CFD analysis of particle shape and Reynolds number on heat transfer characteristics of nanofluid in heated tube

Savas Evran; Mustafa Kurt

doi:10.1515/phys-2024-0046

Article Open Access

CFD analysis of particle shape and Reynolds number on heat transfer characteristics of nanofluid in heated tube

Savas Evran and Mustafa Kurt

Published/Copyright: June 22, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Open Physics Volume 22 Issue 1

Abstract

Various nanoparticles have been used to increase the heat transfer characteristics (HTC) of nanofluids in the heated tube. The use of various shapes of the same nanoparticle can have major impact on the HTC. In this study, computational fluid dynamics (CFD) analysis of the impact of particle shape (Brick and Platelet) and Reynolds (Re) number (4,500, 6,000, 7,500, and 9,000) on the HTC of nanofluid in the heated tube was carried out in accordance with Taguchi method. Heat transfer coefficient, Nusselt (Nu) number, performance evaluation criteria, and average static pressure drop were chosen as HTC. CFD analyses for 1% Fe₃O₄ nanofluids in ANSYS Fluent software were performed in accordance with L8 orthogonal array. Particle shape and Re number were selected as the first and second factors, respectively. Signal/noise analysis was used to decide optimum levels and impact direction on HTC for each factor, whereas analysis of variance was implemented to define the importance levels and percentage impact ratios of the factors. According to the results obtained from the study, the nanofluids with platelet nanoparticles have a higher impact on the heat transfer coefficient compared to Brick nanoparticles. Although the increase in the Re number causes an increase in the heat transfer coefficient, Nu number, and average static pressure drop, it does not have any effect on the performance evaluation criteria. The results obtained from this study can be used as a guidance for experimental studies.

Keywords: ANOVA; CFD; heat transfer; nanofluids; Taguchi method

Nomenclature

b _l: base fluid
CFD: computational fluid dynamics
D b: generation of turbulence kinetic energy due to buoyancy
D k: generation of turbulence kinetic energy due to the mean velocity gradients
F →: external force
g: gravity
HTC: heat transfer characteristics
h: heat transfer coefficient
J → j: diffusion flux
k eff: effective conductivity
k: turbulence kinetic energy
Nu: Nusselt number
n _l: nanofluid
PEC: performance evaluation criteria
Q: heat transfer rate
Re: Reynolds number
S h: energy source
S _k and S _ϵ: user defined source terms
S m: source
S/N: signal/noise
T _i and T _o: inlet and outlet temperature
t: time
U: specific total energy
v →: velocity vector
ρ: fluid density
µ: viscosity
ϵ: rate of dissipation
σ k: turbulent Prandtl numbers for k
τ ̿: stress tensor
Φ: volume fraction
σ ε: turbulent Prandtl numbers for ε
μ t: eddy viscosity
∆ P: pressure drop
Ω ∼ İ Ɉ: mean rate-of-rotation tensor

1 Introduction

The energy needs may increase day by day in many industries and research has been carried out on carbon emissions and energy consumption [1]. Especially in heat energy, the usage of nanofluids has become extensive to increase efficiency. Many researchers [2,3] have analyzed improving the thermal conductivity by using nanofluids. Nanofluids are fluids made of nanoparticles such as metals, carbides, and ceramics using a base liquid [4,5]. Due to the several improved physical properties of nanofluids, they are subject to implementation in many sectors such as energy systems [6] and solar collectors [7], electronics cooling system [8], nuclear power plants [9], heat exchangers [10], microfluidics [11], enhanced oil recovery [12], impingement jets [13], renewable energy [14], combustions [15], porous media [16], etc. Different mathematical methods have been presented on the thermal conductivity of nanofluids and several approach and empirical relations were used for these methods [17,18]. Fe₃O₄ has been used as nanoparticle in many studies [19,20,21] to create nanofluids. Additionally, different nanoparticles such as Cu [22], Al₂O₃ [23], CuO [24], TiO₂ [25], ZnO [26], Ag [27], MWCNT [28], Ni [29], SiO₂ [30], ZrO₂ [31], CNT [32], and Au [33] have been the subject of many studies. Several research works have been done, such as improving performance and heat transfer, using nanofluids. Maddah et al. [34] examined the HTC of nanofluids and determined that thermal conductivity is affected by the change in volume fractions of the particles. They also found that Re number has key effects on the Nusselt (Nu) number and friction factor. Barik et al. [35] reported the impact of duct Re number on the heat transfer performances and decided that increasing duct Re number can also cause an increase in Nu number. Additionally, they detected that increasing nozzle Re number can also lead to a rise in pump power. They applied turbulent flow in the calculations. Mir et al. [36] examined the impact of volume fraction of nanoparticles on overall Nu number and determined that the rise in nanoparticle volume fraction can cause a rise in Nu number of nanofluids. In analysis, they applied Re numbers as 150, 300, 500, and 700. Zhang et al. [37] obtained nanofluids using Fe₃O₄ nanoparticles and water and they examined the variation among mass fraction and convective heat transfer for these nanofluids. They also discussed the trend among magnetic field and convective heat transfer. As an outcome of their studies, they found that the increase in Re number leads to an important increase in the convective heat transfer coefficient. Xuan and Li [38] examined the correlation among thermal conductivity and particle volume fraction for nanofluid formed in alumina–water system and found that the increase in volume fraction caused an increase in the thermal conductivity ratio. Additionally, many studies [39,40,41,42,43,44,45] have applied the Taguchi technique for various analyses of nanofluids. As literature research exhibits, many papers are related to heat transfer performance of nanofluids. Most of these studies have experimental and numerical approaches. However, studies with numerical, theoretical and statistical methods are very limited. In the numerical and statistical paper, the heat transfer characteristics (HTC) of nanofluids depending on various particle shapes and Re numbers were discussed using computational fluid dynamics (CFD) and Taguchi methods. The purpose for selecting various particle shapes in the analyses is to examine various thermal conductivity and viscosity for nanofluids. As understood in Table 4, the thermal conductivity and viscosity of the nanofluid including platelet nanoparticles is higher compared to the nanofluid including Brick nanoparticles. Thus, it can also explain the variations in the HTC of the nanofluid. Therefore, more effective HTC may be obtained by utilizing various shaped particle reinforcement for the same nanofluids. Additionally, ANOVA was implemented to govern the contribution fractions of the particle shape on the HTC. Thus, it may be easier to determine which factor has the maximum efficiency on the systems. With all these advantageous aspects, this study can support many contributions to the literature.

