Anomalous Heat Transfer: Examples, Fundamentals, and Fractional Calculus Models

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The characteristic length scale in heat conduction, a diffusion transport process, changes with the square root of time. As we consider composite structures with increasing complex architectures, however, we find that this classic result is not always the case. When multiscaled heterogeneity is present in the system of interest, our heat transfer calculations may reveal anomalous behaviors. Here, the effects of nonlocality and memory, induced by “fast-paths” or “holdups,” lead to space–time scaling relationships that differ from the square root of time. This article has two themes. The first demonstrates how anomalous transport is induced in heat transfer applications. The second provides the necessary analytical and numerical details of fractional calculus operators and illustrates how these constructs can be used to model anomalous heat transport.

Original languageEnglish (US)
Pages (from-to)333-380
Number of pages48
JournalAdvances in Heat Transfer
StatePublished - 2018


  • Anomalous heat transport
  • Fractional derivatives
  • Numerical models
  • Random-walk simulations


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