Development of a continuous empirical correlation for refrigerant mass flow rate through non-adiabatic capillary tubes

Applied Thermal Engineering 127 (2017) 547–558

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Research Paper

Development of a continuous empirical correlation for refrigerant mass flow rate through non-adiabatic capillary tubes Mehdi Rasti, Jeong Ho Ban, Ji Hwan Jeong ⇑ School of Mechanical Engineering, Pusan National University, Geumjeong-Gu, Busan 46241, Republic of Korea

h i g h l i g h t s
Continuous correlation for mass flow rate through non-adiabatic capillary tube.
Single correlation is applicable to subcooled, two-phase, and superheated regions.
Correlation was validated to be applicable to R-134a, R-410A, R-152a, and R-22.

a r t i c l e

i n f o

Article history: Received 26 January 2017 Revised 5 July 2017 Accepted 12 August 2017 Available online 14 August 2017 Keywords: Refrigerant Non-adiabatic capillary tube Mass flow rate correlation Continuous capillary model Dimensionless parameter Buckingham theorem

a b s t r a c t Capillary tube, suction line heat exchangers (CT-SLHX) are widely used as expansion devices in smallcapacity refrigeration and air-conditioning systems to enhance the refrigeration capacity and ensure that superheated refrigerant vapor enters the compressor. To calculate the mass flow rate through a capillary tube, a reliable non-adiabatic capillary tube model is necessary. Most previous correlations were developed separately for subcooled liquid inlet conditions and for saturated two-phase inlet conditions; so the models are not continuous at the saturated liquid point. An empirical model that is continuous at the saturated liquid point was developed and is introduced herein with a new dimensionless p parameter. This new empirical model is validated using experimental measurements available in the literature for the refrigerants R-134a, R-600a, R-410A, R-152a, and R-22. The new correlation shows good agreement with the experimental data. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction A throttling device connects the condenser outlet to the evaporator inlet, causes a drop in refrigerant pressure, and regulates the refrigerant mass flow rate. A capillary tube is widely used as the throttling device in small-capacity vapor-compression refrigeration systems such as household refrigerators and freezers. It has some advantages including simplicity, low cost, no moving parts, and reduction of the compressor starting torque due to equalization of the condenser and evaporator pressures during the offcycle [1]. A capillary tube is a long and narrow hollow copper tube with a constant cross-sectional area. Generally, the capillary tube inner diameter and its length varies from 0.33 mm to 2.0 mm, and 2 m to 6 m, respectively [2]. Although a capillary tube appears to be quite simple, the refrigerant flow inside the capillary tube is rather complicated because the refrigerant experiences various

⇑ Corresponding author. E-mail address: [email protected] (J.H. Jeong). http://dx.doi.org/10.1016/j.applthermaleng.2017.08.070 1359-4311/Ó 2017 Elsevier Ltd. All rights reserved.

thermodynamic states including subcooled liquid, meta-stable liquid, meta-stable two-phase, and saturated two-phase. Generally, capillary tubes can be categorized as straight or coiled depending on their geometry, and as adiabatic or nonadiabatic depending on the intentional heat transfer to a suction line [3,4]. An adiabatic straight capillary is thermally insulated with negligible heat exchange with the ambient environment. Fig. 1 illustrates a schematic and a P-h diagram of a vapor compression refrigeration system with an adiabatic capillary tube in which the refrigerant is subject to an isenthalpic process. The point 3 in Fig. 1(b) represents that the refrigerant enters the adiabatic capillary tube in a subcooled liquid state. Process 3-4 in Fig. 1(b) is adiabatic and the refrigerant temperature remains constant as far as it is in the subcooled liquid state. Due to flashing of the saturated liquid, a part of total energy converts into kinetic energy. As a result, the enthalpy of refrigerant slightly falls in the latter part of the adiabatic capillary tube. Fig. 2 shows a schematic and a P-h diagram of a vapor compression refrigeration system with a non-adiabatic capillary tube. The non-adiabatic capillary tube is soldered onto the suction line

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Nomenclature A C D f h HX L Nu _ m P Pr Re T V x y z Greeks k

l p q m r

e u2

area, m2 specific heat, J kg1 K1 diameter, m friction factor enthalpy, J kg1 heat exchanger length, m Nusselt number mass flow rate, kg s1 pressure, Pa Prandtl number Reynolds number temperature, K velocity, m/s vapor quality meta-stable mass fraction section length, m

thermal conductivity, W m1 K1 viscosity, kg m1 K1 pi dimensionless parameter density, kg m3 specific volume, m3 kg1 surface tension, N m1

(a) Refrigeration system

wall roughness, mm frictional two-phase multiplier

Subscripts c capillary tube crit critical cond condensing f saturated liquid g saturated vapor hx heat exchanger in inlet l liquid lo liquid only p constant pressure out outlet s suction line sat saturated sc capillary tube inlet subcooled level seg segment sh suction line inlet superheat level sp single phase sup superheat tp two-phase v vapor

(b) P-h diagram

Fig. 1. Vapor compression system employing adiabatic capillary tube.

(a) Refrigeration system

(b) P-h diagram

Fig. 2. Vapor compression system employing capillary tube suction line heat exchanger.

