A comparison of homogenous and separated flow assumptions for adiabatic capillary flow

Applied Thermal Engineering 48 (2012) 186e193

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

A comparison of homogenous and separated flow assumptions for adiabatic capillary flow Thomas W. Furlong, David P. Schmidt* Department of Mechanical and Industrial Engineering, University of Massachusetts Amherst, 160 Governors Dr., Amherst MA 01003, USA

h i g h l i g h t s < A homogeneous and a separated flow model are applied to capillary flows. < A density-based discretization is used for the homogeneous model. < The models are validated for different refrigerants, capillary geometries. < The error shows clear non-normal tendencies and bias. < Non-parametric statistical tests confirm the benefits of the separated flow model.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 11 January 2012 Accepted 4 May 2012 Available online 12 May 2012

Homogenous and separated flow models are investigated for use in modeling one-dimensional adiabatic capillary tube flow. While these methods have been utilized extensively within the literature, the current work provides a rigorous, quantitative comparison of their accuracy using recent experimental data. Simulations utilizing the working fluids R134a, R600a, and R744 are performed for both methods and validated against experimental data. The mean error of the homogenous flow method is 8.55%, 5.4%, and 8.13%, respectively for R134a, R600a, and R744. The mean error of the separated flow method is 5.77%, 4.57%, and 8.03%, respectively for R134a, R600a, and R744. The separated flow method was found to have a smaller mean error and to perform better than the homogenous method as determined by nonparametric statistical tests.
2012 Elsevier Ltd. All rights reserved.

Keywords: Capillary Homogenous Separated

1. Introduction Capillary tubes have been extensively researched over the years due to their multiple uses for fluid expansion and refrigerant control within small and household refrigerator systems, freezers, dehumidifiers, and air conditioners [1e3]. Under adiabatic conditions, the expansion simplifies to Fanno flow, which governed by viscous choking can be categorized as either critical or sub-critical flow. Critical flow occurs when the pressure ratio from the inlet to the outlet of the capillary is sufficiently large enough to cause a Mach 1dor chokeddoutlet condition. Once the critical pressure ratio is reached, further reduction of the outlet pressure will have no effect on the flow conditions within the capillary tube [4]. Subcritical flow occurs when lower pressure ratios are present and the choke point for the expansion would occur beyond the physical outlet of the capillary tube. Within industrial applications, pressure ratios are often sufficiently high enough to ensure critical flow. * Corresponding author. Tel.: þ1 413 545 1393; fax: þ1 413 545 1027. E-mail address: [email protected] (D.P. Schmidt). 1359-4311/$ e see front matter
2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2012.05.007

The expansion of a working fluid and subsequent reduction in pressure along the capillary tube causes the fluid to undergo a flashing process, where the pressure is reduced below the vapor pressure and condensation occurs. Initial experiments conducted by Bolstad and Jordan [5] identified two regimes within the capillary tube, a singlephase and a two-phase regime. Later experiments conducted by Mikol [6], Li et al. [2], and Lin et al. [3] identified an additional flow regime, the metastable region, between the single-phase and twophase regions that consists of a single-phase and a two-phase region. A metastable flow occurs when the system is not in thermodynamic equilibrium, delaying the vaporization of the working fluid such that the pressure of the flow is decreased below the equilibrium saturation pressure before vaporization [7]. The flow regimes present in a typical capillary tube are summarized as: (1) the single-phase region, (2) the single-phase metastable region, (3) the two-phase metastable region, and (4) the two-phase region. The metastable region is typically assumed to be negligibly small, resulting in a flow that is in thermodynamic equilibrium everywhere. Beyond these initial works, the literature has provided only limited results for capillary flow in both quantity of experiments

