chemical engineering research and design 8 6 (2008) 835–868

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Chemical Engineering Research and Design

journal homepage: www.elsevier.com/locate/cherd

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”

abstract

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 ...... 836 2. Two sources of dimensionless numbers ...... 836 2.1. Source one—dimensional analysis ...... 836 2.2. Source two—scaling of equations ...... 837 3. Dimensional analysis ...... 838 3.1. How DA works...... 838 3.2. Comments on DA ...... 839 3.2.1. Choice of variables ...... 839 3.2.2. Variables with independent dimensions...... 840 3.2.3. Similarity and modelling ...... 840 3.2.4. Neglecting variables in DA ...... 841 3.2.5. Limits of DA ...... 841 3.2.6. DA versus SE...... 841 4. Scaling of equations ...... 841 5. Transport of momentum ...... 841 5.1. Mass equation of fluid ...... 842 5.2. Momentum equation of fluid ...... 842 5.3. Energy equation of fluid ...... 844 5.4. Boundary conditions: no slip and free-slip ...... 845 5.4.1. Normal component of free-slip BC ...... 845 5.4.2. Tangential component of free-slip BC ...... 845

∗ Tel.: +420 220 390 299; fax: +420 220 920 661. E-mail address: [email protected]. Received 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.2008.03.007 836 chemical engineering research and design 8 6 (2008) 835–868

5.5. Multi-phase flow ...... 847 5.5.1. Microscale description ...... 847 5.5.2. Mesoscale description (Euler/Lagrange)...... 848 5.5.3. Macroscale description (Euler/Euler) ...... 849 5.5.4. Retention time distribution...... 850 6. Transport of heat ...... 852 7. Transport of mass ...... 853 8. Correlations ...... 854 9. Remark on literature ...... 857 10. Conclusions ...... 859 Acknowledgements ...... 859 Appendix A. Concept of intermediate asymptotics ...... 859 A.1. Motivation example ...... 859 A.2. Two kinds of similarity ...... 859 A.2.1. Complete similarity ...... 859 A.2.2. Incomplete similarity...... 860 A.3. Two kinds of self-similarity ...... 860 A.4. Relation between DA and IA ...... 861 A.5. Beyond IA? ...... 861 A.6. Broader horizons ...... 862 Appendix B. Suggestions for using and teaching DA...... 863 Appendix C. New areas in chemical engineering ...... 864 C.1. Microreactors and microfluidics ...... 864 C.1.1. Microsystems in chemical technology ...... 864 C.1.2. Prevailing forces ...... 864 C.1.3. Governing equations and boundary conditions ...... 864 C.2. Biosystems ...... 865 C.3. Multiscale methodology ...... 865

1. Introduction 2. Two sources of dimensionless numbers (DN) I have always been puzzled with the plethora of dimensionless numbers (DN) occurring in the various branches of chemi- 2.1. Source one—dimensional analysis (DA) cal engineering. From sincere discussions with my peers as well as with students an impression has arose that I am not A general way how to formally describe the surrounding world the only puzzled person in this field. Consequently, the main consists of several steps. First, for the thing under study motivation of this contribution is to try to briefly review DN (‘system’), define all possible qualities of interest. Second, commonly encountered at the transport of momentum, heat, select those qualities that can be quantified, i.e. their amount and mass, as for their origin, physical meaning, interrelation in the thing can be expressed by numbers, or some other and relevance for making correlations. The selection of DN mathematical constructs. Call these measurables the phys- is neither objective nor exhaustive, being biased by working ical quantities. Choose the etalons (measuring sticks, units) mainly in the area of the multi-phase hydrodynamics. to measure each of them. Each physical quantity has four The dimensionless (nondimensional) numbers (criteria, attributes: name, notation, defining relation, and physical unit groups, products, quantities, ratios, terms) posses the follow- that determine its physical dimension (unit and dimension are ing features. They are algebraic expressions, namely fractions, often used interchangeably). The first two are only our labels, where in both the numerator and denominator are powers of the second two are physically substantial. There are seven physical quantities with the total physical dimension equal to basic physical quantities (length, mass, time, electrical current, unity. For example, the Reynolds number, Re = LV/, has dimen- thermodynamic temperature, luminous intensity, amount of sion [1], also denoted as [-]. substance). They are measured by seven basic units of the SI The dimensionless numbers are useful for several reasons. system of units (meter, kilogram, second, ampere, Kelvin, can- They reduce the number of variables needed for descrip- dela, mole), which is canonical nowadays. All other quantities tion of the problem. They can thus be used for reducing are called derived quantities and are composed of the seven the amount of experimental data and at making correla- basics. Depending on the research area, usually only few basic tions. They simplify the governing equations, both by making quantities are used. In mechanics, we have three (length L, them dimensionless and by neglecting ‘small’ terms with mass M, time T), plus one (temperature ) for thermal effects, respect ‘large’ terms. They produce valuable scale estimates, if these are considered. whence order-of-magnitude estimates, of important physical Having the physical quantities, we want to find the rela- quantities. When properly formed, they have clear physical tions among them. We either have the governing equations interpretation and thus contribute to physical understanding (physical laws) or not. The very basic laws (axioms of nature) of the phenomenon under study. Also, choosing the relevant cannot be derived: they must be disclosed or discovered. There scales, they indicate the dominant processes. There are two is nowhere they could be derived from: they are already here, main sources of DN: dimensional analysis and scaling of gov- existing silently, demonstrating themselves through a variety erning equations. of diverse or even disparate effects. Lacking the knowledge chemical engineering research and design 8 6 (2008) 835–868 837

1 Nomenclature ˝ frequency, its scale (s−1); vorticity scale (s−1); a variable in DA; acceleration (m/s2) flow domain with boundary ∂˝ − b variable in DA  Nabla operator (m 1) c concentration (kg/m3) Others cp heat capacity (J/kg K) C coefficient (drag, added mass); concentration BC boundary condition (s) scale (kg/m3) DA dimensional analysis d length, size, particle diameter (m) DN dimensionless number (s) D diameter (m); mass diffusivity (m2/s) Hyb unit of momentum (SI) (kg m/s) = (N s) 2 IC initial condition (s) Dt,ax turbulent/axial dispersion (m /s) e voidage, volume fraction of dispersed phase (-) N number E bulk modulus of elasticity (=1/K) (Pa); energy (J) O(1) order of unity f function symbol (f) SE scaling of equations F force (N); function symbol; factor 1D one-dimensional g gravity (m/s2) [] physical dimension g reduced gravity (=(/)g)(m/s2) * dimensionless, basic DN h height, depth (m) Subscripts j flux, of heat (J/m2 s), of mass (kg/m2 s) a added mass k transfer coefficient (J/m2 s K); rate constant (e.g. − b bulk s 1) − f fluid (continuous phase); flow K compressibility (Pa 1) mix mixture l length (m) p particle (dispersed phase) L length scale (m) r relaxation m mass (kg) s, S surface, interface M mass scale (kg) w wall, interface p pressure (Pa) 0 reference value P pressure scale (Pa); period of oscillation (s) Q flow (m3/s) r radius, position (m); reaction rate (kg/m3 s) about the basic laws, we must try to find them using the other R reaction rate scale (kg/m3 s) available methods. One such a method is DA. S (cross-section) area (m2) Dimensional analysis consists of three steps. First, make a t time (s) list of relevant variables, the physical quantities that describe T time scale (s) the system. Second, convert these dimensional quantities u velocity (dispersed phase) (m/s); master quan- into DN. Third, find a physically sound relationship (scale- tity (u) estimate) of these DN without help of any governing equations v velocity (continuous phase) (m/s) (physical laws). The main problem is to make the list of the rel- V velocity scale, mean speed (m/s) evant variables that is complete and independent. Here the x coordinate, distance, position (m) science meets the art: the choice of the variables is highly z coordinate, distance, position (m) subjective, beyond any rigour. The other two steps are simple in principle owing to the fact that DA is a rigorous math- Greek letters ematical method operating precisely on the lists of chosen ˛ thermal expansivity (K−1) variables. DA relies on several assumptions that are needed ˇ concentrational expansivity (kg−1) for the mathematical proof of its very core, the Pi-theorem. shear rate (s−1) The assumptions are the following. The physical equations are difference, variation dimensionally homogeneous. The physical equations hold for temperature (K) different systems of units. The dimensions of physical quan- temperature scale (K) tities have form of power-law monomials (dimensions like heat diffusivity (/c )(m2/s) p [sin(L) − log(T)+eM] are not allowed). There are quantities with heat conductivity (J/m s K) independent dimensions and their list is complete. Beside capillary length (m) C these, we tacitly presume: the problem is amenable to DA. dynamic viscosity (fluid) (Pa s) kinematic viscosity (fluid), momentum diffu- 2.2. Source two—scaling of equations (SE) sivity (m2/s)

˘i Pi-term The scaling of equations (SE) means nondimensionalization of 3 density (fluid) (kg/m ) the equations describing the system under study (equations 2 interfacial tension (N/m, J/m ) of motion, fundamental equations, governing equations, etc.). 2 time (s); stress (N/m ) It is a technically simple and transparent procedure, which ϕ angle ˚ function symbol (˚) ω frequency (s−1 or 2/s); vorticity (s−1) 1 Since many symbols appear in the text, only those of general use are listed here. At multiple meaning, the context talks. Those of the local meaning, apply usually within one paragraph only, are omitted. 838 chemical engineering research and design 8 6 (2008) 835–868 yields the dimensionless equations and the list of relevant DN. 3. Dimensional analysis (DA) It does not give the relation among these DN. The dimensionless equations have certain advantages. 3.1. How DA works They are independent of the system of units. The dimen- sionless numbers are relevant for the problem. The proportion Choose one physically dependent variable a and choose fur- between individual terms can be seen. These equations apply ther (k + m) independent variables, ai and bi,onwhichwe to all physically similar systems, so they are useful for scale- presume a depends. ai have mutually independent dimen- up/down. sions, of which the dimensions of all remaining variables a The process of scaling proceeds as follows. For instance, and bi can be obtained by combination. We want to find the take the equation of linear mass-spring oscillator: unknown physical law, the function f, we presume it does exist: 2 d x =− 2 m kx [kg m/s ], (2.2.1) = = dt2 a f (a1,a2,...,ak; b1,b2,...,bm)[a] [f ]. (3.1.1)

To reproduce the dimensions of a and bi, we combine ai in where all the four quantities (x, t, m, and k) are dimensional. form of power monomials: Separate them into two classes: parameters (m, k) and variables (x, t). Choose the scales (characteristic values) for the variables, = p01 · p02 · p03··· p0k···≡ [a] [a1] [a2] [a3] [ak] A [a], (3.1.2) length scale L and time scale T. Make substitution x → Lx*, → t Tt*in(2.2.1) to obtain = pi1 · pi2 · pi3··· pik···≡ [bi] [a1] [a2] [a3] [ak] Bi [bi], (3.1.3) 2 ∗ mL d x ∗ where the exponents pij are found for each row by compar- =−(kL)x [kg L/T2]. (2.2.2) T2 dt∗2 ing the dimensions on both sides, based on the dimensional homogeneity of physical equations. Dividing a and bi by the corresponding composites of the same dimension, denoted Although (2.2.2) has dimension, the quantities are sepa- for brevity as A and Bi, and rewriting (3.1.1) in dimensionless rated into dimensionless variables (x*, t*) and dimensional form for another unknown function ˚,weget parameter groups (mL/T2, kL). Dividing (2.2.2) by any parameter group yields the dimensionless equation. Dividing by (kL): a b b b b = ˚ 1 , 2 , 3 ,..., m [-]. (3.1.4) A B1 B2 B3 Bm 2 ∗ m d x ∗ =−x [-] (2.2.3) This equation is usually written in the following notation kT2 dt∗2 as the similarity law for ˘:

2 produces one DN, namely the number N =(m/kT ), the propor- ˘ = ˚(˘1,˘2,...,˘m) [-], (3.1.5) tion (inertia)/(elasticity). Generally, DN show the proportion between the individual where the dimensionless terms (Pi-terms, similarity param- ≡ ≡ terms in an equation correctly only when the dimensionless eters) are ˘ a/A, ˘i bi/Bi. The famous Buckingham variables (*) are scaled so well to be of order of unity ∼O(1). Pi-theorem says: It is possible to get from (3.1.1) to (3.1.5). Then the magnitude of the terms is solely represented by the The main gain is the reduction of the number of variables parameter groups. With the above pendulum, at the choice from (k + m)in(3.1.1) to only (m)in(3.1.5). All k variables ai L ∼ amplitude and T ∼ period P,wehavex* ∈−1, 1 and t* ∈0, are hidden in the denominators of ˘ and ˘i. In mechanics, 1 (one swing), which both are O(1). Consequently, N shows the we have only three basic dimensions, L, M, T, so that k ≤ 3. (inertia)/(elasticity) proportion correctly. Another advantage is that (3.1.5) is dimensionless. According For an equilibrium motion like oscillations, where to experimental convenience, any quantity involved can be the both counter-acting forces are somehow balanced, used to change the value of ˘i. DA merely transforms f into one would expect that their ratio should be unity, i.e. ˚. Rewriting (3.1.5) in the dimensional form, as the similarity N = (inertia)/(elasticity) = 1. However, this is generally not true. law for a: The particular value of N depends on the choice of scales. Realiz- 1/2 ing that (k/m) is the oscillator angular frequency ω, which is a = A · ˚(˘1,˘2,...,˘m)[a]. (3.1.6) defined by ω =(2)/P, then N =(P/2T)2. For the particular choice of the time scale T = P,wehaveN =(P/2P)2 = 1/(2)2 ≈ 0.025. We see a certain progress as compared to (3.1.1): f is written Thus, the actual force ratio is N ≈ 0.025:1 = 1/40, far from unity. as a product of two things, f = A·˚. The first one is the known The reason is that DN is a very rough estimate of the effects it dimensional function A that contains the rough essence of f. compares. For instance, at the laminar–turbulent flow regime It is called the scale estimate (basic scaling, scaling law) for a, transition in pipes, the ratio Re = (inertia)/(viscosity) is not and we write exactly 1, but ∼103. It is therefore better to use a vague ∼ = p01 · p02 · p03··· p0k language and say ‘low Re’ and ‘high Re’, upon strong under- a A a1 a2 a3 ak [a]. (3.1.7) standing that everybody knows what it does mean. Conclude that the diversity of the original four- The second one is the unknown dimensionless function dimensional problem (2.2.1) described by four quantities ˚ that is the ‘fine tuning’ of the scale estimate, to convert (x, t, m, k) is reduced to a single number (m/kT2) at the price of (3.1.7) into the equality a = A·˚. Finding ˚ does not belong to lacking all details that are below the resolution of the scale the frame of DA; this must be done by some other means (e.g. considerations. experimentally or numerically). chemical engineering research and design 8 6 (2008) 835–868 839

As an example, consider a flow in a pipe. The dependent variable is the drag force F. The independent variables are pipe dimensions (D, L), fluid properties (, ), and speed (V). Eq. (3.1.1) now takes form

F = f (D, L, , , V) [N]. (3.1.8)

Variables with independent dimensions are D (length), V (time), (mass). Variables with dependent dimensions are L, F, . Eq. (3.1.4) applied to this example becomes F L = ˚ , [-]. (3.1.9) V2D2 D DV

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

F ∼ V2D2 [N], (3.1.10) Fig. 1 – Definition sketch. Pendulum of mass m and hanger length l swings under gravity g in a medium of viscosity which is the dynamic fluid pressure (V2) times the cross- (or ). section area ∼(D2). The correcting dimensionless term ˚(L/D, /DV)in(3.1.9) is the formula for the friction coefficient C(L/D, Re), which remains for experiments. DA thus produces a great deal of the total solution of the whole problem. variables than necessary, DA either recovers the correct result When no exceptional variable a labelled as ‘physically inde- by eliminating the extras, P(l, g, m) and P(l, g, ), or, DA fails pendent’ is or can be explicated, (3.1.1) is simply written as by insolubility, i.e. having more unknowns than equations, P(l, g, ). A subtle point is considering several quantities of the f = 0, where the variable a becomes bm+1. This notation is suit- able when it is not clear what is the master quantity, or, when same dimension, P(l, d, g). Using the same length scale L for several of them can play this role, depending on our angle of both l and d, DA fails by insolubility. Using one scale L1 for the view at the problem. correct variables (l, g) and another scale L2 for the extra vari- ables (d), DA works and eliminates the extras. However, using 3.2. Comments on DA L1 for one part of the correct variables (l) plus some extras, and another scale L2 for the second part of the correct variables 3.2.1. Choice of variables (g) plus some extras, DA fails by contradiction. The result is It is the main problem of DA, since there is no rigorous pro- summarized in Table 1. cedure for it. The quantities must be physically relevant and To sum up, DA either works or fails. When it works, it gives independent, and their list must be complete. The choice either good or bad result. When it gives good result, either is highly subjective and needs profound understanding of the choice of variables is correct or the extra variables are the problem, experience with usage of DA, intuition, and eliminated. When it fails, it is either by logical contradiction good luck. As a guideline, there are recommendations of (dimension of l.h.s. cannot be made up of dimensions of r.h.s.) what should be taken into account (e.g. system geometry, or by insolubility (extra variables bring more equations but not material properties, kinematic and dynamic aspects, exter- new dimensions). This simple example is purely demonstra- nal conditions, etc.). Very helpful are the governing equations tive; not a general statement proven for all possible situations. related to the problem. Even when we cannot solve them (e.g. Navier–Stokes equation), they indicate the relevant quantities. A simple example demonstrates how the choice of vari- Table 1 – Application of DA for finding period of ables affects the output of DA. Consider the mathematical pendulum, P =2(l/g)1/2 pendulum in Fig. 1. The period P depends on two variables, Variables Features Action Output the length l and gravity g: P(l, g) Proper choice Work Correct result 1/2 l P(l) Missing variable (g) Fail Contradiction P(l, g) = 2 [s]. (3.2.1) g P(g) Missing variable (l) Fail Contradiction P(l, m) Incorrect substitute (m) Fail Contradiction Suppose we do not know it, and we try to find it by DA. Let P(l, ) Incorrect substitute () Fail Contradiction us try different choices of variables. At the proper choice, P(l, P(l, ) Incorrect substitute () Work Wrong result P(l, g, m) Extra variable (m) Work Elimination g), we get the correct result, the basic scaling for the period, P(l, g, ) Extra variable () Work Elimination P ∼ (l/g)1/2. Here, the correcting function ˚ in (3.1.6) contains P(l, g, ) Extra variable () Fail Insolubility no argument and equals 2. With lesser variables than is due, P(l) and P(g), the DA fails. The type of failure is the logical con- P(l, g, d) Extra variable (d) P(L , g(L ), L ) Uniscale L Fail Insolubility tradiction of kind 1 = 0, since it is impossible to make up time 1 1 1 1 P(L1, g(L1), L2) Two scales L1,2 Work Elimination dimension of P (s) from the length l (m) only. When we try P(L1, g(L2), L1) Two scales L1,2 Fail Contradiction to substitute for the missing correct variable (g) another vari- P(L1, g(L2), L2) Two scales L1,2 Fail Contradiction able which is not relevant, the following may happen. DA fails by a contradiction, with P(l, m), P(l, ). DA works but gives an First column: variables chosen for pendulum period P. Second col- umn: features of our choice. Third column: what DA does? Fourth incorrect result, P ∼ l2/, with P(l, ), which is the worst case, column: note on result. Definition sketch in Fig. 1. since there is no indication that things go wrong. With more 840 chemical engineering research and design 8 6 (2008) 835–868

