Introduction

Increasing the efficiency of energetic processes has been in focus in almost all countries worldwide as modern economies and societies require and use more and more energy. At the same time, problems are accumulating with the use of traditional energy resources, including primarily, expanding detrimental impacts on the environment. The highly significant role of fossil fuels in climate change has been made explicit over the course of the past decade. One tool to firmly restrict this impact is to increase the use of renewable energy resources.

Additionally, to be able to achieve significantly lower impact on the climate, energy consumption has to be reduced to the lowest possible extent [1]. This can be reached by optimizing the energy production and energy use [2].

Taking into account the heating systems in buildings, either in industrial or residential areas, it can be concluded that in the near future substantial changes will occur, meaning that the use of heat pumps, especially compressor-driven heat pumps, will increase considerably [3, 4].

The goal of raising the value of heat pump’s coefficient of performance (COP) asks for being able to define as accurately as possible, the structure of the heat pump and the heat pump’s processes, i.e., to further develop and refine the physical and mathematical models of these devices.

The experimental investigation of any heat pump’s system means a complex problem, mainly due to the financial costs and the large number of variables involved. The use of numerical models can reduce the costs and also facilitate understanding the phenomena related to any problem [5]. Therefore, the presented research deals with a science-based operational design of compressor-driven water-to-water heat pumps. The introduced mathematical model serves as a simulation tool, which enables the heat pump system’s optimization regarding its geometric structure and performance. After a thorough literature review, it can be concluded that current research deals with this topic either partially or by disregarding certain aspects of a stationary water-to-water system when describing its processes [6,7,8,9,10,11,12,13,14,15,16,17,18,19].

In the following Table 1, the results of previous research are presented for the assumed conditions.

Table 1 Literature review for heat pump models
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The result of the presented literature review points toward a conclusion that researchers carried out a deep and thorough analysis of the heat pump systems of the interest. However, no overall research can be found, which would incorporate all aspects of a stationary water-to-water heat pump system. Therefore, this study aims at contributing to the research by filling the gap in literature with the development of an overall mathematical model, which incorporates the thermodynamics, fluid mechanics and heat transfer processes that occur in heat pumps more precisely than before, taking also into consideration the pressure drop along the heat exchangers by using the momentum equation.

Additionally, the proposed mathematical model allows the prediction of the parameters of the state at each and every point of the cycle, i.e., of the heat pump.

The development of the mathematical model is presented in several steps (Fig. 1):

Fig. 1
figure 1

Modeling and simulation steps

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  • Presentation of the system model’s structuring with describing the mathematical equations;

  • Solution of the suggested mathematical model. For the solution purposes, the solving algorithm is presented. Based on the developed algorithm, the corresponding C++ program is developed as well;

  • Validation of the mathematical model by performing a comparison of the simulated results with the measured results, which were presented in the previous research [20].

Mathematical model of the heat pump system

The developed mathematical model’s core is represented by the mathematical equation system, which describes the processes in the heat pump. The presented mathematical equations, i.e., the basic equations of the heat exchangers, and the auxiliary equations of the heat exchangers, the equations describing the compressor and the expansion valve, including the initial and boundary conditions, together with the associated dataset, present the overall mathematical model of the heat pump system.

The developed mathematical model is deterministic, stationary and with distributed parameters, with the following assumptions:

  • Thermodynamic properties of the lubricant, which flows in the system together with the refrigerant, are not taken into consideration.

  • Refrigerant’s flow in the evaporator and the condenser is one dimensional and permanent.

  • Refrigerant’s liquid and vapor phases are in thermodynamic equilibrium.

  • In heat exchangers, only axial flows are taken into account.

The observed heat exchangers are horizontal, counter-cross flow, shell and tube type. In the current case, the refrigerant flows through in finned tube bundle, while primary and secondary fluids flow in the shell across the tube bundle. In the shell side of the heat exchangers, baffles are applied for supporting the tubes and for inducing cross flow over the tubes, resulting in improved heat transfer performance.

