Analysis of Thermodynamic Processes in Thermal Energy Storage Vessels

Abstract

To balance the quantity of heat generated and consumed, thermal energy storage systems are crucial for power plants and district heating systems. Particularly when phase transitions and pressure variations are not adequately covered in the existing literature, their work frequently takes place under complicated, changing temperature and fluid dynamic settings. The goal of this research is to create a thermodynamic model that incorporates the effects of steam condensation, steam injection, and heating failures to describe the transient behaviour of temperature and pressure in pressure vessels containing single-phase and two-phase fluids. To account for nonlinear, temperature-dependent steam properties, as well as initial and boundary constraints, the study proposes energy balance models for hot water and saturated steam cases. Numerical simulations evaluating sensitivity to parameter changes are presented alongside analytical solutions for isochoric and isobaric systems. The model also includes direct steam injection heating and the use of a heat exchanger. It explains the changes in temperature and pressure that occur in thermal energy storage systems over time, including significant events such as steam cushion collapse and condensate drainage. According to the sensitivity analysis, the main factors influencing the system’s safety limitations and transient dynamic phenomena are thermal power, heat exchanger capacity, and thermal insulation efficiency. The proposed thermodynamic model closes a major gap in the literature by providing reliable predictions of the transient behavior needed for the safe design and reliable operation of pressure vessels utilized for heat storage in district heating networks. This model can be used by engineers and researchers to optimize system design and steer clear of risky operational situations.

1. Introduction

Today, district heating systems are important for energy efficiency, decarbonisation, and sustainability. Centralised heat production from various sustainable energy sources, such as renewables and waste heat, and its distribution reduce greenhouse gas emissions, costs, and dependence on fossil fuels, while also improving air quality in urban areas.

Thermal energy storage (TES) is important for modern energy infrastructure because it balances the time mismatch between heat production and consumption in various applications [1]. TES provides greater flexibility in managing energy flows in industrial applications, district heating systems, and combined heat and power (CHP) systems [2]. By storing thermal energy during periods of excess supply and releasing it during periods of demand, these systems stabilise supply and improve supply chain efficiency [3,4].

The main categories of TES include sensible heat storage, which uses temperature differences in materials such as water, rock, and oil, and latent heat storage (LTES), which uses phase change materials (PCMs) with high enthalpy of fusion [1,4].

LTES stores heat as the latent heat of fusion of the PCM: during charging, the PCM melts and absorbs heat, while during discharging, it solidifies and releases heat at a nearly constant temperature [5]. This isothermal behaviour provides high energy density in a compact volume, which is superior to sensible heat storage for matching variable supply and demand, especially in domestic and industrial heating systems with renewable energy sources [6]. The low thermal conductivity of most PCMs limits system performance; therefore, techniques to improve heat transfer have been developed, using numerical analysis and optimisation of the geometric parameters of the LTES [7,8].

In district heating plants, the most common heat accumulators are large steam boilers and hot water tanks. In cold climates such as northern Europe and China, pressure vessels serve as thermal energy reservoirs and compensate for the volumetric expansion of water in closed circuits [9]. Hot water and steam are used as working fluids due to their favourable thermodynamic properties, high density, availability, and low cost. In small systems, mechanical expansion vessels with inert gas maintain system pressure, while large industrial systems use saturated steam from auxiliary boilers [9]. Advanced designs include heat exchangers and a control system with advanced algorithms that make it easier to use and combine different energy sources.

Heating systems that use water vapour as the working medium require precise temperature and pressure control to ensure adequate heat transfer. Rapid phase transitions during condensation can damage the vessel structure if the pressure drops below the saturation limit. Therefore, monitoring and control of fluid parameters are essential for safety [9]. In district heating systems, unsteady processes can cause significant accidents, such as pipe bursts, damage to expansion joints, and pump failures, resulting from thermal shocks, leaks, and water hammer (the rapid generation of shock waves from the kinetic energy of the fluid) [10,11]. Two-phase flows, elevated operating temperatures, and closed network configurations that limit pressure relief amplify these effects in district heating systems. A recent study using the Method of Characteristics and the MacCormack approach highlights the crucial role of timely valve closure and system compliance in reducing pressure shocks and preventing damage [10,11].

Non-stationary processes are critical for the safety of pressure vessels, although the literature lacks comprehensive models for atypical conditions. Classical thermodynamics addresses stationary conditions, but dynamics—such as cooling, steam injection, pressure changes, and condensation shocks—require time-dependent modelling. This complicates the design of parameters such as pressure maintenance and overpressure protection in two-phase media systems [9,12].

Pressure control and valve closure strategies are essential for the safe operation of pressurised thermal storage systems. Valve-closing laws—such as linear, concave, or convex profiles—define the rate and timing of valve operations to control flow transients and minimise pressure surges. Emergency shutdown strategies combine rapid yet controlled valve closures (for example, achieving isolation within 30 min as required by regulatory standards) with pressure relief and monitoring to prevent cascading failures such as pipe bursts or pump damage.

In district heating networks, emergency shutdown procedures often require rapid valve closure to prevent thermal shocks and cascading failures. The model developed in this paper explicitly considers transient pressure dynamics during such critical events. Specifically, the isochoric condensation equations (Section 4.1 and Section 4.2) describe the pressure drop dynamics following unexpected cooling of the vapour cushion, while the isobaric condensation equations (Section 4.3) represent the controlled pressure drop. A sensitivity analysis of the heat loss coefficient kA_p (Section Isochoric Condensation—Sensitivity to $k{A}_p$) demonstrates how vessel insulation quality directly influences the time scale of pressure collapse—a key parameter for designing safety valve settings and determining safe closure rates. By providing quantitative predictions of dynamic pressure phenomena, this model enables engineers to establish valve closure protocols that prevent water hammer and maintain system stability.

Transient models indicate that steady-state assumptions introduce errors, particularly during rapid, non-stationary filling and emptying. Developing mass and energy balance formulations that incorporate empirical correlations for evaporation and condensation rates, together with pressure-dependent steam table properties, yields more accurate predictions of transient behaviour than classical equilibrium models [12].

Network modelling in low-temperature district heating (40–50 °C) requires the coupled solution of conservation equations, using graphical representations and methods such as the finite volume method or the method of characteristics. Low-temperature networks, which are increasingly deployed to enhance compatibility with renewable energy sources (solar thermal, heat pumps, waste heat recovery), exhibit substantially different time constants, ranging from minutes to hours, compared to high-temperature networks (80–90 °C) [9,13]. The transient model developed in this work is particularly valuable for low-temperature systems because it:

• captures phase-change dynamics at reduced saturation pressures, which is relevant when solar or heat pump heating provides variable thermal input;
• enables the design of expansion vessels and safety systems scaled for lower temperature differentials; and
• supports the optimisation of insulation and condensation control to improve energy efficiency in systems where renewable sources exhibit intermittent supply.

These capabilities make the model well suited to emerging low-temperature district heating networks that integrate photovoltaic-thermal (PVT) collectors, seasonal thermal storage, and demand-side management with automated valve control.

Nakahara et al. [14] modelled the thermohydraulic behaviour of stratified chilled water in a TES using a one-dimensional model: the lower zone was mixed, while the upper zone was stratified. They presented performance metrics and design implications.

A semi-empirical model developed by Atabaki and Bernier [15] divides the tank into two areas, one defined by an empirical correlation and the other by a quasi-one-dimensional temperature model. Their method showed that internal mixing effects between nodes are significant when applied to a domestic thermal storage tank with a single-coil heat exchanger. Consequently, these effects are considered in the development of their model.

Newton [16] developed a TRNSYS model for tanks with non-uniform cross-sections, using the Crank-Nicolson method (error ≈ 0.005%) as the most accurate approach. Yin et al. [17] investigated condensation in pressure relief tanks using computational fluid dynamics (CFD), analysing the influence of inlet steam and pool temperature, and developed models for both stationary and non-stationary states.

Rodrigues et al. [18] conducted a non-dimensional analysis to represent the transient natural convection model for a domestic storage tank. They found that heat losses through the walls are governed by the Rayleigh number, the overall heat loss coefficient, and the aspect ratio of the tank, while the Prandtl number has a negligible influence.

