1-D thermodynamic model of a water tank

Figure 2a,b shows two kinds of multi-tank models connected in series: with and without vapor-zone connections. Normally, a full cycle consists of charging, stand-by and discharging processes. The stand-by period typically occurs between the charging and discharging stages rather than involving an immediate flow reversal. Therefore, the transient dynamic effects associated with abrupt flow reversal are expected to be negligible in this context. In addition, the main objective of this study is to evaluate whether connecting the steam regions among different tanks is necessary during the charging and discharging processes. To this end, the transient model primarily includes two stages: charging and discharging. Under these conditions, the system can be reasonably simplified as an open system. To further reduce model complexity, several assumptions are introduced. An equilibrium thermodynamic approach is adopted, meaning that at each time step, the water and steam within the tank are assumed to exist in a saturated state. Additionally, the model neglects internal temperature gradients as well as thermal resistances at the interfaces between different metal shells and the insulation layer.

Figure 2.

Schematics of working processes in multi-capsule shaped water tanks: (a) multi-tanks without gas zone connections, (b) multi-tanks with gas-zone connections and (c) an open system analysis of a general capsule shaped tank.

Figure 2. Long description

Panel a shows two rows of tanks labeled Tank 1, Tank 2, Tank 3, and Tank n. In the Charging process, fluid enters Tank 1 from the top right via m dot sub w,in and flows through a bottom pipe connecting all tanks. In the Discharging process, fluid exits from the bottom left of Tank n via m dot sub w,out.

Panel b shows a similar multi-tank setup but with two horizontal pipes connecting the tanks at both the top and bottom. In the Charging process, fluid enters Tank 1 from the middle right. In the Discharging process, fluid exits from the bottom left of Tank n.

Panel c is a detailed schematic of a single tank centered on a coordinate system with a vertical y-axis and horizontal x-axis.

• At the top (y = L/2), an arrow m dot sub v,in enters from the right and Q dot sub w,l exits from the top dome.

• Just below (y = L prime/2), m dot sub v,out exits to the left.

• The upper interior is labeled m sub v,i.

• The center origin o marks the transition to the lower interior labeled m sub w,i, where Q dot sub v,l exits to the right.

• Near the bottom (y = -L prime/2), m dot w,out exits to the left.

• At the base (y = -L/2), m dot w,in enters from the right.

As illustrated in Figure 2c, the change in internal energy of the open system is governed by the combined energy variations of the liquid water and vapor phases between time steps ‘i’ and ‘i + 1’. The system interacts thermally through several mechanisms, including energy input from an external heater, mass transfer of water and vapor during both charging and discharging processes and heat losses through the wet and dry walls of the tank. According to the principle of energy conservation for open systems, the variation in internal energy equals the algebraic sum of energy input, energy output and net heat sources, as expressed in Eq. (1).

(1) $$ \frac{d(mu)}{dt}={\displaystyle \begin{array}{l}\left({\dot{m}}_{w, in}{h}_{in}-{\dot{m}}_{w, out}{h}_{out}\right)+\left({\dot{m}}_{v, in}{h}_{in}-{\dot{m}}_{v, out}{h}_{out}\right)\\ {}+\hskip2px {\dot{Q}}_{heater}-{\dot{Q}}_{loss}\end{array}} $$

in which, $ \dot{m} $ represents the mass flow rate, and the subscripts ‘w’, ‘v’, ‘in’ and ‘out’ refer to water, vapor, inlet and outlet, respectively. The symbols h and u denote the specific enthalpy and specific internal energy. The thermal energy supplied by the external heater and the total heat loss from the system are represented by $ {\dot{Q}}_{heater} $ and $ {\dot{Q}}_{loss} $ , respectively. Since no external heaters are incorporated into the current model, $ {\dot{Q}}_{heater} $ is assumed to be zero throughout the simulation.

To solve the aforementioned differential equation, Equation (1) may first be discretized. During a given time step ∆t, the internal energy term is discretized as corresponding variations of both the vapor and water phases between two consecutive time instants, $ {t}_i $ and $ {t}_{i+1} $ , as shown in Equation (2).

(2) $$ {\displaystyle \begin{array}{l}\frac{m_{w,i+1}{u}_{w,i+1}+{m}_{v,i+1}{u}_{v,i+1}-\left({m}_{w,i}{u}_{w,i}+{m}_{v,i}{u}_{v,i}\right)}{\Delta t}\\ {}\hskip1em ={\displaystyle \begin{array}{l}\left({\dot{m}}_{w, in}{h}_{w, in}-{\dot{m}}_{w, out}{h}_{w, out}\right)+\left({\dot{m}}_{v, in}{h}_{v, in}-{\dot{m}}_{v, out}{h}_{v, out}\right)\\ {}+\hskip2px {\dot{Q}}_{heater}-{\dot{Q}}_{loss}\end{array}}\end{array}} $$

The primary thermophysical properties of water and vapor – such as density, internal energy and enthalpy – are obtained using quadratic interpolation based on data from the saturated water property tables (Çengel et al., Reference Çengel, Boles and Kanoglu2019). Additionally, the changes in the mass of water and vapor are calculated using their respective densities and volumes at time $ {t}_i $ and $ {t}_{i+1} $ . For a capsule shaped tank, the total volume is simplified as a combination of two hemispherical caps and a cylindrical body (as illustrated in Figure 2c). Based on this geometry, a piecewise function is employed to describe the time-dependent variation of the water volume $ {V}_w $ , while the vapor volume at any given time is computed as the difference between the total tank volume and the water volume. Here, the piecewise function is built on the coordinate system shown in Figure 2c, where the origin is located at the geometric center.

