### Table of Contents

# Thermal field

Stan Zurek, Thermal field, Encyclopedia Magnetica, https://e-magnetica.pl/doku.php/thermal_field |

**Thermal field** - distribution of temperature over a certain region that can be mathematically analysed as a scalar field.^{1)}

^{S. Zurek, E-Magnetica.pl, CC-BY-4.0}

The use of vector calculus (divergence, curl, gradient) allows performing finite-element modelling (FEM) of temperature distribution in very complex structures. Such calculations can be combined with solutions for other physical effects.

For example, in a combined magneto-thermal simulation such as induction heating, the power loss generated by electromagnetic excitation is used as the localised heating source. Further parameters such as thermal conductivity of the heated material, as well as the boundary conditions on the surface dictate the local cooling rate of the structure.

Therefore, it is possible to calculate the rate of temperature rise with time during transient conditions, as well as the final temperature of the steady state, dictated by the cooling conditions (with natural air, or forced cooling, including liquid cooling).

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## Temperature as scalar field

In a **scalar field** each point has some scalar value associated with it. Scalar value represents just the magnitude of the value, without any additional attributes such as direction.

The intensity of a scalar field can be shown as intensity of colours in a false-colour image, as used for example in thermal cameras.

An intuitive example of a scalar field is the temperature distribution. Within some volume of solid, liquid, or gas, there is a certain value of temperature present at every point within that volume, and therefore it can be treated mathematically as a kind of field. The exact distribution of temperature is dictated by the heat sources, geometry, material properties, and cooling conditions.

The temperature can be used as an example of calculations which can be performed on a field. The heat flux out of any closed volume is equal to the heat generation within that volume^{2)}. Therefore, the divergence of the field represents the heat source:

$$∇·\vec{F} = q$$ | (W/m^{3}) |

where: $∇·\vec{F}$ - divergence (W/m^{3}) of the vector heat flux $\vec{F}$ (W/m^{2}), q - scalar volume heat generation (W/m^{3}) |

Heat flux is also related to the temperature gradient, through the thermal conductivity of the given material:

$$\vec{F} = k·\vec{G}$$ | (W/m^{2}) |

where: $\vec{F}$ - vector heat flux (W/m^{2}), k - scalar thermal conductivity of the material (W/(m·K)), $\vec{G}$ - vector temperature gradient (K/m) |

and temperature gradient is related to the temperature distribution:

$$\vec{G} = -∇T$$ | (K/m) |

where: $\vec{G}$ - vector temperature gradient (K/m), T - scalar temperature (K) |

Therefore, combination of all these equations results in a second-order partial differential equation, which directly relates the heat sources with material properties, and with the temperature distribution, which is generally of interest in such calculations, and thus can be solved numerically by using the following formulation:^{3)}

$$-∇·(k·∇T) = q$$ | (W/m^{3}) |

where: k - scalar thermal conductivity of the material (W/(m·K)), T - scalar temperature (K), q - scalar volume heat generation (W/m^{3}) |

## Thermal image as illustration of thermal field

^{S. Zurek, E-Magnetica.pl, CC-BY-4.0}

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