Enhancing heat transfer in vortex generator-type multifunctional heat exchangers

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Enhancing heat transfer in vortex generator-type multifunctional heat exchangers

Charbel Habchi

^a^,^b

, Serge Russeil

^a^,^c^,^*

, Daniel Bougeard

^a^,^c

, Jean-Luc Harion

^a^,^c

, Thierry Lemenand

^d

, Dominique Della Valle

^d^,^e

, Hassan Peerhossaini

^d

aEMDouai, EI, F-59500 Douai, France

bFluid Mechanics, Heat and Thermodynamics Group, School of Engineering, Lebanese International University LIU, Beirut, Lebanon

cUniversité Lille Nord de France, F-59000 Lille, France

dThermoﬂuids, Complex Flows and Energy Research Group, LTNeCNRS UMR 6607, École Polytechnique, Nantes University, F-44306 Nantes, France

eONIRIS, rue de la Géraudière, B.P. 82225, 44322 Nantes, France

a r t i c l e i n f o

Article history:

Received 10 August 2011 Accepted 5 January 2012 Available online 11 January 2012

Keywords:

Streamwise vorticity Heat-transfer correlation Protrusions

Vorticity generator High-efﬁciency vortex (HEV) Heat-transfer enhancement

Multifunctional heat exchangers/reactors Numerical simulations

a b s t r a c t

Global and local analysis of the heat transfer in turbulent vortical flows is studied using three- dimensional numerical simulations. Vorticity is generated by inclined vortex generators in a turbulent circular pipeflow with twelve different configurations that fall into three categories. In thefirst category are rows of trapezoidal vortex generators in different arrangements; in the second category the vortex generators arefixed at certain distance from the tube wall, and the third category has vortex generator rows between which a row of small protrusions are inserted on the tube wall. First, a global analysis of the thermal performance is performed for all these configurations, which are also compared with other heat exchangers from the literature. New correlations for the friction factor and Nusselt number are then obtained. Local analysis of the effect of theflow structure on the temperature distribution is carried out for the four configurations showing the best performances. The local analysis involves studying the streamwise vorticityflux to characterize the convective transport process, the turbulent kinetic energy characterizing the turbulent mixing, andfinally the local Nusselt number.

1. Introduction

Artificially generated vorticity is efficient in fluid mixing and heat transfer since it enhances the exchange of fluid particles between the differentflow regions with relatively small increases in pressure loss[1]. Several methods exist for generating vorticity, such as Görtler vortices and Dean cells induced by wall curvature and/or rotation[2,3], embedded vortices generated by jets[4], and turbulence promoters or vorticity generators[5,6]. This last category is the main interest in the present study.

Physical analysis of vorticity effects on heat- and mass-transfer mechanisms is a fundamental issue in optimizing existing heat exchangers and devising new enhanced designs. In fact, two types of vorticity can be distinguished: transverse stagnant vortices that are local recirculations behind the vortex generator and have their axis of rotation perpendicular to the main ﬂow direction, and streamwise vortices advecting in theﬂow direction with a swirling motion [5]. It was shown that most heat- and mass-transfer

enhancement is provided essentially by streamwise vorticity, while the transverse stationary vorticity slightly enhances heat transfer in the region near the vortex generator, generally in the near-wake of the vortex generator[1,5]. This fact is demonstrated by the strong relationship between the streamwise vorticityﬂux and the span-averaged Nusselt number observed downstream from rectangular[7]and trapezoidal[8]vortex generators.

In the present work, pressure-driven longitudinal vorticities are generated in a turbulentflow by using different configurations of vortex generator rows inserted in a circular tube. These vortex generators are based on the trapezoidal mixing tabs used in the high-efficiency vortex (HEV) static mixer[9](seeFig. 1a). Flow past trapezoidal vortex generators has been extensively studied due to its ability to enhance turbulent mixing, mass transfer and phase dispersion by the generation of complex coherent structures [10e12]. Mainly, two types offlow structures are observed downstream from a trapezoidal vortex generator: a counter-rotating vortex pair formed by the pressure difference across the vortex generator, and a periodic sequence of horseshoe-like structures shed from the trailing edges of the vortex generator[13]. These structures have been shown to enhance fluid mixing and heat exchange between low-momentum near-wall regions and the high-momentum zone in theflow core.

*Corresponding author. Ecole des Mines de Douai, Département Energétique Industrielle, 941 Rue Charles Bourseul, BP 10838, 59508 Douai Cedex, France.

Tel.:þ33 3 27 71 23 88; fax:þ33 3 27 71 29 15.

E-mail address:[email protected](S. Russeil).

Contents lists available atSciVerse ScienceDirect

Applied Thermal Engineering

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

doi:10.1016/j.applthermaleng.2012.01.020

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The efficiency of these vortex generators allows the inte- gration of several unit operations such as mixing, chemical reaction, and heat transfer into a single system. This implies the definition of multifunctional heat exchangers-reactors: perform the mixing process and chemical reactions in the same time as the heat exchange [14e16]. The energetic efficiency of multifunctional heat exchanger-reactors is, in great part, determined by the vortex generator geometry that controls theflow structure. Thus, improvement in the mixing process and heat transfer is based on the optimal use of the pressure loss dissipated in the flow.

