Drag Reduction in Turbulent Pipe

Flow Using Polymer Dilute Solutions

CHBE 401

Project Paper

by: Mohammed AlMakhaita

Instructors: Prof. Matteo Pasquali and Dr. Rolf Ryham

Due: December 04, 2009 by 5 PM

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Table of Contents

1. Introduction.....................................................................................................................................3

2. Understanding Drag Reduction.....................................................................................................3Reynold Number...........................................................................................................................3Fanning Factor.............................................................................................................................3Wall Shear Stress..........................................................................................................................3Slope Increment............................................................................................................................5

Maximum Drag Reduction...........................................................................................................6

3. Elastic Sublayer...............................................................................................................................8

4. Commercial Use..............................................................................................................................9

5. Solution to modern day problem.................................................................................................10

6. References......................................................................................................................................11

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Introduction:

B.A Toms discovered the polymeric turbulent flow drag reduction phenomenon in 1948,

upon finding that dilute polymethyl methacrylate solutions have the ability of reducing the

Fanning friction factor of turbulent flow to about the half compared to the pure solvent flow. The

experimental follow-ups of this discovery have shown that adding small concentrations of high-

molecular polymers can reduce the turbulent pipe flow’s drag by 80% (Virk 1975). up to this

date, it is still unclear to us how exactly those additives reduce drag. However, According to

Virk et al (1997), the extension of the macromolecules seem more than likely to be what causing

this phenomenon to occur.

Understanding Drag Reduction:

To grasp the concept behind DR phenomenon, once must understand some important

related parameters. One parameter is the Reynold Number (Re) which is used to measure the

flow intensity. Re is dimensionless and is defined as below

Re=V ´ IDν (I)

Where V is the mean velocity, ID the pipe inner diameter, and ν is the fluid kinematic viscosity.

Another

parameter is the Fanning friction factor, f, which is related to the wall shear stress and is

defined as below

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f 2w

V 2 (II)

Where ρ is fluid density and τw is wall shear stress, which is defined as

tw=DP ´ ID4 DL (III)

Where L is the distance across which the pressure drop is measured.

At laminar flow, (Re<2000), there is no difference in skin friction between dilute polymer

solutions and solvents. The solution follows poiseuille’s law for Newtonian fluids

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√ f=Re⋅f

16 (IV)

Virk (1975) describes three distinct regimes for fully developed turbulent flows in order

of increasing flow rate.

The first regime, called Newtonian regime, experiences no drag reduction, and its flow

follows Prandtl-Karman (P-K) law for Newtonian fluids in turbulent flow

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f4.0log10 Re f 0.4

(V)

The second regime, called the Polymeric regime, depending on the nature of the

polymer solution, it begins at a specific onset and is insensitive to concentration of polymer and

ID (Virk 1975). Relation between friction factor decreases beginning at onset as the flow starts

disobeying the P-K law as shown in (Figure 1)

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f(4.0 )log10 Re f 0.4 log10 2DW

(VI)

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Fig. 1

Where W* is wave number at onset:

W * (w

* /)1/ 2

(VII)

δ, Prandtl-Karman slope increment, denotes the difference in drag reduction between

dilute polymer solution and solvent slopes. δ is directly proportional to concentration and

molecular weight of the polymer and is independent of ID. δ is defined as below

d=kMW C1/2

M w0 (VIII)

Where Mw0 is the molecular weight of an additive monomer.

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Fanning friction factor decreases as flow rate increases in the polymeric regime and the

higher the δ the more drag reduction we get. From (equation VIII) we can see that δ varies as the

square root of concentration. According to (Virk 1975), the onset of drag reduction occurs at a

unique τw, (τw*). The higher the τw* the higher Re √ f the onset will exhibit. This Polymeric

regime will hold true until the flow reaches a point where drag reduction changes its slope and

that takes us to the third regime, called Maximum drag reduction regimeor simply, MDR. all

solutions reach a point where the slope alters to fit Virk’s asymptote governed by the equation

below

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f19.0log10 Re f 32.4

(IX)

The reason behind limited drag reduction will be explained later on page 6.

(Figure 2) is an illustration of flow of polymers of different Mw and C through different

ID. In this case, polyethyleneoxide (PEO) and polyacrylamide (PAM) solutions plotted on P-K

coordinates.

We can see from the plot that laminar flow follows Poiseuille’s law mentioned in (equation IV).

Also, (equation V) represents the Prandtl-Karman law for turbulent Newtonian flow and

(equation IX) representing Virk’s asymptote for MDR. Moreover, the straight lines follow the

Polymeric regime (equation VI) and it is noticed that MDR is a universal value independent of

ID, type of polymer, concentration, or molecular weight.

