Tube Flow Poiseuille Viscometer

As will be shown later, a momentum (force) balance on the fluid in the tube provides a relationships between the shear stress at the tube wall (rw) and the measured pressure drop  

The corresponding shear rate at the tube wall (yw) is given by 

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Two common methods for measuring viscosity are the cup and bob (Couette) and the tube flow (Poiseuille) viscometers. 

The automatic relative viscometer is ideally suited for measuring dilute polymer viscosities. It provides faster analysis and greater precision than is obtainable with conventional glass tube viscometers (Ubbelohde or Cannon-Fenske), which it replaces. The principle of operation is based on measurement of pressure drops due to the continuous forced flow of solvent and sample through two stainless steel capillary tubes placed in series. The pressure drop across each capillary tube obeys Poiseuille s law. The pressme drop is measured by a differential pressure transducer. The sample solution is loaded into a sample loop via a syringe pump and then pushed into one of the two capillaries. A steady-state condition is reached when the sample solution completely fills capillary 2, solvent remaining in capillary 1 at all times. The relative viscosity of the sample solution is determined simply and directly by the ratio of the pressure drops. From the measured relative viscosity, all other solution viscosity measurements can be calculated. Solution viscosities are determined by the viscosity of the sample relative to the reference solvent. The relative viscometer measures the solvent and sample viscosity simultaneously, so errors due to temperatme fluctuation and solvent variations are avoided. The main advantages of this approach are ... 

Earlier experiments involved the collection of SEC effluent aliquots to measure solution viscosity in batches with the very time consuming Ubbelohde drop-time type viscometers. A continuous capillary type viscometer was first proposed for SEC by Ouano. Basically, as shown in Figure 1, a single capillary tube with a differential pressure transducer was used to monitor the viscosity of SEC effluent at the exit of the SEC column. As liquid continuously flows through the capillary (but not through the pressure transducer), the detected pressure drop (AP) across the capillary provides the measure for the fluid viscosity (h) according to the Poiseuille s viscosity law ... 

Capillary Viscometers. Capillary flow measurement is a popular method for measuring viscosity (21,145,146) it is also the oldest. A liquid drains or is forced through a fine-bore tube, and the viscosity is determined from the measured flow, applied pressure, and tube dimensions. The basic equation is the Hagen-Poiseuille expression (eq. 17), where T is the viscosity, r the radius of the capillary, Ap the pressure drop through the capillary, V the volume of liquid that flows in time /, and L the length of the capillary. 

The capillary viscometer. The most common and simplest device for measuring viscosity is the capillary viscometer. Its main component is a straight tube or capillary, and it was first used to measure the viscosity of water by Hagen [28] and Poiseuille [60], A capillary rheometer has a pressure driven flow for which the velocity gradient or strain rate and also the shear rate will be maximum at the wall and zero at the center of the flow, making it a non-homogeneous flow. 

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... 

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. 

The viscosity of a liquid or solution can be measured by using a viscometer whose design is based on the Hagen-Poiseuille law. Essentially, this involves the measurement of the flow rate of the liquid through a capillary tube which is part of the viscometer. Consequently, by measuring the flow time of the solution, t, and that of the pure solvent, to, the relative viscosity can be determined ... 

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 ... 

Obviously, the work of Stokes (1845) was unknown to Rogers and Sabin however, in the same year we find the work of Bingham and White (1911) in which a capillary tube viscometer is described. Thqr remark that ".the well known formula of Poiseuille for the flow of a liquid through a capillary tube may be written as"... 

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. 

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