Search NASA⌕ Search

SEARCH · Search NASA

Results for “wake mixing”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 217 records · Page 12
All sourcesNASA NTRSDOE OSTI
NASA NTRS1966

Wind-tunnel boundary interference for V/STOL TESTING

The wake skew angle used in applying the theory of NASA TR R-124 to data correction should be such that the angular deflection of the wake vorticity from the horizontal is one-half that calculated from momentum theory at the lifting element. This usage is in contrast to that of the original paper which used the angle of the mass flow. Because of large-scale recirculation effects, there is a finite lower limit to the test speed at which reliable and correctable data can be obtained in closed wind tunnels. Although a zero-correction wind tunnel for V/STOL testing has not yet been achieved, it is shown that the use of suitably mixed wind-tunnel boundaries can alleviate boundary effects on V/STOL data.

V/STOL AIRCRAFT↗
NASA NTRS2001

Effects of afternoon "siesta" naps on sleep, alertness, performance, and circadian rhythms in the elderly

STUDY OBJECTIVES: To determine the effects of a 90-minute afternoon nap regimen on nocturnal sleep, circadian rhythms, and evening alertness and performance levels in the healthy elderly. DESIGN AND SETTING: Nine healthy elderly subjects (4m, 5f, age range 74y-87y) each experienced both nap and no-nap conditions in two studies each lasting 17 days (14 at home, 3 in the laboratory). In the nap condition a 90-minute nap was enforced between 13:30 and 15:00 every day, in the no-nap condition daytime napping was prohibited, and activity encouraged in the 13:30-15:00 interval. The order of the two conditions was counterbalanced. PARTICIPANTS: N/A INTERVENTIONS: N/A MEASUREMENTS: Diary measures, pencil and paper alertness tests, and wrist actigraphy were used at home. In the 72 hour laboratory studies, these measures were augmented by polysomnographic sleep recording, continuous rectal temperature measurement, a daily evening single trial of a Multiple Sleep Latency Test (MSLT), and computerized tests of mood, activation and performance efficiency. RESULTS: By the second week in the "at home" study, an average of 58 minutes of sleep was reported per siesta nap; in the laboratory, polysomnography confirmed an average of 57 minutes of sleep per nap. When nap and no-nap conditions were compared, mixed effects on nocturnal sleep were observed. Diary measures indicated no significant difference in nocturnal sleep duration, but a significant increase (of 38 mins.) in 24-hour Total Sleep Time (TST) when nocturnal sleeps and naps were added together (p<0.025). The laboratory study revealed a decrease of 2.4% in nocturnal sleep efficiency in the nap condition (p<0.025), a reduction of nocturnal Total Sleep Time (TST) by 48 mins. in the nap condition (p<0.001) which resulted primarily from significantly earlier waketimes (p<0.005), but no reliable effects on Wake After Sleep Onset (WASO), delta sleep measures, or percent stages 1 & 2. Unlike the diary study, the laboratory study yielded no overall increase in 24-hour TST consequent upon the siesta nap regimen. The only measure of evening alertness or performance to show an improvement was sleep latency in a single-trial evening MSLT (nap: 15.6 mins., no nap: 11.5 mins., p<0.005). No significant change in circadian rhythm parameters was observed. CONCLUSIONS: Healthy seniors were able to adopt a napping regimen involving a 90-minute siesta nap each day between 13:30 and 15:00, achieving about one hour of actual sleep per nap. There were some negative consequences for nocturnal sleep in terms of reduced sleep efficiency and earlier waketimes, but also some positive consequences for objective evening performance and (in the diary study) 24-hour sleep totals. Subjective alertness measures and performance measures showed no reliable effects and circadian phase parameters appeared unchanged.

