## Abstract

Solar air heater (SAH) is one of many applications of solar energy, but it has low thermal performance. To cater to this issue, a novel corrugated jet-impinging channel has been proposed for the absorbing plate of SAHs. The Nusselt number, pressure loss pumping power, and friction factor were computed by experimental and numerical studies for three different arrangements of jets, i.e., 1 × 1 square, 2 × 2, and 3 × 3 jet array with three target plates, i.e., smooth, trapezoidal, and sinusoidal corrugated plates over the Reynolds number range of 1740–2700 and jet-to-target plate spacing of 5.1–14.6. It was observed that jet-to-target plate distance and multiple jets impinging can affect the heat transfer enhancement significantly. An optimization study has been performed using the statistical Taguchi method. The optimum value of Nusselt number was found for the trapezoidal corrugated target plate, with 3 × 3 multiple jet impingement, jet-to-target plate distance of 5.1, and Reynolds number value of 2700, while the optimum value of friction factor was found for 3 × 3 jet impingement. Computational fluid dynamics (CFD) and experimental data are found to be in good agreement. An increase of 11 times was observed in the thermal performance factor as compared to the base case. Trapezoidal corrugation seems to be the effective profile for SAHs.

## 1 Introduction

For the past few decades, the world has been facing an energy crisis because of an increase in the world population and the depletion of fossil fuels [1]. Fossil fuel is a non-renewable source of energy, and it is responsible for global warming, climatic changes, and environmental pollution due to the greenhouse effect [2]. These issues are triggering the world to look for an environmentally friendly source of energy. Solar has the widest scope of all the renewables because it is a clean, convenient, sustainable, safe, abundant, and cheaper source of energy. Solar air heaters (SAHs) are the widely used application of solar energy. It is used for desalination, space heating, drying crops, laundry, and the process industry.

SAHs have generally low efficiencies due to the low thermal capacity of air, friction losses of the absorber plate, and low heat transfer coefficient between air and the absorbing plate [3]. For this reason, scientists and engineers are nowadays facing the challenges of building highly efficient solar air heaters (SAHs). The passive method of heating and their combinations are mostly preferred, as they do not require any mechanical aids [4–6].

One of the efficient and common methods of increasing turbulence and heat transfer is by means of corrugated passages. Corrugation performs two functions. First, it enhances the turbulence; second, it increases the surface area, which increases the heat transfer directly. Yassen et al. [7] studied the effect of corrugation in integral SAH for household use, and it was reported that the temperature rise is higher in the case of corrugated absorber plates. Tokgoz et al. [8] studied the heat transfer enhancement in corrugated ducts for different aspect ratios of corrugation. An increase of 30% was achieved in the thermal performance factor for an aspect ratio of 0.3 and a Reynolds number value of 3 × 10^{3}. Manjunath et al. [9] performed a numerical study of the sinusoidal corrugated absorber plate. It was concluded that due to higher turbulence, an increase of 12.5% can be achieved in heat transfer as compared to the flat plate collector. Khoshvaght-Aliabadi et al. [10] investigated that sinusoidal corrugation has the highest value of heat transfer to the pumping power (PP) as compared to the triangular and trapezoidal type corrugation, while the highest Nusselt number (Nu) and PP were obtained for trapezoidal type corrugation followed by triangular and sinusoidal corrugations.

Impinging jet is one of the most effective and acquiescent methods of heat transfer due to the very high heat transfer rate between the fluid and the wall. The heat transfer coefficient can be increased thrice of conventional convection cooling methods, which makes it an effective mode of heat transfer in gas turbine blades, electronic components, solar collectors, and critical components of machinery [11–17]. Moreover, it proves itself as a feasible solution in the processes of food drying, material-forming, and annealing. Jet impingement causes Choudhury and Garg [18] computationally report the performance efficiency of jet plate SAHs of 19–26.5% over parallel plate heaters for the air flowrate range of 50–250 kg/hm^{2}. Multi-jet impingement cooling systems are favored over single jets because they offer higher and more uniform heat transfer [19]. Strasser et al. studied an isothermal four-jet configuration in a reactor cavity coolant system plenum using a hybrid large eddy simulation (LES)–Reynolds-averaged Navier–Stokes (RANS) approach. It was determined that threshold frequencies lower than 10 Hz are prevalent for thermal stripping. Interferences like asymmetry in the plenum, non-uniformity, and stirring cause premature confluence of jets [20]. Chougule et al. [21] performed the computational fluid dynamics (CFD) study of a 4 × 4 pin fin heat sink with single and 3 × 3 multi-air jet impingement for the Reynolds number range of 7000–11,000. It was concluded that multi-jet impingement shows 3–4 times higher cooling performance as compared to a single jet due to the high convection heat transfer coefficient in the core region of the jet and multiple stagnation points. A uniform temperature distribution and lower temperature were observed in the multi-jet impingent as compared to the single-jet impingement. Chauhan et al. [22] performed the experimental investigation and optimization of jet impingement in SAHs by using the statistical Taguchi method [23]. An optimal design of jet-impinging SAH was obtained with a 37–48.3% increase in thermal performance over the Reynolds number range of 4000–16,000.

For the last three decades, instead of relying on single cooling techniques, many researchers have incorporated compound cooling techniques by selecting a powerful combination of different cooling techniques to increase heat transfer [24]. One such combination is jet impingement on corrugated plates. Aboghrara et al. [25] performed an experimental investigation of circular jet impingement on a corrugated solar absorber plate. An increase of 14% was observed in thermal efficiency as compared to the smooth channel. Ekiciler et al. [26] performed a numerical study on the effect of jet impingement cooling on sinusoidal, flat, and triangular corrugated surface features for the Reynolds number range of 125–500. An increase of 300% and 50% was observed in the values of average and local Nusselt numbers by using corrugated surfaces as compared to flat plates. It was inferred that the shape of the target plate affects the heat transfer substantially.

