Introduction

In many engineering fields, including power generation, chemical processing, the food industry, automotive cooling, HVAC systems, and waste heat recovery, heat exchangers are essential components. To enhance the performances of heat exchangers, several techniques have been used, mainly passive and active methods. Using design changes such as helical coils insertion, the introduction of corrugated or wavy tubes, extended fins, the introduction of nanofluids, etc., are examples of passive methods, where no energy source is used unlikely active methods. In this work, one such passive technique, i.e. brazed helix tube (BHT), is introduced in the inner core portion of the present heat exchanger for performance enhancement and differentiate from others in regard to stated novelty BHT. The BHT integration inside an outer shell converts a double-pipe heat exchanger to a novel multi-fluid heat exchanger (NMFHE), which is presently investigated experimentally to measure thermo-hydraulic and exergetic performance. For the stated performance measures, the following literature is reviewed to acknowledge a better clarity of research in this field.

Three fluid heat exchangers (TFHE) come under multi-fluid heat exchangers (MFHE) and are widely used in specific areas such as cryogenics and various chemical processes. These include systems for separating air, processing helium and hydrogen, and making ammonia gas [1]. One specific design, the triple concentric tube, is popular in the food and pharmaceutical industries. It is mainly used to heat-treat liquids such as milk and fruit juices [2]. There has been a lot of research on these three fluid heat exchangers. Studies have observed their designs, their working principles, and methods of heat transfer. A three-fluid heat exchanger with two thermal communications between the thermally imbalanced fluid streams was given a compact solution for the temperature distribution and temperature cross. There are four different fluid flow configurations that may have been examined [3]. An investigation has been done on analytical relationships between the design variables (the NTU-effectiveness relationships) for a general three-fluid heat exchanger, in which all three streams are in thermal communication [4]. In another work, nonlinear problems and the thermal design theory of multi-fluid heat exchangers (containing more than three fluid streams) and multi-stream plate-fin heat exchangers were not taken into consideration [5]. Based on the conservation of energy principle, the dimensionless governing equations for three fluid heat exchangers are constructed, and they are then resolved using FEM based on the subdomain collocation method and Galerkin’s approach. As can be seen, when the obtained findings are contrasted with the analytical results for the traditional two fluid heat exchangers, the results demonstrate that the subdomain collocation approach is more accurate than Galerkin’s method [6]. The effectiveness-NTU relations were derived in the study as a crucial component of the compressive theoretical investigation carried out by TTHE, and some representatives were shown in graphical form [7]. Counter-flow triple concentric-tube heat exchangers were the subject of a two-part theoretical study. The performance calculations and design calculations included in the case studies were used to show that the three tubes’ relative diameters to one another are the most crucial factors affecting the exchanger’s performance (or size) [8]. Theoretical investigation into TTHE includes the derivation of the governing differential equations and potential solutions under certain circumstances. The work’s generated equations can be applied to both performance and design calculations, in addition to helping to estimate bulk temperature fluctuations along the exchanger [9]. A dairy triple tube heat exchanger’s performance degrades when milk fouling builds up on the heating surface. A simulation model for the precise estimates of milk outflow temperature and fouling thickness. In the study that is being done, the local fouling factor is expressed in terms of the Biot number. It is possible to anticipate the fouling thickness and milk outflow temperature as a function of time and along the full heat exchanger’s length [10, 11]. For the examination of multiple stream heat exchangers, a very effective algorithm has been developed. A stack of (n–1) two stream exchangers divided by diabatic partitions is thought of as an n-stream heat exchanger using this method. The full heat exchanger can be developed non-iteratively starting with the analysis of the fundamental two stream units [12]. With the aid of FEM, the performance of a triple concentric pipe heat exchanger is numerically investigated under steady-state settings for various flow configurations and for insulated as well as non-insulated heat exchanger conditions. Hot water, cold water, and regular tap water are the three fluids being taken into consideration [13]. The heat transfer and fluid flow characteristics of two different hybrid nanofluids are numerically investigated in a helical double-pipe heat exchanger with a curved conical turbulator in a laminar regime using the Fluent software [14]. An off-grid solar poly-generation system is analysed [15] in regard to energy efficiency, exergy efficiency, and economy, and found feasible. The performance of a triple concentric pipe heat exchanger was experimentally investigated under steady-state settings using identical working fluids for two distinct flow arrangements, referred to as N–H–C and C–H–N, and for insulated as well as non-insulated heat exchanger conditions. Normal water flows via the innermost pipe in the N–H–C configuration, hot water flows through the inner annulus, and cold water flows through the outer annulus [16].

Research in the field of triple concentric-tube heat exchangers (TTHE) has led to the introduction of a mathematical model tailored specifically for heat transfer analysis [17]. Notably, the potential of a wood-based home heating system was explored, with emphasis on its temperature outputs and efficiency in heat recovery [18]. Thermo-hydraulic inquiries on the TTHE with two thermal communications were conducted in a steady-state condition [19]. Both experimental and computational analyses of the TTHE accentuated the double-tube heat exchanger specifically [20]. Further, the TTHE equipped with inserted ribs received both experimental and computational analyses [21]. Volume flow rate changes were investigated to determine their impact on TFHE efficiency. As fluid flow rates increased in the TFHE, a corresponding increase in the overall heat transfer coefficient was seen, and efficiencies displayed different behaviours based on combined capacity ratios, R, and NTU [22]. One creative model that used Fortran provided a thorough examination of the TTHE parameters and included gases such as hydrogen, nitrogen, and oxygen [23]. Modern climate control solutions are made possible by the development of an advanced air-conditioning system that can regulate temperature and humidity simultaneously using a TFHE [24]. A spiral tube TFHE infused with various concentrations of graphene/water was specifically studied, with a focus on metrics like heat transfer efficiency [25]. A three-fluid exchanger designed for domestic heating purposes was the subject of another interesting investigation [26]. The limitation of typical air conditioners to change both temperature and humidity at once has been a persistent problem in the sector [27]. A popular three-fluid heat exchanger is the TriCoil system. It can save up to 15.8% on energy, making it an economical option for household settings. Even more impressive, little changes to its design can greatly improve its effectiveness [28].

