Two Phase Flow Phase Change and Numerical Modeling Part 13 docx

350 Two Phase Flow, Phase Change and Numerical Modeling for the ethanol case At large capillary numbers, all data are larger than the Taylor’s law Inertial force is often neglected in micro two phase flows, but it is clear that the inertial force should be considered from this Reynolds number range In Fig 8 (b), dimensionless initial liquid film thickness in 1.3 mm inner diameter tube shows different trend at Ca > 0.12, showing some scattering Reynolds number of ethanol in 1.3 mm inner diameter tube becomes Re ≈ 2000 at Ca ≈ 0.12 Thus, this different trend is considered to be the effect of flow transition from laminar to turbulent Figure 8 (c) shows initial liquid film thickness for water At Re > 2000, initial liquid film thickness does not increase but remains nearly constant with some scattering This tendency is found again when Reynolds number exceeds approximately Re ≈ 2000 The deviation from Taylor’s law starts from the lower capillary number than FC-40 and ethanol Dimensionless initial liquid film thickness of water shows much larger values than that of ethanol and Taylor’s law In the case of 1.3 mm inner diameter tube, dimensionless initial liquid film thickness is nearly 2 times larger than the Taylor’s law at Ca ≈ 0.03 It is clearly seen that inertial force has a strong effect on liquid film thickness even in the Reynolds number range of Re < 2000 3.1.2 Scaling analysis for circular tubes Bretherton (1961) proposed a theoretical correlation for the liquid film thickness with lubrication equations as follows: 2 δ0  3 μU  3 = 0.643   Dh 2  σ  (12) Aussillous and Quere (2000) modified Bretherton’s analysis, and replaced the bubble nose curvature κ = 1/(Dh/2) with κ = 1/{(Dh/2)-δ0} In their analysis, the momentum balance and the curvature matching between the bubble nose and the transition region are expressed as follows:  ρU 2 1  σ   ~  , λ λ  ( Dh 2 ) − δ 0    (13) δ0 σ ~ λ 2 ( Dh 2 ) − δ 0 (14) where λ is the length of the transition region as shown in Fig 9 Eliminating λ from Eqs (13) and (14), they obtained following relation for dimensionless liquid film thickness: δ0 Dh 2 2 ~ Ca 3 1 + Ca 2 3 (15) In Eq (15), dimensionless liquid film thickness asymptotes to a finite value due to the term Ca2/3 in the denominator Based on Eq (15), Taylor’s experimental data was fitted as Eq (11) If inertial force effect is taken into account, the momentum balance (13) should be expressed as follows: 351 Liquid Film Thickness in Micro-Scale Two-Phase Flow  ρU 2 μU 1  σ   , ~  − 2 δ 0 λ  ( Dh 2 ) − δ 0  λ   (16) Using Eqs (14) and (16), we can obtain the relation for initial liquid film thickness δ0/Dh as: δ0 Dh 2 Ca 3 ~ 2 3 2 , (17) Ca + ( 1 − We′ ) 3 where Weber number is defined as We’ = ρU2((Dh/2)-δ0)/σ Equation (17) is always larger than Eq (15) because the sign in front of Weber number is negative Therefore, Eq (17) can express the increase of the liquid film thickness with Weber number In addition, Heil (2001) reported that inertial force makes the bubble nose slender and increases the bubble nose curvature at finite Reynolds numbers It is also reported in Edvinsson & Irandoust (1996) and Kreutzer et al (2005) that the curvature of bubble nose increases with Reynolds and capillary numbers This implies that curvature term κ = 1/{(Dh/2)-δ0} in momentum equation (16) should be larger for larger Reynolds and capillary numbers We assume that this curvature change can be expressed by adding a modification function of Reynolds and capillary numbers to the original curvature term κ = 1/{(Dh/2)-δ0} as: κ= 1 + f ( Re, Ca ) ( Dh 2 ) − δ 0 , (18) Substituting Eq (18) into Eqs (14) and (16), we obtain: δ0 Dh 2 ~ Ca 2 3   We′ Ca + ( 1 + f ( Re, Ca ) )  1 −   1 + f ( Re, Ca )    2 3 2 3 (19) If all the terms with Re, Ca and We can be assumed to be small, we may simplify Eq (19) as: δ0 Dh 2 ~ Ca 2 3 2 3 (20) Ca + 1 + f ( Re, Ca ) − g ( We′ ) In the denominator of Eq (20), f (Re, Ca) term corresponds to the curvature change of bubble nose and contributes to reduce liquid film thickness On the other hand, when the inertial effect increases, g(We’) term contributes to increase the liquid film thickness due to the momentum balance Weber number in Eq (17) includes initial liquid film thickness δ0 in its definition Therefore, in order to simplify the correlation, Weber number is redefined as We = ρU2Dh/σ The experimental data is finally correlated by least linear square fitting in the form as:  δ0  =    Dh steady 2 0.670Ca 3 2 1 + 3.13Ca 3 + 0.504Ca0.672 Re 0.589 − 0.352 We 0.629 (Ca < 0.3, Re < 2000) , (21) 352 Two Phase Flow, Phase Change and Numerical Modeling where Ca = μU/σ and Re = ρUDh/μ and We = ρU2Dh/σ As capillary number approaches zero, Eq (21) should follow Talors’s law (11), so the coefficient in the numerator is taken as 0.670 If Reynolds number becomes larger than 2000, initial liquid film thickness is fixed at a constant value at Re = 2000 Figures 10 and 11 show the comparison between the experimental data and the prediction of Eq (20) As shown in Fig 11, the present correlation can predict δ0 within the range of ±15% accuracy Fig 9 Schematic diagram of the force balance in bubble nose, transition and flat film regions in circular tube slug flow Fig 10 Predicted initial liquid film thickness δ0 by Eq (21) Fig 11 Comparison between predicted and measured initial liquid film thicknesses δ0 353 Liquid Film Thickness in Micro-Scale Two-Phase Flow 3.2 Steady square tube flow 3.2.1 Dimensionless bubble radii Dimensionless bubble radii Rcenter and Rcorner are the common parameters used in square channels: Rcenter = 1 − 2δ 0 _ center , Dh Rcorner = 2 − 2δ 0 _ corner Dh (22) (23) It should be noted that initial liquid film thickness at the corner δ0_corner in Eq (23) is defined as a distance between air-liquid interface and the corner of circumscribed square which is shown as a white line in Fig 2(b) When initial liquid film thickness at the channel center δ0_center is zero, Rcenter becomes unity If the interface shape is axisymmetric, Rcenter becomes identical to Rcorner Figure 12(a) shows Rcenter and Rcorner against capillary number for FC-40 The solid lines in Fig 12 are the numerical simulation results reported by Hazel & Heil (2002) In their simulation, inertial force term was neglected, and thus it can be considered as the low Reynolds number limit Center radius Rcenter is almost unity at capillary number less than 0.03 Thus, interface shape is non-axisymmetric for Ca < 0.03 For Ca > 0.03, Rcenter becomes nearly identical to Rcorner, and the interface shape becomes axisymmetric In Fig 12, measured bubble radii in Dh = 0.3 and 0.5 mm channels are almost identical, and they are larger than the numerical simulation result On the other hand, the bubble radii in Dh = 1.0 mm channel are smaller than those for the smaller channels As capillary number approaches zero, liquid film thickness in a micro circular tube becomes zero In micro square tubes, liquid film δ0_corner still remains at the channel corner even at zero capillary number limit Corner radius Rcorner reaches an asymptotic value smaller than 2 as investigated in Wong et al.’s numerical study (1995a, b) This asymptotic value will be discussed in the next section Figure 12(b) shows Rcenter and Rcorner for ethanol Similar to the trend found in FC-40 experiment, Rcenter is almost unity at low capillary number Most of the experimental data are smaller than the numerical result Transition capillary number, which is defined as the capillary number when bubble shape changes from non-axisymmetric to axisymmetric, becomes smaller as Dh increases For Dh = 1.0 mm square tube, Rcenter is almost identical to Rcorner beyond this transition capillary number