Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “heat engines”

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 271 records · Page 15

Pumped Thermal Energy Storage With Liquid Storage

Pumped Thermal Energy Storage (PTES) uses electricity to power a heat pump; transferring heat from a cold space to a hot space forms a hot and a cold thermal reservoir, thereby storing energy. To discharge, the temperature difference between the two stores is used to drive a heat engine which generates electricity. In this chapter, PTES systems which store energy in liquids will be described. The chapter will concentrate on Joule-Brayton power cycles with molten salt storage. Part-load operation of Joule-Brayton PTES with liquid storage can be managed with "inventory control," and it is demonstrated that this control method leads to relatively good performance over a range of operating points.

Carnot batteries↗

Thermal energy grid storage: Liquid containment and pumping above 2000 °C

As the cost of renewable energy falls below fossil fuels, the key barrier to widespread sustainable electricity has become availability on demand. Energy storage can enable dispatchable renewables, but only with drastic cost reductions compared to current battery technologies. One electricity storage concept that could enable these cost reductions stores electricity as sensible heat in an extremely hot liquid (>2000°C) and uses multi-junction photovoltaics (MPV) as a heat engine to convert it back to electricity on demand hours, or days, later. Furthermore, this paper reports the first containment of silicon in a multipart graphite tank above 2000°C, using material grades that are affordable for energy storage at GWh scales. Low cost molded graphite with particle sizes as large as 10 μm successfully contained metallurgical grade silicon, even with as much as twothirds iron by mass for up to 10 hours and temperatures as high as 2300°C, in tanks as large as two gallons.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Resource assessment of ocean thermal energy conversion in Puerto Rico and U.S. Virgin Islands

Island communities often struggle to establish and maintain traditional electric grids and are therefore heavily reliant on costly imported fossil fuels. In the case of Puerto Rico, these challenges are enhanced by extreme weather and other natural hazards that threaten the local electricity generation and transmission infrastructure. Ocean thermal energy conversion (OTEC) could play an important role in establishing a more resilient electrical grid in the region. Here, in this study, a detailed analysis is conducted to characterize the ocean thermal resource and power potential of OTEC in Puerto Rico based on a 14-year dataset of modeled ocean temperature. The assessment considers seasonal and interannual variability in the region's thermal resource and examines the operational limitations associated with minimal thermal gradients required to run a typical OTEC heat engine. Notably, the local thermal resource is found to be sensitive to El Niño-Southern Oscillation (ENSO) climate patterns, with La Niña conditions linked to greater OTEC power availability. Seven areas of opportunity are identified based on their resource potential and proximity to existing electrical distribution lines, including two that could benefit the nearby U.S. Virgin Islands. The greatest OTEC power potential is observed to the south of the main island of Puerto Rico in the Caribbean Sea with an estimated capacity of 138 MW for a plant pumping cold water from a depth of 1,000 m, or the equivalent amount of electricity required to power 219,000 households.

OTEC↗

Black-hole powered quantum coherent amplifier

Abstract Atoms falling into a black hole (BH) through a cavity are shown to enable coherent amplification of light quanta powered by the BH-gravitational vacuum energy. This process can harness the BH energy towards useful purposes, such as propelling a spaceship trapped by the BH. The process can occur via transient amplification of a signal field by falling atoms that are partly excited by Hawking radiation reflected by an orbiting mirror. In the steady-state regime of thermally equilibrated atoms that weakly couple to the field, this amplifier constitutes a BH-powered quantum heat engine. The envisaged effects substantiate the thermodynamic approach to BH acceleration radiation.

Physics↗

Achieving high tensile strength and ductility in refractory alloys by tuning electronic structure

The energy efficiency of heat engines (gas and steam turbines) for electricity production and propulsion is determined by the Carnot cycle and scales with operating temperature. Commercial nickel- and cobalt-based superalloys melt near 1,500 °C and rapidly lose mechanical strength beyond 1,000 °C. Refractory metals melt well above 2,000 °C but have inherent manufacturability challenges that are barriers to adoption, such as high ductile-to-brittle transition temperatures. Using density functional theory-guided design, we demonstrate tailored local lattice distortions that promote phase-stable, non-equiatomic refractory concentrated solid solutions with both high ductility and strength. Here, we exemplify this for single-phase, body-centred cubic Nb 4 Ta 4 V 3 Ti that exhibits castability, excellent room-temperature tensile yield strength (∼1 GPa) and ductility (approaching 20% uniform strain), and exceptional high-temperature tensile strength (500 MPa at 1,000 °C). These findings illustrate a path for designing materials that hold great potential for advancing next-generation technologies such as Generation IV fission reactors, first-generation fusion-plasma reactors, and more efficient gas turbines for electricity generation and propulsion.

