Numerical simulation of capillary tube expansion devices behaviour with pure and mixed refrigerants considering metastable region. Part I: mathematical formulation and numerical model

A detailed one-dimensional steady and transient numerical simulation of the thermal and fluid-dynamic behaviour of capillary tube expansion devices working with pure and mixed refrigerants has been developed. The discretised governing equations are coupled using an implicit step by step method. A special treatment has been implemented in order to consider transitions (subcooled liquid region, metastable liquid region, metastable two-phase region and equilibrium two-phase region). All the flow variables (enthalpy, temperatures, pressures, vapour quality, velocities, heat fluxes, etc.) together with the thermophysical properties are evaluated at each point of the grid in which the domain is discretised. The numerical model allows analysis of aspects such as geometry, type of fluid (pure substances and mixtures), critical or non-critical flow conditions, metastable regions, adiabatic or non-adiabatic capillary tubes and transient aspects. Comparison of the numerical simulation with experimental data presented in the technical literature will be shown in part II of the present article.     

Numerical simulation of capillary-tube expansion devices behaviour with pure and mixed refrigerants considering metastable region. Part II: experimental validation and parametric studies

A detailed one-dimensional steady and transient numerical simulation of the thermal and fluid-dynamic behaviour of capillary-tube expansion devices working with pure and mixed refrigerants has been developed and presented in part I of this article. The accuracy of the detailed simulation model is demonstrated in this paper by comparison with experimental data from the technical literature.Results presented include both metastable flow modelling and non-metastable flow modelling, homogeneous and separated flow model for metastable flow and the used of different empirical correlation needed in the numerical model. Comparisons of model prediction between various approaches are discussed.Numerical results and parametric studies for concentric capillary tube-suction line heat exchangers have also been presented.     

Effects of coil diameter and pitch on the flow characteristics of alternative refrigerants flowing through adiabatic helical capillary tubes

This paper presents the effects of various geometries of helical capillary tubes on the flow characteristics of alternative refrigerants flowing through adiabatic helical capillary tubes. The theoretical model is based on the conservation of mass, energy and momentum of fluids in the capillary tube. The two-phase flow model developed was based on a homogenous flow assumption. The model was validated by comparing it with the experimental data of published in literature for R-22, particularly various pairs of refrigerants. It was found conventional refrigerants had lower capillary lengths than alternative refrigerants. For all pairs, the numerical results showed that the traditional refrigerants consistently gave lower pressure drops for both single-phase and two-phase flows, which resulted in longer tube lengths. The results show that coil diameter variation (less than 300 mm) for helical capillary tube geometries affected the length of helical capillary tubes. However, pitch variation (more than 300 mm) had no significant effect on the length of helical capillary tubes. This adiabatic helical capillary tube model can be used to integrate system models working with alternative refrigerants for design and optimisation.     

Numerical simulation and validation of HFC-134a fluid flow through short-tube orifices considering metastable conditions

One-dimensional numerical modelling of the fluid-flow behaviour inside short-tube orifices (expansion devices) has been carried out. Governing equations (continuity, momentum, energy, and entropy) for describing the fluid flow have been solved by using a fully implicit step-by-step method. A numerical treatment has been codified for considering thermodynamic and flow transitions (subcooled liquid region, metastable liquid region, metastable two-phase region and equilibrium two-phase region). Sudden contraction and enlargement at inlet and outlet sections were also considered. Fluid-flow variables (e.g., enthalpies, temperatures, pressures, mass fractions, and heat fluxes) and thermophysical and transport properties of fluid were numerically evaluated at each grid point in a discretized domain. The physical model used for solving the fluid-flow problem enabled an analysis of geometry, fluid type, critical and non-critical flow conditions, metastable regions, and transient effects to be performed. A comparison and validation analyses of the simulation results were carried out by using a wide range of mass flow rate experimental data (N_o = 634), which have been reported in the literature for the refrigerant 134a. Using a comprehensive statistical analysis, based on weighted linear regressions with an outlier detection/rejection algorithm at 95% of confidence level, the prediction performance of the numerical model was evaluated. Linear relationships between mass flow rate (predicted) and experimental mass flow rate data were statistically demonstrated. A global statistical evaluation of deviation errors between mass flow rate experimental data and predicted simulation results was also calculated. Average deviation error of ±8.9% was consistently computed between numerical model and experimental data, which demonstrates the good capability of the model developed for predicting the fluid-flow processes.     

