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Horw, 20 September 2016 Page 1/5

Simulation of the ice storage for “SeleCO2” 1. Geometry and input parameters According to the information provided by Fafco, the considered heat exchanger (UW312/8/40) consists of 320 heat exchangers which are placed in parallel. Each heat exchanger consists of 196 tube coils.

The following picture illustrates the Fafco heat exchanger:

The important parameters of the storage unit and the heat exchangers are as follows: • • • • • • • • • •

PCM: water HTF: water/glycol mixture (30%) Total capacity (latent): 1800 kWh Total volume of water: 206 m3 Number of HE: 320 Total number of tube coils: 62’720 Length of a single tube coil: 6.25 m Outer diameter of the tube coil: 6.4 mm Thickness of the tube coil: 0.6 mm Heat conductivity of the tube coil material: 0.23 W/Km

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The thermophysical properties of the HTF are: • Heat conductivity: 0.367 W/mK • Spec. heat capacity: 3245 J/kgK • Viscosity: 919*10^-5 Pas • Density: 1070 kg/m3 The thermophysical properties of the PCM (water) are: • Heat conductivity (solid): 2.25 W/mK • Heat conductivity (liquid): 0.57 W/mK • Density (solid): 917 kg/m3 • Density (liquid): 999.97 kg/m3 • Spec. heat capacity (solid): 2040 J/kgK • Spec. heat capacity (liquid): 4203 J/kgK • Phase change enthalpy: 333’600 J/kg • Viscosity: 100*10^-5 Pas

2. Modeling assumptions: First of all it is worth mentioning that this model should deliver the possibility to be used as a design tool also for applications whereas the detailed geometry of the heat exchanger is not known yet. Therefore it is important that the model can be run with reasonable calculation time. The so called quasi stationary method delivers the possibility to represent the solidification process with simplified equations, which result in a low computational effort. The main assumption of this method is that the temperature distribution in the PCM is assumed to be linear. In other words, it is assumed that the moving boundary problem can be regarded as stationary within a small time step. The advantage of this assumption is, that stationary analytical equations, if such exist, become applicable to describe the heat transfer between the PCM and the HTF during the discharging process. Such analytical equations exist for cylindrical, spherical or plane geometries. Since the considered Fafco heat exchanger consists of cylindrical tube coils, this approach is chosen to model the discharging process. Within the model it is assumed that each heat exchanger consists of identical tube coils which are placed in parallel. Initial and boundary condition are assumed to be the same for each tube coil. The main assumptions of the model are as follows: 1. 2. 3. 4. 5. 6. 7. 8. 9.

Conduction is the only heat transfer mechanism in the solid and the liquid PCM. Within the PCM heat is only transferred in radial direction. Vertical heat transfer in the PCM is neglected. It is assumed that the phase change from solid to liquid occurs isothermal at the known phase change temperature 𝑇𝑇𝑃𝑃𝑃𝑃 Linear temperature profiles are assumed in the solid and the liquid (Quasi-stationary approach). Nucleation difficulties and supercooling effects are assumed not to be present. The thermophysical material properties of the PCM are assumed to be constant (temperature independent) within the solid and the liquid phase. Losses to the ambient are neglected. Overlapping effects of the solid/liquid interface between adjacent tube coils are neglected. The heat transfer resistance due to the forced convection inside the pipe is approximated by a Nusselt-Correlation.

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3. Governing equations A challenge in modeling of latent heat storages is that the temperature profile of the HTF along the tube is unknown (except for the inlet) and needs to be calculated as a part of the solution. The temperature of the HTF increases while it flows through the heat exchanger due to the absorption of heat which is transferred from the PCM to the HTF. Accordingly the temperature of the HTF is dependent on time and location. In order to take that into account a spatial discretization of the heat exchanger is implemented. Each segment consists of a finite length of ∆𝐿𝐿 and an outer radius of 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 . The latter represents the outer boundary of the considered computational domain. 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 is determined through the following geometric relation: 2 2 )𝑙𝑙 − 𝑟𝑟𝑜𝑜𝑜𝑜𝑜𝑜 𝜋𝜋(𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 𝑁𝑁𝑇𝑇𝑇𝑇 = 𝑉𝑉𝑃𝑃𝑃𝑃𝑃𝑃,𝑡𝑡𝑡𝑡𝑡𝑡

Where 𝑟𝑟𝑜𝑜𝑜𝑜𝑜𝑜 is the outer radius of the tube coil, 𝑙𝑙 𝑇𝑇𝑇𝑇 the total length of the tube coil, 𝑁𝑁𝑇𝑇𝑇𝑇 the number of tube coils and 𝑉𝑉𝑃𝑃𝑃𝑃𝑃𝑃,𝑡𝑡𝑡𝑡𝑡𝑡 the Volume of the entire PCM in the storage unit. Rearranging the definition of 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 : 1⁄2 𝑉𝑉𝑃𝑃𝑃𝑃𝑃𝑃,𝑡𝑡𝑡𝑡𝑡𝑡 2 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 = � + 𝑟𝑟𝑜𝑜𝑜𝑜𝑜𝑜 � 𝜋𝜋𝑙𝑙 𝑇𝑇𝑐𝑐 𝑁𝑁𝑇𝑇𝑇𝑇 The following figure illustrates a segment durning the solidification:

The solidification process is divided in two stages as follows: • Stage 1: The PCM is solidifying. The heat which is conducted from the PCM to the HTF results in an axially symmetric solid/liquid front with radius 𝑟𝑟𝑠𝑠 propagating outwards and separating the calculation domain in a liquid and a solid region. • Stage 2: The PCM considered in the calculation domain is completely solidified. Only sensible heat is conducted from the solid PCM towards the HTF.

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The assumption of dividing the process into these two stages allows to consider the sensible heat which is released by the PCM. Stage 1 Once the crystallization process has started a solid/liquid interface 𝑟𝑟𝑠𝑠 propagates from the outer wall of the tube towards 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 , where 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 represents the position of the solid/liquid interface once the storage is completely solidified.

The heat which is released by the liquid PCM results in a decrease of its temperature. This is described by the following energy balance: where 𝑟𝑟𝑠𝑠 (𝑡𝑡) is equal to 𝑟𝑟𝑜𝑜𝑜𝑜𝑜𝑜 since it is assumed that the complete segment is liquid during this stage.

Stage 1 ends as soon as the temperature at the outer wall of the tube 𝑇𝑇𝑊𝑊 is below the phase change temperature of the PCM. Hence it is necessary to define an equation which makes it possible to calculate the time dependent behavior of 𝑇𝑇𝑊𝑊 . Due to the assumption of quasi stationary conditions the heat flux from the PCM to the HTF is equal to the heat flux from the outer wall to the HTF. This leads to the following equation for 𝑇𝑇𝑊𝑊 : 𝑇𝑇𝑊𝑊 (𝑡𝑡) =

𝑄𝑄̇𝐻𝐻𝐻𝐻𝐻𝐻,𝑆𝑆𝑆𝑆𝑆𝑆 (𝑡𝑡) 1 1 𝑟𝑟𝑜𝑜𝑜𝑜𝑜𝑜 � + 𝑙𝑙𝑙𝑙𝑔𝑔 � + 𝑇𝑇𝐻𝐻𝐻𝐻𝐻𝐻 (𝑡𝑡) 𝑟𝑟𝑖𝑖𝑖𝑖 2𝜋𝜋∆𝑙𝑙 ∝𝐻𝐻𝐻𝐻𝐻𝐻 𝑟𝑟𝑖𝑖𝑖𝑖 𝜆𝜆 𝑇𝑇𝑇𝑇

Based on the energy balance at the solid/liquid interface the growth velocity of the solid PCM is defined as: 𝜕𝜕𝑟𝑟𝑠𝑠 (𝑡𝑡) 𝑄𝑄̇𝐻𝐻𝐻𝐻𝐻𝐻,𝑆𝑆𝑆𝑆𝑆𝑆 (𝑡𝑡) − 𝑑𝑑𝑄𝑄̇𝑙𝑙,𝑆𝑆𝑆𝑆𝑆𝑆 (𝑡𝑡) = ℎ𝑃𝑃𝑃𝑃 𝜌𝜌𝑠𝑠 2𝜋𝜋∆𝑙𝑙 𝜕𝜕𝜕𝜕

Where 𝑄𝑄̇𝐻𝐻𝐻𝐻𝐻𝐻,𝑆𝑆𝑆𝑆𝑆𝑆 is the heat flux from the solid/liquid interface to the HTF, 𝑄𝑄̇𝑙𝑙,𝑆𝑆𝑆𝑆𝑆𝑆 is the heat flux from the liquid part of the PCM to the solid/liquid. These are defined as: 𝑄𝑄̇𝐻𝐻𝐻𝐻𝐻𝐻,𝑆𝑆𝑆𝑆𝑆𝑆 (𝑡𝑡) = �𝑇𝑇𝑃𝑃𝑃𝑃 (𝑡𝑡) − 𝑇𝑇𝐻𝐻𝐻𝐻𝐻𝐻 (𝑡𝑡)� �

1

∝𝐻𝐻𝐻𝐻𝐻𝐻 𝑟𝑟𝑖𝑖𝑖𝑖

+

𝑄𝑄̇𝑙𝑙,𝑆𝑆𝑆𝑆𝑆𝑆 (𝑡𝑡) = �𝑇𝑇𝑙𝑙 (𝑡𝑡) − 𝑇𝑇𝐻𝐻𝐻𝐻𝐻𝐻 (𝑡𝑡)� �

1 𝑟𝑟𝑜𝑜𝑜𝑜𝑜𝑜 1 𝑟𝑟𝑠𝑠 (𝑡𝑡) −1 𝑙𝑙𝑙𝑙𝑙𝑙 + 𝑙𝑙𝑙𝑙𝑔𝑔 � 2𝜋𝜋∆𝑙𝑙 𝑟𝑟𝑖𝑖𝑖𝑖 𝜆𝜆𝑠𝑠 𝑟𝑟𝑜𝑜𝑜𝑜𝑜𝑜 𝜆𝜆 𝑇𝑇𝑇𝑇

1 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 −1 𝑙𝑙𝑙𝑙𝑙𝑙 � 2𝜋𝜋∆𝑙𝑙 𝑟𝑟𝑠𝑠 (𝑡𝑡) 𝜆𝜆𝑙𝑙

