THESIS lwfl’Llirqimh‘C-xw A— ~ ‘ ' ‘ “ .. _ I H I- '1', t» F \ ,,w,.tr a! t s ‘. l i . .I": t R ti" t -3 ‘ b 1.... a .7. ‘ 1 " t .' fl I I?“ f‘q‘ {I if .f‘ .. :ur‘ 1" A; ‘3‘ "17‘ h. T ' a; on .a. “(f 4 ‘ m ‘ . {i ‘3» w» I M m This is to certify that the dissertation entitled MATHEMATICAL MODEL PREDICTION OF MOISTURE CONTENT FOR DRY FOOD PRODUCTS STORED AT CHANGING ENVIRONMENTAL CONDITIONS presented by HOWARD CHARLES EC K has been accepted towards fulfillment of the requirements for M.S. . PACKAGING degree in WM Dr. Bruce R. Harte Major professor Date October 26, I983 MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 ° INN! MSU LIBRARIES itilllllilmll RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. MATHEMATICAL MODEL PREDICTION OF MOISTURE CONTENT FOR DRY FOOD PRODUCTS STORED AT CHANGING ENVIRONMENTAL CONDITIONS BY Howard Charles Eek A Thesis Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE School of Packaging 1983 ABSTRACT MATHEMATICAL MODEL PREDICTION OF MOISTURE CONTENT FOR DRY FOOD PRODUCTS STORED AT CHANGING ENVIRONMENTAL CONDITIONS by Howard Charles Eck The shelf life of packaged food products often depends on the product's moisture content. The prOper selection of packaging materials will result in maintaining a pro- duct's moisture content below a critical level for an extended period. The ability to predict a product's moisture content over time can be a valuable tool in selecting a packaging material. There are several methods utilized throughout the food industry to predict shelf life based on moisture absorption. The most scientific methods include math- ematical models that utilize data for the product and package along with environmental conditions to predict moisture content under constant storage conditions. The research described herein has attempted to predict a packaged products moisture content under varying storage conditions. This research has shown that a product's moisture content can be determined for constantly changing storage conditions by using a mathematical model. DEDICATION This thesis is dedicated to my family, especially my parents, in appreciation and thanks for their assis- tance and guidance through all my academic endeavors. Also, to my wife for her patience and support throughout this work. ii ACKNOWLEDGEMENTS The author wishes to thank Dr. Bruce Harte for his efforts and guidance while serving as major advisor. Thanks are also due to Dr. Dennis Heldman and Dr. .Jack Giacin for their criticisms and suggestions as members of the thesis committee. Also, a special note of thanks to Mr. Ray Tucker for his personal involvement and commitment. iii TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES INTRODUCTION LITERATURE REVIEW EXPERIMENTAL METHODS Initial Moisture Content Sorption Isotherms Package Fill Weight Moisture Vapor Transmission Rates Calculation of Package Permeability Preparation of Shelf Life Samples Experimental Shelf Life Testing Calculation of Experimental Moisture Content MATHEMATICAL MODEL FOR MOISTURE ABSORPTION BASED ON VARYING STORAGE CONDITIONS Flow Diagram: Model for Predicting Moisture Content Under Varying Storage Conditions Sample Calculation RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS Future Research APPENDIX BIBLIOGRAPHY iv Page 23 29 31 40 41 43 55 LIST OF TABLES Table Page 1 Initial Moisture Content, Package Fill 11 Weight 2 Moisture Vapor Transmission Rates I 15 3 Difference Between Experimental and 36 Calculated Results 4 Overall Moisture Content Increases as 37 Percent of Original 5 Test Conditions, Salt Solutions and Equili- 43 brium Moisture Content Results for Sorption Isotherms 6 Experimental Test Conditions, Results of Exp- 45 erimental and Calculated Moisture Content --Product A/l 7 Experimental Test Conditions, Results of Exp- 47 erimental and Calculated Moisture Content-- Product A/II 8 Experimental Test Conditions, Results of Exp- 49 erimental and Calculated Moisture Content-- Product B/I 9 Experimental Test Conditions, Results of Exp- 51 erimental and Calculated Moisture Content-- Product B/II LIST OF FIGURES Figure 1 Absorption Isotherm -- Product A 2 Absorption Isotherm -- Product B 3 Experimental and Calculated Moisture Contents - Product A/Material I Experimental and Calculated Moisture Contents - Product A/Material 11 Experimental and Calculated Moisture Contents - Product B/Material I Experimental and Calculated Moisture Contents - Product B/Material II vi Page 12 14 32 33 34 35 INTRODUCTION Shelf life can be defined as the length of time that a product remains of acceptable quality. A product's shelf life can be dependent on numerous factors. One of the most important is the loss of quality due to absorp- tion of moisture from the external environment. In this case, the prOper selection of packaging materials is required to provide a barrier between the internal and external environment. There are numerous methods used throughout the food industry to determine a product's shelf life. Many of these methods attempt to predict moisture absorption, by the product, to a critical level. One commonly used method is accelerated shelf life testing. This technique subjects a product to stressed temperature and humidity conditions. It is assumed that storage under these