M. Rasti et al. / Applied Thermal Engineering 127 (2017) 547–558

to form a capillary-tube-suction-line heat exchanger (CT-SLHX). The advantage of employing CT-SLHX is that it decreases the vapor quality and specific enthalpy at the evaporator inlet by transferring heat from the capillary tube to the suction line, which leads to an increase in the refrigeration capacity of the system with the same refrigerant mass flow rate as shown in process 4-5 in Fig. 2(b). Moreover, it ensures that the superheated vapor in the suction line returns to the compressor as shown in process 5-1 in Fig. 2(b). Thus, slugging and suction line sweating might be eliminated. The disadvantage of employing CT-SLHX is increasing the compressor power consumption due to the higher suction line temperature [5]. Even though two types of CT-SLHX configurations (lateral and concentric configurations) are found in the open literature, the lateral configuration of CT-SLHX is adopted in most household refrigerators and freezers due to the ease of manufacturability and higher heat exchange effectiveness [6]. For CT-SLHX, three different regions can be noted: the inlet adiabatic region, the heat exchanger region, and the outlet adiabatic region. Numerous studies have been carried out to predict the refrigerant mass flow rate through adiabatic and CT-SLHX capillary tubes over the past 60 years. Nevertheless, there are few experimental studies for CT-SLHX. Bolstad and Jordan (1948) were the first researchers to measure temperature and pressure in a CT-SLHX with R-22. Pate and Tree [7–9] published several experimental studies describing details of the experimental apparatus and results of R-12 flow through a CT-SLHX tube. They reported that the mass flow rate through a CT-SLHX was increased over that of the adiabatic capillary by 20%. The complete procedure and results from testing R-134a flow in a CTSLHX was studied by Liu and Bullard [10]. They reported mass flow hysteresis in a CT-SLHX, which was observed to be as large as 7%. Bittle et al. experimentally evaluated CT-SLHX performance with R-134a and R-152a as refrigerants [11,12]. They obtained a mass flow rate for each refrigerant with an applicable range of heat exchanger geometries and operating conditions that included the capillary tube inner diameter (0.66– 0.78 mm), capillary tube length (2.44–3.30 m), heat exchanger length (0.76–1.78 m), condenser temperature (29.44–55.56 °C), evaporator pressure (110–179 kPa), capillary tube inlet condition (8.3 °C subcooled to 5% refrigerant quality), and oil concentration level (0–3%). Their experimental results showed that condenser temperature and capillary tube inner diameter have the greatest effect on the mass flow rate. Furthermore, they suggested two correlations to predict the mass flow rate and suction line outlet temperature. Melo et al. published empirical correlations to predict the mass flow rate and the outlet temperature of the suction line based on the experimental study of a concentric CT-SLHX with R-600a [13]. Also, they experimentally evaluated the mass flow rate and suction temperature of a concentric CT-SLHX with R-134a. Experimental results showed that the inner diameter of the capillary tube had the greatest effect on the mass flow rate. Moreover, the heat exchanger and adiabatic inlet lengths had a significant impact on the suction line outlet temperature. Dirik et al. conducted an experimental study on an adiabatic and a concentric CTSLHX with R-134a [14]. Greenfield [15] presented experimental data in a range of condenser temperatures and subcooled temperatures using a CT-SLHX with R-134a as the refrigerant. Wolf and Pate [16] presented two general correlations based on the Buckingham Pi theorem using experimental data of R-600a, R410A, R-22, and R-134a; for the subcooled and two-phase inlet conditions separately. Mendonca et al. presented experimental data for refrigerant flow inside a CT-SLHX with R-134a for a range of heat exchanger lengths and operating conditions typically found in domestic refrigerators and freezers [17].

549

It should be noted that most of the dimensionless correlations available in the open literature are for the refrigerant mass flow rate through adiabatic capillary tubes [2,18–22]. A few dimensionless correlations are available for predicting the mass flow rate through CT-SLHXs. There are some limitations when using dimensionless correlations. First, most of the researchers reported dimensionless correlations just for the inlet condition of subcooled liquids even though R600a may enter the capillary tube as a two-phase mixture [23]. Second, most of a set of correlations covering subcooled liquid, as well as two-phase mixture inlet conditions, are discontinuous at the saturated liquid point where the subcooled liquid state and saturated two-phase states meet. This discontinuity among of a set of correlations sometimes causes problems obtaining a converged solution in a numerical simulation of a vapor compression refrigeration system [24]. Third, there are few dimensionless correlations that can predict the mass flow rate in the superheat region. Finally, there is no single dimensionless correlation that can predict the mass flow rate through a CTSLHX, and which is continuous at the saturated liquid and saturated vapor points. The objective of this work was to develop an empirical correlation of refrigerant mass flow rate through a CT-SLHX, which is continuous at the saturated liquid point to provide reliable prediction across both subcooled liquid and saturated two-phase inlet conditions. The process was as follows. First, a mechanistic model was used to produce mass flow rate data for R-134a and R-600a through CT-SLHXs considering a wide range of influencing parameters [25]. Second, the mass flow rate data so produced was used as reference data to develop a generalized correlation by applying the Buckingham Pi theorem to generate dimensionless parameters that could be used for the prediction of the mass flow rate through a CT-SLHX. Third, a new dimensionless p parameter and an empirical model were introduced to cover the inlet conditions of both subcooled liquid and saturated two-phase states. The empirical model was validated using the experimental data and the existing correlations available in the literature. 2. Review of dimensionless mass flow rate correlations Some correlations are presented here for the mass flow rate through straight adiabatic capillary tubes and CT-SLHX for both the subcooled and two-phase inlet conditions. The purpose of this review is to assess the existing correlations for continuity of the mass flow rate prediction at the saturation liquid point based on subcooled or two-phase capillary inlet conditions. The corresponding definitions of all the dimensionless p-parameter groups used in the correlations are shown in Table 1. 2.1. Correlations for the mass flow rate through the CT-SLHX and adiabatic capillary tubes The functional form of dimensionless parameters has been widely used to correlate the mass flow rate through capillary tubes with the geometry of the capillary tubes, as well as with the working conditions. The following dimensionless correlations are applicable for R-600a and R-134a to predict the mass flow rate inside lateral CT-SLHX arrangements. In one of the earliest investigations, Wolf and Pate [16] suggested two correlations for a lateral CTSLHX. These two correlations were formulated (based on experimental data of R-600a, R-134a, R-22, and R-410A) for both the subcooled liquid conditions and two-phase mixture conditions at the capillary tube inlet. These researchers developed a set of 15 dimensionless parameters to formulate the mass flow rate inside a CTSLHX. The correlation for the subcooled inlet condition is shown in Eq. (1),

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Table 1 Non-dimensional parameters used in the literature. Equation

p-parameters

Eqs. (1) and (2)

p1

p3

p5

p6

p7subcooled

p7quality

p8

p9

Lc Dc

Lhx Dc

Pc;in D2c

Ps;in D2c

DT sc C pfc D2c

1  xin

DT sh C pfc D2c

_ m Dc lfc

p1

p2

p4

p5

p10quality

p11

p13

Dc Lc

Lhx Lc

ðP in Psat Þ P crit

1  xin

qfs qgs

p12 C pfs C pgs

kfs kgs

p3

p5

p6

p7subcooled

p7quality

p10

Lhx Dc

Pc;in D2c

Ps;in D2c

DT sc C pfc D2c

1  xin

mgc mfc

p12

Lc Dc

p1

p2

p3

p4

p5

p6

p7

D2c qf Pc;in

D2c qf P s;in

Lc Dc

Lhx Dc

D2c q2f C pf DT sc

D2c q2f C pf DT sp

p4

p5subcooled

p5quality

p6

p7

D2c P c;in

D2c C pfc DT sc

xin

Eqs. (3) and (4)

D2c

Eqs. (5)–(7)

Eq. (8)

