The paper covers the temperature thermal control of the water distribution system by predicting the temperature exceeding the safe level at pipes' locations and shutting off the water access to the main distribution pipe. The advantages of predictive thermal control over setting the maximum allowed temperature are provided. The prediction uses a transient heat conduction model with an error function approximation for arbitrary distance from the surface.

It is provided analysis of existing theoretical models for predicting soil temperature and covered complexity of thermal diffusivity surface temperature identification in real time.  The resolution of this problem consists in identifying effective thermal diffusivity and the surface temperature by using least square operator to minimize difference between monitoring and theoretical temperatures as the solution of the transient conduction equation.  Solving such ill-posed problem  with defined operator require solving a nonlinear operator equation. The equations are solved using the theory of hypernumbers. The objectives for using method of hypernumbers for solving nonlinear equation of high complexity are derived from hypernumber sequence convergence. The method can be extended for a new complex thermal conduction model.An example of such thermal control implementation using temperature sensors, minicomputers Raspberry Pi and a solenoid valve is provided.

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1. Meadows, FEATURE: How Wildfire-damaged Plastic Pipes Contaminate Drinking water – and What We Can Do About It. [Intranet] MAVEN’S NOTEBOOK | California Water News Central. 2022. Available from: https://mavensnotebook.com/2022/08/03/feature-how-wildfire-damaged-plastic-pipes-contaminate-drinking-water-and-what-we-can-do-about-it/
2. Metz AJ, Fischer EC, Wham BP. Behavior of Service Lateral Pipes during Wildfires: Testing Methodologies and Impact on Drinking Water Contamination. ACS ES&T Water. 2023 Feb 1;3(2): 275-86.
3. Richter E.Simulation of Heat Transfer Through Soil for the Investigation of Wildfire Impacts on Buried Utilities. : Oregon State University. {Intranet} 2018. Available from: https://ir.library.oregonstate.edu/concern/graduate_projects/kk91ft25w
4. Janssens,ML Predicting Temperature Profiles in Soil and the Effect of Heat Conduction on Buried Thermoplastic Pipes During a Wildfire. Fire10 and Materials 2025 May11. https://onlinelibrary.wiley.com/doi/epdf/10.1002/fam.3301
5. Massman WJ. The challenges of an in situ valuation of a nonequilibrium model of soil heat and moisture dynamics during fires. Hydrology and earth system sciences. 2021 Feb17;25(2):685-709.
6. Robichaud PR, Massman WJ, Bova A, Girona-Garcia A, Alfaro-Leranoz A, and Givson NE. Comparing modeled soul temperature and moisture dynamics during prescribed fires, slash-pile burns and wildfires. International Journal of Wildland Wires, 2025;24:WF22082
7. Brady MK, Dickinson MB, Miesel JR, Wonkka CL, Kavanagh KL, Lodge AG, et al. Soil Heating in Fire (SheFire): A model and measurement method for estimating soil heating and effects during wildland fires. Ecological Applications. 2022 Jun 13;32(6).
8. Lee, S. Transient Conduction in Semi-Infinite Solids. [Internet] Numberanalytics.com. 2025. Available from: https://www.numberanalytics.com/blog/transient-conduction-semi-infinite-solids#google_vignette
9. Miljhovic N. (Transient Conduction. [Intranet] n.d. Available from: https://etrl.mechanical.illinois.edu/pdf/ME320/Lecture%2013%20-%20Transient%20Conduction%20-%20Semi-Infinite%20Body.pdf
10. Cengel Y. Second Edition Heat Transfer A Practical Approach [Internet]. Available from: https://spada.uns.ac.id/pluginfile.php/150729/mod_label/intro/ebook_a_cengel.pdf
11. Mendez P. Transient Conduction. [Internet]. Studocu. 2020. Available from: https://www.studocu.com/row/document/university-of-port-harcourt/petroleum-engineering/transient-conduction-semi-infinite-solid/94774400
12. Pamuk TM. (2024). Numerical Analysis of Transient Soil Temperature Variation During Wildfire. Environmental Research and Technology, 2024;7(4):578-87.
13. Error and Complementary Error Functions. https://www.mhtlab.uwaterloo.ca/courses/me755/web_chap2.pdf
14. Howard RM. Arbitrarily Accurate Analytical Approximation for the Error Function. Mathematical and Computational Applications, 2022 Feb 9;27(1): 14-4.
15. Burgin M. Nonlinear partial differential equations in extrafunctions. Integr. Math. Theory App. 2010; 2(1): 17–50.
16. Burgin M. Hypernumbers and extrafunctions: extending the classical calculus; Springer: New York, NY, USA, 2012.
17. Burgin M, Dantsker AM. A method of solving operator equations by means of the theory of hypernumbers. Not. Nat. Acad. Sci. Kurt. 1995: 8; 27–30.
18. Burgin M, Dantsker A. Pollution information acquisition and discovering pollution sources. Journals Proceedings. 2022; 81(1).
19. Dantsker A, Burgin M. Monitoring Thermal Conditions and Finding Sources of Overheating. The 2021 Summit of the International Society for the Study of Information [Internet]. 2022 Mar 24; Available from: http://dx.doi.org/10.3390/proceedings2022081038
20. Isaacson, KP, Proctor CR, Wang QE, Edwards EY, Noh Y, Shah AD., et al (2021). Drinking water contamination from the thermal degradation of plastics: implications for wildfire and structure fire respons. Environmental Scienc: Water Research &Technology, 2021;7(2): 274-84.

