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The collector efficiency in a downward-type double-pass external-recycle solar air heater with fins attached on the absorbing plate has been investigated theoretically. Considerable improvement in collector efficiency is obtainable if the collector is equipped with fins and the operation is carried out with an external recycle. Due to the recycling, the desirable effect of increasing the heat transfer coefficient compensates for the undesirable effect of decreasing the driving force (temperature difference) of heat transfer, while the attached fins provide an enlarged heat transfer area. The order of performances in the devices of same size is: double pass with recycle and fins > double pass with recycle but without fins > single pass without recycle and fins.

In addition to the essential effects of free and forced convections [

Consider a recycling double-pass solar air heater with n fins attached on the absorbing plate, as shown in _{i}_{fo}

Schematic diagram of a downward-type double-pass external-recycle solar air heater with fins attached.

The steady-state energy balance for a differential section of the absorbing plate with n fins attached, bottom plate and flowing fluid are, respectively:
_{f}_{2}_{c}_{f}_{2} and _{s} are the height of a fin and thermal conductivity of the absorbing plate and fins, respectively, while

Solving Equations (1) and (2) for (_{p} − T_{f}_{1}) and (_{R} − T_{f}_{1}):

The detailed explanation in the derivations of Equations (7) and (8) are expressed mathematically and shown in _{t}_{t}

The result is:

Equation (12) is the temperature distribution of the bulk fluid along the flow direction of subchannel 1. Thus, the fluid temperature at the outlet of subchannel 1 is readily obtained from Equation (12) by substituting the condition: _{f1}_{f}_{c}_{1}

The differential energy-balance equation for the fluid in subchannel 2 is readily obtained by following the same procedure from Equation (1) through Equation (13) with _{f}_{1} and (_{f}_{1}_{f}_{2} and-(_{f}_{2}

Integrating Equation (14) with the use of the boundary condition:
_{f}_{2} = _{fo}

Thus, in addition to Equation (13), the fluid temperature _{f}_{f2}_{f}

The fluid outlet temperature is readily obtained by combining Equations (13) and (17) to eliminate _{f}_{u}

Inspection of Equation (19) shows that the mixed inlet temperature _{i}^{0} due the recycle is not specified a prior. The relation for the mixing effect at the inlet in Equation (20) was used for determination of this value. Thus, a combination of Equations (19) and (20) gives:

The collector efficiency may be defined as:

Substitution of Equation (18) into Equation (22) gives:

Substitution of Equation (21) into Equation (24) with the aid of Equation (18) to eliminate _{i}^{0} and _{fo}

Once the collector efficiency is determined, the fluid outlet temperature is readily obtainable from Equation (23),

The mean fluid and absorbing-plate temperatures are needed for calculating the heat-transfer coefficients. The mean-fluid temperature may be defined as:

Substituting Equations (12) and (16) into Equation (28) and integrating, we have:

With the use of Equations (23) and (24) to eliminate _{fo}_{i}^{0}, respectively, _{fm}_{cf}

The mean absorbing-plate temperature may be defined in term _{cf}

Equations (30) and (32) are the expressions of _{fm}_{pm}_{cf}

The convective heat-transfer coefficient _{w}_{w}

An empirical equation for the loss coefficient from the top of the solar collector to the ambient _{L}

The radiation coefficient between the two air-duct surfaces may be estimated by assuming a mean radiant temperature equal to the mean fluid temperature [

In the study of solar air heaters and collector-storage walls, it is necessary to know the forced convection heat-transfer coefficient between two flat plates. For air, the following correlation may be derived from Kays’ data for fully developed turbulent flow with one side heated and the other side insulated [_{e}/k^{0.8}

Thus, from Equations (37) and (38), one obtains the Reynolds numbers for the rectangular ducts as:

The calculation of prediction values for collector efficiency and outlet fluid temperature is now described as follows. First, with known collector geometries (_{cf}_{fm}_{pm}_{fm}_{pm}_{fm}_{pm}_{fm}_{pm}_{cf}_{cf}

The following flow sheet may be helpful for simply expressing the calculation procedure:

The improvement in performance of an external-recycle double-pass solar air heater with n fins attached on the absorbing plate may be illustrated numerically by using _{c}_{1} = 0.05 m; _{2} = 0.02 m; _{s} = 45 W/mK; _{g} = 0.875; _{p}_{g}_{p}_{R}_{0} = 830 and 1100W/m^{2}; _{i}_{a}^{–8} W/m^{2} K^{4}.