(3) $$ {V}_w={\int}_{-L/2}^x\pi \left({R}^2-{\left({L}^{\prime }/2+x\right)}^2\right) dx,\hskip3.719998em -L/2\le x\le -{L}^{\prime }/2 $$

(4) $$ {V}_w=\frac{2\pi {R}^3}{3}+{\int}_{-{L}^{\prime }/2}^x\pi {R}^2 dx,\hskip5.279997em -{L}^{\prime }/2\le x\le {L}^{\prime }/2 $$

(5) $$ {V}_w=\frac{2\pi {R}^3}{3}+{L}^{\prime}\pi {R}^2+{\int}_{L^{\prime }/2}^x\pi \left({R}^2-{\left(x-{L}^{\prime }/2\right)}^2\right) dx,\hskip0.36em {L}^{\prime }/2\le x\le L/2 $$

To solve the preceding equation, it is crucial to accurately estimate the heat loss at each time step. As depicted in Figure 3, a thermal resistance network is constructed to illustrate the detailed heat transfer pathways. Heat loss from the water tank primarily occurs through two regions: the liquid surface and the vapor surface. The corresponding heat transfer mechanisms include: (1) thermal conduction through the multilayer tank wall, (2) convective heat exchange between the internal fluid and the inner wall surface, (3) convective heat dissipation from the external shell to the ambient air and (4) thermal radiation emitted from the tank surface to the surrounding environment. These mechanisms collectively determine the overall heat loss rate and are integral to the accuracy of the transient energy balance model.

Figure 3.

Thermal resistance of a thermal storage water tank.

Figure 3. Long description

This figure illustrates a thermal resistance network diagram between nodes Ti on the left and Ta on the right. The network splits into two parallel branches. The top branch represents heat transfer from the vapor zone, and the bottom branch represents heat transfer from the water zone. Both branches share an identical series–parallel layout of thermal resistances. Moving from left to right along either branch, heat flows through an ‘Inside convective’ resistance, a ‘Metal’ resistance, and an ‘Insulation’ resistance in series. Following the insulation layer, the path splits into a parallel network consisting of an ‘Outside convective’ resistance on top and a ‘Radiative’ resistance on the bottom. These parallel resistances recombine and connect in series to a final ‘Fouling’ resistance before reaching the node Ta on the far right.

Based on the electrical analogy commonly applied in thermal system modeling, the heat flux is represented as the temperature difference divided by the total thermal resistance, denoted as $ {\psi}_w $ for the water region and $ {\psi}_v $ for the vapor region. Accordingly, Equations (6) and (7) are formulated to quantify the heat losses from the water and vapor interfaces, respectively.

(6) $$ {\dot{Q}}_{w,l}=\frac{A_w\left({T}_t-{T}_a\right)}{\psi_w}={U}_w{A}_w\left({T}_t-{T}_a\right) $$

(7) $$ {\dot{Q}}_{v,l}=\frac{A_v\left({T}_t-{T}_a\right)}{\psi_v}={U}_v{A}_v\left({T}_t-{T}_a\right) $$

in this context, T represents temperature, where the subscripts ‘a’ and ‘t’ denote the ambient environment and the tank surface, respectively. The terms $ {U}_w $ and $ {U}_v $ refer to the overall heat transfer coefficients associated with the water and vapor regions. Correspondingly, the heat losses through the liquid and vapor interfaces are denoted as $ {\dot{Q}}_{w,l} $ and $ {\dot{Q}}_{v,l} $ , respectively, and are determined using these coefficients along with the temperature gradients.

Notations $ {A}_w $ and $ {A}_v $ denote the surface areas of the water and vapor phases, respectively. Given the capsule shaped geometry of the tank (as illustrated in Figure 2c), a piecewise function is employed to estimate the dynamic variation of the liquid surface area over time. The corresponding vapor surface area is then determined by subtracting the liquid surface area from the total internal surface area of the tank.

(8) $$ {A}_w={A}_w=2\pi R\left(x+L/2\right),\hskip2.879999em -L/2\le x\le -{L}^{\prime }/2 $$

(9) $$ {A}_w=2\pi {R}^2+2\pi R\left(x+{L}^{\prime }/2\right),\hskip2.52em -{L}^{\prime }/2\le x\le {L}^{\prime }/2 $$

(10) $$ {A}_w=2\pi {R}^2+2\pi R{L}^{\prime }+2\pi R\left(x-{L}^{\prime }/2\right),\hskip1.2em {L}^{\prime }/2\le x\le L/2 $$

Building upon the series–parallel structure of the thermal resistance network illustrated in Figure 3, the overall thermal resistances associated with the water and vapor surfaces are calculated using Equations (11) and (12), respectively, as outlined in reference (Çengel and Ghajar, Reference Çengel and Ghajar2015). These equations account for the combined effects of conduction, convection and radiation along the respective heat transfer paths.

(11) $$ \frac{1}{U_w}={\psi}_w=\frac{1}{h_{w,i}}+\frac{t_s}{k_s}+\frac{t_i}{k_i}+\frac{1}{h_a+{h}_{rw}}+\frac{1}{h_{fw}} $$

(12) $$ \frac{1}{U_v}={\psi}_v=\frac{1}{h_{v,i}}+\frac{t_s}{k_s}+\frac{t_i}{k_i}+\frac{1}{h_a+{h}_{rv}}+\frac{1}{h_{fv}} $$

in these expressions, $ {k}_s $ and $ {k}_i $ represent the thermal conductivities of the tank’s metallic shell and insulation layer, while $ {t}_s $ and $ {t}_i $ denote their respective thicknesses. The terms $ {h}_f $ and $ {h}_r $ correspond to the heat transfer coefficients accounting for fouling and radiative heat exchange. Meanwhile, $ {h}_i $ and $ {h}_a $ refer to the convective heat transfer coefficients on the inner fluid side and the ambient air side, respectively.

