#
Mathematical Method of Rational Placement of Gas Fire Sensors^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

_{2}and carbon monoxide CO. The sensor is a sensitive element of the gas detector, which registers an increase in the concentration of these gases and gives a signal that the specified limit of the controlled parameter has exceeded. When smouldering, the concentration of CO increases from 20 to 100 mg/m

^{3}, but when a flame appears, it decreases. Also, during the smouldering of organic materials, hydrogen, H

_{2}, is released, the concentration of which, up to 10 mg/m

^{3}, can be detected in the presence of semiconductor sensors. When a small amount of material smoulders, the concentration of hazardous gases is small and they are distributed in the volume of the room due to diffusion. For gas detectors reacting at this stage of the fire, the requirement for the sensitivity threshold of the sensors is from 0.0001% for CO and 0.00001% for H

_{2}[1,2].

_{2}rise to the ceiling of the room where the sensors are located. The physical foundations of the first stage of the process of gas propagation from the heating source to the ceiling of the room were considered in [26,27]. The volume of gas accumulated during the ascent spreads along the ceiling reaches the sensors, which give a signal about the danger and the need to take measures to prevent fire. According to the coordinates of the sensors that gave a signal about the appearance of hazardous gases, it is possible to localize the heating source and automate the warning of a possible fire.

## 2. Location Methods for Gas Fire Sensors

#### 2.1. Problem Statement

_{2}and CO. The sensor signal will allow you to determine the possible localization of combustion products in the room to eliminate heat sources and prevent fire, or take measures to extinguish the fire. The H

_{2}and CO gases that appear as a result of heating are lighter than air, and therefore, will rise upwards. In addition, the density of the heated gas is less than the density of the air, so it is affected by a lifting force equal to the weight of the volume of air displaced by the gas.

_{2}and CO are recorded before the concentration of the gas emitted reaches a dangerous value.

- At the point;
- At several points;
- In the direction of the line (crack);
- On a surface area (surface rupture).

#### 2.2. Methods of Placement of Gas Sensors

#### 2.3. Physical Model of the Gas Movement Process

#### 2.3.1. Movement of Gas from the Source of Danger to the Ceiling of the Room

_{g}ejected in the first time interval (per second) is equal to the density of the gas q multiplied by the volume of the cone V (the radius of the base r and the height h) that the gas will occupy.

_{g}= V q = 1/3 π r

^{2}h q

- m
_{g}—gas mass, mg; - V—cone volume, m
^{3}; - r—cone base radius, m;
- h—cone height, m;
- q—gas density, mg/m
^{3}.

_{g}will be emitted in each unit of time. The main parameters reflecting the state of the gas are the dynamic pressure P, temperature and density q. In this case, the task is to determine the release of gas and the appearance of hazardous substance molecules. Thus, the physical effect of a fire hazardous process is the dynamic pressure created as a result of the emergence and propagation of gas molecules. In addition, the temperature decreases from the source to the GFS location.

_{2}, CO) tends to rise. The pressure on the gas acts in all directions. In addition, the gas molecules are affected by heating and buoyant force directed upwards, so the gas cloud takes on an upward shape. With a good approximation, we can assume that this is the shape of a cone. The top of the cone at each moment of time rests on a source of dangerous gas, where pressure is created.

_{1}= 1/3 h

_{1}π r

_{1}

^{2}= 1/3 h

_{1}S,

_{1}—cone height, r

_{1}—cone base radius, S—cone base area.

^{3}tg

^{2}α.

_{2}. Since the power of the source is constant, then the volume increment, the layer dV

_{2}, which the gas occupies in the second unit of time is equal to the volume that the gas occupied in the first unit of time V

_{1}. The volume increment layer dV

_{2}has the shape of a truncated cone. Its volume is equal to the difference between the volume V

_{2}= 1/3 π r

_{2}

^{3}= 1/3 π h

_{2}

^{3}and the volume of the cone V

_{1}= 1/3π r

_{1}

^{3}= 1/3 π h

_{1}

^{3}. Then, one can write that

_{2}= V

_{2}− V

_{1}= 1/3 π r

_{2}

^{3}− 1/3 π r

_{1}

^{3}= V

_{1},

_{2}is the increase in the volume of gas propagation in the second unit of time; hence we obtain V

_{2}= 2 V

_{1}= 2/3 π r

_{1}

^{3}.

