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This course is intended to utilize the basic numerical methods in problems which are appropriate to a variety of food engineering subject areas.The problems are titled according to the food engineering principles which are used; the problems are arranged according to the numerical methods which are applied as summarized in Table 0.This selection of problems should help food engineering department evaluate which mathematical problem solving package they wish to use in their courses and should provide some typical problems in various courses which can be utilized.Table 0Selection of Problems Solutions Illustrating Mathematical SoftwareWDChapterMathematical modelSoftwareFood Engineering Application0.September 29Introduction1.October 3IntroductionBasic operationsUsing functions Decision FunctionsChartsExcelFluid mechanicsReaction kinetics2.October 13Solving EquationSolution of nonlinear algebraic equation. Bisection method, The Newton-Raphson method, Goal-seek toolSolution of simultaneous equations. Cramer’s Rule, Matrix inversion ExcelThermodynamicsSteady state material balances3. October 20Numerical Differentiation and IntegrationNumerical differentiation Forward, backward and central derivative approachFinite Difference methodEuler’s methodNumerical integration Trapezoid ruleSimpson one-third ruleExcelDryingSimple batch distillationOctober 27First Midterm Exam4.November 3OptimizationLinear programmingNon-linear programmingGoal-seek toolExcel5.November 10Curve FittingLinear regressionNon-linear regressionGoal-seek toolExcel- Sigma plotDrying curve7.November 17Basic StatisticsDescriptive statisticsFrequency distribution Analysis of varianceExcelSensory evaluation8.November 24Statistical Process ControlControl chartsSampling plansProcess capabilityExcelQuality controlDecember 1Second Midterm Exam9.December 8Experimental DesignFactorial designResponse surfaceModelingOptimizationDesignexpertExperimental design10. December 15Process simulationMaterial balancingEconomic analysisSuperpro designerINTRODUCTIONBasic operationsExercise 1. Filling in a Series of NumbersProblem statementImagine we have a laboratory heating apparatus with a thermometer calibrated in degrees Fahrenheit. We need a table to give approximate Celsius values. On completion of the exercise, the worksheet will resemble that in a table.Exercise 2. References: Relative, Absolute and MixedProblem statementApproachTo demonstrate the use of mixed references, we will develop a simple worksheet that displays a multiplication table as shown in a table.Note:Reference Result when formula is copied=A1 the row and the column may change=A$1 the row remains constant, the column may change=$A1 the column remains constant, the row may change=$A$l both the row and the column remain constantHow to Link Cells in Microsoft Excel? 1. To link between cells in the same worksheet, click a new cell and type the equal sign. Click the original cell to which you would like to link. The original cell's location reference is displayed in the formula bar and the value matching the original cell is displayed in the cell.2. To link cells between worksheets, right-click the cell to which you would like to link. Select Copy. Click the desired linked cell. Right-click and select Paste Special. Click Paste Link. The value in the original cell appears in this second cell.3. To link cells between workbooks, open all of the workbooks that will be used to link cells. Click the target cell in the dependent document, which is the workbook in which the information will be updated. Type the equal sign and click the cell in the source workbook, which contains the cell that will be linked. Press Enter on your keyboard.How to Invert Excel Spreadsheets?Users can invert columns and rows of data in a Microsoft Excel spreadsheet without retyping the information. Excel's "Transpose" function allows users to transpose what they copy and paste.InstructionsHighlight your data set.Click "Paste" (or "Edit" in some versions). Select "Paste Special."Check "Transpose" in the dialog box that opens, then click "OK."Using functions Exercise 3. Auto sum and Auto CalculateProblem statementExercise 4. Trigonometric FunctionsProblem statementIn this exercise we experiment with some of the trigonometric functions which occur in many physical problems. These include: SIN, COS and TAN and their inverses ASIN, ACOS and ATAN.Exercise 5: Exponential FunctionsProblem statementDesign a worksheet to show that:(a) =EXP(2) returns e*.(b) =LN(5) returns the natural logarithm of 5.(c) =LOG1 0(5), =LOG(5, I O ) and =LOG(5) all return the logarithm of 5 to base 10.(d) LOG(8,2) returns the value 3, which is the logarithm of 8 to base 2.Exercise 6: Rounding FunctionsExcel provides a number of functions which either truncate or round a value to a required number of digits or to a multiple of some number.Table 1.5. Some rounding functionsApproachReturns the absolute value.=ABS(- 12.55) returns 12.55.Rounds a number down to the nearest integer – cf TRUNC.