Cables in ducts with radiation and convection

New cables in duct

We implemented a new cables in duct calculator to calculate and visualize the thermal resistance of the air between one, two or three cables and the duct through analytic calculation of the convection and radiation. This new tool runs in principle the formulas which are explained in this post.

Thermal Resistance of Air in Cable Ducts: Introduction

When cables run inside an air-filled duct, the air acts as a thermal resistance reducing the cable rating significantly. That “air path” between the cable outer surface and the duct inner surface is exactly what the external thermal resistance term $T'_4$ in IEC 60287-2-1 captures for steady-state current rating calculations.

Reliable and correct calculation of this thermal resistance is critical for accurate dimensioning of power cables. However, IEC turned a complex convection+radiation problem into a single closed-form resistance driven by a geometry proxy ($D_e$) and a temperature proxy ($\theta_m$), with coefficients selected by installation category. The approach is valid only within a defined range of cable and duct dimensions; outside that range, IEC does not provide recommendations. And because this formula uses parameters derived empirically from data the scope of installation types and the constants U,V,Y are limited to those presented in the standard and extrapolating to other geometries/materials is not justified.

In this post we’ll outline a method that computes the thermal resistance of the air region between a cable (or cables) and the inner wall of a duct by explicitly modeling of thermal radiation and natural convection. The calculation is complex as it depends on radiation and convection which are both non-linear and temperature dependent.

We’ll also describe how the result can be compared to the IEC 60287 approach, and why differences are expected.

Thermal Resistance of Air in Cable Ducts: A radiation + convection model

The duct problem can be idealized as two (roughly) concentric surfaces:

• Inner surface: the cable surface (diameter $D_i$, temperature $T_i$)
• Outer surface: the duct inner wall (diameter $D_o$, temperature $T_o$)

Heat leaves the cable and reaches the duct wall through two simultaneous paths:

1. Radiation across the air gap
2. Convection through the air

That means the effective “air thermal resistance” is naturally modeled as a parallel combination: $$T_{4i} = \left(\frac{1}{R_{rad}} + \frac{1}{R_{conv}}\right)^{-1}$$

1) Radiation: Stefan–Boltzmann with a cylindrical exchange factor

Radiation is often non-negligible in ducts because the view between surfaces is good (confined geometry), temperature differences can be tens of °C, and many jackets/polymers have appreciable emissivity. By radiation we mean purely thermal radiation.

Emissivity and geometry coupling

A common way to capture the exchange between two gray, diffuse concentric cylinders is an effective factor: $$F_{rad}=\frac{1}{\left(\frac{1}{\varepsilon_i}\right)+\left(\frac{D_i}{D_o}\right)\left(\frac{1-\varepsilon_o}{\varepsilon_o}\right)}$$ where $\varepsilon_i$ is the cable emissivity and $\varepsilon_o$ is the duct inner-wall emissivity.

Radiative heat flow and an equivalent resistance

Radiative heat flow per unit length can then be written as: $$W_{rad} = \sigma \pi D_i \left(T_i^4 - T_o^4\right) F_{rad}$$ and an “equivalent” resistance follows from: $$R_{rad} = \frac{\Delta T}{W_{rad}}, \Delta T = T_i - T_o$$

2) Convection: compact scaling vs correlation-based Nu approach

Natural convection in enclosed/annular geometries is correlation-heavy and sensitive to geometry and property evaluation. A practical implementation often provides more than one path to compute $R_{conv}$. We consider only natural convection in a duct which is essentially closed at the ends, without any forced convection present (e.g. ventilation).

Option A: an effective-conductivity style scaling

One compact approach is to build a convection coefficient from an effective conductivity term and Rayleigh scaling:

$$h_s = k_{eff}\,Ra_D^{1/4}$$ $$W_{conv} = h_s A_{conv}\Delta T$$ $$R_{conv} = \frac{1}{h_s A_{conv}}$$

Option B: annulus correlation via Rayleigh → Nusselt → h

A more “classic heat transfer” route computes Rayleigh numbers on characteristic diameters, then a Nusselt correlation:

$$Ra_i = \frac{g\beta \Delta T D_i^3}{\nu \alpha}, \quad Ra_o = \frac{g\beta \Delta T D_o^3}{\nu \alpha}$$

After obtaining $Nu$ from a suitable annulus/cylinder correlation, convert to a heat transfer coefficient: $h_{nu} = \frac{k\,Nu}{D_o}$, and finally: $$R_{conv} = \frac{1}{h_{nu}A_i}$$

3) Combine radiation and convection to get the thermal resistance

Once $R_{rad}$ and $R_{conv}$ are known, the effective air-region resistance is: $$T_{4i} = \left(\frac{1}{R_{rad}}+\frac{1}{R_{conv}}\right)^{-1}$$

If you compute multiple convection variants, you effectively get a sensitivity check: how much does your final $T_{4i}$ move when the convection model changes?

What about pure conduction through air?