However, for Dh = 0.3 and 0.5 mm tubes, Rcenter is smaller than Rcorner even at large capillary numbers At the same capillary number, both Rcenter and Rcorner decrease as Reynolds number increases For Ca > 0.17, Rcenter and Rcorner in Dh = 1.0 mm square tube becomes nearly constant It is considered that this trend is attributed to laminar-turbulent transition At Ca ≈ 0.17, Reynolds number of ethanol in Dh = 1.0 mm channel becomes nearly Re ≈ 2000 as indicated in Fig 12(b) Center and coner radii, Rcenter and Rcorner, for water are shown in Fig 12(c) Center radius Rcenter is again almost unity at low capillary number Transition capillary numbers for Dh = 0.3, 0.5 and 1.0 mm square channels are Ca = 0.025, 0.2 and 0.014, respectively These values are much smaller than those for ethanol and FC-40 Due to the strong inertial effect, bubble diameter of the water experiment is much smaller than those of other fluids and the 354 Two Phase Flow, Phase Change and Numerical Modeling numerical results It is confirmed that inertial effect must be considered also in micro square tubes Bubble diameter becomes nearly constant again for Re > 2000 Data points at Re ≈ 2000 are indicated in Fig 12(c) 1.2 (a) x 1.0 D D D D D D R R R R R R R R 1.1 0.9 0.8 0.0 0.1 0.2 0.3 0.4 Ca μU/σ 1.2 (b) D D D D D D x 1.0 R R R R R R R R 1.1 0.9 Re 0.8 0.00 ↑ 0.10 0.20 0.30 Ca μU/σ 1.2 (c) x D D D D D D R R R R R R 1.0 R R 1.1 0.9 0.8 0.00 Re 0.02 ↑ Re 0.04 ↑ 0.06 0.08 0.10 Ca μU/σ Fig 12 Dimensionless center and coner radii, Rcenter and Rcorner, in steady square tubes (a) FC-40, (b) ethanol and (c) water Liquid Film Thickness in Micro-Scale Two-Phase Flow 355 3.2.2 Scaling analysis for square tubes Figure 13 shows the schematic diagram of the force balance in the transition region in square tubes Momentum equation and curvature matching in the transition region are expressed as follows: μU 1 ρU 2 , ~ σ (κ 1 − κ 2 ) − 2 δ0 λ λ (24) δ0 ~κ 1 − κ 2 , λ2 (25) where, κ1 and κ2 are the curvatures of bubble nose and flat film region, respectively In the present experiment, δ0_corner does not become zero but takes a certain value as Ca → 0 Figure 14 shows the schematic diagram of the interface shape at Ca → 0 In Fig 14, air-liquid interface is assumed as an arc with radius r Then, κ2 can be expressed as follows: κ2 = 1 2 −1 = r δ 0 _ corner (26) Fig 13 Schematic diagram of the force balance in bubble nose, transition and flat film regions in square Fig 14 Schematic diagram of the gas liquid interface profile at Ca → 0 If bubble nose is assumed to be a hemisphere of radius Dh/2, the curvature at bubble nose becomes κ1 = 2/(Dh/2) This curvature κ1 should be larger than the curvature of the flat film region κ2 according to the momentum balance, i.e κ 1 ≥ κ 2 From this restraint, the relation of Dh and δ0_corner is expressed as follows: 356 Two Phase Flow, Phase Change and Numerical Modeling 2 2 −1 ≥ Dh 2 δ 0 _ corner (27) From Eqs (23) and (27), the maximum value of Rcorner can be determined as follows: Rcorner ≤ 1.171 (28) From Fig 12, the interface shape becomes nearly axisymmetric as capillary number increases Here, bubble is simply assumed to be hemispherical at bubble nose and cylindrical at the flat film region, i.e Rcorner = Rcenter Under such assumption, the curvatures κ1 and κ2 in Eqs (24) and (25) can be rewritten as follows: κ1 = κ2 = Dh 2 , 2 − δ 0 _ corner (29) Dh 1 2 − δ 0 _ corner (30) We can obtain the relation for δ0_corner from Eqs (24), (25), (29) and (30) as: δ 0 _ corner Dh 2 ≈ 2Ca 3 2 2 , (31) Ca 3 + ( 1 − We′ ) 3 where We′ is the Weber number which includes δ0_corner in its definition Thus, We′ is replaced by We = ρU2Dh/σ for simplicity The denominator of R.H.S in Eq (31) is also simplified with Taylor expansion From Eqs (28) and (31), Rcorner is written as follows: 2 Rcorner ~1.171 − 2 2Ca 3 2 3 (32) 1 + Ca − We The experimental correlation for Rcorner is