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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 415 records · Page 23
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

Wet scavenging of soluble gases in DC3 deep convective storms using WRF-Chem simulations and aircraft observations

In this work, we examine wet scavenging of soluble trace gases in storms observed during the Deep Convective Clouds and Chemistry (DC3) field campaign. We conduct high-resolution simulations with the Weather Research and Forecasting model with Chemistry (WRF-Chem) of a severe storm in Oklahoma. The model represents well the storm location, size, and structure as compared with Next Generation Weather Radar reflectivity, and simulated CO transport is consistent with aircraft observations. Scavenging efficiencies (SEs) between inflow and outflow of soluble species are calculated from aircraft measurements and model simulations. Using a simple wet scavenging scheme, we simulate the SE of each soluble species within the error bars of the observations. The simulated SEs of all species except nitric acid (HNO 3 ) are highly sensitive to the values specified for the fractions retained in ice when cloud water freezes. To reproduce the observations, we must assume zero ice retention for formaldehyde (CH 2 O) and hydrogen peroxide (H 2 O 2 ) and complete retention for methyl hydrogen peroxide (CH 3 OOH) and sulfur dioxide (SO 2 ), likely to compensate for the lack of aqueous chemistry in the model. We then compare scavenging efficiencies among storms that formed in Alabama and northeast Colorado and the Oklahoma storm. Significant differences in SEs are seen among storms and species. More scavenging of HNO 3 and less removal of CH 3 OOH are seen in storms with higher maximum flash rates, an indication of more graupel mass. Graupel is associated with mixed-phase scavenging and lightning production of nitrogen oxides (NOx ), processes that may explain the observed differences in HNO 3 and CH 3 OOH scavenging.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Quantifying modeling uncertainty in simplified beam models for building response prediction

The use of simple models for response prediction of building structures is preferred in earthquake engineering for risk evaluations at regional scales, as they make computational studies more feasible. The primary impediment in their gainful use presently is the lack of viable methods for quantifying (and reducing upon) the modeling errors/uncertainties they bear. This study presents a Bayesian calibration method wherein the modeling error is embedded into the parameters of the model. Here, the method is specifically described for coupled shear-flexural beam models here, but it can be applied to any parametric surrogate model. The major benefit the method offers is the ability to consider the modeling uncertainty in the forward prediction of any degree-of-freedom or composite response regardless of the data used in calibration. The method is extensively verified using two synthetic examples. In the first example, the beam model is calibrated to represent a similar beam model but with enforced modeling errors. In the second example, the beam model is used to represent the detailed finite element model of a 52-story building. Both examples show the capability of the proposed solution to provide realistic uncertainty estimation around the mean prediction.

47 OTHER INSTRUMENTATION↗
DOE OSTI2023

UniKP: a unified framework for the prediction of enzyme kinetic parameters

Prediction of enzyme kinetic parameters is essential for designing and optimizing enzymes for various biotechnological and industrial applications, but the limited performance of current prediction tools on diverse tasks hinders their practical applications. Here, we introduce UniKP, a unified framework based on pretrained language models for the prediction of enzyme kinetic parameters, including enzyme turnover number (k cat ), Michaelis constant (K m ), and catalytic efficiency (k cat / K m ), from protein sequences and substrate structures. A two-layer framework derived from UniKP (EF-UniKP) has also been proposed to allow robust k cat prediction in considering environmental factors, including pH and temperature. In addition, four representative re-weighting methods are systematically explored to successfully reduce the prediction error in high-value prediction tasks. We have demonstrated the application of UniKP and EF-UniKP in several enzyme discovery and directed evolution tasks, leading to the identification of new enzymes and enzyme mutants with higher activity. UniKP is a valuable tool for deciphering the mechanisms of enzyme kinetics and enables novel insights into enzyme engineering and their industrial applications.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2019

Gaussian beam scintillation index for slant path in weak turbulence: closed form expressions

