Problem, closure

Baldyga, J. and Orciuch, W., 1997. Closure problem for precipitation. Transactions of the Institution of Chemical Engineers, 75A, 160-170. 

Generally, the closure problem reflects the idea of a spatially periodic porous media, whereby the entire structure can be described by small portions (averaging volumes) with well-defined geometry. Two limitations of the method are therefore related to how well the overall media can be represented by spatially periodic subunits and the degree of difficulty in solving the closure problem. Not all media can be described as spatially periodic [6,341 ]. In addition, the solution of the closure problem in a complex domain may not be any easier than solving the original set of partial differential equations for the entire system. 

Another important factor in diffusion measurements that is often encountered in NMR experiments is the effect of time on diffusion coefficients. For example, Kinsey et al. [195] found water diffusion coefficients in muscles to be time dependent. The effects of diffusion time can be described by transient closure problems within the framework of the volume averaging method [195,285]. Other methods also account for time effects [204,247,341]. 

Analytical solutions for the closure problem in particular unit cells made of two concentric circles have been developed by Chang [68,69] and extended by Hadden et al. [145], In order to use the solution of the potential equation in the determination of the effective transport parameters for the species continuity equation, the deviations of the potential in the unit cell, defined by... 

Determination of the effective transport coefficients, i.e., dispersion coefficient and electrophoretic mobility, as functions of the geometry of the unit cell requires an analogous averaging of the species continuity equation. Locke [215] showed that for this case the closure problem is given by the following local problems ... 

It is important to note that the closure problem for the species continuity equation requires solutions for the deviations of the potential, i.e the /and g fields, anfi knowledge of the average potential ( ). This result is very similar to that found by the area averaging method in Sauer et al. [345], Utilizing the closure expressions the average species continuity equation becomes... 

As it stands, the last term on the right-hand side of this expression is non-linear in the spatial derivatives and appears to add a new closure problem. However, using the fact that the fluctuation field is solenoidal,... 

Conditional moments of this type cannot be evaluated using the one-point PDF of the mixture fraction alone (O Brien and Jiang 1991). In order to understand better the underlying closure problem, it is sometimes helpful to introduce a new random field, i.e.,15... 

Thus, the closure problem reduces to finding an appropriate expression for the scalar flux (Ujtp). In high-Reynolds-number turbulent flows, the molecular transport term is again negligible. Thus, the scalar-flux term is responsible for the rapid mixing observed in turbulent flows. 

For turbulent reacting flows, LES introduces an additional closure problem due to filtering of the chemical source term (Cook and Riley 1994 Cook et al. 1997 Jimenez el al. 1997 Cook and Riley 1998 Desjardin and Frankel 1998 Wall et al. 2000). For the one-step... 

The chemical-source-term closure problem occurs even for relatively simple isothermal reactions. For example, consider again the simple two-step reaction (5.21) where26... 

The closure problem thus reduces to finding general methods for modeling higher-order moments of the composition PDF that are valid over a wide range of chemical time scales. 

For simplicity, we will use the fictitious chemical species as the subscripts whenever there is no risk of confusion. As shown above, this reaction can be rewritten in terms of two reacting scalars and thus two components for S. The closure problem, however, cannot be eliminated by any linear transformation of the scalar variables. 

Alternatively, a joint velocity, scalar, scalar-gradient PDF could be employed. However, this only moves the closure problem to a higher multi-point level. 

In order to close (Jwe can recognize that because J(0) depends only on the 0, it is possible to replace e by (e The closure problem then reduces to finding an expression for the doubly conditioned joint scalar dissipation rate matrix. For example, if the FP model is used to describe scalar mixing, then a model of the form... 

Baldyga, J. and M. Henczka (1995). Closure problem for parallel chemical reactions. The... 

We face now the standard closure problem To keep going, one should relate in some way or another the current j to the number density. The usual relationship is Pick s law ... 

These loadings are believed to be realistic although a potential closure problem exists (see text). 

On a lattice, so-called crankshaft moves are trivial implementations of concerted rotations [77]. They have been generalized to the off-lattice case [78] for a simplified protein model. For concerted rotation algorithms that allow conformational changes in the entire stretch, a discrete space of solutions arises when the number of constraints is exactly matched to the available degrees of freedom. The much-cited work by Go and Scheraga [79] formulates the loop-closure problem as a set of algebraic equations for six unknowns reducible... 

Kolodny, R., Guibas, L., Levitt, M., Koehl, P. Inverse kinematics in biology the protein loop closure problem. Int. J. Robot. Res. 2005, 24, 151-63. 

Although direct numerical simulations under limited circumstances have been carried out to determine (unaveraged) fluctuating velocity fields, in general the solution of the equations of motion for turbulent flow is based on the time-averaged equations. This requires semi-empirical models to express the Reynolds stresses in terms of time-averaged velocities. This is the closure problem of turbulence. In all but the simplest geometries, numerical methods are required. 

Begin by noting that C() satisfy the Neumann problem given by (5) and boundary conditions whose solution is Cj(x, y, t) = Cj(x, t). We now derive the overall macroscopic mass balance for the species. To this end we begin by deriving the closure problem for Cj. Given Cf(x,t), combine (6) with boundary conditions and neglect the advection induced by du/dt, to obtain the local Neumann problem... 

So far, we are able to construct the constitutive equations for qc, Pc, and y. For moderate solids concentrations, we can neglect the kinetic contributions in comparison to the collisional ones. Thus, we can assume P Pc and qk qc. Substituting the constitutive relations into Eqs. (5.274), (5.275), and (5.281), after neglecting the kinetic contributions, yields five equations for the five unknowns ap, Up, and Tc (or ( 2)). Hence, the closure problem is resolved. 

---