Point kinetics

The simplest model of time-dependent behavior of a neutron population in a reactor consists of the point kinetics differential equations, where the space-dependence of neutrons is disregarded. The safety of reactors is greatly enhanced inherently by the existence of delayed neutrons, which come from radioactive decay rather than fission. The differential equations for the neutron population, n, and delayed neutron emitters, are... 

The energy of the electron gas is composed of two terms, one Hartree-Fock term (T)hp) and one correlation term (Hartree-Fock term comprises the zero-point kinetic energy density and the exchange contribution (first and second terms on the right in equation 1.148, respectively) ... 

A. Establish the point kinetics and equilibrium isotherm for the adsorption step at the site, and the rate coefficients for the various space processes. These are the two types of data required for design. To some extent estimates may be made for the coefficients for the space processes. 

The point kinetics, in terms of a first order, reversible adsorption rate is... 

A simplified transient analysis model of the sulphur iodine and Westinghouse hybrid sulphur cycle was presented by Brown, et al. (2009). This model is utilised in this paper via coupling to a PBMR-268 model and a simple point kinetics model. Some of the key tenants of the analysis model are summarised however interested readers are referred to the original paper for greater detail. The S-I and HyS analysis model is a control-volume model which treats the chemical plant as a closed system. 

The nuclear reactor kinetics was modelled using simple point kinetics. The point kinetics model utilised in the calculation was developed as an analogue to the point kinetics module of the RELAP5 code. The number of delayed neutron groups considered was six. A Doppler feedback coefficient of -0.0095 was used. Xenon feedback was also modelled, although due to the time scales considered in this document the xenon feedback is not relevant and has almost no impact on the results. 

First, a steady-state solution is attained in both the THERMIX model and hydrogen model. Second, a transient is initiated, either in the chemical plant model or in the point kinetics model. Finally, the THERMIX, point kinetics and hydrogen generation models all interact in each time step. The integration scheme is shown in Figure 4. 

Coupling of the codes is performed through the IHX. The data exchanged between the codes consists of the flow rate and temperature of helium through both hot and cold legs of the IHX. The hydrogen model calls THERMIX and the point kinetics model every time step and then new temperature and flow rate values are returned. This process is repeated each time step. 

A transient control volume model of the S-I and HyS cycle is presented. An important conclusion based on the results of this model is that the rate-limiting step of the entire S-I cycle is the HI decomposition section. In the HyS cycle, the rate-limiting step is the H2S04 decomposition. A generalised methodology for coupling these thermochemical cycle models to a nuclear reactor model is overviewed. The models were coupled to a THERMIX-DIREKT thermal model of a PBMR-268 and a point kinetics model. Key assumptions in the PBMR-268 model include flattening of the core and parallelisation of the flow channels. 

Due to its high zero-point kinetic energy, Ps is supposed to dig a cavity, or "bubble" in liquids [56, 57]. Various levels of approximation are possible for a quantum mechanical approach to the problem the potential well (of depth U) constituting the bubble may be considered or not as infinite and/or rigid [58-60]. Some typical values of (rigid) well depth and radius are given in Table 4.2 [61] the bubble radius, Rb, remains in a rather narrow range, about 0.3—0.45 nm, independently of the solvent or temperature. 

For a rigid bubble, the zero-point kinetic energy of Ps is correlated to Rb" 2 which is proportional to y l/4. From the expression of the Ps energy level in the bubble, a constant ratio of 0.329 eVl/2 cm1/4 dyn ,/4 is effectively found between the Ps average momentum and y l/4 for a variety of liquids [61],... 

The most uncertain quantity here is Iq+, for which one obtains the lowest threshold P0+ > -P0- + Pps + 47rP2 —4.3 eV, which occasionally coincides with P. It implies that the above-mentioned relationship Vq P underestimates Vq+ due to neglect of the e+ zero-point kinetic energy contribution. 

When Pg- and the polarization part of VJ," approximately cancel each other, Eq. (14) is reduced to Wl Wq — AX. Thus, the difference Wq — Wl is determined by zero-point kinetic energy AX of electrons in the medium. For example, when S has a maximum of crG(W) close to the thermal region, in the liquid phase it even may lose trapping ability, i.e., WG - AX < 0, because of a lack of electrons with small energies (AX is the lowest possible kinetic energy of the quasi-free electrons in a liquid). 

