Process Modeling methods and tools Lecture 3

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Process Modeling methods and tools Lecture 3
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Lecutre 3 outline

• Traditional dimensional analysis
• Interpretation of dimensionless groups
• Dimensionless groups from physical models
• Model classification (mathematical and physical)
• Model dimensions
• Simulation, design and optimization

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Dimensional analysis

• Buckingham Pi theorem
• Theorem describes how every physically meaningful
  equation involving n variables can be
  equivalently rewritten as an equation of n - m
  dimensionless parameters, where m is the number
  of fundamental dimensions used
• Collect all variables that affect the system and
  analyze their dimensions

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Dimensional analysis

• Example Power consumption in a stirred tank
• variables
• Power
• viscosity
• density
• diameter
• impeller rotational speed

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Dimensional analysis

• Basic dimensions length (L), time (T) and mass
  (M)
• Power W J/s kgm2/s2/skgm2/s3ML-2T-3
• Viscosity Pas kgm/s2/m2s ML-1T-1
• Density kg/m3 ML-3
• Diameter m L
• Impeller speed 1/sT-1
• This analysis is useful even if you are not going
  to formulate new dimensionless groups!

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Dimensional analysis

• Five variables (power, viscosity, density,
  diameter, and impeller rotational speed), and
  three basic dimensions (length, time and mass).
  Therefore there are two independent dimensionless
  groups
• Impeller Reynolds number
• Power number

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Some dimensionless groupsfrom Wikipedia
Abbe number optics (dispersion in optical
materials) Albedo climatology, astronomy
(reflectivity of surfaces or bodies) Archimedes
number motion of fluids due to density
differences Bagnold number flow of grain, sand,
etc. Biot number surface vs. volume
conductivity of solids Bodenstein number
residence-time distribution Bond number
capillary action driven by buoyancy Brinkman
number heat transfer by conduction from the
wall to a viscous fluid Brownell Katz number
combination of capillary number and Bond
number Capillary number fluid flow influenced
by surface tension Coefficient of static friction
friction of solid bodies at rest Coefficient of
kinetic friction friction of solid bodies in
translational motion Colburn j factor
dimensionless heat transfer coefficient Courant-
Friedrich-Levy number numerical solutions of
hyperbolic PDEs Damköhler numbers reaction time
scales vs. transport phenomena Darcy friction
factor fluid flow Dean number vortices in
curved ducts Deborah number rheology of
viscoelastic fluids Drag coefficient flow
resistance Eckert number convective heat
transfer
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Some dimensionless groups
Ekman number geophysics (frictional (viscous)
forces) Elasticity (economics) widely used to
measure how demand or supply responds to price
changes Eötvös number determination of
bubble/drop shape Euler number hydrodynamics
(pressure forces vs. inertia forces) Fanning
friction factor fluid flow in pipes Feigenbaum
constants chaos theory (period doubling) Fine
structure constant quantum electrodynamics
(QED) Fopplvon Karman number thin-shell
buckling Fourier number heat transfer Fresnel
number slit diffraction Froude number wave
and surface behaviour Gain electronics (signal
output to signal input) Galilei number
gravity-driven viscous flow Graetz number
heat flow Grashof number free
convection Hatta number adsorption enhancement
due to chemical reaction Hagen number forced
convection Karlovitz number turbulent
combustion Knudsen number continuum
approximation in fluids Kt/V medicine
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Some dimensionless groups
Laplace number free convection within
immiscible fluids Lewis number ratio of mass
diffusivity and thermal diffusivity Lockhart-Marti
nelli parameter flow of wet gases Lift
coefficient lift available from an airfoil at a
given angle of attack Mach number gas
dynamics Magnetic Reynolds number
magnetohydrodynamics Manning roughness
coefficient open channel flow (flow driven by
gravity) Marangoni number Marangoni flow due
to thermal surface tension deviations Morton
number determination of bubble/drop
shape Nusselt number heat transfer with forced
convection Ohnesorge number atomization of
liquids, Marangoni flow Péclet number
advectiondiffusion problems Peel number
adhesion of microstructures with substrate Pi
mathematics (ratio of a circle's circumference
to its diameter) Poisson's ratio elasticity
(load in transverse and longitudinal
direction) Power factor electronics (real
power to apparent power) Power number power
consumption by agitators Prandtl number forced
and free convection Pressure coefficient
pressure experienced at a point on an
airfoil Radian measurement of angles
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Some dimensionless groups
Rayleigh number buoyancy and viscous forces in
free convection Refractive index
electromagnetism, optics Reynolds number flow
behavior (inertia vs. viscosity) Richardson
number effect of buoyancy on flow
stability Rockwell scale mechanical
hardness Rossby number inertial forces in
geophysics Schmidt number fluid dynamics (mass
transfer and diffusion) Sherwood number mass
transfer with forced convection Sommerfeld number
boundary lubrication Stanton number heat
transfer in forced convection Stefan number
heat transfer during phase change Stokes number
particle dynamics Strain materials science,
elasticity Strouhal number continuous and
pulsating flow Taylor number rotating fluid
flows van 't Hoff factor quantitative analysis
(Kf and Kb) Weaver flame speed number laminar
burning velocity relative to hydrogen gas Weber
number multiphase flow with strongly curved
surfaces Weissenberg number viscoelastic
flows Womersley number continuous and pulsating
flows
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Another approach

