Coupled partial differential equations

Thus the system is defined by two coupled partial differential equations, which can be solved by finite-differencing. 

These coupled partial differential equations can be solved for the temperature and composition at any point in the catalyst bed by using numerical procedures to solve the correspon-... 

Equations 9.1-5 and -7 are two coupled partial differential equations with initial and... 

Specific balance equations for various polymer matrix composites manufacturing processes (i.e., RTM, IP, and AP) have been obtained by simplifying the balance equations. Particular attention has been paid to state all the assumptions used to arrive at the final equations clearly in order to clearly show the range of applicability of the equations. Moreover, appropriate numerical techniques for solution of these coupled partial differential equations have been briefly outlined and a few example simulations have been performed. 

Often, to simplify the analysis, the gas is assumed to be well mixed or to flow as a plug with no diffusion. In the well-mixed system, no gradients exist, and the set of coupled partial differential equations becomes sets of coupled algebraic equations, which is an enormous simplification. In general, however, spatial variations must be considered. 

The tubule is a spatially extended structure, and it presents both elastic properties and resistance to the fluid flow. The dynamic pressure and flow variations in such a structure can be represented by a set of coupled partial differential equations [11]. An approximate description in terms of ordinary differential equations (a lumped model) consists of an alternating sequence of elastic and resistive elements, and the simplest possible description, which we will adopt here, applies only a single pair of such elements. Hence our model [12] considers the proximal tubule as an elastic structure with little or no flow resistance. The pressure P, in the proximal tubule changes in response to differences between the in- and outgoing fluid flows ... 

Spatiotemporal pattern formation at the electrode electrolyte interface is described by equations that belong in a wider sense to the class of reaction-diffusion (RD) systems. In this type of coupled partial differential equations, any sustained spatial structure comes about owing to the interplay of the homogeneous dynamics or reaction dynamics and spatial transport processes. Therefore, the evolution of each variable, such as the concentration of a reacting species, can be separated into two parts the reaction part , which depends only on the values of the other variables at one particular location, and another part accounting for transport processes that are induced by spatial variations in the variables. These latter processes constitute a spatial coupling among different locations. 

The plug flow reactor is increasingly being used under transient conditions to obtain kinetic data by analysing the combined reactor and catalyst response upon a stimulus. Mostly used are a small reactant pulse (e.g. in temporal analysis of products (TAP) [16] and positron emission profiling (PEP) [17, 18]) or a concentration step change (in step-response measurements (SRE) [19]). Isotopically labeled compounds are used which allow operation under overall steady state conditions, but under transient conditions with respect to the labeled compound [18, 20-23]. In this type of experiments both time- and position-dependent concentration profiles will develop which are described by sets of coupled partial differential equations (PDEs). These include the concentrations of proposed intermediates at the catalyst. The mathematical treatment is more complex and more parameters are to be estimated [17]. Basically, kinetic studies consist of ... 

Most challenging are kinetic studies by transient techniques. Here, one has a set of coupled partial differential equations (PDE) to be solved that describe the... 

After some mathematical manipulations [11-14], the one-dimensional system of three non-linear coupled partial differential equations which model the drying process in a thermal equilibrium environment are given by [15-20] ... 

The media which are of interest for this review do not contain macroscopic charge and current densities, hence p = 0 and 7 = 0, and they are not magnetized, so that M = Q. Then Maxwell s equations and the constitutive relations may be combined to yield the following coupled partial differential equation between the electric field E and the dielectric polarization P. 

Systems containing coupled partial differential equations and integro-differential equations, such as (2.1), present significant challenges to mathematical analysis. Much progress on these difficult equations is presented in [MD]. Following Cushing [Cu2] with only minor differences, we assume that (2.1) defines a unique solution S t) and p t,l) for />0 and introduce the moment functions ... 

Since the model equation are coupled partial differential equations, they are solved numerically by using a combination of the orthogonal collocation technique [14] and an ODE integrator [15]. 

Mathematical models of flow processes are non-linear, coupled partial differential equations. Analytical solutions are possible only for some simple cases. For most flow processes which are of interest to a reactor engineer, the governing equations need to be solved numerically. A brief overview of basic steps involved in the numerical solution of model equations is given in Section 1.2. In this chapter, details of the numerical solution of model equations are discussed. 

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. 

Beste et al. [104] compared the results obtained with the SMB and the TMB models, using numerical solutions. All the models used assumed axially dispersed plug flow, the linear driving force model for the mass transfer kinetics, and non-linear competitive isotherms. The coupled partial differential equations of the SMB model were transformed with the method of lines [105] into a set of ordinary differential equations. This system of equations was solved with a conventional set of initial and boundary conditions, using the commercially available solver SPEEDUP. Eor the TMB model, the method of orthogonal collocation was used to transfer the differential equations and the boimdary conditions into a set of non-linear algebraic equations which were solved numerically with the Newton-Raphson algorithm. 

The design and optimization of adsorptive processes typically require simultaneous numerical solutions of coupled partial differential equations describing the mass, heat, and momentum balances for the process steps. Multicomponent adsorption equilibria, kinetics, and heat for the system of interest form the key fundamental input variables for the design. " Bench- and pilot-scale process performance data are generally needed to confirm design calculations. 

Light beams are represented by electromagnetic waves that are described in a medium by four vector fields the electric field E r, t), the magnetic field H r, t), the electric displacement field D r,t), and B r,t) the magnetic induction field (or magnetic flux density). Throughout this chapter we will use bold symbols to denote vector quantities. All field vectors are functions of position and time. In a dielectric medium they satisfy a set of coupled partial differential equations known as Maxwell s equations. In the CGS system of units, they give... 

Equations 5.1.6 represent a set of n - 1 coupled partial differential equations. Since the Fick matrix [ )] is a strong function of composition it is not always possible to obtain exact solutions to Eqs. 5.1.6 without recourse to numerical techniques. The basis of the method put forward by Toor and by Stewart and Prober is the assumption that c and [D] can be considered constant. (Actually, Toor worked with the generalized Fick s law formulation, whereas Stewart and Prober worked with the Maxwell-Stefan formulation. Toor et al. (1965) subsequently showed the two approaches to be equivalent.) With this assumption Eqs. 5.1.6 reduce to... 

Examination of Eqs. 5.1.7-5.1.10 shows that the similarity transformation reduces the original set of n — 1 coupled partial differential equations to a set of n — 1 uncoupled partial differential equations in the pseudocompositions. Equation (5.1.10) for the /th pseudocomponent is... 

Axial or radial temperature and concentration gradients. The following coupled partial differential equations were solved using FEMLAB ... 

The complete solution of these (coupled, partial differential) equations is beyond the scope of this text.5 Here we seek out a simple intuitive solution. 

Computer simulation of explosive fracture of rock can be carried out with finite difference stress wave propagation codes, such as the YAQUI code (2). YAQUI integrates in time the coupled partial differential equations for the conservation of mass, momentum, and energy. For a compressible fluid, these equations are... 

The mathematical model described above involves non-linear, coupled, partial differential equations. The equations were solved using a Finite-Difference method. Details of this mathematical technique have been described elsewhere in the literature (8.9). Figure 2 shows a flowsheet for the numerical solution of these model equations. 

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