Analytical and Experimental Study of Recycling Baffled Double-Pass Solar Air Heaters with Attached Fins

Abstract

The study of the heat transfer of solar air heaters with a new design using an absorbing plate with fins and baffles, which facilitate the recycling of flowing air, is reported. The mathematical formulation and analytical analysis for such a recyclic baffled double-pass solar air heater were developed theoretically. The performance of the device was studied experimentally as well. The theoretical predicted and experimental results were compared with another design, i.e., a downward-type single-pass solar air heater without recycle and double-pass operations reported in our previous work. Significant improvement in heat-transfer efficiency is achieved with the baffle and fin design due to the recycling heating and the extended heat transfer area. The effects of mass flow rate and recycle ratio on the heat-transfer efficiency enhancement as well as on the power consumption increment are also discussed.

1. Introduction

The flat-plate solar air heater [1,2,3,4,5] is considered to be a simple device consisting of one or more glass (or transparent) covers situated above an absorbing plate with the air flowing over [4,5], under [1,2,3] or simultaneously over and under [6] the absorbing plate. The conventional flat-plate solar air heater has been investigated for heat-transfer efficiency improvement by introducing free convection [7] forced convection [8,9] extended heat-transfer area [10,11] and increased air turbulence [12,13]. Moreover, the adoption of the recycle-effect concept can effectively enhance the heat transfer rate and has been applied in many separation processes and reactor designs, as confirmed by the previous works [14,15,16,17].Extending from our previous work [18] this study focuses on a new design featuring recycling as well as turbulence and heat transfer area enhancement, as shown in Figure 1.

Figure 1. Double-pass solar air heater with internal baffles and fins.

Figure 1. Double-pass solar air heater with internal baffles and fins.

This work presents the theoretical mathematical formulations and the experimental verification for the new solar air heater design. The effects of the recycle ratio and air mass flow rate on the heat transfer rate, heat-transfer efficiency improvement, and the power consumption increment are also delineated.

2. Mathematical Formulation

The device proposed is conceptually depicted in Figure 1. An absorbing plate inserted into the air flow channel creates the upper and lower subchannels, which facilitates the double-pass air flow. On both sides of the absorbing plate, baffles and fins guide the air flow and provide turbulence and extra heat transfer area. Like in conventional solar air heaters, two layers of glass with a gap in between are used on top of the upper subchannel to reduce the heat loss to the environment and other sides of the device are thermally insulated.

Figure 2 shows the configuration of the device with recycle. The absorbing plate is inserted in parallel into a parallel-plate channel with width of W and length of L, and the open conduit is divided into two parts both with height of H. Before entering the lower subchannel, the inlet air with mass flow rate and inlet temperature is premixed with part of the air leaving the lower subchannel with mass flow rate and outlet temperature. The steady-state one-dimensional mathematical model was developed with the assumptions of: (1) the temperature of the absorbing plate, bottom plate and bulk fluids are functions of the flow direction only; (2) both glass covers and flowing air do not absorb radiant energy; (3) except the glass cover, all outside surfaces of the solar air collector are well insulated thermally.

Figure 2. Configuration of a double-pass solar air heater with recycle.

Figure 2. Configuration of a double-pass solar air heater with recycle.

2.1. Effect of Attaching Fins with Baffles

The total heat transfer from the absorbing plate with attached fins and baffles to a passing airflow can be written as the following via modifications from Bannett and Myers [19]:

$$q^{″}=h_i\left(A_t+A_f{\eta }_f+A_b{\eta }_b\right)\left(T_p-T_i(z)\right),i=a,b$$

(1)

where the fin efficiency is defined as:

$${\eta }_f=\frac{\text{tanh }{m}_f{h}_f}{m_f{h}_f}$$

(2)

in which:

$$m_f=\sqrt{\frac{2h_i(L+t_f)}{k_f{A}_{tf}}}$$

(3)

The baffle efficiency ${\eta }_b$ is affected by the density of the baffles, i.e., W/D_e and l/L. Following empirical correlations [20,21] the baffle efficiency can be obtained for the fins attached to the lower and upper subchannels as:

$${\eta }_b=26.361{\left(\frac{W_B}{D_e}\right)}^{-0.454}{\left(\frac{l}{L}\right)}^{0.634}\text{ for lower subchannel}$$

(4)

and:

$${\eta }_b=15.583{\left(\frac{W_B}{D_e}\right)}^{0.0518}{\left(\frac{l}{L}\right)}^{0.227}\text{ for upper subchannel}$$

(5)

2.2. Temperature Distributions for the Flowing Air in the Flow Direction

For the steady-state one-dimensional mathematical model, the energy balances for the flowing air streams, glass cover 1, absorbing plate and bottom plate, are given in Equations (6–10).

