Dynamic response of structures with uncertain parameters

This thesis presents a technique for obtaining the response of linear structural systems with parameter uncertainties subjected to either deterministic or random excitation. The parameter uncertainties are modeled as random variables or random fields, and are assumed to be time-independent. The new method is an extension of the deterministic finite element method to the space of random functions.

First, the general formulation of the method is developed, in the case where the excitation is deterministic in time. Next, the application of this formulation to systems satisfying the one-dimensional wave equation with uncertainty in their physical properties is described. A particular physical conceptualization of this equation is chosen for study, and some engineering applications are discussed in both an earthquake ground motion and a structural context.

Finally, the formulation of the new method is extended to include cases where the excitation is random in time. Application of this formulation to the random response of a primary-secondary system is described. It is found that parameter uncertainties can have a strong effect on the system response characteristics.

Fluctuations in a tokamak plasma

An experimental investigation of low frequency floating potential fluctuations (f ≤ 200 kHz) in a research tokamak plasma using two spatially separated electrostatic probes has been performed. The spectra, correlation length, and the phase velocity of the fluctuations in both the radial and azimuthal direction have been determined. The propagation velocity in the toroidal direction was also measured and was found to be in the direction of electron current flow. The waves traveled azimuthally in the ion diamagnetic drift direction, even after the usual E x B rotation was taken into account. The electron density fluctuations associated with these oscillations were large, δn/n ≃ 0.35 - 0.50.

The spectra were found to have regularly spaced peaks which seemed to be related to specific azimuthal modes (m =1,2,3,...,etc. ) A parametric study was made to determine what effect plasma parameters had on these peaks. During periods of high electron density in the first 2 msec of the plasma lifetime, strong sawtooth type oscillations were observed. These oscillations typically had frequencies of approximately 10 kHz and were also present when large amounts of neutral gas were added during the discharge by a process called "gas puffing."

The results are compared with experimental observations made on other plasma devices with electric and magnetic probes and with microwave and CO₂ laser scattering techniques. (The scattering measurements are complimentary to the probe measurements since, in the former case, the wavelength is fixed by the scattering angle, but the oscillations could not be spatially localized.) The oscillations in the Caltech torus were probably related to a drift-tearing type instability which is thought to play a major role in the anomalous particle and energy flux observed in tokamaks. Comparisons are made between current theory and the experimental results. However, the theory for the observed oscillations is still in a rudimentary stage of development, and it is hoped that the present investigation will stimulate future analytical work.

I. The effects of variations in certain parameters upon the efficiency of in vivo hemodialysis with a Kiil dialyzer. II. Interfacial structure of concurrent air-water flow in a two-inch diameter horizontal tube.

Part I

Regression analyses are performed on in vivo hemodialysis data for the transfer of creatinine, urea, uric acid and inorganic phosphate to determine the effects of variations in certain parameters on the efficiency of dialysis with a Kiil dialyzer. In calculating the mass transfer rates across the membrane, the effects of cell-plasma mass transfer kinetics are considered. The concept of the effective permeability coefficient for the red cell membrane is introduced to account for these effects. A discussion of the consequences of neglecting cell-plasma kinetics, as has been done to date in the literature, is presented.

A physical model for the Kiil dialyzer is presented in order to calculate the available membrane area for mass transfer, the linear blood and dialysate velocities, and other variables. The equations used to determine the independent variables of the regression analyses are presented. The potential dependent variables in the analyses are discussed.

Regression analyses were carried out considering overall mass-transfer coefficients, dialysances, relative dialysances, and relative permeabilities for each substance as the dependent variables. The independent variables were linear blood velocity, linear dialysate velocity, the pressure difference across the membrane, the elapsed time of dialysis, the blood hematocrit, and the arterial plasma concentrations of each substance transferred. The resulting correlations are tabulated, presented graphically, and discussed. The implications of these correlations are discussed from the viewpoint of a research investigator and from the viewpoint of patient treatment.

Recommendations for further experimental work are presented.

