First version of simulation model of the thermo-chemical storage system

First version of simulation model of the thermo-chemical storage system Deliverable number: D3.1 Version 1.0 Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or CINEA. Neither the European Union nor the granting authority can be held responsible for them This page is intentionally left blank Basic information on deliverable Project Acronym BEST-STORAGE Project URL http://www.best-storage.eu Responsible partner Dorian Höffner (TUB) Deliverable nature Report (R) Dissemination level Public (PU) Contractual Delivery Date 30th of September 2024 Actual Delivery Date 30th of September 2024 Number of pages 25 Keywords sorption thermalstorage, model, thermal energy storage, heating, cooling, HVAC. Authors Dorian Höffner (TUB), Hossein Gharaee (TUB), Robert Knorre (TUB) Review Jose L. Corrales Ciganda (Tecnalia), Daniel Carbonell (OST) Approval Hugo Grasset (SOLINTEL) Deliverable D3.1 BEST-STORAGE CONSORTIUM SOLINTEL SOLINTEL M&P SL TECNALIA FUNDACION TECNALIA RESEARCH & INNOVATION CERTH ETHNIKO KENTRO EREVNAS KAI TECHNOLOGIKIS ANAPTYXIS TUB TECHNISCHE UNIVERSITAT BERLIN TEKNIKER FUNDACION TEKNIKER NEWTON NEWTON ENERGY SOLUTIONS BV EHPA EUROPEAN HEAT PUMP ASSOCIATION AVAN AVANZARE INNOVACION TECNOLOGICA SL TREA MITTETULUNDUSUHING TARTU REGIOONI ENERGIARGENTUUR GIR GIROA SOCIEDAD ANONIMA OST OST – OSTSCHWEIZER FACHHOCHSCHULE SUPSI SCUOLA UNIVERSITARIA PROFESSIONALE DELLA SVIZZERA ITALIANA Deliverable D3.1 CONTENTS 1 Introduction 1 2 Thermodynamic Model of Sorption Storage System 3 2.1 Steady-StateModel ....................................... 3 2.1.1 ModelDefinition..................................... 3 2.1.2 SimulationResults.................................... 8 2.2 DynamicModel.......................................... 12 3 Detailed Model of the Heat and Mass Exchanger 13 3.1 ModelDescription ........................................ 13 3.2 A/D-Submodel .......................................... 15 3.3 Mathematical Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3.1 A/D-Submodel...................................... 17 3.3.2 Main Conservation Balances in the A/D . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3.3 E/C-Model........................................ 20 3.3.4 Main Conservation Balances in the E/C . . . . . . . . . . . . . . . . . . . . . . . . . 22 4 Conclusion 24 Deliverable D3.1 EXECUTIVE SUMMARY This deliverable introduces and describes two advanced simulation frameworks designed to model sorption thermal storage systems based on Sodium-hydroxide and water. Both models are based on physical equations and allow insights into the systems’ behavior for diverse boundary conditions. The first model uses an integral approach to predict the steady-state working conditions of both the Absorber/Desorber-Unit and the Evaporator/Condenser unit. The second focuses on the detailed, time-dependent internal conditions of the Absorber/Desorber unit. The integral model appliestheNTU-Effectivenessmethod, solvingasetofequationstomaintainsystemstabilityby fulfilling conservation laws. This model, the most advanced in the project, will generate key performance indicators to estimate the performance of a demonstrator unit under certain boundary conditions. It will be further enhanced to account for thermal masses and system dynamics, although this dynamic version is still under development. Once complete, it will simulate the real system’s dynamic behavior. The second model takes a more granular approach, using differential equations based on heat and mass conservation. It captures the dynamic evolution of the system, resolving thermal and concentration fields in detail. This model provides deeper insights into the heat and mass transfer process, offering a local view of system behavior. Both models aim to predict experimental outcomes and provide insights into the absorption process, enabling an understanding of system dynamics and performance potential. Additionally, they will inform control strategies based on input/output conditions and heating system characteristics. Together, these models will support the accurate assessment and optimization of the upscaled system’s performance, each contributing according to their specific strengths. Deliverable D3.1 LIST OF ACRONYMS A/D Absorption/Desorption Unit E/C Evaporator/Condenser Unit HX Solution Heat Exchanger HMX Heat and Mass Exchanger NTU The Number of Transfer Units TCS Thermochemical Storage System LIST OF VARIABLE NAMES Symbol Explanation Unit AArea m2 ˙ CHeat Capacity Flow W K