# Mathematical Modeling of Efficiency Evaluation of Double-Pass Parallel Flow Solar Air Heater

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## Abstract

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## 1. Introduction

_{th}) is decreased due to the low h value between the absorber plate and the airflow. A laminar sub-layer that forms next to the absorber surface when air passes through the duct of an SAH reduces heat transfer to the air stream and, consequently, the η

_{th}. The laminar-to-turbulent conversion of the boundary layer zone is achieved by applying artificial roughness to the absorber plate on one or both sides. That leads to a rise in h values and the rate of heat transfers in the duct.

_{h}) = 0.03375 and the angle of attack (α) = 60

^{o}produced the largest increase in the exergetic efficiency (η

_{exg}) of the rough SAH when compared with the smooth surface, that is, 51%. An exergy analysis of an SAH with double-V incisions in twisted tape was carried out by Kumar [31]. In SAHs with broken arc-ribs with staggering sections, Meena et al. [32] and Saini et al. [8] measured the heat transmission and friction properties. Using numerical simulations for ribbed triangular SAHs, Kumar and Kumar [33] studied the performance enhancements and the correlations for the friction factors and heat transport. Kumar et al. [34]. conducted an experimental investigation of a DPSAH with multiple-C-shaped roughness on an SAH.

## 2. Experimental Setup and Model for DPPFSAH

#### Heat Transfer Modes and Assumptions for Boundary Condition

## 3. Thermo-Hydraulic Performance of DPPFSAH

_{u}) to the amount of sun irradiation (I) attained by the collector’s heated surface is known as the thermal efficiency (η

_{th}) of SAH [1]. It can be expressed as:

_{R}) for incoming air temperature, as follows:

_{0}) for outlet air is the ratio of the actual to maximum feasible heat transfer rate [40] and is expressed as:

## 4. System and Operational Parameters

_{th}) and Efficiency Enhancement Factor (EEF) of the DPPFSAH with perforated multi V-ribs and to determine the optimum values of the parameters that obtain the optimum thermal efficiency, these parameters can be divided into fixed and variable categories.

#### 4.1. Fixed Parameters

#### 4.2. Variable Parameters

## 5. Steps for Efficiency Prediction of DPPFSAH

**Step 1.**During the iterative process, a fixed set of geometrical and roughness values chosen in accordance with Table 2 are used, and varying variables such as W/w, β, Re, ΔT/I, and I values are taken into consideration in accordance with Table 2.

**Step 2.**The plate area can be found as:

**Step 3.**The air temperature at the outlet (T

_{fo}) and the change in air temperature (ΔT) is determined through the air inlet temperature (${T}_{fi}$), as follows:

**Step 4.**The top loss coefficient (U

_{T}) is derived by the Klein [40] and Datta [42] correlation, as follows:

^{2}; ΔT—temperature differential, K; υ—kinematic viscosity, m

^{2}/s; Ra—the Rayleigh number; and α—thermal diffusivity, m

^{2}·s

^{−1}.

**Step 5.**The back loss coefficient (U

_{B}) is the ratio of thermal-conductivity (k

_{ins}) and thickness (t

_{ins}) of insulation [44]:

**Step 6.**The edge loss coefficient (U

_{E}) is calculated using the collector area (A

_{c}), insulator thermal conductivity (k

_{ins}), and thickness (t

_{ins}) as inputs, given as in [44].

**Step 7.**The useful heat in duct is calculated as:

**Step 8.**For the SPSAH and DPPFSAH, the Nu is determined by using empirical correlation developed by Singh et al. [12,14].

**Step 9.**The h value is determined by using the Nu in step 9 using the calculation given below [45]:

**Step 10.**The plate efficiency factor is determined as [42]:

**Step 11.**The heat removal factor is calculated as follows [42]:

**Step 12.**The useful heat gain (Q

_{u}

_{2}) per unit area of the collector calculated as [37]:

**Step 13.**By using steps 7 and 12, Q

_{u}

_{1}and Q

_{u}

_{2}are calculated and compared. If the predicted values of these two terms are not near enough, i.e., $\frac{{Q}_{u1}-{Q}_{u1}}{{Q}_{u1}}$> 0.1%, then the next mean temperature (T

_{pm}) of the absorber plate is revised as:

**Step 14.**Equation (34) uses the value of T

_{pm}derived in Equation (16), and the computations are repeated from step 5 to step 14. Q

_{u}

_{1}and Q

_{u}

_{2}have been iterated until they are near enough, i.e., (Q

_{u}

_{1}− Q

_{u}

_{2}< 0.1% of Q

_{u}

_{1}).