Key innovations in TES include the use of artificial intelligence (AI) for design and optimisation, hybrid configurations, nano-enhanced PCMs, bio-inspired genetic algorithms, assessments of industry decarbonisation technologies, and optimisation of distributor topologies for improved stratification. Mehraj et al. [19] reviewed AI techniques—machine learning, fuzzy logic, and neural networks—to enhance sustainability and performance, providing recommendations. Mehraj et al. [20] developed an AI framework with genetic algorithms, increasing the heat transfer surface by 29%. Palacios et al. [21] evaluated 11 TES technologies (100–300 °C), highlighting the maturity of sensible systems and the potential of latent and thermochemical systems. Kwasi-Effaha and Okpako [22] analysed innovations in sensible, latent, and thermochemical TES with applications in green hydrogen. Wang et al. [23] optimised distributors in thermocline tanks; an octagonal topology reduced losses by 6.4%.

Research Gap and Novelty

Comprehensive models for rigid, closed pressure vessels containing mixtures of steam and water under transient conditions have not been established in the literature. CFD studies (Yin et al. [17]) address the local physics of condensation but cannot provide the rapid parametric sensitivity analysis needed for vessel design. One-dimensional stratified models (Nakahara [14]) do not account for changes in fluid phases. Packed bed models (Trevisan and Guédez [24]) analyse open flow applications in Concentrating Solar Power (CSP), disregarding the closed system pressure constraints essential for district heating expansion vessels.

This investigation addresses this gap by developing an analytical thermodynamic framework that simultaneously encompasses:

• Nonlinear properties of the steam table in differential energy balances (Equations (33)–(39)),
• Dynamics of isochoric pressure collapse (steam cushion rupture, Section 4.1),
• Boundary conditions for condensate drainage (isobaric processes, Section 4.3),
• Design-oriented sensitivity analysis (kA_p, UA effects, Section 3.5).

This research presents four novel contributions not previously identified in the literature:

(a)

An analytical solution for isochoric and isobaric vapor condensation in rigid vessels (Equations (33)–(45)), enabling computation 1000 times faster than CFD while maintaining ±15% accuracy.

(b)

Explicit modelling of vapor cushion collapse—a critical failure mode not addressed by one-dimensional vessel models—quantified by the pressure drop time scale τ_crit.

(c)

Design rules derived directly from sensitivity analysis: maximum kA_p = 7.5 W/K (safe shutdown for 400 h), minimum UA = 6000 W/K (start-up in 30 min).

(d)

The homogeneous equilibrium model is validated against direct contact condensation experiments (Ryu et al. [25], Kim et al. [26]) with a time constant accuracy of ±13%, establishing clear limits of applicability (τ > 10 min).

The specific objectives of this research are as follows: to develop analytical and semi-analytical solutions for saturated steam systems in constant volume (isochoric) and constant pressure (isobaric) processes; to characterise the time course of condensation with and without condensate removal through energy balance analysis; to perform sensitivity analyses to identify the dominant parameters governing transient behaviour; and to demonstrate the application of the model in low-temperature district heating contexts where renewable sources (solar thermal, heat pump systems) introduce variable heating rates, and to validate system designs that ensure safety and reliability while maintaining compatibility with intermittent renewable energy sources.

2. Materials and Method

In a typical district heating system, there are numerous complex components such as heat storage, distribution pipes, expansion vessels and valves. District heating networks comprise three coupled physical domains where thermal storage vessels play distinct but interconnected roles (

Table 1. Three coupled physical domains.

Component	System Dynamic Role	Vessel Model Application
Heat accumulators	Peak shaving buffer: Charge during low demand, discharge during surges [12]	Mass/energy balance (Equation (27)) Boundary condition [27]: Tpipe,out(t) → Tvessel,in(t)
Expansion Vessels (Pressure regulation)	Thermal expansion compensation, steam cushion collapse prevention	Isochoric condensation (Equation (33))
Pipelines (Distributed storage, 20–40% capacity)	Thermal inertia, wave propagation [28]	Boundary condition [27]: Tpipe,out(t) → Tvessel,in(t)

).

Heat accumulators (vessels) do not operate in isolation. They provide state-dependent boundary conditions (T_out, p_out) to 1D pipe solvers and receive disturbance inputs (ṁ_in, valve states) from network hydraulics.

This work examines in detail the thermodynamic processes of pressure vessels used for thermal energy storage in both power plants and district heating systems. These vessels range from large boilers and classical Roots and Marrguere tanks to hot water tanks. The simulated systems contain liquids with either a saturated or unsaturated liquid phase and a vapour phase composed of saturated water vapour, which is essential for heat transfer. The analysis considers two main heating mechanisms: direct steam injection into the liquid and heat transfer via submerged heat exchangers. This approach accurately predicts thermodynamic behaviour and optimises system design for safer and more effective thermal energy storage by reflecting real-world operating conditions.

Each pressure vessel is modelled as a rigid, closed, and spatially homogeneous system. Mechanical work based on deformation is thus neglected.

The governing equations describing thermodynamic processes in closed pressure vessels are derived from first-law energy balance principles applied to unsteady-state systems [29,30]. For systems where heat transfer and work interactions occur but the control volume remains fixed in space, the differential energy balance in its most general form expresses conservation of energy according to Equation (1):

$$dQ-dW=dU$$

(1)

where the change in internal energy depends on net heat transfer and work interactions. For constant-volume processes, characteristic of rigid pressure vessels, mechanical work is negligible (dW ≈ 0) and the energy balance simplifies to:

$$dQ=dU$$

(2)

The resulting differential equations describing the changes in temperature and pressure include heat transfer rates representing losses through the vessel insulation, as well as convective heat transfer mechanisms within the vessel. The convective heat transfer coefficient, h, represents energy transfer between the internal fluid and solid surfaces, with typical values ranging from 50 to 1000 W/(m²·K) for natural convection in liquids and from 5000 to 100,000 W/(m²·K) during steam condensation on cooler surfaces [30,31].

Steam–water mixtures in the two-phase region present additional complexity, requiring explicit modelling of phase transition kinetics.

The thermodynamic quality, x, defined as the mass fraction of vapour in a saturated mixture, is an essential state variable in systems containing both liquid and vapour phases. Latent heat release from condensation processes often dominates the energy balance. In such mixtures, the specific enthalpy is a nonlinear function of temperature because the saturation properties depend on temperature, as determined by rigorous steam table correlations [32]. Two-phase flow dynamics in confined geometries exhibit complex behaviour patterns, with distinct mist flow, annular flow, injection flow, and slug-bubbly flow regimes, each possessing specific thermodynamic and hydrodynamic characteristics. The transition between flow regimes depends on vapour quality, mass flux, system geometry, and the interfacial area between phases [33]. Recent experimental studies on steam condensation in microchannels and heating network applications have documented flow pattern transitions and their effects on local heat transfer coefficients, demonstrating that mist and annular flow regimes have substantially higher heat transfer coefficients than slug-bubbly flow configurations [33,34].

For the non-vapour, hot water phase, the differential energy balance equation takes into account heat losses through vessel insulation, heat input from external sources, and temporal temperature variation. It is given by Equation (3) [35]:

$${\displaystyle \frac{dT}{dt}}=-{\displaystyle \frac{hA}{cM}}\left(T-T_{\mathrm{\infty }}\right)+{\displaystyle \frac{\dot{Q}\left(t\right)}{cM}}$$

(3)

where h is the overall heat transfer coefficient, A is the surface area, c is the specific heat capacity, M is the fluid mass and (t) is the heat input rate. Analytical solutions are derived for cases of free cooling, steady heating, and heating by heat exchangers, enabling assessment of thermal transients under simplifying assumptions.

Saturated two-phase conditions are included in the model, with vapour generation and condensation phenomena described dynamically. The thermodynamic quality, x, or vapour mass fraction, is a key state variable. To model sudden transients such as vapour cooling and condensation, or vapour cushion collapse, two differential equations governing the evolution of temperature and vapour quality are formulated.

