Boundary conditions time dependent

Even when the boundary conditions involve time-dependent functions, the method described in the previous section can be used to good effect. Let us demonstrate this by solving a mass transfer problem in a particle when the bulk concentration varies with time. 

The boundary condition (11.123b) represents the situation where the particle is exposed to a bulk concentration, which decays exponentially with time. 

Equation 11.130 is a first order ordinary differential equation with exponential 

The mean concentration in the particle (of practical interest) is then given by 

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In fact, since the unsteady-state transport equation for forced convection is linear, it is possible in principle to derive solutions for time-dependent boundary conditions, starting from the available step response solutions, by applying the superposition (Duhamel) theorem. If the applied current density varies with time as i(t), then the local surface concentration at any time c0(x, t) is given by... 

The general case of non-uniform initial conditions and time-dependent boundary conditions can also be expressed analytically, and leads to convolution integrals in space and time. Nevertheless, for most practical applications, the assumptions of uniform initial conditions and constant boundary conditions are adequate. 

Solving the diffusion equation in environmental transport can be challenging because only specihc boundary conditions result in an analytical solution. We may want to consider our system of interest as a reactor, with clearly defined mixing, which is more amenable to time dependent boundary conditions. The ability to do this depends on how well the conditions of the system match the assumptions of reactor mixing. In addition, the system is typically assumed as one dimensional. The common reactor mixing assumptions are as follows ... 

A third method—solution by Laplace transforms—can be used to derive many of the results already mentioned. It is a powerful method, particularly for complicated problems or those with time-dependent boundary conditions. The difficult part of using the Laplace transform is back-transforming to the desired solution, which usually involves integration on the complex domain. Fortunately, Laplace transform tables and tables of integrals can be used for many problems (Table 5.3). 

The Laplace transform method is a powerful technique for solving a variety of partial-differential equations, particularly time-dependent boundary condition problems and problems on the semi-infinite domain. After a Laplace transform is performed on the original boundary-value problem, the transformed equation is often easily solved. The transformed solution is then back-transformed to obtain the desired solution. 

Example with Time-Dependent Boundary Conditions. Consider the case where a constant flux, J0, is imposed on the surface of a semi-infinite sample ... 

Equation (4) is thus a time-dependent boundary condition to Eqs. (6, 7), which, supplemented by the remaining boundary conditions (which also involve external constraints resulting from the operation mode of the experiment, s.b.) and possibly by the incorporation of convection, form the most basic Ansatz for modeling patterns of the reaction-transport type in electrochemical systems. However, so far, there are no studies on electrochemical pattern formation that are based on this generally applicable set of equations. Rather, one assumption was made throughout that proved to capture the essential features of pattern formation in electrochemistry and greatly simplifies the problem it is assumed that the potential distribution in the electrolyte can be calculated by Laplace s equation, i.e. Poisson s equation (6) becomes ... 

The interpretation of Eq. [133] for planar Couette flow is as follows at f < 0, the system evolves under normal NVE dynamics (i.e., Vu = 0). At f = 0, the impulsive force 5(f)q, Vu is applied, after which the system continues to evolve under NVE dynamics for f > 0, but the memory of the flow is contained in the definition of q, and p, as specified in Eq. [129]. When periodic boundary conditions are applied on the simulation cell, they must be treated in a way that preserves the flow. This in general will lead to time-dependent boundary conditions for the case of planar Couette flow to be discussed in detail shortly. 

Heat conduction with a constant boundary condition at x =0 was considered in example 4.1. The same technique can be applied for time dependent boundary conditions. Consider the transient heat conduction problem in a slab.[4] The governing equation is ... 

Consider the following heat/mass transfer problem with a time dependent boundary condition,... 

Example 8.9. Heat Conduction with Time Dependent Boundary Conditions... 

Consider that one of the main advantages of the Laplace transform technique is that it can be used for time dependent boundary conditions, also. The separation of variables technique cannot be directly used and one has to use DuhameFs superposition theorem[l] for this purpose. Consider the modification of example 8.7 ... 

In example 8.9 the Laplace transform technique was used to solve a time dependent problem. Inversing the Laplace transform is not straightforward. For complicated time dependent boundary conditions the convolution theorem can be used to find the inverse Laplace transform efficiently. If H(s) is the solution obtained in the Laplace domain, H(s) is represented as a product of two functions ... 

