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NASA NTRS2010

A Terminal Area Analysis of Continuous Ascent Departure Fuel Use at Dallas/Fort Worth International Airport

Aircraft departing from the Dallas/Fort Worth International Airport (DFW) encounter vertical restrictions that prevent continuous ascent operations. The result of these restrictions are temporary level-offs at 10,000 feet. A combination of flow direction, specific Area Navigation (RNAV) route geometry, and arrival streams have been found to be the biggest factors in the duration and frequency of a temporary level-offs. In total, 20% of DFW departures are affected by these level-offs, which have an average duration of just over 100 seconds. The use of continuous descent approaches at DFW are shown to lessen the impact arrivals have on the departures and allow more continuous ascents. The fuel used in a continuous ascent and an ascent with a temporary level-off have been calculated using a fuel burn rate model created from a combination of actual aircraft track data, aircraft manufacturer flight operations manuals, and Eurocontrol's Base of Aircraft Data (BADA) simulation tool. This model represents the average aggregate burn rates for the current fleet mix at DFW. Continuous ascents would save approximately seven gallons of fuel out of 450 gallons used to climb to a cruise altitude of 31,000ft per departure.

Roach, Keenan↗

NASA NTRSDate not supplied

Predicting the Microgravity Performance of Terrestrial Portable Fire Extinguishers

Portable fire extinguishers (PFEs) are a key component of spacecraft emergency response systems. The International Space Station uses custom PFEs to meet the unique microgravity and enclosed space requirements of the vehicle. For future crewed missions to Low Earth Orbit and Deep Space destinations, terrestrial commercial off-the-shelf (COTS) PFEs may offer more economical solutions for the destination as well as the visiting vehicle. However, depending on the design of both the spacecraft and the PFE, the reduced gravity environment is likely to impact PFE performance, especially when multiphase microgravity liquid transport processes are present. Additionally, many terrestrial PFEs now use halocarbons as the suppressant. While studies show these are effective in certain terrestrial fires, some data suggest that low concentrations may actually exacerbate a fire, a phenomenon that would prove particularly concerning in spacecraft. To better characterize the potential of employing COTS hardware, two terrestrial PFEs, one charged with HFC-227ea and Nitrogen pressurant and one charged with carbon dioxide, were modeled to predict performance in both 1-g and microgravity environments. Testing was conducted in 1-g to validate the model. Testing was also conducted to evaluate the effect of sub-extinguishment HFC-227ea concentrations on burn rate in controlled samples. Here we provide a detailed description of the model, report the methods and results of PFE testing, discuss the predicted effects of microgravity on PFE performance, and report the results of material burn rates at sub-extinguishment levels of HFC-227ea.

Morgan B Abney↗

NASA NTRS1997

Combustion of Interacting Droplet Arrays in a Microgravity Environment

This research program involves the study of single droplets and linear arrays of droplets in weakly-buoyant and non-bouyant environments. The primary purpose of the single droplet work was to (1) provide a data base from which to compare droplet array results and (2) to correlate the effects of buoyancy on flame shape. Traditionally convective effects in droplet combustion are represented in terms of the Reynolds number, Re, for forced convection and the Grashof number, Gr, for natural convection. Typically, corrections to the burning rate constant for convective effects are written in terms of Re or Gr(exp 1). The Stefan velocity is not included in these correlations, even though from purely physical reasons, one would expect it to be important, especially at higher burning rates. The flame distortion due to convective effects is less documented quantitatively. Kumagai and Isoda do predict flame shape in natural and forced convective flow fields. Their focus, however, was to predict the actual flame dimensions. Law and co-workers used reduced pressure, high oxidizer ambients to obtain spherical flames. This implies that buoyant flows were reduced at the low pressures, as indicated by a very small Grashof number. Ross et al, however, using scaling arguments showed that reducing the pressure does not have a large effect on the magnitude of the buoyant velocity. Struk et al showed elongated flame shapes during simulated (porous sphere) droplet combustion. The elongation of the flames was due to residual gravity levels aboard the reduced gravity aircraft on which the experiments were conducted. These flame shapes, as well as some data from the literature were interpreted based on a dimensionless grouping called the sphericity parameter, Sp. Sp is the ratio of a characteristic computed buoyant velocity to the Stefan velocity at the flame front. One purpose of the droplet arrays work is to extend the database and theories that exist for single droplets into the regime where droplet interactions are important. The eventual goal being to use the results of this work as inputs to models on spray combustion where droplets seldom burn individually; instead the combustion history of a droplet is strongly influenced by the presence of the neighboring droplets. Recently, Annamali and Ryan have summarized he current status of droplet array, cloud and spray combustion. A number of simplified theories led numerical studies of droplet vaporization/combustion where multiple droplet effects are present are now available. These theories all neglect the effect of buoyancy. Experimentally, most studies to date suffer the effects of buoyancy. It is the dominant transport mechanism in the problem. Only the works of Law and co-worker and more recently by Mikami et al were performed in an environment where buoyancy effects were small. Law and co-workers were limited to high oxygen index, low pressure ambient environments since there studies were conducted in normal gravity.

