Model-Based Condenser Fan Speed Optimization of Vapor Compression Systems

energies

Article Model-Based Condenser Fan Speed Optimization of Vapor Compression Systems

Sebastian Angermeier 1,2,* and Christian Karcher 1 1 Institute of Thermodynamics and Fluid Mechanics, Technische Universität Ilmenau, 98684 Ilmenau, Germany; [email protected] 2 MAHLE GmbH, Pragstr. 26–46, 70376 Stuttgart, Germany * Correspondence: [email protected]; Tel.: +49-1703724441

Received: 1 October 2020; Accepted: 11 November 2020; Published: 17 November 2020

Abstract: Vapor compression systems (VCS) cover a wide range of applications and consume large amounts of energy. In this context, previous research identified the optimization of the condenser fans speed as a promising measure to improve the energy efficiency of VCS. The present paper introduces a steady-state modeling approach of an air-cooled VCS to predict the ideal condenser fan speed. The model consists of a hybrid characterization of the main components of a VCS and the optimization problem is formulated as minimizing the total energy consumption by respectively adjusting the condenser fan and compressor speed. In contrast to optimization strategies found in the literature, the proposed model does not relay on algorithms, but provides a single optimization term to predict the ideal fan speed. A detailed experimental validation demonstrates the feasibility of the model approach and further suggests that the ideal condenser fan speed can be calculated with sufficient precision, assuming constant evaporating pressure, compressor efficiency, subcooling, and superheating, respectively. In addition, a control strategy based on the developed model is presented, which is able to drive the VCS to its optimal operation. Therefore, the study provides a crucial input for set-point optimization and steady-state modeling of air-cooled vapor compression systems.

Keywords: coeﬃcient of system performance; steady-state modeling; set-point optimization; ideal condensing pressure; air-cooled chiller; refrigeration

1. Introduction Vapor compression systems (VCS) are used in many technical applications, such as the air conditioning of buildings, food temperature control, automotive applications, etc., and consume a considerable amount of energy. For instance, air conditioning and refrigeration account for about 24% of household energy consumption in the USA [1] and for hot and high humid countries like Singapore this consumption can represent over 50% [2]. Therefore, increasing the energy eﬃciency of VCS through optimization and control appears to be a key issue [3]. One prospective possibility for energy saving is the quasi-stationary control, which aims to achieve the highest possible eﬃciency within one operating condition, i.e., set-point optimization.

1.1. Ideal Condenser Fan Speed According to Jensen and Skogestad [4], the optimal energy efficiency of a VCS can be obtained by adjusting the five degrees of freedom: refrigerant mass flow rate, superheating, subcooling, condensing, and evaporating pressure. These theoretical degrees of freedom can be influenced by the actuators of the given system. Considering a conventional air-cooled chiller with a thermostatic expansion valve (TXV), as shown in Figure1, the compressor and condenser fan speed are available actuators to set the five degrees of freedom. Hence, an infinite number of combinations of the compressor and condenser

Energies 2020, 13, 6012; doi:10.3390/en13226012 www.mdpi.com/journal/energies Energies 2020, 13, 6012 2 of 26 Energies 2020, 13, 6012 2 of 26 fanactuators speed can to meetset the the five load degrees demand, of but freedom. with di ffHencerente, energyan infinite efficiency number [5,6 ].of Thereby, combinations the energy of the efficompressorciency can beand quantified condenser by fan the speed coeffi cancient meet of system the load performance demand, but with different energy efficiency [5,6]. Thereby, the energy efficiency can be quantified by the coefficient of system performance . COSP = Qe/(Pcom + PF) (1) 𝐶𝑂𝑆𝑃 =𝑄 ⁄(𝑃 +𝑃) (1) . where 𝑄 , 𝑃 , and 𝑃 are refrigeration capacity, compressor power, and condenser fan power. where Qe, Pcom, and PF are refrigeration capacity, compressor power, and condenser fan power. TheThe optimization optimization problem problem of of the the ideal ideal condenser condenser fan fa speedn speed occurs occurs because because of of a tradeoa tradeoffff between between the the compressorcompressor and and fan fan power power consumption consumption [5 ].[5]. For For instance, instance, rising rising fan fan speed speed increases increases the the fan fan power power consumption,consumption, but but decreases decreases the the condensing condensing pressure pressure due du toe to higher higher heat heat rejection, rejection, which which in in turn turn leads leads toto lower lower compression compression work. work. Figure Figure1 shows 1 shows the the power power consumption consumption of compressor, of compressor, fan, andfan, theand sum the ofsum of both against the fan speed at constant cooling load, coolant inlet temperature (𝑇 ), and both against the fan speed at constant cooling load, coolant inlet temperature (Tcool,in), and condenser, 𝑇 aircondenser inlet temperature air inlet (temperatureTair,in) to illustrate ( , the) to tradeoillustrateff. the tradeoff.

air outlet

P fan

Q condenser compressor P Compressor TXV air inlet Fan Compressor + Fan Electric power/ kW evaporator

Condenser fan speed / rpm coolant circuit (a) (b)

FigureFigure 1. Air-cooled1. Air-cooled vapor vapor compression compression system (VCS)system comprising (VCS) comprising compressor, condenser,compressor, thermostatic condenser, expansionthermostatic valve expansion (TXV), evaporator, valve (TXV), and evaporator, condenser fanand ( a).condenser Electric powerfan (a). of Electric the condenser power of fan, the compressor,condenser and fan, the compressor, sum of both forand constant the sum air inlet of temperature,both for constant coolant temperature,air inlet temperature, and refrigeration coolant capacity.temperature, The characteristic and refrigeration of the compressor capacity. The and char condenseracteristic fan of power the compressor results in an and energy condenser eﬃciency fan optimum.power results A low in curvature an energy in efficiency the vicinity optimum. of the minimum A low curvature can be observed in the vicinity (b). of the minimum can be observed (b). The ideal condenser fan speed exhibits a strong dependence on the ambient temperature and the refrigerationThe ideal condenser capacity [7 fan,8]. speed Consequently, exhibits a the strong speed dependence of the condenser on the fanambient must temperature be adjusted inand accordancethe refrigeration with the capacity operating [7,8]. conditions Consequently, in order the to achievespeed of optimum the condenser performance. fan must be adjusted in accordance with the operating conditions in order to achieve optimum performance. 1.2. Optimization of the Condenser Fan Speed Control 1.2.A Optimization key aspect of of the the condenser Condenser fan Fan speed Speed optimization Control is the VCS modeling, which can be separated into blackA key box aspect and component-based of the condenser [ 9fan]. While speed black optimization box models is the use VCS a set modeling, of data to characterizewhich can be theseparated system, component-basedinto black box and models component-based consider physical [9]. While aspects black for box describing models each use componenta set of data of to thecharacterize system. Furthermore, the system, component-based component-based models models can consider be subdivided physical regarding aspects thefor heatdescribing exchanger each modelingcomponent into of empirical, the system. distributed, Furthermore, moving component- boundary,based and models hybrid can orbe lumpedsubdivided models regarding [10,11 the]. Anheat overview exchanger of steady-state modeling VCSinto modelingempirical, is distri givenbuted, by Wan moving et al. [9boundary,] and Zhang and et al.hybrid [12]. or The lumped VCS modelsmodels are [10,11]. usually An assessed overview on of the steady-state basis of the VCS prediction modeling accuracy is given of by the Wan coeﬃ etcient al. [9] of and performance Zhang et al. (COP).[12]. The In this VCS context, models Scarpaare usually et al. assessed [13] have on stated the basi thats of the the mostprediction accurate accuracy models of leadthe coefficient to a COP of estimationperformance error (COP). of 5%, In andthis context,10% can Scarpa be considered et al. [13] ashave a high stated level that of the accuracy. most accurate In fact, models the most lead ± ± modelsto a COP for set-pointestimation optimization error of ±5%, perform and ±10% a COP can estimationbe considered accuracy as a high of about level of10% accuracy. [14–20 ].In fact, the ± most models for set-point optimization perform a COP estimation accuracy of about ±10% [14–20].

Energies 2020, 13, 6012 3 of 26

Using different modeling approaches, various optimizing system controls of varying complexity are proposed in the literature [21,22]. As a simple solution, the ideal condenser fan speed can be determined in advance to derive a linear function [7,8] or a regression [19,23] for control. In 2004, Chan and Yu [8] published a control of the condensing temperature as a linear function of the ambient temperature. In a further publication, Yu and Chang [19] suggested a load-based speed control for a small ambient temperature range. These solutions can provide a significant improvement compared to the traditional constant condensing pressure control, but only achieve suboptimal results for different operating conditions of ambient temperature and cooling load [7,8]. Larsen et al. [24] proposed a linear, online control of the VCS, which calculates the ideal settings iteratively via gradient method and thus obtains high energy efficiency for various operating conditions. Thereby, the condenser fan and compressor speed are considered as actuators. The applied hybrid model is verified by measurements on a supermarket refrigeration system. Although some rough assumptions are made in terms of a lump condenser model and constant compressor efficiency, high accuracy is achieved when calculating the ideal settings. As a drawback of this method, a long time (over 50 min [7]) is required to drive the system to ideal operation. Zhao et al. [10] and Ruz et al. [17] presented steady-state optimization strategies based on the hybrid heat exchanger model suggested by Ding et al. [25]. Zhao et al. used a modified genetic algorithm to obtain optimum set-points, minimizing the total operating costs of the compressor, evaporator fan, and condenser fan. Ruz et al. iteratively optimized the settings of a VCS to achieve the ideal condenser pressure and expansion valve opening. Huang et al. [26] also used a hybrid model to develop an optimizer that incorporates the speed of the compressor, condenser fan, and evaporator fan of an automotive air conditioning system. Further studies employ artificial neuronal networks (ANN) [18,20], sate space modeling [27], exergy-based models [28], and semi-empirical models [16] to predict the ideal condenser fan speed. In addition to these model-based methods, the model-free method of extremum seeking control is discussed for the set-point optimization of VCS [29,30]. The extremum seeking control is a very prospecting solution, since no advanced component models are required. On the other hand, the disadvantages include a long convergence time, until an ideal operation is achieved and there are no predictive capabilities [22]. The most set-point optimization strategies deal with more than condenser fan and compressor speed optimization [10,16–18,26–28], resulting in complicated cross-coupling, especially due to superheating and evaporating pressure [31,32]. Therefore, the published model-based set-point optimization strategies generally need complicated algorithms, whereas black models always require detailed data from measurements for the validation and training [13]. In addition, most solutions for set-point optimization need considerable time to achieve the ideal settings [7,29,30]. To the authors’ knowledge, no method has yet been published that predicts the ideal condenser fan speed according to a single optimization term. The main contribution of the present paper is the introduction of a model-based optimization method for a fast and simple determination of the ideal condenser fan speed under stationary operating conditions. Therefore, a new hybrid model approach is developed to derive a single optimization term. The optimization problem is formulated as minimizing energy consumption of the condenser fan and the compressor subject to constant refrigeration capacity and ambient temperature. To demonstrate the feasibility, a detailed model validation is carried out of 250 measuring points and the model assumptions are discussed. The influences of several parameters on the ideal condenser fan speed are studied to contribute to the understanding of the model accuracy, although some rough simplifications are made. In addition, a control strategy based on the model approach is suggested.