Where ∝𝐻𝐻𝐻𝐻𝐻𝐻 is the convective heat transfer coefficient inside the pipe, 𝑇𝑇𝑙𝑙 the temperature of the liquid PCM, 𝜆𝜆 𝑇𝑇𝑇𝑇 the heat conductivity of the tube coil, 𝜆𝜆𝑙𝑙 the heat conductivity of the liquid PCM, 𝜆𝜆𝑠𝑠 the heat conductivity of the solid PCM, 𝑟𝑟𝑖𝑖𝑖𝑖 the inner radius of the tube coil and 𝑟𝑟𝑜𝑜𝑜𝑜𝑜𝑜 the outer radius of the tube coil. The latter heat flux 𝑄𝑄̇𝑙𝑙,𝑆𝑆𝑆𝑆𝑆𝑆 (𝑡𝑡) decelerates the growth rate of the solid/liquid interface. The heat which is transferred from the liquid PCM to the solid/liquid interface leads to a decrease of the liquid PCM 𝑇𝑇𝑙𝑙 temperature. This is considered with the following equation: 2 ) 𝜌𝜌𝑙𝑙 𝑐𝑐𝑝𝑝,𝑙𝑙 𝜋𝜋𝜋𝜋𝜋𝜋(𝑟𝑟𝑠𝑠 (𝑡𝑡)2 − 𝑟𝑟𝑜𝑜𝑜𝑜𝑜𝑜

𝜕𝜕𝑇𝑇𝑙𝑙 (𝑡𝑡) = 𝑄𝑄̇𝐻𝐻𝐻𝐻𝐻𝐻,𝑆𝑆𝑆𝑆𝑆𝑆 (𝑡𝑡) 𝜕𝜕𝜕𝜕

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Stage 2 Stage2 starts as soon as the solid/liquid interface has reached 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 . It is assumed that during stage 2 only sensible heat is transferred from the PCM to the HTF leading to a decrease of the temperature of the solidified PCM. The analytical equation describing the heat flux from the solidified PCM to the HTF reads: 1 1 𝑟𝑟𝑜𝑜𝑜𝑜𝑜𝑜 1 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 −1 𝑄𝑄̇𝐻𝐻𝐻𝐻𝐻𝐻,𝑆𝑆𝑆𝑆𝑆𝑆 (𝑡𝑡) = �𝑇𝑇𝑠𝑠 (𝑡𝑡) − 𝑇𝑇𝐻𝐻𝐻𝐻𝐻𝐻 (𝑡𝑡)� � + 𝑙𝑙𝑙𝑙𝑙𝑙 + 𝑙𝑙𝑙𝑙𝑙𝑙 � 2𝜋𝜋∆𝑙𝑙 𝑟𝑟𝑖𝑖𝑖𝑖 𝑟𝑟𝑜𝑜𝑜𝑜𝑜𝑜 ∝𝐻𝐻𝐻𝐻𝐻𝐻 𝑟𝑟𝑖𝑖𝑖𝑖 𝜆𝜆 𝑇𝑇𝑇𝑇 𝜆𝜆𝑠𝑠

where 𝑇𝑇𝑠𝑠 represents the temperature of the PCM once the complete domain of the segment is solidified. To take into account the decrease of 𝑇𝑇𝑠𝑠 due to the release of heat the following energy balance is used: 2 2 ) 𝜌𝜌𝑠𝑠 𝑐𝑐𝑝𝑝,𝑠𝑠 𝜋𝜋𝜋𝜋𝜋𝜋(𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑟𝑟𝑜𝑜𝑜𝑜𝑜𝑜

𝜕𝜕𝑇𝑇𝑙𝑙 (𝑡𝑡) = 𝑄𝑄̇𝐻𝐻𝑇𝑇𝑇𝑇,𝑆𝑆𝑆𝑆𝑆𝑆 (𝑡𝑡) 𝜕𝜕𝜕𝜕

Once the temperature of the solidified PCM is equal to the temperature of the HTF the solidification process is completed. These are the main equation to model the solidification process. The equations are numerically solved using a forward differencing scheme. An upwind scheme is used in order to update the boundary condition (Temperature of the HTF) for all the segments. The sum of the heat fluxes per segments delivers the total heat flux of one tube coil. This heat flux is multiplied with the number of tube coils in order to predict the heat flux of the complete storage unit.