Modeling of solidification including supercooling effects in a fin-tube heat exchanger based latent heat storage

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Modeling of solidification including supercooling effects in a fin-tube heat exchanger based latent heat storage Remo Wasera, , Simon Marandaa, Anastasia Stamatioua, Maurizio Zagliob, Joerg Worlitscheka ⁎

a b

Lucernce University of Applied Science and Arts, Competence Center Thermal Energy Storage HSLU, Technikumstrasse 21, CH-6048 Horw, Switzerland Sunamp Switzerland GmbH, Ueberlandstrasse 129, CH-8600 Duebendorf, Switzerland

ARTICLE INFO

ABSTRACT

Keywords: Latent heat storage Modeling Solidification Phase change kinetics Supercooling Fin-tube heat exchanger

Latent heat storages are a promising technology to increase both the efficiency of thermal energy systems and the penetration of renewable energy sources such as solar energy. A major challenge regarding latent storage technology is the relatively high costs resulting from expensive heat exchangers concepts required to transfer thermal energy from the storage material to the heat transfer fluid. Within the presented study, a new modeling method which allows for the optimization of complex heat exchanger designs such as fin-tube concepts in latent storages units is proposed. The mathematical model of the heat transfer in latent heat storages during the solidification process can account for the effects resulting from the supercooling of the phase change material. In order to achieve a low computational effort, the model is composed of two subparts. The first is a 3-dimensional model describing the thermal behavior of the phase change material surrounding the fin structure of the heat exchanger. The second is a 1-dimensional model of the tube coil which allows obtaining the overall heat flow rate in the entire storage unit based on the 3-dimensional phase change material model. The developed model is calibrated and validated using experimental data of a commercial sodium acetate trihydrate based storage unit of manufacturer Sunamp with a phase change temperature of 58 °C. A good agreement between experimental and simulated heat flow rates and heat transfer fluid outlet temperatures could be observed. Moreover, the model is capable of predicting supercooling effects with high accuracy. Due to the acceptable computational effort achieved, the developed model may be used for a heat exchanger geometry optimization in future studies.

2010 MSC: 00-01 99-00

1. Introduction According to the international energy agency IEA in 2014, heat accounts for more than a third of world energy consumption (Eisentraut and Adam, 2014). In 2016, around 50% of the total energy consumption in the EU was used for space heat, domestic hot water, and process heat (An EU Strategy on Heating and Cooling, 2016). With the use of thermal energy storages, the mismatch between supply and demand can be reduced and simultaneously, the end user’s flexibility and self-consumption can be enlarged (Rathgeber et al., 2015). Accordingly, thermal energy storage systems are regarded as a key technology for enabling an increased share of renewable solar energy in the supply system. Latent thermal energy storages use a phase change, usually from liquid to solid to store thermal energy. Hence, they offer the ability to store a considerable amount of energy on a nearly constant temperature level. The resulting volumetric energy density of latent storage is significantly higher than the density reached by sensible

⁎

systems (Zhou et al., 2012). Phase Change Material (PCM) storages have been proposed amongst others as suitable units to absorb solar heat from collectors and provide it directly when needed (Teamah et al., 2018; Elbahjaoui and Qarnia, 2018; Wang et al., 2017; Pereira et al., 2018) or utilized in solar cooling applications Palomba et al.. Furthermore, latent storages were proposed for solar assisted heat pump systems (Youssef et al., 2017; Shirinbakhsh et al., 2018). A major challenge related to latent heat storages is the considerably low thermal conductivity of PCMs which makes the delivery of high thermal power challenging. During solidification, a growing solid PCM layer with a relatively high thermal resistance is formed (Kroeger and Ostrach, 1974). Contrary, during the melting process, free convection in the liquid phase leads to a heat transfer enhancement in comparison to the solidification process (Murray and Groulx, 2014). Consequently, when it comes to the design of a latent heat storage, the process of solidification is more critical than melting. To achieve the heat transfer rates required, the usage of fin-tube heat

Corresponding author. E-mail address: [email protected] (R. Waser).

https://doi.org/10.1016/j.solener.2018.12.020 Received 6 July 2018; Received in revised form 25 October 2018; Accepted 8 December 2018 0038-092X/ © 2018 Elsevier Ltd. All rights reserved.

Please cite this article as: Waser, R., Solar Energy, https://doi.org/10.1016/j.solener.2018.12.020

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Nomenclature

n THTF , Seg, i Tm TPC , s TPC , l TPCM , min TPCM , max TPCM , init TPCM , final Tref Tsup

Roman symbols

Cmax , Seg cp, HTF cp, l cp, s f Fcry h href H Inuc K K1 K2 K3 L Ladj LSeg HPC mPCM , Seg mTc NTc Qcum QHTF QHTF , Sim QHTF , Exp n