The scintillation index (SI) is an important issue in studying the effect of the atmospheric turbulence in the light waves. With the Rytov weak fluctuation theory, the SI is commonly referred to the Rytov parameter. The previous studies dealt with the plane wave and spherical wave models. Each of these gives an idea of how an optical beam will react to turbulence but are not realistic. One is infinitely long, while the other is an infinitesimal point source. The Gaussian beam wave is the most realistic of the optical waves, but also the most complex to use for calculations. In this paper, a closed form expression for Gaussian beam SI propagation in slant path and through weak fluctuation is derived using the polynomial model of the refractive index structure parameter. The slant path SI model of the plane and spherical waves can be generated from the SI expression of Gaussian beam. The analytic expressions are obtained and are then used to analyze the effect of propagation distance, transmitter/receiver heights, and elevation angle on the SI. Signal to noise ratio and bit error rate are calculated for performance evaluation showing a significant enhancement at elevation angles greater than 45°.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

A machine learning approach to thermal conductivity modeling: A case study on irradiated uranium-molybdenum nuclear fuels

A deep neural network was developed for the purpose of predicting thermal conductivity with a case study performed on irradiated nuclear fuel. Traditional thermal conductivity modeling approaches rely on existing theoretical frameworks that describe known, relevant phenomena that govern the microstructural evolution processes during neutron irradiation (such as recrystallization, and pore size, distribution and morphology). Current empirical modeling approaches, however, do not represent all irradiation test data well. We develop a machine learning approach to thermal conductivity modeling that does not require a priori knowledge of a specific material microstructure and system of interest. Our approach allows re-searchers to probe dependency of thermal conductivity on a variety of reactor operating and material conditions. The purpose of building such a model is to allow for improved predictive capabilities linking structure-property-processing-performance relationships in the system of interest (here, irradiated nuclear fuel), which could lead to improved experimental test planning and characterization. The uranium-molybdenum system is the fuel system studied in this work, and historic irradiation test data is leveraged for model development. Our model achieved a mean absolute percent error of approximately 10%. Results indicate our model generalizes well to never before seen data, and thus use of deep learning methods for material property predictions from limited, historic irradiation data is a viable approach. This work challenges the current paradigm in materials science, where material property models are based on limited experimental data fitted to low-dimensionality phenomenological models, or by simulations in which fundamental equations are explicitly solved. The work presented here aims to demonstrate the potential and limitations of machine learning in the field of materials science and material property modeling.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2024

Multi-Fidelity Scheme for Accelerating 4D Finite Element Analysis for Mircoreactors

Next-generation microreactors are currently being designed to be operated terrestrial and extraterrestrial for remote surface power production. These systems will provide an alternative source of carbon-free energy that is versatile and can be utilized for various applications. These applications of microreactors have prompted the usage of high-fidelity unstructured finite element (FE) based approaches to provide time-dependent solutions for multiple physics fields. Computing these high-fidelity (4-D) solutions for multiple design iterations, physics fields, and transient events requires an immense computational cost. These high-fidelity solutions are computational expensive and the cost can be reduced through the implementation of less accurate low-fidelity solutions. Unlike the current fleet of commercial nuclear reactors, these next-gen systems present challenges due to the material and physical limitations required. Due to these constraints, the brute force technique of parameterizing important system characteristics determines whether a design meets the project objective. A system such as a nuclear reactor could have thousands of design parameters that affect system performance. In order to analyze the entire parameter space of such a complex system would require millions of CPU hours and countless design iterations. This costly approach is not practical due to regulatory and budget limitations. In this proposal, an approach that utilizes hybrid high-fidelity and low-fidelity FE models to reduce the computational cost of evaluating these applications in 4-D will be presented. The accuracy of the high-fidelity model and the computational efficiency of the low-fidelity model are taken advantage of to produce a solution that closely resembles the full order high-fidelity solution. By utilizing an unconverged coarse FE mesh, operation limits such as temperature, structural loading, etc., can be evaluated in an accelerated fashion and then can be spatially interpolated onto a finer mesh. The resulting error arising from the coarse mesh can be mitigated with a discrepancy function that actively quantifies and corrects the error in the coarse solution. This discrepancy function can be periodically updated with high-fidelity calculations across the temporal domain thus requiring less iterations on the fine mesh. As a result, the computational cost can be reduced for the evaluation and design iteration of microreactors. The proposal for this research contains three sections: Section 2 provides a literature review, Section 3 outlines the methodology for the proposed multi-fidelity scheme, and Section 4 displays preliminary results of the proposed research.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2026

