MSU LIBRARIES .—;—. RETURNING MATERIALS: P1ace in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. "Flt . ,3 ‘, _. ,..L-H 89;; PARAMETER ESTIMATION METHODOLOGY IN SELECTED MOISTURE DESORPTION MODELS BY Richard Keith Byler A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1983 ABSTRACT PARAMETER ESTIMATION METHODOLOGY IN SELECTED MOISTURE DESORPTION MODELS BY Richard Keith Byler Nonlinear data analysis techniques were used to obtain parameter estimates in moisture desorption models for parboiled rice. A model for equilibriunltmoisture content, EMC, was combined with a thin layer model for moisture content over time and with an Arrhenius form for the drying constant to form a single model of moisture content as a function of time, temperature, relative humidity and initial moisture content. Moisture loss data were collected at twelve combinations of relative humidity and temperature ranging from 17.3 Celsius to 40.6 Celsius and from 8.24 to 0.53 relative humidity. Two samples of approximately 100 grams dry matter content and initial moisture content of between 0.50 and 0.18 dry basis were studied, simultaneously. The dry bulb temperature was maintained to within 0.2 degrees Celsius and the relative humidity to about one-half of one percent of the mean value during each test. Richard Keith Byler The data sets were studied individually, comparing models with from one to four decaying exponential terms, Page's equation, and the diffusion equation for spherical and infinite cylinder geometry. While Page's equation fits the data well, the equation is inadequate. The spherical and infinite cylinder models did not produce acceptable models. The three term exponential was able to predict the data with an error mean square of 0.3 E-6, which was believed to be the approximate accuracy of the data. In the best sets of data the four term exponential was required to explain the measured variation. Data were selected from the complete data sets on an exponentially increasing time interval, over the first 37 hours. The parameter estimates obtained from subsets of 98 data points, following an algorithm described in this dissertation, predicted the complete data sets of over 2460 data points as well as the parameters estimated using the entire data set. These subsets, with constant temperature and relative humidity, were combined and analyzed to produce the final model covering the initial moisture content, from 0.18‘UDIL30. The resulting model, with a residual mean square of 11 E-6, was found to fit the data better than a model with parameters estimated by linear techniques. Richard Keith Byler Approved Major Professor Approved Department Chairman To Patricia Ho Ho ACKNOWLEDGMENTS I wish to express my appreciation to Dr.IL C. Brook for his assistance, inspiration and guidance throughout my period of study at Michigan State University. Appreciation is extended to A. W. and E. L. Farrell for tflue Graduate Fellowship which encouraged my commencement of PhJL level studies. Sincere thanks is expressed to Dr. F. W. Bakker- Arkema, Dr. P. D. Fisher, and Dr. J. B. Gerrish for serving as guidance committee members. The time which they and Dr. C. R. Anderson spent with me on this project was appreciated. iii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . LIST OF FIGURES . . . . . . . . . . . . . . . . . . NOMENCLATURE . . . . . . . . . . . . . . . . . . . . Chapter 1. INTRODUCTION . . . . . . . . . . . . . . . . . . 2. LITERATURE REVIEW . . . . . . . . . . . . . . . . 2.1 Statistics . . . . . . . . . . . . . . . . . 2.1.1 Linear Least Squares Regression . . . . . 2.1.2 Nonlinear Least Squares . . . . . . . . . 2.1.2.1 Nonlinear Regression With Derivatives l 2 1 .2.2 Nonlinear Regression Without Derivatives . . . . . . . . . . 2.1.3 Analysis of Fit . . . . . . . . . . . . . 2.1.3.1 Residual Analysis . . . . . . . . . 2.1.3.2 Correlation Matrix . . . . . . . . . 2.1.4 Examples . . . . . . . . . . . . . . . . 2. 2 Grain Drying . . . . . . . . . . . . . . . 2. 2.1 Equilibrium Moisture Content . . . . . . 2.2. 2 Drying or Diffusion Coefficient . . . . . 2.2.3 Moisture Versus Time . . . . . . 2.2.3.1 Theoretical Thin Layer Equations . . 2.2.3.2 Semitheoretical Thin Layer Equations 2 2.3.3 Empirical Models . . . . . . . . . . 72.2.4 Studies of Rice Drying . . . . . . . . . 2.3 Thin Layer Drying Laboratory Equipment . . . . 2.4 Direct Digital Control . . . . . . . . . . . . 3. OBSERVATIONS AND OBJECTIVES . . . . . . . . . . . 3.1 Constant Relative Humidity and Temperature . . 3.2 Relative Humidity and Temperature Variable . . 3.3 Objectives . . . . . . . . . . . . . . . . . . iv vii ix xii 1 4 5 5 9 l 12 l4 l7 l8 19 32 36 40 43 43 46 50 56 60 63 66 67 71 75 4. EQUIPMENT . . . . . . . . . . . . . . . . . . . . 4.1 Organization of Equipment . . . . . . . . . . 4.2 Microcomputer . . . . . . . . . . . . . . . . 4.3 Transducers and Microcomputer Interface . . . 4.3.1 Temperature Transducer . . . . . . . . . 4.3.2 Temperature Transducer Signal Conditioning . . . . . . . . . . . 4.3.3 Weight Transducer . . . . . . . . . . . 4.3.4 Weight Transducer Signal Conditioning . 4. 4 Air Conditioning Unit . . . . . . . . . . . 4.5 Study Chamber . . . . . . . . . . . . . 4.6 Software . . . . . . . . . . . . . . . . 4.6.1 Data Acquisition . . . . . . . . . 4.6.2 Digital Control . . . . . . . . . . 5. EXPERIMENTAL INVESTIGATION . . . . . . . . . . . 5.1 Product . . . . . . . . . . . . . . . . . . 5.2 Laboratory Methods . . . . . . . . . . . . . . 5.2.1 Equipment Initialization and Initial Data 5.2.2 Treatment During Test . . . . . . . . . . 5.2.3 Sample Post-treatment and Final Data . . 5.3 Data Reduction . . . . . . . . . . . . . . . . 5.4 Variables and Errors . . . . . . . . . . . . . 6. DATA ANALYSIS AND RESULTS . . . . . . . . . . . . 6.1 Analysis of Operation of Equipment . . . . . . 6.2 Exponential Models . . . . . . . . . . . . . . 6.2.1 One Term Exponential Model . . . . . . . 6.2.2 Two Term Exponential Model . . . . . . . 6.2.3 Three Term Exponential Model . . . . . . 6.2.4 Four Term Exponential Model . . . . . . . 6.3 Reduced Data Sets . . . . . . . . . . . . . . 6.3.1 Maximum Time . . . . . . . . . . . . . . 6.3.2 Data Acquisition Interval . . . . . . . . 6.3.3 The Reduced Data Set . . . . . . . . . 6.4 Other Thin Layer Models . . . . . . . . . . 6.4.1 Page' 5 Model . . . . . . 6. 4. 2 Spherical Model . . . . 6. 4. 3 Infinite Cylinder Model . 6.5 Combined Data Sets . . . . . . 5.1 Error Between Data Sets . . .5. 2 Nonlinear Combined Approach . . . . . .5. 3 Traditional Combined Approach . . . . . .5 4 Comparisons of Data and Models . . . . 6. 6 6 6 4.1 Traditional vs. Nonlinear . . . 6. 5. 6. 5.4.2 Comparison With Independent Data . . 77 77 79 82 82 84 86 89 91 95 97 98 100 106 106 107 107 109 111 111 114 117 119 122 124 126 130 144 149 150 152 155 157 157 159 165 170 178 176 185 187 187 189 7. CONCLUSIONS . . 8. SUGGESTIONS FOR FUTURE STUDY APPENDIX A - Listing of Data Acquisition and Control Software APPENDIX B - Listing of BASIC Software APPENDIX C - Listing of Selected Data LIST OF REFERENCES vi 195 199 202 206 213 240 Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table LIST OF TABLES Data for Example One . . . . . . . . . . Data for Example Two . . . . . . . . . . Conversion Factors From Digital To Temperature . . . . . . . . . . Conversion Factors From Digital To Weight Test Conditions for Parboiled Rice . . . Variation in Chamber Conditions . . . . . One Term Exponential Fit to Data Sets . . Two Term Exponential Fit to Data Sets . . Three Term Exponential Fit to Data Sets . Tray Loading Delays . . . . . . . . . . . Four Term Exponential Fit to Data Sets . Time Maximum for Test 18 A . . . . . . . Time Maximum for Test 18 B . . . . . . . Time Maximum for Test 19 A . . . . . . . Time Intervals, Test 18 A . . . . . . . . Time Intervals, Test 20 A . . . . . . . . Time Intervals, Test 20 B . . . . . . . . Reduced Data Sets Fit to Three Term Exponential . . . Page's Equation Fit to Data Sets . . . . Test 19 B With Spherical Model . . . . . vii 19 26 83 88 110 120 124 126 134 145 146 151 151 151 153 153 154 156 158 164 Table Table Table Table Table Table Table Table Table Table viii Test 20 B With Spherical Model Test 18 A With Cylindrical Model Test 19 A With Cylindrical Model Moisture Content at Time Zero Parameter Values, 9 Parameter Parameter Values, 8 Parameter Parameter Values, 7 Parameter Parameter Values, 6 Parameter Two Term Exponential Fit to Test 9 A Parameters For Model 6.24 . Model Model Model Model 165 167 168 177 178 179 179 180 191 191 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure LIST OF FIGURES Predicted and Observed Values of Example Data, Eq. 2.12 O O O O O O O O O O Residuals of Example Data, Eq. 2.12 . Predicted and Observed Values, Example Data, Eq. 2.13 O O O O O O I O O O Residuals of Example Data, Eq. 2.13 . Predicted and Observed Values, Example Data, Eq. 2.15 O O O O O O O O O I Residuals, Example Data, Eq. 2.15 . . Predicted and Observed Values, Example Data’ Eq. 2.17 O O O O O O O O O O Residuals, Example Data, Eq. 2.17 . . Air Flow in System . . . . . . Block Diagram of Process Controller . Circuit Diagram for the Temperature Transducers . . . . . . . . Circuit Diagram for Weight Transducer Circuit Diagram for Air Conditioner Control Study Chamber . . . . . . . . Residuals 19 B vs. Time, Residuals 19 B vs. Predicted, MOdel O O O O O O O O O O 0 ix 1 Term Exp. 1 Term Exp. Model 21 22 23 25 27 29 30 31 78 80 85 90 94 96 127 128 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Residuals 20 Residuals 19 Residuals 19 Model . . Residuals 20 Residuals 18 Residuals 18 Model . . Residuals 18 Residuals 18 Model . . Residuals 19 Residuals 19 Model . . Residuals 20 Residuals 20 Model . . Residuals 19 Model . . Residuals 20 Model . . Residuals l9 Residuals 19 Equation Residuals 20 Residuals 20 Equation Residuals 19 Model . . Residuals 18 B vs. B vs. B vs. B vs. A vs. A vs. B vs. B vs. B vs. B vs. B VS. B vs. B vs. B vs. B VS. B vs. B vs. B vs. Time, Time, Predicted, Time, Time, Predicted, Time, Predicted, Time, Predicted, Time, Predicted, Predicted, Predicted, Time, Predicted, 1 Term Exp. Model 2 Term Exp. Model 2 Term Exp. 2 Term Exp. Model 3 Term Exp. Model 3 Term Exp. 3 Term Exp. Model 3 Term Exp. 3 Term Exp. Model 3 Term Exp. 3 Term Exp. Model 3 Term Exp. 4 Term Exp. 4 Term Exp. Page's Equation . Page's Time, Page's Equation . Predicted, Time, Time, Page's 4 Term Sphere 4 Term Inf. Cyl. 129 131 132 133 135 136 138 139 140 141 142 143 147 148 160 161 162 163 166 169 Figure Figure Figure Figure Figure 6.23 6.24 xi Residuals of Fit to Total Data Set Residuals, Complete Model on Control Data Residuals vs. Time, Page's Model on Control Data 0 O 0 O O O O O O O O O O Residuals, Bakshi's Data Set 1 vs. Residuals, Bakshi's Data Set 2 vs. Model Model 182 184 188 193 194 NOMENCLATURE A'B'CIFIGIHIIIJ = constant parameters a,b,c,d,g = constant parameters [D = diffusion coefficient, lengthz/time [Db = dry bulb temperature, Celsius EDS = change in setting of controller [Dbd = change in moisture content, dry basis, decimal £3 = random errors, Gaussian distribution E [11,E[2].B[3]'E[4] = error history at uniform time intervals I3P4C: = equilibrium moisture content, dry basis, decimal I( = drying constant, l/time I; = typical length 51 = final moisture content of sample, dry basis, decimal I“Cd = moisture content, dry basis, decimal HR = moisture ratio, decimal 511 = initial moisture content, time=0, dry basis, decimal M(t) = moisture content at time=t, dry basis, decimal PI'P2’P3 O O 0 parameters fit by regression w u universal gas constant xii RH RHn RHo YTK {IMP xiii roots of Bessel Function of order 0 coefficient of determination relative humidity, decimal (unless specified) most recently calculated relative humidity, percent previously calculated relative humidity, percent correlation coefficient temperature, degrees C completeness of drying drying air temperature, degrees K time, hours residual weight final weight of can, grams initial weight of can, grams final weight of can and sample, grams initial weight of can and sample, grams water temperature, Celsius weight of sample dry matter, grams total weight of the sample, grams independent variable dependent variable 1 . INTRODUCTION The advent in recent years of the microcomputer has provided researchers with a: tool both to conduct experiments with closer control of important parameters and to collect far more data than was previously possible. The (digital computer, while not really new, has also provided J:esearchers with a powerful tool to analyze data in detail aarmi to evaluate models of biological phenomena with which previously it was not practical to work. In addition, ‘nncare powerful statistical computer programs are available t:c> analyze data, particularly when the models are f1<>rllinear. Modern electronics can provide another valuable t2<><>l in the study of the drying of agricultural products. Drying is one of the oldest methods man has used to Preserve food. Most of the drying is and has been solar drying, but due to weather and other factors artificial Salrydng has been of increasing interest. Engineers have tDecome involved in the design and construction of dryers of aSiricultural products and have been able to improve the alftificial drying process in terms of energy efficiency, {froduct quality, operator time and equipment use. There cOntinues to be a great deal of interest in improving the értificial drying process. Engineers need a mathematical model of the drying process to predict a product's response to different drying conditions. With such a model of the product and a compatible model of the equipment being designed, the engineer can evaluate the design of drying equipment and the response of the product-equipment combination. Engineers have found that the evaluation of alternative designs by testing computer simulation models requires far less time and money than building and testing of the actual (equipment. The parameters of the product drying model :include product moisture content, product maturity, product t:ype, ambient conditions, and time. Much agricultural drying research centers on the Cilqying of cereal grains, which are of extreme importance to mankind as a food source and as feed for animals. In fact the storage of dry grain is our best hedge against famine. Of all the cereal grains, rice is the most important single crzrop. Rice produces more carbohydrates and calories per h9ctare than wheat, maize (corn), barley, oats or millet. :In terms of protein produced per hectare rice is second Only to oats (Luh, 1980). Parboiling is an irreversible hydrothermal process thirfl1has been used, since ancient times, to improve the ITutritional content of polished rice and to increase the Percentage of whole grains after milling. The exact Process varies, and in many cases is a trade secret of the 3 processor. Parboiling can be described, in general terms, as occurring in six steps: preparation of the grain, soaking the grain, cooking, drying, tempering, and milling. The first step in parboiling is to remove foreign material and to sort the grain by size so that only rice of uniform size is parboiled in a given batch. Because the soaking, cooking and drying involve moisture diffusion and heat transfer, the shape and size of the grains affect the process. The goal of the second step is to increase the moisture content of the grains of the rough rice and is usually carried out in large vats of warm water. After the rice contains sufficient water it is heated, usually with steam, to gelatinize the starch. After starch gelatinization the product is dried, often in a series of dryers and then allowed to temper, or rest, for several days. Finally the rice, which up to this point still has the husks attached, is milled (Luh 1980). Little information is available on the modeling of drying of parboiled rice. The goal of this research project was to improve upon cIlllz'rent techniques of thin layer drying data acquisition and modeling. The specific objectives detailed in section 3.30 2. LITERATURE REVI EW The data analysis in this study will utilize several statistical computer programs including nonlinear regression programs. An introduction to the nonlinear regression concepts used in this dissertation is included in section 2.1 along with a discussion of several statistics which are useful in comparing regression analyses. The mathematical relationship between the parameters and the independent variables is referred to, in the regression literature and in this work, as the model. Any model may be used in regression but if the results are to have maximum meaning the model should be related to the Physical phenomena involved. Section 2.2 reviews appropriate models in the literature which are related to the underlying phenomena, of drying, and which other lEESearchers have successfully used. Special emphasis is Placed on the models and parameters used by other researchers pertaining to the thin layer drying of rice and Parboiled rice. The objectives of this study include the design and Construction of equipment. Section 2.3 is devoted to a review of the literature pertaining to the thin layer 4 drying study equipment. The equipment in this study is controlled by a microcomputer. Therefore, in section 2.4 the subject of direct digital control is reviewed. 2.1 Statistics The information contained in this section was obtained largely from Draper and Smith (1981). They provide an excellent presentation of linear regression and an introduction to nonlinear regression. Their book has been .invaluable during the data analysis portion of this investigation. There are two main uses for statistics in this <3:issertation. The first is to fit various nonlinear models t:<> sets of data. The second is to compare the prediction Power of different models to the same set of data. 2«1.1 Linear Least Squares Regression One of the simplest regression problems involves two V‘c‘iriables, the independent variable and the dependent vEiriable, and two parameters. Several pairs of numbers, representing the independent variable and dependent 'Vfiriable, are collected and combined into one data set. What is desired is to obtain an equation of the form: Y=P1*X+P2 2.1 where: Y = the dependent variable X = the independent variable P1,P2 = the parameters to be estimated. Any two numbers may be used for P1 and P2, but some combinations of values for P1 and P2 will predict the measured values of the dependent variable much better than others. The most common way to measure how well an equation using a particular set of parameters predicts the data is to calculate the residual at each value of X. The :residual is simply the observed value minus the predicted xralue. The residuals comprise, therefore, a set of signed raumbers. In a least squares approach to minimizing the Jreasiduals each individual residual is squared and the Squares are then summed producing the residual, or error, EBIJrn of squares. The "best" set of parameters is considered to be the set which produces the lowest possible value for tltie error sum of squares. The error, or residual, mean Square is obtained by dividing the error sum of squares by the degrees of freedom and represents a weighted average of ‘tlie square of the residuals. The square root of the mean SSauare error, often referred to as the standard error of time estimate, or standard deviation, represents the Weighted average residual. The corrected total sum of squares is obtained by Considering the mean value of Y to be the predicted value and then calculating the error sum of squares. The sum of squares due to regression can be calculated by subtracting the error sum of squares calculated for the regression equation from the corrected total sum of squares. Based on the relationship of the paramaters, not the variables, models are divided into three categories: linear, intrinsically linear, and intrinsically nonlinear. The three categories indicate the ease of determining the parameters and do not indicate the complexity of the relationship among the variables. For instance, for two independent variables, X1 and X2, and three parameters, P1, P2 and P3, the following forms are both linear in the parameters: Y=P1*X1+P2*X2+P3 2.2 Y=P1*X1*X2+P2*X12+P3*X2. 2.3 Equation 2.2 is linear in both the parameters and the variables while equation 2.3 is linear in the parameters but not the variables. The category, intrinsically linear, refers to models which appear to be nonlinear in the parameters but which can be transformed into linear form, or linearized. The commonly used model: Z=P3*exp(P2*X) 2.4 8 is an equation of this type. If the logarithm is taken of both sides: ln(Z)=1n(P3)+P2*X 2.5 and two substitutions made: Y=ln(Z) Pl=ln(P3) the equation takes the form: Y=P1+P2*X 2.6 which is linear. Transformations of the dependent variable must be used with caution. The assumption is made in linear regression that the error associated with the dependent variable has a mean of zero and a uniform variance. In the case where these two assumptions hold for the measured dependent variable, Z in equation 2.4, they are violated in the transformed dependent variable, Y in equation 2.6. In this case a nonlinear analysis using equation 2.4 would be more useful than a linear analysis of equation 2.6. The final type, intrinsically nonlinear, includes models which cannot be transformed into models which are linear in the parameters. One example of an intrinsically nonlinear model which will be seen later is: Y=(P1+P2*ln(P3*Xl))*(P4*exp(P5*X2)+ (1.0-P4)*exp(P6*X2)). 2.7 9 There are simpler models which are nonlinear such as: Y=(P1+P2*X)/(l.G+P3*X). 2.8 In the case of linear models, or intrinsically linear models after transformation, there is one unique set of parameters which will produce a unique minimum value for the residual mean square. In addition these parameters, at least in principle, can be determined in closed form. 2.1.2 Nonlinear Least Squares In the case of nonlinear regression the residual sum of squares is again minimized. Most computer programs for nonlinear regression include the provision for weighting the residuals other than by frequency of occurrence. If in the measurement of the variables the researcher feels that some sets of variables are measured more accurately than others, then residuals resulting from them can be weighted more heavily. The algorithm will attempt to reduce the residuals which have the most meaning at the expense of not reducing residuals which have less meaning. If there is no reason to believe that some data are better than other data all weights are unity. The formula for calculating the residuals squared is: 10 RS=W*(Y-f(X,P))2 2.9 where:RS = the weighted residual squared W = the weight for that particular point Y = the dependent variable at that point f(X,P) = the predicted value for the dependent variable using the nonlinear model, the measured values for the independent variables and the current estimate for the parameters. The residual sum of squares is obtained by summing the RS terms. Choosing parameters which minimize the residual sum of squares when the model is nonlinear is much more difficult than in the linear case. The parameters cannot generally be determined in closed form and therefore must be obtained by iteration. In addition there may be more than one set of parameters which will produce the same minimum residual sum of squares. In other situations there may exist a local minimum residual sum of squares with the absolute minimum lying some distance away. If a slight change in any of the parameters increases the residual sum of squares, then that set of parameters is the solution, only if there exists no other set which produces a smaller residual sum of squares. If one or more sets exist which produce a smaller or equal residual sum of squares, then a local minimum is said to exist. If local minimums are present, they may be found by an iterative algorithmto be the solution, when they usually are not the solution sought by the researcher. If local minimums are suspected, alternative starting values for the parameters 11 will often reveal them and certain starting values will lead to the absolute minimum residual mean square. Because the nonlinear functions dealt with in this study are monotonic, continuous anui have Inonotonic, continuous derivatives local minimums are not likely to be a problem. 2.1.2.1 Nonlinear Regression With Derivatives There exist several algorithms which search for increasingly better sets of parameters. Many of them involve the use of the partial derivatives of the model with respect to each of the parameters. The computer program BMDP3R (Dixon 1981) makes use of derivatives amd employs a modified Gauss—Newton algorithm. This algorithm (page 673, Dixon, 1981 and page 462, Draper and Smith, 1981) consists of first carrying out a: Taylor series expansion about the point defined by the most recently calculated set of parameters. If the expansion is limited to the first two terms, the result is: Y=f(X,Pr)+2Zi*(Psi-Pri) 2.10 where: Pr = the most recently calculated set of parameters P5 = the solution set of parameters f(X,Pr) = the model evaluated at X,Pr Zi = the partial derivative with respect to the ith parameter evaluated at the most recent set of parameters. 12 Equation 2.10 is then considered to be a linear regression problem and solved for the terms (Psi—Pri) as the parameters. When the linear regression problem is solved the new estimate for the parameters is the old estimate adjusted as indicated by the results of the linear regression. This process is repeated until either the convergence criterion is met or the maximum number of iterations is reached. The convergence criterion is specified by the user of the program as the relative change in the residual sum of squares from iteration to iteration. The maximum number of iterations is also specified by the user, generally to avoid wasted calculations in cases where the algorithm may not converge. 2.1.2.2 Nonlinear Regression Without Derivatives A second computer program, BMDPAR (Dixon, 1981) searches for improved sets ofjparameters.but does not use the derivative of the model. This program uses a pseudo- Gauss-Newton algorithm. It calculates a linear function L(X,P) equal to f(X,P) at the most recently calculated set of parameters. Because the function is linear the solution is obtained by linear methods. A new linear function is created which is equal to f(X,P) at the improved set of parameters. This algorithm is repeated until convergence or the specified number of iterations is reached. 13 BMDPAR allows the user to eliminate certain of the input data specifying acceptable ranges or identification numbers and allows the user to specify how small the change in the residual mean square is before ending the program. The program informs the user of simple statistics about the input variables comprising the data points which were used in the analysis and the residual sum of squares at each step in the algorithm. The statistics include: 1) the total number of data points 2) the number of data points included in the analysis 3) the mean of each variable 4) the standard deviation of each variable 5) the minimum and maximum value for each variable. When the program ends, the best set of parameters which have been encountered is printed along with the estimated mean square error, the statistic used to compare the various models. The program prints the estimated asymptotic correlation matrix which gives an indication of whether the parameters are independent.ofleach other. It also prints the estimated standard deviation for each parameter, useful in estimating the confidence interval for each parameter. The program prints the estimated value and observed value at each data point if desired and can construct simple graphs of the predicted values and the residuals. The use of these graphs will be discussed under residual analysis. When there are several independent variables a close scrutiny of the list of residuals can be helpful. 14 The term "best fit parameters" refers to the estimate of the parameters which produces the smallest residual sum of squares when that particular set of parameter estimates is used in a prediction equation and the predicted values are compared with the observed values. There is no guarantee that some other set of parameter estimates, which was not tested, might not produce a smaller residual sum of squares. Using the asymptotic standard deviations listed for each parameter estimate, confidence intervals can be calculated so that a range containing the true parameters producing the smallest residual sum of squares is known. 2.1.3 Analysis of Fit As stated earlier any model can be used in a regression analysis but only certain models are of real interest. One question which must be asked during a regression analysis is whether the model was appropriate. The first answer to this question is provided by the analysis of variance table for the regression. This table is an organized way of presenting the significant statistics regarding the regression analysis. The goal is to calculate the error mean square'for the regression and determine the significance of the error mean square. If the regression analysis fails the test, then the model is not appropriate. If it passes there are many steps left before accepting the model. 