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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.
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 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
2.- PROCESS MODELLING
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)
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 (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.
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.
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)
2.- PROCESS MODELLING (PHYSICAL MODEL)
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)
2.- PROCESS MODELLING (PHYSICAL MODEL)
PHYSICAL MODELS FOR THE SOLID PHASE
SELF SEGREGATION MODEL (PLUG FLOW)
BIOMASS BIOMASS
PLUG FLOW
VOLATILES
FINES COARSE
MACROSCOPIC BALANCES
Bubble
EMULSION PHASE
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.
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)
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 (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
2.- PROCESS MODELLING (MATHEMATICAL MODEL)
<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
2.- PROCESS MODELLING (MATHEMATICAL MODEL)
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
2.- PROCESS MODELLING (MATHEMATICAL MODEL)
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
2.- PROCESS MODELLING (MATHEMATICAL MODEL)
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
2.- PROCESS MODELLING (MATHEMATICAL MODEL)
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
2.- PROCESS MODELLING (MATHEMATICAL MODEL)
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:
∙ Cp [ T/ t + vx T/x + vy T/y + vz T/z=k [2T/x2 + 2T/y2+2T/z2] + SR
Accumulation Transport per convection
Transport per thermal
diffusion Generation
2.- PROCESS MODELLING (MATHEMATICAL MODEL)
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
2.- PROCESS MODELLING (MATHEMATICAL MODEL)
∙ Cp [ T/ t + vx T/x + vy T/y + vz T/z=k [2T/x2 + 2T/y2+2T/z2] + SR
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.
2.- PROCESS MODELLING (MATHEMATICAL MODEL)
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
2.- PROCESS MODELLING (MATHEMATICAL MODEL)
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
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 (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
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) = 0f2 (x1, x2, …..., xn) = 0
.
.
.
.
.
.
.
.
fn (x1, x2, …..., xn) = 0
METHODS:
GAUSS ELIMINATION LU DECOMPOSITION AND MATRIX INVERSION SPECIAL MATRICES AND GAUSS-SEIDEL
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.
RUNGE-KUTTA METHODSSTIFFNESS AND MULTISPET METHODS
(EULER’S METHOD, RUNGE-KUTTA)(STIFFNESS AND MULTYISTEP METHOD)
METHODS OF SOLUTION
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 + ∙ 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
2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)
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)
2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)
EFFECT OF REDUCED STEP SIZE ON EULER’S METHOD:
PARTIAL DIFFERENTIAL EQUATION: INVOLVES TWO OR MORE INDEPENDENT VARIABLES.
2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)
(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
2.- PROCESS MODELLING
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
DEVELOP AN ALGORITHM TO SOLVE THE PROBLEM
2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)
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.
2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)
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
2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)
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
2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)
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
9.- ADJUST MODEL PARAMETERS
10.- VALIDATE THE MODEL
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
2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)