Transient simulation of a two-door frost-free refrigerator subjected to periodic door opening and evaporator frosting

Applied Energy 147 (2015) 386–395

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

Transient simulation of a two-door frost-free refrigerator subjected to periodic door opening and evaporator frosting q Bruno N. Borges a, Cláudio Melo a, Christian J.L. Hermes b,⇑ a POLO Research Laboratories for Emerging Technologies in Cooling and Thermophysics, Department of Mechanical Engineering, Federal University of Santa Catarina, 88040900 Florianópolis, SC, Brazil b Laboratory of Thermodynamics and Thermophysics, Department of Mechanical Engineering, Federal University of Paraná, 81531990 Curitiba, PR, Brazil

h i g h l i g h t s  Transient behavior of a refrigerator under periodic door opening is simulated.  The refrigeration loop is modeled following a semi-empirical quasi-steady approach.  Energy and moisture transfer into and within the compartments are modeled.  Key heat and mass transfer parameters were derived from in-house experiments.  Predictions followed closely the experimental trends for power and temperatures.

a r t i c l e

i n f o

Article history: Received 15 September 2014 Received in revised form 20 January 2015 Accepted 21 January 2015 Available online 17 March 2015 Keywords: Household refrigerator Transient simulation Door opening Evaporator frosting

a b s t r a c t This paper describes a quasi-steady-state simulation model for predicting the transient behavior of a two-door household refrigerator subjected to periodic door opening and evaporator frosting. A semiempirical steady-state model was developed for the refrigeration loop, whereas a transient model was devised to predict the energy and mass transfer into and within the refrigerated compartments, and also the frost build-up on the evaporator. The key empirical heat and mass transfer parameters required by the model were derived from a set of experiments performed in-house in a climate-controlled chamber. In general, it was found that the model predictions followed closely the experimental trends for the power consumption (deviations within ±10%) and for the compartment temperatures (deviations within ±2 K) when the doors are opened periodically and frost is allowed to accumulate over the evaporator. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Modern refrigerator design is aimed at energy savings and also at product robustness in relation to evaporator frosting. In this regard, standardized tests [1,2] as well as tests under real usage conditions, that is, with doors opened regularly [3,4] allowing moisture to enter the refrigerated compartment and frost to accumulate on the evaporator [5] are procedures commonly carried out by most manufacturers. Nevertheless, since the experimental procedures for frost-free refrigerators and freezers are costly and time consuming [6,7], simulation models have been devised to improve the product development process [8–15]. None of them, however, can predict

q An abridged version of this manuscript was presented at the 15th International Refrigeration and Air Conditioning Conference at Purdue, July 14–17, 2014. ⇑ Corresponding author. Tel./fax: +55 41 3361 3239. E-mail address: [email protected] (C.J.L. Hermes).

http://dx.doi.org/10.1016/j.apenergy.2015.01.089 0306-2619/Ó 2015 Elsevier Ltd. All rights reserved.

the refrigerator performance degradation due to periodic door opening and consequent evaporator frosting. Recently, Mastrullo et al. [16] put forward a transient simulation model that is suitable to predict the time evolution of the compartment air temperature and the power consumption taking into account the door opening, and the resulting evaporator frosting. The model was developed and validated for a single-door upright freezer, which represents a small niche in the realm of household refrigeration if compared with two-door frost-free appliances, the so-called ‘‘Combi’’ refrigerators [11,14]. To the best of the authors’ knowledge, none of the models available in the literature [8–16] are able to predict the performance of two-door frost-free refrigerators under periodic door opening, which not only affect the sensible and the latent loads, but also allows frost to build-up on the evaporator, thus decreasing the air flow rate supplied by the fan. To advance a simulation model for predicting the transient behavior of a two-door frost-free refrigerator subjected to periodic

