Analytical investigation of exergetic performance on jet impingement solar air heater with multiple arc protrusion obstacles

Abstract

In this present work, an exergetic efficiency of the solar air heater (SAH) with jet impingement on multiple arc protrusion roughened absorber plate is analytically studied. It is examined out for roughness parameters of relative width ratio (W_P/W_AP) of 1–5, relative height ratio (e_P/d_P) of 0.5–2, relative pitch ratio (P_P/e_P) of 8–12 and angle of arc (α_a) of 35°–75°. Performance of SAH based on the second law has been effectively assessed by enumerating the useful exergy gain, irreversibility and losses occurred due to the inefficiency of the system components. Results indicate that the exergetic efficiency has been enhanced by 56.8%. Effects of roughness parameters on exergetic efficiency have been presented based on Reynolds number and temperature rise parameter. The optimized values of roughness parameters are identified, and the maximum exergetic efficiency is found to be 10.5%. Maximum exergetic destruction caused due to temperature difference between absorber plate and Sun is 65.2% at optimized conditions. In order to identify the optimum values of roughness parameters to attain the required temperature rise with maximum exergetic efficiency, design plots have been developed.

Introduction

Thermal energy conversion using solar air heater has been considered as a one of the most promising techniques that can be used for supplying a hot air for space heating, drying of agriculture and industrial products [1]. Properties of air such as low density, small volumetric heat capacity and lower thermal conductivity lead to lower heat transfer coefficient between absorbing surface and working fluid [2,3,4]. Consequently, it results in lower thermal efficiency of the system. To improve its performance, various configurations such as artificially roughened plate SAH with various types of roughness geometries, multi-pass flow configurations, flow combined with recycling, packed bed system have been analyzed experimentally and analytically [5,6,7]. Further, the researchers identified that air jet with impingement on smooth and artificially roughed plate enhances the value of heat transfer coefficient by creating more turbulence underneath the absorber plate. As a result of jet impingement, the performance of SAH was considerably improved by 21.2% [8, 9].

Jet plate design has major impact on thermal performance of SAH, the detailed investigation of its parameters is discussed in the literature. Chauhan and Thakur [10] studied the effect of jet plate design parameters on impinging jet solar air heater with smooth plate and concluded that there was enhancement in 2.67 times of heat transfer and 3.5 times of friction factor when compared it with conventional parallel pass SAH. They also evaluated the thermo-hydraulic performance for the identical configuration of SAH and reported that the maximum effective efficiency was 70% [11]. The performance of jet impingement SAH depends on channel spacing between absorber and jet plate and jet diameter ratio of 0.07 gives maximum thermal performance with cross flow condition [12, 13]. Further, SAH performance is improved by parallel pass jet impingement technique and concluded the maximum energy and exergy efficiency ratio is improved by 2.15 and 3.4 times at Bangalore climatic conditions [14]. Further, micro-jet impingement SAH is developed and its thermal efficiency ranges from 66 to 90% at laboratory testing conditions [15].

Even though the experiments and analytical studies provide performance of SAH, optimization is required for identifying the decisive parameters of the design. The jet plate design parameters are optimized using performance search index and Taguchi methods. It was concluded that a significant effect of stream wise pitch, span wise pitch and jet diameter on thermal performance of SAH is of 9.5%, 41.6% and 48.8%, respectively [16, 17]. A mathematical model developed for predicting the thermal efficiency of PV/T collectors with jet impingement with an accuracy of 10% when compared with experimental results [18].

Even though the thermal systems are analyzed based on energy basis, it is demonstrated that first law- and second law-based exergy analyses have been considered superior for comparison, optimization and performance evaluation. Exergy has been a useful tool to measure the maximum work potential and inefficiencies of individual components that causes the irreversibility [19,20,21]. The exergetic performance of the artificially roughened solar air heaters are compared by Gupta and Kaushik [22] and reported that artificial roughed designs have been suitable for air heaters with higher flow cross sectional area with lower Reynolds number.

Then, further investigations for various configurations of artificially roughened SAH are analytically analyzed and the roughness parameters are optimized to maximize the second law efficiency [23,24,25,26]. Bahrehmand et al. [27] analyzed eight different configurations of SAH based on exergetic performance and evaluated the effect of Reynolds number, channel depth and channel length. They also concluded that thin metal sheet SAH with double glass cover has better performance at lower Reynolds number (Re < 4000) and thin metal sheet SAH with single glass cover has highest exergetic efficiency at higher Re. The exergy-based optimum design and operating conditions have been studied for double-pass V-corrugated SAH by Hedayatizadeh et al. [28, 29] using genetic algorithm. Their analytical investigation shows that low inlet air temperatures, low area ratio and increasing number of glasses leads to higher exergetic efficiency and that the maximum exergy losses occurs at exergy destruction due to temperature difference between sun and absorber plate. Nadda et al. [30] experimentally formulated the correlation for Nusselt number and friction factor for solar air heater with jet impingement on multiple arc shape roughness with accuracy of ± 9% and ± 11%, respectively.

It is concluded that solar air heaters with jet impingement on smooth plate are investigated for various design and operating conditions and the results are reported in the form of thermal efficiency and thermo-hydraulic efficiency. On the other hand, artificially roughed solar air heaters are effectively designed and optimized using exergetic analysis. However, the exergetic performance optimization of solar air heaters with jet impingement on artificially roughened plates has not been investigated.

