Development of New Correlations and Parametric Optimization in Nanofluid Flow through Protruded Roughened Square Channel

Kumar, Sunil; Gupta, Gaurav; Tchier, Fairouz; Xin, Qin; Kumar, Anil; Shami, Faria Ahmed

doi:https://doi.org/10.1155/2022/1603529

Mathematical Problems in Engineering

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Abstract Introduction Results Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Nanofluids and Entropy Analysis with Electroosmotic Phenomenon

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Research Article | Open Access

Volume 2022 | Article ID 1603529 | https://doi.org/10.1155/2022/1603529

Development of New Correlations and Parametric Optimization in Nanofluid Flow through Protruded Roughened Square Channel

Sunil Kumar,¹Gaurav Gupta,¹Fairouz Tchier,²Qin Xin,³Anil Kumar,⁴and Faria Ahmed Shami⁵

Academic Editor: Mohammadreza Safaei

Received29 Nov 2021

Revised13 Feb 2022

Accepted24 Feb 2022

Published06 Apr 2022

Abstract

This study discusses the optimization of heat-transfer parameters in nanofluid flow through a rough square surface channel including a protruded rib compound transverse pattern using response surface methodology (RSM). Flow and geometrical characteristics are optimized, resulting in optimal flow friction and heat transfer performances. The comparison of RSM’s estimated values to experimentally observed values was abandoned. The results demonstrate that the RSM-calculated values agree with the observed values and are within the 5.5 percent uncertainty limitations. Statistical correlations for Nusselt number and friction factor have been developed as functions of protrusion transverse rib height, protrusion transverse rib diameter, X-axis pitch, Y-axis pitch, and Reynolds number. These correlations have been found to predict the values within the error limits of ±8.9% and ±8.7%, respectively. On the basis of correlations developed for Nusselt number and friction factor, an attempt has been made to compare the thermohydraulic performance of protruded roughened square channel.

1. Introduction

Energy conservancy is one of the indispensable issues of the twenty-first century, and it will absolutely be a challenge among the hugest difficulties sooner rather than later. Thus, researchers, engineers, and scientists are significantly trying to address this imperative concern [1].

Utilization of nanofluids (NFs) is one of the approaches to enhance heat transfer (HT), as traditional fluids, like, engine oil, ethylene glycol, and water have moderately low heat transfer rate (HTR) [2, 3]. Turbulence promoters and numerous geometrical shapes of the rib-groove duct are commonly used, since they cause stream division and reattachment, thus breaking the laminar viscous layer and causing enhanced heat transfer [4, 5].

Al-Shamani et al. [6] explored the turbulent flow of water-based Al₂O₃, SiO₂, and ZnO NFs with 1–4% volumes fractions (φ) and 25–70 nm nanoparticle diameter (dnp) through rib-groove duct. Studies reveal that the SiO₂-H₂O NF has a maximum heat transfer coefficient (HTC) in correlation with the other NFs. Kamali and Ali-Reja [7] and Kumar et al. [8] examined turbulent flow in a square conduit with four ribs’ shapes. The outcomes showed that inter-rib distribution affected HTC. Lauriat [9] numerically and experimentally explored whereas Nguyen et al. [10] explored by experimentation the HT behavior of Al2O3-H₂O NF and found that the HTC increases up to 40% as compared to the base fluid.

Hussein et al. [11] numerically and experimentally investigated water-based NFs (CuO-H₂O, Al₂O₃-H₂O, TiO₂-H₂O) for φ that varied from 0 to 2%. The improvement of HT was about 47.3% (CuO), 49% (Al₂O₃), and 45.2% (TiO₂) over pure water. Ahmed and Yusoff [12] concluded that in comparison to the base fluid, Al₂O₃-H₂O NF showed an improvement of 45.7% in HTC at 3 vol. % and Ren = 16,000. Mahian et al. [13] calculated HTC for φ up to 4% and dnp of 25 nm to four different NFs including CuO-H₂O, Al₂O₃-H₂O, TiO₂-H₂O, and SiO₂-H₂O. Findings indicated that Al₂O₃-H₂O NF had the highest HTC at φ = 0.4%. Wen and Ding [14], and Singh and Sarao [15] experimentally found that with an increase in number such as Reynolds Re_n the Nusselt (Nu) and friction factor (f) of Al₂O₃-H₂O NF, heat exchanger improves over the base fluid H₂O.

