8.1 mathematical modelling and systems analysis

469VOLUME V / INSTRUMENTS

8.1.1 Introduction: definition anduse of models

The term model is used in various contexts to refer to aschematic representation of the essential aspects of physical,biological, economic and social phenomena; from a practicalpoint of view, models are tools for simulating thesephenomena, albeit more or less imprecisely.

Technology makes extensive use of models since itis often expensive, complicated or even impossible towork with prototypes of the systems of specificinterest. In the chemical industry, in particular,simulation models are used both to design and controlthe units in which chemical reactions and theseparation of mixtures of several components areconducted, and to manage the networks connectingthese units to one another; in the oil industry,geophysical models are also used in the explorationand development of hydrocarbon reservoirs.

As far as the relationship between a model and a theoryis concerned, it can be observed that a model is oftenconsidered the first stage in a study aiming to obtain moredetailed knowledge of a broad range of phenomena; anexample is the concept of the atom, used by chemists duringthe Nineteenth century as a model to justify Dalton’sempirical law of multiple proportions, which later becamethe Atomic Theory, one of the fundamental theories ofmodern science. Similarly, the concept of the gene,employed as a model to justify Mendel’s laws later took oncrucial importance in molecular biology.

This chapter will concentrate essentially onmathematical models, which consist of a complete andconsistent series of equations (algebraic and differential)which describe a specific object of physical, biological orsocial nature. Often, in terms of their applications and tools,models are a formal expression, using mathematicalequations, of appropriate working hypotheses concerningphenomena which are only partially understood. However,the aim of this discussion is not to survey the mathematicalmodels used in chemical technology, since many of thediscussions in this volume serve this purpose adequately.Rather, some concepts regarding the general properties ofmodels, their formulation and above all their potential in the

development of specific technologies will be discussed ingreater depth.

8.1.2 The characteristics of models

DevelopmentThe formulation of a mathematical model often entails

an interpretation of reality, tracing it back to simplegeometric and physical typologies. This principle followsfrom the teachings of Galileo, who observed that the book ofnature is written in the language of mathematics, whosealphabet consists of circles, squares, triangles, spheres andso forth; this is a simple proposition which has foundconfirmation in various and sometimes unusual applications.For example, the shape of an animal can be represented by aseries of pseudocylindrical elements whose mass is afraction of the total mass M; through simple reasoning basedon geometrical and elastic similarities, it can be seen that thediameter of each of these is roughly proportional to M3/8.This result has been confirmed experimentally, for exampleby comparing the thoracic circumference with the body massof adult primates.

The first step which should be taken to solve a givenscientific or technical problem is the definition of the ordersof magnitude of the main variables involved. This approach,which obviously finds an appropriate application intechnology, can be exemplified effectively by rememberingthat Enrico Fermi managed to estimate with acceptableaccuracy the number of piano tuners in Chicago using easilyavailable information concerning the number of inhabitantsin the city and formulating reasonable hypotheses on theirmusical aptitudes. Another example is a problem consideredby Benjamin Franklin, who observed that if a drop of oilwith a volume of one tenth of a cubic centimetre is allowedto fall onto a lake, it disperses to form a patch which maycover a surface area of up to forty square metres. Franklinnoted that the conservation of volume had to be compatiblewith a thickness of the dispersed layer of about 2.5 nm: thisvalue, corresponding to the size of an oil molecule, wasobtained by simple macroscopic observation, anticipating aresult which the experimental methods of chemicalstructuralism would confirm only many years later.

8.1

Mathematical modellingand systems analysis

Generally speaking, problems concerning the real worldare translated into mathematical terms by identifying themain physical parameters assumed to condition thebehaviour of the system under consideration; these mustlater be formalized as a set of equations reflecting therelations between the variables involved. These equationsderive from the fundamental physical laws which express theconservation of mass and the amount of motion and energy,and from the constitutive equations which reflect the specificbehaviour of the substances or materials involved. Whereaslaws of conservation apply to all situations, constitutiveequations are obviously specific to the type of system underconsideration.

If F is used to indicate the net flow of an object in agiven region, with G being its velocity of generation and Qits total quantity, the principle of conservation provides thefollowing equation:

[1]

where t is time.Generally speaking, the magnitudes defined above

depend on the values of the spatial coordinates; it istherefore helpful to indicate with f the vector whichexpresses the flare per unit of surface area, with g thevelocity of generation per unit volume and q the density ofthe magnitude under examination. As such, considering anarbitrary region of volume V, enclosed by a surface S, wecan write:

[2]

where n is a unitary vector locally perpendicular to thesurface and directed towards the outside. Using Green’stheorem, the conservation equation is thus written in theform:

[3]

with �� being the divergence differential operator. The constitutive equations supply the specific

expressions of f, g and the connections between them. Forexample, with reference to a system subject to materialdiffusion and chemical reactions, g is identical to thereaction velocity r expressed in moles transformed per unitof time and volume, q to the molar concentration C and f isexpressed by �D�C (Fick’s law) where D is the diffusioncoefficient. In this case, [3] becomes:

[4]

A different situation arises when a solid catalyst ispresent on whose surface, reached through a diffusionprocess since it develops inside porous particles, a chemicalreaction takes place; in this case, in addition to concentrationC for the core of the fluid, we must also introduce a surfaceconcentration Cs, expressed in moles per unit of surface area.Applying [3] therefore gives two differential equationsexpressing the balances in the fluid and on the surface, andan algebraic boundary condition expressing the relationship

between the two concentrations and which is generallyformulated by assuming that the adsorption process can beconsidered at equilibrium:

[5]

where DS is the surface diffusion coefficient and rS is thevelocity of the catalytic reaction per unit of surface area. Itcan be observed that the above equations differ from [4] forthe greater impact of the constitutive equations whoseexpressions contain a higher number of specific parameters,in particular the two diffusion coefficients, the parametersinvolved in the adsorption isotherm, f(C), and the reactionvelocity constant. Their values must be obtained either fromindependent measurements or from calculations based on thefundamental physical properties of the system underexamination.

Unfortunately, the values of all the parameters involvedin a model are not always available a priori, so it is at timespreferable to leave some of these open, making it possible togive them the most appropriate value to describe the processunder consideration. This procedure is obviously legitimateas long as it is applied with full awareness of the fact that thepresence in a model of a large number of adaptableparameters, whilst on the one hand giving it greaterflexibility in the description of experimental data, on theother compromises, or at least limits, its predictivecapacities. To bypass these problems, it is a good principle toexamine critically the reliability of the values obtained forthese parameters on the basis of their physical meaning,analysing their orders of magnitude in accordance with thecriteria outlined above.

The subsequent stage is to solve the equations thusobtained; this stage makes considerable use of numericalcalculations and computers, with which problems entailingthe solution of non-linear algebraic equations or systems ofdifferential equations which cannot be integrated analyticallycan be dealt with easily; this approach also allows largeproblems involving hundreds of variables to be solved. Togive two specific examples concerning the study ofindustrial processes, we can remember that determining thecompositions in the various stages of a multi-componentdistillation column involves solving non-linear algebraicsystems consisting of hundreds of equations, whilst thecombustion process of one or more hydrocarbons can bedescribed by integrating tens of non-linear differentialequations which express the conservation of mass of eachcomponent present in the system. In this context, it can bestated that chemical engineering models have reached asignificant level of maturity, since they take intoconsideration numerous and subtle details of the phenomenainvolved; however, it is sometimes legitimate to ask to whatextent such a sophisticated approach is actually necessary inthe simulation of industrial processes.

