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

Authors: Sunil Kumar; Gaurav Gupta; Fairouz Tchier; Qin Xin; Anil Kumar; Faria Ahmed Shami
Journal: Mathematical Problems in Engineering
 (2022)

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## 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.


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## Body

## 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 Ren 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.Table 1
Correlation formulas for Nu andf for different flow regimes.
ResearchersCorrelationsDescriptionSundar et al. [16]Nu=0.215Re0.8Pr0.51+φ0.78f=0.3108Re−0.2451+φ0.423000<Re<220000<φ<0.3%4.50<Pr<6.13Suresh et al. [17]Nu=0.31RePr0.681+φ95.73f=26.44Re−0.87371+φ156.23700<Re<2300;φ=0.1%Pak and Cho [18]Nu=0.021Re0.8Pr0.5104≤Re≤1056.54≤Pr≤12.33Maïga et al. [22]Nu=0.085Re0.71Pr0.35104≤Re≤5X1056.6≤Pr≤13.9Petukhov [23]Nu=RePrf/8/1.07+12.7f/80.5Pr0.66−1f=1.82logRe−1.64−2104≤Re≤5X1050.5≤Pr≤2000Dittus and Boelter [24]Nu=0.024Re0.8Pr0.4for heating0.024Re0.8Pr0.3for  cooling2500≤Re≤1.24X1050.7≤Pr≤120Duangthongsuk and Wongwises [25]Nu=0.074Re0.707Pr0.385φ0.0743000<Re<180000≤φ≤2.0%Madhesh et al. [26]Nu=0.012RePr0.333φ0.0324000<Re<80000.15<φ<2.0%Gnielinski [27]Nu=f/8Re−1000Pr/1+12.7f/80.5Pr0.66−1f=0.079Re−1/42300≤Re≤5X1050.5≤Pr≤106


## 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]:(1)z=fx1,x2,x3,x4,………xn±ε,where z is the response, whereas x1,x2,x3,x4,………xn are the input variables, and ε is the fitting error. The approximation projected using the cubic model of f is as follows:(2)f=a0+∑i=1naixi+∑i<jnaijxixj+∑i=1naixi2+∑i<jnaijxi2xj+∑i<jnaijxixj2+∑i=1naixi3+ε,where ai is the linear/quadratic and cubic effects of xi and between xi and xj [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 concentration0.01≤φ≤0.04, nanoparticle diameter 30≤dnp≤45, stream-wise pitch 1.4≤Xaxis/dprint≤2.6, span-wise pitch 1.4≤Yaxis/dprint≤2.6, ratio of print diameter 0.83≤erib/dprint≤1.67, and Reynolds number 4000≤Renum≤18000 using RSM, and to develop statistical correlations for Nusselt number Nurough and friction factor frough in terms of operating parameters.


## 3. Experimental Details and Preparation of Nanofluid

The experimental setup used for assessingNurough and frough 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].Figure 1
Experimental setup.The four different volume concentrations ofAl2O3/H2O 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 Al2O3 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, Al2O3 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 erib, protrusion transverse rib diameter dprint, X-axis pitch Xaxis, and Y-axis pitch Yaxis 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.Figure 2
Rib shape parameters.Table 2
The governing equations for processing essential parameters [1, 4, 7–10].
S. NoParameterSymbolGoverning equation1Power supplied to the heaterQinQin=I×V2Heat gain by fluidQoutQout=m˙cpTo−Ti3Heat transfer coefficienththt=Qout/ApTpw−Tf4Reynolds numberRenumRenum=ρnfumDh/μnf5Nusselt numberNuroughNurough=htDh/knf6Friction factorfroughfrough=2DhΔp/ρnfLtum2Table 3
Control factors.
Control factorRangeA: Reynolds number,Renum6000–18000B: nanoparticle concentration,φ0.01–0.04C: nanoparticle diameterdnp30–45D:stream-wise pitchXaxis/dprint1.40–2.60E: span-wise pitch,Yaxis/dprint1.40–2.60F: proportion of protrusion height and print diametererib/dprint0.83–1.67


## 4. Validation and Uncertainty Analysis of Investigational Data

TheNusmooth and fsmooth 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 Nusmooth for smooth surface square is as follows:(3)Nusmooth=0.023Renum0.8Prnum0.4.Blasius correlation equation offsmooth for smooth surface square is given as follows:(4)fsmooth=0.079Renum−0.25.For divergence of investigational and the projected data ofNusmooth and fsmooth as a function of Renum, it is evidently illustrated that median (average) deviations of Nusmooth and fsmooth 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 Renum, Nurough, and frough, which are estimated as ±5.37%,±6.18%, and ±5.89%, 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 NumberNurough