3.2.2. Variables with independent dimensions similar but bigger system (prototype P), where it is difficult to Of the physically independent variables of f in (3.1.1),wemust experiment. Without similarity, the model and prototype are choose those with independent dimensions (ai), and leave described by two different relations: the rest for bi. The task reduces to finding k linearly inde- pendent vectors in Rk. For instance, the three mechanical ˘M = ˚M(˘M) (model), i (3.2.2) dimensions generate 3D space (L × M × T) with the base vec- P = P P ˘ ˚ (˘i ) (prototype). tors {1, 0, 0}, {0, 1, 0}, {0, 0, 1}. All mechanical quantities are represented by vectors in this space: speed has coordi- With similarity, the following holds: nates {1, 0, −1}, density has {0, 1, −3}, force has {−1, 1, −2}, etc. We can choose any three non-coplanar vectors for ˘M = ˘P (similarity law for modelling), ai, but the simpler the better. For different choices of ai,DA M = P ∀ M = P produces different DN, but the resulting physical informa- ˘i ˘i i (similarity criteria),˚ ˚ . (3.2.3) tion is identical. Based on the dimensional homogeneity of

(3.1.1), the powers pij in (3.1.2) and (3.1.3) are determined For instance, DA gives the scale estimate (3.2.1) for by the standard routine of solving a set of linear algebraic pendulum period P ∼ (l/g)1/2. Applying the similarity law, equations. 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). 3.2.3. Similarity and modelling Increasing pendulum length 25 times gives only 5 times longer One paradigm says: If systems are similar, DA gives same period, in virtue of (lP/lM)1/2 (scaling coefficient). Knowing this description. This statement forms the basis for the the- may be helpful for designers of big clocks. ory of similarity (similitude) and modelling (scale-up/down). The similarity criteria may not always be met. With a The physical similarity consists in correspondence in geom- simple pendulum, the tuning function ˚ in (3.1.6) has no etry, kinematics, dynamics, etc. One may ask: Can dissimilar arguments ˘i, and is constant, ˚ =2, see (3.2.1). With more systems have same description? Consider four physically dif- complex systems, there can be several Pi-terms (e.g. two in ferent systems shown in Fig. 2. Despite their difference, they (3.1.9)), whose model-prototype equality required by (3.2.3) all share the same description, the formula for drag force given may be difficult to guarantee. The demands of the similarity by DA in (3.1.9). Consequently, different systems can have the criteria for different ˘i may not be fully compatible. Models same description within the framework of dimensional con- with these contradictions are called ‘distorted’, in contrast siderations. DA is ambiguous with respect to physical kind with the ‘true’ model, where we can satisfy all demands. For of systems, it cannot see the physical difference. DA is not instance, in hydraulic engineering, we have modelling based ambiguous with respect to manipulation with symbols repre- on two Pi-terms, namely the Froude and Reynolds numbers, M P M P senting the input variables, owing to the unicity theorem for ˘1 = Re, ˘2 = Fr. Their equality (Fr = Fr , Re = Re ) implies a the solution of linear algebraic systems. severe requirement on the kinematic viscosity of the model Modelling usually means finding a description of a small and real fluids, P/M =(LP/LM)3/2. Considering the great dis- model system on a laboratory scale by DA (model M), where it parity in size of hydraulic models LM and real water works LP, is easy to do measurements, and, to transfer the result on to a it is difficult to find suitable fluids. With LP/LM =102 we need

Fig. 2 – Ambiguity of DA. Four different flow situations with identical description. (Case A) Infinite pipe of diameter D and wall roughness L. (Case B) Finite pipe of diameter D and length L. (Case C) Finite plate of size D × L. (Case D) Liquid piston of length L oscillating in a cylindrical orifice of diameter D. Fluid has density , viscosity and speed V. chemical engineering research and design 8 6 (2008) 835–868 841

P/M =103. The phenomenon of distortion is often qualified basically doable, when little care is taken. As for the output, as a drawback of DA. DA can reproduce all the numbers obtained by SE, if we feed it with the proper variables. On the other hand, DA can gen- 3.2.4. Neglecting variables in DA erate numbers that cannot be obtained by SE, whose physical There are two reasons for neglecting some variables that can relevance may be difficult to assess. In addition, DA gives the apparently play a role in the description of the problem. First, scale-estimate of the unknown relation between the quanti- based on our subjective choice, we do not want some variables ties. It is convenient to apply SE first, to get the proper list in the model. Our experience says, that they may be problem- of DN, and then use DA to find the scaling relations between atic for the smooth operation of DA. Second, we want them, them. but the corresponding ND are either small or large. The tra- dition says that these variables are irrelevant, which is rather 4. Scaling of equations (SE) counter-intuitive. The rationale is, however, the following. The DN usually are ratios of two effects. If this ratio is either too The motivation for SE and the procedure were briefly intro- small or too big, one of the effects is simply negligible. There- duced in Section 2.2. In sake of simplicity, a brief notation fore, it can be neglected as a category, and thus excluded from is used further. The starred dimensionless O(1)-variables are further considerations. For instance, consider the Reynolds omitted, and only the parameters and scales are retained in number. When it is too small or too large, we have either the the ‘scale equations’. These are not the ‘true’ equations, but viscous flow or the potential flow, and this parameter enters relations that indicate the relative proportions between the neither of these two limiting theories. When we neglect the individual terms. As an example, we recover (3.2.1) by SE. The inertia effects completely, there is no sense to compare them governing equation is the conservation of angular momen- with the viscous effects, and vice versa. The act of omitting tum, Fig. 1: some variables looks rather subjective, but there are rules for it, see Appendix A. (ml)x¨ + (mg)x = 0[Nm], (4.1)

3.2.5. Limits of DA where the mass cancels: It was believed that DA is a universal tool whose potential is limited only by the skill of the user. However, like any (l)x¨ + (g)x = 0[m2/s2]. (4.2) other mechanism, DA too can operate only under certain circumstances: there are physical problems that cannot be Choose the variables (x, t) and parameters (l, g). Scaling of solved by DA, in principle. As a brief guideline, few contra- variables by general scales L and T gives indications to application of DA are the following. The problem involves information about the initial and boundary condi- lL + gL = 0 (‘scale equation’) [L2/T2]. (4.3) tions: the system behaviour in the initial times, details of T2 process generation, its behaviour near the system bound- aries, decay via equilibration, energy dispersion or dissipation Choosing the particular scales, L = l and T = P, which are rel- ∼ ∼ 1/2 during the process evolution. There are variables related to evant and make x* and t* O(1), indeed recovers P (l/g) . 2 2 2 2 ∼ the presence of the internal sources/sinks of mass, heat, Here, relation (4.3), l /P + gl = 0, means l /P gl, so that ∼ 1/2 momentum, energy, in the environment in which the process P (l/g) . occurs. Some global conservation characteristics (integrals) Further, in sake of simplicity, the complicated issues of the system are not constant during solution but vary due related to the presence of multiple scales and directional scal- to the sources. Parameters related to variable properties of ing are mostly omitted (except for an example in Section 8). the (micro)structure/texture of the medium carrying the pro- In the same system, different processes can take place, each cess are present. The independent variables are involved in having distinct scales of length, time, speed, etc. Different pro- a complicate way (in exponent, in argument of function, cesses can dominate along different spatial directions, having etc.). Regime transitions occur during the solution of the different scales. The flow domain itself can be highly ani- problem (some variables lose relevance and new ones come sometric, having different dimension in different directions. into play). All these are potentially bad variables, we may We face multiplicity of relevant scales, and their directional not want. Consequently, more sophisticated tools must be dependence. For instance, in a boundary layer different pro- developed and employed to cope successfully with these prob- cesses ‘along’ and ‘across’ can be identified. A body moves lems. Fortunately, there is one, the theory of intermediate along the vertical and its wake grows in the horizontal direc- asymptotics, which can be considered as a generic exten- tion. Likewise, a spiralling bubbles rises up and exerts periodic sion of DA. The basic idea of this concept is presented in deflection in the horizontal plane. We will not resolve the Appendix A. many possible scales and, instead, a single scale will be employed for each quantity. This ‘uniscale’ approach is accept- 3.2.6. DA versus SE able at the general level of description, but must be refined ∼ As for the subjectivity of choices, the comparison between when particular flow situation is analysed. Thus, length x L ∼ ∼ ∼ DA and SE goes in favour of the latter. With DA, one must (but not Lx, Ly, Lz), speed v V (but not Vx, Vy, Vz), nabla  ∼ choose all the quantities, whose relevance and completeness 1/L, etc. is not guaranteed. With SE, these quantities are given by the equations (plus initial and boundary conditions). In equa- 5. Transport of momentum tions, we only must choose the variables to be scaled, and the parameters to be left intact. Also, we must choose suit- Transport of momentum is a synonym for fluid dynamics. able characteristic scales to get the dimensionless variables Conservation of mass and (linear) momentum of fluid are the ∼O(1), to see the proportion between different terms. Both are governing equations. 842 chemical engineering research and design 8 6 (2008) 835–868

5.1. Mass equation of fluid scales (V, P, L, T), and the parameters (, , g), we get by scaling:

The mass balance (continuity equation) for a compressible 2 V V P V 2 2 fluid reads + = + + g [M/L T ]. (5.2.2) T L L L2 ∂ +∇(v) = 0 [kg/m3 s] ∂t Dividing by the convective (inertia) term (V2/L), because of tradition, we get (unsteadiness) + (convection) = 0. (5.1.1)

L P gL Choosing the variables (, v, x, t), their scales (0, V, L, T), and + 1 = + + . (5.2.3) 2 2 the parameters (–), we get by scaling the dimensional ‘scale TV V LV V equation’: The magnitude of the convective (inertia) force is thus V unity. Assigning the four DN their proper names we get 0 + 0 = 0[M/L3T]. (5.1.2) T L 1 1 Dividing by the density scale disappears, and we get Sr + 1 = Eu + + . (5.2.4) 0 Re Fr

1 V + = 0 [1/T]. (5.1.3) T L We can divide (5.2.2) by any term, but DN would not have the usual names. The convection (inertia) term is the most Dividing finally by the convection term (V/L) we get the ‘flow-like’, so it is the natural scaling basis. The four DN have dimensionless ‘scale equation’: a clear physical meaning in terms of forces: L + = L (unsteadiness) 1 0 (5.1.4) Strouhal number,Sr= , VT TV (convection) P (pressure) ∼ Euler number,Eu= , with one DN, namely (L/VT) (unsteadiness)/(convection). V2 (convection) (5.2.5) Usually, we can find suitable scales for length L and speed V. LV (convection) Reynolds number,Re= , The problem often is how to scale the time, i.e. what to take for (viscosity) T. There are few general choices: velocity scaling (T = L/V), peri- V2 (convection) Froude number,Fr= . odic scaling (T = 1/frequency), relaxation scaling (T = relaxation gL (gravity) time), energy scaling (T = L(M/E)0.5), diffusion scaling (T = L2/D), etc. We take the first, since nothing indicates that the flow is The relevance and use of these numbers follow from their periodic or relaxing. Moreover, this choice also follows from definitions and physical content. The Froude number is some- (5.1.3), in the state of balance. Taking the basic scaling T = L/V times (namely in physical literature) defined as (V2/gL)1/2.For turns the ‘scale equation’ into instance, Fr is important where the gravity (buoyancy) and inertia interplay (open channel flow, free surface problems, 1 + 1 = 0. (5.1.5) hydraulic engineering, floating vessels, surface waves at long length scales—not capillary waves; buoyancy driven flows). This should be read: unsteadiness (accumulation) and con- The problem is how to scale time and pressure. Few scal- vection are in balance (same order of magnitude), 1/T = V/L. ings for T were presented at (5.1.4). Others come from the This balance is due to our choice of T, which may not be jus- dynamic equation. We can compare the unsteady term L/TV tified in reality. In case of incompressible fluid, (5.1.1) reduces with the other terms in (5.2.2), to express their similar order ∇ = to v 0, which after scaling gives V/L = 0, meaning that the of magnitude: convection term is zero order, ∼O(0), as expected.

L Convection scaling : T = , → Sr = 1, 5.2. Momentum equation of fluid V VL Pressure scaling : T = , The momentum (force) balance for an incompressible Newto- P (5.2.6) nian single-phase fluid reads L2 Viscosity scaling : T = , ∂v = V + (v.∇)v =−∇p + ∇2v + g [Hyb/m3 s] = [N/m3] Gravity scaling : T . ∂t g (unsteadiness) + (convection) The basic scaling T = L/V follows from the definition of = (pressure) + (viscosity) + (gravity). (5.2.1) Sr = L/TV and also from comparing the unsteady and convec- tion terms, V/T ∼ V2/L. Using the basic scaling makes the Unlike force [Newton] or energy [Joule], momentum does Strouhal number equal to unity, Sr = 1. The unsteady and con- not have a single-word unit: let us call it Hyb [kg m/s] = [N s], vective forces are thus comparable, as a direct consequence provisionally. The two l.h.s. terms, (unsteadiness) and (con- of our choice of scaling. This may not always correspond to vection) are also called the inertial forces, namely the latter reality. one. They are called the Eulerian and Lagrangian (convec- Similar holds for the pressure scale P (P also means P). tive) accelerations, too. Choosing the variables (v, p, x, t), their We can compare the pressure term P/L with the other terms chemical engineering research and design 8 6 (2008) 835–868 843

in (5.2.2), to express their similar order of magnitude: reduced gravity g =(/)g in Fr, which is usual in buoyancy driven flows (see Section 6). The uniform part of the density LV L Unsteady scaling : P = , for T = it is P = V2, field can be absorbed in the gradient pressure term (modified T V pressure p = p + gz), thank to the fact that gravity has poten- → = Eu 1, tial. Then, the buoyancy term is directly proportional to the = 2 → = Convective scaling : P V , Eu 1, (5.2.7) density difference = − 0. The cause of density variations V within a (incompressible) fluid can be, e.g. heat or solute con- Viscous scaling : P = , L centration (see Sections 6 and 7). When the ‘solute’ particles Gravitational scaling : P = gL. are so big that they are endowed with their own momentum, they form the macroscopic dispersion, and we speak about Their use depends on the flow situation, and the rele- ‘multi-phase flow’ (see Section 5.5). Note that (5.2.1) contains vance of the forces involved. Similarly, we can obtain other only two material properties, density and viscosity. The third scaling relations between quantities in (5.2.2), comparing the one, the surface tension, enters only via the boundary condi- corresponding terms. For instance, the so-called gravitational tions (Section 5.4). scaling for speed comes from equating convection and gravity, There are many DN arising in particular flow situations, V2/L ∼ g, which yields V ∼ (g/L)1/2. It also comes from defini- and some of them are mentioned here. For instance, the flow tion of Fr = V2/gL. This scaling is useful, e.g. in sedimentation, through curved pipes and sharp bends is characterized by the 1/2 when particle speed V is not known beforehand, but particle Dean number, Dn = Re(r/rc) , where r is pipe radius and rc size L is. is radius of curvature. This correction to Re accounts for the Note the following. Using the convective (inertial) pres- effects of the secondary flow driven by the centrifugal force sure scaling, which is nothing but the dynamic pressure, the (Dean vortices). Fast liquid flows with large pressure variance Euler number turns to unity, Eu = 1. The pressure and convec- can experience cavitation, where the liquid pressure p comes tive forces are then comparable, as a direct consequence of close to or falls below the vapour pressure pv. Here, the Cavita- our choice. This may not always correspond to reality. Simi- tion number Cv = P/V2 compares the liquid–vapour pressure 2 lar happens when using the convective scaling for both time difference P ∼ p − pv with the dynamic fluid pressure V . T = L/V and pressure P = V2, yielding Sr = 1 and Eu = 1, so that There are numbers related to rotational effects, where cen- the unsteady, pressure and convective forces are comparable. trifugal (˝2R) and Coriolis (2 × v) forces arise in (5.2.1), When some of these DN are ‘missing’ in books and papers, the where is rotation frequency and R distance from rotation authors likely used certain specific scaling that made them axis. The Ekman number Ek =(/˝L2) is the force ratio (vis- unity, perhaps without notifying the reader. To trace it back, cous)/(Coriolis). The Rossby number Ro =(V/˝L) is the force one needs to read carefully what kinds of scales were actu- ratio (inertia)/(Coriolis). Like with the uniform density, also ally employed. The gravity scaling recovers the primary-school the centrifugal force can be absorbed in the pressure term, formula for hydrostatic pressure, p = gh. because it can be put into the gradient form. When Ek and Since we customarily divided (5.2.2) by the convective (iner- Ro are small, a balance between pressure and Coriolis forces tia) term, all numbers in (5.2.5) relate to the inertia force. If we is reached (geostrophic flows). With these numbers, certain want other combinations of forces, we must compose them: phenomena are connected (strong collocations: Ekman lay- ers, Rossby waves), which are important in geophysical fluid pressure = Eu Re, mechanics (meteorology, oceanography). While these num- viscosity bers reflect the forces due to noninertial reference frame (Earth pressure = Eu Fr, (5.2.8) rotation), another rotational effects are studied too, e.g. in gravity viscosity Fr Re2 Taylor–Couette flow between two coaxial cylinders, where we = , → Ga = → Ar = Ga . 2 4 2 gravity Re Fr can meet the Taylor number, Ta =(4˝ L / ). The simplest rota- tional effect is perhaps encountered in the process of mixing, Note the following. The ratio (gravity)/(viscosity) is often with an impellor in a container, where both the gravity (g) and introduced as the Galileo number: centrifugal (ω2r) accelerations interplay in (5.2.1). There are numbers related to rheology of complex flu- Re2 gL3 Ga = = , (5.2.9) ids with non-Newtonian behaviour, where the stress tensor Fr 2 takes complicated forms. Depending on the given stress ten- where the square of Re is used for a practical reason: the veloc- sor and the flow situation, specific numbers can arise. For ity cancels. This is helpful, when V is difficult to estimate, or instance, the Weissenberg number ∼(first normal stress differ- when the speed is a part of the problem solution. For instance, ence N1)/(shear stress), which can also be defined in terms of in many applications (e.g. sedimentation, fluidization, bubble (relaxation time)/(shear rate). Another one is, e.g. the Bingham columns, etc.), we look for speed of bodies and particles mov- number ∼(yield stress)/(viscous stress). There also is a number, ing in fluids (bubbles, drops, solids). When buoyancy effects related to a very general concept, the Deborah number: are relevant, as it usually is in these applications, Ga is cor- rected by (/) and the product is called the Archimedes Tr De = . (5.2.11) number: Tf gd3 Ar = Ga = . (5.2.10) It compares two time scales: material relaxation time Tr 2 and time of flow or deformation Tf. The former scale is a con- This number is also (and more naturally) obtained by scal- stitutive property of the material under study and says how ing the Newton force law for a body falling/rising in a fluid quickly it responds to deformation: Tr = 0 for ‘ideal fluid’ with under gravity (see Section 5.5.2). The buoyant correction (/) instantaneous response, ∼10−12 s for water molecules, ∼10−6 s of Ga is equivalent to replacing in (5.2.9) the gravity g with the for thick lubrication oil (tribology), ∼few seconds for polymer 844 chemical engineering research and design 8 6 (2008) 835–868