From an extensive literature survey, it was perceived that both corrugated absorber plate and jet impingement individually affect the performance of the SAHs significantly. The combination of both cooling techniques is a way that is yet to be explored for a wide range of design and flow conditions by using a multi-objective design methodology. The design of corrugated SAHs with the addition of a jet impingement is proposed in the current study. Therefore, the present work focuses on the investigation of heat transfer as well as friction factor in laminar to turbulent flow regimes, using single and variable multiple jet impingements on smooth, trapezoidal, and wavy corrugated passages over a range of Reynolds numbers and jet-to-target plate spacing by using Taguchi and analysis of variance (ANOVA) method.

The main objectives of this research work include

numerical assessment of flow and thermal properties of fully developed laminar and turbulent, single and multiple jet impingement on smooth, wavy, and trapezoidal corrugated solar collector plates for SAHs;

optimization and selection of optimal configuration using Taguchi and ANOVA method;

development of correlation for Nusselt number and friction factor,; and

Understanding the physics of heat and fluid flow via CFD.

## 2 Research Methodology

The method used in the current study is primarily numerical. Modeling and meshing were performed in ansys followed by a mesh independence study. The mathematical model was formulated and validated with the experimental data and literature. After that, design propositions and modifications were performed. The design of experiment (DoE) technique, i.e., the Taguchi method, was applied to reduce the number of cases, and a CFD study of all the cases was performed. In the final stage, the optimal design was selected, and a parametric study was performed to understand the physics of flow and heat transfer, as shown in Fig. 1.

## 3 Novel Solar Air Heater

A solar collector generally consists of the main absorber plate, a transparent sheet on the exposed side of the absorber plate, a back plate, and insulation below the back plate. Air flows between the passage of the front and back plate, as shown in Fig. 2(a). In the current novel design, instead of the simple airflow in the passage of SAHs, it is impinged in the form of jets on the corrugated absorber plate to enhance the efficiency of SAH, as shown in Fig. 2(b).

## 4 Design Propositions

The profile of the collector plate, jet plate, jet-to-target plate distance, and Reynolds number were varied to enhance the efficiency of SAHs.

### 4.1 Flow and Geometric Parameters

#### 4.1.1 Profile of Target Plate.

Three types of target plates, i.e., smooth, trapezoidal, and sinusoidal corrugations, are considered for the current study, having the same blockage ratio (*e*/*D*_{h}) of 0.164 and pitch-to-groove height ratio (*p*/*e*) of 4, as shown in Fig. 3 and Table 1.

Sr. No. | Pattern of corrugation | Flow area (mm^{2}) | Groove height (mm) | Pitch (mm) |
---|---|---|---|---|

1 | Smooth (f) | $Af=307\xd7212=65,084mm2$ | — | — |

2 | Wavy (w) | $Aw=\u222bab\u222bcd1+(dydx)2dxdy=74,000mm2$a = 0, b = 212 mm,c = 0, d = 307 mm | 2.5 | 10 |

3 | Trapezoidal (t) | $AT=\u222bab(o+2p)Ndy=78,500mm2$a = 0 mm, b = 212 mmN = 30.7 (number of grooves) | 2.5 | 10 |

Sr. No. | Pattern of corrugation | Flow area (mm^{2}) | Groove height (mm) | Pitch (mm) |
---|---|---|---|---|

1 | Smooth (f) | $Af=307\xd7212=65,084mm2$ | — | — |

2 | Wavy (w) | $Aw=\u222bab\u222bcd1+(dydx)2dxdy=74,000mm2$a = 0, b = 212 mm,c = 0, d = 307 mm | 2.5 | 10 |

3 | Trapezoidal (t) | $AT=\u222bab(o+2p)Ndy=78,500mm2$a = 0 mm, b = 212 mmN = 30.7 (number of grooves) | 2.5 | 10 |

#### 4.1.2 Profiles of Jet Plates.

The jet type was varied to single, 2 × 2 array, and 3 × 3 array for the current study. The jets were designed on the base of the same flow area of 224.96 mm^{2}, as shown in Fig. 4.

#### 4.1.3 Jet-to-Target Plate Distance.

*L*is the distance between the plates, and

*D*

_{h}is the hydraulic jet diameter. In the current study,

*L*/

*D*

_{h}was varied to three different values of 5.1, 7, and 14.6.

#### 4.1.4 Reynolds Number.

The main focus of the current work is to study the effect of low Reynolds number. For that purpose, Reynolds number was varied in the laminar to transition flow regimes to three different values of 1740, 2270, and 2700, respectively.

#### 4.1.5 Taguchi Method.

_{9}(3

^{4}) orthogonal array was selected. The number of cases after implementing the Taguchi method was reduced to 9, corresponding to three levels and four factors [27], which is an economical choice. After performing experiments on nine cases, the Taguchi method will give the complete picture of the relation between parameters by using a cross combination of arrays. After selecting the orthogonal array, the next step was to calculate the SNR, which is the logarithmic transformed ratio of the mean and standard deviation of the output parameter. The goal was to maximize the heat transfer “Nu” at the expense of minimum or no rise in

*f*. Therefore, higher-the-better for Nu and lower-the-better characteristics of SNR for

*f*were selected as shown in Eqs. (2) and (3), respectively. The next step was to design the experiments, which are shown in Table 2.

*y*is the observed data at the

_{i}*i*th experiment, and

*n*is the number of experiments.

Sr. No. | Target plate | Re | Jet diameter ratio (D_{h}/D_{hs}) | L/D_{h} |
---|---|---|---|---|

1 | f | 1740 | 0.33 | 5.1 |

2 | f | 2270 | 0.5 | 7.0 |

3 | f | 2700 | 1.0 | 14.6 |

4 | s | 1740 | 0.5 | 14.6 |

5 | s | 2270 | 1.0 | 5.1 |

6 | s | 2700 | 0.33 | 7.0 |

7 | t | 1740 | 1.0 | 7.0 |

8 | t | 2270 | 0.33 | 14.6 |

9 | t | 2700 | 0.5 | 5.1 |

Sr. No. | Target plate | Re | Jet diameter ratio (D_{h}/D_{hs}) | L/D_{h} |
---|---|---|---|---|

1 | f | 1740 | 0.33 | 5.1 |

2 | f | 2270 | 0.5 | 7.0 |

3 | f | 2700 | 1.0 | 14.6 |

4 | s | 1740 | 0.5 | 14.6 |

5 | s | 2270 | 1.0 | 5.1 |

6 | s | 2700 | 0.33 | 7.0 |

7 | t | 1740 | 1.0 | 7.0 |

8 | t | 2270 | 0.33 | 14.6 |

9 | t | 2700 | 0.5 | 5.1 |

Note: *D*_{h} is the hydraulic diameter of the jet, while *D*_{hs} is the hydraulic diameter of a single square jet.

