che410 tubular viscometry

METU Chem. Eng. Dept.

Ch.E. 410 Chem. Eng. Lab II

EXPERIMENT 1.2.

TUBULAR VISCOMETRY

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OBJECTIVE

In this experiment a non-Newtonian solution is characterized in terms of its viscosity

under steady shear-flow condition.

PRELIMINARY WORK

Prior to the experiment carry out the following tasks:

Study the derivation of the stress and velocity profiles for laminar flow of a

Newtonian fluid in a circular tube.

Read up on the classification of fluids based on their viscosity, and the general

characteristics of non-Newtonian liquids. Find out the rheological class of the

fluid used in your experiment (Newtonian/non-Newtonian, shear-thinning or shear

thickening etc.)

Using equations 1-6 below, derive an expression that relates the pressure drop,

volumetric flow rate and the exponent ‘n’ for a power-law fluid. Describe the

linearization of this expression, and how you plan to obtain the parameters n and

K from the slope and the intercept.

Please bring one laptop.

BACKGROUND INFORMATION

Capillary viscometers are widely used instruments for the measurement of viscosity due,

in part, to their relative simplicity, low cost and accuracy. In a typical tubular flow, the

fluid experiences a range of shear rate; the shear rate is zero at the capillary center and

maximum at the wall. Capillary flow data can be used to obtain the viscosity

corresponding to the shear rate evaluated at the wall. To determine the wall shear rate, the

laminar velocity profile is used. Flow data consisting of pressure drop and volumetric

flow rate measurements can be used to obtain the viscosity corresponding to the shear rate

evaluated at the wall as outlined below.

For any rheological behaviour, the shear stress at the wall under steady-state conditions is

given by the expression:

ΔPπR2 = τw2πRL (1)

where P, τw, and R represent pressure, wall shear stress and tube radius, respectively.

a) For Newtonian fluids shear rate at the wall is given by



Experiment 1.2. Tubular Viscometry

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'w = 4Q/ πR3 (2)

which is a unique function of the volume flow rate Q.

b) For non-Newtonian power-law fluids:

3

13

R

Q

n

n

w

(3)

or equivalently, using (2):

wn

n

w

4

13 (4)

Hence the shear rate depends on the power- law index (n) as well as on the flow rate. The

relation between the shear stress and the shear rate at the wall is obtained for a power-

law fluid as:

n

wKw

(5)

nw

n

n

nK

w

4

13 (6)

The power law index n can be obtained from a plot on log-log scales of

τw (which is a unique function of the pressure drop) as a function of 'w, which depends

only on the flow rate Q. The non-Newtonian viscosity at the wall can be evaluated

through:

𝜂 =𝜏𝑤

�̇�𝑤 (7)

End Effects:

The velocity profiles are obtained assuming that flow is fully developed. This assumption

can be quite accurate provided that the entry and exit lengths of the flow are negligible

compared to the total length of the capillary. The complications associated with end

effects should always be considered in viscosity measurements with capillaries of finite

lengths. It is usually safe to neglect end effects as long as the length to diameter ratio

(L/D) of the capillary is on the order of 100-120, at least for purely viscous fluids. There

may be situations when it is not possible to use such large values of (L/D) and therefore,

the resulting shear stress values must be corrected for the end effects.

In order to get the pressure drop associated with the fully developed section of the flow

two experiments can be performed by using two capillaries of two distinct lengths, say LA




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and LB. Both tubes have radius R and are longer than the sum of the entrance (Len) and

exit (Lex) lengths. This requirement ensures that there will be steady fully-developed

shear flow in some section of each tube. In the first experiment, the pressure PA required

for the volumetric flow rate Q through the tube of length, LA is measured. A similar

measurement is made in the second experiment to find PB necessary to obtain the same

flow rate Q.

EXPERIMENTAL

Apparatus : Volumetric flask, glassware, stop watch, peristaltic pump and the tubular

viscometer.

Figure 1. Experimental Set-up

Materials: Distilled water and carboxylmethyl cellulose (CMC) will be used. Before the

lab session, read the MSDS pages of CMC.

PROCEDURE

a) Experiments are to be carried out at least four different flow rates for a 0.5 wt%

CMC-water solution. For each flow rate, once the level in the cylinder becomes

steady, record the liquid level in the cylinder.




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b) Repeat the measurements using the other set-up, which has a different flow length.

The same procedure is applied for the same volumetric flow rates.

CALCULATIONS

Part A:

1. To calculate the power law index (n), take the pressure drop vs flow rate data,

then on a log-log scale plot ΔP/L as a function of volumetric flow rate Q.

2. Find the power law constant (K).

3. Calculate the generalized Re that may be used to check the flow regime. For

power law fluids, it is given by [4]:

Regen = 8[(ρDnV2-n)/K](n/6n+2)n

V = bulk velocity

D = diameter of the capillary

n = power law index

Plot on a log-log scale η as a function of w to observe the non-Newtonian behaviour of

the polymer solution used in the experiment.

Part B: To investigate the end effects on the flow, we wish to make a connection

between Eqn1 and the Bagley end correction factor e. According to Bagley’s scheme, one

can obtain the steady shear flow τw by measuring the pressure drop ΔP in the reservoir for

various flow rates and for tubes of different L/R ratios. For a fixed value of Q, if the

pressure drop ΔP is plotted as a function of L/R, the result is represented by a straight line

with slope 2τw. The line intersects the L/R axis at ‘e’ as given by

ΔP = 2τw(L/R) + 2eτw (8)

As multiple measurements are taken in both part a and part b, the correction factor can be

calculated several times, all of which should compare well with each other.

SUGGESTED READING

1. Bird, R.B., W.E. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, 2nd ed, 2002.

2. Whorlow, R.W., Rheological Techniques, 2nd ed, 1992.

3. Chabbra, R.P., Non- Newtonian Flow and Applied Rheology, 2nd ed, 2008.

4. Metzner, A.B., Reed, J.C., 1955. Flow of non-Newtonian fluids correlation of the

laminar, transition and turbulent flow regime, AICHE J, 434-440.

che410 tubular viscometry

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