melts, and Tr = ∞ for ‘ideal solid’ that does not deform at all. like F =1+Kn·f(Kn), or similar. Also, Kn relates to Re and Mc,by The later scale is a property of the deformation process: the Kn ∼ Mc/Re. Note that the macroscopic no-slip condition on the length of the deformation experiment, the length of obser- wall also breaks, when the flow is highly rarefied and events vation of the material under load, etc. The simple fact that on microscale become relevant. all real materials have Tr < ∞ has an important consequence: everything flows. To witness it, we must perform/watch the 5.3. Energy equation of fluid experiment on large enough time scales Tf, to ensure Tf > Tr, whence low De. Water is ‘solid’ when we hit it very fast, faster The derivation of the full energy equation is cumbersome, than it can open. Thus, any ordinary mortal can walk over and often not necessary. When we neglect the dissipation water, provided one paces faster than ∼10−12 s. This walking and thermal effects and take the steady flow, the conserva- is due to relaxation times of water molecules. Walking sup- tive Bernoulli equation results, which is the first integral of ported by surface tension is mentioned in Section 5.4.1.On the corresponding momentum equation: the other hand, the things we consider ‘solid’ in our everyday ∼ 1 life (Tf hours, days, years), can flow during centuries and mil- v2 + p + gh = const. [J/m3] lennia (e.g. under its own gravity: glass tables in windows of 2 + + = cathedrals, lead pipes in old buildings). The scale Tf for geo- (convection) (pressure) (gravity) const. (5.3.1) logical processes is even longer. To see it, one should span very ∼ ∼ long period, aeons. For instance, the Lord with Tf eternity could see ‘mountains flowing before Him2’. Summing up, we Choosing the variables (v, p, h), their scales (V, P, L), and the see/feel the material liquid at low De (De  1) and solid at high parameters (, g), we get by scaling: De (De  1). 2 2 When dealing with conducting fluids (magnetohydro- V + P + gL = const. [M/LT ]. (5.3.2) dynamics—conducting fluids in magnetic field; metallic and ionic melts, Earth core, interior of stars, plasma), specific Here, we can see the convective and gravitational scaling numbers appear, e.g. N = (magnetic permeability)2 × (magnetic for the pressure directly, P ∼ V2 and P ∼ ␳gL. Dividing by the field)2 × (electrical conductivity) × L2/, or the Alfven number. convective term (V2), because of tradition, we get When dealing with compressible fluids, the Mach number P gL Mc = V/Vs is of paramount importance. Density variations in 1 + + = const. (5.3.3) V2 V2 a fluid can be estimated by / ∼ Kp, where K is the fluid compressibility and p pressure variation, p ∼ V2, using Assigning the DN their proper names we get the inertial pressure scaling. The compressibility effects are important when / > 1, i.e. V > 1/(K)2. Realizing that 1/(K)1/2 1 1 + Eu + = const. (5.3.4) is the speed of sound Vs,wehaveMc > 1. Since 1/K is the Fr bulk modulus of elasticity E, we can introduce the Cauchy number Ch = V2/E, to find that Ch = Mc2. Other related number It is a counterpart of the steady and inviscid (5.2.4), so only 2 Eu and Fr are present. is the Eckert number, Ec = V /cp ∼ (kinetic energy)/(enthalpy), which arises from scaling the full energy equation (tempera- Like the energy, also other quantities can be derived from ture rise by adiabatic compression). A relation holds, Ec ∼ Mc2. the velocity field, and their equations obtained by manipulat- When dealing with rarefied fluids (gases), there is a limit, ing (5.2.1). The vorticity equation is obtained by taking Curl of where our hypothesis about their continuous nature breaks: momentum equation (5.2.1): when the mean free path L of the fluid molecules is compa- m ∂␻ rable with the domain size L . There are not enough mutual + (v.∇)␻ = (␻.∇)v + ∇2␻ [Hyb/m4 s] d ∂t collisions between molecules to ensure the statistically sta- (unsteadiness) + (convection) ble averages. In terms of the Knudsen number, Kn = Lm/Ld, the corrections to rarefication are needed when Kn is large (gas at = (vortex stretching) + (vorticity diffusion), low pressure, in small domain—porous media, membranes, ˝ V˝ V˝ ˝ microchannels). One such correction is the Cunningham (slip) + = + . (5.3.5) T L L L2 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. The pressure and gravity disappear due to their gradient nature (they generate no torque), and the vortex stretch- ing term emerges. It is important for the energy flow down 2 This is often quoted in the reology literature with reference to the turbulent cascade, and is zero in 2D case, causing some the Deborah song (The Old Testament, Book of Judges, Chapter 5, anomalous phenomena. Note that the transport coefficient for Verse 5). Reiner (1964) introduced the concept of De with reference to the ‘flowing mountains’ in this song. It is right that diffusion of both momentum and vorticity is the same (namely the original Hebrew word means ‘to flow’ (root ‘nzl’ = , ). Some estimates of ˝ can be made, e.g. the easiest one, Hedanek,´ 2007). However, the mountains ‘flew’ not because of ˝ = V/L. Note that the convection and vortex stretching terms the very long observation time (only ∼1 (human) day; God are comparable, when the same scales are used for these two subdued Jabin, the king of Canaan, on that day), but because of physically different terms. the short-time and intense anger of God. Therefore, the Related to energy is so-called enstrophy. Energy per unit expressions ‘to quake’ or ‘to melt’, which appear in different mass is the square of velocity ((1/2)v2), enstrophy is the square translations, are closer to the original message. This note is not ␻2 to undermine the great concept of De, but to show that it does of vorticity ((1/2) ). The cross-product is called helicity not have its recourse in the Deborah song (and that the Lord was ((1/2)v·␻). These scalar quantities are often used in turbulence. likely not the very first rheologist). Their equations can easily be scaled and the corresponding chemical engineering research and design 8 6 (2008) 835–868 845

DN formed. It is not sure, whether the terminology (if any) of cles, where the deformation and the motion can strongly numbers thus produces is established and settled. be coupled. Since We stems from the Lagrangian accelera- tion term, it also reflects the particle deformation due to the 5.4. Boundary conditions: no slip and free-slip converging/diverging streamlines. Low We means low defor- mation, and vice versa. Other situation is walking over water Initial and boundary conditions inseparably pertain to the (hydrophobic feet assumed), where the inertia force (dynamic governing equations, and should also be scaled. The initial load) produced by the walker must not exceed the bear- conditions (IC) are given functions of the spatial distribution of ing power of the ‘flexible membrane’ of the surface tension, the variables in the initial time. There is actually nothing to whence low We is required. Note that other mechanisms of scale. On the other hand, the boundary conditions (BC) can be water walking exist, well beyond the bearing capacity of the more elaborate. BC are equations that the variables must fulfil surface tension. For instance, small reptiles (genus Basiliscus) at the boundary ∂˝ of the flow domain ˝, during the whole can run over water using a supporting impulse from a liquid jet process of solution. As such, they must also be scaled. created by a specific shape of their feet. Here, other numbers In single-phase flow, bounded by a rigid wall, the no-slip besides We play a role too. BC applies, which is simple and merely says that the speed is The Capillary number Ca compares the viscous and capillary zero at the wall, v = 0. There is not much to scale, besides v/V, forces. It is relevant when viscous forces dominate, i.e. slow which is kind of unnecessary. flows and small scales. The typical situation is drainage of a In multi-phase flow, where deformable fluid interfaces thin liquid film between two interfaces, at least one of them separate immiscible phases with certain interfacial (surface) is fluid (to have in play). The typical applications abound: tension , the free-slip BC applies, which is far from being interactions of fluid particles with themselves, with solids, trivial. It is a force balance at the fluid–fluid interface. Decom- and with rigid walls (coalescence, bouncing, adhesion), fre- posing the force into normal (normal stress, pressure) and quently encountered in bubble columns, flotation columns, tangential (shear stress) components, BC says: jump in fluid extractors, etc. pressure is balanced by tension , and jump in fluid stress The Bond number Bo and the Eotvos number Eo mean the is balanced by surface gradient of tension S (i.e. gradient same: the ratio of gravity () or buoyancy () forces to the along interface). There are important DN stemming from the capillary forces. The typical situation is deformation of a stag- free-slip BC that relate to deformation of fluid particles and nant drop sitting on a horizontal plane, where Bo compares interfaces. the hydrostatic pressure across the drop (gL) with the capil- lary pressure (/L) inside the drop. Naturally, the length scale 5.4.1. Normal component of free-slip BC L must be the drop size. Similar situation is when bubbles or By scaling, the normal component gives a number that com- drops are entrapped below a horizontal wall, or attached to a pares the fluid pressure and the Laplace pressure produced by needle (L ∼ orifice size). From the equilibrium deformation of the interface tension, N = P/(/L), sometimes called the Laplace bubbles and drops, the static value of can be obtained. Other number, La. It follows from the physical situation, that the situation is standing on the water surface, where the static length scale L cannot be chosen arbitrarily but must relate to load of the stander must not exceed the strength of the surface the curvature of the interface that produces the correspond- tension, whence low Bo is required. An important quantity ing capillary pressure /L [Pa]. Choosing something else would relates to Bo, the capillary length C. Consider that gravity be physical nonsense. Consequently, for L should be taken the forces dominate on large length scales at Bo ≥ 1, and capillary radius of curvature, the size of a bubble or drop, or perhaps forces dominated on small length scales at Bo ≤ 1. Find the crit- even the capillary length. What can be put for the fluid pres- ical (capillary) length scale C, where Bo = 1. Quickly we see that 1/2 sure scale P? Generally, anything from (5.2.7), provided that it the capillarity prevails over gravity when L ≤ C =(/()g) . corresponds to the physical situation in question. Upon differ- The typical situation is the capillary elevation of the water sur- ent scalings for pressure P, the number N takes the following face near the wall of the glass, which occurs in the range up forms and names: to ∼C from the wall. Beyond this, the gravity prevails and the surface is flat. L2V (unsteadiness) Unsteady scaling of P : Un = , Besides the four DN arising from the normal component of T (capillarity) the BC in (5.4.1), another composite number is often used. The ()LV2 (inertia) Convective scaling of P : We = , Morton number Mo is an artificial conglomerate that combines (capillarity) Re and Fr from the governing equations, and We from the BC, V (viscosity) Viscous scaling of P : Ca = , (5.4.1) in such a way, that it contains only the material properties of (capillarity) fluids (plus gravity): ()gL2 Gravitation scaling of P : Bo = Eo = We3 (gravity or buoyancy) Mo = . (5.4.2) . Re4 Fr (capillarity)

Here () means that both and can be used, putting The buoyant version reads Mo =()34g/43 = g()34/3, stress either on gravitational or buoyant aspect of We and Bo. and gravity version reads Mo = g4/3 = g3/3. It can also be The unsteady number Un turns into the Weber number upon obtained by DA, by forming a dimensionless group of (g, , , the convective scaling for time T = L/V. ). The Weber number We compares the inertia and capillary forces. It applies to deformation of bubbles and drops at 5.4.2. Tangential component of free-slip BC free rise or on collisions with an obstacle where dynamic By scaling, the tangential component gives a number that effects are important. Typical situations are, e.g. bouncing compares the surface gradient of the interface tension S at a wall and path instability of rising/falling fluid parti- and the fluid shear stress , N = S/. Since the quantity S 846 chemical engineering research and design 8 6 (2008) 835–868

is called the Marangoni stress Ma, the number N is called the may suspect that effects of the latter are more frequent and Marangoni number Ma: more important. The most of the common liquids (either pure or in mixtures or with surfactants) have in the range Ma = Ma . (5.4.3) ∼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, In case of a flat surface, where a tension difference within orders of magnitude. Moreover, itself influence develops over a length L, the surface gradient can be estimated directly only relatively few hydromechanical processes. Con- as S ∼ /L. The fluid stress can be estimated by the viscous sider for instance, the bubble formation process. The resulting scaling as ∼ V/L, which gives bubble size depends directly on only at the very low-gas flow where the equilibrium between buoyancy and surface force is /L reached during the quasi-steady inflation. This is not the case Ma = = . (5.4.4) V/L V of real gas–liquid contactors, where the bubble size depends on many other effects, involved in violent breakup of irregular It may be difficult to estimate the velocity scale V for com- gas jets produced by the gas distributor. On the other hand, the plex motions near interfaces on the convection basis, and the magnitude of S is difficult to estimate. Taking it ∼/L,we diffusive scaling is therefore applied. V is expressed with help have within the factor of 4 (i.e. almost ‘constant’ for all liq- of the diffusivity of the agent that causes the interface tension uids), but the relevant length scale L can cover very vast range variance (heat, surfactant), V ∼ (diffusivity)/L. It is consistent of values. Consider a bubble of size L. The surface gradient with using the viscous scaling for the fluid stress: in both can develop over any distance, from 0 to ∼L, making /L very cases, the molecular transport is reflected. much varying quantity. Moreover, the gradient formation and The specific formulation depends on the physical process evolution is a dynamic process, with fast temporal changes, that causes the variation of along the interface. Two situa- which depends on the flow situation (unlike , which is mate- tions usually occur: variation of is caused by a difference in rial property of static equilibrium). Also, all real processes temperature or in surfactant concentration c. In the former take place in systems that contain ‘impurities’, which read- case, it is =(d/d)·(∂/∂ϕ) and V = /L. In the latter case, it ily act as surface active agents (most of chemical compound, is =(d/dc)·(∂c/∂ϕ) and V = D/L. The Marangoni number then e.g. reactants). Therefore, no surprise, that bubbling into two reads ‘extremely different’ liquids with = 0.02 and 0.08 N/m gives ⎧ ⎪ L d ∂ almost identical results, while re-distilled water and tap water ⎨ . . (thermocapillarity), d ∂ϕ may differ by tenth of percents in gas holdup. For these rea- Ma = (5.4.5) ⎪ L d ∂c sons, the gas holdup correlations involving are extremely ⎩ . . (surfactants), D dc ∂ϕ unreliable, with respect to this particular variable. Better understanding the surface processes and their where ϕ is the angle along the interface of a bubble or drop, proper scaling is needed. A qualitative sketch of the surfac- and L corresponds to their size. To close the problem, we must tant action is shown in Fig. 3. Based on their tendency to find the dependence of the interface tension on the agent con- gather at the interface, two classes of surfactants can be dis- tent, () and (c), as well as the spatial distribution along the tinguished: positive and negative. The positive surfactants are interface of the agent itself, (x)S and c(x)S. The help comes attracted to the interface, their surface concentration is larger from the thermodynamics and physical chemistry of surfaces, than the bulk concentration (cs > cb). Typically, they are organic where the relation between and (, c)S at the interface is substances (emulsifiers, detergents, tensides, wetting agents, determined, and the relation between (, c)S at the interface etc.) that decrease the surface tension significantly. The neg- and that in the bulk (, c)bulk are established too. Often, the ative surfactants are repelled from the interface, their surface equilibrium between the bulk and interface is assumed (for- concentration is smaller than the bulk concentration (cs < cb). mulas are then called ‘isotherms’). The problem of finding the Typically, they are inorganic substances (salts of mineral acids, interfacial distribution of (x)S and c(x)S is much more compli- electrolytes, ionic solutions, etc.) that increase the surface ten- cated. Under severe assumptions on the flow and transport of sion only slightly. During the bubble rise, its nose feels the bulk

and c we can make simplified theories. Otherwise, we are on concentration (cb). Its tail feels a generally different concen- mercy of numerical experiments by CFD, where all processes tration, resulting from the adsorption/desorption transport are strongly coupled (flow, transport, adsoption/desorption processes occurring between the bulk and the interface, as kinetics). the liquid passed around the bubble. When an equilibrium is The significance of the Marangoni number is that its large reached, the rear concentration can be denoted as ‘equilibrium value indicates the presence of strong Marangoni stresses. concentration’ (ce). Obviously, the positive surfactants have

They are important in several situations. They delay the rise ce > cb, while the negative ce < cb. However, despite the opposite of drops and bubbles in ‘contaminated’ water. They delay concentration profiles along the interface of the positive and the drainage of the film between interfaces, which suppress negative surfactants, their surface tension profile is the same. the coalescence (bubble columns, extractors) and adhesion In both cases, the rear is low and the front is large. This

(flotation). Besides, it generates many other interesting phe- interfacial surface tension gradient (S) is a tensile force that nomena. For instance, at the thermocapillary migration, fine generates the Marangoni stresses. These stresses move the bubbles can ‘rise down’ in a liquid with inverse temperature liquid material elements along the interface, in the direction gradient (hot-bottom cold-top arrangement). Similar is the opposite to the main flow over the bubble. The increase in the electrocapillary motion, where the surface variation of is resistance force is the natural result (retardation of bubble rise caused by the electric charges. in contaminated media). At the first guess, the students (and Surface tension effects are very complex and very tricky. not only them!) would say, that if the positive surfactant delays Surfactants cause large effects even at trace amounts. It is the bubble rise, so the negative surfactant will accelerate it, difficult to separate the effect of and that of S. One which is, however, not so. chemical engineering research and design 8 6 (2008) 835–868 847

Fig. 3 – Positive and negative surfactants in air–water system. (a) Positive surfactants are attracted to surface: nonpolar parts point 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 surfactants have cs > cb, negative surfactants cs < cb. (c) Positive surfactants decrease surface tension (much). Negative surfactants increase 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 have same profile. Surface tension gradient (S) corresponds to rubber sheet with variable thickness: thicker regions shrink, making thinner regions expand (surface gradient is seen in (a)).

