Turbulent direct-contact heat transfer between two immiscible fluids

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Turbulent direct-contact heat transfer between two immiscible ﬂ uids

Thierry Lemenand, Cédric Durandal, Dominique Della Valle, Hassan Peerhossaini

^*

Thermoﬂuids, Complex Flows and Energy Group, Laboratoire de Thermocinétique, CNRS UMR 6607, Ecole PolytechniqueeUniversity of Nantes, Rue Christian Pauc, BP 50609, 44306 Nantes Cedex 3, France

a r t i c l e i n f o

Article history:

Received 24 November 2009 Received in revised form 13 April 2010

Accepted 10 May 2010 Available online 22 June 2010

Keywords:

Direct contact heat exchanger Liquid/liquid dispersion Multifunctional exchanger/reactor Process intensiﬁcation

Heat transfer enhancement Static mixer

High efﬁciency vortex Energy efﬁciency Turbulent mixing

a b s t r a c t

Heat transfer between two immiscible liquid phases in turbulentflow is of great interest in improving the residence time, compactness, and energy cost of cooling and heating processes. The high-efficiency vortex (HEV) device used here as a direct-contact heat exchanger (DCHE) is a generic multifunctional exchanger/reactor in which wall tabs generate longitudinal vortices responsible for convective radial transfer that enhance macro-mixing, phase dispersion and fast temperature homogenization in theflow.

The experiments reported here concern a continuousflow of water in which an immiscible mineral oil is injected. The inlet water temperature ranges from 11 to 13C, and the inlet oil temperature from 40 to 48C; theflow Reynolds number varies between 7500 and 15 000. An algebraic one-dimensional thermal model accounting for the axial evolution of the phase temperatures coupled with drop breakup is developed and validated by the experimental thermal results in the DCHE. This model requires knowledge of the turbulentfield in single-phase conditions; it can be adapted to otherflow geometries and can be used as a sizing tool for engineering design.

Despite the phase separation at the outlet, the DCHE is more efﬁcient than a double-jacketed heat exchanger in terms of global Nusselt number. In addition, the HEV heat exchanger is energetically less costly than the other DCHE for the same heat-transfer capacity.

1. Introduction

Direct-contact heat transfer is based on heat transfer between a primaryﬂow and a dispersed phase (particles, drops, bubbles). A direct-contact heat exchanger (DCHE), even a simple duct, provides cost-effective heat transfer and is thus suitable for preheating and evaporating a working ﬂuid by thermal energy at low temperatures. The main idea underlying this process is to exploit the large interfacial area developed by the dispersed phase, while the absence of a solid wall between the two phases optimizes heat transfer without increasing pressure losses.

DCHE wereﬁrst investigated by Wilke et al.[1]and have since been the topic of numerous works. DCHE have been recommended for water desalination (Sideman and Gat[2], Letan[3]), crystalli- zation (Letan [3], Core and Mulligan [4]), energy recovery from industrial waste (Shimizu and Mori[5]), thermal energy storage (Core and Mulligan [4], Wright [6]) and ice-slurry production (Wijeysundera et al.[7]).

The aim of the present work is to investigate the DCHE process in a high-efﬁciency vortex (HEV) static mixer capable of improving

the mixing in a simple pipeﬂow by generating streamwise vorticity due to vortex generators that intensify radial transfer.

The workingfluids here are oil (hot, dispersed phase), injected at the inlet, and tap water (cold, continuous phase). The oil is dispersed in the waterflow by turbulence, so that no extra energy is needed to produce the droplets (Lemenand et al.[8]): liquid/liquid dispersion and heat transfer take place concomitantly. The oil fraction used in the experiments is 15%, high enough to get a significant temperature change in the water but guaranteeing a low coalescence rate. Oil and water temperatures arefixed at the inlet and water temperature is measured at different locations along the static mixer.

The temperature measurements are compared with the results of a one-dimensional algebraic model for the axial temperature computation that is based on the thermal balance and coupled with a kinetic drop-breakup model. The main difﬁculty in modeling heat transfer in a DCHE arises from the complex interactions between the turbulent continuousﬂow and the droplets. Many heat-transfer models have been developed for the case of constant droplet size during the “life” of the drop (Mitrovic and Stephan [9], Wijey- sundera et al.[7]); however, to the best of our knowledge no heat- transfer model is available for nonequilibrium situations such as drop breakup. In this study, a classical correlation is attempted for the Nusselt number in the case of moving spheres, based on the

* Corresponding author. Tel.:þ33 2 40 68 31 24; fax:þ33 2 40 68 31 41.

E-mail address:hassan.peerhossaini@univ-nantes.fr(H. Peerhossaini).

Contents lists available atScienceDirect

International Journal of Thermal Sciences

j o u r n a l h o m e p a g e : w w w . e l s e v ie r . c o m / l o c a t e / i j t s

doi:10.1016/j.ijthermalsci.2010.05.014

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ﬂuctuating velocity. The well-established properties of HEVﬂow (Lemenand et al.[10], Mohand Kaci et al.[11]) are used to evaluate the dissipation rate involved in the breakup model (Martínez- Bazán et al.[12]).

A crude assumption of constant inner temperature in the drop was tested, but this led to overestimation of the heatflux and hence of the thermal performance, as confirmed by the high Biot number value. The temperature profile in the drop is actually computed by the analytical model of Dombrovsky et al.[13].