QHTF , Seg, i n

QHTF , Tc n

QHTF tsup T THTF THTF , in THTF , out

maximal capacity of a segment, J specific heat capacity of the HTF, J/(kg·K) specific heat capacity of PCM in the liquid phase, J/(kg·K) specific heat capacity of PCM in the solid phase, J/(kg·K) nucleation barrier reduction factor, – crystallization probability function, – sensible heat of the PCM, J/kg reference enthalpy, J/kg enthalpy of the PCM, J/kg nucleation factor, 1/s coverage factor, – constant nucleation pre-exponential factor, – nucleation fitting parameter, K2 fitting parameter for adjacent crystals, J Load, J Load of adjacent segments, J Load of a segment, J latent heat of the material, J/kg PCM mass in a segment, kg HTF mass flow rate in a tube coil, kg/s number of parallel tube coils, – cumulative thermal energy, J heat flow rate between HTF and PCM, W model generated heat flow rate between HTF and PCM, W experimentally determined heat flow rate between HTF and PCM, W heat flow rate between HTF and PCM in segment i at time step n, W totel heat flow rate between HTF and PCM in a tube coil, W totel heat flow rate between HTF and PCM in the entire storage system, W supercooling time, s temperature of the storage material, K temperature of HTF, K if not otherwise specified HTF inlet temperature, K HTF outlet temperature, K

temperature of the HTF in segment i at time step n, K melting temperature, K solidus temperature, K liquidus temperature, K minimum PCM temperature, K maximum PCM temperature, K initial experimental PCM temperature, K final experimental PCM temperature, K reference temperature, K supercooling temperature, K

Greek symbols HTF

l l

l s

convective heat transfer coefficient between the heat transfer fluid and the inner heat exchanger tube wall, W/m2 K phase indicator, – phase indicatior (liquid fraction), – heat conductivity, W/mK heat conductivity of PCM liquid, W/mK heat conductivity of PCM solid, W/mK density, kg/m3 density of PCM in the liquid phase, kg/m3 density of PCM in the solid phase, kg/m3

Abbreviations l cha max min PC init final HEX HTF ref Tc sup

exchangers is a promising approach since such heat exchangers offer a high heat transfer area and their production is standardized resulting in low labor costs during manufacturing. Nevertheless, fin-tube heat exchangers (HEX) have considerable material costs since expensive metals with a high conductivity such as aluminum or copper are usually used. Rathgeber et al. demonstrated in (Rathgeber et al., 2015) that the main cost driver of latent storages is not primarily the PCM but the storage vessel and the charging/discharging device, namely the heat exchanger. This indicates the necessity of approaches to reduce costs of heat exchangers and therefore of latent storages in general. Mathematical models to describe the heat flow during solidification can provide a cost-effective method to tackle this challenge provided that the models are accurate and fast. In the present study, a new modeling approach is presented to predict the heat transfer rate from a PCM to the heat transfer fluid (HTF) during solidification around complex 3-dimensional heat exchanger geometries including supercooling. The model is calibrated and validated using experimental data obtained with a commercial Sunamp heat battery. The Sunamp storage system is composed of a fin-tube heat exchanger which is surrounded by a patented sodium acetate trihydrate based PCM. The melting temperature is 58 °C. The evolution of the heat flow rate during discharge is experimentally obtained for different HTF

liquid charge maximum minimum phase change initial final heat exchanger heat transfer fluid reference tube coil supercooling

inlet temperatures and used for model calibration and validation. 2. Numerical modeling approaches of latent storages 2.1. Modeling of phase change The modeling of latent storages is related to the mathematical description of solid/liquid phase changes. The enthalpy method and the effective heat capacity method (Lamberg et al., 2004) are the two approaches which are usually used to solve phase change problems numerically. Governing equations are introduced that are applied for both, the solid and the liquid phase where enthalpy is described as a temperature dependent algebraic expression (Al-Abidi et al., 2013). A phase change temperature range (the so-called mushy zone) is defined in which the phase change enthalpy is added to or subtracted from the specific temperature dependent enthalpy of the PCM. In the effective heat capacity method, the latent heat is approximated by a large sensible heat over the phase change temperature interval (Mersmann, 2001). Thus, the solid/liquid interface is not explicitly tracked but its location is defined by the mushy zone. Both, the finite volume method and the finite element method can be applied to discretize the resulting energy governing equation from the enthalpy based and effective heat 2

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capacity formulation. The major drawback is the high spatial and temporal resolution required to model complex HEX geometries such as fin-tube heat exchangers. The methods have been applied for the thermal analysis of phase changes around fin-tube heat exchangers (Zauner et al., 2016; Jmal and Baccar, 2015; Khalifa et al., 2014; Waser et al., 2018) spherical capsules (Fan et al., 2016) or within shell-tubes using internal fins (Wang et al., 2016; Rozenfeld et al., 2015; Riahi et al., 2017; Fornarelli et al., 2016; Yuan et al., 2016). Most of the studies exploited the symmetric geometry of the heat exchanger or capsules to reduce the computational effort. However, the prediction of the thermal behavior of the entire storage device remains challenging since an appropriate procedure has to be developed to characterize the overall thermal behavior of the entire system based on the small simulation domain. This challenge has been addresses by Neumann et al. (2017). A reduced computational effort and high accuracy were achieved by coupling a 1-dimensional model for the tubes of the heat exchanger and a reduced 3-dimensional model for the material and fin domains. The approach proposed in this study is similar to the approach proposed by Neumann et al.. However, the proposed model also allows for a prediction of the impact of supercooling on the thermal output of the system during discharging.