After a detailed study on multi-fluid heat exchanger, it is concluded that the present work on multi-fluid heat exchanger with brazed helix tube proposed for enhanced performance is novel, and no information about such heat exchanger is available in the literature. Therefore, an extensive experimental study is conducted presently on NMFHE to predict its thermo-hydraulic and exergetic performance in regard to variations in flow rates, flow configurations, and inlet temperatures. Taguchi–Grey optimization technique is used to predict optimal value of input operating parameters for the optimum performance of the NMFHE, i.e. maximum JF factor and minimum entropy generation number. A confirmation test has been carried out to compare the predicted results with the experimental one and validated. Subsequent sections cover detailed analysis, optimization study, and confirmation test.

Materials and methods

Experimental investigation

Experimental setup

The present research conducts a thorough analysis of a novel multi-fluid heat exchanger (NMFHE), which is an advanced version of a traditional double-tube heat exchanger. The experimental setup of NMFHE is shown in Fig. 1.

Fig. 1
figure 1

Schematic layout of the NMFHE experimental setup. 1—Test section of NMFHE, 2—Shell, 3—Helical coil tube, 4—HF1 tank, 5—Immersion heater (3 kW), 6—Thermostat, 7—Pump (HF1), 8—Flow control valve (HF1), 9—Rotameter (HF1), 10—Inlet pressure gauge (HF1), 11—Outlet pressure gauge (HF1), 12—Fan, 13—HF2 tank, 14—Pump (HF2), 15—Flow control valve (HF2), 16—Rotameter (HF2), 17—Inlet pressure gauge (HF2), 18—Outlet pressure gauge (HF2), and 19—Data logger

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In this design, a copper-made brazed helix tube (BHT) replaces the core inner tube in the double-tube heat exchanger. The unique structure of the BHT enables two separate flow paths; one designated for hot water (denoted as HF1) and the other for air (referred to as HF3). The geometrical arrangement of the BHT permits HF3 to flow through the hollow conduit section, as shown in Fig. 2, while HF1 circulates through the conduit, specifically between the helical tube inlet and outlet.

Fig. 2
figure 2

NMFHE test section. 1—Hollow conduit of BHT, 2—Brazed helix tube (BHT), 3—Outer shell, 4—Shell outlet, 5—End cap, 6—Helical coil tube outlet, 7—Shell inlet, and 8—Helical coil tube inlet

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The cold fluid (HF2) flows through the outer annular shell, which is the space defined between the exterior PVC tube and the BHT in Fig. 2. The NMFHE setup is a total length of 0.871 m, with a PVC outermost shell having an internal diameter of 134 mm and a wall thickness of 9.2 mm. The interior BHT is of 31.62 m in length, 6.3 mm in diameter, 0.81 mm in thickness, a pitch of 7.92 mm, a coil radius of 96.8 mm, and with 104 turns.

For the hydraulic circuit of HF1, a centrifugal pump is fitted for the flow through the BHT. The temperature of HF1 is increased by an electric heater, which is controlled by a thermostat for temperature control. An axial fan of diameter 4 inches supplies HF3 at three different speeds through the hollow conduit of the inner BHT. Furthermore, using K-type thermocouples, those are positioned at four different positions throughout the NMFHE, the temperature gradients of HF1, HF2, and HF3 are accurately obtained at the inlet, intermediate, and outlet stages. For adjusting the flow rates of HF1 and HF2, additional equipment such flow control valves and rotameters is used, for a thorough and controlled analysis of thermo-hydraulic behaviour of the system.

Experimental procedure

This study closely examines the heat transfer characteristics of the novel multi-fluid heat exchanger (NMFHE). In this investigation, the thermo-hydraulic behaviour is tested at different flow rates, varying input temperatures, and at different flow configurations. The experimental investigation comprises of three different volume flow rates of HF1 and HF2 (100 LPH, 150 LPH, and 200 LPH, respectively), three different flow velocities for HF3 (1 m s−1, 2 m s−1, and 3 m s−1), and three different inlet temperatures of HF1 (333 K, 243 K, and 253 K). The inlet temperatures of HF2 and HF3 throughout the experimental procedure are kept constant. As shown in Fig. 3, the investigation makes use of all four flow configurations, including parallel flow (PF) and three different counter-flow configurations (CF1, CF2, and CF3). In the first flow configuration, all three fluids (HF1, HF2, and HF3) flow in the same direction from right to left of the NMFHE test section. As shown in Fig. 3, the flow directions of the other two fluids are unaffected by the counter-flow configurations (CF1, CF2, and CF3); however, the flow directions of HF1, HF2, and HF3 individually are reversed. HF1, HF2, and HF3 are allowed to circulate for 25–30 min inside the NMFHE test section to maintain steady-state condition. Afterward, experimental readings are taken for further analysis, discussion, and optimization study.