2. Experimental apparatus and methods 2.1. Hydraulic loop

Two-phaseflow experiments are carried out by injecting the hot-oil phase into the turbulent cold-waterflow. The experimental setup consists of two feed loops, as shown in Fig. 1. The oil is injected by a centrifugal pump in the center of the HEV section through an injector of inner diameter 3 mm. The water is supplied by a constant-level buffer tank, avoiding the transmission of pump vibrations that might induce parasitefluctuations in theflow. The oil and waterflow rates are controlled by valves and are measured with rotameter-typeflowmeters. The relative experimental errors in the oilflow rate and the waterflow rate are respectively about 11% and 3%. The test section is connected to pre-conditioner and post-conditioner tubes that are both 300 mm straight transparent tubes of circular cross section and inner diameter 20 mm.

2.2. Hydrodynamic characteristics of the multifunctional heat exchanger/reactor

The HEV heat exchanger is made of a straight tube of circular cross section (inner diameter 20 mm) into which are inserted several arrays of tabs that play the role of vortex generators enhancing the mixing capacities of the heat exchanger. As shown in Fig. 2, the tabs are trapezoidal bafﬂesﬁxed to the tube wall at an angle of 30; their dimensions are 7 mm long, 7 mm and 5 mm at the bases, respectively on the wall and at the top of the trapezoid.

The hydraulic diameterDHof the tube equipped with the bafﬂes is 17.1 mm.Fig. 3shows a general view of the test section.

Typical vortices generated by the baffles can be visualized by a laser-induced fluorescence (LIF) technique (Peerhossaini and Wesfreid[14]).Fig. 4shows streamwise vortices visualized downstream from the tabs in the single-phase waterflow for transitional

Reynolds number 1000, which qualitatively represents the flow pattern at higher Reynolds numbers. Four pairs of longitudinal vortices are generated by the tabs as shown inFig. 5, which are broken and recombined at each next tab array. Detailed informa- tion on theflow pattern and turbulence structure of single-phase flow in HEV static mixer can be found in Lemenand et al.[8]and Phillips et al.[15].

These longitudinal vortices contribute by radial convection to intensify the heat and mass transfers, so that to a ﬁrst approximation the temperature and the mean drop size are quite homogeneous in theﬂow cross section. In particular, the high shear zone on top of the tabs ensures the breakup of the dispersed phase (Lemenand et al.[8]).

2.3. Workingﬂuids

The working ﬂuids are water for the continuous phase and vaseline oil with no additives for the dispersed phase; the physical properties of the oil are given inTable 1. The speciﬁc heat of the dispersed phase is measured by a differential scanning calorimeter for temperatures between 10 C and 60 C, and the following expression yields this physical property with relative error below 1%:

C_pdðTÞ ¼ C₁TþC₂ (1) where the constantsC1¼2.98,C2¼2239 andTis the temperature (C). Since the water temperature varies between 11 and 15 C during the experiments, the speciﬁc heat of the continuous phase Cpcis taken constant at 4182 J K¹kg¹.

2.4. Temperature measurement

The continuous-phase temperature along the axis is measured by five K-type thermocouples (labeled K₁ to K₅) with 80 m^m diameter wire. The measurement accuracy is evaluated at about 0.1C after calibration. As seen inFig. 6, two thermocouples (K3and K₄) arefixed near the wall, at longitudinal positionsx¼0.09 m (downstream from the third tab array) andx¼0.17 m (downstream from the seventh tab array). ThermocouplesK1andK2respectively measure the water and oil inlet temperatures at the entrance to the test section. ThermocoupleK5is located at the same axial position as thermocoupleK4(x¼0.17 m) and can be moved in the radial direction; it measures the temperature profile thanks to

Fig. 1.Experimentalﬂow loop.

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a micrometric traversing mechanism that ensures radial displace- ment. These measurements show that the wall temperature is close to the average temperature of the continuous phase, and the temperature can be assumed uniform in the ﬂow cross section.

Fig. 7shows examples of the temperature recordings over time. In these measurements oil temperature is 45C and thermocoupleK5

is located at the center of the test section. Note that theK₄ther- mocouple gives a water temperature near the wall that is constant over time.

The experimental setup does not permit preciseﬁxing of the thermal inlet conditions at constant values for successive runs; the oil temperature increases with the mechanical work of the pump and the water temperature depends on the tank temperature, which is at the ambient temperature. The inlet water temperature lies between 11C and 13C and oil inlet temperature is maintained between 40 and 48C. Insulation of the test section wall is ensured by the 3-cm Plexiglas and by the small temperature difference between water and room.

Finally, the experimental measurements of temperature exhibit a relative dispersion ranging between 6% and 11%. In all experiments, the volume oil fraction is maintained at 15% and the Rey- nolds number isﬁxed at 7500, 10 000, 12 500 and 15 000. Four runs were made for each Reynolds number to ensure reproducibility.

Experiments are numbered from 1 to 16 and are summarized in Table 2.

3. One-dimensional heat transfer model 3.1. Enthalpic model

The one-dimensional model provides the mean axial temperature proﬁle of the two phases, given the inlet conditions. Three main assumptions ground the enthalpic balance:

i) the continuous phase temperature is constant in the tube cross section (experiments validate this hypothesis at the thermocouples precision, see Section2.4), so that the axial dispersion term vanishes (plug-ﬂow model)

ii) local peculiarities due to the tabs are not taken into account, and longitudinal gradients are modeled globally,

iii) in any axial position the dispersed phase is assumed to be a cloud of uniform droplets of Sauter diameter d32 (the surface mean diameter) in order to consider an inter-phase contact surface identical to the actual size distribution.