2. Experimental Methodology Experimental tests have been conducted on an air-cooled vapor compression system to verify the hybrid model proposed in this research. The test setup, calculation backgrounds, and the experimental procedure are described in order to provide an appropriate basis for the interpretation of the ﬁndings. Energies 2020, 13, 6012 4 of 26 Energies 2020, 13, 6012 4 of 26

2.1.2.1. Test Test Bench Bench Description Description TheThe test test specimen specimen of of the the present present study study is is a aseri serialal battery battery cooling cooling unit unit of of the the Mahle Mahle GmbH GmbH [33]. [33 ]. TheThe system system works works as as an an air-cooled air-cooled vapor vapor compress compressionion system system and and consists consists of of a a refrigerant-coolant refrigerant-coolant evaporatorevaporator (chiller), (chiller), a asuction suction gas-cooled gas-cooled electric electric scroll scroll compressor, compressor, an an air-refrigerant air-refrigerant condenser condenser with with integratedintegrated receiver receiver and and dryer, dryer, an anaxial axial condenser condenser fan, fan, and anda thermostatic a thermostatic expansion expansion valve valve (TXV). (TXV). The workingThe working fluid is ﬂuid a mixture is a mixture of R134a of R134aand polyalkylene and polyalkylene glycol glycol(PAG) (PAG)synthetic synthetic oil. A description oil. A description of the VCSof the components VCS components is detailed is detailedin Table inA1. Table The VCSA1. Theis instrumented VCS is instrumented with temperature with temperature (𝑇) and pressure (T) and

(𝑝pressure) sensors ( pbefore) sensors and before after each and aftercomponent. each component. In the liquid In line, the liquidthe speed line, of the sound speed (𝑐 of) is sound measured (cs) is tomeasured calculate tothe calculate oil fraction the oiland fraction a Coriolis and sensor a Coriolis is equipped sensor is to equipped detect the to refrigerant detect the refrigerantmass flow massrate . (𝑚ﬂow ). rateFurthermore, (mR). Furthermore, voltage (𝑈 voltage ,𝑈 ), ( Ucurrentcom,UF), (𝐼 current ,𝐼 ), (Iandcom, IFspeed), and ( speed𝑛 ,𝑛 (n)com are,n Frecorded) are recorded at the at compressorthe compressor and condenser and condenser fan, fan,respectively. respectively. A pictur A picturee of the of theinstrumented instrumented battery battery cooling cooling unit unit is highlightedis highlighted in Figure in Figure A1. A1 The. The experimental experimental setup setup of ofthe the test test specimen specimen and and the the corresponding corresponding test test benchbench is is detailed detailed in in Figure Figure 2.2.

n U I air inlet 16 Δp test chamber T,, T, T, blower condenser refrigerant-oil- m p p , , mixture c 10 T,, p, T T, p, , cooler I thermostatic M n U expansion valve electric p, T, compressor electric heater p, evaporator air T, bypass

V 4T, 4T, coolant

coolant pump electric heater Refrigerant-oil-mixture Coolant Air

FigureFigure 2. 2. SchematicSchematic overview overview of ofthe the experimental experimental setup. setup. The TheVCS VCS is placed is placed in an in air-conditioned an air-conditioned test test chamber. The VCS is instrumented with sensors to measure the temperature (T), pressure chamber. The VCS is instrumented with sensors to measure .the temperature (𝑇), pressure (𝑝), speed (p), speed of sound (cs), and refrigerant mass flow rate (mR) of the VCC, as well as the speed (n), of sound (𝑐), and refrigerant mass flow rate (𝑚 ) of the VCC, as well as the speed (𝑛), current (𝐼), and current (I), and voltage (U) of the compressor and condenser fan. A temperature and volume flow rate voltage. (𝑈) of the compressor and condenser fan. A temperature and volume flow rate (𝑉 )- controlled(Vcool)-controlled coolant coolantflow is pr flowovided is provided at the evaporator. at the evaporator. The VCS itself is placed in an air-conditioned test chamber. Thereby, an electric air heater and a The VCS itself is placed in an air-conditioned test chamber. Thereby, an electric air heater and a cooler are employed to control the air temperature in the chamber. The air temperature is measured cooler are employed to control the air temperature in the chamber. The air temperature is measured at the inlet (𝑇,,) and outlet (𝑇,,) of the condenser. The blower of the test bench supplies the conditioned air into the test chamber, while the air flow through the condenser is provided by the

Energies 2020, 13, 6012 5 of 26

at the inlet (Tc,air,in) and outlet (Tc,air,out) of the condenser. The blower of the test bench supplies the conditioned air into the test chamber, while the air flow through the condenser is provided by the condenser fan of the VCS. To ensure a free suction of the condenser fan, the pressure difference (∆pair) between the test chamber and the environment is monitored and controlled. The secondary mass flow rate through the evaporator is a coolant (50/50 mas.-% water/glycol), which is provided by the coolant pump of the test bench and measured magnetic-inductively (MID). The temperature of the coolant is monitored at the inlet (Tcool,in) and outlet (Tcool,out) by three calibrated type K thermocouples and one Pt100 sensor, respectively. An electric heater is installed in the coolant circuit to adjust the coolant temperature.

2.2. Calculation Background For the assumption of an isenthalpic expansion, the refrigeration capacity

. . Q = mR(he,out h ) (2) e − TXV,in . can be balanced on the refrigerant side of the evaporator, where mR, he,out, and hTXV,in are refrigerant mass flow rate, enthalpy at the evaporator outlet, and at the thermostatic expansion valve (TXV) inlet, respectively. The refrigerant mass flow rate is a direct measured value provided by the Coriolis sensor, whereas the enthalpies must be calculated on the basis of the pressure, temperature, and oil percentage. Therefore, the specific enthalpy of the refrigerant-oil-mixture

h = x h + (1 x )hR + (1 xsteam) ∆hex = f (T, p, x ) (3) oil oil − oil − · oil is determined with respect to Youbi-Idrissi et al. [34] as a function of the oil content of the total mass ﬂow xoil, the steam fraction related to the total mass ﬂow xsteam, the refrigerant enthalpy of liquid and vapor hR, the oil enthalpy hoil, and the excess enthalpy ∆hex (describing the mixing enthalpy of the refrigerant-oil-mixture) [34]. The enthalpy values are taken from Span et al. [35] and measurements of the ILK Dresden mbH on behalf of the Mahle Gmbh [36]. The oil percentage

xoil = f (T, p, cs) (4) can be calculated using a polynomial of temperature, pressure, and sound velocity [37]. The coefficients of the polynomial are carried out by ILK Dresden mbH on behalf of the Mahle GmbH [38]. Based on the aforementioned considerations, the coefficient of system performance can be identified according to Equation (1). Thereby, the compressor and fan power are calculated by the product of the measured current and voltage. The uncertainty of the measurement chain is developed according to the type B of the “Guide to the expression of uncertainty in measurement” (GUM) [39]. Detailed information about the sensors and their corresponding uncertainty are listed in Tables A2 and A3. The extended measurement uncertainty (coverage factor of k = 2) of the derived quantities are calculated with respect to the Gaussian error propagation. Detailed information on the measurement uncertainty calculation is described by Angermeier et al. [40].

2.3. Experimental Procedure and Test Matrix The energy efficient operation of the VCS is investigated for fifteen different set-points, which are oriented to the specifications of battery cooling units for electric buses. The considered set-points are listed in Table1. Energies 2020, 13, 6012 6 of 26

Table 1. Boundary conditions for ﬁfteen set-points regarding the refrigeration capacity, condenser air inlet temperature, and evaporator coolant inlet temperature.

. Qe/kW Tc,air,in/◦CTcool,in/◦C 3.6 35 25 4 20/25/30/35/40 25 5 20/25/30/35 25 6 20/25/30/35 25 6.5 20 25 Energies 2020, 13, 6012 6 of 26

Subject to the conditions in Table1 1,, eacheach set-pointset-point isis measuredmeasured forfor severalseveral combinationscombinations ofof compressor and condenser fan speed to investigate the effect effect on the coefficien coefficientt of system performance (COSP). In sum,sum, 250250 steady-statesteady-state measuring points are conducted. Thereby, a stationarystationary measuringmeasuring point is defineddefined forfor fluctuationsfluctuations belowbelow ±0.10.1 K and ±5050 W W for for the the temperatures temperatures of the secondary fluids fluids ± ± (air andand coolant)coolant) andand thethe refrigerationrefrigeration capacity, respectively.respectively. Within one set-point,set-point, the air volumevolume flowflow rate through the condenser is changed by about 100 rpm of the condensercondenser fan speed between didifferentfferent measuringmeasuring points.points. At the same time, the co compressormpressor speed is adjusted to maintain a constant refrigeration capacity. In this way, thethe determinationdetermination ofof the COSP is covered for the entire application range of the air-cooled VCS to identify the ideal condenser fan speed of a given set-point. Thereby, Thereby, the ideal condenser fan speed for a a set-point set-point is is spec specifiedified as as the the fan fan speed speed leading leading to to the the highest highest COSP. COSP. The results of the COSP and the associated measurementmeasurement uncertainty for one set-point are presented in Figure3 3.. AsAs reportedreported inin literature,literature, aa flatflat optimumoptimum ofof thethe COSPCOSP toto thethe condensercondenser fanfan speedspeed cancan bebe observed. With respect to the uncertainty in measur measurement,ement, only a range of fan speeds can be defined defined as ideal. Consequently, a range of ±300300 rpm is considered to be ideal in this research. ± 4.8

4.5

COSP / - 4.2

3.9 1000 2000 3000 4000 Condenser fan speed / rpm Figure 3. The coefficient coefficient of system performance (COSP) for varying condenser fan speed within a constant set-pointset-point of of 4 kW4 kW refrigeration refrigeration capacity, capa 25city,◦C ambient25 °C airambient temperature air temperature and coolant and temperature. coolant Thetemperature. error bars The define error the bars uncertainty define the in determininguncertainty in the determining COSP and indicate the COSP a range and indicate of 300 rpma range as the of ± ideal±300 rpm condenser as the fanideal speed, condenser due to fan the speed, low curvature due to the around low curvature the optimum. around the optimum. 3. Mathematical Model 3. Mathematical Model The mathematical model is developed for steady-state conditions and is based on a hybrid The mathematical model is developed for steady-state conditions and is based on a hybrid characterization of the air-cooled VCS components. Thereby, the model aims to obtain a tradeoff characterization of the air-cooled VCS components. Thereby, the model aims to obtain a tradeoff between accurate prediction of the ideal condenser fan speed and low calculation effort. between accurate prediction of the ideal condenser fan speed and low calculation effort. 3.1. Calcualtion Procedure 3.1. Calcualtion Procedure . The required model input parameters are the refrigeration capacity (Q ), ambient air temperature The required model input parameters are the refrigeration capacitye ( 𝑄 ), ambient air (T ), evaporating pressure (p ), superheating (∆T ), subcooling (∆T ), thermophysical properties temperaturec,ari,in ( 𝑇 ), evaporatinge pressure ( 𝑝sh), superheating sc( Δ𝑇 ), subcooling ( Δ𝑇 ), of the air and the,, refrigerant, and component specifications. For the model validation, a secondary thermophysical properties of the air and the refrigerant, and component specifications. For the model validation, a secondary mass flow rate through the condenser, i.e., air mass flow rate (𝑚 ) can be inserted to calculate the condensing temperature (𝑇), condensing pressure (𝑝), refrigerant mass flow rate (𝑚 ), air outlet temperature of the condenser (𝑇,), compression power (𝑃), fan power (𝑃), COP, and COSP. Thereby, the model outputs can be compared with the experimental results. In the case of the condenser fan speed optimization, all input parameters must be assumed to be constant within a single set-point. An overview of the calculation procedure is provided in Figure 4.

Energies 2020, 13, 6012 7 of 26

. mass flow rate through the condenser, i.e., air mass flow rate (mair) can be inserted to calculate the . condensing temperature (Tc), condensing pressure (pc), refrigerant mass flow rate (mR), air outlet temperature of the condenser (Tair,out), compression power (Pcom), fan power (PF), COP, and COSP. Thereby, the model outputs can be compared with the experimental results. In the case of the condenser fan speed optimization, all input parameters must be assumed to be constant within a single set-point.

AnEnergies overview 2020, 13 of, 6012 the calculation procedure is provided in Figure4. 7 of 26

Input parameter Output for validation

• 𝑇, 𝑝 Set-point (measured) • 𝑚 • 𝑝  𝑇 Model equations • 𝑇, • 𝑇 ,, • 𝑃, 𝑃 • 𝑄 • 𝐶𝑂𝑃, 𝐶𝑂𝑆𝑃 Constant input Set-point (estimated) parameter • 𝛥𝑇, 𝛥𝑇 Optimization of the Ideal condenser Component specifications condenser fan speed fan speed • Compressor efficiency • Fan characteristic • Condenser heat transfer characteristic

Thermophysical properties Optimization problem • Air formulation • Refrigerant

FigureFigure 4. 4.Overview Overview ofof thethe calculationcalculation procedure.procedure. Inpu Inputsts are are set-point set-point specifications, speciﬁcations, component component characteristicscharacteristics of theof the compressor, compressor, fan, fan, and condenserand condenser heat transfer,heat transfer, as well as as well thermophysical as thermophysical properties ofproperties the air and of the the refrigerant. air and the refrigerant. For the calculation For the ca oflculation the ideal of condenser the ideal condenser fan speed, fan the speed, input the parameters input areparameters assumed toare be assumed constant. to be constant.