obtained by optimizing the coefficients and exponents in Eq (32) with the least linear square method as follows: 2 2.43Ca 3 Rcorner = 1.171 − 2 3 1 + 7.28Ca − 0.255We 1  Rcenter ≅   Rcorner  ( Rcorner > 1) ( Rcorner ≤ 1) (Re < 2000) , (33) 0.215 (Re < 2000) (34) From Eq (34), Rcenter becomes unity at small capillary number However, δ0_center still has a finite value even at low Ca, which means that Rcenter should not physically reach unity Further investigation is required for the accurate scaling of δ0_center and Rcenter at low Ca As capillary number increases, interface shape becomes nearly axisymmetric and Rcenter becomes identical to Rcorner As capillary number approaches zero, Rcorner takes an asymptotic 357 Liquid Film Thickness in Micro-Scale Two-Phase Flow value of 1.171 If Reynolds number becomes larger than 2000, Rcorner becomes constant due to flow transition from laminar to turbulent Then, capillary and Weber numbers at Re = 2000 should be substituted in Eq (33) Figure 15 shows the comparison between the experimental data and the predicted results with Eqs (33) and (34) As shown in Fig 16, the present correlation can predict dimensionless bubble diameters within the range of ±5 % accuracy 1.2 D D D D D D D D D 1.0 R R R R R R R R 1.1 0.9 0.8 0.00 0.10 0.20 0.30 Ca μU/σ Fig 15 Predicted bubble diameter in Dh = 0.5 mm square tube 1.2 1.1 R 1.0 0.9 0.8 0.8 0.9 1.0 1.1 1.2 R Fig 16 Comparison between predicted and measured bubble radii 3.3 Steady flow in high aspect ratio rectangular tubes For high aspect ratio rectangular tubes, interferometer as well as laser confocal displacement meter are used to measure liquid film thickness (Han et al 2011) Figure 17 shows the initial liquid film thicknesses obtained by interferometer and laser confocal displacement meter In the case of interferometer, initial liquid film thickness is calculated by counting the number of fringes from the neighbouring images along the flow direction In Fig 17, error bars on the interferometer data indicate uncertainty of 95 % confidence Both results show good 358 Two Phase Flow, Phase Change and Numerical Modeling agreement, which proves that both methods are effective to measure liquid film thickness very accurately From the analogy between flows in circular tubes and parallel plates, it is demonstrated that dimensionless expression of liquid film thickness in parallel plates takes the same form as Eq (19) if tube diameter Dh is replaced by channel height H (Han, et al 2011) Figure 18 shows the comparison between experimental data and predicted values with Eq (21) using hydraulic diameter as the characteristic length for Reynolds and Weber numbers As can be seen from the figure, Eq (21) can predict initial liquid film thickness in high aspect ratio rectangular tube remarkably well Fig 17 Measured initial liquid film thickness in high aspect ration rectangular tubes using interferometer and laser confocal displacement meter Fig 18 Comparison between measured and predicted initial liquid film thicknesses by Eq (21) in high aspect ratio rectangular tubes 3.4 Accelerated circular tube flow 3.4.1 Acceleration experiment In order to investigate the effect of flow acceleration on the liquid film thickness, measurement points are positioned at Z = 5, 10 and 20 mm away from the initial air-liquid 364 Two Phase Flow, Phase Change and Numerical Modeling Heil, M (2001) Finite Reynolds number effects in the Bretherton problem, Phys of Fluids, 13(9), 2517-2521 Hurlburt, E T & Newell, T A (1996) Optical measurement of liquid film thickness and wave velocity in liquid film flows, Experiments in Fluids, 21, 357-362 Kenning, D B R.; Wen, D S.; Das, K S & Wilson, S K (2006) Confined growth of a vapour bubble in a capillary tube at initially uniform superheat: Experiments and modeling, International Journal of Heat and Mass Transfer, 49(23-24), 4653-4671 Kreutzer, M