Transient simulation of household refrigerators: A semi-empirical quasi-steady approach

Applied Energy 88 (2011) 748–754

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Transient simulation of household refrigerators: A semi-empirical quasi-steady approach q Bruno N. Borges a, Christian J.L. Hermes b,⇑, Joaquim M. Gonçalves c, Cláudio Melo a a

POLO Research Laboratories for Emerging Technologies in Cooling and Thermophysics, Federal University of Santa Catarina, 88040-970 Florianópolis, SC, Brazil Department of Mechanical Engineering, Federal University of Paraná, P.O. Box 19011, 81531-990 Curitiba, PR, Brazil c AC&R Center, Federal Institute of Technological Education of Santa Catarina, Rua José Lino Kretzer 608, 88103-310 São José, SC, Brazil b

a r t i c l e

i n f o

Article history: Received 13 May 2010 Received in revised form 9 September 2010 Accepted 22 September 2010 Available online 20 October 2010 Keywords: Refrigerator Quasi-steady Simulation Energy

a b s t r a c t This paper outlines a simulation model for the cycling behavior of household refrigerators and, therefore, for predicting their energy consumption. The modeling methodology follows a quasi-steady approach, where the refrigeration system and also the refrigerated compartments are modeled following steadystate and transient approaches, respectively. The mass, energy and momentum conservation principles were used to put forward the equation set, whereas experimental data were collected and used to reduce the model closing parameters such as the thermal conductances and capacitances. The experiments were carried out using a controlled temperature and humidity environmental chamber. The model predictions were compared to experimental data, when it was found that the energy consumption is well predicted by the model with a maximum deviation of ±2%. A sensitivity analysis was also carried out to identify opportunities for energy savings. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Governments worldwide are launching more stringent energy consumption policies for household refrigerators and freezers almost steadily. In order to match the new energy regulations most of the manufacturers are seeking alternative ways of improving the thermodynamic performance of their products. In general, design and analysis of household refrigerators follow a trial-and-error procedure, which is costly and time consuming. There are many evidences in the literature showing that numerical simulation and optimization techniques not only may reduce the amount of tests and prototypes needed during the product development phase, but also the final product cost and energy consumption. Many different modeling strategies are available in the literature, although most of them have limitations either in terms of physical modeling or computational performance. Former simulation models for refrigeration systems date back to the early 1980s and were mostly focused on heat pump and air conditioning equipment [1–11]. The development of simulation models for household refrigerators was stimulated by the CFC-12 phase-out in the late 1980s [12–20]. Hermes and Melo [21] carried out a comprehensive literature survey on the models available in the open literature for the transient simulation of household q An abridged version of this manuscript was presented at the 13th International Refrigeration and Air Conditioning Conference at Purdue, 12–15 July, 2010. ⇑ Corresponding author. Tel.: +55 41 3361 3239. E-mail address: [email protected] (C.J.L. Hermes).

0306-2619/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2010.09.019

refrigerators, and concluded that very few approaches are able to simulate the refrigerator cycling behavior (an essential condition for predicting the energy consumption), and none of them were actually validated against energy consumption. Moreover, these models require component-level information (e.g., heat transfer coefficients, friction factors) that depends on proper empirical correlations. In a prior work, Gonçalves et al. [22] investigated the steadystate behavior of a top-mount refrigerator by measuring and modeling the performance characteristics of each of its components. The models were based on the first-principles and also on empirical data. Measurements of the relevant variables were taken at several positions along the refrigeration loop, generating performance data not only for the whole unit but also for each of the cycle components. The experiments were planned and performed following a statistical methodology and considering 13 independent variables that led to over 160 data runs. The model predictions for cooling capacity and power consumption were compared with experimental data when a maximum deviation of ±10% was found. Such an approach, however, is not suitable for predicting the energy consumption due to its steady-state nature. Later, Hermes and Melo [21] introduced a first-principles model for simulating the transient behavior of household refrigerators and freezers. The model was used to simulate a typical top-mount refrigerator, where the compressor is on–off controlled by the freezer temperature, while a thermo-mechanical damper is used to set the fresh-food compartment temperature. Numerical predictions were compared to experimental data, when it was found that the