3.2.3.2. Model Model Description Description

3.2.1.3.2.1. Condenser Condenser Fan Fan AssumingAssuming that that thethe motormotor torquetorque is a quadratic function function of of the the fan fan speed speed and and the the speed speed is is . proportionalproportional to to the the air air mass mass flow flow rate rate ((m𝑚air ),), the the fan fan powerpower 𝑃 =𝜃𝑚. 3 (5) PF = θFmair (5) can be calculated by the third power of the air mass flow rate and the combined fan specification 𝜃. can be calculated by the third power of the air mass flow rate and the combined fan specification θF. 3.2.2. Evaporator and Expansion Valve 3.2.2. EvaporatorNo model of and the Expansion evaporator Valve heat transfer and the expansion valve is required due to the input of theNo refrigeration model of the capacity, evaporator evaporating heat transfer pressure, and theand expansion superheating. valve Instead, is required an energy due to balance the input for of thethe refrigeration evaporator capacity,heat transfer evaporating rate pressure, and superheating. Instead, an energy balance for the evaporator heat transfer rate𝑄 =𝑚 𝑟 +𝑐̅ Δ𝑇 +𝑐̅ Δ𝑇 +𝑐̅ (𝑇 −𝑇) (6) with an isenthalpic expansion. is .applied to derive a relation between the condensing temperature Qe = mR(re + csc∆Tsc + csh∆Tsh + cec(Te Tc)) (6) and the refrigerant mass flow rate. Thereby, 𝑐̅ , 𝑐̅ , 𝑐̅ are the− mean specific heat capacity of Δ𝑇 Δ𝑇 withsubcooling, an isenthalpic superheating, expansion and is appliedboiling curve to derive from a evaporating relation between to condensing the condensing pressure, temperature , and, 𝑇 , 𝑇 , are subcooling, superheating, evaporating and condensing temperature, 𝑟 and 𝑚 are the refrigerant mass flow rate. Thereby, csc, csh, cec are the mean specific heat capacity of subcooling, specific evaporating enthalpy and refrigerant mass flow rate, respectively. The relationship is superheating, and boiling curve from evaporating to condensing pressure, ∆Tsc, ∆T , Te, Tc, are subcooling, demonstrated in Figure 5. sh

Energies 2020, 13, 6012 8 of 26

. superheating, evaporating and condensing temperature, re and mR are speciﬁc evaporating enthalpy andEnergies refrigerant 2020, 13, 6012 mass ﬂow rate, respectively. The relationship is demonstrated in Figure5. 8 of 26

𝑐̅ 𝑇 −𝑇 Δℎ 𝑝

x=0 𝑐̅ Δ𝑇 x=1

Pressure / bar / Pressure 𝑝

𝑟 𝑐̅ Δ𝑇 𝜂Δℎ

Enthalpy / kJ/kg

Figure 5. Simplified log p-h diagram of the vapor compression cycle. The terms employed for the Figure 5. Simplified log p-h diagram of the vapor compression cycle. The terms employed for the energy balance of the refrigeration capacity and the compression work are highlighted. energy balance of the refrigeration capacity and the compression work are highlighted. A modification of Equation (6) leads to the condensing temperature A modification of Equation (6) leads to the condensing temperature . r + c ∆T ̅ +c∆̅ T+c̅ T Q 1 θ e𝑇 =sc sc sh sh ec +e e =𝜃 + , 2 (7) Tc = ̅ +̅ . = θ1 + . , (7) cec cec mR mR as a function of the refrigerant mass flow rate, since the substitutions 𝜃 and 𝜃 merge model input asparameters. a function of the refrigerant mass flow rate, since the substitutions θ1 and θ2 merge model input parameters. 3.2.3. Compressor 3.2.3. Compressor The electric compression power The electric compression power 𝑃 =ℎ, −ℎ,∙𝑚 =Δℎ𝜂𝑚 (8) . 1 . Pcom = (hcom,out hcom,in) mR = ∆hiηcom− mR (8) is a function of the compressor efficiency −𝜂 , refrigerant· mass flow rate 𝑚 , and isentropic . enthalpy difference Δℎ (compare Figure 5). Neglecting the pressure losses due to superheating, is a function of the compressor efficiency ηcom, refrigerant mass flow rate mR, and isentropic enthalpy desuperheating, condensing area, and between the heat exchangers and compressor, the isentropic difference ∆hi (compare Figure5). Neglecting the pressure losses due to superheating, desuperheating, condensingenthalpy is area, defined and as: between the heat exchangers and compressor, the isentropic enthalpy is defined as: Z p Δℎ =c 𝑣𝑑𝑝. (9) ∆hi = vdp. (9) pe To find a reasonable solution for Δℎ, the real gas behavior of the specific volume

To ﬁnd a reasonable solution for ∆hi, the real gas behavior of the speciﬁc volume 𝑣=𝑧𝑅𝑇𝑝 (10) 1 described by the gas constant 𝑅, real gasv =factorzRm Tp𝑧, −temperature, and pressure is applied. For(10) an isentropic state change, the product 𝐾 =𝑧𝑅𝑇 is fairly constant, and thus a possible approach for describedan isentropic by thestate gas change constant is found:Rm, real gas factor z, temperature, and pressure is applied. For an isentropic state change, the product K1 = zRmT is fairly constant, and thus a possible approach for an isentropic state change is found: 𝜈≈𝐾| ∙𝑝 . (11) 1 ν K p− . (11) A second expression of the refrigerant≈ 1|propertiess· combines condensing pressure and temperature:A second expression of the refrigerant properties combines condensing pressure and temperature:

𝑝 =𝐾 exp(𝐾𝑇). (12) pc = K2 exp(K3Tc). (12) Further information of both approaches is provided in Appendix B. Inserting these relations into Equation (8) yields an equation for the compressor power

𝑃 =𝜂𝐾(ln(𝐾) +𝐾𝑇 −ln(𝑝))𝑚 . (13) The combination of Equations (7) and (13) gives a linear expression of the compression power

Energies 2020, 13, 6012 9 of 26

Further information of both approaches is provided in AppendixB. Inserting these relations into Equation (8) yields an equation for the compressor power

1 . Pcom = η− K (ln(K ) + K Tc ln(pe))mR. (13) com 1 2 3 − The combination of Equations (7) and (13) gives a linear expression of the compression power

1 . 1 . Pcom = η− K (ln(K ) ln(pe) + K θ )mR + η− K K θ = θ mR + θ (14) com 1 2 − 3 1 com 1 3 2 3 4 as a function of the refrigerant mass ﬂow rate and the combined input parameters θ3 and θ4.

3.2.4. Condenser For a simple characterization of the complex heat transfer of the condenser, a lumped number of transfer units (NTU) method is chosen. Considering a uniform wall temperature equal to the condensing temperature, and neglecting superheating and subcooling, the condenser heat ﬂow

. . . Q = Q Φc = m c (Tc T )(1 exp( NTU)) (15) c c,max· air p,air − c,air,in − − . . is given, where Qc,max, mair, cp,air, Tc,air,in, Φc, and NTU are the maximal possible condenser heat transfer, air mass ﬂow rate, speciﬁc heat capacity of the air, air inlet temperature, heat exchanger characteristic, and the number of transfer units, respectively. Assuming thermal resistance is only determined by the air side heat transfer, the number of transfer units

. τ kcAc α∗mair NTU = . = . (16) cp,airmair cp,airmair of the condenser can be expressed according to Ding et al. [25], where kc is the overall heat transfer coefficient and α∗ is a constant that depends on the air properties and the heat transfer area. The exponent τ is a constant reflecting the flow characteristic of the heat exchanger. With respect to the external forced convection correlation of Zukauskas [41], the exponent τ = 0.47 is defined.

3.2.5. System An energy balance of the overall cycle

. . . Q = Q + Pcom = m c (Tc T )Φc (17) c e air p,air − c,air,in reveals in combination with Equations (7) and (14), the coupling of the air and the refrigerant mass

ﬂow rate: . . . . maircp,airΦcθ2 Qe + θ3mR + θ4 = maircp,airΦc(θ1 Tc,air,in) + . . (18) − mR Therefore, the refrigerant mass ﬂow rate s 2 . θ6 θ6 . mR = θ = f m , input (19) − 2 − 2 − 7 air with the substitutions

. . . θ4 + Qe + mairΦccp,air(Tc,air,in θ1) mairΦccp,airθ2 θ6 = − , θ7 = (20) θ3 θ3 of the input parameters, is a direct function of the air mass ﬂow rate and the current set-point. Thus, a necessary term for the direct solution of the optimization problem is identiﬁed. Energies 2020, 13, 6012 10 of 26

3.3. Formulation of the Condenser Fan Speed Optimization Problem The optimization of the condenser fan speed intends to minimize the total energy consumption of the VCS at constant refrigeration capacity and secondary ﬂuid outlet temperature at the evaporator, i.e., coolant outlet temperature Tcool,out. The mathematical formulation of the problem is:

. . 3 Min(Pcom + PF) = Min θ3mR + θ4 + θFmair (21) with subject to . Qe, Tcool,out = constant. (22) A reasonable solution is to derive the cost function according to the air mass ﬂow rate of the condenser . ∂PF ∂Pcom ∂PF ∂Pcom ∂mR . + . = . + . . = 0 (23) ∂mair ∂mair ∂mair ∂mR ∂mair and to set it to zero. The chain rule is obtained to calculate the derivation of the compression power to air mass ﬂow rate. Assuming constant input parameters, the derivation

. . ∂PF ∂Pcom ∂mR . 2 ∂mR . + . . = 3θFmair,id + θ3 . = 0 (24) ∂mair ∂mR ∂mair ·∂mair can be expressed using the model Equations (5) and (14). The coupling function derivation

. ∂mR . . = f mair, set point (25) ∂mair − is an implicit function of the air mass ﬂow rate and the operating conditions, and its development is described in AppendixC. Subsequently, an implicit optimization term for the ideal air mass ﬂow rate

. !0.5 . θ3 ∂mR . mair,id = . = f mair, set point (26) −3θF ∂mair − is attained. Indeed, Equation (20) can easily be solved by ﬁx point iteration in Matlab or Microsoft Excel, for example.

4. Model Validation The model validation is demonstrated for the air-cooled VCS described in Section 2.1 and carried out of 250 steady-state measuring points. The deviation of the experimental and the model data are discussed for two diﬀerent cases:

1. Component model validation (varying model inputs) 2. Condenser fan speed optimization validation (constant model inputs)

In detail, this means that for case 1, the input variables are taken from measurement, except the coefficient of the condenser heat transfer, i.e., α∗ and the refrigerant fitting parameters K1, K2, K3, which are constant, respectively. For the validation of the condenser fan speed optimization (case 2), all input parameters are constant within an operation point. Furthermore, the compressor efficiency, subcooling, superheating, and the specific heat capacities of the refrigerant are constant for all set-points. The evaporating pressure, temperature, and enthalpy vary between different set-points. By subdividing into case 1 and 2, the error sources can be evaluated for the components and the optimization scenario, separately. The maximal deviation, the relative root mean square error (R-RMSE), and the coefficient of determination R2 are summarized in Table2. The statistic calculation background is detailed in AppendixD. Energies 2020, 13, 6012 11 of 26

Table 2. Maximum deviation, R-RMSE, and coeﬃcient of determination R2 for the model approach. The ﬁrst value is for varying model inputs (case 1) and the second for constant model inputs (case 2).

. Variable Tc Pcom PF Tc,air,out mR COSP Max deviation in K; kW; 0.6/4.5 0.12/0.2 0/0.02 1.3/- 2.39/4.65 0.34/0.57 kg/h, - R-RMSE in % 0.72/4.36 2.65/7.67 0/4.15 2.73/- 0.69/2.2 2.94/6.12 R2 in % 99.93/97.07 99.69/96.69 1/99.67 99.78/- 99.67/99.29 98.28/92.59

4.1. Condenser Fan

EnergiesThe 2020 condenser, 13, 6012 fan power prediction and the test data are plotted in Figure6. A constant input11 of 26 of the fan characteristic θF induces a coeﬃcient of determination and a maximal deviation of 99.67% andand 0.020.02 kW, whilewhile aa varying input of θ𝜃F results results in in no no error errorand and isis thereforetherefore notnot depicted.depicted. Evidently, thethe modelmodel describesdescribes thethe fanfan powerpower withwith susufficientﬃcient accuracyaccuracyfor fora acycle cycle analysis. analysis.

