Development and validation of a model for an air-to-air air conditioner

Air conditioning units are responsible for a significant amount of global warming, with demand predicted to triple by 2050. Mathematical models aid in designing energy efficient air conditioners that can contribute to climate change mitigation. This study presents a steady-state model for a air-to-air air conditioner. The proposed model can assist in reducing the number of experiments and optimizing the efficiency of air conditioning systems. It utilizes a bottom-up approach to solve for compressor, condenser, capillary tube, and evaporator sub-models. The compressor is modelled using polynomial equations to determine refrigerant mass flow rate and power consumption based on operating temperatures. Heat exchanger models are solved using finite volume approach and reliable correlations are used for void fraction, friction factor, and heat transfer coefficient calculations. The closing equations are the mass conservation, i.e. constant refrigerant charge, and the equivalence between the mass flow rate sucked by the compressor and the one that flows through the capillary tube, while the evaporating and condensing pressures are the independent variables. The capillary tube is modelled using semiempirical correlations. The model is validated against experimental data at various operating conditions and shows a ±7% agreement for predicted cooling capacity.


Introduction
Air conditioning systems are essential for maintaining comfortable indoor environments in various contexts, including residential, commercial, and industrial buildings.The modelling of these systems is crucial to provide the customers and designers with the possibility of comparing different systems and machine configurations in terms of performance.
A significant amount of work on heat pumps and air conditioner modelling is available in the literature and can be divided into three main groups.
1. Equation-fit models: they describe the system as a black-box and the operation is based on equation parameters derived from performance maps.2. General thermodynamic approaches: they describe the heat pump performances according to EN standards.3. Analytical and semi-analytical models: they describe each component through physical analysis.Analytical and semi-analytical models differ from each other as the latter combine empirical relation for the description of the processes in subsystems while the first employs full physical description of each component.In this work a semi-analytical model using the elemental volume approach is presented.Heat exchangers are fully described in an analytical way while the compressor and the expansion device are described through an empirical approach.The proposed approach allows the numerical comparison between the machine performance with different heat exchanger configurations.

Heat exchangers
The model of the fin-and-tube heat exchangers is developed using elemental volume approach.Each heat exchanger is divided into elemental volumes, sufficiently small to provide good computational accuracy.As shown in Figure 1 1) -(3).Splitting and merging components (i.e.bends and headers) can be handled through appropriate balances.

Air-side equations
On air-side, different equations are used depending on tube arrangement: indeed, the shape of elemental volume changes between squared, for the in-line arrangement, and hexagonal, for the staggered arrangement.Generally, for a w elemental volume mass and energy balances take the following form: Air-side momentum balance is solved separately from the previous balances since the influence of pressure variation on heat transfer is negligible: the frictional pressure drop is computed, as reported in equation (6).
The friction factor is computed according to correlations, depending on fin shape, tube arrangement and heat exchanger characteristics.

Heat transfer in elemental volume
In this work, each elemental volume is composed by the refrigerant, the tube, the fins and the air and is analysed as a crossflow heat exchanger in which the air is the unmixed stream and the refrigerant is the mixed one.Two different set of equations are developed to account for the dehumidification process that may occur on the evaporating section of the machine.
The heat transfer of an elemental volume in the condensing section is given by equation ( 7): The effectiveness εw is computed through specific formulation for single and two-phase flow [8].
On the evaporator, firstly the occurrence of dehumidification at elemental volume scale is assessed by comparing the air dew temperature with the average temperature of the finned surface.If the latter exceeds the dew temperature, the elemental volume is modeled according to the methodology outlined for the condenser.In contrast, if dehumidification is present, it is accounted for using the local energy balance.The total heat transferred to the refrigerant is due to the latent heat ( ̇ ) and the sensible heat ( ̇ ), mathematically represented by equations ( 8) - (10).
The humidity ratio X WALL,EXT refers to saturated moist air at T WALL,EXT temperature.The latent heat is expressed as function of the mass transfer resistance, which is computed through the Lewis analogy, assuming a constant Lewis number of 0.845 [9].Once the unknown  WALL,EXT is obtained through an iterative approach, equations ( 8) -(10) are integrated over the elemental volume and the energy balance is applied.The air temperature and humidity at the outlet of each elemental volume w are computed and transferred as input values to the following volume.Table 2 and 3 summarize correlations used for heat transfer coefficients in the present work.
On the evaporator, to account for efficiency reduction due to the presence of condensed water, the Hong and Webb [18] correction is applied.