Basic equations of the heat exchangers

The basic equations of the heat exchangers consist of the mass conservation differential equation of the refrigerant (Eq. 1), the momentum conservation equation of the refrigerant (Eq. 2), the energy conservation equations of the refrigerant (Eq. 3), the pipe wall energy conservation equation (Eq. 5) and the water energy conservation equation (Eq. 6) [21].

$$\frac{{\partial \left( {\rho \cdot w} \right)}}{\partial z} = 0 \to \rho \cdot w = G = {\text{const}}.$$
(1)
$$\frac{{\partial \left( {\rho \cdot w^{2} } \right)}}{\partial z} + \frac{\partial p}{\partial z} + \frac{{f_{\text{t}} }}{{2d_{\text{i}} }} \cdot w^{2} \cdot \rho = 0.$$
(2)
$$\frac{{\partial \left( {G \cdot \left( {h + w^{2} /2} \right)} \right)}}{\partial z} \pm \dot{q}_{\text{i}} \cdot \frac{U}{{A_{\text{i}} }} = 0.$$
(3)
$$\dot{q}_{\text{i}} = \alpha_{\text{ref}} \cdot \left( {T_{\text{t}} - T_{\text{ref}} } \right).$$
(4)
$$\alpha_{\text{cw}} \cdot d_{\text{o}} \cdot \pi \cdot n \cdot \left( {T_{\text{cw}} - T_{\text{t}} } \right) - \alpha_{\text{ref}} \cdot d_{\text{i}} \cdot \pi \cdot n \cdot \left( {T_{\text{t}} - T_{\text{ref}} } \right) = 0.$$
(5)

for: d o ≅ d i

$$\dot{m}_{\text{cw}} \cdot c_{\text{p,cw}} \cdot \frac{{{\text{d}}T_{\text{cw}} }}{{{\text{d}}z}} - \alpha_{\text{w}} \cdot d_{o} \cdot \pi \cdot n \cdot \left( {\overline{{T_{\text{cw}} }} - \overline{{T_{\text{t}} }} } \right) = 0.$$
(6)

Given the basic equations, several transformations of the basic equations can be performed: based on Eq. 1:

$$w = \frac{G}{\rho } = G \cdot v \to G = {\text{const}}.$$

Change in refrigerant’s velocity is:

$$\frac{{{\text{d}}w}}{{{\text{d}}z}} = G \cdot \frac{{{\text{d}}v}}{{{\text{d}}z}} .$$
(7)

Change in the refrigerant’s specific volume is:

$$\frac{{{\text{d}}v}}{{{\text{d}}z}} = \frac{\partial v}{\partial p} \cdot \frac{{{\text{d}}p}}{{{\text{d}}z}} + \frac{\partial v}{\partial h} \cdot \frac{{{\text{d}}h}}{{{\text{d}}z}},$$

and

$$\frac{{{\text{d}}v}}{{{\text{d}}z}} = v_{\text{p}} \cdot \frac{{{\text{d}}p}}{{{\text{d}}z}} + v_{\text{h}} \cdot \frac{{{\text{d}}h}}{{{\text{d}}z}}.$$
(8)

By substituting Eq. 7 into Eq. 8:

$$\frac{{{\text{d}}w}}{{{\text{d}}z}} = G \cdot \left( {v_{\text{p}} \cdot \frac{{{\text{d}}p}}{{{\text{d}}z}} + v_{\text{h}} \cdot \frac{{{\text{d}}h}}{{{\text{d}}z}}} \right).$$
(9)

Based on Eq. 2:

$$G \cdot \frac{{{\text{d}}w}}{{{\text{d}}z}} + \frac{{{\text{d}}p}}{{{\text{d}}z}} + \frac{{f_{\text{t}} }}{{2d_{\text{i}} }} \cdot w^{2} \cdot \rho = 0.$$

By applying Eq. 9:

$$G^{2} \cdot \left( {v_{\text{p}} \cdot \frac{{{\text{d}}p}}{{{\text{d}}z}} + v_{\text{h}} \frac{{{\text{d}}h}}{{{\text{d}}z}}} \right) + \frac{{{\text{d}}p}}{{{\text{d}}z}} + \frac{{f_{\text{t}} }}{{2{\text{d}}_{\text{i}} }} \cdot w^{2} \cdot \rho = 0.$$

Based on previous transformation, the final form of the momentum conservation equation is:

$$\left( {G^{2} \cdot v_{\text{p}} + 1} \right) \cdot \frac{{{\text{d}}p}}{{{\text{d}}z}} + G^{2} \cdot v_{\text{h}} \cdot \frac{{{\text{d}}h}}{{{\text{d}}z}} + \frac{{f_{\text{t}} }}{{2{\text{d}}_{\text{i}} }} \cdot w^{2} \cdot \rho = 0.$$
(10)

