Document [original]

Powder Technology 434 (2024) 119382

Available online 9 January 2024

Review

Analysis of pressure fluctuations in pneumatic conveying systems via

pressure sensors – A review

M. Dikty

Schwedes +Schulze Schüttguttechnik GmbH, Apensen, Germany

HIGHLIGHTS GRAPHICAL ABSTRACT

•Analysis methods for pulsations of hor-

izontal conveying lines.

•Flow mode influence on pressure

pulsations.

•Reasons for emergence of pressure

pulsations.

•Influence of turbulence on pressure

pulsations.

ARTICLE INFO

Keywords:

Review

Pneumatic conveying

Pulsation

Fluctuation

Turbulence

ABSTRACT

Bulk solids can usually be transported mechanically, hydraulically or pneumatically. If the choice is made for

pneumatic transport, the necessary quantity of conveying gas and the necessary conveying pressure for the

dimensioning of the pressure generator are decisive for the dimensioning of the system. The lower the pressure

reserves selected, the more sensitively the pneumatic conveying system also reacts to pressure fluctuations. These

can become so strong that the pressure generator fails due to overload. So, what are the reasons for pulsations

and how can they be evaluated? Influences such as the solids loading ratio (SLR), the conveying gas velocity or

the flow pattern were evaluated with a wide variety of criteria. With the aid of the pressure signal and its

evaluation method, conclusions can be drawn about for example the flow condition or the degree of turbulence.

This article shows the results of these research activities so far.

1. Introduction

The sand dune formation in the desert or the tornado that covers the

house roof are natural occurrences of pneumatic transport. Technically,

however, it is used in many applications to transport bulk solids by

means of a conveying gas, usually air, through a pipeline. This takes

place either in suction or pressure mode. Fig. 1 shows the schematic

structure of both systems. Both systems are characterized by a negative

pressure gradient in the direction of flow. The basic operations of a

suction or pressure system are identical:

* Corresponding author.

E-mail address: [email protected].

Contents lists available at ScienceDirect

Powder Technology

journal homepage: www.journals.elsevier.com/powder-technology

https://doi.org/10.1016/j.powtec.2024.119382

Received 2 October 2023; Received in revised form 27 December 2023; Accepted 2 January 2024

Powder Technology 434 (2024) 119382

•Feeding of the bulk material into the conveying line.

•Transport through the conveying line due to a negative pressure

difference from the material inlet to the outlet.

•Separation of the bulk material from the conveying gas at the

receiving point (exceptions are, for example, direct reactor or burner

feeding in pressure conveying, such as coal firing in the power plant

or cement industry).

•Pressure generation (overpressure with pressure conveyance, nega-

tive pressure with suction conveyance).

The bulk material is fed into the conveying line in over-pressure

operation by means of a rotary valve [2–6], pressure vessel [2,5,6],

screw pump [2,5–8], injector (jet-feeder) [2,5,6,9], flap systems [2,5,6]

or, in the case of exclusively vertical transport, by means of an airlift

[2,5,6]. A suction system can operate without a feeding device. In case

feeding devices are used, then rotary valves of flap systems are installed.

At the receiving point, the bulk material is separated from the transport

gas for further processing. This is done using cyclones and filters. In case

of feeding a burner or a reactor, for example, there is no separation of the

two phases.

Whether a pneumatic conveying system is used or a mechanical one

is preferred depends on many criteria [10]. The decision in favour of

pneumatic conveying is usually made when the conveying route is either

very long or the route between the starting point and receiving point is

characterized by several bends and differences in height. Further criteria

in favour (+) and against (−) of pneumatic conveying are [2]:

•high adaptability of the conveying line to local conditions (+),

•environmental-friendly design (no dust emission) (+),

•feeding several receiving points with one system by conveying line

diverters (+),

•low maintenance for the conveying line (+),

•use of inert gas for air-sensitive solids (+),

•conveyance of toxic and hazardous bulk materials (+),

•broad applicability for a wide variety of solids (+),

•direct transport into systems that are under overpressure (+),

•low CAPEX (capital expenditure) (+),

•less space required (+),

•carrying out chemical or physical processes during conveying (+),

•comparatively high-power requirement (−),

•wear of pipelines and feeding devices (−),

•can be used economically up to a grain size of approx. 10 mm (−),

•product degradation (−),

•dust explosion hazard (−),

•high OPEX (operational expenditure) (−),

•noise emissions from the pressure generators and, in the case of

coarse and hard particles from the pipeline (−).

2. Flow modes

Pressure fluctuations are often linked to the flow mode as referred to

[11–17], which exists in the pneumatic conveying line and characterizes

how the bulk material flows through the conveying pipe. The flow mode

Fig. 1. Pressure system (left) / suction system (right) based on [1] with A storage silo, B filter, C feeding device, D receiving silo, E pressure generator, F vacuum

generator, G conveying line diverter.

Fig. 2. Definition of flow modes in relation to [5,81].

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Powder Technology 434 (2024) 119382

is not always constant in the whole system. While for example the bulk

material still flows through the pipe in the form of dunes (see definition

below) at the beginning of the conveying line, for example, it can leave

the conveying pipe at the end of the conveying line as a dilute flow (see

definition below), depending on the system configuration. The in this

review used definition of the flow modes in horizontal conveying are

chosen in relation to Weber [81] and Klinzing et al. [5] and graphically

depicted in Fig. 2:

Mode A – Dilute flow: No strand or deposit exists. This mode is also called

suspension flow, lean phase flow or homogeneous flow.

Mode B – Blowing

dunes/clusters:

On the pipe bottom slides clusters / small dunes form out.

Between the clusters particles of higher velocity flow.

Mode C – Strand flow: A strand (nearly even surface), moving or not exists on the

ground of the pipe.

Mode D: - Stratified

flow:

A strand is flowing over a lower strand. The lower strand

can move or not.

Mode E - Dune flow: A dune (uneven surface, geometry in leaning to a sand

dune) is flowing through the pipe. A lower strand (flowing

or not) can exist but must not exist.

Mode F – Plug flow: Plugs are flowing with low velocity through the pipe. The

geometry is like a sand dune which fills the complete

vertical pipe cross section. The plugs pick up bulk material

at the front and lose bulk material at the end.

The naming and definition of the flow mode varies from author to

author. Further flow mode overviews are shown e.g., in

[2,35,48,49,82–85]. Depending on bulk material, there are other modes

that can occur between mode A and F. Which flow mode occurs depends

on the solid loading ratio (SLR), the velocity and the physical bulk solids

data, like grain size distribution, particle density or sphericity.

In suspension flow (dilute flow), high flow velocities prevail, fine

bulk materials (d

s,50

<100

m) [6,16,68,86–89] behave here like coarse

bulk materials / granules. They flow almost evenly distributed over the

pipe cross-section. Irrespective of the air turbulence, the coarse particles

perform a transverse movement in addition to the longitudinal move-

ment, which is caused by the particles impacting each other and the

wall. The transverse movement is essentially determined by the hard-

ness and shape of the particles. In the case of fine-grained particles, on

the other hand, the transverse movement occurs also due to the turbu-

lence of the flow [12]. With fine-grained particles, turbulence damping

of the carrier gas is observed, with coarse-grained ones an increase [14].