=INT(-5.6) returns -6.Rounds a number to the nearest even integer.=EVEN(3.25) returns 4.Rounds a number to the nearest odd integer.=ODD(4.25) returns 5.Rounds a number down (towards zero) to the nearest multiple of significance - cf CEILING.=FLOOR( 1.255,0.5) returns 1 .O.Rounds a number to the required number of places.=ROUND( 1.378,l) returns 1.4 (one decimal)=ROUND(123.56,- 1) returns 120 (nearest 10)=ROUND( 123.56,O) returns 124 (nearest integer)Decision FunctionsThe IF function is used when you want a formula to return different results depending on the value of a conditionFigure 1. IF functionExercise 7.Quadratic Equation SolverProblem statementApproachIn this exercise we design a worksheet to solve a quadratic equation in the form ax2 + bx + c = 0 using the quadratic formula: Where Disc=Exercise 8. Horizontal LookupProblem statementApproachFor the purpose of this exercise, a geologist wishes to grade some ore samples based on their rare metal content. Ore with 50 to 59 ppm is to be given a low grade, from 60 to 79 merits a medium ranking, from 80 to 99 is considered high and anything above that is very high.Exercise 9. Vertical LookupProblem statementApproachIn this example, UCL and LCL of x-and R charts gives the limits of any variable data. These limits, of course, depend on the number of observations. The average of averages (x) and average of range (R) are given 42.94 and 2.73 respectively. If number of observation (n) is 7, calculate the UCL and LCL values for both charts. Exercise 10. Conditional Summing and CountingProblem statementExercise 11. Fluid Flow and Reynolds NumberIn problems involving fluid flow, the magnitude of Reynolds number is used as an indicator of the flow conditions. For example, if the Reynolds number is less than 2100, the flow is called laminar. A Reynolds number greater than 10,000 indicates a turbulent flow. Values of Reynolds number between 2100 and 10,000 signify transitional flow. In this worksheet, we will develop a table of Reynolds numbers for flow of water through a pipe. This table will be created with the input of the diameter and the water velocity. The results will be useful to determine at what velocity the flow becomes turbulent.Exercise 12. Friction Factors for Water Flow in a PipeIn designing pumping systems, we require values for friction factors for the liquid flow in a pipe. The friction factors are generally obtained from a diagram commonly referred to as Moody diagram. When the water flow in a pipe is within the laminar region (Reynolds number < 2100), the friction factor varies linearly with Reynolds number. In the turbulent regime, the relationship between friction factor and Reynolds number is nonlinear. In this example, we will use appropriate equations to calculate friction factor from Reynolds number and use the calculations to construct a chart similar to the Moody diagram.ChartsExercise 13. Determining Rate Constants of Zero-Order Reactions636343597679962640759771596203955945119614671580219959947959177595992275133607957723151351559Problem statementApproach25345656782492332965772929261124512193291697565804609189736512110492366805120564927617251078569255760510656092609805907209237976510148492379405912609221128511354492182125108612919870051090449207448510580492029125107928918368859637291322805104040913649259662491150005117936911194051132569739245129600982960512330095232451193409345045115956979252510440098468859903698159251033929666525927009362325848889Exercise 14. First-Order Rate Constants and Half-Life of ReactionsProblem statementApproach12847351730124Exercise 15. GANTT Charts & Project Planning using MS Excel (A bar chart with a difference)Problem statementDataThe techniques learned in this exercise may be applied to make a variety of Bar charts including critical-path (GANTT) charts.TasksStartDuration (days)FinishPlan course04Instructional design5Stroyboarding2Graphics10Simulation3Integration2Testing4Release1Exercise 16. How to create Pareto Chart using MS Excel? (Application of pareto analysis in traditional sucuk processing)Problem statementDataA survey study was done to collect customer complaints about the properties of sucuk products. According to Pareto analysis; the percentage of complaints are listed by the descending order and cumulative percentages are obtained by adding each other to previous ones. The priority problems are being identified by this order.Customer complaints# of complaintsOff flavor30Rancid20Off color10Poor texture8Too oily7Too salty4Without salt3Opening bag1SOLVING EQUATIONSIn this section we examine methods of finding roots of non-linear equations such as polynomial (3x3 – 7x2 - 22x + 40 = 0) and transcendental (exp(-x) - sin(x) = 0). If the equation is written as f(x) then a root of the equation is a value of x such that f(x) = 0. The value of x is sometimes called the zero of the function. Some equations may be solved analytically. The quadratic formula, for example, is used to find the roots of a quadratic equation. With other equations the analytical method may be very complex or not exist at all. In these cases we