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RESEARCH AND TEACHING METHODS. CLASS PROJECT. LECTURER: MANUEL GARCIA-PEREZ , Ph.D. Department of Biological Systems Engineering 205 L.J. Smith Hall, Phone number: 509-335-7758 e-mail: [email protected] OUTLINE. 1.- CLASS PROJECT. 2.- PROCESS MODELLING. PHYSICAL MODEL.

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Lecturer manuel garcia perez ph d

RESEARCH AND TEACHING METHODS

CLASS PROJECT

LECTURER: MANUEL GARCIA-PEREZ , Ph.D.

Department of Biological Systems Engineering

205 L.J. Smith Hall, Phone number: 509-335-7758

e-mail: [email protected]


Lecturer manuel garcia perez ph d

OUTLINE

1.- CLASS PROJECT

2.- PROCESS MODELLING

PHYSICAL MODEL

MATHEMATICAL MODEL

SOLVING THE MODEL AND NUMERICAL METHODS

REFERENCES:

CHAPRA SC, CANALE RP: NUMERICAL METHODS FOR ENGINEERS. WITH SOFTWARE AND PROGRAMMING APPLICATIONS. FOURTH EDITION. McGRAW-HILL HIGHER EDUCATION, 2002

BIRD B.R., STEWART W.E: TRANSPORT PHENOMENA. SECOND EDITION. JOHN-WILEY, 2007

HANGOS K, CAMERON I: PROCESS MODELLING AND MODEL ANALYSIS. ACADEMIC PRESS, 2001.


Lecturer manuel garcia perez ph d

1.- CLASS PROJECT

Goal and Objectives:

(1) Gain basic skills to develop mathematical models describing the behavior of simple processes in which biological materials are converted into food, fuels and chemicals.

(2) Identify suitable numerical methods to solve the mathematical model proposed and develop simple algorithms (programming flow chart) to simulate the process of interest.

(3) Be aware of what kind of experimental data is needed to adjust the parameters of your model.

(4) Propose a strategy to validate the model. How to acquire, process and analyze the information needed for validation.

(5) Explain how to use the computer simulation code developed in this project to study the system of interest.


Lecturer manuel garcia perez ph d

1.- CLASS PROJECT

Tasks

The specific tasks are outlined below:

1.-Make a brief description of the technology you are improving or developing as part of your graduate studies.

2.-Identify a simple component of your technology that you would like to model. Answer the following questions:

What is the intended use of the mathematical model?

What are the governing phenomena or mechanism for the system of interest?

In what form is the model required?

How should the model be instrumented and documented?

What are the systems inputs and outputs?

How accurate does the model have to be?

What data on the system are available and what is the quality of and accuracy of the data?


Lecturer manuel garcia perez ph d

1.- CLASS PROJECT

Tasks

3.-Develop a phenomenological model to describe the behavior of the system of your interest. The phenomenological models should be based on mass and energy balances (Use microscopic, macroscopic or plug flow models).

4.- Identify the most suitable numerical method to solve the model developed in task 3. Try to answer the following questions:

What variables must be chosen in the model to satisfy the degrees of freedom?

Is the model solvable?

What numerical (or analytical) solution techniques should be used?

What form of representation should be used to display the results (2 D graphs, 3D, Visualization)?


Lecturer manuel garcia perez ph d

1.- CLASS PROJECT

5.- Develop an algorithm (programming flow chart) and a computer code (in any high-level computing language) to evaluate how the output variables will change when the input variables are modified. If you decide not to use a high-level computer language you may choose to use Microsoft Excel.

6.- Identify what kind of experimental data should be collected to adjust the parameters of the model proposed.

7.- Suggest a strategy to validate your model.


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING

MODELLING IS NOT JUST ABOUT PRODUCING A SET OF EQUATIONS, THERE IS FAR MORE TO PROCESS MODELLING THAN WRITING EQUATIONS.

A PARTICULAR MODEL DEPENDS NOT ONLY ON THE PROCESS TO BE DESCRIBED BUT ALSO ON THE MODELLING GOAL. IT INVOLVES THE INTENDED USE OF THE MODEL AND THE USER OF THAT MODEL.

THE ACTUAL FORM OF THE MODEL IS ALSO DETERMINED BY THE EDUCATION, SKILLS AND TASTE OF THE MODELLER AND THAT OF THE USER.

THE BASIC PRINCIPLES IN MODEL BUILDING ARE BASED ON OTHER DISCIPLINES IN PROCESS ENGINEERING SUCH AS MATHEMATICS, CHEMISTRY AND PHYSICS. THEREFORE, A GOOD BACKGROUND IN THESE AREAS IS ESSENTIAL FOR A MODELLER. THERMODYNAMICS, UNIT OPERATIONS, REACTION KINETICS, CATALYSIS, PROCESS FLOWSHEETING AND PROCESS CONTROL ARE HELPFUL PRE-REQUISITES FOR A COURSE IN PROCESS MODELLING.


