Coal Removal Using A Vacuum Cleaner Modelling The Suction Process of Coal Particles Using A Vacuum Cleaner

(1)

Coal Removal Using A Vacuum

Cleaner

Modelling The Suction Process of Coal Particles Using A

Vacuum Cleaner

Johan van Jole

Rep

o

rt

(2)

(3)

Coal Removal Using A Vacuum

Cleaner

Modelling The Suction Process of Coal Particles Using A Vacuum

Cleaner

Research Assignmnent

Johan van Jole

(4)

(5)

Assignment Description

This research assignment concerns pre-baked anodes, which are often used in electrochemical processes. These anodes are manufactured in ovens which are filled with coal. The top layer of coal needs to be removed in order to enable lifting the anodes out of the ovens. This research assignment focusses on the removal process of this layer of coal using a vacuum cleaner. The goal of this research assignment is to model the removal of the top layer of the coal. The eﬀects of changing the geometry of the equipment used to remove the coal with respect to the energy consumption of this process are studied using this model.

(6)

(7)

List of Figures

1 A schematic representation of the vacuum cleaner system. . . 4

2 A schematic representation of the calculation procedure. . . 6

3 An example of the intake (source:www.opensourcefoam.net). . . 9

4 Schematic representation of a gas cyclone (source: suvis-gmbh.de). . . 10

5 Deviation between the values for the coeﬃcient of drag. . . 14

6 Relation between the particle size and energy consumption. . . 26

7 Relation between the particle size and energy consumption zoomed in. . . 27

8 Relation between the particle density and energy consumption. . . 28

9 Relation between the inside pipe diameter and energy consumption. . . 30

10 Relation between the cyclone body diameter and energy consumption. . . 31

11 Relation between the time allowed to remove the crust and energy consumption. 32 12 Relation between the time allowed to remove the crust and power of the fan. . . 33

13 Relation between eﬃciency of the fan and the energy requirement. . . 34

(10)

(11)

List of Tables

1 Summary of the assumptions that are made in this model. . . 5

2 Parameters used to check the saltation velocity. . . 11

3 Calculation for the solid-to-wall friction value used in the worked example. . . 12

4 Verification of the solid-to-wall friction in the horizontal section of the pipe. . . . 13

5 Verification of the calculation procedure for the particle Reynolds number. . . 13

6 Verification of the calculation procedure for the drag coeﬃcient of a particle. . . 14

7 Verification of the Darcy friction factor. . . 15

8 Verification of the gas-to-wall friction. . . 16

9 Verification of the terminal velocity . . . 16

10 Calculation of the individual components in the worked example. . . 17

11 Verification of the pressure drop in the vertical ducting sections. . . 17

12 Verification of the characteristic velocity and the Stokes number of a gas cyclone. 18 13 Verification of the Euler number calculation. . . 18

14 Verification of the pressure drop across a gas cyclone. . . 18

15 Verification of the pressure drop across the filter. . . 19

16 Verification of the heat transfer calculation. . . 20

17 Verification for the pressure recovery due to heat transfer. . . 20

(12)

viii List of Tables

26 Values of the variables not changed in the fourth experiment. . . 30

27 Values of the variables not changed in the fifth experiment. . . 32

28 Values of the variables not changed in the sixth experiment. . . 34

29 Values of the variables not changed in the final experiment. . . 35

(13)

Chapter 1 Introduction

This assignment concerns the modelling of the suction process of coal particles using a vacuum cleaner. The components that make up the vacuum cleaner are listed in this chapter. The structure of this research is discussed in this chapter as well.

The top layer of coal in the oven in which the anodes are baked is removed with a vacuum cleaner. The vacuum cleaner consists of several components. The ducting system is used to transport the coal from the oven towards the gas cyclone. A fan creates a pressure diﬀerence across this ducting system which causes air and coal to move into the ducting system. The final component is a gas cyclone which is used to remove the particles from the flow, which deposits the coal into a container mounted below the gas cyclone.

The components of the vacuum cleaner are modeled in Matlab, this model is discussed in Chapter 2. This model calculates the energy requirement for removal of the top layer of coal for any given set-up. Experiments are performed in order to determine eﬀects of changes to the parameters with respect to the energy requirement of the coal removal process. The experiments and the results obtained in these experiments are discussed in Chapter 3. Chapter 4 concludes this research.

(14)

(15)

Chapter 2 Model of the Concept

This chapter discusses the simulation model of the vacuum cleaner that removes the top layer of coal from the ovens in which the anodes are baked. The first section concerns the assumptions on which this model is based. The subsequent sections concern the validation and verification of the model. The results of the simulation are covered in Chapter 3.

The performance of the vacuum cleaner is studied using a model representing all the com-ponents that are used in the system. All comcom-ponents are modelled in Matlab, its code is included in Appendix A. Figure 1 shows a schematic representation of the vacuum cleaner and the position of the components relative to each other.

The vacuum cleaner will be mounted to the crane which lifts the anodes out of the ovens. This enables the vacuum cleaner to reach all the ovens in the production facility. The crane and the frame which connects the vacuum cleaner to the crane are depicted in orange in Figure 1. The oven, anodes and coal can be seen in this figure as well, along with the intake of the system. The air and coal particles enter the system through this intake, after which they continue their path through the ducting system (which is depicted in green). In the gas cyclone (coloured red), the coal particles are removed from the gas stream. The particles are collected at a container below the gas cyclone (which is in blue). The gas stream exits at the top, continues through a filter (in purple) to a fan (which is coloured orange).

(16)

4 Model of the Concept

Figure 1: A schematic representation of the vacuum cleaner system.

2-1

Assumptions

This section covers the assumptions that are at the base of the Matlab model. These assump-tions are listed in order to give the reader an insight into the limits of this model.

In reality, the physical properties of the coal particles are described by a probability distribu-tion. The coal particles in this simulation model are represented by solid spheres which have the same radius, therefore the properties assigned to the particles are constant throughout the model (all particles are equal in size and shape). It is assumed that this simplification doesn’t have an impact on the validity of this model.

The particle density of both the coal as well as the air are considered to be constant. The feed rate of the coal is assumed to be constant as well. Also the ambient pressure at the inlet and exit of the system are considered to be equal.

Another assumption of this model is that the particles and the gas accelerate before they reach the air intake. Therefore there is no pressure drop across the ducting system caused by the acceleration of both the air and the particles.

(17)

2-1 Assumptions 5

The simulation model employs Hinkle’s correlation to estimate the solid-to-wall friction in the horizontal part of the ducting system. Hinkle assumed that the particles lose momentum by colliding with the wall of the pipe. The particles subsequently need to be reaccelerated, creating a pressure drop. The main assumption in his model is that the particles bounce oﬀ the walls immediately after contact, instead of sliding along the wall.

The gas-to-wall friction and the solid-to-wall friction are assumed to be independent. This means that the presence of particles is assumed not to have an eﬀect on the gas-to-wall friction. The flow is assumed to be turbulent and therefore the equations for turbulent flow in a pipe are used to calculate the pressure drop due to the gas-to-wall friction. Any compressibiltiy eﬀects of the air are neglected.

Finally, the exchange of heat between the hot air inside the ducting system and the environ-ment is assumed to be an isentropic process.

The assumptions are summarised in Table 1. Some assumptions need to be checked during the simulation, to which of the assumptions this applies can be found in the third column of the aforementioned table.

Table 1: Summary of the assumptions that are made in this model.

Assumption Description Checked by

Matlab model? 1 Particles all have the same properties, instead of

hav-ing a distribution function.

