Experimental and numerical investigations of local condensation heat transfer in a single square microchannel under variable heat flux

International Communications in Heat and Mass Transfer 71 (2016) 197–207

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Experimental and numerical investigations of local condensation heat transfer in a single square microchannel under variable heat flux☆ H. El Mghari, H. Louahlia-Gualous ⁎ Caen Basse Normandie University, LUSAC Laboratory, 120 rue de l'exode, 50000 Saint Lô, France

a r t i c l e

i n f o

Available online 31 December 2015 Keywords: Condensation Noncircular microchannel Annular flow Modeling Measurements Local heat transfer

a b s t r a c t This paper presents an experimental investigation on the local and average condensation heat transfer in a single noncircular microchannel. Furthermore, it develops one dimensional model for annular condensation flow in a microchannel under variable heat flux condition. The condensate film thickness is calculated for each location in a microchannel including the effects of capillary number, Boiling number, contact angle, heat flux, vapor pressure, and hydraulic diameter. A comparative study shows that the present model well predicts the experimental data concerning local condensation heat transfer coefficient. The mean deviation between the local predictions of the theoretical model with the measurements for local heat transfer coefficient is 20%. It is found that the correlation of Quan et al. (2008) [19] gives the good predictions of the measurements with maximum deviation of 13% at high Reynolds number. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction In recent reviews of the literature, various fundamental researches on two-phase heat transfer and pressure drop characteristics in microchannels have been conducted. Numerous studies have focused on evaporation or boiling inside microchannels since they are basic processes used in microcooling systems at high heat fluxes. However, efficiency of two-phase loop heat pipe used for cooling is mainly influenced by thermal performance of both condenser and evaporator. Analysis of condensation heat transfer in micro/minichannels is very important in development of next generation of ultra-compact and high performance two-phase flow cooling loops. Recent experimental and theoretical results indicate that the physical mechanisms of condensation in microchannels are quite different from those that occurred in conventional scale channels. Gravity and buoyancy are the typically forces governing condensate flow in macrochannels. Reduction of the channel size from macroscale to the microscale channel induces surface tension and shear stress as the dominating forces in microchannels as explained by El Mghari et al. [1]. Condensation in macrochannels involves many engineering applications and it has been widely studied and understood. Recently, systems miniaturization brings new challenges and opportunities particulary, in the flows with phase change applications. Additionally, small channels size can resist to high system pressure allowing the use of high pressure fluids as carbon dioxide in transcritical cycle equipment. However, correlations of condensation heat transfer and pressure ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (H. Louahlia-Gualous).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.12.021 0735-1933/© 2015 Elsevier Ltd. All rights reserved.

drop in macrochannels cannot be directly applied in the microsystems field. Various studies on condensation in microchannels are focussed on two-phase flow patterns. Louahlia-Gualous and Mécheri [2] studied experimentally steam condensation flow patterns in a single capillary glass tube. By varying the inlet pressure and cooling rate, annular flow, stratified flow, bubble flow and elongated spherical bubbles were observed. El Mghari et al. [3] studied nanofluid condensation heat transfer inside a single horizontal smooth square microchannel. The numerical results are compared to previous experimental predictions, and show that the heat transfer coefficient can be improved 20% by increasing the volume fraction of Cu nanoparticles by 5% or increasing the mass flux from 80 to 110 kg/m2 s. Wu et al. [4] investigated condensation flow in wide rectangular silicon microchannels with the hydraulic diameter of 90.6 μm and width/depth ratio of 9.668. Droplet-annular compound flow, injection flow, and vapor slugbubbly flow are observed along the microchannel. Chen et al. [5] studied condensation flow patterns in the silicon triangular microchannels. They proposed correlations of bubbles injection location in the microchannel, bubbles frequency and condensation Nusselt number. Chen and Cheng [6] carried out flow visualization for steam condensation in trapezoidal microchannels with 75 μm hydraulic diameter, cooled by natural air convection at room temperature. Odaymet et al. [7] analyzed relationship between condensation heat transfer and various flow patterns in a single silicon microchannel. They measured local heat transfer through microinstrumentation of the microchannel. Louahlia-Gualous and Odaymet [8] showed that bubbles velocity is variable along the microchannel length and coalescence phenomenon increases flow velocity and reduces bubbles frequency. Concerning condensation heat transfer, Garimella et al. [9] investigated effect of various multimicrochannels shapes and sizes on