By deriving Eq. 3:

$$G^{2} \cdot w \cdot \frac{{{\text{d}}w}}{{{\text{d}}z}} + G \cdot \frac{{{\text{d}}h}}{{{\text{d}}z}} \pm \dot{q}_{\text{i}} \cdot \frac{U}{{A_{\text{i}} }} = 0.$$

Applying Eq. 1 to the derived Eq. 3 and by applying Eq. 9:

$$G^{2} \cdot v \cdot G \cdot \left( {v_{\text{p}} \cdot \frac{{{\text{d}}p}}{{{\text{d}}z}} + v_{\text{h}} \cdot \frac{{{\text{d}}h}}{{{\text{d}}z}}} \right) + G \cdot \frac{{{\text{d}}h}}{{{\text{d}}z}} \pm \dot{q}_{\text{i}} \cdot \frac{U}{{A_{\text{i}} }} = 0.$$

Based on previous reasoning, the final form of the energy equation is:

$$G^{3} \cdot v \cdot v_{\text{p}} \cdot \frac{dp}{dz} + \left( {G^{3} \cdot v_{\text{h}} \cdot v + G} \right) \cdot \frac{dh}{dz} \pm \dot{q}_{\text{i}} \cdot \frac{U}{{A_{\text{i}} }} = 0.$$
(11)

The mathematical model of processes within the heat exchangers is yielded by arranging the equation system consisting of Eqs. 6, 10 and 11 to the derivatives:

$$\frac{{{\text{d}}T_{\text{cw}} }}{{{\text{d}}z}} = \frac{{\alpha_{\text{cw}} \cdot d_{\text{i}} \cdot \pi \cdot n \cdot \left( {\overline{{T_{\text{cw}} }} - \overline{{T_{\text{t}} }} } \right)}}{{\dot{m}_{\text{cw}} \cdot c_{\text{p,cw}} }}.$$
(12)
$$\frac{{{\text{d}}p}}{{{\text{d}}z}} = - \frac{{\left( {\frac{{f_{\text{t}} }}{{2{\text{d}}_{\text{i}} }} \cdot w^{2} \cdot \rho \cdot G^{2} \cdot v \pm \dot{q}_{\text{i}} \cdot \frac{U}{{A_{\text{i}} }} \cdot G} \right) \cdot v_{\text{h}} + \frac{{f_{\text{t}} }}{{2{\text{d}}_{\text{i}} }} \cdot w^{2} \cdot \rho }}{{G^{2} \cdot v_{\text{p}} + G^{2} \cdot v \cdot v_{\text{h}} + 1}} ,$$
(13)
$$\frac{{{\text{d}}h}}{{{\text{d}}z}} = \frac{{\left( {\frac{{f_{\text{t}} }}{{2d_{\text{i}} }} \cdot w^{2} \cdot \rho \cdot G^{3} \cdot v \pm \dot{q}_{\text{i}} \cdot \frac{U}{{A_{\text{i}} }} \cdot G^{2} } \right) \cdot v_{\text{p}} \pm \dot{q}_{\text{i}} \cdot \frac{U}{{A_{\text{i}} }}}}{{G \cdot \left( {G^{2} \cdot v_{\text{p}} + G^{2} \cdot v \cdot v_{\text{h}} + 1} \right)}}.$$
(14)

Determining the tube’s wall temperature from Eq. (5):

$$T_{\text{p}} = \frac{{\alpha_{\text{cw}} \cdot T_{\text{cw}} + \alpha_{\text{ref}} \cdot T_{\text{ref}} }}{{\alpha_{\text{cw}} + \alpha_{\text{ref}} }}.$$
(15)

The presented equations, supplemented by auxiliary equations, together with the equations of state of the refrigerant and the water, and the equations describing heat exchanger characteristics, comprise the stationary mathematical model of divided characteristics of the evaporator. The systems of Eqs. (12)–(15) are suitable for solution by the Runge–Kutta and Adam–Moulton methods.

Knowing the initial and boundary conditions and by introducing the ∆z steps, the equation system gives the thermodynamic and the fluid flow attributes of the water and the refrigerant.

Auxiliary equations of heat exchangers

To be able to solve the basic equations, auxiliary equations are needed as well. The auxiliary equations serve as a correlation toward the determination of the heat transfer coefficients and the pressure drop. However, the existing correlations, which can be found in literature, show great differences [22]. Therefore, parallel to this research, real measurements were carried out to be able to determine the actual heat transfer coefficients both between the refrigerant and the tube, and between the water and the heat exchangers [23, 24]. The results are presented in Table 2.

