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Convection is very hard to model. It most likely changes with increasing temperature difference. (With very a low temperature differences there maybe no convection.) The best link I've found is here, http://www.thermopedia.com/content/660/?tid=110&sn=7 (or go to Thermopedia and look under convection.)

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The equation you state is a very general expression related to heat transfer, and basically everything goes into that constant. Convection of course is one thing, but what about radiative cooling (often important), diffusive cooling (might be important), and heat resistance, since the temperature of your object is not uniform. All these contributions can be ...

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For simple conductive heat transfer, h is $\kappa$, the thermal conductivity, divided by the length over which the temperature gradient exists. You can look this up for a given material. For convective heat transfer, this constant will depend on the details of your problem, including the dynamics of the liquid in question (can't simply look it up, you'd ...

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