15 A statistic is needed in this work to compare the fit of different models to the same set of data and also to compare the fit of a model to a set of data in general. In the linear case the R2 value is often used. The R2 statistic may be defined in different ways. The most useful definition is that it is the square of the correlation coefficient between the predicted and observed values of the dependent variable. However, this statistic is not of great value when comparing the fit of a linear model to different sets of data (Draper and Smith 1981, pages 89-93) and is of less value in the nonlinear case. The statistic that will be used whenever possible is the residual, or error, mean square. This statistic is similar to the average residual squared and is usable for linear and nonlinear models. It is useful when there are few or many data points in the‘analysis and is the best estimate of the variance, if the model is correct. Before the advent of digital computers most of the grain drying regression was done graphically so there was neither a reliable estimate of the goodness of fit to the data nor of what was the probable range on the parameters in the equation “Lg. Hall and Rodriguez-Arias, 1958; Chu and Hustrulid, 1968; and Henderson, 1974). When researchers began using the digital computer in regression, parameter estimates appeared in the literature to four to six significant digits with no indication of the probable 16 confidence interval on the parameter estimates (e.g. Rowe and Gunkel, 1972; Husain, Chen and Clayton, 1973; Zuritz et a1., 1979; Fortes, Okos and Barrett, 1981; and ASAE, 1982). Too frequently, no measure of the goodness of fit was published or if any was it was an R2 value “Lg. Hussain, Chen and Clayton, 1973; Sharaf-Eldeen, Hamdy and Blaisdell, 1979b; Fortes, Okos and Barrett, 1981; and Sharma, Kunze and Tolley, 1982L. This statistic is of limited value in comparing the fit when different sets of data are being considered and does not give much of an indication of the magnitude of the difference between the predicted and observed values. The nonlinear regression techniques do not necessarily produce the best set of parameter estimates in the sense that there exists no set which will result in a lower residual mean square» They can only choose the set which was used in the algorithm and produced the lowest residual mean square. What can be done in addition to choosing the best set which has been tested is to place a confidence interval around the set chosen to give researchers an idea of the range of the true parameters. Usually the parameter estimates with the wider confidence interval are the less reliable parameter estimates in the sense that in varying them a given amount a smaller change in residual mean square results than when the other parameter estimates are varied. 17 A reasonable confidence interval is plus or minus two standard deviations from the estimated value, although in the nonlinear case the exact probability that the actual value will lie within this range is not known. When the regression problem is viewed as an n-dimensional geometric space the confidence intervals are seen as a confidence region (Draper and Smith, 1981 p. 489). This approach can help clarify the nonlinear regression problem. 2.1.3.1 Residual Analysis An examination of the residuals should always be carried out in a regression analysis. If the model sum of squares passed the F test, the residual analysis can still show that the model is not adequate. If thelnodel sum of squares failed the I? test, the residuals can provide guidance in selecting a better model. The assumption is made for the rest of this section that the regression analysis was significant. The residuals are the variation of the measured dependent variable which the model fails to explain. If the model is correct then these are the errors and the residuals provide the best estimate of the error in the data. In the statistical analysis the assumptions about the residuals usually include that they: are independent, have zero mean, have a constant variance and follow a normal distribution. In the examination of the residuals # fi—v—E_ 4qu_~__ 44 18 any evidence that these assumptions are not correct is sought. The residuals are examined graphically by plotting each residual against all reasonable variables. The most likely to plot against include the predicted value, time, (whether or not it is a variable) and each independent variable. Any pattern other than a uniform band about zero for any of these plots is an indication that one or more of the assumptions are violated and may be an indication of inadequacy in the model. A normal plot of the residuals, that is plotted on normal probability paper, will produce a straight line if the variance is constant. 2.1.3.2 Correlation Matrix Onerof the aids in‘working with nonlinear regression is the asymptotic correlation matrix. This is a set of numbers which has been normalized and indicates how independent the parameters are from one another. If the correlation between two parameters is large, near 1 or -1, it indicates that a model with one of these parameters removed may produce a residual mean square almost as low as the present model. It does not indicate that the model is necessarily inappropriate only that the present set of data may not support one of the parameters. 19 2.1.4 Examples The first example is intended to show how residual analysis is used and that R2 values can be misleading if not used with caution. For this example, data were created by using the integers from 1 to 15 as the values for the independent variable and calculating values for the dependent variable from the formula: Y = 8.000+4.000*X+0.100*X2+e 2.11 where e are errors randomly chosen from a normally distributed population with mean 0.000 and standard deviation 0.100. (The assumption is commonly made that errors in measurement are of this type, but the magnitude of the standard deviation depends on the situation). The data are shown in Table 2.1. Table 2.1 Data for Example One Independent Error Dependent Variable Variable 1.0 0.1007 12.2007 2.0 -0.l816 16.2184 3.0 -0.0730 20.8270 4.0 -0.2726 25.3274 5.0 0.0354 30.5354 6.0 0.1161 35.7161 7.0 -0.0454 40.8546 8.0 0.0391 46.4391 9.0 -0.0762 52.0238 10.0 0.0456 58.0456 11.0 0.0370 64.1370 12.0 -0.0292 70.3708 13.0 -0.l086 76.7914 14.0 -0.0641 83.5359 15.0 0.0782 90.5782 20 These data were fit to a model of the form: Y = P1+P2*X 2.12 and it was determined by linear regression that Pl=3.4 and P2=5.6 produces the best fit. The resulting fit is plotted in Figure 2.1. The model seems to work fairly well with an R2 value of 0.995, a value which in most cases is quite acceptable. It might be tempting to stop the analysis at this point and accept the model. However, when the residuals are plotted, as in Figure 2.2, a clear pattern can be seen in them, i.e. the residuals are a function of a variable. The pattern suggests that the model of the form 2.12 is not adequate and another model should be sought. If a model of the form: y = P1+P2*X+P3*X2 2.13 is fit to the data the resulting plot of the predicted and the observed values is as shown in Figure 2.3. In this case it was determined that Pl=7.9, P2=4.0 and P3=0.100 produced the lowest sum of squares. It should be noted that the R2 value isiL99998, higher than before. This is one case where the R2 statistic is of value, in comparing the fit of different models to the same set of data. Viewing the previous R2 value in isolation and not in 023’ ONGOHUMSU'U Dm0;03 (decimal dry basis) are common. What is desired by engineers is a model which predicts the moisture content of a bed of grain many grain thicknesses deep given the air conditions and the duration of exposure of the grain to those conditions. This problem is referred to as the modeling of deep bed drying. 33 One approach to deep bed modeling which has seen increasing use is to divide the volume of grain to be dried into thin layers and model the whole volume by treating each layer separately. The digital computer can then combine the layers into a total picture of the process. The modeling of thernoisture content of these thin layers of grain is referred to as thin layer modeling. For this approach to work the thin layer model for each layer must be accurate over a range of drying temperatures, product moisture contents and air moisture contents. There are generally two approaches to obtaining the thin layer equation: one utilizing moisture diffusion principles and based on an individual particle of grain, and the other obtained from a study of the response of an actual thin layer of grain (Brooker, Bakker-Arkema and Hall, 1974). The process of drying agricultural products of high initial moisture content canknedivided into two periods depending on the drying rate-time relation (Newman, 1932a). The first, at moisture contents of about 0.70 dry basis and above, is called the constant rate period where the drying rate does not change with time. In the second, the falling rate period, the drying rate decreases with time. Grain drying usually occurs only in the falling-rate drying period (Brooker, Bakker-Arkema and Hall, 1974). In the development of thin layer models for moisture relationships in agricultural products several physical 34 parameters have emerged as highly important. The two most important of these parameters are the equilibrium moisture content and the diffusion constant. Each of these parameters has been modeled separately and in general researchers have focused on only one of these parameters at a time. The attempt has been to try to obtain the most useful overall model by obtaining the best model for each of the parameters and then combining the parameters in an overall thin layer model. In this dissertation the moisture contents will be calculated in decimal dry basis. When referring to the work by others the moisture contents will be expressed in decimal dry basis except in formulas which will be kept as published. If the literature refers to the moisture content wet basis the values will not be converted to dry basis. Many models considered here use the variable, moisture ratio, MR, defined as: MR = (M(t)-EMC)/(Mi-EMC) 2.19 where: M(t) = the moisture content, dry basis decimal, at time t EMC = the equilibrium moisture content, dry basis decimal Mi = the initial moisture content, dry basis decimal. The evaporation of moisture requires energy and that energy must come from the air. In addition, the air and the product under study are usually at different 35 temperatures at the beginning of the testing period. These two factors have led to concern over what is the appropriate temperature to use in modeling. Sets of coupled differential equations can be written and solved digitally to determine the relative effect of the temperature and moisture gradients on the drying. Several researchers have done this and then measured the temperature of the product. Rowe and Gunkel (1972) measured the surface temperature of chopped alfalfa during moisture removal and showed that the product was nearly at the air temperature within 6 to 15 minutes. Husain, Chen and Clayton (1973) measured the center temperature of grains of Bluebelle rough rice with a thermocouple and found that the center temperature had changed from the initial grain temperature of 22 C to within 2.5 C of the drying air temperature of 50 C within 15 minutes. Fortes, Okos, and Barrett (1981) measured, with a thermocouple, the center temperature of wheat as it dried. They found that the product changed from the initial temperature of 26C7C2to virtually the drying air temperature of 47.0 C within 3 minutes. They also found that the actual temperature change was faster than the change predicted by a theoretical model. Both studies presented the data in graphic form so it is difficult to make exact conclusions from their data. They assumed particle geometry and created models based on 36 that geometry. In neither case did the researchers defend the validity of their geometric assumptions. 2.2.1 Equilibrium Moisture Content All products display a characteristic water vapor pressure dependent on their moisture content, temperature and physical characteristics. In grain time physical characteristics include the species and variety of the grain and thexnoisture history of the grain. The driving force in drying is considered to be the difference in vapor pressure of the air surrounding the grain and the vapor pressure within the grain. As a result of these two relationships the final moisture content of the grain depends on the water vapor pressure of the drying air. Unfortunately, the vapor pressure within the grain cannot be measured directly. The vapor pressure within the grain is inferred by measuring the vapor pressure of air at whichiru) net moisture exchange takes place. The moisture content of the grain at which there is no net moisture exchange at a given temperature and vapor pressure of the surrounding air is referred to as the equilibrium moisture content (Brooker, Bakker-Arkema and Hall, 1974). The equilibrium moisture content, EMC, is usually determined by measuring the moisture content of grain which has been held in a constant environment for a long period of time. In practice, the grain is held under constant conditions for varying lengths of time from less than a day 37 to overaiyear (Neuber, 1980). In some studies with rice, the product was placed in containers with saturated salt solutions, the rice being held above the salt solution by wire mesh. The weight of the rice was checked periodically until the moisture content stabilized (Karon and Adams, 1949; Breese, 1955; Hogan and Karon, 1955; Zuritz et a1., 1979). This moisture content is known as the static equilibrium moisture content. Allen (1960) discussed the static EMC and a dynamic EMC which is obtained from moisture content vs. time plots. These two equilibrium moisture contents have been observed to be different and Allen proposed that the static EMC should be used in situations involving long term storage but that the dynamic EMC should be used in dynamic situations such as grain drying. Because the dynamic EMC is obtained from graphic or mathematical regresshmn. for which assumptions about the form of the moisture content vs. time relationship must be made, the difference between the static EMC and the dynamic EMC could be a result of either the inadequacy of the model or a result of real physical phenomena. Considering the limited evidence in support of the adequacy of the models available in the literature the simplest explanation, and thus the preferred explanation, is that the models were inadequate. 38 Equilibrium moisture content is not an easily calculated variable but depends on many factors. Neuber (1980) in his review of equilibrium moisture content included the following variables related to shelled corn: 1) Air relative humidity 2) Temperature of both the product and the air 3) Desorption vs. sorption 4) Description of the grain (species, maturity, processing, dimensions, eth 5) Composition of the grain (ash, protein, fat, N.F.E) 6) Pre-treatment of the grain (storage time, MR history) 7) Method of determination of moisture content 8) Air-conditioning method, measurement of air moisture content 9) Differential vs. integral test procedure 10) Moisture equilibrium - definition, frequency of testing While conducting research many variables which are difficult to measure can be included in a model but what is needed for design is an equation of the form: EMC = f(easily measured variables) Pfost et a1. (1976) reported on the static equilibrium moisture content of yellow dent corn from the data of five researchers. They compared five of the most commonly used models: Henderson-Thompson RH = 1—exp(-A*(T+C)*EMC1/B) 2.2a Chung-Pfost RH = exp(-A/R/(T+C)*exp(-B*EMC)) 2.21 39 Day-Nelson RH = l-exp(F*EMCG) 2.22 F = A*TB G = C/TH Chen-Clayton RH = exp(-F*exp(—G*EMC)) 2.23 F = A/TB G = c*TH Strohman-Yoerger RH = exp(A*exp(—B*EMC)*ln(PS)—C*exp(-H*EMC)) 2.24 where: A,B,C,H = empirical constants T = temperature R = universal gas constant RH = relative humidity EMC = equilibrium moisture content, dry basis. PS = saturation pressure of water vapor at T They found all of the equations to be acceptable, but for the equations with three variables (Henderson-Thompson and Chung-Pfost) the standard error of estimate, the square root of thelnean square error, was not appreciably higher than for the other equations. Since these two equations used only three parameters, while the other equations employed more parameters, these two equations were preferred. These two equations have limited theoretical bases. Henderson (1952) proposed the form of the Henderson- Thompson equation and presented some theoretical arguments for the equation with C set to zero. Thompson 40 (1972) found that the addition of C to the equation improved the fit of the equation to the experimental data. Thompson‘s change resulted in biasing the temperature upwards from absolute zero and weakened the theoretical justification for the equation. Chung and Pfost (1967) proposed the Chung-Pfost equation as a two coefficient equation and included theoretical justification for it but the arbitrary addition of an offset from absolute zero improved the fit to the data with this equation also. 2.2.2 Drying or Diffusion Coefficient In all of the diffusion based equations there is a variable concerning the resistance of the grain to moisture diffusion. This variable is the flow rate (mass per time) per unit area (length‘z) with a unit concentration gradient (mass per volume per length). The units reduce to length2 per time~(Newman 1932bL. This variable is referred to as the diffusion coefficient or the diffusivity. Another variable, the drying constant, is closely related but is used in different equations (Brooker, Bakker-Arkema and Hall, 1974). The values of these variables depend on the product and the product temperature. The Arrhenius form is often used for the relationship (Henderson 1974) and is of the form: 41 D = A*exp(B/T) or K=C*exp(H/T) 2.25 where: D = the diffusion coefficient (lengthz/time) K = the drying constant (l/time) T = the absolute temperature A,B,C,H = empirical constants. The diffusion coefficient and the drying constant are related to each other by the geometry of the product. If assumptions are made about the geometry'of the particles and about the moisture time relationship, then values for the diffusion coefficient can be obtained for irregular solids, such as agricultural products, from experimental data of moisture content over time. If no assumptions are made about the geometry of the particles then only the drying constant can be obtained from the data. Some researchers have found it necessary to include a relationship between the drying constant and variables in addition to the product temperature. White, Ross, and Poneleit (1981) modeled the drying constant of popcorn with: K = 0.13+0.0023*exp(0.08*T)-0.0551/DM +0.00235*T/DM 2.26 where: K = the drying constant T = temperature degrees C DM = the change in moisture content from original to final Several researchers have reported evidence that diffusion in agricultural products is concentration dependent (Chu and Hustrulid, 1968a; Husain, Chen and 42 Clayton, 1973L. In both studies the researchers assumed that the moisture dependence followed the relationship: D(t) = A*exp((B*T-C)*M(t)-H/T) 2.27 where: A,B,C,H = regression coefficients M(t) = the moisture content at time t T = the temperature absolute D(t) = the diffusivity at time t However, in neither study did the researchers report on how well the equation fit the data nor the confidence intervals on the parameters. Neither paper included a theoretical justification for the form of the concentration dependent diffusion so the form is empirical. In both studies assumptions were made about particle geometry but no evidence was presented to show that the assumptions were not invalid. (It is not possible to show that the assumptions are valid only that they are invalid). Diffusivity is measured by collecting data on the moisture-time relationship and then fitting the data to a model by some regression technique. The inclusion of variables other than temperature in the model has little theoretical basis and the inclusion of parameters such as initial moisture content of the product, current moisture content or drying air relative humidity could result from using drying models of incorrect form, instead of from such variables' actual effect on the drying being modeled. 43 2.2.3 Moisture‘Versus Time There have been many mathematical models proposed for the falling-rate period of grain drying which can be divided into three categories: 1) Theoretically derived equations, 2) Semitheoretical equations, and 3) Empirical equations. 2.2.3.1 Theoretical Thin Layer Equations. Equations based on theoretical considerations have been derived. Brooker, Bakker-Arkema and Hall (1974) describe a set of general equations developed by Luikov (page 188) and show that they can be reduced to two differential equations in three dimensions relating the product moisture and temperature change with time. Assuming that the temperature of the product does not change, a reasonable assumption in some cases, will result in a single differential equation in three dimensions. If it is further assumed that l) drying is a diffusion process, 2) moisture moves as a vapor, 3) the product is of homogeneous composition, and 4) the product is a regular geometric shape, then the moisture ratio-time ralationship can be solved. Newman (l932a,b) considered three basic shapes, a slab with the edges coated to prevent evaporation except in two opposite faces, a sphere, and a cylinder with no diffusion through the ends. He assumed that the product was 44 homogeneous and of regular geometric shape. He also assumed that dryimg is a diffusion process, that for falling rate drying the initial concentration in the product is uniform, and there is no resistance to evaporation at the air-product interface. He demonstrated that the solution to the problem of diffusion in a porous solid where the diffusivity is constant results in an infinite series of the form: MR = A*exp(-a*D*t)+B*exp(-b*D*t) +C*exp(-c*D*t)+... 2.28 where: t = time in convenient dimensions D = diffusivity, length**Z/time A,B,C..., a,b,c... = Constants related to geometry of the product Some numerical values are, from Newman: slab sphere cylinder thickness=2*L radius=L radius=L a = "E5637 """ 536753" "633%?- B = 0.09006 0.15198 2.13127 c = 2.03242 0.06755 2.05341 a = 2.4674/L2 9.8696/L2 5.7831/L2 b = 22.227/L2 39.48/L2 32.472/L2 c = 61.685/L2 88.83/L2 74.892/L2 The desired form for the equations for the current study is moisture at time t as a function of the other variables. Rearranging this equation produces: 45 M(t) = (Mi-EMC)*(A*exp(—a*D*t)+B*exp(-b*D*t) +C*exp(—c*D*t)+...)+EMC 2.29 where: M(t) the moisture content at time t, dry basis Mi = the initial moisture content, dry basis EMC = the equilibrium moisture content, dry basis D = the diffusivity, length**Z/time The other parameters are as defined above. Newman observed that the first term of the equation should model the diffusion process adequately after sufficient time but that for the initial time, t=0, all of the terms are needed. For the equation to be correct at t=0, the sum of the terms A,B,C,H. must be one. It can readily be seen that at t=0 the sum of A, B, and C is 0.933 for the slab, 0.827 for the sphere and 0.876 for the cylinder of infinite length. However, depending on the relative values of D and L, the effect of the terms past the first one diminishes rapidly due to the exponential decay of the terms. Crank (1975) showed that a series of exponentials of this form is a solution to the general diffusion problem (Crank, 1975 section 2.3). Unfortunately, for these closed form solutions to be directly applicable, the material in which the diffusion takes place must be homogenous and of regular geometry. Although the closed form solution for any regular solid might be obtained by the proper change of coordinate system, only the simplest shapes have yielded to this approach. Clearly most of the biological materials of interest do not fit the standard shapes exactly. 46 Moon and Spencer (1961) showed that the solution in time to the Helmholtz equation (which includes the diffusion equation) is always of the form of a series of decaying exponentials. The solution in space, however, depends on the shape of the particle and the boundary conditions. 2.2.3.2 Semitheoretical Thin Layer Equations Lewis (1921) proposed a relationship considered to be analogous to Newton's law of cooling to describe the diffusion phenomenon. He theorized that the rate of drying is proportional to the driving force, the difference between the air moisture content and the product equilibrium moisture content. This relationship is integrated over time to give: MR = exp(-Kt) 2.30 where: K = drying parameter or constant, 1/hour. Rewriting this equation for moisture content as a function of time and diffusion constant produces: M(t) = (Mi-EMC)*exp(-Kt)+EMC 2.31 In the development of this model the assumption is made that the resistance to moisture flow is concentrated in a layer at the surface of the product; therefore, the geometry of the product particles is irrelevant. The 47 equation is commonly called the exponential or logarithmic model and is probably the most commonly used thin layer drying equation. Sharaf-Eldeen, Hamdy and Blaisdell (1979a) reviewed the literature on the falling rate drying of fully exposed materials and found that many investigators have used the logarithmic model, but many did not have good success in describing the complete drying curve with such a relationship. They state that the experimental evidence shows that the model underestimates the drying rate over the first partcfifthe curve and overestimates it over the last part of the curve. Comparing this equation with Newman's (l932b) equation it is seen that the logarithmic model is the first term of Newman's equation with A=1. If Newman‘s equations were correct, the logarithmic model would be expected to underestimate the drying rate at small time. Also, because regression is used in obtaining the coefficients in the logarithmic model, that model would be expected to overestimate the drying constant at large time. In addition, because of regression, the final moisture content would be predicted to be higher than the observed final moisture content, producing differences between the static EMC and dynamic EMC. Several modifications of the logarithmic equation have been used to compensate for the discrepancy between the 48 observed moisture content and the moisture content predicted by the logarithmic model. Several researchers (Misra and Brooker, 1980; Wang and Singh, 1978a; Agrawal and Singh, 1977; Bakshi, 1979; White, Ross and Poneleit, 1981) have used an equation first proposed by Page, adding an empirical exponent to time in the equation producing: MR exp(-Kta) 2.32 where: an additional empirical constant. (D II Rewriting this equation with moisture content as a function of time produces: M(t) = (Mi-EMC)*exp(-Kta)+EMC. 2.33 Researchers who have used Page's equation have found that it generally fits the data better than the logarithmic model. Misra and Brooker (1980) examined selected data from ten different researchers on the thin layer drying of shelled corn and found Page's equation to fit the data well. They used a sophisticated model for EMC, developed by Bakker-Arkema et al.(1974) and used a five-parameter model of Page's form. They found that the R2 of the curve fit varied from 0.973 down to 0.801 for the individual data sets with an overall R2 of 0.967. Because R2 is not of great value when comparing the fit of an equation to different sets of data the mean square error was calculated from the data Misra and Brooker reported. This statistic 49 varied from 1.5 to 8.0 with a value of 2.15 for the combined data. This mean square error was calculated from the moisture ratio in percent not theinoisturetat time t, as will be done later in this work. Comparing Page‘s equation with the logarithmic equation, Page's equation should be able to fit the data better because the exponent on time can cause the exponential decay to curve slightly, increasing the slope at small time and decreasing the slope at larger time. If the moisture ratio could be measured over time, then Page“s equation could be linearized and for that reason the regression required to choose the parameters with Page's equation would be much simpler mathematically than with Newman's form of the equation. Henderson (1974), Henderson and Henderson (1968), Rowe and Gunkel (1972) and Sharaf-Eldeen, Hamdy, and Blaisdell (1979b) found that a two or three term exponential model produced good results. This model has the strongest theoretical base. Henderson (1974) used graphical methods of regression and thus did not report the goodness of fit of his data. Sharaf-Eldeen, Hamdy, and Blaisdell (1979b) modeled the parts of the total moisture equation separately and obtained regression fits to three equations: EMC, overall two term exponential decay, and drying constant. For the EMC equation they reported an R2 greater than 0.99. For 50 the individual drying curves they reported an R2 "consistently" greater than 0.98. Finally, for the drying constant they reported that both the initial moisture and the temperature were important variables but that the R2 was 0.67. This R2 value is low indicating that the drying constant was not well predicted by their model. They used the same model for the drying constant, equation 2.4, that was used by Chu and Hustrulid (1968a) and by Husain, Chen and Clayton (1973). Unfortunately, they did not report any details of the nature of the lack of fit of this equation to their data nor the technique used in approximating the parameters. 2.2.3.3 Empirical Models Numerous empirical models have been introduced to the literature. However, because in this study'it is desired to use a model with some theoretical basis they will not be reviewed here. Theoretically based equations have the advantage that there is some hope of extrapolating the results from one study to wider ranges of variables and even to other products. There is no justification for the extrapolation of purely empirical equations. 