Results for “Optimal Power Flow” — Page 35

NASA NTRS2005

Towards Rocket Engine Components with Increased Strength and Robust Operating Characteristics

High-energy rotating machines, powering liquid propellant rocket engines, are subject to various sources of high and low cycle fatigue generated by unsteady flow phenomena. Given the tremendous need for reliability in a sustainable space exploration program, a fundamental change in the design methodology for engine components is required for both launch and space based systems. A design optimization system based on neural-networks has been applied and demonstrated in the redesign of the Space Shuttle Main Engine (SSME) Low Pressure Oxidizer Turbo Pump (LPOTP) turbine nozzle. One objective of the redesign effort was to increase airfoil thickness and thus increase its strength while at the same time detuning the vane natural frequency modes from the vortex shedding frequency. The second objective was to reduce the vortex shedding amplitude. The third objective was to maintain this low shedding amplitude even in the presence of large manufacturing tolerances. All of these objectives were achieved without generating any detrimental effects on the downstream flow through the turbine, and without introducing any penalty in performance. The airfoil redesign and preliminary assessment was performed in the Exploration Technology Directorate at NASA ARC. Boeing/Rocketdyne and NASA MSFC independently performed final CFD assessments of the design. Four different CFD codes were used in this process. They include WIL DCA T/CORSAIR (NASA), FLUENT (commercial), TIDAL (Boeing Rocketdyne) and, a new family (AardvarWPhantom) of CFD analysis codes developed at NASA MSFC employing LOX fluid properties and a Generalized Equation Set formulation. Extensive aerodynamic performance analysis and stress analysis carried out at Boeing Rocketdyne and NASA MSFC indicate that the redesign objectives have been fully met. The paper presents the results of the assessment analysis and discusses the future potential of robust optimal design for rocket engine components.

Marcu, Bogdan↗

NASA NTRSDate not supplied

A New Flow-Through Wind Tunnel Balance for Retropropulsion Testing

Supersonic retropropulsion is believed to be an enabling technology for large-scale human Mars lander vehicle concepts. Ground testing of such vehicles requires the development of new flow-through wind tunnel balances because no suitable balances exist. A new flow-through balance is designed that employs membranes in place of bellows, which have historically been used, and that passes the flow through load bearing parts of the balance. Flow through the balance is optimized via the addition of flow guides, which are manufactured as separate parts, and attached to the balance. Calibration of this new flow-through balance demonstrates that it can suitably characterize the load and pressure ranges it was designed for, which provides aero researchers a new tool to directly measure supersonic retropropulsion performance of powered descent vehicles.

Flow through balance↗

NASA NTRSDate not supplied

A New Flow-Through Balance for Retropropulsion Testing

Supersonic retropropulsion is believed to be an enabling technology for large-scale human Mars lander vehicle concepts. Ground testing of such vehicles requires the development of new flow-through wind tunnel balances because no suitable balances exist. A new flow-through balance is designed that employs membranes in place of bellows, which have historically been used, and that passes the flow through load bearing parts of the balance. Flow through the balance is optimized via the addition of flow guides, which are manufactured as separate parts, and attached to the balance. Calibration of this new flow-through balance demonstrates that it can suitably characterize the load and pressure ranges it was designed for, which provides aero researchers a new tool to directly measure supersonic retropropulsion performance of powered descent vehicles.

Flow through balance↗

DOE OSTI2024

Integrated Turbine Component Cooling Designs Facilitated by Additive Manufacturing and Optimization

This research program involved the development of improved gas turbine cooling by optimizing the combined internal and film cooling configurations used for turbine blades and vanes. Improved cooling of turbine airfoils allows the gas turbines to be operated at higher temperatures which improves efficiency for gas turbine-based power generation. This also facilitates operation with hydrogen-based fuels which inherently generate higher operating temperatures. Designs of the new cooling configurations were based on the added geometric flexibility that is available when using metal additive manufacturing techniques. When making these designs, adjustments were made to account for imperfections occurring in the metal additive manufacturing process. New designs for film cooling holes were developed using computational adjoint gradient optimization techniques which generated a unique film cooling hole with 70% greater film cooling effectiveness compared to conventional shaped film cooling holes. A major factor in improving the film cooling effectiveness were small external protrusions on either side of the new hole which generated vortices that pushed coolant towards the surface and increased the lateral spreading of the coolant. Our studies also included optimization of film cooling holes that can be built with conventional manufacturing techniques and optimizing the internal coolant channels that supply coolant to the film cooling holes. Various internal coolant channel shapes were investigated and designs that maximized the cooling of the airfoil while minimizing pressure losses in the channel flows were identified. Testing and verification of the performance of the final configurations was done with a transonic wind tunnel facility which allowed testing at engine realistic high velocities. Final testing of the combined internal and film cooling configurations was done by incorporating the configurations in Penn State’s National Experimental Turbine (NExT) vane. These tests confirmed the enhanced overall cooling effectiveness provided by the new cooling configuration designs.