Compressor
The compressor is modelled using a polynomial function provided by the manufacturer.The compressor model provides the refrigerant flow rate and the power required by the machine as a function of T EVA and T COND .
From the energy balance on the compressor, it is also possible to compute the enthalpy of the refrigerant at compressor outlet.However, to retrieve the correct outlet temperature, oil circulation inside the compressor is considered.The energy balance to retrieve refrigerant outlet enthalpy is expressed in equation (11).

Expansion device
The expansion device considered in the model is an adiabatic capillary tube, therefore, an isenthalpic process is considered.The Shimizu [19] model has been selected in this work.

Refrigerant charge 2.4.1. Charge estimation in heat exchangers
The charge is estimated at the elemental volume scale as shown in equations ( 12) - (13).

Charge estimation solved in compressor oil
Studies available in the literature [21,22] have shown that propane solubility with a mineral oil can reach values up to 40 %, hence it is necessary to account it for.Refrigerant solubility in lubricant oil is considered in this work through experimental data presented in the work of [21], which reports oil solubility at different pressure and temperature.The amount of oil contained in the machine and the oil sump positioning, which determines the oil temperature and pressure, is provided by the manufacturer.

Model structure
The simulation model presented in this work follows the iterative scheme proposed by Domanski and Didion [23].The objective of the model is to characterize the steady state condition at each operating condition.The model is developed using pressure and enthalpy as state parameters, applying three different balances: mass, momentum and energy.A schematic of the cycle is reported in Figure 2.
The simulation process starts with a guessed refrigerant condensing and evaporating pressure and a compressor inlet superheat.Refrigerant mass flow rate and condenser inlet conditions are obtained through the compressor model.The condenser and capillary are simulated, and capillary mass flow rate is obtained: this value is compared with the one obtained in the compressor; if they differ more than the tolerance, the iterative cycle starts from the initial point with a different evaporating pressure.In case of mass flow rate through a capillary tube is higher than mass flow rate through the compressor, evaporating pressure is increased; if it is lower, evaporating pressure is decreased.Once convergence on mass flow rate is reached, the evaporator is solved using as input the data from the capillary tube.Outlet refrigerant temperature is compared with the one found in the previous iteration and is used as input in the following iteration.The entire cycle is simulated again up until convergence on the refrigerant superheat at compressor inlet.Then, the mass inventory is computed and compared with the machine value.If the difference is higher than the tolerance, the condensing pressure is changed, and the cycle is simulated again.In case of computed charge higher than the actual value, condensing pressure is decreased; if it is lower, condensing pressure is increased.

Submodel structure
The model structure for simulating the heat exchangers is the same for the evaporator and the condenser.
First, an initialization of the air temperature profile and all the elemental volume properties is required.Then, the procedure starts from the refrigerant inlet volume.For each volume, balances are solved, and output properties are obtained and transferred to the following volume.Once the thermal profile is solved, the hydraulic one is computed in terms of pressure drop and a check is set up to find convergence on pressure drop between the current and previous iteration.If convergence is not found, pressure profile is updated, and mass flow rate redistributed to find the same pressure drop across parallel circuits.
Results show that the model performed well in predicting both parameters.However, concerning the condenser, most of the data points were overestimated, as demonstrated by the parity plot  On the evaporator, the model performed better, with higher accuracy in predicting both pressure and heat transfer rate than the condenser.However, the evaporator capacity was overestimated due to the higher subcooling degree found at the condenser outlet, as shown in Figure 3 (d), which resulted in a lower enthalpy inlet at the evaporator.Since heat transfer performance at low vapor quality is better than at high vapor quality, the cooling power is overestimated.Additionally, the evaporator outlet refrigerant conditions in Figure 3 (d) showed a smaller superheat with respect to experimental data, supporting the explanation of a lower enthalpy inlet in the evaporator.
Despite the underestimation of evaporating pressure value, as shown in Figure 3 (b), the outlet evaporating absolute humidity was overestimated, indicating an underestimation of the dehumidification process.To improve accuracy, different models of water condensation that have been tested on a wider dataset could be implemented in future works.However, the overestimation of heat power implies the low influence of dehumidification on these experimental data.
As shown in Figure 3 (c), the evaporator outlet air temperature is correctly estimated.The prediction performance on pressures suggests that the model correctly estimated the machine mass inventory and refrigerant flow rate.On the flow rate, it is related only to the compressor behavior, which is not simulated but described through empirical relations, and to the compressor inlet superheat, which is generally correctly estimated as previously explained.Hence, the flow rate and the evaporating pressure, which is the variable associated to the flow rate closure relation, are correctly estimated.On the mass inventory, it is related to the condensing pressure and the operating conditions at which the oil is retained, which at last depend on the oil sump positioning.The estimation on the condensing pressure suggests that the Rouhani Axelsson model for void fraction provides good accuracy.Nevertheless, the computed refrigerant solved in the oil is about 20% of the total mass inventory in all the simulated points, suggesting that a significant error takes place when this is not considered.Future work should always consider this element and develop a wide dataset considering the largest number of refrigerants and oils.