_ pmffiffiffiffiffiffiffiffiffi

p1

Pin qin

_ m Dc lf

Eqs. (9) and (10)

Eqs. (11) and (12)

Eq. (13) Subcooled Two-phase

l2fc mfc

l2fc mfc

l2f

l2f

p1

p2

Lc Dc

D2c hfgc

p1

p2

Lc Dc

D2c hfgc

m l

Dc r mfc l2fc

p1

p2

p3

m l 2 fc

2 fc

1:273m pffiffiffiffiffiffiffiffiffi D2c pin qin

pin psat

1:273m pffiffiffiffiffiffiffiffiffi D2c pin qin

1

2 fc

2 fc

mfc l2fc

p3

qfc qgc qfc qgc

l2fc mfc

l2fc m2fc

l2fc mfc

l2fc m2fc

m l

p4

2 fc

2 fc

D2c Pc;in

mfc l

p4

for 1 K < DT sc < 17 K:

The correlation for the two-phase inlet condition is shown in Eq. (2),

p9 ¼ 0:01960 p0:3127 p1:059 p0:3662 p4:759 p0:04965 for 0:02 1 5 6 7 8 < xin < 0:1:

ð2Þ

Sarker and Jeong [5] proposed a new group of dimensionless parameters to formulate the mass flow rate inside a lateral CTSLHX. They proposed two different correlations for the subcooled and saturated two-phase inlet conditions based on a set of data for R-600a and R-134a. This model was validated against a wide range of experimental data available in the literature. The correlation for the subcooled inlet condition is as in Eq. (3),

p1 ¼ 4:7136 p0:490041 p0:08562 p0:012101 p0:03136 for 1 K < DT sc < 21 K: 2 4 5 12 ð3Þ The correlation for the two-phase inlet condition is shown in Eq. (4),

p1 ¼ 13:9976 p0:4955 p0:6819 p0:3332 p0:2895 p0:3118 for 0:02 2 10 11 12 13 < xin < 0:2:

lgc

p8 _ m Dc lfc

2 fc

pffiffiffiffiffiffiffiffiffi Dc pin qin xin lgc þð1xin Þlfc

ð1Þ

ðlfc lgc Þ

xin

lin

p9 ¼ 0:07602 p0:4583 p0:07751 p0:7342 p0:1204 p0:03774 p0:04085 p0:1768 1 3 5 6 7 8 11

l2f

p5quality

2 fc

pffiffiffiffiffiffiffiffiffi Dc pin qin

Dc Lc

l2fc m2fc

D2c C pfc DT sc

p5

Dc Lc

mgc mfc

hfgc D2c

p5subcooled m l

2 fc

l2f

l2fc m2fc

p6

p7

1

sc 1 þ TTcond

1  xin

1

p11

ðlfc lgc Þ

lgc

p13 lgs lfc

_ m Dc lfc

A correlation specific to R-600a in a CT-SLHX for subcooled inlet condition was also proposed:

p9 ¼ 0:003545 p0:4615 p0:1107 p0:6757 p0:3496 p1:5772 1 3 5 10 15

p1 ¼ 0:0093 p0:6547 p0:0018 p0:3985 p0:1004 p0:1013 p0:0762 for 0 K 2 3 4 5 6 7 < DT sc < 25 K: ð8Þ Numerous correlations to predict the mass flow rate through straight adiabatic capillary tubes were also developed for both the subcooled liquid and two-phase mixture inlet conditions. Some of them are introduced below. Bittle et al. [18] suggested two correlations for straight adiabatic capillary tubes. For the subcooled liquid inlet condition, as shown in Eq. (9):

ð4Þ

ð5Þ

In addition, Zhang suggested a correlation specific to R-134a in a CT-SLHX for the subcooled inlet condition as shown in Eq. (6).

ð6Þ

ð7Þ

Khan et al. [27] experimentally investigated the flow of R-134a inside adiabatic and non-adiabatic capillary tubes. They developed a set of seven dimensionless parameters to formulate the mass flow rate inside a CT-SLHX. They developed a correlation to predict the mass flow rate of R-134a for subcooled inlet condition in the form:

p9 ¼ 0:2307 p0:5027 p0:07255 p0:9258 p0:3293 p0:7258 for 1 K 1 3 5 12 13

p9 ¼ 0:6688 p0:6081 p0:08049 p0:6482 p0:1125 p0:04968 1 3 5 6 7

C pgs C pfc

p8

It should be noted that Eq. (4) had typographical errors in the original article [5] and these errors were corrected in the equation presented in this paper. Zhang [26] updated the refrigerant properties data available from the National Institute of Standards and Technology (NIST) to improve the correlations for predicting the refrigerant mass flow inside a CT-SLHX. In other words, he used the same dimensionless parameters as Wolf and Pate [16] and revised the correlations presented in the ASHRAE Handbook. The general correlation of Zhang based on the experimental data of R-134a, R-600a, R-22, and R-410A; and with the subcooled liquid condition at the CT-SLHX inlet is:

< DT sc < 17 K:

p15

Fig. 3. Prediction of previous of correlations.

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p8 ¼ 1:8925 p0:484 p0:824 p1:369 p0:0187 p0:773 p0:265 for 1 K 1 2 4 5 6 7 < DT sc < 17 K;

ð9Þ

and for the two-phase inlet condition as in Eq. (10),

p8 ¼ 187:27 p0:635 p0:189 p0:645 p0:163 p0:213 p0:483 for 0:03 1 2 4 5 6 7 < xin < 0:25: ð10Þ Zhang [26] presented a general correlation for adiabatic capillary tubes with subcooled inlet conditions as follows:

p8 ¼ 4:002 p0:497 p0:378 p0:369 p0:569 p0:1795 for 1 K < DT sc < 17 K 1 2 3 4 5 ð11Þ A correlation for an adiabatic capillary tube with two-phase inlet condition was also proposed in the form:

p8 ¼ 178:612 p0:6685 p0:513 p0:975 p0:159 for 0:03 < xin < 0:25: 1 2 4 5 ð12Þ Yang and Wang [2] proposed a correlation to predict the mas flow rate for adiabatic capillary tubes which covers both subcooled and two-phase inlet states as follows:

p1 ¼ 4257:9 p0:7338 p0:2220 p0:4671 p0:1226 p1:5956 p0:7061 for 1 K 2 3 4 5 6 7 < DT sc < 16:67 K ð13Þ 2.2. Assessment of the correlations mentioned Fig. 3 presents the mass flow rate predicted by the correlations mentioned above, with a set of the same boundary conditions. For all cases plotted in Fig. 3, capillary tube total length, inlet adiabatic length, heat exchanger length, and inner diameter were assumed to be 2.03 m, 0.38 m, 1.52 m, and 0.86 mm, respectively. The refrigerant R-600a was considered as the working fluid, and the capillary tube inlet pressure, outlet pressure, and suction line inner diameter were assumed to be 549 kPa, 111 kPa, and 6.5 mm, respectively.