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Nomenclature A a b C Cp Dh f h hfg L l  m Nu P P Pr q R Re T U Xtt z

area (m2) width of rectangular microchannels (m) depth of rectangular microchannels (m) coefficient specific heat ( J/kg °C) hydraulic diameter (m), ≡4A=P friction factor heat transfer coefficient (W/m2°C) latent heat (J kg−1) annular condensation length (m) length of the end part in condensation(m) mass flux (kg s−1) Nusselt number, ≡hDh/λ pressure (Pa) perimeter (m) Prandtl number, ≡μCp/λ heat flux (W m−2) curvature radius (m) Reynolds number temperature, (°C) velocity (m s−1) Martinelli parameter axial coordinate (m)

Greek symbols β half of right angle (°) δ film thickness (m) μ viscosity (kg m−1s−1) λ thermal conductivity (W/m°C) φ angle (°) θ contact angle (°) ρ density (kg m−3) σ surface tension coefficient (N m−1) τ shear stress (N m−2) Subscripts c critical g gravity in inlet l liquid lw liquid–wall interface lv liquid–vapor interface red reduced v vapor t total

condensation heat transfer and pressure drop. They developed a pressure drop model predicting 90% of the measured data within 28%. Recently, Quan et al. [10] measured steam condensation heat transfer in two trapezoidal silicon microchannels under three sides cooling conditions. A semi-analytical method for annular condensation is developed basing on turbulent flow boundary layer theory of liquid film. Wang and Rose [11] developed a new model of film condensation in noncircular microchannels including effects of microchannel inclination and shape (square, triangle, inverted triangle, and circular). LouahliaGualous and Asbik [12] developed a numerical model for annular film condensation heat transfer of a binary mixture refrigerants inside a circular miniature tube. The two-dimensional governing equations are solved in the liquid and vapor phases including the interfacial conditions of heat and mass transfer. Chen et al. [13] investigated annular condensation in triangular microchannels. They analyzed the film

curvature radius distribution along the microchannel and the total condensation length under various sizes of the microchannel. From the above literature review, it is clear that there is no reliable model for predicting condensation heat transfer coefficients in a noncircular microchannel including variation of the local coolant heat flux. Also, a major part of the experimental studies on condensation in the microscale channel are conducted through multimicrochannels. In this paper, an experimental setup on condensation heat transfer inside a single silicon microchannel is investigated. An innovative technique to measure local condensation heat transfer coefficient in a silicon microchannel is proposed. This paper reports also the first numerical results of condensation in a noncircular microchannel under variable local heat flux. 2. Experimental setup Experiments on steam condensation heat transfer in a single silicon microchannel are carried out in the test loop shown in Fig. 1a and described in references [7,8]. The working fluid pressure is controlled in a steam generator (1). The saturated vapor was directed to the preheater (3) in order to prevent condensation in tubes (5). A 2 μm filter (2) was used to eliminate any dust in the working fluid. The vapor mass flow rate at the microchannel entrance is adjusted through a regulating valve (4). Flow temperatures at the microchannel inlet and outlet are measured using 75 μm microthermocouples. Pressure is measured by strain gage type pressure transducer. The test section (Fig. 1b) is cooled by water with a controlled inlet temperature. The cooling system consists of the tank containing a thermostatic water (9), a pump (10) for driving the water to the test section inlet and a flow meter (12) with an uncertainty of 4%. Water leaving the test section, is cooled in the heat exchanger (13). The vapor flowing the test section was led into a condenser (6) where it was completely condensed. The collected liquid was weighed during a known period of time with an electronic balance of high precision. The inlet and the outlet temperatures of the cooled water were measured through the microthermocouples (K-type, 75 μm). 2.1. Description of the test section and experimental procedure A 50 mm microchannel is etched in a silicon wafer having a thickness of 1000 μm (Fig. 1b). It is covered using a thin transparent Pyrex glass (500 μm thickness), anodically bonded to the silicon plate top surface. Pyrex glass allows condensate flow visualization in the microchannel. Seven chromel-alumel microthermocouples (50 μm diameter) are placed inside the rectangular microgrooves to measure wall temperature as shown in (Fig. 1b). They are located at 20 μm from the microchannel inner surface. Condensation process is studied using rectangular microchannel with hydraulic diameter of 305 μm (depth of 310 μm and width of 300 μm). Microthermocouples are placed at 1 mm from the channel inlet and being equally spaced with 8 mm up to the channel outlet. Before starting the tests, the vapor generator is degassed and steam pressure is adjusted at the microchannel inlet. Vapor temperature and pressure at the microchannel entrance are measured and controlled in order to make tests at the saturated state. During all tests, the microchannel is illuminated by a cold light source using two optical fiber arms that does not affect the microchannel heat transfer. A high speed camera detecting up to 16,000 frames/second is used to record different flow structures in the microchannel. 2.2. Experimental data and uncertainties Condensation local heat transfer coefficient is estimated at steady state by hz ¼