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING

A MODEL IS AN IMITATION OF REALITY AND A MATHEMATICAL MODEL IS A PARTICULAR FORM OF REPRESENTATION.

IN THE PROCESS OF MODEL BUILDING WE ARE TRANSLATING OUR REAL WORLD PROBLEM INTO AN EQUIVALENT MATHEMATICAL PROBLEM WHICH WE SOLVE AND THEN ATTEMPT TO INTERPRET. WE DO THIS TOGAIN INSIGHT INTO THE ORIGINAL REAL WORLD SITUATION OR TO USE THE MODEL FOR CONTROL, OPTIMIZATION OR POSSIBLE SAFETY STUDIES.

2

1

Real world Problem

Mathematical problem

Mathematical Solution

Interpretation

3

4


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING

SYSTEMATIC METHODOLOGY TO BUILD MATHEMATICAL MODELS

1.- PROBLEM DEFINITION

2.- IDENTIFY CONTROLLING FACTORS

3.- SUITABLE PHYSICAL MODEL

4.- CONSTRUCT THE MATHEMATICAL MODEL

5.- PRELIMINARY EVALUATION OF MODEL

6.- SOLVE THE MATHEMATICAL MODEL (NUMERICAL METHOD)

7.- DEVELOP AN ALGORITM TO SOLVE THE PROBLEM

8.- COMPUTER PROGRAMMING

9.- ADJUST MODEL PARAMETERS

10.- VALIDATE THE MODEL


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (PROBLEM DEFINITION AND CONTROLLING FACTORS)

1.- DEFINE THE PROBLEM: IT FIXES THE DEGREE OF DETAIL RELEVANT TO THE MODELLING GOAL AND SPECIFIES:

A.- INPUTS AND OUTPUTS

B.- HIERARCHY LEVEL RELEVANT TO THE MODEL

C.- THE NECESSARY RANGE AND ACCURACY OF THE MODEL

D.- THE TIME CHARACTERISTICS (STATIC VERSUS DYNAMIC) OF THE PROCESS MODEL.

2.- IDENTIFY THE CONTROLLING FACTORS OR MECHANISMS: THE NEXT STEP IS TO INVESTIGATE THE PHYSICO-CHEMICAL PROCESSES AND PHENOMENA TAKING PLACE IN THE SYSTEM RELEVANT TO THE MODELLING GOAL. THESE ARE TERMED CONTROLLING FACTORS OR MECHANISMS. THE MOST IMPORTANT CONTROLLING FACTORS INCLUDE:

A.- CHEMICAL REACTION, B.- DIFFUSION OF MASS, C.- CONDUCTION OF HEAT D.- FORCED CONVECTION HEAT TRANSFER, E.- FREE CONVECTION HEAT TRANSFER, F.- RADIATION HEAT TRANSFER, G.- EVAPORATION, H.- TURBULENT MIXING, I.- HEAT OR MASS TRANSFER THROUGH A BIUNDARY LAYER J.- FLUID FLOW.


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (PHYSICAL MODEL)

SYSTEMATIC METHODOLOGY TO BUILD MATHEMATICAL MODELS

1.- PROBLEM DEFINITION

2.- IDENTIFY CONTROLLING FACTORS

3.- SUITABLE PHYSICAL MODEL

4.- CONSTRUCT THE MATHEMATICAL MODEL

5.- PRELIMINARY EVALUATION OF MODEL

6.- SOLVE THE MATHEMATICAL MODEL (NUMERICAL METHOD)

7.- DEVELOP AN ALGORITM TO SOLVE THE PROBLEM

8.- COMPUTER PROGRAMMING

9.- ADJUST MODEL PARAMETERS

10.- VALIDATE THE MODEL


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (PHYSICAL MODEL)

3.- CREATE A SUITABLE PHYSICAL MODEL

REALITY

Identified essential process characteristics

Incorrectly identified process characteristics

PHYSICAL MODEL

Identified non-essential process characteristics

THERE ARE STANDARD MATHEMATICAL DESCRIPTIONS FOR EACH OF THE COMPONENTS OF THE PHYSICAL MODEL.


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (PHYSICAL MODEL)

THE LEVEL OF MIXING IS ONE OF THE MOST IMPORTANT PARAMETERS DEFINING THE PHYSICAL MODEL TO BE USED. THIS DETERMINES THE EXISTENCE OF NOT OF GRADIENTS INSIDE THE SYSTEM.

