Theory for the Conductive Thermal Resistor Component
This section presents the underlying theory of the Conductive Thermal Resistor two-port component.
Thermal Resistance Analogy for Steady Conduction
By analogy with Ohm’s law for electric current, the heat rate P (SI unit: W) in a material due to the temperature difference ΔT (SI unit: K) is
(4-89)
where R (SI unit: K/W) is the thermal resistance.
The expression of the thermal resistance used in Equation 4-89 depends on the geometric configuration.
Plane Shell
When considering steady conduction through a plane shell of surface area A (SI unit: m²), thickness L (SI unit: m), and constant thermal conductivity k (SI unit: W/(m·K)), the thermal resistance R is:
Cylindrical Shell
When considering steady conduction through a cylindrical shell of inner radius ri and outer radius ro (SI unit: m), height H (SI unit: m), and constant thermal conductivity k (SI unit: W/(m·K)), the thermal resistance R is:
Spherical Shell
When considering steady conduction through a spherical shell of inner radius ri and outer radius ro (SI unit: m), and constant thermal conductivity k (SI unit: W/(m·K)), the thermal resistance R is:
Steady Conduction Through a Composite Wall
When considering several materials, Equation 4-89 can be updated to get a relationship between the heat transfer rate P and the overall temperature difference ΔToverall (SI unit: K):
where Rtot (SI unit: K/W) is the total thermal resistance.
Thermal Resistances in Series
When considering steady conduction through a composite wall made of n layers of thermal resistances R1,...,Rn, see Figure 4-7, the total resistance is defined by:
with
Figure 4-7: Steady conduction through a composite wall made of n layers
This corresponds to the following serial thermal circuit representation:
where ΔToverall = Tn+1 − T1. Note that by analogy with electrical circuits, the elements placed in series in the circuit share the heat rate (flow variable).
Thermal Resistances in Parallel
When considering steady conduction through a composite wall made of n slabs of thermal resistances R1,…,Rn, see Figure 4-8, the total resistance is defined by:
with
Figure 4-8: Steady conduction through a composite wall made of n slabs.
This corresponds to the following parallel thermal circuit representation:
where ΔToverall = T2 − T1. Note that by analogy with electrical circuits, the elements placed in parallel in the circuit share the temperature difference (effort variable).
Convectively Enhanced Conductivity
See Equivalent Thermal Conductivity Correlations for details about the correlations used to modify the thermal conductivity to account for convective heat flux.
Radiation in Optically Thick Participating Medium
See Rosseland Approximation Theory for details about how to modify the thermal conductivity to account for radiative heat flux when the optical thickness of the medium is large.