Heat transfer between an oil droplet and the water phase is modeled by a convective heat-transfer coefﬁcienth_x. The enthalpic balance on the two phases over a distancedxalong the axisx(Fig. 8) hence yields:

r

cq_mcC_pc

T_c_xþdxT_cx

¼ h_xS_x T_d_xT_cx

(2)

r

dq_mdC_pd

T_d_xþdxT_d_x

¼ hxSx T_d_xTcx

(3) wherercandrdare respectively the densities of the continuous and dispersed phases,qmcandqmdthe continuous and dispersed phase mass ﬂow rates, C_pc and C_pd the continuous and dispersed phase speciﬁc heats, andTcandTdthe continuous and dispersed phase temperatures.Sxis the interfacial area at a given axial posi- tionxdepending on the dispersion development.

The system of two ordinary differential equations(2) and (3)is coupled through the heat-transfer termhxSxðT_d_xTcxÞ. Hence, the heat-transfer term needs two modeling steps: the interfacial areaSx

(see breakup model in Section3.2) and the heat-transfer coefﬁcient hx(discussed in Section3.3).

3.2. Dynamical breakup model

The dynamical breakup model is based on knowledge of the equilibrium drop size, the breakup frequency and the breakup mode, to predict the reduction in drop size along the mixer. These physical parameters are discussed below.

3.2.1. Equilibrium diameter

Drop breakup in turbulent dispersions has attracted considerable interest in recent years. In an agitated dispersion, breakup is deemed to occur when the stress induced by the turbulence exceeds the stabilizing force due to the interfacial tension, so that a maximal sizedmaxof the drop cloud is attained when the resisting Laplace forces are balanced by the turbulent stresses. For diameters belowd_max, the drop can withstand the external forces: breakup can no longer occur. Models for the maximum stable diameter are based on the Kolmogorov theory (Hinze[16]), assuming an inviscid dispersed phase, leading to a maximum diameter varying as3²⁼⁵^. Empirical relations derived from this theory (Sprow[17], Chen and Middleman[18]) are available for prediction of typical drop diameter in a turbulent regime. When the dispersed phases involve highly viscous internal phases (e.g. polymerization processes in Fig. 2.Four tabs in a HEV static mixer section and tab dimensions.

Fig. 3.Overall view of HEV static mixer.

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chemical industrial equipment), Davies[19]proposed adding to the Laplace forces resisting viscous forces, which slightly modify the previous models. Using this idea, numerous authors have proposed expressions for the maximum stable drop diameter for viscous drops in various conﬁgurations (Arai et al.[20], Calabrese et al.[21], Wang and Calabrese[22], Lagisetty et al.[23]).

Let us briefly review the Hinze-Kolmogorov theory. A droplet of diameter d, placed in the turbulentflowfield of the continuous phase, is deformed by the turbulent pressurefluctuations at scale d acting normally between two opposite sides of the droplet interface. This stress can be expressed as a function of the spatial autocorrelation function in collinear mode:

s

1ðdÞ ¼

r

c

D

^u²ðdÞ (4) withD^u²ðdÞthe spatial longitudinal autocorrelation over a distance dequal to the drop diameter. By using the relation between spatial and temporal correlations, one can write:

Fig. 4.Flow visualization of longitudinal vortices cross section downstream from (a)ﬁrst, (b) second, (c) third and (d) fourth tabs arrays forRe¼1000.

Flow

Generated vortices

Tab

Fig. 5.Location of longitudinal vortex generation.

Table 1

Physical properties of the dispersed phase.

Property (at 20C) Value Measurement method Kinematic viscosity 30.010⁶m²s¹ Mettler RM180 rheometer Speciﬁc heat 2297 J K¹kg¹ Differential scanning calorimeter

Density 850 kg m³ Technical data

Thermal conductivity 0.15 W m¹K¹ Technical data Interfacial tension

with water

30.810³N m¹ Krüss (K12) tensiometer by the ring method

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D

^u²ðdÞ ¼ 2u²ðdÞ (5) From the Batchelor analysis[24],u²ðdÞis related to the energy spectrumE(k), withkthe wave number, associated with theﬂuc- tuating velocity in the inertial zone:

u²ðdÞy Z^N

1=d

EðkÞdk ¼

b

ð

3

dÞ²⁼³ (6)

with3the turbulent dissipation rate andban absolute constant depending on theﬂow type. The drop Weber number is usually deﬁned as the ratio between the deforming external turbulent stress and the resisting Laplace pressure, which leads to:

WeðdÞ ¼

r

c

D

^u²ðdÞd

s

⁽⁷⁾

so that:

WeðdÞ ¼ 2

br

cð

3

dÞ²⁼³d

s

⁽⁸⁾

The balance between the terms in this ratio corresponds to a critical Weber number (of order 1, depending on the breakup mode andﬂuid densities):

We_crit ¼ 2

br

c

3

²⁼³^d⁵⁼³max

s

⁽⁹⁾

yielding:

d_max ¼ We_crit

2

b

₃₌₅

r s

c

₃₌₅

3

²⁼⁵ ⁽¹⁰⁾

By gathering of the different constants, we obtain:

dmax ¼ C

s

r

c

₃₌₅

3

²⁼⁵ ⁽¹¹⁾

where the constant value isC¼0.315 (11%) in the present case, as experimentally determined in the HEV static mixer (Lemenand et al.[8]).

