= ‘A ‘ = _,‘ = — = _ —. — — — — _ = THESlS "a“ W .H ; (“fl-4 ' F'- , I 5’”; w; i." ‘ G A ‘fi This is to certify that the thesis entitled A STATISTICAL ANALYSIS OF THE RELATIONSHIP BETWEEN FROZEN FOOD QUALITY AND TEMPERATURE presented by Yi-ding Chu has been accepted towards fulfillment of the requirements for M- S . Jegree in Food Science Major professor Date June lZ/733 0-7 639 MS U is an Affirmative Action/Equal Opportunity Institution MSU RETURNING MATERIgtg: Place in book drop to LIBRARIES remove this checkout from .-3__ your record. FINES will be charged if book is returned after the date stamped below. : :T r I» ’ L;' ti ~ ‘ ‘J‘tf'i I‘,’ / W—fiw’ A STATISTICAL ANALYSIS OF THE RELATIONSHIP BETWEEN FROZEN FOOD QUALITY AND TEMPERATURE By Yi-ding Chu A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Food Science and Human Nutrition 1983 ABSTRACT A STATISTICAL ANALYSIS OF THE RELATIONSHIP BETWEEN FROZEN FOOD QUALITY AND TEMPERATURE By Yi-ding Chu The temperature dependence of most frozen food quality degradation follows the Arrhenius equation, and activation energy has been used to describe the functional relationship between shelf-life and temperature. In this study, statistical methods have been deve- loped for the analysis of shelf-life data and evaluation of activation energies for specific products as well as for commodity groups. Indicator variable with multiple linear regression is the basic technique, and a computer program has been deve- loped to conduct the statistical analysis of shelf-life duration and temperature as input data. Activation energies and standard deviations are obtained as output. The use of activation energy in a shelf-life pre- diction model and the error propagation has been investi- gated also. The absolute error of the predicted shelf- life is smaller when the reference temperature is close to or higher than the storage temperature. ACKNOWLEDGMENTS The author is deeply indebted to her major professor, Dr. Dennis R. Heldman for his guidance, understanding and_ encouragement throughout this work. Appreciations are also expressed to Dr. Martin Hawley, Dr. James Steffe and Dr. Mark Uebersax for their helpful suggestions as members of the graduate committee. The Ministry of Education of Republic of China is thanked for financial assistance which encouraged the author to pursue her studies in the field of food engi- neering. ii TABLE OF CONTENTS LIST OF TABLES .......o.........o................... LIST OF FIGURES ..................o........o........ LIST OF SYMBOLS .................................... INTRODUCTION ...........o.o......o........o......... LITERATURE REVIEW .....o..........o................. Changes During Frozen Storage of Foods ........ Quality and Shelf-life of Frozen Foods ........ Influence of Storage Temperature on Frozen FOOd Quality 00000000000000000000090000000000 Systemization of Time-temperature-tolerance Data eoeeoo000000oooeeeeoeeooeeeeeeeeeeeecoco Modeling of Quality Changes under Variable Storage Temperature oeeeeoeoeooeeeeoeoooeeoco METHODS OOOOOOOOOOOOOOO0.00000000000000...0.0.00...O Theory oeeooeceeeee00000000000000.0000oeeeoeoo. Statistical Analysis of Activation Energy ..... Error Propagation eeoeeoeooe0900000000000...ooo RESULTS AND DISCUSSION OO...OOQOOOOOOOOOOOOOOOOOOOOO Analysis of Activation Energy from Shelf-life Data or a Single PPOdUCt eeeeeooeeoeoeoeeeeoo Comparison or Kinetic Parameters for Shelf-life Data Based on Different Quality Character13t108 00000000000000.0000.000000000 Comparison of Activation Energy for Shelf-life Data of Different Products .................. Error Propagation ooooeeeeoeoeeeeoeeeooeocoocoo SUWRY AND CONCLUSIONS OOOOOOOOOOOOOOOOOCOOOOO0.0.0 RECOMMENDATIONS OOCOOO...OOOOOOOOOOOOOOOOOOOOOOOO00. APPENDIX 00.0.0000...OOOOOOOOOOOOO0.000000000000000Q LIST OF REFERENCES OOOOOOOOOOOOOOOOOOO0.0......0000. iii Page iv vi vii comm w 11 1h 18 18 23 33 35 38 he 51 57 60 61 68 Table 2. 3. h. S. 10. 11. 12. 13. 1h. 15. LIST OF TABLES Examples of kinetic analysis of time-temperature -quality concn or shelf-life data of frozen fOOdS 000000000000000000009.000000000009000... Examples of mathematical model for quality Change or frozen fOOdB oooceoooeeooeooooooeooo Shelf-life or turkey ooooocoooeoooooeoeoeeeeoo Activation energy for shelf-life of turkey ... Kinetic parameters for shelf-life of breaded baconburgers oo00000000000000.0000...coococo-o Maximum storage temperature for frozen peas .. Comparison of kinetic parameters for shelf- life data of frozen peas based on overall quality and 10% vitamin C loss ............... Shelf'life or frozen peas 00000000000000.0000. Comparison of kinetic parameters for shelf- life of frozen peas based on perceptible color difference and 25% vitamin C loss ............ Comparison of activation energy for shelf-life data or frozen p638 0.0000000000000900ooooeooo HQL of frozen berries and activation energy .. Activation energy for HQL of frozen boysen- berries 0000000000000000000.0000.0000000000coo Activation energy for HQL of frozen rasp- berries ooeecoocooco0.0000000000000900...coco. Activation energy for HQL of frozen straw- berries oooooooeeoeooeceecooooooo000000000099o Activation energy for HQL of pie filling berries 00000000000000000000000000000000000000 iv Page 12. 16 36 36 38 39 39 #0 M1 M1 h2 #3 #3 an M7 Table 16. 17. 18. 19. 20. 21. Activation energy for HQL of bulk frozen berries oooooec00000000000000.0000.ooooeoooooo Activation energy for HQL of retail frozen berries ococoaocoo000000000000.000000000000000 HQL of retail berries and regression equations Result of regression analysis for HQL data of retail berries coocoeoooooooeooooeeooooooocooo Activation energy for HQL of retail raspberries and strawberries coeoooeocooeooooooeooooooooco Remaining shelf-life for products stored at -1 C O...0.0.........0.........OOOOOOOOOOOOOO Page 117 us is ‘49 SO 56 Figure 1. 2. 3. h. 5. LIST OF FIGURES Illustration of meaning of regression parameters with indicator variable. ......... Flow chart of the computer program. ......... HQL and PSL for breaded baconburgers. ....... Temperature history for frozen peas exwmple.‘ Influence of errors in activation energy and reference shelf-life on the expected error of predicted shelf-life for frozen peas example. Influence of activation energy and storage temperature on the partial derivative. ...... vi Page 26 32 37 S1 52 56 AEa "'3 Q) "5 Fl p ls D O Dfi'fi‘fi'wao LIST OF SYMBOLS pre-exponential constant in the Arrhenius equation constant (lnB is the intercept of shelf-life vs inverse temperature on semilog paper.) regression coefficients quality concentration initial quality concentration quality concentration at end-point of storage activation energy error of Ea remainkg;shelf-life as a function of storage temperature, time, activation energy and initial shelf-life partial derivative of f with respect to Ea partial derivative of f with respect to Q0 reaction rate constant k attemperature T k at reference temperature remaining shelf-life at reference temperature error of Q initial shelf-life at reference temperature error of Q0 time storage duration at temperature Ti vii shelf-life tQ. tQ,T = shelf-life at temperature T tQ,T+1O = shelf-life at temperature T+1O tR = storage duration at reference temperature tT = storage duration at temperature T T = temperature T1 = constant temperature for storage duration ti TR = reference temperature Xi = independent variables in regression equation Y = dependent variable in regression equation viii INTRODUCTION Freezing and canning are two major methods for long term food preservation. Coupled with quality preservation, freezing has been an important factor in bringing conve- nient foods to consumer. Because freezing properly reduces rate of deterioration reactions responsible for spoilage at room temperature without causing major alternations in appearance, color and flavor, freezing permits much of the work in preparing a food item or an entire meal to be done prior to the freezing step. Shelf-life of frozen foods is determined by properties of raw material, prefreezing history and storage tempera- ture. In the United States, the conventional frozen tempe- rature is -18 0G. Since some fluctuation of temperature will be encountered in practical storage and different standard temperatures are used in other countries, it is desirable to know the temperature dependence of quality loss during frozen storage. Experimental time-temperature-quality change data and mathematical models for quality change under variable storage temperature are available in literature. Statis- tical methods are generally required in solving the mathe- matical models; since only small samples are available from 1 2 large populations being studied. In addition, measurement methods and instruments are not perfect and conclusions drawn from experimental data must take inherent variability into account. Applications of statistical analysis fre- quently require extensive computations. In the past, this has proved to be a major deterrent. Through the use of high speed computers, statistical techniques can be employed without dwelling on computational details. The objectives of this research are: 1. To develop statistical methods for a. computation of kinetic parameters to describe the relationship between frozen food quality change and temperature. b. comparison of kinetic parameters of different quality characteristics of a food product. 