condi- tions is equal to a longer storage period under actual distribution conditions. This method often leads to erroneous results and costly overpackaging (Manathunya, 1976). ' Another method of shelf life prediction has been the use of mathematical models. The models are based on calculations utilizing experimental data for packaging l materials, moisture absorption properties of the product and storage conditions to predict shelf life. A model has been deveIOped to predict moisture absorp- tion of dried food product under constant storage condi- tions (Kliment, 1978). This study subjected the package to constant conditions for a defined time period. At the end of the time period the package was transfered into different storage conditions. This sequence was defined as a time-step. The model was used to calculate moisture content Upon completion of each time-step. The model closely predicted moisture content for a time-step at constant conditions. However, error was introduced due to drastic changes in storage conditions between time-steps. The objective of this study was to utilize a similar calculation to predict moisture content under actual distribution conditions. Storage conditions were cons- tantly monitored throughout the study. The model utilized gradual fluctuation for temperature and relative humidity, therefore, reducing the potential for error noted in pre- vious models under simulated storage conditions. The model was used to predict moisture content under known storage conditions for two products. Each product was repackaged in two packaging materials. LITERATURE REVIEW The prOper selection of packaging materials is critical to ensure a product's quality throughout distri- bution. Several techniques are commonly used by the food industry to aid in the selection of packaging mater- ials. Most methods are based on determining the amount of protection offered by various packaging materials. The amount of protection is commonly measured by the product's moisture content increase versus time. Due to the length of time required for many products to complete the distribution cycle, actual shelf life testing can be prohibitive. Thus, predictive methods have been deveIOped to expedite the evaluation and selec- tion of packaging materials. Accelerated shelf life testing was used to reduce the time required for selecting packaging materials (Easter, 1953). This method subjects a product, packed in various packaging materials, to stress conditions of temperature and relative humidity. The product's quality at accelerated conditions is measured and compared to product quality under normal storage conditions. Product quality can be measured by several techniques including moisture content and analytical methods to measure chemical changes within the product. Once the product under accelerated conditions was judged unacceptable a correlation can be drawn between quality at accelerated conditions versus normal conditions. The product's shelf life can then be predicted by assuming a direct relationship of reactions which determine product quality at normal and accelerated conditions. In many cases this assumption is invalid (Manathunya, 1976). An extension of this method to further reduce time required to select packaging materials, can be the selection of materials for products "similar" to a product previously tested under accelerated conditions. This method was based on the assumption that deterioration reactions will occur at similar rates for "similar" products. Again, in many cases this assumption is invalid (Manathunya, 1976). Another method for shelf life prediction has been the use of mathematical models. A significant amount of research has been conducted and published in this area. In most cases, the models are based on experimental data generated on the food product, packaging material and storage environment. The primary advantage of mathematical models is the ability to significantly reduce the time required to make shelf life predictions. Dried food products can deteriorate through several mechanisms depending upon composition and storage condi- tions. Deteriorative mechanisms include, lipid oxidation, nonenzymatic browning, degradation of proteins and textural change such as the loss of crispness and caking (Mizrahi et al., 1970; Quast and Karel, 1972; Labuza et al., 1972). The rate of deteriorative reactions often depends upon the atmosphere surrounding the product. Therefore, a major function of packaging materials is to provide a barrier between the internal and external environment. By knowing the deteriorative mechanism of the product versus the internal environment and equations to determine the rate of permeation from the external environment a shelf life model can be deveIOped. Numerous studies have been conducted to deveIOp models based on the absorption of moisture to a critical level (Charie et al., 1963; Mizrahi et al., 1970; Iglesias et al., 1975). Labuza et al. (1972) reviewed mathematical models based on deteriorative mechanisms and packaging material prOperties for space rations. This work was extended to include the effect of various storage temperatures on water absorption by Iglesias and Chirife (1976). In this work an equation was deveIOped for several