cpSpecific Heat Capacity J kg K dDiameter m diInner Diameter m doOuter Diameter m lTube Length m ∆hLV Specific Enthalpy of Evaporation J kg ˙ HEnthalpy Flow W mH2O in sol Mass of Water Present in the Solution kg mvap, EC Mass of Water Vapor Present in the EC Vapor Space kg mvap, AD Mass of Water Vapor Present in the AD Vapor Space kg ˙ mAD A/D External Water Mass Flow kg s ˙ mEC E/C External Water Mass Flow kg s ˙ mfluid Fluid Mass Flow (Representative fluid) kg s ˙ msol, in Solution Inlet Mass Flow kg s ˙ msol, out Solution Outlet Mass Flow kg s ˙ mvap Evaporated/Condensed Water Vapor Mass Flow kg s vap, abs Absorbed/Desorbed Water Vapor Mass Flow kg s Nu Nusselt Number - pPressure Pa Continued on next page Deliverable D3.1 Symbol Explanation Unit pAD A/D Pressure Pa pAD, sat, in Solution Saturation Pressure at the Inlet of one Submodel Pa pAD, sat, out Solution Saturation Pressure at the Outlet of one Submodel Pa pEC E/C Pressure Pa Pr Prandtl Number - ˙ QHeat Transfer W RHeat Resistance K W Re Reynolds Number - Sc Schmidt Number - Sh Sherwood Number - tAD, in A/D Water Inlet Temperature K tAD, out A/D Water Outlet Temperature K tEC, in External E/C Water Inlet Temperature K tEC, out External E/C Water Outlet Temperature K TAD, sat Internal Solution Saturation Temperature K TEC Internal EC Temperature K Tsol, in Internal Solution Input Saturation Temperature K Tsol, out Internal Solution Output Saturation Temperature K Twall Wall Temperature of the Pipe K UInternal Energy J UHeat Transfer Coefficient W m2K xAD, sat Internal Solution Saturation Temperature - xin Inlet Fraction of NaOH in Solution - xout Outlet Fraction of NaOH in Solution - Greek Symbols αHeat Transfer Coefficient W m2K βMass Transfer Coefficient m s δFilm Solution Film Thickness m ϵNTU Effectiveness Coefficient - ζFriction Coefficient - λThermal Conductivity Coefficient W m K Continued on next page Deliverable D3.1 Symbol Explanation Unit νKinematic Viscosity m2 s ρDensity kg m3 ΓSpecific Mass Flow Rate of the Solution kg ms Deliverable D3.1 Figure 2.2: Illustration of the cross section view of a single pipe with the falling film and introducing some of the boundary parameters used in the model. TAD,sur f ace =TAD,sat (2.26) TAD,sat =Teq(pAD,eq,xAD,sat)(2.27) To calculate TAD, sat, by applying equation Equation 2.27 we use the previously calculated mass transfer coefficient (β) which is determined using the fluid’s thermophysical properties, the A/D unit pressure, PAD, eq utilizing Equation 2.28, and xAD, sat by applying the NTU-effectiveness method for mass transfer, as outlined in Equation 2.29 to 2.32. pAD,eq =pEC,eq −ζ1 2 m2 vap,EC ρvap A2 pipe (2.28) NTUβ=βAAD ˙ msol,in (2.29) ϵβ=1−exp(−NTUβ)(2.30) ϵβ=xin −xout xin −xAD,sat (2.31) TAD,sat =TNaOH(PAD,xAD,sat)(2.32) Now, that TAD, sat has been determined, we proceed to analyze the heat transfer phenomenon in the A/D unit. First, we calculate the effectiveness of the A/D unit using the same approach applied to the E/C unit. For the A/D unit, the corresponding equations are Equation 2.33 to 2.35. With the value of TAD, sat, now available, we can calculate tAD, out, using Equation 2.36. 7Deliverable D3.1 UAD =1 1 αtube,AD +1 αtubewall,AD +1 αf ilm,AD (2.33) NTUAD =UAD AAD ˙ CAD,ext (2.34) ϵAD =1−exp(−NTUAD)(2.35) ϵAD =tAD,out −tAD,in TAD,sat −tAD,in (2.36) Furthermore, we can calculate the heat transfer from the A/D unit using Equation 2.38 to perform an energy balance assessment of the entire system or to run optimization algorithms for enhancing key performance indicators (KPIs). ˙ QAD =ϵAD ×˙ CAD,ext ×(tAD,in −TAD,sat)(2.37) =˙ CAD,ext ×(tAD,in −tAD,out)(2.38) By performing the modeling, we can obtain the temperature curves for the E/C and A/D units, as well as the solution concentration, which are depicted in Figure 2.3. Figure 2.3: Illustration of the internal and external temperature conditions at the heat exchangers and of the salt concentration conditions in the solution film for the coupled heat and mass transfer NTU-effectiveness method used in this work, depicted for the discharging (absorption) process. The visual representation of the model employed is obtained by considering a one-dimensional, stationary, and linear approach to heat conduction through the film and pipe in the A/D unit, assuming that the film bulk temperature is the arithmetic mean between the liquid surface and the tube wall. In the A/D unit, solution concentration and temperature are coupled, with the concentration decreasing towards the saturation concentration, xAD, sat, while the flow temperature rises