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The response probability density function (pdf) technique developed here utilizes the pdf's of the excitation and sensitivity to obtain the pdf of the response. This technique was used in a statistical model for estimating the probability of glass breakage from sonic booms, as a function of nominal overpressure, window size, aircraft vector, boom duration, and whether the glass was in good condition or cracked. A model window population was devised from available data which includes the pdf's of the dynamic amplification factors and breaking pressures for seven window types. From the histograms of the test data, the distribution of both sonic‐boom stresses and glass strengths were found to be lognormal, which reduced the statistical model to a particularly simple form. It was estimated that there would be 1.1 breaks per million panes in good condition boomed at a nominal overpressure of 1 psf. This agrees well with sonic boom claims data of 0.6 breaks per million panes boomed.

Experimental data from batch vessels on cumulative volumetric drop-size distributions at various times are shown to yield useful information on probabilities of droplet-breakup as a function of drop-size. Such information is sufficient for a priori prediction of drop-sizes in agitated dispersions in batch and continuous vessels. It may also be useful in predicting heat and/or mass transfer in liquid—liquid dispersions by accounting for the simultaneity of transport processes from individual drops and droplet breakage processes.

A random fracture model of one-dimensional breakage has been tested using spaghetti and chalk sticks as the experimental material, and also by sampling from random number tables. One-dimensional breakage data were combined to simulate breakage in two and three dimensions. The experimental size distributions deviate from the random model, suggesting that real breakage is not merely random, and that a proper model must include an evaluation of the mechanism of force application and its effect on the distribution of fracture sites.

A general model has been developed for the process of polymer chain breakage. The molecular weight distributions (MWDs) of the degraded polymer, after a specified number of chain ruptures, can be calculated from the model, given the MWD of the initial material. The calculations can easily be handled numerically on a digital computer. The model is applicable to any process in which polymer chains break, e.g., ultrasonation and high shear mechanical action. The course of degradation is described in terms of the probability that a molecule of a given length will break and the probability that a molecule of a particular length will result from this rupture. Any form of both these probability distributions and the initial MWD can be used in the model.

The effects of ambient pressure and ultrasonic power on the disintegration of yeast suspension have been investigated. The results obtained are, in the main, consistent with the theory that cell breakage is primarily a phenomenon dependent on producing gaseous cavitation in the medium. The importance of the experimental results and techniques applied to Commercial cell disintegrators is briefly discussed. A simple flow system is described which is easily attached to probe-type disintegrators. The use of a crystal pickup for tuning and control purposes is described.

Floc breakup theories are developed for the complex activated sludge floc and they are applied as well to inorganic chemical flocs. Rate expressions were developed for two modes of breakup in two hydraulic regimes. In one type of breakup it was assumed that primary particles are eroded from the floc surface when the surface shearing forces exceed the shear strength of the bonds joining the particles to the floc surface. The analysis of this surface shearing force required the description of floc motion in the turbulent environment. Expressions were developed for the peak floc size as a function of shearing conditions. The rate of floc breakup was described as the product of the number of metastable floc, the frequency of metastable floc disruption, and the number of primary particles eroded per disruption. A second form of floc breakup was observed experimentally with activated sludge floc. It appeared that the overall integrity of the floc was controlled by a filament mesh running through the floc allowing the buildup of very large floc prior to breakage. The high strength of these filaments seemed to allow the buildup of flocs to sizes that would not be possible in the absence of filaments.

The assumptions used in theoretical predictions of the tensile properties of twisted continuous-filament yarns are critically discussed. A modified theory, based on constant yarn volume during extension, is derived, and theoretical and experimental values of modulus are compared. Expressions for yarn extension and breakage at large extensions, and also taking account of deviations from Hooke’s law, are derived and compared with experimental results.

theory has been developed to predict the average inter-fibre pressure in a twisted fibre assembly that is subjected to tension. This has been extended to predict the rupture force in an assembly, such as a worsted roving, where the rupture takes place only by fibre slippage. A wide range of experimental determinations have been made and it is shown that, in general, there is very close agreement between theory and experiment. The theory was also extended to calculate the drafting force in a twisted assembly caused by roller-drafting on the Bradford System. The drafting force was measured experimentally by a newly developed method and, again, it was found that these results agreed closey with the theory. The results presented here were intended primarily to investigate the condition of fibres during drafting, but it would appear that they could also be applied to the problem of yarn strength, by making allowance for the effect of fibre breakage as well as fibre slippage.

