Startseite Mathematical analysis of nanoparticle type and volume fraction on heat transfer efficiency of nanofluids
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Mathematical analysis of nanoparticle type and volume fraction on heat transfer efficiency of nanofluids

  • Savas Evran EMAIL logo
Veröffentlicht/Copyright: 27. Mai 2025
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Open Physics
Aus der Zeitschrift Open Physics Band 23 Heft 1

Abstract

Nanofluids (NFs) have been implemented in several areas to increase heat transfer efficiency. Thus, efficiency for heat energy can be achieved. In this study, the effect of nanoparticle (NP) type, volume fraction, and Re number on the heat transfer efficiency of NFs were analyzed numerically, statistically, and theoretically. Heat transfer coefficient, number of transfer units, wall shear stress, and friction factor were selected as the heat transfer efficiency of NFs. ANSYS Fluent software was utilized to carry out computational fluid dynamics analyses. The numerical calculation scheme was implemented by employing the Taguchi L9 orthogonal array with three decisive factors. NP type, volume fraction, and Re number were assumed as decisive factors with three levels. Signal-to-noise ratio analysis was utilized to determine the direction of impact and ideal levels of each decisive factor on the heat transfer efficiency of NFs. Significance levels and contribution rates of the decisive factors on the heat transfer efficiency were calculated with Analysis of Variance. According to the mathematical responses, the most effective NPs on h and τ w were identified as Gr, Al2O3, and Cu, respectively, whereas the most effective NPs on number of transfer units are found to be Cu, Al2O3, and Gr, respectively. In addition, the increase in Re number from 6,000 to 8,000 causes an increase in h and τ w and a decrease in NTU and f. The data achieved from the mathematical research may be utilized to be a guide paper in experimental analyses.

Keywords: ANOVA; fluid dynamics; heat transfer efficiency; nanoparticle type; Taguchi approach

Nomenclature

A c

cross-sectional area of tube at inlet

ANOVA

analysis of variance

A sr

tube surface area

C 1 ϵ , C 2 , C 3 ϵ , C μ

constants

C p , nf

specific heat of NFs

c p

specific heat

D Tube

diameter of tube

E

energy

F →

external body forces

f

friction factor

G ƙ

emergence of turbulence kinetic energy because of the mean velocity gradients

G b

emergence of turbulence kinetic energy because of buoyancy

g

gravity

HTE

heat transfer efficiency

h

heat transfer coefficient

k nf

thermal conductivity of NF

k bf

thermal conductivity based on base fluid

ƙ

turbulence kinetic energy

k

thermal conductivity of the fluid

M ¯ i

average HTE value at optimum levels for particle

m ̇

mass flow rate of fluid

NFs

nanofluids

NP

nanoparticle

NTU

number of transfer units

Nu ¯ D

Nusselt’s number

Pr

Prandtl numbers

Q .

rate of heat transfer

Re ¯ i

average HTE value at optimum levels for Re number

Re

Reynolds number

r →

location vector for the field point

r → w

combination of all relevant wall boundaries

S ƙ , S ϵ

source terms specified by the user

S/N

signal-to-noise

S m

mass added to the continuous phase from the dispersed second phase

T e , T i

exit and inlet temperature of fluid

T f , ave

average fluid temperature

TI

turbulence intensity

T ¯ OM

overall mean

T s

average temperature of wall surface

t

time

VF ¯ i

average HTE value at optimum levels for volume fraction

ⱴ

fluid velocity

Y M

impact of the fluctuating dilatation for compressible turbulence to the mean dissipation rate

