Journal of Thermal Analysis and Calorimetry https://doi.org/10.1007/s10973-018-7350-4 (0123456789().,-volV)(0123456789().,-volV)

Entropy generation analysis for nanofluid flow inside a duct equipped with porous baffles Hamid Shamsabadi1 • Saman Rashidi2 • Javad Abolfazli Esfahani1 Received: 8 April 2018 / Accepted: 1 May 2018  Akade´miai Kiado´, Budapest, Hungary 2018

Abstract In this research, a numerical simulation is performed to investigate thermal and viscous irreversibilities for Al2O3–water nanofluid inside a duct equipped with porous baffles. The effects of different parameters including Reynolds number, Darcy number, solid volume fraction of nanoparticles, and the number of baffle were investigated on thermal and viscous entropy generation rates and Bejan number. The results indicated that the viscous and thermal entropy generations decrease by increasing the number of baffles for N [ 4. The reductions in the viscous and thermal entropy generations are 32 and 14%, respectively as the number of baffles increases in the range of 4–16. Moreover, the thermal entropy generation is dominant term in most part of the duct except along the centerline of the duct in the space between bottom and top baffles where the velocity gradients are intense. Keywords Entropy  Nanofluid  Baffle  Porous  Duct List of symbols a Baffle height (m) A Surface (m2) Bc Boltzmann constant (–) Be Bejan number (–) CF Forchheimer coefficient (–) CP Specific heat at constant pressure (J kg-1 K-1) df Molecular diameter of base fluid (nm) dp Nanoparticle diameter (nm) Da Darcy number (= K/H2) (–) H Width the duct (m) k Thermal conductivity (W m-1 K-1) K Permeability (m2) lBF Mean free path of water (–) L1 Distance of baffle from input of the duct (m) L2 Distance of baffle from output of the duct (m) N Number of baffle (–) Ng Dimensionless local volumetric entropy generation rate (–)

& Javad Abolfazli Esfahani [email protected] 1

Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad 91775-1111, Iran

2

Department of Mechanical Engineering, Semnan Branch, Islamic Azad University, Semnan, Iran

Nt p Pr Re S Sg T u, v x, y w

Dimensionless total entropy generation rate (–) Pressure (Pa) Prandtl number (= tf/af) (–) Reynolds number (= qfUinH/lf) (–) Baffle spacing (m) Entropy generation rate (W m-3 K-1) Temperature (K) Velocity component in x and y directions, respectively (m s-1) Rectangular coordinates components (m) Baffle thickness (m)

Greek symbols k Binary parameter (–) d Distance between particles (nm) l Dynamic viscosity (kg m-1 s-1) e Porosity (–) q Density of the fluid (kg m-3) u Solid volume fraction (–) Subscripts/superscripts B Brownian eff Effective f Fluid in Inlet nf Nanofluid p Porous

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s w

Solid Wall

Introduction Ducts are widely used in heat exchangers, heaters, and reactors [1, 2]. Heat transfer improvement is a key factor that should be considered when using ducts for these applications. Heat transfer can be improved by active or passive techniques. Active methods need external source of energy, while passive ones change thermal characteristics of the flow or geometry of the system. Accordingly, passive methods have attracted considerable attentions due to easy and affordable operation and no external energy consumption [3]. Using materials with high thermal conductivities such as porous materials or nanoparticles, using swirl flow devices, increasing the heat transfer surface, and using baffle and ribs to disrupt the boundary layer are some passive techniques. Bovand et al. [4] used porous material with high thermal conductivity in a solar heater to improve the heat transfer rate. They found that the heat transfer rate enhances by increasing in the porous layer thicknesses for larger values of Darcy number. Zade et al. [5] used a new eccentric twisted tape insert in a pipe. They found that a swirl influence generated by twisted tape transfers from tube center to near surface by inserting an eccentric twisted tape and this has a more influence on the heat transfer improvement. Li et al. [6] studied numerically the influences of hollow narrow twisted tapes on heat transfer inside a pipe. They concluded that the cross hollow twisted tape is very effective with great-efficiency. Adding nanoparticles to basefluid is a promising technique to achieve a higher heat transfer rate in ducts [7–14]. The thermal characteristics of nanoparticles along with thermophoresis phenomenon and Brownian motion of them are main mechanisms for heat transfer improvement of these materials. Santra et al. [15] improved the heat transfer in a duct by using Cu–water nanofluid. They used both the Newtonian and non-Newtonian nanofluids. They found that for non-Newtonian nanofluid, the surface friction increases rapidly by increasing the nanoparticle concentration for lower values of Reynolds number, while the rate of this increment reduces by increasing the Reynolds number. Bovand et al. [16] used Al2O3–water nanofluid in a duct. They observed that the thermal entrance length increases slightly as the nanoparticle concentration increases. Baffles can be used to disrupt both hydraulic and thermal boundary layers in ducts resulting a higher heat transfer rate. Mousavi and Hooman [17] used staggered baffles in entrance of a duct. They showed that the heat transfer rate increases as the blockage ratio increases.