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Fig. 2

It is shown that drag reduction is bounded between two asymptotes, the Prandtl-Karman

law and the Maximum drag reduction Asymptotes, and based on Polymer molecular weight and

concentration, DR varies accordingly.

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Elastic Sublayer:

The mean velocity profile possesses three distinct zones. The Viscous sublayer zone

occupies the area near the wall. And the Newtonian plug region, at the axis where velocity

profile is shifted upwards from Newtonian law of the wall but remains parallel to it. In between

these two regions there is a region called, Elastic sublayer which is the main characteristic of

drag reduction.

(Figure 3) is a plot of the normalized mean velocity, U+, versus the natural logarithm of

distance from inside wall, y+. the ABCD profile has the three different regions: (1) viscous

sublayer AB where y+ ≤ yv+. B holds the value of yv

+ = 11.6 based on our previous arguments

about the onset of drag reduction. (2) elastic sublayer BC, where ye+ ≤ y+ ≤ R+ , which

originates at onset and grows depending on concentration with increasing drag reduction starting

at zero drag reduction at ( y+ = ye+ ). It is of a varying thickness perhaps that is why it is called

“elastic”. The elastic sublayer continues to grow until it claims the entire cross section of the

pipe at MDR. And since it cannot over expand, maximum drag reduction takes place. (3) the

Newtonian plug region where y+ ≤ R+.

The thickness of the elastic sublayer is related to a value called the effective slip, S+, by

the following equation

S+¿=( Am−An ) ln ¿ ¿

Where Am and An are coefficients of maximum drag reduction and Newtonian plug

region respectively.

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Fig. 3

Commercial Use:

Drag reducing agents were first introduced commercially in 1979 upon adding (DRA) to

the flow of crude oil in the Trans Alaskan Pipeline (Motier et al., 1996).

Trans Alaskan Pipeline System (TAPS) was designed with 12 pump stations along 1287

Km to transport 2 million barrels per day (MBPD) of Alaskan crude oil. However, with the use

of DRA’s, TAP was able to pump 2.1 MBPD of crude oil using only 10 pump stations. The two

remaining pump stations were never built. The TAPS used a DRA of 28 wppm polymer

concentration. Nowadays, globally, crude oil pipelines use DRA mechanics to enhance the flow.

According to Cuenca et al. (2008) the energy reduction estimate upon using 20 wppm polyolefin

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gel in a 307 mm ID and 84 Km pipe line in the Tarragona-Barcelona-Gerona pipeline is 42.6%.

Also, better ways of using drag reduction in marine environments and blood flow are sought.

Solutions to modern day problem:

The exponential increase in India’s population has led to find that the sewer systems are

inadequate to their large numbers. So, DRA’s could be used to enhance the flow as an alternative

to digging out the sewer system and replacing it with pipes of larger diameter. Unlike the latter

solution, using DRA’s is fast and cost effective. Plus, instead of replacing the pipes with larger

ones every time the population becomes too big for the municipal sewer system, DRA’s with

higher concentrations can be added until we reach a point where even maximum drag reduction

does not accommodate the insanely expanding population. But then, I think, inadequate sewer

systems will be the least of our problems. Brostow et al. (2007)

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References:

Brostow, Witold, Haley E. Hagg Lobland, Taruna Reddy, Ram P. Singh, and Leslie White.

“Lowering mechanical degradation of drag reducers in turbulent flow." Lowering

mechanical degradation of drag reducers in turbulent flow 22.1 (2007): 56-60. Materials

Research Society. J. Mater. Res., 3 Aug. 2006. Web. 30 Nov. 2009.

Gomez Cuenca, F., Gomez Marin, M., and Folgueras Diaz, M. B. Energy-Savings Modeling of

Oil Pipelines That Use Drag-Reducing Additives. Energy Fuels. 2008, 22 (5), 3293-3298.

Motier, J. F.; Chou, L-C.; Kommareddi, N. Commercial drag reduction: past, present and future.

International Symposium on Drag Reduction and Turbulence Modification, San Diego,

CA, ASME Fluids Eng. Div. Conf. Proc. 1996, 237 (2) 229-234.

Virk, P. S. Drag Reduction Fundamentals. AIChE J. 1975, 21 (4), 625-656 and references

therein.

Virk, P. S.; Sherman, D. C.; Wagger, D. L. Additive Equivalence During Turbulent Drag

Reduction. AIChE J. 1997, 43 (12), 3257-3259.

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