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The application of chemical kinetics is well known in food engineering, such as the use of Arrhenius plots and D- and z-values to characterize linear microbial inactivation kinetics by thermal processing. The emergence and growing commercialization of nonthermal processing technologies in the past decade provided impetus for the development of nonlinear models to describe nonlinear inactivation kinetics of foodborne microbes. One such model, the enhanced quasi-chemical kinetics (EQCK) model, postulates a mechanistic sequence of reaction steps and uses a chemical kinetics approach to developing a system of rate equations (ordinary differential equations) that provide the mathematical basis for describing an array of complex nonlinear dynamics exhibited by microbes in foods. Specifically, the EQCK model characterizes continuous growth–death–tailing dynamics (or subsets thereof) for pathogens such as Staphylococcus aureus, Listeria monocytogenes, or Escherichia coli in various food matrices (bread, turkey, ham, cheese) controlled by “hurdles” (water activity, pH, temperature, antimicrobials). The EQCK model is also used with high pressure processing (HPP), to characterize nonlinear inactivation kinetics for E. coli (inactivation plots show lag times), baro-resistant L. monocytogenes (inactivation plots show slight lag times and protracted tailing), and Bacillus amyloliquefaciens spores (inactivation plots show protracted tailing; HPP also induces spore activation and spore germination). We invoke further chemical kinetics principles by applying transition-state theory (TST) to the HPP inactivation of L. monocytogenes and develop novel dimensionless secondary models for temperature and pressure (TST temperature and TST pressure) to estimate kinetics parameters (activation energy E a and activation volume ∆V ‡), thereby offering new insights into the inactivation mechanisms of pathogenic organisms by HPP.

Prior to the development of transition state theory, the Arrhenius equation was the principal relationship used in describing the temperature dependence of chemical reaction rates. Research into determining the theoretical basis for the Arrhenius parameters A (pre-exponential factor) and Ea (activation energy) led to the development of transition state theory and the Eyring equation, whose central postulate is a hypothetical transient state called the activated complex that forms through interactions between reactants before they can become products during the process of a chemical reaction. It is from the perspective of transition state theory that we develop two secondary models to reflect the effects of temperature and of high pressure on microbial inactivation by the emerging nonthermal technology of high pressure processing (HPP), and we designate these as transition state (TS) models TST and TSP, respectively. These secondary models are applied to data obtained with two primary models, the enhanced quasi-chemical kinetics (EQCK) differential equation model and the Weibull distribution empirical model, that were used to evaluate nonlinear inactivation kinetics for baro-resistant Listeria monocytogenes in a surrogate protein food system by HPP for various combinations of pressure (207–414 MPa) and temperature (20–50 °C). The mathematical relationships of TST and TSP involve primarily the unique model parameter called “processing time parameter” (t p ), which was developed to evaluate inactivation kinetics data showing tailing. These detailed secondary models, as applied to the parameters of the EQCK and Weibull primary models, have important ramifications for ensuring food safety and the shelf life of food products and support the growing uses of HPP for the safe preservation of foodstuffs.

The enhanced quasi-chemical kinetics (EQCK) model is presented as a methodology to evaluate the nonlinear inactivation kinetics of baro-resistant Listeria monocytogenes in a surrogate protein food system by high-pressure processing (HPP) for various combinations of pressure (P= 207 to 414 MPa) and temperature (T= 20 to 50 °C). The EQCK model is based on ordinary differential equations derived from 6 “quasi-chemical reaction” steps. The EQCK model continuously fits the conventional stages of the microbial lifecycle: lag, growth, stationary phase, and death; and tailing. Depending on the conditions, the inactivation kinetics of L. monocytogenes by HPP show a lag, inactivation, and tailing. Accordingly, we developed a customized, 4-step subset version of the EQCK model sufficient to evaluate the HPP inactivation kinetics of L. monocytogenes and obtain values for the model parameters of lag (λ), inactivation rate (μ), rate constants (k), and “processing time” (tp). This latter parameter was developed uniquely to evaluate kinetics data showing tailing. Secondary models are developed by interrelating the fitting parameters with experimental parameters, and Monte Carlo simulations are used to evaluate parameter reproducibility. This 4-step model is also compared with the empirical Weibull and Polylog models. The success of the EQCK model (as its 4-step subset) for the HPP inactivation kinetics of baro-resistant L. monocytogenes showing tailing establishes several advantages of the EQCK modeling approach for investigating nonlinear microbial inactivation kinetics, and it has implications for determining mechanisms of bacterial spore inactivation by HPP. Practical Application: Results of this study will be useful to the many segments of the food processing industry (ready-to-eat meats, fresh produce, seafood, dairy) concerned with ensuring the safety of consumers from the health hazards of Listeria monocytogenes, particularly through the use of emerging food preservation technologies such as high-pressure processing.

