Integrated and decompositional methods of calculation of CTS.

The essence of integrated methods of calculation of CTS consists in association of systems of the equations describing operation of separate devices in one big system of the equations with the further solution of this system. At a decompositional method of calculation of CTS, it is presented in the form of the separate blocks corresponding to the CTS elements and, calculation of CTS comes down to consecutive calculation of separate blocks. In this case, dimension of each separate system of the equations corresponding to the CTS block is rather small. We will compare characteristics of integrated and decompositional methods of calculation of CTS:

As it has been stated above, the essence of an integrated method consists in association of systems of the equations describing operation of separate devices in one big system of the equations with the further solution of this system. Thus, the linear equations of material and thermal balances unite with the nonlinear equations of balance of chemical reactions, the differential linear and nonlinear equations, the hydrodynamics equations in private derivatives, etc. in uniform "big" system of the equations, for example, in a general view:

 

This system of the equations contains a set of the equations of various type from linear to the differential equations in private derivatives. Such systems of the equations are called mixed and demand special mathematical methods for the decision. Moreover, depending on type of the equations (which complexity is defined by type of modules), methods of the solution of system of the equations can have purely mathematical restrictions and demand special representation of a task. It will lead to the fact that for concrete CTS has to the unique system of the equations is formed. In connection with complexity, the system of the equations can be stubborn and demand application of special mathematical methods. Therefore, before use of an integrated method it is necessary to analyses previously from the mathematical point of view the mathematical dependences which are the cornerstone of CTS modules.

Thus, for use of an integrated method, the designer needs to have rather serious mathematical preparation and special computer programs for the solution of the mixed systems of the equations (linear, nonlinear, differential, in private derivatives, etc.). However even in this case, for the purpose of expeditious obtaining results of calculation, the integrated way of calculation can be recommended only for simple CTS or for CTS where it is necessary to calculate only material balances without kinetics, thermodynamics, etc. (i.e. to solve linear system of the equations).

The essence of a decompositional method of calculation is that CTS is presented in the form of the separate blocks corresponding to the CTS elements. Calculation of CTS comes down to consecutive calculation of separate blocks. In this case, when calculating the separate module it is required to calculate only limited quantity of the equations corresponding to the concrete module i.e. to execute test calculation of concrete process. It should be noted that with the limited number of the CTS possible modules, their algorithms of calculation are developed and given in special literature and in the form of computer programs long ago. For this reason, owing to the universality, the greatest distribution as when calculating difficult, and simple CTS, I have received a decompositional way of calculation.

It is known that the majority of CTS has the recirculation connections forming the closed CTS which direct calculation by means of the decompositional principle is impossible. For the solution of such systems their structure at first needs to be given to the opened look, and, only then to make calculation with use of a decompositional way of calculation. However, in spite of the fact that the theory and analysis algorithms of structure of CTS for the purpose of definition of an optimum set of the broken links with the purpose of the transfer of structure from closed to the opened look, and findings of the optimum sequence of calculation of CTS, are rather well developed, each CTS in itself is unique. In this regard, in a concrete case there can be problems of finding of an optimum set of the broken links and the optimum sequence of calculation in the decompositional way.

There are kinds of a decompositional way of calculation of the closed CTS, the simplest of which is the iterative way of calculation. We will consider an iterative way of calculation of the closed CTS on the example of the elementary scheme submitted in Fig. 4.1.

Fig. 4.1. Illustration of an iterative way of calculation of CTS

 

Apparently in Fig. 4.1a, the elementary closed CTS consists of two modules (And yes In) connected by four technological communications from which communication 4 is recirculation. Proceeding from an initial problem of calculation of CTS, parameters of functioning of elements A and B, and also parameters of the stream number 1 entering CTS will be basic data for calculation of the specified CTS. However, it is impossible to carry out calculation of the module A for the purpose of obtaining parameters of a stream 2 since parameters of a stream 4 are unknown. Calculation of the module B can't also be made since the stream 2 entering this module is unknown. Thus, direct application of a decompositional way of calculation of this closed CTS is impossible.

