Estimating the mixture friction factor for the dense-phase transport of particulates in horizontal pipes with inner longitudinal spiral slots

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Research Paper

Estimating the mixture friction factor for the dense-phase transport of particulates in horizontal pipes with inner longitudinal spiral slots Hossein Rahmanian- Koushkaki a, Seyed Hossein Karparvarfard a,*, Mohammad Hossein Raoufat a, Homayoun Emdad b, Khosrow Jafarpur b a b

Biosystems Engineering Department, College of Agriculture, Shiraz University, Shiraz, Iran Thermo- Fluid Engineering Department, College of Mechanical Engineering, Shiraz University, Shiraz, Iran

article info

This research was devoted to establishing a dimensionless model for estimating mixture

Article history:

friction factor based on the Buckingham Theorem. An experimental test-rig with

Received 4 March 2019

conveying pipeline of 40 mm I.D was designed, assembled and used to collect the data

Received in revised form

needed to model the mixture friction factor. Experiments were carried out in four levels of

6 August 2019

slot depths (0, 0.35, 0.55 and 0.9 mm), three levels of air pressure (100, 200 and 300 kPa) and

Accepted 20 November 2019

three pipeline lengths (3, 6 and 9 m). Conveying tests with three particulate materials of mung bean, corn, and polyethylene pellets were completed using three replications. Static air pressure across the conveyor, mass flow rate of discharge solids and inlet air mass flow

Keywords: Dense-phase

rate were measured during the tests. The model developed can estimate the mixture friction factor within the range of

Dimensional analysis

variables studied. The comparison between predicted and measured mixture friction factor

Longitudinal slots

showed no significant difference. According to values of R2 , RMSE, reduced chi-square and

Modelling

MRD parameters, mixture friction factor modelling of the pneumatic conveyor had good accuracy. Other results showed that the mixture friction factor was 65% less than that of pipelines with inner trapezoidal slots studied earlier, therefore lower upstream pressure was required for dense-phase pneumatic conveying of particulate materials in the new conveyor. The developed model showed acceptable error in predicting pressure drop from the experimental data. © 2019 IAgrE. Published by Elsevier Ltd. All rights reserved.

* Corresponding author. E-mail address: [email protected] (S.H. Karparvarfard). https://doi.org/10.1016/j.biosystemseng.2019.11.015 1537-5110/© 2019 IAgrE. Published by Elsevier Ltd. All rights reserved.

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Nomenclature Symbols A Pipe cross section area (mm2 ) Total cross section area of air-slots in a Aair broached pipe (mm2 ) Total cross section area of lands in a broached Apipe pipe (mm2 ) C Intercept of logarithmic line (constant) D Pipe inner diameter (mm) F Function Fr Froude number (dimensionless) g Gravitational acceleration (ms
2 ) H Depth of inner slots (mm) L Pipe length (m)  Air mass flow rate ( kg s
1 ) M a  Solid mass flow rate ( kg s
1 ) Ms N Slope of logarithmic line (constant) P Air pressure (Pa) Re Reynolds number (dimensionless) Superficial air velocity ( m s
1 ) Va Solid velocity ( m s
1 ) Vs X Small base of a trapezoid slot (mm) Y Long base of a trapezoid slot (mm) Abbreviations ABS Acrylonitrile butadiene styrene DAQ Data acquisition system db Dry basis GUI Graphical user interface MRD Mean relative deviation RMSE Root mean square error SDCV Solid discharge control valve SLR Solid loading ratio (dimensionless) Greek letters D Differential Bed porosity (dimensionless) εb Friction factor due to the air (dimensionless) la lm Mixture friction factor (dimensionless) Solids friction factor (dimensionless) ls ma Air viscosity (Pa s) Air density ( kg m
3 ) ra Bulk density of solids ( kg m
3 ) rb 4 Function Reduced Chi-square c2 j Function

1.

Introduction

A pneumatic conveying system uses air or some other nonreacting gases to transport particulates in pipelines. Gas pressure (either positive or negative), imparts enough energy to the particles to drag them along in scattered, agglomerated or continuous mass of solids (Mills, 2016). This conveying procedure was developed in the middle 19th century for transportation of grains (Klinzing, 2018a). This technology is

