## Abstract

An experimental investigation is carried out to analyze the thermo-hydraulic characteristics of a triangular solar air heater duct having transverse ribs with gaps. The roughness parameters, such as non-dimensional pitch (P/e or $P\xaf$) and non-dimensional height (e/D_{h} or $e\xaf$) are kept in the range of 4.88–20 and 0.021–0.044, respectively. Reynolds number (Re) is kept in the range of 4000–18,000. Two and three gaps of each of 0.01 m are provided to each odd and even number ribs, respectively. Non-dimensional primary width (w_{1}/W) and non-dimensional secondary width (w_{2}/W) are kept constant at 0.29 and 0.4, respectively. A maximum heat transmission of 3.14 times that of the base model is achieved for the transverse ribs with gaps having non-dimensional pitch and height of 9.76 and 0.044, respectively, at Re = 18,000. In the parametric range, the highest friction factor of 3.88 times the base model is encountered for the non-dimensional pitch and height of 4.88 and 0.044, respectively, at Re = 4000. The highest thermal enhancement ratio of 2.31 is reported for the non-dimensional pitch and height of 9.76 and 0.044, respectively, at Re = 18,000. The correlation for the Nusselt number and friction factor is formulated, agreeing with experimental data within ±12% and ±8% deviation, respectively.

## 1 Introduction

Issues like shortage of conventional fuels, population growth, urbanization, and ecological imbalance are among society's most significant energy-related problems. Despite numerous efforts, the energy crisis persists and gets worse. So, the race is on to meet the energy demand using renewable energy sources, such as solar, wind, and geothermal energy. As the most versatile and widely used heat exchanger for converting solar energy into usable heat energy for different heating purposes, the solar air heater duct (SAHD) is the most popular and widely used solar energy generator [1]. The main parts of the SAHD are the absorber plate (AP), glass covers, and insulated walls, as shown in Fig. 1. SAHD has diverse applications, like heating the room in winter, pre-heating the air in the power plant, and drying farming products. The working fluid, like the air, has a low thermal conductivity that suppresses the heat transfer enhancement of the SAHD. Viscous sublayers primarily strengthen the mode of heat transmission by conduction and weaken the convection heat transfer between the AP and air. So, adopting roughness on the surface of the AP is one of the crucial ways to boost the thermal efficiency of the SAHD. Obstruction in the flow due to roughness on the surface of AP reduces the resistance to heat transmission between the air and AP.

Many investigations have been done in recent years using surface roughness as a vital tool to improve the thermal performance of the SAHD. The performance of the SAHD is affected by the shape, orientation, and size of the roughness placed on the AP. The different shapes of the turbulators, such as circular, semi-circular, quarter-circular, square, rectangular, trapezoidal, reverse-L, and inverted-T, are the most preferred to develop a model of roughened SAHD. The performance of a roughened SAHD alters, even if the orientation of the roughness elements changes. The various orientations (patterns) of roughness elements such as transverse, inclined, arc, V-pattern, W-pattern, and S-pattern have been acknowledged by researchers in recent years. When the mainstream flow strikes the rib turbulators placed on the AP, the secondary flow formation accelerates and destroys the viscous sub-layer. This results in strong turbulence in the flow and enhances heat transmission through the SAHD. Thus the height of the rib turbulators and the placement of ribs strongly influence heat transmission. Many investigations have been done by considering the non-dimensional rib pitch (*P/e* or $P\xaf$) and non-dimensional rib height (*e*/*D _{h}* or $e\xaf$) as the effective heat transfer parameter. However, bigger-size rib turbulators also encourage the high-pressure drop across the duct. So rib turbulators with optimized configurations are recommended to place as close as the surface of the AP. From the design point of view, the cross section of air passage through the SAHD is rectangular, triangular, semi-circular, and so on [2,3]. For the simple design and installation, numerous heat transmission analyses for solar air heater ducts with rectangular air passages have been done in recent years.

Conversely, for the same operating conditions, SAHD with triangular air passage performs better than SAHD with rectangular air passage, though it has some difficulty in design and installation [4]. Many investigations have focused on enhancing heat transmission and minimizing the friction penalty. Therefore, the researchers consider the Nusselt number enhancement ratio (NNER), friction factor enhancement ratio (FFER), thermal efficiency (*η*_{th}) [5,6], effective thermal efficiency (*η*_{eff−th}) [7,8], and thermal enhancement ratio (TER) [9,10] as the finding outcomes in their investigations. Jain and Lanjewar [11] examined the thermal performance of a staggered V-ribbed rectangular SAHD (RSAHD) with aligned gaps. The authors addressed a maximum NNER and FFER of 2.73 times the base model within the parametric range. Panda and Kumar [12] numerically investigated the thermo-hydraulic performance of a rectangular RSAHD with a protrusion on the AP. The authors attained a maximum TER of 1.1416 for the RSAHD, with staggered protrusion having a non-dimensional pitch of 1.25. Also, a maximum TER of 1.1534 was addressed for the RSAHD with inline protrusion. Singh et al. [13] attempted to investigate the influence of multiple broken transverse ribs on the thermo-fluid behavior of the rectangular SAHD. They considered *P/e*, *e/D _{h}*, and