## 5 Computational Fluid Dynamics Study

CFD was used for predicting the trends of fluid flow and heat transfer in SAHs. For that purpose, the ansys 14.2 licensed version was used. It is a little bit outdated version, but it was the only licensed version in our lab. It was ensured that the features relevant to us are the same in the available and latest versions.

### 5.1 Geometry and Meshing.

The schematic from Fig. 2(b) was converted to the physical domain in ansys DesignModeler. The flow enters from the inlet side and gets developed in the straight portion of the channel. Then, it gets converted into fine mini jets in the jet plate section and impinged on the heated corrugated plates, which is representative of the absorber plate of the SAHs. The flow then exited from the outlet section after heat transfer with the absorber plate. The specifications of the geometry for a particular sinusoidal case having a 3 × 3 jets array are shown in Figs. 5(a)–5(d). The flow area of the channel at the inlet and outlet is 463 × 401 mm^{2} with a straight length of 960 mm and 100 mm on the rear side of the target plate. The thickness of the jet plate is 5 mm. The length of the inlet blocks is extended so that flow can be fully developed. At the outlet, an extension in length was also provided to compensate for reverse flows. The dimensions of the target plates and jet plates are provided in Table 1 and Fig. 3.

A non-conformal mesh was generated in the ansys Mechanical module, as shown in Figs. 6(a)–6(f). Multi-zone mesh method was used. On the plate and jet side, 30 layers of inflations, with the first layer, were kept at 0.008 to achieve *y* plus less than one. While coarse meshing was formed on the flow straightener section. All the mesh parameters were within the range, i.e., minimum orthogonal were above 0.2. The maximum values of aspect ratio and skewness were below 80 and 0.2, respectively. The average value of wall *y* plus was 0.013, with a maximum value of 0.18 on the corrugated target plate.

### 5.2 Mesh Independence Study.

The grid independence study was accomplished so that the results can be made irrespective of the mesh size. Three different mesh sizes, 3.4 M, 7.0 M, and 8.6 M, were resolved and compared. The part of the fluid domain, which is near the jet and plate section, was finely meshed. The part other than the jet and target plate section has the coarse mesh.

The whole domain was simulated with all the mesh sizes, and the trend of the Nusselt number was plotted in both *x* and *y* directions to get the complete local variation of the Nusselt number in both longitudinal and lateral directions (Fig. 7). The trend of the 7 M mesh size overlapped with 8.6 M meshed domain. So, the mesh of 7 M was selected after mesh independence with a maximum error of 2%, with the next finer meshed geometry at the stagnation point. The corrugated geometries require fine mesh. For that purpose, a mesh size of 0.3 mm, 0.8 mm, 3 mm, and 5 mm was selected on jets, target plate, downstream, and upstream sides, respectively, for all the cases after rigorous study.

### 5.3 Boundary Conditions.

The boundary condition is an integral part of the mathematical model, as the whole solution depends upon these conditions. The boundary conditions used for the analysis are listed in Table 3.

Domain | Conditions | Specifications |
---|---|---|

Inlet | Mass flowrate inlet | 0.00047–0.00212 kg/s,T = 298.9 K |

Outlet | Pressure outlet | P = P_{gage} = 0,$\u2202T\u2202z=\u2202u\u2202z=\u2202v\u2202z=\u2202w\u2202z=0$, T = 299.4 K |

Wall | No-slip boundary condition and adiabatic walls | u = v = w = 0,$\u2202T\u2202x=\u2202T\u2202y=0$ |

Plate | Constant heat flux | T = 305.81 W/m^{2} |

Domain | Conditions | Specifications |
---|---|---|

Inlet | Mass flowrate inlet | 0.00047–0.00212 kg/s,T = 298.9 K |

Outlet | Pressure outlet | P = P_{gage} = 0,$\u2202T\u2202z=\u2202u\u2202z=\u2202v\u2202z=\u2202w\u2202z=0$, T = 299.4 K |

Wall | No-slip boundary condition and adiabatic walls | u = v = w = 0,$\u2202T\u2202x=\u2202T\u2202y=0$ |

Plate | Constant heat flux | T = 305.81 W/m^{2} |

### 5.4 Solution Setup.

The continuity, momentum, energy equations, and turbulence equations were solved for three-dimensional, steady-state, and fully developed laminar to turbulent flow. Besides, turbulence is always a transient phenomenon; we are running a steady-state solver. Due to low turbulence intensity and energetic equilibrium, we can easily resolve turbulence with steady solve, and this is usually associated with RANS formulation. The RANS scheme can fairly predict turbulence up to the extent of primary vortices [21,26,28]. The SIMPLE scheme was used for pressure velocity coupling. The second-order upwind scheme was used for momentum, pressure, turbulent kinetic energy, and turbulent dissipation rate.

The residuals in fluent's cases are problem-specific. Initially, the residuals were set at default. We frequently checked the flux balance and also monitored the change in output parameters. When the change in output parameters is diminished, we assume that the solution has been converged. The residual criteria corresponding to these conditions needed to be the strict one. That is why the residual criteria of 10^{−5} were used for continuity, *x*, *y*, *z* velocities, k, and $\epsilon $. While the energy was set at 10^{−9} to attain complete convergence and flux balance.

### 5.5 Mathematical Modeling.

For solving the fluid domain, continuity, momentum, energy, and k-ε model turbulence equations are solved. The mathematical forms of these equations are presented in Eqs. (4)–(10) in Table 4.