qchannel;z Ts;z −Tsat;z

ð1Þ

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199

(a) Vapour entrance Condensate exit

Microthermocouples

Microchannel

(b) Fig. 1. Experimental setup: (a) schematic diagram, (b) the silicon microchannel instrumentation.

where Ts,z is the local surface temperature given by microthermocouples in the microgrooves, Tsat,z is the local saturation temperature determined from the local pressure of the flow in the microchannel, and qchannel,z is the local heat flux given by qchannel;z

Tc;z −Tw;z ¼ λw Δyw




λsi λw Ts;z þ Tw;z Δysi Δyw

  Pz ¼ Pin −ΔPcon − ΔPm;tp þ ΔPfrict;tp z

ð4Þ

ð2Þ

where λw is the fin thermal conductivity, Tc,z is the contact temperature between the brass fin upper side and the microchannel bottom side as shown by Fig. 2. Tw,z is the wall temperature measured inside the cooling fin at 4.5 mm from the contact surface, and Δyw is the distance between the contact surface and thermocouple location Tw,z. Assuming one dimensional heat transfer in the brass fin, the contact temperature could be given by Tc;z ¼

The condensate flow pressure in the microchannel is given for each z location by




λw λ þ si Δyw Δysi

−1

ð3Þ

Pin is the inlet steam pressure, ΔPcon is the pressure loss resulting from sudden contraction at the microchannel inlet [14]. ΔPm,tp is the momentum pressure drop estimated by Quibén et al. [15] and the void fraction is calculated with the Rouhani and Axelson model [16]. The two-phase frictional pressure ΔPfrict,tp is obtained with Lockhart-Martinelli equation [17]. Pressure sensors and thermocouples calibrations are carried out by comparing responses of each component to those measured by a high precision sensor probes. The uncertainties for different parameters involved in the measurements are ± 0.1 °C for temperature, ±0.2% for the pressure, ±1.5% for the power and ±8% for the thermal resistance.

Fig. 2. The silicon plate on the cooling brass fin.

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3. Condensation heat transfer modeling 3.1. Physical approach In this section, a stationary model for steam condensation in a rectangular microchannel is proposed. The considered geometric configuration is schematized in Fig. 3. The microchannel has a rectangular crosssection with hydraulic diameter Dh. The channel is cooled by an external variable heat flux, with a convective heat transfer coefficient between the channel outer surface and the coolant. A steam mass flow rate is fixed at the microchannel entrance. Upon contact with the channel cold surface, vapor condenses and a liquid film forms. The film thickness varies with the quality, which is itself depends on the axial position. The liquid film end is in an almost hemispherical meniscus when the quality tends towards zero. Its position is one of the unknowns in the problem.