For the airflow in upper subchannel:

$$\frac{-\dot{m}R{C}_p}{W}\frac{d{T}_b(z)}{dz}=h_b{\varphi }_b(T_p-T_b(z))-h_b(T_b(z)-T_{c_1})$$

(6)

For the airflow in lower subchannel:

$$\left[\frac{(R+1)\dot{m}{C}_p}{W}\right]\frac{d{T}_a(z)}{dz}=h_a{\varphi }_a(T_p-T_a(z))-h_a(T_a(z)-T_R)$$

(7)

For the glass cover 1 (inner cover):

$$h_{r,p-c_1}(T_p-T_{c_1})+h_b(T_b(z)-T_{c_1})=U_{c_1-s}(T_{c_1}-T_s)$$

(8)

For the absorbing plate:

$$I_0{\alpha }_p{\tau }_g^2=U_T(T_p-T_s)-U_B(T_p-T_s)-h_b{\varphi }_b(T_p-T_b(z))-h_a{\varphi }_a(T_p-T_a(z))$$

(9)

For the bottom plate

$$h_a(T_a(z)-T_R)+h_{r,p-R}(T_p-T_R)=U_{B-s}(T_R-T_s)$$

(10)

The heat transfer coefficients between airflow and the conduit walls were assumed to be equal, in which:

$${\varphi }_a=1+\frac{A_{f,a}}{A_t}{\eta }_{f,a}+\frac{A_{b,a}}{A_t}{\eta }_{b,a}$$

(11)

$${\varphi }_b=1+\frac{A_{f,b}}{A_t}{\eta }_{f,b}+\frac{A_{b,b}}{A_t}{\eta }_{b,b}$$

(12)

Rearranging Equations (8–10) and solving for (T_p−T_b(z)), (T_b(z)−T_c1), (T_p−T_a(z)) and (T_a(z)−T_R), and then, substituting the results into Equations (6,7), we have the temperature distributions with the dimensionless coordinate in the flow direction as follows:

$$\frac{d(T_b(\xi )-T_s)}{d\xi }=B_1(T_b(\xi )-T_s)+B_2(T_a(\xi )-T_s)+B_3$$

(13)

$$\frac{d(T_a(\xi )-T_s)}{d\xi }=B_4(T_b(\xi )-T_s)+B_5(T_a(\xi )-T_s)+B_6$$

(14)

$$\xi =z/L.$$

(15)

The boundary conditions for solving Equations (13,14) are:

$$\xi =0,T_{a,i}^0=\frac{T_{a,i}+R{T}_{a,L}}{(1+R)}$$

(16)

$$\xi =1,T_a(1)=T_{a,L},T_b(1)=T_{b,L}$$

(17)

By following the similar mathematical treatment performed in our previous work [18], the analytical solutions to this device were obtained after transforming Equations (13,14) into $(D-B_1)(T_b-T_s)-B_2(T_a-T_s)=B_3$ and $-B_4(T_b-T_s)+(D-B_5)(T_a-T_s)=B_6$ with the use of the differential operator notation, in which $D=d/d\xi$. Operate on the second of these with $(1/B_4)(D-B_1)$ and add to the first equation to obtain Equation (18) with using boundary conditions in Equations (16),(17), and hence, Equation (19) was thus obtained. The temperature distributions of the flowing air for both lower and upper subchannels are thus obtained, respectively:

$$T_b(\xi )=\frac{Y_1-B_5}{B_4}{C}_1{e}^{Y_1\xi }+\frac{Y_2-B_5}{B_4}{C}_2{e}^{Y_2\xi }-\frac{B_5(B_3{B}_4-B_1{B}_6)}{B_4(B_1{B}_5-B_2{B}_4)}-\frac{B_6}{B_4}+T_s$$

(18)

$$T_a(\xi )=C_1{e}^{Y_1\xi }+C_2{e}^{Y_2\xi }+\frac{B_3{B}_4-B_1{B}_6}{B_1{B}_5-B_2{B}_4}+T_s$$

(19)

All the coefficients B_i, G_i, M_i, Y_i, C_i, F_i and I_i are referred to the Appendix. Once the inlet and outlet temperatures are known, the useful gain may be estimated from the relation:

$$Q_u=\dot{m}(1+R)C_p(T_{a,L}-T_{a,i}^0)+\dot{m}{C}_p(T_{b,o}-T_{b,L})$$

(20)

or:

$$Q_u=\dot{m}(1+R)C_p(T_{a,L}-T_{a,i})+\dot{m}{C}_p(T_{b,o}-T_{b,L})=\dot{m}(T_{b,o}-T_{a,i})$$

