Basics - Heat transfer isn't what comes to most people's mind when they hear "electrical engineering", but for power systems engineers it's a critical part of their design process. Without an understanding of basic heat transfer mechanics, we can't learn how to size cables correctly.
First off, what is heat transfer? When I say heat transfer, I'm referring to the movement of heat (thermal energy) from one location to another. This transfer of heat coincides with temperature gradients between those points.
Heat transfer occurs through three mechanisms:
Conduction: heat is transferred through a solid material
Convection: heat is transferred via circulation of a fluid (think of the phrase "heat rises")
Radiation: heat is transferred via emission of electromagnetic radiation
Heat transfer via radiation and convection are dominant factors in aboveground conductor sizing. These effects, especially convection, can be difficult to characterize mathematically without empirical analysis. Fortunately, power systems engineers are not as concerned with this situation. Aboveground, the effect of mutual heating between conductors is small. Circuits can be considered to be thermally independent of one another when they are separated by only a couple of cable diameters. Therefore, we can just use ampacity tables taken from the NEC or IEEE. We don't have to calculate ampacities for each specific aboveground configuration.
Underground, the rules are very different. Conductive heat transfer applies for conductors buried underground. The earth acts as a thermal resistance, a medium for preventing the flow of heat. This leads to mutual heating between conductors underground, even at large spacings. Often, spacings of 10' or more are required to achieve effective thermal independence.
Thermal Resistance - Thermal resistance is an analog of electrical resistance to describe the relationship between heat flow per unit time Q' and the temperature difference across the resistance ΔT. Mathematically:
R Q' = ΔT
Where, for a 2-dimensional geometry like a conductor buried in the earth:
R is the thermal resistance of the material (earth, insulation, etc.), typically measured in °C*ft/W
Q' is the heat flow per unit time, typically measured in W/ft
ΔT is the temperature difference between the heat source and the ambient, measured in °C
Although this equation has been referenced to underground burial of conductors, this same physical equation applies to any situation where heat is transferred via conduction, like the insulation in your home.
Insulation has a high thermal resistance
In this analogy, Q' behaves like a current source in an electrical circuit and ΔT is like a voltage difference. Thermal resistances in series add and thermal resistances in parallel offer a lower combined thermal resistance than either individually.
Thermal Circuit with Series Resistances R1, R2
Neher-McGrath Equation - We can take the equation for conductive heat transfer above and expand it to consider the particular case of a cable buried in the earth. For a conductor underground, heat is created by current flowing through the resistance of the cable:
Q' = r I^2
Where:
Q' is the heat flow of the conductor in W/ft
I is the load current in Amperes
r is the AC resistance per unit length of the conductor in Ohms/ft
This particular heat source can be substituted into the conductive heat transfer equation above and re-arranged to obtain the following result:
I = √( ΔT / ( r R ) )
This equation is the simplest form of the Neher-McGrath equation described in the NEC, IEEE, and elsewhere. Instead of solving for temperature gradients based on the heat from a cable, we solve for the maximum load current I given a limit on the temperature rise ΔT. This slightly different formulation of the conductive heat transfer problem is a very common problem for power systems engineers to solve.
The Neher-McGrath equation above is easy to write down, but challenging to apply because of the thermal resistance R. Computing the thermal resistance of any particular cable configuration and construction type is a laborious process outlined by Neher and McGrath in their original 1957 paper.
The thermal resistance is dependent upon the geometry of the situation. Large contact areas over which heat can transfer make thermal resistance smaller. Thicker insulation that the heat has to pass through increases the thermal resistance. Additionally, thermal resistance is dependent upon a material property known as the thermal resistivity ⍴, measured in °C*cm/w. Thermal resistances of things like the earth, conduit walls, and conductor insulation are all proportional to the thermal resistivity ⍴.
Today, numerous software programs exist that can speed up the process of calculating ampacities and thermal resistances.
Example: Solve for the ampacity of a single conductor directly buried in earth with the following properties:
Thermal resistance of the insulation: 10°C*ft/W
Thermal resistance of the earth: 40°C*ft/W
Ambient Earth temperature: 20°C
Conductor maximum operating temperature: 90°C
AC resistance of .0003 Ohms/ft
Solution: Begin by recognizing that heat flowing out of the conductor will be required to pass through both the insulation and the earth, so the thermal resistances are in series. We can define the total system thermal resistance as:
R = 10 + 40 = 50 °C*ft/W
We can solve this problem by applying the Neher-McGrath equation for this particular thermal circuit.
I = √( ΔT / ( r R ) ) = √( (90 - 20) / (50 * .0003) ) = 68 A
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