0. comparison of kinetic parameters of different products and evaluation of kinetic parameters of product groups. 2. To develop a computer program for the statistical analysis. 3. To illustrate the use of kinetic parameters in a shelf- life prediction model and investigate the influence of uncertainty inherent in kinetic parameters on the precision of predicted shelf-life. LITERATURE REVIEW The storage of foods in a frozen condition has been an important preservative method for centuries where sub-. freezing temperatures exist in the ambient environment. With the development of mechanical refrigeration and quick- freezing processes, the frozen-food industry has expanded rapidly. Most commercial storage facilities for frozen materials are at or below ~18 oC, and the storage life for different food items range from months to several years. Changes During Frozen Storage of Foodg In general, freezing cannot improve food quality. It cannot completely preserve the quality; deterioration will take place continually and at rates governed by the storage temperature and type of product. 1. Physical changes The physical changes which occur during freezing and during storage of frozen products include crystallization of ice with expansion of the volume as well as desiccation beginning at the surface of the frozen food. In general, slow freezing will produce large crystals located exclu- sively in extracellular areas while rapid freezing will result in small ice crystals, located both extra- and 3 h intracellularly. Mechanical stress arising from changes in volume can produce damage (or "freezing injury") of frozen foods (Singh and Wang, 1977). Freezing of foods may be associated with weight losses due to moisture evapo- ration. This applies particular to items with relatively high free surface areas and high surface temperatures Just before starting the freezing process. When ice crystals sublime from the surface layer of frozen tissues, a poor appearance is produced. This phenomenon is sometimes referred to as "freezer burn" (Partmann, 1975). 2. Chemical changes Included among the types of chemical changes that can proceed during frozen storage are lipid oxidation, enzymatic browning, flavor deterioration, protein insolubilization and degradation of chlorophyll, other pigments and vitamins. In general, the nutritive values of foods are well preserved by freezing. The changes associated with oxidation can be inhibited by exclusion of oxygen through technical methods including modified gaseous atmosphere, vaccum packaging and treatment by antioxidants. Enzymatic changes are usually controlled by a heat treatment, i.e., blanching, prior to freezing. The destruction of ascorbic acid is widely studied, not only because it is a necessary nutrient, but also for its strong reducing power. Ascorbic acid is considered to serve as an antioxidant in oxidation deterioration reactions. It was found that ascorbic acid loss showed a high 5 correlation with unfrozen water (Singh and Wang, 1977). 3. Microbiological decay A few examples of low-temperature growth of micro~ organisms in food materials have been found. Bacteria have been reported growing at ~S 00 on meat, ~10 °C on cured meats, ~11 0C on fish, ~12.2 0C on vegetables (peas), and ~10 0C in ice cream; yeast at ~S 0C on meats, and ~17.8 DC on oysters; and molds at ~7.8 00 on meats and vegetables and ~6.7 0C on berries. In general, under usual storage conditions of frozen food, microbial growth is prevented entirely. Temperatures of S to 6 °C or less, effectively retard the growth of all food-poisoning organisms except for Type E Clostridium botulinum (food intoxication), which has been reported producing toxin at 3.3 0C (Frazier and Westhoff, 1978), and Yersinia enterocolitigg (food infection), which can grow at ~2 oC (Stern and Pierson, 1979). alit nd She f~li e o o n Foo s The definitions of "food quality" range from the simple, pragmatic test, "that which the public likes best", to the composite of properties or attributes which are thought to be most important to the economic success of a food commodity. The difficulties met in defining the concept "quality" are apparent in all discussions of measure- ment or specification (Labuza, 1982). 1. Quality measurement a. Sensory analysis Evaluating the quality of food by means of the human sense organs is an essential tool in determining the storage stability and ultimately the acceptability of frozen food. All five senses: sight, touch, taste, smell, and sound may be used to analyze the quality as appearance (size, shape, color), flavor (taste, smell), or texture (firmness, chewiness). Sensory tests to determine product storage stability may include the following (IFT, 1981): (1) difference tests to determine whether the storage samples are different from the control. (2) descriptive tests to characterize and/or quantify the changes that may have occurred during storage. (3) acceptance tests to determine the relative acceptance of stored products. b. Objective evaluation Except for the measurement of chemical components ( such as vitamin C), some objective tests are also used to evaluate color, flavor and texture. Objective tests are reproducible, and if simple and rapid, also have the merit of time saving. In the field of frozen foods however, close correlation between subjective perception and objec~ tive tests have not yet been developed. 2. Definition of shelf-life Shelf-life is a rather poorly defined term that fre- quently confuses the consumers, as well as researcher. 7 Some investigators assume that shelf-life ends when it is possible to detect a "just noticeable difference" (JND), as compared to a control product. Usually the control is a sample stored at -h0 °C or below (Dalhoff and Jul, 1963; ABMPS, 1982). Another term used is "high quality life" (HQL), the storage time at which 70-80% of a sensory panel in a triangle test, correctly indicate the quality difference between control and stored samples. In some cases, HQL, JND and "stability time" may refer to the same thing, and the "difference" may be alternatively determined as 1 point decline when a 7-9 hedonic scale is used. "Practical storage life" (PSL) or "acceptability time" is the time a frozen product can be kept and still remain acceptable to the consumer panel. PSL of frozen fruit and vegetables is about 3.1 to 3.5 times longer than HQL and for frozen meats and seafoods, the values are about double (Labuza, 1982; Bzgh-Sorensen et al., 1981; Van Arsdel et al., 1969). 3. Changes determining shelf-life of frozen foods Generally people eat fruit and vegerables for pleasure and as a source of vitamin C and minerals. Therefore, the loss of palatability (color, flavor and texture) and vitamin C degradation are the two main concerns when eva- lusting the shelf-life of frozen fruit and vegatables. The major deterioration of frozen seafoods occurs in the protein and lipid components. Off-flavors develop due to oxidative changes of the fat while protein denaturation 8 causes increased toughness. Deterioration of frozen meats is very similar to that of fish, but occurs more slowly. Protein denaturation, off-flavor production, desiccation of the tissue and myoglobin color changes occur with extended frozen storage . Flavor deterioration is usually the primary factor limiting shelf-life of frozen meats (Labuza, 1982). Influence of Storage Temperature on Frozen Food Quality Storage temperature plays a key role in the preserva- tion of frozen food products. The commercial storage temperature is -18 0C in the United States. In Europe and Japan, the standard temperature used is ~30 °C (ABMPS, 1982). Since some fluctuation of storage temperature would be encountered in practice and there is concern about energy requirements for frozen food storage, the tempera- ture-dependence of frozen food quality is important. Different standards and methods have been used in quality measurement. Generally two types of data appear in the literature: (1) the extent of quality change, such as concentration of a component which is either a reactant or a product of deterioration reaction; percentage of quality loss, and (2) the end-point, length of storage life, such as HQL, PSL, etc. 1. Quality concentration data A two-step procedure is usually used to analyze the experimental quality concentration data as a function of 9 time and temperature. First, the rate of deterioration at fixed temperature is measured, and second, the dependence of rate on temperature is determined. Reaction order with a rate constant (k) and D-value are generally used in the first step. Activation energy (Ea), Z-value, and Q10 are used to describe temperature dependence in the second step (Harris and Karmas, 197k). Amont these parameters, k and Ea are used most often in the studies of food products. Lund (1983) reported a one-step method for determining Ea from quality concentration data. The advantage was a more precise estimation of Ea; a smaller confidence interval due to more degrees of freedom when compared to the results of two-step analysis. 