The final stage is to interpret the results obtained in thecontext of the problem examined, where possible through acomparison with experimental data; this stage is in someways the most difficult because special attention is needed inthe evaluation of the reciprocal compatibility of the resultsobtained. It is also important to be able to evaluate the

∂∂

=∇⋅ ∇( )∂∂

=∇⋅ ∇ +

=

Ct

D C

Ct

D C r

C f C

ss s s

s

( )

( )

∂∂

=∇⋅ ∇( )+Ct

D C r

−∇⋅ + = ∂∂

f g qt

F dS

G gdV

Q qdV

S

V

V

= − ⋅

=

=

∫∫∫

f n

F G dQdt

+ =

MATHEMATICAL AND MODELLING ASPECTS

470 ENCYCLOPAEDIA OF HYDROCARBONS

predictive capacities of the model itself. Often, this analysisevidences a need to refine the values of those parameters inthe model for which a value measured independently orcalculated on the basis of fundamental physical theories isunavailable.

In conclusion, the sequence in which the aboveoperations can be carried out is summarized in the flow chartshown in Fig. 1, where the steps to be taken are explicitlyindicated, showing their logical sequence in the context of aniterative calculation scheme which also aims to define themost appropriate values for adaptable parameters. If theresults obtained by applying the model in one specificformulation turn out to be unsatisfactory, the procedure isrepeated from the beginning to take into consideration amodification of the interpretative hypotheses used toformulate the model.

Conceptual valueIt is important to remember that models often have an

essentially conceptual content, since their purpose is toclarify delicate and difficult points of a given theory. Aninteresting example comes from the study of the statebehaviour of fluids, with the aim of formulating amathematical model able to describe the relationshipbetween three state variables: temperature T, volume V andpressure p. The solution of this problem presents someimportant conceptual implications which particularlyconcern the existence of a critical point characterized bywell-defined values of the three state variables whichdetermine the conditions under which gases can (or cannot)be condensed. This problem has been the subject ofextensive investigation both for its scientific interest and itspractical implications in terms of the condensation ofincoercible gases, especially air; additionally, if this study isbroadened to include mixtures of different components itbecomes extremely important for the design of processes toseparate the components of the mixture. This need has led tothe formulation of numerous equations of state, some ofwhich are mathematically highly complex, whoseapplication is nevertheless limited to specific areas of thestate surface: statistical thermodynamics offers a concise andapparently simple formula to solve the problem:

[6]

where kB is Boltzmann’s constant and F is the potentialenergy of the fluid which depends on the arrangement of the

molecules in space. The drN are elements of volume aroundthe positions occupied by each molecule and the integral iscarried out for all the possible configurations to which theset of molecules may give rise: this is a functional integral,known as the configuration integral, for which onlyapproximate solutions can be obtained applying considerablesimplifications and making use of the numerical simulationmethods deriving, for example, from the Monte Carlotechnique.

Now consider one-dimensional fluids, in other wordsfluids consisting of molecules, punctiform if they are ideal,or in the shape of small segments, and distributed randomlyover a given segment of length L: in the early 1970s it wasdemonstrated that for such fluids it is possible to obtain anaccurate solution of the configuration integral. Using anexpression, albeit fictitious, of the interaction potentialbetween the molecules which also appears when thedistances between the molecules are relatively high, anequation of state is obtained which is similar to the van derWaals model, but containing length L instead of volume andwhose p-L diagram presents horizontal lines indicating thepresence of a phase transition. This operation is performedwithout having to resort to the expedient proposed byMaxwell for identifying the processes from the sinuouspattern taken on by the isotherm curves below the criticalpoint in the van der Waals equation and the equationsderived from it.

Although from a practical point of view this result seemsmeaningless as one-dimensional fluids do not exist, it isnonetheless of considerable conceptual importance since itjustifies the van der Waals model and the use of perturbationmethods to describe the thermodynamic properties of fluids,based on the assumption that the arrangement of themolecules in a liquid depends essentially on forces ofrepulsion whilst those of attraction determine its energy. Inother words, the model confirms that in the description ofthe state behaviour of fluids it is legitimate to separateentropic effects, linked to the distribution of the molecules inspace, from enthalpic effects which depend on theirinteraction energy. Commenting on his work, Marc Kacasked himself if “it is possible that nature is so familiar withmathematics” (Kac et al., 1963), a question reminiscent ofthe more general query posed by Eugene Wigner on the“unreasonable effectiveness of mathematics in the naturalsciences” (Wigner, 1960).

It must be noted, however, that the model describedabove, alalthough it has found significant confirmation in

p k TV

e d d dBk TB= ∂

∂

−∫∫ln/� …Φ r r r

1 2 N

MATHEMATICAL MODELLING AND SYSTEMS ANALYSIS


mathematicalproblem

formulation

equation solutionby means

of analytical ornumerical methods

experimental dataand prediction

comparisonOK

problem interpretationfrom the geometric and

the physical point of view

examinedproblem

resultinterpretation

Fig. 1. Flow chart showing the operations required to develop a model.

the study of fluid thermodynamics, obviously includinghydrocarbons, fails to describe the behaviour of water, themost widespread liquid on Earth; although the molecularstructure of water is apparently extremely simple, the strongimpact of hydrogen bonds on molecular interactionsencourages the formation of specific structures so that theliquid presents peculiar properties, such as the anomalousvalue of the boiling point with respect to that of otherhydrides and the inversion of the density values of the liquidand the solid. It is important to note that these facts areessential for the existence of life on Earth, at least in theform with which we are familiar.

8.1.3 Dimensional analysis

Basics and example of applicationDimensional analysis, applied successfully in different

scientific and technological sectors, has been shown to beextremely useful for simplifying a problem by reducing thenumber of variables involved, grouping them intoappropriate dimensionless groups: if each physical variableis expressed by a suitably selected fundamental magnitudesuch as length L, mass M, time t, temperature q, the key toolof dimensional analysis is the Buckingham theorem,according to which an equation containing n variables and mfundamental units, where m�n, can be reformulated withn�m dimensionless groups. The method limits itself toconfirming the existence of this relationship, withoutidentifying its specific nature.

Having defined the n dimensioned variables of thesystem under examination X1, X2, …, Xn, each of the n�mdimensionless groups is expressed in monomial form asfollows:

[7]

The exponents present in the different expressions areindeterminate with the exception of some specific andappropriately chosen variables, which must not be commonto the different groups, but will be present in each of thesewith an exponent equal to one. The other exponents mustthen be calculated, with the products P being dimensionlesswith respect to the m fundamental dimensions chosen; thiscondition makes it possible to write a series of algebraicequations, the solution of which provides the exponentswhich define the dimensionless groups, linked by an implicitrelationship of the type:

[8]

The actual form of the connection can only beestablished experimentally or through a more detailedanalysis of the system’s physical properties. If only onedimensionless group appears (see the example of the atomicexplosion below), it can be taken as equal to a constant andtherefore a single experiment is sufficient to obtain thedesired relationship. Regardless of its specific form,however, it should be noted that a comparison between any

given system and its model must be carried out with equalvalues of the dimensionless groups characterizing theprocess under examination.

The example mentioned earlier is presented here. In1945, at the height of the Cold War between the UnitedStates and the USSR, «Life» magazine published on its frontcover a series of photographs of the first atomic explosionwhich had taken place in New Mexico that same year (Fig. 2):these showed the variation over time of the size of the area where the explosion was concentrated, thus givingreaders throughout the world an effective description of thepower of a weapon which at that time was held only by theUnited States. This sequence of photographs raised thequestion of whether the value of the energy released by the

ϕ Π Π Π1 2 0, , ,… n m−( )=

Π

Π

1 1 2 1

2 1 2 2

1 1 1

2 2 2

=

=

+

+

X X X X

X X X X

a bmm

m

a bmm

m

…

….......................................Π i

aX i= 1 XX X Xbmm

m ii i

2 … +

.......................................



100 m

0.10 ms

0.24 ms

0.38 ms

0.52 ms

0.66 ms

0.80 ms

0.94 ms

Fig. 2. Progressive growth of the ‘fireball’ produced by a nuclear explosion in New Mexico; the numbers on the left of each photograph indicate the milliseconds from the beginning of the explosion.

explosion could calculated using these images. This task,typical of mathematical modelling, could be accomplishedby simulating precisely the evolution of the event over timeby integrating a series of equations expressing the laws ofthe conservation of matter and the quantity of motion indifferential form.