03 NATURAL GAS↗

DOE OSTI2022

Additive manufacturing and testing of a ceramic heat exchanger for high-temperature and high-pressure applications for concentrating solar power

Heat exchangers with excellent corrosion and oxidation resistances are essential for the next-generation concentrating solar power (CSP) plants using a molten salt heat transfer fluid and a supercritical CO 2 Brayton power cycle that can operate at temperatures > 700 °C for higher efficiencies. Techniques were developed for additively manufacturing ceramic materials for applications involving high temperature, high pressure, and high corrosion resistance as needed for the CSP application. Based on a previous ceramic heat exchanger design with its cross-section geometry of the flow channels optimized for heat transfer and mechanical strength, lab-scale prototype heat exchangers, with integrated headers and incorporated flow channel dimensional compensations, were fabricated by using the binder jetting process. The printed prototype heat exchangers were successfully densified through the processes of liquid polymer infiltration and pyrolysis. In addition, thermophysical properties of the densified silicon carbide parts were measured to provide necessary design information. Experimental heat transfer testing of the lab-scale prototype heat exchanger was conducted, and the experimental data agreed reasonably well with the simulated results. Furthermore, this agreement provides validation of the simulation models for their applicability to the development of a full-scale ceramic heat exchanger.

14 SOLAR ENERGY↗

DOE OSTI2021

AGGREGATE: dAta-driven modelinG preservinG contRollable dEr for outaGe mAnagemenT and rEsiliency (Final Report)

The AGGREGATE project team successfully developed and validated various modules for outage management. Brief summaries of each module are provided to showcase their strength for outage management and restoration for a distribution system with a high penetration of connected distribution energy resources (DERs). In recent years, inverter-based DERs have been widely deployed in distribution system. A most of behind-the-meter (BTM) solar power generation is not visible to the utility. The data-driven DER and load estimation modules are using machine learning (ML) and artificial intelligence (AI) to manage this issue, which provides an opportunity for distribution system operators (DSOs) to operate systems and make decisions in real-time for a distribution system with a high penetration of DERs deployed. Also, the estimated DER and true load can be further leveraged in network aggregation and cold-load pick up estimation for reducing the computing complexity and providing for fast restoration. After load demand and DER power generations have been estimated, the information will support topology and state estimation (SE). The topology estimation module demonstrated the viability of mixed integer linear programming (MILP) formulation to estimate the most likely operational radial topology and outage sections using power flow measurements, historical/estimated load and DERs data and smart meter ping measurements. Formulation includes continuous (power flow, load and DERs data) and binary measurements (smart meter ping measurements) in a single formulation. Errors in continuous data and binary data are modeled as normal distribution and Bernoulli distribution, respectively. In the future distribution grid, the power injection from controllable DERs will be essential for efficient and resilient grid operation. However, determining the optimal DER injections and restoration actions is dependent on knowledge of the system states. State estimation (SE), already the cornerstone of transmission energy management systems, will become commonplace in distribution management systems as more measurements become available from deployment of automated metering infrastructure (AMI). Observability analysis is the first step in SE, as it determines the sufficiency of the available measurements for accurately estimating the current system states. A new type of pseudo-measurement called a Correlational Measurement (CM) is introduced in this module, to enhance the observability of the system to enable more accurate SE. CMs encapsulate knowledge of correlation between demand patterns for similar classes of loads as well as injection patterns for same-technology renewable DERs. During grid contingency scenarios, DERs have been traditionally disconnected, without any fault ride-through capabilities. However, with new regulations and better technology, it is feasible for these resources to contribute to the grid’s restoration after an adverse event and hence enhance resilience. The controllability module proposes a two-step restoration scheme for the power system restoration process by leveraging additional degrees of freedom in power electronics interfaced DERs for mitigating voltage problems. In a resilience mode without the utility system, the distribution grid relies on DERs to serve critical load. In such a severe event with multiple faults on the distribution feeders, actuation of various protective devices (PDs) divides the distribution system into electrical islands. The undetected actuated PDs due to fault current contributions from DERs can delay the restoration process, thereby reducing the system resilience. The Advanced Outage Management (AOM) and the Advanced Feeder Restoration (AFR) modules developed in this project provide improved system resilience with multiple DERs. AOM identifies the faulted sections and actuated PDs in a distribution system with DERs by incorporating smart meter data. The most credible outage scenario including fault locations, PD actuations, and fault indicator (FI) failures is identified by a set of binary integer linear programming incorporating hypotheses. The AFR module serves to restore a distribution system with available energy resources taking into consideration the availability of utility sources and DERs. By partitioning the system into islands, critical load will be served with the available generation resources within islands based on the solution of a MILP. When the utility systems become available, the optimal path will be determined by a spanning tree search algorithm that reconnects these islands back to substations and restores the remaining load. The transmission and distribution (T&D) co-simulation module was used to validate the effect of a control action performed on the distribution side assets as it propagates to the transmission side. This ensures that the control action performed results in a feasible operating point on both the transmission and the distribution system. In addition to validation, the team used the T&D co-simulation module to demonstrate how distribution system assets can be used to mitigate issues on the transmission system. Specifically, the team demonstrated that appropriate switching operations on the distribution side can alleviate the line overload condition on the transmission side without causing new operational constraint violations.

