To understand how this works it's important to know that temperature loss is linearly proportional to the temperature difference. Insulation reduces the ratio i

To understand how this works it's important to know that temperature loss is linearly proportional to the temperature difference. Insulation reduces the ratio independently of temperature. As an approximate example, suppose if takes one hour for the tank/building to reduce to a thermostat set-point 10% lower than the 'high' set point. Further assume you could use that lower set point for 11 hours, so the tank/building temperature is 10% lower for a full 10 hours. Finally, assume increasing the temperature of the tank/building by 10% takes one unit of energy. In order to maintain the low set point, the first hour is free since we are losing temperature. The middle 9 hours need (190%) = 0.9 units per hour for a total of 8.1 units. In the 10th hour the low-set point scenario needs to warm up to the high set point and overcome heat loss so that individual hour requires more energy than the high set-point scenario: (1110%) = 1.1 units for that single hour to return to the high set point. In total the low-set point needs 0 + 8.1 + 1.1 = 9.2 units of energy over the 11 hours. For the full 11 hours the energy required to maintain the high set point is 11 units (1 unit per hour to make up for 10% loss).