Documentation

Equivalent diameter between screen and sheath.

Cable with double layer armour applying CIGRE TB 880 Guidance Point 15

Cable with double layer armour applying CIGRE TB 880 Guidance Point 15

The equation for duplex/triplex and pipe-type cables is according to CIGRE TB 880 Guidance Point 41.

Diameter of the backfill area.

Distance c to calculate the resistance of multi-layer backfill.

Distance d to calculate the resistance of multi-layer backfill.

The following conductor diameters are typical values as a function of the conductor cross-section.

• For IEC sizes, the diameters are taken from IEC 60228 Ed. 3.0 and from HD 603 S2 Part 7 Section A plus HD 620 S2 Part 10 Section F.
• For AWG/kcmil, the diameters are taken from UL 1581 Table 20.1 for solid conductor up to 12 AWG and table 20.3 for round stranded compressed conductors plus ASTM B8-11 for sizes larger than 2000 kcmil.

The user can choose an arbitrary value not limited to the values in the choices below, but a warning is indicated if the conductor diameter seem unreasonably small.

For sector-shaped multi-core cables, the conductor diameter is defined as the diameter of the circle circumscribing the sector-shaped conductors. Based on that value, an equivalent diameter of the conductor $d_x$ is calculated.

For calculation of the cyclic and emergency current rating of cables acc. IEC 60853, the multi-core cable is replaced by an equivalent single-core construction dissipating the same total conductor losses. The equivalent single-core conductor has a diameter as calculated by the formula below where $D_i$ is the value of diameter over insulation and $T_1$ the thermal resistance for the multi-core cable.

The actual conductors are considered to be completely inside the diameter of the equivalent single-core conductor, the remainder of the equivalent conductor being occupied by insulation.

This is the diameter below the insulation.

The maximal diameter of wire in the conductor of flexible types 5 and 6 is given in the IEC 60228 Ed.3.0, more values were taken from ASTM B8-04.

The values for stranded conductors were calculated based on the given minimum number of wires in the conductor $n_{cw}$.

This is the diameter around a source where the soil has dried out. The equation originates from the book 'Kabel und Leitungen für Starkstrom' by L. Heinhold, 5th edition 1999 (in German).

Equivalent diameter between screen/sheath and armour. It becomes the equivalent diameter between screen and sheath when no armour is present.

Definition according to CIGRE TB 531 chap. 4.2.3.3

The equivalent depth of the earth return path of a 50 Hz system with an electric soil resistivity of 100 $\Omega$.m is about 930 m.

For multiple cables twisted or bound together, a combined diameter circumscribing all cables can be calculated. The resulting value is also approximately correct for three loose cables in touching trefoil formation.

The equivalent diameter over the conductors for three core cables without a filler (not round), can be calculated by dividing the circumference with $\pi$. In case of three-core cables, they are called triplex; and in case of two-core cables we call them duplex. The equation is according to CIGRE TB 880 Guidance Point 48 while the circumference of such a cable is according to CIGRE TB 880 Guidance Point 41.

For multiple unequal round objects, possibly non-touching, finding the smallest circle that encompasses all other circles is not easy and we consider following approximation as presented in stackoverflow.com

1. Take the average of all your centers of the circles call this point X
2. Let R1 be the maximum distance from X to a circle center
3. Let R2 be the maximum radius of the circles
4. Then all the circles must fall inside the circle centered at X with radius R1+R2

For spaced ducts inside the air-filled pipe, the above method often results in a circle larger than the inner diameter of the pipe. So instead, we are using another approximation by summation of squared areas of all circular objects inside the pipe to get $A_{eq}$ and then $D_{eq} = \sqrt{A_{eq} / \pi}$.

Outer diameter of the pipe, over the outer protective and insulating cover.

For cables, heat sources and PAC/GIL, we consider $D_{ext}$ to be the outer diameter $D_o$ in meters.

This is the diameter of a round armour wire. In case of flat wires, an equivalent diameter of a round wire which gives the same cross-section with the flat is calculated.

Diameter over the protective jacket over the insulation of a fiber optic cable.

Spacing from centre of hottest cable for a single line source representing the heating effect of all other cables in a group.