In a non-viscous gas which expands adiabatically with aspherical symmetry following an instantaneous injection ofenergy at a central point, these equations take the form:

[9]

In [9], r indicates the distance from the centre of theexplosion, r, u and p are the local values of density,velocity and pressure, whilst g is the adiabatic exponentgiven by the relationship between the specific heats atconstant pressure and volume and M is the mass of the gascontained inside a sphere of radius r. In the third equation,the last term reflects the impact of gravitational attraction,whose universal constant is indicated with G; in theproblem under examination, this term is less important,whilst it gains significance for explosions of cosmicdimensions. The fourth equation expresses the adiabaticconstraint, compatible with the properties of rapidexpansion.

Equations [9] must be integrated assigning as the entrydatum the energy E released by the explosion itself; thismust be attributed the value which makes it possible todescribe satisfactorily the evolution over time of the shockwave shown on the cover of «Life»; at the time, this task wasnot simple, especially because of the problems relating to thediscontinuity present at the boundary of the explosion zone.In any case, the problem could not be solved without usingthe numerical calculation methods which have only recentlybeen made available to modellers.

In 1950, Geoffrey I. Taylor published an important study(Taylor, 1950), in which he proposed a simple solution to theproblem. Starting from the assumption that the explosionwould cause a violent shock wave expanding through anundisturbed medium, it was hypothesized that the radius r(t)of the explosion zone could be expressed as a function of theenergy E released initially, of the density of the air andobviously of time, through a monomial expression consistingof the product of the magnitudes mentioned, each raised to asuitable power. Giving the radius the dimensions of a length,he easily obtained the following:

[10]

more useful when rewritten in logarithmic form:

[11]

This expression could be successfully used to interpolatethe data obtainable from photographs of the explosion; it was

thus possible to calculate the value of the energy released,which amounted to 8.45�1013 J, equivalent to 17,000 tonnesof TNT. In conclusion, one of the most jealously guardedsecrets of the Cold War was published on the cover of amagazine with a wide readership.

Use of differential equationsThe dimensionless groups into which the variables of a

given problem are made to converge can also be identifiedby examining the differential equations which describe theprocesses involved. These equations must be dimensionallyhomogeneous so that, by dividing the variables of one of itsterms by those of another, dimensionless groups areobtained; this approach has the advantage of showing theirphysical significance.

As an example, consider the flow of an incompressiblefluid current in a circular pipe or in contact with a solid andhypothetically spherical object. Its motion can be describedusing the Navier-Stokes equation, which for anincompressible fluid can be written:

[12]

where u is the velocity vector and v is the kinematicviscosity of the liquid. It is then possible to refer to thenatural scale of the magnitudes involved, in other words themean velocity U in the pipe or at an infinite distance fromthe object; the length of the pipe L (or a dimension of theobject, for example the radius of the main section); and atime scale expressed by L/U. If the system variables arerelated to these values, we can write:

[13]

which, when substituted into [12] give:

[14]

where p*�p�u2r.It can be seen that in the above equation only a single

dimensionless group appears, Re�UL/n, in other words theReynolds number which, as is known, characterizes the fieldof flow.

Another interesting example is the model of anagitated reactor with heat exchange, a good referencepoint for a preliminary examination of the behaviour ofreagent mixtures, even although the real situations thathave to be faced are generally more complex than those ofa system in which it is assumed that the concentrations ofthe reagents and the temperature are uniform. The modelaims to demonstrate their evolution using differentialequations expressing the material and heat balance of thereagent mixture flowing through the reaction volume V, inwhich the temperature T and concentration C both have auniform value. In this simplified formulation, thephenomena in play include various physical parameters,especially the heat of reaction (�DH), the specific heat ofthe fluid Cp, the heat transfer coefficient h, the velocityconstant of the reaction k�Aexp(�E��RT), where A is thepreexponential factor and E� is the activation energy. Alsoknown are the heat exchange surface area S, thevolumetric flow rate of the fluid Q, the externaltemperature Te and, obviously, the concentration C0 andtemperature T0 of the fluid upon entry.

∂∂

+ ⋅∇ = −∇ + ∇u*Ret

p*

( ) *u* u* u*1 2

u u U x x L t t L U= = =* * * /

∂∂

+ ⋅∇ − ∇ + ∇ut

p( )u u u=1 2

rν

5

22ln ln lnr E t�

r

+

r t Et( )

/

�2

1 5

r

∂∂

+ ∂∂

+ =

∂∂

=

∂∂

+ ∂∂

+ ∂∂

r r r

r

r

tur

ur

Mr

r

ut

u ur

p

20

4

1

2π

rrGMr

tp u

rp

+ =

∂∂

+ ∂

∂

=

20

0r rγ γ



The large number of physical and operative parametersmakes this analysis difficult and it is therefore helpful to beable to give the equations of the model an appropriate form.If the operating conditions are kept constant, in the spirit ofdimensional analysis this goal can be achieved by using thefeed of the reagent C0 as a reference unit for concentration,the value of the ratio E��R of the activation energy to the gasconstant for temperature, and the ratio V�Q of the volume ofthe reactor to the volumetric flow rate of the feed for time; inthis formulation, the two equations expressing theconservation of mass and energy take the form:

[15]

where C*�C�C0 is the concentration, T*�RT�E� thetemperature and t�tQ/V the time, all dimensionlessquantities. It can be observed that the equations thusobtained contain the following four parameters:

As such, any investigation of the system’s behaviourmust refer to their specific values. However, it can beobserved that under stationary conditions, when both of thefirst members of [15] are taken as equal to zero, twoalgebraic equations are obtained which, when combined withone another, give the following equation containing onlythree parameters:

[16]

Equation [16] expresses the temperature pattern as afunction of the characteristic constants of the system and istherefore particularly suited to studying the thermal stabilityof the reactor. Shown in three-dimensional space by a systemof coordinates Z, T

33

* and x it supplies the surface shown inFig. 3, which forms a good portrait of the thermal behaviourof the reactor: interestingly, this surface presents the cusppattern typical of catastrophe theory, where two zones can beidentified, presenting a low and a high temperaturerespectively; the transition from one to the other can only

occur discontinuously with a sudden variation intemperature.

Corresponding statesThe identification of appropriate algebraic relations

between dimensionless groups sometimes depends on theability to formulate simplified models of the systems underexamination, using appropriate approximations. Theserelations are of universal significance because they refer tocorresponding states of different systems described by thesame physical model; their application, often extremelyuseful, must nonetheless involve a full awareness of theapproximations underlying the reference model.

An example is provided by the study of the stateproperties of fluids which can in principle be obtained byapplying [6]; however, this entails the difficulties connectedwith the calculation of the configuration integral, which wewill indicate with Q(V,T). Consider two fluid substances,1 and 2, whose intermolecular potentials can be expressedwith the equations:

[17]

where u0 and s are the values of an energy and a distance,both specific to the two substances under examination,whereas F indicates a function common to both. Forexample, the Lennard-Jones potential meets theserequirements. Now assume that N molecules of the firstsubstance are confined in a volume V at temperature T andthose of the second in a volume Vs 3

22�s 311 and at a

temperature of Tu022�u0

11. It follows that each configuration ofthe first fluid corresponds to a configuration of the second,so that:

[18]

As such, the integral Q2 can be calculated from the valueof Q1. More generally, considering a reference potential withparameters u0

0 and s0, for any substance a we can write:

[19]

since fa�u0a�u0

0 and ga�sa�s0. It follows that on the basis of[6] the equation of state can be written in the form:

[20] p V Tfg

p Vg

Tfα

α

α α α

, ,( ) =

3 0 3

Q V T g Q Vg

Tf

Nα α

α α

, ,( )=

30 3

φ φ σ σ1 2 22 11… … …, ,... , / ,r r

i

B

i

Bk T k Tu

=( )

222

0

11

0/ u

u r u Fr

u r u Fr

11 11

0 11

22 22

0 22

( )

( )

=

=

s

s

T T e

eZ T

T

T

* **,

/

/

*

*

− =+

=

−

−ξα

αα

1

11

α ξ= =−( )

≠AV Q

H RC

E C/

∆ 0

pp p

pp

hS QC

H C R C E TR QC T

1

0 0

+( )

= −( ) =≠

/

/ *

r

rβ ∆

++( )+( )≠

hST

E QC hSe

pr

dCd

C C e

dTd

T T C e

T

T

* * *

* * * *

τα

β τ ξα

= − −

= − +

−

−

1

1

1

1

/

/

*

**



x

Z

S

U

R

P

T33

*

Fig. 3. Surface showing thestationary states of an agitatedreactor with heat exchange as afunction of the reduced variables Z,T33

*, x. The PURS line, indicating atrajectory along which parameter T

33

*depending from the outsidetemperature is varied, shows that the transition from the lowtemperature zone entails a suddenchange indicated by the dashed line.

