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Non-adiabatic capillary tube flow: a homogeneous model and process description
Applied Thermal Engineering 22 (2002) 1801–1819 www.elsevier.com/locate/apthermeng
Non-adiabatic capillary tube flow: a homogeneous model and process description B. Xu, P.K. Bansal
*
Department of Mechanical Engineering, The University of Auckland, Private bag 92109, Auckland, New Zealand Received 8 February 2002; accepted 23 July 2002
Abstract This paper presents a homogeneous model of refrigerant flow through capillary tube–suction line heat exchangers, which are widely used in small vapour compression refrigeration systems. The homogeneous model is based on fundamental conservation equations of mass, momentum and energy. These equations are solved simultaneously through iterative process. Churchill’s correlation [3] is used to calculate singlephase friction factors and Lin et al. [6] correlation for two-phase friction factors. The single-phase heat transfer coefficient is calculated by Gnielinski’s equation [5] while two-phase flow heat transfer coefficient is assumed to be infinite. The model is validated with previous experimental and analytical results. The present model can be used in either design calculation (calculate the capillary tube length for given refrigerant mass flow rate) or simulation calculation (calculate the refrigerant mass flow rate for given capillary tube length). The simulation model is used to understand the refrigerant flow behaviour inside the non-adiabatic capillary tubes. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Refrigerant flow; Non-adiabatic capillary tube; Two-phase friction; Simulation; Heat transfer
1. Introduction Capillary tube is basically used as a throttling device, which controls the refrigerant flow from high-pressure side to low-pressure side in small household refrigeration systems. It consists of a long hollow tube of drawn copper with an internal diameter between 0.51 and 2 mm and a length from 1 to 6 m. Primarily there are two kinds of capillary tubes, namely adiabatic and
*
Corresponding author. Tel.: +649-373-7599; fax: +649-373-7479. E-mail address: [email protected] (P.K. Bansal).
1359-4311/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 4 3 1 1 ( 0 2 ) 0 0 1 1 0 - 2
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Nomenclature A Cp D f G g h k L m_ NTU Nu Pr p q Re T V U v w x z h e d q s
cross sectional area (m2 ) specific heat (kJ/kg K) diameter (m) friction factor (–) mass flux (Kg/s m2 ) gravity acceleration (m/s2 ) specific enthalpy (kJ/kg) or heat transfer coefficient (kW/m2 K) conductivity (kW/m K) length (m) mass flow rate (kg/s) number of heat transfer units (–) Nusselt number (–) Prandtl number (–) pressure (Pa) heat flux (kJ/m2 ) Reynolds number (–) temperature (K) velocity (m/s) overall heat transfer coefficient (kW/K m2 ) specific volume (m3 /kg) specific work (kJ/kg) or width of solder joint (m) vapour quality (–) distance from capillary tube inlet (m) inclination angle (deg) wall roughness (mm) solder joint thickness (m) density (kg/m3 ) shear stress (N/m2 )
Subscripts c capillary tube cond condenser evap evaporator in inlet adiabatic region HX heat exchanger region j solder joint l liquid o outside out outlet adiabatic region s suction line sp single-phase
B. Xu, P.K. Bansal / Applied Thermal Engineering 22 (2002) 1801–1819
sub sup tp v w
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sub-cooling superheating two-phase vapour wall
Fig. 1. Refrigeration systems with capillary tube as expansion valve. (A) Adiabatic and (B) non-adiabatic.
non-adiabatic tubes (see Fig. 1). The adiabatic capillary tubes expand refrigerant from high pressure to low pressure adiabatically while in the non-adiabatic situation, the capillary tube forms a counter-flow heat exchanger with the suction line that joins the evaporator and the compressor. The design and performance aspects of non-adiabatic capillary tubes are discussed in this paper. Due to their strong influence on the performance of the entire refrigeration system, capillary tubes have been studied extensively over the past 50 years. Dirik et al. [4] performed both an experimental and a theoretical study on non-adiabatic capillary tubes using HFC-134a as the working fluid. Peixoto and Bullard [10] developed a numerical model for capillary tube–suction line heat exchangers. Mezavila and Melo [8] developed a computer model for non-adiabatic capillary tubes using HFC-134a as the working fluid. Although there are several capillary tube models in the open literature, there is very little effort being made to understand the basic physics of the non-adiabatic capillary tube flow. The aim of the present study is to develop a generalized numerical model to understand the refrigerant flow characteristics inside non-adiabatic capillary tubes. Suction line internal diameter is included in this model as an input and the vapour quality distribution in the capillary tube is calculated along the heat exchanger. This computer model can be used to either study refrigerant flow characteristics inside the capillary tube–suction line heat exchangers or design the geometric parameters
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under certain working conditions. The analysis of the refrigerant flow process reveals some essential features of non-adiabatic capillary tube flow and thus provides a basis upon which further studies on the flow can be carried out. The analysis of flow process is of great value in understanding the basic physics of the non-adiabatic capillary tube flow.
2. Assumptions In order to simplify the complexity of the problem, the present model is developed under the following assumptions. • Constant internal diameter and uniform surface roughness of both the capillary tube and the suction line. • Lateral counter-flow heat exchanger arrangement. • Incompressible flow in single-phase region. • Homogeneous two-phase flow. • Pure refrigerant (no oil is entrained). • Steady state flow. • Negligible heat exchange with ambient air. • Thermodynamic equilibrium (i.e. no meta-stable phenomenon).
3. Governing equations The lateral capillary tube–suction line heat exchanger discussed in this paper is shown schematically in Fig. 2. The capillary tube and the suction line form a counter-flow heat exchanger. The capillary tube can be divided into three distinct sections according to the relative position with the suction line, namely the inlet adiabatic region, the heat exchanger region and the outlet adiabatic region. The total length of the capillary tube is given as L ¼ Lin þ LHX þ Lout
ð1Þ
Fig. 2. Capillary tube–suction line heat exchanger.