No

2 properties of air and coal are constant No

3 Ambient pressure at inlet and outlet of system are equal

No

4 Feed rate of coal is constant No

5 Acceleration of particles and gas occurs before they enter the ducting system

No 6 Pressure drop solid-to-wall friction in horizontal

sec-tion of pipe is caused by loss of momentum of the particles due to collisions with the wall.

No

7 Gas-to-wall friction and solid-to-wall friction are inde-pendent

No

8 Gas flow is turbulent Yes

9 Compressibility eﬀects of air can be neglected Yes 10 The exchange of heat is an isentropic process No

(18)

2-2

Validation

This section considers the validation stage of the simulation model that is implemented in Matlab. The measurement uncertainties are listed where applicable and the range in which this model can be used is evaluated. The calculation procedures that are used in this model are explained in this section as well.

The vacuum cleaner is a form of pneumatic transport operating under negative pressure (the pressure at the intake is lower than the ambient pressure). Therefore only dilute phase flow is possible [2]. In dilute phase flow, the particles are fully suspended in the gas flow.

Figure 2 shows a schematic representation of the calculation procedure that is implemented in Matlab. The calculation procedures shown in this figure are explained in this section. Fore readability purposes the pressure drop across the ducting system, the operation of the gas cyclone, the pressure recovery and the parameters relating to the fan are included in seperate subsections.

(19)

2-2 Validation 7

Dilute phase flow transportation dictates that the particles must remain in suspension across the ducting network. Particles remain in suspension as long as the velocity of the fluid is suﬃcient. When the velocity drops below a treshold, not all particles can be suspended and dilute phase flow is not possible anymore. Two of these thresholds exist; one for vertical transport of particles and one for horizontal transport of particles. The vertical transport threshold is called the choking velocity, this is the minimal velocity required to operate the dilute phase flow at a given feed rate of solid material along a vertical pipe [2]. At horizontal transportation, the threshold is called the saltation velocity.

Because this system consists of both horizontal and vertical pipes, both the saltation and chocking velocity need to be taken into consideration. The saltation velocity is always larger than the choking velocity [2], therefore only the saltation velocity is taken into account. However, it is currently not possible to establish the exact conditions under which saltation will occur. The saltation velocity can be estimated using the Rizk correlation. The average error of the Rizk correlation is reported to be± 50 percent [2], therefore the superficial velocity is taken as the calculated saltation velocity multiplied by 1.5. This superficial velocity is the minimal velocity of the gas and particles combined, with this superficial velocity saltation does not occur and dilute phase flow is maintained across the ducting network.

2-2-1 Pressure Loss Across Ducting System

In order to determine the power of the fan which needs to be installed, the pressure loss across the ducting system is calculated. The pressure loss at horizontal sections consists of the pressure loss due to the gas-to-wall friction as well as the pressure loss due to solid-to-wall friction. The solid-to-wall friction is estimated using the correlations provided by Hinkle [2]. Two correlations from Hinkle are used; the correlation to estimate the particle velocity during horizontal transportation and the correlation for the solid-to-wall friction in horizontal pipes. The voidage in the pipe (percentage of volume not occupied by the coal particles) must be known in order to use this correlation. The voidage in the pipe can be established using equations for mass conservation.

Another parameter that needs to be calculated is the drag coefficient of the coal particles. The drag coefficient is used to determine the magnitude of the resistance force acting on the particles. This resistance force is generated by the gas stream. The drag coefficient is found using the Haider and Levenspiel correlation for the drag coefficient for spheres [2]. The correlation for the drag coefficient is valid for any Reynolds number between 0 and 200.000. The solid-to-wall friction can be estimated when the drag coefficient, the voidage in the pipe and the particle velocity are known.

(20)

The gas-to-wall friction in both the horizontal and vertical sections is calculated using the Weisbach correlation for the head loss for turbulent flow in a pipe [3]. In order to use this correlation, the darcy friction factor needs to be calculated. The darcy friction factor can be calculated using the Colebrook equation, which has a reported accuracy of± 15 percent [3]. Because the Colebrook equation is not easy in use the friction factor is often determined using one of two alternative calculation procedures. The first procedure uses the Moody chart for pipe friction (a chart which displays the solution to the Colebrook equation for several values of the Reynolds number and the relative roughness of the pipe). The darcy friction factor can simply be read from this graph. The Reynolds number is a ratio between inertial and viscous forces and is used to charcterize the flow. Another way to obtain the darcy friction factor is to use the Haaland equation. The value for the darcy friction factor calculated by the Haaland approximation varies less than 2 percent compared to the value obtained by the Coolebrook equation [3]. In this model, the Haaland estimation for the Coolebrook equation is implemented to calculate the darcy friction factor.

The pressure drop in the vertical pipe sections consists of the gas-to-wall friction, solid-to-wall friction and the change in static head of the gas and particles. The static head of the gas is the pressure loss due to the forces required to transport the mass of the air upwards. The same holds for the change in static head of the particles. The solid-to-wall friction for the vertical sections is estimated using the Konno and Saito correlation [2]. The gas-to-wall friction for the vertical pipe section is calculated using the same method used to calculate the gas-to-wall friction for the horizontal pipe sections. In order to calculate the pressure drop due to the change in static head of both the gas and the particles, the voidage in the vertical section of the pipe needs to be known. The voidage can be evaluated using the equations for the particle velocity and the mass flux. The mass flux is the amount of mass that is transported across a surface within a certain time interval. The mass flux and the superficial flow velocity in the horizontal and vertical sections are equal because the pipes are equal in dimension. This means that the voidage can be calculated, which is used in the equations for the static head of both the gas and the particles.

The pressure drop caused by bends consists of the pressure drop in the bend itself as well as the influence on the flow both upstream and downstream of the bend [4]. The amount of equations available for the pressure drop caused by bends is limited. One empirical model is able to calculate the pressure drop caused by the bend which includes the eﬀects on the flow both upstream and downstream of the bend [5]. However, this model can only be evaluated at two discrete values of the ratio between the radius of the curve and the diameter of the pipe. This equation cannot be used in this model, because the diameter of the pipe will be changed during the simulation. The pressure drop due to the presence of bends is therefore not expressed with a theoretical model but by an estimate (which is common practice in industry) [2]. The pressure drop around a bend is estimated to be equal to the pressure drop of 7.5 m vertical pipe.

(21)

2-2 Validation 9

2-2-2 Pressure Recovery

The air that enters the system is assumed to be 300◦C, this is considerably higher than the environmental temperature (around 25 ◦C). Therefore heat is exchanged between the ducting system and the environment. This eﬀect is modelled by the cooling laws of Newton [3]. The temperature is evaluated at the inlet and exit of the ducting system. The drop in temperature causes a change in pressure, which is estimated using the ideal gas law for an isentropic process.

Also an intake is used to lower the pressure at the inlet, the shape of this intake is included in Figure 3. The air travels through the intake which speeds up the air, this causes the pressure to drop. This pressure drop helps to sustain the pressure diﬀerence across the system. The pressure at the leading edge of the inlet is assumed to be equal to the ambient pressure.

Figure 3: An example of the intake (source:www.opensourcefoam.net).

2-2-3 Particle Removal

Gas cyclones are often used to separate solid particles from a gas stream. A schematic representation of a gas cyclone is included in Figure 4. In a gas cyclone, a vortex is created, which causes a centrifugal force to act on the particles. The centrifugal force moves the particles outwards until they collide with the cyclone wall and fall down. At the bottom of the gas cyclone, the direction of the vortex is reversed so that the air exits at the top of the cyclone while the particles exit at the bottom. Typically, gas cyclones are used to separate particles larger than 10 µm. Smaller particles can be removed using a filter, located at the gas exit of the cyclone [2].

