Modeling and Experiments for a CO₂ Ground-Source Heat Pump with Subcritical and Transcritical Operation

Abstract

CO₂-based ground-source heat pumps (GSHPs) have the potential to be very environmentally friendly, since GSHPs operate with high energy efficiency, and CO₂ has no ozone depletion potential (ODP) and a low global warming potential (GWP). We developed a prototype CO₂ liquid-to-air GSHP to investigate its performance potential in residential applications. Further, we developed a detailed model of the system that simulates both cooling and heating operation; the model is the primary focus of this report. The model simulates both subcritical and transcritical operation since the system regularly operates near and above the critical temperature of CO₂ (30.98 °C) during heating and cooling operation. The model considered both the refrigerant-side thermodynamic and transport processes in the cycle, as well as the air-side heat transfer and moisture removal.

We performed cooling tests for the prototype CO₂ GSHP that included those from the International Standards Organization (ISO) 13256-1 standard for liquid-to-air heat pumps, as well as extended tests at additional entering liquid temperatures (ELTs). The model predicted the measurements within 0.5 % to 6.7 % for COP, 1.0 % to 3.6 % for total capacity, and 3.3 % to 4.9 % for sensible capacity. We compared the measured cooling performance to published performance data for a commercially-available R410A GSHP and found that for ELTs below 20 °C, the CO₂ GSHP has a higher cooling COP and total capacity than the R410A GSHP. At the ‘standard’ cooling rating condition (ELT 25 °C), the CO₂ GSHP COP was 4.14 and the R410A GSHP COP was 4.43. At ‘part-load’ conditions (ELT 20 °C) the CO₂ GSHP COP was 4.92 and the R410A GSHP COP was 4.99. In the future, the model can be used to investigate methods to improve the CO₂ GSHP performance to meet or exceed that of the R410A system over a wider range of ELTs; possible studies include replacing the electronic expansion valve (EEV) with an ejector, optimizing the charge, and optimizing the heat exchanger geometry and circuiting.

1. Introduction

Ground-source heat pumps (GSHPs) have attracted increasing interest in both research and applications during the last two decades [1] due to their high energy efficiency. GSHPs typically use hydrofluorocarbons (HFCs), R134a or R410A, as the working fluid; however, these refrigerants are scheduled to be phased down due to concerns over climate change. The Kigali Amendment to Montreal Protocol [2] requires the participating parties to reduce HFC use starting in 2019, with reductions of 80 % to 85 % by the late 2040s. It is therefore necessary to look for low global-warming-potential (GWP) refrigerants for GSHPs. As a natural and inexpensive working fluid with a GWP = 1 and no ozone depletion potential (ODP), CO₂ is a promising candidate.

CO₂ has a low critical point (30.98 °C, 7377 kPa), so CO₂-based air-source heat pumps (ASHPs) often operate in the transcritical mode with the high-side pressure above the critical pressure [3]. The high pressure and low efficiency of transcritical CO₂ cycles impedes wider use in heat pumps [4]. Coupled with a ground heat exchanger (GHX), a CO₂ GSHP system could more often operate entirely below the critical temperature (i.e., a subcritical cycle) during the cooling season due to more temperate ground temperatures (compared to air temperatures) [5]. By minimizing operation in the lower-efficiency transcritical cycle, CO₂ GSHPs have potential to be competitive with GSHPs using ‘traditional’ HFC refrigerants (e.g. R410A).

Recently, there have been a number of theoretical and experimental investigations on CO₂ GSHP systems. Lin et al. [6] tested a CO₂ GSHP in the heating mode under various high-side pressures. The COP reached the highest value of 3.4 when the high-side pressure ranged 9 MPa to 10 MPa. Kim and Chang [7] simulated a CO₂ GSHP with a suction-line heat exchanger (SLHX) under various expansion valve openings, compressor speeds, and superheat. The SLHX improved the COP by 2 % to 6 % in the cooling mode, but slightly reduced the COP in the heating mode. Ma et al. [8] simulated conventional R22 and R134a GSHPs and compared them with two transcritical CO₂ GSHPs with an expansion valve and an expander. The COP of the CO₂ system with the expander was comparable with that of R22 and R134a systems when the CO₂ gas cooler outlet temperature was higher than 55 °C.

Direct-expansion CO₂ GSHPs achieve a higher efficiency than indirect systems because they avoid the temperature lift required for a secondary fluid. Eslami-Nejad et al. [9] simulated a direct-expansion CO₂ GSHP in the heating mode, considered the thermal interaction between GHX U-pipe sections, and assessed the pressure, temperature, and quality variations of the CO₂ along the U-pipe. Austin and Sumathy [10] analyzed a transcritical direct-expansion CO₂ GSHP for domestic water heating. There was an optimal gas cooler size that produced the highest COP, with an improvement of 18 % over the baseline. Eslami-Nejad et al. [11] simulated a direct-expansion CO₂ GSHP for space heating and domestic hot water (DHW) in a cold climate. The CO₂ GSHP offered an annual average COP of 2.8; increasing the total borehole length by 25 % only decreased the annual energy consumption by 6 %. Ghazizade-Ahsaee and Askari [12] used a model to study cycle modification for a direct-expansion CO₂ heat pump, and found an ejector increased COP by 16 % and a thermoelectric subcooling generator increased COP by 16.5 %.

A multi-gas-cooler configuration can increase the performance of transcritical CO₂ GSHP by reducing the gas cooler outlet temperature. Jin et al. [13] proposed an improved CO₂ GSHP system with both an air-cooled gas cooler and a water-cooled gas cooler. By increasing the fraction of the heat rejected by the air-cooled gas cooler, the cooling COP could be increased by 8 % to 20 % compared to a basic CO₂ GSHP. Morshed [14] studied the CO₂ GSHP system with an air-cooled and water-cooled gas cooler and an ejector. The ejector increased the COP by 8 % on average, with high-side pressure varying from 8.1 MPa to 11.5 MPa. Hu et al. [15] suggested using the CO₂ GSHP system with an auxiliary gas cooler to reject a portion of heat to the ambient air to eliminate the soil thermal imbalance. Jin et al. [16] analyzed a similar CO₂ GSHP with a SLHX in warm climates, using the ambient air and ground boreholes as heat sinks in the cooling mode, with only the boreholes as a heat source in the heating mode, to maintain a low soil thermal imbalance. Jin et al. [17] introduced a CO₂ GSHP system using city water, the ground, and the ambient air as heat sinks in the cooling mode, while only the ground as heat source in the heating mode. For the studied building, the COP varied from 3.0 to 5.5, and the annual ground thermal imbalance could be 0 %.

Apart from multi-gas-cooler, hybrid CO₂ GSHPs with multiple heat sources can enhance the system performance. Chargui et al. [18] studied a hybrid CO₂ GSHP with two evaporators: a refrigerant-to-liquid heat exchanger to utilize heat from the surface water, and a refrigerant-to-air heat exchanger for use when the surface-water temperature was low. Numerical results showed that the average heating COP was about 3.8. Kim et al. [19] simulated a hybrid solar-assisted CO₂ GSHP for residential hydronic heating. They parametrically varied the undisturbed ground temperature from 11 °C to 19 °C, and the heating COP correspondingly changed from 2.13 to 2.81. Choi et al. [20] compared this solar-assisted CO₂ GSHP system to a conventional R22 GSHP system. With hot water temperature of 44 °C, indoor design temperature of 22 °C, daily solar radiation of 10 MJ/m², and evaporator inlet temperature of 15 °C, the heating COPs of the R22 and CO₂ GSHPs were 3.15 and 2.24, respectively. Ye et al. [21] designed a solar-assisted CO₂ GSHP with an air-cooled gas cooler and a water-cooled gas cooler to maintain a low ground thermal imbalance [22] for heating-dominated applications. They used the simulation to find the solar collector area that eliminated the thermal imbalance for each city.

In the CO₂ heat pump literature cited above: (1) most studies focused on performance analysis or improvement of transcritical cycles, rarely on both subcritical and transcritical cycles; (2) most of the systems were liquid-to-liquid GSHPs for commercial applications, not liquid-to-air GSHPs as would be typical for residences; and (3) the models captured primarily thermodynamic cycle processes, but did not consider transport effects (i.e. pressure drop and heat transfer), and made many simplifying assumptions about the heat exchangers and compressor. The present work addresses each of these three areas to work towards a more precise evaluation of the potential of CO₂ GSHPs.

We developed a detailed model of a residential CO₂ liquid-to-air GSHP that can simulate both the subcritical and transcritical modes depending on operating conditions. Section 2 describes the prototype CO₂ GSHP that is the basis for the model (the unit implements a basic vapor-compression cycle with a SLHX). Section 3 presents the modeling equations that capture the refrigerant-side thermodynamic and transport processes in the cycle, and the air-side heat transfer and moisture removal. The model simulates the cooling operation of the prototype CO₂ GSHP; it also can simulate the heating mode (which is not currently available with the prototype). Section 4 describes the test results of the prototype, which included tests stipulated by the International Standards Organization (ISO) standard for liquid-to-air heat pumps, 13256-1, and additional tests conducted at ‘extended’ entering liquid temperatures (ELTs) to characterize the performance over a wider range of conditions. Section 4 also shows the model predictions of these tests. Section 5 compares the CO₂ GSHP performance with that of a commercially available R410A GSHP, and Section 6 presents the main conclusions.

2. Prototype residential liquid-to-air CO₂ GSHP

The prototype residential liquid-to-air CO₂ GSHP implements a basic vapor-compression cycle with a suction-line heat exchanger, and has a nominal cooling capacity of 7 kW (2 tons). We did not include a reversing valve in the prototype, so it only operates in the cooling mode. The system provides cooling via the air handler, and rejects energy to the heat transfer fluid (water or antifreeze) that circulates through the GHX.

The system (Fig. 1) consists of a semi-hermetic reciprocating compressor (Fig. 2(a), Table 1), a finned-tube evaporator (A-frame, tubes have a rifled inner surface, fins have sine-wave enhancement in the flow direction; Fig. 2(c), Fig. 3 and Table 2), two plate heat exchangers (PHXs, chevron enhanced surface, Fig. 2(b), Table 3), an electronic expansion valve (EEV) and a GHX pump. Each evaporator slab has 4 tube circuits, where each circuit begins at the bottom and ends at the top (Fig. 2(c)). Each tube circuit comprises 16 sequential tubes in the transverse direction, with one skip to an alternate row near the middle of the slab; the row skip balances the heat transfer in the refrigerant circuits since the tubes nearest the air inlet have a higher refrigerant-air temperature difference and therefore greater heat transfer. The volumes of the connecting tubes and auxiliary components (Table 4) were inventoried for future studies that will track and optimize the refrigerant charge. The unit components were attached to an aluminum frame (Fig. 1). A commercially-available air-handling unit (AHU) was modified by installing the finned-tube evaporator (Fig. 1, Fig. 2(c)). The EEV was brazed to the evaporator inlet. An electronic superheat controller was installed inside the AHU close to the fan control board. The refrigeration components were connected using CuFe2P alloy tubes (Fe 2.10 % to 2.60 %, Zn 0.05 % to 0.20 %, P 0.015 % to 0.15 %, Pb ≤ 0.03 %, Cu = balance), rated for 120 bar, and variable-compression-ratio (VCR) fittings. Pressure transducers were attached to the refrigeration tubes using VCR connections. Thermocouples were soldered to the tube surfaces.

Fig 1.

Assembly of the CO₂ GSHP

Fig. 2.

Main components of the CO₂ GSHP

Table 1.

Specifications of the semi-hermetic reciprocating compressor

Parameter	Value
No. cylinders, Ncyl	2
Bore diameter, Db (mm)	22
Stroke, Ls (mm)	22
Displacement @ 50Hz excitation (m3/h)	1.46
Speed @ 50Hz excitation (SC50, rev/min)	1450
Suction valve (mm)	10
Discharge valve (mm)	14
Oil charge (kg)	1.3
Net weight (kg)	73

Fig. 3:

Geometry of the A-frame sine-wave finned-tube heat exchanger

Table 2.

