Entropy production and field synergy principle in turbulent vortical flows

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Entropy production and ﬁ eld synergy principle in turbulent vortical ﬂ ows

Charbel Habchi

^a^,^b

, Thierry Lemenand

^a

, Dominique Della Valle

^a^,^c

, Leonardo Pacheco

^a

, Olivier Le Corre

^d

, Hassan Peerhossaini

^a^,^*

aThermoﬂuids, Complex Flows and Energy Research Group, LTN, CNRS UMR 6607, École Polytechnique University of Nantes, rue Christian Pauc, 44306 Nantes, France

bDépartement Energétique Industrielle, École des Mines de Douai, Boulevard Lahure, 59500 Douai, France

cONIRIS Géraudière, 44322 Nantes, France

dGEPEA, Ecole des Mines de Nantes, CNRS-UMR 6144, B.P. 20722, F-44307, France

a r t i c l e i n f o

Article history:

Received 14 September 2010 Received in revised form 4 July 2011

Accepted 21 July 2011 Available online 25 August 2011

Keywords:

Streamwise vorticity Entropy production Turbulence

Multifunctional heat exchanger/reactor Vortex circulation

Convective heat transfer

a b s t r a c t

The heat transfer in turbulent vorticalﬂows is investigated by three different physical approaches.

Vortical structures are generated by inclined baffles in a turbulent pipeflow, of three different configurations. In thefirst, the vortex generators are aligned and inclined in theflow direction (called the reference geometry); in the second, a periodic 45 rotation is applied to the tab arrays (alternating geometry); the third is the reference geometry used in the direction opposite to theflow (reversed geometry). The effect of theflow structure on the temperature distribution in these different configurations is analyzed. The conventional approach based on heat-transfer analysis using the Nusselt number and the enhancement factor is used to determine the efficiency of these geometries relative to other heat exchangers in the literature.

The effect of vorticity on the Nusselt number is clearly demonstrated, and so as to highlight the respective roles of the coherent structures and the turbulence, a new parameter is defined as the ratio of thevortex circulationto theturbulent viscosity. The relative contribution of the radial convection to heat transfer appears to increase with Reynolds number. The effect of mixing performance on the temperature distribution is investigated by the field synergymethod. A global parameter, namely the intersection angle between the velocity and temperature gradient, is defined in order to compare performances.

Finally, an analysis of energetic efﬁciency by entropy production, involving both heat transfer and pressure losses, is carried out to determine the overall performance of the heat exchangers.

All these approaches lead to the same conclusion: that the reversed geometry presents the best heat transfer coefﬁcient and the best energetic efﬁciency. The reference geometry shows the worst performance, and the alternating array has intermediate performance.

1. Introduction

Forced heat transfer in the turbulent regime is generally controlled by the convective motion of large-scale eddies that appear essentially as transverse and longitudinal vorticity [1,2].

These embeddedﬂow structures, which can be generated by shear instabilities or pressure gradients, play a crucial role in the heat- and mass-transfer mechanisms. The pattern of these vortices in the ﬂow has a decisive impact on the hydrodynamic and thermal performance of thermal devices used in industrial applications [1e11]. A physical understanding of this impact is a fundamental

issue in optimizing the energetic efﬁciency of multifunctional heat exchangers-reactors (MHER) for Green Process Engineering[12].

Several approaches have been used to investigate the heat- transfer mechanisms in the presence of longitudinal vortices in turbulentflow: global approaches using the Nusselt number and an enhancement factor, and also more advanced approaches involving i) the vortex circulation, ii) thefield synergy principle, iii) entropy production, as described below. In the present work, vortices are produced by using three different vortex generator positions in a turbulent pipe flow: the vortex generators are aligned, alternating, or reversed. The global approach is based on the determi- nation of the Nusselt number, which characterizes the convective heat transfer in theflow, and the friction factor, which determines pressure losses. The enhancement factor is defined as the ratio of the convective heat transfer coefficient of the straight-pipeflow over that of the current geometry[13,14]. This parameter allows

*Corresponding author. Tel.:þ33 2 40683124; fax:þ33 2 40683141.

E-mail address:[email protected](H. Peerhossaini).

Contents lists available atScienceDirect

International Journal of Thermal Sciences

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

doi:10.1016/j.ijthermalsci.2011.07.012

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comparison of the efﬁciency of different heat exchangers for the same power consumption.

The intensity of the secondary flow produced by the vortex generators, especially the streamwise vorticity [1], has a direct impact on the heat-transfer process. It is now well established that the circulation of the streamwise vortices is significantly correlated with the Nusselt number[15e17]. However, the vortex circulation characterizes only the convective contribution of the longitudinal vortices and does not account for the fine-grained turbulent structures in the flow. These turbulent structures, which can be characterized by the turbulence kinetic energy (TKE), also play an important role in the heat-transfer mechanism[18]. Hence, a new parameter is defined here to assess the relative effect of the vortex circulation and the turbulent viscosity.