Physical properties of air at 1 atm [

^{3}) |
_{p} |
|||
---|---|---|---|---|

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} |

By substituting the specified values into the appropriate equations with the use of

As might be expected, the outlet air temperature _{fo}_{i}_{cf}

Air outlet temperature (_{i} = 288 K).

Air outlet temperature (_{i} = 298 K).

Collector efficiency (_{i} = 288 K).

Collector efficiency (_{i} = 298 K).

On the other hand, the outlet air temperature decreases when the air flow rate _{0} increases. These results are shown in

The improvements _{f}_{cf}_{c}_{0} obtained in the single-pass device of same size, operated without recycling and fins attached,

The theoretical values of _{f}_{0} [_{i}_{i}_{0} = 830 and 1100 W/m^{2}. It is concluded that the contribution of the desirable effect of increasing fluid velocity by applying the external-recycle operation may be more effective than the undesirable effect of lowering the temperature difference.

The improvement of performance for: (_{0} = 830 W/m^{2}; (_{0} = 1100 W/m^{2}.

_{i} |
_{f} |
|||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

(a) | 288 | 0.01 | 24.60 | 58.94 | 88.30 | 111.92 | 123.02 | 36.49 | 68.91 | 97.68 | 111.97 | |

0.015 | 30.35 | 50.37 | 73.58 | 91.67 | 99.99 | 31.04 | 57.63 | 80.42 | 91.90 | |||

0.02 | 34.65 | 44.47 | 63.88 | 78.66 | 85.35 | 27.32 | 50.30 | 69.63 | 79.00 | |||

293 | 0.01 | 23.28 | 58.94 | 88.40 | 119.22 | 123.34 | 36.52 | 69.34 | 97.93. | 112.71 | ||

0.015 | 28.75 | 50.41 | 73.76 | 92.01 | 100.42 | 31.10 | 58.08 | 80.90 | 92.56 | |||

0.02 | 32.84 | 44.50 | 64.07 | 79.02 | 85.79 | 27.40 | 50.67 | 70.29 | 79.82 | |||

298 | 0.01 | 21.96 | 58.85 | 88.39 | 112.26 | 123.49 | 36.51 | 69.40 | 98.25 | 112.82 | ||

0.015 | 27.14 | 50.30 | 73.77 | 92.15 | 100.63 | 31.12 | 58.09 | 81.16 | 93.01 | |||

0.02 | 31.02 | 44.41 | 64.11 | 79.20 | 86.05 | 27.42 | 50.69 | 70.30 | 79.95 | |||

(b) | 288 | 0.01 | 23.92 | 61.21 | 92.15 | 117.08 | 119.03 | 36.49 | 68.85 | 97.53 | 111.77 | |

0.015 | 29.73 | 52.32 | 76.71 | 95.70 | 104.45 | 31.04 | 57.63 | 80.28 | 91.70 | |||

0.02 | 34.11 | 46.09 | 66.39 | 81.85 | 88.85 | 27.32 | 50.23 | 69.48 | 78.82 | |||

293 | 0.01 | 22.90 | 61.29 | 92.34 | 117.39 | 129.17 | 36.52 | 69.19 | 97.61 | 112.27 | ||

0.015 | 28.48 | 52.42 | 83.39 | 96.01 | 104.92 | 31.10 | 57.94 | 80.60 | 92.15 | |||

0.02 | 32.69 | 46.25 | 66.71 | 82.33 | 89.41 | 27.40 | 50.52 | 69.98 | 79.42 | |||

298 | 0.01 | 21.87 | 61.38 | 92.52 | 117.69 | 129.54 | 36.51 | 69.17 | 97.75 | 112.16 | ||

0.015 | 27.21 | 52.58 | 77.23 | 96.52 | 105.43 | 31.12 | 57.88 | 80.70 | 92.37 | |||

0.02 | 31.26 | 46.33 | 66.92 | 82.69 | 89.85 | 27.42 | 50.49 | 69.86 | 79.38 |

The improvements _{0} = 1100 W/m^{2} and _{f}

Equation (43) may be rewritten using Equations (41) and (42) as:

Some values of _{0}, while decreases when the air flow sate _{i}

Further enhancement in collector efficiency with fin: (_{0} = 830 W/m^{2}; (_{0} = 1100 W/m^{2}.

