MATHEMATICAL MODELING OF HEAT EXCHANGE IN DIRECT FLAT CHANNELS AND DIRECT ROUND PIPES WITH ROUGH WALLS UNDER THE SYMMETRIC HEAT SUPPLY

Objectives. The aim of present work was to carry out mathematical modelling of heat transfer with symmetrical heating in flat channels and round pipes with rough walls.

Methods. The calculation was carried out using the L'Hôpital-Bernoulli's method. The solution of the problem of intensified heat transfer in a round tube with rough walls was obtained using the Lyon's integral.

Results. Different from existing theories, a methodology of theoretical computational heat transfer determination for flat rough channels and round pipes with rough walls is developed on the basis of the principle of full viscosity superposition in a turbulent boundary layer. The analysis of the calculated heat transfer and hydroresistivity values for flat rough channels and round rough pipes shows that the increase in heat transfer is always less than the corresponding increase in hydraulic resistance, which is a disadvantage as compared to channels with turbulators, with all else being equal. The results of calculating the heat transfer for channels with rough walls in an extended range of determinant parameters, which differ significantly from the corresponding data for the channels with turbulators, determine the level of heat exchange intensification.

Conclusion. An increase in the calculated values of the relative average heat transfer Nu/NuGL for flat rough channels and rough pipes with very high values of the relative roughness is significantly contributed by both an increase in the relative roughness height and an increase in the Reynolds number Re. In comparison with empirical dependencies, the main advantage of solutions for averaged heat transfer in rough flat channels and round pipes under symmetrical thermal load obtained according to the developed theory is that they allow the calculation of heat exchange in rough pipes to be made in the case of large and very large relative heights of roughness protrusions, including large Reynolds numbers, typical for pipes of small diameters and narrow flat channels. An increase in the relative heat exchange in air due to an increase in the relative height of the roughness or the Reynolds number is accompanied by an even more significant increase in the hydraulic resistance. Calculated data on averaged heat transfer obtained in the work showed that in the range of determinant parameters for flat rough channels with symmetrical thermal loading, the average heat transfer is higher by (4.811.7)% as compared to round rough pipes – all other things being equal. 

1. Kalinin E.K., Dreytser G.A., Kopp I.Z. i dr. Effektivnye poverkhnosti teploobmena. M.: Energoatomizdat; 1998. 408 s. [Kalinin E.K., Dreytser G.A., Kopp I.Z. et al. Effective heat-exchange suraces. Moscow: Energoatomizdat; 1998. 408 p. (in Russ.)]

2. Lobanov I.E. Matematicheskoe modelirovanie intensifitsirovannogo teploobmena pri turbulentnom techenii v kanalakh. Dissertatsiya na soiskanie uchenoy stepeni doktora tekhnicheskikh nauk. Moskva; 2005. 632 s. [Lobanov I.E. Mathematical modeling of intensified heat-exchange under the turbulent flow in the channels. Doctor of technical sciences dissertation. Moscow; 2005. 632 p. (in Russ.)]

3. Lobanov I.E., Shteyn L.M. Perspektivnye teploobmennye apparaty s intensifitsirovannym teploobmenom dlya metallurgicheskogo proizvodstva. (Obshchaya teoriya intensifitsirovannogo teploobmena dlya teploobmennykh apparatov, primenyaemykh v sovremennom metallurgicheskom proizvodstve.) V 4-kh tomakh. Tom I. Matematicheskoe modelirovanie intensifitsirovannogo teploobmena pri turbulentnom techenii v kanalakh s primeneniem osnovnykh analiticheskikh i chislennykh metodov. M.: Izdatel'stvo Assotsiatsii stroitel'nykh vuzov; 2009. 405 s. [Lobanov I.E., Shteyn L.M. Perspective heatexchange apparatus with intensified heat-exchange for metallurgical production. (General theory of intensified heat-exchange for heat-exchange apparatus used in modern metallurgical production). In 4 volumes. Volume I. Mathematical modeling of intensified heat-exchange under the turbulent flow in channels using major analytical and numeric methods. Moscow: Izdatel'stvo Assotsiatsii stroitel'nykh vuzov; 2009. 405 p. (in Russ.)]

