Simple analytical considerations on a thermal-flow operating parameters of a heat storage tank

Abstract

A heat storage tank (HST), described in this paper, may be applied to numerous systems used for thermal energy generation and storage. Working principle and heat-flow processes of the HST remain the same and are independent of the systems in which the HST is being used. This paper presents a thermal-flow analysis of a heat storage tank (HST) operating in an air compressor heat pump (ACHP) and a dry cooler (DC) combined cycle. The HST is operated under transient conditions for four different ACHP steady states. The HST was analysed using lumped capacitance method. The heating power of a single coil varied from 2.2 to 3.2 kW. 50% increase in mass flow rate of heating water in the coils increased the HST water mean temperature by 5.6 °C. A differential equation for the HST water temperature variation was solved analytically and it was presented with the use of dimensionless numbers. The presented solution is of a general form. An important parameter such as number of the coils may be varied easily. Additionally, a Mathcad computational program was proposed for the estimation of HST-ACHP-DC system thermodynamic and thermal-flow parameters. System dynamics was studied using MATLAB/Simulink. An optimal operating parameters of the HST were found and are presented in the paper. The aforementioned parameters are as follows: mass flow rates of heating and cooling media in all three coils, mass flow rate of freshwater supplying the HST, number of loops in coils and resulting heat transfer area, diameter of the loops of the coils, diameter of the coils, time after which water in the HST reaches a steady value, time period within which the HST may fulfil demand for central heating and domestic hot water (DHW) preparation without being charged simultaneously. The presented analytical model was validated based on the available data in the field literature and a high compliance was reached. For this purpose theoretical model results were compared to results of HST computational fluid dynamics (CFD) simulations. The temperature calculated using the presented model differed from the one obtained by the CFD analysis by 2.96 K.

Highlights

• HST-ACHP-DC design and verification analysis is presented.

• The solution of time-dependent HST water temperature variation is of a general form.

• Optimal thermal-flow operating parameters of the HST coils are proposed.

• An increase in $\dot{m}_{{{\text{air}}}}$ by 48% implies an increase in temperature by 5.6 °C.

Introduction

This article studies the analytical and numerical analysis of a heat storage tank (HST). The HST is coupled with an air compressor heat pump (ACHP) and a dry cooler (DC). This study concentrates on a detailed analysis of the operating parameters of all components of the system, namely the HST, ACHP and DC. However, the HST is the main objective of the presented research. The ACHP and the DC operate as the HST heat supplier and consumer, respectively. The topic of the HST which is taken up within the paper is an important one as increasing the energy efficiency, by means of thermal energy storage, is a desirable effect. The HST operation was therefore studied in terms of the parametric analysis aiming at improving the heat transfer coefficient inside the coils and inside the HST. The HST with two coils was briefly analysed by Dolna [1]. This paper presents further development of HST mathematical model mentioned in [1]. The differential equation describing the HST water temperature variation over time is presented using dimensionless numbers. The HST model assumes that one or two coils, depending on the case, are thermally charging the HST while an additional coil (second or third) operates as a heat sink. In the paper, the analysis of the dimensionless numbers is also included. The differential equation that describes the time-dependent HST water temperature variation was therefore presented in a form including the following dimensionless numbers like Reynolds, Prandtl and Nusselt numbers. However, other dimensionless numbers, such as the Grashof and Rayleigh, were also calculated as they are of a substantial importance in the studied problem. The Grashof number is a measure of the ratio of buoyancy forces to viscous forces acting on a fluid. Additionally, the Grashof number plays the same role in free convection as the Reynolds number plays in forced convection. The Rayleigh number is a product of the Grashof and the Prandtl numbers. As it is known, free convection is not restricted to laminar flow. With the increase in the value of the Rayleigh number, the transition from laminar to turbulent flow occurs. More detailed analysis on dimensionless numbers regarding their values is presented further in the paper. The present paper demonstrates the HST unsteady operation under four distinct steady-state operating conditions of the ACHP. By these means, the HST operation dynamics at the varying heat loads was studied. All calculations of the HST-ACHP-DC system were carried out using Mathcad software. Moreover, the system dynamics was analysed using MATLAB/Simulink environment. The goal was also to couple the HST with the ACHP and DC. Accomplishment of the latter one has led to creating an author’s computational program allowing for a complex thermodynamic and heat-flow analysis of the HST-ACHP-DC system. In work by Chang et al. [2], a single-tank thermocline (STTC) thermal energy storage (TES) was analysed. Authors have proposed an advanced one-dimensional (1D) two- and one-phase (2D) analytical models of the STTC. The STTC was filled up with encapsulated phase change materials (PCM). MicroSol-R plant and STTC TES were studied empirically in PROMES laboratory. The laboratory bench and experimental data are also presented in paper [2]. Both analytical models were validated using the available empirical results provided by PROMS. Computational fluid dynamics (CFD) analysis of the studied system was also carried out. Authors of [2] have stated that the results obtained on the way of 1D-2P analytical model considerations were more exact than CFD results when compared with empirical data. Authors of [2] have introduced dimensionless discharging time and discharging efficiency among others. A multi-parametric optimisation design procedure was proposed as well as numerous cases were studied. Authors of [2] have also introduced a critical value of Peclet number at which the final discharging efficiency increases in a line with the output power of the STTC. In the work by Xu et al. [3], 1D analytical model of a thermocline storage tank (TCST) was presented. The fluid-flow TCST did not contain any heating nor cooling coils. The hot water inlet was localised at the top of the TCST while the cold water outlet was placed at the bottom of the tank. The analytical model presented in [3] captured the temperature variation over time, and the TCST height as transient 1D energy equation was employed. In the paper by Zachár [4], transient heat transfer process as well as temperature distribution in a hot water storage tank was analysed theoretically, among others. Author of [4] has solved the aforementioned problem by applying a coupled system of a first-order partial and a first-order ordinary differential equation (PDE-ODE). Based on the developed analytical solutions, a temperature distribution within the water in the hot storage tank may be obtained. It is commonly known that the HST operates at high thermal energy storage efficiency whenever a thermal stratification within the liquid is obtained. The formation of the thermocline closely depends on such geometric and flow parameters of the HST as its shape, inlet, hydrodynamics and the characteristics of the liquid mixing in the tank [5, 6]. The mixing of liquids occurs due to high-temperature gradients and the buoyancy force acting [7]. Additionally, in many fluid-flow HSTs, the water movement results from charging and discharging processes which is obvious. In the work by Li et al. [6], the Authors concentrate on the HST integrated with the solar installation and they focus on the HST optimisation due to the influence of its external geometry on a thermal stratification in the heated water and on the heat capacity of the tank. It is obvious that the HST, working as a specific heat storing unit, ought to be sensitive and of a proper dynamics in response to the time-dependent solar irradiance. The HSTs studied by Li et al. [6] were of a classical internal geometry; namely, one coil was localised at the bottom of the HST. However, general geometry of studied HSTs differed much due to spherical, cylindrical or circular truncated cone shape of the tank. According to the content of the work by Li et al. [6], the tank geometry has a significant impact on the increase in thermal energy storing efficiency of the HST. In paper [6], the authors have concentrated on the HST optimisation as well as influence of geometry on a thermal stratification in the heated water and on the heat storage capacity. Smusz et al. in paper [8] analyse a coil-type heat exchanger performance in terms of Dean dimensionless number and corresponding Reynolds critical value. They conclude that the heat transfer phenomenon can be intensified in the coil-type heat exchangers by means of inertial and centripetal forces and the viscous forces in the liquid which lead to emergence of reversed flow [8]. The reversed flow is perpendicular to the main flow direction and it can be intuitively concluded that the heat transfer would be intensified. Nowadays, a rapid increase in the interest of the use of PCM in the thermal energy storage units can be noticed [9,10,11]. However, the efficient operation of a system containing the PCM strongly depends on the characteristics of a charging–discharging process dynamics which need to be well known [9]. As it is emphasised in paper by Okten et al. [12], HSTs are widely used in various industries as they improve thermal energy generation, storage and utilisation efficiency. As it is mentioned in [5, 6, 13], the value of the Richardson number in the HST water thermal stratification description is of a significant importance. Its low value denotes for a complete mixing of the liquid masses of diverse temperatures. Consequently, its high value implies a thermal stratification of the liquid which is a desirable phenomenon according to the HST operation efficiency [14, 15]. In paper by Koçak et al. [16], it was reported that a perfect stratification in the HST which was used in a solar collectors system has led to the growth in thermal energy generated by 38% compared to the case of a fully mixed water in the HST. It is stated in the papers [17,18,19] that when the ratio H/D (HST height, H, to its diameter, D) reaches 4 then further increase in this value does not imply any significant thermal efficiency improvement. Furthermore, Lavan et al. [20] indicated that a HST, which fulfils the condition presented through the ratio H/D ϵ〈3, 4〉, is a reasonable compromise between its efficiency and its cost. Castell et al. [5] have presented a relation allowing for estimation of the HST discharging efficiency. The aforementioned efficiency was defined as a ratio of a useful thermal energy, accumulated over time, which can be extracted from the HST, to a total amount of a recoverable thermal energy [5, 21, 22]. This relation allows for an evaluation of the thermal stratification in the HST during a discharging process. As it is mentioned in paper by Fertahi et al. [23], a sensible thermal storage based on the stratified heat storage tank is characterised by technically simplicity which stays in line with low investment and maintenance costs. Therefore, as it is aforementioned, many researchers still concentrate on improving the HST operation parameters such as the proper thermal stratification or a charging/discharging efficiency, different configurations of insulation and other [23]. In the paper by Kurşun [24] an important numerical work on the influence of the HST insulation was carried out. Author of [24] has stated that significant improvement of the thermal stratification was reached for low values of the HST aspect ratio, bottom-to-top insulation diameter ratio and bottom-to-top insulation thickness, with negligible decrease in exergy efficiency as against the HST with a conventional insulation.