towards the asymptotic saturation temperature TAD, sat. In the E/C unit, although no solution concentration is present, a similar principle applies, where the inlet temperature decreases towards the asymptotic temperature TEC, sat. 2.1.2 SIMULATION RESULTS Following the introduction of the steady-state model, the simulation results aim to provide a comprehensive understanding of the system behavior under various conditions. By performing parameter variations, we explore the performance and controllability of the system in detail. These simulations focus on key operational parameters 8Deliverable D3.1 and variables, using a parameter set similar the 1 kW test rig of a sodium hydroxide-based storage system that is build at project partner OST’s facilities. By adjusting these variables, we gain insight into how the system responds under different operational settings. The initial set of simulations are conducted using the specific geometry of the heat exchanger installed in OST’s lab, with the parameters describing the test rig detailed in Table 2.3. Table 2.4 shows the base operational variables. In the following simulations, only a single input variables is varied at a time. This approach allows us to isolate the impact of individual parameters on system behavior. By observing how the system reacts to changes in single variables, we can derive valuable insights into potential control strategies for the system. It is important to note that the simulation results presented should provide a general understanding of the models functionality and are not necessarily perfectly aligned with experiment results. In fact, the trends and curve forms predicted by the model seem to be in line with the experimental results, based on initial comparisons. However, the model seems to slightly underestimate the absorption power. This issue will be addressed in further research. Table 2.3: Geometric and thermal parameters of the test rig. Parameter Value Unit Description di,tube,EC 0.008 m Inner Diameter of EC-tubes do,tube,EC 0.01 m Outer Diameter of EC-tubes ltube,EC 0.3 m Length of EC-tubes nrow,EC 10 - Number of tube rows per EC-column ncol,EC 10 - Number of EC-columns λtube,EC 20 W/mk Thermal conductivity of EC-tubes λH2O0.5918 W/mk Thermal conductivity of water AEC,amb 0 m²Area of heat exchange with ambient (EC) di,tube,AD 0.008 m Inner Diameter of AD-tubes do,tube,AD 0.01 m Outer Diameter of AD-tubes ltube,AD 0.2 m Length of AD-tubes nrow,AD 6 - Number of tube rows per AD-column ncol,AD 1 - Number of AD-columns λtube,AD 45 W/mk Thermal conductivity of AD-tubes λNaOH 0.68 W/mk Thermal conductivity of sodium lye AAD,amb 0 m²Area of heat exchange with ambient (AD) Apipe 0.015 m²Cross-section of pipe between EC and AD Arec 0 m²Recuperator heat exchanger area Table 2.4: Operational parameters of the system. Parameter Name Value Unit Description mext,EC 0.0500 kg/s HTF massflow EC mext,AD 0.0230 kg/s HTF massflow AD msol,in 0.0014 kg/s Solution mass flow xin 0.5000 wt% Solution inlet concentration TAD,from tank 26.0000 °C Solution inlet temperature tEC,in 15.0000 °C HTF temperature at EC-inlet tAD,in 29.0000 °C HTF temperature at AD-inlet mhighC,tank 4584.0000 kg Solution tank capacity Tamb 24.0000 °C Ambient temperature The behavior of the system is significantly influenced by two key external inputs: the temperatures of the heat 9Deliverable D3.1 transfer fluids (HTFs) and the mass flows of the HTFs then proceed to analyze the impact of mass flows. As shown in Figure 2.4, the influence of external inlet temperatures on the system’s performance exhibits a quasilinear trend, provided that all other variables and parameters are held constant. This linear behavior can be attributed to the properties of sodium hydroxide (NaOH). Specifically, there is a quasi-linear relationship between the saturation temperature of water and the saturation temperature of the NaOH solution. This relationship is commonly represented through a Dühring plot, which illustrates the linear correlation between the saturation pressures of different solutions. As the inlet temperature in the EC unit increases, both absorber power and solution concentration change rise, with little deviation from linearity. With rising return temperature from the heating circuit tAD,in the trend is the opposite, but still linear. Consequently, absorber power, concentration change and storage capacity are linearly dependent on the difference between tAD,in and tEC,in. Figure 2.4: Influence of external inlet temperatures on solution outlet concentration and absorber