Partial differential equations time derivative

If the dynamic process behavior has to be considered, Eqs. (3)—(5), (8)-(9) become partial differential equations including derivatives of the hold-up with respect to time (see more details in Section 9.5.2.6). 

We will first consider the simple case of diffusion of a non-electrolyte. The course of the diffusion (i.e. the dependence of the concentration of the diffusing substance on time and spatial coordinates) cannot be derived directly from Eq. (2.3.18) or Eq. (2.3.19) it is necessary to obtain a differential equation where the dependent variable is the concentration c while the time and the spatial coordinates are independent variables. The derivation is thus based on Eq. (2.2.10) or Eq. (2.2.5), where we set xj> = c and substitute from Eq. (2.3.18) or Eq. (2.3.19) for the fluxes. This yields Fick s second law (in fact, this is only a consequence of Fick s first law respecting the material balance—Eq. 2.2.10), which has the form of a partial differential equation... 

For optimization problems that are derived from (ordinary or partial) differential equation models, a number of advanced optimization strategies can be applied. Most of these problems are posed as NLPs, although recent work has also extended these models to MINLPs and global optimization formulations. For the optimization of profiles in time and space, indirect methods can be applied based on the optimality conditions of the infinite-dimensional problem using, for instance, the calculus of variations. However, these methods become difficult to apply if inequality constraints and discrete decisions are part of the optimization problem. Instead, current methods are based on NLP and MINLP formulations and can be divided into two classes ... 

Equations 4.5 and 4.6 are examples of partial differential equations because they contain partial derivatives, i.e., and The d symbol indicates that [C] and T" are functions of several variables. In this case, the variables are time (0 and depth (z), respectively. To evaluate a partial derivative, all but one of the variables must be held constant in the case of depth (z) is held constant and only time (f) is considered a variable. 

The simulation of a continuous, evaporative, crystallizer is described. Four methods to solve the nonlinear partial differential equation which describes the population dynamics, are compared with respect to their applicability, accuracy, efficiency and robustness. The method of lines transforms the partial differential equation into a set of ordinary differential equations. The Lax-Wendroff technique uses a finite difference approximation, to estimate both the derivative with respect to time and size. The remaining two are based on the method of characteristics. It can be concluded that the method of characteristics with a fixed time grid, the Lax-Wendroff technique and the transformation method, give satisfactory results in most of the applications. However, each of the methods has its o%m particular draw-back. The relevance of the major problems encountered are dicussed and it is concluded that the best method to be used depends very much on the application. 

The system of hyperbolic and parabolic partial differential equations representing the ID or 2D model of monolith channel is solved by the finite differences method with adaptive time-step control. An effective numerical solution is based on (i) discretization of continuous coordinates z, r and t, (ii) application of difference approximations of the derivatives, (iii) decomposition of the set of equations for Ts, T, c and cs, (iv) quasi-linearization of... 

Although the partial differential equation Eq. 25-10 is linear and looks rather simple, explicit analytical solutions can be derived only for special cases. They are characterized by the size of certain nondimensional numbers that completely determine the shape of the solutions in space and time. A reference distance x0 and a reference time f0 are chosen that are linked by ... 

The primary purpose of Chapters 2 and 3 is to derive the conservation equations. The conservations equations are partial differential equations where the independent variables are the spatial coordinates and time. Dependent variables are the velocity, pressure, energy, and species composition fields. Inasmuch as we devote some hundred pages to the derivations, it is helpful at this point to have a roadmap for the process. 

Overall our objective is to cast the conservation equations in the form of partial differential equations in an Eulerian framework with the spatial coordinates and time as the independent variables. The approach combines the notions of conservation laws on systems with the behavior of control volumes fixed in space, through which fluid flows. For a system, meaning an identified mass of fluid, one can apply well-known conservation laws. Examples are conservation of mass, momentum (F = ma), and energy (first law of thermodynamics). As a practical matter, however, it is impossible to keep track of all the systems that represent the flow and interaction of countless packets of fluid. Fortunately, as discussed in Section 2.3, it is possible to use a construct called the substantial derivative that quantitatively relates conservation laws on systems to fixed control volumes. 

