In part 1 of this blog series we discussed the duality between the electrical and thermal conduction domains. In this blog, we’ll take a look at the three different modes of heat flow or heat transfer and relate these to thermal resistances. As we will see, heat transfer occurs when there is a temperature difference in or between a medium and always transfers from hot to cold according to the second law of thermodynamics. In the absence of an external energy source for heat, the temperature differences between the mediums will eventually reach thermal equilibrium and thus heat transfer mechanisms will always strive to have its medium and surroundings reach the same temperature. This blog also expands on the concepts presented in our previous blog on Thermal Analysis of Package/PCB Systems: Challenges and Solutions. The figure below shows the three transport mechanisms that are discussed in this blog.

*Figure 2. Three thermal transport mechanisms.*

**Thermal Conduction**

Heat conduction takes place when a temperature gradient or difference exists in a solid or stationary fluid medium. Conduction occurs when heat energy is transferred between particles of the medium in the direction of decreasing temperature. For example, when you touch a hot object, the heat you feel is transferred from the object to your hand by conduction. The rate at which conduction heat flows is described by Fourier’s law of heat conduction as shown below in one dimension:

where

The negative sign in the equation indicates heat flows in the direction of high temperature to low temperature. Thermal conductivity, k, is a measure of how effective the material is at transferring heat, i.e. a good thermal conductor has high thermal conductivity whereas an insulator has low thermal conductance. In conduction, heat flow is within and through the medium body and through direct contact between solid mediums.

The thermal resistance for conduction heat transfer is given by:

where the conduction heat transfer flow, , is equal to the power dissipation through the object, .

**Thermal Convection**

Heat convection occurs when a solid surface is in contact with a flow of fluid at a different temperature. The fluid is typically air in consumer electronic products and the flow can be due to natural convection or forced convection. Natural convection or free convection occurs from temperature differences in the fluid which affects the density and buoyancy of the fluid. The denser components of the fluid (cool air) are heavier and will fall while the less dense components (hot air) are lighter and will rise causing fluid movement. As the hot air surrounding the hot object rises, it is replaced with heavier cool air to remove heat from the object, enabling heat transfer. Forced convection, on the other hand, is fluid movement coming from an external force such as a fan. Heat transfer by convection is described by Newton’s law of cooling:

where

The heat transfer coefficient, h, is a measure of how effective a fluid transfers heat by convection and is determined by its velocity, density, and viscosity. A fan blowing air has a higher velocity than air flow by natural convection and thus will have a higher heat transfer coefficient.

The thermal resistance for convection heat transfer is given by:

**Thermal Radiation**

Heat radiation is a transfer of heat energy from an object by electromagnetic waves and can occur without a medium such as in space or in a vacuum and is emitted by any object with a temperature above 0 degrees Kelvin. An example of radiation heat transfer is the heat or warmth we feel from the sun. Radiation heat transfer depends on the physical surface properties of the object such as its color, orientation, and roughness. Heat transfer by radiation is described by the Stefan-Boltzmann equation:

where

The geometrical shape factor F is the percentage of radiation emitted from the object to the radiation absorbed at the surrounding area’s surface. If an object is enclosed in an enclosure, the geometrical shape factor F = 1.

The thermal resistance for radiation heat transfer can be approximated by a linearization of its heat transfer equation:

where

The table below summarizes the equations for the three heat transfer modes.

The above equations for thermal resistances provide a circuits type method for analyzing the thermal behavior of a system. It should be noted that getting the actual thermal resistance values may require some additional approximations as the geometrical and material properties may not be readily available for your component of interest. Examples of this are the emissivity and geometrical shape factor parameters needed for radiation. Also, non-linearities especially in the radiation resistance are ignored using the resistive network approach but the technique still yields good accuracy for most electronic systems where thermal conduction and convection dominate (notice thermal radiation and convection resistances appear in parallel for systems that have a medium). Furthermore, the network approach provides good insight into the thermal properties of a system and is something that electrical engineers are very comfortable with.

Our next blog will use the concept of thermal resistors to develop a thermal network of a system and our final blog will discuss some cooling techniques electrical engineers can use to cool electronic systems.