Sundar et al. [16] determined convective HTC for a fully developed turbulent NF flow and found a 31.10% increment in Nu at φ = 0.3%. Suresh et al. [17] noticed an improvement of 14.25% in flow friction at φ = 0.1% for Cu-Al₂O₃-H₂O NF flow at a volume flow rate of 2.24 × 10⁻⁵ m³/s. Pak and Cho [18] experimentally observed enhancement in HTC compared to the H₂O for Al₂O₃ and TiO₂ NFs for Ren that ranges from 104 to 105 and φ varies from 0% to 3%. Ho et al. [19] investigated the HTC of Al₂O₃ nanoparticles whose φ varies from 0.1% to 4% in the square duct of various sizes. They proposed a HT correlation showing that improvement in HTC does not exist if φ is larger than 2% and also found around 18% improvement in HTC for the largest area duct.

Li and Peterson [20] experimentally investigated HT characteristics of Al₂O₃-H₂O NF of 25 nm dnp with φ ranging from 0.5% to 6%. A large deterioration percentage in the HTC was observed when φ increases. Ho et al. [21] suggested Nu correlations for the four different models based on two formulas for ≤ φ ≤ 4%, and results confirmed a strong impact about uncertainties related to formulas adopted on the HT characteristics. Correlation formulas for Nu and f given by Sundar et al. [16]; Suresh et al. [17]; Pak and Cho [18]; Maïga et al. [22]; Petukhov [23]; Dittus and Boelter [24]; Duangthongsuk and Wongwises [25]; Madhesh et al. [26]; and Gnielinski [27] are listed in Table 1. Moreover, there are many numerical and experimental studies on heat exchanger using mono and hybrid nanofluids that have also been carried out by different researchers [28–32] and they revealed that the thermal characteristics of nanofluids are found to be higher and better for heat transfer application in comparison to the base fluid.

2. The RSM (Response Surface Methodology)

RSM, an assortment of mathematical/statistical technique, which is helpful for the analysis and modeling of issues, was a reaction of interest that is inspired by design parameters to optimize the boundaries of design on the ideal estimation of the response function, which is the primary goal. Working with the RSM involves the following 3 main steps [33, 34]:(1)Choosing the design or model matrix.(2)Selecting the number of the control factors and responses and accordingly entering the data.(3)Analyzing the responses and conformation of the results.

The optimal set of experimental parameters can be discovered by RSM that delivers a most extreme or least estimation of response and expresses immediate and intelligent impacts of parameters over 2D and 3D plots. The quantitative relation between responses and input variables is translated as follows [33, 34]:where z is the response, whereas are the input variables, and is the fitting error. The approximation projected using the cubic model of is as follows:where is the linear/quadratic and cubic effects of and between and [33, 34].

From literature review, it has been found that there has been no study carried out on thermal performance enhancement of heat exchanger using nanofluid with combined protruded roughened square channel. In other words, the novelty of this study is to study the influence of the combined effect of both nanofluid and protruded roughened square channel. The primary objectives of the current analysis include the following: to optimize the operating parameters, i.e., nanoparticle concentration , nanoparticle diameter , stream-wise pitch , span-wise pitch , ratio of print diameter , and Reynolds number using RSM, and to develop statistical correlations for Nusselt number and friction factor in terms of operating parameters.