Cotton fibres break during ginning if the force applied to overcome the attachment to the seed exceeds the fibre breaking load. The pattern of breakage may be described in terms of mathematical probabilities and frequencies, with parameters which can be estimated from counts of the various types of ends of ginned fibres. Changes in mean length and length distribution caused by ginning were measured directly for one cotton and the results agree well with values calculated from the counts of fibre ends. Further limited experimental results show that the attachment forces may be greater than the breaking loads of the weakest fibres, a feature leading to some inevitable breakage in ginning. The paper concludes with a discussion of some problems associated with the overlap of the frequency distributions of fibre attachment forces and fibre breaking loads.

The note briefly surveys the recent studies on the time variation of particle sizes during such processes as coalescence, heterocoagulation, electrification of aerosols, breakage of particles and emulsification. The general equations for the change of size distributions during coalescence are solved in some cases. A simple model is given for the variation of the mean size when both dispersion and coalescence are present. The results are compared with the experimental data and the stability factor, the limiting size of particles in grinding and emulsification, etc. are quantitatively discussed.

The problem of formation and breakage of liquid threads has been discussed from the theoretical point of view. It has been shown that the length of a liquid thread may be limited by two different break processes: brittle cohesive break and break-up into drops due to development of axisymmetrical capillary waves. The actual break moment and the thread length in given conditions are determined by cohesive break affected by development of capillary waves. The effect of waves capillary dominates in low-viscous liquids at low deformation rates — the practically independent cohesive process in high-viscous materials at rapid deformations. The experimental data are qualitatively consistent with the theory and directly confirm the character of break processes. The theory makes also possible the explaining of all the empirical data concerning the “spinnability” of liquids.

A mathematical model on breakage of cotton tuft in blowing process was introduced in the previous article7). In this paper, the change of size distribution of cotton tuft in the Platt type blowing process was examined experimentally and the probability that a tuft undergoes the breakage action, π was obtained from the data, using the theory developed in the previous paper.Then, using II, the objective function to evaluate the breakage action in blowing is discussed by following equation: where πii is element of II.The application of the function to Platt type blowing process showed that the function appeaas to be resonable.

Abnormal breakage in laboratory mills may be defined as departure from first-order kinetics, and occurs particularly for the larger particle sizes in the mill feed. The reason for this abnormal breakage is that all the particles within a size fraction do not have tire same strength, rather having a distribution of strengths which interacts with a distribution of applied loads on the part of the grinding media. The interrelation between these two distributions, of strengths and forces, is discussed statistically showing the conditions necessary for departure from first-order kinetics. Actual values of the distribution of strengths support the theory, and a final experiment on the grinding of the survival material from previous grinding tests validated the concepts presented.

The strength of a yarn or thread is conventionally estimated from the load which a discrete length of yarn can withstand without breaking. A less frequently used test measures the frequency with which breaks occur when the yarn is unwound from a package under specific conditions of tension, and the purpose of this paper is to investigate the relationship between the single-thread test and the winding test. The distribution of points of breakage along a jute yarn is examined, and shown to follow a random pattern, and these points of breakage are identified with the occurrence of “weak regions” of yarn. The equation connecting the number of weak regions per unit length of yarn, the average length of these weak regions, and the frequency distribution of single-thread breaking load is derived for single-thread test specimens of any length. For jute yarns, it appears that the single-thread frequency distribution is Gaussian as far as 3.5 standard deviation below the mean, but, at greater deviations, the breaka...

The combined tracer and backcalculation method for determining particulate breakage parameters described in Part I of this paper and applied to batch ball milling in Part II is extended here to the open-circuit continuous ball milling of martite iron ore. Breakage parameters are determined by this method in the continuous ball mill in the batch mode of operation. Residence time distributions are obtained in the continuous mode of operation with radioactive tracers. Segregated flow, models based on the dispersion model and the tanks-in-series model are given for use with the breakage parameter and residence time distribution data to predict product weight—size distributions. A particular model based on three perfectly mixed tanks in series is used to describe a mill with entrance and exit trunnions and is compared to both transient and steady-state experimental results. The comparison indicates that this method of simulation will be useful.