Non-NASA Center↗
NASA NTRS2004

NASA Tech Briefs, March 2004

Topics covered include: 1) Advanced Signal Conditioners for Data-Acquisition Systems; 2) Downlink Data Multiplexer; 3) Viewing ISS Data in Real Time via the Internet; 4) Autonomous Environment-Monitoring Networks; 5) Readout of DSN Monitor Data; 6) Parallel-Processing Equalizers for Multi-Gbps Communications; 7) AIN-Based Packaging for SiC High-Temperature Electronics; 8) Software for Optimizing Quality Assurance of Other Software; 9) The TechSat 21 Autonomous Sciencecraft Experiment; 10) Software for Analyzing Laminar-to-Turbulent Flow Transitions; 11) Elastomer Filled With Single-Wall Carbon Nanotubes; 12) Modifying Ship Air-Wake Vortices for Aircraft Operations; 13) Strain-Gauge Measurement of Weight of Fluid in a Tank; 14) Advanced Docking System With Magnetic Initial Capture; 15) Blade-Pitch Control for Quieting Tilt-Rotor Aircraft; 16) Solar Array Panels With Dust-Removal Capability; 17) Aligning Arrays of Lenses and Single-Mode Optical Fibers; 18) Automatic Control of Arc Process for Making Carbon Nanotubes; 19) Curved-Focal-Plane Arrays Using Deformed-Membrane Photodetectors; 20) Role of Meteorology in Flights of a Solar-Powered Airplane; 21) Model of Mixing Layer With Multicomponent Evaporating Drops; 22) Solution-Assisted Optical Contacting; 23) Improved Discrete Approximation of Laplacian of Gaussian; 24) Utilizing Expert Knowledge in Estimating Future STS Costs; 25) Study of Rapid-Regression Liquefying Hybrid Rocket Fuels; and 26) More About the Phase-Synchronized Enhancement Method.

Source record↗
DOE OSTI2024

Probing the atmospheric boundary layer with integrated remote-sensing platforms during the American WAKE ExperimeNt (AWAKEN) campaign

The American WAKE ExperimeNt (AWAKEN) collaboration is an observational-based field campaign in northern Oklahoma intended to analyze the potential influence of onshore wind farms and their collective wakes on wind power production, turbine structural loads, and on the atmospheric boundary layer (ABL). Focusing on the ABL effects, the University of Oklahoma and the Lawrence Livermore National Laboratory collected continuous high-resolution kinematic and thermodynamic profile measurements during 2022 and Summer 2023. The deployment strategy for these campaigns is detailed first, followed by an initial comparison of data from two sites in the AWAKEN domain: a near-farm site to examine collective wake impacts on the ABL, and a far-field site remaining outside the wind farm-waked region. Here, we summarize the datasets available and demonstrate the benefits of these observations and multiple value-added products (VAPs) for investigation of ABL features observed during AWAKEN. We also highlight examples of preliminary analyses, including ABL height detection and nocturnal low-level jet examination, which are produced using novel VAPs based on optimal estimation to retrieve deeper Doppler lidar wind profiles than previously resolved, along with their uncertainty. By including the near-farm and far-field site in these analyses, we identified a pattern of stronger lower-atmospheric mixing at the near-farm site than the far-field site, motivating deeper investigation into the relationship between wind farms and general ABL characteristics. Future analysis will delve deeper into this relationship by examining other ABL characteristics, such as atmospheric stability and convection.

17 WIND ENERGY↗
NASA NTRS2019

Scale–Resolving Simulations of a Fundamental Trailing–Edge Cooling Slot Using a Discontinuous–Galerkin Spectral–Element Method

The accurate prediction of turbulent mixing in high-pressure turbines that incorporate various airfoil surface-cooling strategies is becoming increasing critical to the design of modern gas turbine engines where the quest for improved efficiency is driving compressor overall pressure ratios and turbine inlet temperatures to much higher levels than ever before. In the present paper, a recently developed computational capability for accurate and efficient scale–resolving simulations of turbomachinery is extended to study the turbulent mixing mechanism of a simplified abstraction of an airfoil trailing-edge cooling slot - a plane wall jet with finite lip thickness discharging into an ambient flow. The computational capability is based on an entropy– stable, discontinuous–Galerkin approach that extends to arbitrarily high orders of spatial and temporal accuracy. The numerical results show that the present simulations capture the trends observed in the experiments. Discrepancies between the simulations and experiments are believed to be due to differences in the inflow profiles and tunnel side–wall effects. The thick lip configuration leads to a thicker wake and higher unsteadiness in the wall jet compared to the thin lip. A detailed comparison of the turbulent flowfields is presented to highlight differences arising due to lip thickness variations.