widely spread across a variety of industrial applications, such as livestock feeding (Aarseth, 2004), drying (Bunyawanichakul, Walker, Sargison, & Doe, 2007; Sampaio, Nogueira, Roberto, & Silva, 2007), rice polishing (Someswararao, Mahato, Namgial, Huda, & Das, 2017), pharmaceutical, cement, and transport of other solid products. The applications of pneumatic conveying is expanding due to low maintenance, low manpower costs, ease of control, automation and being environmental friendly (Ji, Liu, & Li, 2018; Klinzing 2018b; Klinzing, Rizk, Marcus, & Leung, 2010). It is expected that growth rate of pneumatic conveying industry will be US $ 30 billion by 2025 (Klinzing, 2018b). These systems, operating in under pressure or vacuum, are generally classified into two major categories, dilute-phase and dense-phase flow. Conventional pneumatic conveyors especially those used for handling of agricultural grains are frequently operated under dilute-phase regime. In this regime the transported material is suspended in the carrier gas with relatively high velocity. This causes pipeline erosion, high power consumption and particle degradation. For overcoming these problems, it is usually recommended that the pneumatic conveying systems are operated under dense-phase regime. In this phase, the mixture velocity is low which results in dramatic improvements in degradation of particles and pipeline erosion when compared to dilute-phase transport. One of the most common criteria for discriminating between dilute and dense-phase pneumatic conveying is solid loading ratio (SLR). It is a dimensionless parameter which is defined as the ratio of solid mass flow rate to gas mass flow rate. When this ratio is less than 15, the system is considered to operate in dilute-phase. Above that value the system operates in dense-phase mode (Mills, 2016). According to the literature there is a broad classification of various dense-phase conveying modes through horizontal pipelines (Dhodapkar, Jacobs, & Hu, 2006). Extrusion flow or ultra-dense plug is a component of this classification. As the name implies, material completely fills the conveying pipeline and moves in an extrusion flow pattern. These pneumatic conveyors are able to convey materials as an ultra-dense plug and the plug extends for the length of conveying pipeline. These systems are suitable for conveying granular materials in which a certain amount of conveying gas can permeate through the material body. By using extrusion flow regime in pneumatic conveying systems, it is possible to obtain solid loading ratios of the order of 500. Using extrusion flow, extremely low gas flow rates are required for transporting materials. In addition, using this technique considerable particle degradation has not been reported for very brittle materials (Klinzing et al., 2010). The equivalent total pressure drop of a pneumatic conveyor through a straight horizontal section of conveying line is a combined result of frictional forces due to flow of gas and solids. These forces include gas-to- pipe wall friction, solids-to-pipe wall friction, solids-to-gas friction, and solidsto-solids friction (Klinzing et al., 2010; Setia, Mallick, Pan, & Wypych, 2016). To estimate pressure drop, two different approaches have been used by the investigators. In the first of these

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approaches, the gas-solid flow was considered as a mixture and the relevant parameters were defined with respect to it (Jafari, 1976; Weber, 1991). Using the mixture friction factor (lm ), the total pressure drop can be presented as below: DP ¼

lm rb LV2s 2D

(1)

According to Eq. (1), reducing the mixture friction factor, will result in reducing the pressure drop. In the second approach, the friction factor of the gas-solid mixture is considered as a combination of two friction components, air only friction factor (la ) and solids friction factor (ls ). In other words the pressure drop is sum of losses due to air and due to solids and can be expressed by Eq. (2) (Barth, 1958). DP ¼

ðla þ SLRls Þra LV2a 2D

(2)

Originally, this equation was employed for dilute-phase conveying of particles. However, a consider number of researchers (Jones & Williams, 2003; Mallick, 2009; Pan & Wypych, 1992; Segmaier, 1978; Weber, 1981) have also applied this expression to predict the pressure drop of densephase pneumatic conveying systems. Ratnayake (2005) has implied that Eq. (1) can estimate the pressure drop in pneumatic conveyors with good accuracy especially for short pipeline lengths. In engineering, one of the most useful analytical methods to develop predictive models for studying a process or a phenomena is dimensional analysis. It is defined as the mathematical theory of functions that is characterised by a generalised type of homogeneity. The law of dimensional homogeneity guarantees that every additive term in an equation shall form a dimensionless term just like other terms forming the equation. In this technique, all relevant physical quantities that influence the phenomena are identified by researcher (Sedov, 1993; Streeter, 1983). These quantities are combined to form dimensionless groups. The dimensionless products are referred to as pi-terms (P- terms). The basis of dimensional analysis technique is Buckingham's Pi-theorem. This theorem states that the number of pi term (s) required to express a relationship between variables is equal to the number of variables involved in the process minus the number of reference dimensions required to express those variables. The importance of dimensional analysis technique is its ability to classify equations, convert equations from one system of units to another and develop prediction equations. In addition this technique can simplify a given problem described by a certain set of variables by reducing the number of variables that need to be considered. The new variables are dimensionless products of the original variables. The outcome of applying dimensional analysis to a problem is savings in both cost and labour during the experimental determination of the functions (Langhaar, 1980; Murphy, 1950). Many investigators have focused on pneumatic conveying of granular materials in horizontal pipelines using dense-phase mode including the works of Behera, Agarwal, and Jones (2015); Setia et al. (2016); Kaur, Mittal, Mallick, Pan, and Jana (2017), are of interest. In most of these studies, the friction factor was

considered as an important parameter in the performance of the system. The solid loading ratio and the Froude number of the air are the most important dimensionless numbers available in the literature for describing the conveying characteristics of pneumatic conveyors (Behera, Agarwal, Jones, & Williams, 2013; Jones & Williams, 2003; Keys & Chambers, 1993; Mallick, 2009; Pan & Wypych, 1992; Wypych, 1989). One of the earliest attempts in this area, was reported by Jafari (1976) and Jafari, Clarke, and Dyson (1981) who identified the parameters that affect the flow in the horizontal pipelines. In their studies parameters affecting the flow were arranged into several dimensionless groups. Finally a dimensionless linear equation was derived based on Buckingham's Theorem which represent mixture friction factor based on independent dimensionless groups. The derived equation was as follow: 2 

8:731 4




2:180

ðM s =M a Þ

f m ¼10

    30:989 L 
6:665 εb
1:69410
3 D
0:5 5  FrÞ  10 (3)

In this equation, the mixture friction factor was shown by fm . Raoufat and Clarke (1998) have conducted further research on packed-bed flow of agricultural particles to a commercial rate by scaling up the rig developed by Jafari (1976). Karparvarfard (1997) looked for a method for pneumatic conveying under dense-phase mode by creating inner slots in pipes along conveying line. Air passes through the slots performs like a lubricant layer. As a result, longer conveying distance for similar head pressure is possible. For this purpose, a test-rig of pneumatic conveyor was designed and developed. Furthermore, a dimensionless equation was presented to predict mixture friction factor. In this formulation, the new dimensionless group (H D) was included into dimensionless groups. This correlation was obtained: fm ¼104:757 2 