y

normal remoteness from the wall at cell centers

μ

viscosity of the fluid

μ t

Eddy viscosity

σ ƙ

turbulent Prandtl numbers according to ƙ

σ ϵ

turbulent Prandtl numbers according to ϵ

ρ

fluid density

ρ ⱴ →

gravitational body force

Γ

blending function

ϵ

rate of dissipation

τ w

wall shear stress

φ

volume fraction

ρ nf

density of NF

ρ bf

density for base fluid

μ nf

viscosity of NFs

μ bf

viscosity of base fluid

μ ¯ HTE

optimal heat transfer efficiency

1 Introduction

Nanofluids (NFs) obtained with nanoparticles (NPs) and base fluids were utilized to increase the heat transmission of the fluid [1,2]. NFs were produced by combining NPs with high thermal conductivity and a working liquid with low thermal conductivity [3]. NPs with high thermal conductivity increase the heat transfer coefficient of NFs [2]. NPs may be grouped under three headings such as metallic, oxide, and carbon [4]. Since each NP has various mechanical, physical, and thermal properties, it can also affect the properties of NFs. Due to these properties of NFs, it may of great potential use in various sectors [5]. Many studies have been reported about various NPs such as Al2O3 [6], TiO2 [7], Cu [8], CuO [9], Cu2O [10], MgO [11], carbon nanotubes [12], multi-wall carbon nanotubes [13], single-walled carbon nanotube [14], ZrO2 [15], SiO2 [16], Si3N4 [3], SiC [17], GO [18], F3O4 [19], Ag [20], Ni [21], SnO2 [22], BN [23], AIN [24], ZnO [25], CB [26], CeO2 [27]. Due to the large number of NPs, NFs have been produced for use in different areas such as nuclear power plant [28], solar energy systems [29], biomedicine [30], heat exchangers [31], jet impingement [32], lubrication machining [33], automotive application [34], electronic cooling systems [35], etc. In addition, different fluids such as water [36], nitride-water [37], engine oil [38], and ethylene glycol [39] have been used as base fluids. Thus, using different base fluids and NPs provide improvements in the thermophysical characteristics of the NFs. Various research works on NFs have been conducted by many scientists in the literature. Among the studies, heat exchangers are particularly prevalent. Calviño et al. [15] stated the heat transfer efficiency of ZrO2 NF using heat exchangers and used four various volume fractions of NPs. Laminar and turbulent flow were chosen for flow. As a result of their study, they found that ZrO2 can have potential effect for heat exchangers. Zhang et al. [40] evaluated the influence of Cu NF in heat exchangers and detected that the increase in Re number and volume fractions of NPs also increased the heat transfer coefficient. Also, they detected that the rise in Re number provides a rise on pressure difference. Fares et al. [41] examined that graphene nanofluids (GNF) NFs on heat exchangers stated a 29% increase in heat transfer coefficient and a 19% increase in average thermal efficiency. They also found that GNF NF has an enhancing impact on thermal characteristic. Fule et al. [42] improved the heat transfer coefficient of heat exchangers by employing CuO NF and observed a 37.3% improvement compared to working fluid, using 0.1% NPs. Said et al. [43] examined CuO NFs using a heat exchanger. They found that NF developed the average heat transfer coefficient by 7% and convective heat transfer by 11.39%. Salameh et al. [44] analyzed the impact of various volume fractions of NPs using NFs including CuO, Al2O3, and TiO2 in heat exchangers. They determined that NFs ensure upper heat transfer coefficient than base fluids. The authors also found that CuO NFs demonstrated the highest performance. Qi et al. [45] determined the impact of TiO2 NF on thermal efficiency and pressure change in heat exchangers and stated that the increase in the volume fraction values of NPs also caused an increase in the heat transfer rate. They observed that NFs exhibited better thermal effectiveness compared to base liquids. Ajeeb et al. [46] stated the heat transfer efficiency using Al2O3 NF in heat exchangers and reached an improvement of 9.1% for Al2O3 in the thermal conductivity. Zheng et al. [47] carried out heat transfer improvement analysis utilizing various NFs in heat exchangers and stated that the most suitable concentration for thermal improvement utilizing CuO NF was 0.5% by weight. Bahiraei and Ahmadi [48] numerically carried out thermohydraulic performance analyses in heat exchangers using alumina-water NFs. They stated that the increase in Re number and concentration play major role on the convective heat transfer and general heat transfer coefficients. In addition to these studies, many studies [49,50,51,52] involving NFs and different methods have been conducted. As can be seen from the mentioned literature summary, there are a lot of studies involving computational fluid dynamics (CFD) approaches. Some of these studies have been confirmed experimentally. However, studies containing statistical and theoretical validations are limited in number. In this study, the impacts of NP type, Volume fraction (VF), and Re number in the heat transfer coefficient, number of transfer units, wall shear stress, and friction factor of NFs were analyzed numerically, statistically, and theoretically.

2 Materials

In the numerical, theoretical, and statistical analyses, three different NPs were chosen to create NFs. Each NP has various properties. Thus, the impact of each NP property in the NFs was also determined. The water in the model was determined as base fluid. The properties such as density, specific heat, thermal conductivity, and viscosity of base fluid (water) were used as 998.2 kg/m3, 4,182 J/kg K, 0.6 W/m K, and 0.001003 kg/m s, respectively [53]. The properties for the NPs utilized are listed in Table 1 [54].

Table 1

Properties of NPs [54]

Properties Symbol Unit Gr Al2O3 Cu
Density ρ kg m−3 2,200 3,700 8,933
Specific heat C p J/kg °C−1 790 880 385
Thermal conductivity k W/mK 5,000 46 401

To analyze the properties of NFs, different volume factions of NPs and base fluid were used. Volume fraction is determined as follows [55]:

(1) φ = V np V bf + V np .

Density of NFs is solved as follows [56]:

(2) ρ nf = ρ bf ( 1 − φ ) + ρ np φ .

The specific heat capacities of NFs may be considered as follows [57]:

(3) C p , nf = ( 1 − φ ) ρbf C p , bf + φ ρnp C p , np ρ nf .

The thermal conductivity of NF depending on working fluid is examined as follows [58,59]:

(4) k nf k bf = ( k np + 2 k bf ) − 2 φ ( k bf − k np ) ( k np + 2 k bf ) + φ ( k bf − k np ) .

The effective viscosity of NFs can be calculated as follows [60]:

(5) μ nf = μ bf 1 ( 1 − φ ) 2.5 .

Properties of NFs in accordance with Eqs. (1)–(5) are defined in Table 2.

Table 2

Properties of NFs

NFs ρ (kg m−3) C p (J kg °C−1) k (W m−1 °C−1) µ (kg/m-s)
0.5% Gr 1004.209 4144.844 0.609042 0.001016
1.0% Gr 1010.218 4108.131 0.618175 0.001029
1.5% Gr 1016.227 4071.851 0.627401 0.001042
0.5% Al2O3 1011.709 4121.620 0.608699 0.001016
1.0% Al2O3 1025.218 4062.831 0.617482 0.001029
1.5% Al2O3 1038.727 4005.572 0.626351 0.001042
0.5% Cu 1037.874 4018.596 0.609005 0.001016
1.0% Cu 1077.548 3867.224 0.618100 0.001029
1.5% Cu 1117.222 3726.603 0.627287 0.001042

As described in Table 2, the increase in the VFs of the NPs provides an increase in the effective density, thermal conductivity, and viscosity of the NFs. However it leads to a decrease in effective specific heat.

3 Numerical analysis

ANSYS Fluent software was utilized for CFD analysis of the heat transfer efficiency of NFs. CFD approach was used for numerical calculations. A tube of 1,000 mm length and 16 mm diameter was used in modeling. The heat flux of 3,500 W/m2K was applied to the tube. In mesh analyses, MultiZone method with uniform surface mesh, Haxa mapped mesh, and free mesh types was selected. In addition, edge sizing, inflation, and face meshing were utilized to perform precise analysis. Absolute criteria for residuals base on continuity, velocity for x, y, z, energy, k, epsilon was selected as 106. Hybrid model was applied as solution initialization. Coupled scheme in solution methods was chosen. For precise analyses, turbulence intensity values were calculated for each Re number. Turbulence intensity may be considered [61]:

(6) T .I = 0.16 Re − 0.125 .