If the equation of state for the reference fluid is

[21]

for a generic substance a we obtain:

[22]

which is simply an expression of the law of correspondingstates which, as is known, plays an important role inthermodynamics.

Another example is provided by the study of thethermodynamic properties of monoatomic solids: thecrystalline system can be compared to a set ofthree-dimensional harmonic oscillators coupled with oneanother such that the vibration of an atom is transmitted tothe surrounding atoms and propagates throughout thecrystalline structure. The motion of each atom is extremelycomplex, but it can be hypothesized that the dynamicbehaviour of the solid can be described by thesuperimposition of harmonic motions, whose frequencies aredistributed in the same way as those of an elastic solid up toa maximum value of νD. It is then shown that thethermodynamic properties of the solid, such as the free andinternal energies and heat capacity, can be expressed byuniversal dimensionless equations as a function of a reducedtemperature (T/qD), where qD�hnD�kB is the Debyetemperature, h is Planck’s constant and kB is Boltzmann’sconstant.

The final example concerns the tangential motion of afluid with respect to a flat solid surface. This situation affectsthe solution of problems concerning the resistanceencountered by the motion of a solid in a fluid and theexchange of matter and heat between the fluid and thesurface. The problem to be solved is the calculation of theprofile of the component ux of the velocity of the fluidtowards the surface; the undisturbed velocity is indicatedwith uo. The solution can be obtained by integrating theNavier-Stokes equation [12] after introducing suitableboundary conditions, in particular assuming that the velocityof the fluid in contact with the wall is zero. If appropriateand reasonable approximations are used concerning the

relative values of the components of the velocity and theirderivatives with respect to the x axis tangential to the surfaceand the y axis, perpendicular to it, and taking into accountthat the variation of pressure in the latter direction isnegligible, we obtain the following relationship betweendimensionless magnitudes:

[23]

where

Using this result it is then possible to express the fluiddynamic equation through the single variable h and tointegrate it numerically; this makes it possible to calculatethe laminar velocity profile of the fluid within the boundarylayer, adjacent to the surface and in which the tangentialcomponent of the velocity is subject to a sudden variation.The transition of the velocity to the value uo outside theboundary layer occurs asymptotically in a thickness that canbe calculated using the following equation:

[24]

Resistance to the transport of the quantity of motion isessentially localized in this thickness.

8.1.4 Scale properties

Recently, interest has turned towards the laws of scale,increasingly present in various sectors of the fundamentaland applied sciences: they are particularly useful inengineering whenever it is necessary to transfer the resultsobtained in the laboratory or pilot plant to the industrialscale.

Laws of scale sometimes take the simple form of apower relationship of the type Y�Xa; an example is [10]which expresses the dimensions of the sphere of fire whichforms after an atomic explosion, or the aforementioned

δ ν≈5 20

. xu

η ν=

ux y0

1 2/

uux

0

=Ψ ( )η

ϕ ϕα

α α αα0

3

3pgf

Vg

Tf

p V T, , , ,

= ( )

ϕ0

0p V T, ,( ) =



Table 1. Examples of phenomena described using laws of scale

SYSTEM OR PHENOMENON IMPORTANT MAGNITUDES LAW OF SCALE

Mean radius of the orbit of a planet t�time taken for one revolutionR � t 2/3

(Kepler)

Extensive thermodynamic magnitudes (Y, X) l�numerical parameter Y(lx)�lY(X)

Random walka�mean distance between two collisionsN�number of steps or collisions R(aN)�a1/2R(N)

Extension of a macromolecule a�distance between two monomers R �a0.588

Turbulencee33

�mean velocity of energy dissipationl�size of a vortexE(k)�energy of a vortex

E(k)�e332/3k�5/3

(Kolmogorov)

Connections between the nodes of a network k�number of connections leaving a nodeP(k)�probability that a node chosen atrandom has k connections

E(k)�k�g

g(k)�1-3

relationship between the size of an animal’s bones and itsmass.

It should be noted, however, that it is not always possibleto identify a specific scale against which to measure thephenomenon: situations of this type are extremely common,as summarized in Table 1. For example, in the case of theturbulent motion of a fluid, a knowledge of its mean velocityis insufficient to describe all its properties, such as itstransportation and mixing properties; it is also necessary totake account of the specific values of the energy associatedwith fluid particles subject to turbulent fluctuations.Specifically, it can be assumed that in turbulent motion, theenergy injected into vortices of size l0 is transferred to theother vortices through a cascade of intermediate scales ln,such that ln�l0�2

n, with n�0,1,2,…, up to the dissipationscale ld. Using this approach, it is shown that the kineticenergy En of the vortices, per mass unit, can be expressed asfollows:

[25] En�3

e2�3ln2�3

where 3

e is the mean value of the velocity of energydissipation. In the context of spectral analysis, used invarious branches of physics, it is shown that [25] iscompatible with the following equation, known asKolmogorov’s law:

[26]

where E(k) is the contribution to the energy deriving fromharmonic components with wavenumbers between k andk�dk; as such, it expresses how the energy is distributed invortices of different lengths in the form of a law of powers.

Among the systems and phenomena in which amultiplicity of scales is present, we could mentionpercolation, which is the slow filtration of a liquid through athick layer of solid porous material. The study of percolationhas provided results of considerable interest in studies ofpetroleum reservoirs, of the spread of epidemics and thepropagation of forest fires. Further spin-offs can be found inthe analysis of the behaviour of ecosystems, the distributionof earthquakes and of networks between interacting centresor nodes, the configuration of polymer solutions andeconomic systems, etc.

It is important to note that the interest in theseproblems became established in the context of studies onthe interactions between elementary particles and thentook on a role of primary importance in the study ofcritical phenomena presenting fluctuations in specificphysical magnitudes such as energy, magnetization anddensity. Traditional physics is based on the use of regularfunctions so that fluctuations are dealt with using themethodologies of classic statistics, ignoring the typicalproperties of self-similarity hidden beneath erraticbehaviour, which is usually attributed to purely randomfactors. The relevant analysis can be conducted byintroducing a scale function which is given the form ofthe aforementioned law of power:

[27]

If variable X is multiplied by a constant l, we obtain:

[28]

As such, if the behaviour of a phenomenon ischaracterized with reference to a specific scale, at a scale

larger by a factor of l similar behaviour is found, butmultiplied by la.

An example is provided by extensive thermodynamicmagnitudes, for which the following equation holds true:

[29]

typical of homogeneous first order functions.Another example is found in the diffusion or random

walk to which molecules are subject as an effect of theirreciprocal collisions; if the mean distance between twocollisions is equal to a, it is shown that the distance travelledafter a number N of steps is expressed by the equation:

[30]

which is of type [28] with a�1�2. It is interesting to notethat the above law of scale also holds true for the descriptionof the distribution of the head-tail distances in a polymermolecule in which it is assumed that the monomer centreshave a negligible volume and are free to rotate around thebonds to which they are attached. If the first condition isremoved by attributing a volume to the monomer centres,their movements are restricted by the fact that they cannot besuperimposed. In this case, the exponent of the law of scalechanges, taking on a value of about 0.588.