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In the present study, the conservation equations of mass, momentum and energy are developed to describe the flow. The model follows the philosophy proposed by Mezavila and Melo [8] and further details can be found in Xu [11]. 3.1. Conservation of mass The refrigerant flow through the capillary tube and the suction line is in steady state and therefore, mass flow rate remains constant. The mass conservation equations for the capillary tube and the suction line can be written as [1] m_ ¼ constant
ð2Þ
Gs ¼
m_ ¼ constant As
ð3Þ
Gc ¼
m_ ¼ constant Ac
ð4Þ
3.2. Conservation of momentum The momentum conservation equation of the capillary tube flow can be written as m_ dVc ¼ sc pDc dz þ Ac dpc þ m_ g sin h
ð5Þ
where, sc is the wall shear stress given by fc qc Vc2 G2 v c ¼ fc c ð6Þ 8 8 The left hand side of Eq. (5) is the change in refrigerant momentum while the terms on the right hand side of Eq. (5) are respectively the friction of the capillary tube wall, force due to pressure difference and force due to change in elevation. Combining Eqs. (5) and (6) yields sc ¼
dpc G2 vc dvc g sin h ¼ fc c þ G2c þ vc dz 2Dc dz
ð7Þ
Similarly, the momentum conservation equation for suction line flow can be given by
dps G2 vs dvs g sin h ¼ fs s þ G2s þ vs dz 2Ds dz
ð8Þ
The local friction factors along the capillary tube and the suction line are calculated as follows. For single-phase flow, which is a part of capillary tube flow and entire suction line flow, Churchill’s equation [3] is employed " #121 12 8 1 þ ð9Þ fsp ¼ 8 1:5 Re ð A þ BÞ
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where ( A¼
"
2:457 ln
B¼
37530 Re
1
#)16
ð7=ReÞ0:9 þ 0:27e=D
ð9aÞ
16 ð9bÞ
For two-phase flow through the capillary tube, the local friction factors are calculated by Lin et al. [6] correlation as given below, vsp 2 ftp ¼ /lo fsp ð10Þ vtp where fsp is calculated by Churchill’s equation given in Eq. (9) and the multiplier, /2lo , is given by 3121 2 12 8 1 þ ðA þB Þ1:5 vv 7 6 Retp tp tp 2 1 ð11Þ /lo ¼ 4 5 1þx 12 vl 8 1 þ 1:5 Resp ðA þB Þ sp
sp
Single-phase flow Reynolds number is given by Re ¼
GD l
ð12Þ
For two-phase flow, an average dynamic viscosity of two-phase flow, ltp , is needed to calculate the two-phase Reynolds number in Eq. (11). Many empirical correlations are available to calculate the two-phase viscosity. As suggested by Bittle and Pate [2], McAdam’s model, is most suitable for the prediction of capillary tube flow, 1 x 1x ¼ þ ltp lv ll
ð13Þ
Therefore, the two-phase flow Reynolds number can be determined by Re ¼
GD ltp
ð14Þ
3.3. Conservation of energy The steady flow energy equations for both the capillary tube and the suction line are based on the first law of thermodynamics. For the flow through the capillary tube it can be given by V2 m_ d hc þ þ gz sin h ¼ dq þ dw ð15Þ 2 The left hand side of Eq. (15) represents the total energy change of the refrigerant flow. On the right hand side of Eq. (15), dq is the heat transfer rate between the capillary tube and the suction line and dw is shaft work input to the capillary tube flow (which is zero).
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Eq. (15) can be rearranged into the following dq G2c dv2c g sin h dz ð16Þ m_ 2 Due to the single-phase flow in suction line side, the energy equation can be written in terms of temperature instead of enthalpy, dhc ¼
Cp m_ dTs ¼ dq
ð17Þ
which can be rearranged into the following dTs ¼
dq Cp m_
ð18Þ
The specific heat is assumed to be constant for each control volume along the suction line. dq in Eqs. (16) and (18) is given by dq ¼ UAðTc Ts Þ:
ð19Þ
It may be noted that simple arithmetic temperature difference has been used in Eq. (19), instead of log mean temperature difference (LMTD), which is usually in association with UA. By using a small incremental length, which is 1 mm in numerical solution process of the current model, the uncertainty caused by this approximation is minimized. The cross section of the non-adiabatic capillary tube is shown in Fig. 3. The heat transfer process from the capillary tube side to the suction line side is illustrated in Fig. 4. This process includes convective heat transfer from the refrigerant inside the capillary tube to the capillary tube wall, conduction from the tube wall to the solder joint, conduction from the solder joint to the suction line wall and from the suction line wall to the refrigerant inside the suction line. Therefore, the overall thermal conductance, UA, can be given by 1 1 lnðDc;o =Dc Þ d lnðDs;o =Ds Þ 1 ¼ þ þ þ þ UA hc Dc p dz 2pkc;w dz kj w dz 2pks;w dz hs Ds p dz
Fig. 3. Cross section of the capillary tube–suction line heat exchanger.
ð20Þ
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Fig. 4. Thermal resistances from the capillary tube to the suction line.
Various resistances in Eq. (20) are determined by geometrical parameters and material properties of the capillary tube–suction line heat exchanger. For single-phase flow, the refrigerant convective heat transfer coefficient in Eq. (19) is calculated using Gnielinski’s equation [5] as Nu ¼
ðf =8ÞðRe 100ÞPr 1 þ 1:27ðf =8Þ0:5 ðPr2=3 1Þ
ð21Þ
For two-phase flow, the main thermal resistance of the heat exchanger lies on the suction line side, and therefore the convective heat transfer coefficient is assumed to be infinite. The validity of this assumption has been confirmed by Mezavila et al. [8].
4. Numerical solution The present model can be used to either optimize the capillary tube length (‘design’ calculation) or simulate the refrigerant mass flow through the capillary tube (‘simulate’ calculation) under certain boundary conditions. 4.1. Inputs and outputs For ‘design’ calculation, inputs to the model include refrigerant mass flow rate, internal and external diameters of the capillary tube and the suction line, evaporating and condensing temperatures, degrees of sub-cooling at the outlet of condenser, degree of superheat at the outlet of evaporator, roughness of inner surface of both the capillary tube and the suction line, and the lengths of the inlet adiabatic region and the heat exchanger region. In addition, thermodynamic and transport properties of the refrigerant are also required for the calculation. Outputs of the
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Fig. 5. Schematic presentation of the elemental layout of the heat transfer mechanism in the capillary tube–suction line heat exchanger.
