Boundaiy conditions

Eigenvalue problems. These are extensions of equilibrium problems in which critical values of certain parameters are to be determined in addition to the corresponding steady-state configurations. The determination of eigenvalues may also arise in propagation problems. Typical chemical engineering problems include those in heat transfer and resonance in which certain boundaiy conditions are prescribed. 

The solution of problems involving partial differential equations often revolves about an attempt to reduce the partial differential equation to one or more ordinary differential equations. The solutions of the ordinary differential equations are then combined (if possible) so that the boundaiy conditions as well as the original partial differential equation are simultaneously satisfied. Three of these techniques are illustrated. 

Diffusion problems in one dimension lead to boundaiy value problems. The boundaiy conditions are applied at two different spatial locations at one side the concentration may be fixed and at the other side the flux may be fixed. Because the conditions are specified at two different locations, the problems are not initial value in character. It is not possible to begin at one position and integrate directly because at least one of the conditions is specified somewhere else and there are not enough conditions to begin the calculation. Thus, methods have been developed especially for boundary value problems. 

Shooting Methods The first method is one that utihzes the techniques for initial value problems but allows for an iterative calculation to satisfy all the boundaiy conditions. Consider the nonlinear boundaiy value problem... 

Convert this second-order equation into two first-order equations along with the boundaiy conditions written to include a parameter. s to represent the unknown value of i (0) = dy/dx 0). 

The parameter s is chosen so that the last boundaiy condition is satisfied y l) = p. Define the function... 

One nice feature of the finite element method is the use of natural boundaiy conditions. It may be possible to solve the problem on a domain that is shorter than needed to reach some limiting condition (such as at an outflow boundaiy). The externally applied flux is still apphed at the shorter domain, and the solution inside the truncated domain is still vahd. Examples are given in Refs. 67 and 107. The effect of this is to allow solutions in domains that are smaller, thus saving computation time and permitting the solution in semi-infinite domains. 

Several Bodies in Series with Heat Generation The simple Fourier type of equation indicated by Eq. (5-15) may not be used when heat generation occurs in one of the bodies in the series. In this case, Eq. (5-5c), (5-5Z ), or (5-5c) must be solved with appropriate boundaiy conditions. 

When temperatures of materials are a function of both time and space variables, more complicated equations result. Equation (5-2) is the three-dimensional unsteady-state conduction equation. It involves the rate of change of temperature with respect to time 3t/30. Solutions to most practical problems must be obtained through the use of digital computers. Numerous articles have been published on a wide variety of transient conduction problems involving various geometrical shapes and boundaiy conditions. 

Laminar Flow Normally, laminar flow occurs in closed ducts when Nrc < 2100 (based on equivalent diameter = 4 X free area -i-perimeter). Laminar-flow heat transfer has been subjected to extensive theoretical study. The energy equation has been solved for a variety of boundaiy conditions and geometrical configurations. However, true laminar-flow heat transfer veiy rarely occurs. Natural-convecdion effects are almost always present, so that the assumption that molecular conduction alone occurs is not vahd. Therefore, empirically derived equations are most rehable. 

Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure bound-aiy condition and two velocity boundaiy conditions (for each velocity component) to completely specify the solution. The no sBp condition, whicn requires that the fluid velocity equal the velocity or any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-... 

Stokes equations, Dirichlet and Neumann, or essential and natural, boundaiy conditions may be satisfied by different means. 

Breakthrough Behavior for Axial Dispersion Breakthrough behavior for adsorption with axial dispersion in a deep bed is not adequately described by the constant pattern profile for this mechanism. Equation (16-128), the partial different equation of the second order Ficldan model, requires two boundaiy conditions for its solution. The constant pattern pertains to a bed of infinite depth—in obtaining the solution we apply the downstream boundaiy condition cf 0 as oo. Breakthrough behavior presumes the existence of... 

The full mathematical model for this problem is Eq. (16-128) with boundaiy conditions... 

The solution to this equation, with initial condition /if= 0 at Ti = 0 and boundaiy condition cf= 1 at = 0, originally obtained for an analogous heat transfer case [Anzelius, Z. Angew Math. Mech., 6, 291 (1926) Schumann, y. Franklin Jn.st., 208,405 (1929)], is... 

There are several possible alternative relationships for E (Lemhch, op. cit.). For simphcity, consider E = K C, where K is not necessarily the same as the equihbrium constant K. Substituting and integrating from the boundaiy condition of C = Cg at / = 0 yield... 

The model that best describes the chemical engineering fundamentals including transport phenomena, rate mechanisms, and the thermodynamics and includes contributions due to equipment nonliuearities and boundaiy conditions should be the model of choice. 

The latter equation is solved for the following initial and boundaiy conditions ... 

Applying Laplace transform to (2.192) and the boundaiy conditions (2.193) to (2.196) one obtains the following solution ... 

Applying the Laplace transform to the boundaiy condition (A.5), and combining the result with (A. 13), gives... 

Applying Laplace transformation to the boundaiy condition (A.4), one finds ... 

Fig. 2.1 Position as a (unction of time for e free particle with no force acting. TVwj boundaiy conditions, e.g. the Initial position and velocity, must be known to specify the trajectory fuly.

In this case, no interfacial conditions need to be specified. Only external boundaiy conditions in terms of operating conditions (gas pressure, gas temperature, chemical composition) and heat transfer from the PEN to the surrounding (adiabatic condition, isothermal condition or a specific heat flux) are required. 

Equations (5) and (8) represent a system of coupled nonlinear equations, which can be solved for

various contributions to the free energy are also calculated in Appendix A. 

Sparrow and Gregg, 1958 [7], obtained a similarity solution for a vertical plate with uniform heat flux boundaiy condition. The range of Pr investigated was from 0.1 to 100. 

In incompressible flow with constant properties and no body forces, the dynamics are independent of the thermodynamics. Once the kinematic flow field is described by the stream function vj/, any number of temperature distributions may be solved with different thermal boundaiy conditions. 

Let us assume that the initial temperature condition in the porous medium is T(x,y, t = 0) = 0. The boundaiy conditions shown are as follows ... 

There is no constant of integration due to the boundaiy condition that both A GIT and A(l/7) are zero at equilibrium. However, AH will be temperature dependent most of the time. For example, in producing ammonia from hydrogen... 

In the present work, a simulation strategy is formulated to study the performance of cathode materials in lithium ion batteries. Here micro scale properties, for example, diffusion of spherical electrode particle within the periodic boundaiy condition, 0electrode particles move in each step to its nearest neighbor distance, employing the condition ir j) > e -dLi lds ), where ir represents the random number, dLil is the nearest neighbor distance for the Li ion in the absence of solvent and ds being the thickness of the sohd phase. The MC codes involve macro scale properties, namely, solvation effects, diffusion coefficients and the concentration gradient... 

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