Flat-Plate Solar Collectors. Optimization of Absorber Geometry

Abstract

A rather involved flat-plate solar collector model is used. The width and thickness of fins is optimized by minimizing the cost per unit useful heat flux. Fins of both uniform and variable thickness are considered. In case of fins with uniform thickness, the optimum distance between tube centre decreases by increasing the operation temperature, while the optimum fin thickness is relatively the same. The best economical performance is obtained in case of fins with optimized thickness. The optimum fin cross-section is very close to an isosceles triangle. The fin width is shorter and the seasonal influence is weaker at lower operation temperatures. Fin width and thickness at base depend on season. The optimum distance between the tubes increases by increasing the inlet fluid temperature and it is larger in the cold season than in the warm season.

Appendices

Appendix 15A

Details are shown about the flat-plate solar collector model used in calculations. The reference is Duffie and Beckman (1974).

1.1 15.A.1 Optical Efficiency

A transparent cover consisting of $N$ identical layers is considered. Radiation is incident on the transparent cover at incidence angle $\theta_{1}$. The relative refractive indexes of transparent layer material and of the medium from where radiation is coming are denoted $n_{2}$ and $n_{1} \;\left( { \approx 1} \right)$, respectively. Then, the refraction angle $\theta_{2}$ of the radiation inside the transparent layer is computed by using the refraction law:

$$ \frac{{\sin \,\theta_{2} }}{{\sin \,\theta_{1} }} = \frac{{n_{1} }}{{n_{2} }} $$

(15.A.1)

The reflectance $\rho$ of the transparent layer is given (for unpolarized radiation) by the following Fresnel formula :

$$ \rho = \frac{1}{2}\left[ {\frac{{\sin^{2} \left( {\theta_{2} - \theta_{1} } \right)}}{{\sin^{2} \left( {\theta_{2} + \theta_{1} } \right)}} + \frac{{\tan^{2} \left( {\theta_{2} - \theta_{1} } \right)}}{{\tan^{2} \left( {\theta_{2} + \theta_{1} } \right)}}} \right] $$

(15.A.2)

The transparent cover transmittance due to reflection, $\tau_{r,N}$, is computed by

$$ \tau_{r,N} = \frac{1 - \rho }{{1 + \left( {2N - 1} \right)\rho }} $$

(15.A.3)

Denote $k_{abs}$ and $a$ the absorption factor and the thickness of one transparent layer, respectively. The actual path of radiation $L_{1}$ through a single transparent layer is given by

$$ L_{1} = \frac{a}{{\cos \theta_{2} }} $$

(15.A.4)

The transparent cover transmittance due to absorption,$\tau_{a,N}$, is computed by Beer-Bouguer-Lambert law :

$$ \tau_{a,N} = \exp \left( { - k_{abs} NL_{1} } \right) $$

(15.A.5)

and the total cover transmittance $\tau$ is given by

$$ \tau = \tau_{r,N} \tau_{a,N} $$

(15.A.6)

Let $\alpha$ be the absorptance of the absorber plate. The transmittance-absorptance product $\left( {\tau \alpha } \right)_{N}$ of the collector takes account of multiple scattering of radiation between the transparent layers and the absorber plate:

$$ \left( {\tau \alpha } \right)_{N} = \frac{\tau \alpha }{{1 - \left( {1 - \alpha } \right)\rho_{d,N} }} $$

(15.A.7)

where $\rho_{d,N}$ is the diffuse reflectance taking the values 0.16, 0.24, 0.29 and 0.32 for 1, 2, 3 and 4 transparent layers, respectively.

All of the solar radiation that is absorbed by a cover system is not lost, since this absorbed energy tends to increase layers temperature and consequently reduce the losses from the plate. Let $\varepsilon_{p}$ be the emittance of the absorber plate. A general analysis for a cover system yields the following expression for the optical efficiency $\eta_{0}$ of the collector

$$ \eta_{0} \left( N \right) = \left( {\tau \alpha } \right)_{N} + \left( {1 - \tau_{a,1} } \right)\sum\limits_{i = 1}^{N} {a_{i} \left( {N,\varepsilon_{p} } \right)\tau^{i - 1} } $$

(15.A.8)

Here $a_{i}$ is the ratio of the overall loss coefficient to the loss coefficient from the i layer to the surroundings, tabulated in Duffie and Beckman (1974, p. 156, Table 7.9.1). The optical efficiency is sometime referred to as the effective transmittance-absorptance product .