Specifications of the A-frame wavy finned-tube heat exchanger

Parameter	Value
Number of slabs	2
Number of tube circuits per slab	4
Number of columns (# of tubes in a row), Ncol	16
Number of rows, Nrow	4
Tube length (mm), Lt (mm)	457
Tube material	CuFe2P alloy
Tube outside diameter, Do (mm)	5.0
Tube inside diameter (fin root diameter), Di (mm)	4.59
Tube inner surface	rifled
Tube rifling fin height (mm)	0.076
Transverse tube pitch, Pt (mm)	19
Longitudinal tube pitch, Pl (mm)	11
Fin material	aluminum
Fin pitch, Pf (mm)	1.59
Fin length, Lf = Nrow·Pl (mm)	44
Fin width, Lfw = Ncol ·Pt (mm)	304
Fin thickness, δf (mm)	0.14
Waffle height, Pd (mm)	0.87
Half-wave length of the sine wave, Xf (mm)	1.6

Table 3.

Specifications of the PHXs

Parameter	Small PHX (SLHX)	Large PHX (condenser/gas cooler)
Number of plates, Np	10	76
Number of channels – liquid side	5	38
Number of channels – vapor/refrigerant side	4	37
Plate length (mm)	377	377
Fluid flow plate length, Lp (mm)	311	311
Plate width, wp (mm)	119.5	119.5
Enlargement factor, fe	1.1	1.1
Fluid flow plate area, APHX = fe· (Np −2)·Lp·wp (m2)	0.327	3.025
Plate thickness, δp (mm) * [24]	0.4	0.4
Mean channel spacing (flow passage height), PPHX (mm) * [24]	2	2
Hydraulic diameter, Dh = 2PPHX (mm)	4	4
Tube connection port diameter (mm)	27	27
Surface enhancement	chevron**	chevron**

^*The plate thickness and channel spacing were taken from [24] as representative values

^**The details of the chevron enhancement are not known.

Table 4.

Dimensions of the connecting tubes and auxiliary components

	Description	Length/height (mm)	Outer diameter (mm)	Thickness (mm)	Internal vol. (cm3)
tubes	Compressor to oil separator	777.9	12.8	0.4	87.98
	Oil separator to condenser	863.6	12.8	0.4	97.67
	Condenser to SLHX	469.9	12.8	0.4	53.14
	SLHX to EEV	2044.7	9.5	0.4	121.55
	Flow meter bypass	822.3	9.5	0.4	48.88
	EEV to evaporator	342.9	9.5	0.4	20.38
	Evaporator to SLHX	1609.7	12.8	0.4	182.05
	SLHX to accumulator	444.5	12.8	0.4	50.27
	Accumulator to compressor	1104.9	12.8	0.4	124.96
auxiliary	Oil separator	120.0	73.0	0.4a	491.30
	Accumulator	250.0	76.1	-	800b

^aEstimated (value not available from the manufacturer)

^bFrom manufacturer’s datasheet

The smaller PHX acts as a SLHX to reduce the enthalpy at evaporator inlet, which increases the refrigeration capacity per unit mass flow. However, the SLHX effectively reduces the mass flow by further superheating the refrigerant exiting the evaporator, which reduces the suction density. The tests and modeling here are in part to determine if the SLHX provides a net benefit to the cycle. The larger PHX functions as a condenser when the ELT is low, < ~27 °C for this prototype, and the CO₂ can reject heat under its critical pressure (7377 kPa [23]) through condensation. When the ELT is high the CO₂ rejects heat above its critical pressure so the larger PHX operates as a gas cooler. The entering liquid temperature (ELT) can range from below to above the critical temperature, depending on the climate zone, time of year, GHX performance characteristics, and GHSP run time. As a result, the CO₂ GSHP operation may change between a subcritical and transcritical cycles.

3. Model of the CO₂ GSHP with subcritical and transcritical operations

The model simulates the cooling operation of the prototype CO₂ GSHP, as well as the heating mode that is not currently available with the prototype. The cooling (Fig. 4(a)) and heating (Fig. 4(b)) configurations of the SLHX cycle are shown as separate systems, however, in a single system both configurations are achieved using two reversing valves (not shown). In the heating mode (Fig. 4(b)), the finned-tube heat exchanger operates as either a condenser or gas cooler. The CO₂ GSHP can operate in the heating mode as a subcritical cycle if the return-air temperature is low and the finned-tube heat exchanger is sufficiency large. Otherwise the heat rejection pressure elevates above the critical pressure, and the GSHP runs in a transcritical cycle. The smaller PHX still acts as a SLHX, while the larger PHX operates as an evaporator.

Fig. 4.

Schematic of the CO₂ GSHP for cooling and heating

The model was divided into four parts based on the operating mode: subcritical cooling (Section 3.1), transcritical cooling (Section 3.2), subcritical heating (Section 3.3) and transcritical heating (Section 3.4). Separate modeling equations are needed for each of the operating modes since the refrigerant heat transfer and pressure drop mechanisms in heat exchangers change when their role changes between evaporator and condenser (or gas cooler) and when the system operation changes between subcritical to transcritical. Additionally, the compressor exhibits significantly different efficiencies under subcritical and transcritical operation.

3.1. GSHP model for subcritical cooling operation

3.1.1. Compressor model

The compressor model was based on data collected using the test facility described in Section 4. These results were compared with manufacturer’s performance data [25], where the efficiencies were inferred using the capacity and power input under various operating conditions. The volumetric efficiency and total compressor efficiency were computed and plotted vs. pressure ratio (Fig. 5 (a) and (b)).

Fig. 5.

Compressor efficiencies and heat loss ratio under subcritical and transcritical conditions

The volumetric efficiency, the ratio of suction volumetric flow (V_suc) to compressor displacement (V_d) [26], was correlated to the pressure ratio within the subcritical range for data collected in the test facility (Eq. (1)). Results for the manufacturer’s data are also shown for comparison Eq. (1a):

$${\eta }_v=\frac{V_{suc}}{V_d}=1.1362-0.1869\left(\frac{p_{dis}}{p_{suc}}\right)$$ (1)

$${\eta }_v=\frac{V_{suc}}{V_d}=1.02118-0.094928\left(\frac{p_{dis}}{p_{suc}}\right)$$ (1a)

where $p_{dis}$ and $p_{suc}$ are respectively the compressor discharge and suction pressures, kPa. Note the measurements cover a narrow pressure ratio range, about 1 to 2, compared to the manufacturer’s data with pressure ratios ranging 1 to 5. The compressor discharge pressure was computed as $p_{dis}=p_{r,c,in}+\mathrm{\Delta }{p}_{dis}$, where $p_{r,c,in}$ is the condenser (or gas cooler) inlet pressure, kPa (Fig. 8), and $\mathrm{\Delta }{p}_{dis}$ is the pressure drop in the discharge line and oil separator (P 1200 – P 1201, Fig. 22) taken from measurement, kPa. Similarly, the suction pressure was computed as $p_{suc}=p_{r,g,out}-\mathrm{\Delta }{p}_{suc}$, where $p_{r,g,out}$ is the SLHX vapor outlet pressure, kPa (Fig. 10), and $\mathrm{\Delta }{p}_{suc}$ is the suction line pressure drop (P 1207 – P 1216, Fig. 22) taken from measurement, kPa.

Fig. 8.

Modeling concept for the large plate heat exchanger as a condenser

Fig. 22.

Test facility for the CO₂ GSHP

Fig. 10.

Modeling concept for the small plate heat exchanger as a SLHX

The compressor displacement rate was determined by the compressor specification and operating speed:

$$V_d=\frac{\pi }{4}{D}_b{}^2{L}_s{N}_{cyl}\frac{S{C}_{50}}{60\text{s}/\text{min}}\frac{fr}{50\text{Hz}}$$ (2)

where $fr$ is the compressor frequency, Hz, which was 50 Hz for all measurements and model results presented here.

The refrigerant mass flow rate through the compressor was calculated by:

$$m_r=V_{suc}{\rho }_{suc}={\eta }_v{V}_d{\rho }_{suc}$$ (3)

where ${\rho }_{suc}$ is the suction refrigerant density (kg/m³) calculated using the temperature and pressure.

The total compressor efficiency was defined as the ratio of power required for isentropic compression to actual power input to the electric motor [26]:

$${\eta }_{com}=\frac{W_i}{W_{com}}=\frac{m_r(i_{dis,i}-i_{suc})}{W_{com}}$$ (4)

where $W_i$ is the power required for isentropic compression, kW; $W_{com}$ is the electrical power input to the compressor motor, kW; $m_r$ is the refrigerant mass flow rate, kg/s; $i_{dis,i}$ is the discharge enthalpy for isentropic compression, kJ/kg; and $i_{suc}$ is the suction enthalpy, kJ/kg.

The total compressor efficiency for subcritical operation was fit to the measurements presented here (Fig. 5(b) and Eq. (5)), and to the manufacturer’s data Eq. (5a) for comparison:

$${\eta }_{com}=-1.4805+2.3392\left(\frac{p_{dis}}{p_{suc}}\right)-0.6783{\left(\frac{p_{dis}}{p_{suc}}\right)}^2$$ (5)

$${\eta }_{com}=-0.606369+1.69123\frac{p_{dis}}{p_{suc}}-0.853205{\left(\frac{p_{dis}}{p_{suc}}\right)}^2+0.182338{\left(\frac{p_{dis}}{p_{suc}}\right)}^3-0.0142022{\left(\frac{p_{dis}}{p_{suc}}\right)}^4$$ (5a)

The compressor discharge enthalpy, $i_{dis}$, was computed using the heat-balance equation for the compressor:

$$m_r(i_{dis}-i_{suc})=W_{com}(1-{\gamma }_{loss})$$ (6)

where ${\gamma }_{loss}$ is the ‘compressor heat loss ratio’, i.e. the compressor heat lost to ambient divided by the electrical power input to the motor. Note that $\left({1-\gamma }_{loss}\right)$ is equivalent to the product of the compressor mechanical and electrical efficiencies (${\eta }_m{\eta }_e$) defined in the ‘Compressors’ chapter of [26].

Measurements of ${\gamma }_{loss}$ were collected using the test facility presented in Section 4. These data are shown as a function of the pressure ratio (Fig. 5(c)), as well as a function of the temperature difference between the compressor (average of discharge and suction temperatures) and the ambient air (Fig. 5(d)). Finally, ${\gamma }_{loss}$ was fit to functions of the pressure ratio (Eq. (7)) and temperature difference (Eq. (7a)), with the former used in the model.

$${\gamma }_{loss}=-0.4446+0.5592\frac{p_{dis}}{p_{suc}}-0.1279{\left(\frac{p_{dis}}{p_{suc}}\right)}^2$$ (7)

$${\gamma }_{loss}=0.03732+0.00246\left(\frac{t_{dis}+t_{suc}}{2}-t_{amb}\right)$$ (7a)

where t_dis, t_suc and t_amb are respectively the discharge temperature, suction temperature and temperature of ambient air surrounding the compressor, °C.

3.1.2. Evaporator model

The evaporator was divided into two sections distinguished by the refrigerant phase: a two-phase section and a superheated section (Fig. 6). The phase-section transition location, determined by the modeling equations in this Section, can occur anywhere in the circuit (i.e. not necessarily at the return bend as shown in Fig. 6(a)). The phase sections were assumed to extend from the front to the back of the evaporator slab in the longitudinal direction (Fig. 3(b)). This implies the refrigerant reaches a saturated vapor state at the same location in each tube circuit, a reasonable assumption considering that the row shift of the tube circuits in the center of the slab attempts to balance the heat load on each circuit (Section 2, Fig. 2(c)). These two sections had the same inlet-air parameters but different outlet-air parameters because of differing heat transfer. The air exits the section at the temperature depicted by the end of the solid ‘air’ arrows (Fig. 6(b)), and the mixing is depicted by the dashed arrows extending from the solid ones. The airflow rates of these two sections were distributed proportional to the section area (Eq. (50)), and the bulk air properties for the outlet air were calculated as the mass-weighted averages of the two airflows leaving the evaporator (Eqs. (51) and (52)). The refrigerant mass flow was assumed to be equally distributed amongst the tube circuits (2 slabs × 4 circuits/slab = 8 circuits).

