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Generalizaciones de funciones convexas

In document TEMAS DE MATEMÁTICAS PARA ECONOMÍA (página 49-69)

A high-pressure system designed for sterilization conditions must be able to withstand high pressures ranging from 600 to 800 MPa and chamber temperatures up to at least 90°C. This can be accomplished by building a pressure chamber of appropriate thickness and a pumping system with fast pressure come-up. Today, sterilization systems range from laboratory scale (0.02–1.5 L), pilot scale (2–50 L), and indus- trial scale (150 L). Existing types of equipment have varied confi gurations (Balasu- bramaniam et al., 2004) that offer different levels of heat retention and effi ciency. A more effi cient use of compression heating would be to lower the process tempera- ture required to achieve sterilization (de Heij et al., 2002).

The following paragraphs will outline the different considerations in designing a system, particularly the vessel layout, pressure transmitting fl uid selection, use of packaging materials, and the food contained within (to understand the heat transfer phenomena). These considerations will be revisited in Section 5.4.1 to establish the design assumptions for developing an analytical or a numerical (computational fl uid dynamic [CFD]) model.

5.2.1.1 Components of a High-Pressure Vessel

A typical (batch) high-pressure machine system, applicable to all temperatures, consists of (a) a thick cylindrical steel vessel with two end closures, (b) a means of restraining end closures (e.g., yoke, threads, and pin), (c) a low-pressure pump, (d) an intensifi er for system compression, using liquid from the low-pressure pump to generate high-pressure process fl uid, and (e) necessary system controls and instru- mentation (Farkas and Hoover, 2000). For HPHT treatment at pressures over 400 MPa, pressure vessels can be built with two or more concentric cylinders made of high-tensile strength steel. An outer cylinder confi nes the inner cylinders such that the wall of the pressure chamber is always under some residual compression at the design operating pressure. In some cases the cylinders and vessel frame are prestressed by winding wire under tension layer upon layer. The tension of the wire compresses the vessel cylinders, reducing the diameter of the cylinders (Hjelmqwist, 2005). This special arrangement allows for an equipment lifetime of over 100,000 cycles at pressures 680 MPa or higher. Preferred practice in the design of high-pressure chambers is to make sure the contained food is in contact with parts made of stainless steel; that way the fi ltered (potable) water can be used as isostatic compression fl uid (Farkas and Hoover, 2000).

For improved insulation, to prevent heat loss through the steel wall, a material with low-thermal conductivity (less than 1 W/m/K) could be used as part of the pres- sure vessel design (de Heij et al., 2003). Vessel materials suggested for this application are polyoxymethylene (POM), polyetheretherketone (PEEK), polytetrafl uoroethylene (PTFE), polypropylene (PP), or ultrahigh-molecular-weight polyethylene (UHMWPE).

Today, high-pressure vessels are regularly insulated with these types of materials in the interior as a liner. A product container (wall thickness 5 mm or more) constructed from these materials can also be used (de Heij et al., 2003). Thermophysical properties (e.g., thermal diffusivity) of these insulation materials under high-pressure condi- tions have not been determined yet, however, empirical data on temperature inside the chamber have proven benefi cial in preventing heat loss at holding times less than 5 min. Size of pressure vessel also plays an important role in compression heating reten- tion (Ting et al., 2002), as larger vessels have been shown to retain more heat during holding times (Hartmann and Delgado, 2003b).

5.2.1.2 Pressure Transmission Fluid

The selected compression fl uid can contribute to compression heating of the food and possible heat gain or loss in the food. Farkas and Hoover (2000) mention food- approved oil or water-containing FDA- and USDA-approved lubricants, anticorrosion agents, and antimicrobial compounds as possible compression fl uids. Water solutions of castor oil, silicone oil, propylene glycol, and sodium benzoate are sometimes used as pressure-transmitting fl uids in laboratory-scale machines (Ting et al., 2002). For HPHT treatment, the typical fl uids used in pressure vessels include water with glycerol, edible oils, and water/edible oil emulsions (Meyer et al., 2000). However, for commercial purposes, the use of potable water is the most recommended com- pression medium for maintaining the cleanliness of product packages (Farkas and Hoover, 2000).

The ideal scenario would be to fi nd a compression fl uid with minimal differ- ences in compression heating behavior comparable to the food sample (Meyer et al., 2000; Balasubramanian and Balasubramaniam, 2003; de Heij et al., 2003). However, this would be impractical in processing foods of varied composition. For instance, foods high in lipid content show higher compression heating than water-based foods (Balasubramanian and Balasubramaniam, 2003). If water is used as the fl uid medium, an increased temperature gradient is created within the lipidic portions of the sample. This phenomenon of temperature increase does not impede achieving the target process temperature (Tp1) in the nonlipidic portions of the sample. Thus,

even though a temperature gradient is developed due to the presence of lipids in the food, the process conditions could be maintained when using water as the pressure medium.