Outer diameter of the insulation around the pipe in the center of a heat source, e.g. a district heat pipe.

Diameter over the protective jacket over the insulation of a heat source, e.g. a district heat pipe.

This is the diameter over the insulation including insulation screen, per phase. This value is used for thermal calculations.

This is used for transient calculation acc. IEC 60853 when considering an equivalent conductor for multi-core cables.

The imaginary layer of soil is used to model a non-isothermal earth surface.

Samson Semenovich Kutateladze introduced the fictitious layer method in his book 'Osnovy Teorii Teploobmena', Mashgiz, Moscow-Leningrad, 1962 (translated by Scripta Technica Inc. as 'Fundamentals of Heat Transfer', Edward Arnold (Publishers) Inc., London, 1963. It is also called the additional wall method, used to model a non-isothermal earth surface with the help of a fictitious earth layer with thickness $d_{im}$. The cable image line source is then placed at distance $L_{cm}+d_{im}$ above the fictitious layer.

Unlike others, we do not just use a constant but actually calculate in an iterative process the thickness of this imaginary layer of soil depending on the losses and ambient conditions. The heat generated from all sources within a certain range near the center $(-3≤x≤3)$ is used to calculate the surface temperature $T_{surf}$ above the center axis $(x=0)$. This temperature is used to calculate the Nusselt number $Nu_{L}$ and the thermal conductivity of the air above ground $k_{gas}$ both used to determine the heat transfer coefficient $h_{tr}$ which is needed to calculate the imaginary layer of soil $d_{im}$. The basics of the method we used was presented in the paper 'Calculation of cable thermal rating considering non-isothermal earth surface' from 2014 by S. Purushothaman, F. de León, and M. Terracciano.

This is the diameter below the insulation screen, which is an important parameter for dimensioning the accessories.

This is the diameter over the insulation screen, per cable.

The twisting of three cores are assumed to occur at the mechanical neutral line which is the location where a centre circle divides the cores into 2 equal large parts.

Diameter in meters used to calculate the thermal resistance of cables, ducts or heat source in air or for calculations where metric dimension is required.

Distances from the centre of the p^th cable to the centre of the reflection of cable k in the ground-air surface.

Distance from the centre of the p^th cable to the centre of cable k.

The IEC method to calculate the temperature at any point in the soil uses the exact analytical solution for a line source in an infinite environment in conjunction with Kennelly's hypothesis to account for the fact that a cable environment is semi-infinite in nature.

However, in reality the entire cable acts as the heat source which deforms the isotherms around the cable, especially with proximity to the earth surface. To consider this physical reality, it is assumed that the cable surface forms an isotherm and that one can imagine that the temperature field is built up by a line source which runs parallel to the cable axis at a distance $d_{psc}$ (eccentricity of the cable) closer to the earth surface according to the book 'Kabel und Leitungen für Starkstrom' by L. Heinhold, 5th edition 1999 (in German).

This yields to a tiny improvement in the ‰ range. Point source correction is optional in Cableizer, and not activated by default.

The U-value of a pipe is often defined as above although the definition may differ from one engineering discipline to another. Insulation manufacturers calculate the U-value at the outer surface of the steel wall, excluding any contribution from internal convection, external convection, and conduction through steel. Pipeline designers typically use the inner diameter of the pipe to define the reference diameter and may exclude any contribution from convection.

The definition of the U-value also differs from one simulation tool to another. In the one-dimensional dynamic multiphase flow simulator OLGA the U-value is based on the inner diameter of the steel wall; user-specified U-values must include the contribution from internal convection. In the one-dimensional steady-state multiphase flow simulator PIPESIM, the U-value is based on the outer diameter of the steel wall; the inside film coefficient can be included in the specified U-value or calculated separately and added to the U-value.

For cables, heat sources and PAC/GIL, we consider $D_{ref}$ to be the diameter $D_{wall}$, same as the insulation manufacturers do.

Equivalent diameter between screen and sheath.

This is the diameter below screen, sometimes called $D_1$.

For multi-core cables with separate screen and common sheath, this is the diameter below sheath, sometimes called $D_shb$.

Outer diameter of soil layer providing a thermal resistance equivalent to $h_{soil}$.

