# If layers of insulation are continuously added to a heated object, will it continue to be better insulated?

If you were to keep adding layers of insulation to something, like blankets to a person, would each blanket continue to improve the insulation? Or do you reach some point at which the next blanket provides no marginal increase in insulation?

Links to explanations of heat or insulation for the layperson are more than welcome.

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The answer is yes, but no, let me explain...

Heat flow is proportional to the thermal conductivity of the insulating material, and to the temperature gradient, $q = -k \nabla T$. If you have temperature $T_i$ inside and $T_o$ outside, and an insulating layer of thickness $h$ and conductivity $k$, the rate of heat loss in constant regime will be

$$q = k\frac{T_i-T_o}{h},$$

which becomes ever smaller as you increase $h$, so extra insulation does have an effect.

On the other hand, if you already have a thickness of insulating material $h$, the effect that extra insulation will have on the heat loss rate is proportional to the derivative of $q$ with regards to $h$,

$$\frac{dq}{dh} = -k\frac{T_i-T_o}{h^2},$$

so the effect of extra insulation is ever smaller, and will eventually become negligible.

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