Traditionally, physics emphasizes the local properties. Indeed, many of its branches are based on partial differential equations, as happens, for instance, with continuum mechanics, field theory, or electromagnetism. In these cases, the corresponding basic equations are constructed by viewing the world locally, since these equations consist in relations between space (and time) derivatives of the coordinates. In consonance, most experiments make measurements in small, simply connected space regions and refer therefore also to local properties. (There are some exceptions the Aharonov-Bohm effect is an interesting example.)... 

A different classification scheme for DEs, short for differential equations, separates those DEs with a single independent variable dependence, such as only time or only 1-dimensional position, from those depending on several variables, such as time and spatial position. DEs involving a single independent variable are routinely called ODEs, or ordinary differential equations. DEs involving several independent variables such as space and time are called PDEs, or partial differential equations because they involve partial derivatives. 

The most common methodology when solving transient problems using the finite element method, is to perform the usual Garlerkin weighted residual formulation on the spatial derivatives, body forces and time derivative terms, and then using a finite difference scheme to approximate the time derivative. The development, techniques and limitations that we introduced in Chapter 8 will apply here. The time discretization, explicit and implicit methods, stability, numerical diffusion etc., have all been discussed in detail in that chapter. For a general partial differential equation, we can write... 

The partial differential Equation 27 can be simplified to an ordinary differential equation in the independent variable z, by the following approximation For typical FCC catalysts and feedstocks, c A - 3 x CpC, and for the MAT test, [C/O] is 3 (Table I). Hence, if we assume that the catalyst and oil temperatures are identical (no heat transfer resistance between oil and catalyst), then as a first approximation, the change in the energy content of the oil and of the catalyst are roughly the same over the length of the run. Thus, the two terms on the left hand side of Equation 27 are approximately the same magnitude. Therefore, the time derivative of T can be lumped with the distance derivative. The right hand side of Equation 27 is divided by 2, and it becomes ... 

The idea of using this technique is that we may transform the partial differential equation in both time and coordinates into an ordinary differential equation in just the coordinates, which is usually easier to solve. The elimination of the derivative with respect to time dc/dt is easily seen by partial integration ... 

Errors and confusion in modelling arise because the complex set of coupled, nonlinear, partial differential equations are not usually an exact representation of the physical system. As examples, first consider the input parameters, such as chemical rate constants or diffusion coefficients. These input quantities, used as submodels in the detailed model, must be derived from more fundamental theories, models or experiments. They are usually not known to any appreciable accuracy and often their values are simply guesses. Or consider the geometry used in a calculation. It is often one or two dimensions less than needed to completely describe the real system. Multidimensional effects which may be important are either crudely approximated or ignored. This lack of exact correspondence between the model adopted and the actual physical system constitutes the basic problem of detailed modelling. This problem, which must be overcome in order to accurately model transient combustion systems, can be analyzed in terms of the multiple time scales, multiple space scales, geometric complexity, and physical complexity of the systems to be modelled. 

The independent variable in ordinary differential equations is time t. The partial differential equations includes the local coordinate z (height coordinate of fluidized bed) and the diameter dp of the particle population. An idea for the solution of partial differential equations is the discretization of the continuous domain. This means discretization of the height coordinate z and the diameter coordinate dp. In addition, the frequently used finite difference methods are applied, where the derivatives are replaced by central difference quotient based on the Taylor series. The idea of the Taylor series is the value of a function f(z) at z + Az can be expressed in terms of the value at z. 

In this explicit scheme, the first-order forward difference approximation is used for the time derivative. The second-order central difference approximation is used for the spatial derivatives. Hence, the finite difference equation (FDE) of the partial differential equation (PDE) Eq. (10.2) is... 

Zel dovich theory — The theory determines the time dependence of the nucleation rate 7(f) = d N (f )/df and of the number N(t) of nuclei and derives a theoretical expression for the induction time T needed to establish a stationary state in the supersaturated system. The -> Zel dovich approach [i] (see also [ii]) consists in expressing the time dependence of the number Z(n,t) of the n-atomic clusters in the supersaturated parent phase by means of a partial differential equation ... 