*W*/

*w*as the roughness parameters to optimize the design configurations for maximum thermal performance. Authors reported a maximum TER of 2.1 and 1.62 for multiple broken transverse ribs and square wave ribs, respectively, with $P\xaf=10$, $e\xaf=0.23$, and non-dimensional width of 7. Singh et al. [14] conducted a numerical study to evaluate the thermo-hydraulic performance of an RSAHD with kite-shape turbulators on AP. As a result of turbulators, authors reported a maximum TER of 2.93 for $P\xaf=56.8$ at Re = 4000. Rathor and Aharwal [15] investigated the aspect of heat transfer augmentation because of a staggering element using liquid crystal thermography in an inclined discrete staggered ribbed SAHD. An enhancement in heat transfer with staggered elements was reported to be 1.2554 times without staggered elements in the same parametric range. Singh et al. [16] presented an evaluation of the thermo-hydraulic performance of an RSAHD with Y-shaped ribs on AP. The authors reported a maximum TER of 2.62 for $P\xaf=10$ and $e\xaf=0.23$ at Re = 18,000 with a rib gap of 6 mm. Kumar et al. [17] evaluated the thermo-hydraulic performance of an RSAHD with NACA 0020 profile ribs on the surface of the AP. The TER resulting from the forward and backward flows is compared to design the optimized roughness configuration. The authors found that the model with backward flow direction results in a maximum TER of 2.2 at Re = 6000. Kumar and Verma [18] used sinusoidal protrusion turbulators on AP to evaluate the thermal performance of the RSAHD. A maximum TER of 2.02 was obtained for $P\xaf=10$ and Re = 4000. Also, the maximum thermal efficiency was 69% in the parametric range. Agrawal et al. [19] studied the thermo-hydraulic performance of a discrete rib-roughened RSAHD. They attained a maximum TER of 2.82 for ribs with $P\xaf=8.33$ and Re = 13, 935. Kumar et al. [20] evaluated the TER of an impinging jet RSAHD having discrete multi-arc ribs on the AP. A maximum TER, NNER, and FFER of 4.1, 7.61, and 6.48 were observed in the parametric range. Jain et al. [21] studied the thermo-hydraulic performance of an RSAHD with single and double discrete arc ribs on the AP. The authors reported the highest TER of 1.6 and 1.56 for the double and single discrete arc ribbed RSAHD at $e\xaf=0.034$, respectively. Nidhul et al. [22] explained the thermo-hydraulic performance of an RSAHD with discrete multiple inclined baffles. The authors obtained a maximum TER of 2.69 within the parametric range. Haldar et al. [23] examined the thermo-hydraulic performance analysis of an RSAHD with wavy surfaces on the surface of the AP for Re = 12,000, and authors reported a maximum TER of 1.96 for the rib with $P\xaf=17.41$. Nanjundappa [24] performed a study to predict the thermo-hydraulic characteristics of an RSAHD with cube-shaped rib-turbulators on the surface of the AP. The authors reported a maximum NNER and FFER of 1.83 and 3.49 within the parametric range. Bezbaruah et al. [25] evaluated the thermo-hydraulic performance of an RSAHD with staggered conical vortex turbulators. A maximum TER of 2.04 was observed for the rib-turbulator with $P\xaf=2.67$ and $e\xaf=0.17$.

Although many studies have been carried out on the thermal performance of the RSAHD [26–30], the investigations on the triangular solar air heater ducts (TSAHD) with artificial roughness are limited. This paragraph describes a few works on the TSAHD. Recently, the TSAHD has been used as the most effective heat transmission device than the RSAHD [31]. The friction penalty involved across the TSAHD is much lower, ensuring the TER enhancement. Bharadwaj et al. [32] developed an experimental model of circular ribbed TSAHD. The authors analyzed the effect of circular ribs on the TER of the TSAHD. Various heat transmission parameters like $P\xaf$, $e\xaf$, attack angle, and Re were considered for the study. They reported a maximum NNER and FFER of 3.13 and 2.96 for $P\xaf=12$ and $e\xaf=0.043$ with an attack angle of 60 deg. Kumar and Goel [33] numerically explained the thermo-hydraulic characteristics of a triangular SAHD with various ribs. The authors obtained a maximum TER of 2.75 for the SAHD with forward chamfered rectangular cross-sectioned ribs (aspect ratio of 2) on the surface of the AP. Kumar and Goel [34] explored the thermo-hydraulic performance of a TSAHD having hemispherical cavities on the AP. The authors reported a maximum heat transmission of 5.33 times that of smooth SAHD. Also, an optimum TER of 3.48 was attained for the non-dimensional pitch of 11. Kumar et al. [35] numerically explored the thermo-hydraulic performance of a TSAHD with forward facing chamfered rectangular ribs. Authors retain a maximum NNER and FFER of 2.64 and 3.14 for the rib aspect ratio of 1.2, respectively. Mahanand and Senapati [36] performed a numerical analysis to study the thermo-hydraulic performance a pentagonal ribbed TSAHD. Authors attained a maximum NNER and FFER of 2.01 and 2.97 for $P\xaf=10$ and $e\xaf=0.05$. Jain et al. [28] numerically explored the thermo-hydraulic performance of a broken inclined ribbed TSAHD. The authors reported a maximum NNER of 2.16 for the relative gap pitch and relative gap width of 0.25 and 1, respectively. This literature expresses that fixing turbulators on the surface of AP significantly strengthens heat transmission through the TSAHD. Limited works have been reported that analyze the triangular SAHD's heat transfer and friction factor characteristics.

After a peer review of the literature, it can be concluded that very few experimental investigations on triangular SAHD are performed using artificial roughness. In many investigations, researchers have concluded that the continuous transverse ribs are thermo-hydraulically more efficient when adequate gaps are provided [13,37–40]. The literature has not reported the provision of gaps to the transverse rib turbulators in the triangular SAHD. Discreteness of the roughness elements on the AP results in the growth of the secondary flow formation [15,41–43]. In most of the previous works, uniform gaps are provided for each rib [11,44–46]. However, for the present study, the uniformity of the gaps is maintained at each odd and even number of ribs, as recommended by Ref. [47]. Providing gaps in rib turbulators is also advantageous as it minimizes the material cost of the ribs. Table 1 presents few literature that highlight the provision of gaps in the rib turbulators. The following points motivate us for the present experimental investigations.

Limited works have been reported that analyze the triangular SAHD's heat transfer and friction factor characteristics.

A few experimental investigations on triangular SAHD are performed using artificial roughness.

The literature has not reported the provision of gaps to the transverse ribs in the triangular SAHD.

