Establishing Dynamic Equation Model

Based on assumptions mentioned in Sect. 3.2.1, let Y =Y t) be the product demand in the moment t, K be the limit value of customer product demand at specific period. 

Y = Y (t) = dY l)/dl indicates that the demand for instantaneous rate of change, based on the above assumptions 3, we can establish the following differential equation  

If t — +0O, then lim, +oo T(t) = K = max 7(r), which indicates that the maximum production amount cannot exceed product market capacity. 

If t = 0, then 7(f) = 7(0) = y, this property shows that at a starting point of the initial forecast, the enterprise products have the initial demand According to this property, we can get the value b. 

If f = the second derivative of 7(f) is zero, we can get the curve inflection point (, f). This means that the growth of demand will experience smaller speed to a larger speed, then from larger to smaller, until the growth rate is zero for such a process. 

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System dynamics steps to solve the problem (Wu J.Z. et al. 1985) is (1) to identify the problem how the coal mine safety input influence factors of coal mine production, which in turn affect coal mine safety performance (2) to determine the system boundary safety production system in coal mine (3) to determine causal graph and define the variable draw causal graph of coal mine safety production system and define the model variable (4) to establish equations, models and analyze the simulation model. 

ABSTRACT In order to reveal the collision behavior regularity between the drill pipe and the coal hole wall in the process of gas extraction drilling, the nonlinear dynamic equations of the drill pipe and the finite element model of the collision between the drill pipe and the hole wall are established. The collision stress variation between the drill pipe and the hole wall with different diameters are analyzed. The results show that with the decrease of the coal hole diameter, the collision stress value between the drill pipe and the hole wall increases and the hole wall is more instability to collapse. When the hole diameter increases to a certain extent, the collision stress variation become gentle and the hole wall become stable. The research provides the theoretical law to select the proper drilling parameters, which can improve the coal hole wall stability in the process of the gas extraction drilling. 

Once the reactor equations and assumptions have been established, and HDS, HDN, HDA, and HGO reaction rate expressions have been defined, the adsorption coefficient, equilibrium constants, reaction orders, frequency factors, and activation energies can be determined from the experimental data obtained at steady-state conditions by optimization with the Levenberg-Marquardt nonlinear regression algorithm. Using the values of parameters obtained from steady-state experiments, the dynamic TBR model was employed to redetermine the kinetic parameters that were considered as definitive values. The temperature dependencies of all the intrinsic reaction rate constants have been described by the Arrhenius law, which are shown in Table 7.4. 

The highest level of integration would be to establish one large set of equations and to apply one solution process to both thermal and airflow-related variables. Nevertheless, a very sparse matrix must be solved, and one cannot use the reliable and well-proven solvers of the present codes anymore. Therefore, a separate solution process for thermal and airflow parameters respectively remains the most promising approach. This seems to be appropriate also for the coupling of computational fluid dynamics (CFD) with a thermal model. ... 

A dynamic model should be consistent with the steady-state model. Thus, Eqs (1) and (4) should be extended to dynamic form. For the better convergence and computational efficiency, some assumption can be introduced the total amounts of mass and enthalpy at each plate are maintained constant. Then, the internal flow can be determined by total mass balance and total energy balance and the number of differential equations is reduced. Therefore, the dynamic model can be established by replacing component material balance in Eq. (1) with the following equation. 

We now derive the time-domain solutions of first and second order differential equations. It is not that we want to do the inverse transform, but comparing the time-domain solution with its Laplace transform helps our learning process. What we hope to establish is a better feel between pole positions and dynamic characteristics. We also want to see how different parameters affect the time-domain solution. The results are useful in control analysis and in measuring model parameters. At the end of the chapter, dead time, reduced order model, and the effect of zeros will be discussed. 

The first attempt to describe the dynamics of dissociative electron transfer started with the derivation from existing thermochemical data of the standard potential for the dissociative electron transfer reaction, rx r.+x-,12 14 with application of the Butler-Volmer law for electrochemical reactions12 and of the Marcus quadratic equation for a series of homogeneous reactions.1314 Application of the Marcus-Hush model to dissociative electron transfers had little basis in electron transfer theory (the same is true for applications to proton transfer or SN2 reactions). Thus, there was no real justification for the application of the Marcus equation and the contribution of bond breaking to the intrinsic barrier was not established. 

The above time-scale decomposition provides a transparent framework for the selection of manipulated inputs that can be used for control in the two time scales. Specifically, it establishes that output variables y1 need to be controlled in the fast time scale, using the large flow rates u1, while the control of the variables ys is to be considered in the slow time scale, using the variables us. Moreover, the reduced-order approximate models for the fast (Equation (3.11)) and slow (the state-space realization of Equation (3.16)) dynamics can serve as a basis for the synthesis of well-conditioned nonlinear controllers in each time scale. 

Considering both the mass balance corresponding to a CSTR and the kinetic equation, a dynamic model was established and validated for steady-state continuous operation of the EMR (Fig. 10.6). Moreover, the model was analyzed to foresee the deviations of the steady-state that occur when short-term changes in operational conditions take place (Fig. 10.7). Higher Orange II loading rates resulting from a more concentrated influent or a variation of HRT caused an increase in dye concentration in the effluent, which was predicted accurately by the dynamic model. The validation was also observed when modifications in the peroxide addition rate occurred. 

The fundamental assumption and equations governing the effect-concentration relationship for each one of the models considered are listed in Table 10.1. The presence or not of an hysteresis loop in the effect-plasma concentration plot of each model is also quoted in Table 10.1. At present, the methodology for performing efficient pharmacokinetic-dynamic modeling is well established [405,456,457],... 

In order to establish the relationship between the static and dynamic fractal dimensions, the initial conditions of the classical static percolation model must be considered for the solution of differential equation (89) which can be written as 0 = Qs = 1 for D = Ds. Here the notation s corresponds to the static percolation model, and the condition s = 1 is fulfilled for an isotropic cubic hyperlattice. The solution of (89) with the above-mentioned initial conditions may be written as... 

Therefore, the consistent study of many-particle system dynamics should start by establishing the Hamiltonian H and then solving the evolution equation (93). Unfortunately, examples of such calculations are very rare and are only valid for limited classes of model systems (such as the Ising model) since these... 

From these time-scales, it may be assumed in most circumstances that the free electrons have a Maxwellian distribution and that the dominant populations of impurities in the plasma are those of the ground and metastable states of the various ions. The dominant populations evolve on time-scales of the order of plasma diffusion time-scales and so should be modeled dynamically, that is in the particle number continuity equations, along with the momentum and energy equations of plasma transport theory. The excited populations of impurities on the other hand may be assumed relaxed with respect to the instantaneous dominant populations, that is they are in a quasi-equilibrium. The quasi-equilibrium is determined by local conditions of electron temperature and electron density. So, the atomic modeling may be partially de-coupled from the impurity transport problem into local calculations which provide quasi-equilibrium excited ion populations and effective emission coefficients (PEC coefficients) and then effective source coefficients (GCR coefficients) for dominant populations which must be entered into the transport equations. The solution of the transport equations establishes the spatial and temporal behaviour of the dominant populations which may then be re-associated with the local emissivity calculations, for matching to and analysis of observations. 

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