Introduction

As a consequence of industrial expansion and population growth, the rate of energy consumption increases. It causes economic distress as well as damage to the environment in the form of acid rain, global warming, ozone layer depletion, and many other phenomena (Perera 2017). Global CO2 emissions from fossil fuels and trade have been essentially flat in recent years, owing to falling coal use globally as well as improved energy efficiency and increased use of renewable energy. The UN’s Framework Convention on Climate Change (UNFCCC2015)’s Paris Agreement formally came into force in 2016. This agreement aims to check global warming temperatures to 1.5–2 °C overhead pre-industrial levels in the twenty-first century, and calls for the development of new technologies to manage CO2 levels in the atmosphere (Arafa 2017). In a near-future energy scenario, technology transfer to 100% of renewable energy is being prioritized (Gielen et al. 2019). For the entire world, the COVID-19 lockdown has provided a vision of a clean air future (Landrigan et al. 2020). This has resulted in an increased understanding of renewable energy's ability to assist in the creation of clean, livable, and equal communities.

Employing renewable forms of energy, of which solar energy is by far the most popular, will help minimize energy demand from conventional, non-renewable sources. The modest and operative way to consume solar energy is to transform it into thermal energy for a variety of applications, including heating and electricity generation. The solar air collector (SAC), which is used for agricultural and industrial drying, building heating, and ventilation applications, is an appropriate technology for utilizing solar energy (Jain et al. 2021b). Because of the thermophysical characteristics of air and the formulation of the viscous sublayer, SAC has a low convective heat transfer rate (Azari et al. 2021). Detailed practical and theoretical studies conducted in the literature have enhanced the performance of SAC. Novel strategies to enhance the performance of SACs presented in the literature are discussed herein.

Sahu et al. (2021) revealed that the arc-shape rib roughened SAC has a maximum thermal efficiency of 71.6% for apex-upstream and 55.8% for apex-downstream than the smooth pass solar air collector (SPSAC) for a mass flow rate ranging from 0.010 to 0.047 kg s−1. Kumar et al. (2021) analyzed the thermohydraulic performance of the SAC with zig-zag-shaped copper tubes and revealed that the maximum value of energy and exergy efficiency as 62.8% and 58.48% was obtained at an airflow rate of 0.05 kg/s. Tabish et al. (2021) analytically evaluated the performance of the protrusion rib-roughened SAC and found that the net effective efficiency of 70.92% was achieved at a rib pitch ratio of 10 and rig height ratio of 0.0289. Kashyap et al. (2021) studied the consequences of a V-ribs with multiple gaps on the absorber plate of SAC with Reynolds number (Re) ranging from 2000 to 21,000. They revealed that due to the production of secondary-flow vortices on the absorber surface, the multiple V-ribs patterns with symmetric gaps resulted in a significant increase in Nusselt number (Nu) and a friction factor (ff). Rahmani et al. (2021) numerically simulated the performance of SAH with wavy and raccoon-shaped fins. They observed that increasing the fin height resulted in a 50% increase in Nu and ff escalation of nearly the same value. Hassan et al. (2021) investigated the thermal performance of multiple arc dimple-roughened SAC and found that the Nu increased by 2.87–5.61% compared to conventional SAC. Chauhan and Thakur (2014) studied the jet impingement SAC for Re ranging from 4000 to 16,000 and revealed that the thermohydraulic performance improves by 34.54–57.89% when compared to the conventional SAC. Chauhan and Kim (2019a, b) examined the thermohydraulic performance of an artificially roughened plate in an SAC, and concluded that the Nu and ff were both increased by 2.24 and 2.65, respectively. Furthermore, they investigated the effect of dimple roughening on effective efficiency and concluded that a pitch ratio of 10, a height ratio of 3, and a dimple arc angle of 60° resulted in an effective efficiency of 69.7%. Salman et al. (2021a, b) experimentally investigated the exergetic efficiency of an indented dimple-roughened absorber plate in a jet-impinged SAC. They revealed that at a temperature rise parameter of 0.03 km2/W, the maximum value of exergetic efficiency is 0.026. Goel et al. (2021) investigated the performance and parametric optimization of jet-finned SAC and found that the optimal values of the fin and jet slot parameters generated the maximum collector efficiency are 0.23 for the fin spacing ratio, 0.46 for the Xi/Dhd, and 0.076 for the Di/Dhd.

Some authors have considered the heat transfer features of combined techniques utilized in single-pass SACs. Goel and Singh (2020b, a, 2022) experimentally compared the performance of a fin-type jet-impinged SAC and found that the thermal efficiency was 8% and 29% improved compared to the conventional and jet-impinged SAC, respectively. Moreover, fin-impinged SAC achieved a maximum enhancement in Nusselt number and effective efficiency of 2.45 and 82.5% at fin spacing ratio = 0.23 and Xi/Dhd = 0.46, respectively. Skullong et al. (2016, 2017) experimented with several artificial roughness configurations to enhance the thermal performance of SAC. It was found that the use of collective delta-wing and wavy-groove vortex creators enhanced the convective heat transfer rate. They concluded that for Re ranging from 4800 to 23,000, the thermohydraulic performance of the combined SAC technique was augmented at approximately 37.7–46.3% higher than that of the grooved SAC. Furthermore, wavy grooves combined with a couple of trapezoidal winglets were utilized to enhance the thermohydraulic performance by 43% over the grooved individual SAC. Salman et al. (2021a, b) experimentally investigated the dimple-roughened absorber plate in jet-impinged SAC and determined that the optimal thermohydraulic performance value of 2.15 is attained at Re = 15,000 for roughness parameters, that is, arc angle = 60°, dimple pitch ratio = 0.269, and dimple height ratio = 0.0267. Goel and Singh (2021a) analytically investigated the performance of jet impinged SAC with fins attached to the absorber plate. They found that the maximum enhancement in temperature rise parameter and thermal efficiency of 20.5% and 16.4% at a Di/Dhd of 0.076, respectively.

