Capillary/tube viscometer 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 ... 

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. 

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. 

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. 

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