The novelty of the presented article is a new simple HST analytical model which enables estimation of the HST water mean temperature variation over time, among others. The model proposed can be used in engineering practice and is of a general character. Hereafter, it can be used to analyse similar technical problems (i.e. coil-type heat exchangers). The model based on which a computational program was prepared allows for a complex heat-flow and thermodynamic analysis of a system containing HST, ACHP and DC. Therefore, the program captures, i.e. the HST response (i.e. water temperature variation, variation of the heat transfer area of the coils, etc.) to the variation of the air mass flow rate in the ACHP primary cycle. The program can be used for design and verification calculations.

Methodology

Theoretical analysis

Within this paragraph, the definition of an investigated problem and simple analytical considerations are presented. As it is commonly known, the left-hand Linde cycle is realised in a compressor heat pump. Therefore, the ACHP cycle was modelled based on a known procedure. All needed calculations were carried out using the Mathcad environment. The Mathcad CoolProp library was used to estimate working fluid thermophysical properties (i.e. thermal conductivity, density, kinematic and dynamic viscosity, specific heat). HST and ambient air served as high- and low-ACHP-heat sources, respectively. Original computational program has been prepared that coupled all subcomponents of the HST-ACHP-HST system. Heat-flow and thermodynamic analysis of the studied system was carried out using the original computational program. This program enables thermodynamic analysis of the ACHP. Additionally, heat-flow analysis of the HST and DC can be carried out. Moreover, this program couples all mentioned devices in a way providing automatic recalculation of all of the resultant parameters of the whole system after applying new input data (i.e. change in mass flow rate of air at the ACHP low-heat source side results in change of the heat transfer area of the heating coils, etc.).

Boundary conditions

In the analytical model, it was assumed that the ambient temperature equals to 25 °C. This temperature value was applied to reflect conditions of a planned HST-ACHP-DC system laboratory bench. The temperature at which the evaporation proceeded was 15 °C lower than the ambient temperature [25]. Condensation temperature was equal to 43 °C or 60 °C. Both cases were compared, and further analysis was carried out for the latter one. The present analysis is about estimating the optimal parameters of the ACHP which supplied thermally the HST. The aim was to provide the most appropriate heat source to supply the HST. Regarding this fact, higher condensation temperature was used for further analysis. R32 working fluid critical temperature and pressure are as follows: $T_{{{\text{crit}}}} = 78.11$ °C and $P_{{{\text{crit}}}} = 57.8$ bars. Therefore, assuming the condensation temperature and pressure at the level of 60 °C and 39.33 bar, respectively, is appropriate. It was assumed that the refrigerant vapour is being superheated by 5 °C at the compressor suction side. The working fluid condensate was also subcooled by 5 °C upstream of the throttle valve. The presented computational program allows for the temperature and pressure variation according to users’ needs. Additionally, the air mass flow rate of the primary cycle of the ACHP may be set to the designated value. This results in change of the R32 and condenser cooling water mass flow rates. The studied HST contains two (coil₁₂, coil₅₆) or three (coil₁₂, coil₃₄, coil₅₆) coils. In the HST with three coils, two of them are heating ones (coil₁₂ and coil₃₄) while coil₅₆ is a cooling one. Both heating coils are coupled with ACHP which operates as the HST heat supplier. Coil₅₆ is coupled with DC to provide a heat intake from the HST. The following temperature boundary conditions were applied to the HST (see Fig. 4): T₁ = 66.64 °C, T₂ = 30 °C, T₃ = 66.64 °C, T₄ = 30 °C, T₅ = 30 °C, T₆ = 50 °C, T₉ = 15 °C, T_i = 15 °C. Temperatures T₁ and T₂ were obtained based on the ACHP cycle thermodynamic analysis. Their values are the same as t₄″ water temperature (see Fig. 3). It was assumed that the heating water is cooled down in the heating coils to 30 °C (T₂ and T₄). Temperatures T₂ and T₄ were also estimated based on the ACHP cycle analysis. The cooling coil outlet temperature was assumed to be 50 °C to provide DHW preparation in real conditions. The analytical model which is presented in the paper is a preliminary study for further experimental work. Therefore, parameters values applied ought to reflect both laboratory and real conditions in which the HST would operate. Hence, the inlet temperature of the cooling coil was set to 30 °C. Mass flow rate of the freshwater $\dot{m}_{11}$ supplying the HST varied from 0.06 to 0.135 kg/s (see Table 7, Fig. 4). Mass flow rate of the cooling water in coil₅₆ was constant and equal to 0.024 kg/s. Mass flow rates $\dot{m}_{12}$, $\dot{m}_{34}$, $\dot{m}_{{{\text{R}}32}}$ and $\dot{m}_{{{\text{air}}}}$ also varied depending on the case applied (see Table 3).

HST-ACHP-DC combined cycle

HST-ACHP-DC combined cycle presented within Fig. 1 reflects a planned laboratory bench. The purpose of the current studies was to investigate it theoretically. In this paragraph, the heat pump cycle calculations were presented intentionally, for the sake of formalities, even though they are of a fundamental character, just to ensure that the whole analysis was performed based on the right theoretical foundations.

Fig. 1

Scheme of the system which was studied analytically

Figure 1 presents the system which was analysed. Starting from the ACHP evaporator outlet (point 1 in Fig. 2), on the R32 side, one can compute a saturation pressure at a given evaporation temperature,$t_{{{\text{ev}}}} = 10\;{ }^\circ {\text{C}}$ when the quality, x = 1. The extensive working fluid parameters, such as enthalpy and entropy, were also obtained at the same conditions.

Fig. 2

Pictorial view of the ACHP cycle presented in P–h diagram for R32

In Fig. 2, the ACHP cycle is presented in the P–h diagram. The figure was obtained using Python and a CoolProp module. The set of formulas presented in Table 1, Eqs. (2)–(3) and Eqs. (6)–(8) reveal a method of calculating an extensive and intensive parameters of the R32 working fluid. Evaporation temperature was assumed to be 15 °C lower than the ambient temperature [25]. Therefore, evaporation occurred in t_ev = 10 °C. Values of evaporation pressure, enthalpy and entropy in point 1 of the Linde cycle (Fig. 2) are presented in Table 1. At the compressor suction (point 1′ in Fig. 2) side, the working fluid is superheated by t_sh = 5 °C. At the compressor discharging side (point 2 in Fig. 2), the enthalpy for an isentropic compression ($h_{{2\_{\text{is}}}}$) was calculated first. The temperature in point 2 could be obtained based on estimated $h_{{2\_{\text{is}}}}$ and known condensation pressure $P_{{{\text{con}}}}$. The CoolProp procedure was used to calculate discussed temperature. However, as the non-isentropic compression was assumed, the estimated temperature at point 2 was set to be 10 °C higher than the temperature at which $h_{{2\_{\text{is}}}}$ was computed. Consequently, the real temperature has reached the value of $t_{2} = 111$ °C.

