CRITICAL MASS FLOW RATE THROUGH CAPILLARY TUBES

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1 Proceedings o the ASME 010 rd Joint US-European Fluids Engineering Summer Meeting and 8th International Conerence FESM-ICNMM010 August 1-5, 010, Montreal, Canada Proceedings o ASME 010 rd Joint US-European Fluids Engineering Summer Meeting and 8th International Conerence August -4, 010, Montreal, Canada FESM-ICNMM010- FESM-ICNMM CRITICAL MASS FLOW RATE THROUGH CAPILLARY TUBES A. Nouri-Borujerdi, P. Jaidmand School o Mechanical Engineering, Shari Uniersity o Technology, Tehran, Iran anouri@shari.edu ABSTRACT This paper presented a numerical study that predicts critical mass low rate, pressure, apor quality, and oid raction along a ery long tube with small diameter or capillary tub under critical condition by the drit lux model. Capillary tubes are simple expansion deices and are necessary to design and optimization o rerigeration systems. Using dimensional analysis by Buckingham s π theory, some generalized correlations are proposed or prediction o low parameters as unctions o low properties and tube sizes under arious critical conditions. This study is perormed under the inlet pressure in the range o 0.8 pin 1.5 Mpa, subcooling temperature between 0 T sub 10 C. The tube diameter is in the range o mm and tube length between 1 L m or water, ammonia, rerigerants R-1, R- and R- 14 as working luids. Comparison between the results o the present work and some experimental data indicates a good agreement. Cluster o data close to the itted cures also shows satisactory results. Keywords: two-phase low, drit lux model, numerical method, capillary tube, dimensionless analysis INTROUCTION Small diameter tubes or capillary tubes hae an extensie implementation as a simple expansion deice in the industrial rerigeration systems to reduce the pressure and temperature o a rerigerant luid low between a condenser and an eaporator. Because o the act that these simple deices reduce the temperature o the low by lashing it into apor phase, it is complicated to analyze the low through them. Typical diameter o a capillary tube is between 0.5 mm and its length is between 1 6 m. Subcooled liquid enters into the capillary tubes and lows as a single phase up to the point where the pressure reaches saturation pressure related to the low temperature. Although it is expected that lashing into the apor phase starts rom the end o single phase region, it happens with some delay in the point where low pressure is a bit less than its saturation pressure. This single phase region is called metastable low region. Howeer, this region is reused in most o numerical calculations. In two-phase region, low starts with bubbly low and ends with annular low pattern i the length o the tube is long enough. By decreasing the pressure at the tube outlet, mass low rate increases up to a critical rate in which the mass low rate remains constant and shock happens at the end o the tube where pressure decreases with a sharp gradient and as a result the liquid lashes into apor by a pretty high rate. The most important models used to study the behaior o two-phase gas-liquid low are homogeneous, separated low, drit lux and two-luid model. In the homogeneous model two phases are considered to low in equilibrium rom dynamic and thermodynamic point o iew, but in the separated low model an experimental correlation or the slip ratio o elocities o two phases is considered through the low. In the drit lux model an experimental correlation is considered to relate drit elocity o the low to the other low characteristics. Wong and Liang [1] proposed a numerical study based on the drit lux model to ind characteristics o R-14 through the capillary tubes under critical conditions. uber s correlation [] or the drit elocity and Lin s two-phase multiplier [] were used in their study. Their results compared predicted low characteristics o R-14 with the predicted low characteristics o R-1. Also some experimental works hae been perormed in order to ind correlations to predict critical low characteristics o the capillary