dimensional analysis scaling

chemical engineering research and design 8 6 ( 2 0 0 8 ) 835–868

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Review

On dimensionless numbers

M.C. Ruzicka ∗

Department of Multiphase Reactors, Institute of Chemical Process Fundamentals, Czech Academy of Sciences,Rozvojova 135, 16502 Prague, Czech Republic

This contribution is dedicated to Kamil Admiral Wichterle, a professor of chemical engineering, who admitted to feel a bit lost in the

jungle of the dimensionless numbers, in our seminar at “Za Plıhalovic ohradou”

a b s t r a c t

The goal is to provide a little review on dimensionless numbers, commonly encountered in chemical engineering.

Both their sources are considered: dimensional analysis and scaling of governing equations with boundary con-

ditions. The numbers produced by scaling of equation are presented for transport of momentum, heat and mass.

Momentum transport is considered in both single-phase and multi-phase flows. The numbers obtained are assigned

the physical meaning, and their mutual relations are highlighted. Certain drawbacks of building correlations based

on dimensionless numbers are pointed out.

© 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Keywords: Dimensionless numbers; Dimensional analysis; Scaling of equations; Scaling of boundary conditions;

Single-phase flow; Multi-phase flow; Correlations

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8362. Two sources of dimensionless numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836

2.1. Source one—dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8362.2. Source two—scaling of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837

3. Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8383.1. How DA works. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8383.2. Comments on DA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839

3.2.1. Choice of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8393.2.2. Variables with independent dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8403.2.3. Similarity and modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8403.2.4. Neglecting variables in DA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8413.2.5. Limits of DA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8413.2.6. DA versus SE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841

4. Scaling of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8415. Transport of momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841

5.1. Mass equation of fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.2. Momentum equation of fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.3. Energy equation of fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8445.4. Boundary conditions: no slip and free-slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845

5.4.1. Normal component of free-slip BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8455.4.2. Tangential component of free-slip BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845

∗ Tel.: +420 220 390 299; fax: +420 220 920 661.E-mail address: [email protected] 19 June 2007; Accepted 2 March 2008

0263-8762/$ – see front matter © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.cherd.2008.03.007

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836 chemical engineering research and design 8 6 ( 2 0 0 8 ) 835–868

5.5. Multi-phase flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8475.5.1. Microscale description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8475.5.2. Mesoscale description (Euler/Lagrange). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8485.5.3. Macroscale description (Euler/Euler) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8495.5.4. Retention time distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 850

6. Transport of heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8527. Transport of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8538. Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8549. Remark on literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85710. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859Appendix A. Concept of intermediate asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859

A.1. Motivation example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859A.2. Two kinds of similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859

A.2.1. Complete similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859A.2.2. Incomplete similarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 860

A.3. Two kinds of self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 860A.4. Relation between DA and IA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 861A.5. Beyond IA? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 861A.6. Broader horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862

Appendix B. Suggestions for using and teaching DA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863Appendix C. New areas in chemical engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864

C.1. Microreactors and microfluidics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864C.1.1. Microsystems in chemical technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864C.1.2. Prevailing forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864C.1.3. Governing equations and boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864

C.2. Biosystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865. . . . .
C.3. Multiscale methodology . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction

I have always been puzzled with the plethora of dimensionlessnumbers (DN) occurring in the various branches of chemi-cal engineering. From sincere discussions with my peers aswell as with students an impression has arose that I am notthe only puzzled person in this field. Consequently, the mainmotivation of this contribution is to try to briefly review DNcommonly encountered at the transport of momentum, heat,and mass, as for their origin, physical meaning, interrelationand relevance for making correlations. The selection of DNis neither objective nor exhaustive, being biased by workingmainly in the area of the multi-phase hydrodynamics.

The dimensionless (nondimensional) numbers (criteria,groups, products, quantities, ratios, terms) posses the follow-ing features. They are algebraic expressions, namely fractions,where in both the numerator and denominator are powers ofphysical quantities with the total physical dimension equal tounity. For example, the Reynolds number, Re = LV/�, has dimen-sion [1], also denoted as [-].

The dimensionless numbers are useful for several reasons.They reduce the number of variables needed for descrip-tion of the problem. They can thus be used for reducingthe amount of experimental data and at making correla-tions. They simplify the governing equations, both by makingthem dimensionless and by neglecting ‘small’ terms withrespect ‘large’ terms. They produce valuable scale estimates,whence order-of-magnitude estimates, of important physicalquantities. When properly formed, they have clear physicalinterpretation and thus contribute to physical understandingof the phenomenon under study. Also, choosing the relevant
scales, they indicate the dominant processes. There are twomain sources of DN: dimensional analysis and scaling of gov-erning equations.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865

2. Two sources of dimensionless numbers(DN)

2.1. Source one—dimensional analysis (DA)

A general way how to formally describe the surrounding worldconsists of several steps. First, for the thing under study(‘system’), define all possible qualities of interest. Second,select those qualities that can be quantified, i.e. their amountin the thing can be expressed by numbers, or some othermathematical constructs. Call these measurables the phys-ical quantities. Choose the etalons (measuring sticks, units)to measure each of them. Each physical quantity has fourattributes: name, notation, defining relation, and physical unitthat determine its physical dimension (unit and dimension areoften used interchangeably). The first two are only our labels,the second two are physically substantial. There are sevenbasic physical quantities (length, mass, time, electrical current,thermodynamic temperature, luminous intensity, amount ofsubstance). They are measured by seven basic units of the SIsystem of units (meter, kilogram, second, ampere, Kelvin, can-dela, mole), which is canonical nowadays. All other quantitiesare called derived quantities and are composed of the sevenbasics. Depending on the research area, usually only few basicquantities are used. In mechanics, we have three (length L,mass M, time T), plus one (temperature �) for thermal effects,if these are considered.

Having the physical quantities, we want to find the rela-tions among them. We either have the governing equations(physical laws) or not. The very basic laws (axioms of nature)cannot be derived: they must be disclosed or discovered. There
is nowhere they could be derived from: they are already here,existing silently, demonstrating themselves through a varietyof diverse or even disparate effects. Lacking the knowledge

chemical engineering research and design 8 6 ( 2 0 0 8 ) 835–868 837

Nomenclature1

a variable in DA; acceleration (m/s2)b variable in DAc concentration (kg/m3)cp heat capacity (J/kg K)C coefficient (drag, added mass); concentration

scale (kg/m3)d length, size, particle diameter (m)D diameter (m); mass diffusivity (m2/s)Dt,ax turbulent/axial dispersion (m2/s)e voidage, volume fraction of dispersed phase (-)E bulk modulus of elasticity (=1/K) (Pa); energy (J)f function symbol (f)F force (N); function symbol; factorg gravity (m/s2)g′ reduced gravity (=(��/�)g) (m/s2)h height, depth (m)j flux, of heat (J/m2 s), of mass (kg/m2 s)k transfer coefficient (J/m2 s K); rate constant (e.g.

s−1)K compressibility (Pa−1)l length (m)L length scale (m)m mass (kg)M mass scale (kg)p pressure (Pa)P pressure scale (Pa); period of oscillation (s)Q flow (m3/s)r radius, position (m); reaction rate (kg/m3 s)R reaction rate scale (kg/m3 s)S (cross-section) area (m2)t time (s)T time scale (s)u velocity (dispersed phase) (m/s); master quan-

tity (u)v velocity (continuous phase) (m/s)V velocity scale, mean speed (m/s)x coordinate, distance, position (m)z coordinate, distance, position (m)

Greek letters˛ thermal expansivity (K−1)ˇ concentrational expansivity (kg−1)� shear rate (s−1)� difference, variation� temperature (K)� temperature scale (K)� heat diffusivity (/�cp) (m2/s) heat conductivity (J/m s K)C capillary length (m) dynamic viscosity (fluid) (Pa s)� kinematic viscosity (fluid), momentum diffu-

sivity (m2/s)˘ i Pi-term� density (fluid) (kg/m3)� interfacial tension (N/m, J/m2)� time (s); stress (N/m2)ϕ angle˚ function symbol (˚)ω frequency (s−1 or 2�/s); vorticity (s−1)

˝ frequency, its scale (s−1); vorticity scale (s−1);flow domain with boundary ∂˝

� Nabla operator (m−1)

OthersBC boundary condition (s)DA dimensional analysisDN dimensionless number (s)Hyb unit of momentum (SI) (kg m/s) = (N s)IC initial condition (s)N numberO(1) order of unitySE scaling of equations1D one-dimensional[] physical dimension* dimensionless, basic DN

Subscriptsa added massb bulkf fluid (continuous phase); flowmix mixturep particle (dispersed phase)r relaxations, S surface, interfacew wall, interface
0 reference value
about the basic laws, we must try to find them using the otheravailable methods. One such a method is DA.

Dimensional analysis consists of three steps. First, make alist of relevant variables, the physical quantities that describethe system. Second, convert these dimensional quantitiesinto DN. Third, find a physically sound relationship (scale-estimate) of these DN without help of any governing equations(physical laws). The main problem is to make the list of the rel-evant variables that is complete and independent. Here thescience meets the art: the choice of the variables is highlysubjective, beyond any rigour. The other two steps are simplein principle owing to the fact that DA is a rigorous math-ematical method operating precisely on the lists of chosenvariables. DA relies on several assumptions that are neededfor the mathematical proof of its very core, the Pi-theorem.The assumptions are the following. The physical equations aredimensionally homogeneous. The physical equations hold fordifferent systems of units. The dimensions of physical quan-tities have form of power-law monomials (dimensions like[sin(L) − log(T) + eM] are not allowed). There are quantities withindependent dimensions and their list is complete. Besidethese, we tacitly presume: the problem is amenable to DA.

2.2. Source two—scaling of equations (SE)

The scaling of equations (SE) means nondimensionalization ofthe equations describing the system under study (equations
of motion, fundamental equations, governing equations, etc.).It is a technically simple and transparent procedure, which
1 Since many symbols appear in the text, only those of generaluse are listed here. At multiple meaning, the context talks. Thoseof the local meaning, apply usually within one paragraph only, areomitted.

and
838 chemical engineering research
yields the dimensionless equations and the list of relevant DN.It does not give the relation among these DN.

The dimensionless equations have certain advantages.They are independent of the system of units. The dimen-sionless numbers are relevant for the problem. The proportionbetween individual terms can be seen. These equations applyto all physically similar systems, so they are useful for scale-up/down.

The process of scaling proceeds as follows. For instance,take the equation of linear mass-spring oscillator:

m

(d2x

dt2

)= −kx [kg m/s2], (2.2.1)

where all the four quantities (x, t, m, and k) are dimensional.Separate them into two classes: parameters (m, k) and variables(x, t). Choose the scales (characteristic values) for the variables,length scale L and time scale T. Make substitution x → Lx*,t → Tt* in (2.2.1) to obtain

(mL

T2

)(d2x∗

dt∗2

)= −(kL)x∗ [kg L/T2]. (2.2.2)

Although (2.2.2) has dimension, the quantities are sepa-rated into dimensionless variables (x*, t*) and dimensionalparameter groups (mL/T2, kL). Dividing (2.2.2) by any parametergroup yields the dimensionless equation. Dividing by (kL):

(m

kT2

)(d2x∗

dt∗2

)= − x∗ [-] (2.2.3)

produces one DN, namely the number N = (m/kT2), the propor-tion (inertia)/(elasticity).

Generally, DN show the proportion between the individualterms in an equation correctly only when the dimensionlessvariables (*) are scaled so well to be of order of unity ∼O(1).Then the magnitude of the terms is solely represented by theparameter groups. With the above pendulum, at the choiceL ∼ amplitude and T ∼ period P, we have x* ∈ 〈−1, 1〉 and t* ∈ 〈0,1〉 (one swing), which both are O(1). Consequently, N shows the(inertia)/(elasticity) proportion correctly.

For an equilibrium motion like oscillations, wherethe both counter-acting forces are somehow balanced,one would expect that their ratio should be unity, i.e.N = (inertia)/(elasticity) = 1. However, this is generally not true.The particular value of N depends on the choice of scales. Realiz-ing that (k/m)1/2 is the oscillator angular frequency ω, which isdefined by ω = (2�)/P, then N = (P/2�T)2. For the particular choiceof the time scale T = P, we have N = (P/2�P)2 = 1/(2�)2 ≈ 0.025.Thus, the actual force ratio is N ≈ 0.025:1 = 1/40, far from unity.The reason is that DN is a very rough estimate of the effects itcompares. For instance, at the laminar–turbulent flow regimetransition in pipes, the ratio Re = (inertia)/(viscosity) is notexactly 1, but ∼103. It is therefore better to use a vaguelanguage and say ‘low Re’ and ‘high Re’, upon strong under-standing that everybody knows what it does mean.

Conclude that the diversity of the original four-dimensional problem (2.2.1) described by four quantities
(x, t, m, k) is reduced to a single number (m/kT2) at the price oflacking all details that are below the resolution of the scaleconsiderations.
design 8 6 ( 2 0 0 8 ) 835–868

3. Dimensional analysis (DA)

3.1. How DA works

Choose one physically dependent variable a and choose fur-ther (k + m) independent variables, ai and bi, on which wepresume a depends. ai have mutually independent dimen-sions, of which the dimensions of all remaining variables aand bi can be obtained by combination. We want to find theunknown physical law, the function f, we presume it doesexist:

a = f (a1, a2, . . . , ak; b1, b2, . . . , bm) [a] = [f ]. (3.1.1)

To reproduce the dimensions of a and bi, we combine ai inform of power monomials:

[a] = [a1]p01 · [a2]p02 · [a3]p03· · ·[ak]p0k· · · ≡ A [a], (3.1.2)

[bi] = [a1]pi1 · [a2]pi2 · [a3]pi3· · ·[ak]pik· · · ≡ Bi [bi], (3.1.3)

where the exponents pij are found for each row by compar-ing the dimensions on both sides, based on the dimensionalhomogeneity of physical equations. Dividing a and bi by thecorresponding composites of the same dimension, denotedfor brevity as A and Bi, and rewriting (3.1.1) in dimensionlessform for another unknown function ˚, we get

a

A= ˚

(b1

B1,

b2

B2,

b3

B3, . . . ,

bm

Bm

)[-]. (3.1.4)

This equation is usually written in the following notationas the similarity law for ˘:

˘ = ˚(˘1, ˘2, . . . , ˘m) [-], (3.1.5)

where the dimensionless terms (Pi-terms, similarity param-eters) are ˘ ≡ a/A, ˘ i ≡ bi/Bi. The famous BuckinghamPi-theorem says: It is possible to get from (3.1.1) to (3.1.5).

The main gain is the reduction of the number of variablesfrom (k + m) in (3.1.1) to only (m) in (3.1.5). All k variables ai

are hidden in the denominators of ˘ and ˘ i. In mechanics,we have only three basic dimensions, L, M, T, so that k ≤ 3.Another advantage is that (3.1.5) is dimensionless. Accordingto experimental convenience, any quantity involved can beused to change the value of ˘ i. DA merely transforms f into˚. Rewriting (3.1.5) in the dimensional form, as the similaritylaw for a:

a = A · ˚(˘1, ˘2, . . . , ˘m) [a]. (3.1.6)

We see a certain progress as compared to (3.1.1): f is writtenas a product of two things, f = A·˚. The first one is the knowndimensional function A that contains the rough essence of f.It is called the scale estimate (basic scaling, scaling law) for a,and we write

a ∼ A = ap011 · a

p022 · a

p033 · · ·ap0k

k[a]. (3.1.7)

The second one is the unknown dimensionless function˚ that is the ‘fine tuning’ of the scale estimate, to convert
(3.1.7) into the equality a = A·˚. Finding ˚ does not belong tothe frame of DA; this must be done by some other means (e.g.experimentally or numerically).

nd design 8 6 ( 2 0 0 8 ) 835–868 839

vd(

F

(F

F

ws

Rd

pfasv

3

3IciitgwmnrN

apt

P

ugPnPtdtabis

Fig. 1 – Definition sketch. Pendulum of mass m and hangerlength l swings under gravity g in a medium of viscosity �

(or �).

Table 1 – Application of DA for finding period ofpendulum, P = 2�(l/g)1/2

Variables Features Action Output

P(l, g) Proper choice Work Correct resultP(l) Missing variable (g) Fail ContradictionP(g) Missing variable (l) Fail ContradictionP(l, m) Incorrect substitute (m) Fail ContradictionP(l, ) Incorrect substitute () Fail ContradictionP(l, �) Incorrect substitute (�) Work Wrong resultP(l, g, m) Extra variable (m) Work EliminationP(l, g, ) Extra variable () Work EliminationP(l, g, �) Extra variable (�) Fail Insolubility

P(l, g, d) Extra variable (d)P(L1, g(L1), L1) Uniscale L1 Fail InsolubilityP(L1, g(L1), L2) Two scales L1,2 Work EliminationP(L1, g(L2), L1) Two scales L1,2 Fail ContradictionP(L1, g(L2), L2) Two scales L1,2 Fail Contradiction

chemical engineering research a

As an example, consider a flow in a pipe. The dependentariable is the drag force F. The independent variables are pipeimensions (D, L), fluid properties (�, ), and speed (V). Eq.

3.1.1) now takes form

= f (D, L, �, , V) [N]. (3.1.8)

Variables with independent dimensions are D (length), Vtime), � (mass). Variables with dependent dimensions are L,, . Eq. (3.1.4) applied to this example becomes

F

�V2D2= ˚

(L

D,

�DV

)[-]. (3.1.9)

Consequently, Eq. (3.1.7) for the basic scaling becomes

∼ �V2D2 [N], (3.1.10)

hich is the dynamic fluid pressure (�V2) times the cross-ection area ∼(D2). The correcting dimensionless term ˚(L/D,/�DV) in (3.1.9) is the formula for the friction coefficient C(L/D,e), which remains for experiments. DA thus produces a greateal of the total solution of the whole problem.

When no exceptional variable a labelled as ‘physically inde-endent’ is or can be explicated, (3.1.1) is simply written as= 0, where the variable a becomes bm+1. This notation is suit-ble when it is not clear what is the master quantity, or, wheneveral of them can play this role, depending on our angle ofiew at the problem.

.2. Comments on DA

.2.1. Choice of variablest is the main problem of DA, since there is no rigorous pro-edure for it. The quantities must be physically relevant andndependent, and their list must be complete. The choices highly subjective and needs profound understanding ofhe problem, experience with usage of DA, intuition, andood luck. As a guideline, there are recommendations ofhat should be taken into account (e.g. system geometry,aterial properties, kinematic and dynamic aspects, exter-

al conditions, etc.). Very helpful are the governing equationselated to the problem. Even when we cannot solve them (e.g.avier–Stokes equation), they indicate the relevant quantities.

A simple example demonstrates how the choice of vari-bles affects the output of DA. Consider the mathematicalendulum in Fig. 1. The period P depends on two variables,he length l and gravity g:

(l, g) = 2�

(l

g

)1/2

[s]. (3.2.1)

Suppose we do not know it, and we try to find it by DA. Lets try different choices of variables. At the proper choice, P(l,), we get the correct result, the basic scaling for the period,∼ (l/g)1/2. Here, the correcting function ˚ in (3.1.6) containso argument and equals 2�. With lesser variables than is due,(l) and P(g), the DA fails. The type of failure is the logical con-radiction of kind 1 = 0, since it is impossible to make up timeimension of P (s) from the length l (m) only. When we tryo substitute for the missing correct variable (g) another vari-ble which is not relevant, the following may happen. DA fails
y a contradiction, with P(l, m), P(l, ). DA works but gives anncorrect result, P ∼ l2/�, with P(l, �), which is the worst case,ince there is no indication that things go wrong. With more
variables than necessary, DA either recovers the correct resultby eliminating the extras, P(l, g, m) and P(l, g, ), or, DA failsby insolubility, i.e. having more unknowns than equations, P(l,g, �). A subtle point is considering several quantities of thesame dimension, P(l, d, g). Using the same length scale L forboth l and d, DA fails by insolubility. Using one scale L1 for thecorrect variables (l, g) and another scale L2 for the extra vari-ables (d), DA works and eliminates the extras. However, usingL1 for one part of the correct variables (l) plus some extras, andanother scale L2 for the second part of the correct variables(g) plus some extras, DA fails by contradiction. The result issummarized in Table 1.

To sum up, DA either works or fails. When it works, it giveseither good or bad result. When it gives good result, eitherthe choice of variables is correct or the extra variables areeliminated. When it fails, it is either by logical contradiction(dimension of l.h.s. cannot be made up of dimensions of r.h.s.)or by insolubility (extra variables bring more equations but notnew dimensions). This simple example is purely demonstra-tive; not a general statement proven for all possible situations.

