Turbulent flow
4.7 WORKED EXAMPLES WORKED EXAMPLE 4.1
A slurry exhibiting power-law flow behaviour is flowing through 15 m of pipe having an inside diameter of 5 cm at an average velocity of 0.07 m/s. The density of the slurry is 1050 kg/m3and its flow index and consistency index aren ¼ 0:4 and k ¼ 13:4 N s0:4=m2, respectively. Calculate the pressure drop in the pipeline.
Solution
First we check to see if the flow is laminar or turbulent by calculating the generalized Reynolds numberRefor the specified flow conditions. We then compare this Reynolds number to the generalized Reynolds number at which the transition from laminar to turbulent flow occurs.
From Equation (4.20)
Retransition¼ 6464 0:4
½1 þ 3ð0:4Þ0:4ð2 þ 0:4Þð Þ2þ0:41þ0:4 1 1þ 3ð0:4Þ
20:4
Retransition¼ 2400
The generalizedRefor the flow conditions in the pipe is found from Equation (4.19),
The pressure drop can then be calculated using Equation (4.8).
P
Using the friction factor method, ff¼ 16
Therefore, the pipeline pressure drop is 48 kPa.
WORKED EXAMPLE 4.2
A slurry with a density of 2000 kg=m3, a yield stress of 0:5 N=m2, and a plastic viscosity of 0.3 Pa s is flowing in a 1.0 cm diameter pipe which is 5 m long. A pressure driving force of 4 kPa is being used. Calculate the flow rate of the slurry. Is the flow laminar or turbulent?
Solution
We will first assume that the flow is laminar and then go back and check this assumption. For laminar flow of a Bingham plastic fluid,
vAV¼Rt0
where the wall shear stress t0
t0¼PR
Then,
Now check to see if the flow is laminar by computingRe and He.
Re ¼rmDvAV
Given thisHe and Re, the flow is laminar.
WORKED EXAMPLE 4.3
The following rheology test results were obtained for a mineral slurry containing 60%
solids by weight. Which rheological model describes this slurry, and what are the appropriate rheological properties for this slurry?
Shear rateðs1Þ Shear stress (Pa)
0 5.80
Solution
The shear stress at zero shear rate is 6.00 Pa. Hence there is a yield stress equal to 6.00 Pa.
In order to determine whether the slurry behaves as a Bingham fluid or if it follows the Herschel–Bulkley model, we need to plot t tyversus shear rate.
This plot yields a straight line with slope equal to 0.05 Pa s which is the value of the plastic viscosity mp.
WORKED EXAMPLE 4.4
A coal–water slurry with 65% volume fraction coal (coal specific gravity¼ 2.5) is pumped at a rate of 3.41 m3/h from a storage tank through a 50 m long, 1.58 cm inside diameter horizontal pipe to a boiler. The storage tank is at 1 atm pressure and the slurry must be fed to the boiler at a gauge pressure of 1.38 bar. If this slurry behaves as a Bingham plastic fluid with a yield stress of 80 Pa and a plastic viscosity of 0.2 Pa s, what is the required pumping power?
Solution
First check to see if flow is laminar or turbulent by computingRe and He. Compute mixture density rm
rm¼ Cwrsþ ð1 CwÞrf
rm¼ ð0:65Þð2500 kg=m3Þ þ ð0:35Þð1000 kg=m3Þ rm¼ 1975 kg=m3
Compute the velocityvAV given the volumetric flow rateQ.
Q ¼ vavpD2
4 ; so vAV¼ 3:41
3600 1
p4ð0:0158Þ2¼ 4:83 m=s
Calculate the Reynolds number Re ¼rmDvAV
mp ¼ð1975 kg=m3Þð0:0158 mÞð4:83 m=sÞ
ð0:2 kg=m=sÞ ¼ 754
t ty(Pa) Shear rateðs1Þ
0.06 1
0.52 10
0.76 15
1.29 25
2.00 40
2.24 45
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Calculate the Hedstrom numberHe
He ¼rmD2ty
m2p ¼1975 0:01582 80
0:22 ¼ 986
GivenRe ¼ 754 and He ¼ 986, the flow is laminar.
Applying Equation (4.13) to the process between points ‘1’ and ‘2’ (Figure 4W4.1)
Hgained by pump hf¼p2 p1
rmg þv2AV
2g
The head loss due to friction in the pipeline alone is related to the pressure drop between points ‘3’ and ‘2’
hf¼p3 p2
Head gained by pump is related to the power required _W Hgained by pump¼ W_
rmgQ¼ 398 m
Therefore; _W ¼ 1975 9:81 ð3:41 3600Þ 398 ¼ 7300 J=s
Therefore, pumping power required is 7.3 kW.