Table 1 Parameters of the cycle in points 1, 1′ and 2 (Fig. 2)

Based on the computed enthalpy values, an isentropic efficiency was calculated according to the definition [26]

$$\eta_{{{\text{is}}}} = \frac{{h_{{2\_{\text{is}}}} - h_{{1^{\prime }}} }}{{h_{2} - h_{{1^{\prime }}} }}$$

(1)

The computed value of the isentropic efficiency was around 0.79 which corresponds well with the real operating conditions [26]. It was assumed that temperature at the compressor discharging side was approximately equal to 111 °C. This result is based on the calculated temperature at point 2 with P_con pressure and assuming an isentropic compression. This temperature was calculated using CoolProp for given enthalpy $h_{{2\_{\text{is}}}}$ and condensation pressure $P_{{{\text{con}}}}$. The resultant value was simply increased by 10 °C which makes that the final value is 111 °C. Usually, the isentropic efficiency is assumed and the temperature at the compressor discharging side is a resulting one.

$$h_{3} (t_{{{\text{con}}}} ,x) = 3.2 \cdot 10^{5} \;[{\text{J/kg}}]$$

(2)

$$s_{3} (t_{{{\text{con}}}} ,x) = 1.4 \cdot 10^{3} \;[{\text{J/(kg K)}}]$$

(3)

where $x = 0$.

$$w_{t} = h_{2} - h_{{1{\prime }}}$$

(4)

The counter flow condenser was divided into three zones (see Fig. 3). In the first one, the superheated R32 vapour was cooled down from the temperature $t_{2}$ to $t_{{{\text{con}}}}$. In the second zone, the condensation process occurred while in the third one the condensate was cooled down from the temperature $t_{{{\text{con}}}}$ to $t_{4}$. The specific thermal load of the condenser was therefore estimated as:

$$q_{{{\text{con}}}} = h_{2} - h_{4}$$

(5)

Fig. 3

Temperature variation of the water and R32 along the condenser length; P_con (R32) = 39.33 [bar], t_con (R32) = 60 [°C]

Finally, the coefficient of performance (COP), $\varepsilon_{{{\text{con}}}} = \frac{{h_{2} - h_{4} }}{{h_{2} - h_{{1{\prime }}} }} = 4.2$ was obtained. The calculated value of the COP stays in line with the COP of the ACHP (KAISAI KHC-10RY1) [27] which was bought for further empirical investigations.

$$P_{4} = P_{{{\text{sat}}}} (t_{{{\text{con}}}} )\;[{\text{bar}}]$$

(6)

$$h_{4} (t_{4} ,P_{4} ) = 3.08 \cdot 10^{5} \;[{\text{J/kg}}]$$

(7)

$$s_{4} (t_{4} ,P_{4} ) = 1.35 \cdot 10^{3} \;[{\text{J/(kg K)}}]$$

(8)

Figure 3 presents the temperature profiles of the condenser cooling water and R32 working medium along the condenser length. The zone numbering starts from the side of the hot R32 vapour inflow (1st zone). At the end of the first zone, there was assumed a minimal temperature difference ${\Delta }t = 5\;^\circ {\text{C}}$. Temperature value of the heated water at the inlet $t_{{3{\prime }}}$ of the condenser was equal to 30 °C.

$$h_{{2{\prime }}} (t_{{{\text{con}}}} ,x = 1) = 4.97 \cdot 10^{5} \;[{\text{J/kg}}]$$

(9)

Working fluid enthalpy in point 2′ (in Fig. 3) was, therefore, calculated using the CoolProp at a given temperature and quality (Eq. 9).

$$h_{{4{\prime \prime }}} = (h_{2} - h_{{2{\prime }}} ) \cdot m_{{{\text{red}}}} + h_{{4{\prime }}}$$

(10)

Equation (10) allows for the calculation of the outlet temperature of the water heated in the condenser

$$t_{4{\prime \prime }} = \frac{h_{4{\prime \prime }} }{Cp_{w}}$$

(11)

which equals $66.64 \;^\circ {\text{C}}$. The water temperature at the end of the second zone $t_{3^{\prime\prime} }$ was calculated in the same way.

$$h_{{3{\prime \prime }}} = (h_{3} - h_{4} ) \cdot m_{{{\text{red}}}} + h_{{3{\prime }}}$$

(12)

$$t_{{3{\prime \prime }}} = \frac{{h_{{3{\prime \prime }}} }}{{Cp_{w} }}$$

(13)

where $m_{{{\text{red}}}} = \frac{{\dot{m}_{{{\text{R}}32}} }}{{\dot{m}_{w} }}$ and $h_{3} ,h_{4}$ apply to the R32 working fluid (Figs. 2, 3) and $h_{{3{\prime \prime }}}$, $h_{{3{\prime }}}$, $t_{{3{\prime \prime }}}$ refer to the heated water (Fig. 3).

The estimation of an appropriate mass flow rate of the R32 working fluid and the water in the secondary ACHP cycle was carried out based on Eqs. (14) and (20), respectively. The air mass flow rate in the primary ACHP cycle was an input value that varied by 0.025 kg/s from 0.06 to 0.135 kg/s. This enabled estimation ACHP operating parameters at four separate steady states. These parameters, namely the heating water mass flow rate and the temperature, were used as input data in the HST analysis.

Assumption of an ideal thermal energy transfer through the evaporator has led to derivation of the equation defining the R32 working fluid mass flow rate:

$$\dot{m}_{{{\text{R}}32}} = \frac{{\dot{m}_{{{\text{air}}}} \cdot (h_{{{\text{ev.air}}{\prime }}} - h_{{{\text{ev.air}}{\prime \prime }}} )}}{{h_{1} - h_{5} }}$$

(14)

The mass flow rate of the cooling water in the condenser was estimated from the following relationship:

$$\frac{{\dot{Q}_{{{\text{con\_ext}}}} }}{{\dot{Q}_{{{\text{ev\_ext}}}} }} = \frac{{\dot{Q}_{{{\text{con\_int}}}} }}{{\dot{Q}_{{{\text{ev\_int}}}} }}$$

(15)

where int denotes an internal ACHP cycle and ext corresponds to the primary or secondary cycle.

$$\dot{Q}_{{{\text{con\_ext}}}} = \dot{m}_{w} \cdot \left( {h_{{4{\prime \prime }}} - h_{{3{\prime }}} } \right)$$

(16)

$$\dot{Q}_{{{\text{ev\_ext}}}} = \dot{m}_{{{\text{air}}}} \cdot \left( {h_{{{\text{ev.air}}{\prime }}} - h_{{{\text{ev.air}}{\prime \prime }}} } \right)$$

(17)

$$\dot{Q}_{{{\text{con\_int}}}} = \dot{m}_{{{\text{R}}32}} \cdot (h_{2} - h_{4} )$$

(18)

$$\dot{Q}_{{{\text{ev\_int}}}} = \dot{m}_{{{\text{R}}32}} \cdot (h_{1} - h_{5} )$$

(19)

By substituting Eqs. (16)–(19) to (15) one may determine the water mass flow rate depending on the air and R32 mass flow rates and the enthalpies of these fluids at relevant points of the cycle,

$$\dot{m}_{w} = \frac{{\dot{m}_{{{\text{air}}}} \cdot (h_{{{\text{ev.air}}{\prime }}} - h_{{{\text{ev.air}}{\prime \prime }}} ) \cdot (h_{2} - h_{4} )}}{{(h_{{4{\prime \prime }}} - h_{{3{\prime }}} ) \cdot (h_{1} - h_{5} )}}$$

(20)

Mathematical model description of the HST

Internal geometry of the HST is presented schematically in Fig. 4. In this paper, the authors concentrate on a detailed analysis of the HST internal geometry as well as its operating parameters and dynamics at varying thermal loads. The heat-flow and the geometrical parameters have been analysed precisely. It has led to the estimation of the optimal operating conditions. The following parameters were analysed among others: coil length, number of the loops in a single coil, height of the coil, heat transfer area of the coil, heat fluxes transferred through the coils, heat transfer coefficient at the heating water side, mass flow rate of the freshwater supplying the HST, air mass flow rate in the ACHP primary cycle and diameter of a single loop of the coil (Tables 5, 6, 7, 8). Two modes of the HST operation were studied, namely continuous and intermittent. Intermittent operation mode of the HST reposed on alternating charging and discharging process occurrence. The main aim was to examine the ability of the HST to fulfil demand for heat for central heating and DHW preparation in a case of not being charged thermally. This paragraph presents the balance equations based on which the HST analysis was carried out. External geometry of the HST is presented in [1] and in Table 2. The internal geometry of the HST varies in terms of the number of loops in the coils. As the HST total volume remains constant, variation in the number of loops in the coils results in variation in the volume of water inside the tank.

