A general non-Fourier Stefan problem formulation that accounts for memory effects

Journal Article · Thu Mar 30 00:00:00 EDT 2023 · International Journal of Heat and Mass Transfer

DOI:https://doi.org/10.1016/j.ijheatmasstransfer.2023.124094· OSTI ID:2422248

Voller, Vaughan R. ^[1]; Roscani, Sabrina ^[2]

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1. Univ. of Minnesota, Minneapolis, MN (United States); OSTI
2. Universidad Austral, Paraguay (Argentina); CONICET, Buenos Aires (Argentina)

The Stefan problem is the classical model of a melting phase change. In heterogeneous systems, such phase changes can exhibit non-Fourier (anomalous) behaviors, where the advance of the melt interface does not follow the expected time scaling. These situations can be modeled by replacing the derivatives, in the governing partial differential equations, with fractional order derivatives. In particular, replacing the time derivatives leads to non-Fourier models that account for memory effects in the system. In this work, by using appropriate time convolution integrals, a general thermodynamic balance statement for melting phase problems, explicitly accounting for memory effects, is developed. From this balance, a general model formulation applicable to problems involving melting over a temperature range (i.e., a mushy region) is derived. A key component in this model is the representation of memory effects through the use of fractional derivative based constitutive models of the enthalpy and heat flux. Further, on shrinking the mushy region to a single isotherm, a general sharp interface melting model is obtained. Here, in contrast to the classic Stefan problem, the fractional derivatives induce a natural regularization, such that the constitutive models for enthalpy and heat flux are continuous at the melt interface; a result confirmed through numerical simulation. To further support the theoretical findings, a physical example of a non-Fourier Stefan problem is presented. Overall the development and results in this paper underscore the importance of explicitly relating the development of fractional calculus models to the appropriate thermodynamic balance statements.

• Research Organization: Energy Frontier Research Centers (EFRC) (United States). Center on Geo-processes in Mineral Carbon Storage; Univ. of Minnesota, Minneapolis, MN (United States)
• Sponsoring Organization: USDOE Office of Science (SC), Basic Energy Sciences (BES)
• Grant/Contract Number: SC0023429
• OSTI ID: 2422248
• Alternate ID(s): OSTI ID: 1968521
• Journal Information: International Journal of Heat and Mass Transfer, Journal Name: International Journal of Heat and Mass Transfer Journal Issue: C Vol. 209; ISSN 0017-9310
• Publisher: Elsevier
• Country of Publication: United States
• Language: English

Related Subjects

42 ENGINEERING
Enthalpy method
Fractional derivative
Memory effect
Sharp interface
Stefan problem