Multi-Dimensional Model

In this diagram, z-coordinate represents the space variation across the cell from the anode side to the cathode side along the thickness of the cell 

---

In contrast to the pseudo 3-D models, tmly multi-dimensional models use, in general, finite element or finite volume CFD (Computational Fluid Dynamics) techniques to solve full 3-D Navier-Stokes equations with appropriate modifications to account for electrochemistry and current distribution. The details of electrochemistry may vary from code to code, but the current density is calculated almost exclusively from Laplace equation for the electric potential (see Equation (5.24)). Inside the electrolyte, the same equation represents the migration of ions (e g. 0= in SOFC), elsewhere it represents the electron/charge transfer. In what follows, we briefly summarize a commonly used multi-dimensional model for PEM fuel cells because of its completeness and of the fact that it also addresses most essential features of SOFC modeling. 

The same approach as used above in the pseudo 2-D approach for stack modeling can also be used in the case of multi-dimensional modeling. The next set of results... 

But there are problems with multi-dimensional models. 

From a purely pragmatic standpoint, even two-dimensional models require too much computer time to be really practicable for optimization purposes. In addition, multi-dimensional models require values for parameters that are difficult or impossible either to estimate with sufficient accuracy, or to measure. 

Gosman, A.D., Lekakou, C., Politis, S., Issa, R.I. and Looney, M.K. (1992), Multi-dimensional modeling of turbulent two-phase flows in stirred vessels, AIChE J., 38, 1946-1956. 

The exploration of the impact on the p-process (and on many other questions) of the multi-dimensional modeling of the pre-SNII and core-collapse SNII or SNIb/c phases. 

The recent application of sparse-matrix techniques combined with computer optimization (vectorization) techniques has, however, improved the speed of Gear s code substantially, so that this advanced algorithm can now be used to study complex problems in multi-dimensional models (see e.g., the SMVGEAR package developed by Jacobson (1995 1998) and Jacobson and Turco, 1994). 

Our approach is to utilize multi-dimensional modeling (KIVA-3V) to design the scavenging system and then to validate the computational predictions for selected operating conditions with the use of a single-cycle scavenging experiment. 

MM Tang, AI Liapis, and JM Marchello. A multi-dimensional model describing the lyophilization of a pharmaceutical product in a vial. Proceedings of the Fifth International Drying Symposium, Cambridge, MA, Vol. 1 (AS Mujumdar, ed.), New York Hemisphere Publishing, 1986, pp. 57-65. 

Nonlinear optimization is one of the crucial topics in the numerical treatment of chemical engineering problems. Numerical optimization deals with the problems of solving systems of nonlinear equations or minimizing nonlinear functionals (with respect to side conditions). In this article we present a new method for unconstrained minimization which is suitable as well in large scale as in bad conditioned problems. The method is based on a true multi-dimensional modeling of the objective function in each iteration step. The scheme allows the incorporation of more given or known information into the search than in common line search methods. 

Liapis, A. L, Bruttini, R., 1995. Freeze-drying of pharmaceutical crystalline and amorphous solutes in vials dynamic multi-dimensional models of the primary and secondary drying stages and qualitative features of the moving interface. Drying Technol. 13 43-72. 

N is too small and a more realistic model of the reactor flow pattern should be sought. Similarly one can calculate p = uL/2D x and use the segregated flow model to calculate by integration the reactor performance while approximating E (0) by the gamma density function. When the dispersion intensity is small this should work. When the intensity is large even the dispersion model itself may not represent physical reality well and a multi-dimensional model is needed. 

---