Approximate Analytical Solution for One-Dimensional Finite Ablation Problem with Constant Time Heat Flux

Braga, Walber; Mantelli, Marcia; Azevedo, João Luiz F.

doi:10.2514/6.2004-2275

Cited by 11 publications

(11 citation statements)

References 9 publications

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“…Previous researchers have chosen an approximating function based on the expansion of a known exact solution and consequently the approximation works well for the chosen boundary condition, but it is not very good for other conditions [4,5,17,16,19,24]. We this in mind from the above analysis we can strictly only deduce that the logarithmic profile is more accurate for h = 1.…”

Section: 2mentioning

confidence: 96%

Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions

Mitchell

Myers

2010

International Journal of Heat and Mass Transfer

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Abstract. In this paper the two main drawbacks of the heat balance integral methods are examined. Firstly we investigate the choice of approximating function. For a standard polynomial form it is shown that combining the Heat Balance and Refined Integral methods to determine the power of the highest order term will either lead to the same, or more often, greatly improved accuracy on standard methods. Secondly we examine thermal problems with a time-dependent boundary condition. In doing so we develop a logarithmic approximating function. This new function allows us to model moving peaks in the temperature profile, a feature that previous heat balance methods cannot capture. If the boundary temperature varies so that at some time t > 0 it equals the far-field temperature, then standard methods predict that the temperature is everywhere at this constant value. The new method predicts the correct behaviour. It is also shown that this function provides even more accurate results, when coupled with the new CIM, than the polynomial profile. Analysis primarily focuses on a specified constant boundary temperature and is then extended to constant flux, Newton cooling and time dependent boundary conditions.

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Section: 2mentioning

confidence: 96%

Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions

Mitchell

Myers

2010

International Journal of Heat and Mass Transfer

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“…In the first we consider the classical problem of the melting of a semi-infinite material at solidus. In the second we re-examine the ablation problem of [3,4,20]. Finally we look at the classical travelling wave solution.…”

Section: Application To Stefan Problemsmentioning

confidence: 99%

“…In the second stage the boundary s(t) moves while the boundary temperature remains at the ablation temperature. Braga & Mantelli [3,4] study this problem using n = π/(4 − π) in the initial heating up phase and subsequently n = 7 during ablation. Mitchell & Myers [13] study the same problem and take n = 4 in both cases.…”

Section: Ablation Due To a Constant Fluxmentioning

confidence: 99%

“…Hristov states that 'the results oscillate around n = 3 '. This ties in with the fact that the majority of studies take n ∈ [2,4]. In a separate paper Hristov [8] uses an entropy minimization technique that also involves matching at x = 0 and so the results reduce to those of the previous paper.…”

Section: Introductionmentioning

confidence: 98%

“…Braga & Mantelli [3,4] took the next logical step and allowed n to be a non-integer. For an ablation problem they chose n based on matching the time to melting predicted by the exact solution and the approximate one.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Optimal exponent heat balance and refined integral methods applied to Stefan problems

Myers

2010

International Journal of Heat and Mass Transfer

104

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Abstract. When using a polynomial approximating function the most contentious aspect of the Heat Balance Integral Method is the choice of power of the highest order term. In this paper we employ a method recently developed for thermal problems, where the exponent is determined during the solution process, to analyse Stefan problems. This is achieved by minimising an error function. The solution requires no knowledge of an exact solution and generally produces significantly better results than all previous HBI models. The method is illustrated by first applying it to standard thermal problems. A Stefan problem with an analytical solution is then discussed and results compared to the approximate solution. An ablation problem is also analysed and results compared against a numerical solution. In both examples the agreement is excellent. A Stefan problem where the boundary temperature increases exponentially is analysed. This highlights the difficulties that can be encountered with a time dependent boundary condition. Finally, melting with a time-dependent flux is briefly analysed without applying analytical or numerical results to assess the accuracy. NomenclatureE n (t) Least squares error e n E n (t) = e n t α error measure n Exponent in approximating polynomial s(t)Position of melt front t 1Time when ablation begins u(x, t) Temperature β Inverse Stefan number δ(t)Heat penetration depth λ Growth rate s = 2λ √ t

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Applying the combined integral method to one-dimensional ablation

Mitchell

2012

Applied Mathematical Modelling

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Approximate Analytical Solution for One-Dimensional Finite Ablation Problem with Constant Time Heat Flux

Cited by 11 publications

References 9 publications

Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions

Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions

Optimal exponent heat balance and refined integral methods applied to Stefan problems

Applying the combined integral method to one-dimensional ablation

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