B.N. Borges et al. / Applied Energy 88 (2011) 748–754

749

Nomenclature Roman C cp h m N Q T v V W UA Greek Dt

U

thermal capacity, J/K specific heat at constant pressure, J/kg K enthalpy, J/kg K mass flow rate, kg/s compressor speed, Hz heat transfer rate, W temperature, K specific volume, m3/kg volume, m3 power, W thermal conductance, W

time-step, s conductance correction factor, W/K

energy consumption was reasonable well predicted by the model with maximum disagreements within a ±10% error band. The compartment air temperatures were also well predicted by the model with maximum deviations of ±1 °C. Such a model, however, requires a considerable amount of CPU time, not being viable for optimization tasks that may require thousands of runs in a row. In order to address these drawbacks, Hermes et al. [23] proposed a simplified methodology for predicting the energy consumption of household refrigerators and freezers with a similar accuracy, but with a substantially lower CPU effort than that required by the dynamic simulation code of Hermes and Melo [21]. The model follows the steady-state, semi-empirical approach introduced by Gonçalves et al. [22], but it is able to predict the refrigerator runtime as a function of the cabinet thermal load and cooling capacity ratio. Such a methodology predicted satisfactorily the energy consumption of refrigerators with fan supplied heat exchangers, but its applicability to refrigerators with natural draft heat exchangers is still under discussion. Furthermore, it is not able to take into account the impact of events such as door openings, defrost strategy and internal loads on the energy consumption. These issues have motivated the development of a novel, but still simplified modeling approach, according to which the refrigeration system and the refrigerated compartments are modeled in steady-state and transient modes, respectively. It should be emphasized that the semi-empirical character of Hermes’ et al. [23] approach was retained as the model closing parameters were derived from experimental data. 2. Mathematical modeling The mathematical model follows that introduced by Hermes et al. [23], according to which the refrigeration loop was divided into the following lumped sub-models: compressor, evaporator, condenser, and internal heat exchanger, as depicted in Fig. 1, which also illustrates the air flow circuit within the refrigerated compartments. In addition, an algebraic transient model was devised for the refrigerated compartments based on the work of Hermes and Melo [21], to predict the time evolution of the compartments temperatures. 2.1. Refrigeration loop sub-model (steady-state) The compressor sub-model provides not only the compression power but also the refrigerant mass flow rate and the refrigerant enthalpy at the compressor outlet. The refrigerant mass flow rate,

e g

heat exchanger effectiveness, dimensionless compressor efficiency, dimensionless

Subscripts a air c condenser e evaporator i inlet k compressor m mullion o outlet f frozen-food r refrigerant, fresh-food s isentropic process x internal heat exchanger

mr, was derived from Eq. (1), where gv stands for the compressor volumetric efficiency:

mr ¼ gv V k N=v 1

ð1Þ

The compression power, Wk, was obtained from Eq. (2), where

gg stands for the compressor overall efficiency: W k ¼ mr ðh2;s  h1 Þ=gg

ð2Þ

The refrigerant enthalpy at the compressor outlet, h2, was calculated from an energy balance at the compressor shell, as follows:

h2 ¼ h1 þ ðW k  Q k Þ=mr

ð3Þ

The compressor heat released rate to the surroundings, Qk, was calculated from

Q k ¼ UAk ðT 2;s  T a Þ

ð4Þ

where Ta is the ambient temperature, T2,s is the isentropic compressor discharge temperature, and UAk is the compressor shell thermal conductance. The refrigerant enthalpy at the evaporator inlet and the refrigerant temperature at the compressor inlet are calculated from an energy balance involving the internal heat exchanger, as follows:

h4 ¼ h3 þ h5  h1

ð5Þ

T 1 ¼ T 5 þ ex ðT 3  T 5 Þ

ð6Þ

The condenser and evaporator sub-models provide not only the refrigerant enthalpies and air temperatures at their respective outlets, but also the heat transfer rates. The evaporator cooling capacity, the refrigerant enthalpy and the air temperature at the evaporator outlet were calculated respectively from:

Q e ¼ ma cp;a ðT i;e  T e Þ½1  expðUAe =ma cp;a Þ

ð7Þ

h5 ¼ h4 þ Q e =mr

ð8Þ

T o;e ¼ T i;e  Q e =ma cp;a

ð9Þ

where Ti,e is the air temperature at the evaporator inlet and Te is the evaporating temperature. It is worthy of note that the term within square brackets in Eq. (7) accounts for the heat exchanger effectiveness, adopted to account for the air-side temperature decrease along the evaporator, which was assumed to be flooded . Similarly, assuming that the condenser subcooling region occupies only a small part of the condenser and also that the vapor provides a lower heat transfer coefficient, but at higher log-mean

750

B.N. Borges et al. / Applied Energy 88 (2011) 748–754

Fig. 1. Schematic representation of the refrigerator.

temperature difference which compensate each other, thus supplying a heat flux with the same magnitude of that calculated assuming two-phase refrigerant, the condenser heat transfer rate and the refrigerant enthalpy at the condenser outlet were calculated respectively from:

Q c ¼ UAc ðT c  T a Þ

ð10Þ

h3 ¼ h2  Q c =mr

ð11Þ

The condensing and evaporating pressures were calculated as suggested by Hermes et al. [23]:

pc ¼ psat ðT 3 þ DT sub Þ

ð12Þ

pe ¼ psat ðT 5  DT sup Þ

ð13Þ

where the refrigerant superheating at the evaporator outlet and the refrigerant subcooling at the condenser outlet were assumed as input data. It worthy mentioning that the compressor volumetric and overall efficiencies, gv (=0.576  0.0162 pc/pe) and gg (=0.860  0.00459 pc/pe), were reduced from the manufacturer’s catalogue. All other empirical parameters (UAk, UAe, UAc, and ex) were derived from experimental tests carried out with the refrigerator under analysis. 2.2. Refrigerated compartments sub-model (transient) The cabinet sub-model calculates the evolution of the refrigerated compartments temperatures over time, and also the air temperature at the evaporator inlet. The time evolution of the air temperatures inside the refrigerated compartments is described by the following ordinary differential equation (ODE):

dT  C ¼ ðUA þ U ÞðT a  T  Þ  UAm ðT r  T f Þ þ m cp ðT o;e dt X  TÞ þ W

ð14Þ

where the asterisk stands for the frozen-food, Tf, or the fresh-food, Tr, compartment temperature. The ‘‘+” signal applies to the fro-

zen-food and the ‘‘” signal to the fresh-food compartment. It is noteworthy that the first and second terms of the right-hand side of Eq. (14) stand for the heat transmission through the cabinet and mullion walls, respectively, whereas the third and fourth terms are related to the cooling capacity and the power generation within the refrigerated compartment (e.g., fans), respectively. The air flow rates are calculated from:

mf ¼ rma

ð15Þ

mr ¼ ð1  rÞma

ð16Þ

where r is the air mass fraction supplied to the frozen-food compartment. It is worthy noting that the thermal capacities, C*, the thermal conductances, UA*, and the correction factors, U*, were all derived from experimental tests carried out with the refrigerator under analysis. The U-factor was introduced to account for variations in the thermal conductances, since the values supplied to the model were obtained in steady-state conditions. Furthermore, assuming that the empirical coefficients do not vary over a time-step, Dt, Eq. (14) can be solved analytically, yielding:

T  ¼ T eq;  ðT eq;  T  Þ expðA Dt=C  Þ

ð17Þ

where Teq,* and A* are given in Table 1 for the ‘on’ and ‘off’ operation regimes. Eq. (17) provides the temperature variation over a time interval Dt based on the temperatures at the end of the previous time-step. The boundary condition for the evaporator sub-model is calculated at each time-step as follows:

T i;e ¼ ð1  rÞT r þ rT f

ð18Þ

2.3. Numerical scheme The solution algorithm is illustrated in the information flow diagram depicted in Fig. 2. The simulation starts assuming that the refrigerator is ‘off’. Then, the temperature rise in each compartment is calculated through Eq. (17) until the thermostat start-up temperature is reached. Thus, the refrigeration loop sub-model is