(Dannemand et al., 2016). Using this technique, the heat of fusion released during solidification can be stored without losses for a long period of time. On the contrary, supercooling is to be avoided in shortterm storages because it prevents the heat of fusion to be released when needed which might affect the performance of the system adversely (Li et al., 2014). Salt hydrates which are due to their low price and high volumetric heat of fusion promising materials for latent storages show supercooling (Kenisarin and Mahkamov, 2016) which may affect the discharging process significantly. Modeling of supercooling in latent storages has been investigated in various studies. Schranzhofer and Puschnig (2006) introduced a definition to calculate a the temperature at which nucleation occurs. The method was adapted by Günther et al. (2007). Both Schranhofer, as well as Guenther, used finite difference method for discretization. Yosef et al. (2017) modeled solidification with supercooling effects using a time and temperature dependent initiation of crystallization. A 2-dimensional finite volume discretization was used and the model was validated with a simple vertical cylinder geometry. Bédécarrats et al. (2009) modeled solidification including supercooling effects using a 1dimensional finite volume discretization in combination with a quasistationary approximation. A nucleation probability in function of supercooling temperature and time is used to determine the point of crystallization. The studies found in literature which model supercooling focus on simple geometries. In the present study, a new modeling approach which allows for a prediction of the heat transfer rate from PCM to the HTF during solidification around complex 3-dimensional HEX geometries including supercooling effects is presented.

2.2. Modeling of supercooling Nucleation is described as the initial process that occurs in the formation of a crystal from a liquid in which a small number of ions, atoms or molecules become arranged in a pattern characteristic of a crystalline solid (Mersmann, 2001). A liquid, which temperature is between the solid/liquid phase equilibrium temperature (herein melting temperature Tm ) and the nucleation temperature is referred to as supercooled Safari et al.. In this study, the quantification of the chemical potential of nucleation is reduced to a temperature difference Tsup . This so-called supercooling temperature difference is defined as follows:

Tsup = Tm

T

3. Methodology and materials 3.1. Numerical model of the storage unit In Fig. 1, a simplified schematic of the investigated PCM surrounded fin-tube heat exchanger is shown. The HTF is flowing through the HEX tubes to charge and discharge the storage. During discharging, heat is transferred from the PCM to the HTF and the phase change material undergoes a phase change from liquid to solid. To mathematically describe this complex process, the proposed model is divided into two

(1)

The effect of supercooling due to poor nucleation can be exploited in long-term storages since it is possible to hold the temperature of the supercooled PCM in equilibrium with the ambient temperature

Fig. 1. Illustration of the basic concept of the model. The phase change is modeled within a small section of the HEX geometry which is half of the smallest symmetric segment. 3

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In the equations above the solidus temperature TPC , s and the liquidus temperature TPC , l are introduced. The liquidus temperature is the lowest temperature at which the material is completely liquid. The solidus temperature is the highest temperature during the PCM is in a fully solid state. A phase change temperature range is introduced, where the part of the PCM that consists of temperatures between the solidus and the liquidus temperature is called the “mushy zone”. The mushy zone helps to avoid discontinuities which may lead to numerical instabilities (Hahn and Özisik, 2012). The latent heat content can now be written in terms of the latent heat of the material, hPC :

H=

Fig. 2. Simulation domain descretized using the finite volume method.

According to Kroeger and Ostrach (1974) or Murray and Groulx (2014), convective effects do not play a significant role during solidification, but may have a great influence on the melting process, particularly if a PCM with a rather low viscosity is used. Since melting is not considered in the present study, mass transfer due to free convection is not modeled.

subroutines as illustrated in Fig. 1. A 1-dimensional tube model describes the time and location dependent HTF temperature evolution along the heat exchanger tube. The thermal behavior of the PCM/HEX compound and the resulting heat transfer from the PCM to the HTF is described within a more complex 3-dimensional PCM model. To minimize the computational effort, the simulation domain of the PCM model is the smallest possible symmetrical HEX segment. The heat flow rate from PCM to HTF QHTF depends on the conditions at the inner tube wall, namely the HTF temperature THTF and convective heat transfer coefficient HTF as well as the current state of the PCM surrounding the tube in terms of enthalpy. To link the two models, two datasets are generated, using the 3-dimensional model. Both datasets contain values for QHTF for a set of discharging processes from 68 °C initial temperature (10 K above the phase change temperature) to a defined boundary temperature at the inner tube wall. The first dataset contains data assuming that the phase change occurs as the melting temperature is reached. The second dataset contains heat flow rates from PCM to HTF when the PCM remains liquid during the entire discharging process. In the 1-dimensional tube model, the heat flow rate from PCM to HTF obtained with the 3-dimensional model is used as source term and depending on the state of the PCM in the symmetric segment (liquid, supercooled or crystallized), the heat flow rate is queried from the first or second dataset. The 3-dimensional PCM model, as well as the 1-dimensional tube model, will be described in detail hereinafter.

