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If water at the same temperature is heated with the same heat power source in two different ambient temperatures (one ambient is low and the other ambient is high) then which water sample will reach the equilibrium state of temperature more quickly, or both will take same time?

This question is related, if we just change the ambient temp. and start heating the object then how will it affect to thermal resistance and thermal time constant?

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The time constant can be defined as a function of density ($\rho$), volume ($V$), heat capacity ($C_p$), surface area ($A_s$), and heat transfer coefficient ($h$).

$$ \tau = \frac{\rho\ V\ C_p}{h\ A_s} $$

The heat transfer coefficient is a function of thermal conductivity and length. $$ h=\frac{k}{x} $$

As temperature of water increases, so does its thermal conductivity. This increases the heat transfer coefficient, which decreases the time constant.

A higher temperature starting point would mean faster heat transfer.

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