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Proceedings of CHT-08 ICHMT International Symposium on Advances in Computational Heat Transfer May 11-16, 2008, Marrakech, Morocco CHT-08-xxx PHASE CHANGE MODEL FOR TWO-PHASE FLUID FLOW BASED ON THE VOLUME OF FLUID METHOD Bitan Shu § , Frank Dammel and Peter Stephan Technische Universität Darmstadt, Department of Mechanical Engineering § Corresponding author. Tel: +49 6151 16 6178 Email: [email protected] ABSTRACT In this paper, a model for the phase change in two-phase fluid flow is presented. The position of the interface is captured implicitly with the volume of fluid (VOF) method. The mass conservation equation and the Navier-Stokes equations are solved over the entire computational domain. Additionally, the energy equation is solved in the area which is occupied by vapor, while the temperature in the liquid and at the interface is assumed to be at a constant saturation temperature. Volumetric source terms are derived in the framework of the finite volume method and introduced into the conservation equations to model the phase change. Test simulation of the 1D Stefan-problem agrees perfectly with the analytical result. The second test case is the 2D axisymmetric film boiling. The results of the numerical simulations agree well with the result calculated with the correlation. 1. INTRODUCTION Two-phase fluid flow can be very often observed in nature and in technical devices, e.g., the smoke in a chimney, liquid in a blender, or boiling flow in heat exchangers. In heat exchangers with boiling flow, a large amount of heat can be transferred at a low temperature difference because of the phase change of the fluid. However, the physics behind this are still not fully understood, although much work has been carried out in this research field. With computational simulations, some of the phenomena that are difficult to observe experimentally can be successfully studied. An important issue in the simulation of a boiling fluid is the motion of the free phase interface. For this free phase interface, there are two different numerical approaches: Lagrangian- and Eulerian- based approaches. The Lagrangian-based approaches work with an adapted grid and have limitations in the case of a large interface motion and changes in interface topology. The Eulerian- based approaches, e.g., the volume of fluid (VOF) method and level set method, overcome this limitation with a fixed grid and an embedded function with which the phase interface is implicitly captured. While the level set method does not conserve mass during the advection of the equation without special measures [Sussman 2000], the VOF method has the advantage of ensuring the mass conservation. To model the phase change in the fluid flow, a theoretical equation for the mass transfer, which is related to the heat transfer at the phase interface, is introduced in [Juric and Tryggvason 1998]. In this approach, the evaporation rate depends on the temperature at the interface. A parameter called

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Proceedings of CHT-08 ICHMT International Symposium on Advances in Computational Heat Transfer

May 11-16, 2008, Marrakech, Morocco

CHT-08-xxx

PHASE CHANGE MODEL FOR TWO-PHASE FLUID FLOW BASED ON THE VOLUME OF FLUID METHOD

Bitan Shu §, Frank Dammel and Peter Stephan Technische Universität Darmstadt, Department of Mechanical Engineering

§Corresponding author. Tel: +49 6151 16 6178 Email: [email protected] ABSTRACT In this paper, a model for the phase change in two-phase fluid flow is presented. The position of the interface is captured implicitly with the volume of fluid (VOF) method. The mass conservation equation and the Navier-Stokes equations are solved over the entire computational domain. Additionally, the energy equation is solved in the area which is occupied by vapor, while the temperature in the liquid and at the interface is assumed to be at a constant saturation temperature. Volumetric source terms are derived in the framework of the finite volume method and introduced into the conservation equations to model the phase change. Test simulation of the 1D Stefan-problem agrees perfectly with the analytical result. The second test case is the 2D axisymmetric film boiling. The results of the numerical simulations agree well with the result calculated with the correlation.