GRAPHIC REPRESENTATION

PHYSICAL MODEL

OBSERVATIONS

MICROSCOPIC BALANCES

ABSENCE OF MACROSCOPIC MIXING IN ALL DIRECTIONS. (ONLY MOLECULAR MIXING, LAMINAR FLOW)

IT IS COMMONLY USED TO DESCRIBE THE BEHAVIOUR OF SYSTEMS IN TURBULENT REGIME.

PLUG FLOW MODEL

MACROSCOPIC BALANCES

MIXING IN ALL DIRECTIONS (IT IS USED TO DESCRIBE THE BEHAVIOUR OF STIRRED TANKS)


Lecturer manuel garcia perez ph d

EXAMPLE (FLUIDIZED BED REACTORS)

2.- PROCESS MODELLING (PHYSICAL MODEL)

PLUG FLOW MODEL

MACROSCOPIC BALANCES

???

FREEBOARD

SPLASH ZONE

BUBBLE PHASE

SOLID PHASE

EMULSION PHASE

EXCHANGE OF HEAT AND MASS

EXCHANGE OF HEAT AND MASS

BUBBLING ZONE

BIOMASS

???

JET ZONE

CARRIER GAS (EMULSION PHASE)

CARRIER GAS (BUBBLE)

CARRIER GAS

BIOMASS

(ONE PHYSICAL MODEL PER PHASE)

SCHEME OF A FLUIDIZED BED REACTOR


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (PHYSICAL MODEL)

PHYSICAL MODELS FOR THE SOLID PHASE

SELF SEGREGATION MODEL (PLUG FLOW)

MACROSCOPIC BALANCES

PLUG FLOW

VOLATILES

FINES

COARSE

Bubble

EMULSION PHASE

BIOMASS

BIOMASS


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (PHYSICAL MODEL)

HOW TO FORMALIZE THE CHEMICAL COMPOSITION OF THE SYSTEM?

OFTEN THE CHEMICAL DESCRIPTION OF THE SYSTEM IS CONDITIONED TO THE KIND OF DATA AVAILABLE IN THE LITERATURE AND BY THE GOALS OF THE MODEL.

k3

ANHYDROCELLULOSE

0.65 GAS + 0.35 CHAR

k1

CELLULOSE

k2

TAR

TYPICAL TERMS USED TO DESCRIBE THE CHEMICAL COMPOSITION OF THERMOCHEMICAL PROCESSES :

BIOMASS, FIXED CARBON (CHARCOAL), VOLATILES, GASES, CO2, CO, H2O, ASH, TARS, BIO-OILS

B

C

D

E

A


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

SYSTEMATIC METHODOLOGY TO BUILD MATHEMATICAL MODELS

1.- PROBLEM DEFINITION

2.- IDENTIFY CONTROLLING FACTORS

3.- SUITABLE PHYSICAL MODEL

4.- CONSTRUCT THE MATHEMATICAL MODEL

5.- PRELIMINARY EVALUATION OF MODEL

6.- SOLVE THE MATHEMATICAL MODEL (NUMERICAL METHOD)

7.- DEVELOP AN ALGORITM TO SOLVE THE PROBLEM

8.- COMPUTER PROGRAMMING

9.- ADJUST MODEL PARAMETERS

10.- VALIDATE THE MODEL


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

CONSTRUCTION OF MATHEMATICAL MODEL

MACROSCOPIC BALANCES

MASS BALANCES SPECIE i:

You should write a mass balance per every component per every phase

1

d mi,tot/ dt = - D( ri <v> S) + wim + ri,av Vtot

Rate of mass accumulation of specie i

Rate of mass generation of specie i by reaction

Net Rate of mass exchange of specie through the interface.

Q

W

ENERGY BALANCE

2

d Etot/dt = - D (ri∙ v ∙ S) [h+ ½ ∙ v2 + F] + Q - W

You should write an Energy balance per phase

Heat

Work

Energy accumulation

Energy associated to each inlet and outlet

MOST COMMON ENERGY BALANCE FOR REACTING SYSTEMS

V ∙ r ∙ cp dT / dt = ∑ Fj∙ cpj∙(Tj - T) + ri,av∙ V ∙(-DHR) + Q + W

Heat

Energy accumulation

Energy associated to each inlet and outlet

Work

rA: Production of compound by chemical reaction (kmol/m3.s) (-) if produced, (-) if consumed

Q: (+) if generated (-) if consumed


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

MEANING OF SOME TERMS:

<v> average velocity (m/s)

S

S: areas of transversal section of inlet and outlet pipes (m2)

<V>

r: density of fluid (kg/m3)

.