food products to be used in predicting shelf life at different storage tempera- tures. Manathunya (1976) developed a model that predicted the moisture content of cereals at two constant storage conditions. The model was based on package permeability and sorption isotherm data generated at the known storage conditions. The model deveIOped proved to be more accurate than accelerated shelf life predic- tions. Several models have been deveIOped to predict shelf life of product that deteriorate through two mechanisms. Quast and Karel (1972) studied potato chip deterioration due to oxidative rancidity and textural changes due to moisture absorption. Karel et al. (1971) deveIOped a similar model to study de- hydrated cabbage. Mizrahi and Karel (1977a) developed an accelerated testing method for predicting the extent of deteriora- tion of moisture sensitive products. The method used data generated from an accelerated test to predict deterioration of the same product for any given package and moisture content combination. The method was later extended to include storage at various tempera- tures by Mizrahi and Karel (1977b). Kliment (1978) developed a mathematical model to predict moisture content under changing storage conditions. The study used a time-step sequence and subjected packaged products to a range of simulated storage conditions. The simulated storage conditions were constant during each time-step. The model was based on data generated for the product and packaging materials over the range of storage conditions. Labuza (1979) reviewed mathematical equations to predict shelf life under fluctuating temperatures in distribution. Zero and first order reaction rates were reviewed. Riemer and Karel (1977) used mathematical models to study vitamin retention of dehydrated tomato juice as a function of time, temperature and moisture content. Villota et al. (1980) extended this work by develOping an equation correlating shelf life of dehydrated vegetables with storage conditions. Labuza (1982) studied the quality loss in whey powders during steady and nonsteady state storage. A comparison was made between the amount of browning and protein quality loss during storage. Aquerre et al. (1983) studied desorption isotherms of rice stored at various temperatures. An equation was reviewed to take temperature into account for sorption isotherms. Paredes et al. (1983) studied the influence of storage on quality of maize meal. The study included comparisons of product quality stored under accelerated humidity conditions. EXPERIMENTAL METHODS The calculation of moisture content based on varying storage conditions requires data generated on the product, package and storage conditions. In most cases, this information can be determined through generally accepted laboratory test methods used throughout the food industry. Initial Moisture Content Numerous acceptable methods exist for determining the moisture content of food products. Most techniques involve removal of water held by the product. Moisture content can be calculated based on weight change. Vacuum oven drying was selected for this work. This technique is widely used throughout the food industry due to the lower oven temperature required. Therefore, the chance of driving off volatile components of the product is reduced. The moisture content was determined for four samples of each product. Approximately ten grams of each sample was weighed into an aluminum dish, and placed into the vacuum oven at 70°C for sixteen hours. The moisture content was calculated from the weight change of the 8 sample. The average moisture content was determined for each product and expressed as 8mg moisture . 100 gms dry product Results are reported in Table l. The following calculation was used to determine the initial moisture content. WI-WFXIOO'Mi WF Where W1 is the initial sample weight, gms. WF is the sample weight after drying, gms Mi is the initial moisture content, 4gms moisture 100 grams dry product Sorption Isotherms A products sorption isotherm can be described as a plot of the amount of water absorbed or desorbed as a function of the equilibrium relative humidity. Two methods of determining sorption isotherms were reviewed. These included, equations for fitting sorption isotherms of foods (Chrife et al., 1978), and determination of sorption isotherms above saturated salt solutions, Wink et al., 1950. The latter method was selected for this work. Approximately 5 grams of each sample was weighed into an aluminum dish and placed over the super saturated salt solution. 