design calculation are pressure and temperature distributions along the capillary tube and suction line and total length of the capillary tube. For ‘simulate’ calculations, all inputs are the same as those for the ‘design’ calculation except that the total length of the capillary tube is an input instead of the refrigerant mass flow rate. Therefore the outputs of the ‘simulate’ calculation are the refrigerant mass flow rate and the pressure and temperature profiles along the capillary tube and suction line. The numerical solution of the present model is carried out by dividing the flow domain into numerous control volumes along the capillary tube, as illustrated in Fig. 5. The length of each control volumes used in the present model is 1 mm. The refrigerant properties are evaluated using REFPROP V 6.0 database [9]. 4.2. Capillary tube The flow diagram of whole numerical solution process is shown in Fig. 6. First of all, the location of the control volume is checked to find out whether the control volume is within the heat exchanger region. Then, from the known enthalpy and pressure at the inlet of each control volume, single-phase or two-phase flow condition is determined. Once these two flow characteristics are determined, thermodynamic and transport properties of the refrigerant flowing through the control volume are evaluated from REFPROP V 6.0 database. Local friction factor and heat transfer coefficient for the non-adiabatic region are calculated from the flow conditions in terms of Reynolds number and Prandtl number. For each control volume, discretized momentum and energy equations are used to calculate enthalpy and pressure at the outlet of the control volume. Because of the interdependence of the enthalpy, pressure and specific volume, the properties at the outlet of the control volume cannot be solved explicitly, and therefore, an iteration process is employed. 4.3. Suction line For each control volume along the suction line, because of the counter-flow heat exchanger arrangement (shown in Fig. 5), all thermodynamic and transport properties are determined based on pressure and temperature at the outlet of the control volume. Friction factor and convective
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Fig. 6. Flowchart for calculations along the capillary tube–suction line heat exchanger.
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heat transfer coefficient can be calculated from these properties in terms of Reynolds number and Nusselt number. The state of the refrigerant at the outlet of the control volume can be solved implicitly by discretized governing equations. 4.4. Iteration for suction line outlet temperature The whole solution process is carried out in the direction of the refrigerant flow through the capillary tube. Because of the counter flow arrangement for the capillary tube–suction line heat exchanger, it is the counter flow direction of the suction line flow. Therefore, the first control volume for the suction line in the solution process is located at the outlet of the suction line, which is unknown at the beginning of the calculation and therefore the suction line properties cannot be solved explicitly. Another iteration process is required to calculate the outlet state of suction line flow. The suction line outlet temperature is initially guessed and then the solution process is carried over the entire heat exchanger region. Then the calculated temperature of suction line inlet is compared with the suction line inlet temperature determined by the evaporating temperature and the superheat at the outlet of the evaporator. This iteration process is repeated until the calculated suction line inlet temperature is equal to the one determined by the working conditions. 4.5. Iteration for mass flow rate When the present model is used for ‘simulate’ calculation, because of the interdependence between the refrigerant mass flow rate and the friction factors, the mass flow rate can not be solved explicitly from the total length of the capillary tube. Therefore, a third iteration process is required for the solution process. Initially mass flow rate is guessed. The guessed mass flow rate is used for a ‘design’ calculation to calculate the total length of the capillary tube. This length is then compared with the inputted length and then the guessed refrigerant mass flow rate is modified and used for ‘design’ calculation again. This iteration process terminates when the calculated capillary tube length is equal to the inputted length. 5. Results and discussion Simulation results from the present model are validated with previous studies, including Mendonca and Melo [8], Dirik et al. [4] and Peixoto and Bullard [10]. Although the present model is capable of handling alternative refrigerant mixtures, the present study is limited to HFC-134a due to the availability of extensive modelling and test data in the literature. 5.1. Validation of the model The ‘simulation’ results from the present model are compared with the experimental results of Mendonca et al. [7] over a wide range of working conditions and geometrical dimensions. For each run, geometric assemblies and working conditions were used to calculate the refrigerant mass flow rate, which were then compared with the measured mass flow rates from their experiments. The comparisons are shown in Fig. 7. It may be observed that the model tends to over predict at
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Fig. 7. Comparison of simulated refrigerant flow rates with the experimental data from Mendonca et al. [7].
lower flow rates and under predict at higher flow rates. However, the maximum disagreement was limited to 20%. Dirik et al. [4] conducted both numerical and experimental study on non-adiabatic capillary tubes that were 5500 and 6500 mm long. The suction line heat exchanger was 1700 mm long and located at 400 mm from the capillary tube exit for each test run. The inner diameter of suction line tube was 5.6 mm. The refrigerant pressure at the capillary tube inlet was adjusted to the saturation value corresponding to the condenser temperatures of 43.3, 48.9 and 54.4 C. Liquid sub-cooling at capillary tube inlet were varied from 5 to 20 K for each condenser temperature. Fig. 8 summaries the comparisons between the test data and the simulation results. The maximum difference was )18.4%, which was largely due to smaller refrigerant mass flow rates through the capillary tubes. Peixoto and Bullard [10] developed a model for capillary tube–suction line heat exchanger that can be used either ‘simulate’ or ‘design’ the refrigerant flow through the capillary tube. The results from Peixoto and Bullard’s model were compared with the present model, as shown in Fig. 9. The input values are listed in Table 1. It may be seen from Fig. 9 that the present model underestimates the mass flow rate for the same sub-cooling. The percentage differences between two models were between 5.6% and 10.8%. 5.2. Description of process The process of refrigerant flowing through the capillary tube–suction line heat exchangers is discussed in this section. Results for the design calculation are used to analyse the flow charac-
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Fig. 8. Comparison of simulated refrigerant flow rates with the experimental data from Dirik et al.
Fig. 9. Comparison of simulated refrigerant flow rates with the modelling results from with Peixoto and Bullard [10].