In the water storage system, the capsule shaped is positioned vertically. To characterize the heat transfer between the tank wall and the internal liquid or vapor phases, a set of empirical correlations are employed to estimate the average Nusselt number, as referenced in (Rohsenow et al., Reference Rohsenow, Hartnett and Cho1998; Majumdar et al., Reference Majumdar, Valenzuela, LeClair and Moder2016). Based on these correlations, the internal convective heat transfer coefficient is subsequently determined using the standard definition of the Nusselt number, linking it to thermal conductivity and characteristic length scales.

The basic idea of these thermal correlations is to first derive an equation for the laminar Nusselt number ( $ {Nu}_l $ ). In the case of laminar flow, the natural convection boundary layer is generally not very thin. Therefore, the laminar Nusselt number can be regarded as a corrected form derived from both the conduction Nusselt number ( $ {Nu}_{cond} $ ) and the laminar thin-layer Nusselt number ( $ {Nu}^T $ ). In this context, the conduction Nusselt number refers to the special case where the Rayleigh number approaches zero and there is no fluid motion. Following the laminar formulation, another equation is developed to describe the turbulent Nusselt number ( $ {Nu}_t $ ), which is based on the assumption that the flow over the surface is in a fully turbulent state. Finally, the average Nusselt number ( $ Nu $ ) is determined by combining the laminar and turbulent Nusselt numbers into a single expression. This combined correlation provides a unified formulation applicable to all flow regimes on the surface, thereby ensuring its validity for a wide range of practical heat transfer conditions.

(13) $$ {Nu}^T=G{\overline{C}}_l{Ra}^{1/4} $$

(14) $$ {Nu}_l={\left[{Nu_{cond}}^n+{\left({Nu}^T\right)}^n\right]}^{1/n} $$

(15) $$ {Nu}_t={\overline{C}}_t{Ra}^{1/3} $$

(16) $$ Nu={\left[{Nu_l}^m+{Nu_t}^m\right]}^{1/m} $$

(17) $$ {h}_i= Nu\frac{k_l}{D} $$

where, $ {\overline{C}}_l $ $ ({\overline{C}}_l=0.671/{\left(1+{\left(0.492/\mathit{\Pr}\right)}^{9/16}\right)}^{4/9}) $ and $ {\overline{C}}_t $ (as listed in Table 1) are constants that depend on the Prandtl number. $ D $ , $ G $ and $ {k}_l $ represent the diameter of the tank, a ratio of integrals (as listed in Table 1) and the thermal conductivity of the liquid, respectively. The Rayleigh number (Ra) characterizes the natural convection process occurring within the liquid phase. In addition, the parameters m and n are determined by fitting to experimental data, as shown in Table 1.

Table 1.

Constant parameters used in empirical correlations for a vertical capsule shaped tank (Rohsenow et al., Reference Rohsenow, Hartnett and Cho1998)

Table 1. Long description

The table features a 2D vertical capsule-shaped tank sketch on the left, demonstrating geometric height L and diameter D. To the right of the sketch is a multi-column data matrix. The vertical columns represent the length-to-diameter ratio (L/D), a Nusselt number parameter, geometric constants G and n, a composite header for parameter C_t under four distinct Pr values (Pr = 0.71, 6.0, 100 and 2000) and a constant m. The horizontal rows correspond to four different aspect ratios ranging from two to five. The matrix shows the qualitative variations of these empirical parameters and constants across different length-to-diameter configurations.

In addition, an empirical correlation is employed to evaluate the natural convective heat transfer between the outer surface of the vertically oriented capsule shaped tank and the surrounding ambient air, as reported in previous studies (Bergman, Reference Bergman2011; Joseph et al., Reference Joseph, Agrawal, Agarwal, Pisharady and Sunil Kumar2017). This correlation primarily characterizes the relationship between the external convective heat transfer coefficient, the Prandtl number and the Reynolds number ( $ \mathit{\operatorname{Re}} $ ).

(18) $$ {h}_a=\left[0.3+\frac{0.62{\mathit{\operatorname{Re}}}^{0.5}{\mathit{\Pr}}^{0.33}}{{\left(1+{\left(\frac{0.4}{\mathit{\Pr}}\right)}^{\frac{2}{3}}\right)}^{0.25}}{\left[1+{\frac{\mathit{\operatorname{Re}}}{28200}}^{\frac{5}{8}}\right]}^{\frac{4}{5}}\right]\times \frac{k_a}{D} $$

here, $ {k}_a $ denotes the thermal conductivity of the ambient air.

Since the heat losses from the water surface and the vapor surface occur simultaneously and independently, they can be considered as parallel heat transfer paths. In addition, the pipelines connecting the tanks are relatively short. Compared with that from the large surface area of the tanks, heat loss along the pipelines is negligible and is therefore neglected in the model. Thus, the total heat loss $ {\dot{Q}}_{loss} $ is calculated as the sum of the heat losses from both interfaces, as expressed in Equation (19).

(19) $$ {\dot{Q}}_{loss}={\dot{Q}}_{w,l}+{\dot{Q}}_{v,l} $$

Mass conservation constitutes a fundamental governing equation crucial for monitoring the changes in both water and vapor masses over each discrete time step. At the subsequent time step, denoted as i + 1, the total mass is determined by summing the water and vapor masses present at time step i, along with the net mass changes of water ( $ {\dot{m}}_w $ ) and vapor ( $ {\dot{m}}_v $ ) occurring during the time interval $ \Delta t $ , as formulated in Eq. (20).

(20) $$ {m}_{w,i+1}+{m}_{v,i+1}={m}_{w,i}+{m}_{v,i}+{\dot{m}}_w\Delta t+{\dot{m}}_v\Delta t $$

Considering that the total volume ( $ V $ ) of the capsule shaped water tank is fixed and does not change, the mass conservation equation can be rewritten by incorporating the relationship between specific volume and mass, which leads to the formulation shown in Eq. (21).

