https://orcid.org/0009-0001-4506-4097
Abstract
In this study, a conjugate laminar Graetz problem in a channel with wall conduction is theoretically studied. The heat exchanger under consideration utilizes a hot fluid stream with embedded heat sources in the internal side of the channel and a boiling liquid at the external side. Because of the nonlinear and nonmonotonic boiling curve, a complex and interesting solution structure exists. When the wall conduction is accounted for the problem admits multiple solutions. For a certain range of the conduction–convection parameter and the heat generation intensity up to five solutions have been determined featuring stable single and multimode temperature profiles. The conduction–convection parameter and the heat generation intensity have a profound effect on the solution structure and the stability since multimode solutions are becoming unstable allowing only the single mode profiles to be stable. An important finding is that when the boiling heat flux is directly imposed as a boundary condition, neglecting wall conduction, it is not possible to capture the multiplicity since a unique temperature profile is predicted. The model developed is applicable to typical heat exchangers with phase change in general and to cryogenic applications in particular where there is a significant temperature difference between the boiling point of the cryogenic liquid and the inlet temperature of the hot fluid in the exchanger.
Issue Section:
Heat Exchangers
Topics:
Boiling,
Heat conduction,
Temperature,
Temperature profiles,
Heat transfer,
Stability,
Heat,
Heat flux
References
1.
Cole
, K. D.
, and Çetin
, B.
, 2017
, “Modeling of Joule Heating and Convective Cooling in a Thick-Walled Micro-Tube
,” Int. J. Therm. Sci.
, 119
, pp. 24
–36
.10.1016/j.ijthermalsci.2017.05.0102.
Cotta
, R. M.
, Naveira-Cotta
, C. P.
, and Knupp
, D.
, 2013
, “Conjugated Convection-Conduction Analysis in Microchannels With Axial Diffusion Effects and a Single Domain Formulation
,” ASME J. Heat Mass Transfer-Trans. ASME
, 135
(9
), p. 091008
.10.1115/1.40244253.
Borah
, A.
, Pati
, S.
, and Baranyi
, S.
, 2021
, “Conjugate Heat Transfer Analysis for Forced Convective Flow Through a Parallel Plate Microchannel: Effect of Nonuniform Asymmetric Heating
,” Numer. Heat Transfer, Part A
, 80
(5
), pp. 210
–233
.10.1080/10407782.2021.19396334.
Sánchez
, S.
, Ascanio
, G.
, Sánchez-Minero
, F.
, Méndez
, F.
, Aguayo
, J. P.
, Ramírez-Jiménez
, E.
, and Alonso-Ramírez
, G.
, 2019
, “Conjugate Thermal-Hydrodynamic Model for the Study of Heavy Oil Transport
,” J. Pet. Sci. Eng.
, 179
, pp. 997
–1011
.10.1016/j.petrol.2019.04.0835.
Gururajan
, V.
, and Som
, S.
, 2022
, “Numerical Solutions to the Reactive Graetz Problem for CO2 Capture
,” Carbon Capture Sci. Technol.
, 5
, p. 100071
.10.1016/j.ccst.2022.1000716.
Shah
, R. K.
, and London
, A. L.
, 1978
, Laminar Flow Forced Convection in Ducts
, Academic Press
, New York
.7.
Papoutsakis
, E.
, and Ramkrishna
, D.
, 1981
, “Conjugated Graetz Problems–I: General formalism and a class of solid-fluid problems
,” Chem. Eng. Sci.
, 36
(8
), pp. 1381
–1391
.10.1016/0009-2509(81)80172-88.
Campo
, A.
, and Schuler
, C.
, 1988
, “Heat Transfer in Laminar Flow Through Circular Tubes Accounting for Two-Dimensional Wall Conduction
,” Int. J. Heat Mass Transfer
, 31
(11
), pp. 2251
–2259
.10.1016/0017-9310(88)90157-39.
Guedes
, R. O. C.
, and Özişik
, M. N.
, 1992
, “Conjugated Turbulent Heat Transfer With Axial Conduction in Wall and Convection Boundary Conditions in a Parallel-Plate Channel
,” Int. J. Heat Fluid Flow
, 13
(4
), pp. 322
–328
.10.1016/0142-727X(92)90002-Q10.
Weigand
, B.
, and Gassner
, G.
, 2007
, “The Effect of Wall Conduction for the Extended Graetz Problem for Laminar and Turbulent Channel Flows
,” Int. J. Heat Mass Transfer
, 50
(5–6
), pp. 1097
–1105
.10.1016/j.ijheatmasstransfer.2006.06.04711.
Adelaja
, A. O.
, Dirker
, J.
, and Meyer
, J. P.
, 2014
, “Effects of the Thick Walled Pipes With Convective Boundaries on Laminar Flow Heat Transfer
,” Appl. Energy
, 130
, pp. 838
–845
.10.1016/j.apenergy.2014.01.07212.
Quintero
, A. E.
, and Vera
, M.
, 2017
, “Laminar Counterflow Parallel-Plate Heat Exchangers: An Exact Solution Including Axial and Transverse Wall Conduction Effects
,” Int. J. Heat Mass Transfer
, 104
, pp. 1229
–1245
.10.1016/j.ijheatmasstransfer.2016.09.02513.
