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Process Modeling methods and tools Lecture 3

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

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

Dimensional analysis

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

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!

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

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

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

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

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

Another approach

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

- Sherwood number
- Ratio of mass transfer film thickness (according

to the film model) to particle diameter - Ratio of convective and diffusive mass transfer

- ? ratio of a circle's circumference to its

diameter - Dimensionless ratio
- Mathematical constant (mathematics does not

involve physical dimensions as such)

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)

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

Dimensionless numbers from physical models

Dimensionless length z

nth order reaction

Dimensionless numbers from physical models

One possible Damköhler number

Dimensionless numbers from physical models

dimensionless concentration

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.

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)

Navier-Stokes

This can be put in a dimensionless form as

Where superscripts denote dimensionless

variables (velocity, time, length)

Navier-Stokes

Important dimensionless parameters for

incompressible Navier-Stokes equations arise from

this process

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

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

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)

- 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

- 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

- 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.

- 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.

- 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

- 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

- 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.

- 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

- 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

- 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

Classical classificationfor 2nd order PDEs

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

Classical classificationfor 2nd order PDEs

Diffusion

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

Classical classificationfor 2nd order PDEs

Newtons law for wave motion

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

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

Classical classificationfor 2nd order PDEs

Laplaces equation for heat conduction

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

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.

Numerical solution of adsorber breakthrough

curves (hyperbolic)

Numerical diffusion (typical for low order

methods)

Oscillations (typical for high order methods)

- 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

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.

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

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

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

Physical classifications

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

Dynamic stirred tank

Plug-flow reactor

are the same

- For steady-state flowsheet simulator

steady state tubular reactor

dynamic batch reactor

are not the same

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?

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

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)

Dimensions

- A new physical dimension
- Spatial discretization in N points in each

direction - Work load ? ND, where D is the number of

dimensions

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

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.

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

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

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

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

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.

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

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

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.

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.

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

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