Energetic Optimization of the Flat Plate Solar Collector

Abstract

In this chapter, the Taguchi method combined with response surface methodology is introduced to energetically optimize various design parameters of the flat plate solar collector. The design parameters of a flat plate collector are the key factors affecting its performance. The effect of the ambient temperature, solar radiation, and wind speed is also included in the design array to achieve a robust configuration. The Taguchi method results showed that the number of collector flow tubes and the back insulation thickness are the most significant factors in the performance characteristics. Since the Taguchi method optimizes only the performance responses individually. Therefore, the data are then preprocessed by the response model approach based on the I-optimal computer design using the coordinate exchange algorithm. The results showed the ability of the proposed method for optimizing the performance of various components of the solar water heating system in particular and renewable energy systems in general.

1 Introduction

The solar collector is the core component of the solar water heating system [1] where the flat plate collector type is the most economical and popular solar collector type since it’s simple in construction and requires less maintenance [2]. Typically the flat plate solar collector consists of an absorber plate painted with a selective coating material to enhance its absorptivity, transparent glass cover (glazing), insulation material in order reduce the losses to the environment, flow tubes welded to the absorber plate, and galvanized steel frame to hold the mentioned components [3] as shown in Fig. 39.1.

Fig. 39.1

The main structure of the flat plate collector [3]

Over the last decades, the researchers have been working on the analysis and development of the flat plate collector. Alvarez et al. [4] conducted an experimental and numerical analysis based on the finite element method for a newly designed flat plate solar collector with a corrugated channel. Besides, Hobbi and Siddiqui [5] performed an experimental study to investigate the impact of various passive heat enhancement devices based on shear-produced turbulence on the thermal performance of a flat plate solar collector. Moghadam et al. [6] investigated the effect of utilizing a mixture of CuO/water nanofluid as heat transfer fluid on the efficiency of the flat plate solar collector. The results illustrated that the usage of this mixture could increase the flat plate collector efficiency by about 21.8%. Furthermore, Zambolin and Del Col [7] presented a comparative study between the glazed standard flat plate solar collector and the evacuated tube solar collector based on EN 12975-2 standard. The daily results showed that the flat plate solar collector was more sensitive to environmental conditions.

Several design parameters affect the thermal performance of the flat plate solar collector. Therefore, in this context, many researchers [8, 9] have strived to obtain the optimum levels of these factors that maximize the thermal performance of the flat plate solar collector. Most of the above researchers utilized the traditional experimental strategy (one factor at time (OFAT) ) approach [10] in order to evaluate the effect of the flat plate solar collector design parameters. This method has a failure in discussing the optimum conditions for a combination of design parameters based on multi-responses. Therefore, Jeffrey Kuo et al. [3] attempted to introduce the Taguchi method combined with the Grey relation analysis to overcome the disadvantage of OFAT in optimizing multi-responses of the flat plate solar collector. Even though this method solves the problem of obtaining the optimum values for the design parameters based on the multi-responses, it evaluates the optimum conditions only based on the main factor analysis without consideration for the interaction between design parameters which leads to different optimum conditions. Therefore, the present study intends to combine the response surface methodology (RSM) from the design of experiment (DOE) approach with the Taguchi method to optimize multi-responses of the flat plate solar collector with consideration for the interaction between the design parameters.

2 Basic Equations Performance

Beckman and Duffie [11] defined the total useful energy output from the flat plate solar collector as given by Eq. 39.1.

$$ {Q}_{\mathrm{net}}={F}_{\mathrm{R}}A\;\left[{\left(\tau \alpha \right)}_{\mathrm{e}}.{I}_{\mathrm{T}}-{U}_{\mathrm{s}}\left({T}_{\mathrm{i}}-{T}_{\mathrm{a}}\right)\right] $$

(39.1)

where F_R is the heat removal factor which represents the actual useful energy gain of the collector to the useful gain if the whole collector surface were at the fluid inlet temperature, (τα)_e is the effective transmittance-absorptance product, and I_T is the hourly solar radiation. U_s represents the total heat loss coefficient of the flat plate solar collector, T_i is the inlet water temperature, and T_a is the ambient temperature . The instantaneous efficiency can be expressed as expressed by Eq. 39.2:

$$ \eta =\frac{Q_{\mathrm{net}}}{I_{\mathrm{T}}.{A}^{.}}={F}_{\mathrm{R}}{\left(\tau \alpha \right)}_{\mathrm{e}}-{U}_{\mathrm{s}}.\frac{\left({T}_{\mathrm{i}}-{T}_{\mathrm{a}}\right)}{I_{\mathrm{T}}} $$

