The Poiseuille Equation and Capillary Viscometers

4 THE POISEUILLE EQUATION AND CAPILLARY VISCOMETERS 4.4a Flow Through Cylinders The Poiseuille Flow 

5 Flow in a cylindrical capillary (a) a volume element in the flowing liquid and (b) the parabolic flow profile. 

The increment in viscous force acting on this element is the difference between the viscous forces on the outer and inner surfaces of the element, with each of these given by Equation (1), with A 2irr(  

we must relate (dv/dr)r+dr to (dv/dr)r. The following expression accomplishes this, provided dr is small  

If this result is substituted into Equation (11), expanded, and only terms linear in dr retained, the expression becomes 

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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. 

Both the capillary viscometer (providing about 0.7% accuracy), the theory of which is based on the Hagen-Poiseuille equation and the oscillating disc viscometer (providing about 0.2% accuracy) are applicable to experimental determination of viscosity at high pressures and temperatures. 

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. 

In a capillary viscometer, a piston of known weight presses the melt through a capillary with a specific diameter and length. The flow of a Newtonian fluid in a capillary obeys the Hagen-Poiseuille equation. 

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 ... 

Several geometries can be used to create velocity gradients in fluids for the computation of tj. The Navier-Stokes equation provides the basis for finding the relationship between tj, the geometry and applied forces. One of the most common arrangements is the capillary viscometer, for which the Poiseuille solution to the Navier-Stokes equation, which has a parabolic flow profile, is used ... 

The most common instrument used for viscosity measurements in low-viscosity liquids is the capillary viscometer. In this instrument, a liquid is made to flow under its own potential head through a narrow capillary with a weU-defined length and cross section. The volumetric flow rate Qota simple Newtonian liquid undergoing laminar flow in a capillary of diameter d and length / is given by Poiseuille s equation ... 

One of the most widely used rheometer configurations is a simple variant of the capillary flow viscometer. In this device, a concentrated polymer solution or melt undergoes Poiseuille flow in a narrow capillary, length L and internal radius R, under the action of an external pressure P. The capillary exit is typically open to the atmosphere, such that a pressure difference Ap = P - produces the driving force that leads to fluid flow (Figure 8.3). If the volumetric flow rate of fluid through the capillary Q is known (measured), Poiseuille s equation can be used to determine the fluid s viscosity if the fluid is Newtonian. 

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