Prandtl Number and Viscosity Correlations of Titanium Oxide Nanofluids

. Many features of nanofuids, such as the Prandtl number and viscosity, are researched as the number of studies conducted in the feld of nanofuids increases. Observations on the Prandtl number and viscosity of titanium oxide nanofuids are made in this study. Tese observations are made at low concentrations of titanium oxide nanoparticles and temperatures ranging from 30.4 ° C to 70.4 ° C. Novel correlations for viscosity and Prandtl number as functions of temperature have been developed and compared to the previously published models for Prandtl number and viscosity. Te results indicate that titanium oxide-ethylene glycol nanofuid has a greater viscosity and Prandtl number than all other titanium oxide nanofuids observed in the study at 0.01 nanoparticle concentration. Te results on viscosity and Prandtl number for the new correlations fall within the same range as those found in the literature, indicating that the new correlations introduced as functions of temperature in this study can be used in future research to establish viscosity and Prandtl number calculations for the diferent types of nanofuids at specifc temperatures.


Introduction
As nanoparticles are added to base fuids such as water, glycols, oils, and refrigerants to generate nanofuids, the thermophysical properties of the resulting fuids improve. Tese nanofuids are utilized in several industrial applications. Te research topic on nanofuids has advanced signifcantly as scientists have been interested in the manufacture of these specialized fuids that contain nanoparticles. As it was discovered that nanofuids possess increased thermophysical characteristics, curiosity emerged. Various nanoparticles have been studied to demonstrate the validity of the tested hypotheses. In this article, the nanofuid viscosity and Prandtl number of titanium oxide-ethylene glycol (40%)/water (60%), titanium oxide-ethylene glycol nanofuid viscosity and Prandtl number, and titanium oxide-water nanofuid viscosity and Prandtl number are analysed. Observations are also made about the viscosity and Prandtl numbers on their respective base fuids, specifcally the enhancement of viscosity and Prandtl number at varied nanoparticle concentrations of 0.004, 0.006, 0.008, and 0.01. It is vital to evaluate how high or low the Prandtl number can be, as a greater Prandtl number indicates a higher viscosity of the nanofuid, while a lower Prandtl number indicates a lower viscosity of the nanofuid. Te titanium oxide nanofuids, at the measured nanoparticle concentration, have the potential for usage in numerous applications due to a minor increase in viscosity for the selected nanoparticle concentrations. Te Prandtl number depends on the nanofuid's specifc heat, viscosity, and thermal conductivity. In addition to detecting the Prandtl number and viscosity of nanofuids using the aforementioned published models, we examine new correlations for Prandtl number and viscosity and compare them to previously published models. It is important to evaluate the efects of temperature on the thermophysical properties of nanofuids, as determined by a thorough analysis of the relevant literature. Te correct analysis of the thermophysical and rheological properties that nanofuids should possess when utilized in a variety of industrial applications necessitates the development of novel correlations that involve temperature. Temperature plays a signifcant role in the thermophysical characteristics of nanofuids. Nanofuids are analysed at a variety of temperatures and nanoparticle concentrations to identify the optimal outcomes. Numerous experiments have been conducted in which researchers have been able to observe these fndings measured at specifc temperatures using measuring devices or theoretical models; however, there is a gap in the literature where more correlations, particularly correlations as functions of temperature, are required to analyse the nanofuids at specifc temperatures using correlations. Hence, theoretical models and experimental investigations may be predicted with more ease, and more precise results can be acquired. Tis study provides more recent relationships between the Prandtl number and viscosity.

Literature Review
Viscosity and the Prandtl number were examined by several experts in their respective felds, and their work and fndings are discussed in Table 1.

Data Reduction
Te Prandtl number depends on the nanofuid's thermal conductivity, specifc heat, and viscosity. Equation (1a) depicts the formula used by Tiandho et al. [21] and other researchers to analyse the nanofuid Prandtl number in scientifc literature.
For nanofuid computations, equation (1a) can be rewritten as shown in the following equation: where P r,nf , μ nf , C p,nf , and k nf are the nanofuid Prandtl number, nanofuid viscosity, and nanofuid specifc heat, respectively. In this study, the Prandtl number and viscosity of three nanofuids are measured at temperatures ranging from 30.4°C to 70.4°C. Te titanium oxide nanoparticles are combined with ethylene glycol (40%)/water (60%), ethylene glycol, and water as the basis of fuids. Equation (2) of the Einstein model is the most popular model for calculating nanofuid viscosity. Manikandan and Baskar [16] were among the researchers who utilised this model.
where μ nf , μ bf , and ∅ indicate the viscosity of the nanofuid and the base fuid, respectively, and ∅ represents the nanoparticle concentration. Te thermophysical properties are observed using ASHRAE (2017) Handbook-Fundamental (SI).

