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Numerical investigation of two-phase flow in a horizontal porous evaporator with localised heating using non-Darcian flow and two equations model

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Abstract

In this article, a new mathematical model, based on the modified enthalpy formulation of Two-Phase Mixture Model (TPMM), is developed by considering non-Darcy flow and Local Thermal Non-Equilibrium (LTNE) conditions to describe numerically the two-dimensional problems of the complete evaporation process inside an asymmetrically heated porous evaporator. The governing equations have been discretised using Finite Volume Method (FVM) on a fixed non-staggered grid layout. The in-house code has been validated against experimental data and the results show good agreement between them. Three different models have been employed for the partitioning of the wall heat flux. Only the effect of the heat flux has been analysed, while all other parameters and properties have been kept fixed during the present investigation. The numerical simulations indicated that the locally high heat flux significantly influences the working fluid flow performances, and lead to extend the superheated vapour region towards the exit of the evaporator. The results also show that the two-phase zone significantly expanded in the axial direction and does not occupy a large portion inside the channel due to the energy transport through the two-phase region is significantly higher in the flow direction as compared to the transverse direction, caused by the decrease in the effective diffusion coefficient in this region and higher axial velocity. Numerical analysis reveal that the different models for the partitioning of the wall heat flux have almost no influence on the overall predictions, except in the vicinity of the wall. The results clearly demonstrate that the numerical simulations with the modified h-formulation by considering non-Darcy law and LTNE model are indispensable for the problem of complete evaporation process, particularly under the high both heat flux and inlet velocity conditions.

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Notes

  1. Difference of about 3.5 °C has been reported by Alomar et al. [6] for the maximum wall heat flux case.

  2. This assumption is valid only for negligible pressure drop through the evaporator, which is true for low mass flow rate applications, as in the present case.

  3. This model assumes that the solid and the fluid phases are in thermal non-equilibrium with each other and they coexist at different temperature in all three regions.

  4. This term an important factor that has the largest impact on growth of the two-phase region, especially in the case that the temperature of the solid phase exceeds the saturation temperature.

  5. α = l and α = v for liquid and vapour phases, respectively.

  6. Based on the pore diameter, the phase velocities and the properties of the local phase.

  7. The numerical values of the recommended parameters may be found in Alomar et al. [2] and hence they are not repeated here for the sake of brevity.

  8. The exit length has been carefully chosen in order to avoid the influence of exit boundary condition on the internal solution.

  9. The lower and the upper values correspond to the properties of vapour (\( {k}_f^{\ast }={k}_v^{\ast } \)) and liquid (\( {k}_f^{\ast }={k}_l^{\ast } \)) phases, respectively [6].

Abbreviations

a s :

Specific surface of the porous medium, 1/m

b :

Body force vector per unit mass, m/s2

\( \tilde{\mathbf{b}} \) :

Normalized body force vector per unit mass =b/g

C p :

Specific heat, J/kgK

d p :

Characteristic pore size of porous matrix, m

D :

Capillary diffusion coefficient, m2/s

ev :

Total evaporated volume fraction

f :

Hindrance function

Fr :

Froude number \( ={u}_{in}/{u}_g={u}_{in}/\sqrt{g\;W}=R{e}_{in}/R{e}_g \)

g :

Acceleration due to gravitational vector, m/s2

h :

Specific enthalpy, J/kg

h fg :

Latent heat of vaporization =hv, sat − hl, sat, J/kg

h sα :

Convective heat transfer coefficient in the pores, W/m2K

j :

Diffusive mass flux vector, kg/m2s

J :

Capillary pressure function

krl, krv :

Relative permeabilities for liquid and vapour, respectively

k :

Thermal conductivity, W/mK

K :

Permeability of porous matrix, m2

l :

Length of individual segments, m

L :

Length of the porous channel, m

n :

Exponent of saturation in the expression for relative permeabilities

p :

Effective pressure, Pa

p c :

Capillary pressure, Pa

Pe :

Peclet number =uinW/α = RePr

Pr :

Prandtl number =μ Cp/k

\( {\dot{q}}^{\hbox{'}\hbox{'}} \) :

Heat flux, W/m2

Q ∗ :