A simulation for predicting the refrigerant flow characteristics including metastable region in adiabatic capillary tubes

Assumptions that no metastable flow phenomenon and flow in two-phase region is homogeneous have been used exclusively to study the flow characteristics in capillary tubes used as an expansion and controlling device in refrigerating systems. However, some experimental results show that due to the delay of vapourization, the onset of vapourization may not take place at the end of the sub-cooled liquid region. The two-phase flow in small diameter tubes may be also not entirely homogeneous due to phase interaction. In this paper, a mathematical model based on conservations of mass, energy and momentum is presented to simulate the refrigerant flow in adiabatic capillary tubes. Different from most previous studies, the metastable flow region is accounted in the model and the annular flow is also assumed to take place in the two-phase region. The model is validated by comparing with the experimental data reported in literature. The agreement between experimental and simulation results indicates that the model with appropriate correlations of pressure at vapourization and slip ratio can be used to predict the two-phase flow behaviour of refrigerant in capillary tubes. Copyright © 2002 John Wiley & Sons, Ltd.     

Two-phase flow model of refrigerants flowing through helically coiled capillary tubes

This paper presents a numerical study of the flow characteristics of refrigerants flowing through adiabatic helically coiled capillary tubes. The theoretical model is based on conservation of mass, energy and momentum of the fluids in the capillary tube. The two-phase flow model developed was based on the homogeneous flow assumption. The viscosity model was also based on recommendations from the literature. The developed model can be considered as an effective tool for designing and optimizing capillary tubes working with newer alternative refrigerants. The model is validated by comparison with the experimental data of Kim et al. (2002) for R-22, R-407C and R-410A, and Zhou and Zhang (2006) for R-22. The results obtained from the present model show reasonable agreement with the experimental data. The proposed model can be used to design helical capillary tubes working with various refrigerants.     

Flow characteristics of pure refrigerants and refrigerant mixtures in adiabatic capillary tubes

This paper provides the results of simulations using an adiabatic capillary tube model which is developed to study the flow characteristics in adiabatic capillary tubes used as a refrigerant control device in refrigerating systems. The developed model can be considered as an effective tool of capillary tubes' design and optimization for systems using newer alternative refrigerants. The model is validated by comparing with the experimental data of Li et al. and Mikol for R12 and Melo et al. for R134a. In particular, it has been possible to compare various pairs of refrigerants. It is found that the conventional refrigerants consistently give longer capillary lengths than the alternative refrigerants. For all pairs, the conventional refrigerant consistently give lower pressure drops for both single-phase and two-phase flow which resulted in longer tube lengths. In addition, an example of capillary tube selection chart developed from the present numerical simulation is shown. The chart can be practically used to select the capillary tube size from the flow rate and flow condition or to determine mass flow rate directly from a given capillary tube size and flow condition. The results of this study are of technological importance for the efficient design when systems are assigned to utilize various alternative refrigerants.     