A study of a new military ration, the Meal, Ready-to-Eat, Individual (MRE) after 2 yr of storage at four different temperatures was conducted. The main emphasis is on the statistical analysis of data on consumer-acceptance, in particular the estimation of shelf-life. The principal finding is that the ration as a whole can be stored for at least 2 yr at temperatures up to 38°C without serious loss of acceptance, although a few individual items may be affected.

The Quasi-chemical differential equation model was used to evaluate the linear and non-linear (shoulders and tailing) inactivation kinetics of Escherichia coli and Listeria monocytogenes in a surrogate food system using the novel non-thermal technology of High Pressure Processing (HPP). Kinetics data were collected over the ranges of 30,0 0-60,000 psi (200-620 MPa) and 20-60 °C. The Quasi-chemical model fit the inactivation data, and secondary models were established by relating the inactivation rate constants with pressure and temperature. These models provide the basis 1) to predict the pasteurization of these pathogens in pressure-treated foods, and 2) to develop predictive models for future work involving spores, that can become incorporated in the ComBase initiative.

Knowledge of the mathematical properties of the quasi-chemical model [Taub, Feeherry, Ross, Kustin, Doona, 2003. A quasi-chemical kinetics model for the growth and death of Staphylococcus aureus in intermediate moisture bread. J. Food Sci. 68 (8), 2530-2537], which is used to characterize and predict microbial growth-death kinetics in foods, is important for its applications in predictive microbiology. The model consists of a system of four ordinary differential equations (ODEs), which govern the temporal dependence of the bacterial life cycle (the lag, exponential growth, stationary, and death phases, respectively). The ODE system derives from a hypothetical four-step reaction scheme that postulates the activity of a critical intermediate as an antagonist to growth (perhaps through a quorum sensing biomechanism). The general behavior of the solutions to the ODEs is illustrated by several examples. In instances when explicit mathematical solutions to these ODEs are not obtainable, mathematical approximations are used to find solutions that are helpful in evaluating growth in the early stages and again near the end of the process. Useful solutions for the ODE system are also obtained in the case where the rate of antagonist formation is small. The examples and the approximate solutions provide guidance in the parameter estimation that must be done when fitting the model to data. The general behavior of the solutions is illustrated by examples, and the MATLAB programs with worked examples are included in the appendices for use by predictive microbiologists for data collected independently.

The slow reaction between peroxodisulfate and formate is significantly accelerated by ascorbate at room temperature. The products of this induced oxidation, CO2 and oxalate (C2O2–4), were analyzed by several methods and the kinetics of this reaction were measured. The overall mechanism involves free radical species. Ascorbate reacts with peroxodisulfate to initiate production of the sulfate radical ion (SO•–4), which reacts with formate to produce carbon dioxide radical ion (CO•–2) and sulfate. The carbon dioxide radical reacts with peroxodisulfate to form CO2 or self-combines to form oxalate. Competition occurring between these two processes determines the overall fate of the carbon dioxide radical species. As pH decreases, protonation of the carbon dioxide radical ion tends to favor production of CO2.

The “quasi-chemical” kinetics model accounts for all 4 phases of the microbial lifecycle based on a proposed series of chemical rate equations. The model fits continuous growth-death kinetics for Staphylococcus aureus in intermediate moisture bread in various conditions of water activity, pH, and temperature. Growth rates evaluated using the quasi-chemical model are compared with values obtained with the Gompertz model. Kinetics data obtained with the quasi-chemical model are integrated with a probabilistic approach to estimate the boundary between growth and no-growth conditions. Continuous modeling of microbial growth/death kinetics in actual foods advances predictive modeling that conventionally separates growth and death models.

Instrumental color measurements were investigated as a method of assessing ration quality after long term storage. Six items were stored at five temperatures from 4 to 60C and withdrawn after 7, 31, 91, 182, 365, 730 and 1095 days. They were assessed for color (CIELAB L*a*b* values, hue angle and chroma), consumer acceptance, and attribute ratings. Color variables varied consistently for only applesauce and cheese spread. L* values were investigated to measure acceptability. Consumer acceptance ratings were divided into two equal sets. The first was used to determine the relationship between the mean score and L* value, then to predict acceptability. Predictions were compared to the second set of scores and type 1 and 2 error rates determined. Although error rates were high by usual statistical standards, they were equal to or lower (although not significantly different) than those based on the mean consumer acceptance score.