Probabilistic Inference of Low-Surface-Brightness Galaxy Morphological Parameters Using Simulation-Based Inference

Low-surface-brightness galaxies (LSBGs) are diffuse, often dark-matter-dominated systems whose faintness makes their structural parameters difficult to measure reliably in wide-field imaging surveys. Robust parameter inference, including uncertainty quantification, is important for population studies and for comparisons with models of galaxy formation, as future surveys are expected to produce increasingly large samples of diffuse galaxies. In practice, LSBG profile modeling is sensitive to sky- background errors, masking choices, contaminating background sources, and the computational cost of obtaining posterior-level uncertainties for large samples. Motivated by these questions, we develop a simulation-based inference (SBI) framework for estimating posterior distributions of LSBG morphological parameters from simulated galaxy images. Using PyImfit, we generate DES-like single-Sersic profile LSBG images with known position angle, ellipticity, Sersic index, effective surface brightness, and effective radius. We then train a normalizing-flow-based neural posterior estimator using the sbi package to infer these parameters from the simulated images. For isolated simulated galaxies, the SBI posterior recovers the true input parameters, produces posterior predictive residuals consistent with the assumed noise model, and shows good empirical calibration in a DES-motivated test regime. We also compare SBI with PyImfit-based MCMC inference and find broadly comparable posterior constraints, while SBI enables substantially faster posterior sampling after training. Finally, we test robustness to compact background contaminants. A model trained only on isolated galaxies produces undercovered posteriors on contaminated images, whereas training on simulations with variable contaminant positions and fluxes improves calibration across contaminated test sets. These results demonstrate the promise of SBI for scalable, uncertainty-aware LSBG morphology inference, while emphasizing that posterior reliability strongly depends on whether training simulations include relevant observational complications.

Batbayar, Bilguun [U. Chicago (main)]↗
DOE OSTI2020

Validation of Hygrothermal Simulations with Wall Performance Experiments in an Environmental Chamber

Oak Ridge National Laboratory is developing a web tool, built on a rule-based expert system, that aids stakeholders in designing energy-efficient moisture-durable walls. Currently, the tool’s expert system database is mostly populated with expert opinion, but work is being done to incorporate durability assessment based on stochastic hygrothermal modeling. Chamber experiments have been conducted to validate the hygrothermal models. The measured temperature and humidity have been compared with those predicted by the hygrothermal model. Experiments and comparison to one-dimensional hygrothermal modeling results were completed with two walls, a Structural Insulated Panel (SIP) based wall and a Concrete Masonry Unit (CMU) based wall. The two walls were succumbed to typical Chicago weather in ORNL’s Heat, Air and Moisture chamber. The walls were sequentially exposed to different scenarios, including diffusion, wetting, air leakage and solar radiation. For the most part, the hygrothermal simulations can be successfully used to predict the performance of these wall systems for the moisture transport phenomenon. Errors between measured and simulated values decreased as measurements got closer to the interior side of the wall. The root mean square error was larger for relative humidity (up to 17.5%-RH for CMU) than for temperature (up to 4.5°C for CMU wall). The errors were larger for the CMU wall than the wood frame wall. The phenomenon, including liquid water, caused large discrepancies between measurement and simulation results, and simulated results showed slower drying of materials than measured results. The one-dimensional nature of the simulation model made simulating air leaks difficult but not impossible.