(22)

Figure 4: Schematic representation of a gas cyclone (source: suvis-gmbh.de).

The pressure drop can be calculated using the equation for the Euler number. The Euler number is a measure of the performance of the gas cyclone. The Euler number can be estimated using an approximate emperical relation between the Euler number and the Stokes number. The pressure drop can be calculated from the Euler number [2].

The filter that is located at the gas exit of the cyclone creates an additional pressure drop. This filter prevents dust being blown into the environment. Because the coal particles are already removed, the pressure drop is only a funtion of the friction caused by the moving gas. The pressure drop across a filter can therefore be modelled by a loss coeﬃcient [3].

2-2-4 Parameters Relating to the Fan

The power required to drive the fan as well as the energy required to remove the layer of coal are calculated. The amount of energy required to remove the coal is used to determine which geometry values (for instance the diameter of the pipes) minimise the energy consumption. The installed power of the fan is calculated in order to be able to select a combination of fan and drive system. The pressure diﬀerence across the fan must be equal to the pressure drop across the system. The power transmitted to the fluid by the fan equals the volume flow rate times the pressure increase across the fan [3]. The power which is needed to drive the fan is equal to the power transmitted to the fluid divided by the eﬃciency of the fan.

(23)

2-3 Verification 11

2-3

Verification

This section concerns the verification of the Matlab model. This process checks whether the model which is described in the previous section is implemented correctly. This section is divided into subsections that cover the verification of the saltation velocity, the pressure drop due to solid-to-wall friction at the horizontal ducting sections, the drag coeﬃcient of the particles, the pressure drop due to gas-to-wall friction, the pressure drop at the vertical ducting sections, the pressure drop at the gas cyclone, the pressure recovery and the fan characteristics.

2-3-1 Saltation Velocity

The saltation velocity is checked using worked example 8.1 from Rhodes [2]. The parameters used in this calculation can be found in Table 2. The parameters entered in the model are carried over from the worked example. The mass flow rate of coal cannot be entered into the Matlab model and therefore the dimensions of the oven need to be adjusted in order to obtain the right mass flow rate.

The mass flow rate in the worked example is 900 kg/hr. The given particle density equals 2500 kg/m3 which makes the volume flow rate of particles 0.36 m3/hr. The time in which the layer of coal must be removed is set at 1 hr, therefore the volume of coal equals 0.36 m3. The thickness of the top layer is set at 0.1 m, this requires the area of the oven to be 3.6 m2_.

The the length and width of the oven are fixed at 3.6 m and 1 m respectively.

The saltation velocity calculated by the Matlab model is equal to the saltation velocity in the worked example. Therefore it is concluded that the equation for the saltation velocity is implemented correctly.

Table 2: Parameters used to check the saltation velocity.

Parameter Value Dimension

Mean particle size 100 µm

Gravitational constant 9.81 m/s2

Inside pipe diameter 78 mm

Particle density 2500 kg/m3

Length of the oven 3.6 m

With of the oven 1 m

Thickness of the top layer 0.1 m

Time in which the coal is removed 1 hr Calculated removal rate of coal 900 kg/hr

(24)

2-3-2 Pressure drop Due to Solid-to-Wall Friction at Horizontal Sections

The equations for the pressure drop due to the solid-to-wall friction are verified using worked example 8.1 from Rhodes as well [2]. The calculation for the pressure drop in the horizontal section in the worked example that is used to verify the model consists of four parts (acceler-ation of particles and gas, solid-to-wall friction and gas-to-wall friction). The acceler(acceler-ation of the particles and the fluid is omitted in this model and the gas-to-wall friction is determined using another method compared to the worked example. Therefore the worked example is only used to verify the solid-to-wall friction.

The worked example only gives the total pressure drop, therefore the four components from the worked example are calculated separately and are added together to check whether the total result matches the one given in the example. The results of this calculation are listed in Table 3. The total calculated value is oﬀ by 1.6 % compared to the printed value. This error is likely introduced by roundoﬀ errors and this deviation is considered to be small. Therefore the value for the solid-to-wall friction will be used to verify the calculation procedure for the solid-to-wall friction in the Matlab model.

Table 3: Calculation for the solid-to-wall friction value used in the worked example.

Pressure drop due to acceleration fluid 132 Pa Pressure drop due to acceleration particles 315 Pa Pressure drop due to gas-to-wall friction 1.01 kPa Pressure drop due to solid-to-wall friction 13.70 kPa Total pressure drop horizontal section 15.1 kPa Given total pressure drop horizontal section 14.9 kPa

In order to calculate the pressure drop due to solid-to-wall friction, the drag coefficient of the particles needs to be known. The method of obtaining this drag coefficient for the particles is different compared to the Matlab model, so for this verification step the Cd value in the model is altered to match the value mentioned in the worked example.

The implemention of Hinkle’s correlations for the particle velocity and the voidage in the horizontal ducting section are checked to ensure the entire procedure performs well. The results of the verification procedure are included in Table 4. The deviation between the value for the solid-to-wall friction in the example and the value calculated by the Matlab model is less than 2 %. The values for the voidage in the pipe and particle velocity match the values specified in the worked example [2]. The implementation of the calculation procedure for the solid-to-wall friction is therefore considered to be correct.

(25)

2-3 Verification 13

Table 4: Verification of the solid-to-wall friction in the horizontal section of the pipe.

Length of horizontal section 30 m

Inside diameter of pipe 78 mm

Mean particle size 100 mum

Coal removal rate 900 kg/hr

Drag coeﬃcient of particles 3.1

-Density of air 1.2 kg/m3

Given voidage in horizontal section 0.998 -Calculated voidage in horizontal section 0.998

-Given particle velocity 11.8 m/s

Calculated particle velocity 11.8 m/s

Given solid-to-wall friction 13.6 kPa Calculated solid-to-wall friction 13.4 kPa

2-3-3 Drag Coeﬃcient

The drag coefficient of the spherical particles is calculated using the Haider-Levensspiel cor-relation. The drag coefficient is a function of the Reynolds number of the particle. This is a different value than the Reynolds number that describes the flow in the ducting system. The Reynolds number of the particle describes the flow around the coal particles when they are suspended in the flow. The resistance force which acts on the particle depends on the slip of the particle, this is the difference between the velocity of the air and the velocity of the particles. The result of the particle Reynolds number verification can be found in Table 5. The results of the hand calculation and the Matlab model are the same, therefore the calculation procedure for the particles Reynolds number is correct.

Table 5: Verification of the calculation procedure for the particle Reynolds number.

Density of air 1.2 kg/m3

Dynamic viscosity of air 18.4 e-6 kg/ms

Diameter of particles 100 µm

Particle velocity 11.8 m/s

Gas velocity 14.9 m/s

Reynolds number calculated by hand 19.63 -Reynolds number calculated by Matlab 19.63

(26)

-14 Model of the Concept

The largest difference between the hand calculation and the value provided by the Matlab model is 0.016, which is equal to a deviation of 4,1 %. Apart from this value, the differences are smaller than 0.0005, which is a deviation smaller than 0.1 %. Therefore it is concluded that the values for the drag coefficient calculated by the Matlab model are in accordance with the values found by hand calculation for all values of the Reynolds number.

Table 6: Verification of the calculation procedure for the drag coeﬃcient of a particle.