The purpose of all the preceding experimental studies was to investigate the features of heat transfer and frictional losses concerning specific use. The increase in heat transfer is attributed to a significant increase in frictional losses within the collector’s duct, and both of these parameters are influenced by the geometric configuration and operational conditions. The heat transfer rate must be higher for a better system efficiency output, and the frictional losses are unpleasant and should therefore be taken into consideration. The goal of identifying the set of conditions under which the desired outcome occurs where the heat transfer is positioned at the highest level, while the undesirable frictional losses are placed at the lowest level, meets a study that arrays down the set of parameters that convey the maximum energy output of the energy conversion system and demonstrate its dominance. The collection of such a best parameter set necessitates a simple, systematic, and practical approach to guide researchers to consider several performance-defining factors and their interconnections.

A large number of test runs should be implemented in both numerical and experimental studies to determine the thermohydraulic performance of an SAC. The layout of the investigation methods can be used to reduce the number of test runs while also obtaining the thermophysical phenomena and optimum solutions. Multicriterion decision-making (MCDM) methods can be used to find solutions consisting of multiple alternatives and performance evaluation criteria that have gained widespread recognition for use in science, engineering, and management, among other fields ( Chauhan et al. 2016; Jain et al. 2021a; Chauhan et al. 2017a, b; Yu et al. 2021; Abdul et al. 2022). An MCDM is a procedure that specifies how attribute data are analyzed to choose the best option among several alternatives. It includes a variety of procedures, such as gray relation analysis, preference ranking arrangement, compromise ranking approach, a method for enrichment evaluation order preference strategy based on resemblance to perfect clarifications, and a successful analytic hierarchy process (AHP) that has been utilized to tackle a slew of decision-making issues. Although several MCDM methods exist in the literature to help designers find the best alternative, most of these methods are difficult to understand and use, requiring a high level of mathematical complexity. Among various MCDM methods (Wang et al. 2017; Rani and Tripathy 2022; Sharma et al. 2017; Singh et al. 2016; Chauhan et al. 2017a, b; Zhao et al. 2021; Abdollahpour et al. 2020), AHP and multi-attributive border approximation area comparison (MABAC) are the most widely utilized ones to address renewable and sustainable energy constraints (Heo et al. 2010; Kahraman et al. 2009). AHP has been utilized by researchers in a variety of situations, such as proposing a public work contract (Bertolini et al. 2006) and selecting a phase change material for thermal storage in solar air systems (Xu et al. 2017). The MABAC approach was employed by a majority of researchers for an assortment of applications, including the selection of hotels on business trip sites (Yu et al. 2017), guided anti-tank missile battery’s defensive operation (Bojanic et al. 2018), and commercially available scooters (Biswas and Saha 2019). The advantage of the integrated AHP-MABAC approach is that it aids in the evaluation of optimal alternatives when there is a disagreement about the relative relevance of the criteria and requires fewer numerical calculations.

The key objective of this study was to determine whether the integrated AHP-MABAC approach can be used to predict the enactment of a jet-impinged SAC with a dimple-roughened absorber plate (JIDRSAC). To accomplish this objective, an investigation was directed to identify the heat transfer rate and frictional losses, which were used to evaluate the optimal constraints. To evaluate the effect of dimple roughness, the optimal set of jet plate parameters, that is, streamwise jet pitch ratio = 1.079, spanwise jet pitch ratio = 1.619, and circular jet diameter ratio = 0.081 were kept constant throughout the test. The purpose of this study was to evaluate the effect of airflow and dimple roughness parameters dimensionalized by the hydraulic diameter (Dh) of the duct, that is, the arc angle (αaa) ranging from 30° to 75°, dimple pitch ratio (pd/Dh) ranging from 0.269 to 1.08, and dimple height ratio (ed/Dh) ranging from 0.016 to 0.0324 on the performance of JIDRSAC. The impact of dimple roughness parameters on the overall performance is also explored to reach an optimal geometrical arrangement that places the desired heat transfer at its highest level while minimizing frictional losses inside the JIDRSAC duct. The quantitative estimation of these parameters is relatively broad and increases the experimental cost and time. Therefore, the trials were designed using the AHP-MABAC, followed by the percentage contribution of the parametric effect over the performance measures. The exergetic efficiency (ηex) performance of the system with respect to the heat transfer rate and frictional losses was considered, and parameters affecting the performance, that is, Re, pd, αaa, and ed, were also investigated. To the authors’ best knowledge, the use of the AHP-MABAC on JIDRSAC has not been investigated before, and it is evident that the optimum set of parameters leads to the maximum ηex of a SAC system.

Experimental setup

As showed in Fig. 1a, an experimental setup was prepared and implemented to investigate dimples on an absorber plate in a jet impingement SAC. It comprises a rectangular-shaped duct measuring 2000 mm × 250 mm × 25 mm, a supporting frame, a pyranometer (SR05-D1A3, Hukseflux), a solar flux simulator panel, a data acquisition panel (National Instruments), a plenum, an orifice meter (Samdukeng), and an air blower (dongkun-DB 301). Figure 1b depicts a schematic of the JIDRSAC duct shape and dimensions. The entry and exit lengths are considered as per the ASHRAE ( ASHRAE standard 93 2003), \(\ge 5\sqrt{{W}_{d}\times {H}_{d}}\) and \(\ge 2.5\sqrt{{W}_{d}\times {H}_{d}}\), respectively. Table 1 lists the characteristics and dimensions of the components used. The flow rate of air was measured using a manometer connected through a Pitot tube in accordance with the guidelines of the International Committee for Measures and Weights (CIPM-81–91) (Picard et al. 2008). By using 50-mm-thick glass wool with thermal conductivity of 0.0343 Wm−1 K−1, the duct is well insulated from three sides to minimize heat loss to the environment. The SAC duct was planned sensibly so that the dimple-roughened absorber plate was fitted at its location. The gap maintained among the backplate, jet plate, dimple-roughened absorber plate, and top glass cover was using acrylic spacers. The absorber plate is made of aluminum with a thickness of 1 mm. Using a circular tip indenter tool, the dimple roughness geometry was manually created beneath the absorber plate (Bhushan and Singh 2011). The dimple-roughened absorber plate is illustrated in Fig. 1c.

