HVAC - whats the math

[boarder42]

So we have an air source heat pump and a furnace for heating.  I have our Programmable thermostat set to use the heat pump if its over 35 or the furnace if its under.  does anyone have the math behind an ideal setpoint, as well as when it may make sense to use the heat pump before it gets too cold outside ie pre heating the house when its warmer or above 35 during the day vs below 35 at night.

[robartsd]

Heat loss of your home is roughly proportional to the temperature difference between inside and outside air. The better insulated your home is the greater the temperature difference your heat pump would be effective at maintaining and longer it will take for your home to lose heat. Since raising your inside air temperature before temperatures drop will mean your home's rate of heat loss will increase, it probably does not make sense to set your indoor temperature any higher during the day in an attempt to reduce nighttime heating costs.

The ideal setpoint for switching between your heatpoint and furnace would be at the point where your the cost of heat from your furnace is less than the cost of heat from your heat pump. The outside air temperature at which this happens will be different for different inside air temperature settings. Lowering your indoor thermostat temperature would lower the temperature where switching to furnace makes sense. Improving your home's insulation would also lower the temperature where switching to the furnace makes sense.

[boarder42]

Heat loss of your home is roughly proportional to the temperature difference between inside and outside air. The better insulated your home is the greater the temperature difference your heat pump would be effective at maintaining and longer it will take for your home to lose heat. Since raising your inside air temperature before temperatures drop will mean your home's rate of heat loss will increase, it probably does not make sense to set your indoor temperature any higher during the day in an attempt to reduce nighttime heating costs.

The ideal setpoint for switching between your heatpoint and furnace would be at the point where your the cost of heat from your furnace is less than the cost of heat from your heat pump. The outside air temperature at which this happens will be different for different inside air temperature settings. Lowering your indoor thermostat temperature would lower the temperature where switching to furnace makes sense. Improving your home's insulation would also lower the temperature where switching to the furnace makes sense.

yeah i understand most of this i just want equations how has an industry been around so long and not have some sort of equation that can be somewhat simplified to calc this with basic assumptions.