Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

1. Advanced Transport Phenomena Module 7 Lecture 30 Similitude Analysis: Dimensional Analysis Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

2. SIMILITUDE ANALYSIS

3. SCALE-MODEL TESTING & EXPLOITATION OF SYMMETRY • Cost of running full-scale, long-duration experiments is very high • Incentive to obtain required info using small-scale models and/ or short-duration (“accelerated”) tests • LH Baeckeland (chemist): “Commit your blunders on a small scale, make your profits on a large scale”

4. SCALE-MODEL TESTING & EXPLOITATION OF SYMMETRY • Under what conditions can one quantitatively predict full-scale (“prototype”) behavior from small-scale (“model”) experiments? • Dynamic, thermal, chemical, geometrical similarity: Can all be obtained simultaneously?

5. SCALE-MODEL TESTING & EXPLOITATION OF SYMMETRY • How will (p) and (m) performance variables be quantitatively interrelated? • If possible, avoid complex relations • Proper choice of test conditions can ensure simple relations • When is blend of small-scale testing & mathematical modeling necessary?

6. SCALE-MODEL TESTING & EXPLOITATION OF SYMMETRY • Single large-scale device is usually more attractive than stringing together many smaller-scale units • Desired capacity at reduced cost • However, reliability, serviceability & availability may be hard to achieve • Redundant arrays can ensure this

7. SCALE-MODEL TESTING & EXPLOITATION OF SYMMETRY

8. SCALE-MODEL TESTING & EXPLOITATION OF SYMMETRY • Usually, geometrical scale factor Lp/Lm >> 1 • To minimize cost of model tests • Sometimes, converse is true • e.g., modeling of nano-devices by micro-devices • Results to be reported in relevant “dimensionless ratios” • Use internal reference quantities (e.g., relevant L, T, t, etc.) • “Eigen measures”  “eigen ratios”

9. TYPES OF SIMILARITY • Geometrical: corresponding distances in prototype & model must be in same ratio, Lp/Lm • Dynamical: force or momentum flux ratios must be same for (p) and (m) • Thermal: corresponding ratios of temperature differences between any two points in (p) and (m) must be equal • Compositional: corresponding ratios of key species composition differences between any two points in (p) and (m) must be equal

10. TYPES OF SIMILARITY • All types of similarity can be simultaneously attained in nonreactive flows over wide non-unity range of Lp/Lm • By making compensatory changes in other system parameters • More difficult to achieve in systems with chemical change, especially under homogeneous/ heterogeneous non-equilibrium conditions

11. NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING • Law of “Corresponding States” • EOS thermodynamic data for rare gases (He, Ar, Ne, Kr, Xe) different => p, V, T data for each gas would plot differently • But, if critical quantities (pc, Vc, Tc) are used as reference values, i.e.:

12. NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING • All have same EOS in terms of reduced quantities: • Works well for other vapors as well (chemically unlike noble gases)

13. NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING Law of “Corresponding States” “Corresponding states” correlation for the compressibility pV/(RT) of ten vapors (after G.-J. Su(1946))

14. NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING • Newtonian Viscosity of a Vapor: • For a pure vapor, viscosity m can be shown to depend on: product of Boltzmann constant and local temperature Mass/molecule Size parameter defined by Energy-well parameter (average volume/molecule) as

15. NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING Newtonian Viscosity of a Vapor: 1 Two-parameter ( ) spherically symmetric intermolecular potential: (a) Dimen- sional; (b) non- dimensional (scaled)

16. NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING • Newtonian Viscosity of a Vapor: • Since this relation must be dimensionally homogeneous, it can be rewritten as: • Viscosity coefficient of all such vapors should correlate as above. • Data for any particular vapor can be used to obtain indicated function for all.

17. NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING • Newtonian Viscosity of a Vapor: • In the perfect-gas limit (s3/v  0), we obtain well-known Chapman-Enskog viscosity law: • This is of earlier “similitude” form • m independent of p • n varies as p-1

18. NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING • Similitude in Biology: Mammal Invariants (Stahl, 1962) • Each mammal, depending on type & size, may have a different:

19. NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING • Similitude in Biology: Mammal Invariants (Stahl, 1962) • Some dimensionless ratios are same for all mammals– mammal invariants or allometric ratios: • Corresponding biological events (e.g., puberty, menopause) occur at appr. corresponding times • All mammals are “models” of one another

20. NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING • “Universal” Drag Law: • Steady drag, D, on smooth sphere of diameter dw in uniform, laminar stream of velocity, U depends on fluid density, r, and Newtonian viscosity, m: • Nondimensional form: • where

21. NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING • “Universal” Drag Law: • All spheres are geometrically similar • If we set: then

22. NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING • “Universal” Drag Law: • i.e., dynamic similarity is also achieved, and: • Quantitative relationship between Dp and Dm • When Re << 1: If both Rep and Rem satisfy this condition, testing need not be done at same Re value ~

23. DIMENSIONAL ANALYSIS • Vaschy (1892), Buckingham (1914): (Pi Theorem) • Any dimensional interrelation involving Nv variables can be rewritten in terms of a smaller number, Np, of independent dimensionless variables • Nv - Np = number of fundamental dimensions (e.g., 5 in a problem involving length, mass, time, heat, temperature) • e.g., drag relation: Nv = 5, Np = 2

24. DIMENSIONAL ANALYSIS • Buckingham Pi Theorem: • Can be used in any branch of science/ technology • Relevant variables must be listed in their entirety • Provides relevant similarity criteria for problems beyond geometrical & dynamical similarity • Relevant dimensionless groups (p’s) are the quantities to be kept invariant in model testing

25. DIMENSIONAL ANALYSIS • Steady heat flow from isothermal sphere in steady uniform (forced) fluid flow • Interrelation between 8 dimensional quantities can be restated in terms of 3 dimensionless groups:

26. DIMENSIONAL ANALYSIS • Steady heat flow from isothermal sphere in steady uniform (forced) fluid flow • Re establishes dynamic similarity • Prandtl number, Pr, represents thermal similarity • Prototype heat fluxes may be determined from model heat fluxes, provided: and

27. DIMENSIONAL ANALYSIS • Geometric similarity for bodies of complex shape requires similarity w.r.t.: • Shape, and • Orientation (relative to oncoming stream, gravity, etc.) • For fluid convection in a constant-property, low-Mach number Newtonian fluid flow:

28. DIMENSIONAL ANALYSIS • In the presence of variable thermophysical properties, forced convection, natural convection, free-stream turbulence:

29. DIMENSIONAL ANALYSIS • For high-speed (compressible) gas flows, add: etc.

30. DIMENSIONAL ANALYSIS • Greatest advantage: • No need to know/ solve underlying equations, subject to ic’s, bc’s, etc. • Weaknesses: • Uncertainty regarding completeness of initial variable list • Inability to exploit info contained in field equations & conditions, which can further reduce # of dimensionless parameters

31. DIMENSIONAL ANALYSIS • Physical significance of dimensionless groups obscured • No insight regarding analogs (other phenomena obeying same laws)