**Step 15.**The roughened double-pass SAH’s ${\eta}_{th}$ is calculated as:

_{u}is the average heat gain, which is calculated as:

**Step 16.**The f value for a DPPFSAH is determined by using the correlation developed by Singh et al. [12,14], which is presented below.

**Step 17.**The pressure drop (ΔP)

_{d}in the duct is calculated as:

**Step 18.**The power requirement of the blower (P

_{m}) is calculated as:

**Step 19.**The thermal efficiency is calculated as:

**Step 20.**The effective efficiency, η

_{eff}, is determined as:

_{f}= 0.65; η

_{m}= 0.88; η

_{tr}= 0.92; and η

_{th}= 0.35).

**Step 21.**The mean fluid temperature (T

_{fm}) is calculated as:

**Step 22.**The Carnot efficiency is determined as:

**Step 23.**The Net exergy-flow ($En)$ is calculated as:

**Step 24.**The Exergy-rate (${E}_{s})$ associated with solar irradiation is calculated as:

**Step 25.**The exergetic efficiency (${\eta}_{exg})$ is determined as:

**Step 26.**To cover the whole range of roughness and operating parameters as shown in Table 1 and Table 2, calculations are performed from step 2 to step 25 for all possible combinations of system and operational parameters. Figure 3 shows the process-flow diagram of the computer program developed in MATLAB that performs all of the computations specified in the preceding sections.

## 6. Results and Discussion

#### 6.1. Effect of Flow and System Parameters on Thermal Efficiency

#### 6.1.1. Effect of Reynolds Number (Re)

_{th}of a DPPFSAH roughened with varied perforated multi-V ribs for various specified roughness levels. The values of β = 0.27 and W/w = 6 were chosen as the optimum roughness parameters. The η

_{th}of the smooth and roughened collectors increase with increase in the Re values in all instances of the SPSAH and the DPPFSAH. The improvement in the h value between the base plate and air caused by the rise in Re can be used to explain these behaviours. Roughness patterns further enhance the SPSAH’s and DPPFSAH’s η

_{th}. In case of SPSAHs and DPPFSAHs, the η

_{th}improve from 69.66% to 80.13% and 74.74% to 86.57%, respectively, for a perforated multi-V rib roughness of e/D = 0.043, β = 0.27, P/e = 10 α = 60°, W/w = 6, W/H = 12, and I = 1000 W/m

^{2}corresponding to an Re from 2000 to 18,000. On the other hand, for a smooth collector, the η

_{th}lies between 27.66% and 65.19%, corresponding to an Re from 2000 to 18,000, respectively.

_{th}values with a relationship of the ΔT/I. As the ΔT/I increased, it was discovered that the η

_{th}of the SPSAH and DPPFSAH rapidly decreased. The temperature of the entering fluid increases, the thermal gradient between the base plate and the airstream decreases. Due to this, the base plate and glass cover’s average temperature increase, which increases the amount of heat lost to the environment while reducing the amount of heat gained, which is actually useful. This lowers the thermal productivity and effectiveness of the SPSAH and DPPFSAH.

#### 6.1.2. Effect of Relative Roughness Width (W/w)

_{th}as a function of ΔT/I for the SPSAH and the DPPFSAH is shown in Figure 5a,b. In both scenarios, the η

_{th}declines and is found to be lowest at W/w = 2 after increasing with the increase in W/w up to 6 and then dropping with the increasing W/w values. The greatest η

_{th}for both the SPSAH and the DPPFSAH was found to be 80.13% for the SPSAH and 86.57% for the DPPFSAH at W/w = 6 and β = 0.27.

