Introduction

Nanotechnology is the use of scientific knowledge in the manipulation and control of matter, often in the nanoscale to exploit phenomena and properties dependent on structure and size. These distinct properties are the properties of individual atoms and molecules and cannot be inferred from the mass form of the same substance [1,2,3,4,5,6,7]. In recent years, it is necessary to manage energy and use new resources because of increasing demand and energy cost [8,9,10,11,12,13,14,15,16]. On the other hand, cooling and heating the energy systems are an integral part of this process [17,18,19,20,21,22,23,24]. Thus, the HT and these systems efficiency could be increased by choosing an appropriate method [25,26,27,28,29,30,31,32,33,34,35,36]. Common liquids such as water, EG, or oil are used to cool and lubrication in conventional systems. However, regarding their low thermal performance, the researchers have suggested adding the nano-sized solid particles known as nanoparticles [37,38,39,40,41,42,43,44] like metals, nitrides, carbon nanotubes and metal oxides to the based liquids like water, EG, or oils that cause better and faster heat transfer than the systems [45,46,47,48,49,50,51,52,53,54,55]. The name of the prepared suspension is NF [56,57,58,59,60,61,62,63,64,65,66] and the preparation process is shown in Fig. 1.

Fig. 1
figure 1

Different stages of NF preparation

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Main function in oil is preventing of friction, but it also has other functions such as sealing, preventing corrosion and cleaning the engine. In addition, the oils are also used to cool the internal parts of the engine [67,68,69,70,71,72,73,74,75,76]. Some other applications of NFs have been displayed in Fig. 2. Thermophysical properties of NF are different from the base fluid [77,78,79,80,81,82,83,84,85,86]. Among the important thermophysical properties, viscosity [87,88,89,90,91,92,93,94], thermal conductivity [95,96,97,98,99,100,101], density [102,103,104] and specific heat capacity [105,106,107,108,109,110] can be mentioned. Knowing the properties of nanofluid will make it used in the right place and it will be planned according to the need and application. Viscosity is defined as the resistance of a fluid to flow. Viscosity is an important parameter for oil pumping. Many scientists examined the adding NPs effect to based fluid. They found that the factors affect the viscosity [111,112,113,114,115,116,117,118,119,120,121,122].

Fig. 2
figure 2

NFs applications in industries

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Specifying rheological behavior of the NF causes to prevent possible damages to the systems such as the car engine [123]. For example, if the NF reveals a non-Newtonian behavior [124,125,126,127,128], the NF has much yield stress because a large amount of yield stress causes to damage the engine parts at the start. The studies about the behavior of the NF can be observed in Table 1.

Table 1 Studies in rheological behavior field of NFs
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Nevertheless, the cost of producing and testing the NFs is relatively high. To reduce costs, modeling can be used to determine the properties of the NFs [133,134,135,136]. In this regard, some scientists like Einstein [137] gave a theoretical model for of relative viscosity prediction of NFs: Einstein’s model is applied for SVF < 0.02 and the sphericity hypothesis of suspended solid particles in based fluid.

$$\frac{{\mu }_{nf}}{{\mu }_{bf}}=(1+2.5SVF)$$
(1)

Wang et al. [138] proposed relative viscosity model that is used according to physical state phases:

$$\frac{{\mu }_{nf}}{{\mu }_{bf}}=1+7.3SVF+123{SVF}^{2}$$
(2)

Moreover, Bruijn [139] proposed a model to predict \(\mu_{r}\):

$$\frac{{\mu }_{nf}}{{\mu }_{bf}}=1-2.5SVF+1.552{SVF}^{2}$$
(3)

But these models are not able for of new generation of the NFs i.e., the hybrid NFs [140,141,142]. In recent years, some scientists presented models for predicting hybrid NFs viscosity to solve this problem [143,144,145,146,147,148,149,150]. Hemmat Esfe et al. [151, 152] have performed many experiments and modeling in this field. According to the importance of NF viscosity in the industry, Asadi and Asadi [153] studied ZnO/MWCNT NF properties of engine oil in based fluid. Their results showed a decrease of 58% in viscosity with rise in T and an increase of 45% with an increase in the SVF. They also found that the increment of the SVF at low temperatures causes a further increase the viscosity. In this study, rheological behavior of MWCNT and ZnO (30:70) with10W40 hybrid NF has been investigated. First, n index of NF was measured and it was determined that NF is a non-Newtonian and pseudo-plastic type of fluid. Relative viscosity and optimal viscosity at different temperatures and SVFs were investigated and compared with other NFs in terms of preparation costs and quality of viscosity. With the help of RSM, a model was presented to predict NF relative viscosity based on SR, SVF and T. Then, the MOD and sensitivity analysis were measured at T = 55 °C, SVF = 1%. The use of RSM improves the accuracy and speed of achieving the result and avoids the costs of preparation of NFs.