Scale Resolving↗
NASA NTRS2008

Distributed Turboelectric Propulsion for Hybrid Wing Body Aircraft

Meeting future goals for aircraft and air traffic system performance will require new airframes with more highly integrated propulsion. Previous studies have evaluated hybrid wing body (HWB) configurations with various numbers of engines and with increasing degrees of propulsion-airframe integration. A recently published configuration with 12 small engines partially embedded in a HWB aircraft, reviewed herein, serves as the airframe baseline for the new concept aircraft that is the subject of this paper. To achieve high cruise efficiency, a high lift-to-drag ratio HWB was adopted as the baseline airframe along with boundary layer ingestion inlets and distributed thrust nozzles to fill in the wakes generated by the vehicle. The distributed powered-lift propulsion concept for the baseline vehicle used a simple, high-lift-capable internally blown flap or jet flap system with a number of small high bypass ratio turbofan engines in the airframe. In that concept, the engine flow path from the inlet to the nozzle is direct and does not involve complicated internal ducts through the airframe to redistribute the engine flow. In addition, partially embedded engines, distributed along the upper surface of the HWB airframe, provide noise reduction through airframe shielding and promote jet flow mixing with the ambient airflow. To improve performance and to reduce noise and environmental impact even further, a drastic change in the propulsion system is proposed in this paper. The new concept adopts the previous baseline cruise-efficient short take-off and landing (CESTOL) airframe but employs a number of superconducting motors to drive the distributed fans rather than using many small conventional engines. The power to drive these electric fans is generated by two remotely located gas-turbine-driven superconducting generators. This arrangement allows many small partially embedded fans while retaining the superior efficiency of large core engines, which are physically separated but connected through electric power lines to the fans. This paper presents a brief description of the earlier CESTOL vehicle concept and the newly proposed electrically driven fan concept vehicle, using the previous CESTOL vehicle as a baseline.

Kim, Hyun Dae↗
NASA NTRS1995

A Numerical Study of the Effect of Wake Passing on Turbine Blade Film Cooling

Time-accurate and steady three-dimensional viscous turbulent numerical simulations were performed to study the effect of upstream blade wake passing unsteadiness on the performance of film cooling on a downstream axial turbine blade. The simulations modeled the blade as spanwise periodic and of infinite span. Both aerodynamic and heat transfer quantities were explored. A showerhead film cooling arrangement typical of modern gas turbine engines was employed. Showerhead cooling was studied because of its anticipated strong sensitivity to upstream flow fluctuations. The wake was modeled as a region of zero axial velocity on the upstream computational boundary which translated with each iteration. This model is compatible with a planned companion experiment in which the wakes will be produced by a rotating row of cylindrical rods upstream of an annular turbine cascade. It was determined that a steady solution with appropriate upstream swirl and stagnation pressure predicted the span-average film effectiveness quite well. The major difference is a 2 to 3 percent overprediction of span-average film effectiveness by the steady simulation on the pressure surface and in the showerhead region. Local overpredictions of up to 8 percent were observed in the showerhead region. These differences can be explained by the periodic relative lifting of the boundary layer and enhanced mixing in the unsteady simulations.

Heidmann, James D.↗
NASA NTRS1980

Simple turbulence models and their application to boundary layer separation

Measurements in the boundary layer and wake of a stalled airfoil are presented in two coordinate systems, one aligned with the airfoil chord, the other being conventional boundary layer coordinates. The NACA 4412 airfoil is studied at a single angle of attack corresponding to maximum lift, the Reynolds number based on chord being 1.5 x 10 to the 6th power. Turbulent boundary layer separation occurred at the 85 percent chord position. The two-dimensionality of the flow was documented and the momentum integral equation studied to illustrate the importance of turbulence contributions as separation is approached. The assumptions of simple eddy-viscosity and mixing-length turbulence models are checked directly against experiment. Curvature effects are found to be important as separation is approached.

Wadcock, A. J.↗
NASA NTRS1973

Dispersion of turbojet engine exhaust in flight

The dispersion of the exhaust of turbojet engines into the atmosphere is estimated by using a model developed for the mixing of a round jet with a parallel flow. The analysis is appropriate for determining the spread and dilution of the jet exhaust from the engine exit until it is entrained in the aircraft trailing vortices. Chemical reactions are not expected to be important and are not included in the flow model. Calculations of the dispersion of the exhaust plumes of three aircraft turbojet engines with and without afterburning at typical flight conditions are presented. Calculated average concentrations for the exhaust plume from a single engine jet fighter are shown to be in good agreement with measurements made in the aircraft wake during flight.