The objective of the present work is focused on analytically evaluating the exergetic performance of the solar air heater with jet impingement on multiple arc protrusion roughened absorber plate. Based on the loss-based exergetic performance evaluation, the values of surface roughness parameters such as relative width ratio (W_P/W_AP), relative height ratio (e_P/d_P), relative pitch ratio (P_P/e_P) and relative angle of arc (α_a) are to be optimized as a function of Reynolds number and temperature rise parameter. The design plots will be prepared in order to identify the optimum values of roughness parameters to attain the desired values of temperature rise. The results, obtained for solar air heater with jet impingement on smooth plate working at identical conditions, will also be compared with multiple arc protrusion roughened plate with jet impingement SAH.

Performance analysis of solar air heater with jet impingement

Thermal performance

The schematic diagram of the solar air heater is shown in Fig. 1a. The arrangements of roughness and their nomenclature and Jet plate configuration have been shown in Fig. 1b, c, d. The thermal performance of a solar air heater with jet impingement is calculated based on first law of thermodynamics which describes that the thermal efficiency of SAH is the ratio between instantaneous rates of useful energy gain over a time period to solar insolation incident on the surface over the same time period.

$$\eta_{{\text{th}}} = \frac{{Q_{{\text{u}}}}}{{IA_{{\text{c}}}}}$$

(1)

Fig. 1

a Schematic layout of SAH with jet impingement on multiple arc-shaped obstacles, b absorber plate, c detailed view of obstacles and d jet plate

The thermal efficiency can also be expressed in terms of heat removal factor (F_R) [31]

$$\eta_{\text{th}} = \frac{{mC_{\text{p}} (T_{\text{o}} - T_{\text{i}} )}}{{IA_{\text{c}} }} = F_{\text{R}} \left[ {\left( {\alpha \tau } \right) - U_{\text{L}} \left( {\frac{{T_{\text{i}} - T_{\text{a}} }}{I}} \right)} \right]$$

(2)

Thermo-hydraulic performance

Thermal performance of the solar air heater is improved due to impinging of higher velocity jets on artificially roughened absorber plate. However, this augmentation of heat transfer also consumes the additional pumping power. Therefore, the thermal energy equivalent of pumping power must be subtracted from the overall heat energy gain to calculate the actual performance of SAH by the following equation [32].

$$\eta_{\text{Eff}} = \frac{{Q_{\text{u}} }}{{I \times A_{\text{C}} }} - \left[ {\frac{{P_{\text{m}} }}{{I \times A_{\text{C}} \times C_{\text{j}} }}} \right]$$

(3)

In the above equation P_m represents the pumping power and C_j represents thermal energy conversion factor and it is equal to 0.2 [32]. The pumping power is calculated using the relation

$$P_{\text{m}} = \rho_{\text{a}} V^{3} f \, L(W + H)$$

(4)

$$\eta_{\text{Eff}} = \frac{{Q_{\text{u}} }}{{I \times A_{\text{C}} }} - \left[ {\frac{{\rho_{\text{a}} V^{3} f \, L(W + H)}}{{I \times A_{\text{C}} \times C_{\text{j}} }}} \right]$$

(5)

Exergy analysis

Exergy analysis of a solar air heater describes the maximum work potential that can be extracted from the energy input. It is a powerful tool for analyzing the system thermodynamically and designing an energy efficient SAH by minimizing exergy destruction or by maximum exergy delivery. By using the exergy analysis, the SAH components can effectively be optimized by considering the both useful energy gain and required pumping power. Figure 2 depicts the exergy balance for a solar air heater with jet impingement. Exergy balance for the solar air heater can be written as

$$\Sigma {\text{ EX}}_{\text{in}} -\Sigma {\text{ EX}}_{\text{out}} =\Sigma {\text{ EX}}_{\text{losses}}$$

(6)

Fig. 2

Exergy balance for SAH

The above equation can be re written as follows:

$${\text{ EX}}_{\text{heat}} - {\text{EX}}_{\text{Work}} + {\text{EX}}_{\text{mass in}} - {\text{EX}}_{\text{mass out}} = {\text{ EX}}_{\text{losses}}$$

(7)

Equation 7 can be expressed in general form of exergy balance is written as:

$$\, IA_{\text{c}} \left[ {1 - \frac{4}{3}\left( {\frac{{T_{\text{a}} }}{{T_{\text{sun}} }}} \right) + \frac{1}{3}\left( {\frac{{T_{\text{a}} }}{{T_{\text{sun}} }}} \right)^{4} } \right] - \dot{m}(h_{\text{out}} - h_{\text{in}} ) - T_{\text{a}} (S_{\text{out}} - S_{\text{in}} ) = {\text{ EX}}_{\text{losses}}$$

(8)

Further, the enthalpy and entropy terms in Eq. 8 is stated as follows:

$$\, IA_{c} \left[ {1 - \frac{4}{3}\left( {\frac{{T_{\text{a}} }}{{T_{\text{sun}} }}} \right) + \frac{1}{3}\left( {\frac{{T_{\text{a}} }}{{T_{\text{sun}} }}} \right)^{4} } \right] - \dot{m}C_{\text{p}} (T_{\text{out}} - T_{\text{in}} ) + \dot{m}C_{\text{p}} T_{\text{a}} \ln \left[ {\frac{{T_{\text{out}} }}{{T_{\text{in}} }}} \right] - \dot{m}RT_{\text{a}} \ln \left[ {\frac{{P_{\text{out}} }}{{P_{\text{in}} }}} \right] = {\text{ EX}}_{\text{losses}}$$

(9)

In this work, the exergy efficiency of the air heater is calculated based on the exergy losses to expose the significance of each cause of loss on the exergy efficiency. The various causes of exergy losses are listed below [32].

