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You are missing that $$h=k\frac{Nu}{L}$$where L is the characteristic length. So, for a given value of the Nusselt number, increasing the thermal conductivity corresponds to an increase in the heat transfer coefficient. In any of the correlations that are out there, if you express Nu as a function of the Reynolds number and Prantdl number, and then solve ...

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Three processes are involved: Conduction: Heat flows from the object to its environment. Removal rate of heat from the interface further away from the object is proportional to the coefficient of conductivity (0.024 for air, 205 for aluminum -see http://www.engineeringtoolbox.com/thermal-conductivity-d_429.html). Convection: The interface between the object ...

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This is in general a very complicated problem. There is three processes by which your object will cool down : conduction and convection which you already mentionned, as well as emission of blackbody radiation. You can treat blackbody radiation with the help of Stefan's law that gives you the power emission with respect to temperature. Conduction and ...

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