First column: variables chosen for pendulum period P. Second col-umn: features of our choice. Third column: what DA does? Fourthcolumn: note on result. Definition sketch in Fig. 1.

and
3.2.2. Variables with independent dimensionsOf the physically independent variables of f in (3.1.1), we mustchoose those with independent dimensions (ai), and leavethe rest for bi. The task reduces to finding k linearly inde-pendent vectors in Rk. For instance, the three mechanicaldimensions generate 3D space (L × M × T) with the base vec-tors {1, 0, 0}, {0, 1, 0}, {0, 0, 1}. All mechanical quantitiesare represented by vectors in this space: speed has coordi-nates {1, 0, −1}, density has {0, 1, −3}, force has {−1, 1,−2}, etc. We can choose any three non-coplanar vectors forai, but the simpler the better. For different choices of ai, DAproduces different DN, but the resulting physical informa-tion is identical. Based on the dimensional homogeneity of(3.1.1), the powers pij in (3.1.2) and (3.1.3) are determinedby the standard routine of solving a set of linear algebraicequations.

3.2.3. Similarity and modellingOne paradigm says: If systems are similar, DA gives samedescription. This statement forms the basis for the the-ory of similarity (similitude) and modelling (scale-up/down).The physical similarity consists in correspondence in geom-etry, kinematics, dynamics, etc. One may ask: Can dissimilarsystems have same description? Consider four physically dif-ferent systems shown in Fig. 2. Despite their difference, theyall share the same description, the formula for drag force givenby DA in (3.1.9). Consequently, different systems can have thesame description within the framework of dimensional con-siderations. DA is ambiguous with respect to physical kindof systems, it cannot see the physical difference. DA is notambiguous with respect to manipulation with symbols repre-senting the input variables, owing to the unicity theorem forthe solution of linear algebraic systems.

Modelling usually means finding a description of a smallmodel system on a laboratory scale by DA (model M), where itis easy to do measurements, and, to transfer the result on to a

Fig. 2 – Ambiguity of DA. Four different flow situations with idenwall roughness L. (Case B) Finite pipe of diameter D and length Llength L oscillating in a cylindrical orifice of diameter D. Fluid ha

design 8 6 ( 2 0 0 8 ) 835–868

similar but bigger system (prototype P), where it is difficult toexperiment. Without similarity, the model and prototype aredescribed by two different relations:

˘M = ˚M(˘Mi

) (model),

˘P = ˚P(˘Pi

) (prototype).(3.2.2)

With similarity, the following holds:

˘M = ˘P (similarity law for modelling),

˘Mi = ˘P

i ∀i (similarity criteria), ˚M = ˚P. (3.2.3)

For instance, DA gives the scale estimate (3.2.1) forpendulum period P ∼ (l/g)1/2. Applying the similarity law,PM/(lM/g)1/2 = PP/(lP/g)1/2, gives the relation between the peri-ods of big and small pendula: PP/PM = (lP/lM)1/2 (scaling rule).Increasing pendulum length 25 times gives only 5 times longerperiod, in virtue of (lP/lM)1/2 (scaling coefficient). Knowing thismay be helpful for designers of big clocks.

The similarity criteria may not always be met. With asimple pendulum, the tuning function ˚ in (3.1.6) has noarguments ˘ i, and is constant, ˚ = 2�, see (3.2.1). With morecomplex systems, there can be several Pi-terms (e.g. two in(3.1.9)), whose model-prototype equality required by (3.2.3)may be difficult to guarantee. The demands of the similaritycriteria for different ˘ i may not be fully compatible. Modelswith these contradictions are called ‘distorted’, in contrastwith the ‘true’ model, where we can satisfy all demands. Forinstance, in hydraulic engineering, we have modelling basedon two Pi-terms, namely the Froude and Reynolds numbers,˘1 = Re, ˘2 = Fr. Their equality (FrM = FrP, ReM = ReP) implies asevere requirement on the kinematic viscosity of the model
and real fluids, �P/�M = (LP/LM)3/2. Considering the great dis-parity in size of hydraulic models LM and real water works LP,it is difficult to find suitable fluids. With LP/LM = 102 we need
tical description. (Case A) Infinite pipe of diameter D and. (Case C) Finite plate of size D × L. (Case D) Liquid piston ofs density �, viscosity � and speed V.

nd de

�

a

3TabiabdcDsffnvniwsi

3IiocsiitpadtmoottcaepindlasA

3ADcietta∼


P/�M = 103. The phenomenon of distortion is often qualifieds a drawback of DA.

.2.4. Neglecting variables in DAhere are two reasons for neglecting some variables that canpparently play a role in the description of the problem. First,ased on our subjective choice, we do not want some variables

n the model. Our experience says, that they may be problem-tic for the smooth operation of DA. Second, we want them,ut the corresponding ND are either small or large. The tra-ition says that these variables are irrelevant, which is ratherounter-intuitive. The rationale is, however, the following. TheN usually are ratios of two effects. If this ratio is either toomall or too big, one of the effects is simply negligible. There-ore, it can be neglected as a category, and thus excluded fromurther considerations. For instance, consider the Reynoldsumber. When it is too small or too large, we have either theiscous flow or the potential flow, and this parameter enterseither of these two limiting theories. When we neglect the

nertia effects completely, there is no sense to compare themith the viscous effects, and vice versa. The act of omitting

ome variables looks rather subjective, but there are rules fort, see Appendix A.

.2.5. Limits of DAt was believed that DA is a universal tool whose potentials limited only by the skill of the user. However, like anyther mechanism, DA too can operate only under certainircumstances: there are physical problems that cannot beolved by DA, in principle. As a brief guideline, few contra-ndications to application of DA are the following. The problemnvolves information about the initial and boundary condi-ions: the system behaviour in the initial times, details ofrocess generation, its behaviour near the system bound-ries, decay via equilibration, energy dispersion or dissipationuring the process evolution. There are variables related tohe presence of the internal sources/sinks of mass, heat,

omentum, energy, in the environment in which the processccurs. Some global conservation characteristics (integrals)f the system are not constant during solution but vary dueo the sources. Parameters related to variable properties ofhe (micro)structure/texture of the medium carrying the pro-ess are present. The independent variables are involved in

complicate way (in exponent, in argument of function,tc.). Regime transitions occur during the solution of theroblem (some variables lose relevance and new ones come

nto play). All these are potentially bad variables, we mayot want. Consequently, more sophisticated tools must beeveloped and employed to cope successfully with these prob-

ems. Fortunately, there is one, the theory of intermediatesymptotics, which can be considered as a generic exten-ion of DA. The basic idea of this concept is presented inppendix A.

.2.6. DA versus SEs for the subjectivity of choices, the comparison betweenA and SE goes in favour of the latter. With DA, one musthoose all the quantities, whose relevance and completenesss not guaranteed. With SE, these quantities are given by thequations (plus initial and boundary conditions). In equa-ions, we only must choose the variables to be scaled, and
he parameters to be left intact. Also, we must choose suit-ble characteristic scales to get the dimensionless variablesO(1), to see the proportion between different terms. Both are
sign 8 6 ( 2 0 0 8 ) 835–868 841

basically doable, when little care is taken. As for the output,DA can reproduce all the numbers obtained by SE, if we feedit with the proper variables. On the other hand, DA can gen-erate numbers that cannot be obtained by SE, whose physicalrelevance may be difficult to assess. In addition, DA gives thescale-estimate of the unknown relation between the quanti-ties. It is convenient to apply SE first, to get the proper listof DN, and then use DA to find the scaling relations betweenthem.

4. Scaling of equations (SE)

The motivation for SE and the procedure were briefly intro-duced in Section 2.2. In sake of simplicity, a brief notationis used further. The starred dimensionless O(1)-variables areomitted, and only the parameters and scales are retained inthe ‘scale equations’. These are not the ‘true’ equations, butrelations that indicate the relative proportions between theindividual terms. As an example, we recover (3.2.1) by SE. Thegoverning equation is the conservation of angular momen-tum, Fig. 1:

(ml)x + (mg)x = 0 [N m], (4.1)

where the mass cancels:

(l)x + (g)x = 0 [m2/s2]. (4.2)

Choose the variables (x, t) and parameters (l, g). Scaling ofvariables by general scales L and T gives

lL

T2+ gL = 0 (‘scale equation’) [L2/T2]. (4.3)

Choosing the particular scales, L = l and T = P, which are rel-evant and make x* and t* ∼ O(1), indeed recovers P ∼ (l/g)1/2.Here, relation (4.3), l2/P2 + gl = 0, means l2/P2 ∼ gl, so thatP ∼ (l/g)1/2.

Further, in sake of simplicity, the complicated issuesrelated to the presence of multiple scales and directional scal-ing are mostly omitted (except for an example in Section 8).In the same system, different processes can take place, eachhaving distinct scales of length, time, speed, etc. Different pro-cesses can dominate along different spatial directions, havingdifferent scales. The flow domain itself can be highly ani-sometric, having different dimension in different directions.We face multiplicity of relevant scales, and their directionaldependence. For instance, in a boundary layer different pro-cesses ‘along’ and ‘across’ can be identified. A body movesalong the vertical and its wake grows in the horizontal direc-tion. Likewise, a spiralling bubbles rises up and exerts periodicdeflection in the horizontal plane. We will not resolve themany possible scales and, instead, a single scale will beemployed for each quantity. This ‘uniscale’ approach is accept-able at the general level of description, but must be refinedwhen particular flow situation is analysed. Thus, length x ∼ L(but not ∼Lx, Ly, Lz), speed v ∼ V (but not ∼Vx, Vy, Vz), nabla�∼ 1/L, etc.

5. Transport of momentum

Transport of momentum is a synonym for fluid dynamics.Conservation of mass and (linear) momentum of fluid are thegoverning equations.

and
5.1. Mass equation of fluid

The mass balance (continuity equation) for a compressiblefluid reads(

∂�

∂t

)+ ∇(�v) = 0 [kg/m3 s]

(unsteadiness) + (convection) = 0. (5.1.1)

Choosing the variables (�, v, x, t), their scales (�0, V, L, T), andthe parameters (–), we get by scaling the dimensional ‘scaleequation’:

�0

T+ �0V

L= 0 [M/L3T]. (5.1.2)

Dividing by �0 the density scale disappears, and we get

1T

+ V

L= 0 [1/T]. (5.1.3)

Dividing finally by the convection term (V/L) we get thedimensionless ‘scale equation’:

(L

VT

)+ 1 = 0 (5.1.4)

with one DN, namely (L/VT) ∼ (unsteadiness)/(convection).Usually, we can find suitable scales for length L and speed V.The problem often is how to scale the time, i.e. what to take forT. There are few general choices: velocity scaling (T = L/V), peri-odic scaling (T = 1/frequency), relaxation scaling (T = relaxationtime), energy scaling (T = L(M/E)0.5), diffusion scaling (T = L2/D),etc. We take the first, since nothing indicates that the flow isperiodic or relaxing. Moreover, this choice also follows from(5.1.3), in the state of balance. Taking the basic scaling T = L/Vturns the ‘scale equation’ into

1 + 1 = 0. (5.1.5)

This should be read: unsteadiness (accumulation) and con-vection are in balance (same order of magnitude), 1/T = V/L.This balance is due to our choice of T, which may not be jus-tified in reality. In case of incompressible fluid, (5.1.1) reducesto ∇v = 0, which after scaling gives V/L = 0, meaning that theconvection term is zero order, ∼O(0), as expected.

5.2. Momentum equation of fluid

The momentum (force) balance for an incompressible Newto-nian single-phase fluid reads

�

(∂v∂t

)+ �(v.∇)v = −∇p + ∇2v + �g [Hyb/m3 s] = [N/m3]

(unsteadiness) + (convection)

= (pressure) + (viscosity) + (gravity). (5.2.1)

Unlike force [Newton] or energy [Joule], momentum doesnot have a single-word unit: let us call it Hyb [kg m/s] = [N s],provisionally. The two l.h.s. terms, (unsteadiness) and (con-
vection) are also called the inertial forces, namely the latterone. They are called the Eulerian and Lagrangian (convec-tive) accelerations, too. Choosing the variables (v, p, x, t), their
design 8 6 ( 2 0 0 8 ) 835–868

scales (V, P, L, T), and the parameters (�, , g), we get by scaling:

�V

T+ �V2

L= P

L+ V

L2+ �g [M/L2T2]. (5.2.2)

Dividing by the convective (inertia) term (�V2/L), because oftradition, we get

L

TV+ 1 = P

�V2+

�LV+ gL

V2. (5.2.3)

The magnitude of the convective (inertia) force is thusunity. Assigning the four DN their proper names we get

Sr + 1 = Eu + 1Re

+ 1Fr

. (5.2.4)

We can divide (5.2.2) by any term, but DN would not havethe usual names. The convection (inertia) term is the most‘flow-like’, so it is the natural scaling basis. The four DN havea clear physical meaning in terms of forces:

Strouhal number, Sr = L

TV

(unsteadiness)(convection)

,

Euler number, Eu = P

�V2

(pressure)(convection)

,

Reynolds number, Re = �LV

(convection)(viscosity)

,

Froude number, Fr = V2

gL

(convection)(gravity)

.

(5.2.5)

The relevance and use of these numbers follow from theirdefinitions and physical content. The Froude number is some-times (namely in physical literature) defined as (V2/gL)1/2. Forinstance, Fr is important where the gravity (buoyancy) andinertia interplay (open channel flow, free surface problems,hydraulic engineering, floating vessels, surface waves at longlength scales—not capillary waves; buoyancy driven flows).

The problem is how to scale time and pressure. Few scal-ings for T were presented at (5.1.4). Others come from thedynamic equation. We can compare the unsteady term L/TVwith the other terms in (5.2.2), to express their similar orderof magnitude:

Convection scaling : T = L

V, → Sr = 1,

Pressure scaling : T = �VL

P,

Viscosity scaling : T = L2

�,

Gravity scaling : T = V

g.

(5.2.6)

The basic scaling T = L/V follows from the definition ofSr = L/TV and also from comparing the unsteady and convec-tion terms, �V/T ∼ �V2/L. Using the basic scaling makes theStrouhal number equal to unity, Sr = 1. The unsteady and con-vective forces are thus comparable, as a direct consequenceof our choice of scaling. This may not always correspond to
reality.
Similar holds for the pressure scale P (P also means �P).We can compare the pressure term P/L with the other terms

nd de

i

vscs�

tws

sEtolTtWauoaf

tw

i

G

wiwiciarn

A

iuo


n (5.2.2), to express their similar order of magnitude:

Unsteady scaling : P = �LV

T, for T = L

Vit is P = �V2,

→ Eu = 1,

Convective scaling : P = �V2, → Eu = 1,

Viscous scaling : P = V

L,

Gravitational scaling : P = �gL.

(5.2.7)

Their use depends on the flow situation, and the rele-ance of the forces involved. Similarly, we can obtain othercaling relations between quantities in (5.2.2), comparing theorresponding terms. For instance, the so-called gravitationalcaling for speed comes from equating convection and gravity,V2/L ∼ �g, which yields V ∼ (g/L)1/2. It also comes from defini-ion of Fr = V2/gL. This scaling is useful, e.g. in sedimentation,hen particle speed V is not known beforehand, but particle

ize L is.Note the following. Using the convective (inertial) pres-

ure scaling, which is nothing but the dynamic pressure, theuler number turns to unity, Eu = 1. The pressure and convec-ive forces are then comparable, as a direct consequence ofur choice. This may not always correspond to reality. Simi-

ar happens when using the convective scaling for both time= L/V and pressure P = �V2, yielding Sr = 1 and Eu = 1, so that

he unsteady, pressure and convective forces are comparable.hen some of these DN are ‘missing’ in books and papers, the

uthors likely used certain specific scaling that made themnity, perhaps without notifying the reader. To trace it back,ne needs to read carefully what kinds of scales were actu-lly employed. The gravity scaling recovers the primary-schoolormula for hydrostatic pressure, p = �gh.

Since we customarily divided (5.2.2) by the convective (iner-ia) term, all numbers in (5.2.5) relate to the inertia force. If weant other combinations of forces, we must compose them:(pressureviscosity

)= Eu Re,(

pressuregravity

)= Eu Fr,(

viscositygravity

)= Fr

Re, → Ga = Re2

Fr→ Ar = Ga

(��

�

).

(5.2.8)

Note the following. The ratio (gravity)/(viscosity) is oftenntroduced as the Galileo number:

a = Re2

Fr= gL3

�2, (5.2.9)

here the square of Re is used for a practical reason: the veloc-ty cancels. This is helpful, when V is difficult to estimate, or

hen the speed is a part of the problem solution. For instance,n many applications (e.g. sedimentation, fluidization, bubbleolumns, etc.), we look for speed of bodies and particles mov-ng in fluids (bubbles, drops, solids). When buoyancy effectsre relevant, as it usually is in these applications, Ga is cor-ected by (��/�) and the product is called the Archimedesumber:

r =(

��

�

)Ga =

(��

�

)gd3

�2. (5.2.10)

This number is also (and more naturally) obtained by scal-
ng the Newton force law for a body falling/rising in a fluidnder gravity (see Section 5.5.2). The buoyant correction (��/�)f Ga is equivalent to replacing in (5.2.9) the gravity g with the
sign 8 6 ( 2 0 0 8 ) 835–868 843

reduced gravity g′ = (��/�)g in Fr, which is usual in buoyancydriven flows (see Section 6). The uniform part of the densityfield can be absorbed in the gradient pressure term (modifiedpressure p′ = p + �gz), thank to the fact that gravity has poten-tial. Then, the buoyancy term is directly proportional to thedensity difference �� = � − �0. The cause of density variationswithin a (incompressible) fluid can be, e.g. heat or solute con-centration (see Sections 6 and 7). When the ‘solute’ particlesare so big that they are endowed with their own momentum,they form the macroscopic dispersion, and we speak about‘multi-phase flow’ (see Section 5.5). Note that (5.2.1) containsonly two material properties, density and viscosity. The thirdone, the surface tension, enters only via the boundary condi-tions (Section 5.4).

There are many DN arising in particular flow situations,and some of them are mentioned here. For instance, the flowthrough curved pipes and sharp bends is characterized by theDean number, Dn = Re(r/rc)1/2, where r is pipe radius and rc

is radius of curvature. This correction to Re accounts for theeffects of the secondary flow driven by the centrifugal force(Dean vortices). Fast liquid flows with large pressure variancecan experience cavitation, where the liquid pressure p comesclose to or falls below the vapour pressure pv. Here, the Cavita-tion number Cv = �P/�V2 compares the liquid–vapour pressuredifference �P ∼ p − pv with the dynamic fluid pressure �V2.

There are numbers related to rotational effects, where cen-trifugal (�˝2R) and Coriolis (2�� × v) forces arise in (5.2.1),where � is rotation frequency and R distance from rotationaxis. The Ekman number Ek = (�/˝L2) is the force ratio (vis-cous)/(Coriolis). The Rossby number Ro = (V/˝L) is the forceratio (inertia)/(Coriolis). Like with the uniform density, alsothe centrifugal force can be absorbed in the pressure term,because it can be put into the gradient form. When Ek andRo are small, a balance between pressure and Coriolis forcesis reached (geostrophic flows). With these numbers, certainphenomena are connected (strong collocations: Ekman lay-ers, Rossby waves), which are important in geophysical fluidmechanics (meteorology, oceanography). While these num-bers reflect the forces due to noninertial reference frame (Earthrotation), another rotational effects are studied too, e.g. inTaylor–Couette flow between two coaxial cylinders, where wecan meet the Taylor number, Ta = (4˝2L4/�2). The simplest rota-tional effect is perhaps encountered in the process of mixing,with an impellor in a container, where both the gravity (g) andcentrifugal (ω2r) accelerations interplay in (5.2.1).