The distribution of fracture intensities occurring during compression of puffed corn meal extrudates was described using an exponential function, and parameters from that analysis were used as fracturability indices. Since “jagged” or oscillating stress-strain functions are typical for porous and brittle materials, fracture intensities were determined by measuring the abrupt, sequential reductions in stress produced during compression. Both distribution exponent and cumulative fracture stress correlated strongly with fracturability measured by other techniques, including fractal and Fourier analysis of stress-strain functions. Distribution parameters also indicated textural differences due to process parameters (structural modification through addition of different levels of sucrose) and storage conditions (equilibration at various relative humidities).

This paper presents a formulation in dimensionless terms of the equations pertaining to first-order processes that follow Arrhenius kinetics, and applies them to problems in food thermoprocessing. the need for it arose from attempts to understand the relation between destruction of pathogenic microorganisms and formation of chemical markers in the food. During processing food is subjected to a thermal pulse that affects the markers and microorganisms. the dimensionless formulation makes it convenient to obtain equations expressing the relation between the decadic reduction in microbial population and the rise in marker concentration for temperature-time profiles of forms hitherto not conveniently calculable. Measurements of chemical markers can be used with these formulas to predict destruction of microorganisms. Examples are given of calculations demonstrating the feasibility of this approach when a single marker is monitored.

Maple-filled French toast (MFFT) enrobed sandwiches are intermediate moisture (IM) bi-layered foods formulated with ‘hurdles’ to prevent the growth of Staphylococcus aureus for product shelf-life (3 years at T = 80 °F ≈ 27 °C). A novel approach to accelerate microbial challenge testing was investigated. The Quasi-chemical model’s secondary model (‘growth-no-growth boundary’) was used to guide the formulation of the MFFT. The novel 3-year test protocol used a five-strain cocktail of S. aureus to challenge MFFT with variations in storage temperature, product formulation, and packaging atmosphere. The inactivation kinetics were evaluated with the modified Quasi-chemical model and determined to be more rapid at T = 35 °C than at T = 25 °C. Removing the oxygen scavenger did not promote S. aureus growth. These results provide a basis for standardizing accelerated microbial challenge tests for new varieties of IM enrobed sandwiches in the pipeline to save time, money, and labor, while ensuring food safety.

Corn meal extrudates were produced using process conditions designed to obtain various product structures. Extrudates were infused with a particle-containing, high melting point lipid suspension, a process used to produce calorically dense components for military rations. Image analysis was used to estimate the degree of particle penetration into the extruded matrices and determine cell size distributions. Infusion uniformity was correlated with structural attributes, such as density, expansion, and average cell size.

Vacuum infusion of porous food matrices (i.e., extruded starch crackers) with calorically dense suspensions of food powders in fat was investigated in an ideal system. Experiments involved infusion of model matrices with lipid suspensions consisting of milled, size-characterized sucrose in corn oil. The influences of pore size/particle size ratio and suspension concentration on the penetration of liquid and particles were determined. A mathematical model that allows prediction of future behavior was fitted to the test results. The model indicates that dry-weight particle penetration is strongly correlated with pore size and independent of initial particle concentration. Infusion of liquid plus particles was increased by raising porosity and lowering particle concentration.

Predictive microbial models generally rely on the growth of bacteria in laboratory broth to approximate the microbial growth kinetics expected to take place in actual foods under identical environmental conditions. Sigmoidal functions such as the Gompertz or logistics equation accurately model the typical microbial growth curve from the lag to the stationary phase and provide the mathematical basis for estimating parameters such as the maximum growth rate (MGR). Stationary phase data can begin to show a decline and make it difficult to discern which data to include in the analysis of the growth curve, a factor that influences the calculated values of the growth parameters. In contradistinction, the quasi-chemical kinetics model provides additional capabilities in microbial modelling and fits growth-death kinetics (all four phases of the microbial lifecycle continuously) for a general set of microorganisms in a variety of actual food substrates. The quasi-chemical model is differential equations (ODEs) that derives from a hypothetical four-step chemical mechanism involving an antagonistic metabolite (quorum sensing) and successfully fits the kinetics of pathogens (Staphylococcus aureus, Escherichia coli and Listeria monocytogenes) in various foods (bread, turkey meat, ham and cheese) as functions of different hurdles (a(w), pH, temperature and anti-microbial lactate). The calculated value of the MGR depends on whether growth-death data or only growth data are used in the fitting procedure. The quasi-chemical kinetics model is also exploited for use with the novel food processing technology of high-pressure processing. The high-pressure inactivation kinetics of E. coli are explored in a model food system over the pressure (P) range of 207-345 MPa (30,000-50,000 psi) and the temperature (T) range of 30-50 degrees C. In relatively low combinations of P and T, the inactivation curves are non-linear and exhibit a shoulder prior to a more rapid rate of microbial destruction. In the higher P, T regime, the inactivation plots tend to be linear. In all cases, the quasi-chemical model successfully fit the linear and curvi-linear inactivation plots for E. coli in model food systems. The experimental data and the quasi-chemical mathematical model described herein are candidates for inclusion in ComBase, the developing database that combines data and models from the USDA Pathogen Modeling Program and the UK Food MicroModel.

