Mathematical Modelling and Numerical Simulation …...ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 15 (2019) No. 3, pp. 213-227 Mathematical Modelling

ISSN 1 746-7233, England, UKWorld Journal of Modelling and Simulation

Vol. 15 (2019) No. 3, pp. 213-227

Mathematical Modelling and Numerical Simulation of Shell-and-Tube HeatExchangers with S-Shaped Turbulators

Younes Menni1*, Ali J. Chamkha2,3, Ahmed Azzi1,4, Boumediene Benyoucef1

1 Unit of Research on Materials and Renewable Energies, Department of Physics, Faculty of Sciences, Abou Bekr BelkaidUniversity, BP 119-13000-Tlemcen, Algeria

2 Mechanical Engineering Department, Prince Sultan Endowment for Energy and Environment, Prince Mohammad BinFahd University, Al-Khobar 31952, Saudi Arabia

3 RAK Research and Innovation Center, American University of Ras Al Khaimah, United Arab Emirates4 Department of Mechanical Engineering, Faculty of Technology, Abou Bekr Belkaid University, BP 230-13000-Tlemcen,

Algeria

(Received June 12 2018, Accepted December 29 2018)

Abstract. This manuscript reports a mathematical modelling and computational fluid dynamic simulationof fluid flow and heat transfer through a duct of rectangular section with bottom and top wall-attached S-upstream/downstream turbulators. Four various turbulator cases were examined in this analysis, which arereferred as cases A, B, C and D. In case A, a turbulator of S-upstream form was investigated. In case B, twoS-upstream turbulators were treated. In case C, a turbulator of S-downstream form was studied and in case D,two S-downstream turbulators were examined. These turbulators are used to enhance the thermal-aerodynamicperformance inside a shell-and-tube heat exchanger. The numerical simulation was carried out using Fluent(CFD code) to solve the problem governing equations. The aerodynamic aspect was presented by stream func-tions, mean, axial, and transverse velocities, dynamic pressure and skin friction coefficients, as well as, kineticenergy, viscosity, and intensity of turbulence. The thermal aspect was presented by temperature fields, as wellas local and average numbers of Nusselt. The thermal-aerodynamic performance was presented by thermalenhancement factors. As expected, the channel provided by turbulators gives significant heat exchange values,especially those turbulators that are oriented towards the rear, case D, due to a decrease in pressure loss. Thesevalues are estimated in order 1.086, 1.459, 1.062, and 1.513 in cases A, B, C, and D, respectively.

Keywords: computational fluid dynamic simulation, turbulators, shell-and-tube heat exchangers, heat.

1 Introduction

Improving the internal structure of the channel by fixing the turbulators on its internal surfaces is an ef-fective way to improve heat exchange coefficients. Numerous studies, both numerical and experimental, havedealt with this subject earlier and recently by including new turbulators in simple or complex structures andin all fields. For example, Singh et al. [1] experimentally investigated the heat and fluid flow characteristicsof rectangular duct having its one broad wall heated and roughened with periodic discrete V-down rib. Theeffect of gap width and position was studied, besides angle of attack, rib pitch and height. Correlations for theNusselt number and friction factor in terms of Reynolds number and rib parameters were developed. Bhutta etal. [2] reported a literature review on the applications of computational fluid dynamics (CFD) in the field ofheat exchangers. They showed that CFD has been employed for the following areas of study in various typesof heat exchangers: fluid flow maldistribution, fouling, pressure drop and thermal analysis in the design andoptimization phase. Yemenici and Sakin [3] conducted a numerical investigation of flows over heated ribbed

∗ Corresponding author. E-mail address: [email protected].

