Identification methodology for stirred and plug-flow reactors

Transcription

1 Identification methodology for stirred and plug-flow reactors A. Bermúdez, J.L. Ferrín, N. Esteban and J.F. Rodríguez-Calo UMI REPSOL-ITMATI November 17, AIChE Annual Meeting

2 Table of contents 1 Introduction 2 Features of reoptim Graphical User Interface Mathematical models for the reactors Mathematical methodology Acceleration procedures 3 Numerical results 4 Conclusions and future work November 17, AIChE Annual Meeting

3 Introduction Problems addressed in this presentation were proposed by the engineers of Repsol (an integrated oil and gas Company), taking into account that they can represent all the situations observed in the experimental setups they are trying to simulate. Acknowledgments Part of this research was developed as an activity in the Joint Research Unit Repsol-ITMATI (code file: IN853A 2014/03) which is funded by FEDER, the Galician Agency for Innovation (GAIN) and the Ministry of Economy and Competitiveness in the framework of the Spanish Strategy for Innovation in Galicia. November 17, AIChE Annual Meeting 1/28

4 Introduction Objectives of the reoptim software tool: 1. Simulation Species concentrations and temperature in the chemical reactor. 2. Identification Functional forms of reaction rates. Numerical values of the parameters defining the above functions. November 17, AIChE Annual Meeting 2/28

5 Introduction Objectives of the reoptim software tool: 1. Simulation Species concentrations and temperature in the chemical reactor. 2. Identification Functional forms of reaction rates. Numerical values of the parameters defining the above functions. November 17, AIChE Annual Meeting 2/28

6 Introduction Data: Mathematical model of the reactor. Experiments at different conditions containing concentrations, temperature, inlet composition of mixture, etc., at some times. A catalogue of kinetic models containing the parameters to be identified. November 17, AIChE Annual Meeting 3/28

7 Introduction Chemical reactors Reactors: 1. Stirred tank reactors (STR): Transient-state Batch STR. Transient-state Batch STR with computed temperature. Transient-state Continuous STR. Transient-state Continuous STR with computed temperature. Steady-state Continuous STR. 2. Plug-flow reactors (PFR): Transient-state PFR. Transient-state PFR with computed temperature. Steady-state PFR. Steady-state PFR with computed temperature. For simplicity we consider constant density for all reactors November 17, AIChE Annual Meeting 4/28

8 Table of contents 1 Introduction 2 Features of reoptim Graphical User Interface Mathematical models for the reactors Mathematical methodology Acceleration procedures 3 Numerical results 4 Conclusions and future work November 17, AIChE Annual Meeting

9 Graphical User Interface reoptim GUI November 17, AIChE Annual Meeting 5/28

10 Mathematical models with computed temperature dy dt = Aδ(θ, y) dθ = H(θ) δ(θ, y) g (θ V ext θ) dt ŵ (θ) y y(0) = y 0 θ(0) = θ 0 November 17, AIChE Annual Meeting 6/28

11 Mathematical models with computed temperature P dv (t) = u p (t) u out (t), dt p=1 dy dt = Aδ(θ, y) + 1 V Wu 1 V y dθ dt = 1 w (θ) y 1 V V(0) = V 0, y(0) = y 0, θ(0) = θ 0. P p=1 u p, [ H(θ) δ(θ, y) g V (θ ext θ) P ( N M i W p i (êi (θ p ) ê i (θ) )) ] u p p=1 i=1 November 17, AIChE Annual Meeting 7/28

12 Mathematical models with computed temperature y + t v y z = Aδ(θ, y), θ + θ t v = 1 z ŵ (θ) y y(0, t) and θ(0, t) are given y(z, 0) = y 0 (z) θ(z, 0) = θ 0 (z) ( H(θ) δ(θ, y) + 2h ) (θ ext θ), R November 17, AIChE Annual Meeting 8/28

13 Mathematical Methodology Simulation Numerical solution of the model equations (Algebraic, ODE or PDE): Finite differences. Second-order Backward Differentiation Formula (BDF2). Newton s method. November 17, AIChE Annual Meeting 9/28

14 Mathematical Methodology Identification Optimization problem: Functional form and parameters involved in it, providing a minimum of: x x e being x and x e, respectively, the numerical and experimental values of the concentration of species and temperature. Solving the optimization problem: Extent-based incremental identification method. Integral (simultaneous) identification method. November 17, AIChE Annual Meeting 10/28

