Continuity and Semicontinuity

Definition 2.2.7 (Continuous function) Let S be a subset of 3 n,jc° 5, and /(x) a real valued function defined on 5. f(x) is continuous at jc° if either of the following equivalent conditions hold  

Remark 3 Convex and concave functions may not be continuous everywhere but the points of discontinuity have to be on the boundary of S. 

Let fi(x) be a family of lower (upper) semicontinuous functions on S. Then (i) Its least upper bound (greatest lower bound) 

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In practice there are two major classifications of cost accounting systems, job costing and process costing. In the former, costs are col-lec ted for each job or batch irrespective of the accounting period. This system is normally used in construction work. Process costing is normally used in continuous and semicontinuous processes. Costs are collected for a specific accounting period. 

Due to this overdetermination problem there exists a strong prejudice in the chemical industry that production planning for batch production is completely different from production planning for continuous production. This leads to the ineffective separation between process engineering and business data processing in continuous and semicontinuous production. 

The design techniques used include both stagewise and continuous-contacting methods and can be applied to batch, continuous, and semicontinuous operations. 

Both continuous and semicontinuous operation can be utilized. This principle has been successfully applied in the resolution of optical isomers in which nucleation must be minimized, and preferably eliminated, to achieve isomer separation, as described by Midler (1970, 1975, 1976) and presented in Examples 7-6 and 11-6 in this book. Tools for monitoring PSD online, as mentioned in Section 2.10.2, are very applicable here. 

The difference between batch, semibatch, continuous, and semicontinuous processing was discussed in Chapter 1. Continuous processes are characterized by parameters which may be geographically distributed within the system but are unchanging with time. Continuous crystallizers are in common use throughout the chemical process industries, but are less so in the pharmaceutical and fine chemical industries because of the typically smaller amounts to be processed. This section on continuous cooling crystallization will discuss continuous operation and point out the differences from batch/semibatch operation described above. It will also illustrate some strategies and equipment types used for these operations. 

Continuous and semicontinuous electrochemical reactors are normally employed for effluent metal ion remediation, where the anode reaction is usually oxygen evolution from water [compare with Equation (26.4)]. After the metal contaminant is captured on the cathode, the cathode can be discarded, the collected metal can be resold, or the deposited metal can be chemically or elecfro-chemically etched into a small volume of a suitable leaching liquor (e.g., water) so as to increase its concentration substantially. 

Design Scope. Stirred vessels, rotor-stator mixers, static mixers, decanters, settlers, centrifuges, homogenizers, extraction colunms, and electrostatic coalescers are examples of industrial process equipment used to contact liquid-liquid systems. Although this chapter emphasizes stirred vessels, the fundamentals of phase behavior are applicable to a broad range of other equipment types. Immiscible liquid-liquid systems are processed in batch, continuous, and semicontinuous modes. 

Continuous and semicontinuous flow reactors are sometimes used in fine chemical and pharmaceutical applications, primarily because some reactions require the high intensity of mixing that can only be achieved in in-line mixers (see Examples 13-3, 13-7, and 13-8a and discussion in Chapter 17). 

Although batch processes are the workhorse in research laboratory environments, continuous (and semicontinuous) reactors predominate for commercial PE production. In a continuous polymerization reactor, all monomers and reagents are constantly fed into the reactor, and the polymer is isolated from the effluent. Flows are adjusted to achieve the desired steady-state conditions as measured by online analytical instruments and polymer analysis. 

Uses Detergent for scouring and bleaching of coUon and its blends in continuous and semicontinuous applic. 

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