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Promvonge et al. [18] used angled baffle in a square duct. They stated that the P-vortex flows induce impinging flows around the baffle. This causes a considerable enhancement in heat transfer rate inside the duct. To achieve larger values of heat transfer improvement in ducts, some researchers combined two augmentation heat transfer techniques. Rashidi et al. [19] reviewed the researches performed on simultaneous usage of nanofluid and inserts for heat transfer improvement. They reported that a good enhancement in the heat transfer rate can be achieved by using vortex generator in base fluid. However, a remarkable enhancement can be observed by simultaneous usage of nanofluids and vortex generator. Targui and Kahalerras [20] employed simultaneously porous baffles and nanofluid in a double-pipe heat exchanger and studied the influences of them on the efficiency of this system. Note that they did not perform an entropy generation analysis for this system. Their findings indicated that in order to reach to the best thermal efficiencies, the nanofluid should be employed in the annular gaps attached to the porous baffles where the cold liquid is circulated. Goudarzi and Jamali [21] employed simultaneously Al2O3-EG nanofluid and wire coil inserts to enhance the thermal efficiency of a car radiator. Their results showed that the thermal efficiency factor is greater than unity for all cases, which means that this combined method can be employed successfully for car radiators to improve the thermal efficiency. Rashidi et al. [22] used simultaneously transverse twisted-baffles and nanofluid in a square duct. They reported that the number of vortices inside the flow increases by increasing the pitch intensity of baffles. The entropy generation analysis is a good method for evaluating a thermal system because the first law only considers conservation of energy and did not discuss about possibility of destruction of the useful work during a heat transfer process. Indeed, the entropy generation analysis can be used for the design, optimization and yield assessment of various systems [23]. Ebrahimi et al. [24] investigated simultaneous influences of nanofluids and vortex generators on the irreversibility in a microduct. They found that usage of nanofluids diminishes the irreversibility level in the microduct equipped by longitudinal vortex generators and also improves the thermodynamic yield of the microduct. Lately, Rashidi et al. [25] performed a second low analysis for a new eccentric twisted tape insert in a pipe. They observed that irreversibility is more serious for larger values of the eccentricity. There are some researches about entropy generation in porous media using both local thermal equilibrium (LTE) and local thermal non-equilibrium (LTNE) models. Akbarzadeh et al. [26] used local thermal equilibrium model to perform first and second laws of thermodynamics analysis for nanofluid flow in a heat exchanger duct with wavy surfaces and a porous plate. Maskaniyan et al. [27] used local thermal equilibrium

