For calculation of heat and mass transfer in more complicated situations (e.g., when fl ow or heat transfer coeffi cients are considered, or variables are used as functions of pressure and temperature) numerical models are needed. As already mentioned, analytical models are not able to contemplate several scenarios in a single expression, especially when modeling the process in two or three dimensions. For this purpose the problem is reduced signifi cantly by requiring a solution for a “discrete” number of points (or grid), rather than for each point of the space–time continuum in which the governing equations of mass, energy, and momentum are applied. These partial differ- ential equations describing the entire system are transformed into a system of equations and solved numerically to approximate the exact solution (Nicolaï et al., 2001).
5.4.3.1 Discretization Methods
Various discretization methods can be used for the numerical solution of high- pressure problems. In all cases, computational grids are tailored to provide a “mesh” independent solution for the numerical approximation of the governing equations. The most commonly used are the fi nite volume method, the fi nite element method, and the fi nite difference method in CFD.
5.4.3.1.1 Finite Volume Method
The fi nite volume method of discretization is most widely used in CFD software packages at the moment. It obeys the clear physical principle of conservation of incoming and outgoing mass, energy, and momentum from each volume element. First, the given computational domain is subdivided into fi nite volume elements. Second, the system of general conservation equations is written in every volume element, independently of the coordinates. Third, the system is integrated over fi nite volume V with surface A. Volume integrals in the governing partial differential equations for all transported quantities in the system (i.e., energy, mass, and momentum), containing a divergence term, are con- verted to surface integrals using the divergence (or Gauss) theorem.* By integrating, these terms are then evaluated as fl uxes at the surface of each fi nite volume. Given that
* The divergence theorem states that outward fl ux of a vector fi eld through a surface is equal to the triple integral of the divergence on the region inside the surface, i.e., it states that the sum of all sources
the fl ux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. More information about the theory behind the fi nite volume method is covered elsewhere (Versteeg and Malalasekera, 1995).
The fi nite volume method has been used by several authors (Hartmann, 2002; Hartmann and Delgado, 2002a,b, 2003a,b; Hartmann et al., 2003, 2004; Ghani and Farid, 2007) to numerically solve conservation laws and thus simulate temperature and fl ow distributions inside high-pressure vessels with different confi gurations (described in Chapter 6).
5.4.3.1.2 Finite Element Method
This method provides an overall solution in the entire computational domain and consists of fi ve steps:
1. The given computational domain is subdivided into a collection as a number of fi nite elements and subdomains of variable size and shape. These are interconnected in a discrete number of nodes. A large number of ele- ment shapes have been suggested in literature and are provided in most fi nite element software packages. A number of 2D and 3D element shapes are shown in Figure 5.7.
2. The solution of the partial differential equation is approximated for each element by a low-order polynomial in such a way that it is defi ned uniquely (i.e., coeffi cients are determined in terms of the “approximate” solution) at the nodes (Nicolaï et al., 2001).
3. The global approximate solution, composed of all solutions at the nodes, can then be written as a series of low-order discrete polynomials.
4. Residuals (or errors) are obtained by substituting the approximate solution into the differential equations.
5. The unknown coeffi cients of the polynomials are found by orthogonaliza- tion of these errors with respect to the polynomials. After the coeffi cients are determined, a system of equations (algebraic or ordinary differential) is solved by certain techniques to obtain predicted temperature fi elds.
The theory of this method escapes the scope of this chapter. More detailed information can be extracted from Nicolaï et al. (2001) and Zienkiewicz (1977). Otero et al. (2007) and Knoerzer et al. (2007) used COMSOL Multiphysics (COMSOL AB, Stockholm, Sweden), which incorporates the fi nite element method, to predict fl ow and temperature distributions in laboratory and pilot scale high-pressure systems.
5.4.3.1.3 Finite Difference Method
The fi nite difference method is based on the approximation of the derivatives in the governing equations by the ratio of two differences (i.e., temperature or velocity over distance or time). For example, the fi rst time derivative of the temperature as a function of time T(t) at time ti can be approximated by
1 ( ) ( ) d d i i i t T t T t T t t + − ≅ Δ (5.41)
with Δt = ti+1 − ti. This expression converges to the exact value of the derivative
when Δt decreases. A fi nite difference approximation can be obtained from a fi rst-order Taylor series approximation of T at time ti applied to the differential
equation describing conservation of energy. Likewise, fi nite difference formulas can be established for second-order derivatives using the Taylor series.
For the purpose of calculating the spatial derivatives, a computational domain is subdivided into a regularly spaced grid of lines that intersect at common nodal points. Subsequently, the space and time derivatives are replaced by fi nite differences.
As shown in Chapter 6, the fi nite difference method was used (Denys et al., 2000a,b) with self-developed Delphi 3 mathematical codes to calculate conductive heat transfer in the whole high-pressure system. Ardia et al. (2004) used the fi nite difference method to predict compression heating of water and other water-based materials using MathCAD (Mathsoft Engineering and Education, Inc., Needham, Massachusetts, USA). Furthermore, Hartmann et al. (2004) applied this method
(a) Prismatic_element Tetrahedral_element Triangle_element Quadrilateral_elements 1 0.5 0 0 0 (b)
FIGURE 5.7 Representation of fi nite elements: (a) typical 2D and 3D fi nite element shapes
(also self-developed) to the vessel wall boundary conditions, to represent the variable temperature due to heat conduction from the hotter compression fl uid used in their fi nite element scheme (Section 5.4.1.1).
5.4.3.2 Computational Fluid Dynamics
Computational Fluid Dynamics (also called Computational Thermal Fluid Dynamics [CTFD]) is a computer-aided analysis of fl uid conservation laws (mass, momentum, and energy) to simulate a process for devices (solid and semisolid structures) that inter- act with fl uid. It allows solving the governing equations for fl uid fl ow and heat transfer (Equations 5.18 through 5.20). The underlying methods for the numerical analysis are the fi nite volume, fi nite element, and fi nite difference methods described before.
Most commercially available software packages include postprocessing features that provide temperature/velocity maps. Further options include the representation of contour, vector, and line plots as well as the visualization of animated fl ow and temperature fi elds (Norton and Sun, 2006). Postprocessing analysis tools in software packages are sometimes not suffi cient, especially when trying to validate the model against real temperature data or, in some cases, when trying to use the predicted output temperature data to calculate microbial inactivation distribution data. In this case, other software routines (e.g., programmed in MATLAB) can extract the data from the solution to perform such tasks.
Numerical simulations of pressure vessels with a vertical pressure fl uid inlet near the center bottom are quite common in the literature (Hartmann, 2002; Hartmann and Delgado, 2002a,b, 2003a,b; Hartmann et al., 2004; Knoerzer et al., 2007; Otero et al., 2007). In this case, 2D cross-sections are used as the computational domain (Figure 5.8) due to rotation symmetry at the central axis.
CFDs software packages such as CFX-4.4 (ANSYS CFX, ANSYS Inc., Southpointe, Canonsburg, Pennsylvania), FLUENT (FLUENT Inc., Lebanon, New Hampshire), PHOENICS (CHAM Ltd., Wimbledon Village, London,), and COMSOL Multiphysics are most commonly used in the high-pressure processing area. These software packages provide numerical algorithms for solving govern- ing equations of fl uid dynamics as well as interfaces for the implementation of special purpose software.
For example, Hartmann and Delgado (2003a,b) enhanced the software per- formance by using their own software routines, covering more than 5000 state- ments of FORTRAN 90 code, and by linking the statements at six different interfaces to the main software code. The geometry of the fl uid volume of the high-pressure cell was digitized by the application of CAD-techniques. Expres- sions for thermophysical properties can be inserted in the software routines rep- resenting the model, by using different equations reported in the literature or determined experimentally, as a function of pressure and temperature (Hart- mann et al., 2003; Knoerzer et al., 2007). Today computer speed and memory capabilities allow for reduced computational times of less than 1 h for a simula- tion of the entire process using a refi ned number of elements in the mesh (Kno- erzer et al., 2007), whereas past publications report computational times of 15 h (Hartmann et al., 2004).
5.4.3.3 Validation of Numerical Models
Numerical methods used for predicting 2D or 3D distributions may converge suggesting solutions that might be plausible, but in fact are not accurate enough (Nicolaï et al., 2001). Therefore the numerical solution must always be validated. The validation process involves the comparison of predicted data (i.e., temperature, velocities, inactivation extent, chemical or physical change, etc.) with measured data.
Temperature validation in a high-pressure vessel has been done by using ther- mocouples adapted to the pressure system. For laboratory-scale systems, one or two thermocouples are suffi cient to measure distribution, by changing their position at predefi ned points. In pilot systems, an array of thermocouples is recommended (Knoerzer et al., 2007) to reduce the number of pressure runs and the resulting inher- ent error. Obtaining reliable temperature measurement with thermocouples remains a challenge; hence, other options are being explored. For instance, a “thermal egg” consisting of a metal shell enclosing a temperature data logger has been developed at Food Science Australia and has successfully demonstrated accurate measurements in all high-pressure processing steps (as validated with thermocouple measurements).
Pressure water inlet Steel wall boundary Carrier Water subdomains
FIGURE 5.8 Computational domain of a rotation-symmetric high-pressure vessel including
a carrier for CFD modeling. (Adapted from Knoerzer, K., Juliano, P., Gladman, S., Versteeg, C., and Fryer, P., AIChE J., 53, 2996, 2007.)
It has the advantage of placing several units in different vessel locations at a single run and even inside sealed food packages without being invasive. Another device being developed to measure temperature under pressure consists of a wireless tem- perature probe, which emits an ultrasound signal that is read by an external data logger (Buckow and Agueeva, 2006).
Once the temperature data is gathered, there are different ways of comparing simulations with measured temperatures. The most common method of verifi cation and validation is to compare measured and simulated temperature curves (Hartmann et al., 2004; Ghani and Farid, 2007; Knoerzer et al., 2007; Otero et al., 2007). In this case, temperature curves predicted and measured in several specifi c locations in 1D, 2D, or 3D can be represented in a parity plot (Knoerzer et al., 2007), where measured temperature and simulated temperatures at identical locations and selected times are represented in a graph.
Temperature curves can be easily plotted using conventional software such as Microsoft Excel. However, comparison of 2D or 3D distributions in form of parity plots requires working in matrices containing temperature data in all locations. Scripts developed in MATLAB can help with this task (Knoerzer et al., 2007). First, temperature vs. time data needs to be gathered for all locations of the mod- eled system. Then, measured and predicted temperature profi les are stored as vectors (generally time–temperature profi les) in 2D or 3D arrays (outlining the system geometry). Measured and predicted profi les are matched in a parity plot from which a correlation coeffi cient is determined.
A further approach, which is also the only possible way to investigate fl ow fi elds inside a high-pressure vessel to date, was identifi ed by Pehl and Delgado (1999, 2002). They developed high-pressure digital particle image thermography (HP- DPIT) and high-pressure digital particle image velocimetry (HP-DPIV). Both appli- cations involve the use of encapsulated thermochromic liquid crystals, which change their color with varying temperature. Transient color fi elds in the pressure/tempera- ture domain document (photographically) the temperature and fl ow fi elds at high pressure. As of today, fl uid dynamic effects can only be studied in laboratory-scale vessels. Given the complexity involved in the addition of a crystal window within the vessel structure, it is not yet possible to determine fl ow effects at pilot scale.
Distributions of inactivation or chemical or physical changes cannot be validated through measurements at selected points. A certain volume in packages containing an initial amount of substance at least needs to be considered. In this case, overall averages for the whole vessel or vessel areas where packages are located are calcu- lated from the predictions in the model. While some authors have validated modeled enzyme inactivation (Hartmann and Delgado, 2003b), none have validated a model for spore inactivation at HPHT conditions. Chapter 6 provides examples of modeling distributions of α amylase enzyme and Escherichia coli solutions, and C. botulinum spores in fl uid and solid “water-like” media.