3.1.1.2. Governing equation without phase change. The second dataset contains heat flow rates assuming that the PCM remains completely liquid during the entire discharging process. To generate the dataset, Eqs. (2)–(4) are applied setting H = 0 in Eq. (3). 3.1.1.3. Geometry and mesh. The simulation domain is based on the smallest symmetric segment of the HEX and is shown in Fig. 2. It consists of one-eighth of the HEX geometry radially and one half of the geometry between two fins. The energy equation (2) is discretized using finite volume method (FVM). Second order upwind spatial and first order implicit time discretization schemes are chosen. 3.1.1.4. Boundary conditions and generation of datasets. A convective boundary condition (BC) at the inner tube surface is applied. At all other boundaries of the simulation domain, symmetric conditions are assumed. To create the datasets, constant boundary conditions are applied with a constant initial temperature of 68 °C. A complete discharging sequence is simulated for free stream temperature between 15 °C and 68 °C and convective heat transfer coefficients between 200 W/(m2·K) and 2000 W/(m2·K) . The heat flow within a segment is a function of the HTF temperature THTF , the convective heat transfer coefficient HTF and the current load of the segment LSeg . The load of the segment is defined as:

3.1.1. 3-Dimensional phase change model 3.1.1.1. Governing equation to model phase change. The first dataset of the 3-dimensional model is generated using the previously described enthalpy method. Commercial CFD software Ansys Fluent is used within this study. The governing energy equation applied to the discretized PCM domain reads:

t

( H) =

·(

T)

LSeg (tc ) =

(2)

T Tref

cp dT

T

TPC , s TPC , s

(4)

Tm ))

3.1.2. 1-Dimensional tube model The tube model is based on a 1-dimensional discretization of a heat exchanger tube coil into n segments as illustrated in Fig. 3. To calculate the temperature of the HTF as a function of time and space, an energy balance is applied for the HTF in the defined segment. Using an upwind discretization scheme, the temperature in segment i at time step n reads:

if TPC, s < T < TPC, l (mushy)

1 if T > TPC , l (liquid)

TPCM , min) + hPC + cp, l (TPCM , max

Here, mPCM , Seg is the mass of the PCM within the simulation domain of the 3-dimensional PCM model. TPCM , max is the maximum PCM temperature which is set to 68 °C and TPCM , min the minimum PCM temperature set to 15 °C. Tm is the phase change temperature and amounts to 58 °C in the present case.

0 if T < TPC , s (solid) TPC , l

(7)

Cmax, Seg

(8)

where cp is the specific heat capacity of the PCM as a function of temperature, Tref an arbitrary reference temperature and href a reference enthalpy. is the so-called phase indicator and represents the liquid fraction within a control volume. The phase indicator may be expressed as:

=

QHTF , Seg (t ) dt

Cmax , Seg = mPCM , Seg (cp, s (Tm

(3)

The sensible heat can further be specified:

h = href +

tc t=0

where tc is the current time of the discharging process and Cmax , Seg is the theoretical maximal capacity of the segment which is approximated using:

where is the temperature dependent density and the temperature dependent heat conductivity. The enthalpy H consists of sensible heat h extended by the latent heat H :

H=h+ H

(6)

hPC

(5) 4

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The probability Fcry is a function of the supercooling time tsup and the nucleation factor Inuc . tsup is the period of time during which the PCM in the segment has been below the phase change temperature in liquid from. Inuc is related to the heterogeneous nucleation rate defined in the classic nucleation theory and is in this study defined as:

Fig. 3. Discretization scheme of the finned tube heat exchanger.

n THTF , Seg, i =

n QHTF , Seg, i

mTc cp, HTF

Inuc (Tsup, f ) = K1·exp n + THTF , Seg, i

1

(9)

where is the heat flow rate from the PCM to the HTF in segment i at time step n, mTc the mass flow rate of the HTF in a tube coil and cp, HTF the specific heat capacity of the HTF. The HTF temperature in the first segment corresponds to the inlet temperature. Thus, following boundary condition is formulated: (10)

Applying (9) and the boundary condition formulated in Eq. (10) for each element and time step, the HTF temperature can be calculated in function of time and space. Finally the heat flow rate of the entire tube coil at time step n is n QHTF , Tc is the sum of the occurring heat flows in all segments, or: I

n

QHTF , Tc =

and the heat flow rate of the entire system at time step n n QHTF

=

f (Ladj ) = 1

(11)

n NTc ·QHTF , Tc

n QHTF

is: (12)

where NTc is the number of parallel tube coils in the storage system. 3.1.2.1. Calculation of convective heat transfer coefficient. Due to the temperature-dependent viscosity of the water, the convective heat transfer coefficient between the HTF and the inner tube wall is a function of temperature. Investigations performed in this study showed that a laminar flow regime might be present at the HEX inlet which then turns into a turbulent flow due to the increased temperature. Thus, to calculate the convective coefficient HTF , the empirical correlations proposed by Gnielinski (2013) is applied which describes convection in laminar, turbulent and transition flow regimes. 3.1.3. Crystallization probability function Due to the stochastic nature of nucleation, the determination of the point of time when crystallization begins is challenging. In the present study, a new time, temperature and location dependent crystallization probability function is proposed.