(21)

Finally, the collector efficiency ${\eta }_D$ of a baffled double-pass solar air heater with external recycle and attached internal fins, relates the actual useful energy gain to the incident solar radiation, was defined as follows Ho et al. [18]:

$${\eta }_D=\frac{Q_u(Usefulgainofenergycarriedawaybyair)}{I_0{A}_c(Totalsolarradiationincident)}$$

(22)

$$=\frac{\dot{m}{C}_p(T_{b,o}-T_{a,i})}{I_0{A}_c}={\tau }_g^2{\alpha }_p-U_L(T_{p,m}-T_s)/I_0$$

(23)

and the airflow outlet temperature is readily obtained from Equation (22), i.e.:

$$T_{b,o}=T_{a,i}+({\eta }_D{I}_0{A}_c/\dot{m}{C}_p)$$

(24)

Equating Equations (22,23), we have:

$$T_{p,m}=T_s+(I_0{\tau }_g^2{\alpha }_p/U_L)-\frac{\dot{m}{C}_p(T_{b,o}-T_{a,i})}{A_c{U}_L}$$

(25)

Substitution of Equation (24) into Equation (25) to eliminate $T_{b,o}$ results in:

$$T_{p,m}=T_s+(I_0/U_L)({\tau }_g^2{\alpha }_p-{\eta }_D)$$

(26)

Equation (26) is the expression of $T_{p,m}$ in terms of ${\eta }_D$.

2.3. Heat Transfer Coefficients

The total heat loss rate from the solar air heater to the surrounding is the summation of the top, edge and bottom loss rates, i.e.:

$$Q_{\text{loss}}=U_L{A}_c(T_{p,m}-T_s)=U_T{A}_c(T_{p,m}-T_s)+U_B{A}_c(T_{R,m}-T_s)+U_E{A}_E(T_{a,m}+T_{b,m}-2T_s)$$

(27)

where the average temperatures $T_{a,m}$, $T_{b,m}$, $T_{p,m}$ and $T_{R,m}$ can be obtained by integration and rearrangement of Equations (10,18,19,26), and $U_T$, $U_B$ and $U_E$ are estimated by Equations (28,32). An empirical equation for $U_T$ was developed [22] following the basic procedure in [23] as:

$$U_T={\left\{\frac{N}{\frac{C_{air}}{T_{p,m}}{\left[\frac{(T_{p,m}-T_s)}{N+f}\right]}^e}+\frac{1}{h_w}\right\}}^{-1}+\frac{\sigma (T_{p,m}+T_s)(T_{p,m}^2+T_s^2)}{\left\{{({\epsilon }_p+0.00591N{h}_w)}^{-1}+\frac{\left[2N+f-1+0.133{\epsilon }_p\right]}{{\epsilon }_g}-N\right\}}$$

(28)

where:

$$f=(1+0.089h_w-0.1166h_w{\epsilon }_p)(1+0.07866N)$$

(29)

$$C_{air}=520(1-0.00005{\beta }^2)\text{ for }{0}^{\circ }<\beta <{70}^{\circ };\text{ for }\beta >{70}^{\circ }\text{ use }\beta ={70}^{\circ }$$

(30)

$$e=0.43(1-100/T_{p,m})$$

(31)

The edge and bottom loss coefficients depend mainly on the insulation thickness:

$$U_B=U_E=k_s/l_s$$

(32)

Moreover, all heat transfer coefficients in Equations (6,7) are estimated by the following empirical and correction equations. The forced convective heat-transfer coefficient for air flowing in the ducts between two flat plates may be estimated by Kay’s data [24,25]:

$$N_{u,i}=h_i{D}_{e,i}/k=0.0158{\text{Re}}_i^{0.8},i=a,b$$

(33)

The modification of McAdams [25] for turbulent flow in a short conduit:

$$N_{u,i}=h_i{D}_{e,i}/k=0.0158{\text{Re}}_i^{0.8}[1+{(D_{e,i}/L)}^{0.7}],i=a,b$$

(34)

The correlation equation for laminar flow was expressed by Heaton et al. [26]:

$$N_{u,i}=4.4+\frac{0.00398{(0.7{\text{Re}}_j{D}_{e,i}/L)}^{1.66}}{1+0.0114{(0.7{\text{Re}}_i{D}_{e,i}/L)}^{1.12}},i=a,b$$

(35)

in which the characteristic length is the equivalent diameter of the air duct:

$$D_{e,0}=\frac{4(2HW)}{2(2H+W)},D_{e,a}=D_{e,b}=\frac{4(HW-5H_f{t}_f)}{2(H+W)+10H_f}$$