Dietrich, Daniel L.↗

NASA NTRS2000

SEM Characterization of Extinguished Grains from Plasma-Ignited M30 Charges

M30 propellant grains that had been ignited in interrupted closed bomb experiments were characterize by scanning electron microscopy (SEM). Previous chemical analysis of extinguished grains had given no indications of plasma-propellant chemical interactions that could explain the increased burning rates that had been previously observed in full-pressure closed bomb experiments. (This does not mean that there is no unique chemistry occurring with plasma ignition. It may occur very early in the ignition event and then become obscured by the burning chemistry.) In this work, SEM was used to look at grain morphologies to determine if there were increases in the surface areas of the plasma-ignited grains which would contribute to the apparent increase in the burning rate. Charges were made using 30 propellant grains (approximately 32 grams) stacked in two tiers and in two concentric circles around a plastic straw. Each grain was notched so that, when the grains were expelled from the bomb during extinguishment, it could be determined in which tier and which circle each grain was originally packed. Charges were ignited in a closed bomb by either a nickel wire/Mylar-capillary plasma or black powder. The bomb contained a blowout disk that ruptured when the pressure reached 35 MPa, and the propellant was vented into a collection chamber packed with polyurethane foam. SEM analysis of the grains fired with a conventional black powder igniter showed no signs of unusual burning characteristics. The surfaces seemed to be evenly burned on the exteriors of the grains and in the perforations. Grains that had been subjected to plasma ignition, however, had pits, gouges, chasms, and cracks in the surfaces. The sides of the grains closest to the plasma had the greatest amount of damage, but even surfaces facing the outer wall of the bomb had small pits. The perforations contained gouges and abnormally burned regions (wormholes) that extended into the web. The SEM photos indicated that a grain from the top tier, which was farther away from the plasma ignition source, sustained more plasma-induced damage to the perforations and the web than did the grains on the bottom tier.

Kinkennon, A.↗

NASA NTRS1995

Combustion of a polymer (PMMA) sphere in microgravity

Polymer combustion is a highly complicated process where chemical reactions may occur not only in the gas phase, but also in the condensed phase as well as at the solid-gas interphase. The chemistry depends strongly on the coupling between the condensed phase and gas phase phenomena. For some polymers, additional complications arise due to the formation of char layers. For others, the behavior of the condensed phase involves swelling, bubbling, melting, sputtering, and multi-stage combustion. Some of these features bear resemblance to the phenomena observed in coal particle combustion. In addition to its relevance to spacecraft fire safety, the combustion of polymeric materials is related to many applications including solid and hybrid rocket propulsion, and of recent interest, waste incineration . The burning rate is one of the most important parameters used to characterize the combustion of polymers. It has been used to rank the polymer flammability under the same experimental conditions and to evaluate various modes of inhibiting polymer flammability. The main objective of this work is to measure the burning rates of a polymeric material in low gravity. Because of inherent logistical difficulties involved in microgravity experiments, it is impossible to examine a wide spectrum of polymeric materials. It is desirable to investigate a polymer whose combustion is less complicated, and yet will lead to a better understanding of the burning characteristics of other more complicated materials. Therefore, a typical non-charring polymer is selected for use in this experimental study. PMMA (polymethylmethacrylate) has been chosen because its thermo-physical properties are well characterized. Although the combustion of PMMA has been extensively studied in 1G experiments, only a limited amount of work has been conducted in low gravity. A spherical sample geometry is chosen in this study because it is the simplest configuration in terms of the microgravity hardware design requirements. Furthermore, a burning PMMA sphere in microgravity represents a one-dimensional flame with overall combustion characteristics expected to be analogous to the combustion of a liquid fuel droplet, a field with many well-developed theories and models. However, differences can also be expected such as the flame-front standoff ratios and the condensed phase processes occurring during combustion.