DFT↗

High Throughput Discovery of Thermoelectric Materials (Final Report)

New materials form the heart of advances in energy technologies; however, the actual synthesis and characterization of new materials remains an enormous bottleneck. In this award, the research team focused on developing new approaches overcome this bottleneck. The approach involved depositing at high (supersonic) velocity multiple crystalline materials at a substrate while controlling the resulting film composition. The film could then be annealed to create a homogeneous alloy that was suitable for characterization. High throughput characterization tools were developed in parallel. The research focused on thermoelectric materials, which convert the flow of heat into electrical power. The development of new thermoelectric materials with high power conversion efficiency could displace traditional mechanical heat engines (e.g. turbines).

30 DIRECT ENERGY CONVERSION↗

Deep Learning and Natural Language Processing for Accelerated Inverse Design of Optical Metamaterials

Optical metamaterial device design has enjoyed a long track of success over the past 50 years leading to the manipulation of light over a wide range of wavelengths spanning the ultraviolet to the far infrared. The manipulation of light over such wavelengths has already led to many technological advancements such as the design of selective radiative absorbers for solar energy, daytime passive cooling using deep space, and optical invisibility cloaks for defense applications. Further disruptive advancements in energy, defense, computing, and biomedical fields could be enabled or enhanced by future optical metamaterial devices. These technologies could lead to increased energy efficiency and hence reduced national primary energy consumption, cheap long duration energy storage, and next generation solid-state heat engines. But historically the methods to invent and develop all of these devices have been time- consuming and based mostly on intuition and iteration. Finding an optimal design can take years.

36 MATERIALS SCIENCE↗

Deep Learning and Natural Language Processing for Accelerated Inverse Design of Optical Metamaterials

Optical metamaterial device design has enjoyed a long track of success over the past 50 years leading to the manipulation of light over a wide range of wavelengths spanning the ultraviolet to the far infrared. The manipulation of light over such wavelengths has already led to many technological advancements such as the design of selective radiative absorbers for solar energy, daytime passive cooling using deep space, and optical invisibility cloaks for defense applications. Further disruptive advancements in energy, defense, computing, and biomedical fields could be enabled or enhanced by future optical metamaterial devices. These technologies could lead to increased energy efficiency and hence reduced national primary energy consumption, cheap long duration energy storage, and next generation solid-state heat engines. But historically the methods to invent and develop all of these devices have been time- consuming and based mostly on intuition and iteration. Finding an optimal design can take years. In this project we developed a machine learning-based algorithm capable of automatically generating device designs to produce desired optical properties, reducing the design cycle life in certain situations to be almost instantaneous.

36 MATERIALS SCIENCE↗

Techno-Economic Analysis and Market Potential of Geological Thermal Energy Storage (GeoTES) Charged With Solar Thermal and Heat Pumps

