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In CFD (Computational Fluid Dynamics) simulations, the space is discretised into a huge number of tiny cells and, therefore, one has the opportunity to use a model of heat transfer that takes into account such amount of detail. Newton's law of cooling gives the rate of heat transfer between a surface and a moving fluid,

$$\dot{Q} = h \cdot A \cdot \Delta T,$$ where $h$ is the heat transfer coefficient (that depends on both thermal and hydrodynamic properties of the flow), $A$ is the surface area and $\Delta T$ is the difference of between the fluid and surface temperatures. The model does not go into the details of local heat transfer between each tiny element of fluid but regards the transfer as a whole (as if both fluid and surface were irreducible). I don't know how the law could be used in a CFD simulation. For instance, which cells of the fluid would be regarded in the formula: the ones immediately in the vicinity, or others some far away? The result would be highly dependent on the chosen cells: the near ones would have a temperature very similar to the one of the wall, while the far ones may not. Also, heat is in reality transferred through collisions between particles or by diffusion, the so-called conduction (as well as by radiation). Convection may correspond to the motion of a big bulk of fluid but the transfer of heat is still made microscopically through conduction. A law for heat conduction such as Fourier's law may describe well local heat transfer, and, therefore, couldn't it be used for any case (heat transfer between two elements of fluid and between an element of fluid and an element of surface)?

The convection law seems to be solely appropriate for back-of-the-envelope calculations where one does not want to deal with the local details of fluid flow. Still, I know that it is used in many CFD codes such as Ansys Fluent. But how do these codes deal with the problem of defining the cells associated with the "vicinity fluid" that transfer heat?

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  • $\begingroup$ What is the "vicinity fluid"? How is if different to the fluid flowing in the pipe for example? Convection can be seen in things small like cigarette smoke and the smoke from burning buildings... $\endgroup$
    – Solar Mike
    Dec 5, 2021 at 12:26
  • $\begingroup$ To apply Newton's cooling law one needs to use the temperature of the approaching fluid. There's the problem of defining this temperature since the fluid occupies a space and not a point, hence my difficulty on understanding how the law may be applied in CFD simulations. I gave the designation "vicinity fluid" to the zone of the fluid whose temperature should be regarded in the formula. $\endgroup$ Dec 5, 2021 at 15:08
  • $\begingroup$ So if the cells have an associated area, volume, density, temperature etc why can't the laws of continuity, convection be applied? $\endgroup$
    – Solar Mike
    Dec 5, 2021 at 15:17
  • $\begingroup$ Laws of continuity and convection of mass may be easily applied. The main problem is convection of heat which requires a model such as Newton's cooling law that use representative temperatures for a whole surface and a whole fluid, while in a CFD simulation temperatures in the cells need to be used. $\endgroup$ Dec 5, 2021 at 15:42
  • $\begingroup$ Please always spell out/introduce the abbreviations for those less educated but interested readers. $\endgroup$
    – r13
    Dec 5, 2021 at 15:44

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You are right, there is a bit of difficulty when it comes to calculating localized heat transfer coefficients in CFD or otherwise. Newton's law of cooling assumes that there is a uniform or constant bulk temperature of the fluid. The temperature difference is the subtraction of the bulk temperature of the fluid from the temperature of the solid surface. The CFD tool I use (SimScale) asks you to set a reference temperature which in the ideal case is your ambient temperature or the bulk fluid temperature. However, in cases where you have a fluid flowing through a cooling channel, the average temperature of the fluid often increases as you move further downstream. In that case, I would choose the inlet temperature of the coolant as the reference temperature. In the end, Newton's law of cooling and the heat transfer coefficient is a simplification based on the assumption of a uniform bulk fluid temperature.

You can find more information on how HTC is calculated in simulation here: https://www.simscale.com/knowledge-base/heat-transfer-coefficients-from-simulation-results/

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