5.2.1.3 Flexible Container

Packaging used for high-pressure treated foods must accommodate more than a 12% reduction in volume, and be able to return to its original volume, without loss of seal integrity and barrier properties (Farkas and Hoover, 2000; Caner et al., 2004). Until now, identifi cation of suitable packages that can survive pressure sterilization condi- tions, i.e., that retain seal and overall integrity as well as adequate barrier properties against oxygen and water vapor, remains a challenge. For example, retort pouches, designed to withstand temperatures of 121°C, may delaminate and blister after ther- mal pressure processing; particularly at chamber temperatures >90°C and pressures >200 MPa (Schauwecker et al., 2002).

For packaged foods, the food-containing material constitutes an intermediate barrier to the transfer of heat from the heating medium into the product or, after thermal processing, from the product to the cooling medium. Few research studies (discussed in Chapter 6) have considered the infl uence of heat transfer between the heating medium, container material, and the product (Larousse and Brown, 1997; Hartmann et al., 2003; Hartmann and Delgado, 2003b). The packaging material selected can affect the preheating rate needed to achieve the initial temperature (Ti),

depending on the composition and thickness of the material.

It is not yet known how the combined application of pressure and internal pressure inside the package can affect heat conductivity during processing of food products. The pressure increase inside the package is not only a result of mass aug- mentation from the pressure medium entering the chamber, but also due to shrinkage of the package under pressure, i.e., its structural response to pressure application and resistance to deformation. In most cases, polymeric laminates provide enough fl ex- ibility as to transfer the pressure inside the package, resulting in pressure equilibrium throughout the vessel.

Hartmann and Delgado (2003b) discussed the mathematical description of pressurization of a liquid (model enzyme solution) inside a package (see Chapter 6), mentioning that pressure increase in the pressure medium and its transmission from the packaging material to the contained enzyme solution is “a highly complex physi- cal process.” This mathematical expression describing the compression of pressure medium represents a fl uid–structure interaction problem. To solve this problem, conservation equations of fl uid dynamics must be applied using expressions from structure mechanics that describe the interface between the fl uids (outside/inside packaging material) and structure (packaging material).

At present, there have been no data reported for the compression heating rates of packaging material layers forming a fi lm. Assuming compression heating of the material gives a higher temperature rise than in both the food and compression medium, there would be a “warmer” layer surrounding the food. This phenomenon is an inherent part of the process and signifi cant if 5% of the vessel is full of packag- ing material (assuming 5% in volume of packaging material per food package). How- ever, testing the compression heating of different types of fi lms would help identify their infl uence on temperature homogeneity inside the package. This is especially important when maximizing the use of vessel space and a large number of packages are included.

5.2.1.4 Food Properties

Food characteristics (e.g., product composition, viscosity, phase state, density, and porosity) determine the pathways of heat transfer in a food during processing (Barbosa-Cánovas and Juliano, 2008). When foods are compressed, their volume is reduced as a function of the imposed pressure. During compression, foods are gener- ally more dense and compacted, and those of a porous semisolid nature have lower porosity up to a certain degree. Thus, foods may experience increased homogeneity and improved temperature distribution during pressurization. However, a common problem is the release of air located within the food to the package, which adds to the initial air headspace. This headspace, located at certain points in the package,

can alter temperature and pressure uniformity, as well as shape because of these two reasons: (a) compressed residual air may not transfer hydrostatic pressure to the food in the same way as the compression medium (Ting et al., 2002), (b) residual air reduces its volume and heats up at higher pressure. Heating up of retained air, as indicated by Boyle’s law of gases, creates an extra temperature gradient in the food package system. The incidence of this effect on product temperature will mainly depend on the product size and amount of headspace in the package. Farkas and Hoover (2000) mentioned that the amount of air in the system is not a critical process factor since it has no effect on high-pressure inactivation kinetics. However, Balasu- bramaniam et al. (2004) indicated that the total headspace, oxygen in particular, should be minimized.

As discussed earlier, compression heating differences between semisolid food and the vessel wall can be signifi cant, thereby causing heat diffusion out of the sam- ple. Consequently, “cold spots” can be located on the surfaces of packages near the vessel wall (Matser et al., 2004). The situation becomes more complex when the food is nonhomogenous in composition or structure. In this case, although transient heat will transfer in different directions according to the location of different components, the main heat fl ux will probably be directed toward the vessel wall. The number of packages in the vessel and their location will determine the extent of temperature reduction inside the total food mass being processed.

In document TEMAS DE MATEMÁTICAS PARA ECONOMÍA (página 49-69)

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