Intro to Practical Fluid Flow Episode 7 pot

Non-Newtonian slurries 5.1 Rheological properties of fluids When the concentration of solid in a slurry is greater than about 50 per cent by volume, the slurry changes its flow character

1 V and Ciare known: this situation arises when the nature of the slurry is known and the quantity that must be transported is fixed

From equation 4.69

Vwˆ V ‡X

i

and from equation 4.70

CiV ˆ Vwqi Tiqi

ˆ V qi‡ qiX

j

Equation 4.72 is true for each particle type and equations 4.71 and 4.72 give as many equations as there are unknowns These are solved most conveniently

by iteration and the following procedure has been found to be effective and rapidly convergent Add each of the equations 4.72

X

i CiV ˆ VXqi‡X

i

qiX

j

Tjqj XTiqi

ˆ Vq ‡ …q 1†X

i

From which

X

i

Tiqiˆ V…C q†

The volumetric fraction for species i is obtained from equation 4.70 as

qiˆ CiV

V ‡ P

i Tiqi Ti

ˆ CiV

1 ‡C qq 1  Ti

V

A convenient iterative solution starts with an estimate of q from which the qi's are calculated using equation 4.75 The assumed value of q is checked using

q ˆP qiwhich also provides a new refined starting value for the next iteration.

At convergence the velocity of the water can be recovered from equations 4.71 and 4.74

Vwˆ V 1 ‡C q

q 1

ˆ V1 C

and the frictional dissipation of energy can be calculated from Vw and the friction factor for the carrier fluid

Transportation of slurries 111

2 Vwand C are known.

When Vwand C are known, a similar iterative solution is possible

From equation 4.72

and substituting for V from equation 4.69

Ci Vw

X

j

Tjqj

0

@

1

A ˆ Vwqi Tiqi …4:78†

By a process similar to that used to develop equation 4.75 it is easy to show that

qiˆCi

C q

C 1

1 VTiw

…4:79†

This equation can be used in place of equation 4.75 to generate the iterative process

At convergence the average slurry velocity can be calculated from equation 4.77 and then the required pipe diameter

Use the FLUIDS toolbox to implement these solution methods

4.6 Practice problems

1 Calculate the flowrate of slurry through 200 m of smooth horizontal pipe, having internal diameter 13 cm, under a pressure drop of 5.4 bar

The friction factor for water can be calculated from fwˆ 0:079Re 0:25

The Durand±Condolios±Worster correlation can be used for the slurry Data:

Particle size 100 m:

Particle density 2670 kg=m3 Density of water 1000 kg=m3 Viscosity of water 0:001 kg=ms Slurry is 30 per cent by volume

1 bar ˆ 105Pa Use the FLUIDS toolbox to calculate the flowrate using the Turian±Yuan correlations

2 The following expressions have been found to apply for calculating the pressure drop when a wood-chip slurry is pumped through a horizontal pipeline Calculate the power required per meter of pipe length to pump

112 Introduction to Practical Fluid Flow

the slurry at a rate of 0.003 m3/s The diameter of the pipe should be chosen to minimize the pressure drop

Pf;sl
Pw

Pw ˆ 2:51C 4gD

V2

 1:42

fwˆ 0:079Re 0:25

The slurry contains 20 per cent wood by volume and has density 980 kg/m3

3 The New Zealand Steel iron sand slurry pipeline has the following design specifications:

Operating velocity ˆ 3:9 m=s Concentration ˆ 48:5 wt per cent solids Solid specific gravity ˆ 4:76

Mean particle size ˆ 120 m Pipeline inside diameter ˆ 187:4 mm The pipeline is constructed in two sections of 9.2 km and 8.8 km respect-ively

Calculate the design tonnage for solids delivered by the pipeline

Calculate the pressure gradient due to friction and the power consump-tion required to overcome fricconsump-tion when the pipeline operates at the design capacity

Use the following values for the properties of water Density of water ˆ 1000 kg/m3

Viscosity of water ˆ 0:001 kg/ms

3 4410 t/h of sand are to transported as a slurry in a 25.6 inch ID pipeline at

a concentration of 20 by volume

Calculate the required slurry velocity and the pressure gradient due to friction

Data:

Density of the sand ˆ 2700 kg=m3: Density of water ˆ 996 kg=m3: Viscosity of water ˆ 0:798 cP:

d50 for the sand ˆ 0:70 mm:

d85 for the sand ˆ 1:00 mm:

4 Your company wishes to design a hydraulic transport system for waste metal stampings in the form of small metal platelets Two shapes are made in the ratio 33 to 67 by mass These are identified as types A and

B respectively Terminal settling velocities for the two shapes were meas-ured and are 199 mm/s and 227 mm/s respectively A 47 per cent by volume slurry of the stampings is pumped upward through a vertical

Transportation of slurries 113

pipeline at an average velocity of 1.7 m/s The specific gravity of the platelets is 7.52 and the sphericity factor for both types is 0.53

Calculate the actual concentration of platelets in the vertical pipeline and calculate the pressure gradient due to friction

Pipe diameter is 2:5 cms Density of water ˆ 1000 kg=m3 Viscosity of water ˆ 0:001 kg=ms Choose a suitable pump for this application and specify the pump motor power required if the stampings must be lifted through 11 m What quantity of stampings can be transported?

5(a) Calculate the rate of energy dissipation due to friction when 800 m3/hr

of water is pumped through 100 m of smooth 20.3 cm ID smooth pipe Data:

Density of water ˆ 1000 kg=m3: Viscosity of water ˆ 0:001 kg=ms:

5(b) A pump having generalized pump characteristic constants

A ˆ 5:8, B ˆ 11:0, C ˆ 700 and impeller diameter 55 cm is used to pump 800 m3/hr of water What is the theoretical power required by the pump if it runs at 1200 rpm?

5(c) At what speed must the pump run to deliver 800 m3/hr into a piping system that is equivalent to 100 m of smooth 20.3 cm internal diameter pipe with a static head of 23 m?

5(d) What power does the pump draw when running under the conditions specified in section 5(c) above?

6 Repeat questions 5(a)±5(d) but the fluid is now a settling slurry con-taining 43 per cent by mass of spherical silica particles of size 0.5 mm

In part (a) the pipe may be considered to be horizontal over its whole length In part (c) the pipe consists of 77 m of horizontal and 23 m of vertical pipe The density of silica is 2700 kg/m3

7 All possible regime boundaries calculated for a slurry in a horizontal pipeline are shown in Figure 4.9 Mark all lines in the figure that represent real regime boundaries

8(a) The two vertical sections in laboratory pumping loop are 294 cm long The fluid under test flows upward in one leg and downward in the other Pressure gauges are installed at the top and bottom of each leg Calculate the difference in pressure registered by the pressure gauges across each leg when water flows in the loop at 2.19 m/s The internal diameter of the pipe is 2.60 cm

8(b) Repeat the calculation of section 8(a) when a limestone slurry flows through the loop The slurry contains 42 per cent by mass limestone which has a density of 2730 kg/m3 The particle size distribution is given in Table 4.1

114 Introduction to Practical Fluid Flow

4.7 Symbols used in this chapter

Ac Cross-sectional area of pipe m2

C Volume fraction of solids in slurry This is the concentration that is discharged from the pipeline Concentrations in individual segments

of the pipe may be different Volume fraction

C

D Drag coefficient at terminal settling velocity

Cr Solid concentration relative to Cvb

Crm Critical deposit concentration relative to Cvb

Csm Critical deposit concentration

Cvb Volume fraction of solids for loosely packed bed

D Pipe diameter m

F Energy dissipated by friction J/kg

Ffr Frictional force between settled bed and pipe wall N

FN Normal force exerted by settled bed on pipe wall N

Fr Froude number

fsl Friction factor for the slurry

fw Friction factor for the carrier fluid

g Acceleration due to gravity m/s2

i Slurry pressure gradient (hydraulic gradient) m water/m pipe

ipg Slurry pressure gradient when flowing as a plug at concentration Cvb

j Slurry pressure gradient m slurry/m of pipe

L Pipe length m

q Concentration of slurry in a vertical segment of the pipe Volume fraction

Q Volumetric flowrate of slurry m3/s

Rab Transition number between regimes a and b

s Specific gravity of solid

V Average velocity of slurry in pipe m/s

Vs Limiting velocity for stationary deposition in a horizontal pipe m/s

Vsm Maximum value of Vsm/s

Vw Average water velocity in vertical pipe segment m/s

T Terminal settling velocity m/s

Pf,w Pressure drop due to friction for carrier fluid alone Pa

 Excess pressure gradient relative to ipg

Table 4.1 Particle size distribution of

limestone

Mesh size (mm) passing (%)