teristics of the refrigerant. The representative geometric parameters and working conditions used in this section are listed in Table 2. The calculation results are shown systematically in Figs. 10–14 respectively for quality, temperature, pressure, local friction factor (as well as local viscosity) and heat transfer rate (as well as
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Table 1 Parameters used for validation with Peixoto and Bullard’s model [10] Property
Value
Capillary tube inlet length ðLinlet Þ Heat exchanger length ðLHX Þ Capillary tube outlet length ðLout Þ Capillary tube internal diameter ðDc Þ Capillary tube external diameter ðDc;o Þ Suction line internal diameter ðDs Þ Suction line external diameter ðDs;o Þ Roughness of capillary tube internal surface ðec Þ Roughness of suction line internal surface ðes Þ Evaporating temperature ðtevap Þ Condensing temperature ðtcond Þ Superheat at suction line inlet ðDtsup Þ
0.96 m 1.4 m 0.4 m 0.84 mm 2 mm 6.6 mm 10 mm 0.00046 mm 0,00046 mm 263.15 K 321.1 K 0K
Table 2 Parameters used in the model for the physical process in capillary tubes Property
Value
Capillary tube inlet length ðLinlet Þ Heat exchanger length ðLHX Þ Capillary tube internal diameter ðDc Þ Capillary tube external diameter ðDc;o Þ Suction line internal diameter ðDs Þ Suction line external diameter ðDs;o Þ Roughness of capillary tube internal surface ðec Þ Roughness of suction line internal surface ðes Þ Refrigerant mass flow rate ðm_ Þ Evaporating temperature ðtevap Þ Condensing temperature ðtcond Þ Superheat at suction line inlet ðDtsup Þ Sub-cooling at capillary tube ðDtsub Þ
0.7 m 1.0 m 0.66 mm 2 mm 6.6 mm 10 mm 0.00046 mm 0,00046 mm 0.001 kg/s 258.15 K 321.1 K 2K 1K
pressure drop rate) profiles along the capillary tube. According to different flow characteristics, all profiles are divided into six different regions, indicated by I, II, III, IV, V and VI on each figure. Region I is an adiabatic single-phase flow region, where the changes in specific volume can be ignored and the pressure varies linearly along the capillary tube due to friction. Because the heat exchange with ambient is ignored, the temperature of the refrigerant remains constant, resulting in constant viscosity, thermal conductivity and specific heat and therefore constant Reynolds and Prandtl numbers in this region. This may explain why the local friction factor and convective heat transfer coefficient remain constant all along this region. Region II is an adiabatic two-phase flow region. Due to the existence of the vapour phase, the refrigerant pressure drops not only due to friction but also to acceleration along this region. The pressure drop causes more liquid flashing into the vapour and therefore the quality increases along this region. Because the specific volume increases with increase in quality, the pressure
B. Xu, P.K. Bansal / Applied Thermal Engineering 22 (2002) 1801–1819
Fig. 10. Variation of refrigerant quality profile along the capillary tube–suction line heat exchanger.
Fig. 11. Variation of the temperature profiles along the capillary tube–suction line heat exchanger.
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Fig. 12. Variation of the pressure profile along the capillary tube–suction line heat exchanger.
Fig. 13. Local friction factor and dynamic viscosity profiles along the capillary tube.
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gradient increases along this region. Because the refrigerant temperature in this region is the saturation temperature corresponding to its pressure, the temperature follows the pressure profile in this region. The viscosity of the refrigerant in this region is lower than that of the refrigerant in Region I due to the existence of vapour phase. With the increase of vapour quality, the refrigerant viscosity decreases all along this region. This decrease in viscosity causes the Reynolds number to increase along this region and therefore, results in a decrease of local friction factor along the capillary tube. Region III is a non-adiabatic two-phase flow region. As it can be seen from Fig. 10, the refrigerant quality within this region decreases due to the heat transfer to the suction line side. The heat exchange rate between the capillary tube and the suction line is determined by the temperature difference between the two sides. The sensible heat from refrigerant temperature drop in capillary tube side is not enough to account for the heat transfer to the refrigerant in suction line side. Therefore, a part of the heat transfer to the suction line side comes from latent heat. The quality profile shown in Fig. 10 reflects the overall result of both the pressure drop and the heat transfer. The effect of the heat transfer is stronger than that of the pressure drop and therefore the quality decreases along this region. This causes the increase in the overall viscosity of the two-phase refrigerant mixture leading to the drop in the Reynolds number and consequently, the increase in the local friction factor. It should be noted that there is a discontinuity in heat transfer profile shown in Fig. 14. This is because different correlations are used for the single-phase and the two-phase flow regimes. When the refrigerant changes from two phases to single-phase, the convective heat transfer in capillary tube side changes from zero to a finite value determined by Gnielinski’s correlation [5]. Therefore, a sharp drop in the heat transfer profile can be observed in Fig. 14.
Fig. 14. Local heat transfer rate and pressure drop profiles along the capillary tube.
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Region IV is a non-adiabatic liquid flow region in which the pressure varies linearly along the capillary tube. The refrigerant temperature drops along this region because of the heat transfer to the suction line side. However, because no phase change is involved in the heat transfer process, refrigerant temperature drops more dramatically in this region than it did in Region III. The Reynolds number of the capillary tube flow decreases because of the increase in refrigerant viscosity caused by the drop in temperature, resulting in an increase in the local friction factor. Region V and VI lie in the adiabatic outlet region. They are adiabatic single-phase flow and adiabatic two-phase flow and the flow patterns within these two regions are similar to those described in Regions I and II, respectively.
6. Conclusions This paper has presented a numerical model for the refrigerant flow through the non-adiabatic capillary tubes. The model was developed based on the finite element solution of conservation equations of mass, momentum and energy. The model predictions agreed with available experimental and analytical data to within 20%. Further the model was used to study the characteristics of non-adiabatic capillary tube flow. An analysis of the flow reveals that the flow characteristics inside a non-adiabatic capillary tube is determined by the interaction between the heat transfer and pressure drop effects of the refrigerant flow. When the effect of the heat transfer is stronger than that of the pressure drop, the refrigerant tends to condense within the heat exchanger region. However, if the pressure drop effect is stronger, the refrigerant flashes within the heat exchanger. The overall flow characteristic in the heat exchange region displays a balance of these two effects.
References [1] P.K. Bansal, A.S. Rupasinghe, A homogeneous model for adiabatic capillary tubes, Applied Thermal Engineering 18 (1998) 207–219. [2] R.R. Bittle, W.R. Stephenson, M.B. Pate, An evaluation of the ASHRAE method for predicting capillary tube– suction line heat exchanger performance, ASHRAE Transaction 101 (1995) 2. [3] S.W. Churchill, Frictional equation spans all fluid flow regimes, Chemical Engineering 84 (1977) 91–92. [4] E. Dirik, C. Inan, M.Y. Tanes, Numerical and experimental studies on adiabatic and non-adiabatic capillary tubes with R-134a in International Refrigeration Conference at Purdue, Purdue University, West Lafayette, Indiana, USA, 1994. [5] V. Gnielinski, New equations for heat and mass transfer in turbulent pipe and channel flow, International Chemical Engineering 16 (2) (1976) 359–366. [6] S. Lin et al., Local frictional pressure drop during vaporization of R-12 through capillary tubes, International Journal of Multiphase Flow 17 (1) (1991) 95–102. [7] K.C. Mendonca et al., Experimental Study on Lateral Capillary Tube–Suction Line Heat Exchanger in International Refrigeration Conference at Purdue, Purdue University, West Lafayette, Indiana, USA, 1996. [8] M.M. Mezavila, C. Melo, CAPHEAT: An homogeneous model to simulate refrigerant flow through non-adiabatic capillary tubes in international refrigeration conference at purdue, Purdue University, West Lafayette, Indiana, USA, 1996. [9] NIST, Thermodynamic and Transport Properties of Refrigerants and Refrigerant Mixtures-REFPROP 1998, sixth ed.
