A function for the percentage of heat lost by an object immersed in a fluid due to conduction versus blackbody radiation Is there a way to derive an equation for the amount of heat lost by an object immersed in a fluid due to blackbody radiation versus conduction using equations such as Planck's law, and the heat loss formula? I ask this as to get a general idea of how efficient different types of heat transfer are (and out of personal curiosity as to how complicated such an equation would be).
 A: The heat loss per unit area (in W/m², say, in SI units) from radiation is often modeled as $\sigma\varepsilon T^4$, where $\sigma$ is the Stefan–Boltzmann constant, $\varepsilon$ is the surface emissivity (assumed to be wavelength independent), and $T$ is the surface temperature.
The corresponding heat gain is often modeled as $\sigma\varepsilon T_\mathrm{env}^4$, where $T_\mathrm{env}$ is an effective or surrounding temperature. This temperature can be challenging to estimate if the body is facing a variety of fluids and distant objects with their own absorptive and reflective characteristics; for example, a "sky temperature" of 270 K, say, may be appropriate for modeling objects facing upward toward a cloudless night.
Nonradiative heat transfer with a fluid can be modeled using a conduction framework, but a challenge is that fluids tend to advect (i.e., exhibit bulk motion) in complex ways that carry thermal energy much faster than this energy would tend to diffuse. As a result, convection models are often used; these model the net heat flux (i.e., per unit area) from the body as $h(T-T_\mathrm{fluid})$, where $h$ is a so-called convection coefficient that is usually found empirically (or modeled as scaling with certain relevant terms such as the Nusselt, Reynolds, Rayleigh, and Prandtl numbers) and $T_\mathrm{fluid}$ is the bulk fluid temperature.
As a simple example, consider a warm (90°C) steel ball, 3 mm in diameter, propelled downward into 10°C water at 1 m/s, which is also the terminal velocity.

*

*When the ball has cooled to a surface temperature of 50°C, what percentage, at most, of the instantaneous cooling is radiative?


*Under what conditions might this percentage be approached?


*What is the instantaneous net heat transfer to the ball in watts?


*What is the instantaneous cooling rate?
Solution. The relevant Reynolds number is $\mathrm{Re}=\frac{vD}{\nu}\approx 3000$, where $v$ is the speed, $D$ is the diameter, and $\nu$ is the kinematic viscosity. Consulting a handbook or equivalent, we estimate the Nusselt number as $\mathrm{Nu}\approx 75$, implying a convection coefficient of $h=\frac{k}{D}\cdot\mathrm{Nu}\approx 15000$, where $k$ is the thermal conductivity of water.
The greatest radiative heat loss occurs if the emissivity is 1, which could be approached if the sphere is oxidized, has a rough surface, and is painted black with a broad-absorption pigment, for instance. The maximum radiative loss is then $q^{\prime\prime}_\mathrm{rad}=\sigma(T^4-T_\mathrm{env}^4)\approx 250\,\mathrm{W\,m}^{-2}$. The convection loss is $q^{\prime\prime}_\mathrm{conv}=h(T-T_\mathrm{fluid})\approx 600000\,\mathrm{W\,m}^{-2}$. Radiative transfer (constituting <0.1%) is thus negligible, and the heat transfer to the ball is $-q^{\prime\prime }A\approx -20\,\mathrm{W}$. The rate of cooling, which depends on the heat capacity $mc=\rho Vc$ (with sphere mass $m$, specific heat capacity $c$, density $\rho$, and volume $V$) is about 400°C/s.
Consider another example: The same steel sphere is removed from a furnace at 1000°C and placed on a thermal insulator to cool evenly. The surroundings are at room temperature (20°C).

*

*Again, compare the convective and radiative losses.


*Estimate the temperature gradient within the sphere.
Solution. The convective setting is now one of natural (density-driven) convection rather than forced (pressure-driven) convection. Again consulting a handbook or equivalent, we estimate a Nusselt number of $\mathrm{Nu}\approx 2$, essentially equal to the quiescent conduction, as the sphere is too small to roil up much advection, even at 1000°C. The corresponding convection coefficient is about $h\approx 20\,\mathrm{W}\,\mathrm{m}^{-2}$, for predicted convective losses of 0.5 W.
The maximum radiative loss ($\sigma A(T^4-T_\mathrm{env}^4)$) at the nearly 1000°C temperature difference between the sphere and surroundings is now predominant: about 4 W. We can expect the sphere to cool at an initial rate of as high as about 100°C/s.
The temperature gradient within the sphere can be estimated by equating the heat flux at the surface with that arising from internal conduction. Since  the conductive heat flux corresponds to $q^{\prime\prime}_\mathrm{cond}=-k\frac{dT}{dx}$ by Fourier's law, where $k$ is now the thermal conductivity of steel, the temperature difference is found from $\frac{q^{\prime\prime}\Delta x}{k}$, where $\Delta x$ corresponds to the radius $D/2$. The internal temperature gradient is thus estimated to be on the order of 0.1 mK; thermal uniformity during cooling is assured.