T.; Kapteijn, F.; Moulijn, J A.; Kleijn, C R & Heiszwolf, J J (2005) Inertial and interfacial effects on pressure drop of Taylor flow in capillaries, AIChE Journal, 51(9), 2428-2440 Moriyama, K & Inoue, A (1996) Thickness of the liquid film formed by a growing bubble in a narrow gap between two horizontal plates, Transactions of the ASME, 118, 132-139 Qu, W & Mudawar, I (2004) Flow boiling heat transfer in two-phase micro-channel heat sink-II Annular two-phase flow model, International Journal of Heat Mass Transfer, 46, 3387-3401 Saitoh, S.; Daiguji, H & Hihara, H (2007) Correlation for boiling heat transfer of R-134a in horizontal tubes including effect of tube diameter, International Journal of Heat Mass Transfer, 50, 5215-5225 Schwartz, L.W.; Princen, H.M & Kiss, A.D (1986) On the motion of bubbles in capillary tubes, Journal of Fluid Mechanics, 172, 259–275 Shedd, T A & Newell, T A (2004) Characteristics of the liquid film and pressure drop in horizontal, annular, two-phase flow through round, square and triangular tubes, Journal of Fluid Engineering, 126, 807-817 Taha, T & Cui, Z F (2006) CFD modelling of slug flow inside square capillaries, Chemical Engineering Science, 61, 665-675 Takamasa, T & Kobayashi, K (2000) Measuring interfacial waves on film flowing down tube inner wall using laser focus displacement meter, International Journal of Multiphase Flow, 26(9), 1493-1507 Taylor, G I (1961) Deposition of a viscous fluid on the wall of a tube, Journal of Fluid Mechanics, 10(2), 161-165 Thome, J R.; Dupont, V & Jacobi, A M (2004) Heat transfer model for evaporation in microchannels Part I: presentation of the model, International Journal of Heat Mass Transfer, 47(14-16), 3375-3385 Tibirica, C B.; Nascimento, F J & Ribatski, G (2010) Film thickness measurement techniques applied to micro-scale two-phase flow systems, Experimental Thermal and Fluid Science, 34 (4), 463-473 Ursenbacher, T.; Wojtan, L & Thome, J R (2004) Interfacial measurements in stratified types of flow Part I: New optical measurement technique and dry angle measurements, International Journal of Multiphase Flow, 30, 107-124 Utaka, Y.; Okuda, S & Tasaki, Y (2007) Structure of micro-layer and characteristics of boiling heat transfer in narrow gap mini-channel system, Transactions of the JSME, Series B, 73(733), 1929-1935 Wong, H., Radke, C J & Morris, S (1995a) The motion of long bubbles in polygonal capillaries Part 1 Thin films, Journal of Fluid Mechanics, 292, 71-94 Wong, H., Radke, C J & Morris, S (1995b) The motion of long bubbles in polygonal capillaries Part 2 Drag, fluid pressure and fluid flow, Journal of Fluid Mechanics, 292, 95-110 16 New Variants to Theoretical Investigations of Thermosyphon Loop Henryk Bieliński The Szewalski Institute of Fluid-Flow Machinery, Polish Academy of Sciences, Gdańsk Poland 1 Introduction The purpose of this chapter is to present three variants of the generalized model of thermosyphon loop, using a detailed analysis of heat transfer and fluid flow (Bieliński & Mikielewicz, 2011) This theoretical investigation of thermosyphon loop is based on analytical and numerical calculations The first variant of thermosyphon loop (HHVCHV) is composed of two heated sides: the lower horizontal and vertical sides and two cooled sides: the upper horizontal and vertical opposite sides This variant is made for conventional tubes and has a one-phase fluid as the working substance The second variant of thermosyphon loop (2H2C) is consisted of two lower evaporators: horizontal and vertical and two upper condensers: horizontal and vertical and is made for minichannels The third variant of thermosyphon loop has an evaporator on the lower horizontal section and a condenser on the upper vertical section This variant contains minichannels and a supporting minipump (HHCV+P) A two-phase fluid is used as the working substance in the second and third