The particles are slowed down by the impact on walls as they bounce

back and must be accelerated again. This creates a pressure loss. In the

case of wall impacts, fine particles are also decelerated aerodynamically

in the boundary layer. This is associated with the tendency to form de-

posits on the wall [15]. Particularly, clusters (Mode B) also flow through

the pipeline as the flow approaches the saltation velocity.

When the transport gas velocity is reduced to/under the saltation

velocity, the two-phase flow begins to separate, which is known as

strand flow. Part of the solid slides as a strand on the pipe bottom, while

the other part is transported flying over the strand. The strand is

essentially driven by impacting particles. Compared to coarse-grained

material (d

s,50

>100

m), fine-grained material tends to form strands

even at higher air velocities [12]. If the air velocity is further reduced,

individual strands can push together to form dunes or plugs. Dune’s flow

through the conveying line like waves, as shearing forces act on the

strand surface flowing on the ground.

When there is a small SLR, the transition from dune to plug

conveying takes place as conveying happens over a dune situated on the

ground. Dune conveyance is in the case of coarse particles and plug

conveyance in the case of fine-grained bulk materials an instationary

flow mode, where the risk of pipe blockages exists. Closed plugs can

form which, if they are longer, eventually clog the conveying line due to

the wedge action and low gas permeability of the bulk material [16].

When conveying with these conveying conditions, safe pneumatic

Fig. 3. Conveying trial for limestone [21], counter pressure 1.0 bar (abs.), flow mode was inspected via glass pipe 2 m after the feeding point, feeding via rotary

feeder into a DN100 pipe. Total conveying distance =152 m including 5 m vertical and seven bends.

M. Dikty

Powder Technology 434 (2024) 119382

conveying can often only be achieved with additional auxiliary equip-

ment, such as an internal aeration pipe to increase the turbulence in the

system [17]. In the case of coarse-grained goods, such wedge actions in

the plug are small, so problem-free conveying can also be achieved with

granules with plug conveying [12].

The flow modes and flow characteristics of fine and coarse bulk

solids listed above lead to more or less strong pressure fluctuations,

which are called pulsations in the following. The previous research re-

sults for pulsations in horizontal conveying pipes are considered in the

following.

3. Reasons for the emergence of pulsations

Pressure fluctuations can be measured in a pneumatic conveying line

with the fluid only and with bulk material. A pressure fluctuation is

called a pulsation and is a rapid (short-term) pressure change in a

pneumatic conveying line at a certain location. The pressure can be

measured during the transition from dilute phase to strand/dune

conveyance, i.e., when the saltation velocity [18–20] is undershot, bulk

material sediments out of the gas stream. This bulk material is either

deposited on the pipe bottom or flows unsteadily over the pipe bottom in

the form of strands forming dunes at their upper surface. The flow modes

and flow characteristics of fine and coarse bulk solids listed above lead

to more or less strong pressure pulsations. Regarding pneumatic

conveying systems, pulsations have been examined many times, pri-

marily in order to characterize the transition from dilute phase to

strand/dense phase conveying, but there is no clear definition of the

term.

The pressure fluctuations/pulsations (measured as absolute or gauge

pressure) in horizontal pipe sections are caused by the following effects,

which are mainly listed by [35,49], which can occur independently of

one another:

1. Fluctuating solids flow rate

•due to unsteady flow from the feeder (screw feeders, rotary airlocks,

gate valves, venturis)

2. instationary accumulation and acceleration of solids at the feed point

depending on the distance to the saltation velocity

3. air supply

•due to the nature of the air supply (e.g., pulsations due to rotating

lobes in roots blower)

•pulsations due to by-pass valves and relief valves in the air supply

system

4. Material transport

•interaction of particle motion with the turbulence structure

5. Line configuration

•presence of bends, diverters

•pipe material (roughness)

•total pipe length (overall pressure of the system)

•inclined pipes and combination of horizontal and vertical pipes

•pipe installations like flaps or valves

•staggered pipes

6. Collector characteristics

7. System abnormalities

•such as leakage or broken components

•electrostatic effects

•external vibrations

Fig. 3 shows an example of a pressure curve for pneumatic conveying

[21]. The pressure fluctuations are explained using limestone powder

s,50

=27

m, ϱ

=1330 kg/m

). The Figure shows the initial conveying

gas velocity on the left ordinate. The right ordinate shows the conveying

line back pressure [bar(g)]. Both values are plotted over time. During

the trial, the average mass flow was kept constant at 1.6 t/h.

The conveying tests were started with a high air velocity and thus a

low SLR. Pulsations in the pressure signal at the beginning of the

conveying line are very low, the flow mode can be classified as “dilute

flow”.

By reducing the speed of the pressure generator, the air volume and

thus the conveying gas velocity can be reduced. The SLR increases as a

result. Visually it could be inspected that due to the reduction in the

conveying velocity, individual particles fall out of the conveying air flow

(saltation velocity is reached) and collect as strands at the bottom of the

conveying pipe, and the two-phase flow separates. The strand continues

to flow at reduced velocity on the pipe bottom, driven by the pressure

difference across the tube element and by the momentum transfer of the

impacting particles of the overflowing dilute phase flow.

It can also be seen that the pressure signal is becoming “more rest-

less”, i.e., that the pressure fluctuations / pulsations are increasing. This

is caused by the unevenly slipping and unevenly developing strands. A

further reduction in the conveying velocity leads to the area of non-

stationary conveying. Dunes form in places, according to the forma-

tion principles listed above. A look at the pressure curve in this time

window shows that the amplitudes of the pressure fluctuations have

increased but can be controlled for pneumatic conveying with sufficient

pressure reserves at the pressure generator.

Fig. 3 clearly shows the results of the measurements and visual in-

spections of the glass pipe section at the beginning of the conveying line

how the pressure fluctuations increase when the conveying gas velocity

is reduced and/or the solid to air ratio increases during the transition

from dilute phase mode to strand conveyance to dune conveyance.

4. Evaluation methods for assessing pulsations except pressure

measurements

The following sections show the results of the efforts which were

made in the field of pulsations, mainly to analyse the transition of flow

modes, but also to analyse the nature of the pulsation for fine and

grained bulk solids. This review gives a detailed overview of the

researched results, which were obtained by using pressure sensors. Be-

sides pressure sensors other methods were applied successfully which

are reviewed here.

In the 1970th the electrical tomography was developed for the

medical diagnostic [64]. Since the end of the last century, it was used to

analyse two-phase flows. Dyakowski et al. [64] presents a review of the

electrical tomography in 2000 where they show that ECT (electrical

capacitance tomography) systems are suitable for monitoring of dry or

nonconducting systems, whereas ERT (electrical resistance tomography)

systems work best for wet or conducting systems. The advantage of the

ERT is, that it is applicable also to dense phase systems. The results

obtained in applying these tomographic devices to dense phase flow

indicate the complete and dynamic patterns that are present in these

systems [77].

In these century further investigations were made. Jaworski and

Dyakowski [63, 64] investigated flow instabilities referred to slugs and

plugs of granular materials in horizontal and vertical channels using ECT

and particle image velocimetry (PIV). Williams et al. [65] investigated

the transient behaviour of the internal flow structure of material pulses

of fly ash.

