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Thermal systems

Find NASA NTRS and DOE OSTI research on heat transfer and thermal systems. Search heat pipes, cooling and thermal management in report titles and abstracts.

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NASA NTRS1999

True Concurrent Thermal Engineering Integrating CAD Model Building with Finite Element and Finite Difference Methods

Thermal engineering has long been left out of the concurrent engineering environment dominated by CAD (computer aided design) and FEM (finite element method) software. Current tools attempt to force the thermal design process into an environment primarily created to support structural analysis, which results in inappropriate thermal models. As a result, many thermal engineers either build models "by hand" or use geometric user interfaces that are separate from and have little useful connection, if any, to CAD and FEM systems. This paper describes the development of a new thermal design environment called the Thermal Desktop. This system, while fully integrated into a neutral, low cost CAD system, and which utilizes both FEM and FD methods, does not compromise the needs of the thermal engineer. Rather, the features needed for concurrent thermal analysis are specifically addressed by combining traditional parametric surface based radiation and FD based conduction modeling with CAD and FEM methods. The use of flexible and familiar temperature solvers such as SINDA/FLUINT (Systems Improved Numerical Differencing Analyzer/Fluid Integrator) is retained.

Panczak, Tim↗

NASA NTRS1999

The Submillimeter-Wave Astronomy Satellite: On-Orbit Thermal Performance and Design Retrospective

A large telescope aperture, stringent thermal stability and temperature range requirements, and a passively-cooled 150 K module presented major challenges in thermal design and hardware fabrication of this Small Explorer satellite. This paper reviews briefly the thermal design of the SWAS science instrument, and examines the first three months of on-orbit thermal history. Measured temperatures for both the science payload and the spacecraft module and solar arrays are compared with those predicted by the correlated analytical model. Similarities and differences are interpreted in terms of the major uncertainties remaining after thermal-balance testing, especially those of MLI performance and telescope aperture properties. Review of the thermal model adequacy and thermal design verification are included to suggest improvements in the thermal design process for future missions.

Boyd, David A.↗

NASA NTRS2001

Spacecraft Thermal Control

Thermal control of the spacecraft is typically achieved by removing heat from the spacecraft parts that tend to overheat and adding heat to the parts that tend get too cold. The equipment on the spacecraft can get very hot if it is exposed to the sun or have internal heat generation. The pans also can get very cold if they are exposed to the cold of deep space. The spacecraft and instruments must be designed to achieve proper thermal balance. The combination of the spacecraft's external thermal environment, its internal heat generation (i.e., waste heat from the operation of electrical equipment), and radiative heat rejection will determine this thermal balance. It should also be noted that this is seldom a static situation, external environmental influences and internal heat generation are normally dynamic variables which change with time. Topics discussed include thermal control system components, spacecraft mission categories, spacecraft thermal requirements, space thermal environments, thermal control hardware, launch and flight operations, advanced technologies for future spacecraft,

Birur, Gajanana C.↗

NASA NTRS2001

Guidelines for the Selection of Near-Earth Thermal Environment Parameters for Spacecraft Design

Thermal analysis and design of Earth orbiting systems requires specification of three environmental thermal parameters: the direct solar irradiance, Earth's local albedo, and outgoing longwave radiance (OLR). In the early 1990s data sets from the Earth Radiation Budget Experiment were analyzed on behalf of the Space Station Program to provide an accurate description of these parameters as a function of averaging time along the orbital path. This information, documented in SSP 30425 and, in more generic form in NASA/TM-4527, enabled the specification of the proper thermal parameters for systems of various thermal response time constants. However, working with the engineering community and SSP-30425 and TM-4527 products over a number of years revealed difficulties in interpretation and application of this material. For this reason it was decided to develop this guidelines document to help resolve these issues of practical application. In the process, the data were extensively reprocessed and a new computer code, the Simple Thermal Environment Model (STEM) was developed to simplify the process of selecting the parameters for input into extreme hot and cold thermal analyses and design specifications. In the process, greatly improved values for the cold case OLR values for high inclination orbits were derived. Thermal parameters for satellites in low, medium, and high inclination low-Earth orbit and with various system thermal time constraints are recommended for analysis of extreme hot and cold conditions. Practical information as to the interpretation and application of the information and an introduction to the STEM are included. Complete documentation for STEM is found in the user's manual, in preparation.