W = <v> ∙ r ∙ S = m = [(m/s)(m2)(kg/m3)] = [kg/s]

wim: transport of component i through the interface per unit of time (kg/s) (+) if it enters to the system and (-) if it exists the system

F : Potential Energy

K: Kinetic Energy

U: Internal Energy


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

PLUG FLOW

CONSTRUCTION OF MATHEMATICAL MODEL

MASS BALANCES SPECIE i:

mi=kc ∙a ∙DC = Ky∙a∙Dy

You should write a mass balance per every component per every phase

Mass balance per unit of volume

dCi / d t + d (vz∙ Ci)/dz = Ri + mi

E

mi

dz

Transport for convection

Rate of mass accumulation of specie i

Net Rate of mass exchange of specie through the interface.

Rate of mass generation of specie i by reaction

Et= (4/D) ∙ U ∙DT

ENERGY BALANCE

r Cp∙ (dT/dt + vz ∙ dT/dz) = SR + Et

You should write an Energy balance per every phase

Heat or work transport through the interface

Energy transport by convection

Rate of Energy accumulation

Units:

Property/vol. time

Heat associated with chemical reactions

RA: Production of compound by chemical reaction (kmol/m3.s) (-) if produced, (-) if consumed

SR: Heat associated with chemical reactions (kJ/m3.s) SR= DHR∙RA (+) if generated (-) if consumed


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

MICROSCOPIC BALANCES

CONSTRUCTION OF MATHEMATICAL MODEL

RECTANGULAR COORDENATES (r, m, D, k, Cp are considered constant)

MASS BALANCES SPECIE i:

dCA/ dt + vxdCA/dx + vydCA/dy + vzdCA/dz=DAB [d2CA/dx2 + d2CA/dy2+d2CA/dz2] + RA

Accumulation

Transport per diffusion

Generation

Transport per convection

ENERGY BALANCE:

r ∙ Cp [ dT/ dt + vxdT/dx + vydT/dy + vzdT/dz=k [d2T/dx2 + d2T/dy2+d2T/dz2] + SR

Transport per thermal diffusion

Transport per convection

Accumulation

Generation


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

BALANCE OF MOMENTUM (FOR NEWTONIAN FLUIDS, CARTESIAN COORDENATES):

NAVIER-STOKES EQUATIONS

DIRECTION X

r ∙ [dvx/ dt + vx d vx/dx + vyd vx/dy + vzd vx/dz = dp/dx + m [d2 vx /dx2 + d2 vx /dy2+d2 vx /dz2] +r gx

Rate of momentum addition by convection per unit volume

Rate of momentum addition by molecular transport per unit volume

Rate of increase of momentum per unit volume

External Force

DIRECTION Y

r ∙ [dvy/ dt + vx d vy/dx + vyd vy/dy + vzd vy/dz = dp/dy + m [d2 vy /dx2 + d2 vy /dy2+d2 vy /dz2] +r gy

Rate of increase of momentum per unit volume

Rate of momentum addition by convection per unit volume

Rate of momentum addition by molecular transport per unit volume

External Force

DIRECTION Z

r ∙ [dvz/ dt + vx d vz/dx + vyd vz/dy + vzd vz/dz = dp/dz + m [d2 vz /dx2 + d2 vz /dy2+d2 vz /dz2] +r gz


Lecturer manuel garcia perez ph d

2.- SINGLE PARTICLE MODELS (MATHEMATICAL MODEL)

CONSTITUTIVE RELATIONS:

TRANSFER RELATIONSHIP:

TE

HEAT TRANSFER

BUBBLE

MASS TRANSFER: mi=K (CEi-CBi)

TB

MASS TRANSFER COEFFICIENT

CBi

MASS TRANSFER

HEAT TRANSFER: E=U ∙ a ∙ (TE-TB)

CEi

EMULSION

HEAT TRANSFER COEFFICIENT

REACTION KINETICS:

Ri = - ko e–E/(RT) Cjn

THERMODYNAMICAL RELATIONS

EQUILIBRIUM RELATIONSHIPS

Raoult’s law model : yi= xj Pjvap/P

PROPERTY RELATIONS

Relative volatility model : yi= aij xi /(1+ (aij -1) xi

Liquid density: rL = f (P, T, xi)

Vapour density: rV = f (P, T, xi)

Liquid enthalpy: h = f (P, T, xi)

Vapour enthalpy: H = f (P, T, yi)

K model : Kj = yj / xj

EQUATIONS OF STATE

Ideal gas, Redleich-Kwong, Peng-Robinson and Soave-Redleich-Kwong equations.


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

WHAT EQUATION SHOULD BE USED?

MASS BALANCES: IF THE PARAMETER OF INTEREST IS RELATED WITH CHANGES IN CONCENTRATIONS

ENERGY BALANCE: IF THE PARAMETER OF INTEREST IS RELATED WITH CHANGES IN TEMPERATURE

BALANCE OF MOMENTUM: IF THE PARAMETER OF INTEREST IS RELATED WITH DISTRUBTION OF VELOCITIES .

WHAT SYSTEM OF COORDENATES SHOULD BE USED?

