Transient problems

Transient problems begin with an initial condition and march forward in time in discrete time steps. We have discussed space derivatives, and now we will introduce the time derivative, or transient, term of the differential equation. Although the Taylor-series can also be used, it is more helpful to develop the ED with the integral method. The starting point is to take the general expression 

In order to approximate the remaining integral, we compute the product, as schematically depicted in Fig. 8.19, /(0(t) t)At which simplifles eqrl. (8.72) t0 

Instead of using the function evaluated at the past, k, we can also construct a product based on the function evaluated at a future point of time, k + 1, as shown in Fig. 8.20. With this we obtain, 

Both methods are first order in time but for practical purposes the explicit Euler is the easiest to apply due to the fact that the only unknown is the value 4 k+l all other terms are evaluated in the Tcth time step, and due to prescribed initial conditions are always known. Hence, the value of cj k+1 can easily be solved for using eqn. (8.74), by marching forward in time. In the implicit Euler case, the whole right hand side of the equation is evaluated in the future, and must therefore be generated and solved for every time step. When marching 

If instead of implementing the values of the parameters either in the past or the future, we evaluate them at both t and t + At and average the results as schematically depicted in Fig. 8.21, the results in an FD expression are given by, 

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Equation (2.106) gives rise to an implicit scheme except for 0 = 0. The application of implicit schemes for transient problems yields a set of simultaneous equations for the field unknown at the new time level n + 1. As can be seen from Equation (2.111) some of the terms in the coefficient matrix should also be evaluated at the new time level. Therefore application of the described scheme requires the use of iterative algorithms. Various techniques for enhancing the speed of convergence in these algorithms can be found in the literature (Pittman, 1989). 

Cardona, A. and Idelsohn, S., 1986. Solution of non-linear thermal transient problems by a reduction method. Int. J. Numer. Methods Eng. 23, 1023-1042. 

The alternating direction method can be used for elhptic problems by using sequences of iteration parameters (Refs. 106 and 221). The method is well suited to transient problems as well. 

Conduction with Change of Phase A special type of transient problem (the Stefan problem) involves conduction of heat in a material when freezing or melting occurs. The liquid-solid interface moves with time, and in addition to conduction, latent heat is either generated or absorbed at the interface. Various problems of this type are discussed by Bankoff [in Drew et al. (eds.). Advances in Chemical Engineering, vol. 5, Academic, New York, 1964]. 

This chapter is a brief diseussion of large deformation wave codes for multiple material problems and their applications. There are numerous other reviews that should be studied [7], [8]. There are reviews on transient dynamics codes for modeling gas flow over an airfoil, incompressible flow, electromagnetism, shock modeling in a single fluid, and other types of transient problems not addressed in this chapter. 

Finally, the post-proeessor helps to visualize the huge amount of data produeed by the flow solver. Veetor, eontour and shaded eontour plots of the veloeity field, the energy dissipation distribution or other variable fields ean be plotted for lines, eross seetions or surfaees of the geometry. Some paekages even offer partiele traeing faeilities and animation for transient problems. 

When transient problems are considered, the time derivative appearing in Eq. (32) also has to be approximated numerically. Thus, besides a spatial discretization, which has been discussed in the previous paragraphs, transient problems require a temporal discretization. Similar to the discretization of the convective terms, the temporal discretization has a major influence on the accuracy of the numerical results and numerical stability. When Eq. (32) is integrated over the control volumes and source terms are neglected, an equation of the following form results ... 

Some transient problems tend to a trivial (and useless) steady-state solution without flux and concentration profiles. For instance, concentration profiles due to limiting diffusion towards a plane in an infinite stagnant medium always keep diminishing. Spherical and disc geometries sustain steady-state under semi-infinite diffusion, and this can be practically exploited for small-scale active surfaces. 