1.2 15.A.2 Overall Heat Loss Coefficient

The overall heat loss coefficient $U_{L}$ is given by

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

(15.A.9)

where $U_{b}$ is the bottom heat loss coefficient , given by

$$ U_{b} = \frac{{k_{b} }}{{L_{b} }} $$

(15.A.10)

where $k_{b}$ and $L_{b}$ are the thermal conductivity and the thickness of the bottom insulation, respectively. For a glazed solar collector , the top heat loss coefficient $U_{t}$ in Eq. (15.A.9) is given by

$$ U_{t} = \hat{U}_{t} \left[ {1 - \left( {s - 45} \right)\left( {0.00259 - 0.00144\varepsilon_{p} } \right)} \right] $$

(15.A.11)

where $\hat{U}_{t}$ is the top heat loss coefficient for a collector tilted 45° while s is collector actual tilt in degrees. The empirical relation proposed in Duffie and Beckman (1974) is used here for $\hat{U}_{t}$:

$$ \hat{U}_{t} = \left[ {\frac{N}{{\frac{244}{{T_{p} }}\left( {\frac{{T_{p} - T_{a} }}{N + f}} \right)^{0.31} }} + \frac{1}{{h_{w} }}} \right]^{ - 1} + \frac{{\sigma \left( {T_{p} + T_{a} } \right)\left( {T_{p}^{2} + T_{a}^{2} } \right)}}{{\left[ {\varepsilon_{p} + 0.0425N\left( {1 - \varepsilon_{p} } \right)} \right]^{ - 1} + \frac{2N + f - 1}{{\varepsilon_{g} }} - n}} $$

(15.A.12)

Here $T_{p}$ and $T_{a}$ are the space averaged absorber temperature and ambient temperature, respectively, $\varepsilon_{g}$ is glass emittance, $h_{w}$ [W/(m²K)] is the convection heat loss coefficient due to the wind speed $w_{wind}$ [m/s]. In practice we used. $h_{w} = 5.7 + 3.8w_{wind}$. Also, $\sigma$ is Stefan-Boltzmann constant and $f = \left( {1 - 0.04h_{w} + 5 \times 10^{ - 4} h_{w}^{2} } \right)\left( {1 + 0.058\,{\text{N}}} \right)$.

Note that in case of collectors with straight fins with rectangular profile $U_{L}$ and $T_{p}$ are computed together by using an iterative procedure shown later in Sect. 15.A.4 of this Appendix 15A. When fins of variable thickness are considered a simpler iterative procedure was used (see Sect. 15.2.5 of the paper). This is possible because the heat removal factor does not enter the calculations in this second case.

1.3 15.A.3 Collector Heat Removal Factor

A register-type collector is considered here. Then, $d$ is tube external diameter and $W$ is the distance between the centers of two neighbor tubes. Let $\delta_{p}$ and $k_{p}$ be plate thickness and its material thermal conductivity, respectively. The standard fin efficiency $F$ for straight fins with rectangular profile is given by:

$$ F = \left[ {\frac{{m\left( {W - d} \right)}}{2}} \right]^{{ - 1}} \tan \left[ {\frac{{m\left( {W - d} \right)}}{2}} \right],\quad \left( {m \equiv \sqrt {\frac{{U_{L} }}{{k_{p} \delta _{p} }}} } \right) $$

(15.A.13,14)

The collector efficiency factor $F^{{\prime }}$ is given by:

$$ F^{{\prime }} = \left( {\frac{1}{{WU_{L} }}} \right)\left\{ {\frac{1}{{U_{L} \left\{ {d + \left( {W - d} \right)F} \right\}}} + \frac{1}{{C_{b} }} + \frac{1}{{\pi d_{i} h_{fi} }}} \right\}^{ - 1/2} $$

(15.A.15)

Here $C_{b}$ is bond conductance, $d_{i}$ is the inside tube diameter and $h_{fi}$ is the heat transfer coefficient between the working fluid and the tube wall. Here $d_{i} = d - 2\delta_{p}$ is used. The working fluid is formally equivalent to water and the following empirical formula was used to evaluate $h_{fi}$ (Carabogdan et al. 1978, p. 54)

$$ h_{fi} = \left( {1430 + 23.3t - 0.048t^{2} } \right)w_{water}^{0.8} d_{i}^{ - 0.2} $$

(15.A.16)

where $t \equiv T_{f,m} - 273.15$ (with $T_{f,m}$ [K]—the average working fluid temperature inside the tube) and $w_{water}$ [m/s] is water speed in the tube. In Eq. (15.A.16) the unit for $d_{i}$ is [m]. The following common value was adopted during calculations

$$ w_{water} = 0.1\,{\text{m/s}} $$

(15.A.17)

