Terms and definitions
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Symbols
h heat dissipation coefficient given in IEC 60287-2-1 for cables in still air W/m 2 ãK 5/4 n number of conductors in a cable - z coordinate corresponding to the tunnel axis m
A t inner tunnel cross-sectional area m 2
C av heat capacity of the air flow W/K
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C vair volumetric heat capacity of air Ws/(m 3 ãK)
D t inner diameter of the tunnel m
F m coefficient for the calculation of radiation shape factor -
I current in one conductor (r.m.s value) A k air thermal conductivity for air W/(mãK)
R alternating current resistance of conductor at its maximum operating temperature Ω/m
T 1 thermal resistance per core between conductor and sheath Kãm/W
T 2 thermal resistance between sheath and armour Kãm/W
T 3 thermal resistance of external serving Kãm/W
T 4t equivalent thermal resistance of cable surrounding Kãm/W
T as convection thermal resistance between cable and air Kãm/W
T at convection thermal resistance between air and inner wall of the tunnel Kãm/W
T st radiation thermal resistance between cable and inner wall of the tunnel Kãm/W
T a equivalent star thermal resistance of air Kãm/W
T e external thermal resistance of the tunnel Kãm/W
T s equivalent star thermal resistance of cable Kãm/W
T t equivalent star thermal resistance of tunnel wall Kãm/W
W a (z) heat removed by the air, at the point z in the cable route W/m
W a (L) heat removed by the air, at tunnel outlet W/m
W c losses in a conductor per unit length, assuming maximum conductor temperature W/m
W d dielectric losses per unit length per phase W/m
The total heat generated by the cable is denoted as \( W \) in watts per meter The ratio of total losses in metallic sheaths to total conductor losses is represented by \( \lambda_1 \), known as the sheath/screen loss factor Similarly, \( \lambda_2 \) indicates the ratio of total losses in the armor to total conductor losses, referred to as the armor loss factor Additionally, \( \nu \) represents the kinematic viscosity of air in square meters per second, while \( \rho \) signifies the soil thermal resistivity in kilopascals per watt.
L 0 reference length (see Formula (16)) m s b Stefan-Boltzmann constant W/(m 2 ãK 4 )
∆θ 0 fictitious increase of ambient temperature to account for the ventilation K θ max maximum permissible conductor temperature °C
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The article discusses various temperature measurements along a cable route, including the air temperature at specific points, such as the inlet and outlet of a tunnel It defines the conductor temperature at a point \( z \) in the cable route, as well as the surface temperature of the cable and the inner tunnel wall at both the inlet and outlet Additionally, it mentions the ground level temperature, providing a comprehensive overview of thermal conditions relevant to the cable system.
General description
The method involves calculating the temperatures of the cable surface, the air within the tunnel, and the tunnel wall, all of which depend on the heat generated by the cables.
For any location along the cable route, a set of formulae is developed, involving:
• heat transfer formulae describing heat transfer mechanisms by radiation and convection between the cables, the air in the tunnel and the tunnel wall;
• energy balance formulae for cables, air in the tunnel and tunnel wall;
• heat transfer formulae for conduction in the surroundings of the tunnel
This set of formulae may be written in such a way that:
• the heat removed by the air, W a (z), is linked to the derivative of the air temperature with respect to the longitudinal coordinate of the tunnel;
Every formula can be approximated using a thermal Ohm’s law, which connects the temperature drop to the heat flow through thermal resistance The heat flow is determined by the heat generated by the cables, denoted as \$W_k\$, and the heat removed by the air, represented as \$W_a(z)\$.
Some of the thermal resistances depend on the air temperature and consequently on the distance along the tunnel
This may be dealt with by dividing the tunnel route into elementary lengths, so that:
• the heat removed by the air is proportional to the difference in the air temperature between elementary length outlet and inlet;
• the thermal resistances may be considered constant for the elementary length
In typical installations analyzed in the CIGRE study, it was found that assuming constant thermal resistances along the tunnel route, based on temperatures at the tunnel outlet, does not result in significant errors.