Outer diameter of the steel pipe excluding outer protective cover.

For cables, heat sources and PAC/GIL, we consider $D_{wall}$ to be the diameter below the outer sheath, the insulation over the pipe or the protective cover respectively.

This equation somewhat underestimates the equivalent diameter of sector-shaped conductors that are not solid.

An alternative approach for multi-core cables with sector-shaped conductors would be to set the equivalent diameter equal to the nominal diameter of a round conductor with the same cross-sectional area. This would still only be an approximation, since the exact degree of compaction of sector-shaped conductors can vary and depends on manufacture. Also, nominal diameters are not available for all cross-sectional areas of conductors and we extrapolated the missing values.

For sector-shaped conductors, CIGRE TB 880 Case study 10 recommends to use the smallest typical conductor diameter as given in IEC 60228 or analogue American standard for AWG/kcmil conductors.

This is the diameter around a source at which the effect of the loss factor commences. Inside the circle of diameter $D_x$, the temperature changes according to the peak value of the losses. Outside this circle, it changes with the average losses. The characteristic diameter is a function of the diffusivity of the medium and the length of the loss cycle. In the majority of cases, the soil diffusivity $\delta_{soil}$ will not be known. In these cases, a value of 0.5*10^-6 m$^2$/s can be used. This value is based on a soil thermal resistivity of 1.0 K.m/W and a soil moisture content of about 7% of dry weight.

The calculation of the characteristic (or fictitious) diameter for sinusoidal load is based on the IEEE paper 'Ampacity calculation for deeply installed cables' by E. Dorison et al, dated 2010. Three different methods can be chosen:

1. Neher McGrath: Neher empirically evaluated constants to best fit the temperature rises calculated over a range of cable sizes. Based on the evaluations by Neher, the characteristic diameter becomes 212 mm for a transient load period of 24 hours (fixed value in Cableizer) and a soil diffusivity of 0.5*10^-6 m$^2$/s.
2. Heinhold: The method by Heinhold is from his book 'Kabel und Leitungen für Starkstrom' by L. Heinhold, 5th edition 1999 (in German). Heinhold defined three equations to calculate the characteristic diameter depending on the type of load curve (sinusoidal, rectilinear, or median).
3. Dorison: The method by Dorison uses the modified Bessel function K of order 0: k0() and the modified Bessel function of first order: k1(). The IEC standard 60853 uses an approximation with an exponential integral valid for an infinite thin wire whereas the formulas presented in the paper the cable diameter influences the diameter of the area affected by load variations. This is particularly important for cables in tunnels, where the tunnel diameter replaces the diameter of the cable.

The characteristic diameter $D_{x_w}$ is the location at which the effect of the weekly loss factor commences.

For more information, refer to the characteristic diameter $D_x$.

The characteristic diameter $D_{x_y}$ is the location at which the effect of the yearly loss factor commences.

For more information, refer to the characteristic diameter $D_x$.

Thickness of tape screen or flat wires or diameter of round wires used to calculate the screening factor.

The equivalent thickness of armour is based on the combined thickness of the first and second armour layer divided by the mean diameter of the first and second armour layer diameter. The error to a separately calculated value for each layer is not significant since the electrical resistance is calculated separately.

Thickness of metallic component, e.g. screen, sheath or armour.

The soil thermal diffusivity is the ratio of the thermal conductivity to the volumetric heat capacity. It is an indicator of the rate of at which a temperature change will be transmitted through the soil by conduction. When the thermal diffusivity is high, temperature changes are transmitted rapidly through the soil. Logically, soil thermal diffusivity is influenced by all the factors which influence thermal conductivity and heat capacity. Thermal diffusivity is somewhat less sensitive to soil water content than are thermal conductivity and volumetric heat capacity. The thermal diffusivity is a particularly useful parameter to aid in understanding and modeling soil temperatures.

For calculation of cyclic rating factors it is not necessary to accurately know the value of thermal diffusivity and it is generally satisfactory to use the tabulated functions given in the standard IEC 60853 which are based on the value of 5 · 10^-7 m$^2$/s, and correspond roughly to a soil having a thermal resistivity of 1 K.m/W, and a moisture content of around 7% of dry weight.

