Heat generation caused by electrical current

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For coupled thermal-electrical and coupled thermal-electrical-structural analyses in Abaqus/Standard the user can introduce a factor, $η$ , which defines the fraction of electrical dissipation in the contact zone converted to heat. The fraction of generated heat into the first and second surface, $f_{1}$ and $f_{2}$ , respectively, can also be defined.

The heat fraction, $η$ , determines the fraction of the energy dissipated due to electrical current that enters the contacting bodies as heat. Heat is instantaneously conducted into each of the contacting bodies depending on the values of $f_{1}$ and $f_{2}$ . The contact interface is assumed to have no heat capacity and may have properties for the exchange of heat by conduction and radiation.

The heat flux densities, $q_{1}$ , going out the surface on side 1, and $q_{2}$ , going out the surface on side 2, are given by

q_{1} = q_{k} + q_{r} - f_{1} q_{g}

and

q_{2} = - q_{k} - q_{r} - f_{2} q_{g},

where $q_{g}$ is the heat flux density generated by the interface element due to electrical current, $q_{k}$ is the heat flux due to conduction, and $q_{r}$ is the heat flux due to radiation.

The heat flux due to conduction is assumed to be of the form

q_{k} = κ (h, \bar{θ}) (θ_{1} - θ_{2}) = κ (h, \bar{θ}) Δ θ,

where the heat transfer coefficient $κ (h, \bar{θ})$ is a function of the average temperature at the contact point, $\bar{θ} = \frac{1}{2} (θ_{1} + θ_{2})$ , and overclosure, $h$ . $θ_{1}$ and $θ_{2}$ are the temperatures of side 1 and side 2, respectively.

The heat flux due to radiation is assumed to be of the form

q_{r} = F [{(θ_{1} - θ^{Z})}^{4} - {(θ_{2} - θ^{Z})}^{4}],

where $F$ is the gap radiation constant (derived from the emissivities of the two surfaces) and $θ^{Z}$ is the absolute zero on the temperature scale used.

The electrical flux density, $J$ , in the interface element is given in terms of the difference in the electric potential, $Δ φ$ , across the interface:

J = σ_{g} (h, \bar{θ}) (φ_{1} - φ_{2}) = σ_{g} (h, \bar{θ}) Δ φ,

where the gap electrical conductance $σ_{g} (h, \bar{θ})$ is a function of the overclosure, $h$ , and the average temperature at the contact point, $\bar{θ}$ . $φ_{1}$ and $φ_{2}$ are the electric potentials of side 1 and side 2, respectively.

In a steady-state analysis the heat flux density generated by the interface element due to electrical current is given by

q_{g} = η σ_{g} (h, \bar{θ}) {(φ_{1} - φ_{2})}^{2} = η σ_{g} (h, \bar{θ}) {(Δ φ)}^{2},

where $η$ is the fraction of dissipated energy converted to heat. In a transient analysis the average heat flux density is given by

\begin{matrix} q_{g} & ​ = \frac{η}{Δ t} \int_{t}^{t + Δ t} σ_{g} (h, \bar{θ}) {(Δ φ)}^{2} d t \\ ​ \approx \frac{1}{3} η σ_{g} (h, \bar{θ}) [{(Δ φ_{t + Δ t})}^{2} + Δ φ_{t + Δ t} Δ φ_{t} + {(Δ φ_{t})}^{2}], \end{matrix}

where $t$ is the time at the start of an increment and $Δ t$ is the time increment.

Using the Galerkin method, the weak form of the equations can be written as

\int_{S} δ θ_{1} q_{1} d S = \int_{S} δ θ_{1} (q_{k} + q_{r} - f_{1} q_{g}) d S, \int_{S} δ θ_{2} q_{2} d S = \int_{S} δ θ_{2} (- q_{k} - q_{r} - f_{2} q_{g}) d S .