2 CFD analysis

CFD analyses for HTC of the nanofluids were carried out using ANSYS Fluent commercial software. In modelling, the pipe was used under the heat flux of 4,000 W/m². The heat flux was implemented homogeneously on the pipe wall surface. The pipe has a length of 1,000 mm and diameter of 15 mm. 3D computational domain was considered for the pipe flow. 3D domain for pipe in CFD analysis was illustrated in Figure 1.

Figure 1

3D domain for tube.

In mesh processing, multi-zone method with free mesh type including hexa-dominant was applied. To ensure homogeneous distribution of the mesh process, face meshing process was applied to the inlet and outlet areas of the pipe. Additionally, inflation method was added. 2,544,000 elements and 2,571,569 nodes were used in the mesh process. Previous study shows that using 425,691 elements was sufficient for the Nu number and fraction factor [46]. Thus, the number of elements and nodes in this study were considered appropriate for mesh operations. Additionally, the enhanced wall treatment including pressure gradient and thermal effects was utilized. Absolute criteria for the continuity, x–y–z velocities, energy, k and epsilon were considered to be 10⁻⁶ and the hybrid method was employed for solution initialization. The realizable k–ϵ approach in ANSYS Fluent was determined to be viscos model and this model can be stated as follows [47]:

(1) ∂ ∂ t ( ρ k ) + ∂ ∂ x ɉ ( ρ k u ɉ ) = ∂ ∂ x ɉ μ + μ t σ k ∂ k ∂ x ɉ + D k + D b − ρ ϵ − Y M + S k ,

(2) ∂ ∂ t ( ρ ϵ ) + ∂ ∂ x ɉ ( ρ ϵ u ɉ ) = ∂ ∂ x ɉ μ + μ t σ ϵ ∂ ϵ ∂ x ɉ + ρ C 1 S ϵ − ρ C 2 ϵ 2 k + υ ϵ + C 1 ϵ ϵ k C 3 ϵ G b + S ϵ ,

(3) C 1 = ma x 0.43 , η η + 5 , η = S k ϵ , S = 2 S i ɉ S i ɉ ,

(4) μ t = ρ C μ k 2 ϵ ,

(5) C μ = 1 A 0 + A s k U * ϵ ,

(6) U * ≡ S İ Ɉ S İ Ɉ + Ω ∼ İ Ɉ Ω ∼ İ Ɉ ,

(7) Ω ∼ İ Ɉ = Ω i ɉ − 2 ϵ i ɉ k ω k Ω i ɉ = Ω i ̅ ɉ ̅ − ϵ i ɉ k ω k ,

(8) A 0 = 4.04 , A S = 6 cos ϕ ,

(9) ϕ = 1 3 cos − 1 ( 6 W ) W = S i ɉ S ɉ k S ki S 3 ∼ S ∼ = S i ɉ S İ Ɉ S i ɉ = 1 2 ∂ u ɉ ∂ x i + ∂ u i ∂ x ɉ ,

(10) C 1 ϵ = 1.44 , C 2 = 1.9 , σ k = 1.0 , σ ϵ = 1.2 .

Equations containing continuity, momentum, stress tensor, and energy for pipe flow in ANSYS commercial software can be calculated as follows, respectively [47]:

(11) ∂ ρ ∂ t + ∇ ( ρ v → ) = S m ,

(12) ∂ ∂ t ( ρ v → ) + ∇ ( ρ v → v → ) = − ∇ p + ∇ τ ̿ + ρ g → + F → ,

(13) τ ̿ = μ ( ∇ v → + ∇ v → T ) − 2 3 ∇ v → I ,

(14) ∂ ( ρ U ) ∂ t + ∇ . [ v → ( ρ U + p ) ] = ∇ . k eff ∇ T − ∑ ɉ h ɉ Ɉ → ɉ + ( τ ̿ eff . v → ) + S h .

3 Statistical analysis

In the statistical analyses, Taguchi method and ANOVA were statistically employed. L8 orthogonal array in Taguchi methodology was used for the grouping of various particle shape and Re number in CFD analyses. Two factors were determined in numerical modeling. The first factor has four levels, whereas the second factor has two levels. Re number was decided to be the first factor, whereas the particle shape was chosen as the second factor. Factors and their levels are established in Table 1.

Table 1

Factors and levels in L8 orthogonal array

Factors	Icon	Levels
Factors	Icon	Level 1	Level 2	Level 3	Level 4
Re number	Re	4,500	6,000	7,500	9,000
Particle shape	PS	Brick	Platelet	—	—

One of the main purposes of this study is to determine the maximum data of the heat transfer characteristic such as heat transfer coefficient, Nu number, performance evaluation criteria (PEC), and average static pressure drop ( ∆ P ) . In this context, “Larger is Better” quality characteristic can be considered to achieve maximum responses. The equation for this quality characteristic can be explained as follows [48]:

(15) ( S / N ) LB for HTC = − 10 log n − 1 ∑ i = 1 n ( y i 2 ) − 1 .

To detect the factors at the optimum levels and their influence directions on the heat transfer characteristic, analysis of signal-to-noise S/N ratio was statistically applied. Minitab R12 statistical program was implemented in all statistical determinations.

4 Nanofluid analysis

In nanofluid characteristic, 1% Fe₃O₄ nanofluids were used. Nanoparticles have brick and platelet shapes in this study. In literature, nanoparticles with brick and platelet shapes have been limited, whereas shapes such as spherical, quasi-spherical, polyhedral, irregular, etc., are generally common. Nanoparticles with different shapes can display various properties and these properties can play a key role on heat transfer characteristic. In the study, the nanofluids with 1% Fe₃O₄ nanoparticle was utilized and water was chosen to be the base fluid. Properties for water [49] and Fe₃O₄ nanoparticles [50] are exhibited in Table 2.

