Solving Unsteady Problems

In transient problems the time coordinate must be discretized in a similar manner as the space coordinates [141, 66, 71, 72, 158, 49, 202]. The major difference between the space and time coordinates lies in the direction of influence. The time dependent variables will affect the flow only in the future, thus there is no backward influence. This has a strong influence on the choice of solution strategy. 

Integration of the convective flux terms over the GCV surface gives  

Introducing these standard finite volume space integrations, and changing the order of integration in the rate of change term we get  

In the framework of the FVM it is generally assumed that the transient term prevails over the whole grid volume. To make further progress we need approximate numerical techniques for evaluating the time integral from to 

It is observed that after the spatial terms have been approximated with appropriate discretizations, as outlined in the subsequent sections, the equation can be cast into the form of an ordinary differential equation on the form  

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The elegant solution of this first example should not tempt the reader to believe that dimensional analysis can be used to solve every problem. To treat this example by dimensional analysis, the physics of unsteady-state heat conduction had to be understood. Bridgman s (2) comment on this situation is particularly appropriate ... 

In the case of dynamic (unsteady) problems, even after the space discretization, we still have to solve a set of ordinary differential equations in time. Therefore, the second step is to discretize the temporal continuum. This is usually done by a finite difference approximation with the same properties of a FDM in space. Depending on the instant in which the information is taken, the time-discretization leads to ... 

Associated with numerical problems is the concept of stability. A numerical scheme is stable when a solution is reached even with large time-steps (unsteady problems) or iteration steps (algebraic system of equations iteratively solved). Therefore the size of the time-step or of the iteration-step is dictated by stability requirements. It must be kept in mind that stability does not mean accuracy an implicit scheme of a dynamic problem is unconditionally stable but the solution obtained with large values of the time step may not be realistic. 

A reverse kinetic problem consists in identifying the type of kinetic models and their parameters according to experimental (steady-state and unsteady-state) data. So far no universal method to solve reverse problems has been suggested. The solutions are most often obtained by selecting a series of direct problems. Mathematical treatment is preceded by a qualitative analysis of experimental data whose purpose is to reduce drastically the number of hypotheses under consideration [31]. 

Implicit methods are preferred for slow-transient flows because they have less stringent time step restrictions than explicit schemes. However, the time step must still be chosen small enough so that an accurate history is obtained. It is further noted that implicit methods can also be used to solve steady problems. In this particular approach, we have to solve an unsteady form of the problem until a steady state is reached. For the artificial time integration, large time steps are often used intending to reach the steady state quickly. 

In order to obtain confidence in the results of numerical simulations, numerical experiments are performed. These involve solving similar problems, for which theoretical or numerical treatments or high-quality experimental data are available. Ultimately, however, simulations of the problem of interest must be verified by increasing the grid density until the solution does not change when additional grid points are added. For an unsteady solution, convergence on time-step size must also be verified. 

Clearly, a DC amplifier may also be used as a differentiator. However, whenever possible this operation should be avoided because of noise. So far, we have seen the use of the DC amplifier as a multiplier, a sign changer, an adder, and an integrator.16 This background is sufficient for solving unsteady lumped problems, which are illustrated next. ... 

It should be noted that all investigations of flow stability of polymerizing liquids are few in number and have been carried out up till now only for unidimensional problems. The problem of stability of steady rheokinetic two-dimensional flows to local hydrodynamic perturbations has not been discussed in the literature yet. Obviously the problem can be solved (the solution is difficult from the technical point of view), for example, by numerical methods solving the problem on unsteady development of the flow of polymerizing mass directly after a forced local change of the profile of the flow velocity. 

In general, it can be concluded that substantial progresses have been made in the experimental and theoretical analysis of trickle-bed reactors under unsteady-state conditions. But until now these results are not sufficient for a priori design and scale-up of a periodically operated trickle-bed reactor. The mathematical reactor models, which are now available are not detailed enough to simulate all of the main transient behavior observed. For solving this problem specific correlations for specific model parameters (e.g. Hquid holdup, mass transfer gas-solid and liquid-solid, intrinsic chemical kinetic, etc.) determined under dynamic conditions are required. The available correlations for important hydrodynamic, mass-and heat-transfer parameters for periodically operated trickle-bed reactors leave a lot to be desired. Indeed, work for unsteady-state conditions on a larger scale may also be necessary. 

Relation between mass- and heat-transfer parameters. In order to use the unsteady-state heat-conduction charts in Chapter 5 for solving unsteady-state diffusion problems, the dimensionless variables or parameters for heat transfer must be related to those for mass transfer. In Table 7.1-1 the relations between these variables are tabulated. For K 1.0, whenever appears, it is given as Kk, and whenever c, appears, it is given as cJK. 

Cg = 0.9375 (insulated surface x = 0.01 m) 7.7 2. Digital Computer and Unsteady-State Diffusion. Using the conditions of Problem 7.7-1, solve that problem by the digital computer. Use Ax = 0.0005 m. Write the computer program and plot the final concentrations. Use the explicit method, M — 2. 

We can solve this unsteady problem by first producing the pressure distribution as just outlined, then updating the front location x = xj and subsequently, repeating this proeess reeursively, as required. The update formula is obtained from the kinematie requirement that... 