The results of RSM model used for Nusselt numberNurough 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 (α=0.05 or 95% confidence). Table 4 presents ANOVA test results for Nurough 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 Nurough 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.Table 4
ANOVA test results for Nusselt number (before exclusion).
From whereSquared sumSquared meanf valueFreedom degree(Prob. > F)Model2.652 E+058036.972690.1033<0.0001SignificantA4052.134052.131356.311<0.0001B81.6381.6327.321<0.0001C59.9159.9120.051<0.0001D428.39428.39143.391<0.0001E449.51449.51150.461<0.0001F1775.651775.65594.341<0.0001AB87.3387.3329.231<0.0001AC40.6640.6613.6110.0004AD29.7129.719.9410.0022AE64.2564.2521.501<0.0001AF48.8948.8916.3610.0001A249.3849.3816.531<0.0001B20.00280.00280.000910.9756C24.284.281.4310.2345D2253.69253.6984.911<0.0001E2306.90306.90102.721<0.0001F2178.64178.6459.791<0.0001A2B0.42670.42670.142810.7063A2C3.943.941.3210.2535A2D4.084.081.3710.2454A2E0.77220.77220.258510.6124A2F8.208.202.7410.1010AB27.117.112.3810.1263AC22.152.150.721310.3979AD28.938.932.9910.0871AE217.4517.455.8410.0176AF26.616.612.2110.1403A³6.436.432.1510.1457B³1.541.540.515610.4745C³3.513.511.1810.2811D³209.08209.0869.981<0.0001E³206.80206.8069.221<0.0001F³1414.791414.79473.551<0.0001Residual280.842.9994Cor total2.655 E+05127Table 5
ANOVA test results of adequacy for Nusselt number (before elimination).
Standard deviationMeanC.V. %R-SquareAdjusted R-squarePredicted R-squarePrecision in terms of adequacy1.73117.041.480.99890.99860.9980198.1663Table 6
Test results ANOVA for Nusselt number (after backward exclusion).
From whereSquared sumSquared meanf valueFreedom degree(Prob. > F)Model2.652E + 0512627.804219.5421<0.0001SignificantA18576.0718576.076207.141<0.0001B1494.371494.37499.341<0.0001C1293.461293.46432.211<0.0001D525.43525.43175.571<0.0001E572.38572.38191.261<0.0001F1801.291801.29601.891<0.0001AB97.7697.7632.671<0.0001AC54.0954.0918.071<0.0001AD35.1735.1711.7510.0009AE72.5272.5224.231<0.0001AF67.3567.3522.501<0.0001A2419.55419.55140.191<0.0001D2263.66263.6688.101<0.0001E2321.46321.46107.411<0.0001F2209.34209.3469.951<0.0001A2F15.6315.635.2210.0243AD217.8017.805.9510.0164AE229.8929.899.9910.0021D³232.76232.7677.781<0.0001E³230.06230.0676.871<0.0001F³1411.281411.28471.571<0.0001Residual317.232.99106Cor total2.655E + 05127Table 7
ANOVA test results of adequacy for Nusselt number (after backward elimination).
Standard deviationMeanC.V. %R-squareAdjusted R-squarePredicted R-squarePrecision in terms of adequacy1.73117.041.480.99880.99860.9980247.5188In Table6, ANOVA test results for Nurough 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 Nurough 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 Nurough in proportion to coded and actual factors is equated in (5) and (6), respectively, as follows:Equation ofNurough in terms of coded factors is as follows:(5)Nurough=109.44+65A+8.46B−7.87C−16.98D−17.72E−48.30F+2.79AB−2.07AC−2.10AD−3.01AE−2.14AF7.35A2−6.07D2−6.71E2−6.03F2+1.50A2F−2.25AD2−2.91AE2+12.08D3+12.01E3+44.39F3.Equation ofNurough in terms of actual factors is as follows:(6)Nurough=−1950.51957+0.010555Renum+272.00413φ−0.615097dnp+676.44264Xaxisdprint+669.11935Yaxisdprint+2795.76986eribdprint+0.026551Renum×φ−0.000039Renum×dnp+0.003069Renum×Xaxisdprint+0.003906Renum×Yaxisdprint−0.002328Renum×eribdprint−2.4093×10−07Renum2−342.56189Xaxisdprint2−339.465Yaxisdprint2−2281.053eribdprint2+7.276Renum2×eribdprint−0.0009Renum×Xaxisdprint2−0.0012Renum×Yaxisdprint2.+55.917Xaxisdprint3+55.592Yaxisdprint3+599.182eribdprint3.


### 5.2. ANOVA Analysis for Friction Factorfrough

The results of the RSM model used for friction factorfrough are presented in Tables 8–11 in the form of ANOVA. Value of “Prob. > F” for used model is not more than 0.05 (α=0.05 or 95% confidence). Table 8 presents the ANOVA test results for frough 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 frough 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 8
ANOVA test results forfrough (before elimination).
SourceSum of squareMean squaref-ValueDegree of freedomProb. > FModel0.08660.00366411.7324<0.0001SignificantA0.00190.00193396.671<0.0001B0.00070.00071331.671<0.0001C0.00080.00081441.611<0.0001D0.00050.0005878.071<0.0001E0.00090.00091531.321<0.0001F0.00140.00142402.011<0.0001AB0.00000.000034.091<0.0001AC0.00000.000059.281<0.0001AD0.00000.000021.741<0.0001AE4.715E-064.715E-068.3810.0046AF5.986E-075.986E-071.0610.3047A20.00010.0001141.991<0.0001B23.048E-063.048E-065.4210.0219C25.826E-075.826E-071.0410.3112D20.00020.0002368.991<0.0001E20.00050.0005918.191<0.0001F20.00080.00081489.901<0.0001A2B3.779E-063.779E-066.7210.0109AB22.548E-062.548E-064.5310.0357AD29.848E-069.848E-0617.511<0.0001A³3.023E-063.023E-065.3710.0224D³0.00020.0002341.701<0.0001E³0.00040.0004663.451<0.0001F³0.00100.00101772.411<0.0001Residual0.00015.625E-07103Cor total0.0866127Table 9
ANOVA test results of adequacy forfrough (before elimination).
Standard deviationMeanC.V. %R-SquarePredicted R-squareAdjusted R-squarePrecision in terms of adequacy0.00080.12860.58300.99930.99890.9992366.2616Table 10
ANOVA test results forfrough (after backward elimination).
SourceSum of squareMean squaref-ValueDegree of freedomProb. > FModel0.08660.00396987.8722<0.0001SignificantA0.00260.00264656.671<0.0001B0.00080.00081459.541<0.0001C0.00080.00081439.341<0.0001D0.00060.00061021.211<0.0001E0.00100.00101770.771<0.0001F0.00140.00142398.821<0.0001AB0.00000.000033.161<0.0001AC0.00000.000061.531<0.0001AD0.00000.000020.901<0.0001AE4.221E-064.221E-067.5010.0073A20.00010.0001141.861<0.0001B24.021E-064.021E-067.1410.0087D20.00020.0002401.801<0.0001E20.00060.0006990.571<0.0001F20.00090.00091532.651<0.0001A2B3.779E-063.779E-066.7110.0109AB23.046E-063.046E-065.4110.0220AD29.275E-069.275E-0616.471<0.0001A³3.023E-063.023E-065.3710.0224D³0.00020.0002396.661<0.0001E³0.00040.0004762.451<0.0001F³0.00100.00101778.151<0.0001Residual0.00015.631E-07105Cor Total0.0866127Table 11
ANOVA test results of adequacy forfrough (after backward elimination).
Standard deviationMeanC.V. %R-SquareAdjusted R-squarePredicted R-squarePrecision in terms of adequacy0.00080.12860.58330.99930.99920.9989382.149Table10 presents ANOVA test results for frough 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 frough 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 frough in proportion to coded and actual factors is equated in equations (7) and (8), respectively, as follows:Equation offrough in terms of coded factors is as follows:(7)frough=0.122−0.033A+.008B−0.007C−0.019D−0.024E−0.041F−0.0012AB+0.0015AC+0.0012AD+0.0007AE+0.0035A2+0.0008B2−0.0057D2−0.009E2−0.0124F2+0.0008A2B+0.0009AB2+0.0015AD2−0.0008A3+0.0123D3+0.017E3+0.0374F3.Equation offrough in terms of actual factors is as follows:(8)frough=−1.8878−5.338×10−06Renum+0.9541×φ−0.0012×dnp+0.7381×Xaxisdprint+1.0035×Yaxisdprint+2.4439×eribdprint−0.00006×Renum×φ+2.8066×Renum×dnp−2.0893×10−06Renum×Xaxisdprint+1.6869×10−06Renum×Yaxisdprint+1.173×10−10Renum2−3.1043φ2−0.3636Xaxisdprint2−0.498Yaxisdprint2−1.9638eribdprint2+1.0939×10−09Renum2×φ+0.0006Renum×φ2+5.9369×10−07Renum×Xaxisdprint2−2.2293×10−15Renum3+0.0569Xaxisdprint3+0.07883Yaxisdprint3+0.5049eribdprint3.