Table 2

Properties for base fluid and nanoparticle

Properties	Symbol	Unit	Base fluid [49]	Nanoparticle [50]
Properties	Symbol	Unit	Water	Fe₃O₄
Density	ρ	kg/m³	998.2	5,200
Specific heat	C _p	J/kg K	4,180	670
Thermal conductivity	ҟ	W/m K	0.6	6
Viscosity	μ	kg/ms	0.001003	—

Volume fraction of nanoparticles in nanofluid can be expressed as follows [51]:

(16) φ = V np V bl + V np .

Density of the nanofluid may be determined as follows [52]:

(17) ρ nl = ρ bl ( 1 − φ ) + φ ρ np .

Specific heat in the nanofluid may be calculated as follows [53]:

(18) C p , nl = ( 1 − φ ) ρ bl C p , bl + φ ρ np C p , np ρ nl .

Thermal conductivity in the nanofluid may be evaluated as follows [54]:

(19) ҟ nl ҟ l = ҟ p + ( n − 1 ) ҟ l + ( n − 1 ) ( ҟ p − ҟ l ) φ ( n − 1 ) ҟ l − ( ҟ p − ҟ l ) φ + ҟ p .

Viscosity of the nanofluid can be solved as follows [55]:

(20) μ nl = μ bl ( 1 + A 1 φ + A 2 φ 2 ) .

Constants in accordance with thermal conductivity and viscosity based on the nanoparticle shapes are described in Table 3 [54].

Table 3

Constants for nanoparticles [54]

Nanoparticle shape	N	A ₁	A ₂
Brick	3.7	1.9	471.4
Platelet	5.7	37.1	612.6

The properties of the nanofluid were found using Eqs (16)–(20) and the results considered are established in Table 4.

Table 4

Properties for nanofluids

Nanofluid	Nanoparticle shape	Density (kg/m³)	Specific heat (J/kg K)	Thermal conductivity (W/m K)	Viscosity (kg/m s)
1 vol% Fe₃O₄/water	Brick	1040.218	4004.537	0.615844	0.001069
1 vol% Fe₃O₄/water	Platelet	1040.218	4004.537	0.621068	0.001437

As can be understood from Table 3, although the nanoparticle shape does not have any impact in the density and specific heat of the nanofluids, it plays a dominant role on thermal conductivity and viscosity. Therefore, variations that may occur in the HTC can be detected in accordance with the thermal conductivity and viscosity of the nanofluids.

5 HTC analysis

In the presented work, the influence of particle shape and Re number on the heat transfer coefficient (h), Nu number, PEC, and ΔP of nanofluid in heated pipe was analyzed as numerically and statistically. Four different Re numbers were considered in the analyses. Re number may be evaluated as follows [56]:

(21) Re = ρ VR µ .

Heat transfer coefficient in nanofluid may be solved as follows [56]:

(22) h = Q A ( T s − T l , ave ) .

Heat transfer rate for nanofluid in pipe may be described as follows [56]:

(23) Q = m l c P ( T o − T i ) .

Nu number of nanofluid may be evaluated as follows [56]:

(24) Nu = hR k .

PEC can be solved as follows [46,57]:

(25) PEC = ( Nu nl / Nu bl ) ( f nl / f bl ) 1 3 .

Pressure drop in accordance with inner pressure and outer pressure can be found as follows [56]:

(26) ∆ P = P i − P o .

6 Results and discussion

This CFD and statistical study is concerned about the evaluation of influence of particle shape and Re number on the h, Nu number, PEC, and ΔP of nanofluid in heated pipe. CFD analyses for 1% Fe₃O₄ nanofluids were carried out in accordance with the L8 orthogonal array and mathematical results achieved are presented in Table 5.

Table 5

CFD results for L8 orthogonal array

Test	Designation	Factors		CFD results
Test	Designation	Re	PS	h (W/m² K)	Nu (−)	PEC (−)	ΔP (Pa)
1	Re₁PS₁	4,500	Brick	1671.9	40.722	0.998	159.44
2	Re₁PS₂	4,500	Platelet	1902.2	45.942	1.128	287.69
3	Re₂PS₁	6,000	Brick	2116.8	51.558	0.998	253.88
4	Re₂PS₂	6,000	Platelet	2409.3	58.189	1.128	458.13
5	Re₃PS₁	7,500	Brick	2547.3	62.044	0.998	366.31
6	Re₃PS₂	7,500	Platelet	2900.6	70.055	1.128	661.06
7	Re₄PS₁	9,000	Brick	2966.5	72.254	0.998	495.77
8	Re₄PS₂	9,000	Platelet	3379.0	81.609	1.128	894.73
Overall mean value				2486.7	60.297	1.063	447.13

As described in Table 5, overall mean values of h, Nu number, PEC, and ΔP were calculated to be 2486.7 W/m² K, 60.297, 1.063, and 447.13 Pa, respectively. S/N ratios corresponding to CFD results based on “the Larger is better” quality characteristic are described in Table 6.

Table 6

S/N ratios of CFD results for L8 orthogonal array

Test	Designation	Factors		S/N ratios (dB)
Test	Designation	Re	PS	Ƞ for h	Ƞ for Nu	Ƞ for PEC	Ƞ for ΔP
1	Re₁PS₁	4,500	Brick	64.4642	32.1966	−0.0169	44.0519
2	Re₁PS₂	4,500	Platelet	65.5851	33.2442	1.0464	49.1785
3	Re₂PS₁	6,000	Brick	66.5136	34.2460	−0.0198	48.0926
4	Re₂PS₂	6,000	Platelet	67.6378	35.2969	1.0468	53.2198
5	Re₃PS₁	7,500	Brick	68.1216	35.8540	−0.0194	51.2770
6	Re₃PS₂	7,500	Platelet	69.2498	36.9088	1.0434	56.4048
7	Re₄PS₁	9,000	Brick	69.4449	37.1773	−0.0200	53.9056
8	Re₄PS₂	9,000	Platelet	70.5758	38.2348	1.0428	59.0338

Velocity and temperature contours of CFD results using L8 orthogonal approach in Taguchi methodology are presented in Figures 2 and 3, respectively.