The first terms in Eq. (7) are the rate of creation of momentum plus the rate of inflow of momentum within the control element. The second terms represent the pressure forces in the control element. The third term represents the wall shear force, where τlw is the liquid–wall shear stress and P lw is the liquid–wall contact perimeter. The fourth term represents the sum of the interfacial shear forces, where τvl is the liquid–vapor shear stress and P vl is the liquid–vapor contact perimeter. The last term represents the rate of momentum generation of phase k due to mass transfer. Rearranging these terms, Eq. (7) can be written as: −Aαk

dpk d _ U Þ −τkw P kw þ τvl P vl −Uk Γk ¼ δz ðm dz k k dz

ð8Þ

Thus for a steady state, the following equation is obtained for the vapor phase

3.2. Mathematical approach The mathematical formulation of the problem is based on the following simplifying assumptions: (i) the thermophysical properties are constant, (ii) the flow in the microchannel is laminar, (iii) gravity is negligible compared to the effects of surface tension, (iv) condensation process is symmetrical in the microchannel, (v) the condensate length can be omitted in the model, (vi) the heat flux is variable as a polynomial trend. The mass conservation equation for the “k” phase in stationary regime is: ∂ ðρ Uk Ak Þ ¼ Γk ∂z k

ð5Þ

where ρk is the density, Uk is the average velocity of phase k, Ak is the  phase k area and Γk is the mass flow rate mk per unit length) to phase k from the various interphase mass transfer, where X X dm _k Γk ¼ ¼0: dz

ð6Þ

The subscripts k = l for the liquid phase, and k = v for vapor phase. For two-phase flow in steady state, the momentum creation rate of phase k and the momentum inflow rate balanced against the sum of the forces acting on that phase plus the momentum generation due to mass transfer, as follows:

3

4

_ l dU1 −Al dpl −τlw P lw dz þ τvl P vl dz−Ul Γl dz ¼ m

τvl ¼

1 ρ f v U2v 2 v

ð11Þ

τlw ¼

1 ρ f U2 2 l l l

ð12Þ

The cross-sections are expressed as

cosðθ þ βÞ cosðθÞ π − þ β þ θ R2 ¼ a R2 sinðβÞ 2

Av ¼ D2h −Al ¼ D2h −a R2

ð13Þ ð14Þ

The local energy balance equation in the liquid phase can be written

5

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ zfflfflfflfflfflffl}|fflfflfflfflfflffl{ zffl}|ffl{ −τ1w P 1w δz þ τv1 P v1 δz þ Uk Γ k

ð10Þ

where pl is pressure in the liquid phase, Al is the liquid cross-section in one corner, P vl is the liquid–vapor perimeter, P lw is the liquid–wall perimeter, τvl is the shear stress at liquid–vapor interface, ṁl is the liquid mass flow rate, τlw is the shear stress at the microchannel heat exchange surface defined as:

Al ¼

ð7Þ

ð9Þ

where Pv is vapor pressure, Av is the vapor cross-section , dSvL is the liquid–vapor interface surface along dz, dSLw is the wet heat exchange surface along dz, τvL is the shear stress at liquid–vapor interface, τw is the wall shear stress, and ṁv is the vapor mass flow rate, Uv is the vapor velocity. For the liquid phase



1

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{
 d _ k Uk Þ −m _ k Uk þ δz ðm _ k Uk m dz 2 zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl

 ffl{ d d ¼ Aαk p− Aαk p þ δz ðAαk pÞ þ p −δz ðAαk Þ dz dz 1

_ v dUv −Av dpv −τvw P vw dz−τvl P vl dz−Uv Γv dz ¼ m

as:

where αk is the k phase void fraction defined as αk = Ak/A.

_ l ¼ qðzÞPdz hfg dm

ð15Þ

Non circular microchannel A’ Vapour entrance

A-A’ Vapour Condensate film

A Fig. 3. Annular condensation flow in a non-circular microchannel.

H. El Mghari, H. Louahlia-Gualous / International Communications in Heat and Mass Transfer 71 (2016) 197–207

3.3. Solution procedure and boundary conditions

where

 z qðzÞ ¼ q0 exp n Dh

ð16Þ

hfg, q0, n et Dh are the latent heat, the constant heat flux density , the number and hydraulic diameter respectively. The total energy equation defined in the length of the total condensation zone as: _ v;in hfg ¼ q0 P Q tot ¼ m

Lnþ1 Dh n

ð17Þ

where L is the length of the total condensation zone and P is the crosssection perimeter. In the vapor phase, the local energy balance equation is _ v ¼ −qðzÞPdz hfg dm

ð18Þ

Introducing the following variables used in non-dimensionalizing the pressures, velocities, and curvature radius Eqs. (22)–(26): R  z  L ρ P  P Dh ; ;z ¼ ;L ¼ ; ρ ¼ v ; P ¼ ;P ¼ Dh Dh Dh Dh ρL 2σ A ρv Uv A ρL UL ; UL ¼ UV ¼ _ v;in _ v;in m m

