NUMERICAL MODELING OF NON-EQUILIBRIUM CONDENSATION IN A HORIZONTAL TUBE by ijosm

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Sadegh Torfi, Ali Ebrahimi

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									                                               (Manuscript No: I12528-10)
                                        April 18, 2012 / Accepted: April 24, 2012




      NUMERICAL MODELING OF NON-
     EQUILIBRIUM CONDENSATION IN A
            HORIZONTAL TUBE
                                             Sadegh Torfi*
                                Department of Mechanical Engineering,
                       Islamic Azad University Susangerd Branch, Susangerd, Iran
                           Phone: (+98) 916 941 7762; fax: (+98) 612 4222324;
                                          (Email: st@siau.ac.ir)

                                             Ali Ebrahimi
                                Department of Mechanical Engineering,
                       Islamic Azad University Susangerd Branch, Susangerd, Iran
                                         (Email: st@siau.ac.ir)

Abstract - Condensation heat transfer inside horizontal tubes is investigated for stratified, cocurrent two phase
of vapor and liquid. The analysis takes into account the effects of interfacial heat transfer and momentum
transfer at non-equilibrium process. The present numerical solution method is based on the forth order Runga-
Kutta. Pressure gradient, heat transfer coefficient, vapor quality and void fraction, liquid phase subcooled
temperature and their corresponding variations throughout tube is studied. The results of the numerical
predictions are found to agree favorably with reported experimental data. In addition, the results show that
neglecting subcooling of liquid phase is reasonable in high vapor quality regions while in low vapor quality
regions effect of liquid phase subcooling is significant. Results shows that for condensation of r114 in 8 mm
diameter horizontal tube, after 5.4 mm from tube inlet, with assuming equilibrium condensation vapor quality is
0.05 and vapor void fraction is 0.21 while with assuming non-equilibrium condensation vapor quality is zero
and vapor void fraction is 0.05.

Keywords: condensation, two phase, none thermodynamically equilibrium, horizontal tube, stratified flow.

                                                   Introduction

Condensation in tubes is encountered in various applications for heat exchangers such as horizontal type
refrigeration systems, domestic air conditioning equipments, and industrial air-cooled and radiation-cooled
condensers. A complete analysis of horizontal tube condenser requires development of reliable predictive
methods which can be used to evaluate heat transfer and pressure gradient within the operating range of these
condensers. The investigation of condensation heat transfer and pressure gradient is the aim of this study. Chato
[1] developed an analytical model for gravity-driven heat transfer of condensation. In his investigation heat
transfer through the accumulated condensate flow at the bottom of tube is neglected. Rosson and Myers [2] and
Jaster and Kosky [3] have developed a method that heat transfer through the accumulated condensate flow at the
bottom of tube takes into account. Rosson and Myers relation for heat transfer through the accumulated
condensate flow is based on laminar-turbulent Lockhart Martinelli parameter and Reynolds number of liquid
phase, while Jaster and Kosky relation for heat transfer through the accumulated condensate flow is found on
turbulent -turbulent Lockhart Martinelli parameter, Reynolds number of liquid phase, liquid Prandtl and Froude
number. Kenuchi Hashizume et al [4] investigated condensate subcooling near the tube exit during horizontal in
tube condensation both experimentally and analytically. Test fluids used were Refrigerants R113, R11, R114,
and R12. Analysis is based on nonequilibrium heat transfer model which predicts experimental data. Y. Chen
and G. Kocamustafaogullari [5] investigated condensation heat transfer inside horizontal tubes for stratified,
concurrent two phase flow of vapor and liquid. Their analysis took to account the effects of interfacial shear,
axial pressure gradient and saturation temperature level. The innovation of this work is considering the effect of



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interfacial heat transfer as well as shear stress in addition to heat transfer through the accumulated condensate
flow through the development of the governing equations.

                                                          Physical Model




The physical system of this study is illustrated in Fig. 1. The present analysis considers a horizontal plain tube
with constant wall temperature, Tw Saturated vapor enters the tube at Z=0 and flows in Z-direction while it
condenses on the tube's surface. It is assumed that vapor temperature along the tube remains constant, while
liquid temperature drops to a subcooled condition. This is because of heat transfer between liquid phase and
tube wall.




    Figure 1 Schematic scheme of a two phase flow of liquid and vapor through a horizontal smooth tube with interfacial heat transfer

    A. CONTINUITY EQUATION

In general, the continuity equation for each phase is presented as:

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Sadegh Torfi, Ali Ebrahimi


    1.    GA

Where      is rate of vapor condensation at the wall per unit length and is obtained as:

    2.