‘1’
TEST YOURSELF
4.1 What are the chief distinguishing characteristics of homogeneous flow and hetero-geneous flow in slurries?
4.2 What is meant by the term ‘critical deposition velocity’ in reference to a setting slurry?
4.3 Name three models that might describe the rheological behaviour of non-setting supensions at high concentrations.
4.4 What is a chief characteristic of a thixotropic fluid?
4.5 Sketch a plot of shear stress versus strain rate for (a) a dilatant fluid and (b) a pseudoplastic fluid.
4.6 Sketch fluid radial velocity profiles within a pipeline carrying a shear-thinning fluid.
4.7 Outline the steps in the procedure for predicting pipeline pressure drop for slurries exhibiting power-law rheology.
4.8 How might one distinguish between a slurry behaving as a Bingham plastic fluid and a Herschel–Bulkley fluid?
4.9 Define the Hedstrom number. How is this number used in prediction of pipeline pressure drop for slurries exhibiting Bingham plastic rheology?
4.10 What steps might be typically involved in preparation of a slurry for transport by pipeline?
4.11 What type of pump would be used in pipeline transport for an abrasive, non-settling slurry requiring pressures up to 60 bar?
4.12 How is erosive wear in slurry pipelines combated?
EXERCISES
4.1 Samples of a phosphate slurry mixture are analysed in a lab. The following data describe the relationship between the shear stress and the shear rate:
Shear rate, _gðs1Þ Shear stress, tðPaÞ
25 38
75 45
125 48
175 51
225 53
325 55.5
425 58
525 60
625 62
725 63.2
825 64.3
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The slurry mixture is non-Newtonian. If it is considered a power-law slurry, what is the relationship of the viscosity to the shear rate?
(Answer: t ¼ 23:4_g0:15.) 4.2 Verify Equation (4.7).
4.3 Verify Equation (4.28).
4.4 A slurry behaving as a pseudoplastic fluid is flowing through a smooth round tube having an inside diameter of 5 cm at an average velocity of 8.5 m/s. The density of the slurry is 900 kg/m3 and its flow index and consistency index are n ¼ 0:3 and k ¼ 3:0 N s0:3=m2. Calculate the pressure drop for (a) 50 m length of horizontal pipe and (b) 50 m length of vertical pipe with the flow moving against gravity.
(Answer: (a) 338 kPa; (b) 779 kPa.)
4.5 The concentration of a water-based slurry sample is to be found by drying the slurry in an oven. Determine the slurry weight concentration given the following data:
Weight of container plus dry solids 0.31 kg Weight of container plus slurry 0.48 kg
Weight of container 0.12 kg
Determine the density of the slurry if the solid specific gravity is 3.0.
(Answer: 1546 kg/m3.)
4.6 A coal-water slurry has a specific gravity of 1.3. If the specific gravity of coal is 1.65, what is the weight percent of coal in the slurry? What is the volume percent coal?
(Answer: 58.6%, 46%.)
4.7 The following rheology test results were obtained for a mineral slurry containing 60%
solids by weight. Which rheological model describes the slurry and what are the appropriate rheological properties for this slurry?
(Answer: Herschel–Bulkley model:k ¼ 0:20, n ¼ 0:81.)
Shear rateðs1Þ Shear stress (Pa)
0 4.0
0.1 4.03
1 4.2
10 5.3
15 5.8
25 6.7
40 7.8
45 8.2
4.8 A mud slurry is drained from a tank through a 15.24 m long horizontal plastic hose.
The hose has an elliptical cross-section, with a major axis of 101.6 mm and a minor axis of 50.8 mm. The open end of the hose is 3.05 m below the level in the tank. The mud is a Bingham plastic with a yield stress of 10 Pa, a plastic viscosity of 50 cp, and a density of 1400 kg/m3.
(a) At what velocity will water drain from the hose?
(b) At what velocity will the mud drain from the hose?
(Answer: (a) 3.65 m/s; (b) 3.20 m/s.)
4.9 A coal slurry is found to behave as a power-law fluid with a flow index 0.3, a specific gravity 1.5, and an apparent viscosity of 0.07 Pa s at a shear rate 100 s1.
(a) What volumetric flow rate of this fluid would be required to reach turbulent flow in a 12.7 mm inside diameter smooth pipe which is 4.57 m long?
(b) What is the pressure drop (in Pa) in the pipe under these conditions?
(Answer: (a) 0.72 m3/h; (b) 19.6 kPa.]
4.10 A mud slurry is draining from the bottom of a large tank through a 1 m long vertical pipe with a 1 cm inside diameter. The open end of the pipe is 4 m below the level in the tank. The mud behaves as a Bingham plastic with a yield stress of 10 N/m2, an apparent viscosity of 0.04 kg/m/s, and a density of 1500 kg/m3. At what velocity will the mud slurry drain from the hose?
(Answer: 3.5 m/s.)
4.11 A mud slurry is draining in laminar flow from the bottom of a large tank through a 5 m long horizontal pipe with a 1 cm inside diameter. The open end of the pipe is 5 m below the level in the tank. The mud is a Bingham plastic with a yield stress of 15 N/m2, an apparent viscosity of 0.06 kg/m/s, and a density of 2000 kg/m3. At what velocity will the mud slurry drain from the hose?
(Answer: 0.6 m/s.)
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