 4ðM s =M a Þ


2:145


0:776

ðFrÞ


0:069ðεb Þ
0:001

10

L D

  30:865


8:213

H D

5

(4)

Experimental results showed that at the same conditions the mixture friction factor was 20% less than that of pipes without inner slots. Karparvarfard (2004) calculated optimum depth of inner rectangular grooves as 2.8 mm. By increasing the depth of grooves, the flow regime was turned from dense into dilutephase. To consider the effect of shape of inner air passage grooves in order to increase the efficiency of pneumatic conveyor, Karparvarfard and Vakili Farahani (2010) designed a new set up of conveying materials in dense-phase regime. The pipelines of this rig had longitudinal trapezoidal slots. By increasing the cross section of air passages, the mixture friction factor was 40% less than in the Karparvarfard (1997) research. In addition, they defined two new parameters, Aair and Apipe , which were cross section area of air-slots and cross section area of lands in the broached pipe, respectively. Finally they derived a dimensionless equation for predicting

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the mixture friction factor based on dimensional analysis. In air ) was used: Eq. (5), there a new dimensionless group (AApipe

referred in the literature (Raheman & Jindal, 2001a; Sharma, Mallick, Mittal, & Pan, 2018; Wypych & Yi, 2003).

2

30:871     L
8:213 H
0:293 Aair 6 7
0:069ðε Þ
0:001 b   D D Apipe 6 7 fm ¼ 105:539 6ð3:6M s =M a Þ
2:145  ðFrÞ
0:776  10 7 4 5

The optimum depth of trapezoidal slots were 1.2 mm. In other words, by changing the shape of slots from rectangular to trapezoidal and at equal upstream air pressure, the mixture friction factor and the optimum depth were decreased. Since internal air-slots are crucial in decreasing mixture friction factor, the aim of the present research was to establish a dimensionless equation for predicting mixture friction factor for pipes with inner longitudinal spiral slots and to compare the resulting friction factor with previous researches at equal upstream air pressure.

2.

 DP D L

lm ¼

0:5rb V2s

(10)



Vs ¼

Ms rb A

(11)

By considering three dimensions of variables in Eq. (6) namely, M (mass), L (length) and T (time), applying Buckingham's Theorem and combination the above parameters, following dimensionless groups or P-terms were found to describe the mixture friction factor in horizontal pipeline. P1 ¼ lm

Theory and development of the model

(5)

(12)



Due to the complex nature of the two-phase flow in densephase conveying of particles and many parameters which had effective roles in this phenomenon, it was decided to use dimensional analysis technique for modelling the mixture friction factor. There were 13 effective quantities in this research which could be presented in the explicit form of Eq. (6).     4 lm ; M s ; M a ; Va ; D; L; g; ra ; εb ; H; ma ; Aair ; Apipe ¼ 0

P2 ¼

Ms 

(13)

Ma Va P3 ¼ pffiffiffiffiffiffiffi gD

(14)

P4 ¼ εb

(15)

(6)

One of the effective parameters in this equation is superficial air velocity (Va ). It was calculated based on Eq. (7) (Mills, 2016). 

Va ¼

Ma ra A

(7)

Schematic view of the cross section of a broached pipe with 20 internal air-slots is shown in Fig. 1. As it can be seen in Fig. 2, the cross section of a spiral slot is nearly like trapezoidal with x and y as bases of a trapezoid. Schematic views of the two parameters Aair and Apipe are shown in Fig. 3. These two parameters were calculated as Eqs. (8) and (9). i hx þ y  1:5 Aair ¼ 20 2 Apipe ¼

 p ðD þ 2HÞ2
D2
Aair 4

(8)

(9)

The mixture friction factor was determined by Eq. (10) (Weber, 1991; Raoufat & Clarke, 1998; Karparvarfard & Vakili Farahani, 2010). This equation is the rewritten form of Eq. (1) ), according to lm and comprised by pressure gradient (DP L pipeline inner diameter (D), solids bulk density (rb ) and solids velocity (Vs ). The solids velocity was calculated by Eq. (11) as

Fig. 1 e Cross section of a broached pipe in which a land and a slot is shown.

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Fig. 2 e View of a slot with 0.35 mm depth. x: small base of the trapezoid, y: long base of the trapezoid Dimensions in mm.

Fig. 3 e Schematic views of Aair and Apipe in a broached pipe.

L P5 ¼ D

(16)

(17)

 L H Aair ; Re ¼ 0 j lm ; SLR; Fr; εb ; ; ; D D Apipe

Aair Apipe

(18)

The P-terms were organised into groups of dimensionless dependent and independent quantities as Eq. (22).

ra Va D ma

(19)

H P6 ¼ D P7 ¼

P8 ¼

ratio, the seventh P-term corresponds to ratio of Aair to Apipe , and the eightieth P-term known as the air Reynolds number. So, it can be rewritten as Eq. (21).

! Va L H Aair ra Va D ; j lm ;  ; pffiffiffiffiffiffiffi; εb ; ; ; ¼0 D D Apipe ma M a gD

(22)

To collect appropriate data set for developing the desired model, a new test-rig was developed and used. The test-rig and equipment used are described in the following section.