5.5. Multi-phase flow solids—no flow inside), and around them in the one con- tinuous carrying fluid. We should write (N + 1) Navier–Stokes 5.5.1. Microscale description (DNS) equations, and solve them simultaneously. They are coupled On the ‘microscopic’ level, the flow is fully resolved, in both by sharing the same boundary condition at the particle–fluid phases: inside N discrete dispersed particles (bubbles, drops; interface. We obtain the flow field in every point of the disper- 848 chemical engineering research and design 8 6 (2008) 835–868

sion. The interface forces move the particles and deform the Note that L must keep the meaning of the particle size d. interfaces. We have the full information about the problem. Dividing mercenarily by 2 = , we get dimensionlessly This fine microscale approach is in the CFD jargon called the ‘direct numerical simulation’ (DNS).  gL3 1  L2V2 = C, (5.5.5) There are no extra forces, as compared with the single- 6 2 2 4 2 phase flow. However, there is a difference: the surface tension force is moved from the boundary condition (Section 5.4) which is nothing but directly into the momentum equation (Section 5.2, Eq. (5.2.1)). 3 Usually, this new term contains the following ingredients: Ar = CRe2. (5.5.6) surface force [N/m3] ∼ interface tension [N/m], interface cur- 4 vature [m−1], interface area [m2/m3]. As for the scaling, this This is an important criterial equation for problems with term could be inertially scaled by (V2/L), like any other term in falling/rising particles, where gravity, buoyancy, and drag play (5.2.1). This would yield (/L)/(V2), which is nothing but We−1. the main role. It indicates a close relation between Re and Ar The specific way how this surface force is converted into a in this type of problems, where an equilibrium is established volumetric force and implemented depends on the modelling between the driving forces (Ar, l.h.s.) and the resistance force approach to the DNS. Currently there are several numeri- (Re, r.h.s.). The former are produced by an external field (e.g. cal strategies, how to describe interfaces and their motion gravity), while the latter involves speed as the main ingredient. (e.g. front tracking method, level set method, volume of fluid Since the speed in Re is often the part of the solution, the grav- method). itational scaling is employed V ∼ (g/L)1/2, which transforms Re into Ga = Re2/Fr. Here, the concept of buoyancy reflected by Ar 5.5.2. Mesoscale description (Euler/Lagrange) has much clearer interpretation than in the single-phase flow: On the ‘mesoscopic’ level, the flow field is resolved only in the effect of particle–fuid density difference is obvious. Note the continuous phase, using the single-phase equation—one that both particle size L and speed V are involved in Re(L, V) and Navier–Stokes (Eulerian view). The dispersed particles (bub- only size in Ar(L3). It leads to an iteration procedure in solving bles, drops, solids) are considered being pointwise, as seen by the sedimentation problem, where the speed depends on the the fluid. Their motion is described by set of N equations of particle size. A usual trick is to separate these two with help motion of system of bodies in space (Largangian view). This of a new suitably defined number. The Lyjascenko number, intermediate-resolution approach is in the CFD jargon called Ly(V3)=Re3/Ar =(V3/g)(/), converts (5.5.6) into the ‘Euler/Lagrange simulation’ (E/L). 3 The many kinds of hydrodynamic forces acting on the par- Ar1/3 = CLy2/3, (5.5.7) ticles are given by various closure formulas, obtained by other 4 means (e.g. experiment, DNS). The forces depend on the local where L stands on l.h.s. and V2 on r.h.s., so that the size flow field near the particles. The particles can also affect the and speed are decoupled. Another problem of course is with fluid motion, as a feedback (coupling). As compared with the C = C(Re(L, V)). single-phase flow, there are many new equations with new Second, consider an unsteady motion, when the resistance forces. All of them can be scaled, and new DN will appear. force consumes the initial momentum of a particle: Consider only the simplest case of a sedimenting particle in a stagnant unbounded fluid, without (particle → fluid) coupling, du 1 d2 which is the paradigm of sedimentation: m =− u2C (IC : t = 0,u= u ) [Hyb/s]. (5.5.8) dt 2 4 f 0

du In the simplest case of the Stokes drag, we have a linear m = gravity − buoyancy − drag [Hyb/s]. (5.5.1) dt relaxation process and (5.5.8) becomes

du Substituting the typical closures for the forces, it reads m =−3du [Hyb/s] (unsteadiness) = (drag). (5.5.9) dt

2 du =  3 −  3 − 1 d 2 This is a rare occasion in two-phase flow when we can m d pg d fg fu C [Hyb/s]. (5.5.2) dt 6 6 2 4 find the relaxation time (original disturbance is reduced by 1/e ≈ 64%; a stable equilibrium presumed, a node). Choosing First, consider a steady particle motion, under the force the variables (u, t), their scales (V, T), and the parameters (m, equilibrium: d, ), we get by scaling:

mV  1 d2 = dV [ML/T2]. (5.5.10) gd3 = u2C ( = − ) [N] T 6 2 4 p f (gravity) − (buoyancy) = (drag). (5.5.3) By turn, we have the relaxation time:

m T = [T]. (5.5.11) r d Choosing the variables (d, u), their scales (L, V), and the 3 parameters (, , g, C), we get by scaling: Substituting for the particle mass m =(/6)d p, the time is

2 2 2  pd pd  3 1 L 2 = ∼ gL = V C [N]. (5.5.4) Tr [T]. (5.5.12) 6 2 4 6 chemical engineering research and design 8 6 (2008) 835–868 849

Another way is to solve the linear problem (5.5.9) directly: stress in the particulate phase, which by the kinetic theory ∼ 2 ∼ is pV , where V d, where is the velocity gradient and d 3d ∼ ∼ u(t) = u exp − t [m/s]. (5.5.13) the particle size, and the viscous fluid stress . With V/L, 0 m and L ∼ d, Ba = pdV/, which actually is St. Equivalently, it can be recasted in terms of the friction forces (particle col- The exponent should be of the form (t/Tr), in order, at t = Tr, lisional)/(fluid viscous). The presence of the interstitial fluid the ratio u/u0 be exp(−1) = 1/e. Accordingly, the relaxation time can be neglected at large Ba, to reach the limit of so-called is ‘dry’ granular flows. 2 2 1 pd pd T = ∼ [T]. (5.5.14) r 18 5.5.3. Macroscale description (Euler/Euler) On the ‘macroscopic’ level, the dispersed particles (bubbles, The variance of (3) between (5.5.12) and (5.5.14) is the mod- drops, solids) are considered as a phase smoothly distributed est price for bypassing the exact solution by the scaling, which in space, forming a ‘pseudo-continuum’. The particles are is acceptable. then assigned continuous concentration and velocity fields. The relaxation time just found is useful for a variety of situ- This approximation is acceptable when we describe the sys- ations, where time scales of different processes are compared, tem on length scales much larger than the discrete scales. The to distinguish between the ‘fast’ and the ‘slow’, which in turn particles and their spacing must be much smaller than the leads to the time decoupling, making the problem easier to system size, and than the smallest scales we want to resolve. handle. One important case is the concept of the Stokes num- In the single-phase flow, estimate the discrete (atomic) scales − − ber St. It compares two time scales: particle relaxation time Tr by 10 9 m, and the beginning of the continuum by 10 6 m, and time of flow change Tf: say. We have three orders of magnitude to bridge the gap. Accordingly, with 1 cm bubbles, the reactor should be of T St = r . (5.5.15) 10-m size, to consider the bubbles as the continuous phase. T f Further, presence of many bubbles is anticipated, for their spacing be small, e.g. comparable with bubble size. Now, the The former says how fast the particle relaxes back to the governing equations should be twice Navier–Stokes: one for steady state, when accelerated with respect to the surround- the continuous phase and one for the dispersed phase (two ing (viscous) fluid. The latter says how fast the flow field interpenetrating continua, twice Euler’s view). These equa- changes. Without a priori knowledge, the basic estimate of Tf ∼ tions are coupled via the interphase momentum transfer. can be made, Tf L/V. Since the particle must feel the changes This coarse macroscale approach is in the CFD jargon called in the fluid, we couple them by the common length scale, the ‘Euler/Euler simulation’ (E/E). clearly given by the particle size, L = d. Then, the Stokes num- The governing equations for single-phase flow are often ber is said to be derived from the ‘first principles’. These mechanical

pdV principles are known for a single continuum, namely in case St = . (5.5.16) of simple fluids, but are only in the process of development for the multi-phase systems. Here, we lack a universally valid In other words, St is the ratio (particle inertia)/(fluid iner- equation, which would be of practical use. There are many tia). More correctly, the particle mass used above should also general equations suggested, but they are too monstrous, 3 contain the added mass Ca, m =((/6)d )(p + Ca), especially and the many closures needed for them are still missing. here, when unsteady effects are considered. Then the particle On the other hand, there also are many simple equations, → density becomes: p (p + Ca). We appreciate it namely in suggested for specific systems and particular flow situations, ≈ −3 case of bubbles in liquids, where p/ 10 , so that the bub- which are practical, yet of limited use. As a compromise, here ble inertia is represented by the added mass, i.e. by the liquid we write the two-phase flow equations purely formally, as two ≈ inertia, p,effective , since Ca is O(1). Then, the Stokes num- Navier–Stokes-like equations: ber becomes the Reynolds number, St = pdV/ ≈ dV/ = Re, meaning ∼ (particle–joint fluid inertia)/(‘viscous fluid inertia’). • Continuous phase: The Stokes number measures the willingness of the car to get off the road when you turn the stirring wheel sud- ∂     ε +∇(ε v) = 0 (mass) [kg/m3 s], (5.5.17) denly. As such, it is used in many situations when we want ∂t to know how much the dispersed particles tend to follow the streamlines of the carrying fluid. The total flow-follower has ∂         ε v + (v.∇)(ε v) =∇ε ␶ + ε f + S St = 0 (passive scalar, tracer). On the other hand, particles with ∂t large St easily hit the wall in bendings of a duct. It is used (momentum) [Hyb/m3 s]. (5.5.18) in devices (impactors) where aerosol particles are sorted out by their value of St, being expelled from the main stream to the wall by their inertia, in multiple progressively narrowing • Dispersed phase: U-bends. The adhesion efficiency of the flotation process also ∂     depends on the value of St, with which a bubble collides with ε +∇(ε u) = 0 (mass) [kg/m3 s], (5.5.19) particles: the lower St, the better for adhesion (no bouncing). ∂t Other numbers appear in specific areas of multi-phase flows, at this mesoscale level of description. For instance, the 2 ∂         Bagnold number in granular flows, Ba = pd /, compares the ε u + (u.∇)(ε u) =∇ε ␶ + ε f − S ∂t effect of the interstitial fluid on the motion of the granules (grains). In terms of stresses, it is a ratio of the collisional (momentum) [Hyb/m3 s]. (5.5.20) 850 chemical engineering research and design 8 6 (2008) 835–868

The continuous phase has density , velocity v, stress ten- • Continuous phase: sor ␶, volume fraction ε. The dispersed phase has density  ␶  ∂ , velocity u, stress tensor , volume fraction ε . The mass +∇(v) = 0 (mass) [kg/m3 s], (5.5.22) conservation for the two-phase mixture needs ε + ε = 1. The ∂t main multi-phase problem is to formulate the stress tensors ∂ ␶ v + (v.∇)(v) =−∇p + ∇2v + f and the interaction term S. As for the scaling, two kinds of ∂t new DN could appear, as compared with single-phase flow. 3 One comes from scaling the stress term ␶, which very likely (momentum) [Hyb/m s]. (5.5.23) will not reduce to simple Newtonian formulation yielding Eu and Re. The other comes from the momentum transfer term While a single dispersed particle does not affect the flow, S. The latter would indicate how strong the phase coupling a large number of them can exert collective buoyancy effects. is. These macroscopic effects consist in fluid density variations caused by distribution of the particle concentration, = (c). If the dispersed phase constitutes a macroscopic dispersion, The interphase coupling is as follows. The (micro) dispersed the particles are the mechanical individuals (small bodies) phase imports the fluid velocity v from (5.5.23) into (5.5.21), and have their own momentum, inertia, gravity, buoyancy, and exports the concentration c from (5.5.21) into the fluid den- whence the momentum equation (5.5.20) applies. If the dis- sity in (5.5.22) and (5.5.23). This is the convection–buoyancy persed phase constitutes a microscopic dispersion, the particles two-way coupling. lack these qualities, and (5.5.20) is needless. The individual Note that Eqs. (5.5.21)–(5.5.23) formally coincide with those particles passively follow the flow (St = 0); very fine particles for heat and mass transport considered in Sections 6 and called ‘passive scalars’ and used for flow visualization (trac- 7. Indeed, these equations represent the non-inertial micro- ers). The true chemical solutions (salt or dye in water) surely disperse limit of the governing equations (5.5.17)–(5.5.20) belong to this category (sub-colloidal). There is a legitimate derived for the macro-dispersed multi-phase mixtures. This question: what is between these two extremes, the macro simple fact opens an interesting window of research: build- and micro. When the dispersed phase earns the right to be ing analogies between the well-understood single-phase flows awarded the full momentum equation? Likely, there are no with heat and mass transport, and much less understood strict and unequivocal criteria, to decide. Probably, it depends multi-phase flows. The buoyant coupling from (5.5.21) to on our choice, what effects and on which scales we wish (5.5.22)–(5.5.23) can be facilitated by any buoyant agent that to study. In the molecular dynamics, very small particles behaves like a passive scalar (true solute, heat, fine particles, (solute/solvent molecules) are moved by force laws of vari- etc.). As the next step, the first-order inertial effects can be ous degrees of resolution. Some forces are derived directly added to this base state. For instance, the buoyancy-driven from molecular potentials, some are modelled as random instabilities in sedimenting layers, fluidized, beds, and bubbly thermal noise to account for Brownian effects, there is a columns may shear certain common features with phenom- resistance force, and the overall formulation can be within ena of thermal convection or halinoconvection. As for the the Langevine ansatz (random forcing, stochastic differential scaling, DN related to heat and mass transport is introduced equations). In the Stokesian (hydro)dynamics, very small par- in Sections 6 and 7, together with the coupling to the hydrody- ticles (fine particles suspended in fluid) are moved by forces namics. They naturally apply also to (5.5.21)–(5.5.23). Another of both hydrodynamic (Stokes limit) and nonhydrodynamic issue is what is the relevant density of a multi-phase mixture, (Brownian, colloidal, interparticle, etc.) origin. Another aspect when evaluating the buoyancy force acting on a submerged is reflected by rheology, where the dispersed phase affects the body. Usually, one takes either the pure fluid density or the intrinsic momentum transport substantially. effective mixture density. It seems however, that both can be For microscopic dispersions, Eq. (5.5.20) is omitted and relevant, depending on the relation between the body size (5.5.19) is modified accordingly. The quantity (ε) is replaced and the size and spacing of the dispersed particle in the with the scalar concentration c. The speed u is identified with mixture. v. The molecular diffusion term is added to the r.h.s., since the macroscopic particles did not have this molecular transport 5.5.4. Retention time distribution mechanism.3 Eq. (5.5.19) then becomes: A brief note is in place, on the retention time distribution (RTD) in equipments. It is the very first thing one must do, • Dispersed phase: before starting any kind of thoughts about the processes in ∂c a given apparatus. We assume that the tracer concentration + (v.∇)c = D∇2c (mass) [kg/m3 s]. (5.5.21) ∂t obeys (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- These modifications must also be reflected by Eqs. (5.5.17) tent with the role of a ‘pure flow follower’. The proper tracer and (5.5.18). Since the dispersion does not exist on the should stick to a fluid particle, and not to diffuse. Therefore, it macroscale, the carrying fluid occupies the whole volume, should obey: ε = 1. Omit the apostrophe at  and ␶. Set S = 0, since the Dc fluid does not receive momentum from the dispersed par- = 0 (mass) [kg/m3 s], (5.5.24) ticles. For Newtonian fluid equations, (5.5.17) and (5.5.18) Dt become which is (5.5.21) with D ≡ 0. This can directly be solved numerically, together with (5.5.22) and (5.5.23), to get the full 3 But there is a concept of ‘hydrodynamic diffusion’ of information about the flow and concentration fields. Then, it macroscopic particles, due to (mostly repulsive) interaction is easy to monitor numerically the tracer content at the exit, to forces, see e.g. Davis (1996). create the RTD response curve. This is not the way we wish to chemical engineering research and design 8 6 (2008) 835–868 851

go in practical applications of RTD. Let us treat the unwanted main flow. Contrasting all these assumptions with the real- diffusion term in (5.5.21). One cure would be the assumption, ity of the actual flows through real technological systems, it that the flow is much faster than the tracer diffusion. This is would rather be naive to expect that Pe, whence the single not usually done, and even if, this is usually not true. Another, scalar quantity Dt (also called: axial dispersion Dax), can con- and more practical cure, is to assume that the tracer is simulta- tain the whole truth about the hydrodynamics. We are in need neously transported by two mechanisms, molecular diffusion of something smarter.