Non-uniform gaps are provided to the ribs (two gaps of 0.01 m are provided to each odd number rib (primary ribs), whereas three gaps are provided to each even number rib (secondary ribs)).

Authors | Flow passage | Types of rib turbulators |
---|---|---|

Singh et al. [13] | Rectangular | Multiple broken transverse ribs and square wave rib turbulators |

Rathor and Aharwal [15] | Rectangular | Inclined rib-turbulators with gaps |

Bharadwaj et al. [32] | Triangular | Continuously inclined rib turbulators |

Lau et al. [47] | Square | Discrete rib turbulators |

Jain et al. [48] | Rectangular | Arc shape ribs with multiple gaps |

Authors | Flow passage | Types of rib turbulators |
---|---|---|

Singh et al. [13] | Rectangular | Multiple broken transverse ribs and square wave rib turbulators |

Rathor and Aharwal [15] | Rectangular | Inclined rib-turbulators with gaps |

Bharadwaj et al. [32] | Triangular | Continuously inclined rib turbulators |

Lau et al. [47] | Square | Discrete rib turbulators |

Jain et al. [48] | Rectangular | Arc shape ribs with multiple gaps |

The present study explores the influence of transverse ribs with gaps on the thermo-hydraulic performance of the TSAHD. Heat transmission parameters such as $P\xaf$, $e\xaf$, and Re are varied in the range of 4.88–19.5, 0.021–0.044, and 4000–18,000, respectively. Two gaps of 0.01 m are provided to each odd number rib (primary ribs), whereas three gaps are provided to each even number rib (secondary ribs). Non-dimensional primary width (*w*_{1}/*W*) and non-dimensional secondary width (*w*_{2}/*W*) are kept constant at 0.29 and 0.4, respectively. This investigation has its novelty on the basis of the flow passage and placement of rib turbulators. The objective of the present experimental analysis can be understood by the following points:

The effect of $P\xaf$, $e\xaf$, and Re on the thermo-hydraulic performance of the TSAHD having transverse ribs with gaps.

The optimum configuration has been reviewed for maximum TER in the parametric range.

Generalized correlations for the Nusselt number and friction factor have been formulated.

## 2 Experimental Setup and Procedure

The experimentation and analysis of the SAHD transverse ribs with gaps are carried out in an open indoor platform. The experimental setup consists of an equilateral triangular duct kept on the stationary table with the help of a rectangular support channel, as shown in Fig. 2. The duct is kept horizontally, and the length of the duct is 2.3 m. Along the duct length, it is split into three parts: entrance, heated, and exit. The length of the heated part is kept at 1 m, whereas the entrance and exit parts are decided as per the ASHRAE standard and kept at 0.65 m each. The height of the TSAHD is considered 0.138 m [32]. The plenum chamber connects the duct with the blower with the help of a fiber pipe of 2 m. The schematic of the front view for the TSAHD and the dimension of each part is shown in Fig. 3. For the study, 1 horse power (HP) blower is used to generate the forced convection through the TSAHD. To avoid the top and bottom losses, the space between the triangular duct and rectangular support channel is provided with glass wool. An electrical heater has been fabricated using nichrome wire on the surface of an asbestos plate of 0.005 m. A mica sheet is provided over the asbestos plate to generate a uniform heat flux of 1000 W/m^{2} all over the asbestos sheet. A constant voltage of electrical energy has been supplied to the heater using a variac transformer. The heated section's upper wall is the AP made of aluminum has been placed in between the heater and triangular duct. The AP is subjected to several rib turbulators on its surface. The flow velocity for different Reynolds numbers has been measured by fitting the hot-wire anemometer at the entrance section, which is normal to the flow. It also measures the inlet temperature of the air before entering the heated section. The flow has been regulated by control valves fitted in the pipeline. Resistance temperature detector (RTD) sensors (resolution: 0.1 °C) are connected to the data logger that shows inlet and outlet temperature whereas K-type thermocouples (resolution: 1 °C) are used to measure the temperature of the AP [49]. Two up-down insulated plates are attached at the departure section for the proper mixing of the fluid. The outlet temperature of the air is the average of the mixed fluid at the departure section. The placing of the thermocouples on the surface of AP is shown in Fig. 4(a). The absorber plate temperature is calculated as the average temperature of the 12 thermocouples inserted into it. The pressure drop across the heated section has been measured with the help of a differential pressure meter (DPM) (resolution: ±0.4 Pa). Three RTD-type thermocouples are inserted at the duct's departure section to measure the air outlet temperature. The hot outlet air is released out of the indoor room, which does not affect the inlet air temperature.

Along with smooth AP, seven ribbed AP are fabricated with various non-dimensional pitches (*P*/*e* or $P\xaf$) and heights (*e*/*D _{h}* or $e\xaf$) to run the experiment for the present study. The transverse ribs with gaps are attached to the surface of the AP in a staggered manner, as shown in Fig. 4(b). The characteristics of the various components and instruments are presented in Table 2 and the pictorial figures are presented in Fig. 5. The system is run for more than 2 h to maintain a steady-state. After the steady-state is attained, data collection for the thermo-hydraulic performance of the staggered ribbed TSAHD starts. The range of Re is kept in the range of 4000–18,000. The piping and instrumentation diagram of the experimental system is displayed in Fig. 6. The following are the collective parameters to analyze the thermo-hydraulic characteristics:

Inlet-flow velocity and temperature using hot-wire anemometer.

The average temperature of the AP using 12 different locations.

The outlet temperature of the air is measured at the exit section of the duct.

Pressure drop across the heated section.