Fig. 6.

Modeling concept of the finned-tube heat exchanger as an evaporator

The evaporator total external surface area (A_e) is the sum of the fin area and the tube area:

$$A_e=A_f+A_t$$ (8)

$$A_f=\left[P_l{P}_t\frac{{\int }_0^{X_f}\sqrt{1+{\left[\frac{P_d}{2}\frac{X_f}{\pi }\cos \left(\frac{\pi }{X_f}x\right)\right]}^2}dx}{X_f}-\frac{\pi }{4}{D}_c{}^2\right]N_{row}{N}_{col}\frac{L_t}{P_f}\times 2$$ (9)

$$A_t=\pi D_o\left(L_t-{\delta }_f\frac{L_t}{P_f}\right)N_{row}{N}_{col}$$ (10)

where D_c is the fin collar outside diameter (mm), which was defined as:

$$D_c=D_o+2{\delta }_f$$ (11)

The evaporator inside tube area, $A_i$, was calculated using the fin-root diameter, $D_i$, without adjustment for the microfins. The average tube area, $A_a$, was evaluated for a diameter halfway through the thickness of the tube.

$$A_i=\pi D_i{L}_t{N}_{row}{N}_{col}$$ (12)

$$A_a=\pi \frac{D_i+D_o}{2}{L}_t{N}_{row}{N}_{col}$$ (13)

For the two-phase section of the evaporator, the refrigerant and air energy balances, and the heat-transfer rate equations are:

$$Q_{r,tp}=m_r(i_{r,tp,out}-i_{r,in})$$ (14)

$$Q_{a,tp}=m_{a,tp}(i_{a,in}-i_{a,tp,out})$$ (15)

$$Q_{e,tp}=Q_{r,tp}=Q_{a,tp}=U_{e,tp}{A}_{e,tp}F\frac{\left(t_{a,in}-t_{r,tp,out}\right)-\left(t_{a,tp,out}-t_{r,in}\right)}{\text{ln}\left[\left(t_{a,in}-t_{r,tp,out}\right)/\left(t_{a,tp,out}-t_{r,in}\right)\right]}$$ (16)

where Fig. 6(b) indicates the variables for mass flow rate, temperature, humidity ratio, pressure, and enthalpy. Additionally, $U_{e,tp}$ is the evaporator overall heat-transfer coefficient of two-phase section, kW/(m²·K); $A_{e,tp}$ is the evaporator area of two-phase section, m²; $F$ is the cross-flow temperature difference correction factor where $F=1$ for the two-phase section [27] because the refrigerant temperature is nearly constant.

The overall evaporator heat-transfer coefficient, $U_{e,tp}$, included the thermal resistances of the air-side overall finned-surface efficiency, the air-side condensate film, the tube, and the two-phase refrigerant convective evaporation [28]. Other useful overall heat transfer coefficients included those between the refrigerant and the tube/fin average condensate surface temperature, $U_{e,s,tp}$ (which is a modified version of the overall heat transfer coefficient for a dry finned-tube heat exchanger on p. 247 of [29]), and between the refrigerant and the tube surface, $U_{e,t,tp}$.

$$U_{e,tp}={\left[\frac{1}{{\xi }_{tp}{h}_{a,tp}{\eta }_o}+\frac{{\delta }_w}{k_w}+\frac{{\delta }_t}{k_t}\frac{A_{e,tp}}{A_{a,tp}}+\frac{\frac{A_{e,tp}}{A_{i,tp}}}{h_{r,tp}}\right]}^{-1}$$ (17)

$$U_{e,s,tp}={\left[\frac{{\delta }_w}{k_w}+\frac{1}{{\xi }_{tp}{h}_{a,tp}}\frac{1-{\eta }_{f,wet}}{{\eta }_{f,wet}+A_t/A_f}+\frac{{\delta }_t}{k_t}\frac{A_{e,tp}}{A_{a,tp}}+\frac{\frac{A_{e,tp}}{A_{i,tp}}}{h_{r,tp}}\right]}^{-1}$$ (17a)

$$U_{e,t,tp}={\left[\frac{{\delta }_t}{k_t}\frac{A_{e,tp}}{A_{a,tp}}+\frac{\frac{A_{e,tp}}{A_{i,tp}}}{h_{r,tp}}\right]}^{-1}$$ (17b)

$${\xi }_{tp}=\frac{i_{a,in}-i_{a,tp,out}}{c_{p,a}(t_{a,in}-t_{a,tp,out})}=1+\frac{i_{lg,w}({\omega }_{a,in}-{\omega }_{a,tp,out})}{c_{p,a}(t_{a,in}-t_{a,tp,out})}$$ (18)

$${\delta }_w=0.005\text{in}\times \frac{25.4\text{mm}}{\text{in}}$$ (19)

where ${\xi }_{tp}$ is the ratio of total heat duty to sensible heat duty (=1 for dry conditions) and captures the additional heat transfer of the condensing water vapor; $h_{a,tp}$ and $h_{r,tp}$ are respectively the air-side and refrigerant-side heat-transfer coefficients, kW/(m²·K); ${\eta }_o$ is the overall air-side finned-surface heat-transfer efficiency, Eq. (25); ${\delta }_t$ and ${\delta }_w$ are respectively the tube thickness of evaporator and the water film thickness of condensate (set to 0.005 in as suggested by p. 260 of [29]), m; $k_t$ and $k_w$ are respectively the tube conductivity and water film conductivity, (0.260 and 0.205) kW/(m·K); $c_{p,a}$ is the specific heat of air, kJ/(kg·K); $i_{lg,w}$ is the vaporization latent heat of water, kJ/kg.

The mass conservation and mass transfer of water on the air side were applied to calculate the humidity ratio (ω):

$$W_{tp}=m_{a,tp}({\omega }_{a,in}-{\omega }_{a,tp,out})$$ (20)

$$W_{tp}=d_{a,tp}{A}_{e,tp}\frac{({\omega }_{a,in}-{\omega }_{s,tp})-({\omega }_{a,tp,out-}{\omega }_{s,tp})}{\text{ln}\left[({\omega }_{a,in}-{\omega }_{s,tp})/({\omega }_{a,tp,out-}{\omega }_{s,tp})\right]}$$ (21)

$$\text{Le}=\frac{h_{a,tp}}{c_{p,a}{d}_{a,tp}}\approx 1$$ (22)

where $W_{tp}$ is the water transfer rate between the bulk air (subscript ‘a’) and the saturated air on the finned-tube surface (subscript ‘s’), kg/s; ${\omega }_{s,tp}$ is the humidity ratio for saturated air at the tube surface (water film surface) temperature in the two-phase section, kg/kg dry air; Le is the Lewis number; $d_{a,tp}$ is the mass transfer coefficient (kg/(m²·s)), which can be estimated using the rule of Le ≈ 1 for air at atmospheric pressure [29]. The air-side condensate surface temperature used to determine ${\omega }_{s,tp}$ was calculated using an average heat flux: $t_{w,s}=\left(t_{r,tp,in}+t_{r,tp,out}\right)/2+Q_{r,tp}/\left(U_{e,s,tp}{A}_{e,tp}\right)$.

The air-side heat-transfer coefficient for the sine-wave finned-tube heat exchanger under dry conditions was calculated using Youn and Kim’s model [30]:

$$j=\frac{\text{Nu}}{\left({\text{Re}\text{Pr}}^{\frac{1}{3}}\right)}$$ (23)

$$j=0.196R{e}_{D_c}^{-0.318}{\left(\frac{P_f}{D_c}\right)}^{0.309}{\left(\frac{X_f}{P_d}\right)}^{0.136}$$ (24)

where j, Nu, Re and Pr are respectively the Colburn factor, Nusselt number, Reynold’s number and Prandtl number; $P_f$ is the fin pitch, m; $P_l$ and $P_t$ are respectively longitudinal and transverse tube pitch, m; $P_d$ is the fin sine-wave waffle height, m; $X_f$ is one half of the sine-wave length, m. The air-side heat transfer coefficient was then $h_a=\mathrm{Nu}{k}_a/D_c$, kW/(m²·K).

The model assumed no condensation occurs if the average external tube surface temperature, $t_t=\left(t_{r,tp,in}+t_{r,tp,out}\right)/2+Q_{r,tp}/\left(U_{e,t,tp}{A}_{e,tp}\right)$, was above the air-inlet dewpoint. For these dry conditions, the overall air-side heat-transfer surface efficiency was calculated as [31]:

$${\eta }_{o,dry}=1-\frac{A_f}{A_e}(1-{\eta }_{f,dry})$$ (25)

where ${\eta }_{f,dry}$ is the fin efficiency, which was calculated by the Schmidt approximation [32]:

$${\eta }_{f,dry}=\frac{\text{tanh}(Y_{m,dry}{R}_o\varphi )}{Y_m{R}_o\varphi }$$ (26)

$$Y_{m,dry}=\sqrt{\frac{2h_a}{k_f{\delta }_f}}$$ (27)

$$\varphi =\left(\frac{R_{eq}}{R_o}-1\right)\left[1+0.35\ln \left(\frac{R_{eq}}{R_o}\right)\right]$$ (28)

$$\frac{R_{eq}}{R_o}=1.27\frac{X_M}{R_o}{\left(\frac{X_L}{X_M}-0.3\right)}^{1/2}$$ (29)

where $k_f$ is the fin conductivity, kW/(m·K); $R_{eq}$ is equivalent radius, m; $R_o$ is tube outside radius, m; $Y_{m,dry}$ is the fin parameter, $X_M$ and $X_L$ are geometric parameters, $X_M=P_t/2$, $X_L=\sqrt{{(P_t/2)}^2+{(P_l)}^2}/2$.

Condensation was assumed to occur if the average external tube surface temperature was below the air-inlet dewpoint. For these wet conditions, $h_a$ was multiplied by the ${\xi }_{tp}$ coefficient obtained from Eq. (18) per [28], so Eq. (25) and Eq. (27) became:

$${\eta }_{o,wet}=1-\frac{A_f}{A_e}\left(1-{\eta }_{f,wet}\right)$$ (30)

$$Y_{m,wet}=\sqrt{\frac{2{\xi }_{tp}{h}_a}{k_f{\delta }_f}}$$ (31)

and ${\eta }_{f,wet}$ was calculated using Eq. (26) with $Y_{m,wet}$ instead of $Y_{m,dry}$. For all these simulations, the air side of the two-phase section was always wet, and the air side of the superheat section was always dry.

The refrigerant-side evaporative heat-transfer coefficient was calculated using the Thome model [33] based on a flow pattern map [34] with the updated formulation [35]. The map considers mass flux and vapor quality to categorize the flow regime as either: annular, intermittent, stratified, slug, stratified wavy, slug-stratified wavy, bubbly, mist, or dryout. These correlations (and all others for the finned-tube heat exchanger in this paper) do not account for the microfins (Table 2) or the small amount of lubricant that circulates with the refrigerant, which would respectively enhance and degrade the heat transfer coefficient.

The flow-boiling heat-transfer coefficient (annular, intermittent, stratified, slug, stratified wavy, slug-stratified wavy, and bubbly regimes) in a horizontal tube was [35]:

$$h_{tp}=\frac{{\theta }_{dry}{h}_g-(2\pi -{\theta }_{dry})h_{wet}}{2\pi }$$ (32)

where ${\theta }_{dry}$ is the dry angle, defining the flow structures and the ratio of the tube perimeter in contact with liquid and vapor, with the detailed calculation method found in [35]; $h_g$ is the vapor- phase heat transfer, as calculated by Dittus-Boelter model [36]:

$$h_g=0.023{\text{Re}}_g^{0.8}{\text{Pr}}_g^{0.4}\frac{k_g}{D_{eq}}$$ (33)

where $D_{eq}$ is the equivalent inside diameter, which is same as inside diameter for circular tubes, Re_g is the vapor-phase Reynold’s number, Pr_g is the vapor-phase Prandtl number, and k_g is the vapor thermal conductivity, W/(m·K).

