Handbook of Food Process Modeling and Statistical Quality Control with Extensive MATLAB® Applications S E C O N D © 2011 by Taylor and Francis Group, LLC E D I T I O N © 2011 by Taylor and Francis Group, LLC Handbook of Food Process Modeling and Statistical Quality Control with Extensive MATLAB® Applications S E C O N D E D I T I O N Mustafa Özilgen Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business © 2011 by Taylor and Francis Group, LLC MATLAB® is a trademark of The MathWorks, Inc and is used with permission The MathWorks does not warrant the accuracy of the text or exercises in this book This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed in the United States of America on acid-free paper 10 International Standard Book Number: 978-1-4398-1486-4 (Hardback) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Library of Congress Cataloging‑in‑Publication Data Özilgen, Mustafa Handbook of food process modeling and statistical quality control / Mustafa Özilgen ‑‑ 2nd ed p cm Rev ed of: Food process modeling and statistical quality control 1998 Includes bibliographical references and index ISBN 978‑1‑4398‑1486‑4 (hardback) ‑‑ ISBN 978‑1‑4398‑1938‑8 (pbk.) Food industry and trade‑‑Production control‑‑Mathematical models Food industry and trade‑‑Quality control‑‑Statistical methods I Özilgen, Mustafa Food process modeling and statistical quality control II Title TP372.7.O95 2011 338.1’9‑‑dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 2011007241 To Sibel and Arda © 2011 by Taylor and Francis Group, LLC © 2011 by Taylor and Francis Group, LLC Contents Preface .ix Author xiii Introduction to Process Modeling 1.1 The Property Balance 1.2 What Is Process Modeling? 14 1.3 Empirical Models and Linear Regression 16 References 36 Transport Phenomena Models 39 2.1 The Differential General Property Balance Equation 39 2.2 Equation of Continuity 40 2.3 Equation of Energy 48 2.4 Equation of Motion 48 2.5 Theories for Liquid Transport Coefficients 49 2.5.1 Eyring’s Theory of Liquid Viscosity 49 2.5.2 Thermal Conductivity of Liquids 53 2.5.3 Hydrodynamic Theory of Diffusion in Liquids 53 2.5.4 Eyring’s Theory of Liquid Diffusion 54 2.6 Analytical Solutions to Ordinary Differential Equations 56 2.7 Transport Phenomena Models Involving Partial Differential Equations 70 2.8 Chart Solutions to Unsteady State Conduction Problems 101 2.9 Interfacial Mass Transfer 108 2.10 Correlations for Parameters of the Transport Equations 113 2.10.1 Density of Dried Vegetables 113 2.10.2 Specific Heat 113 2.10.3 Thermal Conductivity of Meat 114 2.10.4 Viscosity of Microbial Suspensions 115 2.10.5 Moisture Diffusivity in Granular Starch 116 2.10.6 Convective Heat Transfer Coefficients during Heat Transfer to Canned Foods in Steritort 116 2.10.7 Mass Transfer Coefficient k for Oxygen Transfer in Fermenters 117 2.11 Rheological Modeling 124 2.12 Engineering Bernoulli Equation 141 2.13 Laplace Transformations in Mathematical Modeling 151 2.14 Numerical Methods in Mathematical Modeling 158 References 183 Kinetic Modeling 187 3.1 Kinetics and Food Processing 187 3.2 Rate Expression 188 3.3 Why Do Chemicals React? 197 3.4 Temperature Effects on Reaction Rates 200 vii © 2011 by Taylor and Francis Group, LLC viii Contents 3.5 Precision of Reaction Rate Constant and Activation Energy Determinations 202 3.6 Enzyme-Catalyzed Reaction Kinetics 205 3.7 Analogy Kinetic Models 228 3.8 Metabolic Process Engineering 236 3.9 Microbial Kinetics 248 3.10 Kinetics of Microbial Death 269 3.11 Ideal Reactor Design 282 References 306 Mathematical Modeling in Food Engineering Operations 309 4.1 Thermal Process Modeling 309 4.2 Moving Boundary and Other Transport Phenomena Models for Processes Involving Phase Change 343 4.3 Kinetic Modeling of Crystallization Processes 389 4.4 Unit Operation Models 410 4.4.1 Basic Computations for Evaporator Operations 410 4.4.2 Basic Computations for