Fig. 4

Geometry of the studied HST

Table 2 Geometrical parameters of the HST taken from the project assumptions [1]

The radius of a single whorl of the coil,${ }D_{{{\text{coil}}}}$, was assumed to vary from 0.2 to 0.3 m, while the internal diameter of all of the coils was 0.015 m and the pipe thickness was $\delta_{{{\text{coil}}}} = 0.0015$ m.

$$\dot{Q}_{{{\text{hst}}}} = \dot{Q}_{12} + \dot{Q}_{34} - \dot{Q}_{56} - \dot{Q}_{11} - \dot{Q}_{0}$$

(21)

where $\dot{Q}_{{{\text{hst}}}}$ is the overall heat flux transferred to the HST, kW.

$$\dot{Q}_{{{\text{hst}}}} = M_{{{\text{hst}}}} \cdot Cp_{11} \cdot \frac{{{\text{d}}T_{11} }}{{{\text{d}}\tau }}$$

(22)

where $M_{{{\text{hst}}}}$ is the mass of the water in the HST, kg.

The presented HST-ACHP-DC system is described mathematically using lumped parameters model and its energy balance is described through Eq. (21). As it is known, the essence of the lumped parameters model is that the temperature is spatially uniform at any time during the unsteady process. The heat fluxes $\dot{Q}_{12}$ and $\dot{Q}_{34}$ correspond to the heating coils denoted as coil₁₂ and coil₃₄, respectively. The heat source, for two coils 12 and 34 is the ACHP. Therefore, further in the paper, there are listed considerations on the influence of the ACHP varying heating power on the HST operation. The heat flux $\dot{Q}_{56}$ refers to the cooling coil₅₆. The DC is cooling receiver for the HST. The fresh cold water supply, into the fluid-flow HST, at the temperature lower than the mean HST liquid temperature influences the energy balance of the HST. Therefore, supplying the tank with cold water is regarded as the HST heat loss $\dot{Q}_{11}$. The last term in the balance equation $\dot{Q}_{0}$ denotes the heat losses from the HST to its surrounding.

$$\dot{Q}_{12} = \dot{m}_{12} \cdot Cp_{12} \cdot (T_{1} - T_{2} ) = A_{12} \cdot K_{12} \cdot \Delta T_{\log ,12}$$

(23)

$$\dot{Q}_{34} = \dot{m}_{34} \cdot Cp_{34} \cdot (T_{3} - T_{4} ) = A_{34} \cdot K_{34} \cdot \Delta T_{\log ,34}$$

(24)

$$\dot{Q}_{56} = \dot{m}_{56} \cdot Cp_{56} \cdot (T_{6} - T_{5} ) = A_{56} \cdot K_{56} \cdot \Delta T_{\log ,56}$$

(25)

$$\dot{Q}_{11} = \dot{m}_{11} \cdot Cp_{11} \cdot (T_{11} - T_{9} )$$

(26)

$$\dot{Q}_{0} = K_{{{\text{hst}}}} \cdot A_{{{\text{hst}}}} \cdot (T_{11} - T_{0} )$$

(27)

Overall heat transfer coefficient of the coil is calculated as follows:

$$K_{ij} = \frac{1}{{R_{{{\text{coil}},ij\_{\text{hst}}}} }}$$

(28)

Total thermal resistance $R_{{{\text{coil}},ij\_{\text{hst}}}}$ is defined as

$$R_{{{\text{coil}},ij\_{\text{hst}}}} = \frac{1}{{\alpha_{c\_ij} }} + \frac{{d_{{{\text{mean.coil}}}} }}{{\lambda_{r} }} + \frac{1}{{\alpha_{11} }}$$

(29)

where i = 1, 3, 5 and j = 2, 4, 6.

$${\text{Nu}} = 0.023 \cdot {\text{Re}}^{0.8} \cdot \Pr^{0.4}$$

(30)

$$\alpha = {\text{Nu}} \cdot \frac{\lambda }{{d_{{{\text{i.coil}}}} }}$$

(31)

$$\varepsilon_{c} = 1 + 3.54 \cdot \frac{{d_{{{\text{i.coil}}}} }}{{D_{{{\text{coil}}}} }}$$

(32)

$$\alpha_{c\_ij} = \alpha \cdot \varepsilon_{c}$$

(33)

Overall heat transfer coefficient calculated for the HST in terms of the total thermal resistance to heat transfer between the water in the HST and its surrounding air was calculated using Eqs. (34) and (35):

$$K_{{{\text{hst}}}} = \frac{1}{{R_{{{\text{hst}}}} }}$$

(34)

Total thermal $R_{{{\text{hst}}}}$ is presented through a following formula:

$$R_{{{\text{hst}}}} = \frac{1}{{\alpha_{{{\text{air}}}} }} + \frac{{\ln \left( {\frac{{D_{{{\text{ext}}}} }}{{D_{{\text{int}}} }}} \right)}}{{2 \cdot \pi \cdot \lambda_{{{\text{hst}}}} }} + \frac{{\ln \left( {\frac{{D_{{{\text{ext}}}} + 2 \cdot \delta_{{{\text{is}}}} }}{{D_{{{\text{ext}}}} }}} \right)}}{{2 \cdot \pi \cdot \lambda_{{{\text{is}}}} }} + \frac{1}{{\alpha_{11} }}$$

(35)

The heat transfer coefficient $\alpha$ at the heating water side in all three coils was computed using the classical Dittus–Boelter correlation for Nusselt number for turbulent flow (Eq. 30). Additionally, a correction coefficient (Eq. 32) taking into account the coil shape was applied to provide the most accurate heat transfer coefficient value. Finally the heat transfer coefficient of the medium inside the coils was calculated using Eq. (33).

Equations (22)–(27) can be substituted into Eq. (21) and then, the resulting equation can be solved for temperature T₁₁. The time and the temperature were introduced in a dimensionless form defined as $\tau^{*} = \frac{\tau }{{\tau_{0} }}$, $T^{*} = \frac{T}{{T_{0} - T_{i} }}$ respectively. Hence, physical dimensional Eq. (21) was transformed into a non-dimensional mathematical equation which was solved. Appearance of the dimensionless numbers results from the transformation the physical equation into mathematical (Eq. 36):

$$\frac{{{\text{d}}T_{11}^{*} }}{{{\text{d}}\tau^{*} }} = A^{*} + B^{*} - C^{*} - D^{*} - E^{*}$$

(36)

where

$$A^{*} = {\text{Re}}_{12} \cdot \Pr_{12} \cdot \frac{{\pi \cdot d_{{{\text{i.coil}}}} \cdot \lambda_{12} }}{{4 \cdot M_{11} \cdot Cp_{11} }} \cdot \left( {T_{1}^{*} - T_{2}^{*} } \right) \cdot \tau_{0}$$

$$B^{*} = {\text{Re}}_{34} \cdot \Pr_{34} \cdot \frac{{\pi \cdot d_{{{\text{i.coil}}}} \cdot \lambda_{34} }}{{4 \cdot M_{11} \cdot Cp_{11} }} \cdot \left( {T_{3}^{*} - T_{4}^{*} } \right) \cdot \tau_{0}$$

$$C^{*} = {\text{Re}}_{56} \cdot \Pr_{56} \cdot \frac{{\pi \cdot d_{{{\text{i.coil}}}} \cdot \lambda_{56} }}{{4 \cdot M_{11} \cdot Cp_{11} }} \cdot \left( {T_{6}^{*} - T_{5}^{*} } \right) \cdot \tau_{0}$$

$$D^{*} = {\text{Re}}_{119} \cdot \Pr_{119} \cdot \frac{{\pi \cdot d_{{{\text{i.pipe}}}} \cdot \lambda_{119} }}{{4 \cdot M_{11} \cdot Cp_{11} }} \cdot \left( {T_{11}^{*} - T_{9}^{*} } \right) \cdot \tau_{0}$$

$$E^{*} = \frac{{\left( {\frac{{\frac{{L_{{{\text{hst}}}} }}{{\lambda_{11} }}}}{{{\text{Nu}}_{11} }} + \frac{{\ln \left( {\frac{{D_{{{\text{ext}}}} }}{{D_{{\text{int}}} }}} \right)}}{{2 \cdot \pi \cdot \lambda_{{{\text{hst}}}} }} + \frac{{\frac{{L_{{{\text{hst}}}} }}{{\lambda_{{{\text{air}}}} }}}}{{{\text{Nu}}_{{{\text{air}}}} }}} \right)^{ - 1} \cdot A_{{{\text{hst}}}} \cdot \frac{{\mu_{11} }}{{\lambda_{11} }}}}{{M_{11} \cdot \Pr_{11} }} \cdot \left( {T_{11}^{*} - T_{0}^{*} } \right) \cdot \tau_{0}$$

