Tube viscometers

Rheology Measurement

3.1 Tube viscometers

The simplest type of viscometer for the study of fluids of low viscosity is the full capillary apparatus. Most capillary viscometers are modifications of the Ostwald viscometer shown in Figure 3.1.

CHAPTER 3 RHEOWGY MEASUREMENT

---...~'---~

."'-~

Figure 3.1 Ostwald viscometer (Blair (1969»)

The Ostwald viscometer (mounted vertically) depends on the measurement of the time taken for a given volume of liquid to fall through a capillary [Blair (1969) and Whorlow (1980)]. The rate of fall of the liquid depends on the density (0 and the viscosity (Jl.) of the liquid, so that the time of fall is in fact, proportional to Jl./O i.e. the kinematic viscosity of the liquid. Although Ostwald viscometers are very accurate at measuring the viscosity of Newtonian fluids, they are not suitable for the measurement of non-Newtonian fluids. The main difficulty in using capillary viscometers is the fact that the "head" or pressure is falling continuously in the capillary tube. As such, in many modified capillary viscometers the tested material flows under an applied pressure [Blair (1969)].

To demonstrate how the equations governing the forces acting in capillary viscometers are developed, consider the streamline flow of a Newtonian fluid through a capillary tube of length OL, radius 'R' under a pressure difference OP [Blair (1969) and Whorlow (1980)]. The flow through the capillary is illustrated in Figure 3.2.

CHAPTER 3 RHEOLOGY MEASUREMENT

Flow - - -....

- ^, .

/' "'\ 21tIUAL '\ '

/ -.. .... : .•... ~:::: .. :::: .. : .. : ... ~.... . . : R

~\ ~ '\ \ I_{r :}

I I I I , ,i. t

Q' .... _,_._._._._._._._._._._._._._._._._.-

h.~L . .j. ......... ... _ ...... _ . ./

j

\

.

^, " V , ' ^,

\ ... ~,,' .,/

... AL ... +

Figure 3.2 Forces on a short cylinder in capillary [Whorl ow (1980)]

If we consider a short section of the pipe sufficiently far from the ends of the tube, the following assumptions can be made [Whorl ow (1980)]:

1. The flow of the fluid is parallel to the axis.

2. The velocity 'u' of any fluid element is a function of radius 'r' only, giving axial symmetry.

3. The fluid flowing through the capillary is incompressible.

4. The fluid velocity at the wall of the capillary is zero.

5. The normal stress is isotropic.

The velocity gradient, which is the velocity rate of change in the radial direction, diminishes as the r increases. When r

=

0, the velocity gradient is at its maximum. When r

=

R (radius of capillary), the velocity gradient is zero since it is assumed that there is no slip at the wall [Blair (1969)]. The pressure difference OP acts over an area of

ca

^2, so that the force F will be:

F

=

=n> :::R²

(3.1)

Since the area of the wall is 2::RDL, the stresses at the wall and at a distance r respectively are:

CHAPTER 3

r = - -RM

wall 2M

RHEOLOGY MEASUREMENT

(3.2)

Following from Newton's law, which states that the shear stress is proportional to the shear rate, the following equation for velocity gradient is obtained:

du 1 ~Pr

- = - _ . _ - dr Ji 2M

(3.3)

where ~ is the viscosity of the fluid. To get the radial velocity variation Equation 3.3 is integrated with respect to r:

u = M

f

r.dr 2MJi

(3.4)

The boundary condition of Equation 3.4 is that the velocity of the fluid is equal to zero at the wall of the capillary .i.e. when r

=

a. After integration the following equation for radial velocity is obtained:

(3.5)

To obtain the volumetric flowrate (Q) of the fluid in the capillary the velocity in the concentric area is integrated between 0 and R:

R

Q

= fu.27Zrdr

o (3.6)

CHAPTER 3 RHEOLOGY MEASUREMENT

Substituting Equation 3.5 into Equation 3.6 and integrating, the following equation is obtained:

1lR4M

Q=

8~

Equation 3.6 is the Hagen-Poiseuille equation.

(3.7)

For Newtonian fluids the shear rate varies from a maximum near the vicinity of the wall, to zero at the centre of the capillary. Hence, to calculate the viscosity of fluids using capillary viscometers, the conditions at the shear stress and shear rate at the wall are used. For Newtonian fluids the viscosity is given calculated from Equation 3.7:

(3.8)

However, for non-Newtonian fluids the situation is more complex. Correction factors need to be added to account for the deviation of the fluid from Newtonian behaviour. One such correction factor is the Rabinowitsch correction, shown in the equation below:

( ) 1lR4(M/M) J..i

r =

w

8Q[ ~

^{+ 1 dIn}

^{Q )}

4 4 dlnO"w

(3.9)

The term in the brackets is called the Rabinowitsch correction [Barnes et al. (1989)]. When shear- thinning fluids are being tested, the derivative d(lnQ)/d(lnQ.,.) is greater than land for power-law liquids is equal to lin. Figure 3.3 shows the different velocity profiles for the laminar flow of power-law liquids in a capillary tube as 'n' varies. For Newtonian fluids the value of n is equal to one. Note the increase in the wall shear rate and the increasingly plug-like nature of the flow as 'n' decreases.

CHAPTER 3

Power Law index, n 0.2 0.4 1.0

,

"

, \

, \

:

^\

: I

, I

, ^/

' /

,/

RHEOLOGY MEASUREMENT

Figure 3.3 The velocity profile for the laminar flow of power-law liquids, calculated for the same volumetric throughput [Barnes et al. (1989»)

Equation 3.9 shows how the viscosity can be calculated in a capillary viscometer if the flowrate Q, the pressure difference across the capillary, and the dimensions of the capillary are known.

The pressure gradient can be measured using various devices. The most common method is by connecting holes in wall of the capillary to pressure transducers. To avoid end effects the holes on the capillary wall must be far away from the ends [Barnes et al. (1989)). Usually, end effects (entrance effects and exit effects) are avoided by ensuring that the aspect ratio of the capillary is greater than one hundred.