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At least 109 records · Page 6

On recontamination and directional-bias problems in Monte Carlo simulation of PDF turbulence models

Turbulent combustion can not be simulated adequately by conventional moment closure turbulence models. The difficulty lies in the fact that the reaction rate is in general an exponential function of the temperature, and the higher order correlations in the conventional moment closure models of the chemical source term can not be neglected, making the applications of such models impractical. The probability density function (pdf) method offers an attractive alternative: in a pdf model, the chemical source terms are closed and do not require additional models. A grid dependent Monte Carlo scheme was studied, since it is a logical alternative, wherein the number of computer operations increases only linearly with the increase of number of independent variables, as compared to the exponential increase in a conventional finite difference scheme. A new algorithm was devised that satisfies a restriction in the case of pure diffusion or uniform flow problems. Although for nonuniform flows absolute conservation seems impossible, the present scheme has reduced the error considerably.

Hsu, Andrew T.↗

SU-F-I-63: Relaxation Times of Lipid Resonances in NAFLD Animal Model Using Enhanced Curve Fitting

Purpose: The objective of this study is to evaluate the relaxation time of methylene resonance in comparison with other lipid resonances. Methods: The examinations were performed on a 3.0T MRI scanner using a four-channel animal coil. Eight more Sprague-Dawley rats in the same baseline weight range were housed with ad libitum access to water and a high-fat (HF) diet (60% fat, 20% protein, and 20% carbohydrate). In order to avoid large blood vessels, a voxel (0.8×0.8×0.8 cm{sup 3}) was placed in a homogeneous area of the liver parenchyma during free breathing. Lipid relaxations in NC and HF diet rats were estimated at a fixed repetition time (TR) of 6000 msec, and multi echo time (TEs) of 40–220 msec. All spectra for data measurement were processed using the Advanced Method for Accurate, Robust, and Efficient Spectral (AMARES) fitting algorithm of the Java-based Magnetic Resonance User Interface (jMRUI) package. Results: The mean T2 relaxation time of the methylene resonance in normal-chow diet was 37.1 msec (M{sub 0}, 2.9±0.5), with a standard deviation of 4.3 msec. The mean T2 relaxation time of the methylene resonance was 31.4 msec (M{sub 0}, 3.7±0.3), with a standard deviation of 1.8 msec. The T2 relaxation times of methylene protons were higher in normal-chow diet rats than in HF rats (p<0.05), and the extrapolated M{sub 0} values were higher in HF rats than in NC rats (p<0.005). The excellent linear fit with R{sup 2}>0.9971 and R{sup 2}>0.9987 indicates T2 relaxation decay curves with mono-exponential function. Conclusion: In in vivo, a sufficient spectral resolution and a sufficiently high signal-to-noise ratio (SNR) can be achieved, so that the data measured over short TE values can be extrapolated back to TE = 0 to produce better estimates of the relative weights of the spectral components. In the short term, treating the effective decay rate as exponential is an adequate approximation.

60 APPLIED LIFE SCIENCES↗

An analytic method to account for drag in the Vinti Satellite theory

To retain separability in the Vinti theory of earth satellite motion when a nonconservative force such as air drag is considered, a set of variational equations for the orbital elements are introduced, and expressed as functions of the transverse, radial, and normal components of the nonconservative forces acting on the system. In this approach, the Hamiltonian is preserved in form, and remains the total energy, but the initial or boundary conditions and hence the Jacobi constants of the motion advance with time through the variational equations. In particular, the atmospheric density profile is written as a fitted exponential function of the eccentric anomaly, which adheres to tabular data at all altitudes and simultaneously reduced the variational equations to indefinite integrals with closed form evaluations. The values of the limits for any arbitrary time interval are obtained from the Vinti program.

