Eulerian simulation of subcooled boiling flow in straight ... - Springer Link

Journal of Mechanical Science and Technology 27 (5) (2013) 1295~1304 www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-013-0501-4

Eulerian simulation of subcooled boiling flow in straight and curved annuli† Habib Aminfar1, Mousa Mohammadporfard2 and Rasool Maroofiazar1,* 1 Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iran Department of Mechanical Engineering, Azarbaijan Shahid Madani University, Tabriz, Iran

2

(Manuscript Received May 15, 2012; Revised November 5, 2012; Accepted December 3, 2012) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract In this study, two types of water flow, turbulent single-phase flow and low-pressure subcooled boiling flow, in straight and curved horizontal annuli are investigated numerically. The control volume technique is used for discretizing governing equations, the SIMPLEC algorithm for pressure-velocity coupling, and the shear stress transport k-ω model for turbulent flow. A three-dimensional two-fluid model is used for the subcooled boiling flow, the results of which are compared with those of the single-phase flow. The available water boiling experiment results at low pressure are used to validate the numerical results and were found to have good agreement. The inner cylinder surface temperature of the curved annulus in almost all angles is less than that of the straight annulus in both single-phase and subcooled boiling flow. The maxmum and minimum temperatures in the curved annulus occur at defferent points compared to straight annulus ones due to effects of the centrifugal force. Keywords: Centrifugal force; Curved annulus; Low pressure; Subcooled flow boiling; Two-fluid model ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction Boiling in horizontal and vertical ducts under conditions of forced convection is an important subject in many industries because it generates a large heat transfer coefficient. Turbulent boiling flow occurs in many important engineering equipment, such as steam generators, heat exchangers, nuclear reactors, and refrigeration systems. In turbulent subcooled boiling flow, the vapor phase is dispersed as bubbles in the liquid phase. The bubbles are generated at heated surfaces where temperature is sufficiently higher than the local saturation temperature of the fluid. Then, these bubbles leave the heated surfaces under some specific conditions. Possibly the earliest experimental investigation of subcooled flow boiling was conducted by Pierre and Bankoff [1], who observed subcooled boiling in a vertical rectangular channel with heated walls. Pierre and Bankoff measured transverse void fractions over the cross section of the channel at different elevations. Bartel [2] obtained radial profiles of flow parameters at different axial locations in a vertical annulus with a heated inner rod at near atmospheric pressure. Similarly, Lee et al. [3] performed experiments under comparable flow conditions and found radial distributions of flow parameters, such as liquid velocity and void fraction, only at a single axial location. Other studies examined subcooled flow boiling [4-6]. *

Corresponding author. Tel.: +98 9144207211, Fax.: +98 4113392467 E-mail address: [email protected] † Recommended by Associate Editor Gihun Son © KSME & Springer 2013

Among the multidimensional theoretical descriptions of subcooled boiling, the most widely used approach is the twofluid model (for more details, see Refs. [7-9]). This model is an important simulation tool for turbulent boiling flow. In this model, each phase satisfies the conservation equations of mass, momentum, and energy. The phases are coupled at the interface by the balance relations for mass, momentum, and energy [10]. Anglart and Nylund [11], Kurul and Podowski [12], and Roy et al. [13] proposed modifications to the two-fluid model. Roy et al. [13] examined turbulent subcooled boiling flow in a vertical concentric annular channel both experimentally and numerically. The refrigerant R-113 at 2.69 bar was used as a working fluid, and the subcooled boiling model, a twofluid model, was used for the numerical investigation. Then, Roy et al. compared the experimental and simulation results. However, the models of Roy et al. have been applied to either the boiling of water at high pressure or the boiling of refrigerants at low pressure. For subcooled boiling flow of water at low pressure, multidimensional two-fluid models were proposed by JanssnesMaenhout et al. [14], Lee et al. [3], and Tu and Yeoh [15]. Lee et al. [3] successfully applied their model in the simulation of their own experimental results, which includes subcooled boiling flow of water in a vertical annulus under various conditions. The main goal of the present work is to investigate centrifugal force and curvature effects on low-pressure subcooled nucleate flow boiling in curved annulus. To compare and

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highlight such effects, subcooled flow boiling in a straight annulus was also studied. As many parameters affect boiling flow, we focus only on single-phase flow and heat transfer in the first part of this work. After comparing the performance of single-phase flow in both annuli and the effective parameters on the differences between them, we consider subcooled boiling flow in the second part. The rest of the article is organized as follows: in the next section, the numerical procedure of the study, including the governing equations of the two-fluid model and interphase transfer terms, is explained. Results are presented and discussed in section 3. Section 4 concludes the paper.

