LECTURER: MANUEL GARCIA-PEREZ, Ph.D. Department of Biological Systems Engineering 205 L.J. Smith...

LECTURER: MANUEL GARCIA-PEREZ , Ph.D.

Department of Biological Systems Engineering205 L.J. Smith Hall, Phone number: 509-335-7758

e-mail: [email protected]

RESEARCH AND TEACHING METHODS

CLASS PROJECT

OUTLINE

1.- CLASS PROJECT2.- PROCESS MODELLING

MATHEMATICAL MODEL PHYSICAL MODEL

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.

SOLVING THE MODEL AND NUMERICAL METHODS

1.- CLASS PROJECT

Goal and Objectives:

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

(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.

1.- CLASS PROJECT

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?

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:

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).

1.- CLASS PROJECT

Tasks

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)?

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.

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.


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 TO GAIN INSIGHT INTO THE ORIGINAL REAL WORLD SITUATION OR TO USE THE MODEL FOR CONTROL, OPTIMIZATION OR POSSIBLE SAFETY STUDIES.

Real world Problem

Mathematical problem

1Mathematical

SolutionInterpretation

2 3

4


SYSTEMATIC METHODOLOGY TO BUILD MATHEMATICAL MODELS

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

1.- PROBLEM DEFINITION

2.- IDENTIFY CONTROLLING FACTORS

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:

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.- INPUTS AND OUTPUTS B.- HIERARCHY LEVEL RELEVANT TO THE MODEL C.- THE NECESSARY RANGE AND ACCURACY OF THE MODELD.- THE TIME CHARACTERISTICS (STATIC VERSUS DYNAMIC) OF THE PROCESS MODEL.

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.

2.- PROCESS MODELLING (PHYSICAL MODEL)













3.- CREATE A SUITABLE PHYSICAL MODEL

REALITY

PHYSICAL MODEL

Identified non-essential process characteristics

Identified essential process characteristics

Incorrectly identified process

characteristics

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

PHYSICAL MODEL GRAPHIC REPRESENTATION 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)


BIOMASS

JET ZONE

BUBBLING ZONE

SPLASH ZONE

FREEBOARD

CARRIER GAS

SCHEME OF A FLUIDIZED BED REACTOR

(ONE PHYSICAL MODEL PER PHASE)

BUBBLE PHASE

SOLID PHASE

EMULSION PHASE

CARRIER GAS

(BUBBLE)

CARRIER GAS (EMULSION

PHASE)BIOMASS

PLUG FLOW

MODEL

MACROSCOPIC BALANCES

???

???

EXCHANGE OF HEAT AND MASS

EXCHANGE OF HEAT AND MASS

EXAMPLE (FLUIDIZED BED REACTORS)


PHYSICAL MODELS FOR THE SOLID PHASE

SELF SEGREGATION MODEL (PLUG FLOW)

BIOMASS BIOMASS

PLUG FLOW

VOLATILES

FINES COARSE

MACROSCOPIC BALANCES

Bubble

EMULSION PHASE


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.

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

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

k1

k2

CELLULOSE TA

R

k3ANHYDROCELLULOSE 0.65 GAS + 0.35 CHAR

AB C D

E

2.- PROCESS MODELLING (MATHEMATICAL MODEL)













MACROSCOPIC BALANCESCONSTRUCTION OF MATHEMATICAL MODEL MASS BALANCES SPECIE i:

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

Rate of mass generation of

specie i by reaction

Net Rate of mass exchange of specie

through the interface.

You should write a mass balance per

every component per every

phase ENERGY BALANCE

d Etot/dt = - (i ∙ v ∙ S) [h+ ½ ∙ v2 + + Q - W

Rate of mass accumulation of

specie i

Energy accumulation

1

2

QW

You should write an Energy

balance per phase

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

MOST COMMON ENERGY BALANCE FOR REACTING SYSTEMS

Energy accumulation

Energy associated to each inlet and outlet

Energy associated to each inlet and outlet

Heat

Heat

Work

Work

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


<v> average velocity (m/s)

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

: density of fluid (kg/m3)

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

MEANING OF SOME TERMS:

<V>

S

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

.