During isochoric condensation, the process is simulated at constant volume, resulting in a significant reduction in vapour temperature and pressure. The governing equations incorporate empirical correlations for saturation properties, accounting for the nonlinear variation in enthalpy with temperature, based on steam table data [36]. For condensate drainage, the model applies mass flow boundary conditions to simulate the effect of condensate removal on the thermodynamic state.

The model relies on several standard assumptions to enable tractable analysis:

• The vessel is rigid and does not undergo deformation work.
• The system remains spatially homogeneous, with well-mixed temperature and vapour quality.
• Heat transfer is dominated by conduction and convection; radiative effects are negligible.
• The vapour is saturated, and non-equilibrium effects are initially neglected, with local thermal equilibrium assumed between phases.
• Heat losses are modelled as proportional to the temperature difference, and the heat transfer coefficient h is parameterised using empirical correlations.
• If condensate drainage is included, the mass and energy balances can be significantly simplified by assuming complete or partial removal of condensate during transients.

The presented model assumes local thermal equilibrium between the phases, where both the liquid and the vapour instantaneously reach the saturation temperature at the system pressure. This assumption enables analytical solutions and is valid for:

• Slow transients (τ > 10 min): diffusion and mass transfer reach equilibrium within the vessel mixing time.
• Moderate subcooling (ΔT_sub > 10 K): interfacial condensation is rapid and close to equilibrium.
• Normal district heating operation: condensation rates ṁ_cond < 1 kg/(m²·s).
• Expected accuracy: ±10–15% compared with non-equilibrium models.

This assumption does not hold for fast transients:

• Rapid decompression (dp/dt > 0.1 bar/s): Vapour cools faster than the liquid can equilibrate; steam enters a metastable superheated state (T_steam > T_saturation).
• High-velocity jet injection (>5 m/s): Localised superheat or subcooling zones develop before mixing; mass transfer at the phase boundary becomes the rate-limiting factor.
• Scenarios of vapour cushion collapse (emergency release, rupture): Steam crosses the saturation limit into the superheated region; condensation exhibits delayed nucleation.

In these cases, a transient non-equilibrium state develops where T_steam ≠ T_liquid ≠ T_saturation, and the vapour pressure lags behind the saturation pressure. Stevanović et al. [12] showed that rapid pressure cycles create significant non-equilibrium deviations.

Initial conditions specify the temperature T₀, vapour quality x₀, and pressure p₀ within the vessel at the start of the transient. These values are taken from steady operational states before a perturbation, such as shutdown, rupture, or sudden steam injection.

Boundary conditions depend on the process scenario:

• For hot water heating or cooling, the ambient temperature T_∞ is specified, with heat fluxes modelled as functions of the temperature difference.
• For vapour processes, the saturation temperature T_sat(p) and the corresponding saturation vapour quality are used as reference states.
• For heat exchanger interactions, specified inlet heat transfer rates and inlet temperatures serve as boundary conditions during heating processes.

The boundary conditions are applied as initial states, with the transient simulation capturing the evolution based on the differential equations.

The proposed thermodynamic model accurately predicts global energy and mass balances for closed pressure vessels under well-mixed (homogeneous) conditions. However, it cannot capture spatial phenomena such as:

• Thermal stratification caused by high-velocity jets or wall effects,
• Flow regime transitions (from bubbly to annular to mist flows) that alter local heat transfer coefficients,
• Non-equilibrium condensation during rapid depressurisation, where local vapour becomes supersaturated.

The model is suitable for:

• Large industrial thermal storage tanks (volumes > 50 L) with transient timescales exceeding 10 min,
• Simple vessel geometry (cylinders/rectangles without internal baffles),
• Sensitivity analysis and parameter identification (e.g., determining insulation effectiveness kA_p from cooling curves),
• System-level district heating simulations where component-averaged properties are sufficient,
• Design safety margins and relief valve tuning (tolerance ±10–15%),
• Subcooling with ΔT_sub > 10 K (well-mixed, slow equilibration),
• Low injection velocity (<1 m/s).

However, CFD analysis is still required for:

• Emergency scenarios with depressurisation rates greater than 0.1 bar/s (steam cushion collapse with structural resonance risks),
• Complex geometries with internal baffles, heat exchanger coils, or nozzle arrays,
• Validation of local heat transfer coefficients in novel materials or microchannel systems,
• Transient phenomena lasting under 5 min, where diffusion and mixing cannot be assumed complete,
• Subcooling with ΔT_sub < 5 K (rapid, localized condensation),
• High injection velocity (>10 m/s jet injection).

Quantitative comparisons with CFD validate the lumped-parameter approach: analytical solutions (Equations (33)–(45)) achieve approximately 15% accuracy in pressure and time transients compared to CFD for isochoric condensation [17], with computation 1000 times faster. Lumped models predict global pressurisation rates within 10–20% of detailed CFD for homogeneous two-phase flows in pressure vessels, although they lack spatial resolution. Time constants match experiments and CFD within 13–15% (Ryu et al. [17]; Section 4.5), confirming reliability for design transients longer than 10 min.

Future research could develop and apply a two-temperature non-equilibrium model with explicit nucleation kinetics, extending its applicability to emergency scenarios while maintaining computational efficiency for routine design calculations.

3. Investigation of Thermodynamic Processes in Single-Phase Unsaturated Liquids

The design, characteristics, and energy balance of the hot water tank are shown in Figure 1, depending on whether heating is electric or provided via a heat exchanger with auxiliary energy.

We assume that the mass of liquid stored in the tank remains constant. This case represents the behaviour of hot water storage tanks during heating or cooling. The energy balance of the tank is formulated as follows: the change in the energy of the fluid stored in the tank results from heat losses and heating input. The amount of liquid in the tank remains constant. The differential energy balance of the tank can be written as Equation (4) [16]:

$$k{A}_{\mathrm{s}}\mathrm{\Delta }td\mathsf{\tau }=-\mathrm{c}Md\left(\mathrm{\Delta }t\right)+\dot{Q}\left(\tau \right) d\tau ,$$

(4)

where

• $\dot{Q}\left(\tau \right) d\tau$ heat supplied to the tank,
• Δt temperature difference relative to the environment,$\mathrm{\Delta }t = t-t_k$
• M mass of liquid stored in the tank.

Let us define

$$k{A}_{\mathrm{s}}=a_0,$$

(5)

$$cM=a_1.$$

(6)

where

• k heat transfer coefficient between the tank and the environment,
• $A_{\mathrm{s}}$ surface area of the tank insulation,
• c specific heat capacity of the stored fluid.

Then the energy balance equation becomes:

$$a_0\mathrm{\Delta }td\tau =-a_1\mathrm{d}\left(\mathrm{\Delta }t\right)+\dot{Q}\left(\tau \right) d\tau$$

(7)

Rearranging gives the differential form of the energy balance:

$${\displaystyle \frac{d}{d\tau }}\mathrm{\Delta }t+{\displaystyle \frac{a_0}{a_1}}\mathrm{\Delta }t-{\displaystyle \frac{1}{a_1}}\dot{Q}\left(\tau \right)=0.$$

(8)

The solutions of Equation (7) and the temporal evolution of the process are analyzed for the following cases.

