Calculation Corner: A Simple Method for Estimating Radiation Heat Transfer

This article describes a semi-empirical equation that can be used to assess radiation heat transfer in terms of an effective convection coefficient.

The effective convection coefficient due to radiation is shown to be:

Where temperatures are in °C and ε is the emissivity of the radiating surface. This equation is accurate to within ~10% over a temperature range of 0 – 130°C.

EQUATION DERIVATION

The relative effects of heat transfer by radiation are often small enough that they can be ignored in most applications that use liquid cooling or forced air cooling. However, radiation can play a significant role in situations in which the system-level thermal resistance is relatively high, such as natural convection or forced air cooling at high altitudes. Since convection and conduction heat transfer are largely proportional to a temperature difference while radiation is a function of absolute temperatures to the fourth power, it can be difficult to compare the relative impact of each heat transfer mode without resorting to the use of a computer or calculator. This article describes a simple approach for estimating the effects of radiation and comparing them to conduction and convection thermal resistances.

This approach defines a radiation heat transfer coefficient, hrad, as a convection coefficient that would produce the same heat transfer (Q) as radiation if the radiating surface (at temperature TH ) were transferring heat to a fluid at the temperature to which heat is being radiated (TC), as shown in Equation {1}.

This heat transfer is set to be equal to the Stefan-Boltzmann law for radiation heat transfer, which is shown in Equation {2}

Where A is the heat transfer area, σ is Boltzmann’s constant (5.67e-8 W/m² K⁴), ε is the emissivity of the radiating surface, and the subscript “Ab” indicates absolute temperature.

The radiation temperature term can be expanded so that the temperature difference term can then be dropped from both sides

The temperature terms can be made dimensionless by dividing by the difference between Celsius and Kelvin temperature scales:

This leads to the effective radiation heat transfer coefficient being expressed in terms of the dimensionless temperatures as:

Terms can be rearranged as shown below

The terms can be grouped into the form shown in Equation {4}

which can be written as:

in which C₁ and C₂ are functions of H and C. In most electronics cooling applications, surface temperatures are in the range of ~ 0 – 130°C. Thus, the typical values of H and C are likely to be in the range of 0 – 0.47 and the second order terms, (H+C)² and HC, will be relatively small compared to the constants 4 and 6. Since the term σT_*³ = 1.156 W/m²K, at temperatures close to 0°C, C₁ will be ~1.156*4 = 4.6 and C₂ will be ~273.15/(1.156*6) = 40.

Minimizing the error over the entire temperature range of 0-130°C leads to values of C₁ = 4.131 and C₂ = 26.75, which has a maximum error of ~8%. A slightly less accurate, but easier to remember, equation is:

Table 1 shows the error associated with Equation 6 compared to the exact solution for black body radiation between surfaces at two temperatures (with a view factor of 1 between the surfaces).

DISCUSSION

One useful application of Equation {6} is in making a quick determination of whether radiation needs to be considered in a thermal assessment.

For example, if the convection for forced air cooling from a surface is on the order or 100 W/m²K and Equation {6} finds an effective radiation heat transfer coefficient of 8W/m²K, then it is probably safe to neglect radiation. But in free convection air cooling, the heat transfer coefficient will often have a magnitude similar to that of the effective radiation heat transfer coefficient, so radiation does need to be addressed.

A few other issues need to be kept in mind when using Equation {6}:

• Equation {6} is only appropriate for temperatures above 0°C
• You need to account for view factor and relevant areas of the surface that is dissipating power. For example, if the surface is a finned heat sink, then the heat transfer area for convection is the entire fin area (accounting for fin efficiency) while the heat transfer area for radiation is likely to be the planform (base) area.
• The ‘cold’ temperature for convection is not necessarily the same as the ‘cold’ temperature for radiation. For example, convection from an outdoor system will be to the ambient air temperature while radiation will be to the sky temperature or cloud temperature.

ACKNOWLEDGEMENT

Thanks to my colleague John Kramer for recently asking me the simple question “Is it physics based or totally empirical?” about Equation {6}.

I had reverse engineered the equation through curve fitting many years ago; John’s question, after I had suggested that he could use it in an analysis, prompted me to spend a little time to do the algebra necessary to better understand it.