Recently, Guo et al.[19]proposed a new concept for analyzing the convective heat- transfer mechanism based on the combined effects of the temperature gradients and velocity vector and is therefore called the“field synergy principle”. It was shown by Guo et al.[19,20]and Tao et al.[21,22]that heat-transfer enhancement is related to the angles between the streamlines and the isotherms and to the fullness of the dimensionless velocity and temperature profiles. This concept is similar to the alignment of scalar gradients with strain eigenvectors proposed by Lapeyre et al.[23], which can be used to study two- dimensionalflows, as in the alignment dynamics of the temperature gradient in Bénard-Von Kármán vortexflow behind a cylinder[24]. Thefield synergy principle can be used for three-dimensional flows, especially in the design and optimization of MHER[21,22,25,26].

The convective heat-transfer intensification seems to increase with reduction of the intersection angle between the velocity and temperature gradient, and this configuration is the most favor- able to produce uniform velocity and temperature profiles in the flow cross sections. This feature is confirmed in this study;

a global parameter deﬁned as “the average value of the intersection angle” in the cross section is useful in classifying the thermal efﬁciency of the different geometries.

The analysis of the entropy production[27e29]characterizing the amount of the available work in the ﬂow characterizes the overall performance of a MHER by accounting for the heat transfers and pressure losses. Minimizing entropy production provides greater heat-transfer rates and smaller friction factors[2], since less available work is lost in the thermal device. This approach appears to be pertinent in assessing heat-transfer enhancement in different MHER[29]. In the present study, the entropy production is deter- mined by the model equations developed by Kock and Herwig [28,29].

2. Flow conﬁgurations

The three geometries are based on an empty straight pipe equipped with trapezoidal vortex generators, namely a HEV (High Efficiency Vortex) mixer[30]. The HEV static mixer, studied previously as a mixer and heat exchanger[7,8,31e35], shows high efficiency compared to other geometries, especially due to its low energy consumption. In the present work the inner pipe diameter is 20 mm and seven tab arrays, each composed of four vortex generators diametrically opposed and inclined to the wall at a 30 angle, are placed along the 140 mm of the mixer length. More details on the tab geometry are given in Mohand Kaci et al.[34]. The three different configurations of the static mixer are shown inFig. 1.

In configuration (a), the tabs arefixed to the wall and inclined at 30 in theflow direction (reference geometry). In configuration (b), the arrays in the reference geometry are alternately shifted by 45 (alternating geometry). Configuration (c) is similar to configuration (a) but theflow direction is reversed compared to the reference configuration (reversed geometry).

Downstream from each vortex generator, a counter-rotating vortex pair (CVP) is generated due to the pressure difference between the two sides of the tab surfaces[36,37]. The CVP generated in the aligned arrays (Fig. 1(a)) and in the alternating arrays (Fig. 1(b)) produce a common outﬂow in the tab symmetry plane, from the wall toward the centerline, but the reversed arrays (Fig. 1(c)) generate a common inﬂow from the centerline toward the wall[7,31,38].

3. Numerical procedure 3.1. Solver and turbulence models

The numerical simulations are carried out using the CFD code Fluent^Ò6.3[39]. The continuity equations for mass and momentum and the energy equation are solved sequentially with double precision[40], segregated and with second-order accuracy [41].

Pressureevelocity coupling is performed byﬁnite volumes with the SIMPLE algorithm[42].

Mohand Kaci et al.[34]testedfive different turbulence models for theflow dynamics in the HEV mixer with aligned arrays, and concluded that the standard k ³ [43,44] and the RSM models [45e47], associated with a two-layer model to compute the wall region, give a satisfactory description of theflow pattern.

The two-layer model involves solving, in the viscous sublayer ðy^þ<5Þ, the one-equation model of Wolfstein [48], namely the turbulent kinetic energy transport equation where the turbulent

Fig. 1.3D views and longitudinal sections of the geometries studied: (a) aligned, (b) alternating arrays, (c) reversed arrays.

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viscosity and energy dissipation are computed from empirical correlations based on length scales [49]. The two-layer model avoids the use of empirical wall standard functions, which are not valid for three-dimensional complexﬂows. In the present study, the RSM model is used concomitantly with the k ³ model to simulate the hydrodynamics and heat transfer in the three geometries.

The viscosity and thermal conductivity of the water are assumed piecewise linear functions of temperature, as proposed by Rahmani et al.[50]using data from Lienhard IV and Lienhard V [51]. The speciﬁc heat and density, nearly constant for the temperature range, are set respectively at 4182 J kg¹K¹and 998 kg/m³. The thermal conductivity of the tab is taken as constant (100 W m¹K¹).

3.2. Boundary and operating conditions

No-slip boundary conditions are applied at the solid surface of the tabs and at the pipe wall. Most of the simulations are done with a Dirichlet condition for the temperature at the wallTw ¼360 K.

