Two point boundary value problem

This treatment of reaction at the limit of bulk diffusion control is essentially the same as that presented by HugoC 69j. It is attractive computationally, since only a single two-point boundary value problem must be solved, namely that posed by equations (11.15) and conditions (11.16). This must be re-solved each time the size of the pellet is changed, since the pellet radius a appears in the boundary conditions. However, the initial value problem for equations (11.12) need be solved only once as a preliminary to solving (11.15) and (11.16) for any number of different pellet sizes. 

Thus, we solve a two-point boundary value problem instead of a partial differential equation. When the diffiisivity is constant, the solution is the error function, a tabulated function. 

Rigorous error bounds are discussed for linear ordinary differential equations solved with the finite difference method by Isaacson and Keller (Ref. 107). Computer software exists to solve two-point boundary value problems. The IMSL routine DVCPR uses the finite difference method with a variable step size (Ref. 247). Finlayson (Ref. 106) gives FDRXN for reaction problems. 

Elliptic Equations Elhptic equations can be solved with both finite difference and finite element methods. One-dimensional elhptic problems are two-point boundary value problems. Two- and three-dimensional elliptic problems are often solved with iterative methods when the finite difference method is used and direct methods when the finite element method is used. So there are two aspects to consider howthe equations are discretized to form sets of algebraic equations and howthe algebraic equations are then solved. 

They convert the initial value problem into a two-point boundary value problem in the axial direction. Applying the method of lines gives a set of ODEs that can be solved using the reverse shooting method developed in Section 9.5. See also Appendix 8.3. However, axial dispersion is usually negligible compared with radial dispersion in packed-bed reactors. Perhaps more to the point, uncertainties in the value for will usually overwhelm any possible contribution of D. ... 

NUMERICAL SOLUTIONS TO TWO-POINT BOUNDARY VALUE PROBLEMS... 

The forward shooting method seems straightforward but is troublesome to use. What we have done is to convert a two-point boundary value problem into an easier-to-solve initial value problem. Unfortunately, the conversion gives a numerical computation that is ill-conditioned. Extreme precision is needed at the inlet of the tube to get reasonable accuracy at the outlet. The phenomenon is akin to problems that arise in the numerical inversion of matrices and Laplace transforms. 

The gas motion near a disk spinning in an unconfined space in the absence of buoyancy, can be described in terms of a similar solution. Of course, the disk in a real reactor is confined, and since the disk is heated buoyancy can play a large role. However, it is possible to operate the reactor in ways that minimize the effects of buoyancy and confinement. In these regimes the species and temperature gradients normal to the surface are the same everywhere on the disk. From a physical point of view, this property leads to uniform deposition - an important objective in CVD reactors. From a mathematical point of view, this property leads to the similarity transformation that reduces a complex three-dimensional swirling flow to a relatively simple two-point boundary value problem. Once in boundary-value problem form, the computational models can readily incorporate complex chemical kinetics and molecular transport models. 

If we consider the limiting case where p=0 and q O, i.e., the case where there are no unknown parameters and only some of the initial states are to be estimated, the previously outlined procedure represents a quadratically convergent method for the solution of two-point boundary value problems. Obviously in this case, we need to compute only the sensitivity matrix P(t). It can be shown that under these conditions the Gauss-Newton method is a typical quadratically convergent "shooting method." As such it can be used to solve optimal control problems using the Boundary Condition Iteration approach (Kalogerakis, 1983). 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundary or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial value problem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinary differential equations become two-point boundary value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

Absorption columns can be modeled in a plate-to-plate fashion (even if it is a packed bed) or as a packed bed. The former model is a set of nonlinear algebraic equations, and the latter model is an ordinary differential equation. Since streams enter at both ends, the differential equation is a two-point boundary value problem, and numerical methods are used (see Numerical Solution of Ordinary Differential Equations as Initial-Value Problems ). 

For this two point boundary value problem, the following procedure will be used ... 

This is a two-point boundary value problem to which the "shooting method" is applicable, according to this procedure ... 

In summary, formulation of (27) with appropriate placement of finite elements works well for parameter optimization problems. In the next subsection, however, we consider additional difficulties when control profiles are introduced. Stated briefly, the reason for these difficulties lies in the nature of the discretized variational conditions of (16). As shown in Logsdon and Biegler (1989), optimality conditions for parameter optimization problems take the form of two point boundary value problems. For optimal control... 

The YBG equation is a two point boundary value problem requiring the equilibrium liquid and vapor densities which in the canonical ensemble are uniquely defined by the number of atoms, N, volume, V, and temperature, T. If we accept the applicability of macroscopic thermodynamics to droplets of molecular dimensions, then these densities are dependent upon the interfacial contribution to the free energy, through the condition of mechanical stability, and consequently, the droplet size dependence of the surface tension must be obtained. 