Fig. 1
figure 1

a Pictographic view of the test facility, b graphical representation of test facility, c dimple-roughened absorber plate.

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Table 1 Characteristics of apparatus used
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On the top of the SAC duct, the 12 halogen lamps fixed to the iron frame are assembled in four rows with three in each row to produce a solar flux of 1000 W m−2. Each halogen lamp was graded at 500 W and controlled by autotransformers and regulators. A digital solarimeter was coupled to the computer via a data logger to check the consistency of the solar flux along the test section. The temperatures of the inlet air, dimple-roughened heated plate, and outlet air were noted using a K-type thermocouples. A set of three thermocouples each at the entry and exit sections was arranged along the midway of the duct’s height. Along the test section length, fifteen thermocouples were coupled with the absorber plate at different locations as shown in Fig. 2a. The average value of the thermocouples was used to calculate the temperature along the duct. Figure 2b displays the air temperatures inside the duct and along the absorber plate. The absorber plate’s temperature showed linear variation in the airflow direction. Temperature variations in the direction normal to the absorber plate could not be measured because of the narrow duct depth, thus they were assumed to be minimal. A difference in pressure along the test section's length was provided via two pressure taps at the entry and exit sections. The software LabVIEW was designed to store all of the data collected as experimental results in the workstation via a data logger. The range of airflow and dimple-roughened absorber plate parameters, namely pd/Dh, αaa, and ed/Dh, for optimization are given in Table 2. These parameters must be optimized to produce a set of Re, pd/Dh, αaa, and ed/Dh that predict the maximal convective heat transfer with a minimal frictional penalty inside the duct.

Fig. 2
figure 2

a Position of thermocouples, b temperature variation of air inside the duct.

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Table 2 Range of flow and geometric parameters
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The scripts of Duffie and Beckman (2013) were used to acquire the thermo-physical properties of air, such as thermal conductivity (ka), specific heat (Cp), kinematic viscosity (\(v\)), and density (ρa). The obtained values of these important parameters were utilized to calculate the values of the fundamental procedure parameters, which were then used to evaluate the performance-defining criteria (PDCs). The procedure parameters, namely velocity of flowing air (Va), hydraulic diameter (Dhd), Re, hct, Nujd, and ffjd were calculated as follows (Salman et al. 2021a, b):

Va is considered with the assistance of the obtained mass flow rate (ma) through the data logger as

$${V}_{a}=\frac{{m}_{a}}{{\rho }_{a}{W}_{d}{H}_{d}}$$
(1)

where ma is the mass flow rate of air.

The hydraulic diameter is evaluated as

$${D}_{hd}=\frac{2\left({W}_{d}\times {H}_{d}\right)}{\left({W}_{d}\times {H}_{d}\right)}$$
(2)

The Re of airflow in the SAC duct is intended from

$$Re=\frac{{V}_{a}\times {D}_{hd}}{v}$$
(3)

The useful heat gain (Quhg) is given by

$${Q}_{uhg}={m}_{a}{C}_{p}\left({T}_{ao}-{T}_{ai}\right)={h}_{ct}{A}_{ab}\left({T}_{abm}-{T}_{am}\right)$$
(4)

where Aab is absorber plate surface area.

Tabm is bulk mean temperature of the absorber plate \({T}_{abm}=\frac{1}{15}\left(\sum_{i=1}^{15}{T}_{abmi}\right)\); Tam is mean temperature of the absorber plate \({T}_{am}=\frac{{T}_{ai}+{T}_{ao}}{2}\)

The hct for the duct has been evaluated as

$${h}_{ct}=\frac{{Q}_{uhg}}{{A}_{ab}\left({T}_{abm}-{T}_{am}\right)}$$
(5)
$$\mathrm{The\; Nusselt\; number}, {Nu}_{jd}=\frac{{h}_{ct}{D}_{hd}}{{k}_{a}}$$
(6)

Where ka is thermal conductivity of air.

$$\mathrm{The\; friction\; factor}, {ff}_{jd}=\frac{\left(\Delta {D}_{hd}\right)}{2{\rho }_{a}{L}_{t}{V}_{a}^{2}}$$
(7)

where \(\Delta {P}_{d}\) is the differential pressure across the duct, and Lt is the length of the test section.

Experimental validation and uncertainty in measurement

Validation of SPSAC and jet impingement duct

To establish a comparative foundation with heat transfer and pressure losses concerning the Nusselt number and friction factor correlations stated in the literature, experimental tests proving the validity of the SPSAC test rig were performed. The obtained experimental results were compared with the respective correlation predictions. The Dittus–Boelter (Bhatti and Shah 1987) and the modified Blasius equation (Incropera et al. 2011), as presented in Eq. 8 and Eq. 9, were used to correlate the Nus and ffs values determined in contradiction to the data obtained for the smooth absorber plate.

$${Nu}_{s}=0.023{Re}^{0.8}{\mathrm{Pr}}^{-0.4}$$
(8)
$${ff}_{s}=0.085 {\mathrm{Re}}^{-0.25}$$
(9)

Figure 3 depicts an assessment of the experimental and expected values of Nus and ffs as a function of Re. The attributes obtained for Nus and ffs by execution testing on FPSAC and those achieved from the standard correlations are comparable, as shown in Fig. 3. The deviance of the experimental and correlation values of Nus and ffs are within the ± 6–10% range which validates the SPSAC experimental duct.

