CAIRO UNIVERSITY

FACULTY OF ENGINEERING

CHEMICAL ENGINEERING DEPARTMENT

Modeling of Newtonian and Non-Newtonian Fluid

Flow in a Capillary Tube Viscometer using COMSOL

Transport Phenomena I

Under supervision of:

Dr. Ahmed Sherif

Prepared by:

Ahmed Osama Elnabawy

Aly Magdy Aly

Esraa Elmitainy

Mennah Mohamed

Reda Zein

1/1/2011

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Table of Contents

1. Problem Statement ................................................................................................................... 1

2. Solving Procedure.................................................................................................................... 2

3. COMSOL Results .................................................................................................................... 5

4. Conclusion ............................................................................................................................... 9

List of Figures ........................................................................................................................ 10

1

1. Problem Statement

We consider a capillary viscometer, the radius of capillary of which is and the length of which

is , where . It is required to generate the velocity profile across the flow of the fluid in the

capillary; that is, how changes with as shown in

Figure 1. This problem is the development of a simpler case that we have thought of; a case of

falling film on a wall. This problem (i.e. the capillary viscometer) is a little bit more

sophisticated in geometry and represents also an application that has been long sought of.

Of special interest is to measure the pressure drop phenomenon which affects any fluid that

passes inside a capillary tube. The case in hand makes use of the Hagen-Poiseuille equation to

check the consistency of the COMSOL model because we’ll apply our model on water which is a

Newtonian fluid (i.e. Hagen-Poiseuille equation is applicable) and its viscosity is well known

as well.

In case of non-Newtonian fluid, one needs to use more general equation than Hagen-Poiseuille in

order to calculate the pressure drop . The treatment covered by the next few pages shows how

the pressure could be found at the entrance and the exit of the capillary tube, and hence the

difference between them is the pressure drop needed.

Figure 1: A Simple Illustrative Diagram of the Capillary Tube Viscometer Case

Center

line

r

Wal

l

2

2. Solving Procedure

During this model it is assumed that the flowing fluid is water, which is the fluid mostly used to

calibrate viscometers before being used. Also as water is a fully defined fluid it allows the user to

compare the pressure drop from the model to the pressure drop resulting from the Hagen-

Poiseuille equation.

The capillary viscometer is considered as a very small tube with the dimensions of 0.5 mm

radius ( ) and 20 mm in length ( ). Therefore, the model space dimension was defined as Axial

Symmetry (2D) to be able to model half the capillary tube around an axis of symmetry.

The Momentum transfer model used is a steady state analysis for a non Newtonian fluid flowing

in a laminar way. The laminar model is logical since the capillary tube is very thin so there is no

chance of turbulence within the system, also the non Newtonian model is assumed to be able to

use this program later for any type of fluid (Newtonian or non Newtonian) without needing to

specify the whole model again.

As the tube is very small drawing it directly will not give accurate dimensions because the

COMSOL scale is in meters and the model is in millimeters. That was why the model was

defined through specifying an object in the Draw option where the width was specified as 5e-4 m

and the length as 2e-2 m.

The next step was to define the Sub-domain settings, in this window it was checked that the

model used is the power law according to the following relation:

(

)

Where:

: is the shear stress at the pipe wall, Pa

m: flow consistency index, Pa-sn. Where m=µ = viscosity of fluid for Newtonian fluids

n: flow behavior index, dimensionless. Where n= 1 for Newtonian fluids

: Shear rate at the pipe wall, s

-n (in a direction perpendicular to the shear stress)

According to the equation above, and applying it to water, which is a Newtonian fluid. The

values of n and m entered were: n=1 and m=1e-3 Pa-s as shown in Figure 2.

3

Figure 2: Power Law Parameters Definition

The last step before solving was to define the boundaries of the tube. Boundary 1 is the axis of

symmetry, 2is the outlet with a pressure of 0 Pa (used as a reference so the inlet pressure will be

the pressure drop), 3 is the inlet with a z-velocity of 0.01 m/s in the negative z-direction

(calculated through the equation where =0.02 m and =2 s to be satisfy the human

reaction in measuring time) and 4 is a wall with no slip conditions. The model shape just before

solving is shown in Figure 3.

4

Figure 3: Capillary Tube Viscometer Case before Solving

The model is now ready to solve after refining the mesh to contain 960 elements. The results

obtained are displayed in the next chapter.

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3. COMSOL Results

The first result obtained is the surface velocity profile over the whole model; which shows a

maximum velocity at the axis of symmetry and zero velocity at the wall of the tube as shown in

Figure 4.

Figure 4: Surface Velocity Profile

The general velocity profile at a point in the middle of the tube shows a parabolic shape, as

expected, with the maximum velocity at =0 and a zero velocity at =5e-4 as shown in Figure 5

Figure 5: General Velocity Profile

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The pressure change over the whole body can be seen in the surface plot from Figure 6

Figure 6: Surface Pressure Plot

To get specific plots at certain cross sections the Cross-section plot parameters from the post

processing tab was used. In which the z-inlet and outlet were given values inside the domain but

very near the end. Therefore, the values used were as follows z-inlet = 0.019995 and the

z- outlet=1e-6. An example of the data entry is shown in Figure 7 where the pressure profile at

the inlet.

Figure 7: Cross-Section Plot Parameters to Get Pressure Profile at Inlet

7

At the inlet and outlet a velocity profile was obtained for both on the same plot as shown in

Figure 8 which shows a constant inlet velocity and a parabolic outlet velocity as expected.

Figure 8: Inlet and Outlet Velocity Profiles

The outlet pressure profile shows a constant pressure of a very small value (3e-4 Pa) which is

near enough to the zero value specified and the profile shape is as seen in Figure 9.

Figure 9: Outlet Pressure Profile

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The inlet pressure profile is displayed in Figure 10 and the pressure drop can be obtained from it

directly (as the outlet pressure is set to zero) by getting the average of the profile.

Figure 10: Inlet Pressure Profile

This pressure drop is then compared to the one obtained from Hagen- Poiseuille equation as

discussed in the next chapter.

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4. Conclusion

After getting ΔP using COMSOL, We can investigate the accuracy of the result by comparing it

with that of Hagan-Poiseuille equation.

: Average velocity =0.01 m/s.

: Radius of the tube =0.0005 m.

µ: viscosity of water =0.001 Pa.s.

: Length of the tube= 0.02 m.

Pa

Which is approximately equals to that found by COMSOL with an error 2.8%.

This proves the accuracy of this model for calibrating the viscometer and checking the pressure

drop of different non Newtonian fluids.

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List of Figures

Figure 1: A Simple Illustrative Diagram of the Capillary Tube Viscometer Case ......................... 1

Figure 2: Power Law Parameters Definition .................................................................................. 3

Figure 3: Capillary Tube Viscometer Case before Solving ............................................................ 4

Figure 4: Surface Velocity Profile .................................................................................................. 5

Figure 5: General Velocity Profile .................................................................................................. 5

Figure 6: Surface Pressure Plot ....................................................................................................... 6

Figure 7: Cross-Section Plot Parameters to Get Pressure Profile at Inlet ....................................... 6

Figure 8: Inlet and Outlet Velocity Profiles .................................................................................... 7

Figure 9: Outlet Pressure Profile ..................................................................................................... 7

Figure 10: Inlet Pressure Profile ..................................................................................................... 8