3.2.2. Breakup frequency

This approach to diameter determination states that when equilibrium conditions are reached, the drops undergo breakup for a time equal to or greater than the characteristic breakup time. This has been checked by evaluating the present breakup frequency in theﬂow, using the approach due to Martínez-Bazán et al.[12,26]

detailed below. This criterion shows that the ﬁnal drop size is attained at about the middle of the path in the static mixer. Further results show, nevertheless, that the kinetics of drop size reduction is important for the global performance.

The dynamic breakup model is based on the evaluation of the breakup frequency of the droplet by turbulence in the static mixer.

Martínez-Bazán et al.[12]postulate that the time constant for the breakup is related to the balance between the deforming stress and the Laplace forces. In their study of an air/waterﬂow, they suggest:

gð

3

;dÞ ¼ Kgd¹

b

ð

3

;dÞ²⁼³12

s r

cd

₁₌₂

(12) whereK_gis a semi-empirical constant and the maximum equilibrium diameter dmax is hence given by equation (13), since the breakup rate tends to 0 when the drop diameter reaches this value:

d_max ¼ 12

s

br

c

₃₌₅

3

²⁼⁵ ⁽¹³⁾

Thebconstant may depend on the turbulent situation. In wake turbulence, Martínez-Bazán et al.[12]choseb¼8.2 values, given by the Batchelor[24]analysis for equilibrium turbulence. Moreover, by use of equation(10), the critical Weber number in Martínez- Bazán et al.’s work is hence of order 20 (for liquid-gas dispersion).

In the present case of liquid/liquid dispersion, the value of the critical Weber number is taken as of order 1 (Hesketh et al.[25]), Fig. 6.HEV static mixer with location of oil injector and thermocouples.

17 18 19 20 21 22 23

145 150 155 160 165 170 175

Time (s)

Temperat ure (°C )

K5 K4 K3 K1 K5

K4

K3

K1

Fig. 7.Examples of temperature recording by thermocouples,K5is placed at the center of the HEV static mixer cross section; oil inlet temperature is 45C.

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yielding the following practical expression of the breakup rate in our model:

gð

3

;dÞ ¼ K_g

s

r

c

₁₌₂ d³⁼²

"

d dmax

₅₌₃ 1

#₁₌₂

(14) The value of the constantKgmay depend on theﬂuid properties;

for example,Kg¼0.25 for air bubbles in water (given in Martínez- Bazán et al.[12]). From the experimental results detailed in East- wood et al. [27], in which the authors determine the breakup frequency of four liquids in turbulentﬂows,Kgcan be estimated to lie between 4.44 and 5.80, with an average value of 4.85, by iden- tiﬁcation with equation(14). This value can provide a reference case for comparison, but the optimal value ofKgin this study is near 0.75 as discussed in Section4.1.

It should be noted that the breakup model is independent of temperature.

3.2.3. Breakup mode

The breakup mode is needed to compute the diameters of daughter drops resulting from the splitting of a mother drop, at the breakup frequencyg(3,d) as presented in the following section.

In the literature, three main approaches have been used to model thepdfof the daughter particles: phenomenological models based on surface-energy considerations (Tsouris and Tavlarides [28]), statistical models (Coulaloglou and Tavlarides[29], Novikov and Dommermuth[30]) and hybrid models based on a combination of both (Konno et al. [31]). Among the most widely used phenomenological models, Tsouris and Tavlarides [28] propose

a binary breakup mode,m(d₀)¼2: two daughter droplets of sized₁ and d2 are formed, whose most probable sizes are inversely proportional to the amount of surface-energy created in the breakup process.

In the population model elaborated by Martínez-Bazán et al.

[12], the assumption of binary breakage of a drop of sized0into two droplets of conjugated diametersd₁andd₂leads to thepdf f(d₁,d₀) of the daughter drops. The peak of the distribution is always located atd^*:

d^* ¼ d₁ d₀z0:8z

1 2

₁₌₃

(15) which corresponds to the case of two daughter drops of identical volume. Comparisons of the Martínez-Bazán model with experiments are convincing, especially with the results of Konno et al.

[31], which exhibit a peak atd^*¼ 0.8. Other modes have been proposed, giving a maximum probability for the formation of a pair made up of a very large particle and a very small complementary one (Tsouris and Tavlarides [28]). This daughter particle size distribution leads to a U-shapedpdfthat is not in accordance with the experimental distributions measured in stirred tanks and other turbulentﬂows (Hesketh et al.[32]; Sathyagal et al.[33]; Kostoglou and Karabelas[34]; Martínez-Bazán et al.[26]).

In the present work, a binary homogeneous breakup is assumed, as this simple hypothesis is consistent with the objective of a trac- table model.

3.2.4. Final equations for the one-dimensional breakup model The purpose of the dynamic breakup model is to predict the evolution of droplet sizes between injection and the outlet of the test section. By considering the binary breakup, withNxthe number of divisions on dx, we have:

dðxþdxÞ ¼ dðxÞ

2^N^x⁼³ (16)

The number of divisions Nxof the drops between two axial positionsxandxþdxduring the residence time dx/Uis hence given by:

Nx ¼ gdx

U (17)

and therefore,

dðxþdxÞ ¼ dðxÞe^0:231gdx=U (18)

Fig. 8.Heat transfer at droplet surfaces.