3.1. Temperature Change in the Fluid When the Tank Is Not Heated

If the tank is left without heating, i.e., $\dot{Q}\left(\tau \right)=0$, then Equation (7) reduces to a first-order linear homogeneous differential Equation (9) with the initial condition $\tau = 0, \mathrm{\Delta }t = \mathrm{\Delta }{t}_0$

$$\mathrm{\Delta }t\left(\tau \right)=\mathrm{\Delta }{t}_0{e}^{-{\displaystyle \frac{a_0\tau }{a_1}}}$$

(9)

If the tank insulation is designed such that the fluid does not cool below a certain $\mathrm{\Delta }{t}^{*}$ during an idle period ${\tau }^{*}$ the required insulation effectiveness $a_0^{*}$ can be calculated according to Equation (10):

$$a_0^{*}={\displaystyle \frac{a_1}{{\tau }^{*}}}\mathit{ln}{\displaystyle \frac{\mathrm{\Delta }{t}_0}{\mathrm{\Delta }{t}^{*}}}$$

(10)

Conversely, the cooling time ${\tau }^{*}$ for the fluid to reach a specific $\mathrm{\Delta }{t}^{*}$ can be expressed as:

$${\tau }^{*}={\displaystyle \frac{a_1}{a_0}}\mathit{ln}{\displaystyle \frac{\mathrm{\Delta }{t}_0}{\mathrm{\Delta }{t}^{*}}}$$

(11)

3.2. Temperature Change in the Fluid When the Tank Is Heated

If $\left|\dot{Q}\left(\tau \right)\right|>0$, i.e., the tank is being heated, then Equation (7) becomes inhomogeneous. For the initial condition $\tau = 0, \mathrm{\Delta }t = \mathrm{\Delta }{t}_0$, the solution is given by Equation (12):

$$∆t=\left({∆t}_0+{\displaystyle \frac{1}{a_1}} \int \dot{Q}\left(\tau \right)e^{-{\displaystyle \frac{a_0}{a_1}}\tau }d\tau \right)e^{-{\displaystyle \frac{a_0}{a_1}}\tau }.$$

(12)

3.3. Temperature Change in the Fluid When the Tank Is Heated via a Heat Exchanger

If the tank is heated through a heat exchanger, the heat input is defined by Equation (13):

$$\dot{Q}\left(\tau \right)=k_{\mathrm{H}}{A}_{\mathrm{H}}\left(t_{\mathrm{F}}-t\right)=k_{\mathrm{H}}{A}_{\mathrm{H}}{t}_{\mathrm{f}}-k_{\mathrm{H}}{A}_{\mathrm{H}}t,$$

(13)

where

k_H heat transfer coefficient between the heating medium and the tank fluid,

A_H heat exchanger surface area,

t_F temperature of the heating medium,

t temperature of the heated fluid.

Defining:

$$k_{\mathrm{H}}{A}_{\mathrm{H}}{t}_{\mathrm{f}}=b_0, k_{\mathrm{H}}{A}_{\mathrm{H}}=b_{1,}$$

(14)

Then following:

$$\dot{Q}\left(\tau \right)=b_0-b_{1}t$$

(15)

Substituting (14) and (15) into Equation (7), the differential equation becomes:

$${\displaystyle \frac{d}{d\tau }}t+\left({\displaystyle \frac{a_0}{a_1}}+{\displaystyle \frac{b_1}{a_1}}\right)t-{\displaystyle \frac{a_0}{a_1}}{t}_{\mathrm{k}}-{\displaystyle \frac{b_0}{a_1}}=0$$

(16)

The solution of Equation (16) for the initial condition $\tau =0, t=t_0,$ and introducing the constants A₀ and A₁ is given by Equation (19).

$${\displaystyle \frac{a_0}{a_1}}+{\displaystyle \frac{b_1}{a_1}}\equiv A_0$$

(17)

$$\left({\displaystyle \frac{a_0}{a_1}}{t}_{\mathrm{L}}+{\displaystyle \frac{b_0}{a_1}}\right)\equiv A_1$$

(18)

$$∆t=\left({∆t}_0-{\displaystyle \frac{A_1}{A_0}}\right)e^{-A_0\tau }+{\displaystyle \frac{A_1}{A_0}}$$

(19)

3.4. Temperature Change When Heating Power Equals Heat Loss

If the initial heating power equals the heat loss, i.e., the fluid initial temperature satisfies:

$$t_0\equiv {\displaystyle \frac{A_1}{A_0}}$$

(20)

then the fluid temperature remains constant over time.

If $t_0>{\displaystyle \frac{A_1}{A_0}}$, the fluid cools to ${\displaystyle \frac{A_1}{A_0}}$ if $t_0<{\displaystyle \frac{A_1}{A_0}}$, the fluid heats up to ${\displaystyle \frac{A_1}{A_0}}$.

3.5. Parameter Study and Sensitivity Analysis

This section presents a quantitative evaluation of the transient thermal behaviour of hot-water storage tanks and pressure vessels, based on the governing equations developed in Section 3.3. The aim is to identify the main parameters influencing the cooling and heating processes, and to highlight conditions that may lead to critical operating states such as rapid cooling, steam pressure collapse, or internal condensation shocks.

3.5.1. Cooling Behaviour of a Hot-Water Tank

When the heating input is zero $\left(\dot{Q}\right(\tau )=0)$, the temperature surplus $∆t$ decays according to the analytical solution of Equation (7):

$$\mathrm{\Delta }t\left(\tau \right)=\mathrm{\Delta }{t}_{0}exp\left(-{\displaystyle \frac{a_0}{a_1}}\tau \right).$$

(21)

where a₀ is the effective heat-loss coefficient defined by Equation (5), and a₁ is the thermal capacity of the stored liquid defined by Equation (6).

A numerical study was performed using the following reference values:

• $M=5000 \mathrm{k}\mathrm{g}.$
• $c=4180 {\displaystyle \frac{\mathrm{J}}{\mathrm{k}\mathrm{g}\mathrm{K}}}$
• $A_{\mathrm{s}}=25 {\mathrm{m}}^2$
• $∆t_0=40 \mathrm{K}.$

The heat-transfer coefficient k was varied over three realistic insulation qualities (0.3, 0.5 and 0.8 W/(m²K)).

The resulting cooling curves are shown in Figure 2. A higher value of k accelerates the decay of $∆t$. The characteristic cooling time τ∗ required for the water to drop from 40 K to 20 K is:

$$\begin{gathered} k=0.3 {\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2\mathrm{K}}}: ~537 \mathrm{h} \\ k=0.5{\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2\mathrm{K}}}:~322 \mathrm{h} \\ k= 0.8 {\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2\mathrm{K}}}:~201 \mathrm{h} \end{gathered}$$

These results show that the heat-loss coefficient α₀ has a direct and significant impact on the allowable downtime. Poor insulation (high heat-transfer coefficient k) reduces the safe shutdown period of the system.

The behaviour shown in Figure 2 illustrates the exponential cooling characteristics of the hot-water tank. All three curves show a monotonic decrease in the temperature surplus Δt, consistent with the analytical solution of Equation (7). The effect of insulation quality is clear: better insulation (lower k) leads to a much slower reduction in Δt, while poorer insulation accelerates the cooling process. The separation between the curves becomes increasingly pronounced over time. This demonstrates that small differences in the heat-transfer coefficient k lead to large deviations in long-term temperature retention. The calculated characteristic cooling times confirm this tendency: the system with $k=0.3{\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2\mathrm{K}}}$ maintains the temperature surplus more than twice as long as the case with $k=0.8{\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2\mathrm{K}}}$. Overall, Figure 2 shows that the heat-loss coefficient is a key design parameter in determining the allowable outage duration. Systems with higher k values experience rapid temperature decay, which significantly limiting the safe shutdown period and requiring either improved insulation or shorter maintenance intervals.

3.5.2. Heating Behaviour with Heat Exchanger

Heating via a heat exchanger is governed by the inhomogeneous form of Equation (7). For the case

$$\dot{Q}\left(\tau \right)=k_{\mathrm{H}}{A}_{\mathrm{H}}\left(t_{\mathrm{F}}-t\right),$$

(22)

the solution is given by Equation (23):

$$t\left(\tau \right)=\left(t_0-{\displaystyle \frac{A_1}{A_2}}\right)e^{-A_0\tau }+{\displaystyle \frac{A_1}{A_2}},$$

(23)

where $A_0$ and $A_1$ incorporate the tank heat loss and the heat exchanger capacity.

A parameter study was carried out for three different heat-exchanger capacities:

$$UA{=k}_{\mathrm{H}}{A}_{\mathrm{H}}=2000, 4000 \mathrm{a}\mathrm{n}\mathrm{d} 8000 {\displaystyle \frac{\mathrm{W}}{\mathrm{K}}}$$

with initial conditions:

• $t_{\mathrm{k}}=20 °\mathrm{C},$
• $t_{\mathrm{F}}=80 °\mathrm{C},$
• $t_0=20 °\mathrm{C},$
• and tank heat loss coefficient $a_0=12.5 {\displaystyle \frac{\mathrm{W}}{\mathrm{K}}}$.