Fcry (tsup, Ihet ) = 1

e

Inuc·tsup

Ladj K3

(15)

where Ladj is the average load of the crystallized segments adjacent to the considered segment. Ladj is a direct measure for the number of solid crystals present in adjacent segments. As Ladj increases, the number of solid crystals in adjacent segments increases which reduces the nucleation barrier in the considered segment. K3 is used to calibrate the influence of crystals in adjacent segments on the nucleation rate in the considered segment and is used as a second fitting parameter. In Fig. 4(a), the crystallization probability F is plotted against the supercooling time and the supercooling temperature for crystallization barrier reduction factor of f = 0.1. As can be seen, the probability strongly depends on the supercooling temperature whereas the influence of time is less dominant. This is reasonable since in the classic nucleation theory, the nucleation rate is mainly a function of the supercooling temperature (Stefanescu, 2009). In Fig. 4(b), the probability function at a constant supercooling time of tsup = 1 s is plotted for different values for f. As illustrated, the crystallization temperature (temperature when the crystallization probability reaches 1) decreases with increased values for f. Thus, the supercooling temperature decreases as the presence of crystals in adjacent segments increases which is in again good agreement with heterogeneous nucleation theory. In Fig. 5 the heat flow rate from PCM to HTF within a segment is plotted against the load of the segment. The solid black line is an example of a discharging process considering the phase change and the red dashed line is a discharging process neglecting the phase change. In the proposed 1-dimensional model, the dashed red line will be considered as long as the PCM is liquid or supercooled. As crystallization is triggered based on the described approach, QHTF , Seg is queried from the

n

QHTF , Seg, i i=1

(14)

Inuc depends on the supercooling temperature Tsup and a nucleation barrier reduction factor f. The pre-exponential factor K1 is a constant and arbitrary set to 1030 . K2 is a fitting parameter and influences the critical supercooling temperature at which the crystallization probability is increasing. The variable f is defined to account for the fact that segments in the proximity of already crystallized segments have a chance of getting in contact with a crystal which reduced to crystallization barrier. The crystallization barrier reduction factor f is defined as:

n QHTF , Seg, i

THTF ,0 = THTF , in

K2 ·f 2 Tsup

(13)

Fig. 4. (a) Crystallization probability as a function of the supercooling temperature and time for f = 0.1. (b) Crystallization probability as a function of the supercooling temperature for tsup = 1 and various crystallization barrier reduction factors f. 5

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4. Experimental 4.1. Storage unit and PCM A photograph of the investigated commercial latent storage unit of supplier Sunamp is shown in Fig. 6. The total volume of the unit is approximately 24 l and contains 17 l of a patented sodium acetate trihydrate based PCM. The system uses a single port metallic fin-tube heat exchanger to transfer heat from the PCM to the HTF and vice versa. The thermophysical properties of the PCM are outlined in Table 1: 4.2. Experimental design The test facility is designed to discharge the storage unit with a constant inlet temperature THTF , in and constant mass flow mHTF of the HTF (water). A schematic presenting of the experimental setup is shown in Fig. 7. During discharge, cold water flows through the HEX in an open loop. To charge the storage unit water is cycled in a closed loop driven by the pump. The required heat to charge the PCM is delivered by an external thermostat. A screw-on valve is used to adjust the mass flow. Temperature sensors are installed at the inlet and the outlet of the storage unit. Finally, the mass flow meter is placed in front of the inlet. The sensors and their specifications are listed in Table 2.

Fig. 5. Heat flow in a segment in function of the load of the segment for supercooled PCM w/o phase change and crystallizing PCM.

4.3. Quantities of interest and uncertainty analysis The primary objective of the experiments was to determine the time-dependent heat flow rate QHTF which is released by the PCM and absorbed by the HTF during the discharging process. The temperatures at the inlet and the outlet and the mass flow were measured during the discharging process to do so. The heat flow rate is calculated with the following energy balance:

QHTF = mHTF · cp, HTF (THTF , out Fig. 6. Photograph of the Sunamp heat battery used for model validation and calibration.

where mHTF is the mass flow rate of the HTF, THTF , in and THTF , out are the inlet and outlet temperatures of the HTF, cp, HTF is the specific heat capacity of the HTF at the mean temperature between the inlet and the outlet. The cumulative energy Qcum which is transferred from the PCM to the HTF is evaluated by summing the instantaneous heat transfer rates over all time intervals during the discharging process:

Table 1 Thermophysical properties of the PCM used in the experiments provided by Sunamp. Melting temperature Spec. latent heat Density (liquid) Density (solid) Spec. heat capacity (liquid) Spec. heat capacity (solid) Heat conductivity (liquid) Heat conductivity (solid)

Tm hPC l s

cP ,l cP ,s l s

(16)

THTF , in )

58 °C 226 ± 5 J/g 1180 kg/m3 1280 kg/m3 3.5 J/(g K) 2.8 J/(g K) 1 W/(m K) 1 W/(m K)

tend

Qcum =

(QHTF t )