Table 2

Operating conditions and experimental results.

N Re U(m s¹) qmc(l h¹) qmd(l h¹) Tc,inlet(C) Td,inlet(C) Tc,x¼0.09(C) Tc,x¼0.17(C)

1 15 000 0.75 721 127 12.5 46.7 14.5 14.8

2 12.5 44.7 14.2 14.5

3 11.7 47.6 13.9 14.2

4 12.8 43.1 14.3 14.6

5 0 0.63 601 106 12.5 45.1 14.2 14.5

6 12.6 45.0 14.1 14.4

7 11.7 47.5 13.6 13.9

8 12.9 42.0 13.9 14.3

9 10 000 0.50 481 85 11.4 44.9 13.0 13.4

10 12.6 43.1 13.9 14.2

11 11.8 45.2 13.3 13.7

12 12.8 40.4 13.7 14.1

13 7500 0.38 360 64 11.4 43.8 12.5 13.0

14 12.7 42.4 13.5 13.9

15 11.8 45.9 13.2 13.7

16 13.0 41.2 13.9 14.3

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The drop breakup frequency is computed by equation(14), with the local maximum diameter given by equation(11). The evolution of the drop diameter on the axis is presented inFig. 9forKg¼4.85 and 0.75 for different Reynolds numbers. It is interesting to notice that, for all Reynolds numbers, drop breakup is saturated at the same axial coordinate: a“quicker”convective transfer compensates for a higher breakup rate. The equilibrium diameter is attained about 45 mm from the inlet. It will be demonstrated that breakup kinetics plays an important part in the heat transfer.

3.3. Heat transfer model

The one-dimensional model is implicitly based on the assumptions that there is no spatial gradient on the droplet’s surface and that the heat transfer coefﬁcient h is constant.

Convective heat transfer at a surface is represented by the Nusselt number. In order to represent the whole drop size distribution the Nusselt number is scaled by the Sauter diameter,

Nu¼ hd₃₂

l

c (19)

which depends only on the Reynolds number and the Prandtl number, assumed here to be constant in the narrow temperature range of 20C (Sazhin[35]). A correlation for the Nusselt number for non-evaporating droplets is presented in Bird et al.[36]:

Nu¼ 2þC_NuRe¹⁼²_d Pr¹⁼³ (20) with the coefﬁcientC_Nu¼0.6.

Red is the Reynolds number scaled on the droplet’s size and a velocity scale which is the turbulent velocityﬂuctuationu⁰. In fact, the mean slip velocity between the oil drop and the water is negli- gible, so that from the droplet’s point of view, the surroundingﬂuid is stagnant in mean. However, the heat transfer is not purely diffusive withNu¼2, as can be shown by considering the outlet temperature.

The Reynolds and the Prandlt number are then deﬁned by:

Re_d ¼

r

cdu⁰

m

c

(21)

Pr ¼ C_pc

m

c

l

c (22)

wheremcis the continuous phase viscosity andlcis the continuous phase thermal conductivity.

3.4. Temperature proﬁle in the drop

The heat-transfer mechanism inside the droplets is assumed to be of“rigid drop model”type (Kehat and Sideman[38]): there is no deformation of the interface and no internal recirculation, so that heat transfer occurs only by conduction. The Biot number expresses the balance between the heat transfer within the drops and the heat transfer at the drop surface, and is hence representative of the temperature homogeneity inside the drops. It has been shown (Bricard and Tadrist[37]) that if the Biot number is lower than 0.1, the droplet temperature can be assumed uniform (within 5%). The Biot number is maximal for the largest drop of diameterdmax: Bi ¼ hdmax

l

d ¼

l

c

C_d

l

d

Nu (23)

where ld is the dispersed-phase thermal conductivity and the constant Cd ¼ 0.48 is the ratio of the Sauter diameter to the maximum diameter in the HEV static mixer (Lemenand et al.[8]). In this study, the Biot number is close to 1 (as the Nusselt number is around 10), so that the uniform temperature hypothesis is clearly not fulfilled and it is necessary to take internal gradients into account. This is done by using an analytical solution for the temperature profile, providing with reasonable accuracy the temperature distribution inside the droplet, given the initial temperature and the heatflux at the surface.

For the spherically symmetric problem, the transient heat conduction equation inside a droplet (without radiative heat source) can be written:

C_pd

r

dvTd

vt ¼

l

d

r² v vr

r²vTd

vr

(24) whereris the distance from the center of the droplet,tis time, and Tdis the droplet temperature. The boundary conditions are:

vTd

vr

r¼0 ¼ 0 and

l

dvTd

vr

r¼R ¼ h T_cT_dðRÞ

(25) where T_d(R) is the droplet surface temperature and T_c the cross section averaged temperature in the continuous phase. Dom- brovsky et al.[13]suggested a simpliﬁed model for non-isothermal droplets, based on the parabolic approximation of the inner temperature proﬁle:

T_dðrÞ ¼ Tcþ

T_dðRÞT_dð0Þ r R

2

(26) whereRis the droplet radius. Integration of equation(24)along the radius, taking into account the parabolic temperature proﬁle(26) and using the boundary conditions(25), yields:

r

dC_pdR 3

dT_d dt ¼ h

TcT_d

(27) withT_dthe average drop temperature andhthe convective transfer coefficient. The temperature Lagrangian time derivative in one- dimensional coordinates (1D) isðvT=vtÞ þUðvT=vxÞ. In the steady state regime the first term is zero ðvT=vt ¼ 0Þ and the time evolution along the axial coordinate is expressed byðUðdT=dxÞÞ. An explicit first-order discretization scheme allows computing the drop average temperature atxþdx:

T_dðxþdxÞ ¼ T_dðxÞþ 3h

D

^t

r

dC_pdR T_cðxÞT_dðxÞ

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Given the massﬂow ratesq_mdandq_mc, the powerP_xexchanged betweenxandxþdxis computed as:

0,00 0,04 0,08 0,12 0,16 0,20

0 500 1000 1500 2000 2500 3000

Kg=4.85 Re=7500 Re=10000 Re=12500 Re=15000

Droplet diameter, d (µm)

Axial coordinate, x (m)

Kg=0.75 Re=7500 Re=10000 Re=12500 Re=15000

Fig. 9.Axial evolution of the drop diameterdmaxas a function of Reynolds numberRe evaluated in dynamic breakup model.

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Px ¼

r

dC_pdq_md T_dðxþdxÞT_dðxÞ

¼

r

cCpcqmc T_cðxþdxÞT_cðxÞ (29) The continuous-phase temperature axial evolution is hence given by:

T_cðxþdxÞ ¼ T_cðxÞþ 3h

D

^t

r

cCpcR

q_md

qmc T_dðxÞT_cðxÞ

(30) Examples of the evolution of droplet temperature versus time obtained by the parabolic temperature profile model are shown in Fig. 10 for convective heat transfer coefficients 500 and 2500 Wm²K¹and drop diameters 700, 1000, 1500 and 3000m^m, typical of the operating conditions of the HEV static mixer experiments.Fig. 10shows the dimensionless temperatureT_drop^* defined as follows:

T_drop^* ¼ T_dT_dðinletÞ

T_eqT_dðinletÞ (31)

whereT_d(inlet)is the dispersed phase temperature at the inlet of the static mixer andTeqis the“mixing temperature”that is at thermal equilibrium when heat transfer is completed between the two phases (i.e., whenTc¼Td¼Teq). In this way, all the experimental runs for a given Reynolds number can be shown on a master curve, independently ofﬂuid temperatures.

As shown in Fig. 10, the characteristic times of the thermal process are always greater than 100 ms, while the characteristic breakup timetb(the inverse of the breakup frequencyg) is about 18.5 ms. This consideration supports the choice of studying the internal temperature evolution inside the drop.

Some issues must be examined in order to apply the thermal model. The first is to ensure that the heat balance for a typical diameter is representative of all the size distribution. In fact, this desideratum cannot be rigorously fulfilled, since the appropriate diameter for the“mean”surface heat transfer is the Sauter diameter d₃₂ and the scaling diameter for the “mean” temperature profile is the mean algebraic diameter. Nevertheless, the optimal choice in this case isd32, and the bias due to the smaller diameters is“weak”in that it constitutes a small part of the volume.

The second point is the temperature redistribution after the breakup. The crude assumption made here is that after each breakup step, the drop temperature is homogenized at the mean temperature of the “mother” drop. The global balance is hence maintained, unless residual gradients are not considered. The drop proﬁle is reinitialized at each breakup and consequently the time step is determined by the breakup frequency.

4. Results and discussion

4.1. Experimental validation of the dispersion/thermal model A sensitivity study of the Kg constant to the set of “high” modalities is carried out.Fig. 11shows the dimensionless temper- atureT_drop^* versus axial position in the heat exchanger for some values of the constantKg(4.85, 1.50, 1.25, 1.00, 0.75, 0.50, and 0.25) ranging between the extreme values listed in Section 3.2.2. For Kg>1, the temperature is overestimated by the model and breakup is completed near the mixer inlet. ForKg<0.5, heat transfer is underestimated because the breakup is too weak and hence delays the heat transfer. The bestfit is observed forK_g¼0.75, as observed from thefitting coefficient onFig. 12.

Experiments have been performed to test the model assumptions, in particular to see whether the complexity added by taking into account the different physical mechanisms improves the prediction of the DCHE’s capabilities. Two modalities are tested for each model element: breakup kinetics (constant diameter/dynamic breakup), wall heat transfer at the drop surface (constant Nusselt number/convective model), and inner drop temperature proﬁle (uniform temperature/parabolic proﬁle).

The model hierarchy is presented inTable 3, and the predicted axial temperature proﬁles are presented inFig. 13forKg¼4.85, in order to quantify the cross-cutting effects of the basic assumptions.

For this value ofK_g, the effect of the breakup process is not significant, since no difference is observed between the“drop constant size” and the“kinetic breakup” models. It also appears that the uniform drop temperature proﬁle drastically overestimates the heat transfer, so that this assumption must be removed. The“static medium”hypothesis for the heat transfer given byNu¼2 underestimates the heat transfer and seems deﬁnitively inappropriate.

Finally, the high modalities are necessary to account for the heat transfer between the two phases.

4.2. Comparison with experimental results in the HEV static mixer The role of breakup kinetics has not yet been proven because of the high value ofKgin the previous runs. In theﬁnal step, runs are also carried out withK_g¼0.75, which seems more realistic from the considerations in Section 4.1. The predicted axial temperature proﬁles are presented inFig. 14, for different operating conditions.

