Deriving the Radiation Thermal Resistance of a Heat Sink

Key Takeaways

• Radiation heat transfer in heat sinks.

• The electric circuit analogy of heat transfer.

• Deriving radiation thermal resistance.

Radiation is an important heat transfer mode in heat sinks

Heat sinks are common thermal management systems used in electronics to transport heat energy from the circuit to the ambient using heat transfer methods such as conduction, convection, radiation, or a combination of them. Heat transfer in heat sink systems can be described using the electric circuit analogy.

The electrical circuit analogy utilizes the thermal resistance parameter to distinguish between conduction, convection, and radiation mechanisms in heat sinks. In a heat sink heat transfer problem, the conduction thermal resistance is not the same as convection thermal resistance, and both of them are different from radiation thermal resistance. As radiation is an important heat transfer mode in heat sinks, we will explore the electrical circuit analogy of heat transfer emphasizing the radiation thermal resistance.

Radiation Heat Transfer in Heat Sinks

When heat energy is transferred from the hot body to the ambient in the form of electromagnetic waves, it is called radiation heat transfer, or heat transfer by thermal radiation. It is a common heat transfer mode in electronics cooling, especially in heat sinks. The efficiency of heat transfer by thermal radiation in heat sinks is maximum in vacuum. Since thermal radiation does not require a medium for heat transfer and occurs at the speed of light, it is an important mechanism in any thermal management system.

Whether it is conduction, convection, or radiation, heat transfer problems in heat sinks can be analyzed using an electric circuit analogy. 

The Electric Circuit Analogy of Heat Transfer 

The electric circuit analogy of heat transfer is based on Ohm’s law. In this analogy of heat transfer problems, the temperature difference (△T) in the system is analogous to the potential difference (△V) in the equivalent electric circuit. Due to potential differences, current (i) flows from higher potential to lower potential. Likewise, heat flux (q) flows from higher temperature to lower temperature.

We know that according to Ohm’s law, the potential difference can be expressed as:

△V=iR_e

where R_e is the electric resistance in Ohms.

The same concept is put up in the electric circuit analogy. The heat transfer problem can be written as:

△T=qR_t

where R_t is the thermal resistance of the heat transfer mode.

Thermal Resistance

The thermal resistance of a system is the resistance offered to the flow of heat through its boundaries. For a given temperature difference, the thermal resistance is the quantity that influences the rate of heat transfer. The thermal resistance is dependent on the geometry of the system and thermal properties such as thermal conductivity of the medium. The thermal resistance varies with heat transfer processes such as conduction, convection, and radiation.

The concepts of thermal resistance and electric circuit analogy are best-suited for solving steady-state heat transfer problems. The thermal resistance involved in the equivalent electric circuit analogy of heat transfer problems can be a series, parallel, or a series-parallel combination of thermal resistances, depending on the system geometry and modes of heat transfer prevailing in the system. Having an understanding of the thermal resistance value is helpful when calculating the heat flow or temperature on the boundaries of the thermal management system.

Next, we will derive the radiation thermal resistance in heat sinks where the heat energy is radiated to the ambient.  

Deriving Radiation Thermal Resistance 

Consider a heat sink where the thermal exchange takes place from the heat sink surface to the ambient. The thermal radiation between the heat sink surface at temperature T_s and the ambient at temperature T∞ given by:

Note that Q is the heat energy in Watts, ε is the emissivity of the heat sink surface, is the Stefan-Boltzmann constant, and A is the heat transfer area in m².

Rearranging the equation to obtain  △T :

The relationship between heat flux (q) and heat energy (Q) can be given as follows:

The radiation thermal resistance can be written as:

Note that h_rad is the radiation heat transfer coefficient: 

The radiation thermal resistance is dependent on the emissivity of heat sink surfaces. The heat sink fin dimension, heat sink surface texture, and surface color need to be chosen so that they increase the emissivity and decrease the radiation thermal resistance. As the radiation heat transfer co-efficient and heat sink area are inversely proportional to the radiation thermal resistance, steps should be taken to increase both to decrease thermal resistance, thereby increasing the heat transferred.

The heat transfer in heat sinks takes place via conduction, convection, radiation, or a combination of them. In most heat sinks, convection and radiation occur side by side, and their combination contributes to the overall thermal exchange in heat sinks. While analyzing the combined convection and radiation heat transfer in heat sinks using electric circuit analogy, the parallel combination of radiation thermal resistance and convection thermal resistance must be considered.

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