2.2.4 Studies of Rice Drying Allen (1960) studied the deep bed drying of rice and corn (maize). He developed prediction equations of the grain temperature and moisture content over time. Because 51 he concentrated on the deep bed relationships, however, his work is of limited use in this investigation despite the fact that the ultimate goal of thin layer drying studies is to predict the behavior of the grain in a deeper bed. Agrawal and Singh (1977) studied short grain rough rice drying with the relative humidity held constant at 0.26 and varied the temperature from 32 C to 51 C. They also conducted studies with the temperature held constant at 51 C and varied the relative humidity from 0.1875 to 0.85. They used Pagefls equation for the form of the thin layer equation and the Chung—Pfost equation to calculate equilibrium moisture content. They reported that the error mean square comparing the predicted moisture ratio to the observed moisture ratio was 0.0007665. Wang and Singh (1978a) used one set of data from a thin layer study of rice to try to find a suitable prediction equation. They concluded that the theoretical diffusion model did not predict values which agreed well with their experimental values but did not report any statistics. Wang and Singh (1978b) tested one variety of medium grain rough rice at thirty different drying air temperatures and humidities. They compared four equations and chose an equation quadratic in time as the best, partially based on the residual mean square from a nonlinear regression and partially based (”1 practical 52 application. They reported that the three semitheoretical equations: a one term exponential decay with six parameters, Page's equation with six parameters, and a quadratic equation with six parameters all fit the data equally well with an error mean square of about 0.00045 when comparing the predicted moisture ratio to the observed moisture ratio. They also used a theoretical diffusion model based on spherical product geometry and found that the theoretical model was not as good as the other models with an error mean square of 0.000971. Because the focus of their study was to obtain models applicable to drying for short periods of time, only data covering the first forty minutes of drying were included. Sharma, Kunze and Tolley (1982) reported on a "two compartment model" for the moisture relationship in long grain rough rice. They argue that each material in the product displays an independent moisture-time relationship and that each is modeled by one term exponential decay. Noting that there are three parts to rough rice, hull, bran, and endosperm, they proposed a three term exponential model. This is the same form suggested by strictly diffusion arguments for uniform products. Twelve experiments, containingESreplications, were conducted on samples of grain at three temperatures, 24 C, 43 C and 56 C. At the end of 2, 10, 18 and 24 hours they determined the moisture content. They then used nonlinear regression techniques to determine the parameters in a 53 logarithmic model and a two term exponential model. They did not report the relative humidity of the drying air and did not control the relative humidity of the drying air, although they show that it was constant over time. They did not report any measure of how well the curves fit the data but did show graphically the confidence interval on the data. The confidence intervals appeared to be about 0.01 to 0.02 moisture content dry basis. They reported six equations: for drying temperature of 24 C M(t)=15.13+3.60*exp(-0.310*t) 2.34 M(t)=12.01+3.0l*exp(-0.0006*t) +3.97*exp(-0.4*t) 2.35 for drying temperature of 43 C M(t)=8.23+12.38*exp(-0.426*t) 2.36 M(t)=4.99+3.54*exp(-0.0048*t) +12.08*exp(-0.4387*t) 2.37 for drying temperature of 56 C M(t)=6.18+l3.71*exp(-0.528) 2.38 +12.52*exp(-0.6095*t) 2.39 Bhattacharya and Swamy (1967) studied the drying of parboiled rice as it related to breakage of the rice. They presented several plots of moisture content vs. time but did not attempt to describe the drying mathematically. In addition, the drying of the parboiled rice was not well controlled but rather was done by spreading the product in a thin layer and exposing it to ambient conditions. They 54 concluded that drying methods have a substantial impact on the quality of the finished parboiled rice. Only one study of the drying of parboiled rice was found in the literature where a mathematical model of the drying was attempted. In this study (Bakshi 1979), the emphasis was on the parboiling process itself and not on the drying aspects. He dried one variety of short grain rice which had been parboiled in a laboratory parboiling apparatus. He concluded that parboiled rice dries at about the same rate as brown rice and considerably faster than rough rice. He theorized that the husk, which offers considerable resistance to drying in rough rice (Steffe and Singh 1980a) offers little resistance in parboiled rice. Bakshi used a form of Page's equation (2.32) to model the drying of parboiled short grain rice. His study included both parboiled rough rice and parboiled brown rice. He reported that for parboiled rough rice: K a 0.503265+0.0002734*T-0.0001760*RH 2.40 0.064445+0.0046369*T-0.0147194*RH 2.41 and for parboiled brown rice: K = 0.016538+0.000173684*T+0.0064722*RH 2.42 a = -.7766099+0.001417332*T+0.07367*RH 2.43 where: T Temperature C RH = relative humidity, percent. Bakshi reported a residual sum of squares for the regression of the parboiled rough rice data to be 0.331 and the residual sum of squares for the regression of the 55 parboiled brown rice data to be 0.174 when the predicted moisture ratio was compared with the measured moisture ratio and 66 degrees of freedom in each case. Singh has continued the work begun by Bakshi on the thin layer modeling of parboiled rice.* He and P. K. Chandra assumed that the long grain rice behaved as infinite cylinders. They determined the diffusion constant D as a function of temperature. The equation they used for diffusion in an infinite cylinder was: MR = 2 4/(32*Rn2)*exp(—D*Rn2*t) 2.44 where: t = time, hours Rn = roots of the Bessel Function a = radius of infinite cylinder Their model for the diffusion coefficient was: 0 ll 0.0149668*exp(-3748.60/T) 2.45 where: T the air temperature, absolute diffusion coefficient, metersZ/hr. C II For the EMC model they used a model developed by Kachru and Matthes (1976) for long grain rough rice. This model was: * R. P. Singh 1983: personal communication. 56 EMC = 4.510+0.069*RH+8.837*RH9-5-0.015*T*RH0'5 2.46 where: T = temperature, Rankine RH = relative humidity, percent EMC = equilibrium moisture content, percent, d. b. The equilibrium moisture content of rice has been studied by several researchers. Karon and Adams (1949) studied the hygroscopic equilibria of rice using different salt solutions to maintain the environment for the rice. Their tests lasted for forty days. They presented their results in graphic form and did not publish an equation. Hogan and Karon (1955) studied the equilibria of rough rice at three temperatures and several relative humidities. They concluded that the moisture is adsorbed in three different modes, the first from 0.0 to about 0.07 (dry basis) considered to be unilayer adsorption, the second from 0.07 to about 0.14 considered to be a second layer of adsorption, and the third above 0.14 considered to be multilayer adsorption. They did not publish a mathematical model of their data. Juliano (1964) found that the EMC varies considerably with the variety of the rice. He studied four varieties at two temperatures and five relative humidities. He also published his data in tables only. Zuritz et al. (1979) published a study of the EMC values of rough rice in the temperature range from 10 C to 40 C and relative humidity from 0.112 to 0.863. The air moisture content was maintained by salt solutions. They 57 used three equations: Day-Nelson, Chung-Pfost and a semi- empirical equation they originated. They concluded that there was no difference among the predictions of the three equations. They reported an average root mean square error of from 0.24 to 0.31 of the EMC when expressed as whole percent dry basis. Pfost et al. (1976) and ASAE (1982) reported the equilibrium moisture content parameters for eleven products for the two preferred equations. They reported the standard error (root mean square error) to be 0.0097 dry basis for the Henderson-Thompson equation and 0.0096 for the Chung-Pfost equation for rough rice. The equation for rough rice from data collected from five researchers for the Henderson-Thompson equation is: (1n(l-RH)/-l.9187/(T+51.l6l))(1/2'4451)/100 2.47 EMC and the Chung-Pfost equation is: EMC 0.29394-0.046015*1n(—(T+35.703)*1n(RH)) 2.48 where: T temperature in Celsius No information was found in the literature on the modeling of the equilibrium umdsture content of parboiled rice. Bakshi (1979) calculated moisture ratios and therefore used EMC values, when working with parboiled short grain rice, but did not include a prediction equation 58 for the EMC. For the EMC value, he used the moisture content of the product at the end of the data collection period, usually about twelve hours.* Bakshi (1979) in his study of parboiled short grain rice also assumed a spherical particle shape and found the following relationships for the diffusivity over the temperature range from 40.6 C to 56.1 C: raw rough D = 33229. *exp(-8624.3/T) R2 = 2.99 raw brown D = 0.7979*exp(-4933.3/T) R2 = 0.91 parboiled rough D = 411.86*exp(-6977.7/T) R2 = 2.98 parboiled brown D = 401.62*exp(-6743.6/T) R2 = 0.97 Bakshi found that both parboiled rough and parboiled brown rice behaved about the same and at temperatures in the range of 305 K to 320 K both kinds of parboiled rice responded nearly the same as the brown rice. He concluded that the resistance to moisture movement in rough rice was mostly in the hull and that in parboiling, the hulls were split so that the parboiled rice behaved much like brown raw rice. In his conclusions, he stated that there is no constant rate drying in parboiled rice even at moisture contents as high as 0.60 and that the Arrhenius relationship was found to be valid in parboiled rice. * Amarjit Bakshi 1983: personal communication. 59 Steffe and Singh (1980a and 1982) studied the diffusivity of short grain rice endosperm, bran and hulls. They assumed a spherical shape for the rice and found the diffusion coefficient to depend only (”1 the product temperature, thus obtaining the following Arrhenius relationships for rice components over the temperature range of 35 to 55 C: endosperm D = 0.00257*exp(-2880/T) bran D = 0.797 *exp(-5110/T) hull D = 484. *exp(-7380/T) whole rough D = 33.6 *exp(-6420/T) R2 = 0.93 whole brown D = 0.141 *exp(-4350/T) R2 = 0.85 where: T = temperature deg. Kelvin The slopes of the diffusivity-temperature relationships are similar but the diffusivities vary greatly. The hull has the greatest resistance to moisture movement, the bran has an intermediate resistance and the endosperm displays the least. They did not find that the diffusion was concentration dependent in the range of moisture contents studied (0.33 to 0.14). Husain, Chen and Clayton (1973) applied coupled heat and mass diffusion models to the moisture loss in rough rice. They concluded that diffusion depends upon both the temperature of the rice and the concentration of the moisture in the grain. Their model with parameters was: 60 D=P1*exp(P2/T)*exp((P3*T+P4)*M(t)) 2.49 where: P1 = 94.8787 P2 = -7730.65 P3 = 8.833 E—4 P4 = -0.3788 T = product temperature, K ) M(t = moisture content, dry basis percent D - the diffusion coefficient, cal/g F 2.3 Thin Layer Drying Laboratory Equipment Relatively few researchers have described the equipment used in their studies of thin layer drying in any detail. Among those who have are Young and Whittaker (1971), Ross and White (1972), Rowe and Gunkel (1972), Henderson (1974), Agrawal and Singh (1977), and Stone and Kranzler (1981). These researchers used equipment of varying levels of sophistication and described what they used in varying detail. None of them provided much information on lmnv to design thin layer drying study equipment. Rugumayo (1979) described equipment with which he obtained thin layer drying data for shelled corn. His equipment used an Aminco-Aire unit to control drying conditions. The equipment monitored the drying conditions and the weight of the product as it dried and stored the data on paper tape at hourly intervals. His tests ran for 16 to 32 hours and did not reach equilibrium. It was difficult to control the temperature and relative humidity with his equipment and the analysis of his data by Misra 61 and Brooker (1980) showed that the resulting data did not fit Page“s equation as well as had previous data. Solvason and Hutcheon (1965) provided an excellent introduction to the problems encountered in the design of the drying study chamber. They emphasized the psychrometrics involved, as well as the choice of enclosure and conditioning equipment.:h1 the construction of controlled environment cabinets, chambers of less than room size. They noted that mass and energy balances can be written which describe the product, the cabinet, the conditioning equipment and the environment. These balances will often give rough estimates of the important parameters to be considered in the chamber design. From these equations one can conclude that the conditions throughout the chamber can never be uniform as long as there is heat and/or moisture transfer, and that the degree of nonuniformity will depend on the rate of heat and moisture transfer and the air flow rate between the chamber and conditioning equipment. Solvason and Hutcheon discussed the problem of mixing the conditioned air with the stale air around the product, the problems due to radiation if there are significant differences in temperature between the chamber and environment, and the problem of control. They stated that often a simple on/off controller is sufficient. They stated that the simultaneous control of temperature and relative humidity, as is desired in this study, presents 62 special problems. These two parameters are associated through psychrometrics and one cannot be simply controlled independently of the other. An additional problem is that it is impossible to accurately control a parameter which cannot be accurately measured either directly or indirectly. Fisher, Lillevik, and Jones (1981) discussed the measurement of relative humidity and the inherent problems in measuring it. Because the water vapor pressure cannot be measured directly some secondary parameter must be used. Wet and dry bulb temperatures or vapor saturation temperature (dew point) are commonly used. Considering a psychrometric chart and the effect of temperature measurement errors on the calculated relative humidity, it can quickly be determined that the error in relative humidity'due133a temperature measurement error depends upon the section of the chart in which the error is made. For example, consider a case where the dew point is 5 C and a one degree error in dry bulb temperature is made; the resulting error in relative humidity is approximately 0.08 if the air were nearly saturated and only about 0.01 if the air were rather dry. However, for any dry bulb temperature in the range 0 to 40 C, an error in dew point of one degree will result in an error in relative humidity of about 0.08 for any degree of saturation of the air. 63 2.4 Direct Digital Control Man has long striven to control the processes about hinu A major portion of engineering has been devoted to control of processes and machines. For many years the more complicated processes required a human to operate the controls and as engineering progressed the human controller became more powerful. Analog computers were and are able to perform well as controllers, but digital computers are more flexible and easily programmed. However, until recent years the digital computer was too expensive to devote solely to control. The microprocessor, an inexpensive digital computer, has provided engineers a powerful tool in direct digital control. A considerable body of literature and terminology has grown around the branch of engineering concerned with control. Much of this discipline requires an extensive knowledge and complete mathematical description of the process which is to be controlled. Sometimes the expense in time and money is justified but in many cases the mathematical description of the process is too expensive to obtain. In the control literature a process, referred to as the "plant", for which the basic differential equations are not known is considered to be an "unknown plant". The literature applied to the problem, direct digital control 64 of an unknown plant, is much scarcer than the literature applied to digital control of a known plant. Johnson (1977) provided a good introduction to the field of digital control. The first step in control is to define a desired reaction in the plant to be controlled. For example, it may be desired to maintain a water bath at a constant temperature. Ideally, the pertinent variable is measured, either directly or indirectly. In this example, a thermometer may measure the water temperature. The controller then compares the desired output, the water temperature, with the actual output and obtains an error. This error is used to adjust a control element, in this example, a water heater. The controller is designed to maintain nearly zero error. In a digital controller the value of the variable, or variables, is represented by a digital code. The controller compares the code with the desired code, operates on the difference using the control algorithm and then adjusts the control element. The digital controller cannot follow the process continually but must make a measurement of the process, do the necessary computations with that reading, then make the necessary adjustment on the control element. The entire process is repeated, leading to the use of difference equations instead of differential equations. 65 Bibbero (page 161, 1977) described the proportional- integral-differential, PID, algorithm in difference form. If E(l) is the most recently measured error, E(2) the previous error, etc., and I, J, and H are the gains in the proportional, integral, and differential portions of the controller, respectively; the algorithm takes on the form: DS = I*(E(1)-E(2))+J*E(1) +H*(E(l)-2*E(2)+E(3)) 2.50 where: DS = the change in setting of the control element. This algorithm, referred toansa velocity PID algorithm, has several advantages over other possible algorithms. It is relatively simple computationally; the terms have physical significance and are therefore easier to understand than some other algorithms; it is "bumpless" which means that it responds smoothly to step changes in the measured variable; and it is usable with an unknown plant. A relatively simple method can be used to adjust the gain coefficients until nearly optimal response of the system is obtained (Smith 1979). 3. OBSERVATIONS AND OBJECTIVES The goal of this research project is to improve upon current techniques of thin layer drying data acquisition and modeling. Because the effects of temperature and relative humidity on drying are nonlinear, nonlinear data analysis and regression techniques will be used. Transformations on the data, which make the data analysis simpler but distort the effects of the error inherent in all data, will not be used. Rather the relationship of the variables will be used directly to avoid distorting the effects of errors. The parameters resulting from proper data treatment should predict the observed data better than the parameters resulting from the traditional methods. The form of the model under which all thin layer data is collected is: M(t) = f(t,T,Mi,RH) 3.1 where: M(t) the moisture content at time t t = the time T = the temperature Mi = the initial moisture content of the product RH = the relative humidity of the drying air. The researcher measures the moisture content of the product at a known time after it has been subjected to a known relative humidity and temperatureu Thetnoisture content 66 67 data are collected differently by different researchers. The form of the model, however, is the same whether the moisture content is estimated from a weight measured as the drying test is running, or if the weight is measured by removing the sample momentarily from the drying apparatus, or if the moisture content is estimated by removing subsamples periodically from the sample and measuring their moisture content. Usually the relative humidity and temperature are held constant while one sampleris dried and then the procedure is repeated at 21 different relative humidity and temperature. In this case the model for each individual test may be rewritten as: M(t)T,RH = f(t,Mi,EMCT'RH) 3.2 where: EMCT,RH = the equilibrium moisture content. The subscripts emphasize that the model is useful only at one combination of relative humidity and temperature. The subscripts will not be repeated further but the equation should be understood to apply to only one combination of relative humidity and temperature. 3.1 Constant Relative Humidity and Temperature Two of the simpler models for the relationship between the factors in equation 3.2 which have received much attention :hi the grain drying literature are the 68 logarithmic model and Page's model. The logarithmic model produces: M(t) = (Mi-EMC)*exp(-K*t)+EMC 3.3 where: K = the diffusion constant. Some researchers treat EMC as a known constant and others treat it as a parameter to be estimated by regression. Likewise, the quantity (Mi-EMC) is treated as a known constant by some and an unknown parameter by others. If both EMC and (Mi-EMC) are treated as known constants the equation may be rewritten by the transformation: Y = 1n((M(t)-EMC)/(Mi-EMC)) 3.4 where: Y = the transformed dependent variable. The quantity of which the logarithm is taken in equation 3.4 is observed to be the moisture ratio. Using eqution 3.4, equation 3.3 is transformed to: Y = -K*t 3.5 Equation 3.5 is linear and a value for K can easily be obtained from a set of data. However, as observed in section 2.1, transformations on the dependent variable such as equation 3.4 must be done carefully, i.e. examine the residuals for evidence of nonuniform variance. The assumption that the errors in the measured values for M(t) are normally distributed and constant over the range of the independent variable, time, is more reasonable than the 69 assumption that the errors in Y.after the transformation 3.4 are normally distributed and constant over the range of the independent variable. The more reasonable approach, therefore, is to not transform the data but to analyze the data by nonlinear methods assundmmga model of the form of 3.3. In either case, whether model 3.3 or 3.5 is used the residuals should be carefully examined before anything is said about the suitability of the model. If EMC in equation 3.3 is considered to be an unknown parameter, the equation is intrinsically nonlinear and must be treated by nonlinear methods. Therefore, when the logarithmic model is used the use of nonlinear regression techniques is preferable to the use of transformations. Moreover, if EMC is unknown, nonlinear regression techniques must be used. The next model to be considered is Page's equation. Inserting this model in equation 3.2 produces: M(t) = (Mi-EMC)*exp(-K*ta)+EMC 3.6 where: a = the parameter added by Page. In the case of equation 3.6, if either Mi or EMC is unknown the model is intrinsically nonlinear and must be treated by nonlinear regression techniques. In the case where Mi and EMC are considered to be known constants for the particular temperature and relative humidity of the test, a transformation will simplify the 70 estimation of parameter values. First the known Mi and EMC are moved to the same side of the equation as M(t): (M(t)-EMC)/(Mi-EMC) = exp(-K*ta) 3.7 If the logarithm is taken twice and the substitution made: Y = 1n(-ln((M(t)-EMC)/(Mi-EMC))) 3.8 equation 3.7 becomes: Y = ln(K)+a*ln(t) 3.9 Given the data of the value of Y at different times t, the values for K and a may easily be determined. Of course the same comments apply to the model incorporating Page‘s equation as in the model incorporating the logarithmic model, with regard to the errors of the measured dependent variable, M(t), compared with the errors of the transformed dependent variable Y. In fact, because the logarithm of the moisture ratio is taken twice it is even more unlikely that the errors about Y will be normally distributed and uniform over the range of the independent variable, now transformed to ln(t). In the case of Page‘s equation, as in the case of the logarithmic equation, the use of the measured dependent variable is desirable in regression. In the case where either Mi or EMC is to be determined along with the other parameters the use of nonlinear regression techniques is required. 71 The more complex models such as the two term exponential model, equation 3.10, or the even more complex three term exponential model are intrinsically nonlinear and cannot be treated except In! nonlinear regression techniques. M(t) = (Mi-EMC)*(P1*exp(P2*t)+P3*exp(P4*t)) +EMC 3.10 where: P1,P2,P3,P4 = the parameters to be estimated 3.2 Relative Humidity and Temperature Variable In modeling the drying curves obtained ataiconstant relative humidity and temperature, nonlinear regression techniques are desirable and in many cases required for proper treatment of the data. Curves of moisture content vs. time are obtained at discrete values of temperature and relative humidity over the range of temperature and relative humidity of interest. The intent istx>create a model which will be useful at any combination of relative humidity and temperature within the range for which the data were collected. The method common to much of the recent drying literature is for the researcher to have a model for EMC and another model for diffusion constant which is to be inserted in the model for moisture content vs. time. Assuming that the model for EMC is the Chung-Pfost equation (although the argument is the same for any of the currently 72 popular models of EMC) and that the Arrhenius model is to be used for the drying constant, the logarithmic moisture content vs. time model takes the form: M(t) = (Mi-P1+P2*ln(-(T+P3)*ln(RH))) *exp(-P4*exp(P5/T)*t) +P1-P2*ln(-(T+P3)*ln(RH)) 3.11 where: P1,P2,P3,P4,P5 = parameters to be estimated. Equation 3.11 is nonlinear in both the variables, RH, T, t, and M1, and the parameters. Statistically, the most desirable approach is to use the data to fit the parameters P1—P5 with nonlinear regression. The researcher may be working with a product which has a model of EMC where P1, P2 and P3 are known. The effect of temperature on the drying constant as well as the moisture content at time tzlnay be obtained by linear regression. The EMC model and Mi are combined with M(t) to produce the moisture ratio and logarithms are taken of both sides producing: 1n(MR) = -P4*exp(P5/T)*t 3.12 Taking logarithms of both sides again and using the substitution: Y = 1n(-ln(MR)) produces: 73 Y = 1n(P4)+P5/T+1n(t). 3.13 If the same treatment is used with Page's equation, the Arrhenius form may be substituted into equation 3.9 producing: Y = 1n(P4)+P5/T+a*ln(t). 3.14 After the transformations the parameters in equations 3.13 and 3.14 may easily be estimated by linear regression. The only questions about the process are whether, because of the transformations, the assumptions on which linear regression is based are violated, and if they are, in what way the results of the regression are affected. These questions remain unanswered in the grain drying literature. A more important question involves what effect moving the model for EMC (containing the variables temperature and relative humidity) from an unknown parameter to a known variable has on the overall model. There is no reason to assume that the model for EMC produces errors which are significantly lower than the errors elsewhere in the model. In fact, models for EMC for eleven agricultural products have standard errors associated with them from 0.006 to 0.03 (ASAE 1982). With final moisture contents of 0.10 the error is 30 percent. The best way to determine the effect these errors have on the overall model is to collect data of moisture content vs. time for a range of temperatures and relative 74 humidities. Nonlinear regression techniques should then be used to analyze all of the parameters involved even in the cases where it would be mathematically possible to use linear regression. The most reasonable assumptions about the errors in the dependent variable in thin layer moisture relationships of agricultural products require starting with an equation of the form of 3.1. The appropriate substitutions are then made producing an equation such as 3.11 containing only variables which can be measured and the parameters which will be estimated by nonlinear regression techniques. A series of tests are conducted controlling the variables and varying them as appropriate. Finally the data are fit to the nonlinear equation. Following the approach described in the previous paragraph allows much more to be concluded about the model than could be concluded if the traditional approach were used. When the observed data are regressed against a model such as equation 3.11, the model as a whole can be tested for adequacy. In addition, each parameter can be tested to determine its significance and the parameters, if any, which are not significant can be removed from the model. Also, the relative importance of each variable can be observed more easily and possible interactions between the parameters in the model can be observed. The preceding method of experimentation coupled with computer programs for nonlinear regression constitute an improvement over the 75 method of researchers who have had to limit themselves to mathematically simple models, make dubious transformations and make dubious assumptions about errors. All of the currently popular thin layer models are nonlinear in the variables and all are also nonlinear in the parameters if EMC is not known. The errors associated with EMC indicate that (when it is obtained from a model) EMC is a significant source of error and should not be treated as a constant with zero error. In addition, the theoretical diffusion-based model, assuming the resistance to diffusion is not simply located at the surface of the particles, is nonlinear in form. Despite these facts, no study'of the thin layer drying ofauiagricultural product has been published in which the observed data has been regressed against the complete model, including EMC and drying constant with nonlinear methods. 3.3 Obectives In order to accomplish the overall goal of improving current thin layer drying study techniques these specific objectives were formulated. A) Design and construct equipment capable of measuring, controlling and recording the air temperature and relative humidity over time and measuring and recording the moisture content of an agricultural product over time. B) Collect data at small time intervals on an agricultural 76 product, parboiled rice, at several constant relative humidities and temperatures and analyze the data to: 1) determine the most appropriate form for the thin layer equation at a constant relative humidity and constant temperature. Four forms will be considered: a) the logarithmic model, iJL the resistance to moisture movement is at the particle surface, b) a series of decaying exponentials, i.e. the resistance to moisture movement is distributed throughout the particles, c) Page‘s empirical equation, and (D models basedcniregular particle geometry and distributed resistance to moisture movement. 