Numerical simulation of gas-contaminated refrigerant two-phase flow through adiabatic capillary tubes

An unfavorable effect of gas impurities on the throttling process inside a small-diameter tube, i.e. a capillary tube, has been studied in detail. A special testing capillary tube equipped with precise temperature and pressure sensors has been used for an experimental investigation of the capillary flow of a saturated fluorocarbon refrigerant, R218, contaminated by dissolved nitrogen. The gas impurities significantly affected the throttling process, since the two-phase flow started notably earlier than in the case of pure refrigerant flow. Moreover, the gas contamination resulted in a decreased mass flow rate of refrigerant delivered through the capillary tube. A comprehensive numerical model has been developed to simulate the capillary flow of gas-contaminated refrigerant. The model takes into account two coincident thermodynamic events: the throttling process of the refrigerant (solvent) and the gradual release of the dissolved gas impurities (solute) from the refrigerant liquid phase. The gas release is in principle described by using the temperature correlation of the Henry’s law constant. The model considers adiabatic, thermodynamically equilibrated capillary flow with homogeneous two-phase flow. The numerical simulation is in good agreement with our experimental data measured for R218 contaminated by nitrogen.     

Numerical analysis of adiabatic flow of refrigerant through a spiral capillary tube

In the present work, a homogenous model including the metastable liquid region has been developed for the adiabatic flow of refrigerant through the spiral capillary tube. In order to develop the model, both liquid region and two phase region have been discretized into infinitesimal segments to take into account the effect of varying radius of curvature of spiral tube on the friction factor. The effect of the pitch of spiral on the mass flow rate of refrigerant and capillary tube length has been investigated. A comparison of flow characteristics of refrigerant R22 and its alternatives, i.e., R407C and R410A has been made at different operating conditions at the inlet of the capillary tube and it has been found that the flow characteristics of R22 and R407C are almost similar for a given condenser pressure and degree of subcooling at the inlet of capillary tube.     

Analysis on the adiabatic flow of R407C in capillary tube

In this paper the adiabatic flow in the capillary tube is analyzed and modeled for R407C, which is a non-azeotropic mixed refrigerant and one of the alternatives to R22. The equations of energy, continuity and pressure drop through a capillary tube are presented. A mathematical model of the sub-cooled flow region and the two-phase flow region is developed. The results of the calculation show that this numerical model is capable of providing an effective means to analyze components’ performance in optimizing and controlling a R407C air-conditioning system.     

Correlations for sizing adiabatic capillary tubes

This paper presents new correlations for the practical sizing of adiabatic capillary tubes used as an expansion device in small refrigerating and air-conditioning systems. The governing equation based on conservation of mass, energy and momentum is modelled. The developed model is used as an effective tool for studying the effects of relevant parameters on capillary tube length and developing the correlation. In this model, Colebrook's equation is used to determine the friction factor. The two-phase viscosity models are varied depending on the type of refrigerant and are based on the recommendations from past research. By varying the model input parameters, it is possible to show that for all refrigerants, the length decreases as the mass flow rate increases, increases as subcooling increases, increases as tube diameter increases, decreases as tube roughness increases and increases as condensing temperature increases. After the developed model is validated by comparing with existing experimental data, correlations for sizing capillary tubes, which contains the relevant parameters, namely condensing temperature, degree of subcooling, refrigerant mass flow rate, capillary tube inner diameter and roughness, are presented. Different from previous studies, correlations are presented for an extensive number of refrigerants and a wide range of operations. The developed correlations are validated with previous studies and found to agree well with the experimental data. The correlations can be used to integrate with system models working with alternative refrigerants for practical design and optimization. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical analysis of refrigerant flow along non-adiabatic capillary tubes using a two-fluid model

This work presents a numerical model to simulate steady state refrigerant flow along capillary tube-suction line heat exchangers, commonly used in small refrigeration systems. The flow along the straight and horizontal capillary tube is divided into two regions: a single-phase and a two-phase flow region. The flow is taken as one-dimensional and the metastable flow phenomenon is neglected. The two-fluid model is employed for the two-phase flow region, considering the hydrodynamic and the thermodynamic non-equilibrium between the liquid and vapor phases. Comparisons are made with experimental measurements of the mass flow rate and temperature distribution along capillary tube-suction line heat exchangers working with refrigerant R134a in different operating conditions. The results indicate that the present model provides a good estimation of the refrigerant mass flow rate. Moreover, comparisons with a homogeneous model are also made. Some computational results referring to the quality, void fraction and velocities of each phase are also presented and discussed.     