Experimentation

Nanofluid Preparation

MWCNT-ZnO NF was prepared with a volume ratio (30:70) in based fluid 10W40 with the SVF = 0.05%, 0.1%, 0.25%, 0.5%, 0.75% and 1%. The used NPs are visible in Fig. 3. Then, of NPs volumetric concentrations were computed using Eq. 4 and suspended by a 2-step method in based fluid 10W40. The prepared NFs are displayed in Fig. 3.

Fig. 3
figure 3

Applied NPs and prepared NFs in various SVFs

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$${\text{SVF}}(\mathrm{\%})=\frac{\left[\frac{{W}_{MWCNT}}{{\rho }_{MWCNT}}\right]+\left[\frac{{W}_{ZnO}}{{\rho }_{ZnO}}\right]}{\left[\frac{{W}_{MWCNT}}{{\rho }_{MWCNT}}\right]+\left[\frac{{W}_{ZnO}}{{\rho }_{ZnO}}\right]+\left[\frac{{W}_{10W40}}{{\rho }_{10W40}}\right]}\times 10$$
(4)

Results and Discussion

NF Behavior

Rheological behavior of HNF MWCNT-ZnO (30:70) based on 10W40 was researched at T = 5–55 °C and SVF = 0.05–1%. Non-Newtonian properties could be represented by processing the data and using Eq. 5 [154]:

$${\tau }_{OW}=m{SR}^{n}$$
(5)

Figure 4 shows n index in SVF and T. NF behavior is similar to non-Newtonian by increasing T.

Fig. 4
figure 4

n index for MWCNT and ZnO (30:70)/10W40 NF

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Figure 5 shows viscosity versus SR at different SVFs in comparison with the SR. The measurements indicate the dependence of the NF viscosity on SR. The increment of SS is associated with a decrease in viscosity. Given that a rise in the temperature causes to accelerate the movement of the NF particles and reduces the intermolecular force.   Figure 6 shows reduction of NF viscosity with T rise. T and SVF effects on relative viscosity are displayed in  Fig. 7. At higher SVF, maximum amount happens. The relative viscosity changed a little by increasing T because base fluid viscosity also increased proportionally.

Fig. 5
figure 5

Viscosity versus SR at various SVFs

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Fig. 6
figure 6

Viscosity versus temperature at SR = 6665 (1/s)

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Fig. 7
figure 7

Relative viscosity versus T and SVFs at SR = 6665 (1/s)

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Figure 8 shows NF viscosity percentage increment in comparison with SVF at various temperatures. The NF viscosity is decreased slightly by adding small amounts of the NPs to the base fluid. The small amounts of NPs cause the NF layers to displace on each other easily, but the increment of the NP amount causes to increase in the number of collisions, hence the layers move hard. Ultimately, the increment of the collisions causes to increase in the NF viscosity.

Fig. 8
figure 8

Viscosity enhancement versus T and SVFs at SR = 3999 (1/s)

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Compare With Other NF

Presence of four types of NPs has been investigated in the same base fluid 10W40. In this section, the experiment results of MWCNT (30%)-ZnO(70%)/10W40 have been compared with the results of other researches of Hemmat Esfe et al., MWCNT-SiO2(10:90)/10W40 NF [151], MWCNT-CuO(15:85)/10W40 NF [152] and MWCNT-TiO2(55:45%)/10W40 NF [155] at Ts and SVFs. The compared results in Fig. 9 display that NF MWCNT (% 15) -CuO (85%)/10W40 has a lower viscosity than three other NFs at lower SVFs. Moreover, the MWCNT-ZnO (30:70)/10W40 has the lowest viscosity at higher SVFs than other NFs. The optimal NF can be selected according to the use of the NF in Fig. 9 different parts.

Fig. 9
figure 9

Viscosity versus SVF at T = 15 ℃ and SR = 6665 (1/s)

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In this section, viscosity increment percentage and the cost of using the NF MWCNT (30%)-ZnO (70%)/10W40 have been compared with other studies results of Hemmat Esfe et al., MWCNT-SiO2 (10:90)/10W40 NF [151], MWCNT-CuO(15:85)/10W40 NF [152] and MWCNT-TiO2 (55:45)/10W40 NF [155].   Figure 10 shows the lowest cost for preparing the MWCNT-SiO2 (10:90)/10W40 NF. It can be said that it is the best choice according to the approximately equal increment rate of the studied NFs.

