chapter 3 experimental setup and procedure
56 CHAPTER 3 EXPERIMENTAL SETUP AND PROCEDURE 3.1 DESCRIPTION OF THE EXPERIMENTAL SETUP A forced convection and desiccant-integrated solar drying system with and without reflective mirror was fabricated in this work. The system consisted of a solar flat plate air collector, drying chamber, desiccant unit, centrifugal blower and a reversible fan. Solid, CaCl2-based, packed bed of solar regenerated desiccant material was stacked at the inclined roof of the drying chamber. The details of various components, fabrication details, desiccant preparation and the experimental procedure are discussed in this chapter. Figure 3.1 shows the schematic of the forced convection and desiccant integrated solar drying system fabricated for this research work. A forced drought 0.1 kW centrifugal blower was used to supply the ambient air at the required flow rates. A single-pass, single-glazed conventional solar air collector had been used for gaining useful energy from the incident solar radiation. The solar air collector was positioned towards the south at an angle of 30 o with the horizontal. An insulated flexible hose connected the solar air collector and the drying chamber. The drying chamber had ten trays to hold the products for drying. The air heated from the solar flat plate collector was forced through the drying trays to absorb the moisture from the product. A small opening was provided on the eastern side of the drying chamber to take and measure the 57 instantaneous mass of the sample at regular intervals. A solid CaCl2 solar regenerated desiccant unit had been provided at the top of the drying chamber. Cylindrically molded, single-layered, solid desiccants were stacked on the desiccant bed. A reversible fan was used to draw the ambient air through the desiccant bed for its regeneration during the sunshine hours and to circulate the air inside the drying chamber during the off- sunshine hours, respectively. Drying during sunshine hours I 9 I Drying during off sunshine hours I 19 7 11 I 12 18 13 15 8 14 10 16 I 17 7 5 3 1 3 2 6 4 Figure 3.1 Schematic of the forced convection and desiccant integrated solar drying system 1. Blower 2. Flat plate collector 3. Drying chamber 4. Insulation 5. Absorber plate 6. Bottom plate 7. Transparent cover 8. Desiccant bed 9. Plywood (day mode) 10. Air inlet 11. Duct for air exit 12. Drying trays 13. Two-way fan 14. Valve 15. Plywood (night mode) 16. Perforated sheet 17. Duct 18. Horizontal mirror 19. Vertical mirror 58 Figure 3.2 Pictorial view of the experimental setup Copper-constantan thermocouples were fixed to the temperature at different locations in the system. The pictorial view of the experimental setup is shown in Figure 3.2. 3.1.1 Solar flat plate air collector Solar energy collectors are devices employed to gain useful heat energy from the incident solar radiation. For solar energy crop drying applications, the solar flat plate air collectors provided the desired temperature elevations techno-economically than the more complex, concentrating collectors. A simple solar energy collector consisted basically of an absorbing surface usually painted black which absorbed the solar radiation and transmitted it in the form of heat to a working fluid. Provision 59 was made to circulate the air through a duct. For forced circulation solar drying applications, airflow through the collector was achieved by using a fan or a blower. Flat plate air collector of dimension 1.2 m x 2.4 m was fabricated in this work. A 1 mm thick copper sheet painted matte black was used as an absorber plate to absorb the incident solar radiation. The absorber plate was placed directly behind the transparent glass cover with a layer of static air separating it from the cover. The air to be heated flowed between the inner surface of the absorber and the bottom plate made up of the same material. In order to avoid the bulging of the upper rectangular absorber copper plate, tint was provided along its length, which increased the strength as well as the absorbing area. The collector air channel depth was 20 mm and the space between the absorber to the transparent glass cover was 25 mm. To distribute the flowing air uniformly throughout the collector channel, perforated sheets were used at flow channels in the front and back plenums. A 6 mm thick, plain window glass was used as a transparent cover to prevent the upward heat losses parallel to the absorber surface. The cover material prevented convective heat loss from the absorbing plate, reduced long wave radiative heat losses and protected the absorber plate against cooling and by occasional rainfall. The collector frame was made of locally available thick wood and at the bottom, 19 mm plywood was provided to support the absorber plate. A 60 mm thick, fine wooden plank was used as insulation at the bottom of the absorber plate. Polystyrene and silicon rubber of 20 mm thickness was used at the sides to prevent heat losses. 