0.5 Constant input +10 % 0.375

-10 % 0.25

0.125 Modelfan power /kW

0 0 0.125 0.25 0.375 0.5

Measured fan power / kW 𝜃 Figure 6.6. Experimental validation of the condenser fanfan power for aa constantconstant fanfan characteristiccharacteristic θF..

4.2.4.2. Evaporator andand ExpansionExpansion ValveValve TheThe hybridhybrid modelmodel inin SectionSection 3.23.2 neitherneither includesincludes anan evaporatorevaporator heatheat transfertransfer modelmodel nornor anan expansionexpansion valvevalve model, model, as theas the evaporating evaporating pressure pressu andresuperheating and superheatingare inputs are of inputs the model. of the Therefore, model. aTherefore, reasonable a validationreasonable of validation the assumptions of the forassumpti the expansionons for valvethe expansion and the evaporator valve and is the the evaporator evaluation ofis thethe condensingevaluation temperatureof the condensing in Equation temperature (7). The results in Equation for case 1(7). and The 2 are results presented for case in Figure 1 and7. 2 are presentedA maximal in Figure deviation 7. of 0.6 K, an R-RMSE of 0.72%, and a coefficient of determination of 99.93% are performed for varying input parameters. The discrepancy may be caused by the measurement uncertainty of the inputs and the assumption of an isenthalpic expansion. For a constant input, the error 70 70 2 C increasesC to a maximal deviation of 4.5 K, a R-RMSE of 4.36, and a R of 97.07%, respectively. However, ° ° a sufficient predictionVarying input of the condensing temperature is obtained.Constant Consequently, input the input parameters +2 K +2 K in Equation60 (7), i.e., superheating, subcooling, evaporating60 pressure, and coolant load do not change significantly within a constant set-point or barely affect the condensing temperature. In fact, the main source of error for the VCS under consideration is the change in subcooling, whereas superheating-2 K and subcooling50 remain fairly constant. -2 K 50

40 40

30 30 Modelcondensing temperature/ Modelcondensing temperature/ 30 40 50 60 70 30 40 50 60 70 Measured condensing temperature / Measured condensing temperature / °C ° C (a) (b)

Figure 7. Experimental validation of the evaporator and the expansion valve. (a): Varying input parameters (b): Constant input parameters. The accuracy decreases for constant input parameters.

A maximal deviation of 0.6 K, an R-RMSE of 0.72%, and a coefficient of determination of 99.93% are performed for varying input parameters. The discrepancy may be caused by the measurement uncertainty of the inputs and the assumption of an isenthalpic expansion. For a constant input, the error increases to a maximal deviation of 4.5 K, a R-RMSE of 4.36, and a 𝑅 of 97.07%, respectively. However, a sufficient prediction of the condensing temperature is obtained. Consequently, the input

Energies 2020, 13, 6012 11 of 26 and 0.02 kW, while a varying input of 𝜃 results in no error and is therefore not depicted. Evidently, the model describes the fan power with sufficient accuracy for a cycle analysis.

0.5 Constant input +10 % 0.375

-10 % 0.25

0.125 Modelfan power /kW

0 0 0.125 0.25 0.375 0.5

Measured fan power / kW

Figure 6. Experimental validation of the condenser fan power for a constant fan characteristic 𝜃.

4.2. Evaporator and Expansion Valve The hybrid model in Section 3.2 neither includes an evaporator heat transfer model nor an expansion valve model, as the evaporating pressure and superheating are inputs of the model. Therefore, a reasonable validation of the assumptions for the expansion valve and the evaporator is the evaluation of the condensing temperature in Equation (7). The results for case 1 and 2 are presentedEnergies 2020 in, 13 Figure, 6012 7. 12 of 26

70 70 C C ° ° Varying input Constant input +2 K +2 K 60 60

-2 K 50 -2 K 50

40 40

30 30 Modelcondensing temperature/ Modelcondensing temperature/ 30 40 50 60 70 30 40 50 60 70 Measured condensing temperature / Measured condensing temperature / °C °C Energies 2020, 13, 6012 12 of 26 (a) (b) parameters in Equation (7), i.e., superheating, subcooling, evaporating pressure, and coolant load do FigureFigurenot change 7. ExperimentalExperimental significantly validation validationwithin a constant of of the the evaporator set-point or and barely the affect expansion the condensing valve. ( (aa):): temperature.Varying Varying input input In fact, the main source of error for the VCS under consideration is the change in subcooling, whereas parametersparameters ( (bb):): Constant Constant input input parameters. parameters. The The accuracy accuracy decreases decreases for for constant constant input input parameters. parameters. superheating and subcooling remain fairly constant. 4.3. CompressorA maximal deviation of 0.6 K, an R-RMSE of 0.72%, and a coefficient of determination of 99.93% 4.3. Compressor are performedFigure8 depicts for varying the comparison input parameters. of the compression The discrepancy power may predicted be caused by theby the model measurement approach Figure 8 depicts the comparison of the compression power predicted by the model approach uncertaintyand measured of the in theinputs experiments. and the assumption For varying of model an isenthalpic inputs, the expansion. error due Fo tor thea constant refrigerant input, model, the and measured in the experiments. For varying model inputs, the error due 𝑅to the refrigerant model, errorcondensing increases temperature to a maximal calculation, deviation andof 4.5 uncertainty K, a R-RMSE in measurementof 4.36, and a of allof input97.07%, parameters respectively. are However,condensing a sufficient temperature prediction calculation, of the condensing and uncertainty temperature in measurement is obtained. of all Consequently,input parameters the are input included.included. However, However, the the main mainerror error is caused caused by by th thee modeling modeling of the of specific the specific volume, volume, especially especially for for higher pressure. In this context, the validation of refrigerant model is provided in AppendixB. higher pressure. In this context, the validation of refrigerant model is provided in Appendix B. Nevertheless,Nevertheless, the refrigerantthe refrigerant properties properties approach approach in Equations in Equations (11) and (11) (12) and is suitable(12) is forsuitable compression for powercompression calculation power achieving calculation a R-RMSE achieving of 2.65% a R-RMSE and a coe of ffi2.65%cient and of determination a coefficient of ofdetermination 99.69%. If constant of input99.69%. parameters If constant are taken input into parameters account, theare accuracy taken into of theaccount, prediction the accuracy deteriorates, of the particularly prediction due to thedeteriorates, constant compressor particularly eduefficiency, to the constant but still compre leads tossor a coeefficiency,fficient but of still determination leads to a coefficient of 96.69% of and determination of 96.69% and thus can be used for system analysis. This finding is in good agreement thus can be used for system analysis. This finding is in good agreement with the results of Larsen [7], with the results of Larsen [7], who also used constant compressor efficiency and came to the same who also used constant compressor efficiency and came to the same conclusion carrying out the conclusion carrying out the validation of a supermarket refrigerator. validation of a supermarket refrigerator.

2.5 2.5 Varying input Constant input +5 % +10 % 2 2

-5 % 1.5 1.5 -10 %

1 1 Model compressor power / kW / power compressor Model Model compressor power / kW / power compressor Model 0.5 0.5 0.5 1 1.5 2 2.5 0.511.522.5 Measured compressor power / kW Measured compressor power / kW

(a) (b)

FigureFigure 8. Experimental 8. Experimental validation validation of the of compressor the compressor power. power. (a): Varying (a): Varying input input parameters. parameters. (b): Constant(b): inputConstant parameters. input High parameters. prediction High accuracyprediction despite accuracy constant despite constant compressor compressor eﬃciency. efficiency.

4.4. Condenser For the validation of the condenser heat transfer modeling, the air outlet temperature according to Chi and Didion [42] ∗ 𝛼 𝑚 𝑇,, =𝑇 +𝑇,, −𝑇exp− (27) 𝑐,𝑚 can be employed. The results are highlighted in Figure 9. The measured condensing temperature is used as input in Equation (11) to provide an isolated consideration of the condenser model prediction error. Thereby, a R-RMSE and a coefficient of determination of 2.73% and 99.78% are performed, respectively. With respect to the expanded measurement uncertainty of 0.55 K of the air outlet temperature, a maximal deviation of 1.3 K reveals high accuracy despite rough assumptions.

Energies 2020, 13, 6012 13 of 26

4.4. Condenser For the validation of the condenser heat transfer modeling, the air outlet temperature according to Chi and Didion [42]  . τ   α∗m  T = T + (T T )  air  c,air,out c c,air,in c exp .  (27) − −cp,airmair can be employed. The results are highlighted in Figure9. The measured condensing temperature is used as input in Equation (11) to provide an isolated consideration of the condenser model prediction error. Thereby, a R-RMSE and a coeﬃcient of determination of 2.73% and 99.78% are performed, respectively. With respect to the expanded measurement uncertainty of 0.55 K of the air outlet temperature, a maximal deviation of 1.3 K reveals high accuracy despite rough assumptions. Energies 2020, 13, 6012 13 of 26 Energies 2020, 13, 6012 13 of 26

C 60 ° C ° +1.5 K +1.5 K 50 50

-1.5 K -1.5 K 40 40

30 30 Model air outlet temperature / temperature air outlet Model

Model air outlet temperature / temperature air outlet Model 20 20 20 30 40 50 60 20 30 40 50 60 Measured air outlet temperature / °C Measured air outlet temperature / °C

FigureFigure 9. Experimental 9. Experimental validation validation of of the the condensercondenser air air outlet outlet temperat temperature.ure. High Highaccuracy accuracy despite despite Figure 9. Experimental validation of the condenser air outlet temperature. High accuracy despite rough assumptionsrough assumptions is obtained. is obtained. rough assumptions is obtained. 4.5. System4.5. System 4.5. System Model predictions and test data of the refrigerant mass flow rate are provided in Figure 10. With ModelModel predictions predictions and and test test data data of the the refrigerant refrigerant mass mass flow rate flow are rateprovided are providedin Figure 10. in With Figure 10. respect to Equation (19), the mass flow rate calculation includes the errors due to the component With respectrespect to to Equation Equation (19), the the mass mass flow flow rate rate calculation calculation includes includes the errors the errors due to due the tocomponent the component modeling of the evaporator, compressor, and condenser. However, for varying input parameters, the modeling of the evaporator, compressor, and condenser. However, for varying input parameters, the modelingcoefficient of the of evaporator, determination compressor, is 99.67%, and and for constant condenser. inputs However, is 99.27%. for varying input parameters, the coefficoefficientcient of of determination determination is is 99.67%, 99.67%, and and for forconstant constant inputs inputs is 99.27%. is 99.27%.

160 160 160 160 Varying input Constant input Varying input Constant input +2.5 % +5 % +2.5 % +5 % 130 130 130 130 -2.5 % -5 % -2.5 % -5 % / kg/h / / kg/h / / kg/h / 100 kg/h / 100 100 100

70 70 70 Model refrigerant mass flow rate rate flow mass refrigerant Model Model refrigerant mass flow rate rate flow mass refrigerant Model 70 100 130 160 70 Model refrigerant mass flow rate rate flow mass refrigerant Model Model refrigerant mass flow rate rate flow mass refrigerant Model 70 100 130 160 70 100 130 160 70 100 130 160 Measured refrigerant mass flow rate / Measure d re frige rant mass flow rate / Measured refrigerant mass flow rate / Measure d re frige rant mass flow rate / kg/h kg/h kg/h kg/h (a) (b) (a) (b) Figure 10. Experimental validation of the refrigerant mass flow rate. (a): Varying input parameters. Figure 10. Experimental validation of the refrigerant mass flow rate. (a): Varying input parameters. Figure( 10.b): ConstantExperimental input parameters. validation of the refrigerant mass ﬂow rate. (a): Varying input parameters. (b): Constant input parameters. (b): Constant input parameters. Figure 11 highlights the results for the model estimation and the test data of the COSP for Figure 11 highlights the results for the model estimation and the test data of the COSP for varying and constant input parameters. The prediction errors of the component models superpose varying and constant input parameters. The prediction errors of the component models superpose for the COSP calculation. Nevertheless, for varying input, a COSP prediction error within ±10% and for the COSP calculation. Nevertheless, for varying input, a COSP prediction error within ±10% and a R-RMSE of 2.94% are obtained. Hence, the model can be considered as a model with good accuracy a R-RMSE of 2.94% are obtained. Hence, the model can be considered as a model with good accuracy [13]. [13].

