baigiang_mophong

7/29/2019 Baigiang_MoPhong

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Phn 2: m phng my tnhModeling, simulation and optimization for chemical process

Instructor: Hoang Ngoc Ha

Email: [email protected]

B mn QT&TB


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Introduction

Numerical

Analysis

Computer

Programming

SIMULATION


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Computer simulation

Some simulation techniques for solving some

of the systems of equations Solution of (nonlinear) algebraic equations

Ordinary differential equations (ODEs)

Partial differential equations (PDEs)

Numerical methods

Iterative methods Discrete difference methods

Femlab, Fortran, Ansys Matlab/Simulink


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Computer simulation

Computer programming

Assume that you know some computerprogramming language

We are not interested in generating the most

efficient and elegant code but in solving problems(from point of view of engineers) Including extensive comment statements

Use of symbols (the same ones in the equations

describing the systems) Debugging (for mistakes in coding and/or in logic)


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Computer simulation

Example:


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Computer simulation


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Computer simulation


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Computer simulation

Interval halving (chia i khong)


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Computer simulation

This problem can be formulated under the

following form:

The goal is to find the solution of thisnonlinear equations (in ONE VARIABLE)

Tools (Iterative methods) Bisection method (phng php phn on)

Newtons (or Newton-Raphson) method

f(x) = 0, x R


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Iterative method

Intermediate value theorem Iffis a real-valued continuous function on the

interval [a, b], and u is a number between f(a) and

f(b), then there is a such that f(c) = uc [a, b]

Iff(a) and f(b) are of opposite sign, there exist a numberp in [a, b] with f(p)=0


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Iterative method Bisection method

Computer programming: Matlab


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Iterative method Newtons method

Numerical solutions of nonlinear systems of equations (ofSEVERAL VARIABLES) (See Ref.)


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Computer simulation

Interpolation and polynomial approximation

Interpolation and the Lagrange polynomial Cubic spline interpolation

Numerical differentiation and intergration

Numerical differentiation

Richardsons extrapolation

(See Ref.)


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Numerical intergration of OrdinaryDifferential Equations (ODEs)


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Numerical intergration of OrdinaryDifferential Equations (ODEs)

y(t)

ttN

x

y(tN)x

t0

y(t0)x

t1

y(t1)

Interpolation


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Numerical intergration of Ordinary

Differential Equations (ODEs) Tools:

Eulers method

Higher-Order Taylor methods

Runge-Kutta methods


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Differential Equations (ODEs) Eulers method


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Differential Equations (ODEs) Eulers method


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Differential Equations (ODEs) Example

Exact solution?

y(t) = 0.5exp(t) + (t + 1)2

P/p Eulern=10?

y0 = y t2 + 1, t [0 2]y(0) = 0.5

Approximate solution?

n = 10 h = ban

= 0.2



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Differential Equations (ODEs) Local truncation error

The local truncation error in Eulers method is O(h)

Definition


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Differential Equations (ODEs) Higher-Order Taylor methods


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Differential Equations (ODEs) Higher-Order Taylor methods


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Differential Equations (ODEs) Runge-Kutta methods


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Differential Equations (ODEs) Runge-Kutta methods

Xy dng cng thc tnh theo m khngphi o hm tay , cn xp x m khngdng o hm vi

Minh ha qua k=2

wi+1 wiT(k)

O(hk)

f(t, y)

T(2)(t, y) = f(t, y) + h2

f0(t, y)

f0

(t, y) = f0

t(t, y) + f0

y(t, y)y0

(t)


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Differential Equations (ODEs)Nh vy

Cn tm vi sai s a1, 1, 1

a1f(t + 1, y + 1) ' T(2)(t, y)

O(h2

)

T(2)(t, y) = f(t, y) + h2

f0t(t, y) +h2

f0y(t, y)f(t, y)

f(t + 1, y + 1) ' f(t, y) + f0t(t, y)1+f0y(t, y)1


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Differential Equations (ODEs)Cn chn

ng nht hai v

a1f(t, y) + a11f0

t(t, y) + a11f0

y(t, y) =f(t, y) + h2 f

0

t(t, y) +h2 f

0

y(t, y)f(t, y)

a1 = 1

a11 =h

2a11 =

h2

f(t, y)

a1 = 11

= h21 =h2

f(t, y)


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Differential Equations (ODEs)