There are currently five options to choose from to define the thermal diffusivity:

1. If no data are available on the soil the IEC 60853-2 recommends the value 5 · 10^-7 m$^2$/s.
2. If only the thermal resistivity is known the IEC 60853-2 provides values in table D1 from which we have created a best fit power function.
3. If dry density, moisture content in % of dry weight and thermal resistivity are measured the IEC 60853-2 provides equation D-1.
4. Diffusivity is related to density, heat capacity and thermal conductivity of the material.
5. The paper 'Estimation of soil thermal parameters from surface temperature of underground cables and prediction of cable rating' by H.J. Li, 2005 provides an empirical formula based on the publication 'The Transient Temperature Rise of Buried Cable Systems' by J.H. Neher, 1964.
6. The value can also be entered manually. Only the positive real number of the scientific notation shall be entered and the exponent 10^-7 is added automatically.

Correction to consider larger diameter over corrugated sheath as by applying the depth of corrugation $H_{sh}$.
For example, to calculate CIGRE TB 880 Case study 5 a value of 0.25 mm needs to be set.

The gross heat of combustion value (or energy value or calorific value) is used to quantify the energy content of a cable in case of a fire.

The standard gross heat of combustion is the energy liberated when a substance undergoes complete combustion with oxygen at standard conditions (25°C and 1 bar). In thermodynamical terms it is the negative of the enthalpy change for the combustion reaction. The chemical reaction is typically a hydrocarbon or other organic molecule reacting with oxygen to form carbon dioxide and water and release heat. It can be calculated from the standard enthalpy of formation ($\Delta$Hf°) of the substances involved in the reaction, given as tabulated values. The gross heat of combustion is measured in a calibrated oxygen bomb calorimeter. It is usually utilized to quantify the performance of a fuel in a combustion system.

Sources:

• 'Heats of combustion of high temperature polymers' by Richard N. Walters (2000)
• 'Experimental investigation on the heat of combustion for solid plastic waste mixtures' by Liviu Costiuc (2015)
• Prof. Dr. M. Häberlein
• atozplastics.com (used for PS, PIB)
• 'Heats of combustion of oil shale, bitumen, and their mixtures' by V. M. Abbasov (2008) (used for Bitumen)
• 'Determination of calorific value, sulphur and heavy metal concentration in waste oils' by Thinguri Mwangi Thomas (1998) (used for mineral oil wth 9.093 kcal/g)
• 'Formulation and Burning Behaviour of Fire Retardant Polyisoprene Rubbers' by David John Kind (2001) (used for IIR, NR, CR, CSM)
• CIGRE TB 720 'Fire issues for insulated cables in air' (used for PP, EPR, HFFR, Paper)
• 'Calorific values for different raw materials' by C.F. Nielsen (used for Jute)

Note, not all values listed below can be selected in the cable editor.

Fictitious increase of ambient temperature to account for the ventilation in a tunnel.

Temperature rise by crossing heat sources, at the hottest point along the cable.

Temperature rise of the conductor(s) of the rated cable, due to crossing heat source h, at the hottest point in the cable route.

This is the temperature rise of the air inside an air-filled trough or channel.

This is the conductor temperature rise above the ambient temperature respectively above the surface temperature of cable or duct in tunnel and in trough (method by Anders 2010) and the surface temperature of the cable in riser.

For DC, the dielectric losses $W_d$ are zero and the corresponding terms disappear.

Permissible steady-state temperature rise of the conductor due to joule losses.

This is the transient temperature rise of conductor above ambient temperature.

After calculating separately the two partial transients and the conductor to cable surface attainment factor the total transient rise above ambient temperature is obtained

• for buried configurations by simple addition of the cable and modified environmental partial transients;
• for cables in air by modification of the cable partial transient.

This is the corrected transient conductor temperature rise above ambient temperature caused by the change in conductor resistance with temperature during the transient period.

$\theta_{c_0}$ is the conductor temperature at the start of the transient.

This is the conductor steady-state temperature rise above ambient, as defined by Amendment 1 (2008) of the IEC 60853-2.

