Equation mathematical, solving

The need for dimensional consistency imposes a restraint in respect of each of the fundamentals involved in the dimensions of the variables. This is apparent from the previous discussion in which a series of simultaneous equations was solved, one equation for each of the fundamentals. A generalisation of this statement is provided in Buckingham s n theorem(4) which states that the number of dimensionless groups is equal to the number of variables minus the number of fundamental dimensions. In mathematical terms, this can be expressed as follows ... 

For constant fluid density the design equations for plug flow and batch reactors are mathematically identical in form with the space time and the holding time playing comparable roles (see Chapter 8). Consequently it is necessary to consider only the batch reactor case. The pertinent rate equations were solved previously in Section 5.3.1.1 to give the following results. 

Another approach lies in solving of the diffusion equation mathematically as [12]... 

In order to treat these equations mathematically, we use the artifice of generalizing them and solving them for any value of a and for an integral value of /. After such a solution is found, we can specialize it by reintroducing relation (5) between a and Z. Only solutions satisfying Schroedinger s conditions of finiteness will be considered. 

Resolution versus Sensitivity. A quadrupole mass filter can be programmed to move through a series of RF and dc combinations. The Mathieu equation, which is used in higher mathematics, can be used to predict what parameters are necessary for ions to be stable in a quadrupole field. The Mathieu equations are solved for the acceleration of the ions in the X, Y, and Z planes. A selected mass is proportional to (dc x RF x inner radius)/(RF frequency). For a given internal quadrupole radius and radio frequency, a plot can be made of RF and dc values that predict when a given mass will be stable in a quadrupole field. This is called a stability diagram (Figure 13.3). RF and dc combinations follow the value shown... 

With these five equations (Eqs. 23-42 to 23-46), two of them partial differential equations, the limits of the analytical approach and the goals of this book are clearly exceeded. However, at this point we take the occasion to look at how such equations are solved numerically. User-friendly computer programs, such as MAS AS (Modeling of Anthropogenic Substances in Aquatic Systems, Ulrich et al., 1995) or AQUASIM (Reichert, 1994), or just a general mathematical tool like MATLAB and MATHE-MATICA, can be used to solve these equations for arbitrary constant or variable parameters and boundary conditions. 

If the radial equation is solved by using the more conventional power series solution,2 the two mathematically allowed solutions for R(r) have leading terms... 

P 7] The topic has only been treated theoretically so far [28], A mathematical model was set up slip boundary conditions were used and the Navier-Stokes equation was solved to obtain two-dimensional electroosmotic flows for various distributions of the C, potential. The flow field was determined analytically using a Fourier series to allow one tracking of passive tracer particles for flow visualization. It was chosen to study the asymptotic behavior of the series components to overcome the limits of Fourier series with regard to slow convergence. In this way, with only a few terms highly accurate solutions are yielded. Then, alternation between two flow fields is used to induce chaotic advection. This is achieved by periodic alteration of the electrodes potentials. 

Next, we have to solve for the Yj s from the species continuity equations, Equation (32). Unfortunately, these equations cannot be integrated by a similar simple point iteration scheme as they are mathematically "stiff"16 and iterative approaches are unstable. To solve these simultaneous equations, we turn to a perturbation analysis developed by Newman17 where the equations are linearized about an initial guess, and the resulting linear equations are solved numerically. The solution is then used as the next guess, and the linear equations are resolved. The procedure is repeated until the solution no longer changes. 

The mathematical models of the reacting polydispersed particles usually have stiff ordinary differential equations. Stiffness arises from the effect of particle sizes on the thermal transients of the particles and from the strong temperature dependence of the reactions like combustion and devolatilization. The computation time for the numerical solution using commercially available stiff ODE solvers may take excessive time for some systems. A model that uses K discrete size cuts and N gas-solid reactions will have K(N + 1) differential equations. As an alternative to the numerical solution of these equations an iterative finite difference method was developed and tested on the pyrolysis model of polydispersed coal particles in a transport reactor. The resulting 160 differential equations were solved in less than 30 seconds on a CDC Cyber 73. This is compared to more than 10 hours on the same machine using a commercially available stiff solver which is based on Gear s method. 