From the geometrical point of view, self-similarity findsfull confirmation in fractal objects consisting of a set ofpoints distributed in space such that the number of pointswithin a given distance from the centre is expressed by rD,where D is a number between 0 and 3, known as the fractaldimension. An example is provided by the Sierpinski object,the two-dimensional version of which is shown in Fig. 4. Anequilateral triangle is removed from the centre of a two-dimensional triangle of side L, with its corners at the centralpoints of the three original sides; this procedure is thuscontinued to infinity on the triangles remaining after eachoperation. It is easy to show that this process of eliminationfollows the law of scale N�(A�A0)

��, with A being the area

R aN a R N( )= 1 2/ ( )

Y X Y Xλ λ( ) = ( )

Y c X= λα α

Y cX= α

E k k( ) / /� e 2 3 5 3−



Fig. 4. Generation of a Sierpinski figure.

of a cavity, A0 that of the original triangle whilsta�ln3�ln4�0.795. The typical self-similarity of fractalobjects is thus shown by the fact that, observed at differentscales, they present the same characteristics.

Critical phenomenaAn important problem is that of understanding the

physical reasons for the pervasiveness of the power law andself-similar phenomena in natural and social processes: infact there is no unitary explanation and different levels ofinterpretation can be proposed. In the first instance, thatconcerning the study of critical phenomena, typical ofthermodynamics but actually common to various situations,deserves special mention.

For example, percolation concerns various naturalsystems, but also some simple man-made objects such asordinary household coffee makers; its mathematicaldescription offers, as already mentioned, a model able todescribe the behaviour of phenomena which seem to differfrom one another such as the flow of fluids through porousbeds, the propagation of a fire in a forest, the spread ofepidemics, etc. A peculiar feature lies in the fact that inparticular situations a threshold is reached where the liquidflow ceases, fires go out and epidemics stop.

The most suitable physical model to investigate thenature of critical points is provided by second-order phasetransitions used, for example, to study the behaviour ofincoercible gases; when these approach the criticaltemperature they separate into two phases, a liquid phase anda vapour phase. Another example is the behaviour ofmaterials such as iron oxide which become magnetic onlybelow the Curie temperature. These phenomena areassociated with a variation in the scale of fluctuations,respectively in the density of the fluid agglomerates and theclusters of elementary atomic magnets, whose dimensions incorrespondence with the critical point diverge until theyinvade the entire system. In this formulation, criticalityreflects the fact that all the members of a system influenceone another.

The study of these behaviours has made it possible todevelop an effective technique known as the renormalizationgroup, used to study the variation in the interactions betweenthe different parts of a system as the scale at which they areobserved varies. This variation is reabsorbed into theparameters of the model, whose evolution at different scalesis studied using a physical model, known as the Ising model,which consists of a network of interacting elementary units;each unit is characterized by the values �1 and �1 of aspecific variable, known as spin, assuming that interactionsoccur only between adjacent spins, with an interactionenergy that depends on a characteristic parameter and whichmay be positive or negative depending on whether or not thetwo spins have the same sign or different signs. The mostimportant outcome of this study has been the observationthat the temperature dependence of various physicalproperties, such as specific heat, magnetization andcorrelation distance, as they asymptotically approach thecritical point obeys a law of scale and is therefore describedby laws of power which are universal because they do notdepend on the specific properties of the system underconsideration.

By concentrating on the percolation process, thecharacteristics of the aforementioned renormalization

technique can be illustrated in a simple way. Consider aporous medium delineated by a two-dimensional networkconsisting of N elements which each have a probability po ofbeing permeable; the probability that the entire network ispermeable depends on the existence of at least onepermeable horizontal path. With a statistical analysis, forexample using the Monte Carlo method, it is found that acritical value of po exists, equal to 0.59275, below which nopermeable paths are present. The renormalization methodmakes it possible to deal with the problem without the needfor difficult statistical calculations, by studying the evolutionof permeability through different orders of cells, eachcorresponding to a different scale of the system, as shown inFig. 5. Each square indicates a cell whose magnitude iscompatible with the scale under examination, which in turnis subdivided into smaller squares, four in the figure, whichare permeable (grey) or non-permeable respectively; in theright-hand column the probability of the different cellstructures forming is specified. With this premise, it can beseen that only some of these (Figs. 5C, 5D and 5E) presenthorizontal permeability; if account is taken of their number,it is found that the probability of a permeable cell of orderone forming is expressed by:

[31]

The application of the renormalization method entailsiterating over cells of a higher order, so that:

[32]

and in general:

[33]

At the critical point we have pn�1�pn�pc, giving afourth degree equation which, when solved, leads to thefollowing roots �1.618, �1, 0 and 0.618, of which only thelast is acceptable and in satisfactory agreement with thevalue obtained by applying the Monte Carlo method.

8.1.5 Population balances

In many processes of scientific and technologicalimportance, we are interested in describing the behaviour ofa population of individuals that differ from one another as aneffect of one or more characteristic parameters. This

p p pn n n+ = −12 42

p p p2 12

142= −

p p p p p p p p1 02

02

03

0 04

02

042 1 4 1 2= −( ) + −( )+ = −



A

�

B

�

C

�

D

�

�

�

�

�

�

�

�

� � �

�

E

�

p0

p0 (1�p0)

(1�p0)4

4

p0 (1�p0)3

3

p0 (1�p0)22

Fig. 5. Renormalization technique applied to the percolation process.

situation is typical of the materials sector where, forexample, the chains of a polymer differ in length althoughthey have the same composition; the chain lengthdistribution function or molecular weight distributionfunction is therefore an essential element in defining theapplicational properties of a polymer. In the case of acopolymer, the characteristic parameters of the population ofcopolymer chains are at least two in number and include thecomposition of the chain as well as its length. Inprecipitation processes such as crystallization, populationsof particles are produced which differ for their dimensions,which determine their applicational properties. In othersectors, such as economics or sociology, we are interested indescribing populations of individuals classified on the basisof their income or their age, or perhaps a combination of thetwo.

In all these cases, population balances are anappropriate tool for developing mathematical models able todescribe the evolution over time of the relevant population.To this end, we must first introduce a quantitativedescription of these populations: the state of the individualis therefore defined by a magnitude, usually vectorial,containing all the characteristics of the individual to beconsidered; the characteristics are the coordinates throughwhich the individuals in the population are characterized.For example, in the case of the copolymer chains mentionedabove, the state is defined by two variables: the length andcomposition of the chains. Generally speaking, thesevariables are subdivided into external coordinates, whichrepresent the position in three-dimensional space in whichthe individual is found, and internal coordinates, whichrepresent the parameters characteristic of the individual(length, composition, income, etc). In the space of the statesof the individuals the distribution function or probabilitydensity is defined, f(x,t), whose product f(x,t) dVx representsthe number of individuals who at time t have a statebelonging to the small volume dVx centred on state x. Forthe sake of simplicity, consider the case where the state ofthe individuals is defined by a single parameter, such as acrystallization process in which the population consists ofcrystals characterized by different sizes, x: in this case thedistribution function f(x,t) is defined so that the termf(x,t)dx represents the number of crystals whose size at timet has a value between x and x�dx. On the other hand, thetotal number of crystals N(t) present in the system at thegeneric instant of time is given by the integral of thedistribution function extended to the whole possible spaceof the states, which in the case of sizes may range from zeroto infinite:

[34]

The aim is now to describe the evolution over time of thedistribution function of crystal sizes within a discontinuousand uniform (well-mixed) crystallizer, in which thesupersaturation concentration is kept constant over time. Wemust therefore write a balance equation taking intoconsideration all the events leading to a variation in thenumber of crystals with a specific size x at a specific instantof time t. The result is the following population balance:

[35]