S. no. | Instrument details | Remarks |
---|---|---|

01 | Hot-wire anemometer Model: TAM-05SPD Range: 0–30 m/s Resolution: 0.001 Accuracy: ±3% | This device is used to measure the velocity of the air at the entry section of the TSAHD. It is mounted at the entry section of the TSAHD. |

02 | Thermocouple Model: k-type Range: 0–250 °C Accuracy: ±1% | A k-type of thermocouple is used to measure the temperature of the AP. There are 12 thermocouples inserted in the surface of the AP in four sections (three each). |

03 | RTD sensor Model: RTD (PT100) Range: −40 °C to 200 °C Accuracy: ±0.5% | RTD sensors are used to measure the air temperature at the inlet and outlet sections of the TSAHD. There are six sensors inserted into the flow in the inlet and outlet sections (three each). |

04 | Differential pressure meter Model: DPT-priima-AZ-D Range: −50 Pa to 50 Pa Accuracy: ±0.4% | DPM is used to measure the pressure difference across the test section of the TSAHD. It is mounted vertically on the wall of the TSAHD. |

05 | Variable transformer (Variac) Model: SUN006 Range: 0–270 V | The variable transformer maintains a constant solar insolation of 1000 W/m^{2} (heat flux) with the help of a volt meter and ammeter. |

S. no. | Instrument details | Remarks |
---|---|---|

01 | Hot-wire anemometer Model: TAM-05SPD Range: 0–30 m/s Resolution: 0.001 Accuracy: ±3% | This device is used to measure the velocity of the air at the entry section of the TSAHD. It is mounted at the entry section of the TSAHD. |

02 | Thermocouple Model: k-type Range: 0–250 °C Accuracy: ±1% | A k-type of thermocouple is used to measure the temperature of the AP. There are 12 thermocouples inserted in the surface of the AP in four sections (three each). |

03 | RTD sensor Model: RTD (PT100) Range: −40 °C to 200 °C Accuracy: ±0.5% | RTD sensors are used to measure the air temperature at the inlet and outlet sections of the TSAHD. There are six sensors inserted into the flow in the inlet and outlet sections (three each). |

04 | Differential pressure meter Model: DPT-priima-AZ-D Range: −50 Pa to 50 Pa Accuracy: ±0.4% | DPM is used to measure the pressure difference across the test section of the TSAHD. It is mounted vertically on the wall of the TSAHD. |

05 | Variable transformer (Variac) Model: SUN006 Range: 0–270 V | The variable transformer maintains a constant solar insolation of 1000 W/m^{2} (heat flux) with the help of a volt meter and ammeter. |

## 3 Geometrical Details and Working Parameters

The experimental investigation is carried out on an indoor platform. Figure 7 depicts the pictorial setup of the different measuring devices and accessories involved in the experiment. Table 3 lists the variable parameters considered for this experimental investigation.

S. no. | Basic dimensions of the TSAHD | Symbol | Range/Values |
---|---|---|---|

Parameters | |||

01 | Entry section | L_{e} | 0.65 m |

02 | Heated section | L_{h} | 1.0 m |

03 | Departure section | L_{d} | 0.65 m |

04 | Height | H | 0.138 m |

05 | Apex angle | α | $60deg$ |

Roughness parameters | |||

01 | Non-dimensional pitch | P/e or $P\xaf$ | 4.87–20 (seven values) |

02 | Non-dimensional height | e/D or $e\xaf$_{h} | 0.021–0.044 (four values) |

03 | Non-dimensional primary rib width | w_{1}/W | 0.29 |

04 | Non-dimensional secondary rib width | w_{2}/W | 0.4 |

Flow parameters | |||

01 | Reynolds number | Re | 4000–18,000 |

S. no. | Basic dimensions of the TSAHD | Symbol | Range/Values |
---|---|---|---|

Parameters | |||

01 | Entry section | L_{e} | 0.65 m |

02 | Heated section | L_{h} | 1.0 m |

03 | Departure section | L_{d} | 0.65 m |

04 | Height | H | 0.138 m |

05 | Apex angle | α | $60deg$ |

Roughness parameters | |||

01 | Non-dimensional pitch | P/e or $P\xaf$ | 4.87–20 (seven values) |

02 | Non-dimensional height | e/D or $e\xaf$_{h} | 0.021–0.044 (four values) |

03 | Non-dimensional primary rib width | w_{1}/W | 0.29 |

04 | Non-dimensional secondary rib width | w_{2}/W | 0.4 |

Flow parameters | |||

01 | Reynolds number | Re | 4000–18,000 |

## 4 Thermo-Hydraulic Parameters

For the present analysis, to simplify the analysis and avoid the minor losses through the TSAHD, a few assumptions are made as given as follows:

Steady-state operation.

The fluid is incompressible.

Only forced convection can account for the heat transfer.

The test section of the TSAHD is perfectly insulated, i.e., no heat loss through the test section.

Solar insolation of 1000 W/m

^{2}(heat flux) is uniformly distributed on the surface of AP.

*Q*

_{out}) contained by the outlet air is the sum of the energy contained in inlet air (

*Q*

_{in}) and energy received by the air from the hot AP (

*Q*

_{useful}) due to convective heat transfer, as shown in Fig. 8.

*D*

_{h}is the hydraulic diameter of the TSAHD and is calculated as per Eq. (4) [11].

*T*

_{p}and

*T*

_{m}are absorber plate temperature and fluid bulk mean temperature. The value of the

*T*

_{p}has been calculated by averaging the temperature of the 12 locations over the plate surface as given in Eq. (6) and the

*T*

_{m}has been calculated as per Eq. (7) [51].

Duct length (m) | 0.7 | 1 | 1.3 | 1.6 |

Plate temperature (°C) | 70 | 74.5 | 73 | 71 |

Duct length (m) | 0.7 | 1 | 1.3 | 1.6 |

Plate temperature (°C) | 70 | 74.5 | 73 | 71 |

A value of TER greater than one indicates that the use of artificial roughness is effective, and it also facilitates comparing the performance of various surface roughness to pick the optimal one.

## 5 Results and Discussion

The experimentation has been done for TSAHD with staggered ribs to study the performance in terms of the TER. At varied flow velocities, absorber plates having transverse ribs with gaps of different non-dimensional pitches and heights are evaluated to optimize the optimum roughness configuration within the investigated range.

