8.9 Head loss pumping suspensions

Head loss pumping suspensions

Head loss in pipe pumping suspensions or solid particles in suspension in water, behave as non-Newtonian liquids. There are many parameters which effect the flow characteristics of suspensions. Not all of these are known at present. All pumping of solids, i.e., transportation of solid materials in water, is characterized by a certain degree of uncertainty. It is normally recommended that test pumping is carried out in test loops before the main installation is finally developed or commissioned.

Some of the main aspects of the behavior of suspensions are set out below, along with guideline values for the calculation of the more important parameters for certain suspensions.

Head loss when pumping homogeneous suspensions

When the solid particles distribute themselves evenly throughout the cross-section of the pipe, without any concentration gradients, we speak of a homogeneous suspension. In practice, homogeneous suspensions are considered to occur when the particle size is less than about 50μm (0.050 mm).

Homogeneous suspensions with sufficiently high particle content occur as time-independent, plastic, non-Newtonian liquids. These kinds of liquid to some degree follow Binghams law, see also Properties of liquids >>>

τ = τ_o  = tan γ Δv/Δy

Figure 8.9a Homogeneous suspensions with flow characteristics as per Bingham.

The pressure drop in laminar flow in straight piping measured in metres head of liquid is

 

 

 

h_L = head loss    (m)
ρ = density of the suspension (kg/m³)
g = acceleration due to gravity (m/s²)
l = length of pipe (m)
d = diameter of pipe (m)
τ_o = viscous shear stress at boundary (N/m²)
tan γ = plastic dynamic viscosity (Ns/m²)
v = velocity of flow (m/s)

The viscous shear stress at boundary τ_o and plastic viscosity tan can be determined with the aid of a Viscometer or by measuring the pressure drop in a test loop. The loss of head for turbulent flow in straight pipe measured in metres column of liquid (metres suspension) is

 

 

 

i.e. exactly the same formula as for Newtonian liquids.

Expressed in metres liquid column, the head loss pumping suspensions is the same as for pure water in turbulent flow,

Figure 8.9b Examples of loss of head in pipe flow for a homogeneous suspension (ρp = 2600 kg/m³, d=50 mm)

The transition from laminar to turbulent flow occurs, as for water, at R_e ≈ 2300 but with the difference that the apparent (equivalent) viscosity should now be used for Reynolds number. The velocity of flow at transition v_cr for R_e ≈ 2300 is

 

 

 

where

v_cr = critical velocity of flow at transition (m/s)

d = diameter of pipe (m)
τ_o = viscous shear stress at boundary (N/m²)
tan γ = plastic dynamic viscosity (Ns/m²)

Sedimentation in the pipe is avoided completely if the flow velocity in the pipe is greater than v_cr. Thus v_cr is of extraordinary importance when dimensioning pump installations for homogeneous suspensions in long pipes. In short or carefully monitored conduits, V < v_cr can usually be accepted.

There is an optimum particle content for the transportation of solid material by hydraulic methods. If the particle content is too low, the suspension loses its plastic characteristic and with it the possibility of making use of the advantageous “plug” flow. Besides which an unnecessarily large amount of water is pumped per kilogram of solid material. If the particle content is too high the transportation must be carried out at increased velocity due to increased transitional velocity (v > v_cr) so as to avoid sedimentation.

Figure 8.9c Optimum particle content for homogeneous suspensions for the transportation of a given quantity of solid material per unit time.

If, for example, the particle content in the suspension in figure 8.9b were to increase from 60 % to 63%, the velocity of flow would have to increase from about 1.7 to 2,2 m/s. Thereby increasing the transportation capacity by a factor of 1.4, the loss of head by about 1.7 and the pump power consumption, with pipe losses only, by a factor of about 2.3. Where the transportation capacity does not change, by choosing a smaller pipe diameter, the power requirement of the pump increases 1.7 times. The particle content is thus a critical parameter.

 

Head loss when pumping heterogeneous suspensions

Heterogeneous suspensions, that is to say suspensions with concentration gradients across the pipe section, are considered in practice to occur for particle sizes of more than about 50 μm (0.05 mm).

The main difference compared with homogeneous suspensions is that the liquid and the particles in a heterogeneous suspension retain their identity. The flow is a two phase flow. The viscosity of a heterogeneous suspension,  in principle, is the same as that for the pure liquid (the water).

The solid particles are conveyed suspended in the liquid if the particles are small and the velocity high, or by jumps if the particles are large or the speed low.

