Model nodes, mathematical

Multi-node models divided the entire human body into more than two nodes and developed energy balance equation for each node as well the control functions for blood flow rate, shivering metabolic rate, and so on. Stolwijk et al. [20] presented a more complex multi-node mathematical thermal model of the entire human body, in which many efforts are made to flie statement of the thermal controller. This model firstly divided the body into six cylindrical parts of head, trunk, arm, hands, legs and feet and a spherical body part comprising the head. Each part is further... 

Table 4.4 is the summary of the mathematical model and the results obtained for the case study. The model for scenario 1 involves 637 constraints, 245 continuous and 42 binary variables. Seventy nodes were explored in the branch and bound algorithm. The model was solved in 1.61 CPU seconds, yielding an objective value (profit) of 1.61 million over the time horizon of interest, i.e. 6 h. This objective is concomitant with the production of 850 t of product and utilization of 210 t of freshwater. Ignoring any possibility for water reuse/recycle, whilst targeting the same product quantity would result in 390 t of freshwater utilization. Therefore, exploitation of water reuse/recycle opportunities results in more than 46% savings in freshwater utilization, in the absence of central reusable water storage. The water network to achieve the target is shown in Fig. 4.14. 

Subsequently, the condition of complete separation has to be coupled with the material balances derived for the nodes of the SMB unit and implemented in the Equilibrium Theory Model for Langmuir-type systems. That leads to the set of mathematical conditions given below, which the flow rate ratios have to fulfil in order to achieve complete separation, in particular ... 

Another method, less common, will be called here the Pressure Node approach. A company called, TRAX, uses this method in their dynamic modeling software package called ProTRAX . Their literature discusses this method s mathematical derivation. With this approach, the flow elements described previously are connected by either pressure elements (such as a volume) or by what are called, pressure... 

The analysis of polymer processing is reduced to the balance equations, mass or continuity, energy, momentum and species and to some constitutive equations such as viscosity models, thermal conductivity models, etc. Our main interest is to solve this coupled nonlinear system of equations as accurately as possible with the least amount of computational effort. In order to do this, we simplify the geometry, we apply boundary and initial conditions, we make some physical simplifications and finally we chose an appropriate constitutive equations for the problem. At the end, we will arrive at a mathematical formulation for the problem represented by a certain function, say / (x, T, p, u,...), valid for a domain V. Due to the fact that it is impossible to obtain an exact solution over the entire domain, we must introduce discretization, for example, a grid. The grid is just a domain partition, such as points for finite difference methods, or elements for finite elements. Independent of whether the domain is divided into elements or points, the solution of the problem is always reduced to a discreet solution of the problem variables at the points or nodal pointsinxxnodes. The choice of grid, i.e., type of element, number of points or nodes, directly affects the solution of the problem. 

In the steady state simulation and design, the state variables are the flowrates and the pressure drops or terminal pressures for each branch of the net. Each of the branches between two nodepoints are described by a mathematical model of the hydrodynamics relating the pressure drop to the fluid flow between the nodes. The material balance sums up the flow into and out of a node no is... 

A mathematical model for molecular complexity, such as the one developed by Bertz [21], has merit in helping to organize those properties which contribute to a heuristic idea of this subject. Using the language of graph theory [22] such a model will include a hierarchy of types of nodes (atoms) and of types of edges (bonds), as well as various other concepts such... 

The process model is represented by the heat transfer and the mixing process of the two heated parallel passes (figure 3). The input variables of the process are the thermal duties on the two parallel passes, Q-j-i, = 1, 2, and the associated volume flowrates, m, i =, 2. The mathematical model of the furnace is composed of the model associated to each coil and the model of the mixing node with temperature 7), ... 

