Transient solution

Repeat the false transient solution in Example 4.2 using a variety of initial conditions. Specifically include the case where the initial concentrations are all zero and the cases where the reactor is initially full of pure A, pure B, and so on. What do you conclude from these results ... 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

In principle, the steady-state pipeline network problems can always be solved by the transient solution methods after allowing sufficient time steps for the solution to reach steady state. This possibility was discussed by Nahavandi and Catanzaro (Nl) who made a comparison of a transient solution method with the Cross method of balancing flows (R4). For the particular 35-node and 45-branch hydraulic network problem tested, the transient solution method took 108 seconds as compared with the 134 seconds required by the Hardy-Cross method. (See also Section V,A,2.)... 

The transient solution can be obtained, for planar or spherical geometry, using Laplace transforms [26,45], but, for simplicity, we restrict ourselves to the... 

Let us now consider the simple case of steady-state, towards which the transient solution tends when t —> oo but which, for sufficiently small r0, is practically... 

The transient solution to Eq. 7 will be discussed in the next section. Equation 5 can also be directly integrated to relate the remaining polymer thickness to the time dependent oxide thickness... 

This transient solution is depicted in Figure 7.13. For more complex dynamic problems, the use of Laplace or Fourier transforms would be particularly recommended (Wiberg, 1971 Anonymous, 1982). <=... 

The transient solution to this problem, briefly described in Section 8.5.9, has been worked out analytically by Smith et al. (1955) and Hulme (1955), whereas Albarede and Bottinga (1972) calculated numerical solutions for the case where a crystal grows out of a finite amount of melt and discussed the geological implications. 

These new potentials are solutions of wave equations including inside the sources. To obtain the general solution, one must add a particular solution of the inhomogenous potential equations. Usually, the electromagnetic helds Eo, Eo and the potentials , C are discarded for the following reasons. Either (1), they represent transient solutions of Maxwell s equations that decay rapidly to zero or... 

Figure 4.14 illustrates the transient solution to a problem in which an inner shaft suddenly begins to rotate with angular speed 2. The fluid is initially at rest, and the outer wall is fixed. Clearly, a momentum boundary layer diffuses outward from the rotating shaft toward the outer wall. In this problem there is a steady-state solution as indicated by the profile at t = oo. The curvature in the steady-state velocity profile is a function of gap thickness, or the parameter rj/Ar. As the gap becomes thinner relative to the shaft diameter, the profile becomes more linear. This is because the geometry tends toward a planar situation. 

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. 

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... 

In Fig. 15.8 notice that during the time integration, the steady-state residuals increased for a period as the transient solution trajectory climbed over a hill and into the valley where the solution lies. This behavior is quite common in chemically reacting flow problems, especially when the initial starting estimates are poor. In fact it is not uncommon to see the transient solution path climb over many hills and valleys before coming to a point where the Newton method will begin to converge to the desired steady-state solution. 

The difference between the solution-diffusion and pore-flow mechanisms lies in the relative size and permanence of the pores. For membranes in which transport is best described by the solution-diffusion model and Fick s law, the free-volume elements (pores) in the membrane are tiny spaces between polymer chains caused by thermal motion of the polymer molecules. These volume elements appear and disappear on about the same timescale as the motions of the permeants traversing the membrane. On the other hand, for a membrane in which transport is best described by a pore-flow model and Darcy s law, the free-volume elements (pores) are relatively large and fixed, do not fluctuate in position or volume on the timescale of permeant motion, and are connected to one another. The larger the individual free volume elements (pores), the more likely they are to be present long enough to produce pore-flow characteristics in the membrane. As a rough rule of thumb, the transition between transient (solution-diffusion) and permanent (pore-flow) pores is in the range 5-10 A diameter. 

Bubble instability is one of the complications of this process. Only recently did this matter receive theoretical attention. As pointed out by Jung and Hyun (28), there are three characteristic bubble instabilities axisymmetric draw resonance, helical instability, and metastability where the bubble alternates between steady states, and the freeze line moves from one position to another. Using linear stability analysis, Cain and Denn (62) showed that multiple steady state solutions are possible for the same set of conditions, as pointed out earlier. However, in order to study the dynamic or time-dependent changes of the process, transient solutions are needed. This was recently achieved by Hyun et al. (65), who succeeded in quite accurately simulating the experimentally observed draw resonance (28). 