Published by World Academic Press, World Academic Union

214 Y. Menni & A. J. Chamkha & et al.: Modelling and Simulation of Shell-and-Tube Heat Exchangers

walls under the effect of the Reynolds number and rib height. The results indicated that the presence of the ribscan effectively enhance the heat transfer. Sethi et al. [4] carried out an experimental investigation for a range ofsystem and operating parameters in order to analyse the effect of artificial roughness on heat transfer and fric-tion characteristics in solar air heater duct which is having dimple shaped elements arranged in angular fashion(arc) as roughness elements on absorber plate. The experimental data were used to develop Nusselt number andfriction factor correlations as a function of roughness parameters and operating parameters. Liou and Hwang [5]experimentally investigated turbulent heat transfer and friction characteristics in a channel with various-shapedridges mounted on two opposite walls. Experiments were conducted for three ridge shapes, namely triangular,semicircular, and square cross sections. Rao et al. [6] conducted a comparative experimental and numericalstudy on the flow friction and heat transfer of a pin fin dimple channel and a pin fin channel. The friction factor,Nusselt number and the overall thermal performance parameters of the pin fin-dimple and the pin fin channelswere obtained and compared with the experimental data of a smooth rectangular channel and previously pub-lished data of the pin fin channel. Park et al. [7] manufactured a heat exchanger with vortex-generating fins atthe center of each louver. Through 3D computational analysis, they analyzed the air flow characteristics andthe fin temperature in the heat exchanger under dry conditions. In addition, they conducted an experiment forobserving frost behavior and variation in thermal performance of the louvered fin heat exchanger. Mohammedet al. [8] carried out numerical simulation of laminar and turbulent mixed convection heat transfer of nanofluidsflow over backward facing step placed in a horizontal duct having baffle at its wall. In those studies, severalparameters such as different types of nanoparticles (Al2O3, CuO, SiO2 and ZnO), different volume fractionsin the range of 1 % to 4 %, different nanoparticles diameter in the range of 25 to 80 nm were used. Ozgen etal. [9] experimentally investigated a device for inserting an absorbing plate made of aluminium cans into thedouble-pass channel in a flat-plate solar air heater. Three different absorber plates had been designed and testedfor experimental study. In the first type (Type I), cans had been staggered as zigzag on absorber plate, whilein Type II they were arranged in order. Type III is a flat plate (without cans). Thianpong et al. [10] experimen-tally investigated heat transfer, friction factor and thermal performance characteristics in a tube equipped withtwisted-rings. The effects the twist and width ratios on the thermal performance evaluation were also examined.Kumar et al. [11] presented the results of an experimental investigation of heat transfer and friction in the flowof air in rectangular ducts having multi v-shaped rib with gap roughness on one broad wall. The effect of relativeroughness height, relative roughness width, relative gap distance, relative gap width, angle of attack and relativeroughness pitch on the heat transfer coefficient and friction factor was studied. Saini and Verma [12] carriedout an experimental study to investigate the effect of roughness and operating parameters on heat transfer andfriction factor in a roughened duct provided with dimple-shape roughness geometry. Using experimental datacollected during extensive experimental study, correlations for Nusselt number and friction factor were alsodeveloped. Li et al. [13] experimentally studied the flow boiling in TSF and TPF channels. The effects of vol-ume flow rate, heat flux and outlet void fraction on pressure drop and boiling heat transfer performance wereinvestigated. Kamali and Binesh [14] developed a computer code to study the turbulent heat transfer and fric-tion in a square duct with various-shaped ribs mounted on one wall. The simulations were performed for fourrib shapes, i.e., square, triangular, trapezoidal with decreasing height in the flow direction, and trapezoidal withincreasing height in the flow direction. Chu et al. [15] carried out a periodical numerical method based on fluentsoftware to investigate the heat transfer and pressure drop performances of fin and-oval-tube heat exchangerwith five different inlet angles. Tian and Zhao [16] focused on the latest developments and advances in solarthermal applications, providing a review of solar collectors and thermal energy storage systems. Various typesof solar collectors were reviewed and discussed, including both non-concentrating collectors and concentratingcollectors. Kalogirou [17] presented a survey of the various types of solar thermal collectors and applications.Initially, an analysis of the environmental problems related to the use of conventional sources of energy waspresented and the benefits offered by renewable energy systems were outlined. A historical introduction intothe uses of solar energy was attempted followed by a description of the various types of collectors includingflat-plate, compound parabolic, evacuated tube, parabolic trough, Fresnel lens, parabolic dish and heliostat fieldcollectors. Alam and Kim [18] presented a comprehensive literature review of the various heat transfer tech-niques used to increase the performance of double-pass solar air heaters. Literatures on double-pass solar airheaters were reviewed to understand the different techniques used for performance improvement. The different