1. INTRODUCTION

Two-phase fluid flow can be very often observed in nature and in technical devices, e.g., the smoke in a chimney, liquid in a blender, or boiling flow in heat exchangers. In heat exchangers with boiling flow, a large amount of heat can be transferred at a low temperature difference because of the phase change of the fluid. However, the physics behind this are still not fully understood, although much work has been carried out in this research field. With computational simulations, some of the phenomena that are difficult to observe experimentally can be successfully studied. An important issue in the simulation of a boiling fluid is the motion of the free phase interface. For this free phase interface, there are two different numerical approaches: Lagrangian- and Eulerian-based approaches. The Lagrangian-based approaches work with an adapted grid and have limitations in the case of a large interface motion and changes in interface topology. The Eulerian-based approaches, e.g., the volume of fluid (VOF) method and level set method, overcome this limitation with a fixed grid and an embedded function with which the phase interface is implicitly captured. While the level set method does not conserve mass during the advection of the equation without special measures [Sussman 2000], the VOF method has the advantage of ensuring the mass conservation. To model the phase change in the fluid flow, a theoretical equation for the mass transfer, which is related to the heat transfer at the phase interface, is introduced in [Juric and Tryggvason 1998]. In this approach, the evaporation rate depends on the temperature at the interface. A parameter called

the evaporation coefficient is included; however, its measurement is difficult. Other authors [Welch 2000, Emaeeli 2004] assume that the interface has a saturation temperature corresponding to the vapor pressure and derive the evaporation rate from consideration of the heat fluxes on both sides of the phase interface. In this work, we follow this approach and determine the heat flux at the interface explicitly at every time step. From this heat flux, the evaporation rate, contained as a source term in the conservation equations, is calculated. The source terms are volumetric which is more suitable for introduction into the conservation equations in the framework of the finite volume method (FVM). This model is implemented in OpenFOAM, which is written with the programming language C++, and provides a flexible and extensive program library to solve many CFD tasks [OpenFOAM 2007].

2. MATHMATICAL FORMULATION AND NUMERICAL SOLUTION To derive the governing equation for the mass conservation during evaporation, mass changes in both phases are considered. For each phase in general, we have a change of the mass:

( )g

g gev gV

Dmm dV

Dt t

ρρ

∂= = + ∇ ⋅

∂∫ uɺ (1)

and

( )l

l lev lV

Dmm dV

Dt t

ρ ρ∂= − = + ∇ ⋅∂∫ uɺ (2)

In the above equations, D

Dt t

∂= + ⋅∇∂

u denotes the material derivative. The change of mass per

time unit due to evaporation evmɺ is negatively signed for the liquid because the mass of liquid

decreases, and vice versa for the gas phase. Assuming that the density of liquidlρ and the density of gasgρ are constant, and taking advantage

of the relation

V S

dV dS∇ ⋅ = ⋅∫ ∫u u n (3)

we get the following equations for each phase:

g

ev

Sg

mdS

ρ⋅ =∫ u n

ɺ (4)

and

l

ev

Sl

mdS

ρ⋅ = −∫ u n

ɺ (5)

with n , the normal vector at the phase interface. Assuming that the velocities of both phases are the same in the tangential direction of the phase interface, we can add the two equations above and get the following equation for the entire computational cell which contains both phases:

1 1

evSg l

dS mρ ρ

⋅ = −

∫ u n ɺ (6)

The differential formulation of Eq. (6) reads

1 1

evg l

mρ ρ

′′′∇ ⋅ = −

u ɺ (7)

The volumetric evaporation rate evm′′′ɺ has the units [kg/(m3s)]. This equation holds for the entire

computational domain. If there is no phase change in a cell, then evm′′′ɺ has value zero.

Note that the mass conservation equation in [Welch 2000] has a similar formulation for the source term; however, with our formulation, the velocity of the phase interface does not have to be determined explicitly. The Navier-Stokes equation for the momentum transport reads

( ) ( )( )Tp

t

ρ ρ ρ µ σ κ γ∂ + ∇ ⋅ = −∇ + + ∇ ⋅ ∇ + ∇ + ∇ ∂u

uu g u u (8)

Here , , ,µ σ κ and γ are the fluid viscosity, surface tension, curvature of the phase interface, and the liquid volume fraction of the fluid. This so-called volume of fluid is defined as the fraction of the volume of liquid phase to the total volume in a computational cell:

lV

Vγ = (9)

With this equation, we can determine the density of the fluid in a computational cell: ( )l g gρ ρ ρ γ ρ= − + (10)