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[10] R.A. Peixoto, C.W. Bullard, A Simulation and Design Model for Capillary Tube–Suction Line Heat Exchanger in International Refrigeration Conference at Purdue, Purdue University, West Lafayette, Indiana, USA, 1994. [11] B. Xu, A numerical investigation of refrigerant flow through non-adiabatic capillary tubes, ME Thesis, Department of Mechanical Engineering, The University of Auckland, New Zealand, July, 2001.
Non-adiabatic capillary tube flow: a homogeneous model and process description B. Xu, P.K. Bansal
*
Department of Mechanical Engineering, The University of Auckland, Private bag 92109, Auckland, New Zealand Received 8 February 2002; accepted 23 July 2002
Abstract This paper presents a homogeneous model of refrigerant flow through capillary tube–suction line heat exchangers, which are widely used in small vapour compression refrigeration systems. The homogeneous model is based on fundamental conservation equations of mass, momentum and energy. These equations are solved simultaneously through iterative process. Churchill’s correlation [3] is used to calculate singlephase friction factors and Lin et al. [6] correlation for two-phase friction factors. The single-phase heat transfer coefficient is calculated by Gnielinski’s equation [5] while two-phase flow heat transfer coefficient is assumed to be infinite. The model is validated with previous experimental and analytical results. The present model can be used in either design calculation (calculate the capillary tube length for given refrigerant mass flow rate) or simulation calculation (calculate the refrigerant mass flow rate for given capillary tube length). The simulation model is used to understand the refrigerant flow behaviour inside the non-adiabatic capillary tubes. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Refrigerant flow; Non-adiabatic capillary tube; Two-phase friction; Simulation; Heat transfer
1. Introduction Capillary tube is basically used as a throttling device, which controls the refrigerant flow from high-pressure side to low-pressure side in small household refrigeration systems. It consists of a long hollow tube of drawn copper with an internal diameter between 0.51 and 2 mm and a length from 1 to 6 m. Primarily there are two kinds of capillary tubes, namely adiabatic and
*
Corresponding author. Tel.: +649-373-7599; fax: +649-373-7479. E-mail address: [email protected] (P.K. Bansal).
1359-4311/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 4 3 1 1 ( 0 2 ) 0 0 1 1 0 - 2
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Nomenclature A Cp D f G g h k L m_ NTU Nu Pr p q Re T V U v w x z h e d q s
cross sectional area (m2 ) specific heat (kJ/kg K) diameter (m) friction factor (–) mass flux (Kg/s m2 ) gravity acceleration (m/s2 ) specific enthalpy (kJ/kg) or heat transfer coefficient (kW/m2 K) conductivity (kW/m K) length (m) mass flow rate (kg/s) number of heat transfer units (–) Nusselt number (–) Prandtl number (–) pressure (Pa) heat flux (kJ/m2 ) Reynolds number (–) temperature (K) velocity (m/s) overall heat transfer coefficient (kW/K m2 ) specific volume (m3 /kg) specific work (kJ/kg) or width of solder joint (m) vapour quality (–) distance from capillary tube inlet (m) inclination angle (deg) wall roughness (mm) solder joint thickness (m) density (kg/m3 ) shear stress (N/m2 )
Subscripts c capillary tube cond condenser evap evaporator in inlet adiabatic region HX heat exchanger region j solder joint l liquid o outside out outlet adiabatic region s suction line sp single-phase
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sub sup tp v w
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sub-cooling superheating two-phase vapour wall
Fig. 1. Refrigeration systems with capillary tube as expansion valve. (A) Adiabatic and (B) non-adiabatic.
non-adiabatic tubes (see Fig. 1). The adiabatic capillary tubes expand refrigerant from high pressure to low pressure adiabatically while in the non-adiabatic situation, the capillary tube forms a counter-flow heat exchanger with the suction line that joins the evaporator and the compressor. The design and performance aspects of non-adiabatic capillary tubes are discussed in this paper. Due to their strong influence on the performance of the entire refrigeration system, capillary tubes have been studied extensively over the past 50 years. Dirik et al. [4] performed both an experimental and a theoretical study on non-adiabatic capillary tubes using HFC-134a as the working fluid. Peixoto and Bullard [10] developed a numerical model for capillary tube–suction line heat exchangers. Mezavila and Melo [8] developed a computer model for non-adiabatic capillary tubes using HFC-134a as the working fluid. Although there are several capillary tube models in the open literature, there is very little effort being made to understand the basic physics of the non-adiabatic capillary tube flow. The aim of the present study is to develop a generalized numerical model to understand the refrigerant flow characteristics inside non-adiabatic capillary tubes. Suction line internal diameter is included in this model as an input and the vapour quality distribution in the capillary tube is calculated along the heat exchanger. This computer model can be used to either study refrigerant flow characteristics inside the capillary tube–suction line heat exchangers or design the geometric parameters
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under certain working conditions. The analysis of the refrigerant flow process reveals some essential features of non-adiabatic capillary tube flow and thus provides a basis upon which further studies on the flow can be carried out. The analysis of flow process is of great value in understanding the basic physics of the non-adiabatic capillary tube flow.
2. Assumptions In order to simplify the complexity of the problem, the present model is developed under the following assumptions. • Constant internal diameter and uniform surface roughness of both the capillary tube and the suction line. • Lateral counter-flow heat exchanger arrangement. • Incompressible flow in single-phase region. • Homogeneous two-phase flow. • Pure refrigerant (no oil is entrained). • Steady state flow. • Negligible heat exchange with ambient air. • Thermodynamic equilibrium (i.e. no meta-stable phenomenon).