T.W. Furlong, D.P. Schmidt / Applied Thermal Engineering 48 (2012) 186e193

Nomenclature

a ad

˛

m ml mv r rc rv rl s x

CD D fh fi fl fv

Void Fraction Void Fraction of Dispersed Phase Roughness (m) Viscosity (kg/m s) Liquid Viscosity (kg/m s) Vapor Viscosity (kg/m s) Density (kg/m3) Density of Continuous Phase (kg/m3) Vapor Density (kg/m3) Liquid Density (kg/m3) Interfacial Shear (kg/m s2) Wetted Perimeter (m) Drag Coefficient Diameter (m) Hydraulic Friction Factor Wallis Friction Factor Liquid Friction Factor Vapor Friction Factor

and variations in working fluid. Until recently, the working fluids utilized were usually air/water mixtures, R12, and R22 where the results from the experiments were generally given as profiles of pressure and temperature expansions across the capillary tube [2,3,6,8,9]. These results provided significant insight into the flow within a capillary tube, but lacked depth for use in qualitatively evaluating computational models. Recent experiments have provided data for new working fluids (R134a, R290, R407C, R410A, R600a, and R744), while simultaneously providing data more appropriate for use in validating numerical models [10e17]. Utilizing the new sets of data, it is possible to rigorously compare the accuracy of the two prevalent modeling techniques utilized for capillary tube expansion devices. This study looks specifically at R134a, R600a, and R744 in order to avoid any increased error that may be associated with mixture properties. The modeling of capillary tube expansion processes has taken the form of two distinct approaches, where both methods share a common set of basic assumptions: (1) As previously stated, the high length to diameter ratio allows the process to be modeled as a one-dimensional flow. (2) The diameter and surface roughness along the capillary tube is constant. (3) The flow is in thermodynamic equilibrium everywhere, ignoring the metastable phenomenon. (4) The flow is assumed to be steady state. (5) The choked flow condition is enforced at the outlet. (6) Only straight capillary tubes are considered. The homogenous flow method assumes that the thermodynamic properties are defined by the bulk properties of the flow by solving one set of governing equations for the pressure, enthalpy, and density. In contrast, the separated flow method takes into account the differences between the liquid and vapor properties of the flow by solving a set of governing equations for the liquid velocity, vapor velocity, pressure, void fraction, and the liquid enthalpy. This method represents a more realistic view of fluid motion, as the less dense vapor travels at higher velocities than the liquid. This paper is intended to highlight the differences between the two methods in order to better categorize the accuracy of each method for various working fluids. Section 2 provides a brief review of previous modeling work, while Sections 3 and 4 outline the homogenous and separated flow methods utilized for this paper. Section 5 quantitatively compare the two methods in order to better understand their differences.

ftp Flv FWl FWv g h hl hv Mid P Re Rel Rev S Vi Vl Vr Vv w z

187

Two-phase Friction Factor Liquid-Vapor Effects (kg/m2s2) Wall-Liquid Effects (kg/m2s2) Wall-Vapor Effects (kg/m2s2) Mass flux (kg/m2s) Enthalpy (kJ/kg) Liquid Enthalpy (kJ/kg) Vapor Enthalpy (kJ/kg) Drag of Dispersed Phase (kg/m2s2) Pressure (Pa) Reynolds Number Liquid Reynolds Number Vapor Reynolds Number Slip Ratio Interfacial Velocity (m/s) Liquid Velocity (m/s) Relative Velocity (m/s) Vapor Velocity (m/s) Speed of Sound (m/s) Axial Location (m)

2. Review of modeling work The homogenous flow method was implemented several decades ago in the works of Lockhart and Martinelli [18], Whitesel [19], and Fauske [20], which studied pressure drops across tubes and the associated effects of viscosity. Since then, the homogenous method has been implemented under various conditions by several authors. Wong and Ooi [21] assessed the effects of twophase viscosity correlations on the homogenous flow method accuracy, finding the Dukler mixture viscosity expression to perform best for the expansion of R12. Wong and Ooi [22] expanded upon their earlier work by investigating the differences between the separated flow method and the homogenous method. Under the conditions investigated the separated flow method performed better and it was shown that errors tended to increase near the choked region for the homogenous method. Bansal and Rupasinghe [23] utilized the homogenous flow model to create a program designed for the determination of capillary tube geometry within small vapor compression refrigeration systems. Sami and Maltais [24] studied the homogenous flow method utilizing alternative refrigerants R410A, R410B, and R407C, for which the method compared reasonably well with the limited experimental data. Wongwises and Pirompak [25] compared their method to experiments involving R12 and R134a and extrapolated their results to discuss how expansion processes (pressure drops) are effected through the use of R22, R407B, R407C, R410A, and R410B under the same conditions. Gu et al. [26] performed a similar study for R407C as compared to R22. Bansal and Wang [27] provided a study of R22, R134a, and R600a and outlined a comparison between calculated and observed mass flow rates while providing a unique graphical representation of capillary flow. Agrawal and Bhattacharyya [28] utilized the homogenous flow method to analyze the uses of carbon dioxide in the creation of a more efficient transcritical heat pump. More recently the work of Da Silva et al. [17] provided the necessary experimental results for analyzing the accuracy of carbon dioxide simulations in addition to providing a homogenous flow method that consistently produces results 10% error with the experimental results obtained. The separated flow method has been alluded to in literature nearly as long as the homogenous flow method, appearing in an Argonne National Laboratory report in 1962 [20], but has garnered