power. Each plot represents a single parameter variation. All other parameters remain unchanged as depicted in Tables 2.3 and 2.4 Next, we consider the second external factor — the mass flows of the heat transfer fluids (HTFs). Unlike the temperature effects, the behavior of the system in response to varying mass flows is non-linear. The outlet concentration of the NaOH solution follows an exponential saturation function, where the saturation point is determined by the maximum possible dilution based on the return temperature of the heating circuit. Similarly, the absorber power exhibits an exponential saturation function, limited by the power associated with the maximum possible dilution. The non-linear behavior stems from the NTU-effectiveness of the heat exchanger, as described in section 2.1. The NTU-effectiveness relationship leads to diminishing returns in both absorber power and solution dilution as the mass flow increases, explaining the exponential nature of the curves shown in Figure 2.5. Figure 2.5: Influence of external mass flows on solution outlet concentration and absorber power. In addition to external factors, the system’s behavior is also shaped by internal parameters such as the solution mass flow and inlet concentration. These inputs significantly influence the absorber power and storage capacity. Figure 2.6 illustrates the impact of varying solution mass flow on absorber power and storage capacity. As the mass flow increases, absorber power rises asymptotically, approaching a saturation point. However, this increase in power comes at the expense of storage capacity, which steadily declines. The decline in storage capacity can be attributed to the reduced residence time of the solution in the absorber as the flow rate increases, thereby limiting the amount of mass and energy that can be transferred per kg solution. 10 Deliverable D3.1 Figure 2.6: Influence of NaOH solution mass flow on absorber power and storage capacity. Another crucial internal factor is the NaOH solution inlet concentration, as shown in Figure 2.7. A decrease in inlet concentration leads to a quasi-linear reduction in absorber power, storage capacity, and outlet concentration. This phenomenon occurs, because lower inlet concentrations reduce the concentration difference that can be achieved during the absorption process, which in turn decreases the driving force for absorption and heat transfer. Intuitively, the heat and mass transfer is decreased when the solution enters with a concentration closer to saturation conditions. This is visible on the right side of Figure 2.7 by the distance between the two lines. Figure 2.7: Influence of NaOH inlet concentration on absorber power, storage capacity and outlet concentration. In conclusion, the simulation results indicate distinct behaviors of the system in response to both external and internal inputs. The system’s performance under varying external temperatures is relatively linear, largely due to the properties of sodium hydroxide and the relationship between water and NaOH saturation temperatures. In contrast, external mass flows have a non-linear, exponential effect on absorber power and solution concentration, driven by the physical connection between those values, captured by the NTU-effectiveness model of the heat exchanger. Internally, solution mass flow exhibits an asymptotic increase in absorber power but a corresponding decline in storage capacity. Meanwhile, lower inlet solution concentrations lead to a reduction in both absorber power and storage capacity due to the decreased driving potential for absorption. These insights into the behavior of the system provide valuable information for optimizing both control and design, ensuring more efficient operation under various conditions. 11 Deliverable D3.1 2.2 DYNAMIC MODEL Although the dynamic model has not yet been fully validated, the general approach will be introduced in this section. The model equations remain largely unchanged from the steady-state version, with the primary modifications being the addition of heat capacity terms for the Evaporator/Condenser (E/C) and Absorber/Desorber (A/D) units. Furthermore, differential terms for the temperatures are included, as described below. For the dynamic Evaporator/Condenser model, the heat transfer is represented by equation 2.39. Tsteel,EC is assumed to be the temperature at the midpoint of the vessel wall (assumed to follow a linear temperature gradient). ˙ QEC =˙ mvap∆hv(TEC) + msteel,EC ·cp,steel,EC dTsteel,EC dt (2.39) +mwater ·cp,water dTEC dt +˙ Qloss,EC