60 Table 3.1 Details of single-glazed, single-pass, solar flat plate air collector Type of air collector : non-porous conventional type Gross dimensions : 2.4 m x 1.2 m x 0.2 m Area of absorbing surface : 2.2 m x 1.1 m Absorber material : 1 mm Copper sheet Number of glazing : one Thickness of glazing : 6 mm Glazing material : plain window glass Spacing between glazing and absorber : 25 mm Collector tilt angle with horizontal : 30° Mode of air flow : forced air circulation parallel to the absorber Number of air flow channels : one; between absorber and bottom plate Air channel duct height : 20 mm Back insulation : fine wood savings of thickness 60 mm Side insulation : polystyrene and silicon rubber of thickness 20 mm Blower specification : 0.1 kW, three-phase induction motor Air flow rate : upto 300 m3/h Air duct specification : 50 mm diameter, thermally insulated pipe Location of the test : Sathyabama Institute of Science and Technology, Deemed University, Chennai, India Longitude : 80 o18′ E Latitude : 13 o04′ N Height : 16 m above MSL 61 3.1.2 Drying chamber Drying chamber was constructed from wooden materials with inner walls insulated (sometimes without additional insulation). The insulation should be proper in order to minimize the heat losses. The size of the drying chamber fabricated in this work was 2 m x 1.2 m x 1.0 m and had a roof with a slope of 30o. 19 mm thick plywood was used to fabricate the drying chamber. The chamber was insulated with 50 mm polystyrene at the bottom and 30 mm at the sides. The exterior faces of the drying chamber were painted to protect the plywood from deterioration. The drying trays of dimension 1.1 m x 1.1 m were fabricated using wire mesh. The drying chamber housed ten product shelves. Each shelf was kept on a wooden frame fixed to the sidewalls of the drying box, which were 90 mm apart. The drying trays were easily removable to load the products to be dried. The drying chamber also consisted of a two- way fan attached at the exit manifold of the desiccant bed to draw the ambient air through the desiccant bed for its regeneration and to circulate the plenum air in the drying chamber. Numerous holes were provided on the southern side of the drying chamber to provide uniform flow of ambient air through the desiccant bed in the sunshine hours. During the off-sunshine hours, these holes were closed firmly by using flat wooden door covered with polystyrene. The desiccant unit, which was made of perforated, mild steel tray and painted black, was placed at the top of the drying chamber to hold the solid desiccants. Double-glazing was provided at the slanting roof of the drying chamber with an air gap of 50 mm to increase the total energy collected and to reduce the net thermal loss from the desiccant bed. The space between the bottom glass cover and the desiccant bed was 50 mm. The depth of the desiccant bed was 75 mm. A provision was made to include 25 mm thick, insulated plywood just above and below the desiccant bed. 62 During the daytime operation, the plywood was placed at the bottom of the desiccant bed, so that the moist air coming from the drying chamber would not interact with the desiccants. During the off-sunshine hours, the plywood sheet was placed above the desiccant bed to avoid heat losses through the top surface. Horizontal and vertical reflective mirrors of size 1.2 m x 1.4 m were kept at the southern and northern sides of the desiccant bed to increase the intensity of incident solar radiation to enable faster regeneration and to increase the thermal storage capacity of the desiccant. Table 3.2 Details of the drying chamber Type of drying chamber : Indirect type with forced circulation Dimensions of chamber : 1.2 m x 1.2 m x 1 m Number of drying trays : 10 with an area of 1.21 m 2 Material used for construction : 19 mm plywood Size of reflective mirror : 1.2 m x 1.4 m Mode of air flow : Transversal air flow in the upward direction through the drying material Air space between plenum chambers : 70 mm and bottom of drying platform Air duct : Thermally insulated 50 mm diameter pipe connecting the air collector and the drying chamber 63 3.2 DESICCANT PREPARATION The solar drying process could be continued by the use of solar thermal energy storage system during which the solar radiation was not available or partially available. The most attractive form of solar energy storage from the stand point of solar drying would be thermal storage, since it