R ¼

4. Results and discussions

 σ σ dPl ¼ d Pv − ¼ dPv þ 2 dR R R

4.1. Condensation flow structures in microchannel ð19Þ

Using the Eqs. (11)–(16) in the Eq. (8), we have
 σ 1 1 −a R2 dPv þ 2 dR − ρl f l U2l P lw dz þ ρv f v U2v P vl dz 2 2 R  −aρl d R2 U2l ¼ 0

ð20Þ

Introducing Eq. (9), Eq. (17) becomes  0 1 2 2 d ρ A U v v v 1 ρ f U P σ 1 vl 2@ v v v dz þ þ 2 dR A− ρl f l U2l P lw dz −a R 2 2 Av Av R  1 þ ρv f v U2v P vl dz−aρl d R2 U2l ¼ 0 2 ð21Þ  n

2q0 Dzh PAL Uv U

1 1 2 AL 2 f v ρv P vl Uv þ 1 − f l ρl P lw Ul − þ l 2 2 Av hfg Av Al σa−2

ð27Þ

The equations of pressures, velocities, and radius curvature are written in non-dimensional form. The governing equations combined with boundary conditions are solved through an iterative procedure described by El Mghari et al. [1]. At each abscise z, all parameters are calculated particularly, the liquid and vapor velocities, the pressure drop, the liquid film thickness, etc. The convergence criterion used for estimating all parameters is set at 10−12. Calculations are stopped when total condensation is reached in the tube or vapor quality became lower than 1%.

Closure of the system is obtained by writing the Laplace–Young equation as:

dR ¼ dz

201

ρv A2l 2 2 ρl Al 2 U − Ul Av R v R ð22Þ

In this work, numerous experiments are carried out on condensation inside a single microchannel under different inlet vapor mass fluxes. Different condensation flow patterns are identified in the microchannel: mist flow, annular flow, slug flow, and injection/bubbly flow as shown by Fig. 4. In this figure, condensation flow direction in the microchannel is from the left to the right. Mist flow (Fig. 4a) is generally observed at high shear and inertia forces. It is constituted by microdroplets dispersed in the vapor flow and flows along an oscillatory trajectory to form a churn and annular flow (Fig. 4b). Interfacial waves may be appeared and grow during time until breakup of the vapor core and departure of different bubbles sizes and shape (Fig. 4c). Bubbles coalescence leads to formation of elongated bubbly flow with liquid slug separating one bubble from the other (Fig. 4d). Elongated bubbles and liquid slugs having different sizes, travel the microchannel from the entrance to the exit. Bubble size may be reduced and liquid slug size may be increased because of the condensation process on the bubbles surface. In this work, our attention is focussed on the numerical and experimental analysis of the local condensation heat transfer of annular flow that is one of the basis developed condensation flow pattern in microchannels. In the literature, numerical and experimental investigations on heat transfer and pressure drop for condensation flow inside a single microchannel are very limited compared to the large studies on boiling heat transfer in microchannel. 4.2. Condensation heat transfer

The form of the system resolving is shown in system below: q0

 n z Dh

P

dUL ðzÞ UL dR ¼ −2 dz R dz AL ρL hfg  n q0 Dzh P 2 A m dUv L _ v dR ¼− þ dz ρv Av hfg ρv A2v R dz  n z dpL 1 f vL ρv U2v P vL 1 f L ρL U2L P Lw 2q0 Dh PUL 2ρL U2L dR ¼ − − þ 2 2 dz AL AL AL hfg R dz

dPv dUv ρ U2 AL dR f vL ρv U2v P vL − ¼ −2ρv Uv þ2 v v dz dz Av R dz 2 Av

ð23Þ

Annular flow is a stable condensation flow pattern widely occurred in the microchannel at high vapor velocity because condensate film is

ð24Þ

ð25Þ

ð26Þ

Fig. 4. Condensation flow patterns in the microchannel: (a) mist flow, (b) annular flow, (c) injection bubbly flow, (d) slug flow.

