Unsteady state

Consider changing from batch to continuous operation. Batch processes, by their very nature, are always at unsteady state and thus are difficult to maintain at optimal conditions. 

In principle, Chen, given the flux relations there is no difficulty in constructing differencial equations to describe the behavior of a catalyst pellet in steady or unsteady states. In practice, however, this simple procedure is obstructed by the implicit nature of the flux relations, since an explicit solution of usefully compact form is obtainable only for binary mixtures- In steady states this impasse is avoided by using certain, relations between Che flux vectors which are associated with the stoichiometry of Che chemical reaction or reactions taking place in the pellet, and the major part of Chapter 11 is concerned with the derivation, application and limitations of these stoichiometric relations. Fortunately they permit practicable solution procedures to be constructed regardless of the number of substances in the reaction mixture, provided there are only one or two stoichiomeCrically independent chemical reactions. 

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. 

As In the case of the material balance equations, the enthalpy balance can be written in dimensionless form, and this introduces new dimensionless parameters in addition to those listed in Table 11.1. We shall defer consideration of these until Chapter 12, where we shall construct the unsteady state enthalpy and material balances, and reduce them to dimensionless form. 

Chapter 12. MATERIAL AND ENTHALPY BALANCES IN UNSTEADY STATES... 

To write an unsteady state enthalpy balance we require the enthalpy per unit volume of the gas-permeated solid matrix. This is given by... 

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

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

For cakes of high permeabihty, the capillary drain height may be an insignificant fraction of cake thickness, and film drainage becomes the controlling factor in a centrifugal field (7). Under unsteady-state conditions, equation 18 represents the drainable Liquid left in the cake as a function of the centrifugal filtration parameters ... 

There are special numerical analysis techniques for solving such differential equations. New issues related to the stabiUty and convergence of a set of differential equations must be addressed. The differential equation models of unsteady-state process dynamics and a number of computer programs model such unsteady-state operations. They are of paramount importance in the design and analysis of process control systems (see Process control). 

Spreadsheet Applications. The types of appHcations handled with spreadsheets are a microcosm of the types of problems and situations handled with fuU-blown appHcation programs that are mn on microcomputers, minis, and mainframes and include engineering computations, process simulation, equipment design and rating, process optimization, reactor kinetics—design, cost estimation, feedback control, data analysis, and unsteady-state simulation (eg, batch distillation optimization). 

Unsteady-State Direct Oxidation Process. Periodic iatermption of the feeds can be used to reduce the sharp temperature gradients associated with the conventional oxidation of ethylene over a silver catalyst (209). Steady and periodic operation of a packed-bed reactor has been iavestigated for the production of ethylene oxide (210). By periodically varyiag the inlet feed concentration of ethylene or oxygen, or both, considerable improvements ia the selectivity to ethylene oxide were claimed. 

Parabolic The heat equation 3T/3t = 3 T/3t -i- 3 T/3y represents noneqmlibrium or unsteady states of heat conduction and diffusion. 

Equation (5-1) is used as a basis for derivation of the unsteady-state three-dimensional energy equation for solids or static fluids ... 

When temperatures of materials are a function of both time and space variables, more complicated equations result. Equation (5-2) is the three-dimensional unsteady-state conduction equation. It involves the rate of change of temperature with respect to time 3t/30. Solutions to most practical problems must be obtained through the use of digital computers. Numerous articles have been published on a wide variety of transient conduction problems involving various geometrical shapes and boundaiy conditions. 

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. ... 

Diffusion-controlled mass transfer is assumed when the vapor or liquid flow conforms to Tick s second law of diffusion. This is stated in the unsteady-state-diffusion equation using mass-transfer notation as... 

When this is substituted into the previous equation, both sides become functions of T and may be plotted against each other. As Fig. 23-17 of a typical case shows, as many as three steady states are possible. When generation is greater than removal (as at points A— and B-t-), the temperature will rise to the next higher steady state when generation is less than removal (as at points A-t- and B—), it will fall to the next steady state. Point B is an unsteady state, while A and C are steady. 

The nature of batch operations (unsteady-state), frequently involving manual intervention, creates significant issues pertaining to the design of control systems, design of operating procedures, and the interaction between the... 

Volume 42 Laboratory Studies of Heterogeneous Catalytic Processes by E.G. Christoffel, revised and edited by Z. Paal Volume 43 Catalytic Processes under Unsteady-State Conditions by Yu. Sh. Metros... 

Historical data management—This includes the data acquisition and storage capabilities. Present-day prices of storage mediums have been dropping rapidly, and systems with 80 gigabyte hard disks are available. These disks could store a minimum of five years of one-minute data for most plants. One-minute data is adequate for most steady state operation, while start-ups and shutdowns or other non steady state operation should be monitored and stored at an interval of one second. To achieve these time rates, data for steady state operation can be obtained from most plant-wide D-CS systems, and for unsteady state conditions, data can be obtained from control systems. 

The resulting adsorption behavior in an unsteady-state fixed bed adsorber is illustrated in Fig. 7 [32], As the gas stream enters the carbon bed, which is initially free of adsorbate, the adsorbate is rapidly adsorbed, and the gas is essentially free of adsorbate as it continues through the carbon bed. As the adsorbent at the inlet... 

Large condensing loads, if handled on a steady state basis, result in appreciable cooling water and blowdown drum capacity requirements. These loads may be reduced, however, by the use of unsteady state condensing, e.g., by a worm cooler. 

In some cases where condensing loads are high, or where it is required to recover condensed liquid blowdown material for pollution, toxicity or economic reasons, an unsteady state condensing system may be appropriate. Examples or such applications are as rollows ... 

Another example of an unsteady state condensible blowdown system is the design for a phenol condensible blowdown tank. A blowdown tank is used in phenol treating plants to handle streams containing phenol and heavy hydrocarbons (lubricating oil stocks). The blowdown tank is illustrated in Figure 4. The design basis is as rollows ... 

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