2. Numerical procedure Two distinct multiphase flow models are the EulerianEulerian multiphase model, known as the two-fluid model, and the Lagrangian-Eulerian model. The governing equations of the two-fluid model are as follows [15]:

+ ∇. (1 − α)ρl u l  = Γlv   ∂t ∂ [αρ v ]  + ∇.  αρ v u v  = Γlv   ∂t

(1) (2)

where in subcooled boiling flow, the source term Гlv in Eq. (1) represents the mass transfer rate caused by condensation in the bulk subcooled liquid. Momentum equations ∂ (1 − α )ρ l u l    + ∇. (1 − α )ρ u u  (3) l l l    ∂t = −(1 − α)∇p + ∇. (1 − α )( τ l + τlturb ) + (Γlv u v − Γ vl u l ) + f lv     ∂ αρ v u v    + ∇.  αρ u u  = (4) v v v    ∂t −α∇p + ∇. α ( τ v + τ vturb ) + (Γ vl u l − Γ lv u v ) + f vl .  

Energy equations ∂ (1 − α ) ρl H l 

+ ∇(1 − α )ρ lH l u l  =   ∂t   ∇.(1 − α ) Kl ∇Tl  + (Γlv H v − Γ vl H l ) + ql   ∂ [αρv H v ]  + ∇αρv H v uv  =   ∂t ∇. αK v∇Tv  + (Γ vl H l − Γlv H v ) + qv .  

Boundary conditions A constant heat flux boundary condition is applied to the upper part of the inner wall, and the outer wall of the annulus is assumed to be adiabatic. Uniform velocity and temperature profiles are set at the inlet. A pressure boundary condition is applied to the annulus outlet. A free-slip boundary condition for the vapor phase is used at the wall, which accounts for the sliding of bubbles on the laminar sub-layer. The Eulerian-Eulerian model has two different sub-models: homogeneous and inhomogeneous. The inhomogeneous model includes different approaches to multiphase flow modeling, such as the mixture and the particle models. Among these models, the particle model is used in this study. The particle model assumes a continuous-phase fluid which contains particles of a dispersed-phase fluid or solid; thus, it is named as such. In this model, the interphase contact area is Alv =

Continuity equations ∂ (1 − α )ρl 

from the gas phase to the liquid phase.

(5)

(6)

Interfacial transfer terms in the momentum and energy equations represented by Гlv and flv denote the transfer terms

6α db

(7)

where α is the volume fraction of the dispersed phase. Some correlations exist for bubble sauter mean diameter, such as the correlations of Anglart and Nylund [11] and Zeitoun and Shoukri [16, 17]. According to Ref. [15], the model of Zeitoun and Shoukri is in better agreement with experimental results compared with that of Anglart and Nylund. Thus, the model of Zeitoun and Shoukri is employed in this study. The mean bubble diameter db is estimated from the mean Sauter bubble diameter. In multiphase systems, such as boiling flows, some transfer phenomena occur between phases, including interphase momentum, interphase heat, interphase mass, and turbulence. To study these transfer phenomena, the following models are used in this work. 2.1 Interphase momentum transfer model Interphase momentum transfer occurs when interfacial forces act on one phase because of their interaction with another phase. The total interfacial force that acts between the two phases may arise from several independent physical effects, which are considered in this study as follows: flv = − f vl = f D + f L + fTD + fW + fVM .

