# Genetic Algorithm Integrated Fuzzy AHP-VIKOR Approach for the Investigation of W-Cut Insert Heat Exchanger for Cooling of Dielectric Fluid Used in Ultra-High Voltage Transformer

Authors: N. Vasantha Gowri; J. Samson Isaac; T. Muralikrishna; G. Suresh Babu; Melvin Victor Depoures; S. Sekar; P. Sasirekha; M. Ramesh; S. Prabhakar
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

This study proposes a novel W-cut twisted tape to evaluate and enhance the effectiveness of novel grooved tube heat exchanger. The experimental investigations are carried out with W-cut inserts, which possess two main control factors, namely, twist tape ratio (y = 3.5 to 6.5) and width tape ratio (WR = 0.43 to 0.6). The investigation results prove that in all cases the energy transmission and friction factor of W-cut insert along with the first law and second law efficiency are considerably higher than the conventional tube arrangement. The empirical model equations for friction factor, performance enhancement ratio, Nusselt number, and rational efficiency are developed, and they show good affinity with experimental data. Since the abovementioned factors have conflicting terms, it is essential to select the proper operating configuration of the heat exchanger in order to increase the thermal energy transfer rate with reduced consumption of pumping energy. The multi-objective genetic algorithm (GA) tool is used to identify the optimum operating conditions. Further, the hybrid multi-criteria decision-making model, FAHP-VIKOR, is applied to select the perfect model from the set of non-dominated solution. The ranking of alternatives is as follows: A4 > A11 > A2 > A9 > A21 > A17 > A1 > A24 > A11 > A23 > A7 > A18 > A13 > A10 > A20 > A5 > A19 > A14 > A16 > A22 > A25 > A6 > A15 > A3 > A8. The optimized configuration of W-cut insert is compared with the former inserts, and the optimal configuration exhibits supreme performance than other inserts.


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## Body

## 1. Introduction

The global thirst for energy is constantly increasing. The increased demand for the cost of energy and materials has led to the production of high-performance compact heat exchanger systems. Heat transfer augmentation techniques are often used in several thermal system applications such as air conditioning and refrigeration systems, heat recovery process, and chemical reactors to increase the overall performance. Augmentation methods such as eddy flow devices (especially twisted tapes) are extensively used in industrial heat exchangers for enhancing convective heat transfer. The presence of typical twisted tape in the tube leads to increased pumping power, which results in thermal performance below unity, so that proper design of twisted tape is required to increase the heat transfer of the equipment with reduced pressure drop [1]. Many research studies have been carried out in past decades to bring down the friction loss by modifying peripheral of twisted tape with different geometry, which includes double V-ribbed twisted tapes [2], horizontal wing-cut TT [3], perforated V-cut and U-cut TT [4], tapered twisted tape [5], TT consisting of wire nails [6], loose-fit perforated twisted tapes [7], twisted tape with alternate axis [8], ribbed twisted tape inserts [9], finned twisted tape [10], cross hollow TT [11], peripherally cut dual TT [12], helix TT with V-cut [13], square and V-cut twisted tape [14], centre-cleared twisted tape [15], dual TT [16], rectangular-cut TT [17], circular tube with lanced ring insert [18], and perforated helical TT [19]. The double V-cut perforated TT with width ratio and twist ratio of 0.27 and 2 enhances the heat transfer by 3.5 times while operating at the flow Reynolds number ranging from 2000 to 25000 [20]. These modified geometry twisted tapes are generally designed to induce enhanced swirl flow near the wall region, which affects the increase in fluid velocity and reduction in boundary layer thickness. This results in augmented heat transfer when compared to typical twisted tape [21]. In general, the performance enhancement ratios of these tapes are higher than the former typical twisted tapes. For instance, the mean performance ratio of the abovementioned tapes is about 2–4 times higher than the typical one. All these aforementioned researches were carried out in plain tube. Wijayanta et al. [22] utilized short-length twisted tape in concentrated tube heat exchangers that improve heater transfer and friction factor by 0.51 and 2.84 times as compared with the conventional heat exchangers for the Reynolds number ranging from 4500 to 18500. Wijayanta et al. [23] compared the thermo-hydraulic performance of classical and square-cut twisted tapes and resulted that the maximum performance of the heat exchangers is ranging from 74.7 to 80.7%. Then, the researchers modified the tabulators into V shapes and enhanced the performance by 97% [24]. Further modifications are carried out in the turbulators by attaching trapezoidal tape winglet that improves the heat transfer and friction factor by 1.91 and 5.2 times, respectively [25]. Yaningsih et al. incorporated perforated holes in the twisted tapes and improved the heater transfer and friction factor performance in the concentric tube heat exchanger by 32% and 47%, respectively [26].In the recent past, the combined passive method has been extensively used to increase the heat transfer rate. The investigations are carried out with geometrically modified twisted tapes, which are fitted in altered test sections. Most of the researchers revealed that the overall performance of the combined insert performs best compared with the individual ones. Bharadwaj et al. [27] initiated the study on thermal and flow characteristics of the grooved tube fitted with twisted tapes. The grooved tube with twisted tape showed better performance than the simple grooved tube. Kumbhar and Sane [28] made an investigation on heat transfer and pressure drop of the dimpled tube with regularly spaced twisted tape insert. The result showed that full-length twisted tape gave a better performance with dimpled tube. Hong et al. [29] made an experimental study on thermal and friction characteristics of spirally grooved tube fitted with twin overlapped twisted tapes. The function of overlapped tapes in spiral grooved tube was not effective when compared to the simple grooved tube. Verma et al. [30] utilized modified helical coil twisted tapes and improved the heat transfer and friction factor by 3.14 times and 19.9 times as compared with the conventional system. Promvonge et al. [31] studied the effect of twisted ratio on performance characteristics in a helical-ribbed tube with twin TTs. Hong et al. [32] numerically examined the effect of thermal and flow performance of the geometrically modified twin TT’s built-in converging-diverging tube. As mentioned above, the combined twisted tape and grooved tube lead to higher heat transfer rate and friction factor than plain tube. Increased Nusselt number results in energy saving, whereas a rise in friction factor upsurges the pumping cost. Hence, the optimization of grooved tube heat exchanger is essential for obtaining high performance with energy saving and pumping cost reduction [33, 34]. Blade vortex generator inserts, multiple helical tape inserts, and diverging perforated cones are special types of inserts that are used as turbulators to reduce the flow blockage in downstream conditions and improve the Reynolds number [35]. Wijayanta et al. [36] proposed that delta wing tape turbulators raise the heat transfer performance by 177% and their design configurations are optimized using the ANN method [37]. The maximum enhancement using T-wing tape tabulator is 1.15 [38]. The double-side winglet integrated twisted tapes raise the Nusselt number by 269% and improve the performance by 10.1 times as compared with plain tube [39, 40]. Veerabhadrappa Bidari et al. [41] modified the turbulators for heat exchangers by punched delta winglet vortex shapes and improved thermo-hydraulic performance by 1.22 times as compared with other systems.The optimization study to improve the design parameters of concentric heat exchangers has been evoked recently. Han et al. [42] did a multi-objective shape optimization study on corrugated tube double pipe heat exchangers using RSM methods. The maximum performance ratio obtained for the r optimum design parameter is 1.12. Swamee et al. [43] analyzed the optimization of heat exchanger with single degree of difficulty. The solution gives an optimal design for the parameters, which include inner diameter and outer diameter of the pipe, and utility flow rate. Iqbal et al. [44] made an optimization study to determine the optimum shape of longitudinal fins on the outer pipe of the heat exchanger for maximizing the Nusselt number. The result showed that the optimum fin profile offers a Nusselt number, which was about 312% higher than the conventional design.Iqbal et al. [45] designed an optimal configuration of a finned annulus with parabolic fins to enhance the heat transfer. As a result, the optimal configuration of parabolic fins is not compared with triangular and trapezoidal fin in all situations and criteria. Despite various researches being carried out on combined passive techniques, very few researchers focused on the optimization of the insert configurations used in heat exchangers. This shortage creates research gap for upcoming researchers to narrow their investigation. The main objective of this study was to perform optimization on design and operating configurations of WCTT fitted in grooved tube heat exchanger using the genetic algorithm and multi-criteria decision-making model.The genetic algorithm is the searching tool that is available for solving problems when the functional characters are unknown. Some of the researchers used GA to optimize the objective parameters of heat exchangers in order to attain the overall best performance [46, 47]. After a thorough analysis on the performance of the heat exchanger with the effect of WCTT geometrical parameters, the new experimental correlations were developed to predict the Nusselt number and friction factor of the tube under the defined range. These correlations were used as the objective function to find the optimal Pareto front solutions. As the given objectives are conflicting terms, GA-based multi-objective optimization is used to attain optimal configurations. Yet, it was difficult to find the best configuration with MOGA, as all the solutions obtained were optimal. Therefore, this study focuses on achieving the best one using a hybrid MCDM model with four performance criteria. The novel FAHP-VIKOR model is used to evaluate alternative configurations and select the best one with conflicting criteria. The application of MCDM in heat transfer area is very meagre [48, 49]. According to the findings, there is research being done on the selection of twisted tape configurations based on thermal and exergy performance using MCDM models. In this work, the FAHP-VIKOR model is used to select the best from the optimum Pareto front solutions of GA. The structure of the present work is given in Figure 1, and their stages are as follows: (i) initially the experimentation is done and the empirical correlations were developed; (ii) followed by multi-objective GA optimization tool to get set of optimum Pareto solution; (iii) finally, assessment method, which includes FAHP-VIKOR model to rank the optimum Pareto front of GA.Figure 1
Phases of evaluation procedure.


## 2. Experimental Work

### 2.1. Twisted Tapes and Grooved Tube

The WCTT is made of aluminium sheets with a width of 0.0235 m, a thickness of 0.003 m, and a length of tape of 1500 mm. Initially, plain twisted tapes are made by twisting one end of the tape at the desired twist tape ratio (y = 3.5 to 6.4), while the other end is clamped. Then, a W-shaped element is removed from the peripheral of the plain tape with a given width ratio (WR = 0.45, 0.6). In this work, the width ratio is defined as the ratio of the width of the W-cut to the width of the twisted tape. The test section used in this setup is an internally grooved tube, which is composed of internal grooves in the plain copper tube with a specified pitch. These values are fixed based on the boundary layer thickness formulation while maintaining the flow Reynolds number ranging from 3000 to 15000. The cut section of the grooved tube is shown in Figure 2, and the geometry of WCTT is depicted in Table 1 and Figure 3.Figure 2
Cut section of internally grooved test section: (a) grooved copper pipe, (b) 3D model of grooved pipe, (c) schematic of groove.Table 1
Experimental details.
Test tube condition (A)Outer diameter of heat exchanger’s inner tube50.5 mmInner diameter of heat exchanger’s inner tube25 mmReynolds number3000–15000Inlet transformer mineral oil temperature55°CLength (L)1500 mmGroove pitch (P)1.2 mmGroove thickness (X)0.2 mmGroove height (e)0.3 mmInlet cold-water temperature31°CW-cut TT (B)Pitch tape length (H)83, 125, and 150 mmTape width (W)23.5 mmDepth of the cut10 mmConfiguration usedW-cut TTTwist tape ratio (y)3.5, 5.3 and 6.5Tape thickness3 mmWidth ratio (WR)0.45 and 0.6Figure 3
Configuration of W-cut TT: (a)y = 6.4, WR = 0.6; (b) y = 6.4, WR = 0.45; (c) y = 5.3, WR = 0.6; (d) y = 5.3, WR = 0.45; (e) y of 5.3, WR = 0.6; (f) y of 5.3, WR = 0.45.


