## Coil Heat Exchanger Model

#### Parameters

The ice storage model using coil heat exchangers is calculated with the following parameters (Calc indicates the parameter is calculated in the course of the routine):

Parameter | Description | Unit | Value |

m_{ice} | Mass of the ice storage | kg | Calc |

m_{grd} | Mass of the ground surrounding the ice storage | kg | Calc |

t_{shift} | Time distance between the time of the lowest temperature in the course of a year and January 1 | h | 319 |

t_{0} | Number of hours per year for scaling | h/a | 8’760 |

T_{tabelle}(H_{ice}) | Temperature equaling a given heat content H | °C | |

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

f_{ice} | Initial part of the ice storage being frozen (0..1) | – | 0 |

t_{init} | Initial temperature of the ice storage | °C | 15 |

#### Background

The ice storage model is a simple model for a water storage tank as heat exchanger that with temperatures around zero degrees can store or release thermal energy through the phase transition from liquid to solid or from solid to liquid, respectively.

The temperature of the ice storage depends on the heat flows to the wall and the heat flows because of the heat exchangers. The formation of ice on the heat exchangers is not explicitly determined but calculated via the heat content H.

The heat content H of the ice is calculated using

\(H_{ice} = \int\frac{1}{m_{ice}}\left( \sum_{i}^{}\left( {\dot{Q}}_{2ice} \right)_{i} + UA_{tank}\left( T_{wall} – T_{ice} \right) \right)\)

(\(\dot{Q}\) _{2ice})i are the heat flows through heat exchangers and m_{ice} the mass of liquid in the ice storage – volume times the density of water (1,000 \(kg \slash m^{3}\)).

The temperature of the ice storage is subsequently read from an interpolated table containing different temperature values in dependence from the heat content H.

\(T_{ice} = T_{tabelle}(H_{ice})\)

Through the access via the heat content, the thermal energy stored during the melting phase transition can be taken into account, as can the thermal energy released in the solidification phase transition.

The temperature of the ice can be calculated through linear interpolation in dependence from the heat content. The following value pairs between temperature (T) and heat content (E) are used as nodes:

Temperature in °C | Heat Content in \(\mathbf{J/kg}\) |

-10 | -10·2’060 |

-3 | -3·2’060 |

0 | 3.35·10^{5} |

10 | 10·4’182+3.35·10^{5} |

First, the heat content has to be initialized. For that, the initial temperature and the initial ice content are used. They are calculated according to

\(H_{init} = \left( 1 – f_{ice} \right) \cdot 3.35 \cdot 10^{5} + \Theta\left( – f_{ice} \right) \cdot 4^{‘}182 \cdot t_{init}\)

with Θ(x) being the Heaviside function, which results in zero for a negative x and assumes the value one for \(x \geq 0\). f_{ice} is the part of the ice storage that is initially frozen, and is handed over as a parameter.

T_{wall} is initialized with 4°C, the remaining parameters are given.