10 The initial samples had a moisture content equal to the initial moisture content of product used for experimental testing. The samples would either lose or gain moisture depending upon the surrounding rela- tive humidity. Therefore, the isotherms for this work are actually a mixed absorption/desorption iso- therm. Sorption isotherms were determined at three temperatures. The following calculation was used to determine moisture content for the sorption isotherm. The initial dry weight of the sample is determined by, Wd = W1 1 + Mi IOO 11 Table l--Initial Moisture Content % Moisture for experimental Package Fill testing and sorption weight isotherms gms g moisture TIOO g dry product Product A Product B 4.010 520 .645 450 12 Where Wd is the dry weight of the sample, gms. Wi is the initial weight of the sample, gms. M1 is the initial moisture content, gms H20 100 gm dry mix The moisture content for the sorption isotherm, upon equilibrium can now be determined by: ”f - Wd x Wd MC = 100 where Wf is the sample weight after equilibrium, gms. Wd is the initial dry weight of the sample gms. not is the moisture content gms moisture 100 gms dry mix Results are plotted in Figures 1 and 2. The salt solutions and corresponding relative humidities are listed in the Appendix in Table 5 Package Fill Weight Average package fill weight of product for the pouches were used. Results are reported in Table 1. Moisture Vapor Transmission Rates The Moisture Vapor Transmission Rates (MVTR) were determined for each packaging material at four temperatures. A Mocon IRD-2 Infrared Diffusometer was used following ASTM F372-73. Results are in Table 2. 13 < uosvoumnnaumnDOmw segumuom .H ouawwm Aauwuwasm O>Auafiom aauunfiflwaem so an ON on on as om ON ea c o.m o.o o.m o.NH o.mH uosvoua app wooH ousumwoa w l4 m noncommuuaponu0mw acmuauom .N unawwm Amuaeussm m>uuafiom aauunwfluaam so on on so on as on om oH o Illa . II II. o o.H OOOH I.I 0mm ll oNN I . o.~ o.m uoswoua zoo wooH unaumwoa w 15 Table 2-—Moisture Vapor Transmission Rates Temp RH MVTR*' Package (°F) (z) (g/lOOin2/24 hrs) Permeability g/hr/mm dif Material I 100 90 0.41 4.21x10-4 90 90 0.30 4.02x10-4 (104 in2) ' 80 90 0.19 3.47xlo-4 70 90 0.12 3.09x104 Material II 100 90 0.15 1.49xlo-4 90 90 0.11 1.48x10-4 (108 in2) 8O 90 0.03 1.46x10 70 90 0.05 1.29x10-4 * Average of 5 material samples. l6 Calculation of Package Permeability The package permeability was determined from the moisture vapor transmission rates. These values were determined at four temperatures and expressed as gms HZO/hr./mm of water vapor pressure differen- tial/package. Results are in Table 2. The following is the permeability calculation procedure. MVTR X Package Area (in2 ) 24 hr.(AP) lOO Preparation of Shelf Life Samples Two commercially available dry mix products were selected for this work. Both products had rela- tively low initial moisture contents and varied signifi- cantly in composition. Therefore, the product would be expected to absorb moisture from the surrounding atmosphere at differing rates. Product A was a bakery mix product, while B was a dessert topping mix. The primary mode of failure for both products was the absorption of moisture to a critical level. Two packaging materials were selected to be used with each product. Material 11 was four mil coextrusion of high density and low density polyethylene. Material 1 was a two mil low density polyethylene coextrusion. Since one objective of the study was 17 to determine shelf life of each product in various packaging materials, the product had to be repacked in manually designed packages. Therefore, the study did not take into account the affect of packaging equip- ment and distribution handling on finished package quality. Four sets of samples were produced and labeled, AI and All and BI and B11. A set included six manually designed heat sealed pouches. To ensure consistent product mixture prior to producing the test packages, each product was thoroughly mixed and weighed to the approximate declared commercial net weight. Initial product samples were taken for analytical analysis. Finished package weight of each pouch was measured on a tap loading balance and recorded to the nearest one hundredth of a gram. Experimental Shelf Life Testing The finished packages were transferred to an ambient storage area for actual shelf life testing. Under these conditions, temperature and relative humidity are constantly changing. Therefore, the packages would be expected to gain, and possibly lose moisture at varying rates. The packages remained at ambient condi- tions for one month. Storage temperature and relative humidity condi- tions were continuously monitored with a Honeywell 18 (Model 612XO-HT-00-60-7M-L) recorder. Periodically throughout the test, the samples were weighed for the determination of moisture gain or loss, then returned to the ambient storage area. Calculation of Experimental Moisture Content To calculate the moisture content at each weighing, the dry weight of the package must be determined at time t=O. The moisture content at t-O is equal to the initial moisture content. The dry weight at t=0 can be determined by: Wd‘ wi IUD where W1 is the initial weight of product in the package, and M1 is the initial moisture content of the product at t=0, g moisture 100 g dry product Wd is the dry weight of product in the package at time t=0, grams. The moisture content MC at each weighing can be determined by: Me .. Ef_-_Wr1 x 100 Wd 19 where Wf is the weight of product in the package at time t, grams Wd is the dry weight of product in the package at time t=0, grams. MC is the moisture content of the product at any weighing time t, g moisture 100 g dry product MATHEMATICAL MODEL FOR MOISTURE ABSORPTION BASED ON VARYING STORAGE CONDITIONS The stability of many food products depends largely on the product's moisture content. As the moisture content approaches a critical level (an amount above which delivers a product