S. Kase and T. Matsuo, Studies on Melt Spinning, Part II. Steady State and Transient Solutions of Fundamental Equations Compared with Experimental Results, Fundamental Equations on the Dynamics of Melt Spinning, J. Appl. Poly. Sci., 11, 251-287 (1967). 

What initial conditions are imposed on the transient solutions presented in graphical form in this chapter ... 

Describe how one-dimensional transient solutions may be used for solution of two-and three-dimensional problems. 

TABLE 6.1 Full Transient Solution Adjustment Factor, g, in Equation (6.42) as a Function of Volume Precipitate for the Exact Solution to Equation (6.35) as Solved by Nielsen ... 

The population balance simulator has been developed for three-dimensional porous media. It is based on the integrated experimental and theoretical studies of the Shell group (38,39,41,74,75). As described above, experiments have shown that dispersion mobility is dominated by droplet size and that droplet sizes in turn are sensitive to flow through porous media. Hence, the Shell model seeks to incorporate all mechanisms of formation, division, destruction, and transport of lamellae to obtain the steady-state distribution of droplet sizes for the dispersed phase when the various "forward and backward mechanisms become balanced. For incorporation in a reservoir simulator, the resulting equations are coupled to the flow equations found in a conventional simulator by means of the mobility in Darcy s Law. A simplified one-dimensional transient solution to the bubble population balance equations for capillary snap-off was presented and experimentally verified earlier. Patzek s chapter (Chapter 16) generalizes and extends this method to obtain the population balance averaged over the volume of mobile and stationary dispersions. The resulting equations are reduced by a series expansion to a simplified form for direct incorporation into reservoir simulators. 

A simplified one-dimensional transient solution of the bubble population balance equations, verified by experiments, has been presented elsewhere (5) for a special case of bubble generation by capillary snap-off. 

Transient solutions for the Nemst-Einstein equation have been derived assuming a constant source concentration [69]. An approximate solution may be obtained when the decline in source concentration is small over the time period of the study by assuming (a) a constant iontophoretic permeability coefficient Fionp (b) the donor iontophoretic compartment is a well-stirred solution (c) sink conditions exist and (d) the solute concentration in the donor solution Q of volume changes at a rate equal to the total flux JjA out of the compartment into the outer layer of the epidermis ... 

Very often multiphase flow systems show inherent oscillatory behavior that necessitates the use of transient solution algorithms. Examples of such flows are encountered in bubbling gas-fluidized beds, circulating gas-fluidized beds, and bubble columns where, respectively, bubbles, clusters, or strands and bubble plumes are present that continuously change the flow pattern. 

For large X the transient solution gives S [R, X] =0 and the Nusselt number (94) becomes ... 

The relationship between the transient and stationary approaches to the relaxation times has been considered by Eigen and de Maeyer. For any chemical equilibrium a system of nonhomogeneous differential equations which represent the rates of concentration change may be set up. The complete solution of the system is the sum of two solutions. One of these depends on the initial conditions of the dependent variables and upon the forcing function (the transient solution), while the other depends on the differential equation system and on the forcing function (the forced solution). The latter does not depend on the initial conditions of concentration, etc. The step-function methods for studying chemical relaxation experimentally determine the transient behaviour, while the stationary methods determine the steady-state behaviour. 

The formal step-off transient solution of Eq. (172) for t > 0 is obtained from the Sturm-Liouville representation... 

Our objective is to ascertain how anomalous diffusion modihes the dielectric relaxation in a bistable potential with two nonequivalent wells, Eq. (195). The formal step-off transient solution of Eq. (172) for t > 0 is obtained from the Sturm-Liouville representation, Eq. (179), with the initial (equilibrium) distribution function... 

In this chapter, analytical solutions were obtained for parabolic and elliptic partial differential equations in semi-infinite domains. In section 4.2, the given linear parabolic partial differential equations were converted to an ordinary differential equation boundary value problem in the Laplace domain. The dependent variable was then solved in the Laplace domain using Maple s dsolve command. The solution obtained in the Laplace domain was then converted to the time domain using Maple s inverse Laplace transform technique. Maple is not capable of inverting complicated functions. Two such examples were illustrated in section 4.3. As shown in section 4.3, even when Maple fails, one can arrive at the transient solution by simplifying the integrals using standard Laplace transform formulae. 

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