(36)

for downward-type single-pass and double-pass devices, respectively. The average velocities of the downward-type single-pass device and the lower and upper subchannels of the double-pass device are:

$$\overline{v_0}=\frac{m}{2HW\rho },\overline{v_a}=\frac{m(1+R)}{(HW-5h_f{t}_f)\rho },\overline{v_b}=\frac{m}{(HW-5h_f{t}_f)\rho }$$

(37)

Thus; from Equations (36,37); one obtains the Reynolds numbers of downward-type single-pass and double-pass devices; respectively; for the rectangular ducts:

$${\text{Re}}_0=\frac{\rho \overline{v_0}{D}_{e,0}}{\mu },{\text{Re}}_a=\frac{\rho \overline{v_a}{D}_{e,a}}{\mu },{\text{Re}}_b=\frac{\rho \overline{v_b}{D}_{e,b}}{\mu }$$

(38)

The heat transfer coefficients and radiation coefficients in Equations (8–10) were calculated by the following empirical equations. The thermal resistance from glass cover 1 through glass cover 2 to the ambient air, may be expressed as:

$$\frac{1}{U_{c_1-s}}=\frac{1}{U_T}-\frac{1}{h_{r,p-c_1}+h_1}$$

(39)

or:

$$\frac{1}{U_{c_1-s}}=\frac{1}{h_w+h_{r,c_2-s}}+\frac{1}{h_{c_1-c_2}+h_{r,c_1-c_2}}$$

(40)

where the heat-transfer coefficient for free convection of air between two glass covers may be estimated using Hottel’s empirical Equation [23] as:

$$h_{c_1-c_2}=1.25{(T_{c_{1,m}}-T_{c_{2,m}})}^{0.25}$$

(41)

and the convective heat-transfer coefficient for air flowing over the outside surface of glass cover 2, can be calculated using the following empirical equation given by McAdams [25]:

$$h_w=2.8+3.0V$$

(42)

The radiation coefficients between the two air-duct surfaces may be estimated by assuming a mean radiant temperature equal to the mean fluid temperature, viz.:

$$h_{r,p-c1}\approx 4\sigma T_{b,m}^3/[(1/{\epsilon }_p)+(1/{\epsilon }_{c_1})-1]$$

(43)

$$h_{r,p-R}\approx 4\sigma T_{a,m}^3/[(1/{\epsilon }_P)+(1/{\epsilon }_R)-1]$$

(44)

while those between two glass covers and from glass cover 2 to the ambient are, respectively:

$$h_{r,c_1-c_2}=\frac{\sigma (T_{c_{1,m}}^2+T_{c_{2,m}}^2)(T_{c_{1,m}}+T_{c_{2,m}})}{(1/{\epsilon }_g)+(1/{\epsilon }_g)-1}$$

(45)

$$h_{r,c_2-s}={\epsilon }_g\sigma (T_{c_{2,m}}^2+T_s^2)(T_{c_2,m}+T_s)$$

(46)

2.4. Theoretical Calculation Method for Collector Efficiency

The calculation procedures for theoretical predictions of collector efficiency and collector efficiency improvement were described as follows. Firstly, with the given collector geometries ($A_c$ = 0.09 m², H = 0.05 m, h_B = h_f = 0.05 m, L = 0.3 m, W = 0.3 m, l = 0.12 m, l_B = l_s =0.06 m, t_f = 0.002 m, W_B = 0.03 m) and the system properties [${\alpha }_p$ = 0.96, ${\epsilon }_g$ = 0.94, ${\epsilon }_p$ = 0.8, ${\epsilon }_R$ = 0.94, ${\tau }_g$ = 0.875, k_B = k_f = 14.9 W/(m K), k_s = 0.033 W/(m K)], as well as the operating conditions [$\dot{m}$ = 38.52, 57.96 and 77.04 kg/hr, T_in = (30 ± 0.1) °C, T_s = (30 ± 0.1) °C, I₀ = (830 ± 40), and (1100 ± 40) W/m², V = 1.0 m/s], the initial guesses for the average temperatures T_p,m, T_R,m, T_c1,m, T_c2,m are estimated to obtain the heat transfer coefficients and to calculate the temperature distributions of both upper and lower subchannels from Equations (13–17).Secondly, the values of T_p,m, T_R,m, T_c1,m, T_c2,m are then checked by using Equations (10–12,26), respectively, and new iteration values of T_p,m, T_R,m, T_c1,m, T_c2,m may be obtained. If the new calculated values of T_p,m, T_R,m, T_c1,m, T_c2,m are different from the previous iteration values, re-calculated with the proper equations by iteration procedures until the error tolerance is small enough (say less than 10⁻³), and thus the collector efficiencies η_D can be calculated by using Equation (23).The application of recycle-effect concept to a solar air heater actually increases the convective heat-transfer coefficient by producing a preheating effect with the hot outgoing air. Results show that this advantage influence can partially or completely offset the decrement of teat transfer driving force (temperature gradient).The collector efficiency improvement I_D with comparing the collector efficiency between downward-type single-pass device and the present design of solar air heaters is:

$$I_D=\frac{{\eta }_D-{\eta }_S}{{\eta }_S}\times 100\%$$

(47)

in which η_D and η_S, respectively, denote the collector efficiency in baffled double-pass type with fins attached and downward-type single-pass solar air heaters.

3. Apparatus and Experimental Methods

A baffled double-pass solar air heater with external recycle (stainless steel of 7 × 10⁻⁵ m thickness) and with five fins attached on both above and under the absorbing plate for experimental studies, are shown in Figure 1 and Figure 2. The collector (0.3 m long, 0.3 m wide, 0.05 m in height of each subchannel) consists of two glass covers (2.75 × 10⁻³ m thickness and 0.05 m space), a black absorbing plate, baffles welded to the fins, 42 electrical energy supplies (110 V, 125 W) in the part B in Figure 3 and thermal insulation using Styrofoam (0.06 m thickness), as shown in Figure 3 for artificial simulation. The incident solar radiationI₀(arrived irradiation)were measured and recorded with an Epply Laboratory Inc. pyranometer and an Instantaneous Solar Radiation meter Model No. 455 also from Epply Laboratory Inc. Before air entered the lower subchannel, a pre-heater was used to ensure a specified temperature [i.e., T_in = (30 ± 0.1) °C] at the inlet to mix the recycle airflow from the lower subchannel regulated by an adjustable valve. There are two air boxes (0.3 m × 0.3 m × 0.05 m) with 14 holes (0.015 m diameter) were used for air distribution at inlets of both upper and lower subchannels, respectively. Except for the outer glass cover, all parts of the solar air heater were well insulated thermally to minimize the energy loss.

Figure 3. Schematic diagram of a double-pass solar air heater with artificial simulation.

Figure 3. Schematic diagram of a double-pass solar air heater with artificial simulation.

Airflow was steadily supplied by a blower (Model BL model 552, Redmond Co., Owosso, MI, USA) and the mass flow rate controlled by a transformer was measured with Model 24-611 anemomaster from Kanomax Japan Inc. (Osaka, Japan). The air temperatures of absorbing plate and at the inlet and outlet of the solar collector were measured with thermocouples. The ambient temperature [T_s = (30 ± 0.1) °C] measured at a position 0.15 m above the outer glass cover with a thermocouple was monitored by using an air conditioner with a fan to provide wind flow of 1.0 m/s (as indicated by a Chicago Cenco anemometer, TSI Inc., St. Paul, MN, USA) in the experimental apparatus.

4. Results and Discussion

The essential physical properties employed in this study are presented in Table 1, and the operating conditions are T_a,I = 303 K, and $\dot{m}$ = 38.52, 57.96 and 77.04 kg/h. By substituting the specified values into the appropriate equations, theoretical predictions for collector efficiency and airflow outlet temperature with incident solar radiation, mass flow rate and recycle ratio as parameters were obtained. The theoretical predictions and experimental results are plotted in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9.

Table 1. Physical properties of air at 1atm [8].

T (K)	ρ(kg/m3)	Cp [J/(kg K)]	k [W/(m K)]	μ [kg/(m s)]
273	1.292	1006	0.0242	1.72 × 10−5
293	1.204	1006	0.0257	1.81 × 10−5
313	1.127	1007	0.0272	1.90 × 10−5
333	1.059	1008	0.0287	1.99 × 10−5
353	0.999	1010	0.0302	2.09 × 10−5

Table 1. Physical properties of air at 1atm [8].

T (K)	ρ(kg/m3)	Cp [J/(kg K)]	k [W/(m K)]	μ [kg/(m s)]
273	1.292	1006	0.0242	1.72 × 10−5
293	1.204	1006	0.0257	1.81 × 10−5
313	1.127	1007	0.0272	1.90 × 10−5
333	1.059	1008	0.0287	1.99 × 10−5
353	0.999	1010	0.0302	2.09 × 10−5

Figure 4. Effect of recycle ratio on collector efficiency in a double-pass flat-plate solar air heater with recycle; I₀ = 830 W/m², $\dot{m}$ = 38.52 kg/h.

Figure 4. Effect of recycle ratio on collector efficiency in a double-pass flat-plate solar air heater with recycle; I₀ = 830 W/m², $\dot{m}$ = 38.52 kg/h.