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more attention in recent times due to the capabilities of computers. Ishii and Mishima [29] and Ishii and Hibiki [30] provided an extensive overview of the constitutive relations required to implement a solution method capable of distinguishing between the two phases of the fluid. Schwellnus and Shoukri [31] introduced new correlations within a similar method while providing easy comparison to the earlier models of Dobran [32], Richter [33], and Al-Sahan [34]. Escanes et al. [1] utilized a control volume approach to allow critical and sub-critical conditions and transient and steady state conditions to be analyzed for the separated flow method. Seixlack et al. [35] compared the homogenous and separated flow methods and concluded that the better results of the separated flow method were not due to the non-equilibrium between phase velocities and temperatures, but rather due to a better representation of the flow. Liang and Wong [36] implemented the drift flux model, a variation of the separated flow method, for R134a, which compared well with experiments and other separated flow implementations. GarcíaeValladares et al [37] [38]; provided a qualitative comparison of the homogenous and separated flow methods utilizing various assumptions within each method and compiled a limited set of results for R600a. Wongwises and Suchatawut [39] implemented a method that includes the metastable region and provides adequate results as compared to experiments. Agrawal and Bhattacharyya [7] provided a comparative study of homogenous and separated flow methods for carbon dioxide in heat pumps and concluded that there is little difference between the two methods for the case of adiabatic flow, if the fluid tends to be more homogenous than others. Seixlack and Barbazelli [40] implemented a set of governing equations capable of solving for adiabatic or nonadiabatic flow depending on the conditions provided. Numerous other works by Sinpiboon and Wongwises [41], Xu and Bansal [42], and Da Silva et al. [43] consider the effects of non-adiabatic conditions. Khan et al. [44] provided a more inedepth review of capillary tube research in respect to both experimental and simulation work.

3. Homogenous flow Utilizing the homogenous flow method based upon the assumptions outlined in Section 1 the flow through an adiabatic capillary tube can be modeled using Eq. (1), derived from the conservation of energy.

dh g2 ¼ 3 dr r

(1)

Eq. (2) can be similarly derived from conservation of momentum.

dz ¼ dr




g2

r2



dP dr



2rD fh g 2

(2)

Typically a one-dimensional domain is easiest to discretize in space. However, Hermes et al. [45] and Da Silva et al. [17] showed that discretizing with respect to pressure was more appropriate due to the stability of the numerical schemes utilized within the flow domain. As the capillary flow reaches the outlet, jdP=dzj goes to a finite maximum while the inverse, jdz=dPj, goes to a minimum that is approximately zero [20]. Therefore by avoiding a discretization in space, large derivatives can be avoided allowing for a stable solution at the outlet. Eqs. (1) and (2) were arranged to allow discretization by density in order to remove the singularity in the same fashion while allowing pressure to be the dependent variable solved for via the equation of state. It is important to note that in

order to be mathematically consistent, r must be of the form of a monotonic function. The benefits of discretizing by density are not limited merely to the increased stability of the system, but can also increase the accuracy of the solution. With density-based differencing, a large difference between the inlet and outlet density automatically results in more nodes in the domain. Flow properties begin to change rapidly near the outlet, therefore a density discretization naturally provides a greater node concentration at the exit. With the increased node presence near the exit, the sharp changes in the properties of the flow are easier to fully capture, thus increasing the accuracy of the solution. The hydraulic friction factor, which utilizes the Darcy friction factor, for the single-phase and two-phase regions of the capillary tube can be found using Churchill’s equation [46], given by Eq. (3).




8 Re

fh;Churchill ¼ 8

12  3=2 1=12 þ A16 þ B16

(3)

The Reynolds number, Re, for the single-phase and two-phase region is given by Eq. (4), where m is the viscosity.