3. Experimental Details and Preparation of Nanofluid

The experimental setup used for assessing and characteristics of the nanofluid flow constitutes a square duct having length of 340 mm, width of 10 mm, and height of 10 mm [1]. A schematic diagram of the experimental setup is presented in Figure 1, consisting of a test section with an installed heater, inlet and outlet sections, a reservoir tank, a circulating pump, a condenser, a U-tube manometer, and a control valve. The test section length is 108 mm, the outlet section is 32 mm, and the inlet section is 200 mm, which are selected using ASHRAE standard [35] to ensure completely built flow in the test section. The upper wall of the test section was heated by an electric heater providing a constant heat flux of 10 kW/m². A 10 liter reservoir tank of working fluid was mounted in between the 0.75 kW circulating pump and the 1 kW cooling capacity condenser. The fluid was forced to the inlet section and then to the test section to get a fully developed flow before entering the test section. The flow meter was connected between the pump and the inlet section for measuring the flow rates. A T-type thermocouple was used for taking down the temperature reading with the help of data acquisition. A total of 11 thermocouples were attached, out of which 08 were placed on the heated wall, 01 at the inlet, and 02 at the outlet to measure the temperature. When the fluid flow was in a steady-state condition, the pressure drop across the test section, temperature at the inlet and outlet of the test section, and wall temperature of the test section were recorded [1].

The four different volume concentrations of nanofluids were chosen as the sample, then dissolved in 100 ml of distilled water in each case, and then prepared in 4 different clean glass jars. The initial step is to blend nanoparticles in base fluid and the deionized water. The pH is one of the most important parameters, which effect the colloidal stability of oxide nanoparticles, so the pH value of nanofluid is tweaked about to 5.0. In the second step, nanoparticles were mixed with deionized water by an ultrasonic disrupter and homogenized for 2 days. The schematic of the practical setup is shown in Figure 1. The operating parameters are defined by protrusion transverse rib height protrusion transverse rib diameter , X-axis pitch and Y-axis pitch of protrusions transverse rib as shown in Figure 2 [1]. The different processing parameters are determined using equations as presented in Table 2 [1, 4, 7–10] and the control factors, which touch the performance of heat exchanger and are considered in experimentation and optimization are tabulated in Table 3.

4. Validation and Uncertainty Analysis of Investigational Data

The and values are calculated from investigational data for smooth square surface duct, which have been equated with the data obtained from the Dittus-Boelter correlation (3) and (4) [1, 10, 24].

Dittus-Boelter correlation equation [24] of for smooth surface square is as follows:

Blasius correlation equation of for smooth surface square is given as follows:

For divergence of investigational and the projected data of and as a function of , it is evidently illustrated that median (average) deviations of and were within 5%, which shows that experimental approach is acceptable and validates the experimental approach [1].

The uncertainty measurement of various quantities has been evaluated by using computation of uncertainty procedure given by Coleman and Glenn [36] and is in control limits, and the uncertainties values are calculated for , , and , which are estimated as , and , respectively [1].

5. Results

To envisage the impacts of Reynolds number, nanoparticle concentration, nanoparticle diameter, transverse protrusion ribs on HT, and friction factor in a smooth square channel through an experimental/modeling analysis were used. The experimental data and modeling data (RSM) collected for various stream and roughness parameters have been discussed below.

5.1. ANOVA Analysis for Nusselt Number

The results of RSM model used for Nusselt number are presented in Tables 4–7 in the form of ANOVA. The value of (Prob. > F) for used model is not more than 0.05 ( or 95% confidence). Table 4 presents ANOVA test results for before elimination. The value 2690.10 of model F indicates the model significance. P-values less than 0.05 suggest that model terms are significant. In this case, A, B, C, D, E, F, AB, AC, AD, AE, AF, A², D², E², F², AE², D³, E³, and F³ are significant model terms. Values higher than 0.1 suggest insignificance of model terms. Table 5 shows ANOVA test results of adequacy for before elimination. The predicted R-square of 0.9980 is in acceptable arrangement with the adjusted R-square of 0.9986, i.e., the difference is less than 0.2. A ratio greater than 4 is required for adequacy precision measures. In this study, the adequacy precision ratio of 198.166 suggests an adequate signal. Thus, the model can be utilized to circumnavigate the design space.