Viscosity and Prandtl Number's Base Fluid Properties and
Correlations. Prior to introducing the new nanofuid correlations for Prandtl number and viscosity, it is crucial to have examined the base fuid properties for the nanofuid analysis, as base fuids form the basis of the new correlations in this study. Te base fuids are used to compare the fndings obtained using both the theoretical models and the novel correlations to determine the diference between utilizing nanoparticles to enhance thermophysical properties and not using them. Table 2 provides an examination of the base fuids. Te correlations of base fuids are explored at temperatures ranging from 30.4°C to 70.4°C. where P r,nf ,μ bf, ρ p , k p ,C p,bf ,k bf ,∅,C p,p ,ρ bf , and T represent the nanofuid Prandtl number, base fuid viscosity, nanoparticle density, nanoparticle thermal conductivity, base fuid specifc heat, base fuid thermal conductivity, nanoparticle concentration, nanoparticle specifc heat, base fuid density, and temperature, respectively. Tis new association between the Prandtl number and viscosity is applicable to the research of numerous nanofuids that require examination. Zargartalebi et al. [10] When the Prandtl number increased, a reduction in the thickness of the thermal boundary layer was noticed [10] Nasrin [11] Water Aluminum oxide 2% Te rise in the Prandtl number enhanced the heat transfer rate, but it decreased the collector's efciency [11] Nabil et al. [12] Water/ethylene glycol (60 : 40) Titanium oxide-silicon dioxide (50 : 50) 0.5% to 3.0% With an increase in temperature, viscosity was shown to diminish [12] Yu et al. [13] Water MWCNT 0.0047%, 0.023%, 0.0571%, 0.1428%, and 0.2381% Results indicated a small increase in viscosity when temperature rose over the critical threshold [13] Shah et al. [14] Ethylene glycol

Comparison of Theoretical Models with New Correlations for Viscosity and the Prandtl Number
In the analysis, the models described in Sections 3 and 4 are utilized. Results for titanium oxide-ethylene glycol (40%)/ water (60%) nanofuid, titanium oxide-ethylene glycol nanofuid, and titanium oxide-water nanofuid are presented in Section 5. Figure 1 5.1.2. Titanium Oxide-Ethylene Glycol Nanofuid Viscosity Shown in Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 6. Discussion of Results

Prandtl Number.
In accordance with the observations made in Sections 3-5, the Prandtl number for the titanium oxide-ethylene glycol (40%)/water (60%) mixture at various temperatures is 19 at 30.4°C, as shown in Figure 4. Te Prandtl value increases by a small proportion in both theoretical and new correlation results. At 30.4°C, the Prandtl number for titanium oxide-ethylene glycol is shown to range from 137 to 147 in Figure 5. In Figure 6, the Prandtl number for titanium oxide-water nanofuid ranges from 5.50 to 5.79 at 30.40 degrees Celsius. Te Prandtl number of pure ethylene glycol is greater than that of water and ethylene glycol/water mixtures.
Observing the various nanofuids, the Prandtl number increases as 0.004, 0.006, 0.008, and 0.01 nanoparticle concentrations are added to the basic fuids. Yet, as the temperature increases, the Prandtl number similarly falls. In both the theoretical formula for the Prandtl number and the new correlation model, it can be observed that as the viscosity increases, the Prandtl number also increases. As the temperature of the nanofuid increases, the viscosity of the nanofuid lowers, causing the Prandtl number to fall as well.
Te basic fuids have a lower measured Prandtl number than the nanofuids. Compared to ethylene glycol (40%)/water (60%) base fuid and water base fuid, ethylene glycol has a high Prandtl number. Te Prandtl number increases ranging from 1.4% to above 3.45% for the titanium oxideethylene glycol (40%)/water (60%) nanofuid, from 2.4% to 5.8% for the titanium oxide-ethylene glycol nanofuid, and from 1.1% to 2.6% for the titanium oxide-water nanofuid, with reference to the base fuids. Moreover, the Prandtl number determines the thickness of the boundary layer. Te greater the Prandtl number, the greater the nanofuid's high viscosity, which causes nanofuid fow constraints when the nanofuid thickens due to its high viscosity. Kho et al. [22] came to the conclusion that when the Prandtl number increased, the temperature profle continued to decrease.