3. Governing equations The lateral capillary tube–suction line heat exchanger discussed in this paper is shown schematically in Fig. 2. The capillary tube and the suction line form a counter-flow heat exchanger. The capillary tube can be divided into three distinct sections according to the relative position with the suction line, namely the inlet adiabatic region, the heat exchanger region and the outlet adiabatic region. The total length of the capillary tube is given as L ¼ Lin þ LHX þ Lout
ð1Þ
Fig. 2. Capillary tube–suction line heat exchanger.
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In the present study, the conservation equations of mass, momentum and energy are developed to describe the flow. The model follows the philosophy proposed by Mezavila and Melo [8] and further details can be found in Xu [11]. 3.1. Conservation of mass The refrigerant flow through the capillary tube and the suction line is in steady state and therefore, mass flow rate remains constant. The mass conservation equations for the capillary tube and the suction line can be written as [1] m_ ¼ constant
ð2Þ
Gs ¼
m_ ¼ constant As
ð3Þ
Gc ¼
m_ ¼ constant Ac
ð4Þ
3.2. Conservation of momentum The momentum conservation equation of the capillary tube flow can be written as m_ dVc ¼ sc pDc dz þ Ac dpc þ m_ g sin h
ð5Þ
where, sc is the wall shear stress given by fc qc Vc2 G2 v c ¼ fc c ð6Þ 8 8 The left hand side of Eq. (5) is the change in refrigerant momentum while the terms on the right hand side of Eq. (5) are respectively the friction of the capillary tube wall, force due to pressure difference and force due to change in elevation. Combining Eqs. (5) and (6) yields sc ¼
dpc G2 vc dvc g sin h ¼ fc c þ G2c þ vc dz 2Dc dz
ð7Þ
Similarly, the momentum conservation equation for suction line flow can be given by
dps G2 vs dvs g sin h ¼ fs s þ G2s þ vs dz 2Ds dz
ð8Þ
The local friction factors along the capillary tube and the suction line are calculated as follows. For single-phase flow, which is a part of capillary tube flow and entire suction line flow, Churchill’s equation [3] is employed " #121 12 8 1 þ ð9Þ fsp ¼ 8 1:5 Re ð A þ BÞ
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where ( A¼
"
2:457 ln
B¼
37530 Re
1
#)16
ð7=ReÞ0:9 þ 0:27e=D
ð9aÞ
16 ð9bÞ
For two-phase flow through the capillary tube, the local friction factors are calculated by Lin et al. [6] correlation as given below, vsp 2 ftp ¼ /lo fsp ð10Þ vtp where fsp is calculated by Churchill’s equation given in Eq. (9) and the multiplier, /2lo , is given by 3121 2 12 8 1 þ ðA þB Þ1:5 vv 7 6 Retp tp tp 2 1 ð11Þ /lo ¼ 4 5 1þx 12 vl 8 1 þ 1:5 Resp ðA þB Þ sp
sp
Single-phase flow Reynolds number is given by Re ¼
GD l
ð12Þ
For two-phase flow, an average dynamic viscosity of two-phase flow, ltp , is needed to calculate the two-phase Reynolds number in Eq. (11). Many empirical correlations are available to calculate the two-phase viscosity. As suggested by Bittle and Pate [2], McAdam’s model, is most suitable for the prediction of capillary tube flow, 1 x 1x ¼ þ ltp lv ll
ð13Þ
Therefore, the two-phase flow Reynolds number can be determined by Re ¼
GD ltp
ð14Þ
3.3. Conservation of energy The steady flow energy equations for both the capillary tube and the suction line are based on the first law of thermodynamics. For the flow through the capillary tube it can be given by V2 m_ d hc þ þ gz sin h ¼ dq þ dw ð15Þ 2 The left hand side of Eq. (15) represents the total energy change of the refrigerant flow. On the right hand side of Eq. (15), dq is the heat transfer rate between the capillary tube and the suction line and dw is shaft work input to the capillary tube flow (which is zero).
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Eq. (15) can be rearranged into the following dq G2c dv2c g sin h dz ð16Þ m_ 2 Due to the single-phase flow in suction line side, the energy equation can be written in terms of temperature instead of enthalpy, dhc ¼
Cp m_ dTs ¼ dq
ð17Þ
which can be rearranged into the following dTs ¼
dq Cp m_
ð18Þ
The specific heat is assumed to be constant for each control volume along the suction line. dq in Eqs. (16) and (18) is given by dq ¼ UAðTc Ts Þ:
ð19Þ
It may be noted that simple arithmetic temperature difference has been used in Eq. (19), instead of log mean temperature difference (LMTD), which is usually in association with UA. By using a small incremental length, which is 1 mm in numerical solution process of the current model, the uncertainty caused by this approximation is minimized. The cross section of the non-adiabatic capillary tube is shown in Fig. 3. The heat transfer process from the capillary tube side to the suction line side is illustrated in Fig. 4. This process includes convective heat transfer from the refrigerant inside the capillary tube to the capillary tube wall, conduction from the tube wall to the solder joint, conduction from the solder joint to the suction line wall and from the suction line wall to the refrigerant inside the suction line. Therefore, the overall thermal conductance, UA, can be given by 1 1 lnðDc;o =Dc Þ d lnðDs;o =Ds Þ 1 ¼ þ þ þ þ UA hc Dc p dz 2pkc;w dz kj w dz 2pks;w dz hs Ds p dz
Fig. 3. Cross section of the capillary tube–suction line heat exchanger.
ð20Þ
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Fig. 4. Thermal resistances from the capillary tube to the suction line.
Various resistances in Eq. (20) are determined by geometrical parameters and material properties of the capillary tube–suction line heat exchanger. For single-phase flow, the refrigerant convective heat transfer coefficient in Eq. (19) is calculated using Gnielinski’s equation [5] as Nu ¼
ðf =8ÞðRe 100ÞPr 1 þ 1:27ðf =8Þ0:5 ðPr2=3 1Þ
ð21Þ
For two-phase flow, the main thermal resistance of the heat exchanger lies on the suction line side, and therefore the convective heat transfer coefficient is assumed to be infinite. The validity of this assumption has been confirmed by Mezavila et al. [8].