Figure 5. Effect of recycle ratio on collector efficiency in a double-pass flat-plate solar air heater with recycle; I₀ = 830 W/m², $\dot{m}$ = 57.96 kg/h.

Figure 5. Effect of recycle ratio on collector efficiency in a double-pass flat-plate solar air heater with recycle; I₀ = 830 W/m², $\dot{m}$ = 57.96 kg/h.

Figure 6. Effect of recycle ratio on collector efficiency in a double-pass flat-plate solar air heater with recycle; I₀ = 830 W/m², $\dot{m}$ = 77.04 kg/h.

Figure 6. Effect of recycle ratio on collector efficiency in a double-pass flat-plate solar air heater with recycle; I₀ = 830 W/m², $\dot{m}$ = 77.04 kg/h.

Figure 7. Effect of recycle ratio on collector efficiency in a double-pass flat-plate solar air heater with recycle; I₀ = 1100 W/m², $\dot{m}$ = 38.52 kg/h.

Figure 7. Effect of recycle ratio on collector efficiency in a double-pass flat-plate solar air heater with recycle; I₀ = 1100 W/m², $\dot{m}$ = 38.52 kg/h.

Figure 8. Effect of recycle ratio on collector efficiency in a double-pass flat-plate solar air heater with recycle; I₀ = 1100 W/m², $\dot{m}$ = 57.96 kg/h.

Figure 8. Effect of recycle ratio on collector efficiency in a double-pass flat-plate solar air heater with recycle; I₀ = 1100 W/m², $\dot{m}$ = 57.96 kg/h.

Figure 9. Effect of recycle ratio on collector efficiency in a double-pass flat-plate solar air heater with recycle; I₀ = 1100 W/m², $\dot{m}$ = 77.04 kg/h.

Figure 9. Effect of recycle ratio on collector efficiency in a double-pass flat-plate solar air heater with recycle; I₀ = 1100 W/m², $\dot{m}$ = 77.04 kg/h.

This experimental study for recyclic baffled solar air heaters with attaching internal fins was conducted under artificial simulation, as shown in Figure 3. The agreement between the experimental results and theoretical predictions is fairly good, and the operation with recycle-effect device substantially improves the collector efficiency, as confirmed by Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9. In order to validate the accuracy of the experimental results as compared to theoretical predictions, the following definition of error analysis was made:

$$E=\frac{1}{N_{\text{exp}}}{\displaystyle \sum _{i=1}^{N_{\text{exp}}}\frac{\left|{\eta }_{theo,i}-{\eta }_{\text{exp},i}\right|}{{\eta }_{theo,i}}}$$

(48)

where N_exp, η_theo,i and η_exp,i indicate the experimental measurements, theoretical predictions and experimental data of collector efficiencies, respectively. The error analysis may be calculated with the results in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 by using Equation (48). The error analysis of the experimental measurements is of 3.19 × 10⁻² ≦ E ≦ 9.86 × 10⁻² and presented in Table 2.

The theoretical predictions of $I_D$ for the system of the present interest were also calculated by Equation (47), and the results are also listed in Table 3 and Table 4. It is shown that the considerable improvement in collector efficiency is obtained if the operation is carried out with fins and baffles attached under external recycle.

Table 2. The accuracy analysis of the experimental results.

I0 (W/m2)	$\dot{m}$ (kg/h)	Recycle	Recycle + Fins	Recycle + Fins + Baffles
830	38.52	8.79× 10−2	6.80× 10−2	3.20× 10−2
	57.96	3.83× 10−2	5.24× 10−2	3.44× 10−2
	77.04	4.41× 10−2	3.40× 10−2	3.19× 10−2
1100	38.52	9.86× 10−2	8.53× 10−2	3.96× 10−2
	57.96	4.57× 10−2	8.50× 10−2	3.41× 10−2
	77.04	4.41× 10−2	5.10× 10−2	4.18× 10−2

Table 2. The accuracy analysis of the experimental results.

I0 (W/m2)	$\dot{m}$ (kg/h)	Recycle	Recycle + Fins	Recycle + Fins + Baffles
830	38.52	8.79× 10−2	6.80× 10−2	3.20× 10−2
	57.96	3.83× 10−2	5.24× 10−2	3.44× 10−2
	77.04	4.41× 10−2	3.40× 10−2	3.19× 10−2
1100	38.52	9.86× 10−2	8.53× 10−2	3.96× 10−2
	57.96	4.57× 10−2	8.50× 10−2	3.41× 10−2
	77.04	4.41× 10−2	5.10× 10−2	4.18× 10−2

Table 3. Theoretical predictions of heat-transfer efficiency improvement I_D for I₀ = 830 W/m².