Filtration and Membrane Separation Processes 424 4.4.3 Basic Computations for Extraction Processes 450 4.4.4 Mathematical Analysis of Distilled Beverage Production Processes 483 References 504 Statistical Process Analysis and Quality Control 509 5.1 Statistical Quality Control 509 5.2 Statistical Process Analysis 511 5.3 Quality Control Charts for Measurements 576 5.4 Quality Control Charts for Attributes 598 5.5 Acceptance Sampling by Attributes 608 5.6 Standard Sampling Plans for Attributes 620 5.7 HACCP and FMEA Principles .645 5.8 Quality Assurance and Improvement through Mathematical Modeling 656 References 677 Author Index 681 Subject Index 687 © 2011 by Taylor and Francis Group, LLC Preface It has been more than a decade since the first edition of this book appeared on shelves Paperback, hardbound, and e-book versions of the first edition were available in the market More than 130 Internet booksellers included the first book on their lists; I was more than happy with the welcome of the scientific community Students who used the first edition in their classes are now directors of major food establishments and, I am proud of them all The second edition developed by way of an opportunity that presented itself I taught classes at the Massey University in New Zealand; we established our own company in Ankara, Turkey I chaired the Chemical Engineering Department, Yeditepe University in Istanbul, where the most notable contributors were starting a PhD program and a food engineering department Teaching bioengineering classes to the genetic engineering students was one of my most exciting experiences Turkey has the 18th largest economy in the world and the food industry makes up a big part of it There are about 45 food engineering departments in Turkish Universities I was honored to be among the founders of the first and 39th departments The 39th department was the first food engineering department in a foundation (private) Turkish university I appreciate the contributions of Seda Genc, Fatih Uzun, and all of my undergraduate and graduate students, who helped to write the MATLAB® codes through their projects or homework I appreciate the help provided by Dr Esra Sorguven of the Mechanical Engineering Department of Yeditepe University, in the solutions of the examples involving the partial differential equation toolbox I also appreciate the author’s license from MATLAB® (MathWorks Book Program, A#: 1-577025751) The second edition of the book is substantially different from the first edition in the sense that i The title of the book is modified following the recommendations of experts from academia and the industry It is intended to present the book as a compendium of applications within its scope ii The new edition covers extensive MATLAB applications The model equations are solved with MATLAB and the resulting figures are generated by the code The models are compared mostly with real data from the literature Some errors occurred while reading the data; therefore, the model parameters sometimes had different values than those of the references iii Tabular values and plots of mathematical functions are produced through MATLAB codes iv All of the MATLAB codes are given on the CD accompanying the book A summary of the important features and functions of the MATLAB codes used in the book are given in Table 1.1 The readers may refer to this table to locate the functions or syntax they need They may copy lines from the examples and write their own code with them I wrote my own codes by following this procedure I would recommend achieving each task in the code in a stepwise manner and then going on to the next task Each task is usually defined in the examples with phrases such as ix © 2011 by Taylor and Francis Group, LLC 665 Statistical Process Analysis and Quality Control set(H2,’LineStyle’,‘+’) xlabel(‘Time (min)’) ylabel(‘ln(Hunter Lab color parameters