Successive reduction. Integration

A complex integral can often be reduced to one of the standard forms by the method of integration by parts . By a repeated application of this method, complicated expressions may often be integrated, or else, if the expression cannot be integrated, the non-integrable part may be reduced to its simplest form. This procedure is sometimes called integration by successive reduction. See Ex. (5), above. 

This important result enables us to solve integrals of the form Se xndxf for by successive reduction... 

At high temperatures , T > 0d, the Dulong-Petit law applies once again. For T Partial integration allows successive reduction of the x term. At the end the exponential function remains. It disappears at the upper limit d/T 1 and becomes 1 at the lower limit. Hence, Cvib oc T results as a low temperature approximation in agreement with experiment -. Both theories presented have Cvib as a universal function of the reduced temperature... 

The results of the pilot study will be most credible if success criteria are established ahead of time. This avoids the temptation to adjust criteria to demonstrate success. Success criteria should be related to the benefits of integration previously identified. Ideally they should be able to provide an indication of the economic benefit achieved. For example, reduction in... 

As mentioned earlier, a major cause of high costs in fine chemicals manufacturing is the complexity of the processes. Hence, the key to more economical processes is reduction of the number of unit operations by judicious process integration. This pertains to the successful integration of, for example, chemical and biocatalytic steps, or of reaction steps with (catalyst) separations. A recurring problem in the batch-wise production of fine chemicals is the (perceived) necessity for solvent switches from one reaction step to another or from the reaction to the product separation. Process simplification, e.g. by integration of reaction and separation steps into a single unit operation, will provide obvious economic and environmental benefits. Examples include catalytic distillation, and the use of (catalytic) membranes to facilitate separation of products from catalysts. 

Many examples exist for the integration of robotic mechanisms with analysis instruments. Automation is quite successful for sample preparation stages, such as SPE applications for pharmacokinetic studies and for the queuing of samples for instrument analysis. In many cases, significant savings are realized in human labor expense and the reduction of routine operations. Furthermore, consistent robotic operations afford increased precision via reproducibility of operations from sample to sample compared to manual operations. 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

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