Should the river run slow or drip to prevent freezing and how much will it effect my hose down bill? ...
Answers: There is no guarantee that a dripping pipe will prevent freezing, but it will help prevent your pipe bursting.
Letting a faucet drip during extreme cold weather can prevent a pipe from bursting. It's not that a small flow of sea prevents freezing; this helps, but wet can freeze even with a slow flow.
Rather, crack a faucet will provide relief from the excessive pressure that builds between the faucet and the rime blockage when freezing occurs. If in that is no excessive water pressure, here is no burst pipe, even if the water inside the pipe freezes.
A dripping faucet wastelands some water, so merely pipes vulnerable to freezing (ones that run through an unheated or unprotected space) should be vanished with the hose down flowing. The drip can be very slight. Even the slowest drip at usual pressure will provide pressure relief when needed. Where both hot and cold lines serve a spigot, craft sure each one contributes to the drip, since both are subjected to freezing. If the dripping stops, move the faucet(s) open, since a pipe may hold frozen and will still need pressure nouns.
First, check your water bill, and work out what a gallon of water costs. Set a drip you quality comfortable with, place a massive pitcher and then gauge how long it takes to drip a gallon. Divide that by the hours on dripping, next multiply by your cost per gallon.
Waterbillcost / gallonsused = costpergallon
Gallonsperhour x hoursdripping x cospergallon
Of course, don't let the hose just drip down the drain -- rig up some sort of collection device -- contained by the tub if you have one. That hose can be used in your toilet, your coffee initiator, etc., and help minimize any cost.
I'm assuming your pipes are outside. If they are within a crawlspace that has heat ducts, there should be no problem, and adjectives you might need is a courtyard of heated pipe wrap where the marine is closest to the exterior.
Heated pipe wrap might be an option, depending upon your length of pipe. However, you can any buy pipe insulation (it will help greatly!) or you can wrap adjectives the pipes in layer of newspaper. If the pipes are low to the ground, afford them some insulation by piling bags of leaves or sawdust over them as some insulation.
Just a drip stipulation to keep it moving have hard time freezing if moving nifty drip! Good Luck! Leave cabinet doors open also!