In Table 6, ANOVA test results for after backward elimination are tabulated. The value 4219.54 of model F indicates the model significance. There is only a 0.01% chance that an F-value of this large could occur due to noise. P-values less than 0.05 suggest that model terms are significant. In this case, A, B, C, D, E, F, AB, AC, AD, AE, AF, A², D², E², F², A²F, AD², AE², D³, E³, and F³ are significant model terms. Values higher than 0.1 suggest insignificance of model terms. Table 6 shows ANOVA test results of adequacy for after backward elimination. The predicted R-square of 0.9980 is in acceptable arrangement with the adjusted R-square of 0.9986, i.e., the difference is less than 0.2. A ratio greater than 4 is required for adequacy precision measures. In this study, the adequacy precision ratio of 247.519 suggests an adequate signal and the model can be used to circumnavigate the design space. After backward elimination process, the ultimate cubic equation of in proportion to coded and actual factors is equated in (5) and (6), respectively, as follows:

Equation of in terms of coded factors is as follows:

Equation of in terms of actual factors is as follows:

5.2. ANOVA Analysis for Friction Factor

The results of the RSM model used for friction factor are presented in Tables 8–11 in the form of ANOVA. Value of “Prob. > F” for used model is not more than 0.05 ( or 95% confidence). Table 8 presents the ANOVA test results for before elimination. The value 6411.73 of model F indicates that model is significant. P-values less than 0.0500 indicate that model terms are significant. In this ,case A, B, C, D, E, F, AB, AC, AD, AE, A², B², D², E², F², A²B, AB², AD², A³, D³, E³, and F³ are significant model terms. Values higher than 0.1 suggest insignificance of model terms. Table 9 shows ANOVA test results of adequacy for before elimination. The predicted R-square of 0.9989 is in acceptable arrangement with the adjusted R-square of 0.9992, i.e., the difference is less than 0.2. A ratio greater than 4 is required for adequacy precision measures. In this study, the adequacy precision ratio of 366.2616 suggests an adequate signal. Thus, the model can be utilized to circumnavigate the design space.

Table 10 presents ANOVA test results for after backward elimination. The value 6987.87 of model F indicates that the model is significant. There is only a 0.01% chance that an F-value this large could occur due to noise. P-values less than 0.05 suggest that the model terms are significant. In this case, A, B, C, D, E, F, AB, AC, AD, AE, A², B², D², E², F², A²B, AB², AD², A³, D³, E³, and F³ are significant model terms. Values higher than 0.1 suggest insignificance of model terms. Table 11 shows ANOVA test results of adequacy for after backward elimination. The predicted R-square of 0.9989 is in acceptable arrangement with the adjusted R-square of 0.9992, as difference is under 0.2. A ratio greater than 4 is required for adequacy precision measures. In this study, the adequacy precision ratio of 382.149 suggests an adequate signal and the model can be used to circumnavigate the design space. After backward elimination process, the ultimate cubic equation of in proportion to coded and actual factors is equated in equations (7) and (8), respectively, as follows:

Equation of in terms of coded factors is as follows:

Equation of in terms of actual factors is as follows:

5.3. Heat and Fluid Flow

Experimental results of different nanoparticle concentration, particle diameter, and protruded roughness parameters on and in a square duct are presented here. The nanoparticle concentration (φ) of 1%–4% and particle diameter of 30 nm–45 nm are considered in this section. Figure 3(a) represents the with different values of and . The other geometrical and NF flow parameters are fixed, such as and . Modeling results show that the increases as and increase. Also when the values of are 0.02, 0.03 and 0.04, significantly enhanced around 1.075, 1.150, and 1.225 times for of 4000 and 1.053, 1.098, and 1.14 times for equal to 18000, when compared to the value of 0.01. Figure 3(b) exemplifies the value of with different values of and . The other geometrical and NF flow parameters are fixed, such as and . The RSM results are found that the values increase with increase in the values of the . In the protruded rib and cases, the distributions with the of 0.02, 0.03, and 0.04 significantly enhanced approximately around 1.039, 1.078, and 1.127 times for value of 4000 and 1.047, 1.094, and 1.188 for equal to 18000, respectively, when compared to the value of 0.01. This increment is due to the increase in the Prandtl number and thermal conductivity of nanofluid. It is also observed that nanoparticle concentration φ = 0.04 has the best heat transfer rate followed by 0.03, 0.02, and 0.01.

