Can we harvest energy from a Rogowski coil?
In order to harvest 5 W from a conductor carrying 10 A, we need to generate a voltage drop on the conductor of at least 0.5 V. As the primary is a single turn, that means the core must be able to generate 0.5 V per turn.
If we assume a sinusoidal current, at 50 Hz, the core must support a total rms flux of volts/2πf = 0.5/314 = 1.6 mWeber. The flux in Webers is the product of core area, and flux density in Tesla.
If we assume a 1 T rms flux density (a nice round number, and the practical maximum for ordinary transformer steel), the core area therefore needs to be 1.6m m2, or about 40 mm x 40 mm.
Can we actually achieve a 1 T flux density in the core? The cable is said to be 2&quot; in diameter, so has a circumference of about 160 mm. The core will have to sit clear of the cable, so let's estimate 250 mm for the magnetic length of the core. The cable carries 10 Arms, so that generates an H field of 40 A/m in a length of 250 mm. The B field will be Hμ0μr, so the required μr is given by B/Hμ0 = 1/40*4π10-7 = 20k.
20k relative permeability! That's if you want to keep the core area down to 40 mm by 40 mm. You can get that permeability with mu-metal or permalloy, but not the saturation level. Easily available transformer core steel will only deliver μr in the 2k to 5k range, so you'll need significantly more area than 40x40 mm, 40x400 mm if you use 2k steel.
These large numbers are due to restricting the primary to a single turn, of large diameter, while trying to harvest watts.
Once you have a core large enough to intercept the power you need, then you can design the secondary to suit the end voltage you want to produce. It may help to resonate the transformer inductance with a shunt capacitor on the secondary, to save having to make the core even larger to allow for the effect of the finite primary inductance on the power throughput.
During one part of my career, I worked with one of the well known UK energy suppliers, on a clamp-on self-powered energy meter. The maximum primary diameter was 10 mm, so the core length was nearer to 80 mm than 250 mm. The core was about 7 mm x 20 mm. The target harvest was a few mW, not a few watts. The electronics was designed with μA-level opamps and MCU, which then spent most of the time asleep.
If you can engineer your required power down to 5 mW rather than 5 W, you may have a more practical design.