1. RESEARCH AND TEACHING METHODS
CLASS PROJECT
LECTURER: MANUEL GARCIA-PEREZ , Ph.D.
Department of Biological Systems Engineering
205 L.J. Smith Hall, Phone number:

2. SOLVING THE MODEL AND NUMERICAL METHODS
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.

3. 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.

4. The specific tasks are outlined below:
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?

5. 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)?

6. 7.- Suggest a strategy to validate your model.
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.

7. 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.

8. Mathematical Solution
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 1
Mathematical Solution
Interpretation 2 3 4

9. 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

10. 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.

11. 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

12. Incorrectly identified process characteristics
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.

13. 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)

14. (ONE PHYSICAL MODEL PER PHASE)
2.- PROCESS MODELLING (PHYSICAL MODEL)
EXAMPLE (FLUIDIZED BED REACTORS)
PLUG FLOW MODEL
MACROSCOPIC BALANCES
???
FREEBOARD
SPLASH ZONE
BUBBLE PHASE
SOLID PHASE
EMULSION PHASE
EXCHANGE OF HEAT AND MASS
BUBBLING ZONE
EXCHANGE OF HEAT AND MASS
BIOMASS
JET ZONE
???
CARRIER GAS (EMULSION PHASE)
CARRIER GAS (BUBBLE)
CARRIER GAS
BIOMASS
SCHEME OF A FLUIDIZED BED REACTOR
(ONE PHYSICAL MODEL PER PHASE)

15. SELF SEGREGATION MODEL (PLUG FLOW)
2.- PROCESS MODELLING (PHYSICAL MODEL)
PHYSICAL MODELS FOR THE SOLID PHASE
SELF SEGREGATION MODEL (PLUG FLOW)
PLUG FLOW
MACROSCOPIC BALANCES
VOLATILES
FINES
COARSE
Bubble
EMULSION PHASE
BIOMASS
BIOMASS

16. 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
TAR k3
ANHYDROCELLULOSE
0.65 GAS CHAR B C D A E

17. 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

18. CONSTRUCTION OF MATHEMATICAL MODEL
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
Net Rate of mass exchange of specie through the interface.
Rate of mass generation of specie i by reaction 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
Energy accumulation
Energy associated to each inlet and outlet
Heat
Work
MOST COMMON ENERGY BALANCE FOR REACTING SYSTEMS
V ∙ r ∙ cp dT / dt = ∑ Fj ∙ cpj ∙(Tj - T) + ri,av ∙ V ∙(-DHR) + Q + W
Energy accumulation
Energy associated to each inlet and outlet
Heat
Work
rA: Production of compound by chemical reaction (kmol/m3.s) (-) if produced, (-) if consumed
Q: (+) if generated (-) if consumed

19. 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

20. CONSTRUCTION OF MATHEMATICAL MODEL PLUG FLOW
2.- PROCESS MODELLING (MATHEMATICAL MODEL)
CONSTRUCTION OF MATHEMATICAL MODEL
PLUG FLOW
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
Rate of mass accumulation of specie i
Transport for convection
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
Units:
Rate of Energy accumulation
Energy transport by convection
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

21. MICROSCOPIC BALANCES CONSTRUCTION OF MATHEMATICAL MODEL
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 + vx dCA/dx + vy dCA/dy + vz dCA/dz=DAB [d2CA/dx2 + d2CA/dy2+d2CA/dz2] + RA
Accumulation
Transport per diffusion
Transport per convection
Generation
ENERGY BALANCE:
r ∙ Cp [ dT/ dt + vx dT/dx + vy dT/dy + vz dT/dz=k [d2T/dx2 + d2T/dy2+d2T/dz2] + SR
Transport per thermal diffusion
Transport per convection
Accumulation
Generation

22. NAVIER-STOKES EQUATIONS
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 + vy d vx/dy + vz d vx/dz = dp/dx + m [d2 vx /dx2 + d2 vx /dy2+d2 vx /dz2] +r gx
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 Y
r ∙ [dvy/ dt + vx d vy/dx + vy d vy/dy + vz d 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 + vy d vz/dy + vz d vz/dz = dp/dz + m [d2 vz /dx2 + d2 vz /dy2+d2 vz /dz2] +r gz

23. MASS TRANSFER COEFFICIENT HEAT TRANSFER COEFFICIENT
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
PROPERTY RELATIONS
Raoult’s law model : yi= xj Pj vap/P
Relative volatility model : yi= aij xi /(1+ (aij -1) xi
K model : Kj = yj / xj
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)
EQUATIONS OF STATE
Ideal gas, Redleich-Kwong, Peng-Robinson and Soave-Redleich-Kwong equations.