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Nomenclature

Dt Roman A C cp D Dfr G H h Ha hlv hsv k kfr L Le m N NTU p Q r S T UA v V W w W

heat transfer area, m2 thermal capacity, J K1 specific heat at constant pressure, J kg1 K1 inner diameter, m effective vapor diffusivity in frosted media, m2 s1 mass flux, kg/m2 s height, m specific enthalpy, J kg1 Hatta number, dimensionless latent heat of evaporation, J kg1 latent heat of sublimation, J kg1 thermal conductivity, W m1 K1 effective thermal conductivity in frosted media, W m1 K1 length, m Lewis number, dimensionless mass flow rate, kg s1 compressor speed, Hz number of transfer units, dimensionless pressure, Pa heat transfer rate, W air flow ratio, dimensionless compressor swept volume, m3 temperature, K thermal conductance, W specific volume, m3 kg1 volumetric air flow rate, m3 s1 compression power, W humidity ratio, kgv kg1 a width, m

Greek

a d ec ex / Dp

heat transfer coefficient, W m2 K1 frost thickness, m emissivity of the condenser wall, dimensionless heat exchanger effectiveness, dimensionless correction factor, kgv pressure drop, Pa

door opening is therefore the main aim of this study. The proposed model follows a quasi-steady-state approach [14], with a steadystate sub-model for the refrigeration loop and a transient submodel for the energy and moisture transfer into and within the refrigerated compartments. An additional frost growth and densification sub-model was developed to predict the frost accumulation on the evaporator over time.

gg gv q r f

time-step, s global compression efficiency, dimensionless volumetric compression efficiency, dimensionless density, m Stefan-Boltzmann constant, W m2 K4 evaporator dry-out position, m

Subscripts 1 compressor inlet 2 condenser inlet 3 condenser outlet 4 evaporator inlet 5 evaporator outlet a ambient, air c condenser cap capillary tube d door e evaporator f flash-point ff fresh-food fr frost fz freezer g saturated vapor at the evaporating pressure i inlet k compressor l saturated liquid lat latent thermal load m mullion o outlet r refrigerant s isentropic process sat saturation sen sensible thermal load ss steady-state sub subcooling suc suction line sup superheating v saturated vapor x internal heat exchanger

compressor shell thermal conductance (UAk) is also required for the heat transfer calculation [12]. The refrigerant specific enthalpy at the compressor outlet is thus obtained from the following energy balance [13]:

h2 ¼ h1 þ

h2;s  h1

gg



UAk ðT 2  T a Þ mk

2. Simulation model 2.1. Refrigeration loop A 433-liter top-mount refrigerator, running with R-134a and comprised of a 6.76-cm3 hermetic reciprocating compressor, natural draft wire-on-tube condenser, tube-fin evaporator and capillary tube-suction line heat exchanger, illustrated in Fig. 1, was adopted in this study. 2.1.1. Compressor The compressor sub-model uses the volumetric (gv) and overall (gg) efficiencies to calculate the compression power and the refrigerant mass flow rate for a given operating condition. The

Fig. 1. Schematic representation of the refrigeration loop.

ð1Þ

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where (h2,s–h1)/gg is the compression power, whereas the mass flow rate displaced by the compressor, mk, is calculated from [12,13]:

mk ¼ gv

NS

ð2Þ

v1

ð10Þ

where N and S are the compressor speed and the swept volume, respectively. The compression efficiencies were fitted to the experimental data as linear functions of the pressure ratio, and the UAk coefficient was expressed as a linear fit to the surrounding air temperature data [12,13]. 2.1.2. Capillary tube suction line heat exchanger The internal heat exchanger was modeled according to the semi-empirical approach introduced by Hermes et al. [17], who considered the refrigerant flow and the heat transfer as independent phenomena, and derived explicit algebraic expressions for the refrigerant mass flow rate as follows:

 0:145  0:315 mx Lsuc Dsuc ¼ 1:29 e0:285 x mad Lcap Dcap

v f lf v g lg

!0:214 ð3Þ

where mad is the mass flow rate of an adiabatic capillary tube with the same bore and length, calculated as follows:

mad

coefficient was calculated from the correlation proposed by Melo and Hermes [18],  0:60  0:28  0:49  0:08 ac þ arad Aw pt  dt pw  dw Tc  Ta ¼ 5:68 2 arad At þ Aw dt dt Tc þ Ta

