Volumetric flow rate, capillary

Capillary viscometers are useful for measuring precise viscosities of a large number of fluids, ranging from dilute polymer solutions to polymer melts. Shear rates vary widely and depend on the instmments and the Hquid being studied. The shear rate at the capillary wall for a Newtonian fluid may be calculated from equation 18, where Q is the volumetric flow rate and r the radius of the capillary the shear stress at the wall is = r Ap/2L. 

Piston Cylinder (Extrusion). Pressure-driven piston cylinder capillary viscometers, ie, extmsion rheometers (Fig. 25), are used primarily to measure the melt viscosity of polymers and other viscous materials (21,47,49,50). A reservoir is connected to a capillary tube, and molten polymer or another material is extmded through the capillary by means of a piston to which a constant force is appHed. Viscosity can be determined from the volumetric flow rate and the pressure drop along the capillary. The basic method and test conditions for a number of thermoplastics are described in ASTM D1238. Melt viscoelasticity can influence the results (160). 

J The viscosity characteristics of a polymer melt are measured using both a capillary rheometer and a cone and plate viscometer at the same temperature. The capillary is 2.0 mm diameter and 32.0 mm long. For volumetric flow rates of 70 x 10 m /s and 200 x 10 m /s, the pressures measured just before the entry to the capillary are 3.9 MN/m and 5.7 MN/m, respectively. 

A slit die is designed on the assumption that the material is Newtonian, using apparent viscous properties derived from capillary rheometer measurements, at a particular wall shear stress, to calculate the volumetric flow rate through the slit for the same wall shear stress. Using the correction factors already derived, obtain an expression for the error involved in this procedure due to the melt being non-Newtonian. Also obtain an expression for the error in pressure drop calculated on the same basis. What is the magnitude of the error in each case for a typical power law index n = 0.377... 

The selection of the column type is mainly determined by the composition of the sample. In general open-tubular (capillary) columns are preferred for low-density (gas-like) SFC, whereas packed columns are most useful for high-density (liquid-like) SFC. Open-tubular columns can provide a much larger number of theoretical plates than packed columns for the same pressure drop. Volumetric flow-rates are much higher in packed column SFC (pSFC) than in open-tubular column SFC (cSFC), which makes injection and flow control less problematic. 

The shear history simulators operate at a single shear rate during an experiment and do not run shear ramps. For these Instruments, apparent viscosity at a single shear rate Is determined by the relationship of differential pressure (AP), capillary length (L) and radius (r), and volumetric flow rates (Q), as follows. 

Methane is commonly used as a marker for measuring the gas holdup time (tm), which was done on a capillary column 25 m long by 0.25 mm ID by 0.25 pm film thickness. A retention time for methane of 1.76 min was obtained. Determine the average linear gas velocity (v) and the average volumetric flow rate (Fc). Explain how these values differ from the actual velocity and flows at the column inlet and outlet. 

In 1990, Bushey and Jorgenson developed the first automated system that coupled HPLC with CZE (19). This orthogonal separation technique used differences in hydrophobicity in the first dimension and molecular charge in the second dimension for the analysis of peptide mixtures. The LC separation employed a gradient at 20 (xL/min volumetric flow rate, with a column of 1.0 mm ID. The effluent from the chromatographic column filled a 10 pU loop on a computer-controlled, six-port micro valve. At fixed intervals, the loop material was flushed over the anode end of the CZE capillary, allowing electrokinetic injections to be made into the second dimension from the first. 

In a pressure capillary viscometer, such as a rheological die, pressure is used to force fluid through a capillary tube at constant volumetric flow rate Q, as shown in Figure 6.4. The pressure difference AP is measured between points A and B spaced apart a distance 1 along the tube. The basic rheological equations are as follows for shear stress and shear rate taken to be very near the wall in a tube of radius r ... 