_{2}is equal to

_{3}is equal to

_{1}= 1/3 π h

_{1}

^{3}tg

^{2}α,

_{n}becomes equal to the height of the room H. The upward velocity of the gas, v, will decrease in proportion to the increase in the area of the cone layer’s base.

#### 2.3.2. Gas Movement from the Place of Reaching the Ceiling to the Sensors

_{n}is equal (if the angle of the cone is 90°):

_{n}= π r

_{n}

^{2}= π h

_{n}

^{2}= π H

^{2}

_{n+}

_{1}.

_{n}

_{+1}= π r

_{n}

_{+1}

^{2}

_{n+}

_{1}, occupied by the gas for the second unit of time, is equal to the area S

_{1}, occupied by the gas for the first unit of time. The area of the ring is equal to the difference between the area S

_{n+}

_{1}= π r

_{n+}

_{1}

^{2}and the area S

_{n}= π r

_{n}

^{2}. Then, we obtain

_{n}

_{+1}= S

_{n}

_{+1}− S

_{n}= π r

_{n}

_{+1}

^{2}− π r

_{n}

^{2}= S

_{n},

_{n+}

_{1}is the increment of the gas propagation area for the second period of time of movement along the ceiling, from which, we obtain S

_{n+}

_{1}= 2 S

_{n}= 2 π r

_{n}

^{2}. It follows that the radius of the ring r

_{n+}

_{1}is

_{n+}

_{2}.

^{3}. The gas reaches a distance of 6 m to the walls of the room in 80 s. At the distance between the sensors A = 5, the radius of the ring is equal to A at 25 s, the volume of gas will then be 93 m

^{3}.

#### 2.3.3. Selecting Sensor Locations

_{1}; on the right with a higher frequency, with a distance A

_{2}. The black circle corresponds to the boundary of the gas area, which has risen to the top and spreads along the ceiling.

_{g}of a given volume of gas, as shown in (1), with guaranteed registration at a constant density q (which can be considered the maximum, i.e., the upper estimate of the mass of the gas, taking into account diffusion and pressure change)

_{g max}= V

_{g}q

_{min}= V

_{up}for registration occurs when the gas cloud reaches the ceiling height H, (the level of the sensors). Its mass is equal to

_{g min}= V

_{min}q = 1/3 π r

_{n}

^{2}H q

_{x}on the X axis and T

_{y}on the Y axis; i.e., the area of the room S is equal to

_{11}, located in the first row, are equal to

_{12}are

_{1n}in the first row are

_{12}, located in the second row, are

_{2}-th sensor D

_{2n}, located in the second row, are

_{y}= 2T

_{y}/A, where A/2 is the distance between the rows. Thus, it is possible to obtain an estimate of the total number of sensors d

## 3. Results and Discussion

_{m}.

_{m}= 67.02 m

^{3}. The radius of the base of the cone, with the assumptions made, will be 4 m. This volume can be taken as the maximum permissible value, since at lower limit values, the gas will not yet reach the sensors.

_{x}= 20 m, T

_{y}= 10 m; the height of the room is H = 3 m.

_{n}= 3 m, then, according to Formula (13), A = 5.2 m. Let the sensors be located no further than 4.34 m from each other. Let us calculate the coordinates of the sensors and their number.

_{11}, located in the first row, are equal to

_{12}are

_{13}in the first row are

_{x}− x

_{13}= 20 − 17.5 = 2.5, which is less than 3C = 7.5, i.e., the step of placement of sensors, the first row is filled. You can consider the need to fill the remainder to the wall of the room. There are three sensors here.