External exergy loss parameters

• (a) Optical exergy losses $({\text{EX}}_{{{\text{loss}} - {\text{opt}}}} )$.

• (b) Exergy losses due to heat transfer between SAH components and environment $({\text{EX}}_{\text{loss,Qloss}} ).$

Internal exergy loss parameters or Exergy destruction terms

• (a) Exergy losses due to fluid friction $({\text{EX}}_{\text{loss,friction}} ).$

• (b) Exergy losses due to absorption of solar irradiation by the absorber plate $({\text{EX}}_{{{\text{loss,T}}_{\text{ap}} , {\text{T}}_{\text{sun}} }} ).$

• (c) Exergy losses due to transfer of heat to working fluid $({\text{EX}}_{{{\text{loss,T}}_{\text{ap}} , {\text{T}}_{\text{f}} }} ).$

The exergy efficiency of the air heaters are calculated using the equation given below

$$\eta_{\text{ex}} = 1 - \frac{{\Sigma {\text{ EX}}_{\text{loss}} }}{{\Sigma {\text{ EX}}_{\text{inlet}} }}$$

(10)

The net rate of exergy supplied at inlet of the solar air heater is given by:

$$\Sigma {\text{ EX}}_{\text{inlet}} = IA_{\text{c}} \left[ {1 - \frac{4}{3}\left( {\frac{{T_{\text{a}} }}{{T_{\text{sun}} }}} \right) + \frac{1}{3}\left( {\frac{{T_{\text{a}} }}{{T_{\text{sun}} }}} \right)^{4} } \right]$$

(11)

The summation of exergy losses is expressed as:

$$\varSigma {\text{ EX}}_{\text{loss}} = ({\text{EX}}_{\text{loss,opt}} ) + ({\text{EX}}_{\text{loss,Qloss}} ) + ({\text{EX}}_{\text{loss,friction}} ) + ({\text{EX}}_{{{\text{T}}_{\text{ap}} ,{\text{T}}_{\text{sun}} - {\text{loss}}}} ) + ({\text{EX}}_{{{\text{lossT}}_{\text{ap}} ,{\text{T}}_{\text{f}} }} )$$

(12)

The optical loss of the solar air heater is calculated using the relationship shown below:

$${\text{EX}}_{\text{loss,opt}} = IA_{\text{c}} \left( {1 - \alpha_{\text{p}} \tau_{\text{g}} } \right)\left[ {1 - \frac{4}{3}\left( {\frac{{T_{\text{a}} }}{{T_{\text{sun}} }}} \right) + \frac{1}{3}\left( {\frac{{T_{\text{a}} }}{{T_{\text{sun}} }}} \right)^{4} } \right]$$

(13)

The exergy losses due to heat transfer between solar air heater components and environment $({\text{EX}}_{\text{loss,Qloss}} )$ is expressed as:

$${\text{EX}}_{\text{loss,Qloss}} = U_{\text{L}} A_{\text{C}} \left( {T_{\text{ap}} - T_{\text{a}} } \right)\left[ {1 - \left( {\frac{{T_{\text{a}} }}{{T_{\text{ap}} }}} \right)} \right]$$

(14)

where

$$U_{\text{L}} = U_{\text{t}} + U_{\text{b}}$$

(15)

In the above equation $U_{\text{b}}$ is known as bottom loss coefficient and $U_{\text{t}}$ indicates top loss coefficient.

The exergy losses due to absorption of solar irradiation by the absorber plate $({\text{EX}}_{{{\text{T}}_{\text{ap}} ,{\text{T}}_{\text{sun}} - {\text{loss}}}} )$ is given by

$${\text{EX}}_{{{\text{T}}_{\text{AP}} ,{\text{T}}_{\text{sun}} - {\text{loss}}}} = IA_{\text{c}} \alpha_{\text{p}} \tau_{\text{g}} \left[ {1 - \frac{4}{3}\left( {\frac{{T_{\text{a}} }}{{T_{\text{sun}} }}} \right) + \frac{1}{3}\left( {\frac{{T_{\text{a}} }}{{T_{\text{sun}} }}} \right)^{4} - \left( {1 - \frac{{T_{\text{a}} }}{{T_{{_{\text{ap}} }} }}} \right)} \right]$$

(16)

Finite temperature difference between the absorber plate and fluid causing the exergy destruction is given by:

$${\text{EX}}_{{{\text{lossT}}_{\text{AP}} ,{\text{T}}_{\text{f}} }} = IA_{\text{c}} \eta_{\text{th}} T_{\text{a}} \left[ {\left( {\frac{1}{{T_{\text{f}} }}} \right) - \left( {\frac{1}{{T_{\text{ap}} }}} \right)} \right]$$

(17)

The exergy loss due to fluid friction along the length of air flow path is given by:

$$({\text{EX}}_{\text{loss,friction}} ) = \frac{{T_{\text{a}} \dot{m}\Delta p}}{{\rho T_{\text{f}} }}$$

(18)

Analytical model and solution procedure

To evaluate the thermal performance of the multiple arc protrusion roughened plate and smooth plate with jet impingement, SAH uses the identical procedure that was used in artificial roughened plate SAHs [23,24,25,26]. During the analysis, the useful heat gain by the absorber plate, energy transferring to the working fluid and heat losses to the environment are calculated by considering the following assumptions: [26, 31]

• (1) The analysis is conducted at steady-state condition with one dimensional heat flow.

• (2) Heat conduction and leakages along the flow direction are negligible.

• (3) Edge effects are negligible.

