Numerical and Experimental Studies of Two-Phase Flow in Cooled / and Adiabatic Capillary Tubes

A numerical and experimental study was performed to predict the flow and thermal performance of a capillary tube that used in air conditioning and refrigeration systems. In the numerical study, the (CFD) technique was employed to model the problem using the finite volume method for a two-phase, two dimensional flow in the pipe. In the experimental part, an experimental rig was constructed using a split unite to measure the temperature and pressure along the capillary tube. These measurements were taken for (R-22) refrigerant with different ambient temperatures. It was found that for a fixed length and diameter of capillary tube the mass flow rate of (R-22) increases as the inlet temperature increases. The numerical study was then applied to predict the flow and heat transfer along several types of capillary tube, i.e. several lengths, diameters, and refrigerants, for cooled and non cooled tube. In the non cooled capillary results, the capillary tube length of R-407C (R-32/125/134a(23/25/52)) was found to be shorter than that required for (R-22). It was also found that (R-22) vaporized later than its corresponding (R-407C). The same condition was found for (R-12) and its alternative R-134a (CF3CH2F). The numerical results show a large effect of the length of capillary tube on the refrigeration system performance. When the length increases, the drop in pressure, temperature, and density decreases, while the velocity and dryness fraction increases.


Introduction
The capillary tube is a type of expansion device used in small vapour-compression refrigerating and air conditioning systems, located between the condenser and the evaporator.The function of the capillary tube is to reduce the high pressure in the condenser to low pressure in the evaporator.It is also used as an automatic flow rate controller for the refrigerant when varying load conditions and varying condenser and evaporator temperatures are encountered.Its simplicity, low initial cost, and low starting torque of compressors are the main reasons for its use.The capillary tube is commonly used as a constant area expansion device.It is a simple copper tube with an inner diameter of a few millimeters, but the flow inside these tubes is very complex, and The pressure drop has a strong influence on the performance of the whole system.Capillary tubes have been extended to larger units such as unitary air conditioners in sizes up to (10) tons of refrigerant [1].Lin et al. (1990)[2] presented a steady-state, two-phase flow in capillary tubes.Five differential equations based on the drift flux model of two-phase flows were solved by using the runge-kutta method simultaneously.A comparison with experimental data for two-phase flow of R-12 flowing through capillary tubes, shows that the model is practical and can be used for selection of capillary tubes used in refrigerant system.
Chen. (1997)[3] presented a theoretical and experimental study about capillary tube, an experimental test measure the pressure and temperature along the capillary tube to determine the under pressure of vaporization .The flow in the capillary tube was numerically modeled by dividing the flow into two regions, single phase and twophase flow region used one dimensional with metastable phenomenon is account.The result indicated that the vaporization occurred near the capillary tube exit with strong heat transfer between the capillary tube and suction line.Liang and Wong (2001)[4] attempted to exploit the possibility of applying the equilibrium two-phase drift flux model to simulate the flow of refrigerant in an adiabatic capillary tube.Their attempts are to compare predictions with experimental data of Li et al., 1990[5] andMikol, 1963[1] for R12 and R134a.They presented the details of flow characteristics of R134a within a capillary tube, such as, distribution of pressure, void fraction, and dryness fraction and phases velocities.Their results showed a good agreement with the experimental data of Li et al., 1990and Mikol, 1963. Gu et al. (2003)[5] analyzed and modeled the adiabatic flow in the capillary tube for R407C which is a non zetropic mixed refrigerant and one of the alternatives to R22.They presented the equations of energy, continuity, and pressure drop through a capillary tube, and developed a mathematical model of the sub cooled flow region and the two-phase flow region by considering the homogenous model and ignoring the metastable effect.
They conclude that at the same working and geometric conditions, the mass flow rate of R407C is greater than R22 by (4%).Muhsen (2007)[6] carried out an experimental and theoretical study about a capillary tube.He concluded from the experimental results that R600a requires a capillary tube shorter than that for R12 for the same inner diameter and as the ambient temperature increased, the compressor power increased and the capacity decreased.Also, the R600a can be operated with condensing pressures less than that of R12 for different ambient temperatures which makes less power consumption for R600a as compared with R12.This makes the coefficient of performance of the system working with R600a is higher than that of R12.
Many researchers have been investigated the capillary tube performance experimentally and theoretically.Most of these studies treated the flow as one dimensional viscous flow.They used several methods to solve a simplified form of the conservation equation of mass, momentum, and energy.None of the theoretical studies have solved the problem of the capillary tube using two dimensional Navier-stokes equations, and none of the previous experimental work has cooled the total length of the capillary tube.Therefore in the present work, a two dimensional study has been carried out to simulate the theoretical twophase flow of the capillary tube using the conservation equation of mass, momentum, and energy equation as presented by Navierstoke forms.The objective is to study the performance of the capillary tube in an attempt to improve its duty in domestic refrigerator and air conditioning unit.This will be done through building a computer program to simulate the governing equation of flow and heat transfer in two dimensional geometry to predict the effect of capillary tube dimensions on the performance of domestic refrigerator and air conditioning unit.Obtaining a detailed values for temperature , pressure, density, dryness fraction, and velocity distribution in two dimensional geometry, and carrying out an experimental work to use its results to verify the theoretical simulation in order to make the computer code in a common use form and to select a capillary tube for a given application.