Thus, Eq. (6) can be rewritten as Eq. (20): 

Ms

 L H Aair ; Re lm ¼ f SLR; Fr; εb ; ; ; D D Apipe

(21)

(20)

The first P-term, known as mixture friction factor, the second P-term corresponds to the mass flow rate ratio between the solids and the air known as SLR, the third P-term known as Froude number, the forth P-term represents porosity of the bed, the fifth P-term denotes the ratio of pipeline length to the pipeline internal diameter, the sixth P-term represents depth of slots to internal diameter of pipe

3.

Materials and method

The test-rig consisted of a pressurised blow tank, a bend, conveying pipe and a solid discharge control valve (SDCV) at the end. In the following sections the test-rig components are introduced. As the most important objective in this investigation was to see the effect of spiral slots throughout the conveying pipe

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on the flow characteristics and pressure drop, design and fabrication of certain tools to produce desired spiral slots in the inner wall of the pipes was given outmost importance. This section introduces the technique and tools used to produce spiral slots. To carve spiral slots inside the conveying pipelines, a broaching tool was designed and fabricated. Then, by connecting the broached pipes to a pneumatic conveying test-rig of granular materials, the performance of these pipes was evaluated under dense-phase regime. The specifications of broaching machine and test-rig are as follow.

3.1.

Broaching tool

A broaching tool was designed, fabricated and used to cut spiral slots inside the conveying ABS pipes. Figure 4 illustrates the schematic view of the tool. In this machine, a special device named as broach could be pulled inside the pipe fixed in a fixture. The machine basically consisted of three components, carving unit and two units for linear and rotational motions. Carving unit included the broaches made of steel alloy. The broaches are disk type with a root diameter of 37.5 mm and thickness of 8 mm. In addition, the broaches have two angles, attack angle of 15+ and clearance angle of 10+ . Longitudinal bevel slots with angle of 30+ into horizontal axis were made by milling machine along the separated broaches with the minimum and maximum depth of slots. A total of 20 slots were carved on each broach and width of each slot was 1.5 mm. By changing the broaches, the depth of the slots could be selected. The linear and rotational motion units interact with each other. An electric motor with nominal power of 746 W was used as a source of power

generation. A reduction gearbox was attached to the electromotor to increase the output torque of the electric motor. A ball screw unit was attached to the gearbox to convert rotational motion to linear motion. A steel shaft with three spiral machined slots on its periphery was attached on the ball screw. At the end of this shaft, a broach was installed. The spiral pitch of the machined slots was 260 mm. By moving the shaft inside a guide with three balls located in the spiral machined slots, the shaft and consequently the broach could have linear and rotational motions at the same time. So the apparatus could carve and 20 longitudinal spiral slots were made inside the pipe, as it was mentioned in the theory section.

3.2.

Experimental test-rig

Schematic layout of the experimental test-rig is shown in Fig. 5. This rig was operated under positive pressure and in a batch mode. The main components of the rig were conveying pipeline, air supply unit, a blow tank, a SDCV, a receiving hopper and a data acquisition system. The conveying pipeline incorporates a straight section made from ABS polymer with nominal inside diameter of 40 mm and outside diameter of 50 mm and a 90+ bend with radius of curvature of 250 mm. At the beginning of conveying line, a bottom discharge type blow tank having 0.21 m3 volume was used as a feeder to pressurise the conveying material. The air was supplied by a reciprocating compressor with an air tank of 2 m3 and maximum delivery pressure of 700 kPa. The compressor was equipped by a dehumidifier and a pressure control valve to supply air at the desired pressure level for each test run. A SDCV (Fig. 6) similar to the ones which were used in previous

Fig. 4 e Schematic view of the broaching tool. 1- Broach; 2- Pipeline fixture; 3- Steel shaft with spiral machined slots on its periphery; 4- Guide; 5- Ball screw system; 6- Reduction gearbox; 7- Electromotor.

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Fig. 5 e Schematic view of the test-rig. 1- Compressor; 2- Pressure control valve and dehumidifier; 3- Orifice plate; 4- Blow tank; 5- Bend; 6- ABS Pipeline; 7- Pressure transducer; 8- SDCV; 9- Receiving hopper.

Fig. 6 e Layout of the SDCV.

researches (Jafari, 1981; Karparvarfard & Vakili Farahani, 2010; Klinzing et al., 2010; Raoufat & Clarke, 1998) was accommodated at the end of pneumatic conveyor to control the flow of mixture and maintain a very smooth dense flow of solids under a wide range of air pressures and solids flow rates. For each test, the blow tank was filled with desired particulate solid and required air pressure was established throughout the conveyor while the SDCV was at closed position. As soon as air pressure along the pipeline was established, the SDCV is slowly opened to let the mixture flow along the pipeline and exit at the discharge end. Essentially SDCV included a conical probe which could move forward and backward controlled by a handle. At the start of a conveying cycle, the probe was adjusted to the desired position by setting the SDCV, leaving sufficient clearance for air to escape while the solids were retain and no solids could be discharged into the receiving hopper. Once the pipeline was full of materials and desired pressure was established, the conical probe was pulled manually by the handle and the materials was allowed to flow and discharge into the receiving hopper.