L, a and b)’) set(get(AX(2),‘ylabel’), ‘string’, ‘Temperature ( \circ C)’) legend(‘L’,‘a’,‘b’,‘T’,’Location’,‘SouthEast’) 100 50 L a b T 10 20 30 40 50 Time (min) 60 70 Temperature (°C) ln (Hunter lab color parameters L, a and b) When we run the code the Figures E.5.50.1 and E.5.50.2 will appear on the screen 80 90 88 –5 Temperature (°C) ln (Hunter lab color parameters L, a and b) FIGURE E.5.50.1 Variation of the Hunter color parameters L, a, b, and temperature at the can center during thermal treatment of tomato paste L a b T 10 20 30 40 50 Time (min) 60 70 86 80 FIGURE E.5.50.2 Variation of the Hunter color parameters L, a, b, and temperature near the surface of the can during thermal treatment of tomato paste © 2011 by Taylor and Francis Group, LLC 666 Handbook of Food Process Modeling and Statistical Quality Control, Second Edition Example 5.51: Sensory Shelf-Life Predictions by Survival Analysis Hough, Garitta, and Gomez (2006) reported that the times required for unacceptable color development in ground beef samples may be expressed with a log-normal distribution model The standard normal variable of the rejection times is z ln(t ) = ln(t ) − ln(µ ) σ (E.5.51.1) Standard deviation σ is assumed to be independent of the temperature variations, but the populations mean rejection time µ is temperature dependent: E µ = µ exp a RT (E.5.51.2) MATLAB® code E.5.51 estimates the cumulative fraction of the rejected samples as a function of the storage time at two different temperatures MATLAB® CODE E.5.51 Command Window: clear all close all % enter the experimental data tData2=[24 48 96 144 192 240]; % experimentally determined failure times (h) at oC fData2=[0.06 0.16 0.59 0.795 0.86 0.87]; % experimentally determined fractions of failure at oC tData19=[6 12 18 24 36]; % experimentally determined fractions of failure at 19 oC fData19=[0.08 0.23 0.433 0.745 0.90]; % experimentally determined fractions of failure at 19 oC % plot the experimental data loglog(tData2,fData2, ‘x’); hold on loglog(tData19,fData19, ‘o’); hold on legend(‘ T= ^0C’, ‘ T=19 ^0C’,’Location’,’SouthEast’) xlabel(‘Storage Time’) ylabel(‘Fraction Failing’) % enter the model constants lnMu0=-24; % Beta0=muRef-Ea/(R Tref) Er=7820; % Er=Ea/R Temp2=2+273; % T=2 oC expressed in Kelvin Temp19=19+273; % T=19 oC expressed in Kelvin sigma=0.70; % plot the model t=[0:240]; % model times (h) lnMuTemp2=lnMu0+(Er/Temp2); % T=2 oC © 2011 by Taylor and Francis Group, LLC 667 Statistical Process Analysis and Quality Control fModel2=normcdf(log(t),lnMuTemp2,sigma); failures at oC loglog(t,fModel2,‘-’); hold on % model fractions of % plot the model at 19^0C lnMuTemp19=lnMu0+(Er/Temp19); % T=19 oC fModel19=normcdf(log(t),lnMuTemp19,sigma); % model fractions of failures at 19 oC loglog(t,fModel19,‘:’), hold on xlim([0 500]); ylim([0.02 1]); When we run the code Figure E.5.51 will appear on the screen Fraction failing 100 10–1 T = 2°C T = 19°C 100 101 Storage time 102 FIGURE E.5.51 Comparison of the model with the data Experimental data of the fraction failing were determined by visual inspection of the samples by a consumer panel (Adapted from Hough, G., Garitta, L., and Gomez, G., Food Quality and Preference, 17, 468–73, 2006.) Example 5.52: Optimization of the Freeze Drying Conditions During the freeze-drying process, on the interface the thermal energy balance may be expressed as (Lou and Zhou 2008): ∂z (t ) ∂T ∂T kcrust crust + crust (ρcrust c p ,crustTcrust − ρcorec p,coreTcore ) = − kcore core ∂ z z ,crust ∂ z z ,crust ∂t −V ( ∆Hphase change (E.5.52.1) + c p,vaporTphasechange ) Lou and Zhou (2008) assumed quasi steady state conditions on the interface, for example, (dTcrust /dz)z,crust = 0 and (dTcore/dz)z,core = 0 and integrated Equation E.5.52.1 to obtain an expression © 2011 by Taylor and Francis Group, LLC 668 Handbook of Food Process Modeling and Statistical Quality Control, Second Edition