(8)

The forces indicated on the right side of Eq. (8) represent the interphase drag, lift, turbulent dispersion, wall lubrication, and virtual mass forces. The models for these forces used in the present investigation are described as follows. 2.1.1 Interphase drag force Interphase drag force is modeled according to the correlation of Ishii and Zuber [18]


3 C D fD = αρl uv − ul uv − ul 4 db

(

)

(9)

where CD is the drag coefficient and db is the bubble diameter. 2.1.2 Lift force Lift force, which acts perpendicular to the main flow direction, is proportional to the curl of the liquid velocity field and is expressed as follows:

equilibrium across phase interfaces. One of the widely used models is the two-resistance heat transfer model, which is employed in the present study. In this model, separate heat transfer processes in either side of the phase interface are considered, which is achieved by using two heat transfer coefficients defined on each side of the phase interface. Therefore, the sensible heat flux to the liquid phase from the interface is calculated by ql = hl (Ts − Tl )

(

) (

f L = CLαρl uv − ul × ∇× ul

)

2.1.3 Turbulent dispersion force The effect of the diffusion of the vapor phase, caused by liquid phase turbulence, is described by the turbulent dispersion force [20]: (11)

where kl is the turbulent energy of the liquid phase and CTD is the turbulent dispersion coefficient. In this study, CTD is set to 0.1 according to Kurul and Podowski [12]. 2.1.4 Wall lubrication force Wall lubrication force acts in the radial direction by pushing the bubbles away from the wall and prevents their accumulation near the wall. In this study, the model proposed by Antal et al. [21] is used for considering this force: 2   αρl (uv − ul ) d fw = − .max Cw1 + Cw 2 b ,0 n  db yw 

(12)

r where yw denotes the distance from the wall, and n is the normal vector to the wall. The non-dimensional coefficients are set to Cw1 = - 0.01 and Cw2 = 0.05 [21].

2.1.5 Virtual mass force Finally, virtual mass force, which arises because of relative acceleration between the vapor and liquid phases, can be estimated by using the following formulation of Zuber [22]: r r r  ∂u r r   ∂u r r   fVM = CVM αρl  v + uv .∇uv  −  l + ul .∇ul     ∂t   ∂t

(14)

(10)

where CL is the lift force coefficient for which the correlation of Tomiyama [19] is used.

fTD = −CTD ρl kl ∇(1 − α )

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(13)

where CVM is the virtual mass coefficient which is set to 0.5. 2.2 Interphase heat transfer model Interphase heat transfer occurs because of thermal non-

and the sensible heat flux to the vapor phase from the interface is defined as qv = hv (Ts − Tv )

(15)

where hl and hv are the liquid and vapor phase heat transfer coefficients, respectively. Ts is the interfacial temperature, assumed the same for both phases. The liquid heat transfer coefficient could be empirically calculated by Hughmark [23] correlations Nu =

hlv d Kl

= 2 + 0.6 Reb 0.5 Prl1 3 ,

(16a) 0 ≤ Reb < 776.06 , 0 ≤ Prl < 250

hd Nu = lv Kl = 2 + 0.27 Reb 0.62 Prl1 3 ,

(16b) 776.06 ≤ Reb , 0 ≤ Prl < 250 .

In this study, the zero resistance condition on the vapor side of the phase interface is considered. This condition is equivalent to an infinite vapor side heat transfer coefficient. Thus, the interfacial temperature is assumed to be the same as the vaporphase temperature. 2.3 Interphase mass transfer model In this study, interphase mass transfer is described as phase change induced by interphase heat transfer. The tworesistance model for interphase heat transfer is used together with the interphase mass transfer model. The following model is used in the present work. 2.3.1 Wall boiling model Wall boiling starts when wall temperature becomes sufficiently large to initiate the activation of wall nucleation sites. Evaporation starts in the microscopic cavities and crevices. The liquid then becomes supersaturated locally in these nucleation sites, thereby leading to the growth of vapor bubbles at these sites. The bubbles detach from the sites when they are sufficiently large that external forces (i.e., inertial, gravitational, and turbulent forces) exceed the surface tension forces that keep them attached to the wall. As the bubbles leave the wall, they are displaced by superheated liquid in the vicinity of

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occurs during the waiting time and is modeled as simple onedimensional transient heat conduction into a semi-infinite medium, with the liquid at a temperature Tl and the heater surface at a temperature Tw qQ = hQ AQ (Tw − Tl ) .

