Evaluating and Modeling of Parabolic Solar Cooker by RSM

The present paper has dealed with two stages. In the first, an experimental set up of the parabolic cooker with direct use has been made-up and studied at Renewable Energy Research Center-University of Anbar. Based on experimental results, statistical studies by using Response Surface Methodology (RSM) have been done to identify the optimum influential parameters and estimate mathematical temperature model. In this study, parabolic collector parameters that can effect on the Parabolic Collector efficiency are studied in more details. Six amounts of water are used for measuring the increasing of temperature relative to the measuring time. Parabolic Collector parameters are optimised with the consideration of single-response; temperature of the working fluid (in terms of water). The achieved experimental results are analysed by the desirability functional analysis DFA approach, and optimal levels of input factors have been distinguished. ANOVA also has been utilized to recognize the assurance of powerful factors on the response.


Introduction
Solar cooker technology offers a wide variety of applications to exploit this source of renewable energy. In the middle of the thermal applications of solar energy, solar cooking is considered as one of the simplest, the most viable and promising opportunities in terms of the utilization of solar energy. Solar cooker utilises solar energy that is recognized as one of the most choices since it is free and offers clean and environmentally friendly energy. Therefore, it helps in the reduction of the level of greenhouse gas (GHG) emissions and fossil fuel prices, also in the solution of energy reduction in the remote areas such as desert regions [1][2][3]. Solar cookers can be classified into four types [4]: Response Surface Methodology (RSM), is an experimental strategy first described by Box and Wilson in 1951 for determining optimal conditions for multivariable systems, and is considered an efficient technique for process optimization [5]. RSM is useful in the solution of many types of engineering problems. Recently, one of these problems is optimization of the response in solar cookers. One of the earliest mathematical models to test the thermal performance of a Solar Cooker was obtained by Garg et al., [6]. Sinha and Sharma [7] defined a model for parabolic collector as solar Thermal Electric power system, which was established for high temperature uses. Kahrobaian  thermal & optical efficiency, using grey relational analysis. Also, P. Venkataramaiah et. al. [10] applied the desirability functional analysis (DFA) approach to optimize a solar parabolic collector process parameters. They are stated that the silvered mirror strip is the optimum option among the used reflective materials.
This study attempted to apply the RSM approach to evaluate, model and optimize of a solar parabolic collector parameters namely mass of water, and time of the measuring of working fluid temperature (time of trial) with the consideration of single response namely temperature of working fluid.

Solar Cookers Performance
For comparing the different forms of the solar cookers. The characteristic values essential to be well defined, The important values are the terms of power & efficiency. A mean heating power of a cooker is considered as follow: Where; m w is the mass of water (kg) c p is the specific heat capacity at constant pressure (J/kg K),  t ∞-95 is the temperature difference (K)  t is the duration of the measurement (sec) Heating power typically is calculated from ambient temperature up to (95 0 C), to stay away the instability of the precise boiling point. The power of evaporation is determined through the water evaporation at the point of boiling. Solar cooker heatcapacity takes less effect on the thermal performance due to the system works at a const. temperature. The power is considered the measured mass of water evaporated. Where; ṁ:an evaporatation mass of water (kg/s).
The efficiency is calculated in according equation (3). For parabolic concentrators, it is the direct solar radiation on the aperture surface. For flat-plate collectors, the solar radiation is global radiation on the surface [5]. Where; I is the solar radiation (W/ m 2 ).
A is the cooker surface area (m 2 ).
The differences in the tracking mechanism and the surface of various cookers show that efficiencies are appropriate to compare the cookers at the same kind. To compare various kinds of cookers, further parameters, besides efficiency, are required [6].