Fig. 10
figure 10

Comparison between viscosity enhancement and price of the various hybrid NFs

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Correlation

Considering the inability for predicting of MWCNT(30%)-ZnO(70%)/10W40 NF viscosity by using available theoretical models and also the expensiveness of conducting experimental tests, nanofluid viscosity modeling could be suitable way for reducing of costs, high accuracy and increase the speed of obtaining the answer. In this section by RSM, a model has been presented that can predict the relative viscosity at the temperature range and the studied SVF. In the modeling, stepwise method has been selected for the regression analysis. The independent parameters have been added to equation one by one. If an independent parameters doesn’t have a tangible effect on dependent variable during calculations, it will be eliminated from analysis and will not be included in equation. Regression coefficient (Eq. 6) is 0.9529 and model has good accuracy.

$$\frac{{\mu }_{nf}}{{\mu }_{bf}}=0.98+0.71SVF-0.0025T-2.4{\times 10}^{-6}SR+5.69{\times 10}^{-6}SVF*SR-1.21{SVF}^{2}+5.35{\times 10}^{-5}{T}^{2}-5.14{\times 10}^{-6}{SVF}^{2}SR+0.71{SVF}^{3}$$
(6)

The validity and reliability of regression models have been researched using Analysis of Variance (ANOVA). Table 2 displays each parameter in Eq. 6. Table 2 shows the importance of each parameter in Eq. 6. Regression model significance is displayed by high F (172.0563). Prob > F-values less than 0.05 indicate statistical significance of model parameters. Extra model parameters effects are shown greater than 0.05 with Prob > F-values. All irrelevant parameters of model are removed.

Table 2 Presented correlation withANOVA
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Presented model results comparison with experimental results is shown in  Fig. 11. There is an appropriate match between experimental results and prediction, which indicates high accuracy of the model. Thus, we can rely on the results obtained from the model. The MOD versus the SVF at different temperatures and for the two SRs is presented in  Fig. 12. The maximum error of 5% indicates high accuracy. Equation 7 is used to get a MOD (%),

Fig. 11
figure 11

Matching experimental and theoretical results

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Fig. 12
figure 12

MOD versus temperature at SR = 3999 and 6665 (1/s)

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$$MOD \left(\%\right)=\frac{100}{N}\sum_{i=1}^{N}\left|\frac{{\left(\frac{{\mu }_{nf}}{{\mu }_{bf}}\right)}_{Pred}-{\left(\frac{{\mu }_{nf}}{{\mu }_{f}}\right)}_{Exp}}{{\left(\frac{{\mu }_{nf}}{{\mu }_{f}}\right)}_{Exp}}\right|$$
(7)

Sensitivity Analysis

To examine the independent parameters of the temperature and the SVF of the NPs, sensitivity analysis has been used to determine each parameter effect separately. The sensitivity percentage of each parameter on the temperature range and the SVF was obtained by varying ± 5%, ± 10% ± 15%, and ± 20% separately. Equation 8 is used to get the sensitivity percentage.

$$\text{Sensitivity \%}=\left[\frac{\frac{{\mu }_{nf}}{{\mu }_{bf}}\left( SVF=1\%\pm 5\%,\pm 10\%\pm 15\%,\pm 20\%;T=55^\circ C\pm 5\%,\pm 10\%\pm 15\%,\pm 20\%\right)}{\frac{{\mu }_{nf}}{{\mu }_{bf}}\left( SVF=1\%,T=55^\circ C\right)}-1\right] \times 100$$
(8)

Figure 13 shows that SVF effect on NF relative viscosity is higher than T effect according to sensitivity analysis results.

Fig. 13
figure 13

Sensitivity-SVF

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Conclusion

MWCNT -ZnO(30:70)/10W40 NF was investigated with the SVF = 0.05% to 1% and at T = 5–55 °C. The NF behavior was specified as pseudo-plastic type and a non-Newtonian. The following results were obtained by further reviews:

  • At higher (SVFs), NF relative viscosity is much higher because the collisions are increased at higher SVFs of the NPs.

  • The NF viscosity is decreased slightly with small amount adding of NPs to based fluid because low amount of the NPs causes NF layers to displace on each other easily. However, the increment of the NPs causes the layers to move hard; hence, the NF viscosity is increased.

  • The results of the comparison between the different NFs show that the MWCNT-CuO (15:85)/10W40 NF has lower viscosity at lower SVFs than the other three NFs. Moreover, the MWCNT (%30)-ZnO (% 70)/10W40 NF has the lowest viscosity at higher SVFs than other NFs. The optimal NF can be selected according to the use of the NF in different parts.

  • Regarding the comparison of the costs for preparing the different NFs, we concluded that the minimum cost of the preparation is related to the MWCNT(10%)-SiO2 (90%)/10W40 NF. It can be said that it is the best choice according to the approximately equal increment rate of the studied NFs.

  • A model with a maximum error of 5% was presented by Response surface methodology; this shows the high accuracy of the proposed model.

  • NF sensitivity analysis indicated that SVF effect on relative viscosity is higher than of T effect.