24 POWER TRANSMISSION AND DISTRIBUTION↗

NASA NTRS2014

Design Optimization of a Variable-Speed Power Turbine

NASA's Rotary Wing Project is investigating technologies that will enable the development of revolutionary civil tilt rotor aircraft. Previous studies have shown that for large tilt rotor aircraft to be viable, the rotor speeds need to be slowed significantly during the cruise portion of the flight. This requirement to slow the rotors during cruise presents an interesting challenge to the propulsion system designer as efficient engine performance must be achieved at two drastically different operating conditions. One potential solution to this challenge is to use a transmission with multiple gear ratios and shift to the appropriate ratio during flight. This solution will require a large transmission that is likely to be maintenance intensive and will require a complex shifting procedure to maintain power to the rotors at all times. An alternative solution is to use a fixed gear ratio transmission and require the power turbine to operate efficiently over the entire speed range. This concept is referred to as a variable-speed power-turbine (VSPT) and is the focus of the current study. This paper explores the design of a variable speed power turbine for civil tilt rotor applications using design optimization techniques applied to NASA's new meanline tool, the Object-Oriented Turbomachinery Analysis Code (OTAC).

design optimization↗

DOE OSTI2021

Pressure Response Optimization of an Eddy Current-Driven Flyer Plate Valve for the ITER Shattered Pellet Injection System

One technique for mitigating disruptions in a tokamak is shattered pellet injection (SPI). SPI is a process in which a large solid pellet consisting of deuterium, neon, or argon is desublimated in a pipe gun barsrel and launched downstream. Pellets are shattered just before entering the plasma by an impact with an angled tube. Injection of these materials into the plasma radiates stored thermal energy, limits current decay rates, suppresses the generation of runaway electrons, and dissipates runaway electrons if necessary. A critical element of the SPI system is a fast-acting valve that releases high-pressure gas to dislodge and accelerate pellets directly, or indirectly via a mechanical punch. A prototype valve sized for the ITER SPI system has been designed and fabricated. A pulsed high-voltage power supply energizes the valve’s internal magnetic coil, which induces eddy currents in the adjacent flyer plate resulting in a repulsive force between the flyer plate and the coil. The flyer plate action lifts a valve seat, allowing high-pressure gas to flow from the valve plenum to the downstream (breech) location of the pellet or mechanical punch. All of the valve’s internal components are designed to operate in ITER-level static background magnetic fields. Here, a study was conducted to optimize the downstream pressure response for a range of valve sizes and operating pressures. In particular, the study analyzes the breech pressure response associated with varying plenum pressures as well as varying breech volumes. A computational fluid dynamics simulation was built in STAR-CCM+ and validated against data from laboratory experiments. The resulting simulation outputs, in the form of downstream responses for a variety of initial plenum pressures and breech volumes, will be used as a complement to experimental data to ensure the pressure pulse is suitable for pellet survivability. These data, combined with studies on pellet shear strength and shock response, will be applied to optimization of overall operating parameters of the ITER SPI system.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

