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From my heat transfer textbook by Cengel and Ghajar, they propose that you use the average thermal conductivity value as you specified in your last equation: $k_{avg} = k(T_{avg}) = k_0(1+\beta(T_a+T_b)/2)$ where $\beta$ and $k_0$ are material properties, assuming that the material thermal conductivity follows a linear function w.r.t temperature. You ...

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The traditional method is to use a magnetic stirrer However, what are your dimensions? You might also be able to create something along similar lines by wrapping a coil around the containers.

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based on casual observation, believes that coffee will cool faster than ordinary hot tap water Let's revisit the "casual observation" part. Are the initial conditions the same? Water from a kettle will usually be hotter than from a coffee machine. And we tend to do things to coffee that we don't do to water, like add 10% of another liquid (milk) at low ...

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Internal Energy = Heat - Work, or U = Q - W. By definition an adiabatic process has constant heat, so when finding the rate of change of each in a process by taking the time derivative: dU = dQ(0) - dW Q goes to zero, and U = W.

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