variants The new variants reported in the present study is a continuation and an extension of earlier work “Natural Circulation in Single and Two Phase Thermosyphon Loop with Conventional Tubes and Minichannels.” published by InTech (ISBN 978-953-307-550-1) in book “Heat Transfer Mathematical Modelling, Numerical Methods and Information Technology”, Edited by A Belmiloudi, pp 475-496, (2011) This previous work starts a discussion of the generalized model for the thermosyphon loop and describes three variants In the first variant (HHCH) the lower horizontal side of the thermosyphon loop was heated and its upper horizontal side was cooled In the second variant (HVCV) the lower part of vertical side of the thermosyphon loop was heated and its upper part at opposite vertical side was cooled In the third variant (HHCV) a section of the lower horizontal side of the thermosyphon loop was heated and its upper section of vertical side was cooled A one- and two-phase fluid were used as a working substance in the first and in both second and third variants of the thermosyphon loop, respectively Additionally, the first variant was made for conventional tubes and the second and third variants were made for minichannels It was necessary in case of the thermosyphon loop with minichannels to apply some new correlations for the void fraction and the local two-phase friction coefficient in both two-phase regions: adiabatic and diabatic, and the local heat transfer coefficient in flow boiling and condensation Some other variants to theoretical investigations of the generalized model for thermosyphon loop are demonstrated in (Bieliński & Mikielewicz, 2004, 2005, 2010) 366 Two Phase Flow, Phase Change and Numerical Modeling Fluid flow of thermosyphon loop is created by the buoyancy forces that evolve from the density gradients induced by temperature differences in the heating and cooling sections of the loop An advanced thermosyphon loop is composed of an evaporator and a condenser; a riser and a downcomer connect these parts A liquid boils into its vapour phase in the evaporator and the vapour condenses back to a liquid in the condenser The thermosyphon loop is a simple passive heat transfer device, which relies on gravity for returning the liquid to the evaporator The thermosyphon loops are a far better solution than other cooling systems because they are pumpless In such cases, when mass flow rate is not high enough to circulate the necessary fluid to transport heat from evaporator to condenser, the use of a pump is necessary The presented study considers the case where the buoyancy term and the pump term in the momentum equation are of the same order The following applications for thermosyphon loops are well-known, such as solar water heaters, thermosyphon reboilers, geothermal systems, emergency cooling systems in nuclear reactor cores, thermal diodes and electronic device cooling The thermal diode is based on natural circulation of the fluid around the closed-loop thermosyphon (Bieliński & Mikielewicz, 1995, 2001), (Chen, 1998) The closed-loop thermosyphon is also known as a “liquid fin” (Madejski & Mikielewicz, 1971) Numerous investigations, both theoretical and experimental have been conducted to study of the fluid behaviour in thermosyphon loops Zvirin (Zvirin, 1981) presented results of theoretical and experimental studies concerned with natural circulation loops, and modelling methods describing steady state flows, transient and stability characteristics Ramos (Ramos et al., 1985) performed the theoretical study of the steady state flow in the two-phase thermosyphon loop with conventional tube Greif (Greif, 1988) reviewed basic experimental and theoretical work on natural circulation loops Vijayan (Vijayan et al., 2005) compared the dynamic behaviour of the single- and two-phase thermosyphon loop with conventional tube and the different