# Control of Acoustic Energy Input for Cleaning of Industrial Boilers

^{*}

## Abstract

**:**

## 1. Introduction

^{®}Inventor

^{®}2021 student version software sourced from the Autodesk

^{®}US website, readily available information, and reasonable assumptions where applicable. Scripts are utilized in the MATLAB

^{®}environment to perform step function plots and multiple subsystem reduction.

#### Research Outline

- System configuration (plant defined, plant parameter formulation, multiple subsystem reduction).
- Design testing (control system, varied and constant set points, disturbance rejection simulations).
- Research implications.

## 2. System Configuration

#### 2.1. Plant

#### 2.1.1. I-P Transducer

_{o}(t) output as illustrated in Figure 2.

_{o}(s), considering Kirchhoff’s current law yields

#### 2.1.2. Spring, Diaphragm Valve, and Acoustic Horn

#### 2.1.3. Piezoelectric Pressure Sensor

_{o}(s) and considering Kirchhoff’s current law yields the following computations:

#### 2.2. Plant Parameter Formulation

#### 2.2.1. Electrical Systems

#### 2.2.2. Mechanical Translational Systems

^{®}Inventor

^{®}software as shown in Table 3.

^{®}Inventor

^{®}software as shown in Table 5.

^{®}Inventor

^{®}Professional Design Accelerator software.

#### 2.2.3. Subsystems’ Final Transfer Function Representations

#### 2.3. Multiple Subsystem Reduction

^{®}:

#### 2.4. PID Controller

_{sys}, graphical scope display, numeric output display, set pace, and signal lines.

#### 2.4.1. Design Requirements

#### 2.4.2. PID Design

^{−5}. Comparing these results with Table 10 suggests that designing a controller is the next necessary step to achieve the design requirements.

_{p}), integrative (K

_{i}), and derivative (K

_{d}) gains are deliberately set to 1. This is undertaken to analyse the system’s response when the controller executes these gains relative to the input command from signal R(s).

^{−5}at 10 s as shown in the system configuration in Figure 18. Relative to the input command of 1, this suggests that the design requirements are still not met.

^{−5}value, thereafter exponentially ramps up to infinity. At this point, the transient response, steady-state error, and stability design objectives are not met.

_{p}, K

_{i}, and K

_{d}in order to satisfy the design objectives.

#### 2.4.3. PID Tuning

_{p}= 225,775.49, K

_{i}= 208,064.1457, K

_{d}= 21,780.4129, and N = 248.1711 satisfy the design requirements. It is also vital to note that the closed-loop stability is deemed stable at these newly defined gains.

## 3. Testing

_{sys}). The requirements met included a quick transient response, system stability, robustness, and accurate reference tracking.

#### 3.1. Simulation of Control System

#### 3.2. Varied Set Points Simulation

#### 3.3. Constant Set Point Simulation

#### 3.4. Disturbance Rejection Simulation

## 4. Implications

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 16.**PID controller block expanded view [22].