The heat-transfer coefficient on the wet perimeter ($h_{wet}$) was calculated by:

$$h_{wet}={(S\cdot h_{nb}^3+h_{cb}^3)}^{1/3}$$ (34)

where S is the nucleate-boiling heat-transfer suppression factor, with the detailed calculation method found in [35]; $h_{nb}$ is he nucleate boiling heat-transfer coefficient, as calculated by Cooper model [37]:

$$h_{nb}=131{\left(\frac{p}{p_{crit}}\right)}^{-0.0063}{\left[-{\text{log}}_{10}\left(\frac{p}{p_{crit}}\right)\right]}^{-0.55}{M}^{-0.5}{q}^{0.58}$$ (35)

where p is the pressure and $p_{crit}$ is the critical pressure, kPa; M is the molecular weight, g/mol; q is the heat flux, W/m².$h_{cb}$ is the convective boiling heat-transfer coefficient, as calculated by [35]:

$$h_{cb}=0.0133{\left[\frac{4G(1-x){\delta }_l}{{\mu }_l(1-\epsilon )}\right]}^{0.69}{\text{Pr}}_l^{0.4}\frac{k_l}{{\delta }_l}$$ (36)

where G is the mass flux, kg/(m²·s); x is the vapor quality; ${\mu }_l$ is the liquid dynamic viscosity, (N·s)/m²; ε is the vapor void fraction; ${\mathrm{P}\mathrm{r}}_l$ is the liquid Prandtl number; $k_l$ is the liquid thermal conductivity, W/(m·K).

The heat-transfer coefficient in mist regime was calculated by [38]:

$$h_m=2\times {10}^{-8}{\left\{\frac{G{D}_{eq}}{{\mu }_g}\left[x+\frac{{\rho }_g}{{\rho }_l}(1-x)\right]\right\}}^{1.97}{\text{Pr}}_g^{1.06}{\left\{1-0.1{\left[\left(\frac{{\rho }_g}{{\rho }_l}-1\right)(1-x)\right]}^{0.4}\right\}}^{-1.83}\frac{k_g}{D_{eq}}$$ (37)

The heat transfer in the dryout regime was calculated by a linear interpolation [39]:

$$h_{dryout}=h_{tp}(x_{di})-\frac{x-x_{di}}{x_{de}-x_{di}}[h_{tp}(x_{di})-h_m(x_{de})]$$ (38)

where $x_{di}$ and $x_{de}$ are the dryout inception quality and dryout completion quality, respectively. Note the expressions for $x_{di}$ and $x_{de}$ have an error in the reference publication (confirmed by author), the corrected expressions are presented in Appendix A.1.

The refrigerant two-phase friction factors and associated pressure drop for different flow regimes inside the tubes were also calculated using the Thome model, with the detailed methods found in [34]. The heat-transfer coefficient and pressure drop were evaluated at four vapor qualities (0.2, 0.4, 0.6, and 0.8), and the average values were used to calculate the heat-transfer coefficient and pressure drop for the entire heat exchanger. This averaging method was applied for all two-phase heat transfer and pressure drop calculations in this study.

For superheat section, the modeling process was similar to the two-phase section. The air-side heat-transfer coefficient was calculated using Eq. (18) to (31), and the energy balance and heat transfer rate equations had the same form as Eq. (14) to (17). The calculation of moisture removal by the superheat section using Eq. (20) to (22) is a simplification since the equations imply the entire tube and fin surfaces were below the dewpoint (a condition required for dehumidification), but actually a portion would have temperature above the dewpoint. We neglected this effect since the superheat section was small. We also did not include a cross-flow correction factor for Eq. (21) as one does not exist, to our knowledge (for these simulations it was a moot point, since the superheat section always operated dry). The cross-flow correction factor (F) for the heat transfer, Eq. (16), was calculated using the Roetzel expression [40] for a single-pass cross-flow heat exchanger with both fluids unmixed. Values for F for the simulations in Section 4.2 were about 0.9.

$$Y_p=\frac{t_{a,in}-t_{a,sup,out}}{t_{a,in}-t_{r,tp,out}}$$ (39)

$$Y_q=\frac{t_{r,out}-t_{r,tp,out}}{t_{a,in}-t_{r,tp,out}}$$ (40)

$$R=\frac{Y_p}{Y_p}$$ (41)

$$r_{counter}=\frac{Y_p-Y_q}{\ln \left(\frac{1-Y_q}{1-Y_p}\right)}$$ (42)

$$F=1-\sum _{i=1}^m\sum _{k=1}^n{a}_{i,k}{(1-r_{counter})}^k\sin (2i{\tan }^{-1}R)$$ (43)

Where the coefficients $a_{i,k}$ are given in Roetzel et al. [40]. The single-phase heat-transfer and friction factor were calculated using the Gnielinski model [41]:

$$Nu=\frac{\left(\frac{f}{8}\right)(Re-1000)Pr}{1.07+12.7{\left(\frac{f}{8}\right)}^{\frac{1}{2}}(P{r}^{\frac{2}{3}}-1)}$$ (44)

$$f={[0.79\ln (Re)-1.64]}^{-2}$$ (45)

The air exiting the evaporator was the mixture of outlet air from the two-phase and superheat sections, and the fraction of airflow in the two-phase/superheat sections were proportional to the heat exchanger area. Therefore, the overall evaporator model was expressed as:

$$Q_e=Q_{r,tp}+Q_{r,sup}$$ (46)

$$Q_e=Q_{a,tp}+Q_{a,sup}$$ (47)

$$m_a=m_{a,tp}+m_{a,sup}$$ (48)

$$A_e=A_{e,tp}+A_{e,sup}$$ (49)

$$\frac{m_a}{A_e}=\frac{m_{a,tp}}{A_{e,tp}}=\frac{m_{a,sup}}{A_{e,sup}}$$ (50)

$$i_{a,out}=\frac{(m_{a,tp}{i}_{a,tp,out}+m_{a,sup}{i}_{a,sup,out})}{m_a}$$ (51)

$${\omega }_{a,out}=\frac{(m_{a,tp}{\omega }_{a,tp,out}+m_{a,sup}{\omega }_{a,sup,out})}{m_a}$$ (52)

A subprogram for finned-tube evaporator was developed based on Eq. (8) to (52), using the computational flow chart shown in Fig. 7. The state point parameters of the airflow and refrigerant are indicated in Fig. 6(b). Note that many of the equations for the superheat section have the same form as those for the two-phase section, but with appropriate variable substitutions; these equations are indicated with an asterisks in Fig. 7 (e.g. for $Q_{r,sup}$, Eq. (14*), is similar to Eq. (14) but $i_{r,tp,out}$ became $i_{r,out}$, and $i_{r,in}$ became $i_{r,tp,out}$). This notation was used for all computational flow diagrams. The parameters listed as ‘(specified)’ in Fig. 7 (and all other program flowchart figures) are specified according to design parameters or measurements (Table 6). The properties, including temperature (t), pressure (p), and enthalpy (i), were calculated using REFPROP [23]. The properties of moist air were calculated by the ASHRAE method [42]. $t_s$ and $t_w$ are the evaporator surface temperature and water film temperature. ${\Delta p}_{r,sup}^0$ and ${\Delta p}_{r,tp}^0$ denote the refrigerant pressure drops at superheat and two-phase sections, as determined by the guess values for $t_{r,tp,out}$ and $t_{r,in}$; for convergence, these pressure drops must equal the values computed using the correlations (${\Delta p}_{r,sup}$ and ${\Delta p}_{r,tp}$). The secant iteration method [43] was employed to accelerate the convergence speed; an example of the iteration is given for the cycle model in Fig. 12 and Appendix A.2.

Fig. 7. Subprogram flow chart for the finned-tube heat exchanger evaporator.

*Equation has the same form as the one listed, but with appropriate variable substitutions

†Temperature computed using a thermal resistance network with terms from the listed equation

Table 6.

Detailed measured data under subcritical (Standard) and transcritical (ELT-3) conditions

Parameter	Sensor	Subcritical (Standard)	Transcritical (ELT-3)	±Uncert.
Compressor inlet temp. (°C)	TC 1109	24.6	29.5	0.6
Compressor inlet press. (kPa)	P 1216	4525	4609	15
Compressor outlet temp. (°C)	TC 1100	71.6	85.4	0.6
Compressor outlet press. (kPa)	P 1200	7368	8202	20
Compressor power (W)	W 1304	1471	1677	0.2 %
Condenser (gas cooler) inlet temp. (°C)	TC 1101	70.7	84.2	0.6
Condenser (gas cooler) inlet press. (kPa)	P 1201	7313	8156	20
Condenser (gas cooler) outlet temp. (°C)	TC 1102	25.24	30.13	0.6
Condenser (gas cooler) outlet press. (kPa)	P 1202	7321	8161	20
Refrigerant mass flow (g/s)	MF 1400	37.40	35.72	0.25 %
Entering liquid temp. (°C)	RTD 3604	25.01	29.97	0.075
Exiting liquid temp. (°C)	RTD 3602	32.01	36.57	0.075
Entering-exiting liquid diff. press. (kPa)	DP 3318	20.11	22.34	0.1
Liquid mass rate (g/s)	MF 3402	272.3	271.5	0.2 %
Pump power (W)	W 1305	132.2	132.3	0.2 %
SLHX liquid outlet temp. (°C)	TC 1103	20.6	24.0	0.6
SLHX liquid outlet press. (kPa)	P 1203	7315	8150	20
EEV inlet temp. (°C)	TC 1104	20.2	23.3	0.6
EEV inlet press. (kPa)	P 1204	7288	8137	20
EEV outlet temp. (°C)	TC 1105	10.9	11.7	0.6
EEV outlet press. (kPa)	P 1205	4574	4664	15
Evaporator outlet temp. (°C)	TC 1106	15.1	15.8	0.6
Evaporator outlet press. (kPa)	P 1206	4535	4620	15
Entering air dry-bulb temp. (°C)	RTD 3700/3701/3702	26.89/27.06/27.07	26.93/27.34/27.25	0.075
Entering air dew-point temp. (°C)	Dew 3504	14.39	14.38	0.21
Exiting air dry-bulb temp. (°C)	RTD 3703/3704/3705	14.41/14.36/14.35	14.97/14.91/14.90	0.075
Exiting air dew-point temp. (°C)	Dew 3506	12.40	13.01	0.21
GSHP exiting air external static press. (Pa)	DP 3319	59.1	56.9	0.74
Fan power (W)	W 1306	98.4	100.8	0.2 %
Nozzle inlet dry-bulb temp. (°C)	RTD 3706/3707/3607	14.61/14.58/14.63	15.15/15.12/15.17	0.075
Nozzle inlet external static press. (Pa)	DP 3322	43.9	45.1	1
Nozzle inlet-outlet differential press. (Pa)	DP 3320	460.6	451.3	2
SLHX vapor inlet temp. (°C)	TC 1107	15.4	16.1	0.6
SLHX vapor outlet temp. (°C)	TC 1108	24.6	29.5	0.6
SLHX vapor outlet press. (kPa)	P 1207	4534	4617	15

Fig. 12.

Program flow chart for the CO₂ GSHP under subcritical cycle for cooling

3.1.3. Condenser model

The condenser was divided into three phase-dependent sections: superheated, two-phase, and subcooled (Fig. 8(a)). The refrigerant and liquid flow rates of these three sections were the same, and the outlet parameters of one section were the inlet parameters of the subsequent section. The amount of subcooling, ${\mathrm{\Delta }t}_{sub}$, was taken from experimental data (Table 6).