displacement of heater and cooler Misale (Misale et al., 2007) reports an experimental investigations related to rectangular single-phase natural circulation mini-loop The present study provides in-depth analysis of heat transfer and fluid flow using three new variants of the generalized model of thermosyphon loop Each individual variant can be analyzed in terms of single- and two-phase flow in the thermosyphon loop with conventional tubes and minichannels In order to analyse the numerical results of simulation for the two-phase flow and heat transfer in the thermosyphon loop, the empirical correlations for the heat transfer coefficient in flow boiling and condensation, and two-phase friction factor in diabatic and adiabatic sectors in minichannels, are used The analysis of the thermosyphon loop is based on the one-dimensional model, which includes mass, momentum and energy balances The separate two-phase flow model is used in calculations A numerical investigation for the analysis of the mass flux and heat transfer coefficient in the steady state has been done The effect of thermal and geometrical parameters of the loop on the mass flux in the steady state is examined numerically The El-Hajal correlation for void fraction (El-Hajal et al., 2003), the Zhang-Webb correlation for the friction pressure drop of two-phase flow in adiabatic region (Zhang & Webb, 2001), the Tran correlation for the friction pressure drop of two-phase flow in diabatic region (Tran et al 2000), the Mikielewicz (Mikielewicz et al., 2007) and the Saitoh (Saitoh et al., 2007) correlations for the flow boiling heat transfer coefficient in minichannels, the Mikielewicz (Mikielewicz et al., 2007) and the Tang (Tang et al., 2000) correlations for condensation heat transfer coefficient in minichannels has been used to evaluate the thermosyphon loop with minichannels 367 New Variants to Theoretical Investigations of Thermosyphon Loop Finally, theoretical investigations of the variants associated with the generalized model of thermosyphon loop can offer practical advice for technical and research purposes 2 Single phase thermosyphon loop heated from lower horizontal and vertical side and cooled from upper horizontal and vertical side This single-phase variant of thermosyphon loop is heated from below horizontal section  (s 0 ≤ s ≤ s1 ) and vertical section (s 1 ≤ s ≤ s 2 ) by a constant heat flux: q H Constant heat flux  q H spaced in cross-section area per heated length: L H In the upper horizontal section (s 3 ≤ s ≤ s 4 ) and opposite vertical section (s 4 ≤ s ≤ s 5 ) the thermosyphon loop gives heat to the environment The heat transfer coefficient between the wall and environment, αC , and the temperature of the environment, T0 , are assumed constant The heated and cooled parts of the thermosyphon loop are connected by perfectly isolated channels (s 2 ≤ s ≤ s 3 ; s 5 ≤ s ≤ s6 ) S2 COOLED SECTION S3 INSULATION α C ; T0 INSULAT ED SECTION S4 αC T0 S q H H z y ψ x INSULATION INSULAT ED SECTION HEATED SECTION S1 S0 S6 B (a) S5 G (b) Fig 1 The variant of single phase thermosyphon loop heated from lower horizontal and vertical side and cooled from upper horizontal and opposite vertical side (HHVCHV) (a) 3Dimensional, (b) 2D The space co-ordinate s circulates around the closed loop as shown in Fig 1(b) The total length of the loop is denoted by L, cross-section area of the channel is A, wetted perimeter is U Thermal properties of fluid: ρ - density, cp - heat capacity of constant pressure, λ thermal conductivity The following assumptions are used in the theoretical model of natural circulation in the closed loop thermosyphon: 1 thermal equilibrium exists at any point of the loop, 2 incompressibility, because the flow velocity in the natural circulation loop is relatively low compared with the acoustic speed of the fluid under current model conditions, 368 Two Phase Flow, Phase