In this project, we developed a techno-economic analysis (TEA) model that can be used to evaluate the viability of a proposed Geological Thermal Energy Storage (GeoTES) design. This MATLAB-based model integrates distinct subsystem models for the reservoir, wells, power cycle, and solar field to capture their distinct characteristics. It applies this approach in simulating GeoTES storage and dispatch operations for durations ranging from hourly to seasonal. Using cases studies based on GeoTES designs provided by industry partners - Premier Resource Management (PRM) and EarthBridge Energy - we validated the TEA model estimations of system performance and costs (such as thermal and electrical power/energy inflow and outflow, capital costs, and levelized costs of energy and storage) for both concentrating solar thermal (CST) and Carnot Battery (CB) pairings with GeoTES (CST-GeoTES and CB-GeoTES). For the CST-GeoTES case, the model was validated against the proposed system designed by PRM. It showed good agreement with PRM's estimations when well and pump costs derived from PRM's estimations were used. When GETEM-based costs were used, there was a slight overprediction due to GETEM's project/site agnostic assumption of these costs. From a sensitivity analysis perspective, the levelized cost of electricity (LCOE) of the CST-GeoTES case was most sensitive to well flow rate and the charging temperature. An optimal design scenario resulted in an LCOE of 0.11 $\$$/kWhe. CST-GeoTES can also provide a source of heat to meet seasonal demands. With 12-hour and 24-hour levelized cost of heat (LCOH) of 0.018 $\$$/kWhth and 0.022 $\$$/kWhth, respectively, CST-GeoTES could be competitive in the California market with an average industrial price of natural gas in California between 0.041-0.047 $\$$/kWhth. The levelized cost of storage (LCOS) for CST-GeoTES depends on the energy storage duration. Although the LCOS is relatively higher for shorter durations (e.g., ~0.50 $\$$/kWhe for 1 hour of storage), it is an order of magnitude lower (0.06 $\$$/kWhe) for longer storage durations and competitive with lithium-ion batteries (beyond 12 hours of storage) and molten-salt thermal energy storage (beyond 32 hours). Energy. Three options were explored and applied to the EarthBridge case study: (1) A Carnot Battery design using R125 working fluid with both hot and cold storage; (2) A Carnot Battery design using R125 working fluid with only hot storage; (3) A Carnot Battery using a commercially available heat pump with carbon dioxide (CO2) working fluid and hot storage only. The CB-GeoTES with cold storage only had a slight (round-trip) efficiency advantage over the system without (43.4% vs. 42.8%). This is because the cold storage is limited by the freezing point of water, so the cold storage is not much colder than the environment. The system using commercially available technologies was the least efficient - partly because different cycles were used in the heat pump (CO2) and heat engine (binary cycle) which leads to some inefficiencies. Using the commercially available design, the levelized cost of energy (LCOS) from the model (0.10 $\$$/kWhe) was higher than that estimated by EarthBridge (0.068 $\$$/kWhe). This is because of the low round-trip (38.7%) efficiency of the commercially available design. Sensitivity analysis reveals that the model is most sensitive to electricity price. Including electricity price in the TEA for CB-GeoTES leads to an increase in LCOS from the base value to 0.25 $\$$/kWhe. To determine storage sites suitable for GeoTES, we gathered and analyzed geological, petrophysical, and geophysical data of oil and gas reservoir and aquifers in California and Texas. We down-selected possible sites based on cut-off values for site characteristics (e.g., reservoir temperature, formation thickness, permeability, porosity, depth, and brine salinity) and preliminary costs. Using this approach, the Carrizo-Wilcox, Yegua-Jackson, and Dockum brackish aquifers in Texas were identified as having the highest suitability. Similarly, in the central California region, the White Wolf, Belridge South Tulare, and Belridge South Reef Ridge were the most suitable. Going further, we assessed the storage potential in the selected sites. To do this we developed distributions of reservoir characteristic data and applied a Monte Carlo-based analysis to account for intrinsic uncertainty in the acquired data. The analysis revealed that the Carrizo-Wilcox aquifer had the highest storage potential with a mean capacity of 554 TWhth (i.e., 63 TWhe). The estimated capacity serves as an upper limit of storage potential given that not all fields in the basin will be developed. We participated in multiple outreach activities including conference presentations, panel session discussions, and the facilitation of a GeoTES workshop at the NREL Golden campus.

15 GEOTHERMAL ENERGY↗

UHTC-TPMS Heat Exchangers for Concentrating Solar Power Applications with Thermal Energy Storage in Molten Chlorides

Concentrating solar power (CSP) plants convert energy from sunlight into electricity. Generally, a heliostat field is used to reflect sunlight and concentrate it onto a receiver, which heats the receiver and/or the media flowing through it. A flow loop transports the media through the system to a heat exchanger before returning the media back to the receiver to be reheated. The heat extracted by the heat exchanger is then converted into electricity using a conventional heat engine.