The first term on the left represents the accumulationover time of crystals of size x, whilst the second takesaccount of the existence of a crystal growth process, due tothe transportation of matter from the solution to the surfaceof the crystals, whose velocity is indicated with g�dx�dt.This is obviously a term of convective type which developsalong an internal coordinate of the state characterizing thepopulation (crystal size) rather than along a spatialcoordinate. It is worth noting that, to calculate the growthvelocity g(x,t), all the characteristic variables of the crystalson which this velocity may depend must be included in thedefinition of state; in the case under consideration thisvelocity depends on the size of the crystal and thesupersaturation of the solution, which is obviously not acharacteristic parameter of the crystal and can therefore becalculated with an independent material balance. The twoterms in the right-hand side of the equation [35] representthe number of crystals per unit of time which are generatedor removed by effect of a physical phenomenon (crystalaggregation or rupture processes): the term b(x,t) takesaccount of the acts of aggregation which lead to theformation of crystals of size x starting from smaller crystalsand the acts of rupture which lead to the formation ofcrystals of size x starting from larger crystals, whereas theterm d(x,t) takes account of all the events in which a crystalof size x either joins with another crystal or ruptures. It isworth noting that, as already seen for the growth velocityterm g(x,t), all the variables needed to calculate functionsb(x,t) and d(x,t) must be included in the definition of thestate of the individual. A final aspect concerns the definitionof the initial conditions for the differential equation [35].Besides the condition at the initial time which simplydefines the distribution function of any crystals present inthe crystallizer at the beginning of the process, this problemconcerns the initial condition with respect to the coordinate xcharacterizing the population.

Generally speaking, this entails setting the flow ofindividuals for a given value of this coordinate; in the caseunder examination, this flow is given naturally by thenucleation process of the crystals; using n(t) to indicate thenumber of such nuclei formed per unit of time and assumingthat their volume is zero, the corresponding initial conditiontakes the following form:

[36]

Below, two examples of the application of populationbalances to processes of interest in the polymerizationreactions sector will be discussed; in these examples, thecoordinate characterizing the population is not a continuousvariable like the size of crystals in the above example, butrather a discrete variable. For further details on the physicsunderlying the equations described below, see Chapter 6.4.

Polymeric chain length distributionAs noted above, the applicational properties of a

polymer depend on the characteristics and structure of themacromolecules of which it is formed and in particular ontheir length, defined by the number of monomer unitsforming the chain itself. Below, an attempt will be made toformulate a mathematical model able to characterize thepopulation of macromolecules in terms of their length andespecially to calculate the corresponding Chain LengthDistribution (CLD).

g t f t n t( , ) ( , ) ( )0 0 =

∂∂

+∂

∂= −f x t

tg x t f x t

xb x t d x t( , ) ( , ) ( , )( , ) ( , )

N t f x t dx( ) ( , )=∞

∫0



Consider in particular the case of radical chainpolymerization (see Chapter 6.4): since the life of radicalchains is fairly short compared to the polymerization time, itis legitimate to define an instantaneous CLD characterizingthe macromolecules produced at a specific instant of time inwhich all the process variables, such as temperature and theconcentration of monomer, are essentially constant; thenormalized numerical instantaneous chain lengthdistribution, which represents the fraction ofmacromolecules characterized by a length between x andx�dx, will be indicated with fN(x)dx. The length of themacromolecules depends on a complex process includinginitiation, propagation, chain transfer and terminationreactions; these are described in detail in Chapter 6.4 and thediscussion below will therefore be restricted to giving theirstoichiometry and the corresponding expressions of reactionvelocity, all that is needed to formulate population balances:

Rn and Pn represent the concentration of ‘living’ (radical)chains and ‘dead’ chains containing n monomer unitsrespectively; M is the concentration of the monomer, S of thechain transfer agent and I of the initiator (corresponding tothe velocity RI of radical production).

Considering a closed isothermal and uniform(well-mixed) reactor, the population balance for species Rnwhere n>1 gives the following expression:

[37]

where the first term in the second member represents theproduction of radicals of length n starting from radicals oflength (n�1) with the addition of a monomer unit. It isfollowed by various contributions of negative valuerepresenting the consumption of species Rn as an effect ofvarious possible reactions: propagation with a monomermolecule to give species Rn�1, transfer to monomer, M, or tochain transfer agent, S, and bimolecular terminations bycombination or disproportionation with any other radicalspecies present in the system; kp, kfm, kfs, kfp, ktc and ktd arethe kinetic constants of propagation, monomer transfer,chain transfer agent transfer, polymer transfer, bimoleculartermination by combination and bimolecular termination by

disproportionation. For species R1 the corresponding balancemust be written separately, since this is the only specieswhose concentration is influenced by the initiation reaction:

[38]

Considering the high reactivity of radical species, thepseudostationary state approximation can be applied tothem, so that the accumulation term in equations [37] and[38] can be ignored and these can be easily solved to givethe instantaneous CLD of the radical chains:

[39]

where R represents the total concentration of radicals and ais defined by the following sum of characteristic time ratios:

[40]

It should be noted that in each of these addends thenumerator is the characteristic time of the propagationreaction, tp, and the denominator is the characteristic time ofeach of the possible reactions which interrupt the growth ofthe chain (tfm, tfs, ttc, ttd, the characteristic times of thetransfer to monomer, transfer to chain transfer agent,termination by combination and termination bydisproportionation reactions); if one of these is dominantwith respect to the others, the corresponding characteristictime is lower than that of the others and the value ofparameter a is reduced to the ratio of tp to this characteristictime. Additionally, as parameter a increases, the CLD of theradical chains shifts to lower chain length values; this isphysically reasonable since an increase in a corresponds to aslowing of the propagation reaction with respect totermination or chain transfer reactions. [39] represents theCLD in discrete form, since the term Rn/R represents, foreach value of n, the molar fraction of the radical speciescontaining n monomer units present in the system; using theexponential approximation (1�a)�n�exp(�an), valid whena��1, the classic continuous form of this distribution isobtained:

[41]

also known as the most probable distribution or Florydistribution.

For each instant, [39] provides the CLD of the radicalchains present in the system at that moment. Using these,through the kinetics of the various termination reactions, it ispossible to calculate the velocity of the instantaneousproduction of dead chains Pn containing n monomer units:

[42]

where the various terms in the second member represent thecontribution of chain transfer to monomer and to chaintransfer agent reactions and of bimolecular termination bydisproportionation and combination reactions.

dPdt

k M k S k R R k R Rnfm fs td n tc j n j

j

n

= + +( ) + −=

−

∑1

2 1

1

f n n( ) exp= −( )α α

ττ

ττ

= +p

td

p

fmm

p

fs

p

tc

+ +ττ

ττ

α = + + + =k Rk M

kk

k Sk M

k Rk M

td

p

fm

p

fs

p

tc

p

RRn

n=+( )α

α1

k M k k Rp tc td− + +( ) nnn

R=

∞

∑

11

dRdt

R k M k S RI fm fs nn

1

2

= + +( ) −=

∞

∑

dRdt

k MR k M k M k Snp n p fm fs

= − + + +

−1

+ ( ) + +( )=

∞

=∑k mP k k Rfp mm

tc td mm1 1

∞∞

∑

Rn

initiation

propagation

I R

R

��➤�1

2 r fk I Rd I

n

= ≡

+ MM R��➤�kp

n p nr k MR

chain transfer

+ =1

tto monomer

:

R Mn

kfm+ ��➤� RR P1+ =n fm nr k MR

to trasfer agent:

R S R Pn n

kfs+ +��➤�1

rr k SRfs n

n m

ktc

=

+termination

R R ��➤� PP

R R P P

n m tc n m

n m n m

r k R Rktd

+ =

+ +

��➤�

, [ ,... ]

r k R Rn m

td n m==where 1



To calculate the CLD of the dead polymer chains presentin the reactor at the end of the process, it is necessary tointegrate [42] over time; this cannot be done analyticallysince the various terms present in the second member varyduring the process, sometimes in a complex way. It istherefore necessary to use numerical integration. Analternative making it possible to derive simple equationswhich allow for a very effective analysis of the processinvolves the derivation of instantaneous distributions; thiscan be done thanks to the peculiar characteristic of theseprocesses, in which the chains are produced in extremelyshort time intervals with respect to the duration of theprocess.

