Graphical differentiation and integration

Graphical Differential and Integral Logarithmic Extrapolation Graphical Differential and Integral... 

Experimental data of thermod5mamic importance may be represented numerically, graphically, or in terms of an analytical equation. Often these data do not fit into a simple pattern that can be transcribed into a convenient equation. Consequently, numerical and graphical techniques, particularly for differentiation and integration, are important methods of treating thermodynamic data. 

In the first one, the desorption rates and the corresponding desorbed amounts at a set of particular temperatures are extracted from the output data. These pairs of values are then substituted into the Arrhenius equation, and from their temperature dependence its parameters are estimated. This is the most general treatment, for which a more empirical knowledge of the time-temperature dependence is sufficient, and which in principle does not presume a constancy of the parameters in the Arrhenius equation. It requires, however, a graphical or numerical integration of experimental data and in some cases their differentiation as well, which inherently brings about some loss of information and accuracy, The reliability of the temperature estimate throughout the whole experiment with this... 

The distribution function is presented graphically both as integral and differential distribution curves. In the integral distribution F(R) curve the abscissa axis depicts the size and the ordinate axis the fraction or percentage content of the total bubble number or the total volume of those bubbles whose size is bigger or smaller than R. In the differential distribution F(R) curve the abscissa axis depicts again the size but the ordinate axis the fraction content, i.e. number of bubbles entering a definite radius interval. The latter is more often employed. 

Integral (A) Constant temperature and pressure (rate depends upon composition only) Interpretation of integral data usually satisfactory by graphical differentiation or fitting of integral-conversion curves. 

User-defined kinematic laws allow time based simulation. Mechanism motions are sketched or defined using mathematical formulas. Mathematical function manipulations such as addition, subtraction, multiplication, division, scaling, integration, differentiation, and interpolation are important in the customization of calculations. Laws and relationships can be graphically visualized. Simulations analyze the speed and acceleration of coordinated movements of parts in different places of the mechanism as a reaction to specified input movement. 

The two major methods used predominantly in the kinetic analysis of isothermal data on solid-catalyzed reactions conducted in plug-flow PBRs are the differential method and the method of initial rates. The integral method is less frequently used either when data are scattered or to avoid numerical or graphical differentiation. Linear and nonlinear regression techniques are widely used in conjunction with these major methods. 

The partial differential equations that are obtained in most formulations are solved first in the particular simplified cases for which analytical solutions exist, and then more general approximate solutions are proposed through series finally graphical or numerical integrations provide solutions of the general time-dependent cases which are usually physically understandable after representation in graphs or tables. 

---