THIEF Model

Overview

The THIEF (Thermally-Induced Electrical Failure) methodology uses a more detailed calculation than heat soak to estimate cable failures. It uses the cable's dimensions and thermal data to calculate the temperature inside the cable. Assuming the heat transfer in a cable is largely in the radial direction, it calculates the one-dimensional radial conductive heat transfer, with the surface boundary conditions taken as the temperature or heat flux calculated by CFAST. A cable is damaged when the internal cable temperature, after thermal attenuation through the cable's jacket, exceeds a specific threshold for thermoplastic or thermoset insulation.

THIEF models each cable as a homogeneous cylinder: heat is conducted one-dimensionally into the cylinder, with the thermal conductivity (k) and specific heat (c) assumed constant for all cables, and the failure temperature thresholds obtained experimentally.

This method is more realistic than the heat-soak method; however, it requires more computation time and data. It is selected per cable, whenever the modeler assigns the cable's material data in the FRI3D model — so each cable receives its own failure time, and THIEF cables can be freely combined with heat-soak cables (those left with "Null" material data) in the same model.

Governing Equation

Results from the fire simulation are used to calculate heat conduction inside the cable's radial cross-section. The surface temperature is used as the boundary condition to solve the following transient conductive heat transfer equation:

$$\rho c\frac{dT}{dt}=\frac{1}{r}\frac{d}{dr}k\ r\frac{dT}{dr}$$

Where $\rho$, $c$, and $k$ are the effective density, specific heat, and thermal conductivity respectively.

Numerical Solution

The equation is solved by the finite difference method. The cable's radius R is divided into N uniformly spaced radial grids of length dr. A time step dt is selected based on the value of dr as a criterion of numerical stability:

$$dt=\frac{c\rho\ dr^2}{2k}$$

The interior nodes are advanced with the following semi-implicit finite-difference scheme:

$$\rho c\frac{T_i^{n+1}-T_i^n}{dt}=\frac{2k}{\left(r_{i+1}+r_i\right)}\frac{1}{2\ dr}\left[r_i\frac{T_{i+1}^n-T_i^n}{dr}-r_{i-1}\frac{T_i^n-T_{i-1}^n}{dr}+r_i\frac{T_{i+1}^{n+1}-T_i^{n+1}}{dr}-r_{i-1}\frac{T_i^{n+1}-T_{i-1}^{n+1}}{dr}\right]$$

The boundary condition at the cable surface is given by:

$$k\frac{T_{N+1}^n-T_N^n}{dr}={\dot{q}}^n\left(t^n\right)$$

Where ${\dot{q}}^n\left(t^n\right)$ is the net heat flux at the surface due to radiation and convection with the surrounding gas:

$${\dot{q}}^n\left(t^n\right)=\varepsilon\sigma\left(T_g\left(t^n\right)^4-\left(T_s^n\right)^4\right)+h\left(T_g\left(t^n\right)-T_s^n\right)$$

Where $\epsilon$ is the emissivity of the cable surface (assumed to be 0.95 in the THIEF methodology), $\sigma$ is the Stefan-Boltzmann constant, $h$ is the convective heat transfer coefficient (assumed to be 10 $W/m^2/K$), and $T_g\left(t^n\right)$ is the effective gas temperature at the n-th time step. The effective gas temperature may be approximated by the cable's surface temperature obtained from the preliminary CFAST simulation if the radial grid is reasonably small.

Failure Criterion

The cable is assumed to fail when the internal cable temperature exceeds 400°C for thermoplastic cables or 200°C for thermoset cables. The failure time is the moment that threshold is first exceeded.

Using THIEF in FRI3D

To use the THIEF method, add a "Cable" in the model and select the cable material from the dropdown, or create a new one, as shown in the figure below.

What's next?

• FLASH-CAT Methodology — how burning cables become secondary fire sources.
• Heat Soak Model — the simpler, conservative default used when no cable data is available.