Pigeonneau
, F.
, Jaffrennou
, B.
, Letailleur
, A.
, and Limouzin
, K.
, 2016
, “Numerical Investigation of Generalized Graetz Problem in Circular Tube With a Mass Transfer Coupling Between the Solid and the Liquid
,” Int. J. Heat Mass Transfer
, 96
, pp. 381
–395
.10.1016/j.ijheatmasstransfer.2016.01.04014.
Bhowmick
, D.
, and Randive
, P. R.
, 2023
, “Effect of Non-Uniform Heating on Conjugate Heat Transfer and Entropy Generation Characteristics Within a Partially Filled Porous Corrugated Channel
,” Numer. Heat Transfer, Part A
, pp. 1
–23
.10.1080/10407782.2023.225738715.
Yu
, B.
, and Churchill
, S. W.
, 2012
, “Prediction of the Influence of Energetic Chemical Reactions on Forced Convective Heat Transfer
,” Adv. Heat Transfer
, 44
, pp. 1
–117
.10.1016/B978-0-12-396529-5.00001-916.
Higuera
, F. J.
, and Fernández de la Mora
, J.
, 2023
, “The Reactive Graetz Problem for a First Order Homogeneous Decomposition Reaction, and the Limit of Moderately Large Reaction Constants
,” Phys. Fluids
, 35
(9
), p. 093616
.10.1063/5.016889017.
Veeraragavan
, A.
, and Cadou
, C.
, 2010
, “Theoretical Study of Conjugate Heat Transfer Effects on Temperature Profiles in Parallel Flow With Embedded Heat Sources
,” Int. J. Heat Mass Transfer
, 53
(9–10
), pp. 1699
–1711
.10.1016/j.ijheatmasstransfer.2010.01.01918.
Cavazzuti
, M.
, 2020
, “Viscous Heating Effects on Heat Transfer Characteristics of Laminar Compressible Channel Flow
,” Int. J. Heat Mass Transfer
, 153
, p. 119608
.10.1016/j.ijheatmasstransfer.2020.11960819.
Di Marcello
, V.
, Cammi
, A.
, and Luzzi
, L.
, 2010
, “A Generalized Approach to Heat Transfer in Pipe Flow With Internal Heat Generation
,” Chem. Eng. Sci.
, 65
(3
), pp. 1301
–1310
.10.1016/j.ces.2009.10.00420.
Nukiyama
, S.
, 1966
, “Maximum and Minimum Value of Heat Transmitted From a Metal to Boiling Water Under Atmospheric Pressure
,” Int. J. Heat Mass Transfer
, 9
, pp. 1419
–1434
.10.1016/0017-9310(66)90138-421.
Westwater
, J. W.
, 1988
, “Boiling Heat Transfer
,” Int. Commun. Heat Mass Transfer
, 15
(4
), pp. 381
–400
.10.1016/0735-1933(88)90040-122.
Dhir
, V. K.
, 1998
, “Boiling Heat Transfer
,” Annu. Rev. Fluid Mech.
, 30
(1
), pp. 365
–401
.10.1146/annurev.fluid.30.1.36523.
Lüttich
, T.
, Marquardt
, W.
, Buchholz
, M.
, and Auracher
, H.
, 2004
, “Towards a Unifying Heat Transfer Correlation for the Entire Boiling Curve
,” Int. J. Therm. Sci.
, 43
(12
), pp. 1125
–1139
.10.1016/j.ijthermalsci.2004.04.01424.
Haley
, K. W.
, and Westwater
, J. W.
, 1965
, “Heat Transfer From a Fin to a Boiling Liquid
,” Chem. Eng. Sci.
, 20
(7
), pp. 711
–712
.10.1016/0009-2509(65)80010-025.
Klein
, G. J.
, and Westwater
, J. W.
, 1971
, “Heat Transfer From Multiple Spines to Boiling Liquids
,” AIChE J.
, 17
(5
), pp. 1050
–1056
.10.1002/aic.69017050726.
Zhukov
, S. A.
, Barelko
, V. V.
, and Merzhanov
, A. G.
, 1981
, “Wave Processes on Heat Generating Surfaces in Pool Boiling
,” Int. J. Heat Mass Transfer
, 24
(1
), pp. 47
–55
.10.1016/0017-9310(81)90092-227.
Petukhov
, B. S.
, Kovalev
, S. A.
, Zhukov
, V. M.
, and Kazakov
, G. M.
, 1971
, “Measurement of Local Heat Transfer on a Nonisothermal Surface
,” High Temp.
, 9
, pp. 1159
–1161
.28.
Krikkis
, R. N.
, Sotirchos
, S. V.
, and Razelos
, P.
, 2004
, “Analysis of Multiplicity Phenomena in Longitudinal Fins Under Multi-Boiling Conditions
,” ASME J. Heat Mass Transfer-Trans. ASME
, 126
(1
), pp. 1
–7
.10.1115/1.164308829.
Lee
, D. J.
, and Lu
, S. M.