### 2.2. Experimental Apparatus and Test Procedure

Figure4 represents the schematic diagram of experimental setup of double pipe heat exchanger. The system comprises of calming section, test section (smooth or grooved tube) for transformer mineral oil flow, and outer tube for cold-water flow. The calming section and test sections are made of copper tubes with diameter, length, and thickness of 25 mm, 1500 mm, and 1.5 mm, respectively. The outer tube is made of steel tubes with diameter of 52 mm, thickness of 4 mm, and length of 1500 mm. The outer tube is covered with insulation materials such as foam, asbestos, and glass wool to avoid heat loss to the environment. Some other components of heat exchanger include pump, rotameter, control valves, hot and cold-water tank, fan, and control switches. Besides, U-tube manometer and K-type thermocouples were positioned at inlet and outlet of the test section and outer tube to measure the pressure drop and bulk temperatures of the flow. The temperature of the transformer mineral oil is maintained constantly at 55°C using a temperature sensor and relay control unit, while the cold water is maintained at room temperature. The test procedure follows two sections, namely, transformer mineral oil loop and cold-water loop. In the transformer mineral oil loop, the electric heater with 4 KW is used to heat the water. After reaching the specified temperature, the transformer mineral oil is allowed to flow through calming section followed by the test section and it returns to transformer mineral oil tank. As the transformer mineral oil dissipates a certain temperature to the cold water, loss of temperature is noted. To maintain the set temperature, sensor with temperature control relay is used to maintain the transformer mineral oil tank with fixed temperature. Similarly, the same procedure is followed in cold-water loop section. In this, the cold water at room temperature is allowed to flow through the outer pipe and it takes some temperature from the transformer mineral oil. To maintain cold-water temperature, a cooling section is made in which the added temperature nullifies and returns to the cold-water tank with set temperature. The K-type thermocouples with 0.1°C accuracy are used to measure the surface temperature of the internal tube. They are set up at equal distances from the inlet. An 8 mm steel tube is used to protect the thermocouple from direct contact with the water in the annulus section. This tube is brazed to both the outside and inside tubes of the test sections. Experimentation is conducted to determine the thermal performance of heat exchanger at flow rates of 2–10 LPM for transformer mineral oil and constant 10 LPM for cold water. The temperature and pressure of the fluid at inlet and outlet and the surface temperature are measured under steady-state conditions. The details of experimentation are specified in Table 1. Each experiment is repeated three times to ensure its accuracy and repeatability.Figure 4
Schematic diagram of experimental rig.


## 2.1. Twisted Tapes and Grooved Tube

The WCTT is made of aluminium sheets with a width of 0.0235 m, a thickness of 0.003 m, and a length of tape of 1500 mm. Initially, plain twisted tapes are made by twisting one end of the tape at the desired twist tape ratio (y = 3.5 to 6.4), while the other end is clamped. Then, a W-shaped element is removed from the peripheral of the plain tape with a given width ratio (WR = 0.45, 0.6). In this work, the width ratio is defined as the ratio of the width of the W-cut to the width of the twisted tape. The test section used in this setup is an internally grooved tube, which is composed of internal grooves in the plain copper tube with a specified pitch. These values are fixed based on the boundary layer thickness formulation while maintaining the flow Reynolds number ranging from 3000 to 15000. The cut section of the grooved tube is shown in Figure 2, and the geometry of WCTT is depicted in Table 1 and Figure 3.Figure 2
Cut section of internally grooved test section: (a) grooved copper pipe, (b) 3D model of grooved pipe, (c) schematic of groove.Table 1
Experimental details.
Test tube condition (A)Outer diameter of heat exchanger’s inner tube50.5 mmInner diameter of heat exchanger’s inner tube25 mmReynolds number3000–15000Inlet transformer mineral oil temperature55°CLength (L)1500 mmGroove pitch (P)1.2 mmGroove thickness (X)0.2 mmGroove height (e)0.3 mmInlet cold-water temperature31°CW-cut TT (B)Pitch tape length (H)83, 125, and 150 mmTape width (W)23.5 mmDepth of the cut10 mmConfiguration usedW-cut TTTwist tape ratio (y)3.5, 5.3 and 6.5Tape thickness3 mmWidth ratio (WR)0.45 and 0.6Figure 3
Configuration of W-cut TT: (a)y = 6.4, WR = 0.6; (b) y = 6.4, WR = 0.45; (c) y = 5.3, WR = 0.6; (d) y = 5.3, WR = 0.45; (e) y of 5.3, WR = 0.6; (f) y of 5.3, WR = 0.45.


## 2.2. Experimental Apparatus and Test Procedure

Figure4 represents the schematic diagram of experimental setup of double pipe heat exchanger. The system comprises of calming section, test section (smooth or grooved tube) for transformer mineral oil flow, and outer tube for cold-water flow. The calming section and test sections are made of copper tubes with diameter, length, and thickness of 25 mm, 1500 mm, and 1.5 mm, respectively. The outer tube is made of steel tubes with diameter of 52 mm, thickness of 4 mm, and length of 1500 mm. The outer tube is covered with insulation materials such as foam, asbestos, and glass wool to avoid heat loss to the environment. Some other components of heat exchanger include pump, rotameter, control valves, hot and cold-water tank, fan, and control switches. Besides, U-tube manometer and K-type thermocouples were positioned at inlet and outlet of the test section and outer tube to measure the pressure drop and bulk temperatures of the flow. The temperature of the transformer mineral oil is maintained constantly at 55°C using a temperature sensor and relay control unit, while the cold water is maintained at room temperature. The test procedure follows two sections, namely, transformer mineral oil loop and cold-water loop. In the transformer mineral oil loop, the electric heater with 4 KW is used to heat the water. After reaching the specified temperature, the transformer mineral oil is allowed to flow through calming section followed by the test section and it returns to transformer mineral oil tank. As the transformer mineral oil dissipates a certain temperature to the cold water, loss of temperature is noted. To maintain the set temperature, sensor with temperature control relay is used to maintain the transformer mineral oil tank with fixed temperature. Similarly, the same procedure is followed in cold-water loop section. In this, the cold water at room temperature is allowed to flow through the outer pipe and it takes some temperature from the transformer mineral oil. To maintain cold-water temperature, a cooling section is made in which the added temperature nullifies and returns to the cold-water tank with set temperature. The K-type thermocouples with 0.1°C accuracy are used to measure the surface temperature of the internal tube. They are set up at equal distances from the inlet. An 8 mm steel tube is used to protect the thermocouple from direct contact with the water in the annulus section. This tube is brazed to both the outside and inside tubes of the test sections. Experimentation is conducted to determine the thermal performance of heat exchanger at flow rates of 2–10 LPM for transformer mineral oil and constant 10 LPM for cold water. The temperature and pressure of the fluid at inlet and outlet and the surface temperature are measured under steady-state conditions. The details of experimentation are specified in Table 1. Each experiment is repeated three times to ensure its accuracy and repeatability.Figure 4
Schematic diagram of experimental rig.


## 3. Experimental Calculation

### 3.1. Data Reduction

The formulas used to obtain the overall performance of the inserts are as follows.The average of inlet and outlet temperature is taken as bulk temperature, which is expressed as follows:(1)TC=Tci+Tc02,Th=Thi+Tho2.The heat transferred to cold water is provided as follows:(2)QC=mcCpTco−Tci.Heat rejected from the transformer mineral oil is given as follows:(3)Qh=mhcpThi−Tho.Assume that the working fluid has a mean temperature of(4)Qavg=Qc+Qh2.The surface temperature of the tube is given as follows:(5)TW=∑Twi5.The average Nusselt number and average heat transfer coefficient of inner section can be calculated as follows:(6)hi=QavgATh−TW,Nui=hidhk.Equation (7) is used to evaluate the Reynolds number given as follows:(7)Re=Uhdhν.The friction factor due to fluid flow is evaluated as a function of pressure drop given as follows:(8)f=ΔpL/dhρu2/2.The usage of novel W-cut twisted tape and conventional plain tube is compared based on the calculated value of the term performance enhancement factor given as follows:(9)PER=Nu/Nuof/fo1/3.The above equation (9) is used to calculate PER in Section 5.4.


### 3.2. Exergy Analysis

Rational efficiency also known as exergy efficiency or second law efficiency analyses the efficiency of the system by considering the second law of thermodynamics. The rational efficiency of the heat exchanger is defined as the ratio of net exergy output of the system to net exergy input. In this study, second law efficiency is availed to enhance and select the dimensions for the heat exchanger. In general, the qualitative energy of exergy balance is given as follows [50]:(10)Ein−Eout=I.From the equation (10), the net rate of exergy of the heat exchanger is given as follows:(11)Ein=∑i=1N.stream.InmiEstream,iin+∑i=1N.Q.InmiEQ,iin+∑i=1N.W.InmiEW,iin,Eout=∑i=1N.stream.OutmiEstream,iout+∑i=1N.Q.outmiEQ,iout+∑i=1N.W.OutmiEW,iout.The exergy balance for the fluids in heat exchanger [51] can be stated as follows:(12)Ein=Eusefulout+Ewasteout+I.The rate of exergy destruction in terms of generation of entropy is given as follows:(13)I=ToS˙gen.For the concentric heat exchanger, the entropy generation (13) can be written as follows:(14)Sgen=•mhCphlnTh,outTh,in+mcCpclnTc,outTc,in+mhΔPhρhlnTh,out/Th,inTh,out−Th,in+mcΔPcρclnTc,out/Tc,inTc,out−Tc,in.The “exergy output attained” divided by “used exergy” gives the value of rational exergy efficiency stated in the following equation [52]:(15)ε=1−IEused.The overall performance of the system based on the second law is represented by the following equation [53]:(16)Eused=Eobtainedoutput+I.The aforementioned system’s goal to increase the thermal exergy of the hot stream is given as follows:(17)Eobtainedoutput=EhΔT,out−EhΔT,in.Then, by combining equations (16) and (17), the useful value of exergy is given as follows:(18)Eused+I=EcΔp,in+EhΔP,in+EcΔT,in−EcΔT,out+EhΔP,out+EcΔP,out.The system’s overall rational exergy efficiency is given as follows:(19)ηII=∑E•out∑E•in,ηII=mhCphlnTh,out/Th,inmcCpclnTc,out/Tc,in+mhΔPh/ρhlnTh,out/Th,in/Th,out−Th,in+mcΔPc/ρclnTc,out/Tc,in/Tc,out−Tc,in.The above equation is used to calculate rational exergy efficiency in Section5.5.


### 3.3. Uncertainty Analysis

It is possible to calculate the experimental uncertainty of friction factor, Reynolds number, and Nusselt number using the following formulas:(20)ΔReRe=Δmhmh2+Δdidi20.5,(21)ΔNuiNui=Δhihi2+Δdidi2+Δkk20.5,(22)Δff=ΔΔPΔP2+ΔLL2+3Δdidi2+2ΔReRe20.5.The uncertainties and instrument accuracy of the present work are given in Table2.Table 2
Accuracy of the instruments.
AccuracyInstrumentsU-tube manometer±0.0001 mDigital thermocouple±0.1°CRotameter±0.1 L/minTemperature indicators±0.1%Uncertainty (±%)Simple variablePressure head (mm)2.7Temperature (°C)0.08Uncertainty (±%)Compound variablePressure drop (ΔP)5.55Friction factor (f)4.05Nusselt number (Nu)5.38Reynolds number (Re)2.01Heat transfer coefficient (h)5.38Heat transfer (Q)5.24


## 3.1. Data Reduction

The formulas used to obtain the overall performance of the inserts are as follows.The average of inlet and outlet temperature is taken as bulk temperature, which is expressed as follows:(1)TC=Tci+Tc02,Th=Thi+Tho2.The heat transferred to cold water is provided as follows:(2)QC=mcCpTco−Tci.Heat rejected from the transformer mineral oil is given as follows:(3)Qh=mhcpThi−Tho.Assume that the working fluid has a mean temperature of(4)Qavg=Qc+Qh2.The surface temperature of the tube is given as follows:(5)TW=∑Twi5.The average Nusselt number and average heat transfer coefficient of inner section can be calculated as follows:(6)hi=QavgATh−TW,Nui=hidhk.Equation (7) is used to evaluate the Reynolds number given as follows:(7)Re=Uhdhν.The friction factor due to fluid flow is evaluated as a function of pressure drop given as follows:(8)f=ΔpL/dhρu2/2.The usage of novel W-cut twisted tape and conventional plain tube is compared based on the calculated value of the term performance enhancement factor given as follows:(9)PER=Nu/Nuof/fo1/3.The above equation (9) is used to calculate PER in Section 5.4.