The solution of the differential Eq. (36) was found analytically and is expressed by formula (37):

$$T_{11}^{*} = \frac{{\left\{ {\left( {D + G \cdot T_{0}^{*} \cdot \tau_{0} + H \cdot T_{9}^{*} \cdot \tau_{0} + T_{i}^{*} \cdot \tau_{0} \cdot ( - H - G)} \right) \cdot e^{{\tau^{*} \cdot \tau_{0} \cdot ( - H - G)}} } \right\} - D - \tau_{0} \cdot \left( {G \cdot T_{0}^{*} + H \cdot T_{9}^{*} } \right)}}{{\tau_{0} \cdot ( - H - G)}}$$

(37)

where

$M_{11} = M_{{{\text{hst}}}}$ is the total mass of the water in the HST, kg

$$H = {\text{Re}}_{119} \cdot \Pr_{119} \cdot \frac{{\pi \cdot d_{{{\text{i.pipe}}}} \cdot \lambda_{119} }}{{4 \cdot M_{11} \cdot Cp_{11} }}$$

$$G = \frac{{\left( {\frac{{\frac{{L_{{{\text{hst}}}} }}{{\lambda_{11} }}}}{{{\text{Nu}}_{11} }} + \frac{{\ln \left( {\frac{{D_{{{\text{ext}}}} }}{{D_{{\text{int}}} }}} \right)}}{{2 \cdot \pi \cdot \lambda_{{{\text{hst}}}} }} + \frac{{\frac{{L_{{{\text{hst}}}} }}{{\lambda_{{{\text{air}}}} }}}}{{{\text{Nu}}_{{{\text{air}}}} }}} \right)^{ - 1} \cdot A_{{{\text{hst}}}} \cdot \frac{{\mu_{11} }}{{\lambda_{11} }}}}{{M_{11} \cdot \Pr_{11} }}$$

$$D = A^{*} + B^{*} - C^{*}$$

Equations (21) and (36) describe the energy balance for the HST containing three coils, two heating and one cooling. The energy balance for the HST with two coils assumes that one heating coil is removed (in this case it is coil₃₄) and it takes the form as follows:

$$\dot{Q}_{{{\text{hst}}}} = \dot{Q}_{12} - \dot{Q}_{56} - \dot{Q}_{11} - \dot{Q}_{0}$$

(38)

and

$$\frac{{{\text{d}}T_{11}^{*} }}{{{\text{d}}\tau^{*} }} = A^{*} - C^{*} - D^{*} - E^{*}$$

(39)

Shifting between two HST models, namely the first one with three coils and the second one with two coils, can be performed easily in the equations presented above. The variable D contains the terms corresponding to each heating and cooling coil. Therefore, the coil number may be arbitrarily varied. The only limiting condition is the HST height. Therefore, Eq. (37) can also be used for HST water temperature estimation in the case of the HST with two coils. In such case, the variable D should be replaced with $D_{{2\_{\text{coils}}}} = A^{*} - C^{*}$.

In this place, it is worth mentioning dimensionless numbers which play a meaningful role in HST operation analysis, namely the Grashof and Rayleigh numbers. They were calculated for the water inside the studied HST.

$${\text{Gr}}_{{{\text{w.hst}}}} = \frac{{g \cdot L_{{{\text{hst}}}}^{3} \cdot \rho_{{{\text{w.hst}}}} }}{{\mu_{{{\text{w.hst}}}} }} \cdot \beta_{{{\text{w.hst}}}} \cdot \Delta T$$

(40)

where ${\Delta }T$, °C, stands for the temperature difference between the water and the wall of the heating coil. In this case, ${\text{Gr}}_{{{\text{w.hst}}}} = 5.51 \cdot 10^{9}$. The value of the Grashof number $Gr_{w.hst}$ indicates that the buoyancy force dominates the viscous forces. Additionally, the value of the following ratio, $\frac{{{\text{Gr}}_{{{\text{w.hst}}}} }}{{{\text{Re}}_{{{\text{w.hst}}}} }}$, implies that in the HST occurs a free convection as this ratio value is $3.1 \cdot 10^{4}$. Moreover, the calculated Rayleigh number, defined as ${\text{Ra}}_{{{\text{w.hst}}}} = {\text{Gr}}_{{{\text{w.hst}}}} \cdot \Pr_{{{\text{w.hst}}}}$, reached the value of $2.34 \cdot 10^{10}$. If the Rayleigh number is lower than 10⁸, the flow is laminar. However, when the Rayleigh number value exceeds 10¹⁰, the flow is turbulent [7]. Therefore, the Rayleigh number value implies that within the investigated HST the water movement is of a turbulent character.

$${\text{Nu}}_{{{\text{w.hst}}}} = 0.6 \cdot 0.33 \cdot {\text{Re}}_{{{\text{w.hst}}}}^{0.6} \Pr_{{{\text{w.hst}}}}^{0.33}$$

(41)

Equation (41) represents a correlation for the Nusselt number for a shell–tube heat exchanger which was found to be the most suitable for the HST with the internal coils. At the beginning of the correlation Eq. (41), there stands a correction coefficient which equals 0.6. This coefficient accounts for the leakage and a non-perpendicular water inflow to the coils external surface.

$$\alpha_{{{\text{w.hst}}}} = {\text{Nu}}_{{{\text{w.hst}}}} \cdot \frac{{\lambda_{{{\text{w.hst}}}} }}{{d_{{{\text{e.coil}}}} }}$$

(42)

Therefore, the heat transfer coefficient at the HST internal water side was calculated using formula (42).

Validation of the analytical model

The validation of the HST analytical model was carried out based on the data made available by Li et al. [6]. The presented model of the HST is based on the lumped capacitance method approach, as already mentioned, that essentially comes down to the assumption that the temperature is spatially uniform in the whole computational domain. The work by Li et al. [6] presents the CFD analysis assuming that the water velocity through the HST during its charging is set at 0 m/s. Additionally, the HST heating coil velocity inlet boundary condition was set in the commercial computational software. Hence, the heating medium (ethylene glycol) velocity was equal to 0.45 m/s. The geometrical and fluid-flow conditions of the CFD analysis presented in [6] were also used for the present HST analytical model validation. This refers to the height and diameter of the HST as well as to the coil and the heating medium mass flow rates. The validation of the present model was carried out for the case No 1 described in [6]. The reference results for the cylindrical HST are presented in Fig. 6b in [6]. For the purpose of the validation, in the present model it was assumed that there is no fluid flow through the HST and that it is heated by one coil. Regarding that only inlet temperature of the heating medium (ethylene glycol) is known, and that there is no fluid flow through the HST, the balance Eq. (21) simplifies to

$$\dot{Q}_{{{\text{hst}}}} = \dot{Q}_{12} - \dot{Q}_{0}$$

(43)

where $\dot{Q}_{12} = \dot{m}_{12} \cdot Cp_{12} \cdot (T_{12} - T_{11} )$. In this particular case, all parameters indexed with 12 correspond to the ethylene glycol. The terms $\dot{Q}_{{{\text{hst}}}}$ and $\dot{Q}_{0}$ are defined by Eqs. (22) and (27), respectively. Taking this into account, Eq. (43) may be rewritten as

$$M_{{{\text{hst}}}} \cdot Cp_{11} \cdot \frac{{{\text{d}}T_{11} }}{{{\text{d}}\tau }} = \dot{m}_{12} \cdot Cp_{12} \cdot (T_{12} - T_{11} ) - K_{{{\text{hst}}}} \cdot A_{{{\text{hst}}}} \cdot (T_{11} - T_{0} )$$

(44)

Therefore, T₁₁ depends on time while T₁₂ depends on the coil length. T₁₂ is a mean temperature of the ethylene glycol at x length of the coil.