tubes. Melo [4] experimentally studied the eect o the capillary diameter and length, inlet pressure and subcooling and type o rerigerant on critical mass low rates under adiabatic condition. They perormed the study under the inlet pressure in the range o 7 18 bar, subcooling temperature between 16 C and tube diameter between mm or the rerigerants R-1, R-14 and R-600. It was ound that the tube diameter aects critical mass low rate more than other parameters. They presented a correlation by dimensional analysis to estimate critical mass low rate o each rerigerant. The results o their correlations were in agreement with experimental data with 15% error or R-1 and 10% error or R-14. Kim and Choi [5] presented an empirical correlation to predict critical mass low rate under the adiabatic condition. They explored the data aecting the critical mass low rate and generated some dimensionless parameters by using the data o 1 Copyright 010 by ASME

2 Proceedings o ASME 010 rd Joint US-European Fluids Engineering Summer Meeting and 8th International Conerence August -4, 010, Montreal, Canada rerigerants R-1, R-, R-14, R-15, R-407 and R-410with a deiation o 6.5%. Jabaraj et al. [6] conducted an experimental study on low o mixture M0 (R-407c, R-600, R-90) through capillary tubes, They reported a non dimensional correlation to predict critical mass low rate o M0 by using Buckingham s π theory. Their results reealed that approximately 95% o which has a deiation o about 10%. Recently Vins and Vacek [7] perormed an experimental study on the critical low o R-18 through capillary tubes with dierent tube diameters and lengths. They presented a correlation or the critical mass low rate o rerigerant R-18 by both classical power law unction and artiicial neural network. The aerage o deiation by the power law unction was -0.41% and by the artiicial neural network correlation was -0.1%. This study attempts to use a numerical method to predict critical low characteristics or water, ammonia, rerigerants R- 1, R- and R-14 through capillary tubes by drit lux model under dierent operating conditions. Generalized correlations hae been presented or prediction o critical mass low rate, pressure, quality and oid raction distribution through capillary tubes in a wider range o operating conditions. NOMENCLATURE tube diameter, m riction actor G mass elocity, kg/s.m h enthalpy, J/kg p pressure, Pa Re Reynolds number s entropy, j/kg.k T temperature, C u elocity, m/s x apor quality z axial coordinate, m Greek letters α oid raction ε roughness, m φ µ o two-phase riction multiplier iscosity, kg/s.m ν kinematic iscosity, m /s υ speciic olum, m /kg density, kg/m σ surace tension, N/m ψ general ariable Subscripts arcy ex exit liquid g saturated apor liquid dierence o all luid as a liquid i index in inlet s single phase sat saturation sh superheat t turbulent apor MATHEMATICAL FORMULATION Flow through capillary tubes can be diided into single phase, metastable and two-phase low. Single phase low region continues up to the point where low pressure reaches saturation pressure related to its temperature. In this case the length o the single phase low region is. L s p - p in sat G where is the arcy riction actor and or turbulent low is calculated by Colebrook s correlation. (1) log () Reo In the two-phase region, liquid lashes into apor purely because o the reducing pressure. The one-dimensional, steady state goerning equations o mass, momentum and energy or a two-phase low o apor-liquid phases through an adiabatic capillary tube are gien respectiely as ollows. d d ( uα) + [ u (1 α) ] 0 () dz dz d d dp ( o ) G ( αu ) + (1- α) u φo (4) dz dz dz d u uα ( h + ) + dz d u u (1- α ) ( h + ) 0 (5) dz Copyright 010 by ASME