Anderson, B. J.↗

NASA NTRS1995

Effect of Cyclic Aeroconvective Heating on Flexible Thermal Insulations

This paper describes the effect of cyclic aeroconvective heating on the thermal performance of ceramic flexible insulations considered for potential use as thermal protection systems or thermal insulations for future hypersonic vehicles such as the Reusable Launch Vehicles (RLV's) and other applications where structures require thermal protection from severe heating. The thermal response of these materials after exposure to cyclic aeroconvective heating from a plasma arc is described. The thermal insulations evaluated were Composite Flexible Blanket Insulations composed of an outer layer of aluminoborosilicate fabric and alumina insulation. The insulations were evaluated with and without a high emissivity coating. These insulations were exposed to the plasma arc stream for nine minutes reaching surface temperatures of 1150 C and a heat flux of 10.5 W/sq cm. Insulations were exposed three, six, or nine times in order to demonstrate reusability and to determine the effect oi coatings on the surface durability of these insulations. Test results demonstrated the capability of these insulations to protect either composite or metallic structures from high heating environments. It is shown that high emittance coatings reduce backface temperatures. The durability of these insulations when impacted at low velocities was also demonstrated. The interaction of the coatings with the ceramic fibers was characterized. An analytical thermal model was utilized to correlate experimental thermal test results with calculated values.

Kourtides, Demetrius A.↗

NASA NTRS2002

Rocket Motor Joint Construction Including Thermal Barrier

A thermal barrier for extremely high temperature applications consists of a carbon fiber core and one or more layers of braided carbon fibers surrounding the core. The thermal barrier is preferably a large diameter ring, having a relatively small cross-section. The thermal barrier is particularly suited for use as part of a joint structure in solid rocket motor casings to protect low temperature elements such as the primary and secondary elastomeric O-ring seals therein from high temperature gases of the rocket motor. The thermal barrier exhibits adequate porosity to allow pressure to reach the radially outward disposed O-ring seals allowing them to seat and perform the primary sealing function. The thermal barrier is disposed in a cavity or groove in the casing joint, between the hot propulsion gases interior of the rocket motor and primary and secondary O-ring seals. The characteristics of the thermal barrier may be enhanced in different applications by the inclusion of certain compounds in the casing joint, by the inclusion of RTV sealant or similar materials at the site of the thermal barrier, and/or by the incorporation of a metal core or plurality of metal braids within the carbon braid in the thermal barrier structure.

Steinetz, Bruce M.↗

NASA NTRS2003

Aeroheating Thermal Model Correlation for Mars Global Surveyor (MGS) Solar Array

The Mars Global Surveyor (MGS) Spacecraft made use of aerobraking to gradually reduce its orbit period from a highly elliptical insertion orbit to its final science orbit. Aerobraking produces a high heat load on the solar arrays, which have a large surface area exposed to the airflow and relatively low mass. To accurately model the complex behavior during aerobraking, the thermal analysis needed to be tightly coupled to the spatially varying, time dependent aerodynamic heating. Also, the thermal model itself needed to accurately capture the behavior of the solar array and its response to changing heat load conditions. The correlation of the thermal model to flight data allowed a validation of the modeling process, as well as information on what processes dominate the thermal behavior. Correlation in this case primarily involved detailing the thermal sensor nodes, using as-built mass to modify material property estimates, refining solar cell assembly properties, and adding detail to radiation and heat flux boundary conditions. This paper describes the methods used to develop finite element thermal models of the MGS solar array and the correlation of the thermal model to flight data from the spacecraft drag passes. Correlation was made to data from four flight thermal sensors over three of the early drag passes. Good correlation of the model was achieved, with a maximum difference between the predicted model maximum and the observed flight maximum temperature of less than 5%. Lessons learned in the correlation of this model assisted in validating a similar model and method used for the Mars Odyssey solar array aeroheating analysis, which were used during onorbit operations.

