Numerical solution of hyperbolic heat conduction in cylindrical and spherical systems

Lin, Jae Yuh; Chen, Han Taw

doi:10.1016/0307-904x(94)90224-0

Cited by 28 publications

(17 citation statements)

References 12 publications

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“…Thus the present problem is more di cult to be solved than the problem in the rectangular co-ordinate system. Accordingly, a similar technique in a previous work [9] is applied to analyse the present problems. It can be observed from previous works [8,9] that the bad choice of the shape functions will produce severe numerical oscillations in the vicinity of the jump discontinuity.…”

Section: Introductionmentioning

confidence: 98%

“…Accordingly, a similar technique in a previous work [9] is applied to analyse the present problems. It can be observed from previous works [8,9] that the bad choice of the shape functions will produce severe numerical oscillations in the vicinity of the jump discontinuity. Thus, to suppress the numerical oscillations, suitable hyperbolic shape functions in the present study are derived from the associated homogeneous di erential equation in the Laplace transform domain.…”

Section: Introductionmentioning

confidence: 98%

“…It can be observed from these previous works that the major di culty in dealing with such problems is the suppression of numerical oscillations in the vicinity of the jump discontinuity. Chen and Lin [9] have applied the hybrid application of the Laplace transform technique and the control volume formulation in conjunction with the hyperbolic shape functions to analyse the linear HHCP in the solid and hollow cylinders. Results show that this hybrid numerical method has good accuracy.…”

Section: Introductionmentioning

confidence: 99%

“…Various numerical methods have been proposed for solving the partial di erential equation of the hyperbolic type [7][8][9][10][11][12][13][14][15][16]. It can be observed from these previous works that the major di culty in dealing with such problems is the suppression of numerical oscillations in the vicinity of the jump discontinuity.…”

Section: Introductionmentioning

confidence: 99%

“…The non-Fickian di usion equation in the absence of potential ÿeld could be regarded as the hyperbolic heat conduction equation [7][8][9][10][11][12][13][14]. Various numerical methods have been proposed for solving the partial di erential equation of the hyperbolic type [7][8][9][10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

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Study of non‐Fickian diffusion problems with the potential field in the cylindrical co‐ordinate system

Chen

Liu

2003

Numerical Meth Engineering

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SUMMARYThe present study applies a hybrid numerical scheme of the Laplace transform technique and the control volume method in conjunction with the hyperbolic shape functions to investigate the e ect of a potential ÿeld on the one-dimensional non-Fickian di usion problems in the cylindrical co-ordinate system. The Laplace transform method is used to remove the time-dependent terms in the governing di erential equation and the boundary conditions, and then the resulting equations are discretized by the control volume scheme. The primary di culty in dealing with the present problem is the suppression of numerical oscillations in the vicinity of sharp discontinuities. Results show that the present numerical results do not exhibit numerical oscillations and the potential ÿeld plays an important role in the present problem. The strength of the jump discontinuity can be reduced by increasing the value of the potential gradient. The propagation speed of mass wave is independent of the potential gradient and the boundary condition.

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Study of non‐Fickian diffusion problems with the potential field in the cylindrical co‐ordinate system

Chen

Liu

2003

Numerical Meth Engineering

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Comparison of Solutions from Parabolic and Hyperbolic Models for Transient Heat Conduction in Semi-Infinite Medium