The RPI model assumes that the diameter of the bubble influence zone is twice as large as dbw. Therefore, the nondimensional area fraction of bubble influence is calculated by

Fig. 1. Heat flux partition of wall boiling model.

the nucleation sites, after which the nucleation sites are free to create another bubble. In regions of the wall that are unaffected by bubble growth, the wall heat transfer to the liquid may be described by single-phase convective heat transfer. The first and most well known wall boiling model is the RPI model, which was formulated by Kurul and Podowski [24] of the Rensselaer Polytechnic Institute after which the model is named. In the present work, a modified model of Kurul and Podowski [24] is implemented. In this model, the total wall heat flux is split into three components (Fig. 1) qT = qC + qQ + qE .

(18)

All three components of total heat flux and the required closure relations are described below. 2.3.1.1 Single-phase liquid convection heat flux Single-phase liquid convection heat flux can be correlated to the turbulent flow as qC = hlC AC (Tw − Tl )

2  AQ = min 1, N a πd bw

(21)

where dbw is the bubble departure diameter obtained through the correlation that Kurul and Podowski [24] adopted for bubble departure diameter with reference to the work of Tolubinski and Kostanchuk [25]   ∆T dbw = min d ref .exp  − sub ∆Tref  

   , d max   

(22)

(17)

The fluxes indicated on the right-hand side of Eq. (17) represent the single-phase liquid convection heat flux (qC), the quenching heat flux (qQ), and the evaporation heat flux (qE). Vapor is assumed to be saturated everywhere, and no part of the wall heat flux is arranged for the superheating of the vapor phase. In this model, each unit of the heated surface is split into two parts: one unit is influenced by the nucleating bubbles (AQ), and the other one is influenced only by single-phase convection (AC) [12]. In non-dimensional form, AC + AQ = 1 .

(20)

where dmax= 1.4 mm, dref = 0.6 mm, and ∆Tref = 45 K. On the right side of Eq. (21), Na is the nucleation site density, estimated by the following empirical correlation of Lemmert and Chawla [26]:

(

N a = 185 (Tw − Tl )

)

1.805

.

(23)

In Eq. (20), hQ is the quenching heat transfer coefficient, calculated according to Victor et al. [27]: hQ =

2 f tQ K l ρl c pl π

(24)

where f and tQ are the bubble detachment frequency and waiting time, respectively. f is calculated by the modified model of Kurul and Podowski [24], which was first proposed by Cole [28]: f =

4 g ( ρl − ρv ) 3d bw ρl

.

(25)

(19)

where Tw is the temperature of the solid wall and hlC is the liquid single-phase heat transfer coefficient. 2.3.1.2 Quenching heat flux The second component of total heat flux is the quenching heat flux transferred to the subcooled liquid from the bulk flow that fills the volume vacated by departing bubbles. When a bubble lifts off, the boundary layer becomes disrupted, and cold liquid comes into contact with the heated wall, which results in transient conduction on these areas. Conduction

For the bubble waiting time (tQ), Kurul and Podowski [24] employed the model of Tolubinski and Kostanchuk [25]. This model fixes the waiting time between the departures of consecutive bubbles at 80% of the bubble detachment period tQ =

0.8 . f

(26)

2.3.1.3 Evaporation heat flux The last component of the total heat flux is evaporation heat flux, which is consumed for the evaporation of the initially

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Table 1. Dimensions of curved annulus.

Table 2. Working and boundary conditions used in this study for subcooled flow boiling.

Inner diameter (mm)

19

Outer diameter (mm)

37.5

Inlet temperature (°C)

93.9

Curvature radii (mm)

160

Working pressure (MPa)

0.125

Coaxial core length (mm)

1000

Mass flux (kg/m2s)

715.17

Inner wall heat flux (kW/m2)

139.08

Fig. 2 (a) Schematic of curved annulus; (b) used grid; (c) geometrical properties at outlet.

subcooled liquid and is calculated as follows: & lv qE = mH

(27)

where m& is the evaporation mass transfer rate per unit wall area, and Hlv is the difference between specific enthalpies of the saturated vapor and subcooled liquid. 2.4 Turbulence transfer Given the lower vapor density, in the nucleate boiling flow, the motion of the dispersed vapor phase follows the fluctuations of the continuous liquid phase [12]. Accordingly, turbulence stresses are modeled only for the liquid phase, whereas the vapor phase is assumed to be laminar. In the present work, turbulence in the liquid phase is modeled by the shear stress transport model with an additional term that describes bubbleinduced turbulence. Shear and bubble-induced turbulence are linearly superimposed according to the assumption of Sato et al. [29], where the effective viscosity of the continuous liquid phase is expressed as µleff = µl + µlt + µlb .