2. Shelf-life data Kinetic analysis was also applied to storage life data, where the time required to reach some chosen quality level is measured. Van Arsdel and Guadagni (1959) used the reciprocal of time to represent the mean rate of quality change. Lai and Heldman (1983) applied the concept of reaction order and rate constant to illustrate that the inversely proportional relationship between rate constant and shelf-life was valid without regard for the reaction order. This obser- vation represents an advantage of using shelf-life data in that the temperature-dependence of the quality change rate can be evaluated without any assumption of the reaction order. Such assumptions are empirical, when the 10 stoichiometric equation of the mechanism is unknown. Since this is the case in degradation of frozen food quality during storage, assumptions can be avoided. 3. Exceptions to the general temperature-dependence A lower storage temperature will not improve the shelf- life of all products. Lindelzv (1978) has proposed that frozen foods be characterized as (1) normal stability, where storage life increases with lower storage temperature, (2) neutral stability, where storage life is largely unaffected by storage temperature or (3) reversed stability, where storage life decreases with lower storage temperature. Bogh-Szrensen et al. (1981) reported that vacuum-packed cured meats have neutral stability and cured pork (bacon) have reversed stability, when the product was not packed or where the packaging was rather permeable to oxygen. Similar results have been reported previously by Lindel¢v and Poulsen (1975). Fennema (1975) discussed the possible reasons of rate enhancement by freezing based on rates of Amany specific chemical and biological reaction studied at temperatures near subfreezing. Two major factors suggested were enzyme delocalization for cellular enzymatic reaction and a freeze-concentration effect for nonenzymatic reactions as well as noncellular reactions. When overall quality is of concern, several contri- buting factors may be involved and the relative importance may vary with the temperature range considered. In this 11 case, a constant kinetic parameter (such as activation enerSY) cannot be generated because different reaction modes apply to different temperature ranges. A step-wise linear function or curve fitting technique might be used to study the temperature dependence of qulality change. Examples of the types of kinetic analysis used for frozen food quality change are summarized in Table 1. S tem atio of T me-tem a e-to e ance Data Most of the existing quality change data for frozen foods are in terms of time-temperature-tolerance (TTT), in which the allowable or tolerable time and temperature condi- tions for quality retention are specified (Van Arsdel et al., 1969). Basically two types of factors will influence the rate of quality change: (1) environmental factors, such as temperature, pressure, light, etc. and (2) composition factors, such as quality concentration, pH, oxygen, water activity, catalyst, etc. (Saguy and Karel, 1980). 1. PPP-factors In the investigation of frozen food storage life, factors other than temperature are generally referred to as PPP-factors as following: Prodgct. Quality of raw material, including initial quality concentration, pH, presence of catalyst and the final quality concentration defined for the end of storage life. Processing. Heating, blanching and other pre-storage processes influencing enzyme activity and quality 12 .moh pooawa hp cepwawpmo Amm Aommrv m» m + mu. m + a m + m u o . 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Packaging. Oxygen, water activity and light will be affected by the permeability of packaging material (Dalhoff and Jul, 1963; Klose, 1963; Fennema, 197M). 2. Systemization of TTT data Bengtsson et a1. (1972) tried to systemize TTT data and group frozen foods into several predicable categories. The work was based on the assumption that the mode of the deterioration reaction was similar within the frozen food group. The results indicated that when shelf-life data was plotted versus temperature on semilog graph paper, difference foods could be systemized in groups ( with similar slopes and intercepts). It was suggested that improved control of PPP-factors would lead to more precise results: group kinetic characteristics for shelf-life. Modeling of Quality Changes under Variable Storage Tempe- rature Generating models for describing the effect of pro- cessing on food quality is appealing and necessary for process analysis, simulation, prediction and optimization (Saguy and Karel, 1980). The potential for improving food quality through process modeling is significant. Several mathematical models have been developed for the quality change during frozen storage. Factors other than tempe- rature were not incorporated in these models due to the need for extensive experimental work required to generate 15 sufficient information about the correlation between quality and each factor. In addition, no generally valid relationships have been developed to describe the depen- dence of reaction rate on these factors. In most cases only a temperature effect model has been evaluated. Van Arsdel and Guadagni (1959) discussed two conditions necessary and sufficient to assure the mathematical validity: (1) the characteristic in question must be simply additive and (2) the characteristic must be commutative. Most frozen food quality deterioration reactions satisfied these conditions, thus the "total quality change" of products exposed to a variable temperature history could be evaluated through mathematical integration. 1. Models in the literature Van Arsdel et a1. (1969) used Q10 to develop a shelf-life prediction equation for linear temperature history, and a graphical method for irregular temperature history. Singh (1976) reported a model for the concentration of single component in product, where a first order reaction and Arrhenius equation were applied. Heldman and Lai (1983) reported a shelf-life prediction model based on Ea, in which the reaction order need not be considered. Both models can handle irregular temperature history. Details of these models are presented in Table 2. 2. Precision of quality evaluated from a mathematical model The kinetic parameters (Ea, q10) used in the mathe. matical model are estimated from experimental data. Any 16 me can» as uuaauoamnm «ca m o Ammorv seq ma 9.8» pa 0.3.7.32? mfinfigoa "a "ET Er .. .Bmvwmmmuwdwo new .. 0.0 u a S $5.3m papmnoo 3. am 1 I mm ommaoam scams mmoH hpfiamsw "on o Adm vgwo< I up as o “camel Gownshpnoonoo 53.326 no . up “mum p u on nwnwm k hMOpmwn mama Manama N r Orv mod are umevw.o u m a o a o w A no as H a o u . A95» owwnopm ado.“ mo numnoa fins .33 m .3 no mo o as omsaopm no sumsoa usofispfiswo no . AP- cavasowv :m:.o u a accuse qs> m nofipmsvm 903954 mcoom Genoa.“ mo omgno hpwamsw so.“ Hence Hmowpwaonpwa ho moaafiflnm .N 353... 