(Dm) and turbulent dispersion (Dt). Eq. (5.5.21) can be treated as Actually, there has been an attempt is this direction, and a follows. Upon the Reynolds decomposition, the actual veloc- kind of Smart RTD (SRTD) has been suggested. Imagine that a ity field splits into the mean and fluctuating parts, v = V + v. fluid particle has a watch that measures its age , being set to The convective term in (5.5.21) becomes (V·)c +(v·)c, keep- zero at entering the equipment. Thus, is the retention time ing the physical meaning of the convective mass flux, j = J + j. of the fluid particle. At non-relativistic motions, the time on The mean term, J =(V·)c, is harmless. The fluctuating term is the watch coincides with the ‘common’ time t on the labora- modelled as the mass flux driven by the turbulent dispersion, tory clock. The trivial and seemingly useless physical fact that  2 j = Dt c. Eq. (5.5.21) thus reads = t can be recasted into a useful form. Because the watch is fixed to the moving fluid particle, its age must follow the ∂c + (V ·∇)c = (D + D )∇2c (mass) [kg/m3 s]. (5.5.25) convective derivative: ∂t m t D  = 1 (retention time) [-], (5.5.31) Suffice to require that Dm Dt, which can even be realistic, Dt at least in some flow situations. This must, however, very care- fully be checked in slow flows in microchannels. In case of 1D which is already dimensionless. Here, we assume that the dominant main flow, we have V = V = Q/S, and the tracer equa- quantity can be considered to be the field quantity. It is rather tion is uncoupled from the flow equation, and can be solved counter-intuitive, but any fluid particle located at time t in independently: place x can be assigned the amount of time (x, t) it has spent in the equipment. Thus solving the above equation for ‘conser- ∂c ∂c ∂2c + V = D (mass) [kg/m3 s]. (5.5.26) vation of particle age’, we obtain the real ‘distribution’ of the t 2 ∂t ∂x ∂x local retention time within the equipment. Having the field (x, t), we can easily calculate the field of concentration, reaction Fixing the coordinates to the mean flow V deletes the mean rate, conversion, etc. Eq. (5.5.31) can be treated in a similar way convection term and yields like (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 flow ∂c ∂2c = D (mass) [kg/m3 s], (5.5.27) and RTD fields. It can also be simplified, to save the computing t 2 ∂t ∂x power. After the decomposition, V + v, we have the following counterpart of (5.5.25): which is diffusion in stagnant medium. At absence of turbu- lent fluctuations, the axial dispersion is zero, Dt = 0, and the ∂  equation + (V ·∇) + (v ·∇) = 1 (retention time) [-]. (5.5.32) ∂t ∂c = 0 (mass) [kg/m3 s], (5.5.28) · ∂t The mean ‘flux of age’, (V ), is physically plausible, since the stream passing through a given location contains parti- · solves to c = c0 = const. It is the plug flow, where the cross- cles of various age. The fluctuating part, (v ), can either  2 section area marked with c0 is carried through the system with be modelled by a turbulent diffusion term, j = Dts , or left the mean speed V. Anticipating the scaling applied to Eq. (7.1), as it is and take a suitable closure for the fluctuating veloc-  (5.5.26) becomes the RDT analogue of Eq. (7.4): ity v . It can be modelled, e.g. by random functions reflecting truly the local structure of the turbulence. There is wealth of Fo + Pe = 1. (5.5.29) information about the scaling behaviour of turbulent velocity fluctuations in various flow situations, in the literature. This Here, the numbers are defined using the turbulent diffusiv- brings us naturally to the stochastic modelling of RTD. Con- ity: sidering the 1D dominant main flow, where the mean speed is a constant, V = V = Q/S, we have the counterpart of (5.5.26): L2 (unsteadiness) Fourier number,Fo= , DtT (turbulent mass diffusion) ∂  ∂ + (V + v ) = 1 (retention time) [-]. (5.5.33) LV (hydrodynamic convection) ∂t ∂x Peclet number,Pe= . Dt (turbulent mass diffusion)  (5.5.30) In fully 1D case, both V and v are scalars. Fixing the coor- dinates to the mean flow V deletes the mean convection term Two ways leads to the exclusivity of the Pe number occur- and yields the following counterpart of (5.5.27): rence in the RTD problems. First, use the basic time scaling, T = L/V, and the Fourier number becomes the Peclet number, ∂  ∂ + v = 1 (retention time) [-], (5.5.34) Fo = Pe. This makes Eq. (5.5.29): 1+1=1/Pe. Second, fixing the ∂t ∂x coordinates to the mean flow V deletes the convection term (Pe), and by the same scaling Fo becomes Pe. This makes Eq. which can be treated within the framework of the stochas- (5.5.29): Pe +0=1, which corresponds to (5.5.27). Thus, Pe is tic differential equations, with a suitable random v = f (x, t). the only important number in RTD, with the 1D dominant At absence of turbulent fluctuations, v = 0, the equation 852 chemical engineering research and design 8 6 (2008) 835–868

becomes where the convective speed is recasted into the mass flow. ∂ = 1 (retention time) [-] (5.5.35) Since we customarily divided (6.3) with the diffusion term, ∂t both numbers are based on the diffusion rate. If we want other combinations, we must compose them. These compositions and solves to = t. It is the plug flow, where all fluid particles can contain thermal and hydrodynamic quantities to reflect in the moving coordinate system have age = t. This age cor- their coupling. One route leads to the Prandtl number: responds to the position x = Vt from the inlet, measured in the stationary coordinate. With pipe of length L, the particles exit (momentum diffusion) at age = t = L/V. Prandtl number,Pr= . (6.7) (heat diffusion)

6. Transport of heat The Prandtl number is prepared as follows. Take the Peclet number, Pe = LV/. Replace the hydrodynamic convection (LV) For the transport of heat in a moving environment, three bal- with the hydrodynamic diffusion () with help of Re, LV = Re, ances must be considered together: fluid mass balance (5.1.1), to get Pe = Re/. Divide by Re, since it is dimensionless, to get fluid momentum balance (5.2.1), and the heat balance, which Pe = /. Give the product a new name: the Prandtl number, can be written as Pr = Pe/Re. This number is often used in thermal processes, where the ratio of two material properties of fluid determines ∂ + (v ·∇) = ∇2 [K/s] how much the flow is affected by heat diffusion (thermal con- ∂t vection, boiling, etc.). (unsteadiness) + (convection) = (diffusion). (6.1) Another route leads to the heat Grashof number, which is typical for heat-driven buoyancy effects: Choosing the variables (, v, x, t), their scales (, V, L, T), ˛ gL3 (buoyancy force) and the parameters (), we get by scaling: Grashof number,Gr= . (6.8) 2 (viscous force) V + = [/T]. (6.2) Take the hydrodynamic Archimedes number, Ar =(/)Ga. T L L2 Replace the general expression for the density difference Dividing by the temperature scale disappears, and we get with a specific expression for the thermal expansivity of fluid, / = ˛, where ˛ is the coefficient of thermal expansion. 1 V Plug it into Ar to get Ar =(˛)Ga. Give it name the heat + = [1/T]. (6.3) T L L2 Grashof number, alias the thermal Archimedes number. Gr is encountered in thermofluid mechanics. It is essentially a Dividing by the diffusion term (/L2), because of tradition, hydrodynamic number, where the heat enters as the ‘buoyant we get agent’, to produce density gradients. A close derivative of Gr is the thermal Rayleigh number, L2 LV + = 1. (6.4) Ra = Gr Pr: T ˛ gL3 (buoyancy force) Rayleigh number,Ra= . (6.9) We can divide by any term, but then the DN would not (viscous force) have their usual names. The diffusion term is the most typi- This number is decisive for the onset of thermal convection cal one for heat and mass transport phenomena, so it is the and its evolution via series of bifurcation. It compares the driv- natural scaling basis. Assigning the DN their proper names ing thermal disturbance acting on a parcel of fluid, with we get rates of transport processes that tend to smear it out (diffusion of momentum and heat ). Fo + Pe = 1. (6.5) The way the above composite numbers are ‘derived’ by making various combinations and replacements seems to be The following DN arise, having a clear physical meaning in neither transparent nor free from ambiguity and subjectivity. terms of process rates: Actually, they can be obtained correctly, from the correspond- ing governing equations. For instance, Pr, Gr and Ra appear L2 (unsteadiness) Fourier number,Fo= , naturally by scaling the coupled equations for flow and heat T (heat diffusion) transfer, under a useful approximation (Boussinesq), where LV (hydrodynamic convection) Peclet number,Pe= . the only buoyancy-affected density is that at the external (heat diffusion) force field term (f). Often, the linearization near a uniform (6.6) base state is considered (stability studies), or the uniform part of the density field is absorbed in the pressure term, which When using the basic time scaling, T = L/V, the Fourier gives the buoyancy term proportional to the density difference number becomes the Peclet number, Fo = Pe. The unsteady = − 0. Depending on the scales employed, the numbers and convective effects are then comparable, as a direct appear in different places, as either Pr and Ra,orPr and Gr. consequence of our choice of scaling. This may not always cor- Note that, based on the physical analogy and scaling argu- respond to reality. The Peclet number facilitates the coupling ments, it is possible to introduce Ra also for dispersed layers,  3 between the hydrodynamics and heat transfer: it compares e.g. for bubbly layers in bubbly columns, Ra = g eL /mixDhydro the transport by hydrodynamic convection (flow of medium, V) (g – is the reduced gravity, e –volume fraction of bubbles and the molecular diffusion of heat (heat diffusivity ). A mod- (voidage, gas holdup), mix –effective viscosity of bubbly mix- ified version of Pe is the Graetz number, Gr = (mass flow)cp/L, ture, Dhydro –hydrodynamic diffusivity of bubbles). chemical engineering research and design 8 6 (2008) 835–868 853

Other numbers are also related to buoyancy effects. A Choosing the variables (c, v, r, x, t), their scales (C, V, R, L, basic quantity is the buoyant (Brunt–Vaisala) frequency ω T), and the parameters (D), we get by scaling: of the density difference driven oscillator, (d2z/dt2)=ω2z, 1/2 C VC DC where ω =((g/)(d0/dz)) . It is an elementary prototype for + = + R [C/T]. (7.2) 2 internal gravity waves in stratified environments. Here we T L L can encounter the Richardson number, in several variations. Dividing by the diffusion term (DC/L2), because of tradition, The gradient Richardson number Ri = ω2/(∂u/∂z)2. The global we get Richardson number is Ri = gL/V2. Note that it relates to the 2  Froude number Fr’=V /g L =1/Ri, when Fr is corrected for buoy- L2 LV RL2 + = 1 + . (7.3) ancy (apostrophe’). Other kinds of Ri also exist. When both DT D DC buoyancy (stratification) and rotation are present, there are numbers indicating their relative effects (stratification param- We can divide by any term, but then the DN would not have eter, Burger number). The buoyant frequency also follows from their usual names. The diffusion term is the most typical for the gravitational time scaling T = V/g in (5.2.6). Taking V = L/T, heat and mass transport phenomena, so it is natural to take it the time scale becomes T = L/gT, which in terms of frequency as the scaling basis. Assigning the DN their proper names we ω ∼ 1/T reads ω2 = g/L. Employing the reduced gravity g → g to get account for the density variance, and designating /L the Fo + Pe = 1 + Da2. (7.4) scale estimate for (d0/dz), we get what was due. Boundary conditions in heat transfer are of two kinds. Either The following DN arise, having a clear physical meaning in the temperature w is given at the boundary ∂˝ or the heat terms of process rates: flux jw through it. The former case is simple to scale, w/. The latter is given by L2 (unsteadiness) Fourier number,Fo= , =− ∇ 2 DT (mass diffusion) jw ( )w [J/m s], (6.10) LV (hydrodynamic convection) Peclet number,Pe= , where is the heat conductivity. To calculate the flux, the tem- D (mass diffusion) 1/2 perature field must be known, which is not always the case. RL2 (reaction) Damkohler number,Da= . Therefore, another expression for the flux is introduced, with DC (mass diffusion) help of the empirical heat transfer coefficient kh: (7.5)

2 jw = k () [J/m s], (6.11) h w When using the basic time scaling, T = L/V, the Fourier number becomes the Peclet number, Fo = Pe. The unsteady where ()w is the bulk–boundary temperature difference. and convective effects are then comparable, as a direct con- Equating (6.10) and (6.11) yields the formula for the coefficient: sequence of our choice of scaling. This may not always ∇ correspond to reality. The Peclet number facilitates the cou- =−( )w 2 kh [J/m sK]. (6.12) pling between the hydrodynamics and mass transfer: it ()w compares the transport by hydrodynamic convection (flow of A simple scaling of (6.12) by (/L) leads to the Nusselt num- the medium, V) and the molecular diffusion of mass (mass ber Nu (also Biot number, Bi): diffusivity D). Since we customarily divided (7.2) by the diffusion term, k Nu = h , (6.13) the numbers are based on the diffusion rate. If we want other /L combinations, we must compose them. These compositions can contain diffusion and hydrodynamic quantities to reflect which is nothing but the dimensionless heat transfer coef- their coupling. One route leads to the Schmidt number: ficient. The length scale L comes from the near-interface  ∼ temperature gradient in (6.10),( )w 1/L, and should relate, (momentum diffusion) Schmidt number,Sc= . (7.6) e.g. to the thickness of the thermal boundary layer. Although D (mass diffusion) Nu does not present any intellectual challenge on the grounds of scaling, it is the most desired quantity in the heat transfer, The Schmidt number is prepared as follows. Take the Peclet since everyone wants to know how much heat passes through number, Pe = LV/D. Replace the hydrodynamic convection (LV) the interface, without computing the temperature and flow with the hydrodynamic diffusion () with help of Re, LV = Re, fields. Numerous correlations do exist for Nu in the engineer- to get Pe = Re/D. Divide by Re, since it is dimensionless, to get ing literature. Pe = /D. Give the product a new name: the Schmidt number, Sc = Pe/Re (also: mass or diffusion Pr). This number is often used 7. Transport of mass in mass transfer processes, where the transport rate is large enough to affect the flow (e.g. fast-phase changes, boiling, dis- For the transport of mass of a solute in a moving environment, tillation, etc.). In turbulence, the viscosity and diffusivity can three balances must be considered together: fluid mass bal- be not molecular but ‘turbulent’ (eddy viscosity; coefficient of ance (5.1.1), fluid momentum balance (5.2.1), and the solute dispersion), hence turbulent Schmidt number. mass balance, which can be written as Another route leads to the mass Grashof number, which is typical for mass-driven buoyancy effects: ∂c + (v ·∇)c = D∇2c + r [kg/m3 s] ∂t ˇ cgL3 (buoyancy force) Grashof number,Gr= . (7.7) (unsteadiness) + (convection) = (diffusion) + (reaction). (7.1) 2 (viscous force) 854 chemical engineering research and design 8 6 (2008) 835–868

Take the hydrodynamic Archimedes number, Ar =(/)Ga. A simple scaling of (7.11) by (D/L) leads to the Sherwood Replace the general expression for the density difference number Sh (also Sherman number, Sm): with a specific expression for the concentration expansivity of fluid, / = ˇc, where ˇ is the coefficient of concentrational km Sh = , (7.12) expansion. Plug it into Ar to get Ar =(ˇc)Ga. Give it name the D/L mass Grashof number, alias the diffusion Archimedes number. Gr is encountered in flows with ineligible concentration gradi- which is nothing but the dimensionless mass transfer coef- ents, which may interfere with the flow (e.g. halinoconvection, ficient. The length scale L comes from the near-interface  ∼ double diffusive convection, thermosolutal convection). It is concentration gradient in (7.9),( )w 1/L, and should relate, essentially a hydrodynamic number, where the mass enters e.g. to the thickness of the concentration boundary layer. as the ‘buoyant agent’, to produce density gradients. Although Sh does not present any intellectual challenge on A close derivative of Gr is the concentration (salinity, mass) the grounds of scaling, it is the most desired quantity in the Rayleigh number, Ra = Gr Pr: mass transfer, since everyone wants to know how much mass passes through the interface, without computing the concen- ˇ cgL3 (buoyancy force) tration and flow fields. Numerous correlations do exist for Sh Rayleigh number,Ra= . (7.8) D (viscous force) in the engineering literature.

