Numerical simulation and optimization of turbulent nanofluids in a three-dimensional rectangular rib-grooved channel

International Communications in Heat and Mass Transfer 66 (2015) 71–79

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Numerical simulation and optimization of turbulent nanofluids in a three-dimensional rectangular rib-grooved channel☆ Yue-Tzu Yang ⁎, Hsiang-Wen Tang, Bo-Yan Zeng, Chao-Han Wu Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan

a r t i c l e

i n f o

Available online 29 May 2015 Keywords: Turbulent Nanofluids Rib-grooved channel Single-phase model Two-phase model Genetic algorithm Optimization

a b s t r a c t In this study, numerical simulations by single and two-phase models of nanofluids turbulent forced convection in a three-dimensional rectangular rib-grooved channel with constant wall temperature are investigated. The elliptical, coupled, steady-state, three-dimensional governing partial differential equations for turbulent forced convection of nanofluids are solved numerically using the finite volume approach. The standard k − ε turbulence model is applied to solve the turbulent governing equations. The interactive influences of rectangular ribgroove geometrical ratios and nanofluid volume concentration on the average Nusselt number are provided in this study. The average Nusselt number of rib-grooved channel is found to improve more with smaller ribgrooved height ratios, and some ratios of rib-grooved pitch. Furthermore, the numerical results of the single and two-phase models show that there are some differences in simulated flow filed and turbulent convective heat transfer characteristics. In addition, the optimization of this problem is also presented by using the response surface methodology (RSM) and the genetic algorithm method (GA). The objective function E defined as the performance factor has developed a correlation function with four design parameters. It is found that the objective function E is better at Re = 10,000, and rectangular rib-grooved has an 18.2% enhancement. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Rib/groove is one of the commonly used passive heat transfer enhancement techniques in single-phase internal flows in a channel solar air heater by placing the rib/groove periodically in the absorber plate. For decades, several engineering techniques have been developed for enhancing the convective heat transfer rate from the channel surface. The turbulators used for the cooling/heating channel or channel solar air heater such as ribs [1], fins [2,3], grooves [4,5] or baffles [6,7] are often encountered in order to increase the convective heat transfer coefficients leading to the compact heat exchanger and increasing the efficiency. The reason of this may be that the use of ribs/grooves completely makes the change of the flow field and thus the distribution of the local heat transfer coefficient. The application of rib-groove into the channel is to provide an interruption of boundary layer development, to increase the heat transfer surface area and to cause the enhancement of heat transfer by increasing turbulence intensity or fast fluid mixing. Therefore, more compact and economic heat exchanger with lower operation costs can be obtained. In general, the geometry

☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (Y.-T. Yang).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.05.022 0735-1933/© 2015 Elsevier Ltd. All rights reserved.

parameters of ribs in the channel are among the most important factor in the design of channel heat exchangers which have an effect on both local and overall heat transfer coefficients. In particular, the angled rib, rib blockage ratio (BR = b/H), rib pitch ratio (PR) and rib arrangement are all parameters that influence both the heat transfer coefficient and the overall thermal performance. Several studies have been carried out to investigate the effect of these parameters of ribs on heat transfer and friction loss for two opposite roughened surfaces. Promvonge et al. [8] examined numerically the laminar heat transfer enhancement in a square channel with 45° inclined baffle on one wall and reported that a single stream-wise vortex flow occurs throughout the channel and helps to induce impingement jets on the upper, lower and side walls. Again, Promvonge et al. [9,10] also investigated numerically the laminar flow structure and thermal behaviors in a square channel with 30° or 45° inline baffles on two opposite walls. They found that two streamwise counter-rotating vortex flows appear along the channel and vortex-induced impinging jets occur on the upper, lower and side walls. High thermal performance is an important research issue in recent years. The nanofluids compared to the base fluid have better heat transfer, and many researchers have studied nanofluids including experiments and simulations. However, no studies have developed a comprehensive and universal numerical model to investigate the heat transfer effect of nanofluids. Most prior studies assume nanofluids as a single phase flow which is much easier and faster, while the accuracy

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Y.-T. Yang et al. / International Communications in Heat and Mass Transfer 66 (2015) 71–79