may use numerical methods to find approximate roots. One should also remember the usefulness of graphing a function to determine the number and values of its roots.Closed Methods-The Bisection MethodExercise 17. Determining roots by The Bisection methodProblem statementApproachTo demonstrate how we may implement this algorithm in Excel, we shall find the roots of the function exp(-x) - sin(x). Figure shows a plot of this function for values of x from 0 to 4. Step 1. Choose lower xl and upper xu guesses for the root such that the function changes sign over the interval. This can be checked by ensuring that f(xl)f(xu)<0.Step 2. An estimating of the root xr is determined by xr=xl+xu2Step 3. Make the following evaluations to determine in which subinterval the root lies:If f(xa)f(xr)<0, the root lies in the lower subinterval. Therefore, set xu=xr and return to step 2.If f(xa)f(xr)>0, the root lies in the upper subinterval. Therefore, set xl=xr and return to step 2.If f(xa)f(xr)=0, the root equals xr and terminate the computationOpen Methods-The Newton-Raphson MethodExercise 18. Determining roots by The Newton-Raphson MethodProblem statementApproachUse the The Newton-Raphson Method to estimate the root of f(x)=e-x-x employing an initial guess of x0=0. The Newton-Raphson method can be derived on basis of this geometrical interpretation. The first derivative ta x is equivalent to the slope:f'(xi)=fxi-0xi-xi+1which can be rearranged to yieldxi+1=xi-fxif'(xi)which is called the Newton-Raphson formula.Excel’s Goal Seek Tool Exercise 19. Using Excel’s Goal Seek Tool to Locate a Single RootProblem statementApproachEmploy goal seek to determine the root of the transcendental functionfx=x-cos?(x)Step 1.→Exercise 20. Using Excel’s Goal Seek Tool to Locate Multiple RootProblem statementEmploy goal seek to determine the roots of the transcendental functionfx=-1.04lnx-1.26cosx+0.0307exExercise 21. Using Excel’s Goal Seek Tool for a non-linear systemProblem statementEmploy goal seek to determine the roots of the transcendental functionux,y=x2+xy-10=0vx,y=y+3xy2-57=0Solving Simultaneous Equations-Matrix AlgebraCramer's RuleAccording to Cramer's rule, a system of simultaneous linear equations has a unique solution if the determinant D of the coefficients is nonzero. To obtain the solution, each unknown is expressed as a quotient of two determinants: the denominator is D and the numerator is obtained from D by replacing the column in the determinant corresponding to the desired unknown with the column of constants.Thus, for example, for the set of equations2x + y - z = 0x - y + z = 6x + 2y + z = 3the determinant isThe coefficients and constants lend themselves readily to spreadsheet solution. Using the formula =MDETERM(AZ:C4), the value of the determinant is found to be -9, indicating that the system is soluble.Spreadsheet data for three equations in three unknowns.The determinant for obtaining x.For example, to obtain x, the determinant is shown above , and x = 2 is obtained from the formula=MDETERM(A8:C10)/MDETERM(A2:C4)y = -1 and z = 3 are obtained from appropriate forms of the same formula.Exercise 22. Solving Equations Systems of Linear EquationsProblem statementSolve the following system of four simultaneous equations:Solving Simultaneous Equations by Matrix InversionExercise 23. Solving Equations Systems of Linear EquationsProblem statementApproachExercise 24. Solving Equations Systems of Linear EquationsProblem statementApproachExercise 25. Solving Equations Systems of Linear EquationsProblem statementApproachNUMERICAL DIFFERENTIATIONThe simplest method to obtain the first derivative of a function represented by a table of x, y data points is to calculate x and y, the differences between adjacent data points, and use y/x as an approximation to dy/dx.Exercise 26. Problem statementFor the function y=1/(1+x2) create a table of function values with additional columns for the first and second derivatives.Finite Difference MethodThe derivative of a function f at a point x is defined by the limitExercise 27. Derivative of a Worksheet Formula Calculated by Using the Finite-Difference MethodProblem statementThe first derivative of a function y = x3 - 3x2 - 130x + 150 by evaluating the function at x and at x + x. Here a value of x of 1 x 10-8 was used. For comparison, the firstderivative was calculated from the exact expression from differential calculus:F(x) = 3x2 - 6x - 130.Exercise 28. Euler’s MethodEuler developed a method for finding the approximate solution to initial value problems. Let the differential equation to be solved have the form of equation;and let the initial value of y be y0. Let the solution (i.e. the integral of above equation) have the formof equation; New value=old value+slope*step sizeyi+1=yi+f(xi,yi)*hProblem statementUse Euler’s method to numerically integratedydx=-2x3+12x2-20x+8.5From x=0 to x=4 with a step size of 0.01, 0.1, and 0.5. The initial condition at x=0 is y=1.NUMERICAL INTEGRATIONNumerical integration is used to evaluate a definite integral when there is no closed-form expression for