(28)

Bubble-induced turbulence viscosity µlb in the liquid phase depends on the vapor phase volume fraction, the local bubble diameter, and the relative velocity between the phases, as follows: uur ur µlb = C µb ρl αd b uv − ul .

(29)

In the present work, the coefficient Cµb is set to 0.6, as recommended by Sato et al. [29].

Fig. 3. Grid independency results for single-phase liquid flow.

3. Results and discussion In this study, two types of annuli (straight and curved horizontal) are used for the investigation of single- and two-phase flow and heat transfer. Although the main aim of this study is the investigation of flow boiling in curved annulus, fluid flow and heat transfer in straight annulus are also investigated for a detailed study of centrifugal force and curvature effects. In both cases, the dimensions are selected from the work of Lee et al. [3], except for the length of the annuli. Fig. 2 shows a schematic diagram of the curved annulus, and Table 1 lists its dimensions. In the present work, governing equations are discretized through the finite volume method [30]. The SIMPLEC algorithm and the second-order upwind scheme are used for pressure-velocity coupling and the approximation of convection terms, respectively [30]. The solution domain is discretized with uniform spacing in the axial direction. For gridindependent results, different grids are investigated. This grid independence is illustrated graphically in Fig. 3 for singlephase liquid flow in curved annulus. The second and third cases differ only marginally. Therefore, to save computation time, 15 × 40 × 150 grids in the radial, circumferential, and axial directions are chosen. To verify the results of the two-fluid model for boiling flow, the experimental results of the subcooled flow boiling of water in a vertical annulus reported by Lee et al. [3] are used. The boiling test section is a straight vertical annulus with a 0.33 m long unheated inlet part followed by a 1.67 m long heated part.

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Table 3. Working and boundary conditions for single-phase flow. Inlet temperature (°C)

93.9

Working pressure (MPa)

0.125

Mass flux (kg/m2s)

715.17

Inner wall heat flux (kW/m2)

139.08

(a)

Fig. 5. Liquid velocity for heated single-phase liquid flow: (a) midplane of straight annulus; (b) mid-plane of curved annulus; (c) outlet of straight annulus; (d) outlet of curved annulus.

(b) Fig. 4. Validation of numerical results: (a) liquid velocity; (b) void fraction in straight vertical annulus.

The inner diameter, outer diameter, and length of this annulus are 19 mm, 37.5 mm, and 2 m, respectively. For comparison, one of the working conditions of the experimental work noted above is chosen in this study (Table 2). According to these flow conditions, the Reynolds number of flow is approximately equal to 44000, which indicates a turbulent flow. Fig. 4 shows the liquid velocity and vapor void fraction variation with radial direction in an axial location, 1.61 m downstream of the beginning of the heated section. As shown in Fig. 4, the numerical results are in good agreement with experimental ones. 3.1 Single-phase flow To analyze subcooled boiling, detailed studies of the influence of centrifugal force in single-phase flow in the straight and curved annuli are carried out. The flow conditions of single-phase flow are given in Table 3. Fig. 5 shows the velocity contours in the middle and outlet planes of the two annuli. For

Fig. 6. Liquid velocity in radial direction in outlet of two annuli for heated single-phase liquid flow.

the straight annulus, velocity contours are symmetric, but in the curved annulus, asymmetry starts from the inlet, reaches the maximum at mid-plane, and again decreases toward the outlet because of the effect of centrifugal force. Fig. 6 shows the velocity profiles on two radial lines at the outlets of both annuli, and the results confirm the findings in Fig. 5. Based on the velocity profiles in the straight annulus, tem-

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(a)

(b)

(a)

(b)

Fig. 9. Gas phase streamlines in the outlet of (a) straight annulus; (b) curved annulus.