Amundsen, Ruth M.↗

NASA NTRS2004

Chapter 9: MEMS Applications in Spacecraft Thermal Control and Chapter 13: Handling and Contamination Control Considerations for Critical Space Applications

Thermal control systems (TCS) are an integral part of all spacecraft and instruments. To operate properly all spacecraft components must be maintained within a specified temperature band, which in some instances can be quite tight (less than 1 degree C). Traditionally this is accomplished by "passive" designs, but with modem spacecraft and instruments it is increasingly necessary to use active thermal control technologies. Microelectromechanical System (MEMS) can be useful in developing an active TCS, and may be especially useful for small spacecraft. As the MEMS knowledge matures, the applications of MEMS in spacecraft thermal control emerges as a viable technology for thermal engineers. Potential applications include specialized thermal control coatings, thermal switches, and specialized filters for instruments. Although MEMS technology demonstrates benefits, it also poses challenges for thermal en,oineers due to the lack of in-flight MEMS data. As a consequence, in order to design a MEMS thermal control device and receive the full advantage, it is important to understand the potential impact of the space environment on MEMS devices and the design/operational constraints imposed in their use. An entire chapter is devoted to handling and contamination controls for Micro Electro Mechanical Systems (MEMS) in space applications due to the importance of the topic area to final mission success. Handling and contamination control is discussed relative to the full life cycle from the very basic wafer level processing up through on orbit deployment. MEMS packaging will drive the need to tailor the Handling and Contamination Control Plan in order to assure adequacy of the overall program on a program by program basis. Plan elements are discussed at length to assist the user in preparing and implementing effective plans for both handling and contamination control to prevent deleterious effects.

Osiander, Robert↗

NASA NTRS1998

Thermal Effects Modeling Developed for Smart Structures

Applying smart materials in aeropropulsion systems may improve the performance of aircraft engines through a variety of vibration, noise, and shape-control applications. To facilitate the experimental characterization of these smart structures, researchers have been focusing on developing analytical models to account for the coupled mechanical, electrical, and thermal response of these materials. One focus of current research efforts has been directed toward incorporating a comprehensive thermal analysis modeling capability. Typically, temperature affects the behavior of smart materials by three distinct mechanisms: Induction of thermal strains because of coefficient of thermal expansion mismatch 1. Pyroelectric effects on the piezoelectric elements; 2. Temperature-dependent changes in material properties; and 3. Previous analytical models only investigated the first two thermal effects mechanisms. However, since the material properties of piezoelectric materials generally vary greatly with temperature (see the graph), incorporating temperature-dependent material properties will significantly affect the structural deflections, sensory voltages, and stresses. Thus, the current analytical model captures thermal effects arising from all three mechanisms through thermopiezoelectric constitutive equations. These constitutive equations were incorporated into a layerwise laminate theory with the inherent capability to model both the active and sensory response of smart structures in thermal environments. Corresponding finite element equations were formulated and implemented for both the beam and plate elements to provide a comprehensive thermal effects modeling capability.

Lee, Ho-Jun↗

NASA NTRS2004

Atlas V Launch Incorporated NASA Glenn Thermal Barrier

In the Spring of 2002, Aerojet experienced a major failure during a qualification test of the solid rocket motor that they were developing for the Atlas V Enhanced Expendable Launch Vehicle. In that test, hot combustion gas reached the O-rings in the nozzle-to-case joint and caused a structural failure that resulted in loss of the nozzle and aft dome sections of the motor. To improve the design of this joint, Aerojet decided to incorporate three braided carbon-fiber thermal barriers developed at the NASA Glenn Research Center. The thermal barriers were used to block the searing-hot 5500 F pressurized gases from reaching the temperature-sensitive O-rings that seal the joint. Glenn originally developed the thermal barriers for the nozzle joints of the space shuttle solid rocket motors, and Aerojet decided to use them on the basis of the results of several successful ground tests of the thermal barriers in the shuttle rockets. Aerojet undertook an aggressive schedule to redesign the rocket nozzle-to-case joint with the thermal barriers and to qualify it in time for a launch planned for the middle of 2003. They performed two successful qualification tests (Oct. and Dec. 2002) in which the Glenn thermal barriers effectively protected the O-rings. These qualification tests saved hundreds of thousands of dollars in development costs and put the Lockheed-Martin/Aerojet team back on schedule. On July 17, 2003, the first flight of an Atlas V boosted with solid rocket motors successfully launched a commercial satellite into orbit from Cape Canaveral Air Force Station. Aero-jet's two 67-ft solid rocket boosters performed flawlessly, with each providing thrust in excess of 250,000 lbf. Both motors incorporated three Glenn-developed thermal barriers in their nozzle-to-case joints. The Cablevision satellite launched on this mission will be used to provide direct-to-home satellite television programming for the U.S. market starting in late 2003. The Atlas V is a product of the military's Enhanced Expendable Launch Vehicle program designed to provide assured military access to space. It can lift payloads up to 19,100 lb to geosynchronous transfer orbit and was designed to meet Department of Defense, commercial, and NASA needs. The Atlas V and Delta IV are two launch systems being considered by NASA to launch the Orbital Space Plane/Crew Exploration Vehicle. The launch and rocket costs of this mission are valued at $250 million. Successful application of the Glenn thermal barrier to the Atlas V program was an enormous breakthrough for the program's technical and schedule success.

