Discrete time and distance

In practice we wish to use our analysis on digital signals, so our first task is to convert the expressions for volume velocity and pressure in a tube into a digital form, so that we can perform the analysis independent of sampling rate. 

Recall fi om Section 10.2.1 that we can convert time t to a discrete representation time variable n by 

In addition, it is convenient to normalise our measure of distance from the start of the tube, x, with respect to the sampling fi equency. It also helps simplify the analysis if we normahse this with respect to the speed of sound. As with time and frequency normalisation, this allows us to derive general expressions for the system independent of the sampling frequency used and the speed of sound in the medium. Hence normalised distance d is given by 

Using this result we can state the volume velocity and pressure functions, given in Equations 11.6, 11.7 and 11.9, in discrete normalised time and distance as 

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The explicit, finite difference method (9,10) was used to generate all the simulated results. In this method, the concurrent processes of diffusion and homogeneous kinetics can be separated and determined independently. A wide variety of mechanisms can be considered because the kinetic flux and the diffusional flux in a discrete solution "layer" can be calculated separately and then summed to obtain the total flux. In the simulator, time and distance increments are chosen for convenience in the calculations. Dimensionless parameters are used to relate simulated data to real world data. 

The type of diffusion discussed here may be termed normal" or Gaussian diffusion. It arises simply from the statistics of a process with two possible outcomes, which is attempted a very large number of times. In Section 2.1.2.7, the statistical basis of diffusion is enlarged to include random walks in continuous rather than discrete time, and also situations where different distributions of jump distances occur. 

In order to take account of the fact that the solvent is made up of discrete molecules, one must abandon the simple hydrodynamically-based model and treat the solvent as a many-body system. The simplest theoretical approach is to focus on the encounters of a specific pair of molecules. Their interactions may be handled by calculating the radial distribution function, whose variations with time and distance describe the behaviour of a pair of molecules which are initially separated but eventually collide. Such a treatment leads (as has long been known) to the same limiting equations for the rate constant as the hydro-dynamically based treatments, including the term fco through which an activation requirement can be expressed, and the time-dependent term in (Equation (2.13)) [17]. The procedure can be developed, but the mathematics is somewhat complex. Non-equilibrium statistical thermodynamics provides an alternative approach [16]. The kinetic theory of liquids provides another model that readily permits the inclusion of a variety of interactions the mathematics is again fairly complex [37,a]. In the computer age, however, mathematical complexity is no bar to progress. Refinement of the model is considered further below (Section (2.6)). 

Defining a distance function d i,j) = miiipat/is of links (i,j path) and a d -size neighborhood of f as Afd i) = /c such that d i, k) < d, we wish to study the discrete time-evolution of S(f) in which the site variables undergo transitions of the general form... 

In LGCA models, time and space are discrete this means that the model system is defined on a lattice and the state of the automaton is only defined at regular points in time with separation St. The distance between nearest-neighbor sites in the lattice is denoted by 5/. At discrete times, particles with mass m are situated at the lattice sites with b possible velocities ch where i e 1, 2,. .., b. The set c can be chosen in many different ways, although they are restricted by the constraint that... 

Thus the mean square of the distance covered between 0 and t grows proportionally with t, just as in the discrete-time random walk. How could this have been shown a priori (without explicitly solving the M-equation) ... 

The time method of lines (continuous-space discrete-time) technique is a hybrid computer method for solving partial differential equations. However, in its standard form, the method gives poor results when calculating transient responses for hyperbolic equations. Modifications to the technique, such as the method of decomposition (12), the method of directional differences (13), and the method of characteristics (14) have been used to correct this problem on a hybrid computer. To make a comparison with the distance method of lines and the method of characteristics results, the technique was used by us in its standard form on a digital computer. 

The method of characteristics, the distance method of lines (continuous-time discrete-space), and the time method of lines (continuous-space discrete-time) were used to solve the solids stream partial differential equations. Numerical stiffness was not considered a problem for the method of characteristics and time method of lines calculations. For the distance method of lines, a possible numerical stiffness problem was solved by using a simple sifting procedure. A variable-step fifth-order Runge-Kutta-Fehlberg method was used to integrate the differential equations for both the solids and the gas streams. 

Various rate heterogeneity corrections are implemented in several tree-building programs. For nucleotide data, PAUP 4.0 implements both invariants and discrete gamma models for separate or combined use with time-reversible distance and likelihood tree-building methods and invariants in conjunction with the log-det distance method (see below). For nucleotide, amino acid, and codon data, PAML implements continuous, discrete, and autodiscrete models. For nucleotide and amino acid data, PHYLIP implements a discrete gamma model. 

Recall that D is the length of the vocal tract in units of the normalised distance that we used to simplify the equation for discrete time analysis. By use of Equation (11.9), we can determine a value for D from empirically measured values. For instance, if we set the length of the vocal tract as 0.17 m (the average for a man) and the speed of soimd in air as 340 m s , and choose a sampling rate of 10 000 Hz, Equation (11.9) gives the value of D as 5. Hence the transfer fimction becomes... 

In a previous section reference was made to the random walk problem (Montroll and Schlesinger [1984], Weiss and Rubin [1983]) and its application to diffusion in solids. Implicit in these methods are the assnmptions that particles hop with a fixed jump distance (for example between neighboring sites on a lattice) and, less obviously, that jumps take place at fixed equal intervals of time (discrete time random walks). In addition, the processes are Markovian, that is the particles are without memory the probability of a given jump is independent of the previous history of the particle. These assumptions force normal or Gaussian diffusion. Thus, the diffusion coefficient and conductivity are independent of time. 

In order to finite-difference the model partial differential equations, we need values of the state variables at discrete distances, yi,V2, --yN, and zi,Z2,...Zn, and at discrete times, 0, Ai, 2At,..., (n/ — l)At. Here N is the number of grid blocks along the quadrant boundar>% At is the time step, and n/ = t/fAt. The reservoir quadrant is therefore replaced by a system of grid blocks shown in Figure 8.37. The integer i is used as the index in the y direction, and the integer j as the index in the z direction. In addition, the index n is used to denote time. Hence pe use the following notation to identify a process variable... 

Equation (6) is the framework of an explicit finite difference simulation. The electrochemical experiment can be described by a discrete time (of the experiment) and space (distance from the electrode) grid (Pigure 4-2), where t = 0 at the beginning of the experiment and x = 0 at the electrode surface. If the concentration of every species is known for every space and time grid point, then the experiment is completely described. A point on the grid represents the concentration for an entire volume element, the boundaries of which are the midpoints between the grid points. 

Using temperature and retention factor data for 50 and 60°C, we can compute the coefficients A and B in Equation 4.5 for this example, and then apply them to higher temperatures in order to predict isothermal retention times and retention factors. These data can then be used to predict how far the peak will move during each discrete temperature step in our example. A spreadsheet program was used to calculate this data as shown in Table 4.4. The values for k and iR are for isothermal elution at the indicated temperamre step. The values for Zt are the total distances the peak has moved at the end of the indicated step. The last entry for Zt shows that the peak has been eluted from the column (z, = 10.0 m) in just less than 5 min during the 90°C temperature step. 

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