Mathematical modeling
The coordinate system used in this work is the cylindrical coordinate system.The usual radius (r) and longitudinal coordinate (z) are defined in figure (1), where (r) and (z) are the inner radius and the length of the capillary tube, respectively.
In order to solve the governing equations, several assumptions should be used.These are as follows : 1-Straight, horizontal (gravity effects are neglected), and constant inner diameter of the capillary tube is assumed.2-Two dimensional geometry is considered.3-The flow is assumed to be steady , incompressible, and turbulent .4-Thermodynamics equilibrium (i.e.Metastable effect is neglected) [7].5-Heat transfer to the surrounding is negligible.  1 ( ) 2 ( ) ( )

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Where the source terms ( u S) and ( v S ) are given by, 7) Putting (φ ) equals to the enthalpy, equation (1) will represent the energy equation.

( ) ( )
Where, Due to the non-linear behavior of the dryness fraction (x) along the length of the capillary tube , a parabolic profile is assumed following [9]: Where, (x o ) is the dryness fraction at the end of the capillary tube and it is given by, The density for the flow is given by [10], Here (ρ f ) represent the liquid density and ( ρ g ) is the vapor density.The source term (S h ) is set to zero.

S h =0 …(15)
The general modified form of (k-ε) model can be written as follows[8]: For the capillary tube, the following boundary conditions are assumed and problem is solved for the upper half of the tube due to symmetry:

Upstream Boundary Conditions:
The distribution of all flow variables needs to be specified at upstream boundaries [11].

Downstream Boundary Conditions
Normally, the velocities are only known where the fluid enters the calculation domain.At downstream, the velocity distribution is decided by the flow field within the domain.For incompressible flow, the gradients normal to the downstream surface of all quantities are assumed to be zero [11].

Wall Boundary Conditions
The wall is the most common boundary encountered in confined fluid flow problems.The no-slip condition (u = v = 0) is the appropriate condition for the velocity components at solid walls.In the case of turbulent flow, the calculation of shear stress near the wall needs a special treatment.This is due to the existence of boundary layers, across which a steep variation of flow properties occurs and the standard (k-ε) model becomes inadequate.In order to adequately avoid these problems, it would be necessary to employ a fine grid near the wall, which would be expensive.An alternative and widely employed approach is to use a formula which known as "wall function" [11].
The implementation of wall boundary conditions in turbulent flows required the evolution of the following parameters as given below: 21 Where, ( P y ) is the distance of the near wall node (P) to the solid surface and near-wall flow is taken to be laminar if ( + y ≤ 11.63).The wall shear stress is assumed to be entirely viscous in origin.If ( + y >11.63), the flow is turbulent and the wall function approach can be used.(κ) is Von Karman's constant (0.4187) and (E) is an integration constant that depends on the roughness of the wall.For smooth wall (E) has a value of (9.793).Other empirical constants appear in (k-ε) model and wall function are given in Table (1).
A computational technique for the solution of the continuity, momentum, energy, and turbulence quantities, is performed to obtain the discretization form for these equations.
These discretization equations are solved by SIMPLE algorithm with power-law scheme.A computer program based on this algorithm and uses Fortran 90 language, was built to meet the requirements of the problem.A staggered grid in which the velocities staggered midway between the grid intersections, was used to obtain the numerical results.The grid number was (50,10) in (z, r) direction, respectively, see Figure (2).

