Goodbye Fourier, hello Newton

It is roughly the same equation as for Fourier's Law but some ingredients have changed. Newton's Law gives a proportionality relationship between on the one hand the thermal power P given by the solid to the fluid and on the other hand the difference between surface temperature T_s of the solid (the plate which conveyed the heat up to here) and the temperature T_f of the fluid, as well as the area S depending on the fluid-surface contact.
The proportionality factor is h (in W/(m² K) ) and is named the convective transfer coefficient, Newton's Law is thus written :

Generally, surface and fluid temperature vary according to where we're looking at, so it is appropriate to specify what is understood by temperature. In this relation, they are generally defined in the following way :

• T_s is generally an average temperature over the solid surface
• T_f is generally an average temperature of the fluid in contact with the solid

We now can express the thermal resistance associated with convection :

What consequences can we draw from this expression ? Well, we always seek to make it as low as possible. Obviously it is desired to have a high convective transfer coefficient and a large surface area. Concerning the surface area, nothing special except that we will see later how to increase it with fins. Concerning the convective transfer coefficient, things become complicated !
If we want to understand how it is influenced and how to increase it, it is necessary to understand what occurs in the fluid and how it is characterized.
Fluid characteristics are defined by two types of parameters :

• physical properties of the fluid
• nature of the flow