14 SOLAR ENERGY↗

Combined Mesonet and Tracker

Title: Combined Mesonet and Trackers (UNL Mobile Mesonets) Authors University of Nebraska PI: Adam Houston, UNL Professor (ahouston2@unl.edu) Mailing Address: 126 Bessey Hall P.O. Box 880340 Lincoln, NE 68588-0340 CoMeT Overview The University of Nebraska-Lincoln operates three Combined Mesonet and Tracker (CoMeTs). CoMeTs are Ford Explorers (model years 2015, 2017, and 2019) with forward-mounted suites of meteorological sensors and dual moonroofs, combining the capability of a mobile mesonet to collect near-surface observations with the capability of an unmanned aircraft systems (UAS) tracker vehicle, which enables an observer in the second row of seats to see the aircraft and maintain compliance with Federal Aviation Administration policies on UAS operation. The CoMeTs collect observations of slow temperature and humidity at ~2 m above ground level (AGL) using a Vaisala HMP155A, fast temperature at ~2 m AGL using a Campbell Scientific 109SS-L thermistor, pressure at ~2.5 m AGL using a Vaisala PTB210, wind speed and direction at ~3.25 m AGL using an R.M. Young 05103 propeller anemometer, and vehicle heading using a KVH Industries C-100 fluxgate compass (Barbieri et al. 2019). The HMP155A and 109SS-L thermistor are shielded and aspirated within a U-tube (Waugh and Frederickson 2010; Houston et al. 2016). This list of sensors is also included in the CoMeT data file metadata. Manufacturer specifications for these instruments are given in Table 1 of Hanft and Houston (2018). The reported measured quantities are summarized below.CoMeT-3 was funded through an equipment allocation included in the NSF TORUS award (AGS-1824649). Instrument Description The specific sensors included on each CoMeT are summarized in the table at the end of this section. In general each CoMeT collects observations of slow temperature and humidity at ~2 m above ground level (AGL) using a Vaisala HMP155, fast temperature at ~2 m AGL using a Campbell Scientific 109SS-L thermistor, pressure at ~2.5 m AGL using a Vaisala PTB210 barometer with a Gill pressure port, wind speed and direction at ~3.25 m AGL using an R.M. Young 05103 propeller anemometer, position using a Garmin 19x HVS receiver, and vehicle heading using a KVH Industries C-100 fluxgate compass. The HMP155 and 109SS are shielded and aspirated within a U-tube (Waugh and Frederickson 2010; Houston et al. 2016). Fast temperature and corrected RH measurements (using sensors housed within the U-tube) have a time constant of 10-12 s based on data collected across a temperature and RH shock during the CLOUD-MAP 2017 calibration/validation tests on June 26, 2017. Vehicle speed was < 10 kts for this test. CoMeT-1 CoMeT-2 CoMeT-3 Slow Temperature Slow RH Vaisala HMP155A-L20-PT Part #: 22280-7 Vaisala HMP155E Part #: E1AA11A0B1A1A0A Vaisala HMP155E Part #: E1AA11A0B1A1A0A Fast temperature Campbell Scientific 109SS-L20-PT Part #: 21448-3 Campbell Scientific 109SS-L12-PW Part #: 21448-109 Campbell Scientific 109SS-L12-PW Part #: 21448-150 Pressure Vaisala PTB-210 Part #: A1A1B Gill Pressure Port Part #: 61002 Vaisala PTB-210 Part #: A1A1B Gill Pressure Port Part #: 61002 Vaisala PTB-210 Part #: A1A1B Gill Pressure Port Part #: 61002 Wind RM Young 05103-L20-PT Part #: 18435-310 RM Young 05103-L20-PW Part #: 18435-244 RM Young 05103-L20-PW Part #: 18435-244 GPS Garmin GPS 19x HVS (NMEA 0183) Part #: 010-01010-00 Garmin GPS 19x HVS (NMEA 0183) Part #: 010-01010-00 Garmin GPS 19x HVS (NMEA 0183) Part #: 010-01010-00 Compass KVH C-100 Part #: 01-0177-15 KVH C-100 Part #: 01-0177-15 KVH C-100 Part #: 01-0177-15 Logger Campbell Scientific CR6-NA-XT-SW Part #: 28385-9 Campbell Scientific CR6-WIFI-XT-SW Part #: 28385-6 Campbell Scientific CR6-WIFI-XT-SW Part #: 28385-6 Data Collection and Real-Time Processing The reported measured quantities are summarized in the table below. Quantity Units Source Epoch time Seconds GPS Latitude and longitude Degrees GPS Altitude m GPS Pressure hPa PTB210 Temperature (fast) deg C 109SS-L Temperature (slow) deg C HMP155 RH (slow) % HMP155 Vehicle speed m/s GPS Vehicle heading deg C-100 and GPS In addition to the measured variables, several derived variables are calculated. Corrected/fast relative humidity (%) Relative humidity is adjusted to the fast temperature following Richardson et al. (1998) and Houston et al. (2016). Water vapor mixing ratio (g/kg) Dew point temperature (&deg;C) Potential temperature (Kelvin) Virtual potential temperature (Kelvin) Equivalent potential temperature (Kelvin) Regular intercomparisons between all three CoMeTs were performed during TORUS 2019. Comparisons were also conducted between CoMeT-1 and CoMeT-2 during LAPSE-RATE (2018) on 14 July. In these intercomparisons, the vehicles were parked adjacent to each other aligned perpendicular to (and facing into) the wind. To minimize engine heating effects, intercomparisons were only conducted when the wind speed was >10 kts. Data Format Original data files for each deployment are saved as text files and then converted to NetCDF. NetCDF versions have units that are CF compliant and may not match the original units in the txt files. The naming convention for the NetCDF files is as follows: UNL.CoMeT3.