Heat conduction in the tab thickness is accounted for by the coupled option of the two-sided wall model, while the tabs are in contact on each side with theﬂuid.

At the computational domain inlet, a fully developed turbulent flow velocity profile is imposed; the TKE and the turbulence energy dissipation rate are fixed by the turbulence intensity I of the equilibrium turbulent tubeflow[52]. Thefluid temperature at the inlet is 298.15 K. Flow and heat-transfer simulations are carried out in a steady turbulent flow for Reynolds numbers 7500, 10,000, 12,500 and 15,000.

3.3. Meshing and accuracy of the numerical solution

All three geometries are reduced to a 1/8 sector by axial symmetry. A non-uniform unstructured three-dimensional mesh with hexahedral volumes is built and reﬁned at all solid boundaries using the software Gambit^Ò. Mesh size is controlled by adjusting the number of nodes in the radial direction, on the tube periphery and the vortex generators, and on their axial length.

In each of the three geometries, the mesh density is increased until no effect on the quality of the result is detected, i.e., until the relative difference between the numerical results for two consec- utive mesh densities does not exceed 1%. The criterion for grid sensitivity is based on velocity profiles, turbulence dissipation rate, and temperature profile in a location defined by the tab symmetry plane at the outlet. The mesh with the lowest density yielding high- quality results is used to generate and simulate the entire geometry.

More details on the meshing can be found in Mohand Kaci et al.

[34]. Theﬁnal mesh size is 695,178 for the aligned arrays, 724,174

for alternating arrays and 723,778 for reversed arrays.Fig. 2shows the mesh in the symmetry plane of a tab and a cross section in the tab region.

The maximum value of the size of the wall celly^þ_c is less than 2.2 in the three geometries, complying with the conditiony^þ<4, so that the viscous sublayer is properly modeled. Moreover, the more restrictive criterion of Defraeye et al. [53] suggesting that the dimensionless wall normal distance y^* must be in the range 30e500 is also fulfilled, since the maximum value ofy^*is found in the aligned arrays to bey^*_max ¼18:92, for the alternating arrays y^*_max ¼ 18:48 and for reversed arrays y^*_max ¼16:99. Here the dimensionless wall distancesy^þandy^*are respectively obtained from y^þ ¼u_*y=y ând ^y^* ¼C_m¹⁼⁴k¹⁼²^y=y^{, where}û* is the friction velocity,ythe distance to the nearest wall,ythe kinematic viscosity, C_m ¼0:09 a constant used in thek ³ model andk ¼0:42 the von Karman constant.

Series of simulations are carried out with stop-criteria values ranging from 10³to 10⁹. It is found that beyond the convergence criterion 10⁶, no signiﬁcant changes are observed in the temper- atureﬁeld and turbulence quantities, and this 10⁶value is retained for the simulations.

To check the numerical consistency, the global turbulence energy dissipation rate 3, averaged on the whole volume of each geometry, is compared to that obtained from pressure drop expression:

3 ¼ Wm

D

^P

r

^L ⁽¹⁾

withW_mthe meanﬂow velocity,D^Pthe pressure drop between the inlet and the outlet,r^theﬂuid density andLthe test section length.

Results are in good agreement with a relative difference that does not exceed 10⁴for the three geometries.

3.4. Hydrodynamics validation

The numerical simulations of the hydrodynamicfield with the k ³ and the RSM models are compared with the experimental results of Habchi et al. [31] using laser Doppler anemometry (LDA). Velocity profiles at the outlet in the tab symmetry plane for the three geometries are presented in Fig. 3(a). The two turbulent models are in good agreement, and they model the experimental data quite well.Fig. 3(b)presents the turbulence kinetic energy dissipation rate in the three geometries for the same cross section. It is observed that both turbulence models reproduce the experimental results well, especially in theflow coreð0<r=R<0:4Þand in the shear regionð0:4<r=R<0:7Þ. In the wake region, near the wall, the numerical results move slightly away from the experiments. This can be partly attributed

Fig. 2.2D views of the mesh: (a) cross section passing through a vorticity generator, (b) symmetry plan.

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to the relative inaccuracy of the experimental values in this region: the mean axial velocity is close to zero (recirculation ﬂow), which enhances the noise in the LDV measurements, as reported in Habchi et al. [31]. On the other hand, the low convective velocity weakens the validity of the Taylor hypothesis of“frozen turbulence”necessary to compute 3.

4. Results and discussion

4.1. Flow pattern and temperatureﬁeld description

Here theﬂow structures induced by the vortex generators and their effects on the temperature distribution are discussed. Flow charts and contours are shown for Reynolds number 15,000.