Nomenclature List of symbols a acceleration (m/s2) A bottom area of the heating zone (mm2) b rib-grooved width (mm) C1, C2, Cμ, σk, σε closure coefficients specific heat of constant pressure (J/kg K) Cp particle diameter (mm) dp hydraulic diameter (mm) Dh e rib-grooved height (mm) E performance factor drag function fdrag g acceleration of gravity (m/s2) G generation of turbulent kinetic energy (kg/ms3) h enthalpy (J/kg) H channel height (mm) h average convection heat transfer coefficient (W/m K) I turbulent intensity k conduction heat transfer coefficient (W/m K) k turbulent kinetic energy (m2/s2) inlet length (mm) L1 length of the heating zone (mm) L2 L total length (mm) Nu average Nusselt number p rib-grooved pitch (mm) p pressure (Pa) q″ heat flux (W/m2) Re Reynolds number T temperature (K) drift velocity (m/s) Vdr relative velocity (m/s) Vpf u, v, w velocity component (m/s) V, v time-mean and fluctuating velocity (m/s) x, y, z Cartesian x, y, z-coordinate (mm) Greek symbols ρ density of the fluid (kg/m3) μ dynamic viscosity (N s/m2) ν kinematic viscosity (m2/s) ε turbulent kinetic dissipation (m2/s3) τ wall shear stress (Pa) ϕ nanoparticle volume concentration Subscripts 0 smooth channel bf base fluid eff effective f fluid in inlet m mean nf nanofluid p particle w wall

is low in terms of the results of simulation compared to the experimental data, due to a lack of consideration of nanofluid microscopic phenomena. Therefore, many scholars use the two-phase model to simulate nanofluids to improve the accuracy of simulations. Nanofluids were first used by Choi [11] at the Argon national laboratory. Easterman et al. [12] reported that with low nanoparticle concentrations (1–5 vol.%), the effective thermal conductivity of the suspensions can increase by more than 20% for various mixtures. Lee et al. [13] measured four kinds of nanofluids (CuO/water, Cuo/EG, Al2O3/water, Al2O3/EG),

and showed Cuo/EG increases 20% heat transfer by 4% solid volume fraction. At a low solid volume fraction (ϕ b 5%), the thermal conductivity increased linearly with increasing solid volume fraction and the thermal conductivity was also found to be increased with decreasing particle size. Experimental results of Xie et al. [14] show that using the deionized water, ethylene glycol (EG), and pump oil as the base fluid, the heat transfer increased significantly after adding a few nanoparticles (Al2O3). Xuan and Li [15] explored the heat transfer phenomenon of nanofluid (CuO/water) flowing through a pipe with constant heat flux wall, and the Reynolds number ranges from 10,000 to 25,000. Their experimental results showed that nanofluid has a higher heat transfer coefficient than pure water while the nanoparticle concentration is less than 2%. Experiments were conducted to investigate the cooling performance of a microchannel with Al2O3/water nanofluid [16], and the results showed the nanofluid-cooled heat sink performed by examining the heat transfer rate and the pressure drop. Murshed et al. [17] found that the heat transfer effects would be enhanced with the increase of volume concentration of TiO2. Yoo et al. [18] discussed four kinds of nanofluids (TiO2/water, Al2O3/water, Fe/EG, WO3/EG), the results displayed that the thermal conductivity of nanofluids is much better than that of pure water, but they also found that the surface-tovolume ratio was an important factor influencing the thermal conductivity coefficient of nanofluids. Vajjha and Das [19] experimentally determined the thermal conductivity of three nanofluids (CuO, Al2O3, ZnO) and developed new corrections. Maiga et al. [20] showed that heat transfer effect of γ-Al2O3/EG is better than γ-Al2O3/water. Izadi et al. [21] numerically studied the laminar forced convection of Al2O3/water nanofluids in an annulus using a single phase approach, and found that temperature profiles were affected by the particle concentration. Also, a convective heat transfer coefficient increased with the nanoparticle concentration, and the friction coefficient was dependent on the nanoparticle concentration. Heat transfer enhancement due to flows of copper–water nanofluid through a two-dimensional rectangular duct has been studied by Santra et al. [22]. The results showed that there is a little effect of nanoparticles on the flow structure but the isotherms changed and it moved toward the centerline of the channel with an increase in solid volume fraction. The turbulent forced convection flow of a water/ Al2O3 nanofluid in a square tube subject to constant and uniform wall heat flux was numerically investigated by Vincenzo et al. [23]. Heat transfer enhancement increased with the particle volume concentration, but it was accompanied by increasing wall shear stress values. The optimal Reynolds number was analytically determined and it decreased as particles' concentration increased. Three kinds of nanofluids (Cu/water, Al2O3/water and CuO/water) in the two-dimensional wavy channel were numerically studied by Yang et al. [24]. The results showed that the heat transfer could be improved when using nanofluids. The genetic algorithm for multi-objective optimization was performed to obtain the optimal solutions. Akbari et al. [25] compared the CFD predictions of laminar mixed convection of Al2O3/water nanofluids by single phase and three different two-phase models (volume of fluid, mixture, Eulerian). They found that single-phase and two-phase models predicted almost identical hydrodynamic fields but very different thermal ones. The predictions of the three two-phase models were essentially the same. At low volume fractions, comparing with the experimental data, the results simulated by the two phase model were more precious than the single phase model. In the present study, a numerical approach based on the finite difference method is applied to simulate the turbulent flow of Al2O3/water nanofluid in a rib-grooved channel. The main purpose of this study is to optimize the geometries of the forced convective heat transfer in a rib-grooved channel. Effects of volume concentration, rectangular ribgrooved height ratios, rectangular rib-grooved width ratios, rectangular rib-grooved pitch ratios on the fluid flow and heat transfer characteristics are studied and presented. In addition, the optimization of this