: Potential Energy

K: Kinetic Energy

U: Internal Energy


PLUG FLOW

dzTransport for convection

CONSTRUCTION OF MATHEMATICAL MODEL MASS BALANCES SPECIE i:

ENERGY BALANCE

Mass balance per unit of

volume

Ci / t + (vz ∙ Ci)/z = Ri + mi

Cp∙ T/t + vz ∙ T/z) = SR + Et

Rate of mass accumulation of

specie i

Rate of Energy accumulation

Energy transport by convection

Rate of mass generation of

specie i by reaction

Heat associated with chemical

reactions

Net Rate of mass exchange of specie through the interface.

Heat or work transport through the interface

You should write a mass balance per

every component per every

phase

You should write an Energy

balance per every phase

mi

E

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

Units:

Property/vol. time

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

mi=kc ∙a ∙C = Ky∙a∙ y

Et= (4/D) ∙ U ∙T


CONSTRUCTION OF MATHEMATICAL MODEL

CA/ t + vx CA/x + vy CA/y + vz CA/z=DAB [2CA/x2 + 2CA/y2+2CA/z2] + RA

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

MASS BALANCES SPECIE i:

ENERGY BALANCE:

∙ Cp [ T/ t + vx T/x + vy T/y + vz T/z=k [2T/x2 + 2T/y2+2T/z2] + SR

Accumulation

Accumulation

Transport per convection

Transport per convection

Transport per diffusion

Transport per thermal

diffusion

Generation

Generation

MICROSCOPIC BALANCES


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

∙ vx/ t + vx vx/x + vy vx/y + vz vx/z = p/x + [2 vx /x2 + 2 vx /y2+2 vx /z2] + gx

DIRECTION X

∙ vy/ t + vx vy/x + vy vy/y + vz vy/z = p/y + [2 vy /x2 + 2 vy /y2+2 vy /z2] + gy

∙ vz/ t + vx vz/x + vy vz/y + vz vz/z = p/z + [2 vz /x2 + 2 vz /y2+2 vz /z2] + gz

DIRECTION Y

DIRECTION Z

NAVIER-STOKES EQUATIONS

Rate of increase of momentum

per unit volume

Rate of increase of momentum

per unit volume

Rate of momentum addition by convection

per unit volume

Rate of momentum addition by convection

per unit volume

Rate of momentum addition by molecular

transport per unit volume

Rate of momentum addition by molecular

transport per unit volume

External Force

External Force

2.- SINGLE PARTICLE MODELS (MATHEMATICAL MODEL)

CONSTITUTIVE RELATIONS:

MASS TRANSFER: mi=K (CEi-CB

i)BUBBLE

EMULSION

MASS TRANSFER

HEAT TRANSFER: E=U ∙ a ∙ (TE-TB)

MASS TRANSFER COEFFICIENT

HEAT TRANSFER COEFFICIENT

HEAT TRANSFER

TRANSFER RELATIONSHIP:

Ri = - ko e–E/(RT) CjnREACTION KINETICS:

THERMODYNAMICAL RELATIONS

Liquid density: L = f (P, T, xi) Vapour density: V = f (P, T, xi)Liquid enthalpy: h = f (P, T, xi)Vapour enthalpy: H = f (P, T, yi)

PROPERTY RELATIONS Raoult’s law model : yi= xj Pj vap/PRelative volatility model : yi= aij xi /(1+ (aij -1) xi

K model : Kj = yj / xj

EQUILIBRIUM RELATIONSHIPS

EQUATIONS OF STATEIdeal gas, Redleich-Kwong, Peng-Robinson and Soave-Redleich-Kwong equations.

CEi

CBi

TB

TE


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 EQUATION SHOULD BE USED?

WHAT SYSTEM OF COORDENATES SHOULD BE USED?