{deployment date YYYYMMDD}.{start time of observation collection in UTC HHMM}.L2.{post-processing codes}.cdf example: UNL.CoMeT3.20190627.1931.L2.g1.f1.cdf Post-processing codes are included to track modifications to the raw data. These codes are closely connected to error flags associated with each record. Each letter corresponds to a particular instrument: g: GPS p: Barometer tf: Fast temperature ts: Slow temperature rh: Relative humidity f: Compass w: Wind monitor a: All instruments Each number corresponds to a particular post-processing action described more below. Measured and derived variables are included in the following table. Variable Heading Standard Name Units time Time seconds since 00:00:00, 01-01-1970 Alt Altitude meters lat Latitude degrees north lon Longitude degrees east fast_temp Air Temperature kelvin slow_temp Air Temperature kelvin pressure Air Pressure pascals logger_RH Relative Humidity percent calc_corr_RH Relative Humidity percent wind_speed Wind Speed meters per second wind_dir Wind From Direction degrees vehicle_dir Vehicle Direction degrees dewpoint Dew Point Temperature kelvin mixing_ratio Humidity Mixing Ratio g/g theta Air Potential Temperature kelvin theta_v Virtual Potential Temperature kelvin theta_e Equivalent Potential Temperature Kelvin error_flag The error_flag variable is a string that matches the post-processing codes listed above. All instruments will have an associated code, but will have a &ldquo;0&rdquo; if the datum is unchanged from the initial processed value. Error Codes The following table summarizes the error codes for data collected before 2020: Error Code Relevant CoMeT Description g1 1,2,3 Exact correction. GPS position and time reprocessed from raw data g2 1 As far as we can tell this is an exact correction to an error in the GPS time. During the correct time periods the time suddenly went backwards ~250s and stayed at this offset for 750s when it corrected itself. The offset was applied to the &ldquo;time warp&rdquo; period. p1 2 Approximate correction. Hole in the pressure tube connecting the pressure port to the barometer. Resulted in erroneously low air pressure measurements when the vehicle was in motion. Derived variables recalculated (dew point temperature [e depends on qv and p], water vapor mixing ratio, potential temperature, virtual potential temperature, equivalent potential temperature) a1 3 Exact correction. Missing data reprocessed from raw data a2 1 Bug fix to bias correction for ts1, ts2, and rh1: water vapor mixing ratio was off by a factor of 10 and virtual potential temperature was wrong because of this. f1 3 No correction, missing data. Fluxgate compass inoperable. Wind speed and direction calculated using GPS-derived vehicle heading instead. rh1 1 Approximate correction. Constant bias of +1.7% removed from relative humidity. Derived variables recalculated (corrected/fast relative humidity, dew point temperature, water vapor mixing ratio, virtual potential temperature, equivalent potential temperature) ts1 1 Approximate correction. Constant bias of +0.6 K removed from slow temperature. Derived variables recalculated (corrected/fast relative humidity, dew point temperature, water vapor mixing ratio, virtual potential temperature, equivalent potential temperature) ts2 1 Approximate correction. Constant bias of +1.0 K removed. Derived variables recalculated (corrected/fast relative humidity, dew point temperature, water vapor mixing ratio, virtual potential temperature, equivalent potential temperature) References Bolton, D., 1980: The Computation of Equivalent Potential Temperature. Mon. Wea. Rev., 108, 1046&ndash;1053, https://doi.org/10.1175/1520-0493(1980)108<1046:TCOEPT>2.0.CO;2. Hanft, W., and A. L. Houston, 2018: An Observational and Modeling Study of Mesoscale Air Masses with High Theta-E. Mon. Wea. Rev., 146, 2503&ndash;2524, https://doi.org/10.1175/MWR-D-17-0389.1.Wexler Houston, A. L., R. J. Laurence III, T. W. Nichols, S. Waugh, B. Argrow, and C. L. Ziegler, 2016: Intercomparison of unmanned aircraft-borne and mobile mesonet atmospheric sensors. Journal of Atmospheric and Oceanic Technology. 33, 1569-1582, doi: 10.1175/JTECH-D-15-0178.1. Lowe, P. R., 1977: An Approximating Polynomial for the Computation of Saturation Vapor Pressure. J. Applied Meteorology, 16, 100&ndash;103. Richardson, S. J., S. E. Frederickson, F. V. Brock, and J. A. Brotzge, 1998: Combination temperature and relative humidity probes: Avoiding large air temperature errors and associated relative humidity errors. Preprints, 10th Symp. On Meteorological Observations and Instrumentation, Phoenix, AZ, Amer. Meteor. Soc., 278&ndash;283. Waugh, S., and S. E. Frederickson, 2010: An improved aspirated temperature system for mobile meteorological observations, especially in severe weather. 25th Conf. on Severe Local Storms, Denver, CO, Amer. Meteor. Soc., P5.2. [Available online at https://ams.confex.com/ams/25SLS/techprogram/paper_176205.htm.]

