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CHE412 Process Dynamics and ControlBSc (Engg) Chemical Engineering (7th Semester)

Week 2/3Mathematical Modeling

Luyben (1996) Chapter 2-3 Stephanopoulos (1984) Chapter 4, 5

Seborg et al (2006) Chapter 2

Dr Waheed Afzal Associate Professor of Chemical Engineering

Institute of Chemical Engineering and TechnologyUniversity of the Punjab, Lahore

wa.icet@pu.edu.pk

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Test yourself (and Define):• Dynamics (of openloop

and closedloop) systems• Manipulated Variables• Controlled/ Uncontrolled

Variables• Load/Disturbances• Feedback, Feedforward

and Inferential controls• Error• Offset (steady-state

value of error)• Set-point

Hint: Consult recommended books (and google!) Luyben (1996), Coughanower and LeBlanc (2008)

• Stability • Block diagram • Transducer • Final control element • Mathematical model• Input-out model,

transfer function • Deterministic and

stochastic models• Optimization• Types of Feedback

Controllers (P, PI, PID)

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(t)

• Proportional: c(t) = Kc Є(t) + cs

• Proportional-Integral:

• Proportional-Integral-Derivative:

Nomenclature actuating output , error , gain , time constant Consult your class notes onmodelling of stirred tank heater

Types of Feedback Controllers

(Stephanopoulos, 1984)

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• Mathematical representation of a process (chemical or physical) intended to promote qualitative and quantitative understanding

• Set of equations• Steady state, unsteady state (transient) behavior

• Model should be in good agreement with experiments

Mathematical Modeling

Experimental Setup

Set of Equations(process model)

Inputs Outputs

Outputs

Compare

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1. Determine objectives, end-use, required details and accuracy

2. Draw schematic diagram and label all variables, parameters3. Develop basis and list all assumptions; simplicity Vs reality 4. If spatial variables are important (partial or ordinary DEs)5. Write conservation equations, introduce auxiliary equations 6. Never forget dimensional analysis while developing

equations 7. Perform degree of freedom analysis to ensure solution 8. Simplify model by re-arranging equations 9. Classify variables (disturbances, controlled and manipulated

variables, etc.)

Systematic Approach for Modelling (Seborg et al 2004)

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• To understand the transient behavior, how inputs influence outputs, effects of recycles, bottlenecks

• To train the operating personnel (what will happen if…, ‘emergency situations’, no/smaller than required reflux in distillation column, pump is not providing feed, etc.)

• Selection of control pairs (controlled v. / manipulated v.) and control configurations (process-based models)

• To troubleshoot • Optimizing process conditions (most profitable

scenarios)

Need of a Mathematical Model

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• Theoretical Models based on principal of conservation- mass, energy, momentum and auxiliary relationships, ρ, enthalpy, cp, phase equilibria, Arrhenius equation, etc)• Empirical model based on large quantity of experimental data) • Semi-empirical model (combination of theoretical

and empirical models) Any available combination of theoretical principles and empirical correlations

Classification of Process Models based on how they are developed

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Theoretical Models Physical insight into the process Applicable over a wide range of conditions Time consuming (actual models consist of large

number of equations) Availability of model parameters e.g. reaction rate

coefficient, over-all heart transfer coefficient, etc. Empirical model Easier to develop but needs experimental data Applicable to narrow range of conditions

Advantages of Different Models

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State variables describe natural state of a process Fundamental quantities (mass, energy, momentum)

are readily measurable in a process are described by measurable variables (T, P, x, F, V)

State equations are derived from conservation principle (relates state variables with other variables)

(Rate of accumulation) = (rate of input) – (rate of output) + (rate of generation) - (rate of consumption)

State Variables and State Equations

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BasisFlow rates are volumetric Compositions are molar A → B, exothermic, first order Assumptions Perfect mixingρ, cP are constant

Perfect insulation Coolant is perfectly mixedNo thermal resistance of jacket

Modeling ExamplesJacketed CSTR

Coolant

Fi, CAi, Ti

F, CA, T

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Overall Mass Balance (Rate of accumulation) = (rate of input) – (rate of output) Component Mass Balance (Rate of accumulation of A) = (rate of input of A) – (rate of output of A) + (rate of generation of A) – (rate of consumption of A)

Modeling of a Jacketed CSTR (Contd.)

CoolantFci,Tci

VCA

T

Fi, CAi, Ti

F, CA, T

CoolantFco,Tco

Energy Balance (Rate of energy accumulation) = (rate of energy input) – (rate of energy output) - (rate of energy removal by coolant) + (rate of energy added by the exothermic reaction)

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Overall Mass Balance

Component Mass Balance

Energy Balance

Modeling of a Jacketed CSTR (Contd.)

CoolantFci,Tci

VCA

T

Fi, CAi, Ti

F, CA, T

CoolantFco,Tco

Input variables: CAi, Fi, Ti, Q, (F) Output variables: V, CA, T

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Nf = Nv - NE

Case (1): Nf = 0 i.e. Nv = NE (exactly specified system)

We can solve the model without difficulty

Case (2): f > 0 i.e. Nv > NE (under specified system), infinite number of solutions because Nf process variables can be fixed arbitrarily. either specify variables (by measuring disturbances) or add controller equation/s

Case (3): Nf < 0 i.e. Nv < NE (over specified system) set of equations has no solution

remove Nf equation/s

We must achieve Nf = 0 in order to simulate (solve) the model

Degrees of Freedom (Nf) Analysis

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Basis/ Assumptions Perfectly mixed, Perfectly insulated ρ, cP are constant

Stirred Tank Heater: Modeling and Degree of Freedom Analysis

SteamFst

A

Overall Mass Balance

Energy Balance

Degree of Freedom Analysis Independent Equations: 2 Variables: 6 (h, Fi, F, Ti, T, Q) Nf = 6-2 (= 4) Underspecified

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Nf = 4 Specify load variables (or disturbance)Measure Fi, Ti (Nf = 4 - 2 = 2) Include controller equations (not

studied yet); specify CV-MV pairs:

Stirred Tank Heater: Modeling and Degree of Freedom Analysis

SteamFst

ACV MV

h F

T Q

Nf = 2 - 2 = 0 Can you draw these control loops?