Fig. 3
figure 3

Validation of FPSAC duct

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For the optimum parameters of the jet slot, the Nui and ffi statistics for air impinged on a smooth absorber plate revealed through experimental research were compared with the values acquired from Chauhan and Thakur’s correlation (2013), as depicted in Eqs. 10 and 11.

$${Nu}_{i}=0.00166{\left(\mathrm{Re}\right)}^{0.8512}{\left(\frac{{X}_{i}}{{D}_{hd}}\right)}^{0.1761}{\left(\frac{{Y}_{i}}{{D}_{hd}}\right)}^{0.141}{\left(\frac{{D}_{i}}{{D}_{hd}}\right)}^{-1.9854}\mathrm{exp}\left[-0.3498{\left(\mathrm{ln}\left(\frac{{D}_{i}}{{D}_{hd}}\right)\right)}^{2}\right]$$
(10)
$${ff}_{i}=3.475\times {10}^{-1}{\left(\mathrm{Re}\right)}^{-0.5244}{\left(\frac{{X}_{i}}{{D}_{hd}}\right)}^{0.4169}{\left(\frac{{Y}_{i}}{{D}_{hd}}\right)}^{0.5321}{\left(\frac{{D}_{i}}{{D}_{hd}}\right)}^{-1.4848}\mathrm{exp}\left[-0.2210{\left(\mathrm{ln}\left(\frac{{D}_{i}}{{D}_{hd}}\right)\right)}^{2}\right]$$
(11)

Figure 4 shows an evaluation of the experimental and correlated results of Nui and ffi as a function of Re. The maximum deviation of the experimental values of Nui and ffi values was within the range of ± 5–9%. The good agreement shown by the experiment and the correlated values certifies that the accuracy of the data is hoarded with the aid of a test rig. The constant parameters of the jet slot considered are stream-wise pitch ratio (Xi/Dhd) = 1.079, span-wise pitch ratio (Yi/Dhd) = 1.619, and jet diameter(Di/Dhd) = 0.081.

Fig. 4
figure 4

Validation of jet impingement duct

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Uncertainty in measurement

The values acquired through the examination may diverge from their real value since the existence of several factors that come into play during evaluation and recording. The uncertainty in the measurement is the alteration between the measured and real values. The uncertainties related to several instruments used in the data measurement are depicted in Table 3. Considering the errors in the measurements of the flow and geometric parameters, an uncertainty test was performed. The uncertainty was calculated in this investigation using the procedures described by Kline and McClintock (1953) and Moffat (1985). The following is a description of the technique for calculating uncertainty:

Table 3 Range of error of used instruments
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After the parameter is determined by employing thoroughly measured amounts, the uncertainty in the measurement of “y” is provided by

$$\frac{\delta y}{y}={\left[{\left(\frac{\delta y}{\delta {x}_{1}}\delta {x}_{1}\right)}^{2}+{\left(\frac{\delta y}{\delta {x}_{2}}\delta {x}_{2}\right)}^{2}+{\left(\frac{\delta y}{\delta {x}_{3}}\delta {x}_{3}\right)}^{2}+..+{\left(\frac{\delta y}{\delta {x}_{n}}\delta {x}_{n}\right)}^{2}\right]}^{1/2}$$
(12)

where δx1, δx2, δx3, ….. δxn are the possible errors in the measurement of x1, x2, x3,….. xn.

δy is the total, and δy/y is the comparative uncertainty.

The uncertainty in the measurement of duct dimensions and process parameters are evaluated as.

  • A. Hydraulic diameter (Dhd)

    $$\frac{\delta {D}_{hd}}{{D}_{hd}}=\frac{4{\left[{\left({~}^{\delta {D}_{hd}}\!\left/ \!{~}_{\delta {H}_{d}}\right.\times \delta {H}_{d}\right)}^{2}+{\left({~}^{\delta {D}_{hd}}\!\left/ \!{~}_{\delta {W}_{d}}\right.\times \delta {W}_{d}\right)}^{2}\right]}^{1/2}}{2\times \left({W}_{d}\times {H}_{d}\right){\left({W}_{d}+{H}_{d}\right)}^{-1}}$$
    (13)

  • B. Area of the absorber plate (Aab)

    $$\frac{\delta {A}_{ab}}{{A}_{ab}}={\left[{\left(\frac{{W}_{d}\times \delta {L}_{d}}{{W}_{d}\times {L}_{d}}\right)}^{2}+{\left(\frac{{L}_{d}\times \delta {W}_{d}}{{W}_{d}\times {L}_{d}}\right)}^{2}\right]}^{1/2}$$
    (14)

  • C. Reynolds number (Re)

    $$\frac{\delta \mathrm{Re}}{\mathrm{Re}}=\left[{\left(\frac{\delta {V}_{a}}{{V}_{a}}\right)}^{2}+{\left(\frac{\delta {\rho }_{a}}{{\rho }_{a}}\right)}^{2}+{\left(\frac{\delta {D}_{hd}}{{D}_{hd}}\right)}^{2}+{\left(\frac{\delta {\mu }_{a}}{{\mu }_{a}}\right)}^{1/2}\right]$$
    (15)

  • D. Useful heat gain (Quhg)

    $$\frac{\delta {Q}_{uhg}}{{Q}_{uhg}}={\left[{\left(\frac{\delta {m}_{a}}{{m}_{a}}\right)}^{2}+{\left(\frac{\delta {C}_{p}}{{C}_{p}}\right)}^{2}+{\left(\frac{\delta {T}_{am}}{{T}_{am}}\right)}^{2}\right]}^{1/2}$$
    (16)

  • E. Heat transfer coefficient (hct)

    $$\frac{\delta {h}_{ct}}{{h}_{ct}}={\left[{\left(\frac{\delta {Q}_{uhg}}{{Q}_{uhg}}\right)}^{2}+{\left(\frac{\delta {A}_{ab}}{{A}_{ab}}\right)}^{2}+{\left(\frac{\delta {T}_{am}}{{T}_{am}}\right)}^{2}\right]}^{1/2}$$
    (17)

  • F. Nusselt number (Nujd)

    $$\frac{\delta {Nu}_{jd}}{{Nu}_{jd}}={\left[{\left(\frac{\delta {h}_{ct}}{{h}_{ct}}\right)}^{2}+{\left(\frac{\delta {D}_{hd}}{{D}_{hd}}\right)}^{2}+{\left(\frac{\delta {k}_{a}}{{k}_{a}}\right)}^{2}\right]}^{1/2}$$
    (18)