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B.N. Borges et al. / Applied Energy 88 (2011) 748–754 Table 1 Terms of Eq. (17) for the ‘on’ and ‘off’ operation regimes. Compartment Frozen-food

ON T eq;f ¼

OFF UAf T a þUAm T r þrma cp;a T o;e þW v

T eq;f ¼

UAf þUAm þrma cp;a

Af = UAf + UAm + rmacp,Pa Fresh-food

T eq;r ¼

ðUAf þUf ÞT a þUAm T r UAf þUAm þUf

Af = UAf + UAm + Uf

UAr T a þUAm T f þð1rÞma cp;a T o;e

T eq;r ¼

UAr þUAm þð1rÞma cp;a

Ar = UAr + UAm + (1  r)macp,a

ðUAr þUr ÞT a þUAm T f

measured by three thermocouples properly distributed inside the compartments. The air temperature at the evaporator inlet and outlet ports was also measured. The compressor and the evaporator and condenser fans power consumptions were also monitored. Tests were carried out before and after the instrumentation process to verify whether the instrumentation affected the system performance.

UAr þUAm þUr

Ar = UAr + UAm + Ur

3.2. Experimental tests

activated and its equations are simultaneously solved at each timestep via the Newton–Raphson method. The calculation takes place until the thermostat cut-off temperature is reached. This procedure is repeated for ten complete cycles, being the last three used for integrating the energy consumption. Each simulation run took approximately 5 min to be completed in case when a 2.66 GHz IntelÒ Core™ 2 QUAD processor is adopted. The code was implemented using the EES [24] and the REFPROP 7.0 [25] platforms.

3. Experimental work 3.1. Experimental apparatus The system under analysis is a top-mount frost-free refrigerator with a 110-l frozen-food compartment and a 330-l fresh-food compartment running with 100 g of HFC-134a. The frozen-food temperature is controlled by a thermostat that switches the compressor ‘on’ and ‘off’, whereas the fresh-food compartment temperature is controlled by a thermostatic damper that varies the supplied air flow rate. The system is comprised of a 60 Hz, 6.76 cm3 reciprocating compressor, a natural draft wire-and-tube condenser and a finned-tube ‘no-frost’ evaporator. The refrigerator was instrumented and positioned inside an environmental chamber with controlled air temperature and humidity. T-type thermocouples were installed at the compressor suction and discharge ports and at the condenser inlet and outlet sections. Pressure transducers were also installed at the compressor suction and discharge ports. The frozen- and fresh-food compartment temperatures were

Firstly, steady-state tests were carried out at the ambient temperatures of 25 and 32 °C, in order to generate the data needed to reduce the cabinet thermal conductances (UAr and UAf). To this end electrical heaters were placed inside the refrigerated compartments to balance the cooling capacity and therefore to keep the compartments temperatures at the desired levels. Later, energy consumption tests were carried out following the ANSI/AHAM HRF-1 [26] standard at the ambient temperatures of 25 and 32 °C, not only to collect data for deriving all other empirical parameters (UAk, UAc, UAe, ex, r, Uf, Ur, Cf, Cr, DTsub, DTsup, Ton and Toff) but also to be used in the model validation exercise. The empirical parameters were derived by supplying the model with experimental data for temperatures and power consumption. The results thus obtained are summarized in Appendix A. It is worthy noting that the U-factor is nearly nil for the frozen-food, whereas it has a significant impact on the fresh-food, since this compartment is more sensitive to the compressor and condenser heat loads. 4. Results and discussion 4.1. Model validation The model results were compared against energy consumption data obtained at 25 and 32 °C, as summarized in Table 2. It can be seen that the model predictions for the energy consumption and runtime ratio are quite good, with maximum discrepancies of ±2%. Moreover, the refrigerated compartments temperatures are well predicted by the model, with maximum deviations of ±0.4 °C. Fig. 3 shows the compressor power consumption while

Tc Condenser model

Print

Start

UAc, Ta , ΔTsub

yes Tf (t=0),Tr (t=0)

Cycle≥NCYCLES?

t = t +Δt

Te , Ti,e

no

Cycle = Cycle + 1

Evaporator model

UAe , ΔTsup,ma Cabinet model UAf ,UAr ,UAm , r, ma . Cf , Cr , Φf , Φr , Wv,e

Tf ≥Ton ? no yes

T1, h4

t = t +Δt

Tr (t), Tf (t ),Ti,e(t) yes

Cabinet model UAf , UAr,UAm , r, ma . Cf , Cr , Φf , Φr , Wv

.