#### 6.1.3. Effect of Open Area Ratio (β)

_{th}rises with the increase in β, reaches a maximum at 0.27, and then slightly declines as β rises further. It has been discovered that raising the β value results in more turbulence and secondary flow mixing in the vicinity of the perforated ribs, which enhances fluid mixing and lowers the thermal barrier due to the laminar sub-layer, boosting the h value. In addition, as β rises over 0.27, Q

_{u}decreases because secondary air can now be accessed through perforations, and the upper part of the rib starts to behave like a stagger, making it harder for fluid to mix effectively [20,47].

#### 6.2. Efficiency Enhancement Factor (EEF)

_{th}of an SAH with and without artificial roughness operating under similar conditions:

#### 6.3. Effect of Insolation on EEF

^{2}have been considered. From Figure 8, it is evidently observed that the EEF increased with the rise in I value. In addition, as the ΔT/I increased, the EEF increases for a given value of I. The maximum and minimum efficiency improvement factors were found at insolation values of 1000 W/m

^{2}and 600 W/m

^{2}.

## 7. Effective Efficiency (η_{eff}) Criteria for DPPFSAH

_{eff}”, which accounts for the useful energy gain and equivalent heat required to generate equal mechanical energy to overcome pressure losses, as per Cortes and Piacentini [46]. For a DPPFSAH, the optimum value of η

_{eff}was acheived at a W/w of 6 when the ΔT/I was more than 0.01107 Km

^{2}/W. Similarly, for ΔT/I values less than 0.00371 Km

^{2}/W, the DPPFSAH’s smooth collectors perform better than roughened collectors in SPSAHs and DPPFSAHs. Figure 9a,b show that, for a given value of W/w, the DPPFSAH’s η

_{eff}improves as Re increases, reaches an optimum value, and then starts decreasing as the Re rises further. The optimum η

_{eff}is found at W/w = 6, Re = 8527. The DPPFSAH roughened plate has a higher η

_{eff}for Re greater than 19,025. As a result, it is discovered that the roughness geometry in the form of a perforated multi-V shaped rib pattern performs better at lower Re values. While Influence of β on η

_{eff}as a function of ΔT/I and Re for DPPFSAH is shown in Figure 10a,b and the range of parameters β and ΔT/I for highest η

_{eff}for different combination of DPPFSAH is shown in Table 3.

_{eff}. System and design characteristics such as e/D, β, p/e, α, and I are kept constant.

#### Geometric Parameter Optimization Using the Effective Efficiency Criteria

_{eff}) for a given range of design parameters (ΔT/I, I). For different values of solar radiation intensity (I), Figure 11a shows the variation in optimum values of W/w with ΔT/I for different values of I. For ΔT/I < 0.009572 Km

^{2}/W, the best value of W/w is 2 for a DPPFSAH. For a DPPFSAH, the ideal W/w is 6 for ΔT/I > 0.01128 K-m

^{2}/W for the entire range of I. However, the optimum value of W/w for a DPPFSAH is discovered to be a function of ΔT/I (ranging between 0.009572 K-m

^{2}/W and 0.01128 K-m

^{2}/W) and I. Table 5 shows a summary of the findings. While Figure 11b shows the variation of optimum values of β as a function of ΔT/I for different I values. The β = 0.21 represents the optimum settings for ΔT/I < 0.00794 K-m

^{2}/W for a DPPFSAH for the entire range of I studied. The β = 0.27 for ΔT/I > 0.01091 Km

^{2}/W reflects the best conditions for all the selected I values.

## 8. Exergetic Efficiency (η_{exg}) Criterion for DPPFSAH

_{exg}varies with ΔT/I, as shown in Figure 12a. For a DPPFSAH, the optimum value of η

_{exg}was reached at a W/w of 6, corresponding to a ΔT/I value greater 0.15298 Km

^{2}/W, respectively. The smooth DPPFSAH shows a better η

_{exg}compared to roughened DPPFSAH. Figure 12b depicts the variation in η

_{exg}with Re for different values of W/w and fixed values of other parameters, the optimum value of η

_{exg}has been obtained at W/w = 6 for Re < 3685, whereas for Re > 9228, the smooth DPPFSAH shows better η

_{exg}compared to roughened DPPFSAH. The details of optimum range are given in Table 6.