The physical picture is similar like with the thermal Ra: 8. Correlations concentrationally buoyant fluid parcel driven by c moves, and diffusion of momentum () and mass (D) tend to oppose In the preceding sections, many important DN were intro- the motion and to weaken the driving force. Instead of ˇc, duced and commented, and most of them are listed in Table 2. more simple choice c/c0 can also be used. Mass Ra is decisive They can be divided into two classes. First, the basic (primary) for halinoconvection. Both thermal and mass buoyancy effects DN that follows directly from scaling the balance equations are present in double diffusive convection, which, besides oth- and their BC. Second, the other (secondary) DN that are ers, has application in oceanology (hot/cold, more/less salty derived from the basic, or formed by their combinations, or water) and geology (layering in magna chambers). created ‘artificially’. The basic DN are the following: momen- Besides the above coupling between diffusion and hydro- tum transport (Eu, Fr, Re, Sr) and BC (Bo, Ca, We, Ma); heat dynamics via Pe, there is also coupling between the diffusion transport (Fo, Pe) and BC (Nu); mass transport (Da, Fo, Pe) and BC and reaction via Da. A typical situation occurs in heteroge- (Sh). The basic numbers and their link to their closest relatives neous catalysis, where the diffusion and reaction interplay. is shown in Fig. 4. Solving the corresponding equation in case of a model situ- All the numbers obtained by the equation scaling (Section ation (a cylindrical pore in a catalyst, a spherical pellet), we 4) can be reproduced by the dimensional analysis (Section obtain the concentration profile, whose mean value cm nor- 3). However, we must know beforehand the relevant physi- malized by the bulk concentration c0 is the effectiveness factor cal quantities that should be grouped. It seems that most of ∼ F = cm/c0, which also is (mean reaction rate)/(maximum rate). DN in engineering were first obtained by DA. Here, we prefer ∼ The model solution for a pore gives F tanh(Th)/Th, where to relate them to the equations, to give them better physical 1/2 the Thiele number (modulus) is Th = L(k/D) . Here, L is the interpretation. pore length and k the rate constant. Note that for r = kc, the Regardless of their origin, DN are used for making corre- Damkohler number coincides with the Thiele number, Da = Th. lations. There are two main problems encountered. First, to Often, a ‘generalized’ Th is introduced, to retain the last equal- choose suitable DN. We need a complete list of independent ity also for reactions of higher orders. Reaction and mass numbers. Second, to choose suitable characteristic scales to transfer is combined in the Hatta number, Ha. There also are evaluate the DN. Both the numbers and the scales must be numbers typical for the reaction kinetics itself. For instance, relevant for the problem. It is very difficult to choose them the Arrhenius number Ah compares the activation energy and correctly without the sound knowledge of the underlying kinetic energy of molecules. processes and the physical meaning of the numbers. Here, Boundary conditions in mass transfer are of two kinds. Either the numbers generated by scaling of equations have a great the concentration cw is given at the boundary ∂˝ or the mass advantage over those produced by dimensional analysis. They flux jw through it. The former case is simple to scale, cw/C. The contain the correct quantities and have a clear meaning—ratio latter is given by of different effects in terms of common physical quantities (force, rate, time, speed, etc.). However, the problem with the =− ∇ 2 jw D( c)w [kg/m s], (7.9) scales still remains. The procedure of equation scaling can give certain hints what should the proper scales be, but this is not where D is the (mass) diffusivity. To calculate the flux, the con- always sufficient. centration field must be known, which is not always the case. Some choices of scales are apparently wrong. For instance, Therefore, another expression for the flux is introduced, with consider the Bond number, Bo = gL2/, which comes from help of the empirical mass transfer coefficient km: the normal component of the free-slip boundary condition. This number is highly relevant for behaviour of bubbles in = 2 jw km(c)w [kg/m s], (7.10) liquids. Accordingly, it enters numerous correlations for bub- ble size and speed, interfacial area, mass transfer coefficient, where (c)w is the bulk–boundary concentration difference. which are designed for bubble column reactors. Which length Equating (7.9) and (7.10) yields the formula for the coefficient: scale L is appropriate? The physics strongly recommends the bubble size. Despite this, many authors have been using the ∇ D( c)w bubble column size, which is apparently wrong. When we look km =− [m/s]. (7.11) (c)w at books and review papers on bubble columns, it is easy to find chemical engineering research and design 8 6 (2008) 835–868 855

Table 2 – List of dimensionless numbers (rather subjective) argument of ‘importance’. For instance, we can think that the smallest dimension is the most important. 3 2 Archimedes number Ar =(/)gL / (=(/)Ga) 3 2 However, taking L = H and ignoring the lateral walls at a thin Bagnold number Ba = pL / Biot number Bi = Nu layer, simply because A > H, gives a wrong result. The failure is

Bodenstein number Bd = VL/Dax especially evident in stability considerations. The presence of Bond number* Bo =()gL2/ (=We/Fr) walls reduces the spectrum of possible wavenumbers (modes) Capillary number* Ca = V/ substantially, and in effect, stabilizes the layer with respect 2 2 Cauchy number Ch = V /E (=Mc ) to the onset of convection. Therefore, the critical value of Ra 2 Cavitation number Cv = P/V depends on both H and A. The stability issues are very sen- Damkohler number* Da =(RL2/DC)1/2 1/2 sitive to the proper choice of scales. For instance, the critical Dean number Dn = Re(rpipe/rcurv.) value of Ra is different for rectangular and circular finite con- Deborah number De = Tr/Tf 2 Eckert number Ec = V /cp tainers, of the same size. Thus, the argument of ‘importance’ Ekman number Ek = /˝L2 may work, provided that it is physically based: resolve the Eotvos number Eo = Bo processes occurring along different directions, find their inter- 2 Euler number* Eu = P/V relation, and assess their relevance in a given situation. In case Fourier number* Fo = L2/T (heat) of convection, the buoyant rise is vertical, and the heat and Fourier number* Fo = L2/DT (mass) momentum diffusion is horizontal, say, to the first approxi- Froude number* Fr = V2/gL Galileo number Ga = gL3/2 (=Re2/Fr) mation. It is unlikely, that the same choice of scales applies Grashof number Gr = ˛gL3/2 (heat) equally well to convection in infinite horizontal layer and in Grashof number Gr = ˇCgL3/2 (mass) thin vertical slots. Knudsen number Kn = Lmolec/Ldomain Some choices of scales are even more ambiguous. By select- Laplace number La = P/(/L)(=Eu We) ing the scales, we can select the view of the world. For Lewis number Le = /D (=Sc/Pr) instance, imagine a two-phase flow in a long thin electrolysing Ljascenko number Ly =(/)(V3/g)(=Re3/Ar) microchannel of size A × B × C = 100 ␮m × 10 mm × 1m. In Mach number Mc = V/Vsound Marangoni number* Ma = /V these days of the scale-down boom, such an equipment is Morton number Mo = g4 /23 (=We3/Fr Re4) not unusual. At the wall, there are electrodes and bubbles

Nusselt number* Nu = kheat/(/L) are produced by electrolysis. The spectrum of bubbles sizes is Peclet number* Pe = LV/ (heat) quite broad, from few microns, as they are formed, to few cen- Peclet number* Pe = LV/D (mass) timetres, as they coalesce. The point is to choose the proper Prandtl number Pr = Pe /Re = / heat length scale and to define Re for making correlation formulas Rayleigh number Ra = ˛gL3/ (heat) (=Gr Pr) Rayleigh number Ra = ˇCgL3/D (mass) (=Gr Sc) designed for the operational quantities (e.g. liquid flow, bub-  3 Rayleigh number Ra = g eL /mixDhydro (dispersion) ble size and concentration, pressure drop, wall shear, etc.). The Reynolds number* Re = LV/ = LV/ choice L = A means the side view at the channel. We see the Richardson number Ri =(/)(gL/V2) (=(/)/Fr) flow between two virtually infinite parallel horizontal planes, Rossby number Ro = V/˝L 100 ␮m apart. This situation is known as the Poiseuille flow. Schmidt number Sc = /D (=Pe /Re) mass The choice L = B means the top view at a segment of the chan- Sherwood number* Sh = kmass/(L/D) nel. We see the flow between two finite and closely spaced Stanton number Sn =(kheat/V)(/) (heat) (=Nu/Peheat) walls, 1 cm × 1 m in size. This situation is known as the flow Stanton number Sn = kmass/V (mass) (=Sh/Pemass)

Stokes number St = pLV/ through the Helle–Shaw cell. The choice L = C means the global Strouhal number* Sr = L/TV view at the channel. We see the flow between two finite par- Sebestovˇ a´ number Seˇ =1/Mc allel plates, which are narrow and 1-m long. This situation 1/2 Thiele number Th = L(kreac/D) corresponds to the development of boundary layers in a rect- Weber number* We =()LV2/ angular channel. In these cases, different flow profiles along Basic numbers are marked by an asterisk (*) () means or . different directions are relevant. What length scale should Note large diversity both in names and notation of dimensionless then be chosen for the flow correlations? One way around numbers in literature. Those used here are by no means the best or this severe anisometry (1:100:10,000) seems to be the hydraulic obligatory. radius 2AB/(A + B)=99␮m, which, however, leads to nowhere. Taking this figure actually means selecting the picture of a flow through a 99 ␮m dia capillary, which is far from being that this mistake occurs in a great number of correlation for- related to any possible view at our flow situation. So far, the mulas published over more than 30 years, some of them even L for a single-phase flow has only been considered. The case became the classics. These correlations are used for design- with the bubbles is left as a homework for students; enough to ing factories and plants, and they work well. Imagine how they tease them is to ask for introducing the correct Re for a bubble would work, if the correlations would be correct. column. Some choices of scales are ambiguous. We suffer from the In correlations, the composed numbers are often used, presence of more that one candidate for the length scale. For obtained by combining several simple numbers with clear instance, consider the Rayleigh number, Ra = ˛gL3/ ∼ L3, physical meaning. The product, however, can have no clear for a natural convection in a horizontal fluid layer heated from meaning. What can safely be combined? In (5.2.9), we combine below. In case of infinite layer of height H, the obvious choice two numbers, Fr and Re, generated by the same govern- is L = H. In case of a layer confined also by two lateral walls ing equation (5.2.1), to compose Ga. It is acceptable, when separated by distance A, both H and A may be chosen. Thus, they share the same scales. If not, then Ga would, for there are several possibilities, L3 → H3, H2A, HA2, A3. Which instance, be Ga = Re2/Fr = {(V2/L)/(V/L2)}2/{(V2/L)/(g)}, one is correct? Finally, when the layer is inside a finite con- where we correctly discriminate between the inertial () tainer A × B × H, the combinations grow. We can resort to the and viscous () scales for length and speed. Even with the 856 chemical engineering research and design 8 6 (2008) 835–868

Fig. 4 – Flow-chart of scaling governing equations (momentum, heat, mass) and boundary conditions. uniscale (L = L = L), we encounter the problem of DN alge- it, gV =(gL)·(V/L), we can recognized the gravitational and bra. Ga actually is {(inertia)/(viscosity)}2/{(inertia)/(gravity)} = viscous scaling for P, see (5.2.7). (inertia)·(gravity)/(viscosity)2, which is in a variance with the There are combinations of numbers originating from usual interpretation (gravity)/(viscosity), where the powers are different governing equations, which can express their cou- ignored. Can we ignore them? This dubious algebra is also pling (Pr = Pe/Re; Ra = Gr Pr), or only comparison of analogous encountered when we try to eliminate (speed from Fr using material properties (Pr = Peheat/Re = /; Sc = Pemass/Re = /D; Re) or separate (speed and size in sedimentation using Ly) Le = Sc/Pr = /D). some inconvenient quantities. In this spirit, we can create There are combinations of numbers coming from equa- any thinkable combination of DN, since their dimension equals tions and boundary conditions, which sometimes, fortunately, unity [1]. For instance, the combination Eu2 Re Fr would usually belong together (Bo = We/Fr; La = Eu We; Mo = We3/Fr Re4, Stan- · be interpreted as (pressure)/(viscosity) (gravity). Upon insert- ton (Margoulis) number Sn = Nu/Peheat or Sh/Pemass). The ing the physical meanings, it becomes: (p/i) · (p/i) · (i/v) · (i/g) = Morton number is often used in bubbly research. A closer look (p/v) · (p/g) = p2/vg, in brief notation (i – inertia, v – viscos- reveals that the physical meaning is: Mo = (i/c)3/(i/g) · (i/v)4 = ity, g – gravity, p – pressure). With the viscous (p ∼ v) and (gv4)/(c3i2), in brief notation (c – capillarity), which becomes gravitational (p ∼ g) scaling for pressure, we have: (vg/vg) = (gv)/(ci), at the usual ignoring of powers. In literature, Mo is DNU. This means that the physical meaning of the combina- interpreted as capillarity/buoyancy, viscosity/capillarity, etc. tion equals ‘unity’, the dimensionless number unity (DNU), Which one is correct? How then to interpret the Tadaki num- since the physical meanings of the individual numbers ‘can- ber, Td = Re Mo0.23? cel’. These manipulations are formally correct, but where is There can be combinations of numbers belonging to dif- the physics? Actually, the combination Re Fr Eu2 is equal to ferent physical contexts. Why not to eliminate the speed from P2/gV, where the group (gV) is not very transparent, far Fr by the Mach number? Ga = Mc2/Fr thus obtained is formally from the expected (viscosity)·(gravity). Only after unravelling correct, but hardly applies to a free rise of a bubble in a col- chemical engineering research and design 8 6 (2008) 835–868 857

umn. There are no strict rules what can be combined and what explicit length scales is treated, e.g. in Bluman and Cole (1974), cannot, and common physical sense is highly welcome when and, in brief, this topic is covered in many texts on partial making these combinations. differential equations. The uneasy concept of the interme- In correlations, some quantities or effects are often diate asymptotics, which transcends the traditional DA, is neglected, as being ‘small’. It is tempting to say that inertia strongly presented in the book of Barenblatt (1996), which dominates when Re > 1. It is however not completely right. is based on the original Russian edition from the seventies. First, this number is only a scale estimate of two forces, the Its simplified version (Barenblatt, 2003) is especially suitable roughest assessment we can have. There also are possible for students. Besides the books, there is a selection of arti- numerical factors missing in this formula, reflecting the tun- cles on DA and its applications (1.2. Articles on DA): Astarita, ing of Re for a particular flow situation. Its value also depends 1997, Boucher and Alves, 1959, 1963, Buckingham, 1914, 1915, on the choice of scales. Second, this number stands in the Cheng and Cheng, 2004, Dodds and Rothman, 2000, Gunther, flow equation at a dimensionless term that is ∼O(1) but not 1975, Gunther and Morgado, 2003, Klinkenberg, 1955, Lykoudis, exactly = 1. Accordingly, it is better to say ‘small’ and ‘large’ Re, 1990, Macagno, 1971, Prothero, 2002, Rozen and Kostanyan, instead of Re < 1 and Re > 1. Support for this suggestion comes 2002, Sandler, 1970, Sjoberg, 1987, Stephens and Dunbar, 1993, from the following. The laminar–turbulent (viscous–inertial) Vogel, 1998, Wesson, 1980, West, 1984. The references relate transition occurs not exactly at Re = 1, but at about Re ∼ 10n, to both the area of the chemical engineering and also other where n ≈ 3–4 for pipes, 5–6 for flat plate, and varies in a wide research areas, to broaden the horizon, and for inspiration range for flow past different bodies. Likewise, the onset of ther- (biology, cosmology, ecology, economy, geology, medicine, psy- mal convection does not occur exactly at Ra = 1, but at Ra ∼ 102 chology). The topic of DA and scaling is the firm ground of the to 103 (depending on BC, etc.). The flow in microchannels is chemical engineering literacy. Scale-up and design of tech- not completely free molecular at Kn > 1 and no-slip continuous nologies is based on scaling consideration and active use of − at Kn < 1, but at about Kn >101 and Kn <10 2 say. Compress- DA in a variety of particular situations. Therefore, several ref- ibility effects are encountered not exactly at Mc = 1, but the erences on scale-up are also presented (1.3. Books on scale-up): recommended figure is about 0.3. Euzen et al., 1993, Grassmann, 1971, Johnstone and Thring, Conclude that building sensible and reliable correlations is 1957, Stichlmair, 2002, Zlokarnik, 2006. The scale-up is in far from being trivial. Besides choosing the proper dimension- certain respect easier than the scale-down. Considering the less numbers, one must also choose the proper characteristic lab scale ∼1 m, the big equipments are typically ∼101 min scales. Choosing these scales is problematic due to our (either size, i.e. variation within one order only. The microtechnol- intentional or not) ignorance of the underlying physics, pres- ogy goes down to sub-micron ranges, i.e. more than 6 orders ence of processes operating in different spatial directions, and beyond our everyday experience. The chance that new phe- multiscale nature of the system. Care should be taken when nomena will be encountered along this way is very high. combining different numbers, and their origin and physical The importance of surface phenomena, hence surface sci- compatibility should be kept in mind. 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 9. Remark on literature (1981). In the second part of the Reference section, the balance In sake of the text cohesion, there are almost no references equations for the transport phenomena are presented (2. to the literature through the previous chapters, and the com- Transport phenomena). They form the very core of the chemical ments on the main information sources are lumped into this engineering and are in all texts of this sector, so it is needles to part. list them all. There is a selection of sources that relates to this In the first part of the Reference section (1. Dimen- paper. The local standard textbook by Mika (1981) was used sional analysis), there are books fully devoted to DA and the for the equations and some basic scaling, and several other related problems like similarity and modelling (1.1. Books on books were also consulted (e.g. Aris, 1989; Bird et al., 1965; DA): Baker et al., 1973, Barenblatt, 1987, 1996, 2003, Becker, Carslaw and Jaeger, 1947; Crank, 1956; Cussler, 1997; Deen, 1976, Birkhoff, 1960, Bluman and Cole, 1974, Bridgman, 1922, 1998; Rohsenov and Choi, 1961; Slattery, 1972; Slavicek, 1969; Clement-O’Brien and Lawler, 1998, Craig, 2003, Curren, 2005, Thomson, 2000; Welty et al., 1969). De Jong, 1967, Dolezalik, 1959, Duncan, 1953, Focken, 1953, Most space in this paper is devoted to the momentum Gukhman, 1965, Hornung, 2006, Huntley, 1952, Ipsen, 1960, transport, i.e. the fluid dynamics, both single-phase and multi- Isaacson and Isaacson, 1975, Jerrard and McNeill, 1992, Kline, phase (Section 5). This field is close to the author, and also 1965, Kozesnik, 1983, Kurth, 1972, Land, 1972, Langhaar, underlines the remaining two transport processes of heat 1951, Loebel, 1986, Massey, 1971, Murphy, 1950, Palacios, and mass. There are several groups of authors writing about 1964, Pankhurst, 1964, Perry and Chilton, 1973, Porter, 1946, fluid mechanics, according to their background: pure and Schuring, 1977, Sedov, 1959, Skoglund, 1967, Stubbings, 1948, applied mathematicians, physicists and engineers (chemical, Szirtes, 1998, Taylor, 1974, Weast and Astle, 1981, Zierep, 1971, civil, environmental, mechanical, metallurgy, mining, nuclear, Zlokarnik, 1991. These are useful for familiarizing with the urban, etc.). The first choose simple problems and solve them many aspects of this powerful method, as well as with its his- completely on the fundamental level. For the last, complex torical development and many practical examples of its use. problems are chosen, which can be solved only approximately. Two book chapters are also included. One from the Perry’s The engineering books are well-known to our community canonical handbook, and the other from the Birkhoff (1960) (e.g. Cengel and Cimbala, 2006; Massey, 1998; Munson et classical account on fluid mechanics. They represent two lim- al., 1990; White, 1974; Wilkes, 1999), so only the others are its of the spectrum, with the practicality on one side, and the mentioned, when used. A more theoretically minded reader deep theoretical footing on the other. The important concept may wish to consult, e.g. Doering and Gibbon (1995), Sohr of ‘similarity solution’ to spatio-temporal problems lacking (2001), and Temam (2001). The common RTD treatment is pre- 858 chemical engineering research and design 8 6 (2008) 835–868

