Chapter 2 data interpretation

VIETNAM NATIONAL UNIVERSITY – HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY FACULTY OF CHEMICAL ENGINEERING Chemical Reaction Engineering (Homogeneous Reactions in Ideal Reactors) Mai Thanh Phong, Ph.D FCE – HCMC University of Technology Chemical Reaction Engineering Chapter Interpretation of Batch Reactor Data Rates of reaction 1.1 Description of reaction rates Reaction rates depend usually in a complex manner on the concentrations, the temperature and often on the effect introduced by catalysts: r = f ( ci ,T, catalyst) The order of a reaction is related to the concentration dependence Typical examples are: Irreversible Reactions: • First order: A  Products: r = k(T)CA k in [s-1] • Second order: A + B  Products: 2A  Products: r = k(T)CACB r = k(T)CA2 k in [m3 mol-1s-1] k in [m3 mol-1s-1] Mai Thanh Phong - HCMUT Chemical Reaction Engineering Apr 28, 2023 Chapter Interpretation of Batch Reactor Data Reversible Reactions: • First order: A  R: r = k1CA – k2CR k in [s-1] • Second order: A + B  R + S: 2A  2R: r = k1CACB – k2CRCS r = k1CA2 – k2CR2 k in [m3 mol-1s-1] k in [m3 mol-1s-1] 2A  R + S: r = k1CA2 – k2CRCS k in [m3 mol-1s-1] A + B  2R: r = k1CACB – k2CR2 k in [m3 mol-1s-1] The dependency of the reaction rate on the temperature can be described using Arrhenius law: • EA is the activation energy of the reaction  EA  k (T ) k0 exp   • ko is the pre-exponential factor (not dependent RT   on the reaction temperature Apr 28, 2023 Mai Thanh Phong - HCMUT Chemical Reaction Engineering Chapter Interpretation of Batch Reactor Data 1.2 Rate laws of simple reactions In this section, rate equations of simple reactions and the corresponding temporal change of concentration are analyzed A closed system (isothermic, batch reactor) and aconstant volume are assumed .2.1 Irreversible first-order reactions (Decomposition reactions) A  Products The rate equation is: dCi r kCi  i dt (2.1) With i is A and vi = -1, then: dC A r  kC A dt (2.2) CA The integration form: Mai Thanh Phong - HCMUT t dC A or   k dt CA C Ao Chemical Reaction Engineering CA  ln kt(2.3) C A0 Apr 28, 2023 Chapter Interpretation of Batch Reactor Data The eq (2.3) leads to the temporal course of concentration cA With C A C A0 e  kt (2.4) C A0  C A , the eq (2.3) leads to: XA  C A0 (2.5)  ln(1  X A ) ktor X A 1  e  kt (2.6) 2.2 Irreversible bimolecular-type second-order reactions Consider the reaction (A + B  Products) with corresponding rate equation (2.7) r= Mai Thanh Phong - HCMUT Chemical Reaction Engineering Apr 28, 2023 Chapter Interpretation of Batch Reactor Data Noting that the amounts of A and B that have reacted at any time t are equal and given by CA0XA, eq (2.7) can be written in terms of XA as (2.8) r= (2.9) (2.10) Mai Thanh Phong - HCMUT Chemical Reaction Engineering Apr 28, 2023 Chapter Interpretation of Batch Reactor Data After breakdown into partial fractions, integration, and rearrangement, the final result in a number of different forms is (2.11) For the reaction: 2A  Products, the defining second-order differential equation becomes (2.12) r= which on integration yields (2.13) Mai Thanh Phong - HCMUT Chemical Reaction Engineering Apr 28, 2023 Chapter Interpretation of Batch Reactor Data 1.2.3 Empirical rate equations of nth order When the mechanism of reaction is not known, we often attempt to fit the data with an nth-order rate equation of the form (2.14) which on separation and integration yields (2.15) 1.2.4 Zero-order reactions Integrating and noting that CA can never become negative, we obtain directly: Mai Thanh Phong - HCMUT Chemical Reaction Engineering (2.16) Apr 28, 2023 Chapter Interpretation of Batch Reactor Data 1.2.5 Irreversible Reactions in Parallel Consider the simplest case, A decomposing by two competing paths The rates of change of the three components are given by (2.17) (2.18) (2.19) Mai Thanh Phong - HCMUT Chemical Reaction Engineering Apr 28, 2023 Chapter Interpretation of Batch Reactor Data The k values are found using all three differential rate equations First of all, eq (2.17), which is of simple first order, is integrated to give (2.20) Then dividing eq (2.18) by eq (2.19) we