The formulas such as continuity, momentum, stress tensor, and energy in finite element software ANSYS Fluent may be determined, respectively as follows [62]:

(7) ∂ ρ ∂ t + ∇ ( ρ ⱴ → ) = S m ,

(8) ∂ ∂ t ( ρ ⱴ → ) + ∇ ( ρ ⱴ → ⱴ → ) = − ∇ p + ∇ τ ̿ + ρ g → + F → ,

(9) τ ̿ = µ ( ∇ ⱴ → + ∇ ⱴ → T ) − 2 3 ∇ ⱴ → I ,

(10) ∂ ( ρ E ) ∂ t + ∇ · [ ⱴ → ( ρ E + p ) ] = ∇ · k eff ∇ T − ∑ j h j J → j + ( τ ̿ eff · ⱴ → ) + S h .

The k-epsilon realizable and near wall treatment in the viscos model was selected. The considered transport approach for k and ε can be described as follows [62]:

(11) ∂ ∂ t ( ρ ƙ ) + ∂ ∂ x j ( ρ ƙ u j ) = ∂ ∂ x j μ + μ t σ ƙ ∂ ƙ ∂ x j + G ƙ + G b − ρ ϵ − Y M + S ƙ ,

(12) ∂ ∂ t ( ρ ϵ ) + ∂ ∂ x j ( ρ ϵ u j ) = ∂ ∂ x j μ + μ t σ ϵ ∂ ϵ ∂ x j + ρ C 1 S ϵ − ρ C 2 ϵ 2 ƙ + υ ϵ + C 1 ϵ ϵ ƙ C 3 ϵ G b + S ϵ .

The eddy viscosity can be solved as follows [62]:

(13) μ t = ρ C μ ƙ 2 ϵ .

The model constants in ƙ-epsilon model can be solved as follows [62]:

(14) C 1 ϵ = 1.44 , C 2 = 1.9 , σ ƙ = 1.0 , σ ϵ = 1.2 .

Two layer model depending on the enhanced wall treatment can be defined as follows [62]:

(15) Re y ≡ ρ y ƙ µ ,

(16) y ≡ min r w → ϵ Γ w r → − r → w ,

(17) μ t , 2 layer = ρ C m l m ƙ ,

(18) l m = y C l ⁎ ( 1 − e − Re y / A µ ) ,

(19) μ t , enh = λ є μ t ( 1 − λ є ) μ t , 2  layer .

Enhanced wall functions can be determined as follows [62]:

(20) u + = e Γ u lam + + e 1 Γ u turb + ,

(21) Γ = − 0.01 ( y + ) 4 1 + 5 y + ,

(22) d u + d y + = e Γ d u lam + d y + + e 1 Γ d u turb + d y + .

To calculate the heat transfer efficiency of NFs accurately, two-layer model depending on the improved wall treatment was applied. Thus, it was aimed to obtain high precision results in CFD analyses.

4 Statistical analysis

Statistical methods such as Taguchi method and variance analysis were considered to evaluate the impact of decisive factor for various levels on the heat transfer efficiency of the NFs. Nine CFD analyses were performed using different levels of decisive factors such as NP, VF, and Re number. The combination of NP, VF, and Re number according to different levels was determined in accordance with Taguchi L9 orthogonal array with three levels. Decisive factors and different levels are presented in Table 3.

Table 3

Decisive factors at different levels

Decisive factors Symbol Level 1 Level 2 Level 3
Material M Gr Al2O3 Cu
Volume fraction VF 0.5% 1.0% 1.5%
Re number Re 6,000 7,000 8,000

As described in Table 3, the levels of NPs are determined as Gr, Al2O3, and Cu, respectively. VFs of NPs were chosen as 0.5, 1.0, and 1.5%. To decide the effect of the Re number, three different values were selected and considered as 6,000, 7,000, and 8,000, respectively. Signal-to-noise (S/N) ratio analysis was utilized to explain the influence of each decisive factor on the heat transfer efficiency depending on different levels. The results obtained from the CFD data in this analysis were adapted to S/N ratios. Two different characteristics such as “the Larger is Better” and “the Smaller is Better” were utilized in the calculation of S/N ratios. Thus, it was aimed to obtain the highest h, number of transfer units (NTU), and T w values and the lowest f values for NFs. For the maximum h, NTU, and T w values, “the Larger is Better” characteristic was assumed and, “the Smaller is Better” characteristic was considered for the minimum f value. “The Larger is Better” and “the Smaller is Better” characteristics are explained in Eqs. (23) and (24), respectively [63]:

(23) ( S / N ) LB for h , NTU , τw = − 10 . log n − 1 ∑ i = 1 n ( y i 2 ) − 1 ,

(24) ( S / N ) SB for f = − 10 . log n − 1 ∑ i = 1 n ( y i ) 2 .

Minitab R15 program was used for statistical calculation of each characteristic and creation of graphs depending on S/N ratios.

5 Heat transfer efficiency calculation

In mathematical analyses, h, NTU, τ w, and f were chosen as the heat transfer efficiency of NFs. The NFs were passed through a tube exposed to heat flux. The heat flux was kept constant but the levels of NP type, VF, and Re number were changed in each analysis. Thus, h, NTU, τ w, and f values were calculated under constant heat flux.

h for tube can be defined as follows [64]:

(25) h = Q ̇ A sr ( T s − T f , ave ) .

Q ̇ obtained by using working fluid in tube may be evaluated as follows [64]:

(26) Q ̇ = m ̇ c p ( T e − T i ) ,

m ̇ for working fluid may be defined as follows [64]:

(27) m ̇ = ρ ⱴ A c .