IMPORTANT WHEN USING MICROSCOPIC MODELS

CARTESIAN COORDINATE SYSTEM

CYLINDRICAL COORDINATE SYSTEM


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

Simplifications

In steady state the properties do not change with time (dp/dt = 0)

When a property is transported in the same direction by more than one mechanism, you should evaluate the possibility of only taking into account the controlling mechanism. Example: Disregard molecular mechanisms if the property is also transported by turbulent mechanisms.

When the distance to the source that produces the changes is constant in certain direction, then you can consider that there is no gradient of the property of interest along this direction.

Source that produces the changes

z

y

x

Q

Source that produces the changes

dTz/dy = 0


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

EXAMPLE 1:

A viscous fluid is heated as it flows by gravity in a rectangular channel with a moderate slope. Develop a mathematical model that allows you to determine the temperature profiles in the liquid at any position along the channel. The system receives heat from the bottom (Bottom Temperature: 100 oC). The dimensions of the channel are:

Case I: a = 100 cm; h = 5 cm Case II: a = 10 cm, h = 5 cm

a

Y

X

Z

h

vz

HEAT


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

PHYSICAL MODEL:

VISCOUS MATERIAL, FLOWING DUE TO THE ACTION OF GRAVITATIONAL FORCES (MODERATE SLOPE). IT IS LOGICAL TO SUPPOSE THAT IT IS FLOWING IN LAMINAR REGIME. (MICROSCOPIC MODEL)

IN THESE CONDITIONS THE FLOW HAPPENS WITHOUT MIXING IN THE AXIAL DIRECTION. NO MIXING IN THE DIRECTION PERPENDICULAR TO THE FLOW.

PHYSICAL MODEL: MICROSCOPIC MODEL

MATHEMATICAL MODEL:

TEMPERATURE PROFILE

ENERGY BALANCE

COORDENATE SYSTEM: RECTANGULAR (CARTESIAN)

GENERAL MATHEMATICAL MODEL:

r ∙ Cp [ dT/ dt + vxdT/dx + vydT/dy + vzdT/dz=k [d2T/dx2 + d2T/dy2+d2T/dz2] + SR

Transport per thermal diffusion

Transport per convection

Accumulation

Generation


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

SIMPLIFICATIONS:

1.- STEADY STATE: (dT/dt) = 0

2.- THE ONLY COMPONENT OF VELOCITY THAT EXIST IS IN THE DIRECTION OF THE MAIN FLOW (DIRECTION Z): vx = vy = 0

3.- NO CHEMICAL REACTION, SO THERE IS NO HEAT ASSOCIATED WITH THE CHEMICAL REACTION: SR = 0

4.- THERE IS HEAT EXCHANGE ONLY THROUGH THE BOOTOM. THE LATERAL WALLS ARE CONSIDERED INSOLATED: d2T/dx2 = 0

5.- THE HEAT TRANSFER BY CONDUCTION IN THE AXIAL DIRECTION IS NEGLIGIBLE COMPARED WITH THE TRANSPORT OF ENERGY DUE TO THE MOVEMENT OF THE FLUID IT MEANS:

r ∙ Cp vzdT/dz >> k [d2T/dz2]


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

HEAT

a

Y

X

h

Z

vz

HEAT

HEAT

0

0

0

~0

~0

0

r ∙ Cp [ dT/ dt + vxdT/dx + vydT/dy + vzdT/dz=k [d2T/dx2 + d2T/dy2+d2T/dz2] + SR

r ∙ Cp ∙ vz∙dT/dz=k [d2T/dy2]

Case I: a = 1000 cm; h = 5 cm

r ∙ Cp [vzdT/dz=k [d2T/dx2 + d2T/dy2]

Case II: a = 10 cm, h = 5 cm

TO SOLVE THIS EQUATION IT IS NECESSARY TO ESTIMATE THE VALUES OF vz AT DIFFERENT VALUES OF X, Y, Z (MOMENTUM EQUATION). IF THE CHANNEL IS WIDE ENOUGH THEN THE CHANGES OF vz AS A FUNCTION OF X CAN BE CONSIDERED NEGLIGIBLE.


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

EXAMPLE 2:

A GAS IS HEATED IN A TUBULAR HEAT EXCHANGER. BECAUSE OF THE LOW STABILITY OF CERTAIN COMPONENTS THIS STREAM CANNOT REACH TEMPERATURES OVER Ts. DEVELOP A MATHEMATICAL MODEL TO DESCRIBE THE TEMPERATURE PROFILE OF THIS REACTOR.