### 5.3. Heat and Fluid Flow

Experimental results of different nanoparticle concentration, particle diameter, and protruded roughness parameters onNurough and frough 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 Nurough with different values of φ and Renum. The other geometrical and NF flow parameters are fixed, such as dnp=30nm,Xaxis/dprint=1.8Yaxis/dprint=1.8 and erib/dprint=1.67. Modeling results show that the Nurough increases as φ and Renum increase. Also when the values of φ are 0.02, 0.03 and 0.04, Nurough significantly enhanced around 1.075, 1.150, and 1.225 times for Renum of 4000 and 1.053, 1.098, and 1.14 times for Renum equal to 18000, when compared to the φ value of 0.01. Figure 3(b) exemplifies the value of frough with different values of φ and Renum. The other geometrical and NF flow parameters are fixed, such as dnp=30nm,Xaxis/dprint=1.8Yaxis/dprint=1.8 and erib/dprint=1.67. The RSM results are found that the frough values increase with increase in the values of the φ. In the protruded rib and φ cases, the frough distributions with the φ of 0.02, 0.03, and 0.04 significantly enhanced approximately around 1.039, 1.078, and 1.127 times for Renum value of 4000 and 1.047, 1.094, and 1.188 for Renum equal to 18000, respectively, when compared to the φ 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.Figure 3
(a) Variation ofNurough with φ and Renum. (b) Variation of frough with φ andRenum.
(a)(b)Figure4(a) demonstrates the value of Nurough with different values of dnp and Renum. The other geometrical and NF flow parameters were fixed, such as φ=0.04,Xaxis/dprint=1.8Yaxis/dprint=1.8 and erib/dprint=1.67. Modeling results are found that the Nurough increases with decrease in the values of dnp and increase in Renum. When the dnp values are 30 nm, 35 nm, and 40 nm, Nurough significantly enhanced around 1.195, 1.122, and 1.049 times for Renum of 4000 and 1.110, 1.072, and 1.039 times for Renum of 18000 while compared to the dnp value of 45 nm. Figure 4(b) exemplifies the value of frough with different values of dnp and Renum. The other geometrical and NF flow parameters were fixed, such as φ=0.04,Xaxis/dprint=1.8Yaxis/dprint=1.8 and erib/dprint=1.67. Modeling results show that the frough increases with the decrease in dnp and Renum. Also while the dnp values are 30 nm, 35 ,nm and 40 nm, frough significantly enhanced around 1.106, 1.070, and 1.066 times for Renum of 4000 and 1.109, 1.071, and 1.032 times for Renum of 18000 when compared to the dnp 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.Figure 4
(a) Variation ofNurough with dnp and Renum. (b) Variation of frough with dnp andRenum.
(a)(b)Figure5(a) illustrates the value of Nurough for different values of erib/dprint and Renum. The other geometrical and NF flow parameters were fixed, such as Xaxis/dprint=1.8,Yaxis/dprint=1.8,dnp=30nm and φ=0.04. Modeling results are found that the Nurough increases with the increase in the values of erib/dprint up to 1.0 and then increases beyond this, and the value of Nurough starts decreasing with Renum. It also shows that when values of erib/dprint are 0.83, 0.87, 1.0, and 1.25, Nurough significantly enhanced around 1.102, 1.224, 1.306, and 1.143 times for Renum of 4000 and 1.048, 1.118, 1.160, and 1.080 times for Renum of 18000, when compared to the erib/dprint of 1.67. Figure 5(b) exemplifies the value of frough with different values of erib/dprint and Renum. The other geometrical and NF flow parameters were fixed, such as Xaxis/dprint=1.8,Yaxis/dprint=1.8,dnp=30nm and φ=0.04. Modeling results concluded that the frough increases with the decrease in the values of erib/dprint and Renum. It also shows that when the erib/dprint values are 0.83, 0.87, 1.0, and 1.25, frough significantly enhanced around 1.046, 1.127, 1. 156, and 1.093 times for Renum value of 4000 and 1.079, 1.257, 1.306, and 1.168 times for Renum value of 18000 when compared with the erib/dprint value of 1.67.Figure 5
(a) Variation ofNurough with erib/dprint and Renum. (b) Variation of frough with erib/dprint and Renum.
(a)(b)Figure6(a) illustrates the value of Nurough with different values of Yaxis/dprint and Renum. The other geometrical and NF flow parameters were fixed, such as Xaxis/dprint=1.8,erib/dprint=1.0,dnp=30nm and φ=0.04. Modeling results found that the values of Nurough increase with the increase in the value of Yaxis/dprint up to 1.8, beyond this any increase in the value of Yaxis/dprint,Nurough starts decreasing with Renum. Also, when Yaxis/dprint is equal to 1.4, 1.8, and 2.2, Nurough significantly enhanced around 1.150, 1.225, and 1.075 times for Renum value of 4000 and 1.098, 1.141, and 1.053 times for Renum value of 18000 when compared to the Yaxis/dprint value of 2.6. Figure 6(b) exemplifies the value of frough with different values of Yaxis/dprint and Renum. The other geometrical and NF flow parameters were fixed, such as Xaxis/dprint=1.8,erib/dprint=1.0,dnp=30nm and φ=0.04. Modeling results found that the frough increases with the increase in the values of Yaxis/dprint from 1.4 to 1.8 and subsequently any increase in Yaxis/dprint, and the frough starts decreasing with the increase in Renum. While the Yaxis/dprint values are 1.4, 1.8, and 2.2, frough significantly enhanced around 1.115, 1.,170 and 1.057 times for Renum value of 4000 and 1.151, 1.278, and 1.075 times for Renum value of 18000 when compared to the Yaxis/dprint of 2.6.Figure 6
(a) Variation ofNurough with Yaxis/dprint and Renum. (b) Variation of frough with Yaxis/dprint and Renum.
(a)(b)Figure7(a) represents the value of Nurough at different values of Xaxis/dprint and Renum. The other geometrical and NF flow parameters were fixed, such as Yaxis/dprint=1.8,erib/dprint=1.0,dnp=30nm and φ=0.04. Modeling results are found that the Nurough increases with increase in the values of Xaxis/dprint up to 1.8 and then increases beyond this, and Nurough starts decreasing with Renum. While the Xaxis/dprint values are 1.4, 1.8, and 2.2, Nurough significantly enhanced around 1.121, 1.195, and 1.048 times for Renum value of 4000 and 1.072, 1.114, and 1.039 times for Renum value of 18000 when compared to the Xaxis/dprint value of 2.6. Figure 7(b) exemplifies the value of frough at different values of Xaxis/dprint and Renum. The other geometrical and NF flow parameters were fixed, such as Xaxis/dprint=1.8,erib/dprint=1.0,dnp=30nm and φ=0.04. Modeling results are found that the frough values increase with the increase in the values of Xaxis/dprint up to 1.8 and then increases beyond this value, and frough starts decreasing with the increase in the values of Renum. While the Xaxis/dprint values are 1.4, 1.8, and 2.2, frough significantly enhanced around 1.099, 1.135, and 1.046 times for Renum value of 4000 and 1.120, 1.160, and 1.045 times for Reynolds number value of 18000 when compared to the Xaxis/dprint value of 2.6. The deviation in Nurough and frough was because of turbulence, persuaded by the rapidity of Al2O3/H2O. Lesser Xaxis/dprint and Yaxis/dprint values, greater is the velocity of Al2O3/H2O 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 Nurough and frough with that experimental values. The experimental and predicted values are in good considerate with each other, which assures the correctness of the information generated.Figure 7
(a) Variation ofNurough with Xaxis/dprint and Renum. (b) Variation of frough with Xaxis/dprint and Renum.
(a)(b)Figure 8
Experimental results vs RSM results. (a)Nurough. (b)frough (average deviation is ±5.5%).
(a)(b)