Salonvaara, Mikael↗
DOE OSTI2022

Prediction of grain structure after thermomechanical processing of U-10Mo alloy using sensitivity analysis and machine learning surrogate model

Abstract Hot rolling and annealing are critical intermediate steps for controlling microstructures and thickness variations when fabricating uranium alloyed with 10% molybdenum (U-10Mo), which is highly relevant to worldwide nuclear non-proliferation efforts. This work proposes a machine-learning surrogate model combined with sensitivity analysis to identify and predict U-10Mo microstructure development during thermomechanical processing. Over 200 simulations were collected using physics-based microstructure models covering a wide range of thermomechanical processing routes and initial alloy grain features. Based on the sensitivity analysis, we determined that an increase in rolling reduction percentage at each processing pass has the strongest effect in reducing the grain size. Multi-pass rolling and annealing can significantly improve recrystallization regardless of the reduction percentage. With a volume fraction below 2%, uranium carbide particles were found to have marginal effects on the average grain size and distribution. The proposed stratified stacking ensemble surrogate predicts the U-10Mo grain size with a mean square error four times smaller than a standard single deep neural network. At the same time, with a significant speedup (1000×) compared to the physics-based model, the machine learning surrogate shows good potential for U-10Mo fabrication process optimization.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Methods to determine the radiated power in SPI-mitigated disruptions in JET

Here we present techniques for evaluating the radiated power in JET disruptions. Disrupting plasmas are shown to have non-axisymmetric radiation profiles, motivating the re-evaluation of the standard techniques for calculating the total radiated power at JET using bolometry. Four single-channel bolometers at different toroidal locations are exploited to quantify the radiation asymmetry. Toroidal radiation peaking factors integrated over the entire disruption of up to 1.5 have been observed when varying the quantity of neon in pellets used in disruptions mitigated by shattered pellet injection. Using synthetic bolometer diagnostics developed with the Cherab spectroscopy modeling framework, we can estimate the systematic error on total power calculations for relevant radiation profiles and improve estimates of the total radiated power. We show that the component of the systematic error on the total power due to the poloidal radiation profile can be reduced from 70% to 10% with suitable assumptions about the structure of the poloidal profile.

47 OTHER INSTRUMENTATION↗
DOE OSTI2019

A new stabilizing regime of tearing mode entrainment in the presence of a static error field

Uncorrected static error fields in axisymmetric fusion devices are one of few remaining serious obstacles for advancing the present tokamak-based approach to practical reactors. The disruption due to magnetohydrodynamic tearing mode locking is one of them. Recently, a hypothesis has been proposed that there may exist a self-healing stabilized regime in which the static resonant error field is effectively shielded by forcing the mode slowly to rotate inductively by the applied non-axisymmetric field, that has been developed based on non-linear, resistive, reduced MHD simulation with cylindrical single helicity model. Proof-of-principle experiments in the DIII-D device showed that the magnetic mode structure on the plasma surface is qualitatively consistent with the simulation prediction. However, radial mode profiles revealed qualitatively different behavior. This finding led to a revised hypothesis that in actual non-circular toroidal devices, a tearing layer in forced rotation induces the shielding-out process at other rational surfaces when we take into account of multi-resonant Fourier components. The time evolution of the radial penetration was supportive of this hypothesis.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2026

Nonlinear saturation of ballooning modes in stellarators

Ballooning mode saturation is investigated in realistic stellarator configurations using the flux tube approach of Ham et al (2018 Plasma Phys. Control. Fusion 60 075017), Ham et al (2016 Phys. Rev. Lett. 116 235001). The method is adapted to account for the lack of exact force balance in stellarator equilibrium solvers that assume existence of nested flux surfaces. A variational approach for calculating flux tube energy is developed to overcome this force error problem in stellarator numerical equilibria. Saturated (equilibrium) flux tube states that cross 10%–20% of the plasma minor radius are shown to exist for linearly ballooning unstable profiles. It is shown that several features of the displaced flux tube structure in a full nonlinear MHD simulation of Wendelstein 7X are reproduced by our model. Saturated states are found in a compact stellarator equilibrium close but below the marginal ballooning linear instability, i.e. the unperturbed equilibrium is metastable. This suggests that edge-localized-mode-like explosive MHD behavior may be possible in stellarators.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2019

Level-set Modeling Simulations of Chemical Vapor Infiltration for Ceramic Matrix Composites Manufacturing