Watson, J. S.↗

Evaluating the Feasibility of Measuring and Analyzing Strain Caused by Changes in Barometric Pressure in the Vadose Zone

Changes in barometric pressure propagate into the subsurface where they can affect water level measurements in wells and cause vertical strain. Previous strain sensors were developed to measure vertical strain, but they were limited to measurements at a single depth, making it difficult to evaluate strains from a migrating pressure wave. Technology known as Coherence-length-gated Microwave Photonics Interferometry (CMPI) uses optical fiber sensors to measure strain at multiple locations. It has the potential to detect strain caused by variations in barometric pressure at multiple depths in the subsurface. The goal of this study is to evaluate the feasibility of using this technology to record the changes in vertical strain in the vadose zone from small fluctuations in air pressure. Propagation of these air pressure fluctuations is affected by hydrogeological properties such as water content and permeability. A second goal of this study is to evaluate how these properties affect the air pressure distribution, strain, and gas-phase diffusivity with a long-term goal of using strain to monitor the vadose zone. The focus of this study is on a suite of laboratory experiments that used a sand-filled column with an open head space. The CMPI fiber was packaged and embedded along the axis of the column to measure the strain at multiple locations while air pressure transducers were installed through the wall of the column. Small periodic fluctuations in air pressure, similar to barometric pressure fluctuations, were created in the head space of the column using an audio driven speaker driven at 4 Hz using a sinusoidal signal created by a function generator. Then, the fluctuations propagated along the column where they were measured by the transducers and recorded as functions of time. Initial tests were conducted using dry sand, but then additional tests were conducted after injecting water into the column, which changed the hydrologic and mechanical properties. Another experiment was conducted after creating a thin barrier to air flow, which was designed to simulate a thin layer of saturated ground during rainfall. A final experiment was conducted by changing the grain size of the material to create a heterogeneity in the upper half of the column. The suite of six experiments was conducted to highlight effects of variations in water content and heterogeneities on air pressure and vertical strain. Numerical simulations were also created to evaluate the laboratory data and to provide a baseline analysis for the results. The numerical simulations used the governing equations to linear poroelasticity and two-phase flow with boundary conditions representing the experiments. The conceptual model for strain caused by barometric pressure recognizes that a fluctuating barometric pressure causes fluctuations in the air pore pressure that decrease in amplitude and lag in time with increase depth. The pressure distribution results in two different loads that cause strain: 1.) a mechanical load as the barometric pressure acts on the ground surface; and 2.) pressure loading in the pore space. The results show that the total vertical strain is a contribution of both of these loads, and this is the strain that was observed in the laboratory. The lab experiments show that both air pressure and strain vary as sinusoids that propagate with depth. The amplitude of the air pressure and strain decrease as a function of depth in all the conditions that were evaluate. For example, in the dry sand, the amplitude of the air pressure decreases from 25 Pa at the head space to 5 Pa at a depth of 70 cm, whereas the amplitude of the strain decreases from 0.15 to 0.06 µε over the same depth interval. In many cases the amplitudes decrease is approximately negative exponential functions of depth. When water was injected into the sand and the saturation increased, the amplitude of the air pressure increased in the partially saturated sand but decreased sharply to zero where the soil was saturated. The strain decreased with depth, but the transition from partially to fully saturated conditions had little effect on the strain profile, even though it had a major effect on the air pressure. This apparently is because strain is caused by variations in both water and air pressure. The fluctuating air pressure caused the water pressure to fluctuate, which caused strain throughout the column. Phase delay of the air pressure and the strain both increase as linear to bi-linear functions of depth. The phase delays indicate that pressure propagates at a velocity of 7 to 8 m/s in dry or partially saturated sand, but it drops by more than an order of magnitude in the vicinity of the capillary fringe. The phase delays indicate the strain propagates faster than the pressure. The data indicate that the strain propagates from 1.2 to more than 4x faster than the pressure. Switching from sand to silt in the column caused the velocity of the pressure to drop by roughly half for both pressures, but curiously, it had little effect on the velocity of the strain.

54 ENVIRONMENTAL SCIENCES↗

Tracking the dynamic seroma cavity using fiducial markers in patients treated with accelerated partial breast irradiation using 3D conformal radiotherapy