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World Journal of Modelling and Simulation, Vol. 15 (2019) No. 3, pp. 213-227 215

operating and system parameters were categorized to understand their performance behavior. Kabeel et al. [19]aimed to present a review of the literature dealing with improvement methods, design configurations and appli-cations of different types of solar air heaters. Oztop et al. [20] reviewed the previously conducted studies andapplications in terms of design, performance assessment, heat transfer enhancement techniques, experimentaland numerical works, thermal heat storage, effective- ness compassion and recent advances. Rajarajeswari andSreekumar [21] presented an extensive study of the research carried out on matrix solar air heater. Based onliterature review, they concluded that the solar air heaters performed well when packed with porous mediumand this is due to the geometrical parameters of porous material. In addition double pass porous bed solar airheaters performed better than single pass. Various types of matrix materials used in the literature and correla-tions developed for heat transfer and friction factor by different researchers were presented. Skullong et al. [22]experimentally conducted an investigation on heat transfer characteristics in a solar air heater channel usingwavy grooves incorporated with pairs of trapezoidal-winglets (TW) placed on the absorber plate. Ganorkarand Kriplani [23] analyzed the overall performance of suitably designed lateral perforated fins in a rectangularchannel with passive heat transfer augmentation technique. In the rectangular channel, different types of per-forated fins were used. Effect of perforated fins in a rectangular channel was analyzed for different ReynoldNumbers. Murata and Mochizuki [24] numerically simulated the heat transfer in a rib-roughened duct by us-ing the second-order finite difference method in coordinates fitted to transverse or angled ribs. Turbulent andlaminar cases were computed for rib angles of 60◦ and 90◦. The comparison between the laminar and turbu-lent results showed clear differences in heat transfer distribution because the higher momentum fluid of theturbulent case was more disturbed by the ribs as compared to the laminar case. Wang et al. [25] carried out nu-merical simulations and experimental work for water flow forced convection channel with repeated ribs surfacein the range of Reynolds number from 20000 to 60000 at e/D = 0.05. Heat transfer and flow behaviors pastthe single-phase channel were determined. Min et al. [26] applied a simplified conjugate gradient method forthe optimization of the distance of two successive ribs in a two-dimensional channel mounted with ribs on thebottom. The optimal pitch ratio of the ribs was searched under the maximum heat transfer rate. The sensitivityand adjoint problems were not considered but a constant search step size was applied. Other similar works canbe found in the literature as Motlagh and Moghimi [27], Saini and Pratibha [28], Kumar et al. [29], Das andChakrabarti [30], Ahmed and Khan [31], Saini and Katiyar [32], Elangovan and Selvaraj [33], Menni and Azzi[34], and Menni et al. [35]-[37].

This study aims to determine the best internal structure of thermal exchange channels supported by indi-vidual or double S-upstream/downstream shaped turbulators. These turbulators will be installed on the innerwalls of a shell-and-tube heat exchanger. Four approaches will be addressed. Case A is composed of a singleforward-oriented S-turbulator (or S-upstream turbulator). Case B same as the previous channel but in the caseof two overlapping turbulators. Case C, the channel consists of one S-turbulator directed towards the outletthe channel (or S-downstream turbulator). In the fourth case (D), the inclusion of two staggered S-turbulatorsoriented towards the back. Air flow lines, mean, axial, and transverse speed fields, dynamic pressure curves,turbulent kinetic energy and temperature fields, skin friction and heat transfer coefficients, and thermal en-hancement factors will be found in all cases proposed.