of unacceptable quality) the reaction rates for various spoilage mechanisms increase. There are numerous methods available to inhibit spoilage due to high moisture content. Several commonly used food processing techniques include drying, freezing, addition of chemical agents along with many others. In these applications, the preservation method revolves around making water unavailable for the moisture related reactions to occur. In many cases these techniques are used for products with an initial moisture content close to or above the critical level. For many food products, the initial moisture content is below the critical level. For product in this category, the prOper selection of packaging material is often an economical means of maintaining a moisture content below the critical level. 20 21 Numerous methods are available to aid in the selec- tion of packaging materials. These methods range from actual shelf life testing to the use of mathematical models. These techniques evaluate the ability of a package to maintain a product below the critical level. The use of mathematical models is a more scienti- fic, economical and less time consuming method to evaluate the functionality of a package as compared to actual or accelerated shelf life testing (Manathunya, 1976). The models take into consideration certain aspects of the product, package and external environmental conditions to predict a product's shelf life (the length of time required for a product to reach the critical moisture content). The mathematical equations for these models have been widely reported throughout the food industry. External storage conditions, temperature and relative humidity, are constantly changing during distribution and warehousing. Additionally, the internal conditions are constantly changing as the system tries to reach equilibrium. By knowing the internal and external storage conditions for a period of time the change in moisture content of the product can be determined. This research reviews a model deveIOped for determining moisture content of a product stored under actual distri- bution conditions. The model calculates moisture content for a defined period of storage time. The period of 22 time was defined as a time step. The calculation is based on the following assump- tions: (1) (ii) (iii) The moisture in the product and headspace of the package is in equilibrium. The seal is perfect and there is no damage to the side wall of the package. The external conditions for each time step are constant. The rate of water vapor permeation through a packag- ing material at a given time as expressed by Gyeszli (1971), 11! (it where "Ul Po Pi P(PO-Pi) (1) weight of water permeated into the package, gms. , time, hours Permeability constant of the package (gms HZO/hr/mm pressure difference) partial pressure of water in air outside the container (mg Hg) partial pressure of water in air in the container (mg Hg) In many cases it is easier to measure percent relative humidity than to measure the water vapor pressure. 23 Selection of Time Step t, hours i Select Temp and RH Conditions for Time Step 1 Determine initial conditions for moisture, sorption isotherms 1 Calculate sIOpe of sorption isotherm at Temp. (T) 1 Recall data required for calculation, Partial Pressure of Water, package permeability at Temp (T) and weight of dry mix 1 Calculate Internal Relative Humidity (water ACtiVitY)* at the end of time step 1 Determine moisture content at end of time step based on sorption isotherm 1 If upper critical limit has not been reached return to initial step. *Internal Relative Humidity can be used interchangedly with water activity Flow Diagram: Model for Predicting Moisture Content Under Varying Storage Conditions, I. _ r [W -|._ 24 This relationship can be expressed by Equation (2) PST 100 pg .H where p = Water vapor pressure at a temperature (mm Hg) H = % Relative Humidity PST = Saturated vapor pressure of water at tempera— ture (T), (mm Hg) the rate of permeation can be expressed by Equation (3). dM - P T at = P 130(HO - Hi) where Ho = percent relative humidity outside the package Hi percent relative humidity inside the package. The water vapor that has permeated into the package will be absorbed by the food. A very small amount will be contained in the package headspace. Assuming that the water within the package reaches equilibrium between the product and the headspace, the moisture content will be function of the internal relative humidity (Hi)- This function can be described by the sorption isotherm of the product. The sorption isotherm can 25 be described as the amount of water absorbed or desorbed plotted against the equilibrium relative humidity (Labuza, 1968). There are numerous ways to express an absorption isotherm. For most dry food products the curve is constant over the critical range (from the initial moisture content to the critical moisture content). In this case the sorption isotherm can be expressed by the slope of the line, %%-. Where dm = difference between the initial and critical moisture content (percent) dH # difference between the initial and critical equilibrium relative humidity (percent). The amount of water vapor absorbed (K1) by the product over time can be expressed as: Wd . dm