### Convective Heat Transfer Coefficient Equation Review - Thermodynamics

**Thermodynamics Directory**** | Heat Transfer Directory**

*Convective Heat Transfer Coefficient*

The convective heat transfer coefficient (h), defines, in part, the heat transfer due to convection. The convective heat transfer coefficient is sometimes referred to as a film coefficient and represents the thermal resistance of a relatively stagnant layer of fluid between a heat transfer surface and the fluid medium. Common units used to measure the convective heat transfer coefficient are Btu/hr -ft^{2 }-^{o}F.

The formula for heat transfer is:

Q = h * S * (T_{p} - T_{a})

Where:

− Q =heat transferred, J/s = W

− h = heat transfer coefficient, W/(m^{2} K)

− S = transfer surface, m^{2}

− Tp = Plate temperature, K

− Ta = Air temperature, K

For convection we use the convection heat transfer coefficient h_{c}, W/(m^{2} K). A different approach is to define h through the Nusselt number Nu, which is the ratio between the convective and the conductive heat transfer:

Nu = Convective Heat Transfer/Conductive Heat Transfer = (h_{c} * L)/k

Where:

− Nu = Nusselt number

− hc
= convective heat transfer coefficient

− k = thermal conductivity, W/mK

− L = characteristic length, m

The convection heat transfer coefficient is then defined as following:

h_{c} = (Nu * k) / L

The Nusselt number depends on the geometrical shape of the heat sink and on the air flow. For natural convection on flat isothermal plate the formula of Na is given in table 1.

Table 1: Nusselt number formula.

Vertical fins | Horizontal fins | ||

Laminar flow | Nu = 0.59 * Ra ^{0.25} | Upward laminar flow | Nu = 0.54 * Ra^{0.25} |

Turbulent flow | Nu = 0.14 * Ra^{0.33} | Downward laminar flow | Nu = 0.27 * Ra^{0.25} |

Turbulent flow | Nu = 0.14 * Ra^{0.25} |

Where:

Ra = Gr * Pr

is the Rayleigh number defined in terms of Prandtl number (Pr) and Grashof number (Gr). If Ra < 10^{9}the heat flow is laminar, while if Ra > 10^{9} the flow is turbulent.

The Grashof number, Gr is defined as following:

Gr = (g * L^{3} * β * (T_{p} - T_{a})

Where:

− g = acceleration of gravity = 9.81, m/s2

− L = longer side of the fin, m

− β = air thermal expansion coefficient. For gases, is the reciprocal of the temperature in Kelvin:

β = 1 / T_{a}, 1/K

− Tp = Plate temperature, °C.

− Ta = Air temperature, °C

− η = air kinematic viscosity, is 1.5- at 20 °C. 1.6-at 30 °C