Dunlap, Patrick H., Jr.↗

NASA NTRS2005

An Improved Thermal Conductivity Polyurethane Composite for a Space Borne 20KV Power Supply

This effort was designed to find a way to reduce the temperature rise of critical components of a 20KV High Voltage Power Supply (HVPS) by improving the overall thermal conductivity of the encapsulated modules. Three strategies were evaluated by developing complete procedures, preparing samples, and performing tests. The three strategies were: 1. Improve the thermal conductivity of the polyurethane encapsulant through the addition of thermally conductive powder while minimizing impact on other characteristics of the encapsulant. 2. Improve the thermal conductivity of the polyurethane encapsulated assembly by the addition of a slab of thermally conductive, electrically insulating material, which is to act as a heat spreader. 3. Employ a more thermally conductive substrate (Al203) with the existing encapsulation scheme. The materials were chosen based on the following criteria: high dielectric breakdown strength; high thermal conductivity, ease of manufacturing, high compliance, and other standard space qualified materials properties (low out-gassing, etc.). An optimized cure was determined by a statistical design of experiments for both filled and unfilled materials. The materials were characterized for the desired properties and a complete process was developed and tested. The thermal performance was substantially improved and the strategies may be used for space flight.

Pb free↗

NASA NTRS2007

Development of a Cryogenic Thermal Distortion Measurement Facility for Testing the James Webb Space Telescope Instrument Support Integration Module 2-D Test Assemblies

The James Webb Space Telescope Instrument Support Integration Module (ISIM) is being designed and developed at the Goddard Space Flight Center. The ISM Thermal Distortion Testing (ITDT) program was started with the primary objective to validate the ISM mechanical design process. The ITDT effort seeks to establish confidence and demonstrate the ability to predict thermal distortion in composite structures at cryogenic temperatures using solid element models. This-program's goal is to better ensure that ISIM meets all the mechanical and structural requirements by using test results to verify or improve structural modeling techniques. The first step to accomplish the ITDT objectives was to design, and then construct solid element models of a series 2-D test assemblies that represent critical building blocks of the ISIM structure. Second, the actual test assemblies consisting of composite tubes and invar end fittings were fabricated and tested for thermal distortion. This paper presents the development of the GSFC Cryo Distortion Measurement Facility (CDMF) to meet the requirements of the ISIM 2-D test. assemblies, and other future ISIM testing needs. The CDMF provides efficient cooling with both a single, and two-stage cryo-cooler. Temperature uniformity of the test assemblies during thermal transients and at steady state is accomplished by using sapphire windows for all of the optical ports on the radiation shields and by using .thermal straps to cool the test assemblies. Numerical thermal models of the test assemblies were used to predict the temperature uniformity of the parts during cooldown and at steady state. Results of these models are compared to actual temperature data from the tests. Temperature sensors with a 0.25K precision were used to insure that test assembly gradients did not exceed 2K lateral, and 4K axially. The thermal distortions of two assemblies were measured during six thermal cycles from 320K to 35K using laser interferometers. The standard deviation for all of the distortion measurements is less than 0.5 microns, which falls within the ISIM requirement of 3 microns.

Miller, Franklin↗

NASA NTRS2010

James Webb Space Telescope Integrated Science Instrument Module Thermal Balance/Thermal Vacuum Test Configuration and Test Planning at NASA's Goddard Space Flight Center

