Determination of the optimum operating time for batch isothermal performance of enzyme-catalyzed multisubstrate reactions

This communication consists of a mathematical analysis encompassing the maximization of the average rate of monomer production in a batch reactor performing an enzymatic reaction in a system consisting of a multiplicity of polymeric substrates which compete with one another for the active site of a soluble enzyme, under the assumption that the form of the rate expression is consistent with the Michaelis-Menten mechanism. The general form for the functional dependence of the various substrate concentrations on time is obtained in dimensionless form using matrix terminology; the optimum batch time is found for a simpler situation and the effect of various process and system variables thereon is discussed. The reasoning developed here emphasizes, in a quantitative fashion, the fact that the commonly used lumped substrate approaches lead to nonconservative decisions in industrial practice, and hence should be avoided when searching for trustworthy estimates of optimum operation

On the approximate analytical solution of a problem of optimization in the field of immobilized enzymes

The problem of optimizing the thickness of a microporous slab containing an immobilized enzyme is addressed, using an economic criterion as the objective function. The steady-state material balance to the substrate as transported by diffusion and depleted by a biochemical reaction following classical Michaëlis-Menten kinetics within the pellet is obtained. Taking advantage of a number of algebraic manipulations and mathematical artefacts, one is able to solve the resulting second-order, non-linear differential equation by an analytical method, provided that an upper error bound for the solution in the order of 5 per cent is acceptable. The validity of the approximation is tested, and useful applications are reported

Computer aided design of pellets for fixed-bed reactors performing Michaelis—Menten reactions

The problem of designing the pellet characteristics for existing fixed-bed reactors has been studied. The reactor is assumed to be a unidimensional, plug-flow, heterogeneous type. Three particle shapes, two intraparticle mass transfer mechanisms, two lands of film resistance to mass transfer and two poisoning patterns have been considered. An economic balance to both product market price and pellet market cost was taken into account. An interactive program (HIMER) containing all these contributions was thus developed, enabling the investigator to follow the reactor performance along the flow coordinate

The effect of internal thermal gradients on the reliability of surface mounted full-history time-temperature indicators

A simple analytical expression aimed at assessing the theoretical reliability of a full-history time-temperature indicator placed on the surface of a food item as a predictor of the remaining shelf-life in the presence of heat transfer limitations within the food is obtained. Such expression, which depends on three dimensionless parameters only, can be written as a univariate function of a single parameter containing the thermal properties of the food via a suitable algebraic scaling. the derivation of the relevant formulae is based on the integration of a generic quality function across the slab-shaped food item under the assumption that the temperature on the surface of the food undergoes a sinusoidal variation with time. the analysis reported is useful because it provides a quick estimate of the effect of the thermal diffusivity of the food on the relative error associated with the use of a TT1 when the activation energies of the food and the indicator are matched (as should happen in the idealistic case) under realistic environmental conditions of storage

Mathematical design of continuous, isothermal crystallizers with homogeneous nucleation: a simplified approach

A simplified, systematic approach to the mathematical simulation of crystallizers is attempted by using the fundamental principles of mass conservation, via a population balance to the solid phase and a solute balance to both solid and liquid phases. A continuous, isothermal and isochoric crystallizer is assumed to be described by the MSMPR model under transient operating conditions with complete micromixing. The birth and death functions are assumed nil. Homogeneous nucleation is considered at a rate which is independent of the solution supersaturation. The growth rate of the crystals is described by McCabe's law. The possibility of solving the population balance and the mass balance independently is explored, and the conditions of validity for such an approach are found. The maximum linear dimension of crystal and the liquor concentration profile as functions of time are obtained. The approximation is found to be generally good for a period of time right after start-up of the crystallizer. A much wider range of time ensuring a satisfactory approximation is possible provided that the system and operation-dependent parameter takes small values

Double surface-renewal model for the prediction of mass transfer rates during bubble formation with instantaneous reaction on the liquid side

Desorption of a component from a gas mixture being injected through a submerged nozzle, during the bubble formation stage, is theoretically studied. The solute is assumed to suffer an instantaneous reaction on the liquid side, the reaction plane being on the liquid/gas interface. The volume elements on the gas bubble are assumed to result from a forced surface renewal coupled with a natural surface renewal, both achieved via gas elements with fresh composition. A surface residence time distribution is thus obtained, and desorption rates are calculated from the relevant variables, assuming non-steady state, unidimensional diffusion in a semi-infinite medium as the main mass transfer mechanism. Comparison of the model with experimental data is reported, physical evidence being predicted better than with previous models

Estimation or the error implicit in modelling multisubstrate biochemical reactions as lumped-substrate systems

The necessary and sufficient condition associated with the maximum absolute error arising from the simulation of a biochemical reaction occurring in a batch multi-substrate system as if it were equivalent to a system with a pseudo single substrate is presented. The kinetics of the reaction catalyzed by the soluble enzyme is assumed to be described by a Michaelis-Menten rate equation in the absence of enzyme deactivation, whereas the batch system is assumed to be perfectly mixed. The relevance of the analysis is due to the fact that monitoring the concentration of multiple species in a reaction system (as, in principIe, required for controlling purposes) is often technically andfor economically unfeasible especially on the industrial scale. Hence, the strategy reported is useful during the pre-design steps of enzymatic batch reactors as an aid for the decision of whether or not to use complex analytical equipment. According to the analysis described, the appropriateness of the lumped substrate approach needs evaluation at a single point using an algorithm which is of straightforward implementation

Heat removal from heat-sensitive foods: an economic approach to the transient behaviour of finned surfaces

The transient behaviour of a fin used to remove heat from the surface of a rectangular-shaped liquid food container is studied in dimensionless form leading to a single-term equation for the heat transferred in relation to time. For heat-sensitive foods the rate of heat removal is important so an economic value can be ascribed to such removal to balance the captial investment on finning the container. An optimal solution can be found in order to maximize the net profit involved using the fin length as the independent variable

Critical issues affecting the future of dairy industry: Individual contributions in the scope of a global approach

Several constraints that have been affecting the dairy industry are identified in a critical fashion, and directions are given with an emphasis on food processing implemented at the post production level.The rationale for modifications aimed at enhancing the appeal of condensed dairy products should be consubstantiated in strengthening of organoleptic characteristics, improvement of nutraceutical impact,and reduction of polluting power. This enumeration follows an order of increasing time scale required for consumer perception and increasing size scale associated with expected impact. Pursuance of suchstreamlines should lead to manufacture of dairy products that resemble nature more closely in terms of milk coagulation, milk fat modification, milk fermentation,whey fermentation, and starter culture addition. Directions for research and development anticipated as useful and effective in this endeavor, and which have been previously and consistently adopted in the development of an individual research program,are characterization and development of alternative rennets from plant sources, development of starter and nonstarter cultures from adventitious microflora, utilization of probiotic strains as starter cultures, upgrading of whey via physical or fermentation routes, and modification of milk fat via lipasemediated interesterification reactions

Economic criteria in the design of cascades of CSTR’s for the performance of Enzyme -c at a1 yzed reactions

The mass balances on substrate in each unit of a series of CSTRs performing an enzyme-catalyzed reaction described by Michaelis-Menten kinetics (with parameter K,,,)a re written and the necessary and sufficient condition that must be satisfied by the intermediate concentrations in order to obtain a minimum overall capital investment is found on the assumption that the cost of each reactor unit scales up on its capacity according to a fractional factor exponential rule (with parameter n). The asymptotic situations of pseudo-zero order and pseudo-first order behavior are explored. The ratio between consecutive concentrations leading to a minimum overall capital investment decreases as K,,, decreases at a rate that is slower for higher n, and tends to unity as the pseudo-first order situation is approached. If fractional values of n are considered, local minima of the capital investment associated with the overall reactor cascade exist only in certain ranges of substrate conversion; below the lower limits of such ranges, the number of reactor units should actually be decreased. A graphical procedure aimed at obtaining the intermediate optimal concentrations is presented