Viscosity.
When the concentration of nanoparticles increases, the viscosity of titanium oxide nanofuids increases. Considering the previous Sections 3-5, the viscosity increases for all the titanium oxide nanofuids are minimal.
In the fuid analysis, it is essential that the viscosity is kept low. Adding nanoparticle concentrations to nanofuids increases their viscosity. Yet, as demonstrated in Figures 1-3, the viscosity increases slightly when nanoparticle concentrations of 0.004, 0.006, 0.008, and 0.01 are added to a solution. Viscosity increases slightly by fractions of percentage points in all the detected nanofuids, indicating that the necessary results in nanofuid analysis can be obtained. Among the thermophysical parameters of nanofuids, such as thermal conductivity, specifc heat, and density, viscosity increases less than thermal conductivity, specifc heat, and density do at the same nanoparticle concentrations. Maintaining a low viscosity is a desirable outcome in the fuid analysis and makes nanofuids even more suitable for Considering the selected titanium oxide nanoparticle concentrations and how the viscosity increases slightly, titanium oxide nanofuids at lower nanoparticle concentrations could be considered for use in a variety of applications, and careful consideration should also be given to titanium oxide waterbased nanofuid, as the addition of higher nanoparticle    concentrations would afect the decrease percentage of viscosity as shown in this paper. Despite the favourable viscosity of water nanofuids, the addition of nanoparticle concentrations increases the decreasing percentage of the nanofuid relative to the base fuid, as shown in this study.
Tis gives ethylene glycol/water combinations so much promise for usage in industrial applications that the drop in viscosity with increasing temperature for ethylene glycol (40%)/water (60%) nanofuid is identical to the fall in viscosity percentage for ethylene glycol (40%)/water (60%) base  fuid. With the introduction of the new correlation for viscosity, it is confrmed that other studies have seen an increase in viscosity when nanoparticles are added and that the percentage increase in viscosity throughout the nanofuids tested is consistent. It has been demonstrated that a novel correlation as a function of temperature contributes to the research models found in the literature since it produces the desired results. With the results of the new correlation shown in Figures 1-3, we observe a decrease in viscosity, making the usage of this new correlation a viable alternative for analysing viscosity at various temperature ranges. Einstein's model does not include temperature, but with this new correlation, temperature is included, and it has been shown to produce lower viscosity rise fndings when nanoparticles are introduced, making the new correlation preferable for viscosity analysis.

Conclusion
It is always desired that any type of fuid fows freely; therefore, in the observation of the researched study, titanium oxide-ethylene glycol (40%)/water (60%) nanofuid and titanium oxide water nanofuid had the lowest Prandtl number and were preferable to the titanium oxide ethylene glycol nanofuid. In all nanofuids observed, the highest Prandtl number and viscosity were observed at 0.01 nanoparticle concentration and 30.4 degree Celsius. Using the new correlations for Prandtl number and viscosity, this study demonstrated an increase in the Prandtl number and correlated with the previously published data. Te nanofuid research study requires more precise analysis methods, which will aid in the creation of more precise designs for industrial equipment and the rescaling of existing industrial designs. Te additional correlations as functions of temperature augment the literature-based research models that have demonstrated their ability to achieve the required results. With the results of the new correlations shown in Figures 1-6, we observe a decrease in viscosity and Prandtl number with an increase in temperature, making the application of the new correlations a viable alternative for analysing the viscosity at various temperature ranges. Einstein's viscosity model for nanofuid viscosity analysis does not include temperature; with this new correlation for viscosity, temperature is included, as is the case for Prandtl number. With the new correlations introduced in this study, we note that the accuracy of results remains within the same range as when using theoretical models and that the new correlations for Prandtl number and viscosity contribute to the analysis of nanofuids.

Data Availability
Te data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest regarding the publication of this article.