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    D5cap pc  pf pf  pe f gpe þ f ¼ 6:0 þ þ 2 ln g gpf þ f Lcap vf g

ð4Þ

where f = vfpfk, g = vf(1k), and k = 1.67  105p0.72 [17], vf and pf f are the specific volume and pressure at the flash-point, respectively, whereas pc and pe stand for the condensing and evaporating pressures. The correlation is valid for internal heat exchangers with capillary tube inner diameters ranging from 0.553 mm to 2.154 mm, and tube lengths from 2 m to 4 m [17]. The specific enthalpy at the evaporator inlet and the temperature at the compressor inlet are thus expressed as:

h4 ¼ h3  ex cp;1 ðT 3  T 5 Þ

where dt and dw are the tube and wire diameters, respectively, pt and pw are the tube and wire pitches, respectively, and At and Aw are the overall heat transfer surface due to the tubes and the wires, respectively. The correlation is valid for wire-and-tube condensers with tube outer diameter ranging from 4.8 to 6.2 mm, number of tube rows from 13 to 25, and number of wire pairs from 10 to 90 [18]. The radiative heat transfer coefficient is calculated from the following linearized model:

arad ¼ ec rðT c þ T a ÞðT 2c þ T 2a Þ

where r is the Stefan–Boltzmann constant, whereas ec (=0.81) is the emissivity of the condenser walls. 2.1.4. Evaporator The no-frost evaporator, illustrated in Fig. 2, was divided into two domains, namely refrigerant flow and air-side heat and mass transfer. The specific refrigerant enthalpy at the evaporator outlet can be expressed as [12–14]:

h5 ¼ h4 þ



f ¼ fss  ðfss  f Þ expðsDtÞ

NTU ¼

m0:57 Lsuc 1:4 ad0:43 Dsuc

ð6Þ

l

2=3 kg 0:1 c 2=3 p;g g

ð7Þ

2.1.3. Condenser The natural draft wire-and-tube condenser was divided into three domains, namely superheating, saturation and subcooling [12–14]. The overall heat transfer in the condenser is therefore calculated from:

Q c ¼ Q c; sup þ Q c;sat þ Q c;sub ¼ UAc ðT c  T a Þ

s¼

ð1  x4 Þmox hlv q0

ð1  cÞql ð14 pD2e Þhlv q0

Qc mk

q0 ¼

mx ðhv  h4 Þ fo

where f° = Le at t = 0.

ð8Þ

ð9Þ

As the heat transfer is governed by free convection and radiation on the air-side, the thermal conductance was approximated as UAc  (ac + arad) (At + Aw) [12–14]. The combined heat transfer

ð13Þ

ð14Þ

ð15Þ

where mox (=3.8 kg/h) is the initial mass flow rate evaluated at steady-state conditions, c is the mean void fraction of the twophase region, calculated as suggested by Cioncolini and Thome [20], De is the inner diameter of the evaporator coil, and q0 is the heat transfer rate per unit length in the two-phase region, calculated from

where Tc is the condensing temperature, calculated implicitly by the model thus ensuring that the same amount of mass flows through the compressor and the capillary tube, whereas Ta is the surrounding air temperature, which is an input data with a constant value. The specific refrigerant enthalpy at the condenser outlet is expressed as:

h3 ¼ h2 

ð12Þ

where fss is the two-phase boundary position in the steady-state regime, and s is a time constant, calculated respectively from

ð5Þ

where ex = NTU/(1 + NTU) is the heat exchanger effectiveness (0.65), whereas NTU stands for the number of transfer units, calculated as follows [17]:

Q sen þ Q lat mx

where h5 = h(pe, T5) and T5 = Te + DTsup. The evaporator superheating, DTsup, varies with time due to the periodic door opening. To address this issue, a moving-boundary approach, as introduced by Wedekind and Stoecker [19], was adopted:

fss ¼ T 1 ¼ T 5 þ ex ðT 3  T 5 Þ

ð11Þ

Fig. 2. Schematic representation of the ‘‘no-frost’’ evaporator coil.