Poisson equation — In mathematics, the Poisson equation is a partial differential equation with broad utility in electrostatics, mechanical engineering, and theoretical physics. It is named after the French mathematician and physicist Simoon-Denis Poisson (1781-1840). In classical electrodynamics the Poisson equation describes the relationship between (electric) charge density and electrostatic potential, while in classical mechanics it describes the relationship between mass density and gravitational field. The Poisson equation in classical electrodynamics is not a basic equation, but follows directly from the Maxwell equations if all time derivatives are zero, i.e., for electrostatic conditions. The corresponding ( first ) Maxwell equation [i] for the electrical field strength E under these conditions is... 

Making the assumption that the gas stream characteristic curves are indeed horizontal for all practical purposes, is equivalent to setting the time partial derivatives for the concentrations and the energy density equal to zero in the original system of partial differential equations for the gas. Using this approximation reduces the gas equations to a set of steady state equations. 

The continuity equations for mass and energy were used to derive an adiabatic dynamic plug flow simulation model for a moving bed coal gasifier. The resulting set of hyperbolic partial differential equations represented a split boundary-value problem. The inherent numerical stiffness of the coupled gas-solids equations was handled by removing the time derivative from the gas stream equations. This converted the dynamic model to a set of partial differential equations for the solids stream coupled to a set of ordinary differential equations for the gas stream. 

How does one resolve the difficulty associated with partial differential equations The most common way is to reduce the system into a finite number of components. This can be accomplished by lumping together processes based upon time or location, or a combination of the two. One thus moves from partial derivatives to ordinary derivatives, where space is not taken directly into account. This reduction in complexity results in the compartmental models discussed later in this chapter. The same lumping process also forms the basis for the noncompartmental models discussed in the next section, although the reduction is much simpler than for compartmental models. 

The balance equations described in the previous sections include both space and time derivatives. Apart from a few simple cases, the resulting set of coupled partial differential equations (PDE) cannot be solved analytically. The solution (the concentration profiles) must be obtained numerically, either using self-developed programs or commercially available dynamic process simulation tools. The latter can be distinguished in general equation solvers, where the model has to be implemented by the user, or special software dedicated to chromatography. Some providers are given in Tab. 6.3. 

A closed-form analytical solution of this system of partial differential equations and relations (Eqs. 6.58 to 6.64a) is impossible to derive in the time domain. This is due to the extreme complexity of the general rate model, which accormts for the axial dispersion, the film mass transfer resistance, the pore diffusion and a first-order, slow kinetics of adsorption-desorption. 

Steady state mass or heat transfer in solids and current distribution in electrochemical systems involve solving elliptic partial differential equations. The method of lines has not been used for elliptic partial differential equations to our knowledge. Schiesser and Silebi (1997)[1] added a time derivative to the steady state elliptic partial differential equation and applied finite differences in both x and y directions and then arrived at the steady state solution by waiting for the process to reach steady state. [2] When finite differences are applied only in the x direction, we arrive at a system of second order ordinary differential equations in y. Unfortunately, this is a coupled system of boundary value problems in y (boundary conditions defined at y = 0 and y = 1) and, hence, initial value problem solvers cannot be used to solve these boundary value problems directly. In this chapter, we introduce two methods to solve this system of boundary value problems. Both linear and nonlinear elliptic partial differential equations will be discussed in this chapter. We will present semianalytical solutions for linear elliptic partial differential equations and numerical solutions for nonlinear elliptic partial differential equations based on method of lines. 

For nonlinear elliptic partial differential equations, successive relaxation or finite difference approximations can be used in both the coordinates.[7] [12] [13] (Constantinides Mostoufi, 1999 Davis, 1984, Finlayson, 1980) As illustrated by Schiesser (1991),[2] a method of lines was used for 2D and 3D steady state problems by adding a pseudo time derivative, applying finite differences in all the... 

One method to solve partial differential equations using the numerical schemes developed for solving time dependent ordinary differential methods is the method of lines. In this method, the spatial derivatives at time t are replaced by discrete approximations such as finite differences or finite element methods such as collocation or Galerkin. The reason for this approach is the advanced stage of development of schemes to solve ordinary differential equations. The resulting numerical schemes are frequently similar to those developed directly for partial differential equations. 

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