NTU for tube can be calculated as follows [64]:

(28) NTU = h A sr m ̇ c p .

τ w for tube can be solved as follows [61]:

(29) τ w = f ρ ⱴ 2 8 .

Re number can be defined as follows [64]:

(30) Re = ρ ⱴ D μ .

6 Results and discussion

The main goal of the research is to observe the impact of NP, VF, and Re number on the heat transfer efficiency of NFs numerically, statistically, and theoretically. Numerical analyses for the heat transfer efficiency were carried out with CFD approach and the numerical data are listed in Table 4.

Table 4

CFD data depending on L9 orthogonal array

Test Decisive factors CFD data
M VF Re h, (W/m2K) NTU, (−) τ w, (Pa) f, (−)
1 Gr 0.5% 6,000 1994.3 0.3158 0.6474 0.0359
2 Gr 1.0% 7,000 2289.7 0.3097 0.8567 0.0342
3 Gr 1.5% 8,000 2586.4 0.3049 1.0966 0.0329
4 Al2O3 0.5% 7,000 2260.0 0.3085 0.8341 0.0342
5 Al2O3 1.0% 8,000 2546.6 0.3047 1.0598 0.0329
6 Al2O3 1.5% 6,000 2021.4 0.3230 0.6583 0.0359
7 Cu 0.5% 8,000 2502.0 0.3065 1.0209 0.0329
8 Cu 1.0% 6,000 1967.9 0.3298 0.6188 0.0359
9 Cu 1.5% 7,000 2233.7 0.3288 0.7945 0.0342
Overall mean values 2266.9 0.3146 0.8430 0.0343

As can be understood from Table 4, the average values obtained for h, NTU, τ w, and f depending on the L9 orthogonal array are 2266.9 W/m2K, 0.3146, 0.8430 Pa, and 0.0343, respectively. The S/N ratios of the CFD results in Table 4 are tabulated in Table 5.

Table 5

S/N ratios for heat transfer efficiency

Test Decisive factors S/N ratios (ꞑ), dB
M VF Re ꞑ for h ꞑ for NTU ꞑ for τ w ꞑ for f
1 Gr 0.5% 6,000 65.9958 −10.0110 −3.7765 28.9092
2 Gr 1.0% 7,000 67.1956 −10.1823 −1.3439 29.3215
3 Gr 1.5% 8,000 68.2539 −10.3167 0.8010 29.6646
4 Al2O3 0.5% 7,000 67.0822 −10.2148 −1.5754 29.3214
5 Al2O3 1.0% 8,000 68.1192 −10.3222 0.5045 29.6647
6 Al2O3 1.5% 6,000 66.1130 −9.8163 −3.6314 28.9092
7 Cu 0.5% 8,000 67.9657 −10.2712 0.1797 29.6642
8 Cu 1.0% 6,000 65.8801 −9.6340 −4.1697 28.9089
9 Cu 1.5% 7,000 66.9805 −9.6607 −1.9981 29.3211

S/N ratios for h, NTU, and τ w were calculated according to “Larger is Better” quality characteristic and for f was solved in accordance with “Smaller is Better” quality characteristic.

6.1 Selection of optimum levels

In this study, it was aimed to obtain maximum h, NTU, and τ w and minimum f values. The optimum levels of NP, VF, and Re number on the heat transfer efficiency of NFs were calculated utilizing average S/N ratios and CFD data corresponding to each level of each decisive factor. Overall S/N ratios and CFD results are stated in Tables 6 and 7, respectively.

Table 6

Response table for h and NTU

Level h (W/m2 K) NTU (−)
S/N ratios Mean values S/N ratios Mean values
M VF Re M VF Re M VF Re M VF Re
1 67.15 67.01 66.00 2290 2252 1995 −10.170 −10.166 −9.820 0.3101 0.3103 0.3229
2 67.10 67.06 67.09 2276 2268 2261 −10.118 −10.046 −10.019 0.3121 0.3147 0.3157
3 66.94 67.12 68.11 2235 2281 2545 −9.855 −9.931 −10.303 0.3217 0.3189 0.3054
Delta 0.21 0.10 2.12 56 28 550 0.315 0.234 0.483 0.0116 0.0086 0.0175
Rank 2 3 1 2 3 1 2 3 1 2 3 1
Table 7

Response table for τ w and f

Level τ w (Pa) f (−)
S/N ratios Mean values S/N ratios Mean values
M VF Re M VF Re M VF Re M VF Re
1 −1.440 −1.724 −3.859 0.8669 0.8341 0.6415 29.3 29.3 28.91 0.0343 0.0343 0.0359
2 −1.567 −1.670 −1.639 0.8507 0.8451 0.8284 29.3 29.3 29.32 0.0343 0.0343 0.0342
3 −1.996 −1.610 0.495 0.8114 0.8498 1.0591 29.3 29.3 29.66 0.0343 0.0343 0.0329
Delta 0.556 0.115 4.354 0.0555 0.0157 0.4176 0 0 0.76 0 0 0.0030
Rank 2 3 1 2 3 1 2.5 2.5 1 2.5 2.5 1

As described from response table for h and NTU, the optimum h were considered as the first level for NP and the third levels for VF and Re number. For NTU, it was calculated as the third levels of NP and VF and the first level of Re number.

From response table for τ w and f, the optimum τ w were obtained using the first level of NP, the third level of VF, and the third level of Re number. It was determined that the third level of the Re number and all levels of NP and VF can be used to calculate the optimum f value. Main effect plots were drawn using average S/N ratios in Tables 6 and 7 to define the influence of each decisive factor on the h, NTU, τ w, and f depending on the different levels. Main effect plots for h and NTU are presented in Figures 1 and 2.