Part II

The interfacial structure of concurrent air-water flow in a two-inch diameter horizontal tube in the wavy flow regime has been measured using resistance wave gages. The median water depth, r.m.s. wave height, wave frequency, extrema frequency, and wave velocity have been measured as functions of air and water flow rates. Reynolds numbers, Froude numbers, Weber numbers, and bulk velocities for each phase may be calculated from these measurements. No theory for wave formation and propagation available in the literature was sufficient to describe these results.

The water surface level distribution generally is not adequately represented as a stationary Gaussian process. Five types of deviation from the Gaussian process function were noted in this work. The presence of the tube walls and the relatively large interfacial shear stresses precludes the use of simple statistical analyses to describe the interfacial structure. A detailed study of the behavior of individual fluid elements near the interface may be necessary to describe adequately wavy two-phase flow in systems similar to the one used in this work.

Neutron-neutron scattering parameters from the ^3He(d,t)2p and ^3H(d,^3He)2n reactions

The energy spectra of tritons and Helium-3 nuclei from the reactions ³He(d,t)2p, ³H(d,³He)2n, ³He(d,³He)pn, and ³H(d,t)pn were measured between 6° and 20° at a bombarding energy of 10.9 MeV. An upper limit of 5 μb/sr. was obtained for producing a bound di-neutron at 6° and 7.5°. The ³He(d,t)2p and ³H(d,³He)2n data, together with previous measurements at higher energies, have been used to investigate whether one can unambiguously extract information on the two-nucleon system from these three-body final state reactions. As an aid to these theoretical investigations, Born approximation calculations were made employing realistic nucleon-nucleon potentials and an antisymmetrized final state wave function for the five-particle system. These calculations reproduce many of the features observed in the experimental data and indicate that the role of exchange processes cannot be ignored. The results show that previous attempts to obtain information on the neutron-neutron scattering length from the ³H(d,³He)2n reaction may have seriously overestimated the precision that could be attained.

Spontaneous and stimulated light emission due to radiative recombination in forward biased lead telluride P-N junctions

Since the discovery in 1962 of laser action in semiconductor diodes made from GaAs, the study of spontaneous and stimulated light emission from semiconductors has become an exciting new field of semiconductor physics and quantum electronics combined. Included in the limited number of direct-gap semiconductor materials suitable for laser action are the members of the lead salt family, i.e . PbS, PbSe and PbTe. The material used for the experiments described herein is PbTe . The semiconductor PbTe is a narrow band- gap material (E_g = 0.19 electron volt at a temperature of 4.2°K). Therefore, the radiative recombination of electron-hole pairs between the conduction and valence bands produces photons whose wavelength is in the infrared (λ ≈ 6.5 microns in air).

The p-n junction diode is a convenient device in which the spontaneous and stimulated emission of light can be achieved via current flow in the forward-bias direction. Consequently, the experimental devices consist of a group of PbTe p-n junction diodes made from p –type single crystal bulk material. The p - n junctions were formed by an n-type vapor- phase diffusion perpendicular to the (100) plane, with a junction depth of approximately 75 microns. Opposite ends of the diode structure were cleaved to give parallel reflectors, thereby forming the Fabry-Perot cavity needed for a laser oscillator. Since the emission of light originates from the recombination of injected current carriers, the nature of the radiation depends on the injection mechanism.

The total intensity of the light emitted from the PbTe diodes was observed over a current range of three to four orders of magnitude. At the low current levels, the light intensity data were correlated with data obtained on the electrical characteristics of the diodes. In the low current region (region A), the light intensity, current-voltage and capacitance-voltage data are consistent with the model for photon-assisted tunneling. As the current is increased, the light intensity data indicate the occurrence of a change in the current injection mechanism from photon-assisted tunneling (region A) to thermionic emission (region B). With the further increase of the injection level, the photon-field due to light emission in the diode builds up to the point where stimulated emission (oscillation) occurs. The threshold current at which oscillation begins marks the beginning of a region (region C) where the total light intensity increases very rapidly with the increase in current. This rapid increase in intensity is accompanied by an increase in the number of narrow-band oscillating modes. As the photon density in the cavity continues to increase with the injection level, the intensity gradually enters a region of linear dependence on current (region D), i.e. a region of constant (differential) quantum efficiency.