Incorporating Valve Switching Losses Into a Static Optimal Control Algorithm for the Hybrid Hydraulic-Electric Architecture (HHEA)

The Hybrid Hydraulic-Electric Architecture (HHEA) has, in recent years, been proposed as an energy efficient alternative to conventional load-sensing architectures in mobile machines such as excavators and wheel-loaders. HHEA leverages the advantages of hydraulic power and electric power to eliminate throttling valves while also improving the energy and control performance of the system. The architecture utilizes a set of common pressure rails to provide a majority of power and and a small electric motor driven pump to modulate this power to meet the exact demand. Previous work has developed a computationally efficient Lagrange Multiplier approach for determining the optimal pressure rail selections that minimizes the energy losses in the system. The static model used considers only the energy use for each pressure rail selection but not the losses associated with the valves during the transition. This paper presents an approach to include the switching losses in the model and in the optimization procedure. To capture the switching losses, switching events between different rails and at various input and output flow rates were simulated with consideration of valve spool dynamics. A parameterized model that summarizes the losses is then obtained, allowing switching losses to be added to the previous energy analysis. The performance of the switching loss model was compared with reference data obtained from a high-fidelity simulation model. To incorporate the switching losses into optimal control algorithm, an efficient dynamic programming approach that prevents frequent switching is adopted in place of the Lagrange multiplier approach. The overall effect of switching losses on the energy consumption and optimal control decisions is presented. In general, switching losses contribute to about 9–10% of input energy.

24 POWER TRANSMISSION AND DISTRIBUTION↗

NASA NTRSDate not supplied

Transient Optimization of a Gas Turbine Engine

Gas turbine engines are the primary power plants for modern commercial aircraft. Transients prompted by significant changes in thrust or power demand are common and unavoidable. Extreme transient scenarios such as those associated with a go-around during a landing attempt are possible and must be accounted for in the design of the engine and its controller. Engine transients tend to cause a reduction in compressor operability margin, which must be addressed by the engine control system and accounted for in the engine design to prevent events such as compressor stall/surge and combustor blow out. Transient operability concerns typically lead to compromises in the engine design that sacrifice efficiency and/or limit responsiveness. Transient operability is typically managed by logic that limits the fuel flow command. If this logic is not optimized, then the potential for valuable performance could be lost. This study presents a strategy for optimizing the transient limit logic and proposes a strategy for updating the control logic over the lifespan of the engine. The results demonstrate significant improvements in transient operability. For example, of the results at sea level static conditions demonstrated a 31% reduction in the usage of the high pressure compressor operability stack during a snap acceleration transient. Furthermore, a reinforcement learning algorithm is demonstrated to modify the transient logic as the engine degrades to minimize response time while respecting a prescribed compressor operability margin limit. A simple demonstration of the reinforcement learning algorithm resulted in a thrust response time reduction of ~11.8%.

transient↗

NASA NTRSDate not supplied

Transient Optimization of a Gas Turbine Engine

Gas turbine engines are the primary power plants for modern commercial aircraft. Transients prompted by significant changes in thrust or power demand are common and unavoidable. Extreme transient scenarios such as those associated with a go-around during a landing attempt are possible and must be accounted for in the design of the engine and its controller. Engine transients tend to cause a reduction in compressor operability margin, which must be addressed by the engine control system and accounted for in the engine design to prevent events such as compressor stall/surge and combustor blow out. Transient operability concerns typically lead to compromises in the engine design that sacrifice efficiency and/or limit responsiveness. Transient operability is typically managed by logic that limits the fuel flow command. If this logic is not optimized, then the potential for valuable performance could be lost. This study presents a strategy for optimizing the transient limit logic and proposes a strategy for updating the control logic over the lifespan of the engine. The results demonstrate significant improvements in transient operability. For example, of the results at sea level static conditions demonstrated a 31% reduction in the usage of the high pressure compressor operability stack during a snap acceleration transient. Furthermore, a reinforcement learning algorithm is demonstrated to modify the transient logic as the engine degrades to minimize response time while respecting a prescribed compressor operability margin limit. A simple demonstration of the reinforcement learning algorithm resulted in a thrust response time reduction of ~11.8%.