Re ¼

Dg

(4)

m

The variables A and B are given by Eqs. (5) and (6), respectively.

0 B A ¼ 2:457 ln@

B ¼

1 1 C A  7 0:9 ε þ0:27 Re D

37530 Re

(5)

(6)

The Reference Fluid Thermodynamic and Transport Properties Database V8.0 (REFPROP) created by the National Institute of Standards and Technology (NIST) is used as the equation of state to close the system of governing equations [47]. The enthalpy found from Eq. (1) and the discretization variable density are input into REFPROP to determine the pressure for both the single-phase and two-phase regions. REFPROP can also be utilized to calculate the viscosity, thermal conductivity, specific heat, temperature, and the liquid and vapor densities of the flow when necessary. 3.1. Solution methodology The algorithm is a form of root finding, where the mass flow is iteratively calculated in order to produce choking at the specified capillary length. Within this outer loop, the ordinary differential equations are solved from the inlet to the exit of the capillary. Given an initial pressure, temperature, and an estimated mass flow rate at the inlet of the capillary tube the simulation iteratively calculates the conditions at the next node, numerically integrating the governing equations using the first order explicit Euler method, where Eqs. (7) and (8) are solved for the enthalpy and location of the next node, respectively. After each iteration, an under-relaxation parameter is utilized on pressure for stability.

hiþ1 ¼ hi þ Dr

dh dr

(7)

ziþ1 ¼ zi þ Dr

dz dr

(8)

T.W. Furlong, D.P. Schmidt / Applied Thermal Engineering 48 (2012) 186e193

The choked flow condition at the outlet is enforced by checking for the existence of a Mach 1 flow. It is common within the literature to determine the choking condition from the knowledge that the change in entropy, Ds, must be positive in order to be consistent with the laws of thermodynamics for Fanno flow [27]. However, a generalized method that can be applied to non-adiabatic flows is desired. For non-adiabatic flows, Ds is likely to become negative before the choking condition exists due to heat transfer and therefore is not a suitable method [48]. In the single-phase region the REFPROP database is capable of reporting the speed of sound for use in checking the Mach 1 condition, but in the two-phase region the speed of sound is found using Eq. (9) evaluated utilizing a backward finitedifference approximation.

sffiffiffiffiffiffiffiffiffiffi ffi vP w ¼ vr s

(9)

For the adiabatic case it was found to approximately agree with

" ! !# Vl2 d Vv2 ð1  aÞrl Vl hl þ þ arv Vv hv þ ¼ 0 dz 2 2

4. Separated flow The separated flow method utilized the same equations as the homogenous method within the single-phase region, Eqs. (1) and (2). Eq. (2) is rearranged to be associated with a spacial discretization to form Eq. (10) due to density discretization causing unnecessary complexities to the derivation of the separated flow method.

  fh Dz 1 dP ¼ þ g2 d r 2r D g2

(10)

Within the two-phase region, the method outlined by Seixlack and Barbazelli [40] is utilized. Mass conservation is given in Eq. (11).

d ½ð1  aÞrl Vl þ arv Vv  ¼ 0 dz

(11)

The momentum equation for the liquid and vapor phases are given by Eqs. (12) and (13), respectively.

(16)

4.1. Wall friction For bubble and churn flow, a < 0.8, Eqs. (17) and (18) are used to solve for the liquid and vapor wall frictions, respectively [40].

FWl ¼

FWv ¼

ð1  aÞfl rl Vl2 2D

afv rv Vv2 2D

(17)

(18)

The liquid and vapor friction factors, fl and fv, are calculated using the Churchill equation, Eq. (3) using the Reynolds number of the liquid and vapor phases given in Eqs. (19) and (20), respectively.

Ds  0 condition used within the literature.

Once the flow has reached the choked condition, the desired length of the capillary tube must be enforced. If the desired length has not been met, then the simulation will use a root finding secant method to calculate the required mass flow rate to match the correct capillary tube length. Once the correct capillary tube length is established, then the simulation is finished and the results are stored in an output file.

189

Rel ¼

Rev ¼

ð1  aÞrl Vl D

ml

arv Vv D mv

(19)

(20)

Within the annular region, a  0.8, the liquid wall friction is given by Eq. (21),

FWl ¼

ftp g 2  FWv 2rD

(21)

where ftp is given by the correlation of Erth [50], Eq. (22), and the vapor phase is assumed to be in the center of the capillary, ie there are no wall effects (FWv ¼ 0).

