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NASA NTRS1985

Piezoviscous effects in nonconformal contacts lubricated hydrodynamically

The analysis is concerned with the piezoviscous-rigid regime of lubrication for the general case of elliptical contacts. In this regime several formulas of the lubricant film thickness have been proposed by Hamrock and Dowson, by Dowson et al., and more recently by Houpert. However, either they do not include the load parameter W, which has a strong effect on film thickness, or they overestimate the film thickness by using the Barus formula for pressure-viscosity characteristics. The Roelands formula was used for the pressure-viscosity relationship. The effects of the dimensionless load, speed, and materials parameters, the radius ratio, and the lubricant entrainment direction were investigated. The dimensionless load parameter was varied over a range of one order of magnitude. The dimensionless speed parameter was varied by 5.6 times the lowest value. Conditions corresponding to the use of solid materials of steel, bronze, and silicon nitride and lubricants of paraffinic and naphthenic mineral oil were considered in obtaining the exponent in the dimensionless materials parameter. The radius ratio was varied from 0.2 to 64 (a configuration approaching a line contact). Forty-one cases were used in obtaining a minimum film thickness formula. Contour plots indicate in detail the pressure developed between the contacting solids.

Jeng, Y. R.↗

NASA NTRS1987

Piezoviscous effects in nonconformal contacts lubricated hydrodynamically

Jeng, Yeau-Ren↗

NASA NTRS1992

Experimental verification of scaling parameters for thermal stratification

A study to determine important dimensionless parameters associated with thermal stratification of a heated container of liquid, and the pressure history of its vapor, was performed. Analysis of the governing equations identified the modified Grashoff number, the modified Fourier number, and the interface number as the dominant dimensionless parameters associated with this process. A test program designed to verify the validity of these scaling parameters was executed. The quality of scaling was checked by comparing the dimensionless pressures and temperatures for scaled pairs of tests. Results indicate that the bulk liquid temperature, the surface temperature of the liquid, and the tank pressure can be scaled with the three dimensionless parameters. Some deviations were found in the detailed temperature profiles between the scaled pairs of tests.

Ji, H.-C.↗

DOE OSTI2023

Isotope mass scaling and transport comparison between JET Deuterium and Tritium L-mode plasmas

The dimensionless isotope mass scaling experiment between pure Deuterium and pure Tritium plasmas with matched $\rho^*$, $\nu^*$ , β n , q and $T_e/T_i$ has been achieved in JET L-mode with dominant electron heating (NBI+ohmic) conditions. 28% higher scaled energy confinement time $B_t\tau_{E,\mathrm{th}}/A$ is found in favour of the Tritium plasma. This can be cast in the form of the dimensionless energy confinement scaling law as $\Omega_i \tau_{E,\mathrm{th}} \sim A^{0.48 \pm 0.16}$. This significant isotope mass scaling is consequently seen in the scaled one-fluid heat diffusion coefficient $A \chi_{\mathrm{eff}}/B_t$ which is around 50% lower in the Tritium plasma throughout the whole plasma radius. The isotope mass dependence in the particle transport channel is negligible, supported also by the perturbative particle transport analysis with gas puff modulation. The comparison of the edge particle fuelling or ionisation profiles from the EDGE2D-EIRENE simulations show that the absolute density differences that are necessary for the dimensionless match in the confined plasma dominate over any isotope mass dependencies of particle fuelling and ionization profiles at the plasma edge. Local GENE simulation results indicate a mild anti-gyroBohm effect at $\rho_\mathrm{tor}$ = 0.6 and thereby a small isotope mass dependence in favour of Tritium on heat transport and a negligible effect on particle transport. A significant fraction of the isotope scaling and reduced heat transport observed in the Tritium plasma is not captured in the GENE and ASTRA-TGLF-SAT2 simulations by simply changing the isotope mass for the same input profiles.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2024

Correction to “Frank–Kasper Phases of Diblock Copolymer Melts Studied with the DPD Model: SCF Results”