Table 8.9a Normal means of transportation for various particle sizes.

For all practical purposes, particle sizes in the order of 0.05 to 0.2 mm are transported suspended in the liquid. Suspensions with particle sizes down towards the lower limit (0.05 to 0.1 mm) comprise the boundary with the homogeneous suspensions. The surest way of minimizing the risk of sedimentation and blockage of the pipe is to maintain a turbulent pipe flow. The pressure losses, expressed as loss of head in metres suspension, can then be calculated in the same way as for water.

Suspensions of particles with sizes up towards the upper limit (0.1 to 0.2 mm) are on the boundary with the next group where the combined means of transportation prevails. The particle region 0.05 to 0.2 mm is not so well understood as other size ranges.

Suspensions with particles >0.2 mm have been the subject of comprehensive loss of head measurements (Durand, Condolios among others). Figure 8.9d shows an example of a loss diagram for a horizontal pipe.

Figure 8.9d Example of loss of head in heterogeneous suspension, sand ρ_p = 2650 kg/m³, d_pmean≅ 0.4 mm, d = 200 mm

The experimental results shown cover the region.

particle size, d_op = 0.2-25 mm
pipe diameter, d = 40-600 mm
particle density. ρ_p =1500-5000 kg/m³
volumetric concentration K_vol = 2 to 30 %
Liquid = water

The loss of head expressed in metres column of water (mH₂0) is greater than for pure water and exhibits a minimum at the optimum flow velocity v_opt. The loss of head for suspensions of d_p > 0.2 mm flowing in horizontal pipes can be assessed using the formula:

where

h_L = loss of head    (mH₂0)
λ = loss coefficient for water
v = velocity of flow (m/s)
g = acceleration due to gravity (m/s²)
l = length of pipe (m)
d = diameter of pipe (m)
k_f = supplementary loss factor over and above that of pure water

The supplementary losses above those for pure water can be assessed from

where

k_vol = proportion of solid material by volume

k_p = factor for particle characteristics

The particle factor is assessed from

where

V_S= sedimentation velocity for a single particle in still water  (m/s)

d_p= particle diameter (m)

Some text books use the dimensionless “particle resistance coefficient” C_D instead of the particle factor k_p. The relationship between them is k_p = 1/√C_D

The optimum flow velocity v_opt, i.e. the flow velocity for minimum loss of head at any given volumetric concentration, can be assessed by seeking a minimum for Equation 8.9d after setting in Equation 8.9e. Then,

 

For particles > 0.2 mm and at density ρ_p =2650 kg/m³ the text books give

Another velocity of interest is the ” limiting” velocity, below which the particles are in continuous contact with the bottom of the pipe. The limiting velocity can be written

 

where the factor F_lim depends upon the volumetric concentration and size of particles and is determined experimentally.

Figure 8.9e Factor F_lim for the determination of the limiting velocity V_lim

Suspensions with particle contents of different sizes are dealt with as follows:

• Small particles which form a suspension with the pure liquid (water) at the velocity used are considered to be a part of the “carrier” liquid. The water and the small particles (d_p less than perhaps 0.1 mm, certainly < 0.05mm) constitute the carrier liquid with density values and viscosity differing from that of water.
• The larger particles together with the carrier liquid form a heterogeneous suspension. Where there is a variation of size within the group of larger particles, a weighted mean value is used to determine the particle factor k_p

Small particles increase the density of the carrier liquid and reduce the ratio of densities of transported particles and carrier liquid. Because of this both the supplementary losses and the limiting velocity for loss minimum and sedimentation are reduced. Additions of small particles can thus be economically attractive in the transportation of larger particles.

The sedimentation velocity V_S used to calculate the particle factor k_p should be quantified experimentally. The effect of the density of the liquid and its viscosity, together with the geometrical shape of the particle all effect the particle factor. The sedimentation velocity for sand in water is illustrated in figure 8.9f.

Figure 8.9f Measured sedimentation velocity in accordance with Richards for quartz sand (ρ_p =2650 kg/m) in water (10°C, v = 1,3 *10^-6 m²/s) compared with various theoretical expressions.

It is extremely difficult to give general guideline values for the most suitable flow velocity for the hydraulic transportation of solid materials. Ash, however, recommends the following values:

These values are based on pipe diameters between 100 and 200 mm. Note the effect of the pipe diameter on the limiting velocity in Equation 8.9h.