Assuming the given conditions, the temperatures of the cable surface, air, and tunnel wall can be easily calculated based on the cable losses.
The allowable current is calculated using the heat transfer formula for conduction, which relates the temperature difference between the conductor and the cable surface to the energy losses in the cables.
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As temperatures at the tunnel outlet are not known, an iterative process is necessary
The heat produced by a cable, denoted as \$W_k\$, is considered constant throughout its length and is determined based on the maximum allowable conductor temperature, resulting in a conservative estimate of the current rating.
W k is the total heat generated by a cable (W/m); n is the number of conductors in a cable;
The losses in a conductor per unit length, denoted as \$W_c\$ (W/m), are influenced by the maximum temperature of the conductor The ratio of total losses in metallic sheaths to the total conductor losses is represented by \$\lambda_1\$, while \$\lambda_2\$ indicates the ratio of total losses in the armor to the total conductor losses.
W d is the dielectric losses per unit length per phase (W/m);
R is the alternating current resistance of conductor at its maximum operating temperature (Ω/m);
I is the current in one conductor (r.m.s value) (A).
Basic formulae
General
The following heat transfer mechanisms are taken into account:
• radial heat transfer by conduction within the cable,
• heat transfer by radiation from the cable surface to the tunnel wall,
• heat transfer by convection from the cable surface to the air inside the tunnel,
• heat transfer by convection from the air inside the tunnel to the tunnel wall,
• longitudinal heat transfer by convection resulting from the forced or natural flow of air along the tunnel.
Radial heat transfer by conduction within the cable
The conductor temperature is derived from the formula given in IEC 60287-1-1
1 z z θ W T n T n T W T n T T θ λ λ λ (3) where θ(z) is the conductor temperature, at the point z in the cable route (°C); θ s (z) is the temperature of the cable surface, at the point z in the cable route (°C);
T 1 is the thermal resistance per core between conductor and sheath (Kãm/W);
T 2 is the thermal resistance between sheath and armour (Kãm/W);
T 3 is the thermal resistance of external serving (Kãm/W)
The loss coefficients and thermal resistances are defined in IEC 60287-1-1 and IEC 60287-2-1
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Heat transfer by radiation from the cable surface to the inner wall of the
This heat transfer is modelled by Ohm’s thermal law, characterized by a thermal resistance:
D e * is the cable diameter (m); s b is Stefan-Boltzmann constant, 5,67x10 -8 (W/m 2 ãK 4 ); θ s (L) and θ t (L) are the cable surface and tunnel surface temperatures at the tunnel outlet
K t is the emissivity of the cable surface (typically 0,9 for served cable);
K r is the radiation shape factor taking into account the radiation areas
F m is a coefficient given in Table 1 and in Annex C
Table 1 – F m coefficient for radiation thermal resistance calculation
Heat transfer by convection from the cable surface to the air inside the
The convective heat transfer from the cable surface to the air in the tunnel depends on the air flow characteristics, the velocity of the air being the leading parameter
Where laminar air flow occurs, the convection thermal resistance is given by Formula (5):
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(5) where h is the heat dissipation coefficient given in IEC 60287-2-1 for cables in still air
(W/(m 2 ãK 5/4 )); θ at (L) is the air temperature at the tunnel outlet (°C)
Formula (5) applies if the Reynolds number is less than 2 000
If the Reynolds number is higher, the thermal resistance is first assumed to be given by Formula (6), valid for turbulent air flow
Re is the Reynolds number ν e *
Re V⋅D ν is the kinematic viscosity for air (m 2 /s); k air is the thermal conductivity for air (W/(mãK));
K cv is an experimentally determined constant for which values are given in Table 2
Table 2 – Value of parameter K cv
3 cables touching in trefoil 0,070 a to be used where the spacing is larger than 2 x D e * b to be used where the spacing is smaller or equal to 2 x D e *
The values from Formulae (5) and (6) are compared and the higher of the two values is used.