_{exg}was obtained for ΔT/I > 0.0169 Km

^{2}/W at β = 0.27, whereas the maximum value of η

_{exg}for a smooth DPPFSAH were obtained for ΔT/I < 0.01017 Km

^{2}/W. The detailed range of parameters and optimum range of β are given in Table 7. Figure 13b represents the variation in η

_{exg}with β as a function of Re. It is observed that the optimum value of η

_{exg}for smooth DPPFSAH occurs for Re > 8955.

#### Geometric Parameter Optimization Using the Exergetic Efficiency (η_{exg}) Criterion

_{exg}has been drawn in Figure 14a for a DPPFSAH, for a given range of ΔT/I. The W/w value of 2 indicates the best condition for the DPPFSAH, ΔT/I < 0.006051 Km

^{2}/W, for the entire range of I, i.e., from 600 to 1000 W/m

^{2}. Furthermore, for all values of I, ΔT/I > 0.008084 Km

^{2}/W constitutes the optimal condition for a W/w value of 6. For a DPPFSAH, the optimum value of W/w is a function of I for ΔT/I values between 0.006051 Km

^{2}/W and 0.008084 Km

^{2}/W, respectively. Figure 14b depicts the optimum values of β for different values of ΔT/I and I. For the value of β of 0.21, the optimum values are obtained for ΔT/I values up to 0.00794 Km

^{2}/W for DPPFSAH. For the values of ΔT/I above 0.01091 Km

^{2}/W, a β value of 0.31 gives the optimum results for a DPPFSAH. However, for ΔT/I values between 0.00794 Km

^{2}/W and 0.01091 Km

^{2}/W, the optimum value of β is a function of I. The η

_{exg}criterion plays a major role in selecting the optimum values of geometric parameters such as W/w and β based on design parameters such as ΔT/I and I, according to the above discussion. For a particular range of ΔT/I and I, a set of optimum geometric parameters can now be picked from Table 8.

## 9. Comparison of Optimization Criteria

_{th}, η

_{eff}, and η

_{exg}criteria.

_{th}criteria. As a conclusion, Table 9 demonstrate that the best artificial roughness geometries for a DPPFSAH is a combination of the best values of the design variables, namely, a W/w of 6 and a β of 0.27. Although no single pairing of design parameters displays the optimal values for the entire range of the ΔT/I in the case of the η

_{eff}requirements and the η

_{exg}standards, it is noted that no single pairing of design parameters displays the optimal values for all chosen values of the ΔT/I. An artificially roughened DPPFSAH with β = 0.27 and W/w = 6 outperforms all other permutations of DPPFSAHs on all three criteria.

^{2}generated using the η

_{eff}and η

_{exg}criteria, were used to calculate the range of ΔT/I given in Table 9. According to the η

_{eff}and η

_{exg}criteria for various amounts of solar irradiance, Table 10 illustrates the ΔT/I range where the optimal geometric parameter values vary. As shown in the analysis, the optimal values of geometric parameters depend directly on the optimization criteria employed. Therefore, selecting the factors to take into account in order to improve the desired results of SAH becomes crucial. The blower power required to move air through the collector is not included in η

_{th}metrics; rather, they solely take into account gains in thermal energy. In order to maximise the performance of SAH, the thermo-hydraulic considerations, specifically η

_{eff}and η

_{exg}, should be applied.

_{exg}criterion applicable. Due to their narrow temperature range of operation, SAHs are not suitable for work generation. It has been demonstrated that the total exergy flow has negative values in some low-temperature uses. The η

_{exg}criterion is therefore crucial when calculating thermal power at high average temperature. Additionally, the η

_{eff}requirement takes into account the rise in useable heat energy, which is constrained by the energy required to supply blowers the energy to make up for pressure losses. Thus, from the perspective of thermo-hydraulic optimization, the η

_{eff}criterion has been suggested for DPPFSAHs roughened with perforated multi-V ribs.

## 10. Conclusions

_{eff}, the optimal values are shown.