Holdeman, J. D.↗
NASA NTRS1973

A study of ingestion and dispersion of engine exhaust products in trailing vortex systems

Analysis has been made of the ingestion and dispersion of engine exhaust products into the trailing vortex system of supersonic aircraft flying in the stratosphere. The rate of mixing between the supersonic jet and the co-flowing supersonic stream was found to be an order of magnitude less than would be expected on the basis of subsonic eddy-viscosity results. The length of the potential core was 66 nozzle exit radii so that the exhaust gases remain at elevated temperatures and concentrations over much longer distances than previsously estimated. Ingestion started at the end of the potential core and all hot gas from the engine was ingested into the trailing vortex within two core lengths. Comparison between the buoyancy calculations for the supersonic case with nondimensionalized subsonic aircraft contrail data on wake spreading showed good agreement. Velocity and temperature profiles have been specified at various stages of the wake, and the analysis in this report can be used to predict variations of concentrations of species such as nitrogen oxides under conditions of chemical reaction.

Nielsen, J. N.↗
NASA NTRS2006

Planar laser-induced fluorescence (PLIF) investigation of hypersonic flowfields in a Mach 10 wind tunnel

Planar laser-induced fluorescence (PLIF) of nitric oxide (NO) was used to visualize four different hypersonic flowfields in the NASA Langley Research Center 31-Inch Mach 10 Air wind tunnel. The four configurations were: (1) the wake flowfield of a fuselage-only X-33 lifting body, (2) flow over a flat plate containing a rectangular cavity, (3) flow over a 70deg blunted cone with a cylindrical afterbody, formerly studied by an AGARD working group, and (4) an Apollo-geometry entry capsule - relevant to the Crew Exploration Vehicle currently being developed by NASA. In all cases, NO was seeded into the flowfield through tubes inside or attached to the model sting and strut. PLIF was used to visualize the NO in the flowfield. In some cases pure NO was seeded into the flow while in other cases a 5% NO, 95% N2 mix was injected. Several parameters were varied including seeding method and location, seeding mass flow rate, model angle of attack and tunnel stagnation pressure, which varies the unit Reynolds number. The location of the laser sheet was as also varied to provide three dimensional flow information. Virtual Diagnostics Interface (ViDI) technology developed at NASA Langley was used to visualize the data sets in post processing. The measurements demonstrate some of the capabilities of the PLIF method for studying hypersonic flows.

Danehy, Paul M.↗
DOE OSTI2020

Dependence of emittance on the length of an electron bunch during laser-plasma acceleration in guiding structures

We report a theoretical analysis and numerical simulation of the dynamics of transverse emittance of an electron bunch during its acceleration in wake fields generated by a laser pulse in a weakly nonlinear mode. Analytical expressions are obtained for the main factors affecting the emittance growth during acceleration and the case is considered when the characteristic transverse size of the injected bunch exceeds the matched radius determined by the focusing force at the injection point, the initial emittance and the electron bunch energy, and the resulting value of emittance is much larger than the initial one. The dynamics of the emittance growth during acceleration as a function of the length of the electron bunch is described, and the length of the bunch is found at which there occurs a complete phase mixing of betatron oscillations of electrons and the emittance increases to its maximum value determined by the bunch parameters and the focusing force at the injection point. Analytical expressions are in good agreement with the results of numerical simulation. (paper)

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
NASA NTRS2016

Comparison of Structural and Functional Ocular Outcomes Between 14- and 70 Day Bed Rest