In the Shanghai Engineering Research Centre of coal gasification

several investigations were carried out, by analysing the pneumatic

transport of coal, mainly under high pressure. The results of Xu et al.

[66], who analysed the fluctuations of the electrostatic signal of the ECT

in a dense-phase system, shows, that the dominant peak of the power

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Powder Technology 434 (2024) 119382

spectrum moves towards higher frequency with the increasing gas su-

perficial velocity. Pu et al. [67] investigated the motion process of slug

flow and demonstrated that the slug moves in wave motions through the

pipe with ECT and PIV. Fu et al. [69] analysed the dynamic behaviour of

the particles through electrostatic sensor array. They found out, that, as

the superficial gas velocity increases, the peak frequency value of the

electrostatic signal increases linearly in dense-phase region, while that

in dilute-phase region is nonlinear. Cong et al. [38] combined the

pressure signal data and the ECT to identify the flow pattern in a hori-

zontal pipe. With an enlarged team Cong [39] analysed the flow sta-

bilities and instabilities over a big range of flow modes using ECT and

the pressure signal.

Decomposition of fly ash density levels within the pipe cross-

sectional were obtained and statistically analysed by Chen et al. [68].

Later they analysed pulses within a flow by the pulse-growth and decay

segments, which represent the superficial fluidisation and deaeration

processes during conveying [23]. Azzopardi et al. used the ECT to

indicate that for the transport of fine coal there were two types of sys-

tematic fluctuations in the time series of mass flow rate and concen-

tration. Azzopardi et al. [41] analysed the fluctuations in dense phase

pneumatic conveying also of pulverized coal.

As mentioned above, in addition to ECT / ERT, PIV was also used to

evaluate pressure or particle pulsations / fluctuations. Rinoshika and his

teams used PIV technology in a variety of studies [57–61,70–74]. They

measured the particle velocity and concentrations in the acceleration

and fully developed regimes, with and without dune modes. Particle

fluctuation velocity of a horizontal self-excited pneumatic conveying

near the minimum pressure drop or multi-scale particle dynamics of low

air velocity pipe flow were scope of their research. They combined their

analysis for example with wavelet technology, pressure drop measure-

ments or proper orthogonal decomposition (POD). Li et al. [73] used the

PIV to investigate the particle fluctuating intensity during flowing

through bends with coarse polyethylene. Unfortunately, PIV is limited

due to its optical measuring method to low SLR.

The Laser Doppler Anemometer (LDA) or Phase Doppler Anemom-

eter (PDA) is a non-intrusive method to analyse multi-phase flows. Tsuji

and Morikawa [75] used it to analyse the air and solid velocity in a two-

phase flow. They found out, that the particle size has great effect on the

air-flow turbulence. Henthorn et al. [76] investigated the effect on the

pressure drop by the particle characteristics, SLR and Reynolds number.

They also investigated the velocity fluctuations in the single-phase and

multi-phase flow of fine particles.

Hilgraf and Bartusch [44] investigated the mass flow pulsations of a

Geldart group A coal by the evaluation of the load cell signal of the

receiving bin at the end of the conveying line.

Further measuring methods like Positron emission particle tracking

(PEPT), particle tracking velocimeter (PTV), inertial measurement unit

(IMU), or optical fibre probe (OFP), were less used to investigate pul-

sations in horizontal pneumatic conveying systems.

It can be summarized that pulsation / fluctuation measurements are

not only applied to the pressure but also to the air and bulk particles.

There are a variety of measurement methods that have been used in

addition to the pressure measurements considered here. The following

review however concentrates on research related to pressure measure-

ments in horizontal pneumatic conveying systems.

For the pressure measurement calibrated sensors are used. Sensors

Table 1

Summary of all test facility data and bulk solids tested for the research of pulsations in pneumatic conveying systems Test rigs marked with * have vertical downstream

pipe sections included.

Current no.

of test

centre

Reference Pipe

diameter

[mm]

Pipe

lenght

[m]

Vertical

section [m]

Bulk solid d

S,50

[

Particle

density [kg/

]

Bulk

density

[kg/m

]

Geldart-group

of tested

material

SLR [−] Feeding

device

[22,24,50,53] 53 173 12.3* Fly ash 15 2096 724 C 12–50 PV(T)

[24,25] 50 130 12.3* Cement 11 3000 930 C 51–70 PV(T)

[36] 50 130 12.3* Fly ash 15 2096 724 C 34–85 PV(T)

50 130 12.3* Alumina 78 3300 1050 B 25–71 PV(T)

[23] 53 20 – Fly ash 19 2500 – C 19–48 RF

2 [35]

25.4 30 0 Glass bread 450 2340 – B 2–28 SF

25.4 30 0 Glass bread 55 2380 – A – SF

25.4 30 0 Alumina 400 3420 – B – SF

25.4 30 0 PVC 137 1180 – A – SF

3 [37,40]

50.8 25 4.8 PS 3900 1045 735 D 2–29 RF

50.8 25 4.8 PE 3300 1400 892 D 2–29 RF

50.8 25 4.8 Polyolefin 4600 870–940 – D 2–29 RF

[38] 20 34 10.8 Coal dust 36 1382 540 A 230–580 PV(B)

[39]

20 34 10.8 Coal dust 35 1532 507 A 60–540 PV(B)

50 10 4 Coal dust 35 1532 507 A 99–557 PV(B)

5 [41] 36.8 26 8.3 Coal dust 61 1322 537 A 19–98 RF

6 [28–30,61,78] 76 8 0 PE 3500 1210 – D 2–9 PV(B)

[62] 76 8 0 Polystyrene 2900 1400 – D 0.5–10

7 [48] 40 14 0 plastic 190 1000 – A/B 3–10 EF

40 14 0 plastic 2800 1000 – D 1–9 EF

8 [49,56]

50 30 2.1 Polyester 3000 1400 – D 1–6 RF

50 30 2.1 Glass bread 450 2480 – B 1–4 RF

50 30 2.1 Alumina 450 3750 – B 1–4 RF

9 [34,43,46]

69 168 7.0 Fly ash 30 2300 700 A 27–106 TPV(B)

69 148 7.0 White

powder 55 1600 620 A 12–56 TPV(B)

10 [45]

51 70 3.0 Fly ash 45 1950 950 A 16–52 PV(B)

51 70 3.0 Cement 15 3060 1070 C 40–121 PV(B)

63 24 3.0 Fly ash 45 1950 950 A 23–41 PV(B)

63 24 3.0 Cement 15 3060 1070 C 11–44 PV(B)

11 [51]

56.3 52 – Polystyrene 3100 1050 640 D – RF

56.3 52 – Polyamide 2500 1140 646 D – RF

56.3 52 – Polystyrene 860 1100 630 A – RF

12 [79] 10 45 – Coal 36 1350 – A 200–550 PV(B)

SF: screw feeder, RF: rotary feeder, PV(T): pressure vessel top discharge, PV(B): pressure vessel bottom discharge, TPV(B): twin pressure vessel system with bottom

discharge, EF: electromagnetic feeder.