Figure 1 Effects of M, VF, and Re on h.
Figure 1

Effects of M, VF, and Re on h.

Figure 2 Effects of M, VF, and Re on NTU.
Figure 2

Effects of M, VF, and Re on NTU.

Figure 1 shows that the most influential NPs on h were identified as Gr, Al2O3, and Cu, respectively. Increase in the VF of NPs from 0.5 to 1.5% causes an increase in h. This finding was identified by a study by Garud and Lee [65]. In addition, increasing the Re number from 6,000 to 8,000 provides an increase in h. This finding is supported by a previous study [65] and it is stated that increasing Re number also increases h value. As seen in Figure 2, the most effective NPs on NTU were evaluated as Cu, Al2O3, Gr, respectively. An increase in the VF values of NPs from 0.5 to 1.5% causes an increase in NTU, whereas the increase in Re number from 6,000 to 8,000 causes a decrease for NTU. Main effect plots for τ w and f are displayed in Figures 3 and 4.

Figure 3 Effects of M, VF, and Re on τ w.
Figure 3

Effects of M, VF, and Re on τ w.

Figure 4 Effects of M, VF, and Re on f.
Figure 4

Effects of M, VF, and Re on f.

As described from Figure 3, the most dominant NPs on τ w were determined as Gr, Al2O3, and Cu, respectively. Increase in the VF of NPs from 0.5 to 1.5% and Re number from 6,000 to 8,000 causes an increase in τ w. As explained from Figure 4, it was found that NP and VFs did not have any effect on f. This finding can be explained by Eq. (35). According to the Colebrook equation [64], f depends on the relative roughness and Re number. However, increasing the Re number from 6,000 to 8,000 provides a decrease in the f value. A finding is supported by a study and it is stated that an increase in the Re number from 6,000 to 8,000 decreases the f value [65].

6.2 Analysis of variance

ANOVA was implemented according to general approach at 95% confidence interval to decide the percentage contributions and importance levels of NP, VF, and Re number on the h, NTU, τ w, and f. ANOVA data depending on h and NTU are presented in Table 8.

Table 8

ANOVA results for h and NTU

Source h NTU
DF Seq SS Adj MS F P % Impact Seq SS Adj MS F P % Impact
M 2 5,011 2,505 14.74 0.064 1.09 0.000231 0.00012 13.15 0.071 28.04
VF 2 1,216 608 3.58 0.218 0.26 0.000112 0.00006 6.35 0.136 13.54
Re 2 454,669 227,335 1337.36 0.001 98.58 0.000465 0.00023 26.4 0.036 56.30
Error 2 340 170 0.07 0.000018 0.00001 2.13
Total 8 461,236 100 0.000825 100
R-Sq = 99.93% and R-Sq(adj) = 99.71% R-Sq = 97.87% and R-Sq(adj) = 91.47%

According to the F test results for 95% confidence interval in ANOVA, although Re number on h and NTU were determined as significant decisive factors, NP and VF were determined as insignificant parameters. The most effective decisive factors on h were obtained as Re number with 98.58% impact, NP with 1.09% impact, and VF with 0.26% impact, respectively. For NTU, the most powerful decisive factors were decided as Re number with 56.30% impact, NP with 28.04% impact, and VF with 13.54% impact, respectively. ANOVA data for τ w and f are given in Table 9.

Table 9

ANOVA results for τ w and f

Source τ w f
DF Seq SS Adj MS F P % Impact Seq SS Adj MS F P % Impact
M 2 0.00489 0.002445 12.11 0.076 1.82 0 0 ** ** 0
VF 2 0.000387 0.000194 0.96 0.511 0.14 0 0 ** ** 0
Re 2 0.262558 0.131279 650.23 0.002 97.88 0.0000134 0.0000067 ** ** 100
Error 2 0.000404 0.000202 0.15 0 0 0
Total 8 0.268239 100 0.0000134 100
R-Sq = 99.85% and R-Sq(adj) = 99.40% R-Sq = 100.00% and R-Sq(adj) = 100.00%

According to ANOVA data for τ w and f, although the most significant decisive factor on τ w was selected as Re number due to 95% confidence level, NP and VF were determined as insignificant decisive factors. The most dominant decisive factors on τ w were obtained as Re number with 97.88% impact, NP with 1.82% impact, and VF with 0.14% impact, respectively.

7 Statistical validation

The average values obtained from the CFD results corresponding to each level of each decisive factor were used to perform statistical verification of the heat transfer efficiency. These values are given in Tables 6 and 7. The optimum values of h, NTU, and τ w was calculated by employing the optimum levels of important decisive factors. The estimated mean of h, NTU, and τ w can be evaluated as follows [63]:

(31) μ ¯ HTE = M ¯ i + VF ¯ i + Re ¯ i − 2 T ¯ OM .

T ¯ OM represents the overall mean of nine CFD analyses based on the L9 orthogonal array. T ¯ OM for h, NTU, τ w, and f were calculated as 2266.9 W/m2K, 0.3146, 0.8430 Pa, and 0.0343, respectively. M ¯ i denotes the overall data of heat transfer efficiency at the optimum level of NPs. M ¯ i for h, NTU, τ w, and f were found as 2,290, 0.3217, 0.8669, and 0.0343, respectively. VF ¯ i expresses the overall data of heat transfer efficiency at the optimum level of VF. VF ¯ i for h, NTU, τ w, and f were evaluated as 2,281, 0.3189, 0.8498, and 0.0343, respectively. Re ¯ i means the overall data of heat transfer efficiency depending on the optimum level of Re number. Re ¯ i for h, NTU, τ w, and f were evaluated as 2,545, 0.3229, 1.0591, and 0.0329, respectively. Substituting the data of different terms in Eq. (31), estimated heat transfer efficiency was obtained. The comparison of the CFD and predict Taguchi results obtained is presented in Table 10.