dHi “1 ‘ "as ' gt— (4) EE‘ 100 where Wd is the dry weight of the product (grams) The remainder of water vapor within the package (K2) will be contained within the headspace. By using the ideal gas law the weight of water within the headspace can be expressed by the equation, H. K2 = 18 . Ps-fllm . KT (5) 26 K2 = Amount of water in the headspace gms P5 = Saturated water vapor inside the package (mm Hg) Hi = Relative humidity inside the package V = volume of headspace in the package (cm3) R = gas constant T = absolute temperature, °K 18 = molecular weight of water. The amount of water gain at any time can be expressed as: KT = Kl + K2 KT = amount of water gain within the package K1 = amount of water gain within the product K2 - amount of water gain within the headspace Previous studies have found that the amount of water vapor contained in the headspace is negligible (Kliment, 1978). Therefore, for this work the total amount of water within the package at any point in time was assumed to be absorbed by the product. Thus, KT = K1 (7) 27 Substituting Equation (7) into Equation (4) and solving for KT gives: 'm-meee dt 100 dh dt (8) The rate of water vapor permeation equals the change in water vapor content within the package, or, did ' dKT dt dt From equations (3) and (8), we have, dli 1 dm.Wd 3E ( ) (9) . PsT - - TOO The equilibrium relative humidity inside the package is the only factor that changes with time, so, P8T< -n>-dm"d .‘mi 10) P I00H° ' m at— ( By rearranging Equation (10) gives “"1 - (no - Hi) (1’ Parfl ) m 3?— all”? Let Wd am 3° #31? ‘- B (so - Hi) (11) dHi _ Bdt‘ Ho‘Hi 28 Integrate Hi from O to t hours, Hf (In t ___JL_ l = Bdt Hi Ho-Hi o HO‘Hi = eBt ‘ HO'Hf Where Hf = final relative humidity at the end of the time step. or HO-Hf = e'Bt Ho‘Hi Solving for Hf gives: Hf = Ho-(Ho-Hi)e'Bt (12) By knowing the internal equilibrium relative humidity at the end of the time step for the given temperature, the moisture content can be determined from the sorption isotherm. If the calculated moisture content is less than the critical moisture content the calculation can be repeated for the next time step. The following is a sample calculation using the model to determine moisture content. 29 Sample Calculation Determine the moisture content of variable B/I stored at 80°F and 60% relative humidity. Length of the time step is 12 hours. Hf = Ho - (Ho-Hi)e‘Bt where where P So, [)8 Va Hf = Final internal relative humidity at the end of the time step, (%) Ho =External relative humidity (%) t = length of the time step (hours) Wd dm Total package permeability, gms/hr/mm vapor pressure differential/package Inverse of the sIOpe for the sorption isotherm Saturated vapor pressure of water at temp (T). Weight of dry mix in the package, gms. 60% 38.50%, from sorption isotherm at a moisture content of .671% 12 hours .000347 gms/hr/mm vapor pressure differential package 26.221 mm 450 gms 3O dH = 10% (A equilibrium relative humidity from sorption isotherm) dm = .11% (A moisture content from sorption isotherm) Therefore = .000347 (26.221) 10 B 450 Til = 1.838 x 10-3 Now HF = 50_(60-33.50)e-l.838x10‘3(12) HF = 60-(2l.5)e- 022050 HF = 60-(21.5) .978 HF = 38.97 The moisture content at the end of the time step can be determined from the sorption isotherm. In this case, the moisture content corresponding to an equilibrium relative humidity of 38.97% is .679%. Since the critical moisture content has not been attained the calculation is repeated for the next time step. RESULTS AND DI SCUSS I ON The moisture content for all variables was calculated for each time step. The length of a time step varied depending upon the magnitude of change for the storage conditions. As seen from Tables 6,7,8 and 9 the duration of a time step ranged from 6 hours up to 38 hours. Storage conditions were continuously monitored and recorded every two hours. There is some judgment required to determine the length of a time step under changing conditions. In general, a time step was defined as the length of time required for a temperature change of approximately 10°F. The temperature and relative humidity recordings over the period were averaged to arrive at the storage conditions for that time step. Generally, the length of a time step was within the range of 8 to 14 hours. The calculated moisture contents for the products compared very well with the experimental results. The results for these values over the storage period can be seen in Tables 6,7,8 and 9, and plots are in Figures 3,4,5 and 6. The difference between the average experi- mental and calculated moisture contents can be seen in Table 3. The results are plotted in Figures 3,4,5 and 6. 31 H amwuoumz\< uoavoum u mucoucoo ouaumwoa voumaaono one Hmucoawuomxm .m Opawwm Amuaocv oawu amuOH coo com ooe 32 com com OCH 6 woumHDOHmo o Amocoawuoaxm x m.¢ m pontoon app wooH ousumwoa w 33 HH Hmwuoumz\< pompoum : mucoucoo upsumwoa vmumuso~mo one Hmuaoawuomxm .q opawwm Amuzonv wage Hmuoe coo com ooq oom cow coH o ll x: x \ II voumuooamo c we \ \ Hmucoawpomxm a. 9\ m.