1、2008 ASHRAE 239ABSTRACT A mathematical model has been developed to predict theperformance of a helical capillary tube under adiabatic flowconditions. The proposed model can predict the length of theadiabatic helical capillary tube for a given mass flow rate orthe mass flow rate through a given lengt
2、h of capillary tube. Theeffect of parameters like condensing pressure, degree ofsubcooling, pitch of helix and the coil diameter has been stud-ied for the flow of refrigerant R-134a through the adiabatichelical capillary tube. A capillary tube selection chart has beendeveloped, using the proposed mo
3、del, to predict the mass flowrate of refrigerant R-134a through a capillary of size 1.07 mmdiameter and 2 m length. INTRODUCTIONIn a low capacity refrigeration equipment, the expansionof refrigerant between condenser and evaporator is oftenattained by means of a capillary tube. Several researchers h
4、adcarried out a numerous approaches to design the adiabaticstraight capillary tubes. 19Marcy (1949) pioneered the work offinding the length of capillary tube using the Moodys frictionfactor and employing graphical integration method to evaluatethe adiabatic capillary tube length. 14Koizumi et al. (1
5、980)also developed a computational method to calculate the lengthof a capillary tube. They also conducted experiments usingglass capillary tube for flow visualization. 3Bansal et al.(1998) made a detailed analysis of flow of refrigerants throughadiabatic straight capillary tubes. They used the finit
6、e differ-ence method to compute the length of capillary tube. It wasassumed that a homogenous two-phase flow exists inside thecapillary tube. 26Wongwises et al. (2001) also assumed thehomogenous two-phase flow, to compute the length of two-phase region of adiabatic straight capillary tube using then
7、umerical integration technique. The two-phase region wasdiscretized into infinitesimal elements with constant pressuredrop across them. The length of each element is then calcu-lated and finally all lengths are summed up to obtain the twophase length. 23Mikol (1963) established through flow visu-ali
8、zation technique that the flow through the capillary tube isa homogenous two-phase flow. 17Li et al. (1990) has modeledthe adiabatic straight capillary tube for the prediction of meta-stable liquid length using Chen et al.s (1990) correlation forunder pressure of vaporization. 24Wijaya (1990) hascon
9、ducted an experimental study on the flow of R-134athrough the adiabatic straight capillary tube. He has presentedthe effect of inlet subcooling and condensing temperature onthe mass flow rate of R-134a through the adiabatic capillarytube. It has been concluded that the mass flow increases withthe in
10、crease in both inlet subcooling and the condensingtemperature. 22Melo et al. (1999) experimentally investigatedthe effects of the condensing pressure, capillary size anddegree of subcooling at capillary inlet on capillary length fordifferent fluids viz., R-12, R-134a and R-600a. They alsodeveloped s
11、eparate correlations for mass flow rate of theserefrigerants through the adiabatic straight capillary tube. A capillary tube can have the geometries of straight, spiraland helical shape. It has been observed that adiabatic straightcapillary tubes are seldom used in household refrigerators.However, t
12、he orifice tubes are widely used in automotives. Onthe other hand, the helical capillary tubes are widely used indomestic refrigerators and low-capacity air conditioners.Figure 1a shows the schematic diagram of an adiabatic helicalcapillary tube. In helical tubes, a secondary flow perpendicu-lar to
13、the axis is induced by the curvature of the tube, FigureA Homogeneous Flow Model forAdiabatic Helical Capillary TubeMohd Kaleem Khan Ravi Kumar, PhD Pradeep K. Sahoo, PhDAssociate Member ASHRAE Member ASHRAEMohd Kaleem Khan is a PhD Student, Ravi Kumar is an associate professor, and Pradeep K. Sahoo
14、 is an assistant professor in the Depart-ment of Mechanical and Industrial Engineering, Indian Institute of Technology Roorkee, Roorkee, India.NY-08-0302008, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in ASHRAE Transactions, Volume 114
15、, Part 1. For personal use only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAEs prior written permission.240 ASHRAE Transactions1b. This secondary flow has a stabilizing effect on laminarfluid flow, resulting in a higher critic
16、al Reynolds number.However, in the case of two-phase flow, the phase distributionis primarily governed by the centrifugal forces acting on theliquid phase. The secondary flow, also known as Dean Effect,affects heat, mass and momentum transfer in coiled tubes. Although, the research pertaining to the