IMPORTANT WHEN USING MICROSCOPIC MODELS

CARTESIAN COORDINATE SYSTEM CYLINDRICAL COORDINATE SYSTEM

2.- PROCESS MODELLING (MATHEMATICAL MODEL)SimplificationsIn steady state the properties do not change with time (d/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.

x

yz

Source that produces the changes

Source that produces the changesTz/y = 0

Q

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

Z

XY

HEAT

h

a

vz


ENERGY BALANCETEMPERATURE PROFILE

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:

COORDENATE SYSTEM: RECTANGULAR (CARTESIAN)

PHYSICAL MODEL:

GENERAL MATHEMATICAL MODEL:


Accumulation Transport per convection

Transport per thermal

diffusion Generation


SIMPLIFICATIONS:

1.- STEADY STATE: (T/t) = 0

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

∙ Cp vz T/z >> k [2T/z2]

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: 2T/x2 = 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:

Z

XY

h

a

vz



0 0 ~0 ~00 0

∙ Cp ∙ vz ∙ T/z=k [2T/y2]

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.

∙ Cp [vz T/z=k [2T/x2 + 2T/y2]

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

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

HEATHEAT

HEAT

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.

CONDENSATE

SATURATED VAPOUR

GASESGASES

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.


TURBULENT REGIME, A SINGLE PHASE

PHYSICAL MODEL: PLUG FLOW

PHYSICAL MODEL:

MATHEMATICAL MODEL:

PROPERTY OF INTEREST: TEMPERATUREEQUATION: ENERGY BALANCES Cp∙ T/t + vz ∙ T/z) = SR

+ EtSIMPLIFICATIONS:

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

NO CHEMICAL REACTION: SR = 0T/t = 0

Cp∙ vz ∙ T/z = EtTHE VALUES OF Et CAN BE CALCULATED FOR TUBES USING THE FOLLOWING EQUATION:

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

Cp∙ vz ∙ dT /dz = (4/D) U (Tv -T)

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).

A B

Irreversible reaction

rA= K ∙ CA

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

Consider the axial diffusion negligible.

A A+ B

K = A ∙ exp ∙ (-E/RT)

A = 3.00 s-1 E = 4652 kJ/kmol

15 m

Solvent Solvent

INSOLATED SYSTEM


EXAMPLE 3:DATA:

VELOCITY OF FLUID: 3 m/s

ENTALPY OF REACTION: -279.12 kJ/kg

SPECIFIC HEAT: 4.184 J/kg K

PLUG FLOW

EQUATIONS: MASS AND ENERGY BALANCES

PHYSICAL MODEL:

MATHEMATICAL MODEL:

CA / t + (vz ∙ CA)/z = RA + mAMASS BALANCE:

CA / t = 0 (STEADY STATE)

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

0 0

vz ∙ dCA/dz = RA

vz = CONSTANT (INCONPRESSIBLE FLUID)

RA= A ∙ exp (-E/RT) ∙ CA

2.- PROCESS MODELLING (MATHEMATICAL MODEL)EXAMPLE 3: dCA/dz = [A ∙ exp (-E/RT) ∙ CA ]/ vz

ENERGY BALANCE:

Cp∙ T/t + vz ∙ T/z) = SR + Et

T/t = 0 STEADY STATE

E = 0 HOMOGENEOUS INSOLATED SYSTEM

0 0

Cp∙ vz ∙ T/z = SR = -RA

∙ HT/z = -A ∙ exp (- E / RT) ∙ CA ∙ H / (cp ∙ vz)

dCA/dz = [A ∙ exp (-E/RT) ∙ CA ]/ vz T/z = -A ∙ exp (- E / RT) ∙ CA ∙ H / (cp ∙ vz)

MATHEMATICAL MODEL:

???

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

ROOTS OF EQUATIONS:

BRACKETING METHODS:

OPEN METHODS:

GRAPHICAL METHODSTHE BISECTION METHODTHE FALSE-POSITION METHOD

THE NEWTON-PAPHSON METHOD THE SECANT METHOD

SINGLE FIXED POINT ITERATION

f (x) = a ∙ x2+ b∙ x + c = 0

METHODS


LINEAR ALGEBRAIC EQUATIONS:

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

f1 (x1, x2, ……, xn) = 0f2 (x1, x2, …..., xn) = 0

.

.

.

.

.

.

.

.

fn (x1, x2, …..., xn) = 0

METHODS:

GAUSS ELIMINATION LU DECOMPOSITION AND MATRIX INVERSION SPECIAL MATRICES AND GAUSS-SEIDEL


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.