On the other hand, $\dot{Q}_{12}$ can be presented as

$$\dot{m}_{12} \cdot Cp_{12} \cdot {\text{d}}T_{12} = K_{{12\_{\text{e}}}} \cdot O_{{{\text{coil}}{.12}}} \cdot (T_{12} - T_{11} ) \cdot {\text{d}}x$$

(45)

where $O_{{{\text{coil}}{.12}}}$ is the circumference of coil₁₂, m; $T_{12}$ is a temperature of the ethylene glycol at the x length of the coil, K; and $K_{{12\_{\text{e}}}}$ is the overall heat transfer coefficient calculated for the heating coil with ethylene glycol.By solving Eq. (45) for T₁₂ at x = 0, T₁₂ = T₁ boundary conditions applied, one obtains:

$$T_{12} - T_{11} = (T_{1} - T_{11} ) \cdot e^{A \cdot x}$$

(46)

where $A = \frac{{k_{{{\text{coil}}{.12}}} \cdot O_{{{\text{coil}}{.12}}} }}{{\dot{m}_{12} \cdot Cp_{12} }}$.

By substituting Eqs. (44)–(46) and after solving the differential equation, one can obtain a solution allowing for computing temperature variation of the water in the HST, namely $T_{11}$, over time.

$$T_{11} = \frac{{\left\{ {e^{{\tau \cdot \left( { - B \cdot e^{A \cdot x} - W} \right)}} \cdot \left[ {B \cdot T_{1} \cdot e^{A \cdot x} + W \cdot T_{0} + T_{i} \cdot \left( { - B \cdot e^{A \cdot x} - W} \right)} \right]} \right\} - B \cdot T_{1} \cdot e^{A \cdot x} - W \cdot T_{0} }}{{ - B \cdot e^{A \cdot x} - W}}$$

(47)

where $B = \frac{{\dot{m}_{12} \cdot Cp_{12} }}{{M_{11} \cdot Cp_{11} }}$, $W = \frac{{K_{{{\text{hst}}}} \cdot A_{{{\text{hst}}}} }}{{M_{11} \cdot Cp_{11} }}$ and $x$ is the length of the coil, m.

The temperature computed using formula (47) for the second hour of the HST operation time reached the value of 316.96 K. It was compared with the temperature value shown in Fig. 6b in [6] in which the layout of the isothermal surfaces in a longitudinal section of the cylindrical HST is depicted. The results differ by only 2.96 K which is considered satisfactory accordance between the values.

Results and discussion

The dynamics of the HST-ACHP-FC system was carried out using the MATLAB/Simulink. As it is presented in Fig. 5, the HST operated under transient conditions while the ACHP worked at four subsequent steady states (Table 3). The heating coils coil₁₂ and coil₃₄ were provided the same heating power as the mass flow rates $\dot{m}_{12}$ and $\dot{m}_{34}$ were set to be equal at a constant temperature difference at the inlets and outlets of both coils. In this way, the influence of the ACHP heating power variation on the HST operation was studied. The following analysis was conducted for the $D_{{{\text{coil}}}}$ equal to 0.2 m and 0.3 m.

Fig. 5

HST water temperature (T__hst) variation over time at four ACHP steady states for constant values of $\dot{m}_{11}$ = 0.085 kg/s and $\dot{m}_{56}$ = 0.024 kg/s; for each case, the following conditions are fulfilled: $\dot{m}_{w} = \dot{m}_{12} + \dot{m}_{34}$ and $\dot{m}_{12} = \dot{m}_{34} , D_{{{\text{coil}}}} = 0.2\;{\text{m}}$; values of the mass flow rates are presented in Table 3

Table 3 Values of the mass flow rates corresponding to all studied cases

The increase in the mass flow rate of the air in the primary ACHP cycle has obviously resulted in T_hst temperature increase (see Fig. 5). By comparing cases 1 and 4, one may conclude that the increase of $\dot{m}_{{{\text{air}}}}$ = 0.63 kg/s by about 48% leads to increase in T_hst by 5.6 °C, in $\dot{m}_{12}$ and $\dot{m}_{{{\text{R}}32}}$ by 50% and 53%, respectively. When two previously mentioned cases are compared, it is rather clear that the difference in T_hst temperature at the level of 5.6 °C leads to the growth of the stored thermal energy by 4.45 MJ in the case 4 against the case 1. Therefore, by comparing the electric power used to supply the circulation pumps to the benefit of the thermal energy storage one may decide whether it is worthy or not. In the current paper, the presented system was not investigated regarding its economics.

Each of the cases, 1, 2, 3 and 4, depicted in Figs. 5 and 6 is of different HST thermal and geometrical parameters which are listed in Table 5. Equality of $\dot{m}_{12}$ and $\dot{m}_{34}$ causes that all parameters presented in Table 5 are also the same for both heating coils. In the paper, design HST calculations are disclosed; therefore, the coil geometrical parameters differ in all cases, which are presented through Table 5. By comparing all instances, it can be noticed that the highest growth in both temperature T_hst and $\dot{m}_{12}$, $\dot{m}_{34}$ mass flow rates was detected for case 1 and 2 (see Table 4). The 21.4% increase in the mass flow rates has led to increase in the HST water temperature by 2.1 °C.

Fig. 6

Time-dependent HST water temperature (T_hst) variation at four ACHP steady states; $D_{{{\text{coil}}}}$ = 0.2 m

Table 4 An increase in T_hst temperature as an answer to the percentage growth in $\dot{m}_{12}$, $\dot{m}_{34}$ mass flow rates for all studied cases

As it is presented in Table 4, the overall growth in $\dot{m}_{12}$ and $\dot{m}_{34}$ mass flow rates by 50% has resulted in the HST water mean temperature increase by 5.6 °C. At this point, it is to be considered whether the growth in the temperature is worth the growth of demand for electrical energy of the circulating pump.

Within Table 5, design analysis results of the HST heating coils are presented.

Table 5 Thermal and geometrical parameters of coil₁₂ and coil₃₄ of the HST for case 1–case 4 marked in Figs. 5 and 6

Extended calculations were carried out for the geometrical parameters which were previously estimated for four studied cases based on design analysis. In design calculations, the geometrical parameters values were derived. In extended calculations, the values of the geometrical parameters were input data.

Based on the aforementioned, the geometrical parameters such as the heat transfer surface area of the heating coils $A_{{\text{coil12,34}}}$ the length of the coils $L_{{\text{coil12,34}}}$ and their height take now constant values which are case-dependent. Extended calculations were carried out for geometrical parameters presented in Table 5. It is obvious that the growth in the air mass flow rate at the ACHP low-heat source side implies the growth in the heat flux transferred through the HST heating coils. For the purpose of extended calculations, each of four cases can be chosen. Keeping constant geometrical parameters and varying heating medium mass flow rate, one may find the quantity of thermal energy transferred through a particular heating coil. An example of the extended analysis results is presented through Table 6.

Table 6 An example extended analysis results carried out for each of the studied cases

Table 6 presents the results of the extended analysis which was carried out by applying particular geometrical parameters of the coils. The geometrical parameters were constant for each case (case 1–4). Nevertheless, they differed when compared one case to another. As the coil geometry was kept constant for a given case (case 1–4), the overall heat transfer coefficient was varying due to variation of the heat transfer coefficient which resulted from varying Reynolds and Nusselt numbers.

Obviously, the highest heat flux values are observed for the fourth case as the coil heat transfer surface area is the highest. The content of Table 6 allows for choosing an appropriate heating power of the coil which results from the user’s needs.

The case 4 (Table 5) was chosen for further analysis due to highest heat transfer area of the heating coils. Figure 7 shows how the HST mean water temperature varies over time after altering mass flow rate of the freshwater supplying the HST. The mass flow rate of the water ${\dot{\text{m}}}_{11}$ that is heated while flowing through the HST, may vary according to the actual consumer needs. In the case of a minimal usage of the thermal energy stored in the HST by harvesting it through $\dot{m}_{11}$, there is still 4.2 kW transferred by the flow and the HST water temperature is the highest in this case (32 °C). When HST reaches the steady state, the $\dot{m}_{11}$ is increased by 0.025 kg/s that results in T_hst drop by about 4.6 °C. The heat flux which is transferred from the HST reaches 4372 W in this case. Further increase of $\dot{m}_{11}$ to the value of 0.11 kg/s leads to slight growth of the transferred heat flux to 4447 W. The water outflow of $\dot{m}_{11}$ = 0.135 kg/s carries the heat flux at the level of 4.5 kW; however, the HST water temperature decreases significantly in this case as it is only 8 °C higher in value than the initial temperature (15 °C).

Fig. 7

T_hst temperature variation over time and over increasing $\dot{m}_{11}$ at constant $\dot{m}_{{{\text{air}}}}$; $D_{{{\text{coil}}}}$ = 0.2 m

The analysis of the HST operation under varying $\dot{m}_{11}$ mass flow rate was carried for the fourth case which was found to be the most appropriate from an energetic point of view.