2) determine if the results are affected by collecting thin layer data at different intervals and for different lengths of time. There are two questions: a) how frequently should data be collected to obtain parameter estimates and am: what times should the observations be made? b) How long should data be collected to obtain reasonable estimates of EMC? 3) obtain a complete model of the product in the form: M(t) = f(t,T,RH,Mi) 3.1 This model will contain implicitly an EMC model, an Arrhenius equation for the drying constant, and a model of moisture content vs. time. 4 . EQUI PMENT The first objective was to build equipment with which thin layer drying studies could be made. This equipment must be able to control the main independent variables, the drying air temperature and relative humidity. It also must be able to measure the main dependent variable, the moisture content of the product. All of these variables must be recorded over time. 4.1 Organization 3f Equipment Figurer4.l shows the air flow in the equipment. The air entered the Aminco—Aire unit (Aminco-Aire, 1967) and was saturated by'a mist of water. The water temperature was controlled by a refrigeration unit or an electric heater as needed. Next, the air passed into a chamber where it was heated (without adding moisture) by another electric heater. Thus, the air temperature and relative humidity were controlled as the air left the Aminco—Aire unit. The air next passed through a circulation fan and then into the study chamber. After the air left the chamber it recirculated through the Aminco-Aire unit. 77 78 .3523... 54 0.:P0maw Ewumhm c“ 30Hm nee H.e wusmwm #03005.“- £953: ~5th :2: ZOF:90 A sentew a: 52:3 . IIL 3505 3:. 32.30 33o I pica»:- 3530 32> .35... .5.. 81 together. The heart of the microcomputer was a Z-80 microprocessor on a Cromemco Single Card Computer, SCC, (Cromemco, 1980) which also contained 8K of read only memory, ROM, and 1K of random access memory, RAM. Other functions contained on the Cromemco SCC were a serial port (for communication with the printer or video terminal), a parallel port (for control of the air conditioning unit), another parallel port (for communication with the digital cassette deck), and one status port (for control of the communication with the digital tape deck). A second board in the system was a California Computer Systems 16K RAM board (California Computer Systems, undated), providing additional RAM. A third board was a Vector Graphics lZK-ROM/RAM board which has 1K of RAM and up to 12K of ROM (Vector Graphics, undated). The final part of the system was the TecMar analog to digital, A/D, converter unit (TecMar, 1980). This two board set contained a Data Translation 5712 module which provided software controlled gain, multiplexing of 16 input channels and 12-bit analog-to-digital conversion. Because the A/D converter was multiplexed, the eight transducer signals could be connected directly to the module after signal conditioning. The 12-bit A/D converter provided a precision of 1 part in 4096, adequate for the range of values encountered in this study. This unit also contained an Advanced Micro Devices Am9513 timing controller chip, which was used as a "real-time" clock providing time in 82 days, hours, and seconds as well as providing an interrupt to the microprocessor every 40 milliseconds, mS, as required by the control and data acquisition software. 4.3 Transducers and Microcomputer Interface Because only temperatures and weights were measured there were only two types of sensors and two signal conditioning circuits to consider. After the signal conditioning equipment was constructed and the transducers connected to the microcomputer system the transducers were calibrated. 4.351 Temperature Transducer The temperature sensor used was the National Semiconductor precision temperature sensor LM335 (National Semiconductor 1980). This is a solid state integrated circuit device which operates as an huproved zener diode. The voltage output has a linear relationship with absolute temperature of about +l0mV/K. The device operating temperature range is -10 C to +100 C and the corresponding typical nonlinearity over that range is 0.3 C (National Semiconductor 1980). Because time design temperature range for the laboratory equipment was at most 50 C and the majority of the expected nonlinearity is at elevated temperatures (National Semiconductor 1980) the expected nonlinearity in 83 the range 5 C to 50 C was no more than 0.15 C. After the temperature sensor circuits were assembled they were tested against a laboratory mercury thermometer marked in 0.1(L. The sensors were used in a temperature control algorithm during calibration to hold the temperature as constant as possible by controlling the air conditioning unit. The water temperature sensors were calibrated in water and the air temperature sensor calibrations were conducted in air. No nonlinearities were detected in any of the six sensors used. Table 4.1 shows the results of the calibration. In all cases the correlation coefficient, r, was greater than 0.9999. Table 4.1 Conversion Factors from Digital to Temperature Sensor Number A B 2 0.00393 -18.85 3 0.00399 -19.00 4* 0.32 ~15.9 5 0.00392 -l9.05 6 0.00392 ~18.79 7 0.00399 -18.52 * Approximate Predicted Temperature = A*(digital value) + B The conversion factors for sensor 4 are only approximate because its conversion factors were not checked after the initial equipment was set up. Sensor 4 was used to measure the temperature of the air heater and its exact temperature was unimportant. The control algorithm depended on sensor 3 for the chamber temperature control and adjusted the heater temperature as needed to obtain the 84 desired chamber temperature. The conversion factors did not change during the testing. Six temperature transducers were chosen to allow for three to be used in the control algorithm leaving three to measure the air temperature around the samples. Three sensors allowed the temperature to be averaged among the sensors and also provided backup sensors in case of sensor failure. The sensors were inexpensive but the calibration time which was required before their use added considerably to the expense of each data point. 4L3.2 Temperature Transducer Signal Conditioning Figure 4.3 shows the circuit used with the temperature sensors. Because the sensors produce 0V output at 0 K (nominal) and have a slope of 0.01V/K (nominal) they produce at least 2.5V at normal drying temperatures. If a constant voltage ofiLSV’is subtracted from the output of the sensors before the signal is converted to a digital signal, the gain of the A/D converter can be higher, producing a correspondingly higher precision. This idea was implemented vfirfli a National Semiconductor LM336 integrated circuit. voltage reference (National Semiconductor 1980) producing approximately -2.5V to add to the voltage output of the temperature sensors. A small RC 85 muwospmcmue wusumummEme wsu how Emummmo uflsouwu m.v musmflm thmw>zoo D\< h: on >m: ..— unn ecu jvnau .6.) I h 0: DH” E'— 1... e a: man 5: fl.— o l) m _ ex man s: .o v _ . ”mm H _ man s: w.— o 23§( n _ w: _ man s: .o > N e _ man s: JWZZallow two samples to be studied simultaneously. The transducers were located outside the chamber to minimize temperature effects. They would be expected to be affected to some extent by chamber temperature changes, even when located outside the chamber, but should be free of most of the temperature effect. In the case of high humidity conditions within the chamber the weight transducers would also be isolated somewhat from the moisture. The product was placed on trays made of aluminum screen attached to a rod which passed through the bottom of the chamber. There were two trays and one rod attached to each weight transducer. 87 The transducers were designed to be insensitive to moments and forces other than in the one direction of desired loading. The rated error per inch of off—center loading at 1/2 capacity was 0.004% of full scale or 0.049. The transducers were loaded with a 100g. mass alternately at each of the corners and at the center of the screen tray. No discernible correlation between the mass location and the number in the computer was noted. The sample holder was designed so that the sample center of gravity was above the mounting for the support rods. The weight transducers were loaded with known masses and the corresponding number in the microcomputer was noted. Since the relationship between the weight and the number in the microcomputer should be linear (according to the manufacturer's literature) a linear least squares curve fit was run to obtain the conversion factors from the number in the microcomputer to the actual weight, in grams. In all cases the correlation coefficient, r, was greater then 0.9999. Table 4.2 shows the conversion factors for transducer 0 and 1. Although the weight transducer measures force the calculations were made in grams, the unit of mass. Since all data were collected in the same room gravitational effects were constant. 88 Table 4.2 Conversion Factors from Digital to Weight Sensor Number 0 1 A1 0.0366 0.0353 Bl —145.5 -156.8 A2 0.0364 0.0350 82 -l43.8 -155.8 A3 0.0360 0.0348 A3 -140.6 -152.3 A4 0.01627 0.01574 B4 -20.6 -52.9 Sample Weight = A*(digita1 value) + B There were four sets of conversion factors used for the weight transducers. The first covered tests 3 and 4, the second tests 5 through 10, the third tests 11 to 17 and the last tests 18 through 21. In all cases the change in calibration was caused by changes in the system. The numbers associated with A and B refer to the order in time in which the different values were used. The temperature measurement system was adequate from the beginning. At first the weight measurement system was not as accurate as was desired. The change between tests 4 and 5 was due to a change in the sample holder; the change between tests 10 and 11 was again due to a change in the sample holder; and the change between tests 17 and l8‘was due to a change to a better excitation source for the weight transducer. At the end, tests 18 through 21, the weight measurements were more accurate than those at the beginning of the testing. 89 The equipment needed at least one half hour to warm-up properly before any reliable weight data could be collected. It was observed that calibrations completed before adequate warm-up were different than calibrations done after warm-up. However, the conversion factors after warm-up were stable. Because of this warm-up time the equipment was left with the power on for most of the testing period. 4.3u4 Weight Transducer Signal Conditioning Duebto the relatively low level signal obtained from strain gauge type sensors, an Analog Devices, model 2B31J, strain gauge signal conditioner was used as shown in Figure 4.4. This device includes an instrumentation amplifier with gain from 1 to 2000 and a low pass filter with a time constant of 0.58. The excitation of the strain gauge bridge was regulated, at 6.9V, with a National Semiconductor LM399, a temperature stabilized integrated circuit precision voltage reference. A second LM399 was used with a voltage divider to provide an offset voltage which produced nearly 0V out of the strain gauge signal conditioner when there was no sample load on the weight transducer. 90 Kwh¢w>28 s noospmcmu e enmemz Hoe :0) j ngowz(¢h bIOEK .— w22<10 .7. "aha.“ 8w.u-J§( 3(5 :04 tun—530 20:.(h2mglbw2. - ¢w20.h.n—ZOD wOD¢O 2.(¢hn SON D< Ono. UGO Emummflo vacuufiu v.¢ musmfim £34 £3.— R 8 >54 lg EON O< A¢ZOPPED¢ cngwZ<¢b #305: 4.420: .OO< 91 A metal film resister was used to set the gain on the strain gauge signal conditioner at approximately 1880. This gain was near theinaximum allowable on the unit and produced a precision of about 0.016 grams; while the transducer itself had a worst case precision of 0.26 grams. The total range with a gain of 1880 was about 65 grams which was adequate for the testing of grain. This range may not be adequate, however, for the study of products containing more than 50% moisture in which case the gain should be reduced. 4.4 Air Conditioning Unit The unit which was used to condition the air was an Aminco-Aire model J4-5460 which can condition up to 8.5 cubic meters per minute (300 cubic feet per minute). This unit was designed to condition relatively small chambers of less than 1.1 cubic meters (40 cubic feet). The chamber used in this test had a volume of approximately 0.24 cubic meters and about 2.5 cubic meters per minute of conditioned air were circulated through it. In the Aminco-Aire unit, the air moisture and temperature control are achieved in two steps. First the air passes into the larger chamber where there is a mist of water droplets spraying into the air. The droplets fall into a water bath the temperature of which is maintained by a refrigeration unit and electric resistance heater. Each water droplet is surrounded by air and thus heat and 92 moisture transfer occur between the air and water droplet. The air then passes into a second chamber where it is heated by electric heaters. The air finally passes through the test chamber and is recirculated into the first chamber. fmua microprocessor based circuitry controls the temperature of the water bath and the duty cycle of the electric heaters which heat the air. Because the relative humidity is not measured, errors often encountered in measuring humidity are avoided. There is a unique correlation between the Aminco water temperature, air dry bulb temperature and the relative humidity of the air. A chart of this relationship was provided by the manufacturer and a formula obtained by regression was calculated from the chart: DB=P1+P2*ln(RH)+P3*WT*ln(RH)+P4/RH+P5*WT 4.1 where: DB the dry bulb temperature, Celsius RH = the relative humidity, percent WT = the temperature of the water bath, Celsius P1 = 77.4 P2 = —16.98 P3 = -0.0821 P4 = 73.7 P5 = 1.377 Once steady state conditions are reached the test chamber conditions should change very little because the load produced by the drying grain is very slight. Atnuch greater load is the heat loss to the environment. This load does change during the test as the room temperature changes, but the change is relatively minor. By far the 93 greatest load within the air conditioning system is the cooling and saturating of the air followed by the reheating of the air. The Aminco-Aire operates on 208VAC three phase power and so voltage and power amplification were necessary from the SVDC control signals coming from the microcomputer. The circuit is shown in Figure 4JL When the equipment was first built the opto—isolated solid state relays were not readily available so regular opto-isolators, for noise and voltage isolation, were used with relays for power amplification and voltage translation. Later when additional control was desired solid state relays were used. They are much smaller and simpler to use than the series of relays. The switches were operated from an eight bit parallel output port available on the microprocessor board. An additional safety feature was added to turn off all power to the Aminco—Aire if a signal was not received from the microprocessor every 100 ms. This circuitry was a resettable, retriggerable monostable multivibrator attached to a solid state relay. 94 Houpcoo uwcowuwpcou Med uOu Emummwo uflDOueo m.v musmwm COFn¢ m0 mus—IO 44¢ Oh O 4 Lu 2.50:0 cepufizou 2.5.. , .3... Sin» — 82.32 o<>8 n .232. 3 335.0293 0 a3<> <26 u4u< whim n33 h 500.25.. — l\l|/.l 15 oz 03 1 oz 0 no a .1. >15: .353: c... 95.5 943 fl oz 0 on H - . - < OI a . Elm... 1. Id 3 :33: :20 _ F ub(hu 0:8 352.8 82.3. 53.. J 0 00.0.0...’ 8 29:50.52. g . cups... Yu._|.||Av/ I :(o 23 3m: .32. 5.9. 5.55 35:... cups... i\/\|O/9Il.-TL 1.5.. 8.3.. d m D(>°~p _ page 00 CwaO 30:18:38. on..." 95 4.5 Study Chamber A sketch of the study chamber is shown in Figure 4JL The chamber was 0.50m by 0.30m by 1.60m and was constructed mostly of plywood. The wood was painted with several coats of shellac to reduce moisture absorption and was lined in all but the viewing area with 0.25m (1 inch) of rigid foam insulation. A section of top and sides, 1.04m long, was removable for access to the chamber. In this removable section was a piece of clear plastic 0.39m long and extending across the top and down both sides to serve as a viewing area. The test chamber was connected to the air conditioning unit by ducts 0.127m (5 inches) in diameter and about 1 meter long. The duct from the air conditioning unit to the test chamber was insulated with foil-faced fiberglass insulation approximately 302mm thhfl< while the return duct from the chamber totflueair conditioning unit was uninsulated. The center of the load cells was located 1.05m from the front of the chamber and 0.29m behind the center of the load cells was a honey-comb shaped metal grid. This grid acted as a "flow straightener" which made the airflow more uniform around the sample holder. The airflow was found to be parallel to the sides of the chamber in the vicinity of 96 umnEmnu xpsuw m.v musmfim ”moan—$25.: #1033 97 the grain sample and its velocity was measured with a hot- wire anemometer at 21 points in a plane normal to the sides and bottom of the chamber at the rear of the sample holders. The velocity was observed to vary from a minimum of 0.10 m/s to a maximum of 0.70m/s with an average of 0.28 m/s. Henderson and Pabis (1962) found that, in this range, variations in airflow have an "insignificant" effect on the rate of drying. The weight transducers were fastened to a concrete block 0.18m by 0.30m by 0.35m in order to reduce mechanical vibrations and were located 0.25m apart. This block was insufficient to damp out the vibrations and additional filtering was necessary. The vibrations adversely affected the weight measurements. The filtering was included in the strain gauge amplifier described in Section 4JL4. The sample holders consisted of two trays 0.20m by 0.30m made of aluminum screen with a lip of about 10 mm around the edge to hold the sample on the tray. There were two trays attached by an aluminum rod to each weight transducer. The bottom tray was about 0.10m from the bottom of the chamber and the trays were about 0.10m apart resulting in the top tray being located about 0.10m from the top of the chamber. 4.6 Software A considerable amount of software was required for this project. Copies of the programs used in data 98 acquisition and control may be found in the Appendix. The software may be divided into three categories, two of which will be discussed here: the data acquisition software and the digital control software. The software used in data analysis will be discussed in chapter 5. The software for data acquisition and digital control was written in three languages: Z-80 assembly language, BASIC, and Pascal. The microprocessor used was a Z-80 so a few of the most primitive routines, those which were used often and were critical to the timing, were written in Z-80 assembly language. The instructions for data display and storage and the overall control of the equipment were written in integer BASIC. The data acquisition and air conditioning control software was written in Pascal and compiled into Z-80 code. The code written in Pascal was called by an interrupt while programs written in BASIC were running. This made it appear that data storage and control were occurring simultaneously but actually the data acquisition and control functions had priority over data storage and display. 4.651 Data Acquisition The data acquisition software was written in Pascal and called several Z-80 code subroutines. The Pascal program was called every 40 mS. The code contained a variable which counted the number of times it had been 99 called. After this counter reached the value of 5000, which occurs after 200 seconds, it was reset to 0. This time base was originally provided for the control of the refrigeration unit but was not needed with the simple on/off control algorithm which proved adequate. A second timing loop was based on 250 40 mS calls or 10 seconds. At the end of each second for the first 9 seconds, each oftflue8 analog input lines was sampled and converted to digital form. After the 9 values for each of the 8 input lines were obtained they were averaged and the average stored in memory by the end of the tenth second. Also before the end of the tenth second the average values representing the water temperature, heater temperature and air temperature at the entrance to the test chamber were converted to engineering terms for use by the control algorithms. The average of the digital representation of the eight transducer values was placed in memory at the end of the ten second period and a flag was set ininemory. Software in BASIC responded to this flag and reset it before moving the averages to other memory and subsequently calculating averages overaione minute period. The one minute averages were stored on cassette tape and printed at the terminal. The shortest period covered in the recorded data was onerninute, with each data point representing an average of 54 observations. The averages of the weight on each transducer, the water temperature, 100 and the five other temperatures were recorded. The reading of the clock in days, hours, minutes and seconds was recorded with the other data. From the sample weights and the sample dry matter content the moisture content of the product on each sample holder could be calculated. From the Aminco water temperatures and the air temperatures near the product the drying air relative humidity could be calculated. 4.6.2 Digital Control The water temperature was controlled by a simple on/off control algorithnu A dead band ofGLl6 C was used because that was the minimum dead band which did not induce oscillations. At the end of each ten second period the actual water temperature was compared with the desired water temperature. If the actual temperature was more than 0.08 C warmer than the set temperature, the refrigeration unit was activated. If the actual temperature was more than 0.08 C cooler than the set temperature, the water heater was activated. If the actual temperature was within the 0.16 C deadband, then both the refrigeration unit and the water heater were turned off. The heating and cooling of the water was observed to overshoot the goal, especially when large changes in the setting were requested. Therefore, in addition to the previous control algorithm, the set temperature was 101 compared to the actual water temperature at the end of each second. If the two temperatures were within 0.05 C of each other, the heater or cooler was turned off for the remainder of the 10 second period. This adjustment allowed a narrower dead band without undesirable oscillations between the heater and cooler. The control algorithm for the air temperature was first designed as a simple velocity PID controller. Initially the temperature sensors in the chamber were used as the basis of the controller. It became immediately obvious that the heater would be activated and because of the considerable lag in getting heated air to the sensors would become very warm. The thermal masses stored considerable~energyx The algorithm continuously reduced the duty cycle to the heater, when the chamber was too warm, but because of the thermal storage would reduce the setting much too far. When the air in the chamber was too cool the reverse action was taken producing undesirable cycling of the chamber temperature. A simple solution to this problem was to base the control (n1 a sensor located near the heater. It was assumed that under steady state conditions the heat losses would be relatively constant between the heater and the product so that the chamber conditions, while not directly controlled, would be constant. A PID velocity algorithm, equation 3.8, was implemented: 102 DS = I*(E(l)-E(2))+J*E(l) +H*(E(l)-2*E(2)+E(3)). 3.8 One disadvantage of this algorithm is that the derivative term contains what is essentially the second derivative of the input signal. Taking derivatives of variables tends to greatly amplify any noise in the signal so that the derivative action can be badly masked in the noise. A smoothing formulation of the second derivative was used: (3*E(1)-4*E(2)-E(3)+2*E(4)). 4.2 This formula was substituted for (E(1)-2*E(2)+E(3)) 4.3 resulting in the formula: DS = I*(E(l)-E(2))+J*E(l) +H*(3*E(1)-4*E(2)—B(3)+2*E(4)) 4.4 where: DS = the change in the duty cycle of the heaters. In order to use this algorithm, values for the constant coefficients I, J, H must be chosen. There are several methods of choosing the values for these coefficients when control of an unknown plant is desired. All methods involve choosing a particular set of values for the coefficients and then testing the system to see how it will react to changes in the controlled variable. Smith (1979) suggests the following steps: 103 1) Remove all reset and derivative action, i.e. J=H=0, and tune the proportional mode, I, to give the desired response characteristics, ignoring any offset. 2) Increase the proportional gain, and attempt to restore the response characteristics by adjusting the derivative term, IL Repeat until the proportional gain is as large as possible. 3) Adjust the reset time, J, to remove the offset. He adds that the adjustment of H is by far the most difficult and the derivative action is probably not justified in most cases. An algorithm in the form of equation 4.4 was adjusted using the method described by Smith (1979). Controlling the heater temperature did not allow the operator to choose the temperature in the chamber directly but a guess as to the temperature drop from the heater to the chamber would allow the chamber temperature to be set close enough for most applications. Locating the sensor near the source of the energy, the electric heater, made the control reasonably simple. The derivative action of the PID controller was observed to be erratic, due to noise in the temperature signal. Attempts were made to filter the signal both with analog filters of various orders and with digital filters. A simple RC analog filter, with a time constant ofILZ ms, was finally chosen with the smoothing formula described above acting as a digital filter. At the end of each 10 second period the updated error, in air temperature, was computed and an adjustment to the duty cycle for the heater was calculated. During each 104 interrupt, occurring every 40 ms, the current duty cycle was checked to see if the heater should be turned off or on. The appropriate code was then sent to the interface circuit along with data to control the water temperature. The duty cycle was based on 100 of the interrupts so that if the duty cycle were»46%, for example, the heater would be on for 1.84 seconds and off for 2.16 seconds. This period of 4 seconds is short enough so that the heater does not change measurably in temperature due to thermal masses. This algorithm had two main weaknesses: it was difficult to repeat an experiment because it was difficult to obtain a given chamber temperature, and the load in the room was not constant so the chamber temperature tended to drift. Therefore, an addition to the control algorithm was added between test 9 and test 10. Instead of the operator trying to guess what the heater temperature should be to achieve a desired temperature in the chamber, the additional software adjusted the heater temperature setting. This process is sometimes referred to as cascade control. For this second level of control an integral controller was used. The goal of the original PID controller was adjusted by'0.02 times the difference between (fine actual temperature at the entrance of the chamber and the desired temperature. This additional controller makes the system much slower to respond to changes in the setting but the variation of 105 temperature within the chamber was significantly reduced and the actual temperature within the chamber was usually within 0.5 C of the set temperature. 5 . EXPERIMENTAL INVESTIGATION 5.1 Product The parboiled rice which was tested had been processed and collected from a commercial parboiling plant on October 12, 1982 and stored in sealed plastic bags and glass containers at a temperature of 5 C until needed for testing. The rice variety was LaBonnet, a long grain variety, often used in commercial parboiling plants. This particular processing plant had a series of commercial dryers which were used after gelatinization. The first two in this series were rotary dryers operating in the temperature range of 260-320 C. The product was obtained from three places in the material stream in the plant, just before the rotary dryer (initial moisture content from 0.47 to 0.57 dry basis) just after the first dryer (initial moisture fron|0.20 to 0.30 dry basis) and after the second dryer (moisture content from 0.17 to 0.18 dry basis). Most of the rice tested was taken from the exit from the first dryer. Each sample taken from the material stream was about 3909b Several samples of similar moisture content were 106 107 combined and stored in sealed glass containers for at least one week before testing. 5.2 Laboratory Methods The laboratory methods will be described in the order in which they were performed during an individual test. The method was as uniform as possible over the testing of the rice, October 20, 1982 to February 8, 1983. The tests were numbered consecutively from 1 to 21. Several of the tests were not analyzed completely, for various reasons. Tests 1 and 2 were not analyzed at all but were considered to be "warm-up" tests. 