Table 10

Comparison of CFD and predict Taguchi data

Results h for M 1VF3Re3 NTU for M 3VF3Re1 τ w for M 1VF3Re3 f for M allVFallRe3
CFD result 2586.4 W/m2 K 0.3377 1.0966 Pa 0.0329
Taguchi result 2582.2 W/m2 K 0.3343 1.0898 Pa 0.0329
Difference 0.2% 1% 0.6% 0%

Table 10 displays that the differences for h, NTU, τ w, and f are 0.2, 1, 0.6, and 0%, respectively. This finding shows that the low difference between the results increases the accuracy of the analysis.

8 Theoretical validation

Theoretical approach was utilized to define the accuracy of CFD results. Heat transfer efficiency of NFs such as h, NTU, τ w, and f were calculated theoretically. Eqs. (32)–(35) were used to calculate h, NTU, and f, whereas Eq. (29) was utilized for τ w.

Nu number depending on the Pr number and heat flux can be described as follows [64]:

(32) Nu ¯ D = 0.023 Re 0.8 Pr 0.4 .

Pr number can be solved as follows [64]:

(33) Pr = μ C p k .

h obtained by considering the Nu number can be explained as follows [64]:

(34) h = k Nu ¯ D D ,

f can be calculated as follows [64]:

(35) 1 f = − 2 log e / D Tube 3.7 + 2.51 Re f .

The results calculated with the theoretical approach are compared with the results obtained depending on CFD analysis and are presented in Table 11.

Table 11

Comparison of CFD and theoretical results

Results h for M 1VF3Re3 NTU for M 3VF3Re1 τ w for M 1VF3Re3 f for M allVFallRe3
CFD result 2586.4 W/m2K 0.3377 1.0966 Pa 0.0329
Theoretical result 2568.1 W/m2K 0.3381 1.0943 Pa 0.0328
% Difference 0.7% 0.1% 0.2% 0.3%

Table 11 shows that the differences for h, NTU, τ w, and f are 0.7, 0.1, 0.2, and 0.3%, respectively. Thus, it was determined that the CFD analysis was performed with high accuracy.

9 Conclusion

In this numerical, theoretical, and statistical research, the impact of NP type, VF, and Re number in the heat transfer efficiency of NFs were analyzed. Heat transfer coefficient, NTU, wall shear stress, and friction factor were considered as the heat transfer efficiency of NFs. ANSYS Fluent software was employed to perform computational fluid dynamics analyses. The test sequences of the analyses were designed according to the Taguchi method and the L9 orthogonal array was utilized. NP type, VF, and Re number were considered as decisive factors. S/N ratio analysis was applied to determine the optimal levels of decisive factors on the heat transfer efficiency of NFs. Significance levels and contribution rates of the decisive factors on the heat transfer efficiency were evaluated utilizing Analysis of Variance. The important findings obtained from the numerical, statistical, and theoretical study are summarized. The most effective NPs on h and τ w were identified as Gr, Al2O3, and Cu, respectively. For NTU, the most effective NPs are found to be Cu, Al2O3, Gr, respectively. However, no effect of NPs on f could be detected. The increasing Re number from 6,000 to 8,000 causes an increase in h and τ w and a decrease in NTU and f. The increasing volume fractions of NPs from 0.5 to 1.5% causes an increase in h, NTU, and τ w, but no effect was observed on f. The optimum h and τ w data were obtained using 1.5%Gr NFs, and Re number of 8,000. However, the optimum NTU was achieved utilizing 1.5%Cu NFs and Re number of 6,000. According to ANOVA, Re number on h, NTU, and τ w was detected as a significant decisive factor. The most effective decisive factors on h were achieved to be Re number with 98.58% impact, NP with 1.09% impact, and volume fraction with 0.26% impact, respectively. The most powerful decisive factors for NTU were obtained as Re number with 56.30% impact, NP with 28.04% impact, and volume fraction with 13.54% impact, respectively. The most dominant decisive factors on τ w were considered as Re number with 97.88% impact, NP with 1.82% impact, and volume fraction with 0.14% impact, respectively. The differences for h, NTU, τ w, and f were found to be 0.2, 1, 0.6, and 0% in statistical verification, whereas these differences were detected as 0.7, 0.1, 0.2, and 0.3% in theoretical verification.

  1. Funding information: The author states no funding involved.

  2. Author contribution: The author has accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The author states no conflict of interest.

  4. Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2024-12-18
Revised: 2025-04-14
Accepted: 2025-05-07
Published Online: 2025-05-27