The receiving hopper was mounted on a load cell (model: L6G, manufactured by Zemic, China). The solids mass flow rate was monitored and recorded by the load cell and the data acquisition system. Air mass flow rate was measured continuously by an orifice plate with D and D/2 pressure tappings (ISO Standards, 2003). Air temperature and humidity at upstream tapping was also recorded. In summary, average mass flow rates of air and solids were calculated and recorded for each test run. The static air pressures along the pipeline were recorded by piezo-resistive pressure transducers (model: HOTF0010FLCK, manufactured by Hogller, Germany). For uniformity of pressure readings at each section of conveying pipelines, 10 holes of 1 mm I.D were drilled around the pipelines at one metre intervals. A calming collar which was made from polyethylene was installed around each section and covered the holes. Single transducer was then connected to the collar and measured the static air pressure. In addition, one transducer was installed on the blow tank and two were installed across the orifice plate. Specifications of pressure transducers and load cell used are listed in Table 1.

3.3.

Instrumentation system

A user-friendly instrumentation system was developed to monitor and record output signals from the transducers. Basically, it comprised of five components: a power supply unit, a portable data acquisition (DAQ) unit, pressure and load cell transducers, a portable PC and a graphical user interface (GUI). The power supply was used to provide excitation to both the acquisition unit and the transducers. The data acquisition unit consisted of an electronic circuit and a portable PC linked together via a USB cable. The circuit was developed based on STM-32 microcontroller (ARM category,

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Table 1 e Technical specifications of the transducers used in this study. Name of the transducer

Usage

HOTF0010FLCK, Piezo- resistive pressure transducer L6G, Loadcell

Specification

Measurement of air pressure

range: 0e1000 kPa (gauge), output: analogue, 0e5 V (DC), accuracy: % of the range < 0.5, sensitivity: 0.005 V =kPa capacity: 150 kg, full scale output: 2.029 mv =v, precision grade: C3

Measurement of solid material mass

manufactured by STMicroelectronics, Switzerland) with 80 input/output (I/O) ports and 16 analogue to digital (A/D) converters. The transducers were connected to the circuit. By clicking START button on the GUI, sampling data from the transducers were started every 0.1 s. The output signals of the transducers were sent to the microcontroller and after processing operations, the static air pressure and solids mass flow rate quantities were displayed on the GUI unit and saved as an excel file in the working directory of the instrumentation system.

into the hopper. During conveying tests, the static air pressure along the pipeline, mass flow rate of discharge solids and inlet air mass flow rate were recorded by the DAQ.

3.4.

 N    L H Aair f5 f6 f7 ðReÞ lm ¼ C f1 ðSLRÞf2 ðFrÞf3 ðεb Þf4 D D Apipe

Tests procedure

Conveying tests were conducted using three granular materials, mung bean (Vigna radiata L.), corn (Zea mays L.) and polyethylene pellets with different configuration of the testrig. The treatments were four levels of slot depth (0 (without inner slots), 0.35, 0.55 and 0.9 mm), three levels of air pressure (100, 200 and 300 kPa) and three levels of pipeline length (3, 6 and 9 m). Each of the experimental test was conducted in triplicate. Before starting the experiments, the materials were sieved with standard screens to have a fairly uniform particle size distribution. The measured physical properties of the products are listed in Table 2. The initial moisture content was measured using standard hot air oven method and represented based on dry basis (ASABE Standards, 2008). Geometric mean diameter, bulk density, true density and porosity of the test products were measured according to Mohsenin (1986). After filling the blow tank with the particles and sealing the tank top, air flow was started to fill the pipeline with the particles, while the SDCV was set to a position so that no solids could be discharged into the receiving hopper. The air pressure was set to the required level by the pressure control valve and then the SDCV was opened to discharge the solids

Table 2 e Physical properties of materials used for experiment. Physical Properties

Material

4.

For ease of application and simplifying Eq. (22), it was decided to select P-term forms which produce linear relationships with the dependent P-term. Therefore, the following model was considered.

8.20

7.00

… … ….

4.90

7.80

3.70

886.30 1500 40

780 1204.40 34

591 910 35

(23)

In the above equation, two constants C and N were not calculated until septet functions were derived. The linear form of Eq. (23) was obtained by taking logarithm of both sides, as Eq. (24): Logðlm Þ ¼ LogC N     L H Aair f5 f6 f7 ðReÞ þ Log f1 ðSLRÞf2 ðFrÞf3 ðεb Þf4 D D Apipe

(24)

In Eq. (24), N is the slope of logarithmic line and Log C is the vertical intercept. Until all functional relations were determined, the two constants were not calculated. According to the previous studies, each logarithmic functions has special C and N constants are listed in the following (Jafari, 1976; Karparvarfard & Rahmanian- Koushkaki, 2015; Karparvarfard & Vakili Farahani, 2010; Raoufat, 1995): Logf 1 ðSLRÞ ¼ N1 LogðSLRÞ þ C1

(25)

Logf 2 ðFrÞ ¼ N2 LogðFrÞ þ C2

(26)

Logf3 ðεb Þ ¼ N3 Log ðεb Þ þ C3

(27)

Logf 4

  L L ¼ N4 Log þ C4 D D

(28)

Logf 5

  H H ¼ N5 Log þ C5 D D

(29)



Mungbean Corn Polyethylene Pellets Initial moisture content (% db) Geometric mean diameter (mm) Bulk density (kg m
3 ) True density (kg m
3 ) Porosity (%)

Results

Logf6

 Aair Aair ¼ N6 Log þ C6 Apipe Apipe

Logf7 ðReÞ ¼ N7 LogðReÞ þ C7

(30)

(31)

The values of each constant N expresses the effect of each functional group on the value of mixture friction factor. All of the constants C could be combined into one value C for the