ð16Þ

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the case of a uniform coil temperature, i.e. evaporator filled with two-phase refrigerant, the heat and mass balances yield:

   ae Ae Q sen ¼ ma cp;a ðT fr  T i Þ 1  exp  ma cp;a " Q lat ¼ ma hsv ðwfr  wi Þ 1  exp 

ae Ae

ð17Þ !#

ma cp;a Le2=3

ð18Þ

where Qe = Qsen + Qlat, Tfr is the frost surface temperature, wfr = wsat(Tfr) is the humidity ratio at the frost surface, Le is the Lewis number and ae is the air-side heat transfer coefficient calculated as suggested by Barbosa et al. [21], Fig. 3. Schematic representation of the air flow during a door opening event.

ΔP e = 5 Pa ΔP e = 15 Pa ΔP e = 25 Pa ΔP e = 35 Pa ΔP e = 45 Pa

56 54 52

ΔPt [Pa]

Gmax cp

¼ 0:6976  Re0:4842 D

50 48 46

T fr ¼ T e þ 44 42 40 0,000

0,002

0,004

0,006

0,008

0,010



Ae Ato

0:3426

Pr2=3

ð19Þ

where Gmax = ma/Amin is the mass flux at the minimum free flow passage, i.e. the cross-section area obtained from a transversal cut including tubes and fins, and Ato is the surface area of the tube only. One should note that the evaporator of the refrigerator under study is similar to sample #6 tested by Barbosa et al. [21]. In addition, one should note that, in the cases where superheated refrigerant takes place, the terms between brackets in equations (17) and (18) must be modified to account for the refrigerant temperature variation, as described in [22]. The frost formation model was based on the work of Hermes et al. [23], which was originally developed for horizontal flat surfaces and later adapted by [5] for finned-tube heat exchangers (Fig. 2). According to [23], the frost surface temperature is calculated from:

60 58

ae

0,012

Va [m³/s] Fig. 4. Characteristic curve of the fan.

ðQ sen þ Q lat Þd wsat;e hsv qa Dfr þ ð1  cosh HaÞ kfr Ae kfr

ð20Þ

where Ha is the Hatta number, and kfr and Dfr are the effective thermal conductivity and vapor diffusivity of the frost layer, respectively, calculated as described in [23]. In addition, the growth rate of the frost layer of thickness d is calculated from:

2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3  2   dd 2kfr 4 bdQ sen 4bdhsv bdQ sen 5 ¼ þ Gfr  1 þ 1þ dt bqfr dhsv kfr Ae kfr kfr Ae ð21Þ

2.2. Evaporator frosting The evaporator sub-model calculates the cooling capacity from the following heat and mass balances on the air-side (see Fig. 2). In

where Gfr is the total mass flux of the vapor transferred to the frosted medium, and b is an empirical parameter that comes from the frost density correlation in the following form [24]:

Fig. 5. Schematic representation of the wind-tunnel facility.

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qfr ¼ a expðbTfr Þ

ð22Þ

The accumulated frost mass and the frost thickness are calculated, respectively, from mfr = mofr + GfrAeDt and d = do + (dd/dt)Dt [5,23]. In addition, the frost density is obtained from qfr(t > 0) = mofr/Aedo, where the superscript ()o represents the values at the previous time-step. The defrost process has not been accounted for by the model. 2.3. Refrigerated compartments The model for the refrigerated compartments, which was based on the work of Borges et al. [14], is aimed at determining the psychrometric conditions of the moist air inside the fresh-food and freezer (frozen-food) compartments over time by means of transient energy and mass balances, which yield the following expressions for the air temperature and humidity of the fresh-food and freezer compartments, respectively:

  A Dt T ¼ T ss;  ðT ss;  T o Þ exp  C

ð23Þ

  B Dt w ¼ wss;  ðwss;  wo Þ exp  q /

ð24Þ

where T o and wo are the air temperature and humidity ratio of the compartment at the beginning of the time-step, A⁄ = UA⁄ + UAm + md,⁄cp,a + m⁄cp,a and B⁄ = md,⁄ + m⁄. The asterisk ()⁄ indicates either the freezer (fz) or the fresh-food (ff) compartment. The thermal conductance of each compartment, UA⁄, and of the mullion, UAm, and the equivalent thermal capacity and mass of each compartment, C⁄ and /⁄, respectively, were all obtained from experimental data. In addition, the terms Tss,⁄ and wss,⁄ are related to the steady-state condition, and calculated as follows:

T ss; ¼

ðUA þ md; cp;a ÞT a þ UAm T r þ ma; cp;a T o A

ð25Þ

wss; ¼

md; wa  m wo B

ð26Þ

where

T o ¼ rTfz þ ð1  rÞT ff 

Q sen ma cp;a

wo ¼ rwfz þ ð1  rÞwff 

Q lat ma hsv

ð27Þ

ð28Þ

where r is the freezer air flow ratio. The air flow rate entering the cabinet during a door opening event is calculated as described by Wang [25], and shown in Fig. 3, as follows:

md;

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u2gHd; ð1  qa =q Þ 2 ¼ K q W d; Hd; t 3 3 ð1 þ ðq =qa Þ1=3 Þ

ð29Þ

where K⁄ is an empirical discharge coefficient determined from experimental data. The air-side hydrodynamics was modeled according to the methodology outlined in Hermes et al. [13]. The overall pressure drop was thus correlated to the air flow rate through the following expression:

Dpt ¼ cqa V 2 þ Dpe

ð30Þ

where c is an empirical coefficient obtained from experimental data, and Dpe is the pressure drop in the frosted evaporator, calculated as follows:

Fig. 6. Information flow diagram of the simulation model.

2

Dp ¼

fGmax Ae 2qa Amin

ð31Þ

where f is the friction factor calculated from [21]:

f ¼ 5:965  Re0:2948 D

 0:7671  0:4436 Ae Nlo At 2

ð32Þ

The characteristic curve for the fan, depicted in Fig. 4, was fitted to a third-order polynomial using experimental data obtained in a windtunnel facility [13], illustrated in Fig. 5. 2.4. Solution algorithm The model was coded in EES [26]. The solution algorithm, illustrated in Fig. 6, is based on a sequential solution for the models for the refrigerated compartments and the refrigeration loop, whose

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Fig. 7. Summary of the instrumentation.

Table 1 Summary of experimental tests. Experiment

Test facility

Ambient conditions

Empirical parameter

Fan characteristics Cabinet hydrodynamics Refrigeration loop Door opening tests

Wind-tunnel Wind-tunnel Chamber Chamber

21 °C, 50% RH 21 °C, 50% RH 6 runs, doors closed, 25 < T < 38 °C, 8 < w < 32 g/kg 3 runs, door openings, 25 < T < 38 °C, 12 < w < 21 g/kg

3rd-order polynomial c r, gv, gg, UAk, UA⁄, UAm, C⁄ K⁄, /⁄, qofr, mok, b

sub-models are in turn solved simultaneously by the Newton– Raphson technique. The evaporator frosting and dry-out sub-models are solved implicitly in the inner loop, whereas the models for the refrigerated compartments are solved explicitly in an outer loop, as depicted in Fig. 6. The on–off cycles were implemented by means of an IF-THEN-ELSE loop that emulates a thermostat. The door opening patterns are included in the model in an outer loop, as shown in Fig. 6. More information can be obtained in [22].

3. Experimental work The refrigerator was carefully instrumented as illustrated in Fig. 7. The evaporating and condensing pressures were measured by means of strain gauge pressure transducers ranging from 0 to 10 bar (±2 mbar uncertainty) and from 0 to 20 bar (±4 mbar uncertainty), respectively. A Coriolis-type mass flow meter with a measurement uncertainty of ±0.03 kg/h was installed at the compressor discharge. The compressor and fan power consumption were monitored using a digital power analyzer with a measurement uncertainty of ±0.1%. Capacitive relative humidity transducers (±2% uncertainty) were installed at the evaporator inlet and outlet together with pressure takes for air-side pressure drop measurements using a differential pressure transducer ranging from 0 to 62.5 Pa (±0.3 Pa uncertainty). All T-type