In this work, the multiple arc shape protrusion roughened plate design parameters are optimized for yielding the maximum exergetic efficiency as a function of Reynolds number (Re) and temperature rise parameter (ΔT/I).

Step 1 The typical values of design configurations and operating conditions of the solar air heater to be analyzed are selected from Table 1.

Table 1 Typical values of operating and design parameters used in analytical study

Step 2 In this analysis, the inlet temperature (T_i) of the SAH is equal to ambient temperature (T_a). Then, the outlet temperature (T_o) of the air heater is calculated using Eqs. 19 and 20 by considering the desired temperature rise parameter (ΔT/I).

$$T_{\text{o}} = T_{\text{i}} + \Delta T$$

(19)

$$\Delta T = \frac{\Delta T}{I} \times I$$

(20)

Step 3 Based on the inlet and outlet temperatures, the average fluid temperature (T_f) inside a duct and thermo-physical properties of air such as specific heat (C_P), thermal conductivity (K), dynamic viscosity (μ) are calculated by using empirical correlations proposed by [33] as shown below

$$T_{\text{f}} = \frac{{T_{\text{o}} + T_{\text{i}} }}{2}$$

(21)

$$C_{\text{p}} = 1.0057 + 0.000066\left( {T_{\text{f}} - 27} \right)$$

(22)

$$\rho = 1.1774 - 0.00359\left( {T_{\text{f}} - 27} \right)$$

(23)

$$K = 0.02624 + 0.0000758\left( {T_{\text{f}} - 27} \right)$$

(24)

$$\mu = \left[ {1.983 + 0.00184\left( {T_{\text{f}} - 27} \right)} \right] \times 10^{ - 5}$$

(25)

Step 4 The overall heat loss coefficient (U_L) value calculated by summing up of each losses that occurring at top (U_t), back (U_b) and edge (U_e) loss coefficients can be written as:

$$U_{\text{L}} = U_{\text{t}} + U_{\text{b}} + U_{\text{e}}$$

(26)

The back and edge loss coefficients are calculated using Eqs. 27 and 28 as shown below

$$U_{\text{b}} = \frac{{K_{\text{i}} }}{{L_{\text{i}} }}$$

(27)

$$U_{\text{e}} = \frac{{(L + W)ZK_{\text{i}} }}{{LWL_{\text{i}} }}$$

(28)

The top loss (U_t) occurred from the solar air heater is calculated using the expression as stated [34].

$$U_{\text{t}} = \left[ {\frac{N}{{\frac{{C_{\text{t}} }}{{T_{\text{ap}} }}\left( {\frac{{T_{\text{ap}} - T_{\text{a}} }}{{N + f_{\text{t}} }}} \right)^{0.33} }} + \frac{1}{{h_{\text{w}} }}} \right]^{ - 1} + \left[ {\frac{{\sigma (T_{\text{ap}}^{2} + T_{\text{a}}^{2} )(T_{\text{ap}} + T_{\text{a}} )}}{{\frac{1}{{(\varepsilon_{\text{p}} + 0.05N(1 - \varepsilon_{\text{p}} )}} + \frac{{2 \times N + f_{\text{t}} - 1}}{{\varepsilon_{\text{g}} }} - 1}}} \right]$$

(29)

where

$$h_{\text{w}} = 5.7 + 3.8V_{\text{w}} ,C_{\text{t}} = 365.9(1 - 0.00883 \times \beta_{\text{t}} + 0.0001298 \times \beta_{\text{t}}^{2} )$$

$$f_{\text{t}} = (1 - 0.04 \times h_{\text{w}} + 0.0005 \times h_{\text{w}}^{2} )(1 + 0.091 \times N)$$

In the above equation, the initial temperature of the absorber plate (T_AP) can be approximated as:

$$T_{\text{ap}} = \frac{{T_{\text{i}} + T_{\text{o}} }}{2} + 10\;^\circ {\hbox{C}}$$

(30)

Step 5 The rate of useful heat gain (Q_u1) collected by the solar air heater is calculated by Eq. 31

$$Q_{\text{u1}} = A_{\text{c}} \left[ {I\left( {\tau \alpha } \right) - U_{\text{L}} (T_{\text{ap}} - T_{\text{a}} )} \right]$$

(31)

Step 6 The mass flow rate through the channel (m), mass flux (G) and Reynolds number (Re) of the flow are obtained by the following expressions

$$m = \frac{{Q_{\text{u1}} }}{{C_{\text{p}} \Delta T}}$$

(32)

$$G = \frac{m}{WZ}$$

(33)

$$Re = \frac{GD}{\mu }$$

(34)

Step 7 The heat transfer coefficient between the absorber plate and impingement air are calculated by first assessing the Nusselt number. The Nusselt number correlation developed based on the experimental results by Nadda et al. [30] is expressed as:

$$\begin{aligned} Nu =&\, 0.0476Re^{1.0119} \left[ {\frac{{W_{\text{A}} }}{{W_{\text{AP}} }}} \right]^{0.4228} \exp \left[ {0.0529\left( {\ln \left( {\frac{{W_{\text{A}} }}{{W_{\text{AP}} }}} \right)} \right)^{2} } \right]\left[ {\frac{{e_{\text{P}} }}{{d_{\text{P}} }}} \right]^{ - 0.133} \exp \left[ { - 0.228\left( {\ln \left( {\frac{{e_{\text{P}} }}{{d_{\text{P}} }}} \right)} \right)^{2} } \right]\left[ {\frac{{P_{\text{P}} }}{{e_{\text{P}} }}} \right]^{ - 0.1455} \\ & \quad \exp \left[ { - 0.3069\left( {\ln \left( {\frac{{P_{\text{P}} }}{{e_{\text{P}} }}} \right)} \right)^{2} } \right]\left( {\frac{{\alpha_{\text{a}} }}{55}} \right)^{ - 0.7522} \exp \left( { - 1.4876\left( {\ln \left( {\frac{{\alpha_{\text{a}} }}{55}} \right)} \right)^{2} } \right) \\ \end{aligned}$$