_{th}was found to be 86.57% for rough surfaces and 74.74% for smooth ducts. For a DPPFSAH, the optimum design noted EEF = 2.37 at W/w = 6, β = 0.27 and I = 1000 W/m

^{2}. The blower power is considerable at lower ΔT/I values; hence, the EEF increases as the Re and ΔT/I intensities increase. It is also observed that smooth collectors perform better than roughened DPPFSAH collectors for ΔT/I values below 0.00371 Km

^{2}/W, while the best value of η

_{eff}was achieved at W/w = 6, when ΔT/I >0.01107 Km

^{2}/W. Similar to η

_{eff}, which becomes better as Re rises and reaches its peak value for W/w = 6 at Re = 8527, η

_{eff}then starts to fall for all W/w values as the Re values continue to rise. In comparison to the roughened DPPFSAH, the smooth DPPFSAH has a higher η

_{eff}for an Re > 19,025.

^{2}/W, the effect of ΔT/I on η

_{eff}as a function of β attends the highest value of η

_{eff}at β = 0.27, and once the ΔT/I < 0.00359 Km

^{2}/W, the smooth collector outperforms the roughened DPPFSAH. The smooth DPPFSAH has a larger η

_{eff}for an Re > 18,821, and the best η

_{eff}is found for β = 0.27 at Re = 8527. The smooth DPPFSAH also demonstrates higher η

_{exg}than the roughened DPFPSAH with ΔT/I < 0.006985 Km

^{2}/W, with the optimal value of η

_{exg}being attained at a W/w = 6 and at ΔT/I above 0.015298 Km

^{2}/W. The maximum value of η

_{exg}has been reached for Re < 3685 and W/w = 6, but for Re > 9228, the smooth DPPFSAH indicates a higher η

_{exg}in comparison to the roughened DPPFSAH. The optimum value of η

_{exg}has been obtained for 0.0169 < ΔT/I K.m

^{2}/W at β = 0.27, whereas the highest value of η

_{exg}for smooth SPSAH has been obtained for ΔT/I < 0.00359 K.m

^{2}/W. As a consequence, it is found that a perforated multi-V shaped rib patterns roughness architecture works better at smaller Re levels and larger ΔT/I values. The effective efficiency η

_{eff}criterion was found for a DPPFSAH roughened with perforated multi-V ribs. The Re range of 2000–18,000 for DPPFSAH can be designed using the results of the current study.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Details of symbols | Greek symbols | ||

A | Area, (m^{2}) | Δ | Drop/gradient |

C_{d} | Coefficient of discharge | Δ | Partial |

P | Mean static pressure N/m^{2} | η | Efficiency |

H | Height, (m) | ∈ | Emissivity |

h | Heat-transfer coefficient (W/m^{2}·°C) | υ | Kinematic viscosity, (m^{2}/s) |

I | Solar Irradiance (W/m^{2}) | α | Absorptivity |

k | Thermal conductivity (W/m°C) | σ | Stefan–Boltzmann constant (W/m^{2}·K^{4}) |

t | Thickness (m) | ρ | Air density, (kg/m^{3}) |

$\dot{m}$ | Air mass-flow rate, (kg/s) | α | Angle of attack, (^{o}) |

P | Pitch distance (m) | β | Collector slope (^{o}), Open area ratio |

Q | Thermal energy transferred (J) | μ | Dynamic viscosity (N.s/m^{2}) |

𝑞 | Average heat generation (W/m^{3}) | ψ | Circularity |

T | Mean Temperature (°C) | ν | Kinematic viscosity m^{2}/s |

W | Width of channel, (m) | τ | Transmissivity |

w | Width of one set of rib, (m) | ||

V | Velocity of working fluid (m/s) | Abbreviations | |

D_{h} | Hydraulic diameter (m) | DPPF | Double-Pass Parallel Flow |

Subscripts | THPP | Thermohydraulic performance parameter | |

A | Ambient, Air | SAH | Solar Air Heater |

abs | Absorber | ||

Amb | Ambient | m | Mean |

d | Duct/ channel, diameter | u | Useful |

g | Glass cover | t | Thermal |

h | Height, hole | eff | Effective |

Ins | Insulation | ex. | Exergetic |

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**Figure 1.**(

**a**) Schematic view of parallel flow SAH with perforated multi-V rib roughness and (

**b**) Cross-sectional view of the DPPFSAH duct.

**Figure 3.**Flow diagram for the computer program to calculate the thermal efficiency of SPSAH and DPPFSAH.

**Figure 4.**(

**a**) η

_{th}vs. Re and (

**b**) η

_{th}vs. ΔT/I as a function of selected values of parameters of smooth and roughened SPSAHs and DPPFSAHs.