A common application of the multifunctional heat exchanger- reactors studied here is for low viscosity liquideliquid dispersion and thermal homogenization[3,9,14]for moderate turbulentﬂows which may include exothermal chemical reactions, such as acid- catalysed hydrolysis reaction[10]. These geometries enhance the turbulence near the wall and therefore, preventing fouling and impurity[14].

The present paper provides a numerical analysis of the heat and mass transfer in twelve different conﬁgurations of vortex- generator-type multifunctional heat exchanger-reactors. The aim of this study is to analyze the effect of the differentﬂow structures, caused by different vortex generator shapes on the heat and mass transfer and thus to propose an improved geometry based on the HEV static mixer. The numerical method and different multifunctional heat exchanger-reactor geometries are described in section2.

The results are discussed in section3, where global performance analysis is provided, followed by detailed local investigations of the ﬂow structure and transfer phenomena.Section 4is devoted to the conclusions.

2. Problem description

2.1. Governing equations and numerical procedure

The ﬂow ﬁeld is governed by the three-dimensional (3D) steady-state Reynolds-averaged NaviereStokes (RANS) equations.

The continuity and momentum equations for an incompressible Newtonianﬂuid are:

vu_i

vxi ¼ 0 (1)

r

_vx^v

i

u_iu_j

¼ vp vxiþ v

vxj

"

m

^vu_vxⁱ

jþvuj

vxi

d

ij

vuj

vxi

!#

þv

s

ij

vxj

(2)

The last term on the right-hand side of Eq.(2)is the Reynolds stress tensorsij ¼ ru_iu_jresulting from the averaging procedure on the nonlinear convective terms in the momentum equations.

This term is computed by the Prandtl closure hypothesis with the RNG (renormalized group)kε turbulence model[17]. The equations for the turbulence kinetic energykand its dissipation rateε are:

r

^vðku_vxⁱ^Þ

i ¼ v

vxj

a

k

m

effvk vxj

!

þ

r

^Cmk²

εS²

r

ε (3)

r

^vðεu_vxⁱ^Þ

i ¼ v

vxj

a

ε

m

effvε vxj

!

r

^C1εC_mkS²C_2ε

r

^ε_k²R_ε (4) Nomenclature

cp thermal capacity (J/kg K)

d distance between two neighboring vortex generators arrays (m)

D diameter of the heat exchanger (m)

e detachment distance of the vortex generator (m) f friction factor ()

h distance from wall to upper edge of vortex generator (m)

H convective heat transfer coefﬁcient (W/m²K) Jz absolute streamwise vorticityﬂux (1/s) k turbulence kinetic energy (m²/s²)

l distance from vortex generator trailing edge to protrusion center (m)

L length of computational domain (m) Nu Nusselt number ()

_

m massﬂow rate (kg/s) p pressure (Pa) Pr Prandtl number () r radius of protrusion (m)

Re Reynolds number based on tube diameter (¼UmD/y⁾ ()

T temperature (K) u velocity vector (m/s) Um meanﬂow velocity (m/s) (x,y,z) Cartesian coordinate system

Abbreviations

HEV High-efﬁciency vortex mixer

Greek symbols

ak,aε inverse effective Prandtl numbers forkandε respectively

b vortex generator inclination angle () ε turbulence energy dissipation rate (m²/s³) ε global power dissipation rate perﬂuid mass unit

(W/kg)

4 ^{wall heat}ﬂux (W/m²) l thermal conductivity (W/m K)

leff effective thermal conductivity (W/m K) meff effective viscosity (kg/m s)

y ﬂuid kinematic viscosity (m²/s)

<ij deviatoric stress tensor (1/s) r mass density (kg/m³)

sk standard deviation of turbulent kinetic energy (m²/s²) sij Reynolds stress tensor

S modulus of the mean strain rate tensor (1/s) uz streamwise vorticity (1/s)

Subscripts

0 empty pipe

b bulk

f ﬂuid

[ local

s solid

inlet heat exchanger inlet outlet heat exchanger outlet mean mean value

w wall

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where the model constants are C_m ¼ 0.0845, C_1ε ¼ 1.42 and C2ε¼1.68.

The additional termR_εon the right-hand side of Eq.(4)is given by[18]:

R_ε ¼

r

^Cm

h

³¹

h

4:38

1þ0:012

h

³

ε²

k (5)

withh¼kS/ε. It should be noted here that the RNGkεmodel

shows fundamental improvements over the standardkε model [19]since the effects on turbulence of strong streamline curvature, vortices and swirl effect are taken in account, thus enhancing the ﬁnal solution[20].

The heat transfer is computed by solving the energy equation

v

vxi½u_ið

r

^EþpÞ ¼ v

vxj

l

effvT vxjþu_i<_ij

!

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Fig. 1.(a) Three-dimensional view of the HEV mixer and front view of the (b) aligned (Alg) and (c) alternating (Alt) arrays arrangements; here the black trapezoidal tabs correspond to the arrays n and the gray tabs are the tabs in the array nþ1 withq¼45 the tangential rotation angle of the tab rows. (d) The distribution of the protrusions downstream from the tabs array.

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whereEis the total energy,leffthe effective thermal conductivity and<ijthe deviatoric stress tensor representing the viscous heating due to shear effects,i.e.due to velocity gradients for incompressible viscousﬂuidﬂows.