The temperature distribution combined with the velocity vector field and theflow streamlines are presented inFig. 4on a cross section at the outlet of the three geometries. The network of streamlines clearly indicates the presence of primary and secondary longitudinal vortices (inside the red squares), namely CVPs, that have been fully described in Habchi et al.[38]and Le and Papavassiliou[54]. InFigs. 4(a) and 3(b), it can be observed that the CVPs rotate in the same direction in both aligned and alternating arrays since the vortex generators have the same inclination. The common outflow, in the tab symmetry plan, ejects hotfluid from the near-wall region toward the flow core, forming high- temperature “strikes” similarly to what is observed by Le and Papavassiliou[54]who studied the turbulent heat transfer from the wall. The secondary CVPs have opposite vorticity and then a common inflow that reduces the heat pumping from the wall. In the reversed arrays inFig. 3(c), the pressure gradient generating the primary CVP is reversed and so is the sense of rotation of the CVPs.

In this case, the vortex generators induce a common outﬂow in the tab symmetry plane, from the ﬂow core toward the wall. The primary CVP centers in the reversed arrays are more diverted from one another than in the aligned and alternating arrays, where the

CVP centers stay within the limit of the tab. The secondary CVPs are located on the sides and generate a common inﬂow that reduces the convective transport of the common outﬂow, as in the aligned and alternating arrays.

The effect of the common out- and inﬂows on momentum transfer can be quantiﬁed by the convective termT_Cin the transport equation for the TKE:

T_C ¼ !U

$V^/k (2)

with!U the velocity vector andkthe turbulence kinetic energy.

The radial component ofTCis computed in the tab symmetry plane at the outlet of the channels and corresponds to the common out- and inflows. It is observed that the average magnitude ofT_Cin the alternating arrayðT_C_;_altz1:3510²m²s³Þis much greater than that in the aligned arraysðT_C_;_algz6:3410⁴m²s³Þ, leading thus to an estimate of improved radial convective transport in the alternating configuration than the aligned geometry. The intensity of convective transport for the common out- and inflows in reversed arrays is greater than in the two other geometries ðT_C_;_revz3:2410²m²s³Þ, so the radial convective transport in the reversed configuration is much improved over the other two geometries.

The side-view cross-section temperature contours inFig. 5(a) and (b)indicate that the temperature distribution is similar in the aligned and alternating arrays: the thermal boundary layer is renewed by each passage over a tab, and an overheated region is observed directly behind the tab in the wake region. This can be explained by the side view inFig. 6(a) and (b)at radial distance r=R ¼0:8, where the recirculationﬂow is observed directly behind the tabs. These transverse vortices are stagnantﬂows that trap heat and always have negative effects on performance.

Fig. 5(a) and (b) shows clearly that, in the alternating geometry, theﬂow impacts the tab sharply and is instantaneously redirected to the shear region where the streamlines are denser, while in aligned arrays, theﬂow is smoothly oriented toward the shear region. This

0.0 0.2 0.4 0.6 0.8 1.0

-0.5 0.0 0.5 1.0 1.5 -0.5 0.0 0.5 1.0 1.5 -0.5 0.0 0.5 1.0 1.5

Reversed arrays:

k- RSM LDV

r/R Alternating arrays:

k- RSM LDV

Streamwise velocity W (m/s)

Aligned arrays:

k- RSM LDV

0.0 0.2 0.4 0.6 0.8 1.0

0 10 20 30 400 10 20 30 400 10 20 30 40

Reversed arrays:

k- RSM LDV

r/R Alternating arrays:

k- RSM LDV (m2/s3)

Aligned arrays:

k- RSM LDV

a b

Fig. 3.Numerical and experimental radial proﬁles at the outlet for the three geometries in the tab symmetry plane of (a) streamwise velocity and (b) turbulence energy dissipation rate for Re¼15,000 (experiments adapted from Habchi et al.[31]).

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process increases the velocity gradients in the alternating arrays and generates more energetic shear layers characterized by higher TKE and dissipation rates relative to the aligned arrays (Habchi et al., 2009b). This can be observed inFig. 3(b): the TKE dissipation rates in the alternating geometry in the shear region for 0:4<r=R<0:7 are above that in the aligned conﬁguration.

The temperature distribution and flow pattern are strongly modified in the reversed array. As seen inFig. 5(c), the overheated regions behind the tabs do not form because of the absence of any recirculationflow behind the vortex generators (seeFig. 6(c)), and moreover the temperature distribution is more homogeneous in this transverse plane.Fig. 5(d) represents the temperature distribution in the diametral symmetry plan corresponding to the common outflow in the reversed array, illustrating the effect of the efficient renewal of the thermal boundary layer.

4.2. Performance of heat transfer enhancement

The local Nusselt number Nu_[ is computed by a user-deﬁned function (UDF) along theﬂow direction as:

Nu_[ ¼ hD

l

^¼

4

^D

l

^TwT_b;z (3)

where h is the local convective heat transfer coefficient, l ^the thermal conductivity of the workingfluid (water here) and4^the heatflux density computed from the heat balance:

4

¼ mc_ p

p

^D dT_b;z

dz

(4) withm_ the massﬂow rate andT_b;zthe bulk temperatures on the cross sections of axial coordinatez.