5.2.1 Equipment Initialization and Initial Data If the equipment had been turned off previous to the individual test, it was turned on and the water temperature goal and dry bulb temperature goal were set. The equipment was given at least one hour to warm-up and stabilize at the new setting. Before loading the rice on the weight transducers, the data collection computer program was begun. Several initial measurements were taken with known masses placed on the transducers to check the weight conversion factors. All weights, other than those obtained from the weight transducers, were measured with a Mettler, type B5, balance and recorded to the nearest 0.001(L. The rice used in the test was removed from storage and weighed in a container 108 with a lid. The initial product weight was over 100 g. for each weight transducer. Tfimeproduct was then spread over the trays, first on the bottom tray then on the top tray for an individual weight transducer. Any surplus parboiled rice was weighed with the container for a determination of the weight of grain on the transducer. The surplus rice was then returned to storage. Weight transducer 0, also referred to as tray A, was loaded first and transducer 1, tray B, was loaded second. The chamber was left open for one to two minutes while each transducer was loaded and closed for approximately four minutes between the loading of the two transducers. After the product was loaded onto the weight transducers, additional samples were taken from storage to determine the initial moisture content of the grain. The samples for the moisture content determination weighed at least 15 grams and were placed in numbered cans with numbered lids which had been previously oven dried and weighed. The cans and samples were weighed and then placed in an oven at 103 C, with the lids in the oven also but not on the can. The samples were dried for approximately 80 hours. Several prior tests of the moisture determination method showed that the required sample drying time was at least 72 hours. After the samples were dried in the oven the lids were placed on the cans, the samples were cooled in a dessicator, and then weighed again. The rice was 109 removed from the containers and the container and lids were reweighed. The formula used to calculate the moisture content was: MCd = (WSCi-WCi-WSCf+WCf)/(WSCf-WCf) 5.1 where: MCd the moisture content, dry basis WCi weight of the can, initial WCf = weight of the can, final WSCi = weight of the can and sample, initial WSCf = weight of the can and sample, final. 5.2.2 Treatment During Test The microcomputer controlled the drying conditions during each test. The data representing the weight on each transducer, ‘Uua temperature at three places near the samples, the temperature at the entrance of the chamber, the temperature at the air heater, and the temperature of the water were recorded each minute along with the time of recording of the data. Each test was allowed to run for between 23 and 54 hours with most being at least 40 hours. Table 5.1 shows the various tests which were run. The Aminco-Aire water temperature setting as well as the calculated relative humidity and observed chamber temperature and the product moisture content before the test was started are included in this table. 110 Table 5.1 Test Conditions for Parboiled Rice samples taken from the stored grain this analysis Water Relative Chamber Sample Temperature Humidity Temperature Initial Moisture Test Number Celsius Celsius Content* test 3 A 15.0 0.38 34.0 0.276 B 0.229 test 4 A 15.0 0.32 37.8 0.228 B 0.276 test 5 A 7.2 0.24 34.8 0.284 B 0.242 test 6 A 5.2 0.25 31.5 0.505 B 0.286 test 7 A 15.1 0.41 32.5 0.566 B 0.292 test 8 A 20.0 0.51 33.4 0.292 B 0.568 test 9 A 25.0 0.47 40.2 0.576 B 0.294 test 10 A 15.0 0.27 40.6 0.225 B 0.278 test 12 A 12.0 0.50 25.2 0.295 B 0.278 test 13 A 5.0 0.51 17.3 0.302 B 0.296 test 14 A 8.0 0.40 25.0 0.299 B 0.258 test 15 A 15.0 0.40 32.6 0.325** B 0.297 test 16 A 27.0 0.53 39.8 0.324** B 0.299 test 17 A 15.0 0.27 40.6 0.186 B 0.296 test 18 A 14.0 0.27 39.8 0.259 B 0.181 test 19 A 15.0 0.31 38.0 0.252 B 0.183 test 20 A 12.0 0.47 26.3 0.254 B 0.260 test 21 A 12.0 0.47 26.1 0.255 B 0.262 * Moisture content determined by oven method of separate ** Rewetted long grain rough rice, tests not included in 111 5.2.3 Sample Post-treatment and Final Data At the end of a test, the rice~on the two trays of an individual transducer was combined and weighed. A sample was then drawn from this rice, separate samples representing the separate transducers, to determine the final moisture content of the rice. The same procedure was used as was described for determining the initial moisture content of the rice. Since the final moisture content was used to calculate the dry matter weight on each transducer and the dry matter weight affected the results substantially, a duplicate sample was taken for determining the final moisture content starting with test 14. The dry matter weight determined from the duplicate samples agreed to 0.1 g. in all tests and generally agreed to within 0.029. 5.3 Data Reduction In order to proceed with the data analysis and the determination of the proper thin layer model, the moisture content of the rice must be determined. Also the drying air conditions, temperature and relative humidity must be calculated for the same period of time. The most important variable for the determination of the moisture content of the sample was the weight of the dry matter in the sample. This value was calculated from the product weight in grams, averaged over the last five 112 minutes of the test, and the moisture content which had been measured at the end of the test using the following formula: de = Ws*(1.0-M/(M+1.0)) 5.2 where: de Ws M weight of the dry matter in grams weight of the sample in grams final moisture content dry basis decimal In tests 1-13 only one sample was taken from the rice on eachcnfthe weight transducers. Therefore there was only one final dry matter weight to be used. In tests 14-21 duplicate samples were taken from the ricecnieach of the weight transducers. The dry matter content of the rice on each transducer was calculated separately and time two dry matter weights were averaged for each transducer. The data were then converted from integers to engineering terms. The data reduction program converted the time from days, hours, minutes, and seconds, to the hours and seconds since the beginning of the test. It also converted the numbers representing the weights on transducers 0 and l to moisture content dry basis, and converted the numbers representing the temperatures in the system to temperature in Celsius. After the temperatures of the air and water in the Aminco—Aire were known the relative humidity was calculated by the iterative formula, rewritten from equation 4.1: 113 RHn = RHO+P1+P2*WT-DB+Ln(RHO) * (P3+P4*WT)+P5/RHO 5.3 where RHn = the most recent estimate of the relative humidity, percent RHo = the previous estimate of the relative humidity, percent WT = the water temperature in Celsius DB = the dry bulb temperature of the air in Celsius P1 = 77.4 P2 = 1.377 P3 = -l6.98 P4 = -0.0821 P5 = 73.7 The iteration started at the relative humidity which was calculated for the previous time period and ended when the absolute value of (RHo-RHn) was less than 0.0004. After carefully examining the data and physical arrangement of the temperature sensors, it was decided that for tests 3 through 14 the temperatures measured by sensors 5 and 6 best represented the actual temperature of the drying air near the product. Therefore, the average of the values from these two sensors was used in the calculations of relative humidity for the tests 3 through 14 and was also recorded as thelchamber temperature for these tests. Between tests 14 and 15 the distribution of the airflow was improved and temperature sensors 5, 6, and 7 were in the airstream which surrounded the rice. For tests 15 through 21 the average of the temperature readings from sensors 5, 6 and 7 was used as the true air temperature. The calculations were performed with floating point arithmetic, but the results were stored as integers. The 114 moisture contents and the relative humidities were rounded to the nearest 0.001. Temperatures were rounded to the nearest 0.1 degree Celsius. The moisture contents and relative humidities were multiplied by 1000 and the temperatures by 10 to obtain suitable integers for storage. These integer values were converted to ASCII for transmission to the mainframe computer for the data analysis. The data sets were transmitted to the mainframe computer over telephone lines and stored as separate files. 5.4 Variables and Errors In this investigation four variables were important. The first was the temperature of the drying air. Averages for temperature were used in the analysis. The second and third were the other two independent variables, time and relative humidity. The fourth was the dependent variable, the moisture content of the parboiled rice. The temperatures used in the analysis had been rounded to the nearest 0.1 C. The temperature conversion data, calibrated against a laboratory thermometer marked in 0.1 C, was analyzed to obtain the conversion factors in Table 4.1. The estimate of the standard deviation on the predicted temperature is 0.09, so a reasonable estimate of the error in the temperature data is 0.2 C. The time was truncated to a second and was recorded in hours, so the error in time due to truncation could be as large as 0.028%. The clock was controlled by a crystal and 115 was measured to be about 0.0123% too slow. Over a 48 hour test the time would be about 21 seconds too small. When errors in temperature of 0.2 C are entered into the iterative formula for relative humidity (equation 5.3) the resulting uncertainty in the relative humidity is 0.006, in the temperature ranges used in this study. The relative humidities were calculated to a precision of 0.005, but the accuracy was no better than 0.03 because a more accurate standard was not available with which to calibrate the equipment. The formula was tested by measuring the actual moisture content of the air in the chamber and comparing it to the calculated moisture content. Three methods were used to measure the moisture content of the air: a chilled mirror instrument, wet-bulb dry-bulb temperature sensor combination, and a commercial moisture sensor accurate to 0.03. The moisture content of the air from the formula agreed to within the accuracies of the instruments against which it was checked. The largest error in the moisture content measurement calculations came from the weight transducers. The moisture content data for each test were all based on the dry matter content of the samples taken at the end of that test. The errors in theinoisture content oven tests were in the range of 0.0004 moisture content dry basis. However, the errors in the sample moisture content calculations, before test 18, were in the range of 0.002 116 because of the weight transducer error. After the hardware changes between tests 17 and 18 the error due to the transducers dropped to the range offiL0008. Thelnoisture contents of the product were rounded to the nearest 0.001, moisture content dry basis, before the data analysis was begun. Therefore, in the data analysis the error in the moisture content data due to the rounding may be as much as 0.0005 moisture content, dry basis. The error due to transducer error is in the same range for test 18 to 21. One possible source of error was in the determination of exactly when time=0 occurred and what the moisture content of the samples was at that point. Since the equations and models are nonlinear in time the concern over this error is not trivial. In this study time=0 wes considered to occur halfway through the loading of each tray, so this time is not the same for the two trays. The difference was about five minutes. The initial moisture contents were estimated graphically from the data taken at times of less than fifteen minutes. 6. DATA ANALYSIS AND RESULTS The remaining objectives involved determining how well the variables in the drying process had been controlled and the use of available equations to determine appropriate models for the drying relationships. The data analysis consisted of choosing appropriate subsets ofldata and the examination of data with statistical software packages. Four BMDP packages (Dixon 1981) were used in the statistical analyses. BMDPAR and BMDP3R were used to fit various nonlinear models to the data, BMDP6D was used to create plots of the data and residuals, and BMDP2D was used to analyze how much variation there was in the temperature and relative humidity data. The data sets were contained on several files which were analyzed. The original data sets contained data on the moisture content of the samples from every minute for the individual tests, in addition to the temperature and relative humidity data for each minute. Separate files had combined sets of data, with a range of relative humidities and temperatures, but with samples of data at intervals greater than one minute. The first step in the analysis of the moisture loss data was to plot the moisture content vs. time for each 117 118 curve. Visual examination of these plots indicated poor data sets and poor data points. Since all of the plots were at the same scale the different data sets were also compared visually. The moisture content data from the eleventh data set were observed to vary considerably more than the other data sets. The data from this test had sudden changes in the moisture content of£L03 while the other tests had sudden changes of no more than 0.01. The water level in the Aminco Aire unit had become low during this test, the only instance where this problem was encountered. Therefore, data from test eleven was not analyzed further. Upon examination, data from tests three and four were observed to have irregularities in them. These occurred as a result of the sample holders hitting the side of the chamber. This problem was eliminated after the fourth test by pinning the sample trays to the supporting rod. The remaining sixteen data sets were kept for further analysis. These sixteen tests on each of the two trays covered a total of 1308 hours of data with one data point every minute. This amounts to over 78,500 data points of moisture content vs. time. Since each data point represented an average of 54 observations the data sets resulted from over 4 million observations of moisture over time. In contrast, Misra and Brooker (1980) combined data from seven researchers for a total of 15,353 data points. 119 As will be shown later, sheer quantitiescflfdata are not an adequate goal and collecting data every minute about a process which does not change measurably for half an hour is not always needed. However, in order to demonstrate the necessary frequency and duration of data collection some researcher must collect too much data for too long and demonstrate what the limits are. In this study the plan was to collect more data than necessary for longer than necessary to give future researchers some guidance in this matter. 6.1 Analysis 2f Operation gf Equipment The thin layer study equipment maintained conditions as shown in Table 6.1. The distribution of the temperatures and calculated relative humidities did not follow a Gaussian distribution because these two variables were controlled by the digital equipment. The control algorithm requires measurable errors to occur. But after they occur substantially larger errors are unlikely to occur because of the feedback control. Therefore the usual statistic of central tendency, standard deviation, is misleading. What is presented instead is the mean value and the range which includes at least 90% of the data points. The temperature and relative humidity each minute, after the first half hour of the test, was included in this analysis 120 for the remaining period of the run. For most of the tests there were at least 2400 data points in this analysis. During the first half hour the opening of the chamber affected the drying conditions and the results of an indicate the analysis including all data would not effectiveness of the control for the majority of the time. Table 6.1 Variation in Chamber Conditions Relative Humidity Average Chamber Temperature - Degrees Celsius Percentile 5 50 95 5 50 95 test 5 0.23 0.24 0.24 34.4 34.8 35.3 test 6 0.24 0.25 0.26 30.9 31.5 32.2 test 7 0.39 0.41 0.42 32.0 32.5 33.0 test 8 0.50 0.51 0.52 32.9 33.4 33.6 test 9 0.46 0.47 0.49 39.5 40.2 40.7 test 10 0.27 0.27 0.28 40.3 40.6 40.8 test 12 0.49 0.50 0.50 24.9 25.2 25.4 test 13 0.50 0.51 0.52 17.1 17.3 17.6 test 14 0.40 0.40 0.41 24.8 25.0 25.3 test 15 0.40 0.40 0.41 32.3 32.6 32.9 test 16 0.51 0.53 0.54 39.5 39.8 40.4 test 17 0.27 0.27 0.28 40.2 40.6 41.0 test 18 0.26 0.27 0.27 39.5 39.8 40.2 test 19 0.31 0.31 0.31 37.8 38.0 38.2 test 20 0.46 0.47 0.47 26.1 26.3 26.5 test 21 0.47 0.47 0.48 25.8 26.1 26.4 The improvement due to the cascade control introduced between tests 9 and 10 can be seen. Before the inclusion of the cascade control algorithm, the range of chamber temperatures was about 1.0 C and after the change was about 0.5 C. When the chamber temperature distribution for each test is examined, the tests conducted before the addition 121 of the cascade control algorithm show a marked bimodal pattern in time. That is, there were apparently two temperatures about which the temperatures varied. One possible explanation for this could be that diurnal temperature changes in the laboratory affected the chamber temperature. After the control algorithm was improved the temperature-frequency distribution was much more square with a relatively even distribution of several tenths of a degree about the mean and a very rapid decrease in the number of observations at higher and lower temperatures. The water temperature was less variable than the other temperatures, the range including over 90% of the water temperature values generally was 0.1 C. Therefore the variations in the chamber temperature affected the relative humidity in the chamber. The average range of relative humidities before the algorithm change was 0.021 and after the change was 0.010. Because the water temperature was virtually unchanged, the saturation vapor pressure did not change during the tests. The real question about the adequacy of control was whether the variations in the independent variables significantly affected the drying behavior of the product. The best measure of the effect of uncontrolled variation in the independent variables is to study the data from repeat tests. This approach is discussed in Section 6dil. 122 The control of the independent variables, air temperature and relative humidity, was more than adequate for all tests. The control after the improvement in the algorithm between tests 9 and 10 was closer than the control previous to the improvement and removed the bimodal distribution in the air temperature noted in some of the tests. 6.2 Exponential Models In this section the relative humidity and temperature are considered to be constant. Comparisons between tests where relative humidity and temperature have been changed will be covered in later sections. The relative humidity during the test was constant after the first half hour but because of the opening of the chamber varied by as much as 0.15 during the first few minutes. After the chamber had been closed for five minutes the chamber was generally within 0.03 of the mean value. The temperature was constant after the first half hour but was up to 15 C from the mean at the first minute after the chamber was closed. After five minutes had passed the temperature was within 1 C of the mean value for the rest of the test period. The grain temperature was not measured but can be expected to be near the air temperature in the temperature range of this study after the first 10 minutes (Husain, Chen and Clayton, 1973; Fortes, Okos and Barrett, 1981). 123 One of the findings of the literature review was that one»or more decaying exponentials have been used tornodel the drying process. If the resistance to diffusion lies on the surface of the particle, one term should be sufficient. If the resistance is spread throughout the particle then several terms should be required. The set of data with the least error should be used in evaluating the appropriateness of any model. Tests 18, 19 and 20 were used to determine which of the models were most appropriate for thin layer drying modeling of parboiled rice because they had the lowest expected error. Because test 21 was an exact repeat of test 20 it was used only for comparison with test 20. The one term model, also called the logarithmic model was: M(t) = Pl*exp(P2*t)+P3 6.1 The two term exponential model was: M(t) = Pl*exp(P2*t)+P3*exp(P4*t)+P5 6.2 The three term exponential model was: M(t) = P1*exp(P2*t)+P3*exp(P4*t)+P5*exp(P6*t)+P7 6.3 The four term exponential model was: M(t) = P1*exp(P2*t)+P3*exp(P4*t)+P5*exp(P6*t) +P7*exp(P8*t)+P9 6.4 124 The statistical package BMDP3R has a series of exponentials built into it as one of the possible nonlinear models for use in regression. It was, therefore, relatively simple to fit varying numbers of exponential terms to the data. 6.2.1 One Term Exponential Model The one term exponential model was fit with the resulting parameter estimates listed in Table 6.2. The residual mean square, the statistic which will be used for a measureiof "goodness of fitfl'for each data set is also listed. Table 6.2 One Term Exponential Fit to Data Sets Test 18 A 18 B 19 A 19 B 20 A 20 B Pl 0.118 0.067 0.120 0.067 0.107 0.113 P2 -0.215 -0.l6l -0.224 -0.150 -0.161 -0.180 P3 0.078 0.077 0.083 0.081 0.111 0.111 Residual 14.3 5.0 19.1 6.3 10.6 12.7 Mean E-6 E-6 E-6 E-6 E-6 E—6 Square The data sets for these six tests comprise over 2500 data points each, and were believed to be accurate to nearly 0.0005, the roundoff error. Therefore, the expected best case residual mean square should be about 0.25 E—6. The residuals were examined and a very clear pattern was seen in all cases indicating that something was measured in the 125 data which was not accounted for in the model. In other words the model is inadequate. The figures in the tables in this chapter do not necessarily reflect the significant digits in the data. The parameters which represent the moisture content of the grain directly are rounded to the nearest 0.001. The parameters which are in the exponents are listed to three places. The asymptotic standard deviations are known for all parameters in this study but only listed for the final model. Generally the asymptotic standard deviations showed that the parameters directly associated with moisture contents are estimated to 0.001 or better but the parameters in the exponents are only estimated to one or two places. The residual mean square values have six significant digits, but because the intended use of the values is to estimate the variance they are listed to only two or three significant digits. Figure 6.1 shows the residuals in test 19 B plotted against time and Figure 6.2 shows the same residuals plotted against the predicted umdsture content.* In this case the two plots show basically the same thing but in some cases one plot may show a pattern which is not obvious Because so many nonlinear regressions were completed and so much data was available only a few of the residual plots will be included in the dissertation. 126 in the other plot. Figure 6.3 shows the residuals from test 20 B plotted against time. In the plots the numbers refer to the number of data points which should be plotted at the same place on the paper. If more than nine are to be plotted in the same place, the letters of the alphabet are used. If more than 35 points lie on the same place on the plot, the symbol * is used. If any points lie off the plot the symbol * is used at the border of the plot. 6.2.2 Two Term Exponential Model Because the one term exponential model was inadequate and the theory shows that a series of exponentials may be the best model for thin layer drying the two term exponential model was fit to the data. Table 6.3 shows the results from the nonlinear regression of the six data sets to the two term exponential (equation 6.2). Table 6.3 Two Term Exponential Fit to Data Sets Test 18 A 18 B 19 A 19 B 20 A 20 B Pl 0.096 0.043 0.102 0.047 0.074 0.074 P2 -0.647 -0.691 -0.584 -0.510 -0.448 -0.448 P3 0.059 0.045 0.052 0.040 0.058 0.058 P4 -0.102 —0.098 -0.081 -0.070 -0.078 -0.078 P5 0.074 0.075 0.077 0.077 0.106 0.106 Residual 0.39 0.38 0.80 0.44 0.57 0.57 Mean E-6 E-6 E-6 E-6 E-6 E-6 Square 127 *OOOO+O..O+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOO .0125 + .2 . .2 Explanation of symbols, p. 126 .l . .0100 +2 + .2 . .2 . .1 . .1 . .0075 +11 + .12 . . 1 . . 3 . . 4 . .0050 + 2 + . 2 . . 2 . . 3 52 l . . 7 23BB9ASZ9 . .0025 + 4 359JUUNHW9EH 2 + R . 4 lLJWKGCMCBTU *U . E . 4 lSKNQD93433F7E*l *J2 . S . 2 5IKD611 51 CU 6G2 . I . 6 FKB 12 7** . D 0.000 + 6 9NC8 6***7l + U . 33 ZGGS A BRA13 . A . 48 17K 1IN*R . L . 9 17EF 3W**PK6362 . . 6 EL 81 . -.0025 + 6 8LE 14KV*****Z + . A SQBZ . 76BJ4 23C35D . 6QH42 . . 5MJ1 . -.0050 + 68 + -.0075 + + .+....+....+....+....+....+....+....+....+....+... 5. 15 25 35 45 0. 10 20 30 40 TIME, hour Figure 6.1 Residuals 19 B vs. Time, 1 Term Exp. Model .0125 .0100 .0075 .0050 .0025 0.000 F‘S’CUHWFJDU -.0050 Figure 6.2 128 OO+OOOO+OOOO+OOOO+OI.O+**..+..OO+OOOO+OOOO+OOO. + + . 2 . . 2 . . 1 . + 2 + . 2 . . 2 . . 1 . . 1 . + 2 + . 3 . 1 . 3 . . 13 . + 11 + . 2 . . 2 . . 35 3 . . **2 7 . + **8 4 + . *** 4 . . *T*E 13 . . *2PZ2 2 . . *25Z7 6 . + ** 3KSI 6 + *A SS6 15 . *F CE2 57 . . * 9I91 72 . . Q 161 6 . + * BKBl 6 + . 28LD 55 . . * 2EF3481 . . 238DGD . . 17HE8 . + 284 + . Explanation of symbols, p. 126 . + + ...+....+....+....+....+....+....+....+....+.... .06 .10 .14 .18 .22 .08 .12 .16 .20 Predicted Moisture Content, Dec. Dry Basis Residuals 19 B vs. Predicted, 1 Term Exp. Model 129 0*....+....+...O+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OO .0125 +1 .1 . .2 Explanation of symbols, p. 126 . .1 . .0100 +1 . l . .11 . .0075 + l + . 2 . . 3 . . 3 . . 2 . .0050 + 2 3 772 + . 3 CF42LM82 . . 3JRD6NUL93 . . 1 3PGBTSS OYHG3 . . 3 1PO97EJ 15C*O4Pl . .0025 + 3 6P81 44 22G* 1F + R . 3 5K7 l 24 2* . E . 2 CO 6 3**9 . S . 3 4J9 C SDl . I . 5 BJ B*7 . D 0.000 + 4 H5 P**K5 + U . 3 K 2 . A . 3 1A6 L**Y8 . L . 4 401 L6G . . 31 6M 1P**A . -.0025 + 33 84 O**Z**J2 + . 35 1H 19M5 . . 8 7H 2POC****. . C C7 . . 8 I 2191. -.0050 + 215D + . 7BH7 . . 5PP2 . . JC . . 41 . -.0075 + + .+....+....+....+....+....+....+....+....+....+.. 5. 15 25 35 45 0. 10 20 30 40 TIME, hour Figure 6.3 Residuals 20 B vs. Time, 1 Term Exp. Model 130 This model fits the data significantly better than the one term model with the residual mean square at least an order of magnitude lower for the two term exponential than for the one term exponential. In addition the residual mean square values are reasonably near the estimated lower limit of 0.25 E-6. The plots of the residuals were examined for evidence that the model was not adequate. Figure 6.4 shows the residuals from test 19 B plotted vs. time. The plot shows a much improved distribution but there are points within the first several hours which are not satisfactorily modeled. If the residuals are plotted against the predicted values, Figure 6.5, the pattern was even more clear. Figure 6.6 shows the residuals from test 20 B plotted against time. In all six cases the residuals showed patterns such as these and the model must be considered to be inadequate. 6.2.3 Three Term Exponential Model Because the two term exponential was not adequate a three term model (equation 6JH was regressed against the six sets of data. Table 6.4 shows the results of the regression. 131 +OOOO+OOOO+OOOO+OOOO+OOO0+0.CO+OOOO+OOCO+OOOO+OOO + +oooo+009 +00 +0.. +0. + o o o o .0100 + . Explanation of symbols, p. 126 .0075 + .1 .1 .0050 +1 .2 .1 .0025 +2 R .2 2 E .1 5E4 11 7222 l S .2 4HMJ6231 1242A3GHNFH G2271 712 162 I .2 7NNGNBDF5 533669AY18*GVG*9* MZ 1R3 7K62 F D 0.000 +4 JNDBAOQLJLIA9JPUOEPT8R4E4W *H2*K O*I X**K U .4DN9223EHHOIKRSMMFL6B79 8 3 G3CL 6WM B**C 9 A .2P91 15246CHFHB743245 6 BF KK B354 L .1J2 59 312 l 1 14 231 . 3 12 -.0025 + -.0075 + O+OOOO+O0.0+...OO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOO 5. 15 25 35 45 0. 10 20 30 40 TIME, hours Figure 6.4 Residuals 19 B vs. Time, 2 Term Exp. Model 132 OOO+OOOO+OO00+...O+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+ .0100 + + .0075 + + O 1 O . 1 . .0050 + 1 + . 2 . O 1 O .0025 + 2 + R . ll 2 . E . 1F 39B 1 . S . Z*J 1510H72 2 . I . ***DATRLDIE432 2 . D 0.000 + ******086DC987 22 + U . *****YF4222979742223 . A . *5J*OA2 143187535 L . B116E 11 24563 . . 3 111 . -.0025 + + . Explanation of symbols, p. 126 . OOO+OOOO+OOOO+OOOO+OI00+.OOO+OOOO+IOCO+OOOO+OOOO+O .06 .10 .14 .18 .22 .08 .12 .16 .20 .24 Predicted Moisture Content, Dec. d. b. Figure 6.5 Residuals 19 B vs. Predicted, 2 Term Exp. Model rvconmw .0150 .0125 .0100 .0075 .0050 .0025 0.000 -.0025 -.0050 133 +OOOO+OOOO+OOOO+OOOO+OOOO+OO00+0000+OOOO+OOOO+OOO + . Explanation of symbols, p. 126 + I .1 . .1 . + + .1 . .1 O + + 1 O .2 I +1 + .1 O .1 . +2 + .1 13 . .1 2AD711 ll 3 16 6 1 . .1 3DORIA3 3 244216 PB9X3ED1K5 8 C . .3 9PKGOJ1823 3 GUKGJYXEXSGULN* *Y 1V64 1*J2 . +3 HI528JNN7EALG43NONWYGIKFH6IJ03*I ** YZZ U*R+ .3 B211 9CJTQRNREAE2884461 526 I 1P22T*8 3C**L . .236 22AIELA7VR1 4 2 9K 2PL 3I8. .184 43237864 1 216 . . 01 4 1. +K + . 6 . + + .+....+....+....+....+....+....+....+....+....+... 5. 15 25 35 45 0. 10 20 30 40 TIME, hours Figure 6.6 Residuals 20 B vs. Time, 2 Term Exp. Model 134 Table 6.4 Three Term Exponential Fit to Data Sets Test 18 A 18 B 19 A 19 B 20 A 20 B P1 0.028 0.019 0.032 0.019 0.021 0.026 P2 -2.67 —2.60 -2.39 -l.81 -2.68 —2.75 P3 0.087 0.037 0.091 0.042 0.072 0.080 P4 -0.535 -0.459 -0.447 -0.321 -0.359 -0.345 P5 0.054 0.040 0.045 0.034 0.052 0.046 P6 -0.095 -0.088 -0.067 —0.052 -0.069 -0.063 P7 0.074 0.074 0.076 0.075 0.106 0.104 Residual 0.28 0.29 0.45 0.27 0.38 0.25 Mean E-6 E—6 E-6 E-6 E-6 E-6 Square This model fits the data somewhat better than the two term exponential with the residual mean square lower in every case. However, the degree of improvement is not great and the difference in residual mean square is not statistically significant. In several cases the residual mean square is approaching the expected minimum of 0.25 E-6. The plots of the residuals were again examimed. Figure 6.7 shows the residuals from the fit to test 18 A plotted vs. time and Figure 6.8 shows the same residuals plotted against the predicted value. In neither case is there any pattern which was significant. The small dips could easily be ascribed to transducer drift. At time greater than 35 hours the roundoff error can be seen as stripes in the residuals, in Figure 6.7. In Figure 6.8 the slightly triangular distribution is caused because there are far more observations at the lower predicted moisture F‘S’CUHUIFJ'JU .0075 .0050 .0025 0.000 -.0050 135 +OOOO+OOO0+0000+OOOO+OOOO+OOOO+OOOO+OOOO+OOOC+OO + + . Explanation of symbols, p. 126 . + + + + . 1 l 2 2C42 3 1 2 . .285 25899212 2131EGH86315 1 1 3A2 GFBl IH. .3AH5ELNITHBA83874FRNXF5M2U J32E3 3**7 P**N82 . +6JOQNLKNINPJFKJTRFDBDVW6* *7T* J*X U**S V**O+ .5JCMH977BDELQMQHMD2 26GSlR1VL2*Y 1LN5 3BM**L . .14 733211568AC5453 335 8321 55 22 2 3G . . l 2 111 3 11 32 . . l . + + + + I I 0+0.00+OOOC+OOOO+OOOO+OO0......OO+OOOO+OOOO+OOOO+OOO 5. 