The Laplace transform is essential in order to transform a partial differential equation into a total differential equation. After solving the equation the transform is inverted in order to obtain the solution to the mathematical problem in real time and space. 

Equation (1-13) or its body-fixed equivalent is of little use for Van der Waals complexes, as it discriminates one nuclear coordinate, e.g. y = 1. Specific mathematical forms of Hamiltonians describing the nuclear motions in Van der Waals dimers have been developed (7). This point will be discussed in more details in Section 12.4. Here we only want to stress that whatever the mathematical form of the Hamiltonian is used to solve the problem of nuclear motions, the results will be the same, if the Schrodinger equation is solved exactly. However, in weakly bound complexes there is a hierarchy of motions due to the strong intramolecular forces which determine the internal vibrations of the molecules, and to much weaker intermolecular forces which determine their relative translations and rotations. This hierarchy allows to make a separation between the intramolecular vibrations with high frequencies and the intermolecular modes with much lower frequencies. Such a separation of the fast intramolecular vibrations and slow rotation-vibration-tunneling motions can be performed if a suitable form of the Hamiltonian for the nuclear motions in Van der Waals molecules is used. 

These equations are solved for every time step in the simulation. Hence, it is assumed that the system is always on the equilibrium conditions (pseudoequilibrium approach). Model parameters are calculated by mathematical fitting. The good results obtained in the validation, together with the physical meaning of the parameters, are the two main advantages of this model, which can help to understand better the fundamentals of the electrochemical coagulation processes. 

Basic math and algebra skills. Chemistry requires calculations and the manipulation of mathematical equations to solve problems. Review your algebra skills before starting the chemistry lessons and you will find that chemistry will be easier to comprehend. 

Because it is more convenient mathematically, the coordinate system is changed from Cartesian to spherical polar coordinates (see Fig. 12.15) before the Schrodinger equation is solved. In the system of spherical polar coordinates a given point in space, specified by values of the Cartesian coordinates x, y, and z, is described by specific values of r, 6, and < >. 

The initial and boundary conditions under which the diffusion equation is solved define in mathematical form the kind of experiment being performed and the initial conditions of the experiment. It is important to realize that the resulting equations hold true only if these initial and boundary conditions have been maintained. This can be explained with the use of a few examples. 

This paper reports the mathematical modelling of electrochemical processes in the Soderberg aluminium electrolysis cell. We consider anode shape changes, variations of the potential distribution and formation of a gaseous layer under the anode surface. Evolution of the reactant concentrations is described by the system of diffusion-convection equations while the elliptic equation is solved for the Galvani potential. We compare its distribution with the C02 density and discuss the advantages of the finite volume method and the marker-and-cell approach for mathematical modelling of electrochemical reactions. 

Mathematical modeling of batteries can be accomplished successfully through the following steps (1) draw schematic diagram of the battery and define the complexity of the model, (2) formulate the equations, (3) solve model equations, (4) estimate properties and parameters, and (5) validate the model. An explanation for each of these steps is given next. 

Mathematical calculation, based on quantum mechanics, was established in the 1920s and developed with the rapid progress and spread of computers. Mathematical calculation is the field in which the approximated equation is solved by computers. 

Finally, we present an interpretation of our observations in terms of diffusion paths. Basically, the diffusion equations are solved for the case of two semi-infinite phases brought into contact under conditions where there is no convection and no interfacial resistance to mass transfer. Other simplifying assumptions such as uniform density and diffusion coefficients in each phase are usually made to simplify the mathematics. The analysis shows that the set of compositions in the system is independent of time although the location of a particular composition is time-dependent. The composition set can be plotted on the equilibrium phase diagram, thus showing the existence of intermediate phases and, as explained below, providing a method for predicting the occurrence of spontaneous emulsification. 

In order to illustrate the technical problem with the help of the simplest mathematical formalism, and for the sake of simplicity, we first assume that the sensors are linear and that their responses are independent for each investigated chemical species. Therefore the Ay quantities are only calibration constants. From the basic algebra of linear equation systems, it then follows that one needs N independent equations to solve the equation system (1). Therefore, the number M of different sensors has to be larger than or equal to the number TV of chemical species, i.e.,... 

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