4. Numerical solution The present model can be used to either optimize the capillary tube length (‘design’ calculation) or simulate the refrigerant mass flow through the capillary tube (‘simulate’ calculation) under certain boundary conditions. 4.1. Inputs and outputs For ‘design’ calculation, inputs to the model include refrigerant mass flow rate, internal and external diameters of the capillary tube and the suction line, evaporating and condensing temperatures, degrees of sub-cooling at the outlet of condenser, degree of superheat at the outlet of evaporator, roughness of inner surface of both the capillary tube and the suction line, and the lengths of the inlet adiabatic region and the heat exchanger region. In addition, thermodynamic and transport properties of the refrigerant are also required for the calculation. Outputs of the
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Fig. 5. Schematic presentation of the elemental layout of the heat transfer mechanism in the capillary tube–suction line heat exchanger.
design calculation are pressure and temperature distributions along the capillary tube and suction line and total length of the capillary tube. For ‘simulate’ calculations, all inputs are the same as those for the ‘design’ calculation except that the total length of the capillary tube is an input instead of the refrigerant mass flow rate. Therefore the outputs of the ‘simulate’ calculation are the refrigerant mass flow rate and the pressure and temperature profiles along the capillary tube and suction line. The numerical solution of the present model is carried out by dividing the flow domain into numerous control volumes along the capillary tube, as illustrated in Fig. 5. The length of each control volumes used in the present model is 1 mm. The refrigerant properties are evaluated using REFPROP V 6.0 database [9]. 4.2. Capillary tube The flow diagram of whole numerical solution process is shown in Fig. 6. First of all, the location of the control volume is checked to find out whether the control volume is within the heat exchanger region. Then, from the known enthalpy and pressure at the inlet of each control volume, single-phase or two-phase flow condition is determined. Once these two flow characteristics are determined, thermodynamic and transport properties of the refrigerant flowing through the control volume are evaluated from REFPROP V 6.0 database. Local friction factor and heat transfer coefficient for the non-adiabatic region are calculated from the flow conditions in terms of Reynolds number and Prandtl number. For each control volume, discretized momentum and energy equations are used to calculate enthalpy and pressure at the outlet of the control volume. Because of the interdependence of the enthalpy, pressure and specific volume, the properties at the outlet of the control volume cannot be solved explicitly, and therefore, an iteration process is employed. 4.3. Suction line For each control volume along the suction line, because of the counter-flow heat exchanger arrangement (shown in Fig. 5), all thermodynamic and transport properties are determined based on pressure and temperature at the outlet of the control volume. Friction factor and convective
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Fig. 6. Flowchart for calculations along the capillary tube–suction line heat exchanger.
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heat transfer coefficient can be calculated from these properties in terms of Reynolds number and Nusselt number. The state of the refrigerant at the outlet of the control volume can be solved implicitly by discretized governing equations. 4.4. Iteration for suction line outlet temperature The whole solution process is carried out in the direction of the refrigerant flow through the capillary tube. Because of the counter flow arrangement for the capillary tube–suction line heat exchanger, it is the counter flow direction of the suction line flow. Therefore, the first control volume for the suction line in the solution process is located at the outlet of the suction line, which is unknown at the beginning of the calculation and therefore the suction line properties cannot be solved explicitly. Another iteration process is required to calculate the outlet state of suction line flow. The suction line outlet temperature is initially guessed and then the solution process is carried over the entire heat exchanger region. Then the calculated temperature of suction line inlet is compared with the suction line inlet temperature determined by the evaporating temperature and the superheat at the outlet of the evaporator. This iteration process is repeated until the calculated suction line inlet temperature is equal to the one determined by the working conditions. 4.5. Iteration for mass flow rate When the present model is used for ‘simulate’ calculation, because of the interdependence between the refrigerant mass flow rate and the friction factors, the mass flow rate can not be solved explicitly from the total length of the capillary tube. Therefore, a third iteration process is required for the solution process. Initially mass flow rate is guessed. The guessed mass flow rate is used for a ‘design’ calculation to calculate the total length of the capillary tube. This length is then compared with the inputted length and then the guessed refrigerant mass flow rate is modified and used for ‘design’ calculation again. This iteration process terminates when the calculated capillary tube length is equal to the inputted length. 5. Results and discussion Simulation results from the present model are validated with previous studies, including Mendonca and Melo [8], Dirik et al. [4] and Peixoto and Bullard [10]. Although the present model is capable of handling alternative refrigerant mixtures, the present study is limited to HFC-134a due to the availability of extensive modelling and test data in the literature. 5.1. Validation of the model The ‘simulation’ results from the present model are compared with the experimental results of Mendonca et al. [7] over a wide range of working conditions and geometrical dimensions. For each run, geometric assemblies and working conditions were used to calculate the refrigerant mass flow rate, which were then compared with the measured mass flow rates from their experiments. The comparisons are shown in Fig. 7. It may be observed that the model tends to over predict at
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Fig. 7. Comparison of simulated refrigerant flow rates with the experimental data from Mendonca et al. [7].
lower flow rates and under predict at higher flow rates. However, the maximum disagreement was limited to 20%. Dirik et al. [4] conducted both numerical and experimental study on non-adiabatic capillary tubes that were 5500 and 6500 mm long. The suction line heat exchanger was 1700 mm long and located at 400 mm from the capillary tube exit for each test run. The inner diameter of suction line tube was 5.6 mm. The refrigerant pressure at the capillary tube inlet was adjusted to the saturation value corresponding to the condenser temperatures of 43.3, 48.9 and 54.4 C. Liquid sub-cooling at capillary tube inlet were varied from 5 to 20 K for each condenser temperature. Fig. 8 summaries the comparisons between the test data and the simulation results. The maximum difference was )18.4%, which was largely due to smaller refrigerant mass flow rates through the capillary tubes. Peixoto and Bullard [10] developed a model for capillary tube–suction line heat exchanger that can be used either ‘simulate’ or ‘design’ the refrigerant flow through the capillary tube. The results from Peixoto and Bullard’s model were compared with the present model, as shown in Fig. 9. The input values are listed in Table 1. It may be seen from Fig. 9 that the present model underestimates the mass flow rate for the same sub-cooling. The percentage differences between two models were between 5.6% and 10.8%. 5.2. Description of process The process of refrigerant flowing through the capillary tube–suction line heat exchangers is discussed in this section. Results for the design calculation are used to analyse the flow charac-
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Fig. 8. Comparison of simulated refrigerant flow rates with the experimental data from Dirik et al.