1? inherent error will be propagated to the predicted quality value. If a dependent variable y is a known function of the m independent variables x1, x2, ..., xm, i.e., y = f(x1, x2, ..., xm) the expected error in y due to the errors in x is given by 2 = «Ia-I1; 9A 2 (Av) Zi:(ax1 xi) where Ay = maximum error of y %% = partial derivative of f with respect to xi 1 4xi= maximum error of x1 and the expected relative error is given by: <21)2=2:;(___31nf.2ia)2 y alnxi 'xi where §¥ = maximum relative error of y %%%§ = partial derivative of lnf with respect to i lnxi fiai= maximum relative error of xi (Hill and Grieger-Block, 1980) This analysis can be applied in frozen food shelf- life models to demonstrate the uncertainty of quality values due to error of inputs. METHODS The shelf-life model proposed by Heldman and Lai (1983) is being investigated. Experimental time-temperature- quality data from literature have been used to illustrate the estimation of kinetic parameter required in this model. Theory Two theoretical bases were applied in kinetic analysis of frozen food quality change data: (1) a rate expression for reaction at fixed temperature, and (2) influence of temperature as described by the Arrhenius equation. 1. Analysis of reaction order and rate constant a. Quality concentration data Generally, time-temperature-quality concentration data will provide information on changes in a specific quality characteristic as a function of time and temperature. Applying the rate expression: §-%=-kcn (1) where C = concentration of quality characteristic t = time k = reaction rate constant n = reaction order the reaction order and rate constant for a fixed temperature 18 19 can be determined from experimental data. The rate equations for zero, first and second order reaction and the integrated forms are as follows: n=O, %%=-k , C=Co-kt (2) n=1, %%=-kC,lnC=lnCo-kt (3) n=2, %%=-k02, %=% -kt (u) 0 where Co is the initial concentration of the quality charac- teristic. By plotting C, lnC cu' 1/C vs time, reaction order corresponding to the best linear relationship can be used to analyze data. Simple linear regression is applied to estimate the rate constant. Since only one quality characteristic is monitored in most time-temperature- quality change data, and there may be other product components involved in the reaction, the n and k obtained from above procedure should be identified as pseudo-reaction order and a pseudo-rate constant. b. Shelf-life data Usually the shelf-life of a frozen food is measured by sensory evaluation. The quality at the end-point is described qualitatively, such as "acceptable" for PSL or "noticeably different from control" for HQL. Lai and Heldman (1983) expressed the reaction rate constant (k) in terms of shelf- life (tQ), initial quality level (00) and end-point quality level (CQ). Based on equations (2), (3) and (h), the rate constants for zerq,first and second order reaction are expressed as 20 follows: C C n = o, cQ = C0 -th -> k = 'Q'EE'Q' (5) n - 1, anQ = lnCo-'th -9' k = 1n(co/CQ) (6) ts = l. -l 1/0 ‘1/0 n 2 — -kt '—9 _ ' 6Q 00 Q k - __?__30 Q (7) In most cases, quality deterioration of frozen foods. is assumed to follow either zero or first order reaction. However, the reaction order and rate constant cannot be determined from qualitative quality change data. 2. Analysis of activation energy The activation energy in Arrhenius equation is given by: k = A exp (- g%) (8) where k = reaction rate constant A = pre-exponential constant Ea = activation energy R = gas constant T = absolute temperature For quality concentration data, the rate constants (k) at different temperatures are available and Ea can be evaluate by simple linear regression of lnk vs 1/T. For shelf-life data, k values are unknown; Lai and Heldman (1983) reported a functional relationship between tQ and Ea as follows: a. Zero order reaction From equation (5) and (8), C -CQ — E9. tQ " A °"P(‘RT) 21 1 A Ea '—> - ="":"'°XP("") tQ Co CQ RT Ba 1 co- Cg -91ntQ=lnB+-§—(T) ,whereB— A (9) b. First order reaction From equation (6) and (8), ln(C /C ) 0 Q = _.Ea tq Aexp(§-.1.-) 1 .. Es —9 - - exp(- ) tQ ln 00 Q RT ln(C /C) -9 lnt =1ns+§2(1),whereB=——°——9- (10) Q E! T A C. Second order reaction From equation (7) and (8) 1/C - 1/C 0 Q=Aexp(-E-§-) t RT Q 1 Ea -—-> - =7—é—7—exp(--——) tQ 1 Co 1 CQ RT 1/C 1/0 —-) lntQ=lnB+Efig~(JT-) ,whereB= 0A 9"(11) Based on this analysis, the relationship between shelf- life and absolute temperature has been represented in the general form of: .. 329. l lntQ - lnB + R (if) (12) where B is a function of the initial and end-point quality levels (Co and CQ). Shelf-life data from different authors or different papers should not be expected to have similar CO and CQ, due to the lack of a quantitative definition for sensory 22 quality level. However, for a specific experiment, inves- gators have attempted to establish similar initial condi- tions, and maintain a standard end-point quality. Based on this observation, the B value should be constant for a specific experiment. It follows that the activation energy (Ea) can be evaluated from shelf-life data without consi- dering reaction order. 3. Shelf-life model for product under variable storage temperatures Heldman and Lai (1983) used the above-mentioned kinetic analysis to develop a shelf-life prediction model, where the shelf-life of a frozen food product at a reference temperature was used as a quality index. As discussed in equations (5), (6) and (7), shelf-life and rate constant is related as: th = f( Co, CQ) where the function is 00- Cq, ln(Co/CQ) and 1/Co- 1/CQ for zero, first and second order reaction respectively. Thus for quality degradation from Co to some Ct’ the storage time (tT) at any temperature (T) can be related to the storage time (tR) at a reference temperature (TR) as follows: thT = thR = f( Co’ Ct) (13) In other words, the storage time at reference temperature equivalent to the storage time at any temperature can be calculated by: tR = tT ( kT/kR) (1h) By substituting the Arrhenius equation into equation (1h) 23 will give: tR = tT. exp [-EgG-(JT-JT-Rfl <15) The overall quality loss of a product subject to certain time-temperature treatment can be evaluated by integration. The remainhugquality (Q), as expressed as shelf-life at TR’ is given as following: Q=Qo-j: exp[-Efi§(%-%R)]dt (16) where Qo is the initial quality. Statistical Analysis of Activation Energy In addition to the validity of theories applied, the usefulness of a mathematical model also depends on preci- sion of input data. For the above shelf-life model, inputs include temperature history, reference shelf-life (Q0) and activation energy (Ea) as estimated from experimental shelf-life data. I When compared with quality concentration data, the Ea derived from shelf-life data seems to be less reliable. The rate constant at a fixed temperature is determined by only one data value (tQ) while for quality concentration data, k is determined from significant amounts of data (C,t). 1. Use of indicator variable method in evaluation of acti- vation energy Sometimes more that one end-point quality level (CQ) are reported from shelf-life investigation. For each end- point, shelf—life data at several temperatures would be provided. For example, HQL and PSL may be reported 2h simultaneously. Each set of data with the same CO and CQ could be used to evaluate an Ea by equation (12). Since the same mode of deterioration is being described by each data set, the slope of each line (Ea/R) should be the same while the intercepts (lnB) would be different. If data from several data sets can be employed to evaluate the common slope, confidence in the estimated Ea value would increase. Indicator variable method (Neter and Wasserman, 1978) can be applied to quantify the qualitative variable (diffe- rent CQ) in data. If there are N classes in the qualitative variable, N-1 indicator variables which take value of O and 1 will be used. For example, in case of the two end-point standards: HQL and PSL data, we might define 1 indicator variable X2 as follows: x = 1 if HQL 2 0 otherwise (PSL) The regression equation would be: Y = 30 + B1X1 + 82x2 (17) Compared with equation (12), .. $9.1. lntQ—lnB+ R(T) Y is lntq, X1 is 1/T and X2 is the defined indicator variable. To understand the meaning of the parameters in equation (17), consider the case of PSL data. For PSL, X = O and 2 we have: 0 1 1 (PSL) (18) Thus, the equation is a straight line, with intercept BO 25 and slope B1. For HQL data, X2 = 1 and the equation becomes: Y = (BC + Ba) + B x 1 1 (HQL) (19) This also is a straight line, with the same slope B1, but intercept BC + B2, as shown in Figure 1. Multiple linear regression can be applied to evaluate coefficients of equation (17). A pooled activation energy (Ea) for all HQL and PSL data can be calculated from B1. For analysis of shelf-life data of N end-point classes, the regression equation is: Y = so + B1X1 + 32x2 + ... + BNXN (20) where Y is lntQ, X1 is 1/T and the definition of X2, X3, ..., XN is following: 1 if class 1 X2 = 0 otherwise _ 1 if class 2 X3 - 0 otherwise X = 1 if class N-1 0 otherwise 2. Comparison of different quality characteristics of a specific product When more than one quality characteristic of a product has been investigated, it is desirable to determine the relationship among quality changes. For example, shelf- life data were evaluated based on overall sensory quality, sensory flavor quality and concentration of a specific component which was related to flavor change. The purpose of the analysis is to determine whether flavor change is 26 (lntQ) X (1/T) Figure 1. Illustration of meaning of regression parameters for equation (17) with indi— cator variable X2. 