Mass transfer in an airlift reactor with a net draft tube.

Gas-liquid mass transfer in an airlift reactor with net draft tube is investigated. The effects of both the ratio of draft tube to reactor diameter and the reactor pressure on oxygen transfer are considered. The value of the volumetric mass transfer coefficient, kLa, increases with a decreasing diameter ratio at higher air flow rates. The correlation of volumetric mass transfer coefficient with respect to the true superficial air velocity under different reactor pressures is determined. The kLa value decreases with increasing reactor pressure.     

Effects of velocity jumps on blood flow in large arteries

When a human being experiences a sudden velocity change, the blood flow is disturbed. A theoretical analysis to predict the effects of sudden velocity changes on blood flow in large arteries is presented. The situations is modelled as a one-dimensional flow problem in a viscoelastic tube where the fluid viscosity convective term in the equation of motion and nonlinearity in the elastic modulus of the tube wall are neglected. The governing equations of the model are solved by Laplace transformation. The computed results show that relatively high blood pressures, capable of harming circulation, are produced even by relatively moderate velocity jumps.     

Interaction of peristaltic flow with pulsatile flow in a circular cylindrical tube

The effect of pulsatile flow on peristaltic transport in a circular cylindrical tube is analysed. The flow of a Newtonian viscous incompressible fluid in a flexible circular cylindrical tube on which an axisymmetric travelling sinusoidal wave is imposed, is considered. The initial flow in the tube is induced by an arbitrary periodic pressure gradient. A perturbation solution with amplitude ratio (wave amplitude/tube radius) as a parameter is obtained when the frequency of the travelling wave and that of the imposed pressure gradient are equal. The interaction effects of periodic wall induced flow and periodic pressure imposed flow are visualized through the presence of substantially different components of steady and higher harmonic oscillating flow in the first order flow solution. Numerical results show a strong variation of steady state velocity profiles with boundary wave number and Reynolds number and a strong phase shift behaviour of the flow in the radial direction.     

On the propagation of a disturbance through a viscous liquid flowing in a distensible tube of appreciable mass

The velocity of propagation of a disturbance wave in a liquid flowing in a distensible tube is computed. The mathematical model is more general than those used in previous analyses: the tube wall properties are realistic; the convective part of the axial inertia forces is taken into account; radial inertia forces of both the fluid and tube wall are present; viscous stresses are present. Four parameters influencing the velocity of propagation are obtained and discussed. Curves are plotted illustrating the effects of the parameters. Contrary to the results of previous analyses, viscous effects are shown to be appreciable in blood flow. It is also shown that radial inertia effects can be important in laboratory set-ups. The material presented in this paper was adapted from the Ph.D. thesis written by the author at Harvard University.     

The Influence of Drag on Nonlinear Oscillatory Flow through Concentric Annulus

A mathematical model has been developed to study the effect of particle drag parameter and frequency parameter on velocity and pressure gradient in nonlinear oscillatory two phase flow. The main purpose is to apply the model to study the combined effect of introduction of the catheter and elastic properties of the arterial wall on the pulsatile nature of the blood flow. We model the artery as an isotropic thin walled elastic tube and the catheter as a coaxial flexible tube. Blood is modeled as an incompressible particulate viscous Newtonian fluid. Perturbation technique has been applied to find the approximations for velocity and pressure gradient up to second order. Numerical solutions are investigated with graphical presentations to understand the effects of drag parameter, frequency parameter and phase angle on velocity along radial direction and pressure gradient along axial directions. As the drag parameter increases, mean pressure gradient and mean velocity will be decreased. As frequency parameter increases mean velocity profile bends near the outer wall. Due to elastic nature of artery wall, a thin catheter experience small oscillations and a thick catheter remains stationary inside the artery. Finally, the effect of catheterization on various physiologically important flow rate characteristics—mean velocity, mean pressure gradient are studied for a range of different catheter sizes, particle drag parameter and frequency parameters.     