## 5.1. ANOVA Analysis for Nusselt NumberNurough

The results of RSM model used for Nusselt numberNurough 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 (α=0.05 or 95% confidence). Table 4 presents ANOVA test results for Nurough 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 Nurough 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.Table 4
ANOVA test results for Nusselt number (before exclusion).
From whereSquared sumSquared meanf valueFreedom degree(Prob. > F)Model2.652 E+058036.972690.1033<0.0001SignificantA4052.134052.131356.311<0.0001B81.6381.6327.321<0.0001C59.9159.9120.051<0.0001D428.39428.39143.391<0.0001E449.51449.51150.461<0.0001F1775.651775.65594.341<0.0001AB87.3387.3329.231<0.0001AC40.6640.6613.6110.0004AD29.7129.719.9410.0022AE64.2564.2521.501<0.0001AF48.8948.8916.3610.0001A249.3849.3816.531<0.0001B20.00280.00280.000910.9756C24.284.281.4310.2345D2253.69253.6984.911<0.0001E2306.90306.90102.721<0.0001F2178.64178.6459.791<0.0001A2B0.42670.42670.142810.7063A2C3.943.941.3210.2535A2D4.084.081.3710.2454A2E0.77220.77220.258510.6124A2F8.208.202.7410.1010AB27.117.112.3810.1263AC22.152.150.721310.3979AD28.938.932.9910.0871AE217.4517.455.8410.0176AF26.616.612.2110.1403A³6.436.432.1510.1457B³1.541.540.515610.4745C³3.513.511.1810.2811D³209.08209.0869.981<0.0001E³206.80206.8069.221<0.0001F³1414.791414.79473.551<0.0001Residual280.842.9994Cor total2.655 E+05127Table 5
ANOVA test results of adequacy for Nusselt number (before elimination).
Standard deviationMeanC.V. %R-SquareAdjusted R-squarePredicted R-squarePrecision in terms of adequacy1.73117.041.480.99890.99860.9980198.1663Table 6
Test results ANOVA for Nusselt number (after backward exclusion).
From whereSquared sumSquared meanf valueFreedom degree(Prob. > F)Model2.652E + 0512627.804219.5421<0.0001SignificantA18576.0718576.076207.141<0.0001B1494.371494.37499.341<0.0001C1293.461293.46432.211<0.0001D525.43525.43175.571<0.0001E572.38572.38191.261<0.0001F1801.291801.29601.891<0.0001AB97.7697.7632.671<0.0001AC54.0954.0918.071<0.0001AD35.1735.1711.7510.0009AE72.5272.5224.231<0.0001AF67.3567.3522.501<0.0001A2419.55419.55140.191<0.0001D2263.66263.6688.101<0.0001E2321.46321.46107.411<0.0001F2209.34209.3469.951<0.0001A2F15.6315.635.2210.0243AD217.8017.805.9510.0164AE229.8929.899.9910.0021D³232.76232.7677.781<0.0001E³230.06230.0676.871<0.0001F³1411.281411.28471.571<0.0001Residual317.232.99106Cor total2.655E + 05127Table 7
ANOVA test results of adequacy for Nusselt number (after backward elimination).
Standard deviationMeanC.V. %R-squareAdjusted R-squarePredicted R-squarePrecision in terms of adequacy1.73117.041.480.99880.99860.9980247.5188In Table6, ANOVA test results for Nurough 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 Nurough 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 Nurough in proportion to coded and actual factors is equated in (5) and (6), respectively, as follows:Equation ofNurough in terms of coded factors is as follows:(5)Nurough=109.44+65A+8.46B−7.87C−16.98D−17.72E−48.30F+2.79AB−2.07AC−2.10AD−3.01AE−2.14AF7.35A2−6.07D2−6.71E2−6.03F2+1.50A2F−2.25AD2−2.91AE2+12.08D3+12.01E3+44.39F3.Equation ofNurough in terms of actual factors is as follows:(6)Nurough=−1950.51957+0.010555Renum+272.00413φ−0.615097dnp+676.44264Xaxisdprint+669.11935Yaxisdprint+2795.76986eribdprint+0.026551Renum×φ−0.000039Renum×dnp+0.003069Renum×Xaxisdprint+0.003906Renum×Yaxisdprint−0.002328Renum×eribdprint−2.4093×10−07Renum2−342.56189Xaxisdprint2−339.465Yaxisdprint2−2281.053eribdprint2+7.276Renum2×eribdprint−0.0009Renum×Xaxisdprint2−0.0012Renum×Yaxisdprint2.+55.917Xaxisdprint3+55.592Yaxisdprint3+599.182eribdprint3.