This local Nusselt number is normalized by the corresponding Nusselt number for a straight turbulent pipeﬂowNu0. Kakac et al.

[55]examined a large number of correlations for fully developed turbulentﬂow in a circular tube and concluded that the Gnielinski [56]equation (Eq.(5)) agrees with the available data better than any other expression over a range of Prandtl numbers from 0.5 to 200 and Reynolds numbers from 2300 to 510⁶:

Nu₀ ¼ ðf0=8Þ 1þ12:7ðf0=8Þ¹⁼²

Pr

Pr²⁼³1ðRe1000Þ (5) wheref₀is the friction factor in straight-pipeﬂow following the Blasius formula:

f₀ ¼ 0:079Re^0:25 (6)

Fig. 4.Temperature distribution and velocityﬁeld (left) and streamlines (right) at the outlet cross section of the (a) aligned, (b) alternating, and (c) reversed arrays, Re¼15,000.

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where the Reynolds number is based on the pipe diameter Re ¼WD=y^.

It can be seen inFig. 7that the longitudinal evolution of the normalized local Nusselt number Nu_[=Nu₀in the three geometries is spatially periodic due to the tab arrays intervals. When the three configurations are classified by Nusselt number, the reversed- arrays configuration appears to be thermally most efficient:

Nurev>Nu_alt>Nu_alg. The intensification is about tenfold with respect to the straight-pipeflow. In allflow configurations, the local Nusselt number starts to increase from the leading edge of the tab, due to the generation of the secondary flow, and continues to decrease.

The global friction factor in the three geometries is obtained by computing:

f ¼ 2 L=D

D

^P

r

^Wm²

(7) whereD^Pis the pressure drop between the two ends of the test section.

Correlations expressing the friction factor of the three geometries are given inTable 1. The reversed arrays have the highest friction factor of the three flow configurations and the aligned arrays show the smaller pressure drop. The variation of f with Reynolds number in the three geometries is almost constant relative to that in the straight pipeflow.

The global Nusselt number is here deﬁned by:

Nu ¼ mc_ _p

p

^L

l

T_b;outletT_b;inlet

TwTmean (8)

whereTmean ¼ ðT_b;inletþT_b;outletÞ=2.

Fig. 8plots the global Nusselt numbers obtained from Eq.(8)for the three conﬁgurations compared to that in a straight pipeﬂow

(Eq.(5)) for Reynolds numbers between 7500 and 15,000: it is observed that the global Nusselt number in the reversed arrays is greater than that in the two other geometries. The trend slope of Nu versus Reynolds number is the same for the three conﬁgurations.

The correlations for the global Nusselt number of the three geometries are presented inTable 1, afterﬁxing the evolution of the Prandtl number at Pr^0.4so as to obtain the same slope Re^2/3in the three conﬁgurations.

To compare the efﬁciency of these three conﬁgurations for constant pumping power, a thermal enhancement factorh^{is used:}

it is the ratio of the convective heat transferhto that in a straight pipeﬂowh₀, deﬁned as[13,14]:

h

¼ Nu

Nu₀

f

f₀ ₁₌₃

(9) It is observed in Fig. 9 that h is always greater than unity, between 1.8 and 2.8 for these operating conditions. The enhancement factor tends to decrease with Reynolds number, meaning that the vortex generators play a larger role in the thermal enhancement for small Reynolds numbers. The enhancement factorh^for the reversed arrays is 27% higher than for the aligned arrays and 12% higher than for the alternating arrays. The present values ofh are much higher than the other data in the literature, which range between 0.5 and 1.5[13,14].

The Nusselt number is plotted in Fig. 10 versus the power dissipation and compared to some mixer-heat exchange geometries commonly used in the industry for the same Reynolds number interval, 7500<Re<15;000. The straight-pipeflow presents the lowest Nusselt number and power dissipation, since there are no inserts in theflow volume. The Helical KenicsÔhas almost the same performance as the aligned configuration. The intensification is improved in the alternating arrays configuration, and even more Fig. 5.Flow structure and temperature distribution in tab symmetry plane of (a) aligned, (b) alternating, (c) reversed arrays, and (d) in the axial section between two tabs of reversed array, for Re¼15,000.

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in the reversed arrays conﬁguration, with negligible increase in power consumption. The other eight geometries exhibit a lower Nusselt number with power dissipation which varies over two orders of magnitude.

4.3. The vortex circulation approach

In this approach, the streamwise circulation, deﬁned in Eq.(10), is the parameter illustrating the convective intensity of the secondaryﬂow in the pipe cross section[15e17]:

G

uz ¼ Z Z

S

j

u

zjdxdy (10)

whereuzis the streamwise vorticity andSthe surface area of the pipe cross section.

