Analysis Based on Two Kinds of Scales

Conventional dimensional analysis employs single length and time scales. Correlations are thus obtained for the mass or heat transfer coefficients in terms of the minimum number of independent dimensionless groups these can generally be represented by power functions such as 

Here k is the mass transfer coefficient, d is a characteristic length of the system (e.g., the diameter of the tube), U is a characteristic velocity, v is the kinematic viscosity, D is the diffusion coefficient, h is the heat transfer coefficient, k is the thermal conductivity, a is the thermal diffusivity, and A, m, and n are constants. 

Assuming, as usual, a power relation between the variables 

Equation (11) has the same form as Eq. (1). The exponent m of the Reynolds number, 

Since a and P are not independent quantities, being related via Eq. (12), the preceding two inequalities are simultaneously satisfied only if 

---