The solver used for theflow computation is the CFD code Fluent 6.3, which is based on an Eulerian approach to solving the Cauchy equations through cell-centeredfinite volume discretization. The flow equations are solved sequentially with double precision and a second-order upwind scheme for spatial discretization of the convective terms[21]. The diffusion terms are central-differenced and second-order accurate. Pressureevelocity coupling is ach- ieved by the SIMPLE algorithm[22].

The flow in the near-wall region is solved by a two-layer approach in which the near-wall region is divided into a viscous sublayer and a turbulent region. These two regions are delimited by the turbulent wall Reynolds number based on the distance normal to the wally, which is defined as Rew ¼ry ffiffiffi

pk

=m. In the viscous sublayer, for Rew<200, the one-equation model of Wolfstein[23]

is used, where only the momentum and turbulent kinetic energy transport equations are solved (Eqs.(2) and (3)), and the turbulent dissipation rateεis related to the turbulence kinetic energykby an empirical formula defined by Chen and Patel[24]. This approach requires estimation of the wall-adjacent cell size corresponding to an ideal dimensionless wall distancey^þ_c of no more than 5, ensuring that the viscous sublayer is meshed. Therefore, the size of the cells near the wall region is refined using the tool provided in Fluent 6.3, thus refining the near-wall cells for whichy^þ_c >5.

The conduction in the solid tabs is taken into account by solving the heat conduction equation in a solid v²ðlsTÞ=vx²_j ¼0. The material for the vortex generator was chosen as aluminum so that it can be compared with other heat exchangers in the open literature, as shown in section3.2below.

The workingfluid is water, with physical properties assumed independent of temperature[25]. Flow and heat-transfer simulations are carried out for Reynolds numbers 7500, 10,000, 12,500 and 15,000 and for uniform wall temperatureTw¼360 K. The velocity profile for fully developed turbulent pipeflow[26]is prescribed at the inlet of all computational domains with uniform temperature Tin¼298 K. The turbulence kinetic energy and its dissipation rate are estimated by the turbulence intensityIderived from the empty-tube equilibrium state given by Hinze [26]. Atmospheric pressure is prescribed at the outlet. No-slip boundary conditions are prescribed on the vortex generator surfaces and on the pipe wall.

The residual value 10⁶is set as the convergence criterion for the solutions of theflow equations in Eqs.(1e4). Beyond this value no significant changes are observed in the velocity field and turbulence quantities. A value of 10⁸is set for the convergence criterion of the solution of the energy equation (Eq.(6)).

2.2. Computational domain and meshing

In the present study twelve differentflow configurations (HEV static mixer flow geometries) are investigated. The HEV mixer consists of a circular tube of inner diameterD¼20 mm in which seven vortex generator arrays are inserted. Each array is composed of four diametrically opposed trapezoidal vortex generators inclined to the wall at an angleb¼30. The distance of the upper edge of the vortex generator from the wall ish¼3.9 mm. The total length of the tube isL¼140 mm and the distance between two successive tab rows isd¼D¼20 mm. More details on the tab dimensions are provided in Habchi et al.[27].

The difference among the twelveflow configurations lies in modifications of the vortex generator shape; these are summarized inTable 1, which shows the modifications for a single tab. The tabs can be directly inclined in the mainflow direction (Dir) or inversely

inclined (Inv) to the main flow direction. The first configuration presented inTable 1a. AlgeInv is used for numerical validation by comparing the present results for this geometry with those obtained previously with a different numerical model and by laser Doppler velocimetry (LDV)[16]. It should be noticed that two types of arrays arrangements are studied; thefirst is for aligned arrays (Alg), where the seven tabs rows are aligned, and the second is the alternating arrays (Alt), where the tab arrays undergo a periodic q¼45tangential rotation with respect to one another, as shown in Fig. 1. The vortex generator in configuration (b) AlgeInv have the same shape as in (a) Alg-Inv, but the arrays are periodically rotated one with respect to another; seeFig. 1c.

The tabs shown inTable 1(cef) are supported to be at certain distanceefrom the tube wall (this distance is shown in gray) with e ¼1 mm (V1) ore¼ 2 mm (V2). The thickness of the vortex generator legs at their attachment to the wall is 0.25 mm. This distance allows theflow to pass below the vortex generator so as to reduce the recirculationflow region in the near-wake of the vortex generator, and thus to reduce effluent clogging. The tabs presented inTable 1(g, h) are also supported to be at a distancee¼1 mm (D1) from the wall, but the projected surface removed from the bottom part of the tab is added at the top of the vortex generator.

In configurations (i) AlgeInveRev and (j) AlgeDireRev, the trapezoidal vortex generator is reversed where the small base is attached to the wall. The configuration in Table (k) AlgeShf consists of seven aligned arrays in which the odd arrays are inclined in the opposite direction to theflow (Inv) and the even arrays are inclined in theflow direction (Dir).

In the conﬁguration in Table 1 (m) AlgeInveReveProt and Fig. 1d, eight protrusions consisting of hemispheres of radius r¼1 mm are uniformly distributed on the inner wall of the tube at a distancel¼3 mm downstream from each vortex generator array.

In this conﬁguration, the vortex generators are the same as in (i) AlgeInveRev and (j) AlgeDireRev.

A nonuniform unstructured three-dimensional mesh with hexahedral volumes is performed using the software Gambit.

