SENSITIVITY ANALYSIS OF THE TEST PARAMETERS OF A SOLAR FLAT PLATE COLLECTOR FOR PERFORMANCE STUDIES
Subhra Das1, 2 *, Bibek Bandyopadhyay2 and Samir Kr. Saha1
2
Department of Mechanical Engineering, Jadavpur University, Kolkata Solar Energy Centre, Institutional Area, Gurgaon-Faridabad Road (19th Milestone), Gurgaon *Corresponding author: Phone: 0124-4144524, E-mail address: nips.subhra@gmail.com
1
Abstract
The BIS test procedure for performance of solar flat plate collectors allows a certain limit of deviation for the control parameters to define the steady state condition. During actual performance studies, sometimes these limits are exceeded. In this paper an attempt has been made to find out the effect on the performance curve of the collector as a result of exceeding this prescribed limit by a certain small amount. Regression analysis has been used to study the sensitivity of the collector design to the response parameters. It has been concluded that a variation in the range ±0.01 to ±0.04 over and above the prescribed deviation does not significantly affect the response parameters (i.e, thermal efficiency, FRUL and FR(ζα)). Keywords: Flat Plate Collector, Thermal performance test, Sensitivity analysis, Regression analysis.
1. Introduction
The Indian standard procedure for testing solar flat plate collectors and reporting the performance was proposed by Bureau of Indian Standards (IS: 12933: 2003, Part 5, Second Revision). The standard sets limits for environmental conditions, specifies test procedures and calls for the thermal performance to be reported in terms of efficiency of the collector (η) which is expressed as a linear function of (Ti-Ta)/G, where Ti is the fluid inlet temperature , Ta is the ambient temperature and G is the Solar irradiance on the collector plane. The thermal efficiency test is conducted under steady state conditions. To ensure that the collector is operating in steady state conditions over a given test period, it is proposed that none of the experimental parameters deviate from the mean values over the test period by more that the limits prescribed by BIS. It is found that the limits set by the BIS to ensure steady state condition is very difficult to achieve in practice. Our aim in this paper is to study the effect of a small variation in the design parameters on the efficiency of the collector. Various methods have been employed in the literature for modeling the variations of design parameters (or variables). The methods are classified into five major category: fuzzy set theory (Wood and Antonsson, 1987), interval methods (Chen and Ward, 1995), sensitivity analysis methods (Sandgren et al., 1985, Whiting et al., 1999), probabilistic-based methods (Clarke et al., 2001) such as robust design (Sundaresan, et al., 1993; Parkinson, et al., 1993, Xiaoping Du, et al., 1998) and statistical method ( Paul A. Funk, 2000, N.S.Thakur et al., 2003, S.D.Sharma et al.,2005). The difference between the fuzzy set theory and probabilistic methods for dealing with different types of uncertainties are discussed by Wood and Antonsson (1990). While sensitivity analysis is focused on reducing the local rate of change of the design performance, the robust design or Taguchi method (Phadke, 1989; Taguchi, 1993) goes one step further by introducing uncertainty or noise in the system and generating optimal solutions that could reduce the impact of the uncertainty in a global scale. To measure the goodness of fluctuating performance on the variation of the design parameters, a metric needs to be formulated. In this paper we propose to use the “Sensitivity Analysis” to compare the performance of the collector by varying the design parameters with the performance of the collector based on BIS test procedure. In our study we have considered the effect of variation on two design parameters viz., “inlet fluid temperature” and “temperature difference between collector inlet and outlet temperature”
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Advances in Energy Research (AER – 2006)
Nomenclature
∆x FR G Gt Kv m se Mean deviation of control factors, °C. Solar collector heat removal factor, dimensionless. Solar irradiance on the collector plane, W/m2. Solar radiation on the tilted surface, W/m2. Incident angle modifier Mass flow rate, Kg/s. Standard error of estimate
Ta Ti To UL
Ambient temperature, °C. Fluid inlet temperature, °C. Fluid outlet temperature, °C. Overall heat loss coefficient, W/m2°C.
Greek Symbols η Thermal efficiency, dimensionless ζ Transmissivity of cover, dimensionless α Absorptivity of absorber plate, dimensionless δ Small deviation, °
BIS Thermal Performance Test
Thermal tests on solar collectors are performed to evaluate the extent of their capability to provide useful heat under given climatic and operating conditions. The methods of tests as given in this procedure are based on steady state conditions of operation.
2.1 Basic Performance Equations
Thermal efficiency of the solar collector (non_concentrating) is given by
η = FR (τα ) e − FRU L [(Ti − Ta ) / G ]
Equation (1) indicates that the thermal efficiency of a flat plate collector is a linear function of (Ti − Ta ) / G assuming that UL is constant. Graphical representation of the data is made by plotting (Ti − Ta ) / G on X axis and η on Y axis, and adapting statistical curve fitting using the Least Square method. Since the thermal efficiency tests are conducted at or near normal incidence, the value obtained for FR (τα ) e is basically FR (τα ) e , n . But in order to predict the collector performance at other incident angles as would be the case with actual field installations, a multiplying factor called the incident angle modifier, K v is introduced.
(1)
2.2 Steady State Condition
A collector may be considered to be operating in steady state conditions over a given test period if none of the experimental parameters deviate from the mean values over the test period by more than the limits given in the following Table1: Table1: Steady state Condition Sl. No 1 2 3 4 5 Parameter Total Solar irradiance Surrounding air temperature Fluid mass flow rate Collector fluid inlet temperature Temp. difference between collector inlet & outlet Deviation from the Mean value over the test period ±50 W
± 1°C ± 1% ± 0.1°C ± 0.1°C
/ m2
For the purpose of establishing that steady state exists, average value of each parameter taken over successive periods of 30s shall be compared with the mean value over the test period.