Table 2 Mass flow rate prediction based on subcooled correlations. Author Wolf and Pate (CT-SLHX) Bittle et al. (adiabatic)

Eq. 1 9

p for subcooled temperature

Mass flow rate at T sub ¼ 0 (kg h1)



0:03774

0

0:0187

0

0:7061

3.2761 3.2665

D2c C pfc DT sc



l2fc m2fc

D2c C pfc DT sc

m2fc l2fc

Sarker and Jeong (CT-SLHX) Yang and Wang (adiabatic)

3 13

Zhang (adiabatic)

11

– sc ð1 þ TTcond Þ  0:1795

D2c C pfc DT sc

0

m2fc l2fc

Usually, the capillary tube operates under subcooled or saturated liquid inlet conditions but still, at the beginning and end of each ON/Off refrigeration cycle, the refrigerant at the capillary tube inlet becomes superheated. This is because most of the refrigerant is collected in the evaporator, therefore the condenser runs dry [28]. So, the thermodynamic state of the refrigerant at the capillary tube inlet was varied to include subcooled (0.2 to 0.0), two-phase mixture (0.0–1.0), and superheated (1.0–1.2) states so that the thermodynamic qualities of the capillary inlet condition were represented in Fig. 3. This plot demonstrates that all the mentioned correlations have some problems such as discontinuity of the saturated lines, and zero or infinity prediction for the superheated vapor condition. The mass flow rates calculated using the empirical correlations mentioned above, with the capillary tube inlet sub-cooling of 0 °C, are summarized in Table 2. The correlations of Wolf and Pate (CTSLHX), Bittle et al. (adiabatic), and Zhang (adiabatic) predicted that the corresponding mass flow rate would approach zero as the level of sub-cooling approaches zero. This is caused by the p parameters considering the level of subcooling, DT sc . The above three correlations have DT sc in the numerator so that the corresponding p parameter approaches zero as the level of sub-cooling approaches zero, which leads the predictions of mass flow rates to be zero at the saturated liquid state. Table 3 shows the mass flow rates calculated by different adiabatic and non-adiabatic correlations, with the capillary tube inlet quality equal to zero or one. This table also presents the mass flow rate predicted with capillary tube inlet superheating of zero. The mass flow rate predicted by Bittle et al. (adiabatic) and Zhang (adiabatic) goes to infinity as the inlet quality approaches zero. In addition, as the inlet quality approaches one, the mass flow rate predicted by Wolf and Pate (CT-SLHX) and Yang and Wang (adiabatic) goes to zero while the mass flow rate predicted by Sarker and Jeong (CT-SLHX) goes to infinity. All these calculated results are physically incorrect because the mass flow rate should have a finite value greater than zero. Table 3 also shows that Wolf and Pate (CT-SLHX), Sarker and Jeong (CT-SLHX), and Yang and Wang (adiabatic) could not predict the mass flow rate in the superheat region. The physically unreasonable behavior of the correlations is caused by the functional form of the p parameter considering the quality of the refrigerant at the inlet, xin , as shown in Table 3. The continuity of the functional form of the previous empirical correlations is summarized in Table 4. The correlations of Wolf and Pate (CT-SLHX) and Sarker and Jeong (CT-SLHX) are discontinuous at both thermodynamic qualities of ‘0’ (saturated liquid state) and ‘1’ (saturated vapor state). The correlations of Bittle et al. (adiabatic) and Zhang (adiabatic) are discontinuous at the saturated liquid line, while that of Yang and Wang (adiabatic) shows discontinuity at the saturated vapor line. That is, all the dimensionless correlations mentioned above have the discontinuity problems at the saturated lines. The discontinuities of previous correlations are speculated to be caused by the limited range of reference data used for their development. If these correlations are used for a refrigeration cycle simulation, the calculation might not be able

Table 3 Mass flow rate prediction based on two-phase correlations.

p for quality

Mass flow rate at x =0 (kg h1)

Mass flow rate at x =1 (kg h1)

Mass flow rate at T sup ¼ 0 (kg h1)

Author

Eq.

Wolf and Pate (CT-SLHX)

2

ð1  xin Þ

3.3725

0

N/A

Bittle et al. (adiabatic)

10

ðxin Þ0:163

1

0.8785

0.8785

Sarker and Jeong (CT-SLHX)

4

ð1  xin Þ0:6819

4.0375

1

N/A

Yang and Wang (adiabatic)

13

ð1  xin Þ1:5956

3.2665

0

N/A

Zhang (adiabatic)

12

ðxin Þ0:159

1

0.9445

0.9445

4:759

N/A represents that it is not possible to consider the superheated vapor condition.

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M. Rasti et al. / Applied Thermal Engineering 127 (2017) 547–558

ltp ¼ ð1  xÞll þ xlv :

Table 4 Continuity of empirical correlations. Correlation

At saturated liquid

At saturated vapor

Wolf and Pate (CT-SLHX) Bittle et al. (adiabatic) Sarker and Jeong (CT-SLHX) Yang and Wang (adiabatic) Zhang (adiabatic)

Discontinuous Discontinuous Discontinuous Continuous Discontinuous

Discontinuous Continuous Discontinuous Discontinuous Continuous

Heat transfer between a capillary tube and a suction pipe is dominated by thermal resistance of single-phase flow side while two-phase thermal resistance is negligible. A constant Nusselt number, 4.36, is used for laminar flow region while Gnielinski’s [33] correlation is used for turbulent flow region.

