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Application of solution structure theorems to Cattaneo–Vernotte heat conduction equation with non-homogeneous boundary conditions

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Abstract

In this study, a non-Fourier heat conduction problem formulated using the Cattaneo–Vernotte (C–V) model with non-homogeneous boundary conditions is solved with the superposition principle in conjunction with solution structure theorems. It is well known that the aforementioned analytical method is not suitable for such a class of thermal problems. However, by performing a functional transformation, the original non-homogeneous partial differential equation governing the physical problem can be cast into a new form such that it consists of a homogeneous part and an additional auxiliary function. As a result, the modified homogeneous governing equation can then be solved with solution structure theorems for temperatures inside a finite planar medium. The methodology provides a convenient, accurate, and efficient solution to the C–V heat conduction equation with non-homogeneous boundary conditions.

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Abbreviations

\( \mathcal{A} \) :

Constant, Table 1

Table 1 Constants used in the temperature solutions for the two physical problems
Full size table
b :

External heat flux decaying factor

\( \mathcal{B} \) :

Constant, Table 1

\( \mathcal{C} \) :

Constant, Table 1

c :

Wave speed (m/s)

c p :

Specific heat (J/kg K)

\( \mathcal{D} \) :

Constant, Table 1

\( \mathcal{E} \) :

Constant, Table 1

\( \mathcal{F} \) :

Constant, Table 1

\( \Im \) :

Dimensionless internal heat generation

f :

Forcing function

f r :

Reference heat flux (W/m2)

f 1 :

Boundary condition at x = 0

f 2 :

Boundary condition at x = 1

g :

Dimensionless heat generation

\( \mathcal{H}_{1} \) :

Dimensionless initial condition

\( \mathcal{H}_{2} \) :

Dimensionless initial temperature rate of change

k :

Thermal conductivity (W/m K)

n :

Index

q :

Dimensionless heat flux

\( q^{*} \) :

Heat flux (W/m2)

q o :

Dimensionless incident heat flux magnitude

Τ :

Dimensionless temperature

\( T^{*} \) :

Temperature (K)

t :

Dimensionless time

\( t^{*} \) :

Time (s)

u :

Dimensionless temperature solution for the homogeneous problem

u 1 :

Dimensionless temperature solution due to ψ function contribution

u 2 :

Dimensionless temperature solution due to φ function contribution

u 3 :

Dimensionless temperature solution due to f function contribution

w :

Auxiliary function

x :

Dimensionless slab thickness

\( x^{*} \) :

Slab thickness (m)

α :

Thermal diffusivity, k/ρc p (m2/s)

\( \gamma_{n} \) :

Eigenvalue, \( \sqrt {(n\pi )^{2} - 1 } \)

ε:

Relative error

\( \zeta \) :

Dummy variable for time

\( \xi \) :

Dummy variable for space

ρ :

Density (kg/m3)

\( \tau_{CV} \) :

Relaxation time constant (s)

φ :

Initial temperature function

ψ:

Initial temperature rate of change function

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  1. The Aerospace Corporation, M4/908, 2310 E. El Segundo Boulevard, El Segundo, CA, 90245-4691, USA

    Tung T. Lam & Ed Fong

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  1. Tung T. Lam
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  2. Ed Fong
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Correspondence to Tung T. Lam.

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Lam, T.T., Fong, E. Application of solution structure theorems to Cattaneo–Vernotte heat conduction equation with non-homogeneous boundary conditions. Heat Mass Transfer 49, 509–519 (2013). https://doi.org/10.1007/s00231-012-1097-4

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