Where       heat is transfer coefficient between the wall surface and vapor,            the wall perimeter,    the latent
heat of evaporation and       is the saturation temperature.

  is the rate of vapor condensation per unit length at the liquid-vapor interface and is obtained from:

    3.

The heat transfer coefficient is obtained by Nusselt [6]:

                                                  ⁄
                                  (   )
    4.                    [                   ]

This equation is identical to the relation obtained from the classic Nusselt theory for the evaluation of the mean
heat transfer coefficient for laminar condensation on the surface of a horizontal tube of diameter d, except that
the multiplying prefactor has been changed. Chato[1] assumed the value of 0.76 for           in Eq. (4). Later, Jaster
                                   ⁄
and Kosky [3] proposed                as a void fraction to account for the variation with the void fraction. Rosson
and Myers [2] suggested                       .

  is condensate vapor mass regard to interfacial heat transfer which is obtained from:

    5.

  is the average interfacial heat transfer coefficients. Lim and Bankoff [7] proposed for a rectangular duct rate.
Two correlations are:

    6.                                                , rough interface

    7.                                                 ,wavy interface

    B. LIQUID PHASE MOMENTUM EQUATION

Momentum balance for the liquid phase yields:

    8.            [           ]

The interfacial shear stress is defined as:

    9.

Ellis and Gay [8] suggested interfacial friction factor for smooth interface as

    10.

Cohen and Hanratty [9] proposed                         for turbulent liquid-turbulent gas interface with two-dimensional
small amplitude waves.




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      C. VAPOR PHASE MOMENTUM EQUATION

FOR VAPOR PHASE MOMENTUM EQUATION IS DERIVED BY:

      11.         *      +

      D. LIQUID PHASE ENERGY EQUATION

THE LIQUID PHASE ENERGY EQUATION IS :

      12. -GA                                                (         )

JASTERAND       KOSKY    HAVE EXPRESSED HEAT TRANSFER COEFFICIENT FOR THE ACCUMULATED LIQUID AS
FOLLOWS:


      13.

WHERE

      14.        √

      is turbulent-turbulent Lockhart Martinelli parameter that has been defined as:

      15.        ( )     ( )    (    )

For

      16.
      17.

Where

      18.

                                                  Solution procedure

The system of equations is comprised of Eqs. 1, 8, 11 and 12 including five unknown variables such as X, ɑ,Tf,
Pv and Pf . Neglecting surface tension and assuming Pv = Pf. The general form of the system of equation may be
modified as:

      19.

Where                                 i=1,2,3,4

After some manzipulation, the system of equations is
outlined as:
Vapor quality gradient:

      20.

Vapor void fraction Gradient:



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Sadegh Torfi, Ali Ebrahimi



    21.          *                                              *                +                  +    [       ]

Liquid subcooled temperature gradient:

    22.                [     *        (                )                    +               (           )]

Pressure gradient:

    23.          *                        +*       +       *               +,                                *

            +                    -    [        [               ]]


Interfacial vapor condensation rate:

    24.




                                                       Results and discussions

Fig. 2 shows numerical stability versus ΔZ value for subcooled temperature. According to the results it is
obvious that for ΔZ=0.01m, the numerical results are extremely stable. So ΔZ=0.01m is used in this study.




                                     Figure 2 the numerical Results stability versus ΔZ dimension

Fig. 3 and 4 show a typical example of liquid subcooled temperature plotted as a function of Z at various inlet
vapor qualities and is compared with experimental data.

The data are those of Hashizume et al. [5] for R-114a in a 14.8 mm adiabatic tube. According to Fig. 3 and Fig.
4 it is obvious that the numerical method results are consistent with experimental data.




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                         Figure 3 Liquid subcooled temperature variations as a function of Z for R134a




                         Figure 4 Liquid subcooled temperature variations as a function of Z for R134a

Fig. 5 exhibits the variation of subcooled temperature in Z-direction for various wall temperatures for R114 in a
horizontal plain tube during the condensation process. In this case, vapor saturation temperature is 50oC, vapor
mass velocity is 100 Kg/m2.s and tube diameter is 8mm while wall temperature vary form 35 to 45o K As the
diagram shows, the liquid phase subcooled temperature increases while the tube wall temperature decreases.
This behavior arises according to heat transfer between liquid phase and tube.