Figure 8 and Table 7 show the time after which the HST temperature reaches a steady value for selected values of $\dot{m}_{11}$. It is evident that the change of the value of $\dot{m}_{11}$ affects the temperature of the liquid inside the HST. This temperature decreases in time after $\dot{m}_{11}$ is increased which is rather obvious. However, the analysis which was carried out is of a design and a verification character. Therefore, after defining the optimal heat-flow and the geometrical parameters (Table 5, case 4) further analysis was conducted.

Fig. 8

T_hst temperature variation over time at constant $\dot{m}_{\text{air}}$ at four different $\dot{m}_{11}$ values; $D_{\text{coil}}$ = 0.2 m

Table 7 Time required to reach the steady state in the HST with selected values of $\dot{m}_{11}$ mass flow rate (see Fig. 8)

Figure 9 shows the characteristics of an intermittent HST operation. The aforementioned characteristics were obtained for the following operating conditions. Blue dashed line applies to the case in which simultaneous HST charging through both heating coils (coil₁₂, coil₃₄) and discharging through coil₅₆ was assumed. During this HST charging–discharging process $\dot{m}_{11}$ mass flow rate was constant and equal to 0.06 kg/s. After 5 h of charging, the mass flow rates were set to 0 kg/s in both heating coils which naturally resulted in the commencement of the discharging process as both, $\dot{m}_{11}$ and $\dot{m}_{56}$, mass flow rates were kept at constant values, namely 0.06 kg/s and 0.024 kg/s, respectively. The red solid line also presents T_hst temperature variation over time during intermittent charging–discharging process. However, in contrast to the former case the mass flow rate in the coil₅₆, intended to fulfil demand on domestic hot water (DHW) preparation, was turned down to 0 kg/s. In this case, T_hst increased in value by about 7.2 °C relative to the previous case. Nevertheless, Fig. 9 shows that even when the HST prepares DHW during its charging process, it can still accumulate thermal energy at the level that allows for its further continuous operation for the next 1 h without being charged. In this case, the HST water temperature reaches the same value as in the case without DHW preparation which is about 16.3 °C. The HST operation period without being charged may be enlarged to 5 h. Unfortunately, in this case the HST water temperature decreases significantly and its final value reaches 8 °C.

Fig. 9

Intermittent HST charging–discharging process under the conditions of the 4th case (see Table 5) and the lowest $\dot{m}_{11}$ mass flow rate of 0.06 kg/s (see Table 7); DHW, domestic hot water

Further on, a comparison to the HST with the coils diameter $D_{{{\text{coil}}}}$ equal to 0.3 m is presented. The input data are the same as in case 4 described previously. The results are compared to those presented in Table 5 (case 4). As given in Table 5 (case 4) and in Table 8, 0.1 m increase in the coil diameter $D_{{{\text{coil}}}}$ causes a decrease by 100 W/(m² K) in the value of the heat transfer coefficient. In general, lowering the coil diameter implies the growth of curvature of the coil which obviously results in the heat transfer coefficient growth (see Eqs. (31)–(33)). The heat transfer coefficient calculated using Eq. (33) is higher in value than the one calculated using Eq. (31). However, along with the decrease in the coil diameter, its height increases which also must be taken into account if the HST height is kept constant. In the previously studied case, $D_{{{\text{coil}}}} = 0.2\;{\text{m}}$ and this value seems to be the most appropriate from the heat transfer and geometrical points of view.

Table 8 Thermal and geometrical parameters of coil₁₂ and coil₃₄ of the HST for the same input parameters as in case 4 but for the coil diameter $D_{{{\text{coil}}}}$ equal 0.3 m

The comparison of the HST with varying number of the coils was made. In all cases being examined, the initial temperature of the liquid in the HST was equal to 15 °C. The reference case was the one with 3 coils from which two heating coils (i.e. coil₁₂ and coil₃₄) and operated at the same level of heat-flow rate, namely 3.2 kW. In the second instance, one heating coil (coil₁₂) operated at varying heating power of 3.2 kW or 6.4 kW. It was found out that the presented model did capture the change in T_hst when the coil number changed from 2 to 3 in the case of $\dot{Q}_{{{\text{coil}}{. 12}}} = 3.2\;{\text{kW}}$. It was noticed that for this assumption, T_hst for the HST with 2 coils reached the value of 18.47 °C while with three coils the temperature in the HST grew to the value of 27.33 °C. When the heating power of the coil₁₂ was increased from $\dot{Q}_{{{\text{coil}}{. 12}}} = 3.2\;{\text{kW}}$ to $\dot{Q}_{{{\text{coil}}{. 12}}} = 2 \cdot 3.2\;{\text{kW}}$ in the case of the HST containing 2 coils, the water temperature in the HST reached the same value as in the reference case, namely 27.33 °C. In the case of the HST with 2 coils and $\dot{Q}_{{{\text{coil}}{.12}}} = 2 \cdot 3.2\;{\text{kW}}$, the temperature in the HST was the same as in the reference case, namely 27.33 °C. Additionally, in the case of 2 coils, a heat transfer area was somewhat lower than in the reference case which could negatively affect the heat transfer against the reference case. On the other hand, in the case of two coils, the heat transfer coefficient increased by 74% (Table 9) when compared to the reference case which, obviously, improved the thermal energy transfer. Obviously, increasing the heating medium mass flow rate and keeping the same coil diameter d_coil results in the increase in the Reynolds number that implies the increment in Nusselt number and the heat transfer coefficient values.

Table 9 Thermal and geometrical parameters of coil₁₂ for the case of the HST with two coils and $\dot{Q}_{12} = 2 \cdot 3.2\;{\text{kW}}$

To capture a more precise variation in T_hst for the aforementioned case, the HST domain ought to be divided into minimum 3 zones and the T_hst should be calculated for all zones, giving more adequate results. Such analysis was not carried out within the current study as the goal was not to provide a local but general view on the T_hst variation. The aim was also to introduce a methodology allowing for the HST thermal-flow analysis and the presented model is sufficient for it.

Conclusions

HST-ACHP-FC combined cycle was analysed in this paper. The HST response to the varying operating parameters was analysed. The differential equation describing the HST water temperature variation over time was proposed based on the lumped capacitance method. Solution of this equation enables calculation of the time-dependent temperature of the water in the HST using even a calculator. Additionally, a correlation of the aforementioned temperature variation over time and with the coil length was proposed. The analytical model validation was carried out. The model presented gave proper results, and an excellent accordance with the literature data was reached. The results of the analysis led to the following conclusions. Essential information on the HST operation at varying thermal-flow conditions is presented. Hence, when both border cases are compared, namely the case 1 and case 4, it is seen that the increase in the air mass flow rate of the primary cycle and of the heating water mass flow rate in the coil by 48% and 50%, respectively, implied a growth in the HST water temperature by 5.6 °C. It was observed that the change in heating coils mass flow rate from 0.014 to 0.021 kg/s has resulted in the growth of the heat transfer coefficient from 959.1 to 1310 W/m² K. Simultaneously an increase in the heat flux transferred through the heating coils from 2.2 to 3.2 kW was obtained. Additionally, it was noticed that 0.1 m increase in the coil diameter $D_{{{\text{coil}}}}$ caused a decrease by 100 W/(m² K) in the value of the heat transfer coefficient. The proposed solution for the time-dependent temperature variation reveals the dimensionless numbers which dominate in the studied problem. Based on the knowledge about the dimensionless numbers that play a key role in the HST operation, one may identify which parameters should be measured during empirical investigations. Based on the analysis carried out, it was concluded that an increase in diameter of a single loop of the coil $D_{{{\text{coil}}}}$ by 50% implied a decrease in the heat transfer coefficient inside the coil by about 6%. Additionally, it was noticed that the increase in the coil heat transfer surface area by 40% has resulted in the increase in heat flux transferred through the coil by 33–40% depending on the heating medium mass flow rate. Optimal thermal-flow operating parameters of the HST coils were proposed due to highest heat flux transferred to the HST. Based on the analysis carried out, the time after which the HST reaches steady state at varying freshwater mass flow rate was estimated. Intermittent HST charging–discharging process was carried out for case 4 with the freshwater mass flow rate $\dot{m}_{11} = 0.06$ kg/s. The analysis was performed at DHW simultaneous preparation enabled and disabled. When DHW preparation was disabled, the HST water mean temperature increased by 7.2 °C with reference to the other case. Nevertheless, the HST could fulfil demand for heat for central heating and DHW preparation even for 5 h without charging. Future work on the HST may be carried out as there exists a turbulent mixing within the water inside the HST. This phenomena is important due to efficiency of thermal energy storage as it influences the stratification in water. Further works may also concern CFD analysis which would provide valuable data about isothermal surfaces distribution. Additionally, based on CFD analysis multiple geometrical and heat-flow parameters could be more precisely estimated. The speech is about the heat transfer coefficient in the coils and in the HST, the shape and localisation of the freshwater inlet, localisation of eddies and many others.