sented in many standard books (e.g. Levenspiel, 1962). The perform in the dilute viscous limit. The problems come with smart variant of RTD is presented, e.g. by Ghirelli and Leckner strong multiple interactions in dense suspensions (e.g. Brady (2004). and Bossis, 1988; Stickel and Powel, 2005). The exciting area Rarely is found a title with proper and transparent of granular flow and powder technology is a traditional part of treatment of the interface boundary conditions within the engineering (e.g. Brown and Richards, 1970; Nedderman, 1992) engineering literature. The corresponding very important with deep roots in soil mechanics (e.g. Taylor, 1948; Terzaghi, numbers (Bo, Ca, Eo, Ma, Mo, We) are usually presented either 1943). Presently, it witnesses a great boom after being ‘dis- without any comment on their origin (hence physical mean- covered’ by physicists, as a unique and highly specific state ing), or as a result of DA (again lacking the physical meaning). of matter (Duran, 2000; Hinrichsen and Wolf, 2004; Forterre The book by Sadhal et al. (1997) was used, together with that and Pouliquen, 2008). The concept of Ba number is useful in by Edwards et al. (1991). Instructive is the paper by Cuenot et granular flows. al. (1997), which deals with a single bubble rise in a contam- In the third part of the Reference section, several important inated liquid, where Ma and other DN play important roles. particular areas are covered in very brief (3. Other topics). The Macroscopic effects of a surfactant on real bubbly mixtures phenomenon of ‘interface’ becomes increasingly important are demonstrated too (e.g. Ruzicka et al., 2008). nowadays, owing to the progressive scale-down, where the In engineering, the hydrodynamic stability is usually (volume/area) seems to be a more relevant parameter than the neither taught in courses, nor included in the books on hydro- usual (area/perimeter), known as hydraulic diameter. There- dynamics. We tend to assume that things are mean, steady fore, few references to the surface aspects are mentioned and stable, since we want have them like this in appli- (3.1. Surfaces and surfactants): Adamson, 1960, Adamczyk, 2006, cations. However, this is not always the case. Few books Davies and Rideal, 1963, Fawcett, 2004, de Gennes et al., 2004, relate to the stability issue, which is difficult but impor- Israelachvili, 1992. These are useful for multi-phase flows as tant (e.g. Chandrasekhar, 1961; Drazin and Reid, 1981; Drazin, well as for microflows, and were employed at writing about 2002—suitable for students). Equations of vorticity and enstro- these topics. Microtechnology surely deserves a little section phy are often used in turbulence (Davidson, 2004; Frisch, 1995; of references (3.2. Microflows and microsystems), which were Pope, 2000), but not only there, since the vorticity is a subject used for preparing Appendix C: Berthier and Silberzan, 2005, of its own importance (Saffman, 1992). Bruus, 2008, Ehrfeld et al., 2000, Gad-el-Hak, 1999, Hessel et The rotational effects are presented by geophysically al., 2004, van Kampen, 1992, Karniadakis and Beskok, 2002, minded authors (Kundu, 1990; Tritton, 1988; see also Karniadakis et al., 2005, Kockmann, 2006, Li, 2004, Madou, Pedlosky, 1982). Magnetohydrodynamics is not a usual part 2000, Maynard, 2008, Neto et al., 2005, Nguyen and Wereley, of our curriculum, and is mentioned only informatively 2002, Slattery et al., 2004, Stone et al., 2004, Tabeling, 2005, (Chandrasekhar, 1961; Moffat, 2000). On the other hand, com- Thompson and Troian, 1997; see also the periodical “Microfluid pressible and rarefied flows are commonly encountered, and Nanofluid” and others. They are valuable sources of informa- the idea behind Mc and Kn is familiar to engineers (many tion, reflecting the state-of-the-art in this inflating area. The books exist on gas dynamics). Buoyancy effects in fluids are effect of ‘rarefication’ of gases and ‘granulation’ of liquids omnipresent and are covered by several texts, namely the ther- needs the adequate description, where gas/molecular dynam- mal convection (Turner, 1979; Koschmieder, 1993). Analogy ics and stochastic processes are the vital ingredients. DA and between buoyancy effects in single-phase and multi-phase SE on microscale are considered, e.g. in Kockmann (2006, systems has also been pointed out, and multi-phase Ra dis- chapter 2). Several links to biological systems are also offered cussed (Ruzicka and Thomas, 2003). A suitable ‘effective’ (3.3. Biology and biosystems), related to the material presented density should be used to evaluate correctly the Archimedes in Appendix C: McMahon, 1973, Pilbeam and Gould, 1974, force acting on a body immersed in dispersion, depending McMahon and Bonner, 1983, Hjortso, 2005, Nopens and Biggs, on a dimensionless parameter (Ruzicka, 2006). Rheology is a 2006, Thompson, 1943, Perthame, 2006, Ramkrishna, 2000. The vast area, with its own bibliography. The book by Barnes et scaling concept is well known for biologists, which may be al. (1989) is a suitable introduction into the field (De num- surprising for engineers. The population balance modelling is ber included). There are profound texts where dimensional also included into this short section, since the term ‘popula- aspect are treated on the fundamental level (Astarita and tion’ has strong biological connotation. Self-similar aspects Marrucci, 1974) as well as recent textbooks (Macosko, 1994; of these models are treated, e.g. by Ramkrishna (2000).In Morrison, 2001). A set of reviews on the present state of sev- biology, thoughts of hierarchy of scales, structures, and func- eral important branches of fluid mechanics is also included tions are most appealing. Many penetrating perceptions of the (Batchelor et al., 2000). There is a wealth of literature on motion early times were later physically based and formalized. The of animals in fluids (e.g. Childress, 1981; Pedley, 1977; Vogel, pressure from the micro-needs pushes engineers into these 1998, 1994). With curiosities, the water-walking miracle of the dangerous waters, where long-term everyday experience and Basiliscus lizards is also presented (Glasheen and McMahon, intuition, which are our traditional arms, may be near-useless. 1996a,b). Few items were therefore selected, for the engineer to gain the It is not easy to find a general and accessible and phys- inspiration from the philosophers, and to combine it with the ically sound book on multi-phase flow (if any). There are rigour of the physicists (3.4. Multiscale science and hierarchy): only several titles currently available in this growing area Bergmann, 1944, Furusawa and Kaneko, 1998, Garnett, 1942, (e.g. Brennen, 2005; Crowe et al., 1998; Ishii and Hibiki, 2006; Glimm and Sharp, 1997, Henle, 1942, Hoover and Hoover, 2003, Kleinstreurer, 2003; Kolev, 2002; Soo, 1990), where the assis- Li and Ge, 2007, Li and Kwauk, 2004, Koplik and Banavar, 1995, tance was obtained from. Other sources were also consulted Lowry, 1974, Marin, 2005, Pattee, 1973, Sewell, 2002, Simon, for specific areas (Friedlander, 2000; Hinds, 1999; Nguyen and 1965. The last subsection (3.5. Education) is devoted to the Schulze, 2004). The important issue is to formulate the stress pedagogical aspects of DA and relates directly to Appendix tensor and the interphase interaction force. This leads to B: Andrews, 1984, Bloom, 1956, Cadogan, 1985, Calder, 1984, investigation of the microstructure of dispersions, easier to Churchill, 1997, Comenius, 1657, Imrie, 1968, Krantz, 2000, chemical engineering research and design 8 6 (2008) 835–868 859

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

10. Conclusions

A brief guided tour through the field of scaling and dimen- sionless numbers is presented, with respect to their use in chemical engineering. The numbers are listed in Table 2.Two sources of these numbers are considered, the dimensional analysis and the scaling of governing equations with their boundary conditions. The apparent advantage of the equation scaling is twofold: (i) we know the relevant physical quanti- ties and (ii) the numbers thus obtained have clear physical meaning. This meaning must be kept in mind when using the numbers and the scales they involve for making empirical correlations. The mutual relations between different numbers are highlighted. A flow-chart of numbers closely related to the basic equations for momentum, heat, and mass transport is shown in Fig. 4.

Acknowledgements

The author will highly acknowledge the reader’s forbearance Fig. A1 – Demonstration of IA. Surface wave generated by in case of possible mistakes, errors and shortcomings that vibration of triangular element on surface of originally still may occur in this text (like in any other one), and will sin- liquid confined in rectangular container (top view). Wave cerely appreciate comments, suggestions, and constructive moves from generator to boundary. Shape of wave (dotted criticism that could help him to produce something much bet- line): triangular (phase G) → circular (phase I) → rectangular ter in the future. The financial support by GACR (grant nos. (phase B). Circular wave is intermediate asymptotic (IA). 104/06/1418, 104/07/1110), by GAAV (grant nos. IAAX00130702, IAA200720801), by MSMT CR (grant no. KONTAKT ME 952) and by AVCR-CNRS (grant no. 11-20213) is gratefully acknowledged. turbed by either boundary. This happens at the intermediate distance between the container centre and container wall Appendix A. Concept of intermediate (phase I). As the waves begin to feel the container walls, the asymptotics circles turn into rectangles, to accommodate to the shape of the boundary (phase B). The solution describing the ‘happy circles’ is the intermediate asymptotic (IA) of the system: This appendix is to offer the reader a brief exposure of the con- demonstration of pure physics, unaffected by geometrical cept of the intermediate asymptotics (IA). The presentation is constraints, by the past and future. intermediate-precise, in the sense that it is more precise than The three stages (G, I, B) correspond to three different a popular text aimed at the common public, and, at the same regimes of the system behaviour, where different effects, time, less precise than it should be for its physical correctness hence variables, come to play. DA is able to cope with a sin- and mathematical rigour. Only few, hopefully typical, aspects gle regime only. The easiest is the intermediate regime (IA) of IA are mentioned, without claim of generality and exhaus- where besides the physics, which is in all of them, no addi- tiveness. The text is based on the two books by Barenblatt tional variables are present to account for the geometry of (1996, 2003), with some demonstrative examples by the author wave generator and the container boundary. DA may be able (see figures). to describe certain rough aspects of the intermediate stage, by providing scale estimates of some main features of the cir- A.1. Motivation example cular waves. Whether DA can really do so, depends on the physical nature of the problem and on the type of the physi- Vaguely speaking, the intermediate asymptotic is a cal quantities involved. There can be a solution in form of IA, time–space dependent solution of an evolution equation which cannot be obtained by DA. In this sense, the concept of that already forgot its initial conditions, but still does not IA transcends the concept of DA, as pointed out in Section 3.2 feel the limitations imposed by the system boundary or by of the main text. DA applies to problems having the complete extinguishing its internal dynamics. Consider a body of a similarity. still water in a rectangular tank. Let the surface waves be generated at the centre, by an oscillating triangular element, A.2. Two kinds of similarity see Fig. A1. Near the centre, the waves are triangular since they bear the fingerprint of the initial condition, the shape A.2.1. Complete similarity (CS) of the generator (phase G). As they propagate outwards, they The relation between the master quantity and the variables, gradually obtain the natural circular shape,4 being undis- u = f(a1,...,k, b1,...,m) can be converted using DA into the dimen-

4 Why the ‘circular shape’ is natural? A mathematician would say because of the symmetry reason; a physicist would say that a stable configuration; a chemical engineer would argue that one there is no force to make it non-circular and, even if, the circle is has seen only circles, so far. They are all basically right. 860 chemical engineering research and design 8 6 (2008) 835–868

sionless form: about the distribution of special properties within the natu- ral phenomena we observe, and in models we use to describe

˘ = ˚(˘1,˘2,...,˘m), (A.1) them. A rule of thumb says: The ‘better’ the property, the rarer it occurs. Consequently, we may expect that the typical where we want to reduce the number of variables by neglect- case will be the lack of similarity where no further simplifi- ing some Pi-terms. Usually, too small and too large terms cation of (A.1) is possible. It should be attacked directly, via are omitted, without further justifications. Correctly, the experiments, numerical calculations, or approximate analyt- → ∞ behaviour of ˚ in the limit of ˘i 0or must be investigated. ical techniques. The neglection is permitted only when ˚ tends fast enough a finite non-zero limit. Then it is possible to replace ˚ with A.3. Two kinds of self-similarity another function of less variables representing the limit of ˚: In many cases, (A.2) represents the final result produced by = ˘ ˚1(˘1,˘2,...˘h

Since ˚ is usually not known beforehand, at least an a pos- u = U · ˚(˘1,˘2,...˘m) (output of DA). (A.6) 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 similarity In most of our applications, DA finishes here and delivers of the first kind of the phenomenon in the neglected variables the scaling law u ∼ U. The tuning function ˚ is then obtained (h +1,h +2,..., m). In reality, this is a rare situation and a worse by measurements. These applications typically describe case is encountered. steady states of complex processes, in complex geometries, time–space averaged problems with lumped parameters. Less A.2.2. Incomplete similarity (IS) often are studied evolution problems in space and time, prob- If ˚ behaves badly, the condition for CS is not satisfied and lems with distributed parameters. Certain class of problems, the Pi-terms cannot be neglected. They influence the problem in a certain intermediate range, admit self-similar (similarity) regardless how small or big they are. But even here a sim- solutions, in terms of self-similar variables. These variables plification can exist. If our problem possesses certain special for problems with complete similarity can be found by DA. features, it may be possible to take some Pi-terms out of ˚,as Consider a problem described by a complicated governing the power-law factors. In this case, (A.2) becomes equation for function u = u(x, t). The scaling law (A.2), (A.6) obtained by DA can be written as

+ ˘ ˘ = ˛,h 1··· ˛,m 1 h u x ˘ (˘ + ˘m )˚2 ,..., , (A.3) = h 1 ˘ˇ,1 ···˘ı,1 ˇ,h ··· ı,h F ,˘i (self-similar ansatz for CS), (A.7) h+1 m ˘h+1 ˘m U X which can briefly be written as where X(t) and U(t) are the self-similar variables. They are the scale estimates for the spatial coordinate x and the mas- ∗ = ∗ ∗ ∗ ter quantity u(x, t) that ensure the geometrical similarity of ˘ ˚2(˘1 ,˘2 ,...,˘h

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 whole range and exact; for real problem it is approximation in intermediate range.

A.5. Beyond IA?

There may exist even a weaker type of similarity than IS, in the solutions to the general evolution problems on interme- diate ranges of time and space (or even more sophisticated variables) that will be sufficient to earn the status of IA, at least in the intuitive sense, see the grey zone in Fig. A4.Ifthe opposite is true, the grey zone shrinks to zero. Paradoxically, the opposite may be true owing to the terminology reasons. If the notion of IA is anchored in the fact that the weakest simi- Fig. A2 – Guidance for application of DA and similarity larity that IA can bear is IS, there is no room for anything else. analysis. Right branch deals with ‘idealized’ problem, where Such a definition finds its support in the fact that the concept problematic variables are neglected. Dimensionless of similarity is equivalent to the invariance of the governing formulation follows from DA. By complete similarity (CS), equations with respect to certain groups of transformations variables are fewer and similarity law is obtained. With (symmetry). Since the nondimensionalization and scaling governing equations (GE), self-similar solution (first kind) means changing the norm of the measuring units (etalons), can be found. Left branch deals with ‘realistic’ problem. the important role of the renormalization group is a little sur- Incomplete similarity (IS) is encountered and self-similar prise. With such a definition, IA and the group that reflects solution (second kind) is obtained by nonlinear eigenvalue the actual degree of symmetry in the problem are equivalent. problem (NEP) (based on Barenblatt, 1987, 1996, 2003). Consequently, the insolubility of the IS problem in terms of the self-similar solution of the second kind is equivalent to the indeterminacy of the group, and can be interpreted as the tial equations must be solved, with more boundary conditions than is due (overdetermination). Such a special value of is seeked, for which the solution does exist. The qualitative methods for nonlinear systems can be used, and the phase portrait of the problem can be analysed. Due to the overdeter- mination, the solution can correspond a singular objects (e.g. separatrix line). The flow-chart summary is given in Fig. A2.