## 5.2. ANOVA Analysis for Friction Factorfrough

The results of the RSM model used for friction factorfrough are presented in Tables 8–11 in the form of ANOVA. Value of “Prob. > F” for used model is not more than 0.05 (α=0.05 or 95% confidence). Table 8 presents the ANOVA test results for frough 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 frough 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 8
ANOVA test results forfrough (before elimination).
SourceSum of squareMean squaref-ValueDegree of freedomProb. > FModel0.08660.00366411.7324<0.0001SignificantA0.00190.00193396.671<0.0001B0.00070.00071331.671<0.0001C0.00080.00081441.611<0.0001D0.00050.0005878.071<0.0001E0.00090.00091531.321<0.0001F0.00140.00142402.011<0.0001AB0.00000.000034.091<0.0001AC0.00000.000059.281<0.0001AD0.00000.000021.741<0.0001AE4.715E-064.715E-068.3810.0046AF5.986E-075.986E-071.0610.3047A20.00010.0001141.991<0.0001B23.048E-063.048E-065.4210.0219C25.826E-075.826E-071.0410.3112D20.00020.0002368.991<0.0001E20.00050.0005918.191<0.0001F20.00080.00081489.901<0.0001A2B3.779E-063.779E-066.7210.0109AB22.548E-062.548E-064.5310.0357AD29.848E-069.848E-0617.511<0.0001A³3.023E-063.023E-065.3710.0224D³0.00020.0002341.701<0.0001E³0.00040.0004663.451<0.0001F³0.00100.00101772.411<0.0001Residual0.00015.625E-07103Cor total0.0866127Table 9
ANOVA test results of adequacy forfrough (before elimination).
Standard deviationMeanC.V. %R-SquarePredicted R-squareAdjusted R-squarePrecision in terms of adequacy0.00080.12860.58300.99930.99890.9992366.2616Table 10
ANOVA test results forfrough (after backward elimination).
SourceSum of squareMean squaref-ValueDegree of freedomProb. > FModel0.08660.00396987.8722<0.0001SignificantA0.00260.00264656.671<0.0001B0.00080.00081459.541<0.0001C0.00080.00081439.341<0.0001D0.00060.00061021.211<0.0001E0.00100.00101770.771<0.0001F0.00140.00142398.821<0.0001AB0.00000.000033.161<0.0001AC0.00000.000061.531<0.0001AD0.00000.000020.901<0.0001AE4.221E-064.221E-067.5010.0073A20.00010.0001141.861<0.0001B24.021E-064.021E-067.1410.0087D20.00020.0002401.801<0.0001E20.00060.0006990.571<0.0001F20.00090.00091532.651<0.0001A2B3.779E-063.779E-066.7110.0109AB23.046E-063.046E-065.4110.0220AD29.275E-069.275E-0616.471<0.0001A³3.023E-063.023E-065.3710.0224D³0.00020.0002396.661<0.0001E³0.00040.0004762.451<0.0001F³0.00100.00101778.151<0.0001Residual0.00015.631E-07105Cor Total0.0866127Table 11
ANOVA test results of adequacy forfrough (after backward elimination).
Standard deviationMeanC.V. %R-SquareAdjusted R-squarePredicted R-squarePrecision in terms of adequacy0.00080.12860.58330.99930.99920.9989382.149Table10 presents ANOVA test results for frough 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 frough 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 frough in proportion to coded and actual factors is equated in equations (7) and (8), respectively, as follows:Equation offrough in terms of coded factors is as follows:(7)frough=0.122−0.033A+.008B−0.007C−0.019D−0.024E−0.041F−0.0012AB+0.0015AC+0.0012AD+0.0007AE+0.0035A2+0.0008B2−0.0057D2−0.009E2−0.0124F2+0.0008A2B+0.0009AB2+0.0015AD2−0.0008A3+0.0123D3+0.017E3+0.0374F3.Equation offrough in terms of actual factors is as follows:(8)frough=−1.8878−5.338×10−06Renum+0.9541×φ−0.0012×dnp+0.7381×Xaxisdprint+1.0035×Yaxisdprint+2.4439×eribdprint−0.00006×Renum×φ+2.8066×Renum×dnp−2.0893×10−06Renum×Xaxisdprint+1.6869×10−06Renum×Yaxisdprint+1.173×10−10Renum2−3.1043φ2−0.3636Xaxisdprint2−0.498Yaxisdprint2−1.9638eribdprint2+1.0939×10−09Renum2×φ+0.0006Renum×φ2+5.9369×10−07Renum×Xaxisdprint2−2.2293×10−15Renum3+0.0569Xaxisdprint3+0.07883Yaxisdprint3+0.5049eribdprint3.