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F

Z

mB

mD

VB

DxD

BxB

R xD

Reboiler

Condenser

Reflux Drum

(Stephanopoulos, 1984)

Basis/ Assumptions 1. Saturated feed2. Perfect insulation of column3. Trays are ideal4. Vapor hold-up is negligible 5. Molar heats of vaporization of A

and B are similar 6. Perfect mixing on each tray7. Relative volatility (α) is constant 8. Liquid holdup follows Francis weir

formulae 9. Condenser and Reboiler dynamics

are neglected10. Total 20 trays, feed at 10 2, 4, 5 → V1 = V2 = V3 = … VN

(not valid for high-pressure columns)

Modeling an Ideal Binary Distillation Column

= 20

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Reflux Drum Overall Component

V20

DxD

RxD

Reflux Drum

N = 20

V20 R

V19L20Top Tray

Top Tray Overall

Component )Remember V1 = V2 = …. VN = VB

mD

Modeling Distillation Column

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Nth Stage

mN

vNLN+1

LN vN-1

Nth Stage (stages 19 to 11 and 9 to 2) Overall

Component

…. simplify!

Modeling Distillation Column

Feed Stage(10th)

mN

v10L11

L10 v9

F Z

Feed Stage (10th) Overall

Component

…. simplify!

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VBL1

V1 L2

1st Stage Modeling Distillation Column

1st Stage Overall

Component

… simplify!

VB

L1

Column Base

mB

B

Column Base Overall

Component

…. simplify!

VB

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Modeling Distillation ColumnEquilibrium relationships (to determine y)

Mass balance (total and component) around 6 segments of a distillation column: reflux drum, top tray, Nth tray, feed tray, 1st tray and column base.

Solution of ODE for total mass balance gives liquid holdups (mN) Solution of ODE for component mass balance gives liquid

compositions (xN) V1 = V2 = … = VN = VB (vapor holdups) How to calculate y (vapor composition) and L (liquid flow rate) Recall αij is constant throughout the column Use αij = ki/kj , xi + xj =1, yi + yj = 1, and k = y/x to prove

Phase-equilibrium relationship (recall thermodynamics)

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Liquid flow rate can be calculated using well-known Francis weir hydraulic relationship; simple form of this equation is linearized version:

LN is flow rate of liquid coming from Nth stage LN0 is reference value of flow rate LN

mN is liquid holdup at Nth stage mN0 is reference value of liquid holdup mN β is hydraulic time constant (typically 3 to 6 seconds)

Modeling Distillation ColumnHydraulic relationships (to determine L)

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Modeling Distillation ColumnDegree of Freedom Analysis

Total number of independent equations: Equilibrium relationships (y1, y2, …yN, yB) → N+1 (21) Hydraulic relationships (L1, L2, …LN) → N (20)

(does not work for liquid flow rates D and B) Total mass balances (1 for each tray, reflux drum and

column base) → N+2 (22)

Total component mass balances (1 for each tray, reflux drum and column base) → N+2 (22)

Total Number of equations NE = 4N + 5 (85)

44 differential and 41 algebraic equationsNote the size of model even for a ‘simple’ system with several

simplifying assumptions!

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Modeling Distillation ColumnDegree of Freedom Analysis

Total number of independent variables: Liquid composition (x1, x2, …xN, xD, xB) → N+2 Liquid holdup (m1, m2, …mN, mD, mB) → N+2 Vapor composition (y1, y2, …yN, yB) → N+1 Liquid flow rates (L1, L2, …LN) → N Additional variables → 6 (Feed: F, Z; Reflux: D, R; Bottom: B, VB) Total Number of independent variables NV = 4N + 11

Degree of Freedom = (4N + 11) – (4N + 5) = 6

System is underspecified

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Modeling Distillation ColumnDegree of Freedom Analysis

(4N + 11) – (4N + 5) = 6 Specify disturbances: F, Z (Nf = 6-2 = 4) Include controller equations (Recall our discussion on types

of feedback controllers ) General form, of P-Controller c(t) = cs + Kc Є(t)

Controlled Variable

Manipulated Variable

xD R

xB VB

mD D

mB B

R = Kc (xs - xD) + Rs

VB = Kc (xBs-xB) + VBs

D = Kc (mDs-mD) + Ds

B = Kc (mBs-mB) + Bs

Nf = 4 - 4 = 0

Controller Equation(Proportional Controller)

Can you draw these four feedback control loops?

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Feedback Control on a Binary Distillation ColumnCV MV loopxD R 1

xB VB 2

mD D 3mB B 4

R

(Stephanopoulos, 1984)

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Week 2/3Weekly Take-Home Assignment

1. Define all the terms on slide 2 with examples whenever possible.2. Prepare short answers to ‘things to think about’

(Stephanopoulos, 1984) page 33-353. Prepare short answers to ‘things to think about’

(Stephanopoulos, 1984) page 78-794. Solve the following problems (Chapter 4 and 5 of

Stephanopoulos, 1984): II.1 to II.14, II.22, II.23 (Compulsory)

Submit before Friday Curriculum and handouts are posted at:http://faculty.waheed-afzal1.pu.edu.pk/