Yang, Jiann C.↗

NASA NTRS2000

Viscous Analysis of Pulsating Hydrodynamic Instability and Thermal Coupling Liquid-Propellant Combustion

A pulsating form of hydrodynamic instability has recently been shown to arise during liquid-propellant deflagration in those parameter regimes where the pressure-dependent burning rate is characterized by a negative pressure sensitivity. This type of instability can coexist with the classical cellular, or Landau form of hydrodynamic instability, with the occurrence of either dependent on whether the pressure sensitivity is sufficiently large or small in magnitude. For the inviscid problem, it has been shown that, when the burning rate is realistically allowed to depend on temperature as well as pressure, sufficiently large values of the temperature sensitivity relative to the pressure sensitivity causes like pulsating form of hydrodynamic instability to become dominant. In that regime, steady, planar burning becomes intrinsically unstable to pulsating disturbances whose wave numbers are sufficiently small. This analysis is extended to the fully viscous case, where it is shown that although viscosity is stabilizing for intermediate and larger wave number perturbations, the intrinsic pulsating instability for small wave numbers remains. Under these conditions, liquid-propellant combustion is predicted to be characterized by large unsteady cells along the liquid/gas interface.

Margolis, Stephen B.↗

NASA NTRS2000

Viscous and Thermal Effects on Hydrodynamic Instability in Liquid-Propellant Combustion

A pulsating form of hydrodynamic instability has recently been shown to arise during the deflagration of liquid propellants in those parameter regimes where the pressure-dependent burning rate is characterized by a negative pressure sensitivity. This type of instability can coexist with the classical cellular, or Landau, form of hydrodynamic instability, with the occurrence of either dependent on whether the pressure sensitivity is sufficiently large or small in magnitude. For the inviscid problem, it has been shown that when the burning rate is realistically allowed to depend on temperature as well as pressure, that sufficiently large values of the temperature sensitivity relative to the pressure sensitivity causes the pulsating form of hydrodynamic instability to become dominant. In that regime, steady, planar burning becomes intrinsically unstable to pulsating disturbances whose wavenumbers are sufficiently small. In the present work, this analysis is extended to the fully viscous case, where it is shown that although viscosity is stabilizing for intermediate and larger wavenumber perturbations, the intrinsic pulsating instability for small wavenumbers remains. Under these conditions, liquid-propellant combustion is predicted to be characterized by large unsteady cells along the liquid/gas interface.

Margolis, Stephen B.↗

NASA NTRS1980

Analysis of solid propellant combustion in a closed vessel including secondary reaction

A theory for combustion of solid propellants in a closed vessel is presented allowing for residual exothermic chemical reaction in the bulk of the gas in the vessel. Particular attention is given to propellants exhibiting thick gaseous flame zones such as nitrocellulose, double-base and nitramine propellants. For these, the reaction at high pressures is assumed to involve mainly the oxidation of residual hydrocarbons by NO. It is shown that the direct dynamic coupling between the exothermicity, the molecular weight reduction and the changing pressure can influence the dp/dt-p traces obtained, in a manner not directly related to mass burning rate of the solid. Energy and species conservation equations are derived for the bulk of the vessel in differential form; the system is solved numerically. The results show the effect of extended chemical reaction upon measurable combustion characteristics such as dp/dt-p and burn rate pressure exponent, demonstrating its potential importance in interpretation of closed vessel firing data, depending on the pace of the residual gas phase reactions.

Benreuven, M.↗

NASA NTRS1992

The 3D rocket combustor acoustics model

The theory and procedures for determining the characteristics of pressure oscillations in rocket engines with prescribed burning rate oscillations are presented. Analyses including radial and hub baffles and absorbers can be performed in one, two, and three dimensions. Pressure and velocity oscillations calculated using this procedure are presented for the SSME to show the influence of baffles and absorbers on the burning rate oscillations required to achieve neutral stability. Comparisons are made between the results obtained utilizing 1-D, 2-D, and 3-D assumptions with regards to capturing the physical phenomena of interest and computational requirements.

