Watt Reducer Wiring

What does the solenoid manufacturer say in their paperwork/website documentation?

In my experience with low voltage DC devices (sensors etc.) brown is normally positive and blue is 0 volt or negative. However, you should confirm this with the manufacturer's documentation.

I concur.... brown is universally the hot wire in dc, and blue is neutral 0 vdv return...this is probably a capacitor to smooth the inrush current, and improve the power factor...so yeah you want the polarity correct....

..." DC solenoids can be made more efficient by using external circuits which can shape the coil current in such a way that there is an initial current spike required to open the valve. Once the valve opens, the current can be reduced to a maintenance current level, which is just enough to reliably keep the valve open by pulling the armature against the spring tension.

These external circuits can be as simple as connecting the coil in series with a parallel connection of a resistor and a capacitor. In such a circuit, charging the capacitor through the coil provides an initial coil current spike. After the capacitor is charged, the current-limiting resistor passes all the current. The downside of such a simplistic approach is that some of the power is wasted on heating the current-limiting resistor."...

https://tameson.com/choosing-an-ac-or-dc-coil-for-a-solenoid-valve.html

Improving power factor in a DC circuit???

That should be efficiency...same difference...

It may not matter which is which, as often a resistance or an equivalent load is connected in series with the coil which has a lower retaining current than a pull in current. There would likely be smarts in the box to decide when the relay had closed and reduce the current. Unless the relay coil has an included snubber diode then polarity is not an issue for a N to S or a S to N across a relay gap will still pull the relay in unless it is a fancy magnetically biassed relay.

In the old days of relay logic there were a set of make before break changeover contacts on the relay which switched a dropping resistor into the circuit.

This device could actually be the transient supressing diode.

Which would require proper polarity.

Wire in series just before the solenoid. Brown is +ve, blue is -ve. When the valve operates there is a peak surge in the current to pull the plunger into the solenoid coil but once inside less current is needed to hold it in place. The unit contains a capacitor that charges up when the circuit is energized. Once full charge is achieved a zenner diode triggers release of the the stored energy to boost the coil current enough to activate the valve. This allows a smaller solenoid to be installed, one that is big enough to hold the plunger in place but not big enough to pull it into place. Typically energy savings of between 25% and 40% are achievable.

Looks like it might wire similar to this

https://instrumentinabox.com/product/hit-hold-solenoid-valve-driver-1-channel/

"I suppose this is wired in parallel?"

Probably series

"The red arrow goes to the 24 v dc power supply?"

Yes, that is what it says

"The white arrow to the solenoid?"

That's what is left, so yes.