Transportation of slurries 115

s Coefficient of friction between settled bed and pipe wall.

w Viscosity of carrier fluid Pa s

s Density of solid kg/m3

sl Density of slurry kg/m3

w Density of carrier fluid kg/m3

 Fractional increase in frictional pressure drop due to presence of particles

Constant in Durand±Condolios±Worster correlation

Bibliography

The use of the concept of excess pressure gradient as a useful correlating variable originated with Durand in the early 1950s The inclusion of the effect

of particle density is attributed to Worster (Smith 1955, Bain and Bonnington, 1970) The uncertainty in the results that are obtained from the original Durand±Condolios correlation has been frequently discussed in the literature, for example in Govier and Aziz (1972, Chapter 11)

The analysis using separate correlations for the four flow regimes is due to Turian and Yuan (1977)

The stratified flow model is based on the work of Wilson et al (1997) Their comprehensive text covers many aspects of slurry transport and is recom-mended to the student particularly for the many interesting and educational case studies

The treatment of flow of settling slurries in vertical pipes is based on Bain and Bonnington (1970)

References

Bain, A.G and Bonnington, S.T (1970) The Hydraulic Transport of Solids by Pipeline Pergamon Press

Govier, G.W and Aziz, K (1972) The Flow of Complex Mixtures in Pipes Van Nostrand Reinhold Co

Smith, R.A (1955) Experiments on the flow of sand-water slurries in horizontal pipes Trans Instn Chem Engnrs 33, 85±92

Turian, R.M and Yuan, T.-F (1977) Flow of Slurries in Pipelines AIChE Journal 23, 232±243

Wilson, K.C., Addie, G.R., Sellgren, A and Clift, R (1997) Slurry Transportation using Centrifugal Pumps 2nd edition Blackie Academic and Professional

116 Introduction to Practical Fluid Flow

Non-Newtonian slurries

5.1 Rheological properties of fluids

When the concentration of solid in a slurry is greater than about 50 per cent by volume, the slurry changes its flow characteristics and the Newtonian behav-ior of the carrier fluid no longer dominates the flow as it does in the case of the settling slurries that are discussed in Chapter 4 The internal momentum transfer processes must reflect the role that the densely packed solid particles play If the particles are in the colloidal or micron size range, the slurries deviate from Newtonian behavior and they require more complex rheological models to describe their flow behavior The relationship between the local rate

of deformation of the fluid and the shearing stress that is imposed differs from that which characterizes Newtonian fluids This relationship is used to clas-sify the flow behavior of various non-Newtonian fluids

A Newtonian fluid is characterized by a linear relationship between the local shearing stress and the rate of strain within the moving fluid

The proportionality constant is the viscosity of the fluid du/dr is the local spatial gradient of the velocity in the fluid Newtonian fluids have constant viscosity at all stresses and shear rates

Non-Newtonian fluids exhibit various types of non-linearity Four distinct types of non-Newtonian behavior are illustrated in Figure 5.1 which shows how the rate of strain du/dr varies with the shear stress that is applied to the fluid The relationship between shear stress and rate of strain for fluids can be measured in the laboratory and careful measurements can establish the rheological character-istic of any particular fluid Dense slurries made from fine particles often behave

as Bingham plastics at least approximately and this is a useful model for these fluids The Bingham plastic is an idealized model that can be used in theoretical calculations but no real fluids ever behave exactly as a Bingham plastic Figure 5.2 shows experimentally determined rheological behavior for two mineral slurries and it may reasonably be assumed that the Bingham plastic model is an adequate model to describe the flow behavior of these slurries

5.1.1 Bingham plastic fluids

Bingham plastics exhibit a linear relationship between shear stress and rate of strain but, unlike Newtonian fluids, this relationship does not exhibit a zero intercept at zero strain rate The intercept at zero strain rate is called the yield

stress because it is the minimum shear stress that must be applied to the fluid before it will deform at all

The rheological model for the Bingham plastic is

Bdu

drˆ  Y when   Y du

drˆ 0 when  < Y

…5:2†

Bingham plastic

Pseudoplastic

Dilatant

Newtonian

d

dr Rate of strain in the fluid

Figure 5.1 Typical stress-rate-of-strain relationship for non-Newtonian fluids

Rate of strain s–1 0

10 20 30 40 50 60 70 80

Chalcopyrite slurry 69% by weight Chalcopyrite slurry 65% by weight Chalcopyrite slurry 60 % by weight Chalcopyrite slurry 55% by weight Chalcopyrite slurry 50% by weight Nickel laterite slurry 40% by weight Nickel laterite slurry 30% by weight