(a)

(b)

Figure 4(a) demonstrates the value of with different values of and . The other geometrical and NF flow parameters were fixed, such as and . Modeling results are found that the increases with decrease in the values of and increase in . When the values are 30 nm, 35 nm, and 40 nm, significantly enhanced around 1.195, 1.122, and 1.049 times for of 4000 and 1.110, 1.072, and 1.039 times for of 18000 while compared to the value of 45 nm. Figure 4(b) exemplifies the value of with different values of and . The other geometrical and NF flow parameters were fixed, such as and . Modeling results show that the increases with the decrease in and . Also while the values are 30 nm, 35 ,nm and 40 nm, significantly enhanced around 1.106, 1.070, and 1.066 times for of 4000 and 1.109, 1.071, and 1.032 times for of 18000 when compared to the of 45 nm. The 30 nm nanoparticle diameter has the highest heat transfer coefficient. This is because of Brownian motion and good thermal conductivity of nanofluid at smaller nanoparticle diameter.

(a)

(b)

Figure 5(a) illustrates the value of for different values of and . The other geometrical and NF flow parameters were fixed, such as and . Modeling results are found that the increases with the increase in the values of up to 1.0 and then increases beyond this, and the value of starts decreasing with . It also shows that when values of are 0.83, 0.87, 1.0, and 1.25, significantly enhanced around 1.102, 1.224, 1.306, and 1.143 times for of 4000 and 1.048, 1.118, 1.160, and 1.080 times for of 18000, when compared to the of 1.67. Figure 5(b) exemplifies the value of with different values of and . The other geometrical and NF flow parameters were fixed, such as and . Modeling results concluded that the increases with the decrease in the values of and . It also shows that when the values are 0.83, 0.87, 1.0, and 1.25, significantly enhanced around 1.046, 1.127, 1. 156, and 1.093 times for value of 4000 and 1.079, 1.257, 1.306, and 1.168 times for value of 18000 when compared with the value of 1.67.

(a)

(b)

Figure 6(a) illustrates the value of with different values of and . The other geometrical and NF flow parameters were fixed, such as and . Modeling results found that the values of increase with the increase in the value of up to 1.8, beyond this any increase in the value of starts decreasing with . Also, when is equal to 1.4, 1.8, and 2.2, significantly enhanced around 1.150, 1.225, and 1.075 times for value of 4000 and 1.098, 1.141, and 1.053 times for value of 18000 when compared to the value of 2.6. Figure 6(b) exemplifies the value of with different values of and . The other geometrical and NF flow parameters were fixed, such as and . Modeling results found that the increases with the increase in the values of from 1.4 to 1.8 and subsequently any increase in , and the starts decreasing with the increase in . While the values are 1.4, 1.8, and 2.2, significantly enhanced around 1.115, 1.,170 and 1.057 times for value of 4000 and 1.151, 1.278, and 1.075 times for value of 18000 when compared to the of 2.6.

(a)

(b)

Figure 7(a) represents the value of at different values of and . The other geometrical and NF flow parameters were fixed, such as and . Modeling results are found that the increases with increase in the values of up to 1.8 and then increases beyond this, and starts decreasing with . While the values are 1.4, 1.8, and 2.2, significantly enhanced around 1.121, 1.195, and 1.048 times for value of 4000 and 1.072, 1.114, and 1.039 times for value of 18000 when compared to the value of 2.6. Figure 7(b) exemplifies the value of at different values of and . The other geometrical and NF flow parameters were fixed, such as and . Modeling results are found that the values increase with the increase in the values of up to 1.8 and then increases beyond this value, and starts decreasing with the increase in the values of . While the values are 1.4, 1.8, and 2.2, significantly enhanced around 1.099, 1.135, and 1.046 times for value of 4000 and 1.120, 1.160, and 1.045 times for Reynolds number value of 18000 when compared to the value of 2.6. The deviation in and was because of turbulence, persuaded by the rapidity of . Lesser and values, greater is the velocity of nanofluid and less significant is the area of disruption downstream of the transverse with combined protruded roughened square channel. Figure 8(a) and 8(b) show the comparison of predicted values of the and with that experimental values. The experimental and predicted values are in good considerate with each other, which assures the correctness of the information generated.

