Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Thermalization”

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 1,063 records · Page 59
All sourcesNASA NTRSDOE OSTI
NASA NTRS2000

On-Line, Self-Learning, Predictive Tool for Determining Payload Thermal Response

This paper will present the results of a joint ManTech / Goddard R&D effort, currently under way, to develop and test a computer based, on-line, predictive simulation model for use by facility operators to predict the thermal response of a payload during thermal vacuum testing. Thermal response was identified as an area that could benefit from the algorithms developed by Dr. Jeri for complex computer simulations. Most thermal vacuum test setups are unique since no two payloads have the same thermal properties. This requires that the operators depend on their past experiences to conduct the test which requires time for them to learn how the payload responds while at the same time limiting any risk of exceeding hot or cold temperature limits. The predictive tool being developed is intended to be used with the new Thermal Vacuum Data System (TVDS) developed at Goddard for the Thermal Vacuum Test Operations group. This model can learn the thermal response of the payload by reading a few data points from the TVDS, accepting the payload's current temperature as the initial condition for prediction. The model can then be used as a predictive tool to estimate the future payload temperatures according to a predetermined shroud temperature profile. If the error of prediction is too big, the model can be asked to re-learn the new situation on-line in real-time and give a new prediction. Based on some preliminary tests, we feel this predictive model can forecast the payload temperature of the entire test cycle within 5 degrees Celsius after it has learned 3 times during the beginning of the test. The tool will allow the operator to play "what-if' experiments to decide what is his best shroud temperature set-point control strategy. This tool will save money by minimizing guess work and optimizing transitions as well as making the testing process safer and easier to conduct.

Jen, Chian-Li↗
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 NTRS2003

Thermal Property Measurement of Semiconductor Melt using Modified Laser Flash Method

This study further developed standard laser flash method to measure multiple thermal properties of semiconductor melts. The modified method can determine thermal diffusivity, thermal conductivity, and specific heat capacity of the melt simultaneously. The transient heat transfer process in the melt and its quartz container was numerically studied in detail. A fitting procedure based on numerical simulation results and the least root-mean-square error fitting to the experimental data was used to extract the values of specific heat capacity, thermal conductivity and thermal diffusivity. This modified method is a step forward from the standard laser flash method, which is usually used to measure thermal diffusivity of solids. The result for tellurium (Te) at 873 K: specific heat capacity 300.2 Joules per kilogram K, thermal conductivity 3.50 Watts per meter K, thermal diffusivity 2.04 x 10(exp -6) square meters per second, are within the range reported in literature. The uncertainty analysis showed the quantitative effect of sample geometry, transient temperature measured, and the energy of the laser pulse.

Lin, Bochuan↗
NASA NTRS1997

Thermal Design and Analysis for the Cryogenic MIDAS Experiment

The Materials In Devices As Superconductors (MIDAS) spaceflight experiment is a NASA payload which launched in September 1996 on the Shuttle, and was transferred to the Mir Space Station for several months of operation. MIDAS was developed and built at NASA Langley Research Center (LaRC). The primary objective of the experiment was to determine the effects of microgravity and spaceflight on the electrical properties of high-temperature superconductive (HTS) materials. The thermal challenge on MIDAS was to maintain the superconductive specimens at or below 80 K for the entire operation of the experiment, including all ground testing and 90 days of spaceflight operation. Cooling was provided by a small tactical cryocooler. The superconductive specimens and the coldfinger of the cryocooler were mounted in a vacuum chamber, with vacuum levels maintained by an ion pump. The entire experiment was mounted for operation in a stowage locker inside Mir, with the only heat dissipation capability provided by a cooling fan exhausting to the habitable compartment. The thermal environment on Mir can potentially vary over the range 5 to 40 C; this was the range used in testing, and this wide range adds to the difficulty in managing the power dissipated from the experiment's active components. Many issues in the thermal design are discussed, including: thermal isolation methods for the cryogenic samples; design for cooling to cryogenic temperatures; cryogenic epoxy bonds; management of ambient temperature components self-heating; and fan cooling of the enclosed locker. Results of the design are also considered, including the thermal gradients across the HTS samples and cryogenic thermal strap, electronics and thermal sensor cryogenic performance, and differences between ground and flight performance. Modeling was performed in both SINDA-85 and MSC/PATRAN (with direct geometry import from the CAD design tool Pro/Engineer). Advantages of both types of models are discussed. Correlation of several models to ground testing and flight data (where available) is presented. Both SINDA and PATRAN models predicted the actual thermal performance of the experiment well, even without post-flight correlation adjustments of the models.

Amundsen, Ruth McElroy↗
NASA NTRS2005

Miniature Loop Heat Pipe with Multiple Evaporators for Thermal Control of Small Spacecraft

This paper presents an advanced miniature heat transport system for thermal control of small spacecraft. The thermal system consists of a loop heat pipe (LHP) with multiple evaporators and multiple deployable radiators for heat transfer, and variable emittance coatings on the radiators for performance enhancement. Thermoelectric coolers are used to control the loop operating temperature. The thermal system combines the functions of variable conductance heat pipes, thermal switches, thermal diodes, and the state-of-the-art LHPs into a single integrated thermal system. It retains all the performance characteristics of state-of-the-art LHPs and offers additional advantages to enhance the functionality, performance, versatility, and reliability of the system. Steady state and transient analytical models have been developed, and scaling criteria have also been established. A breadboard unit has been built for functional testing in laboratory and thermal vacuum environments. Experimental results show excellent performance of the thermal system and correlate very well with theoretical predictions.

Ku, Jentung↗
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 NTRS2011

Non-Contact Thermal Properties Measurement with Low-Power Laser and IR Camera System

As shown by the Phoenix Mars Lander's Thermal and Electrical Conductivity Probe (TECP), contact measurements of thermal conductivity and diffusivity (using a modified flux-plate or line-source heat-pulse method) are constrained by a number of factors. Robotic resources must be used to place the probe, making them unavailable for other operations for the duration of the measurement. The range of placement is also limited by mobility, particularly in the case of a lander. Placement is also subject to irregularities in contact quality, resulting in non-repeatable heat transfer to the material under test. Most important from a scientific perspective, the varieties of materials which can be measured are limited to unconsolidated or weakly-cohesive regolith materials, rocks, and ices being too hard for nominal insertion strengths. Accurately measuring thermal properties in the laboratory requires significant experimental finesse, involving sample preparation, controlled and repeatable procedures, and, practically, instrumentation much more voluminous than the sample being tested (heater plates, insulation, temperature sensors). Remote measurements (infrared images from orbiting spacecraft) can reveal composite properties like thermal inertia, but suffer both from a large footprint (low spatial resolution) and convolution of the thermal properties of a potentially layered medium. In situ measurement techniques (the Phoenix TECP is the only robotic measurement of thermal properties to date) suffer from problems of placement range, placement quality, occupation of robotic resources, and the ability to only measure materials of low mechanical strength. A spacecraft needs the ability to perform a non-contact thermal properties measurement in situ. Essential components include low power consumption, leveraging of existing or highly-developed flight technologies, and mechanical simplicity. This new in situ method, by virtue of its being non-contact, bypasses all of these problems. The use of photons to both excite and measure the thermal response of any surface material to a high resolution (estimated footprint = 10 square centimeters) is a generational leap in physical properties measurements. The proposed method consists of spot-heating the surface of a material with a low (less than 1 W) power laser. This produces a moderate (5-10 K) temperature increase in the material.

Hudson, Troy L.↗
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↗