Using a house as a model. . .
I’m mostly up to speed with the concept of heat loss being faster the greater the difference, but what I’ve also wondered is this: Is the loss gradient the same regardless of the scale. For example:
Inside temp is 90º F. Outside temp is 70º.
Inside temp is 70º F. Outside temp is 50º.
Inside temp is 50º F. Outside temp is 30º.
Inside temp is 30º F. Outside temp is 10º.
If my math is correct, that’s a 20 degree difference in all four cases.
All other conditions being equal, if I checked the temperature in these various scenarios every half hour for example, would the loss be pretty much the same?
Over that broad of a range, there might be complications. For instance, there might be ice on a surface below freezing, which might make a difference in the insulative value. But other than that, yes.
There are three basic processes for heat transfer: conduction, convection, and radiation. (There is also by some approaches a fourth, evaporative cooling, which is a combination of the first two and can be very complex, but it is not germane here so we’ll ignore it.)
[ul][li]Conduction is strictly dependent upon the thermal difference and material conductivity, and all things being equal it generally conforms to Newton’s Law of Cooling for any real world heat transfer phenomenon, giving an exponential curve. [/li]
[li]Convection depends on the properties of the convective medium, which itself may be altered in heat carrying capability by thermal energy flux (i.e. humid air will carry more heat energy than dry air). More influential is the speed at which the convective medium is moving; a fast blowing wind will carry away energy quickly, while a stilled atmosphere will leave give almost no convective heat transfer. [/li]
[li]Radiation, unlike the other forms, is decidedly non-linear; radiative heat transfer occurs as a fourth order function of temperature, and with a coefficient that changes across the spectrum. This means that the actual temperatures , rather than just the difference between initial and ambient, can have a dramatic influence on the rate of heat transfer.[/ul][/li]
For an above-ground, well-insulated (double pane glass windows, high R-value insulation, et cetera) stick frame house the primary heat loss mechanisms are going to be radiation, convection, and conduction, in that order. (A drafty house may have the first two reversed but most modern homes are pretty well sealed against air loss, sometimes to an unhealthy extent.) So, with a house, no, heat loss will not be the same for the same temperature difference.
It’s all about the same if you are looking at convection and conduction. To make it more accurate you can consider whether conductivity or viscosity are functions of temperature (for gasses the viscosity is about proportional to absolute temperature - yes, gasses get more viscous when they are hotter, the opposite of liquids).
You might use the concept of a heat transfer coefficient. Think of it as the power per area, divided by the temperature difference. The power per area, multiplied by your house’s external area, gives you the total power your house needs to hold a steady temperature. The temperature difference is the difference between the indoor temperature and the outdoor temperature, both taken far enough away from the walls that you can consider them representative of the big spaces and not the surfaces losing the heat. Heat transfer coefficients of maybe 5 or 10 or so, in watts per square meter kelvin, are common for free convective cooling. A breeze will give you more cooling. You might also notice that household insulation is sold according to an R value, and the units of R are the inverse of the units for a heat transfer coefficient.
The very idea of a heat transfer coefficient implies the whole thing works proportionally to temperature difference.
It is a fourth order function, but it’s on the absolute scale, and it works in both directions. Discounting emissivity and reflectance factors, a body at 40 degrees F surrounded by a 30 degree F environment will radiate 1.084 times more energy than it receives, (277.6 K)^4/(272 K)^4
A body at 80 degrees F in a 70 degree F environment will radiate 1.078 times more energy than it receives. (299.8 K)^4/(294.3 K)^4. Pretty much the same number.