The viscosity is approximated with: ( )l g gµ µ µ γ µ= − + (11)

and the curvature is determined with:

γκγ

∇= ∇ ⋅∇

(12)

In Eq. (8), the CFS model is used for the surface tension [Brackbill 1992]. Note that there is no source term in Eq. (8) because the equation is valid for the entire computational cell. To capture the position of the interface, the VOF method is used. We derive the equation for the advection of the VOF function using the general mass conservation equation:

( ) 0t

ρ ρ∂ + ∇ ⋅ =∂

u (13)

We substitute Eq. (10) into Eq. (13) and use Eq. (7), from which we get the following equation for the advection of volume of fluid:

( ) ev

l

m

t

γ γρ′′′∂ + ∇ ⋅ = −

∂u

ɺ (14)

During simulation of the heat transfer in the boiling flow, we assume that one of the phases has the saturation temperature, e.g., the liquid in case of film boiling. We neglect the influence of the pressure on the saturation temperature. The heat equation reads then as follows:

( ) in gas phase

in liquid phase and at interface

p

sat

Tc T k T

t

T T

ρ ∂ + ⋅∇ = ∇ ⋅ ∇ ∂

=

u (15)

Here, , , andsat pT T c λ are the temperature in a computational cell, saturation temperature, specific

heat, and heat conductivity, respectively. This equation is specified for the case in which the heat is conducted in the gas phase. The mass change due to phase change is directly related to the heat flux at the interface on both sides:

( )g l

evlg

q qm

h

− ⋅′′ =

nɺ ɺ

ɺ (16)

Here, evm′′ɺ is the mass transfer through the phase interface, with the units [kg/m2s], gqɺ , lqɺ and lgh

are the heat flux in the gas phase and liquid phase to the interface, and the latent heat of the fluid, respectively. In this work, 0lq =ɺ . As we discretize the equations in the framework of FVM, a

volumetric source term is more suitable for implementation in Eq. (7) and (14):

( )g l I

evlg

q q Am

h V

− ⋅′′′ =

nɺ ɺ

ɺ (17)

where IA means the surface of the phase interface, and V denotes the volume of the computational

cell. Note that the source term appears only in the cells in which a segment of interface is contained, and otherwise it takes on the value of zero. The material properties in the above equations are approximated as follows: ( )p pl p g p gc c c cγ= − + (18)

( )l g gk k k kγ= − + (19)

In summary, we have Eq. (7), (8), (14), and (15) to model the fluid flow with phase. We start every time step with the calculation of the heat transfer, Eq. (15). With the temperature distribution at the phase interface, we can determine the heat flux which leads to phase change. The source term in Eq. (7) and (14) is calculated from these heat fluxes with Eq. (17). The next step is the implicit capture of the phase interface with Eq. (14). A special discretization scheme is used to solve Eq. (14). Details are referred to [Rusche 2002, OpenFOAM 2007]. The material properties, namely the density and the viscosity, are updated according to the new position of the interface. At last, the flow dynamics are simulated with Eq. (7) and (8), which are solved sequentially with the PISO method [Jasak 1996, Issa 1986].

Figure 1. Setup for Stefan-Problem

3. RESULTS AND DISCUSSION 3.1. The Stefan-Problem To test the model, we first simulate the 1D Stefan-problem which is considered in [Welch 2000]. In this problem, a thin film of the gas phase rests at the wall in the beginning. The gas is heated by the wall with a constant temperature. The liquid, adjacent to the gas phase on the other side, is at the saturation temperature. Figure 1 shows the setup of this problem. The length of the computation domain is set to 40 mm.