S. No. | Parameter | Value/Range |
---|---|---|

1. | No. of Pass | Two (DPPF) |

2. | Type of Flow | Parallel Flow |

3. | L | 1.0 m |

4. | W | 0.3 m |

5. | H | 0.025 m |

6. | N | 1 nos. |

7. | k_{ins} | 0.037 W/m-K |

8. | t_{ins} | 0.05 m |

9. | ρ | 1.105 kg/m^{3} |

10. | µ | 1.865 × 10^{−5} kg/s-m |

11. | k | 0.02624 W/m-K |

12. | τα | 0.8 |

12. | _{Hg} | 0.025 m |

13. | β | 0 for horizontal |

14. | 𝜀_{𝑃} | 0.92 |

15. | ε_{g} | 0.88 |

16. | t_{g} | 0.004 m |

17. | T_{a} | 300 K |

18. | V | 1.0 m/s |

Sr. No. | Parameters Notations | Range |
---|---|---|

1. | W/w | 2–10 (five values) |

2. | β | 0.0, 0.21, 0.27, 0.31 (four values) |

3. | Re | 2000–18,000 (Nine Values) |

4. | ΔT/I | 0.002–0.02 Km^{2}/W (Ten Values) |

5. | I | 600–1000 W/m^{2} (Three values) |

Geometric Parameter | ΔT/I (Km^{2}/W) | β |
---|---|---|

β | 0.00359 < ΔT/I < 0.00794 | 0.21 |

0.01091 < ΔT/I | 0.27 | |

0.00794 < ΔT/I < 0.01091 | 0.31 | |

ΔT/I < 0.00359 | Smooth | |

Geometric Parameter | Re | β |

β | Re < 5644 | 0.21 |

5860 < Re < 18,821 | 0.27 | |

5557 < Re < 8527 | 0.31 | |

18,821 < Re | Smooth |

Geometric Parameter | ΔT/I (Km^{2}/W) | W/w |
---|---|---|

W/w | 0.00835 < ΔT/I < 0.01126 | 10 |

0.00815 < ΔT/I < 0.01128 | 8 | |

0.01128 < ΔT/I | 6 | |

0.00823 < ΔT/I < 0.01117 | 4 | |

0.00371 < ΔT/I < 0.01128 | 2 | |

ΔT/I < 0.00371 | Smooth | |

Geometric Parameter | Re | W/w |

W/w | Re < 7239 | 10 |

8527 < Re < 8741 | 8 | |

11,881 < Re < 19,025 | 6 | |

7239 < Re < 8527 | 4 | |

5644 < Re < 8527 | 2 | |

19,025 < Re | Smooth |

Geometric Parameters | ΔT/I (Km^{2}/W) | W/w (Optimum) |
---|---|---|

W/w | ΔT/I < 0.009572 | 2 |

0.009572 < ΔT/I < 0.01058 | Function of ΔT/I of I | |

0.01058 < ΔT/I | 6 | |

Geometric Parameters | ΔT/I (Km^{2}/W) | β (Optimum) |

β | ΔT/I < 0.00794 | 0.21 |

0.00794 < ΔT/I < 0.01091 | Function of ΔT/I of I | |

0.01091 < ΔT/I | 0.27 |

**Table 6.**Range of ΔT/I and Re for different W/w corresponding to highest η

_{exg}range for different combinations of DPPFSAH.