## 3.2. Exergy Analysis

Rational efficiency also known as exergy efficiency or second law efficiency analyses the efficiency of the system by considering the second law of thermodynamics. The rational efficiency of the heat exchanger is defined as the ratio of net exergy output of the system to net exergy input. In this study, second law efficiency is availed to enhance and select the dimensions for the heat exchanger. In general, the qualitative energy of exergy balance is given as follows [50]:(10)Ein−Eout=I.From the equation (10), the net rate of exergy of the heat exchanger is given as follows:(11)Ein=∑i=1N.stream.InmiEstream,iin+∑i=1N.Q.InmiEQ,iin+∑i=1N.W.InmiEW,iin,Eout=∑i=1N.stream.OutmiEstream,iout+∑i=1N.Q.outmiEQ,iout+∑i=1N.W.OutmiEW,iout.The exergy balance for the fluids in heat exchanger [51] can be stated as follows:(12)Ein=Eusefulout+Ewasteout+I.The rate of exergy destruction in terms of generation of entropy is given as follows:(13)I=ToS˙gen.For the concentric heat exchanger, the entropy generation (13) can be written as follows:(14)Sgen=•mhCphlnTh,outTh,in+mcCpclnTc,outTc,in+mhΔPhρhlnTh,out/Th,inTh,out−Th,in+mcΔPcρclnTc,out/Tc,inTc,out−Tc,in.The “exergy output attained” divided by “used exergy” gives the value of rational exergy efficiency stated in the following equation [52]:(15)ε=1−IEused.The overall performance of the system based on the second law is represented by the following equation [53]:(16)Eused=Eobtainedoutput+I.The aforementioned system’s goal to increase the thermal exergy of the hot stream is given as follows:(17)Eobtainedoutput=EhΔT,out−EhΔT,in.Then, by combining equations (16) and (17), the useful value of exergy is given as follows:(18)Eused+I=EcΔp,in+EhΔP,in+EcΔT,in−EcΔT,out+EhΔP,out+EcΔP,out.The system’s overall rational exergy efficiency is given as follows:(19)ηII=∑E•out∑E•in,ηII=mhCphlnTh,out/Th,inmcCpclnTc,out/Tc,in+mhΔPh/ρhlnTh,out/Th,in/Th,out−Th,in+mcΔPc/ρclnTc,out/Tc,in/Tc,out−Tc,in.The above equation is used to calculate rational exergy efficiency in Section5.5.


## 3.3. Uncertainty Analysis

It is possible to calculate the experimental uncertainty of friction factor, Reynolds number, and Nusselt number using the following formulas:(20)ΔReRe=Δmhmh2+Δdidi20.5,(21)ΔNuiNui=Δhihi2+Δdidi2+Δkk20.5,(22)Δff=ΔΔPΔP2+ΔLL2+3Δdidi2+2ΔReRe20.5.The uncertainties and instrument accuracy of the present work are given in Table2.Table 2
Accuracy of the instruments.
AccuracyInstrumentsU-tube manometer±0.0001 mDigital thermocouple±0.1°CRotameter±0.1 L/minTemperature indicators±0.1%Uncertainty (±%)Simple variablePressure head (mm)2.7Temperature (°C)0.08Uncertainty (±%)Compound variablePressure drop (ΔP)5.55Friction factor (f)4.05Nusselt number (Nu)5.38Reynolds number (Re)2.01Heat transfer coefficient (h)5.38Heat transfer (Q)5.24


## 4. Optimization Method

### 4.1. Multi-Objective Genetic Algorithm (MOGA)

Multi-objective optimization (MOO) [54] refers to the solution of problems that deal with more than one objective. In actual engineering problems, most of their objectives are at least in partial conflict with one another. It is difficult to detect the maximum or minimum of multi-objectives simultaneously available for single objective optimization. The term MOO generally refers to the set of non-inferior solution points, known as Pareto optimal solutions [38]. The best solution for the multi-objective problem can be selected from the Pareto set rendering to decision-making methods. MOGA is the multi-objective optimization tool for solving problems with more than one design objective, and it has unique supremacy in multi-objective programming. The steps involved in MOGA are as follows:Step 1: generate random population of chromosomes with a static initial sizeStep 2: assess the fitness of the objective function solutions in the populationStep 3: generate new offspring from parent population using selection, crossover, and mutationStep 4: finally, this process terminates only when the optimal solution is achieved, else it goes to Step 2Evidently, the offspring population will be comparatively good than the parent population as it undergoes the survival of fittest principle. The solution becomes closer and closer to optimal with gradual evaluation. Finally, when GA is convergent, the set of Pareto front optimized solutions is obtained. In this study, hybrid MCDM (FAHP-VIKOR) is employed to select the finest solution from the set of Pareto front solutions.


### 4.2. Multi-Criteria Decision-Making (MCDM)

Multi-criteria decision-making (MCDM) is a tool, which is used for evaluating the trade-offs between several performance criteria to rank, prioritize, or choose the best from the list of alternatives. The MCDM problem selected in this study is to rank the set of Pareto front solutions (taken as alternatives) obtained from GA. The prioritization of these alternatives is done using the following criteria: thermal enhancement ratio (TER), friction enhancement ratio (FER), performance enhancement ratio (PER), and rational exergetic efficiency (REE). Figure5 depicts the various criteria used in performance evaluation. In this MCDM model, at the initial stage, the FAHP is used to give weightage for the criteria, followed by the ranking of alternatives with VIKOR.Figure 5
Different criteria for performance evaluation of grooved tube with W-cut twisted insert.


#### 4.2.1. FAHP Method

Satty and decision [55] proposed the AHP method. In AHP, the decision problem is classified into different levels of hierarchy, each level comprising a finite number of elements [56]. A fuzzy set comprises membership function associated in the range of zero and one. Kwong and Bai [46] proposed the concept of fuzzy AHP. FAHP offers triangular and trapezoidal fuzzy number in accordance with the decision-maker need. The authors of past used FAHP tool to assign weightage to their criteria [57, 58]. The hierarchy of decision-making problem is shown in Figure 6. The steps of FAHP method are as follows:Step 1: complex problem is broken down into simple hierarchy of interconnected criteria.Step 2: the triangular fuzzy number (TFN)M = (l, m, u) is used to fuzzy the pairwise decision matrix A. The membership function of the TFNs used to signify the assessment is M1, M2, M3, M4, M5, M6, M7, M8, and M9 (Table 3). Then, the decision matrix is normalized and the relative weight of each matrix is identified. The expression for eigenvector can be given as follows:(23)Aw=λmaxW.Step 3: consistency index (CI) values are identified:(24)CI=λmaxn−1.Step 4:the consistency ratio (CR) is used to define the consistency of the inputs and is given as follows:(25)CR=CIRCI.Figure 6
Decision hierarchy for performance prediction of W-cut TT.Table 3
Fuzzy numbers’ membership function.
Scale of linguistic importanceFuzzy number(L, M, U)-TFN(1/U, 1/M, 1/L) : reciprocal of TFNEqually: importantM1(1, 1, 1)(1, 1, 1)Equally: moderate importantM2(1, 2, 3)(0.33, 0.5, 1)Weakly: importantM3(2, 3, 4)(0.25, 00.33, 0.5)Moderate: importantM4(3, 4, 5)(0.2, 0.25, 0.33)Moderately: strong importantM5(4, 5, 6)(0.16, 0.2, 0.25)Strongly: importantM6(5, 6, 7)(0.14, 0.16, 0.2)Very strongly: importantM7(6, 7, 8)(0.12, 0.14, 0.16)Very strongly: extreme importantM8(7, 8, 9)(0.11, 0.12, 0.14)Absolutely: importantM9(8, 9, 10)(0.10, 0.11, 0.12)In case the CR value is greater than the acceptable value, inconsistency in judgment occurs and the entire assessment must be reassessed and enhanced. The selection of RCI values based on the number of criteria is given in Table4.Table 4
Random consistency index.
No.12345678910RCI000.520.891.111.251.351.401.451.49


### 4.3. VIKOR Method

Opricovic proposed the VIKOR method to solve decision-making problems with conflicting criteria [59]. In this method, the ranking of the alternatives is influenced by the initial weight of the problem and its closeness to the objective. In the study, many researchers used the VIKOR method to solve multi-objective problems due to its aptness and feasibility. Some are as follows: Ilangkumaran et al. [60–62] used VIKOR along with some hybrid MCDM models to prioritize best treatment for wastewater management. The VIKOR approach was used by Yazdani and Payam [63] to prioritize material selection for MEMS applications. From the above facts, it is evident that the VIKOR model can be used as an effective tool for ranking alternatives [64–66].The VIKOR method is started with the subsequent form Lp metric:(26)LPj=∑i=1nwifi∗−fijfi∗−fi−p1/P1≤P≤∞,j=1,2,…,J.The compromising ranking method of VIKOR is given as follows:Step 1. Matrix Normalization: letXij be the evaluation matrix with values of the jth alternatives and ith criteria, and then, the normalized matrix fij is calculated using(27)fij=xij∑i=1mxij2,i=1,2m;j=1,2,n.Step2.Evaluate the Worst and Best Values (f∗, f−): the best and the worst values are given by the following relation:(28)fi∗=maxjfij,fi−=minjfij.Step 3.Calculate the Sjand Rj Values: the utility measure Sj and regret measure Rj are determined using the relation:(29)Sj=∑i=1nwifi∗−fijfi∗−fi−,(30)Rj=maxifi∗−fijfi∗−fi−.Step 4.Compute the VIKOR Index: the VIKOR index value Qj is calculated using(31)Qj=vSj−S∗S−−S∗+1−vRj−R∗R−−R∗,where(32)S∗=minjSj,S−=maxjSj,R∗=minjRj,R−=maxjRj.Here,v = 0.5.Step 5.Rank the Alternatives: the option with the lowest VIKOR index is chosen as the best value.


## 4.1. Multi-Objective Genetic Algorithm (MOGA)

Multi-objective optimization (MOO) [54] refers to the solution of problems that deal with more than one objective. In actual engineering problems, most of their objectives are at least in partial conflict with one another. It is difficult to detect the maximum or minimum of multi-objectives simultaneously available for single objective optimization. The term MOO generally refers to the set of non-inferior solution points, known as Pareto optimal solutions [38]. The best solution for the multi-objective problem can be selected from the Pareto set rendering to decision-making methods. MOGA is the multi-objective optimization tool for solving problems with more than one design objective, and it has unique supremacy in multi-objective programming. The steps involved in MOGA are as follows:Step 1: generate random population of chromosomes with a static initial sizeStep 2: assess the fitness of the objective function solutions in the populationStep 3: generate new offspring from parent population using selection, crossover, and mutationStep 4: finally, this process terminates only when the optimal solution is achieved, else it goes to Step 2Evidently, the offspring population will be comparatively good than the parent population as it undergoes the survival of fittest principle. The solution becomes closer and closer to optimal with gradual evaluation. Finally, when GA is convergent, the set of Pareto front optimized solutions is obtained. In this study, hybrid MCDM (FAHP-VIKOR) is employed to select the finest solution from the set of Pareto front solutions.