• Dimensionless groups can be specified as ratios
• Ratio of forces
• Ratio of diameters
• etc.
• For example, Reynolds number

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• Sherwood number
• Ratio of mass transfer film thickness (according
  to the film model) to particle diameter
• Ratio of convective and diffusive mass transfer

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• ? ratio of a circle's circumference to its
  diameter
• Dimensionless ratio
• Mathematical constant (mathematics does not
  involve physical dimensions as such)

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Reynolds once more

• Smallest eddy size in turbulent fluid can be
  estimated from the Kolmogorov scale

Reynolds number describes the ratio of largest
and smallest eddy (power 3/4)
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Dimensionless numbers from physical models
Tubular reactor model with axial dispersion.
Constant coefficients
From the previous lecture
Steady state
v m/s c mol/m3 h m D m2/s r mol/m3s
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Dimensionless numbers from physical models
Dimensionless length z
nth order reaction
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Dimensionless numbers from physical models
One possible Damköhler number
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Dimensionless numbers from physical models
dimensionless concentration
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Dimensionless numbers from physical models
Relative difference of various terms can be
estimated based on dimensionless numbers
appearing during non-dimensionalizing
Note that some physical variables affect several
dimensionless numbers. For example velocity.
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Incompressible Navier-Stokes, constant
coefficients
convection of momentum with the flow
momentum flux due to viscous forces
time rate of change of linear momentum
Pressure effect
external force (gravity)
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Navier-Stokes
This can be put in a dimensionless form as
Where superscripts denote dimensionless
variables (velocity, time, length)
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Navier-Stokes
Important dimensionless parameters for
incompressible Navier-Stokes equations arise from
this process
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Model classification (mathematical)

• Several levels are possible
• Easy vs. difficult (subjective)
• Constant coefficients vs. variable coefficients
• Stiff system vs. non-stiff
• Linear vs. non-linear system (algebraic and
  differential). There is also a mathematical
  definition for almost linear
• Homogeneous vs. inhomogeneous

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Model classification(mathematical)

• Number of variables (e.g. binary and
  multicomponent systems)
• Order of differential equations (operators)
• Ordinary differential equations, partial
  differential equations, differential-algebraic
  equations, integrodifferential equations etc.
• Hyperbolic, parabolic and elliptic PDE
• Initial vs. boundary value problem

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Model classification(physical)

• Time dependent vs. steady state
• Classification based on controlling mechanism,
  e.g. diffusion or reaction controlled
• One or several dimensions (physical, i.e. spatial
  dimensions)

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• Easy vs. difficult
• If this is estimated based on required time to
  solve the model with certain computational
  capacities, then this is actually a physical
  classification
• Constant vs. variable coefficients

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• Stiff vs. non-stiff
• Formally ratio of eigenvalues
• In practice, if there are simultaneously very
  fast phenomena dictating step sizes, and very
  slow phenomena dictating simulation time, system
  is stiff.
• Linear vs. non-linear
• In principle, linear systems are easy. Natural
  systems are rarely linear, but often numerical
  solution is based on (local) linearizations

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• Homogeneous vs. inhomogeneous
• If f(?x) ?nf(x) for every ?, then f(x) is
  homogeneous to nth degree.
• Number of variables
• Two-component system one degree of freedom (e.g.
  mole fractions x and 1-x). More components, more
  degrees of freedom. Often one degree of freedom
  leads to scalar equations, more degrees to
  matrices.