  • G. Friction factor (ffjd)

    $$\frac{\delta {ff}_{jd}}{{ff}_{jd}}={\left[{\left(\frac{\delta {V}_{a}}{{V}_{a}}\right)}^{2}+{\left(\frac{\delta {\rho }_{a}}{{\rho }_{a}}\right)}^{2}+{\left(\frac{\delta {D}_{hd}}{{D}_{hd}}\right)}^{2}+{\left(\frac{\delta {L}_{t}}{{L}_{t}}\right)}^{2}+{\left(\frac{\delta \left(\Delta {P}_{d}\right)}{\left(\Delta {P}_{d}\right)}\right)}^{2}\right]}^{1/2}$$
    (19)

Based on the precision of the equipment utilized and the data reduction procedure, the detailed uncertainties in the current examination for the firm and important properties are listed in Table 4.

Table 4 Range of uncertainty in the calculation of process parameters
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Proposed methodology

The integrated AHP-MABAC technique is one of the finest design and optimization methods for evaluating control parameters individually to arrive at the best design values. It entails three main stages of the assessment process, as shown in Fig. 5.

Fig. 5
figure 5

Architecture of the proposed AHP-MABAC methodology

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Stage 1: determination of alternatives, criteria, and performance matrix

In the first stage, the alternatives and criteria for a given MCDM problem are specified. Thereafter, the specified alternatives and criteria are expressed in the form of a performance matrix. Generally, a performance matrix \(\left(U\right)\) for any MCDM problem with M alternatives \(\left({A}_{i},i=\mathrm{1,2},\cdots ,M\right)\) to be evaluated using N criteria \(\left({C}_{j},j=\mathrm{1,2},\cdots ,N\right)\) is constructed as

$$U=\begin{array}{c}{A}_{1}\\ {A}_{2}\\ \begin{array}{c}\vdots \\ {A}_{M}\end{array}\end{array}\begin{array}{c}\begin{array}{ccc}{C}_{1}& {C}_{2}& \begin{array}{cc}{C}_{3}\cdots & {C}_{N}\end{array}\end{array}\\ \left[\begin{array}{c}\begin{array}{ccc}{u}_{11}& {u}_{12}& \begin{array}{cc}{u}_{13}\cdots & {u}_{1N}\end{array}\end{array}\\ \begin{array}{ccc}{u}_{21}& {u}_{22}& \begin{array}{cc}{u}_{23}\cdots & {u}_{2N}\end{array}\end{array}\\ \begin{array}{c}\begin{array}{ccc}\vdots & \vdots & \begin{array}{cc}\vdots & \vdots \end{array}\end{array}\\ \begin{array}{ccc}{u}_{M1}& {u}_{M2}& \begin{array}{cc}{u}_{M3}\cdots & {u}_{MN}\end{array}\end{array}\end{array}\end{array}\right]\end{array}$$
(20)

An element \({u}_{ij}\) of the performance matrix signifies the performance score of the \({i}^{th}\) alternative \({A}_{i}\), concerning the \({j}^{th}\) criterion \({C}_{j}\).

Stage 2: AHP method for weight calculation

AHP is widely used to address the relative importance of activities in an MCDM problem. For this, a pairwise comparison matrix was constructed using a standardized comparison scale of 9 points. For N criteria, \({C}_{j},j=\mathrm{1,2},\dots ,N\), the pairwise comparison matrix \(\left({P}_{NN}\right)\) is constructed as

$${P}_{NN}==\begin{array}{c}{C}_{1}\\ {C}_{2}\\ \begin{array}{c}\vdots \\ {C}_{N}\end{array}\end{array}\begin{array}{c}\begin{array}{ccc}{C}_{1}& {C}_{2}& \begin{array}{cc}{C}_{3}\cdots & {C}_{N}\end{array}\end{array}\\ \left[\begin{array}{c}\begin{array}{ccc}{p}_{11}& {p}_{12}& \begin{array}{cc}{p}_{13}\cdots & {p}_{1N}\end{array}\end{array}\\ \begin{array}{ccc}{p}_{21}& {p}_{22}& \begin{array}{cc}{p}_{23}\cdots & {p}_{2N}\end{array}\end{array}\\ \begin{array}{c}\begin{array}{ccc}\vdots & \vdots & \begin{array}{cc}\vdots & \vdots \end{array}\end{array}\\ \begin{array}{ccc}{p}_{N1}& {p}_{N2}& \begin{array}{cc}{p}_{N3}\cdots & {p}_{NN}\end{array}\end{array}\end{array}\end{array}\right]\end{array} {p}_{ij}=1 if i=j,{p}_{ji}=\frac{1}{{p}_{ij}},{p}_{ij}\ne 0$$
(21)

Every element of the constructed comparison matrix \({p}_{ij}\left(i,j=\mathrm{1,2},\dots ,\mathrm{N}\right)\) is a measure of the criterion weight. As a result, the criterion weights are determined by the right eigenvector (w), which corresponds to the leading eigenvalue.

$${\left({P}_{NN}\right)}_{w}={\lambda }_{\mathrm{max}}\mathrm{w}$$
(22)

It has been reported in the literature that AHP superiority is firmly associated with the reliability of pairwise comparison outcomes. For a completely consistent pairwise comparison matrix, the matrix should have \({\lambda }_{\mathrm{max}}=\mathrm{N}\) and a rank of 1. By stabilizing any of the rows or columns in the table, the weights can be found for the constructed pairwise comparison matrix. Finally, consistency confirmation was performed to ensure that the assessments were adequately reliable. The consistency ratio (CR) is considered as

$$\mathrm{Consistency ration}\left(\mathrm{CR}\right)=\frac{{\lambda }_{\mathrm{max}}-N}{N-1}\times \frac{1}{RI}$$
(23)

where RI is the random index of the matrix, and its value is decided according to the order of the constructed pairwise comparison matrix. The results of the AHP are regarded as reasonable if the assessment of CR is less than the approved maximum limit of 0.1. The pairwise comparison matrix is recreated to increase consistency if the CR value exceeded 0.1.