## 4.2. Multi-Criteria Decision-Making (MCDM)

Multi-criteria decision-making (MCDM) is a tool, which is used for evaluating the trade-offs between several performance criteria to rank, prioritize, or choose the best from the list of alternatives. The MCDM problem selected in this study is to rank the set of Pareto front solutions (taken as alternatives) obtained from GA. The prioritization of these alternatives is done using the following criteria: thermal enhancement ratio (TER), friction enhancement ratio (FER), performance enhancement ratio (PER), and rational exergetic efficiency (REE). Figure5 depicts the various criteria used in performance evaluation. In this MCDM model, at the initial stage, the FAHP is used to give weightage for the criteria, followed by the ranking of alternatives with VIKOR.Figure 5
Different criteria for performance evaluation of grooved tube with W-cut twisted insert.


### 4.2.1. FAHP Method

Satty and decision [55] proposed the AHP method. In AHP, the decision problem is classified into different levels of hierarchy, each level comprising a finite number of elements [56]. A fuzzy set comprises membership function associated in the range of zero and one. Kwong and Bai [46] proposed the concept of fuzzy AHP. FAHP offers triangular and trapezoidal fuzzy number in accordance with the decision-maker need. The authors of past used FAHP tool to assign weightage to their criteria [57, 58]. The hierarchy of decision-making problem is shown in Figure 6. The steps of FAHP method are as follows:Step 1: complex problem is broken down into simple hierarchy of interconnected criteria.Step 2: the triangular fuzzy number (TFN)M = (l, m, u) is used to fuzzy the pairwise decision matrix A. The membership function of the TFNs used to signify the assessment is M1, M2, M3, M4, M5, M6, M7, M8, and M9 (Table 3). Then, the decision matrix is normalized and the relative weight of each matrix is identified. The expression for eigenvector can be given as follows:(23)Aw=λmaxW.Step 3: consistency index (CI) values are identified:(24)CI=λmaxn−1.Step 4:the consistency ratio (CR) is used to define the consistency of the inputs and is given as follows:(25)CR=CIRCI.Figure 6
Decision hierarchy for performance prediction of W-cut TT.Table 3
Fuzzy numbers’ membership function.
Scale of linguistic importanceFuzzy number(L, M, U)-TFN(1/U, 1/M, 1/L) : reciprocal of TFNEqually: importantM1(1, 1, 1)(1, 1, 1)Equally: moderate importantM2(1, 2, 3)(0.33, 0.5, 1)Weakly: importantM3(2, 3, 4)(0.25, 00.33, 0.5)Moderate: importantM4(3, 4, 5)(0.2, 0.25, 0.33)Moderately: strong importantM5(4, 5, 6)(0.16, 0.2, 0.25)Strongly: importantM6(5, 6, 7)(0.14, 0.16, 0.2)Very strongly: importantM7(6, 7, 8)(0.12, 0.14, 0.16)Very strongly: extreme importantM8(7, 8, 9)(0.11, 0.12, 0.14)Absolutely: importantM9(8, 9, 10)(0.10, 0.11, 0.12)In case the CR value is greater than the acceptable value, inconsistency in judgment occurs and the entire assessment must be reassessed and enhanced. The selection of RCI values based on the number of criteria is given in Table4.Table 4
Random consistency index.
No.12345678910RCI000.520.891.111.251.351.401.451.49


## 4.2.1. FAHP Method

Satty and decision [55] proposed the AHP method. In AHP, the decision problem is classified into different levels of hierarchy, each level comprising a finite number of elements [56]. A fuzzy set comprises membership function associated in the range of zero and one. Kwong and Bai [46] proposed the concept of fuzzy AHP. FAHP offers triangular and trapezoidal fuzzy number in accordance with the decision-maker need. The authors of past used FAHP tool to assign weightage to their criteria [57, 58]. The hierarchy of decision-making problem is shown in Figure 6. The steps of FAHP method are as follows:Step 1: complex problem is broken down into simple hierarchy of interconnected criteria.Step 2: the triangular fuzzy number (TFN)M = (l, m, u) is used to fuzzy the pairwise decision matrix A. The membership function of the TFNs used to signify the assessment is M1, M2, M3, M4, M5, M6, M7, M8, and M9 (Table 3). Then, the decision matrix is normalized and the relative weight of each matrix is identified. The expression for eigenvector can be given as follows:(23)Aw=λmaxW.Step 3: consistency index (CI) values are identified:(24)CI=λmaxn−1.Step 4:the consistency ratio (CR) is used to define the consistency of the inputs and is given as follows:(25)CR=CIRCI.Figure 6
Decision hierarchy for performance prediction of W-cut TT.Table 3
Fuzzy numbers’ membership function.
Scale of linguistic importanceFuzzy number(L, M, U)-TFN(1/U, 1/M, 1/L) : reciprocal of TFNEqually: importantM1(1, 1, 1)(1, 1, 1)Equally: moderate importantM2(1, 2, 3)(0.33, 0.5, 1)Weakly: importantM3(2, 3, 4)(0.25, 00.33, 0.5)Moderate: importantM4(3, 4, 5)(0.2, 0.25, 0.33)Moderately: strong importantM5(4, 5, 6)(0.16, 0.2, 0.25)Strongly: importantM6(5, 6, 7)(0.14, 0.16, 0.2)Very strongly: importantM7(6, 7, 8)(0.12, 0.14, 0.16)Very strongly: extreme importantM8(7, 8, 9)(0.11, 0.12, 0.14)Absolutely: importantM9(8, 9, 10)(0.10, 0.11, 0.12)In case the CR value is greater than the acceptable value, inconsistency in judgment occurs and the entire assessment must be reassessed and enhanced. The selection of RCI values based on the number of criteria is given in Table4.Table 4
Random consistency index.
No.12345678910RCI000.520.891.111.251.351.401.451.49


## 4.3. VIKOR Method

Opricovic proposed the VIKOR method to solve decision-making problems with conflicting criteria [59]. In this method, the ranking of the alternatives is influenced by the initial weight of the problem and its closeness to the objective. In the study, many researchers used the VIKOR method to solve multi-objective problems due to its aptness and feasibility. Some are as follows: Ilangkumaran et al. [60–62] used VIKOR along with some hybrid MCDM models to prioritize best treatment for wastewater management. The VIKOR approach was used by Yazdani and Payam [63] to prioritize material selection for MEMS applications. From the above facts, it is evident that the VIKOR model can be used as an effective tool for ranking alternatives [64–66].The VIKOR method is started with the subsequent form Lp metric:(26)LPj=∑i=1nwifi∗−fijfi∗−fi−p1/P1≤P≤∞,j=1,2,…,J.The compromising ranking method of VIKOR is given as follows:Step 1. Matrix Normalization: letXij be the evaluation matrix with values of the jth alternatives and ith criteria, and then, the normalized matrix fij is calculated using(27)fij=xij∑i=1mxij2,i=1,2m;j=1,2,n.Step2.Evaluate the Worst and Best Values (f∗, f−): the best and the worst values are given by the following relation:(28)fi∗=maxjfij,fi−=minjfij.Step 3.Calculate the Sjand Rj Values: the utility measure Sj and regret measure Rj are determined using the relation:(29)Sj=∑i=1nwifi∗−fijfi∗−fi−,(30)Rj=maxifi∗−fijfi∗−fi−.Step 4.Compute the VIKOR Index: the VIKOR index value Qj is calculated using(31)Qj=vSj−S∗S−−S∗+1−vRj−R∗R−−R∗,where(32)S∗=minjSj,S−=maxjSj,R∗=minjRj,R−=maxjRj.Here,v = 0.5.Step 5.Rank the Alternatives: the option with the lowest VIKOR index is chosen as the best value.


## 5. Result and Discussion

From the study [49], it is observed that the modification in peripheral geometry of TT in a smooth tube tends to increase the friction factor, but in terms of the Nusselt number, only a minor improvement is found. The current research looks at the effect of WCTT in grooved tubes. From experimental results, it is observed that different geometrical and operating parameters expose their unique influence on heat transfer, pressure drop, performance enhancement ratio, and rational efficiency. The effects of these parameters are conversed as follows.


### 5.1. Validation Conventional System

To validate the system’s reliability, the plain tube result is compared to the result obtained from the standard correlations given in equations (33) to (36). Once the system’s consistency is established, the plain tube heat exchanger results are recorded and compared to the standard correlation results (shown in Figure 7). The results showed that the thermal and flow performance of the plain tube agreed well with the standard correlation in the range of 5% to 8%, respectively. As a result, the results show that the experimental facility and measurement techniques are reliable. The plain tube results are used in upcoming calculations.Figure 7
Validation of the plain tube.Convective heat transfer correlations—Dittus–Boelter correlation is as follows:(33)Nu=0.023Re0.8Pr0.3forRe>10,000.Gnielinski correlation is as follows:(34)Nu=f/8Re−1000Pr1+12.7f/80.5Pr0.33−1for3000<Re<500,000.Correlations for the friction factor—correlation of Blasius is as follows:(35)f=0.316Re−.025forRe<20,000.Petukhov correlation is as follows:(36)f=0.790lnRe−1.64−2for3000<Re<500,000.


### 5.2. Heat Transfer Investigation of WCTT Insert

The effect of twist ratio (y) on heat transfer features in the grooved tube fitted with WCTT is presented in Figure 8. From Figure 8(a), it is observed that the Nusselt number (Nu) of the tube increases with an increase in Reynolds number and decreases in twist ratio. The WCTT having the value of y = 3.5 provides maximum heat transfer compared with the other twist ratios. This eddy flow generated by the TTs provides better flow mixing between core and wall of the tube, (2) strong turbulence and vortices near the W-cut region lead to thermal boundary layer destruction, and (3) in addition, the groove in tube causes flow separation, and reattachment and recirculation of water result in effective fluid mixing. The abovementioned flow phenomenon stimulates an increase in the turbulent intensity and tangential turbulent fluctuation, which affects the hydrodynamic and thermal boundary layer and therefore increases the heat transfer of the tube. Over the range studied, the WCTT in grooved tube with y = 3.5 provides the mean Nusselt number of about 12%, 20%, and 48% higher than those in the grooved tube with twist ratio of y = 5.3 and y = 6.5, and plain tube, respectively.Figure 8
Nusselt number of WCTT: (a) Nu vs. Re. (b) Nuo/Nup vs. Re.
(a)(b)Similarly, the Nu ratio decreases with increasing Reynolds number in all cases, indicating that at lower Reynolds numbers, the flow cannot generate high turbulence on its own, and thus, introducing TT generates secondary swirl on fluid flow and increases heat transfer. At higher Reynolds numbers, the grooved tube itself creates a massive secondary swirl, and hence, the impact of TT becomes less effective. As expected, the Nu ratio of WCTT in the grooved tube compared with the simple tube is always greater than unity. From Figure8(b), it is obvious that the Nu ratio of the WCTT in the grooved tube is approximately 2–2.5 times higher than in the simple grooved tube and approximately 3.0–3.4 times higher than in the simple tube.Further, the heat transfer is also governed by the WR of the tape, as shown in Figure8(a). It is observed that the effect of WR at a higher range yields a larger value of the Nusselt number compared with the smaller one. This is due to the effect of augmented eddies and vorticities behind the cut, which results in the promotion of turbulent intensity and fluid mixing and additionally increases the heat transfer rate. Figure 8 shows that the mean Nusselt number obtained using a W-cut insert in a grooved tube at WR = 0.6 and y = 3.5, 5.3, and 6.4 is 2.08%, 2.13%, and 2.17% greater than the grooved inner tube at the same condition, WR = 0.4.