Entropy generation analysis for nanofluid flow inside a duct equipped with porous baffles

model to simulate the natural convection and entropy generation in a duct with a porous insert placed as a cooling system. Torabi et al. [28] used local thermal non-equilibrium model to study the heat transfer and entropy generation inside a duct partially filled with a porous plate. Torabi et al. [29] studied theoretically the entropy generation of the nanofluid flow in a porous channel using local thermal non-equilibrium model. Valipour et al. [30] performed a numerical work to study the effects of magnetic field on the heat transfer around a solid cylinder covered with a porous substrate. They employed the least square technique to present two equations for the mean Nusselt number [31–34]. This paper investigates numerically the potentials of porous baffles and nanofluids in a duct from second laws of thermodynamic viewpoint. The porous baffles can be used to decrease the pressure drop associated with these inserts. The literature review shows that the entropy generation analysis for combined usage of porous baffles and nanofluids in a duct has not been investigated by previous researchers. As a result, this topic is covered in this paper.

•

•

nanofluid characteristics with a good accuracy and it is not required to employ discrete and mixture models. All simulations are performed for Reynolds number in the range of 100 \ Re \ 500, solid volume fraction of nanofluid in the range of 0 \ u \ 0.05, Darcy number in the range of 10-7 \ Da \ 10-3, and the number of baffles in the range of 0 \ N \ 16. The porous baffles are isotropic homogeneous. Moreover, the solid zone in porous baffles is in local thermal equilibrium with the liquid phase. Usually, LTE is valid when the local temperature difference between fluid and solid phases in porous media is not important [36]. This assumption fails during fast cooling or heating or problems with large temperature variation inside the porous media. Accordingly, LTE is valid for this problem which the local temperature difference between fluid and solid phases in porous media is not important.

Mathematical modeling Governing equations

Physical model The physical model for the simulated duct is shown in Fig. 1. The Al2O3–water nanofluid with inlet velocity Uin and temperature Tin flows through a duct with length L and width H. The surfaces of the duct have a constant temperature. Some porous baffles with height a, thickness w and space length S are placed inside the duct. The first and last porous baffles are located at the distance of L1 from input and L2 from the output of the duct. The following assumptions are invoked to simulate this problem: • The flow is two-dimensional, incompressible and steady in laminar regime. • The effective single-phase approach is employed to predict the thermophysical characteristics of the nanofluid. Albojamal and Vafai [35] examined effective single-phase, discrete and mixture models to predict the nanofluid characteristics. Their study showed that the single-phase approach is capable to predict the

Uin Tin

The following governing equations can be used to simulate this problem: • Continuity equation: ou ov þ ¼ 0: ð1Þ ox oy • Momentum equations:     qnf ou ou op lnf o2 u o2 u þ v þ u þ ¼  ox oy ox e ox2 oy2 e2  l CF q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  k nf  pffiffiffiffinf u2 þ v2 u; K K  2    qnf ov ov op lnf o v o2 v u þv þ ¼ þ ox oy oy e ox2 oy2 e2 ffi lnf CF qnf pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 k  pffiffiffiffi u þ v v: K K

S

Energy equation:    2  oT oT o T o2 T þv þ qCp;nf u ¼ keff ; ox oy ox2 oy2

L1

ð4Þ

T = const

w H

a

X

ð3Þ

•

L

Y

ð2Þ

T = Const

L2

Fig. 1 Computational domain

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where k is a binary parameter. It is equal to zero for clear fluid region, while it is equal to one for porous region. Moreover, subscript nf shows the characteristics of nanofluid. CP, q and l are specific heat, density and viscosity of the liquid, respectively. Eventually, e and K are porosity and permeability of the porous baffles, respectively. ! The Dupuit–Forchheimer equation (! v ¼ e V ) is used to provide connection between volume-averaged liquid speed ! ( V ) inside the porous baffles and the Darcy velocity (! v ). The Forchheimer coefficient (CF) is calculated as follows: 1:75 CF ¼ pffiffiffiffiffiffiffiffiffiffiffi : 150e3

ð5Þ

The thermal conductivity of porous baffles can be calculated by: keff ¼ eknf þ ð1  eÞks :

ð6Þ

Thermopysical properties of nanofluid The thermopysical properties of nanofluid are presented in this section. The effective density of nanofluid is calculated by [37]: qnf ¼ ð1  uÞqf þ uqp ;

