Discussion: “Hyperbolic Heat-Conduction Equation—A Solution for the Semi-Infinite Body Problem” (Baumeister, K. J., and Hamill, T. D., 1969, ASME J. Heat Transfer, 91, pp. 543–548)

Baumeister, K. J.; Hamill, T. D.

doi:10.1115/1.3449749

Cited by 48 publications

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“…Then Equation ( (9). This is consistent with the conclusion of Baumeister and Hamill (13). However, when t,., the time during which the heat of reaction is being released, is shorter by an order of magnitude, the difference between two sets of solutions becomes important even in the case when C = 8.7 X lo6 cm./sec.…”

Section: Resultssupporting

confidence: 89%

“…The basic difference between the parabolic and hyperbolic equations is that the latter has an additional term which involves a finite propagation speed. Other differences between these two types of equation were recently demonstrated by Baumeister and Hamill (13). The effect of the finite propagation speed may become important when the system is at very low temperature or when the time of interest is very short.…”

Section: --mentioning

confidence: 94%

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Hyperbolic heat conduction in catalytic supported crystallites

1971

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Many investigators (1-9) have attempted to explain the sintering of catalysts which is frequently found during coke burn-off. It is known that this sintering is associated with the anomalous temperature rise which occurs during exothermic reactions (1-3). Cusumano and Low ( 4 ) have recently used an infrared radiometric method to show such high temperature rises during the sorption of O2 on Si02-supported Ni.The early work of Damkohler ( 5 ) determined the maximum steady state temperature rise of a bulk catalyst. Subsequent work by Prater (6) predicted a maximum temperature rise of only 2"C., while the study by Wei (7) predicted a possible rise of 34°C. for a 2 A hot spot. Such small temperature rises do not appear to be comparable to the experimental observations ( 4 ) . More recently, Luss and Amundson (8) employed a shell progressive model to predict a rise of 250°C. for a typical cracking catalyst. Luss (9) has recently proposed two simple analytical models which will give upper and lower bounds on the temperature rise. It appears that the predicted values of the temperature rise tend to be more realistic, but it also seems that they cannot successfully explain the catastrophic catalyst sintering which is occasionally encountered.It is possible that all of these analytical studies may have a fundamental flaw in that they have been based on the conventional Fourier and Fick laws which result in the conventional heat conduction and mass diffusion equation of the parabolic type __ ae + av2e = S at 8, CY and S can be interpreted respectively as temperature, thermal diffusivity, and heat of reaction in the case of energy transport and as concentration, mass dausivity, and rate of reaction in mass transport. The propagation velocities of heat and mass resulting from such a parabolic formulation are infinite, which is of course unrealistic. In fact, Morse and Feshbach (10) --which is a hyperbolic type of equation. The latter equation can also be derived (11) using a truncation of the more general Maxwell equation (12) for an ideal gas. The basic difference between the parabolic and hyperbolic equations is that the latter has an additional term which involves a finite propagation speed. Other differences between these two types of equation were recently demonstrated by Baumeister and Hamill (13). The effect of the finite propagation speed may become important when the system is at very low temperature or when the time of interest is very short. In the present problem of exothermic catalytic reactions, the maximum temperature may occur in a very short time. For example, time periods as short as sec. after reaction were considered by Luss (9). Under such circumstances the hyperbolic type equation may give significantly different results than the parabolic equation. It is the purpose of this paper to investigate the effect of finite speed of heat transfer on the temperature rise of catalytic supported crystallites. AN A LY S I SThe simple yet interesting physical model employed by Luss (9) is adopted here to demonstrat...

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Section: Resultssupporting

confidence: 89%

Section: --mentioning

confidence: 94%

Hyperbolic heat conduction in catalytic supported crystallites

1971

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“…This expansion is good only under the condition that .... Table I. Summary Table for Leidenfrost-Drop Results [5] [Effect of vapor density on drop shape and drop buoyancy has been included in this r' 4…”

Section: Introductionmentioning

confidence: 97%

Anomalous Behavior of Liquid-Nitrogen Drops in Film Boiling

Baumeister

Keshock

Pucci

1971

Advances in Cryogenic Engineering

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An asymptotic expression for the vaporization time of large drops in film boiling has been derived. This expression seems to correlate the nitrogen data. Also, a new expression for the modified latent heat of vaporization is derived. This expression is useful for large temperature gradients occurring within the vapor film supporting the drop o r for fluids having small latent heats such as would occur for cryogens o r ordinary fluids near the thermodynamic critical point.

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“…Sharma [1] proposed a novel transformation to obtain an analytical solution for the damped wave conduction and relaxation equation by the method of relativistic transformation of coordinates. The solution in Baumeister and Hamill [44] is simplified into a useful expression using a Chebyshev polynomial approximation in this study. The solution is compared with the solution from the method of relativistic transformation of the hyperbolic damped wave conduction and relaxation equation quantitavely as well as qualitatively.…”

Section: Introductionmentioning

confidence: 99%

“…Baumeister and Hamill [44] presented an analytical solution for the transient temperature during damped wave conduction and relaxation by the method of Laplace transforms for a semi-infinite medium subject to a constant wall temperature boundary condition. Their solution is for the open interval, τ > X .…”

Section: Introductionmentioning

confidence: 99%

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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Discussion: “Hyperbolic Heat-Conduction Equation—A Solution for the Semi-Infinite Body Problem” (Baumeister, K. J., and Hamill, T. D., 1969, ASME J. Heat Transfer, 91, pp. 543–548)

Cited by 48 publications

References 0 publications

Hyperbolic heat conduction in catalytic supported crystallites

Hyperbolic heat conduction in catalytic supported crystallites

Anomalous Behavior of Liquid-Nitrogen Drops in Film Boiling

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

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