### 5.3. Friction Factor Characteristics of W-Cut TT Insert

Figure9 depicts the magnitude of friction with change in mass flow rate of working fluid passed inside the grooved tube fitted with WCTT and plain tube. It is observed that the friction factor followed the decreasing trend with rise in Reynolds number and showed supremacy at lower twist ratio (y = 3.5) compared with that of higher twist ratios (y = 5.3 and 6.5). This is due to the effect of secondary flow generated by the WCTTs. At lower twist ratio, the turbulence intensity near the wall increases, which augments the fluid friction. In addition, the lower twist ratio TTs blocks the fluid passage in large volume and increases the surface contact, which results in increased static pressure. Also, the eddies and vortices made by W-cut create extensive friction loss that is normally negligible in case of plain tubes. From the graph, it is clear that the friction factor of the grooved tube with WCTT at y = 3.5 is 1.14 and 1.26 times higher than the given tube with y = 5.3 and 6.5, consecutively, and 4.5 times greater as compared with plain tube.Figure 9
Friction factor of W-cut insert: (a)f vs. Re. (b) fo/fp vs. Re.
(a)(b)Figures9(a) and 9(b) illustrate the friction factor ratio (fo/fP) of WCTT with different twist ratios (y) and width ratios (WRs). The result exposed that the friction factor nature increases with the rise in the Reynolds number. Qualitatively, the pressure loss generated by the WCTT is directly related to the results of the heat transfer discussed in Section 5.2 because an effective heat transfer is caused by the strong turbulence in boundary layer, which concurrently increases the interaction of pressure force with inertia force and thus enhances the dynamic pressure loss. At smaller twist ratio (y = 3.5), the friction factor ratio of WCTT is substantially higher than those associated with larger twist ratio (y = 5.3 and 6.5), which are, respectively, in range of 1.10–1.45 times and about 4.95 times as improved with the conventional system.The effect of width ratio (WR) on friction factor follows the same trend as the Nusselt number, whereas the friction factor increases with rise in width ratio. For the present range, the mean friction factor of WCTTs with WR of 0.4 aty = 3.5, 5.3, and 6.4 is higher than that caused by the use of W-cut insert at WR of 0.6 for given twist ratio by around 5%, 6%, and 8%, respectively.


### 5.4. Performance Enhancement Ratio

An equal pumping power comparison is attained to evaluate the heat transfer efficiency in terms of PER of the grooved tube fitted with WCTT. The variation of PER with Reynolds number for WCTT at various twist ratios (y) and width ratios is depicted in Figure 10. In all cases, the value of PER decreases with an increase in the Reynolds number. Also, at a given Reynolds number, the PER of the tube tends to increase with a decrease in y and WR. In all of the above cases, the PER shows how this tape helps save energy, and it also shows how the heat transfer ratio affects the friction penalty.Figure 10
Performance enhancement ratio (ηI) vs. Reynolds number (Re).From the figure, it is observed that the mean enhancement ratio of the grooved tube with WCTT aty = 3.5 is 1.05 times, 1.5 times, and 2.08 times higher than the grooved tube with y = 5.3 and 6.4 and plain grooved tube, respectively. In addition, the WCTT at WR of 0.45 gives a maximum enhancement ratio and is about 0.9–1.4 times higher than the tape with WR of 0.6. This is due to the effect of added pressure drop at higher WR. Hence, it is concluded that the superior performance can be obtained by lower twist ratio and width ratio. However, the above result cannot be optimum, as the pressure drop reaches extreme at this condition. To overcome this problem, MOO techniques are used in this work, to find the optimum results. The parameters such as twist ratio and width ratio of TTs along with the Reynolds number play a major role in selecting optimum thermal performance conditions.


### 5.5. Rational Analysis of Exergy

The exergetic efficiency of WCTT with various geometries is discussed in this segment. Figure11 shows the exergy efficiency of the inserts as a function of Reynolds number for various twist ratios and width ratios. When compared to plain grooved tube, the combined effect of WCTTs with grooved tube demonstrated superior rational exergy efficiency. W-cut causes stronger eddies and vortices, which increases decayed boundary layer thickness and turbulent intensity near the walls, increasing exergetic efficiency. The graph clearly shows that the rational exergetic efficiency of the WCTT with lower twist ratio (y = 3.5) outperformed the inserts with higher twist ratio (y = 5.3 and 6.5), respectively, by 1.04 and 1.10 times. Aside from that, the width ratio, like the PER discussed in Section 5.4, has an effect on rational exergy efficiency. It is difficult to choose the best because its performance in terms of width ratio is contradictory. As a result, the MOO technique is used in this study to achieve the best results.Figure 11
Exergy efficiency (ηII) vs. Reynolds number (Re).


### 5.6. Empirical Correlation

The empirical correlation for Nusselt number, friction factor, performance enhancement ratio, and rational efficiency of the grooved tube with WCTT was developed, as a function of tape geometry (y and WR) and Reynolds number using the least-squares regression analysis. The empirical correlations are as follows:(37)Nu=0.493Re0.635y−0.33Wr0.079,(38)f=12.3Re−0.401y−0.429Wr0.21,(39)ηI=2.81Re−0.071y−0.191Wr−0.0384,(40)ηII=1.69Re−0.095y−0.1041Wr0.06.Figures12(a) to 12(d) depict the predicted data of Nusselt number, friction factor, performance enhancement ratio, and rational efficiency from the above correlations (equations 37–(40)) in comparison with those obtained from the experimental data. Apparently, the predicted data are in good concordance with the experimental data with the discrepancy of ±4%, ±7%,± 3%, and ±2% for Nu, f, ηI, and ηII, respectively.Figure 12
Assessment between the experimental and developed correlation results: (a) Nu, (b) fr, (c)η, (d) ηII.
(a)(b)(c)(d)


## 5.1. Validation Conventional System

To validate the system’s reliability, the plain tube result is compared to the result obtained from the standard correlations given in equations (33) to (36). Once the system’s consistency is established, the plain tube heat exchanger results are recorded and compared to the standard correlation results (shown in Figure 7). The results showed that the thermal and flow performance of the plain tube agreed well with the standard correlation in the range of 5% to 8%, respectively. As a result, the results show that the experimental facility and measurement techniques are reliable. The plain tube results are used in upcoming calculations.Figure 7
Validation of the plain tube.Convective heat transfer correlations—Dittus–Boelter correlation is as follows:(33)Nu=0.023Re0.8Pr0.3forRe>10,000.Gnielinski correlation is as follows:(34)Nu=f/8Re−1000Pr1+12.7f/80.5Pr0.33−1for3000<Re<500,000.Correlations for the friction factor—correlation of Blasius is as follows:(35)f=0.316Re−.025forRe<20,000.Petukhov correlation is as follows:(36)f=0.790lnRe−1.64−2for3000<Re<500,000.


## 5.2. Heat Transfer Investigation of WCTT Insert

The effect of twist ratio (y) on heat transfer features in the grooved tube fitted with WCTT is presented in Figure 8. From Figure 8(a), it is observed that the Nusselt number (Nu) of the tube increases with an increase in Reynolds number and decreases in twist ratio. The WCTT having the value of y = 3.5 provides maximum heat transfer compared with the other twist ratios. This eddy flow generated by the TTs provides better flow mixing between core and wall of the tube, (2) strong turbulence and vortices near the W-cut region lead to thermal boundary layer destruction, and (3) in addition, the groove in tube causes flow separation, and reattachment and recirculation of water result in effective fluid mixing. The abovementioned flow phenomenon stimulates an increase in the turbulent intensity and tangential turbulent fluctuation, which affects the hydrodynamic and thermal boundary layer and therefore increases the heat transfer of the tube. Over the range studied, the WCTT in grooved tube with y = 3.5 provides the mean Nusselt number of about 12%, 20%, and 48% higher than those in the grooved tube with twist ratio of y = 5.3 and y = 6.5, and plain tube, respectively.Figure 8
Nusselt number of WCTT: (a) Nu vs. Re. (b) Nuo/Nup vs. Re.
(a)(b)Similarly, the Nu ratio decreases with increasing Reynolds number in all cases, indicating that at lower Reynolds numbers, the flow cannot generate high turbulence on its own, and thus, introducing TT generates secondary swirl on fluid flow and increases heat transfer. At higher Reynolds numbers, the grooved tube itself creates a massive secondary swirl, and hence, the impact of TT becomes less effective. As expected, the Nu ratio of WCTT in the grooved tube compared with the simple tube is always greater than unity. From Figure8(b), it is obvious that the Nu ratio of the WCTT in the grooved tube is approximately 2–2.5 times higher than in the simple grooved tube and approximately 3.0–3.4 times higher than in the simple tube.Further, the heat transfer is also governed by the WR of the tape, as shown in Figure8(a). It is observed that the effect of WR at a higher range yields a larger value of the Nusselt number compared with the smaller one. This is due to the effect of augmented eddies and vorticities behind the cut, which results in the promotion of turbulent intensity and fluid mixing and additionally increases the heat transfer rate. Figure 8 shows that the mean Nusselt number obtained using a W-cut insert in a grooved tube at WR = 0.6 and y = 3.5, 5.3, and 6.4 is 2.08%, 2.13%, and 2.17% greater than the grooved inner tube at the same condition, WR = 0.4.


## 5.3. Friction Factor Characteristics of W-Cut TT Insert

Figure9 depicts the magnitude of friction with change in mass flow rate of working fluid passed inside the grooved tube fitted with WCTT and plain tube. It is observed that the friction factor followed the decreasing trend with rise in Reynolds number and showed supremacy at lower twist ratio (y = 3.5) compared with that of higher twist ratios (y = 5.3 and 6.5). This is due to the effect of secondary flow generated by the WCTTs. At lower twist ratio, the turbulence intensity near the wall increases, which augments the fluid friction. In addition, the lower twist ratio TTs blocks the fluid passage in large volume and increases the surface contact, which results in increased static pressure. Also, the eddies and vortices made by W-cut create extensive friction loss that is normally negligible in case of plain tubes. From the graph, it is clear that the friction factor of the grooved tube with WCTT at y = 3.5 is 1.14 and 1.26 times higher than the given tube with y = 5.3 and 6.5, consecutively, and 4.5 times greater as compared with plain tube.Figure 9
Friction factor of W-cut insert: (a)f vs. Re. (b) fo/fp vs. Re.
(a)(b)Figures9(a) and 9(b) illustrate the friction factor ratio (fo/fP) of WCTT with different twist ratios (y) and width ratios (WRs). The result exposed that the friction factor nature increases with the rise in the Reynolds number. Qualitatively, the pressure loss generated by the WCTT is directly related to the results of the heat transfer discussed in Section 5.2 because an effective heat transfer is caused by the strong turbulence in boundary layer, which concurrently increases the interaction of pressure force with inertia force and thus enhances the dynamic pressure loss. At smaller twist ratio (y = 3.5), the friction factor ratio of WCTT is substantially higher than those associated with larger twist ratio (y = 5.3 and 6.5), which are, respectively, in range of 1.10–1.45 times and about 4.95 times as improved with the conventional system.The effect of width ratio (WR) on friction factor follows the same trend as the Nusselt number, whereas the friction factor increases with rise in width ratio. For the present range, the mean friction factor of WCTTs with WR of 0.4 aty = 3.5, 5.3, and 6.4 is higher than that caused by the use of W-cut insert at WR of 0.6 for given twist ratio by around 5%, 6%, and 8%, respectively.