Figure 1 is a schematic diagram of the experimental setup. The test section is a horizontal rectangular channel 40 mm in height (H), 160 mm in width (W), and 6,000 mm in length (L). The rectangular channel is completely constructed of transparent acrylic resin, as shown in Figure 2. Tap water and air are used as the gas and liquid phases, respectively. Water is circulated by a 2.2 kW pump fed by a water reservoir 4.2 m away. Air bubbles are injected into the horizontal channel from the upper inner surface of the channel. An array of capillary needles produces bubbles 10-100 mm in length. Before the air and water are mixed, their volumetric flow rates are measured. After leaving the horizontal channel, the gas-liquid mixture is dumped into a tank that acts as a bubble remover when the liquid phase is recirculated it is free of bubbles. At the end of the horizontal channel tracer particles are added to the water to act as ultrasound reflectors. The mean particle diameter is 200 pm and the particle density is 1020 kg/m3. These tracer particles are assumed to...

Capillary Flow Rheometry Next we examine the experimentally obtained results with the capillary flow rheometer shown in Fig. 3.1, which are directly relevant to polymer processing flows, since the attainable shear rate values are in the range encountered in polymer processing. The required pressure drop AP does not increase linearly with increases in the volumetric flow rate Q, as is the case with Newtonian fluids. Rather, increasingly smaller increments of AP are needed for the same increases in Q. The Newtonian Poiseuille equation, relating flow rate to pressure drop in a tube, is linear and given by... 

The volumetric flow rate of laminar Newtonian flow of a liquid through a capillary tube is expressed by Poiseuille s law as ... 

In axial-flow viscometers, the sample is made to flow through a duct of regular cross-section. Capillary (circular cross-section) and slit (rectangular cross-section) viscometers are controlled stress instruments a known pressure difference (which causes shear stress in the sample) is applied over the duct length, and the resulting volumetric flow rate measured. In the extrusion viscometer, a controlled shear rate instrument, the sample is extruded through a capillary tube by the action of a constant speed piston, acting on the sample in a cylindrical reservoir to which the capillary is attached. The pressure difference between the ends of the capillary is measured. 

If both ends of the capillary are connected, allowing a flow circuit as shown in Figure 9.18, the volumetric flow rate, Q, is dependent on the electric current, I, as follows ... 

Streaming Potential Streaming potential is the same phenomena operating in reverse—that is, the flow of electroljdie induces an electric field, E, which is measured. Using transport equations the volumetric flow rate can be related to the pressure drop across the capillary, AjP/L, giving... 

To use the WRM equation, the steps involved are (1) using a tube/pipe/capillary of known diameter (D) and length (L), several sets of volumetric flow rate (Q) versus... 

This expression indicates that the fluid presents a distribution of velocities whose profile corresponds to a paraboloid of revolution the velocity of the fluid reaches its maximum value [v ax = (AP/4ri AL)R ] at the center of the capillary and 0 at the capillary wall. The flow of the fluid through an infinitesimal cylinder defined by r and r + dr is dQ = 2nr dr v(r). Then the volumetric flow rate can be written as... 

The analysis of the flow of fluids through a permeable bed of solids is widely applied in filtration, leaching, and several other processes. A first approach may be made by assuming that the interstices of the bed correspond to a large number of discrete, parallel capillaries. If the flow is streamline, the volumetric flow rate Q is given for a single capillary by Eq. (14) ... 

A second type of capillary viscometer generates a constant volumetric flow rate with a pumping device, and the viscosity is determined by measuring the pressure drop. Such viscometers have been used mostly in research, but some commercial instruments are available. These viscometers may be used over a wide range of conditions for both gases and liquids, and in some cases the uncertainty can be reduced to less than 0.1%. 

Volumetric Flow Rate and a Pressure Gradient Along the Capillary... 

It should be noted that the shear rate is related to the volumetric flow rate. In a capillary rheometer, for example, the shear rate at the capillary wall is related to the... 

It is well known that HDPE and PVC, as well as many polymer suspensions and fllled composites, exhibit slip at the capillary wall. Considering slip at the wall, the volumetric flow rate Q through the die is given by... 

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