would allow the energy to be stored directly as received from the collector. The drying potential in a regenerated, solid desiccant represents one of the most promising mechanisms of thermal storage for the purpose of drying. In this process, the moisture removal from the drying air could be realized by adsorption in a desiccant unit regenerated by solar energy. The heat generated during such exothermic adsorption process was nearly equivalent to the latent heat of vaporization of the removed moisture. The desiccant bed served as an open adsorption–desorption cycle in which the solar energy was stored during the desorption stage as sensible and latent heat. It was recovered later during the adsorption stage where a relatively humid and cold air was drawn through the adsorbent bed, and the exiting warm and dry air could be used for drying. Desiccant drying was more advantageous when solar drying was performed at a temperature slightly greater than the ambient temperature in order to conserve the product quality. It is also of particular importance in the final stage of drying when the drying process becomes very slow. However, changing from normal solar drying during the day to desiccant drying during the night, offered the possibility of continuous daily operation. Based on the research work of Thoruwa et al (2000), a low cost, solar regenerative, CaCl2 based, solid desiccant consisting of 60% bentonite, 10% CaCl2, 20% Vermiculite and 10% cement on dry mass basis was prepared and used in this work. Care was taken to prevent the damage in the size of the vermiculite during mixing and molding. The prepared mixture was added with adequate water and was molded in the shape of cylinders by using 64 a uni-axial, mechanical press. Uniform pressure was applied during the mold preparation in order to maintain uniform porosity and density. These desiccant molds were processed in a vacuum furnace at a temperature of 50oC for 24 h and dried at 200oC for the next 24 h. The dimensions of the desiccant mold were 75 mm diameter and 115 mm length. The mass of each desiccant mold was 400 g. Thermo couple Desiccant molds Figure 3.3 Pictorial view of the desiccant bed Table 3.3 Details of the desiccant unit Solar collector glazed area : 1.32 m 2 Number of glass : 2 Thickness of glass : 6 mm Double glazing air gap : 100 mm Desiccant bed depth : 75 mm Desiccant bed porosity (volumetric) : 0.65 Bulk density of the desiccant : 595 kg/m3 65 3.3 INSTRUMENTATION A calibrated solar meter was used to measure the solar radiation intensity. A hygrometer was used to measure the relative humidity of the ambient air. The mass flow rate of air was measured using calibrated Pitot tube. Copper-constantan thermocouples were used to measure the temperatures at various locations of the system. The inlet and outlet temperatures of air in the solar collector, desiccant chamber and drying cabinet, ambient temperature, wind speed, the amount of solar radiation and mass of the sample tray in the drier were measured at 30 minutes interval during the experiments. Temperature measurements at the cover plate, absorber plate and the desiccants were made by copper-constantan thermocouples. The thermocouples were connected to a multi-channel temperature scanner for continuous monitoring of data. Controlling the speed controller of the centrifugal blower varied the airflow rate. Wind speeds were measured with a vane-type anemometer. Table 3.2 shows the details of the instruments used in the experiments. Table 3.4 Details of the instruments used Quantity Units of Data Input Sensor Type Parameter Measured Electronic balance CopperConstantan Thermocouple Pitot tube Instantaneous mass of product 1 g Cover plate, Absorber plate and product Temperatures 22 °C Mass velocity of air 1 kg/ m2.s Solar meter Total Solar Radiation 1 W/m 2 Anemometer Thermohygrometer Wind Speed 1 m/s Relative Humidity 2 % 66 3.4 EXPERIMENTAL PROCEDURE Drying experiments were conducted for drying the green peas and pineapple slices under six modes: (a) forced convection solar drying; (b) forced convection and desiccant-integrated solar drying; (c) forced convection and desiccant-integrated solar drying with reflective mirror; (d) forced convection and desiccant-integrated solar drying started from 12 noon; (e) drying using desiccant only from 12 noon; and (f) drying using desiccant only after the solar radiation had fallen below 200 W/m2. The solar radiation intensity on the collector surface was measured using calibrated solar meter. Temperature measurements at different locations of the solar dryer were recorded by the copper-constantan thermocouples. A multi-channel temperature scanner with a sensitivity of ± 0.1oC was used to scan the