(c)

(d)

Fig. 7. Liquid temperature for heated single-phase liquid flow: (a) midplane of straight annulus; (b) mid-plane of curved annulus; (c) outlet of straight annulus; (d) outlet of curved annulus.

(a)

Fig. 8. Temperature around inner wall for heated single-phase liquid flow.

perature distribution in this annulus is uniform on the periphery of the inner cylinder wall surface, as depicted in Figs. 7(a), 7(c) and 8(a), 8(c). However, as expected, temperature distribution for the curved annulus in the middle and outlet planes varies on the periphery and is maximum at low-velocity regions (i.e., θ = 180 according to Fig. 2(c)) and minimum at high-velocity regions. According to Fig. 8, for the curved annulus, the temperature in almost all angles in both planes is lower than in the straight annulus, except near θ = 180 , because of the low velocity compared with the straight annulus.

(b) Fig. 10. Bubble detachment frequency in (a) straight annulus; (b) curved annulus.

3.2 Subcooled boiling flow For a detailed investigation of the simultaneous effects of centrifugal force and boiling phenomenon, the subcooled flow boiling in horizontal straight and curved annuli are studied. The obtained results are compared with the single-phase flow.

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(a)

(b)

(a)

(c)

(d)

Fig. 11. Liquid temperature: (a) mid-plane of straight annulus; (b) midplane of curved annulus; (c) outlet of straight annulus; (d) outlet of curved annulus.

As previously mentioned, one of the flow conditions of the experimental work of Lee et al. [3] is chosen in this study for comparison, as seen in Table 2. Fig. 9 shows the gas-phase streamline in both straight and curved annuli in the subcooled boiling. In both cases, bubbles migrate from the left and right sides to the upper surface of the insulated annulus. Streamlines are symmetric in straight annulus and asymmetric in curved annulus because of interactions between centrifugal force acting on the bubbles in the radial direction and buoyancy forces in the upward direction. Fig. 10 also shows bubble detachment frequency versus angle in the annuli. According to Fig. 10(a), for the straight annulus, on the lower section of the inner cylinder surface (i.e., 180 < θ < 360°), this frequency is much higher than on the upper surface (0 < θ < 180°). Thus, bubble nucleation and detachment are faster at the lower section, thereby leading to more vapor generation, more heat transfer, and lower temperature in this region. For straight annulus, the effects of bubble detachment frequency on the temperature contours and profiles are shown in Figs. 11(a), 11(c) and 12(a). At the high detachment region, given the high convective heat transfer coefficient, temperature is lower than in the low detachment region. For the curved annulus, Figs. 10(b), 11(b), 11(d), and 12(b) show the bubble detachment frequency, temperature contours, and surface temperature profile, respectively, of the inner cylinder surface between (0 < θ < 360°). The surface temperature profile follows exactly the bubble detachment frequency curve in a way that temperature at the high-frequency detachment region is lower than at the low-frequency region; thus, it is compatible with the bubble migration streamlines of Fig. 9(b). A comparison among the abovementioned effects in the

(b) Fig. 12. Temperature around inner wall: (a) straight annulus; (b) curved annulus.

curved annulus between single-phase flow (Figs. 8(b) and 8(d)) and two-phase flow (Fig. 12(b)) shows that in singlephase flow, maximum temperature occurs at θ = 180° for both middle and outlet planes. However, for two-phase flow at the same area, according to Fig. 10(b), the bubble detachment frequency is maximum, and consequently, as shown in Fig. 12(b), temperature has minimum value. According to the single-phase flow at this region, high surface temperature increases bubble formation, and consequently, high bubble production and detachment frequency are observed in the twophase flow. This is a very important phenomenon that happens in two-phase flow, but not in single-phase flow, in the curved annulus. The effect of centrifugal force on this area is less than the buoyancy force to affect detachment, and according to Fig. 12(b), it causes a little asymmetry to the temperature profile starting from the middle plane to the outlet planes. To investigate the effect of the Reynolds (Re) number on subcooled flow boiling, another simulation is done with a Re number twice that in the previous simulation. Based on the obtained results, increasing the flow velocity increases heat


fr frL frTD frVM fW g hlC Hlv hQ kl Kl m˙ r n

Fig. 13. Effect of Re number on wall temperature of curved annulus (Re2 = 2Re1).

transfer and decreases the wall temperature of the annulus (see Fig. 13).