Practical Set-up and Measurements
In this section, the practical results in Table 1 are used for evaluating the performance of parabolic solar cooker. The parabolic solar cooker is shown in Fig.1 with a diameter of 1.8m. In this test, 6 amounts of water are used for measuring the increasing of temperature relative to the measuring time. The amounts are 1-litre, 2-litres, 3-litres, 4-litres, 5-litres and 6-litres. During the test, we measure the temperature of water by using the thermometer sensor (k-type 2-channels thermometer). The temperature of water before the test is fixed around 25 degrees then the test started by one liter putted inside the vessel. The thermal wire sensor adjusted inside the vessel for reading the temperature. The measuring of temperature started from 5-seconds because the vessel needs time to be heated. After finishing the measurements, the water will replace by 2-liters and the previous process is repeated till 6-liters.

Response surface methodology
Response surface methodology is a set of statistical and mathematical procedures that are useful for the analyzing and modeling engineering problems. In this route, the main objective is to optimize the response surface that is influenced by various process parameters. Response surface methodology also helps in determining the relationship between the controllable input parameters and the developed response surfaces [11]. The RSM design procedure is as follows: 3) Developing a mathematical model of the 2 nd order response surface with best fittings. 4) Finding the optimal set of experimental factors that offer a (max or min) value of response (Maximum at this study).

5) Representing the direct & interactive effects of process factors through (2D and 3D) plots.
In the RSM, the quantitative form of response surface function stated as below: The main objective is for optimization of the Response (Y). It's assumed that the independent variables are continuous and controllable by the experiments with errors are negligible. It is essential to get an appropriate approximation for a True-Functional Correlation between independent-input variables and the output response. Commonly a 2 nd order model Eq. (5) applied in RSM procedure [11]. ver. 9.0 was utilized for data analyzing, performing RSM and to get 2D & 3D response graphs.
Numerical optimization of input variables depending on the single response was achieved by software (DesignExpert 9.0). In this study, the temperature of the working fluid is considered as the performance characteristic (response) of parabolic solar cooker, maximize of temperature of the working fluid leads to maximize efficiency of solar cooker, so, the desired goal (maximization temperature of the working fluid) was utulized to achieve optimization of input variables & the response.
The levels were specified for each parameter as given in the Table 2. Two process parameters with low and high levels resulted in a total of 48 runs by RSM user defined design. The observations are presented in Table 3 for further analysis and studies.    From Table 1 and equations (1) and (3)

Mathematical Modelling
ANOVA and regression analysis of the experimental results was conducted by (Design-Expert 9.0 software). Estimated regression coefficients of the response surface quadratic model (Eq. 3) are presented in Table 4.    The regression equation of temperature of working fluid (Y) relating to actual levels of input parameters of parabolic solar cooker was found as (Eq. 6):

Response Optimization
The numerical optimization finds a point that maximizes the desirability function, where desirability is an objective function that ranges from zero outside of the limits to one at the goal.
In this work, response optimization is performed by desirability function where the main goal is identified as maximizing of the temperature of the cooking fluid in parabolic solar cooker. The target value of the response is selected as 125 ºC. For the goal of maximum, the desirability will be defined by the following formulas (Eq. 7), where desirability Curves for Goal are shown in Fig. 9 [12]. To optimize the parabolic solar cooker, constraints for desirability function were defined based on the temperature of cooking fluid with emphasis on the mass of water and time parameters as presented in Table 6.  The 2D and 3D response surfaces curves for the single response are plotted in Figs. 10 and 11. The final solutions of desirability function for the response optimization are given in Table 7.
As presented, a maximum desirability (selected) was equal to 0.7956 and the conditions to achieve it were the amount of water of 1 liter and time of 20 minutes.

Conclusions
In this work, based on experimental data, both of mathematical modeling and response optimization have been done to estimate a mathematical model of cooking fluid temperature and identify the optimum influential parameters using Design-Expert 9.0 software. The maximum temperature was selected as the performance characteristic (quality target) of parabolic solar cooker to get the greatest effectiveness; max efficiency. It is identified that the increase in the liters of water up to 6 liters; the value of efficiency of a solar cooker at 20 minutes was 38.7%.
Experimental data of response are analyzed by desirability function, & optimal levels of input variables have been recognized. A maximum desirability was equal to 0.7956 & the conditions to attain it were the amount of water of one liter and time of 20 minutes.