Reynolds number for mixing is a defined quantity, which you have identified correctly as Re=rho*D^2*N/mu where D is the agitator diameter and N is RPMs.  The number of blades on the impeller, impeller type, and other impeller geometry factors have no effect on how Reynolds Number is calculated or the resulting value.
 
The effects of impeller geometry are reflected in the variable correlations defined by the Reynolds Number.  For instance, the values of power number are different for different impeller types, even though the Reynolds Number is the same for impeller diameter and rotational speed.  The turbulent power number for a two-blade, 45-degree-pitched impeller is about 0.80, compared with about 1.37 for a four-blade, 45-degree-pitched impeller.  The Reynolds Number for both impellers is calculated the same way.  Values for the two-blade and four-blade impellers will change in the transition and laminar ranges defined by Reynolds Number.
 
If the cooling coils are installed like plate baffles (same number and similar width), the effect on mixing should be similar.  The power number values are about the same, based on the projected area of the plate or coil baffles.  The process-side heat transfer correlations for the cooling coils are still a function of the same impeller Reynolds Number, but the correlations also include a term for pipe diameter, usually as a ratio with the tank diameter.  A specific correlation for two-blade impellers may not exist, but the coefficient on correlation might be estimated from similar correlations for other impeller types.
 
The problem is not in calculating the Reynolds Number, but in what you do with the Reynolds Number.

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