How to power small solenoid with a 3.7V lithium battery pack

Force from a solenoid = \$(N\cdot I)^2 \dfrac{μ_0\cdot A}{(2 g^2)}\$

Where:

• μ0 = 4π×10-7
• F is the force in Newtons
• N is the number of turns of the coil
• I is the current in Amps in the coil
• A is the area in length units squared (cross sectional area of the coil)
• g is the length of the gap between the coil and the iron.

If you want to check your solenoids pull force and you know the number of turns and have a good estimate of the "gap" and the cross sectional area of the winding then use the formula.

However, it's notable that force is proportional to current-squared and if the 12V you used dropped down to (say) 6V under the load of the solenoid, the pull force would quarter because the current would have halved. It's also worth noting that the force is inversely proportional to the gap-squared - double the gap and the force quarters.

It's most likely a motorised valve.

Figure 1. Motorised valve attached to water hose line.

An industrial version of a motorised valve would have feedback potentiometer so that the actual position of the valve could be confirmed and, perhaps, proportional control could be achieved. Alternatively limit switches could be fitted to stop the motor once full travel had been reached.

In this case and in the case of the garden hose timers you linked I suspect that the motor would be run for a time and if the valve doesn't move then "too bad". With five or ten cycles per day the run time would only be half a minute or so and battery life should be good.

^{simulate this circuit – Schematic created using CircuitLab}

Figure 2. Simplified representation of control circuit for open / close control including optional limit switches. With SW1 and SW4 closed as shown the motor would run CW until the CW limit switch opened.

Figure 2 shows how a simple DC motor would be switched by a 'H-bridge' to effect forward and reverse motion. SW1 to SW4 would, in a practical circuit, be replaced by transistors (with protection diodes).