# Does a higher temperature difference result in a faster rate of heat transfer?

Suppose that I have a thermally insulated tank with hot air at a temperature of 400 C, and the tank is stored in a room with an ambient temperature of 25 C. I also have another identical insulated tank filled with hot air at 50 C that I store in another room with the same ambient temperature. Which tank would have a faster rate of heat transfer? Also, what if a third tank was filled with cold air, perhaps cooled down to 0 C? Would the heat transfer rate for the cold air be same as that for the 50 C since the absolute temperature difference in relation to the ambient temperature is the same?

Yes, It does. Heat transfers in different ways, in your case by conduction through the tank's wall. The insulation will delay heat transfer, but in your first two examples, the larger temperature difference transfers faster.

$$\frac{Q}{t}=\frac{kA(T_{hot}-T{cold})}{d}$$

• Q = heat transfer in time t
• k = thermal conductivity of the tank
• thermal conductivity of the tank insulation
• A =area
• T= temperature
• d= thickness of the insulation

And everything else being equal the third case is correct too.

A concept that can be useful in clearing up relevant concepts to the question is thermal resistance. The good thing is that thermal resistance does not distinguish between conduction and convection and thus, treats a wall in a much simpler way.

The general equation is $$\Delta T = \dot{Q}\times R$$ where:

• $$\Delta T$$: is the total temperature drop (or difference)
• $$\dot{Q}$$ : the heat flow rate
• $$R$$: the absolute thermal resistance.

If you are familiar with Ohm's law ($$V=IR$$) there is a very straightforward analogy between:

• $$\Delta T$$ and $$V$$,
• $$\dot{Q}$$ and $$I$$ and finally
• the resistances.

From the above its obvious that the temperature difference and heat transfer rate are proportional for the cases involving conduction and convection.

Regarding the last part of the question, if the transfer rate is the same in a room with ambient temperature of 25C between two tanks at 50 and 0 C degrees, I'd like to note the following:

1. the absolute heat transfer rate is the same but obviously the direction the heat "travels" is different.
2. For small temperature differences, the heat transfer will be approximately the same, however, as temperatures increase, both heat conductivity and convection can have a temperature dependence, which can have an effect on the values. E.g. the following is an image for steel