The instantaneous distribution of a given propertyrepresents the distribution of that property limited to thechains produced at a given instant of time during thepolymerization process. Specifically, the numericalinstantaneous chain length distribution, fN(n), whichrepresents the numerical fraction of polymer chains of lengthn produced at a given instant of time and is therefore definedby the equation:

[43]

can be calculated from the velocity of the production ofchains of a specific length given by [42] as follows:

[44]

By introducing appropriate dimensionless parameters,the numerical instantaneous distribution can be expressed ina particularly simple form:

[45]

where, in addition to parameter a defined by [40], two newdimensionless parameters have been introduced, also definedas ratios of characteristic times

[46]

[47]

where a�b�g. From this instantaneous distribution, using asimple accumulation process, the so-called cumulativedistribution can be calculated, corresponding to the totalnumber of polymer chains present inside the reactor, asdescribed in Chapter 6.4.

Radical segregation inside polymeric particlesThe emulsion polymerization process is one of the

most frequently used industrial polymerization processes;it is characterized by the fact that the radicals areproduced in the aqueous phase by the decomposition of aninitiator soluble in this phase, whilst polymerizationpredominantly takes place inside the polymer particles,swollen with monomer and fairly small, in the order of

tens or at most a few hundred nanometres. To describe thekinetics of these processes it is therefore necessary tocalculate the number of radicals present inside theseparticles; this number is determined by the interactionbetween the following processes involving the radicalspresent inside a particle:• Entry of a radical into a particle by diffusion from the

aqueous phase: the radicals deriving from thedecomposition of the water-soluble initiator grow in theaqueous phase by adding monomer units which arebarely soluble in the aqueous phase and therefore tend tobecome increasingly hydrophobic and diffuse inside thepolymer particles; the frequency of this process,indicated with r, is proportional to the concentration ofradicals in the aqueous phase and the external surfacearea of the particles.

• Deadsorption of one of the n radicals present in theparticle in the aqueous phase: given their hydrophobicnature, the radical chains have a low propensity todeadsorb; this process normally occurs alongside a chaintransfer to monomer reaction and is the more frequentthe higher the solubility of the monomer in water; it hasbeen shown that even when the monomers are barelysoluble this process must be taken into considerationsince it may significantly influence the distribution ofthe radicals inside particles; the frequency of thisprocess is obviously proportional to the number ofradicals present in the particle and is therefore indicatedwith kdn, where kd is the deadsorption frequency of theradicals from the particle.

• Bimolecular termination between two of the n radicalspresent inside the particle: the frequency of thisprocess is proportional to the probability of each ofthe n radicals encountering another radical n�1 and istherefore indicated by ctn(n�1), where ct is thefrequency of the bimolecular termination reactioninside the particle.Considering these three events, a population balance

equation can be written which describes the variation overtime of the number of polymer particles containing nradicals, Nn:

[48]

The first three terms in the righthand member of thisequation represent the production of particles containing nradicals by the three possible mechanisms: entry of a radicalinto particles containing n�1 radicals, Nn�1; deadsorption ofa radical from particles containing n�1 radicals, Nn�1;bimolecular termination of two radicals in particlescontaining n�2 radicals, Nn�2. The following three negativeterms, by contrast, represent the disappearance of particlescontaining n radicals due to the entry of a new radical,deadsorption and the bimolecular termination of the radicalspresent in the particle.

By applying the pseudostationary state approximationand therefore ignoring the accumulation term in equation[48], its analytical solution can be obtained, first derived byStockmayer in 1957:

[49] N N aI a

n I an m n m n

mn tot=

( )( )

− −( ) + −

−

22

1 3 2 1

1!

dNdt

N k n N c n n Nnn d n t n= + +( ) + +( ) +( ) −− + +r

1 1 21 2 1

− − − −(rN k nN c n nn d n t 1))Nn

γττ

ττ

ττ

= + + = + +k Rk M

kk

k Sk M

td

p

fm

p

fs

p

p

td

p

fm

p

fs

βττ

= =k Rk M

tc

p

p

tc

f nn

N n( ) =+( )

+ −( )+

α

α

γ αβγ β1

10.5

0.5

f n

n

nN

n

( ) =

+ −( ) +( )+( )

+ −( ) +(

γ β β γ

αγ β β γ

0.5

0.5

1

1

1 ))+( )=

∞

∑11 α

nn

f n nN ( ) = chains of length

total number of chaiins=

=

∞∑dP dt

dP dtn

nn 1



where a�(8a)1/2 and the symbol I represents the modifiedBessel function; this equation provides the desired distributionof the radicals present inside particles, Nn, in other words thenumber of particles containing a given number of radicals, n.It can be observed that this distribution is entirely determinedby two dimensionless kinetic parameters alone:

[50]

which represent the relationship between the kineticconstants of deadsorption, kd, and radical entry, r, and thekinetic constant of the bimolecular termination process, ct.The mean instantaneous number of radicals contained withinparticles,

33

n, can be obtained from distribution [49]:

[51]

The logarithm of the value of this parameter is representedas a function of parameter a in Fig. 6 for different values of thedeadsorption parameter m. It can be seen that if deadsorptionis absent (m�0), the mean number of radicals tends towards0.5 when a��1, in other words when the frequency ofbimolecular termination is extremely high with respect to thatof the entry of new radicals. In this case, 0 or 1 radicalsremain inside the particle, depending on whether the initialvalue was even or odd; this corresponds to the case ofmaximum segregation, since at every instant of time there arean equal number of particles containing 0 or 1 radicals,therefore giving a mean value of

33

n�0.5. As a increases, thenumber of radicals within particles increases rapidly, reachingthe asymptotic part of the curve in Fig. 6; this corresponds tothe case of slow terminations, in other words systemscontaining a high number of radicals per particle in which theapproximation n(n�1)�n2 can be applied. In this case, we canwrite a simple balance for the radicals inside particles:

[52]

This balance can be solved analytically, using thepseudostationary state approximation and introducing thekinetic parameters [50], giving:

[53]

High values of a (and constant m) give the asymptoticpattern n��a�2

13

illustrated in Fig. 6; under these conditions,since bimolecular termination is dominant, deadsorption hasno effect and all the curves in Fig. 6 corresponding todifferent values of parameter m fall into the same asymptoticpattern as a tends towards infinity. Deadsorption begins tohave a significant effect on the mean number of radicals perparticle only for values of a comparable to m. In fact, whenam, deadsorption is less frequent than entry (kd�r) andthe number of radicals inside particles would therefore tendto increase; however, bimolecular termination intervenes,being extremely rapid, and brings the mean number ofradicals per particle to 0.5. When a�m, by contrast,deadsorption occurs before the entry of new radicals,preventing two radicals being found in the same particle andthus bimolecular termination. This situation can again bedescribed using [52], but specifying ct�0 and obtaining:

[54]

which corresponds to the different asymptotic patternsshown in Fig. 6 for different values of m when a��1. It isobvious that under these conditions very low values of

33

n canbe obtained, which cannot be reached in the presence ofbimolecular termination alone.

8.1.6 Conclusions

This discussion has shown the enormous potential ofmodelling, which takes advantage both of very large, veryfast computers and advanced and sophisticatedmethodologies of numerical calculation. In their modern

nm

= α

n m m= − ± +2 8

4

α

dndt

c n k nt d= − −r 2 2

nnN

Nnn

nn

= =

∞

=

∞

∑∑

0

0

mkc cd

t t

= =, α r



log n–

m�

10�

6

m�

10�

5

m�

10�

4

m�

10�

3

m�

10�

2

m�

10�

1

m�

1

m�0

m�10

m�

102

1

0

�3

�2

�1

log a�8 �6 �4 �2 0 2

Fig. 6. Logarithmic diagram of the mean number of radicals,

33

n,as a function of the kinetic parameters a and m.

form, models increasingly provide valuable fundamentalphysical content in terms of concepts and data, thanks towhich the results obtained are more reliable than ever before;this is true not only of the description of known facts butalso of predictions regarding the behaviour of physical andchemical systems which are difficult or impossible tomeasure or observe directly.