17、 flow throughhelical tubes of larger diameter has been widely figured in theliterature (1Ali, 2001) yet a little work has been publishedregarding the study of refrigerants flowing inside an adiabatichelical capillary tube. 15Kubair et al. (1963) have suggestedthe pressure drop calculations in spiral
18、ly coiled tubes. Anattempt to study the flow of refrigerant R-22 through an adia-batic coiled capillary tube was made by 16Kuehl et al. (1990).It was concluded that irrespective of percentage of the overalllength coiled or the phase of refrigerant (liquid or two-phasemixture), the resistance to the
19、flow is increased due to coiling.They found that the mass flow rate of the coiled capillary wasabout 5 percent lower than that of a straight capillary tube.Recently, 27Zhou and Zhang (2006) have studied the flowcharacteristics of refrigerant R-22 inside an adiabatic helicalcapillary tube. Not only d
20、id they develop a mathematicalmodel but also validated their model by conducting experi-mentation on the adiabatic helical capillary tube. Their studydid not incorporate the effect of the coil pitch. The present study is focused on the flow of refrigerant R-134a through an adiabatic helically coiled
21、 capillary tube. Theeffect of coil pitch, coil diameter, degree of subcooling atFigure 1 (a) Helical capillary tube.(b) Secondary flow in the cross section.(c) Computational domain of adiabatic helical capillary tube.(d) Forces acting on fluid element.ASHRAE Transactions 241capillary tube inlet and
22、condensing pressure on mass flow rateand consequently on the length of capillary tube has beenundertaken. The proposed model is simple and can coveralmost all the aspects of capillary tube geometry including coilpitch. This model is based on the homogenous two-phase flowmodel. The proposed model is
23、validated with experimentalfindings of 27Zhou and Zhang (2006). The REFPROP 7.0 database (21McLindel et al., 2002),based on the Carnahan-Starling-DeSantis equation of state,has been used in the development of the proposed model todetermine the thermodynamic and transport properties of therefrigerant
24、s.MATHEMATICAL MODELINGThe flow through the helical capillary tube may bedivided in three distinct regions, viz., single-phase subcooledliquid, metastable non-equilibrium flow and two-phase liquidvapor region. In adiabatic capillary tube, the refrigerantexpands from high pressure side to low pressur
25、e side adiabat-ically. The refrigerant usually enters the capillary tube in asubcooled condition. As the liquid refrigerant flows throughthe capillary, the pressure drops linearly due to friction. As thepressure falls below the saturation pressure corresponding torefrigerant temperature, ideally a p
26、art of refrigerant shouldhave been flashed into vapors. But in actual scenario the flowstays in thermodynamic non-equilibrium liquid state, calledmetastable flow, below its saturation pressure under super-heated condition. As a result of it, the vaporization is delayedand flashing point shifts furth
27、er downstream. The possiblereasons of the metastability may be that finite amount of super-heat is required for the formation of first bubble. The directconsequence of metastability is the increase in critical massflow rate which delays the flash point and the choking point.The flow approaches equil
28、ibrium gradually after the vaporiza-tion has occurred. The inception of vaporization gives rise totwo-phase equilibrium flow in the capillary tube. Once thevaporization has occurred, the pressure and temperature dropwill be accelerated. This causes an increase in the vapor qual-ity and fluid velocit
29、y resulting in a higher friction and accel-eration pressure drop. The refrigerant pressure decreasessharply with the increase in vapor quality as an additional pres-sure drop called acceleration pressure drop comes intopicture. With the generation of vapor, the density of refrigerantstarts decreasin
30、g and since the area of cross section of thecapillary tube is constant; the velocity of the refrigerant willstart increasing, which is in accordance with the principle ofmass conservation. In other words, it can be said that the fluidis accelerated. However, the flash gas formed in the capillarytube
31、 because of the gradual expansion of the liquid as its pres-sure is reduced, seriously reduces the flow capacity of the tube.The rapid fall in temperature is because of the rapid fall incorresponding saturation pressure. The gas component of thetwo-phase mixture reduces the mass flow rate of the ref
32、riger-ant considerably. Since, the fluid flow is adiabatic the temper-ature of refrigerant remains constant as long as it is liquid andit falls rapidly as the flashing begins. As shown in Figure 1c, section 1 2 represents pressuredrop due to sudden contraction at capillary inlet, section 2 3is for s