Two-phase flow of refrigerants during evaporation under constant heat flux in a horizontal tube

This paper presents the results of simulations using a two-phase separated flow model to study the heat transfer and flow characteristics of refrigerants during evaporation in a horizontal tube. A one-dimensional annular flow model of the evaporation of refrigerants under constant heat flux is developed. The basic physical equations governing flow are established from the conservation of mass, energy and momentum. The model is validated by comparing it with the experimental data reported in literature. The present model can be used to predict the variation of the temperature, heat transfer coefficient and pressure drop of various pure refrigerants flowing along a horizontal tube. It is found that the refrigerant temperature decreases along the tube corresponding to the decreasing of its saturation pressure. The liquid heat transfer coefficient increases with the axial length due to the reducing thickness of the liquid film. The evaporation rate of liquid refrigerant tends to decrease with increasing axial length, due to the decreasing latent heat transfer through the liquid–vapor interface. The developed model can be considered as an effective tool for evaporator design and can be used to choose appropriate refrigerants under designed conditions.     

Analysis of the laminar flamelet concept for nonpremixed laminar flames

The goal of this paper is to investigate the application of the laminar flamelet concept to the multidimensional numerical simulation of nonpremixed laminar flames. The performance of steady and unsteady flamelets is analyzed. The deduction of the mathematical formulation of flamelet modeling is exposed and some commonly used simplifications are examined. Different models for the scalar dissipation rate dependence on the mixture fraction variable are analyzed. Moreover, different criteria to evaluate the Lagrangian-type flamelet lifetime for unsteady flamelets are investigated. Inclusion of phenomena such as differential diffusion with constant Lewis number for each species and radiation heat transfer are also studied. A confined co-flow axisymmetric nonpremixed methane/air laminar flame experimentally investigated by McEnally and Pfefferle (Combust. Sci. Technol. 116-117 (1996) 183-209) and numerically investigated by Bennett, McEnally, Pfefferle, and Smooke (Combust. Flame 123 (2000) 522-546), Cònsul, Pérez-Segarra, Claramunt, Cadafalch, and Oliva (Combust. Theory Modelling 7 (3) (2003) 525-544), and Claramunt, Cònsul, Pérez-Segarra, and Oliva (Combust. Flame 137 (2004) 444-457) has been used as a test case. Results obtained using the flamelet concept have been compared to data obtained from the full resolution of the complete transport equations using primitive variables. Finite-volume techniques over staggered grids are used to discretize the governing equations. A parallel multiblock algorithm based on domain decomposition techniques running with loosely coupled computers has been used. To assess the quality of the numerical solutions presented in this paper, a verification process based on the generalized Richardson extrapolation technique and on the grid convergence index (GCI) has been applied.     

Numerical investigation of refrigerant flow through non-adiabatic capillary tubes

A mathematical model is developed to study flow characteristics in non-adiabatic capillary tubes. The theoretical model is based on conservation of mass, energy and momentum of fluids in the capillary tube and suction line. The mathematical model is categorized into three different cases, depending on the position of the heat exchange process. The first case is considered when the heat exchange process starts in the single-phase flow region, the second case is determined when the heat exchange process starts at the end of the single-phase flow region, and the last case is considered when the heat exchange process takes place in the two-phase flow region. A set of differential equations is solved by the explicit method of finite-difference scheme. The model is validated by comparing with the experimental data obtained from previous works. The results obtained from the present model show reasonable agreement with the experimental data. The present non-adiabatic capillary tube model can be used to integrate with system models working with alternative refrigerants for design and optimization.     