Continuity | $\u2207.(\rho V)=0$ |

X momentum | $\u2207.(\rho uV)=\u2212\u2202p\u2202x+\u2202\tau xx\u2202x+\u2202\tau yx\u2202y+\u2202\tau zx\u2202z+\rho fx$ |

Y momentum | $\u2207.(\rho \upsilon V)=\u2212\u2202p\u2202y+\u2202\tau xy\u2202x+\u2202\tau yy\u2202y+\u2202\tau yz\u2202z+\rho fy$ |

Z momentum | $\u2207.(\rho wV)=\u2212\u2202p\u2202y+\u2202\tau zx\u2202x+\u2202\tau zy\u2202y+\u2202\tau zz\u2202z+\rho fz$ |

Energy | $\u2207.[\rho (e+V22)V]=\rho q\u02d9+\u2202\u2202x(kc\u2202T\u2202x)+\u2202\u2202y(kc\u2202T\u2202y)+\u2202\u2202z(kc\u2202T\u2202z)\u2212\u2202(up)\u2202x\u2212\u2202(vp)\u2202y\u2212\u2202(wp)\u2202z+\u2202(u\tau xx)\u2202x+\u2202(u\tau yx)\u2202y+\u2202(u\tau zx)\u2202z+\u2202(v\tau xy)\u2202x+\u2202(v\tau yy)\u2202y+\u2202(v\tau yz)\u2202z+\u2202(w\tau zx)\u2202x+\u2202(w\tau zy)\u2202y+\u2202(w\tau zz)\u2202z+\rho f.V$ |

Turbulent kinetic energy | $\u2202\u2202xj(\rho kuj)=\u2202\u2202xj((\mu t\sigma k)\u2202k\u2202xj)+2\mu tEijEij\u2212\rho \epsilon \u2212\rho D$ |

Turbulence dissipation rate | $\u2202\u2202xj(\rho \epsilon uj)=\u2202\u2202xj((\mu t\sigma \epsilon )\u2202\epsilon \u2202xj)+C1\epsilon \epsilon k2\mu tEijEij\u2212C2\epsilon f1\rho \epsilon 2k+\rho E(10)$ σ_{k} = 1.0, and $\sigma \epsilon =1.3$. |

Continuity | $\u2207.(\rho V)=0$ |

X momentum | |

Y momentum | |

Z momentum | |

Energy | |

Turbulent kinetic energy | |

Turbulence dissipation rate | σ_{k} = 1.0, and $\sigma \epsilon =1.3$. |

## 6 Experimental Setup

An indoor experimental setup was established to study the case of a single jet of air impingement on the smooth plate, as shown in Fig. 8.

The working fluid, ambient air, was supplied via a 550-W centrifugal air blower. A globe valve was installed at the downstream side of a blower to regulate the flowrate of air. A sufficient standard length of 20D has been provided so that flow can get fully developed, followed by a flanged type vortex flowmeter (LUGB DN-100, Tianjin, China), having an operating range of 1.6–16 m^{3}/min for an output range of 4–20 mA with an accuracy of 1.5% full-scale reading, was installed to measure the volumetric flowrate. On the downstream side, a 4″ rubber pipe is attached, followed by a conical diffuser-type duct for the pressure recovery. On the downstream side, a flow straighter section is attached so that flow can be smoothened, followed by a jet plate and test section. The test channel was manufactured using duct GI sheets, having a cross-sectional area of 463 mm × 401 mm, which is equal to almost 4 times scale down model of SAHs, as shown in Fig. 9 [29].

Air at high velocity, based upon the value of Reynolds number, ejects from the jet plate, after which it is impinged on the heated plates attached on the front side of the heater. This assembly of the heater and plate can slide in the test section so that the jet-to-target plate distance can be varied. Instead of simulating actual solar radiations, an electric loop heater, made up of Nichrome wire, having a resistance of 58 Ω, was directly attached to the plates to supply a uniform heat flux in the range of 100–1450 W/m^{2} to the absorber plate, which is considered to be reasonably good value of heat energy input for testing SAHs [30,31]. A variable transformer is attached to the heaters to adjust the voltage range according to the requirements. The heating power of the heater is measured by a digital multimeter with 1% accuracy.

The temperature of the heated test section plate is measured by using nine wide-range K-type riveted thermocouples, having an accuracy of 0.75%, arranged in a 3 × 3 array, and the average value was calculated. This is necessary to achieve isothermal conditions. A set of two absolute pressure transducers having a range of 0–0.1 bar with an accuracy of 0.2% were installed, one on the upstream and the other on the downstream side of the jet plate to get pressure difference according to the ISO D-D/2 standard. The air velocity was measured by using a pitot tube attached to the Dwyer (Magnehelic Series-2000, Michigan, USA) differential pressure gauge, with an uncertainty of 1 Pa. Table 5 shows the list of instrumentations installed in the test rig. The experiment was repeated three times, and the mean value was computed.

Sr. No. | Instruments | Description |
---|---|---|

1. | Flow meter | LUGB DN100 vortex flow sensor |

2. | Test section Temp. measurement | Surface thermocouple (K-type) |

3. | Pressure drop measurement | Absolute pressure transducer |

4. | Velocity measurement | Pitot tube and Magnehelic pressure gauge |

Sr. No. | Instruments | Description |
---|---|---|

1. | Flow meter | LUGB DN100 vortex flow sensor |

2. | Test section Temp. measurement | Surface thermocouple (K-type) |

3. | Pressure drop measurement | Absolute pressure transducer |

4. | Velocity measurement | Pitot tube and Magnehelic pressure gauge |

## 7 Data Reduction

*V*is the voltage supplied,

*R*is the resistance of the heating element, and cos

*θ*is the power factor, and it is taken as 1 for heating elements. The heat flux can be calculated by using Eq. (12).

*A*

_{HT}is the heat transfer area, which is equal to 0.065 m

^{2}. The net heat flux can be estimated from the difference between total heat flux $qtot\u2033$ and lost heat flux $qloss\u2033$, as shown in Eq. (13).

*h*) when the system achieves a steady-state condition [32].

*T*

_{ss}is the temperature difference between the average temperature of the test section and ambient air temperature at a steady-state.