_{21}, located in the second row, are equal to

_{22}, located in the second row, are

_{x}− x

_{22}= 20 − 13.75 = 6.25, which is less than 3C = 7.5, i.e., the step of placement of sensors, the second row is also filled, and it is necessary to consider the need to fill the remainder to the wall of the room. There are two sensors here.

_{31}, located in the third row, are equal to

_{32}are equal to

_{33}in the third row are

_{x}− x

_{33}= 20 − 17.5 = 2.5, which is less than 3C = 7.5, i.e., the step of placement of sensors, the third row is filled. You can consider the need to fill the remainder to the wall of the room. There are three sensors here.

_{41}, located in the fourth row, are equal to

_{42}, located in the fourth row, are equal to

_{x}− x

_{42}= 20 − 13.75 = 6.25, which is less than 3C = 7.5, i.e., the step of placement of sensors, the fourth row is also filled, and it is necessary to consider the need to fill the remainder to the wall of the room. There are two sensors here.

_{y}− y

_{42}= 10 − 8.68 = 1.32, which is less than A/2 = 2.17, i.e., the step of placement of sensors along the y-axis, the ceiling surface is filled and the problem is solved.

## 4. Conclusions

_{2}and CO for early detection of fire (accident) is presented. The model-based method for calculating the location of gas analyzers makes it possible to diagnose and predict fire hazard at technological facilities based on gas control, and compare the obtained values with the maximum permissible values. The calculation of the conditions for the rational placement of gas detectors for different heights of technological premises is carried out.

