An experimental–numerical method for estimating heat transfer in a Bridgman furnace

Abstract

Direct measurement of heat flux and heat transfer coefficients in a Bridgman furnace is not always possible using traditional methods. This study characterised a vertical tubular Bridgman furnace using experimental data so that the estimated heat flux and heat transfer coefficients may be used in simulations of future experiments using the same furnace. An experimental-numerical method is presented where a discrete proportional integral derivative controller manipulates the radial heat flux in a front tracking solidification model so that the output temperature profile matches experimental data. The method is applicable for other experimentalists and modellers and its usefulness is demonstrated by example. Highlights A combined experimental-numerical method to estimate heat flux and heat transfer coefficients in a Bridgman furnace is outlined in detail.  The inverse heat transfer problem is solved using a discrete proportional integral derivative controller in series with a front tracking solidification model.  The usefulness of the method is demonstrated by example.

Conditions for CET in a gamma TiAl alloy

“…Any solutal gradient in the melt that may have a destabilising effect is not treated in the current model. Further discussion on the assumptions of the numerical model can be found in reference [14].…”

“…Latent heat release is calculated based on a Thermocalc [17] prediction for solid fraction versus temperature in the full multicomponent alloy, details of which can be found in reference [14]. The solidification conditions at the columnar dendrite tip can be estimated using the model, for example, columnar tip growth rate, temperature and temperature gradient.…”

“…In this way, radial heat flow from the heat source (in the hot zone) or to the heat sink (in the cold zone) is accounted for. The heat transfer coefficients were estimated as a function of axial position through a dedicated characterization study [14] of the same furnace used in the experiments described above.…”

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

“…This yields the explicit finite difference scheme used in the BFFTM. Refer to [15] and [24] for a further description of the BFFTM algorithm.…”

“…The governing heat equation, Eq. (5), is integrated over one CV to give the following [17]; (24) where VCV is the volume of the CV, the subscripts 'e' and 'w' refer to the east and west flat faces of the CV respectively, L is the latent heat generated per unit mass, Vs is the volume of solid material in a CV, and LVs/t=EVCV. Dividing across by A, and introducing Taylor series' expansions for the partial derivative terms (except for Vs/t), we get; 25) where the superscript 'm' is the temporal label, and the subscript 'i' is the spatial label for CVs lined up in the x-direction.…”