ð7Þ

where u is the nanoparticle concentration and subscripts p and f show particle and fluid, respectively. The specific heat of nanofluid is calculated by [38]: ð1  /Þqf Cf þ uqp Cp Cnf ¼ : qnf

leff ¼ lf þ

72Nd

;

VB and d are calculated by [39]:

123

ð12Þ

where KB indicates Boltzmann constant (= 1.3807 9 10-23 J K-1). For thermal conductivity, the equation of Chon et al. [40] is employed as follows:  0:3690  0:7476 kp knf 0:7460 df ¼ 1 þ 64:7  u Pr0:9955 kf dp kf  Re1:2321 ; ð13Þ where df is the molecular size of the water (= 0.3 nm). The Reynolds and Prandtl numbers in above equation are determined by: l Pr ¼ f ; ð14Þ qf af Re ¼

qf KB T ; 3pl2 lBF

ð15Þ

where lBF indicates the mean free path of water (= 0.17 nm). The physical properties of nanoparticles and base fluid are presented in Table 1.

ð8Þ

ð9Þ

The effects of different parameters, including nanoparticle concentration (u), nanoparticle size (dp), nanoparticle density (qp), temperature (T), Brownian velocity of nanoparticles (VB), the nanoparticle and water physical properties, and the distance between particles (d), on the dynamic viscosity of the nanofluid are considered in this equation. N in Eq. 9 is determined by [39]:     N ¼ l1 n1 dp þ n2 / þ n3 dp þ n4 f ð10Þ n1 ¼ 0:000001113; n2 ¼ 0:000002771 n3 ¼ 0:00000009; n4 ¼ 0:000000393;

ð11Þ

Boundary conditions

The theoretical model of Masoumi et al. [39] is employed to determine the dynamic viscosity of the nanofluid. Accordingly, the dynamic viscosity of the nanofluid is determined by: qp VB dp2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 18KB T ; VB ¼ dp pqp dp rffiffiffiffiffiffi p dp ; d¼ 3 6/

The following boundary conditions are considered to solve the above equations: • At the inlet of the duct: u ¼ uin ; T ¼ Tin : ð16Þ • At the wall of the duct: u ¼ 0; v ¼ 0; T ¼ Tw : • At the outlet of duct: ou ov oT ¼ 0; ¼ 0; ¼ 0: ox ox ox

ð17Þ

ð18Þ

•

At the interface between the clear liquid and liquid in porous baffles: uf ¼ up ; v f ¼ v p ; ð19Þ     ou l ou lnf ¼ nf ; ð20Þ oy f e oy p Tf ¼ Tp ;     oT oT knf ¼ keff ; oy f oy p

ð21Þ ð22Þ

Entropy generation analysis for nanofluid flow inside a duct equipped with porous baffles

Material

q/kg m-3

l/kg m-1 s-1

CP/J kg-1 K-1 4182

Water

1000

0.001003

Al2O3

3880

–

where subscripts f and p indicate the liquid and porous regions, respectively, continuity conditions at the interface between the clear liquid and porous zones. According to this condition, the shear stresses of clear liquid and porous zones are equal at the interface between two zones.

Entropy generation analysis The local volumetric viscous and thermal entropy generation rates can be calculated by: 8 knf  2  > > clear region < 2 rT T Sgen;thermal ¼ k in  ð23Þ ;  eff > > : 2 rT 2 porous affles Tin Sgen;viscous 8 ( "    #

) > lnf ov 2 ou 2 ov ou 2 > > > þ 2 þ þ > < Tin oy ox ox oy ( "    # ¼

) 2 2 > > lnf ov ou ov ou 2 l ~2 > > þ þ þ nf V 2 þ > : Tin oy ox ox oy KTin

k/W m-1 K-1 0.597

765

40

1.4

Experimental data of Heyhat et al. [37] Numerical results (present study)