Normalized heat flux \( ={\dot{q}}^{\hbox{'}\hbox{'}}W/\mu {h}_{fg} \)

\( {\dot{q}}_{sf}^{\hbox{'}\hbox{'}\hbox{'}} \) :

Heat exchange term between fluid and solid phases, W/m3

Re :

Reynolds number =uinW/ν

Re g :

Gravitational Reynolds number=ugW/νl

Re p :

Reynolds number based on pore diameter and local phase properties

Re K :

Fluid Reynolds number based on the length scale of the permeability

s :

Liquid saturation

T :

Temperature, K

u :

Velocity component vector, m/s

u g :

Gravitational velocity vector \( \sqrt{\mathbf{g}\;W},\mathrm{m}/\mathrm{s} \)

W :

Height of the duct, m

x, y :

horizontal and vertical Coordinates, respectively, m

α :

Thermal diffusivity =k/ρ Cp, m2/s

β :

Isobaric expansion coefficient, K−1

ΔT :

Temperature difference for relaxation of Γh is in the single-phase regions, K

Δρ :

Difference in densities=(ρl − ρv), kg/m3

γ h :

Advection correction coefficient

Γ h :

Diffusion coefficient in enthalpy equation, kg/m s

ε :

Porosity

λ :

Relative mobility

μ :

Dynamic viscosity, kg/ms

ν :

Kinematics viscosity, m2/s

ρ :

Density, kg/m3

σ :

Surface tension, N/m

\( \tilde{\sigma} \) :

Normalized surface tension coefficient \( ={\rho}_l\sigma W/{\mu}_l^2 \)

eff :

Effective

f :

Fluid

i, e, h :

Inlet, exit and heated, respectively

in :

Inlet

k :

Kinetic

l :

Liquid

max:

Maximum value

min:

Minimum value

s :

Solid

sat :

Saturation

v :

Vapour

w :

Wall

*:

Dimensionless

References

  1. Alazmi B, Vafai K (2002) Constant wall heat flux boundary conditions in porous media under local thermal non-equilibrium conditions. Int J Heat Mass Transf 45:3071–3087

    Article  Google Scholar 

  2. Alomar OR, Mendes MAA, Trimis D, Ray S (2014) Numerical simulation of complete liquid-vapour phase change process inside porous media using smoothing of diffusion coefficient. Int J of Thermal Sciences 86:408–420

    Article  Google Scholar 

  3. Alomar OR, Mendes MAA, Trimis D, Ray S (2014) Simulation of complete liquid-vapor phase change inside divergent porous evaporator. Int J of Material, Mechanics and Manufacturing 2(3):223–229

    Google Scholar 

  4. Alomar OR, Trimis D, Ray S (2015) Numerical simulation of complete liquid-vapour phase change process inside an annular porous evaporator, 10th international conference on Freiberg-St. Petersburg Colloquium of Young Scientists

  5. Alomar OR, Mendes MAA, Trimis D, Ray S (2015) Simulation of complete liquid-vapour phase change inside porous evaporator using local thermal non-equilibrium model. Int J of Thermal Sciences 94:228–241

    Article  Google Scholar 

  6. Alomar OR, Mendes MAA, Trimis D, Ray S (2017) Numerical simulation of complete liquid-vapour phase change inside porous media: A comparison between local thermal equilibrium and non-equilibrium models. Int J of Thermal Sciences 112:222–241

    Article  Google Scholar 

  7. Alomar OR, Mendes MAA, Ray S, Trimis D (2018) Numerical investigation of complete evaporation process inside porous media using staggered and non-staggered grid arrangements. Int J of Thermal Sciences 129:56–72

    Article  Google Scholar 

  8. Alomar OR, Mohammed RR, Mendes MAA, Ray S, Trimis D (2019) Numerical investigation of two-phase flow in anisotropic porous evaporator. Int J of Thermal Sciences 135:1–16

    Article  Google Scholar 

  9. Bear J (1972) Dynamics of Fluids in Porous Media. Elsevier, New York

    MATH  Google Scholar 

  10. Borgnakke C, Sonntag RE (2009) Fundamentals of Thermodynamics. John Wiley & Sons, Hoboken

    Google Scholar 

  11. Easterday OT, Wang CY, Zheng P (1995) A numerical and experimental study of two-phase flow and heat transfer in a porous formation with localized heating from below. Proc ASME Heat Transf Fluid Eng Divisions 321:723–732