In order that the decompositional way could be applied, it is necessary to lead CTS from the closed look to opened. For this purpose, in case of the specified CTS, it is possible "to break off" any stream entering recycling, i.e. a stream 2 or 4. In case of a rupture of a stream 4 (see Fig. 4.1b), going out of the module B and entering the module A, the new stream entering CTS and the module A 4 is formed'. Because division of a stream on 4 and 4' is conditional (CTS applied only to the purpose of the transfer of structure from closed to the opened look), at application of an iterative way of calculation, to the place of a gap the additional module – the iterative block (IB) is located (see Fig. 4.1v). In this case, proceeding from an initial problem of calculation of CTS, basic data for calculation of the specified CTS will be parameters of functioning of elements A and B, and also parameters of the entering streams 1 and 4'. Initial parameters of a stream 4' can decide on application of any algorithm of calculation and on the basis of the set basic data.

With the specified set of basic data there is an opportunity to execute the FIRST calculation of CTS, i.e. to determine parameters of a stream 2, knowing which to calculate parameters of streams 3 and 4. In this case, parameters of a stream 4 will differ from parameters of a stream 4' therefore, the iterative block, having analysed both data sets (streams 4 and 4'), will calculate a total error and will appropriate new values of parameters of a stream 4'. As new values of a stream 4' will be formed by the iterative block taking into account calculated parameters of a stream 4, when performing the SECOND calculation of CTS, the total error will be less, than at the first calculation. Further, cyclic calculations (iterations) are carried out until values of a total error aren't below the required calculation accuracy.

The iterative method of calculation of CTS is usually applied to calculation of rather simple CTS since application of this method for difficult CTS is not rather effective since provides consecutive approximations of required parameters of flows. Because the CTS elements, proceeding from their physical and chemical nature, can function only in the set intervals of change of parameters, application of an iterative method can be sometimes impossible since in the course of convergence of this mathematical method, values of technological parameters can go beyond functioning of the CTS elements. When calculating CTS having several broken-off flows (availability of several recyclings), application of an iterative method in general can be rather problematic since owing to availability of technological communications, iterative processes will be interconnected that will negatively influence achievement of the decision for all system.

When calculating difficult CTS, the having several broken-off flows, the methods of total multidimensional minimization of an error described in special literature are usually applied (for example,/9/). The essence of these methods is that unlike an iterative method, required parameter values of flows are calculated when carrying out calculation, by means of special mathematical methods with restrictions which availability doesn't allow to go beyond functioning of technological operators (in the course of finding of the decision) that allows to reach convergence much quicker and more reliably.

As it was stated above, recycling can be given from the closed type to the opened type by a gap of one of the technological communications entering recycling. In Fig. 4.1g the option of a gap of a flow 2 is provided. In this case, having initial approximations of parameters of a flow 2', the module B will be calculated with determination of parameters of flows 3 and 4, and then the module A with determination of parameters of a flow 2 at first. Unlike the previous option, iteration will be carried out in parameters of a flow 2, but not a flow 4. Questions of the choice of optimal variants of the translation of CTS from closed to the opened type will be considered further.

Comparison of features of integrated and decompositional methods of calculation of CTS are provided in Table 4.1.

Table 4.1.

Comparative characteristics integrated and decompositional of methods of calculation of CTS.

 

Integral method	Decompositional method
Method of submission of the task
Global system of equations the Separate simulating units which are joined to the help of the coordinating program	Global system of equations the Separate simulating units which are joined to the help of the coordinating program
Method of the decision of the task
The joint solution of the equations Sequential calculation with use of an iterative method of calculation, and, with the preliminary analysis of CTS for detection of the optimum sequence of calculation of CTS.	The joint solution of the equations Sequential calculation with use of an iterative method of calculation, and, with the preliminary analysis of CTS for detection of the optimum sequence of calculation of CTS.
Advantages
Possibility of carrying out calculation for any set of unknown variables Smaller number of computation, visualization	Possibility of carrying out calculation for any set of unknown variables Smaller number of computation, visualization
Shortcomings
Big dimensionality of a single system of the equations along with absence of safe methods of the solution of the mixed systems of the linear, non-linear and differential equations of big dimensionality. Uniqueness of each system of equations. Difficulty of creation of an optimum algorithm of calculation of CTS.	Big dimensionality of a single system of the equations along with absence of safe methods of the solution of the mixed systems of the linear, non-linear and differential equations of big dimensionality. Uniqueness of each system of equations. Difficulty of creation of an optimum algorithm of calculation of CTS.
Recommendations
To apply only when calculating of the simplified CTS to Apply to calculation of CTS of arbitrary complexity.	To apply only when calculating of the simplified CTS to Apply to calculation of CTS of arbitrary complexity.

Test questions

1. State an essence of integrated, decompositional, iterative methods of calculation of CTS

2. List merits and demerits of methods of calculation of CTS.

3. Specify scopes of methods of calculation of CTS.