27 the major contributor for overall quality or whether a quantitative definition forinusend-point of sensory flavor quality can be obtained from these types of experimental data. The indicator variable method is used for these types of analyses. If shelf-life data of the N quality characte- ristics are to be analyzed, 2(N-1) indicator variable should be defined. For example, if shelf-life data of two quality characteristics are compared, the 2(2-1)=2 indicator vari- ables are defined as following: 1 if class 1 X2 = 0 otherwise X3 = X1X2(X1) if class 1 0 otherwise The regression equation would be: Y = BC + 31X1 + B2X2 + 33x3 (21) where Y is lntQ and X1is 1/T as before. For the first data set, X2 = 1, X3 = X1 then Y = BC + 81X1 + B2 + 83X1 or, = Y (BC + 82) + (B1 + B3) X1 (class 1) For the second data set, X2 = X3 = 0, therefore Y = Bo + B1X1 (class 2) If the shelf-life - temperature relationships of these two quality characteristics are very similar (B2 = B = O), 3 we can conclude that these two quality parameters may be described by the same temperature influence. If the over- 28 all quality and flavor quality are compared, the conclusion based on shelf-life data will be that flavor is the deter- mining factor for overall quality. In case the mechanism of quality degradation for two quality parameters is the same, but tne end-point chosen for shelf-life evaluating is different, the two linear relationship will have similar slopes while the intercepts will be different. In the regression coefficient for equation (21), this is referred to B3 = O. In summary, two tests will be applied to compare shelf-life data based on two quality characteristics of a product. a. Test of equal slopes Hypothesis: 83 = 0 Test statistic: F = SSE R - SSE F ; SSE(F) df R — df F ' d F where the source of error (SSE) and degrees of freedom (df) comes from analysis of variance of two regression models: Full model: Y BC + B X + B X + B X 1 1 2 2 3 3 B0 + 81X1 + B2X2 Reduced model: Y b. Test of equal slopes and intercepts Hypothesis: 82 = B3 = 0 Test statistic: F = SSE R - SSE F ; SSEEE; R - df F ' df F where Full model: Y = BC + B X 11+B X2 + B X 2 3 3 Reduced model: Y = BC + B1X1 29 Multiple linear regression and analysis of variance are applied to these regression models except for the reduced model of test b, where a simple linear regression is used. Critical value Ex, df(R)-df(F), df(F) can be found in the F-distribution percentage table of any statistics book. 3. Comparison of activation energy of different products Bengtsson et a1. (1972) tried to group frozen foods based on the kinetic characteristics. The kinetic para- meter is the activation energy derived from shelf-life data. It is reasonable to assume that shelf-life of similar pro- ducts is being determined by similar mode of quality degra- dation. Thus the kinetic parameters of different products can be compared and if the difference is not significant, all data can be combined to generate kinetic parameters for product groups. The statistical method for comparison of Ea values or the slopes of lntQ vs 1/T, has been discussed earlier for two sets of data. Here,the general test for N sets of data is presented. The regression equation for N sets of data (N different products) is: Y = BC + 81X1 + 82X2 + ... + BNXN + BN+1XN+1 + + B?_N__1)(2N_.1 (22) where Y is lntq, X1 is 1/T, X are 2(N-1) 2, X3, 000’ XZN-1 indicator variables. X2, x3, ..., XN are defined as: X. = 1 if data set i-1 1 0 otherwise , i = 2, 3, ..., N 30 and XN+1’ XN+2’ ..., X2N-1 are defined as: xN+i = X1. Xi+1 , i =1, 2, coo, N‘1 The test for equal slopes is following: Hypothesis: BN+1 = BN+2 = ... = B21“1 = 0 Test statistic: l(asssa -SSEF _-_ss F d -d .___:dngF where Full model: Y = BC + B1X1 + ... + BZN-1x2N-1 Reduced model: Y = BC + B X + ... + BNXN 1 1 Critical value F“, df(R)-df(F), df(F) with 06:0.1 will be used to judge the hypothesis. If the hypothesis is accepted, i.e., the slopes are similar, than the common slope (B1 of the reduced model) can be used to evaluate the activation energy. If the hypothesis is rejected, this means not all BN+1’ BN+2’ ..., B2N-1 are zero, or not all slopes are the same. If extra- neous data exist and should be excluded from all data sets would be determined next. The existence of extraneous data setcan be detected from regression result of the full model, where information about estimates of each coefficient (Bi)’ their variance, and t-value are provided. By compa- rison with the tabulated t-value, we can get information about whether B1 = O for each i. If all B B N+1’ N+2’°"’ BZN-1 are near zero except one, eliminating the corres— ponding data set may result in equal slopes and provide a common slope for the remaining data sets. Unless the total number of data sets is considerable, it is better to 31 avoid excluding any information. u. Computer program for the statistical analysis A computer program in BASIC programming language has been developed and Hewlett-Packard 85 Personal Computer has been used in this research. The program can be used to analyze shelf-life data to: a. Evaluate activation energy from shelf-life data of more than one end-point quality level. b. Compare kinetic parameters for shelf-life data based on different quality characteristics of a specific product. c. Compare activation energy of different products and evaluate activation energy for product groups. The type of analysis is specified after data (tQ, T) are entered from the keyboard. Temperature unit can be 0C or 0F. Time units are not limited. All units must be consis~ tent for all data. The program is designed for a maximum of 50 data pairs, and the multiple linear regression portion of the program is designed for a maximum of 12 independent variables. Once the type of analysis is selected, values for indicator variables will be defined and the required regre- ssion, analysis of variance, test of hypothesis and/or evaluation of the common activation energy will be performed. The results of the test and/or activation energy with stand- ard deviation will be output. The program is summarized in Figure 2. A program list is presented in the Appendix. 32 (Start) Input data , (tq, T) \ Select type of analysis Data of diffe- rent end-points Data of diffe- rent characte- ristics Calculate Ea rent products Data of diffe-l Compare Ea & lnB I Compare Ea 1 1. result of regression 2. Ea 1’AEa Figure 2. 1. result of test for Ea & lnB result of test for Ea 2. 1. result of] test 2. Ea i’AEa ,,. _,,-.....- .- / ,// \Mo”" Flow chart of the computer program. 33 The keyboard responsmsrequired to run the program are following (in sequence): a. unit of temperature (°C or 0F) b. number of data sets (number of end-point quality levels, number of quality characteristics, or number of products) 0. for each data set, (1) number of data pairs (2) values of each data pairs (tq, T) d. type of analysis (1-compare Ea, 2-compare Ea and lnB, 3-calculate Ea). Error Propagation The shelf-life model (equation 16) is: _ t Q—QO-Ioexp[-1§r—?(T- TRHdt Activation energy evaluated from above statistical analysis will be used, and any inherent uncertainty in Ea will influence the quality value predicted from the model. Equation (16) can be expressed as: Q = f(Qoa E8, T: t) (23) By assuming: (1) the influence of uncertainty in temperature history (T, t) is insignificant as compared to errors from other sources and (2) the dependence between reference shelf-life (Q0) and activation energy (Ea) is negligible. qhe maximum error of Q (AQ) can be evaluated by: AQ= =J(-§-—-f ~50on )2+ +( %%5Ahza)2 (2)1.) 