Stage 3: MABAC approach for final ranking

In the MABAC method, first, the constructed performance matrix is normalized to make different entities comparable using the following equations:

$$\begin{array}{c}{U}_{ij}^{*}=\frac{{u}_{ij}}{{u}_{i}^{+}-{u}_{i}^{-}}, \mathrm{the\; higher\; the\; value\; of\; }j,\mathrm{ the\; better},\mathrm{ and}\\ {U}_{ij}^{*}=\frac{{u}_{ij}-{u}_{i}^{+}}{{u}_{i}^{-}-{u}_{i}^{+}}, \mathrm{the\; lower\; the\; value\; of\; }j, \mathrm{the\; better}.\end{array}$$
(24)

where \({U}_{ij}^{*}\) denotes the normalized value of the ith alternative and jth criterion of the performance matrix. Further, \({u}_{i}^{+}=\mathrm{max}\left({u}_{1},{u}_{2},{u}_{3},\cdots {u}_{M}\right)\), and \({u}_{i}^{-}=\mathrm{max}\left({u}_{1},{u}_{2},{u}_{3},\cdots {u}_{M}\right)\).

Thereafter, weighted normalized performance matrix is structured as

$${\varpi }_{ij}={w}_{j}+\left({U}_{ij}^{*}\times {w}_{j}\right);i=\left(\mathrm{1,2}\dots ..\mathrm{M}\right),j=\left(\mathrm{1,2}\dots \mathrm{N}\right)$$
(25)

The border approximation area matrix is structured by employing the following equation:

$${\mathfrak{R}}_{j}={\left(\prod_{i=1}^{M}{\varpi }_{ij}\right)}^{1/M};j=\left(\mathrm{1,2}\dots \mathrm{N}\right)$$
(26)

The distance amid the weighted normalized performance matrix and border approximation area matrix is calculated as

$${d}_{ij}={\varpi }_{ij}-{\mathfrak{R}}_{j};i=\left(\mathrm{1,2}\dots ..\mathrm{M}\right),j=\left(\mathrm{1,2}\dots \mathrm{N}\right)$$
(27)

Finally, the total distance of the individual alternative from the approximate border area is calculated using the following equation:

$${\Psi }_{i}=\sum_{j=1}^{N}{di}_{j};i=\left(\mathrm{1,2}\dots ..\mathrm{M}\right)$$
(28)

The ranking of alternatives is accomplished on the computed value of \({\Psi }_{i}\), and the alternative with the maximum \({\Psi }_{i}\) value is ranked 1st, while the alternative with the lowest \({\Psi }_{i}\) value is ranked last.

Results and discussion

Experiment results

This investigation aimed to determine the heat transfer characteristics of JIDRSAC. However, dimple-roughened surfaces are more noticeable for enhancing convective heat transfer and keeping friction losses low as compared with conventional SAC; they were therefore selected for investigation. The levels of the selected control factors (CF) used in the experiment are listed in Table 5. These levels are methodically selected from a rational set of CFs that can impact the thermal and hydraulic performance of the JIDRSAC. The results indicate that the dimple roughness arranged in an arc-shaped manner has a robust inclination to release heat away from the roughened absorber plate, yet friction losses occur because of the formation of turbulence inside the duct. Therefore, it is essential to examine both Nujd and ffjd in order to simultaneously take the JIDRSAC performance evaluation into account. When assessing the energy quality associated with a JIDRSAC, it is possible to account for the value of usable energy output, friction losses, and other losses that occur within a JIDRSAC. Altfeld et al. (1988) proposed a method for establishing a link between the usable energy output and pressure losses in a JIDRSAC based on the second law of thermodynamics. The exergetic efficiency (ηex) measures how much work a system can complete with the least amount of energy input, making it important for analyzing a JIDRSAC's performance.

Table 5 Details of system control factors
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$${\eta }_{ex}=\frac{{\eta }_{th}{\eta }_{c}-{P}_{m}\left(1-{\eta }_{c}\right)}{\left(1-{~}^{{T}_{a}}\!\left/ \!{~}_{{T}_{sun}}\right.\right)}$$
(29)

where Pm is the power spent to sustain airflow in the duct and is expressed as

$${P}_{m}=\frac{{m}_{a}\times \Delta {P}_{d}}{{\rho }_{a}}$$

The Nujd, ffjd, and ηex PDCs have been used, and they were all measured and weighed. The combined AHP-MABAC methodology is used to optimize the dimple roughness and flow parameters in a JIDRSAC, where Nujd and ffjd reflect the thermal and hydraulic behavior of the JIDRSAC and ηex provides information on the JIDRSAC's overall performance. The values of Nujd, ffjd, and ηex for sixteen alternatives—parametric sets of dimple-roughened absorber plate taken into consideration for the JIDRSAC at Re = 15,000 are shown in Table 6. Figure 6a and b depicts the selected PDC values for the designated alternatives. The PDCs of the analyzed system is dependent on the CF and flow characteristics, as the alternatives considered are operational deviations of these parameters. The relay behavior of selected PDCs indicates that the modification in the absorber plate configuration has produced a significant change concerning their results.