the integral or when the explicit function is not known and the data is available in tabular form only. Numerical integration (or quadrature) consists of methods to find the approximate area under the graph of the function f(x) between two x values. Trapezoid Rule;The simplest of these uses the trapezoid rule. If we divide the area under the curve into a sufficiently large number of parts, then the area under the curve (the approximate integral) is given by:Exercise 29. The Trapezoid RuleProblem statementSimpson one-third ruleA better approximation to the integral is obtained by taking two adjacent strips and joining the three points on the curve with a parabola. This gives below equation, called the Simpson one-third rule for approximating the area under a curve.This rule requires that there be an even number of equally spaced strips. Exercise 30. The Simpson one-third RuleProblem statementApproximating the curve through four adjacent points to a cubic equation gives the Simpson 3/8 rule shown in below equation.Surprisingly, the 3/8 rule is often less accurate than the 1/3 rule. However, unlike the 1/3 rule, it does not require an even number of strips. This is an advantage when the data is available only in tabular form (the explicit function being unknown) and there is an even number of data points - an odd number of strips.OPTIMIZATIONIn mathematics, computer science and operations research, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a defined domain (or a set of constraints), including a variety of different types of objective functions and different types of domains. Figure 1. The difference between roots and optimaFigure 2. One and multi dimentional optimizationsGolden-Section SearchNewton’s MethodCURVE FITTING "Curve fitting" is frequently used in scientific or engineering applications to obtain the coefficients of a mathematical model that describes experimental data.Linear RegressionLinear regression is not limited to the case of finding the least-squares slope and intercept of a straight line. Linear regression methods can be applied to any function that is linear in the coefficients. Many functions that produce curved x-y plots are linear in the coefficients, including power series, for example,y = a + bx + cx2 + dx3and some functions containing exponentials, such asy = aexSome mathematical models-formulas and their geometric shapesExpressionMathematical modelsGeometric shapeExpressionMathematical modelsGeometric shapeSigmoidal (Gompertz) Exponential growthSigmoidal (Lojistic) Peak (Guassian)Sigmoidal (Sigmoid) Peak (Lorentzian)Sigma (GAB)Polynomial (inverse first order)Polynomial (linear)Exponential decayPolynomial (Quadratic)Least-Squares Fit to a Straight LineThe least-squares line of best fit minimizes the sum of the squares of the y deviations of individual points from the line. This statistical technique is called regression analysis. Regression analysis in the simplest form assumes that all deviations from the line are the result of error in the measurement of the dependent variable y.One of the simplest examples of the method of least squares linear equation are fitted data set.This linear equation:where a0 and a1 are coefficients and e is error or deviation. Error; Sum of the errors for all of the experimental pointsn the number of data. the sum of the errors in the equation "0" emergence of differences between the model and the actual value of the data collection may be possible with negative and positive values??. To calculate the absolute differences in the nature of this remedy and the need to look at the total. In this case, equation is rearranged. Another step is the smallest state of the error sum of squares. This expression will be; The differences between the measured values ??and the model values ??and the sum of squares of the total error will continually show positive values. Errors in the model in order to find the sum of the coefficients a0 and a1 be found at least on the basis of the strategy will help to find these coefficients derivatives to be obtained. These derivatives; Sr in these derivatives to be the minimum "0" must have. In this case, equations are arranged again;Where; . If the equations rearranged;two equations with two unknowns are obtained. If these equations are solved; and Linear regression in the amount of error be discovered in other words, the model equation for the determination of the appropriateness of the data model, data and statistical analysis of measured data is required.Exercise 31. Least-Squares Fit to a Straight Line Using the Worksheet Functions SLOPE, INTERCEPT and RSQProblem statementThe data of the figure conforms to an experimental curve. It can be shown that the vapor pressure waries with the absolute temperature according to the Clausius-Clapeyron equation: Estimate the equation constantsExercise 32. Using “solver” in Excel for non-linear curve fittingWe will use Solver to obtain a best fit to a data set of ocean floor sedimentation rates. Let's work with these data of age of sediment versus depth below the seafloor:Can