4. Lobanov I.E., Shteyn L.M. Perspektivnye teploobmennye apparaty s intensifitsirovannym teploobmenom dlya metallurgicheskogo proizvodstva. (Obshchaya teoriya intensifitsirovannogo teploobmena dlya teploobmennykh apparatov, primenyaemykh v sovremennom metallurgicheskom proizvodstve.) V 4-kh tomakh. Tom II. Matematicheskoe modelirovanie intensifitsirovannogo teploobmena pri turbulentnom techenii v kanalakh s primeneniem neosnovnykh analiticheskikh i chislennykh metodov. M.: Izdatel'stvo Assotsiatsii stroitel'nykh vuzov; 2010. 290 s. [Lobanov I.E., Shteyn L.M. Perspective heat-exchange apparatus with intensified heat-exchange for metallurgical production. (General theory of intensified heat-exchange for heat-exchange apparatus used in modern metallurgical production). In 4 volumes. Volume II. Mathematical modeling of intensified heat-exchange under the turbulent flow in channels using minor analytical and numeric methods. Moscow: Izdatel'stvo Assotsiatsii stroitel'nykh vuzov; 2010. 290 p. (in Russ.)]

5. Lobanov I.E., Shteyn L.M. Perspektivnye teploobmennye apparaty s intensifitsirovannym teploobmenom dlya metallurgicheskogo proizvodstva. (Obshchaya teoriya intensifitsirovannogo teploobmena dlya teploobmennykh apparatov, primenyaemykh v sovremennom metallurgicheskom proizvodstve.) V 4-kh tomakh. Tom III. Matematicheskoe modelirovanie intensifitsirovannogo teploobmena pri turbulentnom techenii v kanalakh s primeneniem mnogosloynykh, supermnogosloynykh i kompaundnykh modeley turbulentnogo pogranichnogo sloya. M.: MGAKKhiS; 2010. 288 s. [Lobanov I.E., Shteyn L.M. Perspective heat-exchange apparatus with intensified heat-exchange for metallurgical production. (General theory of intensified heat-exchange for heat-exchange apparatus used in modern metallurgical production). In 4 volumes. Volume III. Mathematical modeling of intensified heatexchange under the turbulent flow in channels using multi-layer, super-multi-layer and compound models of turbulent boundary layer. Moscow: MGAKKhiS; 2010. 288 p. (in Russ.)]