## 5.4. Performance Enhancement Ratio

An equal pumping power comparison is attained to evaluate the heat transfer efficiency in terms of PER of the grooved tube fitted with WCTT. The variation of PER with Reynolds number for WCTT at various twist ratios (y) and width ratios is depicted in Figure 10. In all cases, the value of PER decreases with an increase in the Reynolds number. Also, at a given Reynolds number, the PER of the tube tends to increase with a decrease in y and WR. In all of the above cases, the PER shows how this tape helps save energy, and it also shows how the heat transfer ratio affects the friction penalty.Figure 10
Performance enhancement ratio (ηI) vs. Reynolds number (Re).From the figure, it is observed that the mean enhancement ratio of the grooved tube with WCTT aty = 3.5 is 1.05 times, 1.5 times, and 2.08 times higher than the grooved tube with y = 5.3 and 6.4 and plain grooved tube, respectively. In addition, the WCTT at WR of 0.45 gives a maximum enhancement ratio and is about 0.9–1.4 times higher than the tape with WR of 0.6. This is due to the effect of added pressure drop at higher WR. Hence, it is concluded that the superior performance can be obtained by lower twist ratio and width ratio. However, the above result cannot be optimum, as the pressure drop reaches extreme at this condition. To overcome this problem, MOO techniques are used in this work, to find the optimum results. The parameters such as twist ratio and width ratio of TTs along with the Reynolds number play a major role in selecting optimum thermal performance conditions.


## 5.5. Rational Analysis of Exergy

The exergetic efficiency of WCTT with various geometries is discussed in this segment. Figure11 shows the exergy efficiency of the inserts as a function of Reynolds number for various twist ratios and width ratios. When compared to plain grooved tube, the combined effect of WCTTs with grooved tube demonstrated superior rational exergy efficiency. W-cut causes stronger eddies and vortices, which increases decayed boundary layer thickness and turbulent intensity near the walls, increasing exergetic efficiency. The graph clearly shows that the rational exergetic efficiency of the WCTT with lower twist ratio (y = 3.5) outperformed the inserts with higher twist ratio (y = 5.3 and 6.5), respectively, by 1.04 and 1.10 times. Aside from that, the width ratio, like the PER discussed in Section 5.4, has an effect on rational exergy efficiency. It is difficult to choose the best because its performance in terms of width ratio is contradictory. As a result, the MOO technique is used in this study to achieve the best results.Figure 11
Exergy efficiency (ηII) vs. Reynolds number (Re).


## 5.6. Empirical Correlation

The empirical correlation for Nusselt number, friction factor, performance enhancement ratio, and rational efficiency of the grooved tube with WCTT was developed, as a function of tape geometry (y and WR) and Reynolds number using the least-squares regression analysis. The empirical correlations are as follows:(37)Nu=0.493Re0.635y−0.33Wr0.079,(38)f=12.3Re−0.401y−0.429Wr0.21,(39)ηI=2.81Re−0.071y−0.191Wr−0.0384,(40)ηII=1.69Re−0.095y−0.1041Wr0.06.Figures12(a) to 12(d) depict the predicted data of Nusselt number, friction factor, performance enhancement ratio, and rational efficiency from the above correlations (equations 37–(40)) in comparison with those obtained from the experimental data. Apparently, the predicted data are in good concordance with the experimental data with the discrepancy of ±4%, ±7%,± 3%, and ±2% for Nu, f, ηI, and ηII, respectively.Figure 12
Assessment between the experimental and developed correlation results: (a) Nu, (b) fr, (c)η, (d) ηII.
(a)(b)(c)(d)


## 6. Optimization Results

From the above experimental study, it is clear that the effects of individual geometric variables on thermal and exergetic efficiency are considered one at a time, but the effects of compound variables are not taken into consideration. It is quite difficult to estimate the optimal working parameter of the WCTT using experimental results, and further, this study has two incompatible objectives (Nu andf). Hence, it is proposed to carry out multi-objective optimization (MOO) and multi-criteria decision-making (MCDM) techniques to determine the best possible design configuration for maximum performance in a grooved tube fitted with WCTT.


### 6.1. Optimization Using Multi-Objective Genetic Algorithm

The main objective of this study is to find the optimum working parameters of the grooved tube with WCTT using GA, which leads to the maximization of the Nusselt number and minimization of pressure drop. Owing to conflicting objectives, the negative Nusselt number (−Nu) is taken as defining parameter, such that lower the value of −Nu gives maximum heat transfer.The multi-objective problem in the optimization of WCTTs is as follows.MinimizationfRe,y,WR=−Nu,f.The abovementioned functions are subjected to the parameters in the range of 2000 ≤ Re ≤ 14000, 3.5 ≤y ≤ 6.4, and 0.45 ≤ WR ≤ 0.6. In the given MOO technique, two geometric variables of WCTTs, namely, twist ratio (y) and width ratio (WR), and Reynolds number (Re) are chosen as the design variables. The maximum and minimum bounds of the design variables are shown in Table 5.Table 5
Range of design variables and the selected values.
Design variablesLower boundHigher boundSelected valuesTwist ratio (y)3.56.43.5, 5.3, 6.4Width ratio (WR)0.450.60.45, 0.6Reynolds number3000140003000, 4500, 6100, 7600, 9200, 10900, 12500, 14000In this study, the GA technique is effectively used to determine the optimal Pareto front for the conflicting objectives. The detailed procedure of GA search method to generate optical Pareto front solutions is provided [50]. The parameter setting of GA is given in Table 6. The optimal Pareto front searched by the GA is shown in Figure 13, and their corresponding values are depicted in Table 7. It is clear that the all points in Pareto front are optimal points and they do not have any supremacy over each other. It is also observed that variation from one optimal point to another leads to the improvement of one objective with inimical change in the other. Thus, in this work, to select the optimal design configuration for heat exchanger with WCTT, the multi-objective genetic algorithm is used, in which the η and ηII are used to evaluate the heat transfer against the friction factor. The corresponding values of η and ηII compared against the two objective functions in optimal Pareto front are illustrated in Figures 14 and 15, respectively. Evidently, the extreme value of η and ηII from the figures gives the optimum geometric parameters for the defined problem.Table 6
Parameter setting for GA.
ParameterValue settingPopulation size100Pareto front population fraction0.7Cross fraction0.8Generation500Fractional tolerance106Figure 13
Optimal Pareto front solution.Table 7
Optimal configurations of W-cut twisted tapes (Pareto front solutions from GA).
PDAsCriteriaREYWR−NufA113509.243.5854870.447139−127.4040.132A213509.223.5449970.594741−130.7960.141A313520.536.3966810.431835−105.0150.103A413510.763.5505410.488913−128.7300.136A513513.594.5842930.43714−117.2940.119A613518.595.7886350.434234−108.5720.107A713511.203.8948120.437794−123.7760.127A813520.536.3966810.431835−105.0150.103A913509.223.5449970.594741−130.7960.141A1013512004.2184270.433286−120.4650.123A1113510.453.5076670.436446−128.0920.133A1213509.703.5305050.533291−129.8530.138A1313511.814.1372160.434913−121.2750.124A1413515.525.1044090.437658−113.2280.113A1513518.516.1926590.435724−106.2100.104A1613518.685.4801630.444343−110.7540.110A1713510.483.7816590.454154−125.3450.130A1813511.814.0122160.434913−122.5090.126A1913514.914.8463000.436005−115.1460.116A2013512.974.5248710.442025−117.9000.120A2113510.403.7613770.479983−126.1170.132A2213516.245.2633900.436984−112.0780.112A2313510.493.9723170.444530−123.1180.127A2413509.493.8354870.447139−124.6030.129A2513518.595.5386350.434234−110.1650.109Figure 14
Multi-objective optimization of Pareto front (η).Figure 15
Multi-objective optimization of Pareto front (ηII).Though the geometric points obtained using Pareto front of MOGA can be taken as the best, one cannot eliminate the remaining solutions, as all the values specified by Pareto front are optimal. Hence, to identify the best operating configuration from the Pareto front set, this study utilizes FAHP-VIKOR model as decision-making tool.


### 6.2. Optimization Using MCDM

This study utilizes the hybrid MCDM model to identify the best operating configuration from the set of Pareto values (taken as alternatives). The performance-defining criteria of the given problem are identified and are given in Table8. In this phase, hybrid MCDM model, FAHP-TOPSIS, is used to rank the alternatives. Here, FAHP is used to assign weightage for the criteria and VIKOR is used to rank the alternatives.Table 8
Details of the different criteria.
CriteriaNotationsPerformance implications of different criteria.Thermal enhancement ratio (TER)Nuo/NupC1-higher-the betterFriction enhancement ratio (FER)fo/fpC2-lower-the betterPerformance enhancement ratio (PER)ηIC3-higher-the betterRational exergy efficiency (REE)ηIIC4-higher-the better


#### 6.2.1. Calculation of Weightage for Criteria

The weightage of criteria used in the study is calculated using the fuzzy analytical hierarchical process (FAHP) method. The nonconformity values of the alternatives associated with predefined criteria are depicted in Figure16. Once the hierarchy diagram is framed, the FAHP computes the weights of each criterion by making pairwise comparison with Satty’s nine-point scale. The fuzzy pairwise comparison results of the criteria are shown in Table 9. The geometric mean values are computed, and the final pairwise comparison matrix is constructed. The weight of each criterion is calculated and tabulated in Table 10 based on the final matrix. Consistency index (CI) and consistency ratio (CR) values are calculated using equations (28) and (29) to check the consistency of the criteria in the pairwise comparison matrix. The CR value is 0.073, which is less than 0.1, according to the results. As a result, the weights assigned to the criteria are satisfactory and can be used to rank the alternatives.Figure 16
Inconsistency of criteria values with alternatives.Table 9
Pairwise comparison matrix for criteria.
TERFERPERREETER(1, 1, 1)(1, 1, 1)(0.25, 00.33, 0.5)(0.25, 00.33, 0.5)FER(1, 1, 1)(1, 1, 1)(0.25, 00.33, 0.5)(0.25, 00.33, 0.5)PER(2, 3, 4)(2, 3, 4)(1, 1, 1)(00.33, 0.5, 1)EE(2, 3, 4)(2, 3, 4)(1, 2, 3)(1, 1, 1)Table 10
Results obtained with FAHP.
CriteriaWeightsλmax, CI, RCICRTER0.1198λmax = 4.201,CI = 0.0669,RCI = 0.90.073FER0.1198PER0.3476REE0.4128