, 1992
, “Two-Mode Boiling on a Horizontal Boiling Wire
,” AIChE J.
, 38
(7
), pp. 1115
–1127
.10.1002/aic.69038071430.
Liaw
, S. P.
, and Yeh
, R. H.
, 1994
, “Fins With Temperature Dependent Surface Heat Flux-II. Multi-Boiling Heat Transfer
,” Int. J. Heat Mass Transfer
, 37
(10
), pp. 1517
–1524
.10.1016/0017-9310(94)90153-831.
Krikkis
, R. N.
, 2015
, “On the Multiple Solutions of Boiling Fins With Heat Generation
,” Int. J. Heat Mass Transfer
, 80
, pp. 236
–242
.10.1016/j.ijheatmasstransfer.2014.09.02032.
Krikkis
, R. N.
, 2021
, “On the Thermal Dynamics of Metallic and Superconducting Wires. Bifurcations, Quench, the Destruction of Bistability and Temperature Blowup
,” J.
, 4
(4
), pp. 803
–823
.10.3390/j404005533.
Krikkis
, R. N.
, 2018
, “A Thermodynamic and Heat Transfer Model for LNG Ageing During Ship Transportation. Towards an Efficient Boil-Off Gas Management
,” Cryogenics
, 92
, pp. 76
–83
.10.1016/j.cryogenics.2018.04.00734.
Al Ghafri
, S. Z. S.
, Perez
, F.
, Heum Park
, K.
, Gallagher
, L.
, Warr
, L.
, Stroda
, A.
, Siahvashi
, A.
, Ryu
, Y.
, Kim
, S.
, Kim
, S. G.
, Seo
, Y.
, Johns
, M. L.
, and May
, E. F.
, 2021
, “Advanced Boil-Off Gas Studies for Liquefied Natural Gas
,” Appl. Therm. Eng.
, 189
, p. 116735
.10.1016/j.applthermaleng.2021.11673535.
Chen
, H.
, Yang
, G.
, and Wu
, J.
, 2023
, “A Multi-Zone Thermodynamic Model for Predicting LNG Ageing in Large Cryogenic Tanks
,” Energy
, 283
, p. 128503
.10.1016/j.energy.2023.12850336.
Smith
, J. R.
, Gkantonas
, S.
, and Mastorakos
, E.
, 2022
, “Modelling of Boil‐Off and Sloshing Relevant to Future Liquid Hydrogen Carriers
,” Energies
, 15
(6
), p. 2046
.10.3390/en1506204637.
Al Ghafri
, S. Z. S.
, Swanger
, A.
, Jusko
, V.
, Siahvashi
, A.
, Perez
, F.
, Johns
, M. L.
, and May
, E. F.
, 2022
, “Modelling of Liquid Hydrogen Boil‐Off
,” Energies
, 15
(3
), p. 1149
.10.3390/en1503114938.
Wang
, J.
, Webley
, P. A.
, and Hughes
, T. J.
, 2024
, “Thermodynamic Modelling of Low Fill Levels in Cryogenic Storage Tanks for Application to Liquid Hydrogen Maritime Transport
,” Appl. Therm. Eng.
, 256
, p. 124054
.10.1016/j.applthermaleng.2024.12405439.
Mohekar
, A. A.
, Tilley
, B. S.
, and Yakovlev
, V. V.
, 2023
, “Transverse Electric-Thermal-Fluid Instabilities in an Electromagnetic Heat Exchanger
,” Int. J. Heat Mass Transfer
, 210
, p. 124187
.10.1016/j.ijheatmasstransfer.2023.12418740.
Weigand
, B.
, 2015
, Analytical Methods for Heat Transfer and Fluid Flow Problems
, 2nd ed., Springer
, New York
.41.
Arpaci
, V. S.
, and Larsen
, P. S.
, 1984
, Convection Heat Transfer
, Prentice Hall
, Englewood Cliffs, NJ
.42.
Zhao
, B.
, Long
, W.
, and Zhou
, R.
, 2021
, “A Convective Analytical Model in Turbulent Boundary Layer on a Flat Plate Based on the Unifying Heat Flux Formula
,” Int. J. Therm. Sci.
, 163
, p. 106784
.10.1016/j.ijthermalsci.2020.10678443.
Speetjens
, M.
, Reusken
, A.
, and Marquardt
, W.
, 2008
, “Steady-State Solutions in a Nonlinear Pool Boiling Model
,” Commun. Nonlinear Sci. Numer. Simul.
, 13
(8
), pp. 1475
–1494
.10.1016/j.cnsns.2006.11.00144.
Pumir
, A.
, Barelko
, V. V.
, and Buryak
, E. V.
, 2007
, “Control of the Boiling Crisis: Analysis of a Model System
,” Eur. Phys. J. B
, 60
(1
), pp. 1
–8
.10.1140/epjb/e2007-00327-845.
Hairer
, E.
, Nørsett
, S. P.
, and Wanner
, G.
, 1987
, Solving Ordinary Differential Equations I, Nonstiff Problems
, Springer-Verlag
, New York
.Copyright © 2025 by ASME
You do not currently have access to this content.