    Google Scholar 

  12. He F, Wang J (2014) Numerical investigation on critical heat flux and coolant volume required for transpiration cooling with phase change. Energy Convers Manag 80:591–597

    Article  Google Scholar 

  13. He F, Wang J, Xu L, Wang X (2013) Modeling and simulation of transpiration cooling with phase change. Applied Thermal Engg 58:173–180

    Article  Google Scholar 

  14. Incropera FP, Dewitt DP, Bergman TL, Lavine AS (2007) Fundamentals of Heat and Mass Transfer. John Wiley & Sons

  15. Kaviany M (1995) Principles of Heat Transfer in Porous Media, Second edn. Springer-Verlag, New York

  16. Li HY, Leong KC, Jin LW, Chai JC (2010) Transient two-phase flow and heat transfer with localized heating in graphite foams. Int J Therm Sci 49:1115–1127

    Article  Google Scholar 

  17. Li HY, Leong KC, Jin LW, Chai JC (2010) Transient behavior of fluid flow and transfer with phase change in vertical porous media. Int J Heat Mass Transf 53:5209–5222

    Article  Google Scholar 

  18. Li HY, Leong KC, Jin LW, Chai JC (2010) Three-dimensional numerical simulation of fluid flow with phase change heat transfer in an asymmetrically heated porous channel. Int J Therm Sci 49:2363–2375

    Article  Google Scholar 

  19. Lindner F, Mundt C, Pfitzner M (2014) Fluid flow and heat transfer with phase change and local thermal non-equilibrium in vertical porous. Transp Porous Media 106:201–220

    Article  Google Scholar 

  20. Mendes MAA, Ray S, Trimis D (2013) A simple and efficient method for the evaluation of effective thermal conductivity of open-cell foam-like structures. Int J Heat Mass Transf 66:412–422

    Article  Google Scholar 

  21. Mendes MAA, Ray S, Trimis D (2013) Determination of Effective Thermal Conductivity of Open-cell Porous Foams for Arbitrary Working Fluid using Single Point Measurement, Proc. 22nd National & 11th ISHMT-ASME Heat and Mass Transfer Conf., Indian Institute of Technology, Kharagpur

  22. Mendes MAA, Ray S, Trimis D (2014) Evaluation of Effective Thermal Conductivity of Porous Foams in Presence of Arbitrary Working Fluid. Int J Therm Sci 79:260–265

    Article  Google Scholar 

  23. Mendes MAA, Ray S, Trimis D (2014) An Improved Model for the Effective Thermal Conductivity of Open-cell Porous Foams. Int J Heat Mass Transf 75:224–230

    Article  Google Scholar 

  24. Najjari M, Nasrallah SB (2003) Numerical study of boiling in inclined porous layer. J Porous Media 6(1):71–81

    Article  Google Scholar 

  25. Najjari M, Nasrallah SB (2005) Numerical study of the effects of geometric dimensions on liquid-vapor phase change and free convection in a rectangular porous cavity. J Porous Media 8(1):1–12

    Article  Google Scholar 

  26. Najjari M, Nasrallah SB (2006) Liquid-vapor phase-change and mixed convection in a porous layer discretely heated. J Porous Media 9(7):671–681

    Article  Google Scholar 

  27. Najjari M, Nasrallah SB (2008) Effects of latent heat storage on heat transfer in a forced flow in a porous layer. Int J Therm Sci 47:825–833

    Article  Google Scholar 

  28. Najjari M, Nasrallah SB (2009) Heat transfer between a porous layer and a forced flow: influence of layer thickness. Dry Technol 27:336–343

    Article  Google Scholar 

  29. Nield DA, Bejan A (2006) Convection in Porous Media, third edn. Springer, New York

  30. Patankar SV (1980) Numerical Heat Transfer and Fluid Flow. Hemisphere, New York

    MATH  Google Scholar 

  31. Peterson GP, Chang CS (1997) Numerical heat analysis and evaluation for two-phase flow in porous-channel heat sinks, Numer. Heat Transfer, Part A 31(2):113–130