IEC 60853-2 Ed.1 defines this to be the transient temperature rise of the conductor above ambient temperature without correction for variation in conductor loss, and based on the conductor resistance at the end of the transient.

This is the transient temperature rise of conductor above outer surface caused by ohmic losses. The transient response of a cable circuit to a step function of load current, considered in isolation, that is with the right-hand pair of terminals of the thermal circuit short-cuircuited, is found with the following formula calculating the transient temperature rise of conductor above outer surface. The parameter $W_c$ is the power loss per unit length in a conductor or an equivalent conductor based on the maximum conductor temperature attained. The power loss is assumed to be constant during the transient.

This is the temperature difference between the conductor and the surface.

This is the conductor temperature rise above the ambient temperature caused by dielectric losses respectively above the surface temperature of cable or duct in tunnel and above the surface temperature of the cable in riser. The temperature rise caused by dielectric losses is relevant for high-voltage cables because the losses are strongly voltage dependent. If the application of system voltage occurs, then an additional transient temperature rise due to the dielectric loss has to be calculated.

• It shall be assumed that the dielectric power factor is constant and equal to an appropriate value in the relevant temperature range.
• If the starting conductor temperature is different from the ambient temperature, then it may be assumed that the dielectric losses are included in the temperature rise obtained at the onset of the transient conditions.
• For cables at voltages up to and including 275 kV, it is sufficient to assume that half the dielectric loss is produced at the conductor and the other half at the screen or sheath (therefore the ratio of losses is equal to 2).The apportioning factor $p_i$ is the same as that used when calculating transient due to joule losses.
• For cables at voltages higher than 275 kV, the dielectric loss can be an important fraction of the total loss (i.e. paper insulated cables) and the factor used to apportion the thermal capacitance of the insulation (dielectric) to the conductor and sheath is replaced.The fraction of the dielectric loss starting from the conductor is still reckoned as 1/2 and therefore the ratio of losses is equal to 2.

The dielectric losses are applied when the cable is energized.

$\Delta\theta_dj$ is a factor having the dimensions of temperature difference, accounts for dielectric losses. It is not used in our calculations but used in CIGRE TB 880.

The transient response of the cable environment is calculated by an exponential integral formula.

Note: The parameters $D_o$ and $L_{cm}$ are in meters.

In the case of HVDC cables, there are two limitations that must be observed. One is the maximum conductor temperature, similarly as in the AC case and the other is the limitation of the electric field strength in the insulation. The latter can be shown to be equivalent to limiting the temperature drop across the insulation.

In an HVDC cable, the electrical field is determined by the resistive properties (governed by the specific electric resistivity $\rho_i$) of the insulation material, contrary to a HVAC cable where this is determined by capacitive properties (governed by the dielectric permittivity $\epsilon_i$). The specific thermal resistivity $\rho_i$ is significantly dependent on temperature, and because this temperature is not constant throughout the insulation material, the electrical field is not constant and changes with changing cable loading situations. In order to ensure that the electrical fields in the insulation material remain within limits, the temperature difference over the insulation material is limited. This limit may be more stringent than the limit on the maximum conductor operating temperature with typical values between 10 and 15 K.

It is noted that the maximum temperature difference over the insulation material of a specific cable is an important parameter during type testing, and should not be surpassed during cable operation. To take this additional limitation into account, the CIGRE TB 880 Guidance Point 9 may be followed.

Cableizer provides a method where the losses generated in the dielectric are considered and the resulting rating will ensure the permissible temperature drop across the insulation.

This is the temperature rise of cable p due to the total power $W_{tot}$ dissipated in cable k.

When both cables p and k are embedded in the same concrete (backfill), the temperature rise by buried object k is adjusted with the difference between the thermal resistivity of the backfill material $\rho_b$ and the thermal resistivity of the soil $\rho_4$: $$\Delta\theta_{kp}=\Delta\theta_b+(\Delta\theta_{kp}-\Delta\theta_b)*\rho_b/\rho_4$$ $\Delta\theta_b$ is the temperature rise by buried object k on the top border of the backfill. With this method, the difference in temperature rise inside the backfill due to the difference in thermal resistivity is taken into consideration.