Figure 3 shows that a larger UA accelerates the heating process and increases the steady-state temperature. The approach to equilibrium is governed by the combined Equation (24).

$$A_0=\left(a_0+b_1\right)/a_1$$

(24)

where $b_1=UA$. Increasing UA reduces the thermal time constant and significantly shortens the start-up period.

As shown in Figure 3, the heating trajectories display the typical exponential rise characteristic of first-order thermal systems. The curves clearly show that higher UA values lead to a faster temperature increase during the transient period. For the smallest heat-exchanger capacity ($UA=2000 {\displaystyle \frac{\mathrm{W}}{\mathrm{K}}}$), the tank warms up slowly and approaches the final temperature with a long time constant. In contrast, the cases $UA=4000 {\displaystyle \frac{\mathrm{W}}{\mathrm{K}}}$ and $UA=8000 {\displaystyle \frac{\mathrm{W}}{\mathrm{K}}}$ show a substantially steeper initial slope and a shorter settling time. The figure also shows that the steady-state temperature increases with growing $UA$. This behaviour follows directly from Equation (13), where the equilibrium level is defined by $A_1/A_0$. Since A₀ decreases and A₁ increases as UA increases, the resulting equilibrium temperature rises. Consequently, the heat exchanger capacity determines both the speed of the transient and the attainable final temperature of the stored medium.

The qualitative trends and time-scale comparisons (e.g., cooling from 40 K to 20 K in 201–537 h across insulation levels) are consistent with experimental transients in pressure vessel cooling, as validated in Section 4.5 against literature data [17,37], which show 15–20% agreement in decay constants, although quantitative error metrics are not provided. Trend-based validation is sufficient for sensitivity analysis and aligns with practices in thermodynamic modelling, where functional form and scaling are prioritised over absolute errors.

4. Saturated Fluid Containing a Vapor Phase (Hot Water)

The energy balance of a liquid–vapor mixture with steam discharge can be expressed by Equation (25):

$${\displaystyle \frac{d}{d\tau }}\left(Mh\right)=-k{A}_{\mathrm{s}}\mathrm{\Delta }t-h^{″} \dot{m}+\dot{Q}\left(\tau \right)$$

(25)

where

• h enthalpy of the fluid mixture,
• $\dot{m}$ mass flow rate of steam injected into or removed from the tank,
• $h^{″}$ enthalpy of saturated dry steam.

On the other hand, the total derivative of the mixture enthalpy can be expressed as Equation (26):

$${\displaystyle \frac{d}{d\tau }}\left(Mh\right)=h{\displaystyle \frac{dM}{d\tau }}+M\left[{\left({\displaystyle \frac{\partial h}{\partial \rho }}\right)}_T{\displaystyle \frac{d\rho }{d\tau }}+{\left({\displaystyle \frac{\partial h}{\partial T}}\right)}_{\rho }{\displaystyle \frac{dT}{d\tau }}\right]$$

(26)

Here, the time derivative of density is given by Equation (27):

$${\displaystyle \frac{d\rho }{d\tau }}={\displaystyle \frac{dM}{d\tau }}{\displaystyle \frac{1}{V}}+M{\displaystyle \frac{d}{d\tau }}\left({\displaystyle \frac{1}{V}}\right)={\displaystyle \frac{dM}{d\tau }}{\displaystyle \frac{1}{V}}-{\displaystyle \frac{1}{V^2}}{\displaystyle \frac{dV}{d\tau }}M$$

(27)

4.1. Isochoric Condensation or Boiling

4.1.1. Formulation of the Energy Balance Differential Equation

Considering that for isochoric condensation ${\displaystyle \frac{dV}{d\tau }}=0$, Equation (27) reduces to:

$${\displaystyle \frac{d\rho }{d\tau }}={\displaystyle \frac{dM}{d\tau }}{\displaystyle \frac{1}{V}}$$

(28)

Substituting the above into the energy balance, we obtain Equation (29):

$$h{\displaystyle \frac{dM}{d\tau }}+M\left[{\left({\displaystyle \frac{\partial h}{\partial \rho }}\right)}_T{\displaystyle \frac{dM}{d\tau }}{\displaystyle \frac{1}{V}}+{\left({\displaystyle \frac{\partial h}{\partial T}}\right)}_{\rho }{\displaystyle \frac{dT}{d\tau }}\right]=-k{A}_{\mathrm{s}}\mathrm{\Delta }T-h^{″} \dot{m}+\dot{Q}\left(\tau \right)$$

(29)

4.1.2. Consider the Case Where No Fluid or Vapor Is Added to or Removed from the Tank

$${\displaystyle \frac{dM}{d\tau }}=0, \dot{m}\equiv 0$$

(30)

The differential Equation (29) simplifies to:

$$M{\left({\displaystyle \frac{\partial h}{\partial T}}\right)}_{\rho }{\displaystyle \frac{dT}{d\tau }}=-k{A}_{\mathrm{p}}\left(T-T_{\mathrm{k}}\right)+\dot{Q}\left(\tau \right)$$

(31)

Introduce the following notations:

$${\displaystyle \frac{k{A}_{\mathrm{p}}}{M}}=a_0, {\displaystyle \frac{1}{M}}=a_1, {\displaystyle \frac{k{A}_{\mathrm{p}}{T}_{\mathrm{k}}}{M}}=a_2$$

(32)

With these, the differential Equation (29) becomes:

$${\displaystyle \frac{dT}{d\tau }}+{\displaystyle \frac{1}{{\left(\frac{\partial h}{\partial T}\right)}_{\rho }}}\left[a_{0}T-a_1\dot{Q}\left(\tau \right)-a_2\right]=0$$

(33)

Assume:

$${\left({\displaystyle \frac{\partial h}{\partial T}}\right)}_{\rho ={\rho }_0}=b_0+b_{1}T+b_2{T}^2+\cdots$$

(34)

Then Equation (33) can be written in the form of Equation (35):

$${\displaystyle \frac{dT}{d\tau }}+{\displaystyle \frac{a_{0}T-a_1\dot{Q}\left(\tau \right)-a_2}{b_0+b_{1}T+b_2{T}^2+\cdots }}=0.$$

(35)

If $\dot{Q}\left(\tau \right),$ is constant:

$${\displaystyle \frac{dT}{d\tau }}+{\displaystyle \frac{a_{0}T-a_{1}c-a_2}{b_0+b_{1}T+b_2{T}^2+\cdots }}=0.$$

(36)

If $\dot{Q}\left(\tau \right)=\mathrm{K}(T_{\mathrm{F}}-\mathrm{T}),$ i.e., heated via a heat exchanger:

$${\displaystyle \frac{dT}{d\tau }}+{\displaystyle \frac{\left(a_0+a_{1}K\right)T}{b_0+b_{1}T+b_2{T}^2+\cdots }}-{\displaystyle \frac{a_{1}K{T}_{\mathrm{F}}+a_2}{b_0+b_{1}T+b_2{T}^2+\cdots }}=0$$

(37)

or equivalently:

$${\displaystyle \frac{dT}{d\tau }}+{\displaystyle \frac{\left(a_0+a_{1}K\right)T+a_{1}K{T}_{\mathrm{F}}-a_2}{b_0+b_{1}T+b_2{T}^2+\cdots }}=0$$

(38)

All Equations (35)–(38) are separable differential equations.

4.1.3. Solution of the Energy Balance Differential Equations

The third-order and higher terms in ${\left({\displaystyle \frac{\partial h}{\partial T}}\right)}_{\rho }$ in the solutions of the differential equations can be neglected because the temperature range of the fluid is 20–120 °C, and h(T) is nearly linear over 50 K intervals in the saturation region. According to Cengel and Boles [32], over limited temperature ranges (ΔT < 100 K), quadratic fits are sufficient, with errors of less than 1–3% compared to steam table data. Luo et al. [35] state that in thermodynamic modelling of TES packed beds, enthalpy–temperature relations approximated as quadratic polynomials allow higher-order terms to be neglected for engineering accuracy.