*h*is the coefficient of heat transfer,

*k*

_{c}is the coefficient of thermal conductivity, and

*A*

_{j}is the flow area of the jet. The same convention of

*A*

_{j}was used by Chougule et al. [21].

*D*

_{h}is the hydraulic diameter of the jet,

*A*

_{j}is the flow area of the jet plate,

*µ*is the dynamic viscosity of air, and

*n*

_{j}is the number of jets.

*P*

_{u}is the upstream pressure, and

*P*

_{d}is the downstream pressure.

*η*. Equation (20) demonstrates that the greater the performance factor value, the more heat is transmitted in contrast to the corresponding pressure drop across the channel [36,37], where

*o*represents the parameters of the base case with the lowest values of thermal performance of all the cases.

## 8 Validation of Computational Fluid Dynamics With Experimental Data

For the purpose of validation, a case of smooth plate corresponding to the 1 × 1 jet array was tested corresponding to the Reynolds number and *L*/*D*_{h} value of 2700 and 14.6, respectively. Various models like k-*ω* and k-*ɛ* were used and compared with the experimental data, as shown in Fig. 10. The values of stagnation values of Nusselt numbers at the absorber plate are plotted. It was decided that the low Re k-*ɛ* model is in good agreement with experimental data, with a percentage error of 12%, hence selected for further study. The uncertainty analysis of experimental data is presented in the Appendix.

## 9 Results and Discussion

In this section, the results of thorough CFD results will be presented to depict the effect of different parameters on flow and heat transfer characteristics.

### 9.1 Analysis of the Mean Values and Signal-to-Noise Ratio.

A total number of nine cases were studied to compute the values of Nu, Δ*P*, PP, and *f*. The mean and SNR values were computed corresponding to each run, as shown in Table 6. The SNR values are plotted for each parameter, as shown in Fig. 10.

Ex. No. | $Nu\xaf$ | $\Delta P\xaf$ | $PP\xaf$ | $f\xaf$ | (SNR)_{Nu} | (SNR)_{ΔP} | (SNR)_{PP} | (SNR)_{f} |
---|---|---|---|---|---|---|---|---|

1 | 67.8 | 29.6 | 0.0450 | 0.0183 | 36.6 | −29.4 | 26.9 | 34.8 |

2 | 53.9 | 22.3 | 0.0296 | 0.0272 | 34.6 | −27.0 | 30.6 | 31.3 |

3 | 14.4 | 8.3 | 0.0068 | 0.0522 | 23.2 | −18.4 | 43.3 | 25.7 |

4 | 47.7 | 13.1 | 0.0132 | 0.0274 | 33.6 | −22.3 | 37.6 | 31.2 |

5 | 33.7 | 6.9 | 0.0048 | 0.0596 | 30.5 | −16.7 | 46.4 | 24.5 |

6 | 96.8 | 70.6 | 0.1598 | 0.0196 | 39.7 | −37.0 | 15.9 | 34.2 |

7 | 20.5 | 4.3 | 0.0023 | 0.0626 | 26.3 | −12.7 | 52.6 | 24.1 |

8 | 67.6 | 50.2 | 0.0968 | 0.0193 | 36.6 | −34.0 | 20.3 | 34.3 |

9 | 116.2 | 31.4 | 0.0487 | 0.0277 | 41.3 | −29.9 | 26.3 | 31.1 |

Ex. No. | $Nu\xaf$ | $\Delta P\xaf$ | $PP\xaf$ | $f\xaf$ | (SNR)_{Nu} | (SNR)_{ΔP} | (SNR)_{PP} | (SNR)_{f} |
---|---|---|---|---|---|---|---|---|

1 | 67.8 | 29.6 | 0.0450 | 0.0183 | 36.6 | −29.4 | 26.9 | 34.8 |

2 | 53.9 | 22.3 | 0.0296 | 0.0272 | 34.6 | −27.0 | 30.6 | 31.3 |

3 | 14.4 | 8.3 | 0.0068 | 0.0522 | 23.2 | −18.4 | 43.3 | 25.7 |

4 | 47.7 | 13.1 | 0.0132 | 0.0274 | 33.6 | −22.3 | 37.6 | 31.2 |

5 | 33.7 | 6.9 | 0.0048 | 0.0596 | 30.5 | −16.7 | 46.4 | 24.5 |

6 | 96.8 | 70.6 | 0.1598 | 0.0196 | 39.7 | −37.0 | 15.9 | 34.2 |

7 | 20.5 | 4.3 | 0.0023 | 0.0626 | 26.3 | −12.7 | 52.6 | 24.1 |

8 | 67.6 | 50.2 | 0.0968 | 0.0193 | 36.6 | −34.0 | 20.3 | 34.3 |

9 | 116.2 | 31.4 | 0.0487 | 0.0277 | 41.3 | −29.9 | 26.3 | 31.1 |

From Figs. 11(a)–11(d), it can be observed that multiple jet impingement has far better heat transfer properties as compared to the single jet. The reason that can be attributed to such a trend is that multi-jet impingement produces small jets of high velocity for the same value of Re, which removes the heat more significantly, and more area is cooled.

Trapezoidal corrugation has far better properties as compared to sinusoidal and smooth plates. This is because the smooth channel has a thick thermal boundary layer, while corrugation produces vortices, due to which efficient mixing of core fluid and hot wall fluid takes place. Hence, more heat transfer occurs. The trapezoidal corrugation has a sharp throat crest region, due to which more flow recirculations occur, and as a result, heat transfer increases, as opposed to the sinusoidal corrugation, which has the sinusoidal profile throughout, resulting in smooth removal of fluid.

It can be seen that as *L*/*D*_{h} increases, the jet expands and loses its ability to cool the plate. If it is decreased below a certain point, then cooling can be reduced as the fluid is in the developing flow regime.

Re has an increasing trend with the heat transfer as it directly increases the turbulence of the fluid, which results in more heat transfer.

### 9.2 Optimized Configuration.

The optimum value of the output parameters was decided based on SNR. The maximum values show the optimized ones in Fig. 10. In the case of Nu, the larger-the-better approach was used. So, from Fig. 11(a), it is observed that the jet plate of 3 × 3 holes, with jets of fluid exited at the Re value of 2700, impinged on the trapezoidal corrugated plate, placed at *L*/*D*_{h} of 5.08, is the optimum condition to maximize the Nu.