_{2}and CO gas control in technological design.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Bywater, D. Detection of Real Fires by Carbon Monoxide Detectors—Foreign Experience. The Results of 10 Years of Research Lead to a Leap in Fire Detection Technology. Available online: https://www.aktivsb.ru/statii/obnaruzhenie_realnykh_pozharov_detektorami_ugarnogo_gaza_zarubezhnyy_opyt.html (accessed on 5 October 2023).
- Gas Detector. Available online: https://en.wikipedia.org/wiki/Gas_detector (accessed on 5 October 2023).
- Bakanov, V. Fire detectors with a gas sensor in the light of current regulatory requirements. Prot. Technol.
**2014**, 4, 71–78. [Google Scholar] - EN 50291-1 2010-04 + A1 2012-06; Electrical Apparatus for the Detection of Carbon Monoxide in Domestic Premises—Part 1: Test Methods and Performance Requirements. iTeh: Etobicoke, ON, Canada, 2012.
- Fedorov, A.; Bytcinskaya, T.; Lukyanchenko, A.; Hung, T.D. Trends in the development of automatic fire detectors. Technol. Technosphere Saf.
**2009**, 23, 111–114. Available online: https://cyberleninka.ru/article/n/tendentsii-razvitiya-avtomaticheskih-pozharnyh-izveschateley-1 (accessed on 5 October 2023). - Fire Alarm Systems and Automation of Fire Protection Systems. Designing and Regulations Rules. CΠ 484.1311500.2020. Available online: https://docs.cntd.ru/document/566249686 (accessed on 5 October 2023).
- Siting Gas Detectors. Available online: https://www.internationalgasdetectors.com/wp-content/uploads/2016/04/SITING-GAS-DETECTORS-BS023.pdf (accessed on 5 October 2023).
- Gas Detector Handbook. Available online: https://hsseworld.com/wp-content/uploads/2022/01/Gas-Detection-Handbook.pdf (accessed on 5 October 2023).
- Fixed Gas Detector. Available online: https://www.safetygas.com/media/productattach/e/n/en-gas_and_flame_detection_gazdetect_2021.ed01-web.pdf (accessed on 5 October 2023).
- Xgard Gas Detectors. Available online: https://evikontroll.ee/pdf/crowcon/Xgard_UM_EN.pdf (accessed on 5 October 2023).
- AP 1027: Cleaning and Sanitizing Portable Gas Detectors. Available online: https://www.gfgsafety.com/fileadmin/templates/img/GfG-Branches/GfG-USA/Application_Notes/AP1027_cleaning_and_sanitizing_portable_gas_detectors_14_MAY_20_lower_res.pdf (accessed on 5 October 2023).
- Available online: https://www.canarysense.com/pdfs/resources/brochures/Honeywell-Gas-Detection-Book2.pdf (accessed on 5 October 2023).
- Guidelines for Placing Sensors on Flammable Gases. Available online: https://www.seitron.ru/assets/docs/dokumenty/razmeshenie%20sensorov.pdf (accessed on 5 October 2023).
- Basic Guidelines for Selection of Fire and Gas Detectors Inst Tools. Fire & Gas System. Available online: https://instrumentationtools.com/selection-of-fire-and-gas-detectors (accessed on 5 October 2023).
- Brattery, A.; Gavelly, F.; Hansen, O.R.; Scott, S.G. Using CFD to analyze gas detector location in process facilities. In Proceedings of the Mary Kay O’Connor Process Safety Center Symposium 2011, College Station, TX, USA, 25–27 October 2011. [Google Scholar]
- Okeke, R.O.; Ehikhamenle, M. Design and simulation of gas and fire detector and alarm system with water sprinkle. Int. J. Eng. Res. Gen. Sci.
**2017**, 5, 216–225. [Google Scholar] - Zhen, T.; Klise, K.A.; Cunningham, S.; Marszal, E.; Laird, C.D. A mathematical programming approach for the optimal placement of flame detectors in petrochemical facilities. Process Saf. Environ. Prot.
**2019**, 132, 47–58. [Google Scholar] [CrossRef] - Güllüce, Y.; Çelik, R.N. FireAnalyst: An effective system for detecting fire geolocation and fire behavior in forests using mathematical modeling. Turk. J. Agric. For.
**2020**, 44, 127–139. [Google Scholar] [CrossRef] - Kaliyev, D.I.; Shvets, O.Y. Convolutional neural networks for solving fire detection problems based on aerial photography. Program Syst. Theory Appl.
**2022**, 13, 195–213. (In Russian) [Google Scholar] - Zhdanova, A.O.; Kopylov, N.P.; Kropotova, S.S.; Kuznetsov, G.V. Characteristics of a Typical Indoor Seat of Fire. J. Eng. Phys. Thermophy
**2023**, 96, 143–149. [Google Scholar] [CrossRef] - Gupta, A.K.; Kumar, R.; Yadav, P.K.; Naveen, M. Fire Safety through Mathematical Modeling. Available online: https://www.researchgate.net/publication/228763095_Fire_safety_through_mathematical_modelling (accessed on 5 October 2023).
- Lukyanchenko, A.A.; Teterin, N.M.; Topolskii, N.G.; Lukyanchenko, A.A.; Fedorov, A.V. Integrated automated system for early fire detection and environmental monitoring “Kassandra”. Nauchn. Tech. Catalog
**2014**, 52. [Google Scholar] - Yutyaev, A.E.; Iakunchikov, E.N.; Oganesyan, A.S.; Agafonov, V.V. Evaluation of design solutions and technological systems of coal mines taking into account the risk. Ugol
**2019**, 7, 52–57. [Google Scholar] [CrossRef] - Petrov, A.E.; Fedorov, A.V.; Kochegarov, A.V.; Lomaev, E.N.; Preobrazhenskiy, A.P. The Analysis of Network Models for the Design of Industrial and Fire Safety Systems for Oil Refineries. IOP Conf. Ser. Earth Environ. Sci.
**2021**, 808, 012024. [Google Scholar] [CrossRef] - Petrov, A.E. Tensor Method and Dual Networks in Electrical Engineering. Russ. Electr. Eng.
**2008**, 79, 645–654. [Google Scholar] [CrossRef] - Luk’yanchenko, A.A.; Petrov, A.E.; Fedorov, A.V.; Denisov, A.N. Method of Rational Location of Gas Sensors for Early Fire Detection Based On Gas Control Technology. J. Adv. Res. Dyn. Control. Syst.
**2020**, 12, 1293–1306. [Google Scholar] [CrossRef] - Petrov, A.; Fedorov, A.; Mintsaev, M.; Ilyukhin, A.; Marsov, V. Theoretical Foundations and Methods for the Rational Location of Gas Fire Sensors Based on Gas Control Technology. Mathematical Modelling of Gas Fire Sensors Location for Early Fire Detection. In Proceedings of the XV International Scientific Conference “INTERAGROMASH 2022”, Rostov-on-Don, Russia, 25–27 May 2022; Lecture Notes in Networks and Systems; Beskopylny, A., Shamtsyan, M., Artiukh, V., Eds.; Springer: Cham, Switzerland, 2023; Volume 574, pp. 1658–1667. [Google Scholar] [CrossRef]