The reduction of iodine by phosphorus(I) (hypophosphorous acid) yields iodide and phosphorus(III) (phosphorous acid). The reaction is autocatalytic in hydrogen ion, autoinhibitory in iodide ion, and zero order in iodine over a wide concentration range. We return to the original article by A. D. Mitchell that presents a mechanism to account for these facts and devise a method to integrate numerically the complex rate equation for this reaction. Integration is combined with nonlinear curve-fitting to determine the rate-limiting rate constant and provide a fit to the data that is considerably improved over the original. The entire exercise provides an excellent case study for introducing the student to numerical integration and curve-fitting of complex nonlinear reaction dynamics.

This paper describes a statistical procedure for estimating the 12D dose in the radiation-sterilization of canned food, using data from an inoculated pack experiment. The method assumes a two-parameter distribution, of which the shifted-exponential is taken as a prototype, and uses the maximum-likelihood principle to estimate the parameters and hence 12D. The procedure is embodied in a computer program which estimates 12D and provides confidence limits on both 12D and the kill at zero dose. The method is described and illustrated by three examples and appears to obtain more information from the data than other methods do.

Average consumer-acceptance scores on the military ration called the Meal, Ready-to-Eat, at four temperatures and storage times through 60 months fitted an Arrhenius-like mathematical model which estimated the dependence of average score on storage time and temperature as well as the effect of temperature on shelf life. Results are presented for individual items and for the most important classes of items.

This paper describes a mathematical model for the enzymatic hydrolysis and fermentation of cellulose by Trichoderma reesei. The principal features of the model are the assumption of two forms of cellulose (crystalline and amorphous), two sugars (cellobiose and glucose), and two enzymes (cellulase and β-glucosidase). An inducer–repressor–messenger RNA mechanism is used to predict enzyme formation, and pH effects are included. The model consists of 12 ordinary differential equations for 12 state variables and contains 38 parameters. The parameters were estimated from four sets of experimental data by optimization. The results appear satisfactory, and the computer programs permit simulation of a variety of system changes.

This paper presents a new mathematical analysis of the inoculated pack method for determining the effectiveness of radappertization (radiation-sterilization) of canned food. A general theory is developed which makes it clear that two interrelated distribution functions, the probability of organism death and the probability of can sterilization, play important parts in the process. A critique is given of the Schmidt-Nank method for calculating the 12D dose, and modifications of both the experimental design and data analysis are suggested. Two examples are worked out, one illustrating the use of the suggested method, the other using the theory and an experiment on canned ham to show that the mode of organism death is unlikely to be exponential.

This paper contains the results of an approximate analysis of the axially symmetric vibrations of deep spherical elastic shells. The approximation is based on the asymptotic formulas for Legendre functions of large degree. The results are relatively simple approximate formulas for the natural frequencies and mode shapes under a variety of boundary conditions at the shell edge. The results agree very well with values obtained by Kalnins, using numerical methods.

This report treats the axisymmetric vibration of thin elastic shells. Estimates of natural frequencies and mode shapes are obtained for a general class of thin shells by applying the approximations obtained in a previous paper by one of the authors. Numerical results are obtained for ellipsoidal shells, and one new theoretical result is found. (Author)

This report analyzes the large deformations of a cylindrical, inflated, single-wall tent due to wind pressure and is based on the membrane theory for large deflections but small strains. The tent cross section is a sector of a circle in the undeformed position, and the wind is blowing on it in the broadside direction. The tent motion is taken as plane, and it is assumed that the wind pressure distribution is known in the deformed state. The problem is solved by numerical analysis and results are presented for the stress, deformed shape, aerodynamic resultants, and anchor forces. The problem is of theoretical interest because the linear membrane theory does not have a unique solution for it, and also because it illustrates that the method of small deformations superposed on large is of little help when the large deformation is of inextensional type.

