Intro to Practical Fluid Flow Episode 9 docx

The friction factor for Sisko fluids can be conveniently obtained by using the Sisko fluid section in the FLUIDS computational toolbox as shown in Figure 5.24.. The friction factor for t

to get X ˆ 2:043 (This requires the use of a non-linear equation solver)

G…0:25; 2:043† ˆ 0:7643

fslˆ 16

Re1

1 ‡ X

G…n; X†ˆ

16

103

1 ‡ 2:043 0:7643 ˆ 6:37  10 2 This can be checked on Figure 5.23

The friction factor for Sisko fluids can be conveniently obtained by using the Sisko fluid section in the FLUIDS computational toolbox as shown in Figure 5.24

The friction factor for the Sisko fluid can be plotted against the variables D*, Q* and V* to facilitate the solution of problems that require the calculation of the flowrate or required pipe diameter when the available pressure gradient is known

D ˆ Re2

Under steady flow conditions

fslˆ D

Substituting for Re1 and fslinto equation 5.97

Dˆ slPGDTF2 2

1

The friction factor is plotted against D* in Figure 5.25 Like the friction factor graph from which it is generated, this plot is also a series of straight lines as can be seen by substituting equations 5.94 and 5.96 into the definition of D*

Figure 5.24 Data input screen for calculation of friction factor of a Sisko fluid

Non-Newtonian slurries 151

In the laminar regime

fslˆ 16G…n; X†1 ‡ X

1

and in the turbulent regime

fslˆ 0: 0798

D3

…5:101† The use of this plot is demonstrated in the following illustrative example

Illustrative example 5.7

Calculate the rate of discharge of solid in the form of a slurry containing 9.5 per cent of solids by volume through a 5-cm diameter pipe under a pressure gradient

of 91.8 Pa/m The particle size in the slurry is 99 per cent < 4 m and independent measurements made in a rotational viscometer indicates that the fluid behaves as

a Sisko fluid with flow index n ˆ 0:16, KSˆ 0:7 Pa s0:16and 1ˆ 2:7  10 3Pa s The density of the solid is 2970 kg/m3and the density of the water is 997 kg/m3 Solution

Since the pipe diameter is known the dimensionless pipe diameter is calculated

Dˆ 1184  91:8

2  …2:7  10 3†2

!1=3 0:05 ˆ 97:67

Dimensionless pipe diameter D *

10–3

10–2

10–1

100

H = 0.15 0.45 1.25 2.5 6 9

Figure 5.25 Friction factor for a Sisko fluid with flow index n ˆ 0:249 Use this plot when the pipe diameter and the available pressure gradient are known

152 Introduction to Practical Fluid Flow

Even with a known value of D*, trial and error methods cannot be avoided entirely for the Sisko fluid because the value of the strain-rate parameter H must be known in order to establish the value of the friction factor

Try H ˆ 45 to start the calculation and get the value of fslfrom a graph of the friction factor against D* such as the one shown in Figure 5.25 Remember that a new graph must be generated for every different value of the flow index n

It is threfore essential to use the FLUIDS toolbox which also provides the value

of Re1 The data input screen for the FLUIDS toolbox is shown in Figure 5.26

fslˆ 0:7685

Re1 ˆ 1101 The density of the slurry is calculated from the densities of the components

slˆ 0:905  997 ‡ 0:095  2970 ˆ 1184 kg=m3

The average velocity is calculated from the Reynolds number

V ˆ 1101  2:7  100:05  1184 3ˆ 0:0502 m=s

8 V

8  0:0502 0:05 ˆ 8:034 s 1

2:7  10 38: 034 0:84ˆ 45:1 which is close enough to the original guess

The discharge rate of slurry is

Q ˆ

4 D2V ˆ 

4 0: 0520:0502 ˆ 9:85  10 5 m3=s

Figure 5.26 Toolbox data input screen to calculate friction factor for a Sisko fluid when the dimensionless pipe diameter is known

Non-Newtonian slurries 153

The discharge rate of solid is

Qsˆ 0:095  9:85  10 5ˆ 9:36  10 6 m3=s The mass discharge rate of solid is

Ws ˆ sQsˆ 2970  9:36  10 6ˆ 0:0278 kg=s The dimensionless flowrate for a Sisko fluid is defined by

Q3 ˆ Re5

1fslˆ 16 1 ‡ X

G…n; X†

1

f4 sl

…5:102† and when the fluid is flowing steadily through a pipe, Q* can be calculated from the flowrate and the available pressure gradient

Q3ˆ32 4slPGDTF

The dimensionless velocity for a Sisko fluid is defined by

V3ˆRef1

sl ˆ16f2

sl

1 ‡ X

and when the fluid is flowing steadily through a pipe, V* can be calculated from the average velocity and the available pressure gradient

V3ˆ 2 2sl

1PGDTFV3

Graphs of the friction factor for Sisko fluids against Q* and V* can be easily generated using the FLUIDS toolbox