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Powder Technology 434 (2024) 119382

can be calibrated and certified by the supplier [34,40,45,46], or the

calibration must be taken place in the test rig. A procedure for the

calibration is shown by Pan [92] or by Mallick [93]. Behera et al. cali-

brated the pressure sensor using a Barnett dead weight tester. [36].

Sensor with wear resistant flush-mounted membrane, like the

PMC131 from Endress and Hauser are used by [34,45,46], they do not

need a purge air connection. If sensors with for example metal mem-

brane are used, the sensors are located reset in the socket, to avoid wear,

which would influence the measured value of the pressure. Such sensors

must be cleaned regularly, or purge air system were used to keep the

measuring socket clean. [35,36,91] The measuring accuracies of such

sensors are <0.1% of the full-scale value. [34,36,45,46] The sensors

should have a sampling rate that meets the conditions of the Nyquist

theorem [37,40].

5. Research results of pulsation obtained via pressure

measurements in horizontal pneumatic conveying lines

As can be seen above, the term pulsation has a wide range of ap-

plications. Besides e.g., air particle pulsations or bulk material particle

pulsations, the gas pressure in a pneumatic conveying line can also

pulsate. The investigations carried out in this field are presented in more

detail below and compared where possible. By way of introduction,

Table 1 shows the test rig configuration used by the respective authors. It

shows the piping data, the bulk materials used, incl. their bulk material

data (bulk density, average particle diameter, particle density, and

Geldart classification). In addition, the Table shows in which solid

loading ratios and with which feeding device the tests were carried out.

In the following, the analysis methods used so far in the analysis of

horizontal pressure pulsations are presented.

5.1. Transient parameter analysis

At first the research results of the direct use of the pressure signal will

be shown by transients. A transient process (lat. Transire “to go by”)

describes a non-stationary process. Fig. 4 shows transients applied to a

static pressure signal waveform that was transient/pulsating. The pres-

sure changes over time. However, the pressure-changes differ from each

other.

The following transients are determined from Fig. 4:

•Pulse duration t

: It is the time interval between a pressure mini-

mum and the subsequent pressure maximum [22].

•Pulse duration t

dec

: It is the time interval between a pressure

maximum and the subsequent pressure minimum [22].

•Pulse amplitude A

: It is the pressure difference between a pressure

minimum and the following maximum [22,24,25].

•Pulse amplitude A

dec

: It is the pressure difference between a pressure

maximum and the following minimum [22].

The following ratios were formed from the transients including the

fluid density

F and the superficial gas velocity vF [22,24,25]:

Slope of pulse growth :sgr =Agr

tgr

(1)

Slope of pulse decay :sdec =Adec

tdec

(2)

Pulse slope ratio :sr=sgr

sdec

(3)

Pulse frequency :fp=1

tgr +tdec

(4)

Non −dimensional pulse amplitude :sIm =Agr

F•v2

(5)

From the pulse frequency (Eq. (4)), an additional dimensionless

pulse frequency was derived, comparable with the Strouhal number. L

is the position of the corresponding pressure sensor in meters from the

feed point [22].

Non −dimensional pulse frequency :Sr =fP•LR

(6)

Transient analysis of the pressure signal was performed by Behera

et al. [22] and Williams et al. [24,25]. They carried out conveying trials

with fly ash and cement, which are both Geldart group C materials, in a

DN50 conveying pipe with lengths of 130 m and 173 m. Three pressure

sensors were installed and evaluated along the conveying line at 27 m,

90 m and 130 m from the feed point.

The course of the pulse slope ratio was evaluated along the

conveying pipe. A clear course of the pulse slope ratio does not result.

The change of the ratio along the pipe is increasing as well as decreasing,

Fig. 4. Transient chart related to a static pressure signal based on [22].

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Powder Technology 434 (2024) 119382

which is explained by Behera et al. [22] with the change of the flow

mode along the pipe and the air holding capacity of the fly ash.

The pulsation frequency tends to decrease along the conveying line

when the conveying starts as dense flow conveying. Like the pulse slope

ratio, the course of the pulsation frequency is influenced by the flow

mode and the air holding capacity according to [22].

The influence of the load on the transients is shown as an example in

Fig. 5.

Fig. 5 shows that the pulse amplitude decreases with increase in the

solids loading ratio. At low solids loading ratios, pulse amplitude is

much higher due to higher pulse growth and vice versa.

Further investigation results are that the pulse slope ratio shows a

decrease with increasing loading. If the SLR increases, the pulse

amplitude difference falls, i.e. the height of the pressure fluctuation is

reduced, which was justified with the aeration behaviour of the bulk

material. The amplitude of the pressure increases along the conveying

line depending on the bulk material and the flow mode, this is due to the

gas expansion along the pipeline [25]. Deposits and dunes which will be

accelerated have to be accelerated to higher velocity. This results into

higher pressure drops and thus high pressure amplitudes. It was

observed that the pulse amplitude decreases when the air mass flow is

increased. The analysis with fly ash and alumina has shown an increase

in pressure fluctuation with an increase in distance from the start of the

pipeline. [24,25], i.e., that the pulsations increases along the conveying

Fig. 5. Variation of pulse amplitude with solids loading ratio [22].

Fig. 6. Wavelet multi-resolution analysis of fluctuating pressure at air velocity of 10 m/s and 0.3 kg/s solids mass flow at locations of x =2 m (left), x =6 m (right)

from feeding point [28].

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Powder Technology 434 (2024) 119382

pipeline, influenced only by the pipeline layout, such as bends. One

reason could be that the re-acceleration of deposits along the conveying

line due to the increasing air velocity and thus increasing relative ve-

locity between air and deposit lead to higher acceleration pressure los-

ses, which are measured in the form of pressure pulsations.

The results of the above shown transient analysis of the static pres-

sure fluctuations with a Geldart group C material shows that with the

help of transients, such as the rate of pressure increase, decrease, pulse

time, pulse frequency, conclusions can be drawn about the flow condi-

tion and the aeration condition of the bulk material in the conveying

pipeline. The strength of the pressure pulses appears to depend on the

length of the pipe and the nature of the material. The solid mass flow

rate showed only a small influence on the pulse structures. Bends in-

fluence the results, due to the rope formation. A turbulent flow will get

smoothed due to the bend and the transients can change significant.

5.2. Frequency analysis

Frequency analysis is a tool for studying periodic signals such as

pressure signals, as it allows the various frequency components of a

signal to be identified and quantified to provide information about its

characteristics.

If the Fourier transformation is used, which is a mathematical

method for decomposing a signal into its individual frequency compo-

nents, then a signal will be converted from the time domain to the fre-

quency domain, allowing the various frequency components of the

signal to be identified [90].

If the wavelet analysis will be used for frequency analysis, unlike

Fourier analysis, which decomposes the signal into sine and cosine fre-

quency components, wavelet analysis uses a basis function called a

wavelet to divide the signal into different frequency ranges. Wavelet

analysis can be useful in analysing signals where the frequency ranges

vary or have sharp temporal changes, as it can capture these changes

better than Fourier analysis [22].