(21) $$ \begin{array}{l}{V}_{w,i+1}=\left({m}_{w,i}+{m}_{v,i}+{\dot{m}}_w\Delta t+{\dot{m}}_v\Delta t-V/{v}_{v,i+1}\right)/\\ {}\left(1/{v}_{w,i+1}-1/{v}_{v,i+1}\right)\end{array} $$

Because both Eqs. (2) and (21) depend on the temperatures at time steps i and i + 1, the temperature at time step i + 1 can be determined using the known temperature at time step i. Following this, the pressure corresponding to the calculated temperature is evaluated by referring to the saturated water property table (Çengel et al., Reference Çengel, Boles and Kanoglu2019).

Furthermore, the capsule shaped tanks connected in series are incorporated into this transient model. In this context, accurately describing the mass flow rate between different tanks becomes crucial during both the charging and discharging processes. Regarding the mass flow within a pipe connecting two capsule-shaped tanks, it can be approximated as a free jet flow, primarily driven by the pressure difference ( $ \Delta P $ ) between the two ends of the pipe. Consequently, the fluid velocity within the pipe ( $ {V}_c $ ) can be estimated using the Bernoulli equation, and the resulting mass flow rate ( $ {\dot{m}}_c $ ) is determined by the product of this velocity and the pipe’s cross-sectional area ( $ {A}_c $ ), as expressed in Eqs. (22) and (23) (Gerhart et al., Reference Gerhart, Andrew and John2016).

(22) $$ {V}_c=\sqrt{2\Delta P/\rho } $$

(23) $$ {\dot{m}}_c={\rho V}_c{A}_c $$

Time step determination

Discretizing the governing equations with respect to time is essential for solving the dynamic model effectively. The chosen size of the time step has a direct impact on both the accuracy and stability of the model. However, selecting excessively small-time steps can lead to a substantial increase in computational cost and time. Therefore, this study investigates a balance between computational efficiency and accuracy by testing four different time step sizes (50 s, 10 s, 5 s and 2 s) using a model of 2 tanks divided into 12 blocks. The model’s performance was assessed by analyzing the temporal profiles of time, temperature and water level throughout the charging and discharging processes.

Additionally, in order to comprehensively assess the thermal performance of the capsule-shaped water tanks within the transient model framework, it is essential to define the relevant physical and geometric input parameters. For this study, a project based in Australia was selected, with key operating and design parameters provided by HYDROSTOR Inc., a CAES system operator. The water tank system considered here is a subsystem of the overall CAES system. Accordingly, several operating parameters, including the inlet and outlet water temperatures and the mass flow rate, were determined based on the requirements for cooling and reheating compressed air during the CAES process. In addition, the system was designed to meet engineering requirements, specifically that the charging process lasts no less than 13 hours and the discharging process no less than 8 hours. These parameters are presented in detail in Tables 2 and 3. Besides, this project is currently at the preliminary design stage. To enable the simulations, the pipeline diameters in both the water and gas zones were set to 10 cm in this study, based on Hydrostor’s initial reference design (Hydrostor, 2025).

Table 2.

Physical parameters in the thermodynamic model (Hydrostor, 2025)

Table 2. Long description

This table consists of a simple two-column data layout. The first column lists various thermodynamic input parameters, including ambient temperature, charging water temperature, initial tank temperature, total mass flow rates for both charging and discharging processes, as well as initial and end operating volume ratios. The second column presents the specified constant values or ratios assigned to each corresponding parameter for the model evaluation.

Table 3.

Initial designing parameters for two thermal storage schemes (Hydrostor, 2025)

Table 3. Long description

This table consists of a simple two-column data layout. The first column lists various initial design parameters for the two thermal storage schemes, including tank quantity, geometric dimensions such as diameter and tangent-to-tangent length, thicknesses of insulation and metal layers, wind speed, thermal conductivities of insulation and shell layers, heat transfer coefficients related to water and vapor fouling effects, and radiation emissivity. The second column presents the specific constant numerical values or ratios assigned to each corresponding design parameter.

Figure 4 illustrates the variations in temperature and water level during five charging–discharging cycles using a time step of 5 seconds for the two-tank model without a gas-zone connection. The results show that after the second cycle, the variations of these thermofluidic parameters tend to stabilize, and no significant cumulative or drift effects are observed. Therefore, to reduce computational cost, only the first three cycles are discussed in the subsequent analysis.

Figure 4.

Thermofluidic performances of a 2-tank system without a gas-zone connection under five cycles: (a) temperature variations and (b) water-level variations. T, P and H denote temperature, pressure and water level, respectively.

Figure 4. Long description

This figure group consists of two line graphs, (a) and (b), illustrating the thermofluidic performances of a 2-tank water tank system without a gas-zone connection under five charging-discharging cycles over a duration of 125 hours. Graph (a) plots temperature variations on the left vertical axis (ranging from 185 to 200 degrees Celsius) and pressure variations on the right vertical axis (ranging from 1000 to 1500 kPa) against time on the horizontal axis. It shows fluctuating cyclical profiles for temperatures T1, T2 and pressures P1, P2. Graph (b) plots water level variations on the vertical axis (ranging from −12 to 12 m) against time, showcasing oscillating cyclical waves for water levels H1 and H2. In both graphs, the variations tend to stabilize after the second cycle, with the subsequent cycles showing highly consistent periodic behavior.

As presented in Table 4, the primary performance metrics for the two-tank model without a gas zone connection were assessed under various time step sizes. It was observed that when the time steps were shorter than 50 seconds, the model outputs – such as operation durations, temperature profiles and water-level variations throughout the three stages – remained consistently stable and showed minimal variation.

Table 4.