Nu ¼ to produce a converged result, depending on the working condition of the cycle. In this regard, it is necessary to develop a correlation of refrigerant mass flow rate through a capillary tube, which is continuous across the subcooled, saturated, and superheated regions. 3. Generation of reference data In order to develop a new empirical CT-SLHX model, it is necessary to have an experimental database with a wide range of influencing parameters. Because the experimental data available in the open literature are limited in terms of the parameters used, a reliable mechanistic model was used in the present work to generate reference data with a wide range of influencing parameters. The mechanistic model developed by Kim et al. [25] was used as an experimental module in this work. This mechanistic model can predict the refrigerant characteristics in adiabatic and nonadiabatic capillary tubes and was validated against numerous set of experimental data. The mechanistic model was developed based on steady-state mass, momentum, and energy conservation equations assuming pure refrigerant, homogeneous two-phase mixture; constant cross-section area and thickness; negligible thermal resistance for two-phase refrigerant flow; and no heat transfer with ambient. This mechanistic model is applicable to all common refrigerants. The thermophysical properties of refrigerants were calculated uses the REFPROP 9.1 [29]. Various correlations are incorporated into this mechanistic model to calculate heat transfer, pressure drop, and the effect of metastable region. Churchill correlation [30] is used to calculate the single phase friction factor, which is applicable over the whole range of roughness and the Reynolds number including laminar and turbulent regions. Churchill’s formula is as follows:

" f ¼8

#1=12 12 8 1 þ Re ðA þ BÞ1:5

ð14aÞ

where

" A ¼ 2:457  ln  0:9 7 Re

B¼

1 þ 0:27 

 16 37; 530 Re

!#16 e

ð14bÞ

D

ð14cÞ

The two-phase pressure drop is calculated using Lin et al.’s [31] correlation,

u2lo ¼

f tp f sp

!

   mv 1 :  1þx

ml

ð15Þ

where f sp and f tp represent single-phase and two-phase friction factor, respectively, and can be obtained using Churchill’s friction factor formula, Eq. (14). In order for calculation of f tp , the Reynolds number for two-phase flow, Retp , is defined as the same as the Reynolds number for single-phase flow with the viscosity of a twophase mixture. Cicchitti et al.’s [32] model is used to provide the viscosity of a two-phase mixture as follows:

ð16Þ

ðf =8Þ  ðRe  1000Þ  Pr 1 þ 12:7  ðf =8Þ

0:5

 ðPr 2=3  1Þ

;

ð17aÞ

where, 2

f ¼ ð0:790  lnðReÞ  1:64Þ

ð17bÞ

A meta-stable region, where vaporization is delayed, may exist between the subcooled liquid region and the equilibrium twophase flow region in a capillary tube. Chen et al. [34] suggested an empirical correlation for the meta-stable flow as follows:

pffiffiffiffiffiffiffiffiffiffi ðPsat  Pv Þ kT sat

r

3=2

      Vv DT sub 0:208 D 3:18 Re0:914 ¼ 0:679 l Vv  Vl Tc D0 ð18Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where D0 (= kT sat =r  104 ) is the reference length, k is the Boltzman constant, and T c is the critical temperature of the refrigerant. The fraction of saturated fluid out of the total fluid is obtained from the relation suggested by Feburie et al. [35], which is as follows:

 0:25   dy P psat  p ¼ 0:02 ð1  yÞ dz A pcrit  psat

ð19Þ

Kim et al. [25] reported that the aformentioned mechanistic model predicted mass flow rate of R-600a measured by Wolf and Pate [16] within discrepancy of ±10%. A laterally soldered counter-flow CT-SLHX is considered in this study and illustrated in Fig. 4. This illustration shows the inlet and outlet adiabatic regions and the heat exchanger region. The capillary tube and suction line are divided into numerous segments of 1 mm long and the calculation proceeds in a section-by-section method. Fig. 5 shows the numerical solution process for capillary tube suction line heat exchangers. Capillary tube, suction line, and heat exchanger geometries, operating conditions and segment number are given as input data. At the beginning of simulation, the pressure drop for each segment is calculated and the refrigerant mass flow rate is assumed. The simulation program checks whether the segment is in the heat exchange region or not. When the section is in the heat exchange region, the suction line outlet temperature is assumed. Based on the assumed mass flow rate, the refrigerant properties, pressure drop and the heat transfer rate of each segment are calculated. The calculation proceeds segmentby-segment up to the calculated capillary tube length is equal to capillary tube inlet adiabatic length and heat exchanger length. The suction line inlet temperature and calculated suction line inlet temperature are compared with each other to judge whether the suction line outlet temperature is obtained. If not, the suction line outlet temperature is modified and the calculation is repeated until the suction line inlet temperature matches the calculated suction line inlet temperature. It should be noted that this simulation program considers delay of vaporization and metastable region if the refrigerant at the segment inlet is in a subcooled liquid state. While the section is in the adiabatic region, the refrigerant properties and pressure drop of each segment are calculated. The process will continue until the iteration number is equal to the segment number. The capillary tube total length and calculated capillary tube length are compared with each other to judge whether the mass flow rate is obtained. If not, the mass flow rate is modified and the calculation is repeated until the capillary tube total length matches to the calculated capillary tube length.

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M. Rasti et al. / Applied Thermal Engineering 127 (2017) 547–558

Lc

From condenser Li

Pc,in hc,in

To evaporator

Lhx Dc

Solder joint

Ds

Ps,in From evaporator

To compressor

(a) Side view of tube arrangement

Solder joint

Dc,out

Dc,in

Ds,in

Ds,out

(b) Cross section Fig. 4. Schematic of a capillary tube-suction line heat exchanger.

For development of a new CT-SLHX empirical correlation for this study, the mechanistic model was used to generate reference mass-flow-rate data with a wider range of parameters. Nowadays, most domestic refrigerators and freezers work with two common refrigerants, R-134a and R-600a [36–38]. For this reason, reference mass flow data were generated for these two refrigerants. The ranges of the parameters examined are given in Table 5. In this study, eight variables were selected as the influential parameters. In order to cover a wide range of conditions with a minimal amount of test runs, a three-level factorial design (L27 array) of the Taguchi method was used for each of the seven different parameters. It should be noted that because of the important role of the capillary inlet condition (p7), five levels were chosen for the capillary inlet refrigerant quality and capillary inlet subcooling. Based on this experimental design, 540 runs were developed to consider a range of various influencing parameters.

