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Lucia, Newton's Law of Cooling applies to heat transfer by convection only, and states that the heat flow rate is proportional to the difference in surface temperature and free stream temperature (ie temperature of the fluid well away from the surface). Note I said convection only, and there are two other ways that heat is transferred in this universe: conduction and radiation. Also, even some types of convection don't *technically* follow Newton's Law of Cooling, as I'll try to explain later. First let's run through the easy ones. Conduction: When heat is transferred through a single solid object or multiple solid objects that are in direct contact with each other, it is called conduction. Conduction is governed by Fourier's Law, which states that the rate of heat transfer is proportional to the temperature gradient (dT/dx) at any point in the solid. This sounds very similar to Newton's Law of Cooling, and in fact it is. The major difference between conduction and convection is that the constant of proportionality in Fourier's Law is based only on one property of the solid, the thermal conductivity, while the constant of proportionality in Newton's Law of Cooling is based on the thermal conductivity of the fluid as well as other properties of the flow field (like pressure, velocity, etc.). We almost always know (or can measure) the thermal conductivity of a substance, but the proportionality constant in Newton's Law of Cooling (or "convective heat transfer coefficient") can be a big, ugly equation involving several variables, or there might not be an analytical solution for it at all! This is a whole subject in-and-of itself, and in fact I spent a whole semester in a graduate-level class on convection deriving the equations for the convective heat transfer coefficient for various cases. Radiation: All things that have a temperature above absolute zero emit thermal energy in the form of radiation. They radiate in the portion of the electro-magnetic spectrum that we call "infra-red" which we cannot see, and the amount of energy they radiate is governed by the Stefan-Boltzmann Law. This law states that the amount of energy radiated per unit surface area is proportional to the absolute temperature to the fourth power. The constant of proportionality in the Stefan-Boltzmann Law is called the Stefan-Boltzmann Constant (what a coincidence!) and, as the name implies, its value does not change. However, the Stefan-Boltzmann Law really applies to so-called "blackbodies" which are imaginary, ideal objects that absorb all radiation that hits them and reflects none. For real objects the amount of energy radiated varies depending on the direction relative to the surface as well as the emittance (or emissivity) of the surface. Note though that the emitted radiation is proportional to the surface temperature to the fourth power, and is not dependant on the temperature of the surroundings. However some heat will also be absorbed from the surroundings, and that will depend on the temperature of objects in the line of sight of the surface. Free Convection: Thermodynamicists differentiate between "free" or "natural" convection (where there is no driving pressure difference or gradient) and "forced" convection (where there *is* a pressure difference that drives the flow), and they do it because the convective heat transfer coefficients (which I will simply call "h" in this paragraph) are different for each case. I said above that the flow field properties help determine h, and this is just one example of that. Without getting too deep in this very complicated subject, let me just point out that if you solved the fluid dynamic equations and figured out the heat transfer for free convection, you would find that it is proportional to the temperature difference between the surface and the free stream *to the 5/4 power* if the flow is laminar, and to the 4/3 power if the flow is turbulent. So strictly speaking, Newton's Law of Cooling does not apply to free convection. However, it is still useful to use the form of Newton's Law of Cooling, and thermodynamicists do this by simply dumping the extra deltaT^1/4 (for laminar flow) or deltaT^1/3 (for turbulent flow) into h. Admittedly, this can lead to confusion since we started by saying that Newton's Law of Cooling was valid for convective heat transfer, then we go on to show that it's not actually valid for *all* convective heat transfer. It might be easier to think of Newton's Law of Cooling as just a way of defining h, ie h is defined as the heat transfer divided by the difference in surface temperature and free stream temperature for all convective flows. In this definition, h might have a small dependance on the temperature difference, and that's just fine. The bottom line is that thermodynamicists can learn a lot about the heat transfer for a given situation if they know h for that case, and this is precisely because we have a rigid definition of h (ie Newton's Law of Cooling). Reference: Heat Transfer, 2nd Ed., by A.F. Mills

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