obtain the following (2.21) which integrated gives simply (2.22) Mai Thanh Phong - HCMUT Chemical Reaction Engineering Apr 28, 2023 10 Chapter Interpretation of Batch Reactor Data 1.2.6 Irreversible Reactions in Series Consider the reaction Rate eqautions for the three components are (2.23) (2.24) (2.25) Mai Thanh Phong - HCMUT Chemical Reaction Engineering Apr 28, 2023 11 Chapter Interpretation of Batch Reactor Data Assuming that at t = 0, concentration of A is CA0, and no R or S present, integration of eq (2.23) gives (2.26) To find the changing concentration of R, substitute the concentration of A from eq (2.26) into the differential equation governing the rate of change of R, eq (2.24); thus (2.27) Solving the above differential equation gives (2.28) Mai Thanh Phong - HCMUT Chemical Reaction Engineering Apr 28, 2023 12 Chapter Interpretation of Batch Reactor Data Noting that which eqs (2.26) and (2.28) gives (2.29) If k2 is much larger than k1, Eq (2.29) reduces to (2.30) If k1 is much larger than k2, then (2.31) Thus, in general, for any number of reactions in series it is the slowest step that has the greatest influence on the overall reaction rate Mai Thanh Phong - HCMUT Chemical Reaction Engineering Apr 28, 2023 13 Chapter Interpretation of Batch Reactor Data By differentiating Eq (2.28) and setting dCRldt = 0, the maximum concentration of R and the time at which it occours can be found: (2.32) (2.33) Mai Thanh Phong - HCMUT Chemical Reaction Engineering Apr 28, 2023 14 Chapter Interpretation of Batch Reactor Data Figure 2.1 shows the general characteristics of the concentration-time curves for the three components: • A decreases exponentially, • R rises to a maximum and then falls, • and S rises continuously, the greatest rate of increase of S occurring where R is a maximum Eq (2.26) Eq (2.29) Eq (2.28) Eq (2.33) Eq (2.30) Figure 2.1 Typical concentration-time curves for consecutive first-order reactions Mai Thanh Phong - HCMUT Chemical Reaction Engineering Apr 28, 2023 15 Chapter Interpretation of Batch Reactor Data 1.2.7 First-Order Reversible Reactions The simplest case is the opposed unimolecular-type reaction (2.34a) Starting with a concentration ratio M = CR0/CA0 th e rate equation is (2.34b) At equilibrium dCA/dt = Hence from Eq (2.34) we find the fractional conversion of A at equilibrium conditions to be (2.35) Mai Thanh Phong - HCMUT Chemical Reaction Engineering Apr 28, 2023 16 Chapter Interpretation of Batch Reactor Data and the equilibrium constant to be (2.36) Combining the above three equations we obtain, in terms of the equilibrium conversion gives (2.37) Taking integration of the above equation, the following relation can be found: (2.38) Mai Thanh Phong - HCMUT Chemical Reaction Engineering Apr 28, 2023 17 Chapter Interpretation of Batch Reactor Data 1.2.8 Second-Order Reversible Reactions For the bimolecular-type second-order reactions (2.39a) (2.39b) (2.39c) (2.39d) with the restrictions that CA0 = CB0, and CR0 = CS0 = 0, the integrated rate equations for A and B are all identical, as follows (2.40) Mai Thanh Phong - HCMUT Chemical Reaction Engineering Apr 28, 2023 18 Chapter Interpretation of Batch Reactor Data Determination of kinetic parameters 2.1 Integral method For the integral method, the concentration-time-course of one commponent is measured A tipycal measurement is depicted in CA Figure 2.2 CA0 * After this measured data are compared with theoretical integrated reaction rate * equation, for example: * C A C A0 e *  kt the unknown parameters can be calculated by non-linear regression or using the linearised form of the integrated rate equation: CA  ln kt C A0 Mai Thanh Phong - HCMUT * * * * * * * * * t Figure 2.2 Typical experimental concentration-time-course Chemical Reaction Engineering Apr 28, 2023 19 Chapter Interpretation of Batch Reactor Data The measured data can be inllustrated in a ln(CA/CA0) vs T diagram The slope of the straight line leads to the reaction rate constant (Figure 2.3) 2.6 of 2.3 Figure 2.3 Determination of the reaction rate constant k using the integrated method in a linearised formulation Mai Thanh Phong - HCMUT Chemical Reaction Engineering Apr 28, 2023 20