In a recent study, (1) we performed the self-consistent field (SCF) calculations of a dissipative particle dynamics (DPD) model of conformationally asymmetric diblock copolymer A-B melts to study the stability of various Frank–Kasper (FK) phases formed by such A-B melts. While all the equations in ref (1) are correct, we did not calculate the dimensionless (mean-field) Helmholtz free energies per chain βf c = βu c,χ + βu c,κ – s c /k B and the dimensionless stresses dβf c /dθ of ordered phases accurately. For example, the dimensionless (nonbonded) internal energy per chain due to the system compressibility βu c,κ = (N/2κ)∑ q βû 0 (|q|)(ϕ̂ A (q) + ϕ̂ B (q) – δ q,0 )(ϕ̂ A (−q) + ϕ̂ B (−q) – δ– q,0 ) should be calculated in the reciprocal space as given (note that βû 0 (|q|) is obtained analytically, and that since ϕ A (r) is real 𝜙̂ A (−𝐪)= $\overline{𝜙̂ A}$(𝐪), i.e., the complex conjugate of ϕ̂ A (q)); this corresponds to numerically evaluating βu c,κ = (N/2κ)∫dr(ϕ A (r) + ϕ B (r) – 1)ψ(r) with ψ(r) ≡ (∫dr′βu 0 (|r–r′|)(ϕ A (r′) + ϕ B (r′) – 1)/V in the real space via the (composite) trapezoidal rule. While the trapezoidal rule is less accurate than the Romberg integration (2) in general, the opposite occurs when the integration domain is periodic; this was pointed out by Matsen in ref (3) and explained in detail in refs (4and5). With our ignorance of this exceptional convergence property of the trapezoidal rule for periodic functions, we calculated all convolutions having the form of ψ(r) in the reciprocal space via the fast Fourier transforms (6) but those having the form of βu c,κ in the real space via Romberg integration; the latter leads to inaccurate results and unfortunately incorrect phase diagrams reported in ref (1), which are corrected here.

He, Juntong↗

DOE OSTI2022

Scaling Behavior of Thermally Driven Fractures in Deep Low‐Permeability Formations: A Plane Strain Model With 1‐D Heat Conduction

Abstract Injection of cold fluids through/into deep formations may cause significant cooling, thermal stress, and possible thermal fracturing. In this study, the thermal fracturing of low‐permeability formations under one‐dimensional heat conduction was investigated using a plane strain model. Dimensionless governing equations, with dimensionless fracture length , aperture , spacing , time , and effective confining stress , were derived. Solution of single thermal fracture was derived analytically, while solution of multiple fractures with constant (or dynamic) spacing were obtained using the displacement discontinuity method (and stability analysis). For single fracture, increases nonlinearly with and then transitions to scaling law , indicating that late‐time fracture length increases linearly with the square root of cooling time. For constantly spaced fractures, deviates from the single‐fracture solution at a later for a larger , showing slower propagation under inter‐fracture stress interaction . For dynamically spaced fractures, fracture arrest induced by stress interaction was determined by the stability analysis; the fully transient solution provides evolution of dimensionless fracture length, spacing, aperture, and pattern; a similar scaling law, with , obtained shows the effect of both stress interaction and fracture arrest. The solution and scaling law provide fast predictions for all reservoir and cooling conditions using (single) model parameter . Application to a geothermal site with demonstrates that thermal fractures reach 0.67, 6.25, and 78.00 m in length, 0.49, 2.30, and 13.00 m in spacing, and 0.43, 2.09, and 12.19 mm in aperture at 1, 100, and 10,000 days.

Chen, Bin↗

DOE OSTI2021

Follow the power—pathways to steady-state tokamak reactors

Abstract Diagramming pathways of dimensionless power is a potent method for extrapolating between operating points on present-day tokamaks and future burning plasma devices. The heat transport power, current drive power, H-mode threshold power and other ‘plasma physics’ powers can be expressed in dimensionally correct (or normalized) form as Pa 3/4 , where P is the power and a is the plasma minor radius, with the relative gyroradius ( ρ * ) dependence ranging from gyro-Bohm-like for transport ( P a 3 / 4 ∝ ρ * − 3 / 2 ), Bohm-like for current drive ( P a 3 / 4 ∝ ρ * − 5 / 2 ) and worse than Bohm-like for H-mode threshold ( P a 3 / 4 ∝ ρ * − 3 ). The D–T fusion power cannot be normalized in the same fashion since it is governed by nuclear physics, but at fixed B T it scales like P a 3 / 4 ∝ ρ * − 9 / 2 . Other ‘mixed physics’ powers can be incorporated into the dimensionless power framework by holding B T fixed in the same manner. Diagramming these dimensionless powers vs ρ * shows how the pathway to a steady-state reactor can be optimized relative to various operational boundaries. Using a steady-state hybrid discharge with β N = 3.2 from DIII-D as the starting point, a multi-parameter optimization finds an attractive pathway to steady-state operation on ITER using 76 MW of current drive power (fusion gain of Q fus = 8), along with a pathway to Q fus = 20 in a JET-sized steady-state reactor with B T = 10 T.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2024

Carbon Capture through Membranes - Leveraging Multiphysics Modeling, Dimensional Analysis and Machine Learning to Scale up and Optimize Devices and Processes for Decarbonization