2 Definition of problem

This paper reports a mathematical modelling and computational fluid dynamic simulation of fluid flow andheat transfer through a duct of rectangular section with bottom and top wall-attached S-upstream/downstreamturbulators, Fig. 1. Four various S-turbulator cases were examined in this analysis, which are referred as casesA, B, C and D. In case A, a turbulator of S-upstream form was investigated, Fig. 1a. In case B, two S-upstreamturbulators were treated, Fig. 1b. In case C, a turbulator of S-downstream form was studied, Fig. 1c, and in caseD, two S-downstream turbulators were examined, Fig. 1d. These turbulators are used to enhance the thermal-aerodynamic performance inside a shell-and-tube heat exchanger. The Reynolds number considered rangesfrom 12,000 to 32,000.

The flow is steady, 2D, turbulent, Newtonian and incompressible. The physical properties of the fluid andsolid are kept constant. The body forces, viscous dissipation and radiation heat transfer are not considered.

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Fig. 1: Geometries under examination: (a) channel with one S-upstream baffle: case A, (b) channel with twoS-upstream baffles: case B, (c) channel with one S-downstream baffle: case C, and (d) channel with two S-downstream baffles: case D.

Based on the above condition, the mathematical model of the turbulent forced-convection airflow is governedby the Reynolds averaged Navier-Stokes (RANS) equations with the k-epsilon turbulence model [38] and theenergy equation. In the Cartesian coordinate these equations can be written in the following compact form

∂

∂x(puφ) +

∂

∂y(pvφ) =

∂

∂x(Γφ

∂φ

∂x) +

∂

∂y(Γφ

∂φ

∂y) + Sφ (1)

where φ is a vector composed of the scalars u, v, T, k and ε; u and v stand for the mean velocities towards thex and y axis respectively; T is the temperature; k and ε stand for kinetic energy and turbulent dissipation rate,respectively; Γφ and Sφ represent the turbulent diffusion coefficient and the source term associated with thegeneral variable φ in this order.

A uniform velocity profile, u = Uin, is introduced at the intake of the channel, x = 0. An atmosphericpressure-outlet profile, P = Patm, is applied at the exit, x = L. No-slip and impermeability boundary conditionsare applied over the solid boundaries. A constant temperature of 375 K was applied on the upper and lower wallsof the channel. The temperature of air was set equal to 300 K at the inlet of the channel. The flow Reynoldsnumber (Re) based on channel aeraulic diameter,

Dh = 2HW/(H +W ) (2)

is given by

Re = ρUDh/µ (3)

The skin friction coefficient (Cf) is given by

Cf =τw

12ρU

2(4)

The friction factor (f) is evaluated from the pressure drop (∆P) as

f =(∆P/L)Dh

12ρU

2(5)

where presents the average axial velocity of the section, and w is the shear stress to the wall. The local Nusseltnumber (Nux) which can be written as



Nux =hxDh

kf(6)

and the average Nusselt number (Nu) can be obtained by

Nu =1

L

∫Nux∂x (7)

The following expression represents the thermal enhancement factor (TEF):

TEF = (Nu/Nu0)/(f/f0)1/3 (8)

The Dittus-Boelter and Petukhov correlations [39] can be used to normalize the average Nusselt numberand friction factor, respectively. The quantities Nu0 and f0 are the average Nusselt number and the frictionfactor of the smooth channel, respectively. The Dittus and Boelter correlation has the form:

Nu0 = 0.023Re0.8Pr0.4 for Re ≥ 104 (9)

The Petukhov correlation has the form:

f0 = (0.79 lnRe− 1.64)−2 for 3 ∗ 103 ≤ Re ≤ 5 ∗ 106 (10)

3 Numerical simulation

The numerical simulation was carried out using Fluent (CFD code) to solve the problem governing equa-tions. The governing flow equations are integrated by the finite volume method [40]. The QUICK numericalscheme developed by Leonard and Mokhtari [41] is employed to discretize the convective terms. The SIMPLEdiscretization algorithm is used for pressure velocity [40]. The numerical results in terms of axial velocity werevalidated with the numerical and experimental results of Demartini et al. [42] for the same structural conditionunder similar operating parameters, as shown in Fig. 2. The evolution of the Nusselt number as a function of

Fig. 2: Numerical validation of axial velocity.

the Reynolds number is also simulated for a smooth air channel and compared with empirical correlation ofDittus-Boelter [39] as shown in Fig. 3. As shown on these figures, a good agreement is obtained.



