question about superficial tension

Our ILB (Immiscible Lattice Boltzmann) is the Rothman -Gunstensen (in D2Q9) .
We would like to introduce superficial tension values in our code.
We thought of converting the physical value of the superficial tension (sigma_p= 0.035 N/m) using converting factors



where U is fluid velocity and L is length, the subscripts p and lb are used to identify physical and lattice units respectively.
or (alternatively) to set a Capillary Number (Ca_N) and calculate the superficial tension in lattice units (sigma_lb) as
sigma_lb= (U_lb.*nuK_lb)./Ca_N;

where nuK_lb is our kinematic viscosity in lattice units and U_lb is the fluid velocity in lb units.

Rothman - Keller - Gunstensen etc suggest to use the relationship:

lambda=1./tau; density=1;

where , as far as we understand, the parameter A is the superficial tension in lattice units.
tau is the relaxation factor.

The two approaches lead to very different values and sigma_lb is orders of magnitude larger than A. Typical values of A are 1E-4. We calculate sigma_lb values of order 1.E2.
Can you help us to understand? why conversion factor does not work in this specific case?