Table 2 Heat transfer coefficient correlations
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In the Dittus–Boelter (Eq. 16) correlation, the exponent of Pr should n = 0.4 for heating and n = 0.3 for cooling, so the correlation is different when it is used in the evaporator and condenser.

The pressure drop distribution in the heat exchangers is also determined based on laboratory measurements [26]. The results are presented in Table 3.

Table 3 Pressure drop correlations
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Description of the piston compressor

The compressor as a system component can be described by the enthalpy change specified by the compression ratio and the internal efficiency of the compressor. The output demand of the compressor is this:

$$W = \dot{m}_{\text{ref}} \cdot \left( {h_{\text{con}} - h_{\text{eva}} } \right) \cdot \frac{1}{\eta } .$$
(22)

Description of the expansion valve

The expansion valve, i.e., the proportioning valve is intended to supply appropriate quantities of refrigerant to the evaporator in function of overheating. Isenthalpic throttling is assumed in operation. The mass flux of the refrigerant flowing through the expansion valve is:

$$G = C \cdot \sqrt {\left( {p_{\text{con}} - p_{\text{eva}} } \right) \cdot \rho_{{{\text{ref}}.}} }$$
(23)

Thermodynamics equations of state of the refrigerant

The thermodynamic and transport properties of the refrigerant are determined by the P–R state equation [27]:

$$p = \frac{R \cdot T}{v - b} - \frac{a\left( T \right)}{{v \cdot \left( {v + b} \right) + b \cdot \left( {v - b} \right)}} ,$$
(24)

with: \(a\left( T \right) = 0.45724 \cdot \frac{{R^{2} \cdot T_{\text{c}}^{2} }}{{p_{\text{c}} }} \cdot \left[ {1 + \kappa \cdot \left( {1 - \sqrt {\frac{T}{{T_{\text{c}} }}} } \right)} \right]^{2} ,\)

$$\kappa = 0.37464 + 1.54226 \cdot \omega - 0.26992 \cdot \omega^{2} ,$$

and \(b = 0.07780 \cdot \frac{{R \cdot T_{\text{c}} }}{{p_{\text{c}} }},\)

where a and b are energy and size parameters, respectively. R is the universal gas constant. ω = 0.3268 is the acentric factor. Subscript c indicates the critical value.

Solving the mathematical model of the heat pump system

By solving the aforementioned mathematical model, the state changes in the refrigerant and the water can be followed. Considering that the physical model is a distributed parameter model, solving the mathematical model allows for the most detailed monitoring of the thermodynamic and fluid flow attributes in the evaporator and the condenser.

The algorithm for solving the system of equations which defines the heat pump system

The basis of the mathematical modeling, besides the equations, the initial conditions, and the associated data set, comprises of the algorithm of the solution, which also has to be connected and evaluated, since the solving algorithm specifies the accuracy of the solution, as well as the applicability of the model. The simplified block diagram of the algorithm is shown in Fig. 2.

Fig. 2
figure 2

Simplified algorithm for solving the mathematical model

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The input data into the simulation is organized in three subgroups:

  • Transport properties (thermodynamic and physical properties of the refrigerant, physical properties of water).

  • Thermodynamic parameters (water mass flow rate, refrigerant mass flow rate, cold water inlet temperature, temperature of the superheated refrigerant vapor).

  • Geometric properties (dimensions of the evaporator and condenser).

The simulation process consists of four sub-processes, which are based on the mathematical models of the compressor, condenser, expansion valve and evaporator. First, the compressor’s mathematical model runs, followed by the condenser’s and expansion valve’s mathematical models. The computation results for the mass flow of the refrigerant in the compressor and the mass flow of the refrigerant in the expansion valve are compared. The temperature of the superheated refrigerant vapor is adjusted until the resulted difference between computation results is smaller than the defined acceptable error. Following this procedure, the evaporator’s mathematical model runs. The computation result output enthalpy from the condenser is compared to the input enthalpy to the evaporator. While the resulted difference between enthalpies is greater than the defined acceptable error, the evaporator’s temperature is adjusted, and the procedure is repeated. Following this, an output is generated, which contains the required data for the analysis of the heat pump system.

