Chapter 5 MATHEMATICAL MODELING OF THE EVACATED SOLAR COLLECTOR. 5.1 Thermal Model of Solar Collector System

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1 Chapter 5 MATHEMATICAL MODELING OF THE EVACATED SOLAR COLLECTOR This chapter deals with analytical method of finding out the collector outlet working fluid temperature. A dynamic model of the solar collector system is prepared to find out the analytical values of temperature. This model is solved by the finite difference method. The Set of single order differential equations are formed to predict the temperature values. The simulated values are compared with experimental values to validate this model. 5.1 Thermal Model of Solar Collector System Introduction A dynamic approach is more interesting in several cases: control strategies, dynamic testing procedures, coupling with others elements. Particularly, predicting the behaviour of solar collectors for a time step, a dynamic modelling brings more information concerning the solar collector. The collector is modelling under hourly step in order to take into account the variation of the geographical parameters. This level of description is useful to understand which parameters have a significant influence on the outlet temperature of the collector. In this model, all physical phenomena are studied separately to describe the model. The dynamic behaviour of the model is verified by numerical tests and measures comparisons. The Two dimensional thermal modeling of a solar evacuated tube is done under transient conditions. The Variations of fluid mass flow rate, ambient temperature, solar radiation, and wind speed are accounted. The semi-analytical model relies on energy conservation equation for small control volumes along the longitudinal axis of the tube. The first order differential equations obtained for each control volume are solved by the use of a fully explicit scheme using a fourth order Runge-Kutta algorithm. An experimental setup has been validated in order to assess the predictions provided by the model. The comparison between simulated and experimentally measured outlet air temperature showed a good agreement. In order to bring a solution to this problem, air is used as the heat transfer fluid instead of water. Using air, 85 P a g e

2 freezing and overheating problems are avoided. Furthermore, air is free, which could be used in an open loop system and presents no risk of contamination in case of leakage of the piping. The Limitation of the air, used as working fluid is, the heat capacity of air is low as compared to that of liquids. But nevertheless it is worth trying to design an air-based collector for specific applications despite this drawback Thermal Model of Solar ETC System The airflow is the most important parameter acting on the defined performance indicators. Higher is the airflow, better is the efficiency and lower is the outlet air temperature. Figure 5.1 shows the thermal model of the evacuated tube exposed to the solar radiation. Fig.5.1Thermal Model of the ETC Exposed to Solar Radiation The large pressure drop caused by 180, changes in the direction of the air at the bottom of evacuated tubes, is observed. Following assumptions are made to solve the set of Equations, Heat transfer is considered one-dimensional along the radial coordinate and axis symmetric. Diffused radiation gained by the actual system is neglected. 86 P a g e

3 Finite differential equation is solved for single evacuated tube and uniformity is consider to all tubes with constant mass flow rate. For a given axial position, heat is equally distributed azimuthally on the receiver and the cover. As the walls are thin, the conduction resistances in the glass walls are considered negligible. There is only one temperature that characterizes the inner and outer surface of glass tubes and the temperature varies axially which is essential because of the heat gained by the fluid. All data for variable thermal properties with air temperature are readily available to account for such variations. Because the flow in an evacuated tube collector is completely enclosed, an energy balance can be applied, to determine the variation in mean temperature of working fluids. Also the temperature with position along this tube and the total convection heat transfer is related to the difference in temperatures at the tube inlet and outlet temperature. For the constant mass flow rate heat gained by the working fluid is given by following equation, + = { ( )} (5.1) The evacuated tube is divided in to n (180) segments along its longitudinal direction, the velocity of the working fluid is more as compared to the width of each segment, and hence change in fluid temperature is negligible. The velocity u of the fluid is assumed to be constant, thus equation above reduces to, = { ( )} (5.2) It is considered that heat transfer coefficient is zero due to convection in between the inner to outer glass as vacuum is present. Heat transfer coefficient due to convection between fluid and receiver is calculated by calculating the Reynolds number and Nusselt number as follows; = In the case of turbulent flow region (Re > 6000), it recommended to use the correlation obtained from the relationship of Gnielinsky, (2005) (5.3) 87 P a g e

4 (5.4) The Darcy s friction factor for the above tubes may be obtained from: f = ( log Re 1.64) -2 (5.5) and = (5.6) Heat transfer to the receiver is given by the equation, ( ) = { + ( ) + ( )} (5.7) Where the product is the transmittance absorptivity of the evacuated tube collector, is Emmisivity of the receiver and is the Stefen Boltzman constant. Temperature of the receiver tube at the end of length L is calculated as; = + (5.8) The outer glass tube temperature is calculated as; = { ( ) + ( ) + ( )} (5.9) Equation (5.9) stats the change in cover temperature which is proportional to the difference between net heat gain rate by the receiver to the heat losses to the environment by radiation and convection. The convective heat transfer coefficient between the atmosphere and the outer glass is given by the equation; = v (5.10) Where, v is ambient air velocity in m/s. 88 P a g e

