Programming mathematical equations

The best quality to be found may be a temperature, a temperature program or profile, a concentration, a conversion, a yield of preferred product, kind of reactor, size of reactor, daily production, profit or cost — a maximum or minimum of some of these factors. Examples of some of these cases are in this group of problems. When mathematical equations can be formulated, peaks or valleys are found by elementary mathematics or graphically. With several independent variables quite sophisticated mathematical procedures are available to find optima. Here a case of two variables occurs in problem P4.12.ll that is solved graphically. The application of Lagrange Multipliers for finding constrained optima is made in problem P4.ll.19. 

The thermodynamic equilibrium models, including surface complexation models, require the solution of a complex mathematical equation system. For this reason, many computer programs (e.g., CHEAQC, CHEMEQL, CHESS, EQ3/6, F1TEQL, Geochemist s Workbench, H ARPHRQ, JESS, MINTEQ and its versions, NETPATH, PHREEQC, PHRQPITZ, WHAM, etc.) have been developed to calculate the concentration and activity of chemical species, estimate the type and amount of minerals formed or dissolved, and the type and amount of sorbed complexes. 

One method of obtaining equilibrium concentrations of the various species involves no simplifying assumptions. In this method as many independent equations relating the unknown quantities are obtained as there are unknowns, in which a mathematical statement for each concentration and a mathematical equation for each equilibrium reaction can be written. These are then solved. For complex systems the calculations may be cumbersome. Sillen developed a computer program HALTAFALL (available in FORTRAN) by which equilibrium concentrations of species can be calculated thout simplifying assumptions. Precise values of equilibrium constants and activity coefficients are required for best results. 

A computer modeling code or program is a set of computer commands that include algorithms to solve a set of mathematical equations describing chemical equilibria,... 

The model method of the forecast is acceptable only at sufficient knowledge of cause and effect associations between initial and final data forecast. This method is based on the application of models, which imitate correlation of groxmd water properties and composition vs. external factors. A clearly formulated problem enables identification of most important factors and processes, maximum simplification of the model and choose the most optimal solution needed mathematical equations or their systems and available programs. 

Divide the spherical coal particle into units and using the finite difference method establish the mathematical equations and make them dimensionless, the purpose of which is to simplify the equations and decrease the number of conditions substitution for the model solution. By means of VB programming and calculation, finally the gas pressure distributions at the different time and the gas desorption amount in the coal particle are obtained. 

This description of the mechanical system is used by special multibody programs to automatically transform it into a set of mathematical equations, mostly ordinary differential equations (ODEs). 

In equation-based simulators, the mathematical equations that describe the physical process are entered into an equation solver that then uses appropriate techniques to solve them. In modular-based process simulators, the mathematical equations that describe the physical process are coded into modules that the user "flow sheets" together. Modular-based process simulators are preferred over equation-based simulators because it is easier for the user to "map" the real world into the virtual one, and programming and debugging of the modules are easier than analyzing sets of equations (Popovic and Bhatkar, 1997). However, equation-based simulators have proved highly successful in the field of optimum process control. Equation-based models handle instrument error and incorrect or errors from modeling simplifications better than modular-based simulators. Modular-based simulators invariably have a data reconciliation step, where the model is nm against values obtained from the instrumentation system and then a least-squares fit is performed to fit the model to the process. 

Some validation protocols use mathematical equations and statistic principles that are relatively complex and that are not discussed here. Computer programs now handle statistical treatment of the data (variance tests, for example). The operator s job essentially consists of entering the experimental results and following a validation protocol established by a specialist in quality procedures. 

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