Effect of time step on the dynamic two-tank model without a gas-zone connection

Table 4. Long description

This table presents a multi-column data layout evaluating sensitivity to temporal resolution. The first column lists several output performance parameters, including total duration, maximum temperatures for both Tank 1 and Tank 2, as well as maximum water levels for both Tank 1 and Tank 2. The subsequent four columns represent different time step scenarios decreasing in size. The horizontal matrix qualitative demonstrates how these dynamic thermofluidic maximums and durations respond and converge as the simulation time step becomes smaller.

Similarly, based on the results in Figure 5, the first three cycles were discussed for the model with a gas zone in the following studies. Table 5 presents a summary of the corresponding results for the two-tank model that includes a connected gas zone. Due to the inclusion of the gas-zone connection, this configuration demonstrates more pronounced dynamic behavior compared to the previous model without the gas zone. Nevertheless, the model responses become stable when the time step is chosen to be less than 5 seconds. Therefore, a consistent time step of 5 seconds is adopted for all subsequent simulations to guarantee both accuracy and numerical stability across both types of models.

Figure 5.

Thermofluidic performances of a 2-tank system with a gas-zone connection under five cycles: (a) temperature and (b) water level.

Figure 5. Long description

This figure group consists of two line graphs, (a) and (b), illustrating the thermofluidic performances of a 2-tank water tank system with a gas-zone connection under five charging-discharging cycles over 125 hours. Graph (a) plots temperature variations on the left vertical axis (185 to 200 degrees Celsius) and pressure variations on the right vertical axis (1000 to 1500 kPa) against time on the horizontal axis. It displays periodic fluctuations for temperatures T1, T2 and pressures P1, P2. Graph (b) plots water level variations on the vertical axis (-12 to 12 meters) against time, showing the cyclical oscillating profiles for water levels H1 and H2. Similar to Figure 4, the variations tend to stabilize after the second cycle, with the subsequent cycles showing highly consistent periodic behavior.

Table 5.

Effect of time step on the dynamic two-tank model with a gas-zone connection

Table 5. Long description

This table presents a multi-column data layout evaluating sensitivity to temporal resolution. The first column lists several output performance parameters, including total duration, maximum temperatures for both Tank 1 and Tank 2, as well as maximum water levels for both Tank 1 and Tank 2. The subsequent four columns represent different time step scenarios decreasing in size. The horizontal matrix qualitatively demonstrates how these dynamic thermofluidic maximums and durations respond and converge for the gas-zone connected system as the simulation time step becomes smaller.

Pressure loss in the connected pipe

Based on the total mass flow rates during the charging and discharging processes, the corresponding mass flow rates for each tank are approximately 11.3 kg/s during charging and 19.5 kg/s during discharging, calculated by dividing the total mass flow rate by the number of tanks. Considering the pipe diameters (8 cm, 10 cm and 12 cm) examined in this study, the water flow in the pipelines remains in a turbulent regime under the present operating conditions, as their Reynolds numbers reach on the order of $ {10}^5 $ . Therefore, it is necessary to examine whether pressure losses in the pipelines have a noticeable impact on the model results.

When pressure losses in the pipelines are considered, the velocity equation (Eq. (22)) can be rewritten as Eq. (24). The pressure loss ( $ {P}_{loss} $ ) is calculated using Eq. (25). In addition, the friction factor ( $ f $ ) is determined using the Haaland equation (Gerhart et al., Reference Gerhart, Andrew and John2016), as expressed in Eq. (26).

(24) $$ {V}_c=\sqrt{2\Big(\Delta P-{P}_{loss}/\rho } $$

(25) $$ {P}_{loss}=\rho g{h}_l=\rho g f\frac{l}{d}\frac{{V_c}^2}{2g} $$

(26) $$ \frac{1}{f^{1/2}}\approx -1.8\log \left[\frac{6.9}{\mathit{\operatorname{Re}}}+{\left(\frac{\epsilon /d}{3.7}\right)}^{1.11}\right] $$

where $ d $ and $ l $ are the diameter and length of the pipeline, respectively. The head loss in the pipe is denoted as $ {h}_l $ . The pipe roughness is represented by $ \varepsilon $ , with a value of $ 4.572\times {10}^{-5}\;\mathrm{m} $ provided by Hydrostor (2025). In addition, the velocity ( $ {V}_c $ ) can be obtained by simultaneously solving Eqs. (22)–(24) using a gradient descent method.

Based on Eq. (25), the pressure loss is strongly related to the pipeline length-to-diameter ratio ( $ L/D $ ), in addition to the flow regime. Therefore, to evaluate the effect of pressure loss on the simulation results, a parametric analysis was performed using the two-tank model without gas-zone connection by varying the pipeline length while maintaining a constant diameter of 10 cm. The results (as shown Figure 6) indicate that the effect of pressure loss on the simulation results becomes non-negligible when the pipeline length-to-diameter ratio ( $ L/D $ ) reaches approximately 100. In the present study, such long pipelines are typically not used to connect tanks within the same block. Therefore, pressure losses are neglected in the subsequent analysis to reduce computational resources.

Figure 6.

Effects of pressure loss on the temperatures of the two-tank model with a gas-zone connection: (a) Tank 1 temperature (diameter = 10 cm) and (b) Tank 2 temperature (diameter = 10 cm).

Figure 6. Long description

This figure group consists of two line graphs, (a) and (b), illustrating the effects of pressure loss on the temperatures of a 2-tank model with a gas-zone connection under four different scenarios over three full cycles (around 75 hours). Graph (a) shows the temperature profile of Tank 1, and graph (b) shows the temperature profile of Tank 2. In both graphs, the vertical axis represents temperature ranging from 185 to 200 degrees Celsius, and the horizontal axis represents time from 0 to 75 hours. Four curves are plotted in each graph, comparing a baseline case with no pressure loss against three different pipeline lengths: l = 1 m, l = 5 m and l = 10 m, while maintaining a constant pipeline diameter of 10 cm. The curves overlap significantly during the first cycle, but a noticeable divergence occurs during the second and third cycles, where longer pipeline lengths result in a visible drift or variation in peak temperatures.