Purpose: The purpose of the present study was to perform an analysis of the changes in the dynamic seroma cavity based on fiducial markers in early stage breast cancer patients treated with accelerated partial breast irradiation (APBI) using three-dimensional conformal external beam radiotherapy (3D-CRT). Methods: A prospective, single arm trial was designed to investigate the utility of gold fiducial markers in image guided APBI using 3D-CRT. At the time of lumpectomy, four to six suture-type gold fiducial markers were sutured to the walls of the cavity. Patients were treated with a fractionation scheme consisting of 15 fractions with a fractional dose of 333 cGy. Treatment design and planning followed NSABP/RTOG B-39 guidelines. During radiation treatment, daily kV imaging was performed and the markers were localized and tracked. The change in distance between fiducial markers was analyzed based on the planning CT and daily kV images. Results: Thirty-four patients were simulated at an average of 28 days after surgery, and started the treatment on an average of 39 days after surgery. The average intermarker distance (AiMD) between fiducial markers was strongly correlated to seroma volume. The average reduction in AiMD was 19.1% (range 0.0%-41.4%) and 10.8% (range 0.0%-35.6%) for all the patients between simulation and completion of radiotherapy, and between simulation and beginning of radiotherapy, respectively. The change of AiMD fits an exponential function with a half-life of seroma shrinkage. The average half-life for seroma shrinkage was 15 days. After accounting for the reduction which started to occur after surgery through CT simulation and treatment, radiation was found to have minimal impact on the distance change over the treatment course. Conclusions: Using the marker distance change as a surrogate for seroma volume, it appears that the seroma cavity experiences an exponential reduction in size. The change in seroma size has implications in the size of the CTV, PTV, and percent of normal breast tissue irradiated when using 3D-CRT.

60 APPLIED LIFE SCIENCES↗

Temperature-dependent model of helium bubble expansion and bursting in tungsten

Continuum-scale models that can reliably predict the behavior of helium in tungsten are of interest to the fusion community due to the projected impact of these materials on fusion reactor operation. In this work, we perform molecular dynamics simulations of spherical helium bubbles of various sizes in tungsten at different temperatures and depths with the goal of determining a mathematical model of the pressure and density at which the bubbles initially expand or burst as a function of depth, size, temperature, and surface orientation. The bubbles are small enough that their loop-punching pressures cannot be accurately predicted with continuum mechanics, and their expansion behavior is important, as it appears to cause many of the features observed on helium-irradiated tungsten surfaces. We vary the temperature, bubble size, bubble depth, and surface orientation in each case, recording the bubble pressure and density that result in bubble expansion. An exponential function with three adjustable parameters is found to fit the results well; the parameters that best fit our results are provided.

36 MATERIALS SCIENCE↗

Accelerated Time-Domain Modeling of Electromagnetic Pulse Excitation of Finite-Length Dissipative Conductors over a Ground Plane via Function Fitting and Recursive Convolution

In this report we overview the fundamental concepts for a pair of techniques which together greatly hasten computational predictions of electromagnetic pulse (EMP) excitation of finite-length dissipative conductors over a ground plane. In a time- domain, transmission line (TL) model implementation, predictions are computationally bottlenecked time-wise, either for late-time predictions (about 100ns-10000ns range) or predictions concerning EMP excitation of long TLs (order of kilometers or more ). This is because the method requires a temporal convolution to account for the losses in the ground. Addressing this to facilitate practical simulation of EMP excitation of TLs, we first apply a technique to extract an (approximate) complex exponential function basis-fit to the ground/Earth's impedance function, followed by incorporating this into a recursion-based convolution acceleration technique. Because the recursion-based method only requires the evaluation of the most recent voltage history data (versus the entire history in a "brute-force" convolution evaluation), we achieve necessary time speed- ups across a variety of TL/Earth geometry/material scenarios. Intentionally Left Blank

36 MATERIALS SCIENCE↗

On the analytical determination of relaxation modulus of viscoelastic materials by Prony's interpolation method

A computer implementation to Prony's curve fitting by exponential functions is presented. The method, although more than one hundred years old, has not been utilized to its fullest capabilities due to the restriction that the time range must be given in equal increments in order to obtain the best curve fit for a given set of data. The procedure used in this paper utilizes the 3-dimensional capabilities of the Interactive Graphics Design System (I.G.D.S.) in order to obtain the equal time increments. The resultant information is then input into a computer program that solves directly for the exponential constants yielding the best curve fit. Once the exponential constants are known, a simple least squares solution can be applied to obtain the final form of the equation.

