Keywords

1 Introduction

Global warming is a phenomenon caused by the accumulation of greenhouse gases, such as CO, CO2 and O2; which sufficiently increases the temperature of the Earth to cause an imbalance in ecosystems. The generation of greenhouse gases is mostly a consequence of human activities as power generation, cooling and heating and transport [1]. In Colombia, 7% of residential energy consumption is destined to water heating [2]. Therefore, harnessing clean energy sources for the operation of heating systems with improved efficiency could contribute to the reduction of greenhouse gas emissions. Currently one of the most efficient systems to produce thermal energy is a Direct Expansion Solar Heat Pump Water Heater (DX - SHPWH) which has a huge potential to increase the efficiency of heat pump systems thanks to the utilization of solar thermal energy in the evaporator [3]. In a DX-SHPWH system, a solar collector acts as the evaporator, absorbing the incident solar radiation and increasing the thermal energy of the refrigerant, which heats water in an accumulation tank as it condenses. The study carried out by Valencia et al. [4] reflect the benefits of the DX - SHPWH systems in reducing energy consumption and environmental impact. In this study, results of a dynamic numerical model indicate that a COP above 3.5 can be obtained using renewable energy sources, exceeding the performance of traditional systems.

The study of the relationships between geometric parameters in this type of systems is relevant to improve the performance as Ma et al. [5] reports. In this article, by means a numerical model, the area ratio between the evaporator and the condenser of a DX-SHPWH is analyzed in order to determine the effects on the systems performance in terms of the COP.

2 System Description

A DX - SHPWH is a type of water heater in which water is stored in a tank and heated using mainly solar thermal energy and thermal energy from the environment by means of a heat pump. The system has four main elements: a compressor, a helical condenser submerged in a water storage tank, a solar collector that acts as an evaporator and an expansion device. A refrigerant circulates cyclically absorbing and transferring heat into the system as it changes phase. The evaporation process occurs directly inside the solar collector, hence, is a direct expansion system.

As shown Fig. 1, at point 1, the refrigerant in vapor phase enters the compressor, where its temperature and pressure are increased to enter point 2 as superheated vapor. Then the refrigerant enters the condenser and changes to liquid phase while transferring heat to the water [6]. From point 3, the liquid flows through the expansion valve to reach point 4 with a lower pressure and temperature. Finally, the refrigerant enters the evaporator, where it absorbs thermal energy from the solar radiation and the surrounding air.

Fig. 1.
figure 1

Thermodynamic cycle of DX - SHPWH scheme.

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Equation (1) states and energy balance for the cooling system.

$${\dot{Q}}_{e}+{\dot{W}}_{c}-{\dot{Q}}_{c}=0$$
(1)

The steady-state system COP is the relation between the useful heat transfer from the condenser and the power consumption, as expressed in Eq. 2.

$${COP}_{HP}=\frac{{\dot{Q}}_{c}}{{\dot{W}}_{c}} $$
(2)

Heat transfer rates in the evaporator and condenser \({\dot{Q}}_{e} \, {\dot{Q}}_{c}\) are indicated in Fig. 1, where \({\dot{\text{m}}}_{\text{r}}\) is the refrigerant mass flowrate and \(\text{h}\) is the refrigerant enthalpy in each point of the cycle, The compressor power \({\dot{W}}_{c}\) is calculated according a characteristic polynomial equation for the compressor selected in this work, Tecumseh Masterflux SIERRA02-0434Y3 in terms of compressor speed (n),condensation temperature (Tc) and evaporation temperature (Te). The equation is presented in the manufacturer’s data sheet [7] and is solved as reported in ref. [4].

3 Mathematical Model

To estimate the different thermodynamic and heat transfer processes that occur within the system, a pseudostationary mathematical model is used that guarantees energy conservation for the components of the cooling cycle assuming steady state, and updates the temperature of the water in the tank considering a time dependent energy balance. The model was developed with the following assumptions: i) the compressor operates at constant speed, ii) negligible pressure drops in the heat exchangers, iii) constant pressure in the water tank and iv) uniform temperature in the water tank.

3.1 Mathematical Model of Collector/Evaporator

In the present study, the evaporator is designed to take advantage of solar radiation (\({\upalpha }{I}_{T})\) and convective heat transfer from the environment \({(U}_{LC}\left({T}_{e} - {T}_{\infty }\right))\). The energy absorbed by the evaporator is estimated by Eq. (3)

$${\dot{Q}}_{e}={\text{F}}^{\prime}{A}_{e}[{\upalpha }{I}_{T} - {U}_{LC}\left({T}_{e} - {T}_{\infty }\right)]$$
(3)

The efficiency factor \({\text{F}}^{\prime}\) can be evaluated using the Hottel Whilliar Bliss model proposed in [8] and shown in Eq. (4), Where F is the fin efficiency, D is the outer diameter of the tube, and W is the pitch of the tube, Cb is bond conductance. F can be calculated from traditional model for fins.