  3:1 S0:25 ftp ¼ pffiffiffiffiffiffiffi exp 1  2:4 Rel

(22)

The amount of slip between the vapor and liquid phases, S, is a value between 0 and 1, assumed to be 1. 4.2. Interfacial force For the bubble and churn flow region, the interfacial force can be calculated using the method outlined by Ishii and Hibiki [30]. The total interfacial shear force can be represented as the sum of two forces, as shown in Eq. (23).

i d h dP ð1  aÞrl Vl2 ¼ ð1  aÞ  FWl þ Flv þ Gl Vi dz dz

(12)

Flv ¼ hMik  Vak $si iz

i d h dP arv Vv2 ¼ a  FWv  Flv þ Gv Vi dz dz

(13)

The first term represents the average drag, while the second term models the effect of interfacial shear. The average drag can be approximated via Eq. (24).

Where, FWk is wall friction for the component k, Flv is the vapor liquid interaction term, and Gk is given by Eq. (14) for the component k.

3C hMid iz ¼  D ad rc Vr jVr j 4 D

(23)

(24)

The interfacial velocity, Vi, is given by Eq. (15), where the weighting factor h is taken to be 0.5 as proposed by Wallis [49].

In the case of one-dimensional flow, the average overall void fraction, ad, simplifies to the void fraction at the current location. It is assumed that the local relative velocity, Vr, is comparatively uniform across the capillary tube and that the magnitude of the relative velocity is smaller than both of the phase velocities. The local relative velocity is defined by Eq. (25).

Vi ¼ hVl þ ð1  hÞVv

Vr ¼ Vd  Vc

Gk ¼

d ½a r V  dz k k k

The mixture energy equation is given by (16).

(14)

(15)

The drag coefficient, CD, is calculated using Eq. (26) [30].

(25)

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T.W. Furlong, D.P. Schmidt / Applied Thermal Engineering 48 (2012) 186e193

CD ¼

4.3. Solution methodology



24 1 þ 0:1 Re0:75 Re

(26)

The Reynolds number is found using Eq. (27).

Re ¼

Drl Vr

(27)

m

The interfacial shear force term can be calculated via Eq. (28) [30].

hVak $siz ¼

fi xi r Vr jVr j 2A c

(28)

Where fi is found using the Wallis correlation, which is applicable for rough wavy films, given by Eq. (29) [30,49].

fi ¼ 0:005½1 þ 75ð1  aÞ

(29)

The wetted perimeter, xi, divided by the area, A, simplifies to 4/D, resulting in Eq. (30) for the interfacial force in the bubble and churn flow region.

Flv ¼

2fi 3C r Vr jVr j  D ad rc Vr jVr j 4 D D c

2fi r Vr jVr j D c

(31)

(32)

The Fauske criterion [20] is utilized to determine when the choked flow condition exists, which states that at the critical point, the change in pressure dP/dz has reached a finite maximum. This is implemented by iterating in the z-direction until dP/dz becomes positive. Once the choked flow condition is reached, a similar root finding method is implemented to determine the mass flow rate required for the desired capillary tube length.

5. Results In order to provide quantitative results, the homogenous and separated flow methods simulated data sets from three different working fluids. The first set of simulations were performed using carbon dioxide in accordance with the experiments conducted by Da Silva et al. [17] with an assumed roughness of 0.001 mm. The results, shown in Fig. 1(a) and (b), produced mean errors of 8.13% and 8.03% for the homogenous and separated flow methods, respectively. The relative mean error was calculated using Eq. (33).

b 15

+/- 10%

10

Mass Flow Rate Relative Error (%)

Mass Flow Rate Relative Error (%)

a 15 5 0 -5 -10 -15 -20 -25

r ¼ ð1  aÞrl þ arv

(30)

It can be seen that the first term accounts for the interfacial forces, while the second term accounts for the drag of the dispersed vapor particles traveling through the capillary tube. The annular flow region is dominated by the interfacial forces and therefore Eq. (30) is reduced to Eq. (31).