4. Development of the empirical correlation 4.1. Influencing parameters In this study, the refrigerant mass flow rate was considered the dependent variable, while geometrical and operational conditions (as well as refrigerant properties) were considered independent variables. The geometrical parameters included the capillary tube total length, the heat exchanger length, the adiabatic entrance length, the capillary tube inner diameter, and the suction line inner diameter. The relevant operational parameters included the capillary tube inlet pressure, the suction line inlet pressure, the capillary tube inlet sub-cooling or refrigerant quality, and the suction line inlet superheat temperature. Furthermore, the relevant properties of the refrigerant moving through the capillary tube and suction line included the latent heat of vaporization, specific volume, viscosity, specific heat, and the capillary tube inlet enthalpy. There-

fore, the dependent variable (the mass flow rate) is a function of all these geometrical and operational conditions, as well as the refrigerant properties. These can all be expressed as follows in Eq. (20),

_ ¼ f ðLc ; Li ; Lhx ; Dc ; Ds ; Pc ; Ps ; x; l; m; C p ; hfg ; hc ; DT sh Þ m

ð20Þ

The effects of significant geometrical and operational parameters had to be examined to develop an empirical model. To find any relationship between the CT-SLHX geometrical and operational parameters with the refrigerant mass flow rate, a quantitative analysis using the mechanistic model based on theoretical equations was performed on the experimental data of Wolf and Pate [16], which is summarized in Table 6. To calculate the refrigerant mass flow rate, each of the parameters was varied by 40%, 20%, 20%, and 40% while the other parameters remained constant. The effects of the aforementioned parameters on the mass flow rate are presented in Figs. 6 and 7. Fig. 6 demonstrates that the inner diameter of the capillary tube is the most influential parameter when the refrigerant is supplied as a subcooled liquid. The second and third most influential parameters were the capillary tube total length and the capillary tube inlet pressure. Results also showed that other parameters have relatively small influence on the mass flow rate. Fig. 7 demonstrates that the inner diameter of the capillary tube is the most influential parameter for the case of two-phase inlet conditions as well, which is followed by the capillary tube inlet pressure. However, the relative impact of the total length of the capillary tube appeared to be reduced in the case of the two-phase inlet conditions, compared with the case of subcooled liquid inlet conditions. Figs. 6 and 7 also demonstrate that the mass flow rate increases with the capillary tube inner diameter, to a power >1.0, for both inlet conditions (subcooled liquid and two-phase mixture), while it increases with the inlet pressure in a proportional manner. Also, these plots show that the refrigerant mass flow rate is inversely proportional to the capillary tube total length. These three param-

554

M. Rasti et al. / Applied Thermal Engineering 127 (2017) 547–558 Table 6 Experimental conditions of Wolf and Pate [16].

START

Input capillary tube and suction line geometry, operating condition, Seg.No.

ΔP = (Pc ,in − Pc ,out )/ Seg. No. Assume mass flow rate

n =1

n = n +1 No

Parameter

Unit

Value

Refrigerant Capillary tube inner diameter Capillary total length Heat exchanger length Suction line diameter Inlet adiabatic length Capillary inlet pressure Suction line inlet pressure Capillary inlet condition: thermodynamic quality Capillary inlet condition: sub-cooling temperature

– mm m m mm m kPa kPa kg kg1 K

R-134a 0.86 3.03 1.52 6.5 0.38 1030 277 0.04 11

20

40

25

Heat Exchanger

Capi. inner diameter Capi. total length SL. inner diameter Capi. inlet length Capi. HX length Capi. inlet pressure SL. inlet pressure Subcooled temperature

Calc. ref. prop. Yes Assume Ts ,out

c ,in − Psat ) < ε

-1

(P

Mass flow rate [kg h ]

No

20

Calc. ref. prop. Yes Calculate Lsubcooled, Lmetastable

No

(P

c ,in

− Psat ) < ε

15

10

5

Yes No

n = Seg.No.

0

Calculate Lsubcooled, Lmetastable

-40

-20

0

% change of parameters

Yes No

(Lcal − Lc ) < ε

Fig. 6. Effects of various parameters on the mass flow rate with the subcooled inlet condition.

No

Lca l > (Lh x + Li )

Yes

Yes

END No

(T

s ,in

− Ts ,ca l) < ε

Yes

Fig. 5. Flow chart for simulation of the capillary tube suction line heat exchanger.

eters are the most influential parameters that have been used in the new correlation to predict the refrigerant mass flow rate. 4.2. Development of a new correlation In the present study, a general mass flow rate correlation for a lateral capillary tube suction line heat exchanger was developed

based on analysis of the variables mentioned in Eq. (20), using the Buckingham Pi theorem. This resulted in the 15 dimensionless p parameters listed in Table 7. The capillary tube inner diameter is inversely related with the capillary tube total length to constitute a dimensionless parameter, p1 because the refrigerant mass flow rate increases with the capillary tube diameter while it decreases with the length of capillary tube. This is due to change in the flow resistance of the capillary tube. The adiabatic upstream length of a capillary tube, the heat exchange length, and the suction line diameter were made dimensionless by the capillary tube inner diameter. They are represented by the dimensionless groups, p2 , p3 , and p4 , respectively. The refrigerant mass flow rate is expected to increase with increases in the adiabatic upstream length and the heat exchange length because the flow resistance is reduced due to the specific volume reduced. In the present work, the effects of the capillary tube inlet pressure, the suction line inlet pressure, and the suction line super-

Table 5 Range of parameters. Parameter

Variable

Unit

R-600a

R-134a

Capillary tube inner diameter Capillary total length Heat exchanger length Suction line diameter Capillary inlet pressure Evaporator inlet temperature Suction inlet temperature Capillary inlet condition: thermodynamic quality Capillary inlet condition: sub-cooling temperature

P1 P2 P3 P4 P5 P6 P7 P8 P8

mm m m mm kPa K K kg kg1 K

0.55–1.16 2–5.2 0.5–1.95 4.15–8.9 410–690 256–268 268–286 0.02–0.99 1–21

0.55–1.16 2–5.2 0.5–1.95 4.15–8.9 680–1380 250–262 262–278 0.02–0.99 1–19

M. Rasti et al. / Applied Thermal Engineering 127 (2017) 547–558

12

-1

Mass flow rate [kg h ]

Table 7 New non-dimensional parameters.

Capi. inner diameter Capi. total length SL. inner diameter Capi. inlet length Capi. HX length Capi. inlet pressure SL. inlet pressure Quality

10

8

p

p1 p2 p3 p4 p5 p6

4

p7 p8

2

p9 -40

-20

Parameters

Description

Lc Dc Li Dc Lhx Dc Ds Dc

Capillary tube length and inside diameter effect

P c;in D2c

Pressure effect of capillary tube inlet

P s;in D2c

Pressure effect of suction line inlet

group

6

0

555

0

20

40

% change of parameters Fig. 7. Effects of various parameters on the mass flow rate with the two-phase inlet condition.