(35)

Based on the Nusselt number, the heat transfer co efficient is calculated as:

$$h = \frac{Nu \, K}{D}$$

(36)

Step 8 The equations shown in Eqs. 37 and 38 are used to calculate the collector efficiency factor (F_P) and heat removal factor (F_R).

$$F_{\text{P}} = \frac{h}{{h + U_{\text{L}} }}$$

(37)

$$F_{\text{R}} = \frac{{mC_{\text{p}} }}{{U_{\text{L}} A_{\text{c}} }}\left[ {\exp \left( {\frac{{U_{\text{L}} A_{\text{c}} F_{\text{P}} }}{{mC_{\text{P}} }}} \right) - 1} \right]$$

(38)

Step 9 Based on the heat removal rate (F_R), the useful energy gained (Q_u2) by the air heater is calculated using the expression:

$$Q_{\text{u2}} = F_{\text{R}} \left[ {I\left( {\tau \alpha } \right) - U_{\text{L}} (T_{\text{o}} - T_{\text{i}} )} \right]A_{\text{c}}$$

(39)

Step 10 The useful heat gains Q_u1 and Q_u2 are calculated using Eqs. 31 and 39. If the difference between the heat gain values exceeds 0.01%, then the temperature of the absorber plate is recalculated and updated by the expression as:

$$T_{\text{AP}} = T_{\text{a}} + \left[ {\frac{{I\left( {\tau \alpha } \right) - {\raise0.7ex\hbox{${Q_{\text{u2}} }$} \!\mathord{\left/ {\vphantom {{Q_{\text{u2}} } {A_{\text{c}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${A_{\text{c}} }$}}}}{{U_{\text{L}} }}} \right]$$

(40)

Then, the iterative process has been continued until the convergence criterion is to be achieved.

Step 11 Then, the thermal efficiency and of the solar air heater is calculated by considering the average values of useful heat gain (Q_u1 and Q_u2) and substituting in Eq. 1.

Step 12 The friction factor calculated by the correlation developed by Nadda et al. [30] based on experimental results can be expressed as:

$$\begin{aligned} f = & 15.601Re^{ - 0.1434} \left[ {\frac{{W_{\text{A}} }}{{W_{\text{AP}} }}} \right]^{0.2569} \exp \left[ {0.1205\left( {\ln \left( {\frac{{W_{\text{A}} }}{{W_{\text{AP}} }}} \right)} \right)^{2} } \right]\left[ {\frac{{e_{\text{P}} }}{{d_{\text{P}} }}} \right]^{ - 0.1708} \exp \left[ { - 0.3957\left( {\ln \left( {\frac{{e_{\text{P}} }}{{d_{\text{P}} }}} \right)} \right)^{2} } \right]\left[ {\frac{{P_{\text{P}} }}{{e_{\text{P}} }}} \right]^{ - 0.2777} \\ & \exp \left[ { - 0.5793\left( {\ln \left( {\frac{{P_{\text{P}} }}{{e_{\text{P}} }}} \right)} \right)^{2} } \right]\left( {\frac{{\alpha_{\text{a}} }}{55}} \right)^{ - 0.9011} \exp \left( { - 1.7618\left( {\ln \left( {\frac{{\alpha_{\text{a}} }}{55}} \right)} \right)^{2} } \right) \\ \end{aligned}$$

(41)

Step 13 The pressure drop occurred at various flow rate are calculated using the following relationship

$$\Delta P = \frac{{4fL\rho V^{2} }}{{2D_{\text{h}} }}$$

(42)

Step 14 Further, the thermo-hydraulic and exergetic efficiency of the solar collector can be calculated using the equations from 3 to 18. The iterative procedure, explained in flowchart, shown in Fig. 3 is repeated for the range of design parameters related to absorber plate roughness and temperature rise parameters.

Fig. 3

Flow chart for analytical solution procedure

Results and discussions

By considering the entire range of design parameters and operating conditions as given in Table 1, the exergetic efficiency is analytically calculated. Figure 4 shows the thermal, thermo-hydraulic and exergetic efficiency as a function of Reynolds number of a multiple arc shape protrusion roughened jet plate solar air heater for the roughness values of P_P/e_P = 8, α_a = 45°, e_P/d_P = 1, W_P/W_AP = 5 and solar insolation of 1000 W m⁻². It is seen from the figure that the thermal efficiency increases monotonically with increase in Reynolds number. This is due to the fact that increase in Reynolds number results in an increase in impingement air velocity and consequently enhances the heat transfer coefficient and heat capacity of air. The thermo-hydraulic efficiency of SAH is equal to thermal efficiency up to Re equal to 2300. Further increase in Reynolds number decreases the thermo-hydraulic efficiency. This happens due to fact that when Re increases, the pressure drop and pumping power also increases as shown in Fig. 5. It is also observed from the figure that when Re > 4200, there is severe rise in pressure drop and pumping power to overcome the fluid friction and turbulence intensity formed inside the duct.