Purpose: To compare structural and functional ocular outcomes in healthy human subjects undergoing 14- and/or 70-day head-down-tilt bed rest (HDTBR). We hypothesized the amount of HDTBR-induced ocular changes be affected by the HDTBR duration. Methods: The studies were conducted at the NASA Flight Analogs Research Unit, The University of Texas Medical Branch at Galveston, Galveston, TX. Participants were selected using NASA standard screening procedures. Standardized NASA screening procedures and bed rest conditions (e.g., strict sleep-wake cycle, standardized diet, continuous video monitoring) were implemented in both studies. Participants maintained a 6deg HDTBR position for 14 and/or 70 consecutive days and did not engage in exercise. Weekly ophthalmological examinations were conducted in the sitting (pre/post-bed rest only) and HDT positions. Ocular outcomes of interest included: near best-corrected visual acuity (BCVA); spherical equivalent, as determined by cycloplegic autorefraction; Goldmann applanation tonometry and iCare (Icare Finland Oy, Espoo, Finland) intraocular pressure (IOP) measurement; color vision; red dot test; modified Amsler grid test; confrontational visual field; stereoscopic color fundus photography; Spectralis OCT (Heidelberg Engineering, GmbH, Heidelberg, Germany) retinal nerve fiber layer thickness (RNFLT), peripapillary and macular retinal thicknesses. Mixed-effects linear models were used to compare pre- and post-HDTBR observations between 14- and 70-day HDTBR for our continuously scaled outcomes.

Cromwell, R. L.↗
NASA NTRS1998

Modeling Macro- and Micro-Scale Turbulent Mixing and Chemistry in Engine Exhaust Plumes

Simulation of turbulent mixing and chemical processes in the near-field plume and plume-vortex regimes has been successfully carried out recently using a reduced gas phase kinetics mechanism which substantially decreased the computational cost. A detailed mechanism including gas phase HOx, NOx, and SOx chemistry between the aircraft exhaust and the ambient air in near-field aircraft plumes is compiled. A reduced mechanism capturing the major chemical pathways is developed. Predictions by the reduced mechanism are found to be in good agreement with those by the detailed mechanism. With the reduced chemistry, the computer CPU time is saved by a factor of more than 3.5 for the near-field plume modeling. Distributions of major chemical species are obtained and analyzed. The computed sensitivities of major species with respect to reaction step are deduced for identification of the dominant gas phase kinetic reaction pathways in the jet plume. Both the near field plume and the plume-vortex regimes were investigated using advanced mixing models. In the near field, a stand-alone mixing model was used to investigate the impact of turbulent mixing on the micro- and macro-scale mixing processes using a reduced reaction kinetics model. The plume-vortex regime was simulated using a large-eddy simulation model. Vortex plume behind Boeing 737 and 747 aircraft was simulated along with relevant kinetics. Many features of the computed flow field show reasonable agreement with data. The entrainment of the engine plumes into the wing tip vortices and also the partial detrainment of the plume were numerically captured. The impact of fluid mechanics on the chemical processes was also studied. Results show that there are significant differences between spatial and temporal simulations especially in the predicted SO3 concentrations. This has important implications for the prediction of sulfuric acid aerosols in the wake and may partly explain the discrepancy between past numerical studies (that employed parabolic or temporal approximations) and the measured data. Finally to address the major uncertainty in the near-field plume modeling related to the plume processing of sulfur compounds and advanced model was developed to evaluate its impact on the chemical processes in the near wake. A comprehensive aerosol model is developed and it is coupled with chemical kinetics and the axisymmetric turbulent jet flow models. The integrated model is used to simulate microphysical processes in the near-field jet plume, including sulfuric acid and water binary homogeneous nucleation, coagulation, non-equilibrium heteromolecular condensation, and sulfur-induced soot activation. The formation and evolution of aerosols are computed and analyzed. The computed results show that a large number of ultra-fine (0.3--0.6 nm in radius) volatile HSO4 - HO embryos are generated in the near-field plume. These embryos further grow in size by self coagulation and condensation. Soot particles can be activated by both heterogeneous nucleation and scavenging of H2SO4-H2O aerosols. These activated soot particles can serve as water condensation nuclei for contrail formation. Conditions under which ice contrails can form behind aircrafts are studied. The sensitivities of the threshold temperature for contrail formation with respect to aircraft propulsion efficiency, relative humidity, and ambient pressure are evaluated. The computed aerosol properties for different extent of fuel sulfur conversion to S(VI) (SO3 and H2SO4) in engine are examined and the results are found to be sensitive to this conversion fraction.