The sensitivity toK_gconstant is evident. As explained above, the

0 20 40 60 80 100 120 140 160 180 200 220 240 260 0,0

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

Dimensionless temperature, T * drop

h =500 Wm^-2K^-1 3000 µm 1500 µm 1000 µm 700 µm t _b

t (ms)

a

0 20 40 60 80 100 120 140 160 180 200 220 240 260 0,0

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

t _b

Dimensionless temperature, T * drop

t (ms)

h =2500 Wm^-2K^-1 3000 µm 1500 µm 1000 µm 700 µm

b

Fig. 10.Thermal model: time evolution ofT_drop^* for different droplet diameters and surface cooling. (a)h¼500 W m²K¹, (b)h¼2500 W m²K¹.

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uniform drop temperature proﬁle assumption must be removed because it overestimates the heat transfer and the Nu ¼ 2 assumption must also be removed because it underestimates the heat transfer. Thus from this point on the “high modality”

conditions in heat transfer calculations are retained.Fig. 15plots the axial evolution of the dimensionless temperature T_drop^* , compared with the temperature measurements for Reynolds numbers 7500, 10 000, 12 500 and 15 000. The model prediction error varies between 0% and 5% at the intermediate axial position (x¼0.09 m) and between 8% and 12% at the outlet axial position (x¼0.17 m). The accuracy of the predicted results appears to be close to that of the experimental measurements.

Finally,Fig. 16shows temperature proﬁles for the dispersed and continuous phases computed with the full model. The decrease in the oil temperature is more pronounced than the increase in the water temperature, mainly due to the volume fraction.

4.3. Comparison with an ideal double-jacketed heat exchanger Here we examine the potential of the HEV static mixer for use as a DCHE. The efﬁciency of a DCHE is hence compared to that of a smooth-wall double-jacketed pipe of identical length and hydraulic diameter, at the same ﬂow rate, and under the most favorable conditions; that is, constant wall temperature over the whole surface. The Colburn correlation is used to evaluate heat transfer at the wall in the empty pipe:

Nu¼ 0:023Re⁴⁼⁵Pr¹⁼³ (32)

The global performance can be evaluated as the energy efﬁ- ciency rateR, the ratio of the thermal power transferred and the pumping (pressure drop) energy lost in the ideal double-jacketed heat exchanger and in the HEV static mixer:

R¼ P_exch

D

^P ⁽³³⁾

0,00 0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00

0,090 0,095 0,100 0,105 0,110

Mean deviation

Kg

Fig. 12.Mean deviation between the model and experiments versus theKgconstant.

Table 3

Physical mechanism models tested on liquid/liquid dispersion, heat transfer and drop temperature proﬁle.

Physical mechanism Low modality High modality Liquid/liquid dispersion

(x3.2)

No breakup Constant diameter (¼dmax)

Dynamic breakup modelKg

Heat transfer (x3.3) Nu¼2 Nu¼2þCNuRe¹⁼²_d Pr¹⁼³ Drop temperature proﬁle

(x3.4)

Uniform model Parabolic model

0,00 0,04 0,08 0,12 0,16 0,20

0,0 0,2 0,4 0,6 0,8 1,0

Dimensionless temperature, T* drop

Experiments - Re=7500 - Kg=4.85 L/L dispersion - Nu - Temperature profile Breakup - Correlation - Parabolic Breakup - Correlation - Uniform Breakup - Nu=2 - Parabolic Breakup - Nu=2 - Uniform No breakup - Correlation - Parabolic No breakup - Correlation - Uniform No breakup - Nu=2 - Parabolic No breakup - Nu=2 - Uniform

Fig. 13.Predicted axial temperature proﬁles for low and high modality conditions, Kg¼4.85, compared to experimental values,Re¼7500.

0,00 0,04 0,08 0,12 0,16 0,20

0,0 0,2 0,4 0,6 0,8 1,0

Dimensionless temperature, T * drop

Axial coordinate, x (m) Experiments - Re =7500

Kg =4.85 Kg =1.50 Kg =1.25 Kg =1.00 Kg =0.75 Kg =0.50 Kg =0.25

a

0,00 0,04 0,08 0,12 0,16 0,20

0,0 0,2 0,4 0,6 0,8 1,0

Dimensionless temperature, T * drop

Axial coordinate, x (m) Experiments - Re =15000

Kg =4.85 Kg =1.50 Kg =1.25 Kg =1.00 Kg =0.75 Kg =0.50 Kg =0.25

b

Fig. 11.Sensitivity test of the Kg constant with the set of high modality conditions, (a) Re¼7500, (b)Re¼15000.

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withP_exchthe thermal power exchanged. The comparison inFig. 17 of the energy efﬁciency rate of the HEV heat exchanger and the ideal double-jacketed heat exchanger shows the superiority of the DCHE: the ratioRfor the HEV static mixer is between 63% and 128%

higher than that of the ideal double-jacketed heat exchanger. This highlights a signiﬁcant advantage of the HEV static mixer, since one of the weak points of DCHE is generally the high pressure drop necessary to ensure mixing of the two phases.