R	Recycle			Recycle + Fins			Recycle + Fins + Baffles
	$\dot{m}$ (kg/h)
-	38.52	57.96	77.04	38.52	57.96	77.04	38.52	57.96	77.04
0.00	29.13	9.19	5.80	30.42	40.54	15.70	88.35	79.19	65.46
0.25	50.16	41.35	35.27	52.75	52.97	36.47	111.97	82.16	67.15
0.50	67.64	54.86	46.38	69.58	60.54	47.10	117.48	84.05	69.81
0.75	78.96	62.97	52.66	80.58	65.68	53.38	120.06	85.14	71.01
1.00	86.73	68.38	56.76	88.35	68.92	57.25	121.68	85.68	71.50
1.25	92.88	72.43	59.66	93.85	72.16	59.90	122.98	86.49	71.98
1.50	97.41	75.14	61.59	98.38	74.05	62.08	123.62	86.76	72.22
1.75	100.97	77.57	63.29	101.62	74.86	63.53	123.95	87.03	72.46

Table 3. Theoretical predictions of heat-transfer efficiency improvement I_D for I₀ = 830 W/m².

R	Recycle			Recycle + Fins			Recycle + Fins + Baffles
	$\dot{m}$ (kg/h)
-	38.52	57.96	77.04	38.52	57.96	77.04	38.52	57.96	77.04
0.00	29.13	9.19	5.80	30.42	40.54	15.70	88.35	79.19	65.46
0.25	50.16	41.35	35.27	52.75	52.97	36.47	111.97	82.16	67.15
0.50	67.64	54.86	46.38	69.58	60.54	47.10	117.48	84.05	69.81
0.75	78.96	62.97	52.66	80.58	65.68	53.38	120.06	85.14	71.01
1.00	86.73	68.38	56.76	88.35	68.92	57.25	121.68	85.68	71.50
1.25	92.88	72.43	59.66	93.85	72.16	59.90	122.98	86.49	71.98
1.50	97.41	75.14	61.59	98.38	74.05	62.08	123.62	86.76	72.22
1.75	100.97	77.57	63.29	101.62	74.86	63.53	123.95	87.03	72.46

Table 4. Theoretical predictions of heat-transfer efficiency improvement I_D for I₀ = 1100W/m².

R	Recycle			Recycle + Fins			Recycle + Fins + Baffles
	$\dot{m}$ (kg/h)
-	38.52	57.96	77.04	38.52	57.96	77.04	38.52	57.96	77.04
0.00	29.22	9.21	7.11	30.19	40.65	17.16	84.42	79.40	67.16
0.25	50.32	41.46	37.25	52.92	53.12	38.48	111.69	82.38	69.36
0.50	67.86	55.01	48.28	69.81	60.70	49.02	117.53	84.28	72.06
0.75	78.90	63.14	54.66	80.52	65.85	55.15	120.13	85.37	73.28
1.00	87.01	68.56	58.82	88.31	69.11	59.31	122.08	85.91	73.77
1.25	92.86	72.36	61.76	93.83	72.36	62.01	123.05	86.72	74.26
1.50	97.40	75.34	63.73	98.38	74.25	64.22	123.70	86.99	74.51
1.75	100.97	77.51	65.44	101.95	75.07	65.69	124.03	87.26	74.75
2.00	103.90	79.40	66.67	104.55	75.61	66.91	124.35	87.53	75.00

Table 4. Theoretical predictions of heat-transfer efficiency improvement I_D for I₀ = 1100W/m².

R	Recycle			Recycle + Fins			Recycle + Fins + Baffles
	$\dot{m}$ (kg/h)
-	38.52	57.96	77.04	38.52	57.96	77.04	38.52	57.96	77.04
0.00	29.22	9.21	7.11	30.19	40.65	17.16	84.42	79.40	67.16
0.25	50.32	41.46	37.25	52.92	53.12	38.48	111.69	82.38	69.36
0.50	67.86	55.01	48.28	69.81	60.70	49.02	117.53	84.28	72.06
0.75	78.90	63.14	54.66	80.52	65.85	55.15	120.13	85.37	73.28
1.00	87.01	68.56	58.82	88.31	69.11	59.31	122.08	85.91	73.77
1.25	92.86	72.36	61.76	93.83	72.36	62.01	123.05	86.72	74.26
1.50	97.40	75.34	63.73	98.38	74.25	64.22	123.70	86.99	74.51
1.75	100.97	77.51	65.44	101.95	75.07	65.69	124.03	87.26	74.75
2.00	103.90	79.40	66.67	104.55	75.61	66.91	124.35	87.53	75.00

