follows different reaction paths with different activation energies, different products will be formed at different temperatures. This makes it difficult to unscramble the data to evaluate the various reaction rate constants because the reaction mechanism changes with changing temperature along the length of the reactor. Figure 5-14 Integral reactor If the catalyst decays significantly during the time an experiment is carried out, the reaction rates will be significantly different at the end of the experiment than at the start of the experiment. In addition, the reaction may follow different reaction paths as the catalyst decays, so that the selectivity to a particular product will vary during the course of the experiment. Consequently, it will be difficult to sort out the various rate law parameters for the different reactions and, as a result, this reactor receives a poor rating in the catalyst decay category. However, this type of reactor is relatively easy and inexpensive to construct, so it receives a high rating in the construction category.

37. Stirred-Batch Reactor In a stirred-batch reactor the catalyst is dispersed as a slurry, as shown in Text Figure 5-15. Although this reactor has better contact between the catalyst and fluid than either the differential or integral reactors, it has a sampling problem. Samples of fluid are usually passed through cyclones or withdrawn through filters or screens to separate the catalyst and fluid, thereby stopping the reaction. However, slow quenching of the reaction in the cyclone or plugging of the filter sampling system by the catalyst particles is a constant concern, thus making the rating in the sampling category only fair. Since the system is well mixed, its isothermality is good. There is good contact between the catalysts and reactants, and the contact time is known since the catalyst and reactants are fed at the same time. However, if the catalyst decays, the activity and selectivity will vary during the course of data collection. Figure 5-15 Stirred-batch reactor

38. Stirred Contained-Solids Reactor (SCSR) Although there are a number of designs for contained-solids reactors, all are essentially equivalent in terms of performance. A typical design is shown in Figure 5-16. In this reactor, catalyst particles are contained in paddles that rotate at sufficiently high speeds to minimize external mass transfer effects and, at the same time, keep the fluid contents well mixed. With this type of operation, isothermal conditions can be maintained and there is good contact between the catalyst and fluid. If the catalyst particle size is small, difficulties could be encountered containing the particles in the paddle screens. Consequently, it receives only a fair rating in the ease of construction and cost category. Although this type of reactor receives a good rating for ease of sampling and analysis of product composition, like the three previous reactors, it suffers from being unable to generate useful data when the catalyst being studied decays. As a result, it receives a poor rating in the catalyst decay category. Figure 5-16 Stirred contained-solids reactor

39. Continuous-Stirred Tank Reactor (CSTR) In this reactor (Figure 5-17), fresh catalyst is fed to the reactor along with the fluid feed and the catalyst leaves the reactor in the product stream at the same rate that it is fed. As a result, the catalyst in the reactor is at the same level of catalytic activity at all times. Thus we are not faced with the problem encountered in the four previous reactors, in which the kinetic parameters evaluated at the beginning of the experiment will be different than those at the end. However, since there will be a distribution of time that the catalyst particles have been in the reactor, there will be a distribution of catalytic activities of the particles in the bed. The mean activity of a catalyst with first-order decay is derived in Example 14-3. If the mean residence time is large, selectivity disguise could be a problem. Since the reactor is well-mixed, isothermality and fluid--solid contact categories are rated as good. However, difficulties can arise in feeding the slurry accurately and, as with the stirred-batch reactor, it is difficult to quench the reaction products. Consequently, it receives only fair ratings in the first (sampling) and fifth (construction) categories. Figure 5-17

40. Straight-Through Transport Reactor Commercially, the transport reactor (Figure 5-18) is used widely in the production of gasoline from heavier petroleum fractions. In addition, it has found use in grain drying operations. In this reactor, either an inert gas or the reactant itself transports the catalyst through the reactor. With this reactor, any possibility of catalyst decay/selectivity disguise is virtually eliminated because the catalyst and reactants are fed continuously. For highly endothermic or exothermic reactions, isothermal operation will be difficult to achieve and it receives a poor-to-fair rating in this category. At moderate or low gas velocities there may be slip between the catalyst particles and the gas so that the gas-catalyst contact time will not be known very accurately. Consequently, this reactor receives only a fair-to-good rating in the gas-catalyst contacting category. This reactor is somewhat easier to construct than the CSTR, but salt or sand baths may be required to try to maintain isothermal operation, and it therefore receives a fair-to-good rating in the construction category. Difficulty in separating the catalyst and reactant gas or in thermally quenching the reaction results in a fair rating in the sampling category. Figure 5-18 Straight-through transport reactor

41. Recirculating Transport Reactor By recirculating the gas and catalyst through the transport reactor (Figure 5-19), a well-mixed condition can be achieved provided that the recirculation rate is large with respect to the feed rate. Consequently, isothermal operation is achieved. Since the reactor is operated at steady state, the kinetic parameters measured at the start of the experiment will be the same as those measured at the end. However, since fresh catalyst is mixed with decayed catalyst from the recycle, the product distribution and the kinetic parameters might not be the same as those measured in a straight-through transport reactor where the gas "sees" only fresh catalyst. The incorporation of a recirculation system adds a degree of complexity to the construction, which gives it a lower rating in this category as well. Figure 5-19 Recirculating transport reactor

42. Summary of Reactor Ratings The ratings of the various reactors are summarized in Table 5-3. ~From this table one notes that the CSTR and recirculating transport reactor appear to be the best choice because they are satisfactory in every category except for construction. ~If the catalyst under study does not decay, the stirred batch and contained solids reactors appear to be the 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. In practice, usually more than one reactor type is used in determining the reaction rate law parameters.

43. Closure ~After studying this section, the student should be able to analyze data to determine the rate law and rate law parameters using the graphical and numerical technique as well as software packages. ~Nonlinear regression is the easiest method to analyze rate- concentration data to determine the parameters, but the other techniques such as graphical differentiation help one get a feel for the disparities in the data. ~The student should be able to describe the care that needs to be taken in using nonlinear regression to ensure you do not arrive on a false minimum for 2. ~It is advisable to use more than one method to analyze the data. ~The student should be able to carry out a meaningful discussion on reactor selection to determine the reaction kinetics along with how to efficiently plan experiments.