Reynolds Number Particle

Hand-Calculation for Drag Coeﬃcient

Matlab Calculation for Drag Coeﬃcient 0.3 86.6 86.6 0.5 53.5 53.5 0.8 34.7 34.7 5 7.25 7.25 50 1.57 1.57 100 1.10 1.10 250 0.724 0.724 750 0.490 0.490 1500 0.401 0.417 10,000 0.420 0.420 60,000 0.470 0.470 100,000 0.471 0.472 150,000 0.470 0.470 200,000 0.469 0.469

(27)

2-3 Verification 15

2-3-4 Pressure Drop Due to Gas-to-Wall Friction

One calculation procedure is used to calculate both the pressure drop due to gas-to-wall friction in the horizontal and vertical sections of ducting. Therefore the verification process can be performed by verifying either the procedure for horizontal or vertical ducting sections. The calculation procedure used by the Matlab model to calculate the pressure drop due to gas-to-wall friction in the horizontal sections is verified in this subsection.

In order to calculate the pressure drop due to gas-to-wall friction, the darcy friction factor must be calculated. The darcy friction factor is calculated by Matlab and verified using the Moody chart [3]. The Moody chart is used as a reference because this chart is based on the same equation used in the Matlab model. A number of values for the roughness of the pipe, pipe diameter and the Reynolds number are evaluated. The corresponding friction factor is read from the Moody chart and is subsequently compared to the value provided by the Matlab model. The result of the verification is displayed in Table 7. The values of the calculation procedure are equal to the values read from the Moody chart and therefore the procedure is correctly implemented.

Table 7: Verification of the Darcy friction factor.

Pipe rough-ness (m) Pipe Diam-eter (m) Roughness/ diameter (-) Reynolds number pipe (-) Friction factor Matlab (-) Friction factor Moody chart (-) 0.0015 0.05 0.03 10,000 0.0603 0.06 0.002 0.05 0.04 200,000 0.0649 0.065 0.002 0.05 0.04 2,000,000 0.0648 0.065 0.0003 0.05 0.006 1,000,000 0.0323 0.032 0.0001 1 0.0001 20,000,000 0.0121 0.012 0.0008 1 0.0008 1,000,000 0.0190 0.019 0.002 1 0.002 200,000 0.0243 0.025

The gas-to-wall friction is verified using example 6.7 from White [3]. The parameters used in this calculation are displayed in Table 8. The example calculates the head loss (an alternative expression for pressure drop with a diﬀerent dimension), while the Matlab model calculates the pressure drop. Therefore the calculated pressure drop needs to be divided by the product of the gravitational constant and the density of the fluid in order to obtain the head loss. Both the head loss and the pressure drop calculated by Matlab can be found in Table 8. The result of the Matlab model is oﬀ by 0.7 % when compared to the value provided in the example.

(28)

Table 8: Verification of the gas-to-wall friction.

Pipe diameter 0.2 m

Roughness of pipe 0.26 mm

Length of pipe 500 m

Density of fluid 900 kg/m3

Flow velocity 6.4 m/s

Given friction factor 0.0227 -Calculated friction factor 0.0226

-Given head loss 117 m Calculated head loss 118 m

Calculated pressure drop 1040 kPa

2-3-5 Pressure Drop at Vertical Sections

When a particle is dropped it will fall towards the earth due to the gravitational forces which act on it. The particle will accelerate until the the gravitational force is equal to the drag force on the particle, then the particle will continue with a constant velocity (called the terminal velocity). The terminal velocity needs to known in order to calculate the particle and fluid velocities in the vertical section of the ducting system. The terminal velocity is calculated by hand and compared to the value generated by the model. The result of this verification procedure is included in Table 9. The terminal velocity calculated by the Matlab model deviates less than 1 % compared to the hand-calculated value, therefore this value is considered to be correct.

Table 9: Verification of the terminal velocity

Particle diameter 100 µm

Coeﬃcient of drag 2.74

-Hand calculation for terminal velocity 0.997 m/s Matlab calculation for terminal velocity 0.998 m/s

The calculation of the pressure drop due to solid-to-wall friction at vertical ducting sections is verified using worked example 8.1 from Rhodes [2]. The pressure drop in this worked example consists of the static head of the fluid, the static head of the solid particles, the solid-to-wall friction and gas-to-wall friction. The components of the worked example are calculated seperately, so the equations in the Matlab model can be verified seperately. The result of this calculation is shown in Table 10. The total calculated value deviates less than 1 % compared to the printed value, therefore the individual components are considered to be correct.

(29)

2-3 Verification 17

Table 10: Calculation of the individual components in the worked example.

Pressure drop gas-to-wall friction 338 Pa Pressure drop solid-to-wall friction 334 Pa Pressure drop static head solid 368 Pa Pressure drop static head fluid 118 Pa Total calculated pressure drop 1156 Pa Given total pressure drop 1148 Pa

These values are used to verify the calculation procedure for the pressure drop in the vertical ducting section. Table 11 shows the result of this verification procedure. The diﬀerences between the calculated values and the values that are given in the example are small (the largest deviation is equal to 0.8 %). Therefore the implementation of the calculation procedure for the pressure drop in the vertical sections is considered to be correct.

Table 11: Verification of the pressure drop in the vertical ducting sections.

Mass flux of solids 52.3 kg/m2_s

Length of vertical section 10 m

Particle density of coal 2500 kg/m3

Diameter of pipe 0.078 m

Given voidage in vertical section 0.999 -Calculated voidage in vertical section 0.999

-Given solid-to-wall friction 334 Pa

Calculated solid-to-wall friction 334 Pa

Given static head of solids 368 Pa

Calculated static head of solids 371 Pa

Given static head of fluid 118 Pa

Calculated static head of fluid 118 Pa

Given pressure drop in vertical section 820 Pa Calculated pressure drop in vertical section 823 Pa

(30)

Table 12: Verification of the characteristic velocity and the Stokes number of a gas cyclone.

Cut size 4.34 µm

Diameter of Cyclone 1.01 m

Viscosity of air 18.3 e-6 Pas

Volume flow rate of gas 2 m3/s

Given characteristic velocity 2.48 m/s Calculated characteristic velocity 2.48 m/s

Given Stokes number 1.4e-4

-Calculated Stokes number 1.40e-04

-The implementation of the approximate relation between the Stokes number and the Euler number is verified next. The examples are taken from Rhodes [2], the Stokes numbers are entered into the Matlab model which calculates the Euler number. The diﬀerence between the given values and the values generated by Matlab is equal to 8.5 %. Therefore this approximate relation might lead to a slight underestimation of the pressure drop across the gas cyclone. The Euler number was also calculated by hand, which gave the same result as the Matlab model. The implementation of this equation is therefore correct, although its results are not accurate.

Table 13: Verification of the Euler number calculation.

Stokes number [-] Euler number [-]

Example 1 1.4e-4 320

Calculated value 1.4e-4 293

Example 2 6e-3 46

Calculated value 6e-3 44.7

The equation for the pressure drop across the gas cyclone is verified using the worked example from Rhodes as well [2]. The parameters entered in the Matlab model are displayed in Table 14. The deviation between these two values is very small, therefore the implementation of this equation is considered to be correct.

Table 14: Verification of the pressure drop across a gas cyclone.