Figure 5.2 Viscometer data for two mineral slurries showing approximate Bingham plastic behavior Data from Huyhn et al (2000) (closed symbols) and Bhattacharya et al (1998) (open symbols)

118 Introduction to Practical Fluid Flow

Bis called the coefficient of rigidity or coefficient of plastic viscosity Yis the yield stress This model is based on the idea that the fluid will not deform, and therefore (du/dr) ˆ 0, unless the shear stress acting on the fluid exceeds a definite critical yield stress Y The parameters B and Y that describe the behavior of these slurries vary with the solids concentration, the particle size, particle shape, and specific surface area and the chemical conditions of the particle surfaces The concentration of any potential-determining ions often has a significant effect on these parameters For example H‡and OH ions are potential determining for many minerals and significant variations in the values of B and Ywith pH have often been reported

This type of behavior is explained physically by the fluid having a 3-D structure associated with the densely packed solids that resists deformation unless the stress is high enough to breakdown this structure Once the structure has been loosened the fluid exhibits Newtonian behavior with a linear relationship between the shear stress and the rate of strain In the absence of any hysteresis effects, the structure is reconstituted as soon as the stress is decreased below Y It is not likely that any real fluid will exhibit Bingham plastic behavior exactly but the model is a useful approximation for many real dense slurries Some typical data measured in laboratory visc-ometers is shown in Figure 5.2

5.1.2 Pseudo plastic fluids

These fluids are characterized by two distinct effective viscosities at the extremes

of low stress and high stress with a smooth transition in between This implies that any structural constraints in the fluid are smoothly overcome as the stress increases and as particles and molecules are aligned more or less in the direction

of flow The effective viscosity decreases as the shear stress increases

The effective viscosity is defined to be the ratio of the shear stress to the rate

of strain

effˆ du dr

Three simple models for pseudo plastic behavior that use the concept of effective viscosity have been used to describe the rheological behavior of some non-Newtonian fluids:

1 Meter Model:

effˆ 1‡ 0 1

1 ‡ 

m

The constants in this model have the following significance

0ˆ viscosity at low shear rate,

1 ˆ viscosity at high shear rate,

Non-Newtonian slurries 119

mˆ the shear stress at which the effective viscosity lies midway between

0 and 1, ˆ a parameter that describes how quickly the effective viscosity varies from its lower shear-rate limit to its high shear-rate value

2 Seely Model:

This model postulates an exponential decrease in the effective viscosity as the shear stress increases

3 Ostwald-de Waele or Power Law Model:

or

 ˆ K dudr

…5:7†

Since du/dr is negative in conduits of circular cross-section, the term inside the brackets is positive which is essential since n is usually not an even integer The parameter K in equation 5.7 is called the fluid consistency coeffi-cient and n the flow behavior index

These models are all based on the idea that the effective viscosity varies

as the local rate of strain in the fluid increases If the effective viscosity decreases as the rate of strain increases (0 < 1 or n < 1) the fluid is shear thinning and behaves as a pseudo plastic If the effective viscosity increases with increasing strain rate (0> 1 or n > 1) the fluid is shear thickening

or dilatant

Some fluids exhibit rheological behavior that appears to include charac-teristics of Newtonian, yield stress and power law behavior to varying degrees as the local rate of strain varies Rather more complex models are required to describe the behavior of such fluids and models which are constructed as combinations of the simpler models have been found to be useful

5.1.3 Pseudo plastic fluids with yield stress

Fluids which exhibit a yield stress but also exhibit a non-linear relationship between shear stress and local rate of strain can be modeled using the equation

dr

ˆ  H when   H

ˆ 0 when  < H

…5:8†

This is commonly referred to as the Herschel±Bulkley model

120 Introduction to Practical Fluid Flow

…5:8†

This is commonly referred to as the Herschel±Bulkley model

120 Introduction to Practical Fluid Flow

or dilatant

Some fluids exhibit rheological behavior that appears to include charac-teristics of Newtonian, yield stress and power law behavior to varying... required to describe the behavior of such fluids and models which are constructed as combinations of the simpler models have been found to be useful

5.1.3 Pseudo plastic fluids with