Menon, Suresh↗
NASA NTRSDate not supplied

An At-Home Evaluation of a Light Intervention to Mitigate Sleep Inertia Symptoms

Introduction: Sleep inertia symptoms typically occur after waking from nocturnal sleep. Under laboratory settings, light exposure upon waking has been shown to improve alertness, mood, and vigilant attention. We investigated whether a field-deployable light-emitting device would help to improve alertness and working memory in a real-world setting. Methods: Thirty-five participants (18 female; 26.4 ± 6.0 y) completed an at-home, within-subject, randomized crossover study. Participants wore actiwatches during their normal sleep-wake schedule for five nights ahead of the adaptation and experimental nights. On the experimental night, participants performed baseline testing before their self-selected bedtime. Forty-five minutes after bedtime, participants received a phone call and were instructed to perform test bouts while wearing light-emitting glasses with the light either on (light condition) or off (control). A 3-minute descending subtraction task (DST) and the Karolinska Sleepiness Scale (KSS) were performed at +7, +17, +27, and +37 minutes after the call. Participants were then instructed to go back to sleep and were called 45 minutes after lights out to repeat the test bouts in the opposite condition. A series of mixed-effects models were performed with fixed effects of condition, test bout, and their interaction, and a random effect of participant. Condition order, sex, and baseline were included as covariates. Results: There was a significant effect of test bout for DST total responses (χ2 [3] = 17.42; p < .001) and total correct (χ2 [3] = 21.29; p < .001) with improved performance at +27 and +37 minutes compared to +7 minutes. Sex was a significant predictor for KSS (F1,30 = 10.26; p = .003), with females (8.20 ± 0.23) rating higher sleepiness than males (7.10 ± 0.25). There were no other significant effects for DST or KSS outcomes (p > .05). Conclusion: These results suggest that the intervention was not able to improve working memory or alertness under naturalistic at-home settings. Further analysis is needed to determine whether these results are applicable to other cognitive performance domains.

light↗
NASA NTRS1957

Investigation of Semivaneless Turbine Stator Designed to Produce Axially Symmetrical Free-vortex Flow

A semivaneless turbine stator designed to eliminate blade wakes and secondary-flow accumulations of boundary-layer air was built and tested. Performance of this stator was evaluated with static pressures measured in the vaneless section and surveys of total pressure and flow angle made at the stator exit. Results are presented in terms of theoretical and experimental velocities and flow angles, boundary-layer parameters, and contours of total-pressure loss across the stator.

TURBINES - MIXED-FLOW↗
NASA NTRS1989

Airfoil profile drag increase due to acoustic excitation

A two-dimensional airfoil (NACA-0009) is subjected to high intensity pure-tone sound over a 1-5 kHz frequency range while immersed in a flow with 240 ft/sec velocity in a quiet flow facility with a Reynolds number of 3 million. Wake dynamic pressures are determined, and the momentum deficit is used to calculate a two-dimensional drag coefficient. Significant increases in drag are observed when the airfoil is subjected to high-intensity sound at critical frequencies. The increased drag is accompanied by movement of the natural transition location. When the transition is fixed by roughness at 10 percent chord, no further transition movement is observed in response to an acoustic Tollmien-Schlichting disturbance. However, a 4 percent increase in the sectional drag coefficient is noted. It is believed to be due to the sound exciting the flow near the airfoil surface (shear layer), thus causing the existing turbulence to become more intense, possess a higher mixing rate (momentum), and increase the skin friction.

Shearin, John G.↗
NASA NTRS1975

Gas sampling method for determining pollutant concentrations in the flame zone of two swirl-can combustor modules

A gas sampling probe and traversing mechanism were developed to obtain detailed measurements of gaseous pollutant concentrations in the primary and mixing regions of combustors in order to better understand how pollutants are formed. The gas sampling probe was actuated by a three-degree-of-freedom traversing mechanism and the samples obtained were analyzed by an on-line gas analysis system. The pollutants in the flame zone of two different swirl-can combustor modules were measured at an inlet-air temperature of 590 K, pressure of 6 atmospheres, and reference velocities of 23 and 30 meters per second at a fuel-air ratio of 0.02. Typical results show large spatial gradients in the gaseous pollutant concentration close to the swirl-can module. Average concentrations of unburned hydrocarbons and carbon monoxide decrease rapidly in the downstream wake regions of each module. By careful and detailed probing, the effect of various module design features on pollutant formation can be assessed. The techniques presently developed seem adequate to obtain the desired information.

Duerr, R. A.↗