Y.-T. Yang et al. / International Communications in Heat and Mass Transfer 66 (2015) 71–79

problem is also presented by using a response surface methodology (RSM) and the genetic algorithm (GA) method. 2. Mathematical formulation A three-dimensional (3-D) numerical analysis on the turbulent nanofluid forced convection in the rib-grooved channels is carried out. A schematic diagram of the geometry and the computational domain is shown in Fig. 1. The computation is composed of the inlet fluid section, test section (a wavy channel), and outlet fluid section. To ensure a fully developed flow at the entrance of test section, the length of the inlet section is L1 (L1 = 80 mm). Also, to prevent the formation of any reversed flows through the test section, the length of the outlet fluid section is 650 mm and the total length of the channel is L (L = 880 mm). The length of the test section is L2 (L2 = 150 mm) and the channel height/width is H (H = 10 mm).

The single-phase model treats the nanofluid as a homogeneous fluid with effective properties and uses the differential equations to express the governing equations. To obtain accurate results with the singlephase model, it is very important to use the most appropriate factors for effective nanofluid properties. The continuity, momentum, and energy equation can be expressed, respectively, as:Continuity equation ð1Þ

Momentum equation

ð6Þ

Thermal conductivity (Maiga et al. [20]) kn f ¼ 4:97ϕ2 þ 2:72ϕ þ 1 kb f

ð7Þ

In the present work the k − ε turbulence model proposed by Launder and Spalding [29] is considered and it has been modified appropriately, in order to take into account the presence of nanoparticles as given in Behzadmehr et al. [30] and Hejazian and Moraveji [31]. It is expressed by the following two equations: 

∇
ðρm VεÞ ¼ ∇


 μ t;m ∇k þ Gm −ρm ε σk

ð8Þ

  μ t;m ε ∇ε þ ðC 1 Gm −C 2 ρm ε Þ σε k

ð9Þ

where Gm represents the generation of turbulent kinetic energy due to mean velocity gradients and μ t;m ¼ ρm C μ

k ; C 1 ¼ 1:44; C 2 ¼ 1:92; C μ ¼ 0:09; σ k ¼ 1:0; σ ε ¼ 1:3: ε ð10Þ

2.2. Mixture model


 ∇
ρn f VV ¼ −∇p þ ∇
τ

ð2Þ

Continuity equation ∇
ðρm V m Þ ¼ 0

Energy equation nf T

μn f ¼ 123ϕ2 þ 7:3ϕ þ 1 μb f

2


 ∇
ρn f V ¼ 0


∇
ρn f VC p;

Dynamic viscosity (Maiga et al. [28])

∇
ðρm VkÞ ¼ ∇


2.1. Single-phase model

73




¼ ∇
kn f ∇T−C p;

n f ρn f vt



ð3Þ

In this model, suitable definitions of thermo physical property relations for nanofluid have a great significance. In the present study, the following equations are considered to calculate the nanofluid thermophysical properties:Density (Pak and Cho [26]) ρn f ¼ ð1−ϕÞρb f þ ϕρp