Rodriguez, Pedro I.↗

Convergence in the temperature response of leaf respiration across biomes and plant functional types

Plant respiration constitutes a massive carbon flux to the atmosphere, and a major control on the evolution of the global carbon cycle. It therefore has the potential to modulate levels of climate change due to the human burning of fossil fuels. Neither current physiological nor terrestrial biosphere models adequately describe its short-term temperature response, and even minor differences in the shape of the response curve can significantly impact estimates of ecosystem carbon release and/or storage. Given this, it is critical to establish whether there are predictable patterns in the shape of the respiration–temperature response curve, and thus in the intrinsic temperature sensitivity of respiration across the globe. By analyzing measurements in a comprehensive database for 231 species spanning 7 biomes, we demonstrate that temperature-dependent increases in leaf respiration do not follow a commonly used exponential function. Instead, we find a decelerating function as leaves warm, reflecting a declining sensitivity to higher temperatures that is remarkably uniform across all biomes and plant functional types. Such convergence in the temperature sensitivity of leaf respiration suggests that there are universally applicable controls on the temperature response of plant energy metabolism, such that a single new function can predict the temperature dependence of leaf respiration for global vegetation. This simple function enables straightforward description of plant respiration in the land-surface components of coupled earth system models. Our cross-biome analyses shows significant implications for such fluxes in cold climates, generally projecting lower values compared with previous estimates.

54 ENVIRONMENTAL SCIENCES↗

Measurement of the total cross section from elastic scattering in pp collisions at s = 8 TeV with the ATLAS detector

A measurement of the total ppcross section at the LHC at √s=8TeV is presented. An integrated luminosity of 500 μb-1 was accumulated in a special run with high-β beam optics to measure the differential elastic cross section as a function of the Mandelstam momentum transfer variable t. The measurement is performed with the ALFA sub-detector of ATLAS. Using a fit to the differential elastic cross section in the -t range from 0.014GeV2 to 0.1GeV2 to extrapolate t→0, the total cross section, σtot(pp →X), is measured via the optical theorem to be σtot(pp→ X) = 96.07±0.18 (stat.)±0.85 (exp.)± 0.31 (extr.) mb, where the first error is statistical, the second accounts for all experimental systematic uncertainties and the last is related to uncertainties in the extrapolation t→0. In addition, the slope of the exponential function describing the elastic cross section at small t is determined to be B =19.74 ±0.05 (stat.) ±0.23 (syst.) GeV-2.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Numerical computation of exponential matrices using the Cayley-Hamilton theorem

A method for computing exponential matrices, which often arise naturally in the solution of systems of linear differential equations, is developed. An exponential matrix is generated as a linear combination of a finite number (equal to the matrix order) of matrices, the coefficients of which are scalar infinite sums. The method can be generalized to apply to any formal power series of matrices. Attention is focused upon the exponential function, and the matrix exponent is assumed tri-diagonal in form. In such cases, the terms in the coefficient infinite sums can be extracted, as recursion relations, from the characteristic polynomial of the matrix exponent. Two numerical examples are presented in some detail: (1) the three dimensional infinitesimal rotation rate matrix, which is skew symmetric, and (2) an N-dimensional tri-diagonal and symmetric finite difference matrix which arises in the numerical solution of the heat conduction partial differential equation. In the second example, the known eigenvalues and eigenvectors of the finite difference matrix permit an analytical solution for the exponential matrix, through the theory of diagonalization and similarity transformations, which is used for independent verification. The convergence properties of the scalar infinite summations are investigated for finite difference matrices of various orders up to ten, and it is found that the number of terms required for convergence increases slowly with the order of the matrix.

Walden, H.↗

Energy Coupling Mechanisms and Scaling Behavior Associated with Laser Powder Bed Fusion Additive Manufacturing

In situ optical absorptivity measurements are carried out to clarify the physics of the laser-material interactions involved and to validate both finite element and analytical models describing laser powder bed fusion processing. Absorption of laser energy is evaluated directly using precise calorimetry measurements and compared to melt pool depths for common structural metal alloys (Ti-6Al-4V, Inconel 625, and stainless steel 316L) as a function of incident laser power, scan velocity and laser beam diameter. Changes in both absorptivity and melt pool depths for all materials are found to vary strongly across the conduction-keyhole mode threshold. A hydrodynamic finite element model is coupled to a ray-tracing-based absorptivity model, yielding excellent agreement with the experimental results and elucidating additional physics. The experimental results are further analyzed using normalized enthalpy ( β ) and normalized thermal diffusion length ( L* th ) relations and demonstrate that the normalized melt pool depth ( d* = d/a , where d is melt pool depth and a is beam radius) is proportional to βL* th , while absorptivity follows an asymptotic exponential function against βL* th . Expressions for melt pool depth and laser absorptivity across different materials and laser scan systems are derived and thus provide useful tools to accelerate the optimization of laser processing parameters for metal 3D printing processes.