Geometric Parameter | ΔT/I (Km^{2}/W) | W/w |
---|---|---|

W/w | 0.05021 < ΔT/I < 0.06134 | 10 |

0.049893 < ΔT/I < 0.06134 | 8 | |

0.015298 < ΔT/I | 6 | |

0.04989 < ΔT/I < 0.06117 | 4 | |

0. 0.0059 < ΔT/I < 0.00871 | 2 | |

ΔT/I < 0.006985 | Smooth | |

Geometric Parameter | Re | W/w |

W/w | 605 < Re < 1075 | 10 |

634 < Re < 1104 | 8 | |

Re < 3685 | 6 | |

783 < Re < 1175 | 4 | |

6207 < Re < 8371 | 2 | |

9228 < Re | Smooth |

**Table 7.**Range of ΔT/I and Re for different β corresponding to the highest η

_{exg}range for different combinations in DPPFSAH.

Geometric Parameter | ΔT/I, Km^{2}/W | β |
---|---|---|

β | 0.00919 < ΔT/I < 0.01279 | 0.21 |

0.0169 < ΔT/I | 0.27 | |

0.04055 < ΔT/I < 0.05271 | 0.31 | |

ΔT/I < 0.00359 | Smooth | |

Geometric Parameter | Re | β |

β | 6182 < Re < 7930 | 0.21 |

Re < 4052 | 0.27 | |

588 < Re < 937 | 0.31 | |

8503 < Re | Smooth |

Rib Roughness Parameter | ΔT/I (Km ^{2}/W) | W/w (Optimum Value) |
---|---|---|

W/w | ΔT/I < 0.006051 | 2 |

0.006051 < ΔT/I < 0.008084 | Function of ΔT/I of I | |

0.008084 <ΔT/I | 6 | |

Rib RoughnessParameter | ΔT/I (Km^{2}/W) | β (Optimum Value) |

β | ΔT/I < 0.00827 | 0.21 |

0.00827 < ΔT/I < 0.0169 | Function of ΔT/I of I | |

0.0169 < ΔT/I | 0.27 |

**Table 9.**ΔT/I range for optimum roughness parameters as determined by the η

_{eff}and η

_{exg}criteria for I = 1000 W/m

^{2}.

Rib Roughness Parameter | ΔT/I (Km^{2}/W) | Rib Roughness Parameter (Optimum Value) |
---|---|---|

W/w | ΔT/I < 0.006051 | 2 |

ΔT/I > 0.01128 | 6 | |

β | ΔT/I < 0.00794 | 0.21 |

ΔT/I > 0.01693 | 0.27 |

**Table 10.**For varied I in DPPFSAH, the range of ΔT/I for optimum roughness parameter values are different to η

_{eff}and η

_{exg}requirements.

Insolation (W/m^{2}) | Roughness Parameter | Range of ΔT/I |
---|---|---|

1000 | W/w | 0.006051 < ΔT/I < 0.008084 |

β | 0.0079413 < ΔT/I < 0.010914 | |

800 | W/w | 0.006719 < ΔT/I < 0.008799 |

β | 0.008753 < ΔT/I < 0.012713 | |

600 | W/w | 0.007528 < ΔT/I < 0.010399 |

β | 0.009325 < ΔT/I < 0.014937 |

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**MDPI and ACS Style**

Singh, V.P.; Jain, S.; Karn, A.; Kumar, A.; Dwivedi, G.; Meena, C.S.; Cozzolino, R.
Mathematical Modeling of Efficiency Evaluation of Double-Pass Parallel Flow Solar Air Heater. *Sustainability* **2022**, *14*, 10535.
https://doi.org/10.3390/su141710535

**AMA Style**

Singh VP, Jain S, Karn A, Kumar A, Dwivedi G, Meena CS, Cozzolino R.
Mathematical Modeling of Efficiency Evaluation of Double-Pass Parallel Flow Solar Air Heater. *Sustainability*. 2022; 14(17):10535.
https://doi.org/10.3390/su141710535

**Chicago/Turabian Style**

Singh, Varun Pratap, Siddharth Jain, Ashish Karn, Ashwani Kumar, Gaurav Dwivedi, Chandan Swaroop Meena, and Raffaello Cozzolino.
2022. "Mathematical Modeling of Efficiency Evaluation of Double-Pass Parallel Flow Solar Air Heater" *Sustainability* 14, no. 17: 10535.
https://doi.org/10.3390/su141710535