The James Webb Space Telescope (JWST) is the next of the great observatories , scheduled to be launched in 2014. Three of the four science instruments are passively cooled to their operational temperature range of 36K to 40K, and the fourth instrument is actively cooled to its operational temperature of approximately 6K. Thermal-vacuum testing of the flight science instruments at the Integrated Science Instrument Module (ISIM) element level will take place within a newly constructed shroud cooled by gaseous helium inside Goddard Space Flight Center s (GSFC) Space Environment Simulator (SES). The test enclosure surrounding the instruments during the integrated ISIM-level thermal balance testing is complex, and is designed to simulate as closely as possible the in-flight conductive and radiative thermal environment around the ISIM. Thermal control and measurement of parasitic sources of heat leak into the test volume is critical, as the dissipation plus known parasitics in the flight ISIM is approximately 454 mW, and additional parasitics attributed to the flight enclosure itself are nearly equal to this, resulting in the energy balance of the inflight ISIM being less than 1.0 W. Sources of test-induced parasitics must be carefully controlled and measured, and the ability to thermally control the test environment is critical to enable accurate thermal balance testing and thermal model correlation. This paper describes the test configuration and plans for the ISIM-level thermal vacuum/thermal balance testing at GSFC.

Glazer, Stuart↗

NASA NTRS2009

A Model of Thermal Conductivity for Planetary Soils: 1. Theory for Unconsolidated Soils

We present a model of heat conduction for mono-sized spherical particulate media under stagnant gases based on the kinetic theory of gases, numerical modeling of Fourier s law of heat conduction, theoretical constraints on the gas thermal conductivity at various Knudsen regimes, and laboratory measurements. Incorporating the effect of the temperature allows for the derivation of the pore-filling gas conductivity and bulk thermal conductivity of samples using additional parameters (pressure, gas composition, grain size, and porosity). The radiative and solid-to-solid conductivities are also accounted for. Our thermal model reproduces the well-established bulk thermal conductivity dependency of a sample with the grain size and pressure and also confirms laboratory measurements finding that higher porosities generally lead to lower conductivities. It predicts the existence of the plateau conductivity at high pressure, where the bulk conductivity does not depend on the grain size. The good agreement between the model predictions and published laboratory measurements under a variety of pressures, temperatures, gas compositions, and grain sizes provides additional confidence in our results. On Venus, Earth, and Titan, the pressure and temperature combinations are too high to observe a soil thermal conductivity dependency on the grain size, but each planet has a unique thermal inertia due to their different surface temperatures. On Mars, the temperature and pressure combination is ideal to observe the soil thermal conductivity dependency on the average grain size. Thermal conductivity models that do not take the temperature and the pore-filling gas composition into account may yield significant errors.

Piqueux, S.↗

NASA NTRS2003

Optimal Feedback Control of Thermal Networks

An improved approach to the mathematical modeling of feedback control of thermal networks has been devised. Heretofore software for feedback control of thermal networks has been developed by time-consuming trial-and-error methods that depend on engineers expertise. In contrast, the present approach is a systematic means of developing algorithms for feedback control that is optimal in the sense that it combines performance with low cost of implementation. An additional advantage of the present approach is that a thermal engineer need not be expert in control theory. Thermal networks are lumped-parameter approximations used to represent complex thermal systems. Thermal networks are closely related to electrical networks commonly represented by lumped-parameter circuit diagrams. Like such electrical circuits, thermal networks are mathematically modeled by systems of differential-algebraic equations (DAEs) that is, ordinary differential equations subject to a set of algebraic constraints. In the present approach, emphasis is placed on applications in which thermal networks are subject to constant disturbances and, therefore, integral control action is necessary to obtain steady-state responses. The mathematical development of the present approach begins with the derivation of optimal integral-control laws via minimization of an appropriate cost functional that involves augmented state vectors. Subsequently, classical variational arguments provide optimality conditions in the form of the Hamiltonian equations for the standard linear-quadratic-regulator (LQR) problem. These equations are reduced to an algebraic Riccati equation (ARE) with respect to the augmented state vector. The solution of the ARE leads to the direct computation of the optimal proportional- and integral-feedback control gains. In cases of very complex networks, large numbers of state variables make it difficult to implement optimal controllers in the manner described in the preceding paragraph.

