Dimensions 1 time 2 distance k dimension of ω 3 2 1

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Dimensions: 1 -time, 2 -distance k (dimension of Ω) = 3 – 2 = 1, base solution = (1, 1, -1) is the only unitless parameter From the Pi-Theorem, Π always satisfies f ( Π )=0, and hence the value of Π is influenced by no other physical quantities; i.e., Π must be a constant. From traffic flow theory, or field measure, we know that Π = 1. Q KV Q V K / 1 1 1 = = Π 0 ) ,..., , ( 2 1 = Π Π Π k f 1 2 0 1 3 ~ 1 1 Rank( ) 2 1 0 K p V Q α α = = M ) 0 , 0 , 0 ( ) , , ( 3 2 1 = M a a a k Π Π Π ,..., , 2 1 Logistics Systems Analysis Dimensional Analysis A Famous Example (Estimating power of nuclear bomb) Several years after the first atomic bomb explosion in 1945, a series of pictures, including the size of the fire ball R and time stamps t were published in a magazine. Based on these photographs a British physicist G.I. Taylor was able to estimate the energy E released by the explosion (which was still a secret at that time). Assumptions: The energy was released at one spatial point. The fireball was spherical. Density of the surrounding air is Dimensional Analysis One base of Ω is (1/5, –1, 2/5, –1/5), hence By Buckingham “pi” theorem, (Small-scale experiments show that constant ~ 1) At t= 0.006 sec, the radius R is approximately 80 meters. Normally = 1.2 kg/m 3 . Then It is known that 1 kg of TNT releases 4×10 6 kg m 2 /s 2 of energy, then E ~ 25 kilo-tons of TNT 4 Rank( ) 3 1 p k = = = M ) 0 , 0 ( ) , , , ( 4 3 2 1 = M a a a a 1/5 2/5 1/5 1 E t R ρ Π = Logistics Systems Analysis (Source: ) 1 2 2 0 1 0 ~ 0 0 1 1 3 0 E
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  • Fall '16
  • Physical Principles of Engineering Analysis

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