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141

final correlation, containing all functional relations. Therefore, the value of C could be neglected as below: Logf1 ðSLRÞ ¼ N1 LogðSLRÞ

(32)

Logf2 ðFrÞ ¼ N2 LogðFrÞ

(33)

Logf3 ðεb Þ ¼ N3 Logðεb Þ

(34)

Logf4

  L L ¼ N4 Log D D

(35)

Logf5

  H H ¼ N5 Log D D

(36)

 Aair Aair ¼ N6 Log Apipe Apipe

 Logf6

(37)

Logf7 ðReÞ ¼ N7 LogðReÞ

(38)

To use dimensional analysis technique, it is necessary to define the following terminology to avoid unnecessary restatements. First Residual ¼ Logðlm Þ
Logf1 ðSLRÞ

(39)

Second Residual ¼ First Residual
Logf2 ðFrÞ

(40)

Third Residual ¼ Second Residual
Logf3 ðεb Þ

(41)

 L D

(42)

Fourth Residual ¼ Third Residual
Logf4

 H D

Fifth Residual ¼ Fourth Residual
Logf5  Sixth Residual ¼ Fifth Residual
Logf6

Aair Apipe

(43)



Seventh Residual ¼ Sixth Residual
Logf7 ðReÞ

(44)

(45)

It is important to consider that the interactions among dimensionless groups were assumed to be neglected. For better understanding the dimensional analysis method, it is helpful to explain the residual concept. For example, in the first residual (Eq. (39)), the effect of the first function on the lm term has been removed for the establishment of the next function. For the other residuals from second through seventh, this concept would be repeated. For determining the functional relations, the following steps should be followed.

4.1.

Fig. 7 e Plot of Log (lm ) against Log (Fr) for fairly equal values of Log (SLR) relating to second replicate trials.

Analysis of the group f1 ðSLRÞ

Inspection of Eq. (24) indicates that the first and the second functional groups are both functions of air mass flow rate and their effects could be separated if one could be held constant while the other is varying. In order to determine the functions, the corresponding data of the second replication was used. It should be noted that this operation was used for

the data of the two remaining replications. Finally, by averaging each functional relation was derived. Further details will follow. Figure 7 shows a typical graph which is established by plotting the values of the Log (lm ) versus the Log (Fr) quantities. Each point of this figure represents a value for Log (SLR). The points of fairly equal Log (SLR) were connected together by solid lines. These lines are parallel to each other. Three solid lines are shown typically in this figure. A perpendicular dotted line connects origin to the solid lines, designated as vertical shift. The vertical spacing is defined as the distance between the origin and the solid lines, because of the effects of Log (SLR) on the Log (lm ) named as Log f1 ðSLRÞ. For determining this function, the vertical shifts could be plotted versus the corresponding values of the Log (SLR) as shown in Fig. 8. It is worth noting that each point on Fig. 8 corresponds to one solid line (one value of SLR) of Fig. 7. By applying same operations on data of the two remaining replications and averaging the constants N, final relationship for the first function is derived as follows: Logf1 ðSLRÞ ¼
2:215LogðSLRÞ

(46)

f1 ðSLRÞ ¼ ðSLRÞ
2:215

(47)

4.2.

Analysis of the group f2 ðFrÞ

For analysis of the second group, the values of the first residual as defined by Eq. (39) were calculated and used for plotting Log f2 ðFrÞ versus the corresponding values of Log ðFrÞ as shown in Fig. 9. By averaging from the slopes of the three replicated measurements, the following relationship was found. Logf2 ðFrÞ ¼
0:615LogðFrÞ

(48)

f2 ðFrÞ ¼ ðFrÞ
0:615

(49)

4.3.

Analysis of the group f3 ðεb Þ

The values of the second residual as defined by Eq. (40) were calculated to plot Log f3 ðεb Þ versus corresponding values of the

142

b i o s y s t e m s e n g i n e e r i n g 1 8 9 ( 2 0 2 0 ) 1 3 3 e1 4 9

Fig. 8 e Relation between Log f1 ðSLRÞ and Log (SLR) for the second replicate trials.

Fig. 9 e Relation between Log f2 ðFrÞ and Log (Fr) for the second replicate trials.

εb (Fig. 10). After averaging the slopes of the three replications, this relation could be shown as: Logf3 ðεb Þ ¼ 1:460ðεb Þ f3 ðεb Þ ¼ 101:460ðεb Þ

4.4.

Analysis of the group f4

(50)



(51)

L D

For the present analysis, the values of the third residual as  defined by Eq. (41) was calculated to plot Log f4 DL against DL for the second replication of the conveying tests (Fig. 11). By averaging the slopes of the three replications, the functional  relationship of f4 DL was calculated as follows:   L L ¼
0:002 Logf4 D D

(52)

 
0:002 DL L f4 ¼ 10 D

4.5.

Analysis of the group f5

(53)

 H D

The values of the fourth residual was found based on Eq. (42)  H and used for plotting Log f5 H D against D(Fig. 12). The following relation was derived after averaging from the slopes of three replications. Logf5

  H H ¼
27:882 D D

 
27:882 H D H ¼ 10 f5 D

(54)

(55)

b i o s y s t e m s e n g i n e e r i n g 1 8 9 ( 2 0 2 0 ) 1 3 3 e1 4 9

Fig. 10 e Relation between Log f3 ðεb Þ and εb for the second replicate trials.

 Fig. 11 e Relation between Log f4

Fig. 12 e Relation between Log f5

L D

and

L D

for the second replicate trials.