Figure 2

CFD velocity contour plots for (a) Re₁PS₁, (b) Re₁PS₂, (c) Re₂PS₁. (d) Re₂PS₂, (e) Re₃PS₁, (f) Re₃PS₂, (g) Re₄PS₁, (h) Re₄PS₂.

Figure 3

CFD temperature contour plots for (a) Re₁PS₁, (b) Re₁PS₂, (c) Re₂PS₁, (d) Re₂PS₂, (e) Re₃PS₁, (f) Re₃PS₂, (g) Re₄PS₁, (h) Re₄PS₂.

As understood from Figure 2, the velocity of the nanofluid rises from the pipe walls to the center of the pipe. This finding can be described by the resistance of the fluid against the pipe walls. As defined in Figure 3, the nanofluid temperature reduces from the pipe walls to the pipe center. This finding can be clarified by the temperature occurring on the pipe walls by the heat flux data. Temperature of pipe walls is high, whereas the temperature decreases towards the center of the pipe. Thus, difference was detected in temperature distribution due to the nanofluid.

6.1 Effect on HTC

To understand the influence of particle shape and Re number on the HTC of nanofluids, CFD analyses were carried out using various levels of different factors. The numerical results achieved by ANSYS Fluent software were evaluated as S/N ratio values to obtain the maximum HTC of nanofluids. Average CFD data corresponding to average S/N ratios were used to define the impact of each particle shape and Re number on the HTC at various levels. Graphs were drawn based on the S/N ratios. Impact of particle shape and Re number on the HTC of nanofluids are displayed in Figure 4.

Figure 4

Main impact plots for (a) h, (b) Nu number, (c) PEC, (d) ΔP.

Figure 4a–d displays that nanofluids containing 1% Fe₃O₄ nanoparticles with platelet shape have a higher influence on the h, Nu number, PEC, and ΔP compared to brick nanoparticles. This finding may be defined by thermal conductivity and viscosity. This situation is clearly determined in Eqs (19) and (20). The thermal conductivity and viscosity for nanofluid including platelet nanoparticle are greater than nanofluid with brick nanoparticle. The increase in thermal conductivity of nanofluids causes an increase in the h, Nu number, and PEC. Although the increase in the Re number from 4,500 to 9,000 leads to an increase in the h, Nu number, and ΔP, it does not have any effect on the PEC. These findings were supported by the study by Garud and Lee [46]. Additionally, another study [58] showed that increasing Re number causes an increase in h, Nu number, and ΔP.

6.2 Optimal levels of particle shape and Re number

ANOVA is one of the statistical methods and it was employed to define the HTC. ANOVA was implemented to achieve the significance level and percent contribution ratio of particle shape and Re number on the HTC of nanofluids. Analyses were implemented in accordance with 95% confidence level. ANOVA results for h and Nu number are shown in Table 7.

Table 7

ANOVA for h and Nu

Source	DF	h					Nu
Source	DF	Seq SS	Adj MS	F	P	% Effect	Seq SS	Adj MS	F	P	% Effect
Re	3	2,132,963	710,988	231.23	0	90.77	1254.05	418.02	263.94	0	91.84
PS	1	207,561	207,561	67.50	0.004	8.83	106.70	106.70	67.37	0.004	7.81
Error	3	9,224	3,075			0.39	4.75	1.58			0.35
Total	7	2,349,749				100	1365.5				100
R ² = 99.61% and R ² (adj) = 99.08							R ² = 99.65% and R ² (adj) = 99.19%

According to the F-test results in Table 5, particle shape and Re number were determined as the most important factors on the heat transfer coefficient. Additionally, Re number has a 90.77% impact on h, whereas particle shape has 8.83% effect. This finding shows that Re number is dominant on h and Nu number compared to the particle shape. For the Nu number, particle shape and Re number were considered as important factors based on F-test results. Re number has a 91.84% influence on the Nu number, whereas particle shape has 7.81% effect. This finding indicates that the influence of turbulent flow on Nu number is more dominant compared to the particle shape. ANOVA outcomes for the PEC and average static pressure drop are displayed in Table 8.

Table 8

ANOVA for PEC and ΔP

Source	DF	PEC					ΔP
Source	DF	Seq SS	Adj MS	F	P	% Effect	Seq SS	Adj MS	F	P	% Effect
Re	3	0	0	**	—	0	248,556	82,852	12.14	0.035	62.04
PS	1	0.0338	0.0338	**	—	100	131,638	131,638	19.29	0.022	32.86
Error	3	0	0		—	0	20,468	6,823			5.11
Total	7	0.0338				100	400,663				100
R ² = 100.00% and R ² (adj) = 100.00%							R ² = 94.89% and R ² (adj) = 88.08%

ANOVA results in Table 6 show that Re number has no impact on the PEC. But, the impact of particle shape is great. Thus, the denominator of F-test was calculated as zero for the PEC. According to this outcome, the one major factor was determined to be the particle shape. Therefore, the particle shape of nanoparticle has a 100% influence on PEC, whereas RE number has a 0% influence. According to the F-test results on the average static pressure drop, both particle shape and Re number were found to be important factors. According to the influence ratios, the Re number had a 62.04% influence on the average static pressure drop, whereas the particle shape had a 32.86% influence. This finding displays that the flow velocity in turbulent flow is greater than the effect of particle shape. The key purpose of the presented work is to determine the optimum levels of factors to obtain the highest HTC of nanofluids. In this context, the numerically calculated h, Nu number, PEC, and ΔP of nanofluid results were converted into S/N ratios. Average CFD results and S/N ratios were calculated for each level of particle shape and Re number. Overall CFD data and S/N ratios at all levels of particle shape and Re number according to h and Nu are exhibited in Table 9.

