Time increment

Figure C2.1.18. Schematic representation of tire time dependence of tire concentration profile of a low-molecular-weight compound sorbed into a polymer for case I and case II diffusion. In botli diagrams, tire concentration profiles are calculated using a constant time increment starting from zero. The solvent concentration at tire surface of tire polymer, x = 0, is constant.

The exact propagator for a Hamiltonian system for any given time increment At is symplectic. As a consequence it possesses the Liouville property of preserving volume in phase space. 

X = X t) sampled at large time increments At. Changing the point of evaluation of from x to A x) brings about some accuracy gains, because is a better description of the quickly varying than... 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

The selection of a time increment dependent on parameter a (i.e. carrying out Taylor series expansion at a level between successive time steps of n and n+Y) enhances the flexibility of the temporal discretizations by allowing the introduction of various amounts of smoothing in different problems. The first-order time derivatives are found from the governing equations as... 

Replace the holdup derivatives in Eqs. (13-149) to (13-151) by total-stage material-balance equations (e.g., dMj/dt = Vj + i + Ej- — Vj — Lj) and solve the resulting equations one at a time by the predictor step of an explicit integration method for a time increment that is determined by stability and truncation considerations. If the mole fraclions for a particular stage do not sum to 1, normalize them. 

Repeat steps 2 through 6 with a corrector step for the same time increment. Repeat again for any further predictor and/or predictor-corrector steps that may be advisable. Distefano (ibid.) discusses and compares a number of suitable explicit methods. 

Repeat steps 2 through 8 for subsequent time increments until the desired amount of distillate has been withdrawn. 

From Table 13-31, a total of 394 time increments were necessary to distill all hut 22.08 Ih-mol of the initial charge of 99.74 Ih-mol following the establishment of total-reflux conditions. If this problem had to he solved by an explicit integrator, approximately 25,000 time increments would have been necessary. 

Over the time increment dt, the force applied on the left-hand side of the element acts over a distance u dt, so the work done on the element from the left is Pu dt. The force on the right-hand boundary of the element k P + (dPIdh), dh, and it travels a distance (u + (dufdh), dh) dt, so the work done by the element on the surrounding fluid to the right is (P + (dP/dh), dh)(u + (dujdh), dh) dt. The net work done on the fluid element is the difference... 

The sputtered mass, Srrii, of element, i, during time increment, St, is described by ... 

Equations 5-110, 5-112, 5-113, and 5-114 are first order differential equations and the Runge-Kutta fourth order numerical method is used to determine the concentrations of A, B, C, and D, with time, with a time increment h = At = 0.5 min for a period of 10 minutes. The computer program BATCH57 determines the concentration profiles at an interval of 0.5 min for 10 minutes. Table 5-6 gives the results of the computer program and Figure 5-16 shows the concentration profiles of A, B, C, and D from the start of the batch reaction to the final time of 10 minutes. 

Cleaning period The total elapsed time that a section of the baghouse is off-stream for cleaning. (This time increment determines the total availability of section fabric for filtration use.)... 

FIG. 33 (a) Evolution of a system of 128 chains quenched at times t — 0 from a state with e = —4.0 (upper left corner) to e — —0.4. Snapshots are shown in time increments At — 65536 MCS (in typewriter fashion from left to right), (b) Evolution of the same system but for time increments At — 524288 MCS [23]. 

The Guggenheim method requires that data be taken at constant time increments equal to At. In the past this was often a disadvantage, particularly when the experiment was not designed to be analyzed by this method, but with modem instm-mental methods of analysis it is common to acquire a continuous record of instrument response as a function of time, so that data can be taken from this record at any desired times. 

Equation (2-59) is also written for time t -I- At, where At is a constant time increment. These equations are subtracted, yielding Eq. (2-60). 

It should be carefully noted that a random variable is only unambiguously defined when the time increments fn and f m are specified... 

We conclude this section by introducing some notation and terminology that are quite useful in discussions involving joint distribution functions. The distribution function F of a random variable associated with time increments fnf m is defined to be the first-order distribution function of the derived time function Z(t) = + fn),... 

In MD simulations we simply solve numerically the classical equations of motion, expressing the changes in coordinates and velocities at a time increment At by... 

Example 2.4 Determine how the errors in the numerical solutions in Example (2.3) depend on the size of the time increment, At. 

The temperature profiles along the x-axis at various times are shown in Figure 4. These values should be compared with the theoretical solution T - erfc [ (l-x)/(2jc t) ]. Some numerical oscillations are noted at the heated boundary at short times due to the inability of the rather coarse mesh and time Increment to capture the thermal boundary layer which forms there. However, this can easily be avoided if desired by using a finer mesh in that region, and also by stepping with shorter time increments initially. 

Clearly, the extent of exotherm-generated temperature overshoot predicted by the Chiao and finite element models differs substantially. The finite element results were not markedly changed by refining the mesh size or the time increments, so the difference appears to be inherent in the numerical algorithms used. Such comparison is useful in further development of the codes, as it provides a means of pinpointing those model parameters or algorithms which underlie the numerical predictions. These points will be explored more fully in future work. 

Sensitivity to step size was thought to be likely due to an unnecessary simplification in the original development of the model. The simplification was to consider initiator concentration constant over a small time increment. When instead the initiator was allowed to vary according to the usual first order decomposition path an analytical solution for the variation of polymer concentration could still readily be obtained and was as follows ... 

Equation (A4) is a first order, linear, ordinary differential equation which can be solved analytically for [PJ assuming X, and X, are constant over a small increment in time. Solving for [PJ from some time ti to tj gives Equation (1) (1). When X, is considered a function of time (i.e., initiator concentration is allowed to vary through the small time increment) while maintaining X, constant over the increment. Equation (A4) can again be solved analytically to give Equation (3). 

A number of parameters have to be chosen when recording 2D NMR spectra (a) the pulse sequence to be used, which depends on the experiment required to be conducted, (b) the pulse lengths and the delays in the pulse sequence, (c) the spectral widths SW, and SW2 to be used for Fj and Fi, (d) the number of data points or time increments that define t, and t-i, (e) the number of transients for each value of t, (f) the relaxation delay between each set of pulses that allows an equilibrium state to be reached, and (g) the number of preparatory dummy transients (DS) per FID required for the establishment of the steady state for each FID. Table 3.1 summarizes some important acquisition parameters for 2D NMR experiments. 

The spectral width SWi associated with the F, frequency domain may be dehned as F, = SWi. The time increment for the ti domain, which is the effective dwell time, DWi for this period, is related to SW as follows DWi = (V2)SW]. The time increments during [Pg.158]

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