Specification of Variables

There are four common ways of expressing the required separation to be made by the distillation column purity of one Or more products, allowable impurities in one or more products, recovery of one or more feed components, and measurable properties of one or more of the products. Examples of these are as 

Parities 99.6 mol % minimum purity of light key in distillate product 

Impurities 0,3 mol % maximum conicot of heavy key in distillate product 

Recovery 95% minimum of light key in feed, to be contained in distillate product 

Properties Distillate product to hove a Reid vapor pressure of 45 fofl in.2 min imam 

These may be in conibination and not limited to one of the k. However, one must be careful not to overspecify or underqiecify dm separation, as discussed below. 

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This effect can be compared to the gas consumption of a car requiring more gas per 100 km being run at a higher average speed. Selective papers address the specifics of variable raw material consumption e.g. in the case of fuel consumption of container ships (Jetlund et al. 2004, p. 1271). Hence, the problem to balance raw material consumption and volatile raw material costs with sales quantities and prices has to be solved in value chain planning. 

Dynamic simulation is more demanding as its steady state counterpart. Firstly, it needs much more sizing elements. Then, the pressure variation cannot be neglected or lumped in the specification of simulation unit. However, in general the specification of variables is more flexible. Any flowsheet variable could be set as, irrespective if this regards input or output streams, or internal unit variables (see later in Chapter 3). 

Input of the starting conditions and specification of variables values in the initial simulation time TIME =0, input of the simulation period TEND. Elements state L(i) = 0, system state S = 0. 

We consider the computation of a trajectory —X t), where X t) is a vector of variables that evolve in time —f. The vector includes all the coordinates of the particles in the system and may include the velocities as well. Unless specifically indicated otherwise X (t) includes coordinates only. The usual way in which such vectors are propagated numerically in time is via a sequence of short time solutions to a differential equation. One of the differential equations of prime concern is the Newton s equation of motion ... 

At present it is not possible to determine which of these mechanisms or their variations most accurately represents the behavior of Ziegler-Natta catalysts. In view of the number of variables in these catalyzed polymerizations, both mechanisms may be valid, each for different specific systems. In the following example the termination step of coordination polymerizations is considered. 

Variable-Area Flow Meters. In variable-head flow meters, the pressure differential varies with flow rate across a constant restriction. In variable-area meters, the differential is maintained constant and the restriction area allowed to change in proportion to the flow rate. A variable-area meter is thus essentially a form of variable orifice. In its most common form, a variable-area meter consists of a tapered tube mounted vertically and containing a float that is free to move in the tube. When flow is introduced into the small diameter bottom end, the float rises to a point of dynamic equiHbrium at which the pressure differential across the float balances the weight of the float less its buoyancy. The shape and weight of the float, the relative diameters of tube and float, and the variation of the tube diameter with elevation all determine the performance characteristics of the meter for a specific set of fluid conditions. A ball float in a conical constant-taper glass tube is the most common design it is widely used in the measurement of low flow rates at essentially constant viscosity. The flow rate is normally deterrnined visually by float position relative to an etched scale on the side of the tube. Such a meter is simple and inexpensive but, with care in manufacture and caHbration, can provide rea dings accurate to within several percent of full-scale flow for either Hquid or gas. 

Utdity power distribution grids normally operate at a fixed frequency of 50 or 60 Hz. These frequencies can be utilized directiy for the induction process if the load characteristics are appropriate. If they are not, specific appHcations can be optimized by the use of variable and higher frequencies produced by soHd-state frequency power converters connected between the supply and the load. 

In order to operate a process facility in a safe and efficient manner, it is essential to be able to control the process at a desired state or sequence of states. This goal is usually achieved by implementing control strategies on a broad array of hardware and software. The state of a process is characterized by specific values for a relevant set of variables, eg, temperatures, flows, pressures, compositions, etc. Both external and internal conditions, classified as uncontrollable or controllable, affect the state. Controllable conditions may be further classified as controlled, manipulated, or not controlled. Excellent overviews of the basic concepts of process control are available (1 6). 

Generally, Httle is known in advance concerning the degree of homogeneity of most sampled systems. Uniformity, rarely constant throughout bulk systems, is often nonrandom. During the production of thousands of tons of material, size and shape distribution, surface and bulk composition, density, moisture, etc, can vary. Thus, in any bulk container, the product may be stratified into zones of variable properties. In gas and Hquid systems, particulates segregate and concentrate in specific locations in the container as the result of sedimentation (qv) or flotation (qv) processes. 

Similarly a hypothetical spHtter must be defined when a stream is spHt. Furthermore, control units ate identified in calculations flow diagrams when temperatures, pressures, and flows of streams ate controlled on the basis of variables in other parts of flow sheets. Sometimes variables such as product specifications and reactor yields can be represented as hypothetical control units. 

In terms of the derived general relationships (3-1) and (3-2), x, y, and h are independent variables—cost and volume, dependent variables. That is, the cost and volume become fixed with the specification of dimensions. However, corresponding to the given restriedion of the problem, relative to volume, the function g(x, y, z) =xyh becomes a constraint funedion. In place of three independent and two dependent variables the problem reduces to two independent (volume has been constrained) and two dependent as in functions (3-3) and (3-4). Further, the requirement of minimum cost reduces the problem to three dependent variables x, y, h) and no degrees of freedom, that is, freedom of independent selection. 

No single method or algorithm of optimization exists that can be apphed efficiently to all problems. The method chosen for any particular case will depend primarily on (I) the character of the objective function, (2) the nature of the constraints, and (3) the number of independent and dependent variables. Table 8-6 summarizes the six general steps for the analysis and solution of optimization problems (Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-HiU, New York, 1988). You do not have to follow the cited order exac tly, but vou should cover all of the steps eventually. Shortcuts in the procedure are allowable, and the easy steps can be performed first. Steps I, 2, and 3 deal with the mathematical definition of the problem ideutificatiou of variables and specification of the objective function and statement of the constraints. If the process to be optimized is very complex, it may be necessaiy to reformulate the problem so that it can be solved with reasonable effort. Later in this section, we discuss the development of mathematical models for the process and the objec tive function (the economic model). 

Specification of the feed stream L (C + 2 variables), the ratio L i i/D, the heat leak q, and the pressure of either stream leaving the divider utilizes these design variables and defines one unique operation of the divider. 

The simple absorber column shown in Fig. 13-23 will be analyzed here to illustrate the procedure. This unit consists of a series of simple equilibrium stages orthe type in Fig. 13-22. Specification of the number of stages N utilizes the single repetition variable and... 

For any specific incident there will be an infinite number of incident outcome cases that can be considered. There is also a wide degree of consequence models which can be apphed. It is important, therefore, to understand the objective of the study to limit the number of incident outcome cases to those which satisfy that objective. An example of variables which can be considered is as follows. 

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