We study the separation performance using membrane modules through dimensional analysis (DA). We formulate the main process equations to identify relevant dimensionless numbers inherent in the physics. In particular, we identify that the critical step in the separation process is mass transfer through the selective layer. Remarkably, the dimensionless feed flow (DFfeed) emerges as a crucial factor in describing this process. Not only does DFfeed directly appear in the governing equations, but it also holds a physical significance associated with the time scales for the mass transfer across the feed side and through the selective layer. Regarding the output performance variables, we consider the recovery, stage cut, productivity and purity. In this context, we profit from experimental data and CFD simulations to evaluate the separation performance of the modules when varying the input flowrate, the scale of the module, and the CO2 permeance. These datasets enable us to establish correlations between performance metrics and the dimensionless feed flow (DFfeed). Using simple power functions of DFfeed, we obtain R2 coefficients exceeding 0.99, indicating the accuracy of the correlations built in the present work. In the future, we wish to use DA to understand key transport mechanisms, predict and control module performance, and challenge the universality of these findings by testing various gas separations across different membrane modules beyond our case study.

Pedrozo, Hector A.↗

DOE OSTI2024

Modeling and Scaling up of Membrane Modules Leveraging Dimensional Analysis

In the current analysis, we study the separation performance using membrane modules through dimensional analysis (DA). We formulate the main process equations to identify relevant dimensionless numbers inherent in the physics. In particular, we identify that the critical step in the separation process is mass transfer through the selective layer. Remarkably, the dimensionless feed flow (DFfeed) emerges as a crucial factor in describing this process. Not only does DFfeed directly appear in the governing equations, but it also holds a physical significance associated with the time scales for the mass transfer across the feed side and through the selective layer. Regarding the output performance variables, we consider the recovery, stage cut, productivity and purity. In this context, we profit from experimental data and CFD simulations to evaluate the separation performance of the modules when varying the input flowrate, the scale of the module, and the CO2 permeance. These datasets enable us to establish correlations between performance metrics and the dimensionless feed flow. Using simple power functions of DFfeed, we obtain R2 coefficients exceeding 0.99, indicating the accuracy of the correlations built in the present work. In the future, we wish to use DA to understand key transport mechanisms, predict and control module performance, and challenge the universality of these findings by testing various gas separations across different membrane modules beyond our case study.

Pedrozo, Hector A.↗

NASA NTRS1972

Optimization of self-acting step thrust bearings for load capacity and stiffness.

Linearized analysis of a finite-width rectangular step thrust bearing. Dimensionless load capacity and stiffness are expressed in terms of a Fourier cosine series. The dimensionless load capacity and stiffness were found to be a function of the dimensionless bearing number, the pad length-to-width ratio, the film thickness ratio, the step location parameter, and the feed groove parameter. The equations obtained in the analysis were verified. The assumptions imposed were substantiated by comparing the results with an existing exact solution for the infinite width bearing. A digital computer program was developed which determines optimal bearing configuration for maximum load capacity or stiffness. Simple design curves are presented. Results are shown for both compressible and incompressible lubrication. Through a parameter transformation the results are directly usable in designing optimal step sector thrust bearings.

Hamrock, B. J.↗

NASA NTRS1974

Turbomachinery design described by similarity considerations

Expressions of operating conditions related to the design of turbomachinery are examined. The dimensionless field of turbomachinery design is discussed. The relationships of dimensionless design parameters such as diameter ratios, flow coefficients, head coefficients, and design choices regarding the fluid and the machine are explained. The core of the design process is said to relate such sets of design parameters to the dimensionless form of the machine. Mathematical models are provided to support the theoretical considerations.

Wislicenus, G. F.↗

NASA NTRS1975

Development of thermal stratification and destratification scaling concepts. Volume 1: Definition of thermal stratification scaling parameters and experimental investigations

The dimensionless parameters associated with the thermal stratification and pressure history of a heated container of liquid and its vapor were examined. The Modified Grashof number, the Fourier number, and an Interface number were parameterized using a single test liquid, Freon 113. Cylindrical test tanks with spherical dome end caps were built. Blanket heaters covered the tanks and thermocouples monitored the temperatures of the liquid, the ullage, the tank walls, and the foam insulation encapsulating the tank. A centrifuge was used for the 6 inch tank to preserve the same scaling parameter values between it and the larger tanks. Tests were conducted over a range of Gr* values and the degree of scaling was checked by comparing the dimensionless pressures and temperatures for each scaled pair of tests. Results indicate that the bulk liquid temperature, the surface temperature of the liquid, and the tank pressure can be scaled with the three dimensionless parameters. Some deviation was, however, found in the detailed temperature profiles between the scaled pairs of tests.

Lovrich, T. N.↗

NASA NTRS1975

Thermodynamic effects on developed cavitation

The results of an investigation of thermodynamic effects are presented. Distributions of temperature and pressure in a developed cavity were measured for zero- and quarter-caliber ogives. A semiempirical entrainment theory was developed to correlate the measured temperature depression in the cavity. This theory correlates the maximum temperature depression expressed in dimensionless form as the Jakob number in terms of the dimensionless numbers of Nusselt, Reynolds, Froude, and Peclet, and dimensionless cavity length, L/D. The results show that in general, the temperature depression increases with L/D and temperature and the cavitation number based on measured cavity pressure is a function of L/D for a given model contour, independent of the thermodynamic effect.