3h or, 2 2 A AQ= -./( Q ) + <-- g—Ea as.) (25) because gab is equal to 1. If the temperature history contains several segments of the constant temperature storage, the shelf-life model (equation 16) can be expressed as: _ $1-1. Q"Qo‘2;°"p[‘a (T. T)]ti (26). The partial derivative (:Ea)’ in equation (25) will be: 'E'IZexPFEITW'lflUt['%('1'1"'1'71I"R)] ‘27) By substitution of equation (27) into equation (25): M2 = M >2 + «xexpt- se- ats-M- we >2 (28) Equation (28) can be used to evaluate the expected error of the predicted shelf-life (AQ), after the temperature history (Ti, ti), activation energy (Ea IpAEa) and reference shelf-life (QO : AQO) have been provided as input. RESULTS AND DISCUSSION Shelf-life data for frozen foods as collected from the literature are used to illustrate the estimation, comparison and use of kinetic parameters to describe the temperature dependence of shelf-life. Analysis of Activation Energy from Shelf-life Data of a Single Product The shelf-life data for turkey (Ristic, 1980) are presented in Table 3. These data are based on three and- point quality levels. In this example, three sets of data (corresponding to the three end-point quality levels) each having two data pairs (tQ, T) have been analyzed. All 6 data pairs are input to the analysis for calculation of activation energy (Ea). The results of the analysis are summarized in Table h. For shelf-life of turkey, the activation energy with standard deviation is 59.0u I 6.53 kJ/mole. A similar example for shelf-life of breaded bacon- burgers (Bogh-Scrensen et al., 1981) are presented in Figure 3. Five temperatures are studied and an optimal storage temperature of about -23 00 is found for shelf-life of breaded baconburgers. Between -10 to -23 00, lower 35 36 Table 3. Shelf-life data for turkey (Ristic, 1980) v.” m End-point Storage temperature (0) quality level "' ~10 _20 (months) (months) Excellent h 1h Good 6 18 Satisfactory 9 21 Table u. Activation energy for shelf-life of turkey Activation Standard Degrees of Correlation Energy Deviation freedom (R2) (Ea) (143a ) (kJ/mole) (kJ/mole) 59.0u 6.53 2 0.980 (shelf life data from Table 3) 37 1; 900. 9///////1r"‘~\\ :3, 700 1 . o\ 'd 1 °\ 0 "1 :2: A H -( £6 at S //’”’ \\\\\‘o- / :0: /’ 100q / -uo -30 -2h -18 -12 Inverse storage temperature (1/K) Figure 3. HQL (O) and PSL (o) for breaded baconburgers (Begh-Serensen et al., 1981). storage temperature retards quality deterioration and improves shelf-life. This is consistent with the Arrhenius equation and activation energy (Ea) can be used to describe the relationship between shelf-life and temperature. Below -23 0C, shelf-life decreases with further decrease of storage temperature, although a linear relationship between lntQ and 1/T still applies, but the slope cannot be inter- preted as activation energy. The results of data analysis are presented in Table 5. For temperature between ~10 and -23 °c, the Ea based on both HQL and PSL is 32.68 1 2.67 kJ/mole. For temperatures below -23 0C, the common slope is ~2678.87 1 93.18. The unit of this slope is the same as that of Ea/R. This kinetic parameter can be used to describe the temperature dependence of shelf-life the same as acti- vation energy except that in this case the value is negative. 38 Table 5. Kinetic parameters for shelf-life of breaded baconburgers Temp range Kinetic Standard Degrees of Correlation (C parameter Deviation freedom (R2) -23 to ~10 32.68(Ea)a 2.67 1 1.000 -u0 to -23 -2678.87 91.38 3 0.999 a unit of Ea: kJ/mole. (shelf-life data from Figure 3) Comparison of Kinetic Parameters for Shelf-life Data Based on Different Quality Characteristics Since the oxidation/reduction reaction is involved in quality changes of food products and vitamin C is an important reducing agent in food material, the association of vitamin C loss with sensory quality change can be evaluated. In the first example (Table 6), shelf-life data based on 10% vitamin C loss and overall quality change are examined by the test of equal slopes and intercepts. Based on the analysis, 10% vitamin C loss and overall quality change are closely associated (Table 7). Both the activation energy(Ea) and the intercept constant (lnB) are similar. These results would suggest that the temperature influence on vitamin C loss and on overall quality are similar and that reaction rate constants have the same general magnitudes. Thus, 10% vitamin C loss might be used as an index for the overall 39 Table 6. Maximum storage temperature for frozen peas (Kramer, 197a) Quality basis Months in storage 6 12 18 2h (C) (C) (C) (C) Overall quality -15.6 -19.h -21.1 -22.2 loss Table 7. Comparison of kinetic parameters for shelf-life data of frozen peas based on overall quality and 10% vitamin 0 loss a.-'~¢o *.~ .q..'~__. .. -. .... ~— v—-———. .. . . H-.. ... . ...—.p..¢....-—u -Inn ...--..- Test hypothesis: the slopes and intercepts for 2 sets of data are equal Analysis of Variance Source Degrees of Sum of F Tabulated-F of error freedom square SSEER) ) 6 0.1fi937 SSE(F) u 0.06727 '25’2’u Conclusion: accept hypothesis, 0‘) 0.25 The Ea and lnB values for the 2 sets of data are similar. (shelf-life data from Table 6) to sensory quality change as used to determine the shelf-life of product. The shelf-life data based on 25% vitamin C loss and perceptible color differences are presented in Table 8. The test of equal slopes and intercepts has been applied to these data. The results (Table 9) indicate that either Ea or lnB, or both are different. The test of equal slopes only is then applied. The shelf-life data based on 25% vitamin 0 loss and perceptible color change have similar slopes and similar activation energies (Table 10). These results suggest that an oxidative reaction is related to the color change of frozen peas and vitamin C, which has strong reducing power, may be involved in that reaction. Table 8. Shelf-life for frozen peas 1 t Perceptible color 25% vitamin C loss difference Temp Shelf-life Temp Shelf-life (C) (me) (C) (mo) -1708 6.73 -1708 21.". “1500 3.27 ‘16.? 18 -1202 1060 ‘1590 12 -90“- 0077 -1298 6 “607 0037 (Kramer, 197a) -309 0017 (Boggs et al., 1960) 81 Table 9. Comparison of kinetic parameters for shelf-life of frozen peas based on perceptible color difference and 25% vitamin 0 loss Test hypothesis: the slopes and intercepts for 2 sets of data are equal Analysis of Variance Source of Degrees of Sum of F Tabulated-F error freedom square SSE(R) 8 2.75965 SSE(R-F) 2 2.78735 670.08 F 001 2 6=27 SSE(F) 6 0.01230 ' ' ' Conclusion: reject hypothesis,“< 0.001 Either the Ea or lnB or both for the 2 sets of data are different (shelf-life data from Table 8) Table 10. Comparison of activation energy for shelf-life of frozen peas Test hypothesis: the slopes for 2 sets of data are equal Analysis of Variance Source of Degrees of Sum of F Tabulated-F error freedom square SSE(R) 7 0.01281 SSE(R-F) 1 0.00011 0.058 F 5 1 6:0.515 SSE(F) 6 0.01230 ° ' ' ...—e .. ..--Mco—n- ...—oo- o—- Conclusion: accept hypothesis,¢¥>0.5 The Ea values for 2 sets of data are similar (shelf-life data from Table 8) 82 Compagison of Activation Energies bgged on Shelf-life Data of Different Products High quality life (HQL) data of five species of berries, with three types of product are listed in Table 11. The activation energy values for these berry products shows that there might be group kinetic characteristics. Three of the five types of berries have more than one set of HQL data. The shelf-life magnitude varies with type of product dramatically. For example, HQL of boysen- berries stored at -17.8 0C are 375: #05 and 605 days for pie filling, bulk and retail products, respectively. The influence of temperature on quality change of the same berries, as described by Ea, may be similar. The results of analysis cfi‘ activation energy for HQL of each berry species are presented in Table 12, 13 and 18. Table 11. High quality life of frozen berries (Guadagni, 1969) and activation energy Product Temperature (0) Activation Standarda Energy Deviation -17.8 -12.2 -6.7 (Ea (IAEa) (days) (days) (days) (kJ/m016)(kJ/mole) Boysenberries(pie) 375 210 us 107.62 29.85 Boysenberries(bulk) 05 125 5 111.92 3.18 Boysenberries(retail) 50 160 35 188.71 5.31 Blueberries(pie) 175 77 18 115.57 20.00 Blackberries(bulk) 630 280 50 128.66 28.37 Raspberries(bulk) 720 315 70 118.u1 21.38 Raspberries(retail) 720 110 18 187.82 .29 Strawberries(bulk) 630 90 18 181.13 7.66 Strawberries(retail) 360 60 10 182.83 2.26 *7 8degrees of freedom = 1 for all Ea. #3 Table 12. Activation energy for high quality life of. frozen boysenberriss Test hypothesis: the slopes of 3 sets of data are equal Analysis of Variance Source of Degrees of Sum of F Tabulated-F error freedom square SSE(R) 5 0.37269 SSE(R-F) 2 0.19679 1.678 F 25 2 3=2.28 SSE(F) 3 0.17590 ° , : Conclusion: accept hypothesis, d:>0.25 3 slopes are similar. Activation Standard Degrees of Correlation Energy Deviation freedom (R2) (kJ/mole) (kJ/mole) 122.76 11.38 5 0.959 ’“ (high quality life data from Table 11) Table 13. Activation energy for high quality life of frozen raspberries "r ”7 Test hypothesis: the slopes for 2 sets of data are equal Analysis of Variance Source of Degrees of Sum of F Tabulated-F error freedom square . SSE(R) 3 0.55275 SSE(R-F) 1 0.86 65 10.588 F 1 1 2=8.53 3 , SSE(F) 2 0.08 10 ' Ah Table 13. (continued) Conclusion: reject hypothesis, ot<0.1 2 Ea values are different. Activationa Standard Degrees of Correlation Energy Deviation freedom (R2) (kJ/mole) (kJ/mole) 153.12 21.85 3 0.9u8 8this is the pooled Ea for 2 sets of data (high quality life data from Table 11) Table 18. Activation energy for high quality life of frozen strawberries Test hypothesis: the slopes of 2 sets of data are equal Analysis of Variance .lm .