Fig. 4

Variation of thermal, thermo-hydraulic and exergetic efficiency with Reynolds number for SAH with jet impingement on multiple arc-shaped obstacles

Fig. 5

Variation of pressure drop and pumping power with Reynolds number for jet impingement on multiple arc-shaped obstacles SAH

The exergetic efficiency follows an ascending trend up to Re 400 and yields the maximum exergetic efficiency of 10.1% and starts to falls on descending trend on further increase in Reynolds number. The exergetic performance of the solar air heater has been low in the entire range of Reynolds number and falls into negative region when the Re ≥ 9000. This happens due to that the various exergy losses, especially exergy required for pumping power to overcome the friction associated with solar air heater duct that outstrips the value of useful thermal exergy collected by solar air heater. The relation between the thermal, thermo-hydraulic and exergy efficiency with the temperature rise parameter (ΔT/I) is shown in Fig. 6. This figure shows that when the temperature rise parameter increases, the exergetic efficiency increases up to the value of 0.135 K m² W⁻¹ then it starts decreasing. During this condition, the thermal efficiency follows a descending trend with the temperature rise parameter. However, the thermo-hydraulic efficiency is lesser than thermal efficiency up to ΔT/I is ≤ 0.024 K m² W⁻¹. Further, increase in ΔT/I deduces the thermo-hydraulic performance and it is equal to thermal efficiency of SAH.

Fig. 6

Variation of thermal thermo-hydraulic and exergetic efficiency with temperature rise parameter with jet impingement on multiple arc-shaped obstacles

The exergy losses related to global irradiation are shown in Fig. 7. It is observed that about 14.5% of exergy losses occur due to the inefficiency of glass cover and this optical loss (EX_loss,Opt/EX_Sun) remains constant and not varies with respect to Reynolds number, absorber plate and fluid temperatures. This happens due to that the effective transmittance absorptance product (α_pτ_g) in Eq. 13 is equal to 0.85. Therefore, the remaining exergy input is continuously loss at all the operating conditions of SAH. The exergy losses occurred between the sun and absorber plate (EX_loss,Tap/EX_Sun) is about 63–84%. This is the major source of exergy loss that increases monotonically with Reynolds number. This happens due to that when the air flow rate increases the absorber plate temperature decreases. Consequently, this enhances the exergy destruction in absorber plate and decreases the exergy loss to the environment (EX_loss,Qloss/EX_Sun). Thus, at lower mass flow rates, EX_loss,Qloss/EX_Sun is high. The exergy losses associated with internal irreversibilities are due to heat transfer between air and absorber plate due to finite temperature difference, (EX_loss, T_AP,Tf/EX_Sun), friction along the duct (EX_{loss,Friction}/EX_Sun). The exergy loss EX_loss,TAP,Tf/EX_Sun gradually decreases with Reynolds number due to reduction in temperature difference between inlet and outlet air. The exergy destruction due to friction increases with increasing in Reynolds number due to higher pumping power and turbulent intensity.

Fig. 7

Various components of exergy losses as a function of Reynolds number for SAH with jet impingement on multiple arc-shaped obstacles

Figures 8 and 9 summarize the effect of relative width ratio (W_P/W_AP) based on Reynolds number (Re) and temperature rise parameter (ΔT/I) on the exergetic performance (η_II) of the solar air heater for solar insolation of 1000 W m⁻², while other roughness parameters are constant. From Fig. 8, it is observed that the relative width ratio (W_P/W_AP) of 5 yields the maximum exergetic efficiency (η_II) for Re < 4800, and for Re > 5900, the smooth plate with jet impingement SAH yields the maximum exergetic efficiency (η_II). When the Re is in the range of 4800–5900, the highest exergetic efficiency occurs for various values of W_P/W_AP in the range of 1–4. This happens due to increase in width ratio leads to formation of leading ends and secondary flow cells. It can be seen from Fig. 9 that the smooth plate with jet impingement SAH performs better by producing higher exergetic efficiency compared with multiple arc shape protrusion roughened plate with jet impingement SAH for ΔT/I < 0.015 K m² W⁻¹. For ΔT/I > 0.033 K m² W⁻¹, the maximum value of exergetic efficiency is obtained for the relative width ratio (W_P/W_AP) equal to 5. When the ΔT/I is in the range of 0.015–0.033 K m² W⁻¹, the highest exergetic efficiency occurs for various values of W_P/W_AP in the range of 1–4. Figure 10 shows the enhancement in exergetic efficiency (η_II/η_S) as a function of temperature rise parameter for various values of relative width ratio (W_P/W_AP). The maximum enhancement has been found at for W_P/W_AP = 5 except at low values of ΔT/I. The rate of enhancement is high when the W_P/W_AP is increased from 1 to 2 as it compared with further increment in the values of W_P/W_AP.

Fig. 8

Exergetic efficiency as function of Reynolds number and relative width ratio (W_A/W_AP)

Fig. 9

Exergetic efficiency as function of temperature rise parameter and relative width ratio

Fig. 10

Enhancement in exergetic efficiency as function of temperature rise parameter and relative width ratio (W_A/W_AP)