### 5.1 Testing of Smooth TSAHD.

_{s}) validation, Disttus–Boelter correlation [53] (as given in Eq. (12)), and Bharadwaj et al. [32] are considered.

From Fig. 9(a), it is observed that the calculated Nu_{s} from present experimentation has a maximum variation of ±7% and ±9% with that of the Disttus–Boelter correlation [53] and Bharadwaj et al. [32]. Additionally, a maximum deviation of ±6% and ±8% is found in the values of *f*_{s}, as it is compared with the modified Blasius equation [54] and Bharadwaj et al. [32] as shown in Fig. 9(b).

### 5.2 Heat Transmission Evaluation.

The influence of experimental parameters (non-dimensional pitch, non-dimensional height, and Reynolds number) on heat transmission through a TSAHD having transverse ribs with gaps has been discussed in this section. Figure 10 illustrates the variation in Nu_{r} for the various flow velocity (Re) at different non-dimensional pitches and non-dimensional heights. High Nu_{r} represents the higher heat removal from the AP. From the figure, it is seen that the heat transmission enhances with the flow velocity. As the flow velocity increases, the accelerated secondary flow penetrates the viscous sublayers, which in turn suppression of thickness of it and causes more excellent heat transmission from AP. Also, at higher Re, the turbulent kinetic energy in the inter-rib region increases, and this leads to higher Nu_{r}. Besides, at lower flow velocity, due to the recirculation of flow behind the ribs, the impinging length of the flow in the inter-rib regions gets shorter, which causes the formation of a poor heat transmission zone. The shorter the length of the impingement, the lower the heat transmission. At significantly lower flow velocity (Re = 4000), the enhancement in heat transfer due to ribbed surfaces is insignificant. For lower flow velocity, turbulent intensity carried by the vortices around the rib turbulators is comparatively the same for all the ribbed surfaces considered. The minimum Nu_{r} is addressed for the Re = 4000. It is also noticed that the heat transmission due to the ribbed surface is much higher than the smooth surface. For all the studied cases, the maximum heat transmission is found for Re = 18,000.

Figure 10 depicts the influence of non-dimensional pitch on Nu_{r} at different Re. Within the range of non-dimensional height from 0.021 to 0.044, it is revealed that as the non-dimensional pitch decreases, heat transmission enhances, i.e., Nu_{r} increases. At the lower non-dimensional pitch, the number of ribs is more on the AP, causing more turbulence in the flow, and the number of impingements in the inter-rib region rises. More impingement represents more contact between the hot surface and air, leading to high heat transmission. The ribs get placed very close to each other for an extremely lower value of non-dimensional pitch. This causes the absence of impingement of the flow in the inter-rib regions, reducing heat transmission. From Fig. 11(a), it is illustrated that the maximum heat transmission occurs for the non-dimensional pitch of 9.76. Nu_{r} reaches its highest value at a non-dimensional pitch of 9.76 and non-dimensional height of 0.04 at an Re of 18,000, and then it declines as the non-dimensional pitch rises. For significantly lower pitch, the distance between the ribs becomes short. This may cause the absence of the impingement of the secondary flow into the inter-rib regions, and heat transmission decreases. Consequently, for a significantly higher pitch, the distance between the consecutive rib turbulators becomes long, so there may be a chance of redevelopment of the viscous layer-layer in the inter-rib regions [50], causing low heat transfer. The impact of non-dimensional height on heat transmission is depicted in Fig. 11(b). According to the figure, for various non-dimensional heights, Nu_{r} steadily rises as Re increases. As the non-dimensional height grows, so does the Nusselt number, and the highest value is seen for the non-dimensional height of 0.044. At the higher non-dimensional height, the secondary flow developed strongly impinges at the inter-rib regions, increasing the length of impingement and enhancing heat transmission. Within the parametric range, the Nu_{r} rises steadily with non-dimensional height and gets its maximum for a non-dimensional height of 0.044. Consequently, it is addressed that a minor modification strongly influences the heat transmission in the non-dimensional pitch and height of the transverse ribs with gaps, as well as changes in the flow velocity of air. The analysis concludes that at Re = 18,000, the highest NNER of 3.15 has been found for the non-dimensional pitch and height of 9.76 and 0.044, respectively.

### 5.3 Friction Factor Evaluation.

Figures 12(a) and 12(b) depict the variation of the friction factor with Re in the parametric range of investigation. From Fig. 12(a), it is depicted that *f*_{r} decreases with increasing Re for various non-dimensional pitches. This is because the boundary layer becomes thinner with increasing velocity. When the fully developed fluid enters the heated section, it strikes the ribs, due to which the secondary flow accelerates and impinges on the heated surface. Accelerated secondary flow destroys the viscous sublayers and enhances heat transmission through the TSAHD. For entire cases, the maximum *f*_{r} is found at Re = 4000. Also, from the figure, it is revealed that the friction factor obtained from ribbed TSAHD is higher than the smooth one. The presence of ribs encourages the higher pressure drop across the heated section. Although the pressure drop across the test section increases with the flow velocity increase, the friction factor maintains indirect proportionality with the square power of the velocity, due to which the friction factor decreases as given in Eq. (10).

For various values of Re, Fig. 13(a) displays the average *f*_{r} as a function of non-dimensional pitch. From the figure, it is experimentally demonstrated that *f*_{r} declines as the non-dimensional pitch increases and vice versa. As the pitch value decreases, the distance between the ribs also decreases. Because of this, more obstruction appears in the flow along the duct length. At low values of non-dimensional pitch, the pressure drop across the heated section rises and causes higher *f*_{r}. However, the distance between the consecutive ribs is greater for a higher pitch, ensuring fewer obstructions in the flow path and lower friction losses. The experimental results show that the maximum friction factor is found for transverse ribs with gaps having a non-dimensional pitch of 4.88. The *f*_{r} versus non-dimensional height graph is displayed in Fig. 13(b). The *f*_{r} is found to rise with non-dimensional height and Re for a constant non-dimensional pitch. This is because increasing the non-dimensional height develops a bigger recirculation zone, which impedes mainstream air movement. From the experimental results, a maximum FFER of 3.88 is found for the transverse ribs with gaps having non-dimensional pitch and height of 4.88 and 0.044 at Re = 4000, respectively.