Data obtained from measurements of the stimulated-mode light-intensity profile and the far-field diffraction pattern (both in the direction perpendicular to the junction-plane) indicate that the active region of high gain (i.e. the region where a population inversion exists) extends to approximately a diffusion length on both sides of the junction. The data also indicate that the confinement of the oscillating modes within the diode cavity is due to a variation in the real part of the dielectric constant, caused by the gain in the medium. A value of τ ≈ 10^-9 second for the minority- carrier recombination lifetime (at a diode temperature of 20.4°K) is obtained from the above measurements. This value for τ is consistent with other data obtained independently for PbTe crystals.

Data on the threshold current for stimulated emission (for a diode temperature of 20. 4°K) as a function of the reciprocal cavity length were obtained. These data yield a value of J’_th = (400 ± 80) amp/cm² for the threshold current in the limit of an infinitely long diode-cavity. A value of α = (30 ± 15) cm^-1 is obtained for the total (bulk) cavity loss constant, in general agreement with independent measurements of free- carrier absorption in PbTe. In addition, the data provide a value of n_s ≈ 10% for the internal spontaneous quantum efficiency. The above value for n_s yields values of t_b ≈ τ ≈ 10^-9 second and t_s ≈ 10^-8 second for the nonradiative and the spontaneous (radiative) lifetimes, respectively.

The external quantum efficiency (n_d) for stimulated emission from diode J-2 (at 20.4° K) was calculated by using the total light intensity vs. diode current data, plus accepted values for the material parameters of the mercury- doped germanium detector used for the measurements. The resulting value is n_d ≈ 10%-20% for emission from both ends of the cavity. The corresponding radiative power output (at λ = 6.5 micron) is 120-240 milliwatts for a diode current of 6 amps.

I. Phosphorescence and the true lifetime of triplet states in fluid solutions. II. Why is condensed oxygen blue?

I. PHOSPHORESCENCE AND THE TRUE LIFETIME OF TRIPLET STATES IN FLUID SOLUTIONS

Phosphorescence has been observed in a highly purified fluid solution of naphthalene in 3-methylpentane (3-MP). The phosphorescence lifetime of C₁₀H₈ in 3-MP at -45 °C was found to be 0.49 ± 0.07 sec, while that of C₁₀D₈ under identical conditions is 0.64 ± 0.07 sec. At this temperature 3-MP has the same viscosity (0.65 centipoise) as that of benzene at room temperature. It is believed that even these long lifetimes are dominated by impurity quenching mechanisms. Therefore it seems that the radiationless decay times of the lowest triplet states of simple aromatic hydrocarbons in liquid solutions are sensibly the same as those in the solid phase. A slight dependence of the phosphorescence lifetime on solvent viscosity was observed in the temperature region, -60° to -18°C. This has been attributed to the diffusion-controlled quenching of the triplet state by residual impurity, perhaps oxygen. Bimolecular depopulation of the triplet state was found to be of major importance over a large part of the triplet decay.

The lifetime of triplet C₁₀H₈ at room temperature was also measured in highly purified benzene by means of both phosphorescence and triplet-triplet absorption. The lifetime was estimated to be at least ten times shorter than that in 3-MP. This is believed to be due not only to residual impurities in the solvent but also to small amounts of impurities produced through unavoidable irradiation by the excitation source. In agreement with this idea, lifetime shortening caused by intense flashes of light is readily observed. This latter result suggests that experiments employing flash lamp techniques are not suitable for these kinds of studies.

The theory of radiationless transitions, based on Robinson's theory, is briefly outlined. A simple theoretical model which is derived from Fano's autoionization gives identical result.

Il. WHY IS CONDENSED OXYGEN BLUE?