SATURATED VAPOUR

GASES

GASES

CONDENSATE

PHYSICAL MODEL

DEPENDING ON THE FLOW REGIME THE TEMPERATURE CAN VARY RADIALLY OR AXIALLY. MOST INDUSTRIAL SYSTEMS OPERATE IN TURBULENT REGIME BECAUSE HEAT TRANSFER COEFICIENTS ARE HIGHER. IT IS REASONABLE TO SUPPOSE THAT THE GAS IS FLOWING IN TURBULENT REGIME.


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

PHYSICAL MODEL:

TURBULENT REGIME, A SINGLE PHASE

PHYSICAL MODEL: PLUG FLOW

MATHEMATICAL MODEL:

PROPERTY OF INTEREST: TEMPERATURE

EQUATION: ENERGY BALANCES

r Cp∙ (dT/dt + vz ∙ dT/dz) = SR + Et

SIMPLIFICATIONS:

EXCEPT DURING STARTUP AND SHUTDOWNS THE SYSTEM WILL BE OPERATING AT STEADY STATE.

dT/dt = 0

NO CHEMICAL REACTION: SR = 0

r Cp∙ vz ∙ dT/dz = Et

THE VALUES OF EtCAN BE CALCULATED FOR TUBES USING THE FOLLOWING EQUATION:

Et = (4/D) U (TV-T)

r Cp∙ vz ∙ dT /dz = (4/D) U (Tv -T)


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

EXAMPLE 3:

Develop a mathematical model to calculate the profiles of temperature and concentration in a steady state for a tubular insolated reactor. This reactor is fed with an homogeneous stream containing component A. Consider an incompressible system (liquid).

Irreversible reaction

A

B

A+ B

A

Solvent

Solvent

INSOLATED SYSTEM

15 m

The dependency of the reaction rate with the temperature can be described by the Arrhenius equation:

E = 4652 kJ/kmol

rA= K ∙ CA

K = A ∙ exp ∙ (-E/RT)

A = 3.00 s-1

Consider the axial diffusion negligible.


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

EXAMPLE 3:

DATA:

VELOCITY OF FLUID: 3 m/s

SPECIFIC HEAT: 4.184 J/kg K

ENTALPY OF REACTION: -279.12 kJ/kg

PHYSICAL MODEL:

PLUG FLOW

EQUATIONS: MASS AND ENERGY BALANCES

0

MATHEMATICAL MODEL:

0

MASS BALANCE:

dCA / d t + d (vz∙ CA)/dz = RA + mA

dCA / d t = 0 (STEADY STATE)

mA = 0 (SINGLE PHASE, NO MASS TRANSPORT THORUGH THE INTERPHASES)

vz = CONSTANT (INCONPRESSIBLE FLUID)

vz∙ dCA/dz = RA

RA= A ∙ exp (-E/RT) ∙ CA


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (MATHEMATICAL MODEL)

EXAMPLE 3:

dCA/dz = [A ∙ exp (-E/RT) ∙ CA ]/ vz

???

ENERGY BALANCE:

0

0

r Cp∙ (dT/dt + vz ∙ dT/dz) = SR + Et

dT/dt = 0 STEADY STATE

E = 0 HOMOGENEOUS INSOLATED SYSTEM

r Cp∙ vz ∙ dT/dz = SR = -RA ∙ DH

dT/dz = -A∙ exp (- E / RT) ∙ CA∙ DH / (cp∙ vz)

MATHEMATICAL MODEL:

dCA/dz = [A ∙ exp (-E/RT) ∙ CA ]/ vz

dT/dz = -A∙ exp (- E / RT) ∙ CA∙ DH / (cp∙ vz)


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING

SYSTEMATIC METHODOLOGY TO BUILD MATHEMATICAL MODELS

1.- PROBLEM DEFINITION

2.- IDENTIFY CONTROLLING FACTORS

3.- SUITABLE PHYSICAL MODEL

4.- CONSTRUCT THE MATHEMATICAL MODEL

5.- PRELIMINARY EVALUATION OF MODEL

6.- SOLVE THE MATHEMATICAL MODEL (NUMERICAL METHOD)

7.- DEVELOP AN ALGORITM TO SOLVE THE PROBLEM

8.- COMPUTER PROGRAMMING

9.- ADJUST MODEL PARAMETERS

10.- VALIDATE THE MODEL


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)

ROOTS OF EQUATIONS:

f (x) = a ∙ x2+ b∙ x + c = 0

METHODS

BRACKETING METHODS:

GRAPHICAL METHODS

THE BISECTION METHOD

THE FALSE-POSITION METHOD

OPEN METHODS:

SINGLE FIXED POINT ITERATION

THE NEWTON-PAPHSON METHOD

THE SECANT METHOD


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)

LINEAR ALGEBRAIC EQUATIONS:

TO DETERMINE THE VALUES OF x1, x2, x3, …… THAT SIMULTANEOUSLY SATISFY A SET OF EQUATIONS:

f1 (x1, x2, ……, xn) = 0

f2(x1, x2, …..., xn) = 0

.

