Capillary flows

It is useful at this point to consider the polymer melt experience as it flows from the reservoir into, through, and out of the capillary tube. In the entrance region shown in 

In careful viscometric studies, this pressure drop is never neglected the pressure reading at zero reservoir height, as found by extrapolation, is used for the value of Pq. 

The polymer melt experience briefly described above is complex and varied it involves steady, accelerating, fully developed, and exit flows and strain recovery. It is not surprising, then, that this apparently simple experiment is used to study not only the viscous but also the elastic nature of polymer melts. 

Entrance and Exit Capillary Pressure Losses the Bagley Correction 

In the fully developed flow region, the assumptions of steady and isothermal flow, constant fluid density, and independence of the pressure from the radius, resulted in the conclusion 

The flow of fluid in capillaries of very small diameter was first discussed by a German engineer named Hagen in 1839. Poiseuille, a Parisian physician, independently made similar experiments concerning the flow of blood through veins of the body and published his results from 1840-1846. The fundamental relationship for the rate of flow through a capillary is known as the Hagen-Poiseuille Law. 

This problem may be approached by dimensional analysis. Before dimensional analysis  

G = pressure gradient along the capillary [FZ ] p = coefficient of viscosity [ZTZ 2] 

The pressure gradient is the pressure from one end of the capillary to the other divided by the capillary length (G = A p/i) 

After dimensional analysis, there is only one nondimensional group 

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Fig. 1. Southern blot analysis of DNA showing (a) step 1, an agarose gel containing separated restriction fragments of DNA, denoted by (—), which is immersed in NaOH to denature the double-stranded stmcture of DNA, and then transferred by capillary flow to a nitrocellulose filter. In step 2, the bound DNA is allowed to hybridize to a labeled nucleic acid probe, and the unbound probe is washed off In step 3, the filter is placed into contact with x-ray film resulting in (b) bands of exposure on the film which are detected after development and correspond to regions where the restriction fragment is...

Reverse osmosis models can be divided into three types irreversible thermodynamics models, such as Kedem-Katchalsky and Spiegler-Kedem models nonporous or homogeneous membrane models, such as the solution—diffusion (SD), solution—diffusion—imperfection, and extended solution—diffusion models and pore models, such as the finely porous, preferential sorption—capillary flow, and surface force—pore flow models. Charged RO membrane theories can be used to describe nanofiltration membranes, which are often negatively charged. Models such as Dorman exclusion and the... 

Capillary Viscometers. Capillary flow measurement is a popular method for measuring viscosity (21,145,146) it is also the oldest. A Hquid drains or is forced through a fine-bore tube, and the viscosity is determined from the measured flow, appHed pressure, and tube dimensions. The basic equation is the Hagen-Poiseuike expression (eq. 17), where Tj is the viscosity, r the radius of the capillary, /S.p the pressure drop through the capillary, IV the volume of hquid that flows in time /, and U the length of the capillary. 

The Ubbelohde viscometer is shown in Figure 24c. It is particularly useful for measurements at several different concentrations, as flow times are not a function of volume, and therefore dilutions can be made in the viscometer. Modifications include the Caimon-Ubbelohde, semimicro, and dilution viscometers. The Ubbelohde viscometer is also called a suspended-level viscometer because the Hquid emerging from the lower end of the capillary flows down only the walls of the reservoir directly below it. Therefore, the lower Hquid level always coincides with the lower end of the capillary, and the volume initially added to the instmment need not be precisely measured. This also eliminates the temperature correction for glass expansion necessary for Cannon-Fen ske viscometers. 

Vapor Pressure. The Shiley Infusaid implantable infusion pump utilizes energy stored in a two-phase fluorinated hydrocarbon fluid. The pump consists of a refillable chamber that holds the dmg and a chamber that holds the fluid. The equiUbrium vapor pressure of the fluid, a constant 60 kPa (450 mm Hg), compresses the bellows, pumping the dmg through a bacterial filter, a capillary flow restrictor, and an infusion cannula to the target body site (56,116). 

Critical moisture content is that obtained when the constant rate period ends and the falling rate periods begin. Second critical moisture content specifies that remaining in a porous material when capillary flow dominance is replaced by vapor diffusion. 

In porous and granular materials, Hquid movement occurs by capillarity and gravity, provided passages are continuous. Capillary flow depends on the hquid material s wetting property and surface tension. Capillarity appHes to Hquids that are not adsorbed on capillary walls, moisture content greater than fiber saturation in cellular materials, saturated Hquids in soluble materials, and all moisture in nonhygroscopic materials. 

Capillary flow is the flow of hquid through the interstices and over the surface of a solid, caused by liquid-solid molecular attraction. 

The structure of the solid determines the mechanism for which internal liquid flow may occur. These mechanisms can include (1) diffusion in continuous, homogeneous solids, (2) capillary flow in granular and porous sohds, (3) flow caused by shrinkage and pressure gradients, (4) flow caused by gravity, and (5) flow caused by a vaporization-condensation sequence. 