Tsteel,EC =TEC −dwall,EC 2·λsteel,EC Qloss,EC Awall,EC (2.40) For the dynamic Absorber/Desorber model, the heat transfer and the temperature are expressed similarly. Additionally, the solution mass is approximated by Equation 2.42. ˙ QAD =˙ mvap ·hv(TEC) + ˙ msol,in ·hsol,in(xin,TAD,in)(2.41) −˙ msol,out ·hsol,out(xout,TAD,out) + ˙ Qloss,AD +msteel,AD ·cp,steel,AD dTsteel,AD dt +msol,AD ·cp,sol(TAD,bulk,xin +xout 2)dTAD,bulk dt msol,AD =AAD ·df ilm ·ρsol(TAD,bulk,xin +xout 2)(2.42) Tsteel,AD =TAD −dwall,AD 2·λsteel,AD Qloss,AD Awall,AD (2.43) These modifications enable the model to account for thermal inertia by considering the heat capacities of the materials involved, thereby providing a more accurate representation of the system’s transient behavior. To make the model dynamic, several steps are still required. First, changes in the data structures for the inputs must be made to accommodate time-dependent variables. Second, adjustments in the solving algorithm are necessary, particularly the incorporation of time steps and the use of finite difference methods for approximating the derivative terms. These enhancements will allow the model to simulate time-dependent processes and capture the dynamic response of the system under varying conditions. 12 Deliverable D3.1 3 DETAILED MODEL OF THE HEAT AND MASS EXCHANGER This chapter presents a comprehensive model of the heat and mass exchanger (HMX). The chapter is organized as follows: section 3.1 Model Description provides an overview of the implemented model, detailing its structure to enable the application of various input signals and the tracking of output signals not only across different tube rows of the A/D unit, but also within specific sections of each tube, referred to as "submodels". This method allows for a more precise and detailed representation of the HMX, particularly in the Absorber/Desorber unit, as opposed to relying solely on generalized variables used for the integral representation of the HMX unit described in chapter 2. Following this, section 3.2 A/D-Submodel, delves into the specifics of the A/D unit, discussing the input, output, and configuration of the system that are essential for composing this detailed model. Finally, Mathematical Modeling Approach, outlines the structure of the program developed, to implement the aforementioned modeling approach (at the moment MATLAB is used). This section provides an general overview of the massand energy balances which were used to formulate the model’s equations, followed by the definition of parameters and a calculation methodology. 3.1 MODEL DESCRIPTION As previously discussed, the objective of this chapter is to present a novel approach for the detailed modeling of the thermo-chemical storage system, with an emphasis on the Absorber/Desorber (A/D) unit. In this approach, the HMX of the A/D unit is divided into several identical submodels that operate independently yet are interconnected, reflecting the staggered tubular geometry of the system. These submodels form the core operational components of our detailed model. Figure 3.1 provides a visual representation of the overall A/D unit, referred to as the "Integral Model" in green, along with its constituent "submodels" in red. Figure 3.1: General overview of the A/D submodel configuration. The overall system outlined with green and the submodel depicted in red. The Integral Model is designed to calculate both temporal and simple localized variations in temperature and concentration within the HMX unit. To achieve this, time-dependent differential equations tailored to the tube geometry are formulated, with boundary conditions adjustable to meet the required resolution of the results. The 13 Deliverable D3.1 Integral Model enables the simulation of an entire tube as whole or divided into sections (referred to as submodels), with the assumption that temperature and concentration fields remain locally uniform within the boundaries of each submodel. To accurately represent a full tube, multiple submodels are interconnected, allowing for the exchange of temperature and concentration information between adjacent submodels. This information exchange resembles the physical connection of the modelled tube sections (submodels). The concept of the submodel and its interactions with neighboring submodels are illustrated in Figure 3.1, and Figure 3.2. In Figure 3.1, submodels aligned side by side (referred to as being in parallel, such as cells [1,1], [1,2], and [1,3]) represent segments of a single tube, where the inputs, states, and outputs are