Nonsteady state

A number of solutions exist by integration of the diffusion equation (7-12) that are dependent on the so-called initial and boundary conditions of special applications. It is not the goal of this section to describe the complete mathematical solution of these applications or to make a list of the most well-known solutions. It is much more useful for the user to gain insight into how the solutions are arrived at, their simplifications and the errors stemming from them. The complicated solutions are usually in the form of infinite series from which only the first or first few members are used. In order to understand the literature on the subject it is necessary to know how the most important solutions are arrived at, so that the different assumptions affecting the derivation of the solutions can be critically evaluated. 

Most solutions of the diffusion equation (7-12) are taken from analogous solutions of the heat conductance equation that has been known for many years  

The selection of diffusion equation solutions included here are diffusion from films or sheets (hollow bodies) into liquids and solids as well as diffusion in the reverse direction, diffusion controlled evaporation from a surface, influence of barrier layers and diffusion through laminates, influence of swelling and heterogeneity of packaging materials, coupling of diffusion and chemical reactions in filled products as well as permeation through packaging. 

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Although equihbrium capacity is the prime concern, few of these closed systems are truly static. Even closed systems have dynamic features or other nonsteady-state aspects, such as temperature fluctuations, moisture ingression, and drydown rates. 

Figure 5 shows conduction heat transfer as a function of the projected radius of a 6-mm diameter sphere. Assuming an accommodation coefficient of 0.8, h 0) = 3370 W/(m -K) the average coefficient for the entire sphere is 72 W/(m -K). This variation in heat transfer over the spherical surface causes extreme non-uniformities in local vaporization rates and if contact time is too long, wet spherical surface near the contact point dries. The temperature profile penetrates the sphere and it becomes a continuum to which Fourier s law of nonsteady-state conduction appfies. 

Storage Tanks The equations for batch operations with agitation may be applied to storage tanks even though the tanks are not agitated. This approach gives conservative results. The important cases (nonsteady state) are ... 

Operator has to observe the process more frequently as a result of nonsteady state operating conditions. This requires more frequent control interventions, leading to an increased potential for human error. 

Since the adsorbent bed must be heated in a relatively short time to reactivation temperature, it is necessary that the reactivation steam rate calculation is increased by some factor that will correct for the nonsteady-state heat transfer. During the steaming period, condensation and adsorption will take place in the adsorbent bed, increasing the moisture content of the adsorbent. A certain portion of the adsorbate... 

Whitby (Ref 7) discovered that in the nonsteady-state with mechanical sieve shakers, the percentage passing versus sieving time curve could be divided into two regions with a transition between (Fig 4). Region 1 exists when there are many particles much less than the mesh size still on the sieve, while region 2 exists when the residue on the sieve consists entirely of nearmesh or larger particles... 

This problem illustrates the solution approach to a one-dimensional, nonsteady-state, diffusional problem, as demonstrated in the simulation examples, DRY and ENZDYN. The system is represented in Fig. 4.2. Water diffuses through a porous solid, to the surface, where it evaporates into the atmosphere. It is required to determine the water concentration profile in the solid, under drying conditions. The quantity of water is limited and, therefore, the solid will eventually dry out and the drying rate will reduce to zero. 

The Ra isotopes in the other decay series can be evaluated similarly. Ra in the series (Table 1) is the product of the third a decay, and so the effects of near-surface deletion or decay of recoiled precursors must be calculated accordingly. Ra in the series is also the product of the third a decay. Further processes that may be considered where circumstances warrant include nonsteady state conditions or removal by precipitation at rates that are fast compared to the decay rate of the Ra nuclides. 

Figure 4. Time series profiles of and temperature, potential density, Chi a, and nitrate (Slagle and Heimerdinger 1991) at 47°N, 20°W (Atlantic Ocean) in April-May 1989. Dashed vertical line represents estimated activity (Chen et al. 1986). The evolution of " Th/ U disequilibrium with time follows that of Chi a and nitrate, confirming the observations illustrated in Figure 3. The series of profiles taken approximately one week apart permits application of a nonsteady state model to the data. [Reprinted from Buesseler et al., Deep-Sea Research /, Vol. 39, pp. 1115-1137, 1992, with permission from Elsevier Science.]...

PF Ni, NFH Ho, JF Fox, H Leuenberger, WI Higuchi. Theoretical model studies of intestinal drug absorption. V. Nonsteady-state fluid flow and absorption. Int J Pharm 5 33-47, 1980. 

R Zipp, NFH Ho. Nonsteady state model of absorption of suspensions in the GI tract Coupling multi-phase intestinal flow with blood level kinetics. Pharm Res 10 S210, 1993. 

Farmer (6) reviewed the various diffusion models for soil and developed solutions for several of these models. An appropriate model for field studies is a nonsteady state model that assumes that material is mixed into the soil to a depth L and then allowed to diffuse both to the surface and more deeply into the soil. Material diffusing to the surface is immediately removed by diffusion and convection in the air above the soil. The effect of this assumption is to make the concentration of a diffusing compound zero at the soil surface. With these boundary conditions the solution to Equation 8 can be converted to the useful form ... 