In turbulentﬂow, the effect of large scale vortical motion on heat transfer enhancement compared to the turbulent diffusion of heat can be examined by estimating the relative effect of the turbulent thermal diffusivity at, which is the ratio of the eddy viscosityytto the turbulent Schmidt number, traditionally set to 1 (therefore, at¼yt). Estimating the eddy viscosity with the k ³ model leads to:

a

t ¼

y

t ¼ C_mk²

3 (11)

where the coefﬁcientC_m ¼0:09[43,44].

Let us deﬁne a new parameter c, which is the ratio of the convective motion of the large-scale vortices (represented by

0.0 0.2 0.4 0.6 0.8 1.0

0 2 4 6 8 10 12 14 16

Nu/Nu0

z/L

Aligned arrays Alternating arrays Reversed arrays

Flow direction

Fig. 7.Longitudinal evolution of normalized Nusselt number in the three geometries, Re¼15;000. The positions of the vortex generators are shown on the x-axis for aligned and alternating conﬁgurations and on the top for the reversed conﬁguration.

Fig. 6.Flow pattern and temperatureﬁeld in a cross section atr=R¼0:8 for (a) aligned, (b) alternating and (c) reversed arrays, Re¼15;000.

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circulation) to the turbulent thermal diffusivity represented by eddy viscosity; the physical meaning of c is similar to that of a Nusselt number:

c

¼

G

uz

y

t (12)

The evolution ofc as a function of Reynolds number, represented inFig. 11(a) indicates that the relative importance of the convective cells increases with turbulence ascincreases with the Reynolds number. InTable 1, theﬁtting functions forc^{versus the} Reynolds number show a power law relation with power 1/3 in the three conﬁgurations; with the highest values for the reversed array and the lowest for the aligned array, just as for the Nusselt number and the thermal enhancement factor.

The heat transfer enhancement factorhis plotted against the parametercof the primary CVP inFig. 11(b). It is observed that the slope for the three geometries is the same and equal to1. From thisfigure, for a given value ofh, the parametercis the highest in the reversed arrays, which leads to conclude that to increase the heat transfer efficiency in the reversed arrays, largercis generated than the other two configurations, and thus, from Eq.(12), larger ratioGuz=ytand hence larger streamwise vorticityGuzor lower eddy viscosityyt.

4.4. Theﬁeld synergy principle

In the ﬁeld synergy principle, the basic assertion is that the intersection angle between velocity and temperature gradients has an important role in the heat transfer[19,21]. This can be brieﬂy demonstrated starting from the energy equation

r

^cp!U

$V^/T ¼

l

V²T (13)

where!U andTare respectively the local velocity and the temperature in Cartesian coordinates. By using Gauss’s theorem to inte- grate Eq.(13)over the domain and neglecting axial conduction in

theﬂuid (since the Péclet number is greater than 100[57]), the energy equation can be written as:

r

^cp

Z Z Z

Vol

ð!U

$V^/TÞdxdydz¼

F

fNu (14)

withFthe total wall heatﬂux, which is proportional to the Nusselt number (see He et al.[58]for more details).

The Nusselt number Nu depends on the dot product

!U$V^/T ¼ j!UjjV^/TjcosðqÞ, where q is the angle of intersection between the velocity vector and the temperature gradient. Hence, forﬁxed velocity magnitude and temperature gradient, the smaller the intersection angleq, the larger the convective heat transfer rate.

The localqcan be obtained from the present numerical simulations by the expression

q

¼arccos 0 BB BB

@

UvT vxþVvT

vyþWvT vz ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

U²þV²þW²

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vT

vx 2

þ vT

vy 2

þ vT

vz 2

s

1 CC CC A (15)

It can be noted that, since the intersection angle varies between 0and 180, two ideal synergy cases exist: forq¼00cosðqÞ ¼1 and q¼1800cosðqÞ ¼ 1. The non-synergy case is q¼900cosðqÞ ¼0, as in channel and straight-pipeﬂows where this angle is maximal (z 90) since the ﬂow streamlines are perpendicular to the temperature gradient. Moreover, regarding Eq.

(14), the global heat transfer depends on the sign of the local synergy values. For heatedfluid, theflow configuration must be designed so as to attain the highest values of positive cosðqÞ; for cooledfluid, theflow configuration must be designed so as to make qas close as possible to 180.

The distribution of the ﬁeld synergy !U

xy$V^/xyT in the cross section of the three geometries is shown inFig. 12. The indexxy denotes the velocity and temperature gradient in theðx;yÞrefer- ence frame, in order to observe the effect of the secondaryﬂow Table 1

Correlations of the friction factor, Nusselt number and parametercin the three geometries.