A simple analytical computational algorithm for the HST is presented. It allows for examining the effect of heat-flow and geometrical parameters variation on the HST operation. The algorithm was employed for few different cases and satisfactory results were obtained.

Appendix

Derivations of A^*–D^* present in Eq. (36):

$$X^{*} = \frac{{\dot{m}_{ij} \cdot Cp_{ij} \cdot \frac{{\mu_{ij} }}{{\lambda_{ij} }}}}{{M_{w} \cdot Cp_{w} \cdot \frac{{\mu_{ij} }}{{\lambda_{ij} }}}} \cdot {\Delta }T_{ij}^{*} \cdot \tau_{0}$$

(48)

where X* = A^*, B^*, C^*, D^*; ${\Delta }T_{ij}^{*} = T_{i}^{*} - T_{j}^{*}$,$i = 1,3,6,11;\;j = 2,4,5,9$

$$\Pr _{{ij}} = Cp_{{ij}} \cdot \frac{{\mu _{{ij}} }}{{\lambda _{{ij}} }}$$

(49)

$$X^{*} = \frac{{\dot{m}_{ij} \cdot Pr_{ij} }}{{M_{w} \cdot Cp_{w} \cdot \frac{{\mu_{ij} }}{{\lambda_{ij} }}}} \cdot {\Delta }T_{ij}^{*} \cdot \tau_{0}$$

(50)

$$\dot{m}_{ij} = \rho_{ij} \cdot A_{{{\text{coil}}}} \cdot w_{ij}$$

(51)

where ${\text{Pr}}_{ij}$ is the Prandtl number, $\dot{m}_{ij}$ mass flow rate, kg/s; $\rho_{ij}$ density, kg/m³; $A_{{{\text{coil}}}} = \pi \cdot \left( {\frac{{d_{{{\text{i.coil}}}} }}{2}} \right)^{2}$ cross-sectional area of the coil and $w_{ij}$ velocity, m/s.

$$\mu_{ij} = \nu_{ij} \cdot \rho_{ij}$$

(52)

where $\mu_{ij}$ is the dynamic viscosity, Pa s, and $\nu_{ij}$ kinematic viscosity, m²/s.

By substituting Eqs. (51) and (52) to (50), a following formula is obtained:

$$X^{*} = \frac{{\rho_{ij} \cdot d_{{{\text{i.coil}}}} \cdot w_{ij} \cdot Pr_{ij} }}{{\nu_{ij} \cdot \rho_{ij} \cdot \frac{{M_{w} \cdot Cp_{w} }}{{\lambda_{ij} }}}} \cdot \frac{{\pi \cdot d_{{{\text{i.coil}}}} }}{4} \cdot {\Delta }T_{ij}^{*} \cdot \tau_{0}$$

(53)

$${\text{Re}}_{ij} = \frac{{d_{{{\text{i.coil}}}} \cdot w_{ij} }}{{\nu_{ij} }}$$

(54)

Consequently Eq. (53) can be presented through Eq. (55):

$$X^{*} = \frac{{{\text{Re}}_{ij} \cdot {\text{Pr}}_{ij} }}{{\frac{{M_{w} \cdot Cp_{w} }}{{\lambda_{ij} }}}} \cdot \frac{{\pi \cdot d_{{{\text{i.coil}}}} }}{4} \cdot {\Delta }T_{ij}^{*} \cdot \tau_{0}$$

(55)

Finally, Eq. (55) takes the following form:

$$X^{*} = {\text{Re}}_{ij} \cdot {\text{Pr}}_{ij} \cdot \frac{{\pi \cdot d_{{{\text{i.coil}}}} \cdot \lambda_{ij} }}{{4 \cdot M_{11} \cdot Cp_{11} }} \cdot (T_{i}^{*} - T_{j}^{*} ) \cdot \tau_{0}$$

(56)

Derivation of E^* present in Eq. (36):

$$E^{*} = - T_{0}^{*} \cdot \tau_{0}$$

(57)

$$E^{*} = \frac{{K_{{{\text{hst}}}} \cdot A_{{{\text{hst}}}} \cdot \frac{{\mu_{11} }}{{\lambda_{11} }}}}{{M_{11} \cdot {\text{Pr}}_{11} }} \cdot (T_{11}^{*} - T_{0}^{*} ) \cdot \tau_{0}$$

(58)

Overall heat transfer coefficient of the HST (water in the HST–HST wall air outside the HST):

$$K_{{{\text{hst}}}} = \frac{1}{{R_{{{\text{hst}}}} }}$$

(59)

Overall thermal resistance of the HST (water in the HST–HST wall air outside the HST):

$$R_{{{\text{hst}}}} = \frac{1}{{\alpha_{{{\text{air}}}} }} + \frac{{\ln \left( {\frac{{D_{{{\text{ext}}}} }}{{D_{{\text{int}}} }}} \right)}}{{2 \cdot \pi \cdot \lambda_{{{\text{hst}}}} }} + \frac{1}{{\alpha_{11} }}$$

(60)

Nusselt number calculated for the water in the HST:

$${\text{Nu}}_{11} = \alpha_{11} \cdot \frac{{L_{{{\text{hst}}}} }}{{\lambda_{11} }}$$

(61)

Nusselt number calculated for the air outside the HST:

$${\text{Nu}}_{{{\text{air}}}} = \alpha_{{{\text{air}}}} \cdot \frac{{L_{{{\text{hst}}}} }}{{\lambda_{{{\text{air}}}} }}$$

(62)

$$R_{{{\text{hst}}}} = \frac{{1 \cdot \frac{{L_{{{\text{hst}}}} }}{{\lambda_{{{\text{air}}}} }}}}{{\alpha_{{{\text{air}}}} \cdot \frac{{L_{{{\text{hst}}}} }}{{\lambda_{{{\text{air}}}} }}}} + \frac{{\ln \left( {\frac{{D_{{{\text{ext}}}} }}{{D_{{{\text{int}}}} }}} \right)}}{{2 \cdot \pi \cdot \lambda_{{{\text{hst}}}} }} + \frac{{1 \cdot \frac{{L_{{{\text{hst}}}} }}{{\lambda_{11} }}}}{{\alpha_{11} \cdot \frac{{L_{{{\text{hst}}}} }}{{\lambda_{11} }}}}\frac{{ - b \pm \sqrt {b^{2} - 4ac} }}{2a}$$

(63)

$$R_{{{\text{hst}}}} = \frac{{1 \cdot \frac{{L_{{{\text{hst}}}} }}{{\lambda_{11} }}}}{{{\text{Nu}}_{11} }} + \frac{{\ln \left( {\frac{{D_{{{\text{ext}}}} }}{{D_{{\text{int}}} }}} \right)}}{{2 \cdot \pi \cdot \lambda_{{{\text{hst}}}} }} + \frac{{1 \cdot \frac{{L_{{{\text{hst}}}} }}{{\lambda_{{{\text{air}}}} }}}}{{{\text{Nu}}_{{{\text{air}}}} }}$$

(64)

After substituting Eqs. (58)–(64) a final version defining $E^{*}$ is obtained:

$$E^{*} = \frac{{\left( {\frac{{\frac{{L_{{{\text{hst}}}} }}{{\lambda_{11} }}}}{{{\text{Nu}}_{11} }} + \frac{{\ln \left( {\frac{{D_{{{\text{ext}}}} }}{{D_{{\text{int}}} }}} \right)}}{{2 \cdot \pi \cdot \lambda_{{{\text{hst}}}} }} + \frac{{\frac{{L_{{{\text{hst}}}} }}{{\lambda_{{{\text{air}}}} }}}}{{{\text{Nu}}_{{{\text{air}}}} }}} \right)^{ - 1} \cdot A_{{{\text{hst}}}} \cdot \frac{{\mu_{11} }}{{\lambda_{11} }}}}{{M_{11} \cdot \Pr_{11} }} \cdot \left( {T_{11}^{*} - T_{0}^{*} } \right) \cdot \tau_{0}$$

(65)