RUNGE-KUTTA METHODSSTIFFNESS AND MULTISPET METHODS

(EULER’S METHOD, RUNGE-KUTTA)(STIFFNESS AND MULTYISTEP METHOD)

METHODS OF SOLUTION


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 + ∙ h

ACCORDING TO THIS EQUATION, THE SLOPE ESTIMATE OF IS USED TO EXTRAPOLATE FROM AN OLD VALUE yi TO 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


EXAMPLE OF EULER’S METHODUSE THE EULER’S METHOD TO NUMERICALLY INTEGRATE THE FOLLOWING EQUATION:

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

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

MATHEMATICAL SOLUTION

NUMERICAL SOLUTION yi+1 = yi + f (xi, yi) ∙ hCOMPARISON OF TRUE VALUE AND APPROXIMATE VALUES OF THE INTEGRAL WITH

THE INITIAL VALUES y= 1 AT x = 0 (h = 0.5)

f (xi, yi)


EFFECT OF REDUCED STEP SIZE ON EULER’S METHOD:

PARTIAL DIFFERENTIAL EQUATION: INVOLVES TWO OR MORE INDEPENDENT VARIABLES.


(cp T)/ t=K {(2T / r2)+((b-1)/r ) T/r)}+(-q)(-/t)

METHODS TO SOLVE PARTIAL DIFFERENTIAL EQUATIONS :

FINITE DIFFERENCE: ELLIPTIC EQUATIONS:

THE CONTROL-VOLUME APPROACH

THE SIMPLE IMPLICIT METHODTHE CRACK-NICOLSON METHOD

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

A (2u/x2)+ B (2u/x y) + C (2u/y2) + D = 0

B2-4AC < 0 (2T/x2)+ (2T/y2) = 0

(T/t)= k (2T/x2)

NUMERICAL SOLUTION

LINEAR SECOND-ORDER EQUATIONS

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.


SYMBOL NAME FUNCTION

TERMINAL REPRESENTS THE BEGINNING OR END OF A PROGRAM

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

PROCESS REPRESENTS CALCULATIONS OR DATA MANIPULATIONS

INPUT/OUTPUTREPRESENTS INPUTS OR OUTPUTS OF DATA AND INFORMATION

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

JUNCTION REPRESENTS THE CONFLUENCES OF FLOWLINES

OFF-PAGE CONNECTOR

COUNT-CONTROLLED LOOP

REPRESENTS A BREAK THAT IS CONTINUED ON ANOTHER PAGE

USED FOR LOOPS WHICH REPEAT A PRESPECIFIED NUMBER OF ITERATIONS


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

xf = 4

‘SET INTEGRATION RANGE’xi = 0

x = xi

‘INITIALIZE VARIABLES’

y = 1

dx =0.5nc = (xf - xi)/dx

‘SET STEP SIZE AND DETERMINE NUMBER OF CALCULATION STEPS’

‘OUTPUT INITIAL CONDITION’

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

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

PRINT x, y

PRINT x, y

DO i = 1, nc

END DOEND


xo, xf, yo, dx

nc = (xf - xi)/dx

i = 0 ….. nc

(dydx)i = - 2∙ xi3 + 12∙ xi

2 – 20 ∙ xi + 8.5 yi+1 = yi + (dydx)i ∙ dx

xi+1 = xi + dx

Xi+1, yi+1

i> nc NoYes

START

END



MS EXCEL, MATLAB, MATHCADPROCESS SIMULATION PROGRAMS :

FORTRAN, BASIC / VISUAL BASIC, PASCAL / OBJECT PASCAL, C / C ++. PROGRAMME LANGUAGE:

COMMERCIAL PACKAGE:

ASPEN, HYSYS, FLUENT

COMPARE THE RESULTS OBTAINED WITH THE MODEL WITH EXPERIMENTAL RESULTS

SIMULATION AND PROCESS ANALYSIS


LECTURER: MANUEL GARCIA-PEREZ, Ph.D. Department of Biological Systems Engineering 205 L.J. Smith...

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Transcript of LECTURER: MANUEL GARCIA-PEREZ, Ph.D. Department of Biological Systems Engineering 205 L.J. Smith...