assumed to remain constant along the local tube length of the segment. Submodels stacked vertically (referred to as being in a row, such as [1,2], [2,2], [3,2], and [4,2]) simulate the effects of the solution dripping down from one section of the tube to the next. The terms "in row" and "in parallel" describe the direction of internal fluid (solution) flow within the system. To address local temperature variations along the tube and differences between tubes, it’s necessary to define and interconnect submodels. Other modeling methods, such as tray columns or partial differential equations, are expected to result in similarly complex calculations. The counter-cross flow configuration further complicates the temperature and concentration fields, but the submodel approach offers a more precise solution. By linking the submodels, this method effectively handles local variations and the complex flow arrangement, allowing for a simplified representation of the naturally distributed parameter system. Figure 3.2: General overview of the complete A/D Model with all incoming and outgoing signals. Some can have negative signs to accound for desorption operation mode. As depicted in Figure 3.1, the A/D unit consists of multiple identical submodels, interconnected by transport delay cells, which account for the physical lenght of each submodel or the distance between two tube rows. Every submodel handles three distinct process flows, with an input and output signal each. Since it is an integral modeling approach, all signals and states are considered constant over the local tube segment, only changing over time. The outputs are determined using the governing ordinary differential equations. A detailed schematic illustration of the flows entering or leaving each submodel is provided in Figure 3.2. In the following subsection, detailed information on the input u(t), state x(t), output y(t), and parameter pconfiguration of a single A/D submodel will be given. 14 Deliverable D3.1 3.2 A/D-SUBMODEL To gain a more detailed understanding of the submodel concept, this section introduces the underlying logic of our approach, by treating the submodel as a control volume and outlining the input and output streams. The governing equations will be introduced in the next section, because the focus lies on providing a more thorough understanding of the signal flows of the model and to present it as clearly as possible to the readers. Each submodel includes three separate flow paths, one of which is the heating/cooling flow, known as the ’External Fluid.’ This flow involves water entering the current submodel from the former submodel (or the backward feed of the heating system) and exiting to the latter submodel (or the forward feed of the heating system). The signals necessary for the mass and energy balance equations for this flow are tH2O(t), and ˙ mH2O(t). As shown in Figure 3.3, the input and output signals of this stream are implemented into the governing equations of the control volume, defining the representative flows. For modeling purposes, the water temperature, tH2O(t), is considered as one of the state variables, which is determined by solving the governing equation represented by ˙ x1. Figure 3.3: Inputs to the submodel from the External Fluid. The second flow entering the submodel’s control volume is the solution flow, referred to as the ’Internal Fluid.’ This flow involves solution droplets descending from the upper tube (or solution distributor) and dropping on the tube below (or the solution pool). Three parameters Tsol(t),˙ msol(t), and xH2O(t)characterize the respective input and output signals. Similar to the external water flow, the parameter Tsol, out(t), is the second state variable, denoted by ˙ x2. The visual representation of this flow is provided in Figure 3.4. Figure 3.4: Inputs to the submodel from the Internal Fluid. The final flow stream considered in the A/D unit is the internal system vapor flow depicted in Figure 3.5. This signal provides each submodel with information about the vapor temperature and pressure in the AD. It is assumed, that this temperature and pressure are constant for the whole A/D unit, such that each submodel receives the same 15 Deliverable D3.1 values from the E/C. In general, the AD pressure is a function of the EC conditions as well as the pressure drop within the vapour tube connecting E/C and A/D. By using the signals TEC(t), and pAD(t)and solving the governing equations, the absorbed or desorbed water in each submodel can be calculated based on the input conditions. Each