    Article  Google Scholar 

  32. Ray S, Alomar OR (2016) Simulation of liquid-vapour phase change process inside porous media using modified enthalpy formulation. Int J of Thermal Sciences 105:123–136

    Article  Google Scholar 

  33. Scheidegger AE (1974) The Physics of Flow Through Porous Media (3rd Ed.). University of Toronto Press, Toronto

    Google Scholar 

  34. Shi JX, Wang JH (2011) Numerical investigation of transpiration cooling with liquid coolant phase change. Transp Porous Media 87:703–716

    Article  Google Scholar 

  35. Voller VR, Swaminathan CR (1993) Treatment of discontinuity thermal conductivity in control-volume solutions of phase change problems. Numer. Heat Transf. Part B 24:161–180

    Google Scholar 

  36. Voss S, Mendes MAA, Pereira JMC, Ray S, Pereira JCF, Trimis D (2013) Investigation on the thermal flame thickness for lean premixed combustion of low calorific H2/CO mixtures within porous inert media. Proc Combust Inst 34:3335–3342

    Article  Google Scholar 

  37. Wang CY (1997) A fixed-grid numerical algorithm for two-phase flow and heat transfer in porous media, Numer. Heat Transfer, Part B 32:85–105

    Article  Google Scholar 

  38. Wang CY, Beckermann C (1993) A two-phase mixture model of liquid-gas flow and heat transfer in capillary porous media, I. Formulation. Int J Heat Mass Transf 36(11):2747–2758

    Article  Google Scholar 

  39. Wang CY, Cheng P (1997) Multiphase Flow and Heat Transfer in Porous Media. In Advances in Heat Transfer, Academic Press, New York 30: 93–196

  40. Wang JH, Shi JX (2008) Discussion of boundary conditions of transpiration cooling problems using analytical solution of LTNE model. ASME J Heat Transfer 130:014504–1–014504-5

    Google Scholar 

  41. Wang CY, Beckermann C, Fan C (1994) Numerical study of boiling and natural convection in capillary porous media using the two-phase mixture model. Numer. Heat Transfer. Part A 26:375–398

    Google Scholar 

  42. Wei K, Wang J, Mao M (2012) Model discussion of transpiration cooling with boiling. Transp Porous Media 94:303–318

    Article  MathSciNet  Google Scholar 

  43. Xin C, Rao Z, You X, Song Z, Han D (2014) Numerical investigation of vapor–liquid heat and mass transfer in porous media. Energy Convers Manag 78:1–7

    Article  Google Scholar 

  44. Yuki K, Abei J, Hashizume H, Toda S (2008) Numerical investigation of thermo-fluid flow characteristics with phase change against high heat flux in porous media. ASME J Heat Transfer 130:012602–1–012602-12

    Article  Google Scholar 

  45. Zhao TS, Liao Q (1999) Mixed convective boiling heat transfer in a vertical capillary structure heated asymmetrically. J Thermophys Heat Transf 13(3):302–307

    Article  Google Scholar 

  46. Zhao TS, Cheng P, Wang CY (2000) Buoyancy-induced flows and phase-change heat transfer in a vertical capillary structure with symmetric heating. Chem Eng Sci 55:2653–2661

    Article  Google Scholar 

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Authors and Affiliations

  1. Engineering Technical College of Mosul, Northern Technical University, Cultural Group Street, Mosul, Iraq

    Omar Rafae Alomar

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Correspondence to Omar Rafae Alomar.

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Highlights

• Complete phase change process inside porous media has been considered.

• Modified enthalpy formulation has been used including the non-Darcian effects along with local thermal non-equilibrium model.

• Conservation equations have been solved by finite volume method on non-staggered grid layout.

• Two dimensional phase change problem is considered as demonstrative example.

• Smoothing of effective diffusion coefficient is required as remedy.

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Alomar, O.R. Numerical investigation of two-phase flow in a horizontal porous evaporator with localised heating using non-Darcian flow and two equations model. Heat Mass Transfer 56, 1203–1221 (2020). https://doi.org/10.1007/s00231-019-02784-x

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