According to the preceding explanation, Nu_{r} and *f*_{r} are greatly influenced by the non-dimensional pitch and height of the transverse ribs with gaps and Reynolds number. Tables 5 and 6 present all the experimentally calculated NNER and FFER in the parametric range. From Table 5, it is observed that the NNER increases with an increase in Re and attains a maximum at Re = 18,000. A maximum NNER of 3.146 is observed for the rib turbulators with $P\xaf=9.76$ and $e\xaf=0.044$ at Re = 18,000. On the other hand, from Table 6 it is noticed that the FFER decreases with increase in Re and encountered maximum at Re = 4000. A maximum FFER of 3.884 is observed for the rib turbulators with $P\xaf=4.88$ and $e\xaf=0.044$ at Re = 4000.

S. no. | Roughness parameters | Flow parameters (Re) | |||||
---|---|---|---|---|---|---|---|

P/e or $P\xaf$ | e/D or $e\xaf$_{h} | 4 × 10^{3} | 8 × 10^{3} | 12 × 10^{3} | 15 × 10^{3} | 18 × 10^{3} | |

1 | 4.88 | 0.044 | 1.867 | 2.335 | 2.58 | 2.879 | 2.93 |

2 | 9.76 | 0.044 | 1.949 | 2.434 | 2.805 | 3.088 | 3.146 |

3 | 14.63 | 0.044 | 1.723 | 2.216 | 2.439 | 2.814 | 2.827 |

4 | 19.51 | 0.044 | 1.543 | 1.85 | 2.107 | 2.324 | 2.346 |

5 | 12.5 | 0.034 | 1.818 | 2.385 | 2.686 | 2.995 | 3.057 |

6 | 15.38 | 0.028 | 1.543 | 2.059 | 2.354 | 2.631 | 2.607 |

7 | 20 | 0.021 | 1.475 | 1.758 | 2.088 | 2.199 | 2.176 |

S. no. | Roughness parameters | Flow parameters (Re) | |||||
---|---|---|---|---|---|---|---|

P/e or $P\xaf$ | e/D or $e\xaf$_{h} | 4 × 10^{3} | 8 × 10^{3} | 12 × 10^{3} | 15 × 10^{3} | 18 × 10^{3} | |

1 | 4.88 | 0.044 | 1.867 | 2.335 | 2.58 | 2.879 | 2.93 |

2 | 9.76 | 0.044 | 1.949 | 2.434 | 2.805 | 3.088 | 3.146 |

3 | 14.63 | 0.044 | 1.723 | 2.216 | 2.439 | 2.814 | 2.827 |

4 | 19.51 | 0.044 | 1.543 | 1.85 | 2.107 | 2.324 | 2.346 |

5 | 12.5 | 0.034 | 1.818 | 2.385 | 2.686 | 2.995 | 3.057 |

6 | 15.38 | 0.028 | 1.543 | 2.059 | 2.354 | 2.631 | 2.607 |

7 | 20 | 0.021 | 1.475 | 1.758 | 2.088 | 2.199 | 2.176 |

Note: The bold value represent the maximum NNER in the parametric range.

S. no. | Roughness parameters | Flow parameters (Re) | |||||
---|---|---|---|---|---|---|---|

P/e or $P\xaf$ | e/D or $e\xaf$_{h} | 4 × 10^{3} | 8 × 10^{3} | 12 × 10^{3} | 15 × 10^{3} | 18 × 10^{3} | |

1 | 4.88 | 0.044 | 3.884 | 3.496 | 2.928 | 2.669 | 2.615 |

2 | 9.76 | 0.044 | 3.75 | 3.401 | 3.038 | 2.796 | 2.513 |

3 | 14.63 | 0.044 | 3.572 | 3.165 | 2.69 | 2.418 | 2.43 |

4 | 19.51 | 0.044 | 3.368 | 3.067 | 2.569 | 2.311 | 2.271 |

5 | 12.5 | 0.034 | 3.654 | 3.362 | 2.896 | 2.669 | 2.513 |

6 | 15.38 | 0.028 | 3.425 | 2.914 | 2.424 | 2.247 | 2.19 |

7 | 20 | 0.021 | 3.167 | 2.86 | 2.46 | 2.196 | 2.123 |

S. no. | Roughness parameters | Flow parameters (Re) | |||||
---|---|---|---|---|---|---|---|

P/e or $P\xaf$ | e/D or $e\xaf$_{h} | 4 × 10^{3} | 8 × 10^{3} | 12 × 10^{3} | 15 × 10^{3} | 18 × 10^{3} | |

1 | 4.88 | 0.044 | 3.884 | 3.496 | 2.928 | 2.669 | 2.615 |

2 | 9.76 | 0.044 | 3.75 | 3.401 | 3.038 | 2.796 | 2.513 |

3 | 14.63 | 0.044 | 3.572 | 3.165 | 2.69 | 2.418 | 2.43 |

4 | 19.51 | 0.044 | 3.368 | 3.067 | 2.569 | 2.311 | 2.271 |

5 | 12.5 | 0.034 | 3.654 | 3.362 | 2.896 | 2.669 | 2.513 |

6 | 15.38 | 0.028 | 3.425 | 2.914 | 2.424 | 2.247 | 2.19 |

7 | 20 | 0.021 | 3.167 | 2.86 | 2.46 | 2.196 | 2.123 |

Note: The bold value represent the maximum FFER in the parametric range.