There are numbers related to rheology of complex flu-ids with non-Newtonian behaviour, where the stress tensortakes complicated forms. Depending on the given stress ten-sor and the flow situation, specific numbers can arise. Forinstance, the Weissenberg number ∼(first normal stress differ-ence N1)/(shear stress), which can also be defined in terms of(relaxation time)/(shear rate). Another one is, e.g. the Binghamnumber ∼(yield stress)/(viscous stress). There also is a number,related to a very general concept, the Deborah number:

De = Tr

Tf. (5.2.11)

It compares two time scales: material relaxation time Tr

and time of flow or deformation Tf. The former scale is a con-stitutive property of the material under study and says how
quickly it responds to deformation: Tr = 0 for ‘ideal fluid’ withinstantaneous response, ∼10−12 s for water molecules, ∼10−6 sfor thick lubrication oil (tribology), ∼few seconds for polymer

and
melts, and Tr = ∞ for ‘ideal solid’ that does not deform at all.The later scale is a property of the deformation process: thelength of the deformation experiment, the length of obser-vation of the material under load, etc. The simple fact thatall real materials have Tr < ∞ has an important consequence:everything flows. To witness it, we must perform/watch theexperiment on large enough time scales Tf, to ensure Tf > Tr,whence low De. Water is ‘solid’ when we hit it very fast, fasterthan it can open. Thus, any ordinary mortal can walk overwater, provided one paces faster than ∼10−12 s. This walkingis due to relaxation times of water molecules. Walking sup-ported by surface tension is mentioned in Section 5.4.1. Onthe other hand, the things we consider ‘solid’ in our everydaylife (Tf ∼ hours, days, years), can flow during centuries and mil-lennia (e.g. under its own gravity: glass tables in windows ofcathedrals, lead pipes in old buildings). The scale Tf for geo-logical processes is even longer. To see it, one should span verylong period, ∼aeons. For instance, the Lord with Tf ∼ eternitycould see ‘mountains flowing before Him2’. Summing up, wesee/feel the material liquid at low De (De � 1) and solid at highDe (De � 1).

When dealing with conducting fluids (magnetohydro-dynamics—conducting fluids in magnetic field; metallic andionic melts, Earth core, interior of stars, plasma), specificnumbers appear, e.g. N = (magnetic permeability)2 × (magneticfield)2 × (electrical conductivity) × L2/��, or the Alfven number.

When dealing with compressible fluids, the Mach numberMc = V/Vs is of paramount importance. Density variations ina fluid can be estimated by ��/� ∼ K�p, where K is the fluidcompressibility and �p pressure variation, �p ∼ �V2, usingthe inertial pressure scaling. The compressibility effects areimportant when ��/� > 1, i.e. V > 1/(K�)2. Realizing that 1/(K�)1/2

is the speed of sound Vs, we have Mc > 1. Since 1/K is thebulk modulus of elasticity E, we can introduce the Cauchynumber Ch = �V2/E, to find that Ch = Mc2. Other related numberis the Eckert number, Ec = V2/cp� ∼ (kinetic energy)/(enthalpy),which arises from scaling the full energy equation (tempera-ture rise by adiabatic compression). A relation holds, Ec ∼ Mc2.

When dealing with rarefied fluids (gases), there is a limit,where our hypothesis about their continuous nature breaks:when the mean free path Lm of the fluid molecules is compa-rable with the domain size Ld. There are not enough mutualcollisions between molecules to ensure the statistically sta-ble averages. In terms of the Knudsen number, Kn = Lm/Ld, thecorrections to rarefication are needed when Kn is large (gas atlow pressure, in small domain—porous media, membranes,microchannels). One such correction is the Cunningham (slip)factor F, which relates the drag coefficients in the dilute
and ‘normal’ (atmospheric pressure, room temperature) flu-ids, Cdilute = Cnormal/F. Not surprisingly, F depends on Kn, e.g.
2 This is often quoted in the reology literature with reference tothe Deborah song (The Old Testament, Book of Judges, Chapter 5,Verse 5). Reiner (1964) introduced the concept of De withreference to the ‘flowing mountains’ in this song. It is right thatthe original Hebrew word means ‘to flow’ (root ‘nzl’ = ,Hedanek, 2007). However, the mountains ‘flew’ not because ofthe very long observation time (only ∼1 (human) day; Godsubdued Jabin, the king of Canaan, on that day), but because ofthe short-time and intense anger of God. Therefore, theexpressions ‘to quake’ or ‘to melt’, which appear in differenttranslations, are closer to the original message. This note is notto undermine the great concept of De, but to show that it doesnot have its recourse in the Deborah song (and that the Lord waslikely not the very first rheologist).

design 8 6 ( 2 0 0 8 ) 835–868

like F = 1 + Kn·f(Kn), or similar. Also, Kn relates to Re and Mc, byKn ∼ Mc/Re. Note that the macroscopic no-slip condition on thewall also breaks, when the flow is highly rarefied and eventson microscale become relevant.

5.3. Energy equation of fluid

The derivation of the full energy equation is cumbersome,and often not necessary. When we neglect the dissipationand thermal effects and take the steady flow, the conserva-tive Bernoulli equation results, which is the first integral ofthe corresponding momentum equation:

12

�v2 + p + �gh = const. [J/m3]

(convection) + (pressure) + (gravity) = const. (5.3.1)

Choosing the variables (v, p, h), their scales (V, P, L), and theparameters (�, g), we get by scaling:

�V2 + P + �gL = const. [M/LT2]. (5.3.2)

Here, we can see the convective and gravitational scalingfor the pressure directly, P ∼ �V2 and P ∼ �gL. Dividing by theconvective term (�V2), because of tradition, we get

1 + P

�V2+ gL

V2= const. (5.3.3)

Assigning the DN their proper names we get

1 + Eu + 1Fr

= const. (5.3.4)

It is a counterpart of the steady and inviscid (5.2.4), so onlyEu and Fr are present.

Like the energy, also other quantities can be derived fromthe velocity field, and their equations obtained by manipulat-ing (5.2.1). The vorticity equation is obtained by taking Curl ofmomentum equation (5.2.1):

�

(∂�

∂t

)+ �(v.∇)� = �(�.∇)v + ∇2� [Hyb/m4 s]

(unsteadiness) + (convection)

= (vortex stretching) + (vorticity diffusion),

�˝

T+ �V˝

L= �V˝

L+ ˝

L2. (5.3.5)

The pressure and gravity disappear due to their gradientnature (they generate no torque), and the vortex stretch-ing term emerges. It is important for the energy flow downthe turbulent cascade, and is zero in 2D case, causing someanomalous phenomena. Note that the transport coefficient fordiffusion of both momentum and vorticity is the same (namely). Some estimates of ˝ can be made, e.g. the easiest one,˝ = V/L. Note that the convection and vortex stretching termsare comparable, when the same scales are used for these twophysically different terms.

Related to energy is so-called enstrophy. Energy per unitmass is the square of velocity ((1/2)v2), enstrophy is the square
of vorticity ((1/2)�2). The cross-product is called helicity((1/2)v·�). These scalar quantities are often used in turbulence.Their equations can easily be scaled and the corresponding

nd de

Dn

5

Igctsmap

Bzw

sttptpiafi

5Bptnltibrescef

sTt

ffea


N formed. It is not sure, whether the terminology (if any) ofumbers thus produces is established and settled.

.4. Boundary conditions: no slip and free-slip

nitial and boundary conditions inseparably pertain to theoverning equations, and should also be scaled. The initialonditions (IC) are given functions of the spatial distribution ofhe variables in the initial time. There is actually nothing tocale. On the other hand, the boundary conditions (BC) can beore elaborate. BC are equations that the variables must fulfil

t the boundary ∂˝ of the flow domain ˝, during the wholerocess of solution. As such, they must also be scaled.

In single-phase flow, bounded by a rigid wall, the no-slipC applies, which is simple and merely says that the speed isero at the wall, v = 0. There is not much to scale, besides v/V,hich is kind of unnecessary.

In multi-phase flow, where deformable fluid interfaceseparate immiscible phases with certain interfacial (surface)ension �, the free-slip BC applies, which is far from beingrivial. It is a force balance at the fluid–fluid interface. Decom-osing the force into normal (normal stress, pressure) andangential (shear stress) components, BC says: jump in fluidressure is balanced by tension �, and jump in fluid stress

s balanced by surface gradient of tension �S� (i.e. gradientlong interface). There are important DN stemming from theree-slip BC that relate to deformation of fluid particles andnterfaces.

.4.1. Normal component of free-slip BCy scaling, the normal component gives a number that com-ares the fluid pressure and the Laplace pressure produced byhe interface tension, N = P/(�/L), sometimes called the Laplaceumber, La. It follows from the physical situation, that the

ength scale L cannot be chosen arbitrarily but must relate tohe curvature of the interface that produces the correspond-ng capillary pressure �/L [Pa]. Choosing something else woulde physical nonsense. Consequently, for L should be taken theadius of curvature, the size of a bubble or drop, or perhapsven the capillary length. What can be put for the fluid pres-ure scale P? Generally, anything from (5.2.7), provided that itorresponds to the physical situation in question. Upon differ-nt scalings for pressure P, the number N takes the followingorms and names:

Unsteady scaling of P : Un = �L2V

�T

(unsteadiness)(capillarity)

,

Convective scaling of P : We = (�)�LV2

�

(inertia)(capillarity)

,

Viscous scaling of P : Ca = V

�

(viscosity)(capillarity)

,

Gravitation scaling of P : Bo = Eo = (�)�gL2

�(gravity or buoyancy)

(capillarity).

(5.4.1)

Here (�)� means that both � and �� can be used, puttingtress either on gravitational or buoyant aspect of We and Bo.he unsteady number Un turns into the Weber number upon

he convective scaling for time T = L/V.The Weber number We compares the inertia and capillary

orces. It applies to deformation of bubbles and drops at
ree rise or on collisions with an obstacle where dynamicffects are important. Typical situations are, e.g. bouncingt a wall and path instability of rising/falling fluid parti-
sign 8 6 ( 2 0 0 8 ) 835–868 845

cles, where the deformation and the motion can stronglybe coupled. Since We stems from the Lagrangian accelera-tion term, it also reflects the particle deformation due to theconverging/diverging streamlines. Low We means low defor-mation, and vice versa. Other situation is walking over water(hydrophobic feet assumed), where the inertia force (dynamicload) produced by the walker must not exceed the bear-ing power of the ‘flexible membrane’ of the surface tension,whence low We is required. Note that other mechanisms ofwater walking exist, well beyond the bearing capacity of thesurface tension. For instance, small reptiles (genus Basiliscus)can run over water using a supporting impulse from a liquid jetcreated by a specific shape of their feet. Here, other numbersbesides We play a role too.

The Capillary number Ca compares the viscous and capillaryforces. It is relevant when viscous forces dominate, i.e. slowflows and small scales. The typical situation is drainage of athin liquid film between two interfaces, at least one of themis fluid (to have � in play). The typical applications abound:interactions of fluid particles with themselves, with solids,and with rigid walls (coalescence, bouncing, adhesion), fre-quently encountered in bubble columns, flotation columns,extractors, etc.

The Bond number Bo and the Eotvos number Eo mean thesame: the ratio of gravity (�) or buoyancy (��) forces to thecapillary forces. The typical situation is deformation of a stag-nant drop sitting on a horizontal plane, where Bo comparesthe hydrostatic pressure across the drop (�gL) with the capil-lary pressure (�/L) inside the drop. Naturally, the length scaleL must be the drop size. Similar situation is when bubbles ordrops are entrapped below a horizontal wall, or attached to aneedle (L ∼ orifice size). From the equilibrium deformation ofbubbles and drops, the static value of � can be obtained. Othersituation is standing on the water surface, where the staticload of the stander must not exceed the strength of the surfacetension, whence low Bo is required. An important quantityrelates to Bo, the capillary length C. Consider that gravityforces dominate on large length scales at Bo ≥ 1, and capillaryforces dominated on small length scales at Bo ≤ 1. Find the crit-ical (capillary) length scale C, where Bo = 1. Quickly we see thatthe capillarity prevails over gravity when L ≤ C = (�/(�)�g)1/2.The typical situation is the capillary elevation of the water sur-face near the wall of the glass, which occurs in the range upto ∼C from the wall. Beyond this, the gravity prevails and thesurface is flat.

Besides the four DN arising from the normal component ofthe BC in (5.4.1), another composite number is often used. TheMorton number Mo is an artificial conglomerate that combinesRe and Fr from the governing equations, and We from the BC,in such a way, that it contains only the material properties offluids (plus gravity):

Mo = We3

Re4 Fr. (5.4.2)

The buoyant version reads Mo = (��)34g/�4�3 = g(��)3�4/�3,and gravity version reads Mo = g4/��3 = g�3/�3. It can also beobtained by DA, by forming a dimensionless group of (g, , �,�).

5.4.2. Tangential component of free-slip BC
By scaling, the tangential component gives a number thatcompares the surface gradient of the interface tension �S�
and the fluid shear stress �, N =�S�/�. Since the quantity �S�

and
is called the Marangoni stress �Ma, the number N is called theMarangoni number Ma:

Ma = �Ma

�. (5.4.3)

In case of a flat surface, where a tension difference ��

develops over a length L, the surface gradient can be estimatedas �S� ∼ ��/L. The fluid stress can be estimated by the viscousscaling as � ∼ V/L, which gives

Ma =(

��/L

V/L

)=(

��

V

). (5.4.4)

It may be difficult to estimate the velocity scale V for com-plex motions near interfaces on the convection basis, and thediffusive scaling is therefore applied. V is expressed with helpof the diffusivity of the agent that causes the interface tensionvariance (heat, surfactant), V ∼ (diffusivity)/L. It is consistentwith using the viscous scaling for the fluid stress: in bothcases, the molecular transport is reflected.

The specific formulation depends on the physical processthat causes the variation of � along the interface. Two situa-tions usually occur: variation of � is caused by a difference intemperature � or in surfactant concentration c. In the formercase, it is �� = (d�/d�)·(∂�/∂ϕ) and V = �/L. In the latter case, itis �� = (d�/dc)·(∂c/∂ϕ) and V = D/L. The Marangoni number thenreads

Ma =

⎧⎪⎨⎪⎩(

L

�

).

(d�

d�

).

(∂�

∂ϕ

)(thermocapillarity),(

L

D

).

(d�

dc

).

(∂c

∂ϕ

)(surfactants),

(5.4.5)

where ϕ is the angle along the interface of a bubble or drop,and L corresponds to their size. To close the problem, we mustfind the dependence of the interface tension on the agent con-tent, �(�) and �(c), as well as the spatial distribution along theinterface of the agent itself, �(x)S and c(x)S. The help comesfrom the thermodynamics and physical chemistry of surfaces,where the relation between � and (�, c)S at the interface isdetermined, and the relation between (�, c)S at the interfaceand that in the bulk (�, c)bulk are established too. Often, theequilibrium between the bulk and interface is assumed (for-mulas are then called ‘isotherms’). The problem of finding theinterfacial distribution of �(x)S and c(x)S is much more compli-cated. Under severe assumptions on the flow and transport of� and c we can make simplified theories. Otherwise, we are onmercy of numerical experiments by CFD, where all processesare strongly coupled (flow, transport, adsoption/desorptionkinetics).

The significance of the Marangoni number is that its largevalue indicates the presence of strong Marangoni stresses.They are important in several situations. They delay the riseof drops and bubbles in ‘contaminated’ water. They delaythe drainage of the film between interfaces, which suppressthe coalescence (bubble columns, extractors) and adhesion(flotation). Besides, it generates many other interesting phe-nomena. For instance, at the thermocapillary migration, finebubbles can ‘rise down’ in a liquid with inverse temperaturegradient (hot-bottom cold-top arrangement). Similar is theelectrocapillary motion, where the surface variation of � iscaused by the electric charges.

Surface tension effects are very complex and very tricky.Surfactants cause large effects even at trace amounts. It isdifficult to separate the effect of � and that of �S�. One

design 8 6 ( 2 0 0 8 ) 835–868

may suspect that effects of the latter are more frequent andmore important. The most of the common liquids (eitherpure or in mixtures or with surfactants) have � in the range∼0.02–0.08 N/m, i.e. differing by factor of 4. The effects pro-duced by surfactants in gas–liquid systems are enormous,within orders of magnitude. Moreover, � itself influencedirectly only relatively few hydromechanical processes. Con-sider for instance, the bubble formation process. The resultingbubble size depends directly on � only at the very low-gas flowwhere the equilibrium between buoyancy and surface force isreached during the quasi-steady inflation. This is not the caseof real gas–liquid contactors, where the bubble size dependson many other effects, involved in violent breakup of irregulargas jets produced by the gas distributor. On the other hand, themagnitude of �S� is difficult to estimate. Taking it ∼��/L, wehave �� within the factor of 4 (i.e. almost ‘constant’ for all liq-uids), but the relevant length scale L can cover very vast rangeof values. Consider a bubble of size L. The surface gradientcan develop over any distance, from 0 to ∼L, making ��/L verymuch varying quantity. Moreover, the gradient formation andevolution is a dynamic process, with fast temporal changes,which depends on the flow situation (unlike �, which is mate-rial property of static equilibrium). Also, all real processestake place in systems that contain ‘impurities’, which read-ily act as surface active agents (most of chemical compound,e.g. reactants). Therefore, no surprise, that bubbling into two‘extremely different’ liquids with � = 0.02 and 0.08 N/m givesalmost identical results, while re-distilled water and tap watermay differ by tenth of percents in gas holdup. For these rea-sons, the gas holdup correlations involving � are extremelyunreliable, with respect to this particular variable.

Better understanding the surface processes and theirproper scaling is needed. A qualitative sketch of the surfac-tant action is shown in Fig. 3. Based on their tendency togather at the interface, two classes of surfactants can be dis-tinguished: positive and negative. The positive surfactants areattracted to the interface, their surface concentration is largerthan the bulk concentration (cs > cb). Typically, they are organicsubstances (emulsifiers, detergents, tensides, wetting agents,etc.) that decrease the surface tension significantly. The neg-ative surfactants are repelled from the interface, their surfaceconcentration is smaller than the bulk concentration (cs < cb).Typically, they are inorganic substances (salts of mineral acids,electrolytes, ionic solutions, etc.) that increase the surface ten-sion only slightly. During the bubble rise, its nose feels the bulkconcentration (cb). Its tail feels a generally different concen-tration, resulting from the adsorption/desorption transportprocesses occurring between the bulk and the interface, asthe liquid passed around the bubble. When an equilibrium isreached, the rear concentration can be denoted as ‘equilibriumconcentration’ (ce). Obviously, the positive surfactants havece > cb, while the negative ce < cb. However, despite the oppositeconcentration profiles along the interface of the positive andnegative surfactants, their surface tension profile is the same.In both cases, the rear � is low and the front � is large. Thisinterfacial surface tension gradient (�S�) is a tensile force thatgenerates the Marangoni stresses. These stresses move theliquid material elements along the interface, in the directionopposite to the main flow over the bubble. The increase in theresistance force is the natural result (retardation of bubble risein contaminated media). At the first guess, the students (and
not only them!) would say, that if the positive surfactant delaysthe bubble rise, so the negative surfactant will accelerate it,which is, however, not so.


Fig. 3 – Positive and negative surfactants in air–water system. (a) Positive surfactants are attracted to surface: nonpolar partspoint to nonpolar air, polar parts © remain in polar water. Negative surfactants are repelled from surface: polar ions © are

repelled by nonpolar air and attracted by polar water. (b) Concentration profile perpendicular to surface: positive surfactantshave cs > cb, negative surfactants cs < cb. (c) Positive surfactants decrease surface tension (much). Negative surfactantsincrease surface tension (little). (d) Concentration profile along surface of rising bubble: positive surfactants have ce > cb,negative surfactants ce < cb. (e) Surface tension profile along bubble surface: both positive and negative surfactants haves ubbm (a)).

5

5Op

ame profile. Surface tension gradient (�S�) corresponds to raking thinner regions expand (surface gradient is seen in

.5. Multi-phase flow

.5.1. Microscale description (DNS)n the ‘microscopic’ level, the flow is fully resolved, in bothhases: inside N discrete dispersed particles (bubbles, drops;

er sheet with variable thickness: thicker regions shrink,

solids—no flow inside), and around them in the one con-tinuous carrying fluid. We should write (N + 1) Navier–Stokes
equations, and solve them simultaneously. They are coupledby sharing the same boundary condition at the particle–fluidinterface. We obtain the flow field in every point of the disper-

and
sion. The interface forces move the particles and deform theinterfaces. We have the full information about the problem.This fine microscale approach is in the CFD jargon called the‘direct numerical simulation’ (DNS).

There are no extra forces, as compared with the single-phase flow. However, there is a difference: the surface tensionforce is moved from the boundary condition (Section 5.4)directly into the momentum equation (Section 5.2, Eq. (5.2.1)).Usually, this new term contains the following ingredients:surface force [N/m3] ∼ interface tension � [N/m], interface cur-vature [m−1], interface area [m2/m3]. As for the scaling, thisterm could be inertially scaled by (�V2/L), like any other term in(5.2.1). This would yield (�/L)/(�V2), which is nothing but We−1.The specific way how this surface force is converted into avolumetric force and implemented depends on the modellingapproach to the DNS. Currently there are several numeri-cal strategies, how to describe interfaces and their motion(e.g. front tracking method, level set method, volume of fluidmethod).