Energies 2020, 13, 6012 14 of 26

Figure 11 highlights the results for the model estimation and the test data of the COSP for varying and constant input parameters. The prediction errors of the component models superpose for the COSP calculation. Nevertheless, for varying input, a COSP prediction error within 10% and a R-RMSE of ± Energies2.94% are2020 obtained., 13, 6012 Hence, the model can be considered as a model with good accuracy [13]. 14 of 26

6 6 Varying input Constant input +10 % +10 % 5 5

-10 % 4 -10 % 4 ModelCOSP / - ModelCOSP / - 3 3

2 2 23456 23456 Measured COSP / - Measured COSP / -

(a) (b)

Figure 11. Experimental validation of the COSP. ((aa):): VaryingVarying inputinput parameters.parameters. (b): Constant input parameters. The prediction accuracy de decreasescreases for constant input parameters.

On thethe otherother hand, hand, the the proposed proposed hybrid hybrid model model does does neither neither simulate simulate the evaporator the evaporator heat transfer heat transfernor the nor expansion the expansion valve characteristics. valve characteristics. As a consequence,As a consequence, the model the model requires requires the inputthe input of the of theevaporating evaporating pressure pressure and and superheating, superheating, which which limits limits the comparability. the comparability. However, However, for constant for constant input, input,parameters parameters over 93% over of 93% the predicted of the predicted COSP are COSP still are within still within10% deviation ±10% deviation and the R-RMSEand the R-RMSE is 6.12%. ± isTherefore, 6.12%. Therefore, the results the are results more are accurate more thanaccurate themodel than the results model of results Chan and of Chan Yu [6 ]and or BraunYu [6] etor al. Braun [16], etfor al. example. [16], for Moreover,example. Moreover, the model the is developedmodel is developed to predict to the predict ideal the condenser ideal condenser fan speed fan within speed a withinconstant a set-point,constant set-point, and in this and respect, in this only respect, small changesonly small in evaporatingchanges in pressureevaporating and pressure superheating and superheatingcan be expected. can be expected.

4.6. Prediction Prediction of of the the Ideal Ideal Condenser Fan Speed The aim of the model approach in Section 33 isis thethe predictionprediction ofof thethe idealideal condensercondenser fanfan speedspeed according to to Equation Equation (26). (26). The The results results for forthe themeas measuredured and the and calculated the calculated ideal condenser ideal condenser fan speed fan arespeed compared are compared in Figure in Figure 12 and 12 andoutline outline a deviation a deviation within within a range a range of of ±300300 rpm, rpm, respectively. respectively. ± Considering the measurementmeasurement uncertaintyuncertainty (compare(compare Section Section 2.3 2.3),), the the predictions predictions appear appear to to be be ideal ideal in interms terms of of energy energy effi efficiencyciency for for varying varying and and constant constant input input parameters. parameters. Based on the results of the model validation in Section 4.5, the high accuracy for the varying input is not particularly4000 surprising. In contrary, the estimation4000 accuracy for the constant input may be surprising, bearingVarying in mind input that the COSP estimation error forConstant constant input and varying input in Figure 11 differs significantly. However, the results of the measured and the predicted+300 COSPrpm for a single set-point 3500 +300 rpm 3500 in Figure 13 point out that the absolute value of COSP vary for a constant input, whereas shape and position in terms of condenser fan speed are comparable. Consequently, some input parameters influence the3000 prediction error with regard-300 rpm to COSP, but3000 influence less the ideal fan-300 speed. rpm In order to explain this circumstance, the impacts of the input parameters are examined in the following. 2500 2500 Model ideal fan speed ideal/ fan Modelrpm Model ideal fan speed / rpm / speed fan ideal Model

2000 2000 2000 2500 3000 3500 4000 2000 2500 3000 3500 4000 Measured ideal fan speed / rpm Measured ideal fan speed / rpm

(a) (b)

Figure 12. Comparison of the measured and the calculated ideal condenser fan speed. (a): Varying input parameters. (b): Constant input parameters. The prediction accuracy is comparable for both cases and is within the uncertainty range of ±300 rpm.

Energies 2020, 13, 6012 14 of 26

6 6 Varying input Constant input +10 % +10 % 5 5

-10 % 4 -10 % 4 ModelCOSP / - ModelCOSP / - 3 3

2 2 23456 23456 Measured COSP / - Measured COSP / -

(a) (b)

Figure 11. Experimental validation of the COSP. (a): Varying input parameters. (b): Constant input parameters. The prediction accuracy decreases for constant input parameters.

On the other hand, the proposed hybrid model does neither simulate the evaporator heat transfer nor the expansion valve characteristics. As a consequence, the model requires the input of the evaporating pressure and superheating, which limits the comparability. However, for constant input, parameters over 93% of the predicted COSP are still within ±10% deviation and the R-RMSE is 6.12%. Therefore, the results are more accurate than the model results of Chan and Yu [6] or Braun et al. [16], for example. Moreover, the model is developed to predict the ideal condenser fan speed within a constant set-point, and in this respect, only small changes in evaporating pressure and superheating can be expected.

4.6. Prediction of the Ideal Condenser Fan Speed The aim of the model approach in Section 3 is the prediction of the ideal condenser fan speed according to Equation (26). The results for the measured and the calculated ideal condenser fan speed are compared in Figure 12 and outline a deviation within a range of ±300 rpm, respectively. ConsideringEnergies 2020, 13 the, 6012 measurement uncertainty (compare Section 2.3), the predictions appear to be15 ideal of 26 in terms of energy efficiency for varying and constant input parameters.

4000 4000 Varying input Constant input +300 rpm 3500 +300 rpm 3500

3000 -300 rpm 3000 -300 rpm

Energies 2020, 13, 6012 15 of 26 2500 2500 Model ideal fan speed ideal/ fan Modelrpm Basedrpm / speed fan ideal Model on the results of the model validation in Section 4.5, the high accuracy for the varying input is not2000 particularly surprising. In contrary, the estimation2000 accuracy for the constant input may be surprising, bearing2000 2500 in mind 3000 that 3500the COSP 4000 estimation2000 error 2500for constant 3000 and 3500 varying 4000 input in Figure 11 differs significantly.Measured ideal fanHowever, speed / rpm the results of the measuredMeasured idealand fan the speed predicted / rpm COSP for a

single set-point in Figure (13a) point out that the absolute value of COSP(b) vary for a constant input, whereas shape and position in terms of condenser fan speed are comparable. Consequently, some inputFigureFigure parameters 12. ComparisonComparison influence of of thethe the predmeasured iction and error the with calculated regard ideal to COSP condenser , but fan influence speed. less( (aa):): Varying Varyingthe ideal fan speed.inputinput In orderparameters. parameters. to explain ( (bb):): Constant Constantthis circumstance, input input parameters. parameters. the im Thepacts The prediction prediction of the input accuracy accuracy parameters is is comparable comparable are examined for for both both in the casescases and and is is within within the the uncertainty uncertainty range range of ±300300 rpm. rpm. following. ±

4.25

3.625 COSP / - Measured Model: Constant input Model: Varying input

3 1000 2000 3000 4000

Condenser fan speed / rpm FigureFigure 13.13.Measured Measured and and predicted predicted COSP COSP of varyingof varying and an constantd constant input input demonstrated demonstrated for one for set-point. one set- Thepoint. model The inputmodel parameters input parameters inﬂuence influence the COSP, the butCOSP, hardly but thehardly ideal the condenser ideal condenser fan speed. fan speed.

5.5. CondenserCondenser FanFan SpeedSpeed OptimizationOptimization 5.1. Influences on the Ideal Condenser Fan Speed 5.1. Influences on the Ideal Condenser Fan Speed The ideal condenser fan speed is influenced by various variables, and according to Equation (26), The ideal condenser fan speed is influenced by various variables, and according to Equation all input parameters of the model in Figure4 have an impact. Strong e ffects have already been reported (26), all input parameters of the model in Figure 4 have an impact. Strong effects have already been for the ambient temperature and the refrigeration capacity, while the influences of the other values like reported for the ambient temperature and the refrigeration capacity, while the influences of the other the superheating, subcooling, compressor efficiency, and the suction pressure are not yet clear. values like the superheating, subcooling, compressor efficiency, and the suction. pressure are not yet To investigate the effect of a parameter x to the ideal air mass flow rate m and consequently to clear. i air,id the ideal condenser fan speed, the formulation of a sensitivity coefficient To investigate the effect of a parameter 𝑥 to the ideal air mass flow rate 𝑚 , and . consequently to the ideal condenser fan speed, the formulation 0.5! of a sensitivity coefficient ∂mair,id ∂ Ψ = . (28) 𝜕𝑚∂ x,i ∂x𝜕i 3θΨF = (28) 𝜕𝑥 𝜕𝑥 3𝜃 with

𝜕𝑃 Ψ= (29) 𝜕𝑚

can be defined. For a constant fan characteristic 𝜃, the sensitivity coefficient . 𝜕𝑚 , Ψ 1 𝜕Ψ =0.5 =𝑓𝑚 ,,𝑥,𝑠𝑒𝑡 𝑝𝑜𝑖𝑛𝑡 (30) 𝜕𝑥 3𝜃 3𝜃 𝜕𝑥

is yielded as a function of the current set-point, the parameter 𝑥, and the ideal air mass flow rate. Thereby, Equation (30) clarifies that only a change of the gradient of the compressor power to the air

Energies 2020, 13, 6012 16 of 26 with ∂Pcom Ψ = . (29) ∂mair can be deﬁned. For a constant fan characteristic θF, the sensitivity coeﬃcient

. 0.5 ! ∂mair,id Ψ − 1 ∂Ψ . = 0.5 = f mair,id, xi, set point (30) ∂xi 3θF 3θF ∂xi is yielded as a function of the current set-point, the parameter xi, and the ideal air mass flow rate. Thereby, Equation (30) clarifies that only a change of the gradient of the compressor power to the air mass flow rate impacts the ideal fan speed. The shape of the compression power in Figure1 illustrates this gradient. It can be seen that an increase in the negative slope would move the ideal fan speed to higher values. The gradient itself depends on the heat transfer rate at the condenser, as an adjustment of the condenser fan speed is only reasonable from an energetic point of view, if the influence of the fan speed on the condensing pressure and thus on the compression power changes. Consequently, the ideal condenser fan speed changes solely, if the input parameter xi modifies the condensing pressure. Thereby, surging condensing pressure causes a higher ideal condenser fan speed. In general, it is fair to assume that the condensing pressure is more influenced by the ambient temperature and the refrigeration capacity than by the subcooling, superheating, compressor efficiency, or the evaporating pressure. However, to quantify this general consideration, the validated model can be used. Therefore, the absolute change of the ideal condenser fan speed is investigated at one reference set-point and highlighted in Figure 14. Here, the adjustment scope of the input parameters is selected with regard to the operating range of the VCS under consideration. With respect to the input values, the total changes of the ideal fan speed are 1100 rpm, 620 rpm and 280 rpm for an increasing refrigeration capacity, ambient temperature, and evaporating pressure, respectively. Changing the compressor efficiency from 0.4 to 0.6 causes a fan speed adjustment of 220 rpm. This small effect is in line with previous results [7]. The modification of the lumped heat transfer coefficient of 30% leads to a change of the fan speed of 130 rpm. Even lower is the effect of the subcooling and the superheating with 120 rpm and 70 rpm. In addition, the common application range for the subcooling and the superheating is much smaller than 15 K. Yet, the model assumption ignores the repercussions of the superheating on the oil circulation, compressor efficiency, and the evaporator heat transfer rate. A change in the heat transfer coefficient of the condenser due to the changing subcooling is also not addressed. However, there is a strong probability that the influence on the condensing pressure is still small, even if the aforementioned cross-coupling is taken into account. In fact, the prediction of the ideal fan speed in Figure 12 provides strong evidence here, as the accuracy is comparable for both constant and varying input of subcooling and superheating. In terms of the hybrid model approach, the distribution in Figure 14 clearly explains the reason for the high prediction accuracy of the ideal condenser fan speed at constant input parameters, since the coolant load and ambient temperature are constant within a constant set-point and the suction pressure is coupled by the secondary side of the evaporator with only slight changes. Moreover, the remaining input parameters barely affect the ideal fan speed and vary only within a small range under real application conditions. Despite the exact values only being valid for the VCS under consideration at the reference point, the distribution in Figure 14 does not significantly change for different references and point to the likelihood that the relative order of magnitude may be generalized to other air-cooled VCS with similar characteristics. In sum, the analysis of the impact of different input parameters on the ideal condenser fan speed reveals high effect of the refrigeration capacity and ambient temperature, moderate impact of the evaporating pressure and compressor efficiency, while the lumped heat transfer coefficient of condenser, subcooling, and superheating have minor impacts. These results are not particularly surprising given the fact that the condensing pressure is the most decisive value for the condenser fan Energies 2020, 13, 6012 16 of 26 mass flow rate impacts the ideal fan speed. The shape of the compression power in Figure 1 illustrates this gradient. It can be seen that an increase in the negative slope would move the ideal fan speed to higher values. The gradient itself depends on the heat transfer rate at the condenser, as an adjustment of the condenser fan speed is only reasonable from an energetic point of view, if the influence of the fan speed on the condensing pressure and thus on the compression power changes. Consequently, the ideal condenser fan speed changes solely, if the input parameter 𝑥 modifies the condensing pressure. Thereby, surging condensing pressure causes a higher ideal condenser fan speed. In general, it is fair to assume that the condensing pressure is more influenced by the ambient temperature and the refrigeration capacity than by the subcooling, superheating, compressor efficiency, or the evaporating pressure. However, to quantify this general consideration, the validated model can be used. Therefore, the absolute change of the ideal condenser fan speed is investigated at one reference set-point and highlighted in Figure 14. Here, the adjustment scope of the input parameters is selected with regard to the operating range of the VCS under consideration. With respect to the input values, the total changes of the ideal fan speed are 1100 rpm, 620 rpm and 280 rpm for an increasing refrigeration capacity, ambient temperature, and evaporating pressure, respectively. Changing the compressor efficiency from 0.4 to 0.6 causes a fan speed adjustment of 220 rpm. This small effect is in line with previous results [7]. The modification of the lumped heat transfer coefficient of 30% leads to a change ofEnergies the fan2020 speed, 13, 6012 of 130 rpm. Even lower is the effect of the subcooling and the superheating with17 of120 26 rpm and 70 rpm. In addition, the common application range for the subcooling and the superheating is much smaller than 15 K. Yet, the model assumption ignores the repercussions of the superheating onspeed the optimization.oil circulation, Nevertheless, compressor theseefficiency, findings and clarify the evaporator that only parametersheat transfer having rate. aA high change impact in the on heatthe condensing transfer coefficient pressure needof the to becondenser considered due for to the the ideal changing condenser subcooling fan speed is prediction. also not addressed. As a result, However,an evaporator there model is a strong is not requiredprobability and that the the compressor influence e ffionciency, the condensing subcooling, pressure and superheating is still small, can evenbe kept if the constant, aforementioned resulting in cross-coupling a simple, high precisionis taken modelinto account. and a directly In fact, solvable the prediction system ofof equations.the ideal fanHence, speed an in explanation Figure 12 provides is provided strong for evidence the high accuracyhere, as the of theaccuracy ideal condenseris comparable fan for speed both prediction, constant andalthough varying some input rough of subcooling assumptions and are superheating. made, and only limited plant specifications are required.

