How the formula for $T'_4$ in IEC was developed

In this article we explain the origins of the IEC 60287 formula for $T'_4$ ($T_{4i}$), its limitations and give an outlook for development of a new method.

Part of the article is a modified version of chapter 9.6.4.1 of the book Rating of Electric Power Cables by G. Anders, published in 1997. The author showed how a general expression for the thermal resistance between cable and duct has been simplified based on the findings of previous publications until he arrived at the equation used by Neher and McGrath (1957) and which was included in a modified form in IEC 287-2-1 (1994).

We marked the simplifications made, which are the reasons for the limited applicability of the formula in the IEC standard as compared to a more general analytic calculation, in bold text.

The general expression considers conduction between cable and duct. Note that for cables in air at ambient pressure, conduction is only relevant in cases with small gap between cable and duct/pipe. However, in case of gas pressure cables conduction becomes more relevant and in case of oil-filled pipe-type cables conduction is a major contributor while radiation becomes negligible.

Note that when we talk about convection only natural convection is meant, no forced convection (e.g. ventilation).

Thermal Resistance Between Cable and Duct/Pipe $T'_{4}$

The development of a rigorous equation for this thermal resistance is quite involved, and the expression depends on the cable surface temperature. This equation can be solved by computer implementation but was considered not suitable for standardization.

An assumption is made that the inner surface of the duct or pipe is isothermal. Considering the outside surface of the jacket under steady-state conditions, the conduction heat flux from its inner surface is equal to the heat loss through free convection and thermal radiation. The energy balance equation takes the form

Convection

Free convection heat transfer in the annular space between long, horizontal concentric cylinders has been considered by Raithby and Hollands (1975). The heat transfer rate per unit length of the duct may be obtained as:

The value of $A_s$ reflects the series connection of two thermal resistances corresponding to the outer surface of the cable and the inner surface of the duct wall and is equal to (Incropera and de Witt, 1990).

The convective heat transfer coefficient is obtained by assuming that the cable and the conduit are concentric cylinders. This assumption is almost always violated in practical installations since cables are usually placed at the bottom of the conduit. However, since the thermal resistance of the gas/liquid surrounding a cable in duct or pipe constitutes a small portion of the total external thermal resistance of the cable, the proposed simplifications have a very small effect on the accuracy of the final results. The heat transfer coefficient represents in this case the effective thermal conductivity of the fluid (gas or oil). The empirical correlation is given by Raithby and Hollands (1985):

When the formula is used for a group of cables in a conduit, $D_e^*$ becomes the equivalent diameter of the group as follows:

• two cables:     $D_e^* = 1.65$ times the outside diameter of one cable, m
• three cables:   $D_e^* = 2.15$ times the outside diameter of one cable, m
• four cables:    $D_e^* = 2.50$ times the outside diameter of one cable, m

Equation (9.44) may be used in the range $10^2 \leq \text{Ra} \leq 10^7$. For Ra < 100, $h_s = 1/\rho$. Denoting by $D_f^*$ the factor representing the cable-duct geometry and substituting (9.43) and (9.44) into (9.42), we obtain

with

The medium between the cable and the enclosure wall is usually air at atmospheric pressure. For this or for other gases and fluids, the material properties can be obtained from the tables found in most of the books on heat transfer, for example, in Incropera and de Witt.

Radiation

The net radiation heat transfer rate between the cable and the inside surface of the conduit is based on the radiative exchange between two surfaces:

This equation is applicable to the case of gas-filled ducts or pipes. The thermal properties which occur in the above equations are temperature dependent.

The thermal resistance between the cable surface and the inner surface of the duct (pipe) is obtained by dividing the temperature drop across the duct (pipe) gap by the total heat emanating from the cable surface. Therefore, by equation (9.41),

Conduction

This chapter is an extension to the article in the book

Equations for conduction as used later in equations (9.54) and (9.55)

All three modes of heat transfer — conduction, convection, and radiation — can be active simultaneously in the geometry of concentric cylinders. Their relative contributions depend strongly on the fluid type and pressure. Conduction in particular is never truly absent in any fluid, though its relative importance varies greatly.

Air at Atmospheric Pressure

All three modes contribute, but their relative weight depends on gap size and temperature difference. At atmospheric pressure, a practical analysis generally requires accounting for all three modes. Conduction dominates only at very small gaps; otherwise, natural convection and radiation are the main contributors.

A common engineering approach for gas-filled annuli is the effective conductivity method, where convection is accounted for by multiplying the fluid's molecular conductivity by the Nusselt number (when Nu = 1, the system is purely conductive).

• Conduction: Always present. In the thin boundary layers adjacent to each cylinder wall, conduction is the primary mechanism. For very narrow gaps or very small temperature differences (low Rayleigh numbers), conduction can dominate across the entire gap. In this regime, the effective thermal conductivity of the air gap approaches the molecular conductivity of air.
• Convection: For moderate-to-large gaps or higher temperature differences, buoyancy-driven flow develops and becomes the dominant heat transport mechanism through the bulk of the fluid. The transition from conduction-dominated to convection-dominated is governed by the Rayleigh number.
• Radiation: Since air is optically transparent to thermal radiation, surface-to-surface radiation occurs directly between the cylinder walls and can be significant — particularly at elevated surface temperatures. At near-ambient temperatures, it is secondary but not negligible.