Semibatch or semiflow processes are among the most difficult to analyze from the viewpoint of reactor design because one must deal with an open system under nonsteady-state conditions. Hence the differential equations governing energy and mass conservation are more complex than they would be for the same reaction carried out batchwise or in a continuous flow reactor operating at steady state. 

For nonsteady-state operating conditions the generalized material balance on reactant A is as follows. 

Analysis of CSTR Cascades under Nonsteady-State Conditions. In Section 8.3.1.4 the equations relevant to the analysis of the transient behavior of an individual CSTR were developed and discussed. It is relatively simple to extend the most general of these relations to the case of multiple CSTR s in series. For example, equations 8.3.15 to 8.3.21 may all be applied to any individual reactor in the cascade of stirred tank reactors, and these relations may be used to analyze the cascade in stepwise fashion. The difference in the analysis for the cascade, however, arises from the fact that more of the terms in the basic relations are likely to be time variant when applied to reactors beyond the first. For example, even though the feed to the first reactor may be time invariant during a period of nonsteady-state behavior in the cascade, the feed to the second reactor will vary with time as the first reactor strives to reach its steady-state condition. Similar considerations apply further downstream. However, since there is no effect of variations downstream on the performance of upstream CSTR s, one may start at the reactor where the disturbance is introduced and work downstream from that point. In our generalized notation, equation 8.3.20 becomes... 

Barrer (19) has developed another widely used nonsteady-state technique for measuring effective diffusivities in porous catalysts. In this approach, an apparatus configuration similar to the steady-state apparatus is used. One side of the pellet is first evacuated and then the increase in the downstream pressure is recorded as a function of time, the upstream pressure being held constant. The pressure drop across the pellet during the experiment is also held relatively constant. There is a time lag before a steady-state flux develops, and effective diffusion coefficients can be determined from either the transient or steady-state data. For the transient analysis, one must allow for accumulation or depletion of material by adsorption if this occurs. 

Other synonyms for steady state are time-invariant, static, or stationary. These terms refer to a process in which the values of the dependent variables remain constant with respect to time. Unsteady state processes are also called nonsteady state, transient, or dynamic and represent the situation when the process-dependent variables change with time. A typical example of an unsteady state process is the operation of a batch distillation column, which would exhibit a time-varying product composition. A transient model reduces to a steady state model when d/dt = 0. Most optimization problems treated in this book are based on steady state models. Optimization problems involving dynamic models usually pertain to optimal control or real-time optimization problems (see Chapter 16)... 

This application fully exploits the dynamic nature of the model since the alarm point will be modified continuously during load changes and other periods of nonsteady-state operation. 

The bulk polymerization of acrylonitrile in this range of temperatures exhibits kinetic features very similar to those observed with acrylic acid (cf. Table I). The very low over-all activation energies (11.3 and 12.5 Kj.mole-l) found in both systems suggest a high temperature coefficient for the termination step such as would be expected for a diffusion controlled bimolecular reaction involving two polymeric radicals. It follows that for these systems, in which radicals disappear rapidly and where the post-polymerization is strongly reduced, the concepts of nonsteady-state and of occluded polymer chains can hardly explain the observed auto-acceleration. Hence the auto-acceleration of acrylonitrile which persists above 60°C and exhibits the same "autoacceleration index" as at lower temperatures has to be accounted for by another cause. 

Figure 2a represents the concentration profile of the tin species during the service life of the coating. The diffusion in the polymer matrix is represented by Fick s second law for nonsteady state flow ... 

Figure 2. Release of organotin from polymer matrix, nonsteady state mass transport in (a) polymer matrix and stationary state mass transport in boundary layer and (b) both media...

For nonsteady-state conditions. Pick s second law of diffusion can be applied... 

Conclusions from Holland (2005) Imbalance within uncertainty of data Nonsteady state supported by data Nonsteady state supported by data Outputs likely underestimated Nonsteady state supported by data Imbalance within uncertainty of data ... 

Whether steady-state or non-steady-state conditions apply depends on the reaction system. Higher concentrations of deactivator decrease normal bimolecular termination by decreasing the concentration of propagating radicals. Steady-state low concentrations of radicals occur when the deactivator/activator ratio is about equal to or greater than 0.1. Non-steady-state conditions occur when the ratio is lower than 0.1. Non-steady-state means non-steady-state conditions for both propagating radicals and deactivator. Under steady-state conditions, both radical and deactivator concentrations are at steady-state, where the radical concentration is lower and the deactivator concentration is higher than for nonsteady-state conditions. 

The solutions of the nonsteady-state expression, Eq. (164), both for single tanks and chains of tanks have been made by Acton and Lapidus (Al), Mason and Piret (M5, M6), and Standart (S22). Aris and Amundson (A15, A16), Bilous and Amundson (B7), Bilous et al. (B9), and Gilles and Hofmann (G3) have studied the stability, control, and response of a stirred tank reactor. 

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