Sharma

2009

Int J Thermophys

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The expression for the transient temperature during damped wave conduction and relaxation developed by Baumeister and Hamill by the method of Laplace transforms was further integrated. A Chebyshev polynomial approximation was used for the integrand with a modified Bessel composite function in space and time. A telescoping power series leads to a more useful expression for the transient temperature. By the method of relativistic transformation, the transient temperature during damped wave conduction and relaxation was developed. There are four regimes to the solution. These include: (i) a regime comprising a Bessel composite function in space and time, (ii) another regime comprising a modified Bessel composite function in space and time, (iii) the temperature solution at the wave front was also developed separately, and (iv) the fourth regime at a given location X in the medium is at times less than the inertial thermal lag time. In this regime, the temperature was found to be unchanged at the initial condition. The solution for the transient temperature from the method of relativistic transformation is compared side by side with the solution for the transient temperature from the method of Chebyshev economization. Both solutions are within 12 % of each other. For conditions close to the wave front, the solution from the Chebyshev economization is expected to be close to the exact solution and was found to be within 2 % of the solution from the method of relativistic transformation. Far from the wave front, i.e., close to the surface, the numerical error from the method of Chebyshev economization is expected to be significant and verified by a specific example. The solution for transient surface heat flux from the parabolic Fourier heart conduction model and the hyperbolic damped wave conduction and relaxation models are compared with each other. For τ > 1/2 the parabolic and hyperbolic solutions are within 10 % of each other. The parabolic model has a "blow-up" as τ → 0, and the hyperbolic model is devoid of singularities. The transient temperature from the Chebyshev economization is within an average of 25 % of the error function solution for the parabolic Fourier heat conduction model. A penetration distance beyond which there is no effect of the step change in the boundary is predicted using the relativistic transformation model. KeywordsDamped wave conduction and relaxation · Laplace transforms · Microscale heat transfer · Parabolic and hyperbolic models · Relativistic transformation Nomenclature erf (r ) Error function erf(r ) = 2 √ π r 0 e −r 2 dr H (τ ) Space integrated temperature I 0 (x) Modified Bessel function of the first kind and zeroth order I 1 (x) Modified Bessel function of the first kind and first order J 0 (x) Bessel function of the first kind and zeroth order k Thermal conductivity of semi-infinite medium (W · m −1 · K −1 ) K 0 (x) Modified Bessel function of the second kind and zeroth order mInitial temperature (K) T s Surface temperature (K) T n (r ) Chebyshev polynomial (Tables 1, 2) uBesse...

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On Analytical Solutions During Damped Wave Conduction and Relaxation in a Finite Slab Subject to the Convective Boundary Condition

Sharma

2010

Int J Thermophys

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This article describes the use of the final condition in the time domain to obtain bounded and physically reasonable solutions for the convective boundary condition for the case of a finite slab. The temperature overshoot problem is revisited for the convective boundary condition. The use of a physically realistic time condition is shown to remove the overshoot and lead to bounded solutions within Clausius's inequality. The ramifications of these findings are discussed. The method of separation of variables was used to obtain the analytical solution for the wave temperature. The governing equation for temperature, a hyperbolic partial differential equation (PDE) is multiplied by exp(τ/2) that results in a hyperbolic PDE less the damping component. The wave temperature can be used to better understand the transient phenomena of heat conduction. For materials with large relaxation times, τ r > ρC p 4h , the temperature can be expected to undergo subcritical damped oscillations. The analytical solution is presented as an infinite Fourier series solution. The solution was found to be bifurcated. For materials with a small relaxation time, the time domain part of the solution was found to be a decaying exponential and for materials with large relaxation times the time domain part of the solution was found to be cosinuous. Analytical solutions for the average temperature of the finite slab were also obtained. The thermal time constant of the material was found from the solution. The average temperature versus time was found to exhibit convex curvature for systems with large Biot numbers and the average temperature versus time was found to exhibit concave curvature for systems with small Biot numbers. The thermal time constant for the finite slab at different Biot numbers were found and tabulated. The thermal time constant versus Biot 123 Int J Thermophys (2010) 31:430-443 431 number was found to exhibit a maxima. When Fourier parabolic equations are used, the thermal constant decreases monotonically with an increase in Biot number.Keywords Convective boundary condition · Damped wave conduction and relaxation · Final condition · Heat waves · Hyperbolic PDE · Infinite Fourier series solution · Method of separation of variables · Subcritical damped oscillations · Taitel paradox · Wave temperature List of Symbols aHalf-width of a finite slab (m)Heat transfer coefficient between the fluid and the slab k Average temperature of the slab = 1Greek Symbols α Thermal diffusivity (m 2 · s −1 ) τ r Relaxation time (s) τ Dimensionless time ( t τ r ) τ c Dimensionless thermal time constant ( t c τ r ) φ Function of space only

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Numerical solution of hyperbolic heat conduction in cylindrical and spherical systems

Cited by 28 publications

References 12 publications

Study of non‐Fickian diffusion problems with the potential field in the cylindrical co‐ordinate system

Study of non‐Fickian diffusion problems with the potential field in the cylindrical co‐ordinate system

Comparison of Solutions from Parabolic and Hyperbolic Models for Transient Heat Conduction in Semi-Infinite Medium

On Analytical Solutions During Damped Wave Conduction and Relaxation in a Finite Slab Subject to the Convective Boundary Condition

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