### 5.4 Thermo-Hydraulic Performance Evaluation.

The formation of secondary flow due to the arrangement of transverse ribs with gaps on the surface of the AP causes augmentation in heat transmission through the TSAHD by suppressing the viscous layer thickness. The accelerated secondary flow impinges on the hot surface in inter-rib regions and minimizes thermal resistance. The interaction due to ribs also significantly raises the pressure drop across the heated section. So it is necessary to adopt the perfect roughness configuration that provides maximum heat transmission and minimum friction loss. This is accomplished by analyzing the functional parameters that describe heat transmission and friction losses. According to the present investigation, the highest NNER has been achieved for the transverse ribs with gaps having non-dimensional pitch and height of 9.76 and 0.044, respectively Re = 18,000. In the same way, the highest NNER has been addressed for the transverse ribs with gaps having non-dimensional pitch and height of 4.88 and 0.044, respectively, at Re = 4000. To analyze the heat transmission and friction losses simultaneously, a thermo-hydraulic parameter has been used which is called TER proposed by Webb and Eckert [52].

The variation of TER with various Re at different non-dimensional pitches and heights has been depicted in Figs. 14(a) and 14(b). The figure shows that for all cases, the TER increases with the flow velocity. The insertion of transverse ribs with gaps causes more heat transmission from the surface of the AP to air, which is more than the smooth AP. This is because an increase in heat transmission has a much more significant effect than an increase in friction penalty, so the TER parameter has a slightly larger value. It is concluded that a maximum TER of 2.31 has been found for the transverse ribs with gaps having non-dimensional pitch and heights of 9.76 and 0.044 at Re = 18,000, respectively. The comparison of the present results with previous works is presented in Table 7. It is observed that the present study has a good TER than Singh et al. [13] and Rathor and Aharwal [15].

Authors | Flow passage | Roughness | Optimum roughness | MaximumTER |
---|---|---|---|---|

Singh et al. [13] | Rectangular | Multiple broken transverse ribs | P/e = 10, e/D = 0.043, _{h}W/w = 7, Re = 15,000 | 2.1 |

Rathor and Aharwal [15] | Rectangular | Inclined ribs with gaps | P/e = 10, e/D = 0.0303, _{h}d/W = 0.25, α = 60 deg, r/g = 2, g/e = 1, p′/P = 0.6, Re = 12,438 | 2.06 |

Jain et al. [48] | Rectangular | Arc shape ribs with multiple gaps | P/e = 10, e/D = 0.043, _{h}g/e = 4, N_{g} = 3, Re = 18,000 | 2.75 |

Present study | Triangular | Transverse ribs with gaps | P/e = 10, e/D = 0.043, _{h}w_{1}/W = 0.29, w_{2}/W = 0.4, Re = 18,000 | 2.31 |

Authors | Flow passage | Roughness | Optimum roughness | MaximumTER |
---|---|---|---|---|

Singh et al. [13] | Rectangular | Multiple broken transverse ribs | P/e = 10, e/D = 0.043, _{h}W/w = 7, Re = 15,000 | 2.1 |

Rathor and Aharwal [15] | Rectangular | Inclined ribs with gaps | P/e = 10, e/D = 0.0303, _{h}d/W = 0.25, α = 60 deg, r/g = 2, g/e = 1, p′/P = 0.6, Re = 12,438 | 2.06 |

Jain et al. [48] | Rectangular | Arc shape ribs with multiple gaps | P/e = 10, e/D = 0.043, _{h}g/e = 4, N_{g} = 3, Re = 18,000 | 2.75 |

Present study | Triangular | Transverse ribs with gaps | P/e = 10, e/D = 0.043, _{h}w_{1}/W = 0.29, w_{2}/W = 0.4, Re = 18,000 | 2.31 |

## 6 Correlations for Experimental Nu_{r} and *f*_{r}

_{r}and

*f*

_{r}are strongly dependent on the non-dimensional pitch (

*P*/

*e*or $P\xaf$), non-dimensional height (

*e*/

*D*or $e\xaf$), non-dimensional primary rib width (

_{h}*W*/

*w*

_{1}), non-dimensional secondary rib width (

*W*/

*w*

_{2}), and Reynolds number. However, the

*W*/

*w*

_{1}and

*W*/

*w*

_{2}are kept constant throughout the experimental analysis. So, mathematically functional relationships for Nu

_{r}and

*f*

_{r}are written as per Eqs. (14) and (15)

The correlations for the Nu_{r} and *f*_{r} is derived using the non-linear regression technique. There are 69 experimental data points considered for the formation of the correlation of the Nu_{r} and *f*_{r}, as shown in Fig. 15. All the experimental data are fitted to a standard non-linear generalized equation in Eqs. (16) and (17). From Fig. 15, it is depicted that most of the data points are lying in the range with a deviation of ±12 and ±8% for the Nu_{r} and *f*_{r}, respectively.

*P*/

*e*≤ 20, 0.021 ≤

*e*/

*D*

_{h}≤ 0.044, 4 × 10

^{3}≤ Re ≤ 18 × 10

^{3}

## 7 Conclusion

Heat transmission and flow friction of an SAHD are affected by the location of the ribs on the absorber plate. The present study focuses on the thermo-hydraulic performance of a TSAHD having transverse ribs with gaps. The effect of roughness arrangement in terms of non-dimensional pitch and height on TER as a function of Nusselt number and friction factor. In this study, the transverse ribs with gaps are configured in seven distinct ways to conduct the experimental analysis. The most important results obtained from the present research are briefly outlined as follows:

Positioning the transverse ribs with gaps on the surface of AP, majorly influences the heat transmission and friction penalty.

In each of the experiments, Nu

_{r}monotonously increases with an increase in Re. Similarly,*f*_{r}declines as Re rises.A maximum NNER of 3.15 is obtained for the configuration with non-dimensional pitch and height of 9.76 and 0.044 at Re = 18,000, respectively.

The highest FFER of 3.89 encountered roughness having non-dimensional pitch and height of 4.88 and 0.044 at Re = 4000, respectively.