Heat transfer by convection from the air inside the tunnel to the inner
This transfer is modelled by Ohm’s thermal law, characterized by a thermal resistance:
If the Reynolds number is greater than 2 500, the air flow is assumed turbulent and the following relationship applies:
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Re is the Reynolds number ν t
Pr is the Prandtl number vair air
C vair is the specific heat of air per unit volume (J/(m 3 ãK));
D t is the inner diameter of the tunnel (m)
If the Reynolds number is less than 2 500, the thermal resistance is considered negligible.
Longitudinal heat transfer by convection resulting from the forced or
flow of air along the tunnel
The heat removed by the air, W a (z), is linked to the air temperature variations according to:
C av is the heat capacity of the air flow (W/K) t vair av C V A
A t is the inner tunnel cross-sectional area (m 2 ).
Radial heat conduction in the soil surrounding the tunnel
For circular tunnels the thermal resistance of the surrounding soil is expressed by:
D u ⋅L ρ soil is the soil thermal resistivity (Kãm/W);
L t is the depth of the tunnel axis (m)
For rectangular tunnels the thermal resistance of the surrounding soil is expressed by:
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For deep tunnels, these formulae will produce conservative results because of soil thermal inertia This subject is under consideration.
Set of formulae
A delta-star transformation is used to derive the following set of formulae:
(12) where z is the coordinate corresponding to the tunnel axis where
T s is the equivalent star thermal resistance of cable;
T t is equivalent star thermal resistance of tunnel wall;
T a is the equivalent star thermal resistance of air; defined as follows: as at st at as a as at st at st t as at st as st s
The delta-star transformation is shown diagrammatically in Annex B.
Solving
The permissible current rating is obtained from Formula (14) which is similar to the classical formula for cable rating given in IEC 60287-1-1:
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∆θ 0 is the fictitious increase of ambient temperature to account for the ventilation (K);
T 4t is the equivalent thermal resistance of cable surrounding (Kãm/W);
L = + + ⋅ (17) θ max is the maximum permissible conductor temperature (°C)
The air temperature θ at (L) at the tunnel outlet is estimated from:
The cable surface temperature and the tunnel wall temperature at the tunnel outlet are derived from the air temperature by:
W a (L) is the heat removed by the air at the tunnel outlet, given by:
Iterative process
The thermal resistances T a , T s and T t are calculated from estimates of the cable surface temperature, the tunnel wall temperature and the air temperature at the tunnel outlet, using Formulae (4), (5) or (6), (7) and (13)
The cable permissible current is derived from Formulae (14) through (15), (16), (17), T e being derived from Formulae (10) and (11) and C av being derived from Formula (9)
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Losses in the cables are calculated with Formulae (1) and (2)
The air temperature at the tunnel outlet is calculated with Formula (18), the cable surface temperature and the tunnel wall temperature are calculated with Formulae (19) and (20), using Formula (21)
The calculation is repeated using these new estimates of the cable surface temperature, the tunnel wall temperature and the air temperature at the tunnel outlet as input, until convergence
As first estimates, the temperatures at the tunnel outlet are taken as the air temperature at the tunnel inlet
Formulae (22) to (25) provide the properties needed for air at the appropriate temperature: Thermal conductivity for air
The volumetric heat capacity of air, C vair , being derived from Pr, k air and ν ν air vair Pr k
Formula (26) gives the air temperature θ at (z) in any location z along the tunnel
W k , T t , T e and L 0 have been determined according to Clause 4
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Cable and installation
The example given in Table A.1 considers 3 single-core cables without armour (T 2 = 0 and λ 2 = 0) spaced vertically within a circular ventilated tunnel (the spacing between the cables being three times their diameter)
Number of conductors in a cable n 1 -
Alternating current resistance of conductor at its maximum operating temperature R 1,28E-05 Ω/m
Dielectric losses per unit length per phase W d 4,0 W/m
Maximum permissible conductor temperature θ max 90 °C
Thermal resistance per core between conductor and sheath T 1 0,341 Kãm/W
Thermal resistance of external serving T 3 0,038 Kãm/W
Soil thermal resistivity ρ soil 1,0 Kãm/W
Air temperature at tunnel inlet θ at (0) 20 °C
Calculated values
The number of significant figures given in Table A.2 does not indicate the accuracy of the calculations but is intended to assist those developing a calculation tool
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Table A.2 – Iterative process for a 1 km long tunnel
Iteration Formula 1 2 3 assumed θ s (L) 20 52,11 52,15 assumed θ t (L) 20 36,83 37,89 assumed θ at (L) 20 36,49 37,30
The temperature profile along the 1 km length of the tunnel is given in Figure A.1
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Figure A.1 – Temperature profile along a 1 km tunnel
In Figure A.1, the thermal properties of air are assessed for the calculated air temperature in the tunnel at each iteration stage When the air thermal properties are evaluated at 30 °C, the current rating is 2,764 A, slightly higher than the previously calculated 2,755 A.
