Flow in a pipe

Fluid flow in a pipe depends on the pressure applied, the radius of the pipe and

the viscosity of the fluid. For a Newtonian fluid, the flow is directly proportional to the

viscosity, which is a constant. For a non-Newtonian fluid having a viscosity that depends

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upon the shearing stress, like grouts and concretes, the flow rate is a complicated function

of the viscosity.

The viscosity () of a fluid is the ratio of the shear stress () to the shear rate (̇):

= /̇. This definition is convenient for Newtonian fluids, and certain non-Newtonian

fluids. In other cases, however, an engineering approach to the description of a fluid can

simplify the analysis. For instance if the fluid is approximated as a power law fluid, it can

be described by Eq. 1 where τ is the shear stress, K the power law consistency index, ̇

the shear rate, n the power law exponent:

n = Kγτ [ 1]
The corresponding velocity profile in a circular pipe is then given by equation 2 [4]:

1 1/

2

(3 1) ( ) 1 ( ) ( 1)

n

p p

Qn r v r

p Rn R

+ + = − +

[ 2]
where v is the fluid velocity as a function of the radial position, r , in the pipe, Q the

volumetric flow rate, and Rp the pipe radius. The fluid power law consistency index, K,

can be calculated using the following equation 3 [4], which requires a pressure drop

measurement over a certain length:

3 3 1/

2

n

n

p

P Q K R

L p

∆ − − =

[ 3]
where ∆P is the pressure drop, and L the distance between the pressure sensors. The

exponent n and the factor K could also be determined via equation 1 from rheological

measurements of the fluid through a rheometer if available. But equations 2 and 3 could

also be used to determine n and K from the pipe flow, in absence of a suitable rheometer.

The shear rate at the wall surface is calculated using the following equation [5, 6]:

3

3 1 ( ) p

p

n Q r R

n R γ

p

+ = = [ 4]
The local shear stress is

τ = ∆rP L / 2 [ 5]
The equations 1 through 5 describe flow of a homogenous fluid in a pipe.

However, concrete is more a complex fluid because it contains aggregates with a wide

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range of sizes. These aggregates interact with the pipe walls and each other, creating

inhomogeneities in the fluid. Thus, concrete flow in a pipe typically occurs in three layers

or regions [5, 6] as shown in Figure 1:

• Slip-layer or lubrication layer,

• The shearing region or layer, and

• The inner concrete or layer, also referred to as a plug flow layer

The thickness of the slip layer depends upon the tribology of the material adjacent

to the pipe material. Tribology is “the science and technology concerned with interacting

surfaces in relative motion, including friction, lubrication, wear, and erosion” [7]. The

thickness of, and the velocity profile within, the shearing layer depends upon the

viscosity and the yield stress. The thickness of the inner layer depends upon the yield

stress.

The composition and physical characteristics of each layer are difficult to know.

Their characterization requires the extraction of material from disparate regions. The

slip/lubrication layer contains mainly cement paste and possibly very small sand particles

[8], while the inner layer contains coarse aggregates. Also, the diameter of the inner layer

or the thickness of the slip-layer is unknown. It is conceivable that prediction of concrete

flow in a pipe will need the characterization of each of the layers.

Figure 1: Profile of flow of concrete in a pipe [6]
2.2. Slip-layer

Several research groups have investigated the slip-layer of concrete flow in a

pipe. Choi et al. [5, 6] measured the thickness of the slip-layer using an Ultrasonic

Velocity Profiler (UVP) in pumping circuits using industrial equipment and found that

there is a 2 mm thick layer along the inner surface of the pipe. However, the layer

thickness could vary depending on the mixture proportions and the pipe configuration.

Kaplan [9] reported that the flow of concrete in a pipe is mainly related to the

viscosity of the slip-layer and that its properties could be measured by tribometry. He

found that the correlation between the properties of the bulk material as measured in a

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rheometer and the properties of the slip-layer was weak. Jacobsen et al. [10] showed by

using colored concrete that the velocity profile of the concrete resembled that of plug

flow in the pipe center, and non-moving slip-layer, similar to that shown in Figure 1.

Kwon et al.[11, 12] measured the rheological properties of concrete before and

after pumping while monitoring the pressure and flow rate and found that while there was

no correlation between bulk concrete rheological properties, e.g., viscosity and yield

stress, and flow rates, there was a strong correlation between properties of the slip-layer

and flow rates. Thus they deduced that the slip-layer is the determining factor to

predicting that concrete will flow in a pipe. They then proceeded to develop a tribometer

that is a coaxial rheometer with a smooth bob made of steel or covered with rubber to

simulate the slip-layer of the pipe.

Ngo et al.[13] observed that the slip-layer is between 1 mm to 9 mm thick, by

visualizing the material flow in the rheometer. He analyzed the layer and found that it

contained sand with a particle size less than 0.25 mm. This would imply that there is a

migration of coarse aggregates from near the wall to the center of the pipe where the

shear rate is lower than that found near the walls.

2.3. Pumping pressure

Another factor in pumping is the pressure applied to the material to move it

through the pipe. Rio et al. [8] showed with a large number of pumping tests that the

relationship between the pressure of the pump and the flow rate of the material is linear:

P k kQ = +1 2 [ 6]
where 1 k and 2 k are two empirical parameters that depend on the material and other

experimental conditions. Rio et al. concluded that the two parameters can be used to

characterize a specific mixture. Rio et al. [8] advocated that the knowledge of these

parameters for a specific mixture and pumping circuit could be used as a quality control

tool to ensure that the applied pressure is sufficient to ensure the desired flow rate.

Feys et al. [14] established an empirical relationship between the plastic viscosity

of the concrete at a shear rate of 10 s-1 and the pressure gradient in a pipe. If the pressure

gradient is too low, the material will not move through the pipe. Feys mentioned two

issues relevant to the prediction of flow in a pipe: 1) the slip-layer influence is very

important, but it is not well understood and is difficult to measure; 2) the shear rates in

the pipe are spatially and temporally varying. One solution for the effect of the slip-layer

would be to measure its rheological properties, if it could be isolated and extracted.

Modeling of the flow in a pipe might help resolve the second issue. Feys et al. [14] also

observed that the pumping of self-consolidating concrete (SCC) requires a higher

pressure, while the yield stress is almost zero, but the plastic viscosity is higher than that

for normal concrete. This could be due to the slip-layer (Figure 1) that would require a

higher shear stress at the same shear rate due to the increased viscosity.

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