5.5.2. Mesoscale description (Euler/Lagrange)On the ‘mesoscopic’ level, the flow field is resolved only inthe continuous phase, using the single-phase equation—oneNavier–Stokes (Eulerian view). The dispersed particles (bub-bles, drops, solids) are considered being pointwise, as seen bythe fluid. Their motion is described by set of N equations ofmotion of system of bodies in space (Largangian view). Thisintermediate-resolution approach is in the CFD jargon calledthe ‘Euler/Lagrange simulation’ (E/L).

The many kinds of hydrodynamic forces acting on the par-ticles are given by various closure formulas, obtained by othermeans (e.g. experiment, DNS). The forces depend on the localflow field near the particles. The particles can also affect thefluid motion, as a feedback (coupling). As compared with thesingle-phase flow, there are many new equations with newforces. All of them can be scaled, and new DN will appear.Consider only the simplest case of a sedimenting particle in astagnant unbounded fluid, without (particle → fluid) coupling,which is the paradigm of sedimentation:

mdu

dt= gravity − buoyancy − drag [Hyb/s]. (5.5.1)

Substituting the typical closures for the forces, it reads

mdu

dt= �

6d3�pg − �

6d3�fg − 1

2�d2

4�fu

2C [Hyb/s]. (5.5.2)

First, consider a steady particle motion, under the forceequilibrium:

�

6gd3�� = 1

2�d2

4�u2C (�� = �p − �f) [N]

(gravity) − (buoyancy) = (drag). (5.5.3)

Choosing the variables (d, u), their scales (L, V), and theparameters (�, ��, g, C), we get by scaling:

�

6gL3�� = 1

2�L2

4�V2C [N]. (5.5.4)

design 8 6 ( 2 0 0 8 ) 835–868

Note that L must keep the meaning of the particle size d.Dividing mercenarily by ��2 = �, we get dimensionlessly

�

6��

�

gL3

�2= 1

2�

4L2V2

�2C, (5.5.5)

which is nothing but

Ar = 34

C Re2. (5.5.6)

This is an important criterial equation for problems withfalling/rising particles, where gravity, buoyancy, and drag playthe main role. It indicates a close relation between Re and Arin this type of problems, where an equilibrium is establishedbetween the driving forces (Ar, l.h.s.) and the resistance force(Re, r.h.s.). The former are produced by an external field (e.g.gravity), while the latter involves speed as the main ingredient.Since the speed in Re is often the part of the solution, the grav-itational scaling is employed V ∼ (g/L)1/2, which transforms Reinto Ga = Re2/Fr. Here, the concept of buoyancy reflected by Arhas much clearer interpretation than in the single-phase flow:the effect of particle–fuid density difference is obvious. Notethat both particle size L and speed V are involved in Re(L, V) andonly size in Ar(L3). It leads to an iteration procedure in solvingthe sedimentation problem, where the speed depends on theparticle size. A usual trick is to separate these two with helpof a new suitably defined number. The Lyjascenko number,Ly(V3) = Re3/Ar = (V3/g�)(�/��), converts (5.5.6) into

Ar1/3 = 34

C Ly2/3, (5.5.7)

where L stands on l.h.s. and V2 on r.h.s., so that the sizeand speed are decoupled. Another problem of course is withC = C(Re(L, V)).

Second, consider an unsteady motion, when the resistanceforce consumes the initial momentum of a particle:

mdu

dt= −1

2�d2

4�fu

2C (IC : t = 0, u = u0) [Hyb/s]. (5.5.8)

In the simplest case of the Stokes drag, we have a linearrelaxation process and (5.5.8) becomes

mdu

dt= −3�du [Hyb/s] (unsteadiness) = (drag). (5.5.9)

This is a rare occasion in two-phase flow when we canfind the relaxation time (original disturbance is reduced by1/e ≈ 64%; a stable equilibrium presumed, a node). Choosingthe variables (u, t), their scales (V, T), and the parameters (m,d, ), we get by scaling:

mV

T= dV [ML/T2]. (5.5.10)

By turn, we have the relaxation time:

Tr = m

d[T]. (5.5.11)

Substituting for the particle mass m = (�/6)d3�p, the time is

Tr = �

6�pd2

∼ �pd2

[T]. (5.5.12)

nd de

u

ti

T

ei

atlhba

S

siccicb

S

tchdcbibm

tdtsSlibtUdp

flBe(


Another way is to solve the linear problem (5.5.9) directly:

(t) = u0 exp(

−3�d

mt

)[m/s]. (5.5.13)

The exponent should be of the form (t/Tr), in order, at t = Tr,he ratio u/u0 be exp(−1) = 1/e. Accordingly, the relaxation times

r = 118

�pd2

∼ �pd2

[T]. (5.5.14)

The variance of (3�) between (5.5.12) and (5.5.14) is the mod-st price for bypassing the exact solution by the scaling, whichs acceptable.

The relaxation time just found is useful for a variety of situ-tions, where time scales of different processes are compared,o distinguish between the ‘fast’ and the ‘slow’, which in turneads to the time decoupling, making the problem easier toandle. One important case is the concept of the Stokes num-er St. It compares two time scales: particle relaxation time Tr

nd time of flow change Tf:

t = Tr

Tf. (5.5.15)

The former says how fast the particle relaxes back to theteady state, when accelerated with respect to the surround-ng (viscous) fluid. The latter says how fast the flow fieldhanges. Without a priori knowledge, the basic estimate of Tf

an be made, Tf ∼ L/V. Since the particle must feel the changesn the fluid, we couple them by the common length scale,learly given by the particle size, L = d. Then, the Stokes num-er is

t = �pdV

. (5.5.16)

In other words, St is the ratio (particle inertia)/(fluid iner-ia). More correctly, the particle mass used above should alsoontain the added mass Ca, m = ((�/6)d3) (�p + Ca�), especiallyere, when unsteady effects are considered. Then the particleensity becomes: �p → (�p + Ca�). We appreciate it namely inase of bubbles in liquids, where �p/� ≈ 10−3, so that the bub-le inertia is represented by the added mass, i.e. by the liquidnertia, �p,effective ≈ �, since Ca is O(1). Then, the Stokes num-er becomes the Reynolds number, St = �pdV/ ≈ �dV/ = Re,eaning ∼ (particle–joint fluid inertia)/(‘viscous fluid inertia’).The Stokes number measures the willingness of the car

o get off the road when you turn the stirring wheel sud-enly. As such, it is used in many situations when we wanto know how much the dispersed particles tend to follow thetreamlines of the carrying fluid. The total flow-follower hast = 0 (passive scalar, tracer). On the other hand, particles witharge St easily hit the wall in bendings of a duct. It is usedn devices (impactors) where aerosol particles are sorted outy their value of St, being expelled from the main stream tohe wall by their inertia, in multiple progressively narrowing-bends. The adhesion efficiency of the flotation process alsoepends on the value of St, with which a bubble collides witharticles: the lower St, the better for adhesion (no bouncing).

Other numbers appear in specific areas of multi-phaseows, at this mesoscale level of description. For instance, the
agnold number in granular flows, Ba = �p�d2/, compares theffect of the interstitial fluid on the motion of the granulesgrains). In terms of stresses, it is a ratio of the collisional
sign 8 6 ( 2 0 0 8 ) 835–868 849

stress in the particulate phase, which by the kinetic theoryis ∼�pV2, where V ∼ �d, where � is the velocity gradient and dthe particle size, and the viscous fluid stress ∼�. With � ∼ V/L,and L ∼ d, Ba = �pdV/, which actually is St. Equivalently, itcan be recasted in terms of the friction forces (particle col-lisional)/(fluid viscous). The presence of the interstitial fluidcan be neglected at large Ba, to reach the limit of so-called‘dry’ granular flows.

5.5.3. Macroscale description (Euler/Euler)On the ‘macroscopic’ level, the dispersed particles (bubbles,drops, solids) are considered as a phase smoothly distributedin space, forming a ‘pseudo-continuum’. The particles arethen assigned continuous concentration and velocity fields.This approximation is acceptable when we describe the sys-tem on length scales much larger than the discrete scales. Theparticles and their spacing must be much smaller than thesystem size, and than the smallest scales we want to resolve.In the single-phase flow, estimate the discrete (atomic) scalesby 10−9 m, and the beginning of the continuum by 10−6 m,say. We have three orders of magnitude to bridge the gap.Accordingly, with 1 cm bubbles, the reactor should be of10-m size, to consider the bubbles as the continuous phase.Further, presence of many bubbles is anticipated, for theirspacing be small, e.g. comparable with bubble size. Now, thegoverning equations should be twice Navier–Stokes: one forthe continuous phase and one for the dispersed phase (twointerpenetrating continua, twice Euler’s view). These equa-tions are coupled via the interphase momentum transfer.This coarse macroscale approach is in the CFD jargon calledthe ‘Euler/Euler simulation’ (E/E).

The governing equations for single-phase flow are oftensaid to be derived from the ‘first principles’. These mechanicalprinciples are known for a single continuum, namely in caseof simple fluids, but are only in the process of developmentfor the multi-phase systems. Here, we lack a universally validequation, which would be of practical use. There are manygeneral equations suggested, but they are too monstrous,and the many closures needed for them are still missing.On the other hand, there also are many simple equations,suggested for specific systems and particular flow situations,which are practical, yet of limited use. As a compromise, herewe write the two-phase flow equations purely formally, as twoNavier–Stokes-like equations:

• Continuous phase:

(∂

∂t

)ε′�′ + ∇(ε′�′v) = 0 (mass) [kg/m3 s], (5.5.17)

(∂

∂t

)ε′�′v + (v.∇)(ε′�′v) = ∇ε′�′ + ε′�′f + S

(momentum) [Hyb/m3 s]. (5.5.18)

• Dispersed phase:

(∂

∂t

)ε′′�′′ + ∇(ε′′�′′u) = 0 (mass) [kg/m3 s], (5.5.19)

(∂)

ε′′�′′u + (u.∇)(ε′′�′′u) = ∇ε′′�′′ + ε′′�′′f − S
∂t

and
The continuous phase has density �′, velocity v, stress ten-sor �′, volume fraction ε′. The dispersed phase has density�′′, velocity u, stress tensor �′′, volume fraction ε′′. The massconservation for the two-phase mixture needs ε′ + ε′′ = 1. Themain multi-phase problem is to formulate the stress tensors�′′ and the interaction term S. As for the scaling, two kinds ofnew DN could appear, as compared with single-phase flow.One comes from scaling the stress term �′′, which very likelywill not reduce to simple Newtonian formulation yielding Euand Re. The other comes from the momentum transfer termS. The latter would indicate how strong the phase couplingis.

If the dispersed phase constitutes a macroscopic dispersion,the particles are the mechanical individuals (small bodies)and have their own momentum, inertia, gravity, buoyancy,whence the momentum equation (5.5.20) applies. If the dis-persed phase constitutes a microscopic dispersion, the particleslack these qualities, and (5.5.20) is needless. The individualparticles passively follow the flow (St = 0); very fine particlescalled ‘passive scalars’ and used for flow visualization (trac-ers). The true chemical solutions (salt or dye in water) surelybelong to this category (sub-colloidal). There is a legitimatequestion: what is between these two extremes, the macroand micro. When the dispersed phase earns the right to beawarded the full momentum equation? Likely, there are nostrict and unequivocal criteria, to decide. Probably, it dependson our choice, what effects and on which scales we wishto study. In the molecular dynamics, very small particles(solute/solvent molecules) are moved by force laws of vari-ous degrees of resolution. Some forces are derived directlyfrom molecular potentials, some are modelled as randomthermal noise to account for Brownian effects, there is aresistance force, and the overall formulation can be withinthe Langevine ansatz (random forcing, stochastic differentialequations). In the Stokesian (hydro)dynamics, very small par-ticles (fine particles suspended in fluid) are moved by forcesof both hydrodynamic (Stokes limit) and nonhydrodynamic(Brownian, colloidal, interparticle, etc.) origin. Another aspectis reflected by rheology, where the dispersed phase affects theintrinsic momentum transport substantially.

For microscopic dispersions, Eq. (5.5.20) is omitted and(5.5.19) is modified accordingly. The quantity (ε′′�′′) is replacedwith the scalar concentration c. The speed u is identified withv. The molecular diffusion term is added to the r.h.s., since themacroscopic particles did not have this molecular transportmechanism.3 Eq. (5.5.19) then becomes:

• Dispersed phase:

(∂c

∂t

)+ (v.∇)c = D∇2c (mass) [kg/m3 s]. (5.5.21)

These modifications must also be reflected by Eqs. (5.5.17)and (5.5.18). Since the dispersion does not exist on themacroscale, the carrying fluid occupies the whole volume,ε′ = 1. Omit the apostrophe at �′ and �′. Set S = 0, since the
fluid does not receive momentum from the dispersed par-ticles. For Newtonian fluid equations, (5.5.17) and (5.5.18)become
3 But there is a concept of ‘hydrodynamic diffusion’ ofmacroscopic particles, due to (mostly repulsive) interactionforces, see e.g. Davis (1996).

design 8 6 ( 2 0 0 8 ) 835–868

• Continuous phase:(∂�

∂t

)+ ∇(�v) = 0 (mass) [kg/m3 s], (5.5.22)

(∂

∂t

)�v + (v.∇)(�v) = −∇p + ∇2v + �f


While a single dispersed particle does not affect the flow,a large number of them can exert collective buoyancy effects.These macroscopic effects consist in fluid density variationscaused by distribution of the particle concentration, � = �(c).The interphase coupling is as follows. The (micro) dispersedphase imports the fluid velocity v from (5.5.23) into (5.5.21),and exports the concentration c from (5.5.21) into the fluid den-sity in (5.5.22) and (5.5.23). This is the convection–buoyancytwo-way coupling.

Note that Eqs. (5.5.21)–(5.5.23) formally coincide with thosefor heat and mass transport considered in Sections 6 and7. Indeed, these equations represent the non-inertial micro-disperse limit of the governing equations (5.5.17)–(5.5.20)derived for the macro-dispersed multi-phase mixtures. Thissimple fact opens an interesting window of research: build-ing analogies between the well-understood single-phase flowswith heat and mass transport, and much less understoodmulti-phase flows. The buoyant coupling from (5.5.21) to(5.5.22)–(5.5.23) can be facilitated by any buoyant agent thatbehaves like a passive scalar (true solute, heat, fine particles,etc.). As the next step, the first-order inertial effects can beadded to this base state. For instance, the buoyancy-driveninstabilities in sedimenting layers, fluidized, beds, and bubblycolumns may shear certain common features with phenom-ena of thermal convection or halinoconvection. As for thescaling, DN related to heat and mass transport is introducedin Sections 6 and 7, together with the coupling to the hydrody-namics. They naturally apply also to (5.5.21)–(5.5.23). Anotherissue is what is the relevant density of a multi-phase mixture,when evaluating the buoyancy force acting on a submergedbody. Usually, one takes either the pure fluid density or theeffective mixture density. It seems however, that both can berelevant, depending on the relation between the body sizeand the size and spacing of the dispersed particle in themixture.

5.5.4. Retention time distributionA brief note is in place, on the retention time distribution(RTD) in equipments. It is the very first thing one must do,before starting any kind of thoughts about the processes ina given apparatus. We assume that the tracer concentrationobeys (5.5.21), being one-way coupled with (5.5.22) and (5.5.23).Strictly speaking, the diffusion term (D) in (5.5.21) is inconsis-tent with the role of a ‘pure flow follower’. The proper tracershould stick to a fluid particle, and not to diffuse. Therefore, itshould obey:

Dc

Dt= 0 (mass) [kg/m3 s], (5.5.24)

which is (5.5.21) with D ≡ 0. This can directly be solvednumerically, together with (5.5.22) and (5.5.23), to get the full
information about the flow and concentration fields. Then, itis easy to monitor numerically the tracer content at the exit, tocreate the RTD response curve. This is not the way we wish to

nd de

gdtnan(fiTiTmj

(

afdti

(

c

(

wle(sst(

F

i

rTFc((t


o in practical applications of RTD. Let us treat the unwantediffusion term in (5.5.21). One cure would be the assumption,hat the flow is much faster than the tracer diffusion. This isot usually done, and even if, this is usually not true. Another,nd more practical cure, is to assume that the tracer is simulta-eously transported by two mechanisms, molecular diffusion

Dm) and turbulent dispersion (Dt). Eq. (5.5.21) can be treated asollows. Upon the Reynolds decomposition, the actual veloc-ty field splits into the mean and fluctuating parts, v = V + v′.he convective term in (5.5.21) becomes (V·�)c + (v′·�)c, keep-

ng the physical meaning of the convective mass flux, j = J + j′.he mean term, J = (V·�)c, is harmless. The fluctuating term isodelled as the mass flux driven by the turbulent dispersion,

′ = Dt�2c. Eq. (5.5.21) thus reads

∂c

∂t

)+ (V · ∇)c = (Dm + Dt)∇2c (mass) [kg/m3 s]. (5.5.25)

Suffice to require that Dm � Dt, which can even be realistic,t least in some flow situations. This must, however, very care-ully be checked in slow flows in microchannels. In case of 1Dominant main flow, we have V = V = Q/S, and the tracer equa-ion is uncoupled from the flow equation, and can be solvedndependently:

∂c

∂t

)+ V

(∂c

∂x

)= Dt

(∂2c

∂x2

)(mass) [kg/m3 s]. (5.5.26)

Fixing the coordinates to the mean flow V deletes the meanonvection term and yields

∂c

∂t

)= Dt

(∂2c

∂x2

)(mass) [kg/m3 s], (5.5.27)

hich is diffusion in stagnant medium. At absence of turbu-ent fluctuations, the axial dispersion is zero, Dt = 0, and thequation

∂c

∂t

)= 0 (mass) [kg/m3 s], (5.5.28)

olves to c = c0 = const. It is the plug flow, where the cross-ection area marked with c0 is carried through the system withhe mean speed V. Anticipating the scaling applied to Eq. (7.1),5.5.26) becomes the RDT analogue of Eq. (7.4):

o + Pe = 1. (5.5.29)

Here, the numbers are defined using the turbulent diffusiv-ty:

Fourier number, Fo = L2

DtT

(unsteadiness)(turbulent mass diffusion)

,

Peclet number, Pe = LV

Dt

(hydrodynamic convection)(turbulent mass diffusion)

.

(5.5.30)

Two ways leads to the exclusivity of the Pe number occur-ence in the RTD problems. First, use the basic time scaling,= L/V, and the Fourier number becomes the Peclet number,o = Pe. This makes Eq. (5.5.29): 1 + 1 = 1/Pe. Second, fixing theoordinates to the mean flow V deletes the convection term
Pe), and by the same scaling Fo becomes Pe. This makes Eq.5.5.29): Pe + 0 = 1, which corresponds to (5.5.27). Thus, Pe ishe only important number in RTD, with the 1D dominant
sign 8 6 ( 2 0 0 8 ) 835–868 851

main flow. Contrasting all these assumptions with the real-ity of the actual flows through real technological systems, itwould rather be naive to expect that Pe, whence the singlescalar quantity Dt (also called: axial dispersion Dax), can con-tain the whole truth about the hydrodynamics. We are in needof something smarter.

Actually, there has been an attempt is this direction, and akind of Smart RTD (SRTD) has been suggested. Imagine that afluid particle has a watch that measures its age �, being set tozero at entering the equipment. Thus, � is the retention timeof the fluid particle. At non-relativistic motions, the time onthe watch coincides with the ‘common’ time t on the labora-tory clock. The trivial and seemingly useless physical fact that� = t can be recasted into a useful form. Because the watch isfixed to the moving fluid particle, its age � must follow theconvective derivative:

D�

Dt= 1 (retention time) [-], (5.5.31)

which is already dimensionless. Here, we assume that thequantity � can be considered to be the field quantity. It is rathercounter-intuitive, but any fluid particle located at time t inplace x can be assigned the amount of time �(x, t) it has spentin the equipment. Thus solving the above equation for ‘conser-vation of particle age’, we obtain the real ‘distribution’ of thelocal retention time within the equipment. Having the field �(x,t), we can easily calculate the field of concentration, reactionrate, conversion, etc. Eq. (5.5.31) can be treated in a similar waylike (5.5.24). It can directly be solved numerically, together with(5.5.22) and (5.5.23), to get the full information about the flowand RTD fields. It can also be simplified, to save the computingpower. After the decomposition, V + v′, we have the followingcounterpart of (5.5.25):

(∂�

∂t

)+ (V · ∇)� + (v′ · ∇)� = 1 (retention time) [-]. (5.5.32)

The mean ‘flux of age’, (V·�)�, is physically plausible, sincethe stream passing through a given location contains parti-cles of various age. The fluctuating part, (v′·�)�, can eitherbe modelled by a turbulent diffusion term, j′ = Dts�2�, or leftas it is and take a suitable closure for the fluctuating veloc-ity v′. It can be modelled, e.g. by random functions reflectingtruly the local structure of the turbulence. There is wealth ofinformation about the scaling behaviour of turbulent velocityfluctuations in various flow situations, in the literature. Thisbrings us naturally to the stochastic modelling of RTD. Con-sidering the 1D dominant main flow, where the mean speedis a constant, V = V = Q/S, we have the counterpart of (5.5.26):

(∂�

∂t

)+ (V + v′)

(∂�

∂x

)= 1 (retention time) [-]. (5.5.33)

In fully 1D case, both V and v′ are scalars. Fixing the coor-dinates to the mean flow V deletes the mean convection termand yields the following counterpart of (5.5.27):

(∂�

∂t

)+ v′

(∂�

∂x

)= 1 (retention time) [-], (5.5.34)

which can be treated within the framework of the stochas-tic differential equations, with a suitable random v′ = f (x, t).At absence of turbulent fluctuations, v′ = 0, the equation

and
becomes(∂�

∂t

)= 1 (retention time) [-] (5.5.35)

and solves to � = t. It is the plug flow, where all fluid particlesin the moving coordinate system have age � = t. This age cor-responds to the position x = Vt from the inlet, measured in thestationary coordinate. With pipe of length L, the particles exitat age � = t = L/V.