4. Conclusions The single-phase and subcooled boiling flow of water in the straight and curved horizontal annuli is investigated numerically. The two-fluid model is used for the simulation of subcooled boiling flow, the results of which are compared with those of the single-phase flow. Good agreement between the numerical and experimental results is obtained. The regions with higher temperature in single-phase flow are the first points where temperature exceeds the saturation temperature in subcooled boiling, and thus vapor generation occurs at the active nucleation sites. Therefore, these regions have more heat transfer and lower wall temperature in a boiling flow. In addition, the surface temperature of the inner cylinder in the curved annulus is lower than in the straight annulus in almost all angles in both single-phase and subcooled boiling flow. As expected, subcooled flow boiling performance improves by increasing flow velocity and Re number.

Nomenclature-----------------------------------------------------------------------AC Alv AQ CD CL cp CTD CVM CW1 CW2 db drbW fD

: Wall fraction influenced by single-phase convection : Interphase contact area : Wall fraction influenced by nucleating bubbles : Drag coefficient : Lift force coefficient : Specific heat capacity : Turbulent dispersion coefficient : Virtual mass coefficient : Non-dimensional coefficient1 : Non-dimensional coefficient2 : Bubble mean diameter : Bubble departure diameter : Drag force

Nu Na Prl qC qE qQ qT Ri Ro Reb t T u tQ

1303

: Frequency : Lift force : Turbulent dispersion force : Virtual mass force : Wall lubrication force : Gravitational acceleration : Liquid single-phase heat transfer coefficient : Difference between specific enthalpies : Quenching heat transfer coefficient : Turbulent energy of liquid phase : Liquid thermal conductivity : Evaporation mass transfer rate : Normal vector to the wall : Nusselt number : Active nucleation site density : Liquid Prandtl number : Single-phase convection heat flux : Evaporation heat flux : Quenching heat flux : Total heat flux : Inner radius : Outer radius : Bubble Reynolds number : Time : Temperature : Velocity : Waiting time

Greek symbols α θ ρ µl µlb µleff µlt

: Void fraction : Angle : Density : Liquid dynamic viscosity : Bubble-induced turbulence viscosity : Effective viscosity : Liquid turbulent viscosity

Subscripts b l v

: Bubble : Liquid : Vapor

References [1] C. C. St Pierre and S. G. Bankoff, Vapor volume profiles in developing two-phase flow, Int. J. Heat Mass Transfer, 10 (1967) 237-249. [2] M. D. Bartel, Experimental investigation of subcooled boiling, M.S.N.E. Thesis, Purdue University, West Lafayette, IN, USA (1999). [3] T. H. Lee, G. C. Park and D. J. Lee, Local flow characteristics of subcooled boiling flow of water in a vertical concentric annulus, Int. J. Multiphase flow, 28 (2002) 1351-1368.

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Habib Aminfar received his Ph.D. from MIT University in 1981. Dr. Aminfar is currently a Professor at the Department of Mechanical Engineering at the University of Tabriz, Iran. His research interests include nanofluid flow and heat transfer, two-phase flows, thermal engineering, and heat transfer. Rasool Maroofiazar is a Ph.D. candidate of the Department of Mechanical Engineering at the University of Tabriz. He received his M.S. in Mechanical Engineering from the University of Tabriz in 2009. His main research interests are two-phase flows, nanofluid flow and heat transfer, and computational fluid dynamics. Mousa Mohammadpourfard received his Ph.D. degree from the University of Tabriz, Iran in 2009. Dr. Mohammadpourfard is currently an Assistant Professor at the Department of Mechanical Engineering at Azarbaijan Shahid Madani University in Tabriz, Iran. His research interests include computational fluid dynamics and its applications in heat transfer and fluid mechanics.