24. 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

25. Simplifications dTz/dy = 0 2.- PROCESS MODELLING (MATHEMATICAL MODEL)
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

26. EXAMPLE 1: 2.- PROCESS MODELLING (MATHEMATICAL MODEL)
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 Y a X Z h vz
HEAT

27. Transport per thermal diffusion
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 + vx dT/dx + vy dT/dy + vz dT/dz=k [d2T/dx2 + d2T/dy2+d2T/dz2] + SR
Transport per thermal diffusion
Transport per convection
Accumulation
Generation

28. 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 vz dT/dz >> k [d2T/dz2]

29. 2.- PROCESS MODELLING (MATHEMATICAL MODEL)
HEAT Y a X Z h vz
HEAT
HEAT ~0 ~0
r ∙ Cp [ dT/ dt + vx dT/dx + vy dT/dy + vz dT/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 [vz dT/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.

30. EXAMPLE 2: 2.- PROCESS MODELLING (MATHEMATICAL MODEL)
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.

31. r Cp ∙ (dT/dt + vz ∙ dT/dz) = SR + Et
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 Et CAN BE CALCULATED FOR TUBES USING THE FOLLOWING EQUATION:
Et = (4/D) U (TV-T)
r Cp ∙ vz ∙ dT /dz = (4/D) U (Tv -T)

32. 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:
rA= K ∙ CA
K = A ∙ exp ∙ (-E/RT)
A = 3.00 s-1
E = 4652 kJ/kmol
Consider the axial diffusion negligible.

33. EXAMPLE 3: dCA / d t + d (vz ∙ CA)/dz = RA + mA vz ∙ dCA/dz = RA
2.- PROCESS MODELLING (MATHEMATICAL MODEL)
EXAMPLE 3:
DATA:
VELOCITY OF FLUID: 3 m/s
SPECIFIC HEAT: J/kg K
ENTALPY OF REACTION: kJ/kg
PHYSICAL MODEL:
PLUG FLOW
EQUATIONS: MASS AND ENERGY BALANCES
MATHEMATICAL MODEL:
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

34. EXAMPLE 3: dCA/dz = [A ∙ exp (-E/RT) ∙ CA ]/ vz
2.- PROCESS MODELLING (MATHEMATICAL MODEL)
EXAMPLE 3:
dCA/dz = [A ∙ exp (-E/RT) ∙ CA ]/ vz
???
ENERGY BALANCE:
r Cp ∙ (dT/dt + vz ∙ dT/dz) = SR Et
dT/dt = 0 STEADY STATE
E = 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)

35. 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

36. 2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)
ROOTS OF EQUATIONS:
f (x) = a ∙ x2+ b∙ x + c = 0
METHODS
BRACKETING METHODS:
OPEN METHODS:
GRAPHICAL METHODS
THE BISECTION METHOD
THE FALSE-POSITION METHOD
THE NEWTON-PAPHSON METHOD
THE SECANT METHOD
SINGLE FIXED POINT ITERATION

37. 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

38. 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 MULTISPET METHODS
(STIFFNESS AND MULTYISTEP METHOD)

39. 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 OF f 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

40. 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 ∙ x ∙ x2 – 20 ∙ x + 8.5
f (xi, yi)
MATHEMATICAL SOLUTION
∫ dy = ∫ (-2 ∙ x ∙ x2 – 20 ∙ x + 8.5) dx
NUMERICAL SOLUTION
yi+1 = yi + f (xi, yi) ∙ h
COMPARISON OF TRUE VALUE AND APPROXIMATE VALUES OF THE INTEGRAL WITH THE INITIAL VALUES y= 1 AT x = 0 (h = 0.5)

41. 2.- PROCESS MODELLING (SOLVING THE MODEL, NUMERICAL METHOD)
EFFECT OF REDUCED STEP SIZE ON EULER’S METHOD:

42. 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:
B2-4AC < 0
(d2T/dx2)+ (d2T/dy2) = 0
THE CONTROL-VOLUME APPROACH
FINITE DIFFERENCE: PARABOLIC EQUATIONS: B2-4AC = 0
(dT/dt)= k (d2T/dx2)
THE SIMPLE IMPLICIT METHOD
THE CRACK-NICOLSON METHOD

43. 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

44. DEVELOP AN ALGORITHM TO SOLVE THE PROBLEM
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.

45. 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
PROCESS
REPRESENTS CALCULATIONS OR DATA MANIPULATIONS
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

46. 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

47. 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

48. SIMULATION AND PROCESS ANALYSIS
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