(17)

t=0

where t stands for the time interval between two measurement points ( t = 5 s ). The cumulative energy is of interest since it may be used to identify the load L of the storage unit during the discharging process. The load is defined as the ratio of the cumulative energy to the theoretical capacity of the PCM. Therefore it is a useful indicator to prove the validity of the measurements. The load is calculated as follows:

dataset generated with the enthalpy method taking the phase change into account. 3.1.3.1. Model assumptions. In summary, the following assumptions are the foundation of the proposed model:

L=

Qcum Cmax

(18)

where Cmax represents theoretical capacity of the storage unit which is defined as the difference of the internal energy of the storage unit between the initial and the final state of the experiment which is approximated with the following equation:

• Convective heat transfer within the PCM is neglected. • The phase change takes place within a temperature range of 1 K, hence from 57.5 °C to 58.5 °C. • No heat transfer between segments in the 3-dimensional model occurs. The only heat transfer is from the PCM to the HTF directly. • The material properties are constant within the solid and liquid phase. • The thermal losses from the PCM to ambient are neglected. • It is assumed that supercooling is influenced by the temperature,

Cmax = mPCM (cp, s (Tm

TPCM , final ) + hPC + cp, l (TPCM , init

Tm))

(19)

In Eq. (8), mPCM describes the entire PCM mass in the storage device, TPCM , final the temperature of the PCM at the end of the discharging process and TPCM , init the temperature of the PCM at the beginning of the discharging process. The load identifies the state of the discharging process between 0 and 1, whereby the latter stands for the state at

time and presence of solid crystals only.

6

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Fig. 7. Schematic of the test facility. Table 2 Sensors used in the test facility. Measurement point

Symbol

Sensor type

Range

Measurement-uncertainty

Temperature of the HTF at the inlet Temperature of the HTF at the outlet Massflow of HTF

TI01/THTF , in TI02/THTF , out FI01/mHTF

PT-100, Class DIN 1/5, Type B PT-100, Class DIN 1/5, Type B Coriolis, Promass 83F, Endress + Hauser

−40–140 °C −40–140 °C 100–1000 kg/h

± (0.06 ° C+ 0.01THTF ) (THTF in °C) ± (0.06 ° C+ 0.01THTF ) (THTF in °C) ± (0.0035m)

heat flow rate as follows (Guide to the Expression of Uncertainty in Measurement, 2003):

Table 3 Experimental matrix. ∗Cmax is calculated using Eq. (19) and the values of 15 °C, 40 °C and 50 °C are used as final temperatures and 68 °C as initial temperature of the PCM.

Experiment 1 Experiment 2 Experiment 3

THTF , in = Tfinal

mHTF

50 ± 1 °C 40 ± 1 °C 15 ± 1 °C

0.1 kg/s 0.1 kg/s 0.1 kg/s

TPCM , init

68°C 68°C 68°C

Cmax ∗

Purpose

2.03 kW h 1.65 kW h 1.53 kW h

Calibration Validation Validation

dQHTF =

QHTF mHTF

2

+

mHTF

QHTF THTF , in

2

+

THTF , in

QHTF THTF , out

2

K

THTF , out

(20) The following calculation of the upper and lower bound of the cumulative energy is used:

dQcum = t

QHTF mHTF

2

+

mHTF

QHTF THTF , in

2 THTF , in

+

QHTF THTF , out

2 THTF , out

K (21)

The resulting uncertainty for the load reads: dL =

t Cmax

QHTF mHTF

2 mHTF

+

QHTF THTF , in

2 THTF , in

+

QHTF THTF , out

2 THTF , out

K

(22) In the equation above K represents the coverage factor. An standardized confidence interval of 95% is chosen (K = 2 ). An investigation on the thermal losses of the unit Qlosses showed that the losses are within the magnitude of the measurement uncertainty of QHTF . Furthermore, losses decrease during the discharging process since the temperature difference between the PCM and the ambient decreases the more heat is released by the PCM. Consequently, it is reasonable to neglect the heat losses.

Fig. 8. Time averaged deviation between experimental and simulated data as a function of fitting parameter K2 and K3 .

4.4. Experimental procedure The experiments performed can be divided into two categories: (i) experiments for model calibration and (ii) validation. For both cases, discharging sequences were measured from an initially fully charged system with a temperature of 68 °C (10 K above the phase change temperature). To charge the system, hot water with an inlet temperature of 68 °C was circulated through the heat exchanger until a constant temperature difference of 0.2 K between the inlet and the outlet was reached. Consequently, it can be assumed that at the beginning of each experiment, the entire temperature of the PCM (Tinit ) was approximately

which the complete energy of the PCM is released. In this work, the primary experimental uncertainty results from the measurement tolerance of the sensors. The measurements uncertainty of the individual sensors mHTF , mHTF ,in and THTF , out are given by the producers and are listed in Table 2. The standard measurement uncertainty with regard to the heat flow rate QHTF is indicated through confidence interval with an upper bound + dQHTF and a lower bound QHTF . Based on the propagation of uncertainty definition, the upper and the lower bound can be calculated using the total differential of the 7

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Fig. 9. (a) Measured and model-generated outlet temperature and (b) corresponding heat flow rate from PCM to HTF during discharging. (c) Measured and modelgenerated load of the storage against discharging time.