thermocouples employed in this study have a measurement uncertainty of ±0.3 °C. More details can be found in [22]. The experimental plan, summarized in Table 1, was designed to provide all of the empirical information required for the sake of model closing. The doors were opened and closed using a purpose-built door-opening device attached to both the freezer and fresh-food doors [3], so that they could be operated independently. The arrangement is placed within a climate chamber, depicted in Fig. 8, which provides a strict control of temperature, humidity and air velocity. The time between door opening events and the length of time of the event are both easily programmed. Patterns comprised of 4 cycles of door opening events, each cycle lasting 1 h and applied in sequence, were adopted, with a 4-h interval between cycles. After this period, the system was kept running for 8 h with the doors closed. The pattern was repeated every 24 h. In a door opening cycle, the freezer door was opened every 12 min for 10 s over a period of 1 h, totalizing 5 opening events per hour. On the other hand, the fresh-food door was opened every 2.5 min for 30 s, totalizing 20 opening events per hour. 4. Results The model predictions were compared with the corresponding experimental data for ambient conditions of 32 °C and 70% RH.

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Fig. 8. Schematic representation of the climate chamber.

25

17 Tff, Experimental

16

Tff, Simulated

1.8 15

Condensing pressure [bar]

15 10

Temperature [°C]

1.9

5 0 Tfz, Experimental

-5

Tfz, Simulated

1.7

14

1.6

13

1.5 1.4

12 1.3 11

1.2

10

1.1

-10

Evaporating pressure [bar]

20

2.0 Pc, Simulated Pe, Simulated

Pc, Experimental Pe, Experimental

1.0

9

0.9

-15

8

-20

0.8

7

0.7 0

-25 0

60

120

180

240

300

360

420

480

540

Time [min]

60

120

180

240

300

360

420

480

540

Time [min] Fig. 10. Time evolution of the condensing and evaporating pressures.

Fig. 9. Time evolution of the refrigerated compartment temperatures.

The simulations were initiated at the compressor start up immediately after a defrost cycle and lasted until the next defrost cycle began. Fig. 9 shows the predicted and experimental results for the refrigerated compartment temperatures for the whole period. Two door opening cycles can be clearly seen, the first from 30 to 90 min and the second from 330 to 390 min. It should be noted that the model follows closely the experimental trends, with an average deviation of around ±2 °C, the maximum discrepancies occurring during door opening.

Fig. 10 compares the predictions of the working pressures with the experimental data. It can be observed that the model predictions for the evaporating pressure are within a ±5% error band, while the condensing pressure is under predicted with an offset of approximately 0.5 bar during the whole period, but with deviations within a ±10% band. Consequently, the power consumption is also reasonably well predicted by the model, with deviations not exceeding ±5%, as shown in Fig. 11. Fig. 12 shows the time-evolution of the evaporator superheating. Five peaks can be observed during the test period, which

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B.N. Borges et al. / Applied Energy 147 (2015) 386–395

200

30

Experimental Experimental Simulated Simulated

180

Experimental

Simulated Simulated 25

Evaporator air-side pressure drop [Pa]

Compression power [W]

160 140 120 100 80 60 40

20

15

10

5 20 0

0 0

60

120

180

240

300

360

420

480

540

0

60

120

180

240

Time [min]

300

360

420

480

540

Time [min]

Fig. 11. Time evolution of the compression power.

Fig. 13. Time evolution of the evaporator air-side pressure drop.