Microcontinuum model for pulsatile blood flow through a stenosed tube

The effects of polar nature of blood and pulsatility on flow through a stenosed tube have been analysed by assuming blood as a micropolar fluid. Linearized solutions of basic equations are obtained through consecutive applications of finite Hankel and Laplace transforms. The analytical expressions for axial and particle angular velocities, wall shear stress, resistance to flow and apparent viscosity have been obtained. The axial velocity profiles for Newtonian and micropolar fluids have been compared. The interesting observation of this analysis is velocity, in certain parts of cycle, for micropolar fluid is higher than Newtonain fluid. Variation of apparent viscosity eta a with tube radius shows both inverse Fahraeus-Lindqvist and Fahraeus-Lindqvist effects. Finally, the resistance to flow and wall shear stress for normal and diseased blood have been computed and compared.     

Accumulation and Conversion of Sugars by Developing Wheat Grains : VII. Effect of Changes in Sieve Tube and Endosperm Cavity Sap Concentrations on the Grain Filling Rate

The extent to which wheat grain growth is dependent on transport pool solute concentration was investigated by the use of illumination and partial grain removal to vary solute concentrations in the sieve tube and endosperm cavity saps of the wheat ear (Triticum aestivum L.). Short-term grain growth rates were estimated indirectly from the product of phloem area, sieve tube sap concentration, and ³²P translocation velocity. On a per grain basis, calculated rates of mass transport through the peduncle were fairly constant over a substantial range in other transport parameters (i.e. velocity, concentration, phloem area, and grain number). The rates were about 40% higher than expected; this probably reflects some unavoidable bias on faster-moving tracer in the velocity estimates. Sieve tube sap concentration increased in all experiments (by 20 to 64%), with a concomitant decline in velocity (to as low as 8% of the initial value). Endosperm cavity sucrose concentration also increased in all experiments, but cavity sap osmolality and total amino acid concentration remained nearly constant. No evidence was found for an increase in the rate of mass transport per grain through the peduncle in response to the treatments. This apparent unresponsiveness of grain growth rate to increased cavity sap sucrose concentration conflicts with earlier in vitro endosperm studies showing that sucrose uptake increased with increasing external sucrose concentration up to 150 to 200 millimolar.     

The effects of wall thickness, axial strain and end proximity on the pressure-area relation of collapsible tubes

Segments of silicone rubber tube were suspended between rigid pipes and subjected to slowly varying transmural pressure covering a range from slight distension to collapse with osculation. The local inside cross-sectional area at a chosen axial site was simultaneously measured via catheter by an electrical impedance method. Pressure-area relations were recorded thus at various axial sites, under varying conditions of axial tube wall tension, in tubes of two different wall thickness (0.3 and 0.4 of mean radius). Unsupported tube segment length was also varied by means of an insert device. The relations were used to calculate the variation of wave velocity with area according to Young's equation. First opposite wall contact during collapse was shown to occur at a smaller fraction of undistended circular cross-sectional area than in the thin-walled tubes investigated previously by others.     

Effect of sparger design on hydrodynamics of a gas recirculation anaerobic bioreactor

The effects of sparger design and gas flow rate on, gas holdup distribution and liquid (slurry) recirculation velocity have been studied in a surrogate anaerobic bioreactor used for treating bovine waste with a conical bottom mixed by gas recirculation. A single orifice sparger (SOS) and a multi-orifice ring sparger (MORS) with the same orifice open area and gas flow rates (hence the same process power input) are compared in this study. The advanced non-invasive techniques of computer automated tomography (CT) and computer automated radioactive particle tracking (CARPT) were employed to determine gas holdup, liquid recirculation velocity, and the poorly mixed zones. Gas flows (Q(g)) ranging of 0.017 x 10(-3) m(3)/s to 0.083 x 10(-3) m(3)/s were used which correspond to draft tube superficial gas velocities ranging from 1.46 x 10(-2) m/s to 7.35 x 10(-2) m/s (based on draft tube diameter). Air was used for the gas, as the molecular weights of air and biogas (consisting mainly of CH(4) and CO(2)) are in the same range (biogas: 28.32-26.08 kg/kmol and air: 28.58 kg/kmol). When compared to the SOS for a given gas flow rate, the MORS gave better gas holdup distribution in the draft tube, enhanced the liquid (slurry) recirculation, and reduced the fraction of the poorly mixed zones. The improved gas holdup distribution in the draft tube was found to have increased the overall liquid velocity. Hence, for the same process power input the MORS system performed better by enhancing the liquid recirculation and reducing the poorly mixed zones.     