The paper covers the temperature thermal control of the water distribution system by predicting the temperature exceeding the safe level at pipes' locations and shutting off the water access to the main distribution pipe. The advantages of predictive thermal control over setting the maximum allowed temperature are provided. The prediction uses a transient heat conduction model with an error function approximation for arbitrary distance from the surface.

It is provided analysis of existing theoretical models for predicting soil temperature and covered complexity of thermal diffusivity surface temperature identification in real time.  The resolution of this problem consists in identifying effective thermal diffusivity and the surface temperature by using least square operator to minimize difference between monitoring and theoretical temperatures as the solution of the transient conduction equation.  Solving such ill-posed problem  with defined operator require solving a nonlinear operator equation. The equations are solved using the theory of hypernumbers. The objectives for using method of hypernumbers for solving nonlinear equation of high complexity are derived from hypernumber sequence convergence. The method can be extended for a new complex thermal conduction model.An example of such thermal control implementation using temperature sensors, minicomputers Raspberry Pi and a solenoid valve is provided.

🔒 This is a subscription article

Full text is available to subscribers and institutional members. Please choose an option below to access it.

1. Meadows, FEATURE: How Wildfire-damaged Plastic Pipes Contaminate Drinking water – and What We Can Do About It. [Intranet] MAVEN’S NOTEBOOK | California Water News Central. 2022. Available from: https://mavensnotebook.com/2022/08/03/feature-how-wildfire-damaged-plastic-pipes-contaminate-drinking-water-and-what-we-can-do-about-it/
2. Metz AJ, Fischer EC, Wham BP. Behavior of Service Lateral Pipes during Wildfires: Testing Methodologies and Impact on Drinking Water Contamination. ACS ES&T Water. 2023 Feb 1;3(2): 275-86.
3. Richter E.Simulation of Heat Transfer Through Soil for the Investigation of Wildfire Impacts on Buried Utilities. : Oregon State University. {Intranet} 2018. Available from: https://ir.library.oregonstate.edu/concern/graduate_projects/kk91ft25w
4. Janssens,ML Predicting Temperature Profiles in Soil and the Effect of Heat Conduction on Buried Thermoplastic Pipes During a Wildfire. Fire10 and Materials 2025 May11. https://onlinelibrary.wiley.com/doi/epdf/10.1002/fam.3301
5. Massman WJ. The challenges of an in situ valuation of a nonequilibrium model of soil heat and moisture dynamics during fires. Hydrology and earth system sciences. 2021 Feb17;25(2):685-709.
6. Robichaud PR, Massman WJ, Bova A, Girona-Garcia A, Alfaro-Leranoz A, and Givson NE. Comparing modeled soul temperature and moisture dynamics during prescribed fires, slash-pile burns and wildfires. International Journal of Wildland Wires, 2025;24:WF22082
7. Brady MK, Dickinson MB, Miesel JR, Wonkka CL, Kavanagh KL, Lodge AG, et al. Soil Heating in Fire (SheFire): A model and measurement method for estimating soil heating and effects during wildland fires. Ecological Applications. 2022 Jun 13;32(6).
8. Lee, S. Transient Conduction in Semi-Infinite Solids. [Internet] Numberanalytics.com. 2025. Available from: https://www.numberanalytics.com/blog/transient-conduction-semi-infinite-solids#google_vignette
9. Miljhovic N. (Transient Conduction. [Intranet] n.d. Available from: https://etrl.mechanical.illinois.edu/pdf/ME320/Lecture%2013%20-%20Transient%20Conduction%20-%20Semi-Infinite%20Body.pdf
10. Cengel Y. Second Edition Heat Transfer A Practical Approach [Internet]. Available from: https://spada.uns.ac.id/pluginfile.php/150729/mod_label/intro/ebook_a_cengel.pdf
11. Mendez P. Transient Conduction. [Internet]. Studocu. 2020. Available from: https://www.studocu.com/row/document/university-of-port-harcourt/petroleum-engineering/transient-conduction-semi-infinite-solid/94774400
12. Pamuk TM. (2024). Numerical Analysis of Transient Soil Temperature Variation During Wildfire. Environmental Research and Technology, 2024;7(4):578-87.
13. Error and Complementary Error Functions. https://www.mhtlab.uwaterloo.ca/courses/me755/web_chap2.pdf
14. Howard RM. Arbitrarily Accurate Analytical Approximation for the Error Function. Mathematical and Computational Applications, 2022 Feb 9;27(1): 14-4.
15. Burgin M. Nonlinear partial differential equations in extrafunctions. Integr. Math. Theory App. 2010; 2(1): 17–50.
16. Burgin M. Hypernumbers and extrafunctions: extending the classical calculus; Springer: New York, NY, USA, 2012.
17. Burgin M, Dantsker AM. A method of solving operator equations by means of the theory of hypernumbers. Not. Nat. Acad. Sci. Kurt. 1995: 8; 27–30.
18. Burgin M, Dantsker A. Pollution information acquisition and discovering pollution sources. Journals Proceedings. 2022; 81(1).
19. Dantsker A, Burgin M. Monitoring Thermal Conditions and Finding Sources of Overheating. The 2021 Summit of the International Society for the Study of Information [Internet]. 2022 Mar 24; Available from: http://dx.doi.org/10.3390/proceedings2022081038
20. Isaacson, KP, Proctor CR, Wang QE, Edwards EY, Noh Y, Shah AD., et al (2021). Drinking water contamination from the thermal degradation of plastics: implications for wildfire and structure fire respons. Environmental Scienc: Water Research &Technology, 2021;7(2): 274-84.