6. Transport of heat

For the transport of heat in a moving environment, three bal-ances must be considered together: fluid mass balance (5.1.1),fluid momentum balance (5.2.1), and the heat balance, whichcan be written as(

∂�

∂t

)+ (v · ∇)� = �∇2� [K/s]

(unsteadiness) + (convection) = (diffusion). (6.1)

Choosing the variables (�, v, x, t), their scales (�, V, L, T),and the parameters (�), we get by scaling:

�

T+ V�

L= ��

L2[�/T]. (6.2)

Dividing by � the temperature scale disappears, and we get

1T

+ V

L= �

L2[1/T]. (6.3)

Dividing by the diffusion term (�/L2), because of tradition,we get

L2

�T+ LV

�= 1. (6.4)

We can divide by any term, but then the DN would nothave their usual names. The diffusion term is the most typi-cal one for heat and mass transport phenomena, so it is thenatural scaling basis. Assigning the DN their proper nameswe get

Fo + Pe = 1. (6.5)

The following DN arise, having a clear physical meaning interms of process rates:


�T

(unsteadiness)(heat diffusion)

,


�

(hydrodynamic convection)(heat diffusion)

.

(6.6)

When using the basic time scaling, T = L/V, the Fouriernumber becomes the Peclet number, Fo = Pe. The unsteadyand convective effects are then comparable, as a directconsequence of our choice of scaling. This may not always cor-respond to reality. The Peclet number facilitates the couplingbetween the hydrodynamics and heat transfer: it compares
the transport by hydrodynamic convection (flow of medium, V)and the molecular diffusion of heat (heat diffusivity �). A mod-ified version of Pe is the Graetz number, Gr = (mass flow)cp/L,
design 8 6 ( 2 0 0 8 ) 835–868

where the convective speed is recasted into the massflow.

Since we customarily divided (6.3) with the diffusion term,both numbers are based on the diffusion rate. If we want othercombinations, we must compose them. These compositionscan contain thermal and hydrodynamic quantities to reflecttheir coupling. One route leads to the Prandtl number:

Prandtl number, Pr = �

�

(momentum diffusion)(heat diffusion)

. (6.7)

The Prandtl number is prepared as follows. Take the Pecletnumber, Pe = LV/�. Replace the hydrodynamic convection (LV)with the hydrodynamic diffusion (�) with help of Re, LV = � Re,to get Pe = � Re/�. Divide by Re, since it is dimensionless, to getPe = �/�. Give the product a new name: the Prandtl number,Pr = Pe/Re. This number is often used in thermal processes,where the ratio of two material properties of fluid determineshow much the flow is affected by heat diffusion (thermal con-vection, boiling, etc.).

Another route leads to the heat Grashof number, which istypical for heat-driven buoyancy effects:

Grashof number, Gr = ˛ ��gL3

�2

(buoyancy force)(viscous force)

. (6.8)

Take the hydrodynamic Archimedes number, Ar = (��/�)Ga.Replace the general expression for the density difference ��

with a specific expression for the thermal expansivity of fluid,��/� = ˛ ��, where ˛ is the coefficient of thermal expansion.Plug it into Ar to get Ar = (˛ ��)Ga. Give it name the heatGrashof number, alias the thermal Archimedes number. Gris encountered in thermofluid mechanics. It is essentially ahydrodynamic number, where the heat enters as the ‘buoyantagent’, to produce density gradients.

A close derivative of Gr is the thermal Rayleigh number,Ra = Gr Pr:

Rayleigh number, Ra = ˛ ��gL3

��


. (6.9)

This number is decisive for the onset of thermal convectionand its evolution via series of bifurcation. It compares the driv-ing thermal disturbance �� acting on a parcel of fluid, withrates of transport processes that tend to smear it out (diffusionof momentum � and heat �).

The way the above composite numbers are ‘derived’ bymaking various combinations and replacements seems to beneither transparent nor free from ambiguity and subjectivity.Actually, they can be obtained correctly, from the correspond-ing governing equations. For instance, Pr, Gr and Ra appearnaturally by scaling the coupled equations for flow and heattransfer, under a useful approximation (Boussinesq), wherethe only buoyancy-affected density is that at the externalforce field term (�f). Often, the linearization near a uniformbase state is considered (stability studies), or the uniform partof the density field is absorbed in the pressure term, whichgives the buoyancy term proportional to the density difference�� = � − �0. Depending on the scales employed, the numbersappear in different places, as either Pr and Ra, or Pr and Gr.Note that, based on the physical analogy and scaling argu-ments, it is possible to introduce Ra also for dispersed layers,e.g. for bubbly layers in bubbly columns, Ra = g′eL3/�mixDhydro

(g′ – is the reduced gravity, e –volume fraction of bubbles(voidage, gas holdup), �mix –effective viscosity of bubbly mix-ture, Dhydro –hydrodynamic diffusivity of bubbles).

nd de

bowicTRFabnettω

as

tflT

j

wpTh

j

wE

k

b

N

wfiteNostfii

7

Ftam


Other numbers are also related to buoyancy effects. Aasic quantity is the buoyant (Brunt–Vaisala) frequency ω

f the density difference driven oscillator, (d2z/dt2) = ω2z,here ω = ((g/�)(d�0/dz))1/2. It is an elementary prototype for

nternal gravity waves in stratified environments. Here wean encounter the Richardson number, in several variations.he gradient Richardson number Ri = ω2/(∂u/∂z)2. The globalichardson number is Ri = g′L/V2. Note that it relates to theroude number Fr’ = V2/g′L = 1/Ri, when Fr is corrected for buoy-ncy (apostrophe’). Other kinds of Ri also exist. When bothuoyancy (stratification) and rotation are present, there areumbers indicating their relative effects (stratification param-ter, Burger number). The buoyant frequency also follows fromhe gravitational time scaling T = V/g in (5.2.6). Taking V = L/T,he time scale becomes T = L/gT, which in terms of frequency∼ 1/T reads ω2 = g/L. Employing the reduced gravity g → g′ toccount for the density variance, and designating ��/L thecale estimate for (d�0/dz), we get what was due.

Boundary conditions in heat transfer are of two kinds. Eitherhe temperature �w is given at the boundary ∂˝ or the heatux jw through it. The former case is simple to scale, �w/�.he latter is given by

w = −(∇�)w [J/m2 s], (6.10)

here is the heat conductivity. To calculate the flux, the tem-erature field must be known, which is not always the case.herefore, another expression for the flux is introduced, withelp of the empirical heat transfer coefficient kh:

w = kh(��)w [J/m2 s], (6.11)

here (��)w is the bulk–boundary temperature difference.quating (6.10) and (6.11) yields the formula for the coefficient:

h = −(∇�)w(��)w

[J/m2 s K]. (6.12)

A simple scaling of (6.12) by (/L) leads to the Nusselt num-er Nu (also Biot number, Bi):

u = kh

/L, (6.13)

hich is nothing but the dimensionless heat transfer coef-cient. The length scale L comes from the near-interfaceemperature gradient in (6.10), (�)w ∼ 1/L, and should relate,.g. to the thickness of the thermal boundary layer. Althoughu does not present any intellectual challenge on the groundsf scaling, it is the most desired quantity in the heat transfer,ince everyone wants to know how much heat passes throughhe interface, without computing the temperature and flowelds. Numerous correlations do exist for Nu in the engineer-

ng literature.

. Transport of mass

or the transport of mass of a solute in a moving environment,hree balances must be considered together: fluid mass bal-nce (5.1.1), fluid momentum balance (5.2.1), and the soluteass balance, which can be written as( )
∂c
∂t+ (v · ∇)c = D∇2c + r [kg/m3 s]

(unsteadiness) + (convection) = (diffusion) + (reaction). (7.1)

sign 8 6 ( 2 0 0 8 ) 835–868 853

Choosing the variables (c, v, r, x, t), their scales (C, V, R, L,T), and the parameters (D), we get by scaling:

C

T+ VC

L= DC

L2+ R [C/T]. (7.2)

Dividing by the diffusion term (DC/L2), because of tradition,we get

L2

DT+ LV

D= 1 + RL2

DC. (7.3)

We can divide by any term, but then the DN would not havetheir usual names. The diffusion term is the most typical forheat and mass transport phenomena, so it is natural to take itas the scaling basis. Assigning the DN their proper names weget

Fo + Pe = 1 + Da2. (7.4)

The following DN arise, having a clear physical meaning interms of process rates:


DT

(unsteadiness)(mass diffusion)

,


D

(hydrodynamic convection)(mass diffusion)

,

Damkohler number, Da =(

RL2

DC

)1/2(reaction)

(mass diffusion).

(7.5)

When using the basic time scaling, T = L/V, the Fouriernumber becomes the Peclet number, Fo = Pe. The unsteadyand convective effects are then comparable, as a direct con-sequence of our choice of scaling. This may not alwayscorrespond to reality. The Peclet number facilitates the cou-pling between the hydrodynamics and mass transfer: itcompares the transport by hydrodynamic convection (flow ofthe medium, V) and the molecular diffusion of mass (massdiffusivity D).

Since we customarily divided (7.2) by the diffusion term,the numbers are based on the diffusion rate. If we want othercombinations, we must compose them. These compositionscan contain diffusion and hydrodynamic quantities to reflecttheir coupling. One route leads to the Schmidt number:

Schmidt number, Sc = �

D

(momentum diffusion)(mass diffusion)

. (7.6)

The Schmidt number is prepared as follows. Take the Pecletnumber, Pe = LV/D. Replace the hydrodynamic convection (LV)with the hydrodynamic diffusion (�) with help of Re, LV = � Re,to get Pe = � Re/D. Divide by Re, since it is dimensionless, to getPe = �/D. Give the product a new name: the Schmidt number,Sc = Pe/Re (also: mass or diffusion Pr). This number is often usedin mass transfer processes, where the transport rate is largeenough to affect the flow (e.g. fast-phase changes, boiling, dis-tillation, etc.). In turbulence, the viscosity and diffusivity canbe not molecular but ‘turbulent’ (eddy viscosity; coefficient ofdispersion), hence turbulent Schmidt number.

Another route leads to the mass Grashof number, which istypical for mass-driven buoyancy effects:

Grashof number, Gr = ˇ �cgL3

�2


. (7.7)

and
Take the hydrodynamic Archimedes number, Ar = (��/�)Ga.Replace the general expression for the density difference ��

with a specific expression for the concentration expansivity offluid, ��/� = ˇ �c, where ˇ is the coefficient of concentrationalexpansion. Plug it into Ar to get Ar = (ˇ �c)Ga. Give it name themass Grashof number, alias the diffusion Archimedes number.Gr is encountered in flows with ineligible concentration gradi-ents, which may interfere with the flow (e.g. halinoconvection,double diffusive convection, thermosolutal convection). It isessentially a hydrodynamic number, where the mass entersas the ‘buoyant agent’, to produce density gradients.

A close derivative of Gr is the concentration (salinity, mass)Rayleigh number, Ra = Gr Pr:

Rayleigh number, Ra = ˇ �cgL3

�D


. (7.8)

The physical picture is similar like with the thermal Ra:concentrationally buoyant fluid parcel driven by �c moves,and diffusion of momentum (�) and mass (D) tend to opposethe motion and to weaken the driving force. Instead of ˇ �c,more simple choice �c/c0 can also be used. Mass Ra is decisivefor halinoconvection. Both thermal and mass buoyancy effectsare present in double diffusive convection, which, besides oth-ers, has application in oceanology (hot/cold, more/less saltywater) and geology (layering in magna chambers).

Besides the above coupling between diffusion and hydro-dynamics via Pe, there is also coupling between the diffusionand reaction via Da. A typical situation occurs in heteroge-neous catalysis, where the diffusion and reaction interplay.Solving the corresponding equation in case of a model situ-ation (a cylindrical pore in a catalyst, a spherical pellet), weobtain the concentration profile, whose mean value cm nor-malized by the bulk concentration c0 is the effectiveness factorF = cm/c0, which also is ∼(mean reaction rate)/(maximum rate).The model solution for a pore gives F ∼ tanh(Th)/Th, wherethe Thiele number (modulus) is Th = L(k/D)1/2. Here, L is thepore length and k the rate constant. Note that for r = kc, theDamkohler number coincides with the Thiele number, Da = Th.Often, a ‘generalized’ Th is introduced, to retain the last equal-ity also for reactions of higher orders. Reaction and masstransfer is combined in the Hatta number, Ha. There also arenumbers typical for the reaction kinetics itself. For instance,the Arrhenius number Ah compares the activation energy andkinetic energy of molecules.

Boundary conditions in mass transfer are of two kinds. Eitherthe concentration cw is given at the boundary ∂˝ or the massflux jw through it. The former case is simple to scale, cw/C. Thelatter is given by

jw = − D(∇c)w [kg/m2 s], (7.9)

where D is the (mass) diffusivity. To calculate the flux, the con-centration field must be known, which is not always the case.Therefore, another expression for the flux is introduced, withhelp of the empirical mass transfer coefficient km:

jw = km(�c)w [kg/m2 s], (7.10)

where (�c)w is the bulk–boundary concentration difference.Equating (7.9) and (7.10) yields the formula for the coefficient:

km = −D(∇c)w(�c)w

[m/s]. (7.11)

design 8 6 ( 2 0 0 8 ) 835–868

A simple scaling of (7.11) by (D/L) leads to the Sherwoodnumber Sh (also Sherman number, Sm):

Sh = km

D/L, (7.12)

which is nothing but the dimensionless mass transfer coef-ficient. The length scale L comes from the near-interfaceconcentration gradient in (7.9), (�)w ∼ 1/L, and should relate,e.g. to the thickness of the concentration boundary layer.Although Sh does not present any intellectual challenge onthe grounds of scaling, it is the most desired quantity in themass transfer, since everyone wants to know how much masspasses through the interface, without computing the concen-tration and flow fields. Numerous correlations do exist for Shin the engineering literature.

8. Correlations

In the preceding sections, many important DN were intro-duced and commented, and most of them are listed in Table 2.They can be divided into two classes. First, the basic (primary)DN that follows directly from scaling the balance equationsand their BC. Second, the other (secondary) DN that arederived from the basic, or formed by their combinations, orcreated ‘artificially’. The basic DN are the following: momen-tum transport (Eu, Fr, Re, Sr) and BC (Bo, Ca, We, Ma); heattransport (Fo, Pe) and BC (Nu); mass transport (Da, Fo, Pe) and BC(Sh). The basic numbers and their link to their closest relativesis shown in Fig. 4.

All the numbers obtained by the equation scaling (Section4) can be reproduced by the dimensional analysis (Section3). However, we must know beforehand the relevant physi-cal quantities that should be grouped. It seems that most ofDN in engineering were first obtained by DA. Here, we preferto relate them to the equations, to give them better physicalinterpretation.

Regardless of their origin, DN are used for making corre-lations. There are two main problems encountered. First, tochoose suitable DN. We need a complete list of independentnumbers. Second, to choose suitable characteristic scales toevaluate the DN. Both the numbers and the scales must berelevant for the problem. It is very difficult to choose themcorrectly without the sound knowledge of the underlyingprocesses and the physical meaning of the numbers. Here,the numbers generated by scaling of equations have a greatadvantage over those produced by dimensional analysis. Theycontain the correct quantities and have a clear meaning—ratioof different effects in terms of common physical quantities(force, rate, time, speed, etc.). However, the problem with thescales still remains. The procedure of equation scaling can givecertain hints what should the proper scales be, but this is notalways sufficient.

Some choices of scales are apparently wrong. For instance,consider the Bond number, Bo = �� gL2/�, which comes fromthe normal component of the free-slip boundary condition.This number is highly relevant for behaviour of bubbles inliquids. Accordingly, it enters numerous correlations for bub-ble size and speed, interfacial area, mass transfer coefficient,which are designed for bubble column reactors. Which lengthscale L is appropriate? The physics strongly recommends the
bubble size. Despite this, many authors have been using thebubble column size, which is apparently wrong. When we lookat books and review papers on bubble columns, it is easy to find

chemical engineering research and de

Table 2 – List of dimensionless numbers

Archimedes number Ar = (��/�)gL3/�2 (=(��/�)Ga)Bagnold number Ba = �p�L2/Biot number Bi = NuBodenstein number Bd = VL/Dax

Bond number* Bo = (�)�gL2/� (=We/Fr)Capillary number* Ca = V/�Cauchy number Ch = �V2/E (=Mc2)Cavitation number Cv = �P/�V2

Damkohler number* Da = (RL2/DC)1/2

Dean number Dn = Re(rpipe/rcurv.)1/2

Deborah number De = Tr/Tf

Eckert number Ec = V2/cp�

Ekman number Ek = �/˝L2

Eotvos number Eo = BoEuler number* Eu = P/�V2

Fourier number* Fo = L2/�T (heat)Fourier number* Fo = L2/DT (mass)Froude number* Fr = V2/gLGalileo number Ga = gL3/�2 (=Re2/Fr)Grashof number Gr = ˛ �� gL3/�2 (heat)Grashof number Gr = ˇ �C gL3/�2 (mass)Knudsen number Kn = Lmolec/Ldomain

Laplace number La = P/(�/L) (=Eu We)Lewis number Le = �/D (=Sc/Pr)Ljascenko number Ly = (�/��)(V3/g�) (=Re3/Ar)Mach number Mc = V/Vsound

Marangoni number* Ma = ��/VMorton number Mo = g4 ��/�2�3 (=We3/Fr Re4)Nusselt number* Nu = kheat/(/L)Peclet number* Pe = LV/� (heat)Peclet number* Pe = LV/D (mass)Prandtl number Pr = Peheat/Re = �/�Rayleigh number Ra = ˛ �� gL3/�� (heat) (=Gr Pr)Rayleigh number Ra = ˇ �C gL3/�D (mass) (=Gr Sc)Rayleigh number Ra = g′eL3/�mixDhydro (dispersion)Reynolds number* Re = �LV/ = LV/�Richardson number Ri = (��/�)(gL/V2) (=(��/�)/Fr)Rossby number Ro = V/˝LSchmidt number Sc = �/D (=Pemass/Re)Sherwood number* Sh = kmass/(L/D)Stanton number Sn = (kheat/V)(�/) (heat) (=Nu/Peheat)Stanton number Sn = kmass/V (mass) (=Sh/Pemass)Stokes number St = �pLV/Strouhal number* Sr = L/TVSebestova number Se = 1/McThiele number Th = L(kreac/D)1/2

Weber number* We = (�)�LV2/�

Basic numbers are marked by an asterisk (*) (�)� means � or ��.Note large diversity both in names and notation of dimensionlessnumbers in literature. Those used here are by no means the best orobligatory.

tmbiw

pifbistot

hat this mistake occurs in a great number of correlation for-ulas published over more than 30 years, some of them even

ecame the classics. These correlations are used for design-ng factories and plants, and they work well. Imagine how theyould work, if the correlations would be correct.