Head tube: a simple device for estimating velocity in running water

Construction and operation of a head tube are described. A head tube is an inexpensive, easily-built alternative to a current meter for measuring water velocity in streams and rivers. When water flowing in a channel is obstructed by an object, its depth increases at the point of zero velocity (stagnation zone). The difference between the flowing-water depth and depth in the stagnation zone is the head (h). A head tube consists of a clear, hollow, 2.5 cm diameter acrylic tube and a sliding sleeve made of clear plastic. The tube is placed vertically on the river bottom. The difference (head) between water level inside the tube and the height of water against the tube's upstream face is measured against a scale (cm s^–1) drawn on the sleeve, which provides a direct reading of velocity. Graduations of the scale are calculated from a hydrodynamic equation relating mean velocity () to head (h): = (2gh)^0.5, whereg is acceleration due to gravity. When tested in a laboratory flume, the head tube gave very precise estimates of velocity (R ² of relationship = 0.98), although the original calibration scale overestimated current meter-measured velocity by 22 percent. The relationship between head tube and current meter estimates of mean velocity determined in a river with stony substrate was less precise than the correspondence observed in the laboratory (R ² = 0.81). However, estimates of discharge based on head tube measurements were within 8 percent of estimates based on current meter readings.     

The flow of a viscous liquid in a converging tube

The problem of the viscous flow of an incompressible Newtonian liquid in a converging tapered tube has been solved in spherical polar coordinates. The method of the solution involves the Stokes' stream function and a technique introduced by Stokes in the study of a sphere oscillating in a fluid. The theory for the flow in a rigid tube includes: (1) the pulsatile flow with both radial and angular velocity components; (2) the steady state flow with both radial and angular velocity components and (3) the very slow steady state flow with only a radial velocity component present. For a tapered elastic tube, the velocity of the propagated pulse wave is determined. The solution given is in terms of the elastic constants of the system and the coordinates for this type of geometry. The pulse velocity is then related to the velocity in an elastic cylindrical tube with the necessary correction terms to account for the tapered tube. Supported in part by the American Heart Association (No. 62F4EG). This work was done during the tenure of an Established Investigatorship of the American Heart Association.     

Spatial velocity distributions in pulse-wave propagation based on fluid–structure interaction

In this paper, spatial velocity distributions in pulse-wave propagation based on a fluid–structure interaction model are presented. The investigation is performed using the assumption of laminar flow and a linear-elastic wall. The fluid–structure interaction scheme is constructed using the finite element method. The results show that velocity distributions embody an obvious time delay in an elastic tube model. Further, the fully developed flow is delayed and the velocity values are increased in comparison with a rigid tube model. The increase in the wall thickness makes the time delay between the velocity peaks of different sites smaller while the time delay between the velocity minima is unchanged. Similarly, the time delay between the velocity bottoms is more easily found when decreasing the internal radius. The model gives valid results for spatial velocity distributions, which provide important information for wave propagation.     

Dynamics of blood flow: modeling of Fåhraeus and Fåhraeus–Lindqvist effects using a shear-induced red blood cell migration model

Blood flow in micro capillaries of diameter approximately 15–500 μm is accompanied with a lower tube hematocrit level and lower apparent viscosity as the diameter decreases. These effects are termed the Fåhraeus and Fåhraeus–Lindqvist effects, respectively. Both effects are linked to axial accumulation of red blood cells. In the present investigation, we extend previous works using a shear-induced model for the migration of red blood cells and adopt a model for blood viscosity that accounts for the suspending medium viscosity and local hematocrit level. For fully developed hematocrit profiles (i.e., independent of axial location), the diffusion fluxes due to particle collision frequency and viscosity gradients are of equal magnitude and opposite directions. The ratio of the diffusion coefficients for the two fluxes affects both the Fåhraeus and Fåhraeus–Lindqvist effects and is found related to the capillary diameter and discharge hematocrit using a well-known data-fit correlation for apparent blood viscosity. The velocity and hematocrit profiles were determined numerically as functions of radial coordinate, tube diameter, and discharge hematocrit. The velocity profile determined numerically is consistent with the derived analytical expression and the results are in good agreement with published numerical results and experimental data for hematocrit ratio and hematocrit and velocity profiles.     