Nusselt number ratio

Table 1 Physical properties of nanoparticles and water

1.2

1.0

0.8

0.6 400

600

800

Re

1000

1200

1400

Fig. 3 Comparison between present results and experimental data of Heyhat et al. [42] clear region : porous baffle

ð24Þ The non-dimensional local volumetric entropy generation rate is defined by: Sgen H 2 Ng ¼ : kf

N¼

1 A

Z

Ng  dA;

ð26Þ

A

where A is the surface of the duct. The Bejan number is defined by: Sgen;thermal : Sgen;thermal þ Sgen;viscous

ð27Þ

Thermal entropy generation

Percentage difference

Be ¼

ð25Þ

The non-dimensional total entropy generation rate is defined by:

Fig. 2 Sample mesh inside the duct Table 2 Effect of grid size on thermal entropy generation at Re = 300, Da = 10-4, u = 0.03, and N = 16

No.

Grid size

1

150 9 20

7.2198

1.8

2

300 9 40

7.3498

1.1

3

600 9 80

7.4307

0.3

4

1200 9 160

7.4530

–

123

Thermal entropy generaon

Fig. 4 Variations of viscous and thermal entropy generations with Reynolds number at Da = 10-4, u = 0.03, and N = 16

Viscous entropy generaon

H. Shamsabadi et al.

0.00015 0.0001 0.00005 0

100

150

200

250

300

350

400

450

500

Re

Ng,thermal-Re Variable Re – Da = 10–4 10 8 6 4 2 0 100

200

300

Re

400

500

Ng,Viscous-Da Variable Da – Re = 300

0.00005

Viscous entropy generaon

Fig. 5 Variations of viscous and thermal entropy generations with Darcy number at Re = 300, u = 0.03, and N = 16

Ng,viscous-Re Variable Re – Da = 10–4

0.00004 0.00003 0.00002 0.00001 0 1.00E–07

1.00E–06

1.00E–05

1.00E–04

1.00E–03

1.00E–04

1.00E–03

Da

Ng,thermal-Da Variable Da – Re = 300

Thermal entropy generaon

10 8 6 4 2 0 1.00E–07

1.00E–06

1.00E–05

Da

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Entropy generation analysis for nanofluid flow inside a duct equipped with porous baffles

Numerical procedure A pressure base finite volume method is employed for solving Eqs. 1–4 along with the boundary conditions. The pressure and velocity terms are stored at cell center and cell faces, respectively, by usage of the staggered grid arrangement. Moreover, pressure and velocity terms are coupled together by using the SIMPLE algorithm [41]. Second-order upwind technique is used to discretize all equations. For all simulations, converge criteria can be passed for the summation of residuals smaller than 10-7.

Grid independence test and validations Sample mesh generated inside the duct is disclosed in Fig. 2. A quadrate and two-dimensional grid is used in this paper. A grid independence test is arranged by considering four grid sizes to insure that the results are independent of grid size. The results of this test are presented in Table 2 for thermal entropy generation at Da = 10-4, u = 0.03, and N = 16. As presented in this table, the difference in the thermal entropy generation between grid sizes of 600 9 80 and 1200 9 160 is about 0.3%. As a result, the grid size of 600 9 80 is used for the rest of simulations.

Results and discussion The results of numerical simulations are presented in this section. The effects of Reynolds number, Darcy number, solid volume fraction of nanoparticles, and the number of baffle on thermal and viscous entropy generation rates and Bejan number are discussed in this section. Figure 4 shows variations of viscous and thermal entropy generations with Reynolds number at Da = 10-4, u = 0.03, and N = 16. As shown in this figure, the viscous Ng,visous-number of baffles

Viscous entropy generaon

0.00006 0.00005 0.00004 0.00003 0.00002 0.00001 0

0

4

8

12

16

Number of baffles Ng,thermal-number of baffles 10

Thermal entropy generaon

Fig. 6 Variations of viscous and thermal entropy generations with number of baffles at Re = 300, u = 0.03, and Da = 10-4

To insure about the accuracy of the numerical method, the current numerical results are benchmarked with the experimental results of Heyhat et al. [42] for laminar aluminum oxide–water nanofluid flow through a horizontal circular duct with a fixed wall temperature. The variation of the Nusselt number ratio, the ratio of the Nusselt number for the nanofluid to that of pure water, with the Reynolds number is plotted in Fig. 3 at u = 0.01 and dp = 40 nm. It can be seen that the current numerical results are in good agreement with the experimental results of the Nusselt number ratio with an average error of 7%. This error is within the error of the experimental results that was 6% [42].