A.4. Relation between DA and IA

DA is a simple effective method for finding the rough scale estimate of a master quantity, u ∼ U, in problems having the complete similarity. This estimate can be used for construct- ing the self-similar variables. IA is a general concept, a kind of universal behaviour that many systems of different origin can produce. It is a spatio-temporal phenomenon, existing within a certain intermediate range of the independent variables. It is an approximate solution to a complex problem, valid in a Fig. A4 – Nesting of similarity concepts. DA: dimensional certain range. It can be represented by the self-similar solu- analysis; CS: complete similarity; IS: incomplete similarity; tion, which is the exact solution to a simplified problem, valid IA: intermediate asymptotic; AI: absence of in the whole range, see Fig. A3. The action radius of the above- intermediateness. Phenomena fall into two classes: AI and discussed concepts is shown in the diagram in Fig. A4. IA. Within IA, further division is possible. 862 chemical engineering research and design 8 6 (2008) 835–868

Fig. A5 – Life cycle of population. Genesis at early times (G), internal dynamics in intermediate range (I), decay due to bounds set on its existence (B). In middle range, IA behaviour is expected. Vertical axis: characteristic parameter describing state of system (large ensemble of interacting units; evolution of biological species, population, culture, civilization, empire). absence of IA, in the given problem. The insolubility can be caused by unavailability of certain integral quantities, either explicit 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 whether they do exist. In mechanics, which is an axiomatic discipline,5 we can prove or disprove their existence, which leads to the consistent statement about the existence of IA. In other sci- ences, where we lack this strong axiomatic footing, we face the undecidability. This links back to the need of an operational definition of IA, suitable also for other and less formalized research areas, where the ‘governing equations’ are either Fig. A6 – Life cycle of individual. IA—unaffected by fully absent, or reflect only a small fragment of truth. Exer- beginning and end (the picture is reproduced with kind cising them would be tempting for applied mathematicians, permission by ‘The Bhaktivedanta Book Trust’). but the results obtained would likely be largely misleading. tain degree of independence and autonomy, while it is still far A.6. Broader horizons from its inevitable end (e.g. internal decay; the Roman Empire in the past, the West in the near future). In his books, Barenblatt mentions that although the concept of What happens with big populations (phylogeny) can repeat IA is a part of the mathematical physics, it has important sig- on the level of a one single individual (ontogeny). This is nificance for variety of general situations that are multiscale demonstrated by the second example. In Fig. A6, we see the in nature. As an example, he presents few situations from our youth, maturity and age. The child bears the fingerprint of its everyday life, to demonstrate this conceptual generality (e.g. birth and the care delivered by the surrounding, to facilitate perception of visual art, visual perception generally, analogy its early existence (phase G). The adult is believed to be able in poetry, intermediate description of historical events, etc.). to behave at least to a certain degree independently of the To contribute to these efforts, the following three exam- experience from the early period, according to one’s free will, ples are introduced, that hopefully comply with the concept if any (phase I). Getting older, the feeling of the presence of of IA, at least on the intuitive level. The first shows the nat- the severe upper bound on the length of the current life cycle ural cycle of a culture or civilization, see Fig. A5. It takes off comes to play, and affects our behaviour strongly (phase B). from zero, builds up, reaches a status quo (‘sustainability’), and To top this appendix on the philosophical level, let us follow eventually degenerates. The intermediate stage can be consid- the third example. By convention, the common people distin- ered as IA, because here the mechanisms driving its internal guish three parts of the time axis: past, present, and future, dynamics are revealed. The culture has already reached cer- see Fig. A7. The present is represented by one singular point only, separating the vast past from the equally vast future. Note that both the past and future do not exist: ‘past’ already 5 From the fundamental point of view, the mechanics is science was, ‘future’ will only be. The practical relevance of this pic- about something that does not exist. There are no elementary ture is that we keep living in a virtual world, generated of particles (‘mechanons’) that would mediate the mechanical our memories, and bounded by our future expectations and interactions. Mechanics merely is a demonstration of the plans. The mental training suggested by many a philosophi- electromagnetic and gravitational forces on the macroscale. On the same argument, the physicists would state that Romeo and cal schools is aimed at expanding the present, since it is the Juliet were not in love: there are no elementary particles only reality we can actually perceive, and the way how to lib- (‘loveons’) mediating this type of interaction. erate ourselves from the diktat of the past and the future. The chemical engineering research and design 8 6 (2008) 835–868 863

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

1. Specify relevant variables (using model equations; choose yourself). 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. Fig. A7 – Mental training: development of extended Check result versus data. Problems? perception of present, in form of IA. Singularity of present 6. Assume incomplete similarity and cancel small/large is unfolded, by suppressing the virtual past and future. terms. Check result versus data. 7. Formulate similarity and scaling laws with fewest vari- extended present is thus our IA, which, in its ultimate form, ables [final output I]. can be experienced as the timelessness. 8. Formulate similarity variables, use model equations, find Appendix B. Suggestions for using and similarity solution [final output II]. teaching DA 9. Relate similarity solution to IA behaviour of your problem, delimitate intermediate region. 10. Does your result comply with data? Can it be generalized? This appendix concerns the problem how to use DA and sim- 11. Improve this recipe, based on your own experience. ilarity analysis, which may also be useful for educational purposes. The basic facts about usage of DA are widely avail- DA—Glossary able in the open literature (see the Reference section). The For the reader’s convenience, a brief glossary of frequent practical recipe below for the application of the similarity anal- terms is also included. The expressions are listed, as they ysis is taken from Barenblatt (1987, 1996, 2003). A selection of appear in the text (place of definition): the educational literature is in References section 3.5. DA—What it is, how it works? • similarity parameter (3.1.5) (also: Pi-term, ˘-term, dimen- It is a very standard issue, with enormous coverage by sionless number); books, handbooks, monographs, etc. • similarity law for ˘ (3.1.5); See Sections 2 and 3 of this paper. • similarity law for a (3.1.6); DA—For which purpose? • scale estimate for a (3.1.7) (also: basic scaling for a, scaling We can use DA for several reasons, differing in the expected law for a); outputs: • similarity theory (Section 3.2) (also: similitude theory, mod- elling, scale-up/down); • scaling law for a master quantity; • similar systems (Section 3.2); • similarity law for modelling and scale up/down; • similarity law for modelling (3.2.3); • similarity variables needed for seeking similarity solution • similarity criterion for modelling (3.2.3); of governing equations; • scaling law: power-law dependence, y = axb; • grouping parameters into DN to reduce experimental data • scaling rule for modelling (3.2.3); and to make correlations. • scale coefficient for modelling (3.2.3); • scale (4.3) (also: characteristic/typical/representative quan- Note: Obtaining the scaling law for a quantity may, but also tity, parameter, value); may not be our ultimate goal. With these scale estimates we • scale equation (4.3); can build valuable models. • complete similarity (A.2); DA versus scaling of equations (SE): • incomplete similarity (A.4); • self-similar: variable, coordinate, solution (A.7) and (A.8) • DA (also: similarity variable/coordinate/solution). ◦ does not give any variables (they must be chosen subjec- tively, relevance not assured); Teaching aspects in general are covered by wealth of ped- ◦ does give dimensionless numbers (all combinations, rel- agogical literature devoted to education at the tertiary level evance not assured); (university), spanning several centuries. Starting with the ◦ does give relation between the numbers (scale-estimate paradigmatic treatment by the father of modern education of master quantity). Comenius (1657), we can arrive at the numerous volumes of • SE the present literature. To name one, Bloom (1956; see also ◦ does give relevant variables (from equations, initial and Krathwohl et al., 1964, Marzano and Kendall, 2006) is espe- boundary conditions); cially useful at preparing the knowledge tests for students. ◦ does give some relevant numbers (maybe not all, equa- The taxonomy indicates how difficult our questions are, and tions are only a ‘model’ of reality); which students skills are required for generating the answers. ◦ does not give relation between the numbers. Students should not only be taught “Know how”, but also “Know why”. Recommendation. First obtain relevant numbers by SE, then Teaching aspects in particular are broadly covered by the apply DA to find similarity laws. quarterly published periodical “Chemical Engineering Educa- 864 chemical engineering research and design 8 6 (2008) 835–868

tion”. There are articles directly related to teaching DA. Written as localized, i.e. concentrated in a plane of zero thickness. We by professional teachers and active researches at the same have the usual ‘macroscopic’ description, with the common time, they are of great didactical value (see e.g. Andrews, 1984; surface effects. When the system size is comparable or smaller Churchill, 1997; Krantz, 2000; Sides, 2002; also Imrie, 1968). To than this figure (nanochannels), the molecular forces cannot attract the students attention to problems of scales, few pop- be considered as localized; they are distributed in space and ular books are available too (e.g. Cadogan, 1985; Calder, 1984; time. We have to abandon the usual ‘macroscopic’ description, Morrison and Morrison, 1994). and develop something smarter. An analogy emerges here. Postscript. Students are forced to produce a number of In macro-hydrodynamics the boundary effects can be local- papers to defend their PhD thesis (“Publish or perish”). To ized into a thin boundary layer negligible with respect to the reduce the increasing information noise in the peer-reviewed bulk volume, while in micro-hydrodynamics the whole bulk literature, remember also the complementary saying: “Better is the boundary layer. In micro-hydrodynamics the molecu- to perish than publish rubbish”. lar forces can be localized into a thin surface layer, while in nano-hydrodynamics, these forces penetrate the whole bulk. Appendix C. New areas in chemical Note that the common shear viscosity increases enormously engineering on nanoscale, when the sheared fluid layer is only several nm thick. The common surface tension decreases, when Far be it from the author to tend to have any kind of visionary the drop size shrinks to nanometric scales. Both and are ambitions. Instead, some comments are presented, related to macroscopic quantities that can loose their usual meaning the current topics suggested as being topical. when transferred from large local-equilibrium ensembles on to sparse families of particles, out of statistical balance. C.1. Microreactors and microfluidics It is useful to know how different forces and physical quan- tities scale with the length L (system size). Expectedly, the body C.1.1. Microsystems in chemical technology or volume forces fall quickly with decreasing L (gravity, iner- Flows and transport phenomena in microscopic and tia, centrifugal, etc.). In contrast, the molecular and surface nanoscale channels are under intense research. The typ- forces become important. Consequently, the force equilibria in ical features are: low Re regimes, strong surface effects, break microsystems are created by a balance of forces others than we down of continuum concept, multiscale nature, presence of are used to. In small systems, intense electrical fields can be many kinds of forces. There are some aspects already known, produces, and important electro-phenomena occur (electroos- that make the micro-world different from ours. Others are mosis, electrophoresis, streaming potential, sedimentation only awaiting their discovery. We still have the well-defined potential, dielectrophoresis). The Debye length becomes the physical quantities and the power of the conservation relevant length scale. The force balance may result from inter- equations, so that both DA and SE can be applied. play between the viscous, pressure, and electrostatic forces. In chemical technology, the microreactors enable better The electrostatic-Ra may then appear. New kinds of flow insta- control of system behaviour. The flow is usually laminar (low bility can also be produced. Re owing to small L and V), with all the advantages of the Stokes equation (linearity hence superposition, reversibility, C.1.3. Governing equations and boundary conditions strong theorems—minimum dissipation, reciprocity). Such The continuum approach is based on the notion of the ‘fluid flows may not be prone to hydrodynamic instabilities. On the particle’, a virtual mesoscale object having the proper number other hand, we lack the advantage of the effective large-scale of molecules, not too small, not too large. This alibistic defi- convective mixing (strong bulk turbulence), and the flow nition may well work for students, but largely fails when we basically is in the boundary layer regime. This increases the have to know how this ‘particle’ compares with the size of our resistance to the transport processes. They must be enhanced microchannel. Since this basic concept is presented in the first by clever design of the system geometry and flow. Thanks to lecture on hydrodynamics, all of us surely know how big it is. the fast heat transfer in systems with large (surface/volume) When it is smaller than the system, the continuum approach ∼ 2 3 ∼ ratios L /L 1/L, we can manage the exo/endo-thermic can be applied, and vice versa. Since gases are thinner than reactions. The mass diffusion can be complemented by liquids, their ‘particles’ must be larger. Consequently, gases micro-convection in several ways (micromixing). Controlling are more prone to discontinuous behaviour in microsystems. the transport and reaction, we can improve the selectivity Note that all fluid properties must be continuous.6 There may and increase conversion. This higher efficiency together be different length scales for kinematics (speed, acceleration), with safe smaller units close to user might compete with thermodynamics (pressure, density), transport (diffusion), etc. the huge volumes produced by present plants on one spot. Also, even when the fluid is continuous, the flow maybenot As ever, common sense should be used to prevent us from (shocks, extreme shearing, etc.). miniaturizing everything. There are some recommendations for microflows in terms of the Knudsen number. One effect is rarefication, which likely C.1.2. Prevailing forces occurs when Kn is larger than ∼10−3 to 10−2. This holds for There are different kinds of forces, with different ranges of gases, where the free path is a well-defined concept. In liq- action. The shortest are the interaction forces between two uids, it is not so, and the molecular interaction distance can small molecules in vacuum. The force range usually increases be taken instead of the free path. This distance could be taken with the molecule size, polarity, number of molecules, and is 10× smaller than the path, based on the density argument, ∼ 1/3 larger in material environment, where strong cumulative and (distance/path) (gas/liquid) . Another effect is the molecu- collective effects can play a role. Starting from few nanome- lar slips of fluid at rigid wall, where the no-slip BC condition tres (nm), these molecular forces in large ensembles can reach to ∼100 nm, say. When the system size is larger than this fig- ure (microchannels), the molecular forces can be considered 6 See e.g. Nguyen and Wereley (2002), Section 2.1.3. chemical engineering research and design 8 6 (2008) 835–868 865

loses validity. The continuum approach and no-slip BC can corresponding matrixes. The population balance models (used − apply for Kn <10 2, say. The continuum approach and free-slip not only for living units, but also for lifeless particles, like bub- − − BC is recommended for Kn ∼ 10 2 to 10 1. The Navier–Stokes bles, drops) are statistical tools, providing us with equations fails for Kn ∼ 100 and other models should be used (e.g. Bur- for probability distribution of certain property within the pop- nett and Woods equations). Statistical approach (Boltzmann ulation (age, size, weight, wealth, health, etc.). It may not be a equation) or molecular dynamics of free discrete particles priori clear what good comes from their possible scaling and should be employed for Kn >101. The Knudsen number can how to use the DN thus obtained. Similarly in other areas (ecol- serve as the expansion parameter in building approximate ogy, economy, medicine, pharmacology, psychology, sociology, models. Different effects can be important for gases (rarefica- etc.), the following points should be made clear. Do we have tion, compressibility, viscous heating, thermal creep, Knudsen well-defined physical quantities? Are they directly related via pump, diffuse/specular reflection at wall) and liquids (wet- certain physical processes? Or even better: Do we have gov- ting, adsorption, eletrokinetics, hydrophobicity). Other effects erning equations? If so, we can try our best at applying DA, can be common (entrance effects, fluid–structure interactions, and possibly also SE. relaxation times upon disturbing, surface roughness). Literature used and recommended: References section 3.3. The diversity and novelty of micro-phenomena need new forces to be included, modified equations and adequate C.3. Multiscale methodology boundary conditions. After scaling, they will produce new or different DN, than we would expect. For instance, the slip Multiscale approach is fashionable currently, but the basic boundary condition on rigid wall for compressible flows leads idea is very old. In your system, choose proper characteris- to Ec, Kn, Pr, Re (cf. with numbers in Sections 5.4.1 and 5.4.2). tic scales for relevant quantities (time, length, force, speed, Other numbers may be encountered too (e.q. Squeeze num- etc.), estimate the magnitude of individual terms in your equa- ber, Bearing number, etc.). Some DN important on macroscale tions, neglect some terms with respect to others, under given may loose their relevance. The common DA is expected to conditions. Change the conditions systematically, to select work also for microsystems, one difficulty being our much individual simple processes and study them thoroughly. This less experience with the world of small, where things often is what physicists have been doing for centuries. The time and go straight against our intuition. It is exactly the experience length scales (T and L) can be obtained by spectral analysis, of and intuition that facilitate the crucial step in DA, the choice either the physical signals (measured or computed by CFD) of relevant variables. or the governing equations (modal dynamics). The physical Literature used and recommended: References sections 3.1 quantity u(x, t) is a function, an element of an abstract func- and 3.2. tion space. It can be represented using the basis functions. One common basis are the harmonic functions (Fourier). Another C.2. Biosystems basis are wavelets (Haar), which basically are wave packets. The force scale relates to L and T, depending how short/long Biology witnesses long-term tradition in using various scal- range it is, and how fast it responds/decays. When the scales ing laws. Observations suggest that animals are small and are separated by a large gap, the corresponding processes are big, slow and fast, eating little and much, etc. Efforts were likely only weakly coupled, and can possibly be studied sepa- spent to relate these properties and to find some rules. rately. When the scales are not separated, the corresponding Resulting empirical correlations shows power-law depen- processes are likely strongly coupled and it may be difficult or dence between various quantities (body size, body weight, even impossible to decompose them, without corrupting the proportions, metabolic rate, heat production, characteristic model (scales in turbulence are typically strongly intertwined). biological times, etc.). Metabolic heat is produced in the Almost all things around us are multiscale systems, at least 3 2 body bulk ∼L and is released by body surface ∼L . The for the simple reason that they are made of atoms ∼10−9 m and (loss/production) ratio is 1/L, indicating that big animals may we are bodies ∼100 m living in space of ∼10xx m. From the hier- have problem with overheating, while small with under- archy of scales follows the hierarchy of functions, as known heating (beyond the limit of thermal regulation, they are from complex systems. Hierarchical systems display one spec- 3 cold-blooded). The weight goes like ∼L and the force gener- tacular feature: existence of ‘emergent properties’. The whole 2 ated by the stress in muscles ∼L . The (force/weight) ratio is is more than the sum of its parts (in symbols: 1 + 1 = 2). The 1/L, indicating that small animals may be relatively stronger additional qualities that do not exist on the level of the indi- than the big ones. The human weight is also expressed by a vidual components are the collective modes of behaviour of 2 2 power-law, as the surface density M/L [kg/m ], with the opti- the whole system that emerges through the interactions of its mum value currently set to 21 (BMI = body mass index). These components. They cannot be predicted from the knowledge of are results based on observations or on elementary scaling the single components itself (if in principle, is a philosophical considerations. question). The quality of ‘houseness’ is not present in each Biotechnology, like chemical technology, applies the bal- single brick the house is composed of. One practical conse- ance equations, which can be scaled. With well-defined quence for microtechnologists: a huge production block built physical quantities, DA can operate. Some specific situations up from many small active elements of various types will do exist at biosystems. There are crucial qualities that are always have potential for producing unpredictable emergent difficult to define and quantify, whence to subject to DA. behaviour. Once you create your LEGO-plant from your micro- For instance, the ‘physiological state’ of living matter is of bricks, it will start its own life. It will be ready to surprise the paramount importance. Further, there are metabolic patterns, creator,7 any time. pathways of such a complexity that a mere set of mass equa- Literature used and recommended: References section 3.4. tions from Section 7 for their description is ineffective. Rather, the graph theory is used to capture their topology and for- malize the problems at least qualitatively, in terms of the 7 There might even be a precedent for this in human history. 866 chemical engineering research and design 8 6 (2008) 835–868

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9 One may ask: Why topologically equivalent drawings do represent biologically different species?