Experimental Apparatus and Rig Layout
The experimental rig was constructed to measure the pressure and temperature along the capillary tube and to use the data from the measurements for comparison with the theoretical part.The rig was constructed to facilitate the measurement with a cooled or non cooled capillary tube, which can be seen in Figure (3).
Two tests were carried out in the present study.The first test was applied to a capillary tube with out cooling i.e. adiabatic capillary tube as found in the normal operating conditions.The second test is applied to a capillary tube with cooling.The cooling was done by placing the capillary tube in contact with return line from the evaporator, and both lines were insulating.

Results and Discussion
The experimental work was directed into two ways, the first is studying the effect of increasing the inlet temperature on the outlet pressure and temperature and the refrigerant effect via the mass flow rate.The second is to study the cooling effect of the capillary tube walls using the suction vapor refrigerant line.
Figure ( 4) shows a comparisons between the CFD results with the experimental data for pressure and temperature profile along the capillary tube.Both results represent the variation of temperature and pressure for a range of inlet temperatures (29 -41 o C).The effect of the inlet temperature is clear in these plots that show the pressure difference along the capillary is increased as the inlet temperature and pressure increased.This may lead to increase the mass flow rate of the refrigerant that pass through the compressor.The temperature drop is derived by the pressure difference  5), ( 6), ( 7) and ( 8) which reveals small differences between both results for different refrigerants.
The present computer code was employed to predict several cases using several refrigerants with different lengths and diameters for both cooled and non cooled capillary tubes.The non cooled tube results are first presented.Figure (10) and (11) show the pressure and temperature distribution a long the capillary tube for different lengths.The results show the pressure drop increases with increasing the length of capillary tube.The reason for the pressure drop is mainly due to strong flow turbulence inside the capillary tube length and also due to the drop of temperature along the tube.The gradient drop in temperature is due to the phase change of the liquid to vapor and the absorption of the latent heat of vaporization.As the capillary tube length increases, the drop in the pressure and temperature increases, but this length is limited by the dryness at exit.The dryness should be minimum to reduce the liberation of the latent heat of flashing of refrigerant liquid and store it to the evaporator to exchange heat with the cooling agent, air in the present study.
Figure (12) shows the velocity distribution along the capillary tube for different lengths.The velocity increases with capillary tube length.The density of refrigerant decreases as the vapor fraction increases and this lead to increase the velocity up to the chock flow.This phenomena is undesirable and it should be prevented by selection the tube length to be shorter.Figure (13) shows the density and velocity distribution along the capillary tube for different refrigerants.The result show that the density decreases as the length increases, and this lead to increase the velocity to satisfy the law of continuity.
Figure (14) shows the relation between pressure and velocity.The result appears that the pressure drop increases with velocity of refrigerant and this leads to increase the drop in density, resulting in an increase in the velocity while the flow advances forward.The velocity for (R-22) is close to (R-407C) because their critical densities are close to each other.The velocity of (R-12) is also found to be close to (R-134a) for similar reasons.
The effect of the cooling on the performance of the capillary tube is presented in Figures (15) ) putting (φ ) equals to unity, and ( φ Γ ) and ( φ S ) are equal zero, the above equation will represent the continuity equation.
) is the ratio of the location of a specific node to the length of capillary tube.

)
PDF created with pdfFactory Pro trial version www.pdffactory.com tube developed due to the throttling action as the liquid-gas moves from condenser to the evaporator.The present numerical results are compared with the numerical predictions of Jung et al.[13]  andOoi et al. [15]  and the experimental data for Sami et al.[14]  for the temperature

Figure ( 9
) shows the velocity vectors distribution along the capillary tube.The results appear that the value of the velocity increased along the tube where more liquid flashes due to pressure drop.The development of the flow along the capillary tube length is clearly shown in sections III and IV.

Figure ( 11 )
Figure (11) Temperature distributions along the capillary tube for different lengths for R-22.

Figure ( 13 )
Figure (13) Density and Velocity distributions along the capillary tube for different refrigerants.

Figure ( 15 )
Figure (15) Effect of cooling part of capillary tube surface on its performance.