Internal heat exchanger model

ΔTsub, ΔTsup, εx yes

Error ≤Tol.?

mr , Wk , h2

no

Compressor model Vk , N, ηg , ηv , UAk

Tr (t), Tf (t ), Ti,e(t) Tf ≤Toff ? no Fig. 2. Information flow diagram of the calculation procedure.

Initial guess

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B.N. Borges et al. / Applied Energy 88 (2011) 748–754

the energy consumption decreases asymptotically as the thermal conductance increases. This occurs because from a certain point the impact of the thermal conductance on the temperature difference between the air and refrigerant streams becomes imperceptible and so the energy savings. It can also be seen that for this particular product the condenser shows more room for improvements than the evaporator. Fig. 6 shows that the energy consumption is almost linearly affected by the compartments thermal conductances. The influence of the thermal capacities of the refrigerated compartments on the energy consumption were also investigated, when it was found that they have a marginal effect on the system energy performance (such an observation is corroborated by results published by Hermes and Melo [27]).

Fig. 4 illustrates the compartments temperatures along three cycles, at an ambient temperature of 32 °C. Once again it is evident that the model predictions are very close to the experimental data. It should be mentioned that the offset between the calculated and measured cycles is due to the runtime ratio prediction error that accumulates over time. 4.2. Parametric analysis A sensitivity analysis to assess the influence of specific design parameters on the overall energy consumption at an ambient temperature of 32 °C was also carried out. The effect of the evaporator and condenser thermal conductances on the system performance is shown in Fig. 5a and b, respectively. As expected [21,23,27],

Table 2 Comparison between model predictions and experimental data. Parameter

Ta = 25 °C

Energy consumption [kW h/month] Runtime ratio [dimensionless] Frozen-food temperature [°C] Fresh-food temperature [°C]

Ta = 32 °C

Exp.

Calc.

Diff.

Exp.

Calc.

Diff.

35.0 0.372 17.7 4.9

35.4 0.378 17.6 4.5

1.3% 1.8% 0.1 0.4

48.8 0.496 17.9 4.7

49.4 0.495 18.0 4.3

1.2% 0.1% 0.1 0.4

Compressor power [W]

200 180

Calculated

160

Experimental

140 120 100 80 60 40 20 0 0

30

60

90

120

150

Time [min] Fig. 3. Comparison between calculated and measured compression power.

15

frozen food evaporator inlet

Air temperatures [°C]

10

fresh food evaporator outlet

5 0 -5 -10 -15 -20 -25 -30 0

30

60

90

120

Time [min] Fig. 4. Comparison between calculated and measured compartment temperatures.

150

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B.N. Borges et al. / Applied Energy 88 (2011) 748–754

Energy consumption variation, %

(a)

Energy consumption variation, %

40

15 30 10 20

5 0

10

-5 0 -10 -10

-15 -20 -50

(b)

(a)

20

-25

0

25

50

75

-20 -50

100

40

(b)

-25

0

25

50

75

100

-25

0

25

50

75

100

75

30 20

50

10 25

0 -10

0 -20 -25

-30 -40 -50

-25

0

25

50

75

100

-50 -50

Fig. 5. Effect of heat exchanger conductance: (a) evaporator and (b) condenser. Fig. 6. Effect of cabinet conductance: (a) fresh-food and (b) frozen-food.