to compute the time required for complete freeze drying of an infinite slab Constants of this integrated theoretical expression were related with the physical properties and the experimental conditions, but, unfortunately the number of these constants were discouragingly high for reliable analysis Lou and Zhou (2008) preferred using the following empirical expression for optimization: t = 1.398L − 0.070P − 0.097T + 0.082L2 + 0.001P + 0.001T − 0.007LP − 0.016LT + 0.001PT + 2.074, (E.5.52.2) where t = freezing times (h), L = thickness of the slab (cm), P = pressure under which the freeze drying occurred (Pa), and T = temperature in the freeze-drying chamber (°C) Constants of Equation E.5.52.1 were determined from the experimental data by regression MATLAB® code E.5.52 computes variation of the model freeze-drying times with pressure, temperature, and thickness Figures E.5.52.1 and E.5.52.2 show that a decrease in the slab thickness, operating pressure, and temperature cause a decrease of the freeze-drying times Therefore, in the present example the shortest freeze-drying time is expected to be obtained when all of these parameters are maintained at their lowest attainable values MATLAB® CODE E.5.52 Command Window: clear all close all format compact L=[4:1:12]; P=[26:4:74]; % constant temperature [L,P]=meshgrid(L,P); for T=50:25:75 t=1.398.*L-0.070.*P-0.097.*T+0.082.*L.^2+0.001.*P.^2+0.001.*T.^ 2-0.007.*L.*P-0.016.*L.*T+0.001.*P.*T+2.074; surf(L,P,t); hold on colormap gray end xlabel(‘L (mm)’) ylabel(‘P (Pa)’) zlabel(‘t’) text(5,80,9, ‘T=75 ^o C’); % insert text to the mesh text(7,20,7, ‘T=50 ^o C’); % insert text to the mesh % constant pressure L=[4:2:12]; T=[53:4:77]; figure [L,T]=meshgrid(L,T); for P=26:48:74; % constant pressures (Pa) t=1.398.*L-0.070.*P-0.097.*T+0.082.*L.^2+0.001.*P.^2+0.001.*T.^ 2-0.007.*L.*P-0.016.*L.*T+0.001.*P.*T+2.074; © 2011 by Taylor and Francis Group, LLC 669 Statistical Process Analysis and Quality Control surf(L,T,t); hold on colormap gray end xlabel(‘L (mm)’) ylabel(‘T (\circC)’) zlabel(‘t (h)’) text(5,80,9, ‘P=74 Pa’); % insert text to the mesh text(5,50,4, ‘P=26 Pa’); % insert text to the mesh When we run the code Figures E.5.52.1 and E.5.52.2 will appear on the screen 20 15 t 10 T = 75°C 80 T = 50°C 60 P (Pa) 40 20 10 12 L (mm) FIGURE E.5.52.1 Variation of the model freeze-drying times with pressure and thickness at 50°C and 75°C 20 t (h) 15 10 P = 74 Pa 80 70 P = 26 Pa 60 T (°C) 50 10 12 L (mm) FIGURE E.5.52.2 Variation of the model freeze-drying times with temperature and thickness at 26 Pa and 74 Pa © 2011 by Taylor and Francis Group, LLC 670 Handbook of Food Process Modeling and Statistical Quality Control, Second Edition Example 5.53: A Desorption Model for Moisture Loss and Color Change on the Surface of the Potatoes During Frying Achir, Vitrac, and Tystram (2008) suggested an asymptotic desorption model for water content of a potato crust in a frying process as x(t ) = xsat − γ (T (t ) − Tsat ), (E.5.53.1) where x(t) = water content of potato surface at time = t (g water/g dry matter), xsat = water content of potato when the capillaries are saturated with (monolayer) water (g water/g dry matter), T(t) = temperature in the crust at time t, Tsat = temperature of the crust when the capillaries are saturated with adsorbed water (the frying process occurs when T > Tsat) and γ is a desorption constant After long mathematical derivations they have ended up with the following moisture loss expression: dw (t ) T − T w (t ) − w ∞ = − oil sat , α 1− w ∞ dt (E.5.53.2) where α = constant (1/°C s), w(t) = x(t)/x0 = reduced water content (g water at time t/g of water at t = 0), w∞ = x(t)/x∞ reduced water content when the water content of the crust is in equilibrium with the water content of the frying oil (g water when t → ∞/g of water at t = 0), Toil = temperature of the frying oil Convective heat balance on the surface (oil-potato interface) leads to the equation: − ∆Hphase change x0 dw = h (Toil − Tsurface ) dt (E.5.53.3) Where h = convective heat transfer coefficient on the crust-oil interface and ΔHphase change is the latent heat of evaporation of water Equation E.5.53.3 may be combined with Equation E.5.53.2 and rearranged to compute Tsurface (temperature of the crust at the crust-oil interface) as T − T w (t ) − w ∞ Tsurface = Toil − α oil sat , α 1− w ∞ (E.5.53.4) where α0 = h/ΔHphase change x0 The color of the crust changes during the frying process due to the Maillard reactions, where the reducing sugars (RS) react with amino acids (A) to produce Shiff’s base (AR), which produce brown pigments (B) after the reaction step named the Amadori rearrangement: RS + A k1 → ← k2 RA → AR → B (E.5.53.5) k−1 The color change of the surface may be assessed by amino acid consumption rate (Achir, Vitrac, and Tystram 2008): © 2011 by Taylor and Francis Group, LLC dc A = k surface c A2 , dt Surface (E.5.53.6) 671 Statistical Process Analysis and Quality Control where ksurface is the reaction rate constant evaluated at the temperature of the crust surface and expressed as E ksurface = kref exp − a − R Tsurface Tref (E.5.53.7) Equation E.5.53.7 is an Arrhenius type expression Where cA is the amino acid concentration, kref is the reaction rate constant at reference temperature Tref Equation E.5.53.6 will be solved as c A (t ) = c A0 1+ ksurfacetc A0 (E.5.53.8) Where MATLAB® code E.5.53 plots the expected reduced water contents, surface temperature, and amino acid consumption rates of a potato slab when all the constants of Equations E.5.53.2, E.5.53.4, and E.5.53.8 are available MATLAB® CODE E.5.53 Command Window: clear all close all format compact global Toil Xinfinity Xo alpha alpha0 % enter the constants of the model and the operating parameters Xinfinity=0.01; % Xinfinity= moisture content attainable at the end of the frying process (kg water/kg dry matter Xo=0.5; % Xo= initial water content (kg water/kg dry matter) alpha=30e3/60; % time constant (1/min C) (adapted from Achir et al., 2008) alpha0=200; for Toil=120:20:180 [t,w]=ode45(‘WaterFraction’,[0 60],1); if if if if Toil==120 Toil==140 Toil==160 Toil==180 plot(t,w, plot(t,w, plot(t,w, plot(t,w, ‘-’); hold on; end ‘:’); hold on; end ‘.-’); hold on; end ‘ ’); hold on; end end xlabel(‘time (min)’) ylabel(‘w (fraction of water remaining)’) legend(‘Toil=120 oC’, ‘Toil=140 oC’ , ‘Toil=160 oC’ , ‘Toil=180 oC’ , ‘Location’,’Best’) % MODELING OF THE SURFACE TEMPERATURES figure for Toil=120:20:180 if Toil==120 Tsat=100; end if Toil==140 Tsat=90; end © 2011 by Taylor and Francis Group, LLC 672 Handbook of Food Process Modeling and Statistical Quality Control, Second Edition if Toil==160 if Toil==180 Tsat=82; end Tsat=75; end [t,w]=ode45(‘WaterFraction’,[0 60],1); Tsurface=Toil-alpha0*((Toil-Tsat)/alpha)*((w-(Xinfinity/Xo))/ (1-(Xinfinity/Xo))); if Toil==120 plot(t,Tsurface, ‘-’); hold on; end if Toil==140 plot(t,Tsurface, ‘:’); hold on; end if Toil==160 plot(t,Tsurface, ‘.-’); hold on; end if Toil==180 plot(t,Tsurface, ‘ ’); hold on; end end ylim([100 200]); xlabel(‘time (min)’) ylabel(‘T (\circC )’) legend(‘Toil=120 oC’, ‘Toil=140 oC’ , ‘Toil=160 oC’ , ‘Toil=180 oC’ , ‘Location’,’Best’) % MODELING THE VARIATION OF THE AMINO ACID SURFACE figure kref=60*2.38e-7; % (1/(m mol kg)) Ea =3e3; % J/mol R=8.31; % J/mol K cA0=0.2; % mol/kg dry matter CONCENTRATIONS ON THE for Toil=120:20:180 if if if if Toil==120 Toil==140 Toil==160 Toil==180 Tsat=100; end Tsat=90; end Tsat=82; end Tsat=75; end [t,w]=ode45(‘WaterFraction’,[0 60],1) time(1:length(t))=t; w1(1:length(w))=w; for i=1:length(time); Tsurface(i)=Toil-alpha0*((Toil-Tsat)/alpha)*((w1(i)-(Xinfinity/Xo))/ (1-(Xinfinity/Xo))); k(i)=kref*(exp(-(Ea/R)*((1/Tsurface(i))-(1/(Tsat+273))))); cA(i)=cA0/(1+k(i)*time(i)*cA0); end if if if if Toil==120 Toil==140 Toil==160 Toil==180 plot(time,cA, plot(time,cA, plot(time,cA, plot(time,cA, ‘-’); hold on; end ‘:’); hold on; end ‘.