Fig. 3: Numerical verification of Nusselt number.

4 Results and discussion

Fig. 4 represents the distribution of streamlines of the air current within the channel in its various internalstructures. Case A represents a channel made up of a turbulator in the S-upstream form attached to the lowerwall. The current lines are straight and regular at the entrance of the channel until the turbulator. The currentdeflects due to its collision with the turbulator and towards the upper section of the channel between the upperface of the turbulator and the lower surface of the top wall of the channel.

Fig. 4: Streamlines for various cases studied: (a) channel with one S-upstream baffle: case A, (b) channel withtwo S-upstream baffles: case B, (c) channel with one S-downstream baffle: case C, and (d) channel with twoS-downstream baffles: case D, Re = 12,000. Stream-function values in kg/s

The dynamic pressure on the lower front of the turbulator decreases, forming a recycling area that is veryweak. The same phenomenon is present behind this turbulator but with a large size until the exit. Case B, thechannel consists of two overlapping S-upstream turbulators, the first on the upper wall and the second on thebottom wall of the channel. As expected, the airfield lines are regular between the channel entrance and the



front side of the first S-upstream turbulator. The pressure decreases next to the turbulator, leading to a deviationof the current towards the bottom of the channel and then dividing it at the upper surface of the turbulator totwo streams. A main stream from left to right, and a secondary stream behind the first turbulator as a resultof the formation of a recycling area. The same phenomenon exists near the second S-upstream turbulator.Low pressure and forming rings for recycling, weak frontal strength, and strong intensity at the back. CaseC, the channel consists of only one turbulator such as case A but this turbulator is directed towards the rear(S-downstream). The same structure and distribution in the field lines except the size of recycling areas next tothe turbulator. Case D relates to a channel consisting of two turbulators, as in case B, but directed towards thechannel outlet. A disturbance in the current due to the presence of these turbulators and a drop in pressure nextto them, causing the formation of currents in the reverse direction represented in the recycling cells.

The mean velocity fields for the different geometrical configurations of the channel are represented inFig. 5. The mean velocity decreases near the S-turbulators especially in the rear areas due to a decrease inthe pressure values, while increasing between the S-turbulators and in the confined areas between the uppersurfaces of the turbulators and the inner walls of the channel. The mean velocity values are very large in thecase B due to the presence of S-turbulators directed towards the inlet of the channel by enlarging the recyclingzones. In this case, mean velocity values are 5.13 m/s, or 4.88 times greater than the input speed, which is29.163, 44.445, and 27.67 percent higher than in cases A, C, and D, respectively.

Fig. 5: Fields of velocity-magnitude for various cases investigated: (a) channel with one S-upstream baffle: caseA, (b) channel with two S-upstream baffles: case B, (c) channel with one S-downstream baffle: case C, and (d)channel with two S-downstream baffles: case D, Re = 12,000. Velocity-magnitude values in m/s.

For a good examination of the aerodynamic structure within the channel, we added the axial velocity fieldsas shown in Fig. 6. In the first case, the axial velocity values increase from the left sharp edge of the turbulatorto its maximum value near the bottom of the upper wall of the channel near its outlet. In the second case, theflow is accelerated from the turbulators, especially at the level of the second turbulator due to the decrease in theflow area within the channel. In the third case, as expected, the axial velocity values rise above the turbulator,but further in the fourth case. The second case gives very significant values for axial velocity compared to othercases A, C, and D, an increase of 29.164, 45.833, and 35,873 % respectively. While low velocities of negativevalues exist near the turbulators, which indicate recycling cells for the back areas.

The transverse velocity fields according to the vertical direction of the channel are present in Fig. 7. Themaximum values of the transverse velocity are located near the sharp edge of the S-turbulators, which areplaced on the lower channel walls in cases A and B, and also next to the upper left section of the lower S-turbulators, but in cases C and D due to extreme deviation of the current towards the upper part of the channel.



Fig. 6: Fields of axial velocity for various cases treated: (a) channel with one S-upstream baffle: case A, (b)channel with two S-upstream baffles: case B, (c) channel with one S-downstream baffle: case C, and (d) channelwith two S-downstream baffles: case D, Re = 12,000. X-velocity values in m/s.