Thermofluidic performances of the two-tank without gas zone connection

In this layout scheme, a pipeline with an inner diameter of 10 cm is used to connect the bottom water zones of the two tanks, enabling free water flow driven by the pressure difference between them. During the charging stage, water at a constant temperature of 195 °C is introduced into Tank 1. Assuming a total of 12 blocks, each consisting of two interconnected tanks (as shown in Figure 2), the total mass flow rate during the charging process is 22.66 kg/s. Simultaneously, due to the existence of the interconnecting pipeline, water stored in Tank 1 can supplement the water in Tank 2. The charging process concludes when the water level in Tank 1 reaches its upper operating limit, corresponding to the specified end operating volume ratio. During the discharging stage, water is released through the outlet of Tank 2 at a mass flow rate of 40.5 kg/s. Again, due to the pressure differential and the interconnection between the tanks, water from Tank 1 can be transferred into Tank 2 to sustain the outflow. The discharging process terminates when the water level in Tank 2 drops below the initial operating volume ratio. The temporal variations of the associated thermofluidic parameters, including temperature, pressure and water level, under this configuration are illustrated in Figure 7.

Figure 7.

Thermofluidic performances of a 2-tank system without a gas-zone connection: (a) temperature and pressure variations and (b) water-level variations.

Figure 7. Long description

This figure group consists of two line graphs, (a) and (b), illustrating the thermofluidic performances of a 2-tank water tank system without a gas-zone connection over three cycles (around 75 hours). Graph (a) plots temperature variations on the left vertical axis (185 to 200 degrees Celsius) and pressure variations on the right vertical axis (1000 to 1500 kPa) against time on the horizontal axis, showcasing cyclical fluctuations for temperatures T1, T2 and pressures P1, P2 with a noticeable time-lag effect between the two tanks. Graph (b) plots water level variations on the vertical axis (-12 to 12 meters) against time, showing asynchronous cyclical profiles for water levels H1 and H2. In both graphs, the variations stabilize and exhibit a highly regular periodic pattern after the second cycle.

The results demonstrate that the thermofluidic performance in both tanks undergoes dynamic variations from the initial condition, primarily due to the alternating processes of water filling and draining. After two complete operational cycles, these variations gradually tend to stabilize and exhibit a more regular pattern. As illustrated in Figure 7a, a distinct delay in the temperature response is observed between the two tanks. Specifically, during the charging phase, the inlet water for Tank 2 is supplied from Tank 1. Unlike Tank 1, which directly receives water at a constant temperature of 195 °C, the temperature of the water entering Tank 2 is subject to fluctuations, depending on the thermal state of the water stored in Tank 1. As a result, Tank 2 exhibits a delayed temperature response. Correspondingly, the saturated pressure inside each tank – being closely linked to the water temperature – also exhibits similar lagging behavior. Since the gas zones of the two tanks are not interconnected, their pressure variations evolve independently, even though they follow broadly similar trends.

In addition to the time-lag effects observed in temperature and pressure, the water-level fluctuations between the two tanks also exhibit asynchronous behavior, as shown in Figure 7b. This lack of synchronization is primarily caused by the differences in the mass flow rates during the filling and draining processes of the two tanks. According to the operating logic of the system, the inflow rate into Tank 1 and the outflow rate from Tank 2 are maintained constant. In contrast, the inflow rate into Tank 2 and the outflow rate from Tank 1 are not fixed, but vary over time. These variable flow rates are primarily determined by the dynamic differences in water level and saturated pressure between the two tanks, as further depicted in Figure 8.

Figure 8.

Variation of water mass flow rate in the connecting pipe between the two water tanks.

Figure 8. Long description

This figure is a single line graph plotting the water mass flow rate against time, spanning three complete operational cycles. The curve exhibits a highly periodic and dynamic cyclical wave profile. At the beginning of each cycle, the flow rate undergoes a rapid and sharp spike to its maximum peak, followed immediately by a steep decline until it reaches a low-level plateau. Midway through each cycle, the flow rate experiences another sudden increase to form a secondary, elevated plateau where it remains stable before dropping back down. This entire fluctuating pattern of sharp peaks, deep valleys, and stable plateaus repeats with high consistency across all three cycles.

Thermofluidic performances of the two-tank system with a gas-zone connection

In an alternative layout scheme, an additional pipeline with an inner diameter of 10 cm is introduced to connect the gas zones of the two tanks. This gas zone interconnection enables vapor to flow between the tanks, with the direction of flow determined by the pressure difference between the respective gas zones. Following the same operating logic as applied in the previous layout, the subsequent analysis investigates the thermofluidic behavior of the two-tank system incorporating the gas zone connection. The evaluation is conducted over three complete charging–discharging cycles to capture the system’s dynamic response and stabilization trends. The corresponding results, including variations in temperature, pressure and water level, are illustrated and analyzed in detail in Figure 9.

Figure 9.

Thermofluidic performances of a 2-tank system with a gas-zone connection: (a) temperature and pressure variations and (b) water-level variations.

Figure 9. Long description

This figure group consists of two line graphs, (a) and (b), illustrating the thermofluidic performances of a 2-tank water tank system with a gas-zone connection over three full operational cycles. Graph (a) plots temperature and pressure variations against time, showing highly consistent, overlapping, and synchronized periodic fluctuations for the parameters across both tanks. Graph (b) plots water level variations against time, displaying identical and overlapping triangular wave oscillations for both tanks. Throughout all cycles, the thermofluidic variations exhibit high symmetry and exceptional stability right from the initial cycle, with no visible time-lag or asynchronous behavior between the two tanks.