15 25 35 45 0. 10 20 30 40 TIME, hours Figure 6.7 Residuals 18 A vs. Time, 3 Term Exp. Model .0075 + + o o o o .0050 +000 .0025 + o o o 0.000 F'S’CIUHUJEIFU + o o -.0025 +0000 -.0050 -.2075 i +0.00+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+O 136 + Explanation of symbols, p. 126 . I; + 36H 21 l *K*44AA752 1121 31211 1 l . ***JR*QODB7 345353221 12 1 111 . ******TIHCEA8A65544233123 411 + *****C968B8B7345123242311411 l . Z*BPJ41322143 11 l l l . 64 32 1 . 1 . + I I O+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OO .075 .125 .175 .225 .275 .100 .150 .200 .250 .050 Predicted Moisture Content, dec. d. b. Figure 6.8 Residuals 18 A vs. Predicted, 3 Term Exp. Model 137 contents and therefore it is likely to have a few observations further from the mean than at the higher moisture contents. In Figure 6.9 the residuals from test 18 B are plotted against time and in Figure 6.10 they are plotted against the predicted value. Figure 6.9 looks much like Figure 6.7 with no pattern which could not easily result from transducer drift. However, in Figure 6.10 there appears to be a slight pattern for the first 40 data points. The residuals for test 19 A are much like those for test 18 B and those for 20 A appear much like those for 18 A and are not included. The residuals from test 19 B plotted vs. time are shown in Figure 6.11 and plotted against the predicted value in Figure 6.12. Figure 6.11 shows no real pattern, but, in Figure 6.12 a pattern is seen similar to that seen in Figure 6.10 only more pronounced. The residuals from 20 B are plotted vs. time in Figure 6.13 and vs. the predicted value in Figure 6.14. Again there is no clear pattern in Figure 6.13 but the pattern seen in Figure 6.10 and 6.12 is repeated in Figure 6.14. On the basis of the pattern in the residuals it was concluded that something was measured during the first half hour of some of the tests which was not accounted for by the three term exponential model. Actually the residuals are all quite low. The roundoff error was 0.00005 and the FCP'CIUHUJL'IIW 138 O+OOOO+OOOO+OO00+...O+OOOO+OOOO+OOOO+OOOO+OOOO+OOO .0075 + Explanation of symbols, p. 126 .0050 + + .0025 + + .l l . . 1 l 2 2371 1 11 . .253 231765135 11 12 4IR36 77 92 34 5C. .1IB4EHAMMH7S88E543C18HYC* *K YUl HLAl MV***M . 0.000 +6MMNORSNLNTIPUGMGHMTBZ2E7*6M*9 **9 V**K1 3*M+ .8CJQGBJBBBE9HIDQSYGD* 1 A 3B GU3 M**A YTlFEW . .5256312 47 54G6869E 2 41 1JC GB6*3 3B . .l l l l 2 12 152 . -.0050 + + -.0075 + + O+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OIOO+OOOO+OOOO+OOO 5. 15 25 35 45 0. 10 20 30 40 TIME, hours Figure 6.9 Residuals 18 B vs. Time, 3 Term Exp. Model 139 +0.00+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OCOO+OO .0075 + + . Explanation of symbols, p. 126 . .0050 + + .0025 + + R . l 1 E . 2E 2 l 1 S . Z*39C831 12131 11 . I . *****PL944646631 . D 0.000 + ******YLHC99655312 + U . ****OPCJHC873 3332 . A . *L*G42143431 l 221 . L . 8331 1 l . -.0025 + + -.0075 + + 0+0.00+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOO .075 .125 .175 .225 .275 .050 .100 .150 .200 .250 Predicted Moisture Content, dec. d. b. Figure 6.10 Residuals 18 B vs. Predicted, 3 Term Exp. Model F‘CD'CIUHUJEISU .0075 .0050 .0025 0.000 -.0050 -.0075 140 O+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOO + Explanation of symbols, p. 126 + +2 . 1 1 l 1 1 1 1 22 .267 266328 21 1 12 1 8 J19132 11 112 334 .1KH16MMAMKLDE54C88AU8II9*B*AED W6 P9 9K3 6*Z +1KILKMJNNNKMMKMMVWKFZIQWCQFSSA*SI*2E*G K** .8BARM6AHB9BDMQPFHEPC6IEB846DCX LR TW E*V SD .9369A1372 7B 6794523A51 4 1 4 128L3 KH C3 .3 12 1 1 121 1 1 14 23 + + + o 0 +00 +oooo+o 000+... + +0 0 O+OOOO+OOOO+OOOO+OO00+.OOO+OOOO+OOOO+OOOO+OOOO+OOO 5. 15 25 35 0. 10 20 30 40 TIME, hours Figure 6.11 Residuals 19 B vs. Time, 3 Term Exp. Model F'CD'CIUHUIL'U'JU .0075 .0050 .0025 0.000 -.0025 -.0050 -.0075 141 +OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OO + + . Explanation of symbols, p. 126 . I I + 2 + . 53 1 l 1 . L*43C78 643 2 . . *****PR312C9A31 1 . + ******RHGB88572 1 + . ****MO7OJE22 45341 . . *PUO661B68 1353 . . A142 1 21 12 . I I + + I I C+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OCOO+OOOO+OOOO+OOO .075 .125 .175 .225 .275 .050 .100 .150 .200 .250 Predicted Moisture Content, dec. d. b. Figure 6.12 Residuals 19 B vs. Predicted, 3 Term Exp. Model 142 +OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OCOO+CCOO+OOOO+O .0050 + + .0025 + + R . . E . 15 3 3 21 2 l3 4 1 . S .145 AA65311 483213115CL58D GS 8 16 22 . I .7GC7HMMLJG3CBGB 1KULFJZRC*KSLRC*4VY 5V5 Z*H 3. D 0.000 +6LJQQKLMMSPPLNMA4MLPXVGOSBP6REZ *N *W C*Z1 2***O+ U .7FELE529ABQHP9KKM915875624 52B L 1P18T* IC* 5. A .451632 2 255697RK1 4 21 9E 2P6 1913. L .2 3B4 1 2 1. . 2 . -.0025 + + . Explanation of symbols, p. 126 . -.0050 + + .+....+....+....+....+....+....+....+....+....+... 5. 15 25 35 45 0. 10 20 30 40 TIME, hours Figure 6.13 Residuals 20 B vs. Time, 3 Term Exp. Model rim-conmw .0050 .0025 0.000 -.0025 -.0050 143 0.00+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+ + + I I . 95 3 3 33 . . **LF 13548942 131 22 l . . ****RKANKJGC7962212532122333142 1 l . + *******XNGBDABC9875533346421213 2 + . **T****AA21384577761144421112 33 l . . * 7VWC7 2 131 2221 4 1 121 . . 4 F3 1 1 . . 2 . + + . Explanation of symbols, p. 126 . + + OOOO+OOOO+OO00+...O+OCOO+OOCO+COOO+OOOO+OOCO+OOOO+ .10 .14 .18 .22 .08 .12 .16 .20 .24 Predicted Moisture Content, dec. d. b. Figure 6.14 Residuals 20 B vs. Predicted, 3 Term Exp. Model 144 transducer error was believed to be in the range of 0.001. Fewer than 2% of the data points in test 19 B, for example, lie outside the band -—0.00125 to +0.00125. Therefore, the three term exponential model fits well for all the tests and is adequate for tests 18 A and 20 A, but does not adequately explain all observations in tests 18 B, 19 A, 19 B and 20 B. 6.2.4 Four Term Exponential Model The curves which show the clearest pattern in the residual with the three term exponential, 19 B and 20 B, are also the ones with the lowest estimated error. The effect measured is the fourth exponential term. The fourth term is smaller than the previous terms, so the measurement error must be lower in order to see the effect of this term. Another observation is that the patterns appear mostly with test B and not with test A. The explanation for this is that the fourth term of the exponential dies away rapidly in time. Test B was always loaded second and none of the data were used until the test chamber had been closed. Therefore, the major effect of the fourth term had died away before the data collection was begun for test A. The delays from the time the trays were loaded until the first data point are shown in Table 6.5 rounded to the nearest half minute. 145 Table 6.5 Tray Loading Delays Test Delay (hours) Test Delay (hours) 5 A 0.117 14 A 0.100 5 B 0.033 14 B 0.033 6 A 0.150 15 A 0.133 6 B 0.033 15 B 0.050 7 A 0.117 16 A 0.100 7 B 0.033 16 B 0.033 8 A 0.117 17 A 0.083 8 B 0.033 17 B 0.017 9 A 0.117 18 A 0.133 9 B 0.033 18 B 0.017 10 A 0.133 19 A 0.117 10 B 0.017 19 B 0.067 12 A 0.183 20 A 0.133 12 B 0.050 20 B 0.050 13 A 0.100 21 A 0.133 13 B 0.017 21 B 0.067 The four tests in which a pattern in the residuals had been noted were fit to a four term exponential (equation 6.4). The other two tests were not fit to a four term exponential because there was no valid reason to do so and the residual mean square had not been reduced significantly by the three term exponential compared to the two term exponential. The results of the fit to a four term exponential model are presented in Table 6.6. 146 Table 6.6 Four Term Exponential Fit to Data Sets Test 18 B 19 A 19 B 20 B P1 0.019 0.023 0.013 0.021 P2 -8.47 -4.66 -8.37 -5.11 P3 0.026 0.034 0.020 0.027 P4 -0.934 -0.982 -0.986 -0.788 P5 0.026 0.075 0.038 0.068 P6 -0.283 -0.386 -0.272 -0.277 P7 0.035 0.044 0.032 0.041 P8 -0.079 -0.063 —0.047 -0.055 P9 0.074 0.075 0.074 0.103 Residual 0.27 0.44 0.25 0.23 Mean E-6 E-6 E-6 E-6 Square The residual mean square was reduced in every case compared to the three term exponential, but not significantly. There was no expectation that the residual mean square would be reduced significantly since the residual due to the points which were not fit well with the three term exponential was very small compared to the total number of data points. However, the pattern in the residuals had been removed by the additional two parameters. Figure 6.15 and Figure 6.16 show the residuals vs. the predicted moisture content for tests 19 B and 20 B respectively. There is no discernible pattern in these plots. A series of decaying exponentials modeled the six data sets adequatelyu In some of the tests it was possible to support a four term exponential model. In the other cases only a three term exponential model was supportable. F>CUHmm5U 147 COO+OOOO+OOOO+OOOO+OOOO+OOOO+OOO0+0000+OOOO+OOOO+ .0075 + + .0050 + + .0025 + + . 73 1 1 l . I*E467757 32 1 1 . . *****YNMJ662785522222 l . 0.000 + ******UMFHIB89975653121 + . ****TNJB8GAB5 4253322 1 . . *ELFH56533151 1 1 . . 91 41 . -.0025 + + . Explanation of symbols, p. 126 . -.0075 + + 0.0+OOOO+OOOO+OOOO+OO00+...OO+OOOO+OOOO+OOOO+OOOO+O .06 .10 .14 .18 .22 .08 .12 .16 .20 Predicted Moisture Content, dec. d. b. Figure 6.15 Residuals 19 B vs. Predicted, 4 Term Exp. Model .0075 + + .0050 + + .0025 + + R . . E . 4 3 2 22 2 . S . **BOB757 6131 111342 21 21 1 . I . *****ZJNIDD96873454632222123123 1 * D 0.000 + *******SOHJE9CD997522725443222 43 l + U . *****QQBBA589447543 14334112 22 . A . *48QD14 5 152 3 l l l . L . 5 9 . . l . -.0025 + + -.0050 + + . Explanation of symbols, p. 126 . -.0075 + + .....+....+....+....+....+....+....+....+....+.... .10 .14 .18 .22 .08 .12 .16 .20 .24 Predicted Moisture Content, dec. d. b. Figure 6.16 Residuals 20 B vs. Predicted, 4 Term Exp. Model 149 Because these six data sets had the lowest expected error, there is no reason to assume that a four term exponential could be supported by the other data sets with greater errors. Therefore, while the four term exponential model has been shown to be useful in some cases, it will not be used when data for different data sets are combined. 6.3 Reduced Data Sets One of the objectives of this study was to obtain guidelines of how often data on thin layer drying should be collected and at what times. The 32 data sets with one data point every minute were too large to fit into the mainframe computer for analysis. Nonlinear regression would have been prohibitively expensive if all of the data had fit in. Parameters may be determined accurately with few data points if a proper model is used and the data are measured with small errors. Of course, variables cannot be measured with zero error and the models do not fit the data exactly. The determination of a reasonable number of observations which are required is always subjective until the errors in each of the variables are known. Three to five observations per parameter are often used providing the observations cover the range of interest. The individual curves and the three term exponential model of seven parameters would require a minumum of 20 to 40 points well spaced in time. 150 6.3.1 Maximum Time If the moisture content model fits the data well it should not be necessary to measure the moisture content of the product at equilibrium but the value for EMC should be obtainable from the model as the moisture content approaches equilibrium. In order to get an indication of the maximum time which is necessary for data collection three of the data sets (18 A, 18 B, and 19 A) were regressed to a three term exponential (equation 6.3) with the last included data point at 24, 30, and 36 hours. The resulting parameter estimates were used to calculate the residual mean square for the entire data set. The value for the residual mean square from the parameter estimates resulting from the regression of all of the data was taken as the "best" answer. In comparing the residual mean square some guideline had to be chosen as to what was a good fit. When the error mean square was more than doubled the fit was considered to be significantly worse. Tables 6.7, 6.8, and 6.9 show the results of this study. 151 Table 6.7 Time Maximum for Test 18 A Time max. (hours) P1 P2 P3 P4 P5 P6 P7 Residual Mean 24 0.07 -3.48 0.086 -0.589 0.058 -0.1086 0.076 0.56 E-6 30 0.031 -2.17 0.085 -0.499 0.052 -0.0884 0.074 0.31 E-6 Square (of total data set) Table 6.8 Time Maximum for Test Time max. (hours) P1 P2 P3 P4 P5 P6 P7 Residual Mean 24 0.018 -6.98 0.035 —0.733 0.048 -0.121 0.077 1.06 E-6 30 0.017 -4.96 0.037 -0.622 0.045 -0.106 0.076 0.47 E-6 Square (of total data set) 36 0.028 -2.68 0.087 —0.535 0.054 -0.0944 0.074 0.28 E-6 18 B 36 0.022 -1.97 0.036 -0.385 0.037 -0.080 0.074 0.29 E-6 Table 6.9 Time Maximum for Test 19 A Time max. (hours) P1 P2 P3 P4 P5 P6 P7 Residual Mean Square (of total data 24 0.027 -3.57 0.086 -0.553 0.053 -0.l09 0.082 2.50 E-6 30 0.034 -2.14 0.090 -0.429 0.044 -0.065 0.076 0.46 E-6 set) 36 0.040 -1.71 0.088 -0.382 0.042 -0.051 0.073 0.56 E-6 47.2 0.028 -2.67 0.087 -0.535 0.054 -0.0946 0.074 0.28 E-6 47.1 0.019 -2.60 0.037 -0.459 0.040 -0.088 0.075 0.29 E-6 43.3 0.033 —2.39 0.091 -0.447 0.046 -0.067 0.076 0.45 E-6 152 The entries in the column headed by the maximum time of over 40 hours are the same as those in Table 6.4. From these tables it is clear that 24 hours of data is not enough, that 30 hours may be enough and that 36 hours certainly is long enough. Therefore, the maximum time for the reduced data set was chosen to be 37 hours, in order to be certain that the maximum time chosen would be large enough for all data sets. 6.3.2 Data Acquisition Interval A series of tests were performed on data sets 18 A, 20 A and 20 B to determine how often data should be collected during 23 study. Traditionally most researchers have collected data on a constant time interval. It would seem logical to collect data only when a measurable change in the independent variable is expected. Because the moisture content may be approximated by exponential decay, an exponentially increasing time interval was used. Subsets of 44, 66, 100 and 150 data points were constructed from the original data sets with evenly spaced points on the predicted moisture ratio (using the logarithmic model and P2 set to -0.10). The value of -0.1 was used because that value was a low estimate of the drying rate. The subsets of data were fit to a three term exponential and the model was compared to the original set of data for the residual mean square calculation. The 153 results of this study are included in Tables 6.10, 6.11, and 6.12. Table 6.10 Time Intervals, Test 18 A Number of 44 66 100 150 2810 Points P1 0.034 0.035 0.020 0.024 0.028 P2 -3.15 -1.54 -3.35 -2.75 -2.67 P3 0.088 0.078 0.088 0.087 0.087 P4 -0.526 -0.459 -0.611 -0.570 -0.535 P5 0.054 0.051 0.059 0.057 0.054 P6 -0.093 -0.086 -0.l06 -0.100 -0.095 P7 0.074 0.074 0.075 0.075 0.074 Residual 0.29 0.33 0.33 0.30 0.28 Mean E-6 E-6 E-6 E-6 E-6 Square (of total data set) Table 6.11 Time Intervals, Test 20 A Number of 44 66 100 150 2810 Points P1 0.016 0.022 0.026 0.020 0.021 P2 -2.12 —3.76 -4.31 -2.97 -2.68 P3 0.067 0.069 0.069 0.069 0.072 P4 -0.385 -0.407 -0.412 -0.392 -0.359 P5 0.055 0.057 0.057 0.055 0.052 P6 -0.079 -0.078 -0.080 -0.078 -0.069 P7 0.107 0.107 0.107 0.107 0.106 Residual 0.50 0.42 0.46 0.43 0.38 Mean E-6 E-6 E-6 E-6 E-6 Square (of total data set) 154 Table 6.12 Time Intervals, Test 20 B Number of 44 66 100 150 2810 Points P1 0.024 0.022 0.022 0.024 0.026 P2 -1.44 -2.80 -2.82 -3.09 -2.75 P3 0.073 0.076 0.078 0.076 0.080 P4 -0.323 -0.378 -0.371 -0.388 -0.345 P5 0.045 0.049 0.049 0.051 0.046 P6 —0.072 -0.078 —0.072 -0.078 -0.063 Residual 0.55 0.47 0.29 0.35 0.25 Mean E—6 E-6 E-6 E-6 E-6 Square (of total data set) The indications are that 44 points are not enough but that 66 points are enough. It would be prudent to include more than 66 points because~of the results shown in table 6.12 where the residual mean square is not double the value for 2810 points but is nearly so with 66 data points. There is no significant difference between 100, 150 and 2810 points with these subsets of data. The serial correlation was between 0.3 and 0.5 when the total data set was fit to the three term exponential model. This coefficient measures whether or not the error at one observation is independent of the error at the previous point. Ideally the coefficient should be near 0.0. If the value is large it means that the estimated errorsfor an individual curve the weight of the dry matter does not matter in determining the parameters and the fit to the data. However, when the individual curves were combined the weight of the dry matter was important. Errors in the 174 weight of the dry matter may have been the reason that the errors between the curves were much higher than the errors within the curves. Iriany case, the residual mean square of 1.3 E-6 was believed to be a more reliable estimate of the actual error in the data for tests 18 through 21 than the residual mean square obtained from the individual data sets. The analysis of the data from 20 A and 20 B gave an estimate of the error in the data with temperature and relative humidity the same, so this error resulted from the experimental method or differences in the weight transducers. The next comparisons were with the transducers and initial moisture content constant, but from different tests with the same settings of relative humidity and temperature. The resulting error will give an indication of how the uncontrolled variations in these variables affect the results. The data from test 20 A and 21 A were combined and fit to a three term emponential. Because all conditions, including initial moisture content, were to have been the same for these two tests a seven parameter model was used. The residual mean square from this analysis wasILAS E-6. The same treatment was given to test 20 B and 21 B with the resulting residual mean square of 1.19 E-6. The individual curves from test 21 were earlier fit to the three term 175 exponential with a residual mean square for test 21 A of 0.34 E-6 and for test 21 B of 0.24 E-6. The residuals of the three combinations were nearly the same so the source of the error was neither the difference between the weight transducers nor the uncontrolled differences iri relative ‘humidity' or temperature. Therefore, the best estimate of the expected mean square error in the data for tests 18 through 21 is then between 1.2 and 1.45 E—6. This will be considered to be unavoidable error, called inherent error, caused by the error of measurement in these sets of data. The inherent error in the tests 1 through 17 was expected to be greater than that for the later tests. To get an estimate of this error, tests as nearly the same as possible were combined. There were two good sets to combine, test 7 B with test 15 B and test 10 B with test 17 IL These combinations were made with an eight parameter fit, the seven parameters of the three term exponential model plus one additional parameter to allow for the initial moisture content to be different. The residual mean square resulting from the combination of test 7 B with 15 B was 3.6 E-6 and for the combination of tests 10 B and 17 B was 2.7 E-6. In both cases the residual mean square was higher than for the later tests as expected. 176 6.5.2 Nonlinear Combined Approach In this section the nonlinear approach was used for the data analysis. The data were selected from the full set of data on an exponentially increasing time interval but with a maximum interval of one hour as explained in Section 6.3.3. Eleven data sets were used (6 B, 8 A, 9 B, 12 B, 13 A, 14 B, 18 A, 18 B, 19 A, 19 B, and 20 B) with 98 points from each data set for a total of 1078 data points. The relative humidity of the drying air ranged from 0.25 to 0.51 and the temperature of the drying air ranged from 17.3 to 40.2 C. The initial moisture content data, as determined from the data plots, were also entered into the data set. The initial moisture content, for each test, which was used in the regression analysis is shown in Table 6.19. Originally the data set had included data from tests with the high moisture rice, initial moisture content greater than 0.45. However, these tests clearly stood out irithe plots oftfluaresiduals indicating that the product was different than the lower moisture content product. The explanation for this is that the high moisture samples were removed from the material streanlat the parboiling plant before entering the first 300 C rotary dryer and the lower moisture samples were removed after the first rotary dryer. The product was evidently changed by the high temperature rotary dryer. There was relatively little data for the 177 high moisture rice, four tests at two temperatures and three relative humidities. Only data from the rice which had been dried in the high temperature rotary dryer was included for the estimation of the parameters in the model. Table 6.19 Moisture Content at Time Zero Test Initial Moisture Test Initial Moistue Content Content 5 A 0.268 14 A 0.298 5 B 0.233 14 B 0.255 6 A 0.500 15 A 0.305 6 B 0.283 15 B 0.292 7 A 0.530 16 A 0.322 7 B 0.293 16 B 0.289 8 A 0.281 17 A 0.174 8 B 0.576 17 B 0.292 9 A 0.572 18 A 0.241 9 B 0.300 18 B 0.178 10 A 0.210 19 A 0.242 10 B 0.277 19 B 0.172 12 A 0.292 20 A 0.253 12 B 0.294 20 B 0.258 13 A 0.309 21 A 0.253 13 B 0.299 21 B 0.260 A general three term exponential form was used as described in Section 6JL3 with the Chung-Pfost form for EMC and an Arrhenius form for drying constant. The form for the EMC was: EMC = Pl-P2*ln(-(T+P3)*ln(RH)) 6.13 The form for drying constant was: 2 = exp(P4/(T+273.2)) 6.14 The three term exponential form was: 178 M(t) = (Mi-EMC)*((1-P6—P8)*exp(P5*Z*t)+P6*exp(P7*Z*t) +P8*exp(P9*Z*t))+EMC 6.15 These equations, with the partial derivatives with respect to each of the parameters, were entered into the nonlinear regression package BMDP3R. The parameter values which were estimated for the model from equations 6.13, 6.14 and 6.15 are shown in Table 6.20. Table 6.20 Parameter Values, 9 Parameter Model Parameter Value Asymptotic Standard Deviation 1 0.170 0.003 2 0.0247 0.0006 3 -8.4 0.7 4 -3160 70 5 -120000 30000 6 0.589 0.009 7 -13000 3000 8 0.27 0.01 9 -2100 500 Residual Mean Square 10.7 E-6 The next step in the regression study was to see what effect the removal of one parameter had on the model. The choice was to either try a two term exponential model or to remove P3. P3 was removed first and the results are shown in Table 6.21. 179 Table 6.21 Parameter Values, 8 Parameter Model Parameter Value Asymptotic Standard Deviation 1 0.187 0.002 2 0.0274 0.0005 4 -3590 60 5 -500000 100000 6 0.586 0.008 7 -50000 10000 8 0.28 0.01 9 -8000 2000 Residual Mean Square 11.4 E-6 The next model with a reduced number of parameters which was fit to the data was the two term exponential model with P3 in the model. The results are shown in Table 6.22. Table 6.22 Parameter Values, 7 Parameter Model Parameter Value Asymptotic Standard Deviation 1 0.182 0.002 2 0.0268 0.0006 3 -9.5 0.6 4 -2730 80 5 -8000 2000 6 0.57 0.01 7 -1200 300 Residual Mean Square 15.5 E-6 The next model was the two term exponential with P3 removed. The results of the regression are shown in Table 6.23. 180 Table 6.23 Parameter Values, 6 Parameter Model Parameter Value Asymptotic Standard Deviation 1 0.204 0.002 2 0.0305 0.0005 4 -3310 70 5 -50000 10000 6 0.53 0.01 7 -7000 2000 Residual Mean Square 17.0 E-6 These alternative models should be compared by the residual mean square values buttflmeF statistic cannot be depended on entirely in the nonlinear case. When the residual mean square values are compared with and without P3 the conclusion can be made that P3 does not improve the fit enough to keep it in the model. When the results from the two term and three term models are compared the three term model is seen to be better than the two term model. The evidence from the study of the individual curves was that the three term model was required to adequately model the data. Therefore, the 8 parameter model was chosen as the appropriate model for the thin layer drying of parboiled rice. The residual mean square value for this model, 11.4 E-6, is considerably higher than was expected from the estimate of the inherent error, between 1.2 E-6 and 3.5 E-6, from Section 6JL2. There are two values which were considered to be parameters in Section €L3.2 and were 181 considered to be variables in this part of the study. One was the initial moisture content and the other was the EMC. Comments have already been made about the difficulty of estimating the initial moisture content and determining the initial time. The initial moisture content was estimated graphically and probably was no more accurate than within 0.003. Published EMC models, the form of which was used in this study, produce estimates no better than within 0.005 (ASAE 1982). The residuals from the regression are shown plotted in Figure 6.23 against time. This figure shows that the errors at small time, due to the initial moisture content variable, and those at large time» due to EMC values, are higher than the errors at intermediate times. The conclusion is then that the larger value for the residual mean square’is due to the problem of obtaining the proper initial moisture content and obtaining suitable values for EMC. The large values are not due to the fact that the three term exponential model is inappropriate. The errors due to the initial moisture variable and the EMC values should be relatively constant for either the two term or the three term exponential, so the error due to these two parts of the equation tends to overshadow the errors due to the difference between the number of terms. Figure26.23 does show that the regression produced a reasonably good fit to the data with fairly constant 182 +OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+ .020 + + .015 + + .010 + l 1 + . l . . ll . .2125431 2 1 l l . .2 533331 1 l 1 . .005 +88311 531 111 l 1 21 1 + R .93274725321111 11 2 2131221 12211111 . E .3657314864531114211 2321 l l l 1 2 . S .4GA 46A5235321 1131221 3 112 11 1 ll . I .6FN9BB1336534423 3 1 11111131 1 11 . D 0.00 +3DHKF8585443442221 2 11 112 l 1 1112 + U .4929A88633l 2122 111 211 11 11 ll 1 . A .3G443766341122 l 11 221112221 11 . L .8045443234642733122412111 121132311 22 . .CA58542412424114211 112223 112 1 32412 . -.005 +2877211121 1111 l 2 2211 12 l + . 34 1111 l 1 1 112 111 . . 12131111 1 . . 11212 1 , . 1 l l . -.010 + 1 + . Explanation of symbols, p. 126 . -.015 + + .+....+....+....+....+....+....+....+....+. 5. 15 25 35 0. 10 20 30 40 TIME, hours Figure 6.23 Residuals of Fit to Total Data Set 183 variance and few points unreasonable far from zero. The slightly hourglass shape has a reasonable physical explanation. A control data set was constructed from data sets which were not used in the determination of the parameters. This data set was created from 9 tests (5 A, 10 A, 10 B, 13 B, 15 B, 16 B, 17 B, 21 A, and 21 B). The same algorithm which was used to choose the data points to include in the data set for estimating the parameters was used to choose the data points for this set. The complete model, obtained by combining equations 6.13, 6.14,(L15 and the parameters in Table 6.21 was: EMC = 0.187-0.0274*1n(-T*1n(RH)) 6.16 Z exp(-3590/(273.2+T) 6.17 M(t) = (Mi-EMC)*((1-0.589-0.27)*exp(—120000*Z*t) +0.589*exp(-13000*Z*t)+0.27*exp(—2100*Z*t)) +EMC. 6.18 The model was used to predict the data of the control data set. The resulting residuals are shown in Figure 6.24 plotted against time. This plot has more scatter than the residuals from the data from which the parameters were determined, as it should have. The initial moisture content variable seems to be even worse for this data set than for the data set from which the parameters were determined. The EMC values do not seem to be as much of a 184 O+OOOO+OOOO+OO00+.0.0+.OOO+OOOO+OOOO+OOOO+OOO0+ .020 + + .015 + + . 1 . 1 . .010 +12 + .341 . . 2 1 R .11 . E .12211 1 1 1 . S .005 + 2613111 + I .32552441 1 1 l . D .2523442442232421 1 1 . U .357B73A3122 1 1 1 l 1 1 . A .523245142221 1 2111 2 1 1 1 121 . L 0.00 +66123621522122122 1 l 111 l 2211 2212 + .452641 1 2332 11 1111 11121 122712 .676 2434 1 1 2222111111 1 61 1 . .521 1212 11 1 12111 1211 l 21 . .52 4322 2 1212112311 11 121332 111 . -.005 +46 32222211231 211222211 1 2 l 1 l l + .4F764243333411 2 2 1 1 l 1 . .1EBBC754321 1121 l 111 1 11 112 . . 8533532122 1 111 l 1 111 l . . 4441221211121111 1 1111 111 . -.010 + 51252 1 121 l 111 + . 58411 1 1 . . 452 11213 . . 13111111 1 . . 11112111 . -.015 + 1231 + . Explanation of symbols, p. 126 . —.020 + + .+....+....+....+....+....+....+....+....+....+. 5. 15 25 35 45 0. 10 20 30 40 TIME, hours Figure 6.24 Residuals, Complete Model on Control Data 185 problem with this data set as they were with the other data set. The residual mean square from this analysis was 26.4 E-6 0 6.5.3 Traditional Combined Approach Realizing that this study included equipment which was believed to produce more accurate data than has been previously avaliable as well as methods of analysis which should produce models with lower overall error, axnethod was sought which would fairly compare the methodology covered in this dissertation with traditional methods. There were two alternatives, either analyze the data from other researchers with complete nonlinear methods or analyze the data from this study with the linearization techniques. The second approach was taken. PageJS equation had proven to be good at fitting the data in previous studies as well as this study. Therefore a model similar to that of Misra and Brooker (1980) was used: M(t) (Mi-EMC)*exp(-K*tn)+EMC 6.19 exp(Pl+P2*ln(T)+P5*V) P3*ln(RH)+P4*Mi the velocity of the air where: <37: uuu In this study the velocity of the air was constant so P1 and P5*V were combined into one parameter'Pl. The Chung- Pfost EMC equation for rough rice was used (ASAE 1982): 186 EMC = 0.294-0.0460*1n(—(T+35.7)*1n(RH)) 6.20 Equation 6.19 was solved for moisture ratio and logarithms taken twice, as shown in Section 3.1, producing: ln(-ln(MR)) = Pl+P2*1n(T)+ln(t)*(P3*ln(RH)+P4*Mi) 6.21 New variables were created for ln(t)*1n(RH) and ln(t)*Mi producing an equation linear in the parameters. Data were then obtained from the same tests which were used to estimate the parameters with the nonlinear method (61L.8 A, 9 B, 12 B, 13 A, 14 B, 18 A, 18 B, 19 A, 19 B, and 20 B). Data points were taken starting at one half hour into the test and at one half hour intervals for the first 12 hours, similar to traditional methods of data treatment. The data set consisted of 264 data points with a temperature range of 17.2 to 40.8 C and a relative humidity range from 0.24 to 0.52. The initial moisture variable was obtained from Table 6.19. The resulting equations for n and K were: n = 2.22*Mi—0.0872*1n(RH) 6.22 K = exp(-3.74+0.879*ln(T)) 6.23 Although the statistical package which was used to obtain the parameters included information of the fit to the data, the statistics which were included lose their physical significance when transformations such as those required to obtain equation 6.21 are performed. The model, from 187 equations 6.21, 6.22 and 6.23, was used to predict the same data set used to test the model from the nonlinear method. The residuals which resulted are plotted in Figure 6.25. The residual mean square was 67.8 E-6. 