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• Order of differential equation
• nth order differential equation
• Can be linear or non-linear, depending on
  parameters a and function f.
• If parameters a depend on x only (not on y), and
  function f is at most first order with respect to
  y, then the equation is linear.

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• Ordinary vs. partial differential equations
• ordinary variables are functions of only one
  independent variable, partial functions of
  several independent variables

Variable c depends on time and on position
Variable c is time invariant ? ? d
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• Differential-algebraic equations
• In addition to the differential equations there
  are algebraic constraints.

for variables x there are both differential and
algebraic equations
for variables y there are only algebraic equations
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• Differential-algebraic equations
• Actually quite common in chemical engineering.

Material balance for flowing phases. Reaction
rate depends on concentrations at catalyst.
Algebraic mass transfer etc. model relates
catalyst and fluid concentrations.
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• Integro-differential equations
• Involve both derivatives and integrals of the
  unknown variable. A reasonably general form

Distributed systems x is a density distribution
with respect to s, and this distribution depends
on t. K is sometimes called a Kernel function
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• Integro-differential equations

Rate of change of density distribution at any
location s depend on the value at that point, and
also on other parts of the distribution
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• Integro-differential equations

Population balances e.g. size distributions are
time dependent. Rate of change of the
distribution (shape) depend on the whole
distribution
x(s,t) under consideration
agglomeration of these may form a particle of
size s
breakage of these may form a particle of size s
breakage of x(s,t) affects the distribution
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Classical classificationfor 2nd order PDEs

• if ? b2 - 4ac
• lt 0 Elliptic
• 0 Parabolic
• gt 0 Hyperbolic

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Classical classificationfor 2nd order PDEs
Diffusion

• aD, b0, c0
• b2 - 4ac 02 - 4?D ?0 0
• Diffusion equation is parabolic.

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Classical classificationfor 2nd order PDEs
Newtons law for wave motion

• a1, b0, c-?
• b2 - 4ac 02 4?1?? 4?
• Wave equation is hyperbolic

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Classical classificationfor 2nd order PDEs
Convection in conservation laws
First order ? ?/?t or ?/?h

• a1, bv, c0 av, b1, c0
• ? b2 - 4ac v2 ? b2 - 4ac 1
• Convection equation is hyperbolic

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Classical classificationfor 2nd order PDEs
Laplaces equation for heat conduction

• a1, b0, c1
• ? b2 - 4ac -4
• Heat conduction equation is elliptic.

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So what?
Suitable numerical methods depend on the equation
type. Parabolic equations are diffusive and
perhaps easiest for numerical point of
view Hyperbolic equations transport information.
Then numerical methods either produce numerical
diffusion or oscillations.
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Numerical solution of adsorber breakthrough
curves (hyperbolic)
Numerical diffusion (typical for low order
methods)
Oscillations (typical for high order methods)
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• Note that most model equations cannot be cast
  clearly into one group

hyperbolic part
parabolic part
Peclet number describes how hyperbolic or
parabolic, i.e. convective or diffusive, the
system is.
Dimensionless time, t/tres
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Initial and boundary value problems

• Initial value problems usually easier start from
  the initial values and march forward in
  position or time.
• Boundary value problems are encountered usually
  in partial differential equations.

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Some boundary conditions

• 1. Variable value specified at the boundary.
• Known as Dirichlet boundary conditions, or first
  type boundary conditions
• Examples
• Catalyst particle surface concentration in case
  of no external mass transfer resistance
• Inlet concentration of a plug flow reactor
  without axial dispersion

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Some boundary conditions

• 2. Derivative specified at the boundary
• Known as Neumann boundary conditions, or second
  type boundary conditions
• Examples
• Symmetry at the particle center
• Danckwerts condition at the reactor exit

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Some boundary conditions

• 3. Linear combination of value and derivative
  specified at the boundary
• Known as Robin boundary conditions, or third type
  boundary conditions
• Examples
• Flux specified in cases of both diffusive and
  convective mass transfer

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Physical classifications

• Time dependent vs. steady state
• From mathematical point of view

Dynamic stirred tank
Plug-flow reactor
are the same
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• For steady-state flowsheet simulator

steady state tubular reactor
dynamic batch reactor
are not the same
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Classification based on controlling mechanism

• This is sometimes related to the mathematical
  classification, but controlling mechanisms may be
  the same from mathematical point of view
• Important analyis when various closures (physical
  models) are evaluated. How much modeling effort
  should be put into each physical closure?