Air or other gases at elevated pressure (5–15 bar)

Higher pressure dramatically changes the balance. At elevated pressures, convection is strongly dominant, radiation is roughly the same as at 1 atm, and conduction becomes relatively minor — though it still operates in the boundary layers.

• Convection — greatly enhanced: For an ideal gas, density scales as $\rho \propto P$, while viscosity and thermal conductivity are nearly pressure-independent. The Rayleigh number scales as $\text{Ra} \propto \rho^2 \propto P^2$ meaning at 5–15 bar the convective intensity is 25–225× stronger than at 1 bar for the same geometry and temperature difference.
• Conduction: Molecular thermal conductivity of gases is essentially pressure-independent (ideal gas behavior), so the absolute conductive contribution does not change — but it becomes relatively less important compared to the now-dominant convection.
• Radiation: Radiation between the surfaces is largely unaffected by gas pressure (for non-radiating gases like air, N₂, Ar, which are optically thin). Radiation therefore remains roughly the same as at 1 atm.

Liquid oil-filled (or water-filled)

The situation changes fundamentally with a liquid such as oil or water: For oil, conduction plays a central role (unlike with gases), convection depends on viscosity, and radiation is effectively absent as a direct inter-surface mechanism.

• Conduction — highly significant: Oil has thermal conductivity roughly 5–7 × higher than air (approx. 0.13–0.17 W/(m.K) vs. 0.026 W/(m.K) for air), and its high viscosity (especially at lower temperatures) can suppress buoyancy-driven convection, leaving conduction as the primary bulk heat transfer mechanism.
• Convection: Whether natural convection is important depends strongly on oil viscosity and temperature. At higher temperatures viscosity drops and convection becomes more active. For high-viscosity oils, the Rayleigh number may be low enough that conduction dominates across the gap.
• Radiation: Oil is essentially opaque to thermal infrared radiation — it absorbs radiation within a very short depth. Therefore, there is no meaningful surface-to-surface radiative heat transfer through the oil bulk. At typical oil operating temperatures (below approx. 150–200 °C), radiation is generally negligible.

Mode	Air @ 1 atm	Gas @ 5–15 bar	Oil
Conduction	Dominant at small gaps; always present in boundary layer	Present but relatively minor	Major contributor
Convection	Significant (moderate–large gaps)	Strongly dominant	Significant only for low-viscosity oil at higher T
Radiation	Significant (especially at high T)	Similar to 1 atm	Negligible

Simplifications

When the first attempts to determine the thermal resistance between the cable and the duct wall were made (Whitehead and Hutchings, 1939; Buller and Neher, 1950), finding the solution of equation (9.49) appeared to be a formidable task. The value of $T'_4$ depends on the unknown cable surface and inner wall temperatures, and the material parameters are also dependent on the mean temperature of the medium. Several iterations are required to compute $T'_4$. In the absence of digital computers, it became apparent that several simplifications were required. The approach proposed by Buller and Neher (1950) provided such simplifications. The same approach was later adopted by Neher and McGrath (1957), and became the basis for North American and IEC standards.

The first approximation concerns the representation of cable/conduit geometry. The effective diameter given in equation (9.46) is approximated by

Next, the following assumptions were made.

1. In the case of inert gas, the physical properties of the medium were assumed to be substantially independent of temperature over the working range, but it was noted that the density is a direct function of the pressure. Thus, if $P$ represents the pressure (in atmospheres), with appropriate numerical values for the material constants, we obtain from equations (9.45) and (9.50)
   $\frac{W_{conv}}{\Delta\theta_{sw}}(gas) = 4.744 \cdot \frac{\left(D_e^*\right)^{3/4}}{1.39 + \frac{D_e^*}{D_d^*}} \cdot P^{1/2} \cdot \Delta\theta_{sw}^{1/4}$ (9.51)
   where $\Delta\theta_{sw} = \theta_s - \theta_w$.
2. When mineral oil is employed as the pipe-filling medium, it was assumed that the physical constants are substantially independent of pressure with the exception of viscosity which, for the type of oil commonly employed, was taken as varying inversely as the cube of the temperature. This leads to the following form of equation (9.45):
   $\frac{W_{conv}}{\Delta\theta_{sw}}(oil) = 2.733 \cdot \frac{\left(D_e^*\right)^{3/4}}{1.39 + \frac{D_e^*}{D_d^*}} \cdot \theta_m^{3/4} \cdot \Delta\theta_{sw}^{1/4}$ (9.52)
   where $\theta_m$ is the mean oil temperature in °C.
3. The radiation component with gas as the medium is assumed to be given with sufficient accuracy by the following expression:
   $\frac{W_{rad}}{\Delta \theta_{sw}}(gas) = 13.21 \cdot D_e^* \cdot \varepsilon_s \cdot (1 + 0.0167 \cdot \theta_m)$ (9.53)
   where $\varepsilon_s$ is the emissivity of the cable outside surface. The radiation component is ignored for pipes filled with oil.