For the two-phase section, the energy balance for the refrigerant and liquid side, as well as the rate equation were:

$$Q_{r,tp}=m_r(i_{r,sup,out}-i_{r,tp,out})$$ (53)

$$Q_{l,tp}=c_{p,l}{m}_l(t_{l,sup,out}-t_{l,tp,out})$$ (54)

$$Q_{r,tp}=Q_{l,tp}=Q_{c,tp}$$ (55)

$$=U_{c,tp}{A}_{c,tp}\frac{(t_{r,sup,out}-t_{l,sup,out})-(t_{r,tp,out-}{t}_{l,tp,out})}{\text{ln}\left[(t_{r,sup,out}-t_{l,sup,out})/(t_{r,tp,out-}{t}_{l,tp,out})\right]}$$

where the variables for mass flow rate, temperature, pressure are indicated in Fig. 8 (b). $c_{p,l}$ is the liquid specific heat, kJ/(kg·K); $A_{c,tp}$ is the area of the condenser two-phase section, m²; $U_{c,tp}$ is the overall heat-transfer coefficient of two-phase section, kW/(m²·K), which was calculated as:

$$U_{c,tp}={\left[\frac{1}{h_{l,tp}}+\frac{{\delta }_p}{k_p}+\frac{1}{h_{r,tp}}\right]}^{-1}$$ (56)

where $h_{l,tp}$ and $h_{r,tp}$ are respectively the liquid-side and refrigerant-side heat-transfer coefficients, kW/(m²·K); ${\delta }_p$ is the plate thickness of condenser, m; $k_p$ is the plate conductivity, kW/(m·K).

The heat-transfer coefficients on both refrigerant side and liquid side for chevron brazed-plate heat exchanger were calculated using the Longo model [44]. For the refrigerant condensation, the heat-transfer coefficient depended on gravity-controlled condensation and forced-convection condensation, according to equivalent Reynolds number:

$$R{e}_{eq}=G\left[(1-x)+x{\left(\frac{{\rho }_l}{{\rho }_g}\right)}^{\frac{1}{2}}\right]\frac{D_h}{{\mu }_l}$$ (57)

where G is the mass flux, kg/(m²·s); x is the thermodynamic vapor quality, kg_g/kg; ${\rho }_l$ and ${\rho }_g$ are respectively liquid and vapor density, kg/m³; $D_h$ is the hydraulic diameter (Table 2), m; ${\mu }_l$ is the liquid viscosity, kg/(m·s).

For vertical, gravity-controlled refrigerant condensation (${Re}_{eq}<1600$), the heat-transfer coefficient was based on [44]:

$$h_{r,tp}=0.943{\left(\frac{k_l^3{\rho }_l^{2}g{i}_{lg}}{{\mu }_l\text{Δ}t{L}_{p,tp}}\right)}^{1/4}$$ (58)

For forced convection refrigerant condensation (${Re}_{eq}>1600$), the heat-transfer coefficient was [44]:

$$h_{r,tp}=1.875\left(\frac{k_l}{D_h}\right)R{e}_{eq}^{0.766}{\mathit{Pr}}_l^{0.333}$$ (59)

where $k_l$ is the liquid conductivity, kW/(m·K); $i_{lg}$ is the condensation latent heat, kJ/(kg); $\mathrm{\Delta }t$ is the temperature difference between the saturated refrigerant and the plate wall, K; $L_{p,tp}$ is the length of the vertical plate in the two-phase section, m;

The two-phase pressure drop in the brazed-plate heat exchanger was calculated by [45]:

$$\text{Δ}{p}_{tp}=2.05\frac{G^2}{2{\rho }_m}$$ (60)

$$\frac{1}{{\rho }_m}=\frac{x}{{\rho }_g}+\frac{1-x}{{\rho }_l}$$ (61)

where ${\rho }_m$ is the quality-weighted average density, kg/m³.

The modeling equations for the superheated and subcooled sections had the same form as those for the two-phase section, Eq. (53) to (56), but used different correlations for the heat transfer and pressure drop. The single-phase heat-transfer coefficient was calculated by the Longo model [46]:

$$h=0.277\left(\frac{k}{D_h}\right){\text{Re}}^{0.766}{\text{Pr}}^{0.333}$$ (62)

and the single-phase friction factor was calculated using the following equation [24]:

$$f=1.18{\text{Re}}^{-0.10}$$ (63)

where Re was based on the hydraulic diameter listed in Table 2. Note that the liquid-side heat transfer and pressure drop were also calculated with Eq. (62) and (63). The phase-dependent sections were combined to form the overall condenser model:

$$Q_c=Q_{r,sup}+Q_{r,tp}+Q_{r,sub}$$ (64)

$$Q_c=Q_{l,sup}+Q_{l,tp}+Q_{l,sub}$$ (65)

$$A_c=A_{c,sup}+A_{c,tp}+A_{c,sub}$$ (66)

A subprogram for PHX condenser was developed using Eq. (53) to (66). The computational flow is shown in Fig. 9.

Fig. 9. Subprogram flow chart for the PHX condenser.

*Equation has the same form as the one listed, but with appropriate variable substitutions

3.1.4. SLHX model

The SLHX model was similar to that of the single-phase section of the condenser, using the same transport equations (Eq. (62) and (63)). The energy balance and heat-transfer equations had the same form as Eq. (53) to (55), e.g. Eq. (53) with variable substitutions for the SLHX was $Q_{r,SLHX}=m_r\left(i_{r,g,out}-i_{r,g,in}\right)$. The component configuration and temperature profile are illustrated in Fig. 10. The subprogram flow chart is detailed in Fig. 11.

Fig. 11. Subprogram flow chart for the PHX SLHX.

*Equation has the same form as the one listed, but with appropriate variable substitutions

3.1.5. EEV model

The EEV was modeled as an isenthalpic throttling device:

$$i_{EEV,out}=i_{EEV,in}$$ (67)

3.1.6. System model

The overall model for the CO₂ GSHP in the subcritical cooling mode combines the component models from Sections 3.1.1 to 3.1.5, where the computational flow is shown in Fig. 12. The program iteratively adjusts $p_{e,out}$ until the $i_{e,in}$ matches $i_{EEV,out}$, and adjusts $p_{c,in}$ until the model-calculated condenser area, $A_c$, equals the actual large plate heat exchanger area $A_{PHX}$ (Table 3). Details of the iteration scheme are shown in Eq. (A.3) to (A.22); the iteration scheme for the other operating modes (heating, transcritical) and component models are not shown here because they follow a similar procedure. The coefficient of performance in the cooling mode (${\mathrm{C}\mathrm{O}\mathrm{P}}_c$) was defined as follows:

$${\text{COP}}_c=\frac{Q_e-W_{fan,int}}{W_{com}+W_{fan,int}+W_{pump,int}}$$ (68)

where $W_{fan,int}$ and $W_{pump,int}$ are the power of fan and pump required to overcome the internal flow resistance in the GSHP, kW.

3.2. GSHP model for transcritical cooling operation

In the transcritical cooling cycle model, the equations governing the evaporator and EEV were the same as those presented for the subcritical cooling model (Section 3.1.2). However, the compressor performance was significantly different and the large PHX operated as a gas cooler (rather than a condenser); models for these components are presented in this section.

3.2.1. Compressor model

The compressor volumetric efficiency and total compressor efficiency in the transcritical range were derived from the Section 4 measurements, Eq. (69) and (70), and the manufacturer’s data are shown for comparison, Eq. (69a) and (70a) (Fig. 5). The volumetric efficiency was fit to:

$${\eta }_v=1.1110-0.1696\left(\frac{p_{dis}}{p_{suc}}\right)$$ (69)

$${\eta }_v=0.95033-0.0756536\left(\frac{p_{dis}}{p_{suc}}\right)$$ (69a)

The total compressor efficiency was fit to:

$${\eta }_{com}=0.5707\left(\frac{p_{dis}}{p_{suc}}\right)-0.1450{\left(\frac{p_{dis}}{p_{suc}}\right)}^2$$ (70)

$${\eta }_{com}=0.47011+0.09649\left(\frac{p_{dis}}{p_{suc}}\right)-0.01673{\left(\frac{p_{dis}}{p_{suc}}\right)}^2$$ (70a)

3.2.2. Gas cooler model

In the gas cooler the refrigerant is the supercritical state throughout, so there was no need to divide the model by phases. However, refrigerant properties are very sensitive to temperature in the supercritical region [5], especially near the critical point. To accurately capture the properties’ influence on the heat transfer, pressure drop and refrigerant charge, the gas cooler (Fig. 13) was divided into 15 equal-temperature change sections (based on a sensitivity study).

Fig. 13.

Modeling concept of the large plate heat exchanger as a gas cooler (N=15)

For the i^th section, the gas cooler energy balance on the refrigerant and liquid sides, as well as the heat transfer rate equation were expressed as:

$$Q_{r,i}=m_r(i_{r,i}-i_{r,i+1})$$ (71)

$$Q_{l,i}=c_{p,l}{m}_l(t_{l,i+1}-t_{l,i})$$ (72)

$$Q_{r,i}=Q_{l,i}=Q_{gc,i}=U_{gc,i}{A}_{gc,i}\frac{(t_{r,i}-t_{l,i})-(t_{r,i+1-}{t}_{l,i+1})}{\text{ln}\left[(t_{r,i}-t_{l,i})/(t_{r,i+1-}{t}_{l,i+1})\right]}$$ (73)

The heat-transfer coefficient and pressure drop on both the refrigerant side and liquid side were calculated using the single-phase correlations from the condenser model in Section 3.1.3 (Eq. (62) and (63)). Then the overall heat-transfer coefficient of the i^th section was calculated as:

$$U_{gc,i}={\left[\frac{1}{h_{w,i}}+\frac{{\delta }_p}{k_p}+\frac{1}{h_{r,i}}\right]}^{-1}$$ (74)

Summing all N sections, the overall gas cooler model was expressed as:

$$Q_{gc}=\sum _1^N{Q}_{r,i}$$ (75)

$$Q_{gc}=\sum _1^N{Q}_{l,i}$$ (76)

$$A_{gc}=\sum _1^N{A}_{gc,i}$$ (77)

A subprogram for PHX gas cooler was developed using Eq. (71) to (77), where the computational flow is shown in Fig. 14.

Fig. 14.

Subprogram flow chart for PHX gas cooler

3.2.3. System model

Figure 15 shows a flow chart for the simulation model of CO₂ GSHP operating in the transcritical cycle for cooling. Most of the processes are similar to those of subcritical cycle; two major exceptions are that the large PHX operates as a gas cooler (instead of a condenser), and the gas cooler inlet pressure, $p_{gc,in}=p_{r,1}$, is specified (Fig. 14) (rather than condenser subcooling, ${\mathrm{\Delta }t}_{sub}$). It would also be reasonable to specify the gas cooler outlet temperature, $t_{gc,out}$, rather than the inlet pressure,$p_{gc,in}$, but that was not done here. In the prototype GSHP, $t_{gc, out}$ or $p_{gc,in}$ can be adjusted by changing the refrigerant charge amount.

Fig. 15.

Program flow chart for CO₂ GSHP under transcritical cycle for cooling

3.3. GSHP model for subcritical heating operation

In a subcritical cycle for heating, the function of the large PHX changes to the evaporator (rather than condenser, for cooling) and the finned-tube heat exchanger becomes the condenser, while the other cycle components remained the same; therefore, this section only discusses the models for the evaporator and condenser in the heating mode.

3.3.1. Evaporator model

The PHX evaporator was divided into two phase-dependent sections; two-phase and superheated, where the entire flows of refrigerant and heat-transfer fluid pass through both sections (Fig. 16).

Fig. 16.

Modeling concept of the large plate heat exchanger as an evaporator

For each section, the energy balance on the refrigerant and liquid sides, as well as the heat transfer rate equation had the same form as Eq. (53) to (55) and Eq. (64) to (66), but with appropriate variable substitutions. The single-phase heat transfer and pressure drop in the PHX evaporator were calculated using the correlations in Eq. (62) and (63). The overall heat-transfer coefficient had the same form as Eq. (56). For the refrigerant evaporation, the heat-transfer coefficient depended on convective boiling and nucleate boiling [46]:

$$h_{cb}=0.122\left(\frac{k_l}{D_h}\right){\text{Re}}_{eq}^{0.8}{\text{Pr}}_l^{1/3}$$ (78)

$$h_{nb}=0.58h_0{\left(\frac{Ro}{0.4}\right)}^{0.1333}Y{\left(\frac{q}{20}\right)}^{0.467}$$ (79)

$$Y=1.2{\left(\frac{p}{p_{crit}}\right)}^{0.27}+\left(2.5+\frac{1}{1-\frac{p}{p_{crit}}}\right)\frac{p}{p_{crit}}$$ (80)

where Re_eq was calculated by Eq. (57); $h_0$ is the reference value of the heat-transfer coefficient, 4.4 kW/(m²·K) (based on measurements with HFC, hydrofluoroolefin (HFO) and hydrocarbon refrigerants in [46], we were unable to find a correlation specifically for CO₂); $Ro$ is the plate mean roughness (set to 0.4 μm based on nominal values for similar heat exchangers; measured value unavailable); q is the heat flux, kW/m².