Within the parametric range, a maximum TER of 2.31 is achieved for the non-dimensional pitch and height of 9.76 and 0.044, respectively.

The correlation for the Nu

_{r}and*f*_{r}is formulated as a function of non-dimensional pitch, non-dimensional height, and Re that satisfy within ±12% and ±8% errors.

Although providing transverse ribs with gaps encounters frictional losses, it is exciting in terms of enhancement of heat transmission. Also, this work can be extended to the comparative evaluation of the thermo-hydraulic performance by inserting the rib elements on AP in an inclined manner with providing gaps. Experimenting with transverse rib turbulators with gaps to explore their behavior in terms of Reynolds number, non-dimensional pitch, and non-dimensional height could be helpful in the future.

## 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*=rib height, m

*h*=heat transfer coefficient, W/m

^{2}-K*k*=thermal conductivity of air, W/m-K

*H*=height of TSAHD, m

*P*=rib pitch, m

*V*=velocity of air, m/s

*W*=width of the TSAHD

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

*A*_{c}=cross-sectional of air passage, m

^{2}*A*_{p}=area of AP, m

^{2}*C*_{p}=specific heat capacity, J/(kg · K)

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

*L*_{d}=length of departure section, m

*L*_{e}=length of entry section, m

*L*_{h}=length of heated section, m

*P*_{a}=atmospheric pressure, Pa

*P*_{w}=wetted perimeter, m

*T*_{i}=inlet temperature of air, K

*T*_{f}=average temperature of fluid, K

*T*_{o}=outlet temperature of air, K

*T*_{p}=average temperature of AP, K

- Δ
*P*= pressure drop in heated section, Pa

### Non-Dimensional Numbers

- $e\xaf$ =
non-dimensional height

- $P\xaf$ =
non-dimensional pitch

*f*_{r}=average friction factor (roughened duct)

*f*_{s}=average friction factor (smooth duct)

*f*_{r}/*f*_{s}=friction factor enhancement ratio

*w*_{1}/*W*=non-dimensional primary width

*w*_{2}/*W*=non-dimensional secondary width

- Pr =
Prandtl number

- Re =
Reynolds number

- Nu
_{r}= average Nusselt number (roughened duct)

- Nu
_{s}= average Nusselt number (smooth duct)

- Nu
_{r}/Nu_{s}= Nusselt number enhancement ratio

### Greek Symbols

### Subscripts

### Abbreviations

## Appendix: Uncertainty Analysis

Every experimentation somehow gets affected by physical situations, like environmental conditions, the state of the measuring device, and measurement accuracy. Thus, it is crucial to calculate the maximum margin of error associated with experimental results. For the present investigation, Moffat [55] uncertainty method is employed to evaluate the uncertainty of the different heat transmission parameters. The estimation of the uncertainty is carried out by following steps:

- Uncertainty in the area of AP (
*A*_{p}):$Ap=Lh\xd7W$$\delta ApAp=[(\delta LhLh)2+(\delta WW)2]0.5=[(0.91000)2+(0.065160)2]0.5=0.000987=0.0987%$ - Uncertainty in the area of air passage (
*A*_{c}):$Ac=0.5\xd7W\xd7H$$\delta AcAc=[(\delta HH)2+(\delta WW)2]0.5=[(0.085138.56)2+(0.065160)2]0.5=0.000735=0.0735%$ - Uncertainty in hydraulic diameter (
*D*_{h}):$\delta DhDh=[(\delta HH)2]0.5=[(0.085138.56)2]0.5=0.000613=0.0613%$ - Uncertainty in Reynolds number (Re):$\delta ApAp=[(\delta VV)2+(\delta \rho \rho )2+(\delta DhDh)2+(\delta \mu \mu )2]0.5=[(0.00151.92)2+(0.00745)2+(0.000613)2+(0.0013\xd710\u221251.7894\xd710\u22125)2]0.5=0.00755=0.755%$
- Uncertainty in density (
*ρ*)$\delta \rho \rho =[(\delta PaPa)2+(\delta ToTo)2]0.5=[(0.15101.25)2+(0.2129.2)2]0.5=0.00745=0.745%$ - Uncertainty in mass flowrate ($m\u02d9$)$\delta m\u02d9m\u02d9=[(\delta \rho \rho )2+(\delta VV)2+(\delta WW)2+(\delta HH)2]0.5=[(0.00745)2+(6.1\xd710\u22127)+(1.65\xd710\u22127)+(3.76\xd710\u22127)]0.5=0.00753=0.753%$
- Uncertainty in useful heat gain (
*Q*_{u})$\delta QuQu=[(\delta m\u02d9m\u02d9)2+(\delta CpCp)2+(\delta \Delta T\Delta T)2]0.5=[(0.00753)2+(1.61006)2+(0.0363.87)2]0.5=0.012=1.2%$ - Uncertainty in heat transmission coefficient (
*h*)$\delta hh=[(\delta QuQu)2+(\delta ApAp)2+(\delta \Delta Tf\Delta Tf)2]0.5=[(0.012)2+(0.00103)2+(0.7517)2]0.5=0.04575=4.575%$ - Uncertainty in Nusselt number (Nu
_{r})$Nur=h\xd7Dhk$$\delta NurNur=[(\delta hh)2+(\delta DhDh)2+(\delta kk)2]0.5=[(0.04575)2+(0.000613)2+(0.1\xd710\u221230.0242)2]0.5=0.04575=4.575%$ - Uncertainty in friction factor (
*f*_{r})$fr=(\Delta P/lh)\xd7Dh2\xd7\rho \xd7v2$$\delta frfr=[(\delta VV)2+(\delta \Delta P\Delta P)2+(\delta DhDh)2+(\delta \rho \rho )2+(\delta LhLh)2]0.5=[(6.1\xd710\u22127)+(0.0011)2+(0.000613)2+(0.00745)2+(8.1\xd710\u22127)]0.5=0.01254=1.254%$