5 Finite difference method was used to solve this system. In this case, the collector was defined as single fluid channel, which was divided into n segments (n=180). The single order differential equation system was solved for each segment in the time domain using a 4 th order Runge-Kutta method. A MATLAB-14B program was developed to solve these sets of equations simultaneously. At first iteration boundary conditions like the glass temperature and air temperature was considered as the same. Equation 5.2, 5.7 and 5.9 are solved simultaneously for the first segment (that is n=0 node). The output of the first segment was passed as an input for the next segment and so on. The last segment gave the output temperature of the outer glass tube, inner glass tube and the outlet temperature of the working fluid. Total tube length was divided in 180 segments. 5.2 SIMULATED COLLECTOR OUTPUT The Table 5.1 given below the detail about the technical specifications of the solar ETC system along with their properties. The collector outlet temperature is calculated by the program for a single evacuated tube and it considered as same for all the tubes. As all the ten tubes are arranged as parallel to each other, together the mass flow rate will be increased and temperature output of all tubes will remain the same as no considerable changes are observed in input parameters to each tube. The details about the program and its output are shown in Appendix A. The Tables 5.2 and 5.3 shows the program calculated sample collector outlet temperature of the working fluid. The variation is observed in the range of 0.932% to 6.36% with average percentage error in calculated and experimental values are 2.459%. Table 5.1 Evacuated Tube Solar Collector setup properties Factor Unit Value Specific heat of working fluid ( air) J/kg k Specific heat of glass cover J/kg k Specific heat of Absorber J/kg k Absorptivity Coefficient Heat Transfer Coefficient Absorber-Fluid W/m 2 k Emissivity of inner glass Stefan Boltzmann Constant x P a g e

6 Table 5.2 Experimental and Analytical Collector Outlet Temperature Solar Radiation (W/m 2 ) Ambient temp ( 0 C) (11/04/2013) Collector Outlet Temp. (Actual) ( 0 C) Collector Outlet Temp.(Analytical) ( 0 C) % Error Time in Hrs Average % error Temperature ( 0 C) y = x x R² = Time (Hrs) Exp. Collector temp Sim. Collector Temp Ambient temp Fig. 5.2 Collector Outlet Temperature Experimentally and Analytically (11/04/2013) Figure 5.2 is a plot of the single day collector outlet working fluid temperature variation. This shows simultaneously the outlet temperature values measured using accurate measuring instruments and analytically calculated values at the same point. Last column in Table 5.2 shows the percentage error in individual reading. It is observed, which is in the range of 1.07 to 6.98 % with the total average percentage error of 2.496% which is well within the acceptable limit. 90 P a g e

7 Table 5.3 Experimental and Analytical Collector Outlet Temperature Time in Hrs Solar Radiation (W/m 2 ) Ambient temp ( 0 C) (12/04/2013) Collector Outlet Temp. (Actual) ( 0 C) Collector Outlet Temp.(Analytical) ( 0 C) % Error Average % error Temperature ( 0 C) y = x x R² = Time (Hrs) Exp. Collector Temp "Sim. Collector Temp" Ambient temp. Fig. 5.3 Collector Outlet Temperature Experimentally and Analytically (12/04/2013) Similar kinds of results were observed for the figure. 5.3, as explained above for the figure. 5.2 with, the percentage error between experimental and analytical collector outlet temperature in the range of 1.5% to 7.06 %. The total average percentage error is 3.55% with R 2 = 0.985, which is also well within the acceptable limit. 91 P a g e

8 Predicted Collector outlet Temperature( 0 C) y = x R² = Predicted Vs Actual Collector temp Actual Collector outlet Temperature( 0 C) Fig.5.4 Collector outlet temperature experimentally and theoretically The above figure 5.4 shows the plot of actual/experimental versus predicted collector outlet working fluid temperature for a single day. It is observed that the predicted and experimental values are in good agreement with R 2 = Similar kinds of results are observed in all experimental and predicted values. It indicated that the simulated mathematical model is accurate. 5.3 RESULT VALIDATION Figures 5.2 and 5.3 present a comparison of experimental results and predictions for a bright sunny day. The simulation presented in Figure 5.4 involved results obtained which, gives a good compromise between the precision of the solution and calculation time. The Results for several days were compared to assess the validity of the predictions over a period of entire test time. Although the model does not exhibit the rapid variations of temperature (probably caused by the turbulence of the flow), the numerical model follows very well the experimental behaviour of the tube. The statistical indicators were also calculated to qualify the difference between the results. The average percentage error of 2.45% to 3.5% in the temperature was obtained which is acceptable. Moreover, this could be explained by the assumption that T a = T sky in the model which slightly reduces the heat losses to the environment as well as the diffusion radiation was not considered in analytical solution. 92 P a g e

9 The Flowchart of the program implemented in MATLAB to calculate the dependent variable that is collector outlet working fluid temperature. Fig.5.5 Flow chart of MATLAB program to calculate collector outlet temperature With a validated model, it is possible to analyze the influence of the environmental conditions (ambient temperature, wind speed, solar radiation) and operation (airflow) parameters on the performances of the tube using this transient model in steady state. 5.4 CONCLUDING REMARKS The performance of solar evacuated tube collector is studied analytically in this chapter. A set of single order differential equations are formed to simulate the performance of the solar collector system. These set of equations are solved numerically by fourth order Runge-Kutta method. To simulate the flow analysis certain assumptions are made and collector out let fluid temperature is calculated for a 93 P a g e

10 single tube, by considering there will not be much variation in physical data like mass flow rate and solar radiations, among all evacuated tubes. The single tube is divided into 180 equal segments and all sets of equations are solved for the first segment, which is at the bottom of tube. Output this segment is passed as an input for the second segment and so on. At the end of 180 th iteration the collector outlet working fluid (air) temperature is calculated by using MATLAB program. The validation of the simulated results is done with the experimental values. It is observed that the average percentage error in the actual collector outlet temperature and simulated or analytically calculated value is in the range of 2.45% to 4.6% for all experimental collector outlet temperature. Hence it is concluded that the actual collector outlet working fluid temperature has good agreement with calculated collector outlet working fluid temperature. 94 P a g e

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