54 ENVIRONMENTAL SCIENCES↗

Experimental investigations of a uranium plasma pertinent to a self-sustaining plasma source

The research is pertinent to the realization of a self-sustained fissioning plasma for applications such as nuclear propulsion, closed cycle MHD power generation using a plasma core reactor, and heat engines such as the nuclear piston engine, as well as the direct conversion of fission energy into optical radiation (nuclear pumped lasers). Diagnostic measurement methods and experimental devices simulating plasma core reactor conditions are discussed. Studies on the following topics are considered: (1) ballistic piston compressor (U-235); (2) high pressure uranium plasma (natural uranium); (3) sliding spark discharge (natural uranium); (4) fission fragment interaction (He-3 and U-235); and (5) nuclear pumped lasers (He-3 and U-235).

Schneider, R. T.↗

The nature of the sunspot phenomenon. II - Internal overstable modes

It had been pointed out by Parker (1974) that the basic cause of the sunspot phenomenon is the enhanced heat transport in the magnetic field of the sunspot. The enhanced transport occurs through convective overstability which operates as a heat engine generating Alfven waves. The characteristics of the convective forces present are investigated along with questions concerning overstability and convectively driven Alfven waves. Relations regarding instability and convectively driven surface waves are discussed and attention is given to individual overstable Alfven modes. It is found that the form of an Alfven wave in the absence of convective forces is entirely arbitrary, so that waves with any arbitrary profile can be fitted into a vertical column of the field without disturbing the fluid outside. With the introduction of convective forces the situation changes so that the presence of lateral boundaries alters the form of the basic wave modes.

Parker, E. N.↗

Steady-state heat transfer in transversely heated porous media with application to focused solar energy collectors

A fluid flowing in a porous medium heated transversely to the fluid flow is considered. This configuration is applicable to a focused solar energy collector for use in an electric power generating system. A fluidized bed can be regarded as a porous medium with special properties. The solutions presented are valid for describing the effectiveness of such a fluidized bed for collecting concentrated solar energy to heat the working fluid of a heat engine. Results indicate the advantage of high thermal conductivity in the transverse direction and high operating temperature of the porous medium.

Nichols, L. D.↗

Gravitational potential energy of the earth: A spherical harmonic approach

A spherical harmonic equation for the gravitational potential energy of the earth is derived for an arbitrary density distribution by conceptually bringing in mass-elements from infinity and building up the earth shell upon spherical shell. The zeroth degree term in the spherical harmonic equation agrees with the usual expression for the energy of a radial density distribution. The second degree terms give a maximum nonhydrostatic energy in the mantle and crust of -2.77 x 10 to the twenty-ninth power ergs, an order of magnitude. If the earth is assumed to be a homogeneous viscous oblate spheroid relaxing to an equilibrium shape, then a lower limit to the mantle viscosity of 1.3 x 10 to the twentieth power poises is found by assuming the total geothermal flux is due to viscous dissipation. If the nonequilibrium figure is dynamically maintained by the earth acting as a heat engine at one per cent efficiency, then the viscosity is ten to the twenty second power poises, a number preferred by some as the viscosity of the mantle.

Rubincam, D. P.↗