Table 9

Mean results for h and Nu

Level	h				Nu
	S/N ratio (dB)		Mean value (W/m² K)		S/N ratio (dB)		Mean value
	Re	PS	Re	PS	Re	PS	Re	PS
1	65.02	67.14	1,787	2326	32.72	34.87	43.33	56.64
2	67.08	68.26	2,263	2648	34.77	35.92	54.87	63.95
3	68.69		2,724		36.38		66.05
4	70.01		3,173		37.71		76.93
Delta	4.99	1.13	1,386	322	4.99	1.05	33.60	7.30
Rank	1	2	1	2	1	2	1	2

The overall results obtained from Table 9 display that the increase in Re number causes an increase in the h and Nu number. Thus, the particle with platelet shape had the optimum impact levels on the h and Nu number, whereas the optimum Re number was determined as 9,000. Therefore, to obtain the maximum h and Nu number, nanofluids including platelet nanoparticles and Re number of 9,000 should be determined. Average data of CFD and S/N ratios at all levels of particle shape and Re number in accordance with the PEC, and average static pressure drop of nanofluid are presented in Table 10.

Table 10

Main results for PEC and ΔP

Level	PEC				ΔP
	S/N ratio (dB)		Mean value		S/N ratio (dB)		Mean value (Pa)
	Re	PS	Re	PS	Re	PS	Re	PS
1	0.51472	−0.01902	1.0630	0.9978	46.62	49.33	223.60	318.90
2	0.51349	1.04484	1.0629	1.1278	50.66	54.46	356.00	575.40
3	0.51200		1.0627		53.84		513.70
4	0.51143		1.0626		56.47		695.30
Delta	0.0033	1.06387	0.0004	0.1300	9.85	5.13	471.70	256.60
Rank	2	1	2	1	1	2	1	2

The results shown in Table 10 exhibits that the nanofluids with platelet-shaped nanoparticles were determined as the optimum since the maximum PEC and static pressure drop were reached while using platelet-shaped nanoparticles. Additionally, because no impact of Re number on the PEC was detected, the ideal level for Re number cannot be analyzed. It indicates that the increase in the Re number in the average static pressure drop leads to an increase in the pressure drop. Therefore, the ideal level for Re number on pressure drop was observed as 9,000. The maximum static pressure drop can be achieved by applying nanofluid including platelet nanoparticles and Re number of 9,000.

6.3 Estimation of optimal heat transfer responses

Particle shape and Re numbers at the optimal levels were selected to achieve the maximum HTC of nanofluids. S/N ratio analysis was applied to decide optimum levels and average results were evaluated. ANOVA was implemented and significant particle shape and Re number based on F-test results were discussed. Determination of predicted HTC regarding S/N ratio and ANOVA results at 95% confidence level can be solved as follows [48]:

(27) μ est = PS opt ̅ + Re opt ̅ − T ave ̅ ,

where PS opt ̅ and Re opt ̅ for the heat transfer coefficient were calculated as 3,173 and 2,648 W/m² K, whereas PS opt ̅ and Re opt ̅ for the Nusellt number were solved as 76.93 and 63.95, respectively. Additionally, PS opt ̅ and Re opt ̅ for the PEC were determined as 1.063 and 1.1278, whereas PS opt ̅ and Re opt ̅ for the average static pressure drop were found as 695.30 and 575.40 Pa, respectively. T ave ̅ for the h, Nu number, PEC, and ΔP were calculated as 2486.7 W/m² K, 60.297, 1.063, and 447.13 Pa, respectively. These data are substituted in Eq. (27) and the predicted Taguchi results are exhibited in Table 11.

Table 11

CFD and estimated optimal heat transfer responses

HTC	h (Re₄PS₂)	Nu (Re₄PS₂)	PEC (Re_allPS₂)	ΔP (Re₄PS₂)
CFD results	3379.0 W/m² K	81.609	1.128	894.73
Taguchi results	3334.3 W/m² K	80.583	1.128	823.57
% differences	1.32	1.16	0	7.95

7 Validation

Numerical and statistical results of Nu number and PEC of nanofluids were compared with theoretical approach. In this context, Nu number and Darcy friction factor were solved theoretically. Calculations for validation were carried out for particle shape and Re numbers under the optimal levels. Theoretical Nu number and Darcy friction factors were considered using Eqs (28) [59] and (29) [60], respectively.

(28) Nu ̅ R = 0.023 Re 0.8 Pr 0.4 ,

(29) 1 f = − 2 log e / R 3.7 + 2.52 R e f .

CFD, estimated Taguchi results, and theoretical results obtained for Nu number and PEC utilizing particle shape and Re number under the ideal levels are exhibited in Table 12.

Table 12

Theoretical validation of CFD results

Nu (Re₄PS₂)		% Difference	PEC (Re_allPS₂)		% Difference
CFD	Formula	% Difference	CFD	Formula	% Difference
81.609	81.624	0.02	1.128	1.132	0.35

Table 12 shows that while the difference among CFD and theoretical data of the Nu number was found to be 0.02%, the difference for the PEC was determined to be 0.35%. These findings prove the precision of the numerical and statistical data.