Fig. 3
figure 3

Flow configuration diagram of the NMFHE

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Uncertainty analysis

Accurate measurements in scientific research are critical for reliable results. However, no instrument is perfect, so it is required to consider possible errors in the measuring instruments. In the context of the present study, three different instruments used for the measurement of temperatures, flow rates, and air velocity are added in Table 1 with their make and the least count.

Table 1 Measuring instruments and least count
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Ensuring the accuracy of the data gathered from the instruments equipped with NMFHE, it is necessary to calculate the uncertainty by using Eq. 1 [28]. The uncertainty values of output parameters are given in Table 2.

$$U_{{\text{f}}} = \sqrt {\left[ {\left( {\frac{\partial f}{{\partial x_{1} }}} \right).Ux_{1} } \right]^{2} + \left[ {\left( {\frac{\partial f}{{\partial x_{2} }}} \right).Ux_{2} } \right]^{2} + \left[ {\left( {\frac{\partial f}{{\partial x_{3} }}} \right).Ux_{3} } \right]^{2} + \ldots + \left[ {\left( {\frac{\partial f}{{\partial x_{{\text{n}}} }}} \right).Ux_{{\text{n}}} } \right]^{2} }$$
(1)
Table 2 Uncertainty in output results
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Data reduction

The entropy generation and JF factor are calculated in the present study using the following formulas as mentioned in [29,30,31].

Entropy generation number

$$N_{{\text{S}}} = \frac{1}{{\pi \cdot {\text{Nu}}}} + \frac{{\pi^{3} \cdot f \cdot {\text{Re}}^{5} }}{{32 \cdot \left( {\frac{{\left( {q^{\prime } } \right)^{2} \rho^{2} \cdot \dot{m}^{2} }}{{K \cdot T \cdot \mu^{5} }}} \right)}}$$
(2)

JF factor

$${\text{JF}} = \frac{j}{{f^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} }}$$
(3)

Verification and validation of the present result

The findings of the present experimental work, i.e. Nusselt number for fluid flow inside the brazed helix tube (BHT), outer shell, and hollow conduit of NMFHE, are compared with Nusselt number correlations provided in the literature.

Figure 4a shows the verification of BHT side fluid (HF1) Nusselt number with the Nusselt number correlations suggested by Rogers and Mayhew [32], i.e. is given in Eq. 4.

$${\text{Nu}}_{{{\text{HF}}_{1} }} = 0.021\left( {{\text{Re}}_{{{\text{HF}}_{1} }} } \right)^{0.85} \left( {\Pr_{{{\text{HF}}_{1} }} } \right)^{0.4} \left( {\frac{{d_{{{\text{c}},{\text{i}}}} }}{{D_{{\text{c}}} }}} \right)^{0.1}$$
(4)
Fig. 4
figure 4

The comparison of a helix tube, b outer shell side Nusselt number, and c hollow conduit side Nusselt number with the literature

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Figure 4b shows the verification of outer shell side fluid (HF2) Nusselt number with the Nusselt number correlations suggested by Coates and Pressburg [33], i.e. is given in Eq. 5.

$${\text{Nu}}_{{{\text{HF}}_{2} }} = 0.6\;{\text{Re}}_{{{\text{HF}}_{2} }}^{0.5} {\text{Pr}}_{{{\text{HF}}_{2} }}^{0.31} \;{\text{for}}\;50 \le {\text{Re}} \le 10,000$$
(5)

Figure 4c shows the verification of hollow conduit side fluid (HF3) Nusselt number with the Nusselt number correlations suggested by Vicente et al. [34], i.e. is given in Eq. 6.

$${\text{Nu}} = 0.374\left( {\frac{{h^{2} }}{{{\text{pd}}}}} \right)^{{0.25}} ({\text{Re}} - 1500)^{{0.74}} \text{Pr} ^{{0.44}}$$
(6)

From Fig. 4a, b, and c, the results of the current experimental investigation appear to agreement reasonably well with those cited in the literature, this similarity helps confirm that our study is accurate. Differences in geometric configurations, thermal interactions, and the intrinsic surface corrugations present in both the shell side fluid (HF2) and the fluid within the inner conduit (HF3) may serve to clarify the minor divergences noted within the observed results.

Taguchi method

The Taguchi method forms the design of experiments technique that primarily tries to reduce the number of experimental repetitions while permitting the prediction of factor interactions, in order to determine particular optimized results. This technique constitutes an orthogonal array by reducing the number of experimental runs, thoroughly evaluating the influence of the investigated elements, providing signal-to-noise (S/N) analysis, and figuring out the best values for the variables under consideration. The performance optimization of heat exchangers can be accomplished successfully by combining the Taguchi method with Grey relational analysis. According to the literature, this collaborative strategy enables the systematic consideration of numerous input variables, producing economic efficiencies through the reduction of costs and experimental test runs [35,36,37].

