Construction of a virtual temperature sensor for application to a holding
furnace in foundry under vacuum conditions
T. Loussouarn1, 2*, D. Maillet1, B. Rémy1, Diane Dan2
1
LEMTA, University of Lorraine & CNRS, Vandoeuvre-lès-Nancy, France
ITKMM Foundry simulation, SNECMA Gennevilliers, Colombes, France
Corresponding author: [email protected]
2
Vacuum holding induction furnaces are used for the manufacturing of turbine blades by loss wax
foundry processes [1]. Heating of the furnace can be implemented by induction, with a control of the
furnace through thermocouples embedded in the heated susceptors, see figure 1. Generally, no
temperature instrumentation is possible in the mould where hot metal is poured before its solidification,
because of its high temperature level (higher than 1000°C). So, indirect estimation has to be made using
transient point temperature measurements in the susceptors. Construction of a virtual sensor requires
the calibration of a reduced thermal model that relates temperatures in the mould to temperatures in the
susceptor first [2].
Figure 1. Heating module configuration studied with
the load (mould) inserted
Once this calibration made, either
through experimental temperature and
electrical heating power measurements
(model identification) or through
comparison
with
corresponding
inputs/outputs of a reference model
(model reduction), the desired mould
temperature can be recovered through
measurements of the temperature of the
susceptors. Here, we consider the
problem of an axisymmetric furnace
centered on a cylindrical hollow mould,
see Figure 1.
Contrary to what happens in other systems such as pollutant advection and diffusion [3] or heat
exchanger modelling [4], presence of radiative transfer between susceptor and mould, in addition to
heat diffusion in the mould, does not allow transient heat transfer modelling using (linear) transfer
functions only.
So, the main objective of this paper is to show how a reduced transient heat transfer model structure
can be constructed (vertical mold considered as a cylindrical fin, with identification of its parameters
starting from a reference detailed numerical model calculated by COMSOL™) and how the transient 1D
temperature distribution in the mould can be recovered next, in this non linear heat transfer (conduction
+ radiation) configuration. This last inversion requires some regularization.
[1] M. Rafique and J. Iqbal, “Modeling and simulation of heat transfer phenomena during investment casting,”
International Journal of Heat and Mass Transfer, 52 (2009), pp. 2132-2139.
[2] T. Loussouarn, D. Maillet, B. Rémy, D. Dan, “Implementation of a numerical holding furnace model in foundry
and construction of a reduced model”, Proceedings of the 7th European Thermal Sciences Conference
(Eurotherm 2016), Krakow, Poland, June 19-23, 2016.
[3] T Maalej, D Maillet and J-R Fontaine, “Estimation of position and intensity of a pollutant source in channel
flow using transmittance functions”, Inverse Problems, 28 (2012) 055010, doi:10.1088/02665611/28/5/055010.
[4] W. Al Hadad, Y. Jannot, D. Maillet, “Convolutive properties of advective and diffusive material systems:
application to heat exchanger characterization”, Physical and Chemical Phenomena in Heat Exchangers and
Multifunctional Reactors for Sustainable Technology, Eurotherm Seminar 106, 10-11 Oct 2016 Paris, France
Keywords: Reduced model, identification, convolution, transient, coupled conduction and
radiation