8 Conclusion

In the presented study, the impact of particle shape and Reynolds (Re) number on the h, Nu number, PEC, and ΔP of nanofluid in heated tube was analyzed in accordance with Taguchi method. Fe₃O₄ was selected as nanoparticle and so 1% Fe₃O₄ nanofluids for CFD analyses was used. Particle shape and Re number were assumed as the first and second factors. Brick and platelet were considered as particle shapes. CFD calculations for HTC of nanofluids were carried out by ANSYS Fluent commercial software regarding L8 orthogonal range utilizing Taguchi methodology. A pipe of 1,000 mm length and 15 mm diameter was implemented in numerical modeling. The heat flux was chosen as 4,000 W/m². S/N ratio analysis was implemented to find optimum levels and impact trend on HTC for each factor, whereas ANOVA was conducted to release the dominant levels and percentage impact rates of the factors. The key results obtained from this study are summarized below:

Increasing Re number from 4,500 to 9,000 causes an increase in HTC of the nanofluid.
The influence of platelet nanoparticles on the HTC of the nanofluid is greater than brick nanoparticles.
The maximum heat transfer characteristic of the nanofluid is achieved by using platelet nanoparticles and Re number of 9,000.
According to the F-test data in the ANOVA results, particle shape is detected as a significant factor. However, Re number was determined to be a significant factor on heat transfer coefficient, Nu number, and average static pressure drop, whereas no impact was found on the PEC.
The contribution ratio of particle shape on the heat transfer coefficient, Nu number, PEC, and average static pressure drop is found to be 8.83, 7.81, 100, and 32.86%, respectively.
The contribution ratio of Re number on heat transfer coefficient, Nu number, PEC, and average static pressure drop is decided to be 90.77, 91.84, 0, and 62.04%, respectively.
The difference between numerical and Taguchi prediction results obtained for heat transfer coefficient, Nu number, performance evaluation criteria, and average static pressure drop was determined to be 1.32, 1.16, 0, and 7.95%, respectively.
In the validation calculations, the difference between the CFD and theoretical results for the Nu number was found to be 0.02%, whereas the difference between the CFD and theoretical data obtained for the performance evaluation criteria was solved to be 0.35%.
The temperature of the nanofluid reduces from the pipe walls to the center of the pipe, whereas the nanofluid velocity increases from the pipe walls to the center.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

References

[1] Waheed R, Sarwar S, Wei C. The survey of economic growth, energy consumption and carbon emission. Energy Rep. 2019;5:1103–15.10.1016/j.egyr.2019.07.006Search in Google Scholar

[2] Indhuja A, Suganthi K, Manikandan S, Rajan K. Viscosity and thermal conductivity of dispersions of gum arabic capped MWCNT in water: Influence of MWCNT concentration and temperature. J Taiwan Inst Chem Eng. 2013;44:474–9.10.1016/j.jtice.2012.11.015Search in Google Scholar

[3] Yu W, Xie H, Chen L, Li Y. Enhancement of thermal conductivity of kerosene-based Fe3O4 nanofluids prepared via phase-transfer method. Colloids Surf A: Physicochem Eng Asp. 2010;355:109–13.10.1016/j.colsurfa.2009.11.044Search in Google Scholar

[4] Lenin R, Joy PA, Bera C. A review of the recent progress on thermal conductivity of nanofluid. J Mol Liq. 2021;338:116929.10.1016/j.molliq.2021.116929Search in Google Scholar

[5] Sajid MU, Ali HM. Thermal conductivity of hybrid nanofluids: a critical review. Int J Heat Mass Transf. 2018;126:211–34.10.1016/j.ijheatmasstransfer.2018.05.021Search in Google Scholar

[6] Cuce E, Cuce PM, Guclu T, Besir AB. On the use of nanofluids in solar energy applications. J Therm Sci. 2020;29:513–34.10.1007/s11630-020-1269-3Search in Google Scholar

[7] Yousefi T, Veysi F, Shojaeizadeh E, Zinadini S. An experimental investigation on the effect of Al2O3-H2O nanofluid on the efficiency of flat-plate solar collectors. Renew Energy. 2012;39:293–8.10.1016/j.renene.2011.08.056Search in Google Scholar

[8] Sohel M, Saidur R, Khaleduzzaman S, Ibrahim TA. Cooling performance investigation of electronics cooling system using Al2O3–H2O nanofluid. Int Commun Heat Mass Transf. 2015;65:89–93.10.1016/j.icheatmasstransfer.2015.04.015Search in Google Scholar

[9] Bafrani HA, Noori-kalkhoran O, Gei M, Ahangari R, Mirzaee MM. On the use of boundary conditions and thermophysical properties of nanoparticles for application of nanofluids as coolant in nuclear power plants; a numerical study. Prog Nucl Energy. 2020;126:103417.10.1016/j.pnucene.2020.103417Search in Google Scholar

[10] Pandya NS, Shah H, Molana M, Tiwari AK. Heat transfer enhancement with nanofluids in plate heat exchangers: A comprehensive review. Eur J Mech-B/Fluids. 2020;81:173–90.10.1016/j.euromechflu.2020.02.004Search in Google Scholar

[11] Yi P, Kayani AA, Chrimes AF, Ghorbani K, Nahavandi S, Kalantar-Zadeh K, et al. Thermal analysis of nanofluids in microfluidics using an infrared camera. Lab a Chip. 2012;12:2520–5.10.1039/c2lc40222jSearch in Google Scholar PubMed

[12] Li K, Wang D, Jiang S. Review on enhanced oil recovery by nanofluids. Oil Gas Sci Technol–Revue d’IFP Energies nouvelles. 2018;73:37.10.2516/ogst/2018025Search in Google Scholar

[13] Balla HH, Hashem AL, Kareem ZS, Abdulwahid AF. Heat transfer potentials of ZnO/water nanofluid in free impingement jet. Case Stud Therm Eng. 2021;27:101143.10.1016/j.csite.2021.101143Search in Google Scholar

[14] Singh T, Atieh MA, Al-Ansari T, Mohammad AW, McKay G. The role of nanofluids and renewable energy in the development of sustainable desalination systems: A review. Water. 2020;12:2002.10.3390/w12072002Search in Google Scholar

[15] Jadeja KM, Bumataria R, Chavda N. Nanofluid as a coolant in internal combustion engine–a review. Int J Ambient Energy. 2023;44:363–80.10.1080/01430750.2022.2127891Search in Google Scholar

[16] Esfe MH, Bahiraei M, Hajbarati H, Valadkhani M. A comprehensive review on convective heat transfer of nanofluids in porous media: Energy-related and thermohydraulic characteristics. Appl Therm Eng. 2020;178:115487.10.1016/j.applthermaleng.2020.115487Search in Google Scholar