Volume flow rate 2 m3/s

Euler number 320

-Stokes number 1.4e-4

-Given pressure drop 1177 Pa Calculated pressure drop 1178 Pa

(31)

2-3 Verification 19

A filter is situated at the exit of the gas cyclone, this causes a pressure drop as well. The equations that are used to calculate the pressure drop across the filter are verified by hand calculation. The results of this verification procedure are listed in Table 15. For any given type of gas cyclone, a fixed ratio between the cyclone body diameter and the cyclone gas exit diameter exists. This is used to calculate the average exit flow velocity. The flow velocity calculated by hand and the flow velocity calculated by Matlab are the same. The values for the pressure drop vary slightly, but this deviation is considered to be negligible. Therefore the implementation of this calculation procedure is considered to be correct.

Table 15: Verification of the pressure drop across the filter.

Volume flow rate 2 m3/s

Cyclone diameter 1 m

Ratio exit duct/cyclone diameter 0.5

-Density of the gas 1.2 kg/m3

Loss coeﬃcient 3

-Exit velocity Matlab 10.2 m/s

Exit velocity hand calculation 10.2 m/s

Pressure drop filter Matlab 187 Pa Pressure drop filter hand calculation 187 Pa

2-3-7 Pressure Recovery

Two sources of pressure recovery exist; heat transfer and the use of an intake. The temperature of the air that enters the ducting system is estimated to be 300◦C, which cools down during transportation. The heat transfer is calculated for the ducting network using the method found in Mills [6]. The results of the verification process of these equations are displayed in Table 16.

(32)

Table 16: Verification of the heat transfer calculation.

Flow velocity 25,5 m/s

Diameter of pipe 84,7 mm

Thermal conductivity of pipe 15 W/mK

Thermal conductivity of gas 0.0447 W/mK

Temperature of gas 600 K

Specific heat capacity of gas 1038 J/kgK

Dynamic viscosity of gas 29,7e-6 kg/ms

Density of gas 0.589 kg/m3

Hand calculated value for heat transfer coefficient 53 W/m2K Heat transfer coefficient calculated by Matlab 53 W/m2K Hand calculated value for total heat transfer coefficient 0.74 W/m2K Total heat transfer coefficient calculated by Matlab 0.74 W/m2K Hand calculated value for heat transfer 1181 W

Heat transfer calculated by Matlab 1181 W

Hand calculated value for the exit temperature 587 K

Exit temperature calculated by Matlab 587 K

The calculation procedure for the pressure recovery due to heat transfer is verified next. The ideal gas law for isentropic adiabatic expansion is used to calculate the pressure diﬀerence between the inlet and exit of the ducting system. This procedure is verified using hand calculations. The result of this verification is included in Table 17. The outcome of the hand calculations is equal to the outcome of the Matlab model, therefore the implementation of this calculation procedure is considered to be correct.

Table 17: Verification for the pressure recovery due to heat transfer.

Inlet temperature 600 K

Temperature at end of ducting system 587 K

heat capacity ratio of air 1.4

-Ambient pressure 1e5 Pa

Pressure diﬀerence hand calculated 7.38 kPa Pressure diﬀerence Matlab 7.39 kPa

The pressure diﬀerence across the intake is verified using hand calculations as well. The results are listed in Table 18. The results of the Matlab model and the hand calculation are equal. Therefore the calculation procedure is considered to be correct.

(33)

2-3 Verification 21

Table 18: Verification of pressure recovery across the intake.

Diameter pipe 84.7 mm

Diameter leading edge of intake 300 mm

flow velocity in pipe 35.4 m/s

Pressure recovery coefficient hand calculated 0.994 -Pressure recovery coefficient Matlab 0.994 -Pressure difference hand calculated 367 Pa

Pressure diﬀerence Matlab 366 Pa

2-3-8 Fan Characteristics

It is assumed that the ambient pressure at the inlet and exit of this system is the same. The pressure diﬀerence across the fan should therefore match the pressure drop across the system. The equations used to calculate the installed power of the fan and the amount energy required to remove the top layer of coal are verified using hand calculations. Table 19 shows the results of the verification of both the fan power and the amount of energy required to remove the coal. The deviation between the values produced by the Matlab model and the hand calculated values is negligible, therefore the implementation of this calculation procedure is considered to be correct.

Table 19: Verification of the power of the fan and energy required to remove the crust.

Volume flow gas 0.0807 m3/s

Pressure drop across system 7514 Pa

Eﬃciency of fan 0.75

-Time of operation 3600 s

Fluid power hand calculated 606 W

Fluid power Matlab 606 W

Fan power hand calculated 808 W Fan power Matlab 808 W Energy required hand calculated 2911 kJ Energy required Matlab 2910 kJ

(34)

(35)

Chapter 3 Results

This chapter concerns the experiments that are performed in order to determine the set-up which minimises the energy consumption of the top layer of coal. The model used to perform the experiments is discussed in Chapter 2. The first section of this chapter covers the parameters that are fixed in all the experiments. The next section concerns the experiments themselves and the final section covers the summarises the findings of the performed experiments.

3-1

Fixed Parameters

This section concerns the parameters that remain constant throughout the experiments. These parameters concern the properties of air, the dimensions of the oven, the dimensions of the ducting system and the properties of the gas cyclone.

The density and viscosity of the air are held constant throughout the experiments. The properties of air are evaluated at 325 K (around 52◦C). The density is set at 0.589 kg/m3 and the viscosity is 2.97e-5 kg/ms [7]. The speed of sound in air is evaluated at 50 ◦C and is equal to 360 m/s [8]. The ambient pressure is set at 1 bar (which is equivalent to 1e5 Pa). The desired pressure diﬀerence between the inlet and the ambient condition is set at 0.01 bar (1e3 Pa). This margin is chosen in order to avoid surge at the pump (pressure gradient

(36)

24 Results

The length of the horizontal and vertical ducting sections are fixed as well. The ducting system is made out of commercial grade steel pipes with a roughness of 0.046 mm [3]. One bend is used in the system. The length of the horizontal section is 5m. and the length of the vertical section is 15 m.

The coal particles are removed from the flow by a gas cyclone and filters. One filter will be installed with a loss factor of 6.0. This loss factor is equivalent to the loss factor of a grilles with the flow area equal to half of the total surface [9]. The type of gas cyclone that is used is the high-eﬃciency Stairmand cyclone, the relation between the dimension of the gas exit and the cyclone diameter is therefore equal to 0.5 [2]. The cut size of the cyclone is set at 5 µm.

The data for the thermal conductivity of the steel (for the ducting) and the air, as well as the heat capacity ratio are given by Mills [6]. The temperature at the inlet is set at 400 K (approximately 100◦C).

Table 20 shows an overview of the parameters that are fixed during the experiments.

Table 20: Overview of the parameters used in the experiments.

Dynamic viscosity of air 2.97e-5 kg/ms

Speed of sound in air 360 m/s

Ambient pressure 1e5 Pa

Desired pressure diﬀerence between inlet and ambient 1e3 Pa

Temperature of environment 300 K

Heat capacity ratio of air 1.4

-Width of the oven 2 m

length of the oven 4 m

Thickness of top coal layer 0.3 m

Length of horizontal ducting section 5 m

Length of vertical ducting section 15 m

Number of bends 1

-Roughness of pipe 4.6e-5 m

Loss factor of a filter 6

-Number of filters 1

-Relation between dimension gas exit and diameter of cyclone 0.5

-Cut size of the cyclone 5 µm

Thermal conductivity of the air 0.0447 W/mK

Thermal conductivity of steel 15 W/mK

Specific heat capacity of air 1038 J/kgK

(37)

3-2 Experiments 25

3-2

Experiments

This section concerns the experiments that are performed in order to determine the set-up for suction process of the top layer of coal which minimises the energy requirement. The eﬀect on the energy consumption due to changes of the particle size, particle density, the inside diameter of the ducting, cyclone body diameter, the time allowed to remove the top layer of coal, the eﬃciency of the fan and the diameter of the leading edge of the intake are investigated in their respective subsections.