1200 Set point: Application range:

900

600

300 condenser fan speed condenser / rpm fan Absolute change of the ideal the ideal of change Absolute

Figure 14. The absolute change of the ideal condenser fan speed for a given application range and Figure 14. The absolute change of the ideal condenser fan speed for a given application range and a a referring set-point. The results are based on the validated model approach. The influence of referring set-point. The results. are based on the validated model approach. The influence of the the refrigeration capacity (Qe) and the ambient temperature (Tc,air,in) are dominant, followed by the refrigeration capacity (𝑄 ) and the ambient temperature (𝑇,,) are dominant, followed by the evaporating pressure (pe) and the compressor efficiency (ηcom). evaporating pressure (𝑝) and the compressor efficiency (𝜂). 5.2. Control Strategy and Energy Saving Potential In terms of the hybrid model approach, the distribution in Figure 14 clearly explains the reason The proposed optimization method can be employed to control the condenser fan speed of for the high prediction accuracy of the ideal condenser fan speed at constant input parameters, since air-cooled vapor compression systems. Therefore, a control strategy with two separated control loops is suggested and depicted in Figure 15. The first loop adjusts the compressor speed to obtain the required refrigeration capacity and the second loop controls the condenser fan speed according to the model approach. Therefore, uncertainties of the model predictions only have an influence on the fan speed (and on the COSP), but not on the refrigeration capacity. During operation, the described input parameters of the model (Figure4) must be partially measured (refrigeration capacity, air inlet temperature and evaporating pressure), partially estimated (subcooling and superheating), and the rest is taken from component specifications and physical properties. However, for the optimization within a set-point, all inputs are assumed to be constant. The proposed control strategy in Figure 15 is applied to the VCS described in Section 2.1 and obtains an energy efficiency loss below 1.5% compared to the ideal operation for all fifteen set-points. The results are highlighted in Figure 16 and outline small deviations for all considered conditions. As a drawback, there is no guarantee that the exact, ideal operation is reached. However, the investigation in Section 5.1 indicates the prediction accuracy to be quite robust. Consequently, the hybrid model is sufficient to be used in optimization control strategies and due to the simplicity of the model it is expected that it can be used for real time applications. Especially for applications with a quasi-stationary coolant demand, the proposed method represents a promising alternative to extremum seeking control or other advanced methods that may be more accurate but have a high convergence time. Furthermore, the method can be easily adapted to other air-cooled vapor compression systems for optimization purposes and only limited input parameters are necessary. In fact, the required input Energies 2020, 13, 6012 17 of 26 the coolant load and ambient temperature are constant within a constant set-point and the suction pressure is coupled by the secondary side of the evaporator with only slight changes. Moreover, the remaining input parameters barely affect the ideal fan speed and vary only within a small range under real application conditions. Despite the exact values only being valid for the VCS under consideration at the reference point, the distribution in Figure 14 does not significantly change for different references and point to the likelihood that the relative order of magnitude may be generalized to other air-cooled VCS with similar characteristics. In sum, the analysis of the impact of different input parameters on the ideal condenser fan speed reveals high effect of the refrigeration capacity and ambient temperature, moderate impact of the evaporating pressure and compressor efficiency, while the lumped heat transfer coefficient of condenser, subcooling, and superheating have minor impacts. These results are not particularly surprising given the fact that the condensing pressure is the most decisive value for the condenser fan speed optimization. Nevertheless, these findings clarify that only parameters having a high impact on the condensing pressure need to be considered for the ideal condenser fan speed prediction. As a result, an evaporator model is not required and the compressor efficiency, subcooling, and superheating can be kept constant, resulting in a simple, high precision model and a directly solvable system of equations. Hence, an explanation is provided for the high accuracy of the ideal condenser fan speed prediction, although some rough assumptions are made, and only limited plant specifications are required.

5.2. Control Strategy and Energy Saving Potential The proposed optimization method can be employed to control the condenser fan speed of air- cooled vapor compression systems. Therefore, a control strategy with two separated control loops is suggested and depicted in Figure 15. The first loop adjusts the compressor speed to obtain the required refrigeration capacity and the second loop controls the condenser fan speed according to Energiesthe model2020, 13approach., 6012 Therefore, uncertainties of the model predictions only have an influence 18on of the 26 fan speed (and on the COSP), but not on the refrigeration capacity. During operation, the described input parameters of the model (Figure 4) must be partially measured (refrigeration capacity, air inlet parameters are usually known during the development process of VCS. Besides real time application, temperature and evaporating pressure), partially estimated (subcooling and superheating), and the the calculation method can be further used to determine the ideal condensing pressure and ideal fan rest is taken from component specifications and physical properties. However, for the optimization speed to adjust the parameters of the control method proposed by Chan and Yu et al. [6], for example. within a set-point, all inputs are assumed to be constant.

Measured n Q , T,,, p,

condenser

ΔT, ΔT T,, Component specification Thermophysical properties

TXV electric M compressor Compressor n Q Control , evaporator p,

Energies 2020, 13, 6012 Q 18 of 26

V T, T, The proposed control strategy in Figure 15 is applied to the VCS described in Section 2.1 and obtains an energy efficiency loss below 1.5% compared to the ideal operation for all fifteen set-points. FigureFigure 15.15. ControlControl strategystrategy forfor thethe set-pointset-point optimizationoptimization ofof aa VCS,VCS, basedbased onon thethe proposedproposed modelmodel toto The results are highlighted in Figure 16 and outline small deviations for all considered conditions. calculatecalculate thethe idealideal condensercondenser fanfan speed.speed. RequiredRequired measuringmeasuring spotsspots areare marked.marked.

5.5

+2.5 % 4.5 -2.5 %

3.5 Model obtained ideal COSP / - ideal COSP obtained Model 2.5 2.5 3.5 4.5 5.5 Measured ideal COSP / -

FigureFigure 16.16. ExperimentalExperimental validationvalidation ofof thethe idealideal COSPCOSP basedbased onon thethe calculatedcalculated idealideal fanfan speedspeed andand thethe measuredmeasured idealideal COSPCOSP forfor fifteenfifteen didiffefferentrent constant constant operation operation conditions. conditions. 6. Conclusions As a drawback, there is no guarantee that the exact, ideal operation is reached. However, the investigationIn this research in Section study, 5.1 a steady-state indicates the model prediction approach accuracy is presented to be toquite predict robust. the ideal Consequently, condenser fanthe speedhybrid of model air-cooled is sufficient vapor compression to be used systemsin optimizati (VCS)on within control constant strategies operation and due conditions. to the simplicity The model of consiststhe model of a it hybrid is expected characterization that it can ofbe the used main for components real time applications. and the optimization Especially problem for applications is formulated with asa minimizingquasi-stationary the totalcoolant energy demand, consumption the proposed by adjusting method the represents condenser a fan promising and compressor alternative speed. to Thereby,extremum the seeking hybrid control model approachor other advanced provides amethod new optimizations that may be method more accurate for the calculation but have a of high the idealconvergence fan speed, time. while Furthermore, only limited the plant method specifications can be areeasily required. adapted The to result other is aair-cooled simple and vapor fast operatingcompression optimization systems for method optimization that does purposes not rely and on algorithms only limited as comparedinput parameters to other are strategies necessary. in the In fact, the required input parameters are usually known during the development process of VCS. Besides real time application, the calculation method can be further used to determine the ideal condensing pressure and ideal fan speed to adjust the parameters of the control method proposed by Chan and Yu et al. [6], for example.

6. Conclusions In this research study, a steady-state model approach is presented to predict the ideal condenser fan speed of air-cooled vapor compression systems (VCS) within constant operation conditions. The model consists of a hybrid characterization of the main components and the optimization problem is formulated as minimizing the total energy consumption by adjusting the condenser fan and compressor speed. Thereby, the hybrid model approach provides a new optimization method for the calculation of the ideal fan speed, while only limited plant specifications are required. The result is a simple and fast operating optimization method that does not rely on algorithms as compared to other strategies in the literature. To demonstrate the feasibility, a detailed model validation is carried out of 250 steady-state measuring points based on fifteen different operating conditions. The validation reveals a prediction error of the coefficient of system performance (COSP) within ±10%. Moreover, the prediction error of the ideal condenser fan speed is negligible with regard to the flat optimum. An explanation for this high accuracy is given by the discussion of the model assumptions, which implies that only parameters with a high influence on the condensing pressure must be considered for the ideal condenser fan speed calculation. Thereby, the analysis confirms the high effect of refrigeration capacity and ambient temperature reported in the literature. However, the results further suggest that the ideal condenser fan speed can be calculated with sufficient precision,

Energies 2020, 13, 6012 19 of 26 literature. To demonstrate the feasibility, a detailed model validation is carried out of 250 steady-state measuring points based on fifteen different operating conditions. The validation reveals a prediction error of the coefficient of system performance (COSP) within 10%. Moreover, the prediction error of ± the ideal condenser fan speed is negligible with regard to the flat optimum. An explanation for this high accuracy is given by the discussion of the model assumptions, which implies that only parameters with a high influence on the condensing pressure must be considered for the ideal condenser fan speed calculation. Thereby, the analysis confirms the high effect of refrigeration capacity and ambient temperature reported in the literature. However, the results further suggest that the ideal condenser fan speed can be calculated with sufficient precision, assuming constant evaporating pressure, compressor efficiency, subcooling, and superheating, respectively. In addition, a set-point optimization control strategy is proposed on the basis of the model approach and obtains the maximal efficiency of the VCS for the fifteen different set-points. Consequently, the hybrid model is suitable to be utilized in optimization control strategies, and is expected to be used in real time applications due to the simplicity of the model. Furthermore, the method can be easily adapted to other air-cooled VCS for optimization purposes. Therefore, the results provide a deep insight into the set-point optimization and hybrid steady-state modeling of air-cooled vapor compression systems.