(39.2)

where the value of F_R(τα)_e, the optical efficiency, and F_RU_s, the total heat loss coefficient, can be calculated using the least square method to get the instantaneous efficiency. Therefore, the performance of the solar collector can be represented by the F_R(τα)_e and F_RU_s.

Based on the instantaneous efficiency model developed by Duffie and Beckman [11], a numerical software called CoDePro is developed to evaluate the performance of the flat plate collector. This software is established according to the test standard [12].

3 Optimization Methodology

The main purpose of the current study is to find out the optimum configuration of the flat plate solar collector design parameters with independency on the environmental factors (Robust design ).

For this purpose, the cross array Taguchi method [10] is presented to investigate the design parameters that optimize the flat plate solar collector responses. This optimization includes maximizing the optical efficiency and minimizing the heat loss coefficient of the solar collector.

3.1 The Design Parameters Levels and the Characteristic Responses

Several controlled factors which include air gap between glass and absorber plate, absorber plate material, absorber plate thickness, absorption film type, insulation thickness, number of collector flow tubes, and inner and outer diameters of the collector tubes might affect the thermal performance of the flat plate solar collector as shown in Fig. 39.1. Furthermore, the performance of the flat plate solar collector is sensitive to the environmental factors Therefore, the effect of these parameters must be also considered in the current problem. The levels of the controlled and uncontrolled factors are summarized in Tables 39.1 and 39.2, respectively. For the characteristic responses of the flat plate solar collector performance , the optical efficiencyF_R(τα)_e and the heat loss coefficient F_RU_s are selected.

Table 39.1 The controlled factors levels

Table 39.2 Uncontrolled factors levels

3.2 Taguchi Method

The Taguchi method is employed to obtain a robust design with a limited number of runs based on the orthogonal array (OA) . The cross-array design approach L27 for the inner array and L8 for the outer array is utilized to obtain the robust design where the inner array is advocated for the controlled factors, whereas the outer array is advocated for the uncontrolled factors (noise factors). The combination of the inner and outer arrays provides information about the interaction between the controlled factors and uncontrolled factors [10].

3.3 The Signal-to-Noise Ratio (S/N Ratio)

The experimental data are then analysed through the “signal-to-noise ratio.” There are several types of the S/N ratio depending on the desired response. However, there are two primary types. The S/N ratio for the maximum and minimum outputs is estimated by [13] as shown in Eqs. 39.3 and 39.4.

Smaller is better:

$$ S/N=10\log \left(\frac{1}{n}\sum \limits_{i=1}^n\frac{1}{y_i^2}\right) $$

(39.3)

Larger is better:

$$ S/N=10\log \left(\frac{1}{n}\sum \limits_{i=1}^n{y}_i^2\right) $$

(39.4)

where y_i is the observed run and n is the total number of the experimental runs.

Since the characteristic responses of the flat plate solar collector at the current study are the optical efficiency and heat loss coefficient. Therefore, the optimum configuration can be found at higher optical efficiency and lower heat loss coefficient where Eq. 39.3 satisfies the goal of achieving a higher optical efficiency, whereas Eq. 39.4 meets the goal of achieving a lower heat loss coefficient.

3.4 Analysis of Variance (ANOVA)

The S/N ratio is utilized to evaluate the experimental responses without obtaining information about the significance of each factor. The ANOVA [14] evaluates the significance of each factor through estimating the experimental error associated with the responses. The ANOVA calculation includes an estimation for several parameters which include (i) degree of freedom, (ii) correction factor, (iii) the sum square, (iv) mean square, and (v) F-ratio, as given by [3].

3.5 Taguchi Method Limitations

Even though the Taguchi method presents a useful manner to obtain a robust design with independency on the noise factors, it has several limitations which include the following:

• I. The cross array design does not present an economical design for investigating a large number of factors [14]; since it crosses the noise factors matrix with the controlled factors matrix, this crossing produces a large number of experimental trials.