[17] Patel HE, Anoop K, Sundararajan T, Das SK. A micro-convection model for thermal conductivity of nanofluids. International Heat Transfer Conference 13. Begel House Inc.; 2006.10.1615/IHTC13.p8.240Search in Google Scholar

[18] Chon CH, Kihm KD, Lee SP, Choi SU. Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3) thermal conductivity enhancement. Appl Phys Lett. 2015;65:89–93.Search in Google Scholar

[19] Ebaid MS, Ghrair AM, Al-busoul M. Investigation of heat transfer enhancement using ferro-nanofluids (Fe3O4/water) in a heated pipe under the application of magnetic field. Adv Mech Eng. 2022;14:16878132221102647.10.1177/16878132221102647Search in Google Scholar

[20] Kichatov B, Korshunov A, Sudakov V, Golubkov A, Kolobov A, Chikishev L. Magnetic nanofluid Fe3O4 in passive cooling system based on thermo-osmotic effect. Int J Therm Sci. 2024;199:108931.10.2139/ssrn.4482241Search in Google Scholar

[21] Afshari F, Muratçobanoğlu B. Thermal analysis of Fe3O4/water nanofluid in spiral and serpentine mini channels by using experimental and theoretical models. Int J Environ Sci Technol. 2023;20:2037–52.10.1007/s13762-022-04119-6Search in Google Scholar

[22] Khamliche T, Khamlich S, Moodley M, Mothudi B, Henini M, Maaza M. Laser fabrication of Cu nanoparticles based nanofluid with enhanced thermal conductivity: Experimental and molecular dynamics studies. J Mol Liq. 2021;323:114975.10.1016/j.molliq.2020.114975Search in Google Scholar

[23] Kong M, Lee S. Performance evaluation of Al2O3 nanofluid as an enhanced heat transfer fluid. Adv Mech Eng. 2020;12:1687814020952277.10.1177/1687814020952277Search in Google Scholar

[24] Nfawa SR, Basri AA, Masuri SU. Novel use of MgO nanoparticle additive for enhancing the thermal conductivity of CuO/water nanofluid. Case Stud Therm Eng. 2021;27:101279.10.1016/j.csite.2021.101279Search in Google Scholar

[25] Hozien O, El-Maghlany WM, Sorour MM, Mohamed YS. Experimental study on thermophysical properties of TiO2, ZnO and Ag water base nanofluids. J Mol Liq. 2021;334:116128.10.1016/j.molliq.2021.116128Search in Google Scholar

[26] Azeez K, Ibrahim ZA, Hussein AM. Thermal conductivity and viscosity measurement of ZnO nanoparticles dispersing in various base fluids. J Adv Res Fluid Mech Therm Sci. 2020;66:1–10.Search in Google Scholar

[27] Pourrajab R, Noghrehabadi A, Behbahani M, Hajidavalloo E. An efficient enhancement in thermal conductivity of water-based hybrid nanofluid containing MWCNTs-COOH and Ag nanoparticles: experimental study. J Therm Anal Calorim. 2021;143:3331–43.10.1007/s10973-020-09300-ySearch in Google Scholar

[28] Lyu Z, Asadi A, Alarifi IM, Ali V, Foong LK. Thermal and fluid dynamics performance of MWCNT-water nanofluid based on thermophysical properties: an experimental and theoretical study. Sci Rep. 2020;10:5185.10.1038/s41598-020-62143-3Search in Google Scholar PubMed PubMed Central

[29] Saleh B, Sundar LS. Experimental study on heat transfer, friction factor, entropy and exergy efficiency analyses of a corrugated plate heat exchanger using Ni/water nanofluids. Int J Therm Sci. 2021;165:106935.10.1016/j.ijthermalsci.2021.106935Search in Google Scholar

[30] Zhang L, Zhang A, Jing Y, Qu P, Wu Z. Effect of particle size on the heat transfer performance of SiO2–water nanofluids. J Phys Chem C. 2021;125:13590–600.10.1021/acs.jpcc.1c02014Search in Google Scholar

[31] Kolappan S, Karthik S, Logesh K, Vasudevan A. Thermal characterisation study of ZrO2/water nanofluid. Int J Ambient Energy. 2020;41:918–21.10.1080/01430750.2018.1492454Search in Google Scholar

[32] Yadav D, Sanserwal M. A comprehensive review of the effects of various factors on the thermal conductivity and rheological characteristics of CNT nanofluids. J Therm Anal Calorim. 2023;148:1723–63.10.1007/s10973-022-11821-7Search in Google Scholar

[33] Ternik P. Conduction and convection heat transfer characteristics of water–Au nanofluid in a cubic enclosure with differentially heated side walls. Int J Heat Mass Transf. 2015;80:368–75.10.1016/j.ijheatmasstransfer.2014.09.041Search in Google Scholar

[34] Maddah H, Alizadeh M, Ghasemi N, Alwi SRW. Experimental study of Al2O3/water nanofluid turbulent heat transfer enhancement in the horizontal double pipes fitted with modified twisted tapes. Int J Heat Mass Transf. 2014;78:1042–54.10.1016/j.ijheatmasstransfer.2014.07.059Search in Google Scholar

[35] Barik AK, Rout S, Mukherjee A. Numerical investigation of heat transfer enhancement from a protruded surface by cross-flow jet using Al2O3–water nanofluid. Int J Heat Mass Transf. 2016;101:550–61.10.1016/j.ijheatmasstransfer.2016.05.077Search in Google Scholar

[36] Mir S, Akbari OA, Toghraie D, Sheikhzadeh G, Marzban A, Mir S, et al. A comprehensive study of two-phase flow and heat transfer of water/Ag nanofluid in an elliptical curved minichannel. Chin J Chem Eng. 2020;28:383–402.10.1016/j.cjche.2019.07.007Search in Google Scholar

[37] Zhang R, Qing S, Zhang X, Li J, Liu Y, Wen X. Experimental investigation and machine learning modeling of heat transfer characteristics for water based nanofluids containing magnetic Fe3O4 nanoparticles. Mater Today Commun. 2023;36:106798.10.1016/j.mtcomm.2023.106798Search in Google Scholar

