Lumped modeling

Figure 10. Comparison of cup average conversion predicted by horizontally lumped model (natural convection included) and axisymmetric model (natural convection neglected) for a = 0, 5 and 10.

Figure 11. Conversion along the vertical tube centerplane as a function of dimensionless length as predicted by the horizontally lumped model for a =15 and 2 = 50.

Desulfurization of two LGO streams was studied using R. erythropolis rKA2-5-l [71] to determine the rate of desulfurization and develop a predictive model for desulfurization of diesel oil. The sulfur removal from various substituted DBTs was modeled using a competitive inhibition and lumping model and the desulfurization of the LGO streams was successfully assessed. The model consisted of a four component system based on... 

Examples for such descriptions were reported recently. Starting with the lumped-model equations explained in Sect. 3.4, a matrix formulation can be found that supports the system optimization [19]. 

Combining Eq. (4.9) with the lumped-model Eq. (3.24), one gets an expression for the temperature on the membrane depending on the source drain-current ... 

Using a lumped model for the reactor metal wall and the simple enthalpy equation h= Cj, T, the energy equations for the reaction liquid and the metal wall... 

Assume that this distributed system can be adequately modeled by a five-lump model of equal lengths. Inside each lump the gas temperature and the composition vary with time, as does the packing temperature. 

Smog-chamber studies are needed for validating both the detailed chemical models and the lumped models. Many of the past chamber studies have not used sufficiently well-defined initial conditions. Measurements of more products and of the reactive intermediates will provide more stringent tests for models. 

The notable feature of the wall-cooled tubular reactor is that there can exist a hot spot near the center of the reactor and near the entrance. We saw this for the lumped model, which allowed only for variations in the direction, but when radial variations are allowed, the effect can become even more severe as both temperature and concentration vary radially. 

Lumped models (usually called lumped parameter models , which is wrong terminology since the state variables are lumped, not the input variables or parameters) described by transcendental equations for the steady state and ODEs for the unsteady state. 

A distributed model is usually described by differential equations. Such a model differs from a lumped model that is generally described by transcendental equations. In chemical and biological engineering distributed systems often arise with tubular equipment. When a one-dimensional model is used for a distributed system there are two types of models ... 

The simplest case of a ROM would be lumped models or zero-dimensional models where the fuel cell is modeled as a single set of control volumes one for each component, e.g. air gas channel, fuel gas channel, PEN, interconnect, etc. (see for example Elizalde-Blancas et al., 2007a). This hides most of the details of what occurs inside the fuel cell but allows for fast simulation times. Lumped models are appropriate for use in system modeling applications where the fuel cell interacts with other devices such as heat exchangers, combustors, turbines, etc. This kind... 

Elizalde-Blancas, F., Pakalapati, S.R., Cayan, F. and Celik, I. (2007a) One-dimensional transient lumped model for fast fuel cell stack analysis, in proceedings of Renewable Energy Symposium, Frostburg, MD, USA, September 14-15. 

Ciano et al. (2006) have used a finite element approach to model a tubular cell 0.3 m long. The equations are available in Ciano et al. (2006). Table 7.2 shows the partial differential equations and the mesh characteristics. This model is computationally demanding and the equations have been solved by adopting an iterative procedure. Initial guess values for temperature and current density are assumed (current density is calculated by means of a lumped model, as the function of the average temperature and the cell voltage). Momentum equation and continuity equation are... 

Typically, a non-linear system dynamic model is made up of individual lumped models of the components which at a minimum conserve mass and energy across the given component, but may also have a momentum equation if pressure drops must also be analyzed. For most dynamic problems of interest in hybrid studies, however, the momentum equation may be taken as quasi-steady (unless the solver requires the dynamic form to perform the numerical solution). Higher fidelity individual models or reduced order models (ROMs) can also be used, where the connection to the system model would be made at each subcomponent boundary. Since dynamic systems modeling is not as common as steady-state modeling, some discussion of modeling approaches will be given. There are two primary methods used to provide solutions for the pressure-flow dynamics of a system model. 

We consider here Equations (9.10), (9.11) and (9.12) in the analysis of a coupled lumped SOFC. Because we now resolve the anode and cathode regions in this analysis (thereby temporally resolving the reactant concentrations, pressures and temperatures), the cell voltage versus, time can also be determined (using the Nemst equation). The anode gas phase lumped model is provided by integrating Equations (9.10) and (9.11) in the x-direction ... 

Cell Heat-up on Load Change — Coupled Lumped Model Analysis... 

The results for a 55 amp load change are shown in Figures 9.10 and 9.11. Figure 9.10 shows the temperature histories for the cell, interconnect, anode exit gas and cathode exit gas. As can be seen the thermal conditions reach their new equilibrium by approximately / = 800 s, resulting in an exponential time constant of r = 266 s. The temperature change for the gas stream is about 150°C. A calculation of the thermal time constant based on the fully lumped model (Equation (9.28) shows... 

Here we solve the same problem as in Section 9.5.1, but now using the One-Dimensional Model developed in the C++ computer language. The node numbers for the calculation domain are shown in Figure 9.12. The setup parameters used for this model are shown in Appendix A9.2 (the parameter names are descriptive so as to define their usage). Unlike the lumped model, the one-dimensional model used a controlled input cell voltage. The cell voltage was specified to be the same as that of the lumped model results in Section 9.5.1 at steady state (circa 0.8 V). The initial conditions assumed zero load, and at time / = 0. the load was applied. 

Fig. 9.17 Comparison of the lumped model with the one-dimensional model results ( A = average over all nodes B = average of inlet and exit nodes).

Fig. 9.22 Coupled lumped model with electrode capacitance.

When Bi zero dimensional or lumped model [2, 11], On the other hand, if Bi 1, the fluid can be considered isothermal and Ts = Too, which changes the convection boundary condition to a thermal equilibrium condition. 

All the works just cited have an important and common assumption the well-known, three-lump Weekman model with three kinetic constants. This three-lump model was a significant achievement, but it lumps together gas and coke, which are clearly different. Therefore, we consider it absolutely necessary to expand this model by using a five-lump model with eight cracking constants and the following definitions A = feedstock (bp > 350 °C), O = gas oil (bp 221-350 °C), E = gasoline (bp 36-221 °C), G = gas (bp < 36 ° C), and C = coke (by combustion). 

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