submodel then feeds back the amount of absorbed water vapor to the E/C, where the results from each submodel can then be summed up to determine the total amount of water absorbed or desorbed in the HMX. The E/C then calculates the new pressure. The total mass of water present in the solution represents the final state in the state space model, denoted by ˙ x3. Figure 3.5: Inputs to the submodel from the E/C. To summarize this subsection, it is important to note that the entire A/D unit is modeled using multiple submodels, each governed by the same set of equations. These submodels are interconnected through delay cells, which account for the transport delays between cells arranged in parallel or in series. The overall schematic of a submodel, illustrating all known and unknown variables as inputs and outputs, is shown in Figure 3.6. By applying the conservation laws and state equations related to the water, solution and vapor, we can determine the representative values of our state vector. Generalized integral balances to show the overall dependencies for each state will be given in the next section. Furthermore, details of the solution structure, along with the input and output vectors are provided. Figure 3.6: Full input and output configuration of one submodel. 16 Deliverable D3.1 Coming to the last state, it is the enthalpy of the water in the overall EC unit. It is employed to determine how much energy is used for sensible and latent heat. The overall energy balance is defined as: ∂Ux5, EC ∂t=˙ Hin, EC −˙ Hout, EC +˙ QHT −˙ Qevap .(3.30) Exchanging the definition of the internal energy for the overall enthalpy and incorporating the transport equations, we find that the enthalpy of the control volume is defined by: ∂x5, EC ∂t=˙ mH2O, int mH2O, int ·cp·(TEC, in −Tref)−˙ mH2O, int −˙ mevap mH2O, int ·cp·(x1, EC (t)−Tref) (3.31) +αEC ·AEC mH2O, int ·((tEC, in −TEC, in)−(x2, EC(t)−x1, EC(t))) log (tEC, in−TEC, in) (x2, EC(t)−x1, EC(t))−˙ mevap ·∆hLV mH2O, int +∂pEC ∂t·VEC To conclude the overall definition of the Absorber/Desorber and Evaporator/Condenser model, it needs to be emphasized, that the model has not yet being testet. The equation formulation is finished however, such that the next step in the development process is to fix the model parameters introduced as pand to test the model simulation. 23 Deliverable D3.1 4 CONCLUSION In this report, two different modeling approaches for the operation of a sodium hydroxide-based thermo-chemical storage system were presented. The first model, an integral thermodynamic model, was developed using a coupled NTU-effectiveness approach to account for both heat and mass transfer limitations. This model demonstrates a good ability to predict steady-state system conditions, though further experimental validation is required to fully assess its accuracy. The second model, a detailed transient model of the heat and mass exchanger (HMX), relies on differential equations and provides insights into both the internal temperature distribution and the system’s transient behavior. Although the models presented were developed for NaOH systems, their methodologies can be extended to other working pairs, such as LiBr-H2O. Future research will focus on applying these models to different system configurations and employing optimization techniques to investigate how geometric factors, particularly the design of the adsorption (A/D) and evaporation-condensation (E/C) heat and mass exchangers, influence system performance and how this can be taken into account in the proposed models. In summary, these two modeling approaches provide valuable tools for understanding and optimizing the operation of thermo-chemical storage systems, and they form the foundation for further advancements in describing, controlling and designing sorption thermal storage systems. 24 Deliverable D3.1 BIBLIOGRAPHY Dorian Hoeffner (2024). propertiesnaoh 0.1.8. https://pypi.org/project/propertiesNaOH/. Accessed: 30.09.2024. Magnus Holmgren (2024). X steam, thermodynamic properties of water and steam. https://www.mathworks.com/matlabcentral/fileexchange/9817-x-steam-thermodynamic-properties-of-waterand-steam. MATLAB Central File Exchange. Accessed: 30.09.2024. 25 Deliverable D3.1 Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or CINEA. Neither the European Union nor the granting authority can be held responsible for them. ©BEST-STORAGE PROJECT. All rights reserved. Any duplication or use of objects such as diagrams in other electronic or printed publications is not permitted without the author’s agreement.