The average boiling heat-transfer coefficient was the greater of the convective and nucleate boiling [46]:

$$h_{tp}=MAX(h_{cb},h_{nb})$$ (81)

The two-phase pressure drop for evaporation inside PHX was [47]:

$$\text{Δ}{p}_{tp}=1.553\frac{G^2}{2{\rho }_m}$$ (82)

The heat-transfer coefficient and pressure drop were evaluated at four vapor qualities (0.2, 0.4, 0.6, and 0.8), and the average values were used for the entire evaporator. Figure 17 shows a flow chart for the subprogram for the PHX evaporator.

Fig. 17. Subprogram flow chart for PHX as the evaporator.

*Equation has the same form as the one listed, but with appropriate variable substitutions

3.3.2. Condenser model

The finned-tube condenser was divided into three sections according to the refrigerant phase (Fig. 18), where the airflow rates of each section were distributed proportionally to the section area. The air from all sections mixed after the condenser. Just as with the evaporator for cooling, the sections were assumed to extend from the front to the back of the evaporator slab in the longitudinal direction. This implies that each of the refrigerant phase-regime transitions (e.g. superheat to two-phase transition) occur at the same location in each tube circuit. The refrigerant mass flow was assumed to be equally distributed amongst the tube circuits.

Fig. 18.

Modeling concept of the finned-tube heat exchanger as a condenser

In each section, the energy balance on the refrigerant side and air side, and the heat transfer rate equations were similar to Eq. (8) to (10) and Eq. (46) to (52) for the finned-tube evaporator in Section 3.1.2. The overall heat-transfer coefficient was similar to Eq. (11), but an additional section for subcooling was included. The air-side heat-transfer coefficient for the wavy finned-tube condenser was the same as that of finned-tube evaporator (Section 3.1.2) except that the coefficient ξ was taken as 1 since there was no water condensate for a condenser. The main differences were on the refrigerant side. The refrigerant condensation heat transfer was calculated using Kondou’s correlation [48], which was modified from Cavallini’s correlation [49] by considering the tube wall temperature:

$$h_{r,tp}=\{\begin{array}{c}\hfill J_G>J_G^T:h_A=h_{lo\_f}[1+1.128x^{0.8170}{\left(\frac{{\rho }_l}{{\rho }_g}\right)}^{0.3685}{\left(\frac{{\mu }_l}{{\mu }_g}\right)}^{0.2363}{\left(1-\frac{{\mu }_g}{{\mu }_l}\right)}^{2.144}P{r}_{l\_f}^{-0.1}]\hfill \\ \hfill J_G\le J_G^T:\left[h_A{\left(\frac{J_G^T}{J_G}\right)}^{0.8}-h_{strat}\right]\left(\frac{J_G^T}{J_G}\right)+h_{strat}\hfill \end{array}$$ (83)

$$J_G=\frac{xG}{\sqrt{g{D}_i{\rho }_g({\rho }_l-{\rho }_g)}}$$ (84)

$$J_G^T={\left[{\left(\frac{7.5}{4.1X_{tt}^{1.111}}+1\right)}^{-3}+{2.6}^{-3}\right]}^{-1/3}$$ (85)

$$X_{tt}={\left(\frac{{\mu }_l}{{\mu }_g}\right)}^{0.1}{\left(\frac{{\rho }_l}{{\rho }_g}\right)}^{0.5}{\left(\frac{1-x}{x}\right)}^{0.9}$$ (86)

$$h_{lo\_f}=0.023{\left(\frac{G{D}_i}{{\mu }_{l\_f}}\right)}^{0.8}{\mathit{Pr}}_{l\_f}^{0.4}\left(\frac{k_l}{D_i}\right)$$ (87)

$$h_{strat}=0.725{\left[1+0.741{\left(\frac{1-x}{x}\right)}^{0.3321}\right]}^{-1}{\left[\frac{k_{l_{\_}f}^3{\rho }_{l_{\_}f}({\rho }_{l_{\_}f}-{\rho }_g){\mathit{gi}}_{lg}}{{\mu }_{l_{\_}f}{D}_i(t_i-t_{\mathit{sat}})}\right]}^{0.25}+(1-x^{0.087})h_{{\mathit{lo}}_{\_}f}$$ (88)

where the subscript ‘_f’ denotes the liquid properties evaluated at the film temperature (the average of tube inside wall temperature $t_i$ and bulk saturation temperature $t_{sat}$).

The condensation pressure drop was predicted using the Friedel correlation [50], which was selected amongst several correlations as the best by Kondou [48]:

$$\frac{\text{Δ}p}{\text{Δ}L}={\text{Φ}}_{lo}^{2}4\times 0.0791{\text{Re}}_{lo}^{-0.25}\left(\frac{G^2}{2D_i{\rho }_l}\right)$$ (89)

$${\text{Φ}}_{lo}^2=C_{F1}+\frac{3.24C_{F2}}{\left(\frac{G}{g{D}_i{\rho }_m}\right)\left(\frac{G^2{D}_i}{\sigma {\rho }_m})\right)}$$ (90)

$$C_{F1}={(1-x)}^2+x^2\left(\frac{{\rho }_l}{{\rho }_g}\right){\left(\frac{{\mu }_g}{{\mu }_l}\right)}^{0.25}$$ (91)

$$C_{F2}=x^{0.78}{(1-x)}^{0.224}{\left(\frac{{\rho }_l}{{\rho }_g}\right)}^{0.91}{\left(\frac{{\mu }_g}{{\mu }_l}\right)}^{0.19}{\left(1-\frac{{\mu }_g}{{\mu }_l}\right)}^{0.7}$$ (92)

where $\frac{\mathrm{\Delta }p}{\mathrm{\Delta }L}$ is the pressure drop per tube length, kPa/m; $\sigma$ is the surface tension, N/m.

The single-phase section differed from the two-phase section in that the heat transfer was calculated using the Gnielinski model, Eq. (44) [41], while the pressure drop was predicted using the Colburn correlation [51], which was selected amongst several correlations as the best by Kondou’s [48]:

$$\frac{\text{Δ}p}{\text{Δ}L}=4\times 0.046{\mathit{Re}}^{-0.2}\left(\frac{G^2}{2D_i\rho }\right)$$ (93)

A subprogram for finned-tube condenser was developed using the flow chart illustrated in Fig. 19.

Fig. 19. Subprogram flow chart for finned-tube condenser.

*Equation has the same form as the one listed, but with appropriate variable substitutions

3.3.3. System model

The component models were assembled into an overall cycle model for the CO₂ GSHP in the subcritical heating mode. The model used a similar computational flow as presented in Fig. 12, but with different input parameters ($m_l,t_{l,in}$ and ${\mathrm{\Delta }t}_{sup}$ for the PHX evaporator; $m_a,t_{a,in},{\omega }_{a,in}$ and ${\mathrm{\Delta }t}_{sub}$ for the finned-tube condenser). In future work these input values listed as ‘specified’ in Figs. 17 and 18 would be determined based on measurements or nominal values from typical equipment. The COP in the heating mode (${\mathrm{C}\mathrm{O}\mathrm{P}}_h$) was defined as:

$${\text{COP}}_h=\frac{Q_c+W_{fan,int}}{W_{com}+W_{fan,int}+W_{pump,int}}$$ (94)

3.4. GSHP model for transcritical heating operation

3.4.1. Gas cooler model

The finned-tube gas cooler (Fig. 20) was divided into N sections (N=15 in this study) of equal refrigerant temperature change. The model was very similar to the finned-tube condenser in Section 3.3.2, but with more sections. The refrigerant-side heat transfer and pressure drop were calculated using Eq. (44) and (93), respectively. The flowchart in Fig. 21 shows the subprogram for the finned-tube gas cooler.

Fig. 20.

Modeling concept of the finned-tube heat exchanger as a gas cooler

Fig. 21. Subprogram flow chart for finned-tube gas cooler.

*Equation has the same form as the one listed, but with appropriate variable substitutions

3.4.2. System model

The component models were combined to form the overall cycle model of CO₂ GSHP under transcritical cycle for heating. The computational flow was similar to the one shown in Fig. 15, with differences in the input parameters ($m_l,t_{l,in}$ and ${\mathrm{\Delta }t}_{sup}$ for PHX evaporator; $m_a,t_{a,in},{\omega }_{a,in}$ and $t_{gc,out}$ for finned-tube gas cooler).

4. Experimental validation of the CO₂ GSHP models

4.1. Measurements of the CO₂ GSHP

To validate the models, we performed cooling tests of the CO₂ GSHP prototype in an environmental chamber with controlled dry-bulb and dew-point temperatures (Fig. 22). The test rig regulated temperature of the GHX liquid (antifreeze liquid: water/ethanol/isopropanol 70/25/5 % by mass [52]) to the targeted ELT using a chiller and a heater. The airflow rate was measured using a nozzle.

We measured the GSHP performance in the cooling mode at test conditions prescribed by ISO 13256-1 [53] (Table 5). We also tested at ‘Extended ELT’ conditions to capture the performance over the entire range of typical ELTs for GSHPs in the cooling mode. Table 6 lists the measurements and instrument uncertainty for two representative data sets: (1) a subcritical condition (‘Standard’ rating test) and (2) a transcritical condition (‘ELT-3’). A more comprehensive listing of test data is in [52].

Table 5.

Target test conditions for the GSHP in the cooling mode [53]

Parameters	Basic tests				Extended ELT tests
	Standard	Part-load	Maximum	Minimum	ELT-1	ELT-2	ELT-3	ELT-4	ELT-5
Return air dry bulb [C]	27	27	32	21	27
Return air dewpoint [C]	14.7	14.7	19.2	11.0	14.7
{Return air wet bulb [C]}	{19}	{19}	{23}	{15}	{19}
Airflow rate [L/s]	307.6				307.6
ELT [C]	25	20	39 {40} a	10	10	15	30	35	37 {40} b
Liquid flow rate [L/s]	0.3476				0.3476
Compressor frequency [Hz]	50				50

^aELT for ‘Maximum’ Cooling test is 39 °C, rather than 40 °C, to keep pressure in pressure transducer range (10,000 kPa)

^bELT for ‘ELT-5’ Cooling test is 37 °C, rather than 40 °C, to keep pressure in pressure transducer range (10,000 kPa)

The airflow rate through the nozzle was calculated by Eq. (95) [54].

$$V_a=C_d{A}_{nozzle}\sqrt{2\text{Δ}p{v}^{\prime }}$$ (95)

$$v^{\prime }=\frac{v}{1+\omega }$$ (96)

where $C_d$ is the nozzle discharge coefficient; $\mathrm{\Delta }p$ is the static pressure difference across nozzle, Pa; $v^{\prime }$ is the specific volume of air at the nozzle, m³/kg (kg of moist air); $v$ is the specific volume of air at the nozzle, m³/kg (kg of dry air); $\omega$ is the humidity ratio at the nozzle, kg/kg (kg of water / kg of dry air).

The nozzles have a throat-to-diameter ratio of 0.6, so the nozzle discharge coefficient was calculated as [54]:

$$C_d=0.9986-\frac{7.006}{\sqrt{\text{Re}}}+\frac{134.6}{\text{Re}}$$ (97)

$$\text{Re}=\frac{D_{\text{nozzle}}}{\mu v^{\prime }}{C}_d\sqrt{2\text{Δ}p{v}^{\prime }}$$ (98)

where $D_{\mathrm{n}\mathrm{o}\mathrm{z}\mathrm{z}\mathrm{l}\mathrm{e}}$ is the nozzle throat diameter, m; $\mu$ is the dynamic air viscosity, kg/(m·s).