-’); hold on; end ‘ ’); hold on; end end xlabel(‘time (min)’) ylabel(‘c (mol/kg dry matter)’) legend(‘Toil=120 oC’, ‘Toil=140 oC’ , ‘Toil=160 oC’ , ‘Toil=180 oC’ , ‘Location’,’Best’) © 2011 by Taylor and Francis Group, LLC 673 Statistical Process Analysis and Quality Control M-File function [dwdt]=WaterFraction(t,w) global Toil Xinfinity Xo alpha alpha0 if Toil==120 Tsat=100; end if Toil==140 Tsat=90; end if Toil==160 Tsat=82; end if Toil==180 Tsat=75; end dwdt=-((Toil-Tsat)/alpha)*((w-(Xinfinity/Xo))/(1-(Xinfinity/Xo))); When we run the code Figures E.5.53.1, E.5.53.2 and E.5.53.1 will appear on the screen When we run the code Figures E.5.53.1, E.5.53.2 and E.5.53.3 will appear on the screen Toil = 120°C Toil = 140°C Toil = 160°C Toil = 180°C w (fraction of water remaining) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 20 30 Time (min) 40 50 60 FIGURE E.5.53.1 Predicted average fraction of water in the crust during the frying process 200 Toil = 120°C Toil = 140°C Toil = 160°C Toil = 180°C 190 180 170 T (°C ) 160 150 140 130 120 110 100 10 20 30 Time (min) FIGURE E.5.53.2 Predicted surface temperatures during the frying process © 2011 by Taylor and Francis Group, LLC 40 50 60 674 Handbook of Food Process Modeling and Statistical Quality Control, Second Edition c (mol/kg dry matter) 0.2 0.2 0.2 Toil = 120°C Toil = 140°C Toil = 160°C Toil = 180°C 0.2 10 20 30 Time (min) 40 50 60 FIGURE E.5.53.3 Predicted amino acid concentrations remaining on the surface during the frying process (Adapted from Achir, N., Vitrac, O., and Tystram, G., Chemical Engineering and Processing, 47, 1953–67, 2008.) Questions for Discussion and Problems A Distribution Model Problems Under what conditions are we permitted to use: i The Gaussian distribution model ii The Poisson distribution model iii The Binomial distribution model If the measurements of a data set are not normally distributed what would you to be able to use the statistical tools you learned in this book? What is the central limit theorem? What are its consequences in sampling? Although the concept of 6σ range is theoretically related with the Gaussian distribution only, it is also used with binomial and Poisson distributions in practice What is the basis for that? What are type and type errors? What is the probability of having exactly seven cans with less than 500 g of contents when eight cans are taken randomly from a population where 90% of all cans have more than 500 g of contents? A fruit juice mix is produced by mixing two different juices The first juice has a sugar content of µΙ = 7% and σ Ι = 0.5%, the other one has the sugar content with µΙΙ = 6% and σ ΙΙ = 1.5% If the juice mix is made with using 30% of the first juice and 70% of the second juice what percentage of the final products contain more than 6.7% sugar? The following set of data with µ = 3.16 g/ml and σ = 0.22 g/ml are pertinent to CO2 measurements in bottled beer Are these data distributed normally? Number of Measurements Measurement (g/ml) © 2011 by Taylor and Francis Group, LLC 3.0 15 3.1 20 3.2 12 3.3 3.4 3.5 3.6 3.8 675 Statistical Process Analysis and Quality Control Vegetable oil samples (A, B, C) were tasted in three laboratories The following iodine numbers (x) were reported Laboratory xA XB XC 139 135 118 135 140 100 140 130 120 i Is there an actual difference in the iodine numbers of the samples? ii Is there an actual difference between the scores of the different laboratories? xv Four inspectors evaluated four products and their scores are listed as Inspector → Lot number ↓ I II III IV 73 82 65 70 75 82 66 73 80 90 73 83 69 80 64 69 i Is there any significant difference between the scores of the products? ii Is there any significant difference between the scores of the inspectors? B Quality Control Charts Five sets of samples were