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G=125kg/m²s

G=75kg/m²s

G=140kg/m²s

G=160kg/m²s

G=110kg/m²s

(a)

(b)

(c) Fig. 5. Surface temperature, heat flux, and heat transfer coefficient for different inlet vapor mass fluxes.

H. El Mghari, H. Louahlia-Gualous / International Communications in Heat and Mass Transfer 71 (2016) 197–207

+20%

[19] [18]

203

[20]

-20%

Fig. 6. Average Nusselt number versus vapor Reynolds number.

entrained from the microchannel under shear stress at the liquid–vapor interface. For annular flow, vapor has enough energy to move up the condensate film. For various inlet vapor mass fluxes, Fig. 5a, b and c show the local surface temperature, heat flux, and heat transfer coefficient respectively measured for annular flow in the microchannel. Experimental results presented in these figures are obtained for various inlet vapor velocity. The surface temperature for annular flow is high for the upstream flow than that of the downstream flow where condensate film is thicker (Fig. 5a). The local heat flux (Fig. 5b) is high near the microchannel inlet and it decreases along the flow direction where the increase of the condensate film leads to the increase in the thermal resistance between the vapor and the coolant. Therefore, local heat transfer coefficient for annular flow becomes very low in the microchannel downstream as shown by Fig. 5c. Fig. 6 shows the average Nusselt number versus the vapor Reynolds number. Experimental results show that condensation average heat transfer increases with increasing inlet vapor velocity. Average Nusselt number is high for high vapor velocity that leads an increase of the liquid velocity and a decrease in the condensate film thermal resistance in the microchannel. Moreover, comparison between the measured average condensation Nusselt numbers and those determined from the correlations of Mishima and Hibiki [18], Quan et al. [19] and Dong and Wang [20] are shown in Fig. 6. It can be seen that all the correlations under predict condensation heat transfer in the microchannel except the The average values from the tested correlations were deduced from the local predictions for vapor quality ranging from 1 to 0 corresponding to the total condensation in the

Fig. 8. Comparison between numerical and experimental condensation local heat transfer coefficient.

channel. Correlation of Mishima and Hibiki [18] underestimates the average condensation heat transfer in microchannel by a mean deviation of 23% obtained at high vapor Reynolds number. It can be seen that the correlation of Quan et al. [19] gives the good predictions of the measurements with maximum deviation of 13% at high Reynolds number. Quan et al. [19] correlation is defined for condensation of steam in the trapezoidal silicon microchannels with the aspect ratio of 3.15. This correlation gives a reasonable prediction of the present measurements. On the other side, correlation of Dong and Wang [20] overestimates the condensation heat transfer coefficient at high vapor Reynolds number where a maximum mean deviation is of 12%. From the literature review, there is no simple correlation proposed to predict average heat transfer for total condensation in square microchannel. For this reason, a new equation of average condensation heat transfer in a square microchannel is proposed basing on the experimental results. Through the obtained results and basing on the correlations presented in the previous works on condensation heat transfer in a macrochannel, average Nusselt number is defined for annular condensation heat transfer in a single microchannel as:

Num ¼ 0:141Re f 0:72 Pr−0:2

Fig. 7. Local condensation heat transfer coefficient in the microchannel.

1:6 X0:85 tt

! ð28Þ

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Fig. 9. Dimensionless condensate film thickness versus vapor Reynolds liquid number.

where Pr ¼

μ f C p; f λf

;

0:9 μ f 0:1 ρ 0:5 ð μ Þ ðρ v Þ , f v

a X tt ¼ ð1−x xa Þ

out and xa ¼ xin þx is the av2

erage quality, Xtt is the lockart Martinelli parameter, Pr is the Prandtl number. The predicted condensation Nusselt number through Eq. (28) are in good agreement with the present measurements within ±8%. 4.3. Verification of the numerical model To validate the numerical model presented above, the local heat transfer coefficients estimated using the proposed model are compared to the measurements for steam condensation in the microchannel. Calculations are conducted for each inlet vapor mass flux using the measured heat fluxes shown in Fig. 4b as a boundary condition. Experimental heat fluxes are interpolated using an exponential form as qz ¼ q0 e−nz

ð29Þ

where q0 and n are defined from the experimental results for each inlet mass flux and z* is dimensionless abscissa defined by Eq. (29).