$${\text{F}}^{\prime}=\frac{1}{{U}_{L}}{\left\{W \left[\frac{1}{{U}_{L} \left(\left(W-D\right)F+D\right)}+\frac{1}{{C}_{b}}\right]\right\}}^{-1}$$
(4)

3.2 Mathematical Model of Helicoidal Condenser and Water Tank

To simulate the water temperature (\({\text{T}}_{\text{w}}\)) evolution during heating, the transient energy balance given by Eq. (5) is numerically solved. \({m}_{w}\), \({C}_{pw}\), \({U}_{Lt}\), \({A}_{t}\), \({T}_{\infty }\) are, respectively, water mass in the tank, water specific heat, overall heat loss coefficient for the tank, tank superficial area and ambient temperature.

$${C}_{pw}\frac{{dT}_{w}}{dt}{m}_{w}={\dot{Q}}_{c} -{U}_{Lt}{A}_{t}\left({T}_{w} - {T}_{\infty }\right)$$
(5)

The heat transferred by the helical condenser \({\dot{Q}}_{c}\) is estimated using the logarithmic mean temperature difference \(\varDelta {T}_{lm}\) as described in Eq. (6):

$${\dot{Q}}_{c}= {U}_{C}{A}_{C}\varDelta {T}_{lm}$$
(6)

The global heat transfer coefficient of the condenser, \({U}_{C}\), is calculated assuming that the conduction resistance through the walls of the condenser pipe is negligible, as indicated in Eq. (7)

$${U}_{c}={\left(\frac{{A}_{c,o} }{{h}_{c,r}{A}_{c,i} }+\frac{1}{{h}_{w}}\right) }^{-1}$$
(7)

The inside (\({h}_{c,r}\)) and outside (\({h}_{w}\)) convection coefficients are calculated via empirical correlations for internal forced condensation and natural external flow in circular pipes. \({A}_{c,o}\) and \({A}_{c,i}\) are the condenser internal and external areas, respectively.

4 Numerical Algorithm

Computational implementation is carried out in Python language, according to the numerical scheme presented in [4]. In this work, the algorithm is modified by neglecting the variable speed in the compressor to avoid possible noise in the geometrical analysis.

After varying the area ratio (AR), between the solar collector/evaporator area and the condenser area for different geometrical configurations, results are obtained and analyzed. The numerical algorithm runs f until the water in the tank reaches the target temperature of 60 ℃.

5 Results and Discussions

The numerical model is used to predict the performance of a proposed DX-SHPWH basic design with the characteristics listed in Table 1. The DX-SHPWH operates under local average weather conditions, using the typical meteorological year (TMY) of Medellín [9]. The initial AR is 15.9 where the evaporator area is 1,4 m2 and the condenser area is 0.087 m2. Starting from the initial areas, a series of variations above and below are made, in equal percentages, for both the evaporator and condenser areas. The variations are made every 20% of the previous value, until obtaining 5 values above and 5 below the initial areas, and the resulting combinations are simulated. Finally, COP for each AR is obtained, and the results are compared.

Table 1. DX-SHPWH basic design for study case.
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Figure 2 shows the influence of AR on the system average COP. Each line represents a specific value of the evaporator area while the condenser area is varied. When the AR increases above 10, a gradual decrease of the COP is observed. The large imbalance between the heat transfer areas increases the superheating at the compressor inlet and causes poor heat transfer at the condenser, which is compensated with a higher condensing pressure. Hence, the power consumption also increases. It can also be observed that the COP is very sensitive to changes in the evaporator area, obtaining higher COP values when the evaporator is larger, because more energy can be absorbed from the environment, which improves the efficiency of the system.

Fig. 2.
figure 2

Impact of AR on system COP for different evaporator areas. Own source.

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Figure 3 shows in detail the influence of the condenser area on the average COP of the system. Initially there is a favorable trend in the COP when the condenser area is increased. However, a stabilization point is reached where the COP approaches a constant value with increasing condenser area. This shows that the condenser area has a smaller impact on system performance in comparison to the evaporator area. A configuration with large evaporator area and AR below 10 is advisable for good system performance.

Fig. 3.
figure 3

Impact of condenser area on system COP for different evaporator areas. Own source

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6 Conclusions

The proposed study demonstrates the importance of geometric parameters on the performance of. An increase of the area of the solar collector or evaporator tends to improve the performance because the system can more readily absorb heat from the environment. However, under a constant collector area, increasing evaporator/condenser area ratio beyond10 causes a gradual decrease of the COP, because the condensing pressure needs to be increased to compensate for the restricted heat transfer in the condenser. In addition, it is observed that the COP tends to a constant value as the condenser area is increased. Hence, after a threshold value for the condenser area the system is mainly sensitivity to changes in the evaporator area. A system with large evaporator area and an evaporator/condenser area ratio below 10 is recommended for good performance.