Dimensional Analysis

Example: (cont)
6.	Form the pi terms by making each π-term dimensionless. i.e. Exponents sum to zero for each dimension and for each dimensionless product. Pi1 = ω B^a V^b ρ^c = 0 or 1/T [L^a (L/T)^b (FT^2/L^4)^c Express the exponents. (For T) ˗1 + 0 - b + 2c = 0 (For F) 0 + 0 + 0 + c = 0 (For L) 0 + a + b - 4c = 0 Solve for exponents a, b, and c. a = 1 b = - 1 and c = 0 So π1 = ω B^-V^-1 result: Pi1 = ω B/V k/ρV^2b^3 →(FL)/[ (FT^2/L^4)(L/T)^2(L^3) = 1 so Pi1 = k/ρV^2B^3
7.	Pi 2 = h B^a V^b ρ^c = 0 or L [L^a (L/T)^b (FT^2/L^4)^c Express the exponents. (For T) 0 + 0 -b + 2c = 0 (For F) 0 + 0 + 0 + c = 0 (For L) 1 + a + b - 4c = 0 Solve for exponents a, b, and c. a = -1 b = c = 0 So Pi2 = h / B
8.	Pi3 = k B^a V^b ρ^c = 0 or FL [L^a (L/T)^b (FT^2/L^4)^c Express the exponents. (For T) 0 + 0 - b + 2c = 0 Solve for exponents a, b, and c. (For F) 1 + 0 + 0 + c = 0 c = - 1 Therefore b = - 2 (For L) 1 + a + b - 4c = 0 and a = - 3 a = -3 b = - 2 c = -1 So Pi3 = k / ρ B^3 V^2
Summary	Pi1 = ω B/V Pi2 = h / B Pi3 = k / ρ B^3 V^2
Check Dimensions	Pi1 = (1/T)(L)(T/L) Pi2 = L/L Pi3 = (FL)/[ (FT^2/L^4) (L^3)(L/T)^2 ] Click here for discussion on similitude.

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