#### 6.2.2. Ranking the Alternatives

In this section, the VIKOR method is proposed for selecting the optimum design among the list of alternatives. The decision matrix of the stated problem is shown in Table11, and the values of the decision matrix are normalized using equation (31). The normalized decision matrix values are calculated and tabulated in Table 12. The best and worst values of each criterion are calculated using equation (32) and are depicted in Table 13, followed by the construction of utility measure relation, regret measure relation, and VIKOR index using equations (33)–(35) shown in Table 14. The ranking of the options is prioritized according to the VIKOR index value, as shown in Table 14 and Figure 17. The alternative A4 among the list of 25 alternatives is the most preferred, as it attains the first rank with concern to overall objectives. According to VIKOR index values, the WCTT configuration rankings are as follows: A4 > A11 > A2 > A9 > A21 > A17 > A1 > A24 > A11 > A23 >  A7 > A18 > A13 > A10 > A20 > A5 >  A19 > A14 > A16 > A22 > A25 > A6 > A15 > A3 > A8. The alternative with twist ratio, y = 3.55, width ratio, WR = 0.489, and Reynolds number, Re = 13511, exhibits the optimal formulation.Table 11
Details of alternatives and criteria (decision matrix).
PDAsCriteriaTERFERPERREEA1−1.764.5141.1490.645A2−1.84.8161.1460.648A3−1.453.4951.0360.632A4−1.774.6181.1570.649A5−1.624.0431.1040.639A6−1.53.6521.0560.633A7−1.714.3371.1390.643A8−1.453.4951.0360.632A9−1.84.8161.1460.648A10−1.664.1821.1220.641A11−1.774.5331.1530.647A12−1.794.7151.1520.648A13−1.674.221.1260.642A14−1.563.8611.0810.637A15−1.463.5511.0420.633A16−1.533.7571.0660.635A17−1.734.4261.1440.644A18−1.694.2761.1330.642A19−1.593.9451.0920.637A20−1.634.0751.1060.639A21−1.744.4881.1450.645A22−1.543.811.0750.636A23−1.74.3141.1340.643A24−1.724.3851.1410.643A25−1.523.7221.0650.635Table 12
Weighted normalized decision matrix.
PDAsCriteriaTERFERPERREEA10.10410.09240.00330.2086A20.11980.11980.02450.0442A30.00000.00000.34760.4128A40.11020.10190.00000.0000A50.05710.04970.14820.3071A60.01650.01430.28940.3784A70.08720.07640.04540.2621A80.00000.00000.34760.4128A90.11980.11980.02450.0442A100.07180.06230.09490.2985A110.10720.09420.00330.2394A120.11540.11070.00760.0108A130.07560.06580.08310.2878A140.03820.03320.21450.3335A150.00560.00510.32960.3907A160.02660.02380.25940.3299A170.09450.08450.03120.1993A180.08130.07090.06360.2798A190.04710.04080.18230.3256A200.05990.05260.14150.2870A210.09810.09010.03490.1147A220.03280.02850.23300.3439A230.08420.07430.06000.2443A240.09110.08070.03830.2262A250.02390.02060.26300.3669Table 13
Best and worst values of criteria.
TERFERPEREEBest value (fi∗)1.804.821.160.65Worst value (fi∗)1.453.501.040.63Table 14
Si, Ri, and Qi values and ranking of alternatives using VIKOR.
PDAsCriteriaSiRiQiRankA10.410.210.07504A20.310.120.07683A30.760.411.000024A40.210.110.00761A50.560.310.487816A60.700.380.906722A70.470.260.202910A80.760.411.000025A90.310.120.07686A100.530.300.344114A110.440.240.03042A120.240.120.04803A130.510.290.282213A140.620.330.641118A150.730.390.934123A160.640.330.779120A170.410.200.13978A180.500.280.268612A190.600.330.619617A200.540.290.484115A210.340.110.08497A220.640.340.707019A230.460.240.212911A240.440.230.19949A250.670.370.780721Figure 17
Ranking of alternatives.To validate the optimum configuration, the obtained result is compared with the other inserts from the past under similar working condition. Figure18 depicted the comparative result of the optimum configuration of WCTT with other inserts. The result showed that the optimum configuration of WCTT showed supremacy over other inserts in the range of 0.5–31.5%, respectively. The WCTT effectively breaks the viscose and thermal boundary layers formulated inside the tubes at turbulent flow conditions. Further, the W-shaped design offers lower flow resistance. This makes the system higher efficient as compared with other configurations of turbulator designs.Figure 18
Performance comparison of optimal configuration WCTT with former tapes.


## 6.1. Optimization Using Multi-Objective Genetic Algorithm

The main objective of this study is to find the optimum working parameters of the grooved tube with WCTT using GA, which leads to the maximization of the Nusselt number and minimization of pressure drop. Owing to conflicting objectives, the negative Nusselt number (−Nu) is taken as defining parameter, such that lower the value of −Nu gives maximum heat transfer.The multi-objective problem in the optimization of WCTTs is as follows.MinimizationfRe,y,WR=−Nu,f.The abovementioned functions are subjected to the parameters in the range of 2000 ≤ Re ≤ 14000, 3.5 ≤y ≤ 6.4, and 0.45 ≤ WR ≤ 0.6. In the given MOO technique, two geometric variables of WCTTs, namely, twist ratio (y) and width ratio (WR), and Reynolds number (Re) are chosen as the design variables. The maximum and minimum bounds of the design variables are shown in Table 5.Table 5
Range of design variables and the selected values.
Design variablesLower boundHigher boundSelected valuesTwist ratio (y)3.56.43.5, 5.3, 6.4Width ratio (WR)0.450.60.45, 0.6Reynolds number3000140003000, 4500, 6100, 7600, 9200, 10900, 12500, 14000In this study, the GA technique is effectively used to determine the optimal Pareto front for the conflicting objectives. The detailed procedure of GA search method to generate optical Pareto front solutions is provided [50]. The parameter setting of GA is given in Table 6. The optimal Pareto front searched by the GA is shown in Figure 13, and their corresponding values are depicted in Table 7. It is clear that the all points in Pareto front are optimal points and they do not have any supremacy over each other. It is also observed that variation from one optimal point to another leads to the improvement of one objective with inimical change in the other. Thus, in this work, to select the optimal design configuration for heat exchanger with WCTT, the multi-objective genetic algorithm is used, in which the η and ηII are used to evaluate the heat transfer against the friction factor. The corresponding values of η and ηII compared against the two objective functions in optimal Pareto front are illustrated in Figures 14 and 15, respectively. Evidently, the extreme value of η and ηII from the figures gives the optimum geometric parameters for the defined problem.Table 6
Parameter setting for GA.
ParameterValue settingPopulation size100Pareto front population fraction0.7Cross fraction0.8Generation500Fractional tolerance106Figure 13
Optimal Pareto front solution.Table 7
Optimal configurations of W-cut twisted tapes (Pareto front solutions from GA).
PDAsCriteriaREYWR−NufA113509.243.5854870.447139−127.4040.132A213509.223.5449970.594741−130.7960.141A313520.536.3966810.431835−105.0150.103A413510.763.5505410.488913−128.7300.136A513513.594.5842930.43714−117.2940.119A613518.595.7886350.434234−108.5720.107A713511.203.8948120.437794−123.7760.127A813520.536.3966810.431835−105.0150.103A913509.223.5449970.594741−130.7960.141A1013512004.2184270.433286−120.4650.123A1113510.453.5076670.436446−128.0920.133A1213509.703.5305050.533291−129.8530.138A1313511.814.1372160.434913−121.2750.124A1413515.525.1044090.437658−113.2280.113A1513518.516.1926590.435724−106.2100.104A1613518.685.4801630.444343−110.7540.110A1713510.483.7816590.454154−125.3450.130A1813511.814.0122160.434913−122.5090.126A1913514.914.8463000.436005−115.1460.116A2013512.974.5248710.442025−117.9000.120A2113510.403.7613770.479983−126.1170.132A2213516.245.2633900.436984−112.0780.112A2313510.493.9723170.444530−123.1180.127A2413509.493.8354870.447139−124.6030.129A2513518.595.5386350.434234−110.1650.109Figure 14
Multi-objective optimization of Pareto front (η).Figure 15
Multi-objective optimization of Pareto front (ηII).Though the geometric points obtained using Pareto front of MOGA can be taken as the best, one cannot eliminate the remaining solutions, as all the values specified by Pareto front are optimal. Hence, to identify the best operating configuration from the Pareto front set, this study utilizes FAHP-VIKOR model as decision-making tool.


## 6.2. Optimization Using MCDM

This study utilizes the hybrid MCDM model to identify the best operating configuration from the set of Pareto values (taken as alternatives). The performance-defining criteria of the given problem are identified and are given in Table8. In this phase, hybrid MCDM model, FAHP-TOPSIS, is used to rank the alternatives. Here, FAHP is used to assign weightage for the criteria and VIKOR is used to rank the alternatives.Table 8
Details of the different criteria.
CriteriaNotationsPerformance implications of different criteria.Thermal enhancement ratio (TER)Nuo/NupC1-higher-the betterFriction enhancement ratio (FER)fo/fpC2-lower-the betterPerformance enhancement ratio (PER)ηIC3-higher-the betterRational exergy efficiency (REE)ηIIC4-higher-the better


### 6.2.1. Calculation of Weightage for Criteria

The weightage of criteria used in the study is calculated using the fuzzy analytical hierarchical process (FAHP) method. The nonconformity values of the alternatives associated with predefined criteria are depicted in Figure16. Once the hierarchy diagram is framed, the FAHP computes the weights of each criterion by making pairwise comparison with Satty’s nine-point scale. The fuzzy pairwise comparison results of the criteria are shown in Table 9. The geometric mean values are computed, and the final pairwise comparison matrix is constructed. The weight of each criterion is calculated and tabulated in Table 10 based on the final matrix. Consistency index (CI) and consistency ratio (CR) values are calculated using equations (28) and (29) to check the consistency of the criteria in the pairwise comparison matrix. The CR value is 0.073, which is less than 0.1, according to the results. As a result, the weights assigned to the criteria are satisfactory and can be used to rank the alternatives.Figure 16
Inconsistency of criteria values with alternatives.Table 9
Pairwise comparison matrix for criteria.
TERFERPERREETER(1, 1, 1)(1, 1, 1)(0.25, 00.33, 0.5)(0.25, 00.33, 0.5)FER(1, 1, 1)(1, 1, 1)(0.25, 00.33, 0.5)(0.25, 00.33, 0.5)PER(2, 3, 4)(2, 3, 4)(1, 1, 1)(00.33, 0.5, 1)EE(2, 3, 4)(2, 3, 4)(1, 2, 3)(1, 1, 1)Table 10
Results obtained with FAHP.
CriteriaWeightsλmax, CI, RCICRTER0.1198λmax = 4.201,CI = 0.0669,RCI = 0.90.073FER0.1198PER0.3476REE0.4128