Similar to the behavior observed in the two-tank system without a gas-zone connection, the thermofluidic performance variations – including temperature, pressure and water level – tend to stabilize after two complete charging and discharging cycles. However, as depicted in Figure 9a, the delays in temperature and pressure responses between the two tanks are significantly reduced. This observation confirms that the presence of the gas-zone connector plays a critical role in promoting more synchronized thermofluidic behavior across the tanks. By enabling unrestricted vapor exchange, the connector effectively facilitates the equalization of pressure and temperature throughout both charging and discharging stages, thereby enhancing the dynamic coupling of the system.

Except to the improved synchronization of temperature and pressure, the water levels in both tanks also remain nearly identical throughout the operational period, as illustrated in Figure 9b. As shown in Figure 10a, vapor is able to flow freely between the tanks via the connecting pipeline, contributing to the continuous balance of saturated pressure. Consequently, unlike the configuration without a gas-zone connection, the water mass flow rates into Tank 2 and out of Tank 1 are now primarily governed by the difference in water levels between the tanks. Furthermore, Figure 10b demonstrates that the net water mass flow rates in both tanks tend to converge over time, indicating increasingly stable flow dynamics. As a result, the water-level variations in the two tanks become progressively more synchronized.

Figure 10.

Variations of vapor and water mass flow rate between the two water tanks: (a) vapor mass flow rate and (b) water mass flow rate.

Figure 10. Long description

This figure group consists of two line graphs, (a) and (b), illustrating the mass flow rates between two water tanks with a gas-zone connection over three full operational cycles. Graph (a) plots the vapor mass flow rate against time, which exhibits a highly symmetrical and periodic high-frequency oscillation forming a fluctuating envelope wave pattern, supplemented by an inset zoom-in view showing rapid periodic micro-oscillations. Graph (b) plots the water mass flow rate against time, showcasing a step-like cyclical waveform with sharp spikes at the beginning of each cycle that gradually damp down and converge, indicating increasingly stable fluid dynamics over time.

Overall, when a multi-tank design does not include a gas zone connection, the temperature, pressure and water-level variations among different tanks, including the end tanks, exhibit significant differences. After connecting the gas zone, these differences are substantially reduced, leading to more consistent thermofluidic performance changes throughout the tanks. Therefore, incorporating a gas-zone connection is necessary in a multi-tank design.

Effect of tank quantity on thermofluidic performances in a multi-tank design

In all three multi-tank configurations, the diameters of the pipelines connecting both the water and gas zones are uniformly set to 10 cm. During the charging phase, hot water at a temperature of 195 °C is introduced into the first tank in each block, while during the discharging phase, water is extracted from the last tank. To ensure consistency in the total mass flow rate of water entering and exiting the system, the charging and discharging mass flow rates per block are adjusted according to the specific design configurations: 22.66–39 kg/s for the 12 × 2-tank setup, 34–58.5 kg/s for the 8 × 3-tank setup and 45.33–78 kg/s for the 6 × 4-tank setup. These rates are selected to achieve comparable system-level throughput (total mass flow rates: 468 kg/s [discharging] and 272 kg/s [charging]) across the different arrangements. The corresponding variations in temperature and pressure for the three multi-tank configurations over three complete charging–discharging cycles are presented in Figure 11, forming the basis for assessing pressure and temperature synchronization among the different designs.

Figure 11.

Variations of temperature and pressure in the three different multi-tank designs: (a) 12 × 2-tank, (b) 8 × 3-tank and (c) 6 × 4-tank.

Figure 11. Long description

Three line graphs labeled a, b, and c share identical axes. The horizontal x-axis represents Time in h from 0 to 75. The primary vertical y-axis on the left represents Temperature in degrees Celsius from 185 to 200. The secondary vertical y-axis on the right represents Pressure in k P a from 1000 to 1500.

* Panel a: Displays four lines. T 1 (solid red) and T 2 (dashed blue) track together, showing three cycles of rapid increase followed by a sharp drop, peaking near 197 degrees Celsius. P 1 (solid black) and P 2 (dashed grey) track together at a higher amplitude, peaking near 1450 k P a.

* Panel b: Displays six lines. T 1, T 2, and T 3 (dotted green) overlap perfectly. P 1, P 2, and P 3 (dotted magenta) also overlap perfectly. The trends follow the same cyclic pattern as panel a, with temperature peaking at 197 and pressure peaking at 1450.

* Panel c: Displays eight lines. T 1 through T 4 and P 1 through P 4 are plotted. All temperature lines (T) are perfectly coincident, and all pressure lines (P) are perfectly coincident. The time scale on the x-axis for this panel ends at 60 h, showing two and a half cycles compared to the three cycles in panels a and b.

Similar thermofluidic performance patterns are observed across the three multi-tank configurations. Specifically, the temperature and pressure variations within each tank stabilize after two complete operating cycles. Furthermore, within each block, the tanks exhibit nearly identical temperature and pressure profiles across all three designs. Since the steam regions are connected, the vapor pressure tends to equalize among the tanks; therefore, increasing the number of tanks within a block would not obviously affect the synchronization of temperature and pressure variations. These findings confirm that the implementation of gas zone connectors effectively balances the pressures and temperatures among tanks in all configurations. However, differences emerge in the water-level variation trends among tanks within a block, as illustrated in Figure 12. As the number of tanks connected in series increases within a block, the synchronization of water-level fluctuations among the tanks decreases markedly. When the pipeline diameter is fixed, the hydraulic exchange capacity among the tanks is limited. As the number of tanks connected in series increases, the difference in inflow and outflow responses between upstream and downstream tanks becomes more pronounced. Consequently, the synchronization of water-level variations decreases as the number of tanks per block increases.

Figure 12.

Variations of water level in the three different multi-tank designs: (a) 12 × 2-tank, (b) 8 × 3-tank and (c) 6 × 4-tank.