Similar transient problems in the CSTR can be easily formulated. If the reactor volume is not constant (such as starting with an empty reactor and beginning to fill it at f = 0) or if the volumetric flow rate v into the reactor is changed), then the variable-densily version of this reactor involving Fa must be used. 

Steinhaus and colleagues (66) have described another version of a stroboscopic instrument that is useful for routine lifetime measurements. They point out that switching a photomultiplier tube on and off by removal and reapplication of the high voltage may at times yield troublesome transient problems. 

For the transient problem, the elliptic character changes to essentially parabolic in time. Thus initial conditions are required for all dependent variables. All the same caveats about... 

It is often the case that after a sufficiently long time, a transient problem approaches a steady-state solution. When this is the case, it can be useful to calculate the steady solution independently. In this way it can be readily observed if the transient solution has the correct asymptotic behavior at long time. 

Solve the transient problem and plot nondimensional velocity and temperature profiles at some representative nondimensional times. 

The transient-solution procedure initially makes very rapid progress in driving the rapidly responding solution components toward their steady-state values. However, while highly stable and reliable, continuing to solve the transient problem to an eventual steady state is... 

From these examples it is apparent that one needs to be cautious when using steady-state methods and continuation procedures near turning points. While the solutions may converge rapidly and even appear to be physically reasonable, there can be significant errors. Fortunately, a relatively simple time-stepping procedure can be used to identify the nonphysical solutions. Beginning from any of the solutions that are shown in Fig. 15.9 as shaded diamonds, a transient stirred-reactor model can be solved. If the initial solution (i.e., initial condition for the transient problem) is nonphysical, the transient procedure will march toward the physical solution. If the initial condition is the physical solution, the transient computational will remain stationary at the correct solution. 

With initial concentrations of A = 0.5, B = 0.2, and C = 0.3 numerically solve the transient problem to predict A(r), B(t), and C(t). Based on the solution, explain in physical terms the short-time behavior and the long-time behavior. Explain the observed behavior in terms of stiffness. 

In this section we describe the spreadsheets used to solve the Stokes problem between a cylindrical shell and an inner rod that rotates with fixed rotation rate, Section 4.8. Both explicit and implicit solution procedures are illustrated. This problem has boundary conditions that are fixed in time, and solves the transient problem to the steady-state solution. Other problems discussed in Chapter 4 have time-varying boundary conditions or time-varying forcing functions. Solving these problems requires only very straightforward modification of the following examples. 

Transport by combined migration—diffusion in a finite planar geometry can achieve a true steady state when only two ions are present, as we saw in Sect. 4.2. The same holds true when there are three or more ions present. Under simplifying conditions [see eqn. (89) below], it is possible to predict the steady-state behaviour with arbitrary concentrations of many ions. However, the corresponding transient problem is much more difficult and we shall not attempt to derive the general transient relationship, as we were able to do in deriving eqn. (82) in the two-ion case. 

To generate the most effective model for a given transient problem, it is wise to work to simplify a dynamic model by identifying which timescales can be ignored. 

Why is an understanding of the transient phenomenon important Large electromagnetic devices such as transformers and motors are practically impervious to the effects of transients. Problems arise because of the sensitivity of the microelectronic devices and circuits that make up the control elements of the power system. The microprocess controller is the nerve center of every present-day manufacturing or commercial facility. Medical electronic instruments used in healthcare facilities are becoming more sophisticated and at the same time increasingly susceptible to... 

The stability of a FD representation deals with the behavior of the truncation error as the calculation proceeds in time or marches in space, typically, transient problems, and problems with convection-convection derivatives. A stable FD scheme will not allow the errors to grow as the solution proceeds in time or space. The issue of stability for transient problems will be analyzed in depth later in this chapter. 

The balance equations used to model polymer processes have, for the most part, first order derivatives in time, related with transient problems, and first and second order derivatives in space, related with convection and diffusive problems, respectively. Let us take the heat equation over an infinite domain as... 

The treatment of transient problems is covered in depth later in this chapter. 

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