Download presentation
Presentation is loading. Please wait.
1
Similitude Analysis: Dimensional Analysis
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
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
12
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
13
NONDIMENSIONAL PRESENTATIONS IN SCIENCE AND ENGINEERING
“Universal” Drag Law: All spheres are geometrically similar If we set: then
14
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 ~
15
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
Similar presentations
© 2025 SlidePlayer.com Inc.
All rights reserved.