Author Contributions: S.A. outlined the structure and content of this article. S.A. performed the validation and investigation. C.K. proved the content. S.A. wrote the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: We acknowledge support for the Article Processing Charge from the Open Access Publication Fund of the Technische Universität Ilmenau. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature

A Area/m2 cs Speed of sound/m/s c Mean specific heat capacity/J/(kg K) · h Specific enthalpy/J/kg I Current/A k Overall heat transfer coefficient/W/(m2 K) . · mR Refrigerant mass flow rate/kg/s n Rotation speed/rpm P Power/W p Pressure/bar . Q Heat flux/W r Evaporating enthalpy/J/kg Rm Specific gas constant/J/(kg K) · R2 Coefficient of determination/% T Temperature/◦C U Voltage/V v Specific volume m3/kg . V Volume flow rate/m3/s x Vapor fraction z Real gas factor/- Greek Letters α Convective heat transfer coefficient/W/(m2 K) · ηcom Compressor efficiency/- θ Substitution ρ Density/kg/m3 Φ Heat exchanger characteristic/- Energies 2020, 13, 6012 20 of 26

Subscripts air Air c Condenser com Compressor cool Coolant Energies 2020, 13, 6012 20 of 26 e Evaporator F Fan 𝑖 Isentropic i Isentropic 𝑖𝑑 Ideal id Ideal 𝑖𝑛 in InletInlet out 𝑜𝑢𝑡 OutletOutlet R 𝑅 RefrigerantRefrigerant sc 𝑠𝑐 SubcoolingSubcooling sh 𝑠ℎ SuperheatingSuperheating AbbreviationsAbbreviations COP𝐶𝑂𝑃 CoeCoefficientfficient of Performance of Performance/-/- COSP𝐶𝑂𝑆𝑃 CoeCoefficientfficient of System of System Performance Performance/- /- MID Magnetic inductive flow meter 𝑀𝐼𝐷 Magnetic inductive flow meter NTU Number of Transfer Units 𝑁𝑇𝑈 Number of Transfer Units PAG Polyalkylene glycol 𝑃𝐴𝐺 RMSE RootPolyalkylene mean square glycol error R134𝑅𝑀𝑆𝐸a 1,1,1,2-TetrafluorethanRoot mean square error TXV𝑅134𝑎 Thermostatic1,1,1,2-Tetrafluorethan expansion valve VCC𝑇𝑋𝑉 VaporThermostatic compression expansion cycle valve VCS𝑉𝐶𝐶 VaporVapor compression compression system cycle 𝑉𝐶𝑆 Vapor compression system Appendix A Appendix A

FigureFigure A1.A1.Instrumented Instrumented air-cooled air-cooled VCS. VCS. 1: Compressor, 1: Compressor, 2: Condenser 2: Condenser (covered), (covered), 3: TXV and 3: Evaporator TXV and (covered),Evaporator 4: (covered), Fan (covered), 4: Fan 5:(covered Coriolis), 5: sensor, Coriolis 6: sensor, Sound 6: velocity Sound sensor,velocity 7: sensor, Air inlet, 7: Air8: inlet, Air outlet, 8: Air 9:outlet, Coolant 9: Coolant inlet, 10: inlet, Coolant 10: Coolant outlet. outlet.

Table A1. Mahle GmbH serial battery cooling unit component description.

Component Description Specification Serial component of MAHLE Behr Plate heat exchanger GmbH & Co. KG Plate amount: 50 Evaporator refrigerant-coolant evaporator Fluid guidance: Refrigerant „S-Flow“, (Chiller) Coolant „I-Flow“ Displacement: 24 cm3 Compressor Suction gas-cooled electric scroll Speed: 2400––6000 rpm Serial component of MAHLE Behr Four passes with 15, 10, 6 und 4 tubes GmbH & Co. KG Condenser in1st, 2nd, 3rd and 4th pass respectively air-refrigerant condenser Receiver between 3rd and 4th pass Fin tube type cross-flow system

Energies 2020, 13, 6012 21 of 26

Table A1. Mahle GmbH serial battery cooling unit component description.

Component Description Speciﬁcation Serial component of MAHLE Behr Plate heat exchanger GmbH & Co. KG Plate amount: 50 Evaporator refrigerant-coolant evaporator Fluid guidance: Refrigerant “S-Flow”, Coolant (Chiller) “I-Flow” Displacement: 24 cm3 Compressor Suction gas-cooled electric scroll Speed: 2400–6000 rpm Serial component of MAHLE Behr Four passes with 15, 10, 6 und 4 tubes in1st, 2nd, GmbH & Co. KG Condenser 3rd and 4th pass respectively air-refrigerant condenser Receiver between 3rd and 4th pass Fin tube type cross-ﬂow system Thermostatic expansion valve of Otto Capacity: 2 ton Expansion valve Egelhof GmbH & Co. KG. Charge: C6 Diameter: 305 mm Fan Serial component of Spal Automotive Speed: 800 to 4000 rpm. Serial component of Sanden Inc. Refrigerant-Oil Polyalkylene glycol (PAG) synthetic oil Product name: SP-A2

Table A2. Sensor description.

Sensor Manufacturer Product Name System: DEWE2-M Data logger Dewetron GmbH Cards: 2402-dSTG, 2402dACC, CNT, 2402-V, EPAD2-TH8, EPAD2-RTD8 Thermocouple–refrigerant Electornic Sensor GmbH IKT 15/10/2KSTS33/2m/ZEH/MFM.K Thermocouple–coolant Electornic Sensor GmbH EST 01 Pt100–coolant SONTEC Sensorbau GmbH TP1051-A0B1C3D1, TE6-G1/2-A0B1C4D2 Pt100–air Electornic Sensor GmbH PT100A 20/050/NB/4 Cu TT17/2 m/LR12.S30 CLAMP-DC-POWER-4-EXT with Current Deweton GmbH PA-IT-65-S-BUNDLE, PA-IT-205-S-BUNDLE Fan speed Keyence Deutschland GmbH FS-N40 Compressor speed Brüel Kjaer GmbH Type 4524 Sound velocity Anton Paar Germany GmbH L-SONIC 6100 with PICO 3000 Pressure ALTHEN GmbH PDCR 5060-TB-A2-CC H0-PC-030B0000G Micro Motion ELITE Coriolis Messsystem: CMF025M176N4FZGZZZMC with Coriolis sensor Emersson Micro Motion 2700 Messumformer: 2700R12CFZGZZZ MID Siemens AG MAG 5100 TW with MAG 6000 IP 67

Table A3. Measurement uncertainty of the instruments of the measuring chain MR: regarding measurement range, No declaration: regarding measurement value.

Element Measuring Chain Uncertainty Distribution

Pt100 Sensor 0.05 ◦C normal - Module 0.08 ◦C rectangular Thermocouple Sensor 0.5 ◦C normal - Module 0.2 ◦C rectangular Pressure Sensor 0.3% normal - Module, Input 0.02%, 0.02% MR rectangular - Module, Output 0.03% rectangular Coriolis Sensor 0.1% normal - Module, Input 0.02%, 0.02% MR rectangular - Module, Output 0.05% rectangular MID System 0.15% normal Sound Velocity Sensor 0.1 m/s normal - Transducer 0.12% rectangular - Module, Input 0.02%, 0.02% MR rectangular Energies 2020, 13, 6012 22 of 26

Energies 2020, 13, 6012 Table A3. Cont. 22 of 26

Element Measuring Chain Uncertainty Distribution - Module, Output 0.05% rectangular -- Module, OutputCalculation 0.05%0.96% rectangularnormal - Calculation 0.96% normal AccelerationAcceleration (Compressor speed)System System 1% 1% normal normal (CompressorFiber optic speed) (Fan speed) System 0.1% normal FiberCurrent optic (Fan (Compressor) speed) System Sensor 0.1% 83 ppm normal rectangular Current (Compressor)- Sensor Module 83 ppm 0.02%, 0.02% MR rectangular rectangular - Module 0.02%, 0.02% MR rectangular Voltage (Compressor) Sensor 83 ppm rectangular Voltage (Compressor) Sensor 83 ppm rectangular -- Module Module 0.3%, 0.02% 0.3%, MR 0.02% MR rectangular rectangular CurrentCurrent (Fan) (Fan) Sensor Sensor 270 ppm 270 ppm rectangular rectangular -- Module Module 0.02%, 0.02% 0.02%, MR 0.02% MR rectangular rectangular VoltageVoltage (Fan) (Fan) Sensor Sensor 270 ppm 270 ppm rectangular rectangular - Module 0.3%, 0.02% MR rectangular - Module 0.3%, 0.02% MR rectangular

Appendix B Appendix B The validation of the refrigerant property models of Equations (11) and (12) is shown in Figure A2. The validation of the refrigerant property models of Equations (11) and (12) is shown in A2. The The error of the property model is within 3%. The coefficient of determination for the specific volume error of the property model is within 3%. The coefficient of determination for the specific volume and and the condensing pressures are 99.99% and 99.85%, respectively. Constant entropy of 1730 J/kgK is the condensing pressures are 99.99% and 99.85%, respectively. Constant entropy of 1730 J/kgK is assumed for the calculation of the specific volume. For a significant change in entropy, the value K1 assumed for the calculation of the specific volume. For a significant change in entropy, the value 𝐾 must be adjusted. However, for the research in this paper, no change of K1 is necessary. must be adjusted. However, for the research in this paper, no change of 𝐾 is necessary.

0.05 1.5 30 2 25 1 0 20 0 0.025 15 -1 bar Error / % / Error -1.5 10 -2 Error/% property data property data model 5 model -3 error error 0 -3 0 -4 Condensingpressure /

Specific volume/m3/kg 250000 1250000 2250000 20 50 80 ° Pressure / Pa Condensing temperature / C (a) (b)

Figure A2.A2. ValidationValidation of of speciﬁc specific volume volume model model approach approach (a). Validation(a). Validation of condensing of condensing pressure pressure model modelapproach approach (b). (b).

The validation of the model is presented for th thee working fluid fluid R134a and the property data are taken from from Span Span and and Wagner Wagner [35]. [35]. For For the the same same pressure pressure boundaries, boundaries, the theworking working fluids fluids R1234yf R1234yf and R290and R290 (Propane) (Propane) have have a maximal a maximal error error of of6% 6% and and 3% 3% for for specific specific volume, volume, and and 3% 3% and and 3.5% 3.5% for condensing pressure prediction, respectively.respectively.

Appendix C The coupling function of the air mass ﬂowflow rate and thethe refrigerantrefrigerant massmass ﬂowflow raterate isis givengiven asas s 𝜃 𝜃 2 . θ6 θ6 . m𝑚R ==− − −𝜃θ ==𝑓f(𝑚m ,, 𝑖𝑛𝑝𝑢𝑡input) (A1)(A1) − 22 − 22 − 7 air with the substitution . 𝜃 +𝑄 +𝐹(𝑚 . )𝑐,̅ 𝑇,, −𝜃 𝜃 = θ4 + Qe + F mair cp,air(Tc,air,in θ1) =𝑥 +𝑥𝐹(𝑚. ) (A2) θ6 = 𝜃 − = x1 + x2F mair (A2) θ3 and

𝑚 Φ𝑐,̅ 𝜃 𝐹(𝑚 )𝑐,̅ 𝜃 𝜃 =− =− =−𝑥𝐹(𝑚 ) (A3) 𝜃 𝜃

Energies 2020, 13, 6012 23 of 26 and . . mairΦccp,airθ2 F mair cp,airθ2 . θ7 = = = x3F mair (A3) − θ3 − θ3 − and . ( ) θ4 + Qe cp,air Tc,air,in θ1 cp,airθ2 x1 = , x2 = − , x3 = . (A4) θ3 θ3 θ3 The derivation of the refrigerant mass ﬂow rate to the air mass ﬂow rate is given as:

. . . F m ∂mR ∂mR ∂ air . = . . (A5) ∂mair ∂F mair · ∂mair

with . . F m = m Φc. (A6) air air· Both gradients of Equation (A5) can be developed separately:

. . x2(x2F(mair)+x1) ∂mR 2 x3 x2 . = r − (A7) − . 2 − 2 ∂F mair (x F(m )+x ) . 2 2 air 1 x F m 4 − 3 air and . ∂F m . τ 1 air α∗(τ 1)mair− (1 Φc) . = − − + Φc. (A8) ∂mair cp,air The combination of the Equations (A7) and (A8) leads to:    .  .  x2(x2F(mair)+x1)  . τ 1   x3 α (τ 1)m − (1 Φ )  ∂mR  2 x2  ∗ air c  . =  r −  − − + Φc. (A9) − . 2 − 2  c  ∂mair  (x F(m )+x ) .  p,air  2 2 air 1 x F m  4 − 3 air Inserting the substitutions from Equation (A4) leads to the derivation of the coupling function:

  .   c (T θ ) c (T θ ) +  p,air c,air,in− 1  p,air c,air,in− 1 . θ4 Qe     F(mair)+    θ3  θ3 θ3  c θ c (T θ )  .  p,air 2 p,air c,air,in− 1  . τ 1   2  α (τ 1)m − (1 Φ )  ∂mR  − θ3 θ3  ∗ air c  . = v  − − + Φc. (A10) m − tu  . 2 − 2  c  ∂ air  c (T θ ) +  p,air   p,air c,air,in− 1 . θ4 Qe     F(mair)+     θ3 θ3  c   p,airθ2 .  2 F mair 4 − θ3

Appendix D The coefficient of determination in percent is defined as    PN ( )2  2  i=1 yi yî  R = 1 −  100, (A11)  P P 2   − N y 1 N y · i=1 i − n i=1 i the root mean square error is defined as s PN 2 = (yi yî) RMSE = i 1 − , (A12) N Energies 2020, 13, 6012 24 of 26 the relative root mean square error in percent is defined as

RMSE R RMSE = 100, (A13) − 1 PN · n i=1 yi where N is the number of validation test data, yi, and yˆi indicate the predicted and the measured value of one test point n, respectively.