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                 Figure 5 Variation of subcooled temperature in Z-direction at various wall temperatures for R114

Fig. 6 shows an example of total heat transfer coefficient variations plotted as a function of vapor quality.
Moreover the results are compared with experimental data. The data are those of Cavallini et al. [10] for R125
and R22 in an 8 mm horizontal plain tube. According to Fig. 6 it is clear that the numerical method results and
experimental data are in good agreement.




                  Figure 6 Variation of heat transfer as coefficient as function of vapor quality for R22 and R125

Fig. 7 shows the variation of pressure gradient in Z-direction at various tube diameters for R114. As it is
expected, the pressure gradient and its corresponding variation slope increases dramatically with decreasing of
tube diameter. The reason for this behavior is that the velocity of each phase, shear stresses between each phase
and tube wall increases while the diameter decreases.




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                     Figure 7 Variation of pressure gradient in Z-direction at various tube diameters for R114

The effect of non-thermal equilibrium assumption on vapor quality is illustrated in fig. 8 for two mass velocities.
Fig.9 shows the effect of non-thermal equilibrium assumption on vapor void fraction for two mass velocities.
These results are obtained by two methods that one, neglects liquid phase subcooled while the other does not. In
last case result is obtained by solving Eqs. 1, 8 and 11 while Eq. 1 redefine as:

    25.

According to the results, it is clear that neglecting subcooling of liquid phase is reasonable in high vapor quality
regions while in low vapor quality regions the effect of liquid phase subcooling is significant.




                              Figure 8 variation of vapor quality at various mass velocities for R114




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                                                        Conclusion

Through this study, the governing equations for stratified two phase flow of liquid and vapor in a horizontal tube
have been developed considering interfacial heat transfer and shear stress. These first order differential
equations were solved numerically using 4th order Runga-Kutta method. Pressure gradient, heat transfer
coefficient, vapor quality and void fraction, liquid phase subcooled temperature, and their corresponding
variations throughout the tube were investigated. The results are in good consistency with the experimental data.




                         Figure 9 variation of vapor void fractions at various mass velocities for R114

In addition, the results show that neglecting liquid phase subcooling is reasonable in high vapor quality regions
while in low vapor quality regions the effect of liquid phase subcooling is significant. Results show that for
condensation of R114 in 8 mm diameter horizontal tube, after 5.4 mm from tube inlet, with assuming
equilibrium condensation vapor quality is 0.05 and vapor void fraction is 0.21 while with assuming equilibrium
condensation vapor quality is zero and vapor void fraction is 0.05.

                                                        References

1.   Chato JC. Laminar condensation inside horizontal and inclined tubes. ASHRAE J , 4, 52–60 (1962).

2.   Jaster H, Kosky PG. Condensation in a mixed flow regime. Int J Heat Mass Transfer, 19, 95–99 (1976).

3.   Rosson HF, Meyers JA. Point of values of condensing film coefficients inside a horizontal tube. Chem Eng
     Prog SympSeries, 61, 190–199 (1965).

4.   K. Hashizume, N. Abe, Condensate subcooling near tube exit during horizontal in-tube condensation, Heat
     transfer engineering, 13, 13-27 (1992).

5.   I.Y. Chen, G. Kocamustafaogullari, Condensation heat transfer studies for stratified, cocurrent two phase
     flow in horizontal tubes, International journal of heat and mass transfer, Issue 6 , 1133-1148 (1987)

6.   W. Nusselt, Die Oberflachenkandensation des Wasser Dampfes, Z. ver. dt. Ing. 60, 541-575 (1916).

7.   I. S. Lim, S. G .Bankoff, Cocurrent steam-water ow in a horizontal channel. Report, Department of
     Mechanical and Nuclear Engineering, Northwestern University, NUREG/CR R1, R2, R4. (1981).


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8.   S. R. Ellis, B. Gay, The parallel flow of two Fluid streams: interfacial shear and fluid-fluid interaction,
     Transactions of the Institution of Chemical Engineers, 37, 506-213 (1959).

9.   L. S. Cohen, T. J. Hanratty, Effects of waves at gas/liquid interface on turbulent air flow. Journal of Fluid
     Mechanics, Vol. 31, 467-479 (1968).

10. A. Cavallini, G. Censi, D. Del Col, L. Doretti, G.A. Longo, L. Rossetto, Experimental investigation on
    condensation heat transfer and pressure drop of new HFC refrigerants (R134a, R125, R32, R410A, R236ea)
    in a horizontal smooth tube, International Journal of Refrigeration, 24 ,73-87 (2001).




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