The blue color of oxygen is mostly derived from double transitions. This paper presents a theoretical calculation of the intensity of the double transition (a ¹Δ_g) (a ¹Δ_g)←(X ³Σ_g^-) (X ³Σ_g^-), using a model based on a pair of oxygen molecules at a fixed separation of 3.81 Å. The intensity enhancement is assumed to be derived from the mixing (a ¹Δ_g) (a ¹Δ_g) _~~~ (X ³Σ_g^-) (X ³Σ_u^-) and (a ¹Δ_g) (¹Δ_u) _~~~ (X ³Σ_g^-) (X ³Σ_g^-). Matrix elements for these interactions are calculated using a π-electron approximation for the pair system. Good molecular wavefunctions are used for all but the perturbing (B ³Σ_u^-) state, which is approximated in terms of ground state orbitals. The largest contribution to the matrix elements arises from large intramolecular terms multiplied by intermolecular overlap integrals. The strength of interaction depends not only on the intermolecular separation of the two oxygen molecules, but also as expected on the relative orientation. Matrix elements are calculated for different orientations, and the angular dependence is fit to an analytical expression. The theory therefore not only predicts an intensity dependence on density but also one on phase at constant density. Agreement between theory and available experimental results is satisfactory considering the nature of the approximation, and indicates the essential validity of the overall approach to this interesting intensity enhancement problem.

Estimation of catalyst activity profiles and deactivation parameters from reactor operating data with applications to naphtha reforming

Techniques are developed for estimating activity profiles in fixed bed reactors and catalyst deactivation parameters from operating reactor data. These techniques are applicable, in general, to most industrial catalytic processes. The catalytic reforming of naphthas is taken as a broad example to illustrate the estimation schemes and to signify the physical meaning of the kinetic parameters of the estimation equations. The work is described in two parts. Part I deals with the modeling of kinetic rate expressions and the derivation of the working equations for estimation. Part II concentrates on developing various estimation techniques.

Part I: The reactions used to describe naphtha reforming are dehydrogenation and dehydroisomerization of cycloparaffins; isomerization, dehydrocyclization and hydrocracking of paraffins; and the catalyst deactivation reactions, namely coking on alumina sites and sintering of platinum crystallites. The rate expressions for the above reactions are formulated, and the effects of transport limitations on the overall reaction rates are discussed in the appendices. Moreover, various types of interaction between the metallic and acidic active centers of reforming catalysts are discussed as characterizing the different types of reforming reactions.

Part II: In catalytic reactor operation, the activity distribution along the reactor determines the kinetics of the main reaction and is needed for predicting the effect of changes in the feed state and the operating conditions on the reactor output. In the case of a monofunctional catalyst and of bifunctional catalysts in limiting conditions, the cumulative activity is sufficient for predicting steady reactor output. The estimation of this cumulative activity can be carried out easily from measurements at the reactor exit. For a general bifunctional catalytic system, the detailed activity distribution is needed for describing the reactor operation, and some approximation must be made to obtain practicable estimation schemes. This is accomplished by parametrization techniques using measurements at a few points along the reactor. Such parametrization techniques are illustrated numerically with a simplified model of naphtha reforming.

To determine long term catalyst utilization and regeneration policies, it is necessary to estimate catalyst deactivation parameters from the the current operating data. For a first order deactivation model with a monofunctional catalyst or with a bifunctional catalyst in special limiting circumstances, analytical techniques are presented to transform the partial differential equations to ordinary differential equations which admit more feasible estimation schemes. Numerical examples include the catalytic oxidation of butene to butadiene and a simplified model of naphtha reforming. For a general bifunctional system or in the case of a monofunctional catalyst subject to general power law deactivation, the estimation can only be accomplished approximately. The basic feature of an appropriate estimation scheme involves approximating the activity profile by certain polynomials and then estimating the deactivation parameters from the integrated form of the deactivation equation by regression techniques. Different bifunctional systems must be treated by different estimation algorithms, which are illustrated by several cases of naphtha reforming with different feed or catalyst composition.