Some choices of scales are ambiguous. We suffer from theresence of more that one candidate for the length scale. For

nstance, consider the Rayleigh number, Ra = ˛ �� gL3/�� ∼ L3,or a natural convection in a horizontal fluid layer heated fromelow. In case of infinite layer of height H, the obvious choice

s L = H. In case of a layer confined also by two lateral wallseparated by distance A, both H and A may be chosen. Thus,here are several possibilities, L3 → H3, H2A, HA2, A3. Which
ne is correct? Finally, when the layer is inside a finite con-ainer A × B × H, the combinations grow. We can resort to the
sign 8 6 ( 2 0 0 8 ) 835–868 855

(rather subjective) argument of ‘importance’. For instance, wecan think that the smallest dimension is the most important.However, taking L = H3 and ignoring the lateral walls at a thinlayer, simply because A > H, gives a wrong result. The failure isespecially evident in stability considerations. The presence ofwalls reduces the spectrum of possible wavenumbers (modes)substantially, and in effect, stabilizes the layer with respectto the onset of convection. Therefore, the critical value of Radepends on both H and A. The stability issues are very sen-sitive to the proper choice of scales. For instance, the criticalvalue of Ra is different for rectangular and circular finite con-tainers, of the same size. Thus, the argument of ‘importance’may work, provided that it is physically based: resolve theprocesses occurring along different directions, find their inter-relation, and assess their relevance in a given situation. In caseof convection, the buoyant rise is vertical, and the heat andmomentum diffusion is horizontal, say, to the first approxi-mation. It is unlikely, that the same choice of scales appliesequally well to convection in infinite horizontal layer and inthin vertical slots.

Some choices of scales are even more ambiguous. By select-ing the scales, we can select the view of the world. Forinstance, imagine a two-phase flow in a long thin electrolysingmicrochannel of size A × B × C = 100 �m × 10 mm × 1 m. Inthese days of the scale-down boom, such an equipment isnot unusual. At the wall, there are electrodes and bubblesare produced by electrolysis. The spectrum of bubbles sizes isquite broad, from few microns, as they are formed, to few cen-timetres, as they coalesce. The point is to choose the properlength scale and to define Re for making correlation formulasdesigned for the operational quantities (e.g. liquid flow, bub-ble size and concentration, pressure drop, wall shear, etc.). Thechoice L = A means the side view at the channel. We see theflow between two virtually infinite parallel horizontal planes,100 �m apart. This situation is known as the Poiseuille flow.The choice L = B means the top view at a segment of the chan-nel. We see the flow between two finite and closely spacedwalls, 1 cm × 1 m in size. This situation is known as the flowthrough the Helle–Shaw cell. The choice L = C means the globalview at the channel. We see the flow between two finite par-allel plates, which are narrow and 1-m long. This situationcorresponds to the development of boundary layers in a rect-angular channel. In these cases, different flow profiles alongdifferent directions are relevant. What length scale shouldthen be chosen for the flow correlations? One way aroundthis severe anisometry (1:100:10,000) seems to be the hydraulicradius 2AB/(A + B) = 99 �m, which, however, leads to nowhere.Taking this figure actually means selecting the picture of aflow through a 99 �m dia capillary, which is far from beingrelated to any possible view at our flow situation. So far, theL for a single-phase flow has only been considered. The casewith the bubbles is left as a homework for students; enough totease them is to ask for introducing the correct Re for a bubblecolumn.

In correlations, the composed numbers are often used,obtained by combining several simple numbers with clearphysical meaning. The product, however, can have no clearmeaning. What can safely be combined? In (5.2.9), we combinetwo numbers, Fr and Re, generated by the same govern-ing equation (5.2.1), to compose Ga. It is acceptable, whenthey share the same scales. If not, then Ga would, forinstance, be Ga = Re2/Fr = {(�V′2/L′)/(V′′/L′′2)}2/{(�V′2/L′)/(�g)},
where we correctly discriminate between the inertial (′)and viscous (′′) scales for length and speed. Even with the


s (mo
Fig. 4 – Flow-chart of scaling governing equation
uniscale (L = L′ = L′′), we encounter the problem of DN alge-bra. Ga actually is {(inertia)/(viscosity)}2/{(inertia)/(gravity)}=(inertia)·(gravity)/(viscosity)2, which is in a variance with theusual interpretation (gravity)/(viscosity), where the powers areignored. Can we ignore them? This dubious algebra is alsoencountered when we try to eliminate (speed from Fr usingRe) or separate (speed and size in sedimentation using Ly)some inconvenient quantities. In this spirit, we can createany thinkable combination of DN, since their dimension equalsunity [1]. For instance, the combination Eu2 Re Fr would usuallybe interpreted as (pressure)/(viscosity)·(gravity). Upon insert-ing the physical meanings, it becomes: (p/i) · (p/i) · (i/v) · (i/g) =(p/v) · (p/g) = p2/vg, in brief notation (i – inertia, v – viscos-ity, g – gravity, p – pressure). With the viscous (p ∼ v) andgravitational (p ∼ g) scaling for pressure, we have: (vg/vg) =DNU. This means that the physical meaning of the combina-tion equals ‘unity’, the dimensionless number unity (DNU),since the physical meanings of the individual numbers ‘can-cel’. These manipulations are formally correct, but where is
the physics? Actually, the combination Re Fr Eu2 is equal toP2/g�V, where the group (g�V) is not very transparent, farfrom the expected (viscosity)·(gravity). Only after unravelling
mentum, heat, mass) and boundary conditions.

it, g�V = (g�L)·(V/L), we can recognized the gravitational andviscous scaling for P, see (5.2.7).

There are combinations of numbers originating fromdifferent governing equations, which can express their cou-pling (Pr = Pe/Re; Ra = Gr Pr), or only comparison of analogousmaterial properties (Pr = Peheat/Re = �/�; Sc = Pemass/Re = �/D;Le = Sc/Pr = �/D).

There are combinations of numbers coming from equa-tions and boundary conditions, which sometimes, fortunately,belong together (Bo = We/Fr; La = Eu We; Mo = We3/Fr Re4, Stan-ton (Margoulis) number Sn = Nu/Peheat or Sh/Pemass). TheMorton number is often used in bubbly research. A closer lookreveals that the physical meaning is: Mo = (i/c)3/(i/g) · (i/v)4 =(gv4)/(c3i2), in brief notation (c – capillarity), which becomes(gv)/(ci), at the usual ignoring of powers. In literature, Mo isinterpreted as capillarity/buoyancy, viscosity/capillarity, etc.Which one is correct? How then to interpret the Tadaki num-ber, Td = Re Mo0.23?

There can be combinations of numbers belonging to dif-
ferent physical contexts. Why not to eliminate the speed fromFr by the Mach number? Ga = Mc2/Fr thus obtained is formallycorrect, but hardly applies to a free rise of a bubble in a col-

nd de

ucm

ndFrniofleiftwrmtnair

flsiemcc

9

Itmp

srD1CDGI111SSZmtTccido


mn. There are no strict rules what can be combined and whatannot, and common physical sense is highly welcome whenaking these combinations.In correlations, some quantities or effects are often

eglected, as being ‘small’. It is tempting to say that inertiaominates when Re > 1. It is however not completely right.irst, this number is only a scale estimate of two forces, theoughest assessment we can have. There also are possibleumerical factors missing in this formula, reflecting the tun-

ng of Re for a particular flow situation. Its value also dependsn the choice of scales. Second, this number stands in theow equation at a dimensionless term that is ∼O(1) but notxactly = 1. Accordingly, it is better to say ‘small’ and ‘large’ Re,nstead of Re < 1 and Re > 1. Support for this suggestion comesrom the following. The laminar–turbulent (viscous–inertial)ransition occurs not exactly at Re = 1, but at about Re ∼ 10n,here n ≈ 3–4 for pipes, 5–6 for flat plate, and varies in a wide

ange for flow past different bodies. Likewise, the onset of ther-al convection does not occur exactly at Ra = 1, but at Ra ∼ 102

o 103 (depending on BC, etc.). The flow in microchannels isot completely free molecular at Kn > 1 and no-slip continuoust Kn < 1, but at about Kn > 101 and Kn < 10−2 say. Compress-bility effects are encountered not exactly at Mc = 1, but theecommended figure is about 0.3.

Conclude that building sensible and reliable correlations isar from being trivial. Besides choosing the proper dimension-ess numbers, one must also choose the proper characteristiccales. Choosing these scales is problematic due to our (eitherntentional or not) ignorance of the underlying physics, pres-nce of processes operating in different spatial directions, andultiscale nature of the system. Care should be taken when

ombining different numbers, and their origin and physicalompatibility should be kept in mind.

. Remark on literature

n sake of the text cohesion, there are almost no referenceso the literature through the previous chapters, and the com-

ents on the main information sources are lumped into thisart.

In the first part of the Reference section (1. Dimen-ional analysis), there are books fully devoted to DA and theelated problems like similarity and modelling (1.1. Books onA): Baker et al., 1973, Barenblatt, 1987, 1996, 2003, Becker,976, Birkhoff, 1960, Bluman and Cole, 1974, Bridgman, 1922,lement-O’Brien and Lawler, 1998, Craig, 2003, Curren, 2005,e Jong, 1967, Dolezalik, 1959, Duncan, 1953, Focken, 1953,ukhman, 1965, Hornung, 2006, Huntley, 1952, Ipsen, 1960,

saacson and Isaacson, 1975, Jerrard and McNeill, 1992, Kline,965, Kozesnik, 1983, Kurth, 1972, Land, 1972, Langhaar,951, Loebel, 1986, Massey, 1971, Murphy, 1950, Palacios,964, Pankhurst, 1964, Perry and Chilton, 1973, Porter, 1946,churing, 1977, Sedov, 1959, Skoglund, 1967, Stubbings, 1948,zirtes, 1998, Taylor, 1974, Weast and Astle, 1981, Zierep, 1971,lokarnik, 1991. These are useful for familiarizing with theany aspects of this powerful method, as well as with its his-

orical development and many practical examples of its use.wo book chapters are also included. One from the Perry’sanonical handbook, and the other from the Birkhoff (1960)lassical account on fluid mechanics. They represent two lim-
ts of the spectrum, with the practicality on one side, and theeep theoretical footing on the other. The important conceptf ‘similarity solution’ to spatio-temporal problems lacking
sign 8 6 ( 2 0 0 8 ) 835–868 857

explicit length scales is treated, e.g. in Bluman and Cole (1974),and, in brief, this topic is covered in many texts on partialdifferential equations. The uneasy concept of the interme-diate asymptotics, which transcends the traditional DA, isstrongly presented in the book of Barenblatt (1996), whichis based on the original Russian edition from the seventies.Its simplified version (Barenblatt, 2003) is especially suitablefor students. Besides the books, there is a selection of arti-cles on DA and its applications (1.2. Articles on DA): Astarita,1997, Boucher and Alves, 1959, 1963, Buckingham, 1914, 1915,Cheng and Cheng, 2004, Dodds and Rothman, 2000, Gunther,1975, Gunther and Morgado, 2003, Klinkenberg, 1955, Lykoudis,1990, Macagno, 1971, Prothero, 2002, Rozen and Kostanyan,2002, Sandler, 1970, Sjoberg, 1987, Stephens and Dunbar, 1993,Vogel, 1998, Wesson, 1980, West, 1984. The references relateto both the area of the chemical engineering and also otherresearch areas, to broaden the horizon, and for inspiration(biology, cosmology, ecology, economy, geology, medicine, psy-chology). The topic of DA and scaling is the firm ground of thechemical engineering literacy. Scale-up and design of tech-nologies is based on scaling consideration and active use ofDA in a variety of particular situations. Therefore, several ref-erences on scale-up are also presented (1.3. Books on scale-up):Euzen et al., 1993, Grassmann, 1971, Johnstone and Thring,1957, Stichlmair, 2002, Zlokarnik, 2006. The scale-up is incertain respect easier than the scale-down. Considering thelab scale ∼1 m, the big equipments are typically ∼101 m insize, i.e. variation within one order only. The microtechnol-ogy goes down to sub-micron ranges, i.e. more than 6 ordersbeyond our everyday experience. The chance that new phe-nomena will be encountered along this way is very high.The importance of surface phenomena, hence surface sci-ences, is not surprising. A long list of various DN is available,e.g. in Boucher and Alves (1959, 1963), Jerrard and McNeill(1992), Land (1972), Johnson (1998) and Weast and Astle(1981).

In the second part of the Reference section, the balanceequations for the transport phenomena are presented (2.Transport phenomena). They form the very core of the chemicalengineering and are in all texts of this sector, so it is needles tolist them all. There is a selection of sources that relates to thispaper. The local standard textbook by Mika (1981) was usedfor the equations and some basic scaling, and several otherbooks were also consulted (e.g. Aris, 1989; Bird et al., 1965;Carslaw and Jaeger, 1947; Crank, 1956; Cussler, 1997; Deen,1998; Rohsenov and Choi, 1961; Slattery, 1972; Slavicek, 1969;Thomson, 2000; Welty et al., 1969).

Most space in this paper is devoted to the momentumtransport, i.e. the fluid dynamics, both single-phase and multi-phase (Section 5). This field is close to the author, and alsounderlines the remaining two transport processes of heatand mass. There are several groups of authors writing aboutfluid mechanics, according to their background: pure andapplied mathematicians, physicists and engineers (chemical,civil, environmental, mechanical, metallurgy, mining, nuclear,urban, etc.). The first choose simple problems and solve themcompletely on the fundamental level. For the last, complexproblems are chosen, which can be solved only approximately.The engineering books are well-known to our community(e.g. Cengel and Cimbala, 2006; Massey, 1998; Munson etal., 1990; White, 1974; Wilkes, 1999), so only the others are
mentioned, when used. A more theoretically minded readermay wish to consult, e.g. Doering and Gibbon (1995), Sohr(2001), and Temam (2001). The common RTD treatment is pre-

and
sented in many standard books (e.g. Levenspiel, 1962). Thesmart variant of RTD is presented, e.g. by Ghirelli and Leckner(2004).

Rarely is found a title with proper and transparenttreatment of the interface boundary conditions within theengineering literature. The corresponding very importantnumbers (Bo, Ca, Eo, Ma, Mo, We) are usually presented eitherwithout any comment on their origin (hence physical mean-ing), or as a result of DA (again lacking the physical meaning).The book by Sadhal et al. (1997) was used, together with thatby Edwards et al. (1991). Instructive is the paper by Cuenot etal. (1997), which deals with a single bubble rise in a contam-inated liquid, where Ma and other DN play important roles.Macroscopic effects of a surfactant on real bubbly mixturesare demonstrated too (e.g. Ruzicka et al., 2008).

In engineering, the hydrodynamic stability is usuallyneither taught in courses, nor included in the books on hydro-dynamics. We tend to assume that things are mean, steadyand stable, since we want have them like this in appli-cations. However, this is not always the case. Few booksrelate to the stability issue, which is difficult but impor-tant (e.g. Chandrasekhar, 1961; Drazin and Reid, 1981; Drazin,2002—suitable for students). Equations of vorticity and enstro-phy are often used in turbulence (Davidson, 2004; Frisch, 1995;Pope, 2000), but not only there, since the vorticity is a subjectof its own importance (Saffman, 1992).

The rotational effects are presented by geophysicallyminded authors (Kundu, 1990; Tritton, 1988; see alsoPedlosky, 1982). Magnetohydrodynamics is not a usual partof our curriculum, and is mentioned only informatively(Chandrasekhar, 1961; Moffat, 2000). On the other hand, com-pressible and rarefied flows are commonly encountered, andthe idea behind Mc and Kn is familiar to engineers (manybooks exist on gas dynamics). Buoyancy effects in fluids areomnipresent and are covered by several texts, namely the ther-mal convection (Turner, 1979; Koschmieder, 1993). Analogybetween buoyancy effects in single-phase and multi-phasesystems has also been pointed out, and multi-phase Ra dis-cussed (Ruzicka and Thomas, 2003). A suitable ‘effective’density should be used to evaluate correctly the Archimedesforce acting on a body immersed in dispersion, dependingon a dimensionless parameter (Ruzicka, 2006). Rheology is avast area, with its own bibliography. The book by Barnes etal. (1989) is a suitable introduction into the field (De num-ber included). There are profound texts where dimensionalaspect are treated on the fundamental level (Astarita andMarrucci, 1974) as well as recent textbooks (Macosko, 1994;Morrison, 2001). A set of reviews on the present state of sev-eral important branches of fluid mechanics is also included(Batchelor et al., 2000). There is a wealth of literature on motionof animals in fluids (e.g. Childress, 1981; Pedley, 1977; Vogel,1998, 1994). With curiosities, the water-walking miracle of theBasiliscus lizards is also presented (Glasheen and McMahon,1996a,b).

It is not easy to find a general and accessible and phys-ically sound book on multi-phase flow (if any). There areonly several titles currently available in this growing area(e.g. Brennen, 2005; Crowe et al., 1998; Ishii and Hibiki, 2006;Kleinstreurer, 2003; Kolev, 2002; Soo, 1990), where the assis-tance was obtained from. Other sources were also consultedfor specific areas (Friedlander, 2000; Hinds, 1999; Nguyen andSchulze, 2004). The important issue is to formulate the stress
tensor and the interphase interaction force. This leads toinvestigation of the microstructure of dispersions, easier to
design 8 6 ( 2 0 0 8 ) 835–868

perform in the dilute viscous limit. The problems come withstrong multiple interactions in dense suspensions (e.g. Bradyand Bossis, 1988; Stickel and Powel, 2005). The exciting areaof granular flow and powder technology is a traditional part ofengineering (e.g. Brown and Richards, 1970; Nedderman, 1992)with deep roots in soil mechanics (e.g. Taylor, 1948; Terzaghi,1943). Presently, it witnesses a great boom after being ‘dis-covered’ by physicists, as a unique and highly specific stateof matter (Duran, 2000; Hinrichsen and Wolf, 2004; Forterreand Pouliquen, 2008). The concept of Ba number is useful ingranular flows.

In the third part of the Reference section, several importantparticular areas are covered in very brief (3. Other topics). Thephenomenon of ‘interface’ becomes increasingly importantnowadays, owing to the progressive scale-down, where the(volume/area) seems to be a more relevant parameter than theusual (area/perimeter), known as hydraulic diameter. There-fore, few references to the surface aspects are mentioned(3.1. Surfaces and surfactants): Adamson, 1960, Adamczyk, 2006,Davies and Rideal, 1963, Fawcett, 2004, de Gennes et al., 2004,Israelachvili, 1992. These are useful for multi-phase flows aswell as for microflows, and were employed at writing aboutthese topics. Microtechnology surely deserves a little sectionof references (3.2. Microflows and microsystems), which wereused for preparing Appendix C: Berthier and Silberzan, 2005,Bruus, 2008, Ehrfeld et al., 2000, Gad-el-Hak, 1999, Hessel etal., 2004, van Kampen, 1992, Karniadakis and Beskok, 2002,Karniadakis et al., 2005, Kockmann, 2006, Li, 2004, Madou,2000, Maynard, 2008, Neto et al., 2005, Nguyen and Wereley,2002, Slattery et al., 2004, Stone et al., 2004, Tabeling, 2005,Thompson and Troian, 1997; see also the periodical “MicrofluidNanofluid” and others. They are valuable sources of informa-tion, reflecting the state-of-the-art in this inflating area. Theeffect of ‘rarefication’ of gases and ‘granulation’ of liquidsneeds the adequate description, where gas/molecular dynam-ics and stochastic processes are the vital ingredients. DA andSE on microscale are considered, e.g. in Kockmann (2006,chapter 2). Several links to biological systems are also offered(3.3. Biology and biosystems), related to the material presentedin Appendix C: McMahon, 1973, Pilbeam and Gould, 1974,McMahon and Bonner, 1983, Hjortso, 2005, Nopens and Biggs,2006, Thompson, 1943, Perthame, 2006, Ramkrishna, 2000. Thescaling concept is well known for biologists, which may besurprising for engineers. The population balance modelling isalso included into this short section, since the term ‘popula-tion’ has strong biological connotation. Self-similar aspectsof these models are treated, e.g. by Ramkrishna (2000). Inbiology, thoughts of hierarchy of scales, structures, and func-tions are most appealing. Many penetrating perceptions of theearly times were later physically based and formalized. Thepressure from the micro-needs pushes engineers into thesedangerous waters, where long-term everyday experience andintuition, which are our traditional arms, may be near-useless.Few items were therefore selected, for the engineer to gain theinspiration from the philosophers, and to combine it with therigour of the physicists (3.4. Multiscale science and hierarchy):Bergmann, 1944, Furusawa and Kaneko, 1998, Garnett, 1942,Glimm and Sharp, 1997, Henle, 1942, Hoover and Hoover, 2003,Li and Ge, 2007, Li and Kwauk, 2004, Koplik and Banavar, 1995,Lowry, 1974, Marin, 2005, Pattee, 1973, Sewell, 2002, Simon,1965. The last subsection (3.5. Education) is devoted to thepedagogical aspects of DA and relates directly to Appendix
B: Andrews, 1984, Bloom, 1956, Cadogan, 1985, Calder, 1984,Churchill, 1997, Comenius, 1657, Imrie, 1968, Krantz, 2000,

nd design 8 6 ( 2 0 0 8 ) 835–868 859

Ka

1

Ascsabstmtcabs

A

Timcct1Ib

Aa

Tciataot((

A

Vttfesgstog

st

Fig. A1 – Demonstration of IA. Surface wave generated byvibration of triangular element on surface of originally stillliquid confined in rectangular container (top view). Wavemoves from generator to boundary. Shape of wave (dottedline): triangular (phase G) → circular (phase I) → rectangular


rathwohl et al., 1964, Marzano and Kendall, 2006, Morrisonnd Morrison, 1994, Sides, 2002.