¢ A uuawoum app wooH unaumwoa w 34 H Haeeaeaz\m soaeoea u mucoucoo ousumfioa voumasofimo can Heucoawuoaxm .m ouswwm Amuaosv wage amuoe coo com ooq com com ooH o mm. in woumgofimo O Hauaoafiummxm x mm. pontoon mum wOOH uuaumwoa w 35 Ha Hmmpoumz\m uoaooum u mucoucoo unaumfioa ooumaoo~mu can Hmuaoawuooxm .c ouawwm Amuaosv mafia Hmuoe coo com oce com com ocH c mm. no. It: ooumfiaofimoo amazon—H.896 x no. nonoouo Rho wcoH ououmfioa w 36 Table 3: Difference Between Experimental and Calculated Results % Difference Product/ Final % Moisture Kexp'Kcalx 100 Material Cal. Exp Kexp A/I 4.325 4.214 2.63 A/II 4.254 4.103 3.68 B/I .785 .780 .64 B/II .717 .724 .97 The calculated moisture content increases were very small between time steps for all products. A more significant percentage increase was noted for the overall storage period due to the additive effect of storage time. The percentage increase for each interval for the four variables can be seen in the appendix in Tables 6,7,8 and 9. The overall moisture content increases can be seen in Table 4. The percent difference between experi- mental and calculated moisture content can also be calculated by determining the percent increase from the initial moisture content. These results can be seen in Table 4. 37 o~.w c~.aa om.¢H o~.~H o.oH HH\m no.m HN.H~ oo.H~ oo.c~ on.oH H\m Nca mo.c < son quoxozv w: willl ooH xAquoxozvuAHanmozv ocH onoozuwz ooa x.mmmzuwz Ago Ago Ago ounvoum mommouocH acouaoc muoumwoz mummuocH ucoucoc ummouocH uaoucoo Hmucoawpooxm one .on0 ousummoz .oHoc Ousumwoz .oxm .>< CQO3UGD mocmhwmm mn— amnawwuc mo unmouom mm mommuuocH uaouaoc ouaumwoz Hamuu>c "q oHnme 38 The moisture gains within a set of packages was very close. This can be exemplified by Product A/II, where the experimental moisture content increase ranged from a low of 2.02% to a high of 2.54%. The calculated moisture content increases corres— ponded to the experimental results for Products B/I and B/II, however, not as close for product All and A/II. This is further demonstrated in Table 4 by calcu- lating the percent difference between the calculated and experimental moisture content increases. Calculating percent difference by this method results in a greater difference for all products as compared to the previous method. It is the author's Opinion that calculating a percent difference based on total moisture content is more realistic than calculating the difference based upon the moisture content increase. In many cases the moisture content increase is quite small and may well be within the experimental error for determining moisture contents. Therefore, this method will result in a greater percentage difference between experimental and calculated results. As expected, Material II provided a superior moisture barrier to both products when compared to Material 1. This was illustrated in Table 4. SUMMARY AND CONCLUSION In many cases, a product's moisture content is a critical factor in determining the product's shelf- life. This is most important when an increase in mois- ture content leads to a reduction in shelf—life. The ability to predict the moisture content of a product enables a packaging scientist to prOperly select materials that will maintain a moisture content below a critical level. The mathematical model discussed in this work proved to be a good method for predicting a product's moisture content under changing storage conditions. The model used actual storage conditions for temperature and relative humidity along with laboratory data for the product and packaging materials to calculate the moisture content increases for a defined time step. By knowing a product's critical moisture content the time required to reach this level can be determined. The model provides a quick, inexpensive means of determining moisture content when compared to commonly used food industry methods such as, accelerated shelf- life and actual storage testing. These methods are often quite time consuming and can inhibit a company's 39 4O ability to rapidly introduce new products. A result of this time constraint may lead to the selection of packages that provide a moisture barrier far greater than required for a product. The model can also be utilized to evaluate the effectiveness of packaging materials for existing pro- ducts, in addition to being used to design packages for new products. In this case, the scientist can readily evaluate numerous products to identify potential cost saving opportunities. Alternate packaging materials can be laboratory tested to determine the package perme- ability; These data can be used in the model to determine the ability to maintain a moisture content below the critical level. Temperature and relative humidity data must be gathered for the distribution system. For this work storage conditions were continuously monitored and recorded every two hours. Selecting the length of a time step involves some subjectivity on the part of the scientist. Several factors must be taken into consideration when selecting a time step. This is necessary because of the assumption that all conditions are constant during a time step. Factors to be considered include; the length of time required for a product to cycle through the distribution system and fluctuation of temperature and relative humidity over time. 41 Shorter time steps will result in a more gradual fluctuation of storage conditions. Therefore, a shorter time step will reduce error because it will be more likely that