Papalexandris, Miltiadis↗

NASA NTRS2003

Improved Thermal-Insulation Systems for Low Temperatures

Improved thermal-insulation materials and structures and the techniques for manufacturing them are undergoing development for use in low-temperature applications. Examples of low-temperature equipment for which these thermal insulation systems could provide improved energy efficiency include storage tanks for cryogens, superconducting electric-power-transmission equipment, containers for transport of food and other perishable commodities, and cold boxes for low-temperature industrial processes. These systems could also be used to insulate piping used to transfer cryogens and other fluids, such as liquefied natural gas, refrigerants, chilled water, crude oil, or low-pressure steam. The present thermal-insulation systems are layer composites based partly on the older class of thermal-insulation systems denoted generally as multilayer insulation (MLI). A typical MLI structure includes an evacuated jacket, within which many layers of radiation shields are stacked or wrapped close together. Low-thermal-conductivity spacers are typically placed between the reflection layers to keep them from touching. MLI can work very well when a high vacuum level (less than 10(exp-4) torr) is maintained and utmost care is taken during installation, but its thermal performance deteriorates sharply as the pressure in the evacuated space rises into the soft vacuum range [pressures greater than 0.1 torr (greater than 13 Pa)]. In addition, the thermal performance of MLI is extremely sensitive to mechanical compression and edge effects and can easily decrease from one to two orders of magnitude from its ideal value even when the MLI is kept under high vacuum condition. The present thermal-insulation systems are designed to perform well under soft vacuum level, in particular the range of 1 to 10 torr. They are also designed with larger interlayer spacings to reduce vulnerability to compression (and consequent heat leak) caused by installation and use. The superiority of these systems is the synergistic effect of improvements in materials, design, and manufacture.

Fesmire, James E.↗

NASA NTRS2011

Thermal Properties of Microstrain Gauges Used for Protection of Lithium-Ion Cells of Different Designs

The purpose of this innovation is to use microstrain gauges to monitor minute changes in temperature along with material properties of the metal cans and pouches used in the construction of lithium-ion cells. The sensitivity of the microstrain gauges to extremely small changes in temperatures internal to the cells makes them a valuable asset in controlling the hazards in lithium-ion cells. The test program on lithium-ion cells included various cell configurations, including the pouch type configurations. The thermal properties of microstrain gauges have been found to contribute significantly as safety monitors in lithium-ion cells that are designed even with hard metal cases. Although the metal cans do not undergo changes in material property, even under worst-case unsafe conditions, the small changes in thermal properties observed during charge and discharge of the cell provide an observable change in resistance of the strain gauge. Under abusive or unsafe conditions, the change in the resistance is large. This large change is observed as a significant change in slope, and this can be used to prevent cells from going into a thermal runaway condition. For flexible metal cans or pouch-type lithium-ion cells, combinations of changes in material properties along with thermal changes can be used as an indication for the initiation of an unsafe condition. Lithium-ion cells have a very high energy density, no memory effect, and almost 100-percent efficiency of charge and discharge. However, due to the presence of a flammable electrolyte, along with the very high energy density and the capability of releasing oxygen from the cathode, these cells can go into a hazardous condition of venting, fire, and thermal runaway. Commercial lithium-ion cells have current and voltage monitoring devices that are used to control the charge and discharge of the batteries. Some lithium-ion cells have internal protective devices, but when used in multi-cell configurations, these protective devices either do not protect or are themselves a hazard to the cell due to their limitations. These devices do not help in cases where the cells develop high impedance that suddenly causes them to go into a thermal runaway condition. Temperature monitoring typically helps with tracking the performance of a battery. But normal thermistors or thermal sensors do not provide the accuracy needed for this and cannot track a change in internal cell temperatures until it is too late to stop a thermal runaway.

Jeevarajan, Judith↗

NASA NTRS2012

Comparison of Observed Surface Temperatures of 4 Vesta to the KRC Thermal Model

In this work, we will compare ob-served temperatures of the surface of Vesta using data acquired by the Dawn [1] Visible and Infrared Map-ping Spectrometer (VIR-MS) [2] during the approach phase to model results from the KRC thermal model. High thermal inertia materials, such as bedrock, resist changes in temperature while temperatures of low thermal inertia material, such as dust, respond quickly to changes in solar insolation. The surface of Vesta is expected to have low to medium thermal inertia values, with the most commonly used value being extremely low at 15 TIU [4]. There are several parameters which affect observed temperatures in addition to thermal inertia: bond albedo, slope, and surface roughness. In addition to these parameters, real surfaces are rarely uniform monoliths that can be described by a single thermal inertia value. Real surfaces are often vertically layered or are mixtures of dust and rock. For Vesta's surface, with temperature extremes ranging from 50 K to 275 K and no atmosphere, even a uniform monolithic surface may have non-uniform thermal inertia due to temperature dependent thermal conductivity.

Titus, T. N.↗