Silicon-carbide (SiC) reinforced ceramic matrix composites (CMCs) are a key enabling technology to reduce fuel consumption and emissions of gas turbine engines. In one manufacturing approach, chemical vapor infiltration (CVI) is limited to only coating SiC fibers. The preform is then fabricated using a lay-up of basic plys or 2-D woven sheets composed of the precoated fibers. At the other extreme, CVI is used to completely densify a 3-D woven preform shaped almost like the gas turbine component itself. The latter approach is more suitable for highly engineered components which sit directly in the gas path of the engine, for example, a high pressure turbine blade. In this case, the geometry is necessarily complex for aerodynamic, stress, and lifing (multi-physics) requirements. Presently, optimizing the CVI-dominated manufacturing approach is largely by trial-and-error. In this work, a first-principles modeling of CVI is performed to realize optimization of SiC/SiC CMC manufacturing. The modeling is based on a level-set framework to describe the interface between the vapor and solid phases. A finite-difference numerical scheme using an immersed boundary method is developed for fixed, structured meshes. Massively parallel direct numerical simulations (DNS) of CVI through fiber-woven geometries are performed using one-step chemistry, and over a range of Thiele moduli. Illustrative applications of the resulting large DNS data sets are given, including the development of fiber-weave specific infiltration models and structure functions for mean-field (porous media) Computational Fluid Dynamics (CFD) simulations of CVI.

36 MATERIALS SCIENCE↗
DOE OSTI2016

Angle-dependent strong-field molecular ionization rates with tuned range-separated time-dependent density functional theory

Strong-field ionization and the resulting electronic dynamics are important for a range of processes such as high harmonic generation, photodamage, charge resonance enhanced ionization, and ionization-triggered charge migration. Modeling ionization dynamics in molecular systems from first-principles can be challenging due to the large spatial extent of the wavefunction which stresses the accuracy of basis sets, and the intense fields which require non-perturbative time-dependent electronic structure methods. In this paper, we develop a time-dependent density functional theory approach which uses a Gaussian-type orbital (GTO) basis set to capture strong-field ionization rates and dynamics in atoms and small molecules. This involves propagating the electronic density matrix in time with a time-dependent laser potential and a spatial non-Hermitian complex absorbing potential which is projected onto an atom-centered basis set to remove ionized charge from the simulation. For the density functional theory (DFT) functional we use a tuned range-separated functional LC-PBE*, which has the correct asymptotic 1/r form of the potential and a reduced delocalization error compared to traditional DFT functionals. Ionization rates are computed for hydrogen, molecular nitrogen, and iodoacetylene under various field frequencies, intensities, and polarizations (angle-dependent ionization), and the results are shown to quantitatively agree with time-dependent Schrödinger equation and strong-field approximation calculations. This tuned DFT with GTO method opens the door to predictive all-electron time-dependent density functional theory simulations of ionization and ionization-triggered dynamics in molecular systems using tuned range-separated hybrid functionals.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Call for Participation: Collaborative Benchmarking of Functional-Structural Root Architecture Models. The Case of Root Water Uptake

Three-dimensional models of root growth, architecture and function are becoming important tools that aid the design of agricultural management schemes and the selection of beneficial root traits. However, while benchmarking is common in many disciplines that use numerical models, such as natural and engineering sciences, functional-structural root architecture models have never been systematically compared. The following reasons might induce disagreement between the simulation results of different models: different representation of root growth, sink term of root water and solute uptake and representation of the rhizosphere. Presently, the extent of discrepancies is unknown, and a framework for quantitatively comparing functional-structural root architecture models is required. We propose, in a first step, to define benchmarking scenarios that test individual components of complex models: root architecture, water flow in soil and water flow in roots. While the latter two will focus mainly on comparing numerical aspects, the root architectural models have to be compared at a conceptual level as they generally differ in process representation. Therefore, defining common inputs that allow recreating reference root systems in all models will be a key challenge. In a second step, benchmarking scenarios for the coupled problems are defined. We expect that the results of step 1 will enable us to better interpret differences found in step 2. This benchmarking will result in a better understanding of the different models and contribute toward improving them. Improved models will allow us to simulate various scenarios with greater confidence and avoid bugs, numerical errors or conceptual misunderstandings. This work will set a standard for future model development.