3 Proceedings o ASME 010 rd Joint US-European Fluids Engineering Summer Meeting and 8th International Conerence August -4, 010, Montreal, Canada φo where is two-phase riction multiplier and Lin [] deried the ollowing expression or that. 1 o o 8 1 ( A + B φ o + x -1) (6) At + Bt Where ε Ao.457 Ln + Reo ε At.457 Ln + Ret Bo, Bt Reo Ret G G Re o, Re t µ µ The apor quality in term o oid raction is as: uα x G rit-flux Model: is essentially a separated-low model in which attention is ocused on the relatie motion rather than on the motion o the indiidual phases. It is particularly useul when the relatie motion is determined by a ew key parameters instead o connection to the low rate o each phase. uber [] presented the ollowing correlation through horizontal tubes. 0.5 ( - ) σg uj 1.48 (8) The drit elocity o apor relatie to the center o mass, u m,is gien by: u m u um uj (9) m where the aerage density and mass elocity are deined as: α + (1- α) (10) u m m α u + (1- α) u m t 16 (7) (11) uring lashing, we assume the equilibrium preails between the inside and outside o the bubble beore growth starts and the initial liquid superheat can be ealuated with Clausius-Clapeyron equation as ollows. 4σνg Tsh T Tsat (1) dsg where σ is the surace tension and sg hg T. From the aailable 5 data, an initial bubble diameter may be assumed as d m, [8]. Combining Eqs. (8-11), you can obtain easily u as a unction o u Ater inserting Eq. (7) into Eq. (6) and u (u ) into Eqs. (-5) and dierentiating the results, the general orm o the dierential equation or three ariables u, α and p is gien by: a du i + b dα dp i + ci di (1) dz dz dz where, the coeicients a,b,c i i i and di are unctions o the physical properties o the liquid-apor phases and are gien in Appendix A. SOLUTION METHO In the aboe analysis, i a subcooled single-phase liquid inters a capillary tube, Eq. (1) is irst applied to the subcooled liquid region with the known initial conditions at the oriice inlet, or instance, pressure and mass low rate. Once the saturated condition is reached, the two-phase low calculations are started until the mass low rate reaches the critical condition. When this case happens, the critical low rate is almost independent o the downstream pressure. In this case Eqs. (1) or i 1,, and are soled simultaneously by the u, α, and ourth order Rung-Kutta method or unknown ariables p under their speciied initial conditions. The existing deriaties o dφ dp or φ,,h,h in Eq. (1) can be obtained by itting the luid properties data ersus pressure rom the thermodynamic table o the matter. RESULTS AN ISCUSSION To predict the critical mass low rate and the critical pressure distribution through the capillary tubes, the present model is examined or a range o inlet pressure o 0.8 pin 1.5 Mpa, subcooling temperature between 0 T sub 10 C. The tube diameter is in the range o mm and tube length between 1 L m or water, ammonia, rerigerants R-1, R- and R-14 as working luids. imensions o the capillary tubes include length o 1 L m, diameter o mm with surace roughness o 0.5 ε µ m. Critical Correlations: In order to obtain the generalized correlations or the critical low parameters, the set o equations (1) or i 1, and is irst soled numerically or the unknown ariables u,α and p, then the critical low Copyright 010 by ASME

4 Proceedings o ASME 010 rd Joint US-European Fluids Engineering Summer Meeting and 8th International Conerence August -4, 010, Montreal, Canada parameters are assumed to be as unctions o physical properties and low geometry as ollows. ψcr L,, ε, pin, Tsub,,, µ, µ (14) where ψcr reers to the dependent ariables and includes: ψcr mcr, Pcr, xcr, Lcr (15) Using Buckingham π theory, the ollowing dimensionless parameters are obtained. m pin Cp Tsub ε π1, π, π, π 4 µ µ ν h g L pin pex π5, π6, π 7, π8 υ p µ in Figure 1 illustrates the mass low rate o water, ammonia, rerigerants R-1, R- and R-14 through the capillary tubes under the critical conditions. The ordinate axis has been dimensionless by the capillary tube diameter and the liquid iscosity. The abscissa axis is indicated by some dimensionless parameters. Comparison between the present results with some experimental data presented by Melo [4], Mikol [9] and Li [10] shows a good agreement. All the results including the numerical results o the present model and the experimental data are also itted by a line with the ollowing dimensionless expression cp T cr in sub 0.7 µ ν m p µ g L h υ µ (16) Figure predicts the pressure distribution