## Model for the heat exchange with the ground

#### Parameters

The ground model is calculated with the following parameters (Calc indicates the parameter is calculated in the course of the routine):

Parameter | Description | Unit | Value |

\(\widetilde{z}\ \) | Average depth of the ice storage under the surface | m | Calc |

G_{t} | Geothermal gradient | K/m | 0.03 |

\(\rho_{grd}\) | Density of the ground | kg/m^{3} | 2’500 |

cp_{grd} | Thermal capacity of the ground | kJ/K | 800 |

\(\lambda\)_{grd} | Thermal conductivity of the ground | W/mK | 2 |

U_{tank} | Thermal conductivity of the storage tank | W/mK | Calc |

U_{earth} | Thermal conductivity of the earth | W/mK | Calc |

V | Capacity of the ice storage | m^{3} | 10 |

z_{boden} | Depth of the bottom of the ice storage | m | 3.2 |

h_{tank} | Height of the ice storage | m | 2.3 |

D_{tank} | Diameter of the ice storage | m | 2.7 |

d_{grd} | Strength of the surrounding ground layer | m | 0.5 |

\(\overline{\theta_{e}}\) | Average outside temperature | °C | 11 |

\(\Delta\theta_{e}\) | Amplitude of the monthly deviations of the outside temperature | °C | 9.3 |

\(\lambda_{wall}\) | Thermal conductivity of the ice storage wall | W/mK | 1.33 |

d_{wall w} | Thickness of the ice storage wall to the sides | m | 0.1 |

d_{wall b} | Thickness of the ice storage wall to the bottom | m | 0.12 |

#### Implementation

The temperature of the undisturbed ground is calculated as a function of the depth in the ground and of time.

\(\zeta_{grd}(t,z) = \overline{\theta_{e}} – \Delta\theta_{e}\exp{\left( – \frac{z}{\delta_{grd}} \right)\cos\left\lbrack 2\pi\frac{\left( t – t_{shift} \right)}{t_{0}} – \frac{z}{\delta_{grd}} \right\rbrack} + G_{t}z\)

With the thermal penetration depth o_{grd} as a measure for the degree to which the temperature in the depth lags behind the surface temperature. It is calculated as follows:

\(\delta_{grd} = \sqrt{\frac{t_{0} \cdot 3600\left\lbrack \frac{s}{h} \right\rbrack \cdot \lambda_{grd}}{\pi \cdot \rho_{grd} \cdot cp_{grd}}}\)

With z as the mean depth of the ice storage z̃ (\(\widetilde{z} = z_{boden} – 0.5 \cdot h_{tank}\)) and the ground temperature around the ice storage assumed to be homogenous.

If the ground has a temperature that is different from that of the ice storage, a thermal flow will start. Thus, the temperature of the wall of the ice storage is calculated using this formula:

\(T_{Wall} = \int\frac{1}{m_{grd}cp_{grd}}\left( UA_{earth}\left( \zeta_{grd} – T_{wall} \right) – UA_{tank}\left( T_{wall} – T_{ice} \right) \right)\mathbb{d}t\)

With UA_{earth} being the thermal transmission coefficient of the ground layer between the undisturbed earth and the ice storage wall. This is calculated according to this formula:

\(UA_{earth} = \frac{\lambda_{grd}}{d_{grd}} \times \left\lbrack \pi\left( \frac{D_{tank} + 2d_{grd}}{2} \right)^{2} + \pi\left( D_{tank} + 2d_{grd} \right)\left( h_{tank} + d_{grd} \right) \right\rbrack\)

UA_{tank} being the thermal transmission coefficient of the ice storage casing and is calculated from the area of the wall or the ground respectively divided by the respective thermal resistance. The heat exchange between ice storage and ground in this model is limited to the lower, cylindrical part of the ice storage. Thermal gains or losses via the conical lid are not taken into calculation.

\(UA_{tank} = \frac{\left( \frac{D_{tank}}{2} \right)^{2}\pi}{\frac{d_{wall\ b}}{\lambda_{wall}}} + \frac{D_{tank\ }\pi\ h_{tank}}{\frac{\mathbb{d}_{wall\ w}}{\lambda_{wall}}}\)

m_{grd} is calculated taking the volume of the ground multiplied by its density – the volume being calculated via the external area of the ice storage and the thickness of the ground layer.