5.3. Wavelet transformation / denoising

The origins of wavelet theory come from signal theory. It is a

transformation of the frequency analysis of signals (time-dependent

functions). This transformation was introduced because the classic

methods for frequency analysis, i.e., the Fourier and the windowed

Fourier transformation, have significant disadvantages in terms of signal

theory. A major disadvantage of the Fourier transformation is that it

insufficiently considers the local properties of a signal [26].

In the world of wavelets, the basic functions are called the scaling

function and the mother wavelet. The wavelet function of the contin-

uous wavelet transform is [26,28]:

Wk(

,s) = 1



|s|

√∫+∞

−∞

f(t)

(t)dt,(7)

where f(t) ist the signal as a function of time (t) with the mother function

/ mother wavelet:

(t) =

*(t−

s),(8)

where s is the scaling parameter as the reciprocal of the frequency. It is

used to set the width of the window function.

is the time variable. If s

and

are defined in binary and discrete form, then the continuous

wavelet transformation becomes the discrete wavelet transformation.

Using the discrete wavelet transform has the advantages of more accu-

rate and faster analysis. In this case the time variable

becomes a and

the scaling factor s becomes b. N is the length of the filter:

Wd[a,b] = 1b

√∑

N−1

i=0

f(t)

[t−a

b](9)

The wavelet transformation is also suitable for noise reduction,

called denoising. The thresholding technique is used for this. Hard and

soft thresholding are the most well-known variants. All small detail

coefficients that can be traced back to the noise and fall below a speci-

fied threshold value

are set to 0 and are not considered in the recon-

struction. Very good results can be achieved with this method if the

output signal has been oversampled [27].

Li et al. [28,29,61,62] carried out conveying tests with polyethylene

pellets (d

s,50

=3500

=1210 kg/m

) in a conveying pipe with a

diameter of 76 mm. Fig. 6 shows the wavelet decomposition of the

pressure signal down to 10 iteration depths (10 frequency bands). The

illustration on the left shows the breakdown of the conveyance with a

SLR of 2, two meters from the feed point. Li et al. classifies the flow mode

as dilute phase. The 200 Hz frequency (band 9) band shows the most

intense deflections. For Li et al., the 200 Hz band is the characteristic

band for dilute phase. The illustration on the right shows the decom-

position of the conveyance six meters from the feed point. Li et al.

classifies the conveyance as dune flow. The 0.78 Hz frequency in band 1

shows the most dominant deflections. 0.78 Hz corresponds to the fre-

quency with which the dunes pass the pressure sensor. Here, too, the

200 Hz frequency band shows more intense deflections than the other

bands. Li et al. interpret this as the dilute flow above the dunes.

Table 2 shows the summary of the authors [28–30,61,62,78] which

indicated the flow pattern by the frequency peaks in the decomposed

frequency bands. The tests were carried out with Geldart group D plastic

granulates.

Mittel et al. [34] used wavelet analysis to assess the static pressure

when conveying Geldart A particles. They presented their results by

means of colour spectra. The colour intensity reflects the energy of the

wavelet coefficients (also called scale coefficient), and the colour bar in

each graph gives the energy values for the different colours. From the

colour spectrum, they can determine that the conveyance at the begin-

ning of the conveying line begins as a dune conveyance and becomes

highly turbulent at the end of the line. They also observed that the

magnitude of the maximum energy increases along the conveying

pipeline, indicating an increase in the amplitude of the pressure fluc-

tuations. Fig. 7 shows an example of the colour spectra for fly ash at a

SLR of 63. At initial locations (i.e., at 6,7 m and 22,6 m from the feed

point, maximum energy is concentrated in the upper region of the plot

and corresponds to high scale coefficients (i.e., scale 50). Since higher

scale coefficients describe the low frequency components of the signal,

therefore it indicates that a signal is composed of large number of low

frequency components, which might be due to reduced turbulence at the

starting of pipe location and in turn indicates dune flow mechanism.

While going from P2 (22,6 m from feed point) to P3 (75,3 m from feed

point), colour intensity is reduced corresponding to higher scales (i.e.,

changes from dark red to light blue) and higher intensity shifts to lower

scales (i.e., scale 20). Thus, parent signals contain comparatively higher

frequency features as compared to the signal obtained at location P1.

With further movement along the pipeline (i.e., from 75,3 m to 115 m

Table 2

Frequency bands of the wavelet analysis of plastic pellets in a swirling gas-solid

flow [28–30,61,62,78].

Frequency band Evaluation

0.5–1 Hz Dune flow [28,29,61,62,78]

1.56–3.13 Hz Periodic sliding clusters [78]

20–80 Hz Heterogeneous suspension flow over dunes [29,61,62]

100–110 Hz Dilute flow over dune flow [28–30,78]

120 Hz Heterogeneous suspension flow [29,62]

20–200 Hz Dilute flow over dune flow [28,29]

200 Hz Only air flow [28–30]

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from feed point) colour intensity almost diminishes at higher scales and

multiple numbers of highly concentrated points are seen at lower scales

(scale 10), which indicate the presence of large number of pressure

fluctuations at different time instants in the signal [34].

5.4. Power spectral density (PSD)

The spectral analysis gives the spectral power density as a result. If it

is applied to the pressure signals p of the pneumatic conveying, the

power of a frequency range in the unit [Pa

/Hz] is obtained. Plotted

against the frequency shows the dominant frequencies that are gener-

ated, for example, by the feeding device, the pressure generator or by the

conveyance.

An analogue signal p(t) is digitally sampled and is now dependent on

the number of samples n. The pressure data series are divided into L

segments with a length of N

for each window-segment x

[n] and rep-

resented as follows [31]:

p[n] = p[n+ⅈNs](10)

with n =1,2,…,NSand i =1,2,…(L−1)

In order to get the lowest variance from a fixed number of samples, it

turns out to be optimal to overlap the window segments by half their

length [32]. Each window-segment xi[n]is multiplied by a window

function w[n][33]:

pwi[n] = pi[n]⋅wi[n](11)

The Fourier transformation of the windowed signal can be expressed

as follows [31]:

Xwi[f] = ∑

n=1

pwi[n]e−2

fn (12)

The corresponding power spectrum is the square of the absolute

value of the amplitude and is according [31] for each segment i:

xx(f) = 1

NSU|Xwi[f]|2(13)

where U is a normalization factor related to the energy in the window

function [31]:

U=1

NS∑

n=1

wi2[n](14)

The average power spectrum is [31]:

Pxx(f) = 1

L∑

i=1

xx(f)(15)

Fig. 7. Variation of energy distribution in different levels along the length (P1 to P5) for fly ash at a SLR of 63 [34].

Fig. 8. PSD observations for dense and dilute-phase conveying of fly ash [36].

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Powder Technology 434 (2024) 119382

The result is a performance with unity [Pa2

Hz ].

The PSD can also be used to distinguish between the periodic and

chaotic nature of the flow process. A wide frequency band in the power

spectrum indicates chaotic signal behaviour, while the periodic nature is

reflected by clear dominant peaks in the frequency band [31].