Reverse heat transfer and re-condensation phenomena in non-adiabatic capillary tubes

A numerical simulation model for lateral capillary tube-suction line heat exchangers is presented here to analyze its performance characteristics in small vapour compression refrigeration systems (e.g. domestic refrigerators). Appropriate heat transfer correlations have been used to illustrate the reverse heat transfer and re-condensation phenomena inside the tubes. Some convergence problems were encountered during the execution of the model when lower vapour temperature inside the suction line caused the two-phase refrigerant inside the capillary tube to re-condense within the heat exchange region. Therefore, a relationship between the re-condensation phenomenon and the divergence problem has been analyzed in the paper. The modelling was performed with two refrigerants, namely HFC-134a and HC-600a. Further, a simple theoretical equation has been developed to express the re-condensation phenomenon in non-adiabatic capillary tubes.     

Flow characteristics of refrigerants flowing through capillary tubes – A review

A comprehensive review of the literature on the flow of various refrigerants through the capillary tubes of different geometries viz. straight and coiled and flow configurations viz. adiabatic and diabatic, has been discussed in this paper. The paper presents in chronological order the experimental and numerical investigations systematically under different categories. Flow aspects like effect of coiling and effect of oil in the refrigerants on the mass flow rate through the capillary tube have been discussed. Furthermore, the phenomenon of metastability and the correlations to predict the underpressure of vaporization have also been discussed. The paper provides key information about the range of input parameters viz. tube diameter, tube length, surface roughness, coil pitch and coil diameter, inlet subcooling and condensing pressure or temperature. Other information includes type of refrigerants used, correlations proposed and methodology adopted in the analysis of flow through the capillary tubes of different geometries operating under adiabatic and diabatic flow conditions. It has been found from the review of the literature that there is a lot more to investigate for the flow of various refrigerants through different capillary tube geometries.     

Numerical modeling of two-phase refrigerant flow through adiabatic capillary tubes

Literature shows that the homogeneous flow assumption has been commonly used in most of the adiabatic capillary tube modeling studies due to its simplicity. The slip effect between the two phases was often not considered in this small diameter capillary tube. This paper attempts to exploit the possibility of applying the equilibrium two-phase drift flux model to simulate the flow of refrigerant in the capillary tube expansion devices. Attempts have been made to compare predictions with experimental results. The details flow characteristics of R134a in a capillary tube, such as distribution of pressure, void fraction, dryness fraction, phase’s velocities and their drift velocity relative to the center of the mass of the mixture are presented.     

Modelling of capillary tubes behaviour with HCFC 22 ternary alternative refrigerants

In this paper, a numerical model is presented for predicting capillary tube performance using new ternary mixtures proposed as alternatives to R 22. The model has been established after the fluid flow conservation equations written for a homogeneous refrigerant fluid flow under saturated, subcooled and two- pase conditions. Numerical results showed that the proposed model in question fairly simulated experimental on ternary refrigerant mixtures and fairly predicted the capillary tube behaviour under the investigated; subcooled, saturated, and two-phase flow conditions. © 1998 John Wiley & Sons, Ltd.     

The effects of improper refrigerant charge on the performance of a heat pump with an electronic expansion valve and capillary tube

For inverter heat pumps and multi-type heat pumps, conventional expansion devices such as capillary tubes, short tube orifices, and thermostatic expansion valves (TXVs) are being gradually replaced with electronic expansion valves (EEVs) because of the increasing focus on comfort and energy conservation. In this study, the effects of off-design refrigerant charge on the performance of a water-to-water heat pump are investigated by varying refrigerant charge amount from −20% to +20% of full charge in a steady state, cooling mode operation with expansion devices of capillary tube and EEV. The characteristics of the heat pump with an EEV are compared with those with a capillary tube. The capillary tube system is more sensitive to off-design charge as compared with the EEV system. Cooling capacity and COP of the EEV system show little dependence on refrigerant charge, while those are strongly dependent on outdoor conditions. In general, for a wide range of operating conditions the EEV system shows much higher performance as compared with the capillary tube system. The performance of the EEV system can be optimized by adjusting the EEV opening to maintain a constant superheat at all test conditions.