3-2-1 Eﬀect of Particle Size

This experiment is performed to determine whether or not the dimension of the particles influences the energy consumption of the suction process of the top layer of coal. Several values of the dimension of the particles are entered into the model. The values of the diameters that are evaluated are listed in Table 21. Table 22 displays the values of the other variables which are held constant during this experiment.

Table 21: Dimension of the spheres used in this experiment.

Diameter of spheres Dimension

20 µm 40 µm 60 µm 80 µm 100 µm 250 µm 500 µm 750 µm 1 mm 2 mm 5 mm 10 mm 25 mm 50 mm

Table 22: Values of the other variables during the first experiment

(38)

26 Results

The outcome of this experiment is displayed in Figure 6. This figure shows that the energy requirement depends on the magnitude of the particles. When the particles grow in size, the solid-to-wall friction decreases (because the number of particles is lower). However the gas-to-wall friction increases, because the flow velocities need to be higher. The pressure drop due to the static head of the solids increases as well, because the voidage in the pipe is reduced. The relation between particle size and energy consumption is zoomed in in Figure 7. The relation between the energy consumption and the particle size for lower values of the particle size can be seen in this figure. It is evident by these figures that a range of particle sizes exists for which the energy consumption is minimised.

(39)

3-2 Experiments 27

Figure 7: Relation between the particle size and energy consumption zoomed in.

3-2-2 Eﬀect of Particle Density

The relation between the particle density and the energy consumption of the suction process of the top layer of coal is evaluated in the second experiment. The density of the crust material in this experiment ranges from 250 to 2500 kg/m3, with intervals of 250 kg/m3. The values of the other variables are listed in Table 23.

Table 23: Values of the variables not changed in the second experiment.

(40)

28 Results

When the particle density increases, the total pressure drop increases as well. This in turn relates to an increase in the amount of energy that is consumed. This eﬀect is nonlinear, as can be seen in Figure 8, which shows the relation between the particle density and the energy consumption.

Figure 8: Relation between the particle density and energy consumption.

3-2-3 Eﬀect of Inside Pipe Diameter

The third experiment concerns the relation between the inside diameter of the pipe and the energy consumption of the crust removal process. The pipe sizes that are evaluated are taken from the DIN standard for schaled pipe [1]. Table 24 shows the standard diameters that are evaluated. The wall thickness is set at schedule 5, which is the lowest value available for the wall thickness. The values of the variables that are not changed are displayed in Table 25.

(41)

3-2 Experiments 29

Table 24: Diameters of pipe used in the experiment [1].

Nominal diameter of pipe [mm] Wall thickness [mm] Inside diameter of pipe [mm]

50 1.7 56.9 80 2.1 84.7 100 2.1 110 125 2.8 136 150 2.8 163 200 2.8 214 250 3.4 266 300 4.0 316 350 4.0 348 400 4.2 398

Table 25: Values of the variables not changed in the third experiment.

Density of coal 1500 kg/m3

Particle size 10 mm

Cyclone body diameter 0.75 m

Time allowed to remove the coal 20 min.

Eﬃciency of the fan 0.66

-Diameter leading edge of inlet 0.3 m

Figure 9 shows the relation between the inside diameter of the pipe and the energy consump-tion. The relation between pipe inside diameter and energy consumption is nonlinear and has a minium at a nominal diameter of 150 mm. This diameter is therefore selected in order to minimise the energy consumption of this system.

(42)

30 Results

Figure 9: Relation between the inside pipe diameter and energy consumption.

3-2-4 Eﬀect of Cyclone Body Diameter

The fourth experiment concerns the relation between the cyclone body diameter and the energy consumption of the system. The diameter of the cyclone body is varied from 0.25 to 2 m, with intervals of 0.25 m. The values of the variables that are not changed in this experiment are listed in Table 26.

Table 26: Values of the variables not changed in the fourth experiment.

Particle size 10 mm

Time allowed to remove the coal 20 min.

(43)

3-2 Experiments 31

The result of this experiment is displayed in Figure 10. When the diameter increases, the average flow velocity drops and the pressure drop across the gas cyclone decreases. At a given point, the pressure drop across the gas cyclone almost diminishes and the energy consumption of the entire process becomes constant. The gains in energy eﬃciency become small after the gas cyclone body is equal to 0.75 m.

Figure 10: Relation between the cyclone body diameter and energy consumption.

3-2-5 Eﬀect of Allowed Time to Remove the Coal Layer

The time that is allowed to remove the top layer of coal will also have an impact on the energy consumption of the crust removal process. Whenever the allowed time decreases, the power

(44)

32 Results

Table 27: Values of the variables not changed in the fifth experiment.

Particle size 10 mm

-Diameter leading edge of inlet 0.3 m

Figure 11 shows the relation between the allowed time and the energy requirement of the removal process. The energy consumption increases slightly at each increase in the time in which the top layer of coal needs to be removed. Figure 12 displays the relation between the allowed time and the installed power of the fan. As the allowed time for the removal process increases, the installed power of the fan is decreased. The decrease in installed power is reduced at each increment in the allowed time.

(45)

3-2 Experiments 33

Figure 12: Relation between the time allowed to remove the crust and power of the fan.

Based on these figures, the time that is allowed to remove the top layer of coal is set at 4 minutes. Four minuts seems a reasonable time to do the job and this gives a fan with a reasonable amount of power requirement. The influence of the allowed time with respect to the energy requirement is very limited and is therefore not taken into consideration when determining the allowed time to remove the coal layer.

3-2-6 Eﬀect of Eﬃciency of the Fan

The eﬃciency of the fan has a direct impact on the energy consumption, because for each decrease in eﬃciency there is an increase in energy consumption. This experiment is performed

(46)

34 Results

Table 28: Values of the variables not changed in the sixth experiment.

Particle size 10 mm

Time allowed to remove the coal layer 4 min Diameter leading edge of inlet 0.3 m

Figure 13 shows the effect on the energy requirement due to changes in the efficiency of the fan. The gains made in energy reduction become smaller when the efficiency becomes larger. This graph can be used in order to determine whether an investment in a more efficient fan is cost-effective. For the remainder of this research, a fan efficiency of 0.5 is assumed to be reasonable. The effect of the efficiency of the fan with respect to the energy requirement of the process is not taken into consideration here because this effect is considered to be small.

(47)

3-2 Experiments 35

3-2-7 Diameter of the Intake

The final experiment concerns the diameter of the leading edge of the intake, which is a conical diﬀuser. The diameter of the intake ranges from 200 to 500 mm, with increments of 25 mm. The values of the other variables held constant in this experiment can be found in Table 29.

Table 29: Values of the variables not changed in the final experiment.

Particle size 10 mm

Time allowed to remove the coal layer 4 min

-The result of this experiment can be found in Figure 14. This figure shows that the geometry of the intake only has a minor eﬀect on the performance of the entire system. With each increase in diameter of the intake, the energy consumption is reduced. These reductions are small, the reduction in energy consumption between the intake of 500 mm and the intake of 200 mm is only 1.5 %.

(48)

36 Results

3-3

Conclusion

The results of the experiments are summarised in this section.

Several parameters were considered during the experiments. From these experiments it can be concluded that the energy consumption is reduced for each increase in the efficiency of the fan and for each decrease in particle density. The energy requirement is also reduced with each increase in cyclone body diameter. However this effect levels out, this means that the gains in energy efficiency obtained by a larger cyclone body diameter diminish as the diameter is increased.