+/- 10%

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

3

6

9

12

15

18

21

-30

24

3

6

9

Mass Flow Rate (kg/hr)

12

18

15

21

24

Mass Flow Rate (kg/hr)

Fig. 1. Comparison of calculated mass flow rates to experiments performed by Da Silva et al. [17] for carbon dioxide (R744) (a) Homogenous (b) Separated.

a 15

b 15 +/- 10%

Mass Flow Rate Relative Error (%)

+/- 10%

Mass Flow Rate Relative Error (%)

Flv ¼

The solution method outlined for the homogenous assumption is utilized for the single-phase region. Once the flow reaches the two-phase region, direct matrix inversion is used to solve Eqs. (11)e(13), (16) for dVl/dz, dP/dz, dVv/dz, and da/dz, respectively. An explicit Euler method is then used to find the value of the variable at the new node. The liquid and vapor densities are calculated from REFPROP given the pressure at the node and then used with the void fraction to find the density via Eq. (32).

10 5 0 -5 -10 -15

2

2.5

3

Mass Flow Rate (kg/hr)

3.5

4

10 5 0 -5 -10 -15 -20

2

2.25

2.5

2.75

3

3.25

3.5

3.75

4

Mass Flow Rate (kg/hr)

Fig. 2. Comparison of calculated mass flow rates to experiments performed by Melo et al. [10,51] for isobutane (R600a) (a) Homogenous (b) Separated.

T.W. Furlong, D.P. Schmidt / Applied Thermal Engineering 48 (2012) 186e193

191

b 25

a 30

+ 10%

Mass Flow Rate Relative Error (%)

Mass Flow Rate Relative Error (%)

+ 10%

25 20 15 10 5 0 -5

0

2

4

8

6

10

12

14

20 15 10 5 0 -5 -10

16

0

2

4

Mass Flow Rate (kg/hr)

8

6

10

12

14

16

Mass Flow Rate (kg/hr)

Fig. 3. Comparison of calculated mass flow rates to experiments performed by Melo et al. [10,51] for R134a (a) Homogenous (b) Separated.

these simulations were experimentally determined by Melo et al. [10] utilizing a standardized measurement technique. The full set of data consists of 189 experiments utilizing isobutane as the working fluid, shown in Fig. 2(a) and (b). The mean errors are 5.4% and 4.57% for the homogenous and separated flow methods, respectively. The third set of data consists of 572 R134a experiments, shown in Fig. 3(a) and (b). The mean errors are 8.55% and 5.77% for the homogenous and separated flow methods, respectively. Further comparisons can be made from García-Valladares et al. [37,38], which performed simulations using seven different methods for 8 experimental cases conducted by Melo et al. [52], shown in Table 1. Three of the methods utilized by García-Valladares et al. [37,38] implement a separated flow method, three the homogenous flow method, and all but the last account for the metastable flow. The separated flow method presented here compares well with the most accurate method presented by García-Valladares et al. [37,38]. The homogenous flow method does not perform as well as the separated flow methods, however it is comparable to the homogenous flow methods in Table 1. The facile conclusion, based upon the mean errors presented for both the homogenous and separated flow methods, is that the separated flow method is more accurate, as evidenced by a smaller mean error. However it is important to look at the statistics of the simulations to see how the two methods relate. Figs. 4, 5, and 6 show the histograms for R134a and isobutane separated into a different histogram for each diameter of capillary tube. As the diameter is utilized as the parameter to distinguish different sets of data, it is the most influential parameter on the mass flow rate [17]. It should be noted that these histograms do not appear to be Gaussian in nature, nor do they have a mean of zero, indicating bias in the error. The histograms shown in Figs. 4 through 6 exhibit

Table 1 Mass flow rate relative error (%) for presented homogenous and separated flow methods compared to methods found in García-Valladares et al. [37,38] using isobutane as the working fluid. Models (I), (II), and (III) utilize a homogenous method while (IV), (V), and (VI) utilize a separated flow method. Model (VII) does not consider the metastable region. H and S represent the homogenous and separated flow methods described in Sections 3 and 4, respectively. Case

(I)

(II)

(III)

(IV)

(V)

(VI)

(VII)