Real Time Optimization. This module receives the values of the variables of the plant, performs reconciliation on these values. This node has a steady state (mathematical, physically based) model of the plant. An optimization is made using that model every hour or so. The optimization results are sent to the lower level, the supervisory control. These results are the new set points of the controlled variables. The best operating point of the plant (which means a set of set points values) is calculated in each optimization. The optimization takes into account constraints on the variables (limited change in manipulated variables, safety, quality, etc. constraints in controlled variables). The node uses as well a historian module with past data of the plant. 

So-called plant dispersion" or extra column effects" have to be taken into account by additional mathematical models rather than including them indirectly in the model parameters of the column, e.g. by altering the dispersion coefficient. The combination of peripheral and column models is easily implemented in a modular simulation approach. In a flowsheeting approach the boundary conditions of different models are connected by streams (node balances) and all material balances are solved simultaneously. 

Mathematically, the SMB model is achieved by connecting the boundary conditions of each column model, including nodes represented by material balances of splitting or mixing models. These so-called node models (Ruthven and Ching, 1989) are given for a component i in the sections I-IV by ... 

Several performance characteristics of rubber such as abrasion resistance, pendulum rebound, Mooney viscosity, modulus, Taber die swell, and rheological properties can be modeled by Eq 7.34. " A complex mathematical model, called links-nodes-blobs was also developed and experimentally tested to express the properties of a filled rubber network system. Blobs are the filler aggregates, nodes are crosslinks and links are interconnecting chains. The model not only allows for... 

The ANN is mathematical model that simulates many characteristics of actual neurons in the brain. Generally, an ANN is a structurally multi-layered network which links aTarge number of nodes (the neuron-like computational elements) and operates dynamically. Although mathematical neurons were conceived as early as 1943, only recently have large-scale real-world applications become practical. 

The most complete mathematical model of a nonuniform adsorbed layer is the distributed model, which takes into account interactions of adsorbed species, their mobility, and a possibility of phase transitions under the action of adsorbed species. The layer of adsorbed species corresponds to the two-dimensional model of the lattice gas, which is a characteristic model of statistical mechanics. Currently, it is widely used in the modeling of elementary processes on the catalyst surface. The energies of the lateral interaction between species localized in different lattice cells are the main parameters of the model. In the case of the chemisorption of simple species, each species occupies one unit cell. The catalytic process consists of a set of elementary steps of adsorption, desorption, and diffusion and an elementary act of reaction, which occurs on some set of cells (nodes) of the lattice. 

Wilders, R., H.J. Jongsma A.C.G. van Girmeken. 1991. Pacemaker activity of the rabbit sinoatrial node. A comparison of mathematical models. Biophys. J. 60 1202-16. 

Bristow, D.G. and J.W. Clark (1982). A mathematical model of primary pacemaking cell in sa node of the heart. Am. J. Physiol. 243,H207-H218. 

Dokos, S., B.G. CeUer, and N.H. Lovell (1996). Vagal control of sinoatrial node rhythm a mathematical model. J. Theor. Biol. 182,21-44. 

Another powerful tool is the cellular automata method invented by John (or Janos) von Neumann and Stanislaw Marcin Ulam (under the name of cellular spaces ). The cellular automata are mathematical models in which space and time both have a granular structure (similar to Monte Carlo simulations on lattices, in MD only time has such a structure). A cellular automaton consists of a periodic lattice of cells (nodes in space). In order to describe the system locally, we assume that every cell has its state representing a vector of N components. Each component is a Boolean variable i.e., a variable having a logical value (e.g., 0 for false and 1 for true ). 

New methodology for exact reliability quantification of highly reliable systems with maintenance was introduced in (Bris 2008a). It assumes that the system structure is mathematically represented by the use of directed acyclic graph (AG), see more details in (Bris 2008b). Terminal nodes of the AG that represent system components are established by the definition of deterministic or stochastic process, to which they are subordinate. From them we can compute a time dependent unavailability function, of individual terminal nodes. Finally a correspondent time dependent imavailability function U(x,t) of the highest node (SS node or top event in classic PRA model) which represents rehabdity behaviour of the whole system may be found. It is clear that U(x,t) < Us(x). 

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