heat level are represented by p5 , p6 , and p8 , respectively. The mass flow rate would be expected to increase with increase of the capillary tube inlet pressure, with decrease of the suction line inlet pressure, and with decrease in the suction line superheat level. Two independent properties are required to specify the thermodynamic state of the refrigerant at the inlet of a capillary tube. The inlet pressure is a familiar and measurable property; so it is widely used as one of the two independent properties. Concerning the second independent property, temperature is usually used to specify the state of the subcooled liquid, while quality is used to specify a saturated two-phase mixture. In order to represent these properties in terms of dimensionless p parameters, various definitions were used in previous work. The previously existing dimensionless p parameters for representing the capillary tube inlet condition of the subcooled liquid and a saturated two-phase mixture, are shown in Table 8. It is seen that most of the dimensionless parameters for the capillary tube inlet condition become zero at the saturated liquid or saturated vapor lines. This leads the predictions of mass flow rates to be discontinuous at the saturated lines. In order to avoid this problem, specific enthalpy was selected as the second independent property to define the dimensionless p-parameter for capillary tube inlet condition as p7 ¼ 1 þ ðhc;in  hf Þ=hfg in the present work. The second term of p7 represents the thermodynamic quality. The dimensionless parameter, p7 , is continuous across a whole range of inlet conditions of the capillary tube and does not become zero for saturated liquid or for saturated vapor conditions. The dependent variable in this study is the refrigerant mass flow rate, represented as a dimensionless parameter, p9. The effects of density, viscosity, and latent heat of vaporization based on the capillary tube inlet condition are represented by p10 , p11 , and p12 , respectively. The effects of viscosity, density, and specific heat based on the capillary tube and suction line inlet conditions are represented by p13 , p14 , and p15 , respectively. The refrigerant mass flow rate in a CT-SLHX is influenced by density, viscosity, and the specific heat of fluids as well. It should be noted; however, that these transport properties vary depending on change in the pressure and temperature, which are already considered by other dimensionless p-parameters. According to the p-parameters, the refrigerant mass flow rate is the dependent variable and is represented by p9 . The general empirical correlation in a nonlinear power form can be represented as follows.

p10 p11 p12 p13 p14 p15

Capillary tube inlet adiabatic length effect Heat exchanger length effect Suction-pipe inside diameter effect

l2fc mfc

l2fc mfc

1þ

hc;in hf hfg

DT sh C pfc D2c 2 2 fc fc

l m

_ m

Dc lfc mgc mfc ðlfc lgc Þ lgc hfgc D2c

l2fc m2fc lgs lfc mgs mfc C pgs C pfc

Enthalpy effect of capillary tube inlet Superheated temperature effect of suction line inlet Refrigerant mass flow rate Density effect of capillary tube Specific volume effect of capillary tube Enthalpy of vaporization effect of the capillary tube Viscosity effect of suction line and capillary tube Specific volume effect of suction pipe and capillary tube Specific heat effect of suction pipe and capillary tube

p9 ¼ Cðp1 Þc1 ðp2 Þc2 ðp3 Þc3 ðp4 Þc4 ðp5 Þc5 ðp6 Þc6 ðp7 Þc7 ðp8 Þc8 ðp10 Þc10 ðp11 Þc11 ðp12 Þc12 ðp13 Þc13 ðp14 Þc14 ðp15 Þc15 ð21Þ The multiple variable regression analysis was performed using the reference database produced by the mechanistic model, as described in the previous section, to obtain the constants that appear in Eq. (21). The criteria of 95% significance level (with small P-values) and maximal R-square value were used to determine which p-parameters remained in the final correlation. The Pvalue indicates the size of an effect. The smaller the P-value, the larger the significance of the parameter is. Results showed that p2 , p3 , p10 , p11 , p13 , p14 , and p15 could be ignored because their P-values were large enough. Figs. 6 and 7 demonstrate that the effect of capillary tube inlet adiabatic length and heat exchange length on the mass flow rate through a capillary tubes is small compared to internal diameter and length of the capillary tube. That is why p2 and p3 could be ignored. The mass flow rate through a capillary tube would be influenced by density and viscosity of the refrigerant. In addition, specific heat affects refrigerant temperature and results in a variation of density and viscosity. The density, viscosity, and specific heat are combined with other various parameters to construct dimensionless groups of p5 , p6 , p8 , and p12 . However, dimensionless groups of p10 , p11 , p13 , p14 , and p15 representing a simple ratio of the same transport properties appeared to be negligible. It is because the transport properties such as density, viscosity, and the specific heat of fluids vary depending on pressure and temperature, which are considered by other dimensionless p-parameters. As a result, one continuous empirical model was developed as Eq. (22), which is applicable to a CT-SLHX for both inlet conditions of subcooled liquid and saturated two-phase mixture.

p9 ¼ 7:6793p0:5741 p0:0186 p0:8174 p0:1733 pc77 p0:0383 p0:0913 1 4 5 6 8 12

ð22Þ

where c7 = 0.9577 for the subcooled-liquid inlet condition and c7 = 2.9832 for the capillary tube saturated two-phase mixture inlet condition. The predicted mass flow rate at saturated liquid state (x = 0) is the same even though the exponent c7 is switched because p7 becomes ‘1’ at thermodynamic quality of zero. This correlation is applicable to wide ranges of common refrigerants for subcooled liquid and saturated two-phase mixture inlet conditions

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M. Rasti et al. / Applied Thermal Engineering 127 (2017) 547–558

Table 8 Dimensionless parameters for the capillary tube inlet condition. Correlation

Subcooled inlet condition

Two-phase inlet condition

p value at T sc ¼ 0

p value at x =0

p value at x =1

Wolf and Pate (CT-SLHX)

DT sc C pfc D2c

1x

0

1

0

Bittle et al. (adiabatic)

D2c cpfc DT sc

x

0

0

1

Sarker and Jeong (CT-SLHX)

1 þ DT Tcritsc

1x

1

1

0

Yang and Wang (adiabatic)

sc 1 þ TTcrit

1x

1

1

0

This work (CT-SLHX)

1þ

1

1

2

l2fc m2fc m2f l2f

hc;in hf hfg

Fig. 8. Comparison of the estimated and experimental mass flow rates for the subcooled inlet condition.