## 5.3. Heat and Fluid Flow

Experimental results of different nanoparticle concentration, particle diameter, and protruded roughness parameters onNurough and frough 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 Nurough with different values of φ and Renum. The other geometrical and NF flow parameters are fixed, such as dnp=30nm,Xaxis/dprint=1.8Yaxis/dprint=1.8 and erib/dprint=1.67. Modeling results show that the Nurough increases as φ and Renum increase. Also when the values of φ are 0.02, 0.03 and 0.04, Nurough significantly enhanced around 1.075, 1.150, and 1.225 times for Renum of 4000 and 1.053, 1.098, and 1.14 times for Renum equal to 18000, when compared to the φ value of 0.01. Figure 3(b) exemplifies the value of frough with different values of φ and Renum. The other geometrical and NF flow parameters are fixed, such as dnp=30nm,Xaxis/dprint=1.8Yaxis/dprint=1.8 and erib/dprint=1.67. The RSM results are found that the frough values increase with increase in the values of the φ. In the protruded rib and φ cases, the frough distributions with the φ of 0.02, 0.03, and 0.04 significantly enhanced approximately around 1.039, 1.078, and 1.127 times for Renum value of 4000 and 1.047, 1.094, and 1.188 for Renum equal to 18000, respectively, when compared to the φ 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.Figure 3
(a) Variation ofNurough with φ and Renum. (b) Variation of frough with φ andRenum.
(a)(b)Figure4(a) demonstrates the value of Nurough with different values of dnp and Renum. The other geometrical and NF flow parameters were fixed, such as φ=0.04,Xaxis/dprint=1.8Yaxis/dprint=1.8 and erib/dprint=1.67. Modeling results are found that the Nurough increases with decrease in the values of dnp and increase in Renum. When the dnp values are 30 nm, 35 nm, and 40 nm, Nurough significantly enhanced around 1.195, 1.122, and 1.049 times for Renum of 4000 and 1.110, 1.072, and 1.039 times for Renum of 18000 while compared to the dnp value of 45 nm. Figure 4(b) exemplifies the value of frough with different values of dnp and Renum. The other geometrical and NF flow parameters were fixed, such as φ=0.04,Xaxis/dprint=1.8Yaxis/dprint=1.8 and erib/dprint=1.67. Modeling results show that the frough increases with the decrease in dnp and Renum. Also while the dnp values are 30 nm, 35 ,nm and 40 nm, frough significantly enhanced around 1.106, 1.070, and 1.066 times for Renum of 4000 and 1.109, 1.071, and 1.032 times for Renum of 18000 when compared to the dnp 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.Figure 4
(a) Variation ofNurough with dnp and Renum. (b) Variation of frough with dnp andRenum.
(a)(b)Figure5(a) illustrates the value of Nurough for different values of erib/dprint and Renum. The other geometrical and NF flow parameters were fixed, such as Xaxis/dprint=1.8,Yaxis/dprint=1.8,dnp=30nm and φ=0.04. Modeling results are found that the Nurough increases with the increase in the values of erib/dprint up to 1.0 and then increases beyond this, and the value of Nurough starts decreasing with Renum. It also shows that when values of erib/dprint are 0.83, 0.87, 1.0, and 1.25, Nurough significantly enhanced around 1.102, 1.224, 1.306, and 1.143 times for Renum of 4000 and 1.048, 1.118, 1.160, and 1.080 times for Renum of 18000, when compared to the erib/dprint of 1.67. Figure 5(b) exemplifies the value of frough with different values of erib/dprint and Renum. The other geometrical and NF flow parameters were fixed, such as Xaxis/dprint=1.8,Yaxis/dprint=1.8,dnp=30nm and φ=0.04. Modeling results concluded that the frough increases with the decrease in the values of erib/dprint and Renum. It also shows that when the erib/dprint values are 0.83, 0.87, 1.0, and 1.25, frough significantly enhanced around 1.046, 1.127, 1. 156, and 1.093 times for Renum value of 4000 and 1.079, 1.257, 1.306, and 1.168 times for Renum value of 18000 when compared with the erib/dprint value of 1.67.Figure 5
(a) Variation ofNurough with erib/dprint and Renum. (b) Variation of frough with erib/dprint and Renum.
(a)(b)Figure6(a) illustrates the value of Nurough with different values of Yaxis/dprint and Renum. The other geometrical and NF flow parameters were fixed, such as Xaxis/dprint=1.8,erib/dprint=1.0,dnp=30nm and φ=0.04. Modeling results found that the values of Nurough increase with the increase in the value of Yaxis/dprint up to 1.8, beyond this any increase in the value of Yaxis/dprint,Nurough starts decreasing with Renum. Also, when Yaxis/dprint is equal to 1.4, 1.8, and 2.2, Nurough significantly enhanced around 1.150, 1.225, and 1.075 times for Renum value of 4000 and 1.098, 1.141, and 1.053 times for Renum value of 18000 when compared to the Yaxis/dprint value of 2.6. Figure 6(b) exemplifies the value of frough with different values of Yaxis/dprint and Renum. The other geometrical and NF flow parameters were fixed, such as Xaxis/dprint=1.8,erib/dprint=1.0,dnp=30nm and φ=0.04. Modeling results found that the frough increases with the increase in the values of Yaxis/dprint from 1.4 to 1.8 and subsequently any increase in Yaxis/dprint, and the frough starts decreasing with the increase in Renum. While the Yaxis/dprint values are 1.4, 1.8, and 2.2, frough significantly enhanced around 1.115, 1.,170 and 1.057 times for Renum value of 4000 and 1.151, 1.278, and 1.075 times for Renum value of 18000 when compared to the Yaxis/dprint of 2.6.Figure 6
(a) Variation ofNurough with Yaxis/dprint and Renum. (b) Variation of frough with Yaxis/dprint and Renum.
(a)(b)Figure7(a) represents the value of Nurough at different values of Xaxis/dprint and Renum. The other geometrical and NF flow parameters were fixed, such as Yaxis/dprint=1.8,erib/dprint=1.0,dnp=30nm and φ=0.04. Modeling results are found that the Nurough increases with increase in the values of Xaxis/dprint up to 1.8 and then increases beyond this, and Nurough starts decreasing with Renum. While the Xaxis/dprint values are 1.4, 1.8, and 2.2, Nurough significantly enhanced around 1.121, 1.195, and 1.048 times for Renum value of 4000 and 1.072, 1.114, and 1.039 times for Renum value of 18000 when compared to the Xaxis/dprint value of 2.6. Figure 7(b) exemplifies the value of frough at different values of Xaxis/dprint and Renum. The other geometrical and NF flow parameters were fixed, such as Xaxis/dprint=1.8,erib/dprint=1.0,dnp=30nm and φ=0.04. Modeling results are found that the frough values increase with the increase in the values of Xaxis/dprint up to 1.8 and then increases beyond this value, and frough starts decreasing with the increase in the values of Renum. While the Xaxis/dprint values are 1.4, 1.8, and 2.2, frough significantly enhanced around 1.099, 1.135, and 1.046 times for Renum value of 4000 and 1.120, 1.160, and 1.045 times for Reynolds number value of 18000 when compared to the Xaxis/dprint value of 2.6. The deviation in Nurough and frough was because of turbulence, persuaded by the rapidity of Al2O3/H2O. Lesser Xaxis/dprint and Yaxis/dprint values, greater is the velocity of Al2O3/H2O 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 Nurough and frough with that experimental values. The experimental and predicted values are in good considerate with each other, which assures the correctness of the information generated.Figure 7
(a) Variation ofNurough with Xaxis/dprint and Renum. (b) Variation of frough with Xaxis/dprint and Renum.
(a)(b)Figure 8
Experimental results vs RSM results. (a)Nurough. (b)frough (average deviation is ±5.5%).
(a)(b)


## 6. Correlations forNurough and frough

As referenced above, the analytical methods are not available for the prediction ofNurough and frough of NF flow through square duct protruded with combined transverse rib; hence, statistical correlations based on experimental data have been developed for Nurough and frough 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 Nurough and frough of a heat exchanger at any specifically required values of variable parameters.


### 6.1. Correlations forNurough

The earlier explained effects of various geometrical parameters onNurough reveal that Nurough monotonously increases with the rise in Renum and strongly depends upon operating parameters ϕ, dnp, Xaxis/dprint, Yaxis/dprint, and erib/dprint and the Renum. The Nurough equation for this dependence can be written as follows:(9)Nurough=NuroughRenum,ϕ,dnp,Xaxisdprint,Yaxisdprint,eribdprint.Figure9(a) represents the plotted Nurough against various Renum and curve fitting using regression analysis through the data point. The resulted regression relation is as follows:(10)Nurough=AoRenum0.864.Figure 9
(a) Plot oflnNurough as a function of lnRenum for all the experimental data. (b) Plot of lnAo as a function of lnϕ. (c) Plot of lnBo as a function of lndnp. (d) Plot of lnCo as a function of lnYaxis/dprint. (e) Plot of lnCo as a function of lnXaxis/dprint). (f) Plot of lnEo as a function of lnerib/dprint.
(a)(b)(c)(d)(e)(f)Here,A0 is dependent on dnp,Xaxis/dprint,Yaxis/dprintanderib/dprint. To correlate A0 and ϕ, lnA0 is plotted on a log-log scale against lnϕ and other fixed roughness parameters. The Figure 9(b) shows relationship of lnA0 with lnϕ, and regression relation is as follows:(11)Nurough=BoRenum0.864φ0.64exp0.065lnφ2.The constantB0 , which depends on dnp,Xaxis/dprint,Yaxis/dprintanderib/dprint, is plotted between B0 and dnp, and fitting curve is represented in Figure 9(c). The relation between these parameters is as follows:(12)Nurough=C0Renum0.864φ0.64exp0.065lnφ2dnp0.784exp−0.168lndnp2.The plot betweenlnCo and lnYaxis/dprint in Figure 9(d) shows the effect of Yaxis/dprint. Thus, the parameters selected yield Nurough that is expressed as follows:(13)Nurough=D0Renum0.864φ0.64exp0.065lnφ2dnp0.784exp−0.168lndnp2Yaxisdprint0.848exp−0.168lnYaxis/dprint2.The data generated from the equation have been plotted betweenlnD0 and lnXaxis/dprint in Figure 9(e) to show the effect of Xaxis/dprint. The curve fitting yields Nurough that is expressed as follows:(14)Nurough=E0Renum0.864φ0.64exp0.065lnφ2dnp0.784exp−0.168lndnp2Yaxisdprint0.848exp−0.168lnYaxisdprint2exp−1.01lnXaxis/dprint2.The data generated from the equation have been plotted betweenlnE0 and lnerib/dprint in Figure 9(f) to show the effect of erib/dprint. The curve fitting by incorporating all the parameters yields final Nurough correlation that is expressed as follows:(15)Nurough=0.054Renum0.864φ0.64exp0.065lnφ2dnp0.784×exp−0.168lndnp2Yaxisdprint0.848exp−0.168lnYaxisdprint2×Xaxisdprint1.046exp−1.01lnXaxisdprint2eribdprint0.235×exp−1.393lneribdprint2.


### 6.2. Correlation forfrough

The effects of various geometrical parameters onfrough discussed earlier for various Renum show dependence of frough on parameters such as ϕ, dnp, Xaxis/dprint, Yaxis/dprint, erib/dprint, and Renum and can be written as follows:(16)frough=froughRenum,ϕ,dnp,Xaxisdprint,Yaxisdprint,eribdprint.Figure10(a) represents the plotted frough against various Renum and curve fitting using regression analysis through the data point. The resulted regression relation is as follows:(17)frough=AoRenum−0.282.Figure 10
(a) Plot oflnfrough as a function of lnRenum for all the experimental data. (b) Plot of lnAo as a function of lnϕ. (c) Plot of lnBo as a function of lndnp. (d) Plot of lnCo as a function of lnYaxis/dprint. (e) Plot of lnDo as a function of lnXaxis/dprint. (f) Plot of lnCo as a function of lnerib/dprint.
(a)(b)(c)(d)(e)(f)ConstantA0 is reliant on parameters φ, dnp, Xaxis/dprint, Yaxis/dprint, and erib/dprint. The functional relationship between A0 and φ is determined by plotting data values of lnA0 against the values of lnφ, as shown in Figure 10(b) through given relation.(18)frough=BoRenum−0.282ϕ0.542exp0.057lnφ2.The constantB0 depends on φ, dnp, Xaxis/dprint, Yaxis/dprint, and erib/dprint. The log-log data plotted between B0 and dnp get the fitting curve as shown in Figure 10(c), which relates between these two parameters as follows:(19)frough=CoRenum−0.282φ0.542exp0.057lnφ2dnp1.32exp−0.219lndnp2.Similarly constantC0 depends on Xaxis/dprint, Yaxis/dprint, and erib/dprint. The data plotted between B0 and dnp get the fitting curve as shown in log-log plot in Figure 10(d), which relates between these two parameters as follows:(20)frough=D0Renum−0.282ϕ0.542exp0.057lnφ2dnp1.32exp−0.219lndnp2Yaxisdprint1.103exp−1.065lnYaxisdprint2.The data generated from the equation have been plotted betweenlnD0 and lnXaxis/dprint in Figure 10(e) to show the effect of Xaxis/dprint. The curve fitting yields frough that is expressed as follows:(21)frough=E0Renum−0.282φ0.542exp0.057lnφ2dnp1.32exp−0.219lndnp2Yaxisdprint1.103exp−1.065lnYaxisdprint2×exp−0.698lnXaxisdprint2Xaxisdprint0.712.Finally after all the parameters are incorporated as shown in the Figure10(f), the final correlation for frough is given as follows:(22)frough=0.53Renum−0.282φ0.542exp0.057lnφ2dnp1.32×exp−0.219lndnp2Yaxisdprint1.103exp−1.065lnYaxisdprint2×exp−0.698lnXaxisdprint2Xaxisdprint0.712eribdprint0.149×exp−0.933lnerib/dprint2.The Figure11(a) displays the comparison of predicted values and experimental values of Nurough and shows maximum ±8.9% deviation. Also, the average absolute deviation in Nurough is found to be 4.82%. Similarly, Figure 11(b) displays the comparison of predicted values and experimental values of frough and shows maximum ±8.7% deviation. Also, the average absolute deviation in frough is found to be 4.53%. Therefore, it can be concluded that correlations developed in this work can predict the values of Nurough and frough with reasonable accuracy.Figure 11
(a) Comparison of experimental and predicted data ofNurough. (b) Comparison of experimental and predicted values of frough.
(a)(b)