This is the temperature rise of cable p due to the total power $W_{tot}$ dissipated in cable k.

Maximum permissible conductor temperature rise above ambient.

This is the temperature rise above ambient at the surface of the p^th cable, whose rating is being determined, caused by the power dissipated by the other (q - 1) cables in the group. The term $\Delta\theta_{pp}$ is excluded from the summation. The influence of touching cables or ducts is exluded from the summation.

For cables in air, $\Delta\theta_p$ is 0.

This is the temperature rise above ambient at the surface of the p^th cable, whose rating is being determined, caused by the power dissipated by the other (q - 1) cables in the group. The term $\Delta\theta_{pp}$ is excluded from the summation. The influence of touching cables or ducts is exluded from the summation.

For cables in air, $\Delta\theta_p$ is 0.

This is the conductor temperature rise above ambient temperature.

$\Delta\theta_R(t)$: Conductor temperature rise above ambient at time t after application of current $I_{c2}$, neglecting variation in conductor resistance.
$\Delta\theta_R(i)$: Conductor temperature rise above ambient, when the magnitude for the step function current is the sustained rated current, and $i$ is expressed in hours.
$\Delta\theta_R(\infty)$: Conductor temperature rise above ambient in the steady state, i.e. the standard maximum permissible temperature rise.

Value of the conductor temperature rise above ambient in the steady-state due to Joule losses, i.e. the standard maximum permissible temperature rise.

It is the difference between the surface temperature of a cable or duct in air and the ambient temperature. This equation is solved by iteration with an initial value for $\Delta\theta_s$ of 16°C.

Peak cyclic temperature rise of the cable surface.

Temperature rise by solar radiation.

Temperature rise by crossing heat sources, at the point z along the cable.

This is the temperature rise of the boundary between the dry and moist zones above the ambient temperature of the soil.

The method is based on a simple two-zone approximate physical model of the soil where the zone adjacent to the cable is dried out whilst the other zone retains the site's thermal resistivity, the zone boundary being on isotherm. This method is considered to be appropriate for those applications in which soil behaviour is considered in simple terms only. The method should be applied to a single isolated cable or circuit only, laid at conventional depths. Installations of more than one circuit shall consider necessary spacing between circuits.

When the permissible current rating is being calculated under conditions of partial drying out of the soil, also calculate a rating for conditions where drying out of the soil does not occur and take the lower rating.

This is the temperature rise of the boundary between the dry and moist zones above the ambient temperature of the soil according to VDE 0276-1000 with continuous load ($LF$ = 1).

This value is fixed to 15 K in VDE 0276-1000 but can be adjusted in Cableizer.

Incremental heat generated due to change of conductor resistance.

Length of the interval used in the calculations.

It has to be verified that $\gamma\times\Delta z<\epsilon$ where $\epsilon$ is a small value, typically 0.01 acc. to IEC 60287-3-3.

Inner diameter of the pipe in the center of a heat source inside which the fluid is flowing, e.g. water in a district heat pipe.

Inner diameter of large enclosing pipe with ducts inside such as it is used for HDD.

Inner diameter of steel pipe of pipe-type cables.

For shapes such as squares or rectangular ducts and tunnels where the height and width are comparable, the characteristic dimension for internal-flow situations is taken to be the hydraulic diameter, defined as $D=4*A/P$, where A is the cross-sectional area, and P is the wetted perimeter. The wetted perimeter for a tunnel is the total perimeter of all inner tunnel walls that are in contact with the flow.

For circular tunnels, the diameter is an input value.

If the cable is not in a duct, $Do_d$ is equal to the cable outer diameter $D_e$.

Outer diameter of the pipe in the center of a heat source inside which the fluid is flowing, e.g. water in a district heat pipe.

Outer diameter of large enclosing pipe with ducts inside such as it is used for HDD.

Outer diameter of steel pipe of pipe-type cables.

For rectangular tunnels, the outer diameter is calculated approximately using formula from Japanese Cable Standard JCS 0501. For circular tunnels, the outer diameter is the inner diameter plus the wall thickness.

The characteristic diameter of a trough is calculated as an equivalent diameter based on a formula by Huebscher, taken from engineeringtoolbox.com.