In 1993, Dhodapkar and Klinzing [35] were the first to apply PSD to

horizontal pneumatic conveyance. They measured the static and dif-

ferential pressure on top and on the bottom of the pipe. According to

Behera [36] the PSD of the signals from top and bottom transmitters is

quite similar, so it is not possible to identify the different flow modes on

the upper and lower sections of a pipeline. According to Cabrejos et al.

[49] both measuring methods provide enough information to identify

the different flow regimes by PSD analysis.

An example of a PSD plot is shown in Fig. 8 for fly ash (d

s,50

=14.91

=2096 kg/m

) with ˙

m as SLR and u

as the gas velocity. The

upper diagram shows dense phase conveyance, the lower one dilute

phase conveyance. With dense phase conveying, the dominant fre-

quency bands are close to the 0 Hz range. Dilute phase conveying

analysis shows several dominant frequencies of smaller PSD altitude

[36].

That the bulk solid has an influence in the PSD is shown in Fig. 9 for

fly ash and alumina under similar flow conditions of SLR, pressure at the

corresponding pressure sensor and superficial gas velocity. The order of

magnitude of power is higher for alumina than that of fly ash [36]. The

peak value of the PSD is at fly ash near 0 and for alumina between 0.6

and 1.0 for the same conditions (SLR, static pressure, velocity, pipe

diameter).

For the following statements it must be considered that the operating

conditions (piping configuration, bulk solids and flow data) on which

the statements are based differs. The letters of the reference-exponent

show the Geldart-group of the trials, on which the statement is based.

Dhodaqkar [35]

A,B

established the following classification based on

his measurement:

•Stratified flow - significant fluctuating components occur near 0 Hz

with lower amplitudes.

•Dune - irregular fluctuations <4 Hz frequency.

•Air flow only - little pressure fluctuations with very low frequency of

∼0.05 Hz.

Mittal et al. [34]

found out that a single frequency component at

0.1 Hz indicates the periodic nature of the pressure signal, which might

have occurred due to periodic rising of dunes near the pipeline entrance.

[35]

A,B

and Cong et al. [38]

analysed, that, if small amounts of bulk

material are added to the pure air flow (small SLR), the power spectrum

is smoothed from over 100 Hz to <2 Hz. If the SLR is further increased,

the power spectrum/pulsations increase again.

The results show, that independent on the Geldart classification the

dominant frequencies shift from the dilute flow (comparatively high

energy content) to the dense phase (comparatively low energy content,

because the particle absorb the energy) towards 0 Hz [34-46

C,A

, 47

, 51

, 62

]. Cong [39]

measured, that the peak of the PSD decreases

with increasing pipe diameter. Increasing bulk solids mass flow of coarse

particles increases the peak value of the PSD.

The PSD peak value depends on the flow condition, test rig set up and

bulk solid. For fine bulk solids the peak of the PSD decreases with

increasing air at the beginning of the conveying line. For the same

measuring position, the dominant frequency shows no tendency with

increasing air velocity, because of the influence of the flow pattern.

Bends influences the data of PSD for pressure sensors close before and

behind the bend, due to strand and rope formation. Usually, the fre-

quency shifts in the direction of larger values.

At the beginning of the conveying pipe, the power spectrum is quite

narrow, and a single dominant peak can be clearly seen in the spectrum.

Along the pipeline and towards the flow direction, the spectrum be-

comes broader, and in addition to the single dominant frequency, other

frequency components also appear in the spectrum [34,36]. At the same

time, the power density of the signal increases from the inlet to the end

of the pipeline.

5.5. Statistics

5.5.1. Standard deviation

The best-known measure of the spread of a distribution is the stan-

dard deviation sor its square, the variance s2. It measures the spread of

the data around its mean. The variance of the pressure around its mean is

[42]:

s2=1

n[(p1−p)2+…+(pn−p)2](16)

The standard deviation is the root of the variance [42]:

s=

s2

√(17)

The standard deviation of the static and/or differential pressure

signal was analysed by Dhodaqkar and Klinzing [35], the teams of Mittal

and Mallick [43,45] and Cabrejos et al. [49]. It could be shown that the

standard deviation is minimum dependent on the bulk solid, pipe

diameter, pipeline configuration (length, bends), flow mode, mass flow

(air, bulk solid) and SLR.

If the experiments from [35,43,45,49] are analysed, it becomes clear

that after a bend measured pressures, have a tendency that the standard

deviation falls in ¾ of the measured data, caused by the strand and rope

formation. Also measuring points 10 m behind the bend shows this

tendency.

In 2014, Mittal and Mallick [43] found that the standard deviation

increases along the conveying line. They disproved this with further

investigations in 2016. [34]

The standard deviation of both absolute and differential signals gives

certain information about the flow regime of the gas-solids flow as it

increases to a maximum value when saltation is approached [49].

Dhodaqkar and Klinzing [35] substantiates this statement for granules to

the effect that the standard deviation only deviates strongly with the

dune flow, whereas it decreases slightly with the stratified flow.

5.5.2. Root mean square (RMS)

In mathematics, the root mean square is a statistical measure of the

Fig. 9. Comparative PSD plot of fly ash and alumina under same flow condi-

tions of conveying [36].

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Powder Technology 434 (2024) 119382

magnitude of a varying quantity. The square mean is a mean value in

which large numbers are weighted more heavily than smaller numbers.

In technology, the root mean square is of great importance for periodi-

cally changing quantities. The effective value, or root mean square is

defined as follows [47]:

RMS =

√=

n∑

i=1

√=

1+p2

2+…+p2

√(18)

Compared to the standard deviation, which accounts for the devia-

tion of individual pressure signal from the mean, RMS accounts for the

absolute magnitude of those pressure points as well.

Only Li [28] and Tsuji and Morikawa [48] analysed the pressure

pulsation signal of a horizontal flow with the RMS. Both researchers

used the static pressure signal of trials with plastic granulate. As an

example, in Fig. 10, Li [28] shows the RMS of the conveying line pres-

sure at 3 measuring points along the conveying line with 76 mm inner

diameter for polyethylene pellets (d

s,50

: 3.5 mm,

: 1210 kg/m

). There

is a kink point / minimum in the trends at an air velocity of 13 m/s. Both

researchers showed, that for Geldart group D particles, the RMS in-

creases sharply after a kink point with decreasing air velocity. For a

Geldart group A/B material the RMS drops after passing this maximum.

The reason is the change in flow mode from blowing dunes (mode B of

Section 2) to strand flow and stratified flow (mode C and D of Section 2).

5.5.3. Probability density function (PDF)

Distributions describe the frequency of an occurring value x and are

usually shown in so-called histograms (Fig. 11). To do this, classes are

formed, and it is counted how often a value falls into a class. The

probability density function PDF(x) is obtained by normalization of the

continuous approximation of the histogram [27]. The following applies:

∫+∞

−∞

PDF(x)dx=1(19)

The probability density function describes the probability that a

value x can assume in a distribution.

A so-called core density estimator is used to determine the PDF, since

histograms can give a different impression due to different classifica-

tions. The equation for the core density estimator for approximating the

PDF reads with b the bandwidth of a histogram window and n the

number of samples [27]:

PDF(xi) = 1

nb 

√∑

i=1

e−1

2(x−xi

b)2

(20)

The direct application of the PDF to the pressure signal found little

use. Cabrejos et al. [49] noted about the absolute wall pressure of the

transport of plastic granulate that the PDF changes from unimodal dis-

tribution for dilute flow and pulsating flow into a bimodal distribution

for moving dunes and saltation back to a unimodal distribution for

settled dunes.