68 °C. Each experiment was performed until the outlet temperature was constant and the difference between outlet and inlet temperature is less than 0.1 K. It is assumed that the final temperature of the PCM Tfinal was equal to the inlet temperature of the HTF. Three different experiments with three different inlet temperatures (50 °C, 40 °C and 15 °C ) were performed. As shown in Table 3, the first experiment was used to calibrate the model estimating the two fitting parameters of the crystallization probability. Experiment 2 and experiment 3 were used to validate the model. Each experiment was performed at least three times to verify the repeatability of the measurements.

5. Results and discussion 5.1. Calibration of supercooling effects The result of the parametric study is shown in Fig. 8. As illustrated, a local minimal deviation of approximately 140 W could be found with K 2 = 7250 K 2 and K3 = 0.7 . The model-generated and measured HTF outlet temperatures against the time with a constant inlet temperature of 50 °C are plotted in Fig. 9 (a). The uncertainties of the experimental results are indicated with error bars for every 20th measurement point. It can be observed that the temperature is maximal at the beginning of the experiment. The temperature drops below 55 °C at around 2 min where it reaches a temporary minimal value. This minimum is due to supercooling when crystallization is inhibited and the phase change enthalpy is not yet released. The PCM is cooled below its phase change temperature without changing its phase. Once the nucleation has started, the released phase change enthalpy results in a sudden temperature increase of the supercooled PCM. This leads to the temporary increase of the outlet temperature which is clearly visible in the figure. Between 10 min and 30 min discharging time, a clear plateau can be observed during which the outlet temperature remains nearly constant. During this time, the PCM solidifies and heat of fusion is released. As shown, a qualitatively excellent agreement between simulation and experimental results is achieved. The model shows the ability to predict supercooling correctly with a temporary reduces overall heat flow rate at around 2 and 10 min discharging time. In Fig. 9(c), the corresponding heat flow rate is plotted. Initially, a thermal output of around 7.5 kW is reached by the system which then drops to less than 2.5 kW. The thermal output

4.5. Model calibration The parameters K1 and K3 were estimated based on the experimental data obtained with an HTF inlet temperature of 50 °C. Supercooling effects are most apparent in this experiment, and hence the experiment appears to be suitable for the fitting of model parameters related to the supercooling effect. An iterative, heuristic parametric study was performed to minimize the time-averaged deviation between experimentally determined heat flow rate from the PCM to the HTF and the simulated heat flow rate QHTF . Mathematically, QHTF is defined as:

QHTF =

N n= 1

n

|QHTF , Sim N

n

QHTF , Exp |

(23)

where N is the total number of measured or simulated data points.

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Fig. 10. (a) Model-generated and experimental outlet temperature and (b) corresponding heat flow rate from PCM to HTF versus discharging time for an inlet temperature of 15 °C and 40 °C. (c) Model-generated and measured load against time for an inlet temperature of 15 °C and 40 °C.

amounts to around 2.5 kW during the solidification plateau. Finally,Fig. 9(c) shows the load of the storage units as a function of the discharging time. The plot confirms the good agreement between model-generated and measured data.

by the same effect. In Fig. 10(b), again, the corresponding heat flow rates are plotted. A significant decrease in the heat flow rate is observed between the discharging process with 15 °C inlet temperature and 40 °C inlet temperature. This is reasonable since the temperature difference between the HTF and PCM is the main driving force of the heat transfer between the HTF and the PCM. The lower the inlet temperature, the more heat is transferred per unit time, which is visible in Fig. 10(b). In Fig. 10(c), the measured and simulated load according to Eq. (18) are plotted as a function of time. The load increases almost linearly with time until it converges to a value of approximately 1. This proves that the cumulative released energy is well aligned with the theoretical storage capacity, which supports the confidence in the results. It is visible that the load slightly exceeds the theoretical maximum of 1. The reason mainly lies in the way the maximal theoretical capacity is defined: Constant heat capacities in both the liquid and the solid phases are assumed and the heat losses are neglected. Furthermore, it is assumed that the initial and final temperatures of the PCM are known values. In the experiment, these temperatures are not measured directly but estimated via the temperatures of the HTF. These assumptions may be erroneous and could, therefore, yield to load values above 1. In this section, it could be proven, that the model predicts outlet temperature and heat flow rate with high reliability for different discharging conditions. It is remarkable that the discharging time, which is a critical requirement in practical applications, is predicted with high accuracy in both cases. Hence, based on the results, the reliability and applicability regarding practical applications with different inlet temperatures could be proven.