For constant roughness parameters, Figs. 11 and 12 summarize the effect of relative height ratio (e_P/d_P) based on Reynolds number (Re) and temperature rise parameter (ΔT/I) on the exergetic performance (η_II) of the solar air heater for solar insolation of 1000 W m⁻². From Fig. 11, it is observed that the relative height ratio (e_P/d_P) of 1 yields the maximum exergetic efficiency (η_II) for Re < 4100, and for Re > 5900, the smooth plate with jet impingement SAH yields the maximum exergetic efficiency (η_II). When relative height ratio increases, the reattachment vicinity is more in the range of 0.5–1. The further increment in height ratio leads to reduction in exergetic efficiency due to less reattachment vicinity. When the Re is in the range of 4100–5900, the highest exergetic efficiency occurs for various values of e_P/d_P in the range of 0.5–2 due to better movement of resulting vortices. It can be seen from Fig. 12 that the smooth plate with jet impingement SAH performs better when the ΔT/I < 0.015 K m² W⁻¹ and for ΔT/I > 0.022 K m² W⁻¹, the maximum value of exergetic efficiency is obtained for e_P/d_P equal to 1. When the ΔT/I is in the range of 0.015–0.022 K m² W⁻¹, the highest exergetic efficiency is observed for various values of e_P/d_P in the range of 0.5–2. Figure 13 reveals the enhancement in exergetic efficiency (η_II/η_S) as a function of temperature rise parameter for various values of relative height ratio (e_P/d_P). The maximum enhancement has been found at e_P/d_P = 1 except at low values of ΔT/I. The rate of enhancement is high when the e_P/d_P is increased from 0.5 to 1 and further increment in relative height ratio (e_P/d_P) leads to considerable decrement on enhancement.

Fig. 11

Exergetic efficiency as function of Reynolds number and relative height ratio (e_P/d_P)

Fig. 12

Exergetic efficiency as function of temperature rise parameter and relative height ratio (e_P/d_P)

Fig. 13

Enhancement in exergetic efficiency as function of temperature rise parameter and relative height ratio (e_P/d_P)

The variation of exergetic efficiency based on Reynolds number (Re) and temperature rise parameter (ΔT/I) at different values of relative pitch ratio (P_P/e_P) and for solar insolation of 1000 W m⁻², relative height ratio (e_P/d_P) of 1, relative width ratio (W_P/W_AP) of 5 and relative angle of arc (α_a) of 45° are shown in Figs. 14 and 15. From Fig. 14, it is observed that the relative pitch ratio (P_P/e_P) of 8 yields the maximum exergetic efficiency (η_II) for Re < 4000, and for Re > 6800, the smooth plate with jet impingement SAH yields the maximum exergetic efficiency (η_II). This happens due to the fact that the flow separation taking place in the downstream of arc-shaped dimple is effectively reattached. When the Re is in the range of 4000–6800, the highest exergetic efficiency occurs for various values of P_P/e_P in the range of 9.5–12 due to better reattachment vicinity. It can be seen from Fig. 15 that the smooth plate with jet impingement SAH performs better when the ΔT/I < 0.015 K m² W⁻¹ and for ΔT/I > 0.024 K m² W⁻¹, the maximum value of exergetic efficiency is obtained for the relative pitch ratio (P_P/e_P) equal to 8. When the ΔT/I is in the range of 0.015–0.024 K m² W⁻¹, the highest exergetic efficiency is observed for various values of P_P/e_P in the range of 9.5–12. Figure 16 reveals the enhancement in exergetic efficiency (η_II/η_S) as a function of temperature rise parameter for various values of relative pitch ratio (P_P/e_P). The maximum enhancement is found for P_P/e_P = 8 except at low values of ΔT/I and further increment in relative pitch ratio (e_P/d_P) causes decrement in enhancement ratio.

Fig. 14

Exergetic efficiency as function of Reynolds number and relative pitch ratio (P_P/e_P)

Fig. 15

Exergetic efficiency as function of temperature rise parameter and relative pitch ratio (P_P/e_P)

Fig. 16

Enhancement in exergetic efficiency as function of temperature rise parameter and relative pitch ratio (P_P/e_P)

The variation of exergetic efficiency based on Reynolds number (Re) and temperature rise parameter (ΔT/I) at different relative angle of arc (α_a) and for solar insolation of 1000 W m⁻², relative height ratio (e_P/d_P) of 1, relative width ratio (W_P/W_AP) of 5 and relative pitch ratio (P_P/e_P) of 8 are shown in Figs. 17 and 18. From Fig. 17, it is observed that the relative angle of arc (α_a) of 45° yields the maximum exergetic efficiency (η_II) for Re < 4400 and for Re > 6100, the smooth plate with jet impingement SAH yields the maximum exergetic efficiency (η_II). This is due to the fact that the flow separation, by leading edges induces more vortices.

Fig. 17

Exergetic efficiency as function of Reynolds number and angle of arc (α_a)

Fig. 18

Exergetic efficiency as function of temperature rise parameter and angle of arc (α_a)

When the Re is in the range of 4400–6100, the highest exergetic efficiency occurs for various values of α_a in the range of 35°–75°. It can be seen from Fig. 18 that the smooth plate with jet impingement SAH performs better when the ΔT/I < 0.016 K m² W⁻¹ and for ΔT/I > 0.03 K m² W⁻¹, the maximum value of exergetic efficiency is obtained for the relative angle of arc (α_a) equal to 45°. When the ΔT/I is in the range of 0.016–0.03 K m² W⁻¹, the highest exergetic efficiency is observed for various values of α_a in the range of 35°–75°. For various values of relative angle of arc (α_a), Fig. 19 reveals enhancement in exergetic efficiency (η_II/η_S) as a function of temperature rise parameter. The maximum enhancement is found for α_a = 45° except at low values of ΔT/I. The rate of enhancement is high when the α is increased from 35° to 45° and further increment in relative angle of arc (α_a) leads to considerable decrement on enhancement ratio.