.

.

.

.

.

.

.

fn(x1, x2, …..., xn) = 0

METHODS:

GAUSS ELIMINATION

LU DECOMPOSITION AND MATRIX INVERSION

SPECIAL MATRICES AND GAUSS-SEIDEL


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)

DIFFERENTIATION

THE DERIVATIVEREPRESENT THE RATE OF CHANGE OF A DEPENDENT VARIABLE WITH RESPECT TO AN INDEPENDENT VARIABLE.

dy/dx = f(x, y)

METHODS TO SOLVE ORDINARY DIFFERENTIAL EQUATIONS :

WHEN THE FUNCTION INVOLVES ONE INDEPENDENT VARIABLE, THE EQUATION IS CALLED AS ORDINARY DIFFERENTIAL EQUATION.

METHODS OF SOLUTION

RUNGE-KUTTA METHODS

(EULER’S METHOD, RUNGE-KUTTA)

(STIFFNESS AND MULTYISTEP METHOD)

STIFFNESS AND MULTISPET METHODS


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)

EULER’S METHOD

SOLVING ORDINARY DIFFERENTIAL EQUATIONS:

dy/dx = f (x, y)

THE SOLUTION OF THIS KIND OF EQUATIONS IS GENERALLY CARRIED OUT USING THE GENERAL FORM:

NEW VALUE = OLD VALUE + SLOPE x STEP SIZE

OR IN MATHEMATICAL TERMS,

yi+1 = yi + f∙ h

ACCORDING TO THIS EQUATION, THE SLOPE ESTIMATE OFfIS USED TO EXTRAPOLATE FROM AN OLD VALUE yiTO A NEW VALUE OVER A DISTANCE h. THIS FORMULA IS APPLIED STEP BY STEP TO COMPUTE OUT INTO A FUTURE AND, HENCE OUT THE TRAJECTORY OF THE SOLUTION.

IN THE EULER METHOD THE FIRST DERIVATIVE PROVIDES A DIRECT ESTIMATE OF THE SLOPE AT xi.

yi+1 = yi + f (xi, yi) ∙ h


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)

EXAMPLE OF EULER’S METHOD

USE THE EULER’S METHOD TO NUMERICALLY INTEGRATE THE FOLLOWING EQUATION:

dy/dx = -2 ∙ x3 + 12 ∙ x2 – 20 ∙ x + 8.5

f (xi, yi)

MATHEMATICAL SOLUTION

∫dy =∫(-2 ∙ x3 + 12 ∙ x2 – 20 ∙ x + 8.5) dx

yi+1 = yi + f (xi, yi) ∙ h

NUMERICAL SOLUTION

COMPARISON OF TRUE VALUE AND APPROXIMATE VALUES OF THE INTEGRAL WITH THE INITIAL VALUES y= 1 AT x = 0 (h = 0.5)


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)

EFFECT OF REDUCED STEP SIZE ON EULER’S METHOD:


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)

PARTIAL DIFFERENTIAL EQUATION: INVOLVES TWO OR MORE INDEPENDENT VARIABLES.

d(cp r T)/ dt=K {(d2T / dr2)+((b-1)/r ) dT/dr)}+(-q)(-dr/dt)

METHODS TO SOLVE PARTIAL DIFFERENTIAL EQUATIONS :

LINEAR SECOND-ORDER EQUATIONS

A (d2u/dx2)+ B (d2u/dx dy) + C (d2u/dy2) + D = 0

NUMERICAL SOLUTION

FINITE DIFFERENCE: ELLIPTIC EQUATIONS:

(d2T/dx2)+ (d2T/dy2) = 0

B2-4AC < 0

THE CONTROL-VOLUME APPROACH

(dT/dt)= k (d2T/dx2)

FINITE DIFFERENCE: PARABOLIC EQUATIONS: B2-4AC = 0

THE SIMPLE IMPLICIT METHOD

THE CRACK-NICOLSON METHOD


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING

SYSTEMATIC METHODOLOGY TO BUILD MATHEMATICAL MODELS

1.- PROBLEM DEFINITION

2.- IDENTIFY CONTROLLING FACTORS

3.- SUITABLE PHYSICAL MODEL

4.- CONSTRUCT THE MATHEMATICAL MODEL

5.- PRELIMINARY EVALUATION OF MODEL

6.- SOLVE THE MATHEMATICAL MODEL (NUMERICAL METHOD)

7.- DEVELOP AN ALGORITM TO SOLVE THE PROBLEM

8.- COMPUTER PROGRAMMING

9.- ADJUST MODEL PARAMETERS

10.- VALIDATE THE MODEL


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)

DEVELOP AN ALGORITHM TO SOLVE THE PROBLEM

WRITING ALGORITHMS USUALLY RESULTS IN SOFTWARES THAT ARE MUCH EASIER TO SHARE, IT ALSO HELPS GENERATE MUCH MORE EFFICIENT PROGRAMS. WELL-STRUCTURED ALGORITHMS ARE INVARIABLY EASIER TO DEBUG AND TEST, RESULTING IN PROGRAMS THAT TAKE A SHORTER TIME TO DEVELOP, TEST AND UPDATE.