Presentation of the numerical method applied for solving the mathematical models of system components

The mathematical model of the heat pump system cannot be solved analytically; therefore, numerical simulation has to be applied. Within the numerical simulation, each variable of the model is discretized. This means that each function of independent variables is represented by discrete numbers in a specific breakdown.

In terms of the numerical solution of differential equations, i.e., the equation system, application of the Runge–Kutta method presents a compromise between simplicity and accuracy. This method has been made more precise by applying the Adams–Moulton predictor–corrector method.

The method essentially involves the first four iterations to be performed using Runge–Kutta method in order to provide an input for the Adams–Moulton method. This latter method can be used for the calculation of solutions starting from the fifth iteration. Adams–Moulton’s method is an iterative method; the desired level of accuracy can be achieved in the given step, usually by performing iteration several times. The number of iterations is n < 10.

The function searched for F(z) = (p(z), h(z), T(z)) is a vector function which each component is a function of the distance z from the starting point of the tube (z = 0). This means that the calculation of the function F actually involves the calculation of three quantities.

The equation F′ = G(p, h, T, z) is to be solved as well.

The Runge–Kutta method is presented as follows:

$$F_{{{\text{k}} + 1}} = F_{\text{k}} + \frac{1}{6} \cdot \left( {k_{1} + 2 \cdot k_{2} + 2 \cdot k_{3} + k_{4} } \right) ,$$
(25)

where:

$$k_{1} = G \left( {p_{\text{k}} , h_{\text{k}} , T_{\text{k}} , z_{\text{k}} } \right),$$
(26)
$$k_{2} = G\left( {p_{\text{k}} + \frac{\Delta }{2} \cdot k_{1} , h_{\text{k}} + \frac{\Delta }{2} \cdot k_{1} , T_{\text{k}} + \frac{\Delta }{2} \cdot k_{1} , z_{\text{k}} + \frac{\Delta }{2}} \right),$$
(27)
$$k_{3} = G\left( {p_{\text{k}} + \frac{\Delta }{2} \cdot k_{2} , h_{\text{k}} + \frac{\Delta }{2} \cdot k_{2} , T_{\text{k}} + \frac{\Delta }{2} \cdot k_{2} , z_{\text{k}} + \frac{\Delta }{2}} \right),$$
(28)
$$k_{4} = G\left( {p_{\text{k}} + \Delta \cdot k_{1} , h_{\text{k}} + \Delta \cdot k_{1} , T_{\text{k}} + \Delta \cdot k_{1} , z_{\text{k}} + \Delta } \right),$$
(29)

where: p 0, h 0, T 0 are the initial values yielded by the model for \(z_{0} = 0,k = 0,1,2,3\). \(\Delta\) is an iteration step. In this case \(\Delta = \frac{z}{j}\), where z is the length of the tube, and j is the number of iteration steps.

The Adams–Moulton predictor–corrector method consists of two steps. First, the value of the equation in demand is calculated using the Adams–Bashforth method in the following step:

$$F_{{{\text{k}} + 1}} = F_{\text{k}} + \frac{\Delta }{24} \cdot \left( {55 \cdot G_{\text{k}} - 59 \cdot G_{{{\text{k}} - 1}} + 37 \quad \cdot G_{{{\text{k}} - 2}} - 9 \cdot G_{{{\text{k}} - 3}} } \right),$$
(30)

where \(G_{\text{k}} = G\left( {p_{\text{k}} , h_{\text{k}} , T_{\text{k}} , z_{\text{k}} } \right)\) and \(k \ge 3\)(“predictor” step).

The result obtained is specified by the Adams–Moulton method, generally by running iteration cycles several times (until achieving the desired accuracy; max. 10 in our case). This is the “corrector” step. Formula of the Adams–Moulton method is:

$$F_{{{\text{k}} + 1}}^{{\text{s}} + 1} = F_{\text{k}}^{\text{s}} + \frac{\Delta }{24} \cdot \left( {9 \cdot G_{{{\text{k}} + 1}}^{\text{s}} \cdot + 19 \cdot G_{\text{k}}^{\text{s}} - 5 \cdot G_{{{\text{k}} - 1}}^{\text{s}} + G_{{{\text{k}} - 2}}^{\text{s}} } \right),$$
(31)

\(s \ge 0, s\)—is the number of steps in the Adams–Moulton method.

