Dr. Rafael M. Digilov

Technion, Israel Instituite of Technology

 

Department of Education in Technology & Science

Technion Cit

Haifa

32000

Israel

972-4-8292742

edurafi@tx.technion.ac.il

 

 

 


Weight-detecting capillary viscometer

 

Abstract

 

An approximate formula for the decay of a liquid-mass with time in a tank draining under gravity through a discharge capillary tube is derived. The advantage of the found solution, compared with the well-accepted exponential expression, is its mathematical consistency modified by the kinetic energy of the liquid flow. Based on this formula, the operation principle of a novel weight-detecting capillary viscometer is designed.

 

Theory

Viscosity is a key concept in fluid mechanics, characterizing the resistance of the fluid to flow. For laminar flow of an incompressible liquid through a pipe of radius r, the pressure drop  caused by viscous losses across a pipe length Lc, is given by the Hagen-Poiseuille law

                                                                                                              (1)

where h is the dynamic viscosity of the liquid, and u is the mean flow velocity.

With Eq. (1), the viscosity of a liquid can be determined using a simple physical system consisting of a tank of constant cross-section S, draining under gravity through a discharge capillary tube placed at depth h below its open to the atmosphere surface. The liquid enters the capillary at the hydrostatic pressure rgh (where r is the density, g the acceleration of gravity) and leaves the capillary at a mean velocity u. Because of viscous losses in the capillary tube, the pressure distribution in the system is given by the Bernoulli-Poiseuille equation

                                                                                                       (2)

where  is the kinetic energy parameter depending on end conditions. With u expressed through the mass loss rate

                                                                                                           (3)

where the negative sign means that the liquid-mass is decreasing, Eq. (2) becomes 

                                                                                               (4)

where t is the characteristic time, depending on the kinematic viscosity  alone

                                                        ,                                                      (5)

and a  is the kinetic energy coefficient

                                                                                                               (6)

With , the quadratic for  Eq. (4), has the solution:

                                                                                           (7)                               

If we assume , we may expand the radical in Eq. (7) and omit the cubic and higher terms to obtain

                                                                                                     (8)

which with the initial condition , is directly integrated to give:

                                                                                   (9)

The advantage of Eq. (9) compared with the expression accepted in physical textbooks

                                                  ,                                              (10)

is its mathematical consistency modified by the kinetic energy correction of the moving liquid that at  is reduced to (10).

Similarly, for a capillary tube connected vertically, one obtains:

                                                      (11)

We will fit Eq. (11) to experimental data  varying t and b  and interpret the results at constant temperature in the term for the dynamic viscosity h using Eq. (5) as follows

                                                                                                                   (12)

  being the meter constant of the apparatus.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Figure1. Computer-based mass-detecting capillary viscometer: possible variations in the design of equipments

Construction of Apparatus: 

We used a precision force sensor to measure the time dependence of the liquid mass  in the liquid column draining under gravity. The apparatus, with possible variations in the design of the equipment is shown in Fig. 1. It consists of a data-acquisition system, a glass cylinder of constant cross section S = 12.93cm2 and length ~0.4m, and a discharge capillary tube of radius r = 1.00mm and length = 19.8mm glued to the bottom of the glass cylinder, so that the liquid can flow freely under gravity.

                                

 

Figure 2. Weight-detecting capillary viscometer

The glass cylinder was hooked directly to a force sensor that in turn was rigidly mounted on a metallic stand and connected through a Data Logger to a personal computer (see Fig. 2). The data were read out to the PC as the mass of the liquid versus time . The resolution of the force sensor was 0.05N, so the accuracy in monitoring the mass was ~ 5 mg. The apparatus doesn't require any calibration once it is set.

Use of Apparatus 

The measurement procedure is as follows.

1.     Hook the empty glass cylinder to the force sensor and measure its weight.

2.     Close the opening of the capillary tube with a plug and fill the tank from the entrance with the tested liquid.

3.     Be sure that air bubbles within the capillary and over the liquid column are removed, as these greatly affect the measurement.

4.     Measure the temperature of the liquid in the tank with the thermometer.

5.     Place the beaker under the opening of the tube and open it, allowing the fluid to flow. The flow should be fully developed.

6.     Adjust the data acquisition system and observe the decay of the liquid weight in the tank versus time for ~ 10 min. Misinterpretation of the experimental data can occur if measurements are made over a short time range.

7.      Save the record data  and fit them to Eq. (11) using the MATLAB software. Figure 2 shows a chart-recording obtained with distilled water, the fit curve that excellent follows an experimental plot and fitting to the exponential function for comparison. 

8.     With the parameter t determined by the fitting procedure and the density of the liquid found in any standard handbook at given temperature, one calculates the viscosity of the fluid by Eq. (12).

 

Teaching objectives

This is a simple and efficient and high precision computer-based laboratory apparatus dealing with a classic problem in fluid mechanics - draining of a column of water through an orifice, which is especially suitable for both measurement and teaching requirements in fluid dynamics courses at various levels. In contrast to most handbooks in which Bernoulli' equation and Hagen-Poiseuille law are introduced separately, this experiment demonstrate the generalize Bernoulli-Poiseuille equation and the limitation of Bernoulli's equation.

The following physics problems can be solved using the weight-detecting viscometer:

                                                     

 

 

Figure 2 Typical plot of the decay of the liquid mass vs time for distilled water.

 

                                             

                                                                                        (13)

                                                                                                         (14)

                                                                                (15)

       being the shear rate:

                                                                                                   (16)