• II. The cross array design does not estimate the effect of the interaction between the controlled factors and the noise factors [10].

• III. The cross array design discards the effect of the quadratic terms in the controlled and noise factors [10].

• IV. The cross array design fails to obtain the optimum design parameters based on multi-responses.

• V. The response model approach is combined with the cross array design to overcome these limitations.

3.6 Response Method Approach

The key issue of obtaining a robust design is to consider the interaction between the controlled and noise factors. Therefore, the response model approach tends to develop a correlation model that involves the main effect, the interaction between the controlled and noise factors, and their quadratic terms with a small number of experimental trials relative to the number of experimental trials required by the cross array design.

3.6.1 Governing Equations

For considering a first-order model including the controlled and noise factors, the model formula is mentioned by Montgomery [10] as shown in Eq. 39.5.

$$ y={\beta}_{\mathrm{o}}+{\beta}_1{x}_1+{\beta}_2{x}_2+{\beta}_{12}{x}_1{x}_2+\gamma {z}_1+{\delta}_{11}{x}_1{z}_1+{\delta}_{21}{x}_2{z}_1+\varepsilon $$

(39.5)

where x₁, x₂ are the controlled factors, z₁ is the noise factor, β_o is the constant coefficient of the response model, β₁, β₂ are the intercept coefficient of controlled factors, β₁₂ is the intercept coefficient of the interaction between the controlled factors, γ is the intercept coefficient of the noise factor, and δ₁₁, δ₁₂ are the intercept coefficients of the interaction between the controlled factors and the noise factor.

According to the assumptions given in [10], the noise factors are random variable even though they are controlled for the experimentation purpose . Moreover, the expected value of noise factors is zero, and its variance is ${\sigma}_z^2$. Based on these assumptions, the mean response can be estimated as shown in Eq. 39.6.

$$ {E}_{\mathrm{z}}(y)={\beta}_{\mathrm{o}}+{\beta}_1{x}_1+{\beta}_2{x}_2+{\beta}_{12}{x}_1{x}_2 $$

(39.6)

For the variance model , Montgomery [10] expanded the response model shown in Eq. 39.7 using the Taylor series, and the result becomes

$$ {V}_{\mathrm{z}}(y)={\sigma}_{\mathrm{z}}^2{\left({\gamma}_1+{\delta}_{11}{x}_1+{\delta}_{12}{x}_2\right)}^2+{\sigma}^2 $$

(39.7)

From the mean and variance models mentioned above, it is noticed that only the controlled factors appear in the response models. This mean provides the ability to achieve the design target with a small variation due to the noise factors. Even through the noise factor doesn’t appear in the responses model, the intercept coefficient between the controlled factors and the noise factor appear in the variance model. This appearance shows the influence of the noise factors in the model response . Also, σ ² presents the mean square of the residuals estimated from the response model.

3.6.2 Choice of the Experimental Design

The selection of the experimental design is vital in the response model approach. In the current problem, we are interested in obtaining a second-order model that includes the interaction between the controlled factors and their quadratic terms. Therefore, the standard response surface designs, such as the central composite design and Box-Behnken design [14], might be applicable especially when the design domain is cube or sphere even though in the current problem the design domain is not a standard one.

The optimal design is a computer-aided design that involves the best subset of all possible experiments based on a certain criterion and a certain algorithm [10]. The optimal design offers an ability to develop a response model based on a nonstandard design domain with a limited number of experimental trials. There are several types of the optimal design such as D-optimal design, G-optimal design, V-optimal design, and I-optimal design. The selection of the appropriate design depends on the order of the required model. The I-optimal design mentioned by Montgomery [14] can satisfy the current problem requirement since it has the ability to form a second-order response model with a limited number of experimental trials. Design matrix can be considered as an I-optimal design when the smallest prediction variance shown in Eq. 39.8 is obtained.

$$ I=\frac{1}{A^{\ast }}\int V\left[\widehat{y}(x)\right]. dx $$

(39.8)

where A ^∗ tends to represent the volume of the design domain and $V\left[\widehat{y}(x)\right]$ is the prediction variance of the design matrix. The minimum prediction variance is calculated based on the coordinate exchange algorithm [10].