In the previous report, we gave an account of the analysis of the mechanism of jet-pulverization observed in a small-scale experimental jet-pulverizing apparatus, and derived some assumptions, of which the proposed relation between mill-power (H) and mill-capacity (G) had the most important meaning. We proposed there the following equation:G∝Hσin which σ_??_2.5}(1)This equation shows that the jet pulverizer becomes more efficient as the mill-power increases. With most mills of other types, value of σ, which shall be called a coefficient of the size effect of pulverizer, is about 1.21.5.In order to confirm this relation and other assumptions, we conducted some experiments with a laboratory-scale jet-pulverizer of practical use (Jet-O-Mizer 202 with 30HP compressor), and the results are given here to make clear the effects of variable factors, such as nozzle pressure, feeding rate of raw materials and diam. of nozzle, etc. on the rate of grinding.This type of jet-pulverizer has a ring-shaped circular tube through which the mixture of air and particle circulates continuously, while some part of the mixture is taken out and sent to the collector successively.The number of circulation, which was inversely proportional to the feeding rate was found to be one of the most important factors in this pulverization. The grinding action was strongest in the jet-stream, and it was negligibly weak on the surface of the mill wall. (Cf. Fig. 7.)From these and other assumptions previously proposed, we derived Eq. 5.X=100KAN1PN1+100(1-KAN1PN1)[1-(1-KAN2PN2)2N] (5)as a fundamental equation for calculating the capacity of the circulating type jet-pulverizer.By comparing the capacity of this type of mill with that of the above-mentioned small-scale experimental apparatus (2HP) and of Jet-O-Mizer 808 (250HP), we substantiated the results of our experiment.Data on Jet-O-Mizer shown in the information by Fluid-Energy Equipment and Processing Co. also verify Eq. 5. (Cf. Fig. 10.)We had assumed, in the previous report that when mill-power became larger, the mechanism of breakage would change from surface grinding to bulk crushing and that the constant for the degree of surface grinding (n) would become small. The results of this experiment shown in Table 4 also verify this assumption.

It is believed that rock breakage by blasting is normally completed both by the dynamic breakage under intense stress waves, and by the quasi-static breakage under the pressure of gas explosion. First in this study the stress condition caused by the time dependent pressure in an elastic medium was computed from time to time as it worked on the spherical cavity, and thereby were inferred what were the stresses in the medium that were affected by the variation of the rise time of the pressure in the cavity. It is concluded that the values of the dynamic stresses in the medium caused by the stress wave decrease with increase of the rise time of the pressure, and that the stress condition in the medium is controlled by the quasi-static stress caused by the pressure in the cavity. It is considered therefore that the breakage of rocks caused by the explosion of the explosives having low velocity of defragration is completed chiefly by the quasi-static pressure of the gas explosion.Next the stress condition caused by the quasi-static pressure was analyzed by using the finite element method as it worked on the circular boreholes, assuming the cases in which the boreholes ranged linearly with constant spacing of 10a, where a was the radius of the hole. At first pressure was applied to every other borehole, and the stress condition around the hole where no pressure was applied was carefully observed, and then the pressure was applied to every hole, and probable change in the stress distribution was studied upon the assumed growth of four radial cracks that would be originated from the inner surfaces of the holes, as two cracks grew along the line which connected the centre of each hole, and two other cracks grew along the line perpendicular to the above. These cracks are supposed to have been produced from the inner surface of each hole when the value of the pressure exceeds the tensile strength of the material. It is clarified as the result that the cracks along the line which connects the centre of each hole will grow further while those which grow perpendicular to the above are repressed. By comparing the results of these stress analyses with the strain measured in the material and the pattern of the crack produced actually by the pressure of the gas explosion, it is concluded finally that the quasi-static gas pressure contributes very much to the completion of smooth wall in the smooth blasting operation.

The dynamic stresses in the materials having various assumed properties caused by detonative loading are computed by using the numerical technique which involves the finite difference in approximation to the equation of motion in the Lagrangian coordinate system, and then, the patterns of the breakage around the explosion, and the behaviour of the stress waves projected into the materials are discussed by referring the results to the above mentioned stress analysis.Concerning the pattern of the breakage around the spherical explosion, it is concluded that the yield zone or shear failure zone is produced in close proximity to an explosion, and this zone is surrounded by the tensile fracture zone consisting of the radial cracks. Finally, the patterns of the breakage of the mild steel and the plexiglass (PMMA) subjected by detonative loading are presented, and it is pointed out that these are explained fairly well by the results of the above mentioned numerical analysis.

