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Cableizer is the only commercially available software able to perform numerical steady-state temperature rise simulations on pressurized air cables like those from Hivoduct.
Posted 2025-01-14
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Cableizer is the only commercially available software able to perform numerical steady-state temperature rise simulations on pressurized air cables like those from Hivoduct. In order to allow for calculation of Hivoduct PAC we had to extend the existing calculation method for gas insulated lines which use nitrogen and SF6 as insulating gas.
The thermal rating of PAC can be calculated analytically using the same methods as are used for GIL, stated in Brochure No. 218 from Cigré [1] and in the IEC 60287 Series [2]. The brochure provides equations to calculate the loss factors for circulating (enclosures are always solid bonded) and eddy currents, skin and proximity effects of conductor and enclosure and for the thermal resistance of the inside gas volume and an optional protective cover.
Note: Several errors were introduced in [1] when copying from its origins [3], [4]. Consider following equations and parameter values (same numbering as in brochure):
\[ b(z_j) = \frac{2}{4\sqrt{2z_j} - 5}, \quad j = c, e \] \[ b_0 = 0.53560 \cdot 10^0 \] \[ y_e = a(z_e) \cdot \left[ 1 + \frac{\beta_e}{2} \right] \] \[ \lambda_{1m} = \frac{R_e}{R_c} \cdot \frac{4}{1} \xi_1 \] \[ W_{\text{rad}} = 5.69 \pi d_c \left[ \left(\frac{\vartheta_c + 273}{100}\right)^4 - \left(\frac{\vartheta_e + 273}{100}\right)^4 \right] \cdot \frac{1}{\frac{1}{\varepsilon_c} + \frac{d_c}{d_w} \left(\frac{1}{\varepsilon_e} - 1\right)} \]Heat transfer: The heat from losses in the conductor is transported through the pressurized air inside the PAC to the enclosure by means of radiation and convection.
For radiation heat transfer $W_{rad}$
\[ W_{\text{rad}} = \sigma \varepsilon_c \pi D_c K_{\text{cc}} (T_c^4 - T_e^4), \quad K_{\text{cc}} = \frac{1}{\frac{1}{\varepsilon_c} + \frac{D_c}{D_{\text{comp}}} \left(\frac{1}{\varepsilon_e} - 1\right)} \tag{1} \]where $\sigma$ is the Stefan Boltzmann constant [W/(m3·K$^4$)], $n_c$ is the number of conductors inside an enclosure, $D_c$ is the external diameter of the conductor [m], $Τ_c$ and $Τ_e$ are the temperatures of conductor and enclosure respectively [K].
$K_{ce}$ is the radiation shape factor from conductor to enclosure, considering $n_c$, where $\epsilon_c$ and $\epsilon_e$ are the effective emissivity of conductor and enclosure, $D_{comp}$ is the inner diameter of the gas compartment [m].
For convection heat transfer $W_{conv}$
\[ W_{\text{conv}} = K_0 [p \cdot (T_c - T_e)^2]^{\frac{1}{3}} \cdot F_{\text{form}} \tag{2} \]where $F_{form}$ is a constant form factor depending on $n_c$, $D_c$, and $D_{comp}$, $p$ is the compartment gas pressure [bar], and $K_0$ is the convection coefficient valid for a specific gas and temperature range.
The brochure contains convection coefficients for SF6 and N2 but not for air. In order to use the method, the coefficient for air has to be derived.
The equation (2) is based on the work of Vermeer as published in [3]. Using equations for gas density $\rho$ [kg/m$^3$], effective coefficient of heat conduction $\lambda$ [W/(m·K)], heat capacity at constant pressure $c_p$ [J/(kg·K)], and dynamic viscosity $\eta$ [Pa·s], the convection coefficient was calculated for temperatures between 30 and 90 °C and the mean value taken. The same procedure was done for dry air.
Following equations from [5] were used:
\[ \rho = \frac{351.99}{T} + \frac{344.84}{T^2} \tag{3} \] \[ \lambda = \frac{2.3340 \cdot 10^{-3} T^{3/2}}{164.54 + T} \tag{4} \] \[ c_p = 1030.5 - 1.9975 \cdot 10^{-1} \cdot T + 3.9734 \cdot 10^{-4} \cdot T^2 \tag{5} \] \[ \eta = \frac{1.4592 \cdot T^{3/2}}{109.10 + T} \cdot 10^{-6} \tag{6} \] \[ K_0 = K \cdot \left[\frac{\rho^2 z_c}{\eta}\right]^{\frac{1}{3}}, \quad K = 0.079 \cdot 2 \pi (g \beta)^{1/3} \tag{7} \]The factor K is calculated from acceleration due to gravity g and the expansion coefficient of the gas $\beta$ [1/K] and is considered a constant $K$ = 0.1638.
Calculating the values for the same temperature range as Vermeer, a value for dry air was derived being $K_0$ = 5.864.
The equivalent resistance is calculated using the thermal resistances for radiation and convection:
\[ T_1 = \frac{1}{\frac{1}{T_{\text{rad}}} + \frac{1}{T_{\text{conv}}}}, \quad T_{\text{rad}} = \frac{\Delta T}{W_{\text{rad}}}, \quad T_{\text{conv}} = \frac{\Delta T}{W_{\text{conv}}} \tag{8} \]This thermal resistance can be used in a calculation according to IEC 60287 for cables. The difference to a cable is, that the thermal resistance is temperature and pressure dependent, which requires an iterative approach.