This stunning agreement does seem somewhat fortuitous. It may be unnecessary to introduce the concept of clusters, which would also involve the lifetime of such clusters and other complicating factors. It is probably more meaningful and rewarding to interpret (16) the isotope factor in terms of Mullen (21) and LeClaire (15), such that a = —AK/2, AK denoting the share of the saddle-point kinetic energy. 

In methods employing two-point kinetics, the enzyme activity is determined by measuring the reflection at two different times. 

Methods in which some property related to substrate concentration (such as absorbance, fluorescence, chemiluminescence, etc.) is measured at two fixed times during the course of the reaction are known as two-point kinetic methods. They are theoreticahy the most accurate for the enzymatic determination of substrates. However, these methods are technically more demanding than equifibrium methods and all the factors that affect reaction rate, such as pH, temperature, and amount of enzyme, must be kept constant from one assay to the next, as must the timing of the two measurements. These conditions can readily be achieved in automatic analyzers. A reference solution of the analyte (substrate) must be used for calibration. To ensure first-order reaction conditions, the substrate concentration must be low compared to the K, (i.e., in the order of less than 0.2 X K, . Enzymes with high K , values are therefore preferred for kinetic analysis to give a wider usable range of substrate concentration. 

Enzymatic methods for the measurement of urea are based on preliminary hydrolysis of urea with urease (urea amidohydrolase, EC 3.5.1.5 main source jack bean meal) to generate ammonium ion, which is then quantified. This approach has been used in end-point, kinetic, conductimet-ric, and dry chemistry systems. ... 

Thermodynamic/equilibrium parameters describe the end point of a process, and are always the same for a given system, regardless of the route taken to arrive at the end point. Kinetic parameters describe a point in time and space, are valid only for that instant in time, and vary with conditions, history, and the many factors that have interacted to produce the system as it is at the instant of its description. At this point, it is appropriate to provide a simple example. Consider the reaction ... 

Pure beta-emitting isotopes exist and may be used for calibration, but only after the energy spectrum is cast into a form called the Kurie plot. The beta spectrum is continuous and extends from zero energy up to a maximum end point kinetic energy (see Fig. 13.12). Because of the shape of the spectrum, it is impossible to accurately determine the end point energy. However, from the... 

D. Betteridge and B. Fields, Two-Point Kinetic Simultaneous Determination of Cobalt(II) and Nickel(II) in Aqueous Solution Using Flow Injection Analysis (FIA). Fresenius Z. Anal. Chem., 314 (1983) 386. 

The most favored decomposition reaction is the cleavage of SiCU, which, after numerous repetitions, should lead to Si02 and SiCU. Considering the discussed decomposition reactions from a molecular view point (kinetic aspect) also leads to a clear preference for decomposition reaction (a) For reaction (a) to occur, only the necessary activation energy from a collision with an energy-rich molecule or the (hot) wall of the reaction vessel has to be supplied. For reactions (b) and (c) to become operative, however, a collision of two molecules of equal composition is necessary. This is rather improbable in the complex reaction mixture. Unimolecular reactions which lead to products under (b) or (c) are not conceivable. 

A two-point kinetic turbidimetric technique using commercially available monospecific antisera has been used for the determination of immunoglobulins G, A, and M in sera. The accuracy of the method when measuring idiotypic monoclonal proteins is greater than that of the radial diffusion method. 

The magnitude of the task of modeling organometallic reactions theoretically means that point-by-point kinetics cannot be expected. However, progress has been made in estimating whether key intermediates predicted experimentally are reasonable computationally. Results from relevant studies will be cited. 

Considering the fact that the MCC is divided into two symmetric loops, a pipe break in one of these directly affects the thermohydraulic and neutronic behaviour on one side of the core. The resulting asymmetry in reactivity and power has to be assessed by a 3-D code. OptionaUy, the information obtained can be compiled as input for use by a point kinetic model, which, however, usually requires some iterations. 

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