Special attention is paid to the near-wall reﬁnement at all solid boundaries (wall and tab surfaces) so as to take into account the high velocity and temperature gradients in these regions.

To determine the appropriate mesh density for solution grid independence, the solver is run with increasing mesh densities until no effect on the results is detected. The criterion for the grid sensitivity test is based on the turbulence kinetic energy and on velocity and temperature profiles at the outlet of the computational domain. The mesh refinement is performed for each of the twelve configurations. The mesh is refined in regions in which large gradients are observed in the turbulence kinetic energy, velocity and temperature profiles. Thefinal mumber of cell ranges between 10M and 17M cells, depending on the flow configuration. The numerical simulations are performed on four parallel processors.

The present work aims to study numerically different flow configurations of a HEV static mixer that mainly consists of seven rows of vortex generator and is generally preceded by a precondi- tioner to produce a fully developedflow at its inlet[9,11]. Seven vortex generator rows are therefore used here for the twelve configurations with fully developed hydrodynamic conditions at the inlet. This choice may be also useful for future experimental laboratory studies.

3. Results and discussion 3.1. Numerical validation

The ﬁrst conﬁguration, (a) AlgeInv, has been studied experi- mentally by using laser Doppler velocimetry (LDV), and

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Table 1

The different tab shapes for the twelve conﬁgurations studied: (a) side view and (b) front view of one tab.

(a) AlgeInv (b) AlgeInv

(c) AlgeInv-V2 (e¼2 mm) (d) AlgeInv-V1 (e¼1 mm)

(e) AlgeDir-V2 (e¼2 mm) (f) AlgeDir-V1 (e¼1 mm)

(g) AlgeInv-D1 (e¼1 mm)

(h) AlgeDir-D1 (e¼1 mm)

(i) AlgeInveRev

(j) AlgeDireRev

(k) AlgeShf

(m) AlgeInveReveProt (r¼1 mm)

(b) (a)

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numerically by using both the Reynolds stress model (RSM) and a standardkεmodel (kεStd)[16].

Fig. 2 shows radial profiles of axial velocity and turbulence kinetic energy from LDV measurements and from numerical simulations with the RSM andkεRNG turbulence models. Good agreement between the different models is observed for the axial velocity. However, a relatively large difference in the turbulence kinetic energy k between the numerical simulations and LDV results can be noticed. This is in fact caused by the use of Taylor’s frozen turbulence theory to calculatekfrom LDV measurements, which is not very efficient, especially in regions close to the wall [29]. Here, only the comparison forRe¼15,000 is presented; in fact, the relative error between the different models decreases for lower Reynolds numbers (12,500e7500), since theflow gradients decrease.

A brief comparison is presented inTable 2between the previous results[16]and those obtained in the present study to validate the choice of the mesh and thekεRNG model.

The global Nusselt numbers presented inTable 2are obtained from the equation

Nu ¼ mc_ p

p

L

l

T_b;_outletT_b;_inlet

TwTmean (7)

wherem_ is the massﬂow rate,Lthe heat exchanger length,l^the thermal conductivity of the workingﬂuid (water here), andTb,inlet

and Tb, outletare the bulk temperatures respectively at the heat exchanger inlet and outlet, and T_mean¼(T_b,_inletþT_b,_outlet)/2. In Table 2,D^pis the pressure difference between the outlet and the inlet cross sections of the heat exchanger.

Table 2suggests that the results from bothkεRNG andkεStd models are in good agreement with the results from the RSM model for both Nusselt number and pressure losses. The relative error for the Nusselt number betweenkεRNG and RSM models is 7.4% and 6.7% for the pressure losses. The relative errors in Table 2 are calculated by:j(kε RNG)(kε Std)/(kε Std)j for relative error between Std and RNGkεturbulence models,

j(kεRNG)(RSM)/(RSM)jfor relative error between RSM and kεRNG turbulence models.

Meanwhile, thekεRNG model gives results in good agreement with the RSM model while using much less computational time.

Therefore, thekεRNG model is used to study the different twelve ﬂow conﬁgurations for four different Reynolds numbers.

3.2. Global performance analysis

In this section a global analysis of the performance of different conﬁgurations is presented. The global Nusselt number obtained from Eq.(7)is presented inFig. 3for all the geometries and for the four Reynolds numbers ranging between 7500 and 15,000.

For single-phaseﬂows, the Nusselt number can be correlated with the Reynolds number via a power-law[14,15]:

Nu ¼ aRe^bPr^0:4 (8)

wherePr¼m^cp/lis the Prandtl number for the workingﬂuid.

It is worth noting that higher coefficient b means a faster increase inNuwithReand thus better heat transfer augmentation with meanflow velocity increase. Generally, for heat exchangers based on circular pipe with inserts or vortex generators, the coef- ficientbin Eq.(8)ranges between 0.3 and 0.8[14,15]. The Nusselt number correlations for the present configurations inTable 3show that the coefficientbranges between 0.6 and 0.85. The application

Table 2

Comparison of Nusselt numberNuand pressure dropDpfor different turbulence models.