 H D and

H D

for the second replicate trials.

143

144

b i o s y s t e m s e n g i n e e r i n g 1 8 9 ( 2 0 2 0 ) 1 3 3 e1 4 9

! Fig. 13 e Relation between Log f6

Aair Apipe

and

Aair Apipe

for the second replicate trials.

! 4.6.

Analysis of the group f6

Aair Apipe

(Fig. 14). As it is clear from Fig. 14, the Reynolds number has no significant effect on the lm . In other words, the effect of this group was not considered in the development of the dimensionless model.

At this stage, with respect to Eq. (43) and calculating the values ! of the fifth residual, the Logf6

Aair Apipe

was calculated. By plotting

! the values of the Log f6

Aair Apipe

4.8.

The overall dimensionless equation

versus the corresponding values Finally by substituting functions obtained in the above steps in Eq. (24), the following equation was obtained:

air (Fig. 13), the relation was established. It should be noted of AApipe

that this relation was derived after averaging from the three slopes of the three replicated measurements.

2



1:460ðεb Þ
0:002

6 Logðlm Þ ¼ LogC þ Log4ðSLRÞ
2:215  ðFrÞ
0:615  10

 Logf6

 f6

Aair Apipe



L D

 Aair ¼
0:715 Apipe

H D

(57)





L D

H D

1:460ðεb Þ
0:002


27:882

3N 7 5

(58)

Final form of the dimensionless correlation

The form of Eq. (58) can be written as follows by rearranging of this equation.

According to Eq. (44), the values of sixth residual was calculated and graph of Log f7 ðReÞ against f7 ðReÞ was plotted

6 lm ¼ 105:108 4ðSLRÞ
2:215  ðFrÞ
0:615  10

Aair Apipe

4.9.

Analysis of the group f7 ðReÞ

2


0:715

In the next step the logarithm values of the quantities in brackets were plotted against the corresponding values of the Log (lm ) and the two constants Log C ¼ 5:108 and N ¼ 0:909 were obtained as shown in Fig. 15. It should be noted that data from all experiments was used to plot this figure.

(56)

 A
0:715 A air Aair pipe ¼ 10 Apipe

4.7.




27:882




0:715

Aair Apipe

30:909 7 5

(59)

145

b i o s y s t e m s e n g i n e e r i n g 1 8 9 ( 2 0 2 0 ) 1 3 3 e1 4 9

Fig. 14 e Relation between Log f7 ðReÞ and Log (Re) for the second replicate trials.

Fig. 15 e Relation between Log (lm ) and dimensionless groups for all experimental trials.

4.10.

Conveyor design limitations

The range of values of dimensionless groups of Eq. (59) are mentioned in Table 3. The obtained dimensionless equation can be used to predict values of mixture friction factor for different conveying conditions but is limited to range of dimensionless groups studied experimentally.

5. 5.1.

Discussion Validation of the dimensionless model

It should be noted that 20% of experimental data were set aside for validation. The predicted data from the

Table 3 e Range of dimensionless groups considered in this research. Dimensionless groups lm SLR Fr εb L D H D Aair Apipe

Range of variation 0.204e304.973 15.218e241.346 4.845e86.026 0.34e0.40 74.441e225 0e0.0225 0e0.310

146

b i o s y s t e m s e n g i n e e r i n g 1 8 9 ( 2 0 2 0 ) 1 3 3 e1 4 9

Fig. 16 e Predicted mixture friction factor versus experimental mixture friction factor.

Fig. 17 e Distribution of residual error of predicted mixture friction factor versus experimental mixture friction factor.

dimensionless model were plotted against this set of experimental data. The points scatted around line 1:1 demonstrates the suitability of the obtained dimensionless model for predicting the mixture friction factor in pipelines with inner longitudinal spiral slots (Fig. 16). Figure 17 shows random distribution of the residual error of predicted mixture friction factor against experimental mixture friction factor. Due to this figure and no specific trend of residual error, the accuracy of the derived dimensionless model is concluded. In addition, four statistical parameters namely coefficient of determination (R2 ), root mean square error (RMSE), reduced chi-square (c2 ), and mean relative deviation (MRD) were used to judge the experimental findings and mode. In other words, the most suitable dimensionless model was identified based

on the highest value of the R2 , least value of the RMSE, least value of the c2 and least value of the MRD. These parameters could be calculated using the following equations.  R2 ¼

n    2 P lm exp;i
lm exp lmpre;i
lmpre i¼1

n  P

lm exp;i
lm exp

m  2 P

i¼1

RMSE ¼

c ¼

2

(60)

i¼1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n  2 1X lm exp;i
lmpre;i n i¼1

n  P 2

lmpre;i
lmpre

lm exp;i
lmpre;i

i¼1

n
r

(61)

2 (62)

b i o s y s t e m s e n g i n e e r i n g 1 8 9 ( 2 0 2 0 ) 1 3 3 e1 4 9

!

n

lmexp;i
lmpre;i 100 X MRD ¼ n i¼1 lmexp;i

(63)

In above equations n is the number of observations, r is the number of constants in dimensionless model. The lm exp;i is the i th experimentally mixture friction factor, lm exp is the average experimental mixture friction factor, lm pre;i is the i th predicted mixture friction factor and lm pre is the average predicted mixture friction factor. The values of R2 , RMSE and c2 were 0.94, 0.84, 1.01, respectively which reflect the accuracy of the dimensionless model. In addition, the value of MRD was 11.16%. According to previous investigations in pneumatic conveying area (Raheman & Jindal, 2001b) and other areas of modelling a process such as drying (Brooker, Bakker- Arkema, & Hall, 1992; Tang, Cenkowski, & Muir, 2004), the acceptable value of this parameter is below than 15%. In the present study, average of this value is 11.16%. Therefore, mixture friction factor modelling of the pneumatic conveyor has good accuracy.