under the critical conditions along the capillary tubes or water, ammonia, R1, R- and R-14 as working luids. The y- and x-axes hae been dimensionless by the inlet pressure and some dimensionless parameters respectiely. The igure also includes a cure itted by all numerical results o the present. The equation o the cure is as ollows. Pin P P in 0.9η-0.6 η +0.7 η (17) cp T sub h g ν υ µ η L cp T in sub h g µ ν υ µ p Fig. 1 Comparison between predicted critical mass low rate and the experimental results through capillary tubes Figures and 4 present the apor quality and oid raction along the capillary tube under the critical low conditions, respectiely. Each igure also includes a cure itted by all the p p c T in sub h g µ ν υ L µ Fig. Prediction o pressure distribution under critical condition through capillary tubes predicted data by the present model and they are expressed respectiely by: 4 Copyright 010 by ASME

5 Proceedings o ASME 010 rd Joint US-European Fluids Engineering Summer Meeting and 8th International Conerence August -4, 010, Montreal, Canada x cr 194 η (18) Where, p p c T in sub µ ν η αcr 0.9 h g υ µ η (19) Where, p p c T in sub µ ν η h g υ µ In these two igures, all data are clustered close to the itted cures and they indicate satisactory results. The aerage amount o oid raction at the tube length is about αext cp T in sub h g µ ν υ µ p Fig. 4 Prediction o oid raction distribution under critical condition through capillary tubes Finally, the critical tube length is also obtained rom the ollowing expression Cp T cr in sub 0.14 µ ν L p g L h υ µ (0) p c in p T sub h g µ ν υ µ Fig. Prediction o quality distribution under critical condition through capillary tubes CONCLUSIONS In this work a numerical method based on the drit lux model was presented to predict the low ariables through an adiabatic capillary tube under the critical conditions or water, ammonia, R1, R- and R-14 as working luids. Using dimensional analysis by Buckingham s π theory, some generalized correlations were proposed or the mass low rate, pressure, quality and oid raction as unctions o low properties and tube sizes. Comparison between the results o the present work and some experimental data show a good agreement. 5 Copyright 010 by ASME

6 Proceedings o ASME 010 rd Joint US-European Fluids Engineering Summer Meeting and 8th International Conerence August -4, 010, Montreal, Canada REFERENCES [1] Liang. S. M, Wong. T. N, 001, Numerical modeling o two-phase rerigerant low through adiabatic capillary tubes, Appl. Thermal Eng, 1, [] uber N, Findlay. J, 1965, Aerage olumetric concentration in two-phase low system, Journal o Heat Transer, 8, [] Lin. S, Kwok. C. C. K, Li. R. Y, Chen.. Y, 1991, Local riction pressure drop during aporization o R-1 through capillary tubes, Int. J. Multiphase Flow, 17, [4] Melo. C, Ferreira. R. T. S, Neto. C. B, Goncales. J. M, Mezaila. M. M, 1999, An experimental analyze o adiabatic capillary tubes, Appl. Thermal Eng, 19, [5] Choi. J, Kim. Y, Chung. J. K, An empirical correlation and rating charts or the perormance o adiabatic capillary tubes with alternatie rerigerants, 004, Appl. Thermal Eng, 4, [6] Jabaraj.. B, Kathirel. A. V, Lal.. M, 006, Flow characteristics o HFC407C/HFC600a/HC90 rerigerant mixture in adiabatic capillary tubes, Appl. Thermal Eng. 6, [7] Vin. V, V. Vacek, 009, Mass low rate correlation or twophase low o R18 through a capillary tube, Appl. Thermal Eng. Article in Press. [8] Richter, H.J, (198), "Separated two-phase low model: application to critical two-phase low, Int. Journal o Multiphase Flow, 9(5), pp [9] Mikol. E. P, 196, Adiabatic single and two-phase low in small bore tubes, Ashrae Journal, 5, [10] Li. R. Y, Lin. S, Chen.. H, 1990, Ashrae Transactions, 96, 54. APPENIX A a 1 α + (1- α) b1 u - u - [ σ( - )] 1-α d σ d c1 u α ( - ) 0.75 dp dp d 1 0 (A1) a αu + (1- α) u 5. u 0.5 b u - u - σ( - ) α c d d u σ αu dp ( - ) + + d dp F w (A) a α h + α u + α h + α u u u b u h + - u h [ σ( - )] ( h + u ) 1- α (1- ) (1- ) d dh u d dh (1 ) c αu h + αu + α + α u dp dp dp dp d d 0.75 ( - ) dp σ h + u (A) 6 Copyright 010 by ASME