36 MATERIALS SCIENCE↗

SU-E-T-86: A Systematic Method for GammaKnife SRS Fetal Dose Estimation

Purpose: Estimating fetal dose is critical to the decision-making process when radiation treatment is indicated during pregnancy. Fetal doses less than 5cGy confer no measurable non-cancer developmental risks but can produce a threefold increase in developing childhood cancer. In this study, we estimate fetal dose for a patient receiving Gamma Knife stereotactic radiosurgery (GKSRS) treatment and develop a method to estimate dose directly from plan details. Methods: A patient underwent GKSRS on a Perfexion unit for eight brain metastases (two infratentorial and one brainstem). Dose measurements were performed using a CC13, head phantom, and solid water. Superficial doses to the thyroid, sternum, and pelvis were measured using MOSFETs during treatment. Because the fetal dose was too low to accurately measure, we obtained measurements proximally to the isocenter, fitted to an exponential function, and extrapolated dose to the fundus of the uterus, uterine midpoint, and pubic synthesis for both the preliminary and delivered plans. Results: The R-squared fit for the delivered doses was 0.995. The estimated fetal doses for the 72 minute preliminary and 138 minute delivered plans range from 0.0014 to 0.028cGy and 0.07 to 0.38cGy, respectively. MOSFET readings during treatment were just above background for the thyroid and negligible for all inferior positions. The method for estimating fetal dose from plan shot information was within 0.2cGy of the measured values at 14cm cranial to the fetal location. Conclusion: Estimated fetal doses for both the preliminary and delivered plan were well below the 5cGy recommended limit. Due to Pefexion shielding, internal dose is primarily governed by attenuation and drops off exponentially. This is the first work that reports fetal dose for a GK Perfexion unit. Although multiple lesions were treated and the duration of treatment was long, the estimated fetal dose remained very low.

60 APPLIED LIFE SCIENCES↗

Development of a ReaxFF potential for carbon condensed phases and its application to the thermal fragmentation of a large fullerene

In this paper, we report the development of a ReaxFF reactive potential that can accurately describe the chemistry and dynamics of carbon condensed phases. Density functional theory (DFT)-based calculations were performed to obtain the equation of state for graphite and diamond and the formation energies of defects in graphene and amorphous phases from fullerenes. The DFT data were used to reparametrize ReaxFF CHO , resulting in a new potential called ReaxFF C-2013 . ReaxFF C-2013 accurately predicts the atomization energy of graphite and closely reproduces the DFT-based energy difference between graphite and diamond, and the barrier for transition from graphite to diamond. ReaxFF C-2013 also accurately predicts the DFT-based energy barrier for Stone–Wales transformation in a C 60 (I h ) fullerene through the concerted rotation of a C 2 unit. Later, MD simulations of a C 180 fullerene using ReaxFF C-2013 suggested that the thermal fragmentation of these giant fullerenes is an exponential function of time. An Arrhenius-type equation was fit to the decay rate, giving an activation energy of 7.66 eV for the loss of carbon atoms from the fullerene. Although the decay of the molecule occurs primarily via the loss of C 2 units, we observed that, with an increase in temperature, the probability of loss of larger fragments increases. Finally, the ReaxFF C-2013 potential developed in this work, and the results obtained on fullerene fragmentation, provide an important step toward the full computational chemical modeling of coal pyrolysis, soot incandescence, high temperature erosion of graphitic rocket nozzles, and ablation of carbon-based spacecraft materials during atmospheric reentry.