3.6.3 Statistical Analysis of the Data

Several statistical criteria are utilized to analysis the experimental results. The statistical analysis includes (i) analysis of variance, (ii) model adequacy checking, and (iii) goodness of fit.

3.6.4 The Optimization in the Response Model Approach

Since the objective of the response model approach is to determine the optimum design parameters levels that maximize the optical efficiency and minimizes the heat loss coefficient, the current experiment can be formulated using the desirability function as:

Maximum	FR(τα)e
Minimum	FRUs, POE
Subjected to
	5 mm ≤ A ≤ 90 mm
	3 mm ≤ B ≤ 13 mm
	C: Steel, copper, and aluminium
	D: Tinox, black chrome, and Thermalox 250
	0.5 mm ≤ E ≤ 2 mm
	10 mm ≤ F ≤ 50 mm
	10 mm ≤ G ≤ 50 mm
	5 ≤ K ≤ 15
	6 mm ≤ L ≤ 20 mm
	0.5 m ≤ M ≤ 2.5 m

where POE is the propagation of error, and it tends to present the standard deviation associated with the responses a function of the controlled factors [10]. Once the optimum design parameters that maximize the optical efficiency and minimize the heat loss coefficient are obtained, a confirmation run is required to verify the optimum configuration.

4 Results and Discussion

4.1 Taguchi Method Results

Based on the cross array design L27-L8, 216 runs of experimental trails are implemented using CoDePro software. The response graph of S/N ratio for the optical efficiency response based on the controlled factors levels is shown in Fig. 39.2.

Fig. 39.2

Response graph for the optical efficiency response

The response graph for the optical efficiency shows that the optimum factor levels that maximize the optical efficiency are A2, B3, C2, D3, E3, F3, G3, K3, L2, and M3. This means that the maximum optical efficiency is found when the absorber plate of the collector is made from copper, with a thickness of 0.2 cm and a length of 2.5 m, and painted with Tinox as an absorption film. For both the side and back insulation, a layer of 5 cm is selected. The distance between the glass and the absorber plate is 4.75 cm, and the distance between the upper and the lower glass layer is 1.3 cm. In terms of the flow tubes, 15 tubes with an inner diameter of 1.3 cm are selected.

To obtain the effect of the controlled factors on the optical efficiency response, the ANOVA is calculated. The results shown in Table 39.3 illustrate that the number of the flow tubes has the most significant effect on the optical efficiency response followed by the back insulation thickness and then the absorption film type , while the remaining factors seem to be insignificant since P-value of these factors is larger than 0.1.

Table 39.3 Response graph for the optical efficiency response

In terms of the heat loss coefficient response, the response graph of S/N ratio based on the controlled factors levels, shown in Fig. 39.3, illustrates that the optimum configuration levels that minimize the heat loss coefficient are A3, B3, C3, D1, E1, F3, G2, K1, L1, and M3.

Fig. 39.3

Response graph for the heat loss coefficient response

This means that the minimum heat loss coefficient is achieved when the absorber plate of the collector is made from steel, with a thickness of 0.05 cm and a length of 2.5 m, and painted with black chrome as an absorption film . A layer of 5 cm is selected for the back insulation, and a layer with a thickness of 3 cm is chosen for the side insulation. The distance between the glass and the absorber plate is 9 cm, and the distance between the upper and the lower glass layer is 1.3 cm. In terms of the flow tubes, five tubes with an inner diameter of 0.6 cm are selected.

The ANOVA for the heat loss coefficient responses illustrates that the back insulation thickness is the most significant factor, followed by the absorption film type, then the number of the flow tubes, and finally the absorber length , whereas the remaining factors seem to be insignificant since P-value of these factors is larger than 0.1 as shown in Table 39.4.

Table 39.4 Analysis of variance of the heat loss coefficient output based on the cross array design

4.2 Results of the Response Model Approach

The obtained results using the cross array Taguchi method show that the optimum optical efficiency response is found at A2, B3, C2, D3, E3, F3, G3, K3, L2, and M3, whereas the optimum heat loss coefficient is found at A3, B3, C3, D1, E1, F3, G2, K1, L1, and M3. Since the optimum combination between the optical efficiency response and the heat loss coefficient is different from the optimum of each single response, the response model approach based on I-optimal design is presented.

