Models & Model development

ThermoTech key-note lecture

by

Tore Haug-Warberg

Department of Chemical Engineering

NTNU

December 4, 2003

What is a model ?

The model (i.e. the abstract concept) is used virtually everywhere e.g. in: Organization theory, economy, politics, planning, computer science, mathematics, physics, psychology, ... Several of these disciplines are represented in this room, and we should really look for a common definition to secure the planning of the ThermoTech project!

Definitions

Law: Regularities existing in objects and events - both observed and posited.

Theory: Postulation system from which laws are deducible as theorems.

Model : Set of constitutional equations turning a theory into a closed equation hierarchy.

2nd lawGravity

Examples (I)

Definitions are OK but examples are some-times better. Let’s have a closer look at the ideal gas (this is the most unifying non-trivial concept I can think of).

Ideal gas (PV=NRT)

Law : R. Boyle (1627-91), J. Charles (1746-1823) and A. Avogadro (1778-1850).

Theory: J. C. Maxwell (1831-79), L. Boltzmann (1844-1906) and J. W. Gibbs

(1839-1903) => statistical mechanics

Model : L. Tisza (1961)

Ideal gas law

The works of Robert Boyle (1627-91), Jaques Charles (1746-1823) and Amedo Avogadro (1778-1850) indicated that PV=NRT is a good approximation for typical gases like air and air components, This investigation took about 150 years of pondering and experimental work!

Ideal gas theory

Statistical mechanics was founded by James Clerk Maxwell (1831-79), Ludwig Boltzmann (1844-1906) and Josiah Willard Gibbs (1839-1903):

Ideal gas is reached by setting: U=0 (model)

V

ZkTp

rdrdeN

Z NkTrrU

NN

ln!

11

/,,3

1

Ideal gas model

Laszlo Tisza formulated in 1961 the Gibbsian thermodynamics as 4 postulates. Within this framework it can be shown that any pV equation-of-state must be on the form p=f(N/V,T). Hence, the simplest conceivable model is p=a(N/V)T. Note that parameter a cannot be resolved by thermodynamic theory alone => measurements (instantition problem)!

Examples (II)

James Clerk Maxwell formulated in 1876 the theory of electromagnetism

which in fact is a closure (no models needed)

dsIudl

dsdl

ds

qds

dtd

dtd

uεB

uBε

uB

uε

0

/

Examples (III)

Fluid mechanics:

Constitutional models:

vΠjqvv

vvΠ

jvvv

:htu

t

iiitc

uh

p

cci

IvvvΠ

j

q

32T

ii cD

T

Examples (IV)

Thermodynamics phase equilibrium:

Saul-Wagner 235-parameter model for water and steam:

ii

iAcrobat Document

Paradigm

A model can be “anything” from a simple statement U=0 to a set of equations with hundreds of parameters. The more advanced a theory is, the less complex the model needs to be:

This follows from the requirement that

ConstantComplexityComplexity TheoryModel

ClosureModelTheory

Conclusion

In mathematical “modeling” it is important to know where the theory stops and the modeling starts, and what are the assumptions and parameters:

TheoryAssumptions

Simplification

Constitutional models

ClosureParameters

Simulation model