temperature data. All the thermocouples were calibrated using standard thermometers. The moisture content of the 100 g sample was measured by weighing the sample using the electronic balance with an accuracy of ±0.001 g. The relative humidity of ambient air was measured using thermo-hygrometer. The oven drying method with temperature fixed at 105oC determined the initial and final moisture contents of each sample. The difference of mass before and after the drying in the oven gave the moisture content. The equilibrium moisture content was determined by drying the products at 60oC in a mechanical dryer until a constant mass was achieved. The corresponding moisture content was reported as the equilibrium moisture content, which was found to be 5% for green peas and 11% for pineapple slices on dry basis. All the readings were recorded at an interval of 30 minutes. 67 In the forced convection solar drying experiments, the set up was tested for air leakages at no load conditions. 20 kg of green peas/pineapple slices with initial moisture content of 80% / 87% were dried at three different mass velocities until the product reached its equilibrium moisture content. A lid was connected on the eastern side of the drying chamber for loading and unloading the drying material. The experiments were started at 8.00 a.m and continued till the solar radiation had fallen below 200 W/m2. Also, all the openings were closed and sealed to prevent heat transfer during the night hours in order to avoid rewetting of the products. Drying experiments were continued in the next day for further removal of moisture from the product. The ambient air was drawn by the centrifugal blower and heated up in the solar flat plate air collector. The heated air entered the drying chamber from the bottom of the tray and was flown upward through the sample, which absorbed the moisture from the product. Air finally exited to the atmosphere. Experiments were conducted with mass velocities of 0.01, 0.02 and 0.03 kg/m2.s. In the forced convection and desiccant-integrated solar drying experiments the desiccants were prepared and processed, as mentioned in Section 3.2. 75 kg of the molded solid desiccants capable of absorbing 30 kg of moisture were stacked in the perforated tray. The plywood, sandwiched by insulated material on both sides, was placed in the provision made just below the desiccant unit. This prevented the mixing of moist air with the desiccant material in the drying chamber during daytime drying. The product to be dried was loaded in the drying trays. The centrifugal blower circulated the hot air from the flat plate air collector to the drying chamber. Controlling the speed of the 68 blower fan regulated the mass velocity. Experiments were conducted with mass velocities of 0.01, 0.02 and 0.03 kg/m2.s. A reversible fan was used to draw the ambient air through the desiccant bed for its regeneration and to circulate the air inside the drying chamber. During the nighttime drying, the insulated plywood was kept above the desiccant bed in order to avoid the heat loss to the ambient. The fan provided in the drying chamber circulated the plenum air at an airflow rate of 0.035 kg/s through the desiccant bed. The moist air from the drying chamber lost its moisture content and gained the sensible heat in the desiccant bed. The drying experiments were continued till the product reached its equilibrium moisture content. In the forced convection and desiccant-integrated solar drying with reflective mirror experiments, the reflective mirrors were provided at the top of the desiccant bed. The amount of solar radiation available and the temperature of the desiccant material were also measured at 30 minutes interval. To study the drying potential of the desiccant material, experiments were conducted by starting the system from 12:00 noon and the time when the solar radiation had fallen below 200 W/m2. The research methodology for the different modes of drying experiments is shown in Figure 3.4. 69 I Reflectors 8 Air inlet 4 Desiccant bed 5 3 7 I Air inlet 1 6 Drying chamber Solar flat plate air collector 2 Observation I, G, V, Ta, Tp, Ti, To, Wo, Wt Data reduction hr, hp, hw, UL, F′, FR, η, DR, SR, ηp, η d, MR, SMER Mathematical Modeling Results and discussion Conclusions Solar drying Desiccant-Integrated Solar Drying Desiccant-Integrated Solar Drying with reflectors Desiccant-Integrated Solar Drying from 12:00 Noon Desiccant Drying from 12:00Noon Desiccant Drying 1-2-3 1-2-3-4-5-6-7 1-2-3-4-5-6-7-8 1-2-3-4-5-6-7-8 4-5-6-7 4-5-6-7 Figure 3.4 Research methodologies for the drying experiments 70 3.5 DATA REDUCTION The means of assessing the thermal performance of solar drying system was extremely useful for improving the system and the drying process. Based on the experimental results, the following parameters were analyzed. 