While the negative weak values are concentrated in contact with the sharp edge of the upper S-turbulator ofthe case B and near the lower section of the top S-turbulator of case C due to the downstream flow direction.The y-velocity values are greater in case B, i.e., 43.794, 60.272, and 25.004 % greater than those in cases A,C, and D, respectively. The dynamic pressure fields of the air current are listed in Fig. 8. The pressure valuesnear the S-turbulators are reduced in all proposed cases, due to the current deflection due to the presence ofthe S-turbulators, and their separation at the upper faces. This leads to the formation of low-intensity rotaryrings on the front, and high-intensity vortices in the back areas. The pressure values in the regions between theinner surfaces of the channel and the top faces of the S-turbulators are increases. The dynamic pressure reachesits maximum value behind the lower S-turbulator near the upper surface of the channel in case B, which isestimated at 16.121 Pa. this value decrease by 49.822 % in case A, 69.115 % in case C while 47.683 % in thelatter case D.

The turbulent kinetic energy values of the air current rise from the upper front of all S-turbulators, due tothe high pressure in these areas due to the extreme deviation in the current due to the turbulator presence anddecreases in flow area, Fig. 9. The case B records the highest values in terms of kinetic energy especially abovethe second turbulator where the maximum values are located near their upper face. The turbulent kinetic energyin this case was 1.95 m2/s2. This value is greater than that given by cases A, C, and D, at about 72.463, 84.806,and 47.683 %, respectively. Fig. 10 shows the turbulent viscosity fields in the different channels studied.Through this figure, different important areas of the viscosity can be identified. These areas are located inregions with high flow velocities, especially as the air pass through the confined space between the S-turbulatorsand the top and bottom walls of the channel, especially above the second S-turbulator in case B, where the valueof turbulent viscosity is 0.0058 Kg/m-s, while 0.0034 Kg/m-s in case A, 0.0021 Kg/m-s in case C, and finally0.0028 Kg/m-s in case D.

The intensity disturbance of the air current was also studied as shown in Fig. 11. This intensity was themaximum value near the sharp edge of the second S-turbulator, as well as near the upper surface of the channelabove the same turbulator and in case B. in this case, the intensity was 113.986 %. In the remaining cases, thisintensity decreases by 47.666 % in case A, 61.005 % in case C, and 20.358% in case D.

The temperature fields in different channel regions are reported in Fig. 12. Hot areas are located near S-turbulators. These areas have low temperature gradients due to low flow velocity in these regions of the channeldue to low pressure. Vortices were generated in those regions at low and negative velocities. Temperature values



Fig. 7: Fields of transverse velocity for various cases examined: (a) channel with one S-upstream baffle: caseA, (b) channel with two S-upstream baffles: case B, (c) channel with one S-downstream baffle: case C, and (d)channel with two S-downstream baffles: case D, Re = 12,000. Y-velocity values in m/s.

Fig. 8: Fields of dynamic pressure for various cases considered: (a) channel with one S-upstream baffle: caseA, (b) channel with two S-upstream baffles: case B, (c) channel with one S-downstream baffle: case C, and (d)channel with two S-downstream baffles: case D, Re = 12,000. Dynamic pressure values in Pa.

are reduced; temperature gradients rise above S-turbulators and in all cases due to the rapid flow of air due tothe extreme deviation in the direction of current due to the presence of S-turbulators, resulting in a decrease inflow area.

Fig. 13a shows the distribution of normalized local Nusselt number profiles at the top axis of the channel inall studied cases. The Nusselt values rise in areas above the lower S-turbulator and in all existing configurations.This is the result of a change in the current direction, from the same turbulator towards the hot top surface of thechannel and thus a good heat exchange. The Nusselt values also rise in the back areas of the upper S-turbulator,due to the presence of cells in continuous rotation, thus mixing well and long enough for heat exchange. TheNusselt numbers decrease in the front of the upper S-turbulators, due to the direction of the air current towards



Fig. 9: Fields of turbulent viscosity for various cases considered: (a) channel with one S-upstream baffle: caseA, (b) channel with two S-upstream baffles: case B, (c) channel with one S-downstream baffle: case C, and (d)channel with two S-downstream baffles: case D, Re = 12,000. Turbulent viscosity values in Kg/m-s.

