Unsteady-state model

Takemasa, Y., S.Togati, and Y. Aral. 1996. Application of an unsteady-state model for predicting vertical temperature distribution to an existing atrium. ASHRAE Transactions, vol. 102, no. 1. 

By using the unsteady-state model, the maximum oxygen penetration depth for highly packed immobilised cells has been reported to be in the range of 50-200 xm. 

Unsteady-State Model for Transfer between Swarms of Bubbles and... 

Chapter 14 and Section 15.2 used a unsteady-state model of a system to calculate the output response to an inlet disturbance. Equations (15.45) and (15.46) show that a dynamic model is unnecessary if the entering compound is inert or disappears according to first-order kinetics. The only needed information is the residence time distribution, and it can be determined experimentally. 

Lapidus, G. Unsteady-state model for gold cyanidation on a rotating disk. Hydrometallurgy 1995, 39, 251-263. 

The complete unsteady-state model for adsorption-desorption and surface reactions of the plug flow laboratory reactor was based on the following equations NH3 mass balance on the catalyst suiface ... 

Unsteady-state models described by ODEs or PDEs. 

An appropriate unsteady-state model is obtained from the microorganism and substrate material balances as follows ... 

Secondly, it is necessary to interpret critical effects recently discovered experimentally and which are of common interest. In the adjacent field, i.e. homogeneous catalysis, a great number of such facts have been accumulated for Belousov-Zhabotinskii reactions. These facts can be interpreted only in terms of the non-linear unsteady-state models. 

Let us perform this study like that carried out for the adsorption mechanism. We will analyze time variations in the solutions of the unsteady-state model (2)-(3). Typical phase patterns are represented in Fig. 13. The heavy closed lines are two isochrones [in this case they are geometric sites of the... 

In the emulsion phase/packet model, it is perceived that the resistance to heat transfer lies in a relatively thick emulsion layer adjacent to the heating surface. This approach employs an analogy between a fluidized bed and a liquid medium, which considers the emulsion phase/packets to be the continuous phase. Differences in the various emulsion phase models primarily depend on the way the packet is defined. The presence of the maxima in the h-U curve is attributed to the simultaneous effect of an increase in the frequency of packet replacement and an increase in the fraction of time for which the heat transfer surface is covered by bubbles/voids. This unsteady-state model reaches its limit when the particle thermal time constant is smaller than the particle contact time determined by the replacement rate for small particles. In this case, the heat transfer process can be approximated by a steady-state process. Mickley and Fairbanks (1955) treated the packet as a continuum phase and first recognized the significant role of particle heat transfer since the volumetric heat capacity of the particle is 1,000-fold that of the gas at atmospheric conditions. The transient heat conduction equations are solved for a packet of emulsion swept up to the wall by bubble-induced circulation. The model of Mickley and Fairbanks (1955) is introduced in the following discussion. 

It is clear that, for this problem, the normal trend is to use the monodimensional and unsteady state model, which is represented by the assembly of relations (3.152)-(3.156). It accepts a very simple numerical solution or an analytical solution made of one of the methods classically recommended such as the variable separation method ... 

However, we cannot a priori use this model without the previous establishment of conditions which accept the transformation of the three-dimensional and unsteady state model into a one-dimensional model. These conditions can be studied using the simulations as a tool of comparison. At the same time, it is interesting to show the advantages of the dynamic (unsteady) methods for the estimation of the diffusion coefficient of the species through the porous membrane by comparison with the steady state methods. 

Concerning the problem of the validity of the monodimensional and unsteady state model for the transport of an entity through the membrane, the simulations with 1/L >100 show that the transport in the Z direction is dominant. At the same time. Figs. 3.55 and 3.56, which give the state of the dimensionless concentration... 

The unsteady state model will be completed by adding the unsteady evolution as dXj /dr, dt/dx and respectively on the left part of the equations... 

Steady versus Unsteady State Models. Until very recently, fluidized bed reactor models have dealt almost exclusively with steady state conditions. Steady state models are unsuitable for control purposes, for load following in fluid bed combustors, and for start-up and shutdown purposes. It is a welcome sign that two of the papers in this symposium (13,15) derive models which are potentially suitable for these purposes. 

Continuons emulsion polymerization is one of the few chemical processes in which major design considerations require the use of dynamic or unsteady-state models of the process. This need arises because of important problems associated with sustained oscillations or limit cycles in conversion, particle number and size, and molecular weight. These oscillations can occur in almost all commercial continuous emulsion polymerization processes such as styrene (Brooks et cl., 1978), styrene-butadiene and vinyl acetate (Greene et cl., 1976 Kiparissides et cl., 1980a), methyl methacrylate, and chloropene. In addition to the undesirable variations in the polymer and particle properties that will occur, these oscillations can lead to emulsifier concentrations too low to cover adequately the polymer particles, with the result that excessive agglomeration and fouling can occur. Furthermore, excursions to high conversions in polymer like vinyl acetate... 

As mentioned in the introduction, the following discussion on modeling results takes as a lead that distinction should be made between steady-state models, unsteady-state models, and dynamic models. The results mentioned focus mainly on automotive exhaust gas treatment, which application has been widely studied, with major emphasis on the oxidation of carbon monoxide. 

Madireddi K., Babcock R.B., Levine B., Kim J.H., and Stenstrom M.K. An unsteady-state model to predict concentration polarization. Journal Membrane Science 157 1999 13-34. 

Goswami, A.N., Gupta, T.C.S.M., Sharma, S.K., Sharma, A. and Krishna, R. (1993). Unsteady-state modeling and analysis for hquid surfactant membrane hydrocarbon separation processes. Ind. Eng. Chem. Res., 32, 634-40. 

Various simplified models can be used with varying degrees of accuracy for the simulation of the transient behaviour of non-porous catalyst pellets. The most suitable unsteady state model for this problem is that with infinite thermal conductivity. This simplified model is quite accurate for metal and metal oxide catalysts. In this model, equation (5.45) disappears and the model becomes strictly lumped parameter described only by ordinary initial value differential equations. 

With regard to the unsteady state behaviour, only the simplest distributed model based on Fickian diffusion with constant effective diffusivities will be considered in this section. However, two important phenomena which are usually neglected in the literature will be included in the unsteady state modelling because of their importance. These are the adsorption mass capacity of the porous catalyst surface and the heat of chemisorption accompanying the steps of the CSD process. 

For a higher degree of generality than the strictly steady state analysis of this book, the unsteady state models are developed to show very briefly, some of the transient characteristics of the system. The steady state models which are the main concern of this book are obtained by omitting the dynamic terms. The model developed in this section is a distributed model. 

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