© 2025 the author(s), published by De Gruyter

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

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  63. Hydraulic fracturing radioactive source tracing technology based on hydraulic fracturing tracing mechanics model
  64. Numerical solution and stability analysis of non-Newtonian hybrid nanofluid flow subject to exponential heat source/sink over a Riga sheet
  65. Numerical investigation of mixed convection and viscous dissipation in couple stress nanofluid flow: A merged Adomian decomposition method and Mohand transform
  66. Effectual quintic B-spline functions for solving the time fractional coupled Boussinesq–Burgers equation arising in shallow water waves
  67. Analysis of MHD hybrid nanofluid flow over cone and wedge with exponential and thermal heat source and activation energy
  68. Solitons and travelling waves structure for M-fractional Kairat-II equation using three explicit methods
  69. Impact of nanoparticle shapes on the heat transfer properties of Cu and CuO nanofluids flowing over a stretching surface with slip effects: A computational study
  70. Computational simulation of heat transfer and nanofluid flow for two-sided lid-driven square cavity under the influence of magnetic field
  71. Irreversibility analysis of a bioconvective two-phase nanofluid in a Maxwell (non-Newtonian) flow induced by a rotating disk with thermal radiation
  72. Hydrodynamic and sensitivity analysis of a polymeric calendering process for non-Newtonian fluids with temperature-dependent viscosity
  73. Exploring the peakon solitons molecules and solitary wave structure to the nonlinear damped Kortewege–de Vries equation through efficient technique
  74. Modeling and heat transfer analysis of magnetized hybrid micropolar blood-based nanofluid flow in Darcy–Forchheimer porous stenosis narrow arteries
  75. Activation energy and cross-diffusion effects on 3D rotating nanofluid flow in a Darcy–Forchheimer porous medium with radiation and convective heating
  76. Insights into chemical reactions occurring in generalized nanomaterials due to spinning surface with melting constraints
  77. Review Article
  78. Examination of the gamma radiation shielding properties of different clay and sand materials in the Adrar region
  79. Special Issue on Fundamental Physics from Atoms to Cosmos - Part II
  80. Possible explanation for the neutron lifetime puzzle
  81. Special Issue on Nanomaterial utilization and structural optimization - Part III
  82. Numerical investigation on fluid-thermal-electric performance of a thermoelectric-integrated helically coiled tube heat exchanger for coal mine air cooling
  83. Special Issue on Nonlinear Dynamics and Chaos in Physical Systems
  84. Analysis of the fractional relativistic isothermal gas sphere with application to neutron stars
  85. Abundant wave symmetries in the (3+1)-dimensional Chafee–Infante equation through the Hirota bilinear transformation technique
  86. Successive midpoint method for fractional differential equations with nonlocal kernels: Error analysis, stability, and applications
https://doi.org/10.1515/phys-2025-0158
Schlagwörter für diesen Artikel
ANOVA; fluid dynamics; heat transfer efficiency; nanoparticle type; Taguchi approach
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Artikel in diesem Heft