To enable the comparison, we describe the analytical solution of this problem below [Welch 2000]. The temperature profile in the liquid phase is uniform and the energy equation in the gas phase is expressed as:

2

2for 0 ( )

p

T Tx t

t c x

λ δρ

∂ ∂= ≤ ≤∂ ∂

(20)

where ( )tδ is the coordinate of the phase interface. The boundary conditions of the problem are:

( ( ), )

( 0, )sat

wall

T x t t T

T x t T

δ= == =

(21)

The velocity of the interface can be calculated with the heat flux at the interface as follows:

( )

gs

x tg lg

k Tu

h x δρ =

∂= −∂

(22)

The analytic solution of the Stefan-problem is given with the following equations [Welch 2000]:

( ) 2t tδ λ α= (23)

where, g

g pg

k

ρ= , erf( )x is the error function, and λ is a solution to the transcendental equation:

Figure 2. Interface position as a function of time

2 ( )exp( ) erf ( ) p wall sat

lg

c T T

hλ λ λ

π−

= (24)

( , ) erferf ( ) 2sat wall

wall

T T xT x t T

tλ α− = +

(25)

In Figure 2, the results, with the material properties given in Tab. 1 and a wall temperature that is 25 K higher than the saturation temperature, are plotted. The results of the simulation agree perfectly with the analytical results. The grid spacing used in the simulation is 0.1 mm. The result with a grid spacing of 0.05 mm shows no considerable difference.

Table 1 Material Properties used in Stefan-Problem

Properties Phase 1 (liquid) Phase 2 (gas)

Viscosity [Pa·s] 610− 51.48 10−⋅

Density [kg/m3] 1000 1

Thermal conductivity [W/(m⋅ K)] 0.589 0.0257

Specific heat [J/(kg⋅ K)] 4181 1007

Latent heat [J/kg] 32200 10⋅ 3.2. Film Boiling On a large horizontal surface, if the wall superheat is high enough, the entire surface is immersed in vapor. This phenomenon is called film boiling [Van Carey 1992]. There are various correlations to

predict the heat transfer coefficient in the case of film boiling. The correlation of Berenson [1961] is based on the assumption that the vapor bubbles are spaced in a square pattern separated by a distance equivalent to the most dangerous Taylor wavelength

( )

1/ 2

0

32

l gg

σλ πρ ρ

=

(26)

The heat transfer coefficient due to forced convection is given by

( )

( )( )

1/ 41/ 23

2

g g l g lg l g

g wall sat

k g h gh C

T T

ρ ρ ρ ρ ρµ σ

′− − = −

(27)

where, C2 = 0.425 and 0.5 ( )lg lg pg wall sath h c T T′ = + − . With these two equations, the Nusselt number

can be obtained: 0 / gNu h kλ= (28)

We consider a two-phase fluid with the properties given in Table 2 which are also used in [Welch 2000]. The computational domain is 2D axisymmetric and has a radius 0 / 2r λ= . The height is set

to 4 r× . At the beginning, the interface has an initial position which is given by the following equation [Esmaeeli 2004]: ( )0 0 00

0.013 0.05 cos 2 /i ty xλ λ π λ

== + (29)

Here, x denotes the coordinate of the computational domain, and the axis is placed at 0x = . The temperature in the liquid phase and at the interface is initialized to the saturation temperature. In the gas phase, it is initialized to increase linearly from the interface to the heated wall. In Figure 3, the spatially averaged Nusselt numbers calculated with different grid resolutions are plotted as a function of time, in comparison to the one calculated by the correlation of Berenson [1961]. The wall temperature is 10 K higher than the saturation temperature. The coarse grid has 64 × 256 cells, the medium grid 96 × 384 cells, and the fine grid 128 × 512 cells. The calculated

Table 2 Material Properties Used in Film Boiling

Properties Phase 1 (liquid) Phase 2 (gas)

Viscosity [Pa·s] 0.1 0.005

Density [kg/m3] 200 5

Thermal conductivity [W/(m⋅ K)] 40 1.0

Specific heat [J/(kg⋅ K)] 400 200

Latent heat [J/kg] 310 10⋅

Surface tension [N/m] 0.1

Figure 3. The Nusselt numbers with different grid resolutions as a function of time in case of wall superheat 10 K higher that the saturation temperature, compared to the result from the correlation of Berenson [1961].