4.4. Comparison with existing direct contact heat exchangers The global heat transfer of a heat exchanger is generally given by the volumetric heat transfer coefﬁcientUV, deﬁned by:

U_V ¼ P_exch V

D

^Tln

(34) whereVis the volume of the heat exchanger andD^Tln the loga- rithmic mean temperature difference deﬁned by:

D

^Tln ¼

T_d;inletT_c;inlet

T_d;outletT_c;outlet ln T_d;inletT_c;inlet

T_d;outletT_c;outlet

! (35)

0, 00 0,04 0,08 0,12 0,16 0,20 0, 0

0, 2 0, 4 0, 6 0, 8 1, 0

Dim ensionless temperature, T * dr op

Ax ial coordinate, x (m)

E x periment s - Re =7500 - Kg =0 .7 5 L/ L dis persion - Nu - T em perat ure prof ile

Break up - Co rrelat ion - Par abolic Break up - Co rrelat ion - Un iform Break up - Nu =2 - Parabolic Break up - Nu =2 - U ni fo rm N o break up - Co rrelat ion - Pa rabolic N o break up - Co rrelat ion - Un iform N o break up - Nu =2 - Parabolic N o break up - Nu =2 - U ni fo rm

Fig. 14.Predicted axial temperature proﬁles for low and high modality conditions, Kg¼0.75, compared to experimental values,Re¼7500.

0,00 0,04 0,08 0,12 0,16 0,20

0,0 0,2 0,4 0,6 0,8 1,0

Experiments - Re=7500

Kg=4.85 Kg=0.75

a

0,00 0,04 0,08 0,12 0,16 0,20

0,0 0,2 0,4 0,6 0,8 1,0

Kg=4.85 Kg=0.75

b

0,00 0,04 0,08 0,12 0,16 0,20

0,0 0,2 0,4 0,6 0,8 1,0

Kg=4.85 Kg=0.75

c

0,00 0,04 0,08 0,12 0,16 0,20

0,0 0,2 0,4 0,6 0,8 1,0

Kg=4.85 Kg=0.75

d

Fig. 15.Prediction of the axial evolution of the temperature in the HEV static mixer following the high modalitiy conditions withKg¼0.75 andKg¼4.85, compared with experimental values (a)Re¼7500, (b)Re¼10,000, (c)Re¼12,500, (d)Re¼15000.

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TheU_Vcoefﬁcient is plotted inFig. 18as a function of Reynolds number. For the actual operating conditions,U_Vranges between 756 and 3008 kWm³K¹. For the same oil-in-water fraction and a higher Reynolds number of 37 000, the DCHE studied by Porter et al. [39] has a U_V of about 3000 kWm³K¹. The concentric annulus used as a DCHE by Shahidi and Özbelge[40]presents aUV

in the range 200e350 kWm³K¹for Reynolds numbers 6650 to 7680. The HEV static mixer thus appears to be an efﬁcient DCHE.

5. Conclusions

Direct-contact heat transfer between two immiscibleﬂuids in a multifunctional heat exchanger/reactor with embedded vorticity in turbulent regime is investigated, on the basis of physical phenomena for the phase dispersion and the heat transfer between the two phases. The phenomenological model is validated by experimental data, in particular the continuous-phase temperature evolution.

The one-dimensional model developed here can be used and adapted to other geometries as a sizing tool for heat transfer in liquid/liquid two-phase operations with concomitant breakup, for instance in energy salvage and thermal plant storage. This model can be transposed to any open-loop geometry (for example those used by Lemenand and Peerhossaini[41], Habchi et al.[42,43]) if theﬂow properties are known, for instance by CFD simulations, in order to size the DCHE at different scales, or to evaluate its performance by determining a global volumetric heat-transfer coefﬁcient. In this semi-global approach, the relative error between predicted and measured temperature is lower than 11%. This means that in the model implementation the model “improvements” could not be assessed rigorously.

Longitudinal embedded vorticity is the main intensiﬁcation factor in the HEV multifunctional heat exchanger, and its properties have been studied intensively in Mokrani et al.[44], Mohand Kaci et al.[45,46], Habchi et al.[47,48], Ajak et al.[49], and Toé et al.[50].

The mixing mechanisms of the HEV static mixer suggest that its performance can be attributed to the simultaneity of heat transfer and droplet breakup, a considerable advantage in terms of energy cost and compactness. Consequently, this static mixer can be considered a good multifunctional heat exchanger (MHE), as described by Ferrouillat et al.[51]. It has been shown that the HEV DCHE is much more efﬁcient than a double-jacketed heat exchanger even under ideal conditions, and seems to be more efﬁcient than other DCHE geometries.

0,00 0,04 0,08 0,12 0,16 0,20

15 20 25 30 35 40 45 50

Dispersed phase temperature, Td (°C)

Re=7500 Re=10000 Re=12500 Re=15000

a

0,00 0,04 0,08 0,12 0,16 0,20

13,0 13,5 14,0 14,5 15,0 15,5 16,0

Continuous phase temperature, Tc (°C)

Re= 15000 Re= 12500 Re= 10000 Re= 7500

b

Fig. 16.Prediction of dispersed phase and continuous temperatures forTc(inlet)¼13C andTd(inlet)¼47C (a) mean drop temperature (b) continuous phase temperature.

7500 10000 12500 15000

0 2 4 6 8 10 12 14

Energy efficiency rate, R (W Pa-1 )

Reynolds number, Re HEV mixer

Ideal double-jacketed exchanger

Fig. 17.Energy efﬁciency rate of HEV heat exchanger and ideal double-jacketed exchanger.

7500 10000 12500 15000

0 500 1000 1500 2000 2500 3000 3500

Volumetric heat transfer coefficient, UV (kW m-3 K-1)

Reynolds number, Re

Fig. 18.Volumetric heat transfer coefﬁcient of HEV heat exchanger.