6. Lobanov I.E., Shteyn L.M. Perspektivnye teploobmennye apparaty s intensifitsirovannym teploobmenom dlya metallurgicheskogo proizvodstva. (Obshchaya teoriya intensifitsirovannogo teploobmena dlya teploobmennykh apparatov, primenyaemykh v sovremennom metallurgicheskom proizvodstve). V 4-kh tomakh Tom IV. Spetsial'nye aspekty matematicheskogo modelirovaniya gidrogazodinamiki, teploobmena, a takzhe teploperedachi v teploobmennykh apparatakh s intensifitsirovannym teploobmenom. M.: MGAKKhiS; 2011. 343 s. [Lobanov I.E., Shteyn L.M. Perspective heat-exchange apparatus with intensified heat-exchange for metallurgical production. (General theory of intensified heat-exchange for heat-exchange apparatus used in modern metallurgical production). In 4 volumes. Volume IV. Special aspects of mathematical modeling of hydro-gas-dynamics, heat-exchange, and heat transfer in heat-exchange apparatus with intensified heat-exchange. Moscow: MGAKKhiS; 2011. 343 p. (in Russ.)] 7. Lobanov I.E., Dotsenko A.I. Matematicheskoe modelirovanie predel'nogo teploobmena dlya turbulizirovannogo potoka v kanalakh. M.: MIKKhiS; 2008. 194 s. [Lobanov I.E., Dotsenko A.I. Mathematical modeling of terminal heat-exchange for turbulated flow in channels. Moscow: MIKKhiS; 2008. 194 p. (in Russ.)] 8. Ievlev V.M. Chislennoe modelirovanie turbulentnykh techeniy. M.: Nauka; 1990. 215 s. [Ievlev V.M. Numeric modeling of turbulent flows. Moscow: Nauka; 1990. 215 p. (in Russ.)] 9. Lyakhov V.K. Metod otnositel'nogo sootvetstviya pri raschetakh turbulentnykh pristenochnykh potokov. Saratov: Izdatel'stvo Saratovskogo universiteta; 1975. 123 s. [Lyakhov V.K. Method of relative correspondence during the calculations of turbulent near-wall flows. Saratov: Izdatel'stvo Saratovskogo universiteta; 1975. 123 p. (in Russ.)] 10. Lyakhov V.K., Migalin V.K. Effekt teplovoy, ili diffuzionnoy, sherokhovatosti. Saratov: Izdatel'stvo Saratovskogo universiteta; 1989. 176 s. [Lyakhov V.K., Migalin V.K. Effect of heat, or diffusion roughness. Saratov: Izdatel'stvo Saratovskogo universiteta; 1989. 176 p. (in Russ.)] 11. Millionshchikov M.D. Turbulentnye techeniya v pogranichnom sloe i v trubakh. M.: Nauka; 1969. 52 s. [Millionshchikov M.D. Turbulent flows in boundary layer and in pipes. Moscow: Nauka; 1969. 52 p. (in Russ.)] 12. Millionshchikov M.D. Turbulentnye techeniya v pristenochnom sloe i v trubakh. Atomnaya energiya. 1970; 28(3):207-220. [Millionshchikov M.D. Turbulent flows in near-wall layer and in pipes. Atomic Energy. 1970; 28(3):207-220. (in Russ.)] 13. Millionshchikov M.D. Turbulentnyy teplo- i massoobmen v trubakh s gladkimi i sherokhovatymi stenkami. Atomnaya energiya. 1971; 31(3):199-204. [Millionshchikov M.D. Turbulent heatand mass-exchange in pipes with smooth and rough walls. Atomic Energy. 1971; 31(3):199-204. (in Russ.)] 14. Kutateladze S.S. Osnovy teorii teploobmena. M.: Atomizdat; 1979. 416 s. [Kutateladze S.S. Fundamentals of heat-exchange theory. Moscow: Atomizdat; 1979. 416 p. (in Russ.)] 15. Dreytser G.A., Lobanov I.E. Issledovanie predel'noy intensifikatsii teploobmena v trubakh za schet iskusstvennoy turbulizatsii potoka. Teplofizika vysokikh temperatur. 2002;40(6):958-963. [Dreytser G.A., Lobanov I.E. Study of terminal heat-exchange intensification in pipes with artificial flow turbulisation. High Temperature. 2002;40(6):958-963. (in Russ.)] 16. Dreytser G.A., Lobanov I.E. Predel'naya intensifikatsiya teploobmena v trubakh za schet iskusstvennoy turbulizatsii potoka. Inzhenerno-fizicheskiy zhurnal. 2003;76(1):46-51. [Dreytser G.A., Lobanov I.E. Terminal heat-exchange intensification in pipes with artificial flow turbulisationJournal of Engineering Physics and Thermophysics. 2003;76(1):46-51. (in Russ.)] 17. Novikov I.I., Voskresenskiy K.D. Prikladnaya termodinamika i teploperedacha. M.: Gosatomizdat; 1961. 548 s. [Novikov I.I., Voskresenskiy K.D. Applied thermodynamics and heat transfer. Moscow: Gosatomizdat; 1961. 548 p. (in Russ.)] 18. Novikov I.I., Voskresenskiy K.D. Prikladnaya termodinamika i teploperedacha. M.: Atomizdat; 1977. 349 s. [Novikov I.I., Voskresenskiy K.D. Applied thermodynamics and heat transfer. Moscow: Atomizdat; 1977. 