Figure 4. The interface positions at different times in case of wall superheat 10 K higher than the saturation temperature. The temperature difference between the isotherms is 2 K.

temporally averaged Nusselt numbers for these different grid resolutions are 21.790, 20.744, and 19.867, respectively. Therefore, it can be concluded that further refinement of the grid leads only to a small change in the Nusselt number. For the fine grid, the time average of the Nusselt number, which is calculated from the first to the last valley in Figure 3 (corresponding to the time interval 0.31 to 3.78 second), is also plotted. The calculated Nusselt number shows a good agreement with the one from the correlation. The local Nusselt number is calculated in the post-processing of the simulation with the following equation:

0

0wall sat y

TNu

T T y

λ

=

∂=− ∂

(30)

In Figure 4, the formulation of the phase interface and the temperature levels at different times are

Figure 5. The Nusselt numbers from the simulation with wall superheat 15 K higher than saturation temperature as a function of time, compared to the result from the correlation of Berenson [1961].

Figure 6. The interface positions at different time in case of wall superheat 15 K higher than the saturation temperature. The temperature difference between the isotherms is 3 K.

plotted. Shortly before the detachment of a bubble, the Nusselt number reaches a maximum because the velocity is large and the forced convection is strong. After the detachment of a bubble, the Nusselt number reaches a minimum because of the weak forced convection in this stage, although the temperature gradient from the heated wall to the phase interface is large. Directly after detachment, the vapor in the bubble is still superheated. Due to the strong convection on both sides of the bubble, it reaches the saturation temperature very quickly. Another simulation with a wall superheat 15 K higher than the saturation temperature is presented in Figure 5. To save CPU time, a medium grid resolution (96 × 384 cells) is used. Because of the higher wall superheat, the evaporation is more intensive and the Nusselt number shows more oscillations. The annulus of the gas phase in the middle of the computational domain at the wall can be formulated instead of a bubble at the axis (t = 0.8 sec in Figure 6).

4. CONCLUSION A model for the phase change in two-phase fluid flow is presented. The conservation equations for fluid flows with phase change are derived from the general conservation equations in the framework of the finite volume method. 2D axisymmetric numerical simulations are carried out. The Nusselt numbers from the simulation agree well with the results calculated with the correlation of Berenson [1961]. The presented model is also valid for 3D simulation in principle, and thus can be further developed to treat more complex boiling systems in future.

REFERENCES Berenson, P.J. [1961], Film-Boiling Heat Transfer from a Horizontal Surface, Journal of Heat Transfer, Vol. 83, pp 351-358. Brackbill, J.U., Kothe, D.B. and Zemach, C. [1992], A Continuum Method for Modeling Surface Tension, Journal of Computational Physics, Vol. 100, pp 335-354. Esmaeeli, A. and Tryggvason, G. [2004], Computations of Film Boiling. Part I: Numerical Method, International Journal of Heat and Mass Transfer, Vol. 47, pp 5451-5461. Issa, R.I. [1986], Solution of the Implicitly Discretised Fluid Flow Equations by Operator-Splitting, Journal of Computational Physics, Vol. 62, pp 40-65. Jasak, H. [1996], Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows, PhD Thesis, Imperial college of Science, Technology & Medicine, Department of Mechanical Engineering, University of London. Juric, D. and Tryggvason, G. [1998], Computations of Boiling Flows, International Journal of Multiphase Flow, Vol. 24, No. 3, pp 387-410. OpenFOAM [2007], Documentation of OpenFOAM. Available from: http://www.openfoam.com [Accessed December 2007]. Rusche, H. [2002], Computational Fluid Dynamics of Dispersed Two-Phase Flows at High Phase Fractions, PhD Thesis, Imperial college of Science, Technology & Medicine, Department of Mechanical Engineering, University of London. Sussman, M. and Puckett, E.G. [2000], A Coupled Level Set and Volume-Of-Fluid Method for Computing 3D and Axisymmetric Incompressible Two-Phase Flows, Journal of Computational Physics, Vol. 162, pp 301-337. Van Carey, P. [1992], Liquid-Vapor Phase-Change Phenomena - An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment, Taylor & Francis, London Welch, S.W.J. and Wilson, J. [2000], A Volume Of Fluid Based Method for Fluid Flows with Phase Change, Journal of Computational Physics, Vol. 160, pp 662-682.