Change and Numerical Modeling 3 4 5 viscous dissipation in fluid is neglected in the energy equations, heat losses in the thermosyphon loop are negligible, ( D L ) 1000 for  ; ( Re G ) > 1000  0.5 ρ  ⋅ G   ρL  0.5 μ  ⋅ L   μG  0.5   ( Re L ) < 1000 for  ; ( Re G ) > 1000  Table 2 Minichannels Correlation for the heat transfer coefficient in flow boiling (18) 377 2 hTPB [ W / m *K ] New Variants to Theoretical Investigations of Thermosyphon Loop 3 2x10 MIKIELEWICZ (2007) SAITOH (2007) 3 1x10 2 3 5x10 1x10 2 qH1 [ W / m ]  Fig 9 Minichannels Heat transfer coefficient in flow boiling h TPB as a function of q H1 in the first evaporator (L=0.2 [m], D=0.002 [m], H=0.07 [m], B=0.03 [m], LH1= LH2=LC1= LC2=0.02 [m], LH1P= LH2P =LC1P= LC2P =0.005 [m]) 3.3 Minichannels The heat transfer coefficient in condensation The condensation heat transfer coefficient for minichannels was calculated using the Mikielewicz formula Eq (17) The term which describes nucleation process in this formula was neglected The heat transfer coefficient for condensation in minichannels was also calculated using the modified Tang formula (Tang et al., 2000) 3 MIKIELEWICZ (2007) TANG (2000) 2 hTPC [ W / m *K ] 2,0x10 3 1,5x10 2 3 5x10 1x10 2 qC1 [ W / m ]  Fig 10 Minichannels Heat transfer coefficient h TPC as a function of q C1 in the first condenser (L=0.2 [m], D=0.002 [m], H=0.07 [m], B=0.03 [m], LH1= LH2=LC1= LC2=0.02 [m], LH1P= LH2P =LC1P= LC2P =0.005 [m] )  The condensation heat transfer coefficient for minichannels h TPC versus heat flux q C1 in the first condenser is presented in Fig 10 378 Two Phase Flow, Phase Change and Numerical Modeling Researcher Mikielewicz et al., (2007) Tang et al., (2000) Correlation JM h TPB = h REF h TANG TPC (R ) M −S n ; 0.836      p SAT       ( − x ) ⋅ ln      λL     pCRIT       = ( Nu ) ⋅   ;  ⋅ 1 + 4.863 ⋅   (1 − x) D                (19) (20) Table 3 Minichannels Correlation for the condensation heat transfer coefficient 3.4 The effect of geometrical and thermal parameters on the mass flux distributions The effect of the internal diameter tube D on the mass flux for the steady-state conditions is presented in Fig 11 The mass flux rapidly increases with increasing internal diameter tube D The GDR (Gravity Dominant Region) decreases with decreasing internal diameter tube D   Fig 11 Mass flux G as a function of q H with internal diameter tube D as the parameter (2H2C) (L=0.2 [m], H=0.07 [m], B=0.03 [m], LH1= LH2=LC1= LC2=0.02 [m], LH1P= LH2P =LC1P= LC2P =0.005 [m] ) The effect of length of the heated section LH2 on the mass flux is demonstrated in Fig 13 If the length of horizontal section B is constant, the mass flux increases with increasing length of vertical section H due to the increasing gravitational pressure drop The increase in length of the vertical section H induces an increase in both the gravitational and frictional pressure drop However, the gravitational pressure drop grows more in comparison to frictional pressure drop The effect of length of the heated section LH2 on the mass flux is demonstrated in Fig 12 The mass flux increases with increasing length of the heated section LH2 in gravity dominant region (GDR) but in friction dominant region (FDR) the mass flux decreases with increasing length of the heated section LH2 ... steady 0.670Ca + 3.13Ca + 0.504Ca0.672 Re 0.589 − 0.352 We 0.629 (Ca < 0.3, Re < 2000) , (21) 352 Two Phase Flow, Phase Change and Numerical Modeling where Ca = μU/σ and Re = ρUDh/μ and We = ρU2Dh/σ... Fig Mass flow rate for laminar flow at the steady state versus modified Rayleigh number at different G/H ratios (HHVCHV) 372 Two Phase Flow, Phase Change and Numerical Modeling MASS FLOW RATE... the water experiment is much smaller than those of other fluids and the 354 Two Phase Flow, Phase Change and Numerical Modeling numerical results It is confirmed that inertial effect must be considered