In Eq. (15.A.16), $T_{f,m}$ was evaluated as a function of the working fluid temperatures at collector inlet and outlet, $T_{f,i}$ and $T_{f,out}$, respectively, by:

$$ T_{f,m} = \left( {T_{f,i} + T_{f,out} } \right)/2 $$

(15.A.18)

The energy balance of the fluid of mass flow rate $\dot{m}$ yields:

$$ T_{f,out} - T_{f,i} = \frac{{Q_{u} }}{{\dot{m}c_{p} }} = \frac{{Q_{u} /A}}{{\left( {\dot{m}/A} \right)c_{p} }} = \frac{{q_{u} }}{{\dot{m}^{{\prime }} c_{p} }} $$

(15.A.19)

Here $Q_{u}$ is the useful heat provided by the collection area $A_{c}$ and $q_{u} \left( { \equiv Q_{u} /A} \right)$ is the useful heat per unit area given by the following formula (15.A.21).

The collector removal factor $F_{R}$ is given by

$$ F_{R} = \frac{{\dot{m}^{{\prime }} c_{p} }}{{U_{L} }}\left[ {1 - \exp \left( { - \frac{{U_{L} F^{{\prime }} }}{{\dot{m}^{{\prime }} c_{p} }}} \right)} \right] $$

(15.A.20)

One reminds that $U_{L}$ entering Eq. (15.A.20) is a function of the unknown space averaged collector temperature $T_{p}$ that may be evaluated from two equivalent expressions of collector energy balance:

$$ q_{u} = \left[ {\eta_{0} - U_{L} \left( {T_{p} - T_{a} } \right)} \right] = F_{R} \left\{ {\eta_{0} - U_{L} \left( {T_{f,i} - T_{a} } \right)} \right\} $$

(15.A.21)

One easily finds:

$$ T_{p} = T_{a} + \frac{{\eta_{0} \left( {1 - F_{R} } \right)}}{{U_{L} }} + F_{R} \left( {T_{f,i} - T_{a} } \right) $$

(15.A.22)

Note that when $T_{f,out} - T_{f,i}$ is given, an iterative procedure is needed to evaluate the quantity $\dot{m}' \equiv \dot{m}/A$. In practice one starts with a guessed value for $\dot{m}^{{\prime }}$. This is used as an entry in Eqs. (15.A.20) and (15.A.21) to evaluate $F_{R}$ and $q_{u}$, respectively. Next, from Eq. (15.A.19) a new value of $\dot{m}^{{\prime }}$ is obtained. This last value is compared with the guessed value and if significantly different a new iteration is performed with the new value of $\dot{m}^{{\prime }}$ as an entry. Finally, note that once $\dot{m}^{{\prime }}$ is known the shape of the collection area may be obtained provided the mass flow rate $\dot{m}$ is also given (Sect. 15.A.5 in this Appendix 15A).

1.4 15.A.4 Iterative Procedure

The quantities $U_{L} ,F,F^{{\prime }} ,F_{R}$ and the temperature $T_{p}$ are evaluated all together through the following iterative procedure with $T_{a}$ and $T_{f,i}$ as input (given) parameters. A guessed value for $T_{p}$ is first adopted. Next, $U_{L} ,F,F^{{\prime }}$ and $F_{R}$ are evaluated from Eqs. (15.A.12), (15.A.13), (15.A.15) and (15.A.20), respectively. Finally, a new value for $T_{p}$ is obtained from Eq. (15.A.22). It is compared with the guessed $T_{p}$ value and if they differ significantly the procedure is repeated by using the new $T_{p}$ value as entry.Note that $T_{a}$ and $T_{f,i}$ in this Appendix A correspond to $T_{a}^{*}$ and $T_{f,i}^{*}$, respectively, in Sect. 15.2.

1.5 15.A.5 Shape of Collection Area

A simple rectangular form of width l and length L may be adopted for the collection surface area $A\left( { = lL} \right)$. The number of parallel tubes on that surface is $l/W$. Note that the mass flow rate in a tube is $\dot{m}_{tube} = \rho_{water} w_{water} \left( {\pi D_{i}^{2} /4} \right)$, where $\rho_{water}$ is the mass density of water. The total mass flow rate on the collection area is $\dot{m} = \dot{m}_{tube} l/W$. With known values for $\dot{m}^{{\prime }}$ and $\dot{m}$, these relationships allow to find $A,l$ and $L = A/l$.

Appendix 15B

Table 15.B.1 gives the values adopted for the flat-plate solar collector system treated in this paper. Quantities not included in this table, such as the distance $W$ between the tubes and the thickness $\delta$ of the absorber plate, are subjected to change and their values are explicitly given in the text.

Table 15.B.1 Values adopted for the flat-plate solar collector