ð4Þ

bf

þ ϕC p;

p

Momentum equation ∇
ðρm V m V m Þ ¼ −∇pm þ ∇
τ þ ∇


n X

! ϕk ρk V dr; k V dr;

k

ð12Þ

k¼1

Energy equation   ∇
ðϕk V k ðρk hk þ pÞÞ ¼ ∇
keff ∇T−C p ρm vt

ð13Þ

Volume fraction equation

Heat capacity (Xuan and Roetzel [27]) C p; n f ¼ ð1−ϕÞC p;

ð11Þ

ð5Þ



 ∇
ϕp ρp V m ¼ −∇
ϕp ρp V dr; p

Fig. 1. Schematic diagram of three-dimensional rib-grooved channel.

ð14Þ

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Y.-T. Yang et al. / International Communications in Heat and Mass Transfer 66 (2015) 71–79

Outlet boundary:In this study, the assumed properties at the outlet are fully developed.

where n X

ρm ¼

ϕ k ρk

ð15Þ

k¼1

keff ¼

n X

ϕk kk

Heated section boundary:Tw = 330K along the heated wall

Vm is the mass average velocity and defined as n X

1 ϕ ρ V ρm k¼1 k k k

u ¼ v ¼ w ¼ 0; k ¼ ε ¼ 0 ð17Þ

Vdr,k is the drift velocity for the secondary phase k, i.e. the nanoparticles in the present study. This is related to the relative velocity n 1 X ¼ Vpf − ϕ ρ V : ρm k¼1 k k f k

V dr;k

ð27Þ

ð16Þ

k¼1

Vm ¼

∂T f ∂u ∂v ∂w ∂k ∂ε ¼ ¼ ¼ 0; and ¼ ¼0 ¼ 0; ∂x ∂x ∂x ∂x ∂x ∂x

ð18Þ

Wall boundary:The no-slip boundary condition and no-penetration at the wall boundary condition are defined for the numerical model. ∂T w ¼ 0; u ¼ v ¼ w ¼ 0; and k ¼ ε ∂n ¼ 0 along the straight section wall boundary:

ð29Þ

In addition, the average Nusselt number and average convection heat transfer coefficient are defined as follows:

The slip velocity (relative velocity) is defined as the velocity of a secondary phase (nanoparticle, p) relative to the velocity of primary phase (water, f)

Nu ¼

V p f ¼ V p −V f :

where

ð19Þ

ð28Þ

hDh kn f

ð30Þ



The relative velocity Vpf proposed by Manninen et al. [32] is determined from
 2 ρp −ρm τ p dp a ð20Þ Vpf ¼ 18μ f f drag ρp while the drag function fdrag proposed by Schiller and Naumann [33]  f drag ¼

1 þ 0:15Re0:687 p 0:0183Rep

where Rep ¼

V m dp ν eff ,

Rep ≤ 1000 Rep N 1000

  q″ ¼m C p T m;in −T m;out =A 

Re ¼

a = g − (Vm ⋅ ∇)Vm, where g is the gravitational

    μ t;m ∇
ρm V m k ¼ ∇
∇k þ Gk; m −ρm ε σk

ð22Þ

     μ t;m ε ∇ε þ C 1 Gk; m −C 2 ρm ε ∇
ρm V m ε ¼ ∇
σε k

ð23Þ

ð32Þ

ρn f uin Dh μn f

ð33Þ

where Dh = H. The thermal performance of the system based on the performance index which considers the heat transfer enhancement besides the increase in pressure drop is used. In order to find the optimum work conditions, the overall performance of the enhancement considering simultaneous effects of heat transfer and pressure drop increment are studied here based on the performance factor. The performance factor E is defined as (Hamid and Mohammad [34])  .



Nu

E¼
Δp

where
 ¼ μ t;m ∇V m þ ð∇V m ÞT :

ð31Þ

Tm,in, Tm,out are the average inlet temperature and outlet temperature, and A is the heating area. The Reynolds number based on the hydraulic diameter Dh of the channel is calculated as follows:

ð21Þ

acceleration. The k − ε turbulence model of Launder and Spalding [29] is adopted for the mixture model. It is expressed by Eqs. (22) and (23)

Gk;m

 T w −T m;in T w −T m;out    h ¼ q″
 T w −T m;in − T w −T m;out ln

Nu0

 :

ð34Þ

Δp0

ð24Þ

The values of the constants are given in Eq. (10).

3. Numerical calculations and validations

2.3. Boundary conditions

The numerical computation is carried out by solving the governing conservation equations with the boundary conditions. This study chooses the non-uniform meshes to build the grid system. The solution procedure is based on SIMPLE algorithm. The convection term of governing conservation equations is discretized by a control volume based finite difference method with a QUICK scheme. A set of different equations is solved iteratively using a line by line solution method in conjunction with a matrix form. The solution is considered to converge when the normalized residual of the algebraic equation is less than a prescribed value of 10−4. Various grids of sizes with 443 × 100 × 15, 443 × 80 × 15, 380 × 75 × 15, 324 × 64 × 15 and 314 × 55 × 15 nodes were employed for checking the mesh independency of the

The boundary conditions for the flow field and thermal boundary are stated as follows. Inlet boundary:The velocity is assumed to have a uniform profile at the inlet, u ¼ uin ;

v ¼ w ¼ 0; T in ¼ 300K

ð25Þ

3=2

kin ¼

k 3 ðIu Þ2 ; ε in ¼ in 0:3H 2 in

ð26Þ

Y.-T. Yang et al. / International Communications in Heat and Mass Transfer 66 (2015) 71–79

solution as shown in Fig. 2. The average Nusselt number ratio, Nu

75

. Nu0

plotted against the Re values is presented in Fig. 2. In the figure, . Nu tends to increase with the increment of Re using the ribNu0

groove. The results show that the difference in the average Nusselt number ratios between the grids with 380 × 75 × 15 and 443 × 80 × 15 nodes is found to be within 1%. For the CPU time and memory storage required which is based on the grid independent study, the grid system with 380 × 75 × 15 nodes is used for the numerical calculations. After the grid independent test, single-phase and two-phase models are also needed to test for the nanofluids. Fig. 3 shows the effect of the single-phase and two-phase on the average Nusselt number ratios. It should be noted that for all Reynolds numbers, the average Nusselt number ratio increases as the Reynold number increases. Furthermore, we compare the average Nusselt number predicted using the two-phase approach. The results clearly show that the single-phase approach gives considerably lower estimates than the two-phase approach for all Reynolds numbers. The results show the average Nusselt number ratio of nanofluid and pure water by two phase models including VOF, mixture, and Eulerian, which are significantly larger than that of the single-phase. In the single-phase model, we have only a virtual fluid with different physical properties, while the mixture model includes the interaction between different phases. If the irregular motion of the particles (Brownian motion) increases, this interaction increases and the effect of the interaction in simulation will be more obvious. Brownian motion is one of the factors influencing the enhancement in heat transfer. Also, the presence of nanoparticles in the base fluid reduces the thickness of thermal boundary layer which has a significant effect on heat transfer improvement. 4. Optimal process This study uses the response surface methodology (RSM) and genetic algorithms (GA) to carry out the optimal process. The response surface methodology is a parameter design for efficient experiments, which can find out the mix effect of parameters by fewer experiments, and create an objective function. RSM was originally developed for the model fitting of physical experiments by Box and Draper [35] and later adopted in other fields. The objective function created by RSM is formulated as f ¼ a0 þ

n X i¼1

ai xi þ

n X n X

ai j xi x j þ …

ð35Þ

i¼1 j¼1

where a0, ai and aij are tuning parameters and n is the number of parameters.

Fig. 2. Grid independent test (h/H = 0.5625, b/H = 0.325, p/H = 0.75, ϕ = 4%).

Fig. 3. Single and two-phase model test (h/H = 0.5625, b/H = 0.325, p/H = 0.75, ϕ = 4%).

After creating the objective function, we use the genetic algorithms (GA) to find out the optimal geometries. GA solves optimization problems iteratively based on biological evolution process in nature. In the solution procedure, a set of parameter values is randomly selected. Set is ranked based on fitness values (i.e. performance factor in this study). A best combination of parameters leading to minimum fitness values is determined. New combinations of parameters are generated from the best combination by simulating biological mechanisms of offspring, crossover, and mutation. This process is repeated until fitness values with a new combination of parameters cannot be further reduced. The final combination of parameters is considered as the optimum solution. In this study, the GA codes have been written in MATLAB programming language [36]. The parameters studied include Reynolds number (5000 ≤ Re ≤ 20,000), volume concentration (2% ≤ ϕ ≤ 4%), rectangular rib-grooved Table 1 Experimental designs and performance factors at Re = 10,000. No

e/H

b/H

p/H

ϕ

E

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

0 0.188 0.188 0.188 0.188 0.188 0.188 0.188 0.375 0.375 0.375 0.188 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.563 0.563 0.563 0.563 0.563 0.563 0.563 0.750 0.563

0 0.325 0.325 0.775 0.775 0.775 0.775 0.325 1 0.55 0.1 0.325 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.775 0.775 0.325 0.325 0.775 0.775 0.325 0.55 0.325

0 0.75 0.75 0.75 0.75 0.25 0.25 0.25 0.5 1 0.5 0.25 0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.75 0.75 0.75 0.75 0.25 0.25 0.25 0.5 0.25

3% 4% 2% 4% 2% 4% 2% 2% 3% 3% 3% 4% 3% 3% 3% 3% 3% 3% 3% 3% 5% 1% 4% 2% 2% 4% 4% 2% 4% 3% 2%

1 1.182 1.151 1.084 1.079 0.910 0.898 0.874 0.643 0.529 0.466 0.782 0.553 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.480 0.480 0.278 0.274 0.240 0.243 0.281 0.277 0.283 0.104 0.273

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Y.-T. Yang et al. / International Communications in Heat and Mass Transfer 66 (2015) 71–79

Table 2 Optimal results. Re

e/H

b/H

p/H

ϕ

E of empirical equation

E of CFD

Error

5000 10,000 15,000 20,000

0.188 0.188 0.188 0.188

0.325 0.325 0.325 0.325

0.75 0.75 0.75 0.75

4% 4% 2% 2%

1.077 1.122 1.009 0.881

1.146 1.182 1.06 0.924

0.06 0.051 0.049 0.047

height ratios (0.1875 ≤ e/H ≤ 0.5625), rectangular rib-grooved width ratios (0.325 ≤ b/H ≤ 0.775), and rectangular rib-grooved pitch ratios (0.25 ≤ p/H ≤ 0.75). The procedures are described as follows. ➢ Use DOE approaches to arrange the e/H, b/H, p/H, and ϕ of RSM system for the objective function as listed in Table 1. ➢ Implement the RSM design models to get the result of the performance factor E by using CFD method as shown in Table 2. ➢ Analyze the regression function from the CFD results of the performance factor E and design the parameters to get the results of objective function as presented in Eqs. (36) and (37). ➢ Use GA method to analyze the objective function and calculate the optimal parameters. From the data curve fitting of Table 1 and CCD predicted results of Table 2, the performance factor E of the present study is chosen as the objective function. The regression analyses are as follows. Re = 10,000,

Fig. 5. Variation of Δp/Δp0 with Re for various rib-grooved width ratios (e/H = 0.1875, p/H = 0.75, ϕ = 4%).

The predicting performance factor E of regression function is in close agreement with those from the CFD computational results within a 4.7% difference. The final combination of parameters is considered as the optimum solution.

5. Results and discussion Eðe=H; b=H; p; ϕÞ ¼ 1:155−2:832  ðe=HÞ þ 0:108  ðb=HÞ

ð37Þ

In this section, variations in pressure drop ratio (Δp/Δp0) with the Reynolds number at different rib-grooved height ratios and ribgrooved width ratios are presented in Fig. 4. It shows that the pressure drop ratio with different rib-grooved height ratios is quite close as shown in Fig. 4. Single-phase and mixture models have no obvious difference for the pressure drop ratio. Rib-grooved height ratio is not an important factor for the pressure drop ratio in the rib-grooved channel. In different width ratio of the rib-grooved channel, the pressure drop ratio coupled with the Reynolds number is discussed in Fig. 5. In this figure, variations in pressure drop ratio are b/H = 0.325 and 0.55. It shows that pressure drop ratio is dramatically influenced by the width ratio of rib-grooved channel, especially at low Reynolds numbers. Therefore, the pressure drop ratio is influenced more by the width ratio than by the height ratio of the rib-grooved channel.

Fig. 4. Variation of Δp/Δp0 with Re for various rib-grooved height ratios (b/H = 0.325, p/H = 0.75, ϕ = 4%).

Fig. 6. Variation of Nu=Nu0 with Re for various rib-grooved height ratios (b/H = 0.325, p/H = 0.75, ϕ = 4%).

þ0:836  p−4:107  ϕ −0:183  ðe=HÞ  ðb=H Þ−1:713 ðe=H Þ  p þ 2:158  ðe=HÞ  ϕ−0:520  ðb=H Þ  p þ2:026  ðb=HÞ  ϕ þ 2:666  p  ϕ þ 2:403  ðe=H Þ2 þ0:231  ðb=HÞ

2

þ 0:178  p2 þ 12:68  ϕ2

ð36Þ

at Re = 20,000, Eðe=H; b=H; p; ϕÞ ¼ 1:134−2:494  ðe=HÞ þ 0:198  ðb=H Þ þ0:253  p−3:476  ϕ −0:085  ðe=HÞ  ðb=HÞ−0:727 ðe=H Þ  p þ 4:369  ðe=HÞ  ϕ−0:337  ðb=HÞ  p þ2:021  ðb=HÞ  ϕ þ 1:92  p  ϕ þ 1:619  ðe=HÞ2 þ0:034  ðb=HÞ

2

þ 0:19  p2 −15:889  ϕ2

Y.-T. Yang et al. / International Communications in Heat and Mass Transfer 66 (2015) 71–79

77

Fig. 7. Variation of Nu=Nu0 with Re for various rib-grooved width ratios (e/H = 0.1875, p/H = 0.75, ϕ = 4%).

Fig. 9. Variation of E with Re for various rib-grooved height ratios (b/H = 0.325, p/H = 0.75, ϕ = 4%).

The average Nusselt number ratio plotted against the Re value for different rib-grooved height ratios is depicted in Fig. 6. In the figure, the average Nusselt number ratio tends to increase with the increment of Re at lower e/H with mixture and single-phase models. But at e/H = 0.1875 and Re N 10,000, the Nusselt number ratio tends to decrease with the increasing Re. It is interesting to note that the Nusselt number ratio of the mixture model is higher than those of the single-phase model. It is visible in the figure that the average Nusselt number ratio for the combined rib and groove turbulator generally is found to be above unity. The rib-groove turbulator on one wall with the mixture model gives the highest average Nusselt number ratio at Re = 10,000. Fig. 7 presents the variation of the Nusselt number ratio with Re values for different rib-grooved width ratios. It can be observed that the Nusselt number ratio is much bigger than the other cases at b/H = 0.325 with the mixture model. It is visible in the figure that the average Nusselt number ratio for the combined rib and groove turbulator generally is found to be above unity at lower b/H. Fig. 8 presents the average Nusselt number ratio of the heated section corresponding to three rib-pitch-to-ribheight (P/H) at Reynolds numbers varying from 5000 to 20,000. At a constant configuration parameter, the average Nusselt number ratio

increases with increasing Reynolds number, except for P/H = 0.75, and Re N 10,000. It can be seen that the average normalized Nusselt numbers for P/H = 0.75 are the highest, and the average Nusselt number ratio for P/H = 0.5 and Re = 5000 is the lowest. The average Nusselt number ratio for P/H = 0.25 is slightly higher than that for P/H = 0.5. Accordingly, it can be concluded that P/H = 0.75 exhibits better overall performance than P/H = 0.5 and 0.25. Fig. 9 presents the variations of the performance factor with the Reynolds number for different rib-grooved height ratios (e/H). The figure shows that the performance factor increases with the decrement of the e/H, increases with the increment of the Reynolds number at Re b 10,000 and decreases with the increment of the Reynolds number at Re N 10,000. It is worth noting that the performance factor at the case of e/H = 0.1875 is much bigger than the others. The performance factor is greater than 1 at the case of e/H = 0.1875 in mixture model with Re ≦ 15,000 and the case of e/H = 0.1875 in single-phase model with Re ≦ 10,000. The variations of the performance factor with the Reynolds number for different rib-grooved width ratios (b/H) are shown in Fig. 10. The figure shows that the performance factor increases with the increment of

Fig. 8. Variation of Nu=Nu0 with Re for various rib-grooved pitch ratios (e/H = 0.1875, b/H = 0.325, ϕ = 4%).