6.5.4 Comparisons of Data and Models In this section the model which was obtained by nonlinear methods will be compared with the model obtained by traditional methods. In addition data from an independent source will be compared with the data obtained in this study. 6.5.4.1 Traditional vs. Nonlinear The parameters for the traditional and nonlinear models were estimated by using selected data from the same tests in this study. They were then compared with selected data from different data sets from this study. The plots of the residuals are shown in Figures 6.24 and 6.25. The fit to the test data by the model resulting from the nonlinear methods is observed to be better than the fit to the same data set by the model resulting from traditional methods. In addition the residual mean square was calculated for the fit by the two models to the test data set. For the model by traditional methods it was 68 E-6 while for the model by nonlinear methods it was 26 E-6. .020 .015 .010 .005 PCP‘CIUHUJBJ'JU -.005 -.010 -.015 -.020 188 O+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+ Explanation of symbols, p. 126 +00 +000 11 1 1 2 11 1 52 211211 1 3323 1 11 21 111 4 1 1 1 . 12221 2 2 1 1 .41 1 1 1 1 1 1 .1 2 111 . 1 12321 +1 132 11 1 1 .22 21 1 1 .12 33 2 2 111 .33 14 1 . 1 11 1 1 11 +2154 33211 1 .62347212122 1 111 . .34C72152511311 111 .296323223221221 1 1 111111 1 .6A664623231 1 1 111 1 11111 +2C45114 21 3212 111 .461 3211 112 11211 11 11 1 . .831153 121 1 21 1 .54654211 222211 112111 .3A98 1 12231 2112 11 1 111 +1B1321 21 12 1 21122 111 1 1 211 9 5344 133 1 11 11 1 111 1 21111 . 25 21 111 1 32421 11111121221 124 421 1 121 31 11 1 2234112 1 1 1 12123211211 11125411 1 111 11 1 13 1 12 1 1 1 1 1 1 1 3 1 11 1 111 111 12 1 1 152 22213 1 1 1 1 1 111 11 1 71 . 12131 1 4 11 13 +0000 +0000 +0000+000 +00 +000 +00 +0000+0000 +0000 +0000 +000 O+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+O 5. 15 25 35 45 0. 10 20 30 40 TIME, hours Figure 6.25 Residuals vs. Time, Page's Model on Control Data 189 Neither fit is very good at low time indicating that the initial moisture variable was in error in several of the tests. The difference in residual mean square was large enough to be considered different. Therefore, the nonlinear method produced a model which fit the data better than the model produced by one of the traditional methods from the same original data. The difference in researcher effort and computer expense was also considerable, with the traditional method much less expensive. If only a good estimate of the moisture content of the product of study is desired, with no intent to use the parameters for anything other than predicting moisture contents, then the traditional method could be used. However, if the researcher desires to base further studies on the results of the analysis, or wishes to learn about the drying process rather than merely modeling it then the nonlinear method must be used, whether or not it produces a better fit to the data. 6.5.4.2 Comparison With Independent Data In this section the data collected in this study are to be compared with the data published by Bakshi (1979) for short grain parboiled rice. As discussed in Chapter 2 Bakshi is the only researcher to have published data on the thin layer drying of parboiled rice. His data were published as time and moisture ratio pairs with 12 data 190 points in the first 1.2 hours. No data at later times were published. He collected two sets of data on the drying of parboiled rough rice at three temperatures for a total of six sets of data and seventy-two data points. The three temperatures he used were 56.1, 51.0 and 40.6 C. The last temperature was near enough to temperatures covered in this study that comparisons could be made. This data set was obtained at 0.79 relative humidity, a wet bulb temperature of 36.7 C, higher than any test in this study. The initial moisture content for the two sets of data was 0.6375. The high moisture rice in this study had been noted to behave differently than the lower moisture rice. The reason for the difference was due to the difference in treatment at the parboiling plant. The wettest rice had not been subjected to the 300 C rotary dryer to which the drier rice had been subjected. The model in this study was developed from data excluding the parboiled rice which had not been partially dried at this high temperature. The EMC values for the rice were not changed substantially but certain of the other parameters had been changed. The data set from this study which most closely resembles Bakshi's with respect to product pretreatment and drying conditions was test 9 A. The temperature in this test was 40.2 and the relative humidity 0.47. Because this data set was from the first sets and therefore contained 191 higher errors it was fit to only a two term exponential. The resulting parameters are shown in Table 6.24. Table 6.24 Two Term Exponential Fit to Test 9 A Asymptotic Parameter Value Standard Deviation 1 0.445 0.004 2 -0.636 0.007 3 0.044 0.003 4 -0.10 0.01 5 0.105 0.001 Residual Mean Square 4.5 E-6 This model was transformed into the form: M(t) = DM*(Pl*exp(P2*t)+P3*exp(P4*t))+P5 6.24 so that it could be used with a different initial moisture content and EMC. The parameters P1 through P4 in equation 6.24 were used to model Bakshi's data and are shown in Table 6.25. P5 is, of course, the EMC. Table 6.25 Parameters For Model 6.24 Parameter Value 1 0.910 2 -0.636 3 0.090 4 -0.0966 The model which was used for EMC was the model developed by the nonlinear technique in this study: EMC = 0.187-0.0274*1n(—T*1n(RH)) 6.25 192 This model, equation 6.24 and 6.25, was used to predict Bakshi's data and the residuals resulting from the prediction to data set 1 are shown in Figure 6.26. The model predicts the data fairly well and is approaching the data at 1.2 hours when the data ends. The shape of the residual plot appears to be similar to that obtained using the two term exponential model with the data from this study; such as Figure Gut. It is interesting to note that the predicted value is greater than the observed value, the short grain rice in Bakshi's study appears to dry slightly more rapidly than the long grain rice in this study. Caution must be used in drawing conclusions from this comparison since the study conditions are different, the product different and his data rather limited. A similar comparison was maderto Bakshi's second set of data. The residuals are shown plotted in Figure 6.27. Because the conditions were the same the residuals should have been the same, however, the moisture content at 0.25 hours appears to differ by about 0.008 in the two tests. The model again predicts Bakshi's data fairly well supporting the premise that the data from this study were basically similar in the time-moisture content relationship to the data that he collected. If these particular two sets of data are a good measure then his data were less repeatable than the data from this study. .005 —0@05 -.010 ["II’CIUHUJM'XJ -.015 -.020 -.025 -.030 -0935 + + + + + + .1 . + + 1 O O 1 O O 1 Q + + . 1 . O l O O l O + + 1 + 1 + O 1 O . 1 . . 1 . + + O+OOCO+OOOO+OOOO+OO00+...OO+OOOO+OOCO+OCOO+OOOO+O .250 .750 1.25 1.75 .2.25 0.00 .500 1.00 1.50 2.00 TIME, hours Figure 6.26 Residuals, Bakshi's Data Set 1 vs. Model .010 .005 -0ggs ["II’CIUHUIIEISU -.010 -0615 -.020 -0g25 -0039 194 +0.00+OO00+.OOO+OOOO+OOOO+OOOO+OOOO+OOOO+OOOO+ + 0 0 +0000 +00 +0000 +0000 + 0 0 0 0 H +0 0 0 O H H 0 +0 +0 +0000 H +0000 +00 +0000 + 0 0 + 0 0 +00 +00 0+0.00+OOOO+OOOO+OOOO+OOOI+OOOO+OOOO+OOOO+OOOO+O .250 .750 1.25 1.75 2.25 0.00 .500 1.00 1.50 2.00 TIME, hours Figure 6.27 Residuals, Bakshi's Data Set 2 vs. Model 7 . CONCLUSIONS In thin layer drying studies nonlinear regression techniques should be used to obtain the parameter estimates because all of the models used are nonlinear in the parameters. For agricultural products EMC is not a known constant. Even when EMC values can be estimated with established models, the errors are in the range 0.005 to 0.03 (ASAE 1982, page 318). The errors in the moisture content data in this study were in the range of 0.001 so the EMC models produce significantly more error than is in the data. The method used in this study was to fit the data to the complete model for moisture content over time, a model which included a model for EMC. This method produced a model with lower total error and allowed the removal of parameters which could not be supported by the data. In addition, this method allowed confidence intervals to be attached to the parameters so that other researchers who may want to use the model can have a better understanding of the parameters in the model. The equipment, which was constructed foriuuein this study, was operated in the range of 0.24 to 0.53 relative humidity and in the temperature range of 17.3 to 40.6 Celsius. The dry bulb temperature was maintained to within 195 196 0.3 C of the mean value for over 90% of the observations (after the initial half hour) during each testing period of over 40 hours. The relative humidity was calculated to have been maintained to within 0.005 of the mean value for over 90% of the observations during the same periods. Models with varying numbers of decaying exponential terms were fit to the data sets believed to have the lowest inherent error. It was concluded that three or four terms were required to adequately model the observations. In reaching this conclusion, plots of the residuals were examined. These plots showed that the errors in models with fewer terms were correlated with the independent variable. The value of the residual mean square alone was not sufficient to show that a two term exponential was an inadequate model. With the three and four term exponential models the weight data of individual curves was predicted with a standard error of 0.0005. Equilibrium moisture content, EMC, was considered to be an unchanging parameter in these curves. Page's empirical equation was found to fit the weight data well, with standard errors as low as 0.0005. In most cases, however, this model does not explain all of the variation in the dependent variable (sample weight) and was rejected as a model for thin layer drying of parboiled rice. 197 Two models based on particle geometry, spherical and infinite cylinder, did not fit the observed data adequately even when the radius of the particle was a parameter. While assumptions about particle geometry cannot be proven to be correct, the evidence from this research showed that these two assumptions were incorrect for parboiled rice. Whenever studies are based on assumptions about particle geometry the validity of the assumptions should be examined. The minimum length of time for data collection was found to be 36 hours for thin layer studies of long grain parboiled rice in the temperature range of 20 to 40 C and the relative humidity range of 0.25 to 0.50. The parameter estimates which result from including data to this maximum time predicted the whole data set (with maximum time of 43 to 47 hours) as well as when the entire data set was used. The model predicts that the EMC value would have been reached in 74 hours. An examination of several data sets was undertaken to determine thetninimum number of observations which were necessary to represent the entire data set. A scheme using exponentially increasing time intervals between data points was examined and found to be useful. The one term exponential model, with a rough estimate of drying rate, was used to choose the time at which data were selected. Data subsets of 98 points for constant relative humidity 198 and temperature were used to estimate the parameters of a three term exponential. The resulting estimates predicted the data as well as estimates obtained from the entire data set of over 2400 data points. The preferred thin layer model which was obtained from this study for parboiled long grain rice in the temperature range of 17 to 40 C, the relative humidity range of 0.25 to 0.51, and initial moisture content from 0.18 to 0.30 was: EMC = Pl-P2*ln(-T*ln(RH)) 7.1 Z = exp(P3/(T+273.2)) 7.2 M(t) = (Mi-EMC)*((l-PS—P7)*exp(P4*Z*t) +P5*exp(P6*Z*t)+P7*exp(P8*Z*t))+EMC 7.3 where: t time, hours T = temperature, celsius RH = relative humidity, decimal Mi = initial moisture content, decimal d. b. M(t) = moisture content at time t, decimal d. b. EMC = equilibrium moisture content, decimal and: Parameter Value Asymptotic Standard Deviation 1 0.187 0.002 2 0.0274 0.0005 3 -3590 60 4 -500000 100000 5 0.586 0.009 6 -50000 10000 7 0.28 0.01 8 -8000 2000 8. SUGGESTIONS FOR FUTURE STUDY 1. The data from other thin layer studies, which has been analyzed with linearization techniques, should be reanalyzed with nonlinear techniques, when the quality of the data warrants further study. This further study would not only clarify the value of the nonlinear approach but could also examine the assumptions of particle geometry made in other studies. 2. There are many products of economic importance which are normally dried or stored at temperatures attainable in the chamber for which there is either no model of the moisture relationships or only limited data from which a model has been constructed. These products should be studied. 3. The closed form solution to diffusion in an ellipsoid would make a much better model for the particle geometry of long grain rice than the infinite cylinder or sphere which werelconsidered:h1this study. The infinite cylinder and spherical models had to be rejected because of the distinct pattern in the residuals. A solution to diffusion in an ellipsoid was searched for in the mathematical and physics literature and was not found. 199 200 4. In this study the individual curves are modeled quite well, but the models used for the combined data do not work nearly as well. This discrepancy is due to problems in measuring the initial moisture content and problems with the form of the EMC equation. Perhaps different experimental techniques could reduce the error in determining the conditions at the initial time. A more theoretically based model may produce a better EMC model. In this study it has been shown that basing the temperature dependence of EMC on the freezing point of water was statistically sound. Further studies of EMC should consider this finding because the form of the model for EMC considered in this study was developed based on perfect gas laws. Considering that the models are nonlinear, the question of initial conditions is not trivial. 5. Further studies of the time interval between data points and the length of data collection in a thin layer drying study with products of vastly different drying rates should be done. The results of these studies would produce guidelines about the length of time and time interval suitable for the study of any agricultural product. 6. Moisture transfer from the air to the product is important in grain drying. Studies of this process, using nonlinear data analysis, should be undertaken. 7. The range of temperature and relative humidity could be expanded for some applications. If the 201 temperature sensor located at the air heater were replaced with a sensor which can operate at temperatures greater than 100(2the present equipment.and algorithms should be capable of maintaining at least 60 C in the chamber. A substantial increase in the relative humidity of the test would require that the study chamber be redesigned because of condensation problems. APPENDICES APPENDIX A This software is for data acquisition and digital control of the Aminco-Aire unit. This software is interrupt driven. PROGRAM TRY60; (*$E+,I+,S-,C-*) { JAN 4, 1983 Byler} { to be assembled with MAINNEW } { and linked with IPUT IGET LIBRKB } CONST STAT=208; LBYTE=210; HBYTE=212; DBSENS=4; WATSENS=2; CHTMP=3; VAR TIME:INTEGER; DBHEAT,WATHEAT,WATCOOL:BOOLEAN; DBDC,LTIME,I1T,JUNK1:INTEGER; ISUM:ARRAY[0..15]OF INTEGER; AVINI:ARRAY[0..15] OF INTEGER; ENGV:ARRAY[0..15] OF REAL; THIST:ARRAY[1..4] OF REAL; DBGOAL,DBSET,WATGOAL,K1,K2,K3:INTEGER; DDC,PVAL,IVAL,DVAL,ADJUST:REAL; d1,d2,d3,d4:integer; PROCEDURE PUT(HBYTE,LBYTE,DATA:INTEGER);EXTERNAL; FUNCTION GET(HBYTE,LBYTE:INTEGER):INTEGER;EXTERNAL; PROCEDURE OUTPUT(VAL,PORT:INTEGER);EXTERNAL; FUNCTION INPUT(PORT:INTEGER):INTEGER;EXTERNAL; FUNCTION ATOD(DEV,GAIN,T1,T2:INTEGER):INTEGER;EXTERNAL; PROCEDURE IPUT(HBYTE,LBYTE,DATA:INTEGER);EXTERNAL; FUNCTION IGET(HBYTE,LBYTE:INTEGER):INTEGER;EXTERNAL; {converts the integers from tflua A/D converter to engineering terms} FUNCTION ITOE(IVAL,ADDR:INTEGER):REAL; BEGIN if addr=4 then itoe.=iget(161,8)/10.0*ival-iget(161,40) else ITOE:=(iget(l61,(2*addr))/l000.0*IVAL -iget(l6l,(2*addr+32)))/l0.0; END; 202 203 {sets the output bits to control the Aminco} PROCEDURE SWITCHES; VAR SWITCH:INTEGER; BEGIN IF (TIME MOD 100)3800 then dbSET:=2500; WATGOAL:=IGET(32,0); if abs(watgoal-2600)>2400 then watgoal:=2000; END; {calls conversion to engineering terms of necessary variables and does water temperature control} PROCEDURE GTENGV; VAR I:INTEGER; BEGIN FOR I:=2 TO 4 DO BEGIN ENGVlI]:=ITOE(AVINI[I],I); IPUT(160,(48+I*2),ROUND(ENGV[I])); END; WATCOOL:=FALSE; WATHEAT:=FALSE; IF ENGV[WATSENS]<(WATGOAL-8)THEN WATHEAT:=TRUE; IF ENGV[WATSENS]>(WATGOAL+8) THEN WATCOOL:=TRUE; END; {stores the current errors in RAM, accessible in BASIC, for debugging of the control algorithms and setting the gains} PROCEDURE PSERR; BEGIN IPUT(161,112,ROUND(WATGOAL-ENGV[WATSENS])); IPUT(161,114,ROUND(DBGOAL-ENGV[DBSENS])); IPUT(32,08,DBGOAL); END; {integral part of cascade control and first part of the PID algorithm} PROCEDURE PIDl; BEGIN DBGOAL:=DBGOAL+ROUND(0.02*(DBSET—ENGVlCHTMP])); THIST[4]:=THIST[3]; THIST[3]:=THIST[2]; THIST[2]:=THIST[1]; THISTll]:=DBGOAL-ENGV[DBSENS]; DDC:=DDC+7.25*THIST[1]-8.0*THIST[2]-THIST[3] +2.0*THIST[4]; END; 205 {second part of the PID control} PROCEDURE PID3; BEGIN IF DDC>15000.0 THEN DDC:=15000.0; IF DDC<-5000.0 THEN DDC:=-5000.0; IF DDC>=0.0 THEN DBDC:=ROUND(DDC/l00.0) ELSE DBDC:=0; PUT(161,84,DBDC); PUT(161,104,255); END; {main program} BEGIN if iget(161,1l6)<0 then begin PASSVAL; DBGOAL:=DBSET+ROUND(0.6*(DBSET-2000)); put(161,ll7,0); thist[3]:=0.0; thist[2]:=0.0; thist[l]:=0.0; end; IF ((TIME<0) OR (TIME>4999)) THEN TIME:=0; TIME:=TIME+1; SWITCHES; 11T:=TIME MOD 250; if (ilt<226) and (ilt mod 25 =0) then getdata; if (i1t>225) and (ilt<235) then avdata; if i1t>241 then case (ilt-241) of l: PASSVAL; 2: GTENGV; 3: PSERR; 5: PIDl; 7: PID3 END END. APPENDIX B This software is for data storage, temperature selection, setting of clock, and initialization of equipment on power up. The most important routines are %40 and %50. >OK >L. 1 REM THIS IS THE MAIN DATA ACQUISITION SOFTWARE 5 FOR S=10 TO l38;@(S)=0;NEXT S 6 IF C0<50 C0=50 10 FOR T=0 TO 224 STEP 32 20 FOR U=0 TO 5 25 IF GET(%A168)<1 G.25 30 FOR S=0 TO 7 35 PUT(%2032)=GET(%A000+S*2) 40 PUT(%2033)=GET(%A001+2*S) 45 @(10+S+T/2)=@(10+S/2)+Z 50 NEXT S 52 PUT(%A168)=0 55 NEXT U 60 RUN%46 70 FOR S=0 TO 9 80 &(42+T+S)=GET(%2014+S) 90 NEXT S 110 P.#%,M,':',L,':',K,' ',O,N, 150 FOR S=10 TO 17 151 P.@(S+T/2), 152 NEXT 5 153 P.E 155 IF D<2 G.210 160 FOR S=1 TO (D—l)*6 170 IF GET(%A168)<1 6.170 180 PUT(%A168)=0 190 NEXT S 210 NEXT T 220 RUN%52 300 G.5 >OK 206 207 >LO.%44 >OK >L. REM SETS THE TIME Q=%DC;R=%DE;OUT(R)=255;OUT(R)=23 OUT(Q)=255;OUT(Q)=138;OUT(R)=1;OUT(Q)=57 OUT(Q)=15;OUT(Q)=0 INPUT'SECONDS'B;OUT(Q)=B 0(Q)=0 INPUT'MINUTES'B;OUT(Q)=B INPUT'HOURS'B;OUT(Q)=B OUT(R)=3;OUT(Q)=57;OUT(Q)=0 INPUT'DAYS-TENSONES'B;OUT(Q)=B INPUT'DAYS—THOUSANDSHUNDREDS'B;OUT(Q)=B OUT(R)=5;OUT(Q)=%31 OUT(Q)=%1C;OUT(Q)=0;OUT(Q)=%40 OUT((R)=9;OUT(Q)=0 OUT(Q)=0;OUT(R)=10;OUT(Q)=0 OUT(Q)=0;OUT(R)=68;OUT(R)=39 OUT(R)=%70 STOP \OCD\)O\U'Iubt»)l\.)i--‘I DOWN NQQ came-w QQQUJ \ocn QM >OK >LO.%46 >OK >L. REM GET THE TIME Q=%DC;R=%DE;OUT(R)=167;OUT(R)=17 J=IN(Q);K=IN(Q) IF(J#0)+(K#0) G.6 OUT(R)=166 OUT(R)=18;L=IN(Q);M=IN(Q) OUT(R)=19;N=IN(Q);O=IN(Q) STOP \O\IO\U14>UJNH >OK >LO.%47 >OK >L. REM PRINTS THE TIME RUN%46 P.#%,'DATE: ',O,N,' TIME: ',M,' STOP O‘U‘INH >OK 'ILI':'IKI'°'IJ 208 TIMER NEXT >LO.%48 >OK >L. 1 REM INITIALIZE 2 REM SET ALL DUTY CYCLES TO 0% ON 3 H=0;H0=0;W0=0 4 REM CLEAR A/D CONVERTER 5 B=IN(%D0) 6 OUT(2)=8;REM SETUP FOR TIMER 9 RUN%44;REM SET CLOCK AND START 40 MS TIM SETUP 10 OUT(3)=4;REM SETUP FOR TIMER 11 OUT(%D0)=%10;REM SETUP FOR TIMER 14 REM CLEAR RAM 15 FOR I=0 TO 512;PUT(%A000+I)=%FF;PUT(%DDAO+I)=%FF; 20 PUT(%ZlFE)=%28;REM LOCK 40 ROOM FOR @() 30 RUN%50;REM RESET TAPE DECK 90 STOP >OK >LO.%4A >OK >L. 1 REM INITIALIZE 2 REM SET ALL DUTY CYCLES TO 0% ON 3 H=0;H0=0;W0=0 4 REM CLEAR A/D CONVERTER 5 B=IN(%D0) 6 OUT(2)=8;REM SETUP EOR TIMER 10 11 90 >OK OUT(3)=4;REM SETUP FOR TIMER OUT(%D0)=%10;REM SETUP FOR TIMER STOP >OK >L. 1 10 40 50 60 70 90 )OK REM SAVES VARIABLES ON TAPE FOR I=0 TO 63;&(20+I)=GET(%A100+I)FNEXT I @(50)=C;@(51)=D;@(52)=E C0=10;D0=0 RUN%52 C0=1 STOP 209 >OK >L. 1 REM GETS VARIABLES FROM TAPE 10 C0=10;D0=0;RUN%5E 20 FOR I=0 TO 63;PUT(%A100+I)=&(20+I);NEXT I 30 C=@(50);D=@(51);E=@(52) 40 C0=1 90 STOP >OK >LO.%4E >OK >L. 1 REM INPUTS CONVERSION FACTORS ON TAPE 10 FOR I=0 TO 15 20 P.'FOR CHANNEL NUMBER',I,' ', 30 INPUT'A='@(0),' B='@(1) 35 PUT(%A100+2*I)=&(0) 36 PUT(%A101+2*I)=&(1) 37 PUT(%A120+2*I)=&(2) 38 PUT(%A121+2*I)=&(3) 40 NEXT I 50 RUN%4C 90 STOP >OK >LO.%50 >OK >L. 1 REM INITIALIZATION OF TAPE 4 FOR I=0 TO 5;I0=IN(10);NEXT I 6 OUT(4)=0;OUT(4)=1;RUN%56 8 H=%18;RUN%54;RUN%56 10 H=%14;RUN%54 12 H=%30;C.%5880;H=%0D;C.%5880 16 RUN%55;RUN%56 20 H=%1A;RUN%54 22 H=%0D;C.%5880;RUN%55;RUN%56 30 ST. >OK 210 >LO.%52 >OK >L. 10 REM SAVES @(10)-@(137) ON TAPE TR=D0,BL=C0 15 RUN%57 20 H=%14;RUN%54;H=(%30+D0);C.%5880 22 H=%0D;C.%5880;RUN%55;RUN%56 24 RUN%5A;H=%17;RUN%54 26 FOR I=0 TO 2;H=(%30+@(I));C.%5880;NEXT I 32 H=%2C;C.%5880;H=%30;C.%5880;H=%0D;C.%5880 38 RUN%55 40 FOR I=0 TO 255 42 H=&(20+I);C.%5880 44 NEXT I 50 RUN%56 52 C0=C0+l 54 P.C0,D0;IF C01000 G.99 56 C0=0;D0=2 99 ST. >OK >LO.%54 >OK >L. 1 REM OUTPUT CONTROL BYTE TO TAPE DECK 2 C.%5800;OUT(10)=H 3 C.%5840;P=IN(10);IFP=94G.6 4 P.'ERROR NO CTRL ECHOED';G.9 6 C.%5840;P=IN(10);IFP=(H+64)G.9 8 P.#%,'ERROR, BE SENT=',H,'BYTE RCVD=',P 9 ST. >OK >LO.%55 >OK >L. 1 REM GET 0D-0A 4 C.%5840;P=IN(10);IFP=13G.6 5 P.#%,P,' RCVD. OD EXPECTED';G.9 6 C.%5840;P=IN(10);IFP=10G.9 7 P.#%,P,' RCVD. 0A EXPECTED‘ 9 ST. >OK 211 >LO.%56 >OK >L. 1 REM GET 07-0D-0A 2 C.%5840;P=IN(10);IFP=7 G.4 3 P.#%,P,' RCVD. 7 EXPECTED';G.9 4 RUN%55 9 ST. >OK >LO.%57 >OK >L. 5 REM CLEAR COMM. FROM TD. 10 GOS.99 20 OUT(10)=27;GOS.99 30 OUT(10)=0;GOS.99 40 H=24;RUN%54 50 RUN%56 90 ST. 99 F.P=1TO9;Q=IN(10);NE.P;R. >OK >LO.%5A >OK >L. 1 REM CONVERTS A BINARY NUMBER C0 TO 3 INTEGERS 4 @(0)=C0/100 6 @(l)=C0/10-@(0)*10 8 @(2)=C0-10*@(1)-100*@(0) 9 ST. >OK >LO.%SB >OK >L. 1 REM INPUTS CONVERSION FACTORS, STORES ON TAPE 10 FOR I=0 TO 15 20 P.'FOR CHANNEL NUMBER',I,' ', 30 INPUT'A='@(0),' B='@(1) 35 PUT(%A100+2*I)=&(0) 36 PUT(%A101+2*I)=&(1) 37 PUT(%A120+2*I)=&(2) 38 PUT(%A121+2*I)=&(3) 40 NEXT I 50 RUN%IE 90 STOP >OK 212 >LO.%5C >OK >L. \lO‘hWNH 8 9 10 12 14 20 >OK REM READ/WRITE TR=D0,BL=C0 CODE STARTING=E0 IF F0=1 G.7 RUN%SE FOR I=0T0255;PUT(E0+I)=&(20+I);NEXT I STOP P.'WARNING THIS SUBROUTINE WILL ERASE WHAT IS CURRENTLY' P.'ON THE TAPE. ENTER 1 TO CONTINUE' INPUT F0 IF F0#1G.20 FOR I=0T0255;&(20+I)=GET(E0+I);NE.I RUN%52 STOP >LO.%5E >OK >L. REM READS @(10)-@(137) FROM TAPE FROM TRDO,BLCO RUN%5A;H=%14;RUN%54 H=(%30+D0);C.%5880 H=%0D;C.%5880;RUN%55;RUN%56 H=%12;RUN%54 FOR I=0 TO 2;H=(%30+@(I));C.%5880;NEXT I H=%2C;C.%5880;H=%30;C.%5880;H=%0D;C.%5880;RUN%55 FOR I=0 TO 255 C.%5840;H=IN(10);IFH#7G.40 C.%5840;P=IN(10);IFP=7G.40 P.'ERROR CODE DETECTED =‘,P;G.99 &(20+I)=H NEXT I RUN%56 ST. APPENDI X C Listing of Selected Data t = time, hours T = temperature of air, Celsius RH = relative humidity, decimal Mi = initial moisture content, decimal d. b. ID = (test number)*2 + tray number it M(t) T RH Mi ID Test 6 A .158333 .467 30.1 .266 .500 12. .175278 .465 30.5 .261 .500 12. .192222 .460 30.9 .257 .500 12. .259722 .449 31.3 .251 .500 12. .343333 .429 31.5 .250 .500 12. .427500 .416 31.6 .248 .500 12. .511389 .398 31.7 .246 .500 12. .594722 .393 31.6 .249 .500 12. .678889 .379 31.7 .248 .500 12. .763056 .368 31.6 .247 .500 12. .846944 .358 31.7 .246 .500 12. .930556 .347 31.7 .247 .500 12. 1.014167 .335 31.7 .246 .500 12. 1.114722 .327 31.7 .247 .500 12. 1.198056 .317 31.7 .247 .500 12. 1.298889 .307 31.7 .247 .500 12. 1.381667 .301 31.7 .248 .500 12. 1.481667 .287 31.8 .246 .500 12. 1.582500 .279 31.7 .247 .500 12. 1.683333 .273 31.8 .247 .500 12. 1.784167 .260 31.8 .245 .500 12. 1.885000 .256 31.8 .246 .500 12. 1.985833 .253 31.8 .248 .500 12. 2.086111 .244 31.8 .247 .500 12. 2.203611 .236 31.8 .247 .500 12. 2.303611 .231 31.8 .246 .500 12. 2.421944 .223 31.7 .248 .500 12. 2.523611 .219 31.7 .247 .500 12. 2.640833 .215 31.6 .248 .500 12. 2.757222 .209 31.6 .247 .500 12. 2.874722 .204 31.7 .247 .500 12. 2.992222 .199 31.6 .250 .500 12. 3.126667 .195 31.6 .247 .500 12. 3.244444 .193 31.6 .247 .500 12. 3.377500 .186 31.6 .248 .500 12. 3.511944 .184 31.7 .249 .500 12. 3.646111 .180 31.6 .248 .500 12. 3.780833 .176 31.6 .249 .500 12. 3.915556 .177 31.6 .248 .500 12. 213 4.066944 4.218333 4.368611 4.517500 4.669722 4.837222 5.005556 5.173333 5.357778 5.541944 5.726111 5.910833 6.111944 6.314167 6.532500 6.750278 6.983889 7.219444 7.455556 7.722500 7.974722 8.261111 8.545833 8.849444 9.169167 9.521111 9.873889 10.257778 10.662222 11.080833 11.549167 12.054167 12.590556 13.196389 13.853333 14.589167 15.429722 16.388611 17.394167 18.403333 19.397222 20.391667 21.391111 22.402222 23.395000 24.402500 25.394167 26.401667 27.388889 28.396111 29.401944 30.391111 31.397500 .173 .169 .163 .161 .162 .159 .158 .150 .150 .147 .147 .140 .142 .143 .138 .137 .136 .134 .133 .130 .131 .126 .126 .126 .124 .123 .125 .117 .120 .122 .117 .119 .116 .112 .115 .109 .106 .110 .110 .108 .104 .104 .103 .108 .102 .099 .101 .098 .099 .094 .096 .096 .096 31.6 31.6 31.6 31.6 31.6 31.6 31.6 31.6 31.5 31.6 31.6 31.6 31.6 31.6 31.6 31.6 31.6 31.6 31.6 31.6 31.7 31.6 31.8 31.7 31.9 31.8 31.8 31.8 31.8 31.9 32.0 31.9 32.0 31.9 31.9 32.0 32.0 31.9 32.1 32.1 32.1 32.2 32.3 32.3 32.3 32.2 31.6 31.5 31.4 31.4 31.4 31.4 31.3 214 .248 .248 .247 .248 .248 .248 .247 .248 .249 .249 .249 .248 .247 .249 .249 .248 .249 .247 .249 .248 .247 .248 .246 .246 .247 .246 .245 .245 .245 .244 .242 .244 .244 .245 .244 .244 .245 .244 .243 .242 .244 .242 .239 .240 .240 .243 .250 .249 .251 .251 .249 .251 .252 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 12. 32.403611 33.393611 34.401667 35.395000 36.405000 37.399167 Test 6 B .041666 .109166 .193611 .260277 .344166 .428055 .511666 .595833 .680277 .764166 .847500 .931111 1.014722 1.115000 1.198611 1.298333 1.381944 1.482500 1.583611 1.684444 1.785000 1.886111 1.985833 2.086944 2.203889 2.305277 2.423611 2.524166 2.640555 2.758055 2.875555 2.993055 3.127777 3.244166 3.378333 3.512500 3.647500 3.782222 3.916389 4.068333 4.218333 4.366944 4.519166 4.670277 4.838333 .094 .092 .092 .092 .092 .093 .276 .267 .256 .249 .243 .238 .234 .228 .224 .221 .217 .214 .212 .208 .204 .201 .198 .197 .194 .192 .189 .185 .184 .181 .179 .175 .175 .172 .170 .168 .166 .164 .162 .160 .157 .157 .154 .153 .151 .150 .148 .147 .144 .144 .142 31.3 31.2 31.2 31.2 31.2 31.3 30.1 31.2 31.4 31.5 31.6 31.6 31.7 31.8 31.6 31.7 31.8 31.7 31.8 31.7 31.7 31.8 31.8 31.8 31.9 31.9 31.7 31.8 31.7 31.8 31.7 31.7 31.7 31.6 31.6 31.7 31.6 31.7 31.6 31.6 31.7 31.6 31.6 31.6 31.6 31.6 31.6 31.6 31.6 31.6 31.7 215 .252 .254 .253 .253 .252 .251 .266 .255 .250 .250 .250 .250 .248 .249 .248 .246 .246 .248 .248 .247 .246 .247 .246 .245 .245 .246 .248 .246 .246 .247 .248 .248 .247 .248 .247 .247 .250 .247 .247 .248 .249 .247 .248 .247 .248 .248 .247 .248 .247 .249 .249 .500 .500 .500 .500 .500 .500 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 12. 12. 12. 12. 12. 12. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 5.006389 5.174166 5.358333 5.541944 5.727500 5.911389 6.113611 6.314444 6.532777 6.750555 6.984722 7.220277 7.456389 7.706944 7.975833 8.260555 8.547777 8.850833 9.168611 9.522500 9.873889 10.259166 10.663055 11.081389 11.550555 12.055277 12.591944 13.197500 13.854444 14.590277 15.430833 16.389722 17.395555 18.403333 19.398889 20.393333 21.392222 22.402222 23.395555 24.403611 25.394444 26.402222 27.389722 28.396944 29.402500 30.391666 31.397777 32.403611 33.394166 34.402777 35.396111 36.390277 37.400277 .141 .140 .138 .137 .135 .134 .133 .132 .130 .130 .127 .126 .126 .124 .123 .121 .120 .119 .120 .118 .116 .114 .118 .114 .112 .110 .110 .108 .106 .107 .105 .104 .103 .101 .101 .099 .099 .098 .097 .095 .095 .094 .092 .093 .094 .093 .091 .090 .089 .091 .090 .091 .088 31.5 31.6 31.5 31.6 31.6 31.6 31.6 31.6 31.6 31.6 31.6 31.7 31.7 31.6 31.6 31.7 31.7 31.7 31.8 31.9 31.8 31.8 32.0 31.9 31.9 31.9 31.9 31.9 31.9 31.9 31.9 32.1 32.0 32.2 32.1 32.1 32.2 32.4 32.2 32.1 31.6 31.4 31.5 31.3 31.5 31.4 31.3 31.3 31.2 31.2 31.2 31.3 31.3 216 .249 .248 .250 .250 .247 .247 .248 .247 .249 .247 .248 .248 .247 .248 .248 .246 .246 .246 .246 .246 .246 .245 .244 .245 .244 .245 .245 .244 .244 .245 .244 .244 .242 .242 .244 .242 .241 .238 .241 .243 .249 .249 .249 .251 .251 .251 .252 .254 .254 .253 .252 .253 .252 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 .283 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. 