3.5.1 Solar flat plate air collector The thermal performance of the solar flat plate air collector was determined by passing the air at its steady state and placing the collector outdoors under clear sunny conditions (Biondi et al 1988). The useful heat gain by a collector is C p(T o T i ) Qu m (3.1) The collector efficiency is given by Qu IA c (3.2) Under steady-state conditions Q u A c FR [ I ( ) e U L ( T o T a )] (3.3) Therefore, the efficiency becomes U F R ( ) e L (To Ta ) I (3.4) 71 where ()e = Effective transmittance-absorptance product for collector = 1.02() (Duffie and Beckman 1991) The collector heat removal factor is related to the collector efficiency factor, F' as FR Cp m A cUL A c U LF ' 1 exp( ) Cp m (3.5) According to Duffie and Beckman (1991) the collector efficiency factor for conventional, flat plate air collector could be estimated using the formula; UL F ' 1 1 1 1 hp [ ] hb hr 1 (3.6) The radiative heat transfer coefficients between the absorber plate and the channel bottom plate could be calculated by using the formula: (Tp2 Tb2 ) (Tp Tb ) hr 1 1 1 p b (3.7) The heat transfer coefficient inside the channel for airflow between the absorber plate and bottom plate might be calculated with the following empirical correlation (Hegazy 1996): 72 4 5 0.0415 L c ( ) k VD E DE h p h b 0.016 1 0 . 808 e D E (3.8) The correlation had the advantage of relating the airflow heat transfer coefficients to the channel depth to length ratio. DE 2DW L (D W ) L DE 4 x flow area wetted perimeter (3.9) (3.10) The radiative heat transfer coefficient from glazing to sky is given by h rgs g (Tg4 Ts4 ) (Tg Ta ) (3.11) Ts = 0.0522 Ta1.5 (3.12) Overall loss coefficient UL= Ub+ Ut (3.13) The collector bottom and side loss coefficient Ub kin 2(D tin t ) 1 tin WL (3.14) An empirical equation given by Duffie and Beckman (1991) is used in the computation of top loss coefficient Ut is given by 73 1 1 Ut c Tp Ta hw )e ( ) Tp 1 f 1 (Tp Ta )(Tp 2 Ta 2 ) 1 f 0.133p (p 0.00591hw ) 1 1 g (3.15) where 3.5.2 f = (1+0.089 hw – 0.1166 hwp) (1+0.07866 N) c = 520 (1 – 0.000051 2) e = 0.43 hw = 5.7 + 3.8 V for 0 70 1 100 Tp Drying Characteristics The quantity of moisture present in the material could be expressed either on wet or dry basis and expressed either as decimal or percentage. The moisture content on wet basis is the mass of moisture present in the product per unit mass of un-dried matter in the product and is expressed as (Ekechukwu 1999) Initial moisture content M wb Wo Wd Wo (3.16) While the moisture content on the dry basis is the mass of moisture present in the product per unit mass of dry matter in the product and represented as 74 Initial moisture content M db Wo Wd Wd (3.17) Final moisture content Mf Wwet Wd Wd (3.18) For drying experiments where mass losses were recorded, the instantaneous moisture content at any given time was computed using the equation ( Modb 1) Wo M tdb 1 100 % Wt (3.19) (1 M owb ) W o M twb 1 100 % Wt (3.20) The main characteristics which were generally used for performance estimation of any solar drying system included drying rate, dryer thermal efficiency, pickup efficiency and specific moisture extraction rate (Leon et al 2002). The drying rate should be proportional to the difference in moisture content between the material to be dried and the equilibrium moisture content. If W is the mass of the wet solid in kg (total moisture plus dry solid ) and Ws is the mass of dry solid in kg Mt W Ws Ws kg total moisture kg dry solid (3.21) 75 For the given drying conditions, the equilibrium moisture content “Me” kg moisture/ kg. dry solid was determined by drying the product at 60oC in a mechanical dryer until a constant mass was achieved. Then, the free moisture “M” in kg free moisture/kg dry solid was calculated for each value of “Mt” as M = Mt - Me (3.22) Using the data calculated from the moisture content, a plot of free moisture content M verses time “t” in hours was made. The rate of drying curve was obtained from the slopes of the tangent drawn to the curve, which gave the values of dM/dt for given values of “t”. Drying rate dM L s ( M t M eq ) dt A dt kg moisture m 2 .h (3.23) The drying process for any material was characterized by the periods of different drying rates. The first period was the constant drying rate, in which the surface moisture was being evaporated at a rate determined by the temperature and humidity of the air, and its rate of circulation. The second period was the falling rate period in which the rate of moisture removal decreased, perhaps because of the migration of the liquid boundary into