we figure out the sedimentation rate from these data? Yes, if we assume a functional form (e.g., linear, quadratic, etc). For sedimentation, let's assume the sedimentation rate stays constant, thus the equation should be linear, i.e., of the form y=mx + b. To analyze these data, we will solve this problem two ways:(1) Use?Add trendline?on a chart made from these sedimentation data(2) Use?Solver?to do the same thing.?Add trendlineCopy the above data into Excel. Make a chart of the data (XY scatter plot). Fit the data with a straight line, using Excel's built-in Add Trendline function. Choose the Linear option. Also, click the "Options" tab and choose to display the equation on the chart. Here is a picture of what that might look like (you'll notice that I changed the Y-axis range so that the data span is more appropriately represented in this dimension - always de-junk your chart!):SolverTo illustrate the concept of Solver, we'll solve the same problem by finding the m and b that minimizes the RMS (root mean square - a measure of the misfit of the model to the data). where?p?is the number of observations,?Hi?is the observed value at?xi, and?Hmodel(xi)?is the model (or calculated value) at?xi. Thus, if we can minimize this measure, the root of the square of the differences between model (prediction) and observation, then we will be getting a best fit according to our description of "goodness" of fit.To do so, follow these instructions:Somewhere on a clean worksheet (e.g., below your freshly copied data table), define two cells, one for slope "m" and another for the Y-intercept "b". Go ahead and stick some number in each (they'll get updated later) and define names for those cells: I recommend naming them m and b. Also, put your estimates in the sheet next to each other (view my spreadsheet in the next image in these notes).Set up a column of model age estimations that uses our guess of m ands b, call this Y_calc which equals m*x + b, where the x values are the depths of the observations, and m and b are our model parameters that describe our best fitting line (see Column C in image below).Determine the RMS between the observed and modeled values: make a column that calculates the difference between the observed ages (Y_obs) and those you just estimated from your guessed m and b (Y_calc) [my column D in image below].?Note that these differences are called the?residuals. In a column to the right of this (Column E), square that difference.Calculate the sum of those squares (thus, simply sum the values in column E), then take the square root of this number, then finally divide by the number of observations. NOTE: count the number of observations using Excel's function count(cell range). If your data set had 100's or 1000's of data, you'd surely do it this way. This is the RMS, and what we will minimize using Solver. Make sure you understand what we're doing: we are computing the misfit of each data point to your line, and then squaring this. Then... we sum these up, square root, divide by the number of observations. Thus, this is essentially a weighted average of our misfits of our model (i.e., the RMS). Hence, minimizing the RMS will minimize our model's misfit. Understand this.Finally, open Solver under the Tools menu, and set it up to make the target cell the one containing your estimation of the RMS (F9 in my sheet) a minimum (click the Min button) by varying the cells that contain m and b (the cell range that is dashed below).Click Solve, Then you will find that after the solver finishes searching for the best fit, it will ask you if you want to keep the result, which of course you do. It will also offer a report (usually on a separate worksheet), select the Answer sheet option.Finally: compare your answers for the best m and b values (i.e., the ones Solver obtained) to those calculated by the Add trendline function.Bonus: Use your final fitted equation to extrapolate to determine what age the sediments should be at 5000 cm, assuming the constant sedimentation rate.Exercise 33. Problem statementExercise 34. STATISTICSMicrosoft Excel is a powerful tool for statistical analysis. In this chapter we look at a very small subset of these tools. The main focus is on the treatment of variability associated with data measurements. Exercise 35. Descriptive StatisticsProblem statementApproachAn experimenter has collected 100 measurements and wishes to know some statistics of the set; for example, the average, the sum, etc. To save the task of entering 100 numbers we will have Excel generate some random numbers. The RAND or RANDBETWEEN functions are not appropriate since they generate uniform distributions of values. So we will use the Random Number generator tool. To simulate the results from an experiment, we will request random numbers with a mean (average) of 10 and a standard deviation of 0.5.Open a new workbook. In A1 of Sheet1 enter the label data. Use the command Tools-Data Analysis and select the item Random Number Generation.To generate the statistics quickly, we will use another Data Analysis