Nu Relative error inNu

Dp(Pa) Relative error inDp

Present results (k-εRNG) 389.4 e 1354.2 e

Habchiet al.[16](k-εStd) 391.5 0.5% 1393.6 2.8%

Habchiet al.[16](RSM) 420.0 7.4% 1451.5 6.7%

6000 8000 10000 12000 14000 16000

100 150 200 250 300 350 400 450 500

(a) Alg-Inv (g) Alg-Inv-D1 (b) Alt-Inv (h) Alg-Dir-D1 (c) Alg-Inv-V2 (i) Alg-Inv-Rev (d) Alg-Inv-V1 (j) Alg-Dir-Rev (e) Alg-Dir-V2 (k) Alg-Shf (f) Alg-Dir-V1 (m) Alg-Inv-Rev-Prot

Nu

Re

Fig. 3.Global Nusselt number versus Reynolds number. The continuous lines represent the power-lawﬁttingNu¼aRe^bPr^0.4.

a

b

Fig. 2.Numerical and experimental radial proﬁles at 3 mm downstream from the 7th tab array in the AlgeInv conﬁguration forRe¼15000 of (a) streamwise velocity U and (b) turbulence kinetic energy k.

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range for these correlations corresponds to Reynolds number between 7500 and 15,000. These Reynolds numbers correspond to those used in industrial applications for this type of multifunctional heat exchangers/reactors.

In bothFig. 3andTable 3, it can be noticed that AlgeInv-V2 (c) and AlgeDir-V2 (e) have the lowest Nusselt numbers. In fact, in these two conﬁgurations, the distance from the wall of the vortex generator is 2 mm from the wall. This distance reduces the contact surface between the heated wall and the vortex generator and thus reduces the conduction heatﬂux through the vortex generator tab;

as a result, convective heat transfer between the vortex generator and the surroundingﬂuid is decreased. Meanwhile, the conﬁgu- rations AlgeInv-D1 (g) and AlgeDir-D1 (h), in which wet surfaces are kept equal to those in the reference geometry AlgeInv (a), have higher Nusselt numbers than AlgeInv-V1 (d) and AlgeDir-V1 (f).

However, the pressure losses are greater, thus reducing the global heat transfer performance of the heat exchanger, as discussed in the next section. The highest Nusselt numbers are obtained for the conﬁgurations AlgeInv (a), AlgeInveRev (i), AlgeDireRev (j) and AlgeInveReveProt (m). Of them all, the AlgeInveReveProt (m) geometry gives slightly higher Nusselt number when the Reynolds number exceeds 12,500, with an increase of about 10% relative to the AlgeInv (a) reference conﬁguration.

The pressure lossesD^pcan be represented by the friction factorf as:

f ¼ 2D L

D

^p

r

^Um²

(9)

withU_mthe meanﬂow velocity,D^pthe computed pressure drop between the heat exchanger inlet and outlet, andrthe working ﬂuid density.

In turbulentﬂows, the global friction factorfis correlated to the Reynolds number as[14,15]:

f ¼ aRe^b (10)

The correlations for fin the configurations studied here are presented inTable 3, andfis plotted versusReinFig. 4. It can be observed from thisfigure that the pressure losses decrease with increasing the distance of the vortex generator from the wall due to the decrease in the projected wet surface. The highest pressure losses are observed for the configurations AlgeInv (a), AlgeInveRev (i), AlgeDireRev (j) and AlgeInveReveProt (m), in which the highest Nusselt numbers were also observed.

The separate analyses of the Nusselt number and the friction factor do not give a clear idea of the global performance of the different geometries. Therefore, to compare the heat-transfer efficiency of the different configurations for constant pumping power, a thermal enhancement factorhis introduced, defined as the ratio

of the HEV convective heat transferHto that in a straight-pipeﬂow H₀[30,31]:

h

¼ Nu

Nu₀ f

f₀ ₁₌₃

(11)

whereNu0is the Nusselt number for a turbulent straight-pipeﬂow.

Kakac et al.[32]concluded that the Gnielinski[33]correlation (Eq.

(12)below) agrees with the available data better than any other expression over a range of Prandtl numbers,Pr, from 0.5 to 200 and Reynolds numbers from 2300 to 5 10⁶ for fully developed turbulentﬂow in circular tubes:

Nu₀ ¼ ðf0=8Þ 1þ12:7ðf₀=8Þ¹⁼²

Pr

Pr²⁼³1ðRe1000Þ (12) where f₀ is the friction factor in straight-pipe turbulent ﬂow following the Blasius formula:

f₀ ¼ 0:079Re^0:25 (13)

Fig. 5 presents the thermal enhancement factor h ^versus Reynolds number for the different ﬂow conﬁgurations. An Table 3

Correlations of Nusselt number and friction factor in the present work.

Nu f

(a) AlgeInv 0.326Re^0.667Pr^0.4 0.761Re^0.004

(b) AlgeInv 0.132Re^0.730Pr^0.4 0.505Re^0.24

(c) AlgeInv-V2 0.050Re^0.814Pr^0.4 0.902Re^0.12 (d) AlgeInv-V1 0.045Re^0.845Pr^0.4 1.134Re^0.10 (e) AlgeDir-V2 0.075Re^0.784Pr^0.4 0.910Re^0.13 (f) AlgeDir-V1 0.376Re^0.630Pr^0.4 1.222Re^0.12 (g) AlgeInv-D1 0.506Re^0.597Pr^0.4 8.883Re^0.29 (h) AlgeDir-D1 0.588Re^0.581Pr^0.4 2.921Re^0.18 (i) AlgeInveRev 0.153Re^0.746Pr^0.4 0.966Re^0.03 (j) AlgeDireRev 0.199Re^0.715Pr^0.4 1.067Re^0.04

(k) AlgeShf 0.253Re^0.666Pr^0.4 0.719Re^0.02

(m) AlgeInveReveProt 0.069Re^0.836Pr^0.4 0.935Re^0.03

6000 8000 10000 12000 14000 16000

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

f

Re

Fig. 4.Friction factor versus Reynolds number. The continuous lines represent the power-lawﬁttingf¼aRe^b.