Variables | Parameter |
---|---|

m_{1} | Valve diaphragm mass |

m_{2} | Acoustic horn diaphragm mass |

k_{1} | Valve spring constant |

k_{2} | Force multiplier |

k_{3} | Acoustic horn spring constant |

c_{1} | Damping coefficient @m_{1} |

c_{3} | Damping coefficient @m_{2} |

**Table 2.**Silicon nitride diaphragm input parameters [11].

Parameter | Value |
---|---|

Material | Silicon Nitride |

Thickness (mm) | 30 |

Diameter (mm) | 297 |

Requested accuracy level | Very high |

General Properties | Value |
---|---|

Material | Silicon Nitride |

Density | 3.180 g/cm^{3} |

Mass | 6.609 kg (Relative Error = 0.000000%) |

Area | 166,549.964 mm^{2} (Relative Error = 0.000000%) |

Volume | 2,078,375.598 mm^{3} (Relative Error = 0.000000%) |

**Table 4.**Titanium diaphragm input parameters [12].

Parameter | Value |
---|---|

Material | Titanium |

Thickness (mm) | 1 |

Diameter (mm) | 367 |

Requested accuracy level | Very high |

General Properties | Value |
---|---|

Material | Titanium |

Density | 4.510 g/cm^{3} |

Mass | 0.477 kg (Relative Error = 0.000000%) |

Area | 212,721.951 mm^{2} (Relative Error = 0.000000%) |

Volume | 105,784.493 mm^{3} (Relative Error = 0.000000%) |

Design Inputs | ||
---|---|---|

Installed length type | Custom | |

Coil length (mm) | 40 | |

Coil direction | Right Direction | |

Spring start | Closed end coils (µL) | 1.5 |

Transition coils (µL) | 1 | |

Ground coils (µL) | 0.75 | |

Spring end | Closed end coils (µL) | 1 |

Transition coils (µL) | 0.75 | |

Ground coils (µL) | 0.5 | |

Calculation Inputs | ||

Minimum load length (mm) | 45 | |

Maximum load length (mm) | 39.270 | |

Working stroke (mm) | 5.730 | |

Spring material predetermined values | Ultimate tensile stress (MPa) | 1860 |

Allowable torsional stress (MPa) | 930 | |

Modulus of elasticity in shear (MPa) | 68,500 | |

Density (kg/m^{3}) | 7850 | |

Utilization factor of material (µL) | 0.9 |

General Properties | Value |
---|---|

$\mathrm{Space}\mathrm{between}\mathrm{coils}\mathrm{of}\mathrm{free}\mathrm{spring}(\mathsf{\alpha}$) | 3.172 mm |

$\mathrm{Pitch}\mathrm{of}\mathrm{free}\mathrm{spring}(\mathrm{t}$) | 3802 mm |

$\mathrm{Stress}\mathrm{concentration}\mathrm{factor}({\mathrm{K}}_{\mathrm{w}}$) | 1 µL |

$\mathrm{Spring}\mathrm{constant}(\mathrm{k}$) | 0.175 N/mm |

$\mathrm{Minimum}\mathrm{load}\mathrm{spring}\mathrm{deflection}({\mathrm{s}}_{1}$) | 29,649 mm |

$\mathrm{Total}\mathrm{spring}\mathrm{deflection}({\mathrm{s}}_{8}$) | 34,379 mm |

$\mathrm{Limit}\mathrm{spring}\mathrm{deflection}({\mathrm{s}}_{9}$) | 60,262 mm |

$\mathrm{Limit}\mathrm{test}\mathrm{length}\mathrm{of}\mathrm{spring}({\mathrm{L}}_{\mathrm{minf}}$) | 16,872 mm |

$\mathrm{Theoretic}\mathrm{limit}\mathrm{length}\mathrm{of}\mathrm{spring}({\mathrm{L}}_{9}$) | 13,388 mm |

$\mathrm{Spring}\mathrm{limit}\mathrm{force}({\mathrm{F}}_{9}$) | 10,517 N |

$\mathrm{Minimum}\mathrm{load}\mathrm{stress}({\mathsf{\tau}}_{1}$) | 377,287 MPa |

$\mathrm{Maximum}\mathrm{load}\mathrm{stress}({\mathsf{\tau}}_{8}$) | 452,744 MPa |

$\mathrm{Solid}\mathrm{length}\mathrm{stress}({\mathsf{\tau}}_{9}$) | 793,600 MPa |

$\mathrm{Critical}\mathrm{speed}\mathrm{of}\mathrm{spring}(\mathrm{v}$) | 10,394 mps |

$\mathrm{Natural}\mathrm{frequency}\mathrm{of}\mathrm{spring}\mathrm{surge}(\mathrm{f}$) | 200,787 Hz |