Empty pipe Aligned arrays Alternating arrays Reversed arrays

f0¼0:079Re^0:25 falg¼0:858Re^0:057 falt¼0:948Re^0:047 frev¼0:761Re^0:004

Nu0 ¼_1þ12:7ðf^ðf⁰^=8Þ

0=8Þ¹⁼² Pr

ðPr²⁼³1ÞðRe1000Þ Nualg¼0:232 Re²⁼³Pr^0:4 Nualt¼0:274 Re²⁼³Pr^0:4 Nurev ¼0:326 Re²⁼³Pr^0:4

c0¼0 calg ¼63:2 Re¹⁼³ calt¼68:5 Re¹⁼³ crev¼88:5 Re¹⁼³

6000 8000 10000 12000 14000 16000

10 100 1000

Aligned arrays Alternating arrays Reversed arrays Empty pipe

Nu

Re 2/3

Fig. 8.Nusselt number versus Reynolds number for the three geometries and for the empty pipe. The continuous curves represent a 2/3 power lawﬁtting.

6000 8000 10000 12000 14000 16000

1.6 1.8 2 2.2 2.4 2.6 2.8 3

-1/3

Thermal enhancement factor

Re -1/3

-1/3

Fig. 9.Thermal enhancement factor versus Reynolds number.

(9)

generated by streamwise vortices on the heat transfer. It can be seen fromFig. 12(a) and (b) that the synergy is negative in the region of the common outﬂow in the aligned and alternating arrays.Fig. 4(a) and (b)shows, in fact, that the velocity vectors of the common outﬂow are opposite to the temperature gradient, leading to a negative angle of intersectionqxy. The same result is

observed in the common outﬂow region of the reversed arrays.

Moreover, in the center of the CVP (indicated by two circles in Fig. 12) in all geometries, the synergy!U

xy$V^/xyT ¼0 because the velocity vector is!U

xy ¼0. The case when!U

xy$V^/xyT ¼ 0 is also observed in regions where the velocity vector is perpendicular to the temperature gradient. The highest values of positive!U

xy$V^/xyT are observed in the regions where the velocity and the temperature gradient are aligned. It should be noticed here that, when making local analysis of theﬁeld synergy, the scalar product!U$V^/Tmust be represented and not the intersection angleqbecause the effect of the velocity magnitude and temperature gradient may have bigger effect on the heat transfer then the intersection angle q^. Therefore,qalone is not a convenient parameter for local analysis for the heat transfer.

For global analysis of the thermal performance, the global intersection angleqVolaveraged on the heat exchanger volume is considered. In the following, the validity of considering the global intersection angleqVolas a criterion for comparing the performance of the different geometries is investigated.

InFig. 13qVolfor the three geometries is plotted versus Reynolds number. Thisfigure shows thatqVol remains constant with Rey- nolds number even that the heat transfer efficiency is increased as observed in section4.2. This leads to conclude thatqVoldoes not feel the heat transfer enhancement due to the increase in Reynolds number because, in this case, the increase inj!Ujandj^/VTjis the raison behind the heat transfer enhancement. However, it can be observed fromFig. 13, thatqVolvaries between the three geometries; the highestqVolare obtained in the aligned arrays, which has the lowest heat-transfer efficiency; the lowestqVolis obtained for the reversed arrays, which are found to have the best heat-transfer performances, as established previously. However, taking for example the relative decrease of qVol between the aligned and reversed arrays, we found it is ðqVol;algqVol;revÞ=qVol;alg ¼5:3%

while the relative heat transfer enhancement between the aligned and reversed arrays is almost 40% suggesting thatqVolalone does not provides a quantitative criterion for the heat transfer.

4.5. The entropy production approach

Entropy production involves both heat transfer and pressure losses and thus allows characterization of the overall performance of a MHER. Entropy production is computed for the three flow configurations by using the model of Kock and Herwig [28,29], where four entropy production mechanisms are identified in the entropy transport equation:

- the rate of entropy production by viscous dissipation, depending on the mean velocity gradients:

S__V ¼

m

T (

2

"

vU vx

₂ þ

vV vy

₂ þ

vW vz

₂# þ

vU vyþvV

vx ₂

þ vU

vzþvW vx

₂ þ

vV vzþvW

vy ₂)

(16)

- the rate of entropy production by turbulence, deﬁned with the TKE dissipation rate:

S__V0 ¼

r

³

T (17)

- the rate of entropy production by heat transfer with mean temperature gradients:

Fig. 10.Nusselt numberversus power dissipation per mass unit of different heat exchangers for Reynolds number range [7500, 15,000].

6000 8000 10000 12000 14000 16000

1000 1200 1400 1600 1800 2000 2200 2400

1/3 1/3

1/3 Aligned arrays

Alternating arrays Reversed arrays

Re

1200 1400 1600 1800 2000 2200 2400

1.6 1.8 2 2.2 2.4 2.6 2.8 3

-1

a

b

Fig. 11.The heat transfer enhancement factorhversus (a) Reynolds number and (b) the parameterc.