Capillary Flow Moisture which is held in the interstices of solids, as liquid on the surface, or as free moisture in cell cavities, moves by gravity and capiUarity, provided that passageways for continuous flow are present. In diying, liquid flow resulting from capiUarity appUes to liquids not held in solution and to aU moisture above the fiber-saturation point, as in textiles, paper, and leather, and to all moisture above the equiUbrium moisture content at atmospheric saturations, as in fine powders and granular solids, such as paint pigments, minerals, clays, soU, and sand. 

The problems experienced in drying process calculations can be divided into two categories the boundary layer factors outside the material and humidity conditions, and the heat transfer problem inside the material. The latter are more difficult to solve mathematically, due mostly to the moving liquid by capillary flow. Capillary flow tends to balance the moisture differences inside the material during the drying process. The mathematical discussion of capillary flow requires consideration of the linear momentum equation for water and requires knowledge of the water pressure, its dependency on moisture content and temperature, and the flow resistance force between water and the material. Due to the complex nature of this, it is not considered here. 

The only approach is to ignore the capillary flow and to assume water vaporization takes place evenly in the thickness of the paper web. With a... 

KapiUarititt, /. capillarity. Kapillaritittsanziehung./. capillary attraction. Kapillar-kraft,/, capillary force, -kreislauf, n. capillary circulation, -rohr, -rohrchen, n, -rohre, /. capillary tube, -spaonung, /. capillary tension, -stromung, /. capillary flow, -versuch, m. capillary test or experiment. -wirkung./. capillary action. 

The purpose of our study was to model the steady-state (capillary) flow behavior of TP-TLCP blends by a generalized mathematical function based on some of the shear-induced morphological features. Our attention was primarily confined to incompatible systems. 

During a steady-state capillary flow, several shear-induced effects emerge on blend morphology [4-6]. It is, for instance, frequently observed that TLCP domains form a fibrillar structure. The higher the shear rate, the higher the aspect ratio of the TLCP fibrils [7]. It is even possible that fibers coalesce to form platelet or interlayers. 

Furthermore, when m interlayers exist within a capillary flow, where m = kRJd and /c is a linear factor, the total flux increase of AQ caused by the slippage of these m interlayers can be written as ... 

As demonstrated, Eq. (7) gives complete information on how the weight fraction influences the blend viscosity by taking into account the critical stress ratio A, the viscosity ratio 8, and a parameter K, which involves the influences of the phenomenological interface slip factor a or ao, the interlayer number m, and the d/Ro ratio. It was also assumed in introducing this function that (1) the TLCP phase is well dispersed, fibrillated, aligned, and just forms one interlayer (2) there is no elastic effect (3) there is no phase inversion of any kind (4) A < 1.0 and (5) a steady-state capillary flow under a constant pressure or a constant wall shear stress. 

Fig. 27. Abrupt contraction cell for flow visualization, birefringence and degradation measurements A inlet (from a peristaltic pump of a pressurized reservoir B outlet (atmospheric pressure or partial vacuum) C interchangeable metallic nozzle with a sapphire tip D capillary flow meter E glass window for flow visualization AP pressure drop (from pressure transducers)...

The second part treats specific problems typical of capillary flow with a distinct interface. It contains four chapters in which steady and unsteady capillary flow are treated. 

Chapter 9 is devoted to regimes of capillary flow with a distinct interface. The effect of certain dimensionless parameters on the velocity, temperature and pressure within the liquid and vapor domains are considered. The parameters corresponding to the steady flow regimes, as well as the domains of flow instability are defined. 

The onset of flow instability in a heated capillary with vaporizing meniscus is considered in Chap 11. The behavior of a vapor/liquid system undergoing small perturbations is analyzed by linear approximation, in the frame work of a onedimensional model of capillary flow with a distinct interface. The effect of the physical properties of both phases, the wall heat flux and the capillary sizes on the flow stability is studied. A scenario of a possible process at small and moderate Peclet number is considered. The boundaries of stability separating the domains of stable and unstable flow are outlined and the values of the geometrical and operating parameters corresponding to the transition are estimated. 

The quasi-one-dimensional model of laminar flow in a heated capillary is presented. In the frame of this model the effect of channel size, initial temperature of the working fluid, wall heat flux and gravity on two-phase capillary flow is studied. It is shown that hydrodynamical and thermal characteristics of laminar flow in a heated capillary are determined by the physical properties of the liquid and its vapor, as well as the heat flux on the wall. 

For a while now, the problem of flow and heat transfer in heated capillaries has attracted attention from a number of research groups, with several applications to engineering. The knowledge of the thermohydrodynamic characteristics of capillary flow with evaporative meniscus allows one to elucidate the mechanism of heat and mass transfer in porous media, to evaluate the efficiency of cooling system of electronic devices with high power density, as well as to optimize MEMS. 

Peles el al. (2000) elaborated on a quasi-one-dimensional model of two-phase laminar flow in a heated capillary slot due to liquid evaporation from the meniscus. Subsequently this model was used for analysis of steady and unsteady flow in heated micro-channels (Peles et al. 2001 Yarin et al. 2002), as well as the study of the onset of flow instability in heated capillary flow (Hetsroni et al. 2004). 

The 1 (0) correlation corresponding to various Weber numbers is shown in Fig. 8.5. The shape of the interface surface in a capillary flow with phase change is presented in Fig. 8.6. As the calculations show, the curvature of the meniscus is not constant and grows toward the periphery. 

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