- -..—...._-l_ . -_ ...- . . v .— _ - .. . - - . -._. --.--........--....._. Source of Degrees of Sum of F Tabulated-F error freedom square SSE(R) 3 0.01287 SSE(R-F) 1 0.00017 0.028 F 5 1 2:0.67 SSE(F) 2 0.01230 - . . Conclusion: accept hypothesis,ot>0.5 2 Ea values are similar. Activation Standard Degrees of Correlation Energy Deviation freedom (R2) (kJ/mole) (kJ/mole) 181.80 3.26 3 0.999 (high quality life data from Table 11) LLS For boysenberries (Table 12), three sets of HQL data are used to determine if the activation energy values are similar, regardless of the product type. The Ea value of retail boysenberries (1h8.711 5.31 kJ/mole) seems to be significantly different from Ea of bulk product (111.92 1 3.18 kJ/mole). The analysis indicates that no significant differences can be judged statistically, because the Ea of pie filling boysenberries (107.62 1 29.u5 kJ/mole) has a very large standard deviation. Analysis of all three sets of data provides an activation energy of 122.76 1 11.38 kJ/mole. In the case of frozen raspberries (Table 13), diffe- rences between activation energy values for HQL of bulk and retail products is quite significant Gx<0.1). A group activation energy (153.12 I 21.85 kJ/mole) could not be used to describe the temperature dependence of HQL for storage of raspberries. The activation energy values for HQL of strawberries (Table 1b) are very close. Based on data for retail and bulk products, the activation energy for HQL of frozen strawberries is 181.80 I 3.26 kJ/mole. The temperature dependence of HQL for frozen berries may be grouped on the basis of the product type (pie fill- ing, bulk and retail). In this case, HQL data of the same product type are input and the test of equal slopes is performed to compare the activation energy values. Conclusions from the analysis (for HQL data of each product type) indicate: (1)The activation energy (Ea) M6 values for pie filling berries are similar indicating that the mechanism of quality degradation for different berries may be the same. The Ea for pie filling berries is 111.59 1 1h.5.kJ/mole (Table 15). (2) Bulk berries may have the same mechanism of deterioration. The activation energy for bulk berries based on four sets of HQL data is 135.02 1 12.38 kJ/mole (Table 16). (3) The Ea values for three retail berries are different 0w<0.01) indicating that the HQL of three retail berries is determined by different mode of deterioration and the group Ea, 172.98 1 7.88 kJ/mole, could not be used to describe the temperature dependence of HQL for storage of retail berries (Table 17). If additional shelf-life data for berries are avails able, an opportunity to identify and eliminate extraneous data and obtain a better estimation for the group Ea value could exist. Data for retail berries are used to illustrate the analysis. From equation (22), the regression equation uSed to fit the three sets of retail HQL data is: Y = BO + 81X1 + 82X2 + 83x3 + Buxu + BSXS (29) The actual regression equation for each set of data is presented in Table 18. If the three sets of data have the same slope, both BM and BS in equation (29) will be zero. The test hypothesis of equal slopes (B8 = BS = 0) is rejected when three sets of HQL data for retail berries are compared (Table 17). By examing the result of regre- ssion coefficients (as provided in the output of the h? Table 15. Activation energy for high quality life of pie filling frozen berries fi‘ v—f ...u .. .v ——.—— Test hypothesis: the slopes of 2 sets of data are equal Analysis of Variance Source of Degrees of Sum of F Tabulated-F error freedom square SSE(R) 3 0.25195 SSE(R-F) 1 0.00613 0.05 F 5 1 2:0.67 SSE(F) 2 0.2u582 ' ' ’ Conclusion: accept hypothesis, dc>0.5 2 Ea values are similar. Activation Standard Degrees of Corrglation R Energy Deviation freedom (kJ/mole) (kJ/mole) 111.59 1h.75 3 0.960 (high quality life data from Table 11) Table 16. Activation energy for high quality life of bulk frozsn berries .1. Test hypothesis: the slopes of u sets of data are equal Analysis of Variance Source of Degrees of Sum of F Tabulated-F error freedom square SSE(R) 7 0.83070 SSE(R-F) 3 0.5739u 2.980 F 1 3 u=u.19 'SSE(F) u 0.25676 ' ’ ’ Conclusion: accept hypothesis,ua>0.1 u Ea values are similar. Activation Standard Degrees of Correlation Energy Deviation freedom (R2) (kJ/mole) (kJ/mole) ~—- 135-02 12.38 7 0.950 (high quality life data from Table 11) he Table 17. Activation energy for high quality life of retail frozen berries ———— —-—~ Test of hypothesis: the slopes of 3 sets of data are equal Analysis of Variance Source of Degrees of Sum of F Tabulated-F error freedom square SSE(R) 5 0.17990 SSE(R-F) 2 0.17350 h0.66u F 01 2 3:30.8 SSE(F) 3 0,006u 0 a : Conclusion: reject hypothesis,at< 0.01 3 Ea values are different. Activationa Standard Degrees of Correlation Energy Deviation freedom (R2) (kJ/mole) (kJ/mole) 172.98 7.88 5 0.990 8this is the pooled Ea for 3 sets of data (high quality life data from Table 11) Table 18. High quality life of retail berries and regression equations v—«-~-.mo—-. .. .. . ... Product Temperature (C) Regression -17.8 -12.2 -6.7 equatiOn (days) (days) (days) Boysenberries 650 160 35 Y (BO+B2)+ (B1+Bu)x1 Raspberries 720 110 18 ,4 ll (BO+B3)+-(B1-+BS)X1 Strawberries 360 60 10 Y — BC + 81X1 (high quality life data from Table 11) M9 computer program), the t-values for Ba and BS indicate that Bk is quite different from zero, while B5 is not (Table 19). Since the three slopes for data of boysen- berries, raspberries and strawberries are 81+Bh’ 811-85 and B1, respectively (Table 18), BS = 0 means that HQL for raspberries and HQL for strawberries have similar slopes (B1), while Bu # 0 means that the slope of HQL for boysene berries is different from the other two. If HQL data of boysenberries are excluded, the result of analysis (for group activation energy) indicate that the remaining retail berries have similar Ea values. The Ea for HQL of retail raspberries and strawberries is 185.18 1 1.80 kJ/mole (Table 20). Table 19. Result of regression analysis for HQL data of retail berries —_ fi: ' 3 = '1' + + + + V 1 1 2 2 Coefficients: F i Bi Variance t Tabulated-t of B. 1 "' 0 8 20066 "' o : h h 5 3 7 17h 13.005,3 5.8u 6 8 0066 . = S u 32 1 1u6 t 15,3 1.25 Conclusion: BM is different from zero (N<:0.005) BS is zero (0L> 0.15) The Ea values of the second and third data sets are similar, while the Ea of the first data set is different from them. (HQL data from Table 18) 50 Table 20. Activation energy for high quality life of retail raspberries and strawberries “...—- Test hypothesis: the slopes of 2 sets of data are equal Analysis of Variance Source of Degrees of Sum of F Tabulated-F error freedom square SSE(R) 3 0.00379 SSE(F) 2 0.00099 ' ’ ’ Conclusion: accept hypothesis, oc>0.1 ‘ 2 Ea values are similar. Activation Standard Degrees of Correlation Energy Deviation freedom (R2) (kJ/mole) (kJ/mole) 185.1u 1.80 3 1.000 .— '_.—- ...—......- .—.--.—-.- -- . m (high quality life data from Table 18) 51 Error Propagation The propagation of error from two sources (activa- tion energy and reference shelf-life) to the predicted shelf-life (Q) is studied by using equation (28). Shelf- life of frozen peas (based on flavor difference) is used as an example. The Ea is 117.11 kJ/mole and the Q0 is 15.2 months at TR = -20 OC (Labuza, 1982). A temperature. history (Figure h) was chosen arbitarily, with T1 = -15 °C for t = 8 months and T = -25 0C for t 1 2 = a months. 2 L1 {'3 Storage time (mo) Figure 8. Temperature history for frozen peas example. Equation (26) was used to evaluate the remainhugshelf- life for frozen peas subject to this storage condition (Q). The result was 2.1 months. Figure 5 shows the influence of error in Ea and Q0 (AEa and«AQo) on the expected error in Q (AQ), as evaluated by equation (28). For example, if 1AEa is 10 kJ/mole andAQO is 1.25 months, AQ will be 1.5 months. It is found that for.AEa within the range of 1 5 kJ/mole, the major part of AQ arises from error of the reference shelf-life (AQO) when the latter has a magnitude of one month or more. On the other hand, if.AQO is con~ trolled within 1 0.5 month, the influence of AEa will be Error of reference shelf-life (AQO) month 52 O i l 1 1 0 5 10 15 20 Error of activation energy (AEa) kJ/mole Figure 5. Influence of errors in activation energy and reference shelf-life on the expected error of predicted shelf-life (4Q) for the example of frozen peas. S3 dominant when its magnitude is larger than 10 kJ/mole. The magnitude of AEa will be more influential if the magnitude of %%z is larger, as shown in equation (25). In this example (frozen peas), %%3 (1)(%%E)t1,T1 and (2)(%%E)t2,T2' From equation (27), it is composed of 2 terms: is found that term (1) is positive, while term (2) is negative (because T2 0H a 4) 'c '3 1.w "-1 .p g. a! (1.. 