### 6.2.2. Ranking the Alternatives

In this section, the VIKOR method is proposed for selecting the optimum design among the list of alternatives. The decision matrix of the stated problem is shown in Table11, and the values of the decision matrix are normalized using equation (31). The normalized decision matrix values are calculated and tabulated in Table 12. The best and worst values of each criterion are calculated using equation (32) and are depicted in Table 13, followed by the construction of utility measure relation, regret measure relation, and VIKOR index using equations (33)–(35) shown in Table 14. The ranking of the options is prioritized according to the VIKOR index value, as shown in Table 14 and Figure 17. The alternative A4 among the list of 25 alternatives is the most preferred, as it attains the first rank with concern to overall objectives. According to VIKOR index values, the WCTT configuration rankings are as follows: A4 > A11 > A2 > A9 > A21 > A17 > A1 > A24 > A11 > A23 >  A7 > A18 > A13 > A10 > A20 > A5 >  A19 > A14 > A16 > A22 > A25 > A6 > A15 > A3 > A8. The alternative with twist ratio, y = 3.55, width ratio, WR = 0.489, and Reynolds number, Re = 13511, exhibits the optimal formulation.Table 11
Details of alternatives and criteria (decision matrix).
PDAsCriteriaTERFERPERREEA1−1.764.5141.1490.645A2−1.84.8161.1460.648A3−1.453.4951.0360.632A4−1.774.6181.1570.649A5−1.624.0431.1040.639A6−1.53.6521.0560.633A7−1.714.3371.1390.643A8−1.453.4951.0360.632A9−1.84.8161.1460.648A10−1.664.1821.1220.641A11−1.774.5331.1530.647A12−1.794.7151.1520.648A13−1.674.221.1260.642A14−1.563.8611.0810.637A15−1.463.5511.0420.633A16−1.533.7571.0660.635A17−1.734.4261.1440.644A18−1.694.2761.1330.642A19−1.593.9451.0920.637A20−1.634.0751.1060.639A21−1.744.4881.1450.645A22−1.543.811.0750.636A23−1.74.3141.1340.643A24−1.724.3851.1410.643A25−1.523.7221.0650.635Table 12
Weighted normalized decision matrix.
PDAsCriteriaTERFERPERREEA10.10410.09240.00330.2086A20.11980.11980.02450.0442A30.00000.00000.34760.4128A40.11020.10190.00000.0000A50.05710.04970.14820.3071A60.01650.01430.28940.3784A70.08720.07640.04540.2621A80.00000.00000.34760.4128A90.11980.11980.02450.0442A100.07180.06230.09490.2985A110.10720.09420.00330.2394A120.11540.11070.00760.0108A130.07560.06580.08310.2878A140.03820.03320.21450.3335A150.00560.00510.32960.3907A160.02660.02380.25940.3299A170.09450.08450.03120.1993A180.08130.07090.06360.2798A190.04710.04080.18230.3256A200.05990.05260.14150.2870A210.09810.09010.03490.1147A220.03280.02850.23300.3439A230.08420.07430.06000.2443A240.09110.08070.03830.2262A250.02390.02060.26300.3669Table 13
Best and worst values of criteria.
TERFERPEREEBest value (fi∗)1.804.821.160.65Worst value (fi∗)1.453.501.040.63Table 14
Si, Ri, and Qi values and ranking of alternatives using VIKOR.
PDAsCriteriaSiRiQiRankA10.410.210.07504A20.310.120.07683A30.760.411.000024A40.210.110.00761A50.560.310.487816A60.700.380.906722A70.470.260.202910A80.760.411.000025A90.310.120.07686A100.530.300.344114A110.440.240.03042A120.240.120.04803A130.510.290.282213A140.620.330.641118A150.730.390.934123A160.640.330.779120A170.410.200.13978A180.500.280.268612A190.600.330.619617A200.540.290.484115A210.340.110.08497A220.640.340.707019A230.460.240.212911A240.440.230.19949A250.670.370.780721Figure 17
Ranking of alternatives.To validate the optimum configuration, the obtained result is compared with the other inserts from the past under similar working condition. Figure18 depicted the comparative result of the optimum configuration of WCTT with other inserts. The result showed that the optimum configuration of WCTT showed supremacy over other inserts in the range of 0.5–31.5%, respectively. The WCTT effectively breaks the viscose and thermal boundary layers formulated inside the tubes at turbulent flow conditions. Further, the W-shaped design offers lower flow resistance. This makes the system higher efficient as compared with other configurations of turbulator designs.Figure 18
Performance comparison of optimal configuration WCTT with former tapes.


## 6.2.1. Calculation of Weightage for Criteria

The weightage of criteria used in the study is calculated using the fuzzy analytical hierarchical process (FAHP) method. The nonconformity values of the alternatives associated with predefined criteria are depicted in Figure16. Once the hierarchy diagram is framed, the FAHP computes the weights of each criterion by making pairwise comparison with Satty’s nine-point scale. The fuzzy pairwise comparison results of the criteria are shown in Table 9. The geometric mean values are computed, and the final pairwise comparison matrix is constructed. The weight of each criterion is calculated and tabulated in Table 10 based on the final matrix. Consistency index (CI) and consistency ratio (CR) values are calculated using equations (28) and (29) to check the consistency of the criteria in the pairwise comparison matrix. The CR value is 0.073, which is less than 0.1, according to the results. As a result, the weights assigned to the criteria are satisfactory and can be used to rank the alternatives.Figure 16
Inconsistency of criteria values with alternatives.Table 9
Pairwise comparison matrix for criteria.
TERFERPERREETER(1, 1, 1)(1, 1, 1)(0.25, 00.33, 0.5)(0.25, 00.33, 0.5)FER(1, 1, 1)(1, 1, 1)(0.25, 00.33, 0.5)(0.25, 00.33, 0.5)PER(2, 3, 4)(2, 3, 4)(1, 1, 1)(00.33, 0.5, 1)EE(2, 3, 4)(2, 3, 4)(1, 2, 3)(1, 1, 1)Table 10
Results obtained with FAHP.
CriteriaWeightsλmax, CI, RCICRTER0.1198λmax = 4.201,CI = 0.0669,RCI = 0.90.073FER0.1198PER0.3476REE0.4128


## 6.2.2. Ranking the Alternatives

In this section, the VIKOR method is proposed for selecting the optimum design among the list of alternatives. The decision matrix of the stated problem is shown in Table11, and the values of the decision matrix are normalized using equation (31). The normalized decision matrix values are calculated and tabulated in Table 12. The best and worst values of each criterion are calculated using equation (32) and are depicted in Table 13, followed by the construction of utility measure relation, regret measure relation, and VIKOR index using equations (33)–(35) shown in Table 14. The ranking of the options is prioritized according to the VIKOR index value, as shown in Table 14 and Figure 17. The alternative A4 among the list of 25 alternatives is the most preferred, as it attains the first rank with concern to overall objectives. According to VIKOR index values, the WCTT configuration rankings are as follows: A4 > A11 > A2 > A9 > A21 > A17 > A1 > A24 > A11 > A23 >  A7 > A18 > A13 > A10 > A20 > A5 >  A19 > A14 > A16 > A22 > A25 > A6 > A15 > A3 > A8. The alternative with twist ratio, y = 3.55, width ratio, WR = 0.489, and Reynolds number, Re = 13511, exhibits the optimal formulation.Table 11
Details of alternatives and criteria (decision matrix).
PDAsCriteriaTERFERPERREEA1−1.764.5141.1490.645A2−1.84.8161.1460.648A3−1.453.4951.0360.632A4−1.774.6181.1570.649A5−1.624.0431.1040.639A6−1.53.6521.0560.633A7−1.714.3371.1390.643A8−1.453.4951.0360.632A9−1.84.8161.1460.648A10−1.664.1821.1220.641A11−1.774.5331.1530.647A12−1.794.7151.1520.648A13−1.674.221.1260.642A14−1.563.8611.0810.637A15−1.463.5511.0420.633A16−1.533.7571.0660.635A17−1.734.4261.1440.644A18−1.694.2761.1330.642A19−1.593.9451.0920.637A20−1.634.0751.1060.639A21−1.744.4881.1450.645A22−1.543.811.0750.636A23−1.74.3141.1340.643A24−1.724.3851.1410.643A25−1.523.7221.0650.635Table 12
Weighted normalized decision matrix.
PDAsCriteriaTERFERPERREEA10.10410.09240.00330.2086A20.11980.11980.02450.0442A30.00000.00000.34760.4128A40.11020.10190.00000.0000A50.05710.04970.14820.3071A60.01650.01430.28940.3784A70.08720.07640.04540.2621A80.00000.00000.34760.4128A90.11980.11980.02450.0442A100.07180.06230.09490.2985A110.10720.09420.00330.2394A120.11540.11070.00760.0108A130.07560.06580.08310.2878A140.03820.03320.21450.3335A150.00560.00510.32960.3907A160.02660.02380.25940.3299A170.09450.08450.03120.1993A180.08130.07090.06360.2798A190.04710.04080.18230.3256A200.05990.05260.14150.2870A210.09810.09010.03490.1147A220.03280.02850.23300.3439A230.08420.07430.06000.2443A240.09110.08070.03830.2262A250.02390.02060.26300.3669Table 13
Best and worst values of criteria.
TERFERPEREEBest value (fi∗)1.804.821.160.65Worst value (fi∗)1.453.501.040.63Table 14
Si, Ri, and Qi values and ranking of alternatives using VIKOR.
PDAsCriteriaSiRiQiRankA10.410.210.07504A20.310.120.07683A30.760.411.000024A40.210.110.00761A50.560.310.487816A60.700.380.906722A70.470.260.202910A80.760.411.000025A90.310.120.07686A100.530.300.344114A110.440.240.03042A120.240.120.04803A130.510.290.282213A140.620.330.641118A150.730.390.934123A160.640.330.779120A170.410.200.13978A180.500.280.268612A190.600.330.619617A200.540.290.484115A210.340.110.08497A220.640.340.707019A230.460.240.212911A240.440.230.19949A250.670.370.780721Figure 17
Ranking of alternatives.To validate the optimum configuration, the obtained result is compared with the other inserts from the past under similar working condition. Figure18 depicted the comparative result of the optimum configuration of WCTT with other inserts. The result showed that the optimum configuration of WCTT showed supremacy over other inserts in the range of 0.5–31.5%, respectively. The WCTT effectively breaks the viscose and thermal boundary layers formulated inside the tubes at turbulent flow conditions. Further, the W-shaped design offers lower flow resistance. This makes the system higher efficient as compared with other configurations of turbulator designs.Figure 18
Performance comparison of optimal configuration WCTT with former tapes.


## 7. Conclusions

To ascertain the best performance configuration of WCTT, the twist ratio, width ratio, and Reynolds number parameters are to be considered, which consists of conflicting objectives with a major focus on meeting enhanced heat transfer. Therefore, an effective optimization concept is essential to resolve the problem. This work uses optimization tools such as genetic algorithm and FAHP-VIKOR on experimental results with varying parameters to identify the best one. Initially, the effect of grooved tube with WCTT is carried out to show the deviance of objectives with the variation of parameters. Then, the set of optimal solutions for the conflicting objectives is identified using multi-objective GA. Finally, the best optimal solution is obtained using the FAHP-VIKOR model. The results of the current work are as follows.(i)
The Nusselt number increases with rise in Reynolds number and width ratio and with decrease in twist ratio. The Nusselt number of the grooved tube with WCTT showed 44–56% higher heat transfer rate than plain tube.(ii)
The friction factor increases with a decrease in Reynolds number and twist ratio and with the increase in width ratio. The rise in friction factor is noticed for the grooved tube fitted WCTT than the plain tube of about 71–78%.(iii)
The performance enhancement ratio of WCTT is in the range of 1.02–1.30 for the given working conditions, and it raises with the reduction in twist tape ratio, width ratio, and Reynolds number.(iv)
The rational exergy efficiency increases with the increase in width ratio, in addition with the reduction in twist tape ratio and flow rate. The mean exergy efficiency of the grooved tube with WCTT is about 1.17–1.28 times higher than the plain tube.(v)
The empirical correlation for the Nusselt number, friction factor, performance enhancement ratio, and rational efficiency was developed, and they showed the discrepancy of ±3.5%, ±6%, ±2%, and ±1.5%, respectively.(vi)
As the design parameters such as twist ratio, width ratio, and Reynolds number strongly influence the overall system performance and to optimize the best working configuration, the integrated GA and FAHP-VIKOR optimization tools are used.(vii)
The genetic algorithm is used to optimize the given data and provides the set of optimal Pareto front solutions. This optimal design leads to trade-off between Nu andf, which results in the use of multi-objective genetic algorithm.(viii)
As all the given solutions of Pareto front were optimum and in the necessity to prioritize the best, the FAHP-VIKOR model is evaluated.(ix)
The order of criteria that dominate the VIKOR index is REE > PER > FER > TER. These criteria are determined to optimize the overall performance of grooved tube heat exchanger employing WCTTs. These performance criteria weights are calculated using FAHP, and their contribution ratio of order is 41.26%, 34.76%, 11.98%, and 11.98%, respectively.(x)
The optimal formulation is A4 with twist ratio (y): 3.55, width ratio (WR): 0.488, and Reynolds number (Re): 13511, which gives outcomes of thermal enhancement ratioNuo/Nup: 1.177; friction enhancement ratiofo/fp: 4.62; performance enhancement ratio: 1.15; and exergy efficiency: 0.649.



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*Source: 2819688-2022-07-31.xml*