$\mathrm{Deformation}\mathrm{energy}({\mathrm{W}}_{8}$) | 0.103 J |

$\mathrm{Wire}\mathrm{length}(\mathrm{l}$) | 509,767 mm |

$\mathrm{Spring}\mathrm{mass}(\mathrm{m}$) | 0.001 kg |

Electrical Systems | |
---|---|

$\mathrm{R}$ | 1 Ω |

$\mathrm{C}$ | 1 F |

$\mathrm{M}$ | 0.01 Ω |

${\mathrm{f}}_{\mathrm{v}}$ | 1 Ns/m |

Mechanical Systems | |

${\mathrm{m}}_{1}$ | 6.61 kg |

${\mathrm{m}}_{2}$ | 0.48 kg |

${\mathrm{k}}_{1}$ | 538.47 N/m |

${\mathrm{k}}_{2}$ | 1 N/m |

${\mathrm{k}}_{3}$ | 175 N/m |

${\mathrm{c}}_{1}$ | 98.55 Ns/m |

${\mathrm{c}}_{3}$ | 15.09 Ns/m |

Parameter | Value |
---|---|

$\mathrm{Rise}\mathrm{time}({\mathrm{T}}_{\mathrm{r}}$) | 0.5 s |

$\mathrm{Settling}\mathrm{time}({\mathrm{T}}_{\mathrm{s}}$) | 1.5 s |

% Overshoot | 0–0.9% |

Closed-loop stability | Stable |

Aggressive/Robust | Robust |

Parameter | Value |
---|---|

$\mathrm{Rise}\mathrm{time}({\mathrm{T}}_{\mathrm{r}}$) | 0.5 s |

$\mathrm{Settling}\mathrm{time}({\mathrm{T}}_{\mathrm{s}}$) | 1.5 s |

% Overshoot | 0–0.9 |

Closed-loop stability | Stable |

Parameter | Value |
---|---|

Controller | PID |

Time domain | Continuous-time |

Form | Parallel |

Source | Internal |

Tuning method | Transfer function based |

Zero-crossing detection | Enabled |

Compensator formula | $\mathrm{P}+\mathrm{I}\frac{1}{\mathrm{s}}+\mathrm{D}\frac{\mathrm{N}}{1+\mathrm{N}\frac{1}{\mathrm{s}}}$ |

Parameter | Tuned | Block |
---|---|---|

P | 183,706.2513 | 1 |

I | 232,966.3411 | 1 |

D | 30,144.1142 | 1 |

N | 206.9837 | 100 |

Parameter | Tuned | Block |
---|---|---|

Rise time | 0.752 s | 2.07 × 10^{5} s |

Settling time | 3.13 s | 3.69 × 10^{5} s |

Overshoot | 5.98% | 0% |

Peak | 1.06 | 1 |

Gain margin | 15.7 dB @ 10.3 rad/s | 109 dB @ 12.4 rad/s |

Phase margin | 69 deg @ 1.81 rad/s | 90 deg @ 1.06 × 10^{−5} rad/s |

Compensator formula | Stable | Stable |

Parameter | Tuned | Block |
---|---|---|

P | 225,775.49 | 225,775.49 |

I | 208,064.1457 | 208,064.1457 |

D | 21,780.4129 | 21,780.4129 |

N | 248.1711 | 248.1711 |

Parameter | Tuned | Block |
---|---|---|

Rise time | 0.562 s | 0.562 s |

Settling time | 1.05 s | 1.05 s |

Overshoot | 0.201% | 0.201% |

Peak | 1 | 1 |

Gain margin | 14.2 dB @ 8.81 rad/s | 14.2 dB @ 8.81 rad/s |

Phase margin | 69 deg @ 2.17 rad/s | 69 deg @ 2.17 rad/s |

Compensator formula | Stable | Stable |

Parameter | Requirement | Actual Controller Design | Actual Control System Design |
---|---|---|---|

$\mathrm{Rise}\mathrm{time}({\mathrm{T}}_{\mathrm{r}}$) | 0.6 s | 0.562 s | 0.562 s |

$\mathrm{Settling}\mathrm{time}({\mathrm{T}}_{\mathrm{s}}$) | 1.5 s | 1.05 s | 1.05 s |

% Overshoot | 0–0.9% | 0.201% | 0.201% |

Closed-loop stability | Stable | Stable | Stable |

Aggressive/Robust | Robust (0.69) | Robust (0.6) | Robust (0.6) |

Set-Point | Pressure |
---|---|

Off | 0 kPa |

Low | 350 kPa |

Mid | 750 kPa |

High | 1000 kPa |

**Table 18.**Acoustic system components [24].

Component | Recommendation |
---|---|

Solenoid valves | 1 per horn, 1 per system |

Manual isolation valve | 1 per horn, 1 per system |

Flow regulator | Air type based regulator |

Flex | Stainless steel material |

Industry Standard Configuration | Research System Components |
---|---|

Solid-state electronic timer | I-P transducer |

Solenoid valve | Spring and diaphragm valve |

No feedback transducer | Piezoresistive pressure sensor |

Industry Standard Configuration | Research System Components |

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**MDPI and ACS Style**

Mafokwane, T.; Kallon, D.V.V.
Control of Acoustic Energy Input for Cleaning of Industrial Boilers. *Acoustics* **2022**, *4*, 609-636.
https://doi.org/10.3390/acoustics4030038

**AMA Style**

Mafokwane T, Kallon DVV.
Control of Acoustic Energy Input for Cleaning of Industrial Boilers. *Acoustics*. 2022; 4(3):609-636.
https://doi.org/10.3390/acoustics4030038

**Chicago/Turabian Style**

Mafokwane, Thabang, and Daramy Vandi Von Kallon.
2022. "Control of Acoustic Energy Input for Cleaning of Industrial Boilers" *Acoustics* 4, no. 3: 609-636.
https://doi.org/10.3390/acoustics4030038