0.0- - h iii '1 00 I l l I A I -30 -25 ~20 -15 -10 Storage temperature Ti (C) Figure 6. Influence of activation energy and storage temperature on the artial derivative. Ea = 80 (o), 120 (o , 160 (a) ld/mole 55 not different, a TR equal to or higher than the storage tem~ perature (Ti) will give more precise prediction of the remaining shelf-life. If TR lower than Ti must be used, the difference between T1 and TR should be as small as possible. The values of activation energy and standard deviation obtained in the examples of statistical analysis have been used to illustrate the influence of storage duration on the precision of predicted shelf-life. The error from reference shelf-life (AQO) is assumed to be negligible. The remaining shelf-life with expected error for products stored at ~15 0C, for 2, h and 6 months are presented in Table 21. All of the reference temperatures are lower than the storage temperature but the differences are within :1 range of 5 0C. The standard deviatiousof activation energies are less than 15 kJ/mole. The results show that the precisionsof remaining shelf-life are within 1 1 month for all examples, with decrease precision as the duration of storage increases. S6 AhHo>Hpocdmma .Nr em .: magma Song sump hm .macwasnuum masMwm .hoMASpucm cache posuoam acme pom om ham or am? .3? none Gprmpwuom .mefieeenuuee eases some muse message assessed (((I‘ “mowahocmmmhv P.0Hm.m e.ofle.me o.oum.me om.e :e.mme m.ee- 0.:m enema assume . 1.. . l . ..I . . . . u . “mowahcoxowanv w 0+0 0 m o+o me m o+o we mm me No mme m we 0 em meson sash . l.. . IV. . l_. . . . u . Amowaacnnomhonv m o+o e m o+: m m 0+0 0 me x? mm Fee m we m we eases msaaeee use I .I as ea 1 N.o+o.p r.o+m.o om.m w.vwr w.ww1 o.mr mcwawwoSWpr I. l I s o.o+m.m :.o+o.e m.o+m.m mm.ee he.mme w.eeu m.me meeeaenmwwhwm l .1 l maewhsn F.o+e.o e.o+m.m o.o+e.: eh.m wh.mm o.weu m.s unseen eeeaeam e.ouw.m :.qfle.~ m.oum.or mm.e :o.mm 0.0m- o.:e hexane Amnpcosv Amnpcoav Anzacosv AmqskmeAeHoE\wxv on Amnpcosv oE o o8 : 08 N wmq H. mm m o .mofluwasp mwwaoum scams hMAono a pm 0 30 + 3 mmfiumaenm mcwfiwSem nowfigflpoe. PEAS—w Hmwufifin panacea )1 I" 0 men as vcaoum posooao pom posse new: eMWHuMchm mnfinwwscm .wN canoe SUMMARY AND CONCLUSIONS Statistical methods have been developed to evaluate activation energy from shelf-life data based on more than. one end-point quality level, to compare kinetic parameters for shelf-life data based on different quality characte- ristics of a product and to compare activation energies for different products and evaluate activation energy for product groups. A computer program in BASIC programming language has been developed to conduct the statistical analysis. The input for the program is shelf-life data as a function of temperature. The result of the comparison and activation energy with standard deviation are obtained as output. Shelf-life data from published literature were used to illustrate the procedure of analysis. Based on these data and the statistical analysis, the conclusions are: 1. The activation energy for shelf-life of frozen turkey based on three end-point quality levels is 59.0h 1 6.53 kJ/mole. 2. For breaded baconburgers, an activation energy of 32.68 1 2.67 kJ/mole can be used to describe the tempera~ ture dependence of both HQL and PSL at temperatures between ~10 to ~23 0C. For temperatures below ~23 0C, the 57 58 kinetic parameter similar to Ea/R is ~2678.87 1 93.18. 3. 10% vitamin C loss may be used as an index for overall quality change to determine the shelf-life of frozen peas. u. Vitamin C may be involved in an oxidative reaction related to color change of frozen peas. 5. Frozen pie filling, bulk and retail boysenberries have similar mechanismscm quality degradation: the activa~ tion energy for HQL is 122.76 1 11.38 kJ/mole. Frozen bulk and retail raspberries have different modesof deterio- ration. Frozen bulk and retail strawberries have similar degradation mechanisms;tfimaactivation energy for HQL is 181.80 1 3.26 kJ/mole. 6. HQL for frozen pie filling boysenberries and blueberries have similar activation energies (111.59 1 18.75 kJ/mole). HQL for frozen bulk boysenberries, blackberries, raspberries and strawberries have similar activation energies (135.02 1 12.38 kJ/mole). HQL for frozen retail raspberries and strawberries have similar activation energies (185.18 1 1.8 kJ/mole) while HQL for frozen retail boysenberries has a different activation energy. The use of activation energy and standard deviation has been investigated also. Conclusions are: 1. The absolute error of predicted shelf-life is equal to or larger than the absolute error of reference shelf— life and increases with the increase of absolute error in activation energy. 2. The magnitude of error in activation energy will be 59 more influential when the partial derivative of the predicted shelf-life with respect to the activation energy has a larger magnitude. 3. The error of predicted shelf-life increases with the increase of storage duration. For a storage temperature lower than or equal to the reference temperature, the absolute error of predicted shelf-life increases with the_ increase of activation energy and storage temperature. RECOMMENDATIONS For future research, if the statistical analysis is applied to study the mechanism of quality change, variance inherent in factors other than those being monitored, such as product, prefreezing history, packaging and members in sensory panel can be taken into account so that the actual process effects can be reflected accurately. If the statistical analysis is used to obtain kinetic parameters required in the shelf-life model, those factors should be well controlled, or considerable and random samples should be investigated to get generally valid kinetic parameters. The error analysis can be further developed to see if an optimal reference temperature can be obtained for periodic and irregular temperature histories in aspects of both absolute and relative errors of the predicted shelf- life. 60 APPENDIX APPENDIX Program list IGD ummmmmewwu -' Cs 1:" 1:: 92,-" 5:: 1:! .1: l3' *1 1:" II_||.'!.'I'._Eb-‘O—‘O-‘f—‘Ht-‘.—‘ H“ —L .- d '0... 8 I r 1' {it IO. - ., -' .. "1 5. U DIN bale .MLIEI-HIVSBJ.TIa101 GLV‘E‘ INTELEn Hr3.15~.b'n'.0e H3=0 a thHF DISP ”TEMPEPHTURE UNIT” 3.?“ INPUT FH' CLEHR DISP “NHH.LIME=I: 92*" INPUT M9 IF m94=1s HHD m9>1 THEM :00 BEEP E GUTD 16b DISP ”MHH UHIH tET: =SU” HF=0 ‘JR [=1 TO M? 013% “A D? DHTH SFIu 0F LIME ”'I IHVUT ultll DISP ”LIME ”;I FUR J=1 ID 611]) H?=H?+1 013P ”T".J." .3”.1.” r““ INPUT TIIHTI.HI'HTW TIPHTJ=LUGIT1VHT'I IF H$=“£“ THEN 330 HIWHFM=1~HLAIKHPluiil I 3+“? 3.15) (213010 34Ez Altmfizl-Hhieri+;TE 13H HEHT s MEET I ELEHP DISP "I.TEET 0F iDUHL :LGPE“ LISP “3.TE$T 0F SLUPE Hmf Ll HE” DISP ”3 CHLCULRTE 'JLp1051.EL35-.Fy35. .r. COMMON 3L0 FE SELEtI DHE Ui TH EH " INPUT G EWIISISUTU 4EIM~+30-450 FFIHT ualhfi “B .1“ “IEEI 0 F EdUHL SLOPE” Q GDIU 4mm PRINT USING “3.11.:=.r.; - ”TEST OF EUUHL BLUPE HUD Li @L“-"H- TEST OF EDUHL SLUFE“ input data select type of analysis 61 1'51 6"}:0'1'7-31 , 2 T} L”;— '11.; ~ 4 02- (U 03 CU II- --.J C". '2." 4-. r ,1 [L'- L.‘ '3' CD 131 6;: Gt 1:: -. - H H I -. - C3128 PRINT USIHfi “3..L“ u ”CHLLUL HIE CIM1NCWI LLUFE" ‘ IF 6:? THEN 430 F“ 1111 HT U'E- I HIT " . I " 1 HEDUFLT HUUP *3 3MB (.1 k' .' :i- + i '3 I -I C .3. ' PEINT U [Uh “3 «I; “ FULL HUULL” PzMEIlul @ HL=IH+EI 3 LI 721 .. LI I:'- =0 U SIIUI63=U FUF l—I ID I“ HIIIII HI: II=I r H L r I DL3J=1 FUR 1:1 TU 1H BI_I.I.EI"[|:1 MEET I T1=3 E D'lfi=i E DILI~H'I +‘ PFIHT UEIHG “EVII- ‘ 'HHL TIPLE LIHEHP PEGHES [HH' DIEP "PLC 3E MHIT" RI=G [11 4". fl HL-~ CI’IH-v—T If i II II . _II 7.. -! '— IF ‘=3 THEN :TBU IF K=M9 THEH iaU GUSUB 2340 IF G21 THEN 3? 30 PRINT USIHL '3: II: “ I “I- TEBT 0F EUUHL LIME” PRINT USING “3 .I.3.-I.2 “ . "III FULL MUUEL n~ EHHE H3 HEIUHE":"'2J FEUULLU HUUEL“ PF‘IHT US IHG ”w.L u . ”SIHP LE LINEHF‘ FEUFE::IHH LIUCI US 5.4er GUSUB 2348 UISP "DUNE” END FDR I=1 TO 012} ELII=CLIXLDKEH+1-II—IJ»EI+1I E'-. 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