Holl, J. W.↗

NASA NTRS1982

Theoretical results for fully flooded, elliptical hydrodynamic contacts

The influence of the ellipticity parameter and the dimensionless speed, load, and materials parameters on minimum film thickness was investigated. The ellipticity parameter was varied from 1 (a ball-on-plate configuration) to 8 (a configuration approaching a line contact). The dimensionless speed parameter was varied over a range of nearly two orders of magnitude. Conditions corresponding to the use of solid materials of bronze, steel, and silicon nitride and lubricants of praffinic and naphthemic mineral oils were considered in obtaining the exponent in the dimensionless materials parameter. Thirty-four different cases were used in obtaining the minimum film thickness formula H min = 3.63U to the 0.68 power G to the 0.49 power W to the -0.073 power 1-e to the 0.68K power). A simplified expression for the ellipticity parameter was found where k = 1.03 (r(y)/r(x)) to the 0.64 power. Contour plots were also shown which indicate in detail the pressure spike and two side lobes in which the minimum film thickness occurs. These theoretical solutions of film thickness have all the essential features of the previously reported experimental observations based upon optical interferometry.

Hamrock, B. J.↗

NASA NTRS1983

Correlating downward flame spread rates for thick fuel beds

Flame spread rates are presented for polymethylmethacrylate fuel beds as a function of gravitational acceleration and ambient pressure and oxygen concentration. The data are correlated by plotting a dimensionless spread rate that is a measure of the heat transferred forward of the flame, required to sustain the flame, compared to the maximum possible forward heat transfer against a Damkoehler number. The latter parameter indicates the approach of the flame to its maximum temperature, where the maximum possible forward heat transfer occurs, such that the dimensionless spread rate approaches unity at large Damkoehler number. Surface regression effects on the flame spread process are incorporated into the dimensionless spread rate, and the final correlation is independent of bed thickness.

Altenkirch, R. A.↗

NASA NTRS1990

Superfluid transport and its applications in space

Various transport modes in superfluid helium are discussed. They include zero net mass flow and finite mass flow for pure superfluid and normal fluid flow. An attempt is made to characterize these transport modes in a common frame of reference. Two dimensionless numbers are used, namely, the dimensionless heat flux number and the dimensionless driving force number. The equations are generalized by the use of a characteristic length so that they can be applied to the transport of HE II in any geometry. The theories are then extended to applications in space. In particular, fountain effect pumps and superfluid management at zero g by vapor-liquid phase separators are discussed in detail.

Yuan, S. W. K.↗

NASA NTRS1995

Consequences of the large ambipolar electric field in the solar wind

The parallel electric field in the solar wind is much smaller than the V x B motional electric field, yet in the proper dimensions units it is very 'large'. At the orbit of earth it is within a few percent of being at the Dreicer limit. This 'large' electric field is required for quasi-neutrality; it will be shown to have interesting consequences for the electron velocity distribution function and the description of transport of heat. Interestingly, a similar dimensionless situation also occurs at the base of the transition region, while below the transition region the dimensionless electric field is very weak. These facts suggest a new way to look at the thermal-suprathermal dichotomy in velocity distributions as the response of a plasma where charge neutrality requires such large dimensionless electric fields.

Scudder, J. D.↗

NASA NTRS1996

Determination of Thermal State of Charge in Solar Heat Receivers

The research project at Howard University seeks to develop analytical and numerical capabilities to study heat transfer and fluid flow characteristics, and the prediction of the performance of solar heat receivers for space applications. Specifically, the study seeks to elucidate the effects of internal and external thermal radiation, geometrical and applicable dimensionless parameters on the overall heat transfer in space solar heat receivers. Over the last year, a procedure for the characterization of the state-of-charge (SOC) in solar heat receivers for space applications has been developed. By identifying the various factors that affect the SOC, a dimensional analysis is performed resulting in a number of dimensionless groups of parameters. Although not accomplished during the first phase of the research, data generated from a thermal simulation program can be used to determine values of the dimensionless parameters and the state-of-charge and thereby obtain a correlation for the SOC. The simulation program selected for the purpose is HOTTube, a thermal numerical computer code based on a transient time-explicit, axisymmetric model of the total solar heat receiver. Simulation results obtained with the computer program are presented the minimum and maximum insolation orbits. In the absence of any validation of the code with experimental data, results from HOTTube appear reasonable qualitatively in representing the physical situations modeled.

Glakpe, E. K.↗