The main steps of the Taguchi method are the selection of independent variables, level determination for each independent variable, selection of an orthogonal array that ensures a balanced comparison of levels, allocation of independent variables to individual columns, execution of the experiments, analytical interpretation of the data, and finally the formulation of conclusive insights. Notably, the minimum number of test runs necessary is derived from the total degrees of freedom (DF) [38] which is given in Eq. 7.

$${\text{DF}} = {\text{Factors}}*({\text{level}} - 1)$$
(7)

Calculated signal-to-noise (S/N) ratio serves as an indicator of the effectiveness of the experimental design by the Taguchi method. The resulting equations outline the formulas for several S/N ratio categories, including “larger-the-better” (LB), “smaller-the-better” (SB), and “nominal-the-better” (NB) conditions which are given in Eqs. 8, 9, and 10, respectively [39].

$${\text{LB}} = \frac{1}{n}\sum {\left( {\frac{1}{{y_{{\text{i}}} }}} \right)}^{2}$$
(8)
$${\text{SB}} = \frac{1}{n}\sum {(y_{{\text{i}}} )}^{2}$$
(9)
$${\text{NB}} = \frac{1}{n}\sum {\left( {y_{{\text{i}}} - y_{{\text{o}}} } \right)}^{2}$$
(10)

Herein, “y” signifies the output variable, and “n” corresponds to the number of responses.

Equation 11 is used to calculate the value of the S/N ratio.

$${\text{S/N}}\;{\text{ratio}} = - 10\log (L_{{\text{i}}} )$$
(11)

where Li is calculated from the formulas given in Eq. 12.

$$L_{{\text{i}}} \left\{ {_{{{\text{Smaller}}\;{\text{is}}\;{\text{better}} = \frac{1}{n}\sum\limits_{i = 1}^{n} {y_{{\text{i}}}^{2} } }}^{{{\text{Larger}}\;{\text{is}}\;{\text{better}} = \frac{1}{n}\sum\limits_{i = 1}^{n} {1/y_{{\text{i}}}^{2} } }} } \right.$$
(12)

In the present study, the combined Taguchi method and Grey relational analysis are used to predict the optimal value of input parameters for the calculated optimized performance of the NMFHE. The control factors are mentioned in Table 3. These control factors include four principal operating flow parameters, specifically the HF1 flow rate, the HF2 flow rate, the HF3 inlet temperature, and the flow configurations. The L18 orthogonal array for control factors, the experimental response, and the signal-to-noise ratio at different input parameters are shown in Table 4. Four control parameters: flow configuration, HF1 flow rate, HF2 flow rate, and HF1 inlet temperature are the input variables that affect the performance of the heat exchanger in terms of entropy generation number (Ns) and JF factor for the optimization work by the Taguchi method. The Ns is analysed with “smaller is better”. The JF factor is analysed with “larger is better”.

Table 3 Control factors with their levels
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Table 4 Taguchi response for JF factor and entropy generation number (Ns)
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The experimental response and S/N ratio for the JF factor and the entropy generation number, Ns are presented in Tables 4 and 5 in L18 orthogonal array.

Table 5 Taguchi S/N ratio (dB) for JF factor and entropy generation number (Ns)
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The evaluation of the percentage contribution of each control parameter to the test performance is a crucial aspect of this study. This evaluation has been conducted using the signal-to-noise (S/N) ratio generated through the utilization of the Taguchi methodology. The resultant findings are graphically represented in Fig. 5. Within this analytical framework, the “delta” for each factor is defined and computed as the variance between the maximum and minimum S/N ratio values. Subsequently, the contribution rate is determined through a more intricate calculation. Specifically, the individual delta of each factor is subtracted from the collective sum of the deltas for all four factors.

Fig. 5
figure 5

Contribution ratio of each operating condition parameter to JF factor and entropy generation number (Ns)

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From Taguchi analysis, the percentage contribution of each factor to the JF factor and Ns was determined for the NMFHE and is given in Fig. 5.

Grey relational analysis

GRA is generally useful for evaluating the relationship degree between sequences and Grey relational grade (GRG). GRA is commonly used to integrate all of the performance characteristics that are analysed into a single number, which is generally the solution to the optimization problem. There are two steps for solving GRA. In the first step, it is needed to convert the data into S/N ratio, and in the second step, the data are pre-processed by normalization of data in the range of zero and one. In the field of heat exchanger performance analysis, the application of GRA facilitates a thorough examination of the interplay between geometric and flow parameters. The Taguchi–Grey relational analysis (GRA) method is employed to investigate parameters such as the Reynolds number, pitch ratio, diameter ratio, etc. [40]. Within a concentric pipe heat exchanger system, corrugated tapes are utilized, and GRA aids in understanding the relationship between Nusselt number and friction factor. This enables the ordering of corrugated tapes based on their effect intensity concerning dimensions such as width, pitch, and thickness. Thickness emerges as the most critical component in terms of the friction factor [41]. They use Taguchi method to investigate various design parameters with a single-point response in order to improve the air-side performance of a wavy fin and tube heat exchanger. For consolidated optimization, which covers all targeted responses concurrently, the idea of Grey relational analysis, a subtle statistical method, is studied [42].

These normalized data are divided into two categories, one in which “larger is better” and in the other “smaller is better”. If the output answer falls into the “bigger is better” group, it is stated using Eq. 13.

$$C_{{\text{i}}} (e) = \frac{{\left[ {x_{{\text{i}}} (e)_{\max } - x_{{\text{i}}} (e)} \right]}}{{\left[ {x_{{\text{i}}} (e) - x_{{\text{i}}} (e)_{\min } } \right]}}$$
(13)

Herein, “xi(e)” represents the original sequence, whereas “Ci(e)” sequence for comparing, with “i” ranging from 1 to n and “e” ranging from 1 to m. If the result falls in “smaller is better” category, then Eq. 14 may be considered

$$C_{{\text{i}}} (e) = \frac{{\left[ {x_{{\text{i}}} (e) - x_{{\text{i}}} (e)_{\max } } \right]}}{{\left[ {x_{{\text{i}}} (e)_{\max } - x_{{\text{i}}} (e)_{\min } } \right]}}$$
(14)