15 Mathematical Methodology Incremental method Main features: The identification problem is decomposed into a sequence of subproblems. Based on algebraic procedures for decoupling the system of equations. Uses the extent concept. Valid for stirred tank reactors. The catalogue of functional forms is defined by the chemist (using experience and knowledge). November 17, AIChE Annual Meeting 11/28

16 Mathematical Methodology Incremental method Details of the incremental methodology can be seen in: References D. Rodrigues, S. Srinivasan, J. Billeter, D. Bonvin. Variant and invariant states for chemical reaction systems, Computers & Chemical Engineering 73 (2015) November 17, AIChE Annual Meeting 12/28

17 Mathematical Methodology Incremental method with computed temperature The model dy = f(θ, y, z, Θ) in [0, T], dt dθ = h(θ, y, z, Θ), dt y(0) = y 0 and θ(0) = θ 0, (1) f = Aδ(θ, y, z, Θ) + Ff 1 + f 2 y. h = H(θ) δ(θ, y, z, Θ) g V (θ out θ) w (θ) w (θ) y ( F P p=1 f 1 p(θ s p θ)e p ). November 17, AIChE Annual Meeting 13/28

18 Mathematical Methodology Incremental method with computed temperature The ODE system (1) is coupled, that is why we work in two stages. The concentrations equations are transformed into the extents equations: de = g(θ, e, z, Θ) in [0, T], dt e(0) = 0. (2) The temperature equation becomes ĥ = d ˆθ = ĥ in [0, T], dt ˆθ(0) = θ 0. ( H( ˆθ) ˆδ g V (θ out ˆθ) w ( ˆθ) w ( ˆθ) ŷ F P p=1 f 1 p(θ s p ˆθ)e p ). (3) November 17, AIChE Annual Meeting 14/28

19 Mathematical Methodology Incremental method with computed temperature We proceed with the following steps: 1. Fitting experimental observations with cubic splines: 1.1 Compute dêe from ê e with e E. dt 1.2 From the expression of g we deduce: ˆδ e = dêe SFf 1 f 2 ê e, e E. dt 1.3 Solve initial value problem (3) to give ˆθ. 2. With ˆθ, we apply the same methodology as for the incremental method without temperature for solving (2). November 17, AIChE Annual Meeting 15/28

20 Mathematical Methodology Integral method Optimization problem (weighted least squares) being J(Θ) := e E N+1 i=1 min Θ J(Θ), ω ien (x e,n i tn S e e e,n (Θ) x i ) 2, where x e,n i (Θ) is the i-th component of the solution of the model at time t n S e. November 17, AIChE Annual Meeting 16/28

21 Integral method Numerical method Optimization: algorithm where the objective function is obtained from the solution provided by the solver. Initial value of Θ: STR - Solution provided by the incremental method. PFR - Multistart. Global optimization: - VNS perturbations. November 17, AIChE Annual Meeting 17/28

22 Acceleration procedures Parallelization of the incremental method Parallelization of the incremental method November 17, AIChE Annual Meeting 18/28

23 Acceleration procedures Parallelization of the integral method Parallelization of the integral method November 17, AIChE Annual Meeting 19/28

24 Acceleration procedures Adjoint method Gradient of the discrete objective function J d (Θ) = J h, t (x h (Θ), Θ) := e E Adjoint method: N+1 i=1 1. Solve the discretized state equation: Bxh = F. ω ien (x e,n (n) d,i tn S e e e,n (Θ) x i ) Compute the adjoint state p as the solution of the following linear system: Bt p = x J h, t (x h (Θ), Θ). 3. Compute the gradient Θ J d (Θ) in terms of the adjoint state: Θ J d (Θ) = Θ Fp + Θ J h, t (x h, Θ). November 17, AIChE Annual Meeting 20/28

25 Table of contents 1 Introduction 2 Features of reoptim Graphical User Interface Mathematical models for the reactors Mathematical methodology Acceleration procedures 3 Numerical results 4 Conclusions and future work November 17, AIChE Annual Meeting

26 Numerical results An example in a PFR@TS with computing temperature We consider a reaction system with 12 species involved in 6 reactions as follows, catalyzed by 1 catalyst: E 1 + E 2 E 3 + E 4, E 2 E E 5, E 2 E 6 + E 7, E 2 E 8 + E 9, E 2 E 10 + E 11, E 4 + E 7 E 12, November 17, AIChE Annual Meeting 21/28

