How many BTU are required to raise the temperature of one gallon of water one degree Fahrenheit in one hour?

This is a pretty straightforward calculation. By definition, a BTU is the amount of energy required to raise one pound of water one degree F. But you have one gallon of water, which weighs approximately* 8.34 pounds. So, you'd need 8.34 BTU to increase one gallon of water one degree F. Note how the amount of time was not important. Whether you heat the water slowly or quickly doesn't matter. You will still require 8.34 BTU to raise the temperature of a gallon of water one degree F. * I say approximately because the weight of water varies slightly with its temperature. Water is at its densest at 4 degrees Celsius (39 degrees F). A gallon of water at temperatures above and below that value will weigh less.