The Reference Temperature in Newton's Law of Cooling

<p dir="ltr">Newton’s law of cooling is a reduced-order model that characterizes convective heat transfer. It relates the heat flux between a fluid and a solid, q_s'', to the difference in temperature between solid’s surface, T_s, and a reference temperature in the flow, T_ref, via a heat transfer coefficient, h. For external flows, the adiabatic-wall temperature, T_aw, is often used as T_ref, which assumes if the T_s on a non-adiabatic surface equals T_awlocally, then q_s'' = 0 at that location. When using T_aw as T_ref, different local values of h are obtained for different T_sdistributions applied to the same convective problem. Therefore, Green’s function has been proposed as an improved reduced model for convection with the claim that it is independent of the T_sdistribution. However, determining the Green’s function influence coefficients is often significantly more expensive than determining T_awand h for use in Newton’s law of cooling.</p><p dir="ltr">In this work, a film-cooling convective problem is studied where a non-adiabatic surface has locations where T_s= T_aw. However, it is demonstrated that q_s'' =/= 0 at these locations and h obtained via T_awis undefined and not meaningful or useful. Therefore, a new framework to interpret Newton’s law of cooling and obtain T_ref and h is developed, named the state-space method, which guarantees T_s - T_ref = 0 when/where q_s'' = 0 and a meaningful and useful h. Using the new state-space method, it is found that is h reasonably independent of the T_sdistribution for laminar flow over a flat plate if it is obtained using T_sdistributions with the same spatial temperature gradient. For these sets of T_s distributions, the spatial gradient of T_refis inversely related to the spatial T_sgradient. Using this insight, a model is developed to predict q_s'' for an arbitrary T_s distribution which is potentially cheaper to obtain for a given convection problem than the Green’s function influence coefficients. Furthermore, it is shown that the Green’s function influence coefficients can depend on the T_s distributions which are used to obtain them.</p>