Density curves are not only described by their position and scatter,

but also characterized by the symmetry and skewness or kurtosis of the

curve. These parameters always describe the deviation from the normal

distribution. The index of skewness is the moment coefficient g

, which

is defined as follows [42]:

gm=m3

s3(21)

With:

m3=1

n∑

i=1(xi−x)3,s3=(

n∑

i=1(xi−x)2

√)3(22a,b)

With:

Fig. 10. RMS of the measured conveying line pressure p from with polyethylene pellets at a mass flow of 0.3 kg/s [28].

Fig. 11. Histogram and density function [42].

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Powder Technology 434 (2024) 119382

gm=0 for symmetric distributions,

gm>0 for left-steep (right-skewed) distributions,

gm<0 for right-steep (left skewed) distributions.

Measures for the kurtosis are intended to characterize how heavily or

weakly the central area and – related to this – the edge areas of the data

are occupied. It should be noted that distributions with the same spread

can have different kurtosis in the middle or different left and right ends

in the edge areas. The kurtosis describes how flat or pointed a distri-

bution is. The kurtosis number of a normal distribution is 3, so Fisher

[42] normalized the kurtosis measure γ with “-3” and defined it as

follows:

γ=m4

s4−3(23)

With:

m4=1

n∑

i=1(xi−x)4,s4=(

n∑

i=1(xi−x)2

√)4(24a,b)

With:

γ=0 for normal distributions,

γ>0 for sharper distributions,

γ<0 for flatter distributions.

Tsuji and Morikawa [48] transported two different plastic pellets of

Geldart group A/B and D. They evaluate their measurements for all

Geldart groups, which were used, for the flow mode A and B (definition

see Section 2). When the fluctuation follows the Gaussian probability

distribution, the skewness and kurtosis factors have the values of 0 and

3, respectively. For bulk solid which is at the boundary of Geldart group

A/B the dune flow has a positive skewness, the strand flow has a

negative skewness. The Geldart group D bulk solid has a positive

skewness in plug flow mode. For the kurtosis, all values are >0 for flow

modes except A and B.

Cabrejos and Klinzing [49] could not determine any clear assignment

of the skewness and kurtosis to the flow patterns when conveying coarse

bulk solids.

5.6. Shannon entropy

The Shannon entropy, also called information entropy, describes the

average amount of information contained in a signal. It comes from

information theory and was developed by Claude E. Shannon in 1948. It

is a measure of the uncertainty or disorder in the system and can be

viewed analogously to entropy in thermodynamics [30]. It is defined as

follows [50]:

Hs= − ∑

i=1

P(xi)logbP(xi),(25)

where P is the probability of any value of x. The logarithmic base b is

either 2, e or 10. The unit of P is with the b =2 Bit, b =e Nat or b =10

Hart [43].

Various investigations with fine bulk solids were carried out to

determine the Shannon entropy using pressure measurements

[43,45,46,50,53,79]. It turns out that the Shannon entropy depicts the

flow conditions in the conveying line well. The Shannon entropy in-

creases along the horizontal conveying line since the turbulence and

thus particle contacts and particle-wall contacts increase for example

due to increase in velocity. In the area of dense phase conveyance, the

Shannon entropy is lower than in the area of dilute conveyance. Like the

standard deviation, the Shannon entropy also changes after a bend. The

Shannon entropy is also reduced because of the deceleration in the bend

and the resulting reduction in turbulence. If the Shannon Entropy has a

high or low level depends on the turbulence of the flow mode itself. The

influence of the SLR or bulk solid mass flow is also linked to the flow

mode. An increasing SLR or mass flow can increase and decrease the

Shannon entropy. In case the increasing SLR or mass flow changes the

flow mode from homogenous flow to strand flow, then the turbulence

reduces, and the Shannon entropy reduces. If the SLR or mass flow

changes from strand flow to dune flow the turbulence increases which

lead to higher Shannon values.

It should be noted that the absolute value of the Shannon entropy

depends on the amount of data, so attempts are only comparable when

the same number of data is considered.

5.7. Rescaled range analysis / Hurst exponent

The rescaled range analysis was developed in 1951 by British hy-

drologist Harold Edwin Hurst to assess and forecast Nile water levels.

The method should detect stability and near instability [51].

The Hurst exponent is calculated in six steps as follows [52]:

1) Calculation of the mean p

2) Calculation of the linear deviation from the mean:

yi=pi−p(26)

3) Calculation of the cumulative deviation from the deviation of the

mean:

zi=yi+yi+1(27)

4) Determining the rescaled range:

R=max(zi)–min(zi)(28)

5) Calculation of the standard deviation s according to Eq. (17).

6) The Hurst exponent is then defined as follows (with n as the time

window of the measurement interval):

S∝nHx→Hx=

lnR

S

lnn (29)

The exponent can be divided into 3 areas, which are generally

interpreted as follows:

0.5

Indicates a time series with long-term alternation between high and low

values in adjacent pairs, meaning that a single high value is likely to be

followed by a low value and that the value will then tend to be high again,

with this tendency to switching between high and low values long into the

future.

0.5 Shows that the values will not show any long-term correlation.

0.5

Indicates a time series with long-term positive autocorrelation, which

means both that a high value in the series is likely to be followed by

another high value, and that values will tend to be high as well long into

the future.

Fig. 12. Course of the Hurst exponent depending on the SLR for the pneumatic

transport of fly ash [53].

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Powder Technology 434 (2024) 119382

Various authors [37,38,40,45,49,51,53] have applied the rescaled

area analysis to pneumatic conveying by using absolute or differential

pressure signals, with a variety of statements that must be considered

very much in relation to the respective test rig.

The final statements are that in a pneumatic conveying system the

Hurst Exponent decreases along the direction of flow in case the flow

mode changes also [40,43,53,59], what implies that there is an increase

in degree of complexity of flow mechanism (or turbulence), comparable

to the Shannon entropy and that the Hurst exponent increases after

passing a bend [43].

[37,40,46,53] found that the Hurst exponent also increases with

increasing SLR, which is attributed to damping of the flow and thus

reduction of flow turbulence.

As an example, the course of the Hurst exponent is shown using fly

ash conveyance as a function of the SLR, see Fig. 12. Shijo and Behera

[53] transported fly ash (d

s,50

=14.91

=2096 kg/m

) through a

173 m long conveying line with a pipe diameter of 53 mm, measuring

the static pressure of the system. It turns out that the Hurst exponent

increases as the SLR increases. That the Hurst exponent increases with

increasing SLR could not be confirmed by all authors, because an

increasing SLR could lead to a change of flow mode, and then also a

change of the Hurst exponent into the opposite direction is possible.

Since not only the SLR influences the Hurst exponent, but also the

change flow mode.