Repeating the calculation using the same data, except for a tunnel length of 10 000 m, results in a current rating of 1 999 A The temperature profile along the 10 km tunnel is shown in Figure A.2
Figure A.2 – Temperature profile along a 10 km tunnel
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If the air thermal properties are determined for a temperature of 30 °C, the permissible current is found to be 2 018 A, instead of 1 999 A This difference is considered to be insignificant
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The heat transfer mechanism in the tunnel and the delta-star given in 4.3 is shown in Figure B.1
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Definition of spacing
The spacing between cables is defined as the distance between cables axis (see Figure C.1)
Calculation of Fm coefficient
The coefficient F m can be calculated with expressions given in Table C.1
Table C.1 – Expression for F m coefficient calculation
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The article discusses the significance of the ratio between spacing and cable diameter, denoted as \( s \) This ratio plays a crucial role in determining the efficiency and performance of cable systems Understanding this relationship is essential for optimizing cable design and ensuring effective electrical transmission.
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[1] Electra n°143 – 144 (1992), CIGRE (International Council on Large Electric Systems), [including Erratum published in Electra n°209 (2003)]
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3 Termes, définitions et symboles 31 3.1 Termes et définitions 31 3.2 Symboles 31
This section provides a comprehensive overview of the method, beginning with a general description It includes essential formulas, covering various heat transfer mechanisms such as radial conduction within the cable, radiation from the cable surface to the tunnel wall, and convection from the cable surface to the air inside the tunnel Additionally, it addresses convection of air within the tunnel to the tunnel wall, longitudinal convection due to forced or natural air flow along the tunnel, and radial heat conduction in the soil surrounding the tunnel The section concludes with a system of formulas, resolution techniques, and an iterative process for effective analysis.
5 Formules relatives aux propriétés de l'air 41
The article includes a temperature profile section, followed by informative appendices detailing calculation examples, cable and installation specifics, and calculated values It also covers the transformation from delta to star configuration and provides a method for calculating the Fm coefficient, including its definition and spacing considerations Finally, a bibliography is provided for further reference.
The temperature profile along a 1 km tunnel is illustrated in Figure A.1, while Figure A.2 depicts the temperature profile for a 10 km tunnel Additionally, Figure B.1 presents the transformation from a triangle to a star configuration, and Figure C.1 provides definitions related to spacing.
Tableau 1 – Coefficient Fm pour le calcul de la résistance thermique de rayonnement 36 Tableau 2 – Valeur du paramètre Kcv 37 Tableau A.1 – Données d'installation 42
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Tableau A.2 – Processus itératif pour un tunnel d'une longueur de 1 km 43 Tableau C.1 – Expression pour le calcul du coefficient Fm 47
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CÂBLES ÉLECTRIQUES – CALCUL DU COURANT ADMISSIBLE – Partie 2-3: Résistance thermique – Câbles posés dans les tunnels ventilés
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