# Heat transfer coefficient

## Newton’s Law of Cooling

Newton’s Law of Cooling is:

$q = h \cdot A \cdot \Delta T$

where q is the heat transfer rate, h is the heat transfer coefficient for the system, A is the area of heat transfer, and ΔT is the temperature difference between the zones of heat transfer. This description of heat transfer applies to convection.

Generally, for a body at some temperature Tbody in a reservoir of fluid Tfluid:

ΔT = Tbody - Tfluid

## Estimating Heat Transfer Coefficient

In principle h can be calculated from measured data for any given system, but it cannot always be derived from theory. h can be estimated for systems of known geometry and known materials using empirical methods, however.

## Nusselt Number

The Nusselt Number of a thermal interface is a dimensionless analog of the heat transfer coefficient. For a system than can be bounded by a finite volume, it can be found using the formula:

$Nu = \fracTemplate:H \cdot D h{{k_{fluid} }}$

where Nu is the Nusselt Number, Dh is the effective diameter of the interface, and kfluid is the thermal conductivity of the convective fluid. However, the Nusselt number can be found using empirical relations for the interface’s materials and geometry as well, and then used to derive the heat transfer coefficient.

For example, for a cylinder parallel to the wind:

$Nu = c \cdot {Re} ^n \cdot \sqrt[3]Template:Pr$

where Re and Pr are the fluid Reynolds Number and Prandtl Number of the cylinder-wind system, and c and n are constants for a given range of Reynolds numbers.

$Re = \fracTemplate:\rho \cdot v \cdot D h{\mu }$

and

$Pr = \fracTemplate:C p \cdot \mu{k }$

where ρ is the air density, v is the wind velocity, Dh is the hydraulic diameter of the cylinder (equal to the diameter for a free floating cylinder, or the diameter of a cylinder with the same surface area in contact with the wind for a half cylinder or buried cylinder), μ is the kinematic viscosity of the air, k is the thermal conductivity of the air, and cp is the specific heat of the air.

The values for most of these constants for the Martian air can be found by assuming the air is equivalent to pure carbon dioxide at 7 mb pressure (the average Martian air pressure). The coefficients for computing the Nusselt number are:

 Reynold's Number Range c Value n Value 1 - 4 0.989 0.330 4 - 40 0.911 0.385 40 - 4000 0.683 0.466 4000 - 40000 0.193 0.618 40000 - 250000 0.0266 0.805

Note that the form of the equation used to find the Nusselt number is empirical. Nusselt Number equations for various systems are simply fitted to experimental data, and there is not necessarily any correspondence between them. If the wind flow is parallel to the cylinder, the Nusselt number is obtained by an entirely different empirical equation:

$Nu = 0.648 \cdot \sqrt Template:Re \cdot \sqrt[3]Template:Pr$

Then, once the Nusselt number of the system is known, the heat transfer coefficient of the wind-cylinder heat transfer surface can be computed.

$h = \frac{{Nu \cdot k_{fluid} }}Template:D h$