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Research Papers: Conduction

A Closed Form Solution of Dual-Phase Lag Heat Conduction Problem With Time Periodic Boundary Conditions

[+] Author and Article Information
Pranay Biswas

Department of Energy Science and Engineering,
Indian Institute of Technology Bombay,
Powai,
Mumbai, Maharashtra 400076, India
e-mail: pranaybiswas@iitb.ac.in

Suneet Singh

Department of Energy Science and Engineering,
Indian Institute of Technology Bombay,
Powai,
Mumbai, Maharashtra 400076, India
e-mail: suneet.singh@iitb.ac.in

Hitesh Bindra

Department of Mechanical and Nuclear
Engineering,
Kansas State University,
3002 Rathbone Hall, 1701B Platt Street,
Manhattan, KS 66506
e-mail: hbindra@ksu.edu

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received July 4, 2018; final manuscript received December 28, 2018; published online February 4, 2019. Assoc. Editor: George S. Dulikravich.

J. Heat Transfer 141(3), 031302 (Feb 04, 2019) (12 pages) Paper No: HT-18-1429; doi: 10.1115/1.4042491 History: Received July 04, 2018; Revised December 28, 2018

The Laplace transform (LT) is a widely used methodology for analytical solutions of dual phase lag (DPL) heat conduction problems with consistent DPL boundary conditions (BCs). However, the inversion of LT requires a series summation with large number of terms for reasonably converged solution, thereby, increasing computational cost. In this work, an alternative approach is proposed for this inversion which is valid only for time-periodic BCs. In this approach, an approximate convolution integral is used to get an analytical closed-form solution for sinusoidal BCs (which is obviously free of numerical inversion or series summation). The ease of implementation and simplicity of the proposed alternative LT approach is demonstrated through illustrative examples for different kind of sinusoidal BCs. It is noted that the solution has very small error only during the very short initial transient and is (almost) exact for longer time. Moreover, it is seen from the illustrative examples that for high frequency periodic BCs the Fourier and DPL model give quite different results; however, for low frequency BCs the results are almost identical. For nonsinusoidal periodic function as BCs, Fourier series expansion of the function in time can be obtained and then present approach can be used for each term of the series. An illustrative example with a triangular periodic wave as one of the BC is solved and the error with different number of terms in the expansion is shown. It is observed that quite accurate solutions can be obtained with a fewer number of terms.

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Figures

Grahic Jump Location
Fig. 1

Temperature profiles at locations r* = 0.7 (at the midpoint) and r* = 0.94 (near the outer boundary) for the DPL heat conduction in spherical coordinates (p = 2) to verify the alternative LT approach (solid lines) with OEEM (dashed lines) in the case of boundary temperature T*r*=1,t*=cos5t*. The temperatures match very well except for small discrepancy at the initial time.

Grahic Jump Location
Fig. 2

Comparisons of temperature distributions for the DPL heat conduction in spherical coordinates (p = 2) obtained by using alternative LT (solid lines) and OEEM (dashed lines) during time (a) 0 ≤ t* ≤ 50 (small time) and (b) 50 ≤ t* ≤ 1000 (large time) in the case of boundary temperature T*r*=1,t*=cos5t*. For large time, the results are identical for two approaches.

Grahic Jump Location
Fig. 3

Temperature profiles at location r* = 0.94 for Fourier and DPL heat conduction in spherical coordinates (p = 2) obtained by using the alternative LT approach in the case of sinusoidal heat flux conditions cosωo*t* with angular frequencies: (a) ωo* = 5, (b) ωo* = 1, and (c) ωo* = 0.1 are applied on the outer boundary

Grahic Jump Location
Fig. 4

Temperature profiles at location r* = 0.94 for Fourier and DPL heat conduction in spherical coordinates (p = 2) obtained by using the alternative LT approach in the case of convective heat loss from the outer boundary to the sinusoidal ambient T∞*t*=cosωo*t* with angular frequencies (a) ωo* = 5, (b) ωo* =1, and (c) ωo* = 0.1

Grahic Jump Location
Fig. 5

Representation of the periodic (nonsinusoidal) boundary function Vot* as defined in Eq. (32) with nondimensional time period tp,o* = 1 and the Fourier series expansion of the specified function having angular frequency ωo,n* = 2nπ (with different number of terms in the series)

Grahic Jump Location
Fig. 6

Temperature profiles at locations r* = 0.46, 0.7, and 1 for the DPL heat conduction in spherical coordinates (p = 2) obtained by using the alternative LT approach in the case of periodic heat flux applied on the outer boundary having time period tp,o* = 1 (ωo,n* = 2nπ)

Grahic Jump Location
Fig. 7

RMS error profiles (calculated at a time over the domain) in DPL solutions with five (solid line) and ten (dashed line) terms in the Fourier series expansion for the time periodic heat flux function. The solution with 15 terms in the Fourier series expansion is considered as the reference solution.

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