Fig. 9. Comparison of simulated refrigerant flow rates with the modelling results from with Peixoto and Bullard [10].
teristics of the refrigerant. The representative geometric parameters and working conditions used in this section are listed in Table 2. The calculation results are shown systematically in Figs. 10–14 respectively for quality, temperature, pressure, local friction factor (as well as local viscosity) and heat transfer rate (as well as
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Table 1 Parameters used for validation with Peixoto and Bullard’s model [10] Property
Value
Capillary tube inlet length ðLinlet Þ Heat exchanger length ðLHX Þ Capillary tube outlet length ðLout Þ Capillary tube internal diameter ðDc Þ Capillary tube external diameter ðDc;o Þ Suction line internal diameter ðDs Þ Suction line external diameter ðDs;o Þ Roughness of capillary tube internal surface ðec Þ Roughness of suction line internal surface ðes Þ Evaporating temperature ðtevap Þ Condensing temperature ðtcond Þ Superheat at suction line inlet ðDtsup Þ
0.96 m 1.4 m 0.4 m 0.84 mm 2 mm 6.6 mm 10 mm 0.00046 mm 0,00046 mm 263.15 K 321.1 K 0K
Table 2 Parameters used in the model for the physical process in capillary tubes Property
Value
Capillary tube inlet length ðLinlet Þ Heat exchanger length ðLHX Þ Capillary tube internal diameter ðDc Þ Capillary tube external diameter ðDc;o Þ Suction line internal diameter ðDs Þ Suction line external diameter ðDs;o Þ Roughness of capillary tube internal surface ðec Þ Roughness of suction line internal surface ðes Þ Refrigerant mass flow rate ðm_ Þ Evaporating temperature ðtevap Þ Condensing temperature ðtcond Þ Superheat at suction line inlet ðDtsup Þ Sub-cooling at capillary tube ðDtsub Þ
0.7 m 1.0 m 0.66 mm 2 mm 6.6 mm 10 mm 0.00046 mm 0,00046 mm 0.001 kg/s 258.15 K 321.1 K 2K 1K
pressure drop rate) profiles along the capillary tube. According to different flow characteristics, all profiles are divided into six different regions, indicated by I, II, III, IV, V and VI on each figure. Region I is an adiabatic single-phase flow region, where the changes in specific volume can be ignored and the pressure varies linearly along the capillary tube due to friction. Because the heat exchange with ambient is ignored, the temperature of the refrigerant remains constant, resulting in constant viscosity, thermal conductivity and specific heat and therefore constant Reynolds and Prandtl numbers in this region. This may explain why the local friction factor and convective heat transfer coefficient remain constant all along this region. Region II is an adiabatic two-phase flow region. Due to the existence of the vapour phase, the refrigerant pressure drops not only due to friction but also to acceleration along this region. The pressure drop causes more liquid flashing into the vapour and therefore the quality increases along this region. Because the specific volume increases with increase in quality, the pressure
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Fig. 10. Variation of refrigerant quality profile along the capillary tube–suction line heat exchanger.
Fig. 11. Variation of the temperature profiles along the capillary tube–suction line heat exchanger.
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Fig. 12. Variation of the pressure profile along the capillary tube–suction line heat exchanger.
Fig. 13. Local friction factor and dynamic viscosity profiles along the capillary tube.
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gradient increases along this region. Because the refrigerant temperature in this region is the saturation temperature corresponding to its pressure, the temperature follows the pressure profile in this region. The viscosity of the refrigerant in this region is lower than that of the refrigerant in Region I due to the existence of vapour phase. With the increase of vapour quality, the refrigerant viscosity decreases all along this region. This decrease in viscosity causes the Reynolds number to increase along this region and therefore, results in a decrease of local friction factor along the capillary tube. Region III is a non-adiabatic two-phase flow region. As it can be seen from Fig. 10, the refrigerant quality within this region decreases due to the heat transfer to the suction line side. The heat exchange rate between the capillary tube and the suction line is determined by the temperature difference between the two sides. The sensible heat from refrigerant temperature drop in capillary tube side is not enough to account for the heat transfer to the refrigerant in suction line side. Therefore, a part of the heat transfer to the suction line side comes from latent heat. The quality profile shown in Fig. 10 reflects the overall result of both the pressure drop and the heat transfer. The effect of the heat transfer is stronger than that of the pressure drop and therefore the quality decreases along this region. This causes the increase in the overall viscosity of the two-phase refrigerant mixture leading to the drop in the Reynolds number and consequently, the increase in the local friction factor. It should be noted that there is a discontinuity in heat transfer profile shown in Fig. 14. This is because different correlations are used for the single-phase and the two-phase flow regimes. When the refrigerant changes from two phases to single-phase, the convective heat transfer in capillary tube side changes from zero to a finite value determined by Gnielinski’s correlation [5]. Therefore, a sharp drop in the heat transfer profile can be observed in Fig. 14.
Fig. 14. Local heat transfer rate and pressure drop profiles along the capillary tube.
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Region IV is a non-adiabatic liquid flow region in which the pressure varies linearly along the capillary tube. The refrigerant temperature drops along this region because of the heat transfer to the suction line side. However, because no phase change is involved in the heat transfer process, refrigerant temperature drops more dramatically in this region than it did in Region III. The Reynolds number of the capillary tube flow decreases because of the increase in refrigerant viscosity caused by the drop in temperature, resulting in an increase in the local friction factor. Region V and VI lie in the adiabatic outlet region. They are adiabatic single-phase flow and adiabatic two-phase flow and the flow patterns within these two regions are similar to those described in Regions I and II, respectively.
6. Conclusions This paper has presented a numerical model for the refrigerant flow through the non-adiabatic capillary tubes. The model was developed based on the finite element solution of conservation equations of mass, momentum and energy. The model predictions agreed with available experimental and analytical data to within 20%. Further the model was used to study the characteristics of non-adiabatic capillary tube flow. An analysis of the flow reveals that the flow characteristics inside a non-adiabatic capillary tube is determined by the interaction between the heat transfer and pressure drop effects of the refrigerant flow. When the effect of the heat transfer is stronger than that of the pressure drop, the refrigerant tends to condense within the heat exchanger region. However, if the pressure drop effect is stronger, the refrigerant flashes within the heat exchanger. The overall flow characteristic in the heat exchange region displays a balance of these two effects.
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[10] R.A. Peixoto, C.W. Bullard, A Simulation and Design Model for Capillary Tube–Suction Line Heat Exchanger in International Refrigeration Conference at Purdue, Purdue University, West Lafayette, Indiana, USA, 1994. [11] B. Xu, A numerical investigation of refrigerant flow through non-adiabatic capillary tubes, ME Thesis, Department of Mechanical Engineering, The University of Auckland, New Zealand, July, 2001.