12 120

Experimental

1st Row 2nd Row 3rd Row 4th Row 5th Row

Simulated Simulated 10

Accumulated frost mass [g]

Evaporator superheating [K]

100

8

6

4

2

80

60

40

20

0 0

60

120

180

240

300

360

420

480

540

Time [min] Fig. 12. Time evolution of the evaporator superheating.

appear when both doors are opened concurrently, thus increasing the thermal loads and pushing the dry-out position downstream of the evaporator. Small variations in the evaporator superheating occur when only the door of the fresh-food compartment is opened. It can also be noted that the model predictions follow the experimental trends satisfactorily, although errors of up to 3 K can be observed in the peaks. Fig. 13 shows the time-evolution of the air-side pressure drop due to evaporator frosting and Fig. 14 explores the air-side prediction capabilities of the model. For a clean evaporator coil, a 5 Pa pressure drop is observed, increasing steadily to 15 Pa during the first cycle of door opening events. This value does not change during the period when the doors are kept closed, since there is no

0 0

60

120

180

240

300

360

420

480

540

Time [min] Fig. 14. Time evolution of the accumulated frost mass along each evaporator row.

moisture infiltration and, therefore, no frost growth. During the second cycle of door openings the evaporator air-side pressure drop increases to around 25 Pa. During the whole period the model predicted satisfactorily the experimental trends, although absolute errors of up to 5 Pa can be observed. Fig. 14 shows the calculated frost mass for each of the five evaporator rows. The rows are numbered from bottom to top, according to the air flow direction, as illustrated in Fig. 2, and comprised 27 (1st), 34 (2nd), 67 (3rd), 66 (4th), and 67 (5th) fins. It can be noted that the frost is mostly accumulated along the first three rows, which is due to the higher humidity gradient at the evaporator inlet. It can also be noted that there is more frost accumulated along the 3rd row than along the 1st row, which is due to the higher heat transfer (finned) area of the former.

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5. Summary and conclusions A quasi-steady-state semi-empirical mathematical model was developed to predict the transient behavior of the key operating parameters (i.e., working pressures, compression power, compartment temperatures, and accumulated frost mass on the evaporator) of a two-door frost-free refrigerator subjected to periodic door opening. The model predictions were compared with a set of in-house experimental data collected in a climate-controlled chamber. The door opening were carried out by a purpose-built apparatus according to a predefined pattern. It was found that the model predictions followed closely the experimental trends, with deviations for the working pressures and power consumption not exceeding the 10% thresholds and predictions for the compartment air temperatures being within ±2 °C error bands. The model was also used to predict the frost distribution over the evaporator coil and it was observed that the frost accumulates mostly in the first three rows, the third row being crucial in terms of frost clogging because of the higher number of fins and thus lower free flow passage of air.

Heat exchanger length: 1.34 m. Approximate heat exchanger effectiveness: 65%. Suction line inner diameter: 7 mm. Evaporador Type/material: no-frost tube-fin/aluminium. Coil length: 7.585 m. Inner diameter: 6.7 mm. Outer diameter: 7.9 mm. Number of tubes: 10 (longitudinal)/2 (transversal). Evaporator height: 189 mm. Evaporator width: 340 mm. Evaporator depth: 59 mm. Number of fins (1st row): 27. Number of fins (2nd row): 34. Number of fins (3rd row): 67. Number of fins (4th row): 66. Number of fins (5th row): 67. Fin dimensions: 35 mm (height)/59 mm (width)/0.125 mm (thickness).

Acknowledgments 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 Brazilian Government Agency CNPq. The authors are grateful to Mr. Rafael Gões for his valuable support in the experiments. Financial support from Whirlpool Latin America S.A. is also duly acknowledged. Appendix A. Refrigerator characteristics Refrigerator Type: Top-mount frost-free. Refrigerant type/charge: HFC-134a/100 g. Cabinet internal volume: 439 liters. Fan power consumption: 7 W. Compressor Type: Hermetic reciprocating compressor. Stroke: 6.76 cm3. Speed: 60 Hz. Condenser Type/material: wire-and-tube/steel. Length of the discharge line: 1600 mm. Condenser height/length of the wires: 1210 mm. Condenser width: 540 mm. Tube outer diameter: 5.1 mm. Bend radius: 28.8 mm. Number of tubes: 21. Wire diameter: 1.4 mm. Number of wires: 90. Surface emissivity: 0.81. Internal heat exchanger Type/material: concentric/copper. Capillary tube outer diameter: 1.90 mm. Capillary tube inner diameter: 0.80 mm. Capillary tube length: 2.55 m.

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