Dynamics of forced and unsteady-state processes
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1 Dynamics of forced and unsteady-state processes Davide Manca Lesson 3 of Dynamics and Control of Chemical Processes Master Degree in Chemical Engineering Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 1
2 Forced unsteady-state reactors Forced unsteady state (FUS) reactors allow reaching higher conversions than conventional reactors. Two alternatives: Reverse flow reactors, RFR; Network of reactors: simulated moving bed reactors, SMBR. Within a SMBR network, the simulated moving bed is accomplished by periodically switching the feed inlet from one reactor to the following one. APPLICATION: Methanol synthesis (ICI patent). Operating temperature: C; Pressure: 5-8 MPa; CO = 10-20%; CO 2 = 6-10%; H 2 = 70-80%. Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 2
3 Forced unsteady-state reactors Catalytic exothermic reactions can be carried out with an autothermal regime. FUS reactors are mainly advantageous when either the reactants concentration or the reactions exothermicity are low. There is an increase of both conversion and productivity that allows: Using smaller reactors, Lower amounts of catalyst. Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 3
4 Methanol synthesis in forced unsteady-state reactors The methanol synthesis reaction is: CO 2H2 CH3OH H R kj/mol The reaction takes place with a reduction of the moles number. Therefore, the reaction is carried out at high pressure. In the past, the methanol plants worked at bar. Nowadays, methanol plants work at lower pressures (50 80 bar). In the low-pressure plants, the following reactions are important too: CO H CO H O CO 3H CH OH H O Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 4
5 Reverse Flow Reactors Valves: V 1, V 2, V 3, V 4 allow periodically inverting the feed direction in the reactor. Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 5
6 RFR: first principles model Equations #Eq. Eq. type Variables Gas phase enthalpic balance 1 Partial derivative T G, T S, yg,i i 1...nComp Solid phase enthalpic balance (catalyst) 1 Partial derivative T G, T S, yg,i, i 1...nComp ys,i i 1...nComp Gas phase material balance ncomp Partial derivative T G, T S, yg,i, i 1...nComp ys,i i 1...nComp Solid phase material balance (catalyst) ncomp Non-linear algebraic T G, T S, yg,i, i 1...nComp ys,i i 1...nComp 2nComp 2 2nComp 2 Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 6
7 RFR: methanol synthesis Reactor diameter Reactor length Void fraction Catalyst mass Apparent catalyst density Catalyst porosity Pellet diameter Inlet temperature Working pressure Surface flowrate Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 7
8 RFR: methanol synthesis Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 8
9 Reverse Flow Reactors Temperature profile once the pseudo-stationary condition is reached Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 9
10 Reverse Flow Reactors Concentration profile once the pseudo-stationary condition is reached Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 10
11 Simulated Moving Bed Reactors Step 2: 3: 1: 1g2g3 2g3g1 3g1g2 Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 11
12 Simulated Moving Bed Reactors Kinetic equations corresponding to a dual-site Langmuir- Hinshelwood mechanism, based on three independent reactions: methanol formation from CO, water-gas-shift reaction and methanol formation from CO2: Velardi S., A. Barresi, D. Manca, D. Fissore, Chem. Eng. J., , 2004 Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 12
13 Simulated Moving Bed Reactors Reaction rates for a catalyst based on Cu Zn Al mixed oxides Velardi S., A. Barresi, D. Manca, D. Fissore, Chem. Eng. J., , 2004 Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 13
14 Simulated Moving Bed Reactors Velardi S., A. Barresi, D. Manca, D. Fissore, Chem. Eng. J., , 2004 Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 14
15 Simulated Moving Bed Reactors After a suitable number of switches the temperature profile reaches a pseudostationary condition. High switch times Low switch times Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 15
16 T GAS [ C] SMBR: the thermal wave s 170 s 220 s 240 s 250 s 260 s Reactors Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 16
17 SMBR: open loop response Velardi S., A. Barresi, D. Manca, D. Fissore, Chem. Eng. J., , 2004 Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 17
18 SMBR: open loop response Velardi S., A. Barresi, D. Manca, D. Fissore, Chem. Eng. J., , 2004 Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 18
19 SMBR: open loop response Disturbance stop after 195 switches Disturbance stop after 310 switches There are more periodic stationary conditions Velardi S., A. Barresi, D. Manca, D. Fissore, Chem. Eng. J., , 2004 Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 19
20 The control problem FEATURES The system may suddenly diverge to unstable operating conditions (even chaotic behavior); The network may shut-down; The reactors may work in a suboptimal region. PROBLEM The reactor network should work within an optimal operating range; Such a range is often narrow and its identification may be difficult. SOLUTION A suitable control system must be synthesized and implemented on-line to avoid both shut-down and chaotic behaviors; An advanced control system is highly recommended; Model based control Model Predictive Control, MPC. Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 20
21 Numerical modeling The model based approach to the control problem calls for the implementation of a numerical model of the network that will be used for: 1. Identification of the optimal operating conditions; 2. Control purposes, i.e. to predict the future behavior of the system. The numerical model is based on a first principles approach: The reactors are continuously evolving (they never reach a steady-state condition) time derivative; Each PFR reactor must be described spatially spatial derivative; The reacting system is catalyzed (therefore it is heterogeneous). Consequently, an algebraic term is required PDAE system. The PDAE system is spatially discretized DAE system. A total of 1067 differential and algebraic equations must be solved to determine the dynamic evolution of the network. Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 21