Following the normalization of the sequence, the deviation of the sequence for all outputs is determined, and the Grey relational coefficient (GRC) was calculated by using the relationship provided in Eq. 15.

$${\text{GRG}} = \frac{{d_{\min } + \psi d_{\max } }}{{d_{{{\text{oi}}}} + \psi d_{\max } }}$$
(15)

In this context, “doi” represents deviation of sequences across all responses. The highest and lowest deviation among all the compared sequences are designated as “dmax” and “dmin”, respectively. The identification coefficient (Ψ) values between 0 and 1 assume a value of 0.5 in the current study. The Grey relational grade (GRG) is calculated by Eq. 16.

$${\text{GRG}} = \frac{{\sum\nolimits_{e = 1}^{e = m} {{\text{GRC}}} }}{m}$$
(16)

Results and discussion

Model prediction and analysis of variance

The optimization software uses linear regression analysis as a function of input parameters to create a linear mathematical model for various output responses. The resultant mathematical equations for each respective response are presented, from Eq. 17 to Eq. 22. These predictive equations were formulated specifically for the JF factor (HF1), JF factor (HF2), JF factor (HF3), Ns (HF1), Ns (HF2), and Ns (HF3).

$$\left\{ {_{{{\text{CF}}2,\;{\text{JF}}\;{\text{factor}}\;{\text{(HF}}_{{1}} {)} = 0.0137 - 0.000048\;{\text{HF}}_{1} \;{\text{flow}}\;{\text{rate}} + 0.00002\;{\text{HF}}_{2} \;{\text{flow}}\;{\text{rate}} - 0.000048\;{\text{HF}}_{1} \;{\text{Inlet}}\;{\text{temp}}.}}^{{{\text{CF}}1,\;{\text{JF}}\;{\text{factor}}\;{\text{(HF}}_{{1}} {)} = 0.01339 - 0.000048\;{\text{HF}}_{1} \;{\text{flow}}\;{\text{rate}} + 0.00002\;{\text{HF}}_{2} \;{\text{flow}}\;{\text{rate}} - 0.000048\;{\text{HF}}_{1} \;{\text{Inlet}}\;{\text{temp}}.}} } \right.$$
(17)
$$\left\{ {_{{{\text{CF2,}}\;{\text{JF}}\;{\text{factor}}\;\left( {{\text{HF}}_{{2}} } \right) = {0}{\text{.07977}} + {0}{\text{.000031}}\;{\text{HF}}_{{1}} \;{\text{flow}}\;{\text{rate}} - {0}{\text{.00009}}\;{\text{HF}}_{{2}} \;{\text{flow}}\;{\text{rate}} - {0}{\text{.000019}}\;{\text{HF}}_{{1}} \;{\text{Inlet}}\;{\text{temp}}{.}}}^{{{\text{CF1,}}\;{\text{JF}}\;{\text{factor}}\;\left( {{\text{HF}}_{{2}} } \right) = {0}{\text{.08047 + 0}}{.000031}\;{\text{HF}}_{{1}} \;{\text{flow}}\;{\text{rate}} - {0}{\text{.00009}}\;{\text{HF}}_{{2}} \;{\text{flow}}\;{\text{rate}} - {0}{\text{.000019}}\;{\text{HF}}_{{1}} \;{\text{Inlet}}\;{\text{temp}}{.}}} } \right.$$
(18)
$$\left\{ {_{{{\text{CF}}2,\;{\text{JF}}\;{\text{factor}}\;{\text{(HF}}_{{3}} {)} = - 0.006 - 0.00008\;{\text{HF}}_{1} \;{\text{flow}}\;{\text{rate}} + 0.000131\;{\text{HF}}_{2} \;{\text{flow}}\;{\text{rate}} + 0.000781\;{\text{HF}}_{1} \;{\text{Inlet}}\;{\text{temp}}.}}^{{{\text{CF}}1,\;{\text{JF}}\;{\text{factor}}\;{\text{(HF}}_{{3}} {)} = - 0.0266 - 0.00008\;{\text{HF}}_{1} \;{\text{flow}}\;{\text{rate}} + 0.000131\;{\text{HF}}_{2} \;{\text{flow}}\;{\text{rate}} + 0.000781\;{\text{HF}}_{1} \;{\text{Inlet}}\;{\text{temp}}.}} } \right.$$
(19)
$$\left\{ {_{{{\text{CF}}2,\;{\text{Ns}}\;{\text{(HF}}_{{1}} {)} = 0.0351 + 0.000083\;{\text{HF}}_{1} \;{\text{flow}}\;{\text{rate}} - 0.00004\;{\text{HF}}_{2} \;{\text{flow}}\;{\text{rate}} - 0.000445\;{\text{HF}}_{1} \;{\text{Inlet}}\;{\text{temp}}.}}^{{{\text{CF}}1,\;{\text{Ns}}\;{\text{(HF}}_{{1}} {)} = 0.0401 + 0.000083\;{\text{HF}}_{1} \;{\text{flow}}\;{\text{rate}} - 0.00004\;{\text{HF}}_{2} \;{\text{flow}}\;{\text{rate}} - 0.000445\;{\text{HF}}_{1} \;{\text{Inlet}}\;{\text{temp}}.}} } \right.$$
(20)
$$\left\{ {_{{{\text{CF2}},\;{\text{Ns}}\;{\text{(HF}}_{{2}} {)} = 0.0046 - 0.000002\;{\text{HF}}_{1} \;{\text{flow}}\;{\text{rate}} - 0.000012\;{\text{HF}}_{2} \;{\text{flow}}\;{\text{rate}} - 0.000025\;{\text{HF}}_{1} \;{\text{Inlet}}\;{\text{temp}}.}}^{{{\text{CF}}1,\;{\text{Ns}}\;{\text{(HF}}_{{2}} {)} = 0.0102 - 0.000002\;{\text{HF}}_{1} \;{\text{flow}}\;{\text{rate}} - 0.000012\;{\text{HF}}_{2} \;{\text{flow}}\;{\text{rate}} - 0.000025\;{\text{HF}}_{1} \;{\text{Inlet}}\;{\text{temp}}.}} } \right.$$
(21)
$$\left\{ {_{{{\text{CF}}2,\;{\text{Ns}}\;{\text{(HF}}_{{3}} {)} = 0.0135 + 0.000003\;{\text{HF}}_{1} \;{\text{flow}}\;{\text{rate}} - 0.000001\;{\text{HF}}_{2} \;{\text{flow}}\;{\text{rate}} - 0.000005\;{\text{HF}}_{1} \;{\text{Inlet}}\;{\text{temp}}.}}^{{{\text{CF}}1,\;{\text{Ns}}\;{\text{(HF}}_{{3}} {)} = 0.0017 + 0.000003\;{\text{HF}}_{1} \;{\text{flow}}\;{\text{rate}} - 0.000001\;{\text{HF}}_{2} \;{\text{flow}}\;{\text{rate}} - 0.000005\;{\text{HF}}_{1} \;{\text{Inlet}}\;{\text{temp}}.}} } \right.$$
(22)