6000 8000 10000 12000 14000 16000

1.5 2 2.5 3

η

Re

Fig. 5.Thermal enhancement factor versus Reynolds number. (Continuous lines represent power-lawfitting curves for the four best configurations to enhance the readability of thefigure).

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interesting observation is that, beyondRe¼10,000, the thermal performance is increased in the AlgeReveInveProt (m) configuration relative to the AlgeInv (a) reference configuration, reaching about 10% forRe¼15,000. Moreover, the slope of the power-law fitting for the AlgeReveInveProt (m) configuration is smaller than those for the other geometries, implying that the decrease in the thermal enhancement factor h with Reynolds number is moderate and thus it is more efficient.

The poorest performance is observed in the AlgeInv (b) configuration, since the periodic alternating arrays arrangement (Fig. 1(c)) increases greatly the pressure losses with no significant increase in the Nusselt number. Note that the values of all thermal enhancement factorshrange between 1.7 and 2.8, meaning that the present configurations enhance the heat transfer performance between 70% and 180% relative to that of an empty pipe. These values of h are higher than those obtained in the literature for different shapes of ribs[30]and twisted tape inserts[31], whose thermal enhancement factors range between 0.5 and 1.5.

Fig. 6shows the global Nusselt number versus the global power dissipation rateεperﬂuid unit mass (W/kg), which is related to the pressure dropD^p^by:

ε¼ Um

D

^p

r

^L ⁽¹⁴⁾

The present configurations are compared inFig. 6to other heat exchangers from the literature[30,31,34e39]for Reynolds numbers ranging from 7500 to 15,000. In this figure, the straight-pipe turbulentflow presents the lowest performance since it lies in region of lowest Nusselt number and power dissipation. The configurations studied here, which are grouped in the region 0:3<ε<8, show better performances than the other heat exchangers using ribs and helical inserts. In fact, the helical KenicsÔ [39] has almost the same performance as the AlgeDir configuration, which is the classical configuration for the HEV static mixer [9,25,28] (as shown in the zoomed inset graph).

However, the performance intensiﬁcation is improved in all the present conﬁgurations, especially in AlgeInveReveProt (m), with only a small increase in the power consumption.

Fig. 6enlarges the region 0:5<ε<10 and 150<Nu<450 to show the most efficient configurations. It is observed that the slopes ofNuversusεdiffer from one configuration to another. In fact, higher slope implies that greater augmentation inNucan be provided with a smaller increase in power dissipation rate ε. Comparing both AlgeInv (a) and AlgeInveReveProt (m) shows that the slope of the latter is 30% higher than that of the former and that it thus provides better performance. Moreover, relative to the classic HEV geometry used in the industry[9], here called AlgeDir, the AlgeInveReveProt (m) configuration shows higher heat- transfer efficiencies, ranging between 40% and 55% with an increase in power consumption ranging between 40% and 45%.

In the next section, the four conﬁgurations presenting the best heat transfer performance are chosen for detailed local analysis. These conﬁgurations are AlgeInv (a), AlgeInveV1 (d), AlgeInveRev (i) and AlgeInveReveProt (m).

3.3. Local analysis of theﬂow structure and heat transfer

Fig. 7shows the temperature distribution and velocity vectors for Reynolds number 15000 at the HEV cross section just downstream from the 7th tabs array,i.e. at the outlet cross section of the heat exchanger. A counter-rotating vortex pair is observed in each configuration that is caused by the pressure difference between the wake of the vortex generator (region with low-momentumflow) and theflow core including high-momentumflow. This counter- rotating vortex pair plays the role of internal agitator and its effect is clearly observed on the radial temperature distribution.

The common inflow in the symmetry plane of the counter-rotating vortex pair,i.e.the radialflow from theflow core towards the wall, reduces the thermal boundary-layer thickness near the wall, thus increasing the temperature gradient and the heat-transfer coefficient. Conversely, the common outflow between two neighboring counter-rotating vortices ejects high-temperature fluid particles from the near-wall region towards theflow core, thus enhancing thefluid mixing process. No significant differences are observed in Fig. 7among theflow topology of the counter-rotating vortex pair in the different geometries; however, the temperature distribution

0.01 0.1 1 10

0 50 100 150 200 250 300 350 400 450

0 1 1

150 200 250 300 350 400

450 Alg-Dir [16]

Alt-Dir [16]

(a) Alg-Inv (b) Alt-Inv (c) Alg-Inv-V2 (d) Alg-Inv-V1 (e) Alg-Dir-V2 (f) Alg-Dir-V1 (g) Alg-Inv-D1 (h) Alg-Dir-D1 (i) Alg-Inv-Rev (j) Alg-Dir-Rev (k) Alg-Shf (m) Alg-Inv-Rev-Prot Staggered triangular ribs [30]

Staggered wedge upstream [34]

Inline wedge downstream [34]

Dual twisted tape [35]

Jagged twisted tape [31]

Twisted tape [36]

V-nozzle [37]

Sulzer SMX [38]

Empty pipe [32]

Helical kenics [39]

Nu ( - )

ε ^(W/kg)

Fig. 6.Nusselt number versus power dissipation per mass unit of different heat exchangers for Reynolds number range [7500, 15,000].