0. Conclusions

brief guided tour through the field of scaling and dimen-ionless numbers is presented, with respect to their use inhemical engineering. The numbers are listed in Table 2. Twoources of these numbers are considered, the dimensionalnalysis and the scaling of governing equations with theiroundary conditions. The apparent advantage of the equationcaling is twofold: (i) we know the relevant physical quanti-ies and (ii) the numbers thus obtained have clear physical

eaning. This meaning must be kept in mind when usinghe numbers and the scales they involve for making empiricalorrelations. The mutual relations between different numbersre highlighted. A flow-chart of numbers closely related to theasic equations for momentum, heat, and mass transport ishown in Fig. 4.

cknowledgements

he author will highly acknowledge the reader’s forbearancen case of possible mistakes, errors and shortcomings that

ay occur in this text (like in any other one), and will sin-erely appreciate comments, suggestions, and constructiveriticism that could help him to produce something much bet-er in the future. The financial support by GACR (grant nos.04/06/1418, 104/07/1110), by GAAV (grant nos. IAAX00130702,AA200720801), by MSMT CR (grant no. KONTAKT ME 952) andy AVCR-CNRS (grant no. 11-20213) is gratefully acknowledged.

ppendix A. Concept of intermediatesymptotics

his appendix is to offer the reader a brief exposure of the con-ept of the intermediate asymptotics (IA). The presentation isntermediate-precise, in the sense that it is more precise thanpopular text aimed at the common public, and, at the same

ime, less precise than it should be for its physical correctnessnd mathematical rigour. Only few, hopefully typical, aspectsf IA are mentioned, without claim of generality and exhaus-iveness. The text is based on the two books by Barenblatt1996, 2003), with some demonstrative examples by the authorsee figures).

.1. Motivation example

aguely speaking, the intermediate asymptotic is aime–space dependent solution of an evolution equationhat already forgot its initial conditions, but still does noteel the limitations imposed by the system boundary or byxtinguishing its internal dynamics. Consider a body of atill water in a rectangular tank. Let the surface waves beenerated at the centre, by an oscillating triangular element,ee Fig. A1. Near the centre, the waves are triangular sincehey bear the fingerprint of the initial condition, the shape
f the generator (phase G). As they propagate outwards, theyradually obtain the natural circular shape,4 being undis-
4 Why the ‘circular shape’ is natural? A mathematician woulday because of the symmetry reason; a physicist would say thathere is no force to make it non-circular and, even if, the circle is

(phase B). Circular wave is intermediate asymptotic (IA).

turbed by either boundary. This happens at the intermediatedistance between the container centre and container wall(phase I). As the waves begin to feel the container walls, thecircles turn into rectangles, to accommodate to the shape ofthe boundary (phase B). The solution describing the ‘happycircles’ is the intermediate asymptotic (IA) of the system:demonstration of pure physics, unaffected by geometricalconstraints, by the past and future.

The three stages (G, I, B) correspond to three differentregimes of the system behaviour, where different effects,hence variables, come to play. DA is able to cope with a sin-gle regime only. The easiest is the intermediate regime (IA)where besides the physics, which is in all of them, no addi-tional variables are present to account for the geometry ofwave generator and the container boundary. DA may be ableto describe certain rough aspects of the intermediate stage,by providing scale estimates of some main features of the cir-cular waves. Whether DA can really do so, depends on thephysical nature of the problem and on the type of the physi-cal quantities involved. There can be a solution in form of IA,which cannot be obtained by DA. In this sense, the concept ofIA transcends the concept of DA, as pointed out in Section 3.2of the main text. DA applies to problems having the completesimilarity.

A.2. Two kinds of similarity

A.2.1. Complete similarity (CS)The relation between the master quantity and the variables,u = f(a1,. . .,k, b1,. . .,m) can be converted using DA into the dimen-

a stable configuration; a chemical engineer would argue that onehas seen only circles, so far. They are all basically right.

and
sionless form:

˘ = ˚(˘1, ˘2, . . . , ˘m), (A.1)

where we want to reduce the number of variables by neglect-ing some Pi-terms. Usually, too small and too large termsare omitted, without further justifications. Correctly, thebehaviour of ˚ in the limit of ˘ i → 0 or ∞ must be investigated.The neglection is permitted only when ˚ tends fast enougha finite non-zero limit. Then it is possible to replace ˚ withanother function of less variables representing the limit of ˚:

˘ = ˚1(˘1, ˘2, . . . ˘h<m) (complete similarity). (A.2)

Since ˚ is usually not known beforehand, at least an a pos-teriori check is in order. The case where we can go from (A.1)to (A.2) is called the complete similarity (CS), or the similarityof the first kind of the phenomenon in the neglected variables(h + 1, h + 2, . . ., m). In reality, this is a rare situation and a worsecase is encountered.

A.2.2. Incomplete similarity (IS)If ˚ behaves badly, the condition for CS is not satisfied andthe Pi-terms cannot be neglected. They influence the problemregardless how small or big they are. But even here a sim-plification can exist. If our problem possesses certain specialfeatures, it may be possible to take some Pi-terms out of ˚, asthe power-law factors. In this case, (A.2) becomes

˘ = (˘˛,h+1h+1 · · ·˘˛,m

m )˚2

(˘1

˘ˇ,1h+1· · ·˘ı,1

m

, . . . ,˘h

˘ˇ,hh+1· · ·˘ı,h

m

), (A.3)

which can briefly be written as

˘∗ = ˚2(˘∗1 , ˘∗

2 , . . . , ˘∗h<m) (incomplete similarity). (A.4)

The number of arguments of ˚ is reduced at the price ofthe power-law factors and several unknown exponents (˛n,ˇn, . . ., ın). Such a case is called the incomplete similarity(IS), or the similarity of the second kind of the phenomenonin the extracted variables (h + 1, h + 2, . . ., m). Although (A.2)and (A.4) are formally similar, there is a substantial differ-ence. The former witnesses the generalized homogeneity ofthe dimensional function f, which is a consequence of thegeneral physical covariance principle (the platform on whichDA operates). The latter witnesses the generalized homogene-ity of the nondimensional function ˚ itself, which is a luckycoincidence, the consequence of the presence of some specialproperties of the problem under study. Neither the extractedvariables nor the exponents in (A.3) can be obtained by DA,in principle. They must be found by some other means, e.g.by numerical solution of the full model, by experiments. Thefollowing two special cases of (A.3) can be met:

˘ = ˘˛m˚2

(˘1

˘ˇm

, ˘2, . . . , ˘m−1

),

˘ = ˘˛m˚2(˘1, ˘2, . . . , ˘m−1),

(A.5)

where only one small or large Pi-term spoils the complete sim-ilarity, namely the term ˘m, which stems from the variable
bm. Note that IS is a much weaker quality than CS, yet morewidespread. On the other hand, even IS is still a rather excep-tional property. It seems to be difficult being more quantitative
design 8 6 ( 2 0 0 8 ) 835–868

about the distribution of special properties within the natu-ral phenomena we observe, and in models we use to describethem. A rule of thumb says: The ‘better’ the property, therarer it occurs. Consequently, we may expect that the typicalcase will be the lack of similarity where no further simplifi-cation of (A.1) is possible. It should be attacked directly, viaexperiments, numerical calculations, or approximate analyt-ical techniques.

A.3. Two kinds of self-similarity

In many cases, (A.2) represents the final result produced byDA, which can be re-written as

u = U · ˚(˘1, ˘2, . . . ˘m) (output of DA). (A.6)

In most of our applications, DA finishes here and deliversthe scaling law u ∼ U. The tuning function ˚ is then obtainedby measurements. These applications typically describesteady states of complex processes, in complex geometries,time–space averaged problems with lumped parameters. Lessoften are studied evolution problems in space and time, prob-lems with distributed parameters. Certain class of problems,in a certain intermediate range, admit self-similar (similarity)solutions, in terms of self-similar variables. These variablesfor problems with complete similarity can be found by DA.Consider a problem described by a complicated governingequation for function u = u(x, t). The scaling law (A.2), (A.6)obtained by DA can be written as

u

U= F

(x

X, ˘i

)(self-similar ansatz for CS), (A.7)

where X(t) and U(t) are the self-similar variables. They arethe scale estimates for the spatial coordinate x and the mas-ter quantity u(x, t) that ensure the geometrical similarity ofthe spatial profile of u. Putting (A.7) into the original govern-ing equation, we get a simplified equation that solves for thefunction F(�), where � is the single compound variable, x/X, pro-vided that the ˘ i-terms do not interfere. The solution u/U = F(�,˘ i) is the exact solution to the simplified equation, and iscalled the self-similar solution of the first kind. Because the sim-ilarity solution holds only in the intermediate range wherethe problem lack internal length scales, it represents IA of theoriginal problem.

When our problem has not the complete similarity, butonly the incomplete similarity, the similarity variables mustbe produced by means other than DA. Namely this holds forthe scaling of the master quantity (A.3). In the simple case of(A.5), we have

u

U= ϕ˛F

(x

X, ˘i

)(self-similar ansatz for IS), (A.8)

where the dimensionless parameter ϕ typically contains infor-mation about the initial stage of the process. Putting (A.8) intothe original governing equation, we get a simplified equationwhose solution is called the self-similar solution of the secondkind. The solution typically leads to the nonlinear eigenvalueproblem (NEP) for the function F:

N(F) − F = 0 (nonlinear eigenvalue problem), (A.9)

where N is a nonlinear operator. The eigenvalue dependson the exponents of IS (˛n, ˇn, . . .). Often, nonlinear differen-


Fig. A2 – Guidance for application of DA and similarityanalysis. Right branch deals with ‘idealized’ problem, whereproblematic variables are neglected. Dimensionlessformulation follows from DA. By complete similarity (CS),variables are fewer and similarity law is obtained. Withgoverning equations (GE), self-similar solution (first kind)can be found. Left branch deals with ‘realistic’ problem.Incomplete similarity (IS) is encountered and self-similarsolution (second kind) is obtained by nonlinear eigenvaluep

ttimpms

A

Deciupaictid

Fig. A3 – Relation between self-similar solution and IA.Real process with three stages: initial generation (G),intermediate stage (I), final stage affected by boundary (B).Self-similar solution for simplified problem is in wholerange and exact; for real problem it is approximation in

the self-similar solution of the second kind is equivalent tothe indeterminacy of the group, and can be interpreted as the

Fig. A4 – Nesting of similarity concepts. DA: dimensionalanalysis; CS: complete similarity; IS: incomplete similarity;

roblem (NEP) (based on Barenblatt, 1987, 1996, 2003).

ial equations must be solved, with more boundary conditionshan is due (overdetermination). Such a special value of

s seeked, for which the solution does exist. The qualitativeethods for nonlinear systems can be used, and the phase

ortrait of the problem can be analysed. Due to the overdeter-ination, the solution can correspond a singular objects (e.g.

eparatrix line). The flow-chart summary is given in Fig. A2.

.4. Relation between DA and IA

A is a simple effective method for finding the rough scalestimate of a master quantity, u ∼ U, in problems having theomplete similarity. This estimate can be used for construct-ng the self-similar variables. IA is a general concept, a kind ofniversal behaviour that many systems of different origin canroduce. It is a spatio-temporal phenomenon, existing withincertain intermediate range of the independent variables. It

s an approximate solution to a complex problem, valid in aertain range. It can be represented by the self-similar solu-ion, which is the exact solution to a simplified problem, valid
n the whole range, see Fig. A3. The action radius of the above-iscussed concepts is shown in the diagram in Fig. A4.
intermediate range.

A.5. Beyond IA?

There may exist even a weaker type of similarity than IS, inthe solutions to the general evolution problems on interme-diate ranges of time and space (or even more sophisticatedvariables) that will be sufficient to earn the status of IA, atleast in the intuitive sense, see the grey zone in Fig. A4. If theopposite is true, the grey zone shrinks to zero. Paradoxically,the opposite may be true owing to the terminology reasons. Ifthe notion of IA is anchored in the fact that the weakest simi-larity that IA can bear is IS, there is no room for anything else.Such a definition finds its support in the fact that the conceptof similarity is equivalent to the invariance of the governingequations with respect to certain groups of transformations(symmetry). Since the nondimensionalization and scalingmeans changing the norm of the measuring units (etalons),the important role of the renormalization group is a little sur-prise. With such a definition, IA and the group that reflectsthe actual degree of symmetry in the problem are equivalent.Consequently, the insolubility of the IS problem in terms of

IA: intermediate asymptotic; AI: absence ofintermediateness. Phenomena fall into two classes: AI andIA. Within IA, further division is possible.


Fig. A5 – Life cycle of population. Genesis at early times (G),internal dynamics in intermediate range (I), decay due tobounds set on its existence (B). In middle range, IAbehaviour is expected. Vertical axis: characteristicparameter describing state of system (large ensemble of

Fig. A6 – Life cycle of individual. IA—unaffected bybeginning and end (the picture is reproduced with kind

interacting units; evolution of biological species,population, culture, civilization, empire).

absence of IA, in the given problem. The insolubility can becaused by unavailability of certain integral quantities, eitherexplicit or implicit, reflecting the conservation principles.Their absence can be for several reasons: they do not exist,they do exist but we do know them, we do not know whetherthey do exist. In mechanics, which is an axiomatic discipline,5

we can prove or disprove their existence, which leads to theconsistent statement about the existence of IA. In other sci-ences, where we lack this strong axiomatic footing, we face theundecidability. This links back to the need of an operationaldefinition of IA, suitable also for other and less formalizedresearch areas, where the ‘governing equations’ are eitherfully absent, or reflect only a small fragment of truth. Exer-cising them would be tempting for applied mathematicians,but the results obtained would likely be largely misleading.

A.6. Broader horizons

In his books, Barenblatt mentions that although the concept ofIA is a part of the mathematical physics, it has important sig-nificance for variety of general situations that are multiscalein nature. As an example, he presents few situations from oureveryday life, to demonstrate this conceptual generality (e.g.perception of visual art, visual perception generally, analogyin poetry, intermediate description of historical events, etc.).

To contribute to these efforts, the following three exam-ples are introduced, that hopefully comply with the conceptof IA, at least on the intuitive level. The first shows the nat-ural cycle of a culture or civilization, see Fig. A5. It takes offfrom zero, builds up, reaches a status quo (‘sustainability’), andeventually degenerates. The intermediate stage can be consid-
ered as IA, because here the mechanisms driving its internaldynamics are revealed. The culture has already reached cer-
5 From the fundamental point of view, the mechanics is scienceabout something that does not exist. There are no elementaryparticles (‘mechanons’) that would mediate the mechanicalinteractions. Mechanics merely is a demonstration of theelectromagnetic and gravitational forces on the macroscale. Onthe same argument, the physicists would state that Romeo andJuliet were not in love: there are no elementary particles(‘loveons’) mediating this type of interaction.

permission by ‘The Bhaktivedanta Book Trust’).

tain degree of independence and autonomy, while it is still farfrom its inevitable end (e.g. internal decay; the Roman Empirein the past, the West in the near future).

What happens with big populations (phylogeny) can repeaton the level of a one single individual (ontogeny). This isdemonstrated by the second example. In Fig. A6, we see theyouth, maturity and age. The child bears the fingerprint of itsbirth and the care delivered by the surrounding, to facilitateits early existence (phase G). The adult is believed to be ableto behave at least to a certain degree independently of theexperience from the early period, according to one’s free will,if any (phase I). Getting older, the feeling of the presence ofthe severe upper bound on the length of the current life cyclecomes to play, and affects our behaviour strongly (phase B).

To top this appendix on the philosophical level, let us followthe third example. By convention, the common people distin-guish three parts of the time axis: past, present, and future,see Fig. A7. The present is represented by one singular pointonly, separating the vast past from the equally vast future.Note that both the past and future do not exist: ‘past’ alreadywas, ‘future’ will only be. The practical relevance of this pic-ture is that we keep living in a virtual world, generated ofour memories, and bounded by our future expectations andplans. The mental training suggested by many a philosophi-
cal schools is aimed at expanding the present, since it is theonly reality we can actually perceive, and the way how to lib-erate ourselves from the diktat of the past and the future. The

chemical engineering research and de

Fig. A7 – Mental training: development of extendedpi

ec

At

Tipapyt

b

o

•••

•

mc

•

•

a

erception of present, in form of IA. Singularity of presents unfolded, by suppressing the virtual past and future.

xtended present is thus our IA, which, in its ultimate form,an be experienced as the timelessness.

ppendix B. Suggestions for using andeaching DA

his appendix concerns the problem how to use DA and sim-larity analysis, which may also be useful for educationalurposes. The basic facts about usage of DA are widely avail-ble in the open literature (see the Reference section). Theractical recipe below for the application of the similarity anal-sis is taken from Barenblatt (1987, 1996, 2003). A selection ofhe educational literature is in References section 3.5.

DA—What it is, how it works?It is a very standard issue, with enormous coverage by

ooks, handbooks, monographs, etc.See Sections 2 and 3 of this paper.DA—For which purpose?We can use DA for several reasons, differing in the expected

utputs:

scaling law for a master quantity;similarity law for modelling and scale up/down;similarity variables needed for seeking similarity solutionof governing equations;grouping parameters into DN to reduce experimental dataand to make correlations.

Note: Obtaining the scaling law for a quantity may, but alsoay not be our ultimate goal. With these scale estimates we

an build valuable models.DA versus scaling of equations (SE):

DA◦ does not give any variables (they must be chosen subjec-

tively, relevance not assured);◦ does give dimensionless numbers (all combinations, rel-

evance not assured);◦ does give relation between the numbers (scale-estimate

of master quantity).SE◦ does give relevant variables (from equations, initial and

boundary conditions);◦ does give some relevant numbers (maybe not all, equa-

tions are only a ‘model’ of reality);◦ does not give relation between the numbers.

Recommendation. First obtain relevant numbers by SE, thenpply DA to find similarity laws.

sign 8 6 ( 2 0 0 8 ) 835–868 863

DA—Recipe for similarity analysis (after Barenblatt; slightlyadapted, see Fig A2):

1. Specify relevant variables (using model equations; chooseyourself).

2. Choose system of units, choose variables with indepen-dent dimension (those most relevant).

3. Apply DA to get similarity law, ˘ = ˚(˘1, ˘2, . . ., ˘m).4. Choose suitable scales for your problem, estimate magni-

tude of Pi-terms (small/large?).5. Assume complete similarity and cancel small/large terms.

Check result versus data. Problems?6. Assume incomplete similarity and cancel small/large

terms. Check result versus data.7. Formulate similarity and scaling laws with fewest vari-

ables [final output I].8. Formulate similarity variables, use model equations, find

similarity solution [final output II].9. Relate similarity solution to IA behaviour of your problem,

delimitate intermediate region.10. Does your result comply with data? Can it be generalized?11. Improve this recipe, based on your own experience.

DA—GlossaryFor the reader’s convenience, a brief glossary of frequent

terms is also included. The expressions are listed, as theyappear in the text (place of definition):

• similarity parameter (3.1.5) (also: Pi-term, ˘-term, dimen-sionless number);

• similarity law for ˘ (3.1.5);• similarity law for a (3.1.6);• scale estimate for a (3.1.7) (also: basic scaling for a, scaling

law for a);• similarity theory (Section 3.2) (also: similitude theory, mod-

elling, scale-up/down);• similar systems (Section 3.2);• similarity law for modelling (3.2.3);• similarity criterion for modelling (3.2.3);• scaling law: power-law dependence, y = axb;• scaling rule for modelling (3.2.3);• scale coefficient for modelling (3.2.3);• scale (4.3) (also: characteristic/typical/representative quan-

tity, parameter, value);• scale equation (4.3);• complete similarity (A.2);• incomplete similarity (A.4);• self-similar: variable, coordinate, solution (A.7) and (A.8)

(also: similarity variable/coordinate/solution).