## 6.1. Correlations forNurough

The earlier explained effects of various geometrical parameters onNurough reveal that Nurough monotonously increases with the rise in Renum and strongly depends upon operating parameters ϕ, dnp, Xaxis/dprint, Yaxis/dprint, and erib/dprint and the Renum. The Nurough equation for this dependence can be written as follows:(9)Nurough=NuroughRenum,ϕ,dnp,Xaxisdprint,Yaxisdprint,eribdprint.Figure9(a) represents the plotted Nurough against various Renum and curve fitting using regression analysis through the data point. The resulted regression relation is as follows:(10)Nurough=AoRenum0.864.Figure 9
(a) Plot oflnNurough as a function of lnRenum for all the experimental data. (b) Plot of lnAo as a function of lnϕ. (c) Plot of lnBo as a function of lndnp. (d) Plot of lnCo as a function of lnYaxis/dprint. (e) Plot of lnCo as a function of lnXaxis/dprint). (f) Plot of lnEo as a function of lnerib/dprint.
(a)(b)(c)(d)(e)(f)Here,A0 is dependent on dnp,Xaxis/dprint,Yaxis/dprintanderib/dprint. To correlate A0 and ϕ, lnA0 is plotted on a log-log scale against lnϕ and other fixed roughness parameters. The Figure 9(b) shows relationship of lnA0 with lnϕ, and regression relation is as follows:(11)Nurough=BoRenum0.864φ0.64exp0.065lnφ2.The constantB0 , which depends on dnp,Xaxis/dprint,Yaxis/dprintanderib/dprint, is plotted between B0 and dnp, and fitting curve is represented in Figure 9(c). The relation between these parameters is as follows:(12)Nurough=C0Renum0.864φ0.64exp0.065lnφ2dnp0.784exp−0.168lndnp2.The plot betweenlnCo and lnYaxis/dprint in Figure 9(d) shows the effect of Yaxis/dprint. Thus, the parameters selected yield Nurough that is expressed as follows:(13)Nurough=D0Renum0.864φ0.64exp0.065lnφ2dnp0.784exp−0.168lndnp2Yaxisdprint0.848exp−0.168lnYaxis/dprint2.The data generated from the equation have been plotted betweenlnD0 and lnXaxis/dprint in Figure 9(e) to show the effect of Xaxis/dprint. The curve fitting yields Nurough that is expressed as follows:(14)Nurough=E0Renum0.864φ0.64exp0.065lnφ2dnp0.784exp−0.168lndnp2Yaxisdprint0.848exp−0.168lnYaxisdprint2exp−1.01lnXaxis/dprint2.The data generated from the equation have been plotted betweenlnE0 and lnerib/dprint in Figure 9(f) to show the effect of erib/dprint. The curve fitting by incorporating all the parameters yields final Nurough correlation that is expressed as follows:(15)Nurough=0.054Renum0.864φ0.64exp0.065lnφ2dnp0.784×exp−0.168lndnp2Yaxisdprint0.848exp−0.168lnYaxisdprint2×Xaxisdprint1.046exp−1.01lnXaxisdprint2eribdprint0.235×exp−1.393lneribdprint2.


## 6.2. Correlation forfrough

The effects of various geometrical parameters onfrough discussed earlier for various Renum show dependence of frough on parameters such as ϕ, dnp, Xaxis/dprint, Yaxis/dprint, erib/dprint, and Renum and can be written as follows:(16)frough=froughRenum,ϕ,dnp,Xaxisdprint,Yaxisdprint,eribdprint.Figure10(a) represents the plotted frough against various Renum and curve fitting using regression analysis through the data point. The resulted regression relation is as follows:(17)frough=AoRenum−0.282.Figure 10
(a) Plot oflnfrough as a function of lnRenum for all the experimental data. (b) Plot of lnAo as a function of lnϕ. (c) Plot of lnBo as a function of lndnp. (d) Plot of lnCo as a function of lnYaxis/dprint. (e) Plot of lnDo as a function of lnXaxis/dprint. (f) Plot of lnCo as a function of lnerib/dprint.
(a)(b)(c)(d)(e)(f)ConstantA0 is reliant on parameters φ, dnp, Xaxis/dprint, Yaxis/dprint, and erib/dprint. The functional relationship between A0 and φ is determined by plotting data values of lnA0 against the values of lnφ, as shown in Figure 10(b) through given relation.(18)frough=BoRenum−0.282ϕ0.542exp0.057lnφ2.The constantB0 depends on φ, dnp, Xaxis/dprint, Yaxis/dprint, and erib/dprint. The log-log data plotted between B0 and dnp get the fitting curve as shown in Figure 10(c), which relates between these two parameters as follows:(19)frough=CoRenum−0.282φ0.542exp0.057lnφ2dnp1.32exp−0.219lndnp2.Similarly constantC0 depends on Xaxis/dprint, Yaxis/dprint, and erib/dprint. The data plotted between B0 and dnp get the fitting curve as shown in log-log plot in Figure 10(d), which relates between these two parameters as follows:(20)frough=D0Renum−0.282ϕ0.542exp0.057lnφ2dnp1.32exp−0.219lndnp2Yaxisdprint1.103exp−1.065lnYaxisdprint2.The data generated from the equation have been plotted betweenlnD0 and lnXaxis/dprint in Figure 10(e) to show the effect of Xaxis/dprint. The curve fitting yields frough that is expressed as follows:(21)frough=E0Renum−0.282φ0.542exp0.057lnφ2dnp1.32exp−0.219lndnp2Yaxisdprint1.103exp−1.065lnYaxisdprint2×exp−0.698lnXaxisdprint2Xaxisdprint0.712.Finally after all the parameters are incorporated as shown in the Figure10(f), the final correlation for frough is given as follows:(22)frough=0.53Renum−0.282φ0.542exp0.057lnφ2dnp1.32×exp−0.219lndnp2Yaxisdprint1.103exp−1.065lnYaxisdprint2×exp−0.698lnXaxisdprint2Xaxisdprint0.712eribdprint0.149×exp−0.933lnerib/dprint2.The Figure11(a) displays the comparison of predicted values and experimental values of Nurough and shows maximum ±8.9% deviation. Also, the average absolute deviation in Nurough is found to be 4.82%. Similarly, Figure 11(b) displays the comparison of predicted values and experimental values of frough and shows maximum ±8.7% deviation. Also, the average absolute deviation in frough is found to be 4.53%. Therefore, it can be concluded that correlations developed in this work can predict the values of Nurough and frough with reasonable accuracy.Figure 11
(a) Comparison of experimental and predicted data ofNurough. (b) Comparison of experimental and predicted values of frough.
(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 ofAl2O3-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 ofNurough was found to be 217, which was found at design parameters values: φ of 4%, dnp of 30 nm, Xaxis/dprint of 1.8, Yaxis/dprint of 1.8, ratio of erib/dprint 1.0, and Renum of 18000.(ii)
The maximum value offrough was found to be 0.132, which was found at design parameters values: φ of 4%, dnp of 30 nm, Xaxis/dprint of 1.8, Yaxis/dprint of 1.8, ratio of erib/dprint 1.0 and Renum of 4000.(iii)
Because of the decrease in reattachment spacing with the increment inRenum, the Nurough increases, whereas the frough 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 forNurough and frough were found to predict the values of Nurough and frough with good accuracy and absolute percentage deviation of 4.82% and 4.53%, respectively.



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*Source: 1603529-2022-04-06.xml*