DIMENSIONAL ANALYSIS BUCKINGHAM PI THEOREM

PROBLEM In the March 1982 issue of the AICHE Journal, Vasalos and coworkers reported experimental findings for particle holdup times in a synthetic fuels reactor. They showed that the Reynolds number at the terminal velocity of a spherical particle was related to the Galileo number. Use the Buckingham Pi Theorem to find these dimensionless groups if the 5 important parameters are diameter D, density , viscosity , velocity  and the buoyant force per unit volume (s - )g. Choose your repeating or core parameters in the order given [that is, begin with diameter, D, then density, , and so on until the appropriate number is obtained].

SOLUTION Parameters: D    (s - )g n=5 parameters Use m, L, and t as primary dimension (r=3). In terms of primary dimensions, units for these parameters are: L m/L3 m/Lt L/t m/L2t2 Three repeating parameters should be selected: D, , and . This means 5-3=2 dimensionless -groups, which are obtained by combining the set of repeating parameters with each of the other remaining parameters one at a time.

SOLUTION CONT’D 1 = Dabc 2 = Defh (s -)g For these to be dimensionless, we must have: (L)a(m/L)b(m/Lt)c(L/t) = m0L0t0 (L)e(m/L)f(m/Lt)h(m/L2t2) = m0L0t0 Exponent equation for m: b + c = 0 b = 1 m: f + h + 1 = 0 f = 1 L: a - 3b - c + 1 = 0 a = 1 L: e - 3f - h - 2 = 0 e = 3 t: -c - 1 = 0 c = -1 t: -h - 2 = 0 h = -2 Hence, 1 = Dabc = D/ (Reynold’s Number) 2 = Defh (s - )g = D3(s - )/2 (Galileo Number)