33、ingle-phase subcooled flow, section 3 4 metastableflow and section 4 5 consists of liquid-vapor two-phaseregion. The present model is based on the following assump-tions: the helical capillary tube is of uniform cross section andsurface roughness, the pure refrigerant flow inside the capil-lary is s
34、teady and one-dimensional and adiabatic as well. Thegoverning equations can be obtained by applying the laws ofconservation of mass, momentum and energy on the fluidelement of length dL inside the capillary tube, as shown inFigure 1d. Applying momentum conservation on the fluid elementof length dL(1
35、)Since m = GA and , on simplification Equation(1) reduces to(2)In subcooled liquid region, the fluid is considered incom-pressible, hence, Integrating Equation (2), the length of single-phase liquidregion(3)The flow through the capillary tube is adiabatic, thetemperature of the subcooled liquid is c
36、onstant as long as it isin liquid state, i.e. T2= T3and the pressure at section 3, i.e. P3,is thus saturated.Considering the entrance effect, the capillary length forsingle-phase flow can be taken as(4)where, k is the entrance loss factor. The value of k istaken as 1.5. (27Zhou et al., 2006)Since, t
37、he flow through the capillary tube is adiabatic,hence, the temperature of liquid refrigerant is constant up tostate 3. The pressure at state 3 corresponds to the saturationtemperature, Ps, of refrigerant.In order to calculate metastable flow length, 6Chen et al.(1990) has proposed the following corr
38、elation for the delay ofvaporization, Ps Pv., or, P3 P4given by the following Equa-tion (5) (5)PA P dP)A wd()dL+() mdV=wfpV28=dL2df-dPG2-d-+=d 0=Lspdfsp-2V2- P2P3()=Lspdfsp-2V2- P1P3()k=PsPv()kBTs1.5- 0 . 6 7 9vgvfg-Re0.914TsubTc-0.208dD-3.18=242 ASHRAE Transactionswhere (6)Equation (5) has been dev
39、eloped for the metastable flowof refrigerant R-12, mass velocity ranging from 1440 kg/m2-s to 5090 kg/m2-s, degree of subcooling ranging from 0 to17C and tube diameter from 0.66 mm to 1.17 mm. However,5Bittle and Pate (1994) have extended the use of 6Chen et al.(1990) correlation for refrigerants R-
40、22 and R-134a with massvelocities up to 11900 kg/m2-s and tube diameters up to 2.29mm. Hence, Chens correlation represented by Equation (5)has been used in the present study to predict the metastableliquid length.The metastable liquid length is determined as (7)The temperature of the refrigerant at
41、state 4 is same as theinlet temperature as the flow is adiabatic and the state of refrig-erant remains liquid till state 4. However, the pressure at state4, Pv, lower than Psis determined by Equation (5) and demar-cates the pressure of inception of vaporization. Thus, thedifference (Ps Pv) is known
42、as underpressure of vaporization.In other words, it can be said that the vaporization is delayedbecause of the presence of metastability. The equation of helix in parametric form is(8)(9)(10)The radius of curvature for helix is given by(11)The arc length of helix is given by(12)In curved tubes, the
43、radial pressure drop causes secondaryflows resulting in a higher critical Reynolds number. (12Ito,1959)(13)9Collier (1972) has proposed the following friction factorcorrelations for the flow through coiled tubes:For laminar flow(14)For turbulent flow with Re (d/D)2 6(15)where, fsis the friction fact
44、or for straight tube, which isdetermined by the following 7Churchill (1977) correlation:(16)where (17)and De is the Dean Number (10Dean, 1927) given by (18)Applying energy equation, for open system of adiabaticflow in the capillary tube section 4 5 (19)Therefore, the vapor quality at any point in th
45、e two-phaseregion is given by(20)The two-phase friction factor ftpcan be calculated using9Collier (1972) correlations. The Reynolds number in two-phase region has to be determined by (21)where = the two-phase dynamic viscosity, depicted in Table 1.Different researchers have suggested the use of diff
46、erentviscosity correlation in the prediction of two-phase length ofcapillary tube. For example, 5Bittle et al. (1994) and 25Wonget al. (1996) have suggested the use of Duklers model forsimulations with R-12 and R-22. Hence, it has been decided touse Duklers model for the prediction of two-phase capi
47、llarytube length. The two-phase region of the tubes, i.e.,4 5, is dividedinto n number of infinitesimal section with a uniform pres-D 10000kBTs-=Lsv2dPsPv()fspV2-=xrcos=yrsin=zp2- =Rr2 p2-2+r-=lr2c2+ =Recrit20000dD-0.32=fcfs111.6De()0.452.221=fcfsRe d D()20.05=fs88Re-121AB+()1.5-+112-=A 2.4571n17 Re
48、()0.90.27 ed()+-16and=B37530Re-16=De RedD-=h3V322-+ hfxhfgG22- vfxvfg+()2+=xhfg G2vfvfg+G2vfvfghfg+()22G2vfg2G2vf22- h3V322- hf+G2vfg2-=RetpVDtpvtp-=tpASHRAE Transactions 243sure differential dP across each section as shown in Figure1d. The pressure at any section i is given byPi= P4- idP (22)whereP4 = the saturation
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