Table 6 Experimental data of the selected PDCs
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Fig. 6
figure 6

Variation of a PDC-1 & PDC-2, and b PDC-3 with alternatives

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As a result, it is essential to examine both Nujd and ffjd to take into consideration the simultaneous concern in performance evaluation. As can be seen from the results presented in Table 5, alternative A-9 has the uppermost value of PDC-1, while the lowest is for A-16. In addition, for PDC-2, the highest value was at A-10 and the lowest at A-16. Therefore, it is impossible to provide an alternative for any absorber in which PDC-1 is the greatest and PDC-2 is the lowest at the same time. Therefore, PDC-3 (ηex) plays a vital role in considering the best alternative for the current study. The analytical results in the form of PDCs show that the dimple roughness parameters pd, αaa, and ed all have a substantial impact on the thermal and hydraulic characteristics of a JIDRSAC. Whereas PDC-1 increases as the value of CF-1 increases from 30° to 60°, it starts to decline and beyond 60°, as observed in the case of the set of alternatives A-1, A-5, A-9, and A-13. This is because of the presence of dimple roughness at the air stream’s departure inside the duct, and the expansion of the secondary vortices mixed uniformly to produce the maximum value of αaa. In addition, the variation in CF-3 shows an enhancement in PDC-1 with an increase in flow rate. This is because the convective heat transfer coefficient is truncated at the fundamental edge of the dimple roughness but high at the succeeding edge. For the set of alternatives A-9, A-10, A-11, and A-12, as the value of CF-2 surges, the value of PDC-3 declines. This is because the number of vortex generations may decline when the value of p/Dhd increases owing to a decrease in heat transfer. The absorber surface inside the stagnation zone affects the flow, and the axial velocity of the jet decelerates and bounces along the radial path. The flow separation zone behaves as a free shear layer, which is very unstable, and the dimples on the absorber plate create a flow detachment zone, which becomes unstable by creating vortices. The vortices will not only support the entrainment of fast mainstream flow from the exterior but also assist in the acquittal of the low heat transfer zone. Owing to the presence of a low-pressure zone at the start of the dimple, the hasty mainstream flow travels downward. After a jet airflow impacts the dimple, the flow splits into two, with one flow salvaging inside the dimple and the former moving out of the dimple. However, as the airflow rate increases, the increase in heat transmission is convoyed by a surge in the friction factor, which has an adverse effect on the performance of the JIDRSAC. The results also showed that with an increase in CF-1, PDC-1 increases, and PDC-2 increases first up to level-3 followed by a decline at the next level. However, PDC-2 increased up to its maximum value with an increase in CF-1. For reasonably good performance of the JIDRSAC system, PDC-1 and PDC-3 must be as high as feasible, while PDC-2 must be as low as possible. Amid the experimented dimple-roughened absorber plate arrangements for Re = 15,000, alternative A-9 consists of a parametric set of pd/Dh = 0.27, αaa = 60°, and ed/Dh = 0.027 holds the optimal value of PDC-3. Although the alternative A-13 holds the maximum value of PDC-2 with a set of parameters pd/Dh = 0.27, αaa = 75°, and ed/Dh = 0.0324, while the lowest PDC-2 is for A-16.

Purpose of PDC weights

In an MCDM problem, the AHP is most commonly utilized to address the relative importance of activities. It provides the decision to be organized hierarchically to reduce complexity and show relationships between different factors. This can also combine qualitative and quantitative aspects in a single-decision framework. As shown in Eqs. 21 and 22, a pairwise comparison matrix is created, and every element of the matrix pij is the measure of standard weight. The PDCs were associated pairwise to assess the intensity of significance for individual PDC using a 9-point scale after establishing the decision hierarchy for the subject, with 1 being correspondingly significant and 9 with the further being particularly more significant. The responses were logged on a comparison matrix, and the priority vectors were calculated using basic matrix algebra procedures (Saaty 2008; Ishizaka and Labib 2011). The pairwise comparison matrix presented in Table 7 is accomplished by allocating each component of the matrix by the sum of its columns. The weight of PDC-1 (0.540) was found to be maximally trailed by the weight of PDC-3 (0.297) and PDC-2 (0.163). To obtain λmax, the average of these values is computed by dividing the respective elements of the weighted sum matrix by their significance vector element. To define the consistency of the matrix, some significant consistency parameters were considered, that is, λmax = 3.009, consistency ratio (CR) = 0.0079, and random consistency index (RI) = 0.58. At the point where the CR appears to be less than 0.1, the matrix judgment becomes significant. As a result, the pairwise comparison matrix was deemed steady; therefore, the weights were incorporated in the MABAC method.

Table 7 Pairwise comparison matrix and results of AHP
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Classification of the final rank

The PDC weights were obtained using the AHP approach, and the remaining steps of the AHP-MABAC process were utilized to rank the created alternatives according to their overall performance. In compliance with the MABAC procedure outlined in stage-3 of “Experimental validation and uncertainty in measurement” section, the individual unit of the decision matrix is standardized in the assortment of 0–1 by Eq. 24. After calculating the normalized matrix, a weighted normalized matrix is created for each criterion by multiplying each column of the normalized decision matrix \({U}_{ij}^{*}\) with the weighted association criterion \({\varpi }_{ij}\) equivalent to that column using Eq. 25. The resulting normalized matrix and weighted normalized matrix for the MABAC approach are listed in Table 8.

Table 8 Normalized and weighted normalized matrix
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The border approximation area matrix is presented in Table 9 and is planned in accordance with the steps followed for the MABAC approach given in Eq. 26. The table specifies that the value of the border approximation area for PDC-1 is the highest (0.7033), and the lowest is 0.2688 for PDC-3. Consequently, the distance between the weighted normalized matrix and the border approximation area matrix of each alternative is evaluated, as shown in Eq. 27. Finally, using Eq. 28, the total distance (\({\Psi }_{i}\)) of each alternative from the approximate border area was determined and is presented in Table 10. Based on the \({\Psi }_{i}\) values against the alternatives, the concluding ranking of the alternatives concerning their overall performance is obtained and shown in Table 10. The best alternative among the 16 alternatives is A-9, with a rank of 1, as shown in Fig. 7. For alternative A-9, the values of the selected PDCs are PDC-1 = 0.3767, PDC-2 =  − 0.0407, PDC-3 = 0.1478, and \({\Psi }_{i}\) is maximum (0.4838). The alternatives can be organized in ascending order as A-9 < A-13 < A-10 < A-5 < A-7 < A-12 < A-3 < A-2 < A-8 < A-15 < A-4 < A-11 < A-6 < A-1 < A-16. Alternative A-16 has the lowest rank among all, with a \({\Psi }_{i}\) value of − 0.2484. From this optimization of JIDRSAC by utilizing the AHP-MABAC technique, it is clear that the dimple roughness parameter, that is, pd/Dh = 0.27, αaa = 60°, and ed/Dh = 0.027 is the best selection under these circumstances, taking into account three PDCs at the same time.