Fig. 10: Fields of turbulent viscosity for various cases considered: (a) channel with one S-upstream baffle: caseA, (b) channel with two S-upstream baffles: case B, (c) channel with one S-downstream baffle: case C, and (d)channel with two S-downstream baffles: case D, Re = 12,000. Turbulent viscosity values in Kg/m-s.

the bottom region of the channel, thus decreasing the contact between the air particles and the top hot surfaceof the channel. The case B, supported by two forward-facing S-turbulators, offers high value of Nusselt, thanksto the expansion of recycling cells in this case. These values decrease in case D, followed by case A, and finallycase C.

The normalized average Nusselt number is increased by increasing the flow velocity at the input of thechannel by increasing the number of Reynolds, Fig. 13b. Also the Nusselt number increases by increasing thenumber of S-turbulators, such as cases B and D. the case B gives important values of about 4.85. This value isgreater than that given in cases A, C, and D at 37.183, 45.045, and 19.414 %, respectively.



Fig. 11: Fields of turbulent intensity for various cases considered: (a) channel with one S-upstream baffle: caseA, (b) channel with two S-upstream baffles: case B, (c) channel with one S-downstream baffle: case C, and (d)channel with two S-downstream baffles: case D, Re = 12,000. Turbulent intensity values in % (* 102).

Fig. 12: Fields of temperature for various cases considered: (a) channel with one S-upstream baffle: case A,(b) channel with two S-upstream baffles: case B, (c) channel with one S-downstream baffle: case C, and (d)channel with two S-downstream baffles: case D, Re = 12,000. Temperature values in K.

The normalized skin friction coefficient profiles are given in Fig. 14a. As expected, the values of frictionare high in areas with high-speed current mean over the bottom S-turbulator and in all cases. The case B has thegreatest loss of energy in terms of friction coefficient. This loss of energy decreases in other cases, especially inthe case of a single S-turbulator such as cases A and C. This observation was confirmed in the following figure,Fig. 14b. Here we find energy loss in terms of average skin friction coefficient for different values of Reynoldsnumber, about 40.527, 57.630, and 53.66 % compared to those in cases A, C, and D, respectively.

Fig. 15 shows the profiles of thermal enhancement factors (TEFs) in the various studied S-turbulatorconfigurations. As expected, the channel with two S-turbulators gives significant TEF values, especially thosethat are oriented towards the back (case D) due to the decrease in energy loss through friction. These values areestimated at 1.086, 1.459, 1.062, and 1.513 in cases A, B, C, and D, respectively.



Fig. 13: Evaluation of normalized (a) local and (b) average Nusselt numbers for various channels.

Fig. 14: Evaluation of normalized (a) local and (b) average friction coefficients for various channels.

5 Conclusion

The main points of this research are summarized as follows:• The inclusion of S-shaped turbulators is an effective way to raise the intensity of heat exchange within thechannel.• Channels with two S-turbulators (cases B and D) are more important than those with only one S-turbulator(cases A and C).• The velocity, skin friction, dynamic pressure, and turbulent kinetic energy values of the air current arelarge next to the upper surface of the bottom wall-mounted S-turbulator, especially in cases with two S-upstream/downstream turbulators, cases B and D, respectively.• Case B (two S-upstream turbulators) gives more heat exchange but is more friction loss due to the intensityof the recirculation zones in it.• The case D (two S-downstream turbulators) is considered to be the best internal structure of the channel dueto high thermal performance coefficients.• This study is numerical and can be tested in the future.



Fig. 15: Thermal enhancement factor.

• In the future, high-tech liquid fluids such as fluids supported by very small solid particles (nanoparticles) canbe studied.

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Mathematical Modelling and Numerical Simulation …...ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 15 (2019) No. 3, pp. 213-227 Mathematical Modelling