Table 4 e Comparison of two sets of mixture friction factors (Present research and Karparvarfard & Vakili Farahani, 2010). Mixture friction factor values in present research

12.139 35.430 5.670 5.037 7.190 5.061 6.142 16.130 6.141

Mixture friction factor Mixture values for the friction factor pipelines with inner reduction (%) trapezoidal cross section 16.596 152.054 96.161 10.375 242.661 32.509 18.536 100.231 6.561

26.856 76.699 94.104 51.451 97.037 84.432 66.864 83.907 6.401

147

The deviation between the predicted and experimental values of the mixture friction factor are probably due to one or more of the following reasons; (1) the errors in measurement of input parameters of the dimensionless groups and actual performance of the conveyor, (2) the lack of accuracy of the equations for calculating of air and solids velocity, (3) inadequate precision of the equations for calculating of Aair and Apipe parameters.

5.2. Comparisons of values of mixture friction factor with those of similar findings Values of mixture friction factor from the dimensionless model were compared with similar findings concluded from a previous study by Karparvarfard and Vakili Farahani (2010). The results showed that on an average the friction factor is 65% less than that of pipelines with inner trapezoidal slots. In other words, the length of conveying materials can be increased without the need to increase upstream air pressure. Summaries of the data concerning the mixture friction factor in the pipelines with spiral slots (present study) and trapezoidal slots (earlier study Karparvarfard & Vakili Farahani, 2010) are presented in Table 4.

5.3.

Pressure drop along the pipes

According to recent research by Rahmanian- Koushkaki and Karparvarfard (2019), the optimum slot depth and length of conveying for the present conveyor was 0.35 mm and 9 m, respectively. Results of pressure distribution along the conveyor for the pipes without inner slots and with slots of 0.35 mm depth during conveying of mungbean seeds in the 9 m long test were compared. As can be seen in Fig. 18, pressure drop becomes larger from the pipeline entrance towards the outlet. Also, these plots indicate that the rate of increase in pressure drop depends on the applied upstream pressure. It could also be seen that for a nearly constant pressure drop, the broached pipeline with 0.35 mm slot depth was able to convey

Fig. 18 e Longitudinal pressure distribution for smooth and broached pipes with 0.35 mm depth of slots in the 9 m long test for conveying mungbean seeds.

148

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Table 5 e Comparison of measured and predicted pressure drop for conveying mungbean (of 9 m length and no inner slots). Serial No.





M a (kg s
1 )

M s (kg s
1 )

SLR

Experimental pressure drop (kPa)

Predicted pressure drop (kPa)

Error (%)

0.005 0.007 0.009

0.146 0.262 0.630

29.2 37.43 70

85 167 234

89.150 129.511 214.636

4.90 22.45 8.27

1 2 3

Table 6 e Comparison of measured and predicted pressure drop for conveying mungbean (of 9 m length and inner air-slots of 0.35 mm depth). Serial No. 1 2 3





M a (kg s
1 )

M s (kg s
1 )

SLR

Experimental pressure drop (kPa)

Predicted pressure drop (kPa)

Error (%)

0.006 0.007 0.010

0.257 0.387 0.842

42.833 55.285 84.2

90 170 251

120.7 206.587 366.266

34.11 21.52 45.92

higher rates of solid, in other words, the conveyor capacity was increased.

5.4. Comparison of predicted pressure drop with measured data The calculated mixture fiction factor from the dimensionless model can be used in Eq. (1) to predict pressure drop required for a specific condition within the range studied. Predicted and measured values of pressure drop for transport of mungbean in the 9 m pipe length without inner air-slots and with airslots of 0.35 mm depth are presented in Tables 5 and 6, respectively. The percentage error between experimental and predicted values of pressure drop is acceptable. A possible reason for such discrepancies is that the dimensionless model includes inlet conditions such as inlet air density and superficial air velocity, whilst these parameters vary along the length of the pipe.

Declaration of interests 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. The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Acknowledgments The work presented in this paper is funded by the Research Council of Shiraz University, Shiraz, Iran. The authors would like to express their appreciation to Shiraz University for the financial supports.

references

6.

Conclusions

A dimensionless model was developed for predicting mixture friction factor in pneumatic conveying of granular materials in horizontal pipelines with inner longitudinal spiral slots. Lower values of friction factor results in lower upstream pressure which is desirable. The results indicated that the model was capable of predicting the mixture friction factor with good accuracy. The developed model could be used for designing pneumatic conveyors of granular materials having characteristics arrangement within the range of dimensionless groups experienced in the present study. Results from comparison of mixture friction factor values calculated from experimental trials for spiral and trapezoidal inner slots showed that for similar conditions, the mixture friction factor in pipelines with spiral slots was 65% less than that for pipelines with inner trapezoidal slots. At the same conditions and pressure drop, the broached pipeline with slot depth of 0.35 mm was able to convey higher rates of solid mass flow compared to pipelines with no inner slots. The model is also used to predict pressure drop and compare with experimental data and shows acceptable discrepancies. The results of this study can be practically applied in to design dense-phase pneumatic conveyors to properly manage energy resources.

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