The potential for implementing models with equationscapable of describing with increasing effectiveness thedetails of the processes under consideration is a result bothof the aforementioned advances in computer technology andthe availability of accurate information on the values of theparameters needed to formulate the equations on which themodel itself is based: as such, the use of lumped magnitudes,or ‘overall’ magnitudes comprising several specificparameters, is becoming increasingly infrequent. It should benoted that this procedure, although sometimes effective,unfortunately cannot entirely avoid some degree ofinaccuracy, or even ambiguity.

The potential for implementation is due on the one handto an improvement in experimental techniques and on theother to the development of computational chemistry andphysics, thanks to which it is now possible to calculate, witha precision sometimes comparable with experimentalmethods, chemical equilibrium constants, the partitioncoefficients of components between different phases, thecoefficients of transportation processes on a molecular scale,bond energies, kinetic parameters and so forth. It is nowpossible to predict with satisfactory results equilibriumdiagrams of the phases of multicomponent systems or todescribe kinetic processes involving tens of species,including unstable radicals, which interact in hundreds ofreactions as is the case, for example, in combustionprocesses. Despite these success stories, however, somedifficult problems remain open; these include, for example,the interaction between turbulent fluctuations and rapidchemical transformations.

Computer modelling is now used with acceptableaccuracy in various technological sectors, and especially inthe oil industry (simulation of the units involved in refiningprocesses, geophysical exploration aimed at identifying thepresence of new oil reservoirs, etc.). Furthermore, once anatural hydrocarbon reservoir has been identified, the exitingflows of oil, water, gas and any chemical substances injectedinto it must be modelled; these are difficult problems,concerning the flow of multiphase fluid mixtures throughporous beds, which must be tackled by solving coupledsystems of non-linear differential equations subject tocomplex boundary conditions, that represent the materialand energy balances of the systems under examination.

In accordance with a proposal formulated by Prater itmay be useful to assess the value of a model on the basis of aparameter known as fundamentality, expressed by the ratiobetween the number of physical and chemical laws involvedin it and the number of adjustable parameters it contains: asshown in Fig. 7, both the utility and the cost of the modeldepend on fundamentality. The former increases following acurve which initially rises rapidly but then tends towards anasymptotic value; the latter follows a curve which initiallypresents low values and then rises rapidly as the value offundamentality becomes increasingly high. The net value ofa model is represented by a curve which takes into accountthe pattern of the preceding curves and is characterized by a

maximum, since for low values of fundamentality utilityprevails, whereas for high values the net value falls due tothe significant rise in cost. Obviously, this interpretationshould be applied with caution since the intrinsic value of amodel actually depends on its nature and the context towhich it is applied; additionally, the specific pattern of thecurves mentioned evolves over time and may depend onvarious incidental circumstances.

To conclude, we cannot fail to note that the use ofmodels is taking on a significant role in resolving problemswhich go beyond the design of industrial appliances,because they involve global decisions regarding the socialand economic impact of human activities. Particularlyworthy of mention is the problem of determining thelimitations on the use of fossil energy sources, especiallyoil, given the restrictions imposed by the presence of anacceptable level of carbon dioxide in the atmosphere: inoperative terms, this problem concerns the calculation ofclimate sensitivity, which expresses the increase in thetemperature of the planet caused by a doubling in theamount of carbon dioxide with respect to current values(370 ppm). It should be observed that in the first half of theNineteenth century this quantity amounted to 280 ppm andthat the increase to current levels has been associated witha rise in temperature of about 0.8°C. The problem is solvedusing global climate models which describe the evolutionover time of the magnitudes characterizing the physical andchemical state of the planet; to be objectively meaningful,these models must take into consideration variousphenomena including atmospheric dynamics, energybalances and the numerous chemical transformationsinvolving the large variety of components present in theatmosphere itself, largely due to human intervention. Theenvironment can in fact be compared to a giganticchemical reactor into which chemical substances producedby industrial or natural activities are fed.



utility

cost

netvalue

0

Fig.7. Utility, cost and net value of a model as a function of fundamentality (number of phenomenological laws/number of adjustable parameters).

Numerous models exist which simulate the behaviourof the atmosphere. These models are developed andapplied at various research centres and, althoughcomparable in terms of the equations used, have provideddifferent results, due mainly to the uncertaintiesconcerning the degree of detail with which the‘geography’ of the Earth’s surface, the role of cloudformation processes, volcanic eruptions, and so forth arerepresented. These problems are dealt with usingadjustable parameters, whose uncertainties, however,have repercussions on the answers obtained: to give justone example, in a comparison between the mostaccredited models conducted by the IPCC(Intergovernmental Panel on Climate Change) in 1995, itemerged that climate sensitivity ranged from 1.5 to4.5°C; this result was too vague to justify decisions whichwould have a significant impact on the parametersconditioning economic development. A recent studyconducted at the Hadley Center for Climate Predictions(United Kingdom), has compared the behaviour of thevarious models using a perturbative criterion aimed athighlighting the importance of the various parametersinvolved; additionally, climate sensitivity was evaluatedon the basis of the results obtained using 53 models,weighting each model depending its capacity to predictknown situations, on the basis of the values assigned tothe adjustable parameters. This analysis made it possibleto restrict uncertainty to values ranging from 2.5 to 3°C:this outcome, which confers greater reliability on studiesof climate change, led to the emergence of a series offurther problems concerning the technical and economicchoices able to ensure a development compatible with astable and acceptable level of carbon dioxide. These areproblems which are inherent to modelling, which thustakes on a central role in the formulation of decisionsaffecting the future of the planet.

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List of symbols

C molar concentration C* dimensionless concentration Cp specific heatCs surface molar concentration D diffusion coefficientDs surface diffusion coefficient f (x) probability density function for variable xfN(n) instantaneous numerical chain length distributionEn energy of a vortexE� activation energyf(x,t) distribution functiong growth velocity (of crystals)h heat exchange coefficientDH heat of reactionI modified Bessel functionk reaction velocity constantkB Boltzmann’s constantl size of a vortexL reference length M mass (of gas from an explosion)n number of radicals inside particles33n mean number of radicals inside particlesNn number of particles containing n radicals per unit

volumeN(t) number of individuals (for example crystals)Ntot total number of particles per unit volumep pressurep* dimensionless pressurePn concentration of dead chains containing n monomer

unitsQ volumetric flow rateQ(V,T) configuration integralr reaction velocity rs catalytic reaction velocity



dr volume elementR ideal gas constantRI velocity of radical productionRn concentration of radicals containing n monomer

units Re Reynolds numberS exchange surfacet timet* dimensionless timeT absolute temperatureT* dimensionless temperatureu local velocity vectoru* dimensionless velocity vectror u11, u°11 intermolecular potential parametersu22, u°22 intermolecular potential parametersV volumex x-axisx* dimensionless x-axisXi dimensional magnitudey y-axis

Greek lettersb dimensionless kinetic parameter,

tp12

ttcg�Cp�Cv ratio of specific heats at constant pressure and

volume d thickness of boundary layerm viscosityr densityn1�m/r kinematic viscosityP dimensionless groupt dimensionless time

Sergio Carrà

Dipartimento di Chimica, Materiali eIngegneria chimica ‘Giulio Natta’

Politecnico di MilanoMilano, Italy

Massimo Morbidelli

Institut für Chemie- und BioingenieurwissenschaftenEidgenössische Technische Hochschule-Hönggerberg

Zürich, Switzerland



8.1 mathematical modelling and systems analysis

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