According to the same experimental conditions, we measured and numerically calculated the local condensation heat transfer coefficient for each inlet vapor mass velocity. Comparison of these results is presented in Fig. 7 showing that the local experimental results are close to those predicted by the numerical model. The small gap between the numerical results and measurements is due to the simplifying assumptions of the model on the one hand and uncertainty in the measurements at microscale channel. The results of the computed local condensation heat transfer coefficient versus the measured values are presented in Fig. 8. It can be seen that the predicted local condensation heat transfer coefficients from the numerical model agree with experimental data with a maximum mean absolute error about ±20%. 4.4. Local characterization of liquid film thickness Numerical results are given in Fig. 9, for steam condensation in the microchannel using experimental conditions corresponding to the tests presented in Fig. 7. Computations were conducted for different mass fluxes tested during experiments corresponding to the vapor Reynolds numbers of 1807, 2402, 2578, 2930 and 3305. Fig. 9 shows numerical predictions of the dimensionless film thickness δ* versus the

Fig. 10. The circumferential distributions of the condensate film thickness at different location along the square microchannel: effect of inlet vapor mass flux.

H. El Mghari, H. Louahlia-Gualous / International Communications in Heat and Mass Transfer 71 (2016) 197–207 Table 1 Void fraction models and correlations. Reference

Model/correlation

Lockhart and Martinelli correlation [17] Chen correlation [21] Zivi model [22] Baroczy model [23] Turner and Wallis two-cylinder model [24] Fauske correlation [25] Spedding and Chen correlation [26]

h i 0:64 ρv 0:36 μ f 0:07 −1 α ¼ 1 þ 0:28ð1−x ðρ Þ ð μ Þ x Þ f v h i 0:6 ρv 0:33 μ f 0:07 −1 ðρ Þ ð μ Þ α ¼ 1 þ 0:18ð1−x x Þ f v i−1 h ρv 0:66 α ¼ 1 þ ð1−x x Þðρ f Þ h i 0:74 ρv 0:65 μ f 0:13 −1 ðρ Þ ð μ Þ α ¼ 1 þ ð1−x x Þ f v h i 0:72 ρv 0:4 μ f 0:08 −1 ðρ Þ ð μ Þ α ¼ 1 þ ð1−x x Þ f

h i−1 ρv 0:5 α ¼ 1 þ ð1−x x Þðρ f Þ h i 0:65 ρv 0:65 −1 ðρ Þ α ¼ 1 þ 2:22ð1−x x Þ

local film Reynolds number Ref = (1 − x)G Dh /μf δ* is defined as the ratio of condensate film thickness and the characteristic length:

δ ¼δ

μ 2f ρ2f g

According to these representations, it is seen that the symmetrical distributions of the condensate film thickness are obtained around the channel perimeter because the gravity has no effect on the condensate flow in the microchannel. Additionally, the condensate film thickness is very thick in the microchannel corners under the effect of the surface tension. The film thickness is to be very thin for distances near the microchannel inlet where the heat transfer is high.

4.5. Local prediction of void fraction

v

f



205

!−1=3 ð30Þ

It can be seen that for each inlet mass flux, the condensation film thickness increases and the total condensation length decreases with decreasing the inlet vapor mass flux. The reason is that the liquid and vapor velocities increase with the inlet mass flux leading an increase of the interfacial shear stress and the condensate film entrainment to the channel exit. As a result, the liquid film thermal resistance decreases and the local heat transfer increases along the channel. Furthermore, it is very interesting to note that doubling the inlet mass flux, reduces by 50% the total condensate length. Fig. 10 shows the condensate film thickness distributions along the circumference of the microchannel cross-section at different locations along the microchannel and for various values of the inlet mass flux. For the same location on the microchannel (z = 9 mm), the condensate film thickness is higher for 75 kg/m2s than other mass flux respectively and the condensation film thickness is about 19 μm for 75 kg/m2s and 4 μm for 160 kg/m2s. This is because the increase of the inlet mass flux decreases effect of capillary force leading to increase the difference between vapor and liquid pressure at the liquid–vapor interface.