transient↗

NASA NTRSDate not supplied

Transient Optimization of a Gas Turbine Engine

Gas turbine engines are the primary power plants for modern commercial aircraft. Transients prompted by significant changes in thrust or power demand are common and unavoidable. Extreme transient scenarios such as those associated with a go-around during a landing attempt are possible and must be accounted for in the design of the engine and its controller. Engine transients tend to cause a reduction in compressor operability margin, which must be addressed by the engine control system and accounted for in the engine design to prevent events such as compressor stall/surge and combustor blow out. Transient operability concerns typically lead to compromises in the engine design that sacrifice efficiency and/or limit responsiveness. Transient operability is typically managed by logic that limits the fuel flow command. If this logic is not optimized, then the potential for valuable performance could be lost. This study presents a strategy for optimizing the transient limit logic and proposes a strategy for updating the control logic over the lifespan of the engine. The results demonstrate significant improvements in transient operability. For example, of the results at sea level static conditions demonstrated a 31% reduction in the usage of the high pressure compressor operability stack during a snap acceleration transient. Furthermore, a reinforcement learning algorithm is demonstrated to modify the transient logic as the engine degrades to minimize response time while respecting a prescribed compressor operability margin limit. A simple demonstration of the reinforcement learning algorithm resulted in a thrust response time reduction of ~11.8%.

transient↗

DOE OSTI2025

A Two-Stage Approach for PV Inverter Engagement in Power Factor Correction and Voltage Regulation

The rapid integration of distributed energy resources, like solar photovoltaics (PVs), can lead to overvolt-age challenges due to reverse power flow and a noticeable decrease in power factor at the substation interface. While existing literature extensively explores utilizing smart inverter capabilities for reactive power flexibility using a volt-var curve (VVC), obtaining time-varying operating points of such curves in real-time is challenging due to computational demands and communication requirements. Similarly, employing optimization-based approaches for reactive power control and active voltage regulation in large-scale distribution feeders is difficult due to the complexity of the problem and the challenges in effectively engaging customer-owned resources. This paper proposes a two-stage strategy to harness smart inverters for reactive power support. The first stage formulates short-term planning by optimally designing VVCs (on a daily or hourly basis) for large-scale solar PVs based on projected system needs and communicating optimal curves to smart inverters in advance. Subsequently, the second stage employs a transactive-based method to involve customer-owned PVs for reactive power support, effectively enhancing overall system performance and addressing real-time demands. In conclusion, the efficacy of this approach will be demonstrated using real-world distribution circuits provided by Vermont Electric Power Company (VELCO) and Vermont Electric Cooperative (VEC).

Poudel, Shiva [Pacific Northwest National Laborato↗

NASA NTRS1985

Performance study of a data flow architecture

Teams of scientists studied data flow concepts, static data flow machine architecture, and the VAL language. Each team mapped its application onto the machine and coded it in VAL. The principal findings of the study were: (1) Five of the seven applications used the full power of the target machine. The galactic simulation and multigrid fluid flow teams found that a significantly smaller version of the machine (16 processing elements) would suffice. (2) A number of machine design parameters including processing element (PE) function unit numbers, array memory size and bandwidth, and routing network capability were found to be crucial for optimal machine performance. (3) The study participants readily acquired VAL programming skills. (4) Participants learned that application-based performance evaluation is a sound method of evaluating new computer architectures, even those that are not fully specified. During the course of the study, participants developed models for using computers to solve numerical problems and for evaluating new architectures. These models form the bases for future evaluation studies.

Adams, George↗

NASA NTRS1993

An overview of the Solar, Anomalous, and Magnetospheric Particle Explorer (SAMPEX) mission

The scientific objective of the NASA Small-class Explorer Mission SAMPEX is summarized. A brief history of the Small Explorer program is provided along with a description of the SAMPEX project development and structure. The spacecraft and scientific instrument configuration is presented. The orbit of SAMPEX has an altitude of 520 by 670 km and an 82 deg inclination. Maximum possible power is provided by articulated solar arrays that point continuously toward the sun. Highly sensitive science instruments point generally toward the local zenith, especially over the terrestrial poles, in order to measure optimally the galactic and solar cosmic ray flux. Energetic magnetospheric particle precipitation is monitored at lower geomagnetic latitudes. The spacecraft uses several innovative approaches including an optical fiber bus, powerful onboard computers, and large solid state memories (instead of tape recorders). Spacecraft communication and data acquisition are discussed, and the space- and ground-segment data flows are summarized. A mission lifetime of 3 years is sought with the goal of extending data acquisition over an even longer portion of the 11-year solar activity cycle.