Table 9 The border approximate area matrix
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Table 10 Distance from border approximate area and ranking of alternatives
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Fig. 7
figure 7

Ranking of selected alternatives

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Sensitivity analysis

The weight of the PDCs used in decision-making has a significant impact on the ranking outcomes. The weighting results of the AHP approach may be subject to change under various conditions, for example, internal changes in the pair-wise comparison matrix. As a result, sensitivity analysis should be considered while evaluating the robustness of the presented MCDM results. The sensitivity analysis of the proposed AHP-MABAC approach was performed by generating new weights of the selected three PDCs (Nujd, ffjd, and ηex) in steps and analyzing their impact on the rankings results. The new weights were determined by adjusting the weight of a PDC in four steps (± 10%, ± 20%, ± 30%, and ± 50%). The weights of the remaining two PDCs were correspondingly changed so that the total weight of all PDCs remained 1. In total, 24 situations were generated, with the relevant ranking results shown in Fig. 8.

Fig. 8
figure 8

Sensitivity analysis

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Analyzing Fig. 8, it can be concluded that changing the weight of the selected PDCs (Nujd, ffjd, and ηex) in four steps will not produce any significant changes in the rankings of the alternatives. In all 24 scenarios, the A-9 remains the first-ranked alternative, whereas the A-16 remains the last-ranked alternative. The alternative A-13 remains the second-ranked except when the weight of Nujd (PDC-1) is reduced by 50%, where second-ranked and the third-ranked alternatives swap places. The ranking results found most sensitive when the Nujd (PDC-1) weight is reduced by 50%, the ranking of the alternative A-8 goes down by two places, while the ranking of alternative A-11 improved by two places. Similarly, alternative A-8 loses its ranking by two places when the weight of ffjd (PDC-2) is increased by 50%, and PDC-3 (ηex) is reduced by 20 to 50%. Apart from these significant changes, it is possible to see that the alternative go up or down by one place. However, these changes do not significantly affect the final results of the model, which is also confirmed by determining the Spearman correlation coefficient value using the following formula,

$$\mathrm{Spearman}\;\mathrm{correlation}\;\mathrm{coefficient}=1-\frac{6\sum_{i=1}^M\nabla_i^2}{M\left(M^2-1\right)}$$
(30)

Here, \({\nabla }_{i}\) = Difference of actual rank and rank achieved with PDC weight variation; M = Alternatives.

The Spearman correlation coefficient remained more than 0.97 for all cases, signifying a strong correlation. It can be concluded from the sensitivity results that the acquired ranking is valid and reliable.

Furthermore, Fig. 9 shows the enhancement in Nujd and ffjd compared with the SPSAC and jet impingement duct. When the jet impingement SAC is compared to a SPSAC, the Nui is enhanced by 2.16–2.81, and the ffi is enhanced by 3.35–5.95. In addition, the present model, that is, JIDRSAC, is compared with the jet impingement SAC, which shows the augmentation in the Nujd and the ffjd in the range of 0.55–0.80 and 0.05–0.15, respectively. Thus, it can be stated that the combined technique of air impinged vertically on the dimple-roughened absorber surface can transmit a greater value of thermal energy from the absorber surface compared to the SPSAC and jet impingement SAC. However, it also leads to a rise in pressure losses, which is anticipated to rise the friction factor.

Fig. 9
figure 9

Augmentation in Nujd and ffjd at optimal CF arrangement

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Conclusions

This study examined the use of AHP-MABAC, an MCDM optimization technique, to determine the best dimple-roughened plate parameter arrangement in a JIDRSAC. The performance of JIDRSAC was investigated with respect to convective heat transfer and friction loss features, and the performance criteria were assessed, revealing a distinct inclination for each of the 16 alternatives considered. The responses were classified into three categories: PDC-1, PDC-2, and PDC-3. The AHP-MABAC model was used to evaluate the best set of CFs to minimize friction losses and increase the convection heat transfer rate from the absorber plate to the flowing air. The CFs, which are the geometric parameters of the dimple roughness, have been characterized as pd/Dh, αaa, and ed/Dh. The outcomes of this study can be useful in a variety of disciplines, including solar-based studies and thermal schemes, for the evaluation and optimization of the combined technique among desirable parameters for maximum performance. Based on the outcomes of the investigation, the following conclusions were drawn:

  1. 1.

    The maximum value of the total distance (\({\Psi }_{i}\)) of individually alternatives from the approximate border area defines the rank of the alternatives. A-9, with the highest value of \({\Psi }_{i}\)=  + 0.4838, has a rank of 1, while A-16 ranked last with the lowest value of \({\Psi }_{i}\)=  − 0.2484.

  2. 2.

    Alternative holds the maximum value of Nujd and ηex with a set of parameters pd/Dh = 0.27, αaa = 60°, and ed/Dh = 0.027.

  3. 3.

    The parameter αaa dominates in predicting the maximum value of the PDCs. The value of Nujd increases as the value of αaa increases from 30° to 60°, and it starts to decline beyond 60°, as observed in the case of the set of alternatives A-1, A-5, A-9, and A-13.

  4. 4.

    Alternatives A-1, A-2, A-3, and A-4 indicate that ffjd increases as the value of parameter ep/Dh increases. This is because a secondary flow was produced along with the appearance of dimples as the value of ep/Dh increased, and the power requisite to pump air through the JIDRSAC increased.

  5. 5.

    JIDRSAC compared to the jet impingement SAC, exhibits an increase in Nujd and ffjd in the range of 0.55–0.80 and 0.05–0.15, respectively.

  6. 6.

    The validation of ranks generated via AHP-MABAC through sensitivity analysis of the PDCs are impermeable by the alteration in criteria weight.