The diameter of the intake has a very limited eﬀect on the energy consumption of the process. With a larger intake, the energy requirement is reduced by a small amount. The eﬀect of changing the pipe diameter is relatively large. For large values of the pipe diameter, the energy requirement increases. A pipe size exists for which the energy consumption of the suction process of the coal particles is minimised.

The particle size has a large influence on the energy consumption, for relatively small and relatively large particles, the energy consumption of the removal process increases. A range of particle sizes exists where the energy consumption is minimised.

The time allowed to remove the top coal layer has a minor influence on the energy requirement. However, the eﬀect on the installed power of the fan is large.

(49)

Chapter 4 Conclusion to this Research

This chapter serves to list the results obtained by the Matlab model. The geometry which minimise the energy requirement for the suction process of the top coal layer are proposed as well.

The removal of the top coal layer at the ovens used to manufacture anodes is performed by a vacuum cleaner. The vacuum cleaner will be mounted to the crane which lifts the anodes out of the cell. The coal particles are deposited in a container which is fixed to the crane as well. The suction process is modeled in Matlab and this model is used to determine the geometry which minimises the energy requirement of the process.

The energy requirement can be reduced when the particle density is reduced. However, the particle density is not a parameter that can be changed easily. The production process of the anodes will likely take place with coal with a fixed specification. The same holds for the particle size, for a particle size between 0.25 and 10 mm the energy consumption is minimised. The particle size cannot easily be modified as well, but when the particle size is way oﬀ this range means have to be found in order to make this possible.

(50)

38 Conclusion to this Research

The energy consumption is minimised with a cyclone body diameter of 0.75 m, combined with ducting with a nominal diameter of 150 mm. The diameter of the leading edge of the intake is equal to 325 mm in order to reduce the energy consumption.

The energy consumption is also reduced for each increase in fan eﬃciency. It is established that the minimum fan eﬃciency should be 0.5, because otherwise the energy consumption of the process becomes too large. The time in which the top layer of coal should be removed is set at 4 minutes, in order to limit the required power of the fan that needs to be selected. Table 30 shows the final result of this research.

Table 30: Conclusion of this research

Nominal diameter of pipe 150 mm

Diameter of leading edge of intake 325 mm

Time in which top coal layer needs to be removed 4 minutes

-Diameter of spheres 10 mm

Particle density 1500

Energy requirement 9.64 MJ Power requirement 40.2 kW

(51)

Appendix A

This appendix displays the code used in the Matlab model. The diﬀerent calculation procedures are displayed in separate frames.

Introduction to the model:

1 %% INTRO

2 % This script will calculate the energy requirement of the removal process

3 % of the top coal layer in ovens used to manufacture anodes. The top coal

4 % layer is removed using a vacuum cleaner.

5

6 % date of last revision: 26-05-2015

(52)

40 Appendix A

Declaration of parameters and variables:

1 %% Parameters and Variables

2 % Variables:

3 r= 100e-6; % diameter of sphere representing the particles [m].

4 rho2= 1500; % particle density of the crust material [kg/m^3].

5 D= 0.0847; % inside diameter of the ducting system [m].

6 Dcycl= 0.75; % cyclone body [m].

7 time= 20; % time in which the crust should be removed [min].

8 eta= 0.66; % efficiency of the fan [-].

9 Din= 0.3; % Diameter of the leading edge of the inlet [m].

10

11 % general parameters:

12 g= 9.81; % gravitational constant [m/s^2].

13 rho1= 0.589; % density of the air [kg/m^3].

14 mu1= 29.74e -6; % dynamic viscosity of air [kg/ms].

15 Ua= 360.3; % speed of sound in air [m/s].

16 Pam= 1e5; % ambient pressure [Pa].

17 Pplus= 1e3; % desired pressure difference between ambient pressure ... and inlet [Pa].

18 Ten= 300; % temperature of the environment [K].

19 gamma= 1.4; % heat capacity ratio of air [-].

20

21 % parameters relating to the cell and the crust material:

22 widthc= 2; % width of the oven [m].

23 lengthc= 4; % length of the oven [m].

24 coalt= 0.3; % average thickness of the top coal layer [m].

25

26 % parameters relating to the ducting system:

27 Lhor= 5; % length of horizontal section of pipe [m].

28 Lvert= 15; % lenght of vertical section of pipe [m].

29 nbend= 1; % number of bends [-].

30 e= 4.6e -5; % roughness of the pipe [m].

31

32 % parameters relating to the particle removal process:

33 Kfilter= 6.0; % loss coefficient of the filter [-].

34 nfilt= 1; % number of filters [-].

35 Ncycl= 0.5; % relation between cyclone body diameter and diameter ... of exit ducting [-].

36 X50= 5e-6; % cut size of the cyclone [m].

37

38 % parameters relating to heat transfer

39 kgas=0.0447; % thermal conductivity of the gas [W/mK].

40 kwall=15; % thermal conductivity of stainless steel [W/mK].

41 cp=1038; % specific heat capacity of air [J/kgK].

(53)

41

Calculation procedure for the saltation velocity:

1 %% Saltation velocity

2 times= time∗60; % time in which the coal is removed [s].

3 Q= (widthc∗lengthc∗coalt)/times; % volume flow of the coal [m^3/s].

4 Mp= Q∗rho2; % mass flow rate of coal [kg/s]

5 a1= (1440∗r)+1.96; % first variable used in the Rizk equation [-].

6 a2= (1100∗r)+2.5; % second variable used in the Rizk equation [-].

7 Apipe= 0.25∗pi∗(D^2); % cross sectional area of the pipe [m^2].

8

9 % the equation for the saltation velocity, derived from the Rizk

10 % correlation [m/s]:

11 Usalt= ((Mp∗(10^a1)∗(g^(a2/2))∗(D^(a2/2)))/(rho1∗Apipe))^(1/(1+a2)); 12 U= 1.5∗Usalt; % the superficial gas velocity [m/s].

Calculation procedure for the pressure loss across the horizontal ducting sections:

1 %% Pressure loss across horizontal section of the ducting system

2 % Acceleration of fluid and particles is assumed to take place before

3 % entering the pipe. Since it is a horizontal pipe, the static head of the

4 % fluid and particles is zero.

5

6 % actual particle velocity according to Hinkles correlation [m/s]:

7 Up1= U∗(1-(0.0638∗(r^0.3)∗(rho2^0.5)));

8 G= Mp/Apipe; % mass flux of solids [kg/m^2∗s].

9 etah= 1-(G/(rho2∗Up1)); % voidage in the horizontal section of the ... pipe [-].

10 Uf1= U/etah; % fluid velocity in the horizontal section of ... the pipe [m/s].

11 Res1= rho1∗(Uf1-Up1)∗r/mu1; % Reynolds number for the airflow around a ... particle [-].

12

13 % Haider and Levenspiel correlation for the drag coefficient of a

14 % sphere [-]:

15 Cd1= ((24/Res1)∗(1+0.1806∗(Res1^0.6459)))+(0.4251/(1+(6880.95/Res1))); 16

17 % Pressure drop due to solid-to-wall friction, Hinkles correlation [Pa]:

18 Fswh= (3/4)∗(Lhor/r)∗rho1∗Cd1∗((Uf1-Up1)^2)∗(1-etah);

19 Rep1= rho1∗Uf1∗D/mu1; % Reynolds number for flow in the horizontal pipe [-].

20

21 % Haaland estimation for the darcy friction factor [-].