taken every other hour from a fruit juice production line The following sugar contents were measured: X1: 5.3, 5.5, 4.0, 4.5, 5.0 g/L X2: 4.7, 4.7, 6.0, 4.0, 5.3 g/L X3: 3.9, 5.1, 5.5, 5.7, 3.0 g/L X4: 4.4, 5.7, 5.4, 5.9, 4.0 g/L X5: 3.9, 5.8, 5.4, 4.0, 3.2 g/L i With the given data, determine the CL, UCL, and the LCL of the quality control chart ii Can we regard the fluctuations in the data as “random process fluctuations”? iii Why we prepare BOTH x and R charts when we prepare the statistical control charts for measurements? iv The USL for the process is 6.0 g/L, the higher sugar content makes the product unacceptably sweet There is no LSL Determine the “process capability index” and discuss your results Power-law fluids have apparent viscosity of the form η = Kγ • n−1 , where η is the apparent viscosity (Pa s), γ is the shear rate (s–1), K is the consistency coefficient (Pa sn), and n is the dimensionless flow behavior index It is required to have K within the limits of 6.40 ± 0.08 and n within the limits of 0.550 ± 0.04 for a mayonnaise of acceptable quality Draw appropriate control charts to determine whether this process with the following K and n values is under control to the given limits © 2011 by Taylor and Francis Group, LLC 676 Handbook of Food Process Modeling and Statistical Quality Control, Second Edition Sample Set 10 K (Pa sn) 6.40, 6.35, 6.45, 6.38, 6.60 6.38, 6.37, 6.35, 6.28, 6.50 6.40, 6.35, 6.45, 6.38, 6.60 6.38, 6.28, 6.35, 6.28, 6.50 6.40, 6.43, 6.45, 6.39, 6.60 6.38, 6.37, 6.45, 6.55, 6.50 6.43, 6.45, 6.47, 6.27, 6.60 6.39, 6.27, 6.39, 6.44, 6.50 6.68, 6.39, 6.35, 6.458, 6.50 6.50, 6.32, 6.55, 6.27, 6.59 n 0.550, 0.530, 0.551, 0.560, 0.558 0.551, 0.532, 0.543, 0.561, 0.547 0.550, 0.530, 0.555, 0.560, 0.558 0.551, 0.532, 0.549, 0.561, 0.547 0.550, 0.530, 0.550, 0.550, 0.558 0.531, 0.533, 0.545, 0.571, 0.549 0.550, 0.530, 0.565, 0.560, 0.558 0.558, 0.539, 0.561, 0.561, 0.547 0.551, 0.532, 0.535, 0.572, 0.547 0.540, 0.520, 0.562, 0.548, 0.580 Hint: Construct K and n charts with UCLK = 0.648, LCLK = 0.632, and UCLn = 0.554, LCLn = 0.546 C Standard Sampling Plans for Attributes Determine the AOQL percentage for the sampling plan with N = 200, n = 5, and c = 0 when the method of analysis is (i) nondestructive and (ii) destructive 100 kg of a food grade chemical will be purchased by a factory The raw material is highly hygroscopic The raw material is provided in 500 g sealed containers There are 20 containers in a box, and boxes in a carton Make a detailed sampling plan D Process Analysis Problems Read the following paragraphs about (i) coffee, (ii) lipstick (Johnson 1999), and (iii) tablet-making processes then answer the following questions separately for each case: i All green coffee fruit are sorted by immersion in water, where the good ripe ones sink while others float A fraction of the skin and pulp are removed by pressing the fruit in water through a screen, the remainder is removed by fermentation for several days followed by washing The water content of the beans are reduced to 12–13% by sun-drying and further down to 10% in driers The dried beans are screened and roasted at 188 to 282°C, which may last up to about 30 minutes The roasters are horizontal rotating drums that tumble the green coffee beans in hot combustion gases The stones, metal fragments, and other wastes are removed from the beans in the “destoners.” The beans are then pneumatically conveyed into hoppers, where they are kept for a while to equilibrate their moisture content After the equilibration stage, the roasted beans are packed either as ground or whole beans ii Lipstick contains a variety of waxes, oils, pigments, and skin smoothers The bee, carnauba plant, and/or candelilla plant wax give lipstick its shape and ease of application The oils and fats used in lipstick include olive oil, mineral oil, castor oil, cocoa butter, lanolin, and petrolatum