Figure 12. Long description

The figure consists of three panels labeled a, b, and c. All panels share the same axes: the X-axis represents Time in h from 0 to 75, and the Y-axis represents Water level in m from -12 to 12.

* Panel a: Contains two lines, H 1 (solid blue) and H 2 (dashed red). Both lines follow an identical triangular wave pattern, oscillating between approximately -10 and 10 meters with three complete peaks at 15, 40, and 65 hours.

* Panel b: Contains three lines, H 1 (solid blue), H 2 (dashed red), and H 3 (dashed orange). H 1 and H 2 are nearly identical. H 3 follows the same triangular frequency but shows slightly lower troughs, reaching approximately -11 meters compared to the -10 meters of H 1 and H 2.

* Panel c: Contains four lines, H 1 (solid blue), H 2 (dashed red), H 3 (dashed orange), and H 4 (dashed purple). The lines show increasing divergence at the troughs. H 1 has the highest troughs at -4 meters, while H 4 has the deepest troughs, returning to the baseline of -10 meters. All four variables maintain synchronized peaks at approximately 10 meters.

Effect of water flow rate on water-level variations in a multi-tank design

In principle, connecting more tanks in series is desirable for multi-tank configurations. However, as shown in Figure 12, increasing the number of tanks also increases the hydraulic resistance in the pipelines, making an obvious gap in the water level within different tanks gradually. To further enhance the synchronization of water-level variations among tanks, a direct method is to adjust water mass flow in and out of the system. To this end, the performance of the 8 × 3 configuration was evaluated under three different water mass flow rates: a reduced flow rate corresponding to that of the 12 × 2 configuration, its original flow rate and an increased flow rate corresponding to that of the 6 × 4 configuration. The results are displayed in Figure 13.

Figure 13.

Variations in the water level under three different water mass flow rates: (a) reduced flow rate, (b) original flow rate and (c) increased flow rate.

Figure 13. Long description

The figure consists of three panels, each with a y-axis representing Water level in meters ranging from -12 to 12 and an x-axis representing Time in hours. Each panel contains three data series: H 1 (solid blue line), H 2 (dashed red line), and H 3 (dashed orange line).

* Panel a: The x-axis ranges from 0 to 110 hours. The three lines are nearly perfectly overlapped, showing a periodic triangular wave pattern. The peaks reach approximately 10 meters at 24, 60, and 96 hours, while troughs drop to -10 meters at 0, 36, 72, and 108 hours.

* Panel b: The x-axis ranges from 0 to 75 hours. The triangular wave frequency is higher than in panel a. H 1, H 2, and H 3 show slight divergence at the peaks and troughs. Peaks occur at approximately 15, 38, and 61 hours. H 1 reaches the highest peaks and shallowest troughs, while H 3 shows the greatest negative displacement at the troughs.

* Panel c: The x-axis ranges from 0 to 50 hours. This panel shows the highest frequency and the most significant divergence between the three series. Peaks occur at approximately 11, 27, and 43 hours. H 1 (solid blue) consistently maintains the highest peaks and highest troughs. H 2 (dashed red) is intermediate. H 3 (dashed orange) shows the lowest peaks and the deepest troughs, reaching nearly -11 meters.

The results indicate that reducing the mass flow rate can significantly improve the synchronization of water levels among different tanks within a block. However, the water mass flow rate is typically constrained by the overall design requirements of the CAES system. This implies that, within this project, the flow rate of the water tank system is not freely adjustable.

Effect of piping diameter on water-level variations in a multi-tank design

To enhance the synchronization of water-level variations among tanks while satisfying the specific design requirements of a CAES system, it is necessary to reduce flow resistance by appropriately increasing the diameter of the connecting pipelines. Under the current calculated conditions, the 8 × 3 configuration represents a balanced design, demonstrating the advantages of a multi-tank arrangement while avoiding the need for excessively large pipeline diameters. Therefore, using this configuration as an example, three pipe diameters – 8 cm, 10 cm and 12 cm – are evaluated in the following study. All other operating parameters remain unchanged, with the mass flow rate of the input hot water (at 195 °C) fixed at 34 kg/s and the output water mass flow rate set at 58.5 kg/s. Figure 14 presents the water-level variations recorded over three complete charging–discharging cycles for each of the tested pipe diameters, allowing a direct comparison of their impact on system performance.

Figure 14.

Variations of water level under three kinds of diameters of connection pipe: (a) 8 cm, (b) 10 cm and (c) 12 cm.

Figure 14. Long description

A three-panel line graph series labeled a, b, and c. All panels share the same axes. The x-axis is Time in h (hours) ranging from 0 to 75. The y-axis is Water level in m (meters) ranging from -12 to 12. Each graph contains three data series: a solid blue line (H 1), a thick dashed red line (H 2), and a thin dashed orange line (H 3). All series follow a periodic triangular wave pattern with peaks at approximately 15, 35, and 55 hours.

* Panel a: The three lines show significant divergence. H 1 reaches the highest peaks (approx. 10) and shallowest troughs (approx. -5). H 3 shows the deepest troughs (approx. -10) and lowest peaks (approx. 8). H 2 sits between them.

* Panel b: The lines are closer together than in panel a. The peaks and troughs of H 2 and H 3 have shifted closer to the values of H 1, though H 3 still shows slightly deeper troughs.

* Panel c: The three lines H 1, H 2, and H 3 are nearly perfectly overlapped, appearing as a single triangular wave oscillating between approximately -10 and 10 meters.

Reduced flow resistance can be effectively achieved by using larger-diameter connection pipes. As the diameter of the piping increases, the variations in water level among the three tanks exhibit greater consistency and synchronization. This trend indicates that enlarging the pipe diameter is a viable and efficient strategy to enhance the coordination of water-level fluctuations across multiple tanks. Improved uniformity in water-level response not only promotes better thermofluidic performance but also contributes significantly to the overall operational stability and reliability of the system.