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Model for the heat exchange with the ground


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

 \(\widetilde{z}\ \)Average depth of the ice storage under the surfacemCalc
GtGeothermal gradientK/m0.03
\(\rho_{grd}\)Density of the groundkg/m32’500
cpgrdThermal capacity of the groundkJ/K800
\(\lambda\)grdThermal conductivity of the groundW/mK2
UtankThermal conductivity of the storage tankW/mKCalc
UearthThermal conductivity of the earthW/mKCalc
VCapacity of the ice storagem310
zbodenDepth of the bottom of the ice storagem3.2
htankHeight of the ice storagem2.3
DtankDiameter of the ice storagem2.7
dgrdStrength of the surrounding ground layerm0.5
 \(\overline{\theta_{e}}\)Average outside temperature°C11
 \(\Delta\theta_{e}\)Amplitude of the monthly deviations of the outside temperature°C9.3
 \(\lambda_{wall}\)Thermal conductivity of the ice storage wallW/mK1.33
dwall wThickness of the ice storage wall to the sidesm0.1
dwall bThickness of the ice storage wall to the bottomm0.12


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 ogrd 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 UAearth 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:

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

UAtank 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.

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

mgrd 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.

Figure: schematic representation of the ice storage model