From Figs. 11(b) and 11(c), the optimum condition for the Δ*P* and PP is achieved for the single jet plate, with a jet of fluid issued at the Re value of 1740, while the target plate and *L*/*D*_{h} seem to be insignificant.

From Fig. 11(d), the optimum value of *f* was achieved for the 3 × 3 multiple jet plate, having a *D*/*D*_{hs} value of 0.33, while all other factors seem to be insignificant. This is due to the fact that jets of smaller size result in less flow recirculation and separation zones. As a result, less friction losses showed up in the system. The smaller-the-better approach was used for the last three parameters. To consider the optimized configuration, the thermal performance factor was calculated from Eq. (20) to incorporate the effect of both heat transfer and friction factor enhancement.

From Taguchi’s design, the optimized configuration was observed to be one with a 3 × 3 jet impinged on trapezoidal corrugation for the Reynolds number value of 2740 and jet-to-target plate distance of 5.1. Table 7 shows the thermal performance factor of all the cases. Confirmation experimentation was performed at the optimized configuration, and a thermal performance factor value of 11.4% was achieved as compared to the base case (case 3), which is the highest among all the cases. Although this configuration is hard to implement due to an increase in design complexity, pressure losses, and manufacturing cost as compared to the conventional design of SAHs, it is believed that this configuration can increase the efficiency of the SAH sufficiently.

### 9.3 Ranking of Contributing Parameters.

The mean values of all the performance parameters are shown in Table 8. The contribution ratio of each control factor on the SNR values of the output parameters is presented in Table 9. In this table, the level is the experimental data point, rank is the grading of the parameter based on significance toward output, and delta is the change in the value of the output parameter from the highest to the lowest value when the input parameters are varied. The optimum conditions are shown in bold letters. From Table 8, it is clear that the dependence of the control factors is different on different output parameters.

Response table of means of Nu | Response table for signal-to-noise ratios of ΔP | ||||||||
---|---|---|---|---|---|---|---|---|---|

Level | Target plate | Re | D/D_{hs} | L/D_{h} | Level | Target plate | Re | D/D_{hs} | L/D_{h} |

1 | 45.38 | 45.32 | 77.38 | 72.52 | 1 | 20.07 | 15.67 | 50.12 | 22.61 |

2 | 59.36 | 51.74 | 72.59 | 57.08 | 2 | 30.17 | 26.47 | 22.26 | 32.40 |

3 | 68.11 | 75.79 | 22.88 | 43.25 | 3 | 28.63 | 36.73 | 6.49 | 23.86 |

Delta | 22.74 | 30.47 | 54.50 | 29.28 | Delta | 10.10 | 21.06 | 43.64 | 9.80 |

Rank | 4 | 2 | 1 | 3 | Rank | 3 | 2 | 1 | 4 |

Response table for signal-to-noise ratios of PP | Response table for signal-to-noise ration of f | ||||||||

Smaller is better | Smaller is better | ||||||||

Level | Target plate | Re | D/D_{hs} | L/D_{h} | Level | Target plate | Re | D/D_{hs} | L/D_{h}1 |

0.027 | 0.020 | 0.101 | 0.033 | 0.033 | 0.036 | 0.019 | 0.035 | 0.033 | |

2 | 0.059 | 0.044 | 0.030 | 0.064 | 0.036 | 0.035 | 0.027 | 0.036 | 0.036 |

3 | 0.049 | 0.072 | 0.005 | 0.039 | 0.037 | 0.033 | 0.058 | 0.033 | 0.037 |

Delta | 0.032 | 0.052 | 0.096 | 0.031 | 0.004 | 0.003 | 0.039 | 0.004 | 0.004 |

Rank | 3 | 2 | 1 | 4 | 2 | 4 | 1 | 3 | 2 |

Response table of means of Nu | Response table for signal-to-noise ratios of ΔP | ||||||||
---|---|---|---|---|---|---|---|---|---|

Level | Target plate | Re | D/D_{hs} | L/D_{h} | Level | Target plate | Re | D/D_{hs} | L/D_{h} |

1 | 45.38 | 45.32 | 77.38 | 72.52 | 1 | 20.07 | 15.67 | 50.12 | 22.61 |

2 | 59.36 | 51.74 | 72.59 | 57.08 | 2 | 30.17 | 26.47 | 22.26 | 32.40 |

3 | 68.11 | 75.79 | 22.88 | 43.25 | 3 | 28.63 | 36.73 | 6.49 | 23.86 |

Delta | 22.74 | 30.47 | 54.50 | 29.28 | Delta | 10.10 | 21.06 | 43.64 | 9.80 |

Rank | 4 | 2 | 1 | 3 | Rank | 3 | 2 | 1 | 4 |

Response table for signal-to-noise ratios of PP | Response table for signal-to-noise ration of f | ||||||||

Smaller is better | Smaller is better | ||||||||

Level | Target plate | Re | D/D_{hs} | L/D_{h} | Level | Target plate | Re | D/D_{hs} | L/D_{h}1 |

0.027 | 0.020 | 0.101 | 0.033 | 0.033 | 0.036 | 0.019 | 0.035 | 0.033 | |

2 | 0.059 | 0.044 | 0.030 | 0.064 | 0.036 | 0.035 | 0.027 | 0.036 | 0.036 |

3 | 0.049 | 0.072 | 0.005 | 0.039 | 0.037 | 0.033 | 0.058 | 0.033 | 0.037 |

Delta | 0.032 | 0.052 | 0.096 | 0.031 | 0.004 | 0.003 | 0.039 | 0.004 | 0.004 |

Rank | 3 | 2 | 1 | 4 | 2 | 4 | 1 | 3 | 2 |

Response table for signal-to-noise ratios of Nu | Response table for signal-to-noise ratios of ΔP | ||||||||
---|---|---|---|---|---|---|---|---|---|

Larger is better | Smaller is better | ||||||||

Level | Target plate | Re | D/D_{hs} | L/D_{h} | Level | Target plate | Re | D/D_{hs} | L/D_{h} |