36 MATERIALS SCIENCE↗

Effects of pulse duration on magnetostimulation thresholds

Purpose: Medical imaging techniques such as magnetic resonance imaging and magnetic particle imaging (MPI) utilize time-varying magnetic fields that are subject to magnetostimulation limits, which often limit the speed of the imaging process. Various human-subject experiments have studied the amplitude and frequency dependence of these thresholds for gradient or homogeneous magnetic fields. Another contributing factor was shown to be number of cycles in a magnetic pulse, where the thresholds decreased with longer pulses. The latter result was demonstrated on two subjects only, at a single frequency of 1.27 kHz. Hence, whether the observed effect was due to the number of cycles or due to the pulse duration was not specified. In addition, a gradient-type field was utilized; hence, whether the same phenomenon applies to homogeneous magnetic fields remained unknown. Here, the authors investigate the pulse duration dependence of magnetostimulation limits for a 20-fold range of frequencies using homogeneous magnetic fields, such as the ones used for the drive field in MPI. Methods: Magnetostimulation thresholds were measured in the arms of six healthy subjects (age: 27 ± 5 yr). Each experiment comprised testing the thresholds at eight different pulse durations between 2 and 125 ms at a single frequency, which took approximately 30–40 min/subject. A total of 34 experiments were performed at three different frequencies: 1.2, 5.7, and 25.5 kHz. A solenoid coil providing homogeneous magnetic field was used to induce stimulation, and the field amplitude was measured in real time. A pre-emphasis based pulse shaping method was employed to accurately control the pulse durations. Subjects reported stimulation via a mouse click whenever they felt a twitching/tingling sensation. A sigmoid function was fitted to the subject responses to find the threshold at a specific frequency and duration, and the whole procedure was repeated at all relevant frequencies and pulse durations. Results: The magnetostimulation limits decreased with increasing pulse duration (T{sub pulse}). For T{sub pulse} < 18 ms, the thresholds were significantly higher than at the longest pulse durations (p < 0.01, paired Wilcoxon signed-rank test). The normalized magnetostimulation threshold (B{sub Norm}) vs duration curve at all three frequencies agreed almost identically, indicating that the observed effect is independent of the operating frequency. At the shortest pulse duration (T{sub pulse} ≈ 2 ms), the thresholds were approximately 24% higher than at the asymptotes. The thresholds decreased to within 4% of their asymptotic values for T{sub pulse} > 20 ms. These trends were well characterized (R{sup 2} = 0.78) by a stretched exponential function given by B{sub Norm}=1+αe{sup −(T{sub p}{sub u}{sub l}{sub s}{sub e}/β){sup γ}}, where the fitted parameters were α = 0.44, β = 4.32, and γ = 0.60. Conclusions: This work shows for the first time that the magnetostimulation thresholds decrease with increasing pulse duration, and that this effect is independent of the operating frequency. Normalized threshold vs duration trends are almost identical for a 20-fold range of frequencies: the thresholds are significantly higher at short pulse durations and settle to within 4% of their asymptotic values for durations longer than 20 ms. These results emphasize the importance of matching the human-subject experiments to the imaging conditions of a particular setup. Knowing the dependence of the safety limits to all contributing factors is critical for increasing the time-efficiency of imaging systems that utilize time-varying magnetic fields.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Estimation of the time for steam generator trip due to cyber intrusions

The time required to trip a pressurized water reactor (PWR) by inserting malicious signals into its steam generator (SG) control system has been studied using the Generic PWR (GPWR) Simulator. A semi-analytical model is developed to approximately reproduce the simulator response and understand the dynamics of the control unit. A series of two proportional-integral controllers determines control action according to preset constants, the readings from the feedwater level sensor, and those from feedwater and steam flowrate transmitters. It is observed that the most important factor that determines whether a trip will occur is how much additional water is added to or withheld from the SG over time compared to normal operating conditions. In order to determine the effects of control action on the SG, changes in mass inventory are considered. This approach models the SG water level as a function of mass inventory and has a backward temporal memory. A Python interface is developed for the GPWR framework to automatically simulate different spoofing scenarios and post-process the related data. We observe that the trip times predominantly depend on flow mismatch and/or level errors. Controller parameters, including the integral time and gain constants, either speed up or slow down the rate of progression to a trip setpoint but do not cause a trip by themselves. The reactor can trip on a high-level signal when the reading crosses above 78%, increased from its reference level of 57%, or a low-level reading when it is below 25%. The present results show roughly how long the operators would have to respond to an attack, given a specific set of spoofing signals within the issue space analyzed. Furthermore, we have generated a simple surface by fitting a combination of exponential functions to the data obtained from the GPWR Simulator. In general, trips on a low level have been observed to occur faster than those on a high level.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