References

1. EIA. Residential Energy Consumption Survey. U.S. Energy Information Administration 2015. Available online: https://www.eia.gov/consumption/residential/index.php (accessed on 30 September 2020). 2. Tiat, S.; Keung, J. Green Building Design Guide. S. Building & Construction Authority, Singapore 2015. Available online: https://www.bca.gov.sg/GreenMark/others/3rd_Green_Building_Masterplan.pdf (accessed on 30 September 2020). 3. UN-Enviorment. The Importance of Energy Efficiency in the Refrigeration, Air-conditioning and Heat Pump Sectors 2018. Available online: https://ozone.unep.org/sites/default/files/2019-08/briefingnote-a_ importance-of-energy-efficiency-in-the-refrigeration-air-conditioning-and-heat-pump-sectors.pdf (accessed on 30 September 2020). 4. Jensen, J.B.; Skogestad, S. Optimal operation of simple refrigeration cycles. Comput. Chem. Eng. 2007, 31, 712–721. [CrossRef] 5. Jakobsen, A.; Rasmussen, B. Energy-optimal speed control of fans and compressors in a refrigeration system. In Proceedings of the Eurotherm 62, Grenoble, France, 17–18 November 1998; pp. 317–323. 6. Chan, K.; Yu, F. Optimum Setpoint of Condensing Temperature for Air-Cooled Chillers. HVAC&RRes. 2004, 10, 113–127. 7. Larsen, L. Model Based Control of Refrigeration Systems. Ph.D. Thesis, Aalborg University, Aalborg, Denmark, 2005. Available online: https://www.osti.gov/etdeweb/servlets/purl/20833740 (accessed on 30 September 2020). 8. Yu, F.; Chan, K. Tune up of the set point of condensing temperature for more energy efficient air cooled chillers. Energy Convers. Manag. 2006, 47, 2499–2514. [CrossRef] 9. Wan, H.; Cao, T.; Hwang, Y.; Oh, S. A review of recent advancements of variable refrigerant flow air-conditioning systems. Appl. Therm. Eng. 2020, 169, 114893. [CrossRef] 10. Zhao, L.; Cai, W.-J.; Ding, X.-D.; Chang, W.-C.; Chang, V.W.-C. Decentralized optimization for vapor compression refrigeration cycle. Appl. Therm. Eng. 2013, 51, 753–763. [CrossRef] 11. Qiao, H.; Radermacher, R.; Aute, V. A Review for Numerical Simulation of Vapor Compression Systems. In Proceedings of the International Refrigeration and Air Conditioning Conference, Purdue University, West Lafayette, IN, USA, 12–15 July 2010; pp. 1–10. Available online: https://docs.lib.purdue.edu/cgi/ viewcontent.cgi?article=2089&context=iracc (accessed on 30 September 2020). 12. Zhang, G.; Xiao, H.; Zhang, P.; Wang, B.; Li, X.; Shi, W.; Cao, Y. Review on recent developments of variable refrigerant flow systems since 2015. Energy Build. 2019, 198, 444–466. [CrossRef] 13. Scarpa, M.; Emmi, G.; De Carli, M. Validation of a numerical model aimed at the estimation of performance of vapor compression based heat pumps. Energy Build. 2012, 47, 411–420. [CrossRef] 14. Zhao, L.; Cai, W.; Ding, X.; Chang, W. Model-based optimization for vapor compression refrigeration cycle. Energy 2013, 55, 392–402. [CrossRef] 15. Jakobsen, A.; Rasmussen, B.D.; Skovrupity, M.J.; Fredsted, J. Development of Energy Optimal Capacity Controlin Refrigeration Systems. In Proceedings of the 8th International Refrigeration Conference, Purdue University, West Lafayette, IN, USA, 25–28 July 2000; pp. 329–336. Available online: https: //docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1498&context=iracc (accessed on 30 September 2020). 16. Braun, M.; Walton, P.; Beck, S. Illustrating the relationship between the coefficient of performance and the coefficient of system performance by means of an R404 supermarket refrigeration system. Int. J. Refrig. 2016, 70, 225–234. [CrossRef] 17. Ruiz, A.G.; Garrido, J.; Vázquez, F.; Morilla, F. A hybrid modeling approach for steady-state optimal operation of vapor compression refrigeration cycles. Appl. Therm. Eng. 2017, 120, 74–87. [CrossRef] Energies 2020, 13, 6012 25 of 26

18. Yang, J.; Chan, K.; Dai, T.; Yu, F.; Chen, L. Hybrid Artificial Neural Network Genetic Algorithm Technique − for Condensing Temperature Control of Air-Cooled Chillers. Procedia Eng. 2015, 121, 706–713. [CrossRef] 19. Yu, F.; Ho, W.; Chan, K.; Sit, R. Logistic regression-based optimal control for air-cooled chiller. Int. J. Refrig. 2018, 85, 200–212. [CrossRef] 20. Shin, J.-H.; Kim, Y.-I.; Cho, Y.-H. Development of Operating Method of Multi-Geothermal Heat Pump Systems Using Variable Water Flow Rate Control and a COP Prediction Model Based on ANN. Energies 2019, 12, 3894. [CrossRef] 21. Goyal, A.; Staedter, M.A.; Garimella, S. A review of control methodologies for vapor compression and absorption heat pumps. Int. J. Refrig. 2019, 97, 1–20. [CrossRef] 22. Behrooz, F.; Mariun, N.; Marhaban, M.; Radzi, M.; Ramli, A. Review of Control Techniques for HVAC Systems—Nonlinearity Approaches Based on Fuzzy Cognitive Maps. Energies 2018, 11, 495. [CrossRef] 23. Yu, F.; Ho, W. Load allocation improvement for chiller system in an institutional building using logistic regression. Energy Build. 2019, 201, 10–18. [CrossRef] 24. Larsen, L.F.S.; Thybo, C.; Stoustrup, J.; Rasmussen, H. A method for online steady state energy minimization, with application to refrigeration systems. In Proceedings of the 43rd IEEE Conference on Decision and Control (CDC), Nassau, Bahamas, 14–17 December 2004; Volume 5, pp. 4708–4713. 25. Ding, X.; Cai, W.; Jia, L.; Wen, C.; Zhang, G. A hybrid condenser model for real-time applications in performance monitoring, control and optimization. Energy Convers. Manag. 2009, 50, 1513–1521. [CrossRef] 26. Huang, Y.; Khajepour, A.; Ding, H.; Bagheri, F.; Bahrami, M. An energy-saving set-point optimizer with a sliding mode controller for automotive air-conditioning/refrigeration systems. Appl. Energy 2017, 188, 576–585. [CrossRef] 27. Yeh, T.-J.; Chen, Y.-J.; Hwang, W.-Y.; Lin, J.-L. Incorporating fan control into air-conditioning systems to improve energy efficiency and transient response. Appl. Therm. Eng. 2009, 29, 1955–1964. [CrossRef] 28. Jain, N.; Alleyne, A.G. Exergy-based optimal control of a vapor compression system. Energy Convers. Manag. 2015, 92, 353–365. [CrossRef] 29. Burns, D.; Bortoff, S.; Laughman, C.; Guay, M. Comparing Realtime Energy-Optimizing Controllers for Heat Pumps. In Proceedings of the 17th International Refrigeration and Air Conditioning Conference, Purdue University, West Lafayette, IN, USA, 9–12 July 2018; Available online: https://docs.lib.purdue.edu/ cgi/viewcontent.cgi?article=3007&context=iracc (accessed on 30 September 2020). 30. Wang, W.; Li, Y. Real-time optimization of intermediate temperature for a cascade heat pump via extreme seeking. In Proceedings of the 13th International Modelica Conference, Regensburg, Germany, 4–6 March 2019; pp. 251–258. 31. Bejarano, G.; Alfaya, J.A.; Ortega, M.G.; Vargas, M. On the difficulty of globally optimally controlling refrigeration systems. Appl. Therm. Eng. 2017, 111, 1143–1157. [CrossRef] 32. He, X.; Liu, S.; Asada, H. Modeling of vapor compression cycles for advanced controls in HVAC systems. In Proceedings of the 1995 American Control Conference—ACC’95, Seattle, WA, USA, 21–23 June 1995; pp. 3664–3668. 33. Hipp, C.; Schauer, M.; Zimmermann, R. System for Controlling the Temperature of an Electric Energy Storage Device. Patent Nr. WO 2019/016021 A2, 2019. Available online: https://patentimages.storage.googleapis. com/77/97/f2/7be732563f0615/WO2019016021A2.pdf (accessed on 28 September 2020). 34. Youbi-Idrissi, M.; Bonjour, J.; Terrier, M.-F.; Marvillet, C.; Meunier, F. Oil presence in an evaporator: Experimental validation of a refrigerant/oil mixture enthalpy calculation model. Int. J. Refrig. 2004, 27, 215–224. [CrossRef] 35. Span, R.; Wagner, W. Equations of State for Technical Applications. I. Simultaneously Optimized Functional Forms for Nonpolar and Polar Fluids. Int. J. Thermophys. 2003, 24, 1–39. [CrossRef] 36. Heide, R. Stoffdatenbestimmung des Gemisches Kältemittel R134a-Öl ND8; Institut für Luft- und Kältetechnik Gemeinnützige Gesellschaft mbH: Dresden, Germany, 1998; Available online: https://www.ilkdresden.de/en/ (accessed on 28 September 2020). 37. Jean-Marc, L.; Louis, V.; Evelyne, M.; Lottin, O. Oil Concentration Measurement in Saturated Liquid Refrigerant Flowing Inside a Refrigeration Machine. Int. J. Thermodyn. 2001, 4, 53–60. 38. Braumoeller, J.; Riedel, R.; Schenk, J. Messung der Schallgeschwindigkeit in R134a-Oel-Gemischen; Institut für Luft- und Kältetechnik gemeinnützige Gesellschaft mbH: Dresden, Germany, 2004; Available online: https://www.ilkdresden.de/en/ (accessed on 28 September 2020). Energies 2020, 13, 6012 26 of 26

39. JCGM, Evaluation of measurement data—Guide to the expression of uncertainty in measurement. Joint Committee for Guides in Metrology 2008. Available online: https://www.bipm.org/en/publications/guides/ gum.html (accessed on 28 September 2020). 40. Angermeier, S.; Föhner, L.; Kerler, B.; Karcher, C. Uncertainty in measurement of a vapor compression system. In Proceedings of the 27th Fachtagung für Experimentelle Strömungsmechanik, Erlangen, Germany, 3–5 September 2019; Available online: https://www.gala-ev.org/images/Beitraege/Beitraege2019/pdf/46.pdf (accessed on 28 September 2020). 41. Žukauskas, A. Heat transfer from tubes in crossﬂow. Adv. Heat Transf. 1972, 8, 93–160. 42. Chi, J.; Didion, D. A simulation model of the transient performance of a heat pump. Int. J. Refrig. 1982, 5, 176–184. [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).