In unsteady states the situation is less satisfactory, since stoichiometric constraints need no longer be satisfied by the flux vectors. Consequently differential equations representing material balances can be constructed only for binary mixtures, where the flux relations can be solved explicitly for the flux vectors. This severely limits the scope of work on the dynamical equations and their principal field of applicacion--Che theory of stability of steady states. The formulation of unsteady material and enthalpy balances is discussed in Chapter 12, which also includes a brief digression on stability problems. 

The equation is most conveniently solved by the method of Laplace transforms, used for the solution of the unsteady state thermal conduction problem in Chapter 9. 

The heat transfer problem which must be solved in order to calculate the temperature profiles has been posed by Lee and Macosko(lO) as a coupled unsteady state heat conduction problem in the adjoining domains of the reaction mixture and of the nonadiabatic, nonisothermal mold wall. Figure 5 shows the geometry of interest. The following assumptions were made 1) no flow in the reaction mixture (typical molds fill in <2 sec.) ... 

The general material balance of Section 1.1 contains an accumulation term that enables its use for unsteady-state reactors. This term is used to solve steady-state design problems by the method of false transients. We turn now to solving real transients. The great majority of chemical reactors are designed for steady-state operation. However, even steady-state reactors must occasionally start up and shut down. Also, an understanding of process dynamics is necessary to design the control systems needed to handle upsets and to enable operation at steady states that would otherwise be unstable. 

The procedure for the solution of unsteady-state balances is to set up balances over a small increment of time, which will give a series of differential equations describing the process. For simple problems these equations can be solved analytically. For more complex problems computer methods would be used. 

Most plant simulations have been steady-state simulations. This is to be expected, since just as a baby must learn to crawl before he can walk, so the simpler steady-state problems must be solved before the unsteady-state ones can be tackled. However, unsteady- state plant simulations are being attempted, and undoubtedly sometime in the future this will be a common tool for plant designers. 

Example 14-7 can also be solved using the E-Z Solve software (file exl4-7.msp). In this simulation, the problem is solved using design equation 2.3-3, which includes the transient (accumulation) term in a CSTR. Thus, it is possible to explore the effect of cAo on transient behavior, and on the ultimate steady-state solution. To examine the stability of each steady-state, solution of the differential equation may be attempted using each of the three steady-state conditions determined above. Normally, if the unsteady-state design equation is used, only stable steady-states can be identified, and unstable... 

Differential equations of the first order arise with application of the law of mass action under either steady or unsteady conditions, and second order with Fick s or Newton-Fourier laws. A particular problem may be represented by one equation or several that must be solved simultaneously. 

Although several different system configurations have been simulated, the focus of this paper will be on the unsteady, compressible, multiphase flow in an axisymmetric ramjet combustor. After a brief discussion of the details of the geometry and the numerical model in the next section, a series of numerical simulations in which the physical complexity of the problem solved has been systematically increased are presented. For each case, the significance of the results for the combustion of high-energy fuels is elucidated. Finally, the overall accomplishments and the potential impact of the research for the simulation of other advanced chemical propulsion systems are discussed. 

We will introduce the product rule through demonstrating its use in an example problem. The product rule can be used to expand a solution without source and sink terms to the unsteady, one-dimensional diffusion equation to two and three dimensions. It does not work as well in developing solutions to all problems and therefore is more of a technique rather than a rule. Once again, the final test of any solution is (1) it must solve the governing equation(s) and (2) it must satisfy the boundary conditions. 

Unsteady state diffusion processes are of considerable importance in chemical engineering problems such as the rate of drying of a solid (H14), the rate of absorption or desorption from a liquid, and the rate of diffusion into or out of a catalyst pellet. Most of these problems are attacked by means of Fick s second law [Eq. (52)] even though the latter may not be strictly applicable as mentioned previously, these problems may generally be solved simply by looking up the solution to the analogous heat-conduction problem in Carslaw and Jaeger (C2). Hence not much space is devoted to these problems here. 

Operation of a batch distillation is an unsteady state process whose mathematical formulation is in terms of differential equations since the compositions in the still and of the holdups on individual trays change with time. This problem and methods of solution are treated at length in the literature, for instance, by Holland and Liapis (Computer Methods for Solving Dynamic Separation Problems, 1983, pp. 177-213). In the present section, a simplified analysis will be made of batch distillation of binary mixtures in columns with negligible holdup on the trays. Two principal modes of operating batch distillation columns may be employed ... 

By using the result of Sparrow and Gregg [36] who solved the unsteady flow problem, the low frequency expressions of f and g can be obtained following our notation ... 

A direct kinetic problem consists of calculating multi-component reaction mixture compositions and reaction rates on the basis of a given kinetic model (both steady-state and unsteady-state) with the known parameters. Reliable solution for the direct problem is completely dependent on whether these parameters, obtained either on theoretical grounds or from special experiments, have reliable values. Modern computers can solve high-dimensional problems. Both American and Soviet specialists have calculated kinetics for the mechanisms with more than a hundred steps (e.g. the reac-... 

Control and monitoring of the chemical reactor play a central role in this procedure, especially when batch operations are considered because of the intrinsic unsteady behavior and the nonlinear dynamics of the batch reactor. In order to meet such requirements, the following fundamental problems must be solved ... 

In this chapter, two-dimensional steady and one-dimensional unsteady heat conduction is discussed. The main method introduced to solve this group of problems is the method of separation of variables. 

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