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Physical dimensions

• Time and spatial dimensions are the same from
  mathematical point of view.
• Independent variables are dimensions from
  mathematical point of view, but not on physical.
• Each physical dimension taken into the model
  increase model complexity a lot

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Dimensions
Mathematical perspective
Physical perspective
c1 c2 c3 c4 ...
a list of component concentrations
c1
a point in concentration space
c1

• A new chemical component (one new mathematical
  dimension) usually increases the problem only
  marginally (N3 at most)

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Dimensions

• A new physical dimension
• Spatial discretization in N points in each
  direction
• Work load ? ND, where D is the number of
  dimensions

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Simulation vs. design

• If feeds and process unit details are given, and
  products are unknown, the problem is called a
  simulation problem
• If there are spesifications for products, and
  some process details are unknown, the problems is
  called design problem

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Simulation vs. design

• In principle, simulation is easier than design.
• Usually (sometimes) simulation can be carried out
  in a straighforward manner by solving unit
  operations starting from the first unit where
  feed is introduced. Then all unit operations are
  solved one by one until the last one gives us the
  products.

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Simulation

• Only the simplest simulation problems acually can
  be solved in such a simple manner. Often an
  iterative solution is necessary also in
  simulation problems

This tear stream needs to be solved iteratively
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Design

• On the other hand, some design problems can be
  solved without iteration

Reactor length (or catalyst mass) unknown
Inlet flow given
Outlet flow (conversion etc.) specified
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Design

• On the other hand, some design problems can be
  solved without iteration

simulated reactor outlet concentration
specified outlet concentration
simulated reactor length
required reactor length
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Design

• Usually design problems require iterative
  solution of unit operation models

Adiabatic tubular reactor, catalyst mass specified
Heat exchanger controls reactor inlet temperature
Outlet flow (conversion etc.) specified
Inlet flow specified
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Design

• Guess heat exhanger duty
• Solve reactor model (initial value ordinary
  differential equation)
• At each location, there may be nonlinear
  algebraic models for reactor operation, e.g.
  reaction rate, estimation (differential-algebraic
  system)
• Compare reactor outlet to the specified. If not
  equal, change exchanger duty and start from 1.

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Optimization
Adiabatic tubular reactor, catalyst mass can be
changed
Heat exchanger controls reactor inlet temperature
Outlet flow (conversion etc.) specified, amount
of a side product should be minimized
Inlet flow specified
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Optimization

• Guess heat exhanger duty and reactor catalyst
  mass
• Solve reactor model (initial value ordinary
  differential equation)
• Compare reactor outlet to the optimization
  constraints and objective function. Go to step 1
  and repeat until no improvement can be obtained

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Optimization

• In this kind of problems, optimum may very often
  be at the limit of the variables (e.g. maximum
  amout of catalyst and just enough heating of the
  feed to get the required conversion)
• Optimization is often part of design problems. Do
  not optimize if the answer is clear. Optimize
  (for optimal design) if it isnt.

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Optimization

• Objective function and constraints are often
  subjective choises
• Specify conversion and minimize side products
• Specify side products and maximise conversion
  (or yield)
• Combine these based on a suitable intuitive or
  economic objective function, e.g. maximise
• Fprod - 5?Fside-prod
• where Fprod is desired product flow rate, and
  Fside-prod is undesired side product flow rate. 5
  is a hat constant given by the design engineer.

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Summary

• Dimensionless numbers can be obtained from
• traditional dimensional analysis
• ratios of two dimensionally similar objects
• by non-dimensionalizing model equations
• Models can be classified in mathematical or
  physical point of view. There are numerous ways
  to classify models.
• Model classification helps to choose the best
  numerical methods for solution
• Simulation, design, and optimization problems are
  different ways of looking at the same physical
  process