1 | 31.5 | 32.1 | 37.7 | 36.2 | 1 | −24.9 | −21.5 | −33.5 | −25.4 |

2 | 34.6 | 33.9 | 36.5 | 33.5 | 2 | −25.4 | −25.9 | −26.4 | −25.6 |

3 | 34.7 | 34.7 | 26.7 | 31.1 | 3 | −25.6 | −28.4 | −15.9 | −24.9 |

Delta | 3.2 | 2.6 | 11.0 | 5.0 | Delta | 0.63 | 6.92 | 17.54 | 0.65 |

Rank | 3 | 4 | 1 | 2 | Rank | 4 | 2 | 1 | 3 |

Response table for signal-to-noise ratios of PP | Response table for signal-to-noise ration of f | ||||||||

Smaller is better | Smaller is better | ||||||||

Level | Target plate | Re | D/D_{hs} | L/D_{h} | Level | Target plate | Re | D/D_{hs} | L/D_{h}1 |

33.6 | 39.0 | 21.1 | 33.2 | 1 | 30.58 | 30.03 | 34.41 | 30.14 | |

2 | 33.3 | 32.4 | 31.5 | 33.0 | 2 | 29.97 | 30.04 | 31.24 | 29.85 |

3 | 33.0 | 28.5 | 47.4 | 33.7 | 3 | 29.84 | 30.32 | 24.74 | 30.40 |

Delta | 0.58 | 10.53 | 26.39 | 0.7 | Delta | 0.74 | 0.29 | 9.67 | 0.55 |

Rank | 4 | 2 | 1 | 3 | Rank | 2 | 4 | 1 | 3 |

Response table for signal-to-noise ratios of Nu | Response table for signal-to-noise ratios of ΔP | ||||||||
---|---|---|---|---|---|---|---|---|---|

Larger is better | Smaller is better | ||||||||

Level | Target plate | Re | D/D_{hs} | L/D_{h} | Level | Target plate | Re | D/D_{hs} | L/D_{h} |

1 | 31.5 | 32.1 | 37.7 | 36.2 | 1 | −24.9 | −21.5 | −33.5 | −25.4 |

2 | 34.6 | 33.9 | 36.5 | 33.5 | 2 | −25.4 | −25.9 | −26.4 | −25.6 |

3 | 34.7 | 34.7 | 26.7 | 31.1 | 3 | −25.6 | −28.4 | −15.9 | −24.9 |

Delta | 3.2 | 2.6 | 11.0 | 5.0 | Delta | 0.63 | 6.92 | 17.54 | 0.65 |

Rank | 3 | 4 | 1 | 2 | Rank | 4 | 2 | 1 | 3 |

Response table for signal-to-noise ratios of PP | Response table for signal-to-noise ration of f | ||||||||

Smaller is better | Smaller is better | ||||||||

Level | Target plate | Re | D/D_{hs} | L/D_{h} | Level | Target plate | Re | D/D_{hs} | L/D_{h}1 |

33.6 | 39.0 | 21.1 | 33.2 | 1 | 30.58 | 30.03 | 34.41 | 30.14 | |

2 | 33.3 | 32.4 | 31.5 | 33.0 | 2 | 29.97 | 30.04 | 31.24 | 29.85 |

3 | 33.0 | 28.5 | 47.4 | 33.7 | 3 | 29.84 | 30.32 | 24.74 | 30.40 |

Delta | 0.58 | 10.53 | 26.39 | 0.7 | Delta | 0.74 | 0.29 | 9.67 | 0.55 |

Rank | 4 | 2 | 1 | 3 | Rank | 2 | 4 | 1 | 3 |

Note: The optimum conditions are shown in bold letters. Ranks are allocated to each parameter separately in descending order of the calculated SNR values. The ranks show the contribution ratio of Nu follows the order *D*_{h}*/D*_{hs} > *L/D*_{h} > target plate > Re, while for the case of *f*, the contributing parameters are listed in the order of *D/D*_{hs} > target plate > *L/D*_{h}*>* Re.

### 9.4 Confirmation Experiment.

*f*came to be 118 and 0.019, respectively. While from Eqs. (21) and (22) and Table 8, the values of Nu and

*f*came to be 120.9 and 0.019 [22].

*o*represents the optimized point of each parameter, and

*η*is the optimized value of the performance parameter. A difference of 2.4% and 0.4% was obtained in Nu and

*f*, which shows the reasonable accuracy of the model.

### 9.5 Flow and Thermal Fields.

The velocity streamlines, contours of Nu, and plots of stream-wise and span-wise Nu on the target plate are presented to understand the effect of Re, target plate, jet plate configuration, and *L*/*D*_{h} on flow and thermal properties, as shown in Figs. 11–18.

From the flow and thermal fields of Fig. 12, it can be seen that a jet of 3 × 3 creates mini jets of large velocities as compared to a single jet of the same Reynolds number. As a result, an increase in the flow profiles is manifested, which, in turn, is responsible for an increase in turbulence. Due to this phenomenon, more vortices are formed, and rigorous mixing of the vortices takes place. The boundary layer will get thinner. The efficient mixing of cold core fluid and hot wall fluid took place. As a result, more area is cooled, and also more cooling took place. Instead of single peaks, multiple peaks of Nu are observed in 3 × 3 and 2 × 2 jet configurations as compared to the single-jet impingement. So 3 × 3 is the optimized jet configuration.

From Fig. 13, it can be seen that an inverse relation of Nu with *L*/*D*_{h} is manifested. It can be seen that when *L*/*D*_{h} is 5.1, the flow jet has more momentum, as shown in the streamlines, and, hence, higher is the ability to cool the plates. When the distance rises to 14.6, the jet expands with high magnitude, and it no longer has the ability to cool the plate, and less value of Nu is manifested. So the *L*/*D*_{h} value of 5.1 is the optimized one.

