05 Modeling Dynamic and Static Behavior of Chemical Processes

7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes

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Modeling Dynamic and Static

Behavior of Chemical Processes

Cheng-Liang ChenPSELABORATORY

Department of Chemical EngineeringNational TAIWAN University


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Chen CL 1

State Variables and State Equations

- State Variables:A set of fundamental dependent quantities whose values

will describe the natural state of a given system

(temperature, pressure, flow rate, concentration )

- State Equations:

A set of equations in the state variables above which

will describe how the natural state of a given system

changes with time


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Chen CL 2

Principle of Conservation of A Quantity S

S =

total massmass of individual components

total energy

momentum


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Chen CL 3

accumulation of S

within a system

time period=

flow of S

in the system

time period

flow of S

out the system

time period

+

amount of S generated

within the system

time period

amount of S consumed

within the system

time period


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Chen CL 4

- Total Mass Balance:

d(V)

dt

= i:inlet iFi j:outlet jFj

- Mass Balance on Component A:

dnAdt

= d(cAV)dt

=i:inlet

cAi

Fi

j:outlet

cAj

Fj rV

-

Total Energy Balance:dE

dt=

d(U + K+ P)

dt=i:inlet

iFihi

j:outlet

jFjhjQWs

C C


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Chen CL 5

Mathematical ModelA Stirred Tank Heater

- Mathematical model of a process

= state equations with associated state variables

Ch CL 6


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Chen CL 6

- Total mass in tank: V = Ah

- Total energy of liquid in tank:

E = U + K+ PdU

dt

dH

dt;

dK

dt=

dP

dt= 0

H = Ahcp T Tref- State variables: h, T

- Total mass balance:

d(Ah)

dt= Fi F

=c

A

dh

dt = Fi

F


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Ch CL 8


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Chen CL 8

- Summary: State equations

Adh

dt= Fi F

AhdT

dt= Fi (Ti T) +

Q

cp

-Summary: variables

state variables: h, T

output variables: h, T

disturbances: Ti, Fi

manipulated variables: Q, F

parameters: A,,cp

Ch CL 9


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Chen CL 9

Mathematical ModelA Stirred Tank Heater (cont)

- Assumed initial steady states:

0 = Adh

dt= Fi,s Fs

0 = AhdT

dt= Fi,s (Ti,s Ts) +

Qscp

Chen CL 10


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Chen CL 10

- Temperature response to a step decrease in inlet temperature:

- Dynamic response to a step decrease in inlet flow rate:

Chen CL 11


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Chen CL 11

Additional Element:Transport Rate Equations

- Transport Rate Equations:

To describe rate of mass, energy, and momentum transfer between

a system and its surroundings

- Example: a stirred tank heater

heat supplied by steam:

Q = U At (Tst T)

Chen CL 12


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Chen CL 12

Additional Element:Kinetic Rate Equations

- Kinetic Rate Equations:

To describe rates of chemical reactions taking place in a system

-

Example: a 1st-order reaction in a CSTR reaction rate equation:

r = k0eE/RTc

A

Chen CL 13


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Chen CL 13

Additional Element:Reaction and Phase Equilibrium Relationships

- Reaction and Phase Equilibrium Relationships:To describe equilibrium situations reached during a chemical

reaction or by two or more phases

- Example: a flash drum temperature of liquid phase= temperature of vapor phase

pressure of liquid phase

= pressure of vapor phase

chemical potential of component i

in liquid phase =

chemical potential of component i

in vapor phase

Chen CL 14


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Chen CL 14

Additional Element:Equations of States

- Equations of States:

To describe the relationshipamong intensive variables

describing the

thermodynamic state

of a system

- Example: a flash drum

Ideal gas law for vapor phase:

pVvapor

= (moles of A + moles of B)RT

= mass of A + mass of Baverage MW

RT

=mass of A + mass of B

yA

MA

+ yB

MB

RT

vapor =mass of A + mass of B

Vvapor

= [yA

MA

+ yB

MB

] p

RT

liquid

= (T, xA

)

Chen CL 15


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Chen CL 15

Dead Time

-Dead Time:

Whenever an input variable of a system changes

there is a time interval (short or long) during which

no effect is obsrved on outputs of the system

- dead time, transportation lag, pure delay,

distance-velocity lag

Chen CL 16


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Chen CL 16

- Example: liquid through a pipe

A: temperature of inlet changes

B: temperature of outlet response

dead time: d

d =volume of pipe

volumetric flow rate=

A L

A Uav=

L

Uav

Tout(t) = Tin(t d)


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Chen CL 19


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Chen CL 19

- Total Mass Balance:

d(V)

dt= iFi F 0

=c= dVdt

= Fi F

- Mass Balance on Component A:

(r: rate of reaction per unit volume)

dnA

dt=

d(cA

V)

dt= c

AiFi cAF rV

Vdc

A

dt

+ cA

dV

dt=FiF= c

AiFi cAF k0e

E/RTcA

V

dc

A

dt=

FiV

cAi c

A

k0e

E/RTcA

Chen CL 20


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Chen CL 20

- Total Energy Balance:total energy E = U + K+ P = U H(T, n

A, n

B) (enthalpy)

dE

dt

=dU

dt

dH

dt

= iFihi(Ti) F h(T)Q (1)

alsodH

dt=

H

TV cp

dT

dt+

H

nA

HA(T)

dnA

dt+

H

nB

HA(T)

dnB

dt

note

dnA

dt =

d(cA

V)

dt = cAiFi cAF rVdn

B

dt=

d(cB

V)

dt= c

BiFi

=0

cB

F + rV

dH

dt= V cp

dT

dt+ H

A

cAi

Fi cAF rV

+ HB cBF rV (2)

Chen CL 21


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(1) = (2) V cpdT

dt =

HA cAiFi cAF rV HB cBF rV+Fi ihi(Ti)

ihi(T) cAi

HA(T)

+icpi(TiT)

F h(T) cAHA(T)+c

BHB(T)

Q

= HA

cAi

Fi (i)

+ HA

cA

F (ii)

+HA

rV

+ HB

cB

F (iii)

HB

rV + FicAiHA (i)

+Fiicpi(Ti T) F cAHA (ii)F c

BH

B (iii)Q

=

HA H

B

Hr

rV + Fiicpi(Ti T)Q

=i,cp=cpi= VdT

dt= Fi(Ti T) +

(Hr)

cp JrV

Q

cp

Chen CL 22


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- Summaries:

state var.s: V, cA

, T

state eqn.s:dV

dt = Fi F

dcA

dt=

FiV

cAi c

A

k0e

E/RTcA

dT

dt =

Fi

V (Ti

T) + Jk0e

E/RT

cA Q

cpV

output var.s: V, cA

, T

input var.s: cAi

, Fi, Ti, Q , F

manip. var.s: Q, F

disturbances: cAi

, Fi, Ti

const. par.s: , cp, (Hr), k0, E , R

Chen CL 23


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Additional Examples of Mathematical Modeling

An Ideal Binary Distillation Column

Chen CL 24


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- Assumptions:

constant vapor holdup:

equal molar heats of vaporization for A and B

negligible heat loss constant relative volativility

100% tray efficiency

V = V1 = = VNyi =

xi1 + ( 1)xi

neglect dynamics of condenser and reboiler

neglect momentum balance for each trayleaving liquid = Li = f(Mi), i = 1, , N

liquid holdup = Mi

Chen CL 25


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- State Equations (1): feed tray (i = f)

total mass:dMf

dt= Ff + Lf+1 + Vf1 Lf Vf

= Ff + Lf+1 Lf

comp A:d(Mfxf)

dt= Ffcf + Lf+1xf+1 + Vf1yf1 Lfxf Vfyf

Chen CL 26


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- State Equations (2): top tray (i = N)

total mass:dM

N

dt= F

R+ V

N1 L

N V

N

= FR LN

comp A:d(M

NxN

)

dt= F

RxD

+ VN1

yN1

LN

xN V

NyN

Chen CL 27


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- State Equations (3): bottom tray (i = 1)

total mass:dM1

dt= L2 L1 + V V1

= L2 L1

comp A:d(M1x1)

dt= L2x2 + V yB L1x1 V1y1

Chen CL 28


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- State Equations (4): ith tray (i = 2, , N 1; i = f)

total mass:dMi

dt= Li+1 Li + Vi1 Vi

= Li+1 Li

comp A: d(Mixi)

dt= Li+1xi+1 Lixi + Vi1yi1 Viyi

Chen CL 29


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- State Equations (5): reflux drum

total mass:dM

RD

dt= V

N F

R F

D

comp A: d(MRDxD)dt

= VN

yN (F

R+ F

D)x

D

Chen CL 30


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- State Equations (6): column base

total mass:dM

B

dt= L1 V FB

comp A:d(M

BxB

)

dt= L1x1 V yB FBxB

Chen CL 31


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- Relationships:

equilibrium relationships:

yi =xi

1 + ( 1)xi i = 1, , f, , N; B

hydraulic relationships: (Francis weir formula)

Li = f(Mi) i = 1, , f, , N

- State Variables:

liquid holdups:

M1, M2, , Mf, , MN; MRD, MB

liquid concentrations:

x1, x2, , xf, , xN; xD, xB

Chen CL 32


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- Summaries:

2N + 4 nonlinear differential equations (state eqn.s)

2N + 1 algebraic equations (equilibrium and hydraulic)

example: N = 20 trays

2N + 4 = 2(20) + 4 = 44 nonlinear diff. eqn.s

2N + 1 = 2(20) + 1 = 41 algebraic equations

Chen CL 33


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Modeling Considerationsfor Control Purposes

- State-variables model

input-output model (convenient for control)

- Degrees of freedom ( df) inherent in the process extent of control problem to be solved

Chen CL 34


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- Input-Output Model:

output = f(input variables)

yi = f(m1, , mk; d1, , dt) i = 1, , m

Chen CL 35


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- Example: Input-Output Model for CSTR

Assumptions: Fi = F dV/dt = 0

Chen CL 36


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Total Energy Balance:

VdT

dt= Fi(Ti T) +

Q

cp

Q = U At(Tst T)

dT

dt+

FiV

+U AtV cp

a1/+K

T =FiV1/

Ti +U AtV cp K

Tst

dT

dt

+ aT = 1Ti + KTst

SS: 0 + aTs =1Ti,s + KTst,s

d(T Ts)

dt+ a (T Ts)

T

= 1 (Ti Ti,s) T

i,s

+K(Tst Tst,s) T

st

dT

dt + aT

=1T

i + KT

st

T

(t) = c1eat +

t0

1

T

i + KT

st

dt

initial: T

(t = 0) = 0 c1 = 0

T

(t) = t

0 1

T

i + KT

st dt

Chen CL 37


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Block Diagram: inputs (T

i(t), T

st(t)) output (T

(t))

This example: output variables = state variables

Chen CL 38

Di ill i bl bl !


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Distillation: output variables = state variables!

State variables:

liquid holdups:

M1, M2, , Mf, , MN; MRD, MB

liquid concentrations:

x1, x2, , xf, , xN; xD, xB

Output variables:

distillate rate and composition: FD

, xD

bottom rate and composition: FB, xB

Chen CL 39

DOF D f F d


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DOF: Degree of Freedom

-

Degrees of Freedom (DOF):# of independent variables that must be specified in order to define

a process completely

DOF = (# Var.s) (# Indep. Eq.s)

Chen CL 40

E l i d k h


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- Example: stirred tank heater

mathematical model: # of eq.s = 2

Adhdt

= Fi F

AhdT

dt= Fi (Ti T) +

Q

cp

# of variables = 6 (h, Ti, T , F , F i, Q) DOF = 6 - 2 = 4

specify Ti, Fi, F , Q h(t), T(t)

in order to specify a process completely

the # ofDoF

should be zero

Chen CL 41

- E l bi di ill i l


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- Example: binary distillation column

DOF = (4N + 11) (4N + 5) = 6

Chen CL 42

D f F d f A P


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Degrees of Freedom of A Process

-

f = DOF = V E = (# Var.s) (# Indep. Eq.s)

- Case 1: DOF = 0

unique values of the V variables

the process is exactly specified

- Case 2: DOF > 0

multiple solutions result from the E equations

can specify arbitrarily f of the V variables

the process is underspecified by f equations

- Case 3: DOF < 0

no solution to the E equations

the process is overspecified by f equations

Chen CL 43

DOF d P C t ll


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DOF and Process Controllers

-

An under-specified process with DOF = f > 0

- Q: how to reduce DOF to zero

to specify system completely with unique behavior ?

from external world: disturbances

to add control loops

- Control loop:

additional equation between MV and CV

additional variable: set-point

same: DOF

difference: specify MV specify set-point

Chen CL 44

- E l ti d t k h t ith t t l l


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- Example: stirred tank heater with two control loops

DOF = 4 DOF = 0 if we specify

Ti, Fi from external world ( disturbances) set-points of the two controllers

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- E l bi di till ti l ( DOF 6)


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- Example: binary distillation column ( DOF = 6)

specification of disturbances (external world):

feed rate (Ff) and feed composition (cf)

DOF = 6 DOF = 4

specification of control objectives ( set-points):

(I) for products:

xD

: distillate composition

xB

: bottom stream composition

(II) for operational feasibility:

MRD: liquid holdup in reflux drum M

B: liquid holdup at base of column

four control loops

DOF = 6 DOF = 4 DOF = 0

Chen CL 46


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Note: other alternative control objectives

(1) keep at desired FD

, xD

, MRD

, MB

(2) keep at desired FB

, xB

, MRD

, MB

05 Modeling Dynamic and Static Behavior of Chemical Processes

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