[38] Xuan Y, Li Q. Heat transfer enhancement of nanofluids. Int J Heat Fluid Flow. 2000;21:58–64.10.1016/S0142-727X(99)00067-3Search in Google Scholar

[39] Mohammadi M, Abadeh A, Nemati-Farouji R, Passandideh-Fard M. An optimization of heat transfer of nanofluid flow in a helically coiled pipe using Taguchi method. J Therm Anal Calorim. 2019;138:1779–92.10.1007/s10973-019-08167-ySearch in Google Scholar

[40] Raei B. Statistical analysis of nanofluid heat transfer in a heat exchanger using Taguchi method. J Heat Mass Transf Res. 2021;8:29–38.Search in Google Scholar

[41] Sobhani M, Ajam H. Taguchi optimization for natural convection heat transfer of Al2O3 nanofluid in a partially heated cavity using LBM. J Therm Anal Calorim. 2019;138:889–904.10.1007/s10973-019-08170-3Search in Google Scholar

[42] Zahmatkesh I, Shandiz MRH. Optimum constituents for MHD heat transfer of nanofluids within porous cavities: A Taguchi analysis in natural and mixed convection configurations. J Therm Anal Calorim. 2019;138:1669–81.10.1007/s10973-019-08191-ySearch in Google Scholar

[43] Bazkhane S, Zahmatkesh I. Taguchi-based sensitivity analysis of hydrodynamics and heat transfer of nanofluids in a microchannel heat sink (MCHS) having porous substrates. Int Commun Heat Mass Transf. 2020;118:104885.10.1016/j.icheatmasstransfer.2020.104885Search in Google Scholar

[44] Javadpour SM, Abadi EAJ, Akbari OA, Goharimanesh M. Optimization of geometry and nano-fluid properties on microchannel performance using Taguchi method and genetic algorithm. Int Commun heat mass Transf. 2020;119:104952.10.1016/j.icheatmasstransfer.2020.104952Search in Google Scholar

[45] Gelis K, Ozbek K, Ozyurt O, Celik AN. Multi-objective optimization of a photovoltaic thermal system with different water based nanofluids using Taguchi approach. Appl Therm Eng. 2023;219:119609.10.1016/j.applthermaleng.2022.119609Search in Google Scholar

[46] Garud KS, Lee M-Y. Numerical investigations on heat transfer characteristics of single particle and hybrid nanofluids in uniformly heated tube. Symmetry. 2021;2–19.10.3390/sym13050876Search in Google Scholar

[47] ANSYS. ANSYS FLUENT 12.0 Theory Guide. Inc, Canonsburg, PA; 2009.Search in Google Scholar

[48] Ross PJ. Taguchi techniques for quality engineering: McGraw-Hill International Editions. 2nd edn. New York, USA; 1996.Search in Google Scholar

[49] Chaurasia SR, Sarviya RM. Comparative thermal performance analysis with entropy generation on helical screw insert in tube with number of strips with nanofluid at laminar flow regime. Int Commun Heat Mass Transf. 2021;122:105138.10.1016/j.icheatmasstransfer.2021.105138Search in Google Scholar

[50] Sheikholeslami M, Shehzad SA. CVFEM for influence of external magnetic source on Fe3O4-H2O nanofluid behavior in a permeable cavity considering shape effect. Int J Heat Mass Transf. 2017;115:180–91.10.1016/j.ijheatmasstransfer.2017.07.045Search in Google Scholar

[51] Khanafer K, Vafai K. A critical synthesis of thermophysical characteristics of nanofluids. Int J Heat Mass Transf. 2011;54:4410–28.10.1016/j.ijheatmasstransfer.2011.04.048Search in Google Scholar

[52] Pak BC, Cho YI. Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles. Exp Heat Transf. 1998;11:151–70.10.1080/08916159808946559Search in Google Scholar

[53] Xuan Y, Roetzel W. Conceptions for heat transfer correlation of nanofluids. Int J heat Mass Transf. 2000;43:3701–7.10.1016/S0017-9310(99)00369-5Search in Google Scholar

[54] Timofeeva EV, Routbort JL, Singh D. Particle shape effects on thermophysical properties of alumina nanofluids. J Appl Phys. 2009;106:1–10.10.1063/1.3155999Search in Google Scholar

[55] Rudyak V, Belkin AA, Tomilina EA, Egorov VV. Nanoparticle Friction Force and Effective Viscosity of Nanosuspensions. Defect Diffus Forum. 2008;273-276:566–71.10.4028/www.scientific.net/DDF.273-276.566Search in Google Scholar

[56] Mauri R. Transport phenomena in multiphase flows. Switzerland: Springer; 2015.10.1007/978-3-319-15793-1Search in Google Scholar

[57] Singh SK, Sarkar J. Thermohydraulic behavior of concentric tube heat exchanger inserted with conical wire coil using mono/hybrid nanofluids. Int Commun Heat Mass Transf. 2021;122:105134.10.1016/j.icheatmasstransfer.2021.105134Search in Google Scholar

[58] Saedodin S, Zaboli M, Rostamian SH. Effect of twisted turbulator and various metal oxide nanofluids on the thermal performance of a straight tube: Numerical study based on experimental data. Chem Eng Process - Process Intensif. 2020;158:108106.10.1016/j.cep.2020.108106Search in Google Scholar

[59] Balaji C, Srinivasan B, Gedupudi S. Heat transfer engineering: fundamentals and techniques. London: Academic Press; 2020.Search in Google Scholar

[60] Cengel Y, Cimbala J. Fluid mechanics fundamentals and applications. 3rd edn. New York: McGraw Hill; 2014.Search in Google Scholar

Received: 2024-03-14

Revised: 2024-04-04

Accepted: 2024-05-22

Published Online: 2024-06-22

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0046

Keywords for this article

ANOVA; CFD; heat transfer; nanofluids; Taguchi method

Creative Commons

BY 4.0