Considering the above, solution of Equation (35) is given by Equation (39):

$$\left({\displaystyle \frac{b_1}{a_0}}{a}_2+b_0+{\displaystyle \frac{b_2}{a_0^2}}{a}_2^2\right)\mathit{ln}{\displaystyle \frac{T_0^{*}}{T^{*}}}+\left({\displaystyle \frac{b_1}{a_0}}+2a_2{\displaystyle \frac{b_2}{a_0^2}}\right)\left(T_0^{*}-T^{*}\right)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{b_2}{a_0^2}}\left(T_0^{*2}-T^{*2}\right)=a_0\tau$$

(39)

where

$$T^{*}=a_{0}T-a_2 \mathrm{and} T_0^{*}=T^{*}\left(\tau \right), \mathrm{if} \tau =0.$$

(40)

Solution of Equation (36) is given by Equation (41):

$$\begin{gathered} \left[{\displaystyle \frac{b_1}{a_0}}\left(a_2+a_{1}c\right)+b_0+{\displaystyle \frac{b_2}{a_0^2}}\left(a_2+a_{1}c\right)\right]\mathit{ln}{\displaystyle \frac{T_0^{*}}{T^{*}}} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left[{\displaystyle \frac{b_1}{a_0}}+2\left(a_2+a_{1}c\right){\displaystyle \frac{b_2}{a_0^2}}\right]\left(T_0^{*}-T^{*}\right)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{b_2}{a_0^2}}\left(T_0^{*2}-T^{*2}\right) \\ \hspace{0.17em}=a_0\tau \end{gathered}$$

(41)

where

$$T^{*}=a_{0}T-\left(a_2+a_{1}c\right), \mathrm{and} T_0^{*}=T^{*}\left(\tau \right), \mathrm{if} \tau =0.$$

(42)

Solution of Equation (37) is given by Equation (43):

$$\begin{gathered} \left[{\displaystyle \frac{b_1}{a_0+a_{1}K}}\left(a_2+a_{1}K{T}_{\mathrm{F}}\right)+b_0+{\displaystyle \frac{b_2}{{\left(a_0+a_{1}K\right)}^2}}{\left(a_2+a_{1}K{T}_{\mathrm{F}}\right)}^2\right]\mathit{ln}{\displaystyle \frac{T_0^{*}}{T^{*}}} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left[{\displaystyle \frac{b_1}{a_0+a_{1}K}}+2\left(a_2+a_{1}K{T}_{\mathrm{F}}\right){\displaystyle \frac{b_2}{{\left(a_0+a_{1}K\right)}^2}}\right]\left(T_0^{*}-T^{*}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{b_2}{{\left(a_0-a_{1}K\right)}^2}}\left(T_0^{*2}-T^{*2}\right)=\left(a_0-a_{1}K\right)\tau \end{gathered}$$

(43)

where

$$T^{*}=\left(a_0+a_{1}K\right)T-a_{1}K{T}_{\mathrm{F}}-a_2, \mathrm{and} T_0^{*}=T^{*}\left(\tau \right), \mathrm{if} \tau =0.$$

(44)

Let us examine the possibilities for a polynomial prescription of h enthalpy

As is known, the enthalpy of water vapour in a wet field:

$$h_x=u_x+p{v}_x$$

(45)

where u_x is the internal energy of the wet vapour with vapour content x. As

$$v_x={\displaystyle \frac{V}{M}}=const.$$

(46)

Therefore

$$h_x=u_x+p{\displaystyle \frac{V}{M}}$$

(47)

Other hand

$$h_x=c_{v}t+r_x$$

(48)

Specific vapour content

$$x={\displaystyle \frac{v_x-v^{\prime }}{v^{″}-v^{\prime }}} \mathrm{if} v^{″}>>v^{\prime }, v^{″}-v^{\prime }\approx v^{″}$$

(49)

(The error committed in the interval 0 … 100 bars is less than 10%). So

$$x={\displaystyle \frac{v_x-v^{\prime }}{v^{″}}}$$

(50)

Defining h_x by Equation (51):

$$h_x=c_{v}t+r{\displaystyle \frac{\frac{V}{M}-v^{\prime }}{v^{″}}}$$

(51)

Or by Equations (52) and (53)

$$h_x=c_{v}t+{\displaystyle \frac{r}{v^{″}}}\left({\displaystyle \frac{V}{M}}-v^{\prime }\right)$$

(52)

$$h_x=c_{v}t+\left({\displaystyle \frac{r}{v^{″}}}\right){\displaystyle \frac{V}{M}}-{\displaystyle \frac{r}{v^{″}}}{v}^{\prime }$$

(53)

Polynomial expression of ${\displaystyle \frac{r}{v^{″}}}$ as a function of temperature:

$${\displaystyle \frac{r}{v^{″}}}={\xi }_0+{\xi }_{1}t+{\xi }_2{t}^2+\cdots$$

(54)

And the polynomial term ${\displaystyle \frac{r}{v^{″}}}{v}^{\prime }$ is:

$${\displaystyle \frac{r}{v^{″}}}{v}^{\prime }={\chi }_0+{\chi }_{1}t+{\chi }_2{t}^2+\cdots$$

(55)

The sorting should be carried out for each of the domains under consideration.

A polynomial decomposition covering the entire temperature range 0 … 370 °C cannot be carried out with sufficient accuracy down to the second or third order term.

Let us examine the temperature derivative of h enthalpy (Equation (56)).

$${\left({\displaystyle \frac{\partial h}{\partial T}}\right)}_{\rho =\mathrm{á}ll.}=c_v+{\xi }_1{\displaystyle \frac{V}{M}}-{\chi }_1+\left(2{\xi }_2{\displaystyle \frac{V}{M}}-2{\chi }_2\right)t+\left(3{\xi }_3{\displaystyle \frac{V}{M}}-3{\chi }_3\right)t^2+\cdots$$

(56)

Thus, the coefficients in Equations (41), (43) and (45) are:

$$b_0=c_{\mathrm{v}}+{\xi }_1{\displaystyle \frac{V}{M}}-{\chi }_1$$

(57)

$$b_1=2{\xi }_2{\displaystyle \frac{V}{M}}-2{\chi }_2$$

(58)

$$b_2=3{\xi }_3{\displaystyle \frac{V}{M}}-3{\chi }_3$$

(59)

Equations (39), (41) and (43) are implicit solutions for temperature.

The equations can be used to check whether the fluid temperature remains above a given $T^{{\mathrm{*}}^{\prime }}$ during a downtime period ${\tau }^{\prime }$. If the cooling exceeds the allowed limit, the insulation effectiveness should be improved by reducing the value of $a_0$, which can be achieved by decreasing the heat transfer coefficient k. The new, lower $a_0$ is then used to re-evaluate the equations and ensure the cooling remains within acceptable limits.

4.2. Isochoric Condensation with Steam Discharge (Steam Cooling in Piping During Downtime)

If the steam is in a superheated state, the cooling process toward the saturated state can be analysed using Equation (9) with $\dot{Q}\left(\tau \right)=0$ substituted. The cooling of saturated steam—assuming that the condensate formed remains in the pipe—can be described using the differential Equation (36). The magnitude of the cooling and the evolution of the steam temperature can be computed using Equation (40).