Substituting (9.47) and (9.51)–(9.53) into equation (9.49) with appropriate values of the thermal resistivities, we obtain

Linearization

Next, Buller and Neher (1950) proposed to linearize equations (9.54) and (9.55). First, they assumed that the second term in both equations and the radiation term in equation (9.54) are constant. Considering equation (9.54), the conduction term constitutes about 14% of the total in the case of a typical cable in duct installation, and about 8% for a typical gas-filled pipe-type installation at 200 psi. The corresponding values for the radiation term are 63 and 43%, respectively. Normal variation of $D_e/D_d$ may produce considerable variation in the conduction term, but the overall effect is small because conduction is such a small part of the total heat flow. In addition, the variation of this ratio has opposite effects on the convection and conduction terms. Buller and Neher concluded that a minimum error should therefore prevail when the conduction term is treated as a constant if the denominator of the convection term is also treated as a constant.

Variation of $\theta_m$ can affect the radiation term by as much as 20% over a sufficiently wide operating range; however, when calculating a cable rating with fixed conductor temperature on the order of 70–80°C, the range of this variable is very small, and an inaccuracy on the order of 3–5% may be expected.

In the case of equation (9.55), the conduction term constitutes about 24% of the total for a typical oil-filled pipe installation. Variation of $\theta_m$ is more important than in the case of gas-filled pipe-type cables, but is still within tolerable limits.

Under the above assumptions, equations (9.54) and (9.55) can be rewritten as (Neher and McGrath, 1957)

The constants $a$, $b$, and $c$ in equation (9.56) were established empirically from data in Buller and Neher (1950) for cables in pipe, and from data in Greebler and Barnett (1950) for cables in fiber and transite ducts. The constants in equation (9.57) were also determined empirically. All these values are given in following table.

Restriction of applicable range

By further restricting the value of $\Delta \theta_{sw}$ to 20°C for cables in ducts and to 10°C for gas-filled pipe-type cables, and restricting the range of $D_e$ to 25–100 mm for cables in ducts and to 75–125 mm for pipes, equation (9.56) can be reduced to its final form.

Final form of equation $T'_4$

in which the values of the constants $U$, $V$, and $Y$ depend on the installation and are given in following table. In newer editions of the IEC 60287-2-1 standard the types of installation were extended with cables in water-filled polymer ducts.

Summary

The detailed physics inside a duct (natural convection, radiation, conduction, temperature-dependent air properties, geometry effects) leads to nonlinear heat-transfer behavior. IEC avoids a full first-principles calculation and instead represents the inside-the-duct contribution as a compact empirical/semi-empirical expression with tabulated constants.

In short: IEC turns 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.

Main simplifications embedded in the IEC approach

• Lumped mechanisms: convection, radiation, and conduction inside the duct are not computed separately; they are combined into one effective resistance term $T'_4$.
• Geometry reduction: real arrangements (eccentricity, multiple cables, spacing) are reduced to an equivalent-diameter representation.
• Linearized: the denominator depends linearly on both $D_e$ and $\theta_m$ via $\left( V+Y\theta_m \right) D_e$ whereas in reality:
  • Radiation scales approximately with a $T^4$ term (Stefan–Boltzmann)
  • Natural convection depends on dimensionless groups (Rayleigh/Nusselt), which are nonlinear functions of temperature and properties.
  IEC’s expression can be seen as a practical linearized correction that approximates these nonlinear effects which provides a stable, easy-to-use formula.
• No explicit air property evaluation: temperature-dependent air properties ($\rho, \nu, k, Pr,\dots$) do not appear explicitly in the $T'_4$ expression; their effect is implicitly absorbed into the fitted constants and the temperature term.
• Uniform temperatures: cable surface and duct wall are represented by single temperatures (no circumferential/3D field).
• Steady-state only: transient heating/cooling (load cycles) is outside the scope.

Restrictions and applicable range

1. The approach is valid only within a defined range of cable and duct dimensions; outside that range, IEC does not provide recommendations.
2. Using parameters derived empirically from data the scope of installation types and the constants $U, V, Y$ are limited to those presented in the standard. Extrapolating to other geometries/materials is not justified.
3. IEC 60287-2-1 is solely applicable to steady-state operation which corresponds to a continuous constant current (100% load factor) producing the asymptotic maximum conductor temperature under constant ambient conditions.

Conclusion and outlook

The restrictions of the Neher & McGrath and IEC 60287 approach to calculate $T'_4$ are key motivation for developing a more accurate but still easy-to-use method using a more explicit model considering convection and radiation.

Since gas-pressure cables are phasing out and oil-filled pipe-type cables are unknown outside of the USA this method will be limited to cables in ambient air under atmospheric pressure. Transient calculation will also not be focus of this method.

New developments like the pressurized air-cables (Hivoduct) can be covered by using the calculation method for gas-insulated lines (GIL) as described in CIGRE TB 218. Cableizer has developed the calculation of GIL already 2018 and later extended the method to cover pressurized dry air (see the blog post and read the publication from 2023).