the material with vapors being formed at the liquid boundary and moving to the surface by diffusion. The falling rate period usually occurred in two stages namely 76 <,,,, 1. the first falling rate period which involved the unsaturated surface drying, and 2. the second falling rate period where the rate of moisture diffusion to the surface was slow and was the determining factor. For agricultural products, the duration of each of these drying regimes depended on the initial moisture content and the safe storage moisture content. All drying had taken place within the falling rate regime if the initial moisture content was below the critical moisture content. Both for hygroscopic as well as non-hygroscopic materials, the nature of the drying rate curves were similar until the unbound moisture within the material was entirely removed. However, in hygroscopic materials, a further decrease in drying rate occurred as some of the bound water was removed. The pickup efficiency determined the efficiency of moisture removal by the drying air from the product and the dryer thermal efficiency was the ratio of the energy used to evaporate the moisture from the product to the energy supplied to the dryer and could be expressed as (Mumba 1996) Pickup efficiency p Wo Wt m a At (h as h i ) (3.24) Dryer thermal efficiency d m w h fg Qs Qf Qd (3.25) 77 Specific moisture extraction rate Rate of moisture removal during the drying process Total energy input to the dryer (3.26) Some structural changes had taken place on drying because of the mass loss and the most important structural variation appeared on crop were the shrinkage ratio which could be expressed as (Midilli 2001) Shrinkage ratio SR Wt Wo (3.27) Moisture ratio MR 3.6 M t M eq M o M eq (3.28) MATHEMATICAL MODELING OF THIN LAYER SOLAR DRYING CURVES Simulation models were helpful in designing new drying systems in improving the existing drying systems or for the control of the drying operation. The drying kinetics of materials could be described completely using their transport properties (thermal conductivity, thermal diffusivity, moisture diffusivity, and interfacial heat and mass transfer coefficients) together with those of the drying medium. In the case of food drying, the drying constant K was used instead of the transport properties. The drying constant combined all the transport properties and might be defined by the thin layer equation. Thin layer equations described the drying phenomena in a 78 unified way, regardless of the controlling mechanism. They were used to estimate the drying times of several products and to generalize drying curves. In the development of thin layer drying models for agricultural products, generally the moisture content of the material at any time after it had been subjected to a constant relative humidity and temperature conditions was measured and correlated to the drying parameters (Midilli and Kucuk 2003). Several thin layer equations were available in the literature. These equations varied widely in nature. Many investigators had successfully used thin layer equations to explain the drying of several agricultural products. Thin layer drying equations contributed to the understanding of the drying characteristics of agricultural products. Thin layer drying models had fallen into three categories namely, theoretical, semi-theoretical and empirical. The theoretical approach concerns either the diffusion equation or simultaneous heat and mass transfer equations. The semi-theoretical approach concerned approximated theoretical equations. The empirical equations were easily applied to drying simulation as they depended on the experimental data. Among these models, the theoretical approaches had taken into account only the internal resistance to moisture transfer while the semi-theoretical and empirical approaches considered only the external resistance to moisture transfer between the product and the air. The drying curves obtained were processed to find the most convenient one among the 13 different mathematical models used for thin layer drying (Yaldiz et al 2001; Lahsasni et al 2004) as shown in Table 3.5. 