Teaching aspects in general are covered by wealth of ped-agogical literature devoted to education at the tertiary level(university), spanning several centuries. Starting with theparadigmatic treatment by the father of modern educationComenius (1657), we can arrive at the numerous volumes ofthe present literature. To name one, Bloom (1956; see alsoKrathwohl et al., 1964, Marzano and Kendall, 2006) is espe-cially useful at preparing the knowledge tests for students.The taxonomy indicates how difficult our questions are, andwhich students skills are required for generating the answers.Students should not only be taught “Know how”, but also
“Know why”.
Teaching aspects in particular are broadly covered by thequarterly published periodical “Chemical Engineering Educa-

and

(distance/path) ∼ (�gas/�liquid)1/3. Another effect is the molecu-lar slips of fluid at rigid wall, where the no-slip BC condition


tion”. There are articles directly related to teaching DA. Writtenby professional teachers and active researches at the sametime, they are of great didactical value (see e.g. Andrews, 1984;Churchill, 1997; Krantz, 2000; Sides, 2002; also Imrie, 1968). Toattract the students attention to problems of scales, few pop-ular books are available too (e.g. Cadogan, 1985; Calder, 1984;Morrison and Morrison, 1994).

Postscript. Students are forced to produce a number ofpapers to defend their PhD thesis (“Publish or perish”). Toreduce the increasing information noise in the peer-reviewedliterature, remember also the complementary saying: “Betterto perish than publish rubbish”.

Appendix C. New areas in chemicalengineering

Far be it from the author to tend to have any kind of visionaryambitions. Instead, some comments are presented, related tothe current topics suggested as being topical.

C.1. Microreactors and microfluidics

C.1.1. Microsystems in chemical technologyFlows and transport phenomena in microscopic andnanoscale channels are under intense research. The typ-ical features are: low Re regimes, strong surface effects, breakdown of continuum concept, multiscale nature, presence ofmany kinds of forces. There are some aspects already known,that make the micro-world different from ours. Others areonly awaiting their discovery. We still have the well-definedphysical quantities and the power of the conservationequations, so that both DA and SE can be applied.

In chemical technology, the microreactors enable bettercontrol of system behaviour. The flow is usually laminar (lowRe owing to small L and V), with all the advantages of theStokes equation (linearity hence superposition, reversibility,strong theorems—minimum dissipation, reciprocity). Suchflows may not be prone to hydrodynamic instabilities. On theother hand, we lack the advantage of the effective large-scaleconvective mixing (strong bulk turbulence), and the flowbasically is in the boundary layer regime. This increases theresistance to the transport processes. They must be enhancedby clever design of the system geometry and flow. Thanks tothe fast heat transfer in systems with large (surface/volume)ratios ∼L2/L3 ∼ 1/L, we can manage the exo/endo-thermicreactions. The mass diffusion can be complemented bymicro-convection in several ways (micromixing). Controllingthe transport and reaction, we can improve the selectivityand increase conversion. This higher efficiency togetherwith safe smaller units close to user might compete withthe huge volumes produced by present plants on one spot.As ever, common sense should be used to prevent us fromminiaturizing everything.

C.1.2. Prevailing forcesThere are different kinds of forces, with different ranges ofaction. The shortest are the interaction forces between twosmall molecules in vacuum. The force range usually increaseswith the molecule size, polarity, number of molecules, and islarger in material environment, where strong cumulative andcollective effects can play a role. Starting from few nanome-
tres (nm), these molecular forces in large ensembles can reachto ∼100 nm, say. When the system size is larger than this fig-ure (microchannels), the molecular forces can be considered
design 8 6 ( 2 0 0 8 ) 835–868

as localized, i.e. concentrated in a plane of zero thickness. Wehave the usual ‘macroscopic’ description, with the commonsurface effects. When the system size is comparable or smallerthan this figure (nanochannels), the molecular forces cannotbe considered as localized; they are distributed in space andtime. We have to abandon the usual ‘macroscopic’ description,and develop something smarter. An analogy emerges here.In macro-hydrodynamics the boundary effects can be local-ized into a thin boundary layer negligible with respect to thebulk volume, while in micro-hydrodynamics the whole bulkis the boundary layer. In micro-hydrodynamics the molecu-lar forces can be localized into a thin surface layer, while innano-hydrodynamics, these forces penetrate the whole bulk.Note that the common shear viscosity increases enormouslyon nanoscale, when the sheared fluid layer is only severalnm thick. The common surface tension � decreases, whenthe drop size shrinks to nanometric scales. Both and � aremacroscopic quantities that can loose their usual meaningwhen transferred from large local-equilibrium ensembles onto sparse families of particles, out of statistical balance.

It is useful to know how different forces and physical quan-tities scale with the length L (system size). Expectedly, the bodyor volume forces fall quickly with decreasing L (gravity, iner-tia, centrifugal, etc.). In contrast, the molecular and surfaceforces become important. Consequently, the force equilibria inmicrosystems are created by a balance of forces others than weare used to. In small systems, intense electrical fields can beproduces, and important electro-phenomena occur (electroos-mosis, electrophoresis, streaming potential, sedimentationpotential, dielectrophoresis). The Debye length becomes therelevant length scale. The force balance may result from inter-play between the viscous, pressure, and electrostatic forces.The electrostatic-Ra may then appear. New kinds of flow insta-bility can also be produced.

C.1.3. Governing equations and boundary conditionsThe continuum approach is based on the notion of the ‘fluidparticle’, a virtual mesoscale object having the proper numberof molecules, not too small, not too large. This alibistic defi-nition may well work for students, but largely fails when wehave to know how this ‘particle’ compares with the size of ourmicrochannel. Since this basic concept is presented in the firstlecture on hydrodynamics, all of us surely know how big it is.When it is smaller than the system, the continuum approachcan be applied, and vice versa. Since gases are thinner thanliquids, their ‘particles’ must be larger. Consequently, gasesare more prone to discontinuous behaviour in microsystems.Note that all fluid properties must be continuous.6 There maybe different length scales for kinematics (speed, acceleration),thermodynamics (pressure, density), transport (diffusion), etc.Also, even when the fluid is continuous, the flow may be not(shocks, extreme shearing, etc.).

There are some recommendations for microflows in termsof the Knudsen number. One effect is rarefication, which likelyoccurs when Kn is larger than ∼10−3 to 10−2. This holds forgases, where the free path is a well-defined concept. In liq-uids, it is not so, and the molecular interaction distance canbe taken instead of the free path. This distance could be taken10× smaller than the path, based on the density argument,

6 See e.g. Nguyen and Wereley (2002), Section 2.1.3.

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Literature used and recommended: References section 3.4.


oses validity. The continuum approach and no-slip BC canpply for Kn < 10−2, say. The continuum approach and free-slipC is recommended for Kn ∼ 10−2 to 10−1. The Navier–Stokesails for Kn ∼ 100 and other models should be used (e.g. Bur-ett and Woods equations). Statistical approach (Boltzmannquation) or molecular dynamics of free discrete particleshould be employed for Kn > 101. The Knudsen number canerve as the expansion parameter in building approximateodels. Different effects can be important for gases (rarefica-

ion, compressibility, viscous heating, thermal creep, Knudsenump, diffuse/specular reflection at wall) and liquids (wet-ing, adsorption, eletrokinetics, hydrophobicity). Other effectsan be common (entrance effects, fluid–structure interactions,elaxation times upon disturbing, surface roughness).

The diversity and novelty of micro-phenomena needew forces to be included, modified equations and adequateoundary conditions. After scaling, they will produce newr different DN, than we would expect. For instance, the slipoundary condition on rigid wall for compressible flows leadso Ec, Kn, Pr, Re (cf. with numbers in Sections 5.4.1 and 5.4.2).ther numbers may be encountered too (e.q. Squeeze num-er, Bearing number, etc.). Some DN important on macroscaleay loose their relevance. The common DA is expected toork also for microsystems, one difficulty being our much

ess experience with the world of small, where things ofteno straight against our intuition. It is exactly the experiencend intuition that facilitate the crucial step in DA, the choicef relevant variables.

Literature used and recommended: References sections 3.1nd 3.2.

.2. Biosystems

iology witnesses long-term tradition in using various scal-ng laws. Observations suggest that animals are small andig, slow and fast, eating little and much, etc. Efforts werepent to relate these properties and to find some rules.esulting empirical correlations shows power-law depen-ence between various quantities (body size, body weight,roportions, metabolic rate, heat production, characteristiciological times, etc.). Metabolic heat is produced in theody bulk ∼L3 and is released by body surface ∼L2. The

loss/production) ratio is 1/L, indicating that big animals mayave problem with overheating, while small with under-eating (beyond the limit of thermal regulation, they areold-blooded). The weight goes like ∼L3 and the force gener-ted by the stress in muscles ∼L2. The (force/weight) ratio is/L, indicating that small animals may be relatively strongerhan the big ones. The human weight is also expressed by aower-law, as the surface density M/L2 [kg/m2], with the opti-um value currently set to 21 (BMI = body mass index). These

re results based on observations or on elementary scalingonsiderations.

Biotechnology, like chemical technology, applies the bal-nce equations, which can be scaled. With well-definedhysical quantities, DA can operate. Some specific situationso exist at biosystems. There are crucial qualities that areifficult to define and quantify, whence to subject to DA.or instance, the ‘physiological state’ of living matter is ofaramount importance. Further, there are metabolic patterns,athways of such a complexity that a mere set of mass equa-
ions from Section 7 for their description is ineffective. Rather,he graph theory is used to capture their topology and for-
alize the problems at least qualitatively, in terms of the

sign 8 6 ( 2 0 0 8 ) 835–868 865

corresponding matrixes. The population balance models (usednot only for living units, but also for lifeless particles, like bub-bles, drops) are statistical tools, providing us with equationsfor probability distribution of certain property within the pop-ulation (age, size, weight, wealth, health, etc.). It may not be apriori clear what good comes from their possible scaling andhow to use the DN thus obtained. Similarly in other areas (ecol-ogy, economy, medicine, pharmacology, psychology, sociology,etc.), the following points should be made clear. Do we havewell-defined physical quantities? Are they directly related viacertain physical processes? Or even better: Do we have gov-erning equations? If so, we can try our best at applying DA,and possibly also SE.

Literature used and recommended: References section 3.3.

C.3. Multiscale methodology

Multiscale approach is fashionable currently, but the basicidea is very old. In your system, choose proper characteris-tic scales for relevant quantities (time, length, force, speed,etc.), estimate the magnitude of individual terms in your equa-tions, neglect some terms with respect to others, under givenconditions. Change the conditions systematically, to selectindividual simple processes and study them thoroughly. Thisis what physicists have been doing for centuries. The time andlength scales (T and L) can be obtained by spectral analysis, ofeither the physical signals (measured or computed by CFD)or the governing equations (modal dynamics). The physicalquantity u(x, t) is a function, an element of an abstract func-tion space. It can be represented using the basis functions. Onecommon basis are the harmonic functions (Fourier). Anotherbasis are wavelets (Haar), which basically are wave packets.The force scale relates to L and T, depending how short/longrange it is, and how fast it responds/decays. When the scalesare separated by a large gap, the corresponding processes arelikely only weakly coupled, and can possibly be studied sepa-rately. When the scales are not separated, the correspondingprocesses are likely strongly coupled and it may be difficult oreven impossible to decompose them, without corrupting themodel (scales in turbulence are typically strongly intertwined).

Almost all things around us are multiscale systems, at leastfor the simple reason that they are made of atoms ∼10−9 m andwe are bodies ∼100 m living in space of ∼10xx m. From the hier-archy of scales follows the hierarchy of functions, as knownfrom complex systems. Hierarchical systems display one spec-tacular feature: existence of ‘emergent properties’. The wholeis more than the sum of its parts (in symbols: 1 + 1 �= 2). Theadditional qualities that do not exist on the level of the indi-vidual components are the collective modes of behaviour ofthe whole system that emerges through the interactions of itscomponents. They cannot be predicted from the knowledge ofthe single components itself (if in principle, is a philosophicalquestion). The quality of ‘houseness’ is not present in eachsingle brick the house is composed of. One practical conse-quence for microtechnologists: a huge production block builtup from many small active elements of various types willalways have potential for producing unpredictable emergentbehaviour. Once you create your LEGO-plant from your micro-bricks, it will start its own life. It will be ready to surprise thecreator,7 any time.

7 There might even be a precedent for this in human history.

and
References8

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in Engineering Dynamics: Theory and Practise of Scale Modeling.(Hayden Book Company, Rochelle Park).

Barenblatt, G.I., (1987). Dimensional Analysis. (Gordon and BreachScience Publishers, NY).

Barenblatt, G.I., (1996). Scaling, Self-Similarity, and IntermediateAsymptotics. (Cambridge University Press).

Barenblatt, G.I., (2003). Scaling. (Cambridge University Press).Becker, H.A., (1976). Dimensionless Parameters. (Applied Science

Publishers, London).Birkhoff, G., (1960). Hydrodynamics: A Study in Logic, Fact and

Similitude. (Princeton University Press) [Chapter 4]Bluman, G.W. and Cole, J.D., (1974). Similarity Methods for

Differential Equations. (Springer, NY).Bridgman, P.W., (1922). Dimensional Analysis. (Yale University

Press, New Haven).Clement-O’Brien, K. and Lawler, G.M., 1998, Applying medication

math skills: a dimensional analysis approach, CENGAGEDelmar Learning.

Craig, G.P., (2003). Quick Guide to Solving Problems Using DimensionalAnalysis. (Lippincott Williams & Wilkins).

Curren, A.M., 2005, Dimensional analysis for meds, CENGAGEDelmar Learning.

De Jong, F.J., (1967). Dimensional Analysis for Economists.(North-Holland, Amsterdam).

Dolezalik, V., (1959). Similarity and Modelling in Chemical Technology.(SNTL Press, Prague, CZ) [in Czech]

Duncan, W.J., (1953). Physical Similarity and Dimensional Analysis:An Elementary Treatise. (Edward Arnold, London).

Focken, C.M., (1953). Dimensional Methods and Their Applications.(Edward Arnold, London).

Gukhman, A.A., (1965). Introduction to the Theory of Similarity.(Academic Press, NY).

Hornung, H.G., (2006). Dimensional Analysis: Examples of the Use ofSymmetry. (Dover, NY).

Huntley, H.E., (1952). Dimensional Analysis. (Macdonald, London).Ipsen, D.C., (1960). Units, Dimensions, and Dimensionless Numbers.

(McGraw-Hill, NY).Isaacson, E. and Isaacson, M., (1975). Dimensional Methods in

Engineering and Physics. (Edward Arnold, London).Jerrard, H.G. and McNeill, D.B., (1992). Dictionary of Scientific Units

Including Dimensionless Numbers and Scales. (Chapman & Hall,London).

Kline, S.J., (1965). Similitude and Approximation Theory.(McGraw-Hill, NY).

Kozesnik, J., (1983). Theory of Similarity and Modelling. (Academia,Prague, CZ) [in Czech]

Kurth, R., (1972). Dimensional Analysis and Group Theory inAstrophysics. (Pergamon Press, Oxford).

Land, N.S., (1972). A Compilation of Nondimensional Numbers, NASASP-274. (NASA Sci. Technol. Info. Office, Washington DC, USA).

Langhaar, H.L., (1951). Dimensional Analysis and Theory of Models.(John Wiley, NY).

Loebel, A., (1986). Chemical Problem-Solving by Dimensional Analysis.(Houghton Mifflin Company, Boston).

Massey, B.S., (1971). Units, Dimensional Analysis and PhysicalSimilarity. (Van Nostrand, London).

Murphy, G., (1950). Similitude in Engineering. (The Ronald PressCompany, NY).

Palacios, J., (1964). Dimensional Analysis. (Macmillan, London).Pankhurst, R.C., (1964). Dimensional Analysis and Scale Factors.

(Chapman & Hall, London).
Perry, R.H. and Chilton, C.H., (1973). Chemical Engineers’ Handbook.
(McGraw-Hill Book Company, NY), pp. 81–85 [Chapter 2]

8 Selection of literature sources is provided below. Since manyof the listed books exist in several editions, to reference is givento the first edition or the edition currently available to the author.

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Porter, A.W., (1946). The Method of Dimensions. (Methuen, London).Schuring, D.J., (1977). Scale Models in Engineering: Fundamentals and

applicatIons. (Pergamon Press, Oxford).Sedov, L.I., (1959). Similarity and Dimensional Methods in Mechanics.

(Academic Press, NY).Skoglund, V.J., (1967). Similitude Theory and Applications.

(International Textbook Company, Scranton).Stubbings, G.W., (1948). Dimensions in Engineering Theory. (Crosby

Lockwood, London).Szirtes, T., (1998). Applied Dimensional Analysis and Modeling.

(McGraw-Hill, NY).Taylor, E.S., (1974). Dimensional Analysis for Engineers. (Clarendon

Press, Oxford).Weast, R.C. and Astle, M.J. (Eds.). 1981, CRC Handbook of Chemistry

and Physics (pp. 306–323 [Chapter F]). (CRC Press, Boca Raton,Florida)

Zierep, J., (1971). Similarity Laws and Modeling. (Marcel Dekker, NY).Zlokarnik, M., (1991). Dimensional Analysis and Scale-Up in Chemical

Engineering. (Springer, Berlin).

1.2. Articles on dimensional analysisAstarita, G., 1997, Dimensional analysis, scaling, and orders of

magnitude. Chem Eng Sci, 52: 4681–4698.Boucher, D.F. and Alves, G.E., 1959, Dimensionless numbers.

Chem Eng Prog, 55(9): 55–64.Boucher, D.F. and Alves, G.E., 1963, Dimensionless numbers-2.

Chem Eng Prog, 59(8): 75–83.Buckingham, E., 1914, On physically similar systems; illustrations

of the use of dimensional equations. Phys Rev, 4: 345–376.Buckingham, E., 1915, Model experiments and the forms of

empirical equations. Trans ASME, 37: 263–296.Cheng, Y.T. and Cheng, C.M., 2004, Scaling, dimensional analysis,

and indentation measurements. Mater Sci Eng R, 44:91–149.

Dodds, P.S. and Rothman, D.H., 2000, Scaling, universality, andgeomorphology. Ann Rev Earth Planet Sci, 28: 571–610.

Gunther, B., 1975, Dimensional analysis and theory of biologicalsimilarity. Physiol Rev, 55: 659–699.

Gunther, B. and Morgado, E., 2003, Dimensional analysisrevisited. Biol Res, 36: 405–410.

Klinkenberg, A., 1955, Dimensional systems and systems of unitsin physics with special reference to chemical engineering.Chem Eng Sci, 4, 130–140, 167–177

Lykoudis, P.S., 1990, Non-dimensional numbers as ratios ofcharacteristic times. Int J Heat Mass Transf, 33: 1568–1570.

Macagno, E.O., 1971, Historico-critical review of dimensionalanalysis. J Franklin Inst, 292(6): 391–402.

Prothero, J., 2002, Perspectives on dimensional analysis in scalingstudies. Perspect Biol Med, 45(2): 175–189.

Rozen, A.M. and Kostanyan, A.E., 2002, Scaling-up effect inchemical engineering. Theor Found Chem Eng, 36(4):307–313.

Sandler, H., 1970, Dimensional analysis of the heart—A review.Am J Med Sci, 260: 56–70.

Sjoberg, L., 1987, Review: Psychometric considerations in thedimensional analysis of subjective picture quality. Displays,210–212.

Stephens, D.W. and Dunbar, S.R., 1993, Dimensional analysis inbehavioral ecology. Behav Ecol, 4(2): 172–183.

Vogel, S., 1998, Exposing life’s limits with dimensionlessnumbers. Phys Today, 51(11): 22–27.

Wesson, P.S., 1980, The application of dimensional analysis tocosmology. Space Sci Rev, 27: 109–153.

West, G.B., 1984, Scale and dimension. Los Alamos Sci,11(Summer/Fall): 2–21.

1.3. Books on scale-upEuzen, J.P., et al., (1993). Scale-Up Methodology for Chemical

Processes. (Gulf Publ. Co, Houston, Texas).Grassmann, P., (1971). Physical Principles of Chemical Engineering.

(Pergamon Press, Oxford).

nd de

J

SZ

2A

A

B

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C

C

C

C

C

C

C

D

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D

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3. Other topics3.1. Surfaces and surfactantsAdamson, A.W., (1960). Physical Chemistry of Surfaces.

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Structure. (Elsevier).Davies, J.T. and Rideal, E.K., (1963). Interfacial Phenomena.

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Capillarity and Wetting Phenomena. (Springer).Israelachvili, J.N., (1992). Intermolecular and Surface Forces.

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1201–1204.Pilbeam, D. and Gould, S.J., 1974, Size and scaling in human

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