In this paper the nonsymmetric, free, elastic vibrations of thin domes of revolution are studied. It is assumed that the frequency is low. The asymptotic approximations previously given by the writer are used to estimate the general solution to the shell vibration equations at low frequencies. Approximations for the low natural frequencies and modes are derived systematically under a variety of edge conditions. Low natural frequencies are found only when the edge conditions impose no forces tangent to the shell surface. When the edge is free (and only then) Rayleigh’s inextensional frequencies are recovered. For certain other edge conditions, new natural frequencies are found that are above Rayleigh’s frequencies but still low compared, e.g., with the lowest membrane frequency. The displacement modes associated with these new frequencies are mostly of inextensional type. The general results are applied to estimate these new frequencies for spherical domes.

A theory is presented for the stress analysis of a flat, circular parachute in steady, vertical descent. Unlike previous treatments of the problem, this theory does not assume that the shape is known. Instead the theory presents relations between the pressure distribution in the opened condition and the shape, drag and stresses in lines and fabric. The theory results in a non-linear third order system of ordinary differential equations with boundary conditions at both vent and skirt. This system was solved by a computer program based on the Runge-Kutta method of numerical integration. The results are in fairly good agreement with measurements on parachutes. The computer program can be used for studies of effects of design changes on shape, drag and stress, and the results of a small study of this sort are included.

In this paper the membrane solutions for nonsymmetric vibration of a spherical dome are studied. The solutions are written in a convenient form, and it is shown how these reduce to the familiar membrane and inextensional solutions in the static limit. This information enables one to perceive several difficulties in finding inextensional frequencies by numerical means and to suggest ways around these difficulties. Also, membrane frequencies are found for circumferential wave numbers up through eight by direct calculations and compared with several different approximations. One type of approximation , due to Jeffreys and Jeffreys, gives quite good results. It is seen that earlier work agrees well with these results except for a few frequencies that are not found at all in the present calculations.

This report describes an investigation of how errors in components of an assembly can affect its performance. In particular the report deals with the situation, uncommon in engineering practice, where the output tolerance of the assembly may be violated even though the tolerances on the components are all met. This situation is analyzed to estimate the probability that the output tolerance will be satisfied given that the component tolerances are met. Three methods are described for estimating this probability. their results are compared in a number of cases, and a best method is chosen. Several simple rules, suitable for preliminary estimates, are also given.

An asymptotic analysis is carried out for the equations of axisymmetric vibrations of thin shells, somewhat analogous to the procedure by which geometrical optics is obtained from electromagnetic theory. It is found that two types of asymptotic solutions are obtained, roughly classifiable as high-frequency membrane solutions and bending solutions. Two explicit representations of these solutions are obtained, one for the shallow region of a shell and the other for the nonshallow region. A number of interesting conclusions emerge from the analysis.

This paper is about the mathematical methods used in calculating the 12D radiation-dose for a food which has been subjected to an inoculated pack. It concerns the case in which only one partial-spoilage data point is obtained from the experiment. A simple, partly graphical procedure is described. This method is based on binomial confidence limits and furnishes an estimate that lies above the (unknown) true 12D value with probability greater than 90%. The method is applied to an inoculated pack for low-level nitrite/nitrate ham described by Anellis et al. (2).

pV\l (10) AP(V)/R02 = 1.867 - 2.399[0.6057 - (V/R03) where R0 is the flat canopy radius, and it is assumed that the vent is not pulled down. The values of the constants appropriate for a horizontally launched, solid, flat canopy are CD = 1.5, A 1 =A 2 =0.5,A 3 = 1, Cmi= 0.375 (11) Cm2 = 0.333, Cgl = 7T/2, Cg2 = 1

The non‐symmetric vibration of shells of revolution is studied by means of classical thin‐shell theory. Approximations are found by asymptotic methods for the eight solutions of the system. The behavior of the solutions when the frequency is either high or low is discussed. The approximations are used to give general descriptions of the types of natural frequencies and modes that can be found. In particular, the occurrence of low natural frequencies associated with inextensional modes is shown to be impossible for many edge conditions and to be calculable solely from membrane theory in certain frequency ranges.