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  2. Single-step fabrication of Ag2S/poly-2-mercaptoaniline nanoribbon photocathodes for green hydrogen generation from artificial and natural red-sea water
  3. Abundant new interaction solutions and nonlinear dynamics for the (3+1)-dimensional Hirota–Satsuma–Ito-like equation
  4. A novel gold and SiO2 material based planar 5-element high HPBW end-fire antenna array for 300 GHz applications
  5. Explicit exact solutions and bifurcation analysis for the mZK equation with truncated M-fractional derivatives utilizing two reliable methods
  6. Optical and laser damage resistance: Role of periodic cylindrical surfaces
  7. Numerical study of flow and heat transfer in the air-side metal foam partially filled channels of panel-type radiator under forced convection
  8. Water-based hybrid nanofluid flow containing CNT nanoparticles over an extending surface with velocity slips, thermal convective, and zero-mass flux conditions
  9. Dynamical wave structures for some diffusion--reaction equations with quadratic and quartic nonlinearities
  10. Solving an isotropic grey matter tumour model via a heat transfer equation
  11. Study on the penetration protection of a fiber-reinforced composite structure with CNTs/GFP clip STF/3DKevlar
  12. Influence of Hall current and acoustic pressure on nanostructured DPL thermoelastic plates under ramp heating in a double-temperature model
  13. Applications of the Belousov–Zhabotinsky reaction–diffusion system: Analytical and numerical approaches
  14. AC electroosmotic flow of Maxwell fluid in a pH-regulated parallel-plate silica nanochannel
  15. Interpreting optical effects with relativistic transformations adopting one-way synchronization to conserve simultaneity and space–time continuity
  16. Modeling and analysis of quantum communication channel in airborne platforms with boundary layer effects
  17. Theoretical and numerical investigation of a memristor system with a piecewise memductance under fractal–fractional derivatives
  18. Tuning the structure and electro-optical properties of α-Cr2O3 films by heat treatment/La doping for optoelectronic applications
  19. High-speed multi-spectral explosion temperature measurement using golden-section accelerated Pearson correlation algorithm
  20. Dynamic behavior and modulation instability of the generalized coupled fractional nonlinear Helmholtz equation with cubic–quintic term
  21. Study on the duration of laser-induced air plasma flash near thin film surface
  22. Exploring the dynamics of fractional-order nonlinear dispersive wave system through homotopy technique
  23. The mechanism of carbon monoxide fluorescence inside a femtosecond laser-induced plasma
  24. Numerical solution of a nonconstant coefficient advection diffusion equation in an irregular domain and analyses of numerical dispersion and dissipation
  25. Numerical examination of the chemically reactive MHD flow of hybrid nanofluids over a two-dimensional stretching surface with the Cattaneo–Christov model and slip conditions
  26. Impacts of sinusoidal heat flux and embraced heated rectangular cavity on natural convection within a square enclosure partially filled with porous medium and Casson-hybrid nanofluid
  27. Stability analysis of unsteady ternary nanofluid flow past a stretching/shrinking wedge
  28. Solitonic wave solutions of a Hamiltonian nonlinear atom chain model through the Hirota bilinear transformation method
  29. Bilinear form and soltion solutions for (3+1)-dimensional negative-order KdV-CBS equation
  30. Solitary chirp pulses and soliton control for variable coefficients cubic–quintic nonlinear Schrödinger equation in nonuniform management system
  31. Influence of decaying heat source and temperature-dependent thermal conductivity on photo-hydro-elasto semiconductor media
  32. Dissipative disorder optimization in the radiative thin film flow of partially ionized non-Newtonian hybrid nanofluid with second-order slip condition
  33. Bifurcation, chaotic behavior, and traveling wave solutions for the fractional (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili model
  34. New investigation on soliton solutions of two nonlinear PDEs in mathematical physics with a dynamical property: Bifurcation analysis
  35. Mathematical analysis of nanoparticle type and volume fraction on heat transfer efficiency of nanofluids
  36. Creation of single-wing Lorenz-like attractors via a ten-ninths-degree term
  37. Optical soliton solutions, bifurcation analysis, chaotic behaviors of nonlinear Schrödinger equation and modulation instability in optical fiber
  38. Chaotic dynamics and some solutions for the (n + 1)-dimensional modified Zakharov–Kuznetsov equation in plasma physics
  39. Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equation
  40. Single-step fabrication of Mn(iv) oxide-Mn(ii) sulfide/poly-2-mercaptoaniline porous network nanocomposite for pseudo-supercapacitors and charge storage
  41. Novel constructed dynamical analytical solutions and conserved quantities of the new (2+1)-dimensional KdV model describing acoustic wave propagation
  42. Tavis–Cummings model in the presence of a deformed field and time-dependent coupling
  43. Spinning dynamics of stress-dependent viscosity of generalized Cross-nonlinear materials affected by gravitationally swirling disk
  44. Design and prediction of high optical density photovoltaic polymers using machine learning-DFT studies
  45. Robust control and preservation of quantum steering, nonlocality, and coherence in open atomic systems
  46. Coating thickness and process efficiency of reverse roll coating using a magnetized hybrid nanomaterial flow
  47. Dynamic analysis, circuit realization, and its synchronization of a new chaotic hyperjerk system
  48. Decoherence of steerability and coherence dynamics induced by nonlinear qubit–cavity interactions
  49. Finite element analysis of turbulent thermal enhancement in grooved channels with flat- and plus-shaped fins
  50. Modulational instability and associated ion-acoustic modulated envelope solitons in a quantum plasma having ion beams
  51. Statistical inference of constant-stress partially accelerated life tests under type II generalized hybrid censored data from Burr III distribution
  52. On solutions of the Dirac equation for 1D hydrogenic atoms or ions
  53. Entropy optimization for chemically reactive magnetized unsteady thin film hybrid nanofluid flow on inclined surface subject to nonlinear mixed convection and variable temperature
  54. Stability analysis, circuit simulation, and color image encryption of a novel four-dimensional hyperchaotic model with hidden and self-excited attractors
  55. A high-accuracy exponential time integration scheme for the Darcy–Forchheimer Williamson fluid flow with temperature-dependent conductivity
  56. Novel analysis of fractional regularized long-wave equation in plasma dynamics
  57. Development of a photoelectrode based on a bismuth(iii) oxyiodide/intercalated iodide-poly(1H-pyrrole) rough spherical nanocomposite for green hydrogen generation
  58. Investigation of solar radiation effects on the energy performance of the (Al2O3–CuO–Cu)/H2O ternary nanofluidic system through a convectively heated cylinder
  59. Quantum resources for a system of two atoms interacting with a deformed field in the presence of intensity-dependent coupling
  60. Studying bifurcations and chaotic dynamics in the generalized hyperelastic-rod wave equation through Hamiltonian mechanics
  61. A new numerical technique for the solution of time-fractional nonlinear Klein–Gordon equation involving Atangana–Baleanu derivative using cubic B-spline functions
  62. Interaction solutions of high-order breathers and lumps for a (3+1)-dimensional conformable fractional potential-YTSF-like model
  63. Hydraulic fracturing radioactive source tracing technology based on hydraulic fracturing tracing mechanics model
  64. Numerical solution and stability analysis of non-Newtonian hybrid nanofluid flow subject to exponential heat source/sink over a Riga sheet
  65. Numerical investigation of mixed convection and viscous dissipation in couple stress nanofluid flow: A merged Adomian decomposition method and Mohand transform
  66. Effectual quintic B-spline functions for solving the time fractional coupled Boussinesq–Burgers equation arising in shallow water waves
  67. Analysis of MHD hybrid nanofluid flow over cone and wedge with exponential and thermal heat source and activation energy
  68. Solitons and travelling waves structure for M-fractional Kairat-II equation using three explicit methods
  69. Impact of nanoparticle shapes on the heat transfer properties of Cu and CuO nanofluids flowing over a stretching surface with slip effects: A computational study
  70. Computational simulation of heat transfer and nanofluid flow for two-sided lid-driven square cavity under the influence of magnetic field
  71. Irreversibility analysis of a bioconvective two-phase nanofluid in a Maxwell (non-Newtonian) flow induced by a rotating disk with thermal radiation
  72. Hydrodynamic and sensitivity analysis of a polymeric calendering process for non-Newtonian fluids with temperature-dependent viscosity
  73. Exploring the peakon solitons molecules and solitary wave structure to the nonlinear damped Kortewege–de Vries equation through efficient technique
  74. Modeling and heat transfer analysis of magnetized hybrid micropolar blood-based nanofluid flow in Darcy–Forchheimer porous stenosis narrow arteries
  75. Activation energy and cross-diffusion effects on 3D rotating nanofluid flow in a Darcy–Forchheimer porous medium with radiation and convective heating
  76. Insights into chemical reactions occurring in generalized nanomaterials due to spinning surface with melting constraints
  77. Review Article
  78. Examination of the gamma radiation shielding properties of different clay and sand materials in the Adrar region
  79. Special Issue on Fundamental Physics from Atoms to Cosmos - Part II
  80. Possible explanation for the neutron lifetime puzzle
  81. Special Issue on Nanomaterial utilization and structural optimization - Part III
  82. Numerical investigation on fluid-thermal-electric performance of a thermoelectric-integrated helically coiled tube heat exchanger for coal mine air cooling
  83. Special Issue on Nonlinear Dynamics and Chaos in Physical Systems
  84. Analysis of the fractional relativistic isothermal gas sphere with application to neutron stars
  85. Abundant wave symmetries in the (3+1)-dimensional Chafee–Infante equation through the Hirota bilinear transformation technique
  86. Successive midpoint method for fractional differential equations with nonlocal kernels: Error analysis, stability, and applications
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Heruntergeladen am 9.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/phys-2025-0158/html
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