8 6 4 2 0

0

4

8

12

16

Number of baffles

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H. Shamsabadi et al.

entropy generation increases with increasing the Reynolds number. This increment is about 96% as the Reynolds number increases in the range of 100–500. The flow inertia and velocity gradient increase with increasing the Reynolds number and this causes greater viscous entropy generation for higher values of Reynolds number. Moreover, it can be seen that the thermal entropy generation increases with increasing Reynolds number. The increment in the thermal entropy generation is about 76% as the Reynolds number increases in the range of 100–500. The heat transfer rate in the duct increases as the Reynolds number increases. This leads to higher temperature gradients in the flow field and, subsequently, thermal entropy generation increases. Figure 5 discloses the variations of viscous and thermal entropy generations with Darcy number at Re = 300, u = 0.03, and N = 16. It can be observed that the viscous entropy generation decreases with increasing the Darcy number. This reduction is about 29% as the Darcy number increases in the range of 10-7 to 10-3. Note that the baffles with lower values of Darcy number impose more obstructions and resistance with larger fluid friction against the fluid flow, as the permeability of a porous matrix

Ng,viscous – Phi

0.000037 0.0000365

Viscous entropy generaon

Fig. 7 Variations of viscous and thermal entropy generations with solid volume fraction of nanoparticles at Re = 300, N = 16, and Da = 10-4

decreases by reducing the Darcy number. This intensifies viscous dissipation and viscous entropy generation in the duct. However, the thermal entropy generation increases with increasing the Darcy number. The increment in the thermal entropy generation is about 88% as the Darcy number increases in the range of 10-7 to 10-3. Note that the convective heat transfer rate and subsequently, the thermal entropy generation in the duct increase with increasing the Darcy number. A porous medium with larger value of Darcy number has more voids and accordingly, fluid can flow easily through the porous matrix. As a result, the flow has higher velocity in the duct and this increases the convection heat transfer rate. Figure 6 plots the variations of viscous and thermal entropy generations with number of baffles at Re = 300, u = 0.03, and Da = 10-4 Generally, the viscous and thermal entropy generations increase by installing baffles inside the duct in comparison with the case of empty duct without using baffles (N = 0). These increments are 57 and 98% for the viscous and thermal entropy generations, respectively at N = 4. Generally, the viscous entropy generation becomes more serious in the duct by installing the

0.000036 0.0000355 0.000035 0.0000345 0.000034 0.0000335 0.000033 0.0000325 0.000032

0

0.01

0.02

0.03

0.04

0.05

Phi

Ng,thermal – Phi 8.6

Thermal entropy generaon

8.5 8.4 8.3 8.2 8.1 8 7.9 7.8 7.7 7.6 7.5

123

0

0.01

0.02

Phi

0.03

0.04

0.05

Entropy generation analysis for nanofluid flow inside a duct equipped with porous baffles Fig. 8 Viscous entropy generation, thermal entropy generation, and Bejan number contours at Re = 300, N = 16, u = 0.03, and Da = 10-4

S-dp-viscous:

0

0.0002

0.0004

0.0006

0.0008

0.001

Viscous entropy generation

S-dt-Thermal:

0

10

20

30

40

50

60

70

80

90

100

Thermal entropy generation

Bejan Number:

0

0.1

0.2 0.3 0.4 0.5

0.6 0.7 0.8 0.9

1

Bejan number

baffles as the baffles create resistance to the fluid flow. Moreover, the baffles enhance the convective heat transfer in duct by disrupting the thermal boundary layer. Accordingly, they increase the temperature gradient and thermal entropy generation. Finally, the viscous and thermal entropy generations decrease by increasing the number of baffles for N [ 4. The reductions in the viscous and thermal entropy generations are 32 and 14%, respectively as the number of baffles increases in the range of 4–16. Figure 7 shows the variations of viscous and thermal entropy generations with solid volume fraction of nanoparticles at Re = 300, N = 16, and Da = 10-4. It can be seen that both viscous and thermal entropy generations increase by using the nanofluid and increasing the solid volume fraction of nanoparticles. These increments are about 11 and 10% for viscous and thermal entropy

generations, respectively, as the solid volume fraction of nanoparticles increases in the range of 0–5%. Increment in viscous entropy generation can be justified by stating that the viscosity of nanofluid increases as the nanoparticles’ solid volume fraction increases, and hence intensifies the flow friction. On the other side, enhancements in heat transfer coefficient caused by employing the nanofluid, especially for high solid volume fraction of nanoparticles are responsible in the increment of the thermal irreversibility in the duct. Note that the surface of the duct is exposed to a fixed temperature. Accordingly, a higher heat transfer coefficient indicates a larger value of temperature gradient between the surface of the tube and fluid. A larger value of temperature gradient causes an increment in the thermal irreversibility inside the heat transferring flow.

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Figure 8 shows the viscous entropy generation, thermal entropy generation, and Bejan number contours at Re = 300, N = 16, u = 0.03, and Da = 10-4. As shown in the contour of viscous entropy generation, the viscous entropy generation has the maximum value along the centerline of the duct in the space between bottom and top baffles. Note that the flow inertia passing through this gap is high due to the reduction of the flow cross section. Accordingly, adequate space for flow expansion and pressure recovery is not provided and the velocity gradient and subsequently, viscous entropy generation increase. The results of Davari and Maerefat [43] showed that there is a significant enhancement of heat transfer rate between the recirculation zone and duct surface. The recirculation zone is formed behind each baffle. Accordingly, the thermal gradient is high at these regions and this leads to intensify the thermal entropy generation at these regions. By comparing the contours for viscous and thermal entropy generations, it can be concluded that the order of viscous entropy generation is negligible in comparison with the thermal one. Note that generally, the viscous entropy generation is important for turbulence regime. As shown in the contour of Bejan number, the thermal entropy generation is dominant term in most part of the duct except along the centerline of the duct in the space between bottom and top baffles where the velocity gradients are intense.

Conclusions In this paper, a numerical simulation was performed to investigate the potentials of nanofluid and porous baffles in a duct from the second low of thermodynamic viewpoint. The effects of different parameters including Reynolds number, Darcy number, solid volume fraction of nanoparticles, and the number of baffle were investigated on thermal and viscous entropy generation rates and Bejan number. The important results of this study are summarized here: • The thermal entropy generation increases with increasing the Reynolds number. The increment in the thermal entropy generation is about 76% as the Reynolds number increases in the range of 100–500. • The viscous entropy generation decreases with increasing the Darcy number. This reduction is about 29% as the Darcy number increases in the range of 10-7 to 10-3. • The viscous and thermal entropy generations decrease by increasing the number of baffles for N [ 4. The reductions in the viscous and thermal entropy generations are 32 and 14%, respectively, as the number of baffles increases in the range of 4–16.

123

•

•

•

Both viscous and thermal entropy generations increase by using the nanofluid and increasing the solid volume fraction of nanoparticles. These increments are about 11 and 10% for viscous and thermal entropy generations, respectively, as the solid volume fraction of nanoparticles increases in the range of 0–5%. The viscous entropy generation has the maximum value along the centerline of the duct in the space between bottom and top baffles. The thermal entropy generation is dominant term in most part of the duct except along the centerline of the duct in the space between bottom and top baffles where the velocity gradients are intense.

Acknowledgements This research was supported by the Office of the Vice Chancellor for Research, Ferdowsi University of Mashhad, under Grant No. 45933.

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