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Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Advanced Transport Phenomena Module 7 Lecture 30 1 Similitude Analysis: Dimensional Analysis
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SIMILITUDE ANALYSIS 2
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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” 3
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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? SCALE-MODEL TESTING & EXPLOITATION OF SYMMETRY 4
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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? SCALE-MODEL TESTING & EXPLOITATION OF SYMMETRY 5
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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 SCALE-MODEL TESTING & EXPLOITATION OF SYMMETRY 6
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Usually, geometrical scale factor L p /L m >> 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” SCALE-MODEL TESTING & EXPLOITATION OF SYMMETRY 8
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Geometrical: corresponding distances in prototype & model must be in same ratio, L p /L m 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 TYPES OF SIMILARITY 9
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All types of similarity can be simultaneously attained in nonreactive flows over wide non-unity range of L p /L m By making compensatory changes in other system parameters More difficult to achieve in systems with chemical change, especially under homogeneous/ heterogeneous non-equilibrium conditions TYPES OF SIMILARITY 10
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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 (p c, V c, T c ) are used as reference values, i.e.: NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING 11
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All have same EOS in terms of reduced quantities: Works well for other vapors as well (chemically unlike noble gases) NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING 12
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Law of “Corresponding States” NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING “Corresponding states” correlation for the compressibility pV/(RT) of ten vapors (after G.-J. Su(1946)) 13
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Newtonian Viscosity of a Vapor: For a pure vapor, viscosity 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 NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING 14
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Newtonian Viscosity of a Vapor: NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING Two-parameter ( ) spherically symmetric intermolecular potential: (a) Dimen- sional; (b) non- dimensional (scaled) 1 15
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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. NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING 16
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Newtonian Viscosity of a Vapor: In the perfect-gas limit ( 3 /v 0), we obtain well- known Chapman-Enskog viscosity law: This is of earlier “similitude” form independent of p varies as p -1 NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING 17
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Similitude in Biology: Mammal Invariants (Stahl, 1962) Each mammal, depending on type & size, may have a different: NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING 18
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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 NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING 19
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“Universal” Drag Law: Steady drag, D, on smooth sphere of diameter d w in uniform, laminar stream of velocity, U depends on fluid density, , and Newtonian viscosity, : Nondimensional form: where NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING 20
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“ Universal” Drag Law: All spheres are geometrically similar If we set: then NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING 21
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“Universal” Drag Law: i.e., dynamic similarity is also achieved, and: Quantitative relationship between D p and D m When Re << 1: If both Re p and Re m satisfy this condition, testing need not be done at same Re value NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING 22
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DIMENSIONAL ANALYSIS Vaschy (1892), Buckingham (1914): (Pi Theorem) Any dimensional interrelation involving N v variables can be rewritten in terms of a smaller number, N , of independent dimensionless variables N v - N = number of fundamental dimensions (e.g., 5 in a problem involving length, mass, time, heat, temperature) e.g., drag relation: N v = 5, N = 2 23
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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 ( ’s) are the quantities to be kept invariant in model testing 24
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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: DIMENSIONAL ANALYSIS 25
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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 DIMENSIONAL ANALYSIS 26
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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: DIMENSIONAL ANALYSIS 27
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In the presence of variable thermophysical properties, forced convection, natural convection, free-stream turbulence: DIMENSIONAL ANALYSIS 28
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For high-speed (compressible) gas flows, add: etc. DIMENSIONAL ANALYSIS 29
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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 DIMENSIONAL ANALYSIS 30
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Physical significance of dimensionless groups obscured No insight regarding analogs (other phenomena obeying same laws) DIMENSIONAL ANALYSIS 31
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