Priem, Richard J.↗

NASA NTRS1992

3D rocket combustor acoustics model

The theory and procedures for determining the characteristics of pressure oscillations in rocket engines with prescribed burning rate oscillations are presented. Analyses including radial and hub baffles and absorbers can be performed in one, two, or three dimensions. Pressure and velocity oscillations calculated using this procedure are presented for the SSME to show the influence of baffles and absorbers on the burning rate oscillations required to achieve neutral stability. Comparisons are made between the results obtained utilizing 1D, 2D, and 3D assumption with regards to capturing the physical phenomena of interest and computational requirements.

Priem, Richard J.↗

NASA NTRS1993

The use of designed experiments in the process development of continuous propellant mixing

A continuous mix pilot plant was constructed at Aerojet Propulsion Division in Sacramento, California to develop a robust propellant mixing process for the full scale plant that was to be built at the NASA Advanced Solid Rocket Motor facility Yellow Creek, Mississippi. The plant was used to conduct dozens of subsystem and full system mixing tests for evaluation of equipment, processing methods, and control schemes for later use at the production plant. As a culmination to this work, a series of designed experiments were conducted using an eight run Taguchi analysis with four factors at two levels each to determine the primary effect of processing parameters on propellant ballistic and mechanical properties. The factors examined in these runs included the propellant production rate (454 (1000) and 622 kg/hr (1371 Ib/hr)), the product temperature out of the mixer (49 (120) and 63 deg C (145 deg F)), mixer screw speed (75 and 90 rpm), and the deaerator excess capacity (20 and 80 percent). Measured response variables included the uncured and cured density, Crawford Bomb liquid strand burning rates, and selected mechanical properties. The experiment revealed that several of the response variables displayed significant changes from run-to-run with the product temperature being the single most important factor. After concluding this experiment, a twenty-six hour confirmation run was conducted to verify the conclusions reached in the designed experiment. The extended run produced over 12,250 kgs (27,000 lbs) of propellant meeting all of the pre-run targeted properties including density (1.803 g/cc (0.065 lb/in(exp 3)) with a 0.12 percent coefficient of variation (CV) at 25 deg C (77 deg F)), liquid strand burn rate (0.889 cm/s (0.350 in/s) with a 0.69 percent CV at 4210 KPa (610 psig), 15.6 deg C (60 deg F)), nominal maximum stress (828 KPa (120 psig) with a 2.84 percent CV, S&E at 25 deg C (77 deg F), 5.08 cm/min (2 in/min)), strain at nominal maximum (47.4 percent with a 3.96 percent CV), and initial tangent modulus (5349 KPa (775 psig) with a 7.26 percent CV).

Campbell, J. A.↗

NASA NTRS1998

Fiber-Supported Droplet Combustion

Individual droplets with diameters ranging from about 2 mm to 5 mm were burned under microgravity conditions in air at 1 bar with an ambient temperature of 300 K. Each droplet was tethered by a silicon carbide fiber of 80 mm or 150 mm diameter to keep it in view of video recording, and, in some tests, a forced air flow was applied in a direction parallel to the fiber axis. Methanol, two methanol-water mixtures, two methanol-dodecanol mixtures, and two heptane-hexadecane mixtures were the fuels. Droplet diameters were measured as functions of time and compared with existing theoretical predictions. The prediction that methanol droplets extinguish at diameters that increase with increasing initial droplet diameter is verified by these experiments. In addition, the quasi-steady burning rate constant of the heptane-hexadecane mixtures appears to decrease with increasing droplet diameter; obscuration consistent with very heavy sooting, but without the formation of soot shells, is observed for the largest of these droplets. Forced convective flow around methanol droplets was found to increase the burning rate and to produce a ratio of downstream-to-upstream flame radius that remained constant as the droplet size decreased, a trend in agreement with earlier results obtained at higher convective velocities for smaller droplets having larger flame standoff ratios. There are a number of implications of the experimental results regarding droplet-combustion theory.