目的 以前に試作した「走行糸負荷装置」に, その後いくつかの問題点を見いだしたので全面的にこれを改良した. 成果 改良型負荷装置により, 糸切れの調査のためのかなりせまい把持間隔の実験までも行なえるようになった.なかでも製品糸の糸切れの記録から, 糸の生成条件を検討するための適当な把持間隔があり得ることが示唆された.

In this report, an analysis of breakage effect of cylinder type- and beater-type machines, which are used in breakage of cotton tufts, was performed by a digital simulation.As cylinder type machine, double Crighton Opener and Superio Cleaner, and also as beater type machine, Double Scutcher were used.On cylinder type machine, the size distribution obtained experimentally agreed with that obtained by the simulation which was made under the assumption that the ratio of a tuft affected by breakage action is propotional to its weight (called size) and the break part was distributed uniformly on tuft.Superio CleanerIt was considered from size distributions by experiment and simulation that breakage of tuft in Superio cleaner is saturated at break stage of S-4 or S-3. Good results were obtained when a by-pass effect that only the tufts with the sizes larger than the max-size estimated from size distribution are broken was introduced into the simulaton model of Superio Cleaner.Crighton OpenerA difference of simulation model between Superio Cleaner and Crighton Opener concerned with the transfer of tuft in the breakage zone and it was supposed in the simulation model that transfer condition of a tuft passing through the breakage zone was propotional to the size.Then, using the transfer condition weighed by a factor decided from the relation between max. length of tuft and size, good results were obtained.In the simulation model of beater type machine, it was assumed in beating zone of Scutcher that the size distribution of tufts cut by beater is exponential and the distance from the beating point to the nip point of a tuft (effective beater guage) was about 30-45mm.The size distributions of the simulaton of beater type machine agreed with experimental results under the above assumptions and the relation between size of a tuft and its max. length.As input data to these simulation models, the size distributions obtained experimentally were used.

A new technique for determining particulate breakage functions is devised. It essentially consists of combining the tracer and back-calculation methods so that both efficiency and accuracy are obtained. A random number (Monte Carlo) method for evaluating the compounded error due to statistical fluctuations in the breakage process, sampling, and the sizing method in back-calculation methods is devised and sample results are given for a particular back-calculation method.

An impact velocity just sufficient to cause immediate breakage in a filament im pacted in tension is called a critical velocity. Theories, of von Karman and Taylor, required for calculating critical velocities are reviewed and extended to include case of shock-wave propagation. Estimates of critical velocity are then calculated for some textile yarns using stress-strain data obtained at impact speeds near 40 m sec. The velocities obtained range in value from approximately 100 m sec for cotton sewing thread and glass fiber yarn to approximately 300 m sec for some vinal, rayon-tire-cord. acrylic, and nylon yarns. Theory is developed for calculating a limiting breaking velocity from the specific breaking energy obtained by high-velocity impact tests. It is shown that the limiting breaking velocity is higher than the critical velocity if the stress-strain curve is predominantly comcave downward, lower if the stress-strain curve is predominantly concave upward, and equal to the cricital velocity if the curve is linear. Critical velocities are also estimated from stress-strain data obtained at 100% min straining rates and compared with calculations from high-velocity-impact data. Good agreement is found in many cases.

The processes analyzed in this paper are the size reduction and size classification of particle assemblies. Particle size distributions are described by vectors, and alterations to size distributions during breakage processes are described by matrices multiplying the vectors. The matrix approximation gives an adequate representation of the processes studied, and the manipulation of the matrices is easy and flexible. The breakage of a particle assembly is thought of as two processes. In the first, the machine breaking the particles is said to select for breakage a proportion of the particles, and the remaining particles are unbroken. To discover a function or matrix which describes the process of selection is to understand how the machine operates. In the second process, the particles selected are broken in a regular way; the proportions of particles of each size formed by the breakage are described by a breakage function or a breakage matrix. The analysis of breakage is in this way given convenient mathematical form. These matrices depend on the characteristics of the machine and on the nature of the particle assembly. After breaking the particles, crushing and grinding machines frequently pass the product assemblies to a classifier from which the larger particles are returned, mixed with fresh material, to the grinding zone. The analysis is extended to the description of such circuits. The experimental work reported concerns the breaking of coal particles in a new grinding machine, ball mills, shatter tests and a beater mill. The selection functions derived throw light on the operation of these machines. Coal breakage has been studied since it is an important field of application, and because coal is typical in breakage of homogeneous rocks. For each of the machines examined and for each particle size, a single breakage function has sufficed to describe the product of breakage: [1 —exp ( —z)]/[ 1 —exp ( — 1)] is the proportion of the product smaller than a fraction z of the original particle size.