A total of 140 numerical trails are implemented based on I-optimal design matrix to investigate the effect of the main controlled factors, their interaction, and quadratic terms on the selected responses.

4.2.1 Regression Models Analysis

Based on the I-optimal design matrix and ANOVA, nonlinear regression models including the main effect, the interaction, and the quadratic terms are developed for both the optical efficiency and heat loss coefficient.

For the optical efficiency model, R-squared is 98.3%, adjusted R-squared is 96.7%, and PRESS criteria is 0.04, while for the heat loss coefficient response, the R-squared is 97.3%, adjusted R-squared is 96.09%, and PRESS criterion is 0.77. These results show the high ability of the proposed regression models in estimating the optical efficiency and the heat loss coefficient within the limits of the design parameters.

4.2.2 Multiple Responses Optimization Using Desirability Function

Based on the desirability function formulation mentioned previously in Sect. 39.3.6.4, the maximum optical efficiency of 71.8% with POE of 0.012 and the minimum heat loss coefficient of 2.76 W/m²-C with POE of 0.059 can be obtained when the distance between the glass and absorber plate is 8.88 cm, the distance between the upper and lower glass cover is 1.29 cm, and the absorber plate is made from aluminium with dimensions of 2.47 m and 0.117 cm for length and thickness, respectively. Furthermore, the absorber plate is painted with Tinox. Regarding the insulation, 4.2 cm and 3.65 cm of thickness are selected for the back and side insulation, respectively. Finally, for the collector flow tubes, 14 tubes with an inner diameter of 1.6 cm are selected. Furthermore, the low value of the POE shows the robustness based on the response model approach can yield to a satisfactory solution.

4.2.3 Validation Run

To confirm the obtained results based on the response model approach, the design parameters at the optimum configuration that maximizes the optical efficiency and minimizes the heat loss coefficient are used to run a validation numerical experiment using CoDePro . The confirmation run shows an offset of 0.5% in predicting the optical efficiency response and about 12.38% in predicting the heat loss coefficient response. These results show the ability of the obtained response models in successfully predicting the characteristics of the flat plate solar collector.

5 Conclusions and Recommendations

The current study attempts to find the design parameters configuration that optimizes the characteristic responses of the flat plate solar collector namely the optical efficiency and the heat loss coefficient, with independency on the environmental factors through using the Taguchi method , and the response model approaches. The results of the study can be summarized as follows:

Based on the analysis of the main factors effect, the Taguchi method shows that the number of collector flow tubes is the most significant factor in the optical efficiency response, while the back insulation thickness is the most significant one for the heat loss coefficient. Furthermore, the optimum design parameters of the optical efficiency are different from the optimum design parameter for the heat loss coefficient. This shows the failure of Taguchi method in optimizing multiple responses.

Based on the analysis of the factors effect and their interaction, the response model approach succeeds in optimizing the multiple responses which include the optical efficiency and the heat loss coefficient with a drop in the POE where it is 0.012 for the optical efficiency and 0.05 for the heat loss coefficient.

The validation run shows the ability of the response models in predicting the optical efficiency and the heat loss coefficient precisely.

As a recommendation for the current study, an experimental test rig based on the optimum design parameters needs to be built in order to validate the optimization results. Furthermore, the same technique of investigation can be extended to various components of the solar water heating system in special and renewable energy systems in general with consideration for the life cycle cost effect in obtaining the optimum configuration for the design parameters.

A :

The air gap between glass and plate (cm)

A ^. :

The solar collector area (m²)

B :

Air gap between cover glass 1 and cover glass 2

C :

Absorber plate material

D :

Absorption film type

E :

Absorber plate thickness (cm)

F :

Back insulation thickness (cm)

F _R :

The heat removal factor

G :

Side insulation thickness (cm)

I _t :

The hourly solar radiation (W/m²)

K :

Number of collector flow tubes

L :

Inner diameter of collector flow tube (cm)

M :

Absorber length (m)

T _a :

Ambient temperature ( °C)

T _i :

Initial solar collector temperature ( °C)

U _s :

The heat loss coefficient (W/m °C)

u _s :

Wind speed (m/s)

Q _net :

The total useful energy output from flat plate solar collector (W)