Fig. 9. Comparison of the estimated and experimental mass flow rates for the twophase inlet condition.

for a wide range of parameters, as mentioned earlier. An example of the mass flow rate calculation of R-600a entering the capillary of a CT-SLHX in a subcooled liquid state is presented in Appendix A. The refrigerant mass flow rate predicted by the new correlation is plotted in Fig. 3, along with predictions from other correlations in a range of ðhc;in  hf Þ=hfg values from 0.2 to 1.2. This plot demonstrates that the mass flow rate predicted in the range of

Fig. 10. Comparison of the estimated and experimental mass flow rates for the subcooled inlet condition of R-410A, R-152a, and R-22.

the subcooled liquid inlet condition (negative thermodynamic quality), and the mass flow rate predicted with the saturated two-phase mixture condition (positive thermodynamic quality), meet each other at the interface; where the thermodynamic quality is zero. Fig. 3 also demonstrates that only this new correlation is able to predict the mass flow rate over all three regions of state (subcooled liquid, saturated two-phase, and superheated regions) and continuously across all three regions. The new correlation was compared with the experimental data of Wolf and Pate [16], Liu and Bullard [10], Mendonça et al. [17], and Melo et al. [13] for the capillary tube subcooled inlet conditions shown in Fig. 8. This plot shows that the present correlation is in good agreement with previous experimental data. Approximately 80% of the experimental data show discrepancy of less than 10% from the prediction by the present correlation. This correlation was also compared with the experimental data produced by Wolf and Pate for capillary tubes with the two-phase inlet condition, and plotted in Fig. 9. This plot demonstrates that the present correlation can also predict very well the mass flow rate of refrigerant through a CT-SLHX for the inlet condition of a saturated two-phase mixture. These two plots also show that the present correlation shows good agreement for both refrigerants (R-134a and R-600a). Moreover, the new correlation was compared with the experimental data of R-410A, R-152a, and R-22 measured by Wolf and Pate [16] for the capillary tube subcooled inlet condition and two-phase inlet condition as shown in Fig. 10 and Fig. 11, respectively. These plots demonstrate that the present correlation can also predict very well the mass flow rate of R-410A, R-152a, and R-22 through a CT-SLHX for the both subcooled and two-phase capillary tube inlet conditions.

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M. Rasti et al. / Applied Thermal Engineering 127 (2017) 547–558

Appendix A. Example of calculating the mass flow rate of R600a See Tables A.1–A.3. Table A.1 Geometry and inlet conditions.

Fig. 11. Comparison of the estimated and experimental mass flow rates for the twophase inlet condition of R-410A, R-152a, and R-22.

5. Conclusions Capillary tube-suction line heat exchangers are widely used in small refrigeration systems to enhance the energy efficiency and ensure that vapor-phase refrigerant enters the compressor. The mass flow rate of a refrigerant flowing through a lateral capillary tube suction line heat exchanger has been explored by numerous previous investigators to develop various empirical correlations. Previous correlations were separately developed for the capillary tube inlet condition of a subcooled liquid state and that of a saturated two-phase mixture state; so a set of two empirical correlations are necessary to predict the refrigerant mass flow rate in a CT-SLHX. These previous empirical correlations available in the open literature have problems such as discontinuity at the saturated liquid line. This shortcoming may cause difficulty in obtaining a converged solution for a numerical simulation of a refrigeration system. In the present work, a new empirical correlation was suggested to remove the defect of the previous correlations. A mechanistic model was used to generate a set of reference data for the mass flow rate through a CT-SLHX. The reference data cover wide ranges of geometrical and operational parameters, and refrigerant properties. Based on these reference data, a generalized empirical correlation for the refrigerant mass flow rate inside a CT-SLHX was developed in a power law form. This involved dimensionless pgroups by implementing a new dimensionless parameter to account for the effects of the capillary tube inlet condition. The new correlation has some advantages such as continuity at the saturated liquid point and prediction of the mass flow rate using only one correlation regardless of refrigerant state at the capillary inlet. The new correlation was compared with experimental data for R134a, R-600a, R-410A, R-152a, and R-22 available in the open literature, and shows good agreement with them. This empirical model can be used in refrigeration cycle simulation tools to reduce the runtime and enhance the prediction accuracy of the refrigerant mass flow rate for various refrigerants. Acknowledgements This work was supported by a Human Resources Development program (No. 20164010201000) and Nuclear Power Core Technology Development Program (No. 2015191010002C) grants from the Korea Institute of Energy Technology Evaluation and Planning (KETEP) financed by the Ministry of Trade, Industry, & Energy, Republic of Korea.

Parameters

Units

Values

Description

Dc Lc Ds Li Lhx

m m m m m

8.63  104 3.3 6.5  103 0.5 1.5

Pc,in Ps,in 4Tsh Tc,in Ts,in

Pa Pa K K K

5.5  105 9  104 15 302 275

Capillary tube inside diameter Capillary tube total length Suction tube inside diameter Capillary tube inlet adiabatic length Capillary tube suction line heat exchanger length Capillary tube inlet pressure Suction line inlet pressure Superheated temperature Capillary tube inlet temperature Suction line inlet temperature

Table A.2 Thermophysical properties at given conditions. Parameters

Units

Values

Description

mfc mgc

qfc qgc mfc

kg m1 K1 kg m1 K1 kg m3 kg m3 m3 kg1

1.452  104 7.559  106 545.78 10.155 1.832  103

mgc

m3 kg1

9.847  102

Cpfc

J kg1 K1

2455.5

hfgc mgs

qgs mgs

J kg1 kg m1 K1 kg m3 m3 kg1

324630 6.909  106 4.525 0.221

Cpgs

J kg1 K1

1631

hc,in hf hfg

J kg1 J kg1 J kg1

268490 299660 309890

Saturated liquid viscosity at Tc,in Saturated vapor viscosity at Tc,in Saturated liquid density at Tc,in Saturated vapor density at Tc,in Saturated liquid specific volume at Tc,in Saturated vapor specific volume at Tc,in Saturated liquid specific heat at Tc,in Enthalpy of vaporization at Tc,in Saturated vapor viscosity at Ts,in Saturated vapor density at Ts,in Saturated vapor specific volume at Ts,in Saturated vapor specific heat at Ts,in Inlet enthalpy at Pc,in & Tc,in Saturated liquid enthalpy at Pc,in Enthalpy of vaporization at Pc,in

Table A.3 Calculated dimensionless parameters.

p group

p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15

Parameters

Value

Lc Dc Li Dc Lhx Dc Ds Dc

3823.87

Pc;in D2c

10605363990

Ps;in D2c 2 fc fc

1735423199

579.37 1738.13 7.53

l2fc mfc

l m

1þ

hc;in hf hfg

DT sh C pfc D2c

l2fc m2fc

_ m

Dc lfc

mgc mfc ðlfc lgc Þ lgc hfgc D2c

l2fc m2fc lgs lfc mgs mfc C pgs C pfc

0.899 387675733765 7869.73a 53.75 18.21 3261707273100 0.048 120.633 0.664

Based on calculated p9 , mass flow rate of R-600a is evaluated to be 9.86  104 kg s1 or 3.55 kg h1. a Calculated using Eq. (22).

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