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differs greatly among the configurations because of the difference in the intensity of the counter-rotating vortex pairs. This intensity can be quantified by the vorticity flux J_z which is obtained by integrating the absolute streamwise vorticityjuzjon theflow cross sectionS[7,8]:

J_z ¼ P_n

i¼1j

u

zjdS P_n

i¼1dS (15)

wherenis the number of cell faces in the given cross sectionS.

The longitudinal variation of the vorticityfluxJzis presented in Fig. 8for different cross sections in the last array of vortex generators. Three configurations, AlgeInv (a), AlgeInveRev (i) and AlgeInveReveProt (m), present similarJz variations with larger values than the AlgeInv-V1 (d) configuration. In these three geometries,J_zstarts increasing from the leading edge of the vortex generator, where the formation of the counter-rotating vortex pair occurs, and reaches its maximum at the trailing edge of the vortex generator. Downstream from the vortex generator, the vorticityflux Jzdecreases due to viscous effects. At the position of the protrusion in the AlgeInveReveProt (m) configuration, a slight increase inJzis observed relative to the other two geometries, due to the local (though weak) vorticity generated by the protrusion; this is shown in Fig. 9, which focuses on the protrusion in the temperature distribution and streamlines in the AlgeInveReveProt (m) configuration.

In fact, the streamwise vorticity in Fig. 9a is caused by the interaction between the acceleratingﬂuid on the periphery of the counter-rotating vortex pair (CVP), produced by the vortex generator, and the low-momentumﬂuid near the stable focus point A.

The recirculation vortex downstream from the protrusion, shown

inFig. 9b, is caused by similar phenomena whereby the accelerating fluid particles in theflow core interact with the slowfluid in the protrusion wake and generate a transverse vorticity centering on focus point B.

The lowest values of the vorticityfluxJzinFig. 8are obtained for the AlgeInv-V1 (d) configuration. In fact, the distance of the tab from the HEV wall reduces the pressure gradients between the wake and theflow core, thus reducing the streamwise vorticity magnitude. FromFig. 8, it is observed thatJzdecreases sharply atz/

Fig. 7.Temperature distribution and velocity vectors on the outlet cross section of (a) AlgeInv, (b) AlgeInv-V1, (c) AlgeInveRev and (d) AlgeInveReveProt forRe¼15000.

0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02

0 100 200 300 400 500

Jz (1/s)

z /L (a) Alg-Inv

(d) Alg-Inv-V1 (i) Alg-Inv-Rev (m) Alg-Inv-Rev-Prot

Fig. 8.Longitudinal variation of the total streamwise absolute vorticity ﬂux for Re¼15,000.

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Lz0.96 for all the conﬁgurations except the AlgeInv-V1 (d), where it seems that the counter-rotating vortex pair persists downstream from the vortex generator for a longer distance than in the other conﬁgurations. However, with its very much decreased intensity it yields a lower performance, as previously mentioned.

It is known that the increase in the vorticityflux Jzenhances heat and mass transfer by increasing the convective transfers among the different flow regions[7,8]. However, the turbulent mixing is in great part caused by velocityfluctuations, and hence can be characterized by the turbulent kinetic energyk. Thefluid mixing mechanism by velocityfluctuations, called meso-mixing, is an important process in using chemical reactions, since it deter- mines the environment for micro-mixing, a mixing process in which reagents are mixed on scales near the Kolmogorov micro- scale[40,41].

Fig. 10shows the axial variation of the total turbulent kinetic energyktotalcomputed by the surface integral of local values ofkin the 7th vortex generator array:

k_total ¼ P_n

i¼1kdS P_n

i¼1dS (16)

When the flow encounters the leading edge of the vortex generator, a shear layer is formed due to the velocity gradient between theflow core and the region below the vortex generator. It was shown [13] that this shear layer undergoes high-velocity fluctuations and thus has high turbulent kinetic energy. This phenomenon is observed in the variation ofk_totalinFig. 10where ktotalincreases from the leading edge and reaches its maximum

slightly downstream from the vortex generator trailing edge. The decrease ink_totalfurther downstream is caused by theflow viscous effects that damp the velocityfluctuations. The difference in thek levels, noted between the AlgeInv-V1 (d) and the three other configurations, is caused by theflow acceleration below the vortex generator, thus reducing the velocity gradients between the wake and theflow core.

The meso-mixing homogeneity can be measured bysk, the ratio of the standard deviation of the turbulence kinetic energy over ktotal:

s

k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P_n

i¼1ðkkmeanÞ² n s

k_total (17)

wherekmeanis the mean value of the turbulence kinetic energy on the HEV cross section.

In fact,skis a measure of the dispersion ofkrelative to the total valuektotal. Low values ofskcan signal good mixing homogeneity.