Dietrich, Daniel L.↗

NASA NTRS1994

The effect of initial diameter in sperically symmetric droplet combustion of sooting fuels

The effect of initial droplet diameter on the burning rate of sooting fuels - n-heptane and 1-chloro-octane - was examined experimentally at low gravity. A 1.2s drop tower provided a low gravity environment to minimize buoyancy and achieve spherically symmetric flames for stationary droplets. Free-floating and fiber-supported droplets were burned, and both techniques gave matching results for droplets of similar initial diameter. Burning rate constants for both fuels were measured for a large number of droplets ranging from 0.4 to 1.1mm in initial diameter.

Jackson, G. S.↗

NASA NTRS1999

Hydrodynamic Instability and Thermal Coupling in a Dynamic Model of Liquid-Propellant Combustion

For liquid-propellant combustion, the Landau/Levich hydrodynamic models have been combined and extended to account for a dynamic dependence of the burning rate on the local pressure and temperature fields. Analysis of these extended models is greatly facilitated by exploiting the realistic smallness of the gas-to-liquid density ratio rho. Neglecting thermal coupling effects, an asymptotic expression was then derived for the cellular stability boundary A(sub p)(k) where A(sub p) is the pressure sensitivity of the burning rate and k is the disturbance wavenumber. The results explicitly indicate the stabilizing effects of gravity on long-wave disturbances, and those of viscosity and surface tension on short-wave perturbations, and the instability associated with intermediate wavenumbers for critical negative values of A(sub p). In the limit of weak gravity, hydrodynamic instability in liquid-propellant combustion becomes a long-wave, instability phenomenon, whereas at normal gravity, this instability is first manifested through O(1) wavenumbers. In addition, surface tension and viscosity (both liquid and gas) each produce comparable effects in the large-wavenumber regime, thereby providing important modifications to the previous analyses in which one or more of these effects was neglected. For A(sub p)= O, the Landau/Levich results are recovered in appropriate limiting cases, although this typically corresponds to a hydrodynamically unstable parameter regime for p << 1. In addition to the classical cellular form of hydrodynamic stability, there exists a pulsating form corresponding to the loss of stability of steady, planar burning to time-dependent perturbations. This occurs for negative values of the parameter A(sub p), and is thus absent from the original Landau/Levich models. In the extended model, however, there exists a stable band of negative pressure sensitivities bounded above by the Landau type of instability, and below by this pulsating form of hydrodynamic instability. Indeed, nonsteady modes of combustion have been observed at low pressures in hydroxylammonium nitrate (HAN)-based liquid propellants, which often exhibit negative pressure sensitivities. While nonsteady combustion may correspond to secondary and higher-order bifurcations above the cellular boundary, it may also be a manifestation of this pulsating type of hydrodynamic instability. In the present work, a nonzero temperature sensitivity is incorporated into our previous asymptotic analyses. This entails a coupling of the energy equation to the previous purely hydrodynamic problem, and leads to a significant modification of the pulsating boundary such that, for sufficiently large values of the temperature-sensitivity parameter, liquid-propellant combustion can become intrinsically unstable to this alternative form of hydrodynamic instability. For simplicity, further attention is confined here to the inviscid version of the problem since, despite the fact that viscous and surface-tension effects are comparable, the qualitative nature of the cellular boundary remains preserved in the zero-viscosity limit, as does the existence of the pulsating boundary. The mathematical model adopts the classical assumption that there is no distributed reaction in either the liquid or gas phases, but now the reaction sheet, representing either a pyrolysis reaction or an exothermic decomposition at the liquid/gas interface, is assumed to depend on local conditions there.

Margolis, S. B.↗

NASA NTRS2000

The Burning of Large N-Heptane Droplets in Microgravity

Experimental results are presented on the burning and sooting behavior of large n-heptane droplets in air at atmospheric pressure under microgravity conditions. The experiments were performed at the Japanese Microgravity Center (JAMIC) 10 sec dropshaft in Hokkaido, Japan. Soot volume fraction, burning rate, flame standoff and luminosity were measured for droplets of 2.6 mm and 2.9 mm in initial diameter. These are the largest droplets for which soot volume fraction measurements have ever been performed. Previous measurements of soot volume fractions for n-heptane droplets, confined to smaller droplet sizes of less than 1.8 mm, indicated that maximum soot volume fraction increased monotonically with initial droplet size. The new results demonstrate for the first time that sooting tendency is reduced for large droplets as it has been speculated previously but never confirmed experimentally. The lower soot volume fractions for the larger droplets were also accompanied by higher burning rates. The observed phenomenon is believed to be caused by the dimensional influence on radiative heat losses from the flame. Numerical calculations confirm that soot radiation affects the droplet burning behavior.