The results from Table 3 and Fig. 13 show that the Hurst exponent is

not a good method to compare conveying systems with each other. The

ranges of the absolute values overlap considerably, so that no clear

assignment of a Hurst exponent to a flow mode is possible. Since a small

Hurst exponent means a large disorder / turbulence, one can recognize

the change of the flow mode in a system based on the Hurst exponent,

which is again comparable with the Shannon entropy.

Fig. 13 show the classification of the results to the flow modes from

Table 3. *Unstable flow is a mode of conveying for coarse particles

where the system is in equilibrium between plug and strand flow [51].

**flow occurs near the pressure drop minimum and is valid for coarse

particles [51].

5.8. Phase space diagram

Phase space diagrams are useful to represent chaotic systems or

deterministic dynamic systems that are extremely sensitive to distur-

bances in their initial conditions. It is possible to construct the 2-dimen-

sional state space plot by means of only one characteristic variable by

using the method of delay coordinates [54]. The values of the variable at

different time delays {0, Δt, 2Δt, …, (d −1)Δt} are used as coordinate

values in the embedding space. 2-dimensional phase space diagrams

have been constructed using the method of delays [55], which involves

plotting original time series X(t) versus its delayed versions, i.e. X(t +

where

is the delay time to collect one data point.

The normalized pressure at time t and its time derivate (dp/dt) at the

same time, (dp/dt) is defined as follows [40]:

dt =P(t+Δt)− P(t)

Δt(30)

The phase space diagram is represented by the normalized value of

pressure drop per unit length at a certain time (t) versus the normalized

value of pressure drop per unit length at (t +dt), where (dt) represented

the time step to collect one data point.

Selection of optimum value of delay time is important for phase

space construction. If the selected time lag is too short, then there are

chances that it would be influenced by noise, on the other hand if too

Table 3

Summary of Hurst exponent evaluation.

Hurst exponent Evaluation

<0.08 Unstable flow [51]*

0.1 … 0.3 Stable flow** [51]

0.35 … 0.5 Only gas flow [49]

0.35 … 1.15 Conveying with bulk solid [49]

<0.40 Only gas flow [37,40]

0.40 … 0.80 Dilute flow, strand flow, unstable flow* [49]

0.40 … 0.99 Conveying with bulk solid [50]

0.57 … 0.92 Conveying with bulk solid [37,40]

0.60 … 1.15 Flow below saltation velocity [49]

>0.74 Flow left of the minimum point of pressure drop [38]

0.70 … 0.90 Conveying with bulk solid [38]

0.80 … 0.91 Conveying with bulk solid [37,40]

0.80 … 1.10 Dune flow [49]

<0.84 Dilute flow [37,40]

Fig. 13. Classification of flow patterns in horizontal pneumatic conveying using the Hurst exponent of pressure signals.

M. Dikty

Powder Technology 434 (2024) 119382

long delay time is chosen, then the phase space attractor would not

reveal the local flow characteristics. The optimum value of delay time is

selected as the first minimum of the mutual information function [52].

Cabrejos [49,56] was in 1994 one of the first, who uses the phase

diagram to judge about a pressure signal. Further investigations were

carried out by [37,40,46,53]. Fig. 14 show as an example phase space

diagrams from [40] for a horizontal pipe section, with air at different gas

velocities. The diagrams shows that the area of the phase space diagram

increases with increasing air velocity, because the turbulence increases.

Independent on the pressure measuring type (static pressure or dif-

ferential pressure) the type of pulsation can be deduced from the geo-

metric position of the data in the phase space diagram. The longer the

horizontal length of the graph, the greater the pressure fluctuations. The

longer the vertical height the faster the pressure fluctuations. An in-

crease in the area means an increase in the turbulence in the conveying

line, caused by e.g., higher fluid velocities or by the transition from

dense flow conveying to dilute phase conveying, as can happen along a

conveying line. However, the geometry of the diagram and its changes

during conveying can only provide qualitative statements on turbulence

and pulsation. It also follows that the influence of the loading allows to

increase or decrease the diagram area, depending on whether the

feeding of bulk material increases or decreases the turbulence. As with

all previous evaluation methods, the measurement after a bend leads to

a change in the direction of turbulence reduction. The phase space plots

indicate that there is no specific range in values for size or area of the

attractor for a dense or dilute mode of flow.

6. Conclusions

The previous scientific work evaluated the pressure signal using

statistical analyses such as PSD, RMS, PDF, standard deviation (SD), root

means square or the skewness or kurtosis. Also, non-statistical methods

like wavelet analysis, transient parameter analysis, Shannon entropy,

rescaled range analysis or the phase space diagram were used. Mainly

qualitative and partially quantitative relationships to flow patterns were

found.

In summary, turbulence is the key factor influencing the analysis

methods. The standard deviation and the transients show the direct

character of the pressure signal. Except for the phase space diagram,

quantitative results are obtained which, however, have little signifi-

cance on their own. In principle, the tendency of the measurement

methods can be shown as a function of the turbulence as follows, if the

turbulence increases, then:

•the pressure pulsations increase,

•the standard deviation of the pressure signal increases,

•the Shannon Entropy increases,

•the Hurst Exponent decreases,

•the area of the phase space diagram increases,

•the dominate frequency of the wavelet analysis increases,

•the power spectral density increases.

The turbulence is a function of the relevant process parameters like

for example air velocity, particle velocity, SLR, mass flow, bulk solids

data, pipe diameter and pipe configuration. All these data results into a

plant specific flow mode. The turbulence can be linked to the flow, for

example acc. [80] mode of Section 2 as follows:

•High turbulence flow modes are: Dilute flow, cluster flow, unstable

dune flow (mode A, B, E)

•Medium and low turbulence flow modes are: Strand flow, stratified

flow (mode C, D), plug flow (mode F)

Fig. 14. Phase space diagram at lower horizontal section (gas only), with different air velocities [40].

M. Dikty

Powder Technology 434 (2024) 119382

Table 4 shown the analysed bulk solids classified into the Geldart

group and linked to the analysis method.

However, the results obtained so far still show room for further in-

vestigations, so the influence of the pipe diameter on the pulsations has

hardly been investigated, and the application of the root mean square

and the probability density function to the pressure signal has only been

used to a limited extent. Depending on the Geldart group or the flow

mode, how does the pressure signal behave below the minimum pres-

sure loss in the direction of the minimum conveying velocity? Can an

analysis method be used to detect blockages before they occur? Are

there other transients that describe the flow modes, the minimum

pressure drops, the minimum conveying gas velocity, such as the ratio of

the pressure rise time to the pressure drop time of a pulsation, or the

ratio of the absolute pressure increase to the absolute pressure drop of a

pulsation. What about the absolute pressure in the conveying line, how

does it influence the pressure signal? The higher density results into a

higher impulse onto the particles and it could result in a change in flow

mode. So further investigations must be carried out, to get a clear and

deep understanding of pressure pulsation in horizontal pneumatic

conveying lines.

CRediT authorship contribution statement

M. Dikty: Writing – review & editing, Writing – original draft,

Validation, Supervision, Project administration, Investigation.

Declaration of competing interest

The authors declare that they have no known competing financial

interests or personal relationships that could have appeared to influence

the work reported in this paper.

Data availability

No data was used for the research described in the article.

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