The R2-value shows the capability of the model to predict output responses, which vary from 0 to 1. The R2-value close to 1 or more than 90% shows a good model fit to relate the independent and dependent variables effectively. The models formulated for the JF factor (HF1, HF2, and HF3), as well as the entropy generation number for HF1, HF2, and HF3, are detailed in the analysis of variance (ANOVA) presented in Table 6. The respective R2-values for these models are 97.93%, 96.76%, 94.05%, 89.51%, 98.88%, and 99.94%. Such results underscore the models’ high precision and dependability in forecasting the output responses.

Table 6 ANOVA analysis and model prediction
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Effect of control factors on output responses

JF factor (HF1)

Figure 6 shows the effect of flow configuration, HF1 inlet temperature, HF1 flow rate, and HF2 flow rate on JF factor (HF1). It is observed in Fig. 6a that the maximum JF factor (HF1) is achieved at CF2 flow configuration, 100 LPH flow rate of HF1, 200 LPH flow rate of HF2, and 333 K inlet temperature of HF1. This finding is consistent with the results presented in Fig. 6b. Notably, the HF1 flow rate has a substantial impact on JF factor (HF1) which is confirmed in Figs. 5 and 6b with a contribution of 58.67%, followed by the HF2 flow rate, HF1 inlet temperature, and flow configuration with contribution of 25.64%, 13.44%, and 2.24%, respectively. The NMFHE performs better in CF2 flow configuration, as the bulk mean temperature of HF1 is higher compared to HF2 and HF3. It is noticed that increment in volumetric flow rate of HF1 and HF2 decreases JF factor (HF1), as increased flow rate leads to more turbulence and more heat transfer. However, the influence of HF1 inlet temperature on JF factor (HF1) is minimal as the variation of temperature is limited to 283 K, which increases the thermo-physical properties of the HF1 slightly. Insignificant changes in thermo-physical properties may be a cause of this result.

Fig. 6
figure 6

Effect of each input parameter on the JF factor (HF1)

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JF factor (HF2)

Figure 7 shows the effect of flow configuration, HF1 inlet temperature, HF1 flow rate, and HF2 flow rate on JF factor (HF2). Figure 7b shows the S/N ratio plot which indicates higher heat transfer performance of the present NMFHE at higher value of S/N ratio. It is observed in Fig. 7b that the maximum JF factor (HF2) is achieved at CF1 flow configuration, 200 LPH flow rate of HF1, 100 LPH flow rate of HF2, and 333 K inlet temperature of HF1. This finding is consistent with the results presented in Fig. 7a. Notably, the HF2 flow rate has a substantial impact on JF factor (HF2) which is confirmed in Figs. 5 and 7b with a contribution of 65.88%, followed by the HF1 flow rate, HF1 inlet temperature, and flow configuration with contribution of 22.94%, 5.88%, and 5.294%, respectively. The NMFHE performs better in CF1 flow configuration, as the bulk quantity of HF2 is higher compared to HF1 and HF3. So, its reversal also affects that the most is stated above. It is noticed that increment in volumetric flow rate of HF1 and HF2 increases JF factor (HF2), as increased flow rate leads to more turbulence and more heat transfer. However, the influence of HF1 inlet temperature on JF factor (HF2) is minimal as the variation of temperature is limited to 283 K, which slightly increases the thermo-physical properties of the HF1. Insignificant changes in thermo-physical properties could be a cause of this result.