_{i} |
||||||
---|---|---|---|---|---|---|

(a) | 288 | 0.01 | 16.45 | 11.48 | 7.20 | 5.21 |

0.015 | 14.75 | 10.12 | 6.24 | 4.22 | ||

0.02 | 13.47 | 9.04 | 5.32 | 3.55 | ||

298 | 0.01 | 16.37 | 11.21 | 7.07 | 5.01 | |

0.015 | 14.63 | 9.92 | 6.07 | 3.95 | ||

0.02 | 13.33 | 8.91 | 5.23 | 3.39 | ||

(b) | 288 | 0.01 | 18.11 | 13.80 | 9.90 | 7.96 |

0.015 | 16.24 | 12.10 | 8.55 | 6.64 | ||

0.02 | 14.74 | 10.76 | 7.30 | 5.57 | ||

298 | 0.01 | 18.22 | 13.80 | 10.08 | 8.19 | |

0.015 | 16.37 | 12.26 | 8.75 | 6.79 | ||

0.02 | 14.84 | 10.92 | 7.55 | 5.84 |

The performance in a double-pass solar air heater with external recycling was investigated in a previous work [_{u}_{cf}_{fo}_{cf}_{fo}_{u}

In addition to the double-pass operation with external recycling, further enhancement in collector efficiency is obtainable if the operation is carried out also with fins attached on the absorbing plate. The further enhancement E based on the device without fins, reaches 13.8% for _{0} = 1100 W/m^{2}, _{i}

The improvements of collector efficiencies, _{f}_{f}

_{c}

surface area of the absorbing plate, ^{2})

the width of absorber surface area, _{1} (m)

_{p}

specific heat of air at constant pressure (J/kg K)

_{e}

equivalent diameter of the channel (m)

further improvement in collector efficiency

efficiency factor of the solar air heater

_{R}

heat-removal factor for the solar air heater

height of the air tunnel in the solar collector, or the distance between glass cover and absorbing plate (m)

convective feat-transfer coefficient for fluid flowing over the plate of duct (W/m^{2} K)

_{p−R}

radiant heat-transfer coefficient between two parallel plates (W/m^{2} K)

_{w}

convective heat-transfer coefficient between glass cover and ambient (W/m^{2} K)

improvement in collector efficiency in the device without fins

_{f}

improvement in collector efficiency in the device with fins attached

_{o}

solar radiation incident (W/m^{2})

thermal conductively of air (W/m K)

_{s}

thermal conductivity of absorbing plate and fins (W/m K)

collector length (m)

mass flow-rate of air (kg/s)

quantity defined by Equation (6)

fin number

Nusselt number

_{u}

useful gain of energy carried away by air per unit time (W)

reflux ratio

Reynolds number of flow channel

temperature (K)

fin thickness (m)

_{t}

loss coefficient from the top of solar collector to the ambient (W/m^{2} K)

average air velocity in the flow channel (m/s)

wind velocity (m/s)

_{1}

distance between fins (m)

_{2}

height of fin (m)

axis along the flow direction (m)

collector efficiency

the Stefan–Boltzmann constant (W/m^{2} K^{4})

_{g}

emissivity of glass cover

_{p}

emissivity of absorbing plate

_{R}

emissivity of bottom plate

_{g}

transmittance of glass cover

dimensionless quantity defined by Equation (4)

_{p}

absorptivity of the absorbing plate

ambient

fluid

inlet

mean value

outlet at subchannel 2 (

absorbing plate

bottom plate

subchannel 1, subchannel 2

mixed

outlet at the first pass or inlet at subchannel 2 (

The authors wish to thank the National Science Council of the Republic of China for its financial support.

The detailed explanation in the derivations of Equations (7) and (8).

One can rewrite Equations (1) and (2), respectively, as follows:

Rearranging Equations (A1) and (A2) into new expressions, one can obtain, respectively:

Combination of Equations (A3) and (A4) gives:

Similarly, combination of Equations (A3) and (A5) gives: