Laplace formulation

Unlike the phenomena and characterization techniques described in Sections 5.3.1 through 5.3.3, the water transport in the PTL lacks adequate theoretical foundations. The current literature questions the assumptions in the original Young-Laplace formulations, and it emphasizes the lack of reliable models. The clarification and consolidation of the relevant terminology, theory, and experimental techniques has been aided by the recent efforts, but finding predictive relationships to PTL performance, degradation, and durability remains a challenge. 

Numerical resolution of mass-transport equations, along with Laplace formulation of the electrical properties of the SECCM system, is an efficient and reliable strategy for the characterization of SECCM mass transport, technique optimization, and the analysis of experimental data with the aim of extracting quantitative information. This is an indispensable tool not only for detailed examination of experimental data but also for developing novel experiments and exploiting new phenomena that can be examined with SECCM. 

Let us first state a few important points about the application of Laplace transform in solving differential equations (Fig. 2.1). After we have formulated a model in terms of a linear or linearized differential equation, dy/dt = f(y), we can solve for y(t). Alternatively, we can transform the equation into an algebraic problem as represented by the function G(s) in the Laplace domain and solve for Y(s). The time domain solution y(t) can be obtained with an inverse transform, but we rarely do so in control analysis. 

We can extend these results to find the Laplace transform of higher order derivatives. The key is that if we use deviation variables in the problem formulation, all the initial value terms will drop out in Eqs. (2-13) and (2-14). This is how we can get these clean-looking transfer functions later. 

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

Inputs + Sources = Outputs + Sinks + Accumulation Formulation of differential equations in general is described in Chapter 1. Usually the ODE is of the first or second order and is readily solvable directly or by aid of the Laplace Transform. For example, for the special case of initial equilibrium or dead state (All derivatives zero at time zero), the preceding equation has the transform... 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

The most convenient procedure is, instead of combining eqns. (18) and (22), to formulate the Laplace transform of the rate equation, eqn. (18), for constant potential... 

We have seen already that the general solution of Fick s second law in the Laplace domain can be formulated in various convenient ways. Here, we choose the notation of eqn. (147)... 

In this section, a description of the state of the art is attempted by (i) a review of the most fundamental types of reaction schemes, illustrated by some examples (ii) formulation of corresponding sets of differential equations and boundary conditions and derivation of their solutions in Laplace form (iii) description of rigorous and approximate expressions for the response in the current and/or potential step methods and (iv) discussion of the faradaic impedance or admittance. Not all the underlying conditions and fundamentals will be treated in depth. The... 

FORMULATION OF THE MASS TRANSFER EQUATIONS AND THEIR SOLUTIONS IN LAPLACE FORM... 

With appropriate adaptations, the other monomolecular reaction schemes can be treated by similar procedures. It may suffice, therefore, to give below the resulting Laplace transforms of the interfacial concentrations in terms of parameters, the meanings of which are given in Table 8. Where possible, references to the literature are given. However, notations and formulations are often quite different due to the personal preferences of the authors. 

General considerations of renormaiizability are best carried through within the field theoretic formulation. We recall from Sect. 7.2 that a Laplace transform with respect to the chain length variables eliminates the segment summations. In the resulting field theoretic formulation a propagator line of momentum k, being part of the m-th polymer line, yields a factor (cf, Eq. (7.17))... 

The boundary conditions are defined in the same way as with the flow analysis network. The nodes whose control volumes are empty or partially filled are assigned a zero pressure, and the gate nodes are either assigned an injection pressure or an injection volume flow rate. Just as is the case with flow analysis network, a mass balance about each nodal control volume will lead to a linear set of algebraic equations, identical to the set finite element formulation of Poisson s or Laplace s equation. The mass balance (volume balance for incompressible fluids) is given by... 

Notice that this is an integral representation of Laplace s equation for temperature. We will need to specify the extra function so it can become a complete representation. It is important to point out here that we have not made any approximation when deriving this formulation, making it an exact solution of the differential equation, V2T = 0. 

The integral formulation for Poisson s equation is found the same way as for Laplace s equation (using Green s second identity, Theorem (10.1.3)), except that now the second volume integral is kept in Green s second identity. For a point xq V the integral formulation... 

Equation (9.17) is solved by a Laplace transformation. In chemical kinetics and diffusion, the problems may often be formulated in terms of partial differential equations... 

The necessity to solve Laplace s equation requires formulating all boundary conditions, and at this point the cell geometry becomes important. Generally, there are two types of boundary conditions that come into play. Any electrically insulating cell wall is mathematically described by zero-flux or von Neumann boundary conditions ... 

This example concerns the typical two-compartment model previously presented under the semi-Markov formulation (cf. Section 9.2.7). By assuming that molecules are initially present in the central compartment, (9.16) is the Laplace transform of the survival function in that compartment. If now the drug molecules are administered by a constant rate infusion between Tg and TE, the Laplace transform of the survival function in the central compartment becomes... 

This transform is widely used to formulate semi-Markov stochastic models, where t and / (t) are the random variable and its probability density function, respectively. In Table F.l, we briefly report some Laplace transform pairs. 

If the obtained formulation E (t) does not have any analytical solution, we can carry out its Laplace transformation. In this case, the images E (s) can be written with the following recurrence relations ... 

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