Since the fan was an integral part of the GSHP, only the portion of the fan power required to overcome the internal resistance was included in the effective power input to the GSHP. The fan power adjustment is:

$$W_{fan,adj}=\frac{V_{\text{fan}}\text{Δ}{p}_{ext}}{\eta }$$ (99)

where $V_{\mathrm{f}\mathrm{a}\mathrm{n}}$ is the fan flow rate, L/s; $\mathrm{\Delta }{p}_{ext}$ is the external static pressure difference (DP 3319 in Fig. 22), Pa; η is 0.3×10³ by convention. The corrected fan power was then computed by subtracting $W_{fan,adj}$ from the fan power measurement (W 1306 in Fig. 22).

The GHX liquid pump was an integral part of the GSHP, so only the portion of the pump power required to overcome the internal resistance was included in the effective power input to the GSHP. The pump power adjustment, was calculated as:

$$W_{pump,adj}=\frac{V_{\text{pump}}\text{Δ}{p}_{ext}}{\eta }$$ (100)

where $V_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}}$ is the pump flow rate of liquid, L/s. The corrected pump power was then calculated by subtracting $W_{pump,adj}$ from the measured pump power (W 1305 in Fig. 22)

The sensible cooling capacity measured by the air-side instruments was increased by the fan adjustment per ISO 13256-1 [53]:

$$Q_{cooling,sens}=\frac{V_a}{v}{c}_{p,a,out}\left(t_{a,in}-t_{a,out}\right)+W_{fan,adj}$$ (101)

The latent capacity was unaffected by the fan adjustment, since the fan inputs only heat and no moisture:

$$Q_{cooling,lat}=\frac{V_a}{v}{i}_{lg,w}\left({\omega }_{a,in}-{\omega }_{a,out}\right)$$ (102)

where ${\omega }_{a,in}$ and ${\omega }_{a,out}$ are the inlet- and outlet-air humidity ratios, kg/kg, and $i_{lg}$ is the latent heat of vaporization of water at 15 °C, 2470 kJ/kg [53]. The total capacity was the sum of the latent and sensible values:

$$Q_{cooling,total}=Q_{cooling,sens}+Q_{cooling,lat}$$ (103)

Note that this calculation neglects the condensate and uses a fixed air specific heat; the resulting capacity is about 0.5 % less than that calculated on the refrigerant side.

The sensible heat ratio was:

$$\text{SHR}=\frac{Q_{cooling,sens}}{Q_{cooling,total}}$$ (104)

The total power input to the GSHP unit was corrected according to:

$$W_{total}=W_{com}+(W_{pump}-W_{pump,adj})+(W_{fan}-W_{fan,adj})$$ (105)

Finally, the cooling COP was defined as:

$${\text{COP}}_c=\frac{Q_{cooling,total}}{W_{total}}$$ (106)

The COP, total capacity, and sensible capacity were measured with nominal uncertainty (k=2, 95 % confidence interval) of ±4 %, ±275 W, and ±150 W, respectively. The uncertainties were calculated by propagating the measurement uncertainties (Table 3), through Eq. (95) to (106).

4.2. Comparison between simulation and experiment

The cooling-mode models described in Section 3.1 and 3.2 were used to predict the cooling COP and capacity of the CO₂ GSHP at the conditions listed in Table 5. Experimental measurements of adjusted pump and fan power were included in the model capacity and COP to make them comparable to the experimental results. The pressure-enthalpy diagrams for a subcritical cycle under the ‘Standard’ test condition and a transcritical cycle under the ‘ELT-3’ test condition are compared in Fig. 23. The thermodynamic process of both the subcritical and transcritical cycles were well predicted by the simulation.

Fig. 23.

Comparison of pressure-enthalpy diagrams for measurements and simulations

Figure 24 compares the COP under various conditions. The simulations agree well with the measurements, with deviations of 0.5 % to 6.7 %. The simulated results are generally within the error bars of the measurements. Fig. 25 compares the total and sensible capacities for the various test conditions. The total capacities agree within the range of 1.0 % to 3.6 %, while the sensible capacities agree within 3.3 % to 4.9 %. The COP deviation was primarily caused by the total capacity deviation, where the experimental total capacity value was likely 2 % to 4 % too low as indicated by the energy balance with the refrigerant in the evaporator [52]. The remaining deviation in COP was largely attributed to the data deviations from the compressor total efficiency curve fit (Fig. 5(b)). The ‘minimum’ test had the largest COP deviation because the experimentally measured ${\eta }_{com}$ value was a significant outlier from the curve fit (Fig. 5(b)).

Fig. 24.

Comparison of measured and simulated COPs

Fig. 25.

Comparison of measured and simulated capacities

The benefits of this model include the demonstrated high accuracy, which gives confidence to use it for future cycle and component design. The inclusion of physics-based semi-empirical correlations enables realistic simulation of: (1) the heat exchangers without simplified assumptions of average temperature difference or change of saturation temperature, (2) with alternate heat exchanger designs that leverage the high heat transfer and low pressure drop of CO₂, and (3) system performance at off-design conditions. By including moisture removal, the validated air-side evaporator model enables design to achieve desired SHR over a range of psychometric conditions and airflow. The limitations of this model include complexity and computation time. A 3.5 GHz computer requires ≈ 3.5 h to run a simulation, though that can be reduced to ≈ 10 s using property data computed using interpolation from pre-computed tables, rather than REFPROP. Further, the model cannot account for non-uniform air or refrigerant flow, as they are assumed to be uniform. The model does not account for frost growth on the evaporator and this phenomenon should be added if the model is used to simulate conditions with refrigerant temperature < 0 °C. Lastly, the heating mode of the model has not been verified.

5. Comparison with commercially-available R410A GSHP

The COP, capacity, and SHR of the CO₂ GSHP were compared to manufacturer’s data [55] for a R410A-based GSHP (Fig. 26 to 28). The R410A system had similar capacity and was commercially available at the entry-level price point (i.e., relatively low cost). The basic information of the two GSHP is compared in Table 7. The comparison is not meant to be technically rigorous, but rather just to provide a rough measure of how the CO₂ system compares with GSHPs currently in the marketplace (a more rigorous comparison would include heat exchangers with equal area, a variety of representative R410A compressors, and an identical fan and pump). Note the CO₂ GSHP is significantly larger and requires more refrigerant because of the A-frame tube-fin refrigerant-to-air heat exchanger. The CO₂ system also required a large cabinet to facilitate constructing the prototype and integrating the instruments.

Fig. 26.

Comparison of COPs between CO₂ GSHP and R410A GSHP

Fig. 28.

Comparison of SHRs between CO₂ GSHP and R410A GSHP

Table 7.

Basic information for the CO₂ GSHP and R410A GSHP

Parameter	CO2 GSHP	R410A GSHP
Capacity (kW) @ ELT 25 °C	6.7	7.0
Cooling COP (kW) @ ELT 25 °C	4.14	4.43
Size (mm×mm×mm)	570 x 670 x 1000	570 x 1020 x 1350
Refrigerant charge (kg)	3.04	1.08
Air heat exchanger type	A-frame tube-fin	flat-frame plate-fin

The R410A system had a slightly higher capacity and COP at the ‘standard’ test condition. At the ‘part-load’ condition the COP and total capacity were very similar, where the R410A values were within the uncertainty bars for the CO₂ GSHP measurements. At lower ELTs (‘ELT-1,2’) the CO₂ GSHP had a higher COP and total capacity than the R410A system; at higher ELTs (‘ELT-3,4,5’) the R410A system had higher values. The R410A unit had a lower sensible capacity and correspondingly a lower SHR across the entire ELT range and was therefore better at removing moisture from the air. If the moisture removal by the CO₂ GSHP was insufficient, the SHR could be reduced by lowering airflow, though this would lower the evaporator saturation temperature and subsequently the COP (per the Carnot efficiency).

These comparisons indicate that the CO₂ GSHP is not necessarily less efficient than the R410A GSHP, though CO₂ is commonly regarded to be less efficient than the conventional refrigerants applied in air conditioners. The study shows that CO₂ GSHP performs better than R410A GSHP with ELTs below 20 °C.

6. Conclusions

This paper showed a detailed model of a residential CO₂ liquid-to-air GSHP with capacity ≈ 7 kW. For cooling, the system generally operates subcritical mode for ELT ≤ 25 °and transcritical mode for ELT 25 °C. Most previous studies for CO₂ heat pumps focused on transcritical cycles, so the present model can contribute to advancing designs of CO₂ GSHPs that operate at the border of subcritical and transcritical cycles. Further, the model considers the refrigerant-side thermodynamic and transport processes, enabling accurate simulation of actual heat exchanger performance rather than relying on simple assumptions such as assumed average temperature difference and saturation temperature drop. Finally, the model includes the air-side heat transfer and moisture removal that are required for designing liquid-to-air systems.

A prototype CO₂ liquid-to-air GSHP was tested per ISO 13256-1, and the data was used to validate the model. The model accurately predicted the measurements within 0.5 % to 6.7 % for COP, 1.0 % to 3.6 % for total capacity, and 3.3 % to 4.9 % for sensible capacity. The high accuracy of the model gives confidence to use it for future cycle and component design.

The CO₂ GSHP had a higher cooling COP than a commercially-available R410A GSHP for ELTs below 20 °C (e.g. COP: 6.00 vs. 5.48 @ ELT 15 °C), where the CO₂ system operated entirely in a subcritical cycle. However, above 20 °C the CO₂ GSHP had lower cooling efficiency (e.g. COP: 4.14 vs. 4.43 @ ELT 25 °C), especially at higher temperatures where the system operated in a transcritical cycle. The system COPs were similar at part-load conditions with ELT 20 °C, the CO₂ GSHP COP was 4.92 and the R410A GSHP COP was 4.99. Future efforts should focus on increasing the CO₂ GSHP efficiency for ELT > 20 °C. Also, an annual simulation could show the fraction of time the GSHP operates at each ELT in various climates.

In the future the model will be used to explore design parameters for the cycle and GHX, different cycle configurations (e.g., one with an ejector or expander rather than an EEV), heating mode design, and comparisons with other refrigerants. Further, the model could be modified to inventory refrigerant charge, aiming to optimize efficiency by regulating the amount of circulating CO₂. This work provides a reference for advancement in using natural refrigerants.

Fig. 27.