The first situation involves two algebraic equations, the second involves an algebraic equation (the mixed phase) and a first-order ordinary differential equation (the unmixed phase), and the third situation involves two coupled differential equations. Countercurrent flow is in fact more compHcated than cocurrent flow because it involves a two-point boundary-value problem, which we will not consider here. 

A simple example of molecular transport for a three-species problem is illustrated here. Consider evaporation of a liquid into a high-aspect-ratio tube open to air. The diffusive transport of species entering the vapor through evaporation can be solved as a onedimensional two-point boundary-value problem. 

Thus the equations that we must solve are 12.196 and 12.197, which comprise a set of two coupled first-order differential equations, subject to the boundary conditions, Xj = 0.01395, and X2 = 0.00712 at z = 0 and Xj = X2 = 0 at z = Z, with the unknown fluxes Ni, N2 that must be found. This equation set could easily be solved as a two-point boundary-value problem using the spreadsheet-based iteration scheme discussed in Appendix D. However, for illustration purposes we choose to solve the equation set with a shooting method, mentioned in Section 6.3.4. We can solve the problem as an ordinary differential equation (ODE) initial-value problem, and iteratively vary Ni,N2 until the computed mole fractions X, X2 are both zero at z = Z. 

Alternatively, you may solve the problem as a two-point boundary-value problem using the TWOPNT software [158],... 

Solve this two-point boundary-value problem by any of the methods mentioned in Section 12.8 or Problem 12.16. Calculate the mixture-averaged diffusion coefficients Dkm using Eq. 12.176. Report the molar fluxes Ni, N2 (mol/m2-s) for SO2 and H2O. 

Keller, H. B., 1976, Numerical Solution of Two-Point Boundary Value Problems. Regional Conf. Series in Appl. Math., SIAM. 

Two point boundary value problems of chemical/biological engineering typically arise from some feedback of information. They can result from any of the following sources ... 

Two point boundary value problems are much more difficult to solve and more demanding than initial value problems for differential equations. One of the strengths of MATLAB is that it has very good and efficient subroutines for solving both IVPs and BVPs. 

Example of an Axial Dispersion Model. Linear and Nonlinear Two-point Boundary Value Problems (BVPs)... 

Solution of the second-order two-point boundary value problem (5.24) for a second-order reaction... 

The axial dispersion model has led to the two-point boundary value problem (5.37) for uj from uj = ujstart = 0 to u> = uJend = 1- DEs are standardly solved by numerical integration over subintervals of the desired interval [uj start, we d. For more on the process of solving BVPs, see Section 1.2.4 or click on the Help line under the View icon on the MATLAB desktop, followed by a click on the Search tab in the Help window and searching for BVP . 

In this section we have presented the first example of two-point boundary value problems that occur in chemical/biological engineering. The axial dispersion model for tubular reactors is a generalization of the plug flow model for tubular reactors which removes some of the limiting assumptions of plug flow. Our model includes additional axial diffusion terms that are based on the simple physics laws of Fick for mass and of Fourier for heat dispersion. 

In this section we have presented and solved the BVPs associated with the diffusion and reaction that take place in the pores of a porous catalyst pellet. The results were expressed graphically in terms of the effectiveness factor rj versus the Thiele modulus d> for two cases One with negligible external mass and heat transfer resistances, i.e., when Sh and Nu —> oo, and another with finite Sh and Nu values. This problem is very important in the design of fixed-bed catalytic reactors. The sample results presented here have shown that for exothermal reactions multiple steady states may occur over a range of Thiele moduli d>. Efficient numerical techniques have been presented as MATLAB programs that solve singular two-point boundary value problems. 

Next we consider the heat balance for the cooling jacket in incremental form as depicted in Figure 7.2. Note first that ours is a case of cocurrent flow in the jacket and reactor. This simplifies the mathematical problem when compared with countercurrent cooling. Countercurrent cooling is physically more efficient, but it transforms the problem mathematically into a more demanding two point boundary value problem which we want to avoid here see problem 3 of the Exercises. 

The first one is based on a classical variation method. This approach is also known as an indirect method as it focuses on obtaining the solution of the necessary conditions rather than solving the optimization directly. Solution of these conditions often results in a two-point boundary value problem (TPBVP), which is accepted that it is difficult to solve [15], Although several numerical techniques have been developed to address the solution of TPBVP, e.g. control vector iteration (CVI) and single/multiple shooting method, these methods are generally based on an iterative integration of the state and adjoint equations and are usually inefficient [16], Another difficulty relies on the fact that it requires an analytical differentiation to derive the necessary conditions. 

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