Previously I showed an equation that showed the ratio of convective and radiative heat transfer. That equation is derived from the ratio of the Nusselt and the radiative heat transfer (RHT) rates. From a practical standpoint the amount of energy transferred from the surface to the atmosphere by each method should be comparable for the situation of the Earth’s surface.
Warning: this article contains significant engineering language and terminology.
Fans are effective at cooling because they induce what is called forced convection. That increases the rate of heat transfer (speeds up cooling) from the warm object to the cool object. Natural convection is what happens without a fan. For instance a warm cookie placed on a counter top. The cookie cools off by radiating heat away and by warming the air that it touches. This warm air rises from the cookie to be replaced by cool air. That is natural convection. That is what happens to the surface of the Earth when it is warmed by the sun. The warm air raises up because it is lighter and cool air replaces it.
There are many situations that the primary equations and dimensionless numbers that are used to determine the convective heat transfer from one object to another. Since air is the medium of transfer from the surface to the atmosphere, that greatly simplifies the situation. For air the Prandtl number is 0.713 (20 °C). When dealing with air, the buoyancy forces will always dominate the viscosity forces (liquids will generally be opposite). This results in a high Rayleigh number. The transition point between laminar and turbulent natural convection will be used to lock the Rayleigh number down. That value for air is 109.
That provides the starting point to derive the ratio between convective and RHT. I will use the same methods as shown by Dr. Physics when he was demonstrating the ratio between the two transfers for the human body and the air. The main difference is that he used the vertical cylinder to represent the human body. The science is the same for transfer from a human as it is from the surface of the Earth. Energy is leaving the warm body to the cooler air.
This is easily rearranged to show the energy transfer over the area as a function of the Nusselt number like this.
Since the area will be the length and the size of interest is 1m and the thermal conductivity of air is 0.257 W/m/K the result is:
Since the Nusselt number is a function of the Rayleigh number it is easy to chart the amount of convective heat transfer over a range of air flows.
Radiative Heat Transfer
Since I want to compare the ratio of the RHT it will also need to be re-arranged from the normal Stefan-Boltzmann transfer I have shown before to the delta form.
Since the area is the same for both, it is only the ratio of the heat transfers that remain.
This is in a different form than the one I used before because I want to show how important the turbulence is to the situation. Air is always moving and in the atmosphere there is very little laminar flow. So the lower region of the turbulent region should be the realm that determines the heat transfer ratio. This would be the region where the Rayleigh Number is 109 < Ra < 1011. The relationship for the Nusselt number and the results for this situation are as follows:
This provides the general guideline for how much energy should be transferred by each method. For the situation of transfer from the Earth’s surface to the atmosphere above, there is not a great temperature difference, as a result the amount of energy transferred by each method is comparable. This is something that anyone with any practical experience in heat transfer should immediately recognize. Radiative is not a dominant method unless there is vacuum or the temperature differences are large. Neither of these situations is true for the surface of the Earth.
When the net LW (23 W/m2) and convective (17 W/m2) are compared in KT97, KT08 or any other energy balance of the Earth, the resultant ratio approximately 0.74. That is exactly what is expected for natural convection at the laminar to turbulent transition. That ratio corresponds to a Rayleigh number of 3.8E9 which is right in the range it should be. Conversely if the convective were to be 4% like warmists that don’t understand heat transfer would have people believe, then the Rayleigh number would be 3.2E4. That is an absurd proposition for the atmosphere. The estimated value for the full atmosphere is 1017 < Ra < 1020. For the sake of a limited scope of a few meters above the surface where the natural convection exists the estimated value of 3.8E9 is very reasonable.
From an engineering point of view it is wholly reasonable and expected that the amount of energy transferred from the surface to the atmosphere by both convective and radiative methods be comparable in magnitude. The purpose of most engineering is to make complex problems solvable. The different dimensionless numbers used in this article provide the method that engineers use to make the estimates that allow the precise complexity to be accounted for in a reasonable manner. Bulk temperature for gases and liquids are used instead of an exact temperature profile. The modern world works because these tried and true methods work. From an engineering perspective, radiative and convective heat transfer from the surface to the atmosphere are comparable in scope and magnitude. That limits CO2 to about 3% of the total and also limits the effect of changing CO2 levels, but more on that later.