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According to "Mathematical Modeling: a Differential Equations approach" by Barnes and Fulford 2nd edition, it should be

$ T' = k (T-T_0)^{5/4} $

where $T_0$ is the temperature of external gas or liquid.

(https://i.stack.imgur.com/c5wE9.jpg)

Is it an empirical approximation, or a theoretical result? What is the basis for the 5/4 coefficient?

Had anybody do experiments to check it? Stangely, I could find only school lab experiments, not scientific experiments. What about computer simulations? This problem should be solved long ago, but I was unable to find the solution.

Obviously, if you test this experimentally, you should take into account several things:

  • you should wait for "equilibrium". Initially, the surface of an object is as hot as it's internal part. With time, the is a distribution of $T-T_0$ may, or may not, become like this:

$ T(\mathbf{x}, t) - T_0 = f(\mathbf{x}) g(t) $ for big $t$

Any experimental check should take into account the following points:

  • you cannot test it on liquid in air, because then there are two convective flows: in the liquid, and in the air. That makes the problem more complicated. Besides, there is evaporation of the liquid, and the glass has its own temperature distribution.
  • you cannot put a body on a surface, because then there is also thermal conductivity to the surface. Besides, the surface changes the convection flow.
  • you should take into account radiation, which is proportional to $T^4$
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The "5/4 relationship" is obtained by analytically solving the partial differential equations for fluid flow (Navier Stokes equations) and conductive/convective heat transfer for the case in which the temperature at surface of the object is held constant at T and the temperature far from the object (in the surrounding fluid) is at $T_0$. It assumes a specific orientation of the surface, and a surface that is flat. The temperature of the object can conceptually be held constant at steady state by generating heat within the object (say by using an electric current). There are other minor assumptions in the analysis. The complete analysis may be found in Transport Phenomena (a real classic that has stood the test of time) by Bird, Stewart, and Lightfoot.

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