The corrector step is canceled in case of \(\left| {F_{\text{k + 1}}^{{{\text{s}} + 1}} - F_{{{\text{k}} + 1}}^{\text{s}} } \right| < \varepsilon\) or otherwise after reaching the maximum number of steps (\(\varepsilon = 10^{ - 12} , \;s \le 10 \;{\text{in our case}} )\). This is followed by the Adams–Bashforth step again by increasing the value of k by one and then by the Adams–Moulton iteration.

Simulation results and discussion

The developed mathematical model is tested by performing a series of numerical simulations to obtain the values for the refrigerant and water regarding their state indicators. The thermodynamics, fluid mechanics and heat transfer processes are examined as functions of changing mass flow of the cooled water (\(\dot{m}_{\text{cw}} = 0.28, 0.42, 0.55\; {\text{and}}\; 0.69 \left[ {{\text{kg s}}^{ - 1} } \right]\)). The used input parameters and initial conditions are presented in Table 4.

Table 4 Input parameters and initial conditions
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The simulation results are presented graphically in the following figures (from Figs. 3–15). Additionally to each figure, a short discussion is given regarding the obtained results.

Fig. 3
figure 3

Prediction of the cooled water temperature distribution in the shell side of the evaporator

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Figure 3 presents the predicted temperature distribution of the cooled water along the shell side of the evaporator. At maximum mass flow levels \(\dot{m}_{\text{cw}} = 0.69 \left[ {{\text{kg s}}^{ - 1} } \right])\), the change in the temperature drop is \(\Delta T = 1.01 \left[ {^\circ {\text{C}}} \right]\). If the mass flow is lowered to 40%, the temperature drop increases to \(\Delta T = 2.45 \left[ {^\circ {\text{C}}} \right]\).

The cooling capacity of the cooled water along the shell side of the evaporator shows a continuous growth. If the cooled water’s mass flow increases, the cooling capacity increases as well (Fig. 4).

Fig. 4
figure 4

Prediction of the cooling capacity of the cooled water

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Along the shell side of the evaporator, the change in the heat transfer coefficient between the cooled water and the tubes is present but slight. At the same time, the average value of the heat transfer coefficient is significantly changing as a function of the cooled water’s mass flow. This is presented in Fig. 5. For higher values of the mass flow, i.e., the higher values of the flow velocity, the heat transfer coefficient show higher values as well, up to 90%.

Fig. 5
figure 5

Prediction of the heat transfer coefficients in the shell side of the evaporator

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In Fig. 6, the cooled water’s pressure drop along the evaporator is presented. The pressure drop is linear. The relation between the pressure drop and mass flow is approximately a quadratic function. However, due to the fact that the flow is not completely turbulent, but in a transition region, instead of \(\Delta p = R \cdot v^{2}\), \(\Delta p = R \cdot v^{1.8}\) is suggested.

Fig. 6
figure 6

Prediction of the pressure drop in the shell side of the evaporator

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Figure 7 shows the evaporation completion at a given evaporator’s length, and the refrigerant’s superheating. In addition, the refrigerant’s pressure drop is presented along the evaporator. The pressure drop is almost linear, slightly convex. If the mass flow is higher, the evaporation pressure increases. Regardless of the mass flow, the pressure drop is always around \(\Delta p = 0.28 \left[ {\text{bar}} \right]\). Due to the pressure drop, the evaporation temperature is not constant. Due to changes in saturation pressure, the saturation temperature changes as well. The temperature change along the evaporation zone is almost \(\Delta T = 2.1 \left[ {^\circ {\text{C}}} \right]\), regardless of the mass flow value. At the superheating phase, the temperature change is \(\Delta T_{\text{sh}} = 4 \left[ {^\circ {\text{C}}} \right]\), which represents an input parameter used in the mathematical model.

Fig. 7
figure 7

Prediction of the refrigerant temperature and pressure in tube side of the evaporator

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In Fig. 8, the change in the refrigerant’s heat transfer coefficient in correlation with the temperature is presented along the evaporator. During the two-phase state, i.e., during phase change, the value of the heat transfer coefficient rises significantly. Maximum values can be observed when annular flow is reached, with the vapor quality around x = 0.8[−]. The total evaporation occurs at 90% of the evaporator’s surface. After that the vapor phase has low values of heat transfer coefficient, which is very disadvantageous from the heat transfer point of view.