Void fraction is defined as the ratio of vapor cross-sectional area and the total microchannel cross-section. It is an important parameter always used to characterize condensate film thickness, heat transfer coefficient, two-phase pressure drop, and flow pattern transition. In the literature review, numerous studies have been carried out on the modeling of the void fraction for two-phase flow. Various models and correlations are proposed for void fraction estimation: Lockhart and Martinelli correlation [17], Chen correlation [21], Zivi model [22], Baroczy model [23], Turner and Wallis two-cylinder model [24], Fauske correlation [25] and Spedding and Chen correlation [26]. These existing correlations, listed in Table 1, are proposed for two-phase flow in a macrochannel. Comparison between numerical estimations to the predictions from these correlations are illustrated in Fig. 11. It can be seen that the correlations of Lockhart and Martinelli correlation [17], Chen correlation [21], Zivi model [22], Baroczy model [23], Fauske correlation [25] and Spedding and Chen correlation [26] give the same predictions and over predict the void fraction in the microchannel. The correlations of Turner and Wallis two-cylinder model [24] have the same trend than the numerical results but also over predict them. No correlations give the satisfactory results in accordance with the numerical results. The new correlation is then proposed taking into account the capillary effect in the microchannel: "



 #−1 1−x 0:7 ρv 0:4 α ¼ 1 þ 0:241 x ρl

ð31Þ

Eq. (32) estimates local void fraction for condensation in the microchannel with average deviation of 3%.

Fig. 11. Void fraction versus vapor quality.

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Fig. 12. Two-phase friction factor versus equivalent Reynolds number for microchannel.

4.6. Pressure drop

5. Conclusions

Pressure drop is an important parameter to characterize two-phase flow. As shown by numerous authors, pressure gradient could be defined thanks to the force balance inside the vapor core assuming no variation of the liquid film thickness and no effect of gas compressibility. Then, the interfacial friction factor is given by [14]:

Experimental investigation on both local and mean heat transfer performances of steam condensation in microchannel is presented and the experimental setup is described in the present paper. A flowpattern-based model was developed to predict the heat transfer in noncircular microchannel condensation under variable heat flux condition. The curvature radius, condensation length and condensate film thickness distribution along the microchannel are obtained using numerical analysis. Some conclusions can be drawn as below: The condensation heat transfer and pressure drop correlations in microchannel are quite different from that in conventional channels. The simple new correlations are proposed basing on the measurements and modeling predictions. The new model for condensation in noncircular microchannel under no uniform coolant heat flux is proposed. It gave the good approximation of local heat transfer coefficient along the microchannel without need more than one iterative loop. The local and average condensation heat transfer depend heavily upon the wall heat flux distribution. A larger wall heat flux leads to a higher heat transfer coefficient and a lower total condensate length. A local comparison of the experimental data and the numerical predictions shows that the developed model gives the good predictions for local condensation heat transfer coefficient inside a noncircular microchannel. Moreover, the model developed in this work provides the best predictions for condensate film thickness distribution around the cross-section and along the microchannel length.

f tp ¼ Dh ρ f


 dP 1 dz F 2G2eq

ð32Þ

where "

Geq


0:5 # ρf ¼ G ð1−xa Þ þ ρv

ð33Þ

Two-phase pressure gradient can be calculated as the sum of three terms: frictional, gravitational, acceleration pressure gradient. The gravitational pressure is neglected in the microchannel. The frictional pressure drop ðdP Þ can be obtained from the following equation: dz F


 dP dP dP ¼− − − dz tp dz F dz ac

ð34Þ

The acceleration pressure gradient can be expressed as " ! !#
 x2in dP x2out ð1−xout Þ2 ð1−xin Þ2 ¼ G2 − − − − ð35Þ dz ac α out ρv ð1−α out Þρ f α in ρv ð1−α in Þρ f where xin is the inlet vapor quality , and xout is the outlet vapor quality Fig. 12 shows the variation of the two-phase friction factor versus the equivalent Reynolds number defined as: Req ¼ Re f þ Rev



0:5 ρf μv μL ρv

ð36Þ

Similar to the Nusselt number and void fraction, the proposed correlations are presented as function of equivalent Reynolds number. The proposed correlations are shown as follows: f tp ¼ 0:0202 Reeq −0:6

ð37Þ

As illustrated in Fig. 11, the results show that approximately 95% of data fall within a deviation of ±5%.

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