A mathematical analysis of the stretch-breaking process used for conversion of endless continuous filaments into staple fiber form is presented. The analysis indicates that the lower the draft or the longer the ratch is, the longer the fibers are. The cumulative fiber length distribution of stretch-broken fibers is obtained assuming a random breakage of filaments. The velocity and acceleration of a point on a single filament in the stretch zone are also obtained in order to be used in the calculation of the effect of temperature of the heater plates used

The basic integro-differential equation describing comminution process is analytically solved by the use of a certain form of selection function and breakage function. The result agrees well with previous experience. The rate constant of the selection function is considered in relation to the stress field produced within a powder bed and to the strength of material crushed. A resulting equation expressing the selection function as a function of particle size, magnitude of force applied and some operational variables as well as material properties is compared with reference data.

Direct assessment of particle breakage, in both natural and artificial environments, is made possible by comparing the strengths of suites of particles which have survived a process with the strengths of suites, otherwise similar, that have not been involved in the process. The `fragmentation load,' the load required to bring about breakage of an individual particle between parallel plane surfaces, is the basic quantity used. It is measured in a slightly modified unconfined compression test machine. Fragmentation of coarse plutonic quartz appears to be a major process in a stream.

CAROME, in a recent letter1 commenting on our article on the mechanical disruption of tobacco mosaic virus2 by laser-induced transients, suggests that the effect we detected may have been caused by a number of complicating factors related to pulse reflexions and/or cavitation of the suspending medium. Although the points made by Carome are certainly valid we would like to clarify the issue. We obtained TMV breakage in all of the cases mentioned by Carome; namely, pulses reflected back through the liquid sample at a water-to-air interface; or totally transmitted without reflexion; or transmitted into a finite thickness water layer and then reflected at a water–air interface. The fact that breakage was obtained with totally transmitted pulses, which corresponds to transmission of a positive pressure transient through the virus suspension, is particularly important. As mentioned in our previous communication2, the effect was enhanced when the reflexions occurred at the liquid–air interface of the virus suspension. Breakage was not, however, significantly increased by reflexions (including negative transients) generated at other interfaces in the system. The details of the hydrodynamical analysis of the stress engendered in the TMV by the laser-induced acoustic transient have been presented in greater detail elsewhere3.

I this note we report preliminary work on a modification of the expansion tube. This modification is suggested by our experiences with secondary diaphragm breakage and its effect on flow quality. It appears that diaphragm fragment size significantly affects flow uniformity. Our experiences imply that large diaphragm "petals" of order of the tube diameter introduce disturbances in the same order of magnitude, and the subsequent expansion process does not sufficiently smooth the flow before the gas reaches the test station. On the other hand, small fragments such as result from shattering by the incident shock introduce disturbances of a correspondingly small order. These disturbances are smoothed out further by the expansion, so that a rather smooth flow results. Accordingly, flow quality is improved by forcing the diaphragm to break into small pieces. If at the same time the diaphragm could be retained, additional improvement in flow contamination could be achieved. Preliminary tests with the modification reported here show that such improvements are indeed achieved. The modification consists of a "nozzle plate" installed at the secondary diaphragm location. The plate is shown in Fig. 1. It contains a large number of individual nozzles that are initially sealed against the driven gas by a common dia

The flow of air between a plate at rest and another one falling onto it either vertically or by folding is studied, and the infrequency of breakage of glass plates colliding in this way is explained. The falling plate may be two‐dimensional, circular, or elliptic, and the results for an elliptic plate give bounds for the motion of a falling rectangular plate.

A technique has been developed for predicting the probability of sonic boom structural damage. This technique utilizes the probability density functions (pdf's) of the sonic boom induced stress and the material strength. Analysis of the data from experiments performed at Oklahoma City, White Sands, and Edwards Air Force Base has shown the stresses produced by sonic booms on windows to have lognormal pdf's. Similarly, analysis of the breaking strength distribution for glass plates has shown it to be log‐normal. From these pdf's a statistical model has been developed for predicting the probability of window breakage as a function of nominal pressure, window size, aircraft flight path direction, and boom duration. Calculations from the model indicate a breakage probability of 1.1 breaks per million panes in good condition boomed at a nominal overpressure of 1 lb/ft2. Results predicted by the model agree well with experimental data at both high and low overpressures. [Work supported by Federal Aviation Admini...