5.5 Practice problems

1 Calculate the pressure gradient due to friction when a power law fluid flows through a 10-cm ID pipe at 20 kg/s

The properties of the fluid are:

Density 1:7  103kg/m3

2 The data given in the table below were obtained in a laboratory pipeline of diameter 0.053 m

Show that these data are consistent with the power-law model and evalu-ate the parameters n and K The density of the fluid is 1000 kg/m3

154 Introduction to Practical Fluid Flow

3 Calculate the pipe diameter that is required to discharge 100 kg/hr of solid in the form of a slurry containing 9.5 per cent of solids by volume under a pressure gradient of 100 Pa/m The particle size in the slurry is 99 per cent < 4m and independent measurements made in a rotational viscometer indicates that the fluid behaves as a Sisko fluid with flow index n ˆ 0:16, KSˆ 7:0 Pa s0:16 and 1ˆ 2:7  10 3Pa s The density of the solid is 2970 kg/m3and the density of the water is 997 kg/m3

5.6 Symbols used in this chapter

D* Dimensionless diameter

fsl Friction factor for non-Newtonian fluid

H Strain-rate parameter for Sisko fluid

K Flow consistency coefficient for power-law fluids Pa sn

KH Flow consistency coefficient for Herschel±Bulkley fluid pa sn

KS Flow consistency coefficient for Sisko fluid Pa sn.

n Flow behavior index for fluids having pseudo-plastic behavior PGDTF Pressure gradient due to friction

Q Volumetric flowrate m3/s

Q* Dimensionless flowrate

r Radial position m

ReB Reynolds number for Bingham plastic

RePL Power-law Reynolds number

Re1 Reynolds number for Sisko fluid

u Fluid velocity m/s

V Average velocity m/s

V* dimensionless velocity

g Rate of strain s 1

gw Rate of strain at the wall s 1

m Viscosity Pa s

mB Coefficient of plastic viscosity for a Bingham plastic Pa s

meff Effective viscosity of a fluid defined as the ratio of local shear stress

to local rate of strain Pa s

m0 Lower limit of effective viscosity Pa s

1 Viscosity at high strain rates Pa s

 Dimensionless variable

r Fluid density kg/m3

rsl Density of non-Newtonian fluid kg/m3

Bibliography

A comprehensive account of non-Newtonian fluids is given by Bird and Wiest (1996) The application of non-Newtonian fluid models to a range of industrial

Non-Newtonian slurries 155

problems is discussed by Chabbra and Richardson (1999) Their text includes many interesting practical exercises that illustrate the application of the tech-niques that are discussed here They also discuss the flow of non-Newtonian fluids in channels of non-circular cross-section The measurement of rheo-logical properties is discussed by Whorlow (1992)

The discussion of Sisko fluids is based on Turian et al (1998a, b) who also present data on the friction loss when Sisko fluids flow through a variety of pipe fittings

Chabbra, R.P (1993) discusses the motion of bubbles, drops and particles in non-Newtonian fluids

References

Bhattacharya, I.N., Panda, D and Bandopadhyay, P (1998) Rheological behavior

of nickel laterite suspensions International Journal of Mineral Processing 53, 251±263

Bird, R.B and Wiest, J.M (1996) Non-Newtonian liquids, Chapter 3 in Handbook

of Fluid Dynamics and Fluid Machinery, Schetz, J.A and Fuchs, A.E (editors),

pp 223±302 John Wiley & Sons, New York

Chabbra, R.P (1993) Bubbles, Drops and Particles in Non-Newtonian Fluids CRC Press

Chabbra, R.P and Richardson, J.F (1999) Non-Newtonian Flow in the Process Industries Butterworth-Heinemann

Darby, R (1988) Laminar and turbulent pipe flows of non-Newtonian fluids Encyclo-pedia of Fluid Mechanics Vol.7 Chapter 2, Cheremisinoff, N.P (editor), Gulf Pub-lishing Company

Dodge, D.W and Metzner, A.B (1959) Turbulent flow of non-Newtonian systems A I

Ch E Journal 5, 189±204

Heywood, N.I and Richardson, J.F (1978) Rheological behavior of flocculated and dispersed aqueous Kaolin Suspensions in pipe flow Journal of Rheology 22, 599±613

Huynh, L., Jenkins, P and Ralston, J (2000) Modification of the rheological properties

of concentrated slurries by control of mineral-solution interfacial chemistry Inter-national Journal of Mineral Processing 59, 305±326

Kemblowski, A and Kolodziejski, J (1973) Flow resistances of non-Newtonian fluids in transitional and turbulent flow International Chemical Engineering 13, 265±279

Ma, T.-W (1987) Stability, rheology and flow in pipes, bends, fittings, valves and venturi meters of concentrated non-Newtonian suspensions PhD thesis, University

of Illinois at Chicago

Metzner, A.B and Reed, J.C (1955) Flow of Non-Newtonian fluids ± correlation of the laminar, transition and turbulent-flow regions A I Ch E Journal 1, 434±440 Thomas, D.G (1960) Heat and momentum transport characteristics of non-Newtonian aqueous thorium oxide suspensions A I Ch E Journal 6, 631±639