Conclusions
A model for the simulation of an air-to-air air conditioner has been developed in this work.The model is based on a semi-analytical approach.The approach provides the possibility of a preliminary analysis on innovative machine configuration performance without the burden of prototyping.The heat exchangers are described through the elemental volume approach and ε -NTU method.The compressor is described through the manufacturer polynomial curves and the capillary tube is described through semiempirical correlations.Mass inventory is computed considering refrigerant dissolved in the oil.The model iteratively solves the components using mass flow rate and mass inventory as closing relations for convergence.The model is validated against experimental data on a air-to-air air conditioner at different air temperature and humidity.Good agreement is found on the heat transfer, with a MARD of 6.8% on the evaporator and 10.3 % on the condenser, and on the operating pressures, with a MARD of 2.1% on the evaporating pressure and 3.2% on the condensing one.Probably, the heat transfer is overestimated due to air flow rate measurement underestimation, while operating pressures show a correct estimation on refrigerant flow rate and mass inventory estimation.Future work should assess the model with a wider database of experimental data with a more accurate air flow rate estimation.

References
, three different indexes corresponding to tube, volume in the tube and row are assigned to each volume.To simplify notation in the present work, a single number called index w is assigned to each elemental volume: the index value increases with the refrigerant circuitation, i.e. with w = 1 at the first tube first volume, w = T•V•R at the last tube last volume.The elemental volume shape is rectangular for an in-line tube arrangement and a hexagonal prism for staggered arrangement.Mass, momentum and energy equations are used in the solution of each elemental volume applying the following assumptions:(i) Steady-state operation regime.(ii) Superheated condensation and subcooled boiling are neglected.(iii) Kinetic and potential energy contributions are negligible.(iv) Return bends, joints, splits, headers are adiabatic.(v) Merging and splitting components are at constant pressure.(vi) Refrigerant properties are provided by RefProp 10.0 [1] 2.1.1.Refrigerant-side balances Refrigerant-side mass, momentum and energy balances are reported in equations (

Figure 1
Figure 1 Fin-and-tube heat exchanger (a) and volume discretization (b) (T = number of tubes per row, V = number of elemental volumes per tube, R = number of rows).

Figure 2 .
Figure 2. Schematic of the iterative model that simulates the air conditioner.At each iteration, the condensing and evaporating pressure are adjusted through the bisection method; the compressor superheat is computed and substituted in the following iteration.
A set of experimental data was available from in-house experiments of a R290 prototype air conditioner carried out at the Department of Energy of Politecnico di Milano.The experiments are in the steady state condition.Refrigerant temperature was measured through T-type thermocouples while pressure was measured at condenser outlet and evaporator outlet, through Huba ® 520 pressure transducer.Air temperature and humidity were measured in 4 points, i.e., both heat exchangers inlet and outlet, through Vaisala ® HMT330 sensors.Air velocity at heat exchangers outlet was measured through a VelociCalc ® 5725 fan anemometer.Nineteen different operating conditions were analysed.The air temperature is in the range [20 -38] °C whereas the air humidity is in the range [4.5 -14.1] g/kg.The geometrical parameters and the refrigerant charge are retrieved from the machine technical data and are used as model input.The model is run at each operating condition and, once convergence is reached, results are collected.Two parameters were considered and reported in Table 3: the Mean Absolute Relative Deviation (MARD) and the Mean Relative Deviation (MRD), expressed for a quantity X as in equations (14.i) -(14.ii)., −  , )  ,   (ii) Figure 3 (a) and the MRD positive value almost equal to MARD one.

Figure 3 .
Figure 3. Parity plot for evaporator and condenser (a) Heat power (b) Operating pressure (±20% error is reported) (c) Air outlet temperature (d) Refrigerant outlet temperature

Table 1 .
Correlations for heat transfer coefficient in single-phase and twophase flow (condensation and evaporation).

Table 3 .
Heat exchange, operating pressure and air conditions MARD and MRD