The collector efficiency improvement increases with increasing incident solar radiation and recycle ratio as well as mass flow rate owing to turbulence increment and forced convective heat-transfer coefficient enhancement, as shown in Table 3 and Table 4, especially for operating at larger air flow rate $\dot{m}$. The improvement in collector efficiency by recycle operation is obtained more than 64% as compared to downward-type single-pass device. In addition, the larger incident solar radiation is associated with the higher absorbing plate temperature of the solar air collector, resulting in the increment of the heat transfer efficiency. For instances, the heat-transfer efficiency improvements with fins and baffles for $\dot{m}=77.04$ kg/h and R = 2, are 72.71% and 75.0% for $I_0$ = 830 and 1100 W/m², respectively. Accordingly, the order of heat transfer efficiencies of flat plate collectors are: collector with fins and baffles> collector with fins > collector without fins and baffles, as indicated in Table 3 and Table 4.

The power consumptions of the solar air heater with fins attached are calculated by the correlation equation. The power consumptions for downward-type single-pass and double-pass solar air heaters without attached baffles, respectively, are as follows:

$$P_S=m{\ell }_{wf,s}=m\frac{2f_F{\overline{v_0}}^{2}L}{D_e}$$

(49)

$$P_D=m{\ell }_{wf,a}+m(1+R){\ell }_{wf,b}=m\frac{2f_F{\overline{v_a}}^{2}L}{D_{e,a}}+m(1+R)\frac{2f_F{\overline{v_b}}^{2}L}{D_{e,b}}$$

(50)

in which:

$$f_F=\frac{24}{\text{Re}},\text{ for laminar flow}$$

(51)

$$f_F=0.0790{\text{Re}}^{-0.25},\text{ for turbulent flow},\text{ Re}<{10}^5$$

(52)

The power consumption increment, $I_P$, may be defined as:

$$I_P=\frac{P_D-P_S}{P_S}$$

(53)

The hydraulic dissipated energy in a single-pass device without recycle are $P_S=7.68\times {10}^{-6}$, $2.36\times {10}^{-5}$ and $5.23\times {10}^{-5}$ W, respectively, for $\dot{m}=38.52$, 57.96 and 77.04 kg/h for mass flow rate. The suitable selections of the operating parameters (R and $\dot{m}$) on considering of both the heat-transfer efficiency improvement, $I_D$, and the power consumption increment, $I_P$, are presented in Figure 10.

Figure 10. I_D/I_P vs. recycle ratio R with mass flow rate as a parameter.

Figure 10. I_D/I_P vs. recycle ratio R with mass flow rate as a parameter.

The economically optimal design must take into account the costs of energy and devices. However, for simplicity, only the power consumption is considered in this study. The index of $I_D$/$I_P$ is adopted for comparing different designs. Operating double-pass solar air heaters with recycle at $R_{optimal}\approx 0.5$ gives the maximum value of $I_D$/$I_P$, as seen from Figure 10, while the values of $I_D$/$I_P$ decrease as the recycle ratio moves away from 0.5. A remark must be stated that even if only energy criterion is to be considered, because thermal energy and electric power are different in terms of the quality of energy, employing exergy analysis can provide different results from those shown in Figure 10.

The present work is actually the extension previous work [18] except the configuration of recycle. Figure 11 illustrates the graphical representation for comparisons with some experimental results and theoretical predictions obtained in [18] under the same design and operating parameters. With those comparisons, the advantage this study is evident for all mass flow rates and recycle ratios.

Figure 11. Comparisons of collector efficiency among the present and previous works [18].

Figure 11. Comparisons of collector efficiency among the present and previous works [18].

5. Conclusions

As demonstrated by theoretical and experimental studies, the introduction of external recycle has positive effects on the heat transfer efficiency of solar air heaters. Moreover, the advantages of baffled solar air heaters with internal fins attached provide higher turbulence and extending the heat transfer area, and thus, the enhancement of heat-transfer efficiency. It is believed that the availability of such a mathematical formulation as developed here for a recycling baffled solar air heater with internal fins attached is an important contribution to the design and analysis of double-pass devices with internal or external recycle. Although the lager mass flow rate and recycle ratio are beneficial to collector efficiency improvement, higher energy dissipation occurs at the same time when operating at a higher mass flow rate and recycle ratio. The value of $I_D$/$I_P$ vs. recycle ratio R was represented with the mass flow rate as a parameter in Figure 10 to recommend an optimal design. After all, both the heat-transfer efficiency improvement and power consumption increment are significant factors to be taken into account simultaneously in designing recyclic double-pass solar air heaters.