Unsteady

Separators working at unsteady conditions produce intermediates with compositions which do not allow them to be recycled. Alternatively, if the intermediate can be recycled, a nonoptimal recycle might produce (additional) unwanted byproducts in the reactor. 

When working at unsteady conditions, separators which normally split useful material from waste streams might lose material unnecessarily to the waste streams. 

Separators working at unsteady conditions produce products which do not meet the required sales specification. 

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... 

An unsteady ain-flow unbalance that alternates between inlets can set up an alternating thrust pattern which can be very damagiag to beariags designed for low thrust load. Mechanical vibration and elastic deformation problems and diagnostic techniques for stmctural iaadequacies ia fan design are discussed ia Reference 16. 

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 ... 

Neurological symptoms result from demyelination of the spinal cord and are potentially irreversible. The symptoms and signs characteristic of a vitamin B 2 deficiency include paresthesis of the hands and feet, decreased deep-tendon reflexes, unsteadiness, and potential psychiatric problems such as moodiness, hallucinations, delusions, and psychosis. Neuropsychiatric disorders sometimes develop independently of the anemia, particularly in elderly patients. Visual loss may develop as a result of optic atrophy. 

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). 

A few smelters have waste heat boilers to recover heat from converter off-gases (48). Most smelters do not have these waste heat boilers because the cycHc operation of converters results in unsteady production of steam. 

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. 

Example The equation 9T/9t = K(d T/dx ) represents the unsteady onedimensional conduction of heat. 

Example The equation for the unsteady transverse motion of a uniform beam clamped at the ends is... 

Phenomena of propagation such as vibrations are characterized by equations of hyperbolic type which are essentially different in their properties from other classes such as those which describe equihb-rium (elhptic) or unsteady diffusion and heat transfer (parabolic). Prototypes are as follows ... 

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

Hyperbolic Equations The most common situation yielding hyperbohc equations involves unsteady phenomena with convection. Two typical equations are the convective diffusive equation... 

Parabolic Equations in Two or Three Dimensions Computations become much more lengthy when there are two or more spatial dimensions. For example, we may have the unsteady heat conduction equation... 

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. 

Velocity The term kinematics refers to the quantitative description of fluid motion or deformation. The rate of deformation depends on the distribution of velocity within the fluid. Fluid velocity v is a vector quantity, with three cartesian components i , and v.. The velocity vector is a function of spatial position and time. A steady flow is one in which the velocity is independent of time, while in unsteady flow v varies with time. 

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