Plant Sciences↗
DOE OSTI2023

Data-driven Whitney forms for structure-preserving control volume analysis

Control volume analysis models physics via the exchange of generalized fluxes between subdomains. Here, we introduce a scientific machine learning framework adopting a partition of unity architecture to identify physically-relevant control volumes, with generalized fluxes between subdomains encoded via Whitney forms. The approach provides a differentiable parameterization of geometry which may be trained in an end-to-end fashion to extract reduced models from full field data while exactly preserving physics. The architecture admits a data-driven finite element exterior calculus allowing discovery of mixed finite element spaces with closed form quadrature rules. An equivalence between Whitney forms and graph networks reveals that the geometric problem of control volume learning is equivalent to an unsupervised graph discovery problem. The framework is developed for manifolds in arbitrary dimension, with examples provided for H(div) problems in $\mathbb{R}$ establishing convergence and structure preservation properties. Finally, we consider a lithium-ion battery problem where we discover a reduced finite element space encoding transport pathways from high-fidelity microstructure resolved simulations. The approach reduces the 5.89M finite element simulation to 136 elements while reproducing pressure to under 0.1% error and preserving conservation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

The Quest for the Inflationary Spectral Runnings in the Presence of Systematic Errors

Cosmological inflation suggests that the scalar spectral index "runs" with scale. Constraints on the values of the spectral runnings, ${\alpha }_{s}\equiv {{dn}}_{s}/d\,\mathrm{ln}\,k$ and ${\beta }_{s}\equiv d{\alpha }_{s}/d\,\mathrm{ln}\,k$, therefore provide a fundamental test of the physics of inflation. In this work, we review the feasibility of measuring the runnings when information from upcoming large-volume galaxy surveys is used to supplement the information provided by a cosmic microwave background (CMB)-S4 experiment, particularly focusing on the effect of including high-k, nonlinear scales. Since these measurements will be sensitive to modeling uncertainties for the nonlinear power spectrum, we examine how three different ways of parameterizing those systematics—introducing zero, two, or several hundred nuisance parameters—affect constraints and protect against parameter biases. Considering statistical errors alone, we find that including strongly nonlinear scales can substantially tighten constraints. However, these constraints weaken to levels not much better than those from a CMB-S4 experiment alone when we limit our analysis to scales where estimates are not strongly affected by systematic biases. Given these considerations, near-future large-scale structure (LSS) surveys are unlikely to add much information to the CMB-S4 measurement of the first running α s . There is more potential for improvement for the second running, β s , for which LSS information will allow constraints to be improved by a factor of 3–4 relative to using the CMB alone. Though these limitations are still above the value predicted by slow-roll inflation, they do probe regions of parameter space relevant to nonstandard inflationary models with large runnings, for example, those that can generate an appreciable abundance of primordial black holes.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Review of analytical studies on seismic fluid-structure interaction of base-supported cylindrical tanks

Analytical solutions for seismic fluid-structure interaction (FSI) of tanks are used by analysts and engineers for a number of purposes, including preliminary sizing, risk assessment, and verification of numerical models. Analytical solutions for seismic FSI of tanks were first developed in the 1930s and have been extended and/or modified in the following nine decades to accommodate different seismic inputs and boundary conditions. This report reviews published analytical solutions for rigid and flexible, base-supported, cylindrical tanks subjected to unidirectional horizontal motion of a small amplitude. The studies parsed the fluid-structure (FSI) response into an impulsive and a convective component, and addressed frequencies of lateral motions of the tank (i.e., impulsive frequencies), hydrodynamic pressures, frequencies of waves (i.e., convective frequencies), wave heights, and reactions at the base. The solutions were categorized into exact and approximate solutions based on the methodologies used. Herein, FSI responses are calculated and compared for a range of tank dimensions using the analytical solutions from the different studies. The responses presented here are normalized to be unitless, and can be used for tanks with different dimensions and mechanical properties, and subjected to different input motions. A sample steel water tank is analyzed using the normalized solutions for two earthquake ground motions with different frequency contents. Ignoring tank flexibility may significantly underestimate seismic responses of the tank and its contained fluid. The approximate solutions, which have been widely used for decades, may not be sufficiently accurate in the modern era. Calculation errors in the reviewed studies are identified and corrected, and differences are quantified for the sample tank.

42 ENGINEERING↗