5. Concluding remarks A simplified quasi-steady methodology for predicting the energy consumption of household refrigerators using a hybrid semi-empirical model (steady-state refrigeration loop and transient refrigerated compartments) was put forward and validated against experimental data. The model predictions were compared with their experimental counterparts, showing maximum deviations of ±2% for the energy consumption and runtime ratio, and maximum differences of ±0.4 °C for the compartment temperatures. It was also observed that the predicted power consumption and compartment temperatures followed closely the experimental trends during the cycling transients. The model was also used to assess the energy performance of a top-mount frost-free refrigerator in terms of the evaporator and condenser conductances, the fresh- and frozen-food compartment conductances and thermal capacities. It was shown that the product energy consumption can be reduced by 5% if the evaporator conductance is increased by 40%, or if the condenser conductance is increased by 20%, or if the fresh-food conductance is decreased by 10%, or if the frozen-food conductance is decreased by 5%. The proposed model is intended to investigate the influence of events like door openings, internal loads and defrost strategy on the energy consumption, which will be addressed in a future publication. It should be emphasized that the model requires only 5 min of CPU time (IntelÒ Core™ 2 QUAD 2.66 GHz processor) to carry out an energy consumption prediction, a value substantially

lower than that required by more sophisticated dynamic simulation models. Finally, it is worthy of note that the model can be easily extended to any other refrigerator if the required empirical coefficients are refitted based on standardized experimental data (from pull-down and AHAM energy consumption tests) available for this particular refrigerator. Acknowledgements This study was carried out at the POLO facilities under National Grant No. 573581/2008-8 (National Institute of Science and Technology in Refrigeration and Thermophysics) funded by the CNPq Agency. Financial and technical support from Whirlpool S.A. is also duly acknowledged. Appendix A. Empirical parameters A.1. Cycling test data UAk = 1.336 W/K. UAc = 8.51 + 0.120 Ta, with Ta in °C, UAc in W/K. UAe = 12.0 + 0.266 Ta, with Ta in °C, UAe in W/K. ex = 0.572 + 0.00434 Ta, with Ta in °C. r = 0.894  0.00181 Ta, with Ta in °C. Uf = 0.013 W/K. Ur = 0.545 W/K.

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Cf = 1.971  104 J/K. Cr = 6.092  104 J/K. DTsub = 0 °C. DTsup = 3 °C. Ton = 15.56  0.0325 Ta, with Ta in °C, Ton in °C. Toff = 16.26  0.0943 Ta, with Ta in °C, Toff in °C. A.2. Steady-state test data UAr = 1.932 W/K. UAf = 0.635 W/K. References [1] Dhar M. Transient analysis of refrigeration system. PhD thesis. West Lafayette (IN, USA): Purdue University; 1978. [2] Chi J, Didion D. A simulation model of the transient performance of a heat pump. Int J Refrig 1982;5(3):176–84. [3] Yasuda H, Touber S, Machielsen CHM. Simulation model of a vapor compression refrigeration system. ASHRAE Trans 1983;89(Part 2):408–25. [4] MacArthur JW. Transient heat pump behaviour: a theoretical investigation. Int J Refrig 1984;7(2):123–32. [5] Murphy WE, Goldschmidt VW. Cyclic characteristics of a typical residential air conditioner: modeling of start-up transients. ASHRAE Trans 1985;91(Part 2):427–44. [6] Sami SM, Duong TN, Mercadier Y, Galanis N. Prediction of transient response of heat pumps. ASHRAE Trans 1987;93(Part 2):471–90. [7] Vargas JVC, Parise JAR. Simulation in transient regime of a heat pump with closed-loop and on–off control. Int J Refrig 1995;18(4):235–43. [8] Rossi T, Braun JE. A real-time transient model for air conditioners. In: IIR international congress of refrigeration, Sydney, Australia; 1999 [CD-ROM]. [9] Browne MW, Bansal PK. Transient simulation of vapour-compression packaged liquid chillers. Int J Refrig 2002;25:597–610. [10] Kim M, Kim MS, Chung JD. Transient thermal behavior of a water heater system driven by a heat pump. Int J Refrig 2004;27:415–21. [11] Lei Z, Zaheeruddin M. Dynamic simulation and analysis of a water chiller refrigeration system. Appl Therm Eng 2005;25:2258–71. [12] Melo C, Ferreira RTS, Negrão COR, Pereira RH. Dynamic behaviour of a vapour compression refrigerator: a theoretical and experimental analysis. In: IIR meeting at Purdue, West Lafayette, IN, USA; 1988. p. 98–106.

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