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3 Assessing the Effect of Variation in the Design Parameters
The factors which affect the performance of the collector are illustrated in Figure1. These can be classified into two categories: “Control factors” and “Noise factors”. The “Control factors” are the parameters which can be specified freely by a designer and the “Noise factors” are the parameters that are not under a designer’s control. “Signal factors” are the intended values of the response of the collector. In this paper, we assume that the signal factors have fixed values. In our study we propose to study the difference in the response caused by variations in the control factors by regression analysis. Let us consider a small variation δ in the design parameters, that is Mean deviation of the control factors, ∆x = ± (0.1 + δ) Where δ = 0, 0.01,…, 0.06 (2)
From the experimental data we select sample which satisfy equation (2) for all the control factors for each value of δ. Thus we get seven samples corresponding to one experimental data. Control Factors
Inlet fluid temperature Difference between inlet & outlet fluid temperature
Solar Flat Plate Collector
Response
Efficiency η
FRUL, FR(ζα)
Noise Factors Solar irradiance Ambient Temperature Figure1: Classification of Design Parameters
3.1 Regression Analysis
Regression analysis is conducted for each sample to obtain relationship between the efficiency η (response) and
(Ti − Ta ) / G (predictor). A simple linear regression is obtained as
η = β 0 + β1 ⎢
where
⎡ Ti - Ta ⎤ ⎥ +ε ⎣ G ⎦
(3)
β1 are referred to as parameters of the regression model and ε is referred as the error term. The error term ε accounts for the variability in η that cannot be explained by linear relationship between η
and and (Ti − Ta ) / G . Then least square method is used to develop values for b0 and b1, the estimates of the model parameters
β0
β0
and
β1
respectively.
The resulting estimated regression equation is
η = b0 + b1 ⎢ i ⎣ G
)
⎡ T − Ta ⎤ ⎥ ⎦
(4)
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The error term ε = η − η is a random variable which is normally distributed with mean zero and variance of same for all value of the predictor.
)
ε
3.2 Measuring the Reliability of the Estimating Equation
From the scatter diagram of the response vs. predictor we realize intuitively that a line will be more accurate as an estimator when the data points lie close to the line than when the points are farther away from the line. To measure the reliability of the estimating equation we compute the “standard error estimate se” which measures the variability or scatter of the observed values around the regression line.
se =
∑ (η − η )
n−2
)
2
Where n is the sample size.
(5)
3.3 Interpreting the Standard Error of Estimate se
Larger the standard error of estimate se, greater the scattering or dispersion of the points around the regression line. Thus if se = 0 we expect the estimating equation to be a “perfect estimator” of the dependent variable. In that case, all the data points would lie directly on the regression line and no points would be scattered around it.
4 Results
Thermal performance test had been carried out at Solar Energy Centre, Gaul Phari, for one Flat Plate Collector (FPC) for seven consecutive days following the BIS test procedure. Testing was done using a Test Stand which is capable of tracking the Sun. Thus, solar radiation is always normal to the surface of the collector. During the experimental period, the following quantities were measured: Ambient temperature (Ta), Solar radiation (Gt), Inlet fluid temperature (Ti), Outlet fluid temperature (To), Wind speed (v) and mass flow rate (m). Mass flow rate is maintained at 0.02 kg/s.m2. Figure2 to Figure5 shows the scatter plot of efficiency (denoted by y) vs. (Ti − Ta ) / G (denoted by x) for each values of δ.
Figure 2: Efficiency Vs (Ti-Ta)/G for δ =0
Figure 3: Efficiency Vs (Ti-Ta)/G for δ =0.01
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Figure 4: Efficiency Vs (Ti-Ta)/G for δ =0.02, 0.03, 0.04
Figure 5: Efficiency Vs (Ti-Ta)/G for δ =0.05, 0.06
The experimental results are shown in the above Figures2-5 and the calculated efficiency, FR (τα ) , FRU L and se are summarized in Table 2. Table 2: Summary of the results Response Factor Min ∆x ± 0.1 ± 0.11 ± 0.12-± 0.14 ± 0.15-± 0.16 0.42 0.42 0.42 0.42 Efficiency Max 0.68 0.68 0.68 0.68 Mean 0.532 0.532 0.533 0.535 5.23 5.25 5.23 5.34 0.684 0.685 0.685 0.69 0.01007 0.01007 0.01009 0.01056
FRU L
FR (τα )
se of efficiency
When the results are compared with the standard regression line (δ = 0); Table 2 shows that the variation in response due change in the design parameters is small for δ = 0.01, 0.02, 0.03, 0.04 compared to those obtained with δ =0.05, 0.06. Standard error of estimate se is large for δ = 0.05, 0.06 compared to that for δ = 0, 0.01, …, 0.04; this suggests that for δ = 0.05, 0.06 the scattering or dispersion of the data points around the regression line is more compared to the rest of the cases and hence is not a good estimate.
5 Conclusions
We find that while testing a flat plate collector, we can accept test data which have deviation upto the range of δ = 0, 0.01,…., 0.04; without causing appreciable variation in the efficiency curve of the collector. Thus the deviation from the Mean value over the test period as given below can be accepted while presenting the test results: Table 3:Summary Result Sl. No 1 2 Parameter Collector fluid inlet temperature Temp. difference between collector inlet & outlet Deviation from the Mean value over the test period
± 0.14°C ± 0.14°C
This would provide designers more flexibility to make decisions based on different design criteria and help them focus on other major hurdles and make design improvements.
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Acknowledgement
The authors acknowledge the help of Shri Pankaj Aggrawal of Solar Energy Centre for providing the detailed data of tests on solar flat plate collectors.
References
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