Unsteady State Methods

I. Experimental determination of diffusivities. Several different methods are used to determine diffusion coefficients experimentally in liquids. In one method unsteady-state diffusion in a long capillary tube is carried out and the diffusivity determined from the concentration profile. If the solute A is diffusing in B, the diffusion coefficient determined is D g. Also, the value of diffusivity is often very dependent upon the concentration of the diffusing solute A. Unlike gases, the diffusivity does not equal Dg for liquids. 

The Measurement of There are two main methods for measuring the unsteady-state method, and the steady-state method. In the... 

Various numerical and graphical methods are used for unsteady-state conduction problems, in particular the Schmidt graphical method (Foppls Festschrift, Springer-Verlag, Berhn, 1924). These methods are very useful because any form of initial temperature distribution may be used. 

When the pulsation amplitude is such as to result in a greater-than-permissible metering error, consideration should be given to installation of a pulsation damper between the source of pulsations and the flowmeter. References to methods of pulsation-damper design are given in the subsection Unsteady-State Behavior. ... 

As outlined earlier, in multizone models, perfect mixing is assumed in the individual zone. The spatial distribution of velocities, contaminant concentrations, and air temperatures in a zone can be determined only by using CFD. On the other hand, wind effects are easily accounted for in multizone models, and unsteady-state simulation is normally performed. On the combined use of the two methods, see Schaelin et al.--... 

In principle, given expressions for the crystallization kinetics and solubility of the system, equation 9.1 can be solved (along with its auxiliary equations -Chapter 3) to predict the performance of continuous crystallizers, at either steady- or unsteady-state (Chapter 7). As is evident, however, the general population balance equations are complex and thus numerical methods are required for their general solution. Nevertheless, some useful analytic solutions for design purposes are available for particular cases. 

As seen in Chapter 7, the operation of bateh erystallizers is inherently unsteady-state. Transient values oeeur of the major operating variables sueh as slurry density, supersaturation, temperature and mean partiele size. Methods of operational eontrol sueh as by use of seeding and temperature programming were also eonsidered in detail. 

Figures 9.17-9.19 clearly show that, as the Biot number approaches zero, the temperature becomes uniform within the solid, and the lumped capacity method may be used for calculating the unsteady-state heating of the particles, as discussed in section (2). The charts are applicable for Fourier numbers greater than about 0.2.

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 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. 

With the exception of this method, all the methods described solve the stage equations for the steady-state design conditions. In an operating column other conditions will exist at start-up, and the column will approach the design steady-state conditions after a period of time. The stage material balance equations can be written in a finite difference form, and procedures for the solution of these equations will model the unsteady-state behaviour of the column. 

Rose et al. (1958) and Hanson and Sommerville (1963) have applied relaxation methods to the solution of the unsteady-state equations to obtain the steady-state values. The application of this method to the design of multistage columns is described by Hanson and Sommerville (1963). They give a program listing and worked examples for a distillation column with side-streams, and for a reboiled absorber. 

Prepare a full instrumentation of flow-sheet of the CO conversion section of the plant, paying particular attention to the methods of controlling liquid levels in the circulating water system and temperatures in the catalyst beds. Derive the unsteady-state equations which would have to be employed in the application of computer control to the CO conversion section of the plant. 

If R is known, it is possible to fit the parameters k, ktCC0, A, At, fcp and fct using kinetic data from a single experiment. Thus, if the reaction diffusion parameter is known from the unsteady state after-effect experiments, the kinetic constant evolution can be determined as a function of free volume, and thus conversion. More details about this method will be published elsewhere (18). 

Apply the method of lines to the solution of the unsteady state dispersion reaction equation with closed end boundary conditions for which the partial differential equation for a second order reaction is,... 

The use of magnetic resonance imaging (MRI) to study flow patterns in reactors as well as to perform spatially resolved spectroscopy is reviewed by Lynn Gladden, Michael Mantle, and Andrew Sederman (University of Cambridge). This method allows even unsteady-state processes to be studied because of the rapid data acquisition pulse sequence methods that can now be used. In addition, MRI can be used to study systems with short nuclear spin relaxation times—e.g., to study coke distribution in catalytic reactors. 

Lazman M., Algebraic geometry methods in analysis of quasi steady state and dynamic models of cataljdic reactions. Proceedings of the 4th International Conference on Unsteady-State Processes in Catalysis USPC-4, Montreal, Quebec, Canada, October 26-29, 2003, Dr. H. Sapoundjiev (Ed.), Natural Resources Canada, 92-93 (2003a). 

Unsteady-State Mass Balance Method One widely used technique for determining Kj a in bubbling gas-liquid contactors is the physical absorption of oxygen or COj into water or aqueous solutions, or the desorption of such a gas from a solution into a sparging inert gas such as air or nitrogen. The time-dependent concentration of dissolved gas is followed by using a sensor (e.g., for O2 or CO2) with a sufficiently fast response to changes in concentration. 

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 ... 

Steady-state periodic heating and unsteady-state methods can be applied to measure the thermal conductivity and diffusivity of coal. Methods such as the compound bar method and calorimetry have been replaced by transient hot-wire/line heat source, and transient hot plate methods that allow very rapid and independent measurements of a and X. In fact, such methods offer the additional advantage of measuring these properties not only for monolithic samples but also for coal aggregates and powders under conditions similar to those encountered in coal utilization systems. 

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]. 

WCo2 = r = h4Pco00 corresponding to the E-R mechanism is not satisfied. At present the pendulum has swung to the opposite side and most research workers [98] are sure that, over a wide range of the reaction parameters (T = 450-950 K, P = 10-7 to 10 5 Torr), only the adsorption mechanism (L-H) is valid. This belief is based on the data obtained in unsteady-state experiments and using modern physical methods, in particular the molecular beam technique [98, 52, 107]. But a fairly good qualitative description on the basis of the L-H mechanism has been obtained in only a few cases [56, 57] and this description concerns rather limited experimental... 

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