Comparison of capacities between CO₂ GSHP and R410A GSHP

Nomenclature

A

area, m²

$c_p$

constant-pressure specific heat, kJ/(kg·K)

d

mass transfer coefficient, kg/(m²·s)

D

diameter, m

Error

relative residual error

f

friction factor

f_e

enlargement factor

fr

frequency, Hz

F

heat exchanger cross-flow correction factor

Fr_g

vapor Froude number $\left[G^2/\left({\rho }_g^{2}gD\right)\right]$

$g$

gravitational acceleration, 9.81 m/s²

G

mass flux, kg/(m²·s)

h

heat-transfer coefficient, kW/(m²·K)

i

enthalpy, kJ/kg

j

Colburn factor, $\text{Nu}/(\text{Re}{\text{Pr}}^{\frac{1}{3}})$

J_G

dimensionless vapor velocity

k

thermal conductivity, kW/(m·K)

L

length/width/depth, m

Le

Lewis number, $h/(c_{p}d)$

m

mass flow rate, kg/s

M

molecular weight, g/mol

N

number of (compressor cylinders, tube rows, heat exchanger plates, sections)

Nu

Nusselt number, $hD/k$

p

pressure, kPa

P

pitch, spacing, m

P_d

waffle height of the sine wave, mm

Pr

Prandtl number, $c_p\mu /k$

Pr_g

vapor Prandtl number, $c_{p,g}{\mu }_g/k_g$

Pr_l

liquid Prandtl number, $c_{p,l}{\mu }_l/k_l$

q

heat flux, W/m²

Q

energy transfer rate, capacity, kW

r

dimensionless heat exchanger parameter

R

radius, m

Re

Reynold’s number, $\rho uD/\mu$

Re_g

vapor-phase Reynolds number, $GxD/\left({\mu }_g\epsilon \right)$

Res

residual

Ro

Roughness, mm

SC₅₀

compressor speed with electrical power at 50 Hz, RPM (rev/min)

S

nucleate boiling heat transfer suppression factor

t

temperature, °C

u

velocity, m/s

U

overall heat-transfer coefficient, kW/(m²·K)

v

specific volume of air, m³/kg (kg of dry air)

v’

specific volume of air, m³/kg (kg of moist air)

V

volumetric flow rate, m³/s

w

width, m

W

water condensation rate per width of fin, kg/(m·s), electric power input, kW

We_g

vapor Weber number $\left[G^{2}D/\left({\rho }_g\sigma \right)\right]$

x

thermodynamic vapor quality, kg_g/kg, distance in the longitudinal direction, m

X

half-wave length of the fin sine wave, m, tube-fin geometric parameter, m, Martinelli parameter

Y

non-dimensional temperature used for cross-flow heat exchangers, non-dimensional nucleate-boiling reduced pressure factor

Y_m

fin parameter

Greek symbols

γ

fraction of electric input to compressor lost as heat to ambient air

δ

thickness, m

Δp

pressure drop computed by correlation

Δp⁰

pressure drop computed with guess value of pressure

Δp/ΔL

pressure drop per unit tube length, kPa/m

Δt

temperature difference

ε

vapor void fraction

η

efficiency

θ

angle, rad

μ

dynamic viscosity, kg/(m·s)

ζ

ratio of total capacity to sensible capacity

ρ

density, kg/m³

σ

surface tension, N/m

ω

humidity ratio, kg/kg dry air

ϕ

dimensionless fin parameter

Φ²

two-phase pressure drop multiplier from Friedel correlation [51]

Subscript and superscript

a

air

A

ΔT independent flow regime from [50]

adj

corrective adjustment to fan power, pump power, or capacity for external airflow resistance

amb

ambient

ao

air-side heat transfer coefficient, not corrected for fin surface effectiveness

b

bore

c

condenser, collar

cb

convective boiling

col

number of tube columns (in the transverse direction)

cold

cold or cold-side

com

compressor

crit

critical

cyl

cylinder

d

displacement, nozzle discharge coefficient

de

dryout completion

di

dryout inception

dis

discharge

e

evaporator

ext

external

eq

equivalent

f

fin, film

_f

properties evaluated at the film temperature (average of wall & bulk fluid)

fw

fin width

g

vapor phase

gc

gas cooler

h

hydraulic

i

isentropic, inside, section number, tube inside wall temperature

int

internal

k

iteration index

l

liquid, longitudinal

L

fin geometric parameter

LB

lower bound

lg

liquid-gas phase change (i.e. latent heat of vaporization)

lo

liquid only

m

mist regime, average

M

fin geometric parameter

n

iteration index

N

number of sections in numerical heat exchanger

nb

nucleate boiling

o

outside

out

outlet

p

plate, cross-flow heat exchanger non-dimensional temperature difference

q

cross-flow heat exchanger non-dimensional temperature difference

r

refrigerant

row

number of tube rows (in the longitudinal direction)

s

stroke, surface

sat

saturated state

strat

stratified flow regime

sub

degrees of subcooling, or heat exchanger section with subcooled refrigerant

suc

suction

sup

degrees of superheat, or heat exchanger section with superheated refrigerant

t

tube, transverse

T

transition

total

total capacity including latent and sensible capacities

tp

two-phase, or heat exchanger section with two-phase refrigerant

tt

turbulent-turbulent Martinelli parameter

UB

upper bound

v

volumetric

w

water

wet

portion of tube wetted by liquid refrigerant

0

reference heat transfer coefficient

1,2,…

numerical heat exchanger index

Abbreviations

AHU

air-handling unit

CI

confidence interval (95 % for all data in this paper)

COP

coefficient of performance

DHW

domestic hot water

ELT

GSHP entering liquid temperature (i.e. borehole outlet temperature)

EEV

electronic expansion valve

GHX

ground heat exchanger

GSHP

ground-source heat pump

GWP

global warming potential

HCFC

hydrochlorofluorocarbon

HFC

hydrofluorocarbon

ISO

International Standards Organization

SLHX

suction-line heat exchanger

PHX

plate heat exchanger

VCR

tube fitting with variable compression ratio (using metal gaskets)

Appendix

A.1. Corrected expressions for dryout inception and completion

The expressions for dryout inception and completion have an error in the reference [39] publication, the corrected equations are:

$$x_{di}=0.58\text{exp}\left[0.52-0.235W{e}_g^{0.17}{\mathit{Fr}}_g^{0.37}{\left(\frac{{\rho }_g}{{\rho }_l-{\rho }_g}\right)}^{0.37}{\left(\frac{{\rho }_g}{{\rho }_l}\right)}^{0.25}{\left(\frac{q}{q_{crit}}\right)}^{0.7}\right]$$ (A.1)

$$x_{de}=0.61\text{exp}\left[0.57-5.8\cdot {10}^{-3}W{e}_g^{0.38}{\mathit{Fr}}_g^{0.15}{\left(\frac{{\rho }_g}{{\rho }_l-{\rho }_g}\right)}^{0.15}{\left(\frac{{\rho }_g}{{\rho }_l}\right)}^{-0.09}{\left(\frac{q}{q_{crit}}\right)}^{0.27}\right]$$ (A.2)

where the ‘${\rho }_g/\left({\rho }_l-{\rho }_g\right)$’ terms needed to be added to the equations in the original reference. This correction has been confirmed with the publication author.

A.2. Iteration example for subcritical cooling

The numerical iteration process for the CO₂ GSHP under subcritical cooling (Fig. 12) is shown here as an illustrative example. The iteration loops shown in the other computational flow diagrams (Fig. 7, 9, 11, 14, 15, 17, 19, 21) used similar iteration processes; they are not discussed here.

The iteration adjusted the condenser refrigerant inlet pressure, $p_{c,in}$, until the computed condenser area, $A_c$, matched the actual large plate heat exchanger area, $A_{PHX}$, within convergence tolerance (1×10⁻⁴ %). Within each $p_{c,in}$ iteration, the evaporator outlet pressure, $p_{e,out}$, was iteratively adjusted until the computed evaporator inlet enthalpy, $i_{e,in}$, matched the computed EEV outlet enthalpy, $i_{EEV,out}$, within convergence tolerance (1×10⁻⁴ %). The upper bound for $p_{c,in},p_{c,in,UB}$, was the CO₂ critical pressure since the cycle was subcritical, and the lower bound, $p_{c,in,LB}$, was set to the CO₂ saturation pressure at the inlet liquid temperature since the CO₂ temperature cannot go below the lowest liquid temperature:

$$p_{c,in,UB}=p_{crit}$$ (A.3)

$$p_{c,in,LB}=p_{sat}\left(t_{l,in}\right)$$ (A.4)

The initial guess for the condenser inlet pressure, $p_{c,in}^{k=1}$ (all superscripts in this section denote iteration index), at the iteration index k = 1, was the average of the upper and lower bounds:

$$p_{c,in}^{k=1}=\left(p_{c,in,UB}+p_{c,in,LB}\right)/2$$ (A.5)

Similarly, the upper bound for $p_{e,out}$ was the CO₂ saturation pressure at the inlet air temperature. The lower bound was established at the saturation pressure of −10 °C since the refrigerant temperature was not expected to be colder for normal air-conditioning operation:

$$p_{e,out,UB}=p_{sat}\left(t_{a,in}\right)$$ (A.6)

$$p_{e,out,LB}=p_{sat}(-10°\text{C})$$ (A.7)

The initial guess, at iteration index n = 1, was again the average of the upper and lower bounds.

$$p_{e,out}^{n=1}=\left(p_{e,out,UB}+p_{e,out,LB}\right)/2$$ (A.8)

Next, per Fig. 12, the equations governing the SLHX, compressor, condenser, EEV, and evaporator were solved. The residual and error of the $p_{e,out}$ iteration loop were computed as:

$$Re{s}^n=i_{EEV,out}-i_{e,in}$$ (A.9)

$$Erro{r}^n=\left|Re{s}^n/i_{e,in}\right|$$ (A.10)

If the error was greater than the tolerance (i.e. $i_{EEV,out}\ne i_{e,in}$) the calculation continued, starting with incrementing the iteration index:

$$n=n+1$$ (A.11)

The upper or lower bound was adjusted according to the residual. If the residual was positive, $i_{EEV,out}$ was too high, the saturation pressure needed to adjust downward. Therefore, $p_{e,out,UB}$ value was reduced to $p_{e,out}^{n-1}$ to force the iteration algorithm to choose a lower saturation pressure. For a negative residual, the adjustment was instead made to $p_{e,out,LB}$:

$$IFRe{s}^{n-1}>0,THEN{p}_{e,out,UB}=p_{e,out}^{n-1}$$ (A.12)

$$IFRe{s}^{n-1}<0,THEN{p}_{e,out,LB}=p_{e,out}^{n-1}$$ (A.13)

A bisection step was used for the first iteration, and a secant step for subsequent iterations:

$$IFn\le 2,THEN{p}_{e,out}^n=\left(p_{e,out,UB}+p_{e,out,UB}\right)/2$$ (A.14)

$$IFn>2,THEN{p}_{e,out}^n=p_{e,out}^{n-1}+Re{s}^{n-1}\frac{\left(p_{e,out}^{n-1}-p_{e,out}^{n-2}\right)}{(Re{s}^{n-1}-Re{s}^{n-2})}$$ (A.15)

‘Adaptations’ to this iteration algorithm included:

1. If the secant step yielded a value outside the upper or lower bounds, a bisection step was taken instead.

2. If the calculated refrigerant-air temperature difference was negative anywhere in the heat exchanger (i.e. at the refrigerant inlet, two-phase outlet, and outlet), then $p_{e,out,UB}$ was set to $p_{e,out}^{n-1}$ to lower range of saturation pressures and temperatures considered in the solution (which increased the air-refrigerant temperature difference). A bisection step was then taken.

3. If the number of iterations exceeded 100 (convergence was typically achieved in 5 to 20 iterations), or if the upper and lower bounds differed by less than 1×10⁻⁴ %, the iteration was stopped and a warning was given. None of the computations presented here generated this warning.

When the error was less than the tolerance, $p_{e,out}$ was converged.

A similar iteration algorithm was employed for adjusting $p_{c,in}$. The residual and error of the $p_{c,in}$ iteration loop were:

$$Re{s}^k=A_c-A_{PHX}$$ (A.16)

$$Erro{r}^k=\left|Re{s}^k/A_{PHX}\right|$$ (A.17)

If the error was greater than the tolerance, the calculation continued, starting with incrementing the iteration index:

$$k=k+1$$ (A.18)

The upper or lower bound was adjusted according to the residual. If the residual was negative, $A_c$ was too small, so the driving refrigerant-air temperature difference was too large; the refrigerant saturation pressure needed to adjust downward. Therefore, $p_{c,in,UB}$ value was reduced to $p_{c,in}^{k-1}$ to force the iteration algorithm to choose a lower saturation pressure. For a positive residual, the adjustment was instead made to $p_{c,in,LB}$:

$$IFRe{s}^{k-1}<0,THEN{p}_{c,in,UB}=p_{c,in}^{k-1}$$ (A.19)

$$IFRe{s}^{k-1}>0,THEN{p}_{c,in,LB}=p_{c,in}^{k-1}$$ (A.20)

A bisection step was used for the first iteration, and a secant step for subsequent iterations:

$$IFk\le 2,THEN{p}_{c,in}^k=\left(p_{c,in,UB}+p_{c,in,UB}\right)/2$$ (A.21)

$$IFk>2,THEN{p}_{c,in}^k=p_{c,in}^{k-1}+Re{s}^{k-1}\frac{\left(p_{c,in}^{k-1}-p_{c,in}^{k-2}\right)}{(Re{s}^{k-1}-Re{s}^{k-2})}$$ (A.22)

This $p_{c,in}$ iteration algorithm used the same ‘adaptations’ as applied to the $p_{e,out}$ algorithm.