From Fig. 14, it can be seen that the trapezoidal plate has the highest value of Nu, followed by the sinusoidal corrugated and smooth plate. This phenomenon can be understood in this way that corrugation causes turbulence in the form of vortices dominant flow. Hence, more heat transfer occurs due to the rigorous mixing of core and wall fluid. It can be seen that in the case of a smooth channel, a single peak is observed at the stagnation point, while in the case of corrugation, multiple peaks are formed due to the effect of increased turbulence with maximum value at the stagnation point and continuously decreasing in the radial direction. At the crest of the corrugation, minima is formed, and maxima is formed on the trough side. As a result, more area is cooled in the case of corrugation and effective heat transfer. If the trapezoidal and sinusoidal corrugation is observed, the trapezoidal profile has a sharp throat area in the crest region. As a result, the local fluid velocity is increased, and a higher temperature gradient exists between the wall and core region. As a result, more mixing occurs between the wall and core fluid due to intense flow recirculations. Hence, the thermal boundary gets thinner, and efficient heat transfer occurs. In the case of a wavy profile, the flow profile is smooth; hence, the boundary layer remains thick. In the case of smooth channels, low heat transfer occurs due to a very thick thermal boundary layer.

It can simply be seen from Fig. 15 that Re increases the flow velocity, which, in turn, increases the turbulence in the form of vortices, as can be seen in the streamlines. The boundary layer will be disturbed. As a result of this, more heat transfer occurs; hence, Nu is increased significantly. So the Reynolds number value of 2700 is the optimized configuration. Hence, an increasing trend was observed.

From the above discussion, it can be inferred that 3 × 3 jets when impinged on the trapezoidal corrugated plate, placed at the jet-to-target plate distance of 5.1 for the Reynolds number value of 2700 is the optimized configuration.

## 10 Conclusion

To improve the thermal efficiency of SAHs, the novel corrugated jet-impinging channel is used for the absorbing plate. A study has been performed for three different arrangements of jets, i.e., 1 × 1, 2 × 2, and 3 × 3 jet array with three channels, i.e., smooth, trapezoidal, and wavy, corrugated over the Re range of 1740–2700 and *L*/*D*_{h} value of 5.1–14.6. An optimization study was performed using the Taguchi method. The accomplished work is summarized below:

The optimum value of Nu was found for the trapezoidal corrugated target plate, with 3 × 3 multiple jet-impinging type,

*L*/*D*_{h}, and Re values of 5.1 and 2700, respectively.An increase of 11.4 times was observed in the thermal performance factor as compared to the base case.

*D*_{h}/*D*_{hs}was found to be the most significant factor for Nu, followed by*L*/*D*_{h}, corrugation profile, and Re, while for the pumping power and pressure loss,*D*/*D*_{hs}followed by Re are the two significant parameters.The optimum value of

*f*was found for 3 × 3 jet impinging, while all other factors were found to be insignificant.CFD results lie within an error band of ±12% of the experimental data.

## Acknowledgment

The authors acknowledge the computational support of Mechanical and Nuclear Department of Pakistan Institute of Engineering and Applied Sciences (PIEAS), Nilore, Islamabad.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*e*=depth of grooves, m

*f*=coefficient of friction (friction factor)

*h*=coefficient of convective heat transfer, W/(m

^{2}K)*k*=turbulence kinetic energy, J/kg

*l*=length of plate, m

*n*=number of jets

*p*=pitch, m

*q*=net heating power, W

*v*=velocity, m/s

*w*=width of plate, m

*A*=area, m

^{2}*N*=number of grooves

*P*=pressure, Pa

*Q*=volumetric flowrate, m

^{3}/s*R*=resistance, Ω

*T*=temperature, °C

*V*=voltage applied across heater, V

- $m\u02d9$ =
mass flowrate, kg/s

*q″*=heat flux, Wm

^{−2}*k*_{c}=thermal conductivity, W/(mK)

*D*_{h}=hydraulic diameter, m

- D
_{h}/D_{hs}= jet diameter ratio

- L/D
_{h}= jet-to-target plate distance ratio

- Nu =
Nusselt number

*p*/*e*=pitch-to-groove height ratio

- Re =
Reynolds number

*e*/*D*_{h}=blockage ratio

*PP*=pumping power, W

### Greek Symbols

### Subscripts

### Abbreviations

### Appendix: Error Propagation Calculations

Due to uncertainty in the measurement of different measuring devices, the error propagates further in the calculations. For that purpose, uncertainty analysis was performed to estimate the reliability of our results and correlations [37]. This propagation of errors in heat transfer coefficient, Nusselt number, friction factor, and thermal performance factor was calculated based on the basic rules of propagation [38].

#### A1 Uncertainty in Nusselt Number

The dimensions were measured using a Vernier caliper, with the least count of 0.05 mm. Since uncertainty is half of the least count. it will be ±0.025 mm.

*l*

_{j}is the length and width of the square jet. The mathematical relation of the heat transfer coefficient is given in Eq. (11). The relative error in the heat transfer coefficient $\delta hh$ can be calculated by Eq. (A3).

To calculate the relative error in the heat transfer coefficient, the relative error in heat transfer rate $\delta qq$, heat transfer area $\delta AHTAHT$, and temperature difference $\delta \Delta T\Delta T$ are needed to calculate.

*q*to the heater was calculated by measuring the electric resistance

*R*and potential drop

*V*across heating coils using a digital multimeter. The relative error in the heat transfer rate can be calculated from Eq. (A4).

A total number of nine thermocouples were used on the test section plate, and the arithmetic mean value was computed. So, uncertainty in temperature values of thermocouples can be calculated from Eq. (A9).

*n*is the number of thermocouples.

#### A2 Uncertainty in Pumping Power

An uncertainty analysis is performed to assess the accuracy of the measurements using the basic error propagation rules. Uncertainties for Nusselt number and friction coefficient are evaluated.

#### A3 Uncertainty in Friction Factor

#### A4 Uncertainty in Thermal Performance Factor

An uncertainty analysis is performed to assess the accuracy of the measurements using the basic error propagation rules. Uncertainties for Nusselt number, friction coefficient, and thermal performance factor are evaluated. An uncertainty of 2.4% and 3.3% was found in the Nusselt number and friction factor, respectively, which shows the error band of our experimental data.