Manzello, Samuel L.↗

NASA NTRS2012

Mechanical and Combustion Performance of Multi-Walled Carbon Nanotubes as an Additive to Paraffin-Based Solid Fuels for Hybrid Rockets

Paraffin-based solid fuels for hybrid rocket motor applications are recognized as a fastburning alternative to other fuel binders such as HTPB, but efforts to further improve the burning rate and mechanical properties of paraffin are still necessary. One approach that is considered in this study is to use multi-walled carbon nanotubes (MWNT) as an additive to paraffin wax. Carbon nanotubes provide increased electrical and thermal conductivity to the solid-fuel grains to which they are added, which can improve the mass burning rate. Furthermore, the addition of ultra-fine aluminum particles to the paraffin/MWNT fuel grains can enhance regression rate of the solid fuel and the density impulse of the hybrid rocket. The multi-walled carbon nanotubes also present the possibility of greatly improving the mechanical properties (e.g., tensile strength) of the paraffin-based solid-fuel grains. For casting these solid-fuel grains, various percentages of MWNT and aluminum particles will be added to the paraffin wax. Previous work has been published about the dispersion and mixing of carbon nanotubes.1 Another manufacturing method has been used for mixing the MWNT with a phenolic resin for ablative applications, and the manufacturing and mixing processes are well-documented in the literature.2 The cost of MWNT is a small fraction of single-walled nanotubes. This is a scale-up advantage as future applications and projects will require low cost additives to maintain cost effectiveness. Testing of the solid-fuel grains will be conducted in several steps. Dog bone samples will be cast and prepared for tensile testing. The fuel samples will also be analyzed using thermogravimetric analysis and a high-resolution scanning electron microscope (SEM). The SEM will allow for examination of the solid fuel grain for uniformity and consistency. The paraffin-based fuel grains will also be tested using two hybrid rocket test motors located at the Pennsylvania State University s High Pressure Combustion Lab.

Larson, Daniel B.↗

NASA NTRS2000

An Overview of Combustion Mechanisms and Flame Structures for Advanced Solid Propellants

Ammonium perchlorate (AP) and cyclotretamethylenetetranitramine (HMX) are two solid ingredients often used in modern solid propellants. Although these two ingredients have very similar burning rates as monopropellants, they lead to significantly different characteristics when combined with binders to form propellants. Part of the purpose of this paper is to relate the observed combustion characteristics to the postulated flame structures and mechanisms for AP and HMX propellants that apparently lead to these similarities and differences. For AP composite, the primary diffusion flame is more energetic than the monopropellant flame, leading to an increase in burning rate over the monopropellant rate. In contrast the HMX primary diffusion flame is less energetic than the HMX monopropellant flame and ultimately leads to a propellant rate significantly less than the monopropellant rate in composite propellants. During the past decade the search for more energetic propellants and more environmentally acceptable propellants is leading to the development of propellants based on ingredients other than AP and HMX. The objective of this paper is to utilize the more familiar combustion characteristics of AP and HMX containing propellants to project the combustion characteristics of propellants made up of more advanced ingredients. The principal conclusion reached is that most advanced ingredients appear to burn by combustion mechanisms similar to HMX containing propellants rather than AP propellants.

Beckstead, M. W.↗

NASA NTRS1991

Propellant development for the Advanced Solid Rocket Motor

The properties of a propellant developed for the NASA Advanced Solid Rocket Motor (ASRM) are described in terms of its composition, performance, and compliance to NASA specifications. The class 1.3 HTPB/AP/A1 propellant employs an ester plasticizer and the content of ballistic solids is set at 88 percent. Ammonia evolution is prevented by the utilization of a neutral bonding agent which allows continuous mixing. The propellant also comprises a bimodal AP blend with one ground fraction, ground AP of at least 20 microns, and ferric oxide to control the burning rate. The propellant's characteristics are discussed in terms of tradeoffs in AP particle size and the types of Al powder, bonding agent, and HTPB polymer. The size and shape of the ballistic solids affect the processability, ballistic properties, and structural properties of the propellant. The revised baseline composition is based on maximizing the robustness of in-process viscosity, structural integrity, and burning-rate tailoring range.

Landers, L. C.↗