22 f= (1/(-1.8∗(log10((6.9/Rep1)+((e/D)/3.7)^1.11))))^2;

23 Fgwh= rho1∗f∗Lhor∗(Uf1^2)/(2∗D); % Pressure drop due to gas-to-wall ... friction [Pa]:

(54)

42 Appendix A

Calculation procedure for the pressure loss across vertical ducting sections:

1 %% Pressure loss across vertical section of the ducting system

2 Ut= sqrt((8∗r∗(rho2-rho1)∗g)/(6∗Cd1∗rho1)); % terminal velocity of the ... particles [m/s].

3 bhelp= -Ut-U-(G/rho2); % variable used in the ... ABC-formula [m/s].

4

5 % voidage in the vertical pipe, calculated using the ABC-formula [-].

6 etav= (-bhelp -sqrt((bhelp^2)-(4∗Ut∗U)))/(2∗Ut);

7 Up2= (U/etav)-Ut; % particle velocity in ... vertical section [m/s].

8

9 % fluid velocity in vertical section (slip velocity equals terminal ... velocity) [m/s].

10 Uf2=Ut+Up2; 11

12 % pressure drop due to solid-to-wall friction, Konno and Saitos ... correlation [Pa].

13 Fswv= 0.057∗G∗Lvert∗sqrt(g/D);

14 Fgwv= rho1∗f∗Lvert∗(Uf2^2)/(2∗D); % pressure drop due to gas-to-wall ... friction [Pa].

15 Fshs= rho2∗g∗Lvert∗(1-etav); % pressure drop due to static head ... of the particles [Pa].

16 Fshg= rho1∗g∗Lvert∗etav; % pressure drop due to static head ... of the gas [Pa].

17 Pvert= Fswv+Fgwv+Fshs+Fshg; % total pressure drop in the ... vertical pipe [Pa].

Calculation procedure for the pressure drop due to bends:

1 %% Pressure loss due to bends

2 Pb= (Pvert/Lvert)∗7.5; % pressure drop across one 90-degree bend [Pa].

(55)

43

Calculation procedure for the pressure drop across the gas cyclone and the filters:

1 %% Solid removal

2 Qgas= Uf1∗Apipe∗etah; % volume flow rate of the gas ... [m^3/s].

3 Vchar= (4∗Qgas)/(pi∗(Dcycl^2)); % characteristic velocity for ... the gas cyclone [m/s].

4 Stk50=((X50^2)∗rho2∗Vchar)/(18∗mu1∗Dcycl); % Stokes number for the gas ... cyclone [-].

5 Eu= sqrt(12/Stk50); % Euler number for the gas ... cyclone [-].

6 Pcycl= Eu∗rho1∗0.5∗(Vchar^2); % pressure drop across the gas ... cyclone [Pa].

7 Uexit= Qgas/(0.25∗pi∗(Dcycl∗Ncycl)^2); % gas velocity at the exit of ... the cyclone [m/s].

8 Pfilt= 0.5∗rho1∗(Uexit^2)∗Kfilter; % pressure loss across a ... filter [Pa].

9 Prem=Pcycl+(nfilt∗Pfilt); % total pressure drop across ... particle removal process [Pa].

Calculation procedure for the equations relating to the pressure recovery:

1 %% Pressure Recovery

2 SPipe= (pi∗D)∗(Lhor+Lvert); % surface area of the pipe [m2].

3 nu= mu1/rho1; % kinematic viscosity of air [m2/s].

4

5 %heat transfer coefficient [W/m2K]:

6 hc= 0.023∗(U^0.8)∗(kgas^0.6)∗((rho1∗cp)^0.4)/((D^0.2)∗(nu^0.4)); 7 Rinv= (1/hc)+((Lhor+Lvert)/kwall);

8 R= 1/Rinv; % total heat transfer coefficient [w/m2K]

9 Q=R∗SPipe∗(Tin-Ten); % heat transfer across ducting system [W].

10 mgas= Uf1∗rho1∗etah∗Apipe; % mass flow of the gas [kg/s].

11 Tout=Tin -(Q/(mgas∗cp)); % temperature of gas at end of ducting system [K].

12 %formula for pressure after isentropic adiabatic expansion [Pa];

13 P2= Pam∗((Tout/Tin)^(gamma/(gamma-1)));

14 PTdiff=Pam -P2; % pressure difference due to isentropic ... expansion [Pa].

15 % Calculation procedure for pressure recovery across the intake:

16 Cp=1-((D^2)/Din^2)^2; % coefficient of pressure recovery at the ... inlet [-].

17 Pin= Cp∗0.5∗rho1∗(U^2); % pressure difference at the inlet [-].

Calculation procedure for the required power of the fan and the energy consumption of the process:

(56)

44 Appendix A

This part of the code is used to verify the validity of some of the equations used in the model:

1 %% checks

2 % Some equations used in this model have a range of operation. When this

3 % range of operation is voilated, a warning is produced. Also, some of the

4 % assumptions are checked.

5

6 % validity of coefficient of drag:

7 if (Res1 > 2e5)

8 msgbox(equation for Cd not valid (Reynolds number outside validity ... range).)

9 end

10

11 % validity of darcy friction factor:

12 if (Rep1 < 4000)

13 msgbox(equation for darcy friction factor is not valid (flow in pipe ... is not turbulent).)

14 end

15

16 % Check for the mass flows in horizontal and vertical section:

17 % (threshold to compensate for eventual roundoff errors is 1e-8 kg/s)

18 Mh= Apipe∗((Up1∗rho2∗(1-etah))+(Uf1∗rho1∗etah)); % mass flow in ... horizontal section [kg/s].

19 Mv= Apipe∗((Up2∗rho2∗(1-etav))+(Uf2∗rho1∗etav)); % mass flow in ... vertical section [kg.s].

20 if (abs(Mh-Mv)>1e-8)

21 msgbox(Continuity relations are violated (mass flows are not equal).) 22 end

23

24 % Compressibility effects of air can be neglected as long as the Mach

25 % number is below 0.3:

26 Ma1=Uf1/Ua; % Mach number for the horizontal section [-].

27 Ma2=Uf2/Ua; % Mach number for the vertical section [-].

28 if (Ma1 >0.3||Ma2 >0.3)

29 msgbox(Compressibility effects of air can not be neglected (Ma>0.3)) 30 end

31

32 % this check ensures that the final pressure is non-negative.

33 if (Pam -Ptot <=0)

34 msgbox(Pressure at inlet incorrect (pressure drop across system is too ... large))

35 end

36 37

38 % this line returns the major deliverables in a messagebox

39 msgbox(sprintf(Energy used = %2.5g kJ \n Fan power = %2.5g W,Energy, ... Power_fan),Result)

Coal Removal Using A Vacuum Cleaner Modelling The Suction Process of Coal Particles Using A Vacuum Cleaner

Coal Removal Using A Vacuum

Cleaner

Modelling The Suction Process of Coal Particles Using A

Vacuum Cleaner

Johan van Jole

Rep

o

rt

Coal Removal Using A Vacuum

Cleaner

Modelling The Suction Process of Coal Particles Using A Vacuum

Cleaner

Research Assignmnent

Johan van Jole

Assignment Description

Table of Contents

List of Figures

List of Tables

Chapter 1

Introduction

Chapter 2

Model of the Concept

2-1

Assumptions

2-2

Validation

2-3

Verification

Chapter 3

Results

3-1

Fixed Parameters

3-2

Experiments

3-3

Conclusion

Chapter 4

Conclusion to this Research

Appendix A

Appendix A