(10)

S__T ¼

l

T²

"

vT vx

₂ þ

vT vy

₂ þ

vT vz

₂#

(18)

- the rate of entropy production by heat transfer withﬂuctuating temperature gradients:

S__T0 ¼

a

t

a l

T²

"

vT vx

₂ þ

vT vy

₂ þ

vT vz

₂#

¼

a

t

a

^S^_^T ⁽¹⁹⁾

The total entropy production is the sum of these four terms:

S_ ¼ S__VþS__V0

|fflfflfflfflffl{zfflfflfflfflffl}

S_viscous

þS__TþS__T0

|fflfflfflfflffl{zfflfflfflfflffl}

S_thermal

(20)

where theﬁrst two terms on the right-hand side are due to viscous (laminar and turbulent) dissipation S__viscous and the second two terms are due to thermal dissipationS__thermal.

Eqs.(16)e(19)are implemented in the post-processing of the CFD to obtain the localS_and the globalS_proentropy production, the latter being the sum over the whole volume:

S__pro ¼ Z Z Z

V

S_dxdydz (21)

In the present study, the wall is maintained at prescribed constant temperature and thus the heatflux is not constant for the different flow conditions and flow rates. Therefore, it is more convenient to use the entropy production numberN_S_{_}

pro, deﬁned by Hesselgreaves[59], where the total entropy productionS_prois non- dimensionalized by the ratioQw=Tw:

NS_pro ¼ S_proTw

Q_w (22)

whereQwandTware respectively the wall heatﬂux and the wall temperature.

The thermal enhancement factorhis presented versusN_S_{_}

proin Fig. 14. It is observed that the entropy production number N_S_{_} decreases when the thermal enhancement factor hincreases. Itpro

appears thatN_S_{_}

prois the lowest for the reversed arrays, which is the more cost-effective conﬁguration.

Moreover, the slopes of the three curves show that, in the reversed arrays, a smaller decrease inN_S_{_}

prois needed to increase the Fig. 12.Distribution of theﬁeld synergy in the tube cross section for (a) aligned, (b) alternating and (c) reversed arrays, Re¼15;000.

6000 8000 10000 12000 14000 16000

83 84 85 86 87 88 89 90

Intersection angle vol (°)

Re

Fig. 13.Intersection angleqvolas function of the Reynolds number.

(11)

thermal enhancement factor h, compared to the reference and alternating arrays (reflecting the fact that this geometry is more efficient for heat transfer than the other two configurations). These observations lead to the conclusion that the entropy production numberN_S_{_}

procan be a good criterion for the thermal enhancement qualiﬁcation in heat exchangers.

5. Conclusions

In the present work, numerical simulations are performed to investigate the influence offlow structure on the heat transfer by using different physical approaches. The numerical procedure and turbulence model are validated by previous experimental results obtained by laser Doppler anemometry. Transverse and streamwise vorticity are produced by using different positions of vortex generators in a static mixer: aligned (the reference geometry), 45 tangential periodical shift (alternating geometry), and the reference geometry with an oppositeflow direction (reversed geometry) which showed the better performance of heat transfer enhancement among other heat exchangers from the open literature.

The global heat transfer appears to be highly correlated with the circulation of the streamwise vortices induced by the vortex generators. A new parameter, defined as the ratio of the vortex circulation (which characterizes the convective motion of large- scale vortices) and the turbulent viscosity, indicates that higher circulation of longitudinal vortices induces very important increase in the heat transfer efficiency due to better radial mixing in theflow section.

Thefield synergy is characterized by an average intersection angle q between the velocity and temperature gradient. It was shown that, when making local analysis of thefield synergy,qâlone is not a convenient parameter for local analysis for the heat transfer.

In fact, the scalar product!U

$V^/Tmust be represented in this case because the combined effect of the velocity magnitude and temperature gradient may have bigger effect on the heat transfer then the intersection angleq. For global analysis of the thermal performance, the global intersection angle qVol averaged on the heat exchanger volume does not feel the heat transfer enhancement when increasing the Reynolds number because, in this case, the increase inj!U

jandj^/VTjis the raison behind the heat transfer enhancement.

As it has been checked in the 3 conﬁgurations of the HEV, the entropy production numberN_S_{_}

proconstitutes a relevant criterion for

classiﬁcation of the overall performance of heat exchangers: the minimum entropy production is concomitant with the maximum thermal efﬁciency.

Acknowledgments

C. Habchi would like to acknowledge fruitful discussions with Dr. M. Gonzalez, Dr. A. Ould El Moctar and Dr. M. Khaled. This work was ﬁnancially supported in part by ADEME (Agence de l’Envir- onnement et de la Maîtrise de l’Énergie). Dr. C. Garnier is gratefully acknowledged for monitoring this grant.

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