Example 8.15. Heat Conduction in a Rectangle with a Time Dependent Boundary Condition... 

Often inversion to time domain solution is not trivial and the time domain involves an infinite series. In section 8.1.4 short time solution for parabolic partial differential equations was obtained by converting the solution obtained in the Laplace domain to an infinite series, in which each term can easily inverted to time domain. This short time solution is very useful for predicting the behavior at short time and medium times. For long times, a long term solution was obtained in section 8.1.5 using Heaviside expansion theorem. This solution is analogous to the separation of variables solution obtained in chapter 7. In section 8.1.6, the Heaviside expansion theorem was used for parabolic partial differential equations in which the solution obtained has multiple roots. In section 8.1.7, the Laplace transform technique was extended to parabolic partial differential equations in cylindrical coordinates. In section 8.1.8, the convolution theorem was used to solve the linear parabolic partial differential equations with complicated time dependent boundary conditions. For time dependent boundary conditions the Laplace transform technique was shown to be advantageous compared to the separation of variables technique. A total of fifteen examples were presented in this chapter. 

In the case of a diffusion-controlled reaction a current-potential curve can be evaluated quantitatively. The diffusion equation has to be solved again by using time-dependent boundary conditions. The mathematics, however, are very complicated and cannot be shown here. They end up with an integral equation which has to be solved numerically [11]. The peak current, /p, for a diffusion-controlled process (reversible reaction) is found to be... 

It has been shown that, for property variations for which superposition of solutions is permitted, a series of solutions corresponding to a step in surface temperature can be utilized to represent an arbitrary surface temperature [22]. This approach is identical with the Duhamel method used in heat conduction problems to satisfy time-dependent boundary conditions... 

Linear elastic fracture mechanics is no longer valid for dynamic fracture mechaiusms since continuous crack propagation addresses time dependent boundary conditions. The energy balance for the dynamic conditions can be written as... 

If the quantity of sorbate adsorbed or desorbed is not negligible compared with the quantity introduced or removed from the ambient fluid phase, the sorbate concentration in the fluid will not remain constant after the initial step, giving rise to a time-dependent boundary condition at the surface of the adsorbent particle. The solution for the uptake curve then becomes ... 

This method is suitable for nonhomogeneous problems when the nonhomogeneity occurs as a time-dependent source term or as time-dependent boundary conditions. 

The second stage is to find a solution to the set of equations. Three basic approaches are possible (a) a complete analytical solution, (b) a partial analytical solution which is completed by a numerical method such as numerical integration, (c) a computer solution either based on a simulation of the experiment or a numerical solution of the set of equations. The first approach is always to be preferred since it leads to an exact equation relating experimental measurables to kinetic parameters. Its range of applicability is, however, limited to relatively simple experiments, and as the experiment becomes more complex it is necessary to deal with time dependent boundary conditions, coupled partial differential equations, and perhaps non-linear equations. Then the computer techniques must be employed, and these generally lead to dimensionless plots. 

Multiple replica simulations can be extended to driven systems (e.g., S5 tems with time-dependent boundary conditions, such as Lees-Edwards or Lagrangian-rhomboid boundary conditions). Each of P processors simulates a replica at a driving rate that is P times faster than the desired... 

To illustrate the procedure, the long-term evolution of the shoreline will be addressed with the aid of the solution of the one-line model with time-dependent boundary conditions proposed by Payo et alf A brief summary of the boundary value problem is presented next. 

It was observed that in some circumstances exposure to ambient fluid is initiated by the formation of a thin molecular layer that adheres to the exposed boundary of the polymer and that the diffusion process involves some time delay before proceeding at full capacity (e.g.. Long and Richman 1960). This observation motivated Long and Richman (1960) to employ a time-dependent boundary condition even under exposure to constant environment, namely... 

It is not possible to solve these equations with unknown time-dependent boundary conditions. However, we can use the equations to identify the important variables that will affect pressure drop, heat, or mass transfer. The velocity, pressure, temperature, and concentration described with these models should be completely described by the dimensionless variables Re, Fr, Pen, P d, and Eu. Keeping all these dimensionless numbers constant will also give the same solution expressed in the dimensionless form. Equations (4.12) and (4.13) contain Re, Fr, and Eu, and we can find correlations for pressure drop by empirically determining the function... 

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