all conditions will be constant. For products that require a significant amount of time (1-2 years) to cycle through distribution, an enormous number of time steps could be required. The feasibility for accumulating the necessary temperature and relative humidity data for the model must be determined. The length of a time step can then be determined. A product that cycles through distribution in a much shorter time frame would require a shorter time step because of the greater potential for error. In both cases the scientist must take into consideration the amount of fluctuation for temperature and relative humidity. The greater the fluctuation in a storage environment the shorter time step would be required to minimize error. Future Research This application of shelf life modeling to con- stantly changing storage conditions is new. The follow- ing identifies several areas for future studies utilizing mathematical models to determine the effect on a product's quality. 1. Shelf life modeling for reaction rates that are moisture related, such as mold and microbial growth 42 and enzymatic activity. 2. Modeling of deteriorative reactions that are not moisture related, such as, vitamin degradation due to temperature fluctuations. 3. Effect of secondary, tertiary and unitized packages on the model for constantly changing storage conditions. 4. DeveIOping a similar model to predict moisture loss for high moisture or liquid food type products. 5. Identify and evaluate alternate means of determining storage conditions for ambient distribution systems, such as United States Weather Bureau data. 6. Deve10p a mathematical model to predict shelf life based on multiple modes of failure. APPENDIX Appendix Table 5: Test Conditions, Salt Solutions, and Equilibrium Moisture Content Results for Sorption Isotherms Equilibrium Moisture Content* Relative Ag moisture Temperature Salt Humidity 100g dry product (°F) Solution (%) Product A B 1000 NaCl 75 15.74 3.20 NaNO 63 7.3 1.25 Mg(N03)2 51 6.04 .75 K2C03 41 5.26 .60 MgClz 32 4.38 .51 K202H302 23 3.52 .45 LiCl 11 2.00 .25 85 NaCl 75 12.36 2.15 NaNOz 64 7.64 1.38 Mg(N03)2 52 6.16 .95 K2CO3 42 5.26 .71 MgClz 32 4.60 .60 KC2H302 23 3.95 .50 LiCl 11 2.20 .35 * Average of three samples 43 Table 5: 44 continued Temperature Salt Relative Humidity Equilibrium Moisture Content* g moisture 100g dry product (°F) Solution (%) Product A B 72 NaCl 75 12.36 3.4 NaNOz 65 8.70 1.6 Mg(NO3)2 52 6.83 1.05 K2C03 44 6.38 .82 MgClz 33 5.26 .70 KC2H3°2 23 4.38 .60 LiCl 11 2.25 .50 * Average of three samples. 45 ecH.< 55H mm «c oH Hoc.¢ cmH co on nH moc.q cmH o: mm «H mo. cMH mm on mH mc.q oNH on no NH Hoc.¢ «HH No mm HH cmc.¢ NcH No No cH nqc.< oo co oo o Hco.¢ «c on No c mmc.¢ us no mm m mo.q co co mo o mo.¢ co on om m omc.q co No «n q «No.q on no mm m HNc.¢ cH no om N ch.q NH mm ow H Ho.q o c owmuo>< anm 304 Aaeaam are so Aaaaam Rae so Aaeaomo Ago Aeoc Ha>eeaae ucoucoc acoucoo eunumHoz oEHH Hooch quoHEom came opoumHoz HmucoaHpooxm O>HuoHom oouoHoonc H\< uooooum nu mucoucoc opoumHoz ooumHooHoo ocmRHmucoaHpooxm mo muHamom .mcoHuHococ umoe HmuaoaHuooxm an mHowH 46 moHcamm me mo owmuo>< r omm. «HN.¢ mo~.¢ coH.c Hem No co mo Hm.¢ on mm Hm No o~.¢ NHm mm «o Ho w~.¢ Hon on HR co mh~.¢ Hoe ow mm on 5N.¢ Rho Ho #5 cm oN.< soc on «c mm mn~.< moo on on on m~.¢ mqq No mm mm moH.¢ Hmo no on on H< stm 304 Aaaaam see so Aaaaam a..5 so Amusemo Ago Amac Haaaaaae ucouaoc ucoucoc ououmHoz oEHB Hooch huHoHaom came ousumHoz HmucoaHpooxm o>HuwHom ooumHaonc HH\< DonoOpm nu mucouooc opaumHoz ooumHoono ocahbmucoaHuooxm mo muHsmom .mcoHuHococ onus HmuaoaHuooxm no oHomH 48 moHoEam me mo owopo>< k cmN.¢ moH.q NHH.¢ Hoc.o Hon No co me mN.q on mm Hm No «N.q NHm mm «c He MN.¢ How on Hm co MN.¢ Hoq me no on NN. who Ho «5 cm NN.¢ Noe on em mm HN.q moo on on on cN.q moo No mm mm ocH.¢ qu no on «m mcH.¢ qu no cc mm cH.¢ HHq mo Nm Nm mNH.q mom on no Hm qu. cmc.¢ oc.¢ ooc.¢ mom on no cm oH.q com on on oN mmH.¢ Hom on co cN mH.q mom no No NN qu.q 5mm co mm oN «H.¢ mNm on «m mN cmH.¢ HHm on HR «N mNH.¢ ooN no No mN NH.¢ ncN on co NN NH.< ooN on on HN cHH.q oQN om cm cN mHH.q omc.¢ ooc.q ch.¢ oMN mo mm oH ocH.¢ ooH mm mm cH cH.< me co mm NH ooaaHucoo no oHan 49 ooo. 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Nom mm co Hm moo. «oN. mHN. coo. mom «m No on moo. oom om oN oN moo. Hom om oo oN moo. m«m mm No NN coo. Nmm oo mN oN ooo. mNm mm «o mN ooo. HHo «m HN «N ooo. ooN m« No mN Noo. NoN «m oo NN oNo. ooN om cN HN mNo. o«N om co oN NNo. Hoo. NoN. mNo. ooN mo mN oH oNo. ooH mm mo oH ooo. ooH co MN NH coacHucoo "o oHan 53 Integration Procedure From Equation (10) (Wd 0 dm) CIHi ..--I I8(I)HO " (I I8(T))' H m T — — LEE. A = Wd ’dm dfi' B = F PS(T)HO 0 ll "UI P3(T) Equation (10) can be rewritten A = - 7FF' B CH Dividing by A gives d“' = 2 -E H at A A or - 9. E-H A (c ) Now multiplying by dt and dividing by (g - H) gives, 0. :1: i e % dt -H olw 54 Let .9 = R P (T A d m H B = F P T H _C. - S( > O a: Ho P Ps(T) Solving for d Hi B C"H Hf Hf I 111_ = -1n(Ho-H)I Hi 2 -Ii Hi = -ln(Ho-Hf ) + ln(Ho-Hi) Now 1n = Ho-Hi = E (t -t1) Substituting for.% gives __ (t) HO-Hf wa dm/dH BI BLI OGRAPHY BIBLIOGRAPHY Anon, 1971. Computer Prediction of Food Storage. Mod. Pkg. 44(8)54. 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