79 Table 3.5 Mathematical models applied to the drying curves (Yaldiz et al 2001; Lahsasni et al 2004) Model No Model Name Model Expression 1 Newton MR = exp (-Kt) 2 Page MR = exp (-Ktn) 3 Modified Page 1 MR = exp ((-Kt) n) 4 Modified Page 2 MR= exp (-(Kt) n) 5 Henderson and Pabis MR= a exp (-Kt) 6 Logarithmic MR= a exp (-Kt) + c 7 Two term MR= a exp (-K0t) + b exp(-K1t) 8 Two term exponential MR= a exp(-Kt) + (1-a ) exp(-Kat) 9 Wang and Singh MR = 1 + at + bt2 10 11 Approximation of MR = a exp (-Kt) + (1-a) exp (-Kbt) Diffusion Modified Henderson and MR = a exp(-Kt) + b exp(-gt) + c Pabis exp(-ht) 12 Verma MR = a exp (-Kt) + (1-a) exp (-gt) 13 Midilli-Kucuk MR = a exp((-Ktn)+bt The correlation coefficient (r) was one of the primary criteria for selecting the best equation to define the solar drying curves of the drying products. In addition to r, the statistical parameter reduced chi-square (χ2) was used to determine the quality of fit. Chi-square could be calculated as: 2 2 N ( MR exp, i - MR pre , i ) i 1 N n (3.29) 80 where MRexp,i is the experimental moisture ratio, MRpre,i is the predicted moisture ratio, N and n are the number of observations and constants, respectively. Modeling the drying behavior of different agricultural products required statistical methods of regression and correlation analysis. Linear and non-linear regression models were important tools to find the relationship between different variables, especially those for which no established empirical relationship existed. The most suitable model was selected by using non-linear regression analyses using the computer program. The constants and coefficients of the best fitting model involving the drying variables such as temperature, velocity and relative humidity of the drying air were determined. The effects of these variables on the constants and coefficients of drying were also investigated by multiple linear regression analyses. 3.6.1 Non-linear Regression Non-linear regression was used to find a non-linear model of the relationship between the dependent variable (Moisture ratio) and a set of independent variables (drying time, model constants and coefficients). Non-linear regression estimated models with arbitrary relationships between independent and dependent variables. This was accomplished using iterative estimation algorithms. The non-linear model to be fitted could be represented by: y = y(x;a) (3.30) The merit function minimized in performing nonlinear regression the following: n y y ( xi ; a ) x 2 (a) i i 1 i 2 (3.31) 81 where σi is the measurement error or standard deviation of the ith data point. As with linear regression, the objective is to minimize the sum of the squares of the distances between the actual data points and the regression line. 3.6.2 Multiple Linear Regressions Multiple regressions were used to predict the variance in interval dependent variables, based on linear combinations of interval, dummy independent variables. Multiple regressions could establish that a set of independent variables explained the proportion of the variance in a dependent variable at a significant level (through a significance test of R2), and could establish the relative predictive importance of the independent variables. Power terms could be added as independent variables to explore curvilinear effects. Cross-product terms could be added as independent variables to explore the interaction effects. The significance of difference of two R2'scould be tested to determine if adding an independent variable to the model helped significantly. Using hierarchical regression, one could see how most variance in the dependent could be explained by one or a set of new independent variables, over and above that explained by an earlier set. Of course, the estimates (b coefficients and constant) could be used to construct a prediction equation and generate predicted scores on a variable for further analysis. The multiple regression equation had taken the form y = b1x1 + b2x2 + ... + bnxn + c (3.32) The b's were the regression coefficients, representing the amount the dependent variable y changed when the corresponding independent changed 1 unit. c was the constant where the regression line intercepted the y axis, representing the amount the dependent y would be when all the independent variables were 0. The standardized versions of b coefficients 82 were the beta weights, and the ratio of the beta coefficients was the ratio of the relative predictive power of the independent variables. Associated with multiple regressions was R2, multiple correlations, which was the percent of variance in the dependent variable, explained collectively by all of the independent variables. Multiple regressions shared all the assumptions of correlation: linearity of relationships, the same level of relationship throughout the range of the independent variable, interval or near-interval data, absence of outliers, and data whose range was not truncated. In addition, it was important that the model being tested was correctly specified. The exclusion of important causal variables or the inclusion of extraneous variables could change markedly the beta weights and hence, the interpretation of the importance of the independent variables.
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