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Research Papers: Heat and Mass Transfer

# On the Relaxation Time Scales of the Classical Thermodynamic Model for Heat Transfer in Quiescent Compressible Fluids

[+] Author and Article Information
Leonardo S. de B. Alves

Laboratório de Mecânica Teórica e Aplicada,
Departamento de Engenharia Mecânica,
Niterói, Rio de Janeiro 24210-240, Brazil
e-mail: leonardo.alves@mec.uff.br

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 27, 2016; final manuscript received April 16, 2016; published online June 7, 2016. Assoc. Editor: Antonio Barletta.

J. Heat Transfer 138(10), 102004 (Jun 07, 2016) (9 pages) Paper No: HT-16-1037; doi: 10.1115/1.4033462 History: Received January 27, 2016; Revised April 16, 2016

## Abstract

An approximate solution of the classical thermodynamic model for compressible heat transfer of a quiescent supercritical fluid under microgravity leads to the well-known piston effect relaxation time $tPE=tD/(γ0−1)2$, where tD is the thermal diffusion relaxation time and γ0 is the ratio between specific heats. This relaxation time represents an upper bound for the asymptotic bulk temperature behavior during very early times, which shows a strong algebraic relaxation due to the piston effect. This paper demonstrates that an additional relaxation time associated with the piston effect exists in this classical thermodynamic model, namely, $tE=tD/γ0$. Furthermore, it shows that tE represents the time required by the bulk temperature to reach steady-state. Comparisons with a numerical solution of the compressible Navier–Stokes equations as well as experimental data indicate the validity of this new analytical expression and its physical interpretation.

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## Figures

Fig. 2

Dimensionless temperature absolute error δΘ versus dimensionless position ξ at dimensionless time γ0 τ=0.001 with γ0=16.2652 [18]

Fig. 3

Dimensionless temperature Θ versus dimensionless position ξ near the left wall for different values of γ0 τ with γ0=50. Lines are analytical solutions and points are numerical solutions.

Fig. 1

Eigenvalue λN convergence with respect to the number of terms N in the summation series solution, showing that λN→1/γ0 in the limit as N→∞

Fig. 5

Least-mean squared error δτE∗ of equilibration time interpolation (58) versus its number of terms I for different γ0 values

Fig. 6

Equilibration time τE∗ normalized by γ0 versus number of terms N in summation series solution for different γ0 values. Points represent the datasets and lines represent their respective interpolation functions based on Eq. (58).

Fig. 7

Relaxation time estimate τE∞ normalized by γ0 versus specific heat ratio γ0. Dashed line represents theoretical value based on expression (50).

Fig. 4

Dimensionless bulk temperatures Θb versus dimensionless time γ0 τ=t/tE from the approximate and generalized integral transform solutions with γ0=50 and the pure thermal diffusion solution

## Errata

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