27 Numerical results An example in a PFR@TS with computing temperature "Exact" expressions of the components of the reaction velocity vector are δ 1 (θ, y) = e 62737/Rθ y 1 y 2 z 1, δ 2 (θ, y) = e 71097/Rθ y 2 2 z0.5 1, δ 3 (θ, y) = e 66914/Rθ y 2 2 z 1, δ 4 (θ, y) = e 85734/Rθ y 2 z 1.1 1, δ 5 (θ, y) = e /Rθ y 2 z 0.5 1, δ 6 (θ, y) = e 63987/Rθ y 4 y 7 z November 17, AIChE Annual Meeting 22/28

28 Numerical results An example in a PFR@TS with computing temperature E 1 E 2 E 3 E 4 E 5 E 6 E 7 E 8 E 9 E 10 y 1 (0, t) [mol/l] y 2 (0, t) [mol/l] y 3 (0, t) [mol/l] y 4 (0, t) [mol/l] y 5 (0, t) [mol/l] y 6 (0, t) [mol/l] y 7 (0, t) [mol/l] y 8 (0, t) [mol/l] y 9 (0, t) [mol/l] y 10 (0, t) [mol/l] y 11 (0, t) [mol/l] y 12 (0, t) [mol/l] θ(0, t) [K] u in (t) [l/s] Table 1: Inlet conditions for the experiments November 17, AIChE Annual Meeting 23/28

29 Numerical results Performance of the adjoint method Computing the gradient of the objective function with K = 256, J = 64 and N p = 68. Performance improvement: Method Adjoint method Incremental quotients Computer time 5 min s 223 min s Table 2: Computer time for evaluating the gradient of the objective function for a PFR@TS with computed temperature, using a Scilab implementation and running in an Intel i at 3.4 GHz With the data used in this demo, the adjoint method is almost 40 times faster than the incremental quotients for calculating the gradient of the objective function. November 17, AIChE Annual Meeting 24/28

30 Numerical results Numerical solution provided by the identification methodology a) Experiments 1 to 5 b) Experiments 6 to 10 2 E1 2 E Composition (mol/l) E 1 E 2 Composition (mol/l) E 6 E E3 0.8 E8 0.6 E4 0.6 E9 E 5 E Time (s) Time (s) Numerical vs Experimental concentrations for species E 1 November 17, AIChE Annual Meeting 25/28

31 Numerical results Numerical solution provided by the identification methodology a) Experiments 1 to 5 b) Experiments 6 to Temperature 390 Temperature Temperature (K) E 1 E 2 Temperature (K) E 6 E E3 340 E8 330 E4 330 E Time (s) E Time (s) E 10 Numerical vs Experimental concentrations for temperature November 17, AIChE Annual Meeting 26/28

32 Numerical results More numerical experiments can be seen in: References A. Bermúdez, N. Esteban, J.L. Ferrín, J.F. Rodríguez-Calo and M.R. Sillero-Denamiel. Identification problem in plug-flow chemical reactors using the adjoint method, Submitted to Computers & Chemical Engineering. M. Benítez, A. Bermúdez and J.F. Rodríguez-Calo. Adjoint method for inverse problems in chemical reaction systems, Submitted. A. Bermúdez, E. Carrizosa, N. Esteban, J. Rodríguez and J.F. Rodríguez-Calo. A two-step model identification for stirred tank reactors: incremental and integral methods, In preparation. November 17, AIChE Annual Meeting 27/28

33 Table of contents 1 Introduction 2 Features of reoptim Graphical User Interface Mathematical models for the reactors Mathematical methodology Acceleration procedures 3 Numerical results 4 Conclusions and future work November 17, AIChE Annual Meeting

34 Conclusions and future work Conclusions: reoptim is being successfully used in reactors with real data by the Spanish energy company Repsol in its Technology Center. Future work: Other chemical reactors can be considered. Extension to reactor networks. November 17, AIChE Annual Meeting 28/28

35 Conclusions and future work Conclusions: reoptim is being successfully used in reactors with real data by the Spanish energy company Repsol in its Technology Center. Future work: Other chemical reactors can be considered. Extension to reactor networks. November 17, AIChE Annual Meeting 28/28