Fig. 7
figure 7

Effect of each input parameter on JF factor (HF2)

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JF factor (HF3)

Figure 8 shows the effect of flow configuration, HF1 inlet temperature, HF1 flow rate, and HF2 flow rate on JF factor (HF3). Specifically, Fig. 8b shows the S/N ratio plot which indicates higher heat transfer performance of the present NMFHE at higher value of S/N ratio. It is observed in Fig. 8b that the maximum JF factor (HF3) is achieved at CF2 flow configuration, 150 LPH flow rate of HF1, 150 LPH flow rate of HF2, and 353 K inlet temperature of HF1. This finding is consistent with the results presented in Fig. 8a. Notably, the flow configuration has a substantial impact on JF factor (HF3) which is confirmed in Figs. 5 and 8b with a contribution of 34.85%, followed by the HF1 inlet temperature, HF2 flow rate, and HF1 flow rate with contribution of 28.19%, 26.16%, and 10.81%, respectively. The NMFHE performs better in CF2 flow configuration, as the bulk quantity of HF2 is higher compared to HF1 and HF3. So, its reversal also affects that the most is stated above. It is noticed that increment in volumetric flow rate of HF1 increases JF factor (HF3), as increased flow rate leads to more turbulence and more heat transfer. However, the influence of HF1 inlet temperature on JF factor (HF3) is minimal as the variation of temperature is limited to 283 K, which slightly increases the thermo-physical properties of the HF1. Insignificant changes in thermo-physical properties may be a cause of this result.

Fig. 8
figure 8

Effect of each input parameter on JF factor (HF3)

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Entropy generation number of HF1

In Fig. 9, the effect of flow configuration, flow rate of HF1, flow rate of HF2, and inlet temperature of HF1 on entropy generation number of HF1, Ns (HF1) is presented. The result indicates that for the “smaller is better” statement, the minimum value of Ns (HF1) is obtained at the HF1 flow rate of 100 LPH, HF2 flow rate of 200 LPH, HF1 inlet temperature of 343 K, and flow configuration of CF2. This finding is consistent with the results presented in Fig. 9b. Notably, the HF1 inlet temperature is found out as the most contributing factor as shown in Figs. 5 and 9b with a contribution of 31.51% followed by the HF1 flow rate, flow configuration, and HF2 flow rate contribute 27.78%, 24.11%, and 16.59%, respectively. The HF1 flow rate has a significant impact on Ns (HF1) as illustrated in Fig. 9. This is because at higher flow rates, the resultant turbulence is more which causes more pressure drop.

Fig. 9
figure 9

Effect of each input parameter on entropy generation number of HF1

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Entropy generation number of HF2

In Fig. 10, the effect of flow configuration, flow rate of HF1, flow rate of HF2, and inlet temperature of HF1 on entropy generation number of HF2, Ns (HF2) is presented. The result indicates that for the “smaller is better” statement, the minimum value of Ns (HF2) is obtained at the HF1 flow rate of 200 LPH, HF2 flow rate of 200 LPH, HF1 inlet temperature of 353 K, and flow conf. of CF2. There is little effect of flow configuration on Ns (HF2). A slight change in value of Ns (HF2) due to a change in the HF1 inlet temperature and HF1 flow rate. This finding is consistent with the results presented in Fig. 10. Notably, the flow conf. is found out as the most contributing factor as shown in Figs. 5 and 10b with a contribution of 84.95% followed by the flow rate of HF2, flow rate of HF1, and inlet temperature of HF1 configuration contribute 13.32%, 1.54%, and 0.19%, respectively. The HF2 flow rate has a significant impact on Ns (HF2) as illustrated in Fig. 10. This is because at higher flow rates, the chaotic motion of HF2 during flow over the corrugated surface of BHT causes more pressure drop.

Fig. 10
figure 10

Effect of each input parameter on entropy generation number of HF2

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Entropy generation number of HF3

In Fig. 11, the effect of flow configuration, flow rate of HF1, flow rate of HF2, and inlet temperature of HF1 on entropy generation number of HF3, Ns (HF3) is presented. The result indicates that for the “smaller is better” statement, the minimum value of Ns (HF3) is obtained at the HF1 flow rate of 200 LPH, flow configuration of CF1, HF2 flow rate of 100 LPH, and HF1 inlet temperature of 353 K. This finding is consistent with the results presented in Fig. 11. Notably, the flow conf. is found out as the most contributing factor as shown in Figs. 5 and 11b with a contribution of 98.44% followed by the inlet temperature of HF1, HF1 flow rate, and flow rate of HF2 contribute 0.81%, 0.62%, and 0.12%, respectively.

Fig. 11
figure 11

Effect of each input parameter on entropy generation number of HF3

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Taguchi–Grey technique for multiple performance optimizations

In the present research, a methodological application of Taguchi–Grey analysis has been employed to perform multiple response optimizations for a NMFHE. The primary objectives encompass maximizing the heat transfer performance, characterized by the JF factor, and minimizing the hydraulic performance, denoted by Ns. A thorough examination of signal-to-noise (S/N) ratios is conducted, and the results are tabulated in Table 5. Subsequent preprocessing and normalization of the data are performed using Eqs. 14 and 15, and the corresponding normalized values and deviation sequences are cataloged in Table 7. Following this preprocessing stage, the deviation sequence Grey relational coefficient (GRC) is systematically calculated for each response using Eq. 15. Additionally, the Grey relational grade (GRG) is determined via Eq. 16, with these computations presented in Table 8. Within this framework, the highest S/N ratio serves as an indicator of the top-ranked factor, specifically correlated to the 12th experimental run.