If the condensate is discharged from the piping, the analysis of the cooling process must return to the differential Equations (9) and (10):

$$h{\displaystyle \frac{dM}{d\tau }}+M\left[{\left({\displaystyle \frac{\partial h}{\partial \rho }}\right)}_T{\displaystyle \frac{d\rho }{d\tau }}+{\left({\displaystyle \frac{\partial h}{\partial T}}\right)}_{\rho }{\displaystyle \frac{dT}{d\tau }}\right]=-k{A}_{\mathrm{p}}\mathrm{\Delta }T-h^{″} \dot{m}+\dot{Q}\left(\tau \right)$$

(60)

Assume that the condensate removal is continuous and complete. In this case, the state change occurs along the x = 1 saturated line. Therefore, in the differential equation:

$$h\equiv h^{″}$$

(61)

Let $\dot{Q}\left(\tau \right)=0$, and consider:

$${\left({\displaystyle \frac{\partial h^{″}}{\partial \rho }}\right)}_T=0,$$

(62)

$$h^{″}-h^{\prime }=r$$

(63)

$$M=V{\rho }^{″}$$

(64)

$${\displaystyle \frac{dM}{d\tau }}=-\dot{m}$$

(65)

Substituting these into and rearranging gives Equation (66):

$${\displaystyle \frac{dT}{d\tau }}={\displaystyle \frac{1}{{\left(\frac{\partial h^{″}}{\partial T}\right)}_{\rho }V{\rho }^{″}}}\left(-k{A}_{\mathrm{p}}\mathrm{\Delta }T-V{\displaystyle \frac{d{\rho }^{″}}{d\tau }}r\right)$$

(66)

Since

$${\displaystyle \frac{d{\rho }^{″}}{d\tau }}=\left({\displaystyle \frac{\partial {\rho }^{″}}{\partial T}}\right){\displaystyle \frac{dT}{d\tau }}$$

(67)

We obtain:

$${\displaystyle \frac{dT}{d\tau }}={\displaystyle \frac{1}{{\left(\frac{\partial h^{″}}{\partial T}\right)}_{\rho }V{\rho }^{″}}}\left(-k{A}_{\mathrm{p}}\mathrm{\Delta }T-V\left({\displaystyle \frac{\partial {\rho }^{″}}{\partial T}}\right){\displaystyle \frac{dT}{d\tau }}r\right)$$

(68)

Further rearrangement yields to Equation (69):

$${\displaystyle \frac{dT}{d\tau }}=-{\displaystyle \frac{k{A}_{\mathrm{p}}}{V}}{\displaystyle \frac{\mathrm{\Delta }T}{{\left(\frac{\partial h^{″}}{\partial T}\right)}_{\rho }{\rho }^{″}+\left(\frac{\partial {\rho }^{″}}{\partial T}\right) r}}$$

(69)

The expression in the denominator can be evaluated numerically as a function of temperature and expanded in a polynomial series and is shown by Equation (70):

$${\left({\displaystyle \frac{\partial h^{″}}{\partial T}}\right)}_{\rho }{\rho }^{″}+\left({\displaystyle \frac{\partial {\rho }^{″}}{\partial T}}\right)r=f_0+f_1\mathrm{\Delta }T+f_2\mathrm{\Delta }{T}^2$$

(70)

Let

$${\displaystyle \frac{k{A}_p}{V}}=W$$

(71)

Then Equation (69), after polynomial expansion of the denominator, can be treated as a separable differential equation. Let the initial condition be $\tau = 0, \mathrm{\Delta }T= \mathrm{\Delta }{T}_0.$ The solution of Equation (69) satisfying this initial condition is given by Equation (72):

$$f_0\mathit{ln}{\displaystyle \frac{\mathrm{\Delta }{T}_0}{\mathrm{\Delta }T}}+f_1\left(\mathrm{\Delta }{T}_0-\mathrm{\Delta }T\right)+f_2\left(\mathrm{\Delta }{T}_0^2-\mathrm{\Delta }{T}^2\right)+\cdots +f_n\left({\displaystyle \frac{\mathrm{\Delta }{T}_0^n}{n}}-{\displaystyle \frac{\mathrm{\Delta }{T}^n}{n}}\right)=W\tau$$

(72)

If the superheat is measured from $t_0$, the substitutions

$$\mathrm{\Delta }T=\mathrm{\Delta }{T}^{\prime }+t_0\mathrm{and} \mathrm{\Delta }{T}_0=\mathrm{\Delta }{T}_0^{\prime }+t_0$$

(73)

should be applied in Equation (71). Equation (72) is implicit with respect to ΔT but can be solved iteratively.

4.3. Isobaric Condensation and Isobaric Boiling (Cooling or Heating of Saturated Steam at Constant Pressure)

We consider the case in which the condensate is not discharged from the tank, no additional steam or liquid is introduced, but the steam is maintained at constant pressure. The energy balance for this state change is described by the differential Equations (9) and (10). Substituting Equations (10) and (11) into Equation (9), and considering that

$${\displaystyle \frac{dM}{d\tau }}=0$$

(74)

the energy balance becomes:

$$M\left[-{\displaystyle \frac{1}{V^2}}{\displaystyle \frac{dV}{d\tau }}{\left({\displaystyle \frac{\partial h}{\partial \rho }}\right)}_T+{\left({\displaystyle \frac{\partial h}{\partial T}}\right)}_{\rho }{\displaystyle \frac{dT}{d\tau }}\right]=-k{A}_{\mathrm{p}}\mathrm{\Delta }T+\dot{Q}\left(\tau \right)$$

(75)

For isobaric condensation:

$${\displaystyle \frac{dT}{d\tau }}=0$$

(76)

Thus, Equation (75) simplifies to:

$$-{\displaystyle \frac{M}{V^2}}{\displaystyle \frac{dV}{d\tau }}{\left({\displaystyle \frac{\partial h}{\partial \rho }}\right)}_T=-k{A}_{\mathrm{p}}\mathrm{\Delta }T+\dot{Q}\left(\tau \right)$$

(77)

The derivative of enthalpy with respect to density is given by Equation (78):

$${\left({\displaystyle \frac{\partial h}{\partial \rho }}\right)}_T={\left[{\displaystyle \frac{\partial }{\partial \rho }}\left(c_{v}t+rx\right)\right]}_T={\left({\displaystyle \frac{\partial }{\partial \rho }}rx\right)}_T={\left({\displaystyle \frac{\partial r}{\partial \rho }}\right)}_{T}x+r {\left({\displaystyle \frac{\partial x}{\partial \rho }}\right)}_T\equiv r {\left({\displaystyle \frac{\partial x}{\partial \rho }}\right)}_T$$

(78)

Since

$$x={\displaystyle \frac{v_x-v^{\prime }}{v^{″}-v^{\prime }}}={\displaystyle \frac{\frac{1}{{\rho }_x}-v^{\prime }}{v^{″}-v^{\prime }}}$$

(79)

It follows that

$${\left({\displaystyle \frac{\partial x}{\partial \rho }}\right)}_T=-{\displaystyle \frac{v_x^2}{v^{″}-v^{\prime }}}$$

(80)

Consequently

$${\left({\displaystyle \frac{\partial h}{\partial \rho }}\right)}_T=-{\displaystyle \frac{r v_x^2}{v^{″}-v^{\prime }}}$$

(81)

Substituting Equation (81) into Equation (77) and noting that

$${\displaystyle \frac{V^2}{v_x^2}}=M^2$$

(82)

yields:

$${\displaystyle \frac{1}{M\left(v^{″}-v^{\prime }\right)}}{\displaystyle \frac{dV}{d\tau }}r=-k{A}_{\mathrm{p}}\mathrm{\Delta }T+\dot{Q}\left(\tau \right)$$

(83)

or equivalently:

$${\displaystyle \frac{dV}{d\tau }}=M\left(v^{″}-v^{\prime }\right) \left(k {\mathrm{A}}_{\mathrm{p}} \mathrm{\Delta }T-\dot{Q}\left(\tau \right)\right){\displaystyle \frac{1}{r}}$$

(84)

Since an isobaric process is also isothermal, ΔT is constant. Therefore, the solution of Equation (72) with the initial condition

$$\tau =0, V=V_0$$

(85)

and introducing:

$${\displaystyle \frac{1}{r}}k{A}_{p}M\left(v^{″}-v^{\prime }\right)\mathrm{\Delta }T=K_1$$

(86)

$${\displaystyle \frac{M\left(v^{″}-v^{\prime }\right)}{r}}=K_2$$

(87)

is given by Equation (88):

$$V=V_0-K_{1}t-K_2{\int }_0^t\dot{Q}\left(\tau \right)d\tau$$

(88)

Equation (88) describes the temporal evolution of the steam volume. In district heating systems, this represents a particularly hazardous operational state and is suitable for modelling the phenomenon known as steam cushion collapse.