A KEY IDEAS BEHIND STRUCTURED PROGRAMMING IS THAT ANY NUMERICAL ALGORITHM CAN BE COMPOSED USING THE THREE FUNDAMENTAL CONTROL STRUCTURES: SEQUENCE, SELECTION, AND REPETITION. BY LIMITING OURSELVES TO THESE STRUCTURES, THE RESULTING COMPUTER CODE WILL BE CLEARER AND EASIER TO FOLLOW.

A FLOWCHART IS A VISUAL OR GRAPHICAL REPRESENTATION OF AN ALGORITHM. THE FLOWCHART EMPLOYS A SERIES OF BLOCKS AND ARROWS, EACH OF WHICH REPRESENTS A PARTICULAR OPERATION OR STEP IN THE ALGORITHM. THE ARROW SHOW THE SEQUENCE IN WHICH OPERATIONS ARE IMPLEMENTED.

NOT EVERYONE INVOLVED WITH COMPUTER PROGRAMMING AGREES THAT FLOWCHARTING IS A PRODUCTIVE ENDEAVOR. IN FACT SOME EXPERIENCED PROGRAMMERS DO NOT ADVOCATE FLOWCHARTS. HOWEVER, I FEEL THAT WE SHOULD STUDY IT BECAUSE IT IS A VERY GOOD WAY TO EXPRESSING AND COMMUNICATING ALGORITHMS.


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)

SYMBOL

NAME

FUNCTION

TERMINAL

REPRESENTS THE BEGINNING OR END OF A PROGRAM

REPRESENTS THE FLOW OF LOGIC. THE HUMPS ON THE HORIZONTAL ARROW INDICATE THAT IT PASSES OVER AND DOES NOT CONNECT WITH THE VERTICAL FLOWLINES

FLOWLINES

REPRESENTS CALCULATIONS OR DATA MANIPULATIONS

PROCESS

REPRESENTS INPUTS OR OUTPUTS OF DATA AND INFORMATION

INPUT/OUTPUT

REPRESENTS A COMPARISON, QUESTION, OR DECISION THAT DETERMINES ALTERNATIVE PATHS TO BE FOLLOWED

DECISION

JUNCTION

REPRESENTS THE CONFLUENCES OF FLOWLINES

OFF-PAGE CONNECTOR

REPRESENTS A BREAK THAT IS CONTINUED ON ANOTHER PAGE

USED FOR LOOPS WHICH REPEAT A PRESPECIFIED NUMBER OF ITERATIONS

COUNT-CONTROLLED LOOP


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)

PSEUDOCODE FOR A “DUMB” VERSION OF EULER’S METHOD

‘SET INTEGRATION RANGE’

xi = 0

xf = 4

‘INITIALIZE VARIABLES’

x = xi

y = 1

‘SET STEP SIZE AND DETERMINE NUMBER OF CALCULATION STEPS’

dx =0.5

nc = (xf - xi)/dx

‘OUTPUT INITIAL CONDITION’

PRINT x, y

‘LOOP TO IMPLEMENT EULER’S METHOD AND SISPLAY RESULTS’

DO i = 1, nc

dydx = - 2∙ x3 + 12∙ x2 – 20 ∙ x + 8.5

y = y + dydx ∙ dx

x = x + dx

PRINT x, y

END DO

END


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)

START

xo, xf, yo, dx

nc = (xf - xi)/dx

i = 0 ….. nc

(dydx)i = - 2∙ xi3 + 12∙ xi2 – 20 ∙ xi + 8.5

yi+1 = yi + (dydx)i∙ dx

xi+1 = xi + dx

Xi+1, yi+1

Yes

No

i> nc

END


Lecturer manuel garcia perez ph d

2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)

PROGRAMME LANGUAGE:

FORTRAN, BASIC / VISUAL BASIC, PASCAL / OBJECT PASCAL, C / C ++.

COMMERCIAL PACKAGE:

MS EXCEL, MATLAB, MATHCAD

PROCESS SIMULATION PROGRAMS :

ASPEN, HYSYS, FLUENT

9.- ADJUST MODEL PARAMETERS

10.- VALIDATE THE MODEL

COMPARE THE RESULTS OBTAINED WITH THE MODEL WITH EXPERIMENTAL RESULTS

SIMULATION AND PROCESS ANALYSIS


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