Fig. 8
figure 8

Prediction of the refrigerant heat transfer coefficients and refrigerant temperature in tube side of the evaporator

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In Fig. 9, the change in the vapor quality in the evaporator is presented in correlation with the temperature. As already discussed, the total evaporation occurs at 90% of the surface.

Fig. 9
figure 9

Prediction of the vapor quality and refrigerant temperature in the tube side of the evaporator

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In Fig. 10, the refrigerant’s temperature change and pressure drop are presented in the condenser. The refrigerant enters the condenser in a superheated state. At around 30% of the total condenser surface, the saturation temperature is reached. This temperature stays constant during condensation. The pressure drop is lower than in the evaporator. This is due to the fact that here the refrigerant has substantially different thermodynamic parameters than earlier. The Reynolds number and the density have different values.

Fig. 10
figure 10

Prediction of the refrigerant temperature and pressure in tube side of the condenser

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In Fig. 11, the change in the refrigerant’s heat transfer coefficient and the vapor quality at the tube side of the condenser are presented. The refrigerant transfers the heat at superheated state at the first 30% of the surface. After that, the condensation phase begins. During the two-phase state, a substantial change in the value of the heat transfer coefficient occurs, from \(\alpha_{\text{ref}} = 81 \left[ {{\text{W\,m}}^{ - 2}\,{\text{K}}^{ - 1} } \right]\) to \(\alpha_{\text{ref}} = 2512 \left[ {{\text{W\,m}}^{ - 2}\,{\text{K}}^{ - 1} } \right]\). After this change, the value gradually decreases as the liquid phase is more and more present, until the entire vapor is condensed.

Fig. 11
figure 11

Prediction of the refrigerant heat transfer coefficients and the vapor quality in tube side of the condenser

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In Fig. 12, the change in the heated water’s heat transfer coefficient along the condenser is presented. The value of the heat transfer coefficient shows a slight increase. This is due to slight increase of the temperature along the condenser, as well as the Reynolds number.

Fig. 12
figure 12

Prediction of the heat transfer coefficients in the shell side of the condenser

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In Fig. 13, the pressure drop of the heated water along the condenser is presented. The pressure change is approximately linear. Slight differences in linearity can be observed as a function of temperature change and thus changes in Reynolds number.

Fig. 13
figure 13

Prediction of the pressure drop in the shell side of the condenser

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In Fig. 14, the change in the heating capacity in the condenser is presented. The heating capacity shows strong increase when the condensation process begins. After this change, the change in the heating capacity in the condenser gradually decreases as the liquid phase is more and more present.

Fig. 14
figure 14

Prediction of the heating capacity

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In Fig. 15, the temperature change in the heated water along the shell side condenser together with the vapor quality is presented. At the first third of the condenser, the temperature change is slight due to low values of heat transfer coefficient. As the condensation prospers, the temperature changes more rapidly.

Fig. 15
figure 15

Prediction of the vapor quality and the heated water temperature in the shell side of the condenser

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Conclusions

This paper presents a developed mathematical model for a steady-state compressor-driven water-to-water heat pump. The solution of the presented mathematical model allows the determination of several characteristics of the refrigerant in the cycle, and of the water flowing in the shell side of the evaporator and the condenser, regarding their thermodynamics, fluid mechanics and heat transfer processes.

The presented mathematical model is simulated by changing the mass flow of the cooled water \(\dot{m}_{\text{cw}} = 0.28, 0.42, 0.55\,{\text{and}}\,0.69 \left[ {{\text{kg s}}^{ - 1} } \right]\), with a referent mass flow of the refrigerant \(\dot{m}_{\text{ref}} = 0.019 \left[ {{\text{kg s}}^{ - 1} } \right]\) and heated water \(\dot{m}_{\text{hw}} = 0.28 \left[ {{\text{kg s}}^{ - 1} } \right]\). The simulation results show the value distribution of the parameters of interest along the evaporator and the condenser.

The presented mathematical model is validated through a series of 118 tests [20]. The test results show that an average difference between the modeled and experimental results for the coefficient of performance is 1.73%. The accuracy of the modeled results is determined by the accuracy of heat transfer correlations and pressure drops correlations.

This points toward the conclusion that the proposed mathematical model provides accurate results, i.e., that the parameters of the state for the heat pump are defined with high accuracy, which is the main scientific contribution of this research.

As a result of the developed mathematical model, the compressor-driven water-to-water heat pump’s process can be optimized, and the maximum value of COP can be obtained with higher accuracy than before, without the need of building the actual prototype system.