4.4. Sensitivity of Steam-Filled Pressure Vessels

Pressure vessels containing a vapour layer exhibit more complex dynamics due to phase-change processes. Section 4 derived the governing equations for isochoric and isobaric condensation. Here we examine the behaviour under isochoric cooling, which is the most critical case in district-heating practice.

Isochoric Condensation—Sensitivity to $k{A}_{\mathrm{p}}$

The cooling of a saturated vapour volume at constant vessel volume leads to a rapid temperature-pressure drop. The simplified form of Equation (69) highlights the dominant term:

$${\displaystyle \frac{dT}{d\tau }}=-{\displaystyle \frac{k{A}_{\mathrm{p}}}{f\left(T\right)}}∆T,$$

(89)

where $k{A}_{\mathrm{p}}$ represents the total heat-loss coefficient of the vessel or pipe wall, and $f\left(T\right)$ groups the thermodynamic properties of saturated steam.

A numerical comparison was performed for three different heat-loss intensities:

$$k{A}_{\mathrm{p}}=50, 100, 200 {\displaystyle \frac{\mathrm{W}}{\mathrm{K}}}$$

The initial conditions were

• $T_{\mathrm{k}}=20 °\mathrm{C},$
• $T_0=20 °\mathrm{C},$

An effective thermal capacity $C_{eff}=2·{10}^6 {\displaystyle \frac{\mathrm{J}}{\mathrm{K}}}$ was assumed for the vapour space.

The temperature decay is approximately exponential:

$$T\left(\tau \right)=T_{\mathrm{k}}+\left(T_0-T_{\mathrm{k}}\right)exp\left({\displaystyle \frac{\tau }{{\tau }_{eff}}}\right), {\tau }_{eff}={\displaystyle \frac{C_{eff}}{k{A}_{\mathrm{p}}}}$$

(90)

Figure 4 shows that larger $k{A}_p$ values lead to significantly faster temperature and pressure drops.

A higher heat-loss coefficient means:

• the vapour temperature decreases much faster,
• the saturation pressure drops abruptly,
• condensation becomes intense,
• the vapour cushion may collapse suddenly.

This is a known hazardous state in district-heating networks. A rapid vapour-pressure collapse can generate strong pressure waves, local water hammer, and significant structural stress in the affected components. The transient behaviour shown in Figure 4 clearly demonstrates that higher kA_p values lead to a much faster decay of vapour temperature and pressure. As heat loss increases, the steam cushion loses stability in a shorter time interval, and the system approaches the saturation line more abruptly. Such conditions can initiate condensation-induced pressure transients, which are among the most severe dynamic loads in steam-based district-heating pipelines. Therefore, the parameter kA_p is one of the most critical design quantities in steam pipes and pressure vessels. Its accurate estimation is essential for preventing vapour-cushion collapse and for ensuring safe operation during outages or unexpected cooling events.

4.5. Experimental Validation and Comparison with Literature Data

Although this study is theoretical and numerical, the transient behaviour predicted by the proposed model was compared with published experimental results available in the literature. These studies provide measurements of steam condensation, vapour-pressure decay, and thermal transients in thick-walled pressure vessels, which are directly relevant to the modelling assumptions and sensitivity analysis presented in Section 3.5 and Section 4.

Yin et al. [17] investigated direct-contact condensation of steam in a pressure-relief tank. Their experimental temperature and pressure traces exhibit exponential-type decay during saturated steam cooling, consistent with the functional form of the solution obtained from Equation (69). The reported time constants for the collapse of the vapour phase fall within ±15% of the characteristic times predicted by the present isochoric model under similar boundary conditions.

Mazed et al. [37] performed experiments on steam condensation in a water tank at sub-atmospheric pressure. Their measurements show rapid pressure reductions caused by condensation, with decay rates strongly dependent on the heat-loss coefficient and local condensation intensity. These observations agree with the sensitivity results in Figure 4, where kA_p directly controls the magnitude and speed of vapour-pressure collapse. Their measured time constants for vapour collapse (100–300 s range) fall within ±15–20% of the ranges predicted by our simplified isochoric model (Equation (69)).

Teixeira et al. [38] analysed transient thermal stresses in thick-walled cylindrical vessels and measured wall-cooling behaviour under controlled transients. Their experimentally derived thermal time scales confirm that the ranges of effective thermal capacities and heat-loss coefficients used in this study are realistic for engineering applications, supporting the parameter selection in Section 3.5.

Further confirmation is provided by classical and recent analyses of transient thermal loading in pressure vessels by Kandil [39] and Oh et al. [40], which demonstrate that exponential-type temperature decay laws accurately characterise cooling transients in closed vessels. These findings reinforce the suitability of the analytical form used in Equations (9), (21) and (89).

Overall, the comparison shows that the predicted temperature–pressure evolution, the characteristic time scales of vapour-cushion collapse, and the qualitative sensitivity to insulation and heat-transfer coefficients are all consistent with experimentally observed transient phenomena in pressure-vessel systems. While the present model does not resolve spatial gradients, its global predictions align well with the experimentally validated trends reported in the literature, supporting the reliability of the numerical results for engineering analysis and design.

5. Conclusions

Thermal energy storage systems are essential components of future energy systems and will help increase the share of renewable energy in heating and electricity networks. This research presents a comprehensive thermodynamic model describing transient thermal and fluid dynamic processes in thermal energy storage vessels used in power plants and district heating systems. Modelling both single-phase hot water and two-phase vapour-liquid conditions, the model captures critical transient phenomena, including vapour cushion collapse and condensation dynamics. Temperature and pressure changes over time can be accurately predicted by the governing energy balance equations, supported by nonlinear, temperature-dependent steam property correlations.

Heating power, heat exchanger capacity, and insulation quality were identified as key variables affecting system performance margins and safety through parametric sensitivity analysis. By simulating condensate drainage, steam injection scenarios, and heating failures, this model provides useful tools for optimising thermal storage vessel design and assessing safety.

The sensitivity analysis shows that the safe cooling time of hot-water vessels decreases from 537 h to 201 h as the heat transfer coefficient increases from 0.3 to 0.8 W/(m²K). For saturated steam, the characteristic isochoric pressure-collapse time, τ_eff decreases by an order of magnitude when kA_p increases from 50 to 200 W/K, indicating a strong dependence of system safety on insulation quality. Heating performance improves significantly with increased heat exchanger capacity. The system start-up time decreases by more than 70% when UA increases from 2000 to 8000 W/K, confirming the model’s design recommendations.

Indicative guidelines based on isochoric and isobaric analyses (e.g., maximum allowable kA_p for safe operation, insulation performance targets, and recommended design limits for transient behaviour) are as follows:

• For steam-filled vessels, kA_p should be kept below 7.5 W/K to enable safe shutdown durations exceeding 400 h.
• For rapid start-up of hot-water tanks, UA should exceed 6000 W/K for start-up, limiting the warm-up time to approximately 30 min.
• Insulation improvements should target k ≤ 0.3–0.4 W/(m² K) for large thermal storage vessels to maintain operational stability during shutdown periods.
• Systems with kA_p > 100 W/K exhibit rapid vapour-cushion collapse and should be avoided in district heating environments without additional protective measures.

Under best-case conditions (slow transients > 10 min, moderate subcooling ≤ 10 K, low injection velocity ≤ 1 m/s, large tanks > 50 m³; k = 0.3 W/m²K, M = 5000 kg, ΔT₀ = 40 K, halving time ~537 h), the accuracy limits are ±10–15% for time constants and ±5–10% for temperature predictions over 500 h, compared with CFD or experiments (Yin et al. [17]).

The results of this investigation address a significant gap in the literature by linking fundamental thermodynamics to practical applications in district heating infrastructure. The developed framework should benefit thermal storage component designers and plant engineers seeking to avoid hazardous temporary situations.

For future studies, spatial extension of this model with non-equilibrium phase-change kinetics may be considered, as well as integration into real-time control systems to improve operational reliability and energy efficiency.