349 p. (in Russ.)] 19. Lobanov I.E. Matematicheskoe modelirovanie predel'nogo teploobmena za schet turbulizatsii potoka pri turbulentnom techenii v ploskikh kanalakh s turbulizatorami. Materialy XXXIV Akademicheskikh chteniy po kosmonavtike ―Aktual'nye problemy rossiyskoy kosmonavtiki‖. Moskva; 2010. S. 200-202. [Lobanov I.E. Mathematical modeling of terminal heat-exchange with flow turbulisation under the turbulent flow in the flat channels with turbulisators. Materials of XXXIV Academic readings about cosmonautics ―Relevant problems of Russian cosmonautics‖. Moscow; 2010. P. 200-202. (in Russ.)] 20. Lobanov I.E., Fleytlikh B.B. Modelirovanie intensifitsirovannogo teploobmena pri turbulentnom techenii v ploskikh kanalakh s periodicheski poverkhnostno raspolozhennymi turbulizatorami potoka na baze semisloynoy modeli turbulentnogo pogranichnogo sloya. Fundamental'nye i prikladnye problemy tekhniki i tekhnologii. 2011;2(286):42-50. [Lobanov I.E., Fleytlikh B.B. Modeling of intensified heat-exchange under the turbulent flow in the flat channels with flow turbulisators, periodically situated on the surface, based on 7-layer model of turbulent boundary layer. Fundamental and Applied Problems of Technics and technology. 2011;2(286):42-50. (in Russ.)] 21. Lobanov I.E., Fleytlikh B.B. Modelirovanie intensifitsirovannogo teploobmena pri turbulentnom techenii v ploskikh kanalakh s periodicheski poverkhnostno raspolozhennymi turbulizatorami potoka na baze semisloynoy modeli turbulentnogo pogranichnogo sloya. Tezisy Mezhdunarodnoy nauchnoy shkoly ―Problemy gazodinamiki i teploobmena v energeticheskikh tekhnologiyakh‖. Moskva; 2011. S. 50-52. [Lobanov I.E., Fleytlikh B.B. Modeling of intensified heat-exchange under the turbulent flow in the flat channels with flow turbulisators, periodically situated on the surface, based on 7-layer model of turbulent boundary layer. Abstracts of International scientific school ―Problems of gas dynamics and heat-exchange in energetic technologies ‖. Moscow; 2011. P. 50-52. (in Russ.)] 22. Lobanov I.E. Teoriya teploobmena pri turbulentnom techenii v ploskikh kanalakh s poverkhnostno raspolozhennymi odnostoronnimi turbulizatorami potoka na baze semisloynoy modeli turbulentnogo pogranichnogo sloya. Moskovskoe nauchnoe obozrenie. 2012;4(1):7-12. [Lobanov I.E. Theory of heat exchange under turbulent flow in flat channels with with single-sided flow turbulisators, situated on the surface, based on 7-layer model of turbulent boundary layer. Moskovskoe nauchnoe obozrenie. 2012;4(1):7-12. (in Russ.)] 23. Lobanov I.E. Analiticheskoe reshenie zadachi ob intensifitsirovannom teploobmene pri turbulentnom techenii v ploskikh kanalakh s periodicheski poverkhnostno raspolozhennymi turbulizatorami potoka na baze semisloynoy modeli turbulentnogo pogranichnogo sloya. Nauchnoe obozrenie. 2012;2:375-387. [Lobanov I.E. Analytical solution of the problem of intensified heat exchange under turbulent flow in flat channels with flow turbulisators, periodically situated on the surface, based on 7-layer model of turbulent boundary layer. Science review. 2012;2:375-387. (in Russ.)] 24. Lobanov I.E. Teploobmen pri turbulentnom techenii v ploskikh kanalakh s ravnomerno raspolozhennymi poverkhnostnymi odnostoronnimi turbulizatorami potoka. Vestnik mashinostroeniya. 2012;8:13-17. [Lobanov I.E. Heat exchange under turbulent flow in flat channels with evenly situated surface single-sided flow turbulisators. Russian Engineering Research. 2012;8:13-17. (in Russ.)] 25. Lobanov I.E. Teoriya gidravlicheskogo soprotivleniya v sherokhovatykh trubakh. Vestnik mashinostroeniya. 2013;7:27-33. [Lobanov I.E. Theory of hydraulic resistanse in rough pipes. Russian Engineering Research. 2013;7:27-33. (in Russ.)] 26. Teploobmen v energeticheskikh ustanovkakh kosmicheskikh apparatov. Pod red. V.K. Koshkina. M.: Mashinostroenie; 1975. 272 s. [Heat exchange in energetic installations of cosmic apparatus. Koshkin V.K. (Ed.). Moscow: Mashinostroenie; 1975. 272 p. (in Russ.)] 27. Zhukauskas A.A. Konvektivnyy perenos v teploobmennikakh. M.: Nauka; 1982. 472 s. [Zhukauskas A.A. Convective transfer in heat-exchangers. Moscow: Nauka; 1982. 472 p. (in Russ.)] 28. Subbotin V.I., Ushakov N.A. Priblizhennye raschety gidrodinamicheskikh kharakteristik turbulentnogo potoka zhidkosti v kol'tsevykh kanalakh. Teplofizika vysokikh temperatur. 1972; 10(5):1025- 1030. [Subbotin V.I., Ushakov N.A. Approximate calculations of hydrodynamic characteristics of liquid turbulent flow in ring channels. High Temperature. 1972; 10(5):1025-1030. (in Russ.)] 29. Lobanov I.E., Paramonov N.V. Predel'nyy teploobmen pri intensifitsirovannom turbulentnom techenii v kanalakh. M.: Izdatel'stvo MAI; 2013. 168 s. [Lobanov I.E., Paramonov N.V. Terminal heat exchange under intensified turbulent flow in channels. Moscow: Izdatel'stvo MAI; 2013. 168 p. (in Russ.)]