The contribution to the variational statement of thermal equilibrium is

δ Π = \int_{S} (δ θ_{1} q_{1} + δ θ_{2} q_{2}) d S = \int_{S} [δ Δ θ (q_{k} + q_{r}) - δ \hat{θ} q_{g}] d S,

where $\hat{θ} = f_{1} θ_{1} + f_{2} θ_{2}$ . The contribution to the Jacobian matrix for the Newton solution is

\begin{matrix} (1) & d δ Π = \int_{S} [δ Δ θ (d q_{k} + d q_{r}) - δ \hat{θ} d q_{g}] d S . \end{matrix}

At a contact point the temperatures can be interpolated with

θ_{1} (s) = M_{1}^{N} (s) θ^{N} and θ_{2} (s) = M_{2}^{N} (s) θ^{N},

where $θ^{N}$ is the temperature at the $N$ th node associated with the interface element. Note that the summation convention will be used for all superscripts. Therefore, the temperature variables can be written as follows:

Δ θ (s) = Δ M^{N} (s) θ^{N}, \hat{θ} (s) = {\hat{M}}^{N} (s) θ^{N}, \bar{θ} (s) = {\bar{M}}^{N} (s) θ^{N},

where ${\hat{M}}^{N} (s) = f_{1} M_{1}^{N} (s) + f_{2} M_{2}^{N} (s)$ and ${\bar{M}}^{N} (s) = \frac{1}{2} (M_{1}^{N} (s) + M_{2}^{N} (s))$ . Substituting the above expressions to Equation 1, we obtain

\begin{matrix} d δ Π = δ θ^{N} \int_{S} [ & Δ M^{N} (\frac{\partial q_{r}}{\partial θ_{1}} d θ_{1} + \frac{\partial q_{r}}{\partial θ_{2}} d θ_{2} + \frac{\partial q_{k}}{\partial Δ θ} d Δ θ + \frac{\partial q_{k}}{\partial \bar{θ}} d \bar{θ}) \\ - {\hat{M}}^{N} (\frac{\partial q_{g}}{\partial Δ φ} d Δ φ + \frac{\partial q_{g}}{\partial \bar{θ}} d \bar{θ})] d S . \end{matrix}

The derivatives of $q_{r}$ , $q_{k}$ , and $q_{g}$ , are as follows:

\frac{\partial q_{r}}{\partial θ_{1}} = 4 F {(θ_{1} - θ^{Z})}^{3}, \frac{\partial q_{r}}{\partial θ_{2}} = - 4 F {(θ_{2} - θ^{Z})}^{3},

\frac{\partial q_{k}}{\partial Δ θ} = κ (h, \bar{θ}), \frac{\partial q_{k}}{\partial \bar{θ}} = \frac{\partial κ (h, \bar{θ})}{\partial \bar{θ}} Δ θ,

and in a steady-state analysis

\frac{\partial q_{g}}{\partial Δ φ} = 2 η σ_{g} (h, \bar{θ}) Δ φ, \frac{\partial q_{g}}{\partial \bar{θ}} = η \frac{\partial σ_{g} (h, \bar{θ})}{\partial \bar{θ}} {(Δ φ)}^{2},

while in a transient analysis

\frac{\partial q_{g}}{\partial Δ φ} = \frac{1}{3} η σ_{g} (h, \bar{θ}) (2 Δ φ + Δ φ_{t}), \frac{\partial q_{g}}{\partial \bar{θ}} = \frac{1}{3} η \frac{\partial σ_{g} (h, \bar{θ})}{\partial \bar{θ}} [{(Δ φ)}^{2} + Δ φ Δ φ_{t} + {(Δ φ_{t})}^{2}] .

Similarly, the contribution to the variational statement of electrical equilibrium is

δ Π = \int_{S} (δ φ_{1} J - δ φ_{2} J) d S = \int_{S} δ Δ φ J d S,

and the contribution to the Jacobian matrix for the Newton solution is

d δ Π = \int_{S} δ Δ φ (\frac{\partial J}{\partial \bar{θ}} d \bar{θ} + \frac{\partial J}{\partial Δ φ} d Δ φ) d S .

The derivatives of $J$ are

\frac{\partial J}{\partial \bar{θ}} = \frac{\partial σ_{g} (h, \bar{θ})}{\partial \bar{θ}} Δ φ and \frac{\partial J}{\partial Δ φ} = σ_{g} (h, \bar{θ}) .