(a)

(b)

(a)

(b)

6. Correlations for and

As referenced above, the analytical methods are not available for the prediction of and of NF flow through square duct protruded with combined transverse rib; hence, statistical correlations based on experimental data have been developed for and of NF flow through square duct protruded with combined transverse rib. The developed correlation is useful for the researchers working in this field in predicting and of a heat exchanger at any specifically required values of variable parameters.

6.1. Correlations for

The earlier explained effects of various geometrical parameters on reveal that monotonously increases with the rise in and strongly depends upon operating parameters , , , , and and the . The equation for this dependence can be written as follows:

Figure 9(a) represents the plotted against various and curve fitting using regression analysis through the data point. The resulted regression relation is as follows:

(a)

(b)

(c)

(d)

(e)

(f)

Here, is dependent on . To correlate and ϕ, is plotted on a log-log scale against and other fixed roughness parameters. The Figure 9(b) shows relationship of with , and regression relation is as follows:

The constant , which depends on , is plotted between and , and fitting curve is represented in Figure 9(c). The relation between these parameters is as follows:

The plot between and in Figure 9(d) shows the effect of . Thus, the parameters selected yield that is expressed as follows:

The data generated from the equation have been plotted between and in Figure 9(e) to show the effect of . The curve fitting yields that is expressed as follows:

The data generated from the equation have been plotted between and in Figure 9(f) to show the effect of . The curve fitting by incorporating all the parameters yields final correlation that is expressed as follows:

6.2. Correlation for

The effects of various geometrical parameters on discussed earlier for various show dependence of on parameters such as , , , , , and and can be written as follows:

Figure 10(a) represents the plotted against various and curve fitting using regression analysis through the data point. The resulted regression relation is as follows:

(a)

(b)

(c)

(d)

(e)

(f)

Constant is reliant on parameters , , , , and . The functional relationship between and is determined by plotting data values of against the values of , as shown in Figure 10(b) through given relation.

The constant depends on , , , , and . The log-log data plotted between and get the fitting curve as shown in Figure 10(c), which relates between these two parameters as follows:

Similarly constant depends on , , and . The data plotted between and get the fitting curve as shown in log-log plot in Figure 10(d), which relates between these two parameters as follows:

The data generated from the equation have been plotted between and in Figure 10(e) to show the effect of . The curve fitting yields that is expressed as follows:

Finally after all the parameters are incorporated as shown in the Figure 10(f), the final correlation for is given as follows:

The Figure 11(a) displays the comparison of predicted values and experimental values of and shows maximum ±8.9% deviation. Also, the average absolute deviation in is found to be 4.82%. Similarly, Figure 11(b) displays the comparison of predicted values and experimental values of and shows maximum ±8.7% deviation. Also, the average absolute deviation in is found to be 4.53%. Therefore, it can be concluded that correlations developed in this work can predict the values of and with reasonable accuracy.

(a)

(b)

7. Conclusions

In this study, the parametric optimization and development of new correlations on heat transfer and flow friction for the turbulent regime of -water-based nanofluid flow in transverse with combined protrusion rib square duct have been discussed, and the following conclusions can be drawn:(i)The maximum value of was found to be 217, which was found at design parameters values: of 4%, of 30 nm, of 1.8, of 1.8, ratio of 1.0, and of 18000.(ii)The maximum value of was found to be 0.132, which was found at design parameters values: of 4%, of 30 nm, of 1.8, of 1.8, ratio of 1.0 and of 4000.(iii)Because of the decrease in reattachment spacing with the increment in the increases, whereas the decreases for all upsides of examined parameters.(iv)The comparison of experimental values and predicted values of RSM model was carried out, and it was seen that the RSM predicted values are as per the experimental values in the in the uncertainties scope of ±5.5%, which shows a decent acceptance between the experimental results and developed RSM model results.(v)The developed correlations for and were found to predict the values of and with good accuracy and absolute percentage deviation of 4.82% and 4.53%, respectively.

Data Availability

The data used to support the findings of the study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This research was supported by the researchers with the supporting project number (RSP-2021/401) and the King Saud University, Riyadh, Saudi Arabia.

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Copyright © 2022 Sunil Kumar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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