Table 7 Normalized and deviation sequences
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Table 8 Grey relation coefficient and grade
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After assigning these rankings, a response table for Grey relational grade (GRG) is formulated, facilitating a thorough examination of GRG across the factors. This involves calculating the mean GRG for each factor by averaging the individual GRG values at their specified levels. As depicted in Table 8, there exists an identifiable correlation between the reference sequence and the GRG compatibility sequence. Higher mean GRG values are indicative of higher correlations, thereby providing significant insights into the interactions of the variables within the NMFHE system. The peak GRG value is recorded at level 3 for the HF2 flow rate, level 2 for the flow configuration, level 1 for the HF1 flow rate, and level 2 for the HF1 inlet temperature, as delineated in Table 9. Consequently, the optimal performance of the NMFHE is achieved with the CF2 flow configuration, an HF2 flow rate of 200 LPH, an HF1 flow rate of 100 LPH, and an HF1 inlet temperature of 343 K.

Table 9 Response table of GRGs and its peak values
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Analysis of variance for GRG

To evaluate the influence of the parameters on the response outcomes, an analysis of variance (ANOVA) was executed for the GRG at a significance level of 5%. Subsequently, the proportional contribution of each parameter was ascertained and is tabulated in Table 10. The findings underscore that the HF1 flow rate exerts the most pronounced effect with a contribution of 34.58%. This is followed by the HF1 inlet temperature, flow configuration, and HF2 flow rate, with contributions of 18.66%, 1.56%, and 0.142%, respectively. Additionally, the R2-value surpassed 92.46%, suggesting a robust correlation between the GRG model and the observed data.

Table 10 Analysis of variance
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Confirmation test and improvement in GRG

The next phase involves anticipating and verifying the quality attributes using Eq. 23 after determining the ideal conditions using Grey analysis:

$${\text{GRG}}_{{{\text{predicted}}}} = {\text{GRG}}_{{{\text{mean}}}} + \sum\nolimits_{i = 1}^{q} {\left( {{\text{GRG}}_{{{\text{oi}}}} - {\text{GRG}}_{{{\text{mean}}}} } \right)}$$
(23)

Herein, q signifies the number of components, GRGoi is indicative of the highest value of the mean GRG, and GRGmean represents the median GRG value.

The GRG prediction is effectuated through Eq. 23, followed by the execution of a validation test to affirm the outcomes. The preliminary condition parameters are discerned by referencing the average value from the GRG dataset, subsequently opting for the GRG value in Table 9 that most closely approximates this mean value.

Table 11 represents that the expected values and the results of the confirmation test are close to each other. This shows that the optimization process is reliable. The Grey relational grade (GRG) is improved by 8.36% which has been confirmed, which indicates that the combine Taguchi technique and Grey relational analysis efficiently improve the performance of the NMFHE.

Table 11 Result of the confirmation test
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Conclusions

A detailed study of the thermo-hydraulic and exergetic performance of the NMFHE is carried out presently by varying different input parameters such as flow configuration, flow rate, and inlet temperature. Currently, the JF factor and entropy generation number, Ns, are determined as the output responses. With the Taguchi method, an L18 orthogonal array is used for conducting experimental runs, and the overall performance of the NMFHE is optimized, i.e. maximum JF factor and minimum Ns determined in regard to the optimal value of input parameters using the Grey relational grade technique in %. Based on the results, the following conclusions have been drawn.

  1. i.

    Variation in flow configuration from CF1 to CF2 resulted higher JF factors for HF1 and HF3, as well as higher Ns for HF1 and HF2. Conversely, the JF factor (HF2) and the Ns value for HF3 decrease with variation in flow configuration from CF1 to CF2. The CF2 flow configuration is identified as the most significant factor to the JF factor (HF3) and the Ns value of HF3 with a contribution of 34.86% and 98.44% of the total variation, respectively.

  2. ii.

    Increasing the flow rate of HF1 resulted in an increased JF factor (HF2) but decreased JF factor (HF1) and Ns (HF1). The JF factor (HF3) showed an initial rise followed by a decline. Minimal changes were seen for Ns (HF2) and (HF3). Notably, the JF factor (HF1) and the Ns value of HF1 were significantly impacted, accounting for 58.67% and 27.78% of the total variation, respectively.

  3. iii.

    With an increase in the flow rate of HF2, the JF factor (HF1), Ns (HF1), and Ns (HF2) increase significantly, while the JF factor (HF2) decreases significantly, with negligible changes observed for Ns (HF3). The JF factor (HF3) increases initially and decreases later. The HF2 flow rate is identified as the most significant factor to JF factor (HF2) and Ns (HF1) with a contribution of 65.88% and 16.59% of the total variation, respectively.

  4. iv.

    With an increase in the HF1 inlet temperature, the JF factor (HF3) increases and the JF factor (HF1) decreases significantly, whereas negligible changes are observed for the Ns (HF2) and Ns (HF3). HF1 inlet temperature is identified as the most significant factor for the JF factor (HF3) and Ns (HF1) with a contribution of 28.18% and 31.51% of the total variation, respectively.

  5. v.

    A confirmation test was performed to validate the results of the optimization analysis. An improvement of 8.36% in performance was observed with the considered GRG model.

  6. vi.

    Performance optimization of the NMFHE was conducted using the Taguchi—GRA method. The CF2 flow configuration, 150 LPH of HF1 flow rate, 150 LPH of HF2 flow rate, and 80 °C of HF1 inlet temperature were found as the optimum input parameters for this study

  7. vii.

    The performance of the NMFHE can be tested with nanofluids for improved heat transfer performance. The numerical analysis could be performed by varying geometrical parameters at different fluid flow conditions. These are the possible future scopes of this study.