13. Test 8 A .125000 .141667 .192222 .259444 .343056 .426944 .511389 .595833 .680278 .765000 .849444 .933611 1.018056 1.102500 1.203889 1.288333 1.389722 1.491111 1.575556 1.676667 1.777778 1.879167 1.980278 2.081667 2.199722 2.301389 2.419722 2.521389 {2.639167 2.757500 2.875278 2.993333 3.127778 3.246111 3.381111 3.516389 3.651111 3.786111 3.921389 4.073056 4.208056 4.359722 4.511389 4.680278 4.848611 5.000000 5.185833 5.354444 5.540000 5.725278 5.910556 .266 .268 .263 .260 .249 .249 .240 .234 .232 .224 .224 .219 .214 .209 .209 .206 .204 .197 .195 .195 .189 .185 .183 .182 .178 .174 .172 .173 .173 .170 .169 .167 .165 .164 .158 .159 .155 .154 .153 .150 .151 .148 .150 .147 .146 .142 .142 .143 .138 .139 .139 31.6 32.1 32.5 32.7 32.9 33.0 33.0 33.0 33.1 33.1 33.1 33.1 33.1 33.1 33.1 33.1 33.2 33.1 33.1 33.1 33.1 33.2 33.2 33.1 33.1 33.1 33.2 33.1 33.2 33.2 33.4 33.3 33.3 33.3 33.2 33.2 33.2 33.2 33.1 33.2 33.1 33.1 33.1 33.1 33.1 33.1 33.1 33.1 33.1 33.0 33.1 217 .551 .543 .531 .522 .520 .516 .516 .513 .512 .512 .512 .513 .512 .512 .513 .512 .511 .512 .510 .510 .510 .510 .511 .510 .511 .511 .512 .510 .512 .510 .504 .506 .505 .507 .510 .510 .509 .508 .509 .511 .511 .510 .510 .511 .511 .514 .512 .510 .513 .512 .512 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 6.113056 6.315000 6.534167 6.753056 6.971667 7.208056 7.460833 7.713611 7.983333 8.253611 8.556667 8.860556 9.180556 9.517222 9.871111 10.259722 10.647778 11.086111 11.558056 12.046944 12.603889 13.195000 13.854167 14.597778 15.425278 16.400833 17.394722 18.389722 19.401667 20.396667 21.401389 22.391111 23.396944 24.403889 25.394167 26.399167 27.388611 28.393611 29.400000 30.405000 31.392500 32.393611 33.395833 34.399722 35.403333 36.390833 37.395278 Test 8 B .041666 .108889 .192500 .259722 .135 .135 .139 .133 .135 .132 .128 .131 .130 .128 .129 .128 .127 .127 .126 .123 .126 .120 .124 .122 .122 .121 .119 .119 .120 .122 .122 .120 .115 .116 .110 .112 .113 .110 .111 .109 .109 .114 .110 .108 .110 .110 .106 .112 .109 .109 .109 .572 .550 .528 .512 33.1 33.1 33.1 33.0 33.0 33.0 32.9 32.9 33.0 32.9 33.0 33.0 33.0 33.1 33.0 32.9 33.0 32.9 33.1 33.1 32.9 32.9 32.8 32.9 32.9 33.0 33.1 33.2 33.3 33.4 33.5 33.5 33.6 33.6 33.7 33.7 33.7 33.7 33.6 33.8 33.7 33.7 33.8 33.7 33.6 33.6 33.7 31.6 32.5 32.7 32.9 218 .510 .513 .511 .512 .514 .515 .516 .515 .515 .516 .514 .513 .513 .512 .513 .516 .515 .517 .515 .512 .516 .517 .517 .514 .516 .514 .513 .510 .507 .505 .502 .500 .501 .498 .498 .500 .497 .498 .501 .495 .497 .497 .495 .497 .501 .500 .498 .551 .531 .522 .520 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .281 .576 .576 .576 .576 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 16. 17. 17. 17. 17. .343611 .428055 .512500 .596944 .681666 .766111 .850277 .934722 1.019166 1.103611 1.205000 1.289444 1.391111 1.492222 1.576389 1.677777 1.779166 1.880277 1.981389 2.082777 2.201111 2.302500 2.421111 2.522500 2.640277 2.758055 2.876389 2.994166 3.128889 3.246944 3.382222 3.517500 3.652222 3.787222 3.922222 4.074166 4.208889 4.360555 4.512222 4.681111 4.832777 5.001111 5.170277 5.355555 5.540833 5.726111 5.911666 6.113611 6.316111 6.534722 6.753889 6.972777 7.209166 .497 .479 .464 .448 .433 .419 .406 .394 .383 .370 .357 .347 .336 .326 .316 .306 .296 .288 .280 .273 .265 .256 .249 .244 .237 .233 .229 .224 .217 .213 .205 .201 .196 .193 .188 .185 .180 .176 .175 .170 .167 .165 .162 .160 .158 .155 .154 .152 .148 .147 .145 .145 .143 33.0 33.0 33.0 33.1 33.1 33.1 33.1 33.1 33.1 33.1 33.1 33.2 33.1 33.1 33.2 33.2 33.2 33.2 33.2 33.1 33.1 33.2 33.2 33.1 33.2 33.2 33.3 33.3 33.2 33.3 33.2 33.2 33.2 33.2 33.2 33.2 33.2 33.2 33.1 33.1 33.1 33.1 33.2 33.1 33.1 33.0 33.1 33.1 33.0 33.0 33.1 33.0 32.9 219 .516 .516 .513 .512 .512 .512 .513 .512 .512 .512 .512 .511 .511 .510 .510 .511 .511 .510 .510 .511 .511 .512 .511 .512 .510 .509 .507 .507 .508 .508 .509 .509 .509 .508 .510 .509 .510 .509 .512 .511 .510 .511 .510 .512 .513 .514 .514 .511 .513 .514 .513 .513 .517 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. l7. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 7.461666 7.714444 7.984444 8.254444 8.557777 8.861666 9.181389 9.518055 9.872222 10.243889 10.649166 11.087222 11.558889 12.048055 12.605000 13.196389 13.855000 14.598889 15.426666 16.401944 17.395833 18.390555 19.402500 20.396389 21.401944 22.391666 23.398333 24.404722 25.394444 26.399444 27.389444 28.393055 29.400555 30.405000 31.392500 32.393055 33.395555 34.400555 35.403055 36.391389 37.395277 Test 9 A .125000 .141944 .191389 .258333 .342500 .426667 .510556 .595000 .679167 .763333 .141 .139 .138 .136 .135 .135 .133 .133 .132 .129 .128 .127 .126 .125 .125 .124 .124 .122 .123 .122 .122 .124 .120 .118 .119 .117 .116 .116 .116 .116 .115 .115 .116 .116 .114 .114 .115 .113 .114 .112 .115 .551 .550 .540 .527 .509 .485 .468 .453 .435 .424 33.0 33.0 32.9 33.0 33.0 33.0 33.0 33.0 33.0 33.0 33.0 32.9 33.1 33.1 33.0 32.9 32.9 32.9 32.9 33.0 33.1 33.1 33.3 33.3 33.4 33.6 33.5 33.7 33.7 33.7 33.7 33.6 33.6 33.7 33.7 33.8 33.7 33.7 33.6 33.6 33.6 35.7 37.4 38.7 39.2 39.4 39.6 39.8 39.9 39.9 40.0 220 .516 .518 .516 .515 .516 .514 .516 .514 .515 .515 .515 .517 .511 .513 .513 .517 .520 .516 .517 .514 .511 .512 .509 .507 .503 .501 .501 .500 .497 .498 .498 .501 .501 .496 .497 .496 .500 .499 .501 .499 .500 .585 .538 .505 .494 .489 .484 .478 .476 .475 .474 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .576 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 17. .17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 17. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. .847778 .931944 1.016389 1.117500 1.201944 1.286389 1.387500 1.488889 1.573056 1.674444 1.775278 1.876389 1.977778 2.095833 2.196944 2.298056 2.416389 2.534444 2.635833 2.753889 2.871944 3.006944 3.124722 3.242500 3.377500 3.512500 3.647500 3.781944 3.917222 4.069444 4.221389 4.373333 4.525278 4.676944 4.845833 5.014167 5.183056 5.351944 5.536944 5.722500 5.908056 6.110556 6.313056 6.532778 6.751944 6.971389 7.207778 7.461389 7.714722 7.984722 8.254722 8.557500 8.860833 .407 .394 .378 .364 .353 .342 .327 .315 .306 .298 .284 .274 .267 .255 .247 .243 .233 .225 .221 .216 .206 .202 .196 .192 .189 .182 .178 .175 .174 .169 .164 .160 .159 .158 .152 .151 .148 .144 .143 .142 .141 .140 .139 .137 .134 .131 .130 .132 .128 .128 .126 .126 .126 40.1 40.1 40.2 40.2 40.2 40.2 40.2 40.2 40.4 40.3 40.3 40.3 40.4 40.4 40.3 40.4 40.4 40.4 40.4 40.4 40.3 40.4 40.3 40.4 40.5 40.6 40.5 40.5 40.6 40.5 40.5 40.6 40.5 40.5 40.5 40.4 40.4 40.5 40.5 40.5 40.5 40.4 40.5 40.5 40.5 40.5 40.6 40.6 40.6 40.6 40.6 40.6 40.6 221 .473 .469 .469 .468 .468 .467 .467 .467 .466 .465 .465 .465 .464 .463 .464 .463 .464 .463 .462 .466 .465 .466 .466 .464 .461 .461 .462 .462 .461 .460 .463 .459 .461 .462 .462 .465 .464 .463 .460 .462 .461 .463 .461 .462 .462 .460 .462 .458 .460 .458 .457 .459 .459 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 9.181944 9.520278 9.874444 10.245278 10.649722 11.088056 11.543611 12.049722 12.589444 13.195278 13.852778 14.595556 15.438333 16.397500 17.393056 18.402778 19.394722 20.405000 21.399722 22.395000 23.390556 24.402222 25.397222 26.391944 27.403889 28.396944 29.393056 30.405000 31.400833 32.395000 33.390556 34.402778 35.396389 36.391389 37.402222 Test 9 B .041667 .108056 .191944 .259167 .343333 .427222 .511667 .595833 .680000 .764444 .848611 .933056 1.017500 1.118611 1.203056 1.287222 .128 .125 .126 .126 .123 .120 .122 .120 .116 .114 .116 .116 .112 .113 .111 .112 .110 .109 .110 .109 .110 .108 .107 .106 .109 .107 .107 .110 .108 .108 .108 .105 .108 .106 .110 .296 .286 .275 .267 .258 .252 .248 .241 .235 .230 .225 .221 .217 .211 .209 .206 40.6 40.5 40.6 40.7 40.8 40.7 40.7 40.6 40.6 40.5 40.1 40.1 40.1 40.0 40.0 40.1 40.2 40.4 40.4 40.6 40.6 40.4 40.3 40.2 40.0 40.0 39.9 39.9 39.7 39.8 39.6 39.6 39.7 39.5 39.5 35.7 38.7 39.2 39.4 39.6 39.8 39.9 39.9 40.0 40.1 40.1 40.2 40.2 40.2 40.2 40.2 222 .458 .461 .460 .458 .453 .457 .457 .460 .459 .460 .471 .473 .473 .471 .471 .470 .468 .464 .462 .460 .459 .463 .467 .470 .472 .476 .475 .477 .481 .478 .481 .479 .482 .485 .485 .585 .505 .493 .489 .484 .478 .476 .475 .474 .473 .469 .469 .468 .468 .467 .467 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .572 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 18. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 1.388611 1.489722 1.574167 1.675000 1.776389 1.877500 1.978889 2.096667 2.198056 2.299167 2.417500 2.535556 2.636944 2.755000 32.873056 3.007778 3.125556 3.243611 3.378611 3.513333 3.648056 3.783056 3.918333 4.070556 4.222778 4.357500 4.526111 4.678056 4.846389 5.015000 5.184167 5.352500 5.537778 5.723611 5.909167 6.111389 6.314167 6.533333 6.753333 6.972500 7.208889 7.462500 7.715833 7.985556 8.255833 8.558333 8.861944 9.183333 9.521389 9.875556 10.246389 10.650278 11.088889 .201 .199 .194 .191 .188 .185 .183 .178 .177 .174 .171 .168 .167 .164 .161 .159 .157 .155 .152 .152 .150 .149 .146 .144 .144 .142 .140 .140 .137 .137 .136 .134 .133 .132 .130 .129 .128 .128 .127 .126 .125 .124 .123 .122 .121 .120 .120 .120 .119 .119 .118 .118 .118 40.2 40.4 40.3 40.3 40.3 40.4 40.4 40.3 40.4 40.4 40.4 40.4 40.4 40.4 40.4 40.4 40.4 40.4 40.5 40.5 40.5 40.5 40.5 40.5 40.5 40.6 40.5 40.5 40.4 40.4 40.4 40.5 40.5 40.5 40.5 40.5 40.5 40.5 40.6 40.5 40.6 40.5 40.6 40.6 40.7 40.7 40.6 40.6 40.7 40.6 40.7 40.7 40.7 223 .467 .466 .465 .466 .464 .464 .465 .464 .462 .462 .464 .462 .464 .465 .464 .464 .466 .464 .463 .462 .461 .459 .459 .461 .460 .460 .462 .462 .463 .462 .464 .462 .461 .460 .462 .460 .462 .462 .461 .461 .459 .462 .459 .458 .458 .457 .457 .458 .459 .458 .456 .455 .457 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 11.544722 12.050556 12.590000 13.196111 13.853889 14.596389 15.422500 16.398611 17.393889 18.402778 19.395833 20.388889 21.400833 22.395833 23.391667 24.403333 25.398333 26.393056 27.404722 28.398056 29.393889 30.389167 31.401389 32.396111 33.391389 34.403889 35.397222 36.392500 37.403056 Test 12 B .058333 .107778 .191389 .258611 .342500 .426389 .510556 .594444 .678056 .762222 .845833 .929722 1.013333 1.114167 1.198056 1.298889 1.382500 1.483333 1.584167 1.685000 1.785556 1.886389 .115 .114 .114 .112 .111 .110 .110 .108 .109 .107 .108 .106 .105 .104 .104 .104 .104 .103 .103 .103 .104 .102 .103 .102 .102 .102 .101 .103 .101 .286 .280 .272 .268 .262 .258 .254 .250 .246 .242 .240 .238 .234 .233 .231 .226 .225 .222 .219 .216 .216 .214 40.7 40.6 40.6 40.2 40.1 40.1 40.1 40.0 40.0 40.1 40.2 40.3 40.3 40.5 40.7 40.3 40.3 40.2 40.1 40.0 39.9 39.8 39.7 39.8 39.6 39.7 39.6 39.5 39.5 24.8 24.8 24.9 24.9 24.9 25.0 24.9 24.9 24.9 24.9 24.9 24.9 24.9 24.9 24.9 25.0 25.0 24.9 24.9 25.0 25.0 25.0 224 .457 .457 .460 .468 .470 .470 .469 .477 .475 .474 .470 .466 .464 .461 .459 .464 .467 .469 .472 .474 .477 .477 .479 .476 .482 .482 .482 .486 .484 .508 .510 .505 .505 .505 .505 .505 .505 .504 .505 .504 .502 .503 .504 .504 .503 .503 .504 .504 .503 .502 .502 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .300 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 1.986667 2.087222 2.204722 2.305556 2.423056 2.523889 2.641389 2.758889 2.876667 2.993889 3.128333 3.245833 3.380278 3.514444 3.648611 3.782778 3.916944 4.068056 4.218889 4.370000 4.520833 4.671667 4.839444 5.007500 5.175278 5.360000 5.544444 5.729167 5.913889 6.115556 6.316667 6.534722 6.752778 6.970833 7.205833 7.457778 7.709167 7.977500 8.262500 8.547222 8.849167 9.168056 9.520833 9.873333 10.258889 10.661389 11.080833 11.551111 12.052500 12.606111 13.193333 13.847500 14.585833 .211 .207 .206 .205 .202 .201 .198 .197 .195 .192 .191 .190 .188 .186 .184 .183 .181 .179 .179 .176 .175 .172 .172 .170 .169 .167 .165 .164 .163 .162 .161 .160 .159 .157 .155 .155 .154 .152 .152 .150 .149 .148 .146 .146 .145 .144 .143 .143 .140 .139 .137 .137 .137 25.0 25.0 25.0 24.9 24.9 25.0 25.0 24.9 25.0 25.0 24.9 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 24.9 25.0 24.9 25.0 25.0 25.0 25.1 25.1 25.2 25.2 25.3 25.4 25.5 225 .504 .504 .502 .503 .503 .503 .503 .504 .502 .503 .503 .503 .503 .504 .504 .503 .503 .502 .504 .503 .502 .503 .503 .502 .502 .502 .501 .500 .502 .503 .502 .503 .503 .502 .503 .502 .502 .502 .504 .503 .503 .502 .504 .503 .501 .501 .502 .501 .498 .498 .496 .493 .491 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 15.424722 16.398333 17.388611 18.395278 19.403056 20.393056 21.400000 22.389444 23.393611 24.399167 25.388611 26.395278 27.402778 28.393611 29.400000 30.390278 31.396389 32.403333 33.394167 34.401944 35.391944 36.398611 37.405278 Test 13 A .108333 .125000 .191389 .257778 .340833 .424722 .508056 .591389 .674167 .758056 .841944 .925556 1.025278 1.107778 1.208611 1.292500 1.393333 1.477222 1.578056 1.678611 1.779444 1.880278 1.981111 2.081944 2.199444 2.300278 2.417778 2.535278 .136 .135 .132 .130 .130 .130 .128 .129 .129 .128 .128 .126 .124 .124 .127 .124 .124 .126 .124 .123 .124 .124 .126 .300 .297 .292 .286 .282 .284 .276 .274 .270 .266 .263 .260 .257 .258 .255 .252 .253 .251 .251 .250 .249 .245 .244 .240 .238 .242 .236 .234 25.5 25.4 25.5 25.4 25.4 25.4 25.5 25.4 25.4 25.4 25.3 25.3 25.4 25.3 25.4 25.4 25.3 25.3 25.2 25.2 25.2 25.3 25.3 20.0 18.5 17.5 17.4 17.3 17.2 17.3 17.3 17.4 17.4 17.4 17.4 17.4 17.5 17.4 17.4 17.3 17.2 17.3 17.3 17.3 17.2 17.2 17.2 17.3 17.3 17.2 17.2 226 .491 .491 .491 .493 .493 .492 .491 .493 .491 .492 .493 .494 .494 .492 .492 .494 .495 .495 .496 .497 .497 .495 .494 .445 .480 .509 .511 .513 .516 .514 .512 .511 .509 .511 .509 .508 .512 .510 .510 .511 .515 .511 .513 .512 .514 .514 .513 .513 .512 .513 .513 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .294 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 25. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 2.635833 2.753056 2.870833 3.005000 3.122500 3.256667 3.374167 3.508333 3.642500 3.776944 3.927778 4.061944 4.213333 4.364444 4.515278 4.683333 4.834444 5.002222 5.186667 5.354444 5.538889 5.723889 5.908056 6.109444 6.310833 6.528889 6.746667 6.981667 7.216389 7.451389 7.719444 ‘7.987778 8.256389 8.558611 8.860556 9.179444 9.514722 9.866944 10.253056 10.656111 11.092500 11.544722 12.047500 12.600556 13.187778 13.859167 14.597222 15.436389 16.392222 17.397778 18.404167 19.394444 20.401389 .234 .231 .229 .227 .226 .224 .226 .226 .224 .215 .210 .211 .208 .210 .206 .203 .201 .200 .198 .199 .198 .198 .195 .194 .194 .188 .186 .188 .186 .182 .181 .179 .179 .175 .172 .172 .171 .168 .168 .165 .164 .162 .161 .158 .156 .156 .152 .145 .146 .146 .145 .148 .141 17.2 17.3 17.2 17.3 17.3 17.2 17.3 17.3 17.3 17.3 17.4 17.5 17.4 17.5 17.5 17.4 17.5 17.5 17.5 17.6 17.6 17.6 17.6 17.6 17.6 17.6 17.7 17.6 17.6 17.7 17.6 17.7 17.7 17.6 17.5 17.5 17.5 17.4 17.4 17.4 17.4 17.3 17.4 17.4 17.3 17.3 17.3 17.3 17.2 17.1 17.1 17.2 17.2 227 .513 .512 .513 .513 .512 .512 .513 .511 .512 .511 .509 .507 .508 .506 .507 .507 .506 .506 .505 .503 .504 .503 .503 .503 .501 .503 .501 .503 .502 .503 .503 .502 .501 .504 .506 .505 .506 .508 .508 .508 .510 .510 .509 .508 .510 .511 .512 .512 .515 .515 .515 .514 .515 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 21.391111 22.396667 23.402778 24.393056 25.399444 26.403889 27.392778 28.399167 29.404167 30.393611 31.399722 32.389444 33.395556 34.402500 35.391944 36.398611 37.404444 Test 14 B .041667 .107500 .190556 .256667 .338889 .421667 .505833 .589444 .673611 .757500 .841389 .925278 1.026111 1.109444 1.210000 1.294167 1.395000 1.478889 1.579444 1.680278 1.781111 1.881667 1.982222 2.082778 2.200278 2.301111 2.418611 2.536111 2.636389 2.753889 2.871111 3.005278 3.123056 3.256944 .145 .142 .144 .146 .143 .142 .143 .138 .145 .139 .138 .138 .140 .137 .136 .134 .137 .254 .247 .240 .235 .232 .226 .224 .222 .220 .218 .214 .213 .210 .208 .204 .203 .201 .198 .196 .194 .193 .191 .189 .187 .187 .185 .182 .180 .178 .178 .176 .173 .172 .170 17.4 17.6 17.7 17.7 17.6 17.6 17.5 17.5 17.5 17.5 17.5 17.5 17.5 17.4 17.5 17.4 17.4 25.3 25.3 25.2 25.3 25.3 25.3 25.3 25.2 25.3 25.3 25.3 25.3 25.3 25.3 25.3 25.3 25.3 25.3 25.3 25.2 25.2 25.2 25.2 25.1 25.2 25.1 25.1 25.1 25.1 25.0 25.0 25.0 24.9 25.0 228 .508 .501 .500 .501 .503 .505 .506 .506 .505 .507 .507 .506 .505 .508 .507 .508 .508 .400 .399 .399 .400 .397 .400 .397 .399 .397 .398 .397 .398 .397 .398 .397 .398 .398 .397 .399 .399 .399 .401 .400 .401 .400 .401 .400 .401 .402 .403 .405 .404 .405 .405 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .309 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 26. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 3.374444 3.508611 3.642778 3.776944 3.927778 4.061944 4.212778 4.363611 4.514444 4.682222 4.833333 5.001111 5.185278 5.352778 5.537222 5.721944 5.923333 6.108056 6.326111 6.527778 6.746111 6.980833 7.215556 7.467500 7.719444 7.987778 8.256389 8.558611 8.860278 9.179167 9.514722 9.867222 10.252778 10.655833 11.091944 11.545000 12.047222 12.601111 13.188611 13.859722 14.598889 15.436111 16.391944 17.398333 18.395000 19.397500 20.404722 21.394167 22.401667 23.392500 24.398611 25.404722 26.392500 .169 .169 .166 .165 .163 .162 .161 .161 .160 .158 .156 .155 .154 .152 .152 .151 .151 .149 .147 .148 .146 .144 .143 .144 .142 .142 .139 .138 .138 .137 .136 .135 .135 .134 .132 .133 .131 .130 .128 .127 .126 .125 .125 .123 .121 .122 .119 .120 .119 .117 .117 .116 .115 24.9 25.0 24.9 24.9 25.0 25.0 25.0 25.0 25.1 25.1 25.0 25.1 25.1 25.1 25.0 25.0 25.0 25.0 25.1 25.0 25.0 24.9 25.0 24.9 24.9 24.9 25.0 25.0 25.0 25.0 25.0 25.0 25.0 24.9 25.0 24.9 24.8 25.0 25.2 25.2 25.2 25.2 25.2 25.2 25.2 25.1 25.1 25.1 25.3 25.5 25.5 25.2 25.0 229 .406 .404 .405 .405 .404 .403 .403 .403 .402 .402 .403 .402 .401 .402 .403 .403 .403 .403 .402 .403 .404 .405 .404 .406 .405 .405 .404 .404 .405 .404 .404 .404 .403 .406 .404 .405 .406 .403 .401 .400 .400 .400 .400 .400 .401 .403 .401 .401 .397 .394 .395 .399 .404 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 27.399444 28.389444 29.396389 30.402778 31.393056 32.400278 33.389167 34.395833 35.402778 36.392778 37.399444 Test 18 A .219722 .236389 .253055 .269722 .336666 .423055 .506666 .590000 .673611 .757500 .840833 .924722 1.025000 1.108611 1.208889 1.292500 1.392777 1.476389 1.576389 1.676944 1.777222 1.877222 1.977500 2.094722 2.195000 2.312222 2.412222 2.529444 2.646389 2.763333 2.880000 2.997222 3.131111 3.248055 3.381666 3.515555 3.649444 3.783333 3.916944 4.067500 .115 .114 .111 .112 .111 .110 .110 .109 .109 .109 .109 .221 .220 .217 .216 .211 .205 .199 .194 .191 .186 .183 .179 .176 .172 .170 .167 .163 .162 .159 .157 .154 .152 .149 .148 .145 .143 .142 .139 .138 .135 .134 .133 .131 .129 .128 .126 .124 .124 .122 .122 24.9 24.9 24.9 25.1 24.9 24.9 25.0 24.9 24.9 25.0 25.0 37.5 37.8 38.0 38.2 38.7 39.1 39.3 39.4 39.5 39.6 39.6 39.7 39.8 39.8 39.7 39.8 39.8 39.8 39.9 39.9 39.8 39.8 39.8 39.7 39.8 39.9 39.8 39.9 39.8 39.8 39.8 39.8 39.7 39.8 39.9 39.7 39.8 39.8 39.8 39.8 230 .405 .406 .405 .403 .406 .406 .405 .406 .405 .403 .402 .299 .295 .291 .289 .282 .276 .275 .272 .272 .269 .269 .269 .267 .267 .269 .269 .267 .267 .266 .268 .267 .268 .268 .268 .266 .266 .267 .267 .268 .268 .267 .268 .268 .268 .267 .268 .268 .267 .267 .268 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .255 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 4.217777 4.368055 4.518611 4.685833 4.836389 5.003611 5.170833 5.354444 5.538333 5.722500 5.922777 6.106389 6.323611 6.524166 6.758055 6.975000 7.209166 7.459444 7.710555 7.978055 8.262500 8.546666 8.847500 9.181666 9.516111 9.866666 10.251389 10.652500 11.087222 11.555277 12.056666 12.591666 13.193333 13.861666 14.597222 15.432222 16.401389 17.404166 18.390555 19.393889 20.396666 21.399166 22.401389 23.404166 24.389444 25.392222 26.395277 27.397500 28.401389 29.404166 30.390000 31.392777 32.395000 .119 .119 .118 .116 .115 .114 .114 .112 .112 .109 .109 .109 .108 .107 .106 .105 .104 .104 .102 .101 .100 .100 .099 .098 .097 .096 .095 .094 .093 .092 .092 .090 .090 .089 .088 .086 .086 .086 .085 .084 .083 .082 .081 .081 .080 .080 .078 .079 .078 .077 .077 .077 .078 39.8 39.8 39.8 39.8 39.8 39.9 39.8 39.9 39.8 39.7 39.8 39.8 39.8 39.7 39.7 39.6 39.6 39.8 39.8 39.8 39.8 39.8 39.8 39.7 39.8 39.8 39.8 39.8 39.7 39.8 39.8 39.7 39.7 39.8 39.8 39.6 39.4 39.6 39.6 39.8 40.0 40.0 40.2 40.1 39.8 39.8 39.9 39.8 39.7 39.8 39.9 39.8 39.9 231 .268 .266 .267 .267 .268 .267 .267 .266 .268 .268 .268 .267 .267 .270 .270 .271 .270 .268 .268 .269 .268 .268 .267 .268 .268 .269 .268 .268 .268 .267 .268 .269 .268 .267 .269 .270 .272 .270 .271 .268 .265 .266 .263 .264 .266 .268 .267 .267 .269 .268 .266 .266 .266 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 .241 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 33.397777 34.401389 35.404722 36.391389 37.394722 Test 18 E .103056 .119722 .186667 .256389 .340000 .423611 .506944 .590556 .674167 .757778 .841389 .925000 1.025278 1.109167 1.209167 1.292778 1.393056 1.476667 1.576944 1.677222 1.777222 1.877778 1.978056 2.095278 2.195556 2.312500 2.412778 2.529722 2.646667 2.763333 2.880556 2.997500 3.131389 3.248333 3.382222 3.516111 3.649722 3.783611 3.917500 4.067500 4.217778 4.368333 4.518889 4.686111 4.836944 5.003889 .077 .076 .077 .077 .076 .166 .164 .159 .156 .152 .150 .147 .145 .143 .141 .139 .138 .136 .135 .133 .131 .130 .129 .128 .126 .125 .123 .123 .121 .121 .120 .119 .118 .117 .116 .115 .114 .114 .112 .112 .111 .110 .110 .109 .108 .107 .107 .106 .105 .105 .104 39.9 39.9 39.8 39.9 39.9 37.5 37.8 38.5 38.9 39.1 39.3 39.4 39.5 39.6 39.6 39.7 39.7 39.8 39.8 39.8 39.8 39.9 39.9 39.8 39.9 39.8 39.8 39.7 39.8 39.9 39.9 39.9 39.8 39.8 39.8 39.8 39.7 39.8 39.9 39.8 39.7 39.8 39.8 39.8 39.8 39.8 39.8 39.8 39.8 39.8 39.8 232 .268 .267 .268 .267 .266 .299 .295 .284 .279 .277 .274 .272 .270 .270 .271 .269 .268 .268 .269 .267 .268 .266 .267 .266. .267 .266 .269 .268 .268 .266 .267 .267 .268 .268 .267 .268 .268 .268 .267 .268 .268 .268 .268 .268 .267 .268 .269 .267 .266 .268 .266 .241 .241 .241 .241 .241 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 36. 36. 36. 36. 36. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 5.171111 5.354722 5.538889 5.722222 5.922778 6.106667 6.323889 6.524444 6.758333 6.975556 7.209444 7.460000 7.710833 7.978611 8.262778 8.547222 8.848056 9.181944 9.516389 9.866944 10.251944 10.652778 11.087500 11.555556 12.056944 12.591944 13.193611 13.861667 14.597500 15.432778 16.401944 17.404722 18.390833 19.394167 20.396944 21.399444 22.401944 23.404444 24.389722 25.392500 26.395278 27.398056 28.401944 29.404722 30.390278 31.393333 32.395556 33.398056 34.401667 35.388611 36.391944 37.395278 .104 .103 .102 .101 .101 .100 .099 .099 .099 .098 .097 .097 .097 .096 .095 .094 .093 .093 .093 .092 .091 .091 .091 .089 .088 .088 .087 .086 .086 .084 .083 .083 .082 .081 .081 .081 .081 .080 .080 .079 .079 .078 .079 .078 .077 .077 .077 .076 .076 .076 .075 .076 39.8 39.8 39.8 39.9 39.7 39.8 39.7 39.6 39.7 39.6 39.7 39.7 39.8 39.8 39.7 39.8 39.8 39.8 39.8 39.8 39.7 39.7 39.8 39.7 39.7 39.8 39.7 39.7 39.7 39.7 39.5 39.5 39.7 39.8 40.0 40.0 40.1 40.1 39.8 39.9 39.9 39.8 39.8 39.8 39.9 39.9 39.9 39.8 39.9 39.8 39.8 39.9 233 .267 .266 .267 .267 .269 .268 .269 .269 .270 .271 .269 .268 .267 .267 .269 .268 .268 .269 .268 .267 .269 .269 .268 .269 .268 .268' .269 .269 .268 .269 .272 .272 .268 .269 .265 .265 .263 .263 .268 .266 .268 .268 .267 .268 .266 .266 .267 .266 .268 .268 .268 .266 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 .178 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. 37. Test 19 A .125000 .141389 .190556 .256667 .340556 .424167 .507222 .590833 .674722 .758333 .841944 .925278 1.025556 1.109167 1.209167 1.292500 1.392778 1.476389 1.576111 1.676389 1.776944 1.877222 1.977778 2.094722 2.194722 2.311667 2.411944 2.528611 2.645556 2.762500 2.879444 2.996389 3.130000 3.246667 3.380278 3.513611 3.647222 3.780556 3.930833 4.064722 4.215000 4.365556 4.515556 4.682778 4.833333 5.000000 5.183889 5.350833 5.534722 5.718333 5.918889 .232 .230 .226 .219 .213 .207 .202 .197 .194 .190 .186 .183 .180 .176 .173 .170 .168 .165 .164 .160 .158 .156 .154 .151 .149 .147 .146 .142 .142 .140 .138 .137 .135 .133 .132 .131 .129 .127 .126 .125 .124 .123 .122 .121 .119 .119 .117 .116 .115 .114 .113 33.9 34.8 36.1 36.9 37.3 37.7 37.8 37.9 38.0 38.0 38.0 38.0 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.2 38.1 38.2 38.2 38.1 38.0 38.0 38.1 38.1 38.1 38.1 38.1 38.2 38.1 38.1 38.1 38.2 38.0 38.1 37.9 38.0 38.0 38.0 38.0 38.0 38.0 38.1 38.2 38.1 38.1 234 .380 .360 .327 .318 .313 .309 .309 .309 .308 .307 .306 .307 .307 .307 .305 .307 .306 .306 .306 .305 .305 .307 .305 .306 .307 .307 .308 .307 .306 .307 .306 .306 .306 .305 .306 .305 .305 .307 .307 .309 .307 .308 .308 .307 .308 .306 .307 .306 .307 .305 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 6.119444 6.320278 6.537500 6.755000 6.971944 7.206111 7.456667 7.707222 7.975000 8.258611 8.542778 8.859444 9.176389 9.510278 9.877778 10.244722 10.661944 11.079722 11.547222 12.048333 12.599722 13.201111 13.852222 14.587222 15.422222 16.391389 17.392500 18.394722 19.396667 20.398611 21.401111 22.404167 23.389444 24.401111 25.404722 26.390833 27.392222 28.394167 29.396944 30.400000 31.403056 32.388889 33.391667 34.394167 35.395833 36.398333 37.400556 Test 19 B .075000 .107778 .190278 .256944 .113 .112 .110 .110 .109 .107 .107 .106 .105 .103 .103 .103 .102 .101 .100 .100 .099 .098 .097 .096 .096 .095 .094 .094 .092 .091 .090 .089 .089 .088 .086 .086 .086 .086 .084 .083 .082 .083 .083 .081 .081 .081 .081 .080 .079 .079 .079 .169 .166 .161 .158 38.0 38.2 38.1 38.0 38.0 38.0 38.1 38.1 38.1 38.1 38.1 38.2 38.0 38.0 38.1 38.1 38.1 38.0 38.0 38.0 38.0 38.1 38.1 38.1 38.0 38.1 38.0 38.0 38.1 38.0 38.0 38.1 38.1 38.0 38.1 38.0 38.0 37.8 37.7 37.8 37.9 37.5 38.2 38.1 38.0 38.0 38.0 33.9 35.3 36.7 37.3 235 .307 .306 .306 .307 .308 .309 .306 .305 .306 .307 .307 .306 .307 .308 .307 .305 .306 .307 .307 .308 .308 .306 .307 .306 .308 .305 .306 .307 .307 .308 .307 .306 .307 .306 .306 .306 .308 .310 .311 .311 .309 .315 .305 .306 .306 .309 .308 .380 .351 .330 .320 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .242 .172 .172 .172 .172 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 38. 39. 39. 39. 39. .340556 .424167 .507500 .591111 .674722 .758611 .842222 .925278 1.025833 1.109167 1.209167 1.292778 1.392778 1.476389 1.576389 1.676667 1.777222 1.877500 1.977778 2.094722 2.194722 2.311944 2.412222 2.528889 2.645556 2.762778 2.879444 2.996389 3.130000 3.246944 3.380278 3.513611 3.647222 3.780556 3.931111 4.064722 4.215000 4.365556 4.515833 4.683056 4.833611 5.000278 5.183889 5.350833 5.535000 5.718611 5.919167 6.119722 6.320278 6.537778 6.755000 6.972222 7.206111 .155 .153 .151 .149 .147 .145 .144 .143 .141 .140 .138 .137 .135 .134 .133 .132 .132 .130 .128 .128 .126 .125 .124 .123 .122 .122 .121 .120 .119 .118 .117 .116 .116 .115 .114 .113 .113 .113 .112 .111 .110 .110 .109 .108 .107 .107 .105 .106 .105 .104 .103 .103 .103 37.6 37.7 37.8 38.0 38.1 38.1 38.0 38.1 38.0 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.2 38.1 38.1 38.1 38.0 38.0 38.0 38.0 38.1 38.1 38.1 38.2 38.1 38.1 38.2 38.2 38.2 38.0 38.1 38.1 38.1 38.0 38.1 38.0 38.0 38.0 38.0 38.1 38.1 38.0 38.1 38.1 38.1 38.0 38.0 38.1 236 .314 .311 .309 .309 .308 .308 .307 .306 .308 .306 .306 .306 .306 .306 .307 .307 .306 .306 .307 .306 .307 .306 .308 .307 .308 .306 .307 .307 .305 .306 .306 .304 .305 .305 .307 .308 .307 .307 .308 .306 .307 .306 .306 .307 .306 .307 .306 .307 .307 .306 .308 .308 .307 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 7.456944 7.707500 7.975000 8.258611 8.542778 8.859722 9.176389 9.510000 9.877500 10.245000 10.662222 11.079722 11.547500 12.048333 12.600000 13.200833 13.852222 14.587500 15.422500 16.391389 17.392500 18.394722 19.396944 20.398889 21.401111 22.404444 23.389444 24.401111 25.404722 26.391111 27.392500 28.394444 29.396944 30.400278 31.403056 32.389167 33.391667 34.394167 35.396111 36.398333 37.400556 Test 20 B .058333 .107778 .189722 .271389 .337500 .420278 .503611 .587222 .670833 .753611 .102 .101 .100 .100 .100 .099 .098 .098 .097 .097 .095 .095 .094 .094 .093 .092 .092 .092 .090 .089 .089 .088 .087 .087 .085 .085 .085 .085 .084 .084 .083 .082 .082 .082 .082 .080 .080 .080 .080 .080 .080 .253 .247 .239 .234 .230 .226 .223 .219 .217 .214 38.0 38.1 38.0 38.2 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.1 38.0 38.0 38.0 38.1 38.0 38.0 38.0 38.0 38.0 38.0 38.1 38.0 38.1 38.2 38.2 38.1 38.0 38.0 38.0 37.9 37.8 37.8 37.9 37.5 38.2 38.1 38.0 38.0 38.0 25.4 25.8 25.9 26.0 25.9 26.0 26.0 26.0 26.1 26.1 237 .308 .306 .307 .305 .307 .305 .305 .308 .307 .306 .307 .306 .307 .309 .307 .305 .308 .307 .308 .306 .307 .307 .307 .309 .306 .304 .305 .306 .307 .306 .310 .310 .310 .311 .308 .314 .305 .306 .308 .309 .308 .494 .486 .481 .478 .479 .477 .478 .478 .476 .476 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .172 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 39. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. .836389 .935556 1.018333 1.117778 1.201111 1.300278 1.383056 1.482500 1.582778 1.816944 1.833889 1.883889 1.984444 2.084722 2.201667 2.301944 2.418611 2.535556 2.636111 2.752778 2.886111 3.003056 3.120000 3.253889 3.387500 3.504722 3.638333 3.788611 3.922222 4.072778 4.206667 4.356944 4.523889 4.673889 4.841111 5.008333 5.175556 5.359444 5.543333 5.727222 5.911111 6.111389 6.311944 6.529444 6.746944 6.980556 7.214167 7.464722 7.715000 7.982500 8.266944 8.550833 8.851944 .211 .208 .205 .203 .201 .198 .196 .194 .192 .188 .188 .187 .185 .184 .182 .180 .179 .176 .175 .173 .172 .170 .169 .167 .165 .164 .163 .162 .160 .159 .157 .156 .155 .154 .153 .152 .151 .150 .148 .147 .146 .145 .144 .143 .142 .141 .139 .139 .139 .137 .136 .135 .134 26.0 26.1 26.0 26.1 26.0 26.0 26.1 25.9 26.0 26.1 26.1 26.2 26.3 26.3 26.4 26.4 26.5 26.4 26.3 26.4 26.3 26.3 26.3 26.3 26.4 26.3 26.3 26.5 26.5 26.4 26.4 26.3 26.4 26.5 26.4 26.2 26.4 26.5 26.4 26.4 26.3 26.1 26.1 26.1 26.2 26.1 26.1 26.1 26.5 26.3 26.2 26.3 26.3 238 .477 .476 .476 .476 .476 .476 .476 .481 .478 .475 .475 .473 .470 .469 .467 .467 .464 .468 .469 .469 .468 .471 .469 .468 .469 .470 .470 .465 .466 .466 .468 .468 .468 .465 .468 .471 .469 .465 .467 .467 .469 .474 .475 .473 .474 .475 .474 .474 .464 .469 .472 .471 .471 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 9.169444 9.520278 9.871389 10.255556 10.656667 11.090833 11.558333 12.059167 12.594444 13.195278 13.846944 14.599444 15.432778 16.393889 17.393333 18.396389 19.398889 20.401944 21.404444 22.390278 23.392500 24.393056 25.395556 26.398333 27.398611 28.400000 29.402222 30.388889 31.391389 32.394167 33.394444 34.395278 35.398611 36.400000 37.402778 .134 .132 .131 .130 .130 .128 .128 .126 .126 .125 .124 .122 .121 .120 .120 .119 .118 .118 .116 .115 .114 .114 .113 .113 .113 .112 .111 .111 .110 .110 .110 .109 .108 .109 .108 26.3 26.1 26.2 26.1 26.3 26.2 26.4 26.3 26.4 26.3 26.3 26.3 26.4 26.4 26.4 26.3 26.5 26.3 26.3 26.4 26.3 26.2 26.3 26.3 26.6 26.4 26.2 26.3 26.3 26.5 26.4 26.4 26.3 26.4 26.4 239 .471 .473 .472 .474 .470 .473 .467 .468 .468 .469 .468 .469 .467 .470 .468 .469 .466 .468 .470 .467 .469 .471 .470 .468 .462 .468 .472 .470 .469 .465 .466 .469 .468 .468 .467 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 .258 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 41. 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