Fig. 10. Variation of E with Re for various rib-grooved width ratio (e/H = 0.1875, p/H = 0.75, ϕ = 4%).

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The optimal velocity distribution between the first and second rib is shown in Fig. 12. We can see that the fluid was accelerated when it flows through the rib and was decelerated when it flows through the groove. This phenomenon is because of cross sectional area's decrement at rib and increment at groove. We can find that there are some vortex occurred in the rear of the rib and in the groove. These phenomena have led to the enhancement of heat transfer effect of the rib-grooved channels. 6. Conclusions

Fig. 11. Variation of E with Re for various rib-grooved pitch ratios (e/H = 0.1875, b/H = 0.325, ϕ = 4%).

the Reynolds number at Re b 10,000, decreases with the increment of the Reynolds number at Re N 10,000 and have a relative minimum value at b/H = 0.55 and Re = 10,000. It is worth noting that the performance factor is greater than 1 at the case of b/H = 0.325 in the mixture model with Re ≦ 15,000, the case of b/H = 0.325 in single-phase model with Re ≦ 10,000, and the case of b/H = 0.775 in the mixture model with Re ≦ 10,000. It also can be found that the difference of the calculations between two models at the case of b/H = 0.775 is much bigger than the others. The variations of the performance factor with the Reynolds number for different rib-grooved pitch ratios (p/H) are shown in Fig. 11. We can observe that the performance factor at the case of p/H = 0.75 is much bigger than the others and the performance factor has a relatively minimum value at the case of p/H = 0.5. The figure shows that performance factor increases with the increment of the Reynolds number at Re b 10,000 and decreases with the increment of the Reynolds number at Re N 10,000. It is worth noting that the performance factor is greater than 1 at the case of p/H = 0.75 in the mixture model at Re ≦ 15,000 and the case of p/H = 0.75 in the single-phase model at Re ≦ 10,000. The big differences are found between the single-phase and the mixture model at p/H = 0.75 and p/H = 0.25.

Numerical computations have been performed to examine the combined effects of rib-grooved turbulators on the turbulent forced convection heat transfer and friction characteristics in a three-dimension channel under a constant temperature boundary condition. Numerical computations are performed with a rectangular rib-grooved channel for the parameters studied including Reynolds number (5000 ≤ Re ≤ 20,000), volume concentration (2% ≤ ϕ ≤ 4%), rectangular rib-grooved height ratios (0.1875 ≤ e/H ≤ 0.5625), rectangular rib-grooved width ratios (0.325 ≤ b/H ≤ 0.775), and rectangular rib-grooved pitch ratios (0.25 ≤ p/H ≤ 0.75). The effect of the above-mentioned parameters on the pressure drop, average Nusselt number ratio, and performance factor are presented and analyzed. The following results are obtained: 1. The numerical results show that the rib-groove width ratio (b/H) leads a considerable difference on pressure drop ratio between the single-phase and the mixture model. Two models have coincident results of the pressure drop ratio of different rib-groove height ratios (e/H). As the rib-groove height ratio (e/H) increased, the predicted pressure drop ratio increased significantly for both the single-phase and the mixture model, and the highest values are at e/H = 0.5625. 2. The numerical results show that the single-phase model gives considerably lower calculations of the Nusselt number ratio than the mixture model for all rib-groove height ratios (e/H), rib-groove width ratios (b/H) and rib-groove pitch ratios (p/H). This is because the single-phase model ignores interactions, thermal equilibrium and drift velocity between the particles and the fluid in the model. 3. Numerical results show that the enhancement in the average Nusselt number depends mainly on the rib-groove height ratio (e/H), the ribgroove width ratio (b/H), and the rib-groove pitch ratio (p/H). In this study, the enhancement in the average Nusselt number ratio has a relatively minimum value at e/H = 0.375, b/H = 0.55 and p/H = 0.5. 4. Through the optimization process and the response surface methodology design and regression, we end up with extending a corresponding equation among the thermal performance factor (E). Four parameters

Fig. 12. Optimal velocity distribution between the first and second rib/grooved (e/H = 0.1875, b/H = 0.325, p/H = 0.75, ϕ = 4%, Re = 20,000).

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