Redhouane Henda Department of Chemical Engineering Lund University, Lund, Sweden Nov , 2006

Transcription

1 Redhouane Henda Department of Chemical Engineering Lund University, Lund, Sweden Nov , Rationale Process simulation successful tool for design, optimization and control of chemical processes Use of simulation expanded due to availability of highspeed computers and software packages Availability of solution techniques further broadened the use of simulation Etc. 2 1

2 Required competency Sound understanding of engineering fundamentals (physical system & mechanisms). Process cannot be viewed as a black box! Modeling skills (sound mathematical relations). Computational skills (proper solution technique, software package, computer, etc.) 3 Impact of simulation on chemical process industry Economic: cheaper to use simulation than to build numerous different-size pilot-plants Operation: Easier to develop alternative operating approaches via a mathematical model than by experimental methods Scale up: First-principles simulations can predict system performance in new and different operating conditions 4 2

3 Problem definition Mathematical modeling of process Equation organization Notice the adaptive nature Computation Results interpretation 5 A good problem definition comes from... What I really want to find out? What are the important consequences of the simulation? Why is simulation work required? What data are available? What form of model is required? What are system inputs, outputs, states,... No precise recipe applies! Creative thought... Imagination

4 Mathematical model... A model (M) for a system (S) and and experiment (E) is anything to which (E) can be applied in order to answer questions about (S). (Minsky) A system (S) can be regarded as an operator mapping its inputs, u, to its outputs, y: S: U Y; y = S[u] 7 Mathematical model... u System S x y x : states Mass, energy, momentum, mechanisms i Model o M Order-of-magnitude x analysis

5 Mathematical model... Type of model Mechanistic Empirical Classification First-principles Trials/experiments Stochastic Deterministic Probabilistic Cause-effect Lumped Distributed Independent of space Dependent on space Linear Nonlinear Superposition applies Superposition does not apply Continuous Discrete Over continuous time For discrete values of time 9 Total mass balance: Rate of Accumulation = of total mass Species mass balance: Rate of Accumulation = of i Energy balance: Rate of Accumulation = of energy Momentum balance: Rate of Accumulation = of momentum Mathematical model... Rate of total - mass in Rate of i in Rate of energy in Rate of momentum in Rate of total mass out Rate of i + out Control vol. Rate of generation of i Control surf. Rate of energy + Rate of generation out of energy Rate of momentum + Rate of generation out of momentum Constitutive Relations (mechanisms, data, correlations...) 10 5

6 Mathematical model... Transport Relations Etc. Etc. MODEL MODEL Reaction Reaction Rates Rates Equipment and and Control Control Constraints Property Property Definition 11 Natural ordering... (Franks) Assemble model relations into solution strategy (variables to solve?) Information-flow diagram Solution strategy parallel cause-and-effect of physical system Example: Heating of a tank 20 C 180 kg/h 150 C, 0.9 m 2 T C 120 kg/h 12 6

7 Natural order... Mass balance: R.o.A Tot. Mass = R.o.M In R.o.M out dm dt = Energy balance: R.o.A Energy = R.o.E In R.o.E out + R.o.G Energy d( M CpT ) = [180CpinTin + U A( Ts T )] 120Cp dt t=0 = 500 kg M dm d ( M Cp T ) = = 180 (1)(20) 120(1) T (0.9)(150 T ) dt dt out T t=0 = 40 C T 13 Computation... Key issues are Is the model solvable? What analytic (if any!) or numerical techniques should be used? Are there any useful on-the-shelf subroutines (NAG, IMSL) or software packages/toolboxes (Matlab, Mathematica, Maple,...) available? What form of visualization is adequate (2D, 3D, etc.)? What can be done to improve speed and/or robustness? COMSOL AB is our tool

8 Let s reason together... Is the model a valid representation of the actual process... How can model implementation be verified? What type of model validation is appropriate? What needs to be changed/added? What level of simplification is necessary? 15 CAVEATS Limitations of process simulation... Lack of data and knowledge of mechanisms Character of computational tools Forgetting assumptions advanced to develop model! 16 8

9 Distributed Parameter Systems (PDEs) Derived from microscopic conservation equations (balances) Incorporate spatial variation of states and/or time changes within volume-at least two independent variables-if one only lumped parameter systems (ODEs) Intensive property T, P, C, ρ, etc. π = π ( r, t) Described by PDEs in 1D, 2D or 3D Initial and boundary conditions apply Time Position vector 17 Distributed Parameter Systems (PDEs) r π r r π π t Convection Diffusion Dynamic Cartesian, cylindrical, spherical systems 18 9

10 Distributed Parameter Systems (PDEs) Coordinate Systems Cartesian Cylindrical Spherical 19 Distributed Parameter Systems (PDEs) Classification (geometric and loose) Lw = aw xx + 2bw xy + cw yy + dw x + ew y + fw = g > 0 Hyperbolic b 2 ac = 0 Parabolic < 0 Elliptic If b 2 ac is not constant (e.g., x, y, etc.) mixed 20 10

11 Distributed Parameter Systems (PDEs) IC and BC BCs Σ 3 Dirichlet: T= f(x,y) on Σ 1 D Neumann: T/ n= g(x,y) on Σ 2 n normal (outward) to Σ 2 Σ 1 Σ 2 Mixed: α(x,y)t + β(x,y) T/ n= γ(x,y) on Σ 3 n normal (outward) to Σ 3 IC T= t = 0 21 Key references M. Minsky Models, minds, machines, Proc. IFIP Congress, pp , R.G. Franks, Mathematical modeling in chemical engineering, Wiley, N.Y., W.F. Ramirez, Computational methods for process simulation, 2nd Ed., Butterworth-Heinemann, Boston, K. Hangos & I. Cameron, Process modeling and model analysis, Academic Press, N.Y.,

12 Course Structure Methods of Weighted Residuals: Distr. Systems Finite Elements: Distr. Systems Tutorial I Solvers for Large Steady-State Pbs. Simulation of Steady-State Distr. Systems Tutorial II Finite Differences: Distr. Systems Tutorial III Advanced Approximations Course Projects 23 12