**Figure 1.**Gas distribution from a source located on the room floor. The upper base of the gas cloud cone is flat.

**Figure 2.**Views from above. Variants of hexagonal arrangement of sensors with a distance between them A

_{1}and A

_{2}. The black circle represents the boundary of the area of gas that has risen to the ceiling.

**Figure 5.**Location of sensors on the ceiling of a room 20 m long, 10 m wide, with a height of 3 m. Overhead view.

Time Since the Beginning of the Ejection, s | The Height of the Gas Cone at Each Moment | Gas Cloud Rise Rate | Room Height | The Difference between the Heights of the Room and the Cone | Accumulated Volume of Gas Cone |
---|---|---|---|---|---|

t, s | r = h, m | v, m/sec | H, m | H − ht | V, m^{3} |

1 | 1.00 | 1.0000 | 4.0 | 3.00 | 1.05 |

10 | 2.15 | 0.4445 | 4.0 | 1.85 | 10.47 |

20 | 2.71 | 0.2482 | 4.0 | 1.29 | 20.94 |

50 | 3.68 | 0.1271 | 4.0 | 0.32 | 52.36 |

60 | 3.91 | 0.1119 | 4.0 | 0.09 | 62.83 |

64 | 4.00 | 0.0209 | 4.0 | 0.00 | 67.02 |

65 | 4.02 | 0.0207 | 4.0 | −0.02 | 68.03 |

Time of Movement on the Ceiling, s | The Radius of the Ring r_{i}, m | The Speed of the Gas Ring | Distance between Sensors, A, m | A − r_{i} | Gas Cloud Volume V + Vup | The Length of the Walls of the Room, L, m | Distance from the Gas Ring to the Walls L − r_{i} |
---|---|---|---|---|---|---|---|

5 | 2.24 | 2.2361 | 6.0 | 3.76 | 72.26 | 15.00 | 12.76 |

10 | 3.16 | 0.9262 | 6.0 | 2.84 | 77.49 | 15.00 | 11.84 |

20 | 4.47 | 0.5992 | 6.0 | 1.53 | 87.96 | 15.00 | 10.53 |

30 | 5.48 | 0.4772 | 6.0 | 0.52 | 98.44 | 15.00 | 9.52 |

40 | 6.32 | 0.4085 | 6.0 | −0.32 | 108.91 | 15.00 | 8.68 |

50 | 7.07 | 0.3629 | 6.0 | −1.07 | 119.38 | 15.00 | 7.93 |

60 | 7.75 | 0.3298 | 6.0 | −1.75 | 129.85 | 15.00 | 7.25 |

70 | 8.37 | 0.3043 | 6.0 | −2.37 | 136.14 | 15.00 | 6.63 |

80 | 8.94 | 0.2840 | 6.0 | −2.94 | 138.23 | 15.00 | 6.06 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Petrov, A.
Mathematical Method of Rational Placement of Gas Fire Sensors. *Sensors* **2023**, *23*, 8349.
https://doi.org/10.3390/s23208349

**AMA Style**

Petrov A.
Mathematical Method of Rational Placement of Gas Fire Sensors. *Sensors*. 2023; 23(20):8349.
https://doi.org/10.3390/s23208349

**Chicago/Turabian Style**

Petrov, Andrey.
2023. "Mathematical Method of Rational Placement of Gas Fire Sensors" *Sensors* 23, no. 20: 8349.
https://doi.org/10.3390/s23208349