tool, namely Descriptive Statistics. Complete the dialog boxThere is a worksheet function corresponding to all but two of the statistics generated by the Data Analysis tool.Exercise 36. Frequency DistributionProblem statementApproachFrequently an experimenter wishes to compare the distribution of experimental data with the normal Gaussian distribution. We will use the data generated in the previous exercise rounded to two decimal places.Use random data in Exercise pute the mean and the standard deviations of the data using the formulas =AVERAGE(data) and =STDEV(data)Enter the values in C5:C 17. These will serve as the bin values (from 8.5 to 11.5 with 0.25 intervals) or value intervals for the frequency formula.With D5:D17 selected, enter =FREQUENCY(data, C5:C17) and use [shift] +[CTRL] + [Enter] to complete the pute the sum of the frequencies with =SUM(DS:D17) in D18.To compute the expected frequencies use these formulas:E5: =$D$18 * NORMDIST(C5, mean, stdev, TRUE)E6: =$D$18 * (NORMDIST(CG,mean,stdev,TRUE) -NORMDIST(C5, mean, stdev, TRUE))Copy this down to row 17, and modify the formula in El 7 to:E17: =$D$18 (1 - NORMDIST(C16, mean, stdev, TRUE))Compute the sum ofthe expected frequencies with the formula=SUM(E5:E17) in E18.Exercise 37. Control ChartsControl charts are commonly used in food manufacturing processes to determine whether a process is operating under control. These charts are useful in finding ways to improve a process and assure production of satisfactory products. In this worksheet, we will use data obtained for some characteristics of a product obtained during a production run. We will create control charts from the given data. In addition, we will learn how to link to another worksheet and look up for some desired information.Problem statementApproachA manufacturing line for production of pie crust is being monitored for variations in the thickness of the pie crust. In total, 50 data points are available. Use 10 subgroups of 5 samples each to develop average and range charts. The data on thickness (mm) of the pie crust are shown in the following table.5.25485.25355.25395.25415.25365.25425.25375.25315.25215.25335.25465.25345.25365.25265.25315.25495.25385.25335.25225.25345.25485.25315.25375.25235.25335.25415.25395.25465.25345.25425.25495.25385.25495.25375.25465.25475.25395.25485.25395.25495.25435.25375.25465.25345.25465.25495.25395.25435.25335.2543First we will calculate average (X) and range (R) of each subgroup. Then we will calculate an average of averages (X) and average of the range values (R). The upper and lower limits for the averages and range are given by following expressions:UCL & LCLMeanX-ChartR-Chart(A2=0.577, D3=0, D4=2.115)Exercise 38. Analysis of Variance: One-Factor, Completely RandomizedDesignIn consumer testing, it is sometimes not possible to use the same judges for testing different treatments. Although, it would be desirable to use the same judges to evaluate samples obtained from different treatments, it may not be economically feasible. In such cases, we have a completely randomized design. Using single-factor ANOVA, we can test to see whether the treatments had any influence on the judges scores; in other words, does the mean of each treatment differ?Problem statementApproachConsider a sensory study where an engineered food with three different flavor compounds, A, B, and C was evaluated for the degree of liking of flavor on a 9-point hedonic scale. The following scores were obtained:For each treatment, 5 samples were evaluated by 5 judges. The same judges were not available for the three treatments. Therefore, the design was completely randomized. Calculate the F value to determine whether the means of three treatments are significantly different.We will use a single factor analysis of variance available in Excel. We will determine the F value at probability of 0.95 and 0.99. These computations will allow us to determine if the means between the three different treatments are significantly different. First make sure that the Data Analysis... command is available under menu item ToolsEXPERIMENTAL DESIGNFOOD PROCESS SIMULATIONFOOD ENGINEERING CALCULATIONSExercise 39. Drying Time CalculationProblem statementDefine the followings a. Define equilibrium moisture content.b. Explain the condition(s) at which drying stops (if hot air is used for drying)c. If you are given a set of data (moisture content vs time), then, how can you estimate drying time for constant and falling rate periods graphically?timeWeight (kg)X04,9440,2626830,44,8850,2470120,84,8080,226561,44,6990,197612,24,5540,15909734,4040,1192564,24,2410,07596354,150,05179374,0190,01699993,9780,006109123,9550Exercise 40. Simple Differential Distillation Problem statementA mixture of 100 ml containing 50 mol % n-pentane an 50 mol % n-heptane is distilled under differential conditions at 1 atm. until 40 mol is distilled. What is the average composition of total the total vapour distillation and the composition of the liquid left? (αAB=7.8) ................
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