Material balance computer codes

The fraction of gas or vapour in the two-phase mixture entering the pressure relief system is an important parameter if the detailed pressure/ time history for the vented runaway reaction is to be predicted. This type of calculation is performed by a number of computer codes (see Annex 4). In order to predict the gas or vapour fraction in the two-phase flow entering the relief system, the "coupling equation", a material balance for the vessel and relief system, needs to be solved iteratively. This is discussed in detail by DIERS[5J and is beyond the scope of this Workbook. 

On the other hand, if you cannot avoid solving a set of simultaneous coupled equations because the set you are working with is tightly coupled in its initial structure and cannot be simplified even by successive substitution of one equation into another, you should read Sec. 2.7, in which computer-aided solutions are discussed. As explained in that section, you can use readily available software in computer center libraries or the computer codes in the pocket in the back of this book to solve sets of independent material balances. 

Many of the problems that you will work with will lead to sets of equations that do have to be solved simultaneously. The purpose of this section is to show you that the formulation of such problems follows the same strategy as used in Sec. 2.3, the strategy outlined in Table 2.4. If only two or three coupled linear material balances are written for a problem, the unknown variables can be solved for by substitution of one equation into another. If the material balances consist of large sets of linear equations, you will find suggestions for solving them in Appendix L, and Fortran computer codes to solve them on the disk in the pocket in the back of this book. A single material balance that is a nonlinear equation of the form... 

If you formulate a material balance problem that results in nonlinear equations, you should review Sec. L.2. There you wiU find an outline of techniques to solve sets of nonlinear equations, and also recommendations as to computer codes that are state of the art. In the pocket in the back of the text is a simple Fortran code to solve sets of nonlinear equations based on Newton s method and another one based on an optimization technique. You can use them to solve one or more nonlinear equations, and they will be effective for most of the problems in this book. 

You are now ready to enter the equations for the internal and boundary nodes. Some spreadsheets may start solving the equations as you enter them leading to all sorts of error messages. To avoid this outcome, define a constant in, say, cell A3 (first column, third row) to be zero, and multiply each equation by A3 as you enter it. Then, when you have completed entering all the equations, you need to solve the problem, change A3 from 0 to 1, and begin iteration. A formula is entered in a cell, but the formulas themselves are not displayed in the cell on the screen. What is displayed is the value given by the formula. For example, to show the product in cell DIO of the feed located in cell F24 and the concentration located in F25, you would enter into cell DIO the formula F24 F25. It is recommended that you set up the overall material balances about specific process units in addition to the equations entered on the main part of the spreadsheet. If you take this step, you can quickly check for errors in the setup of the balances. Also, as you enter equations and data, check interim calculations just as you would check out a computer code as it was written. 

SOLVING MATERIAL AND ENERGY BALANCES VIA COMPUTER CODES... 

Solving Material and Energy Balances via Computer Codes Chap. 5... 

In Sec. 2.4 we discussed combining units from the viewpoint of making material balances. As more and more units are connected together in a plant, you can understand that the degree of complexity requires that the solution of material and energy balances be carried out via a computer code. Such a program can also, at the same time, determine the size of equipment and piping, evaluate costs, and optimize performance. 

The interconnections between the unit modules may represent information flow as well as material and energy flow. In the mathematical representation of the plant, the interconnection equations are the material and energy balance flows between model subsystems. Equations for models such as mixing, reaction, heat exchange, and so on, must also be listed so that they can be entered into the computer code used to solve the equation. Table 5.1 lists the common type of equations that might be used for a single subsystem. In general, similar process units repeatedly occur in a plant and can be represented by the same set of equations, which differ only in the names of variables, the number of terms in the summations, and the values of any coefficients in the equations. 

Once the tear streams are identified and the sequence of calculations specified, everything is in order for the solution of material and energy balances. All that has to be done is to calculate the correct values for the stream flow rates and their properties. To execute the calculations, many computer codes use the method of successive substitution, which is described in Appendix L. The output(s) of each module on interation k is expressed as an explicit function of the input(s) calculated fi om the previous iteration, A - 1. For example, in Fig. 5.16 for module 1,... 

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