Simulations of steady state

Using different types of time-stepping techniques Zienkiewicz and Wu (1991) showed that equation set (3.5) generates naturally stable schemes for incompressible flows. This resolves the problem of mixed interpolation in the U-V-P formulations and schemes that utilise equal order shape functions for pressure and velocity components can be developed. Steady-state solutions are also obtainable from this scheme using iteration cycles. This may, however, increase computational cost of the solutions in comparison to direct simulation of steady-state problems. 

Mathematically speaking, a process simulation model consists of a set of variables (stream flows, stream conditions and compositions, conditions of process equipment, etc) that can be equalities and inequalities. Simulation of steady-state processes assume that the values of all the variables are independent of time a mathematical model results in a set of algebraic equations. If, on the other hand, many of the variables were to be time dependent (m the case of simulation of batch processes, shutdowns and startups of plants, dynamic response to disturbances in a plant, etc), then the mathematical model would consist of a set of differential equations or a mixed set of differential and algebraic equations. 

Shimao, K, Mathematical Simulation of Steady State Isoelectric Eocusing of Proteins using Carrier Ampholytes, Electrophoresis 8, 14, 1987. 

The differential equations are often highly non-linear and the equation variables are often highly interrelated. In the above formulation, yj represents any one of the dependent system variables and, fi is the general function relationship, relating the derivative, dyi/dt, with the other related dependent variables. Tbe system independent variable, t, will usually correspond to time, but may also represent distance, for example, in the simulation of steady-state models of tubular and column devices. 

Shimao, K. (1987). Mathematical simulation of steady state isoelectric focusing of proteins using carrier ampholytes. Electrophoresis 8, 14-19. 

Rohm, H. J., The Simulation of Steady State Behavior of the Dephlegmation of Multicomponent Mixed Vapors, Int. J. Heat Mass Transfer, 23, 141-146 (1980). 

Knowing that the enzyme stability is sufficiently high at pH 3.75 (50% deactivation after 150 h), water was chosen as the reaction medium and the reaction was performed at room temperature. This selection of reaction conditions was followed by a detailed kinetic analysis of the system, the investigation of the reactor kinetics and the simulation of steady state conditions in continuous experiments155, 60]. 

Vanderborght, J., Jacques, D., Mallants, D., Tseng, P.-H., and Feyen, J. Comparison between field measurements and numerical simulation of steady-state solute transport in a heterogeneous soil profile. Hydrology and Earth System Sciences 4, 853-871. 1997. 

We present here classical molecular dynamics simulations of steady-state heat transfer through ID periodic chains coupled to non-Markovian reservoirs in order to elucidate the effect of the contacts on the junction thermal conductance. 

Using Equation (23.27)-Equation (23.33), the mathematical simulation of steady-state and dynamic experiments was carried out. The results of calculation are represented in Figures 23.9, 23.10, 23.14, and 23.15 (lines — calculation, points — experiment). It is easily seen that the model demonstrates a good agreement with experiments in both steady-state and dynamic regimes. This confirms the interpretation of the experiments as the influence of capillary condensation on kinetics and dynamics of catalytic reaction. 

Yang SC (2000) Modeling and simulation of steady-state polarization and impedance response of phosphoric acid fuel-cell cathodes with catalyst-layer microstructure consideration. J Electrochem Soc 147 71-77... 

Alden JA, Compton RG (1997) A general method for electrochemical simulations. 2. Application to the simulation of steady-state currents at microdisk electrodes homogeneous and heterogeneous kinetics. J Phys Chem B 101 9606-9616... 

Alden JA, Hakoura S, Compton RG (1999) Finite difference simulations of steady-state voltammetry at the wall-jet electrode. Effects of radial diffusion and working curves for common electrochemical mechanisms. Anal Chem 71 827-836... 

Bisquert J, Mora-Sero I (2010) Simulation of steady-state characteristics of dye-sensitized solar cells and the interpretation of the diffusion fcaigth. J Phys Chem Lett 1 450-456... 

Hanebuth, M., Dittmeyer, R., Mabande, G.T.P. and Schwieger, W. (2005) On the combination of different transport mechanisms for the simulation of steady-state mass transfer through composite systems using permeation through stainless steel supported silicalite-1 membranes as a model system. Catalysis Today, 104, 352-359. 

Nann and Heinze [39] developed and used an adaptive finite element (AFE) algorithm to carry out simulations of steady-state, chronoamperometric, and voltammetric SECM responses for various nondisk tip geometries, for example, hemispherical, conical, and capillary electrodes. 

FIGURE 5.18 Simulation of steady-state diffusion-limited current at an embedded microdisk electrode by COMSOL 4.0a. (A) The mesh generated by the program. (B) The concentration profile of redox species. (C) The flux distribution over the disk surface. 

---