The longitudinal variation ofskis presented inFig. 11for the last row of vortex generators (HEV exit). It is observed that, in the region corresponding to shear-layer development near the leading edge of the vortex generator,skincreases sharply since the shear layer occupies a small area in the cross section and hence the dispersion ofkvalues is greater. The dispersionskdecreases rapidly downstream from the leading edge as the shear layer and thus the velocity fluctuations are spread in theflow cross section. A local Fig. 9.Temperature distribution and streamlines on (a) aflow cross section, and (b)

a symmetry plane in the AlgeInveReveProt conﬁguration forRe¼15,000.

Fig. 10.Longitudinal variation of the total turbulence kinetic energy ktotal for Re¼15,000.

0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02

0.0 0.2 0.4 0.6 0.8 1.0 1.2

σk(m2s-2)

z/L (a) Alg-Inv

(d) Alg-Inv-V1 (i) Alg-Inv-Rev (m) Alg-Inv-Rev-Prot

Fig. 11. Longitudinal variation of the relative dispersionskforRe¼15,000.

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increase of skis also observed near the protrusion due to large gradients inknear the protrusion boundaries, and thus increased standard deviation.

The local heat transfer can be characterized by the longitudinal variation of the local Nusselt number, presented inFig. 12for the last row of vortex generators, computed as:

Nu_[ ¼ H_zD

l

^¼

f

zD

l

^TwT_b;z (18)

whereHzandfzare respectively the local convective heat transfer coefﬁcient and the local wall heat-ﬂux density at the HEV cross section at axial positionz.

The effect of the protrusion, for the AlgeInveReveProt (m) conﬁguration, on the heat transfer is clearly highlighted inFig. 12 by the local sharp increase (130% over the AlgeInv (a) reference conﬁguration) in the Nusselt number at the leading edge of the protrusion. Another maximum is observed at the trailing edge of the protrusion where the presence of the transverse vortex B, indicated inFig. 9b, increases the local heat transfer by about 80%

relative to the AlgeInv (a) reference configuration, by reduction of the thermal boundary-layer thickness. The Nusselt number decreases slowly downstream from the protrusion towards the leading edge of the vortex generator. From this position on, the counter-rotating vortex pair begins to develop, as it can be seen in the vorticityflux variation inFig. 8, and thus an increase inNu_[is observed up to the trailing edge of the vortex generator, where the vorticity flux is maximum. From this position on the Nusselt number decreases as the vorticityflux decreases. In the configuration AlgeInveRev (i), similar behavior ofNu_[is observed for the region near the vortex generator since it has the same shape as for the AlgeInveReveProt (m) configuration.

InFig. 12, the AlgeInv (a) reference configuration, in the near- wake of the vortex generator, presents larger Nusselt number values than the other configurations with a slower decrease downstream from the tab. The configuration AlgeInv-V1 (d) represents the lowest values of Nusselt number in which the variation is similar to the other geometries. However, for AlgeInv-V1 (d) an inflection point in the variation of Nu_[ is observed at the detachment zone. This inflection may be caused byflow acceleration under the tab caused by the detachment of the vortex generator.

4. Conclusions

Three-dimensional numerical simulations were performed to investigate heat transfer in twelve multifunctional heat

exchangers-reactors. These conﬁgurations are based on the various high-efﬁciency-vortex (HEV) static mixers widely used in industry.

Each configuration consists of a circular tube in which seven successive rows of trapezoidal vortex generators are inserted on the wall. The main difference among the different configurations is in the shape of the vortex generator. The study is conducted for turbulent vorticalflow with Reynolds numbers ranging between 7500 and 15,000 and constant wall temperatureT_w¼360 K.

First, a global analysis of the thermal performance of different configurations is conducted by using the thermal enhancement factor to classify the different configurations at constant power consumption. It is shown that the addition of hemispherical protrusions between the vortex generator arrays greatly enhances the heat transfer with a small increase in the pressure drop. New correlations for the friction factor and Nusselt number obtained from the present results can be used to compare the present geometries to other heat exchangers in the literature. The best comparative representation is obtained by presenting the global Nusselt number versus the total power dissipation. This representation shows that the present configurations bring great improvements to the heat-transfer process with moderate pressure losses.

Numerical simulations show that a streamwise counter-rotating vortex pair is generated downstream from each vortex generator that is caused by the pressure difference between theﬂow core and the wake of the vortex generator. This counter-rotating vortex pair plays the role of internal agitator and thus enhances radial heat and mass transfer in the HEV cross section between the different regions of theﬂow.

To characterize the convective transfer, the absolute streamwise vorticityﬂux in the HEV cross section is calculated: it describes the intensity of the counter-rotating vortex pair. It is shown that the counter-rotating vortex pair starts to develop from the leading edge of the vortex generator and reaches maximum intensity near the trailing edge.

The total turbulence kinetic energyktotalbegins increasing from the leading edge of the vortex generator where the shear layer is generated by the velocity gradients. The total turbulence kinetic energyktotalreaches its maximum near the trailing edge of the vortex generator due to the decrease in the shear layer intensity by viscous forces.

Finally, the longitudinal variation of local Nusselt number computed on different HEV cross sections shows the advantage of using protrusions between two successive vortex generator rows, since they increase the local heat transfer by increasing the temperature gradients and vorticity very close to the heated wall.

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