S trung im (R_K bc 2)

wi+1 = wi + hT(2)(ti, wi)

wi+1 = wi + hh

f(ti +h2

, wi +h2

f(ti, wi))i

w0 = k1 =

h

2

f(ti, wi)k2 = hf(ti + h2 , wi + k1)wi+1 = wi + k2



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Differential Equations (ODEs)S R_K bc 4

w0 = k1 = hf(ti, wi)

k2 = hf(ti + h2 , wi + k12 )k3 = hf(ti +

h2

, wi +k22

)

k4 = hf(ti + h, wi + k3)wi+1 = wi +16 (k1 + 2k2 + 2k3 + k4)

V nh tc R_K cho h v vit chng trnh R_K cho h


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Numerical intergration of Partial

Differential Equations (PDEs) Click here


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Gii thiu chung

u = u(x . . . , t)mt i lng vt l ca h kho st

n chemical species

Inlet material and/orenergetic flux

Outlet material

and/or energetic flux

Pk kSk = 0

dV

{C, T . . . } H PHN BProfile

Local observation


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Ba dng phng trnh o hm ring c

bnng hc bin h thng c

th thuc v cc dng phng trnh sau:

u = u(x . . . , t)

Phng trnh elliptic (tnh-static)2u

x2 +

2u

y2 = f(x, y)

P/t parabolic (b/ton truyn nhit)ut

= a2 2u

x2

Phng trnh hyperbolic (b/ton truyn sng)2u

t2= a2

2ux2


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Ba dng phng trnh o hm ring c

bn Phng php tm nghim

Phng php gii tch Phng php s

tng: xp x sai phn cc o hm ring ti cc im ri rc (kg,tg)

v tnh gi tr ca ti u = u(x . . . , t)

u(x,t)t

u(x,t+t)u(x,t)

t

x

t

(x, t)

(x,t + t)t

u(x, t)


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Xp x sai phn

2u(x,t)x2

u(x+x,t)2u(x,t)+u(xx,t)

(x)2

(x, t)

(x +x, t)(xx, t)

t

x

x


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Xp x sai phn

x

y

P1

P = (x, y)

x P

y P3

P2

P4

2u(P)x2

+ 2u(P)y2

u(P1)+u(P2)+u(P3)+u(P4)4u(P)

h2

h = x = y


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BI TON ELLIPTIC

Bi ton elliptic vi iu kin bin Dirichlet(u =

2ux2

+ 2u

y2= f(x, y), (x, y) R2

u(x, y) = g(x, y), (x, y)


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BI TON ELLIPTIC

Phn hoch v to li : chia nh bi

cc ng thng // vi Oxv Oy

h = x = y

To li bc chia cch u

K hiu P1,P2, P3v P4 l 4

im ri rc x/q P

u(P) u(P1)+u(P2)+u(P3)+u(P4)4u(P)h2

Ln lt thay vo phng trnh elliptic, s

dng cng thc xp x& iu kin bin

Pk = (xk, yk)H PTTT n uk = u(Pk)


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BI TON ELLIPTIC

Example

Li 4 nt n nh s 4 gi tr cn tm. SD PTSP & Gi tr trn bin H PTTT

M phng Matlab


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BI TON PARABOLIC (m hnh

truyn nhit)


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truyn nhit)


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Mc th k: t = kt

Mc th 0: bitu(0)Mc th 1: cha bitu(1)


truyn nhit) Phn hoch v xp x

Chia cch u bc vx

t

u(k) = (. . . uki . . . )

Xp xo hm ring cp 1: TIN - LIu(x,t)t


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truyn nhit) V d

S hin (s/p tin) S n (s/p li)


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truyn nhit tng qut)

Min = {(x, t)|0 x 1, t 0}

Sai phn tin & Sai phn li

Xp x ut

, ux

, 2u

x2& K bin, u Gi tr u ti cc im li


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truyn nhit)

T A A h h


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truyn nhit)K hiu u(k) = [uk1 . . . u

kn], f

(k) = [fk1 . . . f kn ]

BI TON PARABOLIC ( h h


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truyn nhit) S Crank-Nicholson

O li


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Outline

General introduction

Structure and operation of chemical engineeringsystems

What is a chemical process?

Motivation examples Part I: Process modeling

Part II: Computer simulation

Part III: Optimization of chemical processes

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