It’s often useful to compute a conduction-only baseline across the annulus with $A_{cond}=\frac{2\pi}{\ln(D_o/D_i)}$ $$R_{cond}=\frac{1}{kA_{cond}}$$

Even if you don’t use $R_{cond}$ in the final $T_{4i}$, it’s a great sanity check when debugging edge cases.

4) Extending from one cable to multiple cables

With multiple cables, the same mechanisms remain, meaning each cable exchanges radiation with the duct wall (and potentially with other cables) and the the cable set drives natural convection in the confined air space.

In practice, and also according to IEC standard and CIGRE guideance, a group consisting of up to four cables in a common duct are approximated by an equivalent inner cylinder. Theoretically, one can also sum the per-cable contributions with geometric approximations which is more detailed, but more complex and requires assumptions. Either way, the “physics wiring diagram” stays the same: radiation and convection act in parallel and together determine $T'_4$.

Comparing to IEC 60287

The standard IEC 60287-2-1 provides standardized formulae and methods for calculating thermal resistance and is intended for steady-state operation across common installation conditions (including ducts).

When you compare an explicit convection+radiation model to IEC results, systematic offsets are normal because:

• The IEC method is a standardized engineering framework intended to be robust across typical installation ranges.
• A physics-driven model is sensitive to:
  • Emissivity choices ($\varepsilon_i$, $\varepsilon_o$)
  • Natural convection correlations and characteristic lengths
  • Air property evaluation temperature (affecting $Ra$ through $\rho$, $\nu$, $\alpha$, $\beta$)

A useful way to present the comparison

A clear comparison layout is:

• IEC 60287 thermal resistance term (benchmark)
• Analytic model’s split by mechanism:
  • $W_{conv}$ and $W_{rad}$ separately
  • $T_{4i}$ computed from the parallel combination

This decomposition is the main practical advantage: the IEC method gives a result; your model helps explain why you got it (e.g., “radiation dominates” vs “convection dominates”).

Takeaways

• Radiation matters in ducts and is worth modeling explicitly when surface temperatures are elevated. Here using the correct emissivity value of the material is key. However, these values are not defined in an IEC standard for cables nor for ducts.
• Natural convection is correlation-sensitive; keeping an alternate method is a good stability check.
• The duct air resistance is naturally a parallel network of convection and radiation: $T_{4i} = \left(\frac{1}{R_{rad}}+\frac{1}{R_{conv}}\right)^{-1}$
• Comparing against IEC 60287-2-1 remains essential because it is the reference framework used for steady-state cable rating practice.

Air Temperature

Definition

A key sensitivity in natural convection is the choice of the temperature at which air properties are evaluated. In this implementation, air properties are evaluated at an average air temperature $T_{a,av}$ defined using the approach by Sedaghat et.al. in 2018. Therefore, air properties are not taken for a default temperature 20°C or an elevated temperature like 50°C nor for the average temperature between surface of cable and duct as defined by IEC 60287-2-1 but at a temperature derived from the thermal state of the cable(s) and duct wall.

The tool output explicitly lists the temperatures used (e.g., cable surface temperature, duct inner wall temperature, and the resulting average air temperature used for property evaluation).

Air properties

The following correlations (shown here exactly as used) compute air properties from the gas temperature: θgas in °C and Tgas in K. In the duct model, Tgas corresponds to the average temperature as defined in the paper by Sedagat & Leon 2018.

# volumetric thermal expansion coefficient beta_gas [1/K] = beta_gas
beta = 0.003659564-1.33828E-5*theta_gas+4.8053E-8*theta_gas**2-1.5377E-10*theta_gas**3+3.086E-13*theta_gas**4

# specific heat at constant pressure [J/kg.K] = c_p_gas
c_p = 1030.5 - 0.19975 * T_gas + 3.9734e-4 * T_gas**2

# mass density [kg/m3] = rho_gas
rho = 360.77819 * T_gas**(-1.00336)

# kinematic viscosity [m2/s] = nu_gas
nu = 1.32e-5 + 9.5e-8 * theta_gas

# dynamic viscosity [kg/(m.s)] = eta_gas
eta = nu * rho

# thermal conductivity [W/(m.K)] = k_air
k = 2.42e-2 + 7.2e-5 * theta_gas

# thermal diffusivity [m2/s] = alpha_gas
alpha = 9.1018e-11 * T_gas**2 + 8.8197e-8 * T_gas - 1.0654e-5

# Prandtl number = Pr_gas
Pr = 0.715 - 2.5e-4 * theta_gas
  

References

Merkebu Z. Degefa, Matti Lehtonen, Robert J. Millar: 'Comparison of air-gap thermal models for MV power cables inside unfilled conduit', IEEE, 2012
Ali Sedaghat, Haowei Lu, Abdullah Bokhari, Francisco de León: 'Enhanced Thermal Model of Power Cables Installed in Ducts for Ampacity Calculations', IEEE, 2018