Capillary/tube viscometer equation

In a series of experiments on the flow of flocculated kaolin suspensions in laboratory and industrial scale pipelines(26-27-2Sl, measurements of pressure drop were made as a function of flowrate. Results were obtained using a laboratory capillary-tube viscometer, and pipelines of 42 mm and 205 mm diameter arranged in a recirculating loop. The rheology of all of the suspensions was described by the power-law model with a power law index less than unity, that is they were all shear-thinning. The behaviour in the laminar region can be described by the equation ... 

It may be concluded that well-designed rotational and capillary-tube viscometers are generally more useful than any other type. Accordingly, rheological equations for interpretation of data have been presented for these two types of instruments. 

VIt = (p gl + Apjn/ /Bri/ where p is the density of the fluid, g is the force due to gravity, l is the length of the capillary tube, and Ap is the pressure difference between the entrance and exit of the capillary tube. The equation can be rearranged and further simplified to account for all the constants that characterize the viscometer. This also assumes that the difference in height of the two liquid columns is relatively constant during the time required for flow. Thus, the only pressure difference across the liquid is due to the weight of the liquid. With these conditions ... 

The capillary tube viscometer is used to plot the average rate versus the shear stress at the wall of the tube. This is called the pseudoshear diagram, as defined by the Mooney-Rabinovitch equation ... 

The first term on the right hand side of this expression obviously represents the Hagen-Poiseuille law, while the second term (referred to as a kinetic energy correction) represents the influence of the entrance region. Equation 3-2 can be found in many subsequent papers on the use of capillary tube viscometers in which a transient flow process is utilized however, I was never able to locate a derivation of the kinetic energy correction term. 

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 ... 

The Ostwald Viscometer Method There are a number of methods of different kinds of measuring viscosity. The method commonly employed is the Ostwald s viscometer method which is based on Poiseullie s law. This connects the rate of flow of a liquid through a capillary tube with the coefficient of viscosity of the liquid and is expressed by the equation... 

The opportunity to measure the dilute polymer solution viscosity in GPC came with the continuous capillary-type viscometers (single capillary or differential multicapillary detectors) coupled to the traditional chromatographic system before or after a concentration detector in series (see the entry Viscometric Detection in GPC-SEC). Because liquid continuously flows through the capillary tube, the detected pressure drop across the capillary provides the measure for the fluid viscosity according to the Poiseuille s equation for laminar flow of incompressible liquids [1], Most commercial on-line viscometers provide either relative or specific viscosities measured continuously across the entire polymer peak. These measurements produce a viscometry elution profile (chromatogram). Combined with a concentration-detector chromatogram (the concentration versus retention volume elution curve), this profile allows one to calculate the instantaneous intrinsic viscosity [17] of a polymer solution at each data point i (time slice) of a polymer distribution. Thus, if the differential refractometer is used as a concentration detector, then for each sample slice i. 

In both methods, Washburn s equation is used, which was derived from the Poiseuille equation to measure viscosity in capillary viscometers. The rate of volume flow (V/t) through a capillary tube with radius, rc, is given by the Poiseuille equation as... 

The falling-level viscometer is a simple device that may be used to determine flow curves for relatively dilute polymer solutions in a single experiment. It consists of a vertical buret of diameter D connected to a horizontal capillary tube of length L and radius R. The buret is filled with the solution, which is then allowed to drain through the capillary. Data are h vs. t, where h is the height of the fluid in the buret above the capillary. It may be assumed that flow in the capfllary is laminar, that all resistance to flow is in the capillary, and that entrance and exit effects are negligible. The fluid has a known density p. Describe clearly how to proceed from the data, h vs. t, to the flow curve, T vs. y. Show all necessary equations. 

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. 

Solution Equation (16) shows that the velocity gradient is not uniform in a capillary viscometer any more than it is in a concentric-cylinder instrument. The rate of shear dvldr is directly proportional to the radial distance from the axis of the cylinder. At the wall it has its maximum value, which is proportional to Rc] at the center of the tube it equals zero. Some intermediate value, say, the average, might be used to characterize the gradient in a given instrument. This quantity will be different for capillaries of different radii. All of this is similar to the situation in concentric-cylinder viscometers. 

Equation (51) can equally well be applied to a U-tube type viscometer, in which a capillary is connected to essentially a manometer (see, for example, Maron and Belner, 1955). The only modification is that the pressure head, Ah, is calculated from the difference in the decreasing meniscu of the capillary arm and that arising in the manometer arm. 

Perhaps the most familiar technique is the capillary-flow method. The working principle is the Hagen-Poiseuille relationship between the flow rate through a tube of fixed diameter, the pressure drop, and the viscosity. In practice, because the capillary diameter appears to the fourth power in the working equation and is difficult to determine accurately, capillary viscometers are usually calibrated with reference fluids such as water or reference oils that are available from viscometer manufacturers and some national laboratories. 

A flow-through viscometer developed for application as a sensor in automated analyses consisted of a glass capillary connected to the sample flow circuit with thin-walled tubes at both ends. These tubes separate the fluid to be tested from a hydraulic fluid, and this construction ensures the absence of dead space and a minimal test volume. Development of a mathematical model describing the enzymatic degradation of macromolecules resulted in a reciprocal equation allowing rectilinear presentation of the calibration data. The application of the enzyme to the assay of amylase was described. 

Capillary viscometers are the most extensively used instruments for the measurement of viscosity of liquids because of their advantages of simphcity of construction and operation. Both absolute and relative instruments were constracted. The theory of these viscometers is based on the Hagen-Poiseuille equation that expresses the viscosity of a fluid flowing through a circular tube of radius r and length L in dependence of the pressure drop AP and volumetric flow rate Q, corrected by terms for the so-called kinetic-energy and end corrections ... 

In a capillary viscometer with tube radius R, the quantities which are directly measured are pressure drop, AP, for a given volumetric flow rate, Q. If it were known that the fluid under test was a pure power law fluid, then one possible way to determine the K and n parameters is as follows first define the effective viscosity, through the pseudo-Poiseuille equation ... 

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