H

S

1 2 3 4 5 6 7 8 Mean

2.65 1.55 9.56 9.18 5.81 10.21 4.60 7.08 6.33

1.35 0.25 5.39 7.15 2.19 8.58 1.15 5.45 3.94

2.00 0.49 4.62 6.84 1.61 8.36 0.48 5.17 3.70

4.17 1.05 1.92 6.03 0.00 8.40 1.28 4.90 3.51

9.51 2.07 1.84 5.52 2.52 8.08 2.33 4.77 4.58

18.71 7.49 15.52 2.27 14.73 1.84 17.66 2.72 10.12

7.94 3.02 1.06 4.06 4.02 5.59 5.04 2.34 4.13

6.05 2.66 12.09 9.91 7.55 10.19 5.44 6.73 7.58

1.20 0.31 5.08 6.88 1.69 8.08 0.30 4.81 3.54

_ error ¼ m

_ calculated  m _ experiment m  100 _ experiment m

(33)

It should be noted that the data from Da Silva et al. [17] consisted of 66 data points and was simulated in its entirety for the homogenous method. For the separated flow method, one set of initial conditions produced unstable results that could not be overcome. The unstable result is believed to be caused by the location of the initial condition lying too near the critical point of carbon dioxide and presents a possible limitation of the separated flow method that was not seen in the homogenous method. The second and third data sets were provided by Dr. Claudio Melo [51], based on the work of Melo et al. [10], which included a portion of the available experimental data. The roughness for

Number of Occurences

30

b 15

Separated Homogenous

Separated Homogenous

12

Number of Occurences

a 35 25 20 15 10

9

6

3 5 0 -10

-5

0

5

10

Mass Flow Rate Relative Error (%)

15

20

0

-5

0

5

10

15

20

25

30

Mass Flow Rate Relative Error (%)

Fig. 4. Homogenous and separated flow method histograms for R-134a data provided by Melo et al. [10,51] for R134a (a) Diameter A (b) Diameter B.

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T.W. Furlong, D.P. Schmidt / Applied Thermal Engineering 48 (2012) 186e193

a 35

Separated Homogenous

18

Number of Occurences

Number of Occurences

30

b 21 Separated Homogenous

25 20 15 10

15 12 9 6 3

5 0 -10

0

-5

10

5

15

0

0

Mass Flow Rate Relative Error (%)

3

6

9

12

15

Mass Flow Rate Relative Error (%)

Fig. 5. Homogenous and separated flow method histograms for R-134a data provided by Melo et al. [10,51] for R134a (a) Diameter C (b) Diameter D.

and liquid phases. Therefore, by treating the two phases as one homogenous flow, the homogenous method loses important flow properties and results in a less accurate solution. Furthermore a limitation of the separated flow method was produced when the expansion path of the working fluid traveled too near the critical point, resulting in a simulation that would not converge. Future work should consider methods of implementing the separated flow method while overcoming this limitation in order to provide a robust and accurate method. A natural extension of the current work would be to validate against experimental studies of multicomponent refrigerants, such as R410A and R407C [11e13].

18 Separated Homogenous

16

Number of Occurences

14 12 10 8 6 4

Acknowledgments

2 0 -20

-15

-10

-5

0

5

10

15

20

Mass Flow Rate Relative Error (%) Fig. 6. Homogenous and separated flow method histograms for isobutane data provided by Melo et al. [10]).

We acknowledge the financial support of the National Science Foundation under grant #CMMI-1025020. We gratefully acknowledge Prof. Cláudio Melo of the Department of Mechanical Engineering, Federal University of Santa Catarina, for generously sharing experimental data. References

a trend in which the distribution of errors tends to move closer to zero for the separated flow method. Furthermore, the Kolmogorove Smirnov and Wilcoxon Rank Sum tests, which are designed for non-parametric and non-normal distributions, were performed to test if the distributions were statistically and significantly different. For a 95% confidence level the tests determined that the two methods are statistically, significantly different. 6. Conclusion The homogenous and separated flow methods were implemented and validated such that a comparison of the accuracy could be established. The average error for both methods lie within the generally accepted margin of error. However, it is clear that the separated flow method consistently produces more accurate results than its homogenous counterpart. It is important to note that while the mean error was only reduced up to 2.78%, the maximum local error was reduced by as much as 10%, showing a more significant local increase in accuracy. Furthermore, the separated flow method was shown to have a statistically significant increase in accuracy over the homogenous flow method by looking at the histograms and performing the KolmogoroveSmirnov and Wilcoxon Rank Sum Tests. These results provide quantitave evidence that agrees with the results seen within the literature. Within the two-phase region, the expansion process depends on the interactions between the vapor

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