Baker, Daniel N.↗

DOE OSTI2025

Control of core–shell nanoparticles properties through plasma synthesis: a computational study

The improved properties of core–shell nanoparticles (CSNPs) over homogeneous nanoparticles (NPs) have expanded and diversified the applications of these nanomaterials. However, controlling the properties of CSNPs can be a challenging task. Low temperature plasmas have proven to be an effective method of producing NPs with uniform size and morphology, and high yield. That said, NP transport and growth dynamics are sensitive to LTP properties. We report on a computational investigation of the evolution of Ge–Si CSNP properties as a function of operating conditions through the modeling of a flowing, two-zone inductively coupled plasma (ICP) reactor. Ar/GeH 4 and Ar/SiH 4 gas mixtures were supplied to separate plasma zones at a pressure of 1 Torr to promote growth of Ge cores and Si shells. The negatively charged CSNPs are trapped electrostatically in the vicinity of the antennas where the plasma is generated and where the majority of particle growth occurs. Particles that grow to a critical size are then de-trapped by fluid drag due to neutral gas flow. A two-dimensional hybrid plasma model coupled with a three-dimensional kinetic NP transport model were utilized to resolve plasma chemistry and NP growth processes that take place on distinct timescales. The trends in CSNP properties and trapping mechanisms associated with flow rate, applied ICP power and inlet precursor fraction are discussed. While the spatial distribution of plasma produced radical species can have significant impact on the NP growth process, the NP transport dynamics are what ultimately dictates the growth environment that is unique to each particle and so determines their final dimension and composition. The key to optimizing reactor conditions involves controlling the spatial density of growth species and plasma profile as a means to tailor particle trapping dynamics suitable to produce CSNPs for a specific application.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Binder-Coated Carbon Cloth Electrodes for All-Vanadium Redox Flow Batteries

Vanadium redox flow batteries (VRFBs) are a promising solution for integrating intermittent renewable energy sources into the existing power grid. However, enhancing the electrochemical performance of VRFBs is critical for their widespread adoption in grid-scale energy storage. This study investigates the impact of adding a porous binder to a carbon-cloth electrode, with a focus on optimizing thermal activation conditions. The electrochemical performance of the binder-coated electrodes compared to uncoated electrodes is evaluated through electrochemical impedance spectroscopy, polarization curve measurements, and charge-discharge cycling. The surface morphology and structural integrity of the binder-coated electrodes at each activation stage are examined using various material characterization techniques to assess the effects of thermal activation. The results are benchmarked against the experiments using non-coated electrodes to determine the performance improvements offered by the binder coating. Notably, the study reveals that binder-coated electrodes exhibit significantly lower resistance and improved efficiency compared to their uncoated counterparts, with optimal activation conditions enhancing performance metrics crucial for VRFB applications. These findings provide valuable insights for further optimizing electrode design and activation strategies, advancing the development of more efficient VRFB systems for large-scale energy storage.

Caiado, Ashley A.↗

NASA NTRS1987

Optimizing the antenna system of a microwave space power station: Implications for the selection of operating power, frequency and antenna size

A design for a space power station that is to transmit power to the surface of a planet via high powered microwaves should commence with the optimum design of the transmitting and receiving antenna combination to be employed. Once one has assured that the desired amount of power has been transferred (which, after all, is the prupose of any power transmission system), one can, from the constraints imposed by such a design, taylor other parameters of the system such as antenna sizes and weights, power density in the planet's atmosphere (e.g., to avoid electrical breakdown), and frequency of operation. It is the purpose of this brief analysis to provide the working equations of such an optimized antenna system, and to give examples of their use. Related problems that should be analyzed in the future will then be discussed and a flow chart of the indicated order of priority presented. The analysis given here differs from previous work on this subject in that the development given will allow analytical expressions to be obtained for the relevant parameters. This is made possible by employing an approximation procedure to be given during the exposition.

Manning, Robert M.↗