5.2. Model validation Fig. 10(a) illustrates the measured and simulated outlet temperatures THTF , out for the two different inlet temperatures used for model validation (black curve: THTF , in = 15 °C , red curve: THTF , in = 40 °C ). In both cases, an excellent agreement between model-generated and simulated data can be observed. In the 15 °C HTF inlet temperature case, the outlet temperature (and heat flow rate in (b)) during the first four minutes of the discharging process is slightly overestimated by the model. Consequently, the time during which the HTF outlet temperature lies above 45 °C is slightly underestimated. The numerical results for an HTF inlet temperature of 40 °C are also in good agreement with experimental results. Similar to the 15 °C case, the outlet temperature (and heat flow rate) is overestimated in the first part (t < 10 min ) of the discharging process. The measured outlet temperature obtained with an inlet temperatures of 40 °C increases after temporary minimum at around 2 min discharging time. This behavior again is attributed to supercooling of the PCM. The effect is not observed for the experiments with an inlet temperature of 15 °C (black curve), however, based on the numerical results it can be stated that supercooling effects are also present with a HTF inlet temperature of 15 °C, as it will be shown below. It appears that the effect of supercooling is underestimated by the model. Based on this finding it could be derived that the deviation observed in the 15 °C case could be caused 9

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Fig. 11. Comparison between the experimentally obtained heat flow rate from PCM to the HTF for (a) 15 °C (c) 40 °C and (e) 50 °C HTF inlet temperature and numerical results with and without supercooling. In subfigures (b), (d) and (f) the corresponding deviations between experimentally and numerically obtained heat flow rates are plotted versus time for the three cases.

5.3. The effect of including supercooling in the model

the simulations which do not take into account supercooling effects. In the 15 °C case, the plateau is overestimated by approximately 13% if supercooling is neglected. On the other hand, if supercooling is modeled, the deviation amounts to less than 5%. Hence, it appears that supercooling effects are present even if they are not apparent in the heat transfer outlet temperature curve. The 40 °C inlet temperature case points to a similar conclusion as the 15 °C case. The model which neglects supercooling effects shows a deviation of more than 2.4 kW when the plateau occurs whereas the difference between simulated and measured heat flow rate is less than 1 kW for the model which includes supercooling effects. Finally, in the 50 °C, as expected, the difference

In Fig. 11(a), (c) and (e), the model-generated results including and neglecting supercooling effects for all three inlet temperatures as well as the experimentally obtained heat flow rates are plotted versus time. Furthermore, in the plots (b), (d) and (f), the deviation between experimental and numerical results is plotted against time for simulations including and neglecting supercooling effects. Furthermore, in Fig. 12, the corresponding outlet temperatures can be found. As illustrated, in all three cases, the simulations including supercooling effects show a considerably smaller deviation in average than 10

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contrary, the model with supercooling effects is capable of predicting a temporary temperature increase between approximately 3 min and 15 min correctly. Hence, the total discharging time is predicted with high accuracy. 5.4. Computational effort The computational effort for calculating a discharging sequence for a given HEX geometry is less than 2 h on a work station with Intel(R) Xeon(R) CPU E5-1650 v4 @ 3.6Ghz, 3601 MHz, 6 Cores, 12 logical Processors and 128 GB RAM. Hence, the model combines high reliability, high spatial resolution and acceptable computational effort which makes it a suitable design and optimization tool for fin-tube heat exchangers in latent storage units. 6. Conclusion and outlook In the presented study, a latent thermal energy storage unit with a fin-tube heat exchanger was experimentally and numerically investigated. The study focused on the discharging (solidification) process which is more challenging concerning thermal output and heat exchanger design. An experimental study with three different HTF inlet temperatures has been conducted. Furthermore, a novel mathematical model of latent storage units was developed combining accuracy, reliability prediction of discharging times for different discharging conditions, acceptable computational effort, and the possibility to estimate the effects of supercooling. The model is composed of a 3-dimensional phase change model which is combined with a 1-dimensional model of the heat transfer fluid flow in the tube coil. The two models are linked via two datasets containing the heat flow rates from the phase change material to the heat transfer fluid for all possible states and conditions as they are calculated by the 3-dimensional model. Effects of supercooling are considered by the introduction of a crystallization probability function. The model validation showed a good agreement between experimental and model-generated data confirming the reliability of the model. Furthermore, it could be shown that supercooling effects may have a major impact on the heat transfer rate even if they are not apparent in the measured temperature profile of the HTF outlet. This results is of high importance since most of the current numerical studies neglect supercooling in their modeling methods. In future work, the focus will be on the elaboration of the proposed crystallization probability function. Ideally, the function itself could be experimentally validated using a suitable experimental setup to characterize the phase change kinetics of the material. Due to its acceptable computational effort, the developed model will be used for the optimization of the heat exchanger geometry in terms of material costs in commercial Sunamp heat batteries. Conflict of interest We wish to confirm that there are no known conflicts of interest associated with this publication. Acknowledgements The research was conducted within the framework of the Swiss Competence Centers for Energy Research-Heat and Electricity Storage (SCCER-HAE) and was financially supported by the Commission for Technology and Innovation (CTI) Contract No. 1155002545. The authors would also like to thank the Swiss State Secretariat for Education, Research and Innovation (SERI) for the financial support under Contract No. 16.0082.

Fig. 12. Comparison between the experimentally obtained outlet temperatures for (a) 15 °C (c) 40 °C and (e) 50 °C HTF inlet temperature and numerical results with and without supercooling.

between the two models is most obvious since supercooling is the most prominent from all three cases. If supercooling effects are neglected, a high deviation between simulated and measured data can be observed and hence the discharging time is highly underestimated. On the

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