Numerical simulation for the propagation of nonlinear pulsatile waves in arteries.

The behavior of nonlinear pulsatile flow of incompressible blood contained in an elastic tube is examined. The theory takes into account the nonlinear convective terms of the Navier-Stokes equations. The motion of the arterial wall is characterized by a set of linearized differential equations. The region bounded by the flexible arterial wall is mapped into a fixed area in which numerical discretization takes place. The finite element method (Galerkin weighted residual approach) is used for the solution of this nonlinear system. The results obtained are pressure distribution, velocity profile, flow rate and wall displacements along the elastic tube (20 cm long).     

Velocity profiles in central airways with endotracheal intubation: a model study

Steady inspiratory velocity profiles were measured at two flow rates in a 3:1 scale model of the human central airways in the presence of five modes of endotracheal intubation. The presence of an orifice or a short endotracheal tube had no significant effect on the velocity profiles distal to the carina. Long endotracheal tubes change the profiles in both main bronchi. A significant peak occurred in the frontal plane near the walls, and the maximum velocity in the airway was almost identical to the endotracheal tube center-line velocity. The flow impinging on the medial wall of the main bronchus was redirected up around the anterior and posterior walls yielding bipeak velocity profiles in the sagittal plane. A tube placed eccentrically in the trachea over the right main bronchus did not alter the velocity profiles in the left main bronchus, suggesting a redirection of flow over the carina into the left lung. An endobronchial tube at the mouth of the right main bronchus did change the shape of the velocity profiles in the left main bronchus. In the left upper lobar bronchus the presence of trachea intubation had no effect on the velocity profiles. However, in the right upper lobar bronchus, the long endotracheal tube flattened the velocity profiles from the strongly skewed ones seen in the absence of the endotracheal inserts. These results not only are relevant to distribution of ventilation and aerosol particle deposition, but also have strong implications in intrapulmonary gas mixing, especially when high-frequency low tidal-volume ventilation is involved.     

The flow field downstream of an oscillating collapsed tube

A two-component laser Doppler anemometer was used to determine the velocity of aqueous flow in the region from 0.25 to 2.5 diameters downstream of a collapsible tube while the tube was executing vigorous repetitive flow-induced oscillations. The Reynolds number for the time-averaged flow was 10,750. A simultaneous measurement of the pressure at the downstream end of the tube was used to align all the results in time at sixty locations in each of the two principal planes defined by the axes of collapse of the flexible tube upstream. The raw data of seed-particle velocity were used to create a periodic waveform for each measured velocity component at each location by least-squares fitting of a Fourier series. The results are presented as both velocity vectors and interpolated contours, for each of ten salient instants during the cycle of oscillation. In the plane of the collapse major axis, the dominant feature is the jet which emerges from each of the two tube lobes when it collapses, but transient retrograde flow is observed on both the central and lateral edges of this jet. In the orthogonal, minor-axis plane, the dominant feature is the retrograde flow, which during part of the cycle extends over the whole plane. All these features are essentially confined to the first 1.5 diameters of the rigid pipe downstream of the flexible tube. These data map the temporal and spatial extent of the highly three-dimensional reversing flow just downstream of an oscillating collapsed tube.     

Plug Effect of Erythrocytes in Capillary Blood Vessels

As an idealized problem of the motion of blood in small capillary blood vessels, the low Reynolds number flow of plasma (a newtonian fluid) in a circular cylindrical tube involving a series of circular disks is studied. It is assumed in this study that the suspended disks are equally spaced along the axis of the tube, and that their centers remain on the axis of the tube and that their faces are perpendicular to the tube axis. The inertial force of the fluid due to the convective acceleration is neglected on the basis of the smallness of the Reynolds number. The solution of the problem is derived for a quasi-steady flow involving infinitesimally thin disks. The numerical calculation is carried out for a set of different combinations of the interdisk distance and the ratio of the disk radius to the tube radius. The ratio of the velocity of the disk to the average velocity of the fluid is calculated. The different rates of transport of red blood cells and of plasma in capillary blood vessels are discussed. The average pressure gradient along the axis of the tube is computed, and the dependence of the effective viscosity of the blood on the hematocrit and the diameter of the capillary vessel is discussed.