Turian, R.M., Ma, T.-W., Hsu, F.-L.G and Sung, D.-J (1998a) Flow of non-Newtonian Slurries: 1 Friction losses in laminar, turbulent and transition flow through straight pipes International Journal of Multiphase Flow 24, 243±269

156 Introduction to Practical Fluid Flow

Turian, R.M., Ma, T.-W., Hsu, F.-L.G., Sung, D.-J and Plackmann, G.W (1998b) Flow

of non-Newtonian Slurries: 1 Friction losses in bends, fittings, valves and venturi meters International Journal of Multiphase Flow 24, 225±242

Whorlow, R.W (1992) Rheological Techniques, 2nd edition, Ellis Horwood

Wilhelm, R.H., Wroughton, D M and Loeffel, W.F (1939) Flow of suspensions through pipes Industrial and Engineering Chemistry 31, 622±629

Non-Newtonian slurries 157

Sedimentation and thickening

The ratio of water to solids in a slurry usually has a significant impact on the techniques and economics of the transport operations Dilute slurries tend to behave in a fashion that is closer to that of Newtonian fluids while concen-trated slurries can exhibit strong non-Newtonian behavior which the conse-quent effect on the energy that is required to pump the material at the required rate Generally speaking there are usually advantages in reducing the amount of carrier fluid relative to the amount of solids to improve the energy efficiency as measured by the energy required to transport 1 kg of solids Thus dewatering of slurries must always be considered in practice The solids content of a slurry can be increased using sedimentation tech-niques The natural tendency of the solids to settle under the influence of gravity is exploited to remove some of the water from the slurry When the particles that make up the slurry are small, the settling is quite slow and special techniques are required to achieve a separation Because of the slow rates of settling that are commonly encountered, comparatively large equip-ment is required In this chapter the basic principles of batch and continuous settling are discussed and applied to the analysis of the operation of industrial thickeners

6.1 Thickening

Thickening is an important process for the partial dewatering of compara-tively dense slurries Essentially the slurry is allowed to settle under gravity but the particles are so close together that they hinder each other during settling and they tend to settle as a mass rather than individually The rate

of settling is a fundamental characteristic of the slurry which must be deter-mined experimentally for each slurry under appropriate conditions of floccu-lation in order to design and size an appropriate thickener The rate of settling depends on the nature of the particles that make up the slurry and on the degree of flocculation that is achieved For a particular flocculated slurry the rate of settling is determined chiefly by the local solid content of the slurry and will vary from point to point in the slurry as the local solid content varies

6.1.1 Batch thickening

The nature of the thickening process can be observed and the rate of settling determined as a function of the local solid content in a simple batch settling test In this test the position of the liquid-solid interface, called the mudline, is observed as a function of time and the settling rate is determined as a function

of solid content using the so-called Kynch construction which is described below

The batch settling test is illustrated in Figure 6.1 A sample of the floccu-lated slurry is gently and uniformly dispersed in the test cylinder and then allowed to settle undisturbed under gravity If the flocculation is good, the slurry soon develops a well defined interface with clear water above and all the solids below This interface falls as the solid particles below it settle under gravity The sharpness of the interface is maintained because all of the par-ticles at the interface are settling under hindered settling conditions and therefore settle at the same rate

During the test, the position of the interface is measured and plotted as a function of time as shown in Figure 6.1 This simple test provides most of the information that is required to analyze, simulate and design a continuously operating thickener After a long time the settling appears to stop and the interface falls very slowly while the sediment is compressing under its own weight Eventually the settling stops altogether because the sediment reaches its ultimate concentration at which it is essentially incompressible The concen-tration at which settling stops and the bed becomes incompressible is called the ultimate concentration and it is indicate as CUin Figure 6.1 The position of the interface does not change after point A has been reached The ultimate concentration corresponds to the situation when the solid particle that make

up the settling sediment are closely packed in an arrangement that would correspond to hexagonal close packed if the particles were spherical in shape The method used to extract the information from the simple batch settling curve is known as the Kynch construction The Kynch construction is applied

to the batch settling curve and this establishes the relationship between the rate of settling of a slurry and the local solid content The details of the Kynch

Settling slurry

Original slurry level Clear water

Settling mudline

h 1

h 0

Concentration at this point = C 1

h 0

h 1

h

Height

Batch settling curve

Linear segment of the settling curve corresponding

to initial concentration of the slurry.

Slope = ( v C

C

1

1

) = rate at which a layer of concentration is settling.

C U

A

Time

t 1

Slope = ∂X = rate of propagation of a plane

∂t of concentrationC

1

c = c 1

Figure 6.1 The batch settling test A uniform slurry is allowed to settle under gravity and the level of the mudline is plotted as a function of time

160 Introduction to Practical Fluid Flow