Castor oil forms a tough, shiny film when it dries after application However, ingestion of large amounts of castor oil may cause frequent restroom visits Ingredients such as moisturizers, vitamin E, aloe vera, collagen, amino acids, and sunscreen are also added to the lipstick Lipstick gets its color from the added pigments, such as bromo acid, D&C Red No 21, D&C Red No 27, and insoluble dyes known as lakes, such as D&C Red No 34, Calcium lake, and D&C Orange No 17 During the lipstick production process, the mixture of the raw materials is finely ground, the waxes, oils, and lanolin are added and heated up The hot liquid is poured into cold metal molds where © 2011 by Taylor and Francis Group, LLC Statistical Process Analysis and Quality Control 677 it solidifies The formed lipstick is put through a flame for about half a second to create a smooth and glossy finish and to remove imperfections There are frosted, matte, sheer, stain, and long-lasting color lipsticks available in the stores Frosted lipsticks include a pearlizing agent—often a bismuth compound—that adds luster to the color Bismuth oxychloride, which is synthetic pearl, imparts a frost or shine Bismuth subcarbonate is used as a skin protective Most bismuth compounds used in cosmetics have low toxicity when ingested, but they may cause allergic reactions when applied to the skin iii Food supplement or artificial sweetener tablets are produced by pressing the well-mixed ingredients The ingredients must be dry, uniform in particle size, and freely flowing Mixed particle sized powders can segregate and result in nonuniform contents The tablet formulations include a binder to hold the tablet together and give it strength Lactose, dibasic calcium phosphate, sucrose, corn starch, microcrystalline, and modified cellulose are among the preferred binders Ingredients include a disintegrant that hydrates in water to aid tablet dispersion Starch and cellulose, are also excellent disintegrants Small amounts of lubricants are added to the formulations to help the tablets to be easily ejected from the die after pressing Magnesium stearate, stearic acid, hydrogenated oil, and sodium stearyl fumarate are among the common lubricants Tablets can be coated after being pressed with a combination of polymers, polysaccharides, plasticizers, and pigments Coatings must be strong enough to survive handling and prevent the tablets from being sticky Coatings may facilitate printing on tablets, make them easier to swallow and extend the shelf life via protecting the moisture or oxidation sensitive ingredients Coatings with pearlescent effects may enhance brand recognition Tablet machines range from benchtop models that make one tablet at a time to the ones that make millions of tablets in an hour Draw the process flow charts Locate the MCPs and the CCPs on the flow charts What is the difference between the MCPs and CCPs? What kind of a sampling procedure may you apply at these points? 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Arvanitoyannis “Application of failure mode and effect analysis (FMEA), cause and effect analysis, and pareto diagram in conjunction with HACCP to a corn curl manufacturing plant.” Critical Reviews in Food Science and Nutrition 47 (2007): 363–87 © 2011 by Taylor and Francis Group, LLC .. .Handbook of Food Process Modeling and Statistical Quality Control with Extensive MATLAB? ? Applications S E C O N D â 2011 by Taylor and Francis Group, LLC E D I T I O N © 2011 by Taylor and. .. by Taylor and Francis Group, LLC Handbook of Food Process Modeling and Statistical Quality Control with Extensive MATLAB? ? Applications S E C O N D E D I T I O N Mustafa Özilgen Boca Raton London... (pbk.) Food industry and trade‑‑Production control? ??‑Mathematical models Food industry and trade‑? ?Quality control? ??? ?Statistical methods I Özilgen, Mustafa Food process modeling and statistical quality