Order verification of a Bridgman furnace front tracking model in steady state

Mooney, Robin P.; McFadden, Shaun

doi:10.1016/j.simpat.2014.07.005

Cited by 10 publications

(6 citation statements)

References 16 publications

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“…A heat transfer coefficient, h, applies to the radial heat flow at the circumference of the sample. (The BFFTM was recently verified by Mooney and McFadden [18].) Figure 3a shows a schematic drawing of the physical aspects and geometry of the furnace: the cold zone (crystalliser), the baffle and air gap region (shown hatched), the hot zone (heater), and the cylindrical sample moving at some pulling rate, u, as a function of time.…”

Section: Numerical Modelmentioning

confidence: 80%

Directional solidification of a TiAl alloy by combined Bridgman and power-down technique

Mooney¹,

HECHT²,

GABALCOVÁ³

et al. 2016

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TiAl alloys are of interest to the aerospace and automotive industries (particularly for engine components) on account of their relatively low density and good mechanical properties at high temperatures. Processing routes involve melting and solidification of the alloy and require knowledge about the solidification morphology and microstructural texture evolution in the component. Among others, the Columnar-to-Equiaxed Transition (CET) of bcc β(Ti) dendrites is an issue of current interest. This article examines the results from solidification experiments where a combined Bridgman and power-down technique was implemented at four different cooling rates, using cylindrical samples of the TiAl alloy: Ti-45.5Al-4.7Nb-0.2C-0.2B (all at.%). Axial CET was observed in one of the samples and axial columnar to radial columnar microstructural transitions were observed in the others. A Bridgman Furnace Front Tracking Model (BFFTM), tailored specifically for use with the experiment apparatus, was used to estimate the transient thermal conditions and columnar growth conditions for CET and other microstructural transitions. An important link, due to the nature of the power-down technique, between the reversal of radial heat flow in the hot zone of the furnace and unwanted radial columnar growth, is explained using the model. Recommendations are made on how to avoid such growth, viz. use of low cooling rates and large sample diameters. K e y w o r d s : power-down technique, Bridgman furnace, gamma titanium aluminide, columnar to equiaxed transition, radial growth Nomenclature Across sectional area (mm 2) c-specific heat capacity at constant pressure (J kg −1 • C −1) C 0-original alloy composition (at.%) E-latent heat generated per unit volume (W m −3) G-axial temperature gradient (• C mm −1) h-heat transfer coefficient (W m −2 • C −1) p-perimeter (mm) Q-heat flow (W) r-radius (mm) t-time (s) T-temperature (• C) u-pulling rate (mm s −1) V tip-growth rate (mm s −1) x-axial position (mm) X-axial position with respect to the columnar dendrite tip (mm) ∆T tip-dendrite tip undercooling (• C) ρ-density (kg m −3) Sub-scripts H-hot zone l-liquidus s-solidus tip-dendrite tip ∞-infinity (far away)

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Section: Numerical Modelmentioning

confidence: 80%

Directional solidification of a TiAl alloy by combined Bridgman and power-down technique

Mooney¹,

HECHT²,

GABALCOVÁ³

et al. 2016

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“…Complete verification of the numerical code requires a closed form analytical solution (for comparison purposes) as demonstrated in [26]. However, where a closed form solution is not available (as is the case here) verification of the numerical scheme is achievable…”

Section: Verificationmentioning

confidence: 99%

Thermal characterisation with modelling for a microgravity experiment into polycrystalline equiaxed dendritic solidification with in-situ observation

Mooney

Sturz

Zimmermann

et al. 2018

International Journal of Thermal Sciences

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The Multiple Equiaxed Dendrite Interaction (MEDI) experiment was launched on the MASER-13 sounding rocket campaign to investigate polycrystalline equiaxed solidification in the transparent phase change material Neopentylgycol-30wt.%(d)Camphor. This material is of interest as an energy-storage material and as a transparent analogue system of solidification in hypoeutectic metal alloys with a Face Centred Cubic lattice. The liquid sample was cooled under a controlled rate of 0.75 K/min with sufficient time to allow semi-solid conditions to develop under microgravity conditions with the absence of gravity-driven convection. This manuscript provides details on the experimental apparatus and procedures. In addition, the experimental method has been augmented with a numerical model to assist with the characterisation of the essential thermal aspects of the experiment. The model followed a volume-averaging, continuum approach for mushy-zone development. A fast algorithm for computing non-isothermal crystallisation kinetics was applied to the case of binary alloy solidification. To build confidence, a formal numerical verification procedure was performed.The model converged sufficiently with second order accuracy as expected. Application of the model gave close agreement between experimental thermocouple readings and numerical simulation data with a Root Mean Square of 0.2 K for the error. A 2D thermal model is sufficient with a suitable heat loss coefficient at the sample-viewing window boundary.Thermal gradients within the sample and along the growth direction were uniform and approximately 0.3 K/mm. Lateral thermal gradients in the sample ranged from zero at the median plane to 0.3 K/mm at the boundary with the containing structure. However, temperature variation in the lateral direction was predicted to be less than 0.22 K.

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“…Finally we note, that although the practice of grid convergence essentially involves the systematically checking outputs from the code at finer and finer resolutions, a best practice might also need to include an order analysis (e.g. see the recent work by Mooney and McFadden, 2014). With this technique an estimate of the error order of the macrosegregation simulation discretization would clearly highlight convergence issues related to a failure of the simulation code to meet its expected performance.…”

Section: Hff 262mentioning

confidence: 99%

Best practice for measuring grid convergence in numerical models of alloy solidification

Vušanović

Voller

2016

Int Jnl of Num Meth for HFF

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Purpose -When a multi component alloy solidifies the redistribution of solute components leads to the formation of macrosegregation patterns. Blending ideas from a number of recent publications the purpose of this paper is to provide a "best practice" on how grid convergence of a given macrosegregation simulation can be measured and determined. Design/methodology/approach -The best practice is arrived at by considering a benchmark problem consisting of a 2D-casting simulation of an idealized Al-4.5%Cu alloy in a side cooled square (76×76 mm) cavity. The model for this simulation is based on a mixture treatment of the relevant heat and mass transfer equations. Simulations are made using three increasingly refined grid sizes. Findings -The best practice to determine grid resolution involves two steps: first, a visual evaluation of predicted segregation images leading to the evaluation of solute profiles along selected transects; and second, the construction of a cumulative distribution function (CDF) of the predicted segregation field. On application to the benchmark problem, it is concluded that current computer resources are insufficient to grid resolve macrosegregation patterns but that the CDF provides a useful signal of the nature of macrosegregation in a given system.Research limitations/implications -The benchmark is chosen to be representative. Exact convergence behavior, however, may depend on the system chosen. Originality/value -In addition to establishing a best practice for measuring grid resolution of macrosegregation simulations the work also highlights, even in the absence of complete grid convergence, how the recently proposed CDF treatment can inform solidification modeling and process understanding.

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Order verification of a Bridgman furnace front tracking model in steady state

Cited by 10 publications

References 16 publications

Directional solidification of a TiAl alloy by combined Bridgman and power-down technique

Directional solidification of a TiAl alloy by combined Bridgman and power-down technique

Thermal characterisation with modelling for a microgravity experiment into polycrystalline equiaxed dendritic solidification with in-situ observation

Best practice for measuring grid convergence in numerical models of alloy solidification

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