22 H 2 molar fraction Mathematical tricks Stoichiometric closure: ncomp 1 icomp 1 y G,nComp 1 y G,iComp Reactors Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 22
23 Numerical solution of the DAE Boolean matrix that shows the presence indexes of the differential-algebraic system Specifically tailored numerical algorithm for tridiagonal block systems Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 23
24 Numerical solution of the DAE Boolean matrix that shows the presence indexes of the differential-algebraic system Specifically tailored numerical algorithm for banded systems Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 24
25 Numerical solution of the DAE BzzDAEFourBlocks Algebraic equations Algebraic variables Algebraic equations Differential variables Differential equations Algebraic variables Differential equations Differential variables Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 25
26 Numerical solution of the DAE Simulation time: 4000 s Switch time: 40 s Spatial discretization nodes: 97 Number of equations per node: 11 Total number of DAEs: 1067 CPU: Intel Pentium IV 2.4 GHz RAM: 512 MB OS: MS Windows 7 Professional Compiler: COMPAQ Visual Fortran MICROSOFT C Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 26
27 T GAS [ C] CH 3 OH molar fraction Numerical simulation Reactors Reactors 2 3 Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 27
28 T GAS [ C] CH 3 OH molar fraction Numerical simulation Switch time = 40 s Time [s] Time [s] Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 28
29 Parametric sensitivity study Variability interval of the analyzed process variables Variable Interval Inlet gas temperature, T in [K] Inlet gas velocity, v in [m/s] Switch time, t c [s] Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 29
30 Parametric sensitivity study Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 30
31 Parametric sensitivity study Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 31
32 Parametric sensitivity study Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 32
33 Need for speed THE POINT: to simulate 100 switches of the inlet flow with a switch time of 40 s (total of 4,000 s) the DAE system, comprising 1067 equations, takes about 95 s of CPU time on a workstation computer. PROBLEM: the detailed first principles model requires a CPU time that is prohibitive for model based control purposes. SOLUTION A high efficiency numerical model in terms of CPU time is therefore required; Such a model should be able to describe the nonlinearities and the articulate profiles of the network of reactors; Artificial Neural Networks, ANN, may be the answer. Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 33
34 System identification with ANN Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 34
35 ANN architecture Input variables Range Inlet gas temperature, T in [K] Inlet gas velocity, v in [m/s] Switch time, t c [s] Output variables Mean methanol molar fraction, x CH3OH Outlet gas temperature, T GAS [K] Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 35
36 ANN architecture Levels # of nodes Activation function 1 Input 40 Sigmoid 2 Intermediate 1 15 Sigmoid 3 Intermediate 2 15 Sigmoid 4 Output 1 Linear # of weights and biases = 871 Learning factor, a Momentum, b Linear activation constant, m Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 36
37 t c [s] v in [m/s] T in [K] Random input patterns Inlet gas temperature Inlet gas velocity Switch time Input patterns Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 37
38 Levemberg Marquard learning algorithm Initialization of weights + biases SECOND ORDER ALGORITHM Load of the input patterns n ( n) OR n N? av NO YES MAX Forward i N? o i (n) (n) i Random selection YES ERMS (n) x i (n) y i NO av (n) w i (n) b i (n) LM equation Backward new ( n ) av ERMS new av? av Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 38
39 ERMS (log) Algorithms comparison s First order Batch Mode First order Pattern Mode Second order LM s s s s s s s s s s s Iterations Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 39
40 t c [s] CH 3 OH molar fraction The overlearning problem Detailed model I order II order Time [s] Disturbance on the switch time (from 30 to 10 s) Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 40
41 ERMS The overlearning problem th iterat. 31 th iteration Learning Cross Validation Iterations Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 41
42 Relative error CH 3 OH molar fraction ANN cross-validation Detailed model ANN % 0.40% 0.35% 0.30% 0.25% 0.20% 0.15% 0.10% 0.05% 0.00% Pattern number Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 42
43 T in [K] CH 3 OH molar fraction ANN disturbance response Detailed model ANN Time [s] Disturbance on the inlet temperature [K] Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 43
44 t c [s] CH 3 OH molar fraction ANN disturbance response Detailed model ANN Time [s] Disturbance on the switch time [s] Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 44
45 V in [m/s] CH 3 OH molar fraction ANN disturbance response Detailed model ANN Time [s] Disturbance on the inlet velocity [m/s] Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 45
46 ANN CPU times MISO MIMO ANN output variables CH 3 OH T G CH 3 OH+T GAS # of output nodes # of weights and biases = = =887 Jacobian matrix dimensions 4000 x x x 887 CPU time for evaluating J T J [s] Learning procedure CPU time 3h 14 min 3h 13 min 8 h 18 min CPU time for a single ANN simulation [s] 8.08E E E-6 ABOUT 6 ORDERS OF MAGNITUDE IMPROVEMENT BETWEEN THE FIRST PRINCIPLES MODEL AND THE ANN ON-LINE FEASIBILITY OF MODEL BASED CONTROL Davide Manca Dynamics and Control of Chemical Processes Master Degree in ChemEng Politecnico di Milano L3 46
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