Fig. 19

Enhancement in exergetic efficiency as function of temperature rise parameter and angle of arc (α_a)

From the above discussions, it is observed that there are no particular set of geometrical roughness parameter yielding the maximum exergetic efficiency (η_II) for the entire range of Reynolds number and temperature rise parameter (ΔT/I). It can be observed from Figs. 8, 11, 14, and 17 that the exergetic efficiency follows an increasing trend with Reynolds number and reaches the maxima value; then, it follows a diminishing trend. This happens due to the fact that pumping power dominates the useful heat gain at higher values of Reynolds number. From the figures, it is also seen that the multiple arc shape protrusion roughened plate with jet impingement SAH can be effectively operated for Reynolds number in the range of 400–5900 for obtaining the highest exergetic efficiency. Figures 10, 13, 16 and 19 show that the maximum enhancement in exergetic efficiency ratio is 4.84, corresponding to geometrical roughness parameter values of W_P/W_AP = 5, e_P/d_P = 1, P_P/e_P = 8, α_a = 45°.

The exergetic performance for multiple arc protrusion roughened SAH with jet impingement is compared with discrete V-down rib roughness shape [23] and protruded arc shape roughened geometry [24] under identical operating conditions are shown in Fig. 20. From this figure, it is concluded that multiple arc protrusion roughened SAH with jet impingement has better performance when the Re < 4200. From this figure, it is also observed that when Re increases, there is a significant drop in exergetic efficiency. This is due to fact that there is sudden increment in pressure drop and pumping power consumption as observed from Fig. 5. And also, from Fig. 7, it is observed that increase in pressure drop improves the exergy destruction (EX_{loss,Friction}/EX_Sun) due to friction from 1 to 5%. This also results in degrading the exergetic performance of SAH.

Fig. 20

Comparison of exergetic efficiency values for protrusions with other SAH in literature

Design plots for optimum multiple arc protrusion roughness parameters

The highest values of exergetic efficiency yielded by the multiple arc shape protrusion roughened SAH parameters such as relative width ratio (W_P/W_AP), relative height ratio (e_P/d_P), relative pitch ratio (P_P/e_P) and relative angle of arc (α_a) for different temperature rise parameters are shown in Figs. 21–24, respectively. It can be observed from Fig. 21 that for all the values of solar radiation, the relative width ratio (W_P/W_AP) of 5 is optimum for ΔT/I > 0.0286 K m² W⁻¹ and relative width ratio (W_P/W_AP) of 1 is optimum for ΔT/I < 0.015 K m² W⁻¹. When the ΔT/I is in the range of 0.015–0.0286 K m² W⁻¹, the optimum values depends on solar insolation and temperature rise parameter. Figure 22 shows that the relative height ratio (e_P/d_P) of 1 is optimum for ΔT/I > 0.035 K m² W⁻¹ and relative height ratio (e_P/d_P) of 2 is optimum for ΔT/I < 0.019 K m² W⁻¹ for all the values of solar radiation. When the ΔT/I is in the range of 0.019–0.035 K m² W⁻¹, the optimum values depend on solar insolation and temperature rise parameter. For all the values of solar radiation, Fig. 23 shows that the relative pitch ratio (P_P/e_P) of 8 is optimum for ΔT/I > 0.028 K m² W⁻¹ and relative pitch ratio (P_P/e_P) of 12 is optimum for ΔT/I < 0.015 K m² W⁻¹. When the ΔT/I is in the range of 0.015–0.028 K m² W⁻¹, the optimum values depend on solar insolation and temperature rise parameter. For all the values of solar radiation, Fig. 24 reveals that the relative angle of arc (α_a) of 45° is optimum for ΔT/I > 0.035 K m² W⁻¹ and relative angle of arc (α_a) of 75° is optimum for ΔT/I < 0.02 K m² W⁻¹. When the ΔT/I is in the range of 0.02–0.035 K m² W⁻¹, the optimum values depend on solar insolation and temperature rise parameter.

Fig. 21

Optimum values of relative width ratio (W_P/W_AP) on the basis of maximum exergetic efficiency criterion

Fig. 22

Optimum values of relative height ratio (e_P/d_P) on the basis of maximum exergetic efficiency criterion

Fig. 23

Optimum values of relative pitch ratio (P_P/e_P) on the basis of maximum exergetic efficiency criterion

Fig. 24

Optimum values of angle of arc (α_a) on the basis of maximum exergetic efficiency criterion

Conclusions

In this present work, a mathematical model has been developed for predicting the exergetic efficiency of a multiple arc shape protrusion roughened plate with jet impingement solar air heater. The influence of Reynolds number and roughness parameters on exergetic efficiency has been determined. The performance of multiple arc shape protrusion roughened plate with jet impingement SAH has been compared with smooth plate with jet impingement SAH at analogous conditions and following conclusions have been attained.

• The multiple arc shape protrusion roughened plate with jet impingement SAH performs better up to the Reynolds number equal to 5900 and above this value smooth plate with jet impingement SAH is appropriate.

• The maximum exergetic efficiency of multiple arc shape protrusion roughened plate with jet impingement SAH is 10.5%. By using this configuration of solar air heater, the exergetic efficiency is improved up to 56.8% as compared with smooth plate jet impingement SAH.

• The